Seismic Engineering
Jacques Betbeder-Matibet Series Editor Jacky Mazars
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Seismic Engineering
Jacques Betbeder-Matibet Series Editor Jacky Mazars
This page intentionally left blank
Seismic Engineering
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Seismic Engineering
Jacques Betbeder-Matibet Series Editor Jacky Mazars
Part of this book adapted from “Génie parasismique” published in three volumes in France in 2003 by Hermes Science/Lavoisier First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 6 Fitzroy Square London W1T 5DX UK
John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA
www.iste.co.uk
www.wiley.com
© ISTE Ltd, 2008 © LAVOISIER, 2003 The rights of Jacques Betbeder-Matibet to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Betbeder-Matibet, Jacques. [Phenomenes sismiques English] Seismic engineering / Jacques Betbeder-Matibet. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-026-4 1. Earthquake engineering. I. Title. TA654.6.B4813 2008 624.1'762--dc22 2007043949 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-026-4 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.
Table of Contents
Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
Part 1. Earthquakes and Induced Phenomena . . . . . . . . . . . . . . . . . .
1
Chapter 1. Causes of Earthquakes. . . . . . . . . . . . . 1.1. Tectonic earthquakes . . . . . . . . . . . . . . . . . 1.1.1. First attempts at explanation . . . . . . . . . . 1.1.2. From continental drift to plate tectonics . . . 1.1.3. Seismicity of tectonic origin . . . . . . . . . . 1.2. Faults . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Relationship between earthquakes and faults 1.2.2. Classification of faults . . . . . . . . . . . . . . 1.2.3. Focal mechanisms . . . . . . . . . . . . . . . . 1.2.4. Different aspects of rupture . . . . . . . . . . . 1.3. Non-tectonic earthquakes . . . . . . . . . . . . . . 1.3.1. Non-tectonic quakes with natural causes . . . 1.3.2. Artificial earthquakes . . . . . . . . . . . . . . 1.3.3. Induced earthquakes . . . . . . . . . . . . . . .
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5 5 5 9 20 26 27 29 38 45 47 48 49 50
Chapter 2. Parameters Used to Define Earthquakes . 2.1. Elementary theory of elastic rebound . . . . . . . 2.1.1. Description of the elementary model . . . . . 2.1.2. Energy balance . . . . . . . . . . . . . . . . . . 2.1.3. Law of scale . . . . . . . . . . . . . . . . . . . . 2.2. Geometry of the faults . . . . . . . . . . . . . . . . 2.2.1. Length of fault and length of rupture . . . . . 2.2.2. Well documented examples of fault ruptures
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55 56 56 61 65 70 70 78
vi
Seismic Engineering
2.2.3. Correlations of geometric characteristics of ruptures with moment magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Parametric description of earthquakes . . . . . . . . . . . . . . . . . . . . 2.3.1. Source parameters and effect parameters . . . . . . . . . . . . . . . . 2.3.2. Different magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 3. Manifestations of the Seismic Phenomena on the Surface . 3.1. Deformation of superficial terrains . . . . . . . . . . . . . . . . . . . 3.1.1. Deformations linked to tectonics . . . . . . . . . . . . . . . . . . 3.1.2. Deformations linked to vibratory motions . . . . . . . . . . . . 3.2. Seismic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Different types of seismic waves. . . . . . . . . . . . . . . . . . 3.2.2. Ideas on the theory of rays . . . . . . . . . . . . . . . . . . . . . 3.2.3. Attenuation of seismic waves. . . . . . . . . . . . . . . . . . . . 3.3. Induced phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Soil liquefaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Landslides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Tsunamis and seiches . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4. Other seismic manifestations . . . . . . . . . . . . . . . . . . . .
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107 107 107 110 114 114 121 135 143 143 148 154 159
Part 2. Strong Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . .
161
Chapter 4. Strong Vibratory Motions. . . . . . . . . . . . . . . . . . . . . 4.1. Recordings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Examples of accelerograms recorded in the near zone . . . . . 4.1.2. Parametric description of the accelerograms . . . . . . . . . . . 4.1.3. The three components of vibratory motion . . . . . . . . . . . . 4.2. Attenuation laws of peak values. . . . . . . . . . . . . . . . . . . . . 4.2.1. General considerations as regards attenuation laws . . . . . . . 4.2.2. Examples of attenuation laws for peak values . . . . . . . . . . 4.2.3. Recommendations for the use of attenuation laws . . . . . . . 4.3. Directivity effects and site effects. . . . . . . . . . . . . . . . . . . . 4.3.1. Inadequacy of a description based on magnitude and distance 4.3.2. Directivity effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. Presentation of site effects. . . . . . . . . . . . . . . . . . . . . . 4.3.4. Causes of site effects . . . . . . . . . . . . . . . . . . . . . . . . .
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165 165 165 168 178 186 186 188 197 201 201 202 210 212
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223 227
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Chapter 5. Calculation Models for Strong Vibratory Motions . . . . . . 5.1. Orders of magnitude deduced from the basic theory of elastic rebound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Limits of the basic theory of elastic rebound for the calculation of motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Model of elastic rebound with multiple ruptures . . . . . . . . .
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82 93 93 99
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vii
5.1.3. Calculation of the theoretical attenuation laws associated with the model of rebound elasticity with multiple ruptures . . . . . . . . 5.2. Digital source models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. General considerations pertaining to models of digital simulation of the seismic source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Examples of digital simulation of real earthquakes . . . . . . . . . . 5.3. Practical calculations of the site effects . . . . . . . . . . . . . . . . . . . 5.3.1. Models of soil behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Seismic responses of columns of soil . . . . . . . . . . . . . . . . . . 5.3.3. Review of the assessment of site effects . . . . . . . . . . . . . . . .
232 234 240 240 248 267
Part 3. Seismic Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
275
Chapter 6. The Spatial and Temporal Distribution of Seismicity . . . . . 6.1. Data available on the spatial and temporal distribution of seismicity . 6.1.1. Geological data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2. Historical seismicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3. Archeoseismicity and paleoseismicity. . . . . . . . . . . . . . . . . 6.1.4. Instrumental seismicity . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Models of temporal distribution of seismicity . . . . . . . . . . . . . . 6.2.1. Return periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Gutenberg-Richter law. . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3. Model of a characteristic earthquake . . . . . . . . . . . . . . . . . 6.3. Prediction of earthquakes. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1. Seismic precursors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2. Current questions on forecast . . . . . . . . . . . . . . . . . . . . . .
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281 281 281 283 288 294 296 296 300 305 307 308 309
Chapter 7. Assessment of Seismic Hazard . . . . . . . . . . . . . . . . . . . . 7.1. Methods of assessment of seismic hazard . . . . . . . . . . . . . . . . . 7.1.1. General notes pertaining to different approaches . . . . . . . . . . 7.1.2. An example of the deterministic method . . . . . . . . . . . . . . . 7.1.3. Probabilistic methods . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Practices for the evaluation of seismic hazard . . . . . . . . . . . . . . 7.2.1. Normative evaluation and specific evaluation . . . . . . . . . . . . 7.2.2. Zoning for the anti-seismic codes . . . . . . . . . . . . . . . . . . . 7.2.3. Seismic microzoning . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4. Orders of magnitude for hazards due to a fault (vibratory motion and surface rupture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5. Orders of magnitude of vibratory hazard in diffuse seismicity zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6. Effect of the size of the site on the vibratory hazard in a zone of diffuse seismicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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315 315 315 317 321 326 326 327 330
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viii
Seismic Engineering
Part 4. Seismic Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
359
Chapter 8. The Seismic Coefficient . . . . . . . . . . . . . . . . . . . . . 8.1. The seismic coefficient in past earthquake-resistant codes . . . . 8.1.1. Notion of seismic coefficient . . . . . . . . . . . . . . . . . . . 8.1.2. Development of the seismic coefficient . . . . . . . . . . . . . 8.2. The seismic coefficient in current earthquake-resistant codes . . 8.2.1. The structure of current earthquake-resistant codes . . . . . . 8.2.2. The definition of seismic action and the rules of calculation in current earthquake-resistant codes . . . . . . . . . . . . . . . . . .
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365 365 365 366 370 370
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Chapter 9. The Response Spectrum . . . . . . . . . . . . . . . . . . . 9.1. The response spectrum of elastic oscillators . . . . . . . . . . 9.1.1. Response spectrum of elastic oscillators associated with a natural accelerogram. . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2. Response spectrum of elastic oscillators that can be used for designing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Introduction to spectral modal analysis of elastic structures . 9.2.1. Presentation of a simple example to introduce spectral modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2. Calculation model for the chosen example . . . . . . . . . 9.2.3. Non-damped eigenmodes . . . . . . . . . . . . . . . . . . . 9.2.4. Calculation of the response for the chosen example . . . 9.2.5. Calculation of displacements, accelerations and forces for the chosen example . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Structural design spectra . . . . . . . . . . . . . . . . . . . . . . 9.3.1. Reasons for the general consideration of nonlinearities: the behavior coefficient . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Elastic and inelastic design spectrum . . . . . . . . . . . .
Chapter 10. Other Representations of Seismic Action . . . . . . . . 10.1. Natural or synthetic accelerograms . . . . . . . . . . . . . . . . 10.1.1. Types of analyses for which accelerogram representation is necessary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2. Choice of accelerograms for linear analysis . . . . . . . . 10.1.3. Choice of accelerograms for nonlinear analysis . . . . . . 10.2. Random processes . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1. Unfiltered white noise . . . . . . . . . . . . . . . . . . . . . 10.2.2. Filtered white noise . . . . . . . . . . . . . . . . . . . . . . . 10.2.3. Theorem of general Brownian motion . . . . . . . . . . . .
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433 433
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433 435 437 445 446 452 456
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ix
Part 5. The Effects of Earthquakes on Buildings . . . . . . . . . . . . . . . . .
467
Chapter 11. Deformation Effects Sustained by Superficial Ground . . 11.1. Effects of irreversible deformations. . . . . . . . . . . . . . . . . . 11.1.1. Damage directly due to movements on fault surfaces . . . . . 11.1.2. Damage due to irreversible deformations of the ground in a horizontal direction (other than fault movements) . . . . . . . . . 11.1.3. Damage due to irreversible deformation of the ground in a vertical direction (other than fault movements) . . . . . . . . . . 11.2. Effects of reversible deformation . . . . . . . . . . . . . . . . . . . 11.2.1. Details of effects due to reversible deformation with respect to those due to irreversible deformations . . . . . . . . . . . . . . . . . 11.2.2. Static or dynamic character of effects due to reversible deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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473 473 473
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487 490
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Chapter 12. Effects of Vibratory Motions . . . . . . . . . . . . . . . . . . . . . 12.1. Effects at the structure/subsoil contact . . . . . . . . . . . . . . . . . . . 12.1.1. Slipping and tilting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2. Rupture of the ground or foundation system . . . . . . . . . . . . . 12.2. Inertial effects in structures. . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1. General observations on the inertial effects. . . . . . . . . . . . . . 12.2.2. Damage and destruction patterns due to horizontal inertial effects for concrete structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3. Damage and destruction patterns due to horizontal inertial effects for steel structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4. Damage and destruction patterns due to horizontal inertial effects for structures made of masonry or wood . . . . . . . . . . . . . . . . . . . . 12.2.5. Damage patterns due to vertical inertial effect . . . . . . . . . . . . 12.2.6. Effects of shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3. Effects on non-structural elements and supported equipment. . . . . . 12.3.1. Deformations imposed on non-structural elements . . . . . . . . . 12.3.2. Accelerations transmitted to supported equipment . . . . . . . . .
497 498 498 507 512 512
Chapter 13. Effects of Induced Phenomena . . . . . . . . . . . 13.1. Effects of naturally induced phenomena . . . . . . . . . 13.1.1. Effects of liquefaction . . . . . . . . . . . . . . . . . 13.1.2. Other naturally induced phenomena . . . . . . . . . 13.2. Phenomena induced in networks and industrial setups. 13.2.1. Disruption of the functioning of networks . . . . . 13.2.2. Fires . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3. Accidents in industrial facilities . . . . . . . . . . .
573 573 573 575 575 575 578 580
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513 535 546 553 556 564 564 567
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Chapter 14. Scales of Macroseismic Intensity . . . . . . . . . . . . . . . . . . 14.1. Characterization of the force of earthquakes through assessment of their effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1. A summary of the history of scales of intensity . . . . . . . . . . 14.1.2. Description of some scales of intensity . . . . . . . . . . . . . . . 14.1.3. Benefits and limitations of the notion of intensity . . . . . . . . . 14.2. Numerical correlations using intensities . . . . . . . . . . . . . . . . . 14.2.1. Correlations of intensities with parameters of vibratory motion. 14.2.2. Magnitude-intensity relations and attenuation laws of intensity.
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581 581 583 588 594 594 598
Part 6. Seismic Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
603
Chapter 15. Linear Seismic Calculation . . . . . . . . . . . . . . . . . . . 15.1. General observations on linear calculation. . . . . . . . . . . . . . 15.1.1. General formulation with relation to absolute axes . . . . . . 15.1.2. Formulations for block movement of supports . . . . . . . . . 15.1.3. Representation of damping . . . . . . . . . . . . . . . . . . . . 15.1.4. Notes on modeling . . . . . . . . . . . . . . . . . . . . . . . . . 15.2. Modal spectral analysis for block translation of supports . . . . . 15.2.1. Eigenmodes and quantities attached to modes . . . . . . . . . 15.2.2. Number of modes to be retained and combination of modal responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3. Combination of effects with three components. . . . . . . . . 15.2.4. Some properties of stick models working in shear. . . . . . . 15.2.5. Continuous models. Example of a uniform cantilever beam .
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607 607 607 612 619 627 637 638
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653 667 673 685
Chapter 16. Notions on Soil/Structure Interaction. . . . . . . . . . . . 16.1. General observations on soil/structure interaction . . . . . . . . 16.1.1. Presentation of the soil/structure interaction phenomena . . 16.1.2. Kinematic and inertial interaction . . . . . . . . . . . . . . . 16.1.3. Radiative (or geometric) damping . . . . . . . . . . . . . . . 16.2. Practical consideration of the soil/structure interaction . . . . . 16.2.1. General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2. Shallow foundations . . . . . . . . . . . . . . . . . . . . . . . 16.2.3. Cases of deep foundations and linear embedded structures 16.2.4. Winkler type models . . . . . . . . . . . . . . . . . . . . . . .
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703 703 703 709 713 721 721 724 739 746
Chapter 17. Overview of Nonlinear Calculations . . . . . 17.1. General observations on nonlinear calculations . . . 17.1.1. The problem of hypothesis and criteria . . . . . 17.1.2. Methods of giving recognition to nonlinearities 17.2. Some examples of nonlinear calculations . . . . . . 17.2.1. Tilting of the rigid blocks . . . . . . . . . . . . .
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767 767 767 772 781 781
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17.2.2. Basemat uplifts . . . . . . . . . . . . . . 17.2.3. Slipping of massive blocks . . . . . . . 17.2.4. Plasticization of building structures . . 17.2.5. Nonlinear shock absorbers for bridges. 17.2.6. Pipelines going through a fault . . . . .
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xi
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793 800 808 822 827
Part 7. Seismic Prevention Tools . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 18. Technical Aspects of Prevention . . . . . . . . . . . . . . . . 18.1. Tools for learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.1. The analysis of past experience . . . . . . . . . . . . . . . . . . 18.1.2. Test methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.3. Calculation methods . . . . . . . . . . . . . . . . . . . . . . . . 18.2. Earthquake engineering codes for normal risks . . . . . . . . . . . 18.2.1. Area of application and technical objectives of the codes . . 18.2.2. Current and future earthquake engineering codes . . . . . . . 18.3. Special earthquake resistant devices . . . . . . . . . . . . . . . . . 18.3.1. Earthquake resistant supports made of sandwiched elastomer layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2. Other special earthquake resistant devices . . . . . . . . . . . 18.3.3. Active control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4. Earthquake engineering practices for special risk. . . . . . . . . . 18.4.1. Nuclear power plants and facilities. . . . . . . . . . . . . . . . 18.4.2. Chemical, oil and gas plants. . . . . . . . . . . . . . . . . . . . 18.4.3. Dams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5. Seismic diagnosis and reinforcement of the existing framework. 18.5.1. The different aspects of seismic diagnosis . . . . . . . . . . . 18.5.2. Rehabilitation and reinforcement . . . . . . . . . . . . . . . . .
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Foreword
The book written by Jacques Betbeder-Matibet is a work of art in the field of earthquake engineering. Such a thing as an equivalent book to this one does not exist, not even in another language. Sometimes collections of articles have been published but those articles do not go into the subject in as much depth as the reader might wish. For Jacques Betbeder-Matibet’s book the reader does not actually have to study the preface even though it clearly explains the author’s intentions when writing the book. Writing this book has been a real challenge and Jacques BetbederMatibet uses his great talent to cover all fields related to earthquake engineering. This spectrum goes from applied seismology to preventive techniques and their application, i.e. establishing certain regulations for building and constructions. There is no doubt that a large number of readers will be inspired by this book. It will lead to further reflection and increase the readers’ personal knowledge. The book has the advantage that it can also be used by engineers who are just starting to undergo training or work in the field of earthquake engineering as well as by experienced engineers who would like to carry out research in a field with which they are less familiar. Last but not least, this book might even be very beneficial to the people involved in decision-making processes even though the author did not believe them to be part of his target audience. Earthquake engineering is a relatively young science that reached its peak in the 1960s. It is certainly a field of science where engineers are very tempted to rely on mathematics and equations. Dynamic phenomena that dominate the way in which structures are subject to preventive techniques are indeed very well adapted to meticulous processing based on digital supports. All this amplifies the trend towards research which relies on mathematics and equations. However, the models which are used to evaluate a certain piece of information are sometimes far from the physical reality of the phenomena the engineers are trying to describe. When comparing the actual earthquake to our forecasts we, as engineers, realize the weaknesses within their predictions. Should these attempts at predicting the magnitude of earthquakes
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stop and be replaced by learning from “experience” only? If the answer to this question is yes only a small and very privileged group of researchers would be able to predict these phenomena. Jacques Betbeder-Matibet’s approach clearly disagrees with this idea. He uses observations made during post-earthquake missions linked to simple models. His impressive scientific knowledge enables him to explain a “hierarchy of dimensions” and distinguish reasonable predictions from pure speculation. The simplified models the author uses in his book are not simplified in order for them to be processed and analyzed without the use of complex digital models, but represent phenomena within their natural environment without losing any of their preciseness. While developing theoretical aspects the book also reminds its readers of the fundamental change that is introduced by the publication of an entire book devoted to the field of earthquake engineering. Given the fact that not all aspects can be reflected in a model, i.e. in a calculation – even though some people do believe that this is possible – models remain an often very far-fetched description of reality. This is why other constructive measures based on the common sense of an engineer should often be given preference over very sophisticated calculations. Admitting that calculations are not always very useful does not mean that the engineer is showing his/her weaknesses but underlines the fact that an engineer does not only calculate. His/her role cannot be reduced down to calculations only. Common sense and observing physical reality both still remain pillars in this type of job. We have to thank the author for emphasizing this idea throughout his text since this essential fact will be beneficial to all generations of engineers. As a conclusion I would like to add that I have experienced some fantastic moments with Jacques Betbeder-Matibet and other colleagues whilst working on the development of earthquake engineering in France. I was able to witness how much Jacques Betbeder-Matibet has contributed to the progress that has been made in this field of research. The professional experience Jacques Betbeder-Matibet has gained by working in this field for many years, his very demanding attitude towards science, his intellect and last but not least his approach as far as ethics are concerned served both as guidelines and as a source of inspiration throughout our research. There is no doubt that the readers of this book will appreciate its fundamental qualities and benefit from them just as we were able to do before them. Alain Pecker Honorary President of the AFPS (French Association for Earthquake Engineering)
Preface
It is clear that the knowledge that earthquake engineers possess as well as the tools used in order to prevent earthquakes from taking place have considerably improved since the emergence of paraseismic engineering in the 1960s and 1970s. The improvements which have been made include: – a better understanding of the causes and a better evaluation of powerful earthquakes due to the increase of recordings available of such powerful earthquakes, the increase in the number of study programs carried out on site as well as the development and advances made in digital simulation; – a better understanding of seismic hazards for a particular site or for a particular region, i.e. the type and strength of seismic movements which are likely to occur in the future by taking into account the socio-economic importance and the life-span of existing and future buildings and constructions that may be affected by earthquakes; – a better analysis of the behavior of structures which are subjected to strong tremors, thanks to the work carried out during post-seismic investigations, and thanks to the evolution and appearance of new trial methods (vibrating tables, reaction walls, centrifuges) as well as to the remarkable increase in computer processing capacity (particularly in the non-linear domain). Have improvements as regards the knowledge of earthquake engineers and the improvements in study methods which are used to analyze earthquakes led to advances in relation to the prevention of earthquakes? The answer to this question must be explained in detail because the progress that has been made and which has been mentioned above tends to highlight the complexity and variability of the different phenomena affecting earthquakes and therefore uncertainties still remain when such phenomena are used. For example, in relation to the evaluation of ground movements in a seismic risk study a lot of significant uncertainties remain in terms of the actual potential of the earthquakes
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(precise location of the fault, the number of faults and how active they are) as well as for the calculation of vibratory movements. If we only focus on the calculation of vibratory movements it is possible to mention the practical problems brought about by the description of the three terms (mentioned below) which are traditionally used in the field of seismology: – source, which is characterized by magnitude and which, in reality, depends on other factors such as the type of movement of the fault, possible segmentation of the fault map into zones with different characteristics, the temporal course of the rupture outline of these zones, and the constraints associated with these ruptures. These factors can often be identified and described when there are a sufficient number of recordings available for an earthquake which has occurred in a region and in particular in a region which is well-equipped with the material which makes it possible to record the earthquakes. The factors can also be identified and described from certain hypotheses that have been made and which are deemed plausible, i.e. there is a realistic chance that a particular earthquake may occur in the future in a specific region. However, most of the recordings that are available are deliberately ignored in seismic risk analyses because these analyses, by definition, only consider earthquakes which have not yet taken place and which may occur in the future; – propagation, which is characterized by distance (from homes, from the epicenter, from the fault), depends on the type of seismic waves (volume or area), and on the level of inelastic attenuation reached by the sound waves during their propagation as well as on the possible intervention of the effects of directivity or focalization. As is the case for the source, the influence of these factors cannot be taken into consideration in risk studies; – site, which is characterized by the type of soil (rock, closed soil, soft soil), depends on all geotechnical parameters (thickness, inclination and the mechanical characteristics of the layers of the earth) in relation to the make-up of the soil or geological structure (more-or-less hemmed in valleys, sedimentary basins, and also synclinal and anti-clinal basins). These parameters can be evaluated and considered in simple cases of horizontal stratigraphy but cannot be used in the collection of data for risk studies or in the collection of data on a regional scale and even in the collection of data for the study of individual sites, especially when the seismic response of such sites is strongly influenced by topography (of the ground’s surface or underground). Although it would seem necessary for earthquake engineers to understand and be able to access all of the parameters in order to model and calculate virtual seismic movements in their risk analyses, the impossibility of having access to all of the parameters means that they have to use approximate formulae such as magnitude for the source, distance for propagation and type of soil for the site. This means that the standard deviation of these formulae is equivalent to the average value, i.e. in a fail-
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safe approach which is based on the values “average plus standard-deviation”. The parameters of seismic movement (acceleration, speed, displacement) are multiplied by a factor of two in relation to their values when used in an approach which is based on average values. This fact should never be ignored by the different people working within the field of paraseismic engineering. This fact has been confirmed and proved by all earthquakes for which it has been possible to obtain quite a large number of powerful recordings in the epicenter or in the neighboring areas of the epicenter; the large variability in the recorded movements (which not only occurs because of the site effect) must be considered as the rule of thumb rather than as an exception to the rule. There are also a significant number of uncertainties in the area of seismic engineering where the progress which has been made at a theoretical and experimental level deals with simple cases (regular structures, unidirectional excitation). If for such cases the physical significance of the behavior coefficient (i.e. a coefficient greater than one which can be divided by the effort calculated on an elastic model in order to achieve realistic dimension efforts) had been better defined then the transition to more complex cases (irregular structures with a 3D response) can only be calculated with the use of a relevant coefficient coupled with the judgment of an expert who works in this field and not by a scientifically valid approach. These difficulties in the transposition from the study of simple to more complex cases are found in both the design principles as well as in the calculation methods which are used. The “in capacity” dimension, which is a basic principle of the future European code on how to make building and civil engineering structures resistant to earthquakes (known as Eurocode 8), consists of predetermining the concentration of plasticity zones by providing these zones with the appropriate constructive measures which make it possible to control malformations by maintaining an acceptable capacity of resistance. The dimension also involves the increase in size of the other potentially critical zone in order to be sure that the plastifications only occur where we expect them to. This approach cannot be used for irregular structures which are extremely hyperstatic. In such cases the project designer is unable to control the sequence of successive plastifications that result from 3D excitation which can create unpredictable effects such as seismic movements. The “in capacity” dimension can thus become a hazard if the choice of plasticity zone does not correspond to the real outline, this dimension can often be unexpected and even completely unpredictable due to the transfer of force between the structural elements.
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In relation to the calculation methods used, the pushover approach has recently been suggested (it characterizes a structure by an effort-displacement curve which is obtained through a set of non-linear static equations that represent the action of an increasing force). If the pushover approach relies more on displacement criteria (used more in seismic stress) than on criteria related to forces (used mainly in traditional construction codes and standards), then it only applies (in its current form) to structures which are quite sensitive to the 3D character of seismic movements and which are also quite sensitive to torsion efforts. One of the most common errors made, and one in particular which is made by the decision-makers in relation to the prevention of earthquakes is the belief that the main difficulty lies in defining the actual seismic movement from which the engineer has to work (design, calculation and creation) by using well established procedures as the earthquake “is only a question of force amongst other things” and earthquake recognition is a “simple software problem”. It is surprising that such a simplistic speech, which stems from a misunderstanding of the complexity of the phenomenon of earthquakes and of an over appreciation of state of the art technology in relation to the non-linear calculations under 3D dynamic excitation still holds value in certain instances. Will we see the effects of relying only on the capability of computers coupled with a lack of understanding of earthquakes particularly in countries which experience weak or moderate earthquakes, and in which regions will these effects have to be taken into account? The current limits in our ability to analyze non-linear behavior under seismic stress have clearly had repercussions on the reliability of our appreciation of safety margins brought about by dimensioning, regardless of the strategy that has been adopted (acceptance or refusal of material behavior laws in relation to the field of plasticity). For special risk structures, according to the terminology that is used in France, these are structures that pose significant risks to entire towns or even to a region in terms of the damage that they can cause (nuclear power stations, certain chemical factories, large dams, etc.). Research with a high degree of security up until now has led to the creation of dimensions which are primarily based on elastic calculations and on the criteria of static equilibrium between forces. It is the caution taken in this approach (linked to the conservatism of the static character of the criteria used for dynamic charge) along with a stacking up of coefficients in an approach that is carried out in several phases which forms part of the main causes of the obtained level of security and not, as is often thought, the choices made for the calculations of seismic movements. A paraseismic experiment which was carried out more than 25 years ago has enabled me to address different aspects (methods of calculation, paraseismic devices, regulations, post-seismic work) for the different types of structures (nuclear power stations, dams, bridges, tunnels) and has also given me the opportunity to
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work with specialists from several disciplines (geologists, seismologists, soil engineers, as well as civil and mechanical engineers). This work convinced me that it was necessary to make those people involved in the creation of earthquakeresistant designs and structures aware of the factors that they sometimes did not pay enough attention to, or which were deliberately ignored by these very same people. The majority of these factors will probably contradict a more triumphalist view of paraseismic engineering. These factors include: – the recent and incomplete character of the information available on powerful seismic movements and their effects on certain types of constructions. It occurs quite often that a new earthquake which has been recorded and studied, in relation to its consequences, highlights certain things which up until now have been underestimated or completely ignored in terms of both the movement of the ground (such as the killer pulse which is a strong oscillation at low frequency and which is felt at neighboring faults) as well as the behavior and reaction of the structures (for example the reaction of buildings with welded metal frames during the earthquakes at Northridge in 1994 and Kǀbe in 1995); – the importance of experience when analyzing the calculations. There is a tendency to forget that the basis of the paraseismic codes which are applied to everyday constructions are applied for a purely practical reason, and in particular in relation to constructive precautionary methods. The preeminence of feedback must be ensured especially at a time when common sense and critical thinking are being replaced by the use of computers and calculations; – the fundamental role of the detailed design of the different methods used for effectively preventing earthquakes from causing too much damage. Media coverage tends to show the damage caused by an earthquake and prefers to highlight the faults or the refusal to apply preventative regulations which certainly play a part in but which are not the main causes of earthquake disasters. The main causes of these disasters generally come from the fact that the paraseismic codes and standards do not apply to new constructions and only affect a small number of buildings in a town if we take into consideration the recent date for when it became compulsory to apply these standards in the majority of countries worldwide, as well as the vulnerability of the constructions built before this date. The paraseismic codes and standards which have been introduced do not make it possible to pass value judgments on the design of buildings which means that two structures that meet the standard requirements can possess very different safety levels in the sense that one of them can resist powerful earthquakes (which are more powerful than predicted in the codes) without collapsing, whereas the other one which does not have any reserves will collapse; – the current development of the majority of paraseismic codes which on one hand is characterized by increasing complexity. It can be questioned if this increasing complexity is justified because of the current knowledge available in
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relation to the creation of earthquake-resistant designs and structures, and if it will pose or is already posing practical problems in terms of the classification and the correct use of the codes. On the other hand this increasing complexity is characterized by a somewhat dogmatic and illusionary presentation; this type of presentation and the fact that these codes are standardized (which reduces explanations on the required measurements to a minimum) tends to obscure the fundamental importance of the detailed design of the methods used for effectively preventing earthquakes from causing too much damage; – the risks of confusion, in the field of creating earthquake-resistant designs and structures just as in other areas, between research and practical engineering; the unquestionable progress which has been made by researchers in the seismology of strong earthquakes as well as in the analysis of structures are often difficult to echo in operational procedures. These difficulties focus primarily on the availability of necessary data for the implementation of more elaborate procedures, as mentioned earlier for the calculation of earthquakes in a risk study; by way of a comparison the research work carried out in paraseismic engineering is similar to the research work undertaken by a medical examiner which rests on the dissection of the body to be studied (recordings and post-seismic observations, models subjected to trials, results from paraseismic studies on digital models). This essential work does not necessarily have any immediate positive consequences on preventive medicine (“constructive hygiene”, i.e. design) or the vaccination policy (the contents of paraseismic codes and their imposition by statutory means). It is the factors that have been described above coupled with the lack of understanding of such factors by some of the people working in the creation of earthquake-resistant designs and structures and in the minds of the majority of decision-makers which have been my motivation behind the writing of this book whose first edition (in French) was published in 2003. The aim of this book is not to explain what paraseismic engineering is or to explain some of the aspects which form part of this topic (such as the seismology of strong earthquakes, the dynamic calculation of structures or the principles of paraseismic design) for which excellent texts are currently available, but to give a personal point of view on the following three subjects: – the analysis of the current knowledge that earthquake engineers possess. This analysis was created in 2000 and aims at distinguishing between what information is available (in the long term) through results from research, from information which can now be used under certain conditions instead of in the current methods of paraseismic engineering; – the role of the generalist which, in my opinion, is vital. The gaps in our knowledge and the extent of the uncertainties that exist in assessing the level of safety to be researched, depending on the type of structure and on the definition of
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the methods used in order to reach the required level of safety should not result from a series of decisions which are taken only by specialists working in different fields. When paraseismic engineering was as its founding stage, at a time when there were not a lot of recordings of strong earthquakes (and at a time when not a lot of seismologists were interested in this branch of seismology), generalists were recruited alongside the engineers who wrote and edited the codes. Current developments, which are geared towards an ever-increasing specialization of the different people working in this field, make it increasingly difficult for the engineers to improve their career prospects when it comes to working in this job as a multidisciplinary vocation. Career development is just as important nowadays as it was in the past, especially for the co-ordination of studies and for controlling the coherence of the different choices which contribute to the best protection possible against earthquakes; – the importance of being able to remember the size and scale of earthquakes (or to be able to find them through reasoning or simple formulae) which not everyone can remember because earthquakes do not occur that often in our lifetime and when they do they only last for a short period of time. The fact that the majority of people working in paraseismic prevention, at least in countries with moderate seismic activity, have practically no personal experience of earthquakes exposes them to imagine what powerful earthquakes might be like or to make errors when estimating the scale of earthquakes. An understanding of the size and scale of earthquakes is therefore much more essential in paraseismic engineering than in other engineering domains and can be acquired by understanding earthquakes and by comparing some of the earthquake models which have been created in order to simulate earthquake processes. The nature of these three subjects and the limits of my knowledge mean that the text which I have written is subjective and will certainly contain certain caps, questionable judgments or even errors. The approach that I used was to review the different aspects of paraseismic engineering in a logical order (i.e. the phenomena associated with paraseismic engineering, the quantification of their appearance, the description of their effects, the principles and methods used in the prevention of risks). Each aspect has been commented upon in relation to the knowledge that the earthquake engineers have on that specific aspect as well as on determining the orders of magnitude. I have tried to state hypotheses and their limits in terms of their validity as well as stating the pros and cons linked to feedback. Certain parts of this book are rather descriptive and serve as a history to the evolution and development of ideas which in my opinion is very important for the training of generalists. The evaluation of the orders of magnitude relies on analytical calculations on simple models by following a traditional approach which may seem outdated in this era of computing and modern technology, but which forms the basis
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of the engineer’s job, as long as the engineer does not solely rely on the use of computer software. As far as the calculations are concerned I have done my best to only use basic mathematical methods which normally form part of the basics that is taught to engineers. I have also done my best to distinguish between what comes from deductive reasoning taken from hypotheses and the results that come from feedback. The outline that has been adopted and the content of the different chapters have been chosen so that the text can be read by someone who has no previous knowledge of paraseismic engineering on the condition that the reader is prepared to make a certain effort in terms of assimilation. Certain formulae are given without the explanation of their calculations. Some parts of the text include relatively specialized developments which have occurred within the field of paraseismic engineering and which can be omitted by people who are reading the book with the sole aim of having an overall view of paraseismic engineering. The book is divided into seven parts. Part 1 introduces the seismic phenomenon from the point of view of its causes and what the phenomenon appears like on the surface of the Earth. The presentation of this part (for which I was largely inspired by the layout of a large number of popularized books) aims at covering all of the important aspects which should be taken into consideration in relation to paraseismic prevention (and in particular the following resulting phenomena: soil liquefaction, landslides and tsunamis), yet the first part remains essentially qualitative and not too detailed, except where faults and the significance of the magnitude are concerned. As far as these last two subjects are concerned I believe that the majority of engineers have insufficient knowledge regarding them both, which in turn does not enable them to have a clear perception of the ideas of focal depth, distance from the source and the extension of the fault map. A simple mechanical model, based on the theory of elastic rebound by H.F. Reid, and the examination of a certain number of well documented cases of faults have led to the definition of the moment magnitude and to its interpretation in terms of energy, the extension of the fault map and the range of potential damage that earthquakes can cause. Precise indications are also given on seismic waves and their propagation without which it would not be possible to understand both the softening mechanisms of movements and the causes of site effects. Strong vibratory movements, which are the basic elements for the definition of seismic action, are the subject of the second part of this book. In this part there is an introduction to strong-motion recordings (without addressing the issues linked to the instruments used and the processing of the signals). In Part 2 there is also a presentation of the softening laws that are applicable to earthquakes which have been derived from theoretical models, as well as simple diagrams which are used in order to explain site effects and directivity effects and which are also used to
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estimate the dominant frequency of accelerograms (in terms of displacement, speed and acceleration). The current state-of-the-art digital simulation material used for seismic movements is briefly mentioned both for rupture models on the fault map and for the sites’ response in the linear and non-linear domain. Part 3 deals with the seismic risk in relation to the data that characterizes the spatial and temporal distribution of seismicity and its evaluation methods (both determinalistic and probabilistic). Indications on risk studies are given for studies carried out in entire countries (zoning of the paraseismic codes), towns or small local villages (micro-zoning), or on individual industrial structures (especially in the case of nuclear structures). Orders of magnitude are supplied for hazards which occur due to the faults (surface ruptures and vibratory movements) and which occur because of a non-localized seismic zone, so that the influence of certain parameters (e.g. the envisaged maximum magnitude, the depth of homes and residences, and the dimensions of extended sites such as large tunnels and large towns and cities) can be evaluated. Long-term and short-term seismic forecasting is also mentioned. Seismic action, i.e. the characterization of seismic phenomena relating to the calculation of their effects is presented in different forms in Part 4 (seismic coefficients, response spectra, accelerograms and random processes). This subject is undoubtedly one of those subjects that is misunderstood the most, even by some specialists who work in this field and which may be due to the fact that the study of seismic actions is the interface between two different disciplines (seismology and engineering). Seismic action is, on one hand, linked to safety objectives regarding the creation of buildings and structures that are resistant to earthquakes, and on the other hand linked to the calculation methods and verification criteria that are used. Characterizing the seismic phenomena through the use of a response spectrum, which is the most commonly used approach, is linked to the use of linear models for carrying out calculations. Such calculations can be questioned in the case of paraseismic codes that are applied to everyday buildings that are subject to a high level of plastic damage, i.e. due to the non-linear behavior. In the case of using non-linear models, reference to the spectrum is not very appropriate for the choice of entry accelerograms of these models whenever the plastic damage mechanism is cumulative and therefore depends on the duration of excitation (which is poorly represented by the response spectrum). Assessment and evaluation elements are provided and they explain the limits of the use of the spectra and on the selection of accelerograms for both linear and non-linear calculations. Part 4 also introduces the coefficient (known as behavior) which is used in the majority of recent paraseismic codes in the only case where the coefficient can be precisely defined, i.e. when used in regular structures which can be represented by an electro-plastic oscillator (a model with one degree of freedom). This introduction
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makes it possible to highlight the predominance (for security assessment) of the criteria of displacement on the criteria of equilibrium that exist between forces and that are generally used in codes nowadays. Resorting to random and unpredictable processes in order to characterize seismic action is explained in a simple fashion by only referring to the case of pure white or filtered noise in such a way that in Part 6 quadratic combination rules can be justified and the methods of stochastic linearization can be presented. Part 5 describes the effects of earthquakes on buildings and constructions; these effects also form the databases from where the paraseismic codes for the different materials that are commonly used in construction work are taken (i.e. concrete, steel, brickwork and wood). All possible variants of the seismic phenomenon are dealt with; surface ruptures, reversible and irreversible deformations of the ground, vibrating movements which shake buildings or which make them collide with one another, and resulting phenomena (such as liquefaction, effects on traffic, effects on the environment and fires). These descriptions are supplemented by comments on the influence of the overall design and detail of the causes of the damage that is observed as well as on the practical problems which the interpretations of postseismic observations can pose. The use of the effects of earthquakes in order to characterize their level (i.e. the concept of macroseismic intensity) is also presented in Part 5. A short introduction is given to the scales of intensity of the earthquakes, to the abbreviated description of some of the scales and there is also a discussion on the values and limits of this motion of intensity. The digital correlations of intensity along with the parameters of movement (acceleration, speed) and the magnitude of the earthquakes are then studied as well as the softening laws in relation to distance. Part 6 is the most developed part of the book. It is devoted to seismic calculations and is made up of three chapters: – the first chapter (Chapter 15) deals with linear calculations in the form of spectral model analysis which is used for most linear models. Its principles and different phases (such as determining the relevant elements, frequencies or periods used in spectral model analysis, and model distortions, combinations of model responses and directions of excitation as well as the stress calculations used for dimensioning) are presented for general seismic calculations (different ground movements under the supports of structures) and in the case of larger scale movement (translation or translation with rotation) of all the supports. The emphasis is placed on the problems that can be encountered in the selection of the methods used and the use of pseudo-models as well as on certain difficulties that are linked to the application of quadratic combination rules. Attention is given to the risk of errors which is insufficiently understood by the users (especially by those who use and rely
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on black box software). These errors are a result of reciprocal incompatibilities between the displacement fields, the acceleration fields and the internal efforts which are determined by quadratic combinations. In this chapter I also show that the common practices used in the calculation of dimensioning efforts stemming from “maximum” acceleration (in the sense of quadratic combinations) lead to a systematic overestimation that is often moderated (by 15% to 25%) but can sometimes be moderated by more than 100% (which is absurd) for certain, sometimes quite simple structures (such as a continuous section console which is embedded in its base foundations); – the second chapter (Chapter 16) gives an insight into the phenomena surrounding the interaction between the ground and the structures and their consideration in dimension studies. The consideration of such phenomena normally lies with the specialists who work in this discipline (a discipline that can be found at the interface between seismology and the dynamics of soils and structures). However, it seems necessary to me to provide the generalist working in this field with the necessary tools so that they can estimate the size and scale of the earthquakes with the aim of being able to appreciate the interaction phenomena, the influence of the different parameters and the difficulties that can be encountered in relation to radiation, which occurs from the waves that are emitted in the ground from the foundations of the structures; – the third chapter (Chapter 17) introduces non-linear calculations. In this chapter some generalities on the hypothesis and the acceptability of the results can be found (which forms part of the most sensitive issue that needs to be resolved if we want non-linear calculations to become common practice in the dimensioning process). Chapter 17 also gives a brief introduction to the methods used in nonlinear calculations including those methods which rely on linear techniques (stochastic in particular). Six examples of non-linearity are then described and commented upon; these examples have been chosen in order to illustrate the diversity of problems and to establish some formulae in relation to the scale and size of earthquakes and which are relative to phenomena that are widely misunderstood. Amongst these examples we can mention the non-linearities linked to the liaison with the ground (the detachment of concrete slabs, the rocking and sliding of blocks), and the plastic deformations of structures (already mentioned in Part 4 when talking about the behavior coefficient) which also gives the opportunity of introducing the pushover method and the design of shock-absorbers that are used for making bridges resistant to earthquakes (illustration of the stochastic linearization method). By way of a conclusion the different aspects of paraseismic prevention are the subject of Part 7. In relation to the technical aspects, the information and commentaries which have been made in the preceding parts are supported and completed by a brief presentation of the experimental methods used (feedback
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synthesis, trials and experiments using vibrating tables, loaded oscillators, reaction walls or centrifuges, and static experiments with presses). Special paraseismic devices and the strengthening of the existing frame of buildings are also presented (these form part of the key issues in paraseismic prevention in both the short and mid-term, since the application of paraseismic codes only affects new constructions). The principles and methods used in the application of technical texts (standards, practical guidelines, recommendations by professional associations) are also briefly presented and commented upon for both everyday constructions (affected by a normal risk of earthquakes according to the French terminology) and for everyday constructions that are affected by a special risk (once again in accordance with the French terminology which has been used and explained above). Experience has shown that there can be a significant difference between what the engineers actually do and what they think they can do when you take into consideration expressions which come from articles such as the non-collapsing objectives, intrinsic protection or the maximum earthquake. I would like to express my gratitude to the Service of Thermal and Nuclear Studies and Projects (Service Etudes et Projets Thermiques et Nucléaires or SEPTEN) at Electricité de France where I spent the majority of my career and whose material support enabled me to digitize this book. I would also like to thank Danièle Chauvel and Jean-Pierre Touret, the coordinators of the digitization of this book. My sincere thanks also go to Alain Pecker who accepted to write the preface of the book, and I would also like to add to this how honored I was when he accepted to write the preface. The contents of this book come from the many exchanges that I had with the many people who worked with me at Electricité de France, with people from other companies and organizations, with a certain number of work groups or commissions and with people who worked with me on several post-seismic missions. Their points of view may be or may have been different from mine but they have all contributed to the development in my way of thinking. I have tried to thank everyone who has helped me during the creation of this book and I apologize if I have forgotten to mention anyone; I would therefore like to thank N.N. Ambraseys, D. Amir-Mazaheri, D. Aubry, P-Y. Bard, M. Belazougui, P. Bernard, P. Bisch, M. Bouchon, M. Bour, C. Boutin, A. Capra, P. Combes, D. Costes, F. Cotton, G. Czitrom, J. Dalbera, V. Davidovici, J. Despeyroux (+), B. De Vanssay, J-L. Doury, J-Y. Dubié, C. Durouchoux, E. Faccioli, H. Ferrieux (+), F. Gantenbein, J-C. Gabriel, P. Godefroy (+), B. Grellet, Y. Guillon, W. Jalil, M. Kahan, M. Koller, P. Labbé, J. Lambert, A. Levret-Albaret, M. Livolant, P. Mailhé, C. Martin, J. Mazars, J-P. Méneroud, B. Mohammadioun, P. Mouroux,
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N.M.Newmark (+), N. Orbovic, A. Pecker, C. Plichon, J.C. Quéval, J.M. Reynouard, O. Scotti, J.F. Semblat, J-F. Sidaner, P. Sollogoub, R. Souloumiac, P.E. Thévenin, J-P. Touret, E. Viallet, J-P. Walter, F. Wajtkowiak (+), P.Yanev and T. You. 16 January 2008 J. BETBEDER-MATIBET
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Part 1
Earthquakes and Induced Phenomena
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Introduction
The first part of this book deals with earthquakes as a natural phenomenon. However, only the aspects that are of direct interest to those people involved in the field of disaster prevention will be covered. Therefore, this is not a lecture on Earth sciences such as geology, geophysics or seismology which are the basic tools when studying earthquakes. The aims of these fields of study are of greater importance to the general public than the field of study that deals with earthquakes and the reduction of risks. Seismology mainly helps scientists try to understand the internal structure of the globe. Only a minority of researchers in seismology focus on the effects earthquakes have on buildings. In comparison to all other natural disasters, earthquakes are experienced the least often because they do not occur frequently enough when we consider the average lifetime of human beings. Even inhabitants of regions such as California or Japan, which are more likely to be hit by an earthquake, are very unlikely to suffer from the consequences of a major earthquake. Nevertheless, they will certainly experience a high number of smaller earth tremors. In zones that are not very prone to earthquakes the time-span between major earthquakes can be several centuries. This is why there is no collective memory of such events. As a result, the effects of earthquakes are often not very well known. In fact this lack of knowledge does not only concern the general public but also engineers working in the field of earthquake construction. They are not always aware of the dimensions involved and where certain issues are more important than others. Understanding these two concepts is, however, the basis of an engineer’s job. These errors are not rectified but rather emphasized by the media, especially TV which nowadays covers the breaking news of earthquakes or tremors. Regarding computers as omnipotent and assigning a higher importance to their calculations
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rather than observing what actually happens during earthquakes has only made the situation worse. When it comes to the media, only showing what has been destroyed and creating a polemical debate on who is responsible certainly stems from the obligation of delivering and presenting information. However, often simplified and biased versions of complex realities and seismic effects are broadcast as the media only focuses on the relative success or failure of preventive techniques. Computers and the excessive importance that is assigned to them certainly represent a major difficulty when it comes to training the staff who will be in charge of preventive techniques. This part will provide engineers with useful information on the sources of earthquakes and the parameters that are used to describe the effects these phenomena have on the surface of the globe. Physical effects such as phenomena linked to tectonic plates and seismic waves that affect natural sites, as well as resulting phenomena such as soil liquefication, landslides or tsunamis, will also be covered. The impact on the buildings which are affected by the different levels of an earthquake’s intensity is covered in Part 5. The parameters that describe powerful earthquakes as well as the data on them which are stored in a database for preventive engineering will be addressed in Part 2.
Chapter 1
Causes of Earthquakes
1.1. Tectonic earthquakes The great majority of earthquakes are tectonic. Tectonics is the branch of geology that studies the structure and movements of the uppermost parts of the Earth. A coherent theory explaining these movements and the seismic activities that result from it was only formulated and substantiated through experimental observations towards the last quarter of the 20th century. 1.1.1. First attempts at explanation 1.1.1.1. Religious and superstitious beliefs Primitive cosmogonies were based, in many regions, on the idea that the Earth, often represented as a flat plate and not a sphere, was carried by gigantic animals: elephants in India, a water buffalo in China, a frog in Mongolia, a turtle with many of the native peoples of Canada, etc. Earthquakes, according to this concept, were the consequence of movements that these animals made from time to time, tired of carrying their heavy load. Thus we find the following account (see [GOU 94]) reported in 1637 by Jesuit missionaries staying with the Huron of Quebec: “the Father explaining to them some of the circumstances of the passion of Our Lord and speaking about the eclipse of the sun and the quaking of the earth which was felt at that time, they replied that in their country there was spoken of a great earthquake long ago and the sudden eclipse of the sun which was believed to have occurred because the great turtle supporting
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the Earth, changing his posture, turned his shell towards the sun and robbed the world from its sight.” In Japan, a popular belief was held until the end of the 19th century according to which earthquakes were due to the chaotic agitation of namazu, an enormous catfish that lived in the depths. The violent earthquake of 1855, which hit Edo (which today is Tokyo), coincided with the ritual period of the “month without gods”, during which time the divinities go away on a pilgrimage (see [WAL 82]). The god Kashima, whose mission is to force the namazu to keep still by pinning a heavy rock on his head, had gone away for the pilgrimage. The giant cat-fish then took advantage of this by causing earthquakes. We can see in Figure 1.1 a reproduction of an etching on wood made after this earthquake: the inhabitants hurry over to attack namazu, while the smaller figures on the top right, who are builders and artisans profiting from the reconstruction work, run to his defense.
Figure 1.1. Representation of the namazu in an etching on wood (namazu-e) made after the 1855 Edo (Tokyo) earthquake
Divine intervention in the occurrence of earthquakes, which is shown with reference to namazu in Japan, is also found in the West. For the Greeks, the person responsible for earthquakes was not the Titan Atlas, who carried the world on his shoulders, but the god of the sea, Poseidon. This belief, without doubt, lies in the fact that the earthquakes that affect the coastal regions of the eastern Mediterranean often begin in the ocean and are sometimes accompanied by tsunamis.
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Monotheistic religions, particularly Christianity, have emphasized the role of divine intervention, sometimes to the extent of dogmatic delirium. An earthquake is perceived, in most of the chronicles dating from before the 19th century, as a chastisement from God, as punishment for man’s sins. This idea appeared in the Old Testament and was repeated in numerous sacred texts, prayers or sermons. In his monumental work on the history of the decline and fall of the Roman Empire, the first volume of which appeared in 1776, Edward Gibbon concluded his description of the catastrophe of 21 July 365 (a great earthquake accompanied by a tsunami on the coasts of Sicily, Greece and Egypt) with the following ironic declaration: “It was the fashion of the times to attribute every remarkable event to the particular will of the Deity; the alterations of nature were connected, by an invisible chain, with the moral and metaphysical opinions of the human mind; and the most sagacious divines could distinguish, according to the colour of their respective prejudices, that the establishment of heresy tended to produce an earthquake, or that a deluge was the inevitable consequence of the progress of sin and error. Without presuming to discuss the truth or propriety of these lofty speculations, the historian may content himself with an observation, which seems to be justified by experience, that man has much more to fear from the passions of his fellow-creatures than from the convulsions of the elements” (see [GIB 83]). In the middle of the 18th century, the progress of the scientific outlook resulting from the Enlightenment failed to win over the most zealous partisans of the religious theory of earthquakes, who knew how to play on the latest inventions to justify the anathema. Thus in Boston, after a strong earthquake on 18 November 1755, Reverend Thomas Prince did not hesitate to judge that divine anger, manifested through the earthquake, could assume an electric nature and be attracted by the forest of lightning conductors that his famous compatriot Benjamin Franklin had had installed on the roofs of the city (see [WAL 82])! 1.1.1.2. Early non-mythical hypotheses From time immemorial, there have been attempts to explain earthquakes rationally. Certain Babylonian astronomers presented the hypothesis of a causeeffect relationship between these phenomena and the alignment of stars, an idea which has even sometimes been repeated today. Among the Greeks, it is to the philosophers of the city of Miletus, Thales (624-546 BC) and Anaximander (585525 BC) that we owe the first suggestions attributing earthquakes to natural causes. For Anaximander, seismic movements resulted from internal ruptures of the Earth under the effect of its own weight (see [KAG 99]), a concept which is not too far from the idea today, where tectonics due to convection currents at the heart of the mantle (see section 1.1.2) has replaced gravity as the driving phenomenon. It is however the hypotheses of Aristotle (384-322 BC) which had the most lasting influence, until the 17th century in Europe, as in all domains of philosophy
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and science. They are however rather vague and not very credible, since earthquakes are shown to be consequences of the mysterious action of subterranean winds. For Aristotle, in fact, the interior of the Earth was made up of many big caves, from where imprisoned air sometimes violently escaped producing “earthquakeproducing windstorms”. We must nevertheless give credit to the Greek philosopher for an interesting observation: “places where the soil is porous are most disturbed, because of a large quantity of wind that is absorbed” he wrote (see [WAL 82]), which in the light of current knowledge, is perhaps the first mention of the amplifying effect often observed in poor quality land (see section 4.3), even if his explanation of the absorption capacity of wind may make us smile today. In the 17th and 18th century, scientific progress in Europe saw the gradual receding of Aristotelian dogma, which, in the domain of the causes of earthquakes, led to the abandonment of the theory of subterranean winds to be replaced by the chemical origin of seismic phenomena. This doctrine was clearly formulated for the first time by M. Lister (1638-1712) and N. Lemmery (1645-1715): earthquakes were due to internal chemical explosions caused by the mixing of reactive products in keeping with what was then known about making explosives. Great minds, such as, amongst others, Newton (1642-1727) and Buffon (1707-1788), agreed with this theory. It must be recognized, however, that such a hypothesis, even if seemingly more plausible than Aristotle’s, remains the domain of pure conjecture with no experimental justification to support it. Among the significant contributions of this epoch, mention must be made of J. Michell (1761) who, while analyzing the effects of recently occurred earthquakes (England 1750, Lisbon 1755), reached the following conclusion, with an amazing insight considering the means of observation available at the time: “earthquakes are waves provoked by masses of rock which shift many miles under the surface”. He was also the first to estimate the propagation velocity of these waves (see section 3.2). As for the origin of seismic phenomena, the thoughts of Michell remain along the lines of the then commonly established theory of explosion. In the 19th and early 20th centuries, some important advances were made in several domains concerning the comprehension of the seismic phenomenon (see [BEN 85]): – establishment of basic equations of dynamic elasticity (C. Navier, A. Cauchy, D. Poisson from 1821 to 1831), identification of the main types of seismic waves, of volume (D. Poisson 1828) and of surface (Lord Rayleigh 1885, A. Love 1911), studies of vibrations of elastic bodies (D. Poisson 1829, Lord Kelvin 1863, H. Lamb 1882-1889, V. Volterra 1894); – development of the first seismometers (J. Forbes 1841, L. Palmieri 1855, F. Zöllner 1869, J. Milne – J. Ewing – T. Gray 1880, E. Wiechert 1900, B. Galitzine 1906), the first recordings (J. Milne 1880 for a local earthquake, E. Von Rebeur –
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Paschwitz 1889 for a teleseism), first development of a seismograph network (J. Milne 1892-1894 for Japan); – study of the land after destructive earthquakes (R. Mallet 1857 for an earthquake east of Naples, Imperial committee for investigation for the Mino – Owari earthquake in Japan 1891, R. Oldham 1897 for an earthquake in India), publication of the first catalogues and the first seismicity maps (Von Hoff 1840, R. Mallet 1860, J. Milne 1900, F. Montessus de Ballore 1900), establishment of scales of intensity (De Rossi 1874, De Rossi-Forel 1883-1884, G. Mercalli 1902); – observation of land showing the relationship between earthquakes and faults (Kutch earthquake in India 1819, B. Koto 1893 after the Mino-Owari earthquake in 1891, H. Reid 1906-1911 after the San Francisco earthquake in 1906). This remarkable progress in knowledge and the development of means of study contrast with the stagnation in the comprehension of actual causes of earthquakes. Observations establishing a probable correlation between movements of faults on the surface and earthquakes were often received with skepticism by many geologists, essentially because the majority of earthquakes are not accompanied by well characterized permanent deformations of land surfaces. Those that are may therefore appear more as exceptions than as proof of validity of the general theory. The more analytical minds had understood in the mean time, by the end of the 19th century that earthquakes could only be related to forces and displacements responsible for the creation of land form. However, this conviction in itself was not sufficient to support a coherent system able to explain orogeny (formation of mountain ranges) and seismogeny (production of earthquakes). This is what is expressed, with a certain frustration, by F. Montessus de Ballore in the following declaration (see [MON 11]): “If grosso modo, we do know with some certainty that earthquakes result from general geological forces which have created mountains, folded, fractured and thrust one land stratum on top of another, are we better informed about the real origin of seismic phenomena? Certainly not, the solution of the problem has only been extended and transferred to the problem of orogeny or mountain building.” 1.1.2. From continental drift to plate tectonics 1.1.2.1. The intuitive genius of Alfred Wegener In the early 20th century, geology taught that the continents were laterally immobile; that vertical movements were possible, either as a consequence of the old theory of contraction resulting from cooling, or as a result of the concept of isostasy (the continents float on a fluid base made up of denser rock), which was new at the time.
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However, ever since the establishment of the first fairly precise maps of the Americas, the complementarities of the outlines of the coasts of the new continent with that of Europe and Africa have attracted the attention of some curious minds, such as Francis Bacon in 1620. The idea that this cartographic similarity could have been the result of a relative movement of separation between continents which were once joined took a long time to emerge and attempts to explain it were not at all convincing. A. Snider Pellegrini (1858) thus attributed the cause of displacement to the Deluge (see [HAL 79]). It was Alfred Wegener (1880-1930) who for the first time formulated a universal theory of structure and evolution in time of the structure of the earth’s crust. In the beginning, he too emphasized the coincidence between the littoral contours of South America and Africa, he established the continuity between these continents based on evidence/proof on the comparison of geological structures and animal and plant fossils; for example, ancient rock formations (cratons) can be assembled in continuous chains when the two continental blocks are fitted together (see Figure 1.2). The hypothesis accepted at the time to explain this continuity, was the collapse of land bridges which would have once joined America to Africa, considered fixed in the position they occupy currently. This collapse was one of the manifestations of contraction due to global cooling. Wegener pointed out several contradictions in this theory, for example the distribution of mountain ranges, essentially confined in narrow and elongated bands, while a thermal contraction would have produced a uniform distribution, identical to the distribution of wrinkles on the surface of an apple; or again the observation of altitudes on the Earth’s surface (including ocean floors) which shows that its largest part is found at two distinct levels (the majority continental and abyssal plains), which does not seem compatible with a model in which the land form results from occasional vertical movement (the distribution of altitudes should then have been distributed in an approximately Gaussian manner around a mean level) (see [HAL 79]).
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Figure 1.2. Continuity of cratons between the east coast of South America and the west coast of Africa (according to [HAL 79])
Wegener presented his notions in a book titled The Origin of Continents and Oceans, published in German in 1915 and later translated in many languages. Based on arguments in favor of continental mobility, he formulated the hypothesis that the continents were once joined as a single mass which he called “Pangaea” (“one earth” in Greek) and had since drifted apart until they reached their present positions. These movements explained climatic modifications and variations in the distribution of flora and fauna, attested by the study of fossils. The forces responsible for these displacements were attributed to tides, affecting the viscous layer on which the continents floated. This hypothesis on the driving mechanism of continental movements constituted the weakness in Wegener’s theory. H. Jeffreys demonstrated through simple calculations that if this force driven by tides was powerful enough to produce such displacements, it would rapidly modify the rotation of the Earth. This pertinent objection is one of the reasons for the hostility towards the ideas of Wegener expressed by a very large majority of geologists and geophysicists from 1925, whereas the initial response had been quite good, even though a number of specialists remained skeptical. The fact that Wegener was not a trained geologist (after a doctorate in astronomy, he followed a career in meteorology) and thus appeared an “amateur” who made up theories in a field that was not well known to
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him was, without doubt, the main reason for this rejection, along with the disturbing newness of his ideas. Even though Jeffreys’ refutation concerned only one of the aspects of the theory, it was later branded, often with much ill intent, as the proof of absurdity of the whole concept. Hallam reports [HAL 79] that one of the participants in the symposium of the American Association of Petroleum Geologists (1928) cried out: “If we accepted the Wegener hypothesis, we would have to forget everything that we have learned over seventy years and start again at square one!” Like many other pioneers, Wegener’s mistake was being right too early, at a time when people were not prepared to fundamentally question their ideas. We can indeed ask if this era was really one of progress. 1.1.2.2. Sea floor spreading This is the study of ocean floors which, thirty years after the death of Wegener, presented irrefutable evidence of the lateral mobility of continents. Readings from the depths of these floors establish mid-oceanic ridges in all the oceans. These accident lines characterize the ocean floors either by causing valleys 20 kilometers wide and several kilometers deep or topographic ridges that are several hundred meters in height, whose land form was not widely acknowledged as their width varies in general from 5 to 20 kilometers. These ridges extend over more than 70,000 km over the whole surface of the globe, and are situated in the central zones of the oceans, from which we obtain the name mid-oceanic (see [MAD 91]). In the early 1960s, H. Hess and R. Dietz proposed the same hypothesis, independently of each other, to explain the surprising thinness of sediment layers deposited on the ocean floors. With the current rates of sedimentation, it would take 100 to 200 million years to obtain the thickness observed, which is very little, as compared to that of over three billion years, which is admitted as the inside limit of the period of history of continents and oceans. For Hess and Dietz, the fact that the ocean floors are extremely young (confirmed by the age of oceanic islands and submarine volcanoes) is a result of their continual renewal by the addition of new materials in the ridge valleys. The existing floors and their sediment cover was progressively pushed back on the one hand and elevated into ridges on the other through the emergence of new floors (see [HUR 79]). This concept was rapidly verified by the number of anomalies in magnetization of rocks on the ocean floor. It was already known from the study of terrestrial volcanic rocks, on the one hand that they became magnetized when their temperature went below Curie point (around 600°C) and on the other that the direction of the magnetic field on Earth reversed itself numerous times over the course of time in geology.
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Oceanographic measurement surveys demonstrated that ocean floors presented large stripes of unequal width where the magnetic field was sometimes normal (in the same direction as the current field) and sometimes reverse (oriented in the opposite direction). In 1963, F. Vine and D. Matthews demonstrated that the disposition of these opposite stripes of polarity was more or less parallel to the ridges on either side, which was a confirmation of the hypothesis of Hess and Dietz. In fact, if there is a continual addition of materials which fuse along the axis of the ridge, during the cooling process they register the direction of the Earth’s magnetic field, and while progressively moving away, carry with them the imprint of successive changes in polarity, which hence has a configuration parallel to the axis of the ridges. Their observations, carried out on one of the ridges of the East Pacific, were quickly confirmed by studies on other ridges. An important result of these studies was obtaining numeric values for the rate of lateral separation from the ridges. As was already known from the study of lava on land, the age of different inversions of magnetic polarity, the measurement of the width of any given magnetic stripes made it possible to calculate the rate of spreading of the ocean floors. Rates measured in this way are in the order of a few centimeters per year, which is compatible with the idea mentioned earlier of sea sediments being young. Figure 1.3 allows us to visualize the spreading rate of the sea floor for two examples of ridges. It also shows us that the ridges are broken up into numerous segments separated from each other by fractures perpendicular to their axis. These fractures, or transform faults (T. Wilson, 1965), constitute a character trait of the topography of sea floors.
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Figure 1.3. Mid-oceanic ridges in the Atlantic and Eastern Pacific; the present positions of floors formed 10, 20, 30 million years ago are indicated by dotted lines parallel to the ridges (thick lines) (according to [HUR 79])
The spread of the ridges now appears to be an incontestable reality; most of the specialists attribute it to thermal convection currents affecting the upper mantle, which is the part of the globe between the surface crust and a depth of around 700 km. The ascending branches of these currents come up to the ridges, where they would result in a continuous creation of an oceanic crust through partial fusion and hydration of the mantle rocks. This hypothesis is still difficult to reconcile with the fragmentation of ridges in segments separated by transform faults, as thermal convection cells would need to have a very complex geometry to be able to create such an effect. Whatever the case may be, the validity of the theory of sea floor spread and plate tectonics which follows from it is not questioned because of uncertainties regarding the driving mechanism of the observed movements. We must not make the mistake again of completely rejecting Wegener’s theory under the pretext of not understanding the how and why.
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1.1.2.3. Plate tectonics The establishment of the renewal of sea floors would rapidly result in the creation of a model of the globe showing the evolution of the surface of the Earth. In 1968, J. Morgan, D. Mac Kenzie and X. Le Pichon proposed such a model, based on the following principles (see [HUR 79]): – the Earth’s surface is made up of rigid plates of a thickness of around one hundred kilometers; – these plates are created near the mid-oceanic ridges and are progressively separated without any deformations; – the continents, lighter than sea floors, move along with the plates that support them; – the increase in surface, resulting from a continual addition of new material in the ridges, is counter-balanced by the compression of plates in the mantle in certain zones, called subduction zones. Figure 1.4 shows J. Morgan’s model which comprises twelve plates. Their boundaries are made up either of mid-oceanic ridges (marked, with their transform faults, by a thin double line) or “convergence zones”, which correspond to the phenomena of subduction and continental collision, which are discussed hereafter (marked by a thick line). In some cases other types of boundaries are also found (“continental transform faults”, as in western North America or Turkey, and a dotted line in Eastern Siberia), which will be discussed later. It is to be noted that such a model establishes a clear distinction between oceans and sea; the latter often less deep and without ridges, and connected to the continents that they border. In the model in Figure 1.4, there are six main plates, which are, in decreasing order of their surface area: Pacific, American, Eurasian, African, Indo-Australian and Antarctic. It can be demonstrated (Le Pichon 1968) that the minimum number is six, from the kinematical point of view. In 1972, Morgan introduced six other smaller plates: Nazca, Cocos, Caribbean, Philippines, Somalia and Arabia. Other models, containing a larger number of plates (up to around 20) were also presented later on.
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Figure 1.4. A model of 12 plates (according to [MAD 91])
Subduction, already mentioned while talking about the principles of plate tectonics, is complementary to seafloor spreading. Materials arising in the axis of the ridges, after progressively separating from the latter, end up by subducting into the mantle by sliding under another plate. Figure 1.5, which shows a cross-section of the globe for a plane corresponding to latitude –10°, shows several examples of subduction zones between either oceanic plates and a continent (Nazca under South America) or two oceanic plates (Tonga-Kermadec subduction and Indonesian subduction). It is also possible that subduction does not occur in all the contact zones between ocean plates and continents; there are no known subduction zones in the coasts of Africa and it is seen that the African plate which includes portions of the Atlantic ocean in addition to the continent, (to the east of the Mid-Atlantic ridge) and the Indian Ocean (to the west of the Carlsberg ridge) goes through a regular increase.
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Figure 1.5. Cross-section illustrating the expansion mechanism of sea floors and subduction (according to [ALL 79])
The concept of subduction was not clearly defined until the end of the 1960s; it constitutes the last necessary link for a coherent theory on plate tectonics. It also made it possible to interpret earlier observations on the distribution of hypocenters of deep-focus earthquakes (K. Wadati 1935), as we shall see in section 1.1.3. Speeds of plate movements can be calculated by formulae of spherical geometry from measured values of rates of spreading on either side of the ridge. We also find the same values for the rate of sliding of the subducted plates, as for spreading, i.e. a few centimeters per year (for example 9.3 cm/year for the subduction of the Nazca plate under South America). According to the initial ideas of the proponents of this theory, subduction only concerned oceanic plates. Later studies brought to evidence the possibility of the oceanic crust sometimes overlapping the continental crust, instead of sliding under it: this type of phenomenon, observed, for example, in Oman or New Caledonia, is called obduction. In addition, continental subduction had to be admitted in the collision zones between continents.
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Figure 1.6. Collision between India and Eurasia (P. Molnar and P. Tapponnier, 1975) (according to [MAD 91])
The collision zones, of which the best studied example is shown in Figure 1.6 (the India-Eurasia collision), makes up the ultimate stage of the subduction zones of the oceanic plate under a continent (such as the Andean subduction), after the complete disappearance of the concerned portion of the oceanic plate sliding into the mantle. A direct contact is thus established between the continent under which the oceanic place was subducted (in this case, Eurasia) and the continent held up by the plate (here, India). This contact is seen through one or many suture lines and is accompanied by strong compression in the direction of the relative movement between the two continents, which provokes the formation of mountain ranges (here, the Himalayas) by folding and overlapping. The largest part of the southern end of the Eurasian plate, from Maghreb to Burma, is subjected to continental collisions due to the relative movement to the north, to the west of the African plate linked to the Middle East by the Arabic plate, and to the east by the Indo-Australian plate (see Figure 1.4). An almost continuous line of mountain chains marks this collision zone (the Atlas, the Alpine system, the Carpathians, Caucasians, the Turkish and Iranian ranges, and the Himalayan system). The rate of convergence (speed of relative movement) is lesser to the west (about one centimeter per year) than to the east (approximately 5 cm/year for India).
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Figure 1.6 shows that the Himalayan collision produces great fracture lines at the heart of the Eurasian plate and displacement of certain segments along the length of these lines (ejection of China towards the east and Indochina towards the southeast). The distribution of these fracture lines is very similar to that which results from the action of rigid punching on a plane of material having perfectly plastic behavior (see Figure 1.7). The observations show that the concept of a rigid plate first presented by the proponents of the plate tectonics theory is nothing but an initial approximation and that it must be admitted that the intraplate area (inside the plates) may be the seat of deformations, as is, in any case, suggested by the existence of seismic activity in this area (see section 1.1.3).
Figure 1.7. Crushing of a rigid-plastic environment (according to [MOL 79])
Other than ridges, oceanic subduction zones and continental collision zones, another type of plate boundary appears in Figure 1.4: transform continental faults. These are contact zones where the relative movement of plates is a horizontal slip along their lengths. The famous San Andreas Fault system in California (where the Pacific plate slides northwest in relation to North America) and the North-Anatolian fault in Turkey belong to this category. The name “transform faults” initially given to accidents that split the ridges (see section 1.1.2.2) comes from the capacity of this type of fault to transform into other types at its extremities. For example, the San Andreas Fault constitutes a link between the western ridge of the Cocos plate and a small subduction zone (not shown in Figure 1.4) to the north. Figure 1.4, also shows a dotted line in Eastern Siberia, which represents the lesser known boundary between the Eurasian and North American plates. The interaction between plates in the area of Japan is very complex, as is seen in Figure 1.8. According to this drawing, one part of Japan, including the capital Tokyo, would be situated on the North American plate.
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Figure 1.8. Interactions between plates in the area around Japan (according to [EQE 95])
To conclude, the theory of plate tectonics is a definite success of 20th century geology and geophysics, although certain aspects (especially its driving mechanism) remain unclear; it gives a unique, logical, and coherent explanation that links diverse phenomena like the evolution of ocean floors, the movement of continents, the distribution of mountain ranges and volcanoes, and is based upon an impressive amount of experimental proof. We shall now examine its links with the distribution of seismicity. 1.1.3. Seismicity of tectonic origin Interactions between plates, resulting from their relative movements, are the cause of earthquakes we call tectonic. These constitute almost the totality of all observed seismic activity, in terms of energy liberated. These interactions manifest themselves, on the one hand, from localized accumulations of stress and deformations at plate boundaries, and on the other, by the dispersion of these mechanical quantities within these plates. In the first case (plate boundaries), the relative displacements of a few centimeters per year, (or a few meters per century), as we have seen, may sometimes be absorbed “gently, if the surfaces in contact are well lubricated”. However, most often, there is a blockage of the relative slip, which brings about progressively increasing stress until the breaking point is reached: a violent slip is then produced,
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which allows the making up of any “deficit” in movement accumulated since the start of the blockage. It is this violent slip, which is accompanied by vibratory waves that may be felt at great distances, that make up the interplate type of tectonic earthquake (i.e., between plates). According to this plan, interplate seismicity must have a cyclical character, the time interval separating two consecutive earthquakes on the same portion of plate boundaries corresponds to the increase of stress until the rupture threshold is reached; the product of this time interval and the speed of relative movement must be equal to the amplitude of the slip of the earthquake, which is in relation to its size (magnitude). This cyclical character is well verified in a certain number of cases; for example, the subduction of the Philippine plate under Eurasia in southern Japan (see Figure 1.8) appears to produce one major earthquake (magnitude 8; see section 2.1) per century, whose slip amplitude (a few meters) is comparable to the cumulative tectonic movement over a hundred years. On the other hand, the subduction of the Pacific plate under eastern and northern Japan (see Figure 1.8) does not exhibit similar behavior, and an important component of relative displacement is absorbed in an aseismic manner. This hypothesis of seismic cycles is the foundation of the lacunae-based method (see section 6.2) that links the probability of occurrence of a great earthquake to the time elapsed since the last such known event and to the speed of relative movement. In the case of diffusion of stress fields and deformation within the plates, resulting from forces transmitted by the plate boundaries, the seismicity that results, called intraplate, is the consequence of the rupture of the weak zones (faults) on which these deformations are concentrated; we have seen, in Figure 1.6 the example of central and eastern Asia: the earthquakes that are produced here can be violent, but their distribution is more diffused than in the interplate zones, and thus, evidence of their cyclical character is even more difficult to establish. The two types of seismicity of tectonic origin, interplate and intraplate, are shown in Figure 1.9, in which all earthquakes of a size sufficient to be recorded several times, whatever their position, are marked by a point for the period 19681988 (see [MAD 91]).
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Figure 1.9. World seismicity between 1968 and 1988: each point represents an earthquake of magnitude greater than 5 (according to [MAD 95])
We see that the interplate earthquakes define very clearly the plate boundaries of Figure 1.4 and that the intraplate are fewer in number, even absent in certain regions (West Africa, eastern South America, the arctic zones of central and western North America). The distribution of earthquakes is therefore one of the most convincing proofs of the well-established plate tectonics theory. According to the terminology used by seismologists, earthquakes are classified according to their depth as superficial (depth less than 60 km), intermediate (depth between 60 and 300 km), and deep (depth greater than 300 km). This classification is not suitable to the needs of earthquake engineering as earthquakes that have devastating effects are, in the most part, superficial ones, sometimes of the intermediate type, and almost never of the deep type. In the superficial category, we classify together earthquakes that occur at depths of 50 and 10 km, which for example, if they are of the same size (magnitude), do not, generally, have the same destructive potential. As we shall see in section 2.3, we must not speak of the depth of a superficial earthquake in earthquake engineering, without linking it to the size of the rupture zone. A “small sized” earthquake (magnitude 5, for example), at a depth of 15 km is, from the engineer’s viewpoint, clearly deeper than an earthquake of larger size (magnitude 7) whose depth at the start of rupture is estimated to be 25 km.
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According to seismologists, superficial earthquakes are the most frequent, and are observed for all types of plate boundaries and for the entire intraplate area. They correspond to ruptures in the “seismogenic part of the crust”, i.e. the part inside which the temperature of material is low enough for them to have a breakable character; in most regions, the thickness of this earthquake prone portion is about 20 km; at greater depths, an increase in temperature modifies the behavior of rocks and allows them to deform themselves without rupture. The fact that we know of intermediate and deep earthquakes which are produced well below this depth of 20 km (up to 700 km for certain deep earthquakes) is not in contradiction with the earlier hypothesis concerning temperature influence. In fact, we see that the majority of intermediate and deep earthquakes occur in subduction zones and it is likely that the warming of subducted plates, during their plunge, is relatively slow and that the material of these plates conserves its breakable character up to depths of several hundreds of kilometers. Calculation models (N. Toksoz, 1973) show that at a plunging rate of 8 cm/year (average value for subduction zones of the Pacific), the interior of the descending plate remains colder than the surrounding mantle up to a depth of about 600 km (see [MAD 91]). At the slower rate of 1 cm per year (as in the case of the Aegean arc in the eastern Mediterranean), thermal equilibrium would be reached at a depth of 400 km. These theoretical forecasts are confirmed by the determination of the maximum depths of recorded earthquakes, which are greater for subduction zones with high plunge rates than for those with lower rates (approximately 700 km for the Nazca subduction under South America and only 300 km for the Aegean arc). We also know of a few examples of intermediate or deep earthquakes in regions situated at plate boundaries, which are not the seat of currently active subduction, or even within the intraplate domain. This is the case, among others, of southern Spain (where depths of up to 640 km were observed) and in the sector of Vrancea in eastern Romania, where there exists a very active zone of intermediate earthquakes (with depths that vary generally between 100 and 150 km). We consider that these earthquakes bear witness to the existence of older subduction zones in these regions, but could there be a mechanism other than subduction that produces earthquakes at greater depths (see [MAD 91])?
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Figure 1.10. Localization of earthquakes on mid-oceanic ridges and in subduction zones (according to [MAD 91])
Figure 1.10 shows the localization of earthquakes on two types of plate boundaries that represent the longest interplate boundaries, that is, the mid-oceanic ridges and the subduction zones: – on the ridges, seismicity is only superficial: this shows itself on the ridges themselves and on the transform faults that separate the different segments of the ridge from each other. These earthquakes are rarely of large size, and in terms of accumulated energy, they only represent a small percentage of the total accumulated energy liberated in the world. This low percentage, and the fact that the ridges rarely affect inhabited zones (Iceland, East Africa), show that this type of seismicity has only marginal importance in earthquake engineering; – in subduction zones, an example of which is represented on the right in Figure 1.10 of an oceanic plate under a continent, we observe two categories of earthquakes: on the one hand those which are produced within the subducted plate, which aligns itself on an inclined plane, called the Wadati-Benioff surface, and on the other, the superficial earthquakes on the continent: earthquakes of the first type (subducted plate) may be superficial, intermediate or deep according to their distance to the trench that marks the start of the plunge. It is here that we find the biggest earthquakes (up to 200,000 km2 of rupture surface for the great Chilean earthquake of 22 May 1960; see section 2.2). In terms of energy, the seismicity of subduction zones represents almost 80% of the world total (by regrouping contributions from subducted plates and subducting plates) and more than 90% for only the intermediate and deep earthquakes. For the other types of plate boundaries, collision zones and continental transform faults, the characteristics of seismicity are the following:
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– collision zones experience mainly superficial seismicity, although certain regions have seen intermediate earthquakes that correspond to ancient subduction zones, as was mentioned earlier; the energy liberated represents almost 20% of the world total and the stronger earthquakes in these zones can be of greater size (magnitude higher than 8; see section 3.1); but they do not attain the records of the subduction zones; the figure of 20% of total energy that has been cited includes contributions from the collision zones, the zones at the periphery of the plate boundaries, and contiguous intraplate domains; – continental transform faults make up only a small proportion, in terms of length, of the plate boundaries, but they can be very active seismically: the San Andreas Fault system alone (the complexity of this system in Southern California is shown in Figure 1.11) produces about 1 percent of the total world seismic energy. This seismicity is superficial (most earthquakes have a maximum depth of 20 km), and the strongest events (the prototype is the Big One awaited by Californians) are of large size (magnitude of 8; see section 2.1). The North-Anatolian fault in Turkey is comparable in dimension and shows even greater activity. Intraplate seismicity is diffused and superficial (apart from the exception, mentioned above, of some zones of intermediate seismic activity, outside the regions that bear the repercussions of very violent continental collisions as in eastern and western Asia), large intraplate earthquakes appear rarely, but there are a few examples, as in New Madrid (Missouri) in 1811-1812. It is probable that such events show a “return period” (that is the average period between two successive occurrences; see Chapter 6) which is greater than the period for which we have observations. In general, the evaluation of seismic hazard (that is, the probability of occurrence of an earthquake of a given size (see Chapter 7) is more difficult in the intraplate domain than at the plate boundaries although, or because, the level of seismicity there is generally lower; this aspect shall be studied in Part 3.
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Figure 1.11. The San Andreas Fault system in Southern California; the general movement of sections on the Pacific side is from the south-east towards the north-west but undergoes a deviation towards the west on contact with the Sierra Nevada (according to [AND 79])
1.2. Faults Most earthquake engineers have only rudimentary knowledge of faults, because they are considered to be the domain of Earth science specialists and it is sufficient to characterize earthquakes using global parameters (essentially magnitude and distance) so as to be able to calculate the movements that are the input data for these calculations. This remains largely true, in spite of the progress in recording techniques, but such an attitude does not allow the evaluation of the origin and importance of uncertainty, often very great, which affects the determination of seismic movements and which may lead to errors in interpretation, especially about the significance of “obvious” parameters like distance or depth. Thus, the objective of section 1.2 is to give some idea about faults to sensitize the engineer to the limits of the “magnitude-distance” approach and to make it possible for him to communicate with geologists and seismologists. The presentation will remain qualitative as it will be throughout the entire first chapter. Quantitative aspects about geometric characteristics of faults shall be dealt with in section 2.2.
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1.2.1. Relationship between earthquakes and faults As indicated above in section 1.1.1.2., the idea that traces of surface rupture, sometimes observed after violent seismic activity, constitutes the manifestation of the cause of the seismic phenomenon, and not one of its induced effects, took a long time to be understood. Among the perceptive observers who contributed to the promotion of this idea, we can cite, apart from B. Koto and H. Reid who were mentioned earlier, G. Gilbert for the study of escarpments in the Wasatch range near Salt Lake City (1875) and A. McKay for the report on surface ruptures produced by the New Zealand earthquake on 1 September 1888 (see [YEA 97]). It was H. Reid who formulated the first mechanical model, consistent with observations of the terrain and the seismic source. His theory, called the elastic rebound, attributes vibration of the soil, which is what the earthquake really is, to the brutal rupture between two blocks of the Earth’s crust which, in a short time, liberate tectonic stresses accumulated by tectonic deformation. This concept, already mentioned in section 1.1.3, is considered even today as the basic mechanism of tectonic earthquakes; it is shown in Figure 1.12. From the state of rest (part a of Figure 1.12), shearing tectonic deformation produces cumulative accumulation of stress (part b); the drawings remind us of shear in the horizontal plane, like that which is created in the case of transform faults, but it applies also to oblique shear in a vertical plane (in the case of subducted plates). When the breaking point is reached, the two blocks slip brutally one against the other (part c) until they stabilize themselves in a new position of equilibrium (part d), where they are separated by a length that corresponds to the accumulated differential displacement, during phase b, between the faces of blocks farthest away from the rupture surface, called the fault plane.
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Figure 1.12. The four phases of elastic rebound
This basic model, which portrays the essence of the seismic phenomenon, is based on simplified hypotheses of the rupturing process, which are often very complex and may be far removed from the actual conditions. Among these hypotheses are: – geometric simplifications: the blocks are parallelepipeds, the rupture surface is a simple rectangle; the actual faults may be non-plane surfaces and there are several cases, for large-scale earthquakes, where the rupture involves several different faults and not just one; – homogenization of deformations and stresses within the blocks; it may be that the surface of the fault contains zones having very different characteristics and that vibrations emitted during phase c may come from the rupture of some localized hard points and not from a slip of the whole distributed homogenously over the entire fault plane; – omission of dynamic aspects of the rupture; this does not occur instantaneously on the whole surface; it begins at one point and is propagated along the fault plane in a very complex manner. The details of this movement, especially in the stages of slowing down and stopping, appear to play an important role in the emission of seismic vibrations.
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In spite of these simplifications, this model makes it possible to calculate the energy liberated during the rupture in the form of seismic waves (see section 3.1), to explain the cyclical character of the seismic phenomenon, mentioned in section 1.1.3, and also, by way of some supplementary hypotheses, to obtain the values of certain parameters that characterize the emitted vibrations, such as acceleration, velocity, and movement (see section 5.1). H. Reid deduced his theory from observations of the terrain after the great earthquake of San Francisco (18 April 1906), which he had the opportunity to examine closely as director of the scientific commission of investigation constituted the day after this disastrous earthquake. Comparison of geodesic data obtained at different times enabled him to establish slipping of several meters that had occurred on a reach several kilometers along the San Andreas Fault and that this slip had lasted only a few dozen seconds, during which time intense vibrations had been released. The cause and effect relationship between faults and earthquakes was thus made clear after a major event that had produced spectacular effects on the surface. The mechanism of elastic rebound suggested by H. Reid after this earthquake, is considered valid in the almost all the cases, whatever the scale of the seismic phenomenon, which may vary from a few dozen meters (microseism), to a few hundred kilometers, even though the manifestations visible on the surface concern only earthquakes of a certain size (and comparable to their depth), that are seen in emerging portions of the earth. In zones with weak or moderate seismicity, as in the seismic regions of metropolitan France, it is often very difficult to explain the fault-earthquake relationship. We know, on the one hand, many faults, which are shown on geological maps, but which appear not to be seismically active in the current tectonic context, and, on the other hand, hints of paleo-seismicity, (see section 6.1). It is therefore relatively rare to be able to establish, with any degree of certainty, any link between well-identified faults and observed seismicity. We refer the reader back to the concluding observations of section 1.1.3. 1.2.2. Classification of faults Faults are classified into three main types according to the nature of tectonic deformation and the direction of the relative movement of blocks separated by the fault plane; we thus distinguish in Figure 1.13: – normal fault, which corresponds to a tectonic extension; one of the blocks goes down as compared to the other on an inclined fault plane, producing a lengthening of the two blocks;
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– reverse fault, which, as the name indicates, is the opposite of a normal fault: compression tectonics and rise of a block on the other along an inclined fault plane, producing a shortening of both blocks; – strike-slip fault, which is a horizontal motion of two blocks one against the other, corresponding to shearing tectonics in a horizontal plane; the strike-slip motion can be left-lateral (see Figure 1.13), i.e. an observer placed on one of the blocks sees the relative displacement of the other block towards the left, or rightlateral in the opposite case.
Figure 1.13. Three main types of faults
In the majority of cases, the movement of faults does not correspond exactly to one of these three types, but is presented in the form of a combination of two components, one being strike-slip and the other normal or reverse. To describe the movement, we first quote the type of component which has the greatest amplitude and secondly, the type of component with lower amplitude; as the strike-slip motion can be right-lateral or left-lateral, there are eight possible cases for composite movements:
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– right-lateral strike-slip with a normal component; – left-lateral strike-slip with a normal component; – right-lateral strike-slip with a reverse component; – left-lateral strike-slip with a reverse component; – normal fault with a right-lateral strike-slip component; – normal fault with a left-lateral strike-slip component; – reverse fault with a right-lateral strike-slip component; – reverse fault with a left-lateral strike-slip component. The movements of faults are described by the amplitude D of the relative displacement between blocks on the fault plane and by three angles: – the azimuth < is the angle which the fault trace forms in the horizontal plane with the northern direction; – the dip Gis the angle which the fault plane forms with the horizontal plane;
– the slip O measures the angle between the vector displacement and the horizontal in the fault plane. Figure 1.14 reveals the angles for a left-lateral strike-slip with a normal component.
Figure 1.14. Definition of the azimuth dip and slip angles for a left-lateral strike-slip fault with a normal component; horizontal displacement value D cos O in the direction parallel to the trace of the fault and D sin O cos G in the perpendicular direction; vertical displacement value D sin O sin G
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In the above definitions, the fault is assumed to be flat; this assumption does not always correspond to reality. For example, we very often come across, particularly in the case of normal faults, a curvature of the rupture surface whose dip angle decreases when the depth increases (“listric fault”); these deviations from the assumption of flatness especially affect the parts of the fault located close to the surface, whose contribution to the vibratory motion transmitted to the ground is undoubtedly less significant than that of the deeper parts where the land is more resistant thus more “breakable” (see [MAR 88]). We can thus retain the fault plane model by retaining the dip of the deeper parts rather than that of the superficial parts, which frequently appear as quasi-vertical.
Figure 1.15. Diagram of strike-slip orientations (D1, D2), of normal faults (N1, N2) and reverse faults (I1, I2) in a zone subjected to a north-south compression
The type of fault being determined by tectonics, the nature and the orientation of faults in a given area are not unspecified. Figure 1.15 shows in a schematic way that a zone subjected to a north-south compression, as is the case, for example, for parts of Western Europe or of Southern Asia, can present: – normal faults directed north-south, corresponding to the east-west extension which accompanies the north-south compression; – east-west directed reverse faults; – oblique strike-slips, either right-lateral if they are directed NW-SE or leftlateral (SW-NE orientation).
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On the plate boundaries, the main types of faults are as follows: – normal faults on the mid-oceanic ridges and strike-slip faults on the segments of transform faults separating the ridge sections; – reverse faults in the subduction zones, which represent the overlapping of subducted plates; – strike-slip faults on continental transform faults. These characterizations are systematic only in the case of large interplate earthquakes; for occurrences of a smaller size, other mechanisms are possible such as normal faults or strike-slips within the subducted plates or reverse faults in the vicinities of the large continental transform faults. The San Andreas system has the majority of faults in right-lateral strike-slip, but reverse faults also exist, such as White Wolf or Northridge, which were responsible for significant earthquakes on July 21, 1952 (Kern County) and January 17, 1994 (Northridge) respectively. When the earthquake is sufficiently large and superficial so that the fault is expressed on the surface, spectacular effects, some examples of which are given in Figures 1.16, 1.17, 1.18 and 1.19 can be observed. All photographs of these figures, except the top image in Figure 1.16, were taken just after the earthquake; the displacements that they show vary from one to several meters and are thus representative of the amplitude of the fault movements at the time of a fairly big earthquake. The first photograph in Figure 1.16 (Corinthian Channel) highlights a vertical shift of approximately 10 m of ground layers on both sides of the fault, corresponding to the descent of the compartment located on the left against the one on the right. This shift did not occur during the same earthquake but results from the cumulative effects of several earthquakes occurring at different times. One of the difficulties in the interpretation of the paleoseismicity data (see section 6.1), is to determine whether the fault movements noted on the ground are the result of one or several seismic episodes. With the help of well documented examples of recent earthquakes, in section 2.2 we will return to the geometrical characteristics of faults (displacement amplitude, surface and dimensions of the rupture zone).
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Figure 1.16. (Top), normal fault traces, Corinthian Channel (Greece) and (bottom), Fairview Peak earthquake (Nevada) of 16 December 1954
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Figure 1.17. Reverse fault traces: top, the El Asnam earthquake (Algeria) of 10 October 1980; Center, the Spitak earthquake (Armenia) of 7 December 1988; bottom, the Chi-Chi earthquake (Taiwan) of 21 September 1999; the fault formed a waterfall 7 to 8 m in height on the bed of the Tachia river and caused the bridge to collapse
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Figure 1.18. Strike-slip traces: top, the San Francisco earthquake of 18 April 1906; center, the Motagua earthquake (Guatemala) of 4 February 1976; bottom, the Imperial Valley earthquake (Southern California) of 15 October 1979
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Figure 1.19. Strike-slip traces (continued). Top, the Gobi-Altai earthquake (Mongolia) of 4 December 1957: the left-lateral strike-slip reached 10 m in places and cut the hills like a saw; bottom, the Kocaeli earthquake (Turkey) of 17 August 1999 on the north-Anatolian fault; the right-lateral strike-slip sheared the trunk of this fruit tree
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1.2.3. Focal mechanisms First of all, let us remember some definitions (see Figure 1.20); the hypocenter or focus, is the point on the fault plane where the rupture starts; its projection on the surface is the epicenter in the seismological sense of the term, not to be confused with the macroseismic epicentre, which is the point of surface where the strongest effects are noted, which may be different from the seismological epicenter (see section 14.1). Figure 1.20 shows the hypocenter and the seismological epicenter for a reverse fault and a right-lateral or right-lateral fault corresponding to the tectonic situation represented in Figure 1.15 (zone subjected to a north-south compression); the reverse fault ABCD belongs to the I1 type (dip towards the south) and IJKL strikeslip fault to the D2 type (right-lateral or right-lateral) of Figure 1.15; the hypocenters Fi (reverse fault) and Fd (strike-slip fault) were placed in the deepest part (CD and KL respectively) of the fault planes, as is generally the case. For the strike-slip fault, the propagation of rupture starting from the hypocenter Fd occurs following a bilateral mode towards the north-west and south-east; cases of unilateral rupture mode are also known, where the hypocenter is at one end of the fault and the rupture is propagated in only one direction. The examples represented in this diagram are inspired by the Los Angeles area, the strike-slip fault being the San Andreas Fault and the reverse fault that of Northridge.
Figure 1.20. Hypocenter and epicenter for a reverse fault and a right-lateral fault in an area subjected to a north-south compression (the north-east is on the left)
In the diagram, it can be seen that the position of the seismological epicenter can be misleading, in the sense that the surface points located at relatively large epicenter distances are in reality very close to the fault, if the latter is quite large; as is the case with extremity I of the strike-slip fault or the projection on the surface of side AB of the reverse fault.
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This observation is the first explanation of the fact that the macroseismic epicenter can be different from the seismological epicenter (other explanations are related to directivity effects and site effects; see section 4.3). As already mentioned in section 1.1.3, the size of the earthquake, in terms of geometrical dimensions, is an essential concept to which estimates of distance or depth should always be related (see section 2.3). The type of fault movement, normal, reverse or strike-slip motion can be determined from recordings by studying the direction of ground motion for the first arrival of seismic waves to the recording station. In section 1.4.2, we shall see that this first arrival is that of a dilation compression wave and corresponds to an essentially vertical motion, either upwards or downwards. Seismologists had for a long time observed that the polarity of this wave (i.e. its ascending or descending direction) changed with the orientation of the station with relation to the fault. However it took quite some time to work out the theory of the hypocenter mechanism explaining these changes and to develop the techniques to determine this mechanism using seismological observatories.
Figure 1.21. Right-lateral fault with an indication of the beginning of the vertical seismograms on points P1, P2, P3 and P4 framing the epicenter E (according to [MON 97])
Without going into detail (which would call for a lecture on seismology) we can understand the distribution of polarities by considering the diagram in Figure 1.21. Here we see a fault acting as a right-lateral slip, following trace AB on the surface with the hypocenter in F (on the lower side of the presumed vertical fault plane) and the epicenter in E.
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Let us initially examine the point marked P4. While moving towards the right, the block on which P4 is found will tend to compress the matter towards which it is moving, and thus raise the surface; the beginning of the vertical seismogram in P4 consequently comprises a first point towards the top, as indicated in the diagram. Behind the same block (point P3) the matter will undergo a traction, which is accompanied by a surface depression; the vertical seismogram in P3 will thus start with a downward point. These first points of the seismograms in P4 and P3 are produced by the first arrival of seismic waves emitted by the hypocenter F; the path of these waves is carried in a dotted line in the diagram. On the other block, an initial compression in P2 (upward point of the vertical seismogram) and an initial traction in P1 (downward point) will be noticed, for the same reason. The polarities are thus distributed into quadrants: positive (ascending) for EDB and ECA, negative (descending) for EDA and ECB. In order to visualize these quadrants, seismologists use a projection of a sphere on a horizontal plane having the hypocenter as the center, while darkening the parts of the sphere which show compression and lighten those which show dilation. Figure 1.22 shows this sphere in the case shown in Figure 1.21 (right-lateral or rightlateral strike-slip motion on a vertical fault plane). We see in part a) of Figure 1.22 that the path of the waves (or “seismic rays”; see section 3.2.3) is concave towards the top and it leaves the focal sphere through its lower part, while the surface stations are quite far from the epicenter; this results from the variations of propagation velocities of waves with their depth (see section 3.2). We also see in parts b) and c) of Figure 1.22 that the quadripolar distribution of compressions and dilations reveals, in addition to the fault plane, another plane, known as the auxiliary plane, which is perpendicular to it. If no ground observations on the fault movement are available, which is most frequently the case, we cannot choose a priori one of the two planes corresponding to the rupture. Ambiguity can in general be removed by cutting out recording data or by using geological arguments.
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Figure 1.22. Stereographic projection of the lower local hemisphere for a vertical rightlateral strike-slip motion; a) paths of seismic rays 1 and 2 of Figure 1.21; b) fault plane and auxiliary plane-stereographic projection of the lower focal hemisphere showing the quadrants of compression and dilation (according to [MAD 91])
The projection used to obtain diagram c) is stereographic projection, on a horizontal plane located under the sphere, taken with relation to the upper pole S (marked on part b) of the sphere. We limit ourselves, to carry out projection, to the lower hemisphere, which is in general the first to be pierced by the seismic rays, as mentioned. In the case of pure strike-slip motion, which has just been examined, the hypocenter mechanism, i.e. the projection of the parts in compression and dilation is thus very simple, since it is composed of two perpendicular diameters, the quadrants of the same color (dark or light) being in contrast with the top. It should be noted that if the direction of the hypocenter is right-lateral on the fault plane (as in the case of Figures 1.21 and 1.22) it is left-lateral on the auxiliary plane (and vice versa). The case of the normal and reverse faults is a little more complex and the diagram of their focal mechanism reveals arcs of a circle, instead of diameters, as boundaries between compression zones and dilation zones. We consider (see Figure
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1.23) a fault plane perpendicular to vertical plane OY-OZ and presenting a dip angle Gwith relation to the horizontal axis. Part a) of Figure 1.23 shows the focal sphere of the unit radius traced around the hypocenter F, trace FP of the fault plane in the vertical plane OY-OZ, between F and the intersection P with the lower focal hemisphere, and projection Q of P on the horizontal plane tangential to the sphere in its lower pole O. As indicated earlier, a projection is made with relation to the upper pole S of the sphere; trace FP of the auxiliary plane and projection Q’ of P’ is also marked.
Figure 1.23. Diagram of focal mechanism for a normal or reverse fault; a) cross-section of the vertical plane perpendicular to the fault plane and construction of projections Q and Q’ of intersections of the lower focal hemisphere with the fault plane (P) and with the auxiliary plane (P’), b) diagram of focal mechanism in the horizontal plane tangent to the lower pole O of the focal sphere; arcs AQB and AQ’ B are the arcs of circles of respective radii 2/cos G and 2/sin G; the surface between these two arcs is light (dilation) for a normal fault or dark (compression) for a reverse fault
In the OX-OY-OZ system of axes the equations of the right-hand side joining the projection pole S at an unspecified point of the circle obtained as an intersection of the focal sphere of the unit radius by the dip fault plane G are written as:
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x cos I
=
y sin I cos G
=
z2 sin I sin G 1
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[1.1]
I being a variable parameter; the intersection of this perpendicular with the
projection plane Z = 0 is the curve of the following equation, obtained by eliminating I: X2 + Y2 + 4Y tan G = 4
[1.2]
The circle of radius 2/cosG therefore has its center at point X = 0, Y = – 2 tan G. Figure 1.23 part b) shows arc AQB of this circle that appears on the projection as well as the arc of circle AQ’B corresponding to the auxiliary plane (whose radius is 2/sinG). If it is a normal fault, the compressed zones of the focal hemisphere are sides DP and CP’ (see Figure 1.23, part a); in the diagram of focal mechanism the part described by the arcs of the circles AQB and AQ’B must thus appear lighter while the other parts must be darker. The situation is exactly the opposite for an inverse fault. The different diagrams of focal mechanism are summarized in Figure 1.24, with an indication of the movement of blocks on the fault plane and the auxiliary plane: from top to bottom, instances of strike-slip fault, normal fault, reverse fault and oblique fault combining a strike-slip with a reverse component. When the fault movement is pure, that is when it is not a result of a combination of two components, an examination of the diagram of the focal mechanism allows us to visualize the azimuth directly and to determine the dip using a simple calculation. As a result, we arrive at the calculations introduced for the discussion of Figure 1.23 so that the quotient U of maximal width QQ’ of the part between the two arcs of the circle by the radius AB/2 of the projection circle of the equator of the sphere is such that:
ª1 º G = 2 Arc tan « ( U ( U 2)² 8) » ¬2 ¼
[1.3]
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Seismic Engineering
Figure 1.24. The different diagrams of focal mechanism with an indication of two possible fault planes (according to [SHE 99])
The value of G calculated by this formula is less than or equal to S /4; it corresponds to either the fault plane itself or to the auxiliary plane; the values of U vary from 2 ( 2 1 ) = 0.8284 (for G =
S /4) to 1 (for G = R).
Figure 1.25 shows the focal mechanisms of the Harvard CMT catalog (see [SHE 99]) for recent great earthquakes. In it we come across trends mentioned in section 1.2.2 (a predominance of normal faults and strike-slip faults in the region of ridges, reverse faults in the subduction zones and strike-slip faults on the continental transform faults).
Causes of Earthquakes
45
Figure 1.25. Focal mechanisms of great earthquakes from 1977–1994 (according to [SHE 99])
1.2.4. Different aspects of rupture
The focal mechanism gives important information on the type of fault motion in the very first moments of rupture. Is this information sufficient to describe the entire rupture phenomenon? The answer is definitely affirmative for an important incident of a small or average earthquake (up to about 10 or 20 km), but we must be cautious for incidents of a larger scale that are often found to occur as “multiples” (encompassing more than one fault) where the distribution of seismogenic areas on the fault plane is seen to be very diverse. However, temporal aspects such as progression velocity of the rupture front or the chain reaction of consecutive episodes, (the main earthquake and its aftershocks, defined at a later stage), are not normally covered by a general description associated with the theory of focal mechanism. For “normal” earthquakes rupture propagation takes place at a comparable velocity, but slightly lower than the propagation velocity of shear waves in material subject to rupture; this being around 3–3.5 km/s (see section 2.1), the propagation velocity of the rupture is on average 2.5–3 km/s. The total time of rupture, which is the quotient by this travel velocity of the rupture front, therefore varies by a few seconds for medium sized earthquakes, (fault size of 5 to 20 km) a few dozen seconds for big earthquakes (fault size of 50 to 200 km); in the case of great
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earthquakes in subduction zones (fault size of 500 to 1,000 km), rupture time can reach several minutes. These values of rupture time represent scales that are valid for the majority of tectonic earthquakes, but cases where propagation is quicker are observed (the rupture jumping from one point to another on the fault plane) and there are also “slow” earthquakes where propagation velocity is hardly a few hundreds or even a few dozen m/s. These slow earthquakes emit very low vibratory energy in the important frequency ranges for earthquake engineering in present constructions (of a few tenths of a hertz to a few hertz) but may be significantly capable of generating tsunamis (see section 3.3.3). Rupture propagation on the fault plane is one of the causes of directivity effects, that is, the amplifications of vibratory motion often seen in certain directions (see section 4.3.2). If rupture is propagated in a fairly regular manner we may see a phenomenon similar to the Doppler effect in sound (a train whistle is heard at a higher note as it approaches than when it is going away). For the points targeted by rupture propagation the seismic waves emitted by the rupture front tend to accumulate and lead to a reduction of the total duration of the signal and a correlative increase in the amplitude of motion. The process of rupture on a given fault generally contains many episodes; the mainshock (the “earthquake” so to speak) is sometimes preceded by foreshocks that are identified as precursors only in retrospect (see section 7.2 on earthquake prediction) and is almost always followed by a large number of aftershocks, that are smaller earthquakes that are produced on the same fault plane or in its vicinity within hours, days, weeks, and sometimes months or even years following the earthquake. These aftershocks are supposed to be local readjustments that follow the main rupture; they are proof of the geometric complexity and diversity of the affected area. They are often used to gauge the extent of the fault plane with the help of temporary recording systems installed immediately after the earthquake (see section 2.2). Some earthquakes seem to occur with almost no aftershocks at all; this is true of a significant number of quakes of intermediate depth and some intraplate earthquakes with a relatively deep hypocenter (depth in the range of 25-30 km). However, generally speaking, aftershocks are quite numerous (often in several thousands for a given earthquake of fairly large size). The majority of aftershocks are imperceptible to humans and only detected by instruments but it is frequently observed that some aftershocks are strong enough to cause damage in zones already affected by the mainshock. The damage caused by these aftershocks can weaken structures that have seemingly survived and can make them vulnerable to new shocks.
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47
Even though certain statistical laws are available on the temporal distribution, number and scale of aftershocks, prediction of “probable characteristics” of aftershocks is not yet considered reliable enough to be taken into account as an explicit part of earthquake engineering codes, even though the danger they represent is universally accepted. It is widely admitted, as confirmed by the analysis of recurrence and specific studies, that regulations for construction design prescribed in the codes, regarding action of the mainshock provide sufficient safety in case of aftershocks, which do not exceed half the scale of the earthquake in terms of amplitude of vibratory motion and it is thus not necessary to put any specific measures into effect for aftershocks. This limitation of aftershocks admitted to be half the scale of the mainshock is true for the majority of cases, but there are a few cases where two or more earthquakes of comparable amplitudes occurred in the same zone within a period of one or more months: such a phenomenon was observed in Italy in the case of the recent three “earthquake crisis” (Ancona 1972, Frioul 1976, Colfiorito 1997). In this case the term “aftershock” is therefore not valid as these are not “attenuated copies” of the main episode. Seismologists use the term swarm to describe a series of shocks of similar amplitude (normally moderate) in the same fault system. The definition of aftershocks given above is limited in nature as they are related to the return to the state of equilibrium of the fault plane that is destabilized by the rupture. A good number of cases are known where this destabilization apparently results in another earthquake often occurring in a different fault, but close enough to the first one so as to be affected by the stress field of the previous earthquake. This observation contradicts the theory of the seismic cycle (see the beginning of section 1.1.3) which assumes that faults develop independently of each other due to general tectonic deformation alone. It also throws a shadow of doubt over the very foundation, at least in some regions, of the seismic gap method (see [KAG 99]). 1.3. Non-tectonic earthquakes
If tectonic earthquakes are by far the highest in number or most significant from the point of view of earthquake engineering, there are other categories of earthquakes caused by natural or artificial phenomena that will now briefly be presented. A separate section is devoted to earthquakes known as induced earthquakes that are seen, though in an unsystematic manner, in the vicinity of big dams or in areas where gas deposits or oil fields are found. A debate no longer in circulation, though much discussed in the 1960s and 1970s, was the nature of the cause of these earthquakes, natural or artificial.
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1.3.1. Non-tectonic quakes with natural causes
Tectonic quakes have been defined in section 1.1.3 as being the result of mechanical interactions taking place between tectonic plates; these interactions being essentially a result of horizontal motion created by sea floor spreading which is then compensated by subductions. However, certain parts of the plates may also show vertical motion due to causes unrelated to interaction with adjacent plates; this is seen in the case of uplift movement, currently seen in regions known to have been covered by large expanses of ice. The lightening brought about by the melting of such gigantic glaciers leads to upward vertical motion through isostatic compensation as continents float on denser material. This “isostatic rebound” also related to tectonics, seems to occur suddenly in fits and starts, not always in a continuous manner, thus causing earthquakes; this is one of the interpretations to explain the occurrence of some earthquakes in Scandinavia. Earthquakes such as these are tectonic type earthquakes even though they do not fall into the normal category of plate tectonics. Natural earthquakes, called non-tectonic earthquakes, correspond to phenomena different from those that form the basis of the elastic rebound theory, which is the violent rupture of a weak zone (fault plane) after progressive accumulation of shearing stress. Some such phenomena are mentioned below: – motion, often violent and sometimes explosive, of lava, gases and steam during volcanic eruptions; – sudden collapse of natural subterranean cavities due to dissolution resulting from the circulating infiltration of water; – large landslides, involving enormous volumes (about 100,000,000 m3) and high velocities (about 10 m/s). If readings of ground motion produced during such phenomena are available, it is often possible to prove their non-tectonic character by observing the diagrams of their focal mechanisms, which do not show the division into four quadrants of compression and dilation that is a characteristic feature of tectonic earthquakes. Volcanic earthquakes represent the majority of non-tectonic earthquakes. They are a major means of monitoring the activity of restless volcanoes. As far as earthquake engineering is concerned, they are in general considered unimportant, as their levels are normally very low.
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49
However, there are some examples of earthquakes of apparently volcanic nature (that is, not produced by rupture of a tectonic fault) that have wreaked great damage and also taken a life toll; one of the best documented among these being on the island of Ischia (Italy) on 28 July 1883 causing more than 3,000 deaths (see [ROT 72]); the highly localized nature of the damages found on the sides of Mount Epomeo supports the hypothesis of this having been a volcanic earthquake with a very superficial source corresponding to an “aborted eruption” of this volcano (see [ROT 72]). Areas in the vicinity of active volcanoes are also, generally speaking, considered active zones as far as tectonic seismicity is concerned. It is therefore not very easy to determine the cause, volcanic or tectonic, of an earthquake in any of these areas, if it occurs during a period of resurgence of volcanic activity. Earthquakes associated with important landslides, as mentioned above, have often been a subject of debate; if the starting point of an earthquake can be identified with precision through seismographic readings, the same is not the case for landslides which need to be deduced from eye-witness accounts that are often vague or even after the occurrence. Thus we are unable to gauge whether an earthquake is actually the result of a landslide or if it took place slightly prior to it, in which case it may well prove to be the reason for the landslide itself. Many examples of landslides due to earthquakes have been known to occur in mountainous regions (see section 3.3.2). 1.3.2. Artificial earthquakes
Various forms of human activity contribute to underground “ambient sound”, that is to the motion of low amplitudes that can be constantly recorded with appropriate sensitive instruments. These are mainly heavy vehicular traffic (trucks, trains), site-related activities, (drilling, piling, construction of foundations, machines) and the use of industrial installations (vibrations due to machines using revolving or swing movements, shocks associated with the rapid maneuvering of sluice gates and valves, etc.). This motion is not comparable to seismic motion as far as earthquake engineering is concerned, but is however of great interest in the experimental studies of site effects (see section 5.3.4) as it represents free sources of activity. Motion of higher amplitude, reaching in certain cases a level of potentially destructive natural earthquakes, may result from the use of explosives for military purposes (nuclear tests) or use in civil engineering (quarrying, rock extraction, tunneling) or the mining industry (digging and caving). In countries where moderate
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natural seismicity is seen, artificial earthquakes due to explosions represent a large and often predominant part of occurrences recorded by seismographs. Earthquakes of a highly explosive nature correspond to subterranean atomic bomb and thermonuclear bomb tests. The energy released by the most forceful of these weapons is comparable to that released by great earthquakes in the form of seismic waves, but the recordings of corresponding ground motion present particularities that make it possible for seismologists to distinguish them from natural earthquakes. Traditional explosives have been used to simulate natural seismic motion; to obtain a signal duration of a specific length (around ten seconds at least; see section 1.2.4) it is necessary to carry out a series of multiple explosions with a slight time lag so as to remain within reasonable limits where quantities of explosives used are concerned (few hundred kg to a few tons), to be placed at a minimum distance (few tens to a few hundred meters) from the firing zone. Such trials were carried out by Americans in New Mexico and by the Soviets in Tajikistan to observe simulation effects of earthquakes on models of buildings and structures. The use of explosives, particularly nuclear, was proposed to bring about natural earthquakes in exposed areas, in order to release stress accumulated due to tectonic deformations and to thus prevent the occurrence of major earthquakes. Practical difficulties in carrying out such procedures, as well as doubts about its efficiency thankfully led to all such projects being aborted. Apart from these explosions, cavities dug by humans falling in is probably the only other cause of artificial earthquakes of significance. In mining activity, “caving” is quite well known especially by litigations between the operator and the concerned communities. We will now present and discuss the so-called induced earthquakes, especially from the point of view of their natural or artificial character. 1.3.3. Induced earthquakes
The correlation between water reservoirs of a major dam and the occurrence of earthquakes in its vicinity was brought to attention towards the end of the 1930s during the filling of the reservoir of the Hoover Dam in Colorado. Many earthquakes of noticeable amplitude (magnitude 5; see section 2.1) were felt in the period 1935– 1940, at a time when the area was considered practically aseismic.
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51
Other examples of severe earthquakes apparently related to dam reservoirs were observed later, especially at Kariba in Zambezi (more than 2,000 shocks after the completion of the dam in 1958) and at Koyna near Bombay in India (a violent earthquake in 1967, five years after the construction of the structure and two years after the filling of the reservoir). The case of Koyna is always highlighted as the 1967 earthquake caused severe damage, (around 200 deaths) in an area where, as in the case of the Hoover Dam, seismic risk seemed negligible according to historical evidence. These examples led to the theory of induced earthquakes, i.e. an earthquake where human activities are attributed to have triggered it. Among these, apart from use of water reservoirs, (filling, emptying, change in levels) injections and extractions of massive quantities of fluids in oil fields and natural gas reserves are included. Induced earthquakes have been the subject of numerous debates and gave rise to extensive literature in the 1960s and 1970s. The International Commission on Large Dams has designated this issue as a necessary subject of study in projects involving the construction of big dams and has recommended temporary installations of surveillance networks during the first few years of use of the reservoir. Seismicity induced in areas around dams is far from being systematic in nature. Cases where it seems probable or at least plausible to attribute the occurrence of earthquakes to a particular water level of a reservoir or its variations represent only a very small percentage of big reservoirs (maximum depth equal to or higher than 100 m) and occur in only one in 1,000 dams of lesser capacity (see [YEA 97]). Apart from the effect of the depth and volume of the reservoir the following three conditions seem to lead to induced seismicity: – the water level is subject to significant variations in amplitude that are quite frequent; if the level remains more or less stable around the normal mark, induced seismicity, though observed during filling, tends to disappear later on; – the tectonic system of the area is generally the extension type of tectonic system leading to formation of normal or strike-slip faults rather than the compression type (reverse faults); this is confirmed for induced earthquakes occurring during or after filling; – faults show indications of recent activity affecting the top layer of land of the quaternary. Two processes have been suggested to explain the appearance of induced earthquakes: overloading due to the weight of water as well as infiltration of the water into the micro-fractures of the rock, resulting in the reduction of normal stress and thus friction on the surfaces of discontinuity (see Figure 1.26).
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Seismic Engineering
Figure 1.26 shows how these two mechanisms facilitate the play of normal and quasi-vertical faults, which holds true for the second condition enumerated above (a predominance of extension areas and hence of normal faults). Calculations have shown that the first process (overload) had very little influence on stress levels at a depth of several kilometers. Therefore its capacity to trigger a significant earthquake can only affect the balance of existing faults which is already precarious. It is probably the second process (lubrication of faults due to water infiltration) that is responsible for induced seismicity. In any case it seems to be the most plausible one, especially when seismicity is seen after a time gap (many months or even years after the completion of the filling process), which is quite frequent. Modifications of the stress field due to water overload may probably be of an almost instantaneous nature.
Figure 1.26. Combined action of the weight of water and its infiltration into a quasi-vertical network of faults supporting its play in normal faults (collapse of blocks); (according to [MUI 86])
The influence of variations in water pressure on this fractured rocky mass was underlined in certain cases of induced seismicity associated with pumping or massive injection of water into the ground. One such example is that of a 3 km well near Denver (Colorado) where a correlation was clearly observed between pumping and microseisms, and the oil deposits in which injection techniques were put into effect to increase the rate of recovery of hydrocarbons (see [YEA 97]). As against longstanding observations of
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53
seismicity induced by the filling of water reservoirs, those related to fluid extraction are mainly of concern to areas subjected to tectonic compression, the resulting reduction of load due to extraction favors the play of reverse faults (rising up of entire blocks). Today, it is a well-known fact that induced earthquakes are not to be differentiated from all tectonic earthquakes as far as their inherent causes are concerned. It is also widely accepted that accumulation of stress arises out of tectonic deformation but their triggering is precipitated due to the effect of overload and infiltration related to the use of large bodies of fluids. The main problem is not the need for instrumental surveillance, as this is now the rule for all big dams; but the question of whether the knowledge of their existence may have an effect on the assessment of seismic hazard (see Part 2). In highly active seismic areas, the hazard of the strongest shock imaginable is generally quite well covered and the possibility of an induced earthquake does not increase it much more even with a probabilistic approach to the hazard (see Chapter 8). On the other hand, in areas of moderate seismicity, the increased hazard may correspond to earthquakes where a recurrence interval (see section 6.2) is very long (tens of thousands of years) compared to the duration of the observations and induced seismicity may make it possible, or even probable an earthquake could be triggered during the lifetime of the structure and have a significant influence in the assessment of seismic hazard.
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Chapter 2
Parameters Used to Define Earthquakes
After a qualitative description of what causes earthquakes, we must now define the parameters used to describe them. There is a certain confusion regarding this subject, and not just in the minds of the general public; it stems from the mixing of the parameters that define the “size” of the seismic source (magnitude) and others that describe the importance of the effects (intensity). The fact that most types of magnitude presented by the seismologists are based on certain characteristics of signals registered on a particular type of instrument and not on the geometric and mechanical characteristics of the source, also contributes to confusing the issue. That is why this chapter only deals with the classification of the source through parameters with the most clear-cut physical significance, that is the seismic moment (and the moment magnitude that is linked to it) and the static stress drop. These quantities are defined in section 2.1 which gives a simple quantitative description of the model of the elastic rebound mentioned in section 1.2.1 and its consequences from the point of view of the energy report. In section 2.2 we develop, with the help of examples, aspects of the geometry of faults (surface, movement) that form the very basis of the notion of the seismic moment. Instrumental magnitudes are indicated in section 2.3. Classification of damages and intensity scales are dealt with in Part 5.
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2.1. Elementary theory of elastic rebound 2.1.1. Description of the elementary model We take up a very simple version of the elastic rebound model again just as that given in section 1.2.1 (see Figure 1.12) from the geometrical point of view (rectangular fault, parallelepiped blocks). We then consider (see Figure 2.1) a piece of lithosphere in the form of a parallelepiped (length L, breadth B, height H) subjected to a tectonic deformation due to shear forces that are exerted parallel to the length L. When a final deformation state is reached, there is a sharp rupture in the median plane (fault plane) during which the two blocks slide one in relation to the other till a new balanced position is found where they are separated by 'u along their length.
Figure 2.1. Simplified model for the elastic rebound theory
Parameters Used to Define Earthquakes
57
This separation may be unable to completely capture the cumulative differential displacement from the non-deformed state so that the blocks remain subjected to a shear stress in the new state of equilibrium. As indicated in section 1.2, this model can be used for showing shear forces in an inclined plane (case of normal and reverse faults) as well as for shear forces in a vertical plane (in the case of strike-slip faults). The only difference is that the length of the surface rupture, if visible, will be H in the first case instead of L in the second. We assume that: 1) outside the fault plane, the material has an elastic behavior; this hypothesis may seem debatable as we are interested in a break phenomenon which is, in general, the ultimate stage of a plastic behavior; it is justified if the fault is a preexisting one which has already moved during previous earthquakes, because then the fault plane is definitely in a weak zone and the balance before rupture is essentially ensured by the rubbing and meshing of asperities between two blocks. Geological observations show that most of the faults clearly visible on the surface have effectively produced a large number of earthquakes since the accumulated separation can go up to several km even several tens of km, whereas each earthquake only leads to a separation of a few meters maximum (see section 1.2.1). It is possible that the hypothesis of elastic behavior may prove to be incorrect for a fault that did not exist earlier where the rupture concerns healthy material that has not suffered any fractures earlier; 2) at the time of rupture, the shear stress instantaneously passes from the value Vu, final stress point at which the rupture occurs, to the value Vf, final stress point for the new balance, and this occurs over the entire fault plane. As mentioned in section 1.2.1, the model does not reproduce the dynamic aspects of the rupture process linked to the generation of a rupture front on the fault plane (see section 1.2.4); the hypothesis of the instant transition Vu Vf has been chosen as it is the simplest. The stress drop 'V defined by: 'V = Vu – Vg
[2.1]
is an important parameter to describe the seismic source; its direct measurement being quite impossible, it can only be determined in an indirect way and that with reference to a specific source model, we can speak of a stress drop only after having determined the type of model used, the definition of equation [2.1] is the simplest one of static stress drop (i.e. in relation to models that ignore the dynamic aspects of the rupture). The values 'V present two remarkable characteristics:
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– their weakness compared to the values of the gravitational stress due to the weight of the terrains which are of the order of several hundreds of MPa for the actual depths of the seismic focus (10 to 30 km), whereas 'V varies typically from 1 to 10 MPa; – their relatively constant value (variation of a factor of the order of 10, as it has been just shown), whereas the variation of the fault dimensions corresponds to a factor of the order of 104 (see end of section 1.2.1). These two points are discussed in section 2.1.2 for the small values of 'V and in section 2.1.3 for the minimal amplitude of their variations. The elastic behavior hypothesis outside the fault plane is expressed by the equation:
'V =
'u B
P
[2.2]
P being the shear modulus (the second Lamé coefficient) of the material which is related to the density U and the velocity Vs of propagation of the shear waves by the formula (see section 3.2): P = U Vs2
[2.3]
For numerical applications, we take the following values, which are suitable for a compact rock of a depth of several km: P = 3 x 104 MPa; U = 2,700 kg/m3; Vs = 3,333 m/s
[2.4]
Equation [2.2] results from the application of Hooke’s law to simple shearing of one of the blocks, whose distortion (shear deformation) is equal to the quotient of the displacement 'u /2 by the height B/2 subjected to shearing. In addition to the reduction in stress 'V, the fault model is characterized by the seismic moment Mo defined by: Mo = P LH 'u
[2.5]
Mo is definitely a moment since it is by definition equal to the difference between the moments of the forces that act upon the two blocks before and after the rupture, in fact we have the difference 'M: 'M = LH Vu B – LH Vg B = LHB 'V
Parameters Used to Define Earthquakes
59
or as B'V = P 'u according to [2.2] 'M = P LH 'u = Mo Other than the shear modulus P, which is a constant of the material, the definition of Mo only includes the characteristics of the fault, its surface LH and its displacement 'u; in section 2.2 we will go back to the values and the significance of these quantities. From equations [2.2] and [2.5] the result is that the volume LHB of both the blocks is equal to the quotient of the seismic moment by the stress drop: LHB =
Mo
P'u
B=
Mo
[2.6]
'V
For certain applications, it is convenient to replace this volume with that of a sphere; the radius R0 of this equivalent sphere is given by:
§ 3 · 1/3 ¸ © 4S ¹
R0 = ¨
§ Mo · 1/3 = 0.620 ¨ ¸ © 'V ¹
§ Mo · 1/3 ¨ ¸ © 'V ¹
[2.7]
The length L, breadth B, and the height H can be linked to R0 by defining the form factors IL, IB and IH as follows: L = IL R0; B = IB R0; H = IH R0
[2.8]
and where the product must satisfy the equation:
IL IB IH =
4S 3
[2.9]
Mo having the dimensions of a moment (or an energy), its natural unit is the Newton-meter (or Joule) and we must bear in mind that many seismologists still use the CGS system, in which Mo is expressed in dyne-centimeters (or ergs, remember that 1 J= 107 ergs). Whether Mo is measured in Joules or in ergs, its numerical values bring in high powers of 10 (Mo = 3 x 1019 Joules = 3 x 1026 ergs for an earthquake of a large size corresponding to L=50 km, H=20 km, 'u=1 m, which are approximately the characteristics of the earthquake at Kǀbe on 17 January 1995).
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Instead of the seismic moment, we prefer to use the moment magnitude traditionally represented by Mw, and defined by the logarithmic relation: Mw =
2 3
log10 M0 (Nxm) – 6
[2.10]
so that we have for Mo: Mo (Nxm) = 101.5 Mw + 9
[2.11]
The moment magnitude was introduced by Kanamori in 1977, at a time when instrumental magnitudes (see section 2.3.2) had already been around for a long time. At the beginning Kanamori thought that its usage should be reserved for large size earthquakes for which the instrumental scales present a saturation phenomenon (see section 2.3.2), in order to obtain the best possible adjustment of the values provided by the new scale with those of the other scales, he had also proposed constant terms slightly different from those (6 and 9) which figure in relations [2.10] and [2.11]. These “original” coefficients are seen in a number of publications, however, it seems preferable from the didactic point of view, and in order to bring out the most scientifically satisfying nature of the definition of the size of an earthquake by an evaluation of its seismic moment, to take its “round” values (6 and 9) in the relation between Mo and Mw. From the numerical point of view, the differences are minimal (less than 0.1 for the values of Mw) and in practice lower than the uncertainties affecting the determination of magnitudes (see section 2.3.2). The seismic moment (or moment magnitude) and stress drop are the two parameters that characterize the source in the simplified model just described. In equation [2.6] we saw that their quotient was equal to the volume concerned by accumulation and freeing of stress; in the next section 2.1.2, we will see that their product represents a constant factor multiplied by the energy liberated in the form of seismic waves, which enables an initial quantification of the magnitude in terms that are clear to an engineer, a second quantification, also just as important, but lesser known will be presented in section 1.3.2 based on geometric characteristics of the faults (surface and displacement). Finally in section 2.3.1 we will see that by making some additional assumptions, the same model enables us to calculate the parameters of the movement of the Earth (acceleration, velocity, displacement) as a function of Mo and 'V.
Parameters Used to Define Earthquakes
61
2.1.2. Energy balance
The energy balance between the state immediately preceding the rupture and the new equilibrium after rupture is given by the equation: Eu = Eg + Egr + Eg + Ec
[2.12]
in which: – Eu is the elastic energy of deformation stored in the two blocks before rupture (stress is equal to Vu) i.e.: Eu =
1 2
LHB
V U2 P
[2.13]
– Eg is the elastic energy of deformation contained in the two blocks after the rupture (stress is equal to Vg), i.e.: Eg =
1 2
LHB
V ³2
[2.14]
P
– Egr is the energy lost by friction during sliding of the blocks one in relation to the other, the frictional force has a value of LH Vg (hypothesis 2 of the instantaneous stress drop) and the displacement on which this force works has a value of 'u; we therefore have: Egr = LH Vg 'u
[2.15]
– Eg is the gravitational energy which is zero in the model being studied, given the perfect anti-symmetry of the movement of the two blocks (if one rises the other goes down in the same proportion); – Ec is the kinetic energy associated with seismic waves emitted during rupture, and that is what we are trying to calculate. Given that Eg = 0 and equations [2.13] to [2.15] we obtain from [2.12]: Ec =
1 LHB 2
P
V
2 2 u V³
– LH Vg 'u
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Again, by eliminating 'u with the help of [2.2]: Ec =
LHB ª 2 LHB 2 V u V 2V V u V º = 'V2 ³ ³ ³ ¼» 2 P ¬« 2P
Finally, by replacing LHB with Mo/'V (see [2.6]): Ec =
1 2P
Mo 'V
[2.16]
We have the given result, i.e. Ec is the product of an almost constant factor ½ P, of Mo and of 'V. Taking the logarithms to the base 10 of the two members of equation [2.16] and using definition [2.11] of the moment magnitude: Log10 Ec (Nxm) = 1.5 Mw + 9 + log10
'V 2P
[2.17]
The relation between the magnitude and seismic energy has been studied by different authors from recordings; these enable an instrumental magnitude to be determined (see section 2.3.2) but also the energy emitted in the form of waves to be calculated, which is a function of recorded velocity and of the distance between the focus of the earthquake and the recording site. Before the introduction of the moment magnitude, these authors mainly used magnitude Ms according to surface waves (see section 2.3.2) i.e. close to Mw except for very large earthquakes. We can quote for example the correlation proposed in 1956 by Gutenberg and Richter: log10 Ec (Nxm) = 1.5 Ms + 4.8
[2.18]
or that proposed by Bath (see [MAD 91]), which is slightly different: log10 Ec (Nxm) = 1.44 Ms + 5.24
[2.19]
Given the assimilation Ms = Mw, the identification of relations [2.17] and [2.18] leads to: log10
'V 2P
= – 4.2
Parameters Used to Define Earthquakes
63
i.e. with P = 3 x 104 MPa (see [2.4]):
'V = 3.8 MPa
[2.20]
If we interpret the above with the basic model of elastic rebound, the empirical relation [2.18] of Gutenberg and Richter corresponds to this constant value of a stress drop which, as mentioned before, is surprisingly weak when we compare it to the stress due to the weight of the terrain (which has a value, for example, of about 400 MPa, i.e. about 100 times higher, for a typical depth of 15 km).
Figure 2.2. Diagram showing the relation between the seismic moment (or moment magnitude Mw) and the length of the fault; the points corresponding to the different earthquakes studied are distributed between the two lines corresponding to the values 106 Pa (=1 MPa) and 107 Pa (= 10 MPa) the stress drop (according to [MAD 91])
This discovery has led to a number of debates between specialists of rock mechanics; laboratory experiments show that homogenous rock samples resist shear stress of the order of about 100 MPa without breaking when the tests are done at confinement pressures of the same order. The weakness of the value found for 'V (see equation [2.20]) conforms to the arguments given at the beginning of section 2.1.1 and according to which, for a pre-existing fault, the equilibrium before rupture essentially brings in friction and locking between asperities so that the conditions of rupture are very different from those of a homogenous environment.
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Seismic Engineering
These indirect evaluations of 'V are possible from the analysis of certain recordings, they show that the stress drops have quite the same order of magnitude as given in [2.20]. Figure 2.2 shows the relation between the seismic moment (or moment magnitude Mw) and the length L of the fault; according to equations [2.8], [2.7] and [2.11], and we have for this relation: log10 L = log10 (0.620 I L) + 0.5 Mw + 3 –
1 3
log10 'V
i.e., by resolving with relation to Mw: Mw = 2 log10 L +
2 3
Log10 'V + Cte
[2.21]
For a constant value of 'V, the magnitude Mw is thus a linear function of log10 L; we see in the figure that the points representing the earthquakes studied form an elongated scattergram held between the straight lines which correspond to the values 1 MPa and 10 MPa of 'V. The value 3.8 MPa (see [2.20]) is fairly close to the average (arithmetic or geometric) of these two limits. The hypothesis of a constant value of 'V is quite often accepted, this is the case in the Aki scale law (see section 2.1.3). In reality 'V is a variable parameter, but its amplitude of variation is much smaller (factor 10) than that of other characteristics of earthquakes (factor 104 for fault dimensions and factor 1012 for energy). The nature of variations is discussed below (see section 2.1.3). The total energy liberated during the rupture is equal to Eu – Eg, i.e. as Eg = 0, to the sum Ec + E gr of the kinetic energy of the seismic waves and of the energy lost in friction; Ec only represents a fraction K of this total liberated energy. This coefficient K, which measures the “seismic efficiency” of the rupture, only depends upon the relation Vg / Vu as, according to the relations established earlier, we have: K=
Ec Eu E g
=
V u V .g ² 2 V u V g²
=
1V .g / V u 1V .g / V u
[2.22]
K is small if Vg is only slightly lower than Vu (for example K = 0.2 for Vg/Vu = 2/3) which seems believable for most of the earthquakes. The main part of
Parameters Used to Define Earthquakes
65
the energy liberated by the rupture would be lost in friction (and therefore transformed into heat) on the fault plane. 2.1.3. Law of scale
With the accumulation of observation data emerged the idea that the earthquakes were homothetic between themselves, i.e. the physical properties of the source were determined from only one parameter, for example the length of the fault; the other linear geometric parameters (breadth B, height H, displacement 'u) are proportionate to L, the surface of the fault is proportionate to L2, the seismic moment to L3. K. Aki proposed this law of scale in 1967 but it required more than ten years for it to be authenticated through experimentation. This law plays a very important role not only in the interpretation of old earthquakes for which we have data on the length of the fault, but also for the study of seismic hazards (see section 5.1) by providing a method of calculating the maximum seismic potential of a fault with known dimensions. Equations [2.6] and [2.8] express the proportionality constant between the seismic moment and the cube of the length; we obtain: Mo = 'V LHB =
I BI H 'V L3 IL 2
[2.23]
Thus, this constant exists only if the stress drop 'V itself is constant. This hypothesis that has already been mentioned gives only the first approximation. We saw in Figure 2.2 that the large majority of the values of 'V were in the range of 1 to 10 MPa. Is it possible to infer tendencies for attribution of this variation to differences in the type of seismicity or in the type of fault? A frequently expressed opinion is that the intraplate earthquakes have on average a stress drop that is higher than that of the interplate earthquakes. To confirm this statement, precise criteria must be available to determine the inter or intraplate nature, something that is only possible for the interplate earthquakes situated on a plate boundary (subduction zone or large continental transform fault) and for the intraplate earthquakes that are clearly away from the boundaries. Many of the earthquakes occurring in the vicinity of the boundaries between the plates are in an ambiguous position as far as their “inter” or “intra” classification is concerned. Some authors have suggested maintaining the fault type (normal, reverse or strike-slip) as an explanatory parameter of the stress drop values. There appears to be a consensus that the reverse faults seem to be associated, more often than not,
66
Seismic Engineering
with the higher values of 'V. This tendency has been maintained in the formulation of certain attenuation laws (see section 4.2). One of the difficulties related to the usage of stress drops lies in the uncertainty of their determination. As given in section 2.1.1, this determination requires a reference to a model and it is often difficult to obtain a reliable estimate of the characteristics of the model that are necessary for the calculation of 'V, particularly for older earthquakes that were not recorded and for more recent earthquakes recorded with old instruments. When we can estimate the seismic moment Mo and the volume LHB of the two blocks affected by the rupture, 'V can be calculated with formula [2.6]: 'V
Mo
L+%
[2.24]
In practice, we often prefer to calculate 'V with the following formula, deduced from J. Brune’s model [BRU 70], which is a model of a circular fault with radius R: 'V =
7 Mo 16 R 3
[2.25]
To apply relation [2.25] it is sufficient to know the surface LH of the fault plane; the radius R of the circle of the same surface is inferred from it by: R=
LH
S
[2.26]
Although 'V has only a low variation amplitude compared to that of L, it has an important influence on the parameters of the vibratory movement (acceleration, velocity, displacement) (see section 5.1). Then, if, for a general and mainly qualitative description of the seismic source, we can adhere to the law of scale ('V constant), we must be conscious of its limits for the quantitative applications which are of interest to engineers. A physical limit of the validity of the law of scale is made up of the thickness of the part of the earthquake-prone crust, typically about 20 km, described in section 1.1.3. The fault planes are thus obliged to remain within this earthquake-prone part. This constraint has very visible consequences for the large strike-slips whose fault planes are generally almost vertical, their vertical extension being limited to about 20 km. They “make up” by stretching horizontally considerably; the large earthquakes of the continental transform faults such as the San Andreas Fault or the
Parameters Used to Define Earthquakes
67
North-Anatolian fault have fault planes of 300 to 400 km in length and only 15–20 km in height. They deviate greatly therefore from the law of scale which assumes constant relationships between the dimensions of the fault. A question that is still debatable is whether, for this type of earthquake, the displacement 'u remains proportional to L (as given in the law of scale) or if it tends towards a constant value (which would be the case if we suppose it to be proportional to H). The validity limit of the law of scale seems to correspond to the moment magnitudes of the order of 7.5. What is the upper limit of the size of earthquakes? The largest one of those for which we have fairly precise data is the Chilean earthquake of 22 May 1960 whose estimated characteristics are L = 1000 km, H = 200 km, 'u = 30 m, that corresponds to a seismic moment Mo of 1.8 x 1023Nxm or a moment magnitude of 9.5. This is a subduction earthquake (reverse fault) whose fault plane lies entirely in the earthquake-prone part of the subducted plate as it is clearly parallel to the higher edge of this plate. It seems to be close to the maximum possible limit to which most of the seismologists have concurred to fix the magnitude of 10 (higher limit). The logarithmic nature of the scale of moment magnitudes and the relative constancy of the stress drops are such that in practice Mw cannot really go beyond this value even if we consider extreme and totally unreal hypothetical cases. For example, the exaggerated extrapolation of the large strike-slip fault will lead to the following hypotheses: – length L = 40,000 km, i.e. the totality of the Earth’s equator; – height H = 20 km corresponding to the thickness of the earthquake-prone part of the crust; – displacement 'u = 400 m, corresponding to the same rule of proportionality to L as that of the largest known earthquakes ('u = 4 m for L = 400 km). In these conditions we find MO = 9.6 x 1024 Nxm, i.e. magnitude Mw = 10.7. If we accept that the law of scale is valid on the whole range of magnitudes, we can draw up Table 2.1 of the orders of size of the characteristics of the seismic sources. It has been established with the values 'V = 3.8 MPa (see [2.20]) and IB = 2/3 (which is a result, as can be seen in section 2.2, of the adjustment on an empirical correlation between the moment magnitude and the LH surface of the fault plane).
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Seismic Engineering
Mo(Nxm) 15
10
Mw 4
Ec(J)
Ro(km) B(km)
2 LH(km )
LH(km) 'u(m)
6.3 x 10
10
0.40
0.27
1.00
1.00
0.03
1016
42/3
6.3 x 1011
0.86
0.57
4.64
2.15
0.07
1017
51/3
6.3 x 1012
1.85
1.23
21.5
4.64
0.15
1018
6
6.3 x 1013
3.98
2.65
100
10.0
0.33
1019
62/3
6.3 x 1014
8.57
5.71
464
21.5
0.72
1020
71/3
6.3 x 1015
18.5
12.3
2,150
46.4
1.55
1021
8
6.3 x 1016
39.8
26.5
10,000
100
3.33
1022
82/3
6.3 x 1017
85.7
57.1
46,400
215
7.18
1023
91/3
6.3 x 1018
185
123
215,000
464
15.5
1024
10
6.3 x 1019
398
265
1,000,000
1,000
33.3
Table 2.1. Characteristics of seismic sources in relation to the seismic moment (according to the law of scale with 'V = 3.8 MPa, I8 = 2/3)
The values of the seismic moment vary in the table from 1015 to 1024 Nxm, which corresponds to moment magnitudes that range from 4 to 10, the higher limit of 10 representing, as we have seen, the maximum maximorum imaginable for the size of an earthquake, the lower limit of 4 constituting the threshold below which the destruction potential of the earthquake becomes negligible. If the majority of the records obtained by the seismograph networks correspond to magnitudes lower than 4, these events are interesting from the point of view of earthquake-resistant engineering only for studying seismic probabilities; the vibratory movements that are associated with them are too weak to produce damages although they are sometimes felt quite strongly in some areas. In this table we see that the size of the earthquakes, whose most significant measure is the surface LH of the fault plane or the dimension LH of the square having the same area, varies greatly. The surface LH follows quite a simple law,
Parameters Used to Define Earthquakes
69
since it is multiplied by 10 when the magnitude increases by one unit, starting with LH = 1 km2 for Mw = 4. This law is expressed by the equation: LH (km2) = 10Mw – 4
[2.27]
which is undoubtedly the most important equation for the engineer to retain so as to have a physical idea of the significance of the magnitude parameter. Most of the presentations insist more on the correlation of energy with magnitude (multiplication by 1,000 when Mw increases by 2) which is certainly interesting but which hides the fact that energy liberated by volume unit is in reality constant (equal to 'V 2 / 2 P according to relations [2.16] and [2.6]). In section 2.3 we will come back to the significance of the magnitude for the engineer and in section 5.1 to its influence on the amplitude of the vibratory movement. Equation [2.27] results from the choices 'V = 3.8 Mpa and )B = 2/3; we have in fact, according to [2.8], [2.9] and [2.7]: LH = I L I
2 H Ro
1
4S 2 Ro 3IB
1
4S § 3 · 3 § Mo · 3 3IB ¨© 4S ¸¹ ¨© 'V ¸¹
1
1
1 § 4S · 3 § Mo · 3 IB ¨© 3 ¸¹ ¨© 'V ¸¹
That means, taking into account [2.11]: 1/ 3
LH (km2) =
10 4 § 4S · IB ¨© 3 ¸¹
ª¬ 'V MPa º¼
2 / 3
u 10 M w
[2.28]
or, with I B = 2/3 and 'V = 3.8 MPa: LH (km2) = 0.993 x 10Mw – 4
[2.29]
This is identical to [2.27] at about 0.7%; the choice 'V = 3.8 MPa comes from the adjustment on Gutenberg and Richter’s law on energy (see section 2.1.2); that of IB = 2/3 comes from the fact that equation [2.27] is practically equivalent to an empirical equation established by D. Wells and K. Coppersmith (see [2.27]) which will be presented and discussed in section 2.2. The fault surface is a parameter that is more reliable than the length L, although the latter is at the base of the formulation of the law of scale; this statement results from the comparison of the size of samples and standard deviations for the empirical correlations of Wells and Coppersmith (see section 2.2), which is why Table 2.1 does not have a column for the length L, whose size orders can however be inferred
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Seismic Engineering
from studying the values of LH , with the exception of large strike-slips, for which L / LH can reach 4 or 5 (for example, L = 400 km for Mw = 8, whereas the table shows LH = 100km), L typically varying between LH (square fault) and 2 LH . The displacements 'u of the table are in centimeters for the small earthquakes (Mw~4 to 5), in decimeters for the medium sized earthquakes (Mw~6), in meters for the large earthquakes (Mw~7.5 to 8.5), and in decameters for the giants of a magnitude higher than 9. Attention must be drawn to the fact the displacement 'u, that comes in definition [2.5] of the seismic moment, is the average displacement on the fault plane and not on the maximum displacement. The observations on site and numerical inversions of the records (see section 5.2) show that the distribution of the displacements on the fault plane is often very mixed and that the maximum can be much higher than the average. To give just one example, the bottom photo of Figure 1.17 that shows a vertical displacement of 7 to 8m of a thrust fault (earthquake at Chi-Chi, Taiwan on 21 September 1999) is not representative of the average displacement observed on this fault, which is only about 2 m. The energy values (column Ec of the table) are discussed in section 2.3. 2.2. Geometry of the faults 2.2.1. Length of fault and length of rupture
The notion of the length of a fault is very definite when the fault is seen on the surface; it is thus confused with the length of the course of the rupture. After each large earthquake, geologists rush to study the marks and clues visible on the ground in order to map the fault; this is the job of a specialist as the spectacular cases of rupture which can “be seen by the naked eye” are not in the majority. The photos in Figures 1.16 to 1.19 give a slightly deceptive picture of reality as they only deal with major displacements (higher than a meter in most cases) clearly affecting the scenery. It often happens that the courses are difficult to identify as they are hidden by vegetation or snow, or because they are mixed with the secondary ruptures resulting from the landslides or from falling rocks. Figure 2.3 shows to what extent snow can change the surface appearance of a vertical movement fault (thrust), such as the earthquake at Spitak (Armenia) on 7 December 1988 which is the subject of the photo in the middle of the Figure 1.17 taken before heavy snow fall.
Parameters Used to Define Earthquakes
71
Figure 2.3. Members of the French post-earthquake mission organized after the earthquake in Armenia on 7 December 1988 climb towards the fault line, which is hardly visible on this snowy slope
When the outlet of the surface fault lies in a loose ground zone, the appearance of the fault changes very rapidly because of rock falls (in particular for reverse faults with oblique dip where the overhang of the overlapping compartment cannot remain in a state of equilibrium) and over a longer term by erosion due to rain. It is therefore not easy to gauge the displacement of the fault and its dip angle except when the rupture affects rocky terrains or civil engineering works. The reverse fault of Chelengpu, the cause of the earthquake at Chi-Chi (Taiwan) on 21 September 1999, produced vertical displacements of 7 to 8 m in its northern part that we can easily measure on the waterfall and the bridge visible in Figure 1.17 (bottom photo); in its central part, the vertical displacements are weaker, of the order of 1.5 to 2 m and the angle of the dip is not clear where the rupture concerns soft soil as can be seen in Figure 2.4 (photo taken in a sports stadium).
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Seismic Engineering
Figure 2.4. Rock fall of the overlapping section of the reverse fault of Chelungpu (earthquake at Chi-Chi, Taiwan on 21 September 1999) in a loose soil zone
Measuring the length of the fault from the course of the rupture on the surface is only possible for earthquakes of a sufficient size and whose focus is at a depth comparable to the dimensions of the fault plane. In Table 1.1 we can see that below magnitude 6, considering their low values, the displacements have little chance of producing clear effects on the surface and that this is only possible if the earthquake is extremely superficial since LH does not go beyond a few km. For earthquakes of magnitude higher than 6, surface ruptures do not occur systematically but are more probable when the magnitude increases to be almost certain for magnitudes of 7 at least in the regions where the earthquake prone part of the crust does not stretch beyond a depth of 20-25 km. These tendencies do not apply to earthquakes of subducted plates whose inclined fault planes can only reach the surface at the level of the trench (see Figure 1.10), i.e. offshore. On the other hand, the large subduction earthquakes quite often produce phenomena of uplifting or phenomena of subsidence (piling up or caving in of neighboring areas) linked to the movement of the elastic rebound (see section 3.1). Even in the cases where the surface rupture is clear, it is not certain that it represents all of one side of the fault plane. The famous earthquake of Hyogo-ken Nanbu (Kǀbe) on 17 January 1995 of a magnitude close to 7 broke the surface for only about 10 km (to its south-western extremity on the island of Awaiji), whereas the total length of the rupture has been estimated to be about 40 km (see Figure 2.5).
Parameters Used to Define Earthquakes
73
Figure 2.5. Map of faults that may have played a role in the earthquake of Hyogo-Ken Nanbu (Kǀbe) on 17 January 1995; the surface rupture is visible only on the right-lateral fault for about 10 km on the island of Awaiji; the faults marked by a question mark at the north of Kǀbe apparently extended the rupture towards the north-east but did not appear on the surface (according to [COL 95])
As the length of the fault planes can in practice be inferred from the lines on the surface only in a limited number of cases, how do we try to estimate it for general purposes? For recent sizeable earthquakes, the most commonly used method consists of defining the zone where aftershocks are felt with a temporary network of seismographs set up immediately after the earthquake. In the extent where the aftershocks correspond to local readjustments in the vicinity of the fault plane (see section 1.2.4) it is logical to think that the extension of the aftershock zone provides a measure (slightly excessive no doubt) of the dimensions of this plane. It is necessary to resort to a temporary network if we wish to exactly situate the focal points of the aftershocks, in fact the precise location of a focus or specifically of its depth can only be done if we have several instruments framing the source and placed at distances which are in the same order from the depth which is only a very rare occurrence with fixed network stations even in well equipped regions. Figure 2.6 shows a map of aftershocks traced after 15 days of observation for the earthquake at Kǀbe.
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Seismic Engineering
Figure 2.6. Location of aftershocks recorded in the 15 days following the main shock on 17 January 1995 for the earthquake at Hyogo-Ken-Nanbu (Kǀbe) (see [COL 95]). Above, view of plane; below, vertical cut following the SW-NE axis of the rectangle traced on the map above with indication of horizontal distances and depths in km
We notice that the large majority of aftershocks are concentrated in a fairly narrow band of about 50 km in length (in the direction SW-NE) and 20 km in breadth (in a vertical plane). Bringing this band to the fore has enabled us to identify the faults that are likely to have acted in this earthquake (see Figure 2.5), but that have not left visible traces on the surface. In many cases studying aftershocks is the best way to learn the geometric characteristics of the faults. Figure 2.7 represents the results obtained for the earthquake at Loma Prieta on 17 October 1989 which occurred about 100 km south of San Francisco.
Parameters Used to Define Earthquakes
75
This earthquake of a magnitude of 7.1 surprised the Californian seismologists by the absence of rupture on the surface and by the existence of a strong thrust component in the movement of the fault, whereas they expected a practically pure right-lateral fault that is a general characteristic of the earthquakes originating from the San Andreas Fault.
Figure 2.7. Location of the aftershocks for the earthquake of Loma Prieta of 17 October 1989 a) plane view; b) section by the vertical plane AA’; c) section by the vertical plane DD’ (according to [MAD 91])
Study of the aftershocks showed that the fault plane presented a dip of about 70° towards the south-west (part c of Figure 2.7), compatible with the partially reverse character of the mechanism at the hypocenter and that it was thus probable that it did not concern the San Andreas Fault itself, which is perfectly vertical in this region but a secondary fault belonging to the same system [MAD 91]. Systematic usage of aftershocks to estimate dimensions of fault planes has been carried out only since 1970 in most of the developed regions where the seismic zones are relatively easy to access; as, in addition, it basically concerns earthquakes of a sufficient size to justify detailed studies on the ground (magnitudes higher than 5.5-6 to give an idea), it does not give data in the following cases:
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Seismic Engineering
– old earthquakes that were not recorded or recent earthquakes that have not been measured systematically by seismographs on the ground given the practical difficulties of access; – earthquakes with their epicenters in the sea; – small earthquakes (magnitude of the order of 5 or lower than 5). When we do not have observations of surface ruptures or a precise map of the aftershocks, we are reduced to using indirect evaluation methods, the results of which are not very certain, to estimate the fault dimensions. Some of these possible methods are: – distribution of damage, which is generally higher in the zones close to the fault although other factors (particularly site effects; see section 4.3) also play an important role in this distribution; – for subduction earthquakes, the extension of the neighboring zones affected by tsunamis (see Chapter 3 and Chapter 12); this method has been used, for example, to estimate the size of the large historical earthquakes of Peru; – numerical inversion, i.e. adjusting parameters of a numerical model of the seismic source so as to have the best possible reproduction of the characteristics from the recorded signals (see section 2.3.2). After this brief examination of the methods that can be used to learn the dimensions of the faults after the occurrence of the earthquake, it is necessary to say something about a problem that will be discussed in greater detail in Part 2. It concerns estimating the possible size of a future earthquake that is likely to occur on a known fault. The major faults, identified as seismically active and traced on geological maps, generally have a total length that is clearly higher than the rupture lengths associated with the earthquakes that they have produced in the past. The two large continental transform faults already mentioned, San Andreas in California and North-Anatolian in Turkey are more than 1,000 km in length whereas their strongest earthquakes correspond to rupture lengths of 300 to 400 km. For the North-Anatolian fault, whose known history dates back more than a thousand years, we noted episodes of migration of ruptures, either towards the west or towards the east, during which chunks of variable lengths break one after the other (see Figure 2.8).
Parameters Used to Define Earthquakes
77
Figure 2.8. Migration of ruptures on the North-Anatolian fault. The current tendency (map above) is a migration to the west, which is the opposite of that from 1,000 years ago (map below), (according to [COL 99b])
Migration towards the west, which characterized the second half of the 20th century for the western part of the fault, was confirmed in 1999 by the earthquakes that occurred near Izmit (Kocaeli earthquake on 17 August 1999 and Duzce earthquake on 12 November 1999) which do not appear in the map on the top of the figure, which corresponds to the period 1939-1992. It is however to be noted that the second earthquake (12 November 1999) occurred to the east, and not to the west of the first one. The use of migratory tendencies for the prediction of earthquakes will be discussed in section 6.3. Other than this aspect, the question of “maximum magnitude”, i.e. the longest possible rupture on a given fault, is one of the main problems that we come across in the study of seismic hazard (see Chapter 7).
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Seismic Engineering
2.2.2. Well documented examples of fault ruptures
To illustrate the earlier questions on the geometry of faults, we will now present a few examples of earthquakes whose rupture characteristics could be estimated in a fairly reliable manner. These examples have been chosen on the basis of the following criteria: 1) all types of faults must be represented and for each type, the movements must virtually be pure, which means that the secondary component, for example strikeslip for a normal fault or a reverse fault, must have an amplitude distinctly lower than that of the principal component. The reverse faults category has been divided into two, to distinguish between those that affect the superficial part of the continental crust, where the magnitude does not exceed 7.5, and bigger ruptures due to subduction (magnitudes of 7.5 to 9.5); 2) regions where these earthquakes have occurred must represent as wide a sample range as possible for seismic zones in the world; 3) for each type of fault, we must find the whole range of magnitudes that are significant in earthquake-resistant engineering; this condition is difficult to meet for lower magnitudes (less than 5) as it is rare for fault dimensions of such earthquakes to be determined precisely. These criteria have made it possible to record 48 earthquakes (sixteen strikeslips, 12 normal faults and twenty reverse faults, among which there are eight great subduction earthquakes). The sources used to compile this data are essentially the previously mentioned study by Wells and Coppersmith [WEL 94], which concerns earthquakes that have ruptured the surface and for subduction earthquakes, various recent publications on numeric inversions are used. Table 2.2 summarizes the characteristics of the 48 earthquakes, in columns from left to right: – the name of the earthquake and the country where it occurred; – the date (day, month, year); – the dimensions (in km) of the fault plane equated into a rectangle; LRS is the length of the rupture on the surface or parallel to the surface, and LP is the breadth in the perpendicular direction following the dip; – the average displacement 'u (in m); – the moment magnitude Mw calculated with the values of LRS, LP and 'u, with P = 3x104 Mpa; – the stress drop 'V (in MPa) calculated by [2.25] and [2.26].
Parameters Used to Define Earthquakes
Name of earthquake
S T R I K E S L I P
Liège, Belgium Homestead Valley, California Jura Souabe, Germany Tres Pinos, California Kawazu, Japan San Salvador, El Salvador Skopje, Macedonia Parkfield, California Superstition Hills, California Songpan-Huya, China Lancang-Gengma, China Landers, California Dasht-e-Bayaz, Iran Montagua, Guatemala San Francisco, California
Date
LRS x LP
'u
MW
'V
08/11/1983
5x3
4.80
0.661
15/03/1979
6x4
0.03 5
5.04
0.746
0.05 0
5.24
1.27
0.09 0
5.45
0.887
5.54
2.31
5.77
3.60
09/03/1978
4.5 x 6
26/01/1986
11 x 5
17/08/1976
9x4
10/10/1986
6 x 7.5
26/07/1963
17 x 11
0.09 0
6.03
1.07
28/06/1966
35 x 10
0.19
6.29
1.02
24/11/1987
30 x 11
0.33
6.49
2.17
16/08/1976
30 x 12
0.20
6.74
4.62
06/11/1988
80 x 20
0.26
7.02
1.28
28/06/1992
62 x 12
0.54
7.21
7.91
31/08/1968
110 x 20
1.20
7.45
3.58
04/02/1976
257 x 13
0.7
7.61
3.29
18/04/1906
432 x 12
2.95
7.81
3.35
300 x 20
2.30
8.05
6.17
04/12/1957
2.60
Gobi-Altaï, Mongolia
3.30 6.54
N
29/02/1980
3.8 x 5
0.11
5.20
1.84
O
Arudy, France
04/10/1978
7 x 5.5
0.16
5.51
1.88
R
Wheeler Crest, California
29/04/1984
17 x 5
0.14
5.70
1.11
M
Perugia, Italy
30/08/1962
7x8
0.31
5.81
3.03
A
Cache Valley, Utah (USA)
13/09/1986
15 x 14
0.15
5.98
0.757
L
Kalamata, Greece
28/03/1975
15 x 10
0.31
6.10
1.85
Pocatello Valley, Idaho (USA)
13/12/1982
20 x 7
0.87
6.38
5.37
F
Dhamar, Yemen
04/03/1981
26 x 18
0.60
6.62
2.03
A
Corinthe, Greece
23/11/1980
60 x 15
0.64
6.83
1.56
U
Irpinia, Italy
02/03/1987
32 x 14
1.70
6.91
5.87
L
Edgecombe, New Zealand
18/08/1959
45 x 17
2.14
7.13
5.66
T
Hebgen Lake, Montana (USA)
10/11/1946
28 x 30
3.73
7.32
9.41
Ancash, Peru
79
80 R
Seismic Engineering
Goodnow, New York (U.S.A)
07/10/1983
1.5 x 2
0.27
4.92
11.4
South of Niigata, Japan
07/12/1990
6.5 x 5
0.095
5.31
1.22
Miramichi, Canada
09/01/1982
5.5 x 4
0.36
5.58
5.61
Marryat Creek, Australia
30/03/1986
13 x 3
0.50
5.84
5.85
Mont Chenoua, Algeria
29/10/1989
15 x 10
0.23
6.01
1.37
Tennant Creek, Australia
22/01/1988
13 x 9
0.63
6.23
4.26
Frioul, Italy
06/05/1976
19 x 10
1.05
6.52
5.57
San Fernando, California
09/02/1971
17 x 14
1.50
6.69
7.11
Kern County, California
21/07/1952
64 x 19
0.60
6.89
1.26
El Asnam, Algeria
10/10/1980
55 x 15
1.54
7.05
3.92
Tabas-e-Golshan, Iran
16/09/1978
74 x 22
1.50
7.24
2.72
Caucete-San Juan, Argentina
23/11/1977
80 x 30
2.62
7.52
3.91
S
Guam, Mariana Islands
08/08/1993
120 x 40
2.50
7.70
2.64
U
Kanto (Tokyo), Japan
01/09/1923
120 x 50
4.00
7.90
3.77
B
Michoacan-Guerrero, Mexico
19/09/1985
150 x 140
2.20
8.09
1.11
DU
Tokachi-Oki, Japan
16/05/1968
200 x 70
6.00
8.27
3.71
C
Kurile Islands, USSR
13/10/1963
450 x 80
7.00
8.59
2.70
T
Equatorial Coastal Zone, Colombia
31/01/1906
520 x 100
10.00
8.80
3.21
I
Prince William Sound, Alaska
28/03/1964
700 x 150
20.00
9.20
4.51
ON
Valdivia, Chile
22/05/1960
1,000 x 200
30.00
9.50
4.90
E V E R S E
F A U L T
Table 2.2. 48 examples of earthquakes where the dimensions of the fault plane are well known: LRS: length of rupture parallel to the surface (km); LP: breadth of rupture (km); 'u: average displacement (m); 'V: stress drop (MPa)
The magnitude values given in the table are deduced from the seismic moment calculated on the basis of the dimensions of the fault plane and average displacement; they are written with two decimals in order to indicate the difference to instrumental magnitudes which are traditionally given with a single decimal or with a simple fraction (1/4, 1/3, 1/2, 2/3, 3/4) as in Table 2.1; they can show slight deviations from values recorded on instruments, determined at the time of the earthquake; for example the Kern County earthquake on 21 July 1952 (reverse fault) had a magnitude of 7.7 (determined through surface waves; see section 2.3.2) in the traditional catalogs while its moment magnitude Mw is only 6.89 according to the characteristics given in the table. In a large majority of cases, however, the values in the table show a difference lower than 0.3 as against values recorded on instruments.
Parameters Used to Define Earthquakes
81
Large subduction earthquakes constitute a separate category, as their instrumental magnitudes are affected by the phenomenon of saturation (see section 1.3.3.2). For this type of earthquake, only the moment magnitude is representative of the size of the rupture. The stress drop, calculated using Brune’s formula [2.25] on the basis of the radius of the circle having the same surface as the fault plane, shows quite a strong variation amplitude, since the minimum is 0.661 MPa (Liège earthquake on 8 November 1983) and the maximum is 11.4 MPa (Goodnow earthquake on 7 October 1983). We can establish the following tendencies: – earthquakes having a magnitude lower than 5.5 appear to have a low stress drop (often lower than IMPa, especially in the case of strike-slips) with the exception of the Goodnow earthquake (7 October 1983); it is probable that this tendency translates the rather limited range of sample cases (only eight earthquakes) and uncertainties mentioned here above concerning the estimation of fault dimensions for small-scale occurrences, rather than the physical reality; – for medium and large-scale earthquakes (magnitudes above 6), the dispersion of the stress drop is moderate; less than 25% of values (8 out of 33) come from the 1.5–7.5 MPa interval; it is possible that the relative uniformity is the consequence of mean effect; the general character of the stress drop, such as it is calculated in the table, covers the irregularities of a rupture phenomenon better for a large fault plane than for a small one; the occurrence of a smaller dispersion for earthquakes of great magnitude can also be seen in Figure 2.2 where the scatter diagram thins down markedly for magnitudes from around 7; – the 12 cases of continental reverse faults confirm the hypothesis advanced in section 2.1.3 quite well, according to which the stress drop is on average stronger in the case of reverse faults than for other mechanisms (normal fault or strike-slip fault); the average of the 'V values in the table is 4.52 MPa in these 12 cases, as against only 3.36 MPa for the 12 cases of a normal fault; on the other hand, the very large reverse faults (subduction earthquakes) give an average value of 3.32 MPa for the eight cases presented, with a weak dispersion (with the exception of the Mexican earthquake of 19 September 1985 all the other examples are in the range of 2.64– 4.90 MPa). The values of the geometric parameters (dimensions of the fault plane and displacement) correspond to those given in Table 2.1; the tendency of great continental faults to lengthen, already mentioned in section 2.1.3, is evident for strike-slips (Motagua, San Francisco, Gobi-Altai), but is equally significant for certain normal faults (Irprinia, Hebgen Lake) and reverse faults (Kern County, El Asnam, Tabas-e-Golshan) and confirms the validity of the hypothesis of a relatively low thickness (-20 km) of the part of the crust which is prone to earthquakes.
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Seismic Engineering
2.2.3. Correlations of geometric characteristics of ruptures with moment magnitude
2.2.3.1. Wells and Coppersmith correlations In an article mentioned several times [WEL 94], in 1994 D. Wells and K. Coppersmith proposed a set of correlations between different geometric parameters of faults (length of rupture on the surface and in depth, width of rupture according to the dip, surface of rupture, average and maximum displacement) and the moment magnitude. These correlations result from the analysis of a database comprising 244 earthquakes, whose geometric parameters are considered to be fairly precise. Considering the significant amount of data and the fact that future seismic activity will only present a limited number of well-documented new cases, we can take it that these correlations will not significantly improve in the near future. They thus constitute an important reference of the seismological formulae set. As indicated earlier, the validity domain of these correlations cannot be extended very much to include lower magnitudes (around 5 or less than 5) due to uncertainties about the characteristics of smaller earthquakes, which produce surface ruptures only in rare cases. As for higher magnitudes, subduction earthquakes have been excluded because their fault planes are contained entirely within or at the boundaries of subducted plates and so escape observations on site (we note that, for these earthquakes, the dimensions given in Table 2.2 have all been deduced indirectly, by numeric inversion or damage mapping). Tables 2.3 and 2.4 summarize part of the correlations established by Wells and Coppersmith, which concern the following parameters: – LRS: length of rupture on the surface (in km); – SR: surface rupture (in km2), which is shown by LH in Table 2.1; – DM: maximum displacement on the fault plane (in m); – 'u: average displacement on the fault plane (in m, which is the displacement that is part of the calculation of the seismic moment). These correlations are linear relations between the moment magnitude MW and the decimal logarithm of one of these parameters. It is to be noted that inverse relations are also given, i.e. if we have for example: Mw = a + b log10 (LRS)
[2.30]
Parameters Used to Define Earthquakes
83
We do not have the right, in principle, to solve this equation with relation to log10 (LRS) to write: log10 (LRS) = –
a b
+
1 b
Mw
[2.31]
since, as per statistic correlations, the expression of log10 (LRS) on the basis of MW (inverse relation) brings in coefficients different from those that appear in equation [2.31]. This point will be taken up and discussed in section 2.2.3.2. The columns in Tables 2.3 and 2.4 are, from left to right: – the type of correlation; – the type F of fault (S strike-slip, R reverse, N normal, T all types considered together); – the number N of earthquakes used in the analysis; – values a and b of coefficients of linear relation; – value V of the standard deviation of the correlation; – the range 'M of magnitudes for which the correlation is applicable. In Tables 2.3 and 2.4 we see that the consideration of the type of fault, can have a marked influence on the numeric value of certain coefficients (compare, for example, the values of a and b for strike-slips and normal faults in the relations giving log10 (LRS) on the basis of Mw).
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Seismic Engineering
Correlation
MW = a
+ b log10 (LRS)
log10 (LRS) = a + bMW
MW = a + b log10 (SR)
log10 (SR) = a +b MW
F
N
a
b
V
'M
S
43
5.16
1.12
0.28
5.6 – 8.1
R
19
5.00
1.22
0.28
5.4 – 7.4
N
15
4.86
1.32
0.34
5.2 – 7.3
T
77
5.08
1.16
0.28
5.2 – 8.1
S
43
– 3.55
0.74
0.23
5.6 – 8.1
R
19
– 2.86
0.63
0.2
5.4 – 7.4
N
15
– 2.01
0.50
0.21
5.2 – 7.3
T
77
– 3.22
0.69
0.22
5.2 – 8.1
S
83
3.98
1.02
0.23
4.8 – 7.9
R
43
4.33
0.90
0.25
4.8 – 7.6
N
22
3.93
1.02
0.25
5.2 – 7.3
T
148
4.07
0.98
0.24
4.8 – 7.9
S
83
– 3.42
0.90
0.22
4.8 – 7.9
R
43
– 3.99
0.98
0.26
4.8 – 7.6
N
22
– 2.87
0.82
0.22
5.2 – 7.3
T
148
– 3.49
0.91
0.24
4.8 – 7.9
Table 2.3. Correlations between the moment magnitude MW , length LRS (km) and surface SR (km2) of the rupture
In Table 2.4 values concerning reverse faults are shown in brackets, as the quality of these correlations is considered too poor to be able to recommend their use. In practical applications, however, it is these correlations, in particular, established for all types of faults (lines marked T in the tables) that are used. A note about the largest scope that can be made on examining the tables is the importance of the standard deviation V, which varies typically between 0.2 and 0.3 in Table 2.3 and between 0.3 and 0.4 in Table 2.4 (if we do not consider reverse faults, which give values of V which are even greater, but whose reliability is insufficient, as shown above). For relations where the dependent variable is a logarithm, a standard deviation of 0.2 corresponds to a multiplication by 1.58 of the average value; this multiplication reaches 2.00 for V = 0.3 and 2.51 for V = 0.4. A general tendency is seen here, of formulae used in seismology where the standard
Parameters Used to Define Earthquakes
85
deviation is about the same as the average. This fact, which translates the variability of seismic phenomenon and the difficulty in quantifying it using a small number of parameters, should never be lost sight of by those providing earthquake engineering. Correlation
F
N
a
b
V
'M
MW = a + b log10 (DM)
S
43
6.81
0.78
0.29
5.6 – 8.1
(R)
(21)
(6.52)
(0.44)
(0.52)
(5.4 – 7.4)
N
16
6.61
0.71
0.34
5.2 – 7.3
T
80
6.69
0.74
0.40
5.2 – 8.1
S
43
– 7.03
1.03
0.34
5.6 – 8.1
(R)
(21)
(– 1.84)
(0.29)
(0.42)
(5.4 – 7.4)
N
16
– 5.90
0.89
0.38
5.2 – 7.3
T
80
– 5.46
0.82
0.42
5.2 – 8.1
log10 (DM) = a + bMw
MW = a + b log10 ('u)
log10 ('u) = a + b MW
S
29
7.04
0.89
0.28
5.6 – 8.1
(R)
(15)
(6.64)
(0.13)
(0.50)
(5.8 – 7.4)
N
12
6.78
0.65
0.33
6.0 – 7.3
T
56
6.93
0.82
0.39
5.6 – 8.1
S
29
– 6.32
0.90
0.28
5.6 – 8.1
(R)
(15)
(– 0.74)
(0.08)
(0.38)
(5.8 – 7.4)
N
12
– 4.45
0.63
0.33
6.0 – 7.3
T
56
– 4.80
0.69
0.36
5.6 – 8.1
Table 2.4. Correlations between the moment magnitude MW, the maximal displacement DM (m) and average ('u) on the fault plane
Correlations concerning dimensions (length and surface) of the fault plane are distinctly better, from the point of view of their standard deviation, than those related to displacements. While only retaining the correlations marked T in Table 2.3, V has the value 0.22 or 0.28 (according to which we consider the direct or inverse relation) for the relation log10 (SR)-MW; the corresponding values of Table 2.4 are 0.42 (or 0.40) for the relation log10-(DM) MW and 0.36 (or 0.39) for the relation log10 ('u) - Mw.
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Seismic Engineering
Special consideration has been given, in the points developed in section 2.1.3 on the law of scales, to the relation between magnitude and the rupture surface rather than that between magnitude and the length of rupture. This preference does not seem justified based on standard deviation alone, as the law that gives the length of the rupture based on the magnitude (V = 0.22) seems slightly better than the one that gives the surface (V = 0.24); but the tendency is different for inverse laws (magnitude based on the length or the surface), the standard deviation remaining at 0.24 for the law for surfaces while it increases considerably (from 0.22 to 0.28) for the law for lengths. We will see in section 2.2.3.2 that this deviation between direct and inverse laws shows a greater dispersion of data. We see, in addition, in Table 2.3 that the range of sample cases having served to determine these correlations is practically twice as large for the law for surfaces (148 earthquakes against 77), which presents an argument in its favor. The graphic representations of the expressions in Tables 2.3 and 2.4 are given in Figure 2.11 for direct and inverse laws and for the average laws which will be defined in the next section. 2.2.3.2. Considerations about direct and inverse regression correlations; the average correlation making the algebraic inversion possible The Wells and Coppersmith correlations are a linear regression for a variable obtained by the method of least squares; if we consider (see Figure 2.8) a set of N points of coordinates xi, yi (I = 1, 2, ..., N) in the xOy plane, the coefficients a and b of the linear relation, y = a + bx which “best” represents this set is obtained by minimizing the root-mean-square deviation Ey defined by: E2y =
I N
N
¦
(a + bxi-yi)2
[2.32]
i I
This definition implies that we are concerned with y expressed on the basis of x, as the deviations are measured with relation to Oy. If we consider the expression of x on the basis of y by a linear relation, as in x = c + dy, it is necessary to reduce the sum of squares of the deviations measured with relation to Ox or the root-meansquare deviation Ex. For instance: E2x =
I N
N
¦ i i
(c + dyi – xi)2
[2.33]
Parameters Used to Define Earthquakes
87
Figure 2.9. Approximation of a set of points in the plane, by a linear relation; the direct correlation y = a + bx minimizes the sum of squares of deviations measured according to y; the inverse correlation x = c + dy minimizes the sum of squares of deviations measured according to x
As we shall see, the coefficients, c and d resulting from minimization of [2.33] are different from the coefficients – a/b and 1/b that we would find through nonalgebraic inversion of the relation y = a + bx, and this difference, with respect to slopes c and i/b, is a measure for the dispersion of the “scattergram” of given points. Coming back to [2.32], finding the minimum standard deviation Ey leads to cancellation of the partial derivatives with relation to a and b, i.e. the system: N
(
¦x
N
i
)a+(
¦
x
2 i
N
¦
)b =
i I
i I
N
N
¦
Na + (
xiyi
i I
xi) b =
i I
¦
yi
[2.34]
i I
We establish: xg =
N
I
¦
N
i I
N
Ixx =
¦ i I
xi; yg =
I N
N
¦
(xi – xg)2; Ixy =
yi
[2.35]
i I
N
¦ i I
N
(xi–xg)(yi–yg); Iyy =
¦ i I
(yi–yg)²
[2.36]
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Seismic Engineering
This makes it possible to write the solution of the system as follows [2.34]: a = yg –
Ixy Ixx
x g; b =
Ixy Ixx
[2.37]
if we apply equal weight to all the points of the whole, xg and yg representing the coordinates of the center of gravity, and Ixx, Ixy and Iyy representing the elements of inertia tensor around the center of gravity. We state, according to formulae [2.37], that we have: yg = a + b xg
[2.38]
i.e., the straight line in question y = a + bx passes through the center of gravity of N points. Solution [2.37] determines the direct correlation (y considered as a function of x); the inverse correlation x = c + dy obtained through the same formulae allowing only the roles of x and y; we thus have: c = xg –
Ixy Iyy
yg; d =
Ixy Iyy
[2.39]
According to [2.37] and [2.39] the product bd is equal to parameter O defined by: O=
I ² xy IxxIyy
[2.40]
We can easily show, using the Cauchy-Schwarz inequality, that O is less than or equal to one, equality being possible only if (yi – yg)/(xi – xg) has the same value for all the points, i.e. if they are exactly aligned. In practical cases, O is thus less than one, which leads to the point that the slopes of the direct and inverse correlations are different, since the product bd represents their ratio: more precisely in the representation of abscissa x and ordinate y, the inverse correlation is more sloped than the direct correlation; as both pass through the center of gravity, they appear as shown in Figure 2.9 (where the slopes are negative) and Figure 2.11 (where the slopes are positive). Parameter O is a measure of the dispersion of data, which is as large as O is small; we can in fact demonstrate, based on the earlier formulae, that the minima of
Parameters Used to Define Earthquakes
89
standard deviations, corresponding to values [2.37] and [2.39] of the coefficients, are given by the following: Ey, min = (
I O I O I yy)1/2; Ex, min = ( I xx)1/2 N N
[2.41]
We can also show that if, instead of the inverse correlation, the algebraic inversion of the direct correlation is taken, the standard deviation is multiplied by I/
O , hence it increases when O decreases.
The practice of algebraic inversion of correlations is unfortunately quite common, as rarely do we come across authors of correlations who, like Wells and Coppersmith, take the trouble to present the direct and inverse correlations at the same time. This practice can lead to some major errors. In order to avoid these difficulties of inversion, one possibility is the measurement of root-mean-square deviations, not parallel to Ox or Oy, but perpendicular to the line of best fit that we are looking for. Figure 2.10 defines the coefficients D and G for the equation x cos D + y sin D-G = O for this line, called the average correlation.
Figure 2.10. Approximation of a set of points in the plane by a line such that the sum of squares of their distances to the given points is minimum; this line (average correlation) is in between the direct and inverse correlations and also passes through the center of gravity G
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Seismic Engineering
Let the root-mean-square deviation with relation to the distances to this line, Vd, be defined as: Vd2 =
I N ¦ Ni I
(xi cos D + yi sin D – G)2
[2.42]
With notations [2.35] and [2.36] we have values of D and G which minimize Vd: tan 2D =
2Ixy Ixx Iyy
G = xg cos D + yg sin D
[2.43]
The second of relations [2.43] shows that the average correlation, like the direct and inverse correlations, passes through the center of gravity G. If we know the direct correlation y = a + bx and the inverse correlation x = c + dy we can determine coefficients D and G of the average correlation and in fact, the first of equations [2.43] can be written, considering [2.37] and [2.38], as: tan 2D = W =
2Ixy Ixx Iyy
=
2bd d b
[2.44]
from which we can deduce t = tgD as: t = tanD =
1 W
H
I W ² 1
[2.45]
H value + I or – I according to the rule: H = sign (b2 – O)(with O = bd)
[2.46]
We then calculate xg and yg based on a, b, c, and d by writing that the direct and inverse correlations pass through the center of gravity; we thus find: xg =
c ad 1 bd
; yg =
a bc 1 bd
[2.47]
Parameters Used to Define Earthquakes
91
The equation of the average correlation: x + ty = xg + tyg
[2.48]
is entirely determined, since we know t [2.45] xg and yg [2.47]. The application of correlations in Tables 2.3 and 2.4 is presented in Table 2.5 and in Figure xg 2.11; y = a’ + b’x is the equation for the average correlation (a’ = yg + , b’ = t c 1 – ), y = a” + b”x for the algebraic inverse of the inverse correlation (a’ = – , t d 1 b” = ). d y
x
a
b
c
d
O
a’
b’
a’’
b’’
MW
log10 LRS
5.08
1.16
– 3.22
0.69
0.80
4.83
1.34
4.67
1.45
MW
log10 SR
4.07
0.98
– 3.49
0.91
0.89
3.95
1.04
3.84
1.10
MW
log10 DM
6.69
0.74
– 5.46
0.82
0.61
6.68
0.94
6.66
1.22
MW
log10 'u
6.93
0.82
– 4.80
0.69
0.57
6.94
1.12
6.96
1.45
Table 2.5. Average correlations for the correlations in Tables 2.3 and 2.4
Figure 2.11 shows and Table 2.5 confirms (the comparison of values of coefficients b, b’ and b’’) that the dispersion is clearly stronger for correlations that include displacements. This conclusion had already been reached based on the study of standard deviation. For the other correlations (with the length or surface of the rupture), those with the surface appears slightly, but significantly better than those with length (O = 0.89 instead of 0.80 in the relation to extreme slopes b’’/b = 1.12 instead of 1.25). This confirms the choice made for the adjustment of the IB factor (= 2/3) in the basic model of elastic rebound (see section 2.1.3). The fact that coefficients a and b of law M = a + b log10 SR (km2) vary very little (a around 4 and b around 1) between the direct, mean and the algebraic inversion of the inverse correlation justifies the rounded values at 4 and 1 which appear in equation [2.27]. For the laws for displacement, Table 2.5 presents values very close to coefficients a, a’ and a’’, while coefficients b, b’ and b’’ are distinctly different. It is a fortuitous coincidence due to the fact that the displacement which corresponds to the center of gravity G is close to one meter (see Figure 2.11, where this value is shown), thus its logarithm is zero; coefficient a (or a’ or a’’) is thus practically equal to the ordinate of the center of gravity and it is normal to find the same value for the
92
Seismic Engineering
three laws, since they all pass through this point. On the boundaries of the validity domain, on the other hand, the deviations between these laws are great, which confirms the risk of error, referred to earlier, resulting from the use of algebraic inversions.
Figure 2.11. Direct, inverse and average correlations for Wells and Coppersmith correlations presented in Tables 2.3 and 2.4; the least pronounced slope corresponds to the direct correlation, the most pronounced slope to the inverse correlation; the average correlation to an intermediate slope; the rectangle in dotted lines defines the validity domain of the correlations; the center of gravity G of the data is indicated by its coordinates
Parameters Used to Define Earthquakes
93
2.3. Parametric description of earthquakes 2.3.1. Source parameters and effect parameters
When we learn through the media that a notable earthquake has occurred somewhere, we are invariably informed about its “degree” on the Richter scale, which is called an “open-ended” scale, or described as “having nine degrees”. Despite the vague nature of these comments, this information is extremely useful in order to evaluate the importance of the phenomenon and get an idea of the possible damage caused. First of all we must clarify the vocabulary and correct some misuse of the language. The “Richter scale” is nothing but the magnitude in its first definition presented by C. Richter in 1935. Without going into the details, which will be presented and commented upon in section 2.3.2, it can be said that the “degree recorded on the scale” corresponds to the moment magnitude MW defined in section 2.1.1. Table 2.1 makes it possible to evaluate, on the basis of the value of MW, a certain number of characteristics of the source. Among these, the most important is, as indicated earlier, the dimension of the rupture zone, described either by the side LH of the area of the same surface, or by the radius R0 of the sphere of the same volume. In the strict sense, the terminology of scale is incorrect with regard to magnitude, as this term must be reserved for quantities that cannot be measured on an instrument, but simply estimated with reference to observation criteria making it possible to define the degrees of importance of the phenomenon. However, the magnitude (or rather magnitudes, as there are several definitions, as we shall see in section 2.3.2) is determined on the basis of the recording obtained by a seismometer. Thus, it is a measurement and not an evaluation deduced through observations. The so-called “Richter scale” is misleading for the average person who is not used to logarithmic scales. In everyday life, there is no great difference between 6 and 8 for example, whereas an earthquake with the magnitude of 8 represents 1000 times more energy and a fault plane 100 times more extended on the surface, than an earthquake with a magnitude of 6. If it is correct to say that “the scale” is openended, i.e. it has no upper limit, we have seen in section 2.1.3 that the limits imposed by the dimensions of the Earth and the rupture mechanisms on the fault planes do not make it possible to predict that the magnitude may in reality exceed 10. As for the nine degrees that is some times attributed, this is only another way, essentially incorrect but not contradicted by fact, of describing this limitation.
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Seismic Engineering
Magnitude is a source parameter, i.e. it aims at characterizing the size of the rupture zone. Thus, it is only one of the elements that enable the evaluation of the destructive potential of the event; and this depends, in fact not only on the force of the source, but also on other factors, the most important being: – the depth at which the rupture zone is located; – its distance to the “potential targets” (towns, villages, constructed sites, communication pathways, network lines); – the conditions that are eventually unfavorable, and linked to the particularities of certain sites (amplification of vibrations by sediment layers or topographical accidents (see section 4.3), risks of soil liquefaction, landslides or tsunamis (see section 3.3)); – the rather high vulnerability of constructions. We have already highlighted the fact that the notions of depth and distance at the sites make sense in earthquake engineering only if they are linked to the dimension of the source. We shall see in section 5.1 that an approximation, simple, but sufficient to determine the scale of the value of maximum velocity of the vibratory motion, which is undoubtedly (rather than the maximum acceleration) the parameter most representative of the damage potential for modern buildings (see section 14.3), is given by the following rule. The maximum velocity is constant and equals a value V0 that depends only on the stress drop (but is independent of the magnitude), on the inside of the sphere with a radius R0 (which is a function of the magnitude; see Table 2.1) centered on the hypocenter. On the outside of this sphere, the maximum speed decreases in inverse proportion to the distance R at the center V = V0
R0 R
[2.49]
Figure 2.12 shows the consequences of formula [2.49] for earthquakes with a 1 2 for which the depth of the focus could be either 8 km magnitude of 5 , 6 and 6 3 3 or 16 km. As per Table 2.1 the values of R0 corresponding to these magnitudes are 1 2 clearly equal to 2 km (Mw = 5 ) 4 km (MW = 6) and 8 km (Mw= 6 ). 3 3
Parameters Used to Define Earthquakes
95
Figure 2.12. Comparison of the focal spheres (radius R0) and the spheres of potential damage (radius 5.R0) for quakes with a magnitude 5
1 2 , 6, 6 at a depth of 8 or 16 km 3 3
The circles representing the intersections of the spheres of radius R0 by a vertical plane have been traced; the hypocenters are F1 (depth 8 km) and F’1 (depth 16 km) 1 2 , F2 and F’2 for MW = 6, F3 and F’3 for MW = 6 . We 3 3 have also traced in dotted lines circles of radius 5R0 centered on the hypocenters; the radius 5R0 can be considered the “action radius” for the damage potential resulting from the vibratory motion. As per equation [2.49] the maximum velocity is 0.2 V0 for R = 5R0. At distances from the hypocenter higher than 5R0, the maximum velocity V falls below this limit 0.2 V0 (i.e., nearly 0.1 m/s for the “normal” values of the stress drop, see section 6.1) and becomes too weak to cause great damage to reasonably well constructed structures, even if they have not been designed following the earthquake-resistant standards (see section 6.2). Of course, this limit represents only a value for the size and can be questioned, as per the particularities of the source (stress drop) or of the site considered (amplification or directivity effects).
for quakes of magnitude 5
The circles resulting from the intersection by a vertical plane of these “spheres of potential damage” of radius 5R0 have lines B1, B2, B3 on the surface as indicated in 1 2 , 6, 6 3 3 respectively, occurring at a depth of 8 km; for a depth of 16 km, only the magnitudes
Figure 2.12 and correspond respectively to earthquakes of magnitude 5
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2 produce lines B’2 and B’3. Table 2.6 presents the radius and the surface 3 of the “circles of potential of damage” at the surface of the soil, which are obtained from basic calculations.
6 and 6
Magnitude Mw
Depth
Radius CB, CB1
Surface
(km)
(km)
(km2)
5
1 3
C1F1 = 8 km
C1B1 = 6 km
113 km2
5
1 3
C1F’1 = 16 km
X
X
6
C2F2 = 8 km
C2B2 = 18.3 km
1,056 km2
6
C2F’2 = 16 km
C2B’2 = 12 km
452 km2
6
2 3
C3F3 = 8 km
C3B3 = 39.2km
4,825 km2
6
2 3
C3F’3 = 16 km
C3B’3 = 36.7km
4,224 km
2
Table 2.6. Radii and surfaces of the circles of potential of damage on the surface of the soil for the earthquakes in Figure 2.12
Table 2.6 and Figure 2.12, upon which it is based, illustrate the significance of the magnitude and the depth of the hypocenter. Even for a relatively weak variation of MW, there is a big difference in the damage potential since the surface affected is 1 multiplied by a factor close to 10 between MW = 5 and MW = 6, when the depth is 3 1 2 8 km and by a factor higher than 40, between MW = 5 , and MW = 6 (for the 3 3 same depth). The influence of the depth is very noticeable when the magnitude is 1 low (the earthquake with a magnitude 5 no longer has any damaging effects 3 within the validity limits of the earlier hypotheses, when its depth is 16 km) but reduces rapidly as magnitude increases. However, it must be noted from Figure 2.12 that if the affected surfaces (maximum velocity higher than 0.2 V0) do not vary too much when the depth of the
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2 changes from 8 to 16 km, the epicentral zone (around 3 point C3) is subjected to shocks which are clearly stronger in the first case (8 km) than in the second (16 km). In fact, as per [2.49] the maximum velocity in C3 is V0 or 0.5 V0 depending on whether the depth is 8 or 16 km. On the other hand, in the periphery of the affected zones (around points B3 and B’3), the amplitude of the motion is practically the same in the two cases. It can be concluded that the distribution of maximum velocity becomes more uniform on the inside of the affected zone, as the depth increases. This is confirmed by the observations of the damages, which generally decrease much more rapidly for the highly superficial earthquakes than for the relatively deeper earthquakes (see section 1.4.1).
earthquake of magnitude 6
Equation [2.49] also shows that the maximum velocity is only 0.25 V0 at the 1 , that is, a value hardly higher than 3 “limit of damages” 0.2V0. Although the depth of this earthquake is only 8 km, in reality it presents the characteristics of a deep earthquake in terms of earthquake engineering (small variation in amplitude of the motion on the inside of the affected zone). The observations made in section 1.2.4 on the time of rupture show, moreover, that the signals emitted by small earthquakes are of short duration; their damage potential is thus in reality much lower, for the same amplitude of motion, than the damage potential of earthquakes of higher magnitude, whose longer duration signals are more dangerous for the structures which are more easily affected by the cumulative effects of the loading cycles. This is important for the understanding of the significance of magnitude.
epicenter C1 of the quake of magnitude 5
Considering what has been mentioned earlier, it is clear that the absolute value of the depth or the distance is not important. However, what matters more is their relative value obtained by dividing them by the dimension of the source, represented by its equivalent radius R0. The action radius notion of damages is essential for every earthquake engineer; from the practical point of view, when the size of an earthquake has to be judged immediately after the announcement of its occurrence, thus, without having any details in hand, it is easier to calculate this radius as the double of LH , rather than the quintuple of R0, as in the earlier description; we see in Table 2.1 that these two modes of calculation are practically equivalent, but the advantage of LH is that its square LH is obtained by the formula, which is extremely easy to remember, from equation [2.27]. We cannot forget that in such “on the spot” evaluations, these small calculations can provide only average values and that it is better to follow the strict law of seismology, according to which the standard deviation has the same value as the average (see section 2.2.3.1). Take, for
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example, the Mexican earthquake of 19 September 1985 (see Table 2.2), which caused tremendous damage in Mexico City, located approximately 400 km from the hypocenter; however, for an earthquake of magnitude close to 8, formula [2.27] gives LH = 10,000 km2, that is, only 200 km for the action radius taken as 2 LH , as per the rule that has just been mentioned. The fact that the “actual action radius” was twice as high underlines the importance of the site effects (see section 4.3). We must also pay attention to the confusion that is often created regarding the depth of focus. First of all, the precision with which it can be determined depends on whether we can get recordings from a nearby zone; when the earthquake has been recorded only by relatively far-off observatories, they are restricted to choosing, in general, the most plausible value of the depth, taking into account the characteristics of the signals; this choice is made based on a list of standardized values (often 33 km, or 0 km for very superficial earthquakes); the depth values given immediately after the earthquake often come from this type of “calculation”, hence, it is important to consider them as scales rather than measures. The depth announced can then be compared to the dimension of the source (R0 or LH ; if these two lengths are comparable, it means that on the surface there are zones very close to the fault that have felt very violent shocks (this is the case with epicenter C3 in Figure 2.12 when the earthquake of magnitude 6
2
has a depth of 8 km). Just as for distances, 3 the depth (which is none other than the focal distance from the epicenter) becomes significant only if it is linked to the dimensions of the fault. The descriptive parameters of the seismic source, i.e. what we can learn of them just after the earthquake, are the magnitude, which, for all events of a certain size, is communicated systematically to the media, and a clue to the depth, which is generally reserved for the seismologists. The stress drop is not part of the information immediately available. There is another way which, historically, has been the first, and for a long time the only, way to characterize earthquakes. It involves the evaluation of effects (impressions of witnesses, visible manifestations on the ground or in water, damage to buildings) as per the degrees of a scale of intensity. The description of the main scales is given in Chapter 14, for it is logical to discuss it only after having studied the quantitative aspects of the vibratory motions (see Part 2) and the typical methods of destruction resulting from the action of this motion (Chapter 12). Here, we limit ourselves to the following comments: – the notion of intensity is fundamentally different from that of magnitude; the magnitude is a source parameter and hence has in principle only a single value for a given earthquake, the intensity is a part of the effect parameters. Thus, by nature, it
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has as many values as the places where it has been measured; in general the intensity is strongest near the epicenter and decreases as we move further out; – different from the “Richter scale” the intensities are true scales, as per the terminology specified earlier, that is why their degrees must be written in Roman figures, and not Arabic numbers so as to correctly indicate that these are not the numbers on which calculations can be carried out; – determining the intensity requires a very detailed survey of the terrain (analysis of the questionnaires answered by witnesses, evaluation of the characteristics of buildings and other concerned constructions, documenting statistics related to the damages); hence it can only be known a long time after the earthquake has occurred, whereas magnitude is announced by seismological observatories almost immediately; – the tendency today is to rely more on information derived through instruments, and not through intensity evaluations; and apart from the fact that they represent a great volume of work, these are often considered to be superfluous when recordings are available; this is not the correct attitude, for a good survey of intensity provides information complementary to that obtained with the help of instruments, in particular on the site effects or on the vulnerability of constructions; it is also the only way of comparing earthquakes occurring today to those that occurred long ago, for which we only have descriptions given in the archives; – although the intensity evaluations do not directly concern the characteristics of the seismic source, it is possible to give an estimation with the help of empirical correlations deduced from the analysis of recent earthquakes, as we shall see in section 14.2; this is the only method to know, at least approximately, the magnitude and the depth of the old earthquakes. Besides intensities, other effect parameters are being used more and more in regions where instruments are available; these are the parameters of vibratory motion, such as velocity, acceleration or certain general characteristics of time signals (Arias intensity (see section 4.1.3) and response spectra (see Chapter 9)). The development in information technology now enables us, with the help of a sufficiently dense network of stations equipped with modern material, to create “instrumental intensity” cards within minutes of the occurrence of the earthquake; such cards being extremely useful in organizing aid. 2.3.2. Different magnitudes
The notion of magnitude was not introduced in the beginning as a measure of the seismic moment according to a logarithmic scale, although the presentation adopted in section 2.1.1 could make us believe this. In the 1930s, the concept of a seismic
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moment had not really been formed; however, some seismologists were already aware of the fact that it was possible, with the help of an analysis, to take recordings of seismographs to get an idea of the size of the earthquake. This feeling was based on the two following statements: – the time gap between occurrences, at a given site, of the longitudinal and transverse waves (see section 3.2.1) enables us to calculate the distance between this site and the focus of the quake; – the decrease, in relation to this distance ', in the maximum amplitude A measured on the seismograph is clearly the same, whatever the earthquake considered, which is what the Figure 2.13 shows, where a constant difference ML is seen between the curve of the decrease in log10 A (with respect to ') pertaining to any earthquake and the other pertaining to a particular earthquake.
Figure 2.13. The decrease in the decimal logarithm of the maximum amplitude A with respect to the focal distance' is the same whatever the earthquake and enables us to define the local magnitude ML in relation to a particular earthquake
These statements led C. Richter to define the local magnitude ML using the equation: ML = log10
A( ' ) A (') o
[2.50]
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A being the maximum amplitude measured on a calibrated seismograph (Wood and Anderson’s torsion seismometer, of an actual period of 0.8 s and equipped with an amplifier of a coefficient of 2,800). The reference curve A0 (') was chosen in such a way that ML remains positive for very small earthquakes; it corresponds to a magnitude of 1 micron (10-6m) for a distance ' of 100 km. Richter’s proposal in 1935 was elaborated by following the ideas already presented in 1931 by Wadati in Japan. It aimed at providing a means to rapidly estimate through a recording the scale of earthquakes occurring in Southern California. The name local magnitude comes from the fact that these earthquakes are superficial (depth lower than or equal to 20 km) and are recorded at distances reaching, at the most, a few hundred km (in these conditions the focal distance can generally be confused with the epicentral distance). It didn’t take long for this proposal to be recognized and seismologists developed other types of magnitude adapted to conditions different from those that led to the definition of ML (medium or deep earthquakes, teleseismic recordings i.e., those obtained at long distances). These new magnitudes were calibrated so as to almost coincide with ML in its range of validity. Without getting into the details of their definition, which mainly concerns seismologists, the following magnitudes can be quoted as examples: – MS magnitude as per the surface waves (see section 3.2) of a period close to 20s; – mb or mB magnitude as per the volume waves (see section 3.2); – MJMA magnitude as per the definition given by the Japan Meteorological Agency; – Mw moment magnitude, defined in section 2.1.1 using the seismic moment for which techniques were developed to calculate magnitude using recordings. Figure 2.14 presents a comparison of these magnitudes (see [HEA 86]).
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Figure 2.14. Magnitudes ML, MS, mb, mB and MJMA with relation to the magnitude of the moment MW (see [HEA 86])
This figure shows the phenomenon of saturation which affects the instrumental scales for high magnitudes, i.e. the fact that beyond a certain threshold, the value measured no longer increases in proportion to the moment magnitude; the threshold of saturation depends on the type of magnitude (it is around 6 to 7 for mb and ML, and from 7.5 to 8 for the other types); physically, this saturation corresponds to the incapacity of classical seismographs to respond to excitations whose period is clearly higher than the actual period of the instrument. However, the long duration waves emitted by a seismic source are determined by the greatest dimension of the rupture zone (see section 5.1); for larger earthquakes, the duration of these waves reaches several tens of seconds and visibly overtakes that of the instruments. Thus the saturation appears to be faster for the types of magnitude based on the response of the instruments with a shorter period (like the Wood-Anderson seismograph used by Richter to define ML) than for those that require signals with a longer period (like MS, which corresponds to surface waves with a period of 20 s). This statement explains that the moment magnitude Mw should be considered as the only one valid for big earthquakes (magnitudes higher than 7.5-8). To quote an example, the current record of MW = 9.5 (corresponding to the earthquake in Chile on 22 May 1960, already mentioned in section 2.1.3) is given in the catalogs as having a magnitude MS which measures only 8.3. As it can also be used for smaller earthquakes, its use has become widespread, although the transposition in MW of the other magnitudes, according to which the majority of earthquakes listed in the catalogs has been measured, constitutes a rather delicate problem.
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Figure 2.14 shows that around the value MW = 6, the deviation between the different magnitudes is low; that is why the comparison between the attenuation laws of motion (which often use ML or MS, sometimes mb or MJMA, the latter case being systematic under Japanese law) is carried out in section 4.2 for MW = 6. For values significantly different from 6, the differences between the different magnitudes are quite evident. For example, the earthquake that occurred on 15 January 1993 north of Japan was the subject of the following findings: mb = 6.9, MS=7.1 and MW = 7.5 for USGS (United States Geological Survey), MS = 6.7 for the University of Berkeley (California), MJMA = 7.4 (Japan Meteorological Agency), MW = 7.8 for the laboratory of geophysics at Papeete (Tahiti). It can be seen that for the same type of magnitude the values can be different according to the observatories, which can be explained from the fact that the calculations of magnitude involve adjustments linked particularly to the local conditions. Therefore, a bit of confusion may reign soon after the earthquake regarding the value of its magnitude; in general, a few days are enough for the seismologists to give a precise analysis and to agree upon the most plausible value. In the beginning, Richter calibrated his scale so that the smallest earthquakes, that could be detected then with the help of instruments, corresponded to the magnitude 0. Modern day instruments being much more sensitive and a greater density in the network of stations make it possible today to commonly have negative magnitudes (up to –2 or –3) in regions well equipped with instruments. The preceding observations show that determining magnitudes is a job only for specialists. For the engineer, the main thing is to be able to associate their values of parameters that can be used for the appreciation of the risk and the calculation of structures. The parameters of vibratory motion (accelerations, velocities, displacements, periods or dominant frequencies) whose measurement involves not only the characteristics of the source, but also its distance at the particular site and its geotechnical conditions, will be examined in the second part of this book; thus we limit ourselves here to the source parameters which are essentially functions of the magnitude and which are important for the specialists in earthquake engineering, i.e. fault surface, time of rupture and energy. In several places, in the preceding sections, we have stressed the preeminence of the fault surface (or of the length of the side of the square of the same area) to interpret magnitude. It enables us, as we have seen in section 2.3.1, to immediately evaluate the expanse of zones affected by the strong shocks as well as the rather superficial character of the source in terms of earthquake engineering. The considerations on the time of rupture (see section 1.2.4) show that it is, in the majority of cases, proportional to the dimension of the fault; if we retain an average
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velocity of propagation of the rupture of 2.5 km/s, the value of the length of time of the rupture Tr is given by: Tr (s) = 0.4 x 100.5Mw - 2
[2.51]
Formula [2.51] is obtained by taking, for the dimension of the fault, the square root of the fault surface given by equation [2.4]; thus we find that Tr = 1.26 s for Mw = 5, Tr = 4.00 s for MW = 6 and Tr = 12.6 s for MW = 7. Tr represents a lower limit of the duration of the signal recorded on the surface, which generally increases as we move further away from the source due to the separation of different wave forms (see section 3.2.2) and the local ground response (see section 4.3.4). This duration is also an important parameter for the destructive potential of the earthquake (see section 2.3.1) The fault dimension is thus an important parameter, not only because it defines the stretch of affected zones, but also because it governs the duration of the earthquakes, one of the major factors of damage. In most books on earthquake engineering the emphasis is on the energy in order to illustrate the significance of magnitude. It is very important to highlight the extremely rapid variation of energy with the magnitude (factor 1,000 for a difference of 2; see Table 2.1). It mainly shows that the energy produced by small-scale earthquakes is most insufficient to reduce the violence of large-scale potential earthquakes in zones where stress has been accumulating for a long time. Thus, the energy-magnitude relation plays an important role in the application of seismic hazard (see Part 2). From the point of view of earthquake engineering, however, this relation can give rise to interpretations that are too simplistic; this is often the case, for example, for comparisons with the energy produced by other natural or artificial phenomena. When it concerns release of localized and rather instantaneous energy, as for an underground nuclear explosion or the impact of a massive meteorite, the analogy should not be made with a great earthquake, that stretches over a larger area on the surface and occurs very slowly (around ten or a few dozen seconds instead of a very small fraction of a second).. For example, the kinetic energy of a meteorite weighing a million tons and hitting the Earth at a speed of 14 km/s is about 9.8 x 1016 Joules, i.e., according to Table 2.1, the energy Ec of an earthquake of a magnitude very slightly higher than 8. Does that mean that the two phenomena can be compared from the point of view of their effects? Certainly not; even if the comparison is limited to the waves emitted in the ground, their nature (the compression waves override the impact, instead of the shear waves for the earthquake), their mode of emission and propagation (a precisely located source on the surface for the impact, source spread along the depth for the earthquake) and even their total energy (Ec represents the total seismic energy although the energy of the waves emitted by the
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impact is only a fraction of the kinetic energy of the meteorite) present extremely visible differences. As has already been indicated in section 2.1.3, the energy per unit of volume released by the earthquakes is independent of the magnitude if we accept the law of scale. This is so because the magnitude controls the dimensions of the rupture zone, and not because it would act on the very mechanism by which the energy is released that it seems to have a great energy significance. This argument helps us to understand the great importance of the geometrical significance.
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Chapter 3
Manifestations of the Seismic Phenomena on the Surface
3.1. Deformation of superficial terrains 3.1.1. Deformations linked to tectonics The idea that the most terrifying manifestation of earthquakes is people and buildings being swallowed up by cracks suddenly opening up in the ground was largely spread by popular imagery right up to the beginning of the 19th century. The illustration reproduced in Figure 3.1 stands testimony to this. The original is a copper engraving carried out shortly after the earthquake of 26 July 1805 in Naples (see [KOZ 91]). In it we see some unfortunate victims being dragged into the fault, while others run away or call out to the skies. It is clear that this belief continues to haunt certain minds, since the open air museum, close to San Francisco, which was set up to commemorate the great earthquake of 18 April 1906 and inform the Californian population on the seismic risk, devotes a whole panel to the refutation of this fantastic theory, under the catchy title, “Can the San Andreas Fault swallow cities? No!”; a photograph of this panel is shown in Figure 3.2. Section 1.2.2 shows examples of surface rupture caused by faults of various types (see Figures 1.16 to 1.19); these examples show very clear ruptures, marked like staircase steps (normal faults and reverse faults) or saw cuts (strike-slip faults). It often occurs that the trace of the fault is more diffused and presents the aspect of a crushed zone; these less spectacular cases are less easily photographed.
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Figure 3.1. Destruction of Naples by the earthquake of 26 July 1805 (engraving of unknown origin, taken from [KOZ 91])
From the point of view of earthquake engineering, the movements of faults on the surface are the subject of exclusion, i.e. building activity is avoided in the immediate vicinity of the faults recognized as active. Such rules of exclusion imposing very severe constraints on town planning projects, it is advisable to appreciate as precisely as possible the size of the hazard (i.e. the probability of the occurrence of surface rupture) and the width of the bands which it is necessary to neutralize on both sides of the fault. We will return to this discussion in section 7.2 which discussed the aspects of probability and in section 11.1 which deals with the damage caused by the movements of the fault according to the level of displacement and the distance to the trace of the rupture.
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Figure 3.2. Panel from the open air earthquake museum on the San Andreas Fault close to San Francisco; the text on the right relates the disputed story of Mathilda, a cow which was supposedly swallowed by the fault, except for its tail that stuck out even after it closed up
Apart from surface ruptures, large-scale earthquakes can produce overall upward movements (uplifting) or downward movements (subsidence) on wide stretches of the ground’s surface. The consequences are potentially disastrous for coastal areas in the event of subsidence because of the risk of flooding. The Turkish earthquake of 17 August 1999 caused by the North-Anatolian fault close to the eastern end of the Marmara Sea is an example of this; a portion of several kilometers along the edge of the sea caved in by about 2 to 3 m, in the vicinity of the town of Gölcük, causing several districts to be submerged (see Figure 3.3). The flood was brutal, in the shape of a wave that was visible up to 100 m inside the zone that remained above sea level (according to [COL 99b]).
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Figure 3.3. Flooded region of Gölcük after the Kocaeli earthquake (Turkey) on 17 August 1999 (taken from [COL 99b])
3.1.2. Deformations linked to vibratory motions Strong vibratory motions are likely to induce irreversible deformations in the ground resulting in several mechanical processes. The most commonly observed of these processes is compression which affects the rather loose granular soil (sands with an average density or lower, insufficiently compact fills); when they are subjected to several cycles of powerful enough shocks to modify the arrangement of the grains, they tend to evolve to a more compact configuration and this results in a reduction in the thickness of the layer, which can reach several centimeters for a layer of about 10 m. If the ground is dry, compression generally produces an improvement in its mechanical properties. If, on the other hand, it is saturated with water, the increase in compactness results in a rise of the interstitial hydraulic pressure of the liquid; if the porosity of the medium remains low this excess pressure cannot be relieved quickly by intergranular flow, and causes a reduction of friction between grains, which can go on until there is total loss of the capacity of shear strength. This is the phenomenon of ground liquefaction, which is commonly observed in earthquakes of a certain size (magnitudes higher than 6 to get an idea) in zones where the soil has the characteristics described above. It is usual to classify liquefaction as induced phenomena, which are explained in section 3.3 hereafter; this is why deformations of superficial terrain resulting from this phenomenon, which are often the cause of serious damage, are referred to in section 3.3.1.
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Figure 3.4. Compression fill around the abutment of a bridge; Spitak (Armenia) earthquake of 7 December 1988
If dry soil earthworks do not constitute (as liquefaction does) a cause of degradation of the mechanical resistance, they can, however, have detrimental consequences, because of differential displacements that are likely to affect the foundations of the works or their external extensions. Figure 3.4 shows a fill that was subjected to a compression of several decimeters in the vicinity of the abutment of a bridge, at the time of the Armenian earthquake of 7 December 1988. Apart from compression, other irreversible deformations are frequently observed, that result in cracks ranging from a length of few meters to around 10 meters long or more (this should not be confused with traces of faults) or by Localized depressions. Figure 3.5 gives two examples of these.
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Figure 3.5. Examples of cracks and localized depressions: above, a crack in the road surface (Spitak earthquake of 7 December 1988); below, crack and depression in an electric substation (Chi-Chi earthquake, Taiwan, 21 September 1999)
It is easy to recognize the potentially dangerous character of these irreversible deformations because we can observe them after the earthquake has taken place and note the damage that they have caused. It is more difficult to recognize the case of reversible transitory deformations resulting from seismic wave propagation. If we look at some descriptions by eyewitnesses who talk of perfectly visible “earth waves” moving across the ground, the quantification of these transitory deformation
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fields, which is necessary in order to define prevention measures, is quite delicate because, amongst the parameters of vibratory motion, there is a greater uncertainty about displacements in recordings and calculation models than about velocities or accelerations (see Chapter 4). Post-seismic observations show that certain types of damage are clearly due to the effect of transitory differential displacements; the discussion of these types of damage and their interpretation is presented in section 11.2. We will restrict ourselves here to the problems of interpretation that can arise while considering the example of Figure 3.6.
Figure 3.6. Localized deformation of a railway track after the Tangshan earthquake (China) of 28 July 1976
The spectacular deformations of railroads, such as those in this figure, can be attributed to two principal causes: – the movement of a fault having a strike-slip component; this is clearly the case in the photo in the middle of Figure 1.18, where we see that the non-deformed parts of the rails have shifted by approximately a meter in a lateral direction, which corresponds to a left-lateral strike-slip motion of the Motagua fault (Guatemala). This explanation does not appear convincing for Figure 3.6, in which the undeformed portions seem to be in line with each other; – buckling resulting from a compression in the direction of the rails, induced by dephasings of the longitudinal component (parallel to the rails) of the vibratory motion. In the first case it consists of an irreversible deformation of the soil (fault movement) and in the second, of a reversible deformation (dephasing in the longitudinal direction); both are able to produce the same type of effect (Localized irreversible deformation of the rails).
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As for Figure 3.6, it is difficult, with only a single view of the photograph, to choose between the two causes; the explanation according to the fault, even if it seems less plausible for a strike-slip motion perpendicular to the tracks, is compatible with a trace forming a very sharp angle with the direction of the rails (obtained by joining the deformed zones of the two rail tracks); it can thus be the correct explanation if observations of the ground confirm the existence of a fault having this orientation; as for the explanation by the buckling theory, a priori it seems quite acceptable but it remains hypothetical since the transitory differential movements which would have been the cause did not leave a visible and measurable “signature”. This example illustrates the difficulties of an interpretation a posteriori of the seismic damage, which will be taken up in the introduction to Part 5. 3.2. Seismic waves 3.2.1. Different types of seismic waves As indicated in section 1.1.1.2, the concept of seismic waves emerged gradually following observations of the ground (J. Michell 1761, R. Mallet 1857) and with the development of the theory of elasticity in the field of dynamics (D. Poisson 1828, Lord Rayleigh 1885, A. Love 1911). The development of instruments that were able to record these waves, at the end of the 19th century, helped crystallize the concept into reality and to establish seismology as a subject of study. From the mathematical point of view, a wave is a solution of the equations of dynamic elasticity whose dependence with respect to time t only brings one function into play f (t r x/c), x being a co-ordinate of space (distance to a plane for a plane wave, distance to a point for a spherical wave) and c a propagation velocity; for an observer moving at the velocity B c in direction x, the argument of function f remains constant, the result of which is that the amplitude of the motion would seem either to be constant or undergoing a purely geometrical variation, i.e. the relation of the amplitudes in two different points is the same whatever the value of f. In the theory of elasticity (see [TIM 61]), it is seen that in a homogenous and isotropic material, there are two types of volume waves, i.e. they can propagate across the mass (see Figure 3.7): – longitudinal waves or P waves, which produce compressions and dilatations alternating in the direction of the propagation, with a change in volume; – transverse waves or S waves, which produce a distortion perpendicular to the propagation direction, without a change in volume.
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Figure 3.7. Waves of volume in a homogenous and isotropic material: a) longitudinal wave, b) transverse wave
Propagation velocities of these waves play a very important part in seismology and in soil dynamics. They depend on the elastic constants of the medium (Lamé coefficients O and P, Young’s modulus E and the Poisson coefficient Q) using the following formulae:
Vp =
Vs =
O 2P
=
U
P U
=
( I Q )E ( I Q )(1 2Q ) U
E 2( I Q ) U
[3.1]
[3.2]
U being the mass, vp and vs being the respective propagation velocities of P and S waves; the ratio: Vp Vs
=
2( I Q ) 1 2Q
[3.3]
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is an increasing function of Ȟ which ranges from 1.41 for Ȟ = 0 to 1.73 for Ȟ = 0.25 and increases indefinitely when Ȟ tends towards its final value 0.5, which corresponds to an incompressible medium; the velocities of these longitudinal waves are thus greater than those of transverse waves.
Depth (km)
Vp (km/s)
Vs (km/s)
U (kg/m3)
0.0 – 3.0
1.45
0.00
1,020
3.0 – 15.0
5.8
3.20
2,600
15.0 – 24.4
6.80
3.90
2,900
24.4 – 71.0
8.10
4.48
3,380
71.0 – 80.0
8.08
4.47
3,380
80.0 – 171.0
8.05
4.45
3,360
171.0 – 220.0
8.00
4.43
3,360
220.0– 271.0
8.61
4.66
3,450
271.0 – 371.0
8.76
4.71
3,500
371.0 – 400.0
8.88
4.76
3,530
400.0 – 471.0
9.32
5.03
3,770
471.0 – 571.0
9.75
5.29
3,870
571.0 – 600.0
10.08
5.47
3,960
600.0 – 670.0
10.22
5.54
3,990
670.0 – 771.0
10.91
6.10
4,410
Table 3.1. Values for vp, vs and U according to the depth for the PREM (Preliminary Reference Earth Model); the values of the table are the average values for each layer [DZI 81]
In seismology, the values of vp and vs are of the order of several km/s, because we are interested only in the compact materials of the Earth’s crust. The PREM model (PREM, [DZI 81]), largely used by seismologists, thus gives for vp, vs and U in the first 700 km of depth which corresponds to the possible positions of the focus (see Table 3.1). In Table 3.1 we can see that the first layer of a depth of 0 to 3 km corresponds to the water of the oceans (as vs = 0), which covers most of the Earth’s surface; the two following layers (with depths from 3 to 24.4 km) represent the earthquake prone part of the continental crust; here the values of vs and U are close to the values adopted in section 2.1.1 (see [2.4]); situated at a depth of 24.4 km is the Mohorovicic
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117
discontinuity (generally shortened to “Moho” in everyday language) which constitutes the limit between the crust and the terrestrial covering; of course this value of 24.4 km from the PREM model is only an average, the depths of the “Moho” can vary according to the region from 20 to 75 km; at greater depths vp, vs and U increase slowly and steadily. In soil dynamics, we are primarily interested in the velocity vs of transverse waves, which is very variable according to the nature of the superficial terrain; for rocks appearing on the surface, which are generally eroded and definitely less compact than those of the deep layers considered in the PREM model, vs hardly exceeds, except for the odd case, 1,000 to 1,500 m/s; the sedimentary soils of good quality (compact shale, sands and gravel) show values of a few hundred m/s; for poor soils (loose sands, soft clays) vs falls to the range 150–250 m/s; values lower than 100 m/s can be measured in muds. Values of vs will be referred to again in section 16.2, where they will be considered in calculations of interaction of soil-structure. In addition to the volume waves, the equations of dynamic elasticity admit other solutions of an undulatory character, they consist of surface waves, which can appear in the vicinity of surfaces of discontinuity (free surface or interfaces between layers of different materials); in earthquake engineering, the two types of surface waves which are most noteworthy are the Love wave and the Rayleigh wave (see Figure 3.8). The Love wave (part a of Figure 3.8) is a transverse wave (the movement of the particles is perpendicular to the direction of propagation) like the S volume wave but with the difference that the amplitude of this wave is perceptible only close to the surface and decreases quickly with depth. It can exist only in non-homogenous media and is dispersive, i.e. its propagation velocity depends on its frequency.
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Figure 3.8. Surface waves – a) Love wave and b) Rayleigh wave
The Rayleigh wave (part b of the figure) is a type of swell, in which the particles have two components of movement, one vertical and the other horizontal (parallel to the direction of propagation), whose combination produces an elliptical trajectory; as for the Love wave, the amplitude of the movement decreases with depth and the speed of this decrease is greater as the wavelength is small. The Rayleigh wave exists in a homogenous medium, where its propagation velocity is a little lower than the velocity of the S waves (it varies from 0.874 vs for Q = 0 to 0.955 vs for Q = 0.5); it is dispersive in non-homogenous media. In calculations, the seismic waves of volume or of surface are represented by sinusoidal expressions (that, by superposition, reproduce a wave of any form); when there is only one component, of movement (plane P and S waves, as drawn in Figure 3.7, or the Love wave), the latter is shown by: U = D sin Z (t –
x ) c
[3.4]
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119
D is the amplitude of displacement (maximum value of the absolute value of u), Z the angular velocity, and C the propagation velocity in direction x. From these quantities, we define: – the period T =
2S
– the frequency g =
– the wave length
[3.5]
Z
/
1 7
=
Z
= cT =
– the wave number k =
Z C
[3.6]
2S
=
c
[3.7]
g 2S
[3.8]
/
– the amplitude of velocity V = max
wu wt
– the amplitude of acceleration A = Max
= ZD w ²u wt ²
– the amplitude of the deformation H = Max
[3.9]
= Z²D wu wx
=
[3.10]
ZD c
=
V c
[3.11]
In earthquake engineering, the range of significant periods in the broad sense extends (i.e. if all the applications are considered) from 0.02 s to 10 s, but can be restricted within the range 0.1 s to 2 s, in the vast majority of cases; the corresponding range of frequencies goes from 0.1 Hz to 50 Hz (in the broader sense) to 0.5 Hz to 10 Hz (in the restricted range). This results in the shortest wavelengths of about a few hundred meters with the “seismological” values of propagation velocities, whereas in surface soils of poor quality they can go down to some tens of meters and thus be comparable to the sizes of buildings or other constructions. Propagation velocities hitherto defined are characteristic of the medium in which the waves propagate themselves. When we seek to measure them according to the temporal shift between the signals recorded in different points it is necessary to identify the medium whose properties control the propagation; if the signals correspond to surface waves, apparent propagation velocities which are being measured are to be connected to the characteristics of surface soil, at least as long as the wavelengths are not too large (since, as has been indicated previously, the
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penetration depth of the surface waves is of the same order as the wavelength); if they are signals associated with P or S waves, their propagation generally corresponds to the properties of the deep layers and the surface soils only “follow the movement” without influencing its kinematics particularly (they can, on the other hand, increase the amplitude appreciably). Thus, the frequently observed fact can be explained, that the apparent propagation velocities of P and S waves are of the order of km/s, even in zones where vs has much lower values. This observation is important for the study of the soil-structure interaction (Chapter 16). The vibrations emitted by the rupture on the fault plane are an often very complex combination of P waves and S waves; when these waves reach the surface, part of their energy can be converted into surface waves, such as Love or Rayleigh waves, following mechanisms in which the nature of the surface soils plays a large part; the large sedimentary basins, for example, are often traversed by trains of surface waves, which occur on arrival of the incident waves of volume at the limits of these basins. The more we move away from the epicenter the more the preponderance of the surface waves is accentuated, which, as we will see is explained in section 3.2.3, by the lower geometrical attenuation (decrease in I/ R , R being the epicentral distance, instead of I/R for the volume waves); this is why the measurement of magnitude MS is most common when remote recordings of surface earthquakes are available; on the other hand deep earthquakes produce few surface waves and for this reason, are often expressed in magnitudes using volume waves (mb or mB; see section 2.3.2). The difference of propagation velocities for the P and S waves has two important practical consequences: 1) the amplitude of the S waves is definitely higher than that of the P waves, it is seen that, for a seismic source satisfying the assumptions of the theory of elastic rebound, a component u of displacement in a distant field (i.e. at a large distance r considering the dimensions of the source) is given by an expression of the form:
u=
G (T ,I ) I 4SU c
3
r
M
0
§ r· ¨t ¸ © c¹
[3.12]
where U is the density, c the propagation velocity, G (T I) a function of angles T and I which define, with the distance r, the spherical co-ordinates in relation to the the derivative compared to the time of the center of the seismic source and M 0 seismic moment according to the time obtained by replacing in [2.5] 'u by the displacement transitory D (t) on the fault plane. Function G (T I) has different expressions according to the type of wave considered, P or S, but as its values are close (and close to the unit), the relationship
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121
of the amplitudes between the P and S waves is thus approximately equal, at the same point, with the cube of the reverse of the ratio between the velocities vp and vs, and with vp/vs = 1.73 (X = 0.25) we find that the amplitude of the S waves is 5.20 times higher than those of the P waves; 2) the measurement of the time interval '7 separating the arrivals of the P waves and the S waves makes it possible to calculate the distance R between the seismograph and the source, if we know the values of vp and vs; to cover the distance R, the waves have taken the following time: Tp =
R Vp
(P waves); Ts =
R (S waves) Vs
[3.13]
which leads to the relationship between R and 'T = Ts – Tp
R=
VsVp Vp Vs
'7
[3.14]
That is to say, for example, R (km) = 7.14 '7 (S) with the values vp and vs for the 3 – 15 km layer of Table 3.1. It is thus possible, starting from only one recording, to have an estimate of the focal distance R, the measurement of '7 being easy, in general, because of the difference in level between the signals of the two types of waves, which makes it possible to identify the arrival of the S waves well and because of the duration to which it corresponds (at least a few seconds in a close zone, a few tens or hundreds of seconds for a longer distance). This allows the quick localization of the seismic sources, a minimum of three recordings being required for this operation. 3.2.2. Ideas on the theory of rays
The seismic wave notion is based on the idea of a disturbance being propagated in a given direction (plane wave) or emanating from a point (spherical wave) and corresponding to the existence of wavefronts, i.e. surfaces that are orthogonal to the propagation and on which the amplitude of the disturbance is the same on all the points. It seems natural to give an analogy of geometrical optics which will lead us to an analysis of seismic rays that carry the disturbances in the same way as optical beams. So, through fairly simple calculations, we can obtain a certain number of interesting results, particularly on attenuation laws and site effects (see Chapter 5). However, as for optical geometry, application of this theory has its limits when the wavelengths become comparable to the characteristic lengths of the problem under
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study. It is a valid approximate evaluation for relatively high frequencies, but this does not enable us to deal with most of the diffraction problems, the influence of high gradient heterogenities or the related long term effects. The theory of seismic rays presents two aspects. The first, which mainly concerns seismologists, is limited to the calculation of travel time between the emitting source and the receiver; only the propagation velocities play a part (or the opposite of these speeds for which we use the correct term “slowness”) and this is sufficient to resolve almost all the focus localization problems. The second aspect also deals with the characteristics of the disturbances (amplitude and phase for the harmonic depictions); other than the propagation velocities, it brings in density and internal damping parameters. It is this second aspect that is important for significant applications in earthquake engineering. To illustrate the difference between the two aspects, we consider the refractionreflection phenomenon that forms the basis of the theory; a transverse incident wave ui meets a horizontal surface separating two homogenous areas each one having for its density and wave velocity U1, c1 and U2, c2 respectively (see Figure 3.9); the movements of the particles are perpendicular to the plane of the figure; the incident angle (angle between the incident ray and the normal ray at the point of interface) is T1. If we suppose that as in Figure 3.9 we have c1>c2 the refracted ray ut makes an angle T2 lower than T1 with the normal ray, as given in the Snell-Descartes law of refraction: sin T 2 sin T1 = c1 c2
[3.15]
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123
Figure 3.9. Refraction and reflection of transverse plane wave on the interface separating two homogenous areas; the figure describes the case where the propagation speed is higher in the first area than in the second
As in optics, the quotient c1/c2 > 1 represents the refractive index of the second area in relation to the first. Remember that law [3.15] is a result of Fermat’s principle (minimum of the travel time between two points situated on either side of the interface). It suffices for the first aspect of the theory of rays described earlier, i.e. the tracing of rays that help calculate the travel time. Consideration of amplitudes (second aspect of the theory) brings in the necessity of introducing the reflected ray ur, that makes an angle equal to T1 with the normal. In fact, at the interface one must ensure continuity not only of the displacement but also of the shear stress that results from the (perpendicular) displacement of the points on the ground. If we assume that sinusoidal waves [3.4] are expressions of this displacement for the incident wave, the refracted wave and the reflected wave, considering the direction of their propagation, are as follows:
Ui = Di sin Z (t –
Ut = Dt sin Z (t
x sin T1 z cos T1
)
C1 x sin T 2 z cos T 2 C2
)
[3.16]
[3.17]
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Ur = Dr sin Z (t –
x sin T1 z cos T1 C1
)
[3.18]
The continuity of the displacement and the shear stress at point 0 (x = 0, z = 0) is expressed by the system:
ui + ur = ut
[3.19]
wu § wui wur · t ¸ = P2 w z w z w z © ¹
P1 ¨
2
which is written considering [3.15], [3.16] and [3.17] and equations P1 = U1c1 , 2 P2 = U 2 c2 (see [2.3] or [2.53]):
Dt – Dr = Di
[3.20]
Dt cos T2 + O Dr cos T1 = O Di cos T1 where we introduce the impedance ratio O defined by: O=
U1c1
[3.21]
U 2 c2
The solution of system [3.20] is:
Dt
2O cos T1
Di
O cos T1 cos T 2
Dr
O cos T1 cos T 2
Di
O cos T1 cos T 2
If we eliminate T2 by law of refraction [3.15] we obtain:
[3.22]
[3.23]
Manifestations of the Seismic Phenomena on the Surface
ª 1 º Dt = 2 O/ « O 1 tan ²T1 (1 2 ) » Di n ¼ ¬
125
[3.24]
ª º ª º 1· 1· Dr § § = «O 1 ¨1 ¸ tan 2 T1 » / «O 1 ¨ 1 ¸ tan 2 T1 » [3.25] Di © n² ¹ © n² ¹ «¬ »¼ «¬ »¼
where we introduced the index of refraction n: c n= 1 c 2
[3.26]
We see that the tracing of the rays (first aspect of the theory) only requires this index n, moreover the determination of amplitudes (second aspect) brings in the impedance ratio O. The variations of T2 and of Dt/Di according to T1 are represented in Figure 3.10 in the three following cases: – n = 1.15, O = 1.34 corresponding to the Mohorovicic discontinuity at a depth of 24.4 km in the PREM model (see Table 3.1); – n = 2, O = 2.4 corresponding to a moderate contrast between bedrock and surface layer (U1 = 2.4 7/m3, c1 = 1.2 km/s, U2 = 2 T/m3, c2 = 0.6 km/s); – n = 4, O = 5 corresponding to a high contrast between bedrock and surface layer (U1 = 2.25 T/m3, c1 = 1 km/s, U2 = 1.8 T/m3, c2 = 0.25 km/s).
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Figure 3.10. Variations of the angle of refraction T2 and of the ratio Dt/Di (amplitude of the refracted wave divided by the amplitude of the incident wave) in relation to the angle of incidence T1 in the three different cases of contrast: – n = 1.15, O = 1.34 low contrast corresponding to the Mohorovicic discontinuity in the PREM model; – n = 2, O = 2.4 moderate contrast between bedrock and a surface layermade up of good terrain; – n = 4, O = 5 high contrast between bedrock and a surface layer made up of mediocre terrain.
In Figure 3.10, we see the following: – the angle of refraction is low when the contrast is high (high values of index n) and cannot exceed the value T2, A is given by: T2, A = Arcsin
I n
[3.27]
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127
which is attained for T1 = 90° (grazing incidence); the path of the seismic rays is reversible, as is that of optical beams, T2 A represents the maximum angle of refraction; if in fact we invert the direction of the arrows on the rays ui and ut of Figure 3.10, ray ut, which has become incident, can be refracted in area 1 only if we have T2 d T2 A ; for the higher values of T2 the incident ray will be entirely reflexive in area 2; – the amplitude of the refracted wave increases with the contrast and only slightly varies in relation to the angle of incidence as long as the latter is not too high (lower than 50° to give an idea); it quickly shifts towards O when T1 approaches 90°; – the amplitude of the reflected wave is not given in the figure but we can easily deduce it from the refracted wave as we have from the first of equations [3.19]:
Dr
Dt
Di
Di
–1
[3.28]
In this formula we see that the amplitude of the reflected wave is cancelled when Dt = Di, which happens when the angle of incidence has the value T1,0 given by:
T1,0 = Arctan n
O ² 1 n ² 1
[3.29]
i.e. about 61°, 68° and 79° for the three cases of contrast (low, moderate and high) of Figure 3.10. The first of these observations (low value of the angle of refraction when the contrast is high) has important practical consequences; we saw in section 3.2.1 that the velocities vs of the surface terrain were much lower than those of the layers containing the seismic sources; the result is that the seismic rays “stand up” whilst approaching the surface, which they reach in an almost vertical incidence; the wave fronts are thus almost parallel to the surface and there is almost no dephasing of motion at different points of the latter at least if the terrain has a relatively homogenous configuration in the horizontal plane. It is for this reason that the current calculation practice accepts that the motion of the soil is the same at all the points of the foundation of the actual buildings and constructions. This hypothesis has its limitations for sites whose geotechnical structure clearly deviates from a horizontal stratigraphy; in addition it is limited to the case of volume waves and does not evidently apply to surface waves that by definition are propagated horizontally. For these, we must consider the ratio between the wave
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lengths and the foundations that will justify a hypothesis of homogenity of movement in the current cases. Another consequence of the nearly vertical propagation of the seismic rays near the surface is that the beginning of the signal recorded at a given site essentially corresponds to a vertical movement; in fact, this upturning takes place not only for S waves (which have been discussed earlier, (see [3.16] to [3.29]) but also for P waves (although their case is a bit more complicated as the arrival of a P incident ray on an interface produces a total of 4 rays, 2 P and 2 S through refraction and reflection). The beginning of the signal is formed by the P waves and as for these the movement of the particles is parallel to the propagation direction, their almost vertical emergence is mainly evident on the vertical component. This point has already been brought up in the study of the focal mechanisms (see section 1.2.3). Law of refraction [3.15] determines the tracing of the seismic rays when we know the spatial distribution of the propagation velocities; if it corresponds to a set of homogenous layers separated by plane interfaces, the seismic rays are made up of broken lines as shown in Figure 3.11.
Figure 3.11. Tracing of seismic rays in a sedimentary basin with two horizontal layers (velocity c2 and c3) cutting bedrock (velocity c1)
Figure 3.11 shows the path of a seismic ray penetrating from the bedrock in a sedimentary basin consisting of two horizontal layers; after two refractions it reaches the surface at S, where it deflects and turns towards the deeper layers; to make Figure 3.11 clearer, the rays reflected by the interfaces have not been represented.
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129
In environments where mechanical properties vary in a continuous manner the law of refraction provides a differential equation whose integration determines seismic rays. If we consider, for example, an environment where the velocity of propagation is a function c(z) of the depth, to calculate the ray emitted with an angle T0 (in relation to the vertical) through a point where the velocity has a value c0, we have the following system: sin T
sin T 0
c( z )
c0
dx dz
= tg T
[3.30]
[3.31]
where x is the horizontal coordinate in the vertical plane which contains the ray; this system is equivalent to the differential equation: dx dz
=
c ( z ) sin T 0 2 c0 c ²( z ) sin ²T 0
[3.32]
Its integration is easy if c(z) is a linear function; Figure 3.12 shows the result for a linear increase in the velocity with the depth; the rays are arcs of circles centered on the straight line z = zo, (located above the surface) which would correspond to a value of zero for the velocity; the wavefronts are also circles (beam orthogonal to the beam of rays). In Figure 3.12 we see that the rays which reach points on the surface that are distant from epicenter E, such as point P on the left, are emitted downwards from focus F; this specific aspect has been indicated in the notes on Figure 1.22 (see section 1.2.3).
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Figure 3.12. Seismic rays emitted by focus F and wavefronts when the velocity increases linearly to the depth
In homogenous environments, the change in direction of the rays can occur only when there are reflections on free surfaces; their tracing which is quite simple enables us to understand the effect of the topographic site i.e. amplifications of the motion often noted near relief elements such as hills or plateau edges. This effect will be discussed in section 4.3.3 but, as an introduction to this section, a simple example is presented here in Figure 3.13.
Figure 3.13. A plane wave with vertical propagation reflected by a triangular hill with an angle on top of 5 S /8
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131
Figure 3.13 shows the tracing of the seismic rays in a triangular hill with an angle on top of 5 S /8 when the incident field corresponds to an ascending vertical propagation. We see that the play of reflections on the slopes of the hill is such that four distinct rays pass through any point P: – direct ray D which reaches P before undergoing reflection; – rays R1 and R’1 which reach P after one reflection; – ray R2 which reaches P after two reflections. The result is that according to the position of P, these four rays are going to be added together (if they are in phase) or compensated (if there are in opposition of phase), so either the amplitude of the motion in P will be increased, or it will be reduced in relation to that of the incident ray; the phase differences during their passage in P is the result of the differences in length of the path from the initial state (corresponding to any instant before the arrival of the direct ray D at point P). The effect of the multiple reflections is even clearer in the case of a sedimentary layer covering a bedrock; in Figure 3.14 we see that the number of rays passing through point P of this layer is thus infinite; the figure shows four of these rays: – R1 which reaches P after one refraction at the interface; – R2 which reaches P after one refraction and one reflection on the free surface; – R3 which reaches P after one refraction and three reflections (two on the free surface, one on the interface); – R4 which reaches P after one refraction and five reflections (three on the free surface, two on the interface).
Figure 3.14. Multiple reflections on a sedimentary layer
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As for the hill of Figure 3.13, these rays will join together in P to produce an amplication or an attenuation of the motion according to the distribution of the dephasing. The study of this case is given in section 5.3 for an incident field with vertical propagation. Other than the considerations developed earlier on the tracing of the rays and the transmission coefficients at the interfaces, the theory of rays uses explanations based on the concepts of flow of energy and force tubes. The energy transported by a seismic wave is the sum of kinetic energy associated with the motion of particles and of elastic deformation energy. For a sinusoidal wave of shape [3.4], the densities of these two forms of energy whose square of sine or of cosine come into play and whose argument is Z (t-x/c); the average value whether taken on time (on a period) or in space (on a wavelength) is the same in the two cases and we have: Ec = Ed =
1 4
UD2Z2
[3.33]
Ec and Ed respectively being the averages of kinetic energy and deformation
energy densities; the result is for the average Et of the density of the total energy: Et = Ec + Ed = I/2 U D2 Z2 = ½ U V2
V being the amplitude (maximum value) of the particle velocity (see [3.9]).
Figure 3.15. Tube of force formed by the seismic rays pressing on a closed outline c1 (case of P waves)
[3.34]
Manifestations of the Seismic Phenomena on the Surface
133
The seismic rays that press on a closed outline C1 form a tube of force that becomes a conduit through which energy is transported (Figure 3.15); if there are no losses during this transportation, the energy flow is constant in the two sections of the tube. We suppose that outline CI is sufficiently small for the properties of the area (density U1, wave propagation velocity c1) and the particle velocity V1 can be considered as constant in all points of its surface S1 and that it is situated on a plane perpendicular to the seismic rays; the energy flow )1 that penetrates the force tube during a time 't is the energy contained in the cylinder having S1 as base surface and cI 't as height, i.e. according to [3.34]:
M 1 = ½ U1 V²1S1 c1 't
[3.35]
The energy flow )2 coming out of the tube during 't through a closed outline C2 satisfying the same hypothesis as C1 is the same:
M 2 = ½ U2 V²2S2 c2 't
[3.36]
For the ratio of particle velocities, the conservation of flow ()2 = )1) leads to:
V2 U1c1S1 = U 2 c2 S 2 V1
[3.37]
For a tube of force of constant section, such as that obtained for normal incidence on a plane interface (which is the case in Figure 3.9 with T1 = T2 = 0), formula [3.37] gives:
V2 Uc = O (with O = 1 1 as earlier) U 2 c2 V1
[3.38]
Now, for sinusoidal waves, the ratio of velocities must be equal to that of the displacements, i.e. to the ratio Dt/Di given by equation [3.22], which has a value 2O/(O+1) for T1 = T2 = 0; thus, we should have:
V2 2O = V1 O 1
[3.39]
The contradiction between [3.38] and [3.39] comes from the fact we did not include the reflected ray in the approach to energy; the error is not serious if O is
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only slightly higher than one, since the amplitude of this ray is low (see [3.22]); furthermore, we verify that the functions O and 2O/(O + 1) are equal, as well as their first derivative for O = 1. The error is acceptable even for values of O significantly higher than one (6% for O = 2, 15% for O = 3, 25% for O = 4) and it is fairly safe as [3.38] systematically provides values higher than [3.39]; this is why formula [3.38] continues to be used in the site effect studies (see section 5.3) as a first approximate value. The problem of amplification by a soil layer covering a bedrock in reality depends on the frequency of the incident wave since this determines the dephasing between the rays reflected several times within the layer as seen in the discussion of Figure 3.14. If we consider the reflected ray, the energy theory gives the correct value [3.22] for the ratio of velocities. In Figure 3.16 (part a), we see the tube of incident force at the moment where the leading section reaches the interface; the amplitude of the particle velocity (which is parallel to the interface) is vi (= V1) in the notation of equations [3.35] to [3.39], part b of Figure 3.16 shows the situation 't later; the tube of force which has a height of c1 't is divided into two parts, one of height c2 't which continues upwards in the second environment and the other of height c1't which corresponds to the reflection on the interace; the amplitudes of velocity y are represented respectively on it as vt (= V2) and vr.
Figure 3.16. Tubes of incident force, refracted and reflected for a plane interface in the case of normal incidence
The conservation of energy is written as:
U1c1Xi2
U 2 c2Xt2 U1c1Xr2
[3.40]
Manifestations of the Seismic Phenomena on the Surface
135
That is:
O Xi2 Xr2 Xt2
[3.41]
The equation that comes from the continuity of displacements (first of relations [3.19]) must be added to this equation, which leads to that of the velocities:
vi + vr = vt
[3.42]
[3.42] simplifies [3.41] into:
O (vi – vr) = vt
[3.43]
vt and vr are thus determined by the system:
vt + O vr = O vi
[3.44]
vt – vr = vi whose solution is:
vt =
vr =
2O
O 1
O 1 O 1
vi
[3.45]
vi
Thus, for vt/vi and vr/vi we find the same equations as for Dt/Di and Dr/Di ([3.21] and [3.22]) in the case of normal incidence (T1 = T2 = 0), i.e. formula [3.39] for V2/V1. 3.2.3. Attenuation of seismic waves
Attenuation of seismic waves is the term used to express the fact that their amplitude decreases in general when the source is further away. We differentiate geometric attenuation which is the result of the divergence of rays (thus of the increase in the surface of the wavefronts), one example of which is provided by
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Figure 3.12, of the inelastic attenuation that corresponds to different internal dissipation mechanisms of energy during its propagation. 3.2.3.1. Geometric attenuation In a homogenous environment, geometric attenuation is very clearly evident for spherical or cylindrical types of propagation diagrams (pinpointed source); equation [3.36] in fact shows that the amplitude of motion varies in inverse proportion to the square root of the surface of the wavefronts. For spherical wavefronts of radius r, whose surface has a value of 4 S r², the amplitudes decrease in 1/r. This property has already been used in section 2.3.1 (see [2.49]) and is verified in equation [3.11] in section 3.2.1. It is usually used to study vibratory motions in zones close to the source. The case of cylindrical wavefronts corresponds to that of surface waves, which remain confined to the area neighboring the surface; the surface of the wavefront of radius r (distance from the epicenter) is equal to 2 S r p, p being the height of the cylinder (depth of penetration of the wave, which depends on its wavelength, but is independent of r). As indicated in section 3.2.1, the result is that the amplitudes of surface waves decrease in 1/ r , thus attenuating slower than volume waves, which explains their importance at a large distance from the source. These attenuations in 1/r or 1/ r , have been obtained by assuming that the distances from the source are sufficiently high in comparison with the dimensions of the latter in order to accept a spherical or cylindrical propagation diagram for the wavefronts. As we have seen in section 2.3.1, this hypothesis is not acceptable for the epicentral zones of superficial earthquakes of average to high magnitudes; the question then arises of the definition of a representative distance to calculate an attenuation factor and different solutions have been recommended (see section 4.2). Geometric attenuation of volume waves can be affected by the reflection of the rays reflection emitted by the focus on the Mohorovicic discontinuity.
Figure 3.17. Reflection of the waves on the Mohorovicic discontinuity when the incidence angle is higher than the outermost refraction angle
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Figure 3.17 shows a simplified diagram where the “Moho” is a horizontal plane with depth H and where the terrain located above it forms a homogenous layer in which the rays are straight lines. Focus F, whose depth h is lower than H, emits rays in all directions, those which go downwards meet the discontinuity which leads to a refraction (if the incidence angle is lower than the outermost refraction angle given by [3.26]) and a reflection upwards. Figure 3.17 represents a ray FM whose incidence T is higher than the refraction limit. It is entirely reflected and meets the surface at P where it joins the direct ray FP. The reflected ray seems to originate in a fictitious source F’, image of F in relation to the discontinuity, located at depth 2H – h. Considering attenuation in inverse proportion to the distance the ratio between the amplitude vr of the reflected ray and vd of the direct ray is given by: Vr Vd
=
FP F 'P
=
'² h² ' ² (2 H h)²
[3.46]
where ' is the epicentral distance EP. The minimum value ', i.e. ' min, in order to have total reflection can be easily calculated; hence: ' min =
2H h
[3.47]
n² 1
n being the refractive index For this value of ' formula [3.46] leads to:
§ vr · ¨ ¸ © vd ¹ '
= 1 ' min
4 H ( H h) § I · ¨I ¸ 2 ²¹ n (2 H h) ©
[3.48]
With the plausible values H = 30 km, h = 15 km, n = 1.15 (value corresponding to the Mohorovicic discontinuity according to the PREM model; see discussion on Figure 3.10) we find:
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' min = 79 km; (
Vr Vd
) ' = ' min = 0.885
[3.49]
We see that the amplitude of the reflected wave is close to that of the direct wave for ' > ' min, i.e. when there is partial refraction through the “Moho”, vr/vd decreases very fast as the refracted ray becomes important to low incidences (see discussion on Figure 3.10). This reflection phenomenon on the Moho appears at epicentral distances of about 70~80 km, from where they can contribute significantly to vibratory motion; in order that, at these distances, amplitude of the motion is likely to provoke damage, the magnitude must be high (at least equal or higher than 7, according to the rule of 5RO given in section 2.3.1). The influence of the Moho has been mentioned especially for the Californian earthquake at Loma Preita (17 October 1989) described earlier regarding aftershocks (see Figure 2.7), but its quantitative appreciation was, in this case, hidden by the site effects that were responsible for important damages in the San Francisco region about 100 km from the epicenter. To summarize, the geometric attenuation depends on several factors: – the size of the seismic source (dimensions of the fault plane or radius RO of the equivalent sphere; see section 2.1.1); it influences the distance from which we can effectively consider that geometric attenuation exists; we must admit that this distance is of the order of RO, i.e. equal to or greater than about 10 km for a magnitude equal or higher than 7; – depth h of the focus which influences the speed of attenuation from the epicenter and the distance beyond which the surface waves become important; a rapid decrease is basically a sign of a very superficial earthquake (in the sense of section 2.3.1, which means that the depth should be close to RO) and of rather low magnitude (lower than or equal to 6 to give an idea). We can assume that surface waves (with their attenuation in 1/ r ) are prominent in the signals from an epicentral distance of the order of 5 h; for earthquakes of average or low magnitude, this distance is generally higher than the potential damage radius (which has roughly been estimated at 5RO in section 2.3.1); the change in the attenuation type (from 1/r in 1/ r ) thus has no practical incidence for earthquake engineering, except for earthquakes of a magnitude higher than a threshold level of 6.5; – depth H of the Mohorovicic discontinuity, which influences the distance from where we can observe, in the signal an increase in the amplitude of volumetric waves this distance has a value of about 1.8 (2H – h), or of 60 to 100 km in the majority of cases. Likewise for the influence of the surface waves, H can only influence earthquakes of high magnitude; the two distances 5h (important for
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surface waves) and 1.8 (2H – h) (appearance of waves reflected totally on the Moho) are in practice not very different; this makes it difficult to attribute the change in type of attenuation to one particular reason than to another. These considerations can be clearly modified by the existence of inelastic attenuation that will now be presented. 3.2.3.2. Inelastic attenuation No material is perfectly elastic; even if the deformation has a reversible character (return to the non-deformed state after the loading stops) it is always accompanied by effects that tend to slow it down and that dissipate energy. Traditionally we characterize these effects with an energy loss coefficient during a deformation cycle; in seismology this coefficient is the quality factor Q defined by: I Q
=
I
'E
2S
E
[3.50]
E being the maximum value of deformation energy during the cycle, when a loss 'E of this energy occurs; this definition differs from those used in most of the disciplines of mechanics where the inelastic effect coefficient is taken in direct proportion to 'E/E rather than inverse, the factor Q of the seismologists is therefore as high as the damping is low.
To calculate the influence of Q on the attenuation of seismic waves, we consider a tube of force of section S(x), x being the coordinate (rectilinear or curvilinear) that defines the propagation of rays; the energy balance in the volume included between two neighboring sections S(x) and S (x + dx) use the following terms: – flow of energy entering through section x during unit of time [3.35]:
Ix = ½ (U V² S c)x
[3.51]
– flow of energy exiting through section x + dx during unit of time:
Ix + dx = ½ (U V² S c)x + dx
[3.52]
– volume loss of energy during the unit of time equal to the product of the energy lost during a cycle, i.e. according to [3.50]: 2S
1
Q
2
U V² S dx
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by the number of cycles (frequency)
Z 2S
Thus, we have the following equation for the energy balance: 1 1 SdxZ 2 U V² (U V² Sc)x = – ( UV Sc ) x dx – Q 2 2 2
1
[3.53]
i.e. for a homogenous environment (U, c and Q constant): d (S V²) = – Z dx SV ² Qc I
[3.54]
where by integrating: V V0
S0
=
S
e
Zx 2 Qc
[3.55]
V0 being the maximum velocity in section So; the factor So / S represents the geometric attenuation and the exponential inelastic attenuation term. This term can be expressed in a more common way for engineers: [=
1 2Q
[3.56]
[ is the reduced damping, usually expressed as a percentage of the critical damping which will be mentioned again in section 9.1; with this notation the inelastic attenuation factor Fi is expressed by the formula: Fi
e
[ Z x c
[3.57]
In seismology the actual values of the quality factor Q are in hundreds; the PREM model which has already been mentioned (see section 3.2.1) gives the constant value Q = 600 between 3 and 80 km of depth for inelastic attenuation of shear waves; the corresponding values of reduced damping [ are thus of the order of one per thousand, i.e. much weaker than those accepted in earthquake engineering for foundation soil and structures of materials which are typically of the order of several per hundred.
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The result of this difference in size is that the inelastic attenuation of the seismic waves is often higher during the path of some tens of meters covered whilst crossing superficial ground than during several tens of kilometers covered since their emission at the source. For example, with the values of the 3 – 15 km layer from the PREM model (c = 3,200 m/s, Q = 600 i.e. [ = 8.33 x 10-4 for shear waves) a path of x = 20 km of an S wave of 3 Hz (Z = 18.86 rd/s) frequency, which is the right size for the dominant frequency of an earthquake of average magnitude (see section 4.1.3), according to formula (3.57), corresponds to a factor Fi of 0.906; the path by this same wave through a layer of alluvium of a thickness of 50 m, for which we take the typical values c = 300 m/s, [ = 0.05, product Fi = 0.855. This damping effect of the superficial ground is the cause of a false idea that prevailed (and perhaps still prevails) in the minds of certain engineers, according to which the soft soil would provide more favorable foundation conditions than rock in seismic zones. On the contrary, experience shows that damage is almost always worse on terrain with mediocre mechanical characteristics. Wherever there are recordings, the higher values of the parameters of motion (accelerations, velocities) are generally obtained at the surface of the relatively thick (several dozen meters) sedimentary layers (sand, clay). This is simply explained by the fact that the inelastic damping of the layer does not compensate the amplification resulting from the contrast of the impedance over the bedrock. We saw in section 3.2.2 that this amplification can be approximately calculated by taking the square root of the ratio of impedance O (see [3.38]); amplification as at the surface is thus for a layer of thickness h:
as =
O
e
[ Z h c
[3.58]
a formula in which the propagation velocity c in the layer can be replaced by c1/O (c1 = propagation velocity in the bedrock) since the ratio of impedance is nearly equal to the ratio of velocities (the densities vary little between rock and soil); we therefore write: as =
O
e - EO with E = [
Zh c1
[3.59]
By reasoning on given materials for the bedrock and for the layer but bearing in mind that the thickness is variable for the latter, we see that amplification as at the
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surface decreases when the thickness increases and equals one for thickness h, given by: h1 =
c 1
LnO
[Z
2O
[3.60]
With the values considered earlier for waves and layers (Z = 18.85 rd/s, c = 300 m/s [ = 0.05) and for the bedrock c1 = 1,000 m/s (i.e. O = c1/c = 3.33) we find hi = 192 m. A good thick layer is required for it to stop having an amplifying effect at least for the frequency of 3 Hz that has been chosen for the wave. The layers of usual thickness (several tens of meters) can only attenuate higher frequency waves and only on condition that these do not coincide with resonant frequency of the layer (see section 4.3). These calculations have used values of c and [ that correspond to soils of average characteristics required by vibrations of a relatively moderate level, i.e. rather far from that for which their behavior would be highly nonlinear. In section 4.3 we will see that the effect of nonlinearity is to reduce the value of c and increase that of [; these variations can clearly reduce the value of the non-amplifying thickness hI. As we have seen in the example of the numerical calculation of factor Fi given earlier, inelastic attenuation plays rather a minor role if we consider focal or epicentral distances, generally used in earthquake engineering (several tens of kilometers). On the other hand at higher distances it influences the perception of the earthquake in distant field, expression [3.57] of factor Fi shows that the high frequencies are attenuated more quickly than the lower ones, just as in acoustics the sound of deeper instruments carry further than higher sound instruments. Structures or parts of constructions having their own long periods (particularly tall buildings, large bridges, free surface of large reservoirs) can thus respond through visible oscillations to far-off seismic excitations that are hardly felt in the majority of the constructions. Long distance appearances of inelastic attenuation vary according to the region. For example, we noted that the earthquakes of comparable magnitude are felt at far more distant places in the east than in west of the North American continent. These differences in the attenuation factors can lead to difficulties in interpretation when we try to use the perception radius of an earthquake to estimate its characteristics at the source like the magnitude or focal depth (see section 14.2).
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3.3. Induced phenomena 3.3.1. Soil liquefaction
The phenomenon of soil liquefaction was presented in section 3.1.2 as a possible consequence of the vibratory movement on soil saturated with water. For this to occur, there must be, on the one hand, certain texture characteristics and effective vertical stress present, and, on the other, the amplitude and the duration of the vibrations must be sufficient. The characteristics of potentially liquifiable soils have now been clearly established [COL 90]: – for sand, muddy sand and silt, the granulometry must be fairly uniform, with an average granular diameter of about one millimeter, and the effective vertical stress must not be above a maximum limit of 0.2~0.3MPa, which corresponds to a low depth of between 10 to 20 m; – for clayey soil, the granulometry must not be too fine and the plasticity index must be reasonably high.
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Figure 3.18. Liquified soil rises to the surface: above, sand volcanoes after the earthquake at El Asnam (Algeria) on 10 October 1980; below, massive liquefaction at Kǀbe (earthquake of 17 January 1995)
For the vibratory movement, the conditions to be satisfied concern maximum acceleration and the number of cycles, which is an increasing function of rupture time, and thus also of magnitude (see section 4.1.3). Earthquakes of greater magnitude produce a large number of cycles, equal to or greater than 20, and may induce liquefaction effects at considerable distances (up to 200–300 km for a magnitude of 8). Among these effects, the most characteristic is the migration of sand to the surface in the form of small “volcanoes” which, in the case of massive liquefaction, may melt by coalescence to form large stretches; these two aspects are visible in Figure 3.18. These sand migrations occur, generally, after a certain delay (up to several hours after the earthquake has ended) because the excess of interstitial pressure due to vibrations takes time to be reabsorbed by the creation of flows; their rise to the surface is accompanied by a mixture of water and sand being projected upwards, hence the use of the term “volcano” to describe the phenomenon. In Figure 3.18 (top) we see small craters at the tops of these volcanoes through which such projections occur; in certain cases, the violence of this “spitting” is high enough to eject stones of a considerable size. Traces left behind by these sand migrations may constitute indications of old seismic occurrences, sometimes even prehistoric; we then speak of paleoliquefaction (see section 6.1.3). Sand deposits at the surface are the most certain indicator of liquefaction, but it could be shown also through major deformations that may cause great damage to buildings and constructions whose foundations may be damaged by these deformations.
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Figure 3.19 shows the effects of compaction and lateral spreading during the Kǀbe earthquake; the compaction visible in the photograph below, about a meter high, did not damage the building standing on piles; on the other hand, the lateral spreading observed at waterfronts and on river and lake banks may damage buildings even though their foundations may be correctly designed and built. These horizontal and vertical movements often result from the liquefaction of a layer situated at a certain depth (which must be less than the 10 to 20 m limit mentioned earlier). Their amplitude in the horizontal direction may attain, in extreme cases, considerably greater values than those visible in Figure 3.19 (top photo). The liquified layer does not offer any frictional resistance to the sliding of the non-liquified layer and even on weak inclines, this slide may attain several dozen, if not several hundred meters. Such a case was observed at Valdez (Alaska) during the great earthquake of 28 March 1964; a piece of approximately 20 hectares of the top layer slipped more than 100 m towards the sea, taking with it the port and one portion of the city (see Figure 3.20); this sliding was fast enough to produce a wave that caused damage in the neighborhood of this moving piece.
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Figure 3.19. Deformations of the surface due to liquefaction. Below, piling up of about a meter of soil around a building on piles, which stayed in place. Above, a crack opened in a platform by a lateral movement (earthquake at Kǀbe, Japan, on 17 January 1995)
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Another effect of liquefaction is linked to the total loss of resistance to shearing of soil affected by this phenomenon: if the soil bears structures with superficial foundations this loss of resistance can bring about spectacular collapses and toppling. Figure 3.21 shows a group of buildings at Niigata (Japan) after the earthquake of 16 June 1964; certain buildings are severely inclined and one of them is practically lying on the ground; these toppling movements must have occurred “gently” because no apparent damage can be seen to the structures or to the extensions built on the roofs.
Figure 3.20. Ariel view of Valdez (Alaska) after the earthquake of 27 March 1964, showing the extent of the flooded zone, following a slide towards the sea of one portion of the city, due to the liquefaction of an underlying layer.
Figure 3.21. Toppled buildings at Niigata, Japan, after the earthquake of 16 June 1964
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3.3.2. Landslides
From the point of view of seismic effects, the term “landslides” is used to mean a set of phenomena of instability affecting slopes or cliffs. First, there are the “traditional landslides” which we know in hilly and mountainous zones; these are generally activated by non-seismic causes: heavy precipitation, melting snow, unreasonable earthwork at the foot of precariously balanced masses. Seismic vibrations, even of very weak strength, are capable of setting them off, as is seen very frequently in regions with rugged landforms. A typical seismic phenomenon involving terrain movement is the scaling of slopes. It concerns the top layer (thickness of about one meter) of fairly steep slopes, which is dislodged by the action of seismic waves, apparently as a result of sudden decompression (resulting from accelerations perpendicular to the surface) or of shearing (in the direction of the slope). An example is shown in Figure 3.22. Such scaling is generally observed only in the case of strong earthquakes [COL 99a].
Figure 3.22. Scaling of slopes in the epicenter zone of the earthquake on 21 September 1999 at Chi-Chi (Taiwan); the total amount of scaling produced by this earth quake represents a surface area of almost 6,000 hectares
On very steep slopes, we can sometimes observe sudden ruptures of the sides that do not correspond to pre-existing landslides and are different from the caving in of blocks which are described hereinafter.
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These ruptures may involve thicker layers (up to 10 or 20 m) than those of scaling. Figure 3.23 shows such a rupture of the ledge of a cliff that once supported a railway. Rock falls are routinely observed in precipitous relief features due to climatic actions and long-term evolution of equilibrium conditions. In zones that are exposed to this type of risk, a seismic shock may have very serious destabilizing effects and provoke “rock avalanches”. Any damage caused is often completely disproportionate to the direct effect of vibrations. Prudence must therefore be observed in the estimation of the size of ancient earthquakes in mountainous regions, because reported damage, often unreliable in documented archives, may lead us to over-estimate the magnitude of an earthquake if we attribute all damage to the action of vibratory motion (see section 13.1). This is apparently the case, for example, of the earthquake of 20 July 1564 to the north of Nice (the Vesubie valley).
Figure 3.23. Rupture of a cliff-edge during the earthquake of 29 April 1965 near Seattle (State of Washington, United States)
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Figure 3.24. Rock falls on the village of Braulins during the earthquake at Friuli (northern Italy) on 6 May 1976
Figure 3.24 shows the village of Braulins that was partially destroyed by rock falls during the Friuli earthquake (northern Italy) on 6 May 1976; houses saved by these falls suffered only slight damage as a result of vibrations. Figure 3.25 is a reproduction of a 17th century engraving and shows in great detail and with a high degree of realism the destruction of the small city of Piuro (or Plurs), situated in Italy close to the Swiss border, by a major landslide in Monte Canto [KOZ 91]. This catastrophe, which left 1,200 dead, appears to have been the result of an earthquake, although other hypotheses had been put forth earlier (collapse of mine galleries dug in the mountain). If the hypothesis that an earthquake was responsible for the destruction is the correct one, we see that, as in Braulins (see Figure 3.24), this earthquake did not destroy constructions not affected by the passage of the landslide; but the comparison must not be taken too far, for at Piuro, it was not a rockfall but a major landslide involving an entire side of the mountain, with material fine and coherent enough to form a natural dam in the river bed that flows from right to left in the figure.
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Such landslides, whose volume may attain several hundred million cubic meters and velocities of up to 30 to 50 m/s, are known by a certain number of examples which are largely the result of action triggered by an earthquake.
Figure 3.25. Catastrophic landslide at Piuro (northern Italy) on 25 August 1618; this small city is seen before (top) and after (bottom) the landslide, according to [KOZ 91]
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The best-known case of great landslides of seismic origin (which the Japanese call “yamatsunami” or “mountain tsunami”) is the one at Mont Huascaran in Peru which trapped the cities of Yungay and Ranrahirca, causing 60,000 deaths (see Figure 3.26).
Figure 3.26. Catastrophic landslide on the sides of Mount Huascaran (Peru) on 31 May 1970; it was triggered by a strong subduction earthquake (magnitude 7.7) whose epicenter was 130 km away. The slide, consisting of a mixture of ice, rocks and mud, slithered down the slopes making a difference of 4,000 m to the elevation, at velocities estimated at approximately 160 km/h (45 m/s). It affected an area 900 m wide, 1,600 m long (in the direction of the slope) and 15 m thick, i.e. a volume of about 22 million m3 according to [MUI 86]
The energy liberated by these great landslides is comparable to seismic energy Ec considered in section 2.1, for earthquakes of great size (magnitude of the order of 7). Taking data from the landslides of Mount Huascaran (see the caption of Figure 3.26) we find, with an average density of material of 1.5, a variation of potential energy of 1.3 x 1015J, i.e. according to formula [2.18] a magnitude of about 6.9 for an earthquake having an Ec of the same value; the kinetic energy of the landslide represents only a small proportion of the total energy liberated (2.5% for the velocity of 45 m/s estimated for the Peruvian landslide) which is thus mainly dissipated by friction; we conclude from this that landslides of this amplitude are likely to produce a disturbance of the soil, by the effect of friction, whose energy is equivalent to that of a good-sized earthquake.
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As has been indicated in section 1.3.1, it may be difficult to determine if an earthquake recorded at the time of a landslide (whose exact time is difficult to determine if it takes place at night or in an uninhabited region) is the cause of or the effect of the landslide. There is no doubt for the landslide at Mount Huascaran, which was a result of shocks from a major earthquake, but doubt may exist if the recording corresponds to a lower magnitude.
Figure 3.27. Satellite photos taken before and after the quake at Chi-Chi (Taiwan) on 21 August 1999. The lower photo shows a very large mudslide, near Tsaoling, that blocked the valley and created a lake
Figure 3.27 shows a lake being formed as a result of the creation of a natural dam by an earthquake that obstructed a valley in central Taiwan (earthquake of 21 September 1999). Such lakes constitute a delayed risk because such natural dams may give way suddenly if the water level increases: a catastrophic rupture of this nature also occurred in Taiwan after the earthquake of 1941.
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3.3.3. Tsunamis and seiches
Tsunamis (also called earthquake floods) are waves that result from rapid motion of a great volume of water due to different causes: – sudden tectonic movement affecting the bottom of seas and oceans (fault rupture or emergence or subsidence motion); – undersea landslides; – fall of cliffsides along the coast; – explosive eruptions of island volcanoes; – strong underwater explosions. Earthquakes are the most frequent cause of tsunamis, either directly (tectonic movements) or indirectly (induced by underwater landslides); subduction earthquakes particularly are the chief culprits of the large tsunamis in the Pacific, when they are sufficiently superficial for their tectonic movements to affect the trench (see Figure 1.10). Outside the Pacific, examples of tsunamis of seismic origin exist in other regions (eastern Mediterranean, the Caribbean, the Atlantic coasts of Portugal and Morocco at the time of the tsunami that followed the famous earthquake of 1 November 1755, called the Lisbon earthquake). As indicated in section 1.2.4, the “slow earthquakes” (i.e. those whose propagation velocity of the rupture on the fault plane is much lower than the usual values of 2~3 km/s) are, nonetheless, fast enough to produce significant tsunamis. The “tsunami-generation potential” of a tectonic motion of ocean floors appears a priori impossible except if this movement has a vertical component; then and only then does it appear capable of displacing any part of a great water mass. However, we know of tsunamis associated with essentially strike-slip movements (similar to the one of moderate amplitude, for example, apparently due to the undersea rupture of the San Andreas Fault during the 1906 California earthquake); they probably result from undersea landslides brought about by seismic vibrations, or the existence of a secondary component, normal or inverse, on certain reaches of the fault. Landslides appear to play an important role in the generation of tsunamis; which is why the eventuality of a tsunami must be considered for great earthquakes produced by faults close to coastlines, even if their path is purely on land. Tsunami waves have long periods, from 5 to 60 minutes, and a height which, off the coast, does not appear to be over a few decimeters; in these conditions, they are practically imperceptible because the values of the period and height mentioned correspond to a level of variation of velocity of several mm/s. The wavelengths are also very large because they are equal to the product of these periods with the propagation velocity given by the following simple formula:
Manifestations of the Seismic Phenomena on the Surface
V=
155
[3.61]
gh
where g is the acceleration of gravity and h the depth of the water; with g = 10 m/s² and h = 4,000 m (average value of ocean depth) we find that v = 200 m/s, i.e. 720 km/h and wavelengths of several dozen to several hundred kilometers. As they approach the coast the height of the waves increases progressively, which we can explain from the point of view of physics: by stating (as in [3.60]) that the propagation velocity decreases when the depth decreases, hence, the rear of the wave where the depth is greater tends to catch up with the front portion which approaches lower depths on the coast; therefore, there occurs a “piling up” of the wave upon itself and thus an increase in height. From simple hypotheses (linear theory, conservation of energy) we can establish Green’s Law, according to which the height H of the wave is inversely proportional to the fourth root of the depth; we have between heights H1 and H2 of the wave corresponding to depths h1 and h2 respectively: 1/ 4
§ h1 · =¨ ¸ H1 © h2 ¹
H2
[3.62]
With h1 = 4,000 m, h2 = 10 m we find H2/H1 = 4.47; a wave only 50 cm in open seas would therefore attain a height of more than 2 m when the depth falls to 10 m. The amplifications of the height of the wave upon its arrival at the coast may be considerably greater than those obtained by calculations with formula [3.62]; and this is not surprising, as other than the validation limits of the hypotheses (especially that of linearity, which implies that the height of the wave is small compared to the depth) this formula does not take into account the influence of the shape of the coast; the presence of capes and bays is likely to modify the height of the wave as compared to the height of a wave that arrives on a straight coastline. The distribution of amplifications on the coast, in general, is very irregular and shows rapid variations on relatively small distances (of the order of a kilometer). For example, the devastating tsunami that followed the earthquake of Hokkaido-Nansei-Oki (northern Japan) on 12 July 1993 produced wave heights varying from 2 to 30 m, with an average value of 5 m, on the coast of Okushiri island. The tsunamis producing high waves have considerable destructive potential and are thus feared by inhabitants of exposed coastlines. Figure 3.28 shows some of the damage observed at Seward (Alaska) after the tsunami resulting from the great earthquake of Prince William Sound (28 March 1964).
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Figure 3.28. Destruction at Seward due to a tsunami that followed an earthquake in Prince William Sound (Alaska) on 28 March 1964: boat pushed onto the coast, truck destroyed, debris from the ruined port buildings and equipment
The risk of tsunamis in the Pacific led to the creation of an alert system; their propagation velocities (of the order of 700 km/h for an average depth of 4,000 m as we have seen earlier) allows a delay of several hours to alert populations on continental coasts and on archipelagos that may be in danger, when a major earthquake is detected in one of the subduction zones of this ocean. Before this system was set up, many people fell victim to the tsunamis created by faraway earthquakes (up to several thousand kilometers away); they were thus unaware, because the vibratory motion of the Earth is, at those distances, hardly perceptible to humans. Thus, the earthquake of 1 April 1946 (Aleutian Islands-Alaska) produced a tsunami that killed 169 people on the Hawaiian islands (see Figure 3.29).
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Figure 3.29. Arrival of a tsunami wave at Hilo (Hawaiian island) about 3,800 km from the epicenter of the earthquake of 1 April 1946 (Alaska – Aleutian Islands); on the coast of Hawaii, the height of the wave attained a maximum height of 20 m; at the spot where this photograph was taken, the wave was 6 m high; the man seen to the left in the photograph was one of the 169 victims of this tsunami
Coastlines close to undersea seismic sources are affected by tsunamis created within minutes (a few dozen minutes maximum), which barely gives alert systems any time to function efficiently. Hence, reflexes born of prior experiences make up the best protection for populations at risk. Inhabitants of coastlines that may be exposed to tsunamis know that after feeling a particularly strong earthquake, they must immediately move to the nearest high ground; if the vibrations were weak but the sea recedes a few minutes later, that is a sign that the tsunami will arrive soon and that they must move to high ground without delay. This phenomenon of receding sea water is characteristic and has been reported by many witnesses; for example, Rear Admiral Billings, of the American Navy described what he saw at Arica (Chile) on 8 August 1869 [LAN 65]: “Once again the earth trembled, and this time, the sea receded, and as far as we could see, the rocky seabed was exposed to humans for the first time, full of marooned fish and other monsters of the deep. Boats lay on their sides, while the Wateree (the ship on which Billings was sailing) was on the flat seabed; when the waters returned, it was not a wave but an enormous tide that rolled our companion ships over several times, leaving some with their keels in the air, reducing others to flotsam. The Wateree, however, rose up on the waves without the least damage.” Experience also teaches us that there may be many waves and not just one, and that these may arrive on the coast at intervals of up to several tens of minutes, or
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even at intervals of several hours. For the events at Arica, Billings’ report recounts that the strongest tsunami arrived only in the evening, and that it carried the Wateree almost 3 km inland. The people who seek refuge on the hills close to the coast must not, therefore, return immediately after the first wave. The basic precautions (climbing to high ground when the sea recedes and not believing that it is all over after the first wave), which are part of the culture of people living around the Pacific, must be taught wherever the risk of a catastrophic tsunami, albeit infrequent, is nonetheless a reality. Apart from tsunamis, earthquakes whose epicenters lie on the ocean floor may create “sea quakes” which correspond to pressure waves transmitted by the water. The shock of these waves against the hulls of ships can cause damage: one example is of a 32,000 ton tanker that was heavily damaged off Gibraltar on 28 February 1969 by an earthquake of magnitude 8 (see [AMB 85]). If such damage is relatively rare, these shocks are frequently felt by crewmembers with sufficient intensity to warrant an entry into the ship’s log, where the incident is generally described as a meeting with a wreck or with a whale. The term tsunami is reserved, in principle, for marine waves due to the causes mentioned earlier; these same causes may act on lakes and produce destructive waves; examples are cited in mountainous regions after rock falls into lakes. These “lacustral tsunamis” are a seismic risk to dams, especially those with fill work, which are vulnerable if they are submerged by a wave taller than their crest. These lakes may also suffer oscillations of their open surfaces with long periods due to the effects of seismic motion. This phenomenon, called a seiche, is different from a tsunami in that these oscillations result from accelerations transmitted to the bottom of the lake and do not necessitate any intervention from irreversible tectonic displacement (fault rupture or emergence-subsidence movement that may affect the bottom).The longest period T possible for these oscillations can be calculated from formula [3.60] by writing that the dimension L of the free surface is equal to half the corresponding wavelength, i.e. h being the depth: L=
1 2
T
gh where T =
2L gh
[3.63]
For a good-sized lake (L = 10 km, h = 100 m) we find that T = 632 s, i.e. a little over 10 minutes. For a pond or a large reservoir (L = 100 m, h = 10 m) we find T = 20 s, which is again a high value for an oscillation period; the setting into movement of the free surfaces is therefore not possible unless the seismic motion has components with long periods, which implies a strong magnitude. Considering the preponderance of long periods over long distances (see section 3.2.3), this
Manifestations of the Seismic Phenomena on the Surface
159
oscillation effect may be observed several hundred, if not several thousand kilometers from the epicenter, in places where vibratory motion with shorter periods is imperceptible to humans. This is well known to people working on oil wells, who watch level indicators in their large reservoirs; oscillatory variations, unexplained by usual procedures or by local events, indicate an earthquake far away: an example is provided by the Algerian earthquake of El Asnam (10 October 1980) which motivated an oil port chief in Antwerp to call the Belgian seismological authority to warn them of variations seen in his instruments, and thus of an earthquake somewhere far away: the distance from El Asnam to Antwerp is about 1,700 km. 3.3.4. Other seismic manifestations
A certain number of effects felt at the surface have been linked with the occurrence, maybe imminent, of an earthquake. One of the most noteworthy, almost always mentioned by witnesses to great earthquakes, is the noise, which is compared, in different cases, to the passage of a train or a truck, to the rattling of heavy boxes or furniture being moved, to the stamping of a great herd of animals, or to distant artillery fire. These audible manifestations are generally perceived just before the motion of the earth is felt by the witnesses; this leads us to believe that they are due to the effect of P waves of sufficiently high frequency to be audible; such waves, whose vibratory amplitude is lower than S waves which are the most destructive (see section 3.2.1), arrive first. During the phase of strong shocks the noise emitted by the earth is mixed with other sounds (breaking constructions, falling objects, collapsing buildings, cries of panic), but they remain perceptible because of their low frequencies and their often deafening strength. More mysterious are the manifestations of light, which are in no way systematic but whose reality cannot be doubted in certain cases. The following description relates to the earthquake of Idu, in Japan, on 26 November 1930: “Close to the epicenter, glows persisted at least an hour after the start of the earthquake… They varied in form and in color. Most were irradiated like the rays of the rising sun; others were like searchlights and others resembled fireballs. According to one witness, balls of brilliant light were seen in a line. Most were bluish, but others, more yellow or violet, were also seen. They shone with a bright intensity, at 50 km from the epicenter they were brighter than the moon and even in Tokyo, witnesses declared that they could see objects by the light of these lights” [LAN 65]. When the sea receded preceding the tsunami of 15 June 1896 (due to the earthquake at Sanriku, Japan), the uncovered ocean depths showed bluish-white luminescence, which was again observed, this time on the water’s surface, during the earthquake of 3 March 1933 in the same area.
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If this marine luminescence may be attributed to the luminescent microorganisms disturbed by the shocks, the origin of the terrestrial light is difficult to explain: electric phenomena, escape of inflammable gases, confusion with nonseismic related events (storms) or accidents brought about by the earthquake (fires, damage to electric systems)? It must be recognized that the mystery persists, according to one of the rare scientists who studied the question, seismic luminescence remains the darkest chapter of seismology (see [LAN 65]). Among the phenomena that precede the start of an earthquake, abnormal behavior of certain animals has been talked about in the media, especially after studies in China were carried out to predict earthquakes (see section 5.3.1). Figure 3.30 shows a Chinese poster distributed widely in the countryside to bring awareness of the importance of observing such behavior. Different species (horses, pandas, pigs, poultry, rats and fish) show agitation without apparent reason or unexplained listlessness (pandas in a zoo) shortly before the start of an earthquake. If the reality of such behavior is uncontestable, the reliability of interpreting such behavior as predictions of earthquakes is subject to caution. The Chinese used these observations with other observed phenomena as potential precursors (especially variations in subterranean waters; levels in wells, streams, odors or turbidity) to try and set up a prediction system (see section 6.3.1).
Figure 3.30. Chinese poster showing abnormal behavior in certain animals before an earthquake from [WAL 82]
Part 2
Strong Ground Motions
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Introduction
The title of this part was chosen since in the field of earthquake engineering only earth tremors that are strong enough to significantly damage buildings are of further interest to engineers. In other words, the level of acceleration needs to be at least 1 m/s² or a minimum speed of 0.1 m/s needs to be reached. This type of tremor or movement can only be recorded if seismometers are placed close to the epicenter (the minimal distance can be estimated by the 5Ro rule which is described in section 2.3.1). The characteristics of seismometers enable them to record strong earth tremors and still remain very sensitive instruments at the same time. The speed and acceleration conditions were not the primary aims of the forefathers of seismology who were more interested in the description and analysis of the different types of seismic waves and the understanding of the Earth’s inner structure. Traditional seismographs situated in observatories were very sensitive instruments that recorded very weak signals of earth tremors which occurred on the other side of the globe. They provided the researchers with a large amount of data that quickly reached saturation point in the event of stronger tremors. Far field signals are also much “purer” than near field signals. Far field signals have the advantage that their seismic waves are separated into different strands as all of them run at a different speed. Geophysical models which are based on far field signals are therefore very sophisticated, while signals that were recorded close to the epicenter remain rather complex and can thus not be used when analyzing the structure of the Earth. The first seismographs which could be used for strong earth tremors were developed in California in the 1930s. They were first used during an earthquake with a magnitude of 6.3 which took place near to Long Beach in 1933 (its horizontal acceleration was approximately 2.3 m/s²). The El Centro accelerogram (an earthquake that took place in Imperial Valley on 18 May 1940) became known worldwide and was even used as the main basis for seismic calculations over the
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next 40 years. The characteristics of this recording (peaks of acceleration, speed of movement which were recorded at 3.3 m/s², 0.37 m/s and 0.20 m for horizontal components respectively) were considered as being very close to the maximum magnitude value. However, from the 1970s onwards (e.g. the San Fernando earthquake, on 9 February 1971, on its own provided several hundred accelerograms), a progressive increase in the number of recordings showed that there were even higher magnitude values. B. Bolt was very surprised [BOL 99] that only a small number of signals, often those taken from El Centro, were used for research into earthquake engineering up until 1980~1985. In 2000 a database of accelerometric records was made up of several thousand entries which could be used for earthquake engineering. At first glance this number might be impressive but the database does have its weaknesses. For many regions that are not prone to earthquakes only very little data is available. Researchers therefore need to rely on data obtained elsewhere, i.e. under different geological and tectonic conditions. It is thus questionable how representative this data is. Even in regions that are often affected by earth tremors and which are very well equipped with the necessary instruments, there are hardly any or even no records which have been taken close to the epicenter during an earthquake that reached a magnitude of 8 or higher. These problems cannot be solved in countries such as France where the installation of a network of equipment has been delayed and were the frequency and the strength of earthquakes does not enable researchers to produce a complete collection of significant recordings unless the project is carried out over several decades. Despite its weaknesses the data that is currently available shows that there is a great variety of near field signals. Using this data when estimating possible scenarios in the field of earthquake engineering is therefore strongly linked to a high level of uncertainty. The law of standard deviation provides us with an average which has already been mentioned in section 2.2.3. This law applies here and is of even greater importance here than in all other scientific domains.
Chapter 4
Strong Vibratory Motions
4.1. Recordings 4.1.1. Examples of accelerograms recorded in the near zone Figure 4.1 presents 22 examples of accelerograms with horizontal components recorded in the near zone, i.e. at focal or epicentral distances comparable to the dimensions of the source (radius R0 of the equivalent sphere defined in section 2.1.1). These examples have been selected so as to cover the range of significant magnitudes in earthquake engineering from a bigger set established by D. Hudson [HUD 77, HUD 88] who had the very simple but creative idea, of representing the accelerograms on a common scale of time and acceleration (the unit used is g and represents the acceleration of gravity, which is frequently used in earthquake engineering). Table 4.1 specifies the earthquakes and recording stations that correspond to the numbers in the figure, the magnitudes M and the values of maximum acceleration referred to as A (expressed in g).
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Figure 4.1. 22 horizontal accelerograms represented on the same scale indicated below on the right; the numbers correspond to the list in Table 4.1
Strong Vibratory Motions No.
Seism and recording station
M
A [g]
1
Stone Canyon 1972 Melendy Ranch
4.6
0.71
2
Port Hueneme 1957
4.6
0.17
3
Ancona 1972 Rocca
4.9
0.61
4a
San Francisco 1957, Golden Gate Park
5.3
0.12
4b
San Francisco 1957, State Building
5.3
0.10
5
Lytle Creek 1970
5.4
0.20
6a
Parkfield 1966, Temblor
5.6
0.41
6b
Parkfield 1966, Station No. 2
5.6
0.51
6c
Parkfield 1966, Station No. 5
5.6
0.47
6d
Parkfield 1966, Station No. 8
5.6
0.28
7
San Salvador 1986, GIC
5.6
0.69
8
Helena 1935
6.0
0.16
9
Managua 1972
6.2
0.38
10
Coalinga 1983, Pleasant Valley
6.2
0.60
11
Koyna 1967, Koyna dam
6.5
0.63
12
Imperial Valley 1979, Bonds Corner
6.5
0.78
13
Imperial Valley 1940, El Centro
6.7
0.36
14
Montenegro 1979, Petrovac
7.0
0.45
15
Olympia 1949
7.1
0.31
16
Tabas 1978
7.4
0.87
17
Chile 1985, Llolleo
8.0
0.62
18
Mexico 1985, Zacatula
8.1
0.25
167
Table 4.1. Identification of the 22 accelerograms of Figure 4.1; M = magnitude; A = maximum acceleration in g
A joint study of Figure 4.1 and Table 4.1 shows the following observations: – the strength of the signals is very irregular and varies from one example to another; in certain cases, there are only one or two large acceleration peaks which dominate a signal of short duration, as in 1, 6a and 7; in other cases, there are many peaks of amplitude close to the maximum, as in 13, 15, 17 or 18; – the influence of magnitude on maximum acceleration does not appear clearly; high acceleration values (higher than 0.6 g to give an approximate idea) also exist for low magnitudes (examples 1 and 3), average magnitudes (7, 10, 11 and 12) or
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strong magnitudes (16 and 17); this observation is characteristic of the near field, where acceleration is particularly controlled by the stress drop (see section 5.1), but does not apply to the far field where an effect of size appears (Ro/R term of the formulae used in section 2.3.1); – magnitude has, on the other hand, a significant effect over the duration of the signal; all the very short signals correspond to magnitudes lower than 6, the very long signals (more than 20 seconds) to magnitudes higher than 6.5; this point has already been emphasized in the assessment of the damage potential of earthquakes; – for a given earthquake, signals can vary significantly from one station to another, as can been see by comparing 6a, 6b, 6c and 6d, all of which correspond to the same earthquake in Parkfield in 1966, the maximum accelerations almost double between 6d and 6b. In order to go into a more detailed discussion of these examples, we need to specify the parameters that can be used to characterize the accelerograms; however, we can already note, after this rapid overview, that maximum acceleration alone is insufficient to determine the damage potential of a seismic signal. 4.1.2. Parametric description of the accelerograms It is easy to measure the acceleration peak, in absolute value, on an accelerogram. The determination of velocity and displacement peaks requires the integration of the signal once or twice; these integrations use very simple algorithms but can be sensitive to errors related to defects in instruments and errors in readings and the digitization of the acceleration values. Such errors are inevitable, particularly on old seismographs for strong motions, which were equipped with release systems for recordings, when the amplitude of the signal exceeds a certain level; these systems were necessary on one hand to avoid inopportune releases by disturbances of non seismic origin and, on the other hand, because of the dynamics (i.e. extent of the range of amplitudes that can be recorded) of these instruments. Typically the threshold of release was about 0.01g (or 0.1 m/s²) to be able to record accelerations going up to 0.5 g or 1 g. As a result of the existence of this threshold, the beginning of the accelerogram was not recorded and thus, at the initial moment of the recording, the displacement and velocity peaks had non-zero values, but nevertheless values that were unknown. Therefore, if the digital integration of the accelerogram was made by assuming initial rest conditions (zero displacement and velocity), it was necessarily incorrect. Various correction procedures were developed to correct these causes of errors, calling upon elaborate adjustment techniques of the time axis and filtering, which fell within the competence of specialists. We thus distinguish the corrected
Strong Vibratory Motions
169
accelerograms from the uncorrected ones. The discrepancy is not very high in the case of accelerations (not more than 10%) but can be high in the case of velocities and even higher in the case of displacements. For example a variation 'J of only 0.01 m/s² (or 0.001g) in the positioning of the time axis produces, at the end of time T, variations 'v and 'd in velocity and displacement which are given by the formulae: 'v = T 'J; 'd =
T² 'J 2
[4.1]
That is, for T = 20s, 'v = 0.2 m/s and 'd = 2 m; the error in the velocity is thus the same as the measured value, that in the displacement, is higher (see Table 4.2 to compare the above with some measured values). For the same accelerogram, different corrections could be carried out, which explains variations according to the sources used; for example the maximum acceleration of the famous recording of El Centro is 0.36 g in Table 4.1 (source D. Hudson) and only 0.33 g (or 3.3 m/s²) as quoted in the introduction of this second part (source B. Bolt). As indicated before, the variation is moderate for acceleration, but could definitely be much greater for velocity and displacement (section 9.1). In addition to the values of peaks of acceleration, velocity and displacement, which do not give any information on the temporal characteristics of motion, different parameters have been introduced to characterize the totality of the signal. Among these appear spectral parameters which are related to the response of oscillators of different frequencies; they will be defined and discussed in section 9.1; in the present section, we shall only deal with the duration, the number of cycles and the quantities defined by the integrals utilizing the accelerogram. For signals like those seen in Figure 4.1, the definition of duration can only be conventional; indeed, duration in the classic sense of the term, i.e. the time interval between the beginning of the motion and the moment from which the motion is no longer perceptible, depends greatly on the sensitivity of the instrument and its damping properties; it could last several minutes in the case of very sensitive instruments with low damping whereas the significant part of the ground motion would have lasted only a few seconds. It is thus necessary to define a temporal parameter that characterizes the strong part of the signal; the two methods used most usually are (see Figure 4.2):
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Figure 4.2. Two definitions of the duration of an accelerogram J [t] above, the duration TJo separating the first and the last peak above JO; below, the duration TI corresponds to the t interval 5% – 95% of the ratio I[t]/I [T] with I[t] = ³ J ²[W] dW o
The first method defines the duration TJo, as the time interval separating the first and the last peak of acceleration (in absolute value) that is higher than a given value JO (we often takes JO = 0.05 g). The second definition uses the increasing function I[t] defined by:
t
I[t] = ³ o J² [W]dW
[4.2]
J[t] being the accelerogram considered; beyond the time t = T, end of all the signals, I[t] has a constant value I[T]; by dividing I[t], by I[T] we define the duration TI as the time interval separating the reaching of the values 5% and 95% for this ratio. It is this second method [TI] which has been retained in most publications; the integral I[t] which is its base, becomes for t = T equal to a transformation factor, close to the Arias intensity IA defined by [ARI 70]:
Strong Vibratory Motions
IA =
S 2g
I[T] =
S 2g
T ³ o J² [t] dt
171
[4.3]
It has been suggested that this method be used to characterize the damage potential of a signal. The definition TJO has the advantage of being able to be seen on the seismic trace of the accelerogram; taking JO = 0.05 g we get, with a safety margin, a zero duration for accelerograms which is not of any particular interest to earthquake engineering; the definition TI can give a rather long duration if the amplitude of the signal is relatively constant, whatever its level; this definition in itself is not of tangible significance to earthquake engineering and should be used only in connection with an indicative parameter of the level of the shock (value of the peak or Arias intensity). As in the case of duration, the number of cycles of an accelerogram can only be defined in a conventional way since in general, a real signal is quite different from a sinusoid; this is an important parameter in the study of certain seismic effects, in particular with respect to the risks of liquefaction. If the seismic trace of the accelerogram is accurate, we can count, inside the strong part of the accelerogram corresponding to TJO or TI, the number of acceleration peaks exceeding a given fraction (for example, 50% or 75%) of maximum acceleration; half of this number constitutes a first “natural” definition of the number of cycles as well as the visible or the apparent dominant frequency of the strongest oscillations (by dividing this half by the time TJO or TI). Other definitions are possible, as we will see below. Another global parameter in the characterization of the damage potential of an accelerogram is the summation of the absolute value of velocity, or CAV, defined by the relation: CAV = ³
T O
J t
dt
[4.4]
This represents the area between the graph of the accelerogram and the time axis. For the 22 accelerograms already presented (see Figure 4.1 and Table 4.1), Table 4.2 provides the values of a certain number of parameters:
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Seismic Engineering
A
V
D
I[T]
TI
ga1
ga2
gd1
gd2
AD
cm/s²
cm/s
cm
m²/s3
s
Hz
Hz
Hz
Hz
V²
1
697
19.5
0.6
10
2
7.40
5.97
3.97
4.93
1.10
2
167
17.9
4.0
1
9
2.56
2.14
0.41
0.49
2.08
NO.
3
598
9.4
0.7
7
3
19.12
22.04
1.13
0.98
4.64
4a
118
4.6
0.8
6
3
7.68
8.61
0.49
0.43
4.45
4b
98.1
5.1
1.1
6
6
5.73
6.24
0.39
0.36
4.15
5
196
9.6
1.0
2
3
5.62
4.64
0.88
1.05
2.13
6a
402
22.5
5.5
5
5
5.34
5.94
0.35
0.31
4.37
6b
500
77.9
26.3
12
7
1.77
1.50
0.27
0.32
2.17
6c
461
25.4
7.1
10
7
5.48
6.50
0.30
0.25
5.07
6d
275
11.8
3.9
5
13
7.17
10.29
0.25
0.17
7.69
7
677
80.0
11.9
13
4
1.96
1.51
0.74
0.95
1.26
8
157
13.3
3.7
1
2
3.45
3.40
0.31
0.32
3.28
9
373
37.7
14.9
24
10
2.93
3.11
0.22
0.20
3.91
10
589
59.8
28.2
25
8
2.96
3.38
0.18
0.16
4.64
11
618
30.0
10.1
19
5
6.31
8.64
0.25
0.18
6.94
12
765
44.2
16.9
31
11
5.29
7.09
0.22
0.16
6.62
13
353
33.4
10.9
21
25
3.10
3.12
0.21
0.26
3.45
14
441
39.4
13.7
21
13
3.32
3.52
0.25
0.23
3.90
15
304
17.0
10.4
13
22
5.56
9.41
0.13
0.079
10.94
16
853
121.0
94.6
68
17
2.14
2.63
0.11
0.087
5.51
17
608
42.8
17.2
67
38
4.31
5.40
0.21
0.17
5.71
18
245
29.3
8.2
12
45
2.34
2.04
0.32
0.37
2.34
Table 4.2. Parameters characterizing the 22 earthquakes of Table 4.1; NO.: number in Table 4.1; A: maximum acceleration (cm/s²); V: maximum velocity (cm/s); D: maximum displacement (cm); I [T]: integral [4.2] for t = T, total duration of the signal (m²/s3); TI: duration corresponding to the interval 5% – 95% of I [t] [s]; ga1, ga2, gd1, gd2: characteristic frequencies (Hz) defined by equation [4.13], [4.14] and [4.15], AD/V²: dimensionless parameter
The third and fourth columns of the table give the values of the peaks V and D of velocity and displacement (in cm/s and cm), obtained after integrating the corrected accelerograms. Taking into account the reservations that we have as regards the accuracy of these values, particularly in the case of displacements, as mentioned before, we observe that:
Strong Vibratory Motions
173
– the order of magnitude of the velocities is about one tenth or one twentieth of the acceleration (when the same units of time and length are used to define these quantities), only accelerogram no. 3 (Ancona 1972) distinctly goes beyond this range (V/A = 0.016 s); – displacements are in centimeters for earthquakes of low magnitude (M d 5.4); they are in decimeters for those of higher magnitudes; earthquake no. 16 (Tabas 1978) is an exception with a value of about a meter; – the ratio AD/V ² which is the only dimensionless number which can be formed from A, D and V is rather variable (see last column of Table 4.2), but this variation is generally about a few units; it is always higher than one (which is the value that would be obtained for a sinusoidal signal) but is almost 1 for accelerograms 1 (Stone Canyon 1972) and 7 (San El Salvador 1986). The fifth and sixth columns of Table 4.2 relate to the integral I (T) (equation [4.2] with t = T) and to the duration TI of the strong part calculated by taking the average of the two horizontal components [HUD 77, HUD 88]. It can be noted that these durations exceed ten seconds only in the case of magnitudes above 6.5 (accelerograms 12 to 18) with an interesting exception in one of the accelerograms (No. 6d) of the Parkfield 1966 earthquake showing a magnitude of 5.6; the fact that the other accelerograms (6a, 6b, 6c) obtained during this earthquake have durations that are clearly shorter, shows the influence of the site conditions on the characteristics of the signals, as already indicated at the end of section 4.1.1. A comparison of the values of I [T] and TI shows that the acceleration peak A is in general much higher than the quadratic average acceleration AQM than can be defined by: AQM =
0,9 I (T ) TI
[4.5]
where coefficient 0.9 is obtained from the definition of Ti starting from the interval 5%–95% of I[t], the quotient 0.9 I[T]/TI thus represents the average value of the square of the acceleration during the strong part and AQM the root of this average value; with the values of the table (I[T] in m2/s3), formula [4.5] gives AQM in m/s², we find that AQM varies between 0.5 and 2m/s², that is one-quarter or one-fifth of A, except for the accelerograms of the San Francisco 1957 earthquake (no. 4a and 4b). Columns 7 to 10 of Table 4.2 give estimates of a dominant frequency for the accelerations (ga1 and ga2, columns 9 and 90 of Table 4.2). It is obvious that the displacements have an apparent frequency that is much lower than that of the
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accelerations; for example, we can see this in Figure 4.3, which shows the variations according to the acceleration time, velocity and displacement for the north-south component recorded at Tolmezzo during the Friuli earthquake on 6 May 1976.
Figure 4.3. Temporal graphs of acceleration, velocity and displacement (from top to bottom) for the north-south component of the Tolmezzo recording (Friuli earthquake on 6 May)
We can see that the strong part of the accelerogram (at the top in Figure 4.3) is concentrated in the time interval 4 s–8 s and comprises approximately a dozen cycles; this corresponds to an apparent frequency of 3 Hz; in the case of the velocity (middle curve), we find this frequency of 3 Hz in the same interval but a much lower frequency modulation can be observed which is especially visible after 8 s, but which also affects the beginning of the signal (between 0 and 4 s); as regards the displacement (bottom curve), it is this slow oscillation, corresponding visually to a period of about 10 s (frequency of 0.1 Hz), that largely dominates, the effect of the high frequencies being limited to small fluctuations during the strong part of the accelerogram, from 4 to 8 s. These tendencies can be better understood by considering a very simple model, known as the two sines, in which displacement d[t] of the ground is defined by: d[t] = a [k sin Zt – sin kZt]
[4.6]
Z being a pulsation of a comparatively lower value (around 0.5 to 1 rd/s), k a rather large number (from 15 to 30) and a length in millimeters or centimeters. The velocity v[t] and the acceleration a [t] can be obtained by the derivation: v[t] = d [t] = k a Z [cos Z t – cos k Z t]
[4.7]
Strong Vibratory Motions
a[t] = d [t] = – ka Z² [sin Zt – k sin kZt]
175
[4.8]
We note that the initial conditions correspond to a rest state d[o] = O, v[o] = O and that the acceleration a [o] is zero. The parameters a, k and Z of the two sines can be adjusted so that the maximum in absolute value of d[t], v[t] and a[t] is very close to the values D, V and A of the displacement, velocity and acceleration peaks of a given seismic movement; we can indeed easily check the following inequalities: k sin
k [cos
k [sin
(4 n 1)3 2k
3 k
+1d
+ 1] d
(4 n 1)3 2k
1 max t
a
1
max
DZ
t
+ k] d
d t d k + 1
|v[t]| d 2k
1
max
DZ ²
t
[4.9]
| a[t]| d k [k + 1]
where n is the integer nearest to [k + 1] /4, the higher limits are obtained by assuming in [4.6], [4.7] and [4.8] that the sines and the cosines take the value +1 or –1 so that their combination produces the strongest absolute value; the lower limits for displacement and acceleration are obtained by considering the moment [4n – 1] S /[2kZ]; the lower limit for velocity is obtained by considering the moment S /[kZ]. These lower and higher limits are very close to each other when k is equal to or higher than 10 as can be seen in Table 4.3.
176
Seismic Engineering Max |d[t]|/a
Max |v[t]|/aZ
Max |a[t]|/aZ²
k
n
Lower
Higher
Lower
Higher
Lower
Higher
10
3
10.88
11.00
19.51
20.00
109.87
110.00
20
5
20.94
21.00
39.75
40.00
419.94
420.00
30
8
30.96
31.00
59.84
60.00
929.96
930.00
40
10
40.97
41.00
79.82
80.00
1,639.97
1,640.00
Table 4.3. Higher and lower limits of inequalities [4.9]
By retaining the estimates made of the higher limits of [4.9], we use the following system to determine the values of a, k and Z corresponding to the given peaks, D, V and A of the motion: D = [k + 1] a V = 2k a Z
[4.10]
A = k [k + 1] a Z² According to the above equations, the dimensionless ratio r = AD/V² is expressed as: r=
AD V²
=
( k 1)²
[4.11]
4k
This relation constitutes a second degree equation for k, whose solution, which is valid if r t 1, is written as: k = 2r – 1 + 2
r ( r 1)
[4.12]
The dominant frequency ga of the accelerogram corresponds to the pulsation kZ, i.e. according to [4.10] and [4.12]: ga =
1
A r r 1
23 V
r
[4.13]
Strong Vibratory Motions
177
whereas the dominant frequency gd of the displacement, which corresponds to pulsation Z, is given by: gd =
1
V
23 D
r[ r–
r 1 ]
[4.14]
The frequencies thus calculated, appear under the heading 7 and 9 of Table 4.2. We see that the dominant frequency of the accelerations is a few Hertz, that of the displacements, about a fraction of a Hertz. Accelerograms NO. 1 (Stone Canyon 1972) and NO. 3 (Ancona 1972) distinguish themselves from the average, the first because of the small difference between ga1 and gd1 (related to the value of the ratio AD/V2 that is close to one) and the second due to its very high frequency. We thus notice that amongst the four accelerograms of the Parkfield earthquake in 1966, 6b has a frequential content that is very different from that of the others, which once again highlights the variability of the signals for the same earthquake. In a completely different approach, based on the theory of random vibrations, M. Kamiyama proposed [KAM 96] estimating ga and gd D, by using the following formulae: ga =
gd =
1
A
23 V
1
r
V
1
23 D
r
[4.15]
The values calculated by these formulae [4.15] appear under the heading ga2 and gd2 in columns 8 and 10 of Table 4.2. They are generally close to ga1 and gd1, the highest variation corresponding to the cases where the ratio r = AD/V ² has very high values, i.e. accelerograms no. 6d (r = 7.69) and no. 15 (r = 10.94). A hypothesis that was often used for the seismic calculation of high risk installations, is the USNRC (United States Nuclear Regulatory Commission) spectrum that is presented in section 9.1; it corresponds, for a conventional acceleration peak A of 9.81 m/s² (1 g) to velocity peaks V and displacement peaks D equal to 1.22 m/s (48 inches/s) and 0.915 m (36 inches) respectively; we thus obtain r = AD/V² = 6.03 and for the parameters of the model of the two sines, see [4.10] and [4.12]: k = 22.07; a = 0.03966 m; Z = 0.697 rd/s.
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Seismic Engineering
We thus obtain a dominant frequency ga of 2.45 Hz for the accelerations that is close to the value 2.5 Hz for which the acceleration response of a simple oscillator is the strongest (see section 9.1.2), and a frequency gd = 0.11 Hz for the displacements. Formulae [4.15] give slightly different results (ga = 3.14 Hz and gd = 0.086 Hz). In conclusion, we must remember that the peak values of acceleration, velocity and displacement, A, V and D are sufficient to describe the level and the frequential content of the signals. The ratio R = AD/V² indicates the extent of the range of the excited frequencies; it “normally” varies from 2.5 to 7.5 (N.M. Newmark recommended the value 6 for the study of nuclear thermal power stations) but may go beyond this range, particularly in the case of accelerograms where most of the strong motion, as much in the case of velocity and displacement as in acceleration, is produced by only one great oscillation, r is then close to 1 as for a sinusoidal signal (accelerograms nos. 1 and 7 of Table 4.2); of course, it is necessary to keep in mind the fact that the values of D may not be reliable, given the important corrections that its calculation would have required. The quotient A/V, whose values are more reliable than those of AD/V², gives a good indication on the frequential content of the accelerogram, its “normal” values ranging from 10 to 20 (if A and V use the same units of length and time), which corresponds to a dominant frequency of about 2.5~5 Hz (see formula [4.13] or [4.15] with r # 2.5~7.5) Generally speaking, but taking into account exceptions due to the influence of site conditions, signals at high frequency correspond to rock sites and to earthquakes of low or medium magnitude whose source is nearby. As regards the damage potential associated with an accelerogram, the knowledge of peak values is not sufficient, especially if these values are limited to accelerations, as is unfortunately often the case. Such knowledge must be supplemented by information about the duration of the strong motion as well as the response of simple mechanical systems (elastic and non-elastic oscillators, solid bodies likely to slip or rock). 4.1.3. The three components of vibratory motion
The vibratory motions produced by the earthquake have three directions; the seismographs for strong motions are designed and installed in order to be able to record the three components of the motions, two horizontal and one vertical. The orientation chosen, for the horizontal components this often corresponds to the north-south and east-west directions, particularly when the instruments are installed “in free field”, i.e. outside constructions; in the case of instruments installed in
Strong Vibratory Motions
179
buildings (generally on the ground floor or in the foundations, when representative recordings of the ground motions are desired) the axes chosen correspond most often to the main orientation of the building; for example, the horizontal components of the seismograph at Llolleo (Chile), that provided accelerogram no. 17 (Figure 4.1 and Tables 4.1 and 4.2), have been described as S80E and N10E, i.e. the orthogonal directions south-80 degrees towards the east and north-10 degrees towards the east. Seismographs that have mainly one alarm function or start-up function for the automatic safety mechanism of a critical installation, often record only the vertical component that is the first to be detected (see sections 1.2.3 and 3.2.2). Table 4.4 gives the peaks of acceleration, velocity and displacement for the three components of the 22 recordings considered in Tables 4.1 and 4.2, where only the horizontal component having the strongest peak of acceleration had been used; it can be stated that: – the second horizontal component (columns marked H2 in Table 4.4) which, by definition, has a weaker peak of acceleration than that of the other component (columns marked H1) generally has a magnitude that is close to that of the first one; in only three cases (nos. 2, 15 and 17) the ratio H2/H1 of the accelerations falls below 60%, while there are five cases (nos. 4a, 12, 13, 15 and 18) where the ratio H2/H1 of the velocities is higher than one; – the vertical component (columns marked V) is on average significantly weaker than the horizontal components, but there are some rare exceptions to this tendency (recordings no. 11 where the strongest peaks of velocity and displacement are those of the vertical component, and no. 17, where the same can be stated for the peaks of acceleration and displacement).
180
Seismic Engineering No.
A [cm/s²]
V [cm/s]
D [cm]
H1
H2
V
H1
H2
V
H1
H2
V
1
697
549
196
19.5
18.5
4.8
0.6
0.6
0.3
2
167
88.3
29.4
17.9
8.9
1.9
4.0
2.6
0.5
3
598
441
294
9.4
9.4
4.0
0.7
0.7
0.2
4a
118
108
49.0
4.6
4.9
1.2
0.8
2.3
0.7
4b
98.1
68.7
49.0
5.1
4.0
2.3
1.1
0.9
0.6
5
196
147
78.5
9.6
8.7
3.1
1.0
2.1
1.4
275
157
22.5
14.5
4.7
1.4
343
77.9
6a
402
6b
500
6c
461
392
177
25.4
6d
275
275
137
7
677
412
8
157
137
9
373
324
10
589
11 12
4.4
5.5
14.1
26.3
22.5
6.8
7.1
5.2
3.4
11.8
10.8
4.5
3.9
4.4
2.1
383
80.0
61.8
10.9
11.9
14.8
2.3
98.1
13.3
7.3
9.5
3.7
1.4
2.8
324
37.7
30.0
17.5
14.9
6.2
8.7
520
363
59.8
39.2
16.1
28.2
5.0
9.6
618
480
333
30.0
25.2
34.6
10.1
19.4
24.1
765
589
324
44.2
46.7
11.8
16.9
18.5
2.9
13
353
216
275
33.4
36.9
10.8
10.9
19.8
5.6
14
441
304
206
39.4
25.8
17.9
13.7
3.0
8.9
15
304
157
98.1
17.0
21.4
6.8
10.4
8.5
4.0
16
853
804
589
121.0
99.3
38.7
94.6
38.6
11.3
17
608
343
736
42.8
23.6
38.2
17.2
3.7
36.6
18
245
167
128
29.3
33.3
7.7
8.2
2.5
1.2
4.3
Table 4.4. Peaks of acceleration, velocity and displacement for the 22 recordings considered in Tables 4.1 and 4.2; the columns marked H1 correspond to the second, third and fourth columns of Table 4.2 (horizontal component having the strongest peak of acceleration); the columns marked H2 and V correspond to the other horizontal component and to the vertical component respectively; the H2 values are missing for recording 6b (station no. 2 for the earthquake at Parkfield 1966)
The vertical component constitutes “the poor relation” of earthquake engineering; most of the anti-seismic codes for common constructions take into account its effects only in some specific cases; there are mainly two reasons to justify this omission:
Strong Vibratory Motions
181
– the traditional approach of seismic calculation for buildings is based on the concept of equivalent static force for the representation of seismic action; in the horizontal directions, the consideration of the “seismic force” is obviously necessary as the other load cases considered in the dimensioning of the structures do not produce effects that are as important, with the exception of some cases of wind; in the vertical direction, on the other hand, the “seismic force” represents generally only a relatively weak fraction of self weight, except in a few cases (amplification of the vertical response on alternate points or with a far reaching effect between supports, very powerful earthquake) and its omission seems acceptable in view of the usual safety coefficients in gravity dimensioning; – the magnitude of the vertical motion is generally weaker than that of the horizontal motion, as we have seen in Table 4.4; the fact that there can be a few exceptions to this rule is simply due to the law of standard deviation and has no particular significance in engineering that, at least in the case of common constructions, aims at a “statistical protection” (see section 18.2) and hence gives more importance to average tendencies than to extreme tendencies. Obtaining recordings that show a preponderance of the vertical component is sometimes considered to be a recent phenomenon due mainly to the earthquakes at Northridge on 17 January 1994 and at Kǀbe on 17 January 1995 which provided a great deal of instrumental data in the near zone. In fact, as in many similar cases (see, for example, section 4.3 for site effects), the supposedly new characteristic is mainly the consequence of amnesia. In the 1980s, this phenomenon was observed for some recordings such as that of Llolleo in Chile (the earthquake on 3 March 1985, which is no. 17 in Table 4.4); the same year, 1985, is remarkable for having provided the recording at Nahanni in Canada (the earthquake on 23 December 1985) which for a very long time, held the record of being the strongest peak of acceleration, irrespective of the types of components, with its vertical component of 1.73 g (undoubtedly higher in reality due to the saturation of the instrument) as against “only” 0.79 g and 0.99 g for the horizontal components [HUD 88]. It must however be indicated that at Northridge and at Kǀbe, the cases in which the vertical peak is higher than the horizontal peaks are very few in number and do not constitute at all a general tendency as could be believed from some publications (see Figure 4.4). The analysis of the examples with a vertical preponderance has led to the conclusions that two conditions seem to contribute significantly to the possibility of occurrence of this phenomenon: – the proximity of the rupture zones that can be accentuated by directivity effects (amplification of the motions in the direction of propagation of the rupture on the fault plane; see section 4.3.3); this is the case for the above-mentioned readings at Llolleo and Nahanni, as well as for those at Northridge and Kǀbe; this influence is
182
Seismic Engineering
easy to understand when the motion of the fault is mainly vertical as it was at Llolleo (subduction) or at Northridge (reverse faulting), but appears less predictable for strike-slip motions, as at Kǀbe; at epicentral distances that are greater than the dimension of the source, the vertical magnitude is almost systematically lower than the horizontal magnitude, the very rare exceptions resulting it would seem, from directivity effects (recording at Newhall for the earthquake at Northridge, whose three components are practically equal in maximal acceleration, a little lower than 0.6 g; see [COL 94]); – the nonlinear forces in soils having average mechanical characteristics, when they are subjected to strong quakes; in fact, a saturation phenomenon of horizontal accelerations that corresponds to the limitation of shear stresses induced by transversal waves, can occur in such soils; the vertical component caused mainly by longitudinal waves, does not experience this phenomenon and its magnitude can thus overtake that of the horizontal components; this explanation is particularly true for the recordings at Port Island at the time of the earthquake at Kǀbe. In cases where seismic calculations actually take into account the force of the vertical component, the latter is generally supposed to be equal to a fraction (often 2/3 or 70%, sometimes only 50%) of the horizontal seismic action. This approximation is reasonable as regards the values of the peaks of acceleration and truly conservative for the values of the peaks of velocity. Figure 4.4, related to the earthquake at Northridge, enables us to compare the peaks of vertical acceleration (Av) to the peaks of horizontal acceleration (AH); it is seen that there are only two points above the line AV = AH and that most of the points are below the line Av = 0.7 AH. This simple rule of proportionality from AV to AH overlooks the fact that the frequential content of the vertical component is in general different from that of the horizontal components, as we can see in Table 4.5.
Strong Vibratory Motions
183
Figure 4.4. Comparison of the vertical (AV) and horizontal (AH) peaks of acceleration for the high magnitude recordings of the earthquake at Northridge on 17 January 1944
Table 4.5 makes it evident that the dominant frequency of the vertical accelerograms is generally, clearly higher than that of the horizontal accelerograms; the ratio between these two frequencies, whether calculated using formula [4.13] or formula [4.15], is lower than one only in three cases (nos. 8, 11 and 15) and is higher than 1.5 in more than half of the cases, in which values above 2 are frequent (7 times for the calculation with [4.13], 12 times for the calculation with [4.15]). It is this difference in the frequency of the accelerations that results in the fact that the ratio of the peaks of velocity is weaker than that of the peaks of acceleration, as it can be seen in the central columns of Table 4.4, where the ratio V/H1 of velocities is higher than 0.5 in only three cases (nos. 8, 11 and 17). It would therefore be conservative to take for a vertical seismic action, a fraction of the horizontal seismic action when the value of this fraction is deduced solely from the examination of the peaks of acceleration.
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Seismic Engineering
(ga2)v
(ga2)H
Hz
(ƒ a1 )V (ƒ a1 )H
Hz
Hz
(ƒ a2 )V (ƒ a2 )H
11.57
7.40
1.56
10.37
5.97
1.74
4.07
4.60
2.56
1,80
4.97
2.14
2.32
AvDv
(ga1)v
(ga1)H
Hz
2.55
AV
VV
DV
cm/s²
cm/s
cm
2 Vv
1
196
4.8
0.3
2
29.4
1.9
0.5
No.
3
294
4.0
0.2
3.67
21.68
19.12
1.13
22.41
22.04
1.02
4a
49.0
1.2
0.7
23.82
12.86
7.68
1.67
31.72
8.61
3.68
4b
49.0
2.3
0.6
5.56
6.46
5.73
1.13
8.00
6.24
1.28
5
78.5
3.1
1.4
11.44
7.88
5.62
1.40
13.63
4.74
2.88
6a
157
4.4
1.4
11.35
11.10
5.34
2.08
19.13
5.94
3.22
6b
343
14.1
4.3
7.42
7.47
1.77
4.22
10.55
1.50
7.03
6c
177
6.8
3.4
13.01
8.12
5.48
1.48
14.94
6.50
2.30
6d
137
4.5
2.1
14.21
9.52
7.17
1.33
18.27
10.29
1.78
7
383
10.9
2.3
7.41
10.79
1.96
5.51
15.22
1.51
10.08
8
98.1
9.5
2.8
3.04
2.99
3.45
0.87
2.86
3.40
0.84
9
324
17.5
8.7
9.2
5.73
2.93
1.95
8.94
3.11
2.87
10
363
16.1
9.6
13.44
7.04
2.96
2.38
13.15
3.38
3.89
11
333
34.6
24.1
6.70
2.94
6.31
0.47
3.96
8.64
0.46
12
324
11.8
2.9
6.75
8.40
5.29
1.59
11.35
7.09
1.60
13
275
10.8
5.6
13.20
7.95
3.10
2.56
14.72
3.12
4.72
14
206
17.9
8.9
5.72
3.50
3.32
1.05
4.38
3.52
1.24
15
98.1
6.8
4.0
8.49
4.45
5.56
0.80
6.69
9.41
0.71
16
589
38.7
11.3
4.44
4.55
2.14
2.13
5.10
2.63
1.94
17
736
38.2
36.6
18.46
6.05
4.31
1.40
13.17
5.40
2.44
18
128
7.7
1.2
2.59
4.72
2.34
2.02
4.26
2.04
2.09
Table 4.5. Characteristics of the vertical motion for the 22 readings of Table 4.4; the dominant frequency of the accelerogram was calculated by formulae [4.13] (ga1)v and [4.15] (ga2)v and compared to that of the horizontal accelerogram of the strongest acceleration (see Table 4.2)
The original version of the USNRC spectrum, mentioned earlier and presented in section 9.1, takes into account the differences between the frequential content of the horizontal and vertical motions, but, in several practical applications, a simplified version was used, in which, a simple reduction in the 2/3 ratio of the horizontal action is assumed for the vertical action, irrespective of its frequency.
Strong Vibratory Motions
185
The usual practice of seismic calculation supposes that the accelerograms that correspond to the three directions in space are independent, in statistical terms, when they are considered two at a time. This hypothesis is quite well verified in most of the cases; it reflects the complexity of vibratory motion in the near zone, which is the result of the combination of a large number of waves emitted by the different parts of the rupture surface (which is often very heterogenous, see section 1.2.4) and subjected to multiple refractions and reflections by the interfaces between terrains possessing different characteristics; this combination thus presents a random character and the rules for seismic calculations often call for the principle of quadratic cumulation, according to which the expected maximum value of a sum is equal to the square root of the sum of the squares of the maximum values of each of its terms. This rule which is deduced from the addition theorem of variances of random independent variables will be presented and discussed in section 10.2 while studying its application in structure analysis, but will also be used in section 5.1 of the present book in order to determine the attenuation laws associated with the basic model of elastic rebound. It constitutes one of the bases of seismic calculation and frequent difficulties in its interpretation are due to a possible confusion with the Pythagoras theorem (in the combinations of the effects produced by three of the components of seismic motion) and with the loss of signs in the evaluation of action effects on structural elements; these difficulties will be dealt with in section 15.2.3. The hypothesis of statistical independence of the components can be questioned in the immediate vicinity of the fault whose movement strongly influences the form, polarity and frequency of the waves emitted by introducing an important correlation between the components. As some recent earthquakes have provided several recordings at points very close to the fault (Northridge 1994, Kǀbe 1995, Chi-Chi 1999), we have been able to show an impulse of great magnitude with low frequency, acting mainly in the direction perpendicular to the fault; this killer pulse seems to have a great damage potential for flexible structures (tall buildings); it corresponds to velocities of about 1 m/s, displacements of about 1 m and frequencies located in the range 0.3~1 Hz. This phenomenon seems to correspond to powerful earthquakes (magnitudes of about 7 or higher than 7) and shows rapid attenuation as we move away from the fault; it should be studied more closely so as to be able to specify the possibilities of its occurrence and its implications on the definition of calculation of seismic action. It is quite likely that the recording in Iran at Tabas, on 16 September 1978 that appears as no.16 in Tables 4.1, 4.2 and 4.4, constitutes one of the first cases of the killer pulse (component HI giving V = 121 cm/s and D = 94.6 cm) that was recorded by instruments, at a time when this phenomenon had not yet been described. According to [HUD 88], the seismograph was located less than 3 km from the fault trace on the surface.
186
Seismic Engineering
4.2. Attenuation laws of peak values 4.2.1. General considerations as regards attenuation laws
An attenuation law is a formula that enables us to calculate a parameter of seismic motion with respect to a certain number of parameters characterizing the source, the propagation of waves between the source and the site where the motion is of particular interest to us, and the local characteristics of the terrain. The parameters of motion can be values of peaks of acceleration, peaks of velocity or peaks of displacement, spectral ordinates (see Chapter 9) quantities that describe the entire time dimension (duration, Arias intensity, average quadratic acceleration, cumulation of the absolute value of velocity) or certain specific aspects (number of cycles, coefficients of correlation between components, vertical/horizontal ratio). The independent variables of these formulae are at the very least the magnitude (so as to characterize the source) and a parameter of distance (so as to characterize the propagation between the source and the site) but can also include other characteristics of the source (type of fault, stress drop) and parameters linked to the directivity effects (azimuth with respect to the direction of propagation of the rupture) and to the site effects (mechanical properties and stratigraphy of superficial terrains). The term “attenuation law” comes from the fact that these formulae express, in general, a decrease with respect to the distance (although that is not true for the duration or the number of cycles). In this definition, the parameters of seismic motion which appear as a dependent variable are supposed to correspond to the recordings that could be obtained with the help of seismographs installed on the site in question and are thus perfectly defined quantities that one would be able to measure or calculate on the basis of such recordings. The concept of the attenuation law can be stretched to parameters linked to the assessment of the effects produced by seismic motions such as the macro seismic intensity (mentioned in section 2.3.1 and in Chapter 14) or the potential of soil liquefaction, even if these parameters are not, strictly speaking, characteristics of seismic motions. We will limit ourselves, in section 4.2, to attenuation laws of peak values (acceleration, velocity and displacement), those pertaining to intensities shall be taken up briefly in section 9.2.2. All attenuation laws bring into play a magnitude M and a distance R as independent variables and many are satisfied with these two parameters for the prediction of seismic motion. The use of the indefinite article is meant to attract attention to the fact that the choice of the type of magnitude and the definition of distance can vary from one law to another.
Strong Vibratory Motions
187
As we have seen in section 2.3.2, the different magnitudes introduced by seismologists are more or less equal only in a rather small range around magnitude 6 (see Figure 2.14). Thus, it is necessary to identify the type of magnitude for which the law was established and proceed if need be with transpositions, which are rather delicate operations, before using an attenuation law. The current tendency that favors the use of the moment magnitude MW is noticeable only in very few cases as a lot of data from the catalogs has not been transcribed in MW. The definition of distance also plays an equally important role in the application of attenuation laws. Five definitions are usually used (see Figure 4.5) and can lead to values that are noticeably different when the site in question is close to the fault.
Figure 4.5. The five definitions of the distance between the seismic source and a site S: Rh = SH distance from the hypocenter (or focus); RE = SE distance from the epicenter; RC = SC distance from the center of release of seismic energy; Rg = SF the shortest distance between the site and the fault plane; Rp = SP the shortest distance between the site and the surface projection of the fault plane. The fault plane is the rectangle IJKL that projects out to the surface as per ijkl
The fact that there are several definitions of distance goes to show that none of them is truly satisfactory in the very near zone, mainly because the size-of-source effect is not taken into account; we shall see in section 4.2.2 that some attenuation laws involve, for the purposes of calculation of the attenuation, a combination of the distance between the source and the site (defined as per one of the five conventions in Figure 4.5) with a parameter of fault dimension which is an increasing function of the magnitude. Thus, as we have already emphasized several times in the first part of this book, distances must always be related to source dimensions. Attenuation laws are generally established by statistical regression from a database of recordings; in this approach, the coefficients that reflect the decrease
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Seismic Engineering
with respect to the distance, i.e. the attenuation in the true sense of the term, are normally purely empirical, but, in a certain number of cases, the form of attenuation is imposed with a geometrical term in I/R and an exponential factor of non-elastic attenuation (see section 3.2.3); the adjustment through regression concerns only the coefficient of the exponential. 4.2.2. Examples of attenuation laws for peak values
Several attenuation laws for acceleration of velocity and displacement have been proposed for different regions of the world since a significant number of high-level recordings were obtained, i.e. since the 1970s. In an article which, for a long time served as a reference in the subject [CAM 85] K. Campbell analyzed 51 attenuation laws published during the ten years that preceded his study, i.e. the period 1974– 1984. In 2000, 15 years after this article by Campbell, the number of laws that were published was in the hundreds. This section aims at presenting a sample of this abundance of laws in the form of 24 laws, which are chosen using the following criteria: – it is important to cover, as far as possible, all the regions of the world where the study of seismic hazards would be required; this is more wishful than feasible as, in numerous regions, the available data is not sufficient to be able to establish specific laws; – it is important that the laws retained be published in articles that are easily accessible to centers of documentation, as the consultation of these articles is in practice often necessary for the users of the laws, so as to properly define the limits of validity and to what extent the parameters are representative of reality; – the laws retained must, preferably, be the most recently published for a given region, so that we are able to benefit from the most comprehensive databases; relatively older laws, but later than 1980 in all cases, can be kept in the sample if they are still being used or if the most recent laws proposed for the same region digress too much from the usual formalism of description as regards the choice of variables or mathematical expression. This last criterion shows that the constitution of the sample is mainly subjective in nature. In order to facilitate comparisons, only the laws that fall into the same functional framework as regards the relationship between dependant variable and independent variables have been retained; this framework corresponds to the following formula:
Strong Vibratory Motions
1/ k -J ª« R k ( k 1) r º» B DM ª k G M º H, ¼ P=ce « R re e ¬ e » ¬
¼
189
[4.16]
where P is the parameter of predicted motion (dependant variable, i.e. acceleration, velocity or displacement) M the magnitude, R the distance, I an index with a value of 0 or 1 as per the type of fault or the type of soil; the 8 numerical coefficients D, E, J, G, H, c, r and k are the characteristics of each of the laws. Expression [4.16], which can seem very complicated, is in fact deduced from basic considerations on the mathematical structure of the attenuation laws that are the most frequently used; its starting point is the form of the simplest laws, written as: P
D M E
ce
R
[4.17]
An example of this type of law was used in section 2.3.1 where equation [2.49] 1 Ln 10 (which is a result of relations [2.7] and corresponds to the values D = 2 [2.11] defining RO and MO respectively) and E = 1 (purely geometrical attenuation in I/R of the volume waves); we go from form [4.17] to form [4.16] with the help of the following three operations: – addition of an exponential factor of non-elastic attenuation to correct the geometrical attenuation in I/R, coefficient J of this factor corresponds to quotient Z/(2Qc] which multiplies x in the argument of the exponential of equation [3.55]; – correction of the distance R by adding a constant term or a term dependant on the magnitude so as to correct the infinite increase of P when R tends towards zero in [4.17] and to consider the size of source effect; this additive term is constituted by coefficients r and G; coefficient k which also appears in the corrected distance only has two possible values, 1 or 2; it has been introduced so as to adapt to the formalism of certain laws which treat the distance as the hypotenuse of a rightangled triangle of which R is one of the other sides, with the choices k = 2 and E = ½; – addition of a multiplicative factor eHI so as to consider the influence of the type of fault (I = 0 for normal faults and strike-slip faults, I = 1 for reverse faults) or of the nature of the terrain (I = 0 for rock, I = 1 for soil). All the 24 laws retained correspond to horizontal motion, as the laws for vertical motion, which are very few in number, are hardly used in practice since it is sufficient to define vertical seismic action as a fraction of horizontal seismic action
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(see section 4.1.3). There are 18 laws for acceleration, 4 for velocity and 2 for displacement, whose authors, year of publication and the geographical zone for which they were established are given in the following list: 1) laws for acceleration: – Abrahamson-Litehiser 1989 (abbreviated as (AL 89)); zone 1 (all the regions of the world where seismicity has an intraplate character), see [ABR 89]; – Dahle – Bungum – Kvamme 1990 (DBK 90): zone 1, see [DAH 90]; – Joyner Boore 1988 (JB 88); zone 2 (west of the USA), see [JOY 88]; – Mohammadioum – Pecker 1993 (MP 93); zone 2, see [MOH 93]; – Nuttli – Hermann 1984 (NH 84); zone 3 (east of the USA), see [NUT 84]; – Hwang – Huo 1997 (HH 97); zone 3, see [HWA 97]; – Dahle – Climent – Taylor – Bungum – Santos – Ciudad Real – Lindholm – Strauch – Segura 1995 (DCT 95); zone 4 (Mexico and Central America); see [DAH 95]; – Martin 1990 (M 90); zone 5 (Chile); see [MAR 90]; – Ambraseys – Bommer 1991 (AB 91); zone 6 (vast region stretching from the west to the east of Iceland and to Pakistan, passing through the Maghreb (North Africa), Spain, Italy, the Balkans, Greece, Turkey and Iran which is called the “Alpide Belt” by these authors; see [AMB 91]; – Petrovski 1986 (P86); zone 7 (Balkans, Greece); see [PET 86]; – Theodulidis – Papazachos 1992 (TP 92); zone 7; see [THE 92]; – Tento – Franceschina – Marcellini 1992 (TFM 92); zone 8 (Italy); see [TEN 92]; – Xu – Shen – Hong 1984 (XSH 84); zone 9 (North China); see [XU 84]; – Peng – Wu – Song 1985 (PWS 85); zone 9; see [PEN 85]; – Fukushima – Tanaka 1990 (FT 90); zone 10 (Japan); see [FUK 90]; – Iai – Matsunaga – Morita – Sakurai – Kurata – Mukai 1993 (IMM 93); zone 10 ; see [IAI 93]; – Mc Verry – Dowrick – Cousins – Porritt 1993 (MDC 93); zone 11 (New Zealand]; see [MCV 93]; – Munson – Thurber 1997 (MT 97); zone 12 (Hawaii); see [MUN 97].
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191
2) laws for velocity: – Joyner – Boore 1988 (JB 88); zone 2; see [JOY 88]; – Petrovski 1986 (P 86); zone 7; see [PET 86]; – Theodulidis – Papazachos 1992 (TP 92); zone 7; see [THE 92]; – Xu – Shen – Hong 1984 (XSH 84); zone 9; see [XU 84], 3) laws for displacement: – Petrovski 1986 (P 86); zone 7; see [PET 86]; – Theodulidis – Papazachos 1992 (TP 92); zone 7; see [THE 92]. As can be seen in the list, in order to apply the first criteria of choice as mentioned above, we have tried our best to cover the main seismic zones of the world but there are nevertheless shortcomings due to the absence or insufficiency of data in regions that present high seismic hazards such as the North-West of South America (Venezuela, Colombia, Ecuador, Peru, Bolivia), the Caribbean islands or certain countries of South-East Asia (Burma, Indonesia, Taiwan, Philippines). Specific attenuation laws are also missing in most of the regions with moderate seismicity (especially in Western Europe outside the alpine zone), where we must, for lack of anything better, use the “imported” laws, the representativeness of which can be debated, or use “all purpose” laws such as those in zone 1 (intraplate domain). Table 4.6 presents for these 24 laws, the values of the coefficients of formula [4.16] (in cm/s² for acceleration, cm/s for velocity, cm for displacement and km for distance), the type of definition adopted for magnitude and distance, the ratio standard deviation/mean value and, in the last column, the values calculated with these laws in the case M = 6, R = 25 km.
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Seismic Engineering P Z
c
D
k
G
r
E
Jx103
H
M, R
V/m
M=6 R=25
a
1
235.3
0.408
1
1.000
0.284
0.982
0
0.304
SL,p
0.892
DBK 90
a
1
23.0
0.849
1
0
0
1
4.18
0
S, h
1.293
135
JB 88
a
2
110.0
0.530
2
64.00
0
0.5
6.20
0
w, p
0.905
85.6
MP 93
a
2
108.6
0.400
1
0
0
0.712
0
0
SL, h
0.866
121
NH 84
a
3
3.72
1.15
2 3.46x10-4
2.10
0.415
1.59
0
b, e
0.733
230
HH 97
a
3
53.8
0.926
1
0.060
0.700
1.271
3.02
0
w, e
DCT 95
a
4
20.6
0.554
2
36.00
0
0.280
3.02
0.325
w, h
M 90
a
5
71.3
0.830
1
60.00
0
1.030
0
0
s, h
AB 91
a
6
132.3
0.500
1
0
0
1
2.69
0
s, p
P 86
a
7
654
0.544
1
20.00
0
1.333
0
0
SL, h
TP 92
a
7
72.9
1.120
1
15.00
0
1.650
0
-0.41
S, e
AL 89
I sol
I sol
0.362 1.075 1.181 0.905 0.958 1.034
TFM 92
a
8
113.3
0.520
1
0
0
1
2.16
0
L, h
XSH 84
a
9
151.9
0.544
1
8.00
0
1.002
0
0
SL, e
PWS 85
a
9
0.336
1.411
1
0
0
0.873
4.74
0
S, e
FT 90
a 10
20.0
0.944
1
0.032
0.944
1
7.83
0
S, g
IMM 93
a 10
4.31
1.128
1
0
0
1
3.98
0
MDC 93
a 11
213.6
0.493
1
0
0
1
0.875
0
JMA, e
MT 97
a 12
15.4
0.891
2
127.50
0
0.5
5.86
0
w, g
0.888
SL, p
0.888
w, p
1.138
JB 88
94.9 129
I fault
v
2
0.141
1.128
2
16.00
0
0.5
5.6
0.394
0.954 0.429 0.622
I sol
179 85.9 119 107 99.4 107 137 91.2 97.2 120 85.4 139 136 161 100 4.20 6.23
P 86
v
7
4.430
0.936
1
20.00
0
1.379
0
0
TP 92
v
7
0.364
1.410
1
10.00
0
1.620
0
0.221
SL, h
1.103
6.39
1.226
5.42
S, e
I sol XSH 84
v
9
0.142
1.371
1
2.00
0
1.286
0
0
P 86
d
7
0.060
1.288
1
20.00
0
1.343
0
0
TP 92
d
7
0.001
2.080
1
5.00
0
1.850
0
0.966
6.76 7.66
SL, e
I sol
SL, h S, e
Table 4.6. 24 attenuation laws converted into the same form: 1/ k ª º E -J « Rk (k 1)r » ª k º M D M G ¬ ¼ P= c e R re e eH , « ¬
» ¼
1.330
0.82
2.421
0.49 1.28
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– column 1: abridged denomination of the law as per the list; – column 2: predicted parameter P: a acceleration (cm/s²), v velocity (cm/s), d displacement (cm); – column 3: index of the geographical zone according to the list; – column 4 to 11: values of the coefficients D, E, J, G, H, c, r and k;
– column 12: type of magnitude L (local), b (volume waves), S (surface waves), W (moment waves), JMA (Japan Meteorological Agency) and type of distance R (in km) epicentral (e), hypocentral or focal (h), at the fault (g), at the surface projection of the fault (P); – column 13: ratio of the standard deviation V to the mean value m; – column 14: values calculated for M = 6, R = 25 km; for the laws having a nonzero value of coefficient H the first value indicated corresponds to I = O, the second to I = 1; the significance of index I (type of fault or soil) is recalled in the first column. An examination of this table shows that the numerical values of the coefficients of equation [4.16] can vary considerably from one law to another. As for attenuation, the laws of the table can be arranged in three categories: – those that are aligned on a theoretical model of decrease in 1/R (volume waves) with an exponential factor of inelastic attenuation; they correspond to the choices E = 1, k = 1 or E = 0.5, k = 2 with, in the two cases, J>o; this pertains to laws (DBK 90, JB 88, AB 91, TFM 92, FT 90, IMM 93, MDC 93, MT 97) for acceleration and JB 88 for velocity; it is interesting to note that factor J is extremely variable (from 0.875 x 10-3 for MDC 93 to 7.83 x 10-3 for FT 90), which reflects the differences that are seen in non-elastic attenuation; for the sake of comparison, by taking the values of the PREM model (see section 3.2) for the layer located between 3 and 15 km of depth as 3.20 km/s for the speed of propagation of the S waves and 600 for the factor of quality, formula J = Z/[2Qc] gives J = 4.91 x 10-3 if we assume that Z = 6 S rd/s, i.e. a frequency of 3 Hz which is at a plausible magnitude for the dominant frequency of an accelerogram (see Table 4.2); – those that have a non-zero J but do not limit E to a value of 1 (if k = 1) or 0.5 (if k = 2); it concerns laws NH 84, HH 97, DCT 95, PWS 85 for acceleration; the product kE remains quite close to 1 except for DCT 95 where it is hardly higher than 0.5, which would be the value of surface waves; the coefficient J has the value 1.59 x 10-3 for law NH 84, which is a low value (higher however than 0.875 x 10-3 from MDC 93) and reflects the difference in attenuation between the East and West USA (section 3.2.3) where the law JB 88 gives J = 6.20 x 10-3 for acceleration;
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– those that have a zero J , i.e. AL 89, MP 93, M 90, P 86, TP 92, XSH 84 for acceleration, P 86, TP 92, XSH 84 for velocity, P 86, TP 92 for displacement; the values of E have an average that is close to one, while TP 92 is very noticeably different. The additive term r e GM for distance is seen in the majority of the laws but in most cases with G = o; in four cases alone (AL 89, NH 84, HH 97, FT 90) where G z 0 and the corresponding values of G are varied (from 0.284 to 2.10), law FT 90 shows a remarkable feature: that of having G = D, which in turn means that acceleration at zero distance from the source is independent of the magnitude, this being coherent with the notes attached to Table 2.1 (see section 4.1.1); This equality is more due to a choice made by the authors rather than an adjustment by regression. The laws in which r z 0 and G = 0 can be interpreted in terms of the average of the focal depths which is equal to r if k = 1 and to r if k = 2; the values calculated in this manner range from 4 to 20 km, except in the case of M 90 where the depth of 60 km is the result of the impact of subduction zone earthquakes for this Chilean law. It should be noted that the additive limit of distance does not have the same expression in equation [4.16] depending on whether it is applied to the geometric factor or the inelastic factor of attenuation; it contributes to the latter only if k = 2 and does not depend on the magnitude; this independence from the magnitude in inelastic attenuation seems rather logical as in that case it is the length of the distance traveled by the waves (whose average value is the distance from the center of the source) that becomes the decisive factor, whereas as in geometric attenuation, the size of the source has a remarkable effect in very near zones (see section 2.1.3). Apart from the very near zone, where the additive limit of distance plays an important role, the influence of magnitude is essentially seen through factor eDM; in the attenuation laws of velocity, this factor corresponds to the equivalent radius RO of the source that appears in formula [2.49]; as shown above, coefficient D must have a value ½ Ln 10 = 1.15 that is actually close to those of D in the table of the laws of velocity; for the laws of acceleration, the “theoretical” value of D is ¼ Ln 10 = 0.576 (see section 5.1.3), which is not too far from the values of the table in more than half of the cases (AL 89, JB 88, MP 93, DCT 95, AB 91, P 86, TFM 92, XSH 84, MDC 93); the other half correspond to values of D that are much higher, often higher than 1 (NH 84, TP 92, PWS 85, IMM 93). Coefficient H is not equal to zero for 6 of the laws of which one alone (AL 89) brings out the influence of the type of fault and confirms the commonly assumed hypothesis (see section 2.1.3), according to which reverse faults (I = 1) produce motions that are, on average, more violent than normal faults or strike-slip faults (I = 0). The 5 cases where index I corresponds to the geotechnical conditions of the site (I = 0 for rock, I = 1 for soil) shows the amplification by the soil (referred to in
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195
section 3.2) with the exception of law TP 92 for acceleration (but the laws of velocity and displacement established by the same authors confirm the general tendency towards amplification). Of course, such an index only gives a very brief characterization of site conditions and was retained by the authors simply because the knowledge of soil conditions beneath the recording stations did not enable a better description. The difference in the coefficients of the values of attenuation laws are due to the characteristics of the recording databases used by the authors; these often have shortcomings (lack of data in a particular range of magnitudes or distances), uncertainties that could be significant (values of magnitude and location of seismic sources) and other indirect effects (for example, due to the coupling of two parameters that should have been independent, like magnitude and distance). It is hence quite uncertain whether the differences observed in the rapidity of attenuation, for example, reflect tendencies that are significant. The column V/m of the table shows the law of “standard deviation” referred to previously in the introduction to this second part. The ratio of standard deviation to the average (known for all the laws except XSH 84 and IMM 93) is on average about 0.874 for acceleration, 1.156 for velocity and 1.875 for displacement. Law HH 97 remains different with a very low value (V/m = 0.362) that is probably due to the specificity of the method used to write it; to be more specific, its authors tried to compensate for the insufficient number of high magnitude recordings in the eastern part of the USA by completing the database through digital simulations, the parameters of which were adjusted based on the few real signals available; it is possible that this procedure underestimated the variability in seismic phenomena. Very high values of V/m for the last two rows of the table (especially TP 92) confirm the low degree of reliability of displacement calculations (see section 4.1.2). The last column of the table contains the values arrived at with different attenuation laws for M = 6 and R = 25 km. The choice of this pair of parameters is due to the following reasons: – in the case of all the laws, these parameters correspond to the central zone of the domain of validity for which the laws were written; – for M = 6, the difference between the types of magnitude is not very significant (see section 2.3.2 and Figure 2.13); – for R = 25 km, the size of an earthquake of magnitude 6 (R0 = 3.98 km; see Table 2.1) is small enough so that all the different possible definitions of R (see Figure 4.5) are equal, at least for an earthquake whose focus is not very deep (less than 10~15 km);
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– M = 6, R = 25 km is a valid value for the level of seismic hazard possible in metropolitan France (see section 6.2.2). The results shown in the last column were obtained by taking I = 0 and I = 1 for laws where H is not equal to zero. In order to facilitate comparison the values corresponding to I = 0 have been retained in Table 4.7 and the entire list of values has been organized in ascending order with an indication of their relation to the average. A
A
[cm/s²]
mA
PWS 85
85.4
0.694
JB 88
85.6
0.696
DCT 95
85.9
0.698
AL 89
94.9
0.772
TFM 92
97.2
0.790
AB 91
99.4
0.808
MT 97
100
0.813
M 90
107
0.870
P 86
107
XSH 84
LAWS
LAWS
V
V
LAWS
D
D
[cm]
mD
[cm/s]
mv
JB 88
4.20
0.709
0.870
TP 92
5.42
0.916
TP92
0.49
0.748
120
0.976
P 86
6.39
1.08
P86
0.82
1.25
XSH 84
7.66
1.29
MP 93
121
0.984
DBK 90
135
1.10
IMM 93
136
1.11
TP 92
137
1.11
FT 90
139
1.13
MDC 93
161
1.31
HH 97
179
1.46
NH 84
230
1.87
Table 4.7. Values of acceleration (cm/s²), velocity (cm/s) and displacement (cm) calculated for M = 6, R = 25 km, I = 0 with the 24 attenuation laws of Table 4.6; average values are mA = 123 cm/s² for acceleration, mv = 5.92 cm/s for velocity, mD = 0.655 cm for displacement
It is evident that the variability is quite high and reflects the strong influence of the geographic zone; the two laws for the east of the USA are those that lead very distinctly to the highest values of acceleration (NH 84 and HH 97); it is possible that
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197
this trend is the result of very high values for the stress drop for earthquakes in this intraplate zone (see section 5.1.3); it is also seen that the two Japanese laws (FT 90 and IMM 93) give results that are a lot higher than those given by laws established for the entire western coasts of North and South America (JB 88 and MP 93 for the West Coast of the USA, DCT 95 for Central America, M 90 for Chile). However, noticeable variability is also seen in the same zone, as we see when we compare for example, PWS 85 and XSH 84 for North China, or JB88 and MP93 for the West Coast of the USA. Thus, for a correct application of attenuation laws, more than one law is required. 4.2.3. Recommendations for the use of attenuation laws
Attenuation laws, whether deterministic or probabilistic in nature, are extremely important in the study of seismic hazard. Thus, the choice of the same must be a very careful process after consideration of the recommendations given below, some of which have already been touched upon in the preceding section: 1) As far as possible, laws established for tectonic environments similar to that of the site being studied must be selected; it is therefore necessary to consult the database of a law whose use is required; this allows us to eliminate laws whose database is poor as regards ranges of magnitude and distance that are appropriate for the site or where the proportion of earthquakes showing characteristics (type of fault motion, focal depth) significantly different from those under study is too high. 2) Referring to published articles pertaining to these laws that is a prerequisite to consulting the data base referred to in 1) allows us to verify the applicability of the law from the point of view of the extent of its domain of validity, to verify the types of magnitude and distance used and to benefit from the author’s comments. 3) As indicated at the end of section 4.2.2, the use of a single law is not advisable but the criteria for selection described in 1) and 2) are such that they bring the acceptable number of laws down to just two or three, especially in zones of moderate seismicity, where the expected magnitudes would never exceed 6 or 6.5 for the normal levels of hazard. 4) The choice of the distance parameter may represent a few loopholes. In studies of seismic hazard, the seismic source is quite often assumed to be very close to the site; for example, in the case of deterministic methods where “some earthquakes are considered to be under the site” (see section 6.1.2); choosing to consider the epicentral distance (uncompensated by the additive limit of distance) is thus inappropriate; that of the focal distance seems to resolve this problem but difficulties could arise as regards the conditions of the fixing of the lower limit for focal depths, which is often highly arbitrary. Other types of difficulties arise when
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we consider focuses situated at great depths (more than or equal to 40~50 km) in the sense that the majority of laws show extremely low results. Let us thus accept that definitions of distance that require a precise knowledge of the fault (distance Rg to the closest point, distance RP at its surface projection (see Figure 4.5) are often incompatible with the quality of data for the study of seismic hazard. These problems connected to the distance are seen clearly in deterministic methods but may be very easily hidden in probabilistic methods when “black box” software is used. 5) It goes without saying that the importance of standard deviation must not be forgotten while interpreting the results of attenuation laws. Figure 4.6 shows that the range of recordings for the same earthquake may represent more than three times the value of standard deviation, i.e. a ratio almost equal to 10 for extreme values (accelerations recorded at a distance of about 70~80 km for the Loma Prieta earthquake vary between 0.04 g and 0.3 g). The example used in this figure is not an exception; in fact, every time an earthquake takes place in a zone that is wellequipped with seismographs for high intensity motions (as in Northridge in 1994, 1995 and in Chi-Chi in 1999) we observe a scattergram that is as high at a certain distance from the source (in the case of Loma Prieta in Figure 4.6) as it is near the fault (as in the case of the epicentral zone of the Chi-Chi earthquake shown in Figure 4.7); this last example is interesting because it illustrates the dissymmetry of the fronts separated by the fault when it has an oblique dip.
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199
Figure 4.6. Comparison of the accelerations recorded during the Lorna Prieta earthquake on 17 October 1989 using the Joyner and Boore law (JB 88), average [m] and average + standard deviation [m + V] (ref: [COL 89])
In Figure 4.7, we see that values higher than 0.40 g correspond essentially to points located along the trace of the fault (0.78, 1.01, 0.42, 0.79, 0.52, 0.42 and 0.57 from south to north) or in the vicinity of the epicenter to the east of the fault (0.45, 0.59, 1.01, 0.66, 0.60, and 0.49); apart from these two zones, a single point is to be found at 0.41 g to the west (close to the southern extremity of the fault) and another at 0.64 g in continuation of the fault in the south, whereas the majority of stations were located to the west (where there are several urbanized areas while the zone of the epicenter, located in the first foothills of the chain of mountains has a much lower population density). This dissymmetric effect is only reflected in the attenuation laws when they utilize the focal or epicentral distance but cannot be observed if the distance is equal to the surface projection of the fault. This observation makes it clear that the use of the latter type of distance is not recommended, especially in seismic hazard studies of regions where the geometry of the faults is not well known (regions of moderate seismicity).
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Figure 4.7. Horizontal accelerations recorded (in g) for the Chi-Chi (Taiwan) earthquakes on 21 September 1999; the values are much higher in the thrust front (to the east of the trace of the fault) than in the thrusted wall (to the west); The dip of this reverse fault is about 30° towards the east [COL 99c]
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201
A large number of high level recordings for the same earthquake are possible only in an area that is well equipped with seismological instruments and this is only the case in a few of the recent earthquakes. In most cases, information relating to the strong motions of an earthquake, obtained from instruments, relies on a very small number of recordings, (often a single one) for the given earthquake. And this is how the myth that an earthquake can be described by the acceleration value recorded somewhere in the epicentral zone was propagated. Such an idea seemed “natural” and logical at a time when we had not yet become aware of the variability of seismic motions but can no longer have any credibility whatsoever today. 4.3. Directivity effects and site effects 4.3.1. Inadequacy of a description based on magnitude and distance
Though the simplest attenuation laws use it as a working hypothesis, the two examples in 4.2.3 show clearly that seismic motion at a given site cannot be defined on the basis of distance and magnitude alone. The reasons for this are many and have already been mentioned: – the complexity of the process of rupture that cannot be described on the basis of a single parameter such as magnitude; – the simplistic nature of an attenuation model that shows wave propagation using the sole parameter of distance; – the importance of site conditions (geological nature of terrain and topography) whose influence on surface motion is evident but which have been integrated only in a very cursory manner by index I which serves as the basis to some of the attenuation laws. It is of course possible to conceive more elaborate models of attenuation laws by introducing parameters such as stress drop, geometric properties, type of fault motion, the azimuth of the site with respect to fault planes, mechanical properties, the thickness and dip of the superficial layers. Some experiments have been carried out along these lines but without a great degree of success to date, mainly because these supplementary parameters can only be determined by assuming arbitrary hypotheses. Information in the available databases very rarely includes detailed descriptions of the land for recording stations and such information is very often incomplete as regards the faults causing the earthquakes that have been recorded. For a user of an attenuation law who would like to consider parameters such as the stress drop or the angle of azimuth, it would normally be difficult to make an informed choice as regards the values of this. The objective of section 4.3 is to present, along with the phenomena leading to the inadequacy of descriptions based on magnitude and distance, those that can be directly observed such as directivity or
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site effects and to analyze the possible causes. Phenomena related to the process of rupture requiring calculation models shall be discussed in Chapter 5. 4.3.2. Directivity effects
Attenuation laws that use the parameter of distance at a single point (focus, epicenter, center of release of energy; see Figure 4.5) produce circles as isoseismal lines of the predicted motion parameter. Those that use the shortest distance from the site to the surface (fault plane or its surface projection) have isoseismal lines in the form of rectangles with rounded corners. Figure 4.8 shows the different possibilities.
Figure 4.8. Isoseismal lines of the motion parameter predicted by attenuation laws according to the nature of their distance parameter: from the focus or the epicenter (case a), from the surface projection of the fault plane (cases b and c); a) concentric circles around the epicenter; b) a rectangle with rounded corners around the fault projection P1P2P3P4; c) extreme case of b) for a vertical fault
The analysis of recordings and damage caused confirm the validity of these isoseismal lines only in a certain number of well-documented earthquakes, even if we agree to ignore local irregularities produced due to site effects. The tendency towards circular attenuation (case a) is often confirmed for relatively deep earthquakes if their magnitude is not very high whereas the long isoseismal lines “representing the fault” (case c) are generally better suited to big strike-slip faults (for which the reference to the epicenter, which could be situated close to one of the extremities of the fault as assumed in Figure 4.8, is inappropriate). Apart from these
Strong Vibratory Motions
203
cases, we often find that isoseismal lines observed differ considerably from those shown in diagrams a, b, or c in Figure 4.8, showing clearly the amplifications of motion in zones situated along the fault lines. These amplifications are due to the manifestation of directivity effects in the motion and propagation of seismic waves. One simple explanation, similar to that of the Doppler effect in sound technology, can be proposed for the directivity effect associated with the direction of the propagation of rupture on the fault plane, as has been specified in section 1.2.4. We consider a vertical fault (see Figure 4.9) with trace AB on the surface; rupture begins at A and propagates towards B with a rupture velocity vr slightly lower than the velocity of the propagation of shear waves vs (see section 1.2.4).
Figure 4.9. Wave path of waves received at M; the first waves [AM = [AB = 2a] for a rupture propagating from A to B
A 1] and last ones
At a given point M, situated at a distance r and with an azimuth T with respect to the fault, the first shear waves make their appearance after time t1 = A 1/vs required for their propagation after emission at A; the last waves received by M are emitted at B, on termination of rupture on the fault they reach M after a time t2 given by: t2 =
2a
l + 2 Vr Vs
[4.18]
The duration T of the motion observed at M is equal to t2 – t1, i.e.: T=
l l + 2 1 Vr Vs 2a
[4.19]
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Seismic Engineering
Thus, with basic calculations of A 1 and A 2, allowed in this case with the objective of simplification, where the half-length a of the fault is small in comparison with the two other lengths: T=
2a Vr
[1 – E cos T] with E =
Vr Vs
[4.20]
Formula [4.20] shows that duration T will be less than the rupture time 2a/Vr if the angle T is lower than 3/2 and higher in the opposite case. Thus, a decrease in duration must be accompanied by an increase in the level of motion, as waves tend to accumulate. If we assume that the increase is inversely proportional to the duration, we obtain for the motion parameter P (velocity of acceleration), with respect to the geometric attenuation of volume waves: P = P0
R0
1
r
1 E cos T
[4.21]
R0 being the equivalent radius of the fault and P0 being the value of the motion parameter in the rupture zone. Formula [4.21] shows that amplification of motion is considerably high for points “targeted” by rupture propagation since the ratio E = Vr/Vs is only slightly lower than one in a large majority of cases (see section 1.2.4). For example, for E = ¾, the value of P is 4 times greater in front of the rupture (T = 0) than in the direction unaffected by the directivity effect (T = S /2); behind the rupture (T = S ), there is attenuation in the ratio 4/7 = 0.57. This “Doppler” type of directivity effect influences the shape and area of the isoseismal curves. Let us consider a focus F (see Figure 4.10) located at depth h and a direction of propagation of rupture at an angle ) with the horizontal plane.
Strong Vibratory Motions
205
Figure 4.10. Axes and notations for the calculation of the directivity effect; plane x0y corresponds to the surface of the ground, the axis 0z is directed towards the depth and contains focus F [z = h], direction ' of the rupture makes an angle ) with the horizontal plane
Angle ) may coincide with dip G of the fault plane for normal or reverse faults, where the rupture moves from the deepest part to the surface. In the case of strikeslip faults, ) is close to 0, while the dip is often almost vertical (see section 1.2.3). At a point M on the surface, the value of motion parameter P, given by formula [4.21], r being the distance FM and T the TFM angle. The isoseismal lines of P, in plane x0y, are defined by the equation: P r [1 – E cos T ] = R with R = R0 0 P
[4.22]
½ As we G JJJhave G r = [x² + y² + h²] and r cos T = x cos I + h sin I (scalar product '.FM ), after rearranging the terms of the equation, we write it as:
( x xc )² a²
+
y² b²
=1
[4.23]
with: xc =
E cos I 1 E ² cos ²I
(R + E h sin I)
[4.24]
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Seismic Engineering
a=
1 1 E ² cos ²I
[R² + 2 E R h sin I – h² (1 – E ²)]1/2
1
b=
1 E ² cos ²I
[R² + 2 E R h sin I – h² (1 – E ²)]1/2
[4.25]
[4.26]
i.e. the equation of an ellipse with foci x = xc, y = 0, with semi-major axis a and semi-minor axis b. The area S of the isoseismal curve is therefore: S=
S
S
ab=
(1 E ² cos ²I )
3/ 2
[R² + 2 E R h sin I – h² (1– E²)]
[4.27]
The value of R where S = 0 corresponds to the highest possible value Pm of parameter P reached at abscissa point xm of axis 0x; a basic calculation gives us: Pm = Po
xm =
Ro h
1 1 E cos 2 I E sin I
hE cos I 1 E ² cos ²I
2
[4.28]
[4.29]
The factor PoRo/h that appears in the formula for Pm is the value P’m that would be obtained at the epicenter (distance h from the focus) in the absence of the directivity effect. Table 4.8 shows the variations of the ratios Pm/P’m, xm/h and b/a with respect to the angle I for the two values E = 2/3 and E = 4/5.
Strong Vibratory Motions E = 2/3
I
E = 4/5
Pm ' Pm
Xm
b
h
0
1.342
10
Xm
b
a
Pm ' Pm
h
a
0.894
0.745
1.667
1.333
0.600
1.566
0.870
0.754
2.097
1.279
0.616
20
1.813
0.804
0.779
2.592
1.140
0.659
30
2.070
0.707
0.816
3.114
0.961
0.721
40
2.319
0.594
0.860
3.623
0.776
0.790
50
2.546
0.474
0.904
4.085
0.599
0.858
60
2.736
0.354
0.943
4.470
0.436
0.917
70
2.880
0.234
0.974
4.760
0.284
0.962
80
2.970
0.117
0.993
4.939
0.140
0.990
90
3.000
0.000
1.000
5.000
0.000
1.000
(°)
207
Table 4.8. Variations of Pm/P’m, xm/h and b/a with respect to I (degrees) for E = 2/3 and E = 4/5
Table 4.8 shows that the ellipse is quite close to a circle since the ratio b/a is only slightly less than one, but the point at which the motion is maximum, moves with respect to the epicenter except for the vertical propagation of rupture I = 90°; for low values of I, this shift is similar to the focal depth.
Figure 4.11. Isoseismal lines for the same value of the motion parameter; the circle corresponds to the absence of the directivity effect and the ellipse to the case
E = 2 , h = 0 and I = 0 3
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Seismic Engineering
A Doppler type of directivity effect also leads to a visible increase in the “damage potential” of the earthquake, i.e. the area in which the motion parameter exceeds a certain given level is clearly larger than in the absence of this effect, all other factors being equal. Figure 4.11 allows us to visualize such areas in the case of a very superficial earthquake (we shall disregard h before R in formulae [4.24], [4.25] and [4.26]); a circle of radius R centered on the epicenter E contains points where the motion parameter P is higher than P0R0/R when we do not take into account the directivity effect; the ellipse centered on C corresponds to the same criteria of exceedence when the directivity effect is taken into account; the ratio of the surfaces is given by: surface of the ellipse surface of the circle
= [1 – E²]-3/2
[4.30]
The value of this ratio is 2.41 for E = 2/3 (value retained in order to draw Figure 4.11) and goes up to 4.63 for E = 4/5. There is therefore a noticeable increase in the extent of areas that are affected. Formula [4.30] relates to the case of a horizontal propagation of the rupture (I = 0, that corresponds to a strike strip fault); the effect of the increase in the affected areas also exists though the effect is less marked, for a vertical propagation (I = 90°). The isoseismal lines in this case are then circles as in the case where the directivity effect is not taken into account, but the amplification in the direction of the propagation of the rupture (factor 1/(1-E); see Table 4.8) produces a dilatation of these circles for a given level of motion. Using equation [4.21] we can easily show that the ratio of the area of the surfaces in which the motion parameter is higher than P’m/2 (P’m being the epicentral value without directivity that has already been introduced above) is given by: surface of the circle with directivity surface of the circle without directivity
=1+
1 3
E [E + 2]
[4.31]
This is equal to 1.59 where E = 2/3 and 1.75 where E = 4/5, showing a less noticeable increase than in the case of horizontal propagation, but an increase that is nevertheless significant. The influence of angle I can be observed in Figure 4.12, which shows the variation of the ratio P/P’m on axis 0x, with respect to the lower abscissa x/h. The curves have been shown for E = 2/3 taking the three values 0°, 45° and 90° for angle I. We can see that for the vertical propagation (I = 90°) we obtain a very accentuated peak in a rather small area just above the fault and a rapid attenuation when we move away. In the case of an oblique propagation (I = 45°), we observe a
Strong Vibratory Motions
209
rise and a fall in the peak, with the appearance of dissymmetry, tendencies that can also be observed in the case of horizontal propagation (I = 0). The previous calculations relating to the directivity effect are based on an extremely simplified model of the phenomenon of propagation of the rupture that is supposed to occur in a direction and at a velocity that are constant on the total fault plane, and on the hypothesis that the amplitude of motion is inversely proportional to its duration. This simplistic approach is undoubtedly far removed from the actual conditions of rupture whose complexity has already been emphasized. It is therefore probable that the above formulae will give an overestimated evaluation of the directivity effect. However, the reality of the latter is undeniable, as can be seen in the analysis of some recent earthquakes.
Figure 4.12. Variations of P/P’m on the x axis for E = 2/3 and the 3 values 90°, 45° and 0° for angle I. The straight line of the ordinate P/P’m = ½ corresponds to the hypotheses made in formula [4.31] for vertical propagation [I = 90°]
The simplified hypotheses that have been adopted in this section are less debatable for components with low frequency of seismic motion that are associated with large ruptures, than for those with high frequency that result in small-scale ruptures whose distribution over the fault plane and sequence of eruptions are random in nature. The directivity effect should thus be greater for motion parameters influenced by low frequency (displacements and velocity) than for those that essentially translate the effect of high frequency (accelerations). This is what is observed in the killer
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Seismic Engineering
pulse phenomenon that is referred to at the end of section 4.1.3; this oscillation of high amplitude at low frequency was clearly identified in the north of the fault in the case of the Chi-Chi earthquake (see Figure 4.7) where the velocity recorded was much beyond the m/s (up to 3.28 m/s, see [COL 99c]) whereas the accelerations were in the range of 0.4~0.6 g. 4.3.3. Presentation of site effects
The effects of the amplification of seismic motion by superficial layers of land having poor mechanical characteristics have been touched upon briefly in section 3.2.3, mainly to dispel the idea that the damping property of the soil decreases the amplitude of the shocks. We saw in [3.58] and [3.59] that even though this damping effect does exist, it does not compensate the amplification that is due to the impedance contrast between the bedrock and the soil, at least as long as the layer of soil is not very thick. The persistence with which this idea is propagated is rather surprising given that it was proved wrong quite some time ago by perceptive observers. H. Tiedemann tells us [TIE 92]] that after the earthquake in Istanbul on 12 September 1509, a decree was announced by Sultan Bayazid II, prohibiting all reconstruction in areas with soft soil along the coast; his advisors could have taught our so-called modern experts quite a few things! Without going so far back in time, the detailed study of the damage caused by most of the old earthquakes clearly shows the influence of the nature of the soil and reveals almost systematically the relation between the presence of soft soil and the highest damage. We call site effects the changes in the seismic motion in the surface soil that are apparently due to the geotechnical conditions of the local topography of a given site with respect to the motion observed at a neighboring site and to certain reference conditions (rock outcroppings along a horizontal surface). This definition must be accompanied by the following observations: – the name “site effect” in itself implies that the changes observed in the motions are solely due to the local conditions of the site; in reality, other causes related to the characteristics of the incidental wave field (dominant frequency, types of waves, focusing phenomena) could also play a role and thus the use of the adverb “apparently” in the definition; – the reference to these local conditions brings up the question of the scale used to identify the local character; in earthquake engineering, the natural scale is that of the dimensions of the concerned construction, i.e. a few tens of meters in most cases. This scale is sufficient to describe the site effect due to the impedance contrast
Strong Vibratory Motions
211
between a superficial layer of average thickness and the subjacent bedrock; it is not adequate when the changes in the motion result from the collective response of big geotechnical structures (resonance of an entire sedimentary basin or the trapping of waves within a mountainous mass; see Figure 3.13), for which the scale to be considered is kilometric; – the definition assumes the existence of the reference site (horizontal rock outcropping) in the vicinity of the site being studied, that can obviously not be guaranteed in many of the cases; even when the requisite conditions exist (horizontal rock outcropping in the vicinity), in order to be able to make numerical comparisons, we must have available recordings (or record of damage) on the two sites; if this data is not available for the reference site, the characteristics of the motion on the site being studied may not be due to the manifestation of a site effect, but may correspond to a directivity effect;
– the characterization of the reference site lacks a little in precision because the superficial rocks present very variable degrees of alteration and fissuring, which has a strong influence on their mechanical properties, particularly the velocity of the propagation of shear waves; we can assume that this velocity must at least be equal to 800 m/s so that it can be called a reference site, but it is not certain that there is total equivalence between one reference site at 800 m/s and another at 2,000 m/s. The changes in motion that constitute the site effects are different in nature: – change in the amplitude of the signal, most often in the form of an amplification that corresponds quite commonly to a factor ranging from 2 to 3, but that can attain values that are much higher; this ratio of amplification is not necessarily the same for velocity and displacement as for acceleration; it may depend on the level of motion due to nonlinear effects as indicated here below; – appearance of a dominant frequency in the signal that is related to the characteristics of the site and no longer to those of the source; – lengthening of the duration of the signal, particularly in big sedimentary basins where we often observe the creation of surface wave forms that travel to and fro several times between opposite sides of the basins. Figure 4.13 shows four typical configurations that may exhibit different forms of the site effect: a) horizontal stratigraphy with a marked difference in stiffness between the surface layer and the bedrock; b) difference in stiffness with sharp or accentuated underground terrain contours (a relatively narrow and deep valley dug into the rock); c) discontinuity in the horizontal direction;
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Seismic Engineering
d) rugged landforms (hilltop, edge of plateau) as in [COL 93].
Figure 4.13. Four configurations that are often the origin of site effects
Cases a, b and c are characterized by the stiffness contrast (or impedance contrast) between rock and relatively soft soil; case a is the simplest as it can be calculated using unidimensional models of waves with vertical propagation, which constitute the usual hypothesis for the definition of seismic motion (see section 3.2.2), whereas cases b and c bring into play two or three dimensional effects. Case d represents a site effect called topographical in usual terminology, even though the influence of underground topography is also present in case b. 4.3.4. Causes of site effects
The causes of site effects are now rather well understood; they are caused by the manifestation of two phenomena that are easy to understand and that have already been explained in section 3.2.2: – the increase in the amplitude of the wave refracted with respect to that of the incidental wave when there is a marked impedance contrast between the two zones [3.37] and [3.38]; – the trapping of waves by multiple reflections inside a layer (see Figure 3.14) or a landform (see Figure 3.13).
Strong Vibratory Motions
213
The main characteristic of these causes must not lead us to believe that the quantitative prediction of site effects is in reality always very reliable. If it is to be reliable, we must have very accurate knowledge of the geometric and mechanical characteristics of the different layers of the soil whose interfaces are the seat of refractions and reflections, as well as a detailed description of the incident wave field. It is practically only in case a of Figure 4.13 that these conditions can be quite easily satisfied, since the characterization of a horizontal stratigraphy only requires limited geotechnical reconnaissance and since furthermore, the hypothesis of an incident wave field with a vertical propagation constitutes a good representation of reality for this configuration. Apart from this simple case, we must accept that the prediction of site effects often goes beyond the practical possibility of calculation as it is difficult to obtain very accurate data. This is particularly true in the case of the topographical site effect for which discrepancies are often observed between the recordings obtained from instruments and the results obtained from digital models. These discrepancies are sometimes so great (certain recordings show topographical amplification factors that are higher than 20, i.e. 3 to 4 times higher than those calculated) that we might ask ourselves if there aren’t any other causes for landform effects other than those mentioned above. The Cedar Hill Nursery case in Tarzana (north of the agglomeration of Los Angeles) illustrates these difficulties. This site, equipped with a seismograph for powerful motion, had attracted the attention of seismologists at the time of the Whittier Narrows earthquake, on 1 October 1987, because the recordings showed acceleration peaks that were much higher than those observed at other sites located at similar epicentral distances. On the other hand, this phenomenon of amplification had not been observed in the case of the Big Bear and Landers earthquakes that took place on the same day (28 June 1992). However, on 17 January 1994, the Northridge earthquake site once again drew attention as it recorded a “monstrous” acceleration, measuring 1.8 g on one of the horizontal components, with a strong period lasting around ten seconds, without this intense motion being corroborated by the observation of damage in the surrounding areas; the damage caused was not greater than that observed in the vicinity of other sites where recordings were made and where the accelerations were 2 to 4 times lower. After having verified the installation conditions and that the instrument was functioning correctly, the Californian seismologists displayed, shortly afterwards, additional seismographs of the area in order to study the aftershocks. It then appeared [BOU 96] that the amplification phenomena were not repeatable, that is to say, according to the aftershock that was studied, the maximum amplification did not repeat itself for the same seismometer.
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Seismic Engineering
The site effect observed in the case of certain earthquakes in Tarzana is classified as “topographical” as the seismograph is installed on a hilltop (Cedar Hill) that has an altitude of about 20 m with a flat surface of 400 to 500 m, i.e., in reality, a simple hillock like many others that can be found in an agglomeration as big as that of Los Angeles. That a land form as common and insignificant as a hillock can produce amplifications that are that strong, and apparently in an almost random manner depending on the earthquake, is an observation that should lead us to be extremely careful as concerns the present capacity of prediction of the topographical site effect. As indicated earlier, peculiar explanations have been put forward to explain these disconcerting phenomenon; for example, it was suggested that the hill in Tarzana could have been the seat of the beginning of a landslide that would have been the source of vibrations in the immediate vicinity of the seismograph, but there is no evidence to justify this type of speculation. By limiting ourselves to the two causes mentioned at the beginning of this section (impedance contrast and trapping of waves), we can give formulae for the calculation of the site effect by means of a few simple configurations. If we only take into account the impedance contrast, the crossing of waves over an interface that separates two different areas is represented by a factor of change in amplitude of a given motion by the following relation, established in section 3.2.2 (see [3.37]): V2 V1
O [where O =
U1C1 U 2 C2
]
[4.32]
where the impedance ratio O defines the contrast between area 1 and area 2, at the centre of which, the mass density, the velocity of the propagation of the shear waves and the specific velocity are represented respectively by U1, c1, v1 and U2, c2, v2, there is thus amplification if O >1 (penetration of the wave into an area of low impedance). A similar formulation in On (n being an empirical exponent that is not necessarily equal to ½ but that remains pretty close to that value) has been retained in certain approaches that have recently been proposed in order to take into account the site effect (see section 2.3.3.3). As indicated in section 1.4.2.2, the formulae in which the amplitude ratio only depends on the impedance ratio are only suitable when the latter is not very high (lower than 3 to be more specific) because they neglect the influence of the reflections of the waves. These reflections result in the trapping of waves, which is the second of the “primary” causes of the site effect. The amplifications are the collective result of the different reflected waves that arrive on seismic ray paths having the same phase at the concerned point, whereas had they to arrive with opposite phases, they could produce on the contrary, attenuation. In such conditions, the site effect depends not only on the impedance contrast (which determines the
Strong Vibratory Motions
215
amplitudes of the reflected waves; see section 1.4.2.2) but also on the frequency of the waves and the length of their paths (which determine dephasing). In the case of a layer of soil overlying bedrock, calculation is simple when the deformations are relatively small so that we can assume linear elasticity of the soil. Figure 3.14 shows the beam of the trapped waves inside the layer; at a given point inside the layer, there are an infinite number of waves that arrive after having been subjected to refraction (to cross the interface with the bedrock) and a series of reflections on the surface and the base of the layer; for an incident wave with vertical propagation, the primary calculation of dephasing between these waves will allow us to arrive at the series that represents the sum of all these effects at the concerned point in the layer; the summation of this series gives us the following formulae [BET 93]: 1/ 2
K ª º as = [1 + r] « » ¬1 2K r cos T K ² r ² ¼
ab =
1/ 2 1 r ª 1 2K cos T K ² º
2
«1 2K r cos T K ² r ² » ¬ ¼
[4.33]
[4.34]
in which: – as and ab are the amplifications at the free surface and at the base of the layer respectively, with respect to the surface motion that would occur for the bedrock without the layer of soil; –r=
O 1 O 1
–T=2
, O being the impedance ratio bedrock/soil;
Zh
, Z being the pulsation of the wave, h the thickness of the layer and c c the velocity of propagation of the shear waves in the layer; – K e [T , [ being the low damping in the layer, related to the quality factor Q by equation [3.56]. Figure 4.14 shows the variations with respect to parameter T (dimensionless frequency of the wave) the amplifications at the surface and at the base of the layer.
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Seismic Engineering
Figure 4.14. Amplifications as and ab at the surface and at the base of a layer of soil with
respect to parameter T = 2
Zh
(Z pulsation of the wave, h and c, thickness of the layer and
c
velocity of the shear waves in the layer)
We see that as represents a series of attenuating peaks of amplitude for T = S , 3 S , 5 S , etc. to which correspond troughs that are progressively less deep for ab. For T = 2 S , 4 S , 6 S , etc. as and ab are both practically equal to one. The figure represents the values O = 5, [ = 0.02. For small-scale damping lower than one, which is generally the case, we obtain by limited development:
> as @T
¬ªab ¼ºT
> as @T
S
S
2S
ª1 S º «O 2 [» ¬ ¼
1
[4.35]
ª 2 º O «O 1 » S[ ¼ ¬
> ab @T
2S
1
ª S[ º «1 O » ¬ ¼
[4.36]
1
[4.37]
Table 4.9 presents the results obtained using these formulae for two values of [ [0.02 and 0.05] and five values of O (2, 3, 5, 7 and 10).
Strong Vibratory Motions [= 0.02 T= ʌ
217
[= 0.05 T=2ʌ
T= ʌ
T=2ʌ
O
as
ab
as = ab
as
ab
as = ab
2
1.88
0.061
0.970
1.73
0.146
0.927
3
2.74
0.089
0.979
2.43
0.204
0.950
5
4.32
0.139
0.988
3.59
0.299
0.970
7
5.74
0.185
0.991
4.52
0.374
0.978
10
7.61
0.245
0.994
5.60
0.460
0.985
Table 4.9. Amplifications at the surface and at the base [as, ab] of a layer of soil with respect to the impedance ratio O; T = 3 corresponds to the first peak and T = 23 to the first trough of the amplification as at the surface
The amplification at the surface for the first peak (T = S ) has values that are distinctly higher than those that we would obtain by applying formula [4.32]. The frequency g that corresponds to this first peak is no other than the fundamental frequency of the layer of soil; considering T = S and Z = 2 S g, we obtain the following from the definition T = 2 Z h/c: g=
c 4h
[4.38]
and again for the period T = 1/g: T=
4h c
[4.39]
This formula, called the quarter wave resonator formula, as it shows that the thickness of the layer is equal to a quarter of the wave length cT, is one of the most important in earthquake engineering. It is interesting to notice that peak as is accompanied by a marked attenuation of ab, therefore at a weak level of excitation at the base of the layer; this is a characteristic of resonant mechanical systems, excited at a frequency that coincides with their own frequency, that vibrate strongly bringing with them minimal energy (that only compensates the loss of energy due to damping). This trough ab corresponds to the almost total suppression of the depth of certain frequencies that we will deal with in section 5.3.2, when deconvolution calculations are introduced.
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Seismic Engineering
Figure 4.14 shows the very strong dependence of the amplification at the surface with respect to the frequency. The site effect by trapping of waves in a layer of soil is not necessarily repetitive at a given place, if the frequency characteristics of the earthquakes likely to occur there are variable. Several cases of non-repetitivity have been reported, like for example, those of some sites close to San Francisco where strong amplifications were observed in the case of the Loma Prieta earthquake (17 October 1989), whereas nothing unusual had been recorded during the previous earthquake (22 March 1957) whose magnitude was lower (5.3 instead of 7.1) and the frequency content was richer in higher frequencies. A real seismic signal being distinctly more complex than a sinusoid, the amplification calculated with [4.33] overestimates the real amplification, as the waves contained in the signal whose frequency does not coincide with that of the layer are less amplified. In the case of the layers of soil that are most commonly found (having a thickness ranging from ten to several tens of meters, velocities of the shear waves ranging from 200~400 m/s) the resonance frequencies calculated by [4.38] are in the range of 1~10 Hz and are generally located within the seismic frequency field. The influence of the multi-frequency character of the excitation on the amplification at the surface can be estimated with the help of formula [4.33], by considering a sinusoidal wave packet whose pulsations are regularly distributed in the interval (ZC – 'Z, ZC + 'Z) and whose phases are random in nature; in these conditions we can assume the principle of quadratic combination, already mentioned in section 4.1.3, and write the following formulae for the square of amplification as at the surface [BET 93]: a²s =
(1 r )²
T 2 T1
T2
K dT
1
2 2 1 2K r cos T K r
³T
[4.40]
where r, T and K have the definitions given above in [4.33] and [4.34] and where T1 and T2 correspond to the limits of the values of the wave packet, i.e.:
T1
2
h >Zc 'Z @ ; c
T2
2
h >Zc 'Z @ c
[4.41]
As K = e -T[ is rather close to one as long as T is not very large and the damping [ is weak, we can consider it as a constant in equation [4.40], which is easily integrated so as to obtain:
Strong Vibratory Motions
as=
T 2 T1
T
2 § 1K r T ·º tan Arc tan ¨ 1K r 2 ¸» «1K ² r ² © ¹¼T1 ¬
K (1 r )² ª
2
219
2
[4.42]
with:
K
[
e
(T1 T 2 ) 2
[4.43]
By applying formula [4.42] with K = 1 (zero damping) and T2 – T1 = S , we get the values of amplification as of Table 4.10 for O varying from 2 to 10 and TC from S /2 to 3 S /2 (i.e. a fluctuation of an amplitude equal to the width of the band surrounding the resonance Tc = S ). Tc= ʌ
Tc= 2ʌ
Tc= ʌ
Tc= 4ʌ
Tc= 3ʌ
3
Tc= 5ʌ 6
Tc= 7ʌ
2
6
3
2
2
1.414
1.567
1.653
1.679
1.653
1.567
1.414
3
1.732
2.031
2.153
2.184
2.153
2.031
1.732
5
2.236
2.791
2.928
2.957
2.928
2.791
2.236
7
2.646
3.413
3.543
3.569
3.543
3.413
2.646
10
3.162
4.190
4.306
4.328
4.306
4.190
3.162
O
Table 4.10. Amplification at the surface for different values of the impedance ratio O and the central frequency of the wave packet [[ = 0, T2 – T1 = S ]
We see in Table 4.10 that if the amplification is maximum for TC = S (central frequency of the wave packet coinciding with the frequency of the layer) it only reduces very slightly for TC = S /2 or 3 S /2 (frequency of the layer equal to one of the limits of the range of the wave packet). The values of as at resonance are distinctly weaker than those obtained for a monochromatic excitation (as = O for [ = 0 according to equation [4.35]; they correspond, in the hypotheses of Table 4.10, to the formula: 1/ 2
¬ªas ¼ºTc
S
2 1 º ª « 2O (1 S Arc tan O ) » ¬ ¼
[4.44]
which is not very different from [4.32] (that we can find in Table 4.10 for Tc = S /2 or 3 S /2].
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Seismic Engineering
These observations on the values of Table 4.10 confirm the tendencies resulting from the experience feedback: – the amplifications by layers of soil are very frequently observed, even when the dominant frequency of the seismic signal is considerably different from the fundamental frequency of the layer (factor 2 or ½), but does not manifest itself if the difference is too large;
O (resulting from the – the value of the amplifications is closer to conservation of energy in a tube of force; see section 3.2.2), than to O (resulting from multiple reflections of a sinusoidal wave). This effect of amplification is not observed, at least in the case of horizontal components of motion, when the layers are very thick (damping prevails over the impedance contrast] or when the influence of nonlinear behavior becomes prominent for high levels of excitation (see section 5.3.2). The trapping of waves also provides a simple explanation of the topographical site effect, as already mentioned in section 3.2.2 (see Figure 3.13). We shall consider a textbook example shown in Figure 4.15. Any point M of a triangular symmetrical hill having slopes at 45° will have 4 seismic rays passing through it in the case of an incident wave with vertical propagation (the particle motion associated with this wave of which the horizontal motion is perpendicular to the surface plane of Figure 4.15): – direct ray 1 that reaches M before undergoing reflections on the slopes of the hill; – ray 2 that reaches M after two reflections at points D and E; – rays 3 and 4 that reach M after reflection at C and A respectively.
Figure 4.15. Symmetric triangular section of a hill having slopes at 45°; 4 seismic rays 1, 2, 3 and 4 pass through any point M
Strong Vibratory Motions
221
The dephasing of rays 2, 3 and 4 in comparison with ray 1 corresponds to the additional time taken to travel to point M, i.e. if c is the propagation velocity of the shear wave (assuming that the hill is made of homogenous material): t2 =
t3 =
t4 =
1 c 1 c 1 c
[ME + ED + DB] for ray 2
MC for ray 3
MA for ray 4
By naming the coordinates of point M, xo and yo and considering a sinusoidal wave pulsation Z and unit amplitude, we obtain for the superposition u of the four rays through M, after basic calculation of t2, t3 and t4: u = sin Z t + sin Z [t – 2 ray 1
h- xo - yo h+xo - yo h - y0 ] + sin Z [t – ] + sin Z [t– ] c c c
ray 2
ray 3
ray 4
If point M is on the right slope of the hill [therefore if xo + yo = h] we obtain for u: u = 2 [[1 + cos
2Z xo c
] sin Z t – sin
2Z xo c
cos Z t]
[4.45]
The maximum value of um of u is thus: um = [4 [1 + cos
2Z xo c
]² + 4 sin²
2Z xo c
]1/2 = 4 cos
Z xo c
[4.46]
We see that um = 4 at the top of the hill (xo = 0) represents an amplification of 2 with respect to the motion that we would have at the surface of a flat area (where only the reflected wave adds to the incident wave). At the foot of the hill (x0 = h) we could have amplification or attenuation depending on the value of Zh/c; if the height h of the hill is equal to a quarter of the wave length, Zh/c = S /2 and there is total attenuation (um = 0). From this calculation emerge the following tendencies:
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Seismic Engineering
– the amplification at the top of the hill is independent of the frequency, unlike that in the case of the layer of soil; – we find the influence of frequency at points other than the hilltop and in these cases, as in the case of the layer of soil, the influence of frequency is controlled by the ratio of the wave length and a characteristic dimension of the site (height of the hill, thickness of the layer); – this influence of the frequency manifests itself in different ways depending on the position of the point being considered (and could even reach total attenuation). Differential motions can clearly be noticed on the slopes. These tendencies are confirmed by experimental observations, particularly in the case of a “broad band” characteristic of amplification at certain points (independence with respect to frequency) (that is, different from a “narrow band” characteristic often seen in the case of layers of soil) and the importance of differential motions. However, from a quantitative point of view, the “theoretical” formulae of the topographical site effect as in [4.46] are not in keeping with actual measurements in the sense that they tend to underestimate reality, as explained earlier.
Chapter 5
Calculation Models for Strong Vibratory Motions
5.1. Orders of magnitude deduced from the basic theory of elastic rebound 5.1.1. Limits of the basic theory of elastic rebound for the calculation of motions The basic theory of elastic rebound, presented in section 2.1, allows us to offer a simple explanation for the relations that connect the surface of rupture and the energy that is released in the form of seismic waves, to the global parameters that define the source, the seismic moment Mo (or moment magnitude Mw) and the stress drop 'V. The calculation of seismic motion on the basis of these global parameters alone, without introducing any additional hypotheses, is only possible for velocity and displacement; acceleration, on the other hand, is influenced by ruptures of different sizes; the law of distribution of these ruptures on the fault plane must be introduced and explained in the calculation model. The formula of seismic energy (see [2.16]) is given by: Ec =
1 M o 'V P 2
[5.1]
Considering [2.6], it can also be written as: Ec =
/V ² 2P
LHB
[5.2]
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Seismic Engineering
LHB being the volume of the source, we see that the energy released by one unit of volume is equal to 'V²/2P. We can assume that the specific velocity V0 in the rupture zone corresponds to a volumic kinetic energy of the same value, hence that: 1 2
1 'V ²
UV²0 =
[5.3]
P
2
Taking P = U c ² [2.3], we obtain the following for V0: V0 =
'V
[5.4]
Uc
Taking the values [2.4] for U and c and 'V = 3.8 MPa [2.20] deduced from the Gutenberg and Richter law of energy, we have V0 = 0.42 m/s. This value is applied in the rupture zone, i.e. to focal distances R that are lower than or equal to the equivalent radius R0 of source [2.7]; in the case of distances R>R0, we have a decrease in 1/R of the volume waves, hence for particle velocity V we obtain: V = V0
Ro R
=
'V R0
Uc
R
[5.5]
This expression has already been considered in section 2.3.1 [2.49] without specifying the value of V0. It corresponds to an attenuation law of velocity that is written taking into account [2.7] and [2.11]: 2/3 1 § 3 · 1/3 [ 'V ( N / m ²)] x 10 0.5Mw + 3 ¸ 3 4 ( ) S R m © ¹ U ( kg / m )c ( m / s )
V (m/s) = ¨
[5.6]
Particle velocity V thus defined, and having been obtained by taking into account considerations relating to energy, corresponds to the square root of the sum of the squares of the three components of seismic motion; we must therefore apply a reduction coefficient equal to 1/ 3 to particle velocity, so as to compare it to the value of a component; such values being known thanks to recordings on the surface, we must also apply a multiplication coefficient of 2 to take into account the superimposition of the reflected wave on the incidental wave; the equations are hence finally, once and for all, written as [5.5] and [5.6]: V=
2 'V Ro 3 Uc R
[5.7]
Calculation Models for Strong Vibratory Motions
§ 3 · V (m/s) = ¨ ¸ 3 © 4S ¹ 2
1/3
2/3 [ 'V ( N / m ²)] 1 x 10 0.5Mw + 3 3 ( ) R m U ( kg / m )c ( m / s )
225
[5.8]
In order to compare equation [5.8] to the attenuation laws of velocity in Table 4.6, we use the same units (cm/s for V, km for R), we convert the power of 10 into an exponential and take the standard values U = 2,700 kg/m3, c = 3,333 m/s, 'V = 3.8 x 106 N/m², which gives us: V (cm/s) = 0.194 e 1.15M x R-1
[5.9]
This “theoretical” attenuation law, established by very simple reasoning, has numeric coefficients whose orders of magnitude are perfectly comparable to those (c and D) of the four laws of velocity of Table 4.6; the coefficient 1.15 of the exponential is close to the average 1.211 of the four values of D given in the table. The attenuation in R-1 was deliberately chosen and could of course be completed by an inelastic factor, as in the Joyner and Boote 1988 law. As regards the numerical application in the case M = 6, R = 25 km (last column in Table 4.6), we get V = 7.76 cm/s as per [5.9], which is the upper limit of the values in the table. The “theoretical” law [5.7] (or [5.9] when 'V = 3.8 MPa is taken) is therefore acceptable, as much because of the dependence that it implies with respect to the parameters M and R, as because of the orders of magnitude of its numerical values. It is interesting to note that the influence of the stress drop is expressed by a 2/3 power law, that is adequate to note the importance of standard deviation for the attenuation laws (we recall (see section 2.1.2) that we can assume that 'V varies from 1 to 10 MPa, which in the case of this 2/3 power law, corresponds to the factors 0.411 and 1.91 with respect to the values calculated with 'V = 3.8 MPa). Can the values of displacement D and those of acceleration A be deducted from this “theoretical” velocity V? In order to deduce the same, we need to determine a pulsation Z0, linked to the parameters of the rupture, which would allow us to calculate D and A from V by the following: D=
V
Zo
; A = Z0 V
[5.10]
as would be done in the case of a sinusoidal motion. The most natural way to define Z0, is to apply the quarter wave resonator formula [4.39] which we saw in section 4.3.5 for the fundamental period of resonance of a layer of soil; we take for the thickness h, the width B/2 of one of the
226
Seismic Engineering
blocks of the theory of elastic rebound (see Figure 2.1) and we obtain for this fundamental period To: To =
2B
[5.11]
c
Taking into account that B = 2/3 R0 (see section 2.1.3) and the relation between period and pulsation (Z0= 2 S /T0], we obtain for Z0:
Z0 =
3S
c
2
Ro
[5.12]
This pulsation Z0 corresponds to a low frequency value that is all the lower given that R0 is high, hence the magnitude is high. In the case of an earthquake of magnitude 6, R0 is approximately 4 km (see Table 2.1) which with c = 3,333 m/s gives a frequency Z0/2 S of 0.625 Hz; in the case of an earthquake of magnitude 8 (R0= 40 km), the frequency drops to 0.062 Hz. By applying formulae [5.10] with Z0 given by [5.12] and V by [5.7], we get: D=
2 Ro 3S 3 U c ² R
A=
S
'V
4
3
'V
1
U
R
[5.13]
[5.14]
Formula [5.13] provides plausible values for displacement (it gives, for example, 1.99 cm for M = 6 and R = 25 km, with standard values for 'V and P = Uc², which is the order of magnitude of the two last laws in Table 4.6). On the other hand, formula [5.14] for acceleration seems absurd since it does not take into account the magnitude (the factor in R0 has disappeared). Here, we can clearly recognize the limitations of the basic model of elastic rebound that we spoke about the beginning of section 5.1.1 as regards the calculation of accelerations. These accelerations are only slightly influenced by low frequency motions caused by the phenomenon of rupture and therefore by the entire motion of the two blocks associated with the Z0 determined above. In order to obtain an acceptable estimation of accelerations, we
Calculation Models for Strong Vibratory Motions
227
must slightly modify the model of elastic rebound by introducing rupture zones with a higher frequency. 5.1.2. Model of elastic rebound with multiple ruptures
Drawing from an idea suggested by Kanamori and Anderson [KAN 75] for the justification of the Gutenberg and Richter law of distribution of the frequencies of occurrence of earthquakes according to their magnitude (see section 6.2.2), we shall assume that the fault plane consists of multiple rupture zones of different dimensions whose distribution is such that the same area of the fault plane corresponds to that of all the rupture zones of a given size taken together. In other words, the product r2 n (r), where r is the equivalent radius of a source (in the same way that R0 is the equivalent radius for the whole seismic site) and n (r) is such that n (r) dr represents the number of source zones having their equivalent radius between r and r + dr, should be constant. We express this hypothesis, which is of a fractal nature, by the relation: n (r ) = K
R0
[5.15]
r²
K being a numeric coefficient that shall be determined hereafter. Factor R0 was introduced in the interest of dimensional homogeneity and to show the size of the whole source. The equivalent radius r of the basic sources varies between a lower limit r0 and a higher limit equal to kro, k being another numeric coefficient; in the calculations that follow, we shall systematically disregard r0 before kR0 (we will see a little later that r0 is approximately equal to a few tens of meters whereas R0 is, as we have seen, approximately equal to a few kilometers, at least for the earthquakes that are of interest to earthquake engineering). A first relationship is obtained between coefficients K and k by noting that the total volume of the source zones is equal to the combined volume LHB = 4/3 S R30 of the two blocks of the basic model, i.e.:
³
kRo ro
4 3
S
r3 n (r) dr =
4S 3
KR0
³
kRo ro
r dr =
4S 3
R03
[5.16]
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Seismic Engineering
By disregarding r20 before k2 R20, we get: K k² = 2
[5.17]
In order to calculate the motions, we will assume the hypothesis of quadratic combination (this has already been mentioned in section 1.3 and section 4.3.5) of the contributions of the different source zones. These contributions are given by formulae [5.7] (for velocity), [5.13] (for displacement) and [5.14] (for acceleration) by simply replacing R0 with r; we thus obtain for sources with radius r: 2
v (r) =
3
d (r ) =
a (r ) =
'V
r
Uc
R
[5.18]
'V
r²
3S 3 U c ²
R
4
S
3
'V
1
U
R
[5.19]
[5.20]
for velocity V, the principle of quadratic combination is expressed by the relation:
V² =
kR
³ r0 0
§ 2 'V · kR0 ² KR0 dr ¸ ³ r0 © 3 U cR ¹
v² (r) n (r) dr = ¨
[5.21]
or, by disregarding r0 before kRo: V=
2 3
Kk
'V
Ro
Uc
R
[5.22]
In section 5.1.1, we saw that formula [5.7], which is identical to [5.22] to the nearest factor Kk , gave results that were comparable to those of the empirical attenuation laws of velocity. We shall assume, as a second relation between the coefficients K and k, that this factor obtain:
Kk is equal to 1, thus from [5.17] we should
Calculation Models for Strong Vibratory Motions
K=
1
;k=2
2
229
[5.23]
With these values of K and k, the total number N of source zones is:
³
N
kRo
ro
n( r ) dr
Ro
2³
kRo
dr
1 Ro
2
2 ro
r
ro
[5.24]
If we assume that each of these source zones has the same shape factors as the source considered as a whole (see section 2.1.3) and thus that its area on the fault plane is equal to 2 S r², we can check that the source zones occupy the whole of the fault plane, since:
³
2 Ro
ro
2S r 2 n( r ) dr
S Ro ³
2 Ro
ro
dr
2S Ro2
[5.25]
5.1.3. Calculation of the theoretical attenuation laws associated with the model of rebound elasticity with multiple ruptures
The attenuation law for velocities has already been obtained [5.7 or 5.22 with Kk = 1]. Those of displacement D and acceleration A are obtained by the principle of quadratic combination, i.e. with K = ½ and k = 2 using equations [5.18], [5.19] and [5.20]:
D
2
³
2 Ro
³
2 Ro
ro
2
2R 'V · 1 § 4 d ( r )n( r ) dr ¨ Ro ³ r 2 dr ¸ 2 r © 3S 3 U c R ¹ 2 2
o
[5.26]
o
2
A
2
ro
§ S 3'V · 1 2 R dr a ( r ) n( r ) dr ¨ ¸ Ro ³r r2 © UR ¹ 2 2
o
[5.27]
o
from which, by disregarding r0 before R0 in the calculation of the integrals, as carried out earlier, we have:
D
8 'V R02 9S U c 2 R
[5.28]
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Seismic Engineering
A S
3 'V
Ro
2 U ro
R
[5.29]
The only difference between formulae [5.28] and [5.13] is the numeric coefficient that is equal to 0.283 instead of 0.245 representing an increase of 15.5% which is not very significant given the uncertainties that affect the calculation of the displacements on the basis of the accelerograms (see section 4.1.2). We thus observe that the model with multiple ruptures only confirms the results obtained with the basic model for displacements and velocities, as indicated in the beginning of section 5.1.1. In the case of accelerations, on the other hand, the two models are very different, as a comparison between formulae [5.29] and [5.14] shows. Formula [5.29] brings into play the dimension r0 of the smallest source zones; this parameter is directly linked to the maximum frequency gm of the waves that constitute the seismic signal; by following the reasoning used in section 5.1.1 to arrive at formulae [5.11] and [5.12] and by replacing R0 by ro, we obtain the following for gm: gm =
3
c
[5.30]
4 ro
We can thus rewrite equation [5.29] as follows: A=
S
2
'V
gm
U
c
Ro R
[5.31]
By expressing R0 with respect to the stress drop 'V and the moment magnitude Mw, as we did earlier for velocity (see [5.8]), we get the following for D [5.28] and for A [5.31]: 2/3
D(m) =
§ 3 · ¨ ¸ 9S © 4S ¹
S
§ 3 · 2¨ ¸ © 4S ¹
A(m/s²) =
8
1/ 3
[ 'V ( N / m ²)]
1
3
2 U ( kg / m )[c ( m / s ] R ( m) 1/ 6 [ 'V ( N / m ²)]5/6
3
U ( kg / m )
g m( Hz )
x 10Mw+6
1
c ( m / s ) R ( m)
[5.32]
x 100.25Mw+1.5 [5.33]
Calculation Models for Strong Vibratory Motions
231
By carrying out on equations [5.32] and [5.33], the same conversions as were done to go from [5.8] to [5.9], that is, the change in the units and the adoption of standard values for U, c and 'V, we rewrite them in the following form that can be directly compared to the laws in Table 4.6: D (cm) = 0.00566 e2.30 M x R-1
A (cm/s²) = 21.6 e0.576 M x R-1 x
[5.34]
fm ( Hz )
[5.35]
The choice of a plausible value for the maximum frequency gm can be made on the basis of the analysis of recordings, especially by analyzing their spectral properties (see section 9.1); it indicates that gm is generally situated in the range 20~50 Hz, the response spectrum called USNRC, already evoked in section 4.1.2 (and presented in section 9.1), corresponding to the choice gm = 33 Hz or approximately to the geometric average of the limits of this range. By taking the value 33 Hz for gm, equation [5.35] becomes: A (cm/s²) = 124 e0.576M x R-1
[5.36]
This theoretical law is, as in the case of velocity, comparable to the empirical laws of Table 4.6 from the point of view of the values of coefficients c and D of the table; this is, for example, similar to the Joyner and Boore 1988 laws and the Ambraseys and Bommer 1991 laws, that are amongst the most used in practice; if we were to compare it to the other laws (M = 6, R = 25 km), we would find that it gives an acceleration of 157 cm/s², which would place it close to the laws that give the strongest values (see the last column of Table 4.6); this tendency is normal, since the theoretical law does not include a factor of inelastic attenuation, and could be corrected by choosing slightly different values for 'V and gm. The influence of the stress drop has already been emphasized in the discussion about formula [5.9] for velocity; it is still a little more marked in the case of acceleration where the exponent of 'V is equal to 5/6 = 0.833 (see [5.33]) instead of 2/3 = 0.667. The limits IMPa and 10 MPa often assumed as the two limits of the range of the usual variation of 'V, correspond with the power law 5/6, to multiplicative factors equal to 0.329 and 2.24 respectively, with respect to the values calculated with 'V = 3.8 MPa. The implacable character of the “law of standard deviation” is once again underlined, more so since the stress drop is far from being the only cause for the dispersion in the recording data. In conclusion, we note that the value retained for gm justifies disregarding r0 before R0 in the previous calculations; formula [5.30] provides the value r0 = 76 m
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Seismic Engineering
when we take c = 3,333 m/s and gm = 33 Hz, which is very low in comparison with the values of R0 in practical cases (see Table 2.1). 5.2. Digital source models 5.2.1. General considerations pertaining to models of digital simulation of the seismic source
Since the 1970s, we have seen the development of digital models that try to simulate the entire seismic phenomenon: rupture mechanism on the fault plane, propagation of waves emitted by the rupture, local effects due to the geotechnical characteristics of the sites. For a long time, these developments only concerned seismologists, but since about 1990~1995, they have started to play a role in certain studies of seismic hazard for important projects. That is why it would be apt to present below the practical uses and limitations of such simulations. The attempts to digitally simulate real earthquakes that have provided important recordings have constituted the major part of the efforts undertaken in this branch of research. Some examples of these simulations are given in section 5.2.2. These studies have demonstrated the complexity of seismic sources and vibratory motions emitted by ruptures, especially of those components that are of special interest to earthquake engineering, in other words those motions that have frequencies in the range of 1 to 10 Hz. This complexity is due to the fine scale segmentation of the faults (see [MAD 91]) whose scale is all the finer as we consider smaller wave lengths, hence higher frequencies. As a result, at a low frequency, the seismogram shows a smoothened version of the rupture process, while at high frequency, we essentially see bursts of the rupture fronts every time there is stoppage of propagation or rupture of a new fault segment (see [MAD 91]). The model of elastic rebound with multiple ruptures, presented in section 5.1.2, represents a simple approach to understanding this complexity. Since the appearance of digital simulation models, there have been opposing conceptions regarding the causes of the non-uniformity of the rupture mechanism. In particular, two hypotheses, seemingly contradictory, but in reality, complementary, have been proposed: – the barrier model (see [DAS 77]), where we assume the presence on the fault plane, of zones that cannot be crossed (the barriers) by the rupture front; it is the deceleration of the front to stop before the barrier that could be the cause of the emission of intense radiation; – the asperities hypothesis (see [LAY 81]), in which the velocity of rupture propagation is, on the contrary, relatively constant but where the fault plane is
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233
extremely heterogenous since certain zones (the asperities) correspond to hard spots characterized by high values of the stress drop and of the slip produced by the rupture; these heterogeneities are due to the past history of the fault, on which the ruptures produced by previous earthquakes, affected certain zones while sparing others. The random nature of the distribution of barriers or asperities on the fault plane and that of the variation of the velocities of propagation of the rupture front (in the barrier model) or of the stress drop (in the asperities model) is sufficient to explain the complexity of seismic sources. The vibratory motions at the surface, necessarily reflect this complexity, particularly in proximity to the fault, where the deciding element is the distance between the site being studied and the nearest barriers or asperities; as one moves away from the fault, this distance depends less and less on the heterogeneity of the fault and tends towards the usual formalism of the attenuation laws with a single parameter of distance. The principle of digital source simulations is the representation of the fault as a juxtaposition of elements on which the characteristics of the rupture processes are constant (absence of rupture if the element is a barrier, direction and amplitude of the slip produced by the rupture if the element is an asperity). When the simulation attempts to imitate a real earthquake, the choice of the characteristics of each of the fault elements is the result of the minimization of an error function constructed on the basis of the differences between the available recordings and the motions calculated with the model. When the simulation is conducted with a view to predicting the effects of a future earthquake, the construction of the model or the characterization of the fault elements generally relies on similarities with real earthquakes studied earlier. It is thus much less arbitrary since more information on the most plausible rupture modes for the fault being considered is available, such information can be provided by less powerful earthquakes that have occurred previously on the same fault or on neighboring faults that have a comparable tectonic context. These earlier “calibration” earthquakes also allow us to determine the variations of the velocities of wave propagation in areas surrounding the source, for example in the great sedimentary basins. These simulation studies that basically aim at prediction, concern the following areas: – detailed assessment of the seismic hazard associated with the major faults, situated in high risk zones (high risk not only from the point of view of seismic activity, but also from the point of view of the density of the population), mainly in California and Japan; we can take the example of the Kanto basin (district of Tokyo) that was studied to predict the consequences of surface motions of a big earthquake similar to the one that occurred on 1 September 1923 [SAT 99];
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– parametric evaluation of directivity effects associated with the characteristics of rupture propagation on the fault plane, and in particular, the killer pulse phenomenon mentioned in sections 4.1.3 and 4.1.2; – enrichment of the database for the establishment of specific attenuation laws for a region where recordings are rarely available; an example of the same (law of acceleration of Hwang-Huo 1997 for the east of the USA) was given in section 4.2.2. The contribution of such studies towards a better understanding of seismic hazard is bound to increase in the future, but this increase, in most regions of moderate seismicity, is dependent on accurate data of the characteristics of the faults and some important recordings that will enable the calibration of models, all of which will take time. We must also take into account a certain sentiment of distrust that is quite widespread with respect to the approach of all the disciplines of earthquake engineering that is more and more theoretical and “computational” and the validity of which is at the very least contestable. This reluctance is often seen with respect to digital source simulations, even if the approximate nature of the evaluation provided by the standard attenuation laws is now accepted by all. 5.2.2. Examples of digital simulation of real earthquakes
Figure 5.1 shows one of the results obtained by the simulation of the Canadian earthquake at Nahanni on 5 October 1985: moment magnitude 6.6. As regards the distribution of the slip (in cm) on the fault plane, we observe a zone of strong slip above the focus (indicated by a dot) and a rapid attenuation of the amplitude of the slip outside this zone (with a slight rise about 12 km to the south of the focus and in the vicinity of the surface). The slip is therefore very highly variable along the length of the rupture zone and this tendency is confirmed by almost all simulation studies.
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Figure 5.1. Isoseismal curves of the slip for the Nahanni earthquake (5 October 1985); the slip values are in cm, the horizontal dimensions (abscissa) and accordingly the dip (ordinate), in km; the focus is indicated by a dot; taken from [HAR 94]
Figure 5.2. Isoseismal curves of the slip for the Michoacan–Guerrero (19 September 1985); the slip values are in cm, the horizontal dimensions (abscissa) and accordingly the dip (ordinate), in km; the focus is indicated by a dot; taken from [SOM 91]
The decrease that is relatively steady from the focus, which characterizes the Nahanni earthquake, is not the general rule and we frequently observe the presence of several “pockets” of strong slip that corresponds to the hypothesis of asperities. Figure 5.2 shows the same map of the distribution of slips for the subduction earthquake of Michoacan–Guerrero (Mexico, 19 September 1985, moment magnitude 8.1) that presents three pockets that are rather far apart from each other. The variability and irregularity of the dip on the fault plane seems to be the rule even for earthquakes that are distinctly smaller than those presented in Figures 5.1 and 5.2, as illustrated in the case of the New Brunswick earthquake (9 January 1982, moment magnitude 5.6), one of the rare earthquakes of a magnitude lower than 6, that has been the subject of a digital simulation study [HAR 94]. We see the isoseismal curves of the dip and the temporal representation of rupture propagation.
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Figure 5.3. Source model for the New Brunswick earthquake (9 January 1982); a) isoseismal curves of the slip with the same units and slip and graphic representation modalities as in Figure 5.1; b) variation of the seismic moment release with respect to time; c) kinematics of the rupture, during three time intervals each one lasting 0.5 s, taken from [HAR 94]
The rupture zone (part a of Figure 5.3) measures only about 5 x 4 = 20 km² and consists of two slip pockets, one around the focus and the other at 2 km to the south. In part c of the figure, we see that the rupture consists of several episodes: the first is very brief and only concerns the vicinity of the focus; it is followed by the appearance of two sources, above and to the south of the focus from which the rupture propagates horizontally until it joins the focus, in the case of the one coming from the south; this entire complex process only lasts 1.5 s. The main part of the release of the seismic moment takes place after the first phase of rupture around the focus (part b of Figure 5.3). Such graphical representations of the propagation of rupture with respect to time are obtained very frequently as results of simulation studies; the tendency towards a multiplicity of successive sources generally increases with the size of the earthquake. This observation shows us once again how complex the seismic phenomenon can become when we try to describe it in detail.
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Vibratory motions calculated on the basis of digital models of simulation can only be compared to observed motions, in cases where the object of the simulation is a real earthquake. Such comparisons are obviously biased since the parameters of the digital model are adjusted so as to obtain the best possible compatibility between the signals recorded and the signals calculated. They are nevertheless interesting as they give an idea of the quality and precision that may be obtained with these simulations in the best conditions, that is with a model that is well adapted to the specificities of the fault; Figures 5.4 and 5.5 show the horizontal seismograms (acceleration, speed and displacement), observed and calculated on two sites (MOT and KMT) affected by the Hyogo–ken–Nanbu (Kǀbe) earthquake on 17 January 1995. The two sites correspond to soil conditions with very poor mechanical characteristics. The first (MOT; see Figure 5.4) is very close to the fault; we observe in the figure, that the concurrence between the calculations and the measurements is good for the velocity and displacement on the two components as much from the point of view of the form of the signal as from that of the peak values (shown as MAX in the figure, in cm/s for velocity and in cm for displacement); the results of the comparison of accelerations are less satisfactory, especially for the north-south component that represents a very strong isolated peak on the calculated accelerogram and a ratio of maximum values (in cm/s²) that is higher than 2; for the east-west component, the peaks of the accelerogram have closely related values but the gait of the signal is clearly different. For the KMT site (see Figure 5.5) situated at about 45 km from the epicenter, in a direction perpendicular to the fault, the concurrence is good for the component N40E (parallel to the fault), but this not the case for the other component where there are significant differences between the seismograms observed and those calculated, this time in the case of velocity and displacement
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Figure 5.4. Comparison of the seismograms of the two horizontal components, recorded (observed) and calculated (synthetic) at the MOT site, very close to the fault of the Kǀbe earthquake on 17 January 1995 (taken from [KAM 98])
It is possible that the differences observed may be attributed to the site effect, a providential resource for experts hard pressed for explanations, since the site effect was only taken into account in the digital model by way of an empirical formulation (Green functions) adjusted on the recordings of the aftershocks, therefore for excitation levels much lower than those of the main shock (therein lies the question of their validity in the case of significant nonlinear effects; see section 5.3.1). However, this attribution does not seem very probable in the case being studied, since the concurrence is good in certain aspects of the motion (velocities and displacements for MOT, accelerations for KMT) and since the influence of nonlinearity should not be important to KMT, where the level of motions is moderate.
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Figure 5.5. Comparison of the seismograms of the two horizontal components, recorded (observed) and calculated (synthetic) at the KMT site at 45 km from the fault for the Kǀbe earthquake on 17 January 1995 (taken from [KAM 98])
After having seen the above example, it thus seems legitimate to conclude that the methods of digital simulation do not allow us to significantly reduce the uncertainties regarding the determination of the seismic motions with respect to the attenuation law approach, since marked discrepancies may subsist in the “biased” comparisons, such as the one that has just been presented. It is evident that these discrepancies between the calculations and reality can only become higher in the case of the use of digital models constructed, in principle, without any adjustments of the recorded data. The simulation methods however, are of obvious interest in cases where the formulation of attenuation laws reaches its limitations, that is for the zones situated near major faults, as in such zones we generally have at our disposal data that would allow us to make reasonably reliable models of sources.
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5.3. Practical calculations of the site effects 5.3.1. Models of soil behavior
The calculation of site effects calls for the characterization of the mechanical properties of the layers of soil. In the simplest models, which use the hypothesis of linear behavior, there are four properties: – two constants of the theory of elasticity that are usually the shear modulus (i.e. Lamé’s second coefficient P, but for which the notation G is usually used instead of P,) and the Poisson coefficient Q; – the volumic mass U; – a damping viscoelastic parameter for which we can find different notations ([, E or D) when it is the reduced damping coefficient; the quality factor Q, that is inversely proportional to this reduced damping coefficient (see equation [3.55] in section 3.2.3) is used particularly by seismologists. The shear modulus G is linked to the velocity of propagation vs of the shear waves by the relation: G = U vs2
[5.37]
The volumic mass U value being generally known with a fair deal of accuracy (in terms of the soil mechanics), this is in practice equivalent to characterizing soil by its modulus G or by its wave velocity vs; the practitioners of the calculation of site effects generally prefer to use vs, that can be measured by different trials in situ and in a laboratory. The hypothesis of linear soil behavior is often used but constitutes a rather crude approximation of reality. The two most important properties, modulus (or wave velocity) and damping, depend strongly on the level of deformation. Figure 5.6 shows typical curves of variation of the modulus and damping with respect to the distortion strain.
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. Figure 5.6. Curves G/Gmax (G shear modulus, Gmax maximum shear modulus with very low deformation) and D (reduced damping coefficient in %) with respect to the distortion (%) of marls
The shear modulus G shown in the figure is the secant modulus, i.e. the quotient of the shear stress W by distortion (shear deformation), traditionally noted as J in soil mechanics. It is the practice to bring G to its maximum value Gmax obtained for a very low level of deformation (see [PEC 84]). Different formulations have been proposed to describe the laws of behavior of soil. We shall limit ourselves to the simplest one, which is the hyperbolic law defined by:
G Gmax
=
J
r
[5.38]
J J r
J being the distortion and Jr, a reference value of the distortion that characterizes the soil. For values of J that are very low compared to Jr, G is close to its maximum value Gmax. The shear stress W is given by: W = GJ = Gmax Jr
J Jr J
[5.39]
The maximum value Wmax of W is reached asymptotically for very high values of the deformation:
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Seismic Engineering
Wmax = Gmax Jr
[5.40]
The maximum modulus Gmax is often expressed with respect to the maximum velocity vs, max of propagation of the shear waves by means of relation [5.37], we therefore get the following for Wmax: 2
Wmax = U v s1 , max Jr
[5.41]
The numeric values of the reference distortion Jr typically vary from 5 x 10-4 to 10-3 (or from 0.05 to 0.1 when the distortions are expressed in %); the curve in Figure 2.21 corresponds roughly to Jr = 10-3 (since according to [5.38] G/Gmax = 0.5 is obtained for J = Jr). The reduced damping coefficient D can also be represented by a hyperbolic law in the form:
D D max
J
=
[5.42]
J J r
where Dmax is the maximum value of this coefficient (roughly equal to 0.20, in the case shown in Figure 5.6) and Jr the same reference distortion as that of equation [5.38]. The approximate values obtained by relations [5.38] and [5.42] are represented by the dotted curves in Figure 5.6; they are sufficiently close to the experimental curves so that they can be used for the estimation of the orders of magnitude. The level of deformation, i.e. the decisive parameter as regards the choice of the modulus and damping values in linear modules of calculation, depends essentially on the level of seismic excitation; we have seen in section 3.2.1 that for a sinusoidal wave, propagating at velocity c, the maximum value Jmax of the induced deformation in the soil is given by the formula (see [3.11]):
Jmax =
V c
where V is the maximum particle velocity.
[5.43]
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This relation generally results in a low level of seismic deformation. By taking V = 0.3 m/s, which is already a very strong motion (corresponding, for example, according to formula [5.9] to the effect of an earthquake of a magnitude of 7 at 20 km for which we get an acceleration of 0.35 g using formula [5.36]) and a velocity c of 300 m/s (typical for good quality embankments), formula [5.43] gives Jmax = 10-3 (or 0.1%), a value close to that of Jr in Figure 5.6. This remark has two very important practical consequences: – the soil characteristics deduced from static trials of soil mechanics (like, for example, the static penetrometer) are not suitable for seismic calculation models since they generally correspond to very high deformation levels; engineers, in their everyday language, commonly refer to the “dynamic characteristics of soils” (with elasticity modulus values that are often much higher that the “static” values) that would be necessary for seismic calculations; in reality, it is not the static or the dynamic character of the soil that counts but simply the level of deformations; the dynamic aspect comes into play only for certain effects linked to compressibility (Poisson’s coefficient) in saturated soils that are not very permeable, where variations in volume require water migrations that have time constants that are very high; – the level of the motions for which the importance of nonlinear effects becomes essential in soil behavior, is stronger than was thought at the beginning of the calculations of the site effect; we shall see in section 5.3.3 that nonlinearity has an attenuating effect on the amplification of the layers of the soil only if the acceleration of the reference site (rock outcropping on the surface) attains a value of 0.4 g. The generally low level of soil deformation during earthquakes explains the success of the “iterative linear” calculation model, i.e. by a series of linear calculations whose parameters are adjusted according to the results obtained by the previous iteration. This calculation model is commonly used in the study of site effects (see section 5.3.2) and that of soil structure interaction (see section 16.2); it does not of course allow the estimation of irreversible effects, but is sufficient to determine the most important characteristics of seismic response for common levels of excitation. In these iterative linear calculations we generally use the hypothesis that the average level of deformation Jave for which we will calculate the modulus and damping for the following iteration, is a given fraction D of the maximum deformation Jmax [5.43] that is:
Jave = D Jmax = D
V c
[5.44]
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The value of D is generally equal to approximately 2/3. The velocity of propagation c that appears in [5.44] is the velocity vs that itself depends on the level Jave of deformation; taking into account relation [5.37] and admitting approximation [5.38], we rewrite equation [5.44] in the form:
J = E 1J
[5.45]
where we took:
J =
J .moy Jr
;E=
DV J rQ s ' max
[5.46]
Parameter E, which represents the value of the reduced deformation J when we calculate Jave by taking c = vs, max in [5.44], i.e. without taking into account the influence of the deformation on the velocity, constitutes a measure of the importance of nonlinearity. The reduced deformation J is a function of this single parameter that we determine by solving equation [5.45]:
J =
E 2
[E + 4 E ² ]
[5.47]
Instead of this solution as the root of a second degree equation, we can give an iterative solution for [5.45] following the principle of “linear iteration”, given above. This operation that always converges in the case of equation [5.45] is shown in Figure 5.7 for E = 2/3 (which corresponds to the values V = 0.3m/s, vs, max = 300 m/s, Jr = 10-3, D = 2/3 used here above).
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Figure 5.7. Average level of deformation in the case E = 2/3, obtained through iteration
We see that from J = E = 2/3, the solution J = 0.925 given by [5.47] for E = 2/3 is in reality reached after only three iterations. In this case, the average deformation is roughly equal to Jr which corresponds to dividing the modulus G max with very low deformation by 2; this is due to a distinct influence of nonlinearity. In cases where the seismic excitation is lower (accelerations in the range of 0.2~0.3 g, velocities in the range 0.15~0.20 m/s) the reduction factor of the modulus is only in the range of 0.7~0.9. While admitting the arbitrary character of such a definition, we can propose the following criteria for the assessment of the importance of nonlinearity: V > Jr vs, max
[5.48]
which corresponds to the hypothesis that the case previously studied (E = D = 2/3), constitutes an extreme case, separating the area where the nonlinear effects are moderate and can be taken into account by equivalent linear models (V< J r Vs,max) from the area that is clearly nonlinear (V > Jr Vs, max). We can understand the relatively abrupt character of the transition between these two areas by calculating the attenuation factor of a layer of soil by means of a single nonlinear model.
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We shall use equation [3.53] of the assessment of energy during the propagation of a sinusoidal plane wave, i.e., by taking S = Cte and by replacing the quality factor Q by 1/(2[) (see [3.56]), [ being the reduced damping coefficient: (cv²) x + dx – (cv²) x = – 2Z[ v² dx
[5.49]
that is: d dx
(cv²) + 2Z[ v² = 0
[5.50]
We assume that the velocity of propagation c and the damping [ can be calculated by the hyperbolic laws [5.38] and [5.42], for a given level of deformation, with respect to the particle velocity v by a relation in the form given in [5.44]; hence we have: J = Jr
v=
1 w²
[5.51]
w²
Jr
cmax
D
1 w²
[5.52]
w
[ = [ max (1-w²)
[5.53]
w being defined by: w=
c
[5.54]
cmax
i.e. the decrease in the velocity of propagation with respect to its value with very low deformation cmax (= vs, max). By carrying forward relations [5.52], [5.53] and [5.54] to [5.50], we obtain: d ª (1 w²)² º
dx «¬
w
[ max (1 w²) »¼ + 2Z c w² max
3
=0
[5.55]
Calculation Models for Strong Vibratory Motions
Figure 5.8. Variations of F (w) =
2w
-
1
Ln
1 w
1 w ² 2 1 w with respect to w = c/cmax
and of v* =
1
w
247
w
that is:
[ max 4 w² º ª 1 dx «1 w² (1 w²)² » dw = 2Z c ¬ ¼ max
[5.56]
The analytic integration is easy and leads to the relation: F ( w2 ) – F ( w1 ) = G
[5.57]
in which: F (w) =
2w 1 w²
–
1 2
Ln
1 w 1 w
[5.58]
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G=2
[. max Z h, h being the thickness of the layer, w and w2 are the values of w at c max
the limits of the layer. Figure 5.8 shows the variation of the function F (w) and the reduced particle velocity X* defined by: v* =
D
v
J r cmax
1 w
w
[5.59]
In Figure 5.8, we can see the differences between the two areas, almost totally linear and clearly nonlinear, which have been mentioned above: – for values of w that are a little less that 1 (up to 0.8 approximately), F (w) varies very rapidly and this is adequate for equation [5.57], whatever the value of G (therefore of the thickness), with two similar values w1 and w2; the reduction in particle velocity (curve v*) due to the inelastic attenuation is moderate; – for lower values of w (approximately lower than 0.6), F (w) varies slowly whereas we observe the opposite for v*; the attenuation of the particle velocity is very strong, even for layers that are relatively thin. The conventional limit defined by [5.48] corresponds to Q* = D; if we take D = 2/3, as indicated here above, we see in the figure that the associated value of w is approximately 0.7, which indicates that G/Gmax is equal to 0.5. 5.3.2. Seismic responses of columns of soil
The calculation of the seismic response of a column of soil is one of the basic tools of earthquake engineering, not only for the assessment of site effects that is the subject of the present section, but also for the definition of data to be entered for the models of soil-structure interaction by the finite element method (see section 16.2). It consists of representing the soil by a stack of horizontal homogenous layers whose thickness and mechanical properties (modulus or wave velocity, volumic mass, damping) are deduced from in situ reconnaissance (drilling) and laboratory research (tests on samples); this model is subjected to the action of a wave train in vertical propagation, a stress train or a compression train and its response is calculated digitally depending on the hypothesis adopted for the mechanical behavior (linear, iterative, nonlinear) of the soil. This type of calculation constitutes an acceptable simplification of real conditions for sites that have stratigraphy that is mainly horizontal (deep valleys without steep sides, sedimentary basins) where the impedance contrast with the subjacent rock is strong enough to admit an almost vertical incidence of the seismic rays (see section 3.2.2); such sites are very often
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seen in zones that are reserved for urbanization and industrial development. The seismic response of the columns of soil therefore play an extremely important role in the study of site effects, more so as it is, as indicated in section 5.3.3, the only calculation mode of these effects whose reliability can be considered to be assured and which does not require a huge volume of data. In order to understand the principal aspects of these responses of the columns of soil, especially those relating to the boundary conditions to be introduced in the models, it is useful to introduce the notion of characteristic line, which is standard in the theory of the equations of wave propagation. We consider the very simple example of a one-dimensional propagation in a homogenous environment with zero damping; we shall name the particle displacement u (z, t), z being the vertical coordinate (parallel to the direction of propagation) and t, time; this displacement is horizontal (perpendicular to the direction of propagation) for shear waves, vertical for compression waves; here we shall limit ourselves to shear waves for which shear stress W is given by:
W =G
wu wz
= Uc²
wu
[5.60]
wz
(U = volumic mass, c = velocity of wave propagation) The equation of motion is obtained by evaluating the forces acting on an element of a unit section of infinitesimal thickness dz:
U
w ²u wt ²
=
wW
[5.61]
wz
By introducing the velocity v =
wu wt
and by deriving [5.60] with respect to time,
we obtain the system:
U wv wW °° wt wz ® 2 ° wW U c 2 w u °¯ wt wzwt
Uc
2
wv wz
[5.62]
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that can also be written as: w § W · wv c ¨ ¸ °° wt wz © U c ¹ ® ° w § W · c wv °¯ wt ¨© U c ¹¸ wz
[5.63]
from which, by addition and subtraction, we deduce: W · w § W · w § ° wt ¨© v U c ¸¹ c wz ¨ v U c ¸ 0 ° © ¹ ® ° w §v W · c w §v W · 0 °¯ wt ¨© U c ¸¹ wz ¨© U c ¸¹
[5.64]
This system consists of putting down the equations of motion in their characteristic form; this comes from the observation that an equation in the form: w) wt
+a
w) wz
=0
[5.65]
shows that the quantity ) remains constant, when we follow the motion of a point which moves with velocity a; in fact, the variation of ) for such a displacement, for an infinitesimal time dt is written as: d)
) ( z adt , t dt ) ) ( z , t )
dt ( a
w) w) 2 ( z, t ) ( z , t )) ( dt ) wz wt
Thus, if [5.65] is checked, we see that ) is constant (since d)/dt tends to be zero when dto0) for displacement at velocity a. System [5.64] corresponds to the definitions:
)=vr
W ;a= Bc Uc
[5.66]
We thus obtain: v+
W Uc
= Cte on the lines
dz dt
=–c
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v–
W Uc
= Cte on the lines
dz dt
=c
251
[5.67]
dz
= r c are characteristic curves of the system, for a homogenous dt medium, c is constant and the characteristic curves are straight lines r c in the plane of variables t and z. The lines
These properties of conservation along with the characteristic curves make it possible to solve the problem known as deconvolution very easily, i.e. the determination of the downward motion when that on the surface is known.
Figure 5.9. Use of characteristic lines for deconvolution in the case of a homogenous soil a) and for a bilayered soil b)
We shall first consider the case of a homogenous soil a (part a of Figure 5.9); the laws of conservation [5.67] make it possible to write: vM +
vM –
WM Uc WM Uc
= VA +
= VB –
WA Uc WB Uc
[5.68]
with A and B being the points of the free surface (z = H) which are on the characteristics with slope - c and + c that pass through any point M. At these points, the shear stress is zero (WA = WB = 0) since we are on the free surface and the velocity is known by hypothesis, in the form of a given function of time, vs (t); thus, by adding member to member the two equations of system [5.68] we obtain: vM (z,t) =
1 ª
H z H z º ) vs (t ) vs (t « c c »¼ 2 ¬
[5.69]
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In the same way, for bilayered soil (part b of Figure 5.9), we obtain: – in the lower layer (density U2, wave velocity c2): vM +
vM –
WM
= VA +
U 2 c2 WM
= VB –
U 2 c2
WA U 2 c2
[5.70]
WB U 2 c2
– in the upper layer (density U1, wave velocity c1) considering the nullity of the stresses at points C, D, E and F of the free surface:
W vA +
vA –
vB +
vB –
A
U1c1
WA U1c1
WB U1c1 WB U1c1
= vc
[5.71]
= vD
= vE
[5.72]
= vF
From these ratios we can calculate VM with the help of basic calculations; we thus obtain: vM (z,t) =
+
1 4
(1
1 4
(1
U1c1 ) U 2 c2
H h z h H h z h º U1c1 ª ) «vs (t ) vs (t )» U 2 c2 ¬ c2 c1 c2 c1 ¼ H h z h H h z h º ª «vs (t c 2 c1 ) vs (t c2 c1 ) » ¬ ¼
[5.73]
a relation in which vs (t) is the function that defines the given velocity on the surface and h is the thickness of the top layer. We easily generalize for ground containing n number of layers, where we obtain [BET 99]:
Calculation Models for Strong Vibratory Motions
vM (z, t) =
1 2
n
¦
H
n 1
n 1
n 1
i 1
i 1
i 1
(1 + ¦ H i ri ) (vs(t + ¦ H iW i ) + vs (t – ¦ H iW i ))
i
253
[5.74]
with: – Hi = r I for i variant 1 to n-1; – Wi = time taken by the seismic waves to cross-layer i; – ri = impedance ratio between layers i and i + 1 =
Ui ci Ui 1ci 1
;
= sum of all possible 2n-1 combinations of Hi.
– ¦ Hi
Formula [5.74] or its particular cases [5.69] and [5.73] emphasize the general tendency of decrease in amplitude of motion with depth; indeed, velocity vM (z, t) is a linear combination of 2n terms, each using a value of the given velocity at the surface at a moment that is staggered with respect to time t considered at M, and the sum of all the coefficients of this linear combination is equal to one, as can be easily verified. If all these coefficients are positive (as is the case, for example, in [5.73] if the impedance ratio r1 = U1c1/U2c2 is less than 1), the maximum of the vm (z, t) modulus is at most equal to the maximum Vs of the vs (t) modulus and they can only be equal if this maximum Vs has been reached simultaneously by all the points of the surface which intervene in the second member of [5.74], which is not very probable. In these conditions (all positive coefficients) a decrease with depth is thus the rule. If some of the coefficients are negative (which happens in [5.73] if the impedance ratio is more than 1, i.e. if the deep layer is stiffer than the upper layer), it is possible that the amplitude of the motion could be higher at a depth than at the surface. If the time taken to cross the layers Wi is the same or longer than the dominant period of the seismic signal, we can admit that the values of the function vs which intervene in the second member of [5.74] are independent in a statistical sense. We can thus calculate the maximum VM of the velocity at the point M by the quadratic combination rule which has been already used on several occasions (see sections 4.1.3, 4.3.4 and section 5.1); we thus obtain, taking into account the structure of formula [5.74]: 1/ 2
ª 1 n 1 ² º = « n (1 ¦ ri ) » i 1 Vs ¬2 ¼
VM
[5.75]
254
Seismic Engineering
This simple formula was proposed for a bilayered case [SAT 88], then extended to a multilayer case [BET 99]. It predicts a marked reduction in the depth amplitude (as compared to that on the surface) and tallies rather well with the reductions measured in instrumented drillings which were carried out mainly in Japan. Table 5.1 presents the characteristics of 9 of these sites and a comparison of the reductions observed and calculated by [5.75] at 25 recording points (whose depths vary from 15 to 950 m) for a total of 82 recorded earthquakes. Site
Dahan
Etchujima Funabahi
Garner Hachinobe
Iwaki
Kushiro
Tinioka
he
50
40
20
15
100
100
50
22
90
22 220
200
C1
0-7
0-26
0-14
0-18
200, 2000 120, 1800 180, 1600 190, 2000 C2
7-106
26-38
14-32
18-22
Tokyo Tatsumi
Valley 20
0-2
70
100
21
130
251
50
200
660
90
330
950
0-20
20
0.6 .85
0-90
0-22
65, 1550 1000, 2000 125, 1600 520, 1600 140, 1600 2-7
20-290
6.85-17.15
90-140
22-51
400, 2070 230, 1800 300, 1800 600,2200 180, 2600 1450, 2200 310, 1700 700, 1650 210, 1700
C3
106-151
>38
32-77
>22
7-12.5
>290
>17.15
140-475
51-59
500,2080 450, 1800 480, 2000 2000, 2400 145, 1600 2500, 2450 650, 1900 930, 1750 580, 1800 C4
>151
77-83
12.5-15.5
1000, 2100
260, 1800
460, 1850
>83
>15.5
500, 2000
700, 2200
C5
475-830
59-69
1250, 1800 260,1800 >830
69-88
2200, 2450 500, 2000
C6
>88 720, 2200
Ns
23
4
1
2
20
4
16
4
(c-o)1 1.80-2.33
8
1.77-2.69
1.77-2.40
1.41-2.07
2.39-2.89
1.70-2.00
2.46-4.20
1.62-3.02
1.41-1.20
(c-o)2 1.80-2.67
2.28-3.31
2.24-2.72
1.92
1.70-2.00
1.99-4.67
1.69-2.54
2.30-2.90 2.63-2 .69
1.70-1.82
2.51-4.27
3.00-4.20
2.63-4.13
2.19-2.35
3.43-4.37
(c-o)3 2.76-3.73 (c-o)4
Table 5.1. Geotechnological characteristics and comparison between observed and calculated reduction for 9 sites [BET 99], all situated in Japan except for Dahan (Taiwan) and Garner Valley (California)
Calculation Models for Strong Vibratory Motions
255
The different lines of the table represent the following: – he: depth (in m) of the points of recording; – ck: characteristics of the k-th layer of soil; 1st line: limit depth (in m) of the layer. 2nd line: shear wave velocity (m/s) and density (kg/m3). – Ns: number of earthquakes recorded on the site; – (c-o)l: comparison of the values (c) calculated by formula [5.75] with the observed value (o), for the average of the recorded earthquakes, at the l-th recording point (in ascending order of the depths of the line he). We see in this table that the 9 selected sites represent a large variety of conditions from the point of view of the stiffness of the upper layers, the more or less regular nature of the variation in stiffness depending upon the depth and maximum depth of the recording points. The last lines of the table (marked (c-0)l) make it possible to compare the attenuation factor (i.e. the ratio of amplitude on the surface divided by the depth amplitude or VS/VM) with the earlier variations, given that it is calculated by the formula [5.75] or that it is obtained by taking the average of the values measured at this point during the recording of the earthquakes. We find that: – the experimental attenuation factors, apart from very rare exceptions, are close to (point 21 m from Tokyo – Tatsumi, point 200 m from Iwaki), equal to or higher than 2; – the attenuation factor tends to increase with depth; – significant values of the attenuation factor can be achieved at low depths (Etchujima, Funabahi, Hachinohe, Kushiro) that only underline the influence of the stratigraphic profile; – the attenuation factors calculated by [5.75] are lower than the measured factors, with the only exception being the point 21 m from Tokyo – Tatsumi. By adjusting the minimum squares, we can establish the following relationship between the observed attenuation factor ROBS and the calculated attenuation factor RCAL with formula [5.75]:
ROBS = 1.36 RCAL + 0.06
[5.76]
This law of regression is, as usual, characterized by great dispersion since the average quadratic deviation is equal to 0.60. This tendency of dispersion also exists
256
Seismic Engineering
to a lesser extent for a single recording point where the attenuation factor observed can significantly vary from one earthquake to another. Figure 5.10 illustrates the results of Table 5.1; it is a diagram (RCAL (abscissa), ROBS (ordinate)), where the 25 recording points of the 9 studied sites are shown by symbols (white or black circles or triangles) that are variable according to the depth. We observe that all the points except for 2 are inside the beam formed by the two straight lines ROBS = RCAL and ROBS = 2RCAL and that the influence of depth is not very clear. Regression relation [5.76] is represented by dashes. Formula [5.75] thus provides a reasonable estimate of depth attenuation or, inversely, of surface amplification (site effect) in the case of a multi-layered site.
Figure 5.10. Comparison of the observed depth attenuation factor (ROBS) and calculated depth attenuation factor (RCAL) by formula [5.75] for the 25 recording points of Table 5.1
Such a formula can be used to obtain an order of magnitude of the site effect but it is obviously insufficient to undertake detailed studies since it does not take into account the influence of wave frequency. We have already noted in section 4.3.5 the phenomenon of attenuation of certain frequencies at the base of a soft layer which is found on top of compact bedrock, when it is excited with resonance. This phenomenon is a direct consequence of formula [5.69]; if we take in this formula a sinusoidal function for vs (t), we find the following for vM (z,t):
Calculation Models for Strong Vibratory Motions
ª H z º » sin Zt ¬ c ¼
vM (z,t) = cos «Z
257
[5.77]
Z being the pulsation taken for the sinusoidal function. We see in [5.77] that vM (z,t) = 0 for all the pairs of values Z, z such that:
Z
H z c
= (2l – 1)
S 2
, l = l, 2, …
[5.78]
For a given depth z, the suppressed frequencies g are thus:
g = (2l – 1)
c 4( H z )
, l = 1, 2, …
[5.79]
l = 1 corresponds to the fundamental frequency of a layer of thickness H - z (see 4.38), l>1 to the harmonics of this frequency. The deconvolution calculations presented earlier rely on the assumption of zero damping. The presence of a damping term somewhat modifies the preceding conclusions; the frequencies given by [5.79] are no longer completely suppressed but are only greatly attenuated; we can understand the importance of this attenuation by referring to Table [4.9] or equation [4.36]. The existence of these strong attenuations (which for weak damping and moderate impedance contrasts almost result in total suppression; see Table 4.9) implies the definition of a boundary condition of a particular type, at the base of the columns of soil in the calculation models; this question is of great practical importance since the choice of the base of the columns is often arbitrary, more often dictated by need (we need to stop somewhere) rather than by geology, when we have to descend a great depth into the ground to find a well characterized rock substratum. In order that the base of the soil column, such as the one used in the calculation model, enables a correct representation of the phenomenon of attenuation (or strong attenuation) of certain frequencies, this base should be prevented from behaving like a completely reflecting border with respect to descending waves, which would be the case if we force motion on it. We get this result by forcing a viscous damping type condition; for example, if the base of the column is an arbitrarily selected artificial border within a deep layer, whose volumic mass and wave velocity are designated by U and c, it is necessary to force on this border the condition:
258
Seismic Engineering
v=
W
[5.80]
Uc
between velocity v and shear stress W. To justify [5.80], we consider a descending wave, represented in plane t, z by the characteristic SM (see Figure 5.11); we trace the characteristics of the positive slope + c which pass through S and M and cut the artificial border (traced by dashes next to z = zg) into B and A.
Figure 5.11. Definition of an absorbent border condition on plane z = zg located in a homogenous layer
By writing conservation ratios [5.67], we obtain the system:
vM +
vM –
vs –
WM Uc WM Uc Ws Uc
= vs +
= vA –
=–
WB Uc
Ws Uc WA Uc
[5.81]
[5.82]
[5.83]
Calculation Models for Strong Vibratory Motions
259
By adding equations [5.81] and [5.82] and by removing Ws by means of [5.83] we obtain: v M = vs +
1 §
W · 1 § W · VA A ¸ – VB – B ¸ ¨ ¨ Uc ¹ 2 © Uc ¹ 2 ©
[5.84]
We see that it is necessary to force condition [5.80] on the artificial border so as to get vM = vs, i.e. so that the presence of this border does not disturb the propagation of the descending wave. Iterative linear deconvolution calculations are a part of practices prevailing since the mid-1970s; they are carried out with models of columns of soil whose maximum depth is generally about 50 to 100 m, sometimes less if the bedrock is close to the surface. In large sedimentary basins the rock is often located several hundred meters deep and it is common to place the base of the column in relatively deep and compact sedimentary layers by forcing on it an absorbent border condition, defined according to the earlier method. Table 5.2 shows the data and results of a deconvolution calculation for a soil profile whose mechanical characteristics improve steadily with depth; the soil column has a height of 100 m; its base does not correspond with the bedrock, which is located much lower, but with a layer of good quality marl (780 m/s for velocity of shear waves with very low strain).
260
Seismic Engineering
U
Cmax
J max
J ave
G
C
[kg/m3]
[m/s]
[%]
[%]
G max
[m/s]
2.5
2,310
180
0.01142
0.00743
0.91
171.7
5.0
2,310
180
0.04055
0.02636
0.74
154.8
3
7.5
2,000
250
0.03765
0.02448
0.75
216.5
4
10.0
2,000
250
0.05042
0.03277
0.71
210.7
5
13.3
2,440
320
0.02842
0.01847
0.80
286.2
6
16.7
2,440
320
0.03782
0.02458
0.74
275.3
7
20.0
2,440
320
0.04619
0.03002
0.72
271.5
8
23.3
2,230
410
0.03224
0.02096
0.78
362.1
Layer no.
z [m]
1 2
9
26.7
1,810
455
0.03645
0.02369
0.76
396.7
10
30.0
1,500
500
0.03902
0.02536
0.76
435.9
11
35.0
1,860
580
0.02218
0.01442
0.82
525.2
12
40.0
2,000
500
0.03376
0.02194
0.77
438.7
13
46.7
2,140
540
0.02761
0.01795
0.81
486.0
14
53.3
2,140
540
0.03036
0.01973
0.78
476.9
15
60.0
2,140
540
0.03183
0.02069
0.78
476.9
16
67.5
2,080
600
0.03034
0.01972
0.78
529.9
17
75.0
2,110
620
0.03073
0.01998
0.78
547.6
18
85.0
1,930
720
0.02668
0.01734
0.80
644.0
19
92.5
2,050
780
0.02164
0.01406
0.85
719.1
20
100.0
2,050
780
0.02274
0.01478
0.85
719.1
Table 5.2. Iterative linear calculation of deconvolution for a regular soil profile
The columns of the table contain the following information: – column 1: layer number; – column 2: minimum depth of each layer; – column 3: volumic mass of the layer (kg/m3); – column 4: shear wave velocity in the layer for very low strain; – column 5 to 8: deconvolution calculation results for a maximum acceleration of 0.3 g; - column 5: maximum distortion (in %) in the layer,
Calculation Models for Strong Vibratory Motions
261
- column 6: average distortion (in %) in the part equal to 0.65 times the maximum distortion in the layer, - column 7: reduction factor of the modulus in the layer, - column 8: wave velocity for average distortion. Deconvolution calculation has been carried out for a given motion on the surface, with maximum acceleration 0.3 g. It is seen that the reduction factor of the G/Gmax modulus is in the range of 0.7~0.8; the dependence of this factor with respect to the strain corresponds to the curve in Figure 5.6, on which we can check that the strains for this level of reduction of the modulus are about 0.02–0.04%, as in column 6 of Table 5.2 The validity of the consideration of nonlinearity by iterative linear calculation is ensured for this range of strains [MOH 93]. The analysis of the accelerograms obtained at different depths in this deconvolution calculation highlights a rather rapid decrease in the amplitude of the motions when the depth increases. This can be understood by examining the curves of Figure 5.12 which represent function I (t) defined by equation [4.2], i.e. the integral up to time t of the square of acceleration. For a surface signal having a broad frequency band (upper part of Figure 5.12), the decrease in I (t) with depth is relatively steady; maximum acceleration goes from 0.3 g on the surface to 0.12 g at the base of the soil column. For a surface signal having a very short strong part with a very clear dominant frequency of around 8 Hz (lower part of Figure 5.12) we observe a very rapid fall in I (t) from a depth of 5 m and a steady decrease beyond that; however, with c = 154.8 m/s (see last column of Table 5.2 at the base –5 m of the second layer) formula [5.79] gives a frequency of 7.74 Hz for l = 1 and Hz = 5 m, i.e. a value very close to the dominant frequency of the surface accelerogram; the sharp decrease in I (t) is thus a result of the quasisuppression of this frequency at a depth of 0.5 m. We thus observe, as indicated in section 2.2.3.4, the dependence of the response of a column of soil with regard to the frequential content of the excitation.
262
Seismic Engineering
Figure 5.12. Integral I (t) of the square of acceleration for various depths: Ɣ 0 m, ż 5 m, Ÿ10 m, ¨30 m, Ƈ100 m; part a) above: surface accelerogram at 0.3 g presenting a broad frequency band; part b) below: surface accelerogram at 0.3 g having a very marked dominant frequency in the vicinity of 8 Hz and a short duration
Linear iterative calculation, used in this example, reaches its validity limits for very intense excitations of the column of soil; criterion [5.48] given in section 5.3.1 provides a superficial estimate of these limits. Various nonlinear models have been proposed and they make it possible to calculate the response of the columns whatever the level of incidental motion; the one-dimensional character of the model ensures that the cost of these calculations remains moderate. Their results highlight a saturation phenomenon, i.e. the acceleration of the motion on the surface cannot exceed a threshold value which depends upon the properties of the various layers of soil. This results in a modification of the site effect, the amplifying character of the soil response gradually attenuates as the excitation of the bedrock increases until the former disappears (amplification becomes attenuation) at a certain level. We can better understand this phenomenon and determine orders of magnitude by means of the simple model used in section 5.3.1 to calculate the attenuation factor of a layer of soil and arrive at equation [5.57] which is written as:
Calculation Models for Strong Vibratory Motions
F (w2) – F (w1) = G
263
[5.85]
with:
F (w) =
2w 1 w²
–
1 2
Ln
1 w 1 w
;w=
c
[5.86]
cmax
w1, w2 = values of w at the entry into and exit from the layer; G = 2
[ max cmax
Zh, h
being the thickness of the layer and Z, the pulsation of the wave. We introduce as a motion parameter, the dimensionless quantity p defined by: p=
2J rCmax
[5.87]
Dv
where Jr (reference distortion), Cmax (velocity of shear waves with very low strain), D (coefficient introduced in equation [5.44] and v (particle velocity) have been defined in section 5.3.1. This results in equation [5.52] which we can write as: p=
2w
[5.88]
1 w²
from which we can deduce for F (w): F(w) =
2w 1 w²
–
1 2
Ln
1 w 1 w
=p–
1 2
Arcsinh p
[5.89]
The bedrock situated under the layer of soil has a volumic mass Uo, a wave velocity co; the impedance ratio J is thus for very low strains:
O
=
U c
o o
U cmax
[5.90]
264
Seismic Engineering
To determine the w1 value of w at the base of the layer, we use the reasoning proposed in section 3.2.2 (conservation of energy flow in a force tube) that gives the relation:
U o co v o = U1c1v1 2
2
[5.91]
where vo is the particle velocity in the bedrock, v1 is the particle velocity in the soil at the base of the layer where the wave velocity has a value c1 = w1 cmax; taking into account equation [5.52] and definitions [5.87] and [5.90], we can easily show that [5.91] is written as:
Po = O p1 (1 1 p ²1 )
[5.92]
where Po is motion parameter [5.87] in the bedrock and p1 the motion parameter at the base of the layer. According to equations [5.86] and [5.89], motion parameter p2 on the free surface is such that:
P2 –
1
Arcsinh P2 = G + p1 –
2
1 2
Arcsinh P1
[5.93]
It is convenient to take p1 as the calculation parameter; formula [5.92] makes it possible to calculate p0 and transcendental equation [5.93] can be solved by iterations, by writing it in the form:
P2 =
1 2
Arcsinh p2 + G + p1 –
1 2
Arcsinh p1
[5.94]
The borderline case corresponding to the saturation phenomenon evoked earlier is obtained by taking p1 = 0, i.e. infinite amplitude of the motion, as much at the base of the layer as in the bedrock; the threshold value of p2 or P2,l is thus the solution of equation [5.93] when we take p1 = 0:
P2,l
1 2
Arc sin h P2,l
G
[5.95]
Figure 5.13 shows the variation in 1/p2,l which is a parameter proportional to motion amplitude; (see [5.87]) with respect to G.
Calculation Models for Strong Vibratory Motions
265
Figure 5.13. Variation of the maximum amplitude of the motion at the surface of a layer of soil, with respect to parameter G
We see in Figure 5.13 that the saturation phenomenon can be seen for layers of soil whose G parameter is around 1, or higher than 1. Equations [5.92] and [5.94] make it possible to calculate the variation of the amplification produced by the layer of soil with respect to the level of excitation at the bedrock. The curve in Figure 5.14 represents parameter as , quotient of the acceleration at the surface by its maximum value P2,l, with respect to ar , quotient of the acceleration at the bedrock by this same maximum value; the curve was drawn for the values O = 4 and G = 1, that correspond, for example, to the choice Uo = 2,400 kg/m3, co = 1,000 m/s, h = 30 m, U = 2,000 kg/m3, cmax = 300 m/s, Jr = 10-3, [max = 0.25, Z = 20 rd/s (frequency of 3.18 Hz), D = 2/3.
266
Seismic Engineering
Figure 5.14. Variation of acceleration at the surface of a layer of soil, with respect to the acceleration at the bedrock, they are added to the maximum surface acceleration which can be transmitted by the layer
We note in this figure that the layer behaves as an amplifier of the incidental motion for low excitation levels (since curve a s is above the straight line a s ar ; this tendency is inversed when acceleration at the bedrock reaches approximately 0.83 times the maximum acceleration, a value after which there is attenuation. This observation was made in almost all numerical studies using nonlinear models (see, for example, [MOH 92]). In the early 1980s the widespread opinion was that nonlinear effects were strong enough to appreciably attenuate the amplification of the soil layers under the existing thickness and stiffness conditions of these layers. The accumulation of recorded data and the progress made in calculation models have since shown that the amplifying effect persists up to relatively high levels of motion (frequently about 0.3~0.4 g in acceleration). The threshold acceleration al, which can be transmitted at the surface, can be easily calculated from the threshold value p2,l, defined by equation [5.95]; we indeed obtain, according to [5.87]: al = Z vl =
2J c r max Z D P2,l
[5.96]
Calculation Models for Strong Vibratory Motions
267
or again, by removing Z depending upon G: al =
J c ² max G r D[ max hP2 A
[5.97]
The quotient G/P2,l is relatively constant; we can show that it lies between ½ and 1 when parameter G varies from 0 to infinity. With the existing values of the parameters which intervene in its definition ([max = 0.15~0.30, Z = 10~30 rd/s, h = 10~50 m, cmax = 150~400 m/s), G is generally equal to about 1 and we can admit as such, that in terms of the order of magnitude, G/P2,l l = 0.6. Formula [5.97] thus gives al = 10.8 m/s² for the values adopted earlier (Jr = 10-3, cmax = 300 m/s, D = 2/3, [max = 0.25, h = 30 m); the value of al falls to 2.4 m/s² if the layer of thickness 30 m is made up of a material having low mechanical characteristics (Jr = 5x10-4, cmax = 200 m/s). The value of acceleration after which the layer of soil no longer has any amplifying effect is a little lower than a1 (equal to 0.83 al in the case of Figure 2.29); it thus varies from 2 m/s² to 8 m/s² in practical cases, with an average value of about 4~5 m/s². 5.3.3. Review of the assessment of site effects
Studies of the response of columns of soil constitute the majority of the studies of site effects. Conducting such studies is a rather routine affair at present since the calculation software allowing this type of study is widely available and adapted to the existing data processing and calculation means. We should however keep in mind certain precautions while using them: – the data characterizing materials of the various layers of soil should preferably be taken from in situ drillings; resorting to tables or curves obtained from technical literature in order to determine such data can lead to serious errors particularly as regards the velocity of shear waves; – the degree of sophistication of the laws of behavior of soil that are adopted must be proportional to the quality of data on materials available; some of the most elaborate laws need a large number of parameters whose determination is really not possible on the basis of data available from the study presently conducted in soils; the validity of the benefits of these models can thus be deceptive; – the choice of the position of the base of the column must, as indicated in the preceding section, be carefully weighed up, even if the condition of the absorbent border makes it possible in theory to overcome the problem of parasitic reflections
268
Seismic Engineering
on an “artificial” base, it is better to ensure that the results are reasonably stable with respect to a change of position of this base; – the accelerograms which define the excitation should be subjected to a critical examination from the point of view of their validity under the site conditions considered (magnitudes, distance from source); if they are used in a deconvolution calculation (motion imposed on surface), they can be incompatible with the characteristics of the site; this is often the case when we try to impose on a surface, a motion having a large range of frequencies while the response of the site is assessed primarily around the dominant frequency. When simplification of columns of soil is insufficient because of the complexity of the geotechnical structure of the site, we must resort to two- or three-dimensional modeling, as indicated in section 4.3.5; practical difficulties are thus often crippling from the point of view of collection of data and assumptions on the incidental wave field; this is why this type of study was, until now, reserved for research operations on experimental sites and some large projects (important installations or sites in large urban zones). If the application of two- or three-dimensional models to real sites remains limited, many text book examples (valleys or hills having simple shapes are treated mostly as two-dimensional) have been studied by researchers [BAR 83]. This has made it possible to detect certain general tendencies which confirm certain observations that were commented upon earlier (see section 4.3.5): – the appearance of significant differential motions above the thickness variation zones (valleys) or on the slopes of land forms; – the “narrow band” character of the range of frequencies for which we observe amplifications in the valleys, contrary to the “broad band” character observed in land forms; – the fact that, in certain configurations, particularly in the case of deep valleys, amplifications can be very strong (up to four times the level that we would calculate with a one-dimensional model); – the difficulty that is often faced in calculating amplifications due to the topographic site effect, which are generally much higher in experimental observations (recordings) than in digital simulations; – prolongation of the duration of motion in valleys, due to the resonance and the appearance of surface waves. Apart from the calculation approach, the estimation of site effects by experimentation has greatly developed since 1990; it consists of two aspects:
Calculation Models for Strong Vibratory Motions
269
– the establishment of empirical formulae that enable the calculation of modification factors of certain motion parameters on the basis of site characteristics; – the determination of the dominant features of the site response (fundamental frequency, factors of amplification) by in situ measurements using background noise (see section 1.3.2) or micro-seismicity; even if they were not recorded, the analysis of the effects of earlier earthquakes that were felt on the site can also give very useful information. We can give as an example of the first method, the proposals made by Borcherdt [BOR 94] which define amplification factors Fa as:
D
§ v · Fa = ¨ r ¸ ¨v ¸ © s,30 ¹
[5.98]
where vr is a reference velocity (in general about 1,000 m/s), Vs, 30 the average value of the shear wave velocity between the surface and a depth of 30 m and D an exponent. From the recordings obtained for the Loma Prieta earthquake (California, on 17th October 1989), Borcherdt thus proposed: – vr = 997 m/s, D = 0.36 for the amplification of the accelerations; – vr = 1067 m/s, D = 0.64 for the amplification of the velocities. The [5.98] type of formulae are rather close to the amplification in O (O = impedance ratio), evoked earlier in [3.31] and [4.31] for particle velocities. They have the advantage of using just one parameter, vs, 30, for the characterization of the site, whose order of magnitude can be estimated on the basis of the description of the subsurface soil or callow, and which is, without any doubt, the most significant (see section 4.3.4). The simplicity of this formulation obviously implies limitations as regards its validity since it does not take into account the frequential aspect, for excitation as well as for site response. The second method (recordings of motions on the site) became very popular in its alternative approach suggested by Nakamura [NAK 89], which uses ground noise as a source of excitation. In addition to its artificial causes mentioned in section 1.3.2, background noise is also the result of natural phenomena such as wind or waves; the low level of such noise requires the use of highly sensitive seismometers to obtain recordings, but its omnipresence, particularly in urban sites, makes it an “additional resource” that is free and always available; techniques based on the use of the background noise are thus easy to implement and inexpensive.
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Nakamura’s method consists of studying the Fourier spectrum ratios of the background noise recordings for horizontal and vertical components and is thus usually called the H/V method. A study of these ratios shows distinct peaks which apparently correlate well with resonance frequencies of the surface layers. The explanation given by Nakamura for the existence of this correlation did not convince the majority of experts (see [BAR 98, KUD 95]), but it is widely recognized that the method correctly predicts the fundamental frequency of the site. Other explanations based on the properties of Rayleigh waves (see section 3.2.1) justify this coincidence of frequency but question the validity of the amplification ratios deduced from the H/V ratio [COR 99]. Whenever the predictions obtained by the Nakamura method could be compared with measurements made at the time of real earthquakes, the consensus on levels of amplification was not very satisfactory. For the moment, given the present level of comprehension of the foundations of this method, we can say that it is an economical and reliable means of determination of fundamental frequency, but it would be rather adventurous to use the method to try and draw other conclusions as regards the characteristics of the site. The study of site effects by means of in situ recordings can also be done by using low level seismic signals resulting from small relatively frequent earthquakes, from the aftershocks of big earthquakes or from explosions. These methods are more complicated than those using background noise because they generally involve taking recordings over several months, analyzing these recordings and then interpreting the results obtained; indeed, we would need to eliminate the specificities related to the sources and the effects of propagation in order to deduce the site effects of the site being studied from the recordings. Figure 5.15 shows the accelerograms recorded on 5 sites of the Kǀbe center during an aftershock with a magnitude of 4.1 of the earthquake on 17 January 1995. These sites, which are about a few hundred meters from each other, are located approximately 10 km south of the epicenter of this aftershock, the site that is located further to the north (KMC) is on a rock, the others on surface layers having weak mechanical characteristics [COL 95].
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Figure 5.15. Accelerograms of the north-south components for an aftershock of the Kǀbe earthquake on 17 January, 1995); the 5 recording sites are in the center of the city, a few hundred meters away from each other (according to [COL 95])
We observe very strong amplifications (factors of about 5 to 10) with respect to the rock; during the main earthquake, the amplifications on the same sites were much lower, about 2. Thus, the question that arises in the case of a strong motion is that of the transposition of the results obtained for weak motions. The problem of the influence of nonlinear behavior has already been mentioned in section 5.3.2, but it seems that other causes can explain these differences in amplifications. We can, for example, consider the influence of the size of the source: for this aftershock it is small and relatively far away from the recording sites (always the R0/R ratio considered in section 2.3.1) which results in a relatively homogenous wave field in the vicinity of the seismometer; for the main earthquake of a bigger size, the sites considered are in fact very close to the rupture zone and the wave field is more complex (see section 5.2.2) giving in all probability, the effect of apparently multiple sources. The existence of site effects was initially shown by macroseismic observations, i.e. by the inventory of damage caused by the earlier earthquakes. In practically all the cases where such an inventory could be made in a sufficiently detailed manner, it was observed that the distribution of damage generally correlated with the geology and the surface topography of the region in question. Without going back in time to the Istanbul earthquake on 12 September 1509 (see section 4.3.3), we can give
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several examples of important cities located in seismic zones (Japan, California, Mexico, Peru, Chile, Italy, Greece, Iran) where since the beginning of the 20th century, the districts most exposed to seismic risk were identified, after destructive earthquakes. Thus, in Tokyo, a study published in 1913, based on the analysis of the destruction produced by the 1854 earthquake, concluded with the division of the city into three zones with different risk levels [BAR 98]. Such studies however have not had much influence on town planning policies since the return period of major earthquakes at a given place is too long, on the human life scale, for the perception of seismic risk to be a determining factor in the context of decision making related to town planning. Since around 1980–1990, there has been a renewed interest in analysis of macroseismic observations made at the time of earlier earthquakes, this is recognized as a means of obtaining at least a qualitative assessment of the importance of site effects. This tendency reverses the excessive loss of interest in non-instrumental studies of seismic phenomena (see section 2.3.1). Such studies can provide information that is very useful but difficult to obtain through other means, even for earthquakes of a relatively moderate level; we can cite the example of the Liège earthquake on 8 November 1983, when a detailed examination of the insurance files relating to compensation for damages made it possible to draw up a chart of damage that presented a strong correlation with the underground map [JON 90]. The main difficulty lies in the quantitative transcription of these observations in terms of seismic motion, that are to be introduced in dimension calculations (see section 14.2). Given the volume of documents that prove the existence of site effects and the age of these documents and the fact that they show that site effects are the rule rather than the exception, it is surprising and rightly so, that they are still sometimes presented as a recent discovery, due to earthquakes in Mexico City (on 19 September 1985) and Loma Prieta (on 17 October 17, 1989), even though precise records, corresponding to earlier earthquakes (1957 for Mexico City, 1906 for San Francisco) exist for these regions and show the distinctive characteristics of the distribution of damage. However, we should remember that it is truly these two earthquakes of 1985 and 1989 that greatly contributed to the development of the studies of site effects (see section 7.2.3 on microzoning). This development has led to some excesses; a recently observed tendency consists of permuting the roles of causes and effects, so as to hide our inability to understand the causes of certain “abnormal” seismic phenomena as a site effect. We thus have an explanatory panacea where site effect plays the same role as that of evil spirits that are used to explain the causes of accidents or diseases. Likewise, the accuracy of quantitative predictions of the site effect is sometimes presented with too much optimism, especially given the practical difficulties of obtaining precise
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and sufficiently complete data and, in certain cases, the lack of knowledge of the physical causes of this effect. Indeed, we should not conceal the fact that the models currently available cannot explain certain manifestations of the site effect. We have talked about the topographic site effect but this is undoubtedly not the only one. Among the various phenomena whose influence on the site effect can be considered under certain conditions, we can mention “city site interaction”; certain studies [GUE 00, WIR 96] indicate that the presence of a large number of big buildings in a densely constructed area is likely to modify the seismic signal significantly, as compared to that observed in a non-built-up or non-developed area.
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Part 3
Seismic Hazards
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Introduction
The notion of seismic hazard was formed only recently; it had for a long time been confused with the idea of seismic risk, i.e. probabilistic estimation of the consequences of earthquakes in terms of human and material loss. Seismic hazard, according to the terminology used today, is also a probabilistic estimation of these consequences, but limited to ground motion which may occur, without any reference to its impact on constructions. This distinction leads us to the understanding that risk and hazard are not necessarily linked to one another in any given region; zones which are almost desert can present a greater hazard and a lower risk; for example, in some parts of Asia (Tibet, Mongolia) large-scale earthquakes (magnitude 8) are relatively frequent, but cause very little destruction, considering the dispersion and the lifestyle (living in tents) of the nomadic populations concerned. On the other hand, high urban concentrations in a vulnerable area constitute a very high risk even if the hazard is moderate. The confusion between hazard and risk, which undoubtedly persists in the mind of the public and also, perhaps, in the minds of some experts in the field of earthquake engineering and design, rises mainly from the following three causes: – the use of macroseismic intensity scales, not only as a tool for description of earthquakes that have occurred, but also as a reference for safety objectives for certain earthquake engineering codes. Two facts are highlighted at the end of section 14.1.3: firstly, the fact that intensity represents mainly an evaluation of the seismic risk for masonry constructions and secondly, its use in characterizing seismic hazard is not satisfactory; – the link between seismic action and ground motion is not explained, except in some of the most recent earthquake engineering and design codes; this is a result of the fact that the first codes appeared in California and Japan at a time when no good
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quality recording was available (see the introduction to the second part). If later modifications of the Californian and Japanese codes, which serve as a reference to “calibrate” the severity of other national codes, have actually taken into account the progress made in the domain of seismology, they are fundamentally engineering codes, where the coefficient or coefficients representing ground motion are not linked very clearly to the definition of a seismic hazard; – the fact that the main challenge for earthquake engineering and design in most countries which, like France, have only recently imposed the application of earthquake engineering rules for new constructions is the reduction of risk for existing construction. This reduction depends on adopting measures which are practical and economically acceptable and do not require precise knowledge of the hazard. Thus, the formulation of seismic hazard in its strictest sense, i.e. in terms of parameters of ground motion, is a recent tendency rather than a well-established norm in earthquake engineering design codes that are applicable to present constructions. It undoubtedly marks a progress from the point of view of logic and the “traceability” of the functioning of the codes, but one must not get too carried away by its practical scope from the point of view of improvement in prevention. Very often the mistake lies in believing that the main issue, in the policy of earthquake engineering, is to define the hazard. In reality, the uncertainties are equally important while calculating the seismic response of structures (see Part 6 or Chapters 15 to 17) and considering the criteria for the evaluation of safety (as indicated in section 12.1.1 and at the end of section 12.3.2). Some current or future earthquake engineering codes conserve or will conserve a formalism which does not do justice to the definition of the seismic hazard in terms of parameters of ground motion (accelerations, velocities, response spectra of elastic oscillators; see section 9.1). This is due to problems of continuity with the earlier versions (so as not to disorient the “ordinary” users of calculations for structures who are not specialists in earthquake engineering), but also due to the fear that an explicitly “seismological” formulation opens the doors to questioning the code every time the recording goes beyond the “prescribed” level. The extreme variability of parameters of seismic motion, especially accelerations, and their rather weak degree of correlation with the damage observed show that such fears are certainly not without warrant. Let us take just one example, the famous recording at Tarzana during the Northridge earthquake (section 4.3.4), with its horizontal acceleration of 1.8 g, was more than four times more than the “maximum” 0.4 g of the Californian code which was then in use, but the level of damage observed in the vicinity of the seismograph was not at all striking.
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The studies of seismic hazard for a given site initially concerned nuclear power plants (since the beginning of the 1970s), and were later applied to other facilities that were a threat to the environment (large dams, some chemical, petroleum or gas factories). The methods developed at the time of these studies can now be used to define hazard on the scale of a region and no longer merely one site. Traditionally, there is a distinction between probabilistic methods and deterministic methods for determining the hazard; those in favor of these two approaches often faced clashes and “rivalry”. However, in reality, the two methods have some points in common: – the database, i.e., all the geological, seismological and historical information, concerning the seismicity of the region studied; – the importance of expert judgments on the interpretation of this data and hence on the results of the study; – the intervention of an “arbitrary” decision on residual hazard, i.e., events with a probability judged weak enough to be ignored: in the case of probabilistic methods, it is a choice based on the level of probability (or the “return period”) of the parameter of motion in relation to which a protection guarantee is required (for example, Eurocode 8 defines a probability of exceeding 10% for a period of 50 years, or a return period of 475 years; see section 6.2.1). In the case of deterministic methods, it is the choice of defining the rules of the “maximum” event, (for example, the rule of increase in one degree of intensity or half a degree of magnitude with respect to the “greatest historically likely earthquake”; see section 6.1.2). Deterministic methods are in fact, methods with a “Manichean probability” according to which all events other than the one considered to be “maximal” conventionally have a zero probability. To begin with, this part presents the spatial and temporal distribution of seismicity from the point of view of data available and the models proposed for its description; the problem of predicting earthquakes has been briefly described in section 6.3. Chapter 7 is devoted to the methods of evaluation of seismic hazard and their usefulness in establishing zonation and microzonation maps.
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Chapter 6
The Spatial and Temporal Distribution of Seismicity
6.1. Data available on the spatial and temporal distribution of seismicity 6.1.1. Geological data All kinds of geological data serve as the basic element for the evaluation of seismic hazard, whatever the methods used, as long as they can be compared to the geodynamic phenomena that cause earthquakes. In fact, the first stage in the study of hazard consists of making a seismotectonic model, which is a schematization of the mechanism responsible for seismic activity in the region in question [BOU 85]. The description of regional geological structures is a necessary step; it enables the identification of major preexisting fractures and units that are homogenous in nature. These big structural characteristics form a framework within which most of the local observations can be noted and which becomes a preliminary sketch for the seismotectonic model. In fact, experience shows that in most cases, the seismicity observed can be linked either to well defined accidents (faults or fault systems) or to diffused zones. In general, this regional data concerns only those terrains that are situated at depths within the reach of standard geological techniques. Taking into consideration the range of depths of the seismic hypocenters (section 1.1.3), it is necessary to complete them with all the information available on deep structures. These are provided by geophysical techniques: seismic prospecting used by the petroleum industry or carried out by research programs, methods based on the electrical
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conductivity, studies conducted on magnetic or gravimetric anomalies. The quantity, quality and “seismotectonic significance” of these geophysical data vary according to the region under study but they are often the only means of acquiring the knowledge of the basement, when it disappears under thick layers of sediment. The geophysical approach is also necessary for the study of the geological structure of marine areas, even near the coasts. Photographs taken by satellites are often used to show lineaments, which are striking alignments that escape terrestrial or even aerial observations. These lineaments can correspond to major faults whose existence was not suspected based on available data on surface geology. Such “discoveries” have often been made in lesser studied regions and the contribution of satellites seems to have been quite limited in countries possessing good geological maps [BOU 85]. The lineaments that the close study of a satellite photograph suggest can have many causes other than the presence of a fault and give rise to controversies; gathering information from satellites is in any case a part of the constitution of the database for any important study of seismic hazard. The data concerning neotectonics, i.e. the current and recent motion (in geological terms, this could be some five million years corresponding to the Pliocene quaternary period) has become very important in recent studies on seismic hazard, when significant indicators attesting the reality of such motion have been shown. The neotectonic manifestations mainly concern superficial terrains and leave behind visible traces, for those who are aware, in the morphology of terrestrial contours (for example deviations of water courses, discontinuity seen in slopes), or in the topography of terrains exposed during construction work (construction sites, trenches made for the construction of roads, drilling tunnels in subterranean cavities, etc.). The neotectonic indicators that are well characterized, that is, clearly attributed to seismic phenomena and not to other causes, can be of great importance for the evaluation of the activity of a fault and the scale of earthquakes that it is likely to produce. Their quantitative interpretation in the studies on seismic hazard often create difficult problems: – dating seismic events corresponding to these indicators; – attribution of the effect observed during the action of a single or several earthquakes (section 6.1.3); – reconstruction of the real motion of the fault based on regular and often incomplete information (for example, when the indicator is a result of an examination of a trench and does not make it possible to estimate the amplitude of
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the motion component in the direction perpendicular to the plane of the crosssection); – correlation of movements observed with the scale of the earthquake, taking into account the application of empirical laws available (section 2.2.3) and the fact that amplitude of the motion can be quite variable across the fault plane. In addition to the indicators of movement having affected faults in the past, measurements taken on the terrain for a sufficiently long period of time can help acquire knowledge of current movement. This involves geodesic study of leveling and in situ measurements of stress; the evolutions of values measured over a period of several years give an estimation of the rate of deformation of active structures and have been used, in some cases, for the purpose of predicting earthquakes (section 6.3). 6.1.2. Historical seismicity Historical seismicity is the study of past earthquakes based on archived documents describing their effects; the nature and reliability of these documents vary greatly according to the time of occurrence of these events and the historical context of the regions concerned. Thus, we find: – cuttings of articles from local or national newspapers; – extracts from chronicles kept by individuals for a certain period which speak of the life in a city or a region; – reports of surveys concerning an earthquake presented by scientific institutions, commissions formed in the circumstances or simply individuals; – administrative reports on the organization of aid or the cost of repairs of public buildings; – registers of deaths maintained by government or religious authorities; – texts inspired by the occurrence of an earthquake that describe the philosophical or religious aspect of the event; – work by historians relating to the period of the earthquake in the concerned region; – compilations by intellectuals or scientists on natural phenomena. In regions where these archives cover a whole era or several centuries, with the certainty of not having left out a single large-scale seismic event, historical seismicity plays an important role in the evaluation of seismic hazard; as in the case of Europe, the Mediterranean basin and the majority of the countries in southern and
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eastern Asia. This is also the case today, even though recent development of neotectonic studies has attracted attention on earthquakes whose “return period” is clearly greater than the duration of historical observations. In other regions of the world, especially California, the period for which sufficient documentation is available is too short to be able to rely only on historical data. Moreover, since some of these regions are prone to seismic activity due to major faults that are clearly visible on the surface, studies of seismic hazard are mainly based on a “fault-based” approach, which does not give much importance to historical seismicity. Strictly speaking, data from historical seismicity is made up of documents in the archives mentioned earlier. The information on the levels of macroseismic intensity are thus not a part of it, since it is made up of interpretations made on the basis of these documents; however, it is common in the matter of historical seismicity, to refer to catalogs compiled by different people for a country or region. This approach has the advantage of being simple and it is sufficient as a preliminary evaluation of the hazard but we must also be aware of its limitations and the risk of errors linked to blind use of catalogs. As indicated in section 14.1.3, the evaluation of the intensity of an historic earthquake, particularly in the epicenter zone is a delicate operation, as there are practically never any statistics on damage available which it would be necessary to know, in order to strictly apply the rules of assigning degrees of the scale. The “serious” catalogs complete the evaluation of intensity with an indication of the degree of reliability of the evaluation, taking into account the number, precision and credibility of the available documents. We need to be very careful as regards the use of catalogs that do not have such an indication, which often means that the author of the catalog has only copied a value of intensity given in an earlier document, without checking its quality. The use of “first hand” documents is indispensable to get good historical seismicity. We shall see later how the “SIRENE” database was created. This file brings together the data, and its interpretation in terms of intensity, for all the earthquakes felt in metropolitan France. Apart from the difficulties in evaluation of intensity, we must indicate those that are related to the localization of epicenters and to the traces of isoseismal lines. In fact, we must have sufficient reliable data so as to localize the epicenter precisely and sketch the contours of some isoseists. The intensities are, by definition, only known in inhabited areas and their distribution can induce an error regarding the real
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position of the epicenter. As indicated in section 1.2.3, the seismological epicenter (projection of the hypocenter on the surface) is often different from the macroseismic epicenter (place where maximum damage has been seen) that necessarily corresponds to a city or a village and can be influenced by the occurrence of site effects. The difficulties of localization of the epicenter are particularly great in less populated regions, mountainous regions (where the intensities “follow” the valleys) and coastlines (where the seismic source is “somewhere in the sea”). It is often the case that the traces of the intermediate isoseismal lines (level V and VI of the 12 degree scale, as mentioned in section 14.1.3) provide a better indication of the position of the epicenter than the one deduced based on the location of the strongest intensities. As far as data on historical seismicity of a given region is concerned, the question about the level of intensity from which they can be considered complete, is important in the probabilistic approaches to seismic hazard. Very old data, in general, includes only the rather big earthquakes (often described in terms that are too vague to enable any reliable evaluation of intensity), since the memory of weaker events was lost. It is only for the recent period (since approximately 1800 in metropolitan France) that we can evaluate the complete database for the intensities higher than or equal to V. Figure 6.1 shows the distribution per century and per level of intensity of earthquakes felt in France.
Figure 6.1. Number of earthquakes felt per century in France for the intensities V, VI, VII and those higher than VIII (according to [LAM 97])
This figure does not show a continuous increase in seismic activity in France, but simply the fact that the information on the relatively weak earthquakes (level V and VI) is less and less complete as we go back progressively in time. It can be observed that the number of earthquakes causing great damage (intensities higher than VII) is
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noticeably constant (a little lower than 10) for the past four centuries, which goes to show that there has been no significant change in the rate of seismic activity over time. On the contrary, the number of events with weak intensity decreases regularly with the increasing passage of time and, with it, the proportion of lost or forgotten minor earthquakes. This observation is important for the adjustment of coefficients for the laws of temporal distribution of seismicity (section 6.2.2). The data on historical seismicity that concerns France has been put together in the SIRENE database, created in 1979 on the initiative of three organizations that were studying seismic hazard (BRGM, EDF, IPSN) and updated permanently since this date; it includes [LAM 96]: – nearly 6,000 events indexed in the metropolitan territory and neighboring countries, and corresponding to a period of more than 1,000 years; – about 80,000 regular observations noted in terms of intensity for French or foreign localities where earthquakes have been felt; – more than 8,000 bibliographical references, representing about 20,000 pages of texts, copies of documentary sources recorded on CD-ROM. The assembly of this database has necessitated the analysis and critical examination of earlier documents (catalogs on seismicity, files/records on macroseismic surveys), but also research on new documents enabling the improvement in knowledge on certain earthquakes. This work has highlighted some errors (“false earthquakes”, confusion with other natural phenomena such as storms or landslides, errors in dates or in location) and lacuna (earthquakes that have been omitted) in old catalogs, as well as the necessity of revising the intensities (from the point of view of their level or extension of affected zones) for some large-scale earthquakes. The configuration of the database enables the use of macroseismic data with the help of software that enables: – the calculation of parameters characterizing seismic sources (magnitude, depth of the hypocenter); – the determination, for different regions, of laws of temporal distribution of seismicity and attenuation laws of intensity with respect to distance; – the full-scale mapping of data of a site or a region, in different modes of representation. Figure 6.2 shows the main earthquakes in France (some of whose epicenters can be situated outside the boundaries); we can state that they are distributed in several zones of activity (Pyrenees, Alps, Rhine valley, a strip going from the southern Britanny-Vendée to the Massif Central) outside which there are only a few
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earthquakes which are quite spread out (Normandy, North, Burgundy) and two basins (Parisian and Aquitaine) which have practically no earthquakes.
Figure 6.2. Historical seismicity in metropolitan France (intensities higher than or equal to V on the MSK scale) (according to [LAM 97])
The seismicity in France is very weak compared to that of European countries such as Italy or Greece; this is shown by Figure 6.3 in which all the earthquakes with a magnitude higher than four are represented for the period 1963–1993. However, it cannot be ignored in some regions, which in the past have experienced destructive earthquakes and which will continue to experience them in the future
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following the adage “where the earth has quaked, it will quake again”, which speaks of the incommensurability of scales of geological time and human activities.
Figure 6.3. Seismicity of the Mediterranean basin and neighboring regions (earthquakes with a magnitude higher than four for the period 1963–1993); according to [MAR99]
6.1.3. Archeoseismicity and paleoseismicity The period covered by the data on historical seismicity reaches two or three millennia at most in regions where very old archives are available (Greco-Roman Mediterranean, Middle East, northern China) and is reduced to a few centuries (only one or two in countries with a new population like California or Australia) in most
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of the other regions. This period can be too short for the database to be representative of the most violent manifestations of seismic activity. In intraplate zones with an average or moderate seismicity, the return period of large-scale earthquakes can be several thousand, even several tens of thousands of years, or a time interval considerably greater than the period for which historical data is available. Hence, it is important to complete the data in the archives using other indicators of seismicity which enable us to “go back in time”. We talk about archeoseismicity when these indicators are studied through archeological remains and of paleoseismicity if they are observed on natural sites. Archeoseismicity uses traces of damage that are still visible on old monuments, destructive effects having caused the abandonment of the site or proof of reinforcements made during the reconstruction following a seismic catastrophe (see the examples of N.N. Ambraseys quoted in the introduction to Part 5). Figure 6.4 shows a difference in a Tunisian mosaic which bears witness of the play of a strikeslip fault cutting across the foundation of the edifice. It is especially in the periphery of the Mediterranean region that the indicators of archeoseismic events have been found and studied; their identification and dates require the collaboration of seismologists and engineers along with architects, archeologists and historians. In regions such as Egypt or Mesopotamia, archeoseismicity can help in going back 2,000 to 3,000 years BC. In other regions, the remains studied belong to the Greek and Roman civilizations and are thus a little more recent. The contribution of archeoseismicity in these regions is more of a deepening of knowledge on seismic history (importance of destructions caused by known earthquakes through historical documents, highlighting forgotten earthquakes) rather than a large extension of the period covered. On the other hand, in zones where the archives on seismic activity go back only a few centuries, the archeoseismic data is likely to increase this period by a factor of three or four.
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Figure 6.4. Difference of nine centimeters in the mosaic of the tiled floor in a Roman house in Monastir (Tunisia). This difference can be explained only by a horizontal strike-slip motion of a fault underneath (according to [LAM 97])
The quantification (in terms of magnitude or intensity) of archeoseismic events is a delicate question because of the local character in general of the indicators discovered and the possibility of other causes (especially destruction caused by war). It can be effectively completed, in some cases, with the help of analysis of additional observations on structures that are relatively near; if they have apparently resisted the effects of the same seismic activity, the type of estimations described in section 12.1.1 (overturning of block placement) can help in setting a limit which is higher than the amplitude of the ground motion. If archeoseismicity is limited to a relatively short history (a few thousand years), paleoseismicity can go back much farther (a few hundred thousand years); it is based on several types of indicators, for example:
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– traces of old ruptures displacing layers of soil generally observed in trenches (Figure 6.5); – discontinuities in natural structures, such as stalagmites and stalactites found in karstic caves; – traces of past liquefaction.
Figure 6.5. Trace of the inverse fault displacing layers of soil at Courthézon (Vaucluse); according to [LAM 97]
Figure 6.5 shows a motion index that occurred on the fault in Nîmes, observed in a trench made at Courthézon (Vaucluse); this reverse movement is seen by a vertical gap of about 60 centimeters; the study of the layers of soil by this displacement indicates that the corresponding seismic event (which can be multiple, that is, associated with several successive episodes of rupture rather than a single one) occurred less than 100,000 years ago. If such an index confirms the occurrence of an earthquake of considerable scale (magnitude clearly higher than 6), the inexact nature of its dating and quantification in terms of magnitude results in the fact that its consideration, in a “computational” approach for seismic hazard, presents a certain number of difficulties (section 7.1). The attribution of the displacement observed during several earthquakes, rather than a single one, is possible in certain cases with the help of a detailed analysis of the layers of land.
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Figure 6.6. A first rupture (diagram 2) was produced after the end of the deposit of layers b and c (which confirms the equality of their thicknesses in the two compartments. A second rupture (diagram 3) occurred during the deposit of the layer after the disappearance of the thin layer d as a result of erosion on the overlapping compartment. The last rupture, that of 1980 (diagram 4), accentuated the deformation of layers d and e, provoking the formation of a superficial fold in layer f while being deposited, without actually causing any rupture
The interpretation given for Figure 6.6 (drawn after the El Asnam earthquake of 1980) shows three episodes of rupture, of which the first two, earlier than the one in 1980, cannot be dated with precision; we can only give them a range of dates based on the dates estimated for the deposits of different layers of soil. This example illustrates the necessity (and the difficulties) of the interpretation of the neotectonic indicators for their inclusion in studies on seismic hazard. The attribution of a magnitude based on the displacement observed (for example by using the formulae of Wells and Coppersmith quoted in section 2.2.3) crucially depends on the number of episodes of rupture having produced this displacement and remains, in the best of cases, riddled with uncertainties for the following reasons: – the absence of information on the component of fault movement in a perpendicular direction to the walls of the trench;
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– the fact that we do not know if the displacement observed is representative of the mean value of the sliding on the fault plane (i.e. 'u, as per the notation used in section 2.1) or rather an extreme value (maximum or minimum). Let us remember that the maximum displacement is commonly twice or three times larger than the average displacement; – the dispersion (standard deviation) of the correlations used for the calculation of the magnitude with respect to the displacement. Among the indicators other than the traces of fault movements, those found in caves are particularly interesting, as these underground areas are protected against erosion. Figure 6.7 shows a discontinuity in the structure of a stalagmite, which was associated with the effects of a historical earthquake.
Figure 6.7. Recognizing the effects of a historical earthquake in a cave in Italy. The stalagmite in the foreground was ruptured by the earthquake and a new thinner concretion started forming after the rupture. Its dating has made it possible to identify the earthquake responsible for this occurred in 1455 in the region (according to [LAM 97])
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Another type of indicator that provided valuable information in some regions is formed by traces of old liquefactions. The rise of sand which occurs during these phenomena (section 3.3.1) can leave long lasting marks and characteristics that confirm the occurrence of a large-scale earthquake. Its dating is often facilitated by the fact that the upward flow of sand has resulted in organic debris whose age can be determined through isotopic techniques (carbon 14). On the Pallett Creek site (approximately 50 km north-east of Los Angeles), studies of traces of old liquefactions have made it possible to identify nine major earthquakes in the San Andreas Fault, corresponding to the following dates: 1857, 1745, 1470, 1245, 1190, 965, 860, 665, 545 [BOL 90]. The first of these dates has been corroborated by historical documents (called the Fort Tejon earthquake of 9 January 1857); the others are approximations, taking into account margins for error in the dating method. Such information is essential for evaluating the hazard caused in the San Andreas Fault. It shows that the average time interval separating these events is approximately 160 years but there are big gaps with respect to this average (almost 300 years for the longest interval 55 years for the shortest). If the famous Big One expected by the Californians respects the average, it should occur around 2015–2020, or approximately 160 years after the Fort Tejon earthquake, but it could also occur tomorrow or in 100 years. The success of studies on paleoliquefaction depends on the nature of the land, variations in the phreatic layer and the climate; if the climate is too humid or too dry, the indicators can be easily destroyed or distorted (in case of excessive humidity) or be so rare that one needs a lot of luck to discover them (in case of dryness). On the San Andreas Fault, the Pallett Creek site is practically the only favorable site for such studies, at least in southern California [BOL 90]. 6.1.4. Instrumental seismicity Instrumental seismicity, in the classic sense of the term, corresponds to the continuous monitoring of seismic activity in a region, with the help of very sensitive instruments capable of detecting the motion of very weak amplitude. Thus, it has to be distinguished from the recording of strong motion, which was mentioned in Part 2. Furthermore, the types of instruments are different depending on the objective. Seismometers meant for continuous monitoring are generally affected by the saturation of their recording capacity when they are subjected to strong quakes, although the range of sensitivity is considerably wider for new generation instruments than for traditional instruments in seismological observatories.
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In France, the Laboratory for Detection and Geophysics (LDG) network, of the Atomic Energy Commission, was created at the beginning of the 1960s and it had a double mission: military (detection of underground nuclear explosions) and civilian (monitoring of seismic activity) objectives. Several other networks with a local purpose have since been developed by various research organizations, such as the Sism-Alp network set up in the Alps by the observatory at Grenoble (under the guidance of the ReNaSS, National Network for Seismic Monitoring) or the local network around the Moyenne Durance Fault, used by the French Institute for Nuclear Protection and Safety. Other networks have also been set up in French Overseas Departments for the surveillance of volcanic activity (Reunion, Martinique and Guadeloupe) and tectonic activity (Martinique and Guadeloupe). The “Geoscope” network, created and in use since 1980 by the Institute for Earth Physics in Paris, aims at providing recordings of high quality for all earthquakes of a certain scale occurring in the world; for this purpose it has stations (23 at present) equipped with very sensitive instruments with a wide frequency band, and spread out all over the world. Temporary surveillance networks can be installed in the neighborhood of the site of a large construction work or an installation to widen the scope of knowledge on local seismicity. Even low magnitude earthquakes, recorded during the period of use (a few years in general) provide useful information on the current tectonic activity through their focal mechanism (see section 1.2.3) and an image of the activity of well known or lesser known faults due to their spatial-temporal distribution. These temporary networks enable the monitoring of induced seismicity for big dams during and after the priming and filling of the structure (section 1.3.3). Figure 6.8 shows earthquakes with magnitudes higher than 3.5 for metropolitan France recorded during the period 1962–1994; the lower limit of 3.5 was chosen in order to have a complete sample, i.e., the density of the stations and the sensitivity of the seismometers guarantee that all the earthquakes with magnitudes higher than this limit have been detected.
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Figure 6.8. Earthquakes with magnitudes higher than 3.5 recorded in France and its neighboring countries for the period 1962-1994 [LAM 97]. The dark circles indicate magnitudes higher than 5, clear circles include magnitudes ranging between 3.5 and 5
This diagram highlights the coherence of instrumental seismicity data with that of historical seismicity (Figure 6.2) with regard to identification of active zones. Due to the relatively short period of instrumental observations, the sample used for Figure 6.8 is comprised of few events of a significant size, but the zones exposed to seismic hazard (and “aseismic” zones, Parisian and Aquitainian basins) appear as clearly as in Figure 6.2. The comparison with Figure 6.3, where the lower limit is a magnitude of 4 instead of 3.5, shows that the majority of the earthquakes in Figure 6.8 must have a magnitude ranging between 3.5 and 4, since we do not find them in the corresponding part of Figure 6.3. 6.2. Models of temporal distribution of seismicity 6.2.1. Return periods The expression “return period” has already been used on several occasions; it has in fact entered everyday language, which is regrettable because it causes confusion. The Tr return period is defined as the interval of average time separating two occurrences of an earthquake of a given size in an area or on a fault; the example of the San Andreas Fault given at the end of section 6.1.3 shows that the return period of sufficiently strong earthquakes in order to induce effects of liquefaction is
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approximately 160 years along this fault, but that it is only an “average periodicity” and the interval of time separating two successive events can vary by a factor of two as compared to this average. The existence of a return period, in the meaning as given before, simply expresses the stationary nature of the seismic process, which makes it possible to assimilate probability to the frequency of occurrence observed. The basic theory of elastic rebound, presented in section 2.1, supposes that the production of earthquakes by a given fault obeys a perfectly periodic process whose period Tr corresponds to the necessary time for the accumulation of stresses until the rupture threshold; with Vd indicating the rate of tectonic deformation and continuing with the notations used in section 2.1, we must thus have: Vd Tr = 'u = B
'V
P
= IB RO
1/3 1/3 'V 'V = IB §¨ 3 ·¸ §¨ Mo ·¸ P P © 4S ¹ © 'V ¹
i.e. for Tr: 1/3
Tr = IB §¨ 3 ·¸ © 4S ¹
M O1/3 'V 2/3 PVd
[6.1]
With the standard values considered in section 2.1 (IB = 2/3 'V = 3.8 Mpa, P = 3 x 104 Mpa) and by expressing the seismic moment Mo on the basis of the moment magnitude Mw, we have: Tr (years) = 0.0336
100.5 MW V (cm / year ) d
[6.2]
If we apply this formula for the San Andreas Fault, by taking Mw = 8 (earthquakes of the same order as those of San Francisco in 1906 or Fort Tejon in 1857) and Vd = 2 cm/yr (plausible order of magnitude for the rate of deformation, according to studies conducted on the ground) we find Tr = 168 years, i.e. a value very close to the return period deduced from paleoliquefaction analyses. In spite of the apparent success of formula [6.2] in the case of the San Andreas Fault, the basic model of elastic rebound stands on much too simplistic assumptions to account for the random nature of the production of earthquakes by a given fault because of the heterogenity of rupture zones and stress fields (which in particular can be modified following the movement of faults located in the vicinity). We can however accept that these assumptions are closer to reality for big earthquakes (whose size is sufficient to produce a “smoothing” of heterogenities) than for the small ones (which are more dependent on local hazards of the rupture mechanism).
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This idea has led to the theory of the characteristic earthquake (section 6.2.3) according to which a fault produces an earthquake close to the maximum limit of which it is capable, at rather regular intervals, and also smaller earthquakes, which follow another law of temporal distribution. The return period Tr is often replaced in calculations by its inverse which is the probability of occurrence per unit of time. In the probabilistic methods of evaluation of the seismic hazard, the levels of protection concerned correspond to criteria fixing the probability of the limit not being exceeded for a given duration, corresponding to the operating time envisaged for the work to be built. Such criteria lend themselves to simple calculations when the assumption is made that the occurrences are independent from each other (which is highly debatable and contradicts the “mechanical” models of earthquake production such as that of elastic rebound). With this assumption of independence, the probability Po (D) of there being no occurrence during duration D (counted in years) is given by: Po (D) = (I – p) D
[6.3]
where p is the annual probability of occurrence, i.e. the inverse of the corresponding return period Tr. In earthquake engineering codes applicable to current constructions, the desired level of safety often corresponds to a probability of the limit not being exceeded of 90% during a lifespan D of 50 years; the annual probability p of the event exceeding the limit must thus be such that: Po (50) = 0.9 = (I – p) 50
[6.4]
We find p = 0.0021, i.e. one return period Tr = I/p of 475 years. Countries using a probabilistic approach to seismic hazard for nuclear power stations in general maintain a return period of 10,000 years for exceeding the specified level for designing; according to formula [6.3], this criterion corresponds, for example, to a probability of the limit not being exceeded of 99% for a duration of 100 years. These values of 475 years and 10,000 years for the return periods are now to some extent “enshrined”, so much so that it has become very difficult to propose different values for studies of seismic hazard. It is thus important to remember that the choice of a return period really determines the level of protection only if we specify, at the same time, the degree of confidence (average or average plus a standard deviation) of the evaluation; indeed
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the formulae used for the calculation of hazard (in particular, attenuation laws, section 4.3.2) are for the majority, subjected to the law of standard deviation equal to the average and their dispersion is naturally reflected in the results of the study of the hazard. Equation [6.3] was obtained by writing that probability Po (D) of no occurrence for duration D is equal to the product of probability I-p of no occurrence for all the years of this duration. The same principle, resulting from the assumption of independence, makes it possible to obtain the following expressions for probabilities P1 (D) (one occurrence for duration D) and P2 (D) (two occurrences for duration D): P1 (D) = D p (I – p) D-1 P2 (D) =
1 D (D – 1) p2 (I – p) D – 2 2
[6.5] [6.6]
In addition, for an unspecified number of occurrences n for the duration D: Pn (D) =
1 D (D – 1) (D – 2) … (D – n + 1) pn (1 – p) D – n n!
[6.7]
As in practice D is large compared to n and p is small with respect to one, the following approximations can be made: D (D – 1) (D – 2) ... (D – n + 1) # Dn (I – p) D – n = [(I – p) I/P]P (D -n) # e -P (D – n) # e -pD Which gives, bringing forward in [6.7]: Pn (D) =
1 (pD)n e- pD n!
[6.8]
i.e. Poisson’s Law of Distribution. Pn (D) thus depends on the product pD (which is equal to the quotient of duration D by return period Tr = I/P) and presents a maximum for pD = n or an equal duration of n times the return period; this maximum does not correspond to a very high probability, since: Pn (n Tr) =
1 n -n I n e # n! 2S n
[6.9]
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Factorial approximation using Stirling’s formula is used. For the example of the San Andreas Fault mentioned earlier, the probability of the phenomenon observed (nine occurrences for the duration of around 1,500 years or 9 times the return period) is thus only 0.133 according to this formula [6.9]. This does not call into question the calculation of the return period as the average time interval separating two occurrences because the maximum is quite “peaked”, in the sense that the probability of having n occurrences falls quickly if we consider different durations of n times the return period, as we can verify with equation [6.9]. The model of occurrences independent of each other, on which preceding calculations are based, is certainly quite far removed from reality. Even for an isolated fault, the history of past ruptures influences the accumulation of stresses and thus the occurrence of future earthquakes. For a fault system, if an earthquake of significant size occurs on a fault, it modifies the state of stress and thus the evolution of the nearby faults, upon which the triggering of future earthquakes will either be facilitated, or opposed. 6.2.2. Gutenberg-Richter law
The Gutenberg-Richter law on the frequency of occurrence of earthquakes in a given region is one of most important laws of seismology. It expresses that the number N (M) of earthquakes of magnitude higher or equal to M, occurring on average each year in this region is given by the relation [GUT 44]: log10 N (M) = a – b M
[6.10]
which can also be written as: N (M) = 10 a-b M
[6.11]
The constants a and b of [6.10] and [6.11] vary according to the area studied but it is noteworthy that the values of coefficient b have a very narrow range of variation of around a year; it is very rare that b goes beyond the interval 2/3~3/2 (0.67~1.50) and it is often very close to 1, which is the “theoretical” value, as we shall see hereafter. Coefficient a, on the other hand, varies greatly according to the surface of the area and its seismic activity. The centennial magnitude M100 is that which is reached or exceeded on average, every 100 years, i.e. it corresponds to the value – 2 of the decimal logarithm N (M100); we thus have:
The Spatial and Temporal Distribution of Seismicity
M100 =
a2 b
301
[6.12]
For example, for the surface area 280,000 km² corresponding to northern and central California, the analysis of seismic data recorded between 1949 and 1983 leads to the following form of the Gutenberg-Richter law: log10 N (ML) = 4.23 – 0.815 ML
[6.13]
ML being the local magnitude (see section 2.3.2). For this area, the centennial magnitude calculated by [6.12], is 7.6, which translates its very strong seismicity. For the whole world, we can admit the approximate values a = 8 and b = 1 in order to calculate the orders of magnitude; we thus have each year, on average: – 10 earthquakes of a magnitude higher than or equal to 7 (i.e. the magnitude of the Kǀbe earthquake of 17 January, 1995), i.e. approximately one per month; – 100 earthquakes of a magnitude higher than or equal to 6 (i.e. the magnitude of the Macedonian earthquake at Skopje of 26 July, 1963), i.e. approximately two per week; – 1,000 earthquakes of a magnitude higher than or equal to 5 (i.e. the magnitude of the Epagny-Annecy earthquake of 15 July, 1996) i.e. approximately three per day. These figures are averages and the annual readings can deviate significantly. They show that earthquakes are more frequent than imagined, at least for a majority of people, who only hear about them when there are victims or when the quake is widely felt. The values a = 8, b = 1 constitute only a first approximation; a more precise study of world seismicity leads to the following relations [MAD 91]: log10 N (M) = 7.74 – 0.968 M (6 d M d 7.6)
[6.14]
log10 N (M) = 10.1 – 1.28 M (M > 7.6
[6.15]
The decrease of N (M) with relation to M is thus faster for greater magnitudes; it is probable that this change of slope corresponds to the modifications in the law of scale (2.1.3) when certain dimensions of the fault plane reach the limits fixed by the thickness of the seismogenous part of the crust (see section 15.1.3).
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It is interesting to note the lower limit of 6, taken for the magnitude in equation [6.14]. It is indeed necessary to be sure that we have listed all the earthquakes if we want to count them; however below a certain size, some earthquakes go unnoticed because they occur in places far away from recording stations (desert areas, oceanic zones). As explained in section 6.1.2 regarding historical seismicity, it is very important to ensure the exhaustiveness of samples in order to determine the coefficients of a statistical relation. This applies to the calculation of coefficients a and b of the Gutenberg-Richter laws, particularly in zones of moderate seismicity; there are cases where two different teams, working on the same area with the same data, ended up with different sets of coefficients for a and b, because of a difference in the judgment on the limit of exhaustiveness. In practical applications, the Gutenberg-Richter law is often used, not in [6.10] or [6.11] form, but in a truncated form, in which the magnitude can only vary in interval M1~M2; the introduction of the minimal magnitude M1 corresponds to the preoccupation of exhaustiveness mentioned earlier, that of the maximum magnitude M2 to the limitation fixed at the size of the earthquakes likely to occur in the area in question after taking into account its tectonic mode and of the dimension of faults which it contains. The truncated Gutenberg-Richter law is expressed by the following relation: N (M) = 10a
10 bM 10 bM 2 1 10
b M 2 M1
[6.16]
which leans towards form [6.11] when M2 indefinitely increases; for M = MI [6.16] gives exactly the same value as [6.11] for N (M1). The value chosen for M1 in [6.16] has practically no importance because the power of 10 which appears in the denominator is still extremely low (about a thousandth for M2 – M1 = 3). For M = M2 we obtain the desired truncation since N (M2) = 0. Apart from N (M), which corresponds to earthquakes of magnitude higher or equal to M, we also use the density n (M) such as the product n (M) dM represents the number of earthquakes whose magnitude is between M and M + dM; we thus have, from the definition of N (M): n (M) dM = N (M) – N (M + dM)
[6.17]
which gives for n (M): n (M) = –
dN ( M ) dM
[6.18]
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For N (M) given by [6.16] we thus have: n (M) =
b1n10 1 10
b M 2 M1
x 10a – bM
[6.19]
This expression of n (M) makes it possible to calculate the total released seismic energy on an average each year; we thus use expression [2.18] of energy released in the form of seismic waves by an earthquake of magnitude M: Ec (N x m) = 10 1.5M + 4.8
[6.20]
For all the earthquakes (M varying from M1 to M2), the total energy liberated Et is thus: M2
Et (N x m) =
³
n (M) Ec dM
[6.21]
M1
i.e., taking into account [6.19] and [6.20], we have: Et (N x m) =
bLn10 x10a +4.8 -b M -M 1 -10 2 1
M2
³
10 (1.5 – b)M dM
[6.22]
M1
From where, by integrating: Et (N x m) =
b 10a 4.8 [10(1.5-b)M2 – 10(1.5-b)M1] 1.5 b 1 10 b M 2 M1
[6.23]
As indicated before, we can ignore terms where M1 intervenes, which gives: Et (N x m) =
b 1.5 b M 2 10a + 4.8 x 10 1.5 b
[6.24]
By taking a = 8, b = 1 and M2 = 10 for the whole world (2.1.3) we find Et = 1.26 x 1018 N x m which is the order of magnitude obtained on the basis of recordings, which show that the annual average energy was 4.5 x 1017 N x m since the beginning of the 20th century, the year 1906 having been the most active with five earthquakes of magnitude higher than 8 and an Et of approximately 2 x 1018 N x m [MAD 91]. The total energy is especially due to the earthquakes of strong magnitude; for example, for the period 1975–1989, the earthquakes of magnitudes ranging between
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6 and 6.4 represent more than 70% of the total number of earthquakes having a magnitude higher than 6, but barely 5% of the released energy, whereas earthquakes of magnitude higher than 7.5 (3% of the total number) contributed nearly 70% of the total energy [MAD 91]. These observations show that the idea often presented, that small earthquakes release enough energy to be able to delay larger occurrences, unfortunately has no serious foundation. From the theoretical point of view we can find the Gutenberg-Richter law by means of simple hypotheses on the distribution of the rupture zones according to their size. Kanamori and Anderson [KAN 75] have, for example, assumed that the product of the number of earthquakes of magnitude close to M by the fault surface corresponding to this magnitude is constant; with the notations used in section 2.1 and by introducing density n (M) defined before, this assumption is expressed by the relation: n (M) IL IH R²0 = Cte i.e. taking into account [6.18] and relation [2.7] between R0 and the seismic moment: dN = – K x 10 -M dM
[6.25]
K being a constant if the power law of scaling is admitted. By integrating [6.25] with a constant of zero integration because N (M) must lean towards O when M increases indefinitely we thus have: N (M) =
K x 10-M Ln10
[6.26]
This relation is identical to [6.11] if b = 1. The Kanamori and Anderson hypothesis which has just been used is what made it possible to build the model of elastic rebound with multiple ruptures (section 5.1.2) and to obtain theoretical attenuation laws in accordance with the experiment. The Gutenberg-Richter law is generally only incorrectly verified for an isolated fault, except for events of a small size. On the other hand it is applied satisfactorily to areas sufficiently large to contain faults of different dimension. In the probabilistic methods of evaluation of the seismic hazard, which use a division of source-zones (section 6.1.3), it is therefore necessary to raise the question of the applicability of this law to certain small size source-zones.
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6.2.3. Model of a characteristic earthquake
The concept of a characteristic earthquake has already been mentioned in section 6.2.1; it relates to the distinction between two different types of seismicity on a given fault; this would produce, on one hand, relatively small earthquakes according to the Gutenberg-Richter law, and on the other hand, a definitely stronger earthquake (characteristic earthquake) at quite regular intervals. This idea only emerged at the beginning of the 1980s [WES 94], i.e. nearly 40 years after the proposal by Gutenberg and Richter, which goes back to 1944; it was indeed necessary to wait for the accumulation of a sufficient amount of instrumental data covering a rather wide range of magnitudes to realize that certain faults deviated from Gutenberg and Richter’s model for strong magnitudes. The use of paleoseismic data also contributed to supplement the sample while making it possible to estimate return periods for events of a significant size, as we saw in section 6.2.1. Several faults located in Southern California (Figure 1.11) were thus studied by Wesnousky [WES 94]. Figure 6.9 shows the results obtained for two of them, Elsinore and Newport-Inglewood. We note a good alignment to the GutenbergRichter law for the magnitudes going from 3 to a value a little lower than 5. For larger magnitudes, the number N(M) is constant, which simply means that there is no earthquake listed between the magnitude of 5 and the magnitude of the strongest earthquake produced by the fault (which is a little higher than 7). We also note that, and this has important consequences for the evaluation of the seismic hazard, if we elongate the Gutenberg-Richter lines in the diagram log10 N (M) in relation to M (drawn in dotted lines in the diagram) we underestimate the hazard for stronger magnitudes since we would thus find magnitudes of only about 6 for frequencies of occurrence of about 10-3 per year, which correspond to the maximum values observed, close to 7. For faults that are very large in size, like that of San Andreas, it is necessary to perform an analysis in segments because they cannot be considered homogenous along their entire length. As indicated in section 2.2.1, the appreciation of “maximum” magnitudes likely to be produced by a fault depends obviously on its overall length but also on the homogenity of its characteristics.
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Figure 6.9. Variation of the annual number of earthquakes of magnitudes higher than M for the Elsinore and Newport-Inglewood faults (according to [WES 94])
In regions with moderate seismicity where instrumental data often only shows very weak earthquakes (with magnitudes lower than 4), at least in the case of certain fault zones, the reliability of the Gutenberg-Richter laws deduced only on the basis of this data can be a problem. It may then be judicious to take the help of models of a characteristic earthquake, adjusted based on historical data for their definition. Figure 6.12 highlights the difference between the two types of model (GutenbergRichter with truncation to a maximum magnitude and Gutenberg-Richter with the characteristic earthquake equal to the maximum earthquake of the first model).
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307
Figure 6.10. Differences between the Gutenberg-Richter law with truncation and a characteristic earthquake
6.3. Prediction of earthquakes
In the field of natural sciences, the prediction of earthquakes is certainly one of the most publicized subjects in the media. After the Kǀbe earthquake (17 January, 1995) a survey showed that more than half of the Japanese people had been surprised that it had not been possible to sound an alarm, which proves the “communication” skill of certain specialists who deal with this problem. It has to be recognized indeed, that after a period of hope and enthusiasm during the 1970s, the objective of a reliable and exact prediction still appears very distant and it is at the very least misleading, not to mention irresponsible and intellectually dishonest, to lead people to believe that it is within our reach. From the point of view of terminology, we usually distinguish between relatively long-term forecasts (a few years or a few tens of years) and short-term predictions (a few weeks or a few days). The first, which seems rather well controlled in certain areas of strong seismic activity, can be very useful to define priorities in preparative actions (informing concerned populations, training teams in civil safety, reinforcement of backup facilities and reduction in vulnerability of the existing habitat). The second, which today is only a mere hope, aims at making it possible to take emergency measures (evacuation of buildings, temporary provision of shelter, food and water for a large number of people, stopping public transport, security measures for networks and sensitive industrial equipment).
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We limit ourselves in the following part to a very brief description of the problems of prediction and forecast. It presents the policy of prevention of seismic risk rather than earthquake engineering and design in the true sense of the term (which only relate to the provisions in architecture and construction that aim to minimize seismic risk). In addition, we must emphasize that if the prediction of earthquakes becomes operational one day, it will not eliminate the need for earthquake engineering and design, so as to reduce economic losses in areas affected by earthquakes. 6.3.1. Seismic precursors
The idea that large-sized earthquakes are necessarily preceded by some precursory evidence, detected by appropriate monitoring, appears obvious, when the extent of rupture zones and the released quantities of energy are considered. Some such evidence, or seismic precursors, has been proposed: – abnormal variations of geophysical parameters (propagation velocity of seismic waves, electric potential, etc.); – measurable deformations of the ground surface; – movements of fluids (variation of the water level in wells, emission of gases such as radon, etc.); – modifications of seismic activity, either in the direction of its increase (“premonitory” swarm) or in the direction of its decrease (the “calm which precedes the storm”); – surprising behavior of domestic or wild animals (see section 3.3.4). Several of these precursors have been used to try to develop methods of prediction and have enjoyed certain success with the media; we can cite for example: – propagation velocities of seismic waves (velocity of P waves, relation Vp/Vs velocities of P and S waves), deduced from the measurement of travel time of the waves for small natural or artificial earthquakes (explosions); the study of the variation of these velocities resulting from the increase in microfracturing when stresses approach rupture point (theory of dilatancy) gave a lot of hope to the “predictors” during the 1970s, but the analysis of all the data now available on this subject presents serious misgivings on its reliability as a precursor; – anomalies of electric potential, measured by means of electrodes planted in the ground at a distance of one hundred to a thousand meters. The VAN method (Varotsos-Alexopoulos-Nomikos), proposed in Greece since the beginning of the 1980s, was widely publicized in the media and continues to be the subject of debate;
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taking into account the rather broad range of “predictions” asserted by this method in terms of the time limit of occurrence and magnitude ratio, and very strong seismic activity in Greece, a relatively high rate of “success” can be simply attributed to the laws of chance. Attempts to apply this method outside Greece have, it seems, brought no conclusive evidence. If the VAN electric signal is perhaps a valid precursor, the physical causes which would explain it are still not very clear; – the behavior of animals frequently reported by witnesses at the time of great earthquakes became famous following the success of the Chinese in 1975 (successful prediction of the Haicheng earthquake, having a magnitude of 7.3, on 4 February 1975), although the Chinese method is founded on observation of several indications, and not only on that of the animals. The fact that the terrible Tangshan earthquake (27 July 1976), only 300 km from Haicheng, could not be predicted shows that this precursor is no more “infallible” than the others; – the leveling measurements carried out over a rather long period, highlighting vertical movements of the ground close to Niigata (west coast of the main island of the Japanese archipelago), where a regular variation was observed for approximately 60 years and was replaced, in June 1964, by a sudden variation just before the 16 June 1964 earthquake. This indication could have been used to sound an alert but we know of other cases where similar phenomena occurred, without being correlated with the occurrence of an earthquake. The Palmdale area to the north of Los Angeles thus experienced a remarkable uplift (more than 20 cm on a surface of several thousand km²) during the 1960s–1980s without a cause-effect relation between this “swelling” and the earthquakes coming to the south of this zone (San Fernando 1971 and Northridge 1994) being established. All these dashed hopes show that all “mono-parametric” approaches of prediction, even if based on a valid precursor, are not sufficient a priori, taking into account the great variability of the seismic phenomenon. P. Bernard concludes [BER 95] that “the observation, of only a single anomaly just before a great earthquake does not teach us anything on the underlying physical mechanisms and does not allow the development of an applicable predictive model in other tectonic regions; moreover, from the practical point of view, it would take centuries of such observations to establish a reliable statistical law”. 6.3.2. Current questions on forecast
Short-term forecast, or prediction, making it possible to trigger alarms and to make emergency arrangements, remains a fundamental objective of Earth sciences, but the large majority of specialists who work on this subject today refuse to present any prognosis on any practical chances of realization, let alone the necessary time to reach it. Important research programs exist on various sites equipped with
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instrumentation adapted to the study of precursors, and where the occurrence of important earthquakes is considered probable in the relatively near future. These sites correspond to the principal tectonic regimes of the most active zones: – great strike-slip faults: San Andreas Fault in California, North-Anatolian Fault in Turkey, Haiyuan Fault in northern China near Lanzhou; – subduction: Tokai region to the south-east of Tokyo, interaction of the Nazca and South American plates in northern Chile; – extension: Corinthian gulf in Greece. The parameters subjected to instrumental monitoring on these sites vary greatly: microseismicity at stations on land and underwater (in the subduction zones), ground deformation, migration and temperature of ground water, emission of radon gas, electromagnetic field, etc. Although the awaited earthquakes have not yet taken place, interesting results have already been obtained, for example, on the correlation between the temperature of ground water and the evolution of microseismicity on the North-Anatolian Fault [BER 95]. Along with these in situ experiments, many studies are being carried out in laboratories, in order to understand, on small-scale models, the mechanisms of the physical phenomena, in particular when there is coupling of two effects, such as, for example, between the circulation of water in permeable rocks and the generation of electric potentials. Attempts at numerical modeling are systematically made by interpreting the experimental results. Pilot sites for study of precursors have been selected based on the criterion of immediacy of an earthquake of strong magnitude. This criterion is based on the idea of the seismic cycle which has already been discussed for interplate seismicity (section 1.1.3) and which led to the modeling of a characteristic earthquake for an isolated fault (section 6.2.3). If the time passed since the last great earthquake is such that its product by the rate of tectonic deformation is comparable to the fault displacement associated with an earthquake of this size, it is to be expected that the rupture will occur in the immediate future since the stresses have reached a critical level. For example, for the North Chile site, the last great earthquake, having a magnitude of about 8.5~8.7 (displacement of approximately 10 m), was in 1877; taking into account the deformation rate of subduction (9 cm/yr), reached in 2000, there is a “deficit” of displacement of 0.09 x 123 = 11.07 m which indicates that the end of the cycle has been reached. This very simple approach is the basis of the “seismic gaps” method (Figure 6.11) for the evaluation of the most exposed zones of subduction.
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Figure 6.11. Map of seismic gaps for subduction zones of the Pacific (according to [MAD 91])
On the San Andreas Fault, a rather short but regularly active segment was identified near Parkfield and was chosen as the experimentation site for the study of precursors; this segment produced earthquakes of magnitude close to 6 in 1881, 1901, 1922, 1934 and 1966, i.e. with a return period of 22 years. Figure 6.12 shows estimates of the probability of occurrence of earthquakes along the entire fault and emphasizes the Parkfield area as the most significant for in situ experimentation.
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Figure 6.12. Probability of occurrence of earthquakes along the San Andreas Fault during the period 1988-2018. The probability is strongest in a segment approximately 30 km long close to Parkfield in central California (according to [MAD 91])
The fact that the earthquake is “late” at Parkfield, as compared to the estimates deduced from past seismicity, illustrates the difficulty of long-term forecast, whose objectives are however, much less ambitious than those of prediction. Certain recent publications even call into question the principle of the seismic gap method. For example Kagan and Jackson [KAG 99] emphasize that it is relatively frequent to observe “pairs” of great earthquakes, i.e. two ruptures having a common part and occurring at a definitely shorter interval of time than that required for the accumulation of stresses to reach their critical point. Such observations, which represent fifteen cases out of 70 for earthquakes having a magnitude higher than 7.5 during the period 1976–1998, contradict the basic assumption of the seismic gap method, according to which it is necessary to wait for the reconstitution of the state of stress in order to produce a new major rupture in a recently ruptured zone. An even more radical query relates to the possibility of a forecast, according to recently obtained results in chaos theory for nonlinear systems [YEA 97]. This theory results from the observation that the response of certain systems to extremely weak disturbances presents a completely random character at the end of a certain time. In meteorology, this phenomenon received the name of “butterfly” effect (the fact that one of these insects beating its wings somewhere at a certain moment can in the long term have a considerable influence on the weather at the other end of the world).
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In seismology, the model of slipping blocks connected by springs was widely used to make digital simulations of the occurrence of earthquakes; it comprises (Figure 6.13) a line of blocks placed on a rigid and motionless support; each block is connected by a spring to the blocks located in front of and behind it in the line on the one hand, and to a rigid bar which moves with a constant velocity Vd on the other.
Figure 6.13. The model of slipping blocks connected to each other by springs and pulled by a bar which moves at constant velocity
If we consider a single block, this model is identical to the basic model of elastic rebound described in section 2.1.1; the spring which binds it to the mobile bar bends gradually and the block abruptly slips on the support while the effort transmitted by the spring exceeds the resistance of friction; this slipping quickly stops and the cycle begins again. In the models having several blocks, a large variety of block behaviors and of slip and stop sequences can be observed; in certain cases the blocks appear to slip in a relatively independent way; in other rarer cases, the slip of a block gradually leads to a few adjacent blocks slipping. Chaotic behavior could be highlighted even for some models with two blocks [NAR 92]. Do these observations imply that forecasting is impossible? At least two arguments can be advanced to question an equally discouraging assertion: 1) The fact that certain block-spring models are chaotic should not let us believe that this is a general tendency; the majority of these systems present, at least during a certain interval of time, a behavior compatible with the model of the characteristic earthquake: the unpredictability resulting from chaotic behavior could thus relate only to certain fault zones or very long-term behaviors.
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2) The block-spring models are too diagrammatic to enable the observation of precursors; the comparison with the issue of meteorology is interesting; although weather cannot (and will undoubtedly never) be predicted in the long run, the forecasts for a few days are increasingly reliable because the precursory phenomena (variations of pressure, temperature or moisture in the air) are increasingly better supervised and modeled. We can thus think that relatively long-term seismic forecast, on the basis of well understood and well supervised precursors, is not contradictory to very long-term chaotic behavior. The study of seismic precursors is thus a research topic whose need can hardly be contested. We should however not have illusions concerning the immediate consequences which a major discovery in this field would have. It is indeed, hard to imagine, that the development of a prediction method in a given area of strong seismicity (California, Japan, Greece, etc.) can be transposed to another area without a certain period of calibration considering qualitative and quantitative diversity of tectonic contexts; the duration of such a period of calibration will necessarily be rather long (several decades) in zones of moderate seismicity.
Chapter 7
Assessment of Seismic Hazard
7.1. Methods of assessment of seismic hazard 7.1.1. General notes pertaining to different approaches In the introduction to Part 3 of this book, some indications were given as regards the differences between the probabilistic and deterministic methods of assessment of seismic hazards. The debate between the partisans of these two methods has occupied and continues to occupy an important place in discussions on the definition of a policy of prevention of seismic risk and the development of regulations. This debate of which certain aspects have often assumed a polemic nature, has glossed over some of the most fundamental questions pertaining to the objectives and the importance of the assessment of seismic hazards. We owe the following classification that explains the meaning of certain terms and challenges of the studies on seismic hazards to H.B. Seed [SEE 82]; it distinguishes between: – the Maximum Credible Earthquake (MCE), that is, the most powerful or maximum earthquake conceivable, based on rational knowledge of the tectonic context of the region; the adjective “credible” is not related to the personal conviction of any particular expert but is aimed at an objective assessment of the maximum seismic potential of the faults of the region, based on available data. Such an estimation goes beyond textbook examples, where the tectonic context would be perfectly known and understood as compared to a real situation; the estimation of the MCE is therefore in practice marred by uncertainty but, the concept in itself is well defined. The estimation of the MCE is the responsibility of geologists and seismologists;
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– the Seismic Safety Evaluation Earthquake (SSEE) which is the maximum earthquake for which anti-seismic protection is required. The SSEE may be of the same intensity as that of the MCE but it is not rare for it to be much lower, for example, if the return period of the MCE is incomparably longer than the life of the installation or if the cost of the anti-seismic protection against the MCE seems unreasonable or enormous in comparison with the total cost of the project. The definition of the SSEE is the responsibility of the government or of the client after consultation with seismologists and engineers; – the Seismic Engineering Design Earthquake (SEDE) which is the set of hypotheses characterizing seismic action in the analysis of the safety needs of the installation; the SEDE depends of course on the SSEE but also on the type of installation and the choices made at different stages in the designing of the installation; according to the specifics, in terms of vulnerability of the installation to be constructed, the level of adequacy and accuracy of the calculation models, the level of caution as regards verification or testing criteria, the characterization of the SEDE could be different for the same SSEE. The definition of the SEDE is the responsibility of engineers. This distinction is important and brings to light the ambiguities of certain notions that are regularly used, particularly those that use the adjective “maximum”. As mentioned in the introduction, the conventional anti-seismic codes for current constructions have a form that is essentially like the SEDE type, without any specific reference to an SSEE level, and even more so, to an MCE level. The recent codes (of Eurocode 8 type) introduce an SSEE (that generally corresponds to a return period of 475 years; see section 5.2.1) but without linking it to an MCE. This introduction of an SSEE stems from a need for communication, but it is true, as indicated in the introduction, that it will hardly change the current practice of determination of the SEDE, that favors continuity and coherence with the reference codes. In the case of critical installations, the current practices concede a prominent role to the SSEE, even if certain terms using the adjective “maximum” can give the impression of an MCE. The SSEE is fixed by probabilistic criteria (return period of 10,000 years for nuclear power stations; see 6.2.1), or by overall rules that link it to the “maximum” earthquake deduced from the history of seismicity. Depending on the tectonic context and the seismic activity of the region of the site being studied, these criteria or rules determine the SSEE/MCE ratios that may vary greatly; this ratio may practically be equal to one (for example, if we have used a return period of 10,000 years in a region that is very active, where the major faults develop their maximum seismic potential at intervals that are much shorter) or even higher than one (if the overall rules taking into consideration a safety margin with
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respect to the historical “maximum” are applied in these same active regions, since this “maximum” is equal to the MCE); it may, on the other hand, be significantly less than one in zones with moderate seismicity, where the return period of the MCE is higher than 10,000 years (implying a very low probability than that it was observed in the historical period). The change from the SSEE to the SEDE, for these critical installations is generally simpler than for regular constructions as only elastic calculations based on the response spectrum of the ground motion (see Chapter 9) and the static criteria of force equilibrium for safety justification are used; in “ordinary” anti-seismic codes, this transition from the SSEE to the SEDE is far from being the most critical element in the dimensioning chain (section 9.3), to the extent that the specification of the SSEE has often been glossed over in conventional codes. It is surprising to observe that the “clash of faiths” as regards the advantages and disadvantages of the probabilistic or deterministic methods has taken precedence over the basic debate concerning the choice of the SSEE level, its transcription into SEDE and the management of the uncertainties that should be considered as more important. The slightly dualistic distinction between the probabilistic and deterministic method has undoubtedly played a role in the “religious war” character of certain discussions and opinions. In reality, there are several intermediary degrees between the extreme versions of probabilism and determinism, we could, for example, introduce a dose of determinism in certain elements of the probabilistic method (for example, in models of characteristic earthquakes, corrected by historical data; section 6.2.3). 7.1.2. An example of the deterministic method The French Basic Rule of Safety (RFS) I.2.c was established in 1981 by the Central Safety Service for nuclear installations in order to encode the determination of seismic motion on French nuclear powerstation sites. It was updated in 1998 so as to take into account the changing knowledge landscape, particularly in the neotectonic and paleoseismic fields. Its general framework that is the same for both its versions, defines a method that consists of several stages [COL 81]: 1) the collection of geological and seismic data in the region of the site and its eventual completion by specific studies (detailed analysis of the effects of a historical earthquake, detailed reconnaissance of a fault); 2) the delimitation of “tectonic fields” and “seismogenous accidents”, in other words, the delimitation of source zones that are surfacic and lineic respectively,
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producing seismicity; these source zones are homogenous, i.e., a historical earthquake identified at a point in the zone is considered as likely to occur at any point of the same zone; 3) the determination of the maximum historically probable earthquake (SMHV), in other words, the earthquake producing the greatest intensity on the site, assuming that the most powerful historical earthquake of each source zone would occur at the point in this zone that is closest to the site. In particular, the most powerful historical earthquake of the source zone to which the site belongs, is brought back under the site. There may be several SMHV for the same site, for example, a “big” earthquake far away and a “small” earthquake close by, both having the same intensity on the site; 4) the definition of the safe maximum earthquake (SMS) associated with each SMHV, i.e. a conventional earthquake, producing on the site an intensity that is higher (according to the Medvedev-Sponheuer-Karnik (MSK) scale of seismic intensity) by one degree than that of the corresponding SMHV, but having the same epicentral position and focal depth; 5) calculation of the parameters of ground motion (spectrum of elastic response; see section 9.1) associated with the SMHV and the SMS, with respect to their magnitude (calculated according to their intensities, using formulae [14.3] or [14.4] and abiding by the 1981 or 1998 versions of the Fundamental Rules of Safety or RFS) and their distance from the site. Figure 7.1 illustrates stage 3) that deals with the determination of the SMHV in a text book example where there are six source zones.
Figure 7.1. Determination of the maximum historically probable earthquakes (SMHV) for site S
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Zone I, which contains site S, also contains a lineic seismogenous accident (fault) that constitutes zone 2; the epicenters and the levels of intensity of the strongest historical earthquakes are indicated for each zone, as well as the movement of the epicenters (arrows) from their historical position to the point that is closest to the site; an earthquake of intensity level VII at the N-NW of the site in zone I is brought back under the site; that of intensity level VIII in zone 2 is displaced along the fault up to the point P2; the earthquakes of zones adjacent to zone I are displaced to points P3, P4, P5 and P6. The intensity of the SMHV of site S is at least, of intensity level VII, corresponding to the earthquake brought back under the site, but one must study the effects on the site of an earthquakes of level VII-VIII at P3 and of an earthquake of level VIII at P2 that could have an intensity equal to or higher than level VII; this study calls for the use of the attenuation laws of intensity such as [14.15]. Assuming that the effect of P2 is stronger than that of P3 and corresponds precisely to that of intensity level VII on the site, we can quote the example mentioned above where there are two SMHV of intensity level VII corresponding to a far-away earthquake (P2) and to a close-by earthquake (under the site). In this example, we can see just how crucial the delimitation of the source zones can be; the decision to relate an earthquake of intensity level VIII to the seismogenous accident that constitute zone 2 means that we displace it to P2, that is relatively far from the site; if we relate it to zone I, or if we do not retain zone 2 as a well identified seismogenous accident, we will have to displace it under the site and the level of the SMHV will become VIII instead of VII. We can clearly observe that despite a codification that is apparently very precise as regards the method to be adopted, the RFS I.2.c method relies essentially on the opinions of experts in stage 2 (delimitation of the source zones). If we also take into account the uncertainties inherent to the determination of epicentral intensities of historical earthquakes (section 14.1.3) it is not surprising that two different teams working on the same basic data can well arrive at results that are very different in terms of the motion parameters that define the seismic hazard. This holds true for all the methods of determination of seismic hazard that all have to make certain choices that are extremely subjective at certain stages in the method. It is only when a consensus has been reached as concerns the interpretation of reference documents (seismo-tectonic zone map defining the source zones, seismic history file) that the possibility of divergence in the results is minimized; in other words, the precedents of application of the rule is as important, if not more important, than the rule itself. The margin of a unit of intensity considered at stage 4 in order to define the SMS corresponds more or less to double for motion parameters influenced by velocity (frequencies from one to several hertz) and to a little less than double for those that
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are related to acceleration (frequencies higher than 10 Hz) as we shall see in section 14.2.1. It is well adapted to zones of moderate seismicity, as in metropolitan France, where the period of historical observations is undoubtedly shorter than the return period of the strongest earthquakes. It can, on the other hand, lead to results that are extremely disadvantageous in very active zones where there is no reason to overestimate the strongest known earthquake that should be close in value to the “maximum credible earthquake” (MCE). We find this situation in some French territories, in the subduction zone in Guadeloupe and Martinique, where the earthquake in 1843, of a magnitude of 8, is without a doubt representative of this maximum. The RFS I.2.c method favors the notion of intensity in the definition of its different stages; in reality, when we examine the procedure in detail, we realize that the parameters of magnitude and focal depth play a very important role. If the SMHVs are considered in terms of intensity, they are characterized by magnitude and depth for the calculation of the parameters of ground motion (since the correlation of these parameters with intensity are now considered with suspicion; see section 14.2.1); focal depth in particular has a strong influence on the results of these calculations when the earthquakes are brought under the site. The values of the magnitude and focal depth of the SMHVs are obtained from the isoseismal lines of the associated historical earthquakes, following the methods described in section 14.2.2. The sensitivity of the results with respect to focal depth, explains why the level of intensity of the SMHV or SMS does not in itself determine the values of acceleration or velocity that define the seismic hazard. As indicated at the end of section 14.1.3, choosing to favor the notion of the level of intensity in the definition of the seismic hazard is not satisfactory from a technical point of view. It can only be justified as having been made in the interests of communication, in the sense that the information that is accessible to the general public refers particularly to the intensity of past earthquakes. In regions of strong seismic activity, where the faults responsible for the major earthquakes are well identified, the deterministic assessment of the hazard is generally based on the characterization of a maximum earthquake for each of the major faults. This characterization is done in terms of magnitude, on the basis of the lengths of “maximum” rupture that may correspond to the total length of the fault or only to a fraction of this length, when the fault is made up of several segments that have different characteristics. The calculation of the motion parameters, which define the hazard for a given site, is done with the help of the attenuation laws of the type presented in section 4.2, on the basis of these maximum magnitudes and the distances between the site and the faults.
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In this “fault” approach, there is no need to consider an additional safety margin (as in the transition from the SMHV to the SMS in RFS I.2.c) if the estimations of the maximum magnitudes were reasonably conservative. On the other hand, it is wise to consider the possible occurrence, in the immediate vicinity of the site, of an earthquake of a relatively low magnitude, corresponding to a fault that is rather short, that may not have been detected by geological studies. A “floating” earthquake such as this has been included, for example, in Japanese regulations applicable to nuclear power stations; conventionally, it corresponds to a magnitude of 6.5 having its focus under the site that in the case of a number of sites leads to motion parameters that are more severe than those originating from maximum magnitudes on well identified faults. 7.1.3. Probabilistic methods The probabilistic methods of assessment of the seismic hazard vary greatly according to the nature and number of probabilistic parameters, the type and method of obtaining the basic data and the formulae used for the calculation of motion parameters. Nevertheless, they relate to a common framework which was proposed by Cornell in 1968 [COR 68], which is: 1) the definition of source zones (well identified faults or zones of diffused seismicity); 2) the description of the conditions of occurrence of earthquakes for each of the source zones (distribution of the focal depth, models defining the frequency of occurrence of earthquakes according to their magnitude as in the Gutenberg-Richter earthquake or the model of a characteristic earthquake, maximum magnitude); 3) the choice of a certain number of attenuation laws that enable the calculation of motion parameters with respect to a given magnitude and distance; 4) the calculation of motion parameters corresponding to a given probability of exceedance on the site being studied. The principle of this calculation may be described in the following manner: we consider (Figure 7.2) a site S whose seismic hazard is governed by two source zones, a fault F and a zone Z of diffuse seismicity. The frequency of occurrence of earthquakes in the two source zones is governed by the Gutenberg-Richter law, with truncation at a maximum magnitude Mm that is expressed by the formula: N (M) = 10a (10 -bM – 10 -bMm)
[7.1]
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which is identical to [6.16] when we ignore, as usual, the power of 10 in the denominator. The value of a, b and Mm may be different for the two source zones.
Figure 7.2. Model with two source zones (fault F and diffuse zone Z) for the calculation of the seismic hazard of site S
The motion parameter P that is of interest to us is given by an attenuation law in the form: P = C R-E e
DM
[7.2]
which is the simplest version of equation [4.16]; R is the focal distance of the site, C, D and E three constants; from [7.2] we obtain: § ¨ PR E bM 10 = ¨ ¨ C ¨ ©
b
Ln10
·D ¸ ¸ ¸ ¸ ¹
[7.3]
The existence of a maximum magnitude Mm fixes an upper limit Rm for the focal distance for a given value of the parameter P; according to [7.2] Rm is given by: § CeD M m Rm = ¨ ¨ P ©
1/ E
· ¸ ¸ ¹
[7.4]
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In each of the source zones, the distribution of seismicity is assumed to be homogenous, that is, an earthquake of a given magnitude has the same probability of occurrence at any point in the zone; the annual rate of exceedance of the value P on site S, due to the seismic activity of the zone Z is thus given by the formula: nZ (P) =
³
Z ( R d Rm )
N ( Mp )
dV ¦t
[7.5]
in which: – Z (R d Rm) represents all the points in zone Z whose distance from the site is lower than or equal to Rm; – Mp is the value of the magnitude that produces P at site S for a given distance R (lower than or equal to Rm); – dV is a surface element of zone Z; – 6t is the total area of zone Z. Considering [7.1] and [7.3], [7.5] is written as: b
10a § C · D nz (P) = ¦ t ¨© P ¸¹
Ln10
³
Z ( R d Rm )
R
E bLn10 D
dV 10abMm
¦m ¦t
[7.6]
where 6m is the area of the part of Z whose points are at a distance from the site that is lower than or equal to Rm. We see that the calculation of nz (P) calls for the evaluation of a geometrical integral; in some simple cases, this evaluation can be made analytically (see section 7.2.4); however, in general, it is made numerically. The annual rate nF (P) of exceedance of value P on site S, due to the seismic activity of fault F, is given by the same formula [7.6] if we agree that dV represents an element of length on the fault and 6t the total length of the fault. The sum of nz (P) and nF (P) represents the annual rate of exceedance of the value P that is the inverse of the return period Tp of this exceedance, on site S. Generally, when there is any number of source zones, identified by an index i, we get the following for Tp: Tp = [ 6i ni (P)] -1
[7.7]
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ni (P) being the annual rate of exceedance due to the source zone i, that is given by a formula of the [7.6] type. In practice, the calculation is done “in reverse” since the initial data corresponds to the return period Tp. We generally prefer to trace the variation curve of Tp (or of its inverse which is the annual rate of exceedance) depending on the value of P; this enables us to appreciate the influence of the choice of Tp as against that of the motion parameters. It is also the practice to probabilize certain parameters, that are characterized by a high level of uncertainty, such as, for example, the focal depths or the maximum magnitudes, and to determine the confidence intervals (standard deviation) of the evaluations. These “complications” make it necessary to use rather complex software that is often “black box”. That is why some formulae that enable the estimation of the orders of magnitude will be presented in sections 7.2.4 and 7.2.5. It is interesting to identify the source zones that have a prominent influence on the result, that is, on the value of P; it a matter of a de-aggregation operation that consists of the simple calculation, for each source zone of the ratio ri defined by: ri = ni (P) / [6i ni (P)] = Tp ni (P)
[7.8]
This ratio indicates the importance of the role of source zone i; we often observe that it varies considerably according to the motion parameter being studied; for example, accelerations are more strongly influenced by local earthquakes of relatively low magnitude than velocities. Source zones that are very close to the site, even though of moderate magnitudes, can thus be decisive for accelerations, whereas velocities will be controlled by source zones that are further away but that have a stronger seismic potential. These factors lead us to believe that the delimitation into source zones has, as in the determinist method, a strong influence on the result. Generally speaking, it is not in the interests of safety to take into account diffused source zones (surfacic) spread over a large area, as they produce a “dilution of seismicity”. This observation shows that the delimitation into source zones in a probabilistic approach is not necessarily the same as that of a deterministic approach. In a deterministic approach, in fact, the increase in the area of the tectonic regions results in the transfer of a greater number of historical earthquakes towards the site and therefore we observe a tendency towards an increase in the SMHV (section 7.1.2). An exercise conducted on a test site showed that a choice of source zones adapted to the application of the deterministic method of RFS I.2.c (section 7.1.2) produced, when used in the probabilistic approach, motion parameters that were almost two times lower than those obtained from a precise delimitation considering the same geological and
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seismic data, but conceived by practitioners of the probabilistic study of the seismic hazard. Factors other than the delimitation of source zones have an important influence on the results of the probabilistic methods; they are essentially the coefficient b of the Gutenberg-Richter laws and those of the attenuation laws used for the calculation; it is therefore practical to conduct sensitivity studies of the results by envisaging different values for these coefficients. Contrary to what we would believe at first sight, the maximum magnitude values of the source zones do not play an important role, at least not in the case of return periods that are relatively short, such as the 475 years often used in anti-seismic codes in regular constructions (section 6.2.1); some orders of magnitude related to this subject will be given in sections 7.2.4 and 7.2.5. Probabilistic methods require a much larger volume of data than deterministic methods; in simpler terms, we could say that deterministic methods require only “big” earthquakes whereas the “small” earthquakes influence the value of certain parameters used in the probabilistic approach. Probabilistic methods are therefore more cumbersome to put in place; on the other hand, they are generally less sensitive to important differences in the opinions of experts as regards crucial aspects such as the linking of a historical earthquake to a seismogenous accident or a tectonic region (see the notes on Figure 7.1). When the seismic hazard is defined by several ground motion parameters (for example, an acceleration-velocity-displacement triplet or the ordinates of a response spectrum for different frequencies; see section 9.2.1), we often target the same probability of exceedance for each of them; one of the consequences of such an approach, called a uniform hazard, is that it is generally impossible to make them correspond to an earthquake that is physically plausible characterized by its values of magnitude and distance from the site. This constitutes a disadvantage in cases where it is necessary to complete the definition of the hazard in order to carry out certain types of calculations, for example, when we need to represent seismic motion by one or several accelerograms so as to conduct a nonlinear analysis. This disadvantage does not generally arise, in deterministic methods where the reasoning is based on hypothetical earthquakes, but that can be defined in seismological terms, as the SMS of the RFS I.2.c (see section 7.1.2).
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7.2. Practices for the evaluation of seismic hazard 7.2.1. Normative evaluation and specific evaluation The methods of evaluation of seismic hazard presented in sections 7.1.2 and 7.1.3 were described for a specific site. Such specific evaluation, which requires a large amount of study, is justified for critical installations that involve special risks (as explained in decree 91-461, dated 14 May 1991), i.e. those installations for which the effects of even minor damage resulting from an earthquake on people, goods and the environment are not restricted to the immediate vicinity of the installation. They include factories that present the risk of escape of toxic and dangerous products (chemical, nuclear) or those that present the risk of floods (big dams). Hazard studies for a given site were originally undertaken for nuclear power stations; they were then extended to other special risk installations. For regular constructions presenting normal risks (seismic damage restricted to the perimeter of the site or to its immediate vicinity), the definition of seismic hazard is generally statutory and normative in nature and in reference to preestablished zoning, and hence does not require any specific study. The establishment of seismic zoning on a national scale requires a synthesis of seismic hazards, based on all the local studies available, using the general methods described above, but whose degree of complexity must be proportionate to the relatively rough features of the cartography or terrains being studied. The purpose of zoning is to specify the conditions of application of the antiseismic codes; it must thus be adapted to their format. As indicated in the introduction to this part and in section 7.1.1, a certain number of anti-seismic codes do not specify a precise definition of seismic hazard, in terms of ground motion; zonings associated with these codes, are thus not, strictly speaking, synthesized seismic hazard maps. The term zoning implies that the parameter or parameters that characterize the hazard have a constant value in each zone. If, for example, for practical reasons, the number of zones is reduced (from three to five as is the case in a large majority of national codes), we obtain a rather simplified representation of the hazard, which in reality has continuous variation. A specific problem arises when the government decides not to impose any antiseismic precautions in zones where the hazard is the lowest. Such a “zero” zone (according to French zoning terminology from 1985), which simply means that the hazard is judged low enough to be neglected for regular constructions, is often taken to be a zone where the seismic hazard is zero.
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It is therefore very difficult to convince owners of installations at special risk located in this zone that they must carry out a seismic hazard study for their site. We should hope that the term “zero zone” disappears from future zonings. Generally speaking, there may be apparent contradictions between certain specific evaluations of hazard and normative evaluations (zoning). The return periods associated with these two types of evaluation are indeed very different (10,000 years instead of 475 years when we use the probabilistic approach; see section 5.2.1). The zoning hierarchy thus primarily corresponds to relatively frequent events and could be debated when we consider much rarer earthquakes. Certain sites located in the zero zone of the French zoning of 1985 have a “special risk” hazard that is higher than that of other sites located in zone one. 7.2.2. Zoning for the anti-seismic codes The zonings associated with anti-seismic codes were introduced and dealt with briefly, in section 7.2.1. It is important that they are clearly distinguished from hazard maps, as they necessarily incorporate elements whose basis is as much, if not more, political as it is technical. Take for example, the decision to have a zero zone or the preoccupation of continuity with former zonings, which reflect the reluctance to reverse the decisions of the authorities in charge of regulation. Given the extent of the studies required to be undertaken for the establishment of zonings, these zonings are revised, in the same country, only after large intervals of time (about 10 to 20 years in the majority of cases); these intervals therefore correspond to the average “lifespan” of the codes. It is of course possible that the experience drawn from a major earthquake could lead to such revisions being made earlier as compared to the “normal time period”. Figure 7.3 shows the zoning established in 1985 for France [LAM 96].
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Figure 7.3. Seismic zoning from 1985 for metropolitan France (excluding Corsica, which is in zone 0)
It consists of five zones (0, Ia, Ib, II and III) of which the one representing the highest hazard (zone III) concerns only the overseas departments of Martinique and Guadeloupe. The above zoning was established by an essentially deterministic method which gave great importance to historical data of seismicity. It can be seen that the zero zone accounts for approximately 85% of the metropolitan territory. A comparison with Figure 6.2 shows that the most active zones (the Pyrenees, the Alps, Alsace) can easily be found, but that the vast diffuse seismicity zone that extends from the west to the center of France is represented only by some isolated pockets corresponding to places where historical earthquakes reached a relatively high level so as to cause significant damage (intensity higher than VII MSK).
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This zoning will soon be revised for the application of Eurocode 8. The new zoning will be based on the probabilistic method and use of a return period of 475 years common to a lot of national anti-seismic codes, but will also comprise, as is usually the practice, elements of a political nature. Probabilistic seismic hazard maps are generally presented in the form of a network of isovalue curves of a parameter like acceleration or velocity. Figure 7.4 shows a seismic hazard map in acceleration (return period of 475 years), established in 1985 for Turkey and the zoning map adopted in 1995 in this country.
Figure 7.4. Seismic hazard map in acceleration (above) and seismic zoning map (below) for Turkey; according to [MAR 99]
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A comparison of the two maps reveals relatively good consistency, but there are significant differences, in particular, for the zone that is most exposed to seismic hazards (in black) in the south-west. These differences result from the fact that the zoning considered the major historical earthquakes many of which have left an indelible mark on the collective memory of the Turkish people; such arguments will always hold great weight for decision makers, even if they sometimes appear somewhat contradicted by the results of the “technocratic” approach of return periods. Furthermore, the probabilistic criteria of hazard maps gives rise to a fundamental question, that the same return period is retained for two zones having different values for coefficient b of the Gutenberg-Richter laws, but receiving the same level of acceleration for the return period of 475 years. This equality of the hazard for this period should not hide the fact that, of the two zones being considered, the zone with a lower value of coefficient b is actually more dangerous than the other one, since the probability of an earthquake producing stronger effects at the “statutory” level is stronger there. It is advisable to supplement hazard studies carried out for a given return period (475 years for example) with an analysis of the sensitivity of the results to variations during this period. 7.2.3. Seismic microzoning Seismic zonings are established on a regional scale and cannot incorporate the influence of local conditions that are likely to modify the hazard. For example, the French zoning of 1985 (Figure 7.3) uses the “canton” (administrative unit in the French system of local government, each canton being made up of several “communes”) as a unit of delimitation, which means that all the territories belonging to “communes” falling under the same “canton” are subjected to the same level of risk. Moreover, zoning assesses the hazards only in terms of vibratory motions and does not apply to the prevention of risks resulting from the ground rupture by a fault or induced effects (liquefaction, landslides). It is thus necessary to supplement zoning by more detailed studies, known as microzoning, whose objective is to map, on a scale generally varying from 1/5,000 to 1/25,000, the various aspects of the local seismic hazard, namely: – active tectonic structures (faults likely to appear on the surface); – induced phenomena (liquefaction, ground movements, possibly tsunamis); – modifications in the vibratory motion due to local geomorphologic conditions (site effects; see section 4.3) or due to the proximity of the faults (for example, overlapping compartments of reverse faults; section 4.2.3).
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With respect to the fault, microzoning studies aim at determining the neutralization belts on both sides of the traces of rupture. The width of these belts depends on the type of fault, uncertainties as regards its location and the nature of the terrain. The methodological guide to microzoning established by the French Association of Earthquake Engineering [COL 93a] fixes for example, a minimum of 50 meters on both sides for competent ground and a maximum of 200 meters for friable ground, except for the overlapping part of a reverse fault, where the maximum can be much larger. The choice of these widths is the subject of some notes in section 11.1.1, so as to draw attention to the fact that tectonic risk (ruins of buildings due to ground ruptures) was undoubtedly highly overestimated in areas of moderate seismicity such as metropolitan France, at least within the framework of normal risk. In section 7.2.4, we return to the probabilities of surface ruptures. The recognition of zones that could be affected by induced phenomena can lead to the definition of other spaces of neutralization, insofar as the prevention of risks associated with these phenomena by the reinforcement of constructions is often impossible or extremely expensive (section 13.1). The hazard corresponding to the induced effects results from the combination of the sensitivity of the site to the occurrence of these phenomena (liquefaction, instability of the slopes) and the “strength” of vibratory motions considered as likely to occur. This combination thus depends, amongst other factors, on the level of probability associated with the regional seismic hazard. We must draw attention to the fact that the possibility of liquefaction is, on certain sites, primarily controlled by far-away earthquakes of strong magnitude, which contribute very little to the vibratory seismic hazard for regular constructions. Modifications in the vibratory motion due to local conditions can be evaluated in various ways: – calculation of the site effect from data describing the mechanical characteristics and the geometry of surface formations; this data is obtained from the synthesis of ground reconnaissance already carried out and could be supplemented by a campaign of specific measures; – experimental studies using background noise or other natural or artificial sources of excitation in order to determine the natural period of the sites and their factor of amplification (as indicated in section 5.3.3 the latter is generally not very reliable); – digital models on a local scale (a few dozen km²) take into account the geometry and the type of fault motions. Figure 7.5 shows the results of such a model for the city of Catania in Sicily [FAC 99].
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Figure 7.5. Seismic microzoning carried out for the city of Catania (Sicily). On the left, characterization of surface terrains by their S wave velocity; on the right, map of accelerations calculated by a fault model considered as representative of the historical earthquake of 1693 (according to [FAC 99])
Microzoning studies are carried out at various levels of sophistication depending on the level of risk involved. The AFPS Guide mentioned in [COL 93a] distinguishes between the following three levels: – A, where we are limited to the compilation and the interpretation of the data available; – B, where available information is supplemented by a limited number of complementary investigations (drillings, trenches, in situ measurements); – C, where we undertake very detailed studies implying ground reconnaissance and the development of models of calculation. The scales adopted for cartography vary, obviously, according to the level of the studies to be conducted (1/10,000 to 1/25,000 for A and B, 1/5,000 to 1/1,000 for C). The costs naturally increase from A to C.
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Although the usefulness of seismic microzoning is hardly contestable and the majority of the techniques used in these studies are operational at a reasonable cost, actually taking into account the results thus obtained often proves difficult, even in the most vulnerable countries (California, Japan), because certain municipalities do not really accept the idea that construction is prohibited in certain zones or that it can be noticeably more expensive in certain districts than in others. In France, it is the Plan for Prevention of Risks (PPR) that should include a section on seismic microzoning. 7.2.4. Orders of magnitude for hazards due to a fault (vibratory motion and surface rupture) Following the method described in section 7.1.3, the annual rate of exceedance nF (P) of value P of a parameter of ground motion due to the seismic activity of fault F, can be calculated analytically on the following assumptions: – the fault is assimilated to a straight line, of length 2l, situated at a depth h; – the frequency of occurrence of earthquakes on the fault is governed by a Gutenberg-Richter law with truncation at a maximum magnitude Mm, that is: N (M) = 10a (10-bM – 10-bMm)
[7.9]
and we assume that coefficient b has its “theoretical” value b = 1; – motion parameter P follows an attenuation law of the form: P=c e
DM
R-E
[7.10]
where we assume that E = 1 (geometrical attenuation of volume waves; see section 3.2.3) and that D is either equal to ½ Ln 10 or to ¼ Ln 10; these values correspond respectively to the “theoretical” values of this exponent for velocity and acceleration (section 6.1.3) Under these conditions, the annual rate of exceedance nF (P) of value P in site S is given by formula [7.6] which is written: X
10a § C · nF (P) = 2l ¨© P ¸¹
³
x2 x1
R X dx – 10
a Mm
x 2 x1 2l
[7.11]
where: – X is an integer equal to 2 if P is the velocity and to 4 if P is the acceleration;
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– R is the distance between the site and a point on the fault; – x1 and x2 correspond to the extreme positions of the points on the fault that contribute to the possibilities of exceedance, i.e. those whose distance from the site is lower than or equal to the limit Rm defined by equation [7.4] which becomes: Rm =
Ln10 c Mm eX p
In the system of axes defined by Figure 7.6, where site S has x = xO, y = yO, z = 0 as co-ordinates and where the fault goes from point F1 (x = – l, y = 0, z = – h) to point F2 (x = l, y = 0, z = – h), [7.11] is written as: a
nF (P) =
X
10 § c · ¨ ¸ 2l © p ¹
³
x2 x1
ª x xO 2 yO2 h 2 º ¬ ¼
X / 2
dx 10
a Mm
x2 x1 2A
[7.12]
x being the abscissa of the variable point on the fault.
Figure 7.6. System of axes Ox, Oy, Oz for the calculation of seismic hazard due to fault F1 F2 on any site S on the surface
We can assume that xo t 0; the x1 and x2 limits of the integral that intervenes in [7.12] can have different expressions which are recapitulated in Figure 7.7 where D represents the distance from the site to the straight line that represents fault F1 F2 and is expressed as follows: D=
yO2 h 2
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Figure 7.7. Various cases for the x1 and x2 limits in the calculation of the integral of formula [7.12]
The calculation of the integral of formula [7.12] is basic for X = 2 (velocity) and X = 4 (acceleration); in order to present the results, the following notations are introduced: [=
xO ;O D
l ; U D
Rm with D D
yO2 h 2
[7.13]
Therefore, we have: 2
If [ d O and 1 d U d 1 O [ (case c of Figure 7.7) [7.14] 2
U ² 1 10 § c · nF (V) = 10 Arc tan U ² 1 ¨ ¸ AD © v ¹ O a
nF (A) =
If
10a § c · 2AD 3 ¨© A ¸¹
4
ª « Arc tan U ² 1 «¬
a Mm
U ² 1 º U ² 1 10 » 2 U O »¼
a Mm
1 (O [ )² d U d 1 (O [ )² (case d and f of Figure 7.7)
[7.15]
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nF (V) =
nF(A)=
a
2
O [ U ² 1 aMm §c· ª 10 ¨ ¸ ¬ Arc tan U ² 1 Arc tan(O [ ) º¼ 2 AD © V ¹ 2O
10
4ª §c· « Arc tan U ² 1 ¨ ¸ 3 4lD © A ¹ « ¬ 10
a
¼
2
a
nF (V) =
10
º U ² 1 O [ » [7.17] Arc tan(O [ ) 2 2 U 1 O [ »
1 O [ d U (case e and g of Figure 7.7)
If
nF(A)=
[7.16]
a
4lD
3
2
a Mm 10 § c · ª¬ Arc tan O [ Arc tan O [ º¼ 10 ¨ ¸ 2lD © V ¹
4 º O [ O [ §c· ª » ¨ ¸ « Arc tan O [ Arc tan(O [ ) 1 O [ ² 1 O [ ² ¼ © A¹ ¬
[7.18]
– 10a-Mm
[7.19]
For the numerical application of these formulae, we consider the case of a fault of length 100 km (l = 50 km), of depth h = 10 km, with an activity parameter equal to 3.5 (which corresponds, according to formula [6.12] to a centennial magnitude of 5.5 for an untruncated Gutenberg-Richter law). Such a choice of parameters is representative of the most active faults of metropolitan French territory. We first study the influence of maximum magnitude by calculating the inverse of nF (A), i.e. the return period of acceleration, for the value A = 2 m/S² (# 0.2g) at the most vulnerable point which is at the center of the surface fault projection (xo = 0, yo = 0). For coefficient C of attenuation law [7.10] we take the value 1 m/s² which is close to that of theoretical law [5.36]: A (m/s²) =
e0.576 M R (km)
100.25 M R (km)
[7.20]
This law, the numerical coefficients of which are quite easy to remember, is more than sufficient for the estimation of the orders of magnitude.
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For maximum magnitudes varying from 6 to 7.5 parameter U (with D = 10 km for a given point) varies from 1.58 to 3.75 and, taking in to account that O = 5, formula [7.15] is applied and we thus obtain the following values for return period TA (Table 7.1). Mm
Rm (km)
TA (years) for A = 2m/s²
6
15.8
514
6¼
18.3
431
6½
21.1
387
6¾
24.3
363
7
28.1
348
7¼
32.5
338
7½
37.5
333
Table 7.1. Variation in the return period with respect to Mm
We observe that the effect of the maximum magnitude is relatively low; the last value calculated in the table (for Mm = 7.5), i.e. TA = 333 years, is very close to the threshold value (for Mm raised to infinity) that is equal to 323 years when we apply formula [7.19]. This conclusion as regards the influence of Mm is related to the choice of parameters in this example (moderate activity of the fault, consideration of the average value of acceleration); this holds true for seismic hazard studies based on the return period of 475 years in zones of low to average seismic activity (furthermore, it is evident that the values of TA in the table are close enough to this value of 475 years). This value could be questioned in other contexts, for example, if longer return periods (10,000 years as shown in section 5.2.1) in zones of moderate or high seismic activity were considered. After the sensitivity study regarding maximum magnitude, it is possible to examine the influence of the distance from the fault. Let us consider the following (Figure 6.8) three profiles: – CD perpendicular to the surface fault projection, from its center C; – EF, parallel to CD, from extremity E of the fault projection; – EL, in the extension of the fault from the same extremity E.
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Figure 7.8. Profiles considered for the study of the influence of the horizontal distance from the fault
Calculations are based on a value of 6.5 for the maximum magnitude Mm and the same acceleration as before (2 m/sL); the extremities D, F and L of the three profiles corresponding to the maximum dm = Rm² h ² of the horizontal distance d; for d > dm, the acceleration must necessarily be lower than 2m/s²; with the hypotheses that have been retained, we obtain dm = 18.6 km.
The results are shown in Figure 7.9 in the form of curves showing the variation of the decimal logarithm of the return period with respect to the horizontal distance d. We observe that the return period of acceleration 2m/s² increases rapidly with the distance from the fault and tends towards infinity when d reaches its threshold value dm = 18.6 km. As we could have expected, the hazard is highest (and the return period shortest) for profile CD, followed by EF and EL. The fact that EL appears to be the least vulnerable profile is a natural consequence of the hypotheses of the calculations. In reality, we have numerous examples of the directivity effects (section 4.3.2) that show that the extension of the fault can be subject to a higher degree of hazard; such effects were not taken into consideration while drawing up formulae [7.14] to [7.19].
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Figure 7.9. Variation of the return period TA (in years) with respect to the horizontal distance d from the trace of the surface fault (in km) for the three profiles CD, EF, EL defined in Figure 7.8; these variations have been calculated for a maximum magnitude of 6.5 and an acceleration of 2 m/s²
To conclude the study on sensitivity, the variation curve of the return period of acceleration can be traced according to its level, with the same hypotheses as before (l = 50 km, h = 10 km, Mm = 6.5, a = 3.5) considering the center C of the trace of the fault in Figure 7.10. The variation of TA is quite rapid, this being due to the presence of the term in A-4 in formulae [7.15], [7.17] and [7.19]. The return periods correspond respectively to the accelerations of about 2.1 m/s² and 3.5 m/s² for the selected example; we observe that for TA = 10,000 years, we are placed on a part of the curve that is influenced by maximum magnitude (vertical asymptote associated with the
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maximum value of acceleration that is 4.21 m/s² in the hypotheses retained); this tallies with the observation drawn above about the influence of Mm.
Figure 7.10. Variation of the return period TA with respect to acceleration A at the center of a fault having the characteristics l = 50 km, h = 10 km, a = 3.5, Mm = 6.5
The hazard represented by the surface rupture may be estimated by calculations similar to those used for the vibratory hazard. We shall first consider a fault with a vertical dip (Figure 7.11), in such a way as to maximize, for a given dimension, the probability of it reaching the surface. The center of the fault plane has a depth h, of which the probability is considered to be uniform between limits h1 and h2; the half-height W of the fault plane is defined with respect to magnitude M by means of the equation: W (km) = l x 10 M/X
[7.21]
l and X being two constants; the “normal” value of X is 2 if we assume the scaling law (see section 2.1.3) in numerical applications, we take l = 0.005, which corresponds to a square fault plane whose area would be given by equation [2.27]; thus, we get W = 5 km for M = 6; this hypothesis of the square fault plane was selected to maximize the probability of surface rupture; in reality, the length of vertical fault planes is usually larger than their height (see Table 2.2).
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Figure 7.11. Fault with vertical dip to study the “surface rupture” hazard. W is the half-height of the fault plane; depending on the depth of the center C of this plane, there is no rupture (position C1) or surface rupture (position C2)
The frequency of occurrence of earthquakes with respect to their magnitude is, as before, given by the Gutenberg-Richter law with the truncation at the maximum magnitude Mm (equation [7.9]), but coefficient b could have any value (whereas we had taken b = 1 in the study of vibratory hazard). The term hm represents the depth that the center of the fault plane should have for it to reach the surface when the magnitude is equal to its maximum value Mm; taking into account [7.21], we thus obtain: hm = l x 10 Mm /X
[7.22]
If hm is less than the lower limit h of the depths, surface rupture is impossible; consequently there are only two cases that need to be retained: c
h1 d hm d h2
d
h2 < hm
In case c the depth at which surface rupture may occur varies between h1 and hm; taking into account the hypothesis of equiprobablity of depths, in the interval h1 – h2, , we obtain the following for the annual rate nR of surface ruptures: nR = ³ N M h hm
h1
dh h2 h1
[7.23]
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Mh being the magnitude whose associated fault plane just about reaches the surface when the depth of its center is equal to h; we thus obtain the equation: h = l x 10 Mh/X
[7.24]
Given [7.9] and [7.24], equation [7.23] is written as: a
nR =
10 h2 h1
³
hm
h1
ª§ l ·Xb º bMm «¨ ¸ 10 » dh ¬«© h ¹ ¼»
[7.25]
The same equation [7.25] is also applied in case d (h2 < hm) by replacing hm with h2 in the higher limit of the integral. After basic integrations, we obtain the results: in case c (h1 d hm d h2): nR =
X b 1 Xb §h · §h · º 1 ª «1 X b ¨ 1 ¸ X b 1 ¨ 1 ¸ » TO « © hm ¹ © hm ¹ »¼ ¬
[7.26]
in case d (h2 < hm) Xb X b 1 § h2 h1 · § h1 · º 1 ª § h1 · nR = «1 ¨ ¸ (X b 1) ¨ ¸¨ ¸ » TO « © h2 ¹ © h1 ¹ © hm ¹ »¼ ¬
[7.27]
To being the reference period defined by: Xb
§ h h ·§ h · To = (Qb – 1) ¨ 2 1 ¸ ¨ 1 ¸ x 10-a © h1 ¹ © l ¹
[7.28]
For the numerical application of the above, we take the “normal” values Q = 2, b = 1, a distribution of the depths corresponding to the two cases h1 = 5 km, h2 = 10 km and h1 = 3 km, h2 = 10 km, and finally for the fault activity, value a = 3.5 (centennial magnitude of 5.5, as in the case of the evaluation of the vibratory hazard earlier). The calculations of the inverse of nR, i.e. the return period TR of the surface rupture, by formulae [7.26] and [7.27] lead to the results of Table 7.2 where different values of the maximum magnitude Mm have been considered:
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Mm
hm (km)
TR (years), h1 = 5 km
TR (years), h1 = 3 km
6
5
f
1660
6¼
6.67
5055
878
6½
8.89
1651
605
6¾
11.86
981
482
7
15.81
791
431
7¼
21.08
713
407
7½
28.12
675
394
Table 7.2. The influence of the maximum magnitude on the return period of a surface rupture by a fault of activity a = 3.5 for the two cases of minimum depth h1 = 5 km and h1 = 3 km
We see that TR decreases very rapidly when Mm is slightly higher than the threshold value at which rupture is possible (hm = h1), then decreases at a much slower rate up to the point that it starts tending asymptotically towards the threshold value TR, lim given by the following equation, deduced from [7.27] and [7.28]: Xb ª § h ·Xb 1 º §h ·§ h · TR, lim = (Xb – 1) ( ¨ 2 1¸ ¨ 1 ¸ x 10-a / «1 ¨ 1 ¸ » «¬ © h2 ¹ »¼ © h1 ¹© l ¹
[7.29]
In “standard” cases, Xb = 2, this formula takes a very simple form: (TR, lim) Xb = 2 =
h1h2 u 10 a l2
[7.30]
In the two cases studied in Table 7.2, the values of TR, lim, calculated by [7.30], are equal to 632 years (h1 = 5 km) and 379 years (h1 = 3 km). The influence of minimal depth h1 is noticeable though not surprising, since low values of h1 are associated with surface ruptures due to earthquakes of low magnitudes, hence quite frequent; the tectonic hazard represented by these “small” earthquakes is not very significant as the amplitudes of the corresponding displacements are very low. Thus, in the application of the preceding formulae, it would be more realistic to take h1 values comparable to the half-heights of the fault planes, corresponding to magnitudes high enough to produce significant displacements. We can assume that magnitude 6, where displacement is supposed to be between 10 and 50 cm (see Table 2.2), represents the lower limit of significant tectonic hazard; the corresponding value of h1 is 5 km with a choice of parameters as
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above (X = 2; l = 0.005 km), and a study of Table 7.2 shows that the return period of a surface rupture is counted in thousands of years for a fault whose centennial magnitude is 5.5 and whose maximum magnitude is at most 6.5. As these conditions constitute a plausible explanation for the most active faults in metropolitan France, we observe, as indicated in section 11.1.1, that the tectonic hazard is relatively insignificant in the case of return periods adapted for installations at normal risk. The preceding formulae ([7.26] to [7.28]) show distinctly that it is the parameter of activity that proves most critical for the return period of surface ruptures; the presence of factor 10-a in equation [7.28] of TO shows that the return period is divided by 10 when the centennial magnitude increases by a single unit (with b = 1). 7.2.5. Orders of magnitude of vibratory hazard in diffuse seismicity zones
Diffuse seismicity zones, i.e. zones where observed seismicity cannot be associated with well identified faults, play an important role in the study of seismic hazard in moderately active intraplate regions. We therefore assume, for lack of a better alternative, that the probability of occurrence of earthquakes of a given size would be the same at each point in these areas. This hypothesis allows us to calculate in a rather simple manner, the return period exceeding a given value of a soil motion parameter. We will consider the same set of hypotheses as those that were used earlier for the faults, i.e.: – a uniform distribution of focal depths h between the lower limit h1 and the higher limit h2; – a Gutenberg-Richter law for the frequency of occurrence of earthquakes in the zone being studied, expressed by equation [7.9]: N (M) = 10a (10-bM – 10-bMm)
[7.31]
Mm being the maximum magnitude; coefficient b could have any value, and is not equal to 1 as in the beginning of section 7.2.4, – an attenuation law for the motion parameter P having the form according to [7.10]: P = C eD M R-E
[7.32]
where coefficients c, D and E have a priori any value; henceforth we will replace D by the coefficient X, so that:
Assessment of Seismic Hazard
X=
Ln10
345
[7.33]
D
in such a way that, if the values of the theoretical attenuation laws are adopted (section 5.1.3), we will get X = 2 if parameter P is the velocity and X = 4 if it is the acceleration. We shall introduce values P1 and P2 of parameter P corresponding to an earthquake of maximum magnitude Mm that occurs respectively at distances h1 and h2: P1 = c 10Mm / X h1 E
[7.34]
§h · P2 = c 10Mm/X h2 E = P1 ¨ 1 ¸ © h2 ¹
E
[7.35]
For a given value of P, the higher limit of possible focal distances or Rm (P), is given by equation [7.4]: 1/ E
§c · Rm (P) = ¨ eD Mm ¸ ©P ¹
1/ E
§p · h1 ¨ 1 ¸ © P¹
[7.36]
The calculation of the annual rate nz (P) of exceedance of value P, at a surface site is obtained by formula [7.5], by carrying out the necessary transpositions so as to adapt it to the case of a volumic zone (layer thickness h2 – h1) instead of a surfacic zone. As a variable of integration, it is logical to use the distance R from the site to the focal point that varies between h1 and Rm (P), the only case of practical interest is where the chosen value of P is lower than P1; thus, for nz (P) we have: nz (P) =
³
Rm ( P )
h1
N M p1 R
V R dR
¦ h t
2
h1
[7.37]
where MpR is the necessary magnitude to produce P at distance R, V (R) the area of the part of the sphere of radius R, centered on the site that is situated in the seismogenic layer of thickness h2 – h1 and 6t the total area (plane) of the zone of diffuse seismicity. Given the information in [7.31], [7.32] and [7.33], N (MP1R) is expressed by: ª§ p · Xb EXb bMm º N (MP1R) = 10a «¨ ¸ R 10 » ¬«© c ¹ ¼»
[7.38]
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For the value of V (R), there are two possible cases, depending on whether R is higher or lower than h2 (Figure 7.12).
Figure 7.12. Determination of V (R) in the two cases h1 < R < h2 and b) h2 < R
If h1 < R < h2 (case a) of the figure) V (R) is the area of a spherical cap: V (R) = 2 S R (R – h1)
[7.39]
If h2 < R (case b) of the figure) V (R) is the area of a spherical zone: V (R) = 2 S R (h2 – h1)
[7.40]
With formulae [7.38], [7.39] and [7.40], the calculation of integral [7.37] is done in a basic manner. In order to present the results, we shall introduce the following notations: x=
p1 x t 1 P
[7.41]
G=
h2 G t 1 h1
[7.42]
J = EXb To (reference period) =
[7.43] ¦ t h2 h1 2S h13
10
bMm a
[7.44]
Assessment of Seismic Hazard
Tz (return period) =
1 nz P
347
[7.45]
and for ratio To/Tz: – if I d x d GE: TO TZ
x
J /E
2 J 3 J
Jx
3/ E
3 3 J
Jx
2/ E
2 2 J
1 6
[7.46]
– if GE d x: 3J J G 1 2 / E 1 3 J /E TO 1G x x G 1 TZ 2 J 3 J 2 2 J 6
[7.47]
These formulae present a singularity for the value 2 of the parameter that is of practical importance as t corresponds to the choices E = 1 (geometric attenuation of volume waves), X = 2 (theoretical attenuation law of velocity) and b = 1 (“normal” value for the Gutenberg-Richter law). By going towards the threshold, [7.46] and [7.47] become the following, when J tends towards 2: – if I d x d GE: TO TZ
· 2/ E 1 2 3/ E 1 § 2 x ¨ 1 Lnx ¸ x 3 2© E 6 ¹
[7.48]
– if GE d x: TO TZ
ª1 1 G LnG º 2 / E 1 3 Lnx x G 1 6 G 1 »¼ ¬2 E
G 1 «
[7.49]
Figure 7.13 shows the variation of log10 (TZ/TO) with respect to x in the following cases: – E = 1, X = 4 (which corresponds to the acceleration for parameter P in the “theoretical” law) and b = 0.75, 1 or 1.25; – E = 1, X = 2 (which corresponds to the velocity for parameter P in the theoretical law) and b = 0.75, 1 or 1.25.
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Figure 7.13. Variation of the return period with respect to the motion parameter (velocity or acceleration) in a zone of diffuse seismicity with the theoretical attenuation laws and b = 0.75, 1 or 1.25
In a numerical application, we consider a zone of diffuse and moderate seismicity, of a total area 6t = 10,000 km ² and where the focal depths vary between h1 = 5 km and h2 = 20 km; the centennial earthquake is of magnitude 5 and the maximum magnitude Mm is equal to 6.5. Thus, as per relation [6.12], we get the following values for the activity coefficient a and the reference period TO:
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349
– if b = 0.75, a = 1.75 and TO = 254,700 years; – if b = 1, a = 3 and TO = 604,000 years; – if b = 1.25, a = 4.25 and TO = 1,432,000 years. From the curves of Figure 7.13 we obtain, for accelerations corresponding to a return period of 475 years (by taking as in 7.2.4, c =1 m/s², D = ¼ Ln 10 and E = 1 in the attenuation law [7.10]): – for b = 0.75, log10 TZ/TO = – 2.73, x = 8.50 and A = 0.99 m/s²; – for b = 1, log10 TZ/TO = – 3.10, x = 7.87 and A = 1.07 m/s²; – for b = 1.25, log10 TZ/TO = -3.48, x = 7.27 and A = 1.16 m/s². We observe that in this case, the influence of parameter b of the GutenbergRichter law is rather low; this occurs due to the moderate character of the seismicity of the diffuse zone being studied and due to the fact that accelerations are often controlled by earthquakes of relatively small magnitudes occurring close to the site. In the example being considered, they must be of a lower magnitude than the centennial magnitude, since with M100 = 5, it is necessary that the focus be at approximately 18 km so as to produce on the site, similar accelerations to those found, i.e. 1 m/s². This interpretation is confirmed by the observation that A increases with b, which in turn corresponds to the same tendency for the number of occurrences N (M) when M is lower than the “pivot” value M100; when M > M100, N (M) decreases when b increases and this decrease would be reflected on the acceleration, if it was strongly influenced by these “large” earthquakes. In order to be able to obtain orders of magnitude with very simple formulae, we could transform the preceding formulae [7.46] to [7.49], by considering a case where the coefficients have their “normal” values (E = 1, X = 2 or 4 depending on whether the motion parameter is velocity or acceleration, b = 1) and where focal depth has a single value h (instead of varying in the interval h1 – h2); we then obtain: 2
TO' TA
2 º 1 ª§ Am · 1» «¨ ¸ 2 «¬© A ¹ »¼
TO' Tv
§ Vm · ª Vm 1 º 1 ¨ V ¸ « Ln V 2 » 2 © ¹ ¬ ¼
[7.50]
2
[7.51]
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Seismic Engineering
where T’O is a reference period defined by: T’O =
¦t 10Mm a 2S h 2
[7.52]
TA and TV are the return periods associated respectively with values A and V of acceleration and velocity. Am and Vm are the maximum possible values for these motion parameters, corresponding to the earthquake of maximum magnitude Mm at distance h (focal depth) from the site, i.e. by using the rounded “theoretical” attenuation laws (section 5.1): Am (m/s²) =
10 Mm / 4 h(km)
Vm (m/s) = 0.002
[7.53]
10 Mm / 2 h(km)
[7.54]
Formula [7.50] can be easily inverted to give the following expression of A/Am with respect to T’O/TA: A Am
ª T' º «1 2 0 » TA »¼ «¬
1/ 2
[7.55]
or again, taking into account [7.52] and [7.53], and by introducing the centennial magnitude: M100 = a + 2 (see [6.12]), instead of a: 10 Mm / 4 A (m/s²) = h
ª 100 ¦ t 1 MmM 100 10 «1 S h ² TA «¬
º » »¼
1/ 2
[7.56]
where h is in km, 6t is in km² and TA is in years. Table 7.3 shows the results obtained by formula [7.56] with 6t = 10,000 km², h = 10 km, TA = 475 or 10,000 years, and different values of Mm and M100.
TA = 10,000 years
T4 = 475 years
Assessment of Seismic Hazard M100 = 5
M100 = 5.5
M100 = 6
Mm = 6.5
1.07
1.39
1.78
Mm = 7
1.08
1.43
1.86
Mm = 7.5
1.09
1.45
1.90
Mm f
1.11
1.47
1.97
Mm = 6.5
2.06
2.53
2.98
Mm = 7
2.18
2.75
3.37
Mm = 7.5
2.26
2.91
3.67
Mm f
2.37
3.16
4.21
351
Table 7.3. Values of A (m/s²) calculated by [7.56] for 6t = 10,000 km², h = 10 km and different values of TA, Mm and M100
We observe that the maximum magnitude Mm has a significant influence only in the case, TA = 10,000 years; when Mm increases indefinitely, acceleration tends towards a threshold value Alim given by: § S TA Alim = ¨ 10 M100 ¨ 100 ¦ h² t ©
1/ 4
· ¸¸ ¹
[7.57]
The passage from TA = 475 years to TA = 10,000 years leads roughly to a doubling of acceleration. The graphic representation of [7.50] and [7.51] is given in Figure 7.14. We see that for the same return period TZ = TA = TV, acceleration is closer to its maximum value than velocity. The methods of uniform hazard, mentioned at the end of section 7.1.3, thus favor, in the description of ground motion, the components of relatively high frequency (those which are related to acceleration) rather than those of medium frequency (related to velocity). If we limit ourselves to the two parameters acceleration and velocity, in a probabilistic approach to uniform hazard, it is possible to determine values MT and
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RT of the magnitude and focal distance of an earthquake which would produce on the site values A and V, calculated for the same return period T.
Figure 7.14. Graphic representation of relations [7.50] and [7.51]
We can easily show, with the help of rounded theoretical attenuation laws, that we should have: § V (m / S ) · MT = 4 log10 ¨ 500 ¸ A(m / s ²) ¹ ©
[7.58]
Assessment of Seismic Hazard
RT (km) = 500
V (m / s) A²(m / s ²)
353
[7.59]
If, for example, we take the case Mm = 6.5, M100 = 6 of Table 7.3 we get: – T = 475 years, A = 1.78 m/s², V = 0.099 m/s (as in Figure 7.3), MT = 5.78, RT = 15.6 km; – T = 10,000 years, A = 2.98 m/s², V = 0.216 m/S, MT = 6.24, RT = 12.2 km. 7.2.6. Effect of the size of the site on the vibratory hazard in a zone of diffuse seismicity
In a zone of diffuse seismicity, the annual rate of exceedance of the given value of a parameter of ground motion is necessarily higher for a large site (i.e. spread out over a certain area) than that for a specific site (point). The question that thus arises is whether is it necessary to raise or overestimate the assessment of the seismic hazard for a point (as assumed in section 7.2.5) when the installation which is the subject of the seismic hazard study involves a big area (large agglomeration, very vast industrial facility) or an elongated area (very large tunnel or another big linear structure). The answer to this question can be given in relatively simple terms if the following assumptions are verified: – the focal depths are uniformly distributed between h1 (lower limit) and h2 (higher limit); – the frequency of the earthquakes is governed by a non-truncated GutenbergRichter law: N (M) = 10a-bM
[7.60]
in which coefficient b has only three possible values 0.75, 1 or 1.25 – the motion parameter considered is the acceleration which is assumed to follow the theoretical attenuation law: A = c 10M/4 R-1
[7.61]
where, as was done earlier, we shall take c = 1 m/s² for A in m/s² and R in km; – the studied site in the form of a convex polygon, whose area is given by S and semi-perimeter by L.
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Figure 7.15. Composition of area V (R) for a polygonal site in the case h1 d R d h2
The annual rate of exceedance nZ (A) of acceleration A is given by the general formula [7.37], adapted to the preceding assumptions (Mm o f thus Mro f, E = 1, X = 4), i.e.: nZ (A) =
10a § A · -4b ¦t (h 2 h1) ¨© C ¸¹
³
f
h1
R 4b V (R) dR
[7.62]
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355
It is also necessary to adapt the expression of the area V (R) to the hypothesis of the shape of the site; there are two cases depending upon the focal distance R which can be lower or higher than h2: 1st case, h1 d R d h2 (Figure 7.15) V (R) thus consists of a horizontal area, identical to the polygonal site (pentagon P1P2P3P4P5, in the figure), of N cylindrical elements (N = number of sides of the polygon) denoted as C1, C2… Cn and of n spherical section elements S1, S2… Sn, whose union forms a complete spherical cap; thus, by introducing the angle T1 marked on the figure, we obtain: V (R) = S + 2 LRT1 + 2 S R² (1 – cos T1)
[7.63]
or again, since R cos T1 = h1: V (R) = S – 2 S R h1 + 2 S R² + 2 LR Arc cos
h1 R
[7.64]
2nd case h2 < R (Figure 7.16) Thus there is no longer a horizontal surface and V (R) is made up of n cylindrical elements and n spherical section elements whose union forms a complete spherical cap; thus, by introducing angles T1 and T2 marked on the figure, we have: V (R) = 2 LR (T1 – T2) + 2 S R² (cos T2 – cos T1)
[7.65]
or again, since R cos T1 = h1 and R cos T2 = h2: V (R) = 2 S R (h2 – h1) + 2 LR (Arc cos
h1 h – Arc cos 2 ) R R
[7.66]
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Figure 7.16. Composition of the area V (R) for a polygonal site in the case h2 < R
Equation [7.62] which determines nZ (A) is thus written as: § A· nZ (A) 6t (h2 – h1) 10-a ¨ ¸ ©c¹
=
h2
³h
1
4b
f f ª¬ SR 4b 23R 2 4b º¼ dR – 2Sh1 ³ R1-4b dR + 2Sh2 ³ R1-4b dR h1 h2 [7.67]
+ 2L
f
³h
1
R1-4b Arc cos
f 1-4b h1 h dR – 2L ³ R Arc cos 2 dR R R h2
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357
The integrals of ArcCos are obtained analytically in the three cases b = 0.75, b = 1 and b = 1.25 and we get: – for b = 0.75: 3
10a § c · ª Lnh 2 / hI 1 h h º L nZ (A) = 2 S 1 2 22 » « 2S ¨ ¸ h2 h1 h1h2 2 h1 h2 ¼ 6t © A ¹ ¬
[7.68]
– for b = 1: nZ (A) =
4 S L(h1 h2 ) S h12 h1h2 h22 º 10a § c · ª S « » 3 6t ¨© A ¸¹ ¬ h1h2 4 h12 h22 h13 h23 ¼
[7.69]
– for b = 1.25 5
nZ (A) =
10a § c · ª S h1 h2 4 L h12 h1h2 h22 S ( h1 h2 )(h12 h22 ) º « » 6t ¨© A ¸¹ ¬ 3 h12 h22 9 h13h23 4 h14 h24 ¼
[7.70]
In the case where L = 0 and S = 0 (point specific site), we can verify that these formulae are identical to the general formula [7.47] for E = 1, X = 4 and infinite Mm. In the case of a large site (spread out area) (L > 0, S > 0), we thus have the following expressions of amplification factor FA by which we must multiply the acceleration evaluated for a point specific site so as to take into account the increase in the hazard when the same return period is kept [BET 96]: – for b = 0.75: 1/ 3
ª 1 L(h2 h1 ) 1 S (h22 h12 ) º FA = «1 » 2 2 ¬ S h1h2 Lnh2 / h1 4S h1 h2 Lnh2 / h1 ¼
[7.71]
– for b = 1: ª 1 L(h1 h2 ) 1 S (h12 h1h2 h22 ) º 1/4 FA = «1 » h12 h22 3S ¬ 4 h1h2 ¼
[7.72]
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– for b = 1.25: 1/ 5
ª 4 L(h12 h1h2 h22 ) 3 S (h12 h22 ) º FA = «1 » 4S h12 h22 ¼ ¬ 3S h1h2 (h1 h2 )
[7.73]
SITES SITES
SQUARE
LINEAR
Table 7.4 gives the values of factor FA for some examples of linear sites (length L > 0, area S = 0) and of square sites (side L/2, area S = L²/4) for values h1 = 5 km, h2 = 20 km. L (km)
S (km²)
FA, b = 0.75
FA , b = 1
FA, b = 1.25
2
0
1.022
1.030
1.034
5
0
1.054
1.070
1.078
10
0
1.104
1.129
1.138
20
0
1.191
1.225
1.230
50
0
1.397
1.425
1.409
4
4
1.047
1.062
1.070
10
25
1.118
1.152
1.167
20
100
1.240
1.294
1.308
40
400
1.480
1.547
1.541
Table 7.4. Factor of amplitude FA of acceleration for different large spread-out linear or square sites
We observe that the effect of the size of a large spread-out site becomes significant when its size is comparable to, or bigger than, the focal depth. The increase in acceleration as compared to that of a point specific site is about 40% for a 50 km tunnel and about 50% for a large area of 20 X 20 = 400 km². The influence of coefficient b of the Gutenberg-Richter law is low.
Part 4
Seismic Action
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Introduction
In the traditional meaning of the term, action refers to the set of parameters that serve as entry data for the calculation of effects of a natural or artificial phenomenon. Thus it has to be linked to the mode of calculation used. The more the action is simplified to make it accessible to non-specialized designers, the more the definition is reduced to quite crude diagrams which hide the physical reality of the phenomenon being studied, which is often very complex. This type of situation is quite the normal rule in construction equations applicable to the current works. Before moving on to seismic action, where there is an enormous difference between reality (vibratory ground motion) and calculation diagrams, perhaps it would be useful to clear up the problems by using the “statutory” practice for a more familiar phenomenon such as wind. In the current equations (such as the wind and snow load-regulations in France) the action of wind is essentially represented by static pressure fields applied to the walls of the structure. The dynamic aspects of the stimulus (temporal variations of the module and the direction of speeds), their interactions with the structures (response from the vibration’s own modes, aeroelastic phenomena) and the influence of the adjacent structures (obstacle or concentration effects) are considered only for exceptional structures. Taking these elements into account will require a much more precise definition of the action and implementation of extremely elaborate methods of study (large digital models in three-dimensional dynamics, experimenting with wind tunnels on reduced-scale models). A similar method is found in earthquake engineering. For a long time the current equations have been limited to “lateral resistance”, i.e., checking the resistance to horizontal static forces, which were often fixed without clear reference to a movement of earth parameter. It is only recently that more “seismological” definitions have been used in earthquake-resistant codes (see the introduction to Part 3) in relation to the
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development of dynamic studies for hazardous installations particularly nuclear power plants (transient calculations, tests on vibrating tables). Currently, seismic action is mostly defined in terms of the elastic response spectrum. This notion, which will be described in Chapter 9, has its origin in a basic result of linear dynamics of undamped structures, which means that there is an absence of coupling between the responses of their own modes of vibration. The elastic response spectrum as a characteristic of the seismic action is thus linked to the adequacy of linear calculations for gauging security. This adequacy is generally quite difficult to establish as the failure schemes are conditioned by the post-elastic capacities of structures (fragile or ductile behavior, dissipative or non-dissipative nature of energy during loading cycles). The elastic response spectrum is not a very good indicator of the damage potential of seismic movement, particularly because it is not very sensitive to the influence of the duration of the accelerogram. The elastic response spectrum is usually used to define the seismic action for risky installations which have very stringent safety rules which strictly monitor any chances of being damaged. Maintaining the seismic responses in the elastic field which in theory corresponds to an absence of damages, is thus a simple way of achieving these objectives and it does not appear necessary to go further than the elastic spectrum to define the action. Such an approach is, to say the least, debatable for earthquake-resistant codes that can be applied to current structures; in fact, the economic constraints limit the potential of these codes intended to prevent collapse during a violent earthquake. On the other hand, they are highly ambitious on a technical plane as it is unbelievably more difficult to aim for “limiting ruin” for a three-dimensional dynamic stimulus with certain unpredictable characteristics such as seismic stimulus than to check the stability in the elastic domain. In addition, as this technical challenge can only be taken up by using simple calculation methods so as to be able to codify and implement them without risking that they are misinterpreted by “lay” people calculating the structures, the earthquake-resistant codes, where action is an elastic response spectrum, must “catch up with” this definition, which is not quite appropriate using the “behavior coefficient” approach. This notion is not particularly satisfactory and in the current status of knowledge much more empirical (almost “a rough guess”) than scientific. In order to have a clear understanding of post-elastic behavior, the failure schemes must be as physically plausible as possible. It is necessary to complete the definition of the seismic action if the latter is given in terms of the elastic response spectrum. In fact, the design response spectra do not represent a real movement of the Earth, but a set of possible movements corresponding to different seismological conditions (magnitude, distance to the site). Their transcription in accelerogram terms, which is basic data for nonlinear studies, is a rather delicate operation which
Introduction
363
requires knowledge of the tectonic context of the region. In particular, using synthetic accelerograms adjusted to the spectrum has led to a lot of incorrect results. This third part deals successively with the three versions used to define seismic action: seismic coefficients, response spectra and accelerograms. The response spectra will be described in relation to the modal spectrum analysis from which they cannot be separated as we can hardly talk of seismic action without referring to the calculation mode as the seismic action is the basic entry data. Very brief notes are given on the representation of seismic action by random processes.
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Chapter 8
The Seismic Coefficient
8.1. The seismic coefficient in past earthquake-resistant codes 8.1.1. Notion of seismic coefficient The seismic coefficient is a dimensionless number whose product by weight, supported by a structural element, represents a static horizontal force reputed to be equivalent to the forces of horizontal inertia acting on this element. It was introduced in the first attempts at anti-seismic codification that followed the “founding” earthquakes: San Francisco 1906 in California, Messina 1908 in Europe and Kanto 1923 in Japan. The analysis of the destruction had brought to light, as the primary cause, an insufficient resistance of the buildings to the horizontal forces; these inertial forces, thus proportional to the mass, could be as a first surmise represented by a fraction of the weight; the choice of this fraction called seismic coefficient was originally purely empirical, as in those days there was no registration of strong movements (see introduction to the second part). The values retained for the seismic coefficient in the first equations were of the order of 0.1 (0.075 to 0.1 in the American Uniform Building Code of 1927, 0.08 in the Los Angeles city code of 1933) and were applied in the same manner to all buildings whatever their height. The later developments of seismic coefficient equations, a summary of which is given in the following section, took into account the influence of a certain number of parameters in the expression of this coefficient and in particular: – the level of seismic hazard; – the fundamental period of free oscillations of the structure;
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– the quality of the foundation soil; – the height above the ground of the point where the “equivalent seismic force” is calculated; – the likelihood of deformation of the structural system in the post-elastic domain. The progressive introduction of these complications has been necessitated by the awareness of the fact that the notion of the seismic coefficient, if it is to be defined simply is in reality an amalgamation of several different notions, each one of which presents specific problems, more or less resolved, in the current state of knowledge: – the seismic hazard, in terms of vibratory movements of the earth, whose, as we saw in Part 2, uncertain characteristics define its nature; – the seismic response of the structures in the elastic domain (fundamental period of oscillation) and post-elastic (capacity of deformation before failure); – equivalence, supposed to be taken for granted between the effect of a static force and that of an action in dynamic reality (we will see in section 12.1.1 the inadequacy of static criteria to judge the stability of blocks against toppling over). From the point of view of the person using the earthquake-resistant codes, the formula in terms of seismic coefficient presents the advantage of a greater simplicity, since the “seismic forces” are supposed to act in a static manner; the calculation methods used for implementation are thus similar to those used for the other cases of loads and the verification criteria are the usual criteria of construction rules. This argument of simplicity in usage carries a lot of weight in favor of the seismic coefficient approach; that is why it continues to be preferred in certain countries. 8.1.2. Development of the seismic coefficient As indicated in the previous section, the seismic coefficient of the equations quickly ceased to be a simple number with a single value assigned to appear like a product of several factors, each one having a precise significance. This development can be summarized in Table 8.1 for the American codes [BER 89].
The Seismic Coefficient DATE
EQUATION
SEISMIC COEFFICIENT
1927
UBC
0.075~0.10
1933
LACC
0.08
1952
ASCE-SEAONC
1959
SEAOC
K , K = 0.015~0.025 TI
1976
UBC
K/T11/3, K # 0.05
1988
SEAOC
ZIS K/T12/3
367
ZIS (K/T12/3)/Rw UBC: Uniform Building Code; LACC: Los Angeles City Code; ASCE: American Society of Civil Engineers; SEAONC: Structural Engineers Association of Northern California; SEAOC: Structural Engineers Association of California; TI: basic period (s); K: numerical coefficient; Z: seismic zone coefficient; I: importance coefficient; S: earth coefficient; RW: reduction factor of the structural system. Table 8.1. A brief history of the development of the seismic coefficient in American codes (according to [BER 89])
We see that the influence of the basic period T1 of the structure was first introduced in the 1950s, with the formulae favoring flexible structures (high values of TI); in practice, T1 was calculated by simple formulae bringing in the number of floors or the height of the building. An example of these formulae is given hereafter for the Japanese code of 1981 [8.4]. The form adopted in 1976 by the Uniform Building Code presents, apart from the influence of the period TI, influence of the seismic activity zone (Z), of the importance of risk in accordance with the function of the building (I) and the nature of the foundation soil (S). This way of expressing the seismic coefficient is found in the codes of several countries adopted in the 1970s. The following stage shown in the table by equation SEAOC 1988, consisted of adding an extra coefficient (RW) enabling us to take into account the smaller or larger capacity of the structural system to undergo large post-elastic deformations. The introduction of this coefficient, called a behavior coefficient in France, characterizes the last generation of the seismic coefficient equations which will be discussed in section 8.2. The seismic coefficient which is expressed in the last column of Table 8.1 is related to the foundations of the building, which means that it is multiplied by the
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total weight of the latter which gives the horizontal shear force at ground level (base shear). The shear force in the different floors is obtained by multiplying the force at the base by a new coefficient which takes into consideration the deformation profile of the building (which generally amplifies the accelerations in the higher stages). As a typical example of the seismic coefficient, that does not invoke a behavior coefficient, we can study the formulation of Rules PS 69/82 [COL 69] which were used in France until fairly recently (1997), the seismic coefficient V enabling us to calculate the shear force acting on any floor is the product of four factors:
V=DEJG
[8.1]
in which: – D is a coefficient characterizing the hazard. In the ministerial decrees that order the application of rules PS 69/82, the values maintained for D depend not only on the zone of seismic activity (zonal map of Figure 7.3) but also on the importance classification of the building; thus D represents coefficient Z and coefficient I of the American equations of Table 8.1; its values vary from 0.5 to 2, the pivotal value of I corresponding to the usual apartment buildings located in zone II of the zones drawn up in 1985 (see Figure 7.3); – E is a response coefficient of the building, which is a function of fundamental period T and of the rate of internal damping of the structure; it can be compared to coefficient KTI-1/3 of equation SEAOC 1959; for a normal rate of damping, its values vary between 0.05 and 0.10; – J is a distribution coefficient according to the floors; for a perfectly regular building of N floors, the value Jn of J at the nth floor level is given by the equation:
Jn =
3n 2N 1
[8.2]
If N is quite high, J is about 1.5 at the top of the building. – G is a foundation coefficient varying from 0.9 to 1.3, that indicates the fact that constructions are more vulnerable on second-rate ground (4.3.3). Certain codes have introduced two levels of verification, the first corresponding to a relatively moderate earthquake, where we assume that there are no significant structural damages, and the second to a very violent earthquake, which should not lead to the construction collapsing. This is the case of the Japanese code of 1981,
The Seismic Coefficient
369
where the seismic coefficient Cn of floor level n is given by the following formula, valid for a moderate earthquake: Cn = C0 Z Rt An
[8.3]
with: – C0: seismic coefficient of reference, with a general value of 0.20 (sometimes 0.30 for certain specific buildings); – z0: coefficient of zone whose modulation is rather weak, as it varies from 0.7 to 1.0 on Japanese territory (the lowest value, 0.7, being reserved only to the island of Okinawa); – Rt: coefficient of structural response, dependent on fundamental period T, and the nature of the ground. Its values are at most equal to I and decrease when the stiffness of the ground increases (by a factor that could go up to 2 between soft soil and rocks); basic period T1 can be calculated by the very simple formula: T1 (s) = (0.02 + 0.01r) x H (m)
[8.4]
where H is the total height (in meters) and r the ratio between the height of all the floors constructed in steel and H; the value of r is thus 0 for concrete buildings and I for buildings with an entirely metallic structure. – An: coefficient of distribution according to the floors, given by the equation § 1
An = 1 + ¨
© Dn
· 2T 1 Dn¸ ¹ 1 3T 1
[8.5]
Dn being the ratio between the weight supported by the floor n and the total weight, the value of An is thus I at the base [DI = I] and can have high values at the top (“whipping” effect). For very violent earthquakes, where we only plan for the construction not to collapse, the shear forces are increased in relation to those calculated earlier (for a moderate earthquake] with the help of the formula: Vu = Ds Fe Fs x 5 Vm
[8.6]
in which: – Vu and Vm are respectively the ultimate shear force and the shear force under a moderate earthquake;
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–Ds is a coefficient dependent on the ductility which varies from 0.25 (high ductility structure) to 0.55 (low ductility structure); –Fe and Fs are coefficients which depend respectively on the eccentricity and the regularity of the distribution of the stiffness; they can vary from 1 to 1.5. This formula greatly penalizes fragile and irregular structures, for which Vu = 6.19 Vm (resulting from Ds = 0.55, Fe = Fs = 1.5) with relation to ductile and regular structures (Vu = 1.25 Vm with Ds = 0.25, Fe = Fs = 1). The fact that in the Japanese code of 1981 there are coefficients linked to the ductility and the regularity of the structures (Ds, Fe, and Fs) shows that this code is ranked among the most recent codes. However, the simple formulation of this code for the user is typical of the older codes which “hide” the difficult problems given by the exact definition of the ground movement and by the detailed determination of the structural response; in this approach founded on a reasonable empirical judgment, experience comes back into play (observation and analysis of the behavior of structures which have been subjected to real earthquakes) and it is this experience only which enables us to judge the validity of the contents of the equations, even if these are deducted from studies of extreme cases considering the seismological data and all the possibilities of calculation and trial methods. 8.2. The seismic coefficient in current earthquake-resistant codes 8.2.1. The structure of current earthquake-resistant codes Compared to earlier codes, based on the use of a simple expression of a seismic coefficient, in which the influence of the structure in question is limited to that of the fundamental period, a large majority of current codes are characterized by: – the introduction of coefficients aiming to translate behavioral differences in the post-elastic domain; different notations are used (reduction factor Rw in the USA, ductility coefficient Ds in Japan, as seen in section 8.1.2); in Europe the term behavior coefficient has been retained, with the notation q; – the limitation of the use of a seismic coefficient, in the traditional sense of the term, in the case of regular structures from the point of view of distribution of stiffness and inertia; – the recourse to calculation of seismic response of the structure in general cases (irregular structures); this calculation is performed on the basis of a linear model and its results are divided by the behavior coefficient to determine the equivalent static forces used for verifications;
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371
– the significant development of “detailing rules”, i.e. the construction design details for structural elements and their joints; such rules were already in existence in earlier codes, but the introduction of the behavior coefficient has necessitated a detailed study of conditions making it possible to obtain a good capacity of postelastic deformation under cyclic loads. These evolutions have been motivated by the progress in knowledge, as much in the field of seismology of strong movements as in the field of nonlinear dynamic materials. In fact it cannot be contested, that the main fault of the first codes lay in the absence of any differentiation between ductile structures and fragile structures, resulting in disparities in obtaining effective security, certain structures presenting significant misgivings in case of the occurrence of an earthquake exceeding the statutory level while others do not possess appreciable safety margins. Moreover, the importance of construction practices cannot be insisted too strongly; past experience indicates without debate that a very high proportion of the damage seen is to be attributed to bad planning of details or defective construction. The judgment on the evolution of calculation codes needs to be more critical, due to the disadvantages resulting from certain aspects of the introduced modifications. In the meantime, while a more in-depth discussion will be given in section 18.2, the following section aims to draw attention to a few important points. 8.2.2. The definition of seismic action and the rules of calculation in current earthquake-resistant codes In many codes in current use, reference to the notion of a seismic coefficient has practically disappeared. In the code currently in use in France (PS 92 rules, [COL 96b]) it is only found in a subsection of four lines in the presentation of all the seismic actions and in sections on local actions and the stability of slopes. The user of such codes thus has the impression that the process is fundamentally different from that in the older codes. In reality, the basic hypothesis, i.e. judging safety through the balance of static forces, is still the same. The only difference lies in the calculation of seismic forces to be introduced into this balance; in older codes this calculation is immediate (it is sufficient to multiply the supported weight by the seismic coefficient, which is given in a very simple explicit formula; see section 8.1.2), while in the new codes we have to proceed in the general case to a modal spectrum analysis (see section 9.2), which is a technique pertaining to the dynamics of structures and is not part of the usual
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background or training of designers in civil engineering. The necessity for them to learn methods in the area of dynamics is often introduced as an important advantage in modern earthquake-resistant codes. It is beyond doubt that the notion of the mode of vibration and the knowledge of methods of calculating their period and their deformation are very important in earthquake engineering. Determining the modes is a necessary prerequisite to any attempt to describe and predict the dynamic response of a structure. We cannot forget, however, that the modes of vibration are associated with linear elastic behavior, which is not in general typical of the damage process which can be seen. The design envisaged by the codes corresponds to conditions far removed from the elastic domain, as their aim is non-collapse, i.e. a state where significant damage is accepted. The significant deviation from elastic conditions is evident in the high values of the reduction coefficient of strains (Rw or q) in current earthquake-resistant codes, which frequently vary between 2.5 and 5 and can sometimes exceed 10 (the highest possible value of Rw in the American code is 12). As the choice of these values is greatly affected by empirism, we could quite correctly ask ourselves if linear calculations are adequate enough to establish a determining base for seismic strains and forces. This adequacy has been studied in detail through nonlinear calculations in the time domain and experimentation on vibrating tables, for simple structures made from models with a single degree of freedom; an overview of these studies will be given in Chapter 17. Their main result has been to establish approximate equality between maximal displacements, whatever their mode of calculation (linear or nonlinear), at least for structures whose natural period is sufficiently large (this is the “Newmark theorem” mentioned hereafter in section 9.3.1). This displacement constancy validates the use of a dividing coefficient of strains having values in the order of those mentioned above for ductile structures, in order to calculate the seismic strain on the basis of a result of a calculation of elasticity. This calculation framework, which is thus validated by studies mentioned earlier, is found in the codes for regular structures, that is those whose dynamic response is essentially controlled by their fundamental mode; the elastic calculation is “already done” as the definition of seismic action in terms of the elastic response spectrum corresponds precisely to the case of an oscillator with one degree of freedom. It is sufficient to divide it by the behavior coefficient to get a formulation of the seismic coefficient type. For irregular structures, whose response is influenced by several modes of vibration, the codes adopt the same method (elastic calculation through modal
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373
spectrum analysis and division of results by the behavior coefficient). The adequacy of seismic strains calculated in this way is thus highly debatable, since validation studies which will have to be carried out cannot be expected in practice, considering their complexity and specific character. The nonlinear behavior of irregular structures under tridirectional dynamic forces, evidently cannot be represented in a satisfactory manner with an over simplified “recipe”. The supposed “seismic strains” obtained by dividing the strains calculated on an elastic model by q have practically no clear mechanical significance outside of cases of very simple structures, such as systems with one degree of freedom, wherein the sequence of formation of plastic hinges can be controlled by the designer. The apparent precision of calculations obtained through the equations (modal spectrum analysis on a linear model) is thus largely wishful thinking, considering that the representation of nonlinearities by the behavior coefficient is extremely crude. The linear calculation of the seismic response is however very useful as it makes it possible to understand the transmission path of the stresses and to clearly indicate the critical areas for frequency. Thus it is a tool for architectural design, making it possible to identify the structural elements subjected to the strongest strains and to avoid difficulties related to frequency coincidences (resonance) between the excitation (movement of the ground) and the response (modes of vibration). Its use to determine the structural strains through a simple behavior coefficient must be presented for what it represents in reality, i.e. a very crude procedure adopted because we cannot do any better in the current conditions of study. Most of the current earthquake-resistant codes used, which are written on the basis of routine standard conventions (i.e. practically without any explanatory notes of required practices), make the mistake of presenting on the same plane, results of a calculation technique which has its own rules and logic (modal spectrum analysis) and an over simplified approach (behavior coefficient). This biased presentation contains the grave error of making the calculation seem like the main element in paraseismic prevention where construction is concerned which in reality is based on the following trilogy: design, construction practices and execution. The definition of seismic action in terms of elastic response spectrum is adapted to linear calculation through modal spectrum analysis. As indicated in the introduction to Part 4, it is necessary to complete this definition with hypotheses of the regional seismic activity if we are to undertake calculations in a realistic manner in the nonlinear domain. This definition of action with a spectrum which is often presented as significant progress as compared to the “seismological black out” of the seismic coefficient, is thus essentially related to a mode of calculation which we
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have just seen as being quite contestable, at least where its application is concerned in the determination of strains in a structure and design of complex structures. We must not lose sight of a fundamental point, which was mentioned at the start of section 8.2.2; current earthquake-resistant codes are, as their predecessors, based on a “force” approach (static balance between external forces and their resistant internal forces). Even though the term “static equivalent” often has unfavorable connotations nowadays, it is in reality the underlying dogma of safety verifications required by the codes. The use of very sophisticated models (three-dimensional dynamics with finite elements) eventually results in a field of static strains which must be able to be “boxed in” by the structure; thus everyone uses the “static equivalent” whether they know it or not. This “force” approach which presents great practical advantages, is not very well adapted, in the majority of cases, to the realistic judging of the safety of the construction subjected to effects of seismic activity. We have seen that for very simple structures, it is the displacement criteria which best describes the ultimate state achieved in the course of movement. This occurrence conforms to the general principle according to which we should always use deformation criteria, rather than stress criteria, to characterize the effects of a dynamic phenomenon. Moreover we have been helping, since the mid-1990s, with the emergence of propositions aiming to develop earthquake-resistant codes taking a more realistic approach of nonlinear behavior [LEE 99], notably by using displacement criteria. The definition of seismic action must thus be better adapted than a simple elastic response spectrum in the study of these behaviors.
Chapter 9
The Response Spectrum
9.1. The response spectrum of elastic oscillators 9.1.1. Response spectrum of elastic oscillators associated with a natural accelerogram We consider (Figure 9.1) a simple oscillator made up of mass m, linked to rigid support S by a spring of stiffness k and a dashpot of coefficient c (which when multiplied by the velocity represents the viscous force which opposes the displacement). From time t = 0, the support is driven by transient motion s(t) which induces relative displacement x(t) of the mass m in relation to the support. The accelerogram that excites the support is the second derivative of displacement s(t); we denote the derivative in relation to the time by inserting a dot above the symbol; the velocity of the support is thus s (t), its acceleration s (t). The displacement of mass m from the absolute axis is s (t) + x(t), its acceleration s(t) + x(t) ; thus, we have the dynamic equation: m ( s x ) = – kx – c x
[9.1]
as the opposing force of the spring and damping force of the dashpot only depend on . the relative displacement x (t) and its derivative x(t)
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We introduce the notations:
Z=
k 1 ;[= m 2
c km
[9.2]
and equation [9.1] is rewritten as: x + 2[ Z x + Z²x = – s
[9.3]
Figure 9.1. Diagram of a simple oscillator; the support S is driven by an imposed motion s (t), provoking a relative displacement x (t) of mass m of the oscillator in relation to the support
Z is the actual angular frequency of the oscillator, which means the angular frequency of the oscillations without damping; in fact, with [ = 0 and s = 0, equation [9.3] is reduced to: x + Z²x = 0
[9.4]
whose solution is a simple linear combination of functions sin Zt and cos Zt. From the angular frequency Z we define: eigenperiod T =
23
Z
[9.5]
The Response Spectrum
eigenfrequency g
Z 23
=
1 T
377
[9.6]
[ is the rate of reduced damping, usually expressed as a percentage; in seismic calculations, it generally has low values of about a few percent; the name reduced damping comes from the fact that [ = I (100%) marks the limit between the oscillatory or non-oscillatory nature of the free movements of the mass; the equation of these movements, i.e. equation [9.3] without second member ( s = o): x
2 [Z
x
Z² x o
[9.7]
has solutions that are linear combinations of functions: e[Zt
sin cos
(Zt 1 [ ² ) if [ < 1 [9.8]
e
( [ r [ ² 1) Zt
if [ > 1
It is only in the case [ I (almost always present in practice) that this solution is a damped sinusoid. [ = I corresponds by definition to critical damping. In the first of equations [9.8] we see that the presence of a non-zero damping modifies the period of free oscillation (the pulse becomes Z I [ 2 instead of Z for zero damping); as [ is low in practical cases, this modification is not significant (about 0.5% for [ = 10% which is already a high value for structural damping) and it is usually not taken into account. Practically the only significant influence of the damping is the reduction of the amplitude of free oscillations. We can easily show that the ratio of the amplitudes of the two successive crests of the oscillation only depend on reduced damping [; the Napierian logarithm of this ratio called the logarithmic decrement is given by the formula: į = Ln
Dn Dn1
2S[ 1[ 2
[9.9]
Dn being the amplitude of the crest of range n. Table 9.1 gives the values of G, eG = Dn/Dn+1 and the angular frequency variation
Z‘/Z = I-[ ² for a certain number of values of reduced damping.
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eG =
Dn Dn 1
Z' Z
[
[%
G
0.005
0.5
0.0314
1.0319
1.0000
0.01
1
0.0628
1.0649
1.0000
0.02
2
0.1257
1.1339
0.9998
0.05
5
0.3146
1.3696
0.9987
0.1
10
0.6315
1.8804
0.9950
0.2
20
1.2825
3.6058
0.9798
0.5
50
3.6276
37.6223
0.8660
1[ 2
Table 9.1. Influence of reduced damping on the decrease of amplitude of successive crests and the variation of the angular frequency of the oscillations
For any excitation accelerogram s (t) of its support, the solution of equation [9.3] of the motion of the simple oscillator corresponding to the initial state of rest x(0) = 0, x (0) = 0 is given by Duhamel’s integral: x (t) = –
1
Zc ³
t o
s (W )e[Z (t W ) sin Zƍ (t – W) dW
[9.10]
where we have put: Zƍ = Z 1 [ ²
[9.11]
We have considered the only interesting case in practice of under-critical damping ([ I). Equation [9.10], which we can establish easily by the integration of the second order linear differential equation using standard techniques, is a particular case of the more general solution: rn(t) = –
with T =
Zn Z'
S 2
t
³ os (W )e
[Z ( t W )
Arc sin [
sin [Z‘ (t – W) + nT] dW
[9.12]
[9.13]
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379
It can be shown that relative displacement x(t), relative velocity x (t) and s (t ) x(t ) of the oscillator are respectively equal to functions absolute acceleration rn(t) defined by [9.12] for the values 0, 1 and 2 of index n: x (t) + s (t) =r2(t) x (t) = r0(t); x (t) = r1(t);
[9.14]
Duhamel integrals [9.10] or [9.12] are convenient to present the theoretical results; they can also be used for the numerical calculation of the response of the oscillator when oscillation accelerogram s(t) is given in a numerical form, which is generally the case; but there are other numerical integration algorithms of equation [9.3] which are more efficient in practice. The response spectrum in relative displacement, for a given excitation accelerogram s (t), is the maximum of the modulus of x(t) during the movement of the oscillator; as the latter is characterized by its angular frequency Z and its reduced damping [, the response spectrum in relative displacement is a function of the two variables Z and [ so that it is noted Sd (Z,[); thus, according to [9.10], we have: Sd(Z[ =
1 Max Z' t
t
³ s (W )e
[Z ( t W )
0
sin Z '(t W )dW
[9.15]
From Sd (Z[ , we have the following quantities: Sv (Z[ = Z Sd (Z[)
[9.16]
Sa (Z,[) = Z² Sd (Z[) = Z Sv(Z[)
[9.17]
which are the pseudo-velocity and the pseudo-acceleration respectively; the physical significance of these quantities is easy to determine; – the pseudo-velocity Sv (Z[) is the value of the velocity which would give a kinetic energy equal to the maximum value of the elastic energy stored in the spring; in fact according to [9.16] and [9.2] we have: 1 m Sv2 (Z , [ ) 2
2 1 mZ ² S (Z , [ ) d 2
2 1 k S (Z,[) d 2
[9.18]
Sd (Z[) being by definition the maximum of the modulus of relative displacement which activates the spring, the last expression of [9.18] represents the maximum elastic energy of the latter and we have the property given for Sv (Z[); the pseudo-
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velocity can be quite different from the maximum of the modulus of relative . velocity x (t), thus the usage of the prefix pseudo-: – pseudo-acceleration Sa (Z[) represents, for low damping rates, a good approximation of the maximum of the modulus of the absolute acceleration of the oscillator; in fact for zero damping we can rewrite equation [9.3] in the following form: s x = – Z²x
[9.19]
from which we deduce: Max ª.. .. º Max s x» = (Z² x) = Z² Sd (Z[) = Sa (Z[) t «¬ t ¼
[9.20]
For [ , there is thus a strict equality between the pseudo-acceleration and the maximum of the modulus of absolute acceleration. As, in practice, the damping coefficients are low, we can accept that these two quantities are nearly equal. Similarly, as for velocity, using the prefix pseudo- shows that it is only an approximate value, but this approximation is generally better for the acceleration. The response spectrum in displacement, pseudo-velocity or pseudo-acceleration can be represented in different ways; as they are functions of the two variables Z and[, we generally choose to trace a series of curves related to Z (or the quantities derived from Z), period or frequency (see [9.5] and [9.6]), each curve corresponding to a specific value of damping coefficient [. The values of the abscissa and the ordinate in the graphs are most often: – in abscissa, frequency f or period T; angular frequency Z is almost never used, – in ordinate pseudo-velocity Sv (Z[) or pseudo-acceleration Sa (Z, [); or the real maximum “non-pseudo” of the velocity or of the acceleration which we can obtain from equation [9.12]. In addition, if we consider the logarithmic or linear scale options, either in abscissa or in ordinate, or on both axes, we come to a wide variety of representations, some examples of which are given below. Figure 9.2 shows the response spectrum of the north-south component of the recording at Tolmezzo (northern Italy) during the Friuli earthquake (6 May 1976). This is a representation in pseudo-acceleration (ordinate) as a function of the period (abscissa) with linear scales on both axes, for reduced damping of 0, 2, 5, 10 and 20%.
The Response Spectrum
381
Figure 9.2. Example of the spectrum in pseudo-acceleration as a function of the period (spectrum of the north-south component of the recording at Tolmezzo (northern Italy) during the Friuli earthquake (6 May 1976)
Its appearance is typical of real accelerogram spectra, i.e. highly irregular with sharper peaks and valleys for lower values of the damping coefficient. We notice in particular a high peak around the period 0.25 s (frequency 4 Hz). This irregularity shows that the seismic action cannot be correctly represented by just one accelerogram of a real earthquake (or by its spectrum); in the example of Figure 9.2 structures having a fundamental period of 0.25 s would be more activated than those having basic periods of 0.20 s or 0.30 s. Considering the random nature of these spectrum peaks on the axis of the period on the one hand, and uncertainties in modeling which affect the precision of the calculation of actual periods, on the other hand, it is evident that these large differences in the amplitude of the responses for these relatively close periods have no practical significance for designing. Designing can only be done through a smoothed spectra, resulting from statistical study, as presented in section 9.1.2.
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Figure 9.3. Response spectrum in a quadri-logarithmic diagram of the north-south component recorded at El Centro (California) during the Imperial Valley earthquake (18 May 1940). This accelerogram corresponds to number 13 of Figure 4.1 (according to [HUD 77]); damping coefficients: 0, 2, 5, 10, 20%; units: inch, inch/s, g
Another frequently used spectral representation is the quadri-logarithmic diagram, which consists of carrying the pseudo-velocity in ordinate and the period (or frequency) in abscissa with logarithmic scales on the two axes; taking into account equations [9.16] and [9.17], by taking logarithms, we have:
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383
Log Sv(Z[) = log Sd(Z[) + log Z = log Sd (Z[) + log 2S – log T Log Sa (Z[) = log Sv (Z[) + log Z = log Sv (Z[) + log 2S – log T From this we deduce that in the quadric-logarithmic diagram with log T in abscissa, the parallel straight lines at the first bisector (log Sv (Z[) = log T + Cte) correspond to constant values of pseudo-acceleration while those parallel to the second bisector (log Sv (Z[) = – log T + Cte) correspond to constant values of displacement. An example of a quadri-logarithmic diagram is given in Figure 9.3. The name quadri-logarithmic diagram comes from the fact that we plot on it, in addition to the lines parallel to the axes, lines that are parallel to the first and the second bisectors giving effect to the note that has just been made. Thus, we obtain on the same figure information on three spectral parameters (displacement, pseudovelocity and pseudo-acceleration), at the price of the usual difficulty in reading and interpolation of the logarithmic scales. The example represented in Figure 9.3 corresponds to the famous El Centro recording, already mentioned in Part 2. As in Figure 9.2, we observe the presence of several peaks separated by well-marked valleys. The quadri-logarithmic diagram can also be plotted with frequency in abscissa; Figure 9.4 shows such a representation (without oblique scales, to simplify the graph) for the same component of the El Centro recording. The passage in logarithmic co-ordinate from the period to the frequency is shown by a simple reversal of the abscissa axis (since log f = – log T). Comparisons between Figures 9.3 and 9.4 show certain differences between two spectra; particularly around frequency 0.1 Hz (10 s period) we notice that the curves corresponding to different damping coefficients merge into one in Figure 9.4, though they remain quite separate in Figure 9.3. These small differences at low frequencies are very probably due to different choices in the correction procedures of basic recorded data; as indicated in section 4.1.1, the low frequency content of the seismic signals can be greatly affected by the displacement drift, resulting from the loss of the beginning of this signal on old recording apparatus. In the quadri-logarithmic diagram with frequency in abscissa, the straight lines parallel to the second bisector are those that correspond to a constant pseudoacceleration. In Figure 9.4 we noticed the straight line which appears to constitute an asymptote to the high frequencies by the curves corresponding to different damping coefficients, with the indication “maximum ground acceleration”.
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Figure 9.4. Quadri-logarithmic diagram with frequency in abscissa for the response spectrum of the north-south component of the El Centro recording
This asymptotic value of the pseudo-acceleration is in fact equal to the maximum acceleration of the support of the oscillator; to be assured we must remember that a high-frequency oscillator corresponds to an extremely stiff spring. The mass therefore practically follows the movement of the support as it is linked to it by an almost rigid bar; by passing to the limit (infinitely stiff spring, infinite frequency) the maximum acceleration of the mass is necessarily equal to that of the support, quite independently of the damping. We often indicate this acceleration as zero period acceleration (ZPA). The frequency from which we can consider that the
The Response Spectrum
385
pseudo-acceleration becomes practically equal to ZPA is called cut off frequency, which typically varies from 25 to 40 Hz (we often use the value of 33 Hz). On the other hand, the low frequencies correspond to oscillators with extremely flexible springs; the mass thus remains almost immobile in relation to absolute axes since the force transmitted to it by the spring is almost zero and its maximum relative displacement in relation to the support is thus very close to the maximum absolute displacement of the latter. By reaching the limit (infinitely flexible spring, zero frequency or infinite period), the mass does not move in the absolute axes and there is strict equality between these maximum displacements (relative for the mass, absolute for the support) independent of the damping. This tendency is observed in the left part of Figure 9.4, but not in the right part of Figure 9.3 which was undoubtedly drawn up using a version of the El Centro accelerogram which was not corrected very well. An accelerogram response spectrum has extremely useful overall information on the amplitude of and the frequencies contained in the signal; generally it has three parts: – a low frequency zone where the response of the oscillator is essentially controlled by the displacement of the support; the central frequency of this zone corresponds effectively to frequency gd defined by equations [4.14] or [4.15]; for a frequency of about half or one-third of gd we arrive at a convergence of different damping curves towards a unique value of the displacement which is equal to the maximum absolute displacement of the support; the tracing of the spectra often does not go down low enough in frequency (or does not rise high enough in period) in order to reach this convergence; – a high frequency zone where the response of the oscillator is controlled essentially by the acceleration of the support. This zone goes from a frequency between 3 ga and 5 ga (ga being defined by equations [4.13] or [4.15]) at about 10 ga, which constitutes an approximation of the cut off frequency, the response of the oscillator is rigid (it moves together with its support) and there is an equality between pseudo-acceleration and maximum acceleration of the support whatever the damping; – a zone of intermediate frequencies is placed between the two earlier zones where the responses in pseudo-acceleration and pseudo-velocity are generally amplified in relation to maximum values of acceleration and velocity of the support; this zone presents peaks and valleys distributed in an apparently random manner; the order of the size of the highest amplifications in accelerations can be estimated by the following formula:
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Seismic Engineering
A0 As
5 5 2 [ (%)
[9.21]
A0 being the pseudo-acceleration of the reduced damping oscillator [ (expressed as a percentage) and As the maximum acceleration of the support; the inverse proportion of the square root of [ will be shown in section 9.2.1 for an excitation of the “white noise” type; this has been well verified for real accelerograms for damping coefficients between 1% and 20%. As indicated in the introduction to this chapter, the elastic response spectrum does not respond much to the influence of the duration of the accelerogram; this influence should reflect on the distance between the curves corresponding to the different damping but this effect is difficult to appreciate in practice as it can be hidden by other effects related particularly to the frequency content. The spectrum gives the maximum reached by the response of the oscillator but does not say if this maximum has been approached, and how many times, by other oscillations during the movement; such data is however very important as the structural damage is often incremental, i.e. it increases quickly with the number of loads: the elastic spectrum is thus a bad indicator of the damage potential of an accelerogram. As far as the choice between the different representations of the spectrum is concerned, and particularly of the variable in abscissa, behavior is different according to the application domain; the earthquake engineering of the building and civil engineering works (and thus the earthquake-resistant codes in the usual sense of the word) prefer the formulation in period, as the periods of the actual important modes are more often of the order of a second or a fraction of a second, but are never very short. In industrial earthquake engineering where we are interested in the behavior of machines, some of which have high frequencies (of the order of the cut off frequency), it is customary to keep frequency as a variable in abscissa. 9.1.2. Response spectrum of elastic oscillators that can be used for designing
In the previous section we saw that the spectra associated with real accelerograms are not acceptable for design calculations as they have very high variations of response for relatively low variations of frequency (or of period). The spectra of the earthquake-resistant codes or calculation specifications for industrial installations are always smoothed spectra, whose usage does not have such random risks of underestimation hazards of the response.
The Response Spectrum
387
reduced dampings (%)
Figure 9.5. Calculation response spectrum of the USNRC (United States Nuclear Regulatory Commission) represents pseudo-acceleration in ordinate and frequency in abscissa in a bi-logarithmic diagram
In order to obtain these smoothed spectra, we must put together a group of real accelerograms representative of seismic conditions of the region and of the geotechnical specificities of the site studied. This work is only possible in wellequipped areas where the seismic activity is sufficiently high for us to get a large number of high level recordings. In a number of regions, one of which is France, this data is lacking, either because it is not well-equipped or because the observation period is too short due to the moderate level of seismicity. Therefore, these lacunae must be rectified by using accelerograms recorded elsewhere but which we consider to be reasonably similar to those which could occur in the region.
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From the response spectra of the accelerograms of this collection, a statistical analysis (average or average with a fraction of a typical deviation) determines the calculation response spectra; Figure 9.5 gives an example of such a spectrum. It is made up of straight line segments whose exact coordinates at extremities are given in the table below the figure. The cut-off frequency is 33 Hz. This is a spectrum published in 1973 by the USNRC (United States Nuclear Regulatory Commission) [COL 73]. Initially meant for the calculation of American nuclear power plants, it has been, and is still largely, used throughout the whole world, not only in the nuclear domain but also in other industrial installations. For example, it has been retained as a standard fixed spectrum for designing certain nuclear power plants and special atrisk installations in France [COL 93b]. This is an average spectrum with a standard deviation obtained from about 30, mainly Californian, accelerograms. To facilitate its introduction in software programs, a “dotted line” form has been maintained. Unlike the spectra discussed in section 9.1.1, the spectrum in Figure 9.5 does not correspond to a real accelerogram but to a set of recordings. Although it is possible to determine the synthetic accelerograms whose spectrum is similar to in Figure (see section 9.1.2), it is possible that such accelerograms only distantly resemble real signals; the consequences of their usage for nonlinear calculations will be dealt with in section 9.1.3. The earthquake-resistant codes for actual constructions, when they contain a definition of an elastic calculation spectrum, use representations by segments similar to that of the USNRC spectrum. For example, in the appendix of the PS92 rules used in France [COL 96b] the spectra reproduced in Figure 9.6 can be found. There are four spectra corresponding to different site conditions (So rock, S1, S2, S3 soils of decreasing quality), for the same value (5%) of the reduced damping. The scales chosen are linear in both abscissa (period) and in ordinate (acceleration); the descending branches for the large periods are branches of equilateral hyperbola (which would become straight line segments in logarithmic scale) in 1/T.
The Response Spectrum
389
Figure 9.6. Standard elastic spectra corresponding to different site conditions, from the PS 92 rules
The spectra of Figures 9.5 and 9.6 are standard spectra, which means that they are calibrated on a standard value of acceleration at zero period (1 for the USNRC spectrum, 1, 0.9 or 0.8 according to the type of soil for the PS92 spectrum). In order to use them in the calculations we must multiply them by the acceleration of calibration corresponding to the site or the region being studied. This approach through standard spectra calibrated by means of a parameter expressing the level of seismicity of the region (most often it is the maximum acceleration of the soil) was proposed during the 1960s and 1970s and continues to be used in most earthquake-resistant codes. Spectra defined by the attenuation laws appeared more recently (from about 1980 for the first attempts and particularly after 1990). In these spectra the spectral ordinates are given for a certain number of frequencies (or periods), as a function of the magnitude of the earthquake and of its distance from the considered site. As for the attenuation laws of peak values (see section 4.2), different functional forms are possible, the simplest being: S (g) = C (g) e D(g) M R -E(g)
[9.22]
In this equation, analogous to law [4.17] for peak values S(f) is a spectral ordinate (generally pseudo-velocity or pseudo-acceleration) for frequency f, magnitude M and distance R from the site; coefficients C, D and E are functions of
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Seismic Engineering
the frequency. Most often these functions are given in the form of numerical value tables for a certain number of frequencies (this number normally has a value of several dozen so as to provide a good description of the variation of these functions). The connection between the attenuation laws of peak values is specified at least for acceleration which corresponds, as we have seen, to an infinite value of frequency f). Numerous relations of attenuation in spectral ordinates are now available in different parts of the world. In [BET 02] we find the description and analysis of 17 of these relations. Figure 9.7 shows the comparison of the four laws established for Europe in the conditions M = 6, R = 20 km and for reduced damping of 5%, it is presented in the quadrilogarithmic diagram with the pseudo-velocity (ordinate) in cm/s and frequency in hertz (abscissa).
Figure 9.7. Spectra at 5% calculated for M = 6 and R = 20 km with four attenuation laws in spectral ordinates valid for Europe (according to [BOU 98])
In relation to standard spectra of Figures 9.5 and 9.6, we notice the rounded form of the spectra resulting from the attenuation laws (this form comes from the large number of frequency values used to define the coefficients); the agreement between the four laws is satisfactory for average and high frequencies, but decreases slightly at low frequencies.
The Response Spectrum
391
The interest of the spectra calculated by attenuation laws is that their form is adapted better to specific seismic conditions of the built-up site than the form of the standard spectra. The influence of the variations of the magnitude and distance in fact reflects not only on the amplitude level but also on its distribution between low and high frequencies. This effect is clearly visible for spectra corresponding to the same macro-seismic intensity value, which is a result of the application of RFS 1.2.c (section 6.1.2) at nuclear power stations in France. Figure 9.8 shows four spectra (at 5% reduced damping) corresponding to the same level of intensity (VIII MSK) but at different conditions of magnitude and distance.
Figure 9.8. Variation of spectrum, for a same intensity, according to magnitude and distance
These spectra have been calculated by the attenuation laws of RFS 1.2.c for the following pairs of magnitude and distance: M = 5.8, R = 15 km; M = 6.5, R = 30 km; M = 7.0, R = 50 km and M = 7.7, R = 100 km. These pairs correspond to VIII MSK intensity when we use Mohammadioun equation [4.3]. We see that the four curves pass through the same point (for a frequency of 1.5 Hz) to the right of which the spectrum increases when the distance diminishes, while the inverse tendency is observed for low frequencies. This “fan like” structure is a result of the choice of form [9.22] for the attenuation law of spectral ordinates; in fact, D (g) is a decreasing function of the frequency since at low frequency it is the displacement of the soil which controls the response of the oscillator (D = Ln 10 in theory [5.34]) while at high frequency it is
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Seismic Engineering
acceleration (D = ¼ Ln 10 in theoretical law [5.35]); E (g) is on the other hand nearly constant ((E = 1 in theory). However, if we reason that the intensity is constant, then the magnitude is a linear function of log10 R whose slope has a value of 2 (see [14.3] or [14.10]); thus as per [9.22] we have: ª 2 º D ( ³ ) E ( ³ ) » log10 R Log10 S (g) = F (I,g) + « ¬ ln10 ¼
[9.23]
F (I,g) being independent of R; and taking into consideration previously defined variations of functions D (g) and E (g), there is a value of the frequency such that the coefficient of log10 R in [9.23] is zero (this coefficient varies from about 1 at low frequencies to – ½ at high frequencies). For this frequency value, the spectral ordinate no longer depends on the distance, only on the intensity. For attenuation laws with forms which are more complicated than [9.22], we do not generally find this remarkable property of the pivotal point but we always notice a very high reduction of the deviation between the different spectra in the vicinity of a frequency of 1 to 2 Hz. Evidently we would obtain good correlations of the intensity with the response parameters of an oscillator around this frequency but as indicated in section 14.1.3, the correlations using the intensity no longer provoke much interest. To conclude on the elastic response spectra that can be used in the calculation for designing, we must mention floor spectra which correspond not to the movement of the ground but to the movement inside a building. They are required for the calculation of equipment and materials whose supports are fixed to parts of civil engineering structures; the term floor spectra is used in practice even when these elements are not floors. Floor spectra, as compared to ground spectra, are characterized by high peaks in the vicinity of natural frequencies of the building itself and often by the high amplifications of the acceleration levels. Figure 9.9 shows a diagrammatic example of the floor spectrum in a building having a basic frequency f of about 4 Hz.
The Response Spectrum
393
Figure 9.9. Floor spectrum in a high level of a building having fundamental frequency f1 of 4 Hz, the ground spectrum (in dotted lines) has zero period acceleration of 2 m/s² and a peak of 5 m/s², for a frequency of about 2.5 Hz. The floor spectrum (in full lines) has zero period acceleration of 4 m/s² resulting from amplification in height in the building and a peak of 20 m/s² around frequency f1. These high values are typical of floor spectra calculated in nuclear power plants
The peaks of the floor spectra are higher and narrower than those of the ground spectra; taking the excitation theory in white noise presented in section 9.2.1 we can show that the amplification of the peak i.e. the ratio A0 (acceleration of the oscillator having the peak frequency) on As (acceleration of the support) is effectively equal to: A0 =5 AS
50 [ ([1 [ )
[9.24]
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Seismic Engineering
[ and [I respectively being the reduced damping (expressed in percent) of the
oscillator and the load bearing structure (building). The comparison with form [9.21] shows that the amplification is twice as high for the floor spectrum in the case [ = [1 = 5. 9.2. Introduction to spectral modal analysis of elastic structures 9.2.1. Presentation of a simple example to introduce spectral modal analysis
We consider (Figure 9.10) a perfectly regular building of N levels whose load bearing structure is made up of identical floors and columns. At the foundation the columns are set in a rock assumed to be non-deformable. The structure is constructed of reinforced concrete. h
k
h
k
h
k
h
k
h
k
h
k
m
uN
m
uN -1
m m m
u2
m
u1
s(t) Figure 9.10. Perfectly regular building of N levels (N = 6 in the figure), with a floor-column structure. On the left, position at rest; on the right, position deformed under action of a horizontal accelerogram s (t ) acting upon the foundation; in the center, spring mass model outlining the building
By m we indicate the mass of a floor, by h the distance between two consecutive floors and by EI (E = Young’s modulus, I = moment of inertia) the bending stiffness of a column. The following hypotheses are adopted: 1) floors can be considered as infinitely rigid with respect to horizontal forces acting upon the structure; 2) columns work in pure bending; we neglect the deformations due to shear force and normal force;
The Response Spectrum
395
3) deformations remain in the linear elastic domain; 4) mass can be concentrated at the level of the floors; 5) resistance to non-structural elements (façades, internal partitions) can be neglected; only the columns contribute to the lateral resistance; 6) seismic shock acting upon the foundation rock leads to a movement of the whole medium, without dephasing between the different points and this movement of the whole medium is a tridirectional translation (two horizontal components, one vertical component); taking hypothesis 3) into account with a linear elastic behavior, we can study the action of each one of these components separately and for the rest we limit our study to a horizontal component defined by accelerogram s (t); 7) distribution on the horizontal plane of the columns and mass is sufficiently symmetrical with relation to the direction of the excitation component so that we can assume that the response of the structure is reduced to displacements parallel to this direction, without the appearance of twisting movements around a vertical axis; the rotations of the horizontal axis are not permitted by the selected hypotheses of behavior for floors and columns; the building is therefore deformed by pure shear stress. These hypotheses are typical of the current practice of seismic calculation and require some notes on their degree of comprehensiveness and their validity limits. 1) Behavior of the floors The floors must play their role of a diaphragm, i.e. ensure the transmission of horizontal forces to the bracing elements (columns) with a distribution that is as even as possible between these elements; this general principle of earthquakeresistant design is respected by all floors of apartment or office buildings; it is not so for floors that are largely bare (warehouse floors) of industrial buildings; this hypothesis of an infinitely rigid behavior of floors in their plane is typical in seismic calculation. The floors undergo a bending deformation in the perpendicular direction to their plane, as shown in Figure 9.11.
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Seismic Engineering
Figure 9.11. Joint bending deformations of floors and columns, corresponding to a horizontal offset Gu between two consecutive floors
The stiffness regarding horizontal deformations depends at the same time on resistance to bending of the columns and of the floors; it is frequently assumed that resistance of the floors can be ignored. 2) Behavior of the columns The hypothesis of deformation in pure bending is typical for elongated elements such as the usual building columns. On the first level, to simplify matters, we suppose that the columns are embedded at their foundation in rocky terrain or an inflexible raft. This corresponds to the current practice of calculations required by earthquake-resistant codes, to avoid difficulties of the models taking into account the soil-structure interaction (see Chapter 16) and also because this hypothesis is considered as being on the side of safety (we will see in section 16.1.1 what we must consider regarding this statement). For all the columns to function as those of the first level, in order to simplify the resolution to the maximum, we are going to suppose that the rotations are prevented at the head and at the foot for all the columns, that corresponds to an infinite stiffness of the floors perpendicularly to their plane, thus to a less plausible hypothesis and opposite to the usual practice (see earlier the discussion of hypothesis 1). The deformation diagram between two floors will thus be that of Figure 9.12 (instead of Figure 9.11).
The Response Spectrum
397
Figure 9.12. Deformation of double-set bending of columns to an offset Su between two consecutive floors, for which bending stiffness is supposed to be infinite
This hypothesis enables us to have an entirely analytical resolution of the problem. 3) Elastic nature of deformations As already mentioned several times, this hypothesis does not correspond to reality in earthquake engineering of actual constructions; the elastic calculation required by the codes does not aim to reproduce the real response of the structure, when this is highly strained but to provide elements to arrive at a reasonable design. 4) Concentration of the mass at floor level This practice is usual for the actual buildings, where the own mass of the floors is high if we compare it to that of the partitions and of the furniture. It could be different for industrial buildings (heavy material, large amount of empty space in the floors). In certain cases, it may be necessary to include heavy equipment in the seismic calculation model, not only for their mass but also for the stiffness of their supports. In cases where it is necessary to consider the horizontal axis rotations, the effects of inertia of rotation must be represented in the model. It is usual to add a fraction of temporary mass to permanent mass in the building. 5) Omission of the resistance of non-structural elements It is evident that this omission is on the safer side for the calculation of static effects; it is not necessarily the same for the seismic calculation, and experience shows that the presence of non-structural elements can have an unfavorable influence on the behavior of the structure, for example, by opposing certain
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Seismic Engineering
deformation modes of structural elements or by making some dissymmetries appear following ruptures distributed in a random manner (see discussion on the twisting effect in section 12.2.2); the case of a concrete framework with masonry filling is a typical case. The processing of non-structural elements is one of the finer points of modeling from the seismic calculation point of view. 6) Hypothesis on the movement transmitted to the foundations The hypothesis of tridirectional translation is the usual rule; besides it seems to be plausible except in certain cases of motion where the surface waves dominate (section 2.2.2) or where the plane dimensions of the constructions are comparable to the wavelengths of the dominant frequencies. As far as the inflexible nature of the foundation ground is concerned, it is a part of the commonly practiced simplification hypotheses (see the previous notes on hypothesis 2). 7) Plane symmetry of mass and stiffness This plausible and commonly adopted hypothesis for regular buildings has been retained in the example in order to simplify the model; in the case of irregular buildings where it would not be verified, we must have recourse to threedimensional models, a fact that does not, in principle, present difficulties, but does effectively burden the task of calculation. These observations, which are referred to again in Part 6 appear here to underline the importance and difficulty of the modeling work which constitutes the first stage of the seismic calculations 9.2.2. Calculation model for the chosen example
In the scope of the hypotheses described in section 9.2.1 the calculation model is a simple mass-spring model; it is represented in the central part of Figure 9.10. Each level of the building, of mass m, has only one degree of freedom of horizontal translation; it is linked to the neighboring levels by a spring of stiffness k. By taking into account the hypothesis of blocked rotation at the head and foot of each column of height h (Figure 9.12) using a basic calculation of the strength of materials, we find for k: k = 12
EI NP h3
Np being the total number of columns between two consecutive levels.
[9.25]
The Response Spectrum
399
A simple model such as this, often called a “stick model” may appear extremely rudimentary. However, if the building satisfies the hypotheses previously presented and if the concentration of the mass and estimation of the stiffness has been carried out according to the rules of the art, we get, from this model, sufficiently precise data to calculate seismic response; more elaborate models (three-dimensional finite element beam models for example) are justified only in the cases where distribution irregularities of the mass and stiffness make it difficult to establish representative stick models. Sometimes in such models we introduce dampers in parallel to the springs to have a mechanism likely to dissipate a part of the vibratory energy. In practice, most often, we limit ourselves to introducing a general term of damping in the equations of the modal responses (see section 9.2.3) without effectively representing damping in the model. By u1, u2,…, un we indicate the relative displacements of levels 1 at N in relation to the ground; these displacements are unknown functions of time t, which must be determined when we use accelerogram s (t) which acts upon the foundation; we must therefore start by establishing the differential equation system verified by functions un (t) (n = 1, 2,…, N). With relation to absolute axes, the acceleration of the floor of number n is the sum of the driving acceleration s (t) imposed by the ground and the relative acceleration of the floor in relation to the ground, that means un (by taking the same convention seen in section 9.1.1 for the temporal derivatives); the force of inertia In acting on the nth floor is thus: s un In = – m
[9.26]
The force Fn + 1, n exerted by the level n + 1 on level n is the product of stiffness k between stages [9.25] by the shifting un+1 – un: Fn+1, n = k (un+1 – un)
[9.27]
In the same way, we have for force Fn-1, n exerted by level n – 1 on level n: Fn-1,n = – k (un – un-1)
[9.28]
The equilibrium of forces acting on level n: In + Fn+1,n + Fn-1, n = 0 considering [9.26], [9.27] and [9.28] is expressed as:
[9.29]
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Seismic Engineering
m ün + k (– un-1 + 2un– un+1) = – m s
[9.30]
Equation [9.30] is applied as it is to floors for which 2 d n d N – 1; for the first floor (n = 1) we can apply it by considering a fictitious floor of number 0 joined to the ground for which u 0 = 0; using [9.30] we thus have: m ü1 + k ( 2u1– u2) = – m s
[9.31]
For the last floor (n = N), we can similarly consider a fictitious floor number N + 1, which would not exert any force on floor number N, which means we must take uN+1 = uN from which: mün + k (-uN-1 + uN) = – m s
[9.32]
The set of equations [9.30], [9.31] and [9.32] can be written in matrix form: [M] {ü} + [K] {u} = – s [M] {'}
[9.33]
Figure 9.13. Mass and stiffness matrices
>M@ is the mass matrix, which is diagonal in this example, >K@ the stiffness matrix (tridiagonal in the example), {u} the column vector of the degrees of freedom of the structure and {'} the column vector of the direction of excitation, i.e. a vector whose components have a value 1 for the degrees of freedom of translation parallel to this direction; if the model contains degrees of freedom of rotation, the corresponding components of {'} are zero. If, as mentioned earlier, we had incorporated dampers parallel to the springs in the model, equation [9.33] would have taken the form: [M] {ü} +[C] {ü} + [K] {u} = – s [M] {'} where >C@ is the damping matrix.
[9.34]
The Response Spectrum
401
At the initial moment t = 0, the displacements and velocities are zero:
^u`t = 0 = 0; ^u `t = 0 = 0
[9.35]
Equation [9.33] (or its form [9.34] with damping) with the initial related conditions [9.35] constitute the basic formula for the seismic calculation; it has been established in the simple case of the chosen example, but its validity is general. 9.2.3. Non-damped eigenmodes
The spectral modal analysis method is based on the eigenmode notion, which provides a framework specially adapted to the study of the dynamic response in the linear domain. The non-damped eigenmodes are modes of deformation of the structure which, in the absence of damping and any external excitation, correspond to internal forces which exactly balance the forces of inertia associated with a sinusoidal variation during the time of this deformation and therefore can be maintained indefinitely. This definition implies that they correspond to solutions of the form: {u} = {v} sin Zt
[9.36]
of equation [9.33] through which we have arrived at s = 0; by putting [9.36] in [9.33] thus modified, we have the equation: ([K] – Z² [M]) {v} = 0
[9.37]
In order to be able to find the non-zero vectors {v} verifying this condition, Z must correspond to one of the eigenangular frequencies of the structure, i.e., the values of Z which cancel the determinant of matrix >K@ – Z² >M@. It can be shown [CLO 75] that for a structure at N degrees of freedom, there are exactly N eigenangular frequencies and that to each one of them corresponds an eigenmode, i.e. a vector {v} verifying [9.37]. In certain cases, there can be multiple eigenangular frequencies to which are associated eigenvectors in number equal to their order of multiplicity. The eigenangular frequencies Zi (i = 1, 2,…, N) are arranged by increasing values; with them, we associate eigenfrequencies gi and eigenperiods Ti defined by:
gi =
Zi 2S ; Ti = 2S Zi
1 (i 1, 2,....., N ) ³i
[9.38]
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Seismic Engineering
The consequence of equation [9.37] is that the eigenmodes verify the important relation of orthogonality in relation to matrices >K@ and >M@: {vi}T [K] {vj} = 0
[9.39]
{vi}T [M] {vj} = 0
[9.40]
{vi} and {vj} being two different eigenmodes, corresponding to eigenangular frequencies Zi and Zj and T being the symbol of the transposition (permutation of rows and columns) in matrix calculation. In fact, by writing [9.37] for index i of the eigenmode (Z = Zi, {v} = {vi}) and by pre-multiplying by the transpose of {vj}, we obtain: {vj}T [K]{vi} = Zi2 {vj}T [M] {vi}
[9.41]
from which, by permuting indexes i and j: {vi}T [K]{vj} = Z 2j {vi}T [M] {vj}
[9.42]
Matrices [K] and [M] are symmetric which leads to the following equalities between scalars: {vj}T [K] {vi} = {vi}T [K] {vj}
[9.43]
{vj}T [M] {vi} = {vi}T [M] {vj}
[9.44]
Thus, by subtracting [9.42] from [9.41] we have: 0 = ( Zi2 Z 2j ) {vj}T [M] {vi}
[9.45]
which, for Zi z Zj, leads to [9.40]; [9.39] thus results from [9.41] (or from [9.42]). If we explain equation [9.37] with expressions [9.33] of matrices [M] and [K] we find the following system for components v1, v2,…vN of one of the eigenmodes: Dv1 – v2 = 0 – vn-1 + D vn – vn+1 = 0, n = 2, 3,…,N-1
[9.46]
The Response Spectrum
403
– vN-1 + (D -1) vN = 0 where we have introduced the notation: D=2–
mZ ² k
[9.47]
System [9.46] accepts solutions in this form: Vn = E sin nI
[9.48]
E being an arbitrary constant (system [9.46] being homogenous, its solutions are defined with a multiplicative constant value) and I a quantity related to D by the equation: D = 2 cos I
[9.49]
In fact we verify without difficulty that the first equation of system [9.46] is satisfied if we substitute expressions [9.48] and [9.49] for it, then cumulatively that if [9.48] is verified for Vn-1 and Vn, it is also verified for Vn + 1; the last equation of the system then provides the following condition which determines I: – sin (N-1) I + (2 cos I – 1) sin N I = 0 i.e. sin NI = sin (N+1) I
[9.50]
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Seismic Engineering
Figure 9.14. Eigenmodes of a perfectly regular building being deformed by pure shear and embedded at its base in the case N (number of levels) = 4. The convention for standardization of modes means putting the same value of 1 for displacement at the last level in all the modes. The eigenangular frequencies Zi are given for each mode as a function of k / m (k stiffness between two levels, m mass of a level)
The Response Spectrum
405
This equation has two possible solutions that correspond: – either at NI = (N + 1) I – 2iS (i is any integer) a solution without interest as thus vn = 0 whatever n is; – or at (N + 1) I = S – NI + 2(i – 1) S (i is any integer) i.e.:
I=
2i 1 S, i = 1, 2,…, N 2N 1
[9.51]
We can limit ourselves to these N values of i as we find I = S (thus vn = 0 for all n) for i = N + 1 and for i = N + 2, N + 3,…, 2N + 1 the values of T are obtained by subtracting from 2S those that are calculated by [9.51] for i = N, N –1,…,2,1; this means changing the sign of vn. Considering equations [9.47], [9.49] and [9.51] for the N eigenangular frequencies of the model being studied we have:
Zi
2
k § 2i 1 S · sin ¨ ¸ , i = 1, 2,…, N m © 2N 1 2 ¹
[9.52]
In addition, for components Vi,n of eigenvector {vi} corresponding to the eigenangular frequency Zi (by choosing constant E of equation [9.48] so that vi,N = 1 for all the modes): § 2i 1 · § 2i 1 · vi,n = sin ¨ nS ¸ / sin ¨ N S ¸ ,in i = 1, 2, …, N © 2N 1 ¹ © 2N 1 ¹
[9.53]
Figure 9.14 indicates the eigenmodes calculated by [9.53] in the case N = 4, with indication of the values of eigenangular frequencies (equation [9.52]). We can note on the aspect of the eigenmodes in this figure, and this applies for all one-line models (any mass set along one same line and interlinked by any stiffness): – the first mode, or fundamental mode, has a deformation whose displacements regularly increase with height; – the higher modes present vibration nodes, i.e. points where the displacement is zero; the number of nodes (including the base) is equal to the mode number; – the points where the modal deformation reaches a maximum or a minimum, called anti-nodes of vibration, are also equal in number at the mode number (for the second mode of the figure, the minimum – 1 of the first two levels count as only one extreme).
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Seismic Engineering
The fundamental mode that gives the lowest frequency (or the highest period), generally has (section 9.2.4) a decisive influence on the response of the whole building. For the shear type of deformation considered in the chosen example (which is characteristic of the beams and columns framework), the fundamental mode period is proportional to the height of the building (or to the number of floors). We notice in fact that as soon as the number N of levels is higher or equal to 2, the argument of the sine in formula [9.52] is for I = 1 (fundamental mode) small enough for us to write:
Z1 # 2
k S m 2(2 N 1)
Thus, for the period T1 = 2S/Z1: T1 # 2(2N + 1)
m k
[9.54]
Thus, for a fairly high N, T is practically proportional to N, i.e. to the height of the building, which justifies the empirical formulae of codes such as [8.4] (Japanese code 1981). By the same reasoning we can arrive at formula [4.39] of the basic period of a layer of homogenous soil. A layer such as this can be associated with a building that is deformed in pure shear each level of which corresponds to a lamina of soil of thickness e, with the following values of parameters of mass and stiffness: m = USe
[9.55]
GS k= e U and G being the density and shear modulus of the soil respectively and S the surface of the right section of the soil column; thus we have, by taking expressions [9.55] into [9.54] and by ignoring I in light of 2N since the number of lamina of soil must be high: T1 = 4Ne
U G
[9.56]
This is identical to [4.39] since Ne is the total thickness h of the layer and G / U the propagation velocity c of the seismic shear waves.
The Response Spectrum
407
9.2.4. Calculation of the response for the chosen example
The non-damped eigenmodes that have just been determined for the given example, constitute the most natural coordinate base to study the dynamic response of the model; thus we look for solution {u} of equation [9.33] under the form of a development on the basis of the eigenmodes: N
¦
{u} =
ri (t) {vi}
[9.57]
i 1
The N functions of time ri (t), coefficients of the development of {u}, characterize the response of each mode to the excitation represented by accelerogram s (t); by putting [9.57] into [9.33] we obtain: [M]
N
¦
r i {vi) + [K]
i 1
N
¦
ri {vi} = – s [M] {'}
[9.58]
i 1
i.e., as the eigenmodes {vi} verify equation [9.37]: N
¦
( r i + Zi2 ri) [M] {vi} = – s [M] { ' }
[9.59]
i 1
We pre-multiply this equation by transpose {vj}T of any mode; the orthogonality relation [9.40] is such that in the first member, all the terms of the sum are zero except the one that corresponds to i = j; we thus obtain: rj Z 2j rj
p j s , j = 1, 2,…, N
[9.60]
By setting: ª v j º > M @^'` Pj = ¬ ¼T ^v j ` > M @^ v j ` T
[9.61]
coefficients Pj thus defined are the participation factors of the eigenmodes. Differential equations [9.60] determine functions rj (t), in relation to the initial conditions of rest: rj (0) = 0; r j (0) = 0, j = 1, 2,…, N which result from [9.35] and [9.57].
[9.62]
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Seismic Engineering
It is observed that rj (t) are decoupled, i.e. they can be calculated independently of each other on the basis of s (t). This remarkable property is the very foundation of modal analysis and shows the importance of the notion of a simple oscillator. We have used form [9.33] as an equation of motion without any damping. In reality there are always dissipative forces which produce progressive attenuation of oscillations, and it would have been more appropriate to consider equation [9.34], which takes this effect into account through the use of damping matrix >C@, but the decoupling of modal responses would then be lost. In fact it is only for very specific forms of matrix >C@ that it produces orthogonal relations between analogous modes in [9.39] and [9.40], as we shall see in section 15.1.3. In the meantime, taking into account significant uncertainties about the nature of damping (which casts important doubts on the validity of the viscous model of the latter) and the fact that the rates of damping are low, we have to be satisfied in current practice to assume that there is a decoupling of modal responses and to introduce a general term of damping in equation [9.60] which becomes: r 2[ j Z j rj Z 2j r j j
p j s
[9.63]
We have introduced, as for the simple oscillator studied in section 9.1.1, the reduced damping coefficient [j. The value may vary according to the mode in question, if the structure is heterogenous, but for regular buildings as in the chosen example, it is normal to maintain a single value which depends only on the construction material; often it is 5% for buildings made of reinforced concrete. The only difference between equations [9.63] and [9.3] for the simple oscillator, lies in the presence of participation factor pj in the second part of [9.63]. As in both the cases, the initial conditions are of rest, factor pj is simply the factor of proportionality between the solutions, and we have: rj (t) = pj x (t), j = 1, 2, …, N
[9.64]
if response x(t) of the simple oscillator is calculated for angular frequency
Z = Zj and reduced rate of damping [ = [j.
We can thus write, using the formula of Duhamel’s integral [9.10]: rj (t) = –
with Z ' j
Pj t s W e [ jZ j (t W ) sin Z cj (t W )dW Z c j ³o
Z j 1 [ j2
[9.65]
The Response Spectrum
409
Bringing forward [9.65] into development [9.57] of solution {u} we get: {u} =
Pj
N
[Z ¦ Z ' ^v ` ³ s W e t
j
j 1
j
0
j
j
( t W )
sin Z ' j t W dW
[9.66]
i.e. by allowing the signs for summation and integration: {u} = –
ª
N
¬
J 1
³ s (W ) ««¦ t
0
Pj ^v j `
Z 'j
e
[ j Z j ( t W )
º sin Z ' j (t W ) » dW »¼
[9.67]
We thus have a formula for calculating solution {u} on the basis of accelerogram s (t) and modal parameters (Zj, [j, {vj} and pj). The factors of participation pj, defined by general formula [9.61], have the following expression in the case of the chosen example: § N · § N · pj = ¨ ¦ v j , n ¸ / ¨ ¦ v² j , n ¸ ©n I ¹ ©n I ¹
[9.68]
On the basis of expression [9.53] of components vj1n of eigenmodes, basic calculations of trigonometric sums make it possible to get the following formula: Pj = 1
j 1
2 § 2 j 1 · § 2 j 1 S · sin ² ¨ N S ¸ / sin ¨ ¸ 2N 1 2 1 N © ¹ © 2N 1 2 ¹
[9.69]
Eigenmodes being defined with a close multiplicative arbitrary constant, due to the structure of formula [9.61], the participation factors are inversely proportional to this constant. Their values do not have any physical significance. It is product Pj {Vj} of the eigenvector by its participation factor which has significance for the seismic response of the structure; these products are seen to appear in formula [9.67] which explains this response. They verify the general formula N
¦ P ^v ` ^'` j
j
[9.70]
j 1
{'} being the direction vector of the excitation defined by [9.33]. The relation in [9.70] is demonstrated very simply by pre-multiplying it by line vector {vi} T [M] where {vi} is any given eigenvector; orthogonality relation [9.40] thus gives us:
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Seismic Engineering
pi {vi}t [M] {vi} = {vi}T [M] {'}
[9.71]
which follows immediately from definition [9.61]. In the case N = 4 of the chosen example, the eigenmodes were presented in Figure 9.14. Table 9.2 completes this information by giving values of participation factors pi and products pi vi,n for all the storeys of the building (n = 1, 2, 3, 4). 1st mode i
1
2nd mode i
2
3rd mode i
3
4th mode i
4
pi
1.2411
– 0.3333
pivi,1
0.4310
0.3333
0.1837
0.0520
pivi,2
0.8101
0.3333
– 0.0638
– 0.0796
pivi,3
1.0914
0.0000
– 0.1615
0.0701
pivi,4
1.2411
– 0.3333
0.1199
– 0.0277
Zi ZO
0.3473
1.0000
1.5321
1.8794
0.1199
– 0.0277
Table 9.2. Participation factors pi and products pi {vi} for eigenmodes
for the chosen example in the case N
4
It is observed that products pi vi,n have a tendency to decrease in modulus (with a few exceptions) when we pass from mode i to mode i I; as at a given level their sum is equal to I [9.70], we realize that the contribution of the first mode to this sum is highly significant, at least for levels 2, 3 and 4. These considerations on the importance of the first mode for regular structures will be taken up in the next section.
9.2.5. Calculation of displacements, accelerations and forces for the chosen example
Formula [9.67] makes it possible to calculate vector {u} of relative s t . This calculation, which does not displacements for a given accelerogram present any specific numerical difficulties, provides the complete history of the response for all the instants contained between the start and end of the movement. Such detailed knowledge of the response is necessary only in certain applications; most often in the case of earthquake engineering and design of a building, we are only interested in maximum values of displacements and forces.
The Response Spectrum
411
To determine them, it is better to use the following formula deduced from [9.57] and [9.64], instead of [9.67]: N
¦
{u}
x (t, Zi, [i) pi {vi}
[9.72]
i 1
where x (t Zi, [i) shows the solution of equation [9.3] of a simple oscillator having Zi for angular frequency and [i for rate of damping. To simplify this, we write [9.72] for a given component un of vector {u}: N
¦
un
x (t, Zi, [i) pi vi, n
[9.73]
i 1
We assume that the response spectrum of accelerogram s t has been determined; this provides an immediate estimation of an upper bound of the modulus of un: N
Max un d ¦ Sd Zi ,[i pi vi , n t
[9.74]
i 1
since a sum necessarily has an upper bound through the sum of maximum values of the moduli of its different terms. As is generally the case, if it is not the spectrum for displacement Sd(Z, [) which is provided, but rather the pseudo-acceleration spectrum Sa (Z, [), then relation [9.74] is written as: N
Max un d ¦ t
i
1
2 1 Zi
Sa Zi ,[i pi vi , n
[9.75]
Combinations [9.74] or [9.75], called arithmetic combinations of modes, are almost never used, as they correspond to the least probable scenario in which all the modal responses would reach their maximum value at the same time. We prefer to use the quadratic combination, which has already been presented several times (sections 4.1.3, 4.3.4, 5.1 and 5.3.2) and which consists of writing that the square of the maximum of a sum is equal to the sum of squares of the maximum of its terms. This rule of combination, often abbreviated as SRSS (Square Root of the Sum of the Squares) will be justified in section 15.2.2, in the context of the hypothesis of excitation in white noise.
412
Seismic Engineering
The application of SRSS rules to equation [9.75], in the case where the pseudoacceleration spectrum is known, leads to the formula: Max t
un2
N
1
¦Z i 1
4 i
S a2 Zi ,[i pi2 vi2, n
[9.76]
The presence of the fourth power of angular frequency as the denominator strongly accentuates the importance of the fundamental mode in the response, which was highlighted in the notes on Table 9.3. In order to make a numerical application, the following values are given: m (mass of a level)
106 kg
h (height of a storey)
3m
E (Young’s modulus for concrete) Np (number of columns)
3 x 104 MPa
[9.77]
32
a (side of the assumed square section of the columns)
0.5 m
We thus find for stiffness k between the storeys (formula [9.25] with I for a square column): k = 12
3x1010 x (0.5) 4 x 32 = 2.22 x 109 N/m 12 x 33
from which for eigenperiods Ti k m
a4 / 12
[9.78]
2S / Zi (see Table 9.2):
47,1rd / s, T1 = 0.38 s, T2 = 0.13 s, T3 = 0.0875 s, T4 = 0.071 s
To simplify, we assume that the response spectrum is such that the four modes correspond to the same value of pseudo-acceleration, 2.5 m/s2. This value 2.5 m/s2 is associated with a maximum ground acceleration of 1 m/s2 (see formula [9.21]) with a damping of 5%, i.e. an earthquake strong enough to be felt but not destructive, except possibly in case of very vulnerable constructions or elements. In these conditions, the application of formula [9.76] leads to the following values for maximum displacement of storeys (Table 9.3).
The Response Spectrum Floor
Maximum displacement (mm)
n=1
4.0
n=2
7.6
n=3
10.2
n=4
11.6
Table 9.3. Maximum displacements of levels in the case N
413
4
It can easily be verified that these values are determined in quasi-totality by the contribution of the fundamental mode. It would be necessary to provide more significant numbers to see the appearance of other modes; for example on the first floor, there is a displacement of 4.038 mm after taking all the modes and 4.019 mm on taking only the first mode. This shows, considering the modest precision in seismic calculations we have to be satisfied with, that higher modes are totally negligible for the calculation of displacements in this example. This result is typical for regular structures. The maximum shear force at base Vo,max is calculated on the basis of the maximum displacement u1, max of the first level: Vo, max
ku1, max
[9.79]
With the numerical values previously determined, we find Vo, max 8.97 x 106 N, i.e., for a total mass of 4 x 106 kg, an average acceleration in the structure of 2.24 m/s2. On the basis of Vo, max we can calculate the maximum bending moments at the base of the columns; in the hypotheses corresponding to the calculation of stiffness with formula [9.25] the moment Mo, max at the base of the column of the first level is given by: M0, max =
h 1 V0, max = 0.42 x 106Nxm 2 Np
[9.80]
For this moment value, cracking would be significant as the associated bending stress V ³ given by:
Vg =
a Mo, max 2 I
6 Mo, max a3
[9.81]
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Seismic Engineering
is equal to 20 MPa in the present case, whereas the static stress under the action of eigenweight is only 5 MPa. The calculation of column stiffness in the hypothesis of a non-fissured section is thus not compatible with the results obtained for seismic response. However, it is common practice in earthquake engineering. As indicated in section 7.2.2, the approach of code-based calculation is essentially conventional. In the case of a regular building as in the chosen example, this approach can be validated through nonlinear analyses (see section 9.3.1), but the case of irregular structures is far less clear. Instead of calculating shear force at the base on the basis of displacement on the first level, it could have been determined through the sum of forces of inertia acting on the structure. As all the modes in the chosen example correspond to the same pseudo-acceleration of 2.5 m/s2, this sum can be calculated by multiplying the total mass 4 x 106 kg by the acceleration; and thus finding Vo,max 107 N or a value higher by 11% than the one previously determined. We shall see in section 15.2.4 that this is the general tendency in “stick models”, i.e., internal forces calculated on the basis of maximum displacements are lower than those deduced for maximum accelerations when these maxima are determined through the SRSS rule of quadratic combination. This difference results from the use of this rule, which does not make it possible to obtain perfect coherence between the deformation fields and the stress fields. The difference is low for regular structures, as in this example, but it can be significant in case of irregularities. Considering the importance of the fundamental mode in seismic response of regular structures, we can ask ourselves about the practical necessity of calculating all the eigenmodes, as we have done for the chosen example. There are approximate methods which are simple to use which make it possible to determine period T1 and the deformation of the single fundamental mode; the best known is the Rayleigh method for which the starting point is equation [9.37] which defines angular frequencies and eigenmodes; by pre-multiplying it by transpose {v}T of an eigenvector, we find the following expression for Z²:
^v` > K @^v` T ^v` > M @^v` T
Z²
[9.82]
If this formula is applied by taking an approximate expression of the deformation of the fundamental mode for {v}, we can hope that the value found for Z will be close to the eigenangular frequency of this mode. For the approximation of {v1} we use static deformation {w} obtained by imposing a uniform acceleration * for all the masses of the structure, i.e., the solution of the equation:
The Response Spectrum
[K] {w}
* [M] {'}
415
[9.83]
We thus have, according to [9.82] and [9.83], the following approximate expression for period T1 2S/ZI of the basic mode: 1/ 2
T1 #
ª ^w`T > M @^w` º 2S « » T «¬ * ^w` > M @^'` »¼
[9.84]
The calculation of {w} is basic in the case of a perfectly regular building of N levels; for its components wn we find: wn =
1 m* n 2 N 1 n , n = 1, 2, …, N 2 k
[9.85]
On the basis of [9.85] for the approximation of T1 by applying formula [9.84] we find: T1 # 2 S
m 2 N ( N 1) 1 k 5
[9.86]
This approximation is excellent, as it can be verified in Table 9.4 where the exact value of T1/To (with To 2S m/k ), deduced from [9.52], is compared to the approximate value, deduced from [9.86], for different values of N.
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Seismic Engineering
N
§ T1 · ¨ ¸ © TO ¹exact
§ T1 · ¨ ¸ © TO ¹ approx.
Error (%)
1
1.000000
1.000000
0
2
1.618034
1.612451
0.34
3
2.246979
2.236066
0.49
5
3.513333
3.492852
0.58
7
4.783384
4.753936
0.62
10
6.690753
6.648317
0.63
15
9.871863
9.808153
0.64
20
13.053978
12.969159
0.65
30
19.418984
19.292356
0.65
Table 9.4. Comparison between exact values and approximate values by the Rayleigh method of basic period for a perfectly regular building of N levels deforming in pure shear
It is observed that the error is always less than one percent and the approximation is by default; it can be shown that this tendency is systematic (see section 15.2.1). As for the deformation, maximum displacements calculated by quadratic combination of modes can be compared (see Table 9.3) to those resulting from formula [9.85] with * 2.5 m/s2; the results of this comparison are given in Table 9.5. Level
Maximum displacement (mm) by quadratic combination
Maximum displacement (mm) by static deformation
n=1
4.0
4.5
n=2
7.6
7.9
n=3
10.2
10.1
n=4
11.6
11.3
Table 9.5. Comparison of displacements calculated by quadratic combination and by static
deformation ( *
2.5 m/s2) for a perfectly regular four storey building
It is observed that the two modes of calculation of displacements are in concordance.
The Response Spectrum
417
This valid observation for regular buildings and the use of the Rayleigh method is permitted in earthquake-resistant codes. The latter generally allow the use of simple formulae to calculate the deformation (power law behavior on the side above the ground) instead of static deformation {w} under uniform acceleration. There may be an impression that the quasi-equality of displacements in Table 9.5 comes from the assumption of a flat spectrum, i.e. producing the same pseudoacceleration for all the modes. The dynamic excitation thus seems very close to static action and it seems natural to have almost identical displacements in both cases. In fact it is not and this quasi-identical nature of results comes from the predominance of the fundamental mode in dynamic response and the low difference between the deformation of this mode and static deformation. This can be understood by considering the case of a spectrum for which pseudo-accelerations are inversely proportional to the period, as on the descending branches of the PS 92 spectra in Figure 9.6. To facilitate the comparison we keep the same value of 2.5 m/s² for the pseudo-acceleration of the first mode; higher modes of shorter periods thus have stronger pseudo-accelerations. Table 9.6 gives the new displacements of levels calculated by quadratic combination of the four modes and by limiting oneself to the first mode 2.50 m/s² as pseudoAs in Tables 9.3 and 9.5, by taking the value A1 acceleration of the first mode, the values for other modes are thus: A²
7.20 m/s², A3
11.03 m/s² and A4
13.53 m/s²
We observe that the displacements vary very little compared to those in Tables 9.3 and 9.5, whereas the dynamic excitation seems much more severe, as the accelerations of the last two modes exceed 10 m/s². Thus, it is certainly the importance of the basic mode which explains the effectiveness of the Rayleigh method. Level
Maximum displacement (mm)
Maximum displacement (mm)
by quadratic combination
by a single first mode
n=1
4.18
4.03
n=2
7.65
7.55
n=3
10.20
10.19
n=4
11.64
11.61
Table 9.6. Maximum displacements of levels for a perfectly regular four storey building in the case where the pseudo-accelerations of modes are inversely proportional to their period
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Seismic Engineering
9.3. Structural design spectra
In earthquake-resistant codes applicable to constructions at present, the term design spectrum was until now reserved for a modified elastic spectrum, meant as the basis for a general calculation of nonlinear effects. In the case of industrial establishments with special risk (especially nuclear power stations), this term refers to an elastic spectrum for calculation of the type presented in section 9.1.2. These terminological differences are regrettable but we can reasonably hope that they will disappear in the future, as recent evolutions of codes such as Eurocode 8 seem to indicate. The design spectrum as per earthquake-resistant codes is in fact quite a mixed notion, arising more from the seismic coefficient rather than structural dynamics and incorporating well established considerations in a non-transparent manner, but which need to constitute a precise presentation of objectives and clear treatment through appropriate means; it concerns the ductile strength of stiff structures and consideration of the effects of the second order in flexible structures. These points will be explained and discussed in section 9.3.2. Before this, section 9.3.1 will be devoted to the presentation of the behavior coefficient, which is the basis of structural design in current earthquake-resistant codes. 9.3.1. Reasons for the general consideration of nonlinearities: the behavior coefficient
In a publication which is still famous [NEW 60], N.M. Newmark established that a simple oscillator, subjected to excitation of a seismic accelerogram, always had about the same value of its maximum relative displacement, whatever the perfect elastoplastic model in question to describe nonlinear behavior of its spring, at least as far as its eigenperiod (corresponding to the elastic behavior of the spring) was the same as (or greater than) the dominant period of the excitation signal. The nonlinear model considered by Newmark was the simplest: elastic-perfectly plastic behavior with discharge parallel to the elastic phase (Figure 9.15). He varied the value of force limit Fu corresponding to plastification and conserving the same elastic phase slope and the same excitation accelerogram. A typical example of the results thus obtained is presented in Figure 9.16.
The Response Spectrum
419
Figure 9.15. Elastic-perfectly plastic model
The maximum displacement (in modulus) is observed to be around 70 mm for the four cases under consideration, the first of which (Fu f) corresponds to purely elastic behavior; when the plastification force is reduced the number of plastic cycles increases but the maximum displacement reached is practically constant. This remarkable result which can be called “Newmark’s theorem” is the basis of modern earthquake-resistant codes, which have sought to take into consideration nonlinear behavior of structures in a simple manner.
Figure 9.16. Response of an elastic-perfectly plastic oscillator for different values of force Fu of plastification Fu = f corresponds to elastic behavior [FIL 96]
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Seismic Engineering
Plastification making the structure more deformable, an increase in displacement would be expected as compared to cases without plastification (elastic behavior). If this increase is not produced, it means that there is a compensatory mechanism which attenuates the response, i.e. the dissipation of energy associated with hysteretic cycles. A very simple explanation can be given for the phenomena in question, in the following way: When an elastic-perfectly plastic oscillator completes a hysteretic cycle during which its maximum displacement reaches the value of P Ge (Ge being the displacement corresponding to the yield strength), it can be admitted that it behaves like an elastic oscillator having reduced stiffness (with relation to the elastic phase of the elastoplastic oscillator) and damping as large as the area of the cycle. Figure 9.17 shows that the reduction of stiffness returns to division by P and the area of the hysteretic cycle is proportional to P-1.
Figure 9.17. Equivalence between an elastoplastic oscillator and an elastic oscillator with reduced stiffness and increased damping
Thus, for period T (which is inversely proportional to the square root of the stiffness) and for damping [ of the equivalent elastic oscillator, we can write: T
TO
P
[9.87]
[
[o + [1 (P – 1)
[9.88]
The Response Spectrum
421
To and [o being values for the period and damping respectively when the behavior of the elastoplastic oscillator is linear (P 1); [1 characterizes the increase of damping through the effect of hysteretic cycles. To determine the response of the equivalent elastic oscillator, we assume the response spectrum for pseudo-acceleration of the excitation signal to be known, with the expression in the form: Sa
§T · Ao ¨ ¸ © To ¹
n
[o [
[9.89]
Factor [ o / [ corresponds to equation [9.21] which gives the dependence of amplification in acceleration on the basis of damping. Factor (T/To)n corresponds to the three parts of the spectra in Figure 9.6, i.e.: – for n
1, linear rise for low values of the period;
– for n
0, maximum acceleration plateau;
– for n period.
– 1, the descending branch of the hyperbola for high values of the
Displacement G is by definition the quotient of the pseudo-acceleration by the square of the angular frequency i.e. according to [9.89]:
G
1 S aT ² 4S ²
§T · 1 AoTo2 ¨ ¸ 4S ² © To ¹
n2
[o [
[9.90]
from which, expressing T and [ on the basis of P through relations [9.87] and [9.88]:
G
n2 1 AoTo2 P 2 4S ²
ª [1 º «1 P 1 » ¬ [o ¼
1/ 2
[9.91]
For [1 [o (which is a plausible order of magnitude corresponding to the damping doubling for P 2), [9.91] takes the simple form:
G Go
P
n 1 2
[9.92]
where Go is displacement Ao To2 /4 S 2 calculated on the basis of the elastic hypothesis for the elastoplastic oscillator.
422
G
Seismic Engineering
We see that for n Go whatever P is.
– 1 (descending branch of spectra for long periods) we have
This “demonstration” of the Newmark theorem thus provides an assessment of its validity limitations, which corresponds to sufficiently flexible oscillators (in their elastic phase) so that their period is situated in the descending branch of the spectrum, and thus it is higher than the dominant periods of the seismic signal (which are associated with the spectrum plateau). Formula [9.92] shows that in the other parts of the spectrum, the displacement of the elastoplastic oscillator increases in relation to that of the elastic oscillator, proportionally to P on the plateau (n 0) and for P on the left of the plateau (n 1). These tendencies were noticed by Newmark who proposed a rule for energy equivalence between the elastoplastic oscillator and its elastic phase for intermediary period (region of the spectrum plateau). This is shown in Figure 9.18.
Figure 9.18. Equivalence of energy between the elastic response and elastoplastic response; the highlighted areas (triangle Po P1 E and rectangle D1 P1 P2 D2) are equal
Maximum deformation P Ge reached by the elastoplastic oscillator corresponds to the same deformation energy as for a purely elastic oscillator. This condition (equality of marked areas in the diagram) implies, as seen in a simple calculation, that the maximum deformation obtained by the elastic calculation is equal to 2P 1 Ge; the relation of deformations (plastic/elastic) is thus equal to P / 2P 1 which is a little lower, but quite comparable to factor basis of equation [9.92].
P previously found on the
The Response Spectrum
423
Coefficient P used above is the ductility coefficient, which characterizes the level of deformation of behavior distribution that can be reached on the plastic part of the constitutive law; it is thus acceptable a priori as a deformation criteria, according to the general principle mentioned in section 8.2.2, to gauge the safety in relation to dynamic action such as an earthquake. However, habits acquired are such that its interpretation in earthquake-resistant codes is generally presented in terms of forces, in the following manner: – for structures that are quite flexible (whose period is in the descending branch of spectra) the forces calculated on the elastic model can be divided by P, as we are in the domain of validity of Newmark’s theorem (equality of displacements whatever P is); – for intermediate structures (period in the spectrum plateau), elastic forces can be divided by 2 P 1 (or P ) as we see in Figure 9.18; – for stiff structures (period on the left of the spectrum plateau) it is necessary to use elastic forces, as the ratio of (plastic/elastic) deformations is precisely equal to P. Of course the design of a structure with reduced stresses with relation to those resulting from elastic calculations is meaningful only if appropriate constructive measures are taken, making it possible to master plastic deformations accepted in this way, from which follows the emphasis of the codes on the importance of these measures. It can be estimated in the mean time, that better comprehension of codes, thus better conditions for their appropriate application, would be obtained by “laying the cards on the table”, i.e. by “confessing” that the only real interest of any elastic calculation is to determine displacements and on the basis of this, the project designer must think of possible options to accommodate these displacements, taking into account safety issues and economic constraints. The foreseeable evolution of codes towards an approach based on displacement criteria, discussed at the end of section 8.2.2, certainly represents progress from this point of view. The study of seismic response of nonlinear oscillators has been carried out by several authors, following that by Newmark. It mainly concerns the influence of the type of accelerogram and the plastic behavior model of the oscillator spring (especially models better adapted than the simple elastic-perfectly plastic diagram cyclical behavior of elements made from reinforced concrete). Figure 9.19, taken from [MIR 94], shows some examples of laws/distribution of variation of the RP factor for reduction of force on the basis of the ductility coefficient P and period T.
424
Seismic Engineering
Figure 9.19. Variation of coefficient RP of reduction of force on the basis of ductility coefficient P and period T of the oscillator (according to [MIR 94])
These different propositions, which correspond to a large variety of seismic signals and elastoplastic models, agree upon the main conclusions which we have already arrived at during the previous discussion: – beyond a period which may vary according to the type of accelerogram, but which is generally about 0.5 s, RP is more or less equal to P whatever the period; – at zero period, RP is equal to 1 whatever the ductility coefficient.
The Response Spectrum
425
Among the laws shown in Figure 9.19 the simplest expressions of factor RP are those presented by Hidalgo and Arias (case d) [HID 90] and Vidic, Fajfar and Fischinger (case f) [FAJ 84] which are respectively written as: §T 1 · RP 1 ¨ o ¸ 1 ¹ T P ©
R P 1 P 1 RP
1
(with To = 0.02 s)
[9.93]
T for 0d T d T1 (with T1 = 0.5 s) T1
P for T > T1
[9.94]
These studies present a solid base for the structural design approach, maintained by earthquake-resistant codes, with the means of a reduction factor of elastic forces, at least for simple structures, which can be assimilated reasonably to a simple oscillator. This coefficient, called the behavior coefficient in the majority of codes, depends a priori on a number of factors, whose influence is more or less well explained: – the characteristics of seismic signals, notably from the point of view of their frequency content and their duration. Parameters To [9.93] or T1 [9.94] take their influence into account in a simple manner, which certainly varies according to the nature of the ground; – the basic period of the structure or, to be more precise, its ratio to a characteristic period of the seismic signal (such as To or T1); – the bracing mode, according to whether it favors dissipation of vibratory energy or not through cycles of plastic deformation, without compromising the capacity to bear vertical charges; – the nature of materials used for bracing structures; – the regularity of the structure, in plane or in elevation. The factors mentioned in this list are currently those that are taken into consideration, either explicitly or implicitly, in the majority of earthquake-resistant codes. This consideration is explicit in the case of the mode of bracing, the nature of materials and regularity. It is often implicit for periods of the seismic signal or the structure, which are involved in the construction of the structural design spectrum (see section 9.3.2) but may not appear in tables giving the values of the behavior coefficient. These values are essentially fixed through expert judgments based on experience (post-seismic missions and laboratory testing) and comparison with codes having a reference value.
426
Seismic Engineering
Some believe that it may be possible to determine them through calculation based on validated nonlinear models. This type of research should certainly be encouraged, but it currently seems to be limited to cases where the cause of nonlinearity is unique and well identified, such as, for example, the formation of a plastic hinge in a structural element or the uplift of part of the foundation. In more complex cases (multiple causes of nonlinearity, high degree of hyperstaticity of the structure, three-dimensional nature of the response) the calculation of the behavior coefficient is not to be expected in practice. In fact, in such cases, the very notion of the behavior coefficient, in the normal sense of the term, needs to be questioned, as indicated in section 7.2.2. The dogma of a single behavior coefficient for a given structure, making it possible to determine structural design forces through a simple division, is not justified except for regular structures. The desire to extend it to cases of complex structures is without doubt an acceptable compromise, given the current status of knowledge (on condition that the penalty coefficients for irregularity are established with due caution) but must not be presented as the achievement of a scientifically validated approach. Certain modern earthquake-resistant codes (PS92 rules, Eurocode 8) may be strongly criticized in this respect, as they lead us to believe that the calculation methods which are described have the same degree of validation and reliability for all types of structures, which is not at all true. The difficulties linked to the uniqueness of the behavior coefficient clearly appear for constructions having different systems of bracing in both horizontal directions. In effect, the behavior coefficient being dependent on the mode for bracing, will have different values for both these directions and we still do not know how to combine the effects of horizontal components for coupled eigenmodes which have a two-dimensional deformation. In spite of these limitations, the behavior coefficient is important as it forces the project designers to think, at least if they are conscious of their responsibilities that go beyond the mechanical application of rules of calculation. To have the right to divide the determined stresses by three or five for a dynamic model which is often fairly elaborate necessarily implies the task of scrupulously monitoring the provisions and rules of construction which are indispensable so as to benefit from this “miraculous abatement”. In older codes with the seismic coefficient, this aspect was completely hidden and the user of the codes could, in good faith, believe that the effect of earthquakes was reduced to mere horizontal forces of quite modest amplitude (around one tenth of the weight) and its consideration did not pose any problem in a large majority of cases. This biased vision of things especially led to the establishment of the belief, still rooted today in the minds of a number of engineers, according to which seismic movements only rarely exceed levels of acceleration of 0.1 or 0.2 g. The accumulation of recordings highlighting definitely
The Response Spectrum
427
higher levels in most of the epicenter zones often provoked reactions of skepticism or consternation among engineers and made it even more difficult for them to collaborate with seismologists, however necessary it was. The fact that we can practice good quality earthquake engineering and design with moderate effort, but while accepting to “pay the price” in terms of provisions and rules for construction, is very clearly underlined in the method known as “capacity design”, developed in New Zealand and now implemented by most of the recent earthquake-resistant codes (especially Eurocode 8). The method consists of pre-determining the zones where plastification should take place, and taking special construction measures in these zones (especially for setting reinforcement of parts in reinforced concrete) enabling plastic deformations to occur without risk of damage, and overdesigning other potentially critical sections in order to be sure that plastification will occur only where they are expected and not elsewhere. This very logical approach is practical in cases where the structure is sufficiently simple so that pre-determination of plastification zones can be made with certainty. For complex structures, which are highly statically indeterminate and whose response has a three-dimensional character, the reliability of this pre-determination can be doubtful. Once again, we observe the practical difficulties of extension of notions and methods, which were essentially developed for simple cases, to more complex cases. 9.3.2. Elastic and inelastic design spectrum
Elastic design spectra, which serve as the basis for seismic studies for special risk industrial establishments (see section 18.2) were presented in section 9.1.2 under the title of response spectra for calculation. This detail in meaning is important, as design spectra in earthquake-resistant codes, which are presented later on, are not, in fact, response spectra at all, in the sense that they do not represent the response of a well defined mechanical system, as is a simple oscillator with an elastic or elastoplastic spring. Information given in section 9.1.2 on elastic spectra must be completed here with regards to the cut off frequency and the influence of damping. The cut off frequency which was defined in section 9.1.1 as the frequency based on which the pseudo-acceleration is equal to the maximum acceleration of the support, independently of the rate of damping, does not appear in the spectra in Figures 9.6, 9.7 and 9.8. This omission is without consequence for current calculations where we are satisfied with the response of some modes with relatively
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low frequency (less than 10 Hz, as an example) which is generally the case in earthquake-resistant codes. On the other hand it can be quite constraining in very detailed modal analyses, where it is necessary to distinguish rigid modes of response (which follow the motion of the support; see section 9.1.1.) from others. It is therefore desirable for the cut off frequency to be indicated in spectra, if they are to be used for complete modal analysis. This observation applies especially to spectra obtained through attenuation laws. For example one of the spectra in Figure 9.7 stops at 20 Hz, which is without doubt below the cut off frequency (which typically varies from 25 to 40 Hz, as indicated in section 9.1.1). As for damping, the spectra representing ground motion correspond most often to cases where damping is at 5% (Figures 9.6, 9.7 and 9.8). The influence of damping other than 5% is regularly taken into account by correction formulae of the form:
U
§ 5 D · ¨ ¸ ©[ D ¹
E
[9.95]
[ being the reduced damping (expressed in percent) for which we wish to evaluate spectral ordinates, D and E are both constants; factor U defined by this formula is applied in a multiplicative manner to the spectral value at 5%; for example, we find the following choices: – D E . in the PS92 rules [COL 96b]; – D E . in the AFPS 92 guide for earthquake-resistant protection of bridges [COL 95]. Such correction factors independent of the period must not be used at the end points of the spectrum (very short or long periods) as they are contradictory to the property of convergence towards acceleration (very short periods) or the displacement (very long periods) of the support. They are thus only useful for calculations using modes with their periods in the central area of the spectrum (from 0.1 s to 2 or 3 s to get a fair idea). For complete modal analyses, corrections at the end points need to be modified in order to respect the condition of convergence, or completely traced spectra must be used for different damping, as in the USNRC spectrum in Figure 9.5.
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429
Figure 9.20. Standardized design spectra, corresponding to different site conditions, from PS 92 rules
Design spectra, as far as earthquake-resistant codes are concerned, are modified elastic spectra which are generally presented as prerequisites before the use of a behavior coefficient. The modifications with relation to elastic spectra, concern the following points: – the maximum acceleration plateau is extended towards the left up to the zero period; this extension replaces the rising part of the short period; – the descending branch for long periods is raised to correspond to a decrease in T –2/3 instead of in T –1 that is found in elastic spectra. Figure 9.20 shows the design spectra for the PS 92 rules [COL 96]), deduced from elastic spectra in Figure 9.6 with these two operations, for the four types of sites considered in this norm (Go rocky site for reference, S1, S2 and S3 sites corresponding to ground with decreasing quality). According to the use of these norms, which do not explain why the rules are implemented, the reasons for modifications changing from elastic spectra to design spectra have never been clearly explained. The extension of the plateau essentially corresponds to the concern of limiting the reduction of forces calculated in the elastic domain in periods higher than a certain value, of around 0.25–0.5 s (see the curves in Figure 9.19); for periods
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smaller than this limit and particularly for zero period, this reduction is not justified. Thus, if we divide the results of the elastic calculation by q (the behavior coefficient ! 1), the forces corresponding to modes for short periods will be greatly underestimated if their response is determined on the basis of the elastic spectrum. Its replacement by the design spectrum reduces the risk of underestimation, but it is not sufficient to ensure that the elastic force is found at the zero period, whatever the value of the behavior coefficient. The PS 92 rules have chosen to replace coefficient q, with the modified value q’ for periods less than value T% (left side of plateaus in the elastic spectrum) according to the expression: q’ =
2.5 T § 2.5 · 1 ¨1 ¸ TB © q ¹
[9.96]
Figure 9.21 illustrates this operation. This manipulation makes it possible to return to the value 1 for a standardized spectrum for T 0, at least for its lower values. It brings us back to admitting that the behavior coefficient does depend on the period.
Figure 9.21. Replacement of the behavior coefficient q by a modified value q’ for periods T less than T%; RE elastic spectrum, RD design spectrum, RD/q design spectrum divided by q, RD/q’ design spectrum divided by q’
Other reasons can be given to justify the extension of the plateau:
The Response Spectrum
431
– a certain lack of confidence with regard to high frequency components in signals recorded by old seismographs for strong motion, considering the imperfections of these instruments; – the desire to do away with the rising part in the spectrum, so as to avoid the underestimation of the eigenperiod for structures leading to a reduction of the response. There is in fact a fairly general tendency towards such underestimation in current calculations, especially due to the frequent omission of effects of soilstructure interaction (buildings are considered as embedded into their foundations on non-deformable ground). The raising of descending branches indicates the desire not to encourage project designers to create structures that are too flexible which could be sensitive to second order effects (additional bending moments due to off centering of vertical charges resulting from horizontal displacement). The choice of the power as 2/3 of the period to fix the speed of the decrease of the spectrum already constitutes a relief as compared to earlier codes, in which a power of 1/3 was used (see coefficient E of the PS 69/82 rules [8.1]) with a constant value above a certain period. If we can understand the motivations that have led to the raising of the branches, the present tendency is to yield to it while, simultaneously, imposing a systematic verification of the second order effects. It has been realized that, for structures with a very long fundamental period (high-rise buildings, large bridges) the displacements associated with these artificially raised branches was becoming completely unrealistic, to the point where it was more advantageous for the project designer to perform a purely elastic calculation (without the behavior coefficient) with the elastic spectrum. Thus the following paradox was reached: an a priori pessimistic calculation (elastic behavior thus without damage) can give less constraining results than a calculation which accepts damage (behavior coefficient higher than one). Modifications carried out on elastic spectra to transform them into design spectra are definitely dictated by considerations which do not have much to do with the physics of seismic motion. Design spectra are thus not response spectra, in the real sense of the term, since they do not represent the response of a well identified mechanical system; their use to determine modal responses in an elastic analysis is more like “cooking” than a logical approach in which all hypotheses would be clearly explained. As indicated earlier, (see introduction to Part 4 of this book and section 8.2.2), we can certainly question the wisdom of the method of earthquakeresistant codes (relatively detailed elastic calculations followed by a simplistic attempt to consider nonlinearities by dividing by the behavior coefficient). However, if we accept it, we must separate what can be dealt with in a rigorous manner (elastic modal analysis) from that which originates in an essentially empirical reasoning (dividing by the behavior coefficient). We should therefore not use anything other
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than an elastic spectrum for elastic calculation and then, if required, “manipulate” the results of such a calculation for designing. It is more convenient to give all the details of the “recipe” to arrive at the values of the dividing coefficients of the forces by taking into account the influence of all the parameters (bracing mode, material, regularity, as well as the fundamental period of the structure and frequency characteristics of the seismic signal). Design spectra of current codes are nothing but remnants that date back to the era of formulations of seismic coefficients, and we can only hope that they disappear quickly.
Chapter 10
Other Representations of Seismic Action
10.1. Natural or synthetic accelerograms 10.1.1. Types of analyses for which accelerogram representation is necessary Transient seismic calculation on the basis of an accelerogram is as old as, if not older than, modal calculation based on a response spectrum. As indicated in the introduction to Part 2, a very small number of real accelerograms (the most famous recording being El Centro on 18 May 1940) served as the basis for most study on earthquakes in the 1960–1970s. The means of computerized calculation available at the time was very limited in terms of rapid memory capacity and thus better suited for explicit transient schemes (in which the solution is “moved forward” step-bystep in time) than for schemes that need manipulation of large size matrices (implicit transient diagrams or modal analysis methods). For this reason, explicit transient calculations were widely used in early seismic studies using numeric methods, before being gradually superseded by modal calculation, following the “increase in power” of software programs. Today transient seismic calculation is maintained only in domains where it is necessary to obtain temporal information to be able to conduct studies in design and engineering. These domains are as follows: – detailed nonlinear analyses where the aim is to modalize physical reality and to obtain the complete history of seismic response, as against simplified nonlinear analyses (linear models reputed to be “equivalent” or the use of “adjustment” coefficients such as the behavior coefficient) which only aim at the approximation of maximum values of the response;
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– certain linear analyses whose models present features which make it difficult to use regular modal methods; this is the case, for example, for systems comprising localized dampers with significant viscous force, whose damping matrices have a structure that does not make it possible to admit decoupling of modal responses even in an approximate manner; – certain linear analyses of structures for which it is not enough to know maximum values of the response to realistically evaluate the risk of damage. This is the case, for example, for complex metallic structures with thin shells, where a proper evaluation of the risk of buckling instabilities necessitates a precise description of variations in deformation in the course of time; – certain linear analyses whose results serve as input data for calculations of materials or fixed appendices for the structure under study. It may therefore be necessary to know the complete history of the motion of attachment points. This deals with determining the floor spectra (see the end of section 9.1.2), which can result from a transient calculation of the load-bearing structure (but which can also be obtained through other techniques, especially spectrum transfer, section 10.2.2). The use of transient calculation implies the choice of accelerograms to be used for the calculation. It is a subject that has provoked many discussions and which cannot be considered as settled at present, due to various aspects (natural or synthetic character of accelerograms, their number, coherence criteria in the seismotectonic context of the site, adjustment criteria for a spectrum, statistical independence criteria for the different components acting simultaneously) and its repercussions on the validity and cost of study. For applications in linear analyses, the choice of accelerograms is not very critical as the basic definition of seismic action is then naturally represented by an elastic response spectrum calculation. It is therefore sufficient to make sure that the elastic spectra of accelerograms used give a satisfactory approximation of the base spectrum, either taken separately (if the calculation is done for a single accelerogram) or together (if several calculations are made and the mean or the upper bound of the results is taken). The case of nonlinear analyses is much more difficult; the elastic response spectrum, even if often considered the definition of seismic action, is, as indicated in the introduction to Part 4, a poor indicator of the damage potential of an accelerogram. We shall see in section 10.1.3 that it is possible to obtain an increase in nonlinear response for a change in the accelerogram corresponding to a decrease in elastic spectrum. The type of nonlinearity, particularly from the point of view of the cumulative or non-cumulative nature of damage, is crucial to an appropriate choice of accelerogram. These sensitive issues are brought up in section 10.1.3.
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The much simpler case of transient calculations in the linear domain is discussed in the following section. 10.1.2. Choice of accelerograms for linear analysis The practice of using synthetic accelerograms adjusted so that their elastic spectrum for a certain damping value (generally 5% for ground spectra and 2% for floor spectra) is very close to a given elastic design spectrum has been largely used in the nuclear industry since the 1970s. An example of such an accelerogram is given in Figure 10.1. The determination of these compatible synthetic accelerograms with a given spectrum has necessitated the updating of specialized software programs, which generally proceed in an iterative manner, through progressive improvement of a set of parameters (for example, the amplitudes for a truncated Fourier series, whose periods are predetermined and phases are random) based on an initial choice often corresponding to a real accelerogram. We must understand that this adjustment is not a mathematically well established problem because, if a given accelerogram is associated in a unique way to a response spectrum, its reciprocity is not true. This is the reason why the adjustment can only be approximate and for a single reduced value for damping. In practice, for “normal” elastic design spectra, a fairly good approximation can be obtained, as in the case of Figure 10.1. Very precise codification of adjustment criteria have been defined by the USNRC (United States Nuclear Regulatory Commission). If spectrum adjustment can be considered satisfactory, the same cannot be said about the shape of accelerograms. The accelerogram in Figure 10.1 presents about 15 positive peaks and negative peaks whose amplitudes are close to maximum acceleration and are uniformly distributed inside a “strong section” of more than 15 seconds duration. The number of peaks of strong amplitude and the duration are too high for the type of seismicity corresponding to the “target” spectrum (moderate seismicity where the highest magnitudes are around 6–6.5). Such faults are typical of synthetic accelerograms adjusted for design spectra and are not surprising as these spectra do not correspond to a single earthquake but to an average (if possible matched by standard deviation) for a set of earthquakes (section 8.1.2). Spectra of real accelerograms are never smooth over a very large range of frequencies, but rather present a series of distinct peaks and troughs (see Figures 9.2 and 9.3).
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Seismic Engineering
Figure 10.1. Adjustment of a synthetic accelerogram so that its spectrum can reproduce the spectrum correctly at 5% damping for a project in a nuclear power station (above). The accelerogram (below) has a duration of 25 s and contains 16 positive peaks and 14 negative peaks whose moduli are higher than or equal to 80% of the maximum acceleration of 0.25 g
In spite of these defects, the use of synthetic accelerograms compatible with calculation spectra is generally considered acceptable for linear analysis, since it is after all the spectrum which constitutes the natural and appropriate measurement of the severity of seismic actions. The consequences of the defective nature of accelerograms are either negligible (when we are concerned essentially with maximum values of the response) or reasonably within the sense of security (when we are concerned with the complete history of the response) since there is an increase in the number of high level cycles. Instead of synthetic accelerograms, natural accelerograms can also be used for seismic calculations on linear models. It is then necessary to take several of them so that together their spectra correspond to the spectrum at the start. The calculation time is thus much higher than for a single synthetic accelerogram, but the results are
Other Representations of Seismic Action
437
more realistic from the point of view of the number of response cycles, which can be significant for some applications, for example for the evaluation of the effects of oligocyclic damage. One option which is almost never used due to its complexity is to consider the set of natural accelerograms serving as the basis for the definition of the calculation spectrum, to calculate the structure for each of them and apply the same statistic processing to the obtained results (mean or mean plus a fraction of standard deviation) as the one used for spectrum definition. This is the only perfectly rigorous procedure and can help clarify the problem of calculation margins in difficult cases. Even when synthetic accelerograms are used, it is normal to consider several (typically three) for linear analysis. This makes it possible to take action against any local defects of adjustment which may coincide with the frequency of an important mode for a structure under consideration, especially if different rates of damping are seen for the rate chosen for adjustment. It is also a necessity for two or three dimensional analyses where accelerograms are applied simultaneously on different axes. Even if the spectra have the same form in different directions, the accelerograms must be statistically independent (section 4.1.3) for all the components taken two by two; with three synthetic accelerograms fulfilling this condition of independence, six acceptable possibilities can be obtained by permutation for the calculation of a three dimensional structure. 10.1.3. Choice of accelerograms for nonlinear analysis As already mentioned repeatedly, the fact that the elastic response spectrum in design is not a good measurement of potential damage can be seen while studying the response of very simple nonlinear systems on synthetic accelerograms adjusted for the spectrum (such as that in Figure 10.1). Such a study was, for example, carried out by Radicchia, Mezzi and D’Ambrisi [RAD 92]; these authors have defined 30 synthetic accelerograms adjusted for the same spectrum and corresponding to the same duration (20 s). These 30 signals were applied to 24 oscillators with one degree of freedom having elastic-perfectly plastic behavior (as those considered in 9.3.1), where the plastic yield limit is a quarter of the maximum force calculated in the elastic domain (which remains the same whatever the accelerogram used, since they all have the same elastic spectrum); the periods of these oscillators varied from 0.2 s to 2.5 s with a time step of 0.1 s. Figures 10.2 and 10.3 summarize the results obtained.
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Seismic Engineering
Figure 10.2. Variation of displacement according to the period for three levels of probability (normal distribution); according to [RAD 92]
Figure 10.2 shows the variation of maximum displacement on the basis of the period for three levels of probability (16%, 50%, 84%) corresponding to a normal distribution. The displacement increases with the period and the deviation between the curves for 16% and 84% is about half of the ordinates of the curve 50%. Figure 10.3 gives the ratio for each period between the strongest response (obtained for one of the 30 accelerograms used) and the mean value (r,m,s or quadratic mean). It is observed that this ratio often exceeds the value 3, without any noticeable influence in the period. The dispersion is considerable, which shows that the calculation using a single adjusted accelerogram for the spectrum is more like gambling than reasonable engineering practice. In other words, whatever is admitted for linear analyses (calculation with a single synthetic accelerogram on the condition that it is well adjusted for the spectrum) can become absurd in the case of nonlinear analyses.
Other Representations of Seismic Action
439
Figure 10.3. Relation of maximum displacement to mean displacement (rms) for 24 oscillators under study; according to [RAD 92]
In order to approximate normal conditions of execution of transient calculations (with a fairly small number of accelerograms) Radicchia, Mezzi and d’Ambrisi then studied responses in groups of four accelerograms (or 27,405 possible combinations based on a set of 30) and eight accelerograms (5,852,925 combinations). The objective was to define amplification factors F, to be applied to the mean response calculated with four or eight accelerograms, for a given confidence coefficient Pc, aiming for a non-exceeding probability PND. Figure 10.4 gives these factors F in the case where four accelerograms are used (the results obtained with eight accelerograms are practically the same) for the values 0.50, 0.84 and 0.95 for levels of probability Pc, and PND. It is observed that F is significantly independent of the period and that, if PND the probability of non-exceeding of 0.84 is targeted with a good confidence coefficient (Pc 0.84), a safety margin of around 50% in relation to the mean calculated with only four accelerograms needs to be taken. This margin is reached or even exceeds 100% when higher probabilities (Pc PND 0.95) are considered.
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Seismic Engineering
Figure 10.4. Amplification factor F to be applied to the mean response calculated with four accelerograms when a non-exceeding probability PND is aimed at with confidence coefficient Pc = 0.50, 0.84 and 0.95; according to [RAD 92]
These results cast a doubt on the practice of nonlinear calculation with a very small number of synthetic accelerograms adjusted for a design spectrum. The reference to spectrum seems to provide a guarantee of safety and this practice is often admitted without question. We must not forget that the nuclear industry, which has largely contributed to the popularity of the use of synthetic accelerograms, has limited itself to essentially linear calculations (generation of floor spectra and calculation of soil-structure interaction; see Chapter 16).
Other Representations of Seismic Action
441
There is another reason not to trust synthetic accelerograms adjusted for a spectrum for nonlinear analysis; it concerns the often unrealistic and extremely misleading nature of nonlinear results obtained with these accelerograms, at least in cases where the damage mechanism has a marked cumulative character. Several recent studies have attracted attention to this point, for example, studies by Naeim and Lew [NAE 95]. These authors have defined 24 synthetic accelerograms adjusted for the same spectrum using two different methods for the generation of these signals. After having examined the problems of displacement drift (one of the accelerograms showed a drift of four meters after double integration in the absence of any specific precautions) and distribution of energy (which has a density too high and too uniform over the range of frequencies as compared with real accelerograms), they have calculated the response of a building with nonlinear behavior (due to frictionbearing systems in the foundation). In addition to the dispersion of results with the accelerogram used (comparable to those obtained in the study of Radicchia, Mezzi and d’Ambrisi), the calculations carried out show extreme conservatism resulting from the use of synthetic accelerograms; for example the displacement is divided by a factor of around 3 when the synthetic accelerogram is replaced with a real one scaled so that its spectrum at 5% covers the design spectrum in the range of periods 1–4 s which is significant for this low frequency structure. Figure 10.5 shows this comparison of spectra for the Taft recording (Kern County earthquake of 21 July 1952). With this real accelerogram set to scale (which corresponded to the multiplication of recorded values by 3.48) the displacement calculated is 22 cm (8.7 inches) while in the case of synthetic accelerograms adjusted to the design spectrum, it varies from 60 cm (23.7 inches) to 73 cm (28.9 inches). Similar results were obtained for other real accelerograms set to scale in the same way.
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Seismic Engineering
Figure 10.5. Scaling of the spectrum of the Taft recording so that it covers the design spectrum (according to [NAE 95])
These tendencies at first seemingly paradoxical (the hierarchy of transient results is inverse to that of the spectrum) are obviously not systematic and hold for the particularities of the structure studied by Naeim and Lew. Meanwhile, they constitute a salutary warning of the difficulties of transient nonlinear analyses, which are not limited to laws of behavior and their numeric simulation, but also have to do with the representation of action. At the risk of stating the obvious, it suffices to remember that the elastic spectrum is basically meant to serve as input data for elastic calculations and that it does not represent anything more. Other nonlinear studies using synthetic accelerograms adjusted for the spectrum have given less discouraging results, as much from the point of view of dispersion as conservatism. For example, it is the case of basemat uplift (temporary loss of contact between the base of a building and the soil in the foundation, on part of the contact surface) when calculation models do not take the irreversible deformations of the soil into account. This uplift can be produced, under the effect of horizontal accelerations of the soil, for buildings that are relatively slender when the stresses due to the overturning moment and weight cancel each other on one of the sides of the bearing. Figure 10.6 shows rocking without uplift (Part ) and rocking with uplift (Part ) on elastic ground. Part of the figure shows the effects of ground rupture, by punching instabilities under one corner of the base. These different behaviors will be commented upon in section 17.2.2. They are briefly presented here
Other Representations of Seismic Action
443
to draw attention to the importance of the type of nonlinearity in the choice of accelerograms for a transient calculation. Uplift without irreversible degradation of the ground (part of Figure 10.6) corresponds to a case of nonlinear elasticity, in which the discharge occurs along the same path as the charge. In these conditions there is no cumulative effect of charge cycles and the greatest uplift reached in the course of motion depends essentially on the highest acceleration peak around the rocking frequency of the construction. Two accelerograms having such a peak of the same amplitude will noticeably produce the same maximum value of uplift, even if they show significant differences otherwise. Accelerograms adjusted for the design spectrum will thus give similar results and these results will be comparable to those obtained with natural accelerograms presenting a similar peak in the same frequency zone. The choice of accelerograms is generally less critical for transient calculations with nonlinear elastic models.
Figure 10.6. Uplift of a building on elastic ground (parts and ) and on ground susceptible to rupture by punching (part )
The situation is quite different when the nonlinear model corresponds to a cumulative damage mechanism. The response then shows a more marked random character and becomes very sensitive to the number of cycles and the total duration of the signal.
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Seismic Engineering
These two effects (random character and influence of the number of cycles) are the consequence of the differences between the charge and discharge paths in the stress-deformation diagram. Figure 10.7 makes it possible to compare these paths in the case of the moment rotation curve associated with the uplift on elastic ground and of an elastoplastic system of the type considered in section 9.3.1.
Figure 10.7. Comparison between nonlinear elastic behavior (moment rotation curve for the uplift on elastic ground, part and cumulative behavior cycles of elastic-perfectly plastic behavior, part )
In the first case, the stress (moment)-deformation (rotation) relation is bijective; the state of the system is thus highly constrained by this relation and the maximum response is not very sensitive to the number of excitation peaks and their temporal links. In the second case, on the contrary, the state of the system sweeps the band situated between the positive (Fu) and negative (– Fu) plastic yield limit under the effect of fluctuations of excitation; if the number of cycles is high, the minimal phase shifts in the changes in signs for the rate of deformation can result in very significant differences in the positions reached at the end of the movement. These positions thus have a certain random character (whence the dispersion observed in the study of Radicchia, Mezzi and d’Ambrisi with accelerograms supposedly “equivalent”, since they are adjusted for the same spectrum) but must, in statistical terms, be as far away from the initial rest position as the number of cycles is large (from which comes the excessive conservatism as compared to natural accelerograms in the study by Naeim and Lew), as we shall see in the generalization of Brownian motion presented in section 10.2.1. It follows from these considerations that the use of synthetic accelerograms adjusted for the spectrum in nonlinear calculations must be limited to the case of non-cumulative models. In the case of cumulative models, natural accelerograms or possibly synthetic accelerograms which show “good tendencies” must be used
Other Representations of Seismic Action
445
(which implies that they are not adjusted for all of the spectrum). The number of accelerograms to be used and the criteria that they must verify are issues which do not currently have definite answers. We can imagine that a limited number (about 5 to 10) of accelerograms that correspond well to characteristics of regional seismicity should be adequate in the majority of cases. The criteria to be applied should correspond to some adjustment to the elastic spectrum (for example of the average spectra associated with the chosen accelerograms) and also, similarly, to conditions taking into consideration the duration of the strong part (like Arias intensity (see equation [4.3]) or the cumulative CAV of the absolute value of velocity (see [4.4]). 10.2. Random processes The representation of seismic action by random processes has been the subject of numerous research projects but has not really penetrated current calculation practices. Many earthquake engineering presentations do not even mention it. There are two main reasons for this “exclusion”: – mathematical formalism of random processes is not normally part of the “baggage” of structural engineers; earthquake engineering codes in general are supposed to be accessible in a way that they can be applied by “ordinary” practitioners of civil or mechanical engineering; – habits acquired in the matter of seismic calculations as much for special as for normal risk, favor a deterministic approach, which is in line with that of most building codes for structures in reinforced concrete or steel, even if they claim to be “semi-probabilistic”. Probabilistic methods are allowed for the determination of response spectra for calculation, but the use of these spectra to determine designaction effects is perfectly deterministic in appearance. The representation of seismic action through random processes would imply questioning these practices. In reality, the calculation rules for earthquake engineering codes can only be justified for certain aspects (combination of modal responses, combination of different components of excitation, influence of damping) while considering the simulation of movements through simple random processes (white noise). This section is thus devoted to the presentation of consequences of a white noise simulation (unfiltered in section 10.2.1, filtered in section 10.2.2) of seismic ground motion on the response of a simple linear oscillator. The basic tool for this presentation is the theorem of “generalized Brownian motion”. A basic demonstration (i.e., not drawing from knowledge of the theory of random processes) of this is given in section 10.2.3. The consequences of such a simulation on the response of a system with many degrees of freedom (rules of combination of modal responses) will be described in section 15.2.2. The use of general methods of
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Seismic Engineering
stochastical dynamics, on the basis of the characterization of the signal by its power spectral density, will not be discussed. 10.2.1. Unfiltered white noise We consider (Figure 10.8) a signal Jb (t) obtained by taking a constant value of acceleration for each time interval 't determined by random selection following a probability distribution with zero mean and variance VJ², possibly variable on the basis of time t.
Figure 10.8. Model of unfiltered white noise
Such a signal, in the case where the variance VJ² is constant, corresponds to what we call unfiltered white noise in the theory of random vibrations; it is a very rudimentary model of seismic signals, which suffices to realize certain properties of response spectra and to justify the rules of quadratic combination (as with the SRSS rule used in section 9.2.5). The law of probability chosen for random selection and their number N (thus the total duration N't of the signal) determine the probable distribution of the acceleration peak reached during the realization of this process. It is assumed that the law of random selection is the normal distribution of zero mean, whose distribution function )(x) is defined by: )(x) =
1 2S
³
x
f
e
u ² / 2
du
[10.1]
Other Representations of Seismic Action
447
)(x) is the probability of non-exceeding the value x VJ, VJ being the standard deviation (square root of the variance). Table 10.1 gives values ) for x varying from 0 to 4.5. x
) (x)
x
) (x)
x
) (x)
0.0
0.500000
1.6
0.945201
3.1
0.999032
0.1
0.539828
1.7
0.955435
3.2
0.999313
0.2
0.579260
1.8
0.964070
3.3
0.999517
0.3
0.617911
1.9
0.971283
3.4
0.999663
0.4
0.655422
2.0
0.977250
3.5
0.999767
0.5
0.691462
2.1
0.982136
3.6
0.999841
0.6
0.725747
2.2
0.986097
3.7
0.999892
0.7
0.758036
2.3
0.989276
3.8
0.999928
0.8
0.788145
2.4
0.991802
3.9
0.999952
0.9
0.815940
2.5
0.993790
4.0
0.999968
1.0
0.841345
2.6
0.995339
4.1
0.999979
1.1
0.864334
2.7
0.996533
4.2
0.999987
1.2
0.884930
2.8
0.997445
4.3
0.999991
1.3
0.903200
2.9
0.998134
4.4
0.999995
1.4
0.919243
3.0
0.998650
4.5
0.999997
1.5
0.933193
Table 10.1. Values for normal distribution function ) (x) for x varying from 0 to 4.5
1 – ) (x) thus represents the probability of exceeding of the value xVJ and also, due to symmetry, the probability of having negative values less than – xVJ. The probability Px of exceeding xVJ by the absolute acceleration value is thus, for random choice: Px
2 [I – ) (x)]
[10.2]
In order that this value xVJ corresponds to the probability of non-exceeding PA during N independent random choices, we must have: (I – Px)N
I – PA
[10.3]
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Seismic Engineering
which, considering [10.2], implies: 1 [1 + (1 – PA)I/N] 2
) (x)
[10.4]
Table [10.2] gives values of ) (x), for PA 0.50, 0.10 or 0.01 and N 100, 200, 500 or 1,000, calculated by relation [10.4] and corresponding values of x, deduced from Table 10.1.
PA = 0.50 PA = 0.10 PA = 0.01
N = 100
N = 200
N = 500
N = 1000
) (x)
0.996546
0.998270
0.999307
0.999654
x
2.7
2.9
3.2
3.4
) (x)
0.999473
0.999737
0.999895
0.999947
x
3.3
3.5
3.7
3.9
) (x)
0.999950
0.999975
0.999990
0.999995
x
3.9
4.1
4.3
4.4
Table 10.2. Values of x and ) (x) for a certain number of values for N and PA
We see that for the values of N in the order of a few hundreds, the acceleration peak is about three or four times the standard deviation, with a weak influence of the targeted confidence level (i.e. of parameter PA). Velocity v (t) and displacement d (t) associated with the accelerogram Jb (t) of Figure 10.8 are the following: t
v (t) =
³J
d (t) =
³ (t W )J
0
b
(W )dW
t
0
b
[10.5] (W )dW
[10.6]
These integrals are particular cases of the relation: X (t) =
³
t
0
f (t W )J b(W )dW
[10.7]
When function f is “slowly variable”, i.e. quasi-constant for a variation interval 't of W, we can replace the integral for the second member of [10.7] with the sum:
Other Representations of Seismic Action
X (t) = 't
n
¦fJ k
449
[10.8]
b, k
k 1
t ; fk = f (t – k't); Jb, k = value randomly chosen for acceleration during 't the kth interval of time 't.
with: n =
We see that X (t) follows a “general Brownian motion” type of random process, i.e. a succession of random jumps to the right or to the left from an initial position that coincides with the origin. The amplitude of each jump is the product of the determininistic factor 't fk (that depends on index k) by the result of random choice for Jb, in the kth interval; the variance in amplitude of the jumps is thus equal to 't² f²k VJ². In classic Brownian motion, the jumps have constant amplitude and their random nature concerns only their direction (to the right or to the left). It can be seen in these conditions that if we write: F (t) = 't
t
³ V J (W ) f ²(t W )dW 2
[10.9]
0
the random variable X(t)/ F (t ) follows normal distribution defined by the function of distribution [10.1]. The demonstration of this “theorem of general Brownian motion” is given in section 10.2.3; where we establish the following relations in particular: probability {|X(t)|, 0d t d T, < 1.15
F (T ) } = 0.50
[10.10]
probability {|X(t)|, 0d t d T, < 1.96
F (T ) } = 0.90
[10.11]
probability {|X(t)|, 0d t d T, < 2.81
F (T ) } = 0.99
[10.12]
We can thus write that the maximum Xm of the modulus of X (t), for t varying from 0 to T, is given by: Xm = gp
F (T )
[10.13]
gp being a coefficient we call peak factor, which is all the larger as the targeted probability of non-exceeding the value Xm is low (gp = 1.15 for 50% chances of non-exceeding, gp = 1.96 for 10% and gp = 2.81 for 1%).
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Seismic Engineering
We can apply these results to equations [10.5] and [10.6] for velocity and displacement; the hypothesis of slow variation of function f is evidently verified for [10.5] (where f = 1) and can also be admitted for [10.6] (where f = t-W) if the interval of time 't is short with respect to t. For a constant variance, VJ² for random choice of accelerations, we thus find, according to [10.9] and [10.13]: Vm = gp VJ 't
N
[10.14]
Dm = gp VJ 't2
N3 3
[10.15]
Vm and Dm being the respective maximum and minimum values of velocity and displacement and N being the total number of time intervals (the total duration is thus N't). We consider the values VJ = 1m/s², 't = 0.01s and N = 500 (i.e., a duration N't of 5 s). For a probability of non-exceeding of 50%, we find a peak acceleration of 3.2 m/s² (case PA = 0.50, N = 500 of Table 10.2) and, with gp = 1.15 [10.10] a peak velocity Vm of 0.257 m/s and peak displacement Dm of 0.742 m. The value found for Vm is coherent with that for acceleration (see Table 4.2) while the displacement is a little too great. If we take a non-exceeding probability of 10% (PA = 0.10 in Table 10.2, gp = 1.96 according to [10.11]) we find 3.70 m/s² for acceleration, 0.438 m/s for velocity and 1.265 m for displacement, and thus, once again, there is coherence between velocity and acceleration and displacement is too high. These orders of magnitude show that the very simple model of unfiltered white noise is reasonably comparable to real accelerograms from the point of view of peak motion values. It remains to extend the comparison to elastic response spectra. To do so, we use relation [9.12] by replacing the accelerogram with white noise Jb: rn (t) =
Zn Zc
t
³J 0
b
(W )e [Z ( t W ) sin[Z c(t W ) nT ]dW
[10.16]
We remember that the formula gives for n = 0 relative displacement, for n = 1 relative velocity and for n = 2 absolute acceleration of a simple oscillator with angular frequency Z and reduced damping [. Parameters Z’ and T which also appear in [10.16] can thus be expressed:
Other Representations of Seismic Action
Z’ = Z 1 [ ² ; T =
S 2
+ arc sin [
451
[10.17]
Relation [10.16] is of the form [10.7] with the following expression for function f: f (t-W) = –
Z n [Z (t W ) e sin[Z c(t W ) nT ] Zc
[10.18]
In order that this function might be considered to be slowly variable at the scale of 't, we must limit ourselves to relatively low angular frequencies (i.e., such that Z't is not too high, lower than 0.5 to take an example). We then have, for maximum Rn of the modulus of rn (t) at interval (O,T), according to relations [10.13], [10.9] and [10.18].
Z 2n Z c2
Rn = gp ['t V²J
³
T
0
e
2 [Z ( T W )
sin ²[Z '(T W ) nT ]dW ]1/ 2
[10.19]
The calculation of the integral is basic and we find: R²n = g²p V²J 't
2[ZT 1 Z 2 n 3 ª 1 º cos(2n 1)T e ( cos[2Z ' T (2n 1)T ]) » « [ 4(1 [ ²) ¬ [ ¼
For reasonably large T, we can ignore the term containing the exponential and we find, considering definition [10.17] of T: R0 = gp
R1 = gp
R2 = gp
VJ 't 2Z [Z VJ
't
2
[Z
VJ
Z't (1 4[ ² ) [
2
[10.20]
[10.21]
[10.22]
R0 is the displacement response spectrum, R1 = ZR0 the spectrum in pseudovelocity; R2 is the spectrum in absolute acceleration that is equal to the pseudo-
452
Seismic Engineering
acceleration spectrum Z 2 R0 multiplied by the factor 1 4[ ² ; since [ is generally very small compared to 1, factor
1 4[ ² is close to 1 and we find that pseudo-
acceleration is almost equal to the maximum absolute acceleration (section 9.1.1.). Equations [10.20] to [10.22] show that response spectra are inversely proportional to the square roots of the damping, independent of the frequency. This was discussed in section 9.1.1 (see equation [9.21]) and can be verified on real accelerogram spectra. Formula [10.22] provides an acceleration spectrum proportional to Z , which does not reproduce the asypmtotic relation to the maximum acceleration of support when Z increases infinitely (see section 9.1.1); this is due to the fact that the slow variation hypothesis of function f [10.18] is no longer verified for such values of Z. In conclusion, we can say that the unfiltered white noise model constitutes an acceptable representation of certain aspects of real seismic movements (peak values of acceleration, velocity, and displacement, layering of different damping curves for response spectra) but that it also presents certain obvious faults, which is not surprising given its rudimentary nature. Various improvements have been proposed, especially to account for the non-stationary character of real seismic signals. In the following section, we limit ourselves to the improvement resulting from filtering stationary white noise. 10.2.2. Filtered white noise
A better simulation of seismic movement is obtained by taking for the accelerogram J (t), the response in absolute acceleration of a simple oscillator with angular frequency Z0 and damping [0 to excitation of unfiltered white noise Jb(t), i.e., according to [10.16]: J(t) = –
with Z01
Z02 Z0'
t
³J 0
b
(W )e
Z0 1 [ 02 ; T0 =
[oZo (t W )
S 2
sin[Z0' (t W ) 2T 0 ]dW
Arc sin [ o
[10.23]
[10.24]
This white noise, filtered by the oscillator (Z0 [0) was introduced to earthquake engineering by Kanaï and Tajimi ([KAN 67] and [TAJ 60]) who proposed the following values to represent ground motion:
Other Representations of Seismic Action
Z0 = 15.7 rd/s (frequency of 2.5 Hz); [0 = 0.60
453
[10.25]
Thus, this involves a greatly damped oscillator, for which Z0c = 0.8 Z0 and T0 = 2.214 rd. Response rn (t) (n = 0 relative displacement, n = 1 relative velocity, n = 2 absolute acceleration) of a simple oscillator with angular frequency Z and with damping [ is given by the following formula obtained by replacing Jb (W) with J (W) in [10.16] defined by [10.23]: n
Z 2Z rn (t) = 0' ' Zo Z
³
t
o
e
[Z ( t W )
sin >Z '(t W ) nT @
W [oZo (W W c) sin ª¬Zo' (W W c) 2T o º¼ dW cº dW x ª³ J b W 1 e ¬« 0 ¼»
This expression is in fact the double integral: ([Z [0Z0 )W [0Z0W ' Z02 Z [Zt J (W ')e ' ' e ³ ³ D Z0Z n
rn (t)=
b
x sin [Zƍ(t-W) + n T] sin [Zƍ0 (W-W1) + 2T0]dW dWƍ
[10.26]
extended to the triangular domain D, defined by 0 d W d t, 0 d W’d W; it can also be written as: n
Z 2Z e rn (t) = 0 Z '0 Z '
[Zt
³
t
0
J (W ')e
[0Z0W '
b
[10.27] ª x «³ e ¬ W' t
([Z [0Z0 )W
º sin[Z '(t W ) nT ]sin[Z '0 (W W ') 2T o ]dW » dW c ¼
This is the same as form [10.7] with a somewhat complex expression of function f; which will not be explained here, but which is determined by the simple integration of products between exponential and trigonometric functions. Formula [10.13] thus gives for the maximum Am of acceleration (n = 2):
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Seismic Engineering 1/ 2
ª º V J2Z0 't r ([ 0 r[ ) » A m = gp « 2 « 4[[ 0 (1 [ 02 )(1 [ 2 ) » 2 1 2r[ 1 [ 0 1 [ [ 0[ ] r ² »¼ «¬
[10.28]
Where r stands for ratio Z/Z0; equation [10.28] was established with the hypothesis that [ (but not necessarily [0) is low with regard to 1, which enables some simplifications. Limit A0 of Am when r increases infinitely is given by: 1/ 2
ª º V J2Z0 't A 0 = gp « 2 2 » ¬« 4[ 0 (1 [ 0 )(1 [ ) ¼»
[10.29]
A0 does not represent the maximum value of the support acceleration, because the hypothesis of the slow variation of function f is no longer verified when r is taken to infinity. It is simply a reference value that makes it possible to present results without dimensions (quotient Am/Ao in Figure 10.9). This figure shows the variation curves of Am/A0 on the basis of r = Z/Z0 calculated by [10.28] in the two following cases: – [0 = 0.6; [ = 0.05 which corresponds to an oscillator with 5% damping excited by Kanaï-Tajimi filtered white noise; – [0 = 0.05; [ = 0.05 which corresponds to an oscillator with 5% damping excited by the movement of a supporting structure, which is itself damped at 5% and excited at its base by unfiltered white noise; this is a schematization of the problem of floor spectra mentioned at the end of section 8.1.2.
Other Representations of Seismic Action
455
Figure 10.9. Response spectra of oscillators with 5% damping excited by filtered white noise; the case [q = 0.60 corresponds to the Kanaï-Tajimi spectrum; [q = 0.05 corresponds to floor spectra
We see that the Kanaï-Tajimi spectrum shows a gradual slope that can be favorably compared to elastic design spectra shown in section 9.1.2 (even when these are schematized, for reasons of convenience for introduction of data, by angular lines). We can show [LAB 90] that in a mean plus standard deviation approach it is close to the USNRC spectrum (Figure 9.5) over a large range of frequencies, by taking the value in [10.25] recommended by Kanaï and Tajimi, for Zo (Zo = 15.7 rd/s, i.e. a frequency of 2.5 Hz). The maximum amplification of the Kanaï-Tajimi spectrum corresponds to the value 5.30 of the ratio Am/Ao, which would be too high a value (for damping [ of 5%) if Ao were to represent the maximum ground acceleration, but we have already indicated above that this was not the case. As for higher frequencies, the use of this
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Seismic Engineering
spectrum must thus be limited to values of r = Z / Zo lower than a certain limit (about 5 to 10, for example). The curve corresponding to the floor spectrum ([q = 0.05) confirms the notes made in 8.1.2; we can no longer call it a hill, as for ground spectra, because we can observe a narrow peak around r = 1 whose amplification is considerable (Am/Ao = 14.2). From [10.28] and [10.29] we can easily show that this amplification peak is, for [ and [o which are small with respect to one, given by the expression. § Am · ¨ ¸ r=1 = © A0 ¹
1
[ ([ 0 [ )
[10.30]
This is coherent with formula [9.24] (where [I plays the role of [o, i.e., the damping of the supporting structure). The differences in coefficients between the two formulae follow conventions on damping values (expressed in percentages in formula [8.24] and the fact that Ao in [10.30] does not represent the support acceleration, as As does in formula [9.24]. The filtered white noise model thus definitely marks a net progress in terms of validity as compared to the unfiltered white noise model. Its use however, remains limited because, as mentioned earlier, methods of stochastic dynamics have not yet entered current practices in earthquake engineering. Rules of quadratic combination of modal responses, in particular, continue to be based on the simpler, unfiltered white noise model (section 15.2.2). 10.2.3. Theorem of general Brownian motion
General Brownian motion that we can associate with integrals of form [10.7], when function f is slowly variable on a scale of time interval 't, has already been described in section 10.2.1. It corresponds to the sum X (t) in form [10.8]: x (t) = 't
n
¦
fk Jb,k
[10.31]
k 1
where each term contains a deterministic factor 't, fk and one random factor Jb,k which is the value of the randomly chosen acceleration for the kth interval.
V J2 being the variance of acceleration distribution, the variance for jumps in Brownian motion are given by: S² = 't² f²k V J2
[10.32]
Other Representations of Seismic Action
457
The presence of factor f²k in [10.32] shows that the variance of jumps is variable in time; we can also, in any case, admit that VJ² is itself variable in time. To find the properties of X (t), we reason, to begin with, in the discrete case; the law of probability of jumps is defined by the data of I + 1 (I any given integer) simple probabilities pi that represent the probabilities of amplitude jumps i ' x to the right or to the left; po is the probability of a zero amplitude jump (“on the spot” jump) and p1 the probability of jumps ± 'x, p2 the probability of jumps ± 2 ' x, etc.
Figure 10.10. Passage of time n ' t to the time (n+1) 't for general Brownian motion with I = 2
This law of probability is evidently, as it must be, with zero mean; its variance S² is given by: S² = 2 ' x²
I
¦
i² pi
[10.33]
i I
The pi values may depend on the number n of the jump, but must always verify the relation: po + 2
I
¦
pi = I
[10.34]
i I
We indicate by Wm, n the probability that at time n ' t the abscissa of the particle must be lesser than or equal to m ' x. It is easy to establish a recurrence relation between Wm, n; this will be demonstrated to simplify in the case of I = 2 (Figure 10.10); the generalization at any given value I is immediate.
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Seismic Engineering
Wm, n+1 is, by definition, the probability that at time (n+1) 't, the particle shall be at point P (abscissa m ' x) or to the left of this point; and in order that this be so, it is necessary and sufficient that at the preceding time n ' t, the particle must be: – either in A (abscissa (m – 2) 'x) or to the left of A, because then the particle cannot go beyond P to the right during the jump between n ' t and (n+1) 't (because we have taken I = 2); this situation corresponds to probability Wm-2,n; – or in B (abscissa (m-1) 'x), which corresponds to probability Wm-1,n – Wm-2,n, and that the jump at + 2'x does not occur (probability I – p2); – or in C (abscissa m'x) which corresponds to probability Wm, n – Wm-1, n, and that the jumps at + 2'x and + 'x do not occur (probability I – P2 – P1); – or in D (abscissa (m + 1)'x), which corresponds to probability Wm+1, n –Wm,n, and that the jumps at + 2'x, + 'x, and the jumps on the same spot do not occur (probability 1 – p2 – p1 – po); – or in E (abscissa (m + 2)'x), which corresponds to probability Wm+2, n – Wm+1, n and that the jumps at + 2'x, + 'x, the jump on the spot and the jump at – 'x do not occur (probability I – p2, – p1, – po – p1 = p2). Thus, we have, for Wm, n+1: Wm, n+1 =
Wm-2,n + (I – p2) (Wm-1, n – Wm-2, n) + (I – p2 – p1) (Wm,n – Wm-1,n) + (I – p2 – p1 – po) (Wm+1,n – Wm,n) + (I – p2 – p1 – po – p1) (Wm+2,n – Wm+1,n)
or, by developing further: Wm,n+I = po Wm,n + p1 (Wm-I,n + Wm+I,n) + p2 (Wm-2,n + Wm+2,n)
[10.35]
It is clear that for any I, we shall find the same: Wm, n + I = po Wm, n +
I
¦
pi (Wm-i, n + Wm + i, n)
[10.36]
i I
Now let us go from the discrete to the continuous by supposing that the amplitude of jumps and the time steps 't are small enough to consider that Wm, n are point values of a continuous and derivable function W(x, t) which, close to point x = m'x, t = n't can be replaced by its development in a Taylor series limited to the second order; we can therefore write:
Other Representations of Seismic Action
459
Wm, n + I = W (x, t) + 't
wW 't 2 w 2W ( x, t ) ( x1t ) 2 wt ² wt
[10.37]
Wm-i, n = W (x, t) – i'x
wW 'x 2 w 2W ( x, t ) i ² ( x1t ) 2 wx 2 wx
[10.38]
Wm+i, n = W (x, t) + i'x
wW 'x 2 w 2W ( x, t ) i ² ( x1t ) 2 wx 2 wx
[10.39]
from which, by referring to the recurrence relation [10.36]: wW 't ² w ²W . 2 wt ² wt
W + 't
I ª º ª I º w ²W Po 2 Pi W ¦ « » « ¦ i ²'x ² pi » wx ² i I ¬ ¼ ¬i I ¼
[10.40]
According to [10.34] the W coefficient of the second member is equal to 1; the terms in W are thus eliminated and, by dividing by 't we have: wW wt
't w ²W 2 wt 2
'x 2 ª I 2 º w 2W ¦ i pi »¼ wx 2 't «¬ i I
[10.41]
and, by making 't and 'x tend towards 0 by retaining a constant value of the ratio 'x²/'t: wW wt
'x 2 ª I º w 2W i ² pi » ¦ « 2 't ¬ i I ¼ wx
[10.42]
Taking [10.33] and [10.32] into consideration, this equation can be written as:
wW wt
1 w ²W 'tV J2 f ² 2 wx ²
The coefficient of
[10.43]
w ²W of the second member contains two factors which are wx ²
functions of present time t: the variance VJ² of randomly chosen acceleration and the square of function f; as f depends upon the argument T – t (T is the total duration of the signal) we can rewrite [10.43] as follows:
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Seismic Engineering
wW wt
1 w ²W F '(t ) 2 wx ²
[10.44]
F’(t) being from the derivative of function F (t) defined by: F (t) = 't
t
³ V J (W ) f ²(T W )dW 2
0
[10.45]
The following initial and boundary conditions are associated with the partial derivative equation [10.44]: W = 1 for x ! 0 for t = 0
[10.46]
W = 0 for x 0 for x o + , W = 1
[10.47]
for x o -, W = 0
[10.48]
which show that the initial position of the particle is the origin x = 0 and that W represents a probability and thus is necessarily between 0 and 1. With these conditions, equation [10.44] is resolved by changing variable: [ = x [F(t)] -½
[10.49]
F (t) being the function defined by [10.45]; from [10.49] we deduce: w[ wx
[ F (t )] -1/2;
w[ wt
w ²[ wx ²
0
[10.50]
x 3 / 2 F (t ) @ F '(t ) > 2
[10.51]
A function of the single variable [ is taken for W and thus, for its partial derivatives with relation to x and t, considering [10.50] and [10.51] we have:
Other Representations of Seismic Action
wW ([ ) wx
dW w[ d [ wx
> F (t )@
w ²W ([ ) wx ²
> F (t )@
d ²W w[ d [ ² wx
1/ 2
dW w[ d [ wt
wW wt
1/ 2
461
dW d[
> F (t )@
1
d ²W d[ ²
x dW 3 / 2 F (t ) @ F '(t ) > 2 d[
[
> F (t )@ 2
1
F '(t )
dW d[
Hence, by bringing forward into [10.44]: d ²W dW [ d[ ² d[
[10.52]
0
The integration is immediate and, A and B being any two constants, gives: W([) = A
³
[
f
u2 / 2
e
du B
[10.53]
The initial and boundary conditions [10.46] to [10.48] are satisfied by taking: A=
1 2S
;B=O
[10.54]
In fact, considering the relation: 1 2S
³
f
f
e u ² / 2 du = 1
[10.55]
and by the fact that for t tending towards 0, [ tends towards + f or – f according to whether x is positive or negative (because F(t) tends to 0 when t tends to 0), we see that: W ([) =
1 2S
³
[
f
eu ² / 2
[10.56]
verifies conditions [10.46] to [10.48] very well. W ([) is thus identical to the normal distribution function defined by [10.1] and we have demonstrated the theorem of general Brownian motion according to which
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Seismic Engineering
the random value X(t) / F /(t ) , F(t) being defined by [10.45], follows normal distribution. We notice that this result does not depend upon the form of the law of probability chosen for random accelerations, which must simply fulfill the condition of the zero mean. In the remainder of this section, we shall use the usual notation ) (instead of W) to denote the function of normal distribution. We immediately deduce the following results from the properties of this function: probability { X(t) ! w
F (t ) } = 2 [1 – ) (w)]
[10.57]
(this relation is equivalent to [10.2]). probability { X(t) w
F (t ) } = I – 2 [I – ) (w)] = 2 ) (w) – I
[10.58]
The most interesting probabilities to consider are those related to extreme values of X(t) for t varying from 0 to T; by denoting by X+ and X- respectively the maximum and minimum values of X(t) during this time, we have: probability {X+ ! w probability {X- – w
F (T ) } = 2 [1 – ) (w)] F (T ) } = 2 [1 – ) (w)]
[10.59] [10.60]
These results can be demonstrated by the following reasoning: we consider the entire group of trajectories that reach a given value xO (positive) of the abscissa between the time 0 and the time T at least once; this group contains all the trajectories that end to the right of xO, because, due to the continuity of movement from the initial position x = 0, they have necessarily passed at least once in xO. This group also contains trajectories that, after reaching xO, end to the left of xO at time T; the two types of trajectories are equally probable because in starting from xO, the particle has “forgotten” the earlier history of its motion (mutual independence of jumps) and has equal chances of finishing at time T to the right as to the left of xO (symmetry of the law of probability of jumps); thus we have: probability {X(T) t x0} =
1 probability {reaching x0 for o d t d T} 2
Other Representations of Seismic Action
463
However, the probability that the particle reaches x0 between t = 0 and t = T is equal to the probability that X+ is at least equal to xO, which is expressed by: – probability {X + ! w
F (T ) } = 2 probability { x (T) > w
– probability {x (T) > w
F (T ) };
F (T ) } = 2 [1-) (w)].
Relation [10.59] is thus established; the right-left symmetry brings the verification of [10.60]. We now consider the probabilities P1 = probability {X- t – xo and X+ d xo}
[10.61]
P2 = probability {X- d – xo and X+ t xo}
[10.62]
P3 = probability {X- t – xo and X+ t xo}
[10.63]
P4 = probability {X- d – xo and X+ d xo}
[10. 64]
They verify the relations: P1 + P2 + P3 + P4 = 1
[10.65]
P3 = P4
[10.66]
P1 + P4 = 2) (xo /
F (T ) ) – 1
[10.67]
[10.65] results from the fact that the four types of trajectories that enable the definition of P1, P2, P3 and P4 cover all possible cases and are disjoint sets; [10.66] is an immediate consequence of the right left symmetry of the law of probability of jumps. As for [10.67], it is deduced simply from [10.59], because: P1 + P4 = probability {X+ d xO} = I – probability {X+ t xo} = 1 – 2 [1-) (x0/ F (T ) ] = 2 ) (xo /
F (T ) ) – 1
From these three relations, by eliminating P3 and P4 we obtain: P1 = 4 ) (xo /
F (T ) ) – 3 + P2
[10.68]
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Seismic Engineering
P2 being positive, we have for P1 the lower bound: P1 ! 4 ) (xo / xo/
F (T ) ) – 3
[10.69]
Lower bound of P1 [9.69]
Upper bound of P1 [9.71]
1.0
0.3654
0.3798
1.1
0.4573
0.4649
1.2
0.5397
0.5435
1.3
0.6128
0.6146
1.4
0.6770
0.6778
1.5
0.7328
0.7331
1.6
0.7808
0.7810
1.7
0.8217
0.8218
1.8
0.8563
0.8563
1.9
0.8851
0.8851
2.0
0.9090
0.9090
2.1
0.9285
0.9285
2.2
0.9444
0.9444
2.3
0.9571
0.9571
2.4
0.9672
0.9672
2.5
0.9752
0.9752
2.6
0.9814
0.9814
2.7
0.9861
0.9861
2.8
0.9898
0.9898
2.9
0.9925
0.9925
3.0
0.9946
0.9946
F (T )
Table 10.3. Bounds of probability P1 that X t is lower than xo during the entire interval 0 t T
An upper bound can be obtained by observing that P2 is less than the product of the probability that X+ is at least equal to xo by the probability that X- is less than –2xo, because the trajectories that reach xo at a certain time between O and T have fewer chances of ending up to the left of – xo at time T than those which would leave
Other Representations of Seismic Action
465
from xo at time t = o; these latter trajectories follow the same laws of probability as the trajectories that leave from the origin at time t = 0 and such that X- d – 2xo. Thus, according to [10.59] and [10.60], we have: P2 2 [1 – ) (xo/ F (T ) )] x 2 [1 – ) (2xo/ F (T ) ]
[10.70]
from which, for P1 the upper bound, according to [10.68]: P1 1 – 4 ) (2xo
F (T ) ) [1 – ) (xo/ F (T ) ]
[10.71]
Bounds [10.69] and [10.71] obtained for P1 are sufficiently close to make the calculations practical, as shown in Table 10.3, deduced from Table 10.1 of numerical values of function ). We see that the bounds are equal (to the fifth decimal place) from xo/ F (T ) greater than or equal to 1.8; the error committed in taking the arithmetic mean of the bounds is always less than 1%, with the only exception being the first line of the Table (xo / F (T ) =1). Table 10.3 enables, by interpolation, the peak factors from equations [10.10], [10.11] and [10.12] to be determined, i.e.: – gp = 1.15 for a 50% probability of non-exceeding; – gp = 1.96 for a 90% probability of non-exceeding; – gp = 2.81 for a 99% probability of non-exceeding.
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Part 5
The Effects of Earthquakes on Buildings
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Introduction
Destruction caused by earthquakes and the resulting loss of lives have always shocked the senses. The suddenness, unpredictability and amplitude of seismic catastrophes have often led to fatalistic attitudes of the populations concerned and religious “atonement” (see section 11.1.1). However, we also know of examples of enhancement of construction techniques in certain regions, aiming to ensure better resistance in buildings to seismic shocks. N.N. Ambraseys thus refers [AMB 76] to special precautions taken during the reconstruction of the city of Taxila (Northern Pakistan) after the earthquake in 25 AD: deepening of the foundations, increasing the thickness of walls. He also mentions modifications described in Anatolia and Syria in the Byzantine era: reduction of the height of houses and reinforcement of walls with wooden frameworks [AMB 76]. It seems to prove that these changes in modes of construction resulted from the analysis of the effects of destructive earthquakes. In other cases, the link between the choice of a particular technique and the concern for earthquake resistance has not been established. Hu Shiping concludes his study [HU 91] on palaces, temples and pagodas in Northern China with the affirmation that the resistant behavior of these constructions during earthquakes results from the choice of good construction methods, but that this choice was not dependent upon the consideration of seismic risk. In the same way, the hypothesis which is sometimes presented according to which the peculiar structure of Inca monuments (wall faces formed of irregular blocks arranged with great care) corresponded to the objective of being resistant to earthquakes remains the domain of conjecture. Description and research of causes of damage brought about due to earthquakes have thus had a definite influence on the art of construction in some regions exposed to seismic risk. Even today, observations deduced from past experience continue to
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Seismic Engineering
be the very foundation of earthquake-resistant engineering. We cannot imagine a time where operations of an all-powerful computer will render unnecessary the often criticized but indispensable role of “the engineer’s judgment” on the importance of experimental aspects in the prevention of risks linked to earthquakes. Purely “computational” approaches, which have known great success in many fields of engineering, are not sufficient in the current state of earthquake engineering, due to significant uncertainties which affect not only the characteristics of seismic movement (see Part 2) but also mechanisms for the appearance and development of structural damage in transient and often three-dimensional situations. The calculations prescribed by earthquake engineering codes, which must necessarily remain simple enough to enable codification, are generally only “structural dimensioning calculations” making it possible to verify that a certain level of safety has been achieved. However, in most cases they are incapable of predicting the real mode of destruction in cases where seismic action significantly exceeds the statutory level. The study of seismic effects on constructions, the subject of Part 5, is thus a basic element for anyone who wishes to arrive at an adult appreciation of antiseismic prevention. The study has many practical difficulties: – detailed observation of damage is possible only for visible or observable parts of the structures in question. We must forego it for hidden parts (foundations) and parts where access is prohibited due to the risk of partial or total collapse in the event of a strong aftershock; – cases where there is complete destruction of the construction do not often give very much information on the chain of events leading to total collapse. We must contend with biased interpretations which depend more on conjecture or assumptions rather than the objective analysis of observations; – obtaining precise data on the characteristics of damaged structures (nature of materials, results of tests carried out for their characteristics, reinforcement of parts in reinforced concrete, rules applied and calculations carried out at the time of construction, or subsequent transformations) and of their bearing soil is often very difficult; – information on the seismic movement affecting the construction to be observed is always rather brief. In the best of cases, we can hope for a recording from a relatively short distance but most often we have to be satisfied with signals recorded several kilometers away or with a simple characterization of the source in terms of magnitude and focal distance; the uncertainties about the excitation motion then become significant.
Introduction
471
Despite these difficulties, the collection of observation data through post-seismic missions has made it possible to establish systematic tendencies on which the principles of earthquake engineering concepts are based. For clarity of presentation, the data is arranged hereafter in the order of physical causes which are responsible for the observed effects (deformation of superficial ground, vibratory movements and induced phenomena). Chapter 14 is devoted to the description of macroseismic intensity scales.
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Chapter 11
Deformation Effects Sustained by Superficial Ground
11.1. Effects of irreversible deformations 11.1.1. Damage directly due to movements on fault surfaces It is relatively rare for the foundations of a construction to be affected by fault movement. The appearance of the seismic rupture on the surface is in fact produced only in certain conditions (dimensions of the fault plane in the same order as the focal depth; see section 11.2) and in principle the rules of good construction practices do not allow the foundations of a building to be constructed on a fault trace that is known to be active. However, it is possible that the surface rupture may occur in a zone where the indications attesting the presence of a fault were either not detected or were deliberately ignored during construction. In case of fault movement in the foundation area, the eventual survival of the building or the structure in question depends essentially on the nature and the amplitude of the movement. Horizontal movements (faults where the predominant component is a strike-slip) appear less destructive than vertical movements (normal or reverse faults). Collapse is systematic only for very significant displacement amplitudes (more than or equal to 1 meter for vertical movements and between two and three meters for horizontal movements). Figures 11.1 and 11.2 show the destruction of a bridge and a sluice dam, after movement of the Chelungpu fault (Chi-Chi earthquake, Taiwan on 21 September 1999).
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Seismic Engineering
This reverse fault was the subject of Figure 1.17, where vertical displacements of approximately seven to eight meters, produced at its north end, caused the formation of a waterfall. We can see in Figure 11.1 the separation between the bridge pier situated in the raised section (in the foreground on the left) and the pillars of the intact portion (in the background). It is observed that the vertical movement of the fault was accompanied by a right-lateral strike-slip component of comparable amplitude (several meters). Due to the exceptional magnitude of these displacements obviously neither the bridge nor the dam had any chance of survival. When the amplitude of the fault movement is clearly lower (less than one meter), the damage can be very serious, making the structure irreparable, but complete collapse is far from being systematic. Figure 11.3 relates to another section of the Chelungpu fault, where the vertical displacement did not reach one meter.
Figure 11.1. Collapse of a bridge due to movement of the Chelungpu fault (Chi-Chi earthquake, Taiwan, on 21 September 1999)
Deformation Effects Sustained by Superficial Ground
475
Figure 11.2. Collapse of the Shih-Kang dam due to the movement of the Chelungpu fault (Chi-Chi earthquake, Taiwan, 21 September 1999). The dam was situated about 1 km from the bridge shown in Figure 11.1, on the Tachia River. The vertical movement of the fault reached nearly ten meters [COL 99b]
The fault passed through the center of the largest part of this building and its movement produced a tilting of the whole building (photo on the left) and a deformation concentrated in one span (photo on the right) in line with the ground rupture trace. We can see that the beams were able to withstand this deformation, with vertical joints near the extremities for the higher stories but there was rupture on the first storey. The building is, of course, damaged beyond repair and would have to be demolished but it did not collapse. Such observations have often been made during other earthquakes that produced surface ruptures in urban areas. From this observation it follows that the risks related to fault movements must not be blown out of proportion in regions with moderate seismicity due on the one hand to the very low probability of surface rupture (section 7.2.4) and on the other to relatively low displacement values which can be expected for magnitudes of about 6 (see Table 11.1). The rule adopted by all earthquake engineering codes with respect to active fault zones consists of neutralizing strips of land on either side of fault traces where construction is banned. If this rule is imposed without contest for major faults where there is activity beyond any doubt, it could lead, in case of its “strict”
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application, according to the principle of precaution, to bans for very wide spread sectors of urban zones, representing a rather illusory gain of the level of prevention of seismic risk. In fact, there are often significant uncertainties about the location of the possible surface rupture by faults of moderate size and activity, which has the consequence of a ban on construction in zones that are much larger for a weak risk than for high risk, if the rule for neutralization of land strips is to be applied.
Figure 11.3. Behavior of a multi-story building situated on the Chelungpu fault (Chi-Chi earthquake Taiwan, 21 September 1999); view from a distance (on the left) and detail of damage (on the right) The question of the width of land strips to be neutralized also needs to be raised. It is typically between 100 and 200 m in the majority of earthquake engineering codes but can be more in certain configurations (overlapping section of reverse faults). Past experience clearly indicates that damage due to fault movement is
Deformation Effects Sustained by Superficial Ground
477
generally confined to the immediate neighborhood (within a few meters or, at the most, tens of meters) of the fault. It is also evident that the intensity of vibratory motion is no higher on the fault than a distance away, probably a result of the fact that the most superficial parts of the fault plane contribute less to high frequency radiation (accelerations) than parts situated at greater depth where materials are “more brittle” (section 11.2.2). It is in fact frequently observed that buildings very close to the fault suffer very little or no damage at all. These reasons are in favor of a strict limitation of the width of land strips to be neutralized when a fault trace is precisely located. If the ban on construction on a well identified active fault can be applied to a building or structure, which can generally be constructed in another location, the same cannot apply to network lines (rail-roads, electric lines, pipelines) which, by nature, must at some point cross a fault zone. Appropriate construction methods must be followed in such areas based on the nature of the line and the characteristics of the fault (probable type and amplitude of the movement). Past experience of earthquake effects on underground pipelines is significant, but well documented cases of pipelines having sustained fault movement are relatively fewer in number. A study undertaken in California for the period 1933–1994 gives the figure as less than 10% for damage observed in pipe works which can be attributed to irreversible displacements of the ground, among which displacements induced by liquefaction and landslides are cited more frequently than fault movements [COL 98]. Available observations indicate that the most significant damage corresponds to cases where the type of fault movement and the angle at which the latter is crossed by the pipeline are such that it results in axial compression of the tube. Faults showing a reverse component are thus more dangerous than those that are predominantly normal. For strike-slips, the influence of the angle of crossing is crucial. Figure 11.4 shows the case of a reverse fault (displacement 'i according to the dip of the angle G on the horizontal) having a component 'd of left-lateral strikeslip; if the pipeline has the angle T with the fault trace on the ground surface, displacements 'a (axial direction) and 'p (perpendicular to its axis) sustained by the pipeline are given by:
'a = 'd cos T – 'i cos G sin T
[11.1]
'p = ( ('d sin T + 'i cos G cos T)² + '²i sin² G)1/2
[11.2]
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Seismic Engineering
These formulae are also applied in case of a fault having a normal component 'n (in the direction of the dip) by replacing 'i with – 'n.
Figure 11.4. Crossing of a reverse fault having a strike-slip component by a pipeline
Table 11.1 summarizes the data (components of the fault movement, angle of crossing and diameter of the pipe, axial and perpendicular displacements calculated by [1.1] and [1.2]) for four cases of pipelines having suffered damage but not rupture through fault movements of around one meter. The damage observed in these pipelines, even in cases where a compression appeared (negative 'a for Kern County and Tennant Creek), did not lead to loss of leak tightness, even though significant deformations were observed. In the case of the pipe works of Tennant Creek, an excavation was conducted to examine the pipe along part of its length and to release the strains. The tube immediately became twisted taking the form of a sinusoidal along the entire stretch of the trench.
Deformation Effects Sustained by Superficial Ground
EARTHQUAKE Kern County California, 1952 Imperial Valley California, 1979 Edgecumbe New Zealand, 1987 Tennant Creek Australia, 1988
T
479
Diameter
'a
'p
(mm)
(m)
(m)
100
864
– 0.37
1.01
0
30
100
0.52
0.30
0
1.6
90
100
0.80
1.39
0.9
0
40
350
– 0.06
0.95
'd
'i
'n
(m)
(m)
(m)
degree s
1.0
0.4
0
0.6
0
0
0.3
Table 11.1. Four examples of pipelines that underwent fault movement without rupture; displacements 'a and 'p were calculated using [11.1] and [11.2] assuming that G = 60° [COL 98]
In the case of welded seams, the wall showed wrinkling resulting from the phenomenon of buckling. A 100 m run at the level of the fault was replaced as a safety measure; the new segment had a length 970 mm shorter than the older one [COL 98]. Past experience concerning the effect of fault movements on underground pipelines is thus limited to displacement amplitudes of about one meter. It indicates that steel pipelines made with modern welding techniques usually show good response in these conditions. On the other hand, older procedures (unprotected oxyacetylene welding and arc welding) have frequently resulted in ruptures. In addition to the quality of welding, the good responses of pipes in fault zones depends on the adoption of appropriate measures meant to facilitate deformation of the tube while minimizing the risks of dangerous forces appearing (compression). Such measures, traditionally called detailing measures, constitute an essential element in earthquake engineering; we shall encounter them hereafter with reference to all the effects of earthquakes, whatever the damage mechanism.
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Seismic Engineering
Figure 11.5 A residence situated on the Hayward Fault, east of San Francisco Bay; this fault with a right-lateral strike-slip appears to have been the source of two historic earthquakes (in 1836 and 1868) but the current deformation is aseismic, at the rate of a few mm/year
In the case where pipelines cross faults, construction details to be studied concern: – the orientation of the pipeline with relation to the fault, which must be chosen so as to favor the bending motion and eventual moderate tension on the tube. Normal faults and strike-slips must in general be crossed in a perpendicular manner; reverse faults must preferably be free with a weak angle (about, or less than, 30°); the perpendicular section is the best practical solution when the type of fault movement is difficult to foresee a priori; – the suppression of singular points that can create anchoring of the pipeline into the ground (elbow joints, tapping, sluices and valves) over a distance of about 100 to 200 m on either side of the fault;
Deformation Effects Sustained by Superficial Ground
481
– precautions meant to facilitate movements of pipe works such as: fill work of loose materials placed with low cover height, trenches with inclined rather than vertical walls, lining the pipeline with a low friction coefficient; – local increase in resistance of the pipeline in the fault zone, by increasing its thickness; – finally, as mentioned earlier, volume control at 100% of the quality of welding of runs crossing the fault. To conclude on the effects of fault movements, the case of aseismic soil creep, can be mentioned, i.e. the very slow sliding movement (with tectonic deformation velocities of at most a few cm/year; see section 11.1.2), without the production of earthquakes, of two sections separated by the fault, for which some examples are known, particularly in California. Buildings constructed on such faults become progressively deformed and must be repaired regularly (Figure 11.5). The deformation of the door frame is quite visible. 11.1.2. Damage due to irreversible deformations of the ground in horizontal direction (other than fault movements) Outside of fault zones, irreversible horizontal deformations of the ground are frequently observed for earthquakes of strong intensity. They correspond to the opening up of cracks or triggering of landslides which can be seen even on very shallow slopes; Figure 3.6 gives two such examples. The phenomenon of liquefaction of an underlying layer (see section 3.3.1) is often the cause of such deformation; and is translated through lateral spreading of land bordering the sea or on shores of lakes or watercourses. Although liquefaction was presented in Chapter 3 as part of induced phenomena, its consequences in terms of irreversible movements (horizontal in the present section, vertical in the following section) will be discussed here, without having to wait for Chapter 13 which is devoted to these phenomena, so as to group the descriptions of observed effects, whatever their cause. The collapse of isostatic spans of bridges constitutes some of the most spectacular damage caused by great earthquakes. They are relatively frequent, but can be due to causes other than the intervention of irreversible horizontal displacements. Three cases are presented in Figures 11.6, 11.7 and 11.8.
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Seismic Engineering
Figure 11.6. Collapse of a span of the Nishinomiya Bay Bridge after the Hyogo-Ken-Nanbu earthquake (Kǀbe, Japan) on 17 January 1995
Figure 11.7. Collapse of a span of the Million Dollar Bridge on the Copper River (Alaska) after the Prince William Sound earthquake (28 March 1964)
Deformation Effects Sustained by Superficial Ground
483
Figure 11.8. Collapse of a span of the Bay Bridge in San Francisco after the Loma Prieta earthquake (17 October 1989). This structure, one of the longest (13 km in total) and the most frequented in the world, did not suffer any other significant damage, but had to be closed to traffic for more than a month, resulting in considerable disruption in the region
Among these three examples only the one in Figure 11.6 (Kǀbe earthquake) was clearly attributed to the effect of irreversible displacements of the ground (due to liquefaction). The two others apparently resulted from the slipping of the portal leg caused either by transient differential displacements between bridge piers (due to phase displacement in ground movement) or by movements of the decks under the effect of forces of inertia (leading to their sliding on the bridge bearing or rupture of the latter). However, the prevention of these accidents is easy. It consists of simply ensuring the maintenance of the support for significant displacement of the deck elements
484
Seismic Engineering
which can generally be done a low cost. According to regulations in Japan which were in effect before the Kǀbe earthquake, this maintenance had to be guaranteed, for a horizontal displacement dh of the deck given by the formula: dh = 0.5 L + 20
[11.3]
(dh in centimeters for length L of the span of the bridge written in meters). The results obtained are probably not on the safe side for short bridge spans in the case of violent earthquakes (dh = 30 cm for L = 20 m). Other than the effects on bridges, irreversible horizontal displacements can affect linear structures such as railways (see Figure 3.6) or pipelines. Figure 11.9 shows deformations caused by a lateral movement of the fill work at an electricity transformer control station.
Figure 11.9. Deformations in conduits caused by lateral movement of the fill work at an electric control station (Chi-Chi earthquake, Taiwan, on 21 September 1999)
Deformation Effects Sustained by Superficial Ground
485
Figure 11.10. Damage to quays due to lateral spreading following liquefaction (Kǀbe earthquake on 17 January 1995); container port (above) and areas south of the city (below)
Lateral movements observed in land fills or embankment slopes are generally manifested in a localized and apparently rather random manner. The consequences of liquefaction in terms of lateral spreading are much more systematic. Built up embankments and quays in box piles rotate and are drawn towards water, frequently
486
Seismic Engineering
causing rupture of the covering land fill. Almost the whole of the port area of Kǀbe, constructed essentially on land reclaimed from the sea by hydraulic filling, was thus devastated during the earthquake on 17 January 1995. The resulting damage can be observed in Figure 11.10 (and also on the bottom part of Figure 3.19). The prevention of this damage can pose a problem for which a practical solution can prove difficult if the surfaces in question are large: reduction of the risk of liquefaction through appropriate measures (section see 3.1.1), anchoring in layers unaffected by liquefaction, stabilization through walls in soil cement [COL 99a]. The effects of irreversible horizontal movements which were described earlier depend on extension mechanisms (opening up of cracks, lateral spreading). Compression mechanisms can also be seen sometimes, two examples of which are given in Figure 11.11.
Figure 11.11. Effects of horizontal compression; on the left buckling and upward ejection of metal plates covering a concrete gutter (Chi-Chi earthquake, Taiwan, 21 September 1999); on the right buckling of a metal slatted covering around a building (Kǀbe earthquake, 17 January 1995)
Deformation Effects Sustained by Superficial Ground
487
11.1.3. Damage due to irreversible deformation of the ground in a vertical direction (other than fault movements) As indicated in section 3.1.2, settlement is the most frequently observed manifestation of irreversible deformation induced by vibratory motion of seismic origin. It affects the natural layers of soil; layers of land fill work and structures made from earth or related material (slopes, embankments, dams). The amplitudes of settlement vary according to the thickness of the layer, the loose or compact nature of the material, the possible intervention of liquefaction phenomena and the nature of seismic motion (level and number of vibratory cycles). In the absence of liquefaction, these amplitudes most often do not exceed 1% of the layer thickness subjected to settlement. The damaging consequences are thus limited and essentially concern the effect of differential displacements between elements which follow soil settlement and those which do not follow it (for example structures or parts of structures founded on piling whose lower ends rest on a hard and deep layer); such effects have been reported for junctions of various networks around a structure on piling (Figure 11.12).
Figure 11.12. Settlement of fill work around a water filtration factory building in Los Angeles (Northridge earthquake, 17 January 1994); the amplitude of settlement (about 15 cm) is visible on the deformed lines and the traces at the base of the building
488
Seismic Engineering
Figure 11.13. Examples of settlement due to liquefaction after the Hyogo-Nanbu earthquake (Kǀbe, Japan) on 17 January 1995; above, settlement around a bridge pier founded on piling; below, top of the piling bared under the base of a building
Deformation Effects Sustained by Superficial Ground
489
In cases where liquefaction is a concern, vertical movements can reach definitely significant values as they correspond, not to the increase in the compactness of the layer in question, but simply to compensation of the volume of soil that has undergone lateral spreading on the edges of the liquefied segment. Settlement amplitudes of several decimeters are frequently observed in zones subjected to massive liquefaction. In addition to the example presented in the top part of Figure 3.19, Figure 11.13 shows two cases also taken from the Kǀbe earthquake.
Figure 11.14. The tilting of a small tank, without deep foundations, due to liquefaction of the underlying soil, (Kǀbe earthquake, 17 January 1995). The tilting motion damaged the pipe work which connected the tank to the rest of the installation which was founded on piling and did not follow the soil settlement
490
Seismic Engineering
As for settlements which do not result from liquefaction, damage caused by the downward motion of the ground in liquefied zones can be moderate if it is not accompanied by lateral spreading. At Kǀbe, where liquefaction affected a very significant stretch of reclaimed land, especially on the artificial islands of Port Island and Rokko Island, the phenomenon of lateral spreading only affected parts bordering the sea, for a stretch several tens of meters in width. The central parts of the artificial islands, where a number of residential buildings had been constructed, saw little damage, the buildings having remained in place on their pilings whereas the surrounding ground underwent settling. Only the effects of differential displacements were noted, for similar reasons as those explained for Figure 11.12; Figure 11.14 shows a small tank of demineralized water in an installation, which, given its light weight had not been founded on piling like the adjacent heavier equipment; it was subjected to tilting which damaged the connecting pipe work. A similar incident was reported in the case of a valve of a gas tank; the valve not being founded on piling was affected by soil settlement, causing the loss of leak tightness of the linking clamp and leading to evacuation of the entire area as a safety measure. 11.2. Effects of reversible deformation 11.2.1. Details of effects due to reversible deformation with respect to those due to irreversible deformations Irreversible deformations, resulting, for example, from fault movements, lateral spreading or settlement, leave visible traces on the land; and it is thus quite easy to make designers aware of the necessity of adopting appropriate construction practices to prevent risks associated with these deformations. Moreover, past experience of earthquakes makes it possible to appreciate the effectiveness of these methods and to justify quantitative criteria retained for their application. Reversible deformations resulting from the propagation of seismic waves and the dynamic response of superficial land present a more difficult problem, as we can only try to imagine them at a later time, without any available indications about their manifestation in reality, other than indirect indications such as, for example, the collapse of bridge spans which do not seem to be due to fault movements or effects of liquefaction. Eye-witness descriptions, described in section 3.1.2, of the clearly visible waves on the surface of the ground during earthquakes, have often provoked reactions of incredulity when these deformations did not leave any marks (cracks, residual curvature) on the land. A widespread opinion attributes these descriptions to the fact
Deformation Effects Sustained by Superficial Ground
491
that perception of movements in the environment is disturbed when the observer is himself part of the shaking of the ground and this disturbance leads to an exaggeration of the apparent amplitude of the movements. Whatever the case, the existence of reversible deformations cannot be questioned. Some sites equipped with heavy instrumentation, i.e., having several dozen seismographs arranged according to horizontal or vertical lines with a very short length step (10–20 m) showed significant differences in movements recorded at points quite close together (relative distances in the order of a few tens of meters). These differences translate the reality and make possible the determination of reversible deformations in the horizontal plane and in the vertical direction. Their most frequent causes are as follows: – differential phasing of movements associated with the phenomenon of wave propagation; section 3.2.1 presents simple formulae for the calculation of differential phasing effects for a sinusoidal wave; equation [3.11] is important to be retained in practice:
H max =
V C
[11.4]
Hmax being the maximum reversible deformation, V the maximum modulus of the particular velocity and C the velocity of wave propagation. This formula has already been used in section 5.3.1 (equation [5.43]) to show that the level of reversible deformation is generally low, at least in homogenous terrains; – progressive loss of signal coherence in the course of its propagation, probably due to diffraction phenomena for singular points on a small-scale (boulders included locally in materials with homogenous appearance on a larger scale, cracks and fissures); – influence of discontinuity zones: large-scale heterogenity, transitions between soil and rock, abrupt variations of geometric characteristics (thickness, dip) of layers of land. Compared to irreversible deformations, which are generally limited to some localized zones (fault traces, liquefiable terrains and insufficiently compacted fill work), reversible deformations affect the entire region subjected to strong vibratory motions. Even if they are most often intrinsically less damaging, the incomparably larger extent of their zone of action consequently leads to the fact that the overall effects are often more significant in terms of damage caused, than irreversible deformations. Thus, as indicated in section 11.1.1, more than 90% of damage to underground pipelines has been attributed to them in California for the period 1933–1994 [COL 98].
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Seismic Engineering
Another difference between irreversible deformation and reversible deformation concerns difficulties in their quantification, whether details of input data for a calculation or definition of construction methods. While we can refer to post-seismic observations on land for irreversible deformations, the estimation of reversible deformations is based only on calculation models. We have seen in section 4.1.2 that uncertainties about the determination of movements were greater for displacements than for accelerations or velocities. This increase in uncertainties affects the assessment of reversible deformations (which are linked to differential displacements). If the orders of magnitude can be well defined for deformations due to differential phasing, those resulting from influences of diffraction and discontinuity zones are much more difficult to estimate due to lack of sufficient data. The practical consideration of reversible deformations must thus be coherent with the degree of uncertainty which affects its determination. 11.2.2. Static or dynamic character of effects due to reversible deformations The comparison of effects of reversible and irreversible deformations can be discussed from a different point of view than in the earlier section. This is the static or dynamic nature of their action. It is widely admitted that dynamic aspects do not need to be considered in order to study the effects of irreversible deformations. This attitude is a priori completely justified for consequences of liquefaction or settlement; the movements that they produce in surface land are in fact relatively slow and do not present inversions of the direction of velocity. Thus, they are not able to excite the response of eigenmodes of vibration of the structures in question. It is not necessarily the same in the case of fault movements. We have seen in section 5.3.1 that the order of magnitude for velocity VO on the fault plane is given by formula [5.4].
VO =
'V Uc
[11.5]
'V being the stress drop, U and c the mass density and the velocity of wave propagation of the material subjected to rupture; with the standard values U = 2,700 kg/m3, c = 3,333 m/s [2.4] and the average value 'V = 3.8 Mpa [2.20], we find VO = 0.42 m/s. This order of magnitude is corroborated by very rare reports of fault movements on the surface, which evoke very rapid phenomena whose duration is about one second. The dynamic effects seem possible for certain structures with such values of velocity. However, as the consideration of faults in earthquake engineering codes
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consists of simple rules of exclusion (see section 11.1.1) for buildings or construction work, the question of calculation of the effect of fault movements is not raised and the question of the static or dynamic nature is not to be dealt with. Cases where we cannot make do with rules of exclusion (as in the case of pipelines discussed in section 11.1.1) are generally treated without calculation through the adoption of appropriate construction methods. As far as reversible deformations are concerned, their action is most often considered as essentially static as the ground displacements correspond to much lower frequencies (or to much longer periods) than those that characterize accelerations (see section 4.1.2 and Figure 4.3). This is especially the case for calculations of underground pipelines (section 16.2.3) where, in general we have to be satisfied in admitting that conduits follow ground movement. The verification of their behavior consists simply of studying the consequences of the deformation given by formula [11.4], when it is applied in a static manner to pipelines. We shall see in section 15.1.1 that in the most general formulation of linear seismic calculation, in which the structure to be studied has many support points with differential movements, the effect of differential displacements between supports can be shown by a static term which “depends on time” and does not include eigenmodes of vibration. The question of the static or dynamic nature of reversible deformations of the ground thus seems pointless, but it is frequently asked in practice either because different models have been used to calculate vibration on the whole and for differential displacements of supports, or because the hypothesis of linearity (from which the result mentioned earlier follows) does not correspond to reality. Cases where reversible deformations can contribute to the excitation of eigenmodes have been retained in some parts of Eurocode 8 [EUR 94]. It concerns the action of Rayleigh waves (see section 3.2 and Figure 3.9) on tall thin structures but where the width is sufficient for the undulating ground motion to induce a rotational excitation; this is illustrated in Figure 11.15.
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Figure 11.15. Excitation in rotation induced by Rayleigh waves for a building whose width b is about a quarter the length of wave
If the reality of such a phenomenon seems incontestable, the question can be raised on introducing it into earthquake engineering codes when the recorded data which could make it possible to quantify this rotational excitation is limited or unavailable except for a few rare sites. The determination of seismic movements, particularly for applications in current buildings, is currently based on not very sophisticated methods (attenuation laws; see section 4.2) which do not make it possible to estimate the importance of the contribution of Rayleigh waves (see section 15.1.2). Another case highlighting a possible dynamic aspect for reversible deformations is that of torsion oscillations around a vertical axis. Torsion is a frequent cause of damage in buildings whose different aspects are discussed in section 12.2. It can result, among other explanations, from the non-uniform ground motion under the foundations of a fairly large sized building, hence from the reversible deformations. Figure 11.16 gives an example of damage due to effects of torsion to which reversible deformations of the ground have perhaps contributed.
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Figure 11.16. High rise building having suffered damage due to torsion: cracking due to shearing forces forming Xs on wall elements resisting the torsion moment around the vertical axis; see section 12.2.2 (Michoacán Guerrero earthquake in Mexico, 19 September 1985)
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Chapter 12
Effects of Vibratory Motions
Vibratory motions of the ground are, in the literal sense, the essence of an earthquake; they are transmitted to the entire mass of the structures through links that these masses have with the foundation ground causing transient acceleration fields and deformations which can induce different modes of damage ranging from mild (microcracks) to those causing total destruction (collapse, tilting of the entire structure). It is therefore essential to keep in mind this chain of events if we want to interpret correctly the post-seismic observations and understand the design practices actually in use or the rules which are liable to be included in future in earthquake engineering codes. This becomes all the more essential as progress in seismic engineering has been much slower than is imagined and is limited to already existing knowledge. It is to be noted that the principle of verification of seismic safety using the capacity of resistance to the static forces deemed “equivalent”, though still in use, dates back to an era when no high intensity recordings were available and when it was almost impossible to carry out studies in nonlinear dynamics for want of proper testing and suitable calculation methods. In fact, it took a long time to get to know the complexity of the relationship between the movement of the ground and that of the structure on the one hand and the diversity of the damage mechanism on the other. Gradually, the following factors were identified and studied based on their influence on the observed effects: – three-directional nature and time-dependent sequence (intensity, duration, frequency ranges, non-stationary character) of the ground movement;
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– transmission of movement to the foundations keeping in mind the interaction at the interface (diffraction of the incident wave field, emission from the foundations energized by the response of the superstructure) and the capacity limits of the linkages (partial uplift and slipping of the basemats, local punching, ovality around the boreholes of the piles); – dynamic response of the structure in the linear (fundamental vibration modes) and the nonlinear (effects of plastic deformation, cracking, impacts at the boundaries of certain zones) domains; – heterogenous distribution of weak points (fragile elements with weak deformation capacity before rupture, insufficient connections to maintain the link between the different parts of the structure, zones exposed to shocks against adjacent structures, etc.). Contrary to Chapter 11, which described the seismic effects related mainly to the static aspect of differential displacements, Chapter 12 shall deal mainly with those effects produced by the vibratory character of ground motion. The interface effects (contact between the ground and the foundations of the structure), the inertial effects (due to forces of inertia) and the effects on non-structural elements and supported equipments shall be analyzed successively. 12.1. Effects at the structure/subsoil contact 12.1.1. Slipping and tilting Elements and structures that are simply placed on their supports are able to move under the action of seismic vibration. These displacements which are generally observed only at the final stage (after the earthquake) correspond to two hypothetical cases: – slipping in a horizontal plane often of the order of a few centimeters can often be measured from the mark that it leaves on the support corresponding to the initial position; – total tilting or overturning. During displacement the element or the structure behave like solid bodies; the most common observations concern small objects (less than a meter and at the most two or three meters) like vases, furniture, tombstones, statues or industrial equipment; to this list certain natural objects such as boulders which balance precariously, which are quite common in some of the mountain masses, can be added.
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Figure 12.1 shows, in a very schematic manner, the slipping and the overturning of a parallelepiped block of height h and width b subjected to a horizontal acceleration J.
Figure 12.1. Slipping and overturning of a parallelepiped block under the action of a horizontal acceleration; line GG’ shows the displacement of the center of gravity of the block
By assuming that acceleration J is applied on the support in an impulsive manner at time t = 0 and is maintained constant for a time t > 0, a basic logic of statics shows that if P refers to the friction coefficient below the base of the block, the conditions that define the behavior of the block are determined by the dimensionless number P h/b, i.e. taking g as the acceleration due to gravity, we have:
Ph < 1 there is slipping if J > P g b [12.1]
b Ph > 1 there is overturning if J > g b h This type of reasoning was used right from the early beginnings of seismic engineering to estimate the order of magnitude of seismic accelerations when no recordings were available. The fact that the blocks did not overturn was considered to be “proof” that the acceleration could not exceed the ultimate value b g/h given by [12.1]. In reality, in the case of an excitation of the support corresponding to an
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accelerogram and not to a static acceleration, if the maximum value of the reading J taken on the accelerogram is greater than bg/h, it signifies that the block, in the course of its movement will go through incipient tilting around an edge of the base; for the block to overturn completely the energy communicated to it must be sufficiently high; it is clear from the right side of Figure 12.1 that in the initial phase of the movement towards overturning, the center of gravity should first move upwards leading to a reduction in the kinetic energy of tilting. If this is significantly lower than the increase in potential energy due to gravity necessary to bring the center of gravity right above the pivoting edge, the instability limit will not be attained and the block will fall back on its base. In the real case of a seismic movement, overturning depends not only on the maximum horizontal acceleration but also on other characteristics of the signal (dominant frequency, duration, influence of vertical excitation). The relative importance of these factors shall be discussed in section 17.2.1; this chapter will be limited to giving the non-overturning criteria established by Ishiyama [ISH 82] for slender blocks: V<
k gr O
[12.2]
where V is the maximum horizontal ground velocity, k a numerical coefficient of about 0.4-0.5, O the slenderness ratio of the block (O = h/b) and r the distance between the center of gravity and the pivoting edge (i.e. close to h/2 for a slender block). Criterion [12.2] involves the slenderness ratio and the size of the block simultaneously (through the intermediary of r), whereas static condition [12.1] depends only on the slenderness ratio; it is clear from this that for the same slenderness ratio, the smaller blocks overturn much more easily than the big blocks; the analysis of past experience confirms beyond doubt that the type of overturning discussed here (tilting of solid bodies) can affect only relatively small blocks. Cases of tilting of buildings corresponding to other mechanisms (rupture of the load bearing elements of the first floor or destruction of the foundations) are discussed hereunder. The non-overturning conditions of blocks are very important in seismic engineering because they have led to underestimating the seismic accelerations on the one hand (at a time when the validity of equation [12.1] was trusted), which had for a long time influenced the mindset of the engineers, and on the other hand because they highlight the inadequacy of the static criteria to assess the safety aspect; however, such criteria are still adopted by current earthquake engineering codes to maintain the seismic load case as a part of the routine construction rules, at
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least in the field of civil engineering; this utilization implies that the “seismic forces” to which the criteria are applied are “tampered with” (mainly through the famous “behavior coefficient”; see section 18.1.4); this has given rise to quite a few misunderstandings even amongst certain professionals of seismic calculation.
Figure 12.2. Overturning of a small building in the center of Kǀbe (Hyogo-ken-Nanbu earthquake of 17 January 1995); the building was stopped from falling by the building opposite on the narrow street
In the case of a block equal to a standard statue (O = 3, h = 2 m, that is r # 1 m) non-overturning criterion [12.2] is written with k = 0.4: V < 0.418 m/s
[12.3]
For acceleration A in the case of a “normal” seismic movement (i.e., for which A/V varies from 10 to 20, section 4.1.2), this corresponds to the inequalities: A < 4.18 m/s² or A < 8.35 m/s²
[12.4]
These ultimate values are significantly higher than the one given by static criterion [2.1] (A < g/O = 3.27 m/s²).
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For bigger blocks of size equal to that of small buildings, criterion [12.2] indicates that the type of overturning similar to “tilting of solid bodies” is not possible in practice, even under extremely violent seismic movements. However, examples of overturned buildings such as the one represented in Figure 12.2 are known. Such tiltings are due to the rupture of the columns on the first level of the building which was weakened by the big open spaces provided to facilitate parking of vehicles or commercial activities (buildings with “transparent” first floors). Figure 12.3 shows the tilting pattern which involves two stages.
Figure 12.3. Tilting pattern of a building with a transparent first floor; the trajectory of the center of gravity is entirely below its initial position
The building consists of two portions: the first transparent level represented by columns AAO and DDO and the upper levels forming a rigid block ABCD); the rupture of column DDO causes tilting around the top A of the other column which has remained intact; this first phase ends when the top D of the rectangle hits the ground (the block of the upper levels occupies the position (A’B’C’D’); the tilting can eventually become total when the block pivots around D’ until it is completely laid down on its side (position A’’B’’C’’D’’); it can also stop and the block remains inclined in the position A’B’C’D’; the figure shows the trajectory GG’G’’ of the center of gravity; a simple calculation can show that the trajectory is completely
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Figure 12.4. Slipping of a statue in the center of Kǀbe (Hyogo-Ken-Nanbu earthquake of 17 January 1995); the initial position of the base of the pedestal has left a mark on the ground indicating that the slipping movement consisted of two components: a translation of around 25 cm and a vertical axis rotation of around 15°; the pivoting events around the corners of the base could contribute towards the movement (see the notes in the text below)
lower than G, even at Gv (right above the pivoting edge D’), if the following condition is fulfilled: b<2
l (h l )
[12.5]
l being the height of the columns of the first level, h and b the height and the width of the upper block; this condition is typical of the “self-overturning” buildings which are completely unstable in case of rupture of the lower columns because their total tilting does not require any external energy input.
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The partial or total overturning of a building, apart from being due to the collapse of a transparent first floor, can also be due to the rupture of foundations; this aspect shall be studied in section 12.1.2. Coming back to the phenomenon of slipping or tilting of objects much smaller than buildings, the examples given in Figures 12.4, 12.5 (slipping) and 12.6, 12.7 (tilting) can be analyzed.
Figure 12.5. A 4 cm slip of a section of piping simple placed on the floor of an industrial unit (Spitak earthquake, Armenia, 7 December 1988)
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Figure 12.6. Bicycles in a garage tilted after the Hyogo-Ken-Nanbu earthquake on 17 January 1995; it can be observed that the general idea of deducing the direction of the epicenter from the direction of fall of the overturned objects can have certain exceptions...
For the Ashiya tombstones (Figure 12.7) having dimensions h = 0.700 m, b = 0.275 m, we have O = 2.54, r = 0.376 m; using these values and with k = 0.4, the value of V2, ground velocity at the limit of overturning [12.2] becomes: Vl = 0.30 m/s
[12.6]
Since only some of the stones were overturned, it could be that the ground velocity V on this site was around Vl; by using the same rule as in the case of [12.4], that is A (m/s²) = 10 to 20 times V (m/s), the accelerations were found to be about 3 to 6 m/s²; this seems to be plausible according to the data recorded by an accelerometer in this zone. Analysis of well documented cases of overturning of objects enables us to estimate the order of magnitude of seismic movements by using the method which was just followed ([12.2] and the given relation between velocity and acceleration). In the particular case of the Ashiya tombstones, it can be observed that static criterion [12.1] would give the same value (A = g/O = 3.86 m/s²) but this is because the stones are small.
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For bigger objects static criteria should not be used so as not to run the risk of an obvious underestimation of the amplitude of the movements. Unlike in the case of overturning, in slipping it is difficult to estimate the level reached by the velocities or the accelerations of the ground. This is due to major uncertainties regarding the values of the friction coefficient, to the fact that the final slipping noticed is a result of a number of successive unknown events and not just one, and finally in the case of sufficiently slender bodies, to the possible interventions of the pivoting phases around the corners of the base following “oblique” incipient tilting induced by the combination of two horizontal components; Figure 12.8 shows how these pivotings can produce a “crab steer” whose final position can be wrongly interpreted as due to slipping.
Figure 12.7. Effects of the Hyogo-Ken-Nanbu earthquake (17 January 1995) in a cemetery in Ashiya (north of the town of Kǀbe); the tombstone at the front overturned (after the rupture of the four corners of the base at the points where it was stuck) whereas the others remained in place. These differences in behavior of objects of the same size and subjected to the same excitation because of their closeness indicate the random nature of seismic destruction which will be discussed in section 17.2
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Figure 12.8. Successive pivotings around the corners of the base of a block; initial position ABCD; pivoting around the corner D to the position A’B’C’D’ (= D) then pivoting around the corner A’ to the final position A’’ (= A’) B’’C’’D’’
12.1.2. Rupture of the ground or foundation system The phenomena of rigid body type slipping or tilting described in section 12.1.1 can occur only if the support (ground or floor of the building) is sufficiently resistant against major deformation. In particular, right from its initial phase (incipient pivoting around an edge of the base), the tilting movement reflects a concentration of forces transmitted to the support; thus, to stop the tilting from continuing and possibly overturning it is essential that the support does not give way under this concentrated load. This condition is generally fulfilled for objects of small size which do not depend much on their support (gravitational stresses of about a few hundredths of MPa); the situation is different for buildings with several floors where the stress at the support attains or exceeds generally one-tenth of MPa. In areas where the terrains have mediocre mechanical characteristics, an earthquake can cause ruptures of the ground that lead to the foundations sinking on one side causing the building to incline.
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Figure 12.9 shows such an inclined building of modest size (three levels):
Figure 12.9. Punching of the ground and inclination of a three storey building following an earthquake in Caracas (29 July 1967)
These ground ruptures are usually due to its liquefaction (see Figure 3.21 showing certain buildings in Niigata, Japan, after the earthquake of 16 June 1964). These figures (3.21 and 12.9) bring out two characteristics of these ruptures irrespective of their mechanisms: – the inclination can attain significant values (angles of about ten or more degrees) but can rarely correspond to total tilting (building lying on its side); the building at the center of Figure 3.21 holds a record in inclination; in all the cases the inclination is accompanied by a significant sinking (almost equal to or more than a meter) of a portion of the foundation; this is different from the case of tilting due to the rupture of a transparent first floor (Figure 12.2); – the structural and the non-structural elements of the building itself seem to be often intact or just a little damaged; this indicates that the inclination motion is relatively slow and that its stoppage does not correspond to a violent shock.
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The above two observations show that surface rupture is not a major seismic risk for the safety of human lives which is the main objective of current earthquake engineering codes applied to buildings for human habitation; however, it can present major risks for industrial installations (leakage of toxic substances, explosives or inflammables which can result from circuit failures due to inclination movements and sinking of buildings or big equipment) and can entail considerable economic losses (cost of repairs, interruption of activity for several months leading to temporary or permanent loss of market). The near-total destruction of the port at Kǀbe due to massive liquefaction of the earth filling (see section 11.1.2) following the earthquake on 17 January 1995 is a typical example; apart from the incidence of leakage from the valve of a gas tank (section 11.1.3 and Figure 11.14) which would have had serious consequences, the damages due to liquefaction entailed direct losses (reconstruction of the quay and other port installations) to the tune of a few billion dollars and indirect losses (production cut off, loss of contracts, transfer of activities to other ports mainly to the port of Osaka) which could be much higher and are difficult to estimate correctly. Section 12.1.1 dealt with the problems of wrong interpretation of the causes for the tilting of buildings with a transparent first floor (Figure 12.2 and its notes in the text). The same risk also exists in interpreting the effects of surface ruptures on overturning or inclination of buildings, structures or equipment. In the end, cases of surface ruptures are rather rare and they generally present specific characters mentioned earlier (sinking in the ground and apparent absence of structural damages). Cases of rupture of load bearing elements at the base of the structure are much more frequent; Figures 12.10 and 12.11 show two examples of such cases.
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Figure 12.10. Lateral overtipping of a section of several hundreds of meters of the Hanshin expressway (Kǀbe earthquake of 17 January 1995) as a result of the rupture at the base of a set of piers [DAL 95]
Figure 12.11. Tilting of a cylindrical tank on the horizontal axis due to the rupture of a load bearing cradle in reinforced concrete (Spitak earthquake, Armenia, 7 December 1988)
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These cases of rupture at the base are similar, both in their effects and in their causes, to the overturning produced by the destruction of a portion of the columns of the transparent first floors (Figure 12.2). They can be easily identified because the ruptured portions are visible above the ground surface. On the other hand, it is difficult to identify cases where the rupture affects the elements of foundation (footing, basemats, piles); Figure 12.12 presents an example of the above case.
Figure 12.12. Collapse of the foundations of a ten storey building in Mexico (Michoacan Guerrero earthquake of 19 September 1985)
This building constructed on floating piles tilted almost completely. The piles made of prefabricated concrete elements were pulled out during the tilting movement; the detail regarding one of the piles pulled out shows that they were apparently in good condition [COL 85]. The most plausible scenario for the failure of the foundations given in this figure is [COL 85]: lateral friction along the floating piles situated at an extremity of the building and which undergo the largest stresses due to the swinging oscillations of the building, was reduced under the effect of the vibratory motion of the ground causing the tilting of the building which rests on the surface of the ground through the intermediary of its basemat; considering the weight of the building (ten stories)
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and the weak ground resistance in an area which was originally a lake dwelling in Mexico City (which explains the need for piles), there was a surface rupture below the basemat which was working as a punch; this rupture was confirmed by the appearance of a ridge pushed back in the zone towards which the tilting occurred; the edge piles of the opposite zone, which were only subjected to a weak frictional resistance, followed the movement without damages other than those due to the formation of a hinge and ball joint on their head. Generally, the damages suffered by the deep foundations seem very significant when these are not supported on a hard layer of ground; this is the case with the floating piles in the example given above; it is also the case with compensated foundations which are very common in Mexico for buildings with a few stories; this type of foundation consists of pushing the building into the ground up to a certain depth in such a way that the overloading due to its weight compensates the weight of the earth removed thereby resetting the preexisting conditions of static gravitational equilibrium; the action of the seismic vibrations on the ground characteristics (see section 5.3.1) is able to strongly disturb these conditions, as those of the lateral friction of the floating piles in the previous example. A study of the important damage imposed on the foundations on floating piles has led the Mexican authorities to take urgent measures immediately after the earthquake of 1985 including significantly reducing the friction coefficient taken into account in the design of these piles; this value was reduced from 0.60 times the undrained ground cohesion before the quake to 0.35 times. In practice this amounts to banning this type of foundation for buildings of a certain size; the reason being that in order to balance the weight the length of the piles would have to be such that they can reach the first relatively hard layer at about 35–40 m depth [COL 85]. 12.2. Inertial effects in structures 12.2.1. General observations on the inertial effects Inertial effects are those that can be attributed to the action of the forces of inertia induced into the structure by the vibratory excitation movement transmitted to its base by seismic waves; their different forms in the order of increasing degrees of damage are: – damage marks such as cracking of partitions, falling of plaster, breaking of glazing, damage to façades, displacement or overturning of furniture etc. on nonstructural elements;
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– visible damage on structural elements without appreciable permanent deformation of the entire structure and without localized ruptures or collapse: cracking of load bearing walls, columns or beams in concrete, stripping to bare reinforcements, incipient buckling in the bars of the steel frames, traces of disintegration of the load bearing masonry, etc.; – major damage seriously affecting the structural integrity: major permanent deformation due to rotation of plasticized sections, rupture of certain structural elements causing relative displacements between parts of the structure, partial collapse in certain zones, etc.; – total destruction of a section of the structure in a horizontal (destruction of a floor in a building) or in a vertical (separation of a building into two sections) plane which can lead to complete collapse. In certain cases these effects can mainly be due to shock against the adjacent structures rather than to the forces of inertia associated with the basic movement of the structure considered; however, the effects of shock should be classified among the inertial effects because the shock results from the amplitude of the oscillations induced by the vibratory excitation; these are dealt with in section 12.2.6. Other parts of section 12.2 are defined by the nature of the materials used for the construction of the structural elements because most of the damage patterns have a specific character linked to the special features of behavior and usage of these materials. Emphasis is on the effects due to the horizontal forces of inertia which play an important role in seismic engineering (see section 4.1.3). Section 12.2.5 deals with the effects that can be attributed to the vertical component of the seismic movement; these effects are relatively few and are sometimes difficult to interpret. Finally, the inertial effects on the non-structural elements and the supported equipment shall be analyzed in section 12.3. 12.2.2. Damage and destruction patterns due to horizontal inertial effects for concrete structures X-shaped shear cracks are the most commonly observed damage to walls irrespective of whether they are made of concrete or stonework, concrete blocks or bricks; the way they are formed is explained in Figure 12.13, for a masonry panel used as filler in a concrete column-beam structure.
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Figure 12.13. Formation of “x-shaped” shear cracks in a masonry panel used as filler in a concrete framework: a) initial state without cracks; b) appearance of a diagonal crack under the action of a horizontal acceleration directed towards the right (the crack is perpendicular to the disturbed diagonal of the panel deformed into a parallelogram); c) appearance of a second crack when the acceleration changes direction; d) final state observed after the earthquake
Formation of a regular network of x-shaped cracks is relatively rare because the cracks appear selectively at specific points formed by the angles of the window frames or corbels, as shown in Figure 12.14. Similarly in buildings whose façades have several openings, the shear cracks are observed mainly on the portions of the walls between two openings (Figure 12.15).
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Figure 12.14. Shear cracks in a building in Marina District of San Francisco (Loma Prieta earthquake 17 October 1989)
Figure 12.15. Shear cracks on walls between two windows: on the left, systematic cracking of pillars in a building in Kǀbe (earthquake on 17 January 1995); on the right, details of a cracked wall between two windows (earthquake on 21 September 1999 in Chi-Chi, Taiwan)
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The severity and possibilities of repairing shear cracks vary depending on the slenderness ratio and the structural function of the element that is affected.
Figure 12.16. Reinforced, concrete, load-bearing wall destroyed by shear effect in a building in Kǀbe (earthquake on 17 January 1995). This twelve storey building did not collapse in spite of this rupture and thus saved the lives of its occupants. This example illustrates the superiority of concrete structures with load bearing walls, as opposed to with columns and beams, against the risk of collapse
In sufficiently big and relatively slender walls of reinforced concrete, the cracking can lead to complete rupture without losing the bearing capacity; the building is then irreparable and should be demolished; but since it has not collapsed, it fulfills the main objective of an earthquake-resistant code for constructions used for human habitation; Figure 12.16 shows an example of a completely sheared load-
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bearing wall of a building which remained erect after the Kǀbe earthquake (17 January 1995).
Figure 12.17. Shear rupture of the vertical load bearing elements; on the left, bridge pier sheared half way up (Chi-Chi earthquake, Taiwan, 21 September 1999); on the right, building column showing incipient buckling of the longitudinal reinforcements stripped due to the scaling of the embedding concrete induced by the rupture (Spitak earthquake, Armenia 7 December 1988)
Shear cracking is also observed with varying degrees of damage in load bearing elements other than walls: columns, bridge piers.
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The slenderness ratio is still an important factor for assessing the risk of collapse resulting from a crack due to shear forces; but, compared to the case of walls, there can be aggravating circumstances with respect to maintaining the function of carrying the loads. Figure 12.17 shows two cases of vertical load bearing elements (a relatively stiff bridge pier and a column of a building) after a shear rupture. Even though the pier and column represented in this figure were ruptured, the portions of structure whose weight were supported by them did not fall but were just short of falling; it is all the more important to notice the incipient buckling of the longitudinal reinforcements (vertical) of the column resulting from their stripping because of the destruction of the embedding concrete and their compression due to the transfer of a portion of the vertical load which was almost totally absorbed by the concrete before the rupture; when the buckling exceeds that in the case of the column in Figure 12.17, the longitudinal reinforcements deform considerably towards the exterior producing a significant shortening of the column which in turn may lead to the collapse of the structure; the entire scenario and an example from the Spitak earthquake as in Figure 12.17 is given in Figure 12.18.
Figure 12.18. Destruction pattern of a pillar with buckling of the longitudinal reinforcements (on the left) and an example of a structural joint of an industrial building (on the right) after the Spitak earthquake of 7 December 1988, in Armenia
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This destruction mechanism of the compressed vertical elements is one of those most often mentioned in the database of seismic pathology. Post-seismic missions around the world have taken thousands of pictures relating to this. Figure 12.19 gives three examples of this, chosen from countries (California and Japan) considered to be references in the field of seismic engineering; our aim is not to question this reputation but simply to bring out the fact that structural details considered essential today were totally ignored in the 1960s and 1970s when the structures concerned were actually constructed. From the figures it is clear that the ruptures occur generally either on the top or at the bottom of the vertical load bearing elements; cases of ruptures at mid-height are also known but are rather rare. Prevention of these often devastating ruptures involves respecting the two following principles which are absolutely basic for all constructions in concrete in seismic zones: – avoid vertical load bearing elements which are not sufficiently slender (or “short columns” as per the terminology in use) if they are subjected to shear loading by the horizontal seismic action; – foresee a sufficiently dense and resistant transversal reinforcement (frames, fastenings) in the most stressed zones (top and bottom) of the elements so as to improve the resistance to shear and to ensure the containment of the concrete thereby preventing it from falling once it is cracked and allowing buckling of the longitudinal reinforcements. These principles are all the more important when the vertical load bearing elements have a cross-section (through a horizontal plane) of small surface; in these conditions, the shear rupture and the simultaneous buckling of the longitudinal reinforcements can lead to axial crushing of these elements (see Figure 12.19 at the top and in the middle) or to a lateral shift of the portions on both sides of the severed zone (Figure 12.19 at the bottom) whose amplitude is sufficient to cause partial or total collapse of the structure being supported. When the cross-section is significant (load bearing walls or large bridge piers), in a majority of cases the rupture is not accompanied by a loss of the transmission function of the vertical loads, as was indicated earlier (Figure 12.16 and its comments). That is why buildings, whose bracing (i.e. the structural elements which provide lateral resistance) is at least partially ensured by concrete load bearing walls, behave better than the ones with frameworks. M. Fintel, while summarizing the analysis of past experience from the period 1960–1988, concludes that no cases of the collapse of structures having load bearing walls have been found, whereas there have been hundreds of cases of collapse of columns-beams structures [FIN 94].
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Figure 12.19. Three examples of vertical load bearing elements in reinforced concrete destroyed due to shear rupture followed by buckling of the longitudinal reinforcements. At the top, the base of a column of a six storey building in El Centro (California), completed in 1971 and constructed following the earthquake norms in force at that time (Imperial Valley earthquake 15 October 1979). In the middle, the top of a pier of the Fairfax bridge (Northridge earthquake 17 January 1994, California); this structure, constructed in 1962, should have gone through a seismic retrofitting in April 1994; a similar bridge situated close by had been reinforced before the earthquake and resisted the earthquake well. At the bottom, rupture of the pier heads of a Shinkansen structure (fast train) east of Kǀbe (17 January 1995). The construction finished in 1971 was in accordance with the earthquake-resistant rules in force at that time
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The inadequacies of the transversal reinforcement are striking in the cases of significant buckling of the longitudinal reinforcements (Figures 12.18 and 12.19). They refer to the very high importance of the spacing of frames and to the disproportion between the diameters of longitudinal and transversal reinforcements. Their consequences can be more damaging if in addition the longitudinal reinforcements have design or implementation faults (inadequacy of section, presence of hooks in critical zones). Figure 12.20 shows three examples that should be avoided.
Figure 12.20. Defective reinforcements of vertical load bearing elements taken from three bridge infrastructures or buildings of Kǀbe (earthquake of 17 January 1995). At the top, disproportion between the longitudinal and transversal reinforcements; below left, inadequate transversal reinforcement and presence of longitudinal hooks; below right, general inadequacy (transversal and longitudinal)
It is noteworthy to observe that short columns are often used in cases where modifications are carried out after the completion of the construction work whereas the initial design was sound. Figure 12.21 shows an industrial hall where a small
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cabin with a certain level of sound proofing for the security staff of the installation was added; the person responsible for this addition decided, with the best will in the world, to place this cabin against one of the columns supporting the hall; in effect he has transformed this column into a short column stiffer than the others; as a result, during an earthquake a major part of the horizontal forces will be taken by this column which is not designed to take more than the seismic load foreseen for a normal column because in the beginning all the columns were supposed to work in the same manner. Therefore, there is an obvious risk of rupture for this column. The correct solution would have been to move the cabin a little away from the column to enable it to behave as expected. This example shows that sometimes apparently insignificant details can cause damage or even destruction of the building.
Figure 12.21. Creation of a short column due to irresponsible addition of a small sufficiently rigid cabin against one of the columns of an industrial hall; a) installation drawing of the cabin; b) rupture of the short column thus created during an earthquake and repercussion of its ruin on the rest of the structure (as per [DAV 88])
A similar, but not the only possible, explanation was given regarding the collapse of a parking structure at the University of Northridge. This apparently well constructed recent building collapsed from inside during the earthquake on 17 January 1994 causing a very spectacular curvature of the edge columns (Figure 12.22); only a part of this can be attributed to the instantaneous effects (the rest arising out of a progressive increase of the sag under the action of the weight).
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Figure 12.22. Collapse of the central portion of the parking structure at the University of Northridge dragging one of its façades with it (above) and detail of a bent column (above) showing the uniform distribution of the cracking which reflects the good quality of construction
It has been suggested that the access ramps to the upper levels of the parking structure, generally treated as non-structural elements in the calculations, would have played a role similar to the cabin in the previous example by increasing the stiffness of the columns to which they were hooked. However, it may be more appropriate to attribute the cause of collapse to the design accepted by the Californian earthquake-resistant code for this type of building; according to this design the load carrying and bracing functions were ensured by two independent structural elements; the columns and beams laid out on the façades were designed to provide lateral resistance while the interior columns had no function other than balancing the gravitational loads; such a structural system can ensure safety in case of an earthquake only if the interior columns can follow the displacements imposed by the deformation of the façades under the horizontal seismic action, without losing their load bearing capacity; the code demanded a proper accounting of these displacements for regular earthquakes but the Northridge earthquake produced very high accelerations in the epicentral zones, significantly exceeding the “maximum” expected; considering its location this must have been the case for this parking lot. A different design of the structural behavior could have enabled this excess to be adapted without causing destruction; this is seen from the proper behavior of another parking structure situated about a kilometer away and whose bracing was stiffer and consisted of a concrete load bearing wall.
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If this explanation is valid, there is a basic question which has no clear answer in practically all the codes: does the intensity of the earthquake, as prescribed by the code, have a maximum (for whatever be the meaning of this term), in which case the question of its being exceeded should not arise, or should we be prepared for the consequences of a high intensity earthquake through appropriate structural and architectural details? In the first hypothesis the “good” structures (those which have allowances for resistance beyond the design) are penalized because they could have been designed in a less demanding manner and yet fulfill the no collapsing objective of the code; in the second hypothesis it is the “bad” structures which should be over designed since they have few reserves of resistance. Apart from the shear ruptures of the load bearing elements, the most common cause for partial or total destruction of concrete structures is the rupture of the links between the different parts of the construction; a frequent case is where the joints of the construction rupture and they cut the building into two, as is shown in Figure 12.23. Ruptures of links are the main cause of the major damage during the Armenian earthquake of Spitak (7 December 1988); a large number of prefabricated buildings with column-beam structures having masonry filling were destroyed due to the weak links at the junctions of the frame work and at the contact between the walls and the floors; these prefabricated floor elements were just placed on top of the walls without any linking force other than that resulting from friction. Collapsing occurs either because of the rupture of certain junctions leading to the destruction of the bracing system, or because of the fall of certain elements on the floor setting off “a pack of cards” action. Figure 12.24 shows the typical case of a four storey building.
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Figure 12.23. Collapse of a portion of a factory at Campania (earthquake on 23 November 1980 north of Naples) following a rupture of the vertical joint; the portion that stood erect (in the background) has lost some masonry fillings
Figure 12.24. Destruction of the telephone exchange at Spitak (Armenia) by the earthquake of 7 December 1988; the masonry fillings have remained in place on the pinions while they have disappeared on the long rear wall; the three upper levels have been completely destroyed over three quarters of the surface of the building
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This defective monolithism, when it did not cause complete destruction, produced dislocation or partial collapse; the amplitude of deformations resulting from dislocation has sometimes been so great that it is difficult to imagine how one portion of a building can still be standing. Figure 12.25 shows a residential building that has gone through a deformation such as this and whose extremity has collapsed.
Figure 12.25. Dislocation, torsion and collapse of an extremity of a building in Leninakan (Armenian city currently known as Gumri) during the earthquake of 7 December 1988
Partial collapse often affects the angles situated at the intersection of two roads, probably due to the combination of the two components of horizontal seismic action; Figure 12.26 shows an example of this. This “corner” effect is observed quite frequently; it seems to result from the combination of two perpendicular “extremity” effects, i.e., the fact that in a row of adjoining buildings, those that are at the extremities are the most affected. The poor behavior of these buildings does not imply that the principle of prefabrication in seismic zones should be condemned. During the same earthquake in Spitak, all the prefabricated buildings using load bearing panels instead of the infilled masonry framework sustained the shock; these panels made of light concrete (expanded aggregates of volcanic tuff) have truss type reinforcements on two faces and are equipped with interpenetrating buckles on their vertical edges ensuring
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wedged keying of the concrete poured in situ, after the stringing of vertical reinforcements; the connections implemented in this way have an excellent resistance and guarantee a monolithic character to the construction; Figure 12.27 shows one of these buildings which suffered only minor damages [COL 88a].
Figure 12.26. Partial collapse due to lack of monolithism of an angle of the building situated at the intersection of two roads (Spitak earthquake, Armenia, 7 December 1988)
The shear ruptures of vertical load bearing elements and the dislocations due to destruction of the links can occur with variable configurations according to the structure of the buildings, the position of their weak points and the particular aspects of seismic movements. Apart from tilting as a result of the collapse of a transparent first floor (see Figure 12.2) and collapses due to lack of monolithism (Figures 12.23, 12.24, 12.25 and 12.26), there have been cases where the destruction was limited to just one floor or a group of floors which could be situated at the base, middle or top of the building; examples of these different patterns of collapse are given in Figures 12.28, 12.29 and 12.30.
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Figure 12.27. Building with prefabricated load bearing panels at Leninakan (Gumri) after the earthquake in Armenia on 7 December 1988; the damage was limited to a few cracks and to the fall of part of the facade (below right). These panels are shown in Figure 12.44
It can happen that different modes of destruction are observed for identical buildings belonging to the same construction group; an example of this is seen in Figure 12.31. Buildings with highly dissymmetric bracing systems (i.e., those with a significant distance in the horizontal plane between the center of inertia and the center of torsion) are subjected to torsional movements around a vertical axis; these movements induce shear stresses in the bracing elements which are farthest from the axis of torsion. Figure 11.16 shows the x-shaped shear cracks formed in the load bearing walls by such stresses. In addition to the influence of dissymmetry the stress due to torsion can in certain cases be attributed to the phase shifts of the ground motions below the foundation if this is quite big (section 11.2.2); it is often aggravated by the collapse of the masonry panels of the façades which increases the dissymmetry of distribution of the masses and the stiffnesses. These last two causes (phase shift of movements and creation of dissymmetries following the destruction of certain elements) make it imperative to verify the resistance capacity to torsion
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even if the structure had a perfectly symmetric design. Examples of destruction of dissymmetric buildings due to torsion are shown in Figures 12.32 and 12.33.
Figure 12.28. Ruptures of the first floor: above, additional collapse of a bay at the end of an office building in Northridge (earthquake of 17 January 1994); below, residential building in Kǀbe (earthquake of 17 January 1995) where it can be seen that the upper floors suffered very little damage
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Figure 12.29. Ruptures of a floor at mid-height during the earthquake of Hyogo-Ken-Nanbu (Kǀbe) on 17 January 1995; it can be seen that as in the case of the building at the bottom of Figure 12.28 the other floors seem to be intact; the building shown below is the old town hall of Kǀbe; it was brought down to the level at which it was destroyed and recommissioned in this configuration; this proves that the lower floors had only limited damage
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Figure 12.30. Ruptures of upper floors during the Michoacan-Guerrero earthquake (Mexico) of 19 September 1985; above, the Continental Hotel, constructed in 1950; below, the telecommunications center which shares this building with the Ministry of Communications and Transport; destruction of the upper floors has been quite frequent in Mexico for buildings with around ten floors
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Figure 12.31. A group of residential buildings of twelve floors which suffered various degrees of damage during the Chi-Chi earthquake, Taiwan, of 21 September 1999; the building in the front tilted following the rupture of the first floor; the one just behind, on the left, split into two (see the column of balconies pulled out due to this rupture and hanging in midair after the total collapse of one of the two pieces); the building right at the back, on the right, seems to have been relatively safe
Figure 12.32. Destruction of a school in Mexico during the earthquake of 19 September 1985
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School buildings are often vulnerable because the functional requirements impose a dissymmetric lay out; as a result, providing an efficient bracing around the big class rooms becomes a delicate task.
Figure 12.33. Partial collapse of a five storey building during the Alaska earthquake (28 March 1964); in spite of a layout very closely resembling a square, the bracing was highly dissymmetric consisting mainly of concrete load bearing walls on the south and west elevations and nothing on the north and east elevations (which can be seen in the picture); this partial collapse certainly aggravated the dissymmetry
To conclude this long section on concrete structures, it should be noted that the appearance of the buildings that stood erect is often deceptive when seen from a distance because the level of damage is not obvious. Figure 12.34 shows a tall building which from a distance gives the impression of having resisted the shock whereas it has actually suffered a significant deformation distributed uniformly between the floors and corresponding to a horizontal displacement of the top by more than a meter. Inside the building, the central core which ensured bracing was almost close to rupture and most of the floors were devastated due to the destruction of the partitions. This building, damaged beyond repair, had to be eventually destroyed. For proper assessment of the behavior of buildings during earthquakes, it is imperative to have a close look at the structures, concrete or otherwise, irrespective of the mode of construction.
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Figure 12.34. Sixteen storey building in Leninakan (Gumri) after the Armenian earthquake of 7 December 1988; above, general view and detail showing the shift between the floors induced by the quake; below, destruction of the partitions in one of the floors
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12.2.3. Damage and destruction patterns due to horizontal inertial effects for steel structures Steel structures present a large variety because of their presence in the field of civil engineering (buildings, bridge infrastructure), in networks (pylons, pipelines) and in industry (atmospheric or pressurized tanks, circuits, handling equipment, various machines and instruments). Their main damage patterns due to earthquakes correspond to either instability of form (buckling of slender bars, lateral buckling of I-shaped beams, local buckling of plates or thin shells) or to plastic deformation that can lead to rupture, mainly for the fasteners (weldings, bolts). Buckling of the oblique elements of the steel frame which undergo a compression loading due to the horizontal seismic action is frequently observed (Figures 12.35 and 12.36).
Figure 12.35. Buckling of an element of bracing of an industrial installation (Kǀbe earthquake of 17 January 1995)
The appearance of these buckling deformations signifies that the critical stress of compression of the diagonal bars is lower than the rupture stress of the fasteners; this reflects good design; the total stability of the construction is not generally compromised and the repairs are limited to the replacement of the deformed bars.
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While still respecting this principle (buckling of the diagonals should precede the rupture of the fasteners) it is possible, by optimizing the section of the diagonals, to achieve structural integrity even in the case of extremely violent earthquakes; Figure 12.37 shows an example.
Figure 12.36. Generalized buckling of the diagonals in the metallic frame of a building (Kǀbe earthquake of 17 January 1995)
Figure 12.37. Sugar refinery in Spitak (Armenia) after the earthquake of 7 December 1988; this building went through very strong tremors (0.6~0.8 g?) but its well braced framework with diagonals of ladder-beams perfectly resisted the shock while all the masonry fillings collapsed
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When the weak point of the structure consists of fasteners (i.e. their stress due to rupture is lower than that corresponding to buckling or plasticization of the bracing beams) the ruptures of linkages which can occur under a strong seismic load often cause severe damage, sometimes even collapse. Figure 12.38 shows a building in Kǀbe which is significantly inclined due to ruptures of linkages in the metallic framework.
Figure 12.38. Inclination of around 15° with respect to the vertical in a building in the center of Kǀbe (earthquake of 17 January 1999) as a result of ruptures of elements of the bracing system (mainly certain fasteners of the diagonal tie rods)
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After the Northridge earthquake (17 January 1994) many ruptures of weldings in the connections of metallic frames were observed. Amongst these ruptures the first to be spotted were in the buildings with visible damages (remnant deformations, destruction of structural or non-structural elements) and they needed detailed expert examination. Following this discovery which highlighted a major design or manufacturing problem (because in principle the designs and the quality controls of weldings should make the linkages more resistant than the elements that they connect), it was decided to verify the state of the weldings in other buildings apparently intact from outside; this verification revealed that once the “claddings” of the framework were removed, there were several other cases of rupture of weldings; as a result, a generalized check of all the metallic buildings in the zone subjected to strong tremors was carried out; the cost of this verification operation was considerable given the number of weldings and the work involved in dismantling each one of them to have better access. This problem related to the weldings was confirmed by later earthquakes (mainly that of Kǀbe) and it was one of the major lessons learnt from the Northridge earthquake. In a majority of cases this problem seemed to be due to faulty welding practice at the work site and due to lack of proper inspection. When the design and manufacture of the fasteners have been correctly executed, the structures having metallic frameworks behave extremely well even during high intensity earthquakes. Figure 12.39 shows two examples taken from the Mexican earthquake of 19 September 1985. Bridges using metallic framework for their entire structure or just for a portion of it can show the same damage as the frame work of buildings (buckling, rupture of fasteners). There can also be displacements or falling of parts of deck elements. Figures 11.7 and 11.8 showed two cases where these movements resulted probably from partial displacements between points of the ground but the role of an inertial effect on the supporting devices of the deck plates leading to their slipping cannot be excluded.
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Figure 12.39. This photo taken in the center of Mexico City after the earthquake of 19 September 1985 shows the Latin American tower right at the back, a 44 storey building made of metallic framework which was totally unaffected. On the left there is a tall telecommunications tower which also stayed intact. The building in the front which collapsed was an office building whose reinforced concrete parts presented several defects mentioned in section 12.2.2.
This explanation of displacement of deck elements due to inertial effects is most plausible when these displacements occur in the transversal direction (perpendicular to the axis of the bridge). Figure 12.40 shows such a case observed in Kǀbe on a big metallic structure using box girders (for the piers as well as for the deck elements) and not frames. The seismic damage for the metallic structures used in networks (power transmission or telecommunication poles, pipelines and railways) can be due to inertial effect but in a majority of cases this damage results from the ground effects (differential movements, foundation disorders due to ground ruptures). Figures 12.41 and 12.42 give two illustrations in addition to what was said in sections 3.1.2 (Figure 3.6 and its notes), 11.1.2 and 12.1.2.
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Figure 12.40. Transversal displacement of over a meter of the upper deck plate of a big curved bridge over the port area of Kǀbe (earthquake of Hyogo-Ken-Nanbu of 17 January 1995); this displacement which occurred at the level of a supported joint separating the curved portion of the structure (in the back ground) from its rectilinear part (towards the right), can explain the high amplitude of the movement. Minor damage can be seen below one of the two piers in the front probably due to an incipient buckling under vertical seismic action (section 12.2.5)
Figure 12.41. Buckling of rails after the San Fernando earthquake, California, on 9 February 1971 due to differential movements of the ground in the direction of the track. The rails on the left which had the same problem were repaired at the time the photo was taken
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Figure 12.42. Slight inclination of an electric pylon following a ground rupture around its foundation (Spitak earthquake, Armenia, 7 December 1988)
For lines consisting of several hundreds or even thousands of pylons the ground exploration cannot be done in detail under every pylon; foundation problems such as the one shown in Figure 12.42 are statistically possible at certain points. A case where the inertial effect could contribute to the damage or even could be the main cause for ground rupture is given in Figure 12.43. Incipient ground rupture indicated by the inclinations of the concrete supports could have been caused by the force of inertia along the axis of the piping; this force should have been high for this long rectilinear section and has caused the rupture of the clamps; if this explanation is valid the ground rupture would have been more pronounced if the clamps had been more resistant because the stress transmitted to the supports would have been greater.
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Figure 12.43. Damage suffered by a large water pipeline (Northridge earthquake, California, 17 January 1994); the concrete supports are slightly inclined (photo on the right) and the clamps which ensure connection of the piping with these supports were severed in several places
For a large majority of metallic structures used in industry their good performance under seismic loading depends mainly on the holding capacity of their anchoring on the ground or on the structures of the buildings where they are located. These anchorings should therefore be designed and manufactured with great care by keeping in mind the three-directional character of the seismic action and the fact that the efficiency of an anchoring depends not only on its constituent elements (plates, weldings, bolts) but also on the capacity of its support medium (ground or concrete of a building element) to absorb the transmitted stress. In certain cases the functional necessities do not allow anchoring of the structure to its supports; a typical case is that of the overhead traveling cranes which are used on an almost permanent basis in certain industries and which can be “taken unawares by an earthquake” while they are working. The analysis of past experience, which is available in abundance for such commonly used materials, indicates that the rare cases of these cranes falling should be attributed either to the anchorage rupture of their travel tracks or to the deformation of the structures holding these travel tracks.
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Figure 12.44 shows two overhead heavy duty traveling cranes that withstood the Spitak earthquake (Armenia) of 7 December 1988, without any damage.
Figure 12.44. Two overhead traveling cranes belonging to industrial installations of Leninakan (Gumri) which survived the Armenian earthquake of 7 December 1988; the one in the photo below was used to handle prefabricated panels in concrete like those used in the building given in Figure 12.27
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During the earthquake, these two cranes would have undergone relative displacements of rolling and slipping with respect to their travel tracks; the accelerations transmitted to the structural elements of the bridges (beams, load trolleys) were limited by the occurrence of these displacements; the structural details to be foreseen should include ways to control these displacements: for example providing anti-lifting clips to prevent a possible derailment rather than trying to improve the conditions for the structure itself to absorb the stresses. The principle which ensures seismic safety of industrial materials using the resistance of their anchorings accepts another exception apart from that mentioned above (mobile materials cannot be anchored); this exception concerns big materials whose structure consists of thin metallic shells incapable of resisting without damage the stress concentrations which would appear around a resistant anchoring point. The typical case is that of unpressurized tanks whose sheet metal thickness is only about a few millimeters; because of this thinness they are exposed to risks of buckling at common zones (local “sharp edged” buckling and “elephant foot” ridges) and of tearing near the fasteners. These intrinsically vulnerable materials present a rich and diversified “seismic pathology”: other than the damage illustrated in Figure 12.45 the following damage is observed: destruction of the supports leading to the destruction of the thin shell (Figure 12.46), overflowing of the contained liquid due to the oscillation of the free surface causing the rupture of the roof (Figure 12.47).
Figure 12.45. Dynamic buckling of a thin hemispherical shell obtained through an experiment carried out on a shaking table whose movement, guided by servocontrolled jacks, reproduces a recorded seismic signal (as per [COL 80])
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Figure 12.46. Tilting of a thin tank due to the rupture of the concrete columns of its base slab (earthquake of 7 December 1988, Spitak, Armenia)
The tilting has led to local buckling of the wall. When comparing with Figure 12.11 where the casing was not damaged due to the tilting of the pressurized reservoir, the vulnerability of unpressurized reservoirs is brought out. Preventing such damage involves a set of measures which should be studied on a case to case basis; to overcome the risk of buckling the thickness of the lower ferrules which are the most exposed, should be increased. The supports and the foundations should be designed to resist the inertial stresses acting on the mass of the contained liquid (which is generally higher than the basic mass of the reservoir); the choice of their linkage mode with the shell should be the result of a compromise between the risk of tearing off when anchored and that of an excessive displacement (capable of destroying the drain and the inlet pipes) for a reservoir simply placed on its support; the risks of overflowing can be kept to a minimum by working on the design of the roof or on the level of the liquid. Attention should be drawn to the fact, realized out of experience, that the leakages observed are due more to the piping being pulled out than to the damage of
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the shell even when it is highly deformed due to buckling; but using flexible piping to eliminate the risk of pulling out may prove to be easier and (cheaper) than trying to reduce the buckling of the shell.
Figure 12.47. Spillage of the liquid contained in a tank at the sugar factory in Spitak, during the Armenian earthquake of 7 December 1988; this spillage results from the oscillation (sloshing) of the free surface induced by ground movement and the loss of water tightness of the roof
12.2.4. Damage and destruction patterns due to horizontal inertial effects for structures made of masonry or wood Masonry constructions played a big role in the initial developments of seismic engineering because their ability to withstand tremors has helped in establishing the intensity scales (see Chapter 14). Even though, in theory, the most modern versions of these scales tried to take into account the special characteristics of concrete and steel structures which are widely used in recent constructions, in practice, because of the problems posed by the diversity of these structures and a certain (regrettable; see section 2.3.1) alienation with respect to the notion of intensity, the commonly used scales are still those that are valid for traditional dwellings made of masonry mostly constructed between 1800 and 1950 in rural or urban zones. The materials used for the masonry elements (natural stones or “artificial stones” such as bricks and solid or hollow concrete blocks) present, if they are of good quality, a perceptible resistance to shear stresses whose value can vary from one to a few MPa. Walls constructed using these elements can therefore resist the significant horizontal stresses if the grouting was done properly with a good mortar. Figure 12.48 shows shear stresses affecting “the solid block” of materials.
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Figure 12.48. Rupture at the base of a granite pier (on the left) and crack due to shear on a brick wall (on the right). The pillar element of a torii (gateway at the approach to a Shinto shrine) situated in the center of Kǀbe, was clearly sheared during the earthquake of 17 January 1995, despite the excellent quality of granite
As for the wall, it is interesting to note that the crack starts from the angle of a window (see Figure 12.14 and its notes) and crosses the bricks instead of following the joints, as is seen in Figure 12.13; this “solid block” cracking of the bricks reflects very good workmanship (earthquake of San Salvador, of 10 October 1986, as per [TIE 87]). The analysis of past seismic experience related to masonry is abundant and indicates that stone or brick walls without chaining reinforcement (made from metal or poured concrete) are vulnerable to horizontal inertial effects. Their workmanship is variable (very rarely comparable to the level of the example represented on the right of Figure 12.48) and their resistance becomes considerably reduced due to ageing and lack of maintenance. In addition they are often exposed to risks of disintegration under the effect of seismic action perpendicular to their plane (which can displace or make certain elements fall) and to repercussions of damages to heavy roofs as will be seen later. Figure 12.49 shows a typical example of seismic damage in a building with non-reinforced masonry.
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Figure 12.49. Damage suffered by a building made of masonry without chaining (earthquake of 21 June 1990, Manjil, Iran)
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The load bearing wall in the above picture was severed due to shear and an entire side of the building collapsed. Total destruction was avoided due to the presence of a concrete beam connecting the top of the walls and to a certain load bearing capacity of the sheared wall; this aspect regarding the walls was already discussed in section 12.2.2 (see Figure 12.16); this residual resistance of masonry walls is however uncertain and there have been cases of total collapse of similar buildings during the same earthquake. A frequent cause of destruction of masonry buildings is the weight and the fragile nature of the roofs which can either collapse as a block after their connections with the walls rupture (Figure 12.50) or can come apart like a “pack of cards” when they are made of jointed elements with weak interconnection between them (Figure 12.51).
Figure 12.50. Monolithic fall of a very heavy roof (earthquake of 9 February 1971, San Fernando, California)
The structural details to be considered for masonry structures relate mainly to the use of reinforcing elements (reinforcements, chaining) which ensure a monolithic behavior of the entire construction and its resistant parts both in the vertical (lowering of loads) and in the horizontal (bracing) directions. Reinforcements are necessary along the periphery of the masonry panels, the frames of the openings (doors, windows) and for the entire or for a portion of the panels (in order to control the displacements perpendicular to their plane) in case they are quite big. These reinforcements are given in detail in specialized works [ZAC 96].
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Figure 12.51. A typical destruction of a roof during the Manjil earthquake (Iran) of 21 June 1990; these roofs, consisting of a juxtaposition of small brick arches supported on steel columns, were responsible for a majority of the 40,000 victims of this earthquake
For small buildings (one, two or three levels) these details enable us to obtain a satisfactory performance (limited damage) even in the case of violent earthquakes. In the case of big, old structures such as churches or palaces of high heritage value, post-seismic observations highlight a fragile type of behavior; i.e., the damage is limited or even minor up to a certain level of intensity of the tremors but beyond this limit their severity increases rapidly leading to a partial or total collapse following a disintegration of the load bearing elements (walls, pillars, buttresses) to the destruction of arches of a wide span or even to the fall of an extension or of heavy and voluminous ornaments (towers, bell towers). Figure 12.52 shows the destruction of a church during the earthquake in Armenia on 7 December 1988.
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Figure 12.52. Near total destruction of a church in the center of Leninakan (Gumri) during the earthquake of 7 December 1988
The analysis of past seismic experience shows that structures made of wood usually show a better behavior than those made of masonry. This is due to the fact that wooden buildings are often light and that this material offers an excellent resistance to actions along its grain. On the other hand, wood is fragile against actions perpendicular to the grain and does not have any plastic deformation capacity. In California, where numerous houses have wooden structures the behavior is generally good even though cases of destruction of roofs and peripheral elements (canopies and verandahs) without sufficient bracing are not unheard of (Figure 12.53). The analysis of past seismic experience is insufficient when it comes to big wooden structures using the latest techniques (glue-laminated portal frames, wirebraced frames, curved shear walls). Good performances are seen to be obtained by ensuring that the seismic loads are not exerted in an adverse direction (shear perpendicular to the grains) and that the connections (and the gluing of the gluelaminated elements) are properly designed and manufactured; different tests carried out alternately under dynamic or static loads confirm the above details.
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Figure 12.53. Damage suffered by houses with wooden frames during the Coalinga earthquake (California) of 2 May 1983; collapse of the roof (above) and deformation due to torsion of a colonnade while the main structure seems intact (below)
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12.2.5. Damage patterns due to vertical inertial effect As indicated in section 4.1.3, the earthquake engineering codes consider the effect of the vertical component of seismic movement only in certain cases. Analysis of post-seismic observations sometimes highlights their influence, as will be seen later in this section, but quite often their role in damage does not seem to be major. The seismic vibration is almost always three-directional and the effects recognizably due to horizontal actions (shears, displacements or lateral tiltings) are rarely zero or so weak that the effects of the vertical action can be “isolated”. The example in Figure 12.54 seems to fit into this diagnosis.
Figure 12.54. Destruction of a column in a prefabricated concrete parking lot (Northridge earthquake, California, 17 January 1994); this rupture in the middle of one side of the building, without the appearance of any horizontal displacement, seems to be due to the effect of vertical movement
In metallic structures, the vertical elements can buckle as was indicated in the text below Figure 12.40. Another example regarding a bridge pier is shown in Figure 12.55. These incipient bucklings are definitely due to the effect of the vertical component of the movement; they have remained controlled and have not caused any major damage.
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A case where the buckling due to vertical action is the likely cause of total destruction is shown in Figure 12.56.
Figure 12.55. Ridge due to buckling at the base of a bridge pier (Hyogo-Ken-Nanbu earthquake, Japan, 17 January 1995)
Figure 12.56. Destruction of a tank in Spitak (Armenian earthquake, 7 December 1988)
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The destroyed tank which was identical to those remaining intact (on the right) seems to have “exploded” due to the increased pressure of liquid inside as a result of the vertical acceleration and buckling effect. Vertical accelerations are also manifested by the projection into the air of structures or objects that are just resting on the ground; eye witnesses during violent earthquakes or post-seismic observations have often mentioned this fact. During the great Assam earthquake, in the north of India (12 June 1897), a reliable witness reports that boulders were “dancing like peas on a drum head which is being beaten”. Even though such a phenomenon does not necessarily mean that the vertical acceleration due to the ground has exceeded that of gravity, certain recordings (section 4.1.3) show that it is very much possible. In addition to the cases where vertical projections have been directly observed by witnesses, a close examination of the position and state of damage of structures and concerned objects after the earthquake suggests the possibility of such occurrences; however, such interpretations cannot always be considered as undisputable proof. An example pertaining to a building is shown in Figure 12.57.
Figure 12.57. Shift observed between the structure and foundation in a fire station after the San Fernando earthquake, California (9 February 1971). The undamaged appearance of wood laths at the base of the facade which covered the foundation before the quake suggests rather a lifting mechanism than slipping (which would have damaged the laths) [KOZ 91]
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The most plausible scenario for this example seems to be a projection into the air with vertical acceleration followed by a drop in a shifted position due to horizontal displacement of the ground during the “levitation” phase. However, it cannot be excluded that an incipient tilting (section 12.1.1) has produced the result observed. Vertical projection often concerns small objects (decorative objects, household gadgets, industrial equipment). Figure 12.58 shows two examples from the San Salvador earthquake of 10 October 1986 [TIE 87].
Figure 12.58. Damage to household equipment apparently due to vertical projection during the San Salvador earthquake (10 October 1986). On the left, the cistern cover would have been lifted up before falling into the toilet bowl. On the right, the vase appears to have broken as the result of falling onto the floor after projection into the air [TIE 87]
12.2.6. Effects of shocks Quite a few cases of damage or partial destruction due to shocks between adjacent buildings were observed during the Mexico earthquake, on 19 September 1985. Generally they correspond to interactions, due to the amplitude of horizontal oscillations, between buildings of different height and rigidity without sufficient space between them. The typical case is that of the shock to a relatively flexible building against a neighboring shorter and more rigid building causing destruction of the floors situated above the roof level of the latter either because of the tilting of these floors after rupture at this level (Figure 12.59) or because of their downward telescoping movement (Figure 12.60).
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Figure 12.59. Tilting of the upper part of a building after the rupture at the roof level of the neighboring building (on the right) during the Michoacan-Guerrero earthquake of 19 September 1985. The cause of the destruction is clearly the shock between the two buildings
Risks linked to these lateral shocks can be prevented by adjusting sufficient play between adjacent structures; earthquake engineering codes specify rules regarding spacing to be respected depending on the level of seismicity and the building height; for example, the Mexican code of 1977 (which was in force at the time of the earthquake of 1985) recommended a spacing at least equal to 0.008 times the height with a minimum of five centimeters. It is obvious that this rule is quite binding (which allows a minimum play of 24 cm for a building of 30 m height that is roughly around ten floors) and was hardly applied even though an earlier (1957), less violent earthquake compared to the one of 1985, but still fairly devastating, drew the attention of the Mexican engineers to the risks due to shocks. Experience (not only in Mexico) shows that meaningful rules regarding spacing are, in practice, very difficult to respect for technical (it is difficult to achieve very large joints) as well as economical (land costs in urban zones are very high) and social (non-esthetic “lanes” thus created between buildings risk being converted into
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garbage spaces or even dangerous back-alleys) reasons. Given the pressures to reduce as far as possible the minimum spacing required by the rules they are often relinquished, dictated by the fatalistic sentiment that if stringent rules are fixed for ensuring safety, they are not going to be respected.
Figure 12.60. This office building, relatively flexible because of its column-beam structure, was jammed between two low and rigid buildings; the floors situated above the roof of these buildings (earthquake of 19 September 1985 in Mexico City) collapsed like a pack of cards
In the majority of cases of shocks between buildings that are very close, there is a likelihood that these buildings would have survived or suffered less damage had the spacing been sufficient to eliminate the risk of violent impact. Thus, not respecting the rules regarding spacing given in the codes, provided these rules have not totally sacrificed the safety aspect in trying to reach a compromise, is an irresponsible way of aggravating the seismic risk; the disadvantages mentioned above in having more spacing between buildings than what is given now can be overcome if sincere efforts are made to look for solutions. Risks due to interaction between buildings can exist even with correct spacing if the joint sealing materials have not been removed at the end of construction or if functional requirements impose connections (access gangways, liquid circuit connections) of sufficient stiffness to transmit significant forces from one building to
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another. For example, it is possible that a floor of the town hall in Kǀbe (see Figure 12.29) was destroyed at least partially due to the forces transmitted to this floor by a gangway connecting it to the high rise building which can be seen in the background of the photo. Normally these connections are considered secondary elements and are not represented in the seismic calculation while in reality their contribution to horizontal stress can be significant. In addition to these lateral interactions due to impact or thrust, effects of shocks induced by earthquakes correspond more often to the fall of elements or entire parts of structures situated close by. Figure 12.61 shows the destruction of a tank by the fall of a chimney in an industrial installation.
Figure 12.61. Destruction of a tank by the impact of part of a chimney in a factory in Leninakan (Gumri) in Armenia during the earthquake of 7 December 1988
The rupture pattern of the chimney causing its lower portion to split in two is very uncommon; the most common pattern is rupture of a horizontal section situated somewhere between one quarter to half the height.
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In the analysis of seismic safety of industrial installations presenting risks to the environment, it is accepted that when structures which were not designed following earthquake engineering regulations fall, they can damage any other structure situated within a radius equal to their height; this rule is valid in the case of chimneys or slender structures resembling an “inverted pendulum” such as water tanks (Figure 12.62).
Figure 12.62. This water tank situated in the epicentral zone of the Manjil earthquake (Iran) of 2 June 1990, was empty and suffered only minor damage (crack at the base of the shaft). Cases of rupture of full water tanks due to a lateral fall are known even though it is not systematic even during violent earthquakes
On the other hand, this rule is not very realistic for other types of structures; analysis of past experience indicates that they collapse “on themselves” without
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appreciable lateral projection with the exception of well defined cases where tilting is possible (transparent first floor, destruction of foundations; see section 12.1). Buildings subjected to strong tremors are a danger to the neighborhood because parts of roofs (tiles, rain gutters, chimneys) or façades (panels, balconies, glazings) falling from these buildings can cause injury or even death; this risk due to the fall of objects is second only to the risk due to total or partial collapse of structures (Figure 12.63). Since these risks remain during aftershocks, the ban on approaching the buildings that are damaged by the main shock is justified.
Figure 12.63. Car crushed on a road in Kǀbe (Hyogo-ken-Nanbu earthquake of 17 January 1995)
In the case of medium intensity earthquakes which do not generally cause the collapse of buildings, falling of roof elements becomes the main danger. The Epagny earthquake (near Annecy) of 15 July 1996 is a typical example of this case [COL 96a]; more than 500 chimneys were damaged and the debris was scattered on
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the streets which were fortunately deserted because the earthquake occurred during the night (2:13 am). Reducing seismic vulnerability of these fragile and less maintained elements should be the priority in exposed zones even before all applications of earthquake codes (which concern only buildings for construction) but due to lack of coercive measures prescribed by regulations (such as a construction license) this requirement cannot be imposed on already existing constructions. Certain equipment, by virtue of its mode of suspension or of the fragile nature of its components, is very sensitive to collision effects induced by seismic vibrations. Under this category fall the ceramic insulators of electric control houses whose vulnerability is well known (Figure 12.64).
Figure 12.64. Electric insulator graveyard in a transformer station (Chi-Chi earthquake, Taiwan, 21 September 1999)
Another effect of shock that is more difficult to visualize than the previous effects but nevertheless plays a major role in certain damage patterns, affects structures which fall back at their base after going through an incipient tilting. This impact produces a compression wave which propagates towards the top of the
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structure and is transformed, after reflection at the top, into a traction wave with amplitude sufficient to crack the load bearing elements. The order of magnitude of the corresponding tensile stresses can be estimated from the results given in the tables of section 5.3.2 on the theory of characteristic lines; these tables show (see equation [5.67]) that these stresses are close to the product Ucv where U and c are respectively the mass density and the velocity of propagation of the traction-compression wave of the structural material and v the velocity of the impact. It will be seen in section 17.2.1 that, for a perfectly regular building, the calculation of the tensile stress V, due to the shock of falling back, using this product Ucv can be expressed as a first approximation by the formula: V=
v
O
3EV O gH
V being the maximum horizontal velocity of the ground movement, O and H the slenderness ratio (ratio of height/width) and the height of the building, E Young’s modulus of the load bearing material elements and VO the static stress in these elements resulting from the self weight and g the acceleration due to gravity. For load bearing elements made of concrete (E = 3 u 104Mpa), working under static stress of Vo = 5 Mpa, it is found that V = 7.8 Mpa with the set of values V = 0.4 m/s, O = 2 and H = 30 m; it is clear that a high intensity earthquake (V = 0.4 m/s corresponds to the accelerations of 4 to 8 m/s²; see section 4.1.2) can induce shock stresses higher than the static stress which means cracking of the load bearing elements. To summarize the effects of shock it can be recalled as indicated in section 3.3.3 that earthquakes having their epicenters in the sea can generate pressure waves whose impact against the hull of a ship is capable of causing damage. The order of magnitude of the dynamic overpressure 'p due to these “sea quakes” can also be estimated through product Ucv; by taking the relative values of water (U = 1,000 kg/m3, c, velocity of sound = 1,400 m/s) and a particular velocity v of 0.2 m/s, we find 'P = 0.28 Mpa, i.e. an over-pressure more than the atmospheric pressure that can then deform the thin sheet metals and produce cavitation effects on the hull when the over-pressure sign is inversed during the sinusoidal propagation.
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12.3. Effects on non-structural elements and supported equipment 12.3.1. Deformations imposed on non-structural elements According to the terminology used in the construction field, non-structural elements are those which do not contribute to the resistance for the bracing as well as for carrying loads but form an integral part of the building and are, in principle, very rarely modified after the completion of construction; they are the elements of the façade, of the roof and of the internal layout (stairs, partitions); often, some of these elements contribute to the resistance particularly against the lateral forces in virtue of their connections with the structural elements which support them; this is the case with the masonry fillings of the façades of buildings having a structure of columns and beams or with the stairways solidly connected to their enclosures which are a part of the bracing system. The problem of participation of the non-structural elements in the lateral resistance is one of the most difficult aspects of earthquake-resistant designs; this is mentioned in section 15.1.4 with respect to modelization towards calculations. In any case even if this participation is negligible, the influence of the non-structural elements on the damping of the seismic response cannot be questioned. Irrespective of this delicate question on the possible role of non-structural elements, the common approach in seismic engineering consists simply of verifying their performance with respect to the deformations imposed on them by the bracing system. Earthquake engineering codes have rules to limit the storey drift and the aim of these rules is not only to ban excessively flexible structures but also to reduce the damage suffered by non-structural elements; these rules can be more binding for the project manager than those concerning the lateral resistance. The analysis of past experience indicates clearly that non-structural damage can be considerable even in cases where structures have resisted. This observation applies in the way as mentioned earlier to non-structural elements of buildings but also to the equipment of industrial installations with more serious consequences from an economical point of view (see section 12.3.2) because the cost of this equipment is often much higher than that of the buildings where they are installed. The most visible non-structural damage is obviously that suffered by elements of façades; mainly glass windows. Figure 2.65 shows quite a paradoxical case noticed in Kǀbe (earthquake of 17 January 1995).
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Figure 12.65. These apparently similar buildings experienced different luck during the Kǀbe earthquake (17 January 1995); the building on the right has relatively minor structural damages but has lost all the glass panes whereas the one on the left has retained them while its first level has collapsed
Figure 12.66 gives an example of non-structural damage with serious consequences; it concerns a hospital whose structure resisted well but it could not be used because its stairs fell down. The building itself was not damaged very much but the loss of the stairs interrupted the hospital functions; the destruction of the stairs can be due to incompatibilities of deformation or due to the ruptures of the links with the structure. Apart from their consequences in terms of loss of function (as in the case of Figure 12.66) the fall of non-structural elements constitutes a direct menace to safety, not only for people who are close to the building (see section 12.2.6 and Figure 12.63), but also for its occupants. Figure 12.34 shows the near total destruction of the partitions inside a building that did not collapse. In individual houses several cases of the roof being perforated due to the fall of a heavy chimney are known; Figure 12.67 shows one such case.
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Figure 12.66. Fall of a staircase in a hospital in Leninakan (Gumri), Armenia, during the earthquake of 7 December 1988
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Figure 12.67. In this house where the walls and the chimney were built of brick, the fall of the chimney perforated the light wooden structure of the roof and seriously injured the occupant of this room (Whittier Narrows earthquake, California, 1 October 1987)
12.3.2. Accelerations transmitted to supported equipment In addition to the non-structural elements mentioned in the previous section, buildings normally have a number of pieces of equipment for either collective (elevators, central heating units) or individual (furniture, lighting devices) use; these are often simply placed on the floors or fixed using fixtures that have not been designed to resist seismic stresses. As such they can be easily overturned even in the case of moderate earthquakes; it is essential to realize that the horizontal accelerations are generally amplified in the upper floors by a factor of about 2 to 4 compared to those felt on the ground floor. Figure 12.68 shows two examples of damage to office furniture or documentation centers.
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Figure 12.68. Overturning of office and library furniture in buildings which have otherwise resisted seismic tremors. On the left, effects of the Livermore earthquake, California, 24 January 1980. On the right, effects of the Chi-Chi earthquake, Taiwan, 21 September 1999, in the premises of the University of Chi Nan; other than the furniture the damage also concerned the false roofs and the computer equipment
This overturning of objects can obviously have serious consequences in hospitals and in industrial installations using inflammable or toxic products. Figure 12.69 shows the storage condition in a paint factory after the Motagua earthquake (Guatemala, 4 February 1976). Certain barrels ruptured due to the impact on the floor and their contents spread thus creating a fire hazard.
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Figure 12.69. Falling of barrels and cans of paints in a factory in Guatemala Ciudad (Motagua earthquake, 4 February 1976)
After the earthquake of 17 January 1995, in Kǀbe, several post-seismic fires broke out due to the overturning of paint cans and solvents used by light engineering industries in the city and also due to kitchen stoves. While it is difficult to expect to eliminate the risks connected to these movements totally because many of these devices and containers are movable and cannot be fixed to their supports, it should at least be possible to reduce them to a great extent in store houses using some sensible practices (secure fixing of furniture and frames, removable cross bars or straps preventing the tilting of barrels and bottles). Similar precautions should be taken for certain museums where invaluable collections are exposed on the shelves without any special protection totally at the mercy of earthquakes of a significant level, while simple “handyman’s tricks” (fixing points, connections in transparent plastic) would be sufficient to prevent these generally light objects from falling. As indicated in section 12.2.3 in the discussion about metallic structures, producing resistant anchorages is the basic rule of the seismic safety of equipment, except for that which has to be mobile (such as overhead traveling cranes) or which cannot resist the stress concentrations (such as large tanks with thin walls).
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Forgetting this rule can not only destroy the equipment itself but also interrupt the production of the entire installation. Figure 12.70 shows a situation where wrongly fixed emergency batteries caused a long interruption in the production of an AC turbo generator power plant.
Figure 12.70. A group of emergency batteries of the power generating unit in Kirovakan (Armenia) affected by the earthquake of 7 December 1988. The tremors provoked the triggering of the AC turbo generator which is a common phenomenon during high intensity earthquakes (due to the alarm devices in the case of excessive vibrations of bearings). The emergency batteries should have ensured the continuous functioning of the lubrication pump of the bearings but this did not happen due to the damage caused by faulty anchoring; lack of lubrication resulted in the jamming of the rotor during its slowdown following the triggering. The above picture was taken after the batteries were put back in their places
For materials of command-control that are generally installed in metallic frame cabinets, their fixing to floors or walls of the rooms where they are located should be completed by devices that can eliminate shocks between adjoining cabinets; these shocks can disturb the functioning of these materials by creating power line disturbances in the electrical circuits or by triggering false alarms of system failure. Often to prevent risks due to shock, it is sufficient to link up the top of the cabinets which are placed side by side. Using silent blocks at the base of the materials to protect them from high frequency vibrations can have adverse effects in the case of seismic vibrations where the frequency levels are clearly much lower (a few Hz or at the most between ten and hundred Hz instead of several hundreds).
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The fixing rule should be applied carefully when it comes to preparing for the effects of differential displacements; this is the case with piping which connects equipment of significant mass (pumps, tanks, exchangers); these can have different seismic responses and thus impose their basic displacement onto the extremities of the connecting pipes whose supports should be designed to enable them to follow these movements without becoming ruptured. It is important to note that the cases of seismic destruction to piping are mainly due to the incompatibility of the extremity to move given the rigidity of the section considered. Only one case of rupture due to inertial effect [LAB 98] is known whereas the usual method of seismic design of piping gives importance to this type of effect by establishing the safety requirement on the limitation of the stresses due to inertia; as already mentioned in section 12.1.1 regarding the overturning of rigid blocks, this observation illustrates the inadequacy of the static criteria of forces or of stresses used by earthquake engineering codes.
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Chapter 13
Effects of Induced Phenomena
Naturally induced phenomena (liquefaction, landslides, tsunamis and seiches) were presented in section 3.3 not only from the point of view of their physical causes, but also their effects on constructions. Moreover, damage due to deformation of liquefied soil has been commented upon in section 11.1.2. Hence it is not necessary to dwell on these phenomena in this chapter. On the other hand, “artificially” induced phenomena (disruption of networks, fire, industrial accidents) have not yet been discussed and deserve further development, considering the importance of some of their consequences. 13.1. Effects of naturally induced phenomena 13.1.1. Effects of liquefaction The damaging effects of liquefaction which were described earlier (see Figures 3.19, 3.20, 3.21 and 11.6, 11.10, 11.13 and 11.14) concerned spectacular cases corresponding to extreme manifestations of this phenomenon (Niigata and Alaska 1964, Kǀbe 1995). The most frequently observed effects, which can occur during earthquakes of average magnitude (of about 6), are relatively limited damage to the foundation, which is not always easy to distinguish from other cases of partial destruction of the ground (punching, differential settlements). Figure 13.1 shows the effect of tilting and opening up of a vertical fissure in the wall of a small building, which expert opinion has attributed to liquefaction.
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Figure 13.1. Damage caused to a small building by soil liquefaction (Manjil earthquake, Iran, 21 June 1990)
It is possible to significantly reduce the risks related to liquefaction through soil improvement techniques or reinforcement of the foundations. The suggested improvements mainly deal with either compacting of the liquefiable layer (often carried out through preloading of the land) or vertical drains (shafts filled with gravel) to restrict the rise of interstitial pressure under the effect of seismic vibrations, which is the cause of the phenomenon (section 3.1.2). Of course, the combination of both techniques can also be considered. At Kǀbe, the earthquake of 17 January 1995 demonstrated that these procedures made it possible to very significantly reduce settlement and tilt induced by liquefaction [COL 99a]. Their relatively high cost restricts their use to limited surfaces, such as the foundation of a building or a construction work, as it can become prohibitive in case of the entire area. In addition, their localized nature is such that they are practically of no use in zones affected by lateral spreading over extended areas near embankments or shorelines. It is then necessary to reinforce the foundation system to make it possible for it either to resist displacement by anchoring it to non-liquefied soil (a solution which can be considered if the liquefiable layer is relatively thin) or follow the displacement while maintaining a
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monolithic character. Different systems for reinforcement (ring-shaped walls around groups of piles, networks of diagonal and vertical panels coupled in soil-cement) have been suggested and experimented [COL 99a]. Radical solutions that are generally very costly can be adopted in some cases to eliminate the risk of liquefaction: through soil substitution or permanent lowering of the ground water level so as to maintain it in soil which is not susceptible to liquefaction. Temporary sinking of ground water can also improve the effectiveness of compacting through preloading. 13.1.2. Other naturally induced phenomena As opposed to risks related to liquefaction, which, in some cases, can be reduced through the corrective measures which have just been mentioned, those resulting from large landslides or tsunamis several meters in height do not generally allow for any defense other than the exclusion of the site (if it is a structure under construction) or evacuation (if it involves an existing city) when indications from monitoring systems (for landslides) or warning systems (in case of tsunamis; see section 3.3.3) show that danger is imminent. Localized protection against tsunamis can be considered with the help of dams or breakwaters (which are efficient for waves that are a few meters in height), as we see in some ports of the Pacific, but this is not possible for the whole of an exposed shoreline and is of no use in extreme cases (waves 30 m in height, of which some examples are known). Certain special risks associated with landslides can be seen in a deferred way. In section 3.3.2, the rupture of natural barriers resulting from the obstruction of valleys by great landslides were mentioned. The risk of contamination by suspended dust particles due to landslides or stripping of slopes and wind action has been observed; among the 61 deaths caused by the Northridge earthquake (California, 17 January 1994), three were attributed to the spores of a poisonous mushroom which is found in the mountains north of the Los Angeles area. This poisoning, which usually only affects a few hikers who frequent the mountains, affected more than 150 people after the earthquake. 13.2. Phenomena induced in networks and industrial setups 13.2.1. Disruption of the functioning of networks Transport, communication and distribution networks are vulnerable in the case of an earthquake of high intensity. Consequences of this vulnerability can be very
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damaging for the organization and routing of aid and can be a contributing factor in the toll on the loss of lives. Damage caused to road and rail transport networks (collapse of bridges, obstruction of routes due to landslides, roads blocked due to fallen buildings or debris, destruction of railways) contributes to the state of confusion which generally reigns just after the catastrophe and can considerably delay assistance and rescue operations. Certain urban settlements are particularly unfavorable, such as elongated port cities that lie between the mountains and the sea, which become isolated if the access structures at their extremities are destroyed. This was the case in Kǀbe, where the loss of access by land to the east or west (see Figure 2.5), not compensated by the sea route due to extensive destruction of quays (see Figure 11.10), made it very difficult not only for the arrival of heavy back up facilities, but also for the awareness of the extent of the disaster in the hours following the earthquake on 17 January 1995. Simultaneously, with the reduction of vulnerability of strategic structures for continued access, it is necessary to establish a plan for organization of aid adapted to different possible scenarios, to ensure the greatest redundancy in the rapid handling of teams and rescue operations. Feedback from post-seismic experience clearly highlights the importance of preparation, which is an essential part of prevention. In Japan, preparation efforts have been undertaken for a long time in the Tokyo area, where exercises with public participation are carried out every first of September (the anniversary of the terrible earthquake of 1923). The state of preparation was far less in Kansaï (Osaka-Kǀbe-Kyoto region), where the risk was wrongly perceived to be a lot lower. Telephone communication networks are often greatly disrupted after earthquakes, not only because of damage suffered in the communication stations or lines, but also due to saturation resulting from the substantial influx of calls from the neighborhood of the disaster affected area. Civil security organizations must therefore have at their disposal, a back up network protected from disruptions affecting the general network. Most often, damage to telephone equipment can be quickly repaired, and among all the networks, it is usually the telephone system that is re-established first. Distribution of water is very frequently interrupted after great earthquakes, as the network of pipes of different diameters covers a very large area and comprises often heterogenous elements from the point of view of age and vulnerability. Therefore, high risks of rupture necessarily exist in areas exposed to land movements (fault slip, liquefaction and settlement). Figure 13.2 shows flooding in some streets in
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Northridge, caused by water pipes bursting during the earthquake of 17 January 1994.
Figure 13.2. Flooded streets after the rupture of water pipes (Northridge earthquake, California, 17 January 1994)
The loss of distribution of water makes it very difficult to fight against postseismic fires (section 13.2.2) and necessitates the establishment of a transport system of truck mounted tanks, which often adds to the saturation of any remaining free urban thoroughfare, if the population of the disaster area is significant. The need to provide water for a large number of people can arise even if the distribution network is still partly operational, as the water is often unsuitable for consumption from being polluted due to local ruptures. The vulnerability of the electricity distribution system has already been highlighted in sections 12.2.6 (Figure 12.64 and notes) and 12.3.2 (Figure 12.70). It is due to the fragile nature of certain equipment in sub-stations (especially ceramic isolators) and the fact that groups of alternating current turbogenerators are provided with protection, which cause them to be triggered when they detect abnormal vibrations on the bearings. The triggering of some groups and disruption due to incidents in the transformers can be sufficient for a breakdown in the electricity network, as we have been able to observe in the case of a number of earthquakes of
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strong intensity. Loss of external electric power is thus a basic rule in the analysis of the safety of industrial installations in the case of earthquakes (see section 13.2.3). 13.2.2. Fires Post-seismic fires are greatly feared, especially in Japan, where the memory of the 1 September 1923 catastrophe (majority of the 140,000 deaths due to a gigantic fire) is still vivid. Figure 13.3 shows a small building showing the remnants of a fire after the Kǀbe earthquake (17 January 1995).
Figure 13.3. Burnt-down remains after fires at Kǀbe (earthquake of 17 January 1995)
The photo in Figure 13.4 was taken a few hours after the Loma Prieta earthquake, California, on 17 October 1989. It shows fire raging in the Marina District of San Francisco.
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Figure 13.4. Fire in the Marina District of San Francisco (Loma Prieta earthquake of 17 October 1989)
There are many causes of these fires: – rupture of gas pipelines in the distribution network to households; pipelines ensuring the transport of gas are generally resistant, even in the case of violent earthquakes; – accidents affecting electric material under tension (especially lighting equipment and fittings); – spills from containers of flammable liquids (see Figure 12.69 and its notes) or kitchen equipment; – accidents affecting chemical factories or storage units of liquid or gaseous hydrocarbons. The work of firefighters against these fires is often very difficult due to the lack of water resulting from burst pipes, the large number of households and numerous obstacles in the movement and deployment of teams. It is quite common that new
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fires are produced by aftershocks or that fires thought to be extinguished or under control show a surge in activity. Firefighting measures can thus last several days, or even several weeks in extreme cases where some fires take on huge proportions. 13.2.3. Accidents in industrial facilities Other than fires, some industrial establishments can be the cause of post-seismic accidents with serious consequences on the environment (pollution, leakage of toxic products) even though past experience has not shown any major events of this nature. Seismic risk must be part of the assessment of danger of the establishment. The assessment of risks and the definition of preventive measures must be based on real modes of damage and destruction and not on “computational” modes (see the note at the end of section 12.3.2 on pipelines). The proposed scenarios for these studies must take into consideration plausible disruption of operating conditions, especially where external power supply for electricity and water is concerned. Automated mechanisms related to triggering of seismic instruments (such as shut down mechanisms of chemical or nuclear reactions, closing of valves or mechanical locking) can constitute effective preventive measures in certain cases. They must be studied with great care from the point of view of feasibility of their functioning and possible unfavorable consequences of their action in certain conditions. This precaution in particular is intended for systems with “seismic pins”, which are meant to break when the “seismic force” exceeds a certain level. The concept of such systems is based too often on the miscomprehension of real modes of destruction and an underestimation of the extreme variability of seismic forces in real conditions.
Chapter 14
Scales of Macroseismic Intensity
The notion of macroseismic intensity was briefly presented at the end of section 2.3.1. Even though this “naturalistic” concept is a little outdated in the eyes of many modern seismologists, it continues to play a major role in the study of seismic hazard, particularly in regions with moderate seismicity, where our assessment of past earthquakes rests almost entirely on descriptions of damage. It should also be of some interest for certain studies in microzoning (see section 7.2.3) and for the diagnosis and reinforcement of existing structures built with masonry (section 18.5), where the behavior constitutes the very basis of the scales of intensity which are used the most (see the beginning of section 12.2.4). 14.1. Characterization of the force of earthquakes through assessment of their effects 14.1.1. A summary of the history of scales of intensity It appears that the first mention of the notion of intensity, using a very rudimentary classification of observed effects, was made by an Italian doctor, Domenico Pignataro [WAL 82]. The starting point was a report prepared at the request of the Academy of Science and Literature in Naples, after a series of six destructive earthquakes ravaged Calabria between 5 February and 28 March 1783. On the basis of this report, which contained a number of eye-witness accounts and detailed descriptions on the damage, and information on the effects of other earthquakes in Italy, Pignataro established a scale for classification of four degrees: light, moderate, strong and very strong.
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In his research study in two volumes devoted to the earthquake on 16 December 1857 to the east of Naples, Robert Mallet used a scale with four degrees, like Pignataro’s, but more precise in terms of classification criteria. He had the idea of tracing the isoseismal lines on a map of the region, i.e. the lines which separate the zones where the levels of intensity differ by one degree, which made it possible for him to localize the epicenter. Later work (De Rossi 1874, De Rossi-Forel 1883–1884, Mercalli 1902), mentioned in section 1.1.1.2, developed towards a greater refinement in description, the number of degrees of the scale going up to ten (De Rossi-Forel) and then to 12 (Mercalli). The criteria for classification were detailed, by underlining the natural differences in the observed effects (impressions of eye-witnesses during the earthquake, damage observed after the earthquake on constructions, changes sustained on land or in water) and the influence of the type and quality of constructions. In the 20th century, several scales with 12 degrees were developed in different parts of the world following the principles of Mercalli (nature of effects, influence of the type of construction), while trying to take into account the evolution of building techniques. The Mercalli scale itself has known several versions after the initial proposition in 1902 (“modified” Mercalli in 1931 then in 1956, the New Zealand version in 1965), which have been widely used except in Europe and Japan. In Europe it is the MSK scale (Medvedev-Sponheuer-Karnik) of 1964 which is most extensively used, even though older scales (such as the MCS scale) MercalliCancani-Sieberg) continue to be used in some countries so as to maintain coherence with earlier assessments in catalogs on historic seismicity. A new method (EMS, European Macroseismic Scale, from 1992) was formulated recently. These scales with 12 degrees are presented in section 14.1.2. In Japan, the specific nature of traditional habitation, as compared to that in Europe or America, has led to the adoption of a scale of intensity (JMA, Japan Meteorological Agency, from 1951) which is quite different from scales in the West, as it consists of only eight degrees; it is also presented briefly in section 14.1.2. As indicated earlier, scales of intensity are currently out of favor in part of modern seismological domains. This is in fact no surprise, as the necessary training to enable accurate assessment of intensity is nothing like that received by seismologists. The assessment of damage in fact requires sound knowledge of current and past construction modes, which evidently corresponds more to profiles of architects and engineers having a penchant for history rather than profiles of specialists in Earth sciences.
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Recent earthquakes are generally not studied from the point of view of the distribution of intensities but rather in a somewhat superficial manner. This is regrettable as this is how information, which may be useful for comparisons with past earthquakes or for the identification of site effects, is lost. 14.1.2. Description of some scales of intensity “Modern” scales of intensity are based on damage statistics according to the type of building. For example, the MSK 1964 scale defines five degrees of damage: – 1st degree: light damage: cracking of plastering, falling of small plaster debris; – 2nd degree: moderate damage: cracking of walls, falling of rather big blocks of plaster, falling tiles; cracking of chimneys or falling of parts of chimneys; – 3rd degree: serious damage: wide and deep meandering cracks in walls, falling of chimneys; – 4th degree: destruction: breaches in walls: possible partial collapse; destruction of connection between different parts of a construction; destruction of fillings or interior partitions; – 5th degree: total damage: total collapse of the construction; and three classes of constructions: – type A: residences made of clay, tar, sun-dried mud bricks; rural residences, constructions made of uncarved stone; – type B: constructions made of ordinary bricks or blocks of concrete, constructions with a mix of masonry and wood; constructions made of cut stone; – type C: reinforced constructions; quality constructions made of wood. For damage statistics, the scale is limited to three levels of assessment of the percentage of constructions affected, with the following equivalence: – some, a few:
about 5%
– many, numerous:
about 50%
– most:
about 75%
On the basis of these statistics, the degrees of intensity are attributed according to the following grid (Table 14.1).
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Degree of damage and type of construction
Some (~ 5%)
Many (~ 50%)
Most (~ 75%)
1A
V
VI
VI-VII
1B
VI
VI-VII
VII
1C
VI-VII
VII
VIII
2A
VI
VI-VII
VII
2B
VI-VII
VII
VIII
2C
VII
VIII
IX
3A
VI-VII
VII
VIII
3B
VII
VIII
IX
3C
VIII
IX
X
4A
VII
VIII
IX
4B
VIII
IX
X
4C
IX
X
XI
5A
VIII
IX
X
5B
IX
X
XI
5C
X
XI
XII
Table 14.1. Attribution of degrees of the MSK 1964 scale according to the degree of damage (1 to 5), the type of structure (A, B or C) and the percentage of constructions affected (5%, 50%, 75%); according to [GOD 85]
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585
A study of this table calls for a few notes: – the degrees of intensity are indicated in Roman numerals; as noted in section 2.3.1, this convention should always be used in order to demarcate the difference from a number expressing the value of a measurable quantity, which can be used in calculations. The intensities are not measurable values but only observable values; – the intermediate degree VI-VII appears in several places in the table; it reflects common practice in cases where there may be some uncertainty between two consecutive degrees of the scale and the aim to correct the particularity in the gradation of damage between degrees VI and VII of the MSK scale, as it is described in its original version [GOD 85]. The introduction of this additional degree VI-VII reinstates a perfect coherence in the progression based on percentage and the type of structure in each of the related sub-tables, each having one degree of damage; – the degrees of intensity less than V correspond to the total absence of damage; these degrees are thus defined only on the basis of eye-witness accounts; the first significant damage (level 3) begins at VI-VII, VII or VIII degrees according to type A, B or C of the construction. Figure 14.1 summarizes the MSK 1964 scale with the help of an abridged description and diagrams for II, IV, VI, VIII, X and XII degrees.
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Figure 14.1. Brief description of 12 degrees of the MSK 1964 scale and diagrams to illustrate the main effects for II, VI, VIII, X and XII degrees [MUI 86]
The other scales of intensity with 12 degrees differ very little from the MSK 1964 scale, as we can see in Table 14.2 which compares six scales of 12 degrees to the Japanese scale JMA 1951 which is presented thereafter:
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587
Table 14.2. Comparison of several scales of intensity
– MCS (Mercalli-Cancani-Sieberg) 1942, still used in Italy for reasons of homogenity with catalogs on historic seismicity; – JMA (Japan Meteorological Agency) 1951, Japanese scale with 8 degrees; – Geofian 1953; – MM (modified Mercalli) 1956, version updated by C. Richter; – MMNZ (New Zealand version of modified Mercalli) 1965; – EMI (international macroseismic scale), aborted attempt to impose one single scale; – MSK (Medvedev-Sponheuer-Karnik) 1964; according to [GOD 85]. The Japanese JMA 1951 scale is noticeably different from the other scales in this table, not only because it has only eight degrees (seven degrees from I to VII for intensities felt, and 0 degree for quakes that are too weak to be perceived by man)
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but also because it depends largely on the behavior of traditional Japanese residences. Table 14.3 gives us a brief description.
Table 14.3. Brief description of the eight degrees of the Japanese JMA (Japan Meteorological Agency) scale 1951
We notice in Table 14.2 that there are only three degrees corresponding to serious damage in the Japanese scale (V, VI and VII) as against six in the “Western” scales (VII to XII). Post-seismic experience shows that the Japanese approach is definitely more realistic, as in practice, degrees XI and XII of scales with 12 degrees are not used at all (the detailed description of the MSK 1964 scale recognizes moreover that for these two ultimate degrees “the determination of the intensity of shaking requires special investigations”, without really explaining what that may consist of). Degree XII, for which “nothing man-made remains standing” (see Figure 14.1), appears more poetic rather than scientific. 14.1.3. Benefits and limitations of the notion of intensity Intensity constitutes a synthetic assessment of the severity of shaking in a given place.
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589
If its evaluation was conducted on the entire affected zone by experienced people using all the available data (questionnaires filled out by witnesses, report of damage on land, cost of repairs), a good amount of very useful information is then available which in general enables a fairly reliable estimation of the characteristics of the seismic source (focal depth, extent of the rupture zone) and highlights particularities related to site effects or directivity effects. The normal manner of showing this information is through a tracing of a map of isoseismal lines. Figure 14.2 shows an example of such a tracing for an Algerian earthquake of average magnitude (M = 6).
Figure 14.2. Map of isoseismal lines. MSK 1964 scale, Chenoua earthquake (29 October 1989), magnitude 6.0
In this figure a regular decrease of intensities is observed when we move further away from the epicenter zone where level VIII was reached. The isoseismal lines are along the east-west direction and of course could not have been extended into the sea. The regular decrease is often destroyed by the presence of “pockets” of higher or lower intensity inside the same zone. Figure 14.3 gives such an example for the epicenter zone of the Irpinia earthquake north of Naples (23 November 1980).
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Figure 14.3. Isoseismal lines in the epicenter zone of the Irpinia earthquake (23 November 1980) according to the MCS 1942 scale. The zone of intensity IX consists of two pockets of intensity X and one with intensity VIII surrounds a pocket of intensity VII in [ORS 99]
The presence of these pockets (also called “exclaves”) is an indicator of the quality of the survey on intensity. It most often reflects the manifestation of a site effect that is due to local land conditions; a well known example is that of San Francisco where a narrow correlation between observed intensities and surface geology was evident in the great earthquake of 18 April 1906 (Figure 14.4). The appearance and extension of isoseismal lines gives information on the seismic source, especially of its depth. As indicated in section 2.3.1, very superficial earthquakes show a more rapid decrease in the level of damage than deep earthquakes. This is shown in Figure 14.5 where the tracings of isoseismal lines are compared for two earthquakes of similar magnitude but different depths.
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Figure 14.4. Distribution of intensities (modified Mercalli 1931 scale) in San Francisco during the earthquake of 18 April 1906 (at the top); the geological map (below) allows us to observe the correlation between the nature of superficial terrains and observed intensities. This correlation was confirmed by the Lorna Prieta earthquake (17 October 1989) (according to [BOL 78])
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Figure 14.5. Comparison of isoseismal lines, traced on the same scale, for the Vrancea earthquake (Romania) on 4 March 1977 (on the left) and that of El Asnam (Algeria, 10 October 1980). The first is an intermediate earthquake (depth of about 100 km), the second a very superficial earthquake (which caused surface rupture; see Figure 1.17) but they are of similar magnitude, a little more than seven (according to [GOD 85])
It is seen in this figure that the macroseismic radius, i.e. the maximum distance of perception by observers (which corresponds to intensity III for scales with 12 degrees), is three to four times larger for the Romanian earthquake. Empirical correlations, which are examined in section 14.2.2, make it possible to evaluate the depth and magnitude of an earthquake for which a reliable tracing of isoseismal lines was possible. This is the only possible method to estimate characteristics of older earthquakes which are known only through texts describing their effects. A good map of isoseismal lines is thus a very useful document, even for recent earthquakes whose characteristics have been determined through instruments. Its establishment necessitates a certain quality of data which is often difficult to obtain due either to omissions or contradictions in archives (for historic earthquakes) or
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593
because, currently, studies of intensity are no longer carried out in detail (for recent earthquakes). By its very nature, intensity is a statistical notion; its evaluation is thus applied to an area and not to an isolated structure. It follows that it is frequently necessary to interpret, rather than strictly apply, the rules of the scale when precise information is available for only one “prestigious” structure (church or castle). This situation is the rule for the majority of historical earthquakes. The estimations of intensity for these earthquakes are thus essentially interpretations and we must not be surprised about the differences that may exist between two authors who have studied the same earthquake. These differences in interpretation are particularly sensitive for the higher degrees of the scale that are associated with partial destruction or collapse. We often encounter great difficulty in deciding the type (A, B or C) of construction whose damage is described in archived documents; in some cases, manifestations of induced phenomena (especially landslides or rock falls) are probably the main cause of destruction and can be misleading if these are attributed to the sole effects of vibratory motion (thus the earthquake on 20 July 1564 in the Vesubie valley north of Nice, gave rise to overestimations of intensity). High level intensities at the epicenter (higher than VIII) are thus often questionable. Weak or undestructive intensities, but which are strongly felt and provoke alarm or fear, i.e. V, VI and VII degrees of scales with 12 degrees, are more reliable as they are essentially based on eye-witness accounts (which are generally quite numerous since the shaking is felt by everyone) and do not require precise statistics of damage. The corresponding isoseismal lines thus constitute the best choice for the application of correlations enabling the determination of magnitude and depth. Weaker level intensities (IV and below) are less significant, as they are not felt by all the people and are thus less reliable. The use of the epicenter intensity to characterize seismic hazard has been the rule in several fields (earthquake engineering codes for current constructions, safety specifications for certain installations presenting risk to the environment) at least in countries with moderate seismicity. This practice can be justified from the point of view of the communication policy, since most of the popular literature accessible to the wider public and catalogs on historic seismicity give special weight to the notion of intensity. In the meantime it presents serious drawbacks from a technical aspect for the following reasons:
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– lack of reliability, as discussed before, for epicenter intensities of a high level in the case of historic earthquakes, which often corresponds to the manifestation of a site effect in an area where constructions are especially vulnerable, or of induced phenomena (liquefaction, landslides); – the fact that the classification of constructions (types A, B and C in the MSK 1964 scale) of scales in current use is basically adapted for buildings in masonry of a few stories at most and does not suit the majority of modern constructions, especially those that are not meant as residences. Attempts to introduce a more diversified classification, as in the EMS 1992 scale, are too recent to have had practical effects, especially in the context of a general lack of interest in the intensity-based approach; – inadequacy of the sole intensity level in order to characterize seismic action while dealing with calculation in methods of determining seismic hazard based on safety objectives formulated in terms of intensity (section 6.1.2). In practice, it is mandatory to arrive at a characterization in terms of magnitude and distance from the site, to be able to determine the parameters of seismic action. In reality, the intensity is as much, if not more, an assessment of risk for buildings in masonry work than an evaluation of hazard, which would be used whatever the nature of construction. It is recalled that hazard characterizes the probability of occurrence of a certain level of vibratory motion in a given site, whereas risk describes the consequences of this motion on construction. Intensity is essentially, as indicated earlier, a descriptive tool of the effects of an earthquake (which has its value, even today when it seems “outmoded” to some seismologists). Wanting to make it a safety reference is misuse of the language and is misleading for the user, who cannot make deductions based only on this data of basic assumptions of calculations that he is asked to carry out. We saw in section 9.2 that seismic action (in terms of response spectra) differing significantly according to the type of earthquake can correspond to the same level of intensity. It is hoped that the reference to intensity in the regulations is given only as an informative remark and is no longer presented as a presentation of safety objectives. 14.2. Numerical correlations using intensities 14.2.1. Correlations of intensities with parameters of vibratory motion Correlations of intensity with parameters of vibratory motion (mainly maximum horizontal acceleration) were quite frequently used until the early 1980s. We can cite as an example the correlation of Murphy and O’Brien [MUR 77]:
Scales of Macroseismic Intensity
log10 Ah = 0.25 I + 0.25
595
[14.1]
Ah being the maximum horizontal acceleration (in cm/s²) and I the intensity according to the modified Mercalli scale. We can see in this formula that an increase of one degree of intensity corresponds to the multiplication of acceleration by the factor 101/4 = 1.778; we thus have Ah = 56 cm/s² for I = VI, Ah = 100 cm/s² for I = VII, Ah = 178 cm/s² for I = VIII, Ah = 316 cm/s² for I = IX, etc. Medvedev, Sponheuer and Karnik, to establish the MSK scale, had the idea of doubling the parameters of movement for an increase of one degree of intensity; this rule of doubling is presented in Table 14.4. Intensity
Ah (cm/s²)
Vh (cm/s)
Dh (cm)
V
12-25
1-2
0.05-0.1
VI
25-50
2-4
0.1-0.2
VII
50-100
4-8
0.2-0.4
VIII
100-200
8-16
0.4-0.8
IX
200-400
16-32
0.8-1.6
X
400-800
32-64
1.6-3.2
Table 14.4. Correspondence between degrees of the MSK 1964 scale and parameters of ground motion, Ah: peak of horizontal acceleration (0.1 s d T d 0.5 s), Vh: peak of horizontal velocity (0.5 s d T d 2.0 s), Dh: amplitude of a pendulum of eigenperiod 0.5 s and damping at 8% of critical damping (according to [GOD 85])
It is to be noted that the peaks in movement Ah, Vh and Dh in this table are defined with reference to a certain type of instrumentation and are thus not exactly comparable to peak values that were discussed in the second part of this book. The fact that the authors of the MSK 1964 scale defined ranges of values for parameters of ground motion shows that they were aware of the dispersion of this type of correlation. This dispersion arises from the very notion of intensity, which is not a physical measurement but a synthetic qualitative assessment (and hence often subjective), whose link with the recorded data can be expressed only in terms of orders of magnitude. However, dispersion is in practice increased considerably by necessity, in order to establish these intensity-motion parameter correlations, to associate the observed intensities with acceleration or velocities recorded on seismometers often situated at
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a certain distance (several kilometers) from the place where the intensity was assessed. For example, the strongest acceleration recorded during the Friuli earthquake (northeastern Italy) on 6 May 1976 is the so-called Tolmezzo acceleration; in reality the seismograph was not installed in Tolmezzo village, but near the Ambiesta arch dam several kilometers from there, and on a rocky crag dominating the dam and lake. Considering what we know today about the importance of site effects and the great spatial variability of seismic movement (see Figures 4.6 and 4.7) it is completely illusory to want to relate the acceleration recorded in Ambiesta with the intensity observed in Tolmezzo. Most of the available data at the time where there was an attempt to establish this type of correlation was of the same “quality”, but it is probable that the proponents of correlations were not aware of it and did not sufficiently follow rigorous selection of data to exclude data corresponding to differences in the nature of sites or distances which were too great between the seismograph and place of observed damage. Such a selection would moreover have caused the usable database to shrink away to nothing. The consequences of the addition of a dispersion of method (due to insufficient verification of data) to intrinsic dispersion (due to the very nature of intensity) are visible in Figure 14.6. This famous diagram, by N.N. Ambraseys [AMB 73], shows the “clouding of points” obtained by plotting maximum horizontal accelerations based on intensity (modified Mercalli scale) for all the data for the period 1933–1973. The dispersion is such that it seems extremely adventurous to trace a correlation line in the midst of this “cloud”. The dispersion is a little less strong for velocity than for acceleration (undoubtedly because “intrinsic” dispersion is weaker) but is still significant. Diagrams such as the one in Figure 14.6 have discredited the very principle of correlations between intensity and the parameters of movement which are practically no longer used today. It is likely that we could, through the thorough selection of intensity-movement parameter couples from the currently available database, establish new correlations that are distinctly better, but such work no longer draws much interest.
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Figure 14.6. Diagram by N.N. Ambraseys [AMB 73] where intensities (according to the modified Mercalli scale) are plotted in the abscissa and maximum horizontal accelerations (in % of the acceleration of gravity) as ordinates. All the data obtained during the period 1933–1973 has been represented in the diagram, without any critical examination of their validity from the point of view of the similarity in site conditions and distance between the seismograph and the place where damage was observed
Even though the correlations that use intensity are no longer used, the starting point of Medvedev, Sponheuer and Karnik, i.e., the rule of doubling, continues to be part of the reference guide to earthquake engineering. It seems to be better applied to velocities than to accelerations, for which the multiplication factor corresponding to an increase by one degree of intensity is more about 1.5~1.8 (1.78 according to the Murphy-O’Brien law quoted earlier).
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14.2.2. Magnitude-intensity relations and attenuation laws of intensity Different proponents have put forward relations making it possible to calculate the magnitude based on the intensity and the distance separating the seismic source from the place where the intensity was estimated. We can quote for example: – Karnik’s relation (1969): M = 0.5 IO + log10 h + 0.35
[14.2]
IO being the intensity at the epicenter and h the focal depth in km; – Mohammadioun’s relation (1982): M = 0.55 I + 2.2 log10 D – 1.14
[14.3]
I being the intensity at distance D (in km) from the epicenter; – Levret-Albaret’s relation (1998): M = 0.44 I + 1.48 log10 D + 0.48
[14.4]
I and D having the same significance as for [14.3]. These empirical relations have been established by adjustment to the method of least squares on the basis of a database. V. Karnik used European earthquakes for which the epicenter intensity and depth were considered well known at the time of the study. B. Mohammadioun used data obtained mainly in California. A. LevretAlbaret took into account 73 relatively recent French earthquakes (20th century) for which macroseismic and recorded data of good quality is available. The differences in the coefficients between these three relations result from the particular nature of the database of course, but also from the methods used to estimate the parameters, particularly depth, determining which is often a delicate matter (section 2.2.1). These relations establish a link between the two most utilized “measures” to characterize the “force” of an earthquake. They are, as is the case in seismology, affected by a significant dispersion, as their standard deviation is around 0.4 to calculate the magnitude. Their structure with a single parameter for distance (h or D) shows that they suppose that the seismic source can be considered punctual, which excludes their application in epicenter zones for earthquakes of fairly strong magnitude.
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We can use equations [14.3] or [14.4] to check the “action radius of damage” rule discussed in section 2.3.1. If it is admitted that serious damage starts at a level of intensity of about VII–VIII it can be deduced from [14.3] in which I = 7.5: DI = 7.5 = 10 0.45M-1.36
[14.5]
The rule of 5 RO or 2 LH given in section 2.3.1 to determine the distance limit of serious damage, is expressed according to [2.27] in [BET 03a] by: 5 RO or 2 LH = 100.50 M-1.70
[14.6]
Equations [14.5] and [14.6] give similar results as can be verified in Table 14.5. M
DI = 7.5 [4.5]
5 RO or 2 LH [4.6]
5
7.8 km
6.3 km
5.5
13.0 km
11.2 km
6
21.9 km
20.0 km
6.5
36.7 km
35.5 km
7
61.7 km
63.1 km
Table 14.5. Comparison of equations [14.5] and [14.6] for the evaluation of the action radius of serious damage
At the theoretical level the magnitude-intensity-distance relation can be demonstrated on the basis of the two following hypotheses: – intensity I is related to a parameter P of ground motion (acceleration, velocity, spectral ordinate, etc.) by a multiplication expression translating the assumption that, an increase by one degree of intensity corresponds to the multiplication by a factor k of the value of this parameter; this is expressed by the formula: P = PO x k I
[14.7]
kPO being the value of the parameter for I = 1; – parameter P is given on the basis of magnitude M and focal distance D by the attenuation law having the regular form 4.2.2: P = C eDM D-E e- JD C, D, E and J being constants.
[14.8]
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By equating expressions [14.7] and [14.8] for P and passing to decimal logarithms, we get: log10 PO + I log10 k = log10 C + D M log10e – E log10 D – JD log10e or, by solving for M: M=
log10 k log10 ( PO / C ) E J I log10 D D D log10 e D log10 e D D log10 e
[14.9]
This relation gives results quite similar to [14.3] if it is supposed that parameter P is the velocity given by law [5.9] (D = 1.15, E = 1, J = 0 and C = 0.194 when the velocity is expressed in cm/s) and that intensity VII corresponds to a velocity of 10 cm/s (which gives Po = 0.078 cm/s by admitting the rule of doubling, where k = 2); we thus find: M = 0.60 I + 2.0 log10 D – 0.79
[14.10]
i.e. coefficients close to those in [14.3]. If, instead of the velocity, acceleration is chosen as the parameter related to intensity through equation [14.7], coefficient D is divided by two (see [5.36]) and the value of multiplicative coefficient k needs to be reduced to conserve the coefficient of about 0.5-0.6 in terms of intensity. This observation confirms the fact, mentioned at the end of section 14.2.1, that the rule of doubling is better applied to velocity than to acceleration. Equation [14.9] can be used to find the attenuation laws of intensities; if it is written for conditions at the epicenter (I = IO, epicenter intensity, and D = h focal depth), we obtain: M=
log10 k log10 ( PO / C ) E J IO log10 h h D log10 e D log10 e D D log10 e
[14.11]
and, by subtracting [14.9] from [14.11]: IO – I =
log10 e D log10 J D h log10 k h log10 k
E
[14.12]
It is noteworthy that this relation does not include coefficient D; the choice of the parameter associated with intensity thus does not have any influence on the
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601
attenuation of intensities based on distance, since the other coefficients E and J of law [14.8] depend on the propagation mode of seismic waves. Admitting that E = 1 (geometric attenuation of volume waves) and that J corresponds to equation [3.55] (inelastic attenuation), [14.12] is written as: IO – I =
D 1 log10 e Z log10 D h log10 k h 2 log10 k Qv s I
[14.13]
Taking k = 2 (rule of doubling), Q = 600 and Vs = 3.2 km/s (layer between 3 and 15 km depth of the PREM model; see Table 3.1 and notes on equation [3.57]) we get with Z = 18.85 rd/s (dominating frequency of 3 Hz): IO – I = 3.32 log10 D + 0.00708 (D – h) h
[14.14]
focal distance D and depth h being expressed in km. Empirical attenuation laws of intensities have been put forward by different authors; the most commonly used is the Kovesligethy-Sponheuer law which is written as: IO-I = 3.30 log10
D G (D – h) h
[14.15]
It is thus practically identical to [14.14] where the multiplying coefficient of the logarithm is concerned; coefficient G which translates inelastic attenuation always has about the same magnitude as those in [14.14] but can vary depending on the region under study; furthermore, its influence is significant only at great distances. The attenuation laws of intensities are used especially to estimate the focal depth. We look for the value of h which produces the best adjustment for “theoretical” attenuation (equations [14.14] and [14.15]) on the observed attenuation. Focal distances D are in general calculated on the basis of the surface S of isoseismal lines of intensity I by assimilating it to a circle (or an ellipse when the isoseismal lines have a marked tendency to be extended in one direction); in the assumption of a circle, it is expressed with the relation: D=
S
S
h²
[14.16]
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For the study of the best adjustment, it is recommended that the most reliable isoseismal lines be favored, i.e., as indicated earlier, those that correspond to average intensities, felt by all the eye-witnesses but that do not produce serious destruction, i.e. essentially V and VI and, to a lesser extent, IV and VII.
Part 6
Seismic Calculations
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Introduction
In 1907, one year after the major earthquake that hit San Francisco on 18 April 1906, the American professor Charles Derleth made the following statement: “It is rather trivial to try and calculate the strains that lie at the origin of an earthquake. These calculations cannot provide us with conclusions of any practical value”. The amount of knowledge which was available on seismic movements and the nonlinear dynamics of structures justified, and at the same time emphasized, this somewhat negative comment. This certainly had an impact on the formal criteria that were adopted for the first earthquake construction techniques (see section 8.1) which were based on the simple calculation of lateral resistance. This type of resistance does not represent the real impact of an earthquake but only focuses on a certain level of security in the event of an earthquake. The conclusions that were drawn from observing earthquakes and not the moment magnitude determine whether the level of safety is sufficient or not. Almost a century later, progress in the field of seismology has been made, and studies (which include calculations and tests) on the movement of tectonic plates have been carried out. Derleth’s statement finally seems outdated nowadays. Today, there is a large amount of very diverse and detailed data that defines the movement of tectonic plates. This data is precise in most cases, even though some important research points such as zones that are close to the epicenter or site effects still remain unclear (see Part 3). Dynamic calculation in the field of linearity has even become increasingly common for complex structures that require models of several hundred or even thousand degrees of freedom. Non-linear phenomena are becoming easier to understand and we can see continuous improvements in relation to the creation of models. This is especially the case for phenomena that have a seismic impact (plastifications, frictional resistance). This rather optimistic point of view, however, needs to be dealt with carefully. The introduction to Part 3 already emphasizes the excessive importance that is
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assigned to calculations, even though the comprehensiveness of calculations and its contribution to the efficient prevention of earthquakes, as indicated in section 15.2.2, mainly depends on the design of a building as a whole, details of construction (devices used during the construction process) and the quality of the work carried out as well as the quality of the material being used on the construction site. This is the case particularly if the comprehensiveness is based on a linear calculation which is used in structural engineering plans and relies on the use of the behavioral coefficient. Relying on the calculations as being the standards to follow might even lead to the absurd conclusion that a building’s poor design can be justified by a very detailed calculation. This part of the book deals mainly with linear calculations as they are currently used as a basic tool in structural engineering plans. The aim is not to give a lecture on dynamic linear structures but to explain the difficulties that result from this application in the field of seismology and show how these problems could be resolved. These problems are, for example, differential movements between supports, absorption, the consequences of losing figures in quadratic equations or the often misunderstood concept of interaction between the ground and the building’s structure. Some examples of non-linear calculations will be given at the end of this part.
Chapter 15
Linear Seismic Calculation
15.1. General observations on linear calculation 15.1.1. General formulation with relation to absolute axes The formulation adopted in section 9.2.1 for the presentation of a simple example of modal analysis – i.e., in terms of relative displacements with relation to a non-deformable base where seismic movement is sustained – is the most frequent in seismic calculation. It is possible only if all the support points of the structure sustain the same movement. In the general case, they can sustain different movements and it then becomes necessary to adopt a formulation with relation to absolute axes. A discretized structure having Ns structural points and NA support points is considered. Each point has n degrees of freedom. The vector for degrees of freedom is noted as {u} marked with relation to an absolute system of axes. The vector has (Ns + NA) n components; it can be partitioned into two sub-vectors: – {us} corresponding to the degrees of freedom of structural points (nNs components); – {uA} corresponding to the degrees of freedom of support points (nNA components). The structure is subjected to excitation by movements at the support points, i.e., the sub-vector {uA} is given on the basis of time. We establish:
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Seismic Engineering
{uA} =
¦u i
ai
(t ) ^G
i
`
[15.1]
uai (t) is the scalar function which defines the movement of the i-th degree of freedom of supports and {Gi} a vector with dimension nNA all the components of which are zero except the one corresponding to the i-th degree of freedom which is equal to 1. uai (t) functions can vary from one support point to another (nonsynchronous excitation). The structure is characterized by its mass and rigidity matrices, [M] and [K], which are square matrices of the order (Ns + NA) n, symmetric and defined to be positive. They can be partitioned to show the parts which operate on the sub-vectors {us} and {uA}. Figure 15.1 shows the structure of the rigidity matrix (the black squares correspond to non-zero terms) and its partition into sub-matrices [Kss] (square nNs x nNs), [KAA] (square nNA x nNA) and [Ksa] (rectangular nNs x n NA), on a simple example (bridge with two spans and a central pier).
Figure 15.1. Model with 7 points for a bridge with two spans. There are 4 structural points (1, 2, 3 and 4) and 3 support points (5, 6 and 7); taking two degrees of freedom per point (horizontal displacement u and vertical displacement v) there are a total of 14 degrees of freedom; the part on the right of the figure shows the structure of the rigidity matrix [K] (non-zero terms are marked by black squares) and its partition into [Kss] (8x8), [KAA] (6x6) and [Ksa] (8x6)
To identify the non-zero terms of the matrix (black squares) it is assumed, that the deck elements (5-1, 1-2, 2-3 and 3-7) and pier elements (2-4 and 4-6) work under normal force (for example, the horizontal force at the extremities of element 1-2 of the deck depends only on the difference u2 – u1) and bending (for example, the vertical force at point 1 only depends on vertical displacements at points 2, where there is a joint on the top of the column, and 5, where there is fixed support on the abutment).
Linear Seismic Calculation
609
As the displacements which constitute the components of vector {u} are taken with relation to absolute axes, vector {ü} represents the absolute acceleration vector and we have, for the equation of movement: [M] {ü} + [K] {u} = {F}
[15.2]
Vector {F} for external forces applied is reduced to reactions of the support, as there is no force exerted on the structural points. Considering the partitioning of vector {u} into {us} and {uA}, of matrix [K] into [Kss], [KAA] and [KsA], and matrix [M], into [Mss], [MAA] and [MSA], in developing equation [15.2] we find: [Mss] {üs} + [Kss] {us} = – [MsA] {üA} – [KsA] {uA}
[15.3]
[MSA]T {üs} + [KsA]T {us} + [MAA] {üA} + [KAA] {uA} = {FA}
[15.4]
Equation [15.3] determines {us} as {uA} and thus {üA} is known. Equation [15.4] thus makes it possible to calculate the reactions of the support {FA}. To resolve [15.3] we look for {us} in the form: {us} = {u’s} +
¦
uai (t) {wsi}
[15.5]
i
where vectors {wsi} are static deformations of the structure when a displacement equal to 1 is imposed on the degree of freedom i of supports, while simultaneously blocking the other degrees of freedom of the supports. According to [15.1] we have {uA} = {Gi} when this displacement is applied to the degree of freedom i, from which the following equation is deduced from [15.3] to determine {wsi}: [Kss] {wsi} = – [KSA] {Gi}
[15.6]
which leads to: {wsi} = – [Kss] –1 [KsA] {Gi}
[15.7]
We thus have the following, on developing [15.5]: {us} = {u’s} –
¦ i
uai (t) [Kss]–1 [KsA] {Gi}
[15.8]
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Replacing {us} in [15.3] by expression [15.8] and {uA} by its expression [15.1], we find: [Mss] {ü’s} + [Kss] {u’s} = ¦ üai(t) ([Mss] [Kss]–1[KsA] – [MsA]) {Gi}
[15.9]
In fact, the terms which bring in uai (t) are eliminated, which is the basic reason for changing the unknown quantity [15.5]. To determine {u’s} we have thus obtained an equation in a form similar to the example shown in section 9.2 (with a formulation in relative displacements), the only difference being the replacement of vector [M] {'} in the second member of equation [9.33], with a more complicated expression, but whose calculation does not present any difficulties. In [15.8] we see that the solution {us} of the initial problem is the superimposition of a dynamic term ({u’s}) – which depends on the accelerations of support points and a static term dependent on time – which only involves support displacements and the static deformations under displacement imposed on the supports. The resolution of [15.9] is carried out according to the same two stage scheme as that used in section 9.2, i.e., finding non-damped eigenmodes followed by development coefficients for the solution on the base made up of these modes. Eigenmodes to be considered correspond to cases where all supports are blocked, i.e., to solutions of the form: {us} = {vs,k} sin Zkt; {uA} = 0
[15.10]
for equation [15.3] they thus verify the following relation, identical to [9.37]: ([Kss] – Z²k [Mss]) {vs,k} = 0 , k = 1.2,… nNs
[15.11]
Modal reactions of the support are associated with these eigenmodes, i.e. vectors {RA,k} such that: {FA} = {RA,k} sin Zkt
[15.12]
verifies equation [15.4] when {us} and {uA} are replaced by their expressions in [15.10], i.e.: {RA,k} = ( [KsA]T – Z²k [MsA]T ) {vs,k}, k=1, 2,…, nNs
[15.13]
It is demonstrated, as in section 9.2.3, that the eigenmodes {vs,k} defined by [15.1] verify very important relations of orthogonality: {vs,k}T [Mss] {vs,A} = 0
[15.14]
Linear Seismic Calculation
{vs,k}T [Kss] {vs,A} = 0
611
[15.15]
for different mode indices, k and A . As in section 9.2.4, we now look for the solution {u’s} of [15.9] in the form of the development:
¦
{u’s} =
rk (t) {vs,k}
[15.16]
k
From which, by developing [15.9] and considering [15.11]:
¦
( rk + Z²k rk) [Mss] {vs,k} =
¦ u
ai
(t ) ([MSS][KSS]-1[KSA]-[Msa]){Gi}
[15.17]
i
k
On premultiplying by {vs,A}T and using orthogonal relation [15.14], we arrive at: rA ZA2 rA ) {vs,A}T [Mss] {vs,A} (
=
¦ u
(t ) ^v s ,A ` (MSS][KSS]-1[KSA]-[Msa]){Gi} T
ai
i
[15.18]
Or again, in the second member, by using the fact that, a scalar equal to a matrix product does not change when this product is transposed: rA ZA2 rA ) {vs,A}T [Mss] {vs,A} (
=
¦ u
(t ) ^G i`
T
ai
i
> K
SA
T 1 @ > K SS @ > M SS @ > M @T {vs,A} SA
[15.19]
now we have, according to [15.11]: {vs, k} = [KSS]-1[MSS] Zk2 ^vs , k ` , k = 1, 2,…, nNs This makes it possible for [15.19] to be rewritten as follows: rA ZA2 rA ) {vs, A}T [Mss] {vs, A} (
=
¦ u
ai
i
T § 1 · T T (t ) ^G i ` ¨ 2 ¸ > K SA @ > M SA @ Z © A ¹
Finally according to [15.13]:
^v
s ,A
`
[15.20]
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^ `
rA ZA2 rA ) {vs,A}T > M SS @ vs ,A (
1
Z
2 A
¦ u t ^G ` ^RA,A ` T
ai
i
[15.21]
i
We have thus obtained a differential equation similar to [9.60] for the development coefficient r A (t) [15.16], i.e.: rA + ZA2 rA = –
¦p
i ,A
uai (t )
[15.22]
i
where there are as many participating factors pi,A as there are support points; the latter given by the formula: Pi ,A = –
^RA,A ` ZA2 ^v ` > M ss @^v ` s ,A s ,A 1
^G i `
T
[15.23]
T
As we have seen in section 9.2.4, it is at this stage that a damping term is usually introduced by rewriting equation [15.22] as follows: rA + 2[A ZA rA + ZA2 rA
¦p
i ,A
uai (t )
[15.24]
i
We see that the dynamic part (determination of {u’s}) of the resolution of the problem of multi-support excitation is very similar to the case of single support excitation presented in section 9.2. The static part dependent on time (second term of equation [15.5]) simply requires the calculation of deformations {ws,i} under displacement imposed on supports, which does not present any difficulties. The general formulation (case of multiple supports) thus does not seem to be significantly more complex than the normal formulation in relative displacements (case of single support). However, it is rarely used, no doubt because the data concerning displacements uai(t) is not considered very reliable (see Part 2). When it is necessary to take differential displacements between supports into account, usually the use of simple design rules is preferred for the calculation of their effects, or even limiting oneself to construction practices. 15.1.2. Formulations for block movement of supports
The assumption of block translation of supports is almost always utilized in seismic calculation; we are often satisfied with the consideration of horizontal components of this translation motion (case of earthquake engineering codes for regular constructions), as in the example discussed in section 9.2. Even in cases where the three-directional nature of seismic excitation is taken into account, the theorem of superposition of load cases in linear analysis makes it possible to carry
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out a separate calculation for each of the three components. The general presentation of modal spectral analysis for the translation of supports will be carried out in section 15.2 on soil-structure interaction (where the movement of supports is part of the unknown factors of the problem) in Chapter 2. The current section aims to present the case of block rotation of supports, which was mentioned in section 11.2.2 (see Figure 11.15). A building whose basemat is supposed to be non-deformable, is animated by a movement of horizontal translation displacement s(t) of center C of the basemat with relation to a fixed point 0) and rotation motion T(t) around C (see Figure 15.2). The system of absolute axes is denoted as XO Z, and the mobile system attached to the basemat as xCz.
Figure 15.2. Building whose basemat is subjected to excitation of horizontal translation and rotation
The coordinates in the absolute axes of a point of coordinates x and z in the mobile system are given by the relations: X = s + x cos T – z sin T
[15.25]
Z = x sin T + z cos T If s, T, x and z are functions of time t, it can easily be verified that components Ax and Az for absolute acceleration in the mobile system have the expression:
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Ax = X cos T Z sin T
Az = X sin T Z cos T
s cos T x 2 zT zT xT 2
[15.26]
s sin T z 2 zT xT zT 2
For small movements, cos T = 1 can be taken and the terms of the second order ignored. We thus get: Ax = s x zT
[15.27]
Az = z xT In a formulation in relative displacements with relation to system xCz related to the basemat, it is seen in the expression of Ax that the influence of rotation is translated simply by a term proportional to the z axis. Noting vector {Z} whose components are equal to the height above the basemat of structural points for degrees of freedom parallel to the direction of excitation, the equation for movement is thus written as: [M] ^u` > K @^u`
s > M @^'` T> M @^Z `
[15.28]
The presence of rotation excitation only modifies the second member with relation to the case of block translation of supports. Its resolution by development on the basis of eigenmodes is thus very similar to the presentation in section 9.2 (and mentioned in most general cases in section 15.1.1) and results in the following equations for development coefficients rA (t ) : rA 2[ AZA rA ZA2 rA
qAT pA s
[15.29]
which are identical to [9.63], aside from the fact that the second member now carries a complementary term which is the product of the angular acceleration T by participation factor qA in rotation of mode A; the expression of factor qA is obtained simply by replacing {'} with [Z] in formula [9.61] for the participation factor in translation:
^v A ` > M @ ^ Z ` T ^ v A ` > M @^ v A ` T
qA
[15.30]
Just as participation factors in translation are development coefficients of vector {'} on the basis of the eigenmodes (see [9.70]), factors qA are development factors of vector {Z}:
Linear Seismic Calculation
{Z} =
¦ qA ^vA `
615
[15.31]
l
The demonstration of this property, the immediate consequence of the orthogonality of eigenmodes, can be completely traced on the one given at the end of [9.2.4]. In the case of the perfectly regular building studied in [9.2], {Z} axis vector is such that: {Z}T = (h, 2h,…, Nh)
[15.32]
and we find, for factors ql, considering the relations [9.53] verified by modal deformations: qA
h § 2A 1 · § 2A 1 S · N S ¸ / sin ² ¨ sin ² ¨ ¸ N 2N 1 2 1 © ¹ © 2N 1 2 ¹
[15.33]
On the basis of this formula, we can, in the considered case N = 4 [9.2], put forth Table 15.1, which we can compare with Table 9.2.
1 q h A 1 q vA.1 h A 1 q vA.2 h A 1 q vA.3 h A 1 q vA.4 h A
ZA Zo
st 1 mode A=1
2nd mode A=2
3rd mode A=3
4th mode A=4
3.5736
0.3333
0.0783
0.0148
1.2410
–0.3333
0.1199
– 0.0276
2.3326
–0.3333
– 0.0416
0.0423
3.1426
0.0000
– 0.1054
– 0.0372
3.5736
0.3333
0.0783
0.0148
0.3473
1.0000
1.5321
1.8794
Table 15.1. Participation factors in rotation and products of these factors by components of eigenmodes in the case of a perfectly regular building with four stories working under pure shear
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We can verify that the sums of lines of terms qA vA,n (n=1, 2, 3 or 4) are equal to nh, in accordance with formula [15.31]. We also observe, comparing with Table 5.2, that these products qA vA,n decrease more rapidly with rank A of the mode, than similar products with the participation factor in translation. The predominance of the first mode is thus even greater for rotation excitation than for translation. This point will be discussed in section 15.2.1. What is the practical significance of rotation excitation" Orders of magnitude can be estimated while considering the propagation of a sinusoidal wave producing a vertical movement uz from ground surface (see Figure 11.15); for uz we take the expression: x ) c
uz = G sin Z (t –
[15.34]
G being the amplitude, Z the angular frequency and c the velocity of propagation in the horizontal direction x. For a building of width b in this direction, rotation T is given by: T=
1 ª u z x b u z x 0 º¼ b¬
[15.35]
On the basis of [15.34], after a basic calculation, we find for T: T=–2
G b
sin
Zb
b · § cos Z ¨ t ¸ 2c © 2c ¹
[15.36]
from which, for the maximum angular acceleration:
T max
2
Z ²G b
sin
Zb 2c
[15.37]
Z² G represents the maximum Az of vertical acceleration associated with wave [15.34]. We thus have:
T max
2
Az Zb sin 2c b
[15.38]
For a building of total height H, the horizontal acceleration resulting from angular acceleration T varies from 0 at the base to H T at the top. Its average value is thus H T /2, from which for maximum Ax of this average value according to [15.38]:
Linear Seismic Calculation
Ax =
1 H T max 2
Az
H Zb sin b 2c
617
[15.39]
or further, by introducing the wavelength / (= 2Sc/Z) of the wave [15.34]: Ax = Az
H § b· sin ¨ S ¸ b © /¹
[15.40]
The sinusoidal wave [15.34] thus produces, by effect of rotation, a horizontal acceleration proportional to its vertical acceleration, the proportionality coefficient depending on dimensions H and b of the building and the wavelength /. When it corresponds to relatively long periods (of about a second, or more than a second) these values are expressed in hundreds of meters, or even in kilometers (/ # 300 m for a Rayleigh wave having 1 s for the period propagated on the surface of a layer of alluvium; see section 3.2) and we can, in equation [15.40], replace the sine by its argument, which gives for the Ax / Az ratio: Ax Az
S
H /
[15.41]
For waves having quite long periods, the question of rotation excitation thus seems to apply only in the case of buildings of very great height (H more than or equal to 100 m) or in case of certain high-rise structures (large industrial chimneys, telecommunications towers). However, such structures have basic periods of several seconds (see formula [8.4] taken from the Japanese earthquake engineering code) and, so that they are significantly excited by a rotation movement (for which, as indicated earlier, the response of the fundamental mode is largely preponderant), it would require wave [15.34] to have a period of the same size, which would correspond to wave lengths in kilometers rather than hectometers and thus to Ax / Az ratios significantly smaller than one. Taking this discussion into account, it is logical to examine the case of a surface wave having the same period TI as the fundamental mode of the building, which represents the situation where rotation excitation is potentially the most dangerous. We suppose that TI is related to height H by the following formula, similar to [8.4]: T1 = T0
H H0
[15.42]
With T0 = 1s, we must take H0 = 50 m so that [15.42] is identical to [8.4] in the case of buildings with concrete structures (r = 0 in [8.4]) and H0 = 33m to obtain this identity in the case of metal structures (r = 1 in [8.4]). The wavelength / is, for
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condition of resonance, equal to the product of T1 by the propagation velocity c and we have, by using [15.40]: Ax Az
§ H b· H sin ¨ S 0 ¸ b © cT0 H ¹
[15.43]
or, by introducing notations:
O
Ax AZ
H (slenderness ratio of building); D b
§ O · §D · ¸ sin ¨ ¸ ©D ¹ © O ¹
D¨
S
H0 cT0
[15.44]
[15.45]
We see that D shows the upper boundary of the Ax / Az ratio since, in modulus, sine is always smaller than its argument. Therefore, with the values given earlier (T0= 1s, H0 = 50 m or 33 m depending on whether the structure is made of concrete or steel), D is generally distinctly lower than one (D = 0.52 for H0 = 50 m and c = 300 m/s). This upper boundary D is reached in reality when slenderness ratio O is rather large (more than or equal to 2, to give an example). We thus find, through this approach, that potentially dangerous rotation excitation depends essentially on slenderness or elongation (more than height) and that it corresponds at the most (in cases where superficial land is relatively soft) to half of the vertical excitation due to surface waves. The conclusion is as follows: the need to ask the question about rotation excitation for slender structures founded on soils of a mediocre nature arises only in cases where these waves (especially Rayleigh waves) represent an important contribution of seismic movement. Current practice of assessment of seismic hazard does not make it possible to distinguish the part which is due to surface waves in the characterization of movement (response spectra or accelerograms). This shortcoming is obvious in most earthquake engineering codes where movements are defined with reference to zoning and the type of soils, thus not precisely indicating data which would be necessary (magnitude, type of fault, focal depth, distance to the site in question) to evaluate the importance of surface waves. The inclusion of rotation excitation in certain parts of Eurocode 8 [CEN 00] is thus rather misleading (see section 12.2.2). It seems reasonable to limit the study of possible consequences of such excitation to exceptional buildings or structures, where the conditions of slenderness ratio and foundation, as well as location (for example in great sedimentary basins, some distance from the epicenter of earthquakes of strong magnitude), reveal that this risk cannot be excluded.
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15.1.3. Representation of damping
We have seen earlier (see sections 9.2.4 and 15.1.1) that in practice, the influence of damping was taken into account in a lump-sum fashion by introducing a term for viscosity in differential equations for functions rl (t), development coefficients of the solution on the basis of eigenmodes. A more “scientific” approach consists of constructing a damping matrix [C], based on hypotheses of a structural model, and we thus get, as the equation for movement: [M] {ü} + [C] { u } + [K] {u} = {F}
[15.46]
where {F} is the vector of external forces applied (forces of inertia resulting from ground acceleration in formulations in relative displacements, reactions of supports in formulations in absolute displacements).
Figure 15.3. Model with two masses with the expressions of matrices for mass, damping and rigidity
The resolution of equation [15.46] by the method of development on a modal base results in decoupled equations for coefficients of this development only if (see section 9.2.4) eigenmodes are orthogonal with relation to matrix [C], i.e., if: {vk}T [C] {vA} = 0 for k z A
[15.47]
{vk} and {vA} being two different eigenmodes. For matrices [C] constructed on the basis of a mechanical model, by introducing viscous damping between certain structural points there is no reason why relation [15.47] should be verified. For example, in the very simple case of the model with two masses represented in Figure 15.3, it is easily shown that the verification of this
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relation implies that dashpot coefficients are proportional to the stiffness of springs (c2 / c1 = k2 / k1). Considering the low value of structural damping and lack of detailed knowledge about their cause, it is acceptable to “overlook” the fact that [15.47] is generally not verified and consider that modal responses are definitely decoupled, which is the justification of the method of modal analysis. It is thus not necessary to explain the damping matrix in normal applications of this method. On the contrary, for transient calculations that do not call upon a development of a modal base, it is required to have an expression of this matrix [C]. Apart from the construction of [C] on the basis of a model, common practice consists of using the Rayleigh hypothesis which supposes that [C] is a linear combination of matrices of mass and rigidity: [C] = D [M] + E [K]
[15.48]
This formulation shows the advantage of simplicity (as the construction of [M] and [K] is a prerequisite for all linear calculations of seismic response) and in addition, which is important in certain applications, it ensures the verification of equation [15.47]. In fact, since two different eigenvectors are orthogonal with respect to matrices [M] and [K] (see [9.39] and [9.40] or [15.14] and [15.15]), they are also with relation to form [15.48] of [C]. From the physical point of view, the justification of this expression is not very clear, at least where the term proportional to the mass matrix is concerned. Rayleigh’s hypothesis is thus essentially a useful device making it possible to obtain a plausible expression of the damping matrix at little cost (and preserving the important property of decoupling of modal responses). In practice, the determination of coefficients D and E of combination [15.48] is carried out in the following manner. When we look for the solution of equation [15.46] by the development: {u} = ¦ rk (t ) ^vk `
[15.49]
k
we find, after the replacement of [C] by [15.48] and premultiplication by {vl}T, the following equation for the function rA (t):
^vA ` ^ F ` T ^ v A ` > M @^ v A ` T
rA ª¬D EZA2 º¼ rA ZA2 rA
[15.50]
If we use the coefficient of rA in canonical form 2[A ZA, we find, for reduced damping [A:
Linear Seismic Calculation
[A =
· 1§ D EZA ¸ ¨ 2 © ZA ¹
621
[15.51]
We determine D and E by writing that [A is equal to a given value [o for A = 1 (fundamental mode) and A = n (high mode, possibly the latter if we consider n equal to the total number of degrees of freedom of the structure). Thus, we find:
D = [o
2Z Z ;E Z1 Zn 1
n
[o
2
Z1 Zn
[15.52]
With this choice, reduced damping [A is lower than [o (which is in terms of safety) for all modes of indices included between 2 and n – 1, as the variation of [A based on ZA has the appearance indicated in Figure 15.4 which shows that the minimum value of [A is reached for ZA =
Z1Zn , thus between Z1, and Zn, and is
equivalent to:
[A, min = [o
2 Z I Zn
Z Zn
< [o
[15.53]
1
For modes whose eigenangular frequency is more than Zn, the reduced damping is higher than [0. Therefore, to be sure of remaining within the scope of safety, it is necessary that these modes do not significantly contribute to the response, which is generally true for the overall response (which is controlled largely by the first modes and especially by the fundamental mode) but cannot be verified for certain local responses. A good practice is to take the angular frequency of the first mode whose frequency exceeds the cut-off frequency of the elastic design spectrum for Zn. In effect we have seen in section 9.1.1 that beyond the cut-off frequency, eigenmodes have rigid behavior (i.e., they follow the movement of the support) which does not depend on damping.
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Figure 15.4. Variation of reduced damping in Rayleigh’s hypothesis
Rayleigh’s hypothesis [15.48] is a special case of the more general formula: [C] = [M]
¦ D > M @ > K @ 1
n
n
n
[15.54]
where integers n and coefficients Dn can be arbitrary. It is easily shown that matrices [C] defined in this way, verify the condition of orthogonality of modes [15.47] and that the following relation, analogous to [15.51], is obtained:
[A =
1 ¦ D ZA2n 1 2 n n
[15.55]
It is thus possible, in principle, to find a set of values for n and Dn making it possible to obtain, for each mode, a given value of the reduced damping coefficient, while conserving the decoupling of modal responses. Considering the burden of the corresponding heavy calculations (for products and power of matrices which intervene in equation [15.54]), this possibility is practically never used and only has theoretical value (other than cases corresponding to Rayleigh’s hypothesis for which we accept n = 0 and n = 1). The representation of the damping, which is carried out in a fixed manner in equations of modal responses or by the construction of Rayleigh’s matrix, rests on the reference to a given value [o, judged as appropriate after taking into account the characteristics of the structure and the level of applied forces. The choice of [o plays an important role as the response is approximately proportional to the inverse of its square root (see sections 9.1.1 and 10.2.1) but is very difficult in practice because of:
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– the fact that damping depends not only on the structural system (load bearing and bracing elements) but also on non-structural elements whose deformations and internal frictions can contribute significantly to the dissipation of vibratory energy; – the influence of the level of excitation on that of damping; in structures sustaining strong forces, mechanisms which are not manifested during weak levels of vibrations appear (filling up of chinks, friction in assemblies, plastic behavior in certain zones); – the intervention of external causes on the apparent damping in the structure in question, resulting from its relations with the ambient environment. The phenomenon of damping of radiation is particularly noteworthy (see section 16.1.3) which corresponds to waves emitted in the ground by vibrations of the foundations. These waves, whose characteristics depend on the nature of ground on the site, bring with them a part of the vibratory energy of the structure and thus contribute to the damping of its response; – the lack of a comprehensive diagram for viscous damping (proportional to relative velocities) to describe the dissipation phenomena which are rather of a hysteretic nature (proportionality to plastic displacements). These difficulties explain that recourse to damping measures do not make it possible, in most cases, to resolve the problem of the choice of [o, even though measurement techniques and identification of apparent damping are well mastered in the majority of cases. If work is carried out in situ, i.e. by subjecting an existing structure to a given dynamic excitation and measuring its response, the damping that is determined is generally not usable for seismic calculation as the level of excitation is too weak and the part due to radiation damping is not known a priori. If laboratory tests are carried out, elements for the estimation of purely structural damping linked to the nature of materials can be obtained, but it is very difficult to reproduce real conditions of dissipation of vibratory energy for complex structures (non-structural elements, assemblies). Tests at low level on very simple elements (for example, beams of reinforced concrete) typically give very low values for reduced damping (less than 1%) while “regulatory” values are about a few percent.
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PS92
Non-reinforced concrete Reinforced concrete Prestressed concrete
3 4 2
Welded steel Bolt steel Pipelines (I ! 305) Pipelines (I 305)
2 4
Reinforced masonry Reinforced concrete masonry
6 5
Glue-laminated wood Bolted wood Nailed wood
4 4 5
RG 1.61 OBE
RG 1.61 SSE
Japanese Nuclear Stations
4 2
7 5
5
2 4 2 1
4 7 3 2
1 2 0.5 0.5
Hard rock (Vsa1.5km/s) Weathered rock (Vsa0.5km/s)
5 10
Table 15.2. Comparison of reduced damping (in %) in different documents with regulatory prescriptions
The most reliable damping evaluations are deduced from past experience of earthquakes in cases where recordings inside a building are available; we can then determine the value of damping which makes it possible to get the best approximation of measured responses through calculation. Most earthquake engineering codes and other documents with regulatory status have lists of damping values, classified according to the type of material. Table 15.2 gives some examples of such lists, taken from the following references: – PS92 Rules, currently in use in France [AFN 95]; – Regulatory Guide 1.61 of the United States Nuclear Regulatory Commission; this document distinguishes between the case of OBE (operating basis earthquake), for which the stress is limited to half of the elastic limit, and SSE (safe shut down earthquake), where the elastic limit is reached [USA 73]; – Japanese practice for seismic analysis for nuclear power stations [RG 79].
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The study of this table calls for the following notes: – the two columns related to Regulatory Guide 1.61 show that the doubling of the level of stress from half of the elastic limit (OBE) to the elastic limit (SSE) more or less corresponds to a doubling of the damping value; – significant differences can exist (between American and Japanese regulations for pipelines), which are not explained by differences in the technology of the equipment in question but differences of estimation of safety margins. The damping coefficient is often, more than an objective reality, a means to calibrate the severity of the case of seismic load and its contribution to design; – values retained in the PS92 Rules, in cases where we can compare them with those in Regulatory Guide 1.61, are in line with OBE values and not SSE values. This confirms, as indicated earlier, that the elastic calculation required by earthquake engineering codes is only a reference calculation (meant to prepare the division by the behavior coefficient) as it is done with damping values corresponding to moderate stress levels (half of the elastic limit according to the Regulatory Guide) while we aim for structural design in the plastic domain; – differences between “welded steel” and “bolted steel” values clearly demonstrate that it is how the jointing is carried out more than the nature of the material, that has an influence on the damping value; – damping given for rock in the Japanese regulations correspond to quality factors of 5 or 10 (see [3.50]), very low when compared to those admitted in seismology in compact materials of the Earth’s crust, which can reach several hundreds. These observations show that “regulatory” values for damping are not scientifically validated precise values but simply orders of magnitude judged as plausible by expert opinion, in a given context of choice and “the desire to display” safety coefficients. Lists such as those in Table 15.2 have the advantage of being available and making it possible for the designer to quantify a parameter that he has no means to estimate on his own. They constitute only one element, amongst others, in the search for a certain level of safety and especially must not be dissociated from other rules of calculation and structural design according to the regulations of which they form a part. This implies that, in principle, we do not have the right to “fish around” in these lists at will to take the damping values that seem most appropriate (in a favorable or prudent manner) for the application that is being processed. Ultimately, damping being only one of the parameters which determines the level of linear elastic response, its choice is really important only if this response is
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used as such for structural design. In the approach of earthquake engineering codes, where elastic forces are divided by a behavior coefficient of about a few units, and whose values are essentially empirical, it is rather illusory to “refine” damping further. Eurocode 8, based on this notion, favors the value of 5%, which is a suitable order of magnitude for a large majority of current constructions. In certain cases, the damping can have significantly different values from the average structural level of 5%; as, for example: – oscillations of free surface of liquids contained in large tanks, basins or pools. These oscillations, whose calculation is important to assess the risk of overflow or impact against the roof of the tank, have very low damping and they are associated with values of [ in the order of 0.5%. – cable trays which, when subjected to strong shaking, are the seat of slip and friction between cables on the one hand and between the cables and their metallic supports on the other hand, which has a strong effect of damping ([ of about 10 to 15% according to laboratory tests). – structures equipped with special damping devices, such as certain bridges where some of the deck-abutment or deck-bridge pier joints carry hydraulic or mechanical jacks developing significant forces in case of rapid differential transient movements; equivalent linear damping of several tens of percent can then appear (see section 17.2.5). – effects of radiation in soil-structure interaction, already mentioned earlier (and which will be further detailed in section 16.1.3); for certain modes of interaction, especially pumping (vertical movement of the foundation) this effect corresponds to apparent reduced damping that can reach 50% or even 100% in some cases (overcritical damping). In the eventuality of very high damping manifested in a localized manner (special devices or soil-structure interaction), conditions [15.47] of decoupling of modal responses are generally no longer verified, even in an approximate manner. Numerical integration configurations for equation [15.46] are then used to “advance” the solution {u} from time step to time step. Such configurations have a certain “numerical” rate of damping which is added to the “physical” damping introduced in the model through the medium of matrix [C]. This “camouflaged” cause of damping needs to be carefully controlled which, in some cases, can be more significant for attenuation of the response than “real” damping. Certain calculations with finite elements of soil-structure interaction are very debatable from this point of view (see section 16.2). Earlier considerations concern cases of damping of a homogenous structure or a homogenous part of a composite structure. The case of structures having different
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parts made from different materials or having different configurations for bracing elements is mentioned in section 15.2.1 (equation [15.117]). 15.1.4. Notes on modeling
Modeling work, which was discussed as an example in section 9.2, is the noble aspect of seismic calculation. At a time (in the 1970–1980s) when means of computerized calculation were much less often used or effective as they are today, designers were naturally aware of the importance of this work as it was necessary to “economize the degrees of freedom” and thus choose those that enabled a good representation of real deformation capacities with a limited number of structural points. Current software used for calculation is able to easily process large-scale models (several hundreds, even thousands of degrees of freedom) and is generally equipped with aids for modeling, which can lead us to believe that the stage of model construction is less crucial than before. In reality, this is still an essential stage, as it implies a certain number of choices related to the final calculation and the necessity of adopting simplified hypotheses for certain aspects; in addition, it largely conditions the practical possibilities of design optimization. The first issue to be considered is the level of complexity of the model. If we expect seismic calculation to directly supply elements enabling the detailed verification of the structure, by combining with the other load cases, it seems natural to choose a sufficiently detailed model to reach this objective. It is now common practice, for complex structures, to consider models with finite elements having several thousands degrees of freedom. However, very often there may be two ways in which to proceed: – dynamic seismic calculation on a relatively simple model on the basis of which a field of static forces considered equivalent to the effects of seismic action is deduced; – verification of the structure with the help of a detailed static model taking into account the totality of load cases, including the earthquake represented by the previously determined force field. This procedure in two steps is applied in the following cases: – at the designing stage, when different variants for certain parts of the structure are studied; – when the variation of certain parameters needs to be considered, for example the mechanical characteristics of the ground, when a standard design is envisaged for a structure which can be constructed at different sites;
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– the study of margins, where a nonlinear model is traced on a linear model making it possible to explore modes of damage and destruction for different types of accelerograms. This operation is possible at a reasonable cost with relatively simple models. Very detailed models are thus normally reserved for the final stage of study, when the characteristics of the structure are considered to be definitive. However, it may happen that the complexity of the structure is such that there is no possibility other than making a detailed model, even at the preliminary stage of the study. If “stick” type models (see section 9.2) make it possible to correctly process most of the buildings, they can be very difficult to define when the structure has several coupled parts, especially with relation to the effects of torsion. The construction of simple models, sufficiently representative of real behavior, is generally more difficult than that of complex models and requires a level of expertise which can only be acquired through experience. Apart from cases of regular buildings, where “stick” models can be easily constructed by following some simple rules, the definition of such models is based on a sense of “mechanical synthesis” which implies sound knowledge of strength of materials, which unfortunately is becoming a dying skill. In addition to the choice of complexity level, the establishment of the model is the result of an entire set of decisions about elements (structural and non-structural) that must be represented, their representation mode (mass only or mass and stiffness) and their characterization in mechanical terms (elasticity modulus, taking into account the degree of cracking or not, possible need to consider a range of values rather than a single value). These different aspects will now be discussed. The first question is about the representation of secondary structures and the equipment contained in the structure of interest, which is most often unlikely to significantly influence the overall response, through their own response. We thus have to be satisfied, in the majority of cases, with representing them in the calculation model in terms of their mass. However, it is possible that some elements of significant mass have sufficiently supple supports for the question of dynamic coupling between these elements and the main structure to be raised. Criteria regarding the ratio of the mass of the secondary structure to the main structure are available, as well as the ratio of fundamental frequencies of vibration, that clarify cases in which coupling is to be taken into account. A rule that is commonly used in the nuclear industry consists of (see Figure 15.5): – overlooking the coupling if the ratio of mass is less than or equal to 1%; – taking coupling into account if the ratio is more than 10%;
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– also taking coupling into account, for ratios of mass between 1 and 10%, if the frequency ratio is between 0.8 and 1.25.
Figure 15.5. Criteria for dynamic coupling between the secondary structure and main structure in the nuclear industry
The consideration of dynamic coupling generally tends to attenuate the response of the main structure as, if there is a coincidence of frequencies, the secondary structure behaves in the same way as a tuned-mass damper, i.e., it attracts a significant part of the total vibratory energy for its own vibration. This principle is used in high rise buildings to reduce their dynamic movements (due to wind or seismic origin) by installing a mass on top of it of about one percent of the total mass, with supports designed to ensure the coincidence of frequencies (see section 18.3.2). The amplifying effects of coupling thus essentially concern the secondary structure and are generally overlooked in the “normal risk” approach. The representation of mass in the model may require special consideration in the following cases: – structures where the mode of concentration of masses can have a significant influence on the deformation of eigenmodes, as well as their period. This situation is normal for modes with very short periods, which do not generally have an important role in the overall response, but can also concern significant modes with fairly long periods, when the forces of inertia due to movements of masses produce considerable moment effects on the distribution of deformations; masses of insufficient number or placed too close to the potential rotation axes are likely to lead to an underestimation of these effects; – an example of modeling of a metal pylon [CAP 85] shows a variation of a factor of about 2 for eigenperiods according to the chosen option for the concentration of masses. The introduction of degrees of freedom for rotation and associated inertia moments can be a solution but it is easier to implement for effects
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of torsion around a vertical axis than for bending of the horizontal axis (in which deformations of horizontal elements, beams and floors, often have the appearance shown in Figure 9.11); – structures in which certain parts do not sustain forces applied in the same manner according to the direction considered for seismic excitation. For example, the moving trolley of an overhead traveling crane can be considered as joined to the beams with respect to the horizontal excitation perpendicular to them and vertical excitation. However, in the direction parallel to the beams, the joint depends only on the friction between the wheels and their tracks (which only exists in case of drive wheels, as non-driving wheels are “rotating” and only attract rolling friction which is quite negligible). This friction joint is not sufficient to maintain stability of the clamp with its track in case of a violent earthquake. In the model for seismic calculation, the mass of the trolley will thus have its full value for horizontal excitation perpendicular to the beams and for vertical excitation, and a reduced value, or even zero, for the other horizontal direction. Software often comprises of commands enabling the activation or non-activation of masses according to the direction of the excitation; – structures consisting of significant liquid mass susceptible to oscillation under the effect of seismic excitation. In cases where wall elements containing these masses can be considered rigid, the theory of incompressible ideal fluids [DAV 82] makes it possible to show that a correct representation of the effects of horizontal excitation is obtained by plotting a diagram of the liquid in the following manner (Figure 15.6): – mass Mi (i for impulsion) joined rigidly to the wall elements; – mass Mc (c for convection) joined elastically to the wall elements and representing the part of the oscillating liquid near the free surface.
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Figure 15.6. A seismic model for a tank with rigid walls
Masses Mi and Mc (it is to be noted that the sum Mi + Mc can be different from the total mass of the liquid) and their distances hi and hc to the base of the tank are given by analytical formulae for tanks with a simple form (cylindrical or parallelepiped) [DAV 82], as well as the oscillation period of mass Mc; for a cylindrical tank with vertical axis, of radius R and height H, we have for period Tc: g H ª § Tc = 2S «1.837 tank ¨ 1.837 R R © ¬
·º ¸» ¹¼
1/ 2
[15.56]
These oscillation periods are long for tanks of current dimensions (we find for example through [15.56] Tc = 3.39 s for H = R = 5 m). The result is that corresponding accelerations on the response spectrum for calculation are low, which leads to the fact that a model in which free surface oscillation would not generally be taken into account provides pessimistic results for forces exerted on the walls. The effects of oscillation can thus be overlooked, if a conservative approach is adopted to verify resistance. However, it is still necessary to estimate the height of the wave to evaluate the risk of shock against the roof of the tank, which is a frequently observed cause of damage. As indicated in section 15.1.3, the calculation of the height of the wave must be carried out with an appropriate value of damping, which is very low, about 0.5%. If the liquid is on the outside of the structure under study (totally or partially submerged structures), it is necessary to consider it in the modeling through the introduction of added masses representing the mass of liquid which is set into
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motion by the displacements of the structure. We show for example that, for a cylinder which is displaced perpendicularly to its axis, the added mass is equal to the liquid mass having the same volume as the cylinder [LAN 59]. As the name added mass indicates, it is necessary to add the liquid mass and the cylinder’s mass in writing the matrix for mass, taking care to see that, normally, added masses in reality form a tensor and thus show non-diagonal terms in this matrix. For submerged structures in large volumes of water (harbor works) added mass is in the order of that of the volume of displaced water during immersion (and even equal to this volume in case of a cylinder as mentioned above). Structures containing relatively thin layers of liquid between metal shell elements can show effects of added mass beyong comparison with the mass of the layers. In order to understand this, a circular plate of radius R placed at a rather low distance h from a rigid wall (Figure 15.7) is considered. If the plate is displaced towards the plate with velocity G , a radial flow is created whose velocity v, at distance r from center 0, is obtained by volume balance (incompressible liquid):
Sr² G = 2Sr h v
[15.57]
A pressure field p (r,t) is associated with this flow which verifies the following equation [AFP 90]: wv wv I wp v wt wr U wr
[15.58]
0
U being the mass density of the liquid; overlooking the nonlinear term vwv/wr and taking into account [15.57], we have: wp wr
1 r U G 2 h
[15.59]
and, by integration with the condition p = 0 for r = R: p=
1 G U R² r ² 4 h
[15.60]
This pressure distribution produces the force Fp exerted on the plate: Fp =
³
R
O
2S r p dr =
S 8
U
R 4 G h
[15.61]
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Figure 15.7. Radial flow created by moving a circular plate towards a close by fixed wall – this flow produces a force on the plate which has the character of the force of inertia (proportional to acceleration G and opposite to the movement)
Force Fp is a force of inertia, as it is proportional to G and is opposite to the movement of drawing closer. Added mass M which corresponds to it is according to [15.61]: M=
S 8
U
R4 h
1 R² M0 8 h²
[15.62]
where Mo = SR² hU is the liquid mass between the plate and the wall. It is seen that the ratio M/Mo can have very high values if h is slow compared to R. Added mass is thus very significant for certain industrial equipment made up of fitted vessels with annular spaces filled with liquid. Their consideration can necessitate rather complex formulations when the walls are deformed or when the assumption of incompressibility of the fluid is not justified [GIB 85]. To draw a conclusion on the representation of mass in models for seismic calculation, we must point out the question of the effective presence of temporary mass in current operating situations of buildings and structures. This question is treated in earthquake engineering codes through simultaneity coefficient data
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indicating what part of temporary mass has to be taken into account in the calculation. In the case of industrial structures, where we may encounter rare situations where certain elements support significant loads (handling of heavy materials, special storage), we need to think about whether it is necessary to consider the occurrence of an earthquake during these periods of short duration. It is up to the owner of the construction to provide the answer considering the relevant safety objectives. The representation of stiffness also raises two questions, one about the choice of mechanical properties of materials (influence of cracking, and possibly, the dynamic nature of seismic load) and the other about the contribution of non-structural elements. It does not seem justified, for materials used currently in construction work (concrete, steel, masonry and wood), to consider “dynamic” values of the elasticity modulus. These values were highlighted during tests in rapid dynamics (for example impact of projectiles), but are not applied in dynamics of relatively slow seismic loads. The option chosen by almost all earthquake engineering regulations is thus to use normal values of elasticity modulus (static modules) in seismic calculation, even though certain publications [RAP 84] claim to have results of tests showing the increase in modulus of concrete under seismic forces. For certain special material such as elastomeric materials used for supports or earthquake-resistant damping devices (see section 18.3), the modulus can depend on the frequency of loading cycles and its determination generally requires tests on samples (these tests also help to provide reference values to monitor the effects of aging). For soils, as already indicated in section 5.3.1, the modules to be used in seismic calculation, often wrongly called “dynamic”, are in fact static modules corresponding to relatively low deformations, certainly much lower than those resulting from static tests for identification of soils (as in penetration tests). “Seismic” modules are typically about 60 to 80% of the modulus for low deformation, that can be deduced from geophysical tests such as crosshole testing and can be much higher (of a factor of 10 or more) than static modules. Where the influence of cracking is concerned, there are two cases that can be brought up: – if the results of linear seismic calculation are used directly for structural design, i.e., if we aim to obtain nearly elastic behavior of the structure, then the evaluation of stiffness should be as realistic as possible and the effects of cracking need to be taken into account, which often involves processing through iterations, as the level of cracking depends on the response;
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– if the linear seismic calculation is only a reference calculation, based on which the real nonlinear response is evaluated by means of a behavior coefficient, then the evaluation of stiffness is purely conventional and earthquake engineering codes have chosen the convention of uncracked sections. The question of contribution to the stiffness of non-structural elements, already discussed in section 9.2.1, mainly concerns filling structures with portal frames with masonry. If particular dispositions are not taken to loosen the masonry panel from the basic frame, a diagonal compression rod develops in the panel under the action of horizontal force F on one of the corners (see Figure 15.8).
Figure 15.8. Transmission of horizontal force F through a diagonal rod of compression in a masonry panel
A simple calculation shows that horizontal stiffness kh of this rod i.e., the relation between force F and horizontal displacement Gh of its point of application, is given by the formula: kh = Ed Sd
A cos T = Ed Sd h² A² d
[15.63]
h and A being the height and width of the panel, d = h² A ² the length of the rod Ed and Sd Young’s modulus for masonry and the cross-section of the rod. For current masonry (made with stones, bricks or concrete blocks) the values for Ed are about 3,000–4,000 Mpa (with strong dispersion), or about a tenth of the modulus for concrete. Earthquake engineering codes give details of dimensions to be taken to calculate the cross-section of rod Sd. For example, PS92 Rules [AFN 95] propose formulae such as those given below:
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Sd = e x min (4e, d/10)
[15.64]
e being the panel thickness. With these values for Ed and Sd, we observe that formula [15.63] provides stiffnesses generally greater than values that correspond to the bending of columns of the structural framework. For example, by taking expression [9.25] for bending stiffness kp of a column of square cross-section a², we find, for the stiffness ratio: kh/kp =
4 Ed e ² A h 3 E h² A² a 4
[15.65]
By assuming in [15.64] that 4e d/10; we thus have, with plausible choices A = 2h and a = 2e: kh/kp =
1 Ed § h · ² ¨ ¸ 10 E © e ¹
[15.66]
With Ed/E = 0.1 this ratio is higher than 1 if h is greater than 10e, which is normally the case. Masonry panels are therefore at least as stiff as the columns of the structural framework and must be taken into account in modeling. If the panels are stiff in their planes, they are generally less resistant with respect to perpendicular forces applied to their plane. It can thus be asked if the component of seismic action in this perpendicular direction does not distort the panel to the extent of preventing the development of compression rods. Even if this question may seem legitimate, it is preferable to take into account the fillings in the stiffness model, considering the importance of their contribution. Once the elements influencing stiffness are determined and modulus of their composing material is chosen, the calculation of stiffness is carried out by methods of strength of materials or, in complex cases, with the help of models with finite elements. Great care must be taken to define the position and fixing conditions of joints (articulation, fixed support, elastic joint) at the ends of linear elements for which stiffness is calculated. It must be remembered that the stiffness of beams working under bending motion is inversely proportional to the cube of their length (see [9.25]) and can therefore vary a lot for a relatively low variation. In the case shown in Figure 15.9, the joining beam between rigid slabs D1 and D2 must not be represented according to diagram (2) (joint between neutral axes of slabs) but
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diagram (1), which restores the actual length subjected to bending, through rigid joining elements.
Figure 15.9. Diagram of joint working under bending motion between two rigid elements: correct (1) and incorrect (2) representation; according to [CAP 82]
15.2. Modal spectral analysis for block translation of supports
From the point of view of the hypothesis described in section 9.1, i.e., a formulation in terms of relative displacements with relation to a rigid support on which a translation motion is imposed corresponding to accelerograms given in the three directions in space (two horizontal and one vertical). Considering the linearity, the effects of these three directions of excitation can be considered separately; we thus reconsider, for the equation of movement, formulation [9.34]: [M] {ü} + [C] ^u ` + [K] {u} = – s [M] {'}
[15.67]
with notations which have already been used: [M] mass matrix, [C] damping matrix (often omitted and replaced by the direct introduction of a term for damping in equations of modal responses), [K] stiffness matrix, {u} relative displacement vector, s function of time, defining the accelerogram in a given direction characterized by vector {'}, whose components are equal to 1 for the degrees of freedom parallel to this direction and 0 for the other degrees of freedom. Solution {u} of equation [15.67] corresponds to development on the basis of eigenmodes {vk}: {u} =
¦r
k
k
( t ){v k }
[15.68]
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which verify the relations: ([K] – Z²k [M]) {vk} = 0
[15.69]
{vA}T [M] {vk} = {vA}T [K] {vk} = 0, k z A
[15.70]
15.2.1. Eigenmodes and quantities attached to modes
Non-damped eigenmodes {vk} defined, at a close multiplying constant, by equation [15.69] are elements which are naturally essential for the description of linear dynamic response. We have seen in section 9.2 that the essential part of this response was in the fundamental mode for regular structures. Differential decoupled equations which define development coefficients rk(t) [15.68] bring in angular frequencies and participation factors of modes and have the form (see [9.63]): r + 2[ A Z A rA A
ZA2 rA = – p As
[15.71]
with pAfor the participation factor (see [9.61]):
^vA ` > M @^'` T ^ v A ` > M @^ v A ` T
pA =
[15.72]
As indicated in section 9.2.4 participation factors do not themselves have any physical significance as their values depend on the arbitrary multiplying constant chosen to define modes. It is the product of pA by the vector {vA} which is meaningful; we refer to relation [9.70]:
¦p
k
{vk} = {'}
[15.73]
k
This relation makes it possible to define modal mass PA which plays an important role in the selection criteria of modes (see section 15.2.2). The following scalar quantity is calculated: Mt = {'}T [M] {'} by replacing {'} by its expression [15.73], that is:
[15.74]
Linear Seismic Calculation T · § § · Mt = ¨ ¦ pk ^vk ` ¸ > M @ ¨ ¦ pA ^vA ` ¸ © A ¹ © k ¹
639
[15.75]
Taking the relations of orthogonality into account [15.70], the only non-zero terms of the product are those for which k is equal to A and we have: Mt = ¦ pA2 {v A }T > M @^v A `
[15.76]
A1
The quantities PA defined by:
^v ` > M @^'` ² [M] {v } = T
PA =
2 pA
{vA}T
A
^ v A ` > M @^ v A `
A
T
[15.77]
are called modal masses. Their expression shows that they are independent of the multiplying constant of modes and that their sum is equal to Mt, which is nothing but the total mass of the structure. This is evident for diagonal matrices for mass, such as the one obtained in the example seen in section 9.2, as with:
We have: N
{'}T [M] {'} = Mt =
¦m
k
[15.78]
k I
In the general case (non-diagonal [M]), the identity between Mt and the total mass comes from the following expression of kinetic energy associated with relative velocities: Ec =
1 { u }T [M] { u } 2
[15.79]
This relation follows from the equation of free oscillations without damping, i.e., equation [15.67] in which [C] = 0 and S = 0:
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Seismic Engineering
[M] { u } + [K] {u} = 0
[15.80]
Premultiplication by { u }T, gives: { u }T [M] { u } + { u }T [K] {u} = 0
[15.81]
Therefore, from the symmetry of matrices [M] and [K] it follows that:
^u` > M @^u`
1 T T ^u` > M @^u` ^u` > M @^u` 2
^u` > K @^u`
1 T T ^u` > K @^u` ^u` > K @^u` 2
T
T
d §1 T · ¨ ^u` > M @^u` ¸ dt © 2 ¹ d §1 T · ¨ ^u` > K @^u` ¸ dt © 2 ¹
We can thus write [15.81] as follows: d §1 T T · d §1 · ¨ ^u` > M @^u` ¸ ¨ ^u` > K @^u` ¸ dt © 2 ¹ dt © 2 ¹
0
[15.82]
from which, by integration: 1 1 T T ^u` > M @^u` ^u` > K @^u` 2 2
Constant
[15.83]
The second term of the first member being the energy of elastic deformation Ed, the first term is kinetic energy Ec as in non-damped free oscillations; the sum of these two energies is constant. Formula [15.79] is thus established and, in applying it to a movement of constant velocity equal to 1, we find: Ec =
1 2
{'}T [M] {'} = ½ Mt
[15.84]
This shows that Mt, defined by [15.74], is the total mass of the structure. Modal mass PA (see [15.77]) is important to evaluate the contribution of different modes of the global response, as modes of large modal mass are most significant a priori. Most often in practice, where the multiplying constant of modes is chosen so that we have: {vk}T [M] {vk} = 1
[15.85]
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We see in [15.77] that modal mass is then equal to the square of participation factors. In the case of excitation in rotation of the structure support, which was studied in section 15.1.2, we saw that the process is quite similar to that of translation excitation, on the condition that direction vector {'} is replaced by height vector {Z} whose components are equal to the heights of structural points above the base. This vector verifies the following relation, similar to [15.73] (see [15.31]): {Z} =
¦q ^v ` k
[15.86]
k
k
qk being the participation factor in rotation for mode k, which has the following expression (see [15.30]): qk =
^v k `T ^M`^Z` ^v k `T ^M`^v k `
[15.87]
By analogy with the presentation given earlier for modal mass, the following scalar quantity is defined: It = {Z}T [M] {Z}
[15.88]
and it is shown using [15.86], that it is written as: It =
¦q ^v ` ^M ` ^v ` T
2 A
A
[15.89]
A
A
It is the inertia moment with relation to the plane defining the zero height, as we see it in [15.79] by taking {u} to be a rotation movement at constant angular velocity equal to the unit, i.e.{ u } = {Z}. Quantities iAdefined by:
^ v ` > M @ ^ Z ` q ^ v ` > M @^ v ` = T
iA
2 A
A
T
A
A
^v A ` > M @^v A ` T
2
[15.90]
are modal inertia moments, independent of the multiplying constant of modes and whose sum is equal to It. For perfectly regular buildings studied in section 9.2, the expressions for participation factors pA and qA have already been given (see [9.69] for pA and [15.33]
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here for qA). For modal mass and modal inertia moments, we find, after basic calculations of trigonometric sums:
PA Mt iA It
1 § 2A 1 · § 2A 1 S · N S ¸ / sin ² ¨ sin² ¨ ¸ , A = 1, 2,…, N N 2 N 1 © 2N 1 ¹ © 2N 1 2 ¹ § 2A 1 S · A 3 § 2A 1 · sin ² ¨ N S ¸ /sin4 ¨ ¸, 2 N N 1 2 N 1 ² © 2N 1 2 ¹ © 2N 1 ¹
1, 2,…, N
[15.91]
[15.92]
In the case where N = 4, considered for numerical applications in section 9.2.5, values calculated by these formulae are as follows (Table 15.3). It is observed that the predominance of the first mode, which is already very pronounced for modal mass, becomes overwhelming for modal inertia moment. This note, which was made in section 15.1.2, has general scope, at least for relatively regular buildings. A
μA Mt
iA It
1
0.8934
0.9876
2
0.0833
0.0111
3
0.0196
0.0011
4
0.0037
0.0002
Table 15.3. Reduced modal mass and reduced modal inertia moments in the case of a perfectly regular building with 4 stories under pure shear
The importance of modes can also be appreciated by considering their contribution to the deformation energy Ed of the response, which is given by the relation: Ed = ½ {u}T [K] {u}
[15.93]
in which solution {u} verifies equation [9.72]: {u} =
¦ x t, Z , [ p ^v ` i
i
i
i
[15.94]
i
x (t, Z, [) being the solution of equation [9.3] for the definition of the response spectrum associated with accelerogram s (t):
Linear Seismic Calculation
x 2[Zx Z² x
s
643
[15.95]
By developing [15.94] in [15.93], considering the orthogonality of modes with respect to matrix [K] (see [15.70]) we get: Ed =
1 T ¦ pA2 x² t , ZA , [A ^vA ` > K @^vA ` 2 l
[15.96]
Or again, taking into account [15.69] and [15.77]: Ed =
1 ¦PAZA2 x² t , ZA , [A 2 A
[15.97]
Quantities eA defined by: 1 PAZA2 x ²(t , ZA , [ A ) 2
eA
[15.98]
are modal energies. Unlike modal masses or modal inertia moments, which depend only on characteristics of modes, modal energies also depend on the response of modes thus on excitation imposed on the support. Function x (t , ZA , [ A ) has, by definition [9.15] of the response spectrum, the maximum value Sd ZA , [ A , which leads to: Max eA t
1 PAZA2 S d2 ZA , [ A 2
1 PA S v2 ZA , [A 2
[15.99]
Sv (Z, [) being the pseudo-velocity [9.16], where we again verify, as in section 9.1.1, its significance in terms of energy. The fact that modal energy depends on excitation complicates its use in evaluating the relative importance of modes with relation to the use of modal mass which does not depend on it. The selection criteria of modes (see section 15.2.2) that are commonly practiced thus especially call upon modal mass. We can however give two rules for the upper boundary of deformation energy Ed to make it possible to evaluate the orders of magnitude. These rules are expressed by the following two very simple relations: 1 Ed M tV ² 2
[15.100]
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Seismic Engineering
Ed Ed, s (A)
[15.101]
V being the maximum for pseudo-velocity of the response spectrum and Ed,s (A) the deformation energy associated with static deformation {w} under the action of uniform acceleration equal to A, maximum in pseudo-acceleration of the response spectrum. Rule [15.100] follows immediately from [15.97] and [15.99], as we can write: Ed
1 Max eA V ² ¦ PA t 2 A
¦ A
[15.102]
which brings us back to [15.100], as the sum of modal mass is equal to the total mass Mt. To establish [15.101], the definition of static deformation {w} under uniform acceleration is taken, which has already been used in section 9.2.5 to present Rayleigh’s approximation of the fundamental mode: [K] {w} = A [M] {'}
[15.103]
A relation identical to [9.83], apart from the change in notation for acceleration (A instead of *). We look for development coefficients sA of {w} based on eigenmodes, i.e. according to [15.103]: {w} =
¦s ^v ` A
A
A
A> K @
1
> M @^'`
[15.104]
from which, by premultiplication by {vk}T [K] and considering the orthogonality of modes with respect to matrix [K] (see [15.70]): sk {vk}T [K] {vk} = A {vk}T [M] {'}
[15.105]
and according to ([15.69] and [15.72]): sk =
A
Zk2
Pk
[15.106]
We thus have for deformation energy Ed,s (A) associated with deformation {w}:
Linear Seismic Calculation
1 T ^w` > K @^w` 2
Ed,s (A) =
§ Pk PA A² § T · ¨ ¦ ^vk ` ¸ > K @ ¨ ¦ 2 2 © k Zk2 Z © A A ¹
·
^vA ` ¸ ¹
645
[15.107]
i.e., due to the orthogonality relation: Pk2 A² T v K @^v k ` ¦ 4 ^ k` > 2 k Zk
Ed,s (A) =
[15.108]
and finally, according to [15.69] and [15.77]:
Pk A² ¦ 2 k Zk2
Ed,s (A) =
[15.109]
On the other hand, if A is the maximum pseudo-acceleration, pseudo-velocity is less than or equal to A/Z and we have, according to [15.99]: Ed
¦Max A
t
eA <
PA A² ¦ 2 A ZA2
[15.110]
The reconciliation of [15.109] and [15.110] establishes inequality [15.101]. If the period of the fundamental mode is situated in the spectrum zone which corresponds to the maximum value A of acceleration, the upper bound [15.101] is very close to the maximum elastic energy associated with this mode. In the case of a perfectly regular building of N stories studied in section 9.2, where the components of static deformation {w} have the expression [9.85], it can easily be shown that deformation energy Ed,s (A) associated with {w} is given by: Ed,s (A) =
1 Mt A² N 1 2 N 1 12 Z02
[15.111]
with Z o2 = k/m, while the maximum deformation energy of the fundamental mode is, according to [15.99]: Max e1 = t
1 A² P1 2 Z12
[15.112]
Considering expressions [15.91] (for modal mass) and [9.52] (for modal angular frequencies), we find that:
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Seismic Engineering
Maxe1 3 NS S = sin ² / sin 4 2 Ed , s ( A) 2 N N 1 2 N 1 2N 1 2 2 N 1
[15.113]
i.e., considering [15.92]: Maxei Ed , s ( A)
iI It
[15.114]
This equality does not have general scope (it is due to the particular nature of the building in question) but the tendency that it indicates, i.e. a very strong preponderance of energy of the first mode, is typical of all regular buildings. Table 15.3 shows that the fundamental mode almost represents 99% of the upper limit of energy. This observation indicates that the energy criteria for the selection of modes favor the first mode and are thus more difficult to apply in practice than criteria for modal mass, all the more because they depend on excitation, as indicated earlier. Relation [15.106] obtained for development coefficients of static deformation {w} make it possible to show that Rayleigh’s approximation for the fundamental mode is in excess for the frequency (or by default for the period). In fact, through easy calculation using orthogonality relations of modes we find (see [9.84]): § 2S · ¨ ¸ © T 'I ¹
2
* ^w` > M @^'` T
^w` > M @^w` T
=
PA
PA
¦Z /¦Z 2 A
A
[15.115]
4 A
A
The approximation 2S/T’1 = Z1c of the basic angular frequency Z1 is thus such that by stating D A
Z '12 Z12
PA / ZA4 : 1
¦D A A
¦D Z A
A
2 A
Z12
1
¦D A
¦D Z A
A
2 A
Z12
[15.116]
l
All the terms of the sum of the second member are positive or zero (for A = 1); and we thus have, as declared, Z1c ! Z1. Modal damping is determined, as indicated in section 15.1.3, either by reference to tables of pre-established values (based on the type of materials and the nature of the bracing elements) or on the basis of an explicit expression of the damping matrix. The use of value tables is immediate if the structure is homogenous. For a structure made up of different parts with different materials or having different systems of bracing, the following rule is very commonly used, which is expressed by the formula:
Linear Seismic Calculation
[A
§ · § · ¨ ¦[i Ed ,A ,i ¸ / ¨ ¦Ed ,A ,i ¸ © i ¹ © i ¹
647
[15.117]
where [i is the reduced damping applicable to part i of the structure according to the value table and Ed ,A ,i the deformation energy corresponding to mode A for the same part i. Damping [ A of mode A is thus the average damping of different parts, weighted by deformation energy produced by this mode in these parts. This rule, which is easy to apply as the calculation of deformation energy of different parts is immediate on the basis of modal deformation, is not theoretically justified but seems plausible due to the significance of damping in terms of energy. To illustrate the notion of modal mass, the most important among those that have been presented, it is of interest to study a simple example of a different type from the one considered in [9.2], which concerned a regular multistorey building, where the frequencies of modes are quite separate and where the first mode is highly predominant. We consider a low and rigid building (such as a pumping station of an electric power station) founded on a set of N pilings penetrating the ground in rigid bedrock and subjected to the action of an earthquake having two horizontal components of accelerograms sx and sy , according to two rectangular axes (see Figure 15.10).
Figure 15.10 Low and rigid building founded on pilings whose seismic behavior is that of a solid with 3 degrees of freedom; displacements U and V of the center of gravity G with relation to its initial position G0 (rest) and rotation T around a vertical axis passing through G
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The following assumptions are made: – the building is sufficiently low and the vertical stiffness of piling sufficiently high so that the effects of balancing around a horizontal axis can be ignored; – the response of the building, assumed to be infinitely rigid, is only in the horizontal plane and has a translation component (U according to X, V according to Y) for the center of gravity with relation to its initial position G0 and a rotation T around the center of gravity. These movements are assumed to be small (terms of the second order are negligible, sin T T, cos T = 1); – the distribution of mass in the building and piling in the foundation is not perfectly symmetric; thus, there is no coincidence between the center of gravity G and the center of rigidity of the piling system; – the piling is sufficiently high in number for it to be treated as “continuous”, by introducing density n of piling (number of piles per unit of surface of the foundation); – all the piles have the same horizontal stiffness kh (which depends on characteristics of piling and the soil) and negligible torsion stiffness. xGy being a system of axes related to the building, the following notations are defined: – n (x,y): number of piles per unit of surface; –N=
³³ n x, y
–a=
1 N
³³
x n (x, y) dx dy: eccentricity in x of the center of rigidity of piling;
–b=
1 N
³³
y n (x, y) dx dy: eccentricity in y of the center of rigidity of piling;
– r² =
1 N
³³ x² y ²
dx dy: total number of piles;
n (x, y) dx dy: square of the radius of gyration of the piling
system; – P (x,y): mass of the building per unit of surface; –M=
³³ P x, y
– U² =
1 M
dx dy: total mass of the building;
³³ x² y ² P x, y
dx dy: square of the radius of gyration of the
building; – I = MU²: inertia moment of the building with relation to the center of gravity.
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649
By definition of the center of gravity, we have, moreover, the relations:
³³ xP (x,y) dx dy = 0; ³³ y P
(x, y) dx dy = 0
[15.118]
On the basis of these hypotheses and definitions, the equations are obtained without any difficulty through the laws of dynamics and we arrive at the following system: M U sx Nkh (U - bT) = 0 M V s y Nkh (V + aT) = 0
[15.119]
IT Nkh r ²T Nkh aV bU = 0
or, in the canonical form [15.67]:
> M @^u` > K @^u`
s > M @^'`
[15.120]
with:
^u`
§U · ¨ V ¸ ;>M @ ¨T ¸ © ¹
§M O O· ¨ O M O ¸ ;> K @ ¨O O I ¸ © ¹
§ Nk O bNk · ¨ h h¸ ¨ O Nk aNk ¸ h h ¸ ¨ ¨ bNk aNk Nk r ² ¸ h h h ¹ ©
[15.121]
and: s sx or sy ; ^'`
^' x `
§1· §O· or ^' y ` = ¨ 1 ¸ ¨O ¸ ©O¹ ©O¹
[15.122]
It is noted, as declared in section 9.2.2, that the component of {'} corresponding to the degree of freedom of rotation is zero. The equation that determines eigenangular frequencies is obtained by canceling the determinant of matrix [K] – Z² [M], or, considering [15.121]:
Z
2 O
Z ² r ²ZO2 U ²Z ² a ² b² ZO4 ZO2 Z ² 2
0
[15.123]
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Seismic Engineering
where we stated:
ZO2
N
kh I ; U² = M M
[15.124]
There is thus an obvious root Z = Zo; the two other roots are calculated by resolving the equation of the second degree in Z²: U² Z4 – (r² + U²) Ȧ02 Z² + (r² – a² – b²) Ȧ04 = 0
[15.125]
For simplification, it is assumed that the radii of gyration r and U are equal (which is close to reality, as the distribution of piling is normally dictated by that of the mass). Equation [15.125] is then written as:
Z²Z02 2
a²b² Z4 U² 0
[15.126]
The three eigenangular frequencies of the system are thus: Z1 = Z 0 1 D ;
Z2
Z0 ; Z3
Z0 1 D
[15.127]
by establishing:
D
1
U
a ² b²
[15.128]
With the help of the Cauchy-Schwartz inequality, it can easily be demonstrated that the definitions given earlier for a, b and U = r imply that D is less than one. In practice D is generally much smaller than one as in principle we aim to reduce the eccentricities of the center of rigidity of piling in relation to the center of gravity. We can then write, instead of [15.127]:
Z1
§ ©
Z0 ¨ 1
D·
§ D· ¸ ; Z2 = Zo ; Z3 = Zo ¨1 ¸ 2¹ 2¹ ©
[15.129]
The three eigenfrequencies are thus close; for a distribution of piling strictly similar to that of mass (a = b = 0, D = 0) they could even be confused and we would encounter the case described in section 9.2.3, of a multiple root of order 3.
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651
The components of eigenmodes are easily determined on the basis of expressions [15.121] of matrices [M] and [K] and values [15.129] found for eigenangular frequencies; we get through an appropriate choice of the mode multiplicative constant: 1st mode
U1 = b; V1 = –a; T1 = D
[15.130]
2nd mode
U2 = a; V2 = b; T2 = 0
[15.131]
3rd mode
U3 = b; V3 = – a; T3 = –D
[15.132]
The eigenmodes are shown in Figure 15.11. nd
rd
Figure 15.11. Eigenmodes for a building founded on pilings shown in Figure 15.10
We see that the 2nd mode corresponds to a pure but oblique translation with relation to the axes, whereas the 1st and 3rd modes are coupled modes having both translation (which is perpendicular to that of the 2nd mode) and rotation (of the same amplitude but with opposite signs for these two modes). The participation factors and modal mass are calculated by formulae [15.72] and [15.77]. There are two cases to be distinguished in direction X or Y for excitation, as the expression of vector {'} depends on it (see [15.122]); we find
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Seismic Engineering
– for direction X: P1, x =
b M b² ; P1, x = 2 a ² b² 2 a ² b²
[15.133]
P2, x =
a a² ; P2, x = M a ² b² a ² b²
[15.134]
P3, x =
b M b² ; P3, x = 2 a ² b² 2 a ² b²
[15.135]
– for direction Y: P1, y =
P2, y =
a M a² ; P1, y = 2 a ² b² 2 a ² b²
b b² ; P2, y = M a ² b² a ² b²
P3, y = –
a M a² ; P3, y = 2 a ² b² 2 a ² b²
[15.136]
[15.137]
[15.138]
We verify that the sum of modal mass, for each direction, is equal to M. The purpose of this example is to show that tendencies derived for regular multistorey buildings (see the example in section 9.2), i.e., the preponderance of the fundamental mode and the separation of eigenfrequencies, are not necessarily valid for buildings sensitive to effects of torsion. It is in fact observed that in expressions of P1,x and P2,x if b is lower than a, the first mode has a modal mass distinctly lower than that of the second. The mode of the lowest frequency thus may not be the most important for the response. The same phenomenon can be observed for structures having flexible appendices. In this case, for the first mode, it is frequent to have a simple deformation of one of these appendices which does not have any significant influence on the response of the whole unit. The sensitivity to torsion equally affects the precision in determining the lowest frequency by Rayleigh’s method. If formula [15.115] is applied, with expressions [15.127] for eigenangular frequencies (which are valid even if D is not much smaller than one) and expressions [15.133] to [15.135] for modal masses in direction X, we find for approximation Z1c of Z1:
Linear Seismic Calculation
a ² 1 D ² b ² 1 D ²
653
2
Z1c
2
Z
2 0
[15.139]
a ² 1 D ² b 2 1 D ² 2
We thus have, for the ratio Z1c / Z1 , according to [15.127]:
Z1c Z1
1/ 2
ª º a ² 1 D ² b² « 1 D » 2 a ² 1 D ² b² 1 D ² »¼ «¬
[15.140]
In the case where the eccentricity of the piling system only exists in the perpendicular direction to the excitation (a = 0), we get:
Z1c Z1
1D 1 D ²
[15.141]
This expression presents a maximum of 1.099 for D 2 1 0, 414 (which corresponds to significant eccentricity of the center of rigidity of piling). We see that Rayleigh’s approximation can, in extreme cases of sensitivity to torsion, produce an error of excess in the order of 10%, while it is typically exact to a close percentage (see Table 9.4) for regular multistorey buildings. 15.2.2. Number of modes to be retained and combination of modal responses
Apart from very simple models, with a few degrees of freedom, such as the examples presented in sections 9.2 and 15.2.1, the calculation of the totality of eigenmodes is practically never carried out. For complex models comprising several hundreds or even thousands of degrees of freedom, such a calculation would create a considerable amount of data, a large majority of which is either of no use for the calculation of the response (modes of zero or negligible modal mass) or highly riddled with uncertainties (modes of high frequency whose deformations are too complicated to be correctly shown through the chosen model). The question is thus how many modes should be retained and which criteria enable this selection. The criteria are in practice based on the cut-off frequency and cumulative modal mass. We have seen in section 9.1.1 that the cut-off frequency (generally contained between 25 and 40 Hz) is the frequency beyond which the simple oscillator spring can be considered a rigid bar which compels the mass to follow the movement of the support. Frequency modes greater than or equal to the cut-off frequency are called modes for rigid response; in equation [9.3] (or [15.95]), used for the definition of the response spectrum associated with the accelerogram s (t), we can ignore the “dynamic” terms which bring in the derivatives x and x , and we get for x:
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Seismic Engineering
x=
s Z²
[15.142]
i.e., the relative displacement corresponding to the static application of the support acceleration. All the modes with rigid response thus follow the accelerogram applied to the support without dephasing and we shall see hereafter that they can be regrouped in a rigid pseudo-mode. Whereas, modes of frequency less than the cut-off frequency, called modes with dynamic response, respond independently of each other and must be combined by using other rules. The correct practice for modal spectral analysis thus consists, in principle, of calculating only the modes for dynamic response and taking into account the set of all modes with rigid response by using a pseudo-mode. For large models, however, this procedure can lead to a very high number of modes and it is often the case that we have to be satisfied with a certain number of modes where the last one has a frequency that may be significantly less than the cut-off frequency. In such cases, cumulative modal mass is generally used to decide whether the number of modes is sufficient or not. We have seen in section 15.2.1 that the sum of modal mass of all the modes is equal to the total mass of the structure. The criteria for cumulative modal masses impose the sum of modal masses for retained modes to be at least equal to a certain percentage (often the figure 90% is given) of the total mass. Thus, there is assurance that the mass associated with the neglected modes represents only a small part of the mass of the structure, therefore the retained modes, completed if necessary by a pseudo-mode, are normally sufficient for the description of the response of the whole structure. However, it must be recognized that certain local responses can be greatly influenced by higher rank modes (of higher frequency than that of the last retained mode). It is sometimes difficult to follow the 90% criterion for cumulative modal mass, especially for structures presenting elements with significant mass showing very stiff behavior with respect to the forces acting in the direction in question (this is the case, for example, in buildings comprising very heavy and rigid low parts and light superstructures). We thus have to be satisfied with a lower percentage, for example 60% or 70%, and compensate the mass default by overestimating the pseudo-mode, as we shall see later. Criteria other than cumulative modal mass have been proposed for the selection of modes. They are based, for example, on modal energy, which was discussed in section 15.2.1, or on the study of the influence of certain additional modes (recommendations from the American Nuclear Regulatory Commission). These
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655
criteria, which are more difficult to apply than cumulative modal mass, do not seem to still be in use today. Pseudo-modes that were referred to earlier follow from the expression [15.94] of development of the solution based on eigenmodes and formula [15.142] which characterizes the modes for rigid response. Nd denotes the number of modes for dynamic response and Nr the number of modes for rigid response. The total number of degrees of freedom is thus equal to Nd + Nr. There are two cases to be distinguished depending on whether displacements or accelerations are taken into account. For displacements, equation [15.94] is rewritten by separating the modes according to their type of response, dynamic or rigid: Nd
Nd Nr
i 1
i Nd 1
{u} = ¦ x t , Zi , [ i p i ^v i ` ¦ x t , Zi , [ i p i ^v i `
[15.143]
Considering [15.142], the second sum of the second member of [15.143] can be written as:
Pi ^v i ` 2 Nd 1 Z i
Nd Nr
Nd Nr
i Nd 1
i
¦ x t , Zi , [ i p i ^v i ` s t ¦
[15.144]
Therefore, we have, according to [15.104] and [15.106], for static deformation {w1} under unit acceleration: Nd Nr p Pi i ^ ` ^v i ` v ¦ i 2 2 1Z i Nd 1 Z i i
Nd
{w1} = ¦ i
[15.145]
from which for [15.144]: Nd p ª º ¦ x t , Zi , [ i p i ^v i ` s t «^w 1 ` ¦ i2 ^v i `» Nd 1 i 1Z i ¬ ¼
Nd Nr i
[15.146]
and for [15.143]: Nd
ª
Nd
º pi ^v i `» 2 1Z i ¼
i 1
¬
i
{u} = ¦ x t , Z i , [ i p i ^v i ` s t «^w 1 ` ¦
[15.147]
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Seismic Engineering
It is seen in this expression that the response is made up of the sum of modal responses of modes for dynamic response and by an additional term, proportional to the acceleration of excitation s (t), and where the calculation only brings in static deformation under unit acceleration and the Nd dynamic modes. This additional term, which groups the entire set of modes for rigid response, is the pseudo-mode in displacement, sometimes also called residual mode or rigid mode. For absolute accelerations {ü}, we write that they are the sum of ground and relative accelerations, i.e., according to [15.94]: Nd Nr
{ü} = s( t )^'` ¦ x t , Zi , [ i p i ^v i `
[15.148]
i 1
For modes for rigid response, we have seen that x is zero; as a result, for {ü}: Nd
{ü} = s t ^'` ¦ x t , Zi , [ i p i ^v i `
[15.149]
i 1
which can again be written as:
ª «¬
º »¼
Nd
Nd
>
@
{ü} = s t ^'` ¦ p i ^v i ` ¦ x t , Zi , [ i s t p i ^v i ` i 1
i 1
[15.150]
As for formula [15.147] for displacements, we see that modes for rigid response are grouped into one pseudo-mode proportional to the acceleration of excitation, whose calculation only brings in modes for dynamic response. According to [15.73] we see that this pseudo-mode is zero if all the modes are of the dynamic response type. Pseudo-modes in equations [15.147] and [15.150] are genuine pseudo-modes whose weight in these equations is as great as the mass default is significant. Current earthquake engineering codes admit the use of simplified pseudo-modes which simply correspond to the first term of real pseudo-modes ({w1} for displacements and {'} for accelerations) multiplied by a correcting factor which depends on the mass default. For example, AFPS90 recommendations [AFP 95] advocate the use of the following multiplying coefficient:
C = 0.9M t ¦ Pi / M t ¦ Pi i
i
[15.151]
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657
Pi being modal masses of retained modes. This formula assumes that the sum of this modal mass does not exceed 90% of the total mass Mt. Formulae [15.147] and [15.150] enable the calculation of the temporal development of the response in relative displacement and in absolute acceleration. In the modal spectral analysis, for functions s (t) and x (t, Z, [) only their maximum values which enable the construction of the response spectrum are known. The question regarding the combination of these values is thus raised for the estimation of maximum values of the response. We have seen in section 9.2.5 that the so-called arithmetic combinations, where we take the sum of maximum values for each term, are practically never used, due to their very low probability (they suppose in fact that all the terms reach their maximum and with the same sign at the same time). The commonly used rule is that of simple quadratic combination (which has previously been presented in this book under the abbreviation SRSS). This rule of combination – which expresses that the square of the maximum value of the response can be taken equal to the sum of the squares of maximum values of each of the terms – is a particular case of complete quadratic combination (CQC) which will now be presented. We consider equation [15.147], without the pseudo-mode, using Duhamel’s integral [9.10] for the expression of x (t, Zi, [i); we thus have for a component uk of the vector {u}: ª pv t uk = – ³ s W « ¦ i i , k e [iZi t W sin Zi' t W O « i Z' i ¬
º » dW » ¼
[15.152]
with: vi,k = component of the rank k of mode {vi} and Zic = Zi 1 [ i2 , [ i 1 It is stated: – that accelerogram s is an unfiltered white noise Jb which was presented in section 10.2.1. – that the angular frequencies of modes which intervene in equation [15.152] are not too high, so that the sinusoidal factor can be considered as being slowly variable to the scale of time step 't associated with the definition of white noise. This hypothesis excludes in particular modes for rigid response (hence the omission of the pseudo-mode) and also, strictly speaking, modes for dynamic response whose frequencies are relatively close to the cut-off frequency.
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Seismic Engineering
The generalized theorem of Brownian motion can thus be applied and written for the maximum of uk (see [10.13]): F T
Max | uk | = gp
[15.153]
gp being a peak factor, large as we target the lowest probability of exceeding and F (T) the following quantity (see [10.9]): 2
ª pv º [iZi T W sin(Zi' (T W )) » dW 'tV J ³ « ¦ i i , k e O « i Z' » i ¬ ¼ T
2
F (T) =
[15.154]
where V J2 is the assumed constant variance for random selection of accelerations used to define unfiltered white noise. Considering the case of a stationary excitation, i.e., by making T tend towards infinity, we find by developing the square in [15.154]: f p2 'tV J2 ¦¦ K i , j pi vi , k p j v j , k
Max uk2
i
[15.155]
j
with: f
Ki,j =
1 ³e Z 'Z ' o i
[i Zi [ j Z j T
sin(Zi'T ) sin(Z 'jT ) dT
[15.156]
j
The basic calculation of this integral leads to:
Ki,j =
[iZi [ j Z j ª« 2Zi'Z 'j
1
« [ Z [ Z ² (Z ' Z ' )² j j i j ¬ i i
º » [iZi [ jZ j ² (Zi' Z 'j )² »¼ 1
[15.157] which gives, for i = j: Kii =
1 4[iZi3
[15.158]
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The maximum Ri of the response in displacement of the simple oscillator serving to define the response spectrum for angular frequency Zi of the mode i was calculated in section 10.2.1 (equation [10.20]): Ri = gpVJ
't 4[iZi3
[15.159]
Thus, by using [15.158], [15.155] can be rewritten in the form: Max u²k =
¦ ¦Q
i, j
i
pi vi , k Ri p j v j , k R j
[15.160]
j
where coefficients Qi,j are given by: Qi,j =
Ki , j
[15.161]
K ii K jj
Formula [15.160], symmetrical in i and j, shows that within the framework of the hypotheses adopted, the square of the maximum of the response in displacement is expressed by a complete quadratic combination of maxima for modal responses. It is to be noted that if Ri and Rj are necessarily positive, products pi, vi,k, can have any given sign. The method used to establish [15.160] guarantees that the complete quadratic combination always gives a positive result for which we can take the square root. On the basis of [15.157] and [15.158] we show that coefficients Qij of the complete quadratic combination have the expression: Qi,j =
8Zi Z j [iZi [ j Z j [iZi [ jZ j
(Z Z )² 4[i[ jZiZ j (Zi2 Z 2j ) 4([ i2 [ j2 )Zi2Z 2j 2 i
2 j
[15.162]
This formula was established by Wilson, Der Kiureghian and Bayo [WIL 82]. In frequently encountered cases, where the damping is the same for all the modes ([i = [j = [), [15.162] can be written in the form: 3
Qi,j =
8[ ² r 2 (r 1)[(r 1)² 4[ ² r ]
where r denotes the ratio Zi/Zj.
[15.163]
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This formula gives the same value when we change r to 1/r; it shows that Qij decreases rapidly when r goes farther from the value 1 (Zi = Zj) but that this decrease is less rapid when the damping increases. Table 15.4 gives the values for Qij, calculated by [15.163], for some values of r and in the case [ = 0.05 (current value for structures) and [ = 0.20 (value that may be encountered in the calculations of soil- structure interaction – see section 16.1).
r
Qi,j ([ = 0.05)
Qi,j ([ = 0.20)
1.0
1.000
1.000
1.02
0.962
0.998
1.05
0.807
0.985
1.1
0.523
0.945
1.2
0.230
0.824
1.3
0.125
0.692
1.4
0.079
0.575
1.5
0.055
0.479
1.6
0.041
0.404
1.7
0.032
0.345
1.8
0.026
0.297
1.9
0.022
0.259
2.0
0.018
0.229
Table 15.4. Coefficients of complete quadratic combination [15.163] for different values of ratio r for frequencies
The observation that the increase of damping tends to couple modes is typical in structural dynamics. The complete quadratic combination, defined by [15.162] or [15.163], often denoted by its abbreviation CQC, was established with the hypothesis of an excitation in stationary unfiltered white noise which, as we have seen in section 10.2, constitutes an acceptable though rudimentary approximation of real seismic movements, at least for certain aspects. The validity of its application in modal spectral analysis has been confirmed by an entire set of comparisons between results that it provides and those for temporal calculations of response. The rapidity of the decrease of Qij when r goes further away from 1 justifies the use of simple quadratic combination (SRSS) when the eigenfrequencies of the structure are well separated. In fact, coefficients of the rectangular terms of the
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complete quadratic combination are then very small and can be ignored while only retaining square terms. Other complete quadratic combinations have been proposed. For example, for the expression of coefficients Qij Rosenblueth and Elorduy [ROS 69] give: ª § Z" Z" i j Qij = «1 ¨ ' « ©¨ [i Zi [ 'j Z j ¬
· ¸¸ ¹
2
º » » ¼
1
[15.164]
[’i and Zs being defined by: [ƍi = [i +
2 ZiT f
Z i'' Zi 1 [i'
[15.165]
2
[15.166]
where Tf is the duration of the strong part of the accelerogram during which the signal is assimilated to white noise. In the case where Tf is taken towards infinity, [’ values are identical to [ and we obtain the following expression for Qij when the damping is the same for all modes: 2 ª § 1 · § r 1 · º Qij = «1 ¨ 2 1¸ ¨ ¸ » ¹ © r 1 ¹ ¼» ¬« © [
1
[15.167]
with, as in [15.163], r = Zi/Zj. Combination [15.167] was recommended by Humar [HUM 84] for the calculation of buildings strongly influenced by the effects of torsion, as the example considered in section 15.2.1. It yields results very close to those for [15.163], as we can see in Table 15.5. With relation to Table [15.4] a slight increase of coefficients Qij is observed, insignificant for [ = 0.05 and a little more pronounced for [ = 0.20 (but which exceed 10% only when r = 1.6).
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r
Qi,j ([ = 0.05)
Qi,j ([ = 0.20)
1.0
1.000
1.000
1.02
0.962
0.998
1.05
0.808
0.986
1.1
0.525
0.948
1.2
0.233
0.834
1.3
0.128
0.710
1.4
0.083
0.600
1.5
0.059
0.510
1.6
0.045
0.439
1.7
0.036
0.383
1.8
0.030
0.338
1.9
0.025
0.302
2.0
0.022
0.273
Table 15.5. Coefficients of complete quadratic combination [15.167] for different values of ratio r for frequencies
SRSS was first introduced in the modal spectral analysis. As seen earlier, it remains perfectly valid if eigenfrequencies are separated (a value of 1.2–1.3 of ratio r is sufficient to assure the validity of the SRSS option). The problem of close eigenfrequencies appeared when relatively complex models began to be made, to take into account the two or three dimensional character of structures and their sensitivity to effects of torsion. Before the introduction of complete quadratic combinations based on white noise simulations of seismic signals (such as [15.162] or [15.164]), this problem was tackled through simplistic methods, consisting of grouping close frequency modes into “packets”. Inside the same “packet”, the modes are combined arithmetically and then the simple quadratic combination is done for the results of different “packets”. These rules are not very satisfactory and have only historic value today, as they are based on arbitrary criteria (to define the proximity of frequencies and the constitution of “packets”) and introduce discontinuities in the results for minimum variations of data. To illustrate the comparison of the different rules of combination in cases where frequencies are close, once again the example of the building discussed in section 15.2.1 (see Figure 15.10) is taken with the following assumptions: – damping is 5% for all three modes;
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– the spectrum for pseudo-acceleration (for 5% damping) varies proportionally to the frequency and is 2 m/s² for a frequency of 1 Hz; – their numerical values are as follows for the quantities defined in section 15.2.1: Zo = 2Srd/s, a = 2 m, b = 1 m, D = 0.1. We thus find, for coefficients Qij of complete quadratic combination ([15.163] or [15.167]): Q11 = 1; Q22 = 1; Q33 = 1; Q12 = 0.792; Q23 = 0.808; Q31 = 0.501. The comparison of the three rules of combination SRSS, CQC and ABS (arithmetic sum of the three modes considered as belonging to the same “packet”) is considered. The results are presented in Table 15.6 for displacements U and V (expressed in mm) of the center of gravity in directions X and Y respectively, and rotation T (expressed in milliradians).
Earthquake X
Earthquake Y
U (mm.)
V (mm.)
T (10-3rd)
SRSS
41.1
24.8
0.717
CQC
48.7
7.92
0.508
ABS
50.6
40.5
1.011
SRSS
24.8
30.5
1.432
CQC
7.92
44.7
1.012
ABS
40.5
50.7
2.023
Table 15.6. Comparison of SRSS, CQC and ABS rules of combinations of modes for the building in Figure 15.10
It is observed that the SRSS option underestimates responses parallel to the direction of excitation (U for the earthquake in X, V for the earthquake in Y) with relation to the CQC option; on the other hand, it highly overestimates (factor of about 3) the responses perpendicular to this direction (V for the earthquake in X, U for the earthquake in Y) and overestimates rotation (around 40%). The study of a certain number of structures having eigenmodes shows that these tendencies are very quasi-general and are not related to the particular nature of the example in question. The ABS option obviously gives the highest results, the deviation being particularly great for the perpendicular responses to the direction of excitation. In conclusion, it is the CQC rule which constitutes the best choice. It is easy to introduce into calculation software and its validity has been established, as indicated earlier, by comparisons with temporal analyses. However, care has to be taken about
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certain practices, dictated by the obsessional study of safety margins, which have led all the terms to be taken with the same sign in this combination, whatever the real sign of the different modal responses. The real CQC combination for modal spectral analysis conserves the signs appropriate to these responses. Differences between CQC results with and without signs can be considerable; for example, for the structure in three modes studied earlier (Table 15.6) the perpendicular response V for the earthquake in X changes from 7.92 mm (CQC with signs) to 37.1 mm (CQC without signs), i.e. the multiplication of the SRSS result by 1.5, which is itself highly overstated. The CQC combination without signs represents hasty decisions which, under the pretext of obtaining better safety, cast a veritable shadow over margins. It thus needs to be combated wherever it may persist. SRSS or CQC combinations apply only to dynamic response modes. When the expression of the structural response brings in a pseudo-mode, as in equations [15.147] and [15.150], it is quadratically combined with the SRSS or CQC combinations of dynamic response modes by taking the acceleration reading on the response spectrum for the last calculated mode as the maximum value for s (t ) . By denoting the pseudo-acceleration spectrum by Sa (Z[) angular frequency by Z and damping by [ and n being the indicator of the last calculated mode, we thus get for maximum Uk of the component uk of the vector {u} ([15.147]): n
U
2 k
n
= 6 6 Qij pi vi , k
S a (Zi ,[i )
i 1j 1
Zi2
p j v j,k
S a (Z j ,[ j )
Z 2j
²
n p ª º S (Zn,[ n ) « w1, k 6 A2 vA , k » A 1Z »¼ A ¬« [15.168] 2 a
in the case of the CQC combination and: U k2
n
Sa2 (Zi , [i )
i 1
Zi4
= 6 pi2 vi2, k
n p ª º S a2 (Zn , [ n ) « w1 ,k 6 2A vA , k » l 1Z A ¬ ¼
2
[15.169]
in the case of the SRSS combination. For maximum * k of absolute acceleration for degrees of freedom of k range, we get, according to [15.150]: * 2k
2
n n n ª º 6 ¦ Qi , j pi vi , k Sa (Zi , [i ) p j v j , k S a Z j , [ j S a2 (Zn , [ n ) « ' k 6 pA vA , k » A 1 i 1j 1 ¬ ¼ [15.170]
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in the case of the CQC combination and: * k2
n n ª º 6 pi2 vi2, k Sa2 (Zi , [i ) Sa2 (Zn , [ n ) « ' k 6 pA vA , k » 1 A i 1 ¬ ¼
2
[15.171]
in the case of SRSS combination. The comparison of combinations in displacement and acceleration show that the influence of higher modes is much more sensitive for accelerations (due to the 4 2 2 denominator Zi Z j or Zi in [15.168] or [15.169] which ensures the preponderance of the first mode or modes for displacements). This observation has significant consequences in the calculation of forces in the structure according to which it is carried out based on modal forces or accelerations (see section 15.2.4). In equations [15.168] to [15.171], taking spectral acceleration Sa(Zn,[n) of the last mode retained as maximum acceleration s (t ) indicates a preoccupation of conservatism as Sa(Zn,[n) is higher than the real maximum of s (t ) which is the spectral acceleration beyond the cut-off frequency, as we have seen in section 9.1.1. This brings us back to taking Zn/(2S) as the new cut-off frequency and Sa (Zn, [n) as the new asymptotic acceleration (see Figure 15.12), thus defining a spectrum that is more penalizing than the real spectrum, but not enough to guarantee that one or more dynamic response modes do not exist among modes where the angular frequency is higher than Zn, which produce a response higher than the one calculated with n retained modes and the pseudo-mode, at least locally. However, such an eventuality is very unlikely if, with the frequency Zn/2S, we have exceeded the zone of highest amplification of spectral accelerations.
Figure 15.12. Difference in the cut-off frequency for the consideration of the pseudo-mode in quadratic combinations
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In conclusion on the combination of modes, the following diagram can be presented (see Figure 15.13) as a summary of the points presented in section 15.2.2.
Figure 15.13. Diagram defining modal combinations according to cumulative modal mass and its ratio to cut-off frequency
The axes in this diagram are cumulative modal mass in the abscissa ¦Pi divided by total mass Mt and the frequency quotient g of the last calculated mode by the cutoff frequency gc in the ordinate. There are thus four possible cases: – ¦Pi 0.7Mt and g gc (shaded area). The extraction of modes has not been taken far enough to be able to apply the normal rules of combination. It needs to be pursued further or the calculation model needs to be modified (in general towards a simplification); – 0.7Mt Pi 0.9Mt and g gc. We thus have the case of application of formulae [15.168] to [15.171] denoted by the symbol C + P which signifies that the quadratic combination C (simple or complete) applies to all the calculated modes, and also the pseudo-mode P; – 0.9Mt ¦Pi Mt and g gc. We can thus be satisfied with the quadratic combination C by itself (without the pseudo-mode P) but, of course, the option C + P still remains valid; –any given ¦Pi and g!gc. The combination C’ + P’ is applied where the quadratic combination C’ applies only to modes whose frequency is lower than the cut-off frequency; the subtracted part of the pseudo-modes must also, of course, be limited to the same modes. We draw attention to the case where we have at the same time, ¦Pi ! 0.9 Mt; it is the C’ + P’ combination that is applied and not combination C.
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15.2.3. Combination of effects with three components
Seismic calculation is carried out most often by separating the effects of the three components of excitation (two horizontal and one vertical). In the hypothesis of linear behavior, this separation is a simple consequence of the principle of superimposition. In the assumption of nonlinear behavior, it is also frequently used, even if it is not justified a priori. A response U of the structure (displacement of a point in a given direction or particular component of a force in one section or of a stress tensor at one point) is generally made up of three terms Ux, Uy and Uz which are maximum values for U for excitations in X, Y and Z respectively. The problem of combination of directions consists of determining the maximum Umax for U when the three components of excitation are applied simultaneously, which is the case during a real earthquake. The most natural rule for the calculation of Umax is again that of quadratic combination:
Umax = U x U y U z 2
2
2
1
2
[15.172]
The justification of this rule rests on quasi-independence, in the statistical sense of the term, of the earthquake components taken two by two; as instantaneous values, response U is the sum of the three non-correlated terms between them and its maximum value is obtained by the square root of the sum of squares of the maxima of each of these terms, in the same way as the sum of contributions of separate modes (thus non-correlated) is obtained by the SRSS rule. The validity of the hypothesis of statistic independence of the three components of movement was discussed in section 4.1.3, where, among phenomena likely to question it, the phenomenon called the killer pulse by the fault community has found special mention. Data available at the project stage is not always sufficient in general, to decide if such effects must be taken into account and if so, how. The hypothesis of independence of components and the soundness of their quadratic combination are thus admitted without discussion by earthquake engineering codes. An unfortunate coincidence is that formula [15.172] reminds us of the Pythagoras theorem, which has often led to confusion. It must therefore be insisted that Ux, Uy and Uz are not components of the response U on three axes but are related to a single response component considered successively from the point of view of excitation in X, in Y and in Z. For example, if (see Figure 15.14) a cylindrical tank in vertical axis is considered, the horizontal component in direction X brings about an overturning moment Mx of axis OY which induces at point P
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(abscissa x, ordinate y) at the base of the cylinder a vertical stress of tractioncompression given in the formula: Vx = M x
x I
[15.173]
I being flexion inertia of the cross-section by a horizontal plane. Similarly, excitation in Y produces an overturning moment My of axis OX and vertical stress: Vy = My
y I
[15.174]
Figure 15.14. Tank in vertical axis subjected to three directional excitation
Finally, the vertical component produces normal force N which corresponds to the stress: Vz = N
S
S being the cross-section area.
[15.175]
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If it is supposed usual that excitations in X and Y have the same intensity, overturning moments Mx and My have the same maximum value M and according to [15.172], for maximum vertical stress Vmax at point P, considering equations [15.173] to [15.175], we have: V²max =
M2 2 N2 2 x y ( ) I2 S2
M 2 R2 N 2 2 I2 S
[15.176]
R being the radius of the cylindrical casing. The result obtained for Vmax is independent of the orientation of the horizontal axes. It can be used to evaluate the risk of buckling of the casing, which is a mode of damage often observed in thin tanks (see section 12.2.3). A similar calculation can be performed to determine the traction stress in an anchoring bolt of the tank, if there is one. Formula [15.172] must only be used when the studied response is expressed in terms of instantaneous value, under the form of a sum of effects of the three components; it is often the case that the response is not additive with respect to the components. It is then advisable to use Newmark’s empirical rule which consists of supposing that when the effect of a component is at its maximum, the effects of the two other components is 40% of their maximum, considering all the possible combinations of signs to find the most unfavorable case. This rule, of course, can also be applied in the case of an additive combination and is then expressed by: r Ux r 0.4Uy r 0.4Uz Umax = Sup
r 0.4Ux r Uy r 0.4Uz
[15.177]
r 0.4Ux r 0.4Uy r Uz This rule is, in all the cases, slightly more conservative than [15.172]; for example, if Ux = Uy = Uz it gives Umax = 1.8 Ux (to be compared with Umax = Ux 3 = 1.73 Ux obtained by [15.172]). On the contrary for Ux = Uy and Uz = 0, the value Umax = 1.4 Ux calculated by [15.177] is slightly lower than the one resulting from [15.172] (Umax = Ux 2 = 1.41 Ux). If rule [15.177] needs to be overestimated in all cases with relation to (6.172) 0.4 must be replaced by 2 – 1 = 0.414; this is demonstrated easily by getting the smallest coefficient D such that: [Ux + D (Uy + Uz) ]² t U²x + U²y + U²z
[15.178]
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Seismic Engineering
Ux, Uy, Uz being such that Ux t Uy t Uz t 0 (this is a valid supposition, as all the permutations in [15.177] must be considered). By developing this, we can write [15.178] in the form: Uy [2DUx – (1 – D²) Uy] + Uz [2DUx – (1 – D²)Uz] + 2D²UyUz t 0
[15.179]
As Ux t Uy t Uz t 0, we see that the inequality is satisfied if we have: 2D = 1 – D² or D =
2 – 1 = 0.414
[15.180]
This sufficient condition is equally necessary as the inequality becomes equality for Ux = Uy and Uz = 0. In the majority of earthquake engineering codes (PS92 Rules, Eurocode 8), the coefficient 0.4 has been reduced to 0.3. Rule [15.177] is thus conservative, with relation to quadratic combination [15.172], only if perpendicular responses (Uy and Uz if Ux is preponderant) are not too high (Uy = Uz 0.73 Ux and Uy 0.66 Ux if Uz = 0), which is by far the most frequent case in practice. In the case of a non-additive combination, Rule [15.177] (possibly with coefficient 0.3 instead of 0.4) makes it possible in general to provide a satisfactory solution to the problem of combination of directions, which is often a source of practical difficulties. For example, we consider a tank anchoring bolt for which the shear force that it can sustain during an earthquake needs to be calculated. This shear force V is the result of force Vx according to Ox and force Vy according to Oy and verifies the relation: V=
Vx2 Vy2
[15.181]
This time, it is the Pythagoras theorem! We admit that due to dissymmetry (influence of pipe tapping and ovalling) Vx and Vy both depend on excitations in X and in Y, with the following values: – VxX maximum of Vx due to excitation in X = 0.9; – VxY maximum of Vx due to excitation in Y = 0.2; – VyX maximum of Vy due to excitation in X = 0.1; – VyY maximum of Vy due to excitation in Y = 1.0. When the effects of excitation in X are maximal, those for excitation in Y are 40% of their maximum; we thus have at this moment:
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– Vx = 0.9 + 0.4 x 0.2 = 0.98; – Vy = 0.1 + 0.4 x 1.0 = 0.50. from which we obtain the result: V=
0.98
2
0.50
2
[15.182]
1.10
Similarly we find, when effects of excitation in Y are maximal: – Vx = 0.4 x 0.9 + 0.2 = 0.56; – Vy = 0.4 x 0.1 + 1.0 = 1.04. and for V: V=
0.56
2
1.04
2
[15.183]
1.18
In this example, the maximum to be retained for V is the highest of the two values in [15.182] and [15.183], that is, the second (1.18). By taking the coefficient 0.3 instead of 0.4, this value is reduced to 1.13. The method that has just been explained is the correct way to use Newmark’s hypothesis, but sometimes erroneous interpretations are observed as, for example: V = Sup (VxX + 0.4 VyY , 0.4 VxX + VyY) = 1.36 V = Sup
VxX2 VyX2 0, 4 Vy2x Vy2y , 0, 4 VxX2 Vx2y Vx2y VyY2
1.38
which give very penalizing results. Newmark’s rule [15.177] was recently [WIL 95] the subject of debate due to the fact that, apart from quadratic combination, its results are not invariant with respect to the orientation of horizontal axes. This is not a difficulty in the case of structures with simple forms where the choice of axes is obvious but can become so for complex structures (for example curved bridges) where different choices are possible. This rule however remains an indispensable part for non-additive combinations in practice. When we combine the effects of different directions of excitation, whether it is through quadratic combination [15.172] or Newmark’s rule [15.177], care has to be taken in case a particular response of the structure has more than one term for the same component of seismic movement. Figure 15.15 illustrates such a situation.
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Figure 15.15. A portal frame with rigid cross bracing showing both terms (MH and NH), due to the horizontal component, which contributes to stress in the columns
This figure shows a portal frame made up of rigid cross bracing and two beams with fixed support at the head (in the cross bracing) and at the foot (in the ground considered non-deformable). We denote by h the height of the center of gravity G of the cross bracing (where all mass is supposed to be concentrated) above the base, e its thickness and l the distance between the axes of the columns. The action of the horizontal component of seismic movement produces an inertial force FH applied to G and the foot of the columns is subjected to the following reactions of the soil: 1 shear force: VH = FH 2 1 2h e normal force: NH = FH 4 l 1 bending moment: MH = FH 2h e 8
[15.184] [15.185] [15.186]
These relations result from conditions of equilibrium of the whole structure on the one hand and each of the columns (with the same moment MH on the head and foot) on the other. Vertical stresses associated with normal force NH at bending moment MH have well defined signs. On the external faces of the columns, i.e. those situated towards the exterior of the portal frame, they have the same sign (compression for the column on the right and traction for the column on the left in the diagram) while they have the opposite sign on the internal faces. If the width of a column is denoted by a, its section and its bending inertia by S and I, the correct
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application of the quadratic combination of directions leads to the following expressions of vertical stresses on the faces of the columns: 2
§M a N · §N · external face Ve² = ¨ H H ¸ ¨ V ¸ S ¹ © S ¹ © 2I 2
internal face V i2
§ M H a N H · § NV · ¨ 2I S ¸ ¨ S ¸ © ¹ © ¹
2
[15.187]
2
[15.188]
where NV is the normal force due to the vertical component of the movement. Therefore, in such a case, normal force
N H2 NV2 resulting from the combination
of directions must not be used to calculate compound flexion. 15.2.4. Some properties of stick models working in shear
Buildings comparable to the one studied in section 9.2 are considered, in the sense that the overall deformation under the effect of a horizontal component of seismic movement is reduced to shear deformation parallel to excitation without the appearance of torsion around the vertical axis. However, other than the example cited earlier, the distribution of mass and stiffness between the stories is supposed to be arbitrary. Buildings corresponding to structures with columns and beams, regular in the horizontal plane but that can be irregular in the vertical direction, and represented by stick models resembling the one in Figure 9.10 with mass mn and stiffness kn varying between the first storey (n = 1) to the top (n = N). Figure 15.16 shows such a model, excited at the base by an accelerogram s (t), the corresponding mass and rigidity matrices, vectors {u} (displacement with relation to the ground) {Gn} and {hn} which will come in for the calculation of shear forces and bending moments acting on storey n.
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Seismic Engineering
Figure 15.16. Stick model working in shear (on top left) with notations for mass mn, stiffness between stories kn, heights above base zn and the accelerogram for excitation s t , and expressions of matrices for rigidity and mass and vectors {u} (relative displacements), {Gn} and {hn}
It is assumed that the distribution of eigenfrequencies is such that we can calculate the responses through simple quadratic combination (SRSS) without having to use a pseudo-mode, i.e., according to Figure 15.13, where enough modes have been extracted with a frequency less than the cut-off frequency, to reach a cumulative modal mass equal to at least 90% of the total mass.
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Noting x (t, Z, [) as before, the solution of equation [9.3] used for the definition of the response spectrum associated with the accelerogram s t , we state: ai (t) = – Z²i x (t, Zi, [i)
[15.189]
Ai = Max [ai (t)] = Sa (Zi, [i)
[15.190]
Zi and [i being the angular frequency and reduced damping of the mode of rank i and Sa (Z, [) the response spectrum for pseudo-acceleration. As an instantaneous value, the response for relative displacement is given by formula [15.94]: {u} = 6 x t , Zi , [i pi ^vi ` i
[15.191]
i.e., for the component un (relative displacement of the n-th storey of the building), by introducing functions ai (t) defined by [15.189]: un = – 6 i
pi
Zi2
vi , n ai t
[15.192]
vi,n being the n-th component of the mode {vi}. The participation factors pi are development coefficients for direction vector {'} on the basis of eigenmodes (see [9.70] or [15.73]). Development coefficients pi,n of the vector {Gn} defined in Figure 15.16 are also introduced, as they come in later. They are given by:
^vi ` > M @^G n ` T ^ v i ` > M @^ v i ` T
Pi,n =
[15.193]
i.e., since matrix [M] is diagonal: § N · §N · pi,n = ¨ 6 mA vi ,A ¸ / ¨ 6 mA vi2,A ¸ A n © ¹ ©A 1 ¹
[15.194]
pi,1 is no other than the normal participation factor pi. We obtain, for the maximum of un by applying the SRSS combination to [15.192] and taking into account [15.190]: M a x un2 t
6 i
Ai2
Zi4
Pi 2,1 vi2, n
[15.195]
676
Seismic Engineering
As an instantaneous value, vector {Ü} for absolute accelerations is given by formula [15.150] in which we have removed the pseudo-mode: {Ü} = 6 ª¬ x t , Zi , [i s t º¼ pi ^vi ` i
[15.196]
For a weak rate of damping, the usual approximation for pseudo-acceleration can be made, which consists of replacing the absolute acceleration x s by ai (t) defined by [15.189]; we thus have, by also writing pi = pi,1: {Ü} = 6 pi ,1 ^vi ` ai t
[15.197]
i
and for maximum SRSS of a component Ün: M a x Un2 t
6 Ai2 pi2,1vi2, n
[15.198]
i
Now the responses in shear force and bending moment at level n of the building are considered, or respectively Vn and Mn. The shear force Vn in instantaneous value can be calculated in two ways: (1) by taking the resultant inertia forces at storey n and levels situated above, i.e.: Vn = {Gn}T [M] {Ü}
[15.199]
or, considering approximation [15.197]: Vn = 6 ai t pi ,1 ^G n ` > M @^vi ` T
i
[15.200]
or again, according to the definition of pi,n in [15.193]: Vn = 6 ai t pi , I pi , n ^vi ` > M @^vi ` T
i
[15.201]
(2) on the basis of the rigidity and displacement matrix, i.e.: Vn = – {Gn}T [K] {u}
[15.202]
i.e., taking into account [15.192], by replacing pi with pi,I: Vn = 6 ai t i
Pi ,1
Z
2 i
^G n ` > K @^vi ` T
[15.203]
Linear Seismic Calculation
677
Thus, we have the relations: {Gn} = 6 pi , n ^vi `
[15.204]
i
I
Zi2
> K @^vi ` > M @^vi `
[15.205]
The first is similar to [9.70] or [15.73], and the second is a rewritten form of [15.69]. From which, on developing [15.203] and considering the orthogonality of modes (see [15.70]): Vn = 6 ai t pi ,1 pi , n ^vi ` > M @^vi ` T
[15.206]
i
This is the same expression as [15.201]. The two modes of calculation of Vn are thus coherent as long as the reasoning is based on instantaneous values. We shall see later that this coherence is lost if we use SRSS values of the maxima and that, more precisely, shear force Vn calculated on the basis of forces of inertia associated with SRSS acceleration is systematically greater than shear force calculated by performing the SRSS combination in [15.201] or [15.206], which is written as: M a xVn2 t
6 Ai2 pi2,1 pi2, n ª^vi ` > M @^vi `º ¬ ¼ i T
2
[15.207]
Modal mass for storey Pi,n can be defined by the relations:
Pi,n = pi2,n ^vi ` > M @^vi ` T
[15.208]
similar to equation [15.77] used to define modal mass in the regular sense of the term (which is identical to Pi,I). It can easily be demonstrated through reasoning based on that in section 15.2.1 that the sum of modal mass for a storey n is equal to the sum of masses of the structure situated at this level and above: 6 Pi , n i
^G n ` > M @^G n ` T
N
6 mA
A n
[15.209]
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Seismic Engineering
By introducing modal masses for storeys, [15.207] is rewritten in the form: M a xVn2 t
6 Ai2 Pi ,1 Pi , n
[15.210]
i
Maximum shear force VI,max at the base of the structure is obtained by making n = 1 in [15.210] or, with regular modal mass Pi: 2 Vi ,max
6 Pi2 Ai2
[15.211]
i
This formula shows the physical significance of modal mass. If it is supposed in [15.211] that accelerations Ai for modes all have the same value A, as in the numerical application carried out in section 9.2.5 we have for V1,max: V1,max = A 6 Pi2
[15.212]
i
V1,max is thus lower in this case, than the product of A by the total mass of the structure that we would be tempted to consider as the “real” value when all modal accelerations are equal. This option of total mass, often adopted by earthquake engineering codes in simplified methods, thus constitutes a safety margin. For regular structures, to which these simplified methods are applied, V1,max is generally much closer to the contribution of the first mode than the upper limit associated with the consideration of total mass. For example, for the building with four stories studied in section 9.2, V1,max is 0.8975 MtA according to formula [15.212] while the contribution of the first mode is 0.8934 MtA. For the calculation of bending moment Mn at storey n, we consider the moment of inertia forces acting above this storey, i.e.: Mn = {hn}T [M] {Ü}
[15.213]
where the vectors {hn}, defined in Figure 15.16, verify the following relations: {hn} = 6 qi , n ^vi ` with qi , n i
^hn ` > M @^vi ` T ^ v i ` > M @^ v i ` T
[1.214]
For n = 1, {h1} is identical to vector {Z} introduced in 15.1.2 and qi,1 is equal to the participation factor in rotation qi of mode i defined by equation [15.30]. Considering [15.197] and [15.214], due to the orthogonality of modes relation [15.213] is written as:
Linear Seismic Calculation
Mn = 6 ai t pi ,1qi , n ^vi ` > M @^vi ` T
679
[15.215]
i
The same formula is obtained by working on the rigidity matrix and the displacement field, as we have seen before for shear forces. The SRSS combination, applied to [15.215], gives: 6 Ai2 pi2,1qi2, n ª^vi ` > M @^vi `º ¬ ¼ i T
M a x M n2 t
2
[15.216]
We introduce modal inertia moments for the storey defined by: qi2, n ^vi ` > M @^vi ` T
ii , n
[15.217]
from which it is demonstrated, as for modal mass for the storey, that their sum represents the inertia moment of stories situated above the storey n in relation to the plane of this storey: 6 ii , n i
^hn ` > M @^hn ` T
N
6 mA zA zn
2
A n 1
[15.218]
We can thus rewrite [15.216] as follows: 6 Ai2 Pi ,1ii , n
M a xM n2
i
t
[15.219]
which gives, for the maximum M1,max of the moment at the base (n = 1): 2 M 1,max
6 Ai2 Pi ii i
[15.220]
This formula illustrates the physical significance of modal inertia moments; taking into account the very high predominance of the fundamental mode for modal inertia moments (see Table 15.3), the role of higher modes is still weaker for M1,max than for V1,max. Formulae obtained earlier for the maxima of Vn and Mn make it possible to demonstrate the affirmation according to which the use of SRSS accelerations for the calculation of forces produces results that are systematically overestimated with relation to the use of deformations (SRSS combination of modal forces). On the basis of expression [15.207] for maximum of V²n, considering expression [15.194] of pi,n:
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Seismic Engineering
M a x Vn2 t
N §N · 6 Ai2 pi2,1 ¨ 6 ml vi ,l ¸ i 1 l n © ¹
2
[15.221]
i.e. again: M a x Vn2 t
N § N · 6 ¨ 6 fi ,A ¸ i 1© A n ¹
2
[15.222]
with:
gi,A = Aipi.1mAvi,A
[15.223]
For shear force V’n calculated on the basis of SRSS accelerations, we have according to [15.198]: '2 n
Ma x V t
1 ªN §N · 2º « 6 mA ¨ 6 Ai2 pi2,A vi2,A ¸ » A n ©i 1 ¹ ¼» ¬«
2
[15.224]
i.e. again: '2 n
Ma x V t
ª N § N 2 · 12 º « 6 ¨ 6 fi ,A ¸ » «¬ A n © i 1 ¹ »¼
2
[15.225]
[15.222] can be rewritten as follows: M a x Vn2 t
N N ªN º 6 « 6 f i ,2A 2 6 fi , A f i ,O » A² O n i 1 ¬A n ¼
[15.226]
i.e. again by permutation of the order of summations: M a x Vn2 t
N
N
N
6 6 f i ,2A 2 6 ƒi,A ƒi,O
A ni 1
[15.227]
A² O n
On the other hand by developing [15.225] we have: 1
M a xVn' 2 t
N N N §N · 2§N · 6 6 ƒ i2, A 2 6 ¨ 6 ƒ i2,A ¸ ¨ 6 ƒ i2, O ¸ A ni 1 A² O n © i 1 ¹ ©i 1 ¹
from which, by subtracting [15.227] from [15.228]:
1
2
[15.228]
Linear Seismic Calculation
M a xVn' 2 M a xVn2 t
t
1 1 º N ª N § · 2§N · 2 N 2 6 «¨ 6 ƒ i2,A ¸ ¨ 6 ƒ i2,O ¸ 6 ƒ i ,A ƒ i ,O » i i A² O n © i 1 1 1 ¹ © ¹ «¬ »¼
681
[15.229]
Thus we have, using the Cauchy-Schwarz inequality:
§ N 2 ·§ N 2 · §N · ¨ i61 ƒ i ,A ¸¨ i61 ƒ i ,O ¸ t ¨ i61 ƒ i , A ƒ i , O ¸ © ¹© ¹ © ¹
2
[15.230]
The result is that the terms of the second member of [15.229] are all positive (or exceptionally zero) and we have, as declared: Max |V’n| t Max |Vn|
[15.231]
We similarly show that the bending moments calculated on the basis of SRSS accelerations are higher than those calculated by formula [15.219]. Formula [15.210] makes it possible to establish the Matsushima theorem [MAT 84] according to which, for a response spectrum for constant pseudo-velocity, shear force is proportional to the square root of the mass of stories above the level in question. This result comes from the observation that, for stick models working in shear, modal mass is proportional to the square of eigenperiods, either strictly, or more or less closely. This proportionality is exact for models in power laws where mass and stiffness vary according to power laws of the height above base with any index of power [BET 92]. In most cases, it is only approximately verified but with rather low deviations (about 10% to 20% at the most). We thus write, whatever the rank of mode i:
Pi Z²i = C
[15.232]
C being constant. On developing [15.210] we obtain, since Pi = Pi,1: M a x Vn2 t
C6 i
Ai2
Zi2
Pi , n
[15.233]
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Seismic Engineering
For a spectrum for constant pseudo-velocity Sv (branch in I/T for spectra in PS92 Rules in Figure 9.6), we have Ai = Sv Zi and [15.233] is written: M a x Vn2 t
CSv2 6 Pi , n i
[15.234]
i.e. again, according to [15.209]: M a x Vn2 t
§ N · CSv2 ¨ 6 mA ¸ ©A n ¹
[15.235]
which establishes Matsushima’s theorem. A similar result is obtained for bending moment since [15.219] is written under the same hypotheses (Pi Z²i = C and Ai = Sv Zi): M a x M n2 t
CSv2 6 ii , n i
[15.236]
i.e., according to [15.218]: M a x M n2 t
2 · § N CSv2 ¨ 6 mA zA zn ¸ © A n 1 ¹
[15.237]
Matsushima’s theorem implies the “whiplash” effect at the top of the building. The distribution of shear forces defined by formula [15.235] corresponds to a distribution of equivalent static acceleration which is inversely proportional to the square root of the sum of masses situated above the storey in question, thus reaching elevated values for the last stories. This effect is translated by the term in 1 / D n in formula [8.5], which is taken from the Japanese earthquake engineering code. For stick models in shear for which condition [15.232] represents a good approximation (i.e. practically all the cases which do not present noticeable discontinuity in the distribution of mass and stiffness), a general theory of SRSS combinations of modal responses can be developed for response spectra having simple forms, especially corresponding to constant values of pseudo-velocity or pseudo-acceleration [BET 92]. For example, for maximal shear force at the base V1,max, which is the most significant quantity in the calculations prescribed by earthquake engineering codes, we can establish the following formulae, which only bring in mass mn and stiffness kn at different levels (see Figure 15.16): – for a spectrum with constant pseudo-velocity Sv:
Linear Seismic Calculation
§N ·ª N 1 § N ·º V1,max = Sv ¨ 6 mn ¸ « 6 ¨ 6 mA ¸ » © n 1 ¹ ¬ n 1 kn © A n ¹ ¼
1
683
2
[15.238]
– for a spectrum with constant pseudo-acceleration Sa: §N · V 1,max = Sa ¨ 6 mn ¸ ©n 1 ¹
1
2
2 ªN 1 § N · º « n61 ¨ A6n mA ¸ » ¹ »¼ «¬ kn ©
1
2
ªN 1 § N ·º « n61 ¨ A6n mA ¸ » ¹¼ ¬ kn ©
1
2
[15.239]
As any given response spectrum can be enveloped by a spectrum with constant pseudo-velocity or by a spectrum with constant pseudo-acceleration, formulae [15.238] and [15.239] make it possible through a very simple calculation based on characteristics of the model, to determine the upper limit for shear force at the base. It is sufficient to take for Sv the value for pseudo-velocity which corresponds to the branch in I/T of the spectrum (see Figure 9.6), for Sa the value of pseudoacceleration of the spectrum plateau, and retain for the upper limit of V1,max the lowest value calculated by [15.238] and [15.239]. This method in general provides an approximation which is completely acceptable for the real maximum of shear force at the base, the error of excess thus committed being mostly about a few percent. The fact that this error is so low comes from the fundamental mode, which is largely preponderant in the overall response, and in practice is always situated on one of the two branches (in I/T or the plateau) used to define the envelope spectra. In the case of the perfectly regular building with N stories studied in section 9.2 where mass is equal everywhere to m and all stiffnesses equal to k, the application of formula [15.238], after a basic calculation, leads to: V1,max = Mt
Sv 2 2(2 N 1) T1 N ( N 1)
[15.240]
Mt (= Nm) being the total mass of the building and T1 its fundamental period which is given by formula [9.54]: T1 = 2 (2N + 1)
m k
[15.241]
while [15.239] gives, for a spectrum with constant pseudo-acceleration Sa: V1,max = Mt Sa
2N 1 3N
[15.242]
684
Seismic Engineering
Formula [15.242], applied to the example discussed in section 9.2.5 (Mt = 4 u 106 kg, Sa = 2.5 m/s², N = 4), gives for V1,max the value 8.66 u 106 N, which is slightly smaller than that (8.97 u 106N) calculated on the basis of maximal displacement u1, max of the first level. This slight difference comes from the fact that formula [15.211], from which expression [15.242] follows, for perfectly regular buildings, was established by the SRSS combination of modal forces (see [15.201] or [15.206]). It shows the general principle of non-coherence between the SRSS distributions of deformations, forces and accelerations which especially implies that: – forces associated with SRSS deformations, those resulting from the SRSS combination of modal forces and those calculated on the basis of forces of inertia corresponding to SRSS accelerations are different. We have seen earlier that forces deduced from inertia forces are systematically greater than those obtained on the basis of modal forces; the deviation is generally moderate (10 to 20%) for regular structures whose principal modes are excited at comparable levels, but can be considerable in other cases, as we shall see in section 15.2.5. – displacements and deformations that are calculated by applying forces determined either by the combination of modal forces or by transcription of SRSS accelerations in forces of inertia to the structure, are not the same as those resulting from the SRSS combination of displacements and deformations associated with different modes. The correct practice of modal spectral analysis thus consists, in principle, of using only quadratic combinations (simple or complete) of modal contributions of the same nature as those of the quantity that we wish to calculate, i.e., it is necessary to combine: – modal displacements to calculate the response for displacement; – modal forces to calculate the response in terms of forces; – modal accelerations to calculate the response for acceleration; and to not be allowed to use the results of these combinations as input data for the calculation of certain elements of the response. This principle is unfortunately often ignored. Numerous software programs for seismic calculation use SRSS or CQC accelerations to calculate the forces, which is certainly oriented towards safety, but can lead to absurdities (overstatement of more than 100% with relation to the combination of modal forces), in particular for complex structures. In reality, the calculation of accelerations should not be undertaken unless it is necessary to verify the performance of anchorage of equipment installed in the structure. Saftey verification for the structure should be
Linear Seismic Calculation
685
carried out by determining the forces on the basis of modal forces and the displacements on the basis of modal displacements, without any reference to accelerations, for which the convergence of the series of modal contributions is moreover the worst (see sections 15.2.2 and 15.2.5). 15.2.5. Continuous models. Example of a uniform cantilever beam
Continuous models, i.e. those where we conserve a formulation in terms of equations with partial derivatives (as opposed to discrete models which adopt a matricial formulation of equations of movement), only have an analytical solution in simple cases. Some of these solutions however present didactic interest. The case of shear beams verifying the Matsushima condition [15.232] was discussed in section 15.2.4. It is developed in references [BET 89, BET 92]. The current section is limited to cases of uniform cantilever beams (i.e. whose characteristics are constant along the entire height) working under bending, which is an idealized model for long structures such as towers or chimneys.
Figure 15.17. Uniform cantilever beam
A cantilever beam with uniform characteristics is considered (see Figure 15.17), fixed at its base in non-deformable ground which is subjected to action of a horizontal accelerogram s (t). The height of the cantilever beam is denoted by H, its total mass by Mt, its bending inertia by I and the Young’s modulus of the building material by E. For a response in pure bending, we have the following equation for the horizontal displacement u(z,t) with relation to its base:
686
Seismic Engineering
EI
w 4u M t w 2u wz 4 H wt 2
M t s t H
[15.243]
with the boundary conditions corresponding to fixed support (z = 0) and at the free extremity (z = h) where the bending moment and shear force are zero: z=0
u
z=H
w 2u wz 2
wu wz
0
0 and
w 3u wz 3
0 and
[15.244] 0
The initial conditions correspond to the state of rest: t=0
u = o and
wu wt
0
[15.245]
We establish:
]=
z H
Z0 =
EI Mt H 3
[15.246]
and equation [15.243] takes the form: w 4u 1 w 2u w] 4 Z02 wt 2
1
Z02
s t
[15.247]
This equation is resolved by the same method as in the case of the discrete model, i.e. by a development based on eigenmodes; these in turn are determined by the following choices in [15.247]: u (],t) = vi (]) sin Zit ; s (t) = 0
[15.248]
which gives: 2
d 4 vi § Zi · ¨ ¸ vi d ] 4 © Z0 ¹
0
[15.249]
by stating:
Di
Zi Z0
[15.250]
Linear Seismic Calculation
687
The solution of [15.249] which verifies the boundary conditions [15.244] is given (to a close multiplying constant) by: vi (]) = cosh Di] – cos Di] –
cosh D i cos D i sinh D i] sin D i] sinh D i sin D i
[15.251]
and eigenangular frequencies Zn are determined by the following equation, that we get by writing the condition of zero shear force at the free end for expression [15.251]: cosh Di cos Di + 1 = 0
[15.252]
Eigenmodes vi (]) present the property of orthogonality in the interval (0.1) i.e.: 1
³ v ] v ] d ] i
0
0 for i z j
j
[15.253]
which is demonstrated simply by multiplying [15.249] by vj and taking the integral from 0 to 1: 1
D i4 ³ vi v j d ] 0
³
1
0
4
vj
d vi d] 4 d]
[15.254]
from which, by integrating in parts the second member and considering the boundary conditions:
D
4 i
³
1 0
vi v j d ]
2
3
dv j d v i d] ³ 3 0 d] d] 1
³
1
0
d v j d 2 vi d] 2 d] 2 d]
[15.255]
then, by permutation of indices i and j and subtraction:
D
4 i
D 4j ³ vi v j d ] 1
0
0
[15.256]
which leads to [15.253] for Di z Dj. Once the eigenmodes are determined, we look for the solution of [15.247] in the form: u (],t) = 6 rA t vA ] A
[15.257]
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Seismic Engineering
As in the discrete case, we find (by developing [15.257] into [15.247], by multiplying by vi and integrating from 0 to 1, taking into account the orthogonality [15.253]) that the development coefficients rl (t) [15.257] verify the equations: pi s t
ri Zi2 ri
[15.258]
i.e. again, if a term for damping is introduced: ri 2[iZi ri Zi2 ri
pi s t
[15.259]
with the following expression of participation factor pi:
³
pi
I
O
I
vi d ] / ³ vi2 d ]
[15.260]
O
These factors pi are the development coefficients of the unit on the modal base; we thus have: 6 pi vi
[15.261]
1
i
a relation which is similar to equation [15.73] for the direction vector {'}. By integrating [15.261] from 0 to 1 reduced modal mass Pi/Mt is brought in: 2
1 ª 1 º 6 « ³ vi d ] » / ³ vi2 d ] i ¬ 0 0 ¼
1
1 = 6 pi ³ vi d ] i
Pi
0
[15.262]
2
Mt
1 ª 1 º 2 «¬ ³ 0 vi d ] »¼ / ³ 0 vi d ]
[15.263]
whose sum is thus equal to 1. With expression [15.251] for eigenmodes, it can be easily demonstrated that:
³
1 0
vi2 d ]
1 ; pi
³
1 0
vi d ]
2 cosh D i cos D i Pi ; D i sinh D i sin D i M t
pi2
By resolving equation [15.252] we find for the first ten modes.
[15.264]
Linear Seismic Calculation i
Di
Įi2 = Ȧi /Ȧ0
pi
Pi/Mt = p²i
1
1.875104069
3.516015270
0.782991755
0.613076089
2
4.694091133
22.03449157
0.433935895
0.188300361
3
7.854757438
61.39721441
0.254425295
0.064732231
4
10.99554074
120.9019162
0.181898018
0.033086889
5
14.13716839
199.8595301
0.141470838
0.020013998
6
17.27875953
298.5555309
0.115749056
0.013397844
7
20.42035225
416.9907860
0.097941502
0.009592538
8
23.56194490
555.1652475
0.084882630
0.007205061
9
26.70353756
713.0789182
0.074896443
0.005609477
10
29.84513021
890.7317973
0.067012607
0.004490689
689
Table 15.7. The first ten modes of a uniform cantilever
The form of equation [15.252] shows that cos Di, should be very small when index i is large, to compensate for the high value of hyperbolic cosine; we thus have, with a very good approximation:
D i # 2i 1
S 2
; pi #
4
2i 1 S
[15.265]
For example, for i = 6, we have 11 S/2 = 17.27875959 (to be compared with D6 = 17.27875953) and 4/11 S = 0.115749049 (to be compared with p6 = 0.115749056). It is observed in Table 15.7 that the predominance of the first mode in modal mass (61% of the total mass) is less remarkable than in the example of the model in shear discussed in section 9.2. To reach a cumulative modal mass of 90% of the total mass, it is necessary to retain the first five modes. For the calculation of the response, it is supposed that the spectrum has the form shown in Figure 15.18, i.e. a constant plateau of acceleration A0 for periods less than T0 and a descending branch in I/T for periods greater than T0.
690
Seismic Engineering
Figure 15.18. Response spectrum in pseudo-acceleration considered by the studyof a uniform cantilever
We suppose that the number Nd of modes whose period is situated in the descending branch is at the most equal to 2 (values of Nd more than 2 would correspond to structures that are excessively flexible that are never seen in practice) and that responses can be calculated by simple quadratic combination (SRSS) without the use of a pseudo-mode, which is justified by the separation of eigenfrequencies (see Table 15.7) and the cumulation obtained (more than 90% of the total mass) for modal mass of the retained modes. By denoting by Ri, the response of mode i (in displacement, rotation, acceleration, bending moment and shear force), we thus have for the global response R: ª S 2 Z º R = « 6 pi2 Ri2 a 4 i » i Zi ¼» ¬«
1
2
[15.266]
Considering the hypothesis made on the spectrum and relation [15.250], this expression of R can be put in the form: A ª Z 2T 2 R2 R2 º R = 02 « 0 20 6 pi2 i4 6 pi2 i8 » Z0 ¬ 4S i d Nd D i i ² Nd D i ¼
1
2
[15.267]
Modal responses Ri are defined on the basis of the deformation vi (]) of mode i by the relations:
Linear Seismic Calculation
– in displacement Ri = vi (])
691
[15.268]
1 dvi H d]
[15.269]
– in acceleration Ri = – Z²i vi (])
[15.270]
– in rotation Ri = Ti (]) =
2
– in bending moment Ri =
EI d vi 2
H d]
[15.271]
2
3
– in shear force Ri = –
EI d vi 3
H d]
[15.272]
3
Based on expression [15.251] of vi ([) and equation [15.252], these modal responses Ri can be calculated at the base (] = 0), at mid-height (] = ½) and at the top (] = 1). We find: – at the base (] = 0): - in bending moment Ri = 2
- in shear force Ri =
– at mid-height ] =
[15.273]
EI piD i4 H3
[15.274]
1 ): 2
- in displacement Ri = 1
- in rotation Ri =
EI 2 Di H2
Di H
1 4
i
pi2D i2
[15.275]
1 2 2 i pi D i 1 4
- in acceleration Ri = Z02D i4 1
1 4
[15.276] i
pi2D i2
[15.277]
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Seismic Engineering
1 2 2 EI 2 pi D i Di 1 2 H 4 i
- in bending moment Ri =
[15.278]
EI 3 1 2 2 i pi D i 1 Di 3 H 4
- in shear force Ri =
[15.279]
– at the top (] = 1): - in displacement Ri = 2 x (-1) i 1 - in rotation Ri =
1 H
[15.280]
i 1
piD i2
[15.281]
- in acceleration Ri = 2 Z02D i4
[15.282]
Considering expressions [15.273] to [15.282] and that reduced modal mass is equal to the square of participation factors (see [15.264]) from [15.267] we obtain by establishing:
I
Z02To2 ; Pi* 4S 2
Pi Mt
pi2
[15.283]
the following expressions for responses: – in displacement:
Z02 u 1 A0
Z02 u 1 2 A0
ª P* P* º 2 «I 6 i4 6 i8 » ¬ i d Nd D i i ² Nd D i ¼
1
2
[15.284]
i i ª Pi* § 1 * 2 · Pi* § 1 * 2 · º «I 6 4 ¨ 1 Pi D i ¸ 6 8 ¨ 1 Pi D i ¸ » ¸ i ² Nd D i ¨ ¸» 4 4 «¬ i d Nd D i ¨© ¹ © ¹¼
1
2
[15.285]
– in rotation:
Z02 H T 1 A0
ª Pi*2 º *2 I P 6 6 « i d Nd i i ² Nd 4 » Di ¼ ¬
1
2
[15.286]
Linear Seismic Calculation
Z02 HT 1 2
ª Pi* § 1 * 2 Pi* § 1 * 2 i· i ·º «I i d6Nd 2 ¨ Pi D i 1 ¸ i ²6Nd 6 ¨ Pi D i 1 ¸ » Di © 4 Di © 4 ¹ ¹¼ ¬
A0
1
2
693
[15.287]
– in acceleration: A 1 A0
2
A 1 A0
2 ªI 6 Pi*D i4 6 Pi* º i ² Nd ¬« i d Nd ¼»
1
2
[15.288]
ª § 1 i * 2 · § 1 i * 2 · º «I 6 Pi*D i4 ¨1 Pi D i ¸ 6 Pi* ¨1 Pi D i ¸ » ¨ ¸ i ² Nd ¨ ¸» 4 4 «¬ i d Nd © ¹ © ¹¼
1
2
[15.289]
– in bending moment: M 0 M t A0 H
2
M 1
M t A0 H
ª P* º 2 «I 6 Pi* 6 i4 » i ² Nd D i ¼ ¬ i d Nd
1
2
[15.290]
i ª § 1 i * 2 · P * § 1 * 2 · º «I 6 Pi* ¨1 Pi D i ¸ 6 i4 ¨1 Pi D i ¸ » ¸ i ² Nd D i ¨ ¸» 4 4 «¬ i d Nd ¨© ¹ © ¹¼
1
2
[15.291]
– in shear force: V 0 M t A0
2
V 1
M t A0
ªI 6 P *²D 4 6 P *2 º «¬ i d Nd i i i ² Nd i »¼
1
2
[15.292]
ª Pi* § 1 * 2 i· i ·º * 2§1 * 2 «I i d6Nd Pi D i ¨ Pi D i 1 ¸ i ²6Nd 2 ¨ Pi D i 1 ¸ » Di © 4 ©4 ¹ ¹¼ ¬
1
2
[15.293]
These expressions make it possible to assess the influence of higher modes. In fact, as indicated before (see [15.265]) product pi Di is practically constant (equal to 2) when index i of the mode is equal to or higher than 3 (p3D3 = 1.998448978 according to Table 15.7). This leads to the product Pi*D i2 / 4
piD i
2
/ 4 , which
comes in the response expressions at mid-height, being practically equal to one for i t 3. We thus observe that, in formulae [15.284] to [15.293], the sums which represent the contribution of higher modes (i ! Nd) are from the series:
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Seismic Engineering
– in Pi* / D i8 , i.e. in (2i-1) 10 for displacements; – in Pi*2 / D i4 or Pi* / D i6 , i.e. in (2i-1) 8 in both cases, for rotations; – in Pi* , i.e. in (2i-1) 2 for accelerations; – in Pi* / D i4 , i.e. in (2i-1) 6 for bending moments; – in Pi*2 or Pi* / D i2 , i.e. in (2i-1) 4 in both cases, for shear force. The influence of higher modes is related to the convergence velocity of these series; in the order of the increasing influence of these modes, the responses are to be ranked as follows: displacement, rotation, bending moment, shear force and acceleration. We find, as mentioned earlier, that the convergence of the quadratic combination is much slower for accelerations than for displacements. Relations [15.284] to [15.293] correspond to the general formula: R* (]) = [IBR (], Nd) + CR (], Nd)]½
[15.294]
in which R* (]) denotes the adimensional responses corresponding to the first members of equations [15.284] to [15.293] and BR (], Nd) and CR (], Nd) two numerical coefficients which depend on the type of response, the position of the considered point (] = 0, ½ or 1) and the number Nd of modes situated on the descending branch of the spectrum (Nd = 0, 1 or 2). Coefficients BR (], Nd) are given in Table 15.8. Nd = 0
Nd = 1
Nd = 2
Bu (1, Nd)
0
0.198368570
0.199919904
Bu (½, Nd)
0
0.022867124
0.023657247
BT (1, Nd)
0
0.375862290
0.411319316
BT (½, Nd)
0
0.268332303
0.268650850
BA (1, Nd)
0
30.31627756
396.0097528
BA (½, Nd)
0
3.494737519
189.7493429
BM (0, Nd)
0
2.452304356
3.205505800
BM (½, Nd)
0
0.282691700
0.666311445
BV (0, Nd)
0
4.646546219
21.86159957
BV (½, Nd)
0
3.317221443
3.471882046
Table 15.8. Coefficients BR (], Nd) for Nd = 0, 1 or 2
Linear Seismic Calculation
695
For the calculation of coefficients CR (], Nd), in principle it would have to be limited to modes whose frequency is less than the cut-off frequency, as we have seen in 15.2.2. The number of these modes depending on the structure in question and the chosen spectrum, the simplest option is to retain all the modes for the calculation, thus to make summations of infinite series. We thus obtain an overestimation of the effect of higher modes, which is insignificant in the majority of cases, considering the rapid convergence of these series. The case where convergence is slowest is that for accelerations where, as indicated earlier, the terms in the series are equal to reduced modal mass M i . The excess error made by taking all the modes is thus in the order of the default of cumulative modal mass corresponding to the cut-off frequency or, according to Table 15.7, 13.4% if there are only three modes before this frequency, 8.1% if there are five and 5.1% if there are eight. Table 15.9 gives coefficients CR ([, Nd) calculated with all the modes in cases Nd = 0, 1 or 2. Nd = 0
Nd = 1
Nd = 2
Cu (1, Nd)
0.016049382
0.000003213
0.000000018
Cu (½, Nd)
0.001851364
0.000001627
0.000000000
CT (1, Nd)
0.030477973
0.000074216
0.000001187
CT (½, Nd)
0.021706795
0.000001212
0.000000556
CA (1, Nd)
4.000000000
1.547695644
0.794494200
CA (½, Nd)
0.821257268
0.538565568
0.154945823
CM (0, Nd)
0.200000000
0.001631425
0.000080092
CM (½, Nd)
0.023662165
0.000795040
0.000004917
CV (0, Nd)
0.417434424
0.041572134
0.006115108
CV (½, Nd)
0.271025275
0.002692971
0.002374424
Table 15.9. Coefficients CR ([, Nd) for Nd = 0, 1 or 2
In applications, the determination of the number Nd of modes situated on the descending branch of the spectrum in Figure [15.18] is carried out on the basis of I defined by [15.283]; if Nd = 0 it signifies that:
Z1 t
2S EI 2S 1 or D12 t or again I t 4 3 D1 T0 T0 Mt H
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Seismic Engineering
If Nd = 1, it signifies that:
Z1
2S 2S 1 1 and Z2 t or 4 d I < 4 T0 T0 D2 D1
If Nd = 2 it signifies that:
Z2
2S 2S 1 1 and Z3 t or 4 d I < 4 T0 T0 D3 D2
We thus have, in numerical values: Nd = 0 if I t 0.080890681 Nd = 1 if 0.002059652 dI 0.080890681
[15.295]
Nd = 2 if 0.000262705 dI 0.002059652 We now dispose of all the elements to make numerical applications. For example, (see Figure 15.19 on the left) a cantilever made up of a hollow cylinder of height H = 120 m, external diameter D = 10 m and whose wall thickness e is equal to 0.4 m is considered. This cantilever beam can represent a summary model of a factory chimney (real chimneys generally have a diameter which decreases slightly based on the height and a smaller thickness at the top than at the base).
Linear Seismic Calculation
697
Figure 15.19. Cantilever beam modeling a factory chimney and two cases of spectra of the type in Figure 15.18 (drawn here in the diagram for acceleration in ordinate value, angular frequency in the abscissa)
For characteristics of the material (concrete), by taking a density U = 2,500 kg/m3 and Young’s modulus E = 3 u 104 Mpa we find that for total mass Mt and bending inertia I: ª§ D · § D · Mt = S «¨ ¸ ¨ e ¸ ¹ «¬© 2 ¹ © 2 2
I=
2
º 6 » HU = 3.62 x 10 kg »¼
4 S ª§ D · § D
4 · º 4 e «¨ ¸ ¨ ¸ » = 139 m 4 «¬© 2 ¹ © 2 ¹ »¼
[15.296]
[15.297]
from which for angular frequency Zo defined by [15.246]:
Zo =
EI = 0.816 rd/s Mt H 3
and, for eigenfrequencies and periods (Table 15.10).
[15.298]
698
Seismic Engineering Index i of mode
Frequency gi (Hz)
Period Ti (s)
1
0.457
2.19
2
2.34
0.428
3
8.02
0.125
4
15.7
0.064
5
26.0
0.038
Table 15.10. Frequencies and periods of the first five modes of the chimney shown in Figure 1.19 (H = 120 m, D = 10 m, e = 0.4 m)
In the table we are limited to the first five modes which are the only ones to have a frequency lower than the cut-off frequency (of about 30 Hz) For the spectrum defining seismic excitation, two cases (see Figure 15.19 on the right) are considered: – Ao = 7.5 m/s², T0 = 0.3s, which corresponds to spectrum So (rock) in Figure 9.6 for an acceleration in zero period of 3 m/s² and prolongation of the plateau at 7.5 m/s² until zero period (infinite frequency); – Ao = 4.5 m/s², To = 0.6 s, which corresponds to spectrum S2 (average soil) in Figure 9.6 for an acceleration at zero period of 2 m/s² and prolongation of the plateau at 4.5 m/s² until zero period (infinite frequency). With these values for To and one obtained for Zo (see [15.298]) we find for parameter I defined by [15.283]: – in the 1st case (To = 0.3s), I = 0.00152, which shows, according to [15.295] that we have Nd = 2; – in the 2nd case (To = 0.6s), I = 0.00607 from which Nd = 1. With Tables 15.8 and 15.9, formula [15.294] enables the calculation in both cases of the responses of the cantilever beam in displacement (u (1) and u (½)), rotation (T (1) and T (½)), acceleration (A (1) and A (½)), bending moment (M (0) and M (½)) and shear force (V (0) and V (½)); the results obtained are presented in Table 15.11.
Linear Seismic Calculation
Type of
First case
Second case
All
1st mode
All
1st mode
modes
only
modes
only
u (1) (m)
0.196
0.195
0.235
0.234
u (½) (m)
0.067
0.066
0.080
0.079
T (1) (rd)
0.0023
0.0022
0.0027
0.0027
T (½) (rd)
0.0019
0.0019
0.0023
0.0023
A (1) (m/s²)
8.86
1.61
5.92
1.93
A (½) (m/s²)
4.99
0.547
3.37
0.656
response
(106 Nxm)
229
199
251
239
M (½) (106 Nxm)
104
67.6
98.0
81.1
V (0) (106 N)
5.39
2.28
4.30
2.74
(106 N)
2.38
1.93
2.46
2.31
M (0)
V (½)
699
Table 15.11 Calculation of responses for the cantilever beam and the two cases of spectra represented in Figure 15.19 (H = 120 m, D = 10 m, e = 0.4 m)
We observe: – that the first mode is sufficient to realize displacements and rotations; – that, on the contrary, the consideration of a single first mode would lead to a significant underestimation of moments and forces, particularly for the moment at mid-height and for the force at the foot; – that accelerations depend essentially on higher modes; – that average acceleration corresponding to shear force at the base (or the quotient of V (0) by total mass Mt), which is 1.49 m/s² in the first case and 1.19 m/s² in the second, is of a much lower level than that for response in acceleration A (1) and A (½); – that the two cases considered for the spectrum lead to rather similar results (the first case being the most penalizing for accelerations, the moment at mid-height and the force at the foot, and the second for displacements, the moment at the foot and force at mid-height), which is not surprising as the two spectra give very close accelerations for the first two modes (see Figure 15.19 where the corresponding angular frequencies Z1 and Z2 are indicated). The deviations between the complete solution and that with the single first mode are lower in the second case than the first.
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Seismic Engineering
This example illustrates tendencies indicated earlier on the influence of higher modes and we find the classification: displacement, rotation, bending moment, shear force and acceleration, in the order of increasing influence. The difference from the example discussed in section 9.2, where the structure in question was also perfectly regular, is that here it is a structure with a long period for which the first mode is situated in a spectrum zone that is not very amplified. Higher modes are in the amplified zone, which reduces the predominance of the fundamental mode, particularly for shear force. We also come across the incoherence, and this is the main lesson to be learnt from this example, between accelerations and forces and moments calculated by quadratic combination. Forces, that could be calculated by taking a distribution of accelerations corresponding to SRSS accelerations all acting in the same direction, are around three times higher here than those obtained by SRSS combination of modal forces. This incoherence does not mean that SRSS accelerations are meaningless; they represent a reasonable approximation of real maximal values for absolute acceleration at different points of the structure. What produces incoherence, is the hypothesis that consists of taking all the accelerations with the same sign to calculate forces. If the reasoning is correct by taking the consequence of inertia forces as an instantaneous value, as in section 15.2.4, positive and negative alternations of distribution of acceleration corresponding to higher modes (which present many nodes and anti-nodes of opposing signs) produce an effect of compensation and we reach, in cases where all the modes are excited with the same pseudo-acceleration A0 of the plateau, formula [15.212] for shear force V0 at the base: 6 Pi2
V0 = A
[15.299]
i
With SRSS accelerations all taken with the same sign we would have (see [15.288] and [15.289]): V’0 = A 6 Pi
[15.300]
i
The deviations between formulae [15.299] and [15.300] can be considerable; by taking modal mass Pi inversely proportional to (2i –1)² (which is the case for high modes for the uniform cantilever beam; see [15.265]) we find in fact, that: – if summations are applied to all the modes, we have V0 / V’0 = 0.817 as: 1
1 3
4
1 5
4
...
S
4
96
1 1 § · 0.817 ¨1 2 ² " ¸ © 3 5 ¹
0.817
S
2
8
Linear Seismic Calculation
701
– if summations are applied to all modes except the first, we have V0/V’0 = 0.518 as: 1 3
4
1 5
4
...
S
4
§ 1 1 · 1 0.518 ¨ 2 2 ... ¸ 96 ©3 5 ¹
§S · 0.518 ¨ 1¸ - ¨ 8 ¸ © ¹ 2
– if summations are applied to all modes except the first two, we have V0/ V’0 = 0.394 as: 1 1 ... 54 7 4
S4 96
1
1 81
§1 1 · 0.394 ¨ 2 2 ... ¸ 5 7 © ¹
§S2 1· 0.394 ¨ 1 ¸ 8 9 © ¹
We can only repeat the principle put forward at the end of section 15.2.4: the results obtained by quadratic combination should not be used as data to calculate certain elements of the response and, particularly, SRSS or CQC accelerations must not serve to determine the forces.
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Chapter 16
Notions on Soil/Structure Interaction
16.1. General observations on soil/structure interaction 16.1.1. Presentation of the soil/structure interaction phenomena It has been indicated several times, particularly section in 9.2.1, that the hypothesis commonly accepted by earthquake engineering codes for a perfectly embedded foundation of a building on non-deformable ground with all its points acted upon by the same movement is apparently not applicable for buildings with a significant mass constructed on grounds which are not rocky. The influence of the deformability of the ground can be estimated in a simple manner using the following method. A building linked to the ground through a rigid basemat and having a typical shear type deformability is taken for study (see Figure 16.1); this link is supposed to be elastic and can be represented by a system of springs having a stiffness Kx with respect to a horizontal force (which produces a horizontal displacement ux) and a stiffness KT with respect to an overturning moment about the horizontal axis (which produces a rotation T). Under the action of a horizontal force F applied at a height h above the basemat, the displacement with respect to the portions of the ground unaffected by the deformation of the link with the basemat consists of three terms which are added together: – displacement ux = F/Kx of the basemat; – displacement uT = hT produced at a height h by the rotation of the basemat; – displacement ud = F/k due to the building’s own deformability, k being its stiffness at height h.
704
Seismic Engineering
Figure 16.1. Deformation of a building whose basemat is elastically linked to the ground
Rotation T is determined using the equilibrium of the moments: Fh = KTT
[16.1]
Total displacement u becomes: § 1 h² 1 · u = ux + uT + ud = F ¨ ¸ © K x KT k ¹
[16.2]
from which the following is derived: ª k § K x h2 F = ku / «1 ¨1 KT «¬ K x ©
·º ¸» ¹ »¼
[16.3]
Compared to the case of non-deformable ground (Kx and KT tending towards infinity) it is observed that the apparent stiffness of the building is reduced; this reduction has its repercussions on natural frequency T (which is inversely proportional to the square root of the stiffness); this leads to:
Notions on Soil/Structure Interaction
T Tf
1
k § K x h2 · ¨1 ¸ Kx © KT ¹
705
[16.4]
Tf being the frequency of the building embedded in non-deformable ground. Stiffness Kx and KT of the soil springs can be calculated using the following formulae derived from the theory of elasticity for a rigid circular basemat of radius r, linked to the surface of a homogenous half-space characterized by its shear modulus G and its Poisson’s coefficient v: Kx =
8 Gr 2 Q
[16.5]
KT =
8 Gr 3 3 1 Q
[16.6]
For natural stiffness k of the building, its relation with the embedded base frequency Tf is used, i.e.: k=
4S 2 m Tf2
[16.7]
m being the mass of the building which can be linked to the pressure p exerted by the basemat on the ground under the effect of self-weight: mg = Sr²p
[16.8]
g being the acceleration due to gravity. For frequency Tf, its proportionality to height h is accepted according to the formula: Tf = T0
h h0
[16.9]
derived from the Japanese earthquake engineering code (see [8.4]) and which has already been used in section 15.1.2 (see [15.42]) where it was observed that when T0 = 1 s, h0 = 50 m corresponds to the case of concrete structures and h0 = 33 m to the steel structures). By transposing equations [16.5] to [16.9] in [16.4], we have:
706
Seismic Engineering
T Tf
ª S3 h 2 pr § 3 1 Q h 2 2 Q 0 2 2 ¨ 1 «1 2 gT0 Gh © 2 Q r 2 ¬«
·º ¸» ¹ ¼»
1
2
[16.10]
i.e. again by assuming:
O
T Tf
h ;V r
h02 p gT02 rG
ª S3 § 1 3 1 Q · º «1 2 Q V ¨ 2 ¸» 2 2 Q ¹ ¼» ©O ¬«
1
2
[16.11]
This fairly simple expression enables the assessment of the influence of these three parameters Q,V and O; the expression referring to Poisson’s coefficient is as usual not very significant because in practice its range of variation is limited; i.e., about 0.25 ~ 0.3 (rock) to 0.4 a 0.45 (relatively soft ground); the slenderness ratio O which intervenes through the inverse of its square ceases to have any significant influence beyond 2 (because 1 / O² becomes small as against 3 (1-Q)/2-Q, which is slightly higher than 1); the most important parameter is V, which brings in a (h0) type of structure, the dimension of the foundation (r) and the quotient of the static pressure by the shear modulus of the ground (p/G). The highest values of V are of the order of 0.1, corresponding to a concrete structure (h0 = 50 m) exerting a strong pressure (p = 0.4 a 0.5 MPa) on ground with mediocre characteristics (G = 100 a 200 Mpa i.e., shear wave velocities of 200 a 300 m/s which in turn means modules just enough to ensure static equilibrium under such high pressures) through the intermediary of a small size basemat (r = 5 m). Table 16.1 shows the values of T/Tf, calculated using [16.11] for Q = 0.4, O = 1.2 or 4 and V varying from 0.001 to 0.1.
Notions on Soil/Structure Interaction
V
§T · ¨ ¸Ȝ= 1 © T¥ ¹
§T · ¨ ¸O=2 © Tf ¹
§T · ¨ ¸O=4 © Tf ¹
0.001
1.026
1.017
1.015
0.005
1.125
1.083
1.072
0.01
1.237
1.159
1.139
0.02
1.436
1.299
1.262
0.03
1.611
1.425
1.375
0.04
1.768
1.541
1.479
0.05
1.912
1.649
1.576
0.06
2.046
1.750
1.668
0.07
2.172
1.846
1.754
0.08
2.291
1.936
1.837
0.09
2.404
2.023
1.916
0.10
2.512
2.107
1.992
707
Table 16.1. Numerical applications of equation 16.11 for Q = 0.4
It is obvious that taking the ground deformability into account has a significant effect on the frequency of the structure as soon as parameter V reaches 0.02; this limit is easily exceeded for large buildings constructed on alluvial grounds even if their quality is reasonably good; as an example, for a concrete building of ground area 300 m² (r = 10 m) exerting a pressure 0.3 MPa on ground of modulus G equal to 320 MPa (corresponding to a specific mass of 2,000 kg/m3 and a shear wave speed of 400 m/s), we have V = 0.023. Steel buildings are less sensitive to this effect because the shift from h0 = 50 m (concrete) to h0 = 33 m (steel) divides parameter V by 2.30, all other things being equal. The increase in the periods (and, along with it, the increase in the displacements of the building) are the most easily understandable consequences of the deformability of the ground; the above formulae enable an estimation of their orders of magnitude and show that this effect is significant for big buildings constructed on grounds with medium or mediocre mechanical characteristics. However, the term soil/structure interaction given generally to the study of phenomena related to the transmission of movement from the ground to the foundations also consists of some less intuitive aspects; such as:
708
Seismic Engineering
– the fact that the stiffness of the soil springs that control the extension of the periods are not constants but functions of the frequency at which the oscillatory movements are carried out; formulae [16.5] and [16.6] used earlier to represent these stiffnesses are valid under static conditions and therefore are acceptable a priori for sufficiently low frequencies but when the frequencies are higher they can introduce serious errors. It will be seen in section 16.1.3 that it is the parameter: a0 =
Zr vs
[16.12]
(Z = pulse of the oscillation, vs = shear wave velocity of the soil) which is suitable for assessing the influence of the frequency; low values of this parameter correspond to the low stiffness variations with respect to their static values; for higher values of a0 the influence on KX is practically negligible but significant for KT and for Kz (vertical pumping movement); – appearance of a damping effect, called radiative or geometric, which has already been mentioned in section 15.1.3; this corresponds to the waves emitted on the ground by the vibrations of the foundations which carry part of the vibratory energy of the structure far and thereby play the role of a damper; this effect depends equally on the frequency and shows up in different ways depending on the ground configuration; it will be seen in section 16.1.3 that it can reach very high values for translation movements (horizontal and vertical) when the ground is homogenous but that it becomes zero below a certain frequency in the case of a soil layer on top of compact bedrock; – the need to define the ground movement not just at a point as in the case of block motion but in the entire volume involved in the soil/structure interaction phenomena by using a wave model. Characterizing the movement in terms of response spectra or accelerograms is in general in relation to the movement of a point of the surface on a vacant site, i.e., before the construction of the structure under study; this movement is known as the free-field ground motion. The data on this movement should be completed by establishing a wave model defining the movement of all the points of the ground and it should be compatible with the free field specification at the surface. The most commonly used model is that of plane waves of vertical propagation; shear waves for the horizontal components, tractioncompression waves for the vertical component; the equations describing this model and the deconvolution techniques used to determine the movement deep below when it occurs on the surface were given in section 5.3.2; – the difference, sometimes very significant, that can exist between the free field movement culminating from the study of seismic hazard and the real movement of the foundation of the structure, taking into account the soil/structure interaction effects; this difference is the result of not just the deformability of the ground but
Notions on Soil/Structure Interaction
709
also of the point to point coupling of the ground imposed by the basemat, if it is sufficiently rigid; this latter effect has a tendency to filter out the high frequencies of the movement transmitted to the structure with respect to the free field movement; this fact is confirmed by comparing the recordings obtained inside and outside a building; however in common practice, this effect is still not being considered in the numerical models (see section 16.2.4); – existence of the unilateral link type nonlinearities, i.e., corresponding to a momentary loss of contact between the ground and a portion of the surface of the foundation; this is the main problem with the basemat uplift (see section 17.2.2) which can intervene even for a linear elastic behavior of the ground; other examples of unilateral link require the appearance of irreversible deformations such as the ovalizing of the drilled holes around the piles which results from a horizontal seesaw movement and modifies the resistance pattern of the foundation. The above observations show that the soil/structure interaction is a field for specialists. The fact this interaction has been deliberately ignored in most of earthquake engineering codes confirms more this fact, than the affirmation (which anyway needs a little more refinement, as shall be seen in section 16.2.2) according to which ignoring this interaction would be on the side of safety. It hardly seems possible to codify its calculation procedures in a simple manner which would enable “ordinary” engineers in design offices to treat this topic in a reliable manner. The aim of this chapter is just to provide an explanation for the physical phenomena and some indications on the simplest methods and their validity limits; this should explain the use of the term “notions” in the title. 16.1.2. Kinematic and inertial interaction
It is common to distinguish between two forms of soil/structure interaction: – kinematic interaction which corresponds to the modifications of the incident field of waves (free field movement) due to the presence of the foundation taking its shape and rigidity into account but disregarding its mass; – inertial interaction which corresponds to the effects of the forces of inertia associated with the movement produced after taking into account the kinematic interaction and with the real masses of the foundation and the superstructures. This distinction may seem a bit academic; it will be seen later in this chapter that in practice this distinction is very important for justifying the impedance functions method for structures with shallow foundations (see section 16.2.2). It results from a theorem by Kausel [KAU 78] on a possible decomposition of the solution to general equation [15.67] of the formulation on relative displacements:
710
Seismic Engineering
[M] { u } + [C] { u } + [K] {u} = – s (t) [M] {'}
[16.13]
Using such a formulation to take into account the soil/structure interaction warrants some notes since it corresponds to the hypothesis of a block translation movement of the supports; it is therefore necessary to specify these supports and the extension of the model associated with matrices [M], [C], [K] and with vector {u}. In order to identify the supports for which the hypothesis of a block motion is acceptable, it is necessary to move away from the foundation so that the disturbances due to the deformations of the ground in the vicinity of the foundation can be considered negligible. The ground should therefore be divided into two parts: – a disturbed volume around the foundation, which forms part of the model associated with equation [16.13], along with the structure under study; – an undisturbed volume at a certain distance from the foundation which forms the supports of the model and to which a command acceleration defined by the function s (t) is applied. Determining the limit between the disturbed and the undisturbed volume depends on the size of the building and the characteristics of the layers of the terrain; it therefore requires some experience (the division between the disturbed and the undisturbed volume is in reality conventional since the effect of the disturbances is felt in an increasingly attenuated manner up to infinity) and a good knowledge of the advantages and disadvantages, as well as the limitations of the different possible methods. The dimensions of the disturbed volume are typically about two to three times that of the foundation of the structure.
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Figure 16.2. Establishing a discretized model for a soil/structure interaction calculation: 1) definition of the disturbed volume, 2) deconvolution on the basis of the surface to define the accelerogram s (t) at the base of the disturbed volume, 3) definition of the absorbing boundary conditions at the vertical limits of the disturbed volume, 4) discretization of the structure and the ground within the disturbed volume
As indicated earlier the generally adopted seismic wave model to define the free field movement is vertical propagation; the movement is therefore the same at all the points on a same horizontal plane (consisting of a wavefront) but varies according to the depth considered; as a result, the hypothesis of block movement at the limits of the disturbed volume is possible only if these limits are in the same horizontal plane. Regarding the limits of volume contained in the vertical planes, as it is not possible to consider them as supports, other conditions called the absorbing boundary conditions are in general applied to them; these help in preventing the seismic waves from being reflected on these boundaries; these conditions are similar to the ones studied in section 5.3.2 for the ground columns (see [5.65]). The construction of a discretized model (of finite elements or finite differences) for calculating the soil/structure interaction involves work to be carried out in several stages; Figure 16.2 gives a schematic representation of this work: 1) estimating the volume of the disturbed ground taking into account its characteristics and those of the structure; in practice the boundaries of this volume consist of basic horizontal plane and vertical lateral limits;
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Seismic Engineering
2) determining the accelerogram s (t) to be imposed at the base of the disturbed volume through deconvolution calculation (see section 5.3.2) based on the data on free field motion at a point S on the surface of the undisturbed volume; 3) defining the absorbing boundary conditions at the limits of the disturbed volume; these conditions correspond in general to the viscous dampers distributed over its limits; 4) establishing a discretized model for the ground and structure assembly in the disturbed volume. The practical problems involved in establishing and implementing such models shall be dealt with in section 16.2.1 while this section will be limited to proving the theorem by Kausel based on equation [16.13]. The matrix of mass [M] is decomposed as: [M] = [Ms] + [Mo]
[16.14]
[Ms] being the part of [M] that corresponds to the masses of the ground and [Mo] the one that corresponds to the masses of the structure. The solution {u} as a sum of two terms is: {u} = {u1} + {u2}
[16.15]
By transposing [16.14] and [16.15] in [16.13], we have: 1} + [C] { u 1} + [K] { u1 } + [Ms] { u 2} + [M0] ({ u 1} + { u2 }) + [C] { u2 } + [K] { u2 } [Ms] { u =–
s (t) [Ms] {'} – s (t) [M0] {'}
[16.16]
{u1} should satisfy the relation: [Ms] { u1 } + [C] { u 1} + [K] {u1} = - s (t) [Ms] {'}
[16.17]
Therefore, {u2} should be the solution of: [M] { u 2 } + [C] { u 2 } + [K} {u2} = – [M0] ({ u1 } + s (t) { ' })
[16.18]
The interpretation of equations [16.17] and [16.18] is very simple and justifies the distinction announced earlier: – {u1} is the movement that would be obtained with a fictitious building without mass (but which would have retained its rigidity and damping) by imposing excitation s (t) at the base of the model;
Notions on Soil/Structure Interaction
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– {u2} is the movement that would be obtained with the complete model (building equipped with its mass) by imposing the forces of inertia corresponding to the absolute movement (drive plus relative displacement) that the fictitious building had in the first stage, on the only masses of the building. The total movement {u} = {u1} + {u2} of the real building is therefore the sum: – of the movement that the fictitious building of zero mass would be subjected to under the action of the seismic excitation imposed at the base of the model (kinematic interaction); – of the movement that the real building would be subjected to, in the absence of the seismic excitation at the base, had it been subjected to the forces of inertia corresponding to the previous movement (inertial interaction). Generally this type of decomposition of the solution into two terms is not of practical interest for determining {u} because each of the calculations necessary to determine {u1} then {u2} is as heavy as the direct calculation of {u} based on [16.13]. In the case of shallow foundations, i.e., the case of buildings that are just placed on the surface of the ground, it is found that the kinematic interaction is zero for the wave model with vertical propagation; in effect, considering the zero mass fictitious building does not disturb the movement of the ground in any way. The acceleration vector that appears as the second term of [16.18] is the same for all the points of the building and is equal to the absolute free field motion on the surface of the ground. It is therefore sufficient in the case of buildings with shallow foundations to consider applying this free field motion on their masses directly. It should be noted that this method is not applicable if the foundations are sunk to a certain depth because the rigidity of the fictitious building will then disturb the wave propagation; even in cases where the foundations are superficial this method cannot be applied if the seismic wave model is different from that of vertical propagation. 16.1.3. Radiative (or geometric) damping
The radiative (or geometric) damping has already been discussed (see sections 15.1.3 and 16.1.1). Its physical causes are easy to understand; to prove them in the simplest manner, a building that is moved from its equilibrium position can be studied (see Figure 16.3); for this, a horizontal static force F is applied on the building which is allowed to oscillate freely after suddenly stopping this force.
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Figure 16.3. Oscillations of a building after cutting off the force F; these oscillations produce a series of radial waves on the ground which move away from the foundation of the building
The oscillations of the building give rise to waves in the ground and these waves propagate by moving away from their point of origin towards infinity; they carry with them a part of the deformation energy which was accumulated in the initial position (before the release of the force F). This effect of energy transfer is equivalent to a damping effect and leads to a progressive diminution of the amplitude of oscillations even if there exists no other internal damping (viscosity of the ground, friction at the ground-foundation interface). A similar effect can be observed with regard to objects floating on vast stretches of water whose oscillations produce a divergent system of waves on the surface. Though the causes of this radiative damping have nothing mysterious about them, the quantification of this phenomenon calls for rather complex calculations even with simple configurations. To get a simple idea of this situation, a rigid disc of radius r placed on the surface of a homogenous elastic half space can be examined; this disc is subjected to the action of a variable harmonic vertical force of pulse Z (see Figure 16.4).
Figure 16.4. Action of a harmonic vertical force on a rigid disc placed on a homogenous elastic half space
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Normally, a formulation with complex numbers is used for this type of calculation; the harmonic vertical force P is therefore written as:
P = P0 e iZt i
1
[16.19]
Vertical displacement G of the disc under the action of P can be determined in an analytical manner and it is expressed as:
G
1 Q Po e 4Gr
iZ t
F1 iF2
[16.20]
G being the shear modulus of the ground, Q its Poisson’s coefficient, F1 and F2 two dimensionless functions of Q and the parameter a0 = Zr/vs already defined ([16.12]), whose expressions being very complicated will not be given here. The variations of F1 and F2 based on the parameter a0 for a Poisson coefficient Q equal to 1/3 are given in Figure 16.5 (F2 being negative, it is –F2 that is represented).
Figure 16.5. Functions F1 and F2 of equation [16.20] for Q = 1/3 and a0 varying from 0 to 4
For other values of Q and a0 the numerical tables or the abacus can be referred to [SIE 91]. By differentiating [16.20] with respect to time, we obtain:
G
1 Q Po e 4Gr
iZ t
iZ F1 Z F2
[16.21]
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Seismic Engineering
and, by multiplying [16.20] by ZF1, [16.21] by –F2 and through member to member addition:
Z F1G F2G
1 Q iZ t P0 e Z F12 F22 4Gr
[16.22]
i.e., for the exciting force P = Poe iZt: P = KG + C G
[16.23]
with: K=
FI 4Gr I Q F12 F22
C=–
F2 4Gr 2 1 Q Z F1 F22
[16.24]
4
U vs r 1 Q
2
F
a0 F2
2 1
F22
[16.25]
where the relations G = Uv 2s and a0 = Zr/vs are used in the second expression of C. Relation [16.23] shows that the ground on which the disc of radius r is placed can be represented by a spring of stiffness K acting in parallel with a dashpot of coefficient C (C is positive since F2 is negative); K and C depend on the excitation frequency through the intermediary of parameter a0. Figure 16.5 shows that for low values of a0, F1 can be considered equal to 1 and F2 proportional to a0 (F2 = – 0, 85 a0); therefore according to [16.24] and [16.25], the values K0 and C0 of K and C are: K0 =
4Gr 1 Q
C0 = 3.4
U vs r 2 1 Q
[16.26]
[16.27]
K0 is the static vertical stiffness (similar to expressions [16.5] and [16.6] of the horizontal translation stiffness and the rocking stiffness). These values K0 and C0 can be used to normalize values [16.24] and [16.25] of K and C; Figure 16.6 shows the variations of K/K0 and of C/C0 as a function of a0, for Q = 1/3:
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Figure 16.6. Variations of the stiffness K/Ko and the damping C/C0 normalized for Q = 1/3 and a0 varying from 0 to 4
This figure shows that the influence of the excitation frequency is significant for the values of a0 higher than 2, more so for the normalized stiffness. The damping which corresponds to coefficient C (see [16.25]) represents the radiative damping for the vertical excitation of a rigid disc on a homogenous ground; from [16.24] and from the first expression of C in [16.25], it can be easily shown that the associated reduced coefficient of damping [ can be given by:
[=
1 F2 2 F1
[16.28]
Analysis of Figure 16.5 shows that this formula [16.28] leads to values of [ higher than 50% once a0 exceeds the value (slightly more than 1) for which –F2 = F1; these very high values of damping are much stronger than the structural damping (which are of the order of several percent; see section 15.1.3); this implies that the radiative damping has a very strong influence on the vertical excitation response, at least under the conditions defined earlier (rigid circular basemat, homogenous elastic ground). Expressions [16.24] and [16.25] of K and C bring out the real and imaginary parts of impedance function which by definition is the quotient of the force applied P = PoeiZt (see [16.19]) by the displacement G that it produces; according to [16.20] we thus have:
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Seismic Engineering
P
G
4Gr 1 1 Q F1 iF2
4Gr F1 iF2 1 Q F12 F22
[16.29]
from which by comparing with [16.24] and [16.25]: K = Re ; C =
1
Z
Im ()
[16.30]
The formalism of the impedance functions is the one which is generally adopted for the study of the soil/structure interaction, mainly in the case of shallow foundations (see section 16.2.2). The radiative damping also reaches high values for the horizontal translation movement (still known as the sieving movement); on the other hand it is clearly lower for rotational movements (rocking around a horizontal axis and torsion around a vertical axis). The formulae for calculation applicable to these cases shall be presented in section 16.2.2. The hypothesis of a homogenous half space to represent the ground constitutes a text book case which is rarely observed in reality. The real grounds are generally stratified with contrasts of mechanical properties which can be significant from one layer to the other; in cases where there are layers of high thickness corresponding to the same type of materials, a progressive increase of modules with depth is generally observed. The modulus variations, discontinuous or continuous, resulting from these heterogenities have a significant influence on the propagation of the divergent waves towards infinity; these waves are responsible for the radiative damping to the extent of suppressing the possibility of an effective transfer of energy in certain cases; the radiative damping is then zero at least for certain values of the excitation frequency. A typical case of such a situation is provided by a layer of ground on top of a rigid bedrock. The waves emitted by a disc placed on the surface of this layer are reflected on the bedrock and sent back towards the surface according to a pattern represented on the left side of Figure 16.7.
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719
Figure 16.7. Propagation pattern of the waves emitted by the vibration of a disc on the surface of a layer of ground on top of a rigid bedrock (above); equivalent cone pattern (on the right)
This pattern can be associated with an equivalent pattern (right side of the figure) where the wave propagation follows the generators of a cone of revolution whose sections through equidistant planes (of distance h equal to the thickness of the ground layer) represent alternatively elements of bedrock or of free surface [WOL 80]. In this conical propagation where the wavefronts are spherical caps centered at the top of the cone, the amplitude u of the movement verifies the law of spherical attenuation (see section 3.2.3): u = u0
ro r
[16.31]
r being the distance covered along the generator of the cone and uo the value of u for r = ro; distance r is linked to horizontal distance R through relation r = R / sinT. [16.31] can then be rewritten in the form: u = u0
r0 sin T R
[16.32]
The energy E transported per unit of time outside the cylinder of radius R and of height h is proportional to the product 2SRh u² which according to [16.32] becomes: 2SRh u² = 2Shu 02 r02
sin 2 T R
[16.33]
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Seismic Engineering
This relation shows that the energy E transported towards infinity is zero because 2SRh u² tends to zero when R increases indefinitely. Trapping the waves inside the ground layer apparently cancels the radiative damping irrespective of the angle of incidence T considered for the waves emitted by the disc. The above observations have not brought into play the frequency of excitation. While studying the propagation of waves, when the phases of movements and the effects of addition and subtraction that they produce at the same point for two waves that have followed different paths are considered, it is observed that the result mentioned earlier (absence of energy transfer towards infinity, and thus radiative damping) is correct only if the frequency of excitation is lower than the fundamental frequency gc of the ground layer given by formula [4.38]:
gc =
c 4h
[16.34]
c being the velocity of the waves to be considered according to their type (P or S) and h the thickness of the layer. For excitation frequencies greater than gc, the transfer of energy towards infinity and thus the presence of a radiative damping becomes possible. The detailed calculation justifying this possibility is too complicated to be given here; however a general idea can be obtained by referring to [WOL 85] and [WOL 97]; the sharp change in behavior when the excitation frequency crosses limit gc is due to the fact that in the absence of internal damping the resonance phenomenon of the mechanical system does not require an energy input from outside. [WOL 97] gives the results of a numerical calculation for a disc subjected to a sinusoidal movement of pulse Z in the two cases Z = 0.95 Zc and Z = 1.05 Zc, Zc being the pulse gc/2S associated with the fundamental frequency of the layer; in the first case (Z = 0.95 Zc), the force exerted on the disc is practically in phase with the displacement of the disc, which indicates a negligible dissipation of energy; on the other hand, in the second case (Z = 1.05 Zc), there is almost exactly a 90q phase shift between the force and the displacement; the force is therefore almost proportional to the displacement derivative and reproduces the behavior of a dashpot which dissipates nearly the total injected energy in the form of radiative damping. The radiative damping represented by a reduced damping coefficient [r (formula [16.28] for vertical excitation) is added to internal damping [i of the ground resulting from its visco-elastic behavior which was presented in section 5.3.1 (see Figure 5.6 and equation [5.42] which show that this internal damping depends on the deformation level sustained by the ground and that it is typically around 5%).
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721
In the calculations for soil/structure interaction, the total damping [ to be considered is therefore:
[ = [r + [i
[16.35]
In the cases where [r is very high (vertical pumping movements or horizontal sieving movement) the contribution of [i becomes negligible; incidentally, it is common practice to limit the value of [ to an upper limit of about 30%; these limiting rules, which were introduced for the sake of conservatism in the design practices of nuclear power plants, create coherence problems in the methods of calculation through discretization (finite elements) and the impedance functions method, as shall be seen in section 16.2.1. A common practice consists of calculating the radiative damping by assuming a homogenous half space as the ground as simple analytical solutions are then available (section 16.2.2) which, though established for circular or rectangular basemats can be extrapolated to foundations of any form by retaining an acceptable level of approximation. It is then unusual to consider that the real radiative damping (under real ground conditions) represents only a fraction of this theoretical damping; the following rule is often used:
[=
1 [ r [i 2
[16.36]
[r being the radiative damping calculated on homogenous ground. The coefficient ½ of formula [16.36] relates to the fact that the radiative damping for grounds whose modulus increases regularly with the depth is significantly weaker than in the case of homogenous ground because this increase in the modulus produces a concave curvature towards the top of seismic rays which are sent back towards the surface (Figure 3.12), somewhat like the case of a layer on top of a rigid bedrock (see Figure 16.7). The imposed coefficient of ½ seems acceptable in most ground conditions but is not suitable for the cases of complex stratigraphies with well marked contrasts where the radiative damping can be zero for certain frequencies as in the case analyzed before and where the frequency dependency of the impedance functions can present vast irregularities. 16.2. Practical consideration of the soil/structure interaction 16.2.1. General case
In the most general case (foundations of any form, superficial or deep), the solution to the problem of soil/structure interaction can be carried out based on a
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discretized model of the type described in section 16.1.2 (see Figure 16.2) which corresponds to equation [16.13]. Such models are necessarily large because they contain a large volume of ground whose representation should be sufficiently sharp so as to ensure a correct simulation of the wave propagation in the range of significant frequencies; this condition fixes an upper limit to the size 'A of the ground cells used in the discretization (finite elements or finite differences), whose ratio to the wave length c/g (c = velocity of propagation, f = frequency) should be sufficiently small to ensure the approximation of a sinusoid by a broken line; for example, the value of this ratio should not exceed 1/5; this leads to:
'A d
1c 5 f
[16.37]
If waves of frequency 20 Hz have to be represented in a relatively soft ground (c = 300 m/s for shear waves), elements whose size does not exceed 3 m should be taken; this condition leads to a large number (several thousands) of ground elements since the disturbed ground volume (see Figure 16.2) has dimensions (about double the dimensions of the foundation as indicated in section 16.1.2) which often reach 100 m. Dimension ' A of the ground elements also intervenes in the stability condition of the numerical integration pattern of equation [16.13], when this is of the explicit type, i.e., the calculation of the solution at instant t + 't is done directly from the solution at time t (and possibly solutions at earlier instants, t –'t, t –2't, etc.); time step 't should then satisfy the relation:
't d
'A c
[16.38]
which gives 't d 0.01 s with the previous values of 'A and c ('A[= 3 m and c = 300 m/s); a time step equal to 1/100th of a second is commonly used to describe natural or artificial accelerograms representing the seismic excitation but it can happen that condition [16.38] imposes time steps 't clearly smaller than this limit; this makes the calculation significantly exhaustive. These observations on the time step and its consequences on the total time of calculation implies that it is a time dependent method (integration over time) which is generally used for solving equation [16.13] and not the spectral modal analysis method; the reasons for this choice are as follows: – the large size of the models due to a large number of ground elements would make modal analysis very exhaustive to implement (a large number of modes to calculate);
Notions on Soil/Structure Interaction
723
– the presence of dampers concentrated on the lateral boundaries (step 3 of the construction of the model in Figure 16.2) leads to a damping matrix [C] for which the decoupling conditions of the modal responses accepted implicitly in the spectral methods are far from being fulfilled; – the temporal method enables us to take into account at least partially, the nonlinear effects which are produced in certain types of ground; generally, the iterative linear approximation which was presented in section 5.3 is used; carrying out the iterative approach is easier with the time analysis rather than with the spectral analysis (better estimation of the average level of stress in an element to adjust its properties in view of the following iteration); – we generally want to establish that the lateral boundaries have been placed sufficiently far from the structure; such a justification is easy to make based on the results of a temporal calculation, by comparing the movement of a point on the free surface near the boundaries to a given movement in open space (these two movements should be in close agreement); this verification is only partially possible with a spectral analysis, which only allows cross checking of the maximum values. The calculation of soil/structure interaction in the general case culminates in a temporal solution of a very large system of equations (several thousands of degrees of freedom); thus the heterogenity of the ground as well as its nonlinearities can be taken into account. The disadvantages of this approach are: its cumbersome nature which frequently forces the use of plane models and limits the possibilities of parametric studies and the “black box” nature of the corresponding softwares which can be used only by specialists. The two dimensional approximation (plane models) of the three-dimensional problems of soil/structure interaction is generally used but it presents risks of overestimation of radiative damping [WOL 99]. In a homogenous medium the waves of a plane model have cylindrical wavefronts, i.e., a geometric attenuation to 1/ r instead of 1/r for spherical wavefronts; as a result of this if the reasoning outlined in section 16.1.3 is taken for the ground layer on top of rigid bedrock, there is no suppression of the radiative damping in this configuration since the second member of equation [16.33] no longer contains the factor 1/R. This observation confirms the tendency to exaggerate the radiative effects which characterize the plane models. In general, the verification of the effective damping is one of the delicate points in using large finite element models to calculate the soil/structure interaction; other than the defects in representing the radiative damping while simplifying the models (plane approximation of a three-dimensional problem) the potentially damaging effects from the point of view of the quality of damping modeling should not be ignored:
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Seismic Engineering
– absorbing boundary conditions imposed by the vertical limits of the disturbed volume; these dashpot type conditions though suited to a particular type of wave may prove inefficient for other types; for example, if the characteristics of dashpot are adjusted to avoid shear wave reflections which propagate perpendicularly at the boundary, its absorbing effects will be lower for shear waves that propagate obliquely or for traction-compression waves or more so, for surface waves (mainly Rayleigh waves) whose particle movement is very different from the volume wave movements. These defects of the absorbing boundaries which can compromise the representation of the radiative damping (in this case with a tendency to underestimate due to the parasitic reflection of waves insufficiently absorbed at the boundary; this is contrary to the previously mentioned influence of model simplification) do not have any significant consequences if the lateral boundaries have been placed sufficiently far from the foundations of the structure because the free field motion that can be found in the undisturbed volume does not contain any waves that can be defectively absorbed at the boundary; – imperfections of the numerical integration patterns of equation [16.13] which in general contain a certain rate of numerical damping whose sensitivity goes up as the frequencies concerned increase; this has already been indicated at the end of section 15.1.3. This numerical damping contributes to the effective damping in the model in a proportion which is often difficult to estimate and is therefore another reason for its overestimation. These notes are meant to draw attention to the care and the degree of expertise which should direct the modelization and the calculation of problems related to soil/structure interaction using big discrete models; the first examples of such calculations which date back to the 1970s were often carried out without sufficient awareness about the underlying difficulties because of the then existing cut-throat competition between software providers. All regulation attempts published at that time, mainly by the American nuclear commission, reflect the controversies that developed and they are only of historical interest today. Apart from finite element discretization methods the general problems of soil/structure interaction have also been addressed by boundary elements methods which correspond to dicretizing the integral equations into which the elastodynamic problem can be formulated. These methods offer the advantage of a rigorous treatment of radiation conditions towards infinity but are, in practice, limited to cases where the ground can be considered homogenous in large volumes. 16.2.2. Shallow foundations
It was observed in section 16.1.2 that in the normal case of free field motion corresponding to vertically propagating waves, the structures whose foundations can
Notions on Soil/Structure Interaction
725
be considered as just placed on the ground surface are calculated simply by subjecting them to a command acceleration equal to that obtained on the surface for the free field motion. All the procedures described in Figure 16.2, mainly the assessment of the extension of the disturbed volume and the deconvolution of the movement based on the surface, become useless. It is sufficient to calculate the inertial interaction resulting from the movement of the masses of the structure under the action of the free field acceleration; in the hypothesis of a linear behavior this calculation may be carried out, frequency by frequency, using a Fourier series decomposition of the response in relative displacement. This possibility is much simpler than the general case presented in section 16.2.1 and emphasizes the interest of the impedance functions which, for a given frequency, enable the link of the basemat with the ground through a set of springs and dampers adapted to different types of movement (vertical translation or pumping, horizontal translation or sieving, rotation of the horizontal axis or rocking, rotation of the vertical axis or torsion) to be represented. The calculation of the response should be done, in principle, in the frequency space and should end by returning to the time space through a reciprocal Fourier transformation. In practice, the soil spring method, which is another name given to the use of impedance functions, is often implemented with constant values (independent of the frequency) of the stiffness of the springs and of the damping of the dashpots. It can be shown [PEC 86] that an excellent approximation can thus be obtained for systems with one degree of freedom (rigid foundations subjected to harmonic excitations by the operation of rotary machines). For more complex structures which should be schematized by models with several degrees of freedom, it is common to adjust the stiffness and the dampers of the basemat-ground link by taking those provided by the impedance functions for the fundamental frequency of the system; since this frequency is not known a priori and since it depends not only on the characteristics of the structure but also on those of the link with the ground, it is necessary to proceed by iterative methods. This method offers satisfactory results, except in cases where the irregular stratigraphy leads to sharp variations of the impedance functions for small variations of the excitation frequency. The impedance functions have been calculated analytically or numerically in a certain number of cases of shallow foundations with simple configurations (rigid circular or rectangular basemats on a homogenous half space or on a layer on top of a homogenous half space). For a circular basemat of radius r on a homogenous half space, Table 16.2 gives the equations of the stiffness and the dampers which can be used with a good degree of approximation for values of a0 = Zr/vs [16.12] included between 0 and 4; the worst approximation corresponds to the case of rocking.
726
Seismic Engineering Vibration mode
Stiffness of the spring
Coefficient of the dashpot
Vertical
Kz =
4 Gr 1 Q
Cz = 0.85 Kz
r vs
Horizontal
Kx =
8 Gr 2 Q
Cx = 0.576 Kx
r vs
Rocking
Torsion
KT =
8 Gr 3 3 1 Q
K< =
16 3 Gr 3
CT =
C< =
0.30 1 BT
KT
0.30 B\ 1 B\
r vs
K\
r vs
Table 16.2. Springs and dashpots for a disc of radius r on a homogenous half space of shear
modulus G and of Poisson’s coefficient Q; v s velocity of the shear waves = G / U with U = mass density; IT, I< moments of inertia of the structure for rocking and torsion; BT = 3 (1-Q) IT / (8Ur5), B< = 2 I< / (Ur5)
This table gives the expressions for the static stiffness of the various movements: pumping (vertical), sieving (horizontal) and rocking which were used earlier (see [16.26], [16.5] and [16.6]). For the coefficients of dashpots, only the expressions for the pumping and sieving movements have been taken directly from the values of the impedance functions in near static conditions (low frequencies of excitation); the equation of Cz, for example, is strictly equivalent to constant Co defined by [16.27]. Regarding the rotational movements (rocking and torsion), the coefficients of dashpots were modified based on those derived from the impedance functions by introducing coefficients (BT and B<) which are proportional to the moments of inertia (IT and I<) of the masses of the structure taken respectively with respect to horizontal axis of rocking and the vertical axis of torsion. This difference in treatment between the movements of translation (pumping and sieving) and the movements of rotation (rocking and torsion) is due to the importance of the radiative damping; it was indicated in section 16.1.3 that this damping was much lower for the rotational movements; therefore to obtain an acceptable approximation of the real response by taking the frequency-independent values for the characteristics of the springs and the dampers, the equation adopted for the coefficients of dashpots should take into account the inertia of the structure because it is the damping that controls the response in the vicinity of the resonance frequency. For highly damped systems (case of translation movements), the resonance effects being very much attenuated, it should be sufficient to take values
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727
which do not depend on the structural characteristics for the equation of the dashpots coefficients. It has to be indicated that the exact calculation of the impedance functions brings out a coupling term between the sieving and the rocking movements in the sense that applying a horizontal force at the centre of the disc of radius r induces not only a displacement parallel to this force but also a rotation; this effect is due to Poisson’s coefficient (the compressed ground in front of the force goes up while the stretched ground at the back descends), but its amplitude is weak and is often neglected. A different expression for the stiffness Kx for the horizontal moment is found in the literature; it is written as: Kx =
32 1 Q 7 8Q
Gr
[16.39]
This formula is in fact very much like the one given in Table 16.2 (the deviation, which is zero for Q = 0.5, does not exceed 5% for Q = 0.25). As a simple example of the soil spring method, the case of the building outlined in Figure 16.1 can be taken; this shows the sieving and rocking springs (Kx and KT) and the three terms of displacement u at the height h: ux (translation of the basemat), uT (solid body displacement due to the rotation of the basemat) and ud (displacement resulting from the deformation of the building). These notations are completed by defining: – coefficients Cx and CT of the dashpots of the sieving and the rocking movement; – constants k and c characterizing the stiffness and the dashpot of the building with respect to its own deformation; – masses m and M associated respectively with the building (which are supposed to be concentrated at height h) and with the basemat; – moment of inertia IT of the total mass (building and basemat) with respect to the horizontal axis passing through the center of the basemat (same notation as in Table 16.2); – accelerogram s (t) characterizing the force-field ground motion. The equations of the movement are obtained by the basic law of dynamics; displacements u and ux of the masses m (building) and M (basemat) as well as rotation T are chosen as the degrees of freedom. For mass m, the restoring force due to the link with the basemat is:
728
Seismic Engineering
F = k u d + c u d
[16.40]
Taking into account [16.2] and uT = hT, the following equation is obtained: m ( u + s ) = -k (u – ux – hT) – c ( u u x hT )
[16.41]
For the mass M of the basemat, the forces acting are those of the link with the building (see [16.40]) and the link with the ground, which gives the equation:
M ( ux + s ) = k (u – ux – hT) + c u u x hT – Kx ux – Cx u x
[16.42]
For the equilibrium of the moments, the contribution of the forces of inertia consists of the term IT T (solid body movement in rotation) and an additional term corresponding to the part of the force of inertia of mass m which is not due to the rotational movement; these inertial moments are balanced by the reaction moment developed by the link with the ground; we therefore arrive at: IT T + mh [ u + s – h T ] = –KTT – CT T
[16.43]
i.e., considering [16.41]: (IT-mh²) T – h [k (u – ux – hT) + c( u u x – h T )] = –KTT – CT T
[16.44]
Equations [16.41], [16.42] and [16.44] can be written in the routine matrix form (see [15.67] or [16.13]): [M] { u }+ [C] { u } + [K] {u} = – s (t) [M] {'}
[16.45]
with:
>M @ ^u`
§m ¨0 ¨0 ©
· 0 0 ¸; K M 0 0 I mh 2 ¸ ¹ T
§u · ¨ u ¸ ; > '@ ¨¨ x ¸¸ ©T ¹
> @
§ ¨ k ¨ k ¨ kh ©
k k Kx kh
· ¸ ¸ ; >C @ 2 kh K ¸ T¹ kh kh
§ ¨ c ¨ c ¨ ch ©
c c Cx ch
· ¸ ¸ 2 ch C ¸ T¹ ch ch
[16.46]
§1· ¨1¸ ¨0¸ © ¹
The moment of inertia IT – mh², which appears as the last diagonal element of the matrix of mass [M], is nothing but the moment of inertia I’T of the basemat taken
Notions on Soil/Structure Interaction
729
with respect to its rocking axis. The eigenangular frequencies of the undamped modes of the system are determined by the equation:
det > K @ Z
2
>M @
k k Z 2m kh k k Kx Z2M kh kh kh kh 2 K Z 2 I '
T
0 [16.47]
T
i.e., by developing the determinant: mMI’TZ6 – [mM (kh² + KT) + mI’T (k + Kx) + MI’Tk] Z4 + [m (kKT + kh²Kx + KTKx) + MkKT + I’TkKx] Z² – kKxKT = 0
[16.48]
If the masses of the basemat are ignored (M = 0 and I’T = 0) this equation is reduced to: m (kKT + kh²Kx + KTKx) Z² – kKxKT = 0
[16.49]
which is identical to [16.4]. Expressions similar to those in Table 16.2 were given for the rectangular basemats on homogenous ground [SIE 91]. In practice, the formulae used for circular basemats can also be used for rectangular basemats by giving r a value which produces the same surface (for the pumping and the sieving modes) or the same moment of inertia (for the modes of rocking and of torsion). This approximation is acceptable for basemats of any form provided it is simply connected and not too stretched; for an basemat of surface S and of moments of inertia IT (rocking) and I< (torsion), it amounts to defining the following values of the equivalent radii: 1
§ 4I · §S· 2 r z = rx = ¨ ¸ ; r T = ¨ T ¸ S © ¹ © S ¹
1
4
; r\
§ 2I\ · ¨ ¸ © S ¹
1
4
[16.50]
and using them in place of r in the formulae of Table 16.2. The validity limit of this rule can be linked to the ratio L² / S between the length L of the perimeter and the surface S; the value of this ratio is 4 S = 12.57 for a circle, 16 for a square and 18 for a rectangle twice as long as it is wide; the limiting value of this ratio can be fixed as 20; this corresponds to a rectangle with sides in the ratio (3 + 5 ) /2 = 2.62.
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Seismic Engineering
For the shallow foundations consisting of several footings, the stiffness of the soil springs can be calculated by adding the contributions of each one of the footings for the degrees of freedom of translation (pumping and sieving) and by considering the total rotation of the foundation for rocking and torsion (as a result the restoring moment is influenced by the vertical stiffnesses in the case of rocking and by the horizontal stiffnesses in torsion); thus, the following formulae are obtained: Kz = 6 K z ,i ;Kx = 6 K x ,i i
[16.51]
i
KT = 6 KT ,i xi2 K z ,i ;K< = 6 K\ ,i ri 2 K x ,i i
i
[16.52]
where xi and ri indicate respectively the distance from the center of the index i footing i to the axes of rocking and torsion; similar formulae can be written for the coefficient of dashpots. Expressions [16.51] and [16.52] assume that the footings are interlinked in a perfectly rigid manner; this assumption can be a bit far from reality when their links are made of a simple system of interlocking joists (which are imposed by earthquake engineering codes as compulsary structural details) or of beams acting as supports to the superstructures of the building. Incidentally, the hypothesis of infinite rigidity of foundations which is generally adopted for calculating the impedance functions (for example, in Table 16.2) is roughly verified only for buildings with strong bracing using walls. In effect, even thick basemats are relatively flexible once their surface is large; in fact, their rigidity comes from the walls that they carry just as with the stiffeners of metallic plates. Several studies have dealt with the problem of impedance functions for flexible foundations [IGU 81, WOL 85] and have demonstrated significant differences with the results obtained with rigid foundations, mainly for the high frequency excitations. However, in practice the soil/structure interaction is calculated using the hypothesis of infinite rigidity approach because, in most of the cases, including the flexibility factor in the calculation does not modify significantly the results in the range of relatively low frequencies which correspond to the fundamental frequencies of the structures, and in addition it complicates the analysis. Using the soil spring method in its most common form (foundations supposed to be rigid and considered as circular discs of equivalent radii, simplification of the real ground conditions to bring back the case of a homogenous half space or a single layer on bedrock, use of frequency-independent values for the stiffness of springs and the coefficients of dashpots) can be imagined in practice only as part of the parametric study. It is necessary to assess the influence of the simplification of the
Notions on Soil/Structure Interaction
731
hypothesis and the uncertainties over the values of certain parameters (particularly the ground modules). Designing “equivalent” homogenous ground with authentic site conditions requires experience and may need auxiliary equations (for example to adjust the values of settlement between the real ground and the equivalent half space); it can thus occur that different values of the shear modules G for different modes of vibration (pumping, sieving, rocking and torsion) of the foundation are adopted. The parametric studies are often limited to the effect of the ground moduli (that can be made to vary from ±50% with respect to its most plausible value). As was indicated previously (see for example sections 9.2.1 and 16.1.1) the effects of the soil/structure interaction are normally ignored by earthquake engineering codes applicable to regular constructions. The reason generally given to justify such a practice is that the extension of the fundamental frequencies ([16.10] or Table 16.1) resulting from the interaction compared to the case of the embedded base, produces a diminution of the acceleration transmitted to the building, considering the nature of the spectra used for designing. This argument is valid but it is necessary to be aware that when the acceleration diminishes, the displacement increases, which can, in certain cases, be important for the assessment of safety; for example if there is a risk of shock against adjacent structures or if the effects of the second order (P –') are critical for a very flexible building. Another consequence, less intuitive than that of the displacement increase as well as of the omission of the soil/structure interaction effects, is the overestimation of the structural damping. In fact it has been shown by Veletsos [VEL 97] that the reduced damping [ of the fundamental mode, resulting from the damping (radiative and internal) of the ground and that of the structure, is given by the expression: §T · [ = [g + [s ¨ ¸ © Tf ¹
3
[16.53]
where [g represents the ground contribution, [s the structural damping (which is defined in earthquake engineering codes by tables such as Table 15.2) and T/Tf the ratio of the extensions of the periods due to the soil/structure interaction [16.4]; this formula [16.53] shows that for high values of this ratio (last lines in Table 16.1) the structural damping contribution becomes negligible (the value of the second term of [16.53] is only 0.6% for [s = 5% and T/Tf = 2). If this diminution of the influence of the structural damping is not compensated by the damping due to the ground (for example, in the case of a layer on bedrock studied in section 16.1.3, for the frequencies lower than the fundamental frequencies
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Seismic Engineering
of the layer), it is seen that ignoring the soil/structure interaction can lead to overestimation of the effect of damping and hence under estimation of the response. This potentially dangerous consequence of the routine practice of seismic design can appear when the values of [s are very high; this corresponds to cases where special dampers are introduced in the structure to attenuate its seismic response. For the “normal” values of [s (about 5%), the high diminution of its influence is compensated by the internal damping of the ground (which is also about 5%; see notes before [16.35]) which always exists even in cases where the radiative damping is zero. Formula [16.53] can be simply established by applying the rule defined in section 15.2.1 for the damping of composite structures (see [15.117]); the deformation energies of the springs represented in Figure 16.1 are, as per [16.1] and [16.2]: Ex =
1 K x u x2 2
1 F2 2 Kx
[16.54]
ET =
1 KT T 2 2
1 F 2 h2 2 KT
[16.55]
Ed =
1 2 kud 2
1 F2 2 k
[16.56]
Therefore, applying [15.117], by indicating the reduced damping associated with sieving and rocking movements and with the building’s own deformation by [x, [T and [d, we have:
[
§ [ x [T h 2 [ d · § 1 h2 1 · ¸/¨ ¸ ¨ KT k ¹ © K x KT k ¹ © Kx
[16.57]
from which, by multiplying by k and by using [16.4]: § ·§ T · k kh [ = ¨[x [T [d ¸ ¨ ¸ ¨ Kx ¸ © Tf ¹ KT © ¹ 2
2
[16.58]
In the routine hypothesis of a reduced structural damping proportional to the pulse (second term of equation [15.51] which corresponds to a damping matrix proportional to the rigidity matrix), we can write:
Notions on Soil/Structure Interaction
[d = [s
Tf T
733
[16.59]
[s being the structural damping defined by the codes and would be applicable for an embedded base calculation (T = Tf); by transposing [16.59] in [16.58], we have: 2
§ §T · k kh 2 ·§ T · [T [ = ¨[x ¸¨ ¸ [ s ¨ ¸ KT ¹ © Tf ¹ © Tf ¹ © Kx
3
[16.60]
which is identical to [16.53] with the explanation of the contribution [g of the ground to the global damping. In short, it is generally safe to ignore the soil/structure interaction except in cases where safety is controlled by the displacements and in cases where the radiative damping is zero and special damping devices are used to reduce the seismic load. When the soil/structure interaction is taken into account the soil spring method seems to be relatively simple to use and provides satisfactory results provided a parametric study of the influence of the ground modulus is carried out and its validity limits are considered (shallow foundations, soil profile relatively regular). In more complicated cases (very irregular stratigraphy, deep foundations), it is necessary to take the help of specialists in this particular field of seismic engineering. The difficulties arising as a result of properly considering the effects of interaction between the ground and the foundations can be illustrated by adopting a simple method in order to show the variations of impedance functions when the foundations of two structures are joined. This effect of structure-soil-structure interaction is rarely taken into account in seismic studies, even though its consequences, as indicated in the following analysis, cannot be ignored in a certain number of cases. Figure 16.8 shows two basemats of surfaces respectively S1 and S2 placed side by side and subjected to horizontal static forces F1 and F2 which produce horizontal displacements U1 and U2
734
Seismic Engineering
Figure 16.8. Interaction between two adjacent foundations under the action of static horizontal forces
If the interaction between the two basemats is ignored, the following relations are obtained: F1 =
8
2 Q S
G S1 .U1 ; F2 =
8
2 Q S
G S2 .U 2
[16.61]
which result from the stiffness expression Kx in Table 16.2 by taking the values of rx (see [16.50]) for r, corresponding to surfaces S1 and S2 of the two basemats. These relations, valid for isolated configurations, should be replaced by the following equations in the case where the interaction is taken into account: F1
F2 =
8
G K11U1 K12U 2
[16.62]
8
G K12U1 K 22U 2
[16.63]
2 Q ) S 2 Q S
where the identity of the non-diagonal coefficients (K12) come from the reciprocity theorem of the theory of elasticity. Coefficients K11, K22 and K12 can be determined by considering the following three cases: (1) F2 = 0, F1 z 0
Notions on Soil/Structure Interaction
735
According to [16.63] and [16.62], we have: U2 =
§ K2 · K12 8 G ¨ K11 12 ¸ U1 U 1 ; F1 = K 22 ¹ K 22 2 Q S ©
[16.64]
It is accepted that the relationship between F1 and U1 is identical to the relationship valid for isolated configuration (first of equations [16.61]), i.e., the second foundation, under no load condition (F2 = 0), does not have any influence on the first; this hypothesis (only an approximation because in reality the rigidity of the second foundation hampers the surface deformations of the ground) can be referred to by the equation: K11 –
K122 K 22
[16.65]
S1
(2) F1 = 0, F2 z 0 By using the same hypothesis of non-influence of the unloaded foundation, the following relation, similar to [16.65], can be derived: K22 –
K122 K11
[16.66]
S2
(3) U1 = U2 = U The set of two basemats behaves as a single basemat of surface S1 + S2; the total force F = F1 + F2 can then be represented by: F = F1 + F2 =
8
2 Q S
G S1 S 2 U
[16.67]
which, by calculating F1 and F2 using [16.62] and [16.63] and by assuming U1 = U2 = U, leads to the relation: K11 + K22 – 2K12 =
S1 S2
[16.68]
Equations [16.65], [16.66] and [16.68] enable the calculation of coefficients K11, K22 and K12 based on surfaces S1 and S2; we obtain: K11 = D S1 ; K22 = D S 2 ; K12 = E S1 S 2
1
4
[16.69]
736
Seismic Engineering
Taking coefficients D and E as:
D
1
2U U /
D D 1
1 U ² 2U ; E
[16.70]
where: U=
S2 S1
[16.71]
It can be verified that D an E are invariants with respect to the transformation of U in 1/U, which is logical because this transformation only permutes the numbering of the two basemats. Table 16.3 gives the numerical values of D and E for U varying from 0 to 1. This table gives coefficients D and E which vary very little once ratio U² of the surfaces exceeds 0.1. When the ratio of the surfaces is small compared to one, the importance of the interaction between the basemats is very different depending on whether we consider the case of the big or small basemat; this importance can be estimated through the ratios K12/K11 and K12/K22 (last columns of the table); for example, for S2/S1 = 0.25, the value of K12/K22 is exactly double the value of K12/K11. In other words, the large foundation influences the movement of the small foundation strongly but the reverse is not true. S2 S1
D
E
K 12 K 11
K 12 K 22
0.0
0.00
1.0000
0.0000
0.0000
1.0000
0.1
0.01
1.0654
0.2640
0.0784
0.7836
0.2
0.04
1.1091
0.3478
0.1402
0.7012
0.3
0.09
1.1407
0.4007
0.1924
0.6413
0.4
0.16
1.1638
0.4366
0.2373
0.5932
0.5
0.25
1.1803
0.4614
0.2764
0.5528
0.6
0.36
1.1918
0.4781
0.3107
05179
0.7
0.49
1.1994
0.4890
0.3411
0.4873
0.8
0.64
1.2040
0.4956
0.3682
0.4602
0.9
0.81
1.2064
0.4990
0.3924
0.4360
1.0
1.00
1.2071
0.5000
0.4142
0.4142
ȡ=
S2 S1
Table 16.3. Values of coefficients D and E based on U =
(see equations [16.70] and [16.71])
S 2 / S1
Notions on Soil/Structure Interaction
737
To assess these influences quantitatively, the free vibrations of the coupled system formed by two buildings with foundations on basemats of surfaces S1 and S2 are studied; the equations of these vibrations are obtained from [16.62] and [16.63] by replacing F1 and F2 with the forces of inertia and by taking into account relations [16.69] for coefficients K11, K12 and K22; they can be written as: m1 U1
D k1U1 E k1k 2 U 2
[16.72]
m2 U2
E k1k2 U1 D k2U 2
[16.73]
m1 and m2 being the masses of the buildings, k1 and k2 the stiffness of the sieve springs corresponding to the isolated configurations, i.e., stiffness Kx of Table 16.2 with r1 =
S1 / S and r2 =
S 2 / S where pulses Z1 and Z2 of the free oscillations are
such that: Z1 = k1 / m1 ; Z2
[16.74]
k2 / m2
The basic modes of the coupled system have pulses Z which should verify the following relations, deduced from [16.72] and [16.73]: m1 DZ12 Z 2 U1
E m1m2 Z1Z2U2
[16.75]
m2 DZ22 Z 2 U 2
E m1m2 Z1Z2U1
[16.76]
from which the equation that determines Z taking E 2 Z4 – D Z12 Z22 Z 2 DZ12Z22
D 2 D (see [16.70]):
0
[16.77]
whose roots are: 2 1ª º Z2 = «D Z12 Z22 r D 2 Z12 Z22 4DZ12Z22 » 2¬ ¼
[16.78]
Taking the case where the masses are proportional to the support surfaces (same pressure on the ground) and the sieving stiffnesses to the root of these surfaces (result of the formulae of Table 16.2) equation [16.78] can then be rewritten by introducing the parameter U defined by [16.71] as:
738
Seismic Engineering
§Z · ¨ ¸ © Z1 ¹
2
2 ª § 1« § 1· 1· Dº D ¨1 ¸ r D 2 ¨1 ¸ 4 » 2« © U¹ U» © U¹ ¬ ¼
[16.79]
Components U1 and U2 of the eigenvectors deduced from [16.75] can be taken as equal to: 2 ª §Z · º «D ¨ ¸ » U1 = 1; U2 = E U «¬ © Z1 ¹ »¼
1
[16.80]
Table 16.4 gives, depending on U, the values of the dimensionless eigenfrequencies and the components of the eigenvectors. The hypotheses adopted for the masses and the stiffnesses in isolated configuration are such that the smaller building has the highest fundamental frequency (second column of the table which gives the ratio Z2/Z1). The third and fourth columns show that, in the coupled system, the eigenfrequencies move away from one another more so when the ratio of the surfaces is high. Components U1 and U2 of the first mode are practically equal while for the second mode they are of opposite signs, the absolute value of U2 being higher than that of U1 for the low values of U ( U2 / U1 is approximately equal to 1/U², i.e., reciprocal of the ratio of the masses). The first mode concentrates almost all the masses, as is shown in the last column which gives the reduced modal mass which is expressed as (by applying formula [15.77]):
2
U1 U U 2 1 = 1 U 2 U12 U 2U 22 m1 m2
P1
2
[16.81]
Notions on Soil/Structure Interaction
U
Ȧ2 1 = Ȧ1 ȡ
Ȧ st 1 Ȧ1
Ȧ nd 2 Ȧ2
MODE
739
MODE
U 1
U2 Ist MODE
U2 2nd MODE
μ1 m1 + m 2
0.1
3.1623
0.9966
1.0357
1
0.8647
–115.73
0.9998
0.2
2.2361
0.9883
1.0656
1
0.8510
–29.372
0.9992
0.3
1.8257
0.9762
1.0941
1
0.8554
–12.983
0.9984
0.4
1.5811
0.9611
1.1224
1
0.8695
–7.1905
0.9979
0.5
1.4142
0.9438
1.1511
1
0.8875
–4.5047
0.9979
0.6
1.2910
0.9247
1.1806
1
0.9093
–3.0547
0.9983
0.7
1.1952
0.9044
1.2110
1
0.9324
–2.1889
0.9989
0.8
1.1180
0.8834
1.2421
1
0.9556
–1.6342
0.9995
0.9
1.0541
0.8621
1.2741
1
0.9784
–1.2618
0.9999
1.0
1.0000
0.8409
1.3066
1
1.0000
–1.0000
1,0000
Table 16.4. Eigenangular frequencies and modes for two adjoining buildings coupled
through the ground depending on U =
S 2 / S1
These observations confirm that when two adjoining buildings are of significantly different masses, it is the heaviest that imposes its movements on the lightest. If the two masses are almost equal, the two buildings oscillate as a whole like a single mass at a slightly lower frequency compared to the one obtained in isolated configuration. In both cases, the second mode of the coupled system (for which the lighter building goes through a much greater displacement than the heavier one) is completely negligible because its reduced modal mass is almost zero. This leads to the conclusion that the soil/structure interaction has a marked influence on the response of the adjacent buildings; this influence diminishes rapidly when the distance between foundations increases and becomes negligible for distances of about half the size of the biggest foundation ([WAL 85]). 16.2.3. Cases of deep foundations and linear embedded structures
If taking the soil/structure interaction into account can be handled in a relatively simple manner in most cases of shallow foundations (with the conditions mentioned at the end of section 16.1.2), it is not the same when the foundations are deep (piles,
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Seismic Engineering
wells, underground walls, partially or totally embedded structures). This difference in the level of difficulty is due to the following reasons: – the effects of kinematic interaction (see section 16.1.2) are important and should be of interest for a specific modelization; – the deep foundations can include a large number of elements (cases of piles); this can lead to practical problems of modelization (if all of them have to be represented) or to delicate questions of assessment of the group effect in dynamic situations (if homogenization techniques are used); – in general, choosing the deep foundations option corresponds to mediocre conditions from the point of view of mechanical characteristics of shallow terrains; the case then becomes “pathological” from the point of view of the impedance functions varying with the excitation frequency (see section 16.1.3 for possible consequences on the radiative damping); choosing the ground moduli to be retained for the calculation also turns out to be difficult because the real behavior is highly nonlinear for strong seismic movements. The warning given earlier (at the end of section 16.1.1 and in section 16.2.2) on the need to consult specialists for problems related to soil/structure interaction, which cannot be treated with the simplest variants of the soil springs method, becomes essential mainly in the case of deep foundations. That is why the influence of the depth of sinking of the massive foundations and the interaction of the ground with the embedded linear structures (piles, piping, and tunnels) are going to be dealt with in a basic manner. For the first of these problems (partially embedded massive structures) the influence of sinking is to increase the stiffness of the link with the ground and the radiative damping with respect to the case of foundations that are simply placed on the surface. For relatively low values of relative deepening (quotient of the depth of sinking by the equivalent radius r = S / S of the foundation of surface S) the influence can be obtained by multiplying coefficients K and C of Table 16.2 by the following factors [KAU 78]: It = 1 +
2e for the coefficients of translation (Kx, CX, Kz, Cz) 3r
[16.82]
e for the coefficients of rotation (KT, CT, K\, C\) r
[16.83]
Ir = 1 + 2
Applying these corrections to the stiffness and to the coefficients of dashpot of the soil springs is only a rough approximation, more so because the kinematic interaction is no longer zero for the foundations embedded in the ground; this has already been indicated; the calculations having culminated in these corrections
Notions on Soil/Structure Interaction
741
suppose that the vertical walls of the basemats are connected to the ground, which is probably not very realistic when there is traction at the interface given the compacting difficulties of the earth filling around the foundations. For the second of the problems mentioned earlier (embedded linear structures) simple calculations are commonly carried out by representing the soil/structure interaction by the distributed springs acting in the longitudinal direction (parallel to the axis of the linear structure) and in the transversal direction (perpendicular to this axis). Values K A (longitudinal) and Kt (transversal) of the stiffness of the springs, per unit length along the axis of the structure, are proportional to the shear modulus G of the ground: K A OA G; Kt = Ot G
[16.84]
Different expressions of coefficients O A and Ot, bringing in Poisson’s coefficient Q, have been proposed [AFP 98], leading to varying numerical results (from 0.5 to 2 or 3) for these dimensionless factors. Given this dispersion, the French association for seismic engineering suggests the simple equation: O A = Ot = 1 i.e. K A = Kt = G
[16.85]
for first approximation calculations. To illustrate this type of analysis, the case of a pile crossing a layer of ground of thickness H to penetrate into the bedrock at the base of the layer is studied (see Figure 16.9).
Figure 16.9. Pile of diameter I pegged in at the base of a layer of ground of thickness H which has a seismic movement Us in the horizontal direction; the pile carries a mass m on top
742
Seismic Engineering
By ignoring the forces of inertia in the pile, the equation of its deflected shape U (z) when the ground has a displacement Us(z) is written as: EI
d 2U dz 2
H
M H VH H z ³ K t z ' z ª¬U s z ' U z ' º¼ dz’ z
[16.86]
E being the Young’s modulus of the material of the pile, I the moment of inertia, MH and VH the bending moment and the shear stress applied on top of the pile and Kt the transversal stiffness ([16.84]) of the ground-pile interaction. By differentiating twice with respect to z, equation [16.86] takes the form: d 4U 4E 4U d] 4
4 E 4U s
[16.87]
by taking: ]=
z ; 4E4 H
Kt H 4 EI
Ot
G H4 E I
[16.88]
The boundary conditions associated with [16.87] are: ] = 0, U = 0 and
d 2U d]
2
= 0 (pivoting at the tip)
d 3U dU = 0 and d] d] 3 shear force VH)
]=1
,
[16.89]
H 3VH (embedding in mass m which exerts a EI [16.90]
To solve equation [16.87], it is assumed that Us (]) corresponds to the deflected shape of the fundamental mode of the ground layer, i.e.: Us (]) = G sin
S 2
]
[16.91]
Displacement G of the ground surface (] = 1) being such that the corresponding acceleration Z² G is equal to a given value *s with, as an expression of pulse Z:
Notions on Soil/Structure Interaction
Z
S vs
743
[16.92]
2H
which comes from relation [16.34] with the notation vs for the shear wave velocity in the layer. Therefore G is: G=
*s
4
Z2
S2
*s
H2 vs2
[16.93]
In these conditions, the solution U (]) of equation [16.87] is determined using basic calculations; we have the following equation: 3 4 H VH 64E S U(]) = G sin ] ª cJ sV cos E] sinh E] cJ sV sin E] cosh E] º¼ 4 4 3 § 2 2 ·¬ 64E S 4E EI ¨ J s ¸
©
¹
[16.94] after having taken: c = cos E; s = sin E; J = cosh E; V = sinh E
[16.95]
from which the displacement ' = U (1) on top of the pile: 3
4
'=
64E G 4
64E S
4
H VH 3
2
4 E EI J s
2
JV cs
[16.96]
Assuming that the building placed on the piles is sufficiently rigid it behaves like a mass which follows the movement imposed by the ground layer and mass m of Figure 16.9 represents the quotient of the total mass of the building by the number of piles. Acceleration *H on top of the pile is therefore: *H = Z²'
[16.97]
and the shear force VH exerted on top of the pile is: VH = m *H = m Z² '
[16.98]
from which, referring to [16.96], the following expression of the ratio *H/*S of the accelerations is arrived at:
744
Seismic Engineering
*H *S
ª S m JV cs º / «1 E» 4 4 3 2 2 64 E S «¬ 4Ot U H J s »¼ 64 E
4
2
[16.99]
where the specific mass U of the ground (= G/v 2s ) is introduced. The bending moment MH on top of the pile is calculated from the second differentiation of V (]) (see [16.94]): 4
MH =
16Ot U H * s 4
64 E S
4
HvH
2
2E J s
2
JV cs
[16.100]
from which, by replacing VH with m *H (see [16.98]) and by using formula [16.99]: MH =
3 2 § S 3 JV cs 16m* s H ª U H m JV cs · º 2 / 1 [16.101] O E E 2 « ¨¨ 4 4 2 2 3 2 ¸» t m 64 E S ¬« J s © 4Ot U H J s ¸¹ ¼»
In the simplified methods of pile designs under seismic load, most earthquake engineering codes accept that the stress and moments can be calculated by assuming that the piles follow the deformation of the ground exactly; moment Ms on top is thus given by: Ms =
S
2
EI
4 H
2
G
m* s H
Ot m 4E
UH
3
4
[16.102]
from which, by dividing [16.101] by [16.102]: MH MS
2 ª 2 m § S 3 JV cs m JV cs ·º 1 E 2 2 / ¨1 E . 2 2 ¸» [16.103] 4 4 « 3 3 ¨ 64E S «¬ Ot U H J s © 4Ot U H J s ¸¹»¼
64E
4
formulae [16.99] and [16.103] enable us to highlight the main characteristics of the soil-pile interaction. The presence of a denominator that can a priori be cancelled for certain values of the parameters introduces a resonance risk (amplitude of the response becomes infinite in the absence of damping); the resonance condition is written as:
Notions on Soil/Structure Interaction
UH
3
S ² JV cs E 4Ot J ² s ²
m
745
[16.104]
To assess the significance of this, let us specify the range of the numerical values of the factors that intervene in this equation. For cylindrical piles of diameter I , taking Ot = 1 [16.85] we have: I
S 64
4
I 4 ; 4E 4
64 G § H · ¨ ¸ ;E S E©I ¹
§ G · 2¨ ¸ ©SE ¹
1
4
H
I
[16.105]
by taking E = 3 u 1010 N/m² (concrete), U = 1,800 kg/m3, vs = 100 m/s or 200 m/s, E is found to be:
E
0.235
E
0.333
H
I H
I
for vs = 100 m/s
for vs = 200 m/s
[16.106]
The ratio H/I varying typically from 5 to 20, E seems to vary from 1 to 10; for these values of E the coefficient (JV – cs) / (J² – s²) which intervenes in [16.104] is practically equal to 1 and the resonance condition [16.104] can be written in the following manner:
I H
8 U gH 3 P S
§E· ¨ ¸ ©G¹
1
4
[16.107]
By introducing static pressure p (= 4 mg/SI²) in the piles, with the same values of E, U and vs as before, g = 10 m/s² and p = 5 u 106 N/m²:
I H
I H
0.00593H m for vs = 100 m/s 0.00420 H m for vs = 200 m/s
[16.108]
These conditions can be verified in some plausible cases (for example I = 0.593 m for H = 10 m). The resonance risk forms a part of the verification criteria of a system of piles in the seismic zone, as is the case in the PS92 Rules
746
Seismic Engineering
[AFN 95]. The length H of the piles being fixed by the site characteristics, the resonance risk can be avoided only by acting on the diameter I. Formula [16.103] shows that even if the resonance is far away, the stress and the moments to which the piles are subjected can be significantly increased with respect to those calculated by taking into account only the deformations imposed by the ground. Therefore, while designing piles the part of the stress that results from the forces of inertia acting on the supported structure should be kept in mind. On the other hand, for completely embedded linear structures (pipelines and tunnels) which are not loaded by the inertial reactions coming from the portions in free air, the common and fully justified practice is to calculate them based on just the action due to ground deformations; this means ignoring the effects of the soil-structure interaction under conditions where such an omission is for the sake of safety. 16.2.4. Winkler type models
In section 16.2.3, use of distributed springs to represent the soil-structure interaction in linear structures was studied. The same method, called the Winkler springs method, has been used for a longtime for shallow foundations (footings, basemats) in the static domain and roughly since 1960 in the dynamic domain [PEC 86]. In its simplest version, the method involves schematizing the link with the ground through a bed of independent springs acting in the vertical direction. Thus using simple analytical calculations it is possible to obtain the expressions for pumping and rocking stiffnesses not only in the linear case (purely elastic springs) but also in the nonlinear cases corresponding to uplift (partial loss of link between the basemat and the ground assuming that the Winkler springs can work only under compression) or to the plasticization of the ground (if an elasto-plastic law is adopted for the behavior of the springs). The case of uplift with only elastic springs under compression is represented in Figure 16.10, since these springs do not work under traction.
Figure 16.10. Bed of Winkler springs under a basemat of width 2a on which a constant weight P and an increasing moment M act; we move from the elastic behavior center)to uplift (right)
Notions on Soil/Structure Interaction
747
For a rectangular basemat, basic calculations not shown here enable us to determine the behavior when the springs, assumed to be identical, follow an elastic, perfectly plastic law in compression and cease to transmit the forces as soon as they are under traction. The following notations are introduced: – 2a: length of the foundations perpendicular to the axis of rocking; – 2b: width parallel to the axis of rocking; – k: stiffness of a spring in its elastic phase; – Vu: ultimate stress corresponding to the plastic yield limit of the behavior pattern of the springs; – Vo: static stress under the effect of self-weight; – P: self-weight (= 4abV0); – M: overturning moment; – Go: static sinking under the action of self-weight; – G: sinking of the center of the foundation; – G*: reduced sinking (= G/Go); – T: rotation under the action of the overturning moment; – T : reduced rotation (= aT / Go); – V : safety coefficient under self-weight (=Vu/Vo); – M*: reduced overturning moment (= M/ (Pa); – S*: fraction of the uplifted surface; – W*: reduced energy (=2 ³ MdT / (PGo). There are five distinct areas of behavior (see Figure 16.11): (1) the domain INS (instability) corresponding to: V 1
[16.109]
In this case where there is no practical interest, the ground is too weak to take the self-weight; (2) the domain PWU (plasticization without uplift) corresponding to the two conditions:
748
Seismic Engineering 2
1 V 2; V* – 1 d T* d
V* 4 V * 1
[16.110]
We thus have the following relation: ª 2 V * 1 º M* = (V* – 1) «1 » T * ¼» ¬« 3 8 ª W* = (V* – 1) «V * 1 2T * 3 ¬
G* = V* + T* – 2
V
*
[16.111]
V
*
º 1 T * » ¼
1 T * ; S* = 0
[16.112]
[16.113]
(3) the domain PAU(plasticization and uplift) corresponding to the conditions:
T* t
V *2 4(V * 1)
if V * 2 [16.114]
T* t
V
*2
if V * t 2
4
We thus have the relations: 1 V *3 48 T *2
[16.115]
1 · 1 V *3 § W* = 1 – V* + 2 ¨1 * ¸ T * 24 T * © V ¹
[16.116]
M* = 1 –
G*
1
V*
V* § 2 1 1 V* · ¨ * 1¸ T * ; S * 1 * 2 ©V 4 T* V ¹
[16.117]
(4) the domain EWU (elasticity without uplift) corresponding to the conditions: 0 dT dV – 1 if V 2
[16.118]
Notions on Soil/Structure Interaction
749
0 dT d 1 if V t 2 We thus have the equations: 1 * M* = T ; W * 3
1 *2 * T ; G =1; S * =0 3
[16.119]
(5) the domain EAU (elasticity and uplift) corresponding to the two conditions: V* t 2; 1 d T * d
V *2
[16.120]
4
We thus have the relations: M* = 1 –
2 1 3 T*
W* = 1 + 2T –
[16.121] 8 * T 3
G = – T + 2 T * ; S * 1
[16.122]
1
T*
Figure 16.11. The five domains of behavior of a bed of Winkler springs
[16.123]
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Seismic Engineering
These relations shall be used in section 17.2.2 to estimate the importance of the effects of uplift for relatively slender buildings; the appearance of uplift corresponds to the value 1 of the dimensionless parameter T*. As the safety coefficient under static loads is typically equal to at least 3, in Figure 16.11, the uplift normally precedes plasticization (when the moment and thereafter the rotation are increased, we move successively through the states EWU, EAU and possibly PAN if V* is about 3). Though the Winkler model for shallow foundations provides a convenient method to estimate certain nonlinear effects, it has certain defects: – the fact that the springs work independently of one another does not allow reproductions of stress concentrations on the edges of the basemats, a process which characterizes the solutions obtained by the theory of elasticity; – the uniform distribution of the stress obtained by the Winkler model is in fact closer to a plastic behavior of the ground than the elastic behavior; – as a result of this difference in behavior the ratio of the stiffness of rocking and pumping calculated with the Winkler model cannot be coherent with the ratio obtained using the theory of elasticity; for a circular basemat of diameter r, using this theory, KT and Kz of Table 16.2 become: KT =
K 8 4 Gr 3 ; Kz = Gr from which T 3 1 Q 1 Q Kz
2 2 r 3
[16.124]
whereas with the Winkler springs of stiffness k, distributed uniformly at the rate of n per unit of surface the stiffnesses KT and Kz are: K’T = nk
S 4
r 4 ; K’z = nkSr 2 from which
K 'T K 'z
1 2 r 4
[16.125]
By comparing [16.125] and [16.124] it can be observed that if k is adjusted to obtain K’z = Kz, K’T is only 3/8 = 0.375 of KT; this significant difference affects the value Md of the overturning moment which corresponds to the beginning of uplift; rotation Td is then equal to quotient Go, the sinking under the effect of the selfweight P by the radius r of the basemat; since P = Kz Go we have: Md = KT Td = KT
Gq r
KT P Kz r
[16.126]
Notions on Soil/Structure Interaction
751
2 1 Pr according to [16.124] (theory of elasticity) and Md = Pr according 3 4 to [16.125] (Winkler springs). In other words the Winkler model is uplifted more easily than the model using elastic theory.
i.e. Md =
This tendency observed here for a circular basemat is systematic for foundations of any form, at least in the area of validity of the rules of equivalence with circular basemats (see [16.50]); it has been seen in 16.2.2 that this tendency corresponds to the condition L² 20S, L being the perimeter and S the section; by applying these rules of equivalence it can be easily shown that with the same notations as those used in equations [16.124] and [16.125] for the circular basemat, the following result is obtained: K 'T KT / K 'z K z
3S
1
4
IT
4 2 S
1 1
4
0.706
2
IT S
1
4
1/ 2
[16.127]
for a basemat with any surface S and inertia IT; for a rectangular basemat of sides 2a and 2b (S = 4ab, IT = 4a3b/3) this leads to the equation: K 'T KT / K 'z K Z
§a· 0.379 ¨ ¸ ©b¹
1
4
[16.128]
For a square (a=b), this ratio is not very different from that of the circle (0.375) and goes up to 0.483 for a rectangle corresponding to the limit mentioned above: L² = 20S (i.e. a = 2.62b). Stiffness k of the Winkler springs does not depend only on the characteristics of the ground (G and Q ) but also on the size of the foundations; for a circular basemat of radius r, the comparison of expressions [16.124] and [16.125] leads to the values of: k=
32 G 4 G or k = S 1 Q nr 3S 1 Q nr
[16.129]
depending on whether the adjustment is made on the stiffness of rocking or pumping. The presence of r in the denominator of these expressions show that the significance of k is not purely local; this poses a representativeness problem when a plastic yield limit is introduced in the behavior law of the springs because this limit is determined a priori only by the mechanical characteristics of the ground.
752
Seismic Engineering
The defects linked to the non-coherence of the solutions of the theory of elasticity can be corrected by foregoing the uniformity of the distribution of the springs or by modifying the size of the spring bed; such strategies however do not give a proper interpretation of the results thus obtained for nonlinear behaviors (uplift or plasticization). In practice, it is the basic model (uniform distribution under the entire surface of the basemat) which continues to be in use mainly because it is simpler and it produces safe results in estimating the uplift, as has been seen earlier. To significantly improve Winkler’s model, it is necessary to introduce couplings between the springs so as to distribute the deformations under the action of a concentrated force. The simplest case of a model of coupled springs which can be applied to the case of a continuous footing is represented in Figure 16.12.
Figure 16.12. Single line model of coupled Winkler springs
This figure shows a line of coupled elements each consisting of: – a rigid piston (in white) vertically guided into a non-deformable well (hatched) and supported at the bottom of the well by a spring of stiffness k; – an elastic link of stiffness K with each of the adjacent pistons; this link absorbs the shearing force (like a neoprene support);
Notions on Soil/Structure Interaction
753
By denoting the vertical force applied to the piston head of index i as Fi, its displacement as ui (Fi and ui are taken as positive when acting towards the base), the equation of equilibrium can be written as: Fi – kui + K (ui-1 – ui) – K (ui – ui+1) = 0
[16.130]
which can be rewritten as: ui 1 2ui ui 1
O2
k ui KO2
Fi KO2
[16.131]
where O denotes the spacing of the elements in the line; the first term of [16.131] is nothing but the conventional approximation of the second derivative using the finite differences method; by moving from discontinuous to continuous this equation can be rewritten as: d 2u u dx 2 c 2
F kc 2
[16.132]
where x is the coordinate in the direction of the line and where c is the length defined by: c=O
K k
[16.133]
This constant c that characterizes the couplings between the Winkler springs is independent of the spacing of elements O because k is proportional to O (section of the column of ground modelized by this spring) while K is inversely proportional to O (thickness of the neoprene support). From differential equation [16.132], it can be concluded that: – when the displacement u(x) at the piston heads is imposed by the support of plane and rigid foundations, the second derivative is zero and F becomes: F = ku
[16.134]
i.e. the same relation as with the uncoupled springs because the shear forces are canceled in pairs, – to the left and to the right of the support zone of the foundation; since F is zero, we have:
754
Seismic Engineering
ue
u=
§¨ x x2 ·¸ / c ©
2
u = u e 1
¹
for x > x2
for x < x 1
x x1 / c
[16.135] [16.136]
u1 and u2 being respectively the values of the foundations sinking at their left extremity (x1) and at their right extremity (x2). The length of coupling c (see [16.133]) thus controls the extent of disturbance of the free surface of the ground due to the presence of the foundations. From relations [16.134] to [16.136] the equilibrium conditions of plane and rigid foundations of width 2a (see Figure 16.13) can be studied. By taking again the notations used for the uncoupled Winkler model (self-weight P, overturning moment M, sinking G of the center of the foundations and rotation T).
Figure 16.13. Action of plane and rigid foundations on a line of Winkler elements coupled as in the drawing of Figure 16.12
When the uplift is absent (on top of the figure), the sinking u(x) of the foundation is given by the following relations, deduced from equations [16.134] to [16.136]: u = G – Tx (–a d x d +a)
[16.137]
u = (G – Ta) e-(x-a)/c (a d x d +f)
[16.138]
u = (G + Ta) e(x+a)/c (–f d x d – a)
[16.139]
Notions on Soil/Structure Interaction
755
Below the foundation (–a x +a) force F exerted on the Winkler element situated on the x abscissa is given by equation [16.134] (F = ku = k (G – Tx)); by denoting the number of elements per unit of length as n and the total number of elements as N (N = 2an), the resultant P0 and the moment M0 of the totality of the forces are: a
P0 = ³ k G T x ndx a
2ankG
M0 = ³ k G x T x 2 ndx a
a
NkG
2 3 a nkT 3
[16.140] 1 Nka 2T 3
[16.141]
For the Winkler element situated to the right of the foundations (x = +a), there is equilibrium between the following forces: – F2: force exerted by the foundations (towards the bottom); – k (G-Ta): reaction of the spring at the bottom of the well (towards the top); § du · – KO ¨ ¸ © dx ¹ x § du · – KO ¨ ¸ © dx ¹ x
= KOT: shear force on the left (towards the bottom); a
KO a
G T a c
shear force on the right (towards the top).
The equilibrium equation is then written as: F2 – k (G – Ta) + KOT – KO
G T a c
0
[16.142]
i.e. because KO = kc²/O (see [16.133]) and O = 2a/N: F2 = k (G – Ta) +
Nk ªc G T a c 2T º¼ 2a ¬
[16.143]
N being large, the first term is negligible compared to the second; finally we have: F2 =
Nk 2
c ªc º « a G a a c T » ¬ ¼
[16.144]
For the force FI exerted on the Winkler element situated to the left of the foundations (x = -a), we have similarly:
756
Seismic Engineering
F1 =
Nk 2
c ªc º « a G a a c T » ¬ ¼
[16.145]
It is seen that the elements at the edge are subjected to concentrated forces; this is coherent with the results deduced from the theory of elasticity (stress distribution tending towards infinity at the edges); the self-weight P (total force acting on the foundation) and the overturning moment M are obtained by adding to P0 and M0 calculated earlier, the contributions of these edge reactions; we obtain: c P = Nk G Nk G a
§ c· Nk ¨ 1 ¸ G © a¹
1 M = Nka 2T Nk a c cT 3
§ a2 · Nk ¨ ac c 2 ¸ T 3 © ¹
[16.146] [16.147]
The stiffness of rocking and pumping, KsT and Ksz, resulting from the single line model of coupled springs are thus: § c c2 · 1 3 3 ¨ ¸ a a2 ¹ © § c· Ksz = Nk ¨1 ¸ © a¹
KsT = Nk
a2 3
[16.148] [16.149]
These formulae show that the coupling of the springs (c non-zero) results in a greater increase for KsT than for Ksz (compared to the values corresponding to the decoupling); this enables us to compensate for premature uplift of the decoupled model. The appearance of the uplift corresponds to the cancellation of force F2, i.e. according to [16.144], to a rotation Td given by: Td =
G ac
[16.150]
This condition indicates the equality of the slopes between the foundation and the unloaded ground at the right edge of the foundations. The rotation Td is reduced by the coupling (with respect to the value Td = G/a obtained in case of decoupling) because this coupling increases the pumping stiffness but the uplift is nevertheless delayed due to very strong increase in the rocking stiffness.
Notions on Soil/Structure Interaction
757
Due to the uplift (at the bottom of Figure 16.13) from a point D of abscissa [, the sinking u(x) of the foundations becomes: u = G – Tx (-a d x d [)
[16.151]
u
G T[ e ( x [ ) / c [ d x d f
u
G T a e
x a / c
f d x d a
[16.152] [16.153]
The abscissa [ of the point of uplift is determined by the equality condition of the slopes between the foundations and the free ground, i.e.: T=
G T[
[16.154]
c
As earlier, the result P0 and the moment M0 of the forces acting under the foundation (excluding the extremities) are: P0
Mq
[
³ k G T x ndx a
³
[
a
T ª º nk «G a [ a 2 [ 2 » 2 ¬ ¼
k G x T x 2 ndx
T ªG º nk « a 2 [ 2 a 3 [ 3 » 3 ¬2 ¼
[16.155]
[16.156]
The concentrated force F2 in D is zero; the concentrated force F1 on the right edge is given by the same expression [16.145] as earlier; therefore the result and the total moment are: P
M
§ a2 Nk ª [ 2 ·º 2 «G a c [ T ¨ ac c ¸ » 2a ¬ 2 ¹¼ © 2
Nk ª § a 2 [2 «G ¨ ac 2a ¬ © 2 2
· § a3 [ 3 ·º 2 2 T a c ac ¸ ¨ ¸» 3 ¹¼ ¹ © 3
[16.157]
[16.158]
By eliminating G using relation [16.154] and by introducing as before, the reduced moment M* = M/ (Pa) and the reduced rotation T = T / Td (with expression [16.150] for Td), we arrive at the following equation after a basic calculation:
758
Seismic Engineering
M* =
1 J 3
3 ª § J · *º 2 ¨ «3 ¸ T » «¬ T * © 1 J ¹ »¼
[16.159]
J indicating the ratio c/a. When there is no uplift (T Td or T 1) the relation M* – T* becomes: M* =
1 3J 3J 2 3 1 J
2
T*
[16.160]
Contrary to the decoupled model, the final rotation Tu, with a total uplift and a moment M* equal to 1, has a finite value. It can easily be shown that: 2
§ 1 J · * Tu = T d ¨ ¸ or T u © J ¹
§ 1 J · ¨ ¸ © J ¹
2
[16.161]
Tu therefore remains small for plausible values of parameter J (corresponding to a significant effect of the coupling). The Winkler model with coupled springs is thus much more realistic from this point of view than the decoupled model (where Tu is infinite). For a regular increase of the overturning moment, the tilting is attained in three phases: (1) absence of uplift: 0 dT d; 0 d0 d
1 3J 3J 2 3 1 J
2
;
(2) uplift: 2
§ 1 J · 1 3J 3J 2 d M* d1; 1 dT d ¨ ¸ ; 2 3 1 J © J ¹
(3) rotation around the left edge of the foundation; the state of ground deformation remains constant, identical to the one attained at the end of the previous phase (T (1 +J) / J²). The previously described reduced energy W* and the fraction S* of the detached surface are easily calculated to obtain: – in the absence of uplift:
Notions on Soil/Structure Interaction
W* =
1 3J 3J 2 3 1 J
2
T
*2
759
[16.162]
– when there is uplift 3 ª 8 1 § J · *2 º W* = 1 J «1 T * 2T x ¨ ¸ T » 3 © 1 J ¹ »¼ ¬« 3
[16.163]
§ 1 · S* = 1 J ¨¨1 ¸¸ T* ¹ ©
[16.164]
For J = 0 (decoupled model) relations [16.160], [16.159], [16.162], [16.163] and [16.164] become respectively identical to [16.119] (1st equation), [16.121], [16.119] (2nd equation), [16.122] and [16.123] (2nd equation). The Winkler model with coupled springs, which was presented in the case of a line of elements (corresponding to a rectangular elongated footing, or a continuous footing) can also be applied to basemats of any form [NOG 96]; differential equation [16.132] is then replaced by an equation with partial derivatives whose analytical solutions can be obtained for simple forms (circle or rectangle). The representation of the soil/structure interaction using springs distributed under the foundation is not limited to the vertical Winkler springs. Scanlan [WOL 85] studied the dephasing influence linked to wave propagation on translation and torsion responses of rigid basemats using beds of horizontal springs. The effect of a shear wave propagating along direction Ox on a rectangular basemat (see Figure 16.14) is studied; this wave causes a sinusoidal displacement of the ground in the Oy direction.
760
Seismic Engineering
Figure 16.14. Effect of a shear wave with horizontal propagation on a rigid rectangular basemat
For a pulse Z and a velocity of propagation c of the sinusoidal shear wave, the components sx and sy of the ground displacement are given by: § x· sx = 0; sy = s0 sin Z ¨ t ¸ © c¹
[16.165]
If ux and uy are the displacement components of a point on the basemat with respect to the absolute axes x0y, the shear stresses Vx and Vy at this point, resulting from the link with the ground, are supposed to be: Vx = k (ux – sx); Vy = k (uy – sy)
[16.166]
k being a characteristic stiffness factor per unit area of the ground; in the horizontal plane this hypothesis is similar to that of the independent Winkler springs in the vertical direction. The basemat of sides 2a and 2b along the axes [GK linked to its center of gravity G, has a rigid body displacement which reduces to a translation V along oy and a rotation T around G; this rotation is produced from the position at rest in which the axis G[ makes the angle Io with Ox; a point on the basemat with coordinates [, K in the system of axes [GK thus has the following coordinates in the absolute system xOy: x = [ cos (Io +T) – K sin (Io +T)
Notions on Soil/Structure Interaction
y = V + [ sin (I0 +T) + K cos (I0 +T)
761
[16.167]
For low values of rotation T, the displacement of a point with coordinates x0, y0 in the position of rest is: ux = – Ty0; uy = V + Tx0
[16.168]
from where, using the hypothesis [16.166], the resultant Fy and the torsion moment MG, of the ground reactions become: Fy = ³³ V y d [ dK
ª
³³ k «¬V T x
0
Z x ·º § s0 sin ¨ Zt 0 ¸ »d [ dK c ¹¼ ©
MG = ³³ ª¬ xV y y V V x º¼ d [ dK ª § Zx § k ³³ « y0 T x0 T y0 x0 T y0 ¨ V T x0 s0 sin ¨ Zt 0 c © © ¬
· ·º ¸ ¸ »d [ dn ¹ ¹¼
i.e., by developing the sine function and by taking into account the fact that the odd functions of [ and of K do not contribute anything to the double integrals: Fy = kS [V – K0 s0 sin Zt]
[16.169]
MG = kSU [UT + [K1 (cos I0 – T sin I0) + K2 (sin I0 + T cos I0)] s0 cos Zt][16.170] where surface S and radius of gyration U of the basemat have been introduced; these are given by the following expressions: S = ³³ d [ dK
U² =
4ab
[16.171]
2 2 1 [ K d [ dK ³³ S
2 1 2 a b 3
[16.172]
and notations K0, K1 and K2 corresponding to the integrals: K0 =
1 §Z · §Z · cos ¨ [ cos I0 ¸ cos ¨ K sin I0 ¸ d [ dK S ³³ c c © ¹ © ¹
K1 =
1 SU
§Z
³³ [ sin ¨© c [ cos I
0
· §Z · ¸ cos ¨ K sin I0 ¸ d [ dK c ¹ © ¹
[16.173]
[16.174]
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Seismic Engineering
K2 =
§Z
1 SU
³³K cos ¨© c [ cos I
o
· §Z · ¸ sin ¨ K sin Io ¸ d [ dK ¹ ©c ¹
[16.175]
The basic calculation of these integrals leads to: K0
sin D sin E
D
1 § sin D
a 3
K1
[16.176]
E · sin E cos D ¸ ¹ E
[16.177]
· sin D 1 § sin E cos E ¸ ¨ ¹ D
[16.178]
¨ a b D © D 2
2
b 3
K2
a b E © E 2
2
D and E being defined as: D
Za c
cos I0 ; E
Zb c
sin I0
[16.179]
Coefficient K0 is always lower than or equal to one in absolute value (the value one is attained only for D = E = 0, i.e. for a zero pulse Z); from this note it is clear that the phase shifts due to wave propagation below the basemat attenuate the excitation sustained by the basemat during the translation motion; on the other hand they produce a torsional excitation which does not exist in the usual hypothesis of the in phase movement below all the points of the foundation. To assess the practical consequences of these phase shift effects, the case of a rigid building of total mass m and of moment of inertia I = mU², U being the radius of gyration defined earlier in the case of basemats (see [16.172]), is taken; this amounts to assuming that the building is a homogenous parallelepiped having the basemat as base; the angle Io between the systems of axes xOy and [GK of Figure 6.14 is taken as zero; therefore E = 0 (2nd equation [16.179]; this leads to: K0
sin D
D
; K1
1 § sin D · cos D ¸ ; K2=0 ¨ ¹ a b D © D a 3 2
2
[16.180]
Taking into account [16.169] and [16.170], the equations of the movement of the building (displacement V of the center of gravity in direction Oy and rotation T around the center of gravity) are written as:
Notions on Soil/Structure Interaction
763
kS >V K 0 s0 sin Z t @
[16.181]
mU² T = – kSU [UT + K1 s0 cos Zt]
[16.182]
M V
The pulse of the free oscillation mode (both translation and torsion) of the building is marked as : and is given by the relation: kS m
:²
[16.183]
from where by transposing in [16.181] and [16.182], we have: V + :²V = :²K0s0 sin Zt
T + :²T = – :²K1
s0
[16.184]
cos Zt
U
[16.185]
The forced excitation solutions of these equations are: V=
:
2
2
: Z
T
:
2
K 0 s0 sin Zt
[16.186]
s0
[16.187]
2
2
: Z
2
K1
cos Zt
U
The movement V1 of an extremity of the basemat is to be noted ([ = a), i.e.: V1 V aT
2
ª º a s0 « K 0 sin Zt K1 cos Z t » U : Z ¬ ¼
:
2
2
[16.188]
whose maximum absolute value is given as: Max VI =
: 2
2
: Z
2
s0 K K 2 0
2 1
a
2
U
2
[16.189]
Compared to the excitation without phase shift which corresponds to K0 = 1, K1 = 0 (see [16.180]) the amplitudes of V and V1 are then multiplied by the following coefficients Cv and Cv1:
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Seismic Engineering
Cv = K0
Cv1
sin D
[16.190]
D
K 02 K12
a
2
U
2
sin ²D
D
2
ª 3a ² 1 § sin D ·º « 2 cos D ¸ » 2 ¨ ¹¼ ¬a b D © D
2
[16.191]
The variation of these coefficients, for D ranging from 0 to S, is represented in Figure 16.15 (Cv1 has been calculated by ignoring b² as compared to a² in the second term under the root).
Figure 16.15. Coefficients Cv and Cv1 of the response modification due to phase shift for a rectangular building. Cv is applied to the center of gravity, Cv1 to the extremity of the basemat; the variable D is equal to the product by S of the ratio between the length of the basemat and the wave length of the sinusoidal movement of excitation
It is observed that Cv decreases rapidly with increasing D, till it becomes canceled when D = S; this is not surprising because the length 2a of the basemat is then equal to the wave length 2S c/Z; on the contrary Cv1 remains higher than 1 for D lower than 3, reaching a maximum of 1.39 for D = 1.83. The phase shift can therefore reduce the movement of the center of gravity considerably if the excitation consists of dominant components with wavelengths close to the dimensions of the basemat; at the same time it can also amplify the movement of the extremities of the building under the torsion effects that it creates.
Notions on Soil/Structure Interaction
765
As indicated in section 16.1.1, the comparisons between the recordings obtained from inside the buildings and those carried out in free field but near the building highlighted a significant effect on the basemat due to filtration of high frequencies of the signals. According to the drawing given above, this effect may partially be due to the influence of the phase shift because the high frequencies correspond to the small wavelengths which can be similar or even lower than the dimensions of the foundations. However, if this effect is to be considered using the sin D/D factor of the previous formulae, fairly low values (for example c = 300 m/s with Z = 2S u 5 rd/s, i.e. a frequency of 5 Hz, a = 20m and Io = 0; as a result D = 2.09 and sin D /D = 0.413) should be given for the velocity of propagation c which intervenes in the definition of D (see [16.179]). Whereas study of the recordings shows that the apparent velocities of propagation are of the order of km/s, even when the superficial terrains have shear wave velocities of 200 or 300 m/s (see section 3.2.1). When c is about 1,000 or 1,500 m/s a significant reduction due to the phase shift effect would only be possible for very high frequencies (according to earthquake engineering), of about 20 Hz or more. These observations regarding the values to be retained for the velocities of propagation along with the importance of the effects of torsion (which counteract the reduction of the movement of the assembly near the edges) probably explain that the phase shifts due to excitation have not been included in current practice, at least not for the design of buildings even if they are of significant dimensions. Certain examples of application to long bridges are known however; in such cases the phase shifts are due either to a deterministic model of wave propagation (following the same principles as mentioned above) or to a random model based on the semiempirical formulations of the loss of coherence of the signals based on the distance. The choice of the type of model depends on the excitation frequency as the deterministic approach corresponds more to the relatively low frequencies; this approach may seem totally unrealistic for higher frequencies (above 10 Hz, to give an idea).
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Chapter 17
Overview of Nonlinear Calculations
17.1. General observations on nonlinear calculations 17.1.1. The problem of hypothesis and criteria The choice of accelerograms for nonlinear analysis, already discussed in section 10.1.3, will not be discussed here except to mention the following three points: – accelerograms deemed equivalent (as they are adjusted to the same design spectrum) can produce visibly different responses when they are used in nonlinear calculations (see Figures 9.16 and 9.17); – it can happen that a natural accelerogram, set to scale in such a manner that its spectrum exceeds a given design spectrum, produces a response much lower than that obtained from a synthetic accelerogram adjusted for this same design spectrum (see Figure 9.19 and its notes); – the choice of an accelerogram suitable to study the response of a certain type of nonlinear model can be questioned if this model is a little complicated (see Figure 9.20 and its notes where the differences between the cumulative and the noncumulative models are indicated in terms of sensitivity of the choice of accelerogram). These three observations reveal the same difficulty: inadequacy of the elastic response spectrum to characterize the seismic action when the nonlinear response presents a cumulative character, i.e., the rate of damage depends strongly on the number of cycles of excitation. This undisputable observation (easy to understand through basic arguments; see section 10.1.3) is, however, largely ignored in practice as the forces of inertia due to routine and blind submission to the rules of the
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regulations are so large even though these were drafted with only linear calculation in mind. Choice of accelerograms is not the only problem that we are faced with in the nonlinear models. The results can also be greatly influenced by the combination of excitation components in different directions, depending on the manner in which it is considered; there is a general tendency to forget that the principle of superposition, which enables us to calculate the effects of the three components of seismic movement separately and then combine their results using the SRSS or Newmark’s rule (see section 15.2.3), applies only to linear behavior. Most nonlinear analyses are carried out considering only the horizontal direction of excitation; however the influence of the vertical excitation can be significant; for example, when the curvature-moment law of a reinforced concrete column which bends under the effect of horizontal forces, depends largely on the normal stress and thus on the vertical component of the movement. The above observation on the importance of normal stress emphasizes the difficulties in modeling the nonlinear behavior both from the point of view of mechanical hypothesis and also from that of the numerical algorithms used to represent it. Experts in nonlinear modeling know the traps to be avoided in a certain number of simple cases as in the conventional elasto-plastic models for linear elements (columns, beams), slipping due to Coulomb friction at the interface or shocks due to the filling of spaces. In most cases these traps have well identified causes (fineness of the meshing which has repercussions on the extent of plasticized zones and so also on the concentration of deformations, representation of the behavior discontinuities, numerical instability phenomena linked to the choice of the time step or to the number of iterations necessary to converge towards the solution at every time step) but the measures to be adopted to avoid falling into those traps have to be defined on a case by case basis and the calculator would require solid practical experience. In more complex cases (nonlinear behavior of structural elements other than beams, three-dimensional forces, detailed representation of the rheology of the soil) the initialization of an adequate modeling requires in general a certain number of numerical tests (to define the conditions of use of the algorithms and to assess the sensitivity of the results to the parameter variations), along with laboratory tests (to measure certain parameters and evaluate the representativeness of the model). The above approach relates more to research than to practical engineering; the widely known practice of a benchmark (i.e., a competition between research teams where, using his own numerical model, each participant should try to reproduce the experimental results based on the data on the structure tested and the excitation
Overview of Nonlinear Calculations
769
applied) enables us to group the research inputs by making the large test programs “cost effective” given the high cost of these tests [AFP 00]. These observations show that nonlinear calculations are still far from being accepted as tools for designing in most of the applications related to earthquake engineering. Until now their contribution has only been towards validating the simplified methods of the codes (notably the use of the behavior coefficient to represent the nonlinear effects) and the study of margins. These limitations to the outcome of the nonlinear analysis result from the difficulties mentioned above in choosing the excitation mode (type of accelerogram and number of components) and in finalizing the modeling (with reference to its mechanical and numerical aspects) but these limitations are also justified given the problem of verification criteria. In the fields of application which have detailed building codes, mainly for structures made of concrete (reinforced or prestressed) or steel, these very codes pose many obstacles for the use of nonlinear calculations in designing. The criteria used for justifying the need for safety are almost completely based on the static equilibrium of forces; thus they are not well suited for seismic loads for which it is the level of deformation and not the level of stress that should be used to evaluate safety. This aspect has been mentioned several times (see sections 9.3.1 and 12.1.1). Applying the criteria of the rules to the seismic events entails representing the seismic effects by static forces deemed “equivalent”; the fact that these static forces are often obtained through more or less complex dynamic calculations does not change anything because these calculations relate to real behavior only for the levels of excitation that are clearly lower than those corresponding to the risk of total collapse. Assessment of the safety margins obtained using this design to the static equivalent based on linear calculations is possible only if the real level of deformations at near ruin conditions (with a strong nonlinear behavior) can be evaluated based on the level associated with the static forces of calculation and these levels of deformation linked to the intensity of excitation. As indicated in section 9.3.1, using the behavior coefficient according to the earthquake-resistant codes corresponds precisely to this approach, Newmark’s theorem being supposed to enable control of the level of non-elastic deformations corresponding to displacements equal to those calculated with an elastic model and the values of the behavior coefficient being in principle adjusted to satisfy the non-collapse condition (limit close to ruin) for the seismic excitation specified by the code. Nonlinear calculations are used more in the areas of applications where the verification criteria and the calculation methods are less regulated as in soil
770
Seismic Engineering
mechanics and in techniques involving special materials (shock absorbing devices or certain anti-seismic supports mentioned in section 18.3.2). For example, it is common practice to study the seismic behavior of dams and dikes constructed by earth filling through dynamic models of finite elements taking into account the complex rheological laws of the soil and the influence of the water; the two dimensional character (plane deformation) and the relatively simple geometry enable us to treat these nonlinear problems at reasonable calculation costs. In the same way, the shock absorbing devices presently used in the seismic protection of a certain number of bridges are in general designed using nonlinear calculations using accelerograms even though there is a different approach using the stochastic linearization technique (see sections 17.1.2 and 17.2.5). For this type of nonlinear calculation used in designing, the verification criteria generally depend on the displacement limits. In the case of dams constructed with earth filling, these limits can correspond to the maintenance of certain watertight conditions (for example whether this sealing is guaranteed by a flow-retarding facing whose elements need not undergo significant differential displacements) or to the extension of a liquefied zone (which should remain sufficiently small so as not to jeopardize the total stability of the structure). For the shock absorbing devices it is the stroke of the jack that fixes the maximum acceptable limit of displacement. In both cases the criteria that fix the boundary between what is admissible and what is not are therefore determined by the characteristics of certain elements of the structure and they result from a dialog between the designer and the project manager. This dialog is arbitrated by the administrative authority of the trustee who ensures public safety. This is a completely different situation compared to the one which is used in the case of concrete or steel structures where the static criteria of the building codes have, in practice, statutory authority even if they are not adapted to the case of a dynamic loading like an earthquake. This irrelevance is particularly striking for the criteria of stability with respect to the risk of overturning, as indicated in section 12.1.1; this problem shall be dealt with more in detail in sections 17.2.1 and 17.2.2. In short, the applications of nonlinear calculation to the designing process require a set of precautions of which the designer is not always aware. This explains the lack of confidence that they create; these precautions should concern: – the choice of accelerograms: we can just repeat the conclusions drawn in section 10.1.3, i.e., the preference to be given to the natural accelerograms chosen based on the seismic activity of the region where the concerned site is situated; synthetic accelerograms tuned to the elastic design spectrum according to a practice which is as well known as it is questionable in its principle should not be used in the case of nonlinearities with cumulative character, at least as long as these
Overview of Nonlinear Calculations
771
accelerograms continue to present high value cycle counts and very long durations of strong motion portion; – the assessment of the influence of the three directional character of the seismic excitation, mainly that of the vertical component; this influence which should be expected not only while selecting the degrees of freedom of the model but also from the point of view of modeling the nonlinear phenomena (see the example, given earlier, of the normal stress in the columns), can only be understood through numerical tests conducted taking the variability of the seismic movement into consideration; – the care to be taken while defining the proper conditions of use of the numerical model modeling the nonlinearity (extension of the concerned zones, fineness of the meshing, choice of the time step, convergence speed of the iterative procedures); special care should be taken of the “mild instabilities ” that can affect certain algorithms when these conditions are not respected (inadequate time step, insufficient number of iterations); the numerical solution thus obtained can be completely incorrect while still retaining a plausible form; this type of risk is typical of certain nonlinear behaviors (in the linear models, the numerical instability due to the choice of a time step that is too high in certain partially or completely explicit models results in a solution “explosion” that is easily detectable); the study of these proper conditions of use involves a series of tests to check the stability of the numerical solution and to judge its precision. – the definition of the verification criteria of the results of the model depending on the safety objectives aimed at; the simplest criteria to be used are those which introduce displacements (as mentioned in the examples above – dams built with earth filling or the shock absorbers of bridges) but they are unsuitable in the case of localized damages; the solution may then be to depend on the deformation criteria; this may lead to the problem of representativeness of the deformations calculated on the model which depends on its description fineness in the zones where deformations concentrate; this difficulty is encountered more often in the performance, in terms of local deformations, of the plastic hinges at the extremities of the columns and the beams, which depending on the fineness of the meshing, are distributed over a varying number of elements; other types of criteria, bringing into play parameters typical of the plurality of the damage at different scales (local and semi-local) were proposed to solve these difficulties. These indications show that the question of the criteria that should be retained has no single answer and should be studied on a case by case basis.
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Seismic Engineering
17.1.2. Methods of giving recognition to nonlinearities The previous paragraph dealt with some of the recommendations for using nonlinear calculations mainly in relatively less frequent cases where their results are used in areas other than research, i.e., for designing structures for construction or for the seismic diagnosis of existing installations. These general recommendations are just an advisory note meant to bring out the particularity of the nonlinear problems which exists in the three domains where a case by case choice has to be made (excitation mode, modeling and verification criteria). The fact that there is a great deal of diversity in nonlinear problems has to be emphasized even though it may sound a bit repetitive. According to P. Labbé, if linearity is a well defined mathematical property which leads to, apart from other consequences, the superposition principle of load cases, nonlinearity is a nonproperty; it cannot therefore be part of a general theory; at the most an outline of the classifications can be made (as in the natural sciences) and illustrated with examples. Different classification principles can be used such as the nature of the nonlinear phenomenon, the localized or spread out character of its manifestations or the methods adopted to consider its use in calculation methods. The two frequently encountered cases of nonlinearity under seismic excitation correspond to: – the reach of the zone beyond the elastic domain of behavior of the materials with the appearance of irreversible deformations which can add up at every load cycle or end up creating stable hysteresis loops capable of dissipating an important quantity of energy while limiting the extent of damage to the structure and the degradation of its resisting capacities; – the modifications in the type of links the structure has with its exterior; they can refer to the phenomena of slipping or uplift of the foundation at the interface with the ground which has already been mentioned in section 12.1.1, or to shocks against adjacent structures (see section 12.2.6). For very flexible structures (tall buildings, structures with guys or long cables) certain geometrical nonlinearities (P – ' effect, sag effect in the cables) also have to be considered but this is relatively rare and concerns only certain exceptional structures. “Plasticity” type nonlinearities having a priori a cumulative character belong to the cases where the choice of accelerograms of excitation is critical. In the “connection” type nonlinearities, non- or little cumulative cases (uplift on firm ground, mainly elastic shocks) and cases that can become cumulative (uplift associated with the punching of the ground, shocks involving irreversible
Overview of Nonlinear Calculations
773
deformations of portions subjected to impacts, slipping of the structures that are simply placed particularly if there is a frictional dissymmetry along the direction of the relative speed at the interface) are encountered. As mentioned in 10.1.3, the cumulative or non-cumulative character is a crucial element for selecting accelerograms. From the point of view of extension of the affected zones by the nonlinear phenomena, the case of localization, i.e., the concentration of effects, is by far the most frequent; this is obvious for the nonlinearities of links but also applies to the effects of plasticity in most of the structures. The distributed plastic deformation that is seen in the columns of Figure 12.22 represents an exceptional case (probably not due to the instantaneous effects of the quake but due to progressive collapse of the structure of this car lot after certain columns in its central part gave way under the action of the weight). In general for concrete or steel structures the plastic effects (cracking and deformation of framework in the reinforced concrete elements, permanent deformations in the steel elements) affect only critical zones of limited extension (extremities of columns and beams). The idea, sometimes defended by certain designers, that it is possible to obtain a relatively uniform distribution of plastic deformations under seismic excitation is disproved by the analysis of past experience that clearly indicates that the rule is almost always the concentration of damage in certain loaded sections. It is only for the ground or for the earth fillings of dikes and dams that there can be a relative homogenity of the field of deformations, except of course in the vicinity of certain odd points like the edges of the basemats or the pile extremities. It is a common practice to take advantage of this localization of the nonlinear effects in establishing the numerical model; for the finite element models, knowing the critical zones enables us to limit the number of elements by adopting a coarse meshing outside these zones; it is also possible to limit the usage of elements capable of simulating nonlinear behavior to the critical zones; the rest of the structure being treated with purely conventional elements this practice helps the calculation costs significantly. For the nonlinearities of links the current practice is to use special interface elements that are capable of representing the effects of slipping or loss of contact along with the standard elements of elastic behavior for the structure and the ground. For spring-mass type models (such as those used in the examples dealt with in section 16.2.2 or 15.2.4), the recognition of nonlinearities depends only on the processing of the laws of force-displacement or momentrotations for some of the springs corresponding to the elements that can constitute the critical zones (plastic hinges) or to the connections with the ground. The predetermination of the zones, where the nonlinear phenomena appear, forms the basis of the principle of capacity design that is already mentioned at the end of section 9.3.1. In the numerical models it is therefore logical to limit to these
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Seismic Engineering
zones the use of finite elements or of springs capable of simulating behaviors outside the linear elastic domain because outside these zones this principle leads to certain over dimensioning meant to restrict the damage to “what is expected”. These models in which the nonlinearities are kept “under house arrest” do not help in assessing the ultimate behavior until complete destruction which occurs most of the time when the different points, where the stresses have been successively transferred, have exhausted their capacity to resist; this is almost always true at least in structures having a certain degree of redundancy; predicting this transfer process through calculation goes beyond the present practical possibilities because it has a certain randomness (linked to the unavoidable manufacturing defects with respect to the implementation plans) and corresponds more often to the three dimensional configurations, not only for the deformation modes but also for the influence of the different components of excitation. It has been observed (see section 17.2.1) that, in the simple case of tilting of the rigid blocks, the influence of the vertical component is relatively negligible as long as the excitation levels are moderate (rocking oscillations corresponding to weak rotations) but becomes very significant when the risk of overturning increases. The models with predetermined nonlinearities, while they help in clarifying the choice of behavior coefficients in designing to capacity, can only evaluate the safety margins for conditions which are far from total ruin. This limitation can lead, as the case may be, to overestimation or underestimation of the real margins obtained through designing. The notes made above in this section refer to actual nonlinear calculations, i.e., calculations whose model consists of special elements that enable us to simulate the real behavior of materials or structural elements. Here, we limit ourselves to these notes without going into the details of the modeling methods and the numerical algorithms which fall outside the scope of this book. On the other hand, it may be useful to give some information on the linearization methods that are still used in a certain number of applications. These methods consist of adapting the characteristics of a linear model (mainly the rigidity matrix, possibly the damping matrix and exceptionally the mass matrix) so that its response is equivalent to that of a real nonlinear model. It is obvious that the “equivalence” thus obtained can only be partial and that certain aspects of the nonlinear response, mainly the residual deformations, will not be reproduced by the linearized model. Strictly speaking, the linearization methods should be distinguished from the all-inclusive methods like that of the behavior coefficient in which the model used for the linear calculation corresponds to a conventional reference state (non-cracked sections for the structures in concrete) and where the definition of excitation could have been “tampered with” (design spectra according to earthquake-resistant codes; see section 9.3.2). In the linearization methods neither
Overview of Nonlinear Calculations
775
the results (these are not divided by a behavior coefficient) nor the excitation (elastic design spectra are used) of the linear model are modified. The linear iterative approximation has already been discussed in sections 5.3 and 16.2; it consists of adjusting the characteristics of the model based on the results obtained during the previous iteration to reproduce a law of nonlinear behavior as well as possible. It is commonly used in the case of grounds where the relation between shear modulus G and distortion J cannot be considered as linear in practical cases; Figure 5.7 is an illustration of this iterative process. An acceptable convergence level of the iterations is generally obtained in a few steps (2 to 5). This method is now accepted by certain earthquake-resistant codes for concrete structures. Thus the AFPS 92 guide for the seismic protection of bridges [AFP 95] specifies that for piers the secant stiffness in cracked concrete can be taken into account. This recognition is in practice necessarily iterative since the state of cracking depends on the response of the structure and thus on the choice of stiffness. The AFPS 92 guide is limited to the stiffness variation while the cracking also affects the structural damping; we have seen in section 9.3.1 that the “proof” of the Newmark theorem gives as much importance to the increase of damping as to the extension of the period to justify the displacement invariance. The stand taken by this guide was mainly for the sake of conservatism and also due to the fact that the commonly used damping coefficients take into account a certain degree of cracking of the concrete elements. With this assumption of constant damping, taking into consideration the effect of cracking leads to an increase in displacements compared to the case without cracking; this is in contradiction to the Newmark theorem. The method called the structure of replacement [SHI 76] went one step further by suggesting an increase of the damping while reducing the stiffness; the rules used to define these modifications based on the excitation level and the accepted degree of non-elastic deformation are clearly established only in the case of relatively flexible concrete structures (fundamental period of about one second or more) that are sufficiently regular so that the first mode is dominant in the response. Once again the same practical application limits already mentioned in section 9.3 regarding the dogma of the uniqueness of the behavior coefficient or the difficulties in implementing the design to capacity for irregular structures were observed The method of stochastic linearization corresponds to a different approach that consists of looking for the stiffness and damping parameters of the equivalent linear model by minimizing the mathematical expectation of the square of the error committed by replacing the nonlinear law by a linear law; to define the
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Seismic Engineering
mathematical expectation we should understand the laws of probability (in general normal laws of zero average) for the distribution of displacements and velocities or determine these laws based on the hypothesis of the nature of the random process in play (for the Markoff processes, the Fokker-Planck equation that can be solved analytically in some simple cases enables us to obtain the equation of the laws we are looking for [MIG 93]). For this linearization method, the definition of seismic excitation is therefore more naturally associated with a random formulation (see section 10.2) than with a deterministic one like the response spectrum. A rough idea of the method can be obtained by referring to an oscillator of one degree of freedom whose nonlinear behavior corresponds to the hysteresis loops in the displacement (abscissa)/ force (ordinate) diagram; Figure 17.1 shows a symmetric loop with respect to a “skeleton” I1 (x) in such a manner that the restoring force I x, x can be expressed as:
I x, x I1 x S x I2 x
[17.1]
x being the displacement, x the velocity and S the sign function (S x = + 1 if x ! 0 and S x = –1 if x 0 )
Figure 17.1. Symmetric hysteresis loop for an oscillator with one degree of freedom
The error H committed while replacing the nonlinear equation of the movement: m x + I x, x = – m s
[17.2]
Overview of Nonlinear Calculations
777
(m mass of the oscillator, s accelerogram of excitation of the support) by the linear equation: m x + c x + k x = – m s
[17.3]
is obtained by subtracting corresponding members, using [17.1]: H = I1 (x) – k x + I2 (x) S x – c x
[17.4]
The mathematical expectation E {H2} of the square of the error is expressed as:
³³ ª¬I x kx I x S x cx º¼ ²dp x dp x
E ^H²` =
1
2
[17.5]
with, in the hypothesis of a normal law for the distribution of x and x : x²
1
dp (x) =
2S
dp x =
1 2S
V x2 dx Vx
2
e
x ²
e
[17.6]
V x2 dx V x
2
[17.7]
V x2 and V x2 being the respective variances of x and x . In integral [17.5] the terms corresponding to the odd functions of x have zero contribution; therefore: E ^H²` =
³³ ª¬I x kx º¼ 1
2
2
dp x dp x ³³ ª¬I2 x S x cx º¼ dp x dp x
[17.8]
The values of k and c which minimize E ^H²` are obtained by canceling their partial derivatives: w w E ^H²` = 0; E ^H²` = 0 wk wc
[17.9]
Taking into account [17.8] this gives: k=
³³ x
I1 (x) dp (x) dp x /
³³ x²
dp (x) dp x
[17.10]
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Seismic Engineering
c= The
³³ x S x I x dp x dp x / ³³ x ²dp x dp x
[17.11]
2
double
integrals
being
taken
over
the
entire
plane
f d x d f, f d x d f are products of simple integrals; by using the equations:
³
f
f
dp x
³
f
dp x 1 ;
f
³
f
f
x ²dp x V x2 ; ³
f
f
x ²dp x V x2
[17.12]
[17.10] and [17.11] can be written as: 1
k
V
c=
2 x
1
V x2
³
f
³
f
f
f
xI1 x dp x
[17.13]
I2 x dp x ³
f
f
x dp x xS
[17.14]
The second integral, which plays a role in the expression of c, is easily calculated:
³
f
f
x dp x xS
f
x
0
2S
2³
e
x ² 2V x2
dx
2
V
S
x
f
V x ³ ue 0
u² 2
du
2
S
V x
[17.15]
and formula [17.14] becomes: c=
2 1
S V x
³
f
f
I x dp x 2
[17.16]
As an example, the case of Coulomb friction corresponds to:
I1 x 0 ; I2 x P mg
[17.17]
P being the friction coefficient and g the acceleration due to gravity; formula [17.16] enables simulation of the friction by a dashpot whose coefficient is: c=
2
S
P
mg
V x
[17.18]
It is logical that the more intense the excitation movement (large V x ) the smaller the coefficient c. The determination of V x requires representing the excitation
Overview of Nonlinear Calculations
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through a random process. Considering [17.2] and [17.17] the nonlinear equation of the movement is written as: mx P mgS x
ms
[17.19]
This corresponds to the assumption that the excitation is strong enough for the slipping to go on without stopping, the friction being insufficient to stop the relative movement between the mass and its support. The linearization operation described earlier has led to the replacement of this equation by the linear equation: x c ' x
s
[17.20]
with, according to [17.18]: cƍ =
c m
2 Pg
[17.21]
S V x
For the initial rest conditions (x (0) = 0, x (0) = 0) the solution to [17.20] is given by the following formula, similar to the Duhamel integral [9.10]: x (t) =
1 t s (W ) ª¬1 e c '(t W ) º¼ dW c ' ³0
[17.22]
by taking s W as an unfiltered white noise, as defined in 10.2.1, the generalized theorem of Brownian motion can be applied and written for x and x using equations [10.7], [10.9] and [10.13]: Max x t , t H [O, T] = gp, x
Max x t , t H [O, T] = f p , x
ª 2 2c ' T 3 4e c 'T e 2 c 'T º «V J 't » 2c '3 ¬ ¼
ª 2 1 e 2 c 'T º «V J 't » 2c ' ¼ ¬
1
2
[17.23]
12
[17.24]
gp, x and f p , x being the peak factors, V J2 the variance of the law of random sorting of white noise accelerations and 't the time step. Taking f p , x = 1, the desired expression for the variance V x is obtained, i.e., by assuming that T is big enough so that the exponential term in [17.24] can be ignored:
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Seismic Engineering
V x2
V J2 't
[17.25]
2c '
from which, considering [17.21], the following can be derived: c’ =
4 P²g² S V J2 't
[17.26]
The target is achieved, i.e., getting a value of c’ which represents the best approximation of nonlinear equation [17.19] through the linear equation [17.20]. The numerical values of c’ are high in the practical cases, as is seen by taking the typical values μ 0.3 , V J 1m / s ² (which correspond to a maximum acceleration of about 3 to 4 m/s²; see [10.2.1]), 't = 0.01s which with g = 9.81 m/s² gives c’ = 1103 s-1. This proves that the exponential term was ignored in [17.24] and shows that in [17.23] only the first term of the numerator of the fraction is to be considered. This gives for the maximum amplitude of relative slipping between the mass and its support:
V J3 3 1 Max x t , t H [O, T] = gp, x 't 2T 2 4 P²g²
S
[17.27]
This formula will be used in section 17.2.3 to evaluate the slipping of the blocks laid. To establish [17.26] it was assumed that the distribution of the values of x followed a normal law; in reality this case of Coulomb friction is simple enough to make it possible to solve the Fokker-Planck equation mentioned earlier and determine the real law of probability of x [CON 84]; this is found to replace 4/S = 1.27 by 1 in formula [17.26], which is not of any consequence when confined to estimate the orders of magnitude. The stochastic linearization method calls for techniques which are not generally used by structural engineers contrary to iterative methods by adjusting mechanical properties which are more intuitive and do not require any special training. That is why an important part of this section has been devoted to practical application principles and modalities of the stochastic linearization method with a view to show that they are easy to understand. As against the iterative methods, stochastic linearization provides analytical expressions of the coefficients of the equivalent linear model thereby facilitating the calculation of the orders of magnitude and with little cost the study of sensitivity on the influence of the parameters. This possibility is extremely useful in the studies at the preliminary
Overview of Nonlinear Calculations
781
project level. In the cases where its implementation is relatively easy it is an option to be considered in place of the approaches with which the engineers are more familiar. An example of this will be given in section 17.2.5. The other parts of section 17.2 deal with different cases of nonlinear analysis corresponding to the important aspects of seismic response. The selected examples show modelings which are as simple as possible in order to bring out clearly the gap between the “reality” thus schematized and the “fiction” contained in the routine rules of calculations when they are made to “say things” for which they have not been designed, particularly when it comes to safety margins. 17.2. Some examples of nonlinear calculations 17.2.1. Tilting of the rigid blocks
This subject has already been dealt with in section 12.1.1 because of its historical and didactic importance in seismic engineering. Its modeling for calculation purposes, though it appears simple, is not devoid of difficulties if we have to keep in mind all the possibilities (rocking by pivoting around a side of the base, “oblique” rocking around a corner – see Figure 12.8 – jumping phases during which the block does not have any contact with its support). Several authors have published results obtained with different models (for example in the references [ISH 82, SHA 99, ZHU 82]) and their comparison with the experimental results obtained using shaking tables. To have a rough idea of this type of calculation the case of a slender parallelepiped pivoting around a side of its base is taken (Figure 17.2).
Figure 17.2. Rocking oscillation of a slender parallelepiped
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Seismic Engineering
The following are the notations defined partially in the figure: H (height), B (width of the base), r (half-diagonal), O (slenderness ratio = H/B), D (angle whose tangent is B/H = 1/O), m (mass), I0 (moment of inertia with respect to the pivoting side O), J (horizontal acceleration), g (acceleration due to gravity), T (rotation); the following equations are obtained: tanD =
1
O
;r
1 H ² B² 2
H 1 ; Io 2 cos D
4 mr ² 3
[17.28]
of which the last one corresponds to the hypothesis of a homogenous distribution of the mass. The equation of motion is obtained by the equilibrium of the moments taken with respect to 0: ș = m J r cos (D –T) – mgr sin (D –T) I0
[17.29]
For small values of D (high slenderness ratio) and T, the routine approximations can be made (cos D = cos T = 1, sin D =D, sin T =T) which, along with equation [17.28] for I0, enable [17.29] to be rewritten in the form: 3 ș g DJ ș 4r
3 J D g 4r
It is seen that from the idle state (T = 0, ș when: J > Dg
[17.30] 0 ) movement can be activated only
[17.31]
which is the static instability condition mentioned in section 12.1.1 (second of the equations [12.1]). From [17.30] it is also observed that this type of oscillation does not correspond to the simple oscillator models examined earlier because the period of free oscillations (J = 0) of low amplitude is not constant and tends towards 0 when this amplitude diminishes. It is clear from [17.30] that this period T0 (calculated as the quadruple of time necessary for the block to fall back on its base when it is released from an initial rotation T0) is given by: T0 =
8 3
§ D · r Arc cos h ¨ ¸ g © D T0 ¹
[17.32]
Overview of Nonlinear Calculations
783
which for T0/D << 1, gives, by neglecting second order terms in the Taylor series development: To = 8
2r T o 3g D
[17.33]
Therefore, the period of oscillation becomes smaller with the diminishing amplitude. This phenomenon is known to all those who have played with small slender objects placed on a table. To integrate the differential equation [17.30] let us consider the simple case where acceleration J of the support, supposed to be constant and verifying condition [17.31], is applied only for a duration T; the following notations are introduced:
Z²
3g ; :² 4r
§ 3 J· g DJ Z ² ¨1 D ¸ 4r g¹ ©
[17.34]
The motion has two phases: (1) constant acceleration J is applied for time t included between O and T and the block starts from the idle position (T = 0, ș 0 ) for t = 0; equation [17.30] is then written as:
T :²T
:²
J D g g DJ
[17.35]
and its solution as:
T
J D g J D g sinh :t cosh :t 1 ; T : g DJ g DJ
[17.36]
(2) for t > T the acceleration of the support is zero; the equation of the movement becomes:
T Z ²T
Z ²D
[17.37]
and its solution, which is linked to the previous solution at time T, is written as:
T
D ¬ªD T T ¼º cosh Z t T
T T sinh Z t T Z
[17.38]
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Seismic Engineering
T Z ª¬D T T º¼ sinh Z t T T T cosh Z t T
[17.39]
T (T) and T T being given by equations [17.36]. It can be assumed that this second phase of movement ends at the instability limit, i.e., rotation T only reaches value D which brings the center of gravity G directly over the pivoting edge and this is achieved with zero speed. Using basic calculations which will not be reproduced here, it can be shown from equations [17.36], [17.38] and [17.39] that the end of this second phase corresponds to an infinite time and that the following equations can be arrived at:
T T 1/ ª¬I I ² D ²I º¼ D V G
ª º 1 D ²I I Arc tanh « » 3 1 D ²I ¬« I 1 I ² D ²I ¼»
4
V ² / 2I
[17.40]
[17.41]
[17.42]
The non-dimensional quantities I, V and G being defined by:
I
J ;V Dg
JT D gr
;G
1 JT ² 2 Dr
[17.43]
The significance of these quantities is as follows: – I is the ratio, higher than one, between the acceleration J applied during the time T and the acceleration Dg which corresponds to the limit of static stability [17.31]; – V is the ratio between velocity JT reached by the support at the end of the first phase (and which remains constant during the second phase because the acceleration there is zero) and a reference velocity vr of the block, which is given by the equation: vr = D gr
[17.44]
vr being the velocity that corresponds to a kinetic energy m vr2/2 equal to potential energy when the center of gravity is at the vertical 0 (instability limit). Equating these two energies leads to:
Overview of Nonlinear Calculations
1 2 mv r 2
mgr 1 cos D
785
[17.45]
which for small D (cos D = 1- D²/2) results in [17.44]; – G is the ratio between displacement J T ² / 2 of the support at the end of the first phase and product D r which represents the half-width of the base for a slender block. Table 17.1 gives the values of T (T)/ D , V and G for a certain number of values of 1 . I and D 4 I
ș T /Į
1/ 2I
V
1
0.4924
0.5000
1.2
0.4114
1.4
4
I / 2I - 1
G
f
2.3094
f
0.4167
2.4518
1.9795
2.5047
0.3532
0.3571
2.0056
1.7962
1.4366
1.6
0.3095
0.3125
1.8056
1.6796
1.0188
1.8
0.2754
0.2778
1.6739
1.5988
0.7783
2
0.2481
0.2500
1.5912
1.5396
0.6330
2.5
0.1988
0.2000
1.4676
1.4434
0.4308
3
0.1658
0.1667
1.3992
1.3856
0.3263
3.5
0.1422
0.1429
1.3555
1.3472
0.2625
4
0.1245
0.1250
1.3251
1.3197
0.2195
5
0.0997
0.1000
1.2858
1.2830
0.1653
7
0.0713
0.0714
1.2441
1.2435
0.1106
10
0.0499
0.0500
1.2153
1.2155
0.0738
15
0.0333
0.0333
1.1941
1.1945
0.0475
20
0.0250
0.0250
1.1839
1.1843
0.0350
f
0.0000
0.0000
1.1547
1.1547
0.0000
3
Table 17.1. Values of T (T) /D, V and G depending on I, for D
1
4
This table made for a slenderness ratio O of 4 D 1 4 enables us to bring out the following tendencies which are valid for all slender blocks ( D 13 or O = 3 being the lower limit of the slenderness ratio):
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Seismic Engineering
– ratio I of the accelerations can reach values much higher than one which forms the static criterion, without jeopardizing the stability in case the excitation duration is sufficiently short; – at the stability limit, ratio V between the maximum velocity of the support and the reference velocity vr (see [17.44]) is reasonably constant for the higher values of I and tends asymptotically towards a limit whose value according to equation [17.41] is 2 / 3 1.1547; for I higher than 1.5, V can be calculated with less than 10% error using the following simple formula:
V#
4 3
I / 2I 1
[17.46]
In the table, the column situated to the right of V enables us to judge the approximation thus obtained which improves as I increases; – the fact that the limit of V, for I tending towards infinity, is higher than one shows that the kinetic energy of rotation communicated to the block at the moment the pivoting started is lower than that which corresponded to the translation movement having the velocity of the support; it is in fact true that the excitation then resembles an instantaneous shock (duration T tends towards zero when I tends towards infinity) which introduces a velocity but does not produce a rotation due to lack of time (T(T) tends towards zero as is seen in the table); if this velocity initiation occurred without any loss of energy during the translation transfer of the support-rotation of the block, we would necessarily have V = 1 since definition [17.44] of vr corresponds to an equality between the work done by the weight (which consumes all the kinetic energy of rotation communicated to the block) and the kinetic energy of translation associated with vr; in reality in this transfer of one type of movement to another it is the conservation of the kinetic moment that applies and not the conservation of energy; a simple calculation shows that velocity Vs of the support at the time of activation of the rotational movement should be equal to: Vs =
vr 3 cos D
2
which, for small D, corresponds to the limit 2/ 3 found for the ratio V
[17.47] Vs / vr ;
– ratio G of the displacements decreases rapidly when I increases (and tends towards zero for I infinite) but corresponds to significant displacement of the support (higher than one-third of the width of the base for I < 2 and one-tenth of this width for 2 < I <4) in practical cases; – rotation T(T) produced when the excitation stops is practically equal to D / (2I) as shown in the third column of the table; it becomes very small compared to D
Overview of Nonlinear Calculations
787
when I is big; the main part of the rotation necessary to reach the stability limit is produced only after the end of excitation. These observations are pertinent to a very simple model in which the excitation is extremely schematic and cannot represent a real seismic movement. However, the numerous theoretical and experimental studies related to this problem clearly show that these observations are qualitatively valid. In particular, the reference velocity vr (see [17.44]) represents the natural unit to be used to measure the maximum excitation velocity V when the stability limit is reached. Ishiyama has defined the following criterion of the stability block [ISH 82]: V < k Vs
[17.48]
where Vs is the critical velocity of the support and k a numerical coefficient of about 0.4 (this value is obtained from the statistical study of a set of calculations and trials corresponding to different conditions of slenderness ratio and excitation); this critical velocity, which gets converted into a rotation movement of the block (conservation of the kinetic moment), corresponds to the point of reaching the stability limit (center of gravity directly over the pivoting edge) at the end of this movement. Considering equations [17.47] and [17.45], whatever the value of angle D condition [17.48] can be written as: V<
2 3
k 2 gr 1 cos D / cos D
[17.49]
In practice as the problem of stability with respect to overturning arises only for slender blocks (i.e. for the lower values of D) and by rounding it to ½ the product 2k / 3 (which is 0.462 with k = 0.4, a value given by Ishiyama) the stability condition takes the form: V<
1 D gr 2
[17.50]
i.e., since D is the inverse of slenderness ratio O, condition [12.2], where it is indicated that it is this criterion in velocity and not the static criterion (acceleration lower than Dg) that applies for the blocks of a certain size. Several studies using numerical simulations and trials on shaking tables have confirmed its validity; Figure 17.3 shows the results obtained in one of these studies [SHA 99]. This figure brings out, on the one hand, the good similarity between criterion [17.50] and the results of the study (the broken line marks the boundary above
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Seismic Engineering
which no block has been overturned; it practically coincides with the smooth curve corresponding to the criterion) and on the other hand the weak influence of the vertical excitation (there is little difference between the two diagrams). It is equally interesting to note that certain blocks whose values of H and B place them in the unstable zone, have remained upright. This observation underlines the partially random character of seismic destruction. This has been confirmed by the observation carried out after certain trials on the shaking table according to which blocks having identical characteristics and which are tried simultaneously (subjected to the same excitation) can face different consequences, some surviving the tremor while others are overturned.
Figure 17.3. Stability with respect to overturning of a set of blocks having different heights H and base widths B which were submitted to the accelerogram recorded at El Centro during the Imperial Valley earthquake in 1940 (no. 13 of Tables 4.1 and 4.2). The diagram on top corresponds to the action of the only horizontal component, the one below, to the two components (horizontal and vertical). The symbols used are + for the absence of rocking, for rocking without overturning and z for overturning. The straight line starting from the origin represents the static criterion and the curve situated below, the velocity criterion defined by equation [17.50] (according to [SHA 99])
Overview of Nonlinear Calculations
789
This random nature of the destruction is also found in structures more complex than rigid blocks. During post-seismic missions such differences in behavior were observed frequently (ranging from maintaining the stability of the structure despite certain damages to total collapse) in groups of buildings considered similar (see Figure 12.31). Amongst the possible causes for these differences, such as the variation of seismic movement over relatively small distances or the influence of fraudulent mistakes in the execution or bad workmanship, the one that should not be forgotten is the fact that the series of events producing more and more serious damage is sensitive to marginal differences in the mechanical properties of the structures and in the details of the dynamic excitation. When the situation is close to complete destruction, certain parameters whose influence was weak for moderate degrees of damage become crucial for the evolution of the dynamic response which confirms its random nature. For example, in the case of rigid blocks the influence of the vertical component which is otherwise negligible for low or medium amplitude rocking oscillations becomes sensitive in near overturn conditions [WEI 94]. Considering this random nature, certain authors have published stability criteria with respect to overturning for different levels of probabilities. Table 17.2 gives an example of this, in the form of acceleration values associated with a probability of overturn of 0.16 [WEI 94]. O 4 3.5 3 2.5 2
r = 1.22m
r = 1.83m
r = 2.44m
r = 3.05m
r = 3.66m
(4gt)
(6gt)
(8gt)
(10gt)
(12gt)
3.41
4.79
5.89
6.87
7.93
(4.32)
(5.30)
(6.11)
(6.84)
(7.49)
3.97
5.52
6.73
7.99
8.91
(4.94)
(6.05)
(6.99)
(7.81)
(8.56)
4.67
6.36
7.91
9.50
10.61
(5.77)
(7.06)
(8.15)
(9.12)
(9.99)
5.69
7.67
9.03
1.160
13.08
(6.92)
(8.47)
(9.78)
(1.094)
(11.98)
7.26
9.22
11.45
1.351
16.08
(8.65)
(10.59)
(12.23)
(1.367)
(14.98)
Table 17.2. Acceleration (in m/s²) causing overturn with 16% probability, for different values of the slenderness ratio O and of the half-diagonal r of parallelepiped blocks (as per [ZHU 99])
As 2.45 2.80 3.27 3.92 4.91
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Seismic Engineering
The first line of the cells of this table show the value of acceleration (derived from statistics based on the study of 75 real accelerograms) which corresponds to a 16% overturn probability for a block having a slenderness ratio O (left column) and the dimension r (half-diagonal, top line) associated with the cell under study. The figures within brackets in the second line of each cell are the values calculated with criterion [17.50] by applying the following rule (see section 4.1.2), to move from velocities to accelerations: acceleration (m/s²) = 10 x velocity (m/s)
[17.51]
The two values in every cell are absolutely comparable and this confirms the validity of criterion [17.50]. The last column of the table with the heading As gives the value of the acceleration (As = g/O) corresponding to the static criterion of stability; as indicated earlier this criterion tends to overestimate the risk of overturning more so when the size of the blocks increase. As indicated in section 12.2.6, the rocking oscillations which do not lead to overturning give rise to compressive-tractional stresses due to shocks which are produced when the block falls back flat on its base. For a unidirectional drawing of wave propagation, the value of these stresses V is given by the following (see [5.67]): V=Ucv
[17.52]
U being the mass density, c the propagation velocity of the waves and v the striking velocity. To connect v with the velocity V of the movement of the support, an equivalence between the kinetic energy of rotation linked to the angular velocity T and the kinetic energy of translation mV²/2 is used, i.e. the relation: 1 I oT ² 2
1 mV ² 2
[17.53]
i.e., by using the last of equations [17.28]: r T =
3 V 2
[17.54]
The velocity v at the time of striking reaches its maximum value at the extremity of the base which is opposite the pivoting edge with the value of V as: V = 2 r T sin D
[17.55]
Overview of Nonlinear Calculations
i.e. for small D (sinD = D =
1
O
791
, O being the slenderness ratio) and considering
[17.54]: v=
3
V
[17.56]
O
To apply formula [17.52] the following equations are used: E
c=
U
;U=
Vo gH
[17.57]
valid for a homogenous block and where E is the Young’s modulus for the material, H the height of the block and V o the static stress due to the weight. Hence, by transposing [17.56] and [17.57] in [17.52]: V=
V
O
3EV o gH
[17.58]
Section 12.2.6 shows that this formula leads to stress values which can be higher than the static stress V o for a strong excitation and a relatively weak slenderness ratio; the vertical shocks caused by the rocking can produce cracks on the load bearing elements of the concrete building. The calculations developed in this section assume that the support of the block is non-deformable. It can be shown that by abandoning this hypothesis the results concerning the stability with respect to overturning undergo only slight modifications; this conclusion arises from the fact that in the energy balance used to define the reference velocity vr (see [17.45]), the additional term corresponding to the deformation energy of the support is very small compared to the work done by the weight during the rise of the center of gravity to its ultimate position directly over the pivoting edge, at least for normal values of the slenderness ratio. In effect, in the most realistic of the uplift models presented in section 16.2.4, the model of the coupled Winkler springs, the total uplift corresponds to a relatively weak rotation (see [16.161]) and precedes a rotation phase around the edge of the base very similar to the case of a non-deformable support and during which time the deformation of the support does not evolve.
792
Seismic Engineering
The deformation energy of the support Wu, during the total uplift state, is given by equation [16.163] in which the ultimate value Tu* (see [16.161]) of the reduced rotation T* is taken, i.e., after an easy calculation: Wu =
§ 3 2· 1 mgG o ¨1 ¸ 2 © J J²¹
[17.59]
J being the parameter c/a of the coupled springs model whose values are of the order of one and Go the sinking of the block in the support under the action of the weight mg. The ratio between this energy Wu and the work Wp of the weight (see [17.45]) in the case of a slender block (small D = 1/O) is thus: Wu Wp
G § 3 2· § 3 2· 1 mgG o ¨ 1 ¸ / ª¬ mgr 1 cos D º¼ = o ¨ 1 ¸ 2 ² J J r D² © J J ² ¹ © ¹
[17.60]
By taking J = 1 and a slenderness ratio of 4 (D = ¼), this ratio is of the order of 100 Go / r, that is less than 0.1 in practical cases where Go is measured in mm when r is measured in m. The reference velocity vr which determines the stability criterion of Ishiyama is subjected to very little modification when the deformability of the support is taken into account. It is only recently, that is since 1980–1990, that the studies related to the overturning of blocks have reached a certain level of precision and they continue to be of interest in the field of research mainly in finding out the level of movements during the earlier unrecorded earthquakes. During the major San Francisco earthquake (18 April 1906), the detailed documentation regarding the overturning of a locomotive enabled us to conclude that the acceleration should have reached the level of 1g in the vicinity of the San Andreas Fault [ANO 89]. For a lesser known earthquake (14 December 1872 in the Northwest of the USA near the Canadian border), the study carried out on a boulder (Omak rock, [WEI 94]) apparently unstable but untouched by the tremors has shown that for the boulder to have overturned there should have been movements stronger than those estimated during an earlier study; this can raise doubts about the previously accepted location of the epicenter. These examples show that the subject is of current interest.
Overview of Nonlinear Calculations
793
17.2.2. Basemat uplifts
The models of coupled or non-coupled Winkler springs mentioned in section 16.2.4 are frequently used to study the uplift of basemats [BIS 85, KOB 84]; they can be easily included in the software and are an acceptable simulation of real behavior despite certain principle defects, which are more prominent in the decoupled models. More elaborate models using finite elements have also been used. These call for interface elements especially designed to eliminate the contact between the ground and the foundations during traction. The nonlinear calculations carried out with these models, either spring type or finite element type, are generally not very sensitive to the choice of accelerograms except in cases where the irreversible deformations of the ground are taken into account (see Figure 10.6). These calculations are therefore done using synthetic accelerograms tuned to the elastic design spectrum. The calculations on uplift are done only for important structures for which a behavior simulation, as close to reality as possible, is sought. However it cannot be said positively that this objective has been achieved considering that the problem of uplift persists. It has never been a topic of meaningful experimental study because of its furtive nature (it happens during an earthquake and in general does not leave any visible traces afterwards) and the smallness of the displacements and the rotations which it produces. This is why simplified methods which aim only at determining the orders of magnitude have been proposed. Tseng and Liou (see [TSE 81]) have used a principle of equivalence of energy between the linear case (without uplift) and the nonlinear case (with uplift). The application of this principle can be visualized in Figure 17.4.
Figure 17.4. Equivalence of energy between the linear behavior (without uplift) and the nonlinear behavior (with uplift) on the curve M* (reduced moment)-T* (reduced rotation)
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Seismic Engineering
When there is no uplift, the rotation between the reduced moment of overturning M* and the reduced rotation T* is linear (1st of equations [16.119]); in the presence of uplift on an elastic ground, this relation is represented by a curve (see [16.121]) which tends asymptotically towards one. The principle of equivalence of energy is expressed by the equality of the hatched surfaces on the figure; the reduced rotation
T*2 of the nonlinear case can thus be determined from the rotation T 1* calculated
based on the reduced moment resulting from the conventional seismic analysis, i.e., linear. It is obvious that when uplift is taken into account the rotation is increased but the stresses are reduced. The relations established in section 16.2.4 for the reduced moment M* and the reduced energy W* enable a simple application of the principle of equivalence proposed by Tseng and Liou. In the case of the Winkler model with independent springs the reduced energy in the presence of uplift is expressed [16.122] as: W* = 1 + 2 T *
8 * T 3
[17.61]
whereas for the case without uplift, it is (see [16.119]): W*
1 *2 1 T ; M* = T * so W * 3M *2 3 3
[17.62]
Based on a linear calculation of the seismic response of the structure under study, the maximum value of the reduced moment M o* is determined; if this value is lower than 1/3 the second part of equation [17.62] shows that the reduced rotation T * has not reached the value 1, therefore there is no uplift. On the contrary if M o* is greater than 1/3, the energy 3 M o*2 that would be calculated ignoring the uplift (third part of equation [17.62]) would be that obtained if the uplift was taken into account. Thus, from [17.61], we can have: 3 M o*2 = 1 + 2 T *
8 * T 3
[17.63]
The solution given by this second degree equation in T * is:
T*
2 3 *2 1 Mo 3 2 18
[17.64]
Overview of Nonlinear Calculations
795
To this reduced rotation corresponds fraction S* of the detached surface given by equation [16.123]: S* 1
1
T*
1 3 / ª2 ¬«
27M
*2 o
1 / 2 º ¼»
[17.65]
This fraction S* enables us to assess the importance of the uplift of the basemat and its damaging consequences (punching of the ground, high stresses in the foundations and the lower portions of the building). More often, in the decision process the acceptable values of S* are limited to around 30%. It is very important to specify the calculation mode of the detached fraction; a widely known earlier practice consisted of writing the equality of the reduced moments (instead of the equality of the reduced energy) to determine this fraction; considering relations [16.121] and [16.123], we have: S* =
1 3M o* 1 2
[17.66]
Compared to [17.65], this equation leads to an increased value of the results; this increase is marginal when M o* is just a little higher than the limit 1/3 which indicates the starting of the uplift (S*= 0.25 using [17.66] instead of S* = 0.19 using *
*
[17.65] for M o = 0.5), but is significant when M o approaches its static limit (S* = 0.70 by [17.66] instead of S* = 0.38 by [17.65] for M o* = 0.8). For M o* = 1, instability, in the static sense, is achieved (S* = 0’) using [17.66], whereas [17.65] indicates a uplift which can still be acceptable (S* = 0.465) if the ground quality is good and if the lower portions of the building have a high degree of resistance. Values of M o* greater than one are even acceptable (S* = 0.598 for M o* = 1.5) in the case of foundations on a rock, whereas to a large extent they break the static criterion which is again found to be inadequate under seismic conditions. Several authors have compared the estimation of the uplift using the energy equivalence method (see [17.65]) to the results obtained through different nonlinear models. In [TSE 81], Tseng and Liou mention a slight underestimation of the effects of uplift derived from the equivalence in energy with respect to “reality” (i.e., nonlinear calculation) and suggest a remedy. Using a model where the frequency dependence of the soil springs (see sections 16.1.3 and 16.2.2) was taken into account, Kobori et al. [KOB 84] reach a more refined conclusion according to which there could be a tendency to underestimate for the higher rocking frequencies and overestimate for the lower frequencies. Other studies [BET 86] indicate that the energy method seems to provide an acceptable approximation, given the uncertainties in most of the parameters which play a role in this lesser known
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phenomenon. Table 17.3 shows the results of one of these comparative studies carried out using Winkler nonlinear models. Excitation
Eú (MPa)
* (M*/ M o )NL
(M*/ M o* )EE
S*NL
S*EE
NRC 0.2 g
6,000
0.800
0.813
0.28
0.30
NRC 0.3 g
6,000
0.640
0.655
0.43
0.46
NRC 0.4 g
6,000
0.527
0.549
0.53
0.57
NRC 0.5 g
6,000
0.453
0.477
0.58
0.63
NRC 0.2 g
500
0.973
0.977
0.085
0.085
NRC 0.2 g
1,000
0.892
0.857
0.24
0.24
NRC 0.2 g
2,000
0.834
0.849
0.28
0.30
NRC 0.2 g
4,000
0.826
0.835
0.27
0.28
NRC 0.2 g
8,000
0.806
0.821
0.28
0.29
Table 17.3. Comparison of the uplifts calculated using energy equivalence (EE) and a nonlinear model (NL) for different excitations and ground conditions (Eú = Young’s modulus); according to [BET 86]
The model in this study represented a building of a nuclear power plant and its seismic excitation corresponded to the spectrum of the USNRC (see Figure 9.5) or to the synthetic accelerograms tuned to this spectrum; two sets of calculations were performed; the first one for an increasing level of the maximum acceleration of the ground (from 0.2 g to 0.5 g) with the same ground (Young’s modulus of 6,000 Mpa i.e., a good quality terrain without being rocky), the second for a constant excitation (0.2g) but by expecting different ground conditions (Young’s modulus varying from 500 to 8,000 MPa). The results obtained through nonlinear calculations using accelerograms (index NL) and using energy equivalence (index EE) show a good correlation as much for the ratio M*/ M o* (reduced moment with uplift divided by the reduced moment without uplift) as for fraction S* of the detached surface. There is a likelihood that the differences in the assessment mentioned earlier of the validity of the approximation provided by the energy equivalence is explained on the one hand by the differences in the nonlinear models and on the other hand by the different choices of excitation accelerograms (for example, Tseng and Liou used natural accelerograms at El Centro and Taft and their results are qualitatively different depending on the accelerogram used). In any case, for the preliminary evaluations the energy equivalence method is sufficient; its usage with the Winkler
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model with non-coupled springs (that has led to relation [17.65]) seems quite conservative since this model is uplifted more easily compared to the more realistic models, as was indicated in 16.2.4. The energy method can also be applied without much complication to the case where the plastic deformations of the ground are taken into account. It is sufficient to replace expression [17.61] or [16.122] of the reduced energy W* in the second member of equation [17.63] with the expression (see [16.116]) which corresponds to the elastic, perfectly plastic model of independent Winkler springs; we thus have: 3M o*2
1 · 1 V *3 § 1 V * 2 ¨ 1 ¸T * 24 T * © V *¹
[17.67]
where V* is the safety coefficient under the effect of self weight, i.e., the ratio of the final stress Vu of the plastic zone of the behavior law of the springs and the static stress Vo due to the weight acting alone. Equation [17.67] is a second degree equation in T* with a solution written as:
T*
2 ª º § M o*2 M o*2 · 1 V *2 » ¨1 3 * ¸ 1 3 * 4 « V 1 © V 1 ¹ 3 V * 1 » ¬ ¼
V *«
[17.68]
Therefore, by transposing [17.68] in the second of equations [16.117] which defines fraction S* of the detached surface, we have: ª § 3M *2 · 1 V *2 º» M *2 V * 1 1/ «1 3 * o ¨ 1 * o ¸ « V* V 1 © V 1 ¹ 3 V * 1 » 2
S* =
¬
[17.69]
¼
Taking V* = 3, which is a common safety coefficient in soil mechanics for the calculations related to foundations, it is found that S* = 0.453 for M o* 1 , i.e., a value very close to the one found earlier (S* = 0.465) in the case of elastic behavior of the ground; such values of uplift, if they are acceptable for very good grounds or for rocks, are probably not acceptable for soft ground because they correspond to significant irreversible deformations and therefore to visible sinking and inclinations of the foundations if the seismic excitation lasts very long. If no plasticization of the springs of the model is desired (given its simplicity this does not imply that the real deformations will remain reversible but that they should be limited to a behavior zone where they cannot increase rapidly for a slight increase
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of the loads), the final value of the reduced moment is obtained by transposing the value T* = V *2 / 4 in [17.67] which corresponds to the boundary between the EAU (elasticity and uplift) and PAU (plasticization and uplift) domains of Figure 16.11. Thus, the following condition is arrived at: M o*2 d
1 3V *2 8V * 6 18
[17.70]
In terms of fraction of the uplifted surface, the corresponding condition is written (as per [16.117]): S* = 1 –
2
[17.71]
V*
For the current value V* = 3, the final values of M o* and S* are equal to 0.707 and 0.333 respectively; we get back to the criterion, mentioned above, of 30% of the surface uplifted. Uplift is least influenced by vertical excitation which acts at the same time as horizontal excitation. This observation has already been mentioned with reference to the tilting of rigid blocks (see section 17.2.1). This is due to the higher dominant frequency of the vertical movement (see Table 4.5) and to the importance of radiative damping for the pumping (see section 16.1.3). On the other hand, uplift has a significant influence on the vertical response of a building which consists of a component whose frequency is double that of the rocking oscillations; this component corresponds to the lifting of the center of gravity as a result of uplift; a complete cycle of this lifting movement (going up and coming down) is actually produced when the uplift takes place only on one side of the building, i.e., during a half cycle of the rocking movement. The order of magnitude of the amplitude of this component can be estimated using the equations established in section 16.2.4 for the Winkler model with purely elastic non-coupled springs. In this hypothesis, the first of equations [16.123] enables us to calculate the reduced sinking G , i.e., the ratio between the sinking G of the center of the basemat and its value Go under static conditions (weight P acting alone): G* = –T* + 2 T *
[17.72]
The upwards displacement w of the center of gravity is therefore: w = Go (1 – G*) = Go
T * 1
2
[17.73]
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799
This can be expressed based on fraction S* of the detached surface using the second of equations [16.123]: § S* · w = Go ¨ ¸ © 1 S * ¹
2
[17.74]
The pulsation of the lifting movement is 2Zb (Zb = angular frequency of the rocking movement) and the static sinking Go is equal to mg/Kz (m = mass, g = acceleration due to gravity, Kz vertical stiffness), i.e. to g/ ZV2 (Zv = angular frequency of the pumping movement); the vertical acceleration Ad due to uplift is thus: Ad = 4 Zb2 w = 4 g
Zb2 § S * · Zv2 ¨© 1 S * ¸¹
2
[17.75]
If this “parasite” vertical acceleration has to be limited to a given value Av (for example, the value corresponding to the pumping movement), using equation [17.75] a simple limitation criterion of the detached surface can be obtained and can be written as: ª Z S* < 1 / «1 2 b Zv ¬
g º » Av ¼
[17.76]
In the case of foundations on a hard ground (rock) it is the actual deformations of the building itself that determine pulsations Zb and Zv rather than the effects of the soil-structure interaction; this criterion leads to the ultimate values of S* which are still about 30%; we then typically have Zb / Zv = 0.5 for buildings with strong wall bracing using basemats – this with Av = g/4 leads to the same limit of 33% for S* as criterion [17.71] with V* = 3. This order of magnitude of the acceptable uplift is valid both on soft (for which criterion [17.71] applies) as well as on hard ground, where it is the considerations regarding the structures (parasite vertical accelerations, high stresses in the basemat and the base of the load bearing elements) that fix the limit that should not be exceeded. The important point for the application of this 30% criterion is to have a realistic but safe estimate of fraction S* of the uplifted surface (nonlinear calculations, energy equivalence method [17.65]); formula [17.66] should not be used any more even at the pre-designing stage.
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17.2.3. Slipping of massive blocks
Estimating the amplitudes of the slipping of the blocks placed on a support subjected to a seismic excitation was dealt with at the end of section 17.1.2 at the time of presenting the stochastic linearization methods; the phenomenon of slipping concerns blocks with insufficient slenderness ratio which makes the horizontal forces of inertia incapable of causing first a tilting and eventually an overturn. A basic static reasoning has enabled us to establish condition [12.1] which is given as: PO<1
[17.77]
P being the friction coefficient and O the slenderness ratio, and ensures that the sliding precedes the tilting; considering the current values of the friction coefficient (from 0.2 to 1 depending on the smooth or rough character of the surface in contact); this condition [17.77] is verified for all the massive blocks (slenderness ratio O less than one). For an unfiltered white noise excitation (see section 10.2.1), characterized by a variance V J2 of the law of random sorting of the accelerations and by time step 't, the amplitude of slipping X is given by formula [17.27]: X=
S 4
f p, x
V J3 't 3 / 2T 1/ 2 P²g²
[17.78]
T being the duration of excitation, gp,x a peak factor defining the probability of exceeding this amplitude X and g the acceleration due to gravity. By taking plausible values for the different parameters, this formula predicts the displacements in centimeters of the same order of magnitude as those observed often in reality (see Figure 12.5) for example, with VJ = 3 m/s², 't = 0.01s, T = 10 s, P = 0.2 and gpx = 1.96 (probability to exceed is 10% (see [10.11]), we find X = 3.3 cm. If the order of magnitude is correct, the tendency of formula [17.78] is towards underestimation with respect to the effects observed during post-seismic missions, as the previous numerical application corresponds to a powerful excitation (VJ = 3 m/s² is associated with a maximum acceleration close to 1 g; see [10.2.1]) and with a low value of the friction coefficient (in practice, μ = 0.2 is obtained only for machined surfaces). The reasons for this underestimation are to be studied not so much in the white noise schematization of the seismic excitation (not having taken factor S/4 = 0.785 instead of 1, which is the exact value for a rigorous stochastic linearization as indicated at the end of section 17.1.2) but more in the hypothesis of the symmetric
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friction (same friction coefficient irrespective of the direction of slipping). Experience shows that this symmetry is more an exception than a rule and can be obtained only if special precautions are taken in preparing the contact surfaces; for “ordinary” surfaces which remained “stuck” to one another during a long period of time (several months or even years), the direction of the first episode of slipping seems to determine, in a majority of cases, the future evolution of most of the other episodes probably because this direction induces a preferential orientation in the forms of rupture of the micro-asperities which ensured a certain meshing of the surfaces before the first slipping. If the friction coefficient has different values (μ1 and μ2) along the direction of slipping, the Brownian motion produced (see sections 10.2.1 and 10.2.3) due to white noise excitation becomes dissymmetric; it is then observed [CON 84] that a deterministic drift whose velocity vd is given by the expression below gets added to the random diffusion effect of the Brownian motion: Vd =
V J2 't 2 1 2g
1 2
[17.79]
By taking the same values as previously for VJ (3m/s²) and 't (0.01s), μ1 = 0.2 and μ2 = 0.3 we find vd = 7.6 mm/s, i.e., for an excitation duration T of 10 s a drift of 7.6 cm, which is more than double the displacement calculated before (3.3 cm) for a symmetric friction corresponding to the coefficient μ = 0.2. The slipping amplitudes observed after the earthquakes are probably due to this effect of drift rather than to the random fluctuations caused by symmetric slipping. This is one of the reasons for which the measurement of these amplitudes hardly helps in estimating the level of seismic excitation except in the case of a lower acceleration limit which might have exceeded the value μg to activate the slipping. Several studies on the slipping of the blocks [BET 92] using digital simulation have been carried out; most of them dealt with the case of symmetric friction either with a view to apply it to certain structures or to certain elements of the structures (handling equipment capable of slipping on its travel tracks, anti-seismic supports with slipping plates; see section 18.3.2) with machined contact surfaces, or for the theoretical models of rupture of the seismic faults (line models of slipping blocks, see Figure 16.13, linked by springs was mentioned in section 6.3.2). The case of dissymmetric friction has made it possible to deal with the problems of slipping of the ground on natural slopes or the stability of dikes and dams in earth filling, in a simple but realistic manner. In these applications to geotechnical structures, the hypothesis of extreme dissymmetry of friction is adopted, i.e., the slipping is supposed to be possible only in the downward direction along the greatest slope line.
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In fact, acceleration *r necessary to move the block up the slope is much higher than *d which is sufficient to make it descend; Figure 17.5 shows the equilibrium of the forces in the two situations by referring, as is usually the case in soil mechanics, to the friction coefficient μ by tanI, I being the angle of friction of the materials.
Figure 17.5. Slipping of a block on a slope: i angle of inclination, * acceleration parallel to the slope
It is easily shown that accelerations *r and *d are given by the expressions: *r = g sin (I + i) / cos I; *d = g sin (I – i)/cos I
[17.80]
i being the angle of inclination of the slope. The ratio *r/*d is therefore sin (I + i) /sin (I – i), i.e., a very high value if the angle of friction is hardly higher than the inclination (static equilibrium almost close to instability); this often occurs in the case of natural slopes. In the case of artificial slopes (dikes and dams) the static coefficient of safety is normally more important but the upstream-downstream dissymmetry is high and the fact that it is considered as total (slipping impeded towards upstream) is obviously in the sense of safety in estimating the amplitude of slipping. In this hypothesis of total dissymmetry, the displacements due to slipping corresponding to a given accelerogram can be determined using a simple calculation as shown by Newmark [NEW 65]; his principle is shown in Figure 17.6.
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Figure 17.6. Schematic Newmark diagram for calculating the slipping of a rigid block in the hypothesis of completely dissymmetric friction. Every time acceleration A exceeds critical value J (start of slipping) in the direction where slipping is possible, the block slides until relative velocity Vr is canceled (equality of the surfaces of the hatched zones on accelerogram A (t)), causing an increase in relative displacement Dr between the block and its support
The total displacement ' at the end of the earthquake is obtained by adding the increments of all the slipping phases; easy to program for any accelerogram, this method has been very successful in practice. In the text book example of an accelerogram consisting of identical triangular notches, the orders of magnitude can be calculated using simple formulae; the notations used in Figure 17.7 are: – *: maximum acceleration of the triangular spike; – J: critical acceleration (start of slipping); – W: time gap between the start of slipping and the reach of maximum value *; – Tg: notch cycle, corresponding to the dominant frequency of the accelerogram that is to be represented by the sequence of notches (see section 4.1.2).
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Figure 17.7. Accelerogram with triangular spikes for the application of the Newmark method for the slipping of blocks
By taking the start of slipping as the initial time, the relative displacement G(t) of the block with respect to its support is given by: (1) for 0 d t d W :
G G
G
* J
W
t
1 * J 2 t 2 W 1 * J 3 t 6 W
(2) for W d t d W
G * J
[17.81]
Tf 2
* J
W
t W
G
1 1 * J 2 * J W * J t W t W 2 2 W
G
t W 1 1 1 1 2 * J W 2 * J W t W * J t W * J 6 2 2 6 W
[17.82] 3
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805
If during this phase (2) the velocity is canceled, i.e.:
W
Tf tW 2 2 2
[17.83]
the final displacement G f based on equations [17.82] is: 1§ 2 · § J· 2 ¸ *Tf 2 ¨ 1 ¸ ¨1 16 © 3 ¹ © *¹
G³
be:
3
[17.84]
In order to verify condition [17.83], it is easily shown that the ratio J / * should
J *
t
2 1 2 1
[17.85]
0,172
If J / * is smaller than this limit, the cancellation of the relative velocity is produced in a third phase of the movement t ²W T f / 2 ; then using basic calculations which are not explained here, Gf can be written as:
Gf
3 2 1 2ª J §J · 8 §J · 2º *T f «1 4 ¨ ¸ 2¨ ¸ » * ©*¹ 3 8 © * ¹ »¼ «¬
[17.86]
Expressions [17.84] and [17.86] of Gg match, thus also their first two derivatives, for the ultimate value of J / * defined by [17.85]. The total displacement ' due to slipping at the end of the earthquake is the product of Gf and the number of cycles Nc which is equal to the ratio between duration T and period Tg . Hence, by taking expression [17.84] of Gg ( J / * ! 0.172 ), we have:
'
1§ 2 · 2 ¸ *T 7 f ¨1 16 © 3 ¹
§ J· ¨1 ¸ © *¹
3
§ J· 0.121*TT f ¨1 ¸ © *¹
3
[17.87]
This equation gives acceptable orders of magnitude for amplitude ' of the final slipping; for example by taking * 5 m / s 2 , Tg = 0.33 s (i.e. a dominant frequency of 3 Hz for the accelerogram, which is a plausible value for a fairly strong
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Seismic Engineering
earthquake (see Table 4.2) and T = 5 s (i.e. with Tf = 0.33 s a number of cycles Nc equal to 15, which corresponds to the magnitudes close to 7), we have ' = 51.2, 21.6, 6.4 and 0.8 cm for the values 0.2, 0.4, 0.6 and 0.8 of the ratio J / * ; in Figure 17.8 these results are compared to the statistics obtained by applying the Newmark method to a set of real accelerograms corresponding to earthquakes of magnitudes between 6.3 and 6.9 [AOU 00].
Figure 17.8. Variation curves of displacement due to slipping ' (in cm) based on the ratio J/* between the critical acceleration (starting of the slipping) and the maximum acceleration of the signal. The X symbols correspond to the application of formula [17.87] with * = 5 m/s², T = 5s, Tf = 0.33s (according to [AMB 88])
The value 0.2 of the ratio J / * corresponds for * = 5 m/s2 to a friction coefficient of 0.1 that is obtained by taking an angle of friction of 30° and a slope of 25° in the calculation * d / g with the second of formulae [17.80]; Figure 17.8 shows that in these conditions, slipping ' varies from a few decimeters to a few meters depending on whether a medium or low value of the probability of overshooting is considered. The dispersion is thus important and this is not surprising because the completely dissymmetric slipping is a cumulative phenomenon which is sensitive to the duration of the accelerogram (number of cycles). The very simple application of the Newmark method (see Figure 17.6) is justifiable only if the vertical component (which modifies the value of the critical acceleration J) is negligible, a fact that has been verified in the studies taking into
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account a two-directional excitation, and if the slipping does not modify the conditions which control the friction at the interface; the constant friction coefficient model reaches its validity limits for slipping of significant amplitude fast enough, especially when the contact surfaces are situated in a zone saturated with water. For big landslides that hurtle down the entire slope at great speeds (or yamatsunamis; see section 3.3.2), it has been assumed that the heat dissipated by the friction could cause evaporation of water on the contact surface and cover the surface with a cushion of vapor completely cutting out friction at the interface. The Newmark method, which is the simplest nonlinear calculation that can be performed in the field of earthquake engineering should therefore be used only for the slipping of limited amplitude (a few decimeters at the most) affecting structures made by earth filling consisting of materials, whose mechanical properties and method of compacting are well characterized. Under these conditions, this method enables the assessment, in a realistic manner, of the risk of degradation of the structure under the action of earthquakes. In particular, it shows that the safety coefficient of a slope can go below one during tremors without necessarily reaching a catastrophic situation. Figure 17.8 (or formula [17.87]) shows that displacement ', at the end of the earthquake does not exceed a few centimeters as long as the ratio J / * (which is just the static safety coefficient) does not fall below 0.5. Displacements of this size normally have no serious damaging consequences for well designed and constructed structures. That is why in general the earthquakeresistant codes accept that the stability of the slopes under seismic action be verified under static conditions with ground accelerations of about half their real value (for example, the reduction coefficient of accelerations vary from 0.5 to 0.4 depending on the type of ground, in [AFN 95]). Whereas to verify the stability of a retaining wall it is necessary to take the real acceleration (not reduced) while calculating the earth pressure. Can the stochastic linearization method of slipping discussed at the beginning of this section in the case of symmetric friction be applied to the case of completely dissymmetric friction? Formula [17.79] gives the velocity of the drift vd resulting from dissymmetry; if the hypothesis of total dissymmetry is considered, in this formula, P2 should be made to tend towards infinity and we have: vd
V J2 't 2P g
[17.88]
I
By multiplying this velocity by the signal duration T and by adding the contribution of the random fluctuation (which is smaller than that of the drift, as
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indicated earlier), a formula for calculating the slipping ' which is as simple as [17.85] is obtained but it is not well suited for low friction coefficients. In the extreme assumption of a zero friction, the displacement due to slipping ' depends only on the accelerogram used and corresponds to the displacement associated with the positive alternations of the ground speed, i.e., to an infinite value, while the drift velocity vd (see [17.79]) tends towards infinity when the coefficient 1 is made to tend towards 0. A very simple version of the stochastic linearization method which was introduced with an unfiltered white noise excitation is not suitable for simulating completely dissymmetric slipping. The reason being that this excitation model does not properly represent the displacements despite its capacity to reproduce the accelerations and velocities reasonably well (see section 10.1.1). For this slipping pattern, the application of the Newmark method is so much simpler that the need for a linearization technique seems out of place. 17.2.4. Plasticization of building structures
This topic has already been discussed in sections 9.3.1 and 10.1.3 in the case of simple oscillators whose springs have an elasto-plastic behavior pattern; this case that can represent only regular structures whose response mainly follows the fundamental mode forms the basis of the current earthquake-resistant code formalism where nonlinear effects are taken into account through a unique behavior coefficient. As indicated in section 9.3.1, this strategy is justified using the Newmark theorem which shows the equality of the displacements of the entire structure irrespective of the model, be it linear or nonlinear, used in the calculation. This theorem is fairly well verified if applied to relatively flexible structures whose capacity to resist does not degrade much during the plastic deformation cycles (thereby enabling the dissipation of energy) and highlights the main importance of displacement for the assessment of security. The routine approach used in the calculation of structures and the regulations governing construction based on equilibrium of forces are such that it is difficult to use the criteria in displacement even though these relate more to natural seismic effects. However since 1990 there has been a progressive evolution towards adopting methods in displacement, mainly the use of the push-over method [KIM 99]. The name push-over comes from the fact that the basis of this method consists of plotting a unique force-displacement curve to represent the behavior of the structure when its upper part is subjected to more and more “push”. It involves a static stress analysis using a nonlinear model in which the forces applied horizontally have a distribution similar to that of the displacements of the basic mode of vibration and
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are multiplied by an increasing factor O till a plastic state of damage is reached; this state indicates the acceptable limit for safety (Figure 17.9).
Figure 17.9. Principle of plotting the curve V-' (V shear force at the base, ' displacement at the top) in the push-over method
The curve representing the behavior of the structure is plotted by marking the displacement ' of the top along the abscissa and the shear force V at the base along the ordinate (product of factor O by the sum of the forces of stages proportional to the displacements of the first mode); it has a linear portion (' d 'e) and a portion of decreasing slope until the final displacement 'u which corresponds in general to reaching the final criterion of rotation in one of the plasticized sections. It is supposed to represent an intrinsic characteristic of the structure from the point of view of the effect of the horizontal actions irrespective of whether these actions are static or dynamic; considering the method used to establish this curve, the assumption on its intrinsic character will be as close to the reality of the seismic excitations as will be the importance of the fundamental mode and the regularity of the structure from the point of view of distribution of masses and stiffnesses. This curve is associated with an estimation of the rate of damping which would correspond to a hysteresis loop after reversal of the loading direction; this estimation obviously depends on the level of deformation at the end of the cycle.
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To verify the design, the capacity of the structure represented by this curve V-' has to be compared to the demand of the seismic excitation which is characterized by the elastic response spectrum after taking into account the increase in damping due to the effects of plasticization. This comparison is carried out in a diagram A-D (A acceleration along the ordinate, D displacement along the abscissa); in this diagram the comparison requires a transposition of the response spectrum on the one hand and the curve V-' on the other. The transposition of the spectrum is carried out simply by expressing the period T (or the frequency g as per the selected representation mode) based on the ratio A/D since according to the definition of the pseudo-acceleration (see [9.17]) we have: A D
Z2
4S 2 from which T T2
2S
D A
[17.89]
For example, for a given spectrum in the diagram A-T, like those in Figure 9.6, we have the following expressions of A as a function of T: 0 d T d T1 ; A
§ ·T [ A0 ¨ A1 0 A0 ¸ ¨ ¸T [ © ¹ 1
T1 d T d T2 ; A T2 d T ; A
A1
[17.90]
[0 [
A1
[17.91]
[O [
T2 T
[17.92]
where the rule proved in section 10.2.1 is used for a white noise excitation for which the spectral amplitude is inversely proportional to the square root of the rate of damping [ ([0 is the reference damping, in general 5%, for which the spectra are traced in the earthquake-resistant codes). By applying formula [17.89], equations [17.90] to [17.92] become: D T2 0d d 12 ;D A 4S
2
ª T1 § ·º [ « A A0 / ¨¨ A1 0 A0 ¸¸ » A [ «¬ 2S © ¹ »¼
T12 D T22 d d ;A 4S 2 A 4S 2
A1
[0 [
[17.93]
[17.94]
Overview of Nonlinear Calculations
T22 D d ; AD 4S 2 A
811
2
§ A1T2 · [ 0 ¨ 2S ¸ [ © ¹
[17.95]
It is seen that for the two parts of the spectrum which, in practice, are important (the plateau T1 d T d T2 and the descending branch T t T2 ), the spectrum has the same form in the A-D diagram where the straight lines that pass through the origins are those that correspond to a given value of the period; the plateau with constant acceleration is not modified in zone T1 d T d T2 and for T t T2 we also have equilateral hyperbolic branches since AD has a constant value. Figure 17.10 shows a spectrum (limited to these two parts) plotted in the A-D diagram with T1 = 0.3 s and T2 = 0.6 s.
Figure 17.10. Response spectrum in the acceleration-displacement diagram and the capacity spectrum of a structure
The same figure shows a curve marked as spectrum of capacity that corresponds to the transposition of the curve V-' (moving from forces to accelerations and relation between the displacement of the last floor and the center of gravity). The comparison between capacity and demand involves determining the intersection of the capacity spectrum by the response spectrum curve associated with a damping compatible with the level of deformation. The comparison is in general iterative starting from a displacement considered acceptable from the point of view of its consequences in terms of structural damage.
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If for the damping corresponding to this displacement there is an intersection of the two spectra for a displacement close to the one selected at the beginning, the design can be taken as satisfactory; if that is not the case, the value of the chosen displacement has to be changed to get this compatibility; if this cannot be obtained under sufficient safety conditions, the design has to be modified. With respect to the behavior coefficient method (where the stresses calculated on a reference elastic model are divided by this coefficient; see section 9.3.1) the pushover method has the following advantages: – assessment of the nonlinear behavior is much more realistic because it follows the plotting of the curve V-', specific to the structure studied, whereas the behavior coefficient is an imposed value taken from a table which can be a rough approximation valid for a given structure category at most; – the pattern of progressive damage until the final displacement 'u indicates the weak points of the structure better than the elastic deformation field resulting from the reference calculation, which, after being divided by the behavior coefficient, is used to determine the design stresses; – the need for compatibility between the displacement level on the capacity spectrum and the damping considered on the response spectrum curve is clearly highlighted; this forces the engineer to think; though this necessity also exists for the behavior coefficient method, those using earthquake engineering codes are often unaware of this (using a behavior coefficient higher than one is justified only if the stresses thus determined correspond to the non-elastic deformations in certain critical zones) and are not always motivated to verify either. These advantages are significant and the price to pay to obtain them (i.e., performing a nonlinear stress analysis under increasing load) will be even more within the reach of design offices, thanks to easily available software to do this type of calculation. However, the practical limitations of the push-over method in its present form are the same as those of the behavior coefficient; only relatively regular structures with little sensitivity to torsion, which are reasonably represented by their fundamental mode, can be treated by the above described procedure. For more complex structures certain steps involved in this procedure are either difficult to implement (for example, the pattern of increasing the thrust only in one direction if the fundamental mode has torsion components) or have disputable principles (such as the intrinsic character of the curve V-' or the comparison of capacity and demand using a simple intersection of curves in the A-D diagram). Several studies comparing the results obtained by this method with those of the nonlinear dynamic calculations have been published recently [KIM 99, LEE 99]. They show that the “real” damage pattern (i.e., corresponding to nonlinear
Overview of Nonlinear Calculations
813
calculations using accelerograms) can be somewhat different from that obtained by the push-over calculation, mainly for the top floors, even if the structure is regular. For example, [KIM 99] referred to the case of a six storey office building of regular metallic structure subjected to three powerful accelerograms (Newhall and Sylmar recorded during the Northridge earthquake of 17 January 1994 and recordings of the Kǀbe earthquake of 17 January 1995). Certain results of these calculations, and the push-over analysis, are given in Table 17.4 and in Figure 17.11. Push-over Maximum acceleration
Newhall
Sylmar
Kǀbe
0.60 g
0.84 g
0.83 g
Displacement ' (cm)
34.95
31.25
35.31
35.59
Acceleration V/m
0.22 g
0.23 g
0.24 g
0.24 g
TMax columns (rad)
0.016
0.014
0.021
0.019
TMax beams (rad)
0.014
0.018
0.014
0.016
Distortion floor 1 (%)
2.00
1.83
2.49
2.01
Distortion floor 2 (%)
1.85
1.63
1.88
1.59
Distortion floor 3 (%)
1.78
1.62
1.87
1.67
Distortion floor 4 (%)
1.34
2.14
1.34
1.76
Distortion floor 5 (%)
0.85
2.00
1.03
2.21
Distortion floor 6 (%)
0.39
0.77
0.75
1.44
Table 17.4. Comparison of the push-over results with those of the calculations using accelerograms for a regular building (according to [KIM 99]); V indicates the base sheer force and m the mass of the building
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Seismic Engineering
Figure 17.11. Comparison of the position of the plastic hinges in a push-over calculation and in three calculations using accelerograms for a regular building (according to [KIM 99])
This building constructed in 1977 following the UBC-73 code (for a maximum ground acceleration of 0.4 g) was thus tested digitally for excitation levels visibly higher than that retained for designing (the first row of Table 17.4 shows that the three accelerograms had a maximum acceleration varying from 0.60 to 0.84 g). There is a similarity between displacements ' at roof level (second row of the table) and shear force V at the base (transformed into equivalent acceleration by dividing by the total mass, third row). The push-over analysis has been pushed to reach an ultimate rotation of 0.020 radian corresponding to a displacement 'u of around 50 cm, clearly higher than the value of 35 cm which coincides with the values of the transient calculations. It can however be noticed (fourth row of the table) that this ultimate rotation is reached for certain columns using Sylmar accelerograms, and closely followed by that of Kǀbe. The comparison of the level distortions (i.e., the quotients of the relative ceilingfloor displacement by the height of the floor) brings out (the last six rows of the table) a good similarity push-over/accelerograms for the first three levels but also a clear tendency to underestimate the push-over results in the higher storeys. This observation can be visualized in Figure 17.11, which shows the position of the
Overview of Nonlinear Calculations
815
plastic hinges near the extremities of certain elements (beams and columns) in the different calculations; it can be seen that the last level of the building remains elastic in the push-over calculation, while it is strongly plasticized (mainly in the columns of the central lines) in the transient calculations. These observations show that the displacement criterion ' of the roof cannot translate completely the real state of structural damage but that it correctly identifies, at least for regular structures, the level of deformations in the lower levels of the building which are more critical with respect to the risk of total collapse. Regarding the application of the push-over method to the safety verification for this example, the result is positive for the Newhall and Sylmar accelerograms; but much more disappointing for that of Kǀbe, as is seen in Figure 17.12.
Figure 17.12. Intersections of the capacity spectra and the response spectra at 30% for the example in Figure 17.11 (according to [KIM 99])
For a displacement of 35 cm of the roof of a building the rate of estimated damping is 30%; the curves corresponding to the response spectra of the Newhall and Sylmar accelerograms break the capacity spectrum at points whose abscissa are close to this value of 35 cm (32 cm for Newhall, 33 cm for Sylmar) indicating thereupon a good similarity. On the other hand, for the Kǀbe accelerogram, the displacement associated with the intersection is only 16 cm. This difference with respect to the other two accelerograms seem to arise from the pulsed character of the signal of Kǀbe which can be attributed to a killer pulse type effect on this recording carried out close to the fault (see section 4.1.3). When the transient response of the
816
Seismic Engineering
building is analyzed at this signal, it is found that the structure goes through a high plastic excursion and never comes back to the origin [KIM 99]. The subsequent cycles of movement are carried out from the new origin and their surface in the force-displacement diagram (i.e. the energy dissipated) is clearly smaller than if the first sign inversion of the loading had enabled the initial plastic displacement to be regained; Figure 17.13 illustrates this behavior difference.
Figure 17.13. Hysteresis loops centered (left) and shifted (right) depending on whether the initial plastic excursion is caught up or not
If a push-over type method should consider such a phenomenon, the calculation rules of the equivalent damping corresponding to a given level of plastic deformation on the curve V-' should be made dependent on the type of seismic excitation and in particular, they should reduce the damping for the pulse type excitations. In the case of the Kǀbe accelerogram used for the above mentioned study, by referring to Figure 17.12, the damping retained for the response spectrum should move from 30% to about 10% so that the curve shifts sufficiently towards the right and produces an acceptable intersection (a displacement of around 30 cm) with the capacity spectrum. It is not difficult to frame the rules that produce such a decrease but, as of now, the earthquake-resistant codes have hardly accepted any differentiation in the seismological nature of excitation, if it is not for the level, the frequency content and possibly the duration (for the number of equivalent cycles referred to in the studies on liquefaction). The introduction of considerations of the type of excitation into the codes could lead to problems regarding training of designers and more so regarding the data accessibility that enables us to decide whether a particular aspect should or should not be taken into account. In addition to the question of the possible pulsed character of the load, the question of contribution of the surface waves, mainly Rayleigh waves, can be important in certain cases, as has been seen in section 15.1.2 for rotation excitation.
Overview of Nonlinear Calculations
817
Regarding the plastic deformations associated with a pulsed loading, simple formulae can be given in the case of an acceleration step of finite duration T whose effect has already been analyzed in section 17.2.1 for the tilting of the blocks. A simple oscillator (Figure 17.14) of elastic, perfectly plastic force-displacement law whose plateau corresponds to the value 2J of static acceleration and a pulsed acceleration constant DJ (D! 1) acting between time 0 and time T are considered.
Figure 17.14. Oscillator of elastic, perfectly plastic system and acceleration step of duration T
The differential equation of displacement u (t) is written as: u I u
*
[17.96]
I (u) being the reaction (n terms of acceleration) of the elasto-plastic spring that is defined by:
I u Z 2u
0 d u d ue
I u 2J
u t ue , u ! 0
I u 2J Z 2 u u0
u d u0
[17.97]
with * DJ for 0 d t d T , * 0 for t > T and the initial conditions u(0) = 0, u (0) = 0, the solution is basic but a little bit laborious given the number of cases to be observed (depending on whether the response is plastic or elastic and whether the plasticization starts or ends during or after the loading). Here we limit ourselves to give the results by introducing the notations:
E
ZT 2
; P
u0 ue
[17.98]
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Seismic Engineering
E is the product using S of the ratio between the duration T of the step and the period of the oscillator in its elastic phase, P is the ductility demanded, i.e., the ratio between the maximum plastic displacement and the displacement that corresponds to the yield point. Figure 17.15 gives the expressions of P based on D and E in the five possible cases.
Figure 17.15. Ductility demanded by an acceleration step
The zones of the figure corresponding to these five cases have the following characteristics: – zone (1) defined by: 0 d D d1; E d
S
[17.99]
2
1 d D ; E d Arc sin
1
D
Overview of Nonlinear Calculations
819
The duration of the step even for the higher values of D is too short to reach the plastic domain; ductility demand P is lower than one (since the response is elastic) and is expressed by the formula:
P D sin E
[17.100]
– zone (2) defined by: 0 d D d1; E t
S
[17.101]
2
The response remains elastic but contrary to case (1), reaches its maximum before the end of the step according to the formula:
P D
[17.102]
Equations [17.100] and [17.102] define the elastic response spectrum of the step for zero damping; – zone (3) defined by: 1 d D ; Arc sin
1
D
d E d Arc sin
1
D
[17.103]
The behavior becomes plastic only after the end of the step with a ductility demand:
P
1 1 D 2 sin 2 E 2
[17.104]
– zone (4) defined by: 1 d D d 2 ; E t Arc sin
1
D
D 1 2 D
[17.105]
The plasticization and the appearance of a periodic movement of unloadingreloading takes place before the end of the step with a ductility demand independent of its duration according to the formula:
P
1 2 D
[17.106]
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Seismic Engineering
– zone (5) defined by: 1 d D d 2 ; Arc sin
2 d D; Arc sin
1
D
1
D
d E d Arc sin
1
D
D 1 2 D
[17.107]
dE
The plasticization starts before the end of the step but its maximum is only reached after; the ductility demanded depends simultaneously on D and E; this is expressed as:
P
1D 1 ·ª D 2§ 1 ·º § D ¨ E Arc sin ¸ « D 1 ¨ E Arc sin ¸ » [17.108] 2 2 D ¹¬ D ¹¼ © ©
As a result of formulae [17.104], [17.106] and [17.108] the ductility demand can be considerable for a step duration of the order of the period of the oscillator, as can be seen in Figure 17.16.
Figure 17.16. Ductility demands for a step acceleration
The three curves plotted on the figure correspond to the durations of the step S· S· 1§ 1§ whose ratio to the period of the oscillator is 1 (E S), 2 ¨ E ¸ or 4 ¨ E ¸. 2¹ 4¹ © ©
Overview of Nonlinear Calculations
821
The spreading out between these curves is striking; for example, for D = 3, i.e., a level of excitation which exceeds the plastic level only by 50% (Figure 17.14), the ductility demand P moves from 2.7 for E S/4 to 7.5 for E S/2 and exceeds 20 for E S. The influence of the duration of the step is therefore quite high. In step loading, the energy communicated to the oscillator is always higher than or equal to the energy which would correspond to purely elastic behavior; the equality occurs only if the plasticization occurs after the end of the step; because in that case the energy incidental to the loading depends only on the elastic properties of the oscillator; this can be verified easily using expressions [17.100] and [17.104] of the ductility demand in the zones (1) and (3) of Figure 17.15. In fact if Pp represents the plastic demand in zone (3) and Pe the elastic demand that could be had by stretching the validity domain of zone (1), the following relation between the two demands is obtained:
Pp
1 1 Pe2 that is Pe 2
2P p 1
[17.109]
The situation described in Figure 9.18 for the energy equivalence between the elastic and the plastic behavior reappears. In the other cases where the plasticization starts before the end of the step, the energy corresponding to the plastic solution is higher than that of the prolonged elastic solution; for example, between zones (2) and (4), we have:
Pp
1 1 ! 1 Pe2 2 D 2
because 2 D 1 D 2
1 1 D 2 2
[17.110]
2 D D 1 is always smaller than 2 for D ! 0. 2
The previous observations on the effect of an acceleration step were mainly aimed at drawing attention to the highly damaging nature of accelerograms which have a high initial pulse causing a substantial plastic excursion of the structure in a direction which is not regained by the subsequent cycles of movement. The situation is therefore far from the conditions of application of the Newmark theorem, which uses hysteresis loops centered around the origin (see Figure 9.17 for “proving” the theorem in section 9.3.1), to arrive at the equality of plastic and elastic displacements and thereupon at a decrease in the plastic energy with respect to the elastic energy. The killer pulse phenomenon close to large faults constitutes a limited but significant analysis of past experience which clearly indicates the harmful nature of this phenomenon for fairly long period structures. It is therefore
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Seismic Engineering
very important that the conditions of appearance of this phenomenon be specified (limits on the fault size and the distance to it, depending on the type of fault movement). As indicated in sections 17.1.1 and 10.1.3, the choice of accelerograms is critical for nonlinear calculations particularly when these aim at simulating plastic behaviors and more so because the structure is irregular. Given the current state of knowledge available, it should be possible to find a way to introduce this type of calculation into earthquake-resistant codes with a sufficiently precise codification of its validity limits and its implementation conditions. The push-over method (where the required nonlinear calculation is static with a monotonous loading and therefore is relatively simple in its principle and input data) definitely constitutes progress compared to the behavior coefficient methods but needs to be improved, in particular from the point of view of its application to structures with a noticeable degree of irregularities in the distribution of mass and stiffness; this type of structure is not handled satisfactorily using the current codes and is not suitable, at least in its earlier versions, for the practical use of the push-over method. 17.2.5. Nonlinear shock absorbers for bridges
Using localized shock absorbers to reduce the response of civil engineering structures subjected to high level dynamic actions (wind, earthquakes, external impacts or explosions) has become common practice since 1990. These shock absorbers, available in a variety of types (adopting different principles of both mechanical and hydraulic design and covering a wide range of capacity both in force and stroke), produce a restoring force I x, x which depends on the relative displacement x between their extremities and on its own time derivative x (relative velocity). For those shock absorbers which equip large bridges, the force I x, x can be expressed using the following form: I x, x = kx +c S x x
D
[17.111]
where S is the sign function introduced in section 17.1.2 (S x = +1 if x > 0, S x = –1 if x < 0), k, c and D are constants; the exponent D is included between 0 and 1. For D = 1, the classic case of linear viscosity is retrieved whereas for D = 0 the already analyzed dry Coulomb friction is retrieved (see sections 17.1.2 and 17.2.3).
Overview of Nonlinear Calculations
823
With expression [17.111] of the recall force, the stochastic linearization can be handled using the method developed in section 17.1.2; for the stiffness k’ and the dashpot constant c’ of the equivalent linear oscillator the following expressions similar to [17.10] and [17.11] respectively are obtained: k'
³³ kx dp x dp x / ³³ x dp x dp x
c'
x x ³³ cxS
2
2
D
2
dp x dp x / ³³ x dp x dp x
[17.112] [17.113]
Equation [17.112] shows that k’ = k. Whereas considering [17.16] and [17.7] c’ becomes: c'
2
c
S V x
1D
f
³u
1D
eu
2
/2
du
[17.114]
0
The integral is expressed with the help of gamma function * and it enables us to write: c'
D § D· c 2 2 * ¨ 1 ¸ 1D 2 ¹ V x S ©
2
[17.115]
For D = 0 (solid friction) this formula is identical to [17.18], c being replaced by Pmg. Contrary to what has been done in section 17.1.2 for the linearization of slipping, here V x is not going to be determined based on a white noise excitation. In fact, since the shock absorber defined by equation [17.108] contains a term referring to elastic springs, it is logical to link V x to V x , square root of the relative displacement variance which is associated with the elastic response spectrum, i.e., to the routine mode of definition of the seismic excitation; this reference to the response spectrum could not be used for slipping for which the linearization accepts only one dashpot term. Equations [10.20] and [10.21] show that:
V x
ZV x
[17.116]
k / m if m is Z being the pulsation associated with stiffness k of the spring ( Z the mass); the following equation can therefore be rewritten as [17.115].
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Seismic Engineering
c'
c
ª 2ZV x º ¬ ¼
1D
f D
[17.117]
by introducing the notation: § D· * ¨1 ¸ S © 2¹
2
f D
[17.118]
In practice, this function f (D) can be considered equal to one (it decreases from 1.128 to 1 when D varies from 0 to 1). The relationship between Vx and the response spectrum introduces a peak factor, function of the targeted probability of not exceeding (see section 10.2.1) and the rate of damping obtained with the help of the shock absorber. Recourse to the peak factor can be avoided by using a partially deterministic approach in the stochastic linearization [KAH 00]; therefore, instead of [17.117] we have: c'
c
Z S d
1D
g (D )
[17.119]
where function g (D) is defined by: g D
§ D · §3 D · * ¨1 ¸ / * ¨ ¸ S © 2¹ ©2 2¹
2
[17.120]
and where Sd refers to the response spectrum in motion; ZSd is thus the pseudovelocity, i.e., the pseudo-acceleration S divided by Z. Function g (D) is, like f (D), quite close to one (it decreases from 1.273 to 1 when D varies from 0 to 1). By taking g(D) = 1 and by accepting that the spectral ordinates are inversely proportional to the square root of the rate of damping [ (see section 10.2.1) we can write: c
'
§Z c ¨¨ © Sa
[ [0
· ¸¸ ¹
I D
[17.121]
[0 being the reference damping (in general 5%) for which spectrum Sa in pseudoacceleration is given. By using relation c’ = 2[Zm, m being the mass the movement of which should be dampened (see [9.2]), the following expression is obtained for c:
Overview of Nonlinear Calculations
c
2m[
1D / 2
[0
1D / 2
1D
§S · Z¨ a ¸ ©Z¹
2mSa [
1D / 2 1D / 2 §
[0
Z· ¨ ¸ © Sa ¹
825
D
[17.122]
which enables us to design the nonlinear shock absorber when the desired rate of damping [ is fixed, i.e., the importance of response reduction. For example, [ = 20% (starting from [0 = 5%) should be attained to divide the response by 2, since it varies as 1/ [ . To test the validity of this approach, Kahan [KAH 00] carried out comparisons between the displacements expected by this approach and those calculated by the nonlinear time-dependent simulation. He used 25 synthetic accelerograms spectra whose mean reproduced the spectrum which conforms to the PS92 Rules [AFN 95] and studied their action on a set of 17 oscillators corresponding to law [17.111] with a mass of 5,600 tons (typical of the deck of a big bridge), elastic eigenperiods varying between 0.1 s and 4 s and two possible values (0.1 and 0.3) for the exponent D. The value of c was determined with a view to divide by 2 ([ = 20%) or by 3 ([ = 45%) the response for a period of 1 s (for which acceleration Sa of the spectrum prescribed in the PS92 Rules, at 5% damping is 2.25 m/s² when the nominal acceleration is 3 m/s²). The results are given in Figure 17.17 for S0 sites (rock; see Figure 9.6). The values of c indicated in the figure below each diagram have been determined without going through the simplifications which have led to formula [17.122] (i.e., the functions (D) and g(D) were not considered equal to one and that a natural damping of 5% which adds to the effects of the nonlinear shock absorber was taken into account); determining c requires an iterative procedure which can be implemented easily. Application of formula [17.122] with m = 5.6 u 106 kg, [0 = 0.05, [ = 0.20 or 0.45 based on the response reduction factor U is 1/2 or 1/3 , Sa = 2.25 m/s² and Z = 2Srd/s for a period of 1 s leads to values that are not very different: – c = 2.993 u 106 N (m/s) –0.1 (instead of 2.543 u 106) for D = 0.1, [ = 0.20; – c = 4.222 u 106 N (m/s) –0.3 (instead of 3.789 u 106) for D = 0.3, [ = 0.20; – c = 4.675 u 106 N (m/s) –0.1 (instead of 4.707 u 106) for D = 0.1, [ = 0.45; – c = 7.152 u 106 N (m/s) –0.3 (instead of 7.607 u 106) for D = 0.3, [ = 0.45.
Seismic Engineering
Displacement (m)
826
Figure 17.17. Comparison of the displacements through nonlinear time analysis (medium curve marked as n1 and indication of a standard deviation of this curve on both sides) and through stochastic linearization (curve marked sto for the purely stochastic version, [17.117], and curve marked det for the partially deterministic version, [17.119]). The curve marked sre is the response spectrum in motion for a 5% damping corresponding to the average of the 25 accelerograms used (according to [KAH 00])
A close analysis of the figure shows a reasonably good similarity between the nonlinear analysis and the linearization methods for periods less than 2–2.5 s; the deviation becomes significant for higher periods which can probably be due to the very low ratio between the duration of 10 s chosen for the synthetic accelerograms and the period of the oscillators; the number of cycles defined by this ratio should be sufficiently high so that certain assumptions of the stochastic linearization approach can be verified.
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827
17.2.6. Pipelines going through a fault
As indicated in section 11.1.1, there is no question of taking up the calculations for estimating the effects of fault movement on buildings and civil engineering structures because the presence of an active fault in the right-of-way or near the foundation constitutes a factor for disqualifying the site. The problem is different for pipelines of a fluid transport network (water, oil, gas) which might have to go through faults in seismic zones. The potentially damaging consequences of these crossings can be minimized by adopting correct structural details mainly in choosing the angle of intersection based on the type of movement of the fault; but it is also interesting to determine the order of magnitude of the movements likely to cause ruptures of pipelines depending on their characteristics (nature of the constituent materials, diameter and thickness, stiffness of the ground under which they are buried). This determination can be carried out through nonlinear stress analysis (the dynamic aspects are a priori negligible) taking into account the plastic deformation not only of the walls of the pipelines but also of the ground. The modeling of the effects of interaction at the ground-conduit interface (thrust and pullback, slipping with friction) is full of uncertainties which imply assessing the sensitivity of the results to the parameter variations characterizing these interactions. An example of such a parametric study is given in [AFP 00]; some of these results are given below. A simple formula to calculate the maximum deformation induced in a pipeline crossing a pure strike-slip fault perpendicularly can be obtained in the following manner (see Figure 17.18). It is assumed that the pipeline is completely plasticized in traction (stress equal to yield point Vy over the entire section) along a length L on both sides of the fault trace; in this state of total plasticization the pipeline does not offer any resistance to bending. The ground is also plasticized along the length 2L of the section studied; it counteracts the displacement of the pipeline with a force of resistance per unit of length equal to DVs, D being the diameter and Vs the shear stress limit of the ground.
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Seismic Engineering
Figure 17.18. Deformation of a pipeline due to the left-lateral movement of a strike-slip fault crossed perpendicularly at its profile
By taking the origin of the axes at the left extremity of the deformed segment, y(x) refers to the deflected shape of the pipeline when the compartment to the right of the fault drifts from quantity '; this deflected shape is anti-symmetric with respect to the plane of fault (x = L); we thus have: y (0) = 0; y(L) =
' ; y(2L) = ' 2
[17.123]
Mechanical energy W spent to produce the deflected shape y(x) consists of two terms: – the plastic deformation work of the pipeline whose volume density is equal to V y H, H being the strain deformation (deformation due to elongation); – work of the ground resistance force (DVs per unit of length) whose displacement from the point of application along the abscissa x is equal to y(x). Taking the thickness of the tube as e, the volume per unit of length is equal to SD e; considering only the half-segment 0 d x d L due to the anti-symmetry of the deflected shape, W becomes: W
³
L
0
L
V y HS Dedx ³ DV s ydx 0
[17.124]
Overview of Nonlinear Calculations
829
Strain deformation H is connected to derivative y’ (x) of the deflected shape through the relation:
H
1 y ' 2
1
2
1
[17.125]
Equation [17.124] is then written as: W
ª
1
º
SV y De « ³ 1 y '2 dx L » V s D ³ ydx ¬
L
2
0
¼
L
0
[17.126]
i.e., by introducing length R defined by: R
W
Se
Vy Vs
[17.127]
1 ª L º DV s « ³ ª« R 1 y '2 2 y º»dx RL » ¼ ¬ 0¬ ¼
[17.128]
To determine y(x), an energy minimizing principle W is going to be used (or principle of least force, a nice law whose applications in mechanics are numerous), which amounts to writing that y(x) should be the solution of Euler’s equation for the calculation of the variations [COU 53]: d § wF · wF ¨ ¸ dx © wy ' ¹ wy
[17.129]
0
where F (y, y’) is the functional which appears under the integral in [17.128], i.e.: F (y, y’) = R (1+y’²)
1
2
+y
[17.130]
With this expression of function F, relation [17.129] leads to the following differential equation for y(x): R
d ª 1 y ' 1 y ' ² 2 º 1 , ¼» dx ¬«
[17.131]
from which, by integrating once with y’ (0) = 0 (which amounts to ignoring the elastic deformations of the pipeline outside the plasticized segment):
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Seismic Engineering
y’ 1 y '²
1
2
x R
[17.132]
This equation is solved with respect to y’ to arrive at: y'
x§ x2 · ¨1 2 ¸ R© R ¹
1
2
[17.133]
and, by integrating with y (0) = 0 (first of equations [17.123]): 1 ª § 2 · 2º x y = R «1 ¨ 1 2 ¸ » « © R ¹ » ¬ ¼
[17.134]
which can be written as: x 2 y 2 2 Ry
0
[17.135]
The deflected shape is therefore an arc of a circle of radius R centered at point x = 0, y = R. For x = L, y should be equal to half of the displacement of fault ' (second of equations [17.123]); therefore, for half-length L of the plasticized segment, according to [17.135] we have: L
1 R' ' 2 4
[17.136]
This relation has meaning only if ' is less than 4R; this condition has always been verified in practice because according to [17.127] the smallest plausible value for R is about 3 m (obtained by taking e = 4 mm, Vy = 250 MPa and Vs = 1MPa, (i.e., an ordinary steel tube of minimum thickness sunk in a very rigid earth filling); the upper limit of 4R for fault displacement ' is thus equal to at least 10 m, which can possibly be attained only by the ground faults having completely exceptional characteristics (see Table 2.2 and Figure 2.11). Maximum strain deformation Hm is attained for x = L, i.e., on the fault trace; according to [17.125], [17.133] and [17.136] this can be expressed as: Hm =
' 2R '
[17.137]
Overview of Nonlinear Calculations
831
This relation shows 2R as the upper limit of ' instead of 4R according to [17.136]; this limit ' = 2R, for which Hm is infinite, corresponds to L = R, i.e., the deflected shape in the form of an arc of a circle has an angle at the center of 90° and becomes a tangent to the fault trace. As indicated above, the plausible values of R are such that this limit does not have any practical importance at least not more than that of 4R; which means, in the denominator of [17.137], ' can be neglected against 2R which, considering [17.127], gives rise to:
Hm
1 ' Vs 2S e V y
[17.138]
The parametric study mentioned above [AFP 00] has enabled the comparison of results obtained by this formula with those of nonlinear calculations with finite elements; eight types of pipelines (diameter D of 114 to 1,016 mm, thickness e of 3.2 to 18 mm, yield points Vy of 250 to 480 MPa) as well as different ground conditions (stiff clay, loose or dense sand with shear stress values Vs ranging from 0.2 to 1.4 MPa) were analyzed. Fault displacement ' varied from 0.1 to 1 m; the results of this comparison are given in Figure 17.19.
Figure 17.19. Comparison of the maximum deformations calculated using formula [17.138] to those obtained using finite elements models (FE symbol)
832
Seismic Engineering
As is clear from the figure, formula [17.138] has a tendency to underestimate the deformation (compared to the results obtained from finite element models that are supposed to reflect “reality”) when this is about 2 to 3%. Beyond this range, the tendency is less systematic and even the opposite (i.e., it becomes an overestimation) for minor deformations (Hm < 1%); this can probably be attributed to the fact that the elastic deformations have been neglected while establishing the formula. In terms of order of magnitude, equation [17.138] can be considered acceptable because the underestimation, whenever it is produced, always corresponds to a factor less than two. Therefore, with a safety coefficient 2, this formula can be applied to decide whether the risk of rupture is significant considering the amplitude estimated for the fault movement. This also confirms the appropriateness of the structural details indicated in section 11.1.1 regarding the increase of resistance (greater thickness and a steel grade of higher yield point) and the decrease of ground stiffness (loose materials filling) for the segments of pipelines which cross the fault.
Part 7
Seismic Prevention Tools
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Introduction
Resisting and reducing the risks of future earthquakes and their destructive forces is now commonly known as seismic prevention. It is obvious that this term excludes studies related to the prediction of earthquakes with a certain degree of precision based on their location (position of epicenters and affected zones), their magnitude and intensity and their date of occurrence that enable timely evacuation of the population. Short-term forecast, though a fundamental objective of seismology, is currently far from being operational and it is doubtful if it will come into being in the near future (section 7.2). In any case, even if it was possible to precisely predict future tremors, the need for seismic preventive measures to avoid economic disaster cannot be ignored. Long-term forecast, which aims “only” at identifying the most vulnerable zones on a scale of a few decades, forms part of the studies related to natural hazards and finds its place in the prevention arsenal, mainly in defining priorities for vulnerability reduction programs and investments in policies related to civil safety. Seismic prevention involves multiple aspects and calls for the involvement of competent people having varied experience. The above-mentioned aspects and the role of the intervening agents can be classified into the following three areas: – the technical field involves acquiring the required knowledge, defining the preventive measures to be adopted in constructions, finalizing the technical aspects of manufacture and certification (calculations and testing) and codifying (drafting standards and guidelines). The previous chapters dealt with the presentation and discussion of the main points concerning different themes which are either resolved or are still controversial. The next part of this book will therefore concentrate on summarizing the contents of these chapters and completing the information by covering those topics which were either not dealt with or were developed
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Seismic Engineering
superficially such as experimental methods, special earthquake-resistant devices or techniques to assess and strengthen the seismic resistance of existing constructions; – the political aspect deals with the regulatory systems (promulgation of legal documents defining the obligations of the owners of constructions with respect to the application of technical texts, norms and guidelines, and the verification of this application), the preparation (planning the preventive actions based on the risks and available resources, organizing disaster relief, having a strategy for crisis management) and public information; – the social aspect, i.e. the awareness amongst the population of the seismic risk, the importance given to such risks and the degree of confidence with respect to the decisions taken by the government to handle them. Of course people living in zones of intense seismic activity actually “live with earthquakes” and react differently to such situations compared to those living in calmer zones such as metropolitan France. In such zones, “seismic ignorance” is so great that reactions are outrageous varying from total denial of such risks (“earthquakes do not occur in our country and the earthquake prevention rules imposed on us are just the whims and fancies of the technocrats”) to almost becoming superstitious in accepting them (“nothing can be done against earthquakes and to construct potentially unsafe structures in earthquake prone zones is totally unacceptable”). The sociological aspects contribute a lot in accepting, on the one hand, the need for earthquake prevention measures and on the other hand, in realizing that such a need involves just respecting some of basic design and manufacturing principles in the majority of new constructions without in any way substantially increasing the cost of the project. It is symptomatic that the financial implication of the application of seismic prevention codes are, at least in France, traditionally called “supplementary cost for seismic prevention”; the use of such terminology creates a feeling in people’s minds that it is superfluous expenditure that can be avoided, says G. Czitrom [CZI 99]. In fact, this famous “supplementary cost” (just a small percentage of the cost of the structural work of the building) apart from being another item of the cost estimate (in urban areas the structural work cost is only 10 to 25% of the total construction cost) does not produce any real and measurable economic consequences considering the uncertainties related to other factors (such as land costs). On the contrary, the costs are very high if existing housing constructed without any protection against earthquakes needs to be modified to become reasonably resistant to earthquakes; this is why in France as elsewhere, the application of earthquake-resistant codes is required only for new constructions. This differential treatment creates problems in accepting the seismic prevention measures in countries with moderate seismic activity.
Introduction
837
Lack of seismic knowledge impedes the dissemination of the guidelines to be observed during and after an earthquake. The recent seismic events in France like that experienced in Epagny near Annecy (15 July 1996) shows that the event was not considered to be an earthquake [DEV 96] by most of those who witnessed it. This has lead to a variety of more or less rational reactions which can only increase the state of confusion during and after a major tremor. The aspects related to the state of preparation and the psychology of the people going through a high intensity earthquake are beyond the scope of this chapter even though their contribution in successfully implementing the seismic prevention policy is invaluable. Communicating their description and the actions to be taken to improve the situation comes under sociology; an overview of these problems is given in [DEV 99].
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Chapter 18
Technical Aspects of Prevention
18.1. Tools for learning 18.1.1. The analysis of past experience As indicated several times, prevention measures and earthquake-resistant codes in particular were and still are based on lessons learnt from the observation and analysis of seismic effects. This experimental nature of seismic engineering should be strongly highlighted; more so at a time when resorting to the computer and blindly following its predictions have become sacrosanct for certain policy makers. The analysis of past seismic experience refers to the summary of knowledge acquired on the ground and has always depended on the personal involvement of certain individuals. Currently it is organized in a systematic manner, at least with respect to information gathering in the form of post-seismic missions sent to the zones affected by major earthquakes. These missions are the responsibility of public bodies (ministries, research institutes), professional associations or learned bodies (like the AFPS in France). In addition to the contribution of these missions towards obtaining qualitative or quantitative data on the effects of earthquakes and towards spreading the information thus collected through public contacts, their role in training and enriching the experience of their participants should not be forgotten. This aspect is
840
Seismic Engineering
often misinterpreted, mainly by certain sponsors of these missions; but it is essential, writes N.N. Ambraseys [AMB 98], “the investigations carried out on the ground are the only way to objectively assess seismic reality and to bridge the gap between this reality and the image created by theoretical models”. The fact that post-seismic missions bring together specialists from different disciplines and give them an opportunity to discuss their points of view regarding a common observation is extremely useful in starting off a constructive dialogue between seismologists and engineers. These missions thus provide an unforgettable experience for those who have had the chance to participate in at least one of them. The analysis of past experience has had a great influence on the philosophy of the earlier earthquake-resistant codes which were drawn up at a time (1930 to 1970) when there existed very few recordings of high intensity quakes (see section 8.1); these codes essentially adopted an empirical approach, in the sense that the levels of the seismic coefficients were not directly linked to those of ground movements and their increasing severity resulted mainly from observations made during a real earthquake. Section 14.2 indicates that the influence of such an approach is still clearly noticeable in the Californian and Japanese codes. This does not mean that this approach is now obsolete and that earthquakeresistant codification results entirely from a Cartesian process in which it is the seismologist who defines the movement and the engineer, the rules for calculation and designing. In fact, the indisputable developments in the field of strong motion seismology have helped more in understanding the phenomenon than in reducing the uncertainties in the evaluation of seismic hazard. Earthquakes can no longer be characterized by a certain level of acceleration based on only one or two recordings. All earthquakes, for which a large number of accelerometric signals from epicentral zones (Northridge 1994, Kǀbe 1995, Chi-Chi 1999) are available, bring out the extreme variability in the seismic movements and the difficulty in explaining it with numerical models (Figure 4.7). Earthquakes of moderate intensity, such as can be expected in France, for example, those in Colfiorito 1997 (near Assise) or in Mount Parnès (near Athens), confirm this variability tendency (which is not just due to the influence of the site effects; section 4.3). The analysis of past seismological experience has shown, in addition to the large variability in movements, certain special phenomena in the vicinity of faults, such as the famous killer pulse (section 4.1.3), which were unknown until recently. It is therefore as necessary as the analysis of past experience on the behavior of the structures, a domain that interests mainly the engineers. In this domain, it is the
Technical Aspects of Prevention
841
observations made on the ground and not the statements made based on numerical calculations that have helped in bringing out the notion of structural details that play an important role in the safety of constructions as was seen in Part 5. Thus, the analysis of past experience, under its different aspects, appears to be the basis of earthquake-resistant codes. This is as true today (when various data on the recordings and powerful methods of calculation and trials are available) as it was previously (when an empirical approach was the only possibility). Gathering information on the analysis of past experience through post-seismic missions was presented at the beginning of this section. Using the data thus obtained poses practical problems as they are more qualitative than quantitative in nature. It is therefore difficult to use them to validate the numerical models. The amplitude of seismic movements that have produced certain effects in a construction is, in particular, rarely known with precision (lack of recording in the proximity), as are the characteristics of the structure in question (material properties, importance of the steel reinforcement of the concrete). The fact that most of the post-seismic missions do not have a follow-up (after the publication of the mission report) is due to the low usability nature of the data and also due to the common lack of effort made to make the most of the collected information; additional investigations, definition and execution of a study program to clarify the behavior of a particular structure represent significant investment involving financial participation which is difficult to obtain, particularly from companies that have “leased” the services of one of their employees for the mission. The practical impacts of the analysis of past seismic experience are sometimes questioned due to their qualitative nature. The American nuclear industry has made a significant effort to formalize the analysis of past experience so as to give it the potential to certify certain widespread equipment (mechanical or electrical); if this equipment has successfully stood the test of a major earthquake (for which the ground movements could be estimated with reasonable accuracy) then similar materials could be considered fit for an earthquake as long as this does not exceed, with a safety margin, the level of the reference earthquakes. This method of rating using the analysis of past experience can be applied to equipment found in many industrial set-ups. This includes common mechanical equipment such as pumps, motors, tanks, traveling cranes of medium size and capacity, electrical equipment such as batteries, transformers, control cabinets, etc. The method reaches its limits when the equipment, though standard is of an exceptional size, or is specific to certain industries that are rarely found in the database prepared based on past earthquakes. Its application to the equipment of the American nuclear power plants was carried out during a test campaign on shaking tables of certain specimens and shall be briefly presented in section 18.1.2.
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Seismic Engineering
Systematic gathering of information on the seismic behavior of the industrial equipment started only after 1980. It is therefore more recent than the information collected from the observations relating to the civil engineering constructions and buildings. It is also more difficult as it involves getting permission from the owners of the industries to enter the sites and gather information; some of the owners hesitate to receive post-seismic missions especially if there has been damage. Table 18.1 gives an idea of the scope of the analysis of past experience on equipment based on the lessons from 25 earthquakes [LAB 00]. The acceleration values indicated within brackets in this table are not from the actual recordings but are only estimated values based on the available factors (level of damage, extrapolation of the values recorded at a certain distance from the site under study). Taking into account what has been said above regarding the variability of seismic movements, the accelerations in the table are only plausible orders of magnitude even in cases where they correspond to recordings (values without brackets), because the seismographs were seldom placed exactly at the point where the observations on the behavior was made. This uncertainty over the excitation has been observed earlier as one of the main practical difficulties in using the analysis of past experience. The last ten columns of the table indicate, using a symbol (X or O depending on whether the observations indicate significant damage or not), a rough estimate of the behavior of the installations in their totality or of certain specific equipment. The objective of this schematic representation is just to give an idea of the importance of the available database on the industrial installations that have suffered high intensity earthquakes. Certain columns such as those corresponding to tanks, electrical stations or transformers highlight a visible vulnerability, but for practical conclusions it is preferable to refer to the detailed descriptions given in the reports of the post-seismic missions.
6.1 6.0 6.9 7.1 7.8 7.5 6.8 7.0 7.4 7.8 8.1 6.7 7.0 7.9
1987 1987 1988 1989 1990 1991 1992 1992 1992 1993 1993 1994 1995 1995
California
California
Armenia
California
Philippines
Costa – Rica
Turkey
California
California
Japan
Mariana islands
California
Japan
Mexico
Whittier Narrows
Superstition Hills
Spitak
Loma Prieta
Luzon
Valle de la Estrella
Erzincan
Cape Mendocino
Landers
Hokkaido – Nansei-Oki
Guam
Northridge
Hyogo – ken – Nanbu
Manzanillo
X O X O
X X X
X X O
O X X
O O X
0.4
O
X
X
X
X
X
X X
0.3 – 0.8
X
X
0.4 – 1.8
X
X O X
(0.25)
X
0.5 – 0.8
O
X
0.4 – 1.2
X
X
0.6 – 1.5
X
X
0.5
X
X X X
X
0.2 – (0.6)
X
0.2 – 0.4
X
(0.2) – (0.4)
X X X
0.2 – (0.6)
X
0.3
X
0.3 – 0.45
X
O
0.28 – (0.9)
O O O
0.4 – 0.6
Thermal power plants
6.3
5.4
O O
(0.4)
Hydraulic power plants
0.50 – 0.97
X
1987
6.0
1986
California
Chalfont Valley
Diesel generators
0.23
X X X X O X
1986
5.9
1986
California
North Palm Springs
Tanks
0.18
X
El Salvador
7.7
1986
Alaska
Adak
Chemical factories
0.15 – 0.3
Piping
New Zealand
5.0
1986
Ohio
Leroy
X
Edgecombe
8.1
1985
Mexico
Michoacan-Guerrero
Underground conduits X
X
0.3 – 0.6
X
Electric control stations
0.3 – (0.4)
Transformers
0.3 – 0.6
(0.2)
(g)
ACCELERATIONS
Electrical cabinets
San Salvador
7.18
1985
Chile
Valparaiso
6.2
1984
California
6.7
Morgan Hill
7.4
M
1983
California
DATE
1978
Japan
Miyagi-Ken-Oki
REGION
Coalinga
EARTHQUAKE
Technical Aspects of Prevention 843
X X X
X X X
O X
X X O
O X O
X
Table 18.1. Summary of the analysis of past experience on equipment based on 25 earthquakes [LAB 00]. The presence or the absence of damage is indicated by the symbols X and O respectively. The accelerations within brackets are estimated values
844
Seismic Engineering
The case of tanks, which has already been dealt with at the end of section 12.2.3 (Figures 12.45, 12.46 and 12.47 and their notes), brings out a clear difference between: – tanks under pressure, generally of small size, which behave excellently even during violent earthquakes provided their anchorage and supports are properly designed and manufactured; – and thin-walled tanks, where the free surface of the liquids is at atmospheric pressure and which go through a large variety of disorders (buckling, deformation at the base of the supports, lifting of pegs, damage to floating roofs) sometimes leading to total destruction. Similarly for the piping, the corresponding column in the table which shows a moderate report (as many X as O), does not illustrate the fact that the majority of the piping networks are not affected by seismic actions even though they have been designed and laid without any special earthquake-resistant precautions (this is true for most of the installations visited in order to establish a database). As indicated at the end of section 12.3.2, the rare cases of seismic damage to piping were due to bad installation conditions (lack of anchorages, connection to heavy equipment susceptible to moving during earthquakes, usage of motorized valves whose masses and sizes are disproportionate with respect to those of the pipings). If used discretely and with complete knowledge of the details of the post-seismic observations, the analysis of past experience on equipment is extremely useful in assessing the real vulnerability of the industrial installations which otherwise cannot be studied applying a purely “computational” approach. We shall come back to the topic of piping in the following section. 18.1.2. Test methods A great variety of experimental techniques is now available to simulate the effects of earthquakes or to determine certain characteristics of the dynamic response. The most direct simulation method is by using shaking tables which are rigid plates whose movements are controlled by a set of jacks driven by a computer. Thus, any type of dynamic excitation (sine curves, accelerograms of real earthquakes) can be reproduced to one or more degrees of freedom (up to a maximum of 6 corresponding to 3 translations and 3 rotations of a solid body), depending on the number of jacks and their effects on test bodies fixed on the plate. With a square or rectangular plate and a set of 8 jacks (4 vertical at the vertices of the plate and 2
Technical Aspects of Prevention
845
horizontal on two adjoining sides), as represented in Figure 18.1, all possible movements are obtained by actuating: – in a synchronous manner on the two horizontal jacks on one side for translations along X and Y; – in a synchronous manner on the four vertical jacks for translation along Z; – in an asynchronous manner on the four horizontal jacks for rotation around Z; – in an asynchronous manner on the four vertical jacks grouped in pairs on the opposite sides for rotations around X and Y. There are very few three axis shaking tables as indicated in this figure and they are rarely used in practice with a large number of degrees of freedom. This restriction is due to the following two reasons: – when the mass of the test body is large, the large number of jacks having different drives and the coupling between the axes through moments (tilting moments for X-Z and Y-Z couplings and torsion moment for the X-Y coupling) complicate the servo-control command to obtain a given movement considerably; – the multiaxial trials are more difficult to carry out and require in general a study of the influence of the choice of accelerograms for the different excitation components. This makes them less suitable for research and development activities. The trials on shaking tables are often carried out in monoaxial, horizontal excitation or in biaxial, horizontal-vertical excitation. The monoaxial type corresponds more to trials of investigative nature with more emphasis on easy interpretation and repeatability whereas the biaxial types correspond to the qualification tests of the given equipment. This explains why most of the shaking tables do not have triaxial capacity and are limited to monoaxial (for which one jack suffices) or to biaxial (with three jacks, one horizontal and two vertical, if a rotational excitation along the horizontal axis is required).
846
Seismic Engineering
Figure 18.1. Triaxial shaking table with a load bearing capacity of 100 tons at Takasago Technical Institute (Mitsubishi) (as per [LIV 86])
Technical Aspects of Prevention
847
The shaking tables have some limitations which have important repercussions on the validity of the tests that they enable us to carry out. The first limitation is the size. Due to technical and economic reasons the size cannot exceed certain values for a given plate size and a loaded mass (that is the mass of the test body). Currently, the biggest shaking table belongs to NUPEC (Nuclear Power Engineering Test Center) at Tadotsu in Japan, which was commissioned in 1982 for the Japanese nuclear industry; it has a 15 m square table with a loaded mass of 1,000 tons. The sizes of tables of all other plants are far behind; the biggest shaking table in Europe AZALEE of the TAMARIS laboratory of CEA at Saclay, has a 6 m2 plate and a bearing capacity of 100 tons; the triaxial table in Figure 18.1 has similar characteristics. More commonly used tables (in most of the earthquake prone countries), which are sufficient for testing most equipment have plates of around 10m2 area and can carry 10 to 20 tons. Finally, in certain laboratories we can find “microtables” of 1 or 2 m2 area and a loaded mass of a few hundred kilograms. As a result of this size limitation the weight of the equipment which can be tried on one to one scale should not exceed a few tons in order to be accepted by most testing facilities. This limit is suitable for the majority of electrical equipment (mainly control cabinets) and the mechanical components (valves and taps, medium sized motors and pumps, shock absorbing devices) but is grossly insufficient for large equipment and civil engineering buildings and structures however small they may be. Tests on shaking tables are carried out more often on scaled down models with the exception of certification programs for sufficiently light equipments. Accepting a reduced scale implies respecting similarity conditions and it poses scale effect problems (that is, effects that we cannot correctly reproduce if the scale is too small). For example, in the case of reinforced concrete structures, it is generally accepted that the scale effects for representing the behavior of the frames, fix a lower limit of the order of 1/3-1/4 to the scale of the models; at still smaller scales the diameter of the reinforcing bars and the size of the aggregate become too small so that interactions between the reinforcement and the concrete correspond to the same mechanisms as in one to one scale. Several similarity rules can be used to define the characteristics of the models; the most common corresponds to the conservation of velocities; if the scale ratio is 1/O (that is, the dimensions of the models are O times smaller than those of the real object), this rule implies that the time should be reduced by the same O ratio.
848
Seismic Engineering
The accelerations in the case of the models are therefore O times higher than in reality. This means that the ratio of the forces of inertia (masses by 1/O3) to the section of the structural elements (by 1/O²), i.e. the mechanical stress due to the seismic action, is retained in this similarity. This invariance of the seismic stresses ensures an easy transposition of the results obtained with the models to the real structure, notably with respect to nonlinear behavior, except when these are highly influenced by the stresses due to dead load. In effect, the gravity acceleration is not obviously multiplied by O, as the seismic accelerations are. As a result of this the static stresses of the model are O times smaller. This representation defect can have important consequences, for example, in assessing the behavior of the columns, where the normal stresses due mainly to the weight, play an important role (section 17.3.1). This defect can be rectified in different ways, for example, by introducing additional masses to the gravity action or through prestressed cables. A more radical solution, not without other disadvantages, is to artificially increase the apparent gravity by placing the model in a centrifuge; it will be seen later that this technique is commonly used in tests on soil behavior and foundations. A second type of limitation corresponds to the characteristics of the table itself and the jacks which control its movements. The performances of the shaking tables, in terms of maximum possible displacement, speed and acceleration are limited respectively by the stroke, the oil flow rate and the dynamic force of the jacks [LIV 86]. The frequency range, i.e. the range of usable frequencies, is conditioned by the rigidity of the plate and by certain resonance phenomena which can appear in the driving system (servo valve, jacks); the more we increase the stroke of the jacks to be able to reproduce movements having a high value of displacement, the more we reduce the frequency range because the resonance frequency of the oil column diminishes with respect to its length. It is therefore necessary to look for compromises between contradictory demands. This leads to strokes generally of about r 100 to 200 mm, which are not sufficient to reproduce certain seismic movements rich in low frequencies. Table 18.2 gives some performance characteristics of the NUPEC table by Tadotsu and the AZALEE table by the CEA, which were mentioned above.
Technical Aspects of Prevention LuL (m)
M (103kg)
NUPEC (Tadotsu)
15 u 15
1000
AZALEE (Saclay)
6u6
100
TABLE
F (kN)
D (mm)
V (m/s)
A (g)
X = 3u104
X = r 200
X = 0.75
X = 1.84
Z = 3.3u104
Z = r 100
Z = 0.375
Z = 0.92
X = 1.2u103
X = r 125
X = 0.70
X = 1.00
Y = 1.2u103
Y = r 125
Y = 0.70
Y = 1.00
849
'g (Hz) 0-30
0-100
Table 18.2. Performances of the NUPEC and AZALEE shaking tables. L side of the table; M maximum loaded mass; F force of the jacks; D displacement; V velocity; A acceleration with the maximum loaded mass; 'g frequency range; X, Y horizontal excitations, Z vertical excitation ([COL 83] and [COR 90b])
Apart from the difference in size of the two installations, the order of magnitude of the parameters is the same in both cases. The limit values in acceleration may appear “comfortable”, but it should not be forgotten that for scaled down trials conducted on the basis of the similarity rules described earlier, the acceleration to be achieved with the models should be more than in real cases. For building models with a scale of about 1/3 (refer to the previous considerations on the limit due to the scale effect in the reinforced concrete structures), the acceleration limit 1 g on the table corresponds to a real acceleration of about 0.33 g which can be insufficient to reach significant levels of damage. A third type of limitation results from the installation conditions and from the environment of the materials or models tested on the table. The table is a rigid metallic structure (in order to obtain a reasonably large frequency range) on which the test bodies are either just placed or fixed with bolts; it is therefore not possible to reproduce the support conditions of structures built on relatively deformable ground and the soil-structure interaction phenomena. Interposition of a layer of soil between the model and table can be foreseen but due to the reduction of gravitational stresses validity problems arise. For testing mechanical equipment it can also prove difficult to correctly simulate the operating conditions that exist in reality (pressure and temperature, link with adjacent equipment) on a table. In spite of these different limitations, tests on shaking tables contribute much in enhancing knowledge about earthquake-resistant engineering and in assessing the safety of a large amount of equipment, components and structural elements. It is noteworthy that such data cannot be obtained by other means. This aspect is especially important for equipment that should be functional during and after an earthquake. In fact, it is not possible to verify the continuity of this capacity simply by using calculations (which are mainly suitable for verifying mechanical resistance) as good
850
Seismic Engineering
functioning implies movement of mobile parts (opening or closing of valves, operation of rotating machines) or non-interruption of electrical circuits (without the appearance of electrical noise such as micro-ruptures or untimely triggering of alarms). It is thus obvious that trials using shaking tables decide the earthquakeresistant nature of safety equipment of the industrial installations. These certification procedures which were introduced by the nuclear industry in the 1970s are defined in standard documents specifying the representation of seismic excitation (monoaxial or multiaxial, using synthetic accelerograms or modulated sinusoids), the mounting conditions of the specimens on the table and the instrumentation to be installed. The equipment concerned is mainly electrical command-control equipment and the mechanical components of valves and taps; their relatively weak masses (a few hundred kilograms or a few tons at most) allow the use of medium capacity shaking tables. Figure 18.2 shows two examples of these certifications.
Figure 18.2. Seismic certification of materials carried out on a 10 ton capacity biaxial table. On the left, testing an electrical cabinet; on the right, testing a neutron flux chamber [COL 84b]
As indicated in section 18.1.1, the trial programs on shaking tables along with the analysis of past seismic experiences are used to study the behavior of certain equipment and specify the verification criteria to be used for designing. For example, the case of piping in the American nuclear industry whose calculation
Technical Aspects of Prevention
851
rules favored stresses due to inertia whereas the analysis of past experience shows without ambiguity that differential displacements are much more important and responsible for almost all the damage observed, which incidentally is rare. The test program conducted by the EPRI (Electric Power Research Institute) concerned the piping components (mainly the elbows) and two complete lines [COL 91b]; the main conclusions drawn from the experimental results are as follows: – the components were destroyed completely only after several earthquake simulations of amplitude at least equal to ten times the level accepted by the design criteria; – for the components whose design is controlled by pressure, the destruction pattern, which was not anticipated by the numerical models, corresponds to the ratchet-stress phenomenon (association of a low cycle fatigue effect and a plastic deformation build-up); – for the components without pressure there is no destruction pattern in the real sense (sealing loss), but functionality losses due to narrowing of the passage for the fluid (bending of elbows) are observed (generally for the excitation levels much higher than that of the criteria).
Figure 18.3. Principle of the pseudo-dynamic method of seismic simulation
852
Seismic Engineering
These conclusions bring out the importance and the invaluable nature of the tests necessary for a realistic assessment of the safety aspect which is often carried out based on a so-called “computational truth” which is totally unsuitable for seismic situations as it is highly influenced by the way current load cases are treated. Using shaking tables in this study results more in the second role (researchdevelopment activities) of this method and these trials than the certification procedures mentioned earlier. In the field of research, the shaking tables have contributed much towards finalizing numerical models (see section 17.1.1 regarding international benchmark on the CAMUS program, [COL 00c]) and validating special earthquake-resistant devices (section 18.3). The simulation of seismic effects can also be carried out by the pseudo-dynamic method using a reaction wall. Introduced by Tanaka in 1975, its principle consists of combining a mathematical model and a series of static trials in which the forces exerted by the jacks represent the forces of inertia induced by a seismic excitation and are determined at every time step using the numerical model (Figure 18.3).
Figure 18.4. Reaction wall of BRI (Building Research Institute) at Tsukuba (Japan); according to [COR 86]
Technical Aspects of Prevention
853
This schematic diagram of the principle shows that the method uses a displacement approach, the structural deformation in time t + 't being derived from its calculation at time t and from the forces in action; the new deflected shape is imposed on the test structure through jacks pressed against the reaction wall. The new status of the reaction forces is measured and this helps in restarting the calculations for the following time increment and so on. Figure 18.4 shows the installation at the Building Research Institute at Tsukuba (Japan). This is the biggest installation currently used in the world. In Europe, the biggest installation of this type is found in the ELSA (European Laboratory for Structural Assessment) laboratory of the EC Joint Research Center at Ispra (Italy). Table 18.3 shows its characteristics as compared to those of the one in Tsukuba. BRI
ELSA
(Tsukuba)
(Ispra)
Height of the reaction wall (m)
25
16
Maximum shear force (MN)
40
20
720
200
8
8
1 000
500
Maximum bending moment (MN.m) Number of jacks Maximum force of a jack (kN) Stroke of the jacks (m) and number (within brackets)
r 0.5 (6)
r 0.25 (6)
r 1.0 (2)
r 0.5 (2)
Table 18.3. Comparison of the characteristics of the BRI (Building Research Institute, Tsukuba, Japan) and ELSA (European Laboratory for Structural Assessment, Ispra, Italy) reaction walls
These installations help in testing the multi-storey buildings at full scale (up to 7–8 for the BRI wall and 4–5 for the ELSA wall) by subjecting them to horizontal forces of several MN which are of the same order of magnitude as their weight. Information regarding the post-elastic behavior of the buildings can thus be obtained. These are otherwise difficult to obtain through shaking tables (except
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Seismic Engineering
possibly at Tadotsu). The fact that the tests are a series of static conditions facilitates considerably the investigations on the progression of damages (for example, the starting and progress of the cracks in the reinforced concrete elements). As against these advantages, the pseudo-dynamic method presents a certain number of disadvantages with respect to the tests on a shaking table: – in the test, since the influence of time on the rate of deformation and material properties together with the visco-elastic mechanisms of damping is not taken into account, it has to be simulated in the calculation for jack driving; – a test corresponding to the complete procedure of an accelerogram can last for several days due to the smallness of the time step 't (to avoid numerical instability in the case of an explicit scheme for the calculation of displacements) on the one hand and due to the time necessary for carrying out detailed investigations at every time step on the other; such a long duration can introduce the phenomenon of deferred parasitic deformation (stress relaxations), mainly when the damage reaches notable levels. – the principle itself of the pseudo-static test supposes that the forces of inertia can be simulated by using a relatively low number of jacks acting in the same direction; this hypothesis is suitable only for regular structures whose fundamental mode is dominant in the elastic response; the study of irregular structures and the consideration of a bi or three-directional excitation would pose problems which could be very difficult to solve in practice; these limits are the same as those mentioned in connection with the push-over method (section 17.2.4). Thus, the reaction wall method appears to be complementary to that of the shaking tables; the preferred field of application of the pseudo-dynamic method is that of regular buildings tested at full-scale; this method is essentially a research tool to formalize the earthquake-resistant codes (the possibilities offered by the ELSA laboratory have thus been used for a part of the pre-norm research work associated with Eurocode 8). However, when it comes to aspects such as the seismic certification of materials or investigation of the multiaxial effects in excitation and response, the shaking tables appear to be the only possible approach. In addition to shaking tables and reaction walls the simulation of seismic effects can depend on simpler and easier experimental methods; they are mainly: – in-structure shakers, which enable imposing frequency and amplitude controlled dynamic structural loading on structures (generally sinusoidal); with these instruments it is possible to determine in situ the eigenfrequencies and modes of existing buildings and structures (with weak loading levels compared to those corresponding to the actual seismic actions); using these vibrators it is also possible to produce, in certain types of equipment, stresses comparable to those of the
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earthquakes; this possibility is interesting for “thread-like” equipment (piping, control rods and combustible elements of nuclear reactors) which are not well suited to tests on shaking tables due to their length and differential displacements between their support points (the TAMARIS laboratory of the CEA is equipped with a twelve meter deep pit enabling vibratory tests for elongated structures with the help of vibrators fixed on its walls); – static presses which enable subjecting structural elements (beams, columns, walls) to applied forces or deformations; thus by varying the level and loading signs it is possible to establish behavior laws of these elements under cyclic loading alternating from negative to positive loads; static working is not a disadvantage because in seismic reality the forces of inertia do not correspond to the own masses of these elements but to the masses that they support; the main difficulty is in controlling the forces transmitted ultimately to the element under study, taking into account its fixing conditions and the deformability of the frame of the press. Most of our knowledge on nonlinear behavior of the reinforced concrete elements was obtained from these alternating static tests. A highly sophisticated test method with a specific application field is represented by the centrifuge [LUO 86]. While presenting conditions of similitude for the tests on shaking tables the lack of validity which affect the gravity stresses (reduction of these stresses in the scale ratio) and the corrective measures commonly taken (additional masses, prestressed cables) were mentioned; these measures which reestablish the static stress through localized loads are not sufficient to simulate properly the behavior of the materials, such as soils, where the influence of gravity is strikingly obvious for the totality of the stress tensor. The centrifuge enables us to create an artificial gravity and thereby to respect totally the similitude conditions for the soil behavior. The compensation is that in practice we have to work at extremely reduced scales, of about 1/100, given the limited size of the nacelles that can be fixed at the end of the rotating arm of the centrifuge. The seismic movement simulation is carried out using small explosive charges whose detonation sequence is programmed to reproduce the characteristics of the seismic signal on a required scale (i.e. at very high frequencies of about a few hundred Hz). It is therefore an extremely delicate testing technique calling for the intervention of specialists. Interesting results were obtained from the study of the soil-structure interaction of rigid buildings [LUO 86] or massive foundations (oil platforms).
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As indicated in section 1.3.2, using explosives for simulating seismic movements was started mainly in New Mexico and in Tajikistan for experiments on building models. This type of simulation is not very popular nowadays given the practical difficulties in obtaining a good simulation and a good repeatability from one test to the other. In certain cases, the in situ experiment depends on the effects of natural earthquakes; this type of test consists of instrumenting a terrain densely (in the surface and in depth) with working models built over it and “waiting” for earthquakes. It is thus a test program which can stretch over several years even in active earthquake zones and is generally carried with international cooperation. On the eastern coast of Taiwan, the sites of Lotung (where the research objectives were mainly seismological) and Hualien (which has a model one-quarter of the scale of a nuclear power plant) serve as good examples. 18.1.3. Calculation methods Calculation methods, i.e., seismic analysis software is not traditionally considered to be a learning tool, which is the subject of this section. This attitude is beginning to change due to the following reasons: 1) experimental studies are now systematically associated with the numerical models either because the execution of the tests depends on the results obtained by these models (as in the pseudo-dynamic models described in section 18.1.2), or because the object of the tests is to provide data which enable us to compare or validate the numerical simulations, mainly in order to assess whether any calculation method is ready to be codified in the earthquake-resistant rules. The international benchmark CAMUS [COL 00c] already mentioned in 18.1.2 has referred to the test results, using shaking tables, of a model of a regular six storey building; it has helped in comparing the capacities of different nonlinear models to reproduce these experimental results from the data characterizing the tested structure and the applied excitation; Table 18.4 regroups the results obtained by 10 teams from 8 different countries. The table shows that while the answers in terms of global efforts and displacement at the top are correctly predicted by certain teams (mainly 4 and 5), the deformation of the floors at the upper levels are largely underestimated; this observation shows that it is premature to introduce and formalize the utilization of nonlinear dynamic models in earthquake-resistant codes even for regular structures. This point has already been mentioned in sections 17.1.1 and 17.2.4:
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TEST
1
2
3
4
5
6
7
8
9
10
'
4.33
2.28
1.72
1.32 3.37
4.73
3.18
2.96
3.61
5.60
2.59
1.33
M
345
353
581
225 322
358
299
413
398
169
1 297
352
V
111
126
186
95 131
133
117
159
235
201
473
121
N
300
157
163
293
250
267
734
390
H4
10.4
0.62
5.38
0.13
0.06
0.17
H3
> 25
1.08
7.26
5.30
4.21
2.18
2.63
H2
2.64
1.59
2.98
1.77
2.27
2.93
2.03
H1
2.85
2.42
2.05
1.40
2.58
2.54
2.17
5.30
7.90
Table 18.4. Comparison of the results of 10 nonlinear models (columns numbered 1 to 10) of an experimental test program CAMUS (column TEST). The following are the comparison parameters (1st column): ' displacement on top end (cm), M maximum bending moment (kN.m), V maximum shear force at the base (kN), N maximum normal force in compression at the base (kN) including the self weight effect, H1, H2, H3, H4 maximum deformation (‰) at the levels 1, 2, 3 and 4 [COL 00c]
2) numerical models are commonly used as experimental models to obtain data on the influence of certain parameter variations. Examples of this approach which obviously requires a preliminary validation (generally obtained through comparison with some test results) have been given several times in the previous chapters; they are: – certain attenuation laws, such as that of Hwang-Huo [HWA 97] (see section 4.2.2), which have been established from a database consisting of some real and a certain number of “simulated” recordings obtained with a numerical modeling of the source and of the propagation of seismic waves; – studies on the response from oscillators of elastoplastic law, like that by Radicchia, Mezzi, and Ambrisi [RAD 92] which was introduced in section 10.1.3 and which aimed at specifying the influence of the choice of accelerogram;
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– studies on the response from other simple nonlinear systems such as blocks which are likely to tilt (Figure 17.3) or slide (Figure 17.8) have enabled us to bring out easy criteria for use in safety applications (refer to the Ishiyama criterion, [17.49], [17.50], for the stability of the blocks laid). 3) calculations can be the only practically usable approach to study certain phenomena whose onset conditions are very difficult to reproduce in the tests; this is the case with the basemat uplifts (see section 17.2.2). Using calculations to complete or even replace the experimental data in the behavior related studies capable of defining the safety criteria, implies thoroughness in the establishment of models and in initiating the validation procedures. The drifts which can result from the “computational terrorism” and the lack of critical approach have been mentioned several times in the previous chapters. Those who have tried honestly (i.e. without adjusting certain parameters for “convenience”) to reproduce test results through calculation or observations obtained from analysis or from past experience, know that it is difficult and that no one has ever done it at the first attempt. In particular the design calculations recommended in the earthquakeresistant codes or in accordance with the commonly accepted “good practice” are often only a method to obtain an acceptable level of safety without real capacity to predict the real destruction patterns (see, for example, the observations developed in sections 12.1.1, 17.2.1 and 17.2.2 on the overturn stability or those of section 18.1.2 on piping); these design calculations are not suitable for the seismic diagnosis of an existing structure (see section 18.5). 18.2. Earthquake engineering codes for normal risks The previous chapters presented in a detailed manner the main points of earthquake-resistant codes, i.e. zoning (see sections 7.2.1 and 7.2.2) and different design coefficients (general seismic coefficient – see Chapter 8 – and behavior coefficient; section 9.3.1). This section will be limited to certain observations on the technical philosophy of the codes. This aspect is probably not sufficiently explained in their write up and can lead to different interpretations. 18.2.1. Area of application and technical objectives of the codes In all earthquake prone countries a clear distinction between treatment of common buildings and structures and that of potentially dangerous installations spread over a population cluster or even a region is observed. According to French regulations (see the act of 14 May 1991 [COL 91c]), buildings, equipment and installations fall under “normal risk” category (where the consequences of an
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earthquake concern only the occupants and the immediate neighborhood) and under “special risk” (where the consequences may not be limited to the neighborhood). The implementation orders of this decree related to buildings (decree of 16 July 1992 and of 19 May 1997) and bridges (decree of 15 September 1995) for normal risks, to classified installations (decree of 10 May 1993) for special risks. These texts place much importance on the distinction between the two categories: – for normal risk the seismic prevention follows a normative approach, i.e. the obligations of the owner of the construction are limited to the proper application of technical standards; – for special risk the approach is “exacting”, i.e. the operating permit is given only when the concerned ministry approves a special report prepared under the responsibility of the owner of the construction. This difference in approach is not “notified” officially by the decree of 14 May 1991, which has adopted the same text to announce the publication of the orders defining the preventive measures to be applied to buildings, equipments and installations irrespective of whether they belong to the normal or special risk category. It is the absence of any reference in the order of 10 May 1993 to a norm on classified installations that brings out the fact that the two types are not treated in the same fashion. This order has only confirmed the practice followed for a long time in the nuclear field that corresponds to the strict approach (critical examination of the safety report of the user by the authority responsible for issuing authorization). We shall discuss seismic prevention as a part of special risk installations in section 18.4. In the normal sense, earthquake-resistant codes are standards defining the rules of calculation and construction to be used in seismic zones and they are enforced by the government. In practice they are reserved for the normal risk category giving priority to buildings; in fact the generalized reference to “buildings, equipment and installations” made by the decree of 1991 should not give the impression that the earthquake-resistant standardization will cover all types of equipment used in industry; this equipment is so diverse and are often considered to be special risk items that drafting earthquake-resistant standards for all of them cannot be foreseen within a short or medium term. At the global level there are very few texts giving seismic precautions for certain types of equipment. These are mainly from the USA and Japan; most of these documents do not have the status of a norm and are only technical guides published under the sponsorship of professional associations. Classification of risks as normal and special results from either an official decision by a government body (in the form of a statutory text, decree or ministerial order giving a list of structures subjected to its application) or from a practice considered as jurisprudence (followed by the corporate services in charge of
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processing the safety applications). The definition of the two types of risks given above may not be sufficient to classify a building or a special structure in one or the other of the two categories; for example buildings such as hospitals, fire stations and bridges which are essential during an earthquake do not, strictly speaking, belong to the normal risk category because their destruction or loss of functionality can have serious consequences in the management of the crisis during an earthquake and may affect a larger group of people than just neighbors and occupants. For this reason the 1991 decree introduced four classes A, B, C and D within the normal risk according to the importance of the risk to people and the socioeconomic role. Class D corresponds to “buildings, equipment and installations whose functioning is important for civil safety, defense and maintaining law and order. The orders of 16 July 1992 and 29 May 1997 for buildings and that of 15 September 1995 for bridges have defined the contents of the various classes and the calculation rules to be applied; the level of seismic action varies according to the classes and the seismic zones (Table 18.5). ZONES
CLASSES A
B
C
D
0
0 (0)
0 (0)
0 (0)
0 (0)
Ia
0 (0)
1.0 (0.5)
1.5 (0.5)
2.0 (0.75)
Ib
0 (0)
1.5 (0.5)
2.0 (0.75)
2.5 (1.0)
II
0 (0)
2.5 (1.0)
3.0 (1.2)
3.5 (1.5)
III
0 (0)
3.5 (1.5)
4.0 (1.7)
4.5 (2.0)
Table 18.5. Definition of seismic action according to class A, B, C or D and seismic zone 0, Ia, Ib,II or III (7.2.2 and Figure 7.3); the first value is the nominal acceleration aN in m/s² (order of 29 May 1997 for buildings and of 15 September 1995 for bridges), the second, coefficient D of the PS 69/82 rules ([14.1]) (order of 16 July 1992 for buildings, currently revoked and replaced by the order of 29 May 1997); this second value is given in parentheses
The buildings and structures of class D are therefore subjected to stricter rules than those of the other classes, but remain within the framework of the normative approach; the difference with the special risk is obvious in zone 0 where the seismic risk is neglected irrespective of the class while recognizing it is mandatory for the classified installations (section 7.2.1); in other zones the level of seismic action as defined in Table 18.5 can be comparable with the class D buildings and structures of special risk. However, the calculation rules are different in the two cases (using a
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behavior coefficient greater than one in the normative approach while elastic dimensioning being practically necessary in the exacting approach. This difference in treatment between the normal and special risk which is found in most countries along with other terminologies mainly corresponds to disputes on prerogatives between administrations and can have ill effects when communicating with the public. A much more realistic process, inspired by the Japanese nuclear regulations, would be to apply the same set of relatively simple rules to all installations and particularly to those in the special risk group either by simply increasing the safety coefficients or by imposing specific additional studies (section 18.4). In view of the above considerations, the area of application of earthquakeresistant codes corresponds, in practice, to buildings and certain civil engineering structures (bridges, retaining structures, towers, big chimneys). Only new structures and buildings are governed by statutory documents which impose the use of the codes; this restriction, on the one hand, is due to economic observation (seismic prevention is possible at reasonable costs for new constructions, but may prove to be very expensive for existing buildings; refer to the introduction of Part 7) and on the other, due to the code structure itself which is adapted more to a project logic than to the diagnostic requirements from the point of view of reinforcements. It is sometimes difficult to decide whether the transformations carried out on an existing building are sufficient to make it a “new” building and require the code to be applied. The order of 29 May 1997 specifies the conditions on the nature of transformations (additions by way of increasing the height or by juxtaposition, structural modifications) that should enable the majority of cases to be handled (situations where ambiguity persists become targets of arbitration by adopting the most demanding option). The earthquake-resistant codes applied to buildings and structures have safety targets which are defined in very general terms. The PS 92 rules, which are the application standards mentioned by the order of 29 May 1997, declare in their foreword that “the main objective is to protect human lives, reduce the destruction due to the collapse of the buildings to a minimum under tremors of nominal level. A second important objective is to limit material damage but given the significant incursions of the materials in their plastic range a noticeable proportion of buildings may be reparable after an earthquake of nominal acceleration”. Using such vague expressions is probably unavoidable but the jurists have difficulty with the case of the collapse of a building calculated according to the norm, at least about the interrogation regarding exceeding or not exceeding the
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nominal stress level due to seismic movements actually sustained by the building concerned. It does not serve any purpose to speculate on what could be “low probability of destruction” or “a noticeable proportion” of non-reparable buildings because the assessment of the “acceptable rate of loss” depends mainly on political and sociological factors in the current context which is mainly oriented towards refusal of all risks by the public and obsessional legalism. It is more interesting to think about the significance of the level of nominal stress which is often perceived as “maximum”. This does not make any sense in the field of seismology, more so in zones of moderate activity where the interval between occurrences of strong earthquakes could be considerable (several thousands of years). The probabilistic formulation of the hazard in modern codes (such as Eurocode 8 in Europe), with an “enshrined” value of 475 years for the recurrence period of the level for which protection is required (section 6.2.1), may be considered as a progress from this point of view. Whether exceeding the nominal level should be foreseen or not is an important point to consider. The PS 92 Rules specify that if the seismic action is considered an accident, “we want to avoid brittle fracture in the vicinity of the nominal acceleration using partial and complementary safety coefficients and to penalize structures with irregularities that can increase the risk of badly controlled behavior”. This formulation clearly shows that the intension of the law makers was to fight what is known as “the cliff effect”, i.e. the risk of total destruction when the nominal level of seismic action is slightly exceeded. However the warning is slightly demanding because the codes cannot have value judgements on the design principles of the structures and thereby incur the wrath of promoters of certain construction techniques. If the rules insist on specific behavior requirements beyond the calculation level, these promoters may feel unduly penalized. As far as the differences between bracing modes are concerned, the codes are limited to design details (the famous “structural details”), which are anyway very important, but they cannot go to the extent of writing that concrete structures with load-bearing walls are more resistant than those with column-beam framework to the risk of collapse. This has been shown by the analysis of past experience (see section 12.2.2 and [FIN 94]). This timorous attitude of the codes in the field of design results, in the fact that any two structures for which the criteria are verified and which are apparently “equivalent” from an earthquake engineering point of view, can actually present different levels of effective safety in the sense that one can be resistant to quakes that are much stronger than those foreseen by the code whereas the other one has little “reserves” in case the seismic action exceeds that taken into account in the
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calculation. The question of behavior beyond the nominal level is important in choosing this level. If the earthquake level of the code is treated as maximum with no possibility of overshooting, the “good structures” are at a disadvantage because these could have been designed in a less demanding manner and still satisfy the noncollapse objective; if an earthquake of intensity stronger than that of the code is taken into account through the adoption of adequate structural details, then the “bad structures” will have to be over dimensioned. Such observations will probably find their natural place one day in earthquake engineering codification and in the legal texts that define their applicability. 18.2.2. Current and future earthquake engineering codes The evolution of earthquake engineering codes in France and in the world, mentioned in section 8.2.2, is currently going through a turning point. Should the characterization of the “seismic load case” by the equivalent static forces (irrespective of the complexity level and the static or the dynamic nature of the calculations enabling the definition of these forces) be maintained or should we move towards formulations in displacement for which the first example is given by the push-over method (section 17.2.4)? The issue is important because it determines the position of the anti-seismic codes in the general framework of the rules governing construction and the degree of specialization (involving need for training) of the designers. Some of the codes of the present day, such as the PS 92 Rules or Eurocode 8, which arise from the conventional approach, already pose or will pose practical application problems due to their complexity which probably goes beyond the capacity of assimilation of “ordinary” professionals of structural analysis. Amongst the reasons that explain this complexity, without however justifying it, the following can be mentioned: – the number and diversity of points of view of the code writers which involve compromising on certain points; this search for consensus often results in multiplication of the variants or the exceptions thereby making it difficult for the user to understand; – an excessive attachment to the notions introduced at a certain stage of development of the codes when its principle itself is disputable in a more general framework; as is the case of the unique behavior coefficient in one direction, as indicated in section 9.3.1, which makes sense only for regular structures that are practically monomodal, and no longer correspond to a rational mechanical drawing for irregular structures. It results from the formulation adopted for the calculation of the equivalent static forces (multimodal elastic analysis and division by the behavior
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coefficient, following a possibly iterative procedure to check the coherence of the hypotheses; see section 17.2.4) that the designer should follow a path full of hurdles which, though it may give the impression of thoroughness, depends actually on hazardous extrapolation based on a process validated only for regular structures; – a highly restrictive definition of the operating conditions of the simplified methods (which corresponds to the case of structures resembling an elasto-plastic oscillator of one degree of freedom, i.e., for which the behavior coefficient has a reasonably clear meaning); the structures qualified as “regular” by these codes and which can be analyzed using these simplified methods, are in reality rare exceptions because they have totally unrealistic criteria for definition; – a turn, which can be seen in Eurocode 8, towards attitudes which are often dogmatic and “intellectual” (in the wrong sense) and which reflect a misconception of the reality of construction projects and some of the lessons from the analysis of past experience. Amongst numerous examples of this deviation the excessively complex structural details (very difficult to be carried out on the work site) in reinforced concrete walls, the rejection of any inelastic behavior of tanks which should remain waterproof or the partiality in favor of concrete frames can be mentioned. The increasing complication in construction rules in practically all fields can probably be considered a factor of progress only if it stems from significant improvement in the methods of design, dimensioning and manufacture. In earthquake engineering this improvement arising out of the analysis of past experience has dealt more with the role of ductility and the importance of structural details necessary for obtaining it than with the capacity of numerical models to reproduce the real behavior of complex structures, at least for the models that remain sufficiently simple to be codified. This relative failure of the calculation comes from the fact that earthquakeresistant codes applicable to normal risk are in reality much more ambitious than the rules used in special risk where just the elastic behaviors are considered because it is more difficult beyond comparison to achieve a non-collapse condition (i.e. a state close to that of destruction) than stability in the linear field. It is therefore disputable, to say the least, to give preference to the calculation method mentioned above (multimodal elastic analysis and division by the behavior coefficient) for the simple application of seismic coefficients, if these include in their formulation a factor that acts as behavior coefficient to penalize the irregular or less ductile structures. As indicated in section 8.1.2, the Japanese code of 1981 has retained such a formulation and nothing indicates that the heavy damages observed during the Kǀbe
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earthquake (17 January 1995) mainly to the buildings and structures constructed before the enforcement of this code can be attributed to the “simplistic” nature of the calculation using seismic coefficients. As long as the evaluation of the behavior coefficients remains essentially empirical or “a thumb rule” and where the dogma of the uniqueness of this coefficient is not justified for the irregular structure it is just an illusion to believe that the reference elastic calculation is a mandatory gateway for determining the forces for designing. It was seen in section 8.2.2 that the main interest of this calculation of reversible deformation is to highlight sensitive frequencies (fundamental mode for the overall response and “local” modes for the response of certain elements) and the distribution of the forces in the bracing system (mainly with the possible appearance of torsional stress). The present day codes mentioned earlier (PS 92 and EC 8) are therefore far from representing an outcome; with respect to the earlier codes, if they can take the credit for explicitly recognizing the ductility and for laying emphasis on the structural details, their practical application is based unnecessarily on complex procedures possibly leading to misinterpretation by users without this complexity being considered as progress or simply a necessity from a safety point of view. Considering the future enforcement of Eurocode 8 at a European level, it may not be very constructive to only criticize this code. Two types of measures can be foreseen to “limit the damage” when it comes into effect: – considerably stretch the area of application of the simplified methods which should be usable in a majority of common buildings for housing purposes; – promote the “design” aspect while training the users by insisting on the “good” method to analyze the results of the linear calculation (identification of critical frequencies, functioning of the bracing, sensitivity to torsion; section 8.2.2) and by acting outside the code on the practical impacts of microzonation studies (section 7.2.3) to fit the buildings and structures to the characteristics of their construction sites (eigenfrequencies, soil properties). As indicated several times in the previous chapters there is a possibility that the future evolution of earthquake engineering codes will favor an approach in displacement rather than under forces. The difficulties of such a change on a technical level (for example, for the assessment of the ultimate rotational capacities of the plasticized sections) or on a philosophical level (the present training for designers is such that the static equilibrium of the forces represents an unshakeable dogma in the matter of safety of constructions) should not in any case be underestimated.
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18.3. Special earthquake-resistant devices “Conventional” seismic prevention depends on the principles of an overall approach for the design and on the technical rules (reinforcement of bracing, structural details) whose implementation calls for only routine construction techniques. The idea that prevention can be made to depend, either totally or in part, on the action of special devices is quite old (the first propositions seem to have been made in the beginning of the 20th century) but it was only in the 1970s and more so since 1980–1985 that this idea was concretized through a certain number of constructions. It corresponds to different mechanisms that can be linked to the following three categories: – the shift in the frequency more towards relaxation to place the response in a zone where the excitation is less intense; the supports made of sandwiched elastomer layers which are the most popular of the special devices belong to this type and are dealt with in section 18.3.1; – increase in the damping to attenuate the response; localized (as for bridges; see section 17.2.5) or spread out (dissipative bracing) shock absorbers can be used and they can be made to act alone or in combination with other systems, as will be seen in 18.3.2; – the energy transfer either by converting part of the kinetic energy of the oscillations into gravitational energy (raising the center of gravity) or by shifting it into a specially designed accessory system (dynamic “shock absorbers” tuned into the frequency of the fundamental mode of the structure); this type of device shall also be studied in section 18.3.2. As a conclusion to this part, section 18.3.3 briefly gives the active control systems in which the reduction of the seismic response results from the guided action of an energy transfer mechanism (generally through a mass linked to the structure whose movements are controlled by jacks servo-slaved to the time history of the excitation, which is recorded in real time); these “futuristic” systems depend more on research than on practical engineering. 18.3.1. Earthquake-resistant supports made of sandwiched elastomer layers The form of elastic spectrums of typical design (Figures 9.5 and 9.6) shows that the amplification of the response in pseudo-acceleration diminishes rapidly at low frequencies (less than 1–2 Hz, i.e., periods more than 0.5–1 s, according to the type of ground).
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It is thus interesting to design foundation systems that enable the fundamental frequency to be lowered sufficiently so that it remains in this less amplified zone (for example in the range of 0.5–1 Hz, or 1–2 s in period). The earthquake-resistant supports made of sandwiched elastomer layers represent the simplest solution to obtain this relaxation; Figure 18.5 shows how these supports can be used and the detail of a support. The supports are made of a pile of elastomer layers (of a 1–2 cm thickness) separated by steel plates; they are generally circular or rectangular (square) with planar dimensions of about a few decimeters. They are installed on concrete pedestals linked to a lower basemat (or possibly to natural terrain if it is a rock of good quality) so as to create a “cave” of the height of a person for easy inspection and if necessary to replace certain supports. Through the intermediary of a basemat on top, the building rests on a set of supports which are its only link with the ground for the gravitational load as well as for the horizontal forces.
Figure 18.5. Foundations of a building on supports made of sandwiched elastomer layers: overall cross-sectional view through a vertical plane (above) and detail of a support (below)
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Seismic Engineering
This type of support, which has been used for a long time in civil engineering structures, mainly in bridges, to enable displacements due to thermal expansion without creating excessive forces, deforms easily due to shear forces of the elastomer layers under stresses parallel to their plane but remains stiff against the forces in the perpendicular direction, thanks to the presence of metallic plates. Figure 18.6 shows a thick circular support being subjected to a shear test in the laboratory.
Figure 18.6. Thick circular supports. On the left, installation below a strategic building for organizing emergency help in Los Angeles; on the right, shear test of a similar support in a Japanese laboratory
The stiffness characteristics of a support made of sandwiched elastomer layers can be determined, as a first approximation, by assuming the elastomer to be an elastic non-compressible material (Poisson’s coefficient is equal to 0.5). Thus, for a layer of thickness h and area s jammed between two supposedly non-deformable plates, [RAJ 76] the following formulae are obtained: kH = G
s ; kV h
1 s2 G 3 ; kT KP h
1 s3 G 3 KM h
[18.1]
where: – kH, kV and kT are respectively the stiffness of the layer with respect to a horizontal force ( parallel to the plane of the layer), a vertical force (perpendicular to the plane of the layer) and a moment about the horizontal axis (which could produce a rotation of the top plate with respect to the lower plate); – G is the shear modulus of the elastomer whose value is typically about 1 Mpa;
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– KP and KM are the numerical coefficients which depend on the form in the plane of the layer and are calculated either numerically for any form or analytically for a circle or rectangle; here, only the values for the circle or square are given: KP = 2.09 (circle) or 2.37 (square)
[18.2]
KM = 79.0 (circle) or 86.2 (square) For a complete support consisting of N layers of thickness h and whose fixing conditions prevent any rotation of the support head with respect to its base, the expressions of horizontal and vertical stiffness KH and KV, if it can be assumed that the layers are sufficiently numerous to go from discrete to continuous in the formulation of the equations of equilibrium, are as follows: Gs ª§ e · tan D § e · º ¨1 ¸ » I / « I 1 ¸ Nh ¬¨© h¹ D © h ¹¼
KH =
1 Gs ² KV = NK h3 P
[18.3]
[18.4]
Other than G, s, N and h which have already been defined, the parameters I, e and D which come into the picture have the following significance: I = ratio between the normal stress of compression of the support and the elastomer modulus (i.e. I = P/ (Gs) if P is the vertical force applied at the support); e = thickness of a plate. D=
N h² e· § KM I ¨I ¸ h¹ 2 s ©
[18.5]
The form of equation [18.3] shows that horizontal stiffness KH is the product of Gs/(Nh), which is the value obtained using a basic calculation of simple shear using a coefficient which is smaller than 1 (because tan D/D is always greater than 1). In practical cases this coefficient is closer to one because parameter D is small (which leads to tan D/D # 1) and I of the order of some units, as will be seen later. Using the following limited development: tanD
D
# 1
D² 3
[18.6]
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Seismic Engineering
from [18.3] it can easily be shown that to limit the horizontal stiffness reduction, with respect to the “normal” value Gs/ (Nh), to less than 10%, the compression of the support (represented by the parameter I) should be limited according to the condition: I+
e h
1 4 s² 1 4 3K M N ² h 4 2
[18.7]
For very high values of I, the parameter D can get closer to S/2; the stiffness KH tends to 0 and the compression of the support becomes instable, i.e., the buckling diagram as represented in Figure 18.7.
Figure 18.7. Buckling during compression of the support in sandwiched elastomer layers having a large slenderness ratio
From expression [18.5] of D, the following stability condition for the support is found: I+
1e 1 2h 2
e² 4S ² s ² h² K M N ² h 4
[18.8]
This condition, as well as that of [18.7], mainly brings into the picture the parameter s² / (N2h4). For a given total thickness Nh of the elastomer, the smaller the slenderness ratio and the greater the number of layers, the larger will the parameter be. Therefore it is by acting on these two factors that a satisfying operating mode for the supports can be obtained, that is, without appreciable influence of the compression on the horizontal stiffness and correlatively, without any risk of instability. As an example Table 18.6 gives the extreme values of I as a function of the slenderness ratio Nh/a for a square support of side a, with N= 4 and e/h = 0.3.
Technical Aspects of Prevention Slenderness ratio Nh/a
Ii instability limit
1
1.21
*
1/2
5.27
1.25
1/3
12.03
3.71
1/4
21.51
7.18
1/5
33.69
11.65
871
I90% (90% of the stiffness Gs/(Nh))
Table 18.6. Influence of the slenderness ratio for a square support in sandwiched elastomer layers of side a (N = 4, e/h = 0.3)
The commonly adopted slenderness ratios for square supports are small, about 1/4~1/5, in order to remain in the same manufacturing range as the bridge supports and to benefit from the analysis of past experience in the field of durability; the table shows that for such slenderness ratios the instability limit corresponds to very high compressions of values greater than those seen normally, which are a few Mpa in size (Ii = 33.69 corresponding to the 1/5 slenderness ratio gives, with G # 1 Mpa, a compression stress greater than 30 Mpa). The limit I90%, corresponding to a loss of stiffness lower than 10% with respect to the value Gs/ (Nh), can, on the contrary, be reached for the supports subjected to compressions found in the higher range of acceptable values (for example, I90% = 7.18, corresponding to the ¼ slenderness ratio gives roughly a stress of 7 Mpa, which is possible in certain applications). For smaller slenderness ratios if it is possible to maintain sufficient safety with respect to instability, it becomes difficult to respect the I90% limit in the case of the table calculated with N = 4; therefore the number N of layers must be increased because, according to [18.7] and [18.8] the extreme values of I are roughly proportional to N. The thick circular supports like those of Figure 18.6 have greater slenderness ratios (in general in the range of 1/2~1/3) and a greater number of layers (10 to 20). Table 18.7 gives the extreme values of I calculated with N=10 and e/h=0.3 for the same slenderness ratios as those considered in Table 18.6.
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Seismic Engineering
Ii Instability limit
overhang Nh/a
I 90% (90% of the stiffness Gs ) Nh
2.63
0.34
1/2
10.95
3.31
1/3
24.83
8.40
1/4
44.27
15.53
1/5
69.25
24.71
1
Table 18.7. Influence of the slenderness ratio for a circular support in sandwiched elastomer layers of diameter a (N = 10, e/h = 0, 3)
These limits under compression are distinctly higher (of a factor at least equal to 2) than those of Table 18.6; this arises from the choice of the number N which, as indicated above, has an almost linear influence on Ii and I90%; an influence even more sensitive results from the slenderness ratio; the form of inequalities [18.7] and [18.8] shows that the extreme values of I are practically proportional to the inverse of the square of slenderness ratio. This is confirmed by an analysis of Tables 18.6 and 18.7. In the presentation of the supports made of sandwiched elastomer layers in the beginning of this section, it was indicated that their vertical stiffness was much higher than their horizontal stiffness; in fact according to [18.3] and [18.4] by taking the “normal” case where the influence of the compression can be ignored in the expression of the horizontal stiffness (that is simply taken as equal to Gs/(Nh)) we have:
KV KH
1 s K p h²
[18.9]
i.e. for square support of side a, according to [18.2]: KV KH
§ a · 0.42 N ² ¨ ¸ © Nh ¹
2
[18.10]
This ratio of stiffnesses is therefore proportional to the square of the quotient of the number of layers per slenderness ratio and in practice attains values of about 100 to 200 (KV/KH = 168 with N = 4 and Nh/a = 1/5).
Technical Aspects of Prevention
873
The trend is the same for thick circular supports (KV/KH = 152 with N = 10 and Nh/a = ½). Formula [18.10] shows that for a given slenderness ratio it is the presence of the plates (that determine the number N of layers) that ensures a high vertical stiffness. It is important that it is so because the operating mode of the support system should be the one represented on the left side of Figure 18.8 i.e. the horizontal seismic actions should induce a pure translation of the building in the direction of excitation, without giving rise to a rotary motion. This will be the case with flexible supports in the vertical direction (right side of the figure).
Figure 18.8. Seismic response of a building on sandwiched elastomer layer supports. On the left, required behavior (pure translation with shear deformation of the supports); on the right, unwanted behavior (translation coupled with rotation, due to an insufficient vertical stiffness of the supports
To reduce the seismic response, the sandwiched elastomer layer supports are therefore effective only against the action of the horizontal components of the movement. The influence of the vertical excitation on the structures equipped with this system of support will be studied below. From the point of view of calculation of the seismic response, the buildings on sandwiched elastomer layer supports behave exactly like the one in the example given in section 15.2.1 (Figures 15.10 and 15.11), i.e. like a non-deformable solid placed on a bed of relatively flexible springs in the horizontal directions and extremely stiff ones in the vertical direction. The movements of this solid under the action of the horizontal components of excitation correspond to three degrees of freedom in a horizontal plane (translation parallel to the two axes and torsional rotation around a vertical axis). From this observation (and from the results obtained in section 15.2.1) it is clear that it is highly desirable to choose a layout plan for the supports which practically eliminates the possibility of torsion (because otherwise
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Seismic Engineering
the supports situated at the edges of the building are clearly more stressed due to shear than the central supports) i.e. to make the position of center of gravity in the plan coincide with that of the center of torsion of the support system. Taking this condition as fulfilled, the calculation of the response and the designing of the support systems become extremely simple. This is an undisputable advantage (and safety insurance) to the credit of the system; following is the process: 1) it is assumed that the compression in the supports remains limited to such values that the horizontal stiffness could be calculated without taking into account the influence of I; we thus have for the angular frequency Z: Z² =
GS Hm
[18.11]
G being the shear modulus of the elastomer layers, S the total area of the supports, H the thickness of the elastomer layers in a support (this is the product Nh used earlier) and m the mass of the building; 2) let the admissible distortion in the supports be G (in general about 1), i.e. the ratio between the displacement of the support head (which is also the displacement of the building) and the thickness H of the elastomer layers; as the displacement of the building is determined with relation to angular frequency Z defined earlier [18.11] and the rate of damping [ by the design elastic spectrum, the following equation is obtained: GH =
1
Z²
Sa Z , [
[18.12]
where Sa (Z,[) is the spectrum in pseudo-acceleration; 3) let the safety coefficient be J for maintaining the compression in the supports; in fact, independent of the effect of the vertical component of movement, the compression is influenced by the tilting moment Mh due to horizontal forces of inertia developed in the building; for a building of length L in the direction of excitation and whose center of gravity is situated at height Z above the plane of supports (see Figure 18.8 on the left), according to the hypothesis of linear profile of vertical stress in the support system, Mh can be derived as:
Mh
1 LSV 1 6
[18.13]
Technical Aspects of Prevention
875
V1 being the stress variation in the edge supports with respect to its static value Vo (which corresponds to the previously used product IG); the safety coefficient J in compression leads to the equation:
V1
1
J
1 mg
Vo
[18.14]
J S
g being the gravity acceleration. As Mh is also equal to the product of lever arm Z and the total force of inertia m Sa (Z,[), taking into account [18.13] and [18.14] we have: g = 6 JO Sa (Z,[)
[18.15]
where O is the slenderness ratio Z/L; 4) determining the area S of the supports based on equations [18.11], [18.12] and [18.15] enables us to write: S=
m S a Z , [ G G
HmZ ² G
1 mg 6 JGO G
[18.16]
It is remarkable that S does not depend on the seismic excitation defined by the spectrum Sa (Z,[); it is also observed that parameter I introduced earlier (which represents the compression level of the supports and can influence the horizontal stress) can be expressed in a simple manner as follows: I=
mg GS
6JGO
[18.17]
which shows that it depends only on safety coefficients J and G as well as on the slenderness ratio O. In the same way, based on [18.15] it is found that the acceleration Sa (Z,[) experienced by the building depends only on g and coefficients J and O according to the formula:
Sa (Z [
g 6JO
[18.18]
Therefore, irrespective of the excitation it is the same and does not depend on the mechanical properties of the supports;
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Seismic Engineering
5) the angular frequency Z which can be determined for a given spectrum and damping verifies equation [18.18]; the thickness H of the elastomer layers is derived using equation [18.11]; for example, if the following expression is taken for Sa (Z,[): Sa (Z,[) = Ao Z
Zo
[o [
[18.19]
which has already been used in the previous chapters (see section 9.3.1) and correspond to the constant pseudo-velocity branch of the spectrum (low frequencies), Ao, Zo and [o being the reference values; using [18.18] we get:
Z Zo
1 g 6 JO AO
[ [o
[18.20]
and using [18.11] for H and the previously obtained equation [18.16] for S: H=6
JO A02 [o G gZ02 [
[18.21]
6) the detailed characteristics of the supports (form, surface, number of layers) are chosen by respecting the values found for H and S and by choosing a case where the influence of the compression on the stiffness of the supports is negligible. By way of numerical application, the following data are considered: – for spectrum Ao = 3 m/s²; Zo = 2S rd/s (either a frequency of 1 Hz or a period of 1s); [o = 0.05; these values correspond to the spectrum S1 (firm ground) of Figure 9.6, with the zero period acceleration set at 3 m/s²; – for the building m = 4.5 x 107 kg (45,000 tons) and O = 0.5 (height equal to the length if the masses are distributed homogenously); – for the supports G = 106 N/m²; [ = 0.07 (common use value with synthetic elastomer layers); – for the safety coefficients G = 1 and J = 1.5. Taking g = 10 m/s2 the previous formulae lead to: S = 100 m²; H = 0.0733 m;
Z Zo
0.876 ; I = 4.50
[18.22]
Technical Aspects of Prevention
877
400 square supports with sides 50 cm, which with the thickness H = Nh = 7.33 cm produces a slenderness ratio of the support of 1/6.82; with N = 4 and I = 4.5, it is found that condition [18.7] is widely verified (the second member of this inequality is 22.65), and that the compression has no appreciable influence on the stiffness (the multiplier of Gs/ (Nh) in formula [18.3] has a value of 0.994). The average compression of the supports is of 4.5 MPa and varies from 1.5 MPa to 7.5 MPa for the edge supports. These values are absolutely acceptable. The fundamental frequency is 0.876 Hz (period 1.14 s), which is in the appropriate range of utilization for this system of supports (very low values of this frequency, lower than 0.3–0.4 Hz could have practical problems of sensitivity to wind); for this frequency the spectral acceleration is 2.22 m/s² [18.19]. This is much less than the response that could have been had with conventional foundations (the spectrum plateau used is at 7.5 m/s² for the reference damping [o = 0.05). The sandwiched elastomer layer supports solution enables us to obtain a very perceptible reduction of the response for this building. In order that the design layout described above corresponds to the reality it is necessary to take a certain number of precautions for installing the supports and for going ahead with the verifications. For installing the supports, it was mentioned earlier that it is necessary to reduce the gap between the projection of the center of gravity and the center of torsion of the support system to a minimum. It should also be verified that all the supports work in a similar manner. This implies a strict control of their horizontality and the uniformity of the distribution of the compressions under static loads. This point requires good knowledge of the distribution of vertical static loads at the base and shows that the utilization of the system is delicate a priori if the ground has mediocre characteristics due to differential settlements which can show up. Reclamping of the supports using jacks can be foreseen to make the compressions uniform. The mechanical properties of the elastomer layer which were used in the previous calculations are the modulus G and the reduced damping [; these parameters depend on a certain number of factors amongst which: – the precise composition of the elastomer layer and the manufacturing condition of the supports (mainly the vulcanization procedure); their influence on damping is particularly important. The supports, made of natural rubber for example, have very low damping (which makes it necessary to add complementary systems to produce a damping effect by friction or plastic deformation of metallic pieces like the lead core or steel bars outside the support; see Figures 18.12 and 18.14) whereas certain synthetic elastomer layers enable values of [ of the order of 15–20% to be reached;
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Seismic Engineering
– the dynamic character of the applied load which even for the relatively low frequency of oscillation can have tendencies to increase the modulus (typically by a factor of 1 to 1.4) with respect to its static value, at least for certain types of elastomer layers; – the nonlinear behavior of the supports which shows up for high distortion levels. In general the increase of distortion produces an increase of stiffness and thus of the apparent modulus; the choice of coefficient G in the designing process implies a reflection on the choice of the most representative value for modulus G; the nonlinear effect is not so sensitive if G is set close to I, as in the example dealt with before, but can become so for G = 1.5–2 (adopted values in certain applications and that remain clearly lower than the extreme limits corresponding to the destruction of the support due to tearing up of the elastomer layer); – effects of aging during the life of the structure which tend to increase the modulus and diminish the damping in proportions that can move at the most a few dozen percent. These observations show that the use of the sandwiched elastomer layer supports requires qualifying tests which allow only one reasoned assessment of the choice of the calculation parameters G and [ in relation to an evaluation of the influence of aging; these tests should take into account the operating conditions (mainly temperature). For critical facilities a monitoring program of the supports in place, consisting of periodic checks on test samples is necessary to ensure that the evolution of the characteristics does not question the options of the designer. Another type of verification which should be carried out right from the preliminary project stage, concerns the building itself; the calculation plan which was presented assumes that the building behaves like a rigid block and that the deformations are concentrated at the supports. Such a plan can correspond to reality only if the deformability of the structure is clearly lower than that of the supports i.e., the fundamental frequency of the building, calculated as if it has been embedded at the base, should be perceptibly higher than that corresponding to angular frequency Z determined earlier. To give a quantitative significance to the expressions that were just used (“deformability clearly lower”, “frequency perceptibly higher”), a perfectly regular building similar to a beam of constant section deforming in pure shear strain (like the example dealt with in section 9.2) is taken. The building is placed on sandwiched elastomer layer supports. The numerical model is presented in Figure 18.9.
Technical Aspects of Prevention
879
Figure 18.9. Shear beam placed on anti-seismic supports which are represented by a horizontal spring
A formulation with a continuous variable is used. Its unknown is the function of u (z, t) that represents the horizontal displacement with respect to the height above ground z and time t; if k represents the stiffness of the support system, U, P and S (density, shear modulus and area of the section) the characteristics of the beam, H the height and s (t) the accelerogram of excitation, the equation of the movement becomes: § w ²u · U¨ s t ¸ © wt ² ¹
P
w ²u wz ²
[18.23]
and, for the boundary conditions: z = 0, PS
z = H,
wu ku wz
wu wz
0
0
[18.24]
[18.25]
Equation [18.24] gives the equilibrium of the forces at the base and [18.25] the zero force condition at the free end. The resolution is easily carried out by the modal method presented in Part 4 and Part 6; we limit ourselves to a few results on eigenfrequencies and modes while the
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Seismic Engineering
supporting calculations are left to the reader; the following is the expression for the deflected shape vn (z) of the nth eigenmode:
z · § vn (z) = cos Dn ¨1 ¸ © H¹
[18.26]
Dn being a non-dimensional angular frequency linked to the angular frequency Zn of this mode through the equation:
U P
Dn = Zn H
[18.27]
The Dn s are the roots of the transcendental equation:
D n tan D n
D 02
[18.28]
where Do is the non-dimensional angular frequency that could be calculated by taking the beam as a rigid block placed on the supports (as in the functional diagram represented on the left side of Figure 18.8), i.e.: D0 =
k U H m P
[18.29]
if the total mass of the beam is represented as m. From [18.26], [18.28] and [18.24] the following equations for the fundamental mode can be easily established: uI,0 = cos DI
P1*
ª § 1 ·º 2sin ²D1 / «D1 ¨ D1 sin 2D1 ¸ » 2 ¹¼ ¬ ©
(Ep/Ea)I =
· 1 § 2D1 1¸ ¨ 2 © sin 2D1 ¹
[18.30]
[18.31]
[18.32]
uI,0 being the displacement at the base (i.e. at the level of the support heads), P*I the fraction of the total mass which is associated with the modal mass and (Ep/Ea)I, the ratio of the deformation energies between the beam (Ep) and the supports (Ea). Table
Technical Aspects of Prevention
881
18.8 gives the values of these quantities for D1 varying from 0.1 at S/2, as well as those of Do [18.29] and ratio r between the frequency at the embedded base (infinitely stiff supports) and the frequency corresponding to D0 (very stiff beam); this ratio is given by the equation: r=
S 2D 0
S 2
/ D1 tan D1
[18.33]
The values of this table enable us to assess the validity conditions of the designing method spelt out earlier for buildings on sandwiched elastomer layer supports [18.11]-[18.22]. This corresponds to the hypothesis of the deformation concentrated on the supports. The most used parameter for quantifying these conditions is ratio r (fourth column), which can be calculated easily based on the characteristics of the building and the supports. The last three columns show that for r higher than 3 the deformation of the beam remains limited because its energy of deformation represents less than 10% of that of the supports, with a deflected shape (which varies from u1, 0 at the base to 1 on top) close to pure translation; the fundamental mode represents at least 99.9% of the total mass, which ensures that the effect of the higher modes is perfectly negligible. These observations remain relatively true for a value of r between 2 and 3 (ratio of the deformation energies at the most equal to 21%) but the situation deteriorates rapidly when r reduces from 2; for r = 1 (corresponding to D1 = 1.139) the deformation energies are practically equal and the displacement at the base falls to 0.42 times its value at the top; on the contrary the importance of the fundamental mode, even though decreasing ( P1 = 0.954 for r = 1), remains ensured however. This is normal for a regular building. *
A study of the first three columns of the table shows that for r higher than 2 the decrease of the fundamental frequency D1 with respect to Do (hypothesis of the rigid block on supports) remains lower than 10%. This is within the usual margin of uncertainty of the current frequency calculations, considering the precision with which the characteristics of the materials are known. The deviation in frequency increases rapidly for the smaller values of r and obviously tends to infinity when we are closer to the conditions of the embedded base (infinitely stiff supports).
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Seismic Engineering D1
Do
D1/Do
r
uI,o
P1*
(Ep/Ea)I
0.1
0.100
0.999
15.682
0.995
1.000
0.003
0.2
0.201
0.995
7.801
0.980
1.000
0.014
0.3
0.305
0.984
5.156
0.955
1.000
0.031
0.4
0.411
0.973
3.820
0.921
0.999
0.058
0.5
0.523
0.956
3.006
0.877
0.999
0.094
0.6
0.641
0.936
2.452
0.825
0.997
0.144
0.7
0.768
0.911
2.046
0.765
0.994
0.210
0.8
0.908
0.881
1.731
0.697
0.990
0.300
0.9
1.065
0.845
1.475
0.622
0.983
0.424
1.0
1.248
0.801
1.259
0.540
0.974
0.600
1.1
1.470
0.748
1.068
0.454
0.960
0.861
1.2
1.757
0.683
0.894
0.362
0.942
1.277
1.3
2.164
0.601
0.726
0.267
0.917
2.022
1.4
2.849
0.491
0.551
0.170
0.885
3.679
1.5
4.599
0.326
0.342
0.071
0.845
10.129
S/2
f
0
0
0
0.811
f
Table 18.8. Variations of certain parameters based on the frequency for the fundamental mode of a regular building on sandwiched elastomer layer supports
These observations show that the method of designing based on the hypothesis of rigid building is a good approximation of the reality for r t 3 and remains acceptable for r between 2 and 3. For other values of r, this method is not sufficient and is not of any interest because, as indicated in the beginning of the section, recourse to earthquake-resistant supports is based on the idea of a shift in the frequency. A last type of verification while using sandwiched elastomer layer supports concerns the link the building has with the exterior (access, connection with the
Technical Aspects of Prevention
883
outside networks). These have to be designed to be adjusted without damage to the relative displacements between the building on supports and its environment which follows the motion of the ground. These displacements which are a few centimeters in size for applications in zones of moderate seismicity (as in the design given above where the value of displacement is 7.33 cm) can attain 30 to 40 cm when it comes to protection against an earthquake of magnitude 8 at a distance closer to the fault. Such displacement values require a special design for the fluid system inlets (such as using bellows) and for the construction of access passages which enable pedestrians or vehicles to enter the building. During the Northridge earthquake of 17 January 1994, a coverplate slab of the main entrance to a building on supports caused a series of shocks whose effects are visible on the recordings obtained inside the structure. This very rigid slab was put in place to replace the one specified initially in the project which was designed like a “fuse” and had actually given way during a previous earthquake (Landers 1992). This “accident” which had no serious consequences emphasizes once again the importance of details in seismic prevention planning [COL 94a]. The question of sensitivity of the structures on supports to the action of the vertical component of seismic movement has sometimes been raised. It is known that the vertical stiffness of the supports is much higher than their horizontal stiffness; as a result, the frequency due to the supports in the vertical direction (assuming that the building is a rigid block) typically varies from 10 to 20 Hz. Such a value may coincide with the vertical fundamental frequency of the building. This resonance risk does not have any distinct consequences if the link with the ground through the lower basemat (Figure 18.5) enables us to obtain an important radiative damping as is normally the case with pump stroke (section 16.1.3). It is therefore necessary that the lower basemat be sufficiently rigid. This condition, as indicated above, is often imposed to reduce differential settlements to make the compressions uniform in the supports. The earthquake-resistant solution represented by the sandwiched elastomer layer supports is now part of the designer’s paraphernalia and it has been used quite frequently (several hundred in the year 2000) confirming that it is no longer just an idea but a reality [COL 90c]. Even though the analysis of past experience has its own reservations regarding its validity, the few known examples confirm the genuineness of the concept whose qualities are its simplicity and the possibility of a complete and permanent control of safety elements. What stops it from being put in to common use is its additional cost as against conventional solutions. Not only are the supports themselves fairly expensive but also the complementary structural parts which are necessary for their installation (lower basemat, retaining walls around the “cave”, special devices to establish contact with the exterior). This explains the use of these supports only in the special risk category or normal buildings of the highest
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Seismic Engineering
risk category and not in common constructions; as shall be seen later in section 18.5.2, these supports can also be used in seismic rehabilitation of historical buildings of heritage value. One of the reasons why it is less economical to put an “ordinary” normal risk building on supports is that it does not seem possible to design the structure according to routine practice for this type of risk; i.e. by accepting encroachments in the plastic range (behavior coefficient higher than one). A certain number of numerical models have proved the importance of ductility demand in buildings on supports for relatively weak levels of excitation (double the one that corresponds to the elastic limit), for which the designing of a building on conventional foundations would cause no problems (behavior coefficient of the order of 2). Using the diagram of Figure 18.10 a simple explanation of this observation which is surprising a priori can be given. The top part of the figure shows a very simple model to which a building on supports can be reduced when plasticization of its structure is envisaged, i.e., a mass m linked to its support through two springs in series. The first (relatively weak stiffness k) represents the supports; the second (stiffness K clearly much higher) represents the building; the two springs are connected through a pad which slides when the force transmitted by the second spring exceeds the value corresponding to the plastic plateau of its law of behavior. The displacements of the support heads and that of the center of gravity of the building are noted as x and X respectively. The lower part of the figure shows the force-displacement diagrams for the behavior relations of the supports (linear law of slope k) and of the building (bilinear law consisting of an elastic part of slope K k/ (K + k) and a plastic plateau).
Technical Aspects of Prevention
885
Figure 18.10. Model of a building on earthquake-resistant supports whose structure is susceptible to undergoing plastic deformations
When the elastic limit for the building is reached, its displacement is Xe (point A) while that of the support head is xe; the equality of the forces in the two springs gives rise to the equations: k xe = K (Xe – xe); xe =
K X e ; X e xe ; K k
k Xe K k
[18.34]
from where the ratio of the deformation energies stored respectively in the structure and supports (i.e. the same ratio as the one found in the last column of Table 18.8) becomes equal to k/K (i.e. the inverse of the square of r in the same table). A reasonably good correlation between this very simple model and the one that has helped to write Table 18.8 (for example, for r = 3, k/K = 1/9 = 0.111 whereas the table gives 0.094 for the ratio of the deformation energies) can be noticed. If, based on this elastic limit stage, the intensity of the accelerogram of excitation is doubled, the response calculated elastically shifts from point A to point B; Assuming that the common rule (Newmark “theorem”) of retention of the displacements apply, the real state of the system corresponds to point C with the plastic displacement Xp. This shift from A to C is very “intensive” in ductility requirement. In fact the energy increase (corresponding to the area of the rectangle
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Seismic Engineering
ACXeXp) should be absorbed by the only structure, the load applied and thus the energy absorbed do not change in the supports (triangle xe AXe). The situation is very different from what the building would experience without supports (left side of the force-displacement diagram) where the doubling of the excitation with respect to the elastic limit state would simply shift from A’ to C’ with an energy absorption equal only to double (rectangle A’C’X’eX’p) the elastic energy (triangle OA’X’e). In other words, in the case of supports, the structure that plasticizes should absorb the same amount of energy as is absorbed in the supports, which is much higher than its own elastic limit energy; this is only possible by calling for a high degree of ductility. For a multiplication of the intensity of excitation by O with respect to the state of elastic limit, the ductility requirement P is given as:
O X e xe P= X x e e
[18.35]
i.e., considering equations [18.34]: P = (O – 1)
K O k
[18.36]
or, since k/K = 1/r², as indicated earlier: P = (O – 1) r² + O
[18.37]
With the typical value r = 3, which corresponded to a good use of the supports, this formula shows that a modest doubling of the excitation (O = 2) requires a ductility of 11. The numerical simulations with nonlinear models excited by accelerograms have confirmed this order of magnitude. The designing of structures on supports should therefore be carried out in elastic (or possibly with the behavior coefficients slightly higher than one, [BET 92]). As a result there is no hope of any saving on this item in spite of the reduction of the elastic response brought by the system. The difference, in economic terms, is in the cost of repairs which should be almost zero in the case of the solution with supports whereas the costs can be substantial in the case of conventional solutions in which the main aim is to avoid collapse.
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18.3.2. Other special earthquake-resistant devices
A large number of earthquake-resistant devices as the main objects of patents have been proposed since 1980–1990. World conferences on earthquake engineering which take place every four years have now collected more than 100 papers on this subject which is usually called seismic isolation. The choice of this expression is not particularly suitable because the structures equipped with these devices are not really isolated from seismic actions but simply have a response wherein the potentially dangerous aspects are more or less attenuated. Many of these systems are complementary to the sandwiched elastomer layer supports and aim at increasing the apparent damping through different mechanisms of dissipation of energy or mobilizing a progressive effect of blocking against severe earthquakes going beyond the design limits foreseen. The most commonly proposed damping mechanisms are friction (which requires precise machining of the surfaces in contact and a high quality of manufacture to ensure good durability and to avoid the effects of dissymmetry mentioned in section 17.2.3) and the plastic deformation of the metallic rods (whose form is often optimized to obtain a progressive plasticizing distributed over a large part of the component). Figure 18.11 shows the support system made from a sandwiched elastomer layer with sliding plates manufactured by French companies for the nuclear power plant at Koeberg (South Africa).
Figure 18.11. Sandwiched elastomer layer supports whose head consists of a sliding plate with friction. At the top, a general diagram (left) and support detail (right); at the bottom, principle of operation (according to [COL 84c])
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In this system the sliding with friction comes into the picture only when the seismic movement exceeds a certain level (acceleration higher than the product Pg, P being the friction coefficient of about 0.2 and g the acceleration due to gravity). At levels lower than this limit the behavior is that of conventional supports [COL 84c]. In other systems based on the utilization of friction there is a damping mechanism (Figure 18.12) active from the beginning.
Figure 18.12. Support systems of sliding with friction consisting of a damping mechanism effective from low levels of loading
In the top part of Figure 18.12 the sandwiched elastomer layer supports that are used in parallel with the sliding plates (instead of in series as in Figure 18.11) can be seen. These supports develop a horizontal restoring force while the supports equipped with friction plates (which should take the main part of the vertical loads) ensure the dissipation of energy. At the bottom of the same figure the system which is represented gets its spring force due to gravity as a result of the concave shape of the support surfaces; friction is also mobilized on these surfaces as well as on the ball and socket heads which enable the horizontality of the structure during its movement to be maintained. The accuracy of machining necessary for the manufacture of such a system limits its use to mechanical applications rather than civil engineering structures. For buildings, systems easier to manufacture and based on the principle of spring force due to energy transfer were proposed and carried out in Crimea. They use eggshaped supports (Figure 18.13) which lift the supported structure when they roll without sliding under the effect of a horizontal displacement of this structure.
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Figure 18.13. Operating principle of the egg-shaped supports in the case where they are made of two spherical hoods conjoined at their base
The figure shows such a support formed by the union of two spherical hoods of radii R1 and R2 whose centers C1 and C2 are at a distance of R1 + R2 – h, h being the height of the support (i.e. also the distance between the upper and lower basemats). The right side of the figure refers to a certain number of parameters (angle of rotation D of the line which joins the centers, horizontal and vertical displacement u and v of the upper basemat with respect to the lower, normal and tangential reactions Ni and Ti developed at the points of contact) in the case of horizontal seismic loading. In the hypothesis of rotation without sliding, basic geometric observations lead to the following expressions of displacements u and v: u = (R1 + R2) D – (R1 + R2 – h) sin D
[18.38]
v = (R1 + R2 – h) (1 – cos D)
[18.39]
To determine the angular frequency Z of the system when it oscillates freely, the equations of movements in the absence of external excitation are studied, i.e.: mu ¦ Ti
[18.40]
i
mv mg ¦ N i i
[18.41]
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Seismic Engineering
m being the mass of the supported structure (considered as a rigid block according to the same hypothesis as that used for the sandwiched elastomer layer supports in section 18.3.1), g the acceleration due to gravity and by summing up the reactions for all the supports; the equilibrium in moment of a support leads to the equation: Ti [R1 + R2 – (R1 + R2 – h) cosD] = Ni [R1 + R2 – h] sin D
[18.42]
which enables us to eliminate the reactions between [18.40] and [18.41] to obtain:
v g sin D = 0 [R1 + R2 – (R1 + R2 – h) cosD] u + (R1 + R2 – h)
[18.43]
By calculating u and v using [18.38] and [18.39], the following differential equation [18.44] can be finally arrived at to determine D with relation to time: [h² + 2 (R1 + R2) (R1 + R2 – h) (1 – cos D)] D + (R1 + R2) (R1 + R2 – h) D ² sin D g R1 R2 h sin D [18.44] whose first integral is found by taking w = D ² as unknown which is considered as a function of D; we thus have: dw dt
dw D from which D dD
2DD
1 dw 2 dD
[18.45]
and after transposing in [18.44] and integrating: [h² + 2 (R1 + R2) (R1 + R2 – h) (1 – cosD)] w + 2g (R1 + R2 – h) (1 – cosD) = h² wo [18.46] wo being the value of D 2 for D = 0, that is according to once derived [18.38]:
uo2 wo = h²
D 02
[18.47]
u0 being the velocity when the oscillation goes through the rest position D = 0. When the amplitude of the oscillations (D << 1) is weak, equation [18.46] takes the form:
D ² g
R1 R2 h ² D h²
D 02
[18.48]
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which characterizes a linear oscillator whose angular frequency Z is given by:
Z
1 g ( R1 R2 h) h
[18.49]
It is clear that with this system it is possible to obtain sufficiently low oscillation frequencies that reduce the seismic response significantly as in the case of the sandwiched elastomer layer supports; for example with R1 = R2 = h = 0.5 m a frequency of 0.71 Hz is found. The disadvantages are linked to the onset of a vertical oscillation (the frequency being double that of the horizontal oscillations as observed in section 17.2.2 in relation to the uplift of the basemats) and particularly at a very low value of damping because the rolling friction is practically negligible. These gravitational systems thus require the addition of complementary devices, just as with some sandwiched elastomer layer supports, such as those using natural rubber to obtain sufficient damping.
Figure 18.14. Sandwiched elastomer layer supports equipped with additional shock absorbers using the plastic deformation of metallic components: central lead core (on the left) and external steel bar in the form of an hour glass (on the right)
Amongst the complementary devices which do not call for friction mechanisms the more common ones are based on the plastic deformation of bars of steel or other metals. A type of sandwiched elastomer layer support developed in New Zealand uses (left side of Figure 18.14) a lead core placed in the axis of the support. The thick, circular support systems (Figure 18.6) produced in Japan prefer the solution of shock absorbers outside the elastomer layer supports (this facilitates their inspection and their replacement after a sufficiently strong earthquake that could have deformed them) with steel bars either of constant or optimized sections to increase the volume of the plasticized zone and thus the dissipation of energy (the right side of Figure 18.14 shows a bar in the form of a hourglass). Increasing the damping through friction or plasticity has also been used with multiple variants within the bracing system itself. For example (Figure 18.15), for a
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metallic bracing in the form of a St Andrew’s cross, every intersection of the diagonals can be equipped with mechanical shock absorbers taking advantage of the elongation of the stretched diagonals (left side), or for a concrete frame the different levels can be connected by a cable in series with a prestressed shock absorber connected to the anchoring point on the ground [COL 00a] as represented on the right side of the figure. For weak stresses, the shock absorber, being prestressed, behaves like a rigid element and only the cable works in its elastic field; under strong stresses the shock absorber is mobilized and contributes towards dissipating the energy. Such systems belong to the category of dissipative bracing which continues to be of interest in several researches. The system with cables (Figure 18.15) uses a localized shock absorber which is subjected to forces resulting from the deformation of the bracing assembly. In most cases where the localized shock absorbers are installed, their function is to attenuate solid body type movements of entire structures or heavy equipment. A typical example is that of bridges. Section 17.2.5 gives an idea of the designing of their shock absorbers through stochastic linearization. Figure 18.16 shows two possibilities using a shock absorber anchored on an abutment to reduce the longitudinal movements of the basemat.
Figure 18.15. Two examples of dissipative bracing. On the left, shock absorbers at the crossings of diagonals of the St Andrew’s crosses; on the right, connecting the floors by a cable anchored to the ground by a prestressed shock absorber
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Figure 18.16. Shock absorbers for bridges in the longitudinal direction. On top of the shock absorber is the only control element of the movement of the deck, all the pier heads being equipped with sliding supports (with negligible friction). At the bottom, the shock absorber is meant to reduce the load of the two central piers which are rigidly connected to the deck (according to [CAP99])
Figure 18.17. Heavy equipment shock absorber in an industrial installation
Figure 18.17 shows a hydraulic shock absorber in an industrial installation to dampen the movement of a heavy component. In such a case the influence of the
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ambient conditions during operations (temperature, possible physico-chemical aggressivity of the surroundings, possible radioactive exposure in nuclear power plants) should certainly be taken into account in the studies related to aging and they can be the deciding factors for the choice of materials (particularly in selecting the oil used in hydraulic shock absorbers). In addition to gravitational systems mentioned above (Figure 18.12 on the left and Figure 18.13) several devices using the transfer of mechanical energy have been proposed [BET 93]. One of the simplest which is commonly used in high-rise buildings is the dynamic shock absorber tuned to the frequency of the fundamental mode of the building (TMD for Tuned Mass Damper) which consists of suspending near the top a pendulous mass whose frequency of oscillation coincides with the first frequency of the structure (Figure 18.18); in practice the pendular installation which requires a lot of free space in the vertical direction is often replaced by other systems that connect with the building (for example, the sandwiched elastomer layer supports). These help in obtaining this tuning through less cumbersome means. The ratio r between the pendulous mass and the total mass m of the building is typically about 1/100. To ensure frequency tuning, the same ratio r should exist between the stiffness of the spring and that of the building.
Figure 18.18. Dynamic shock absorber tuned to the frequency of the fundamental mode of the building; diagram of the pendular installation (on the left) and schematic representation (on the right)
The reason for the efficiency of this device in lessening the dynamic response comes from the peculiarities of the eigenmodes, where the displacement x of the pendulous mass is much higher than that of the center of gravity of the building, denoted X; based on the equations of the non-dampened eigenmodes [15.69] that are written as:
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[(1 + r) k – Z²m] X – r k x = 0
[18.50]
– r k X + (r k – Z² r m) x = 0
[18.51]
the equation for the eigenfrequencies Z of the system is arrived at: 4
2
§Z · §Z · ¨ : ¸ 2 r ¨ : ¸ 1 0 © ¹ © ¹
[18.52]
k / m is the angular frequency of the building without the pendulous where : mass; the roots are:
Z² :²
1
r 1 r 4 r B 2 2
[18.53]
Basic calculations enable the following expressions for the deflected shapes and modal masses to be found: First mode:
X1 =
1 2
r 4 r r ; x1 = 1
[18.54]
P1 =
mª r º «1 r 3 r » 2 ¬« 4 r ¼»
[18.55]
Second mode:
X2 =
1 2
P2 =
mª r º «1 r 3 r » 2 ¬« 4 r ¼»
r 4 r r ; x2 = – 1
[18.56]
[18.57]
In practice as r is very small during 1, these formulae can be simplified as:
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First mode: 1 § · r ¸ ; X1 Z1 = : ¨ 1 2 © ¹
r ; xI
1 ; P1
r ; x2
1 ; P 2
§1 3 · m¨ r¸ ©2 4 ¹
[18.58]
Second mode: 1 § · r ¸ ; X2 Z2 = : ¨ 1 2 © ¹
§1 · 3 m ¨¨ r ¸¸ 4 ©2 ¹
[18.59]
It is seen that the presence of the pendulous mass “splits” the fundamental mode of the building into two modes of near equal characteristics in frequency and modal mass; it is interesting to note that for these two modes the displacement of the pendulous mass is high (in the ratio 1/ r ) against that of the building. As a result the deformation energies of the two springs are visibly equal in each of the two modes (as the stiffnesses are in the ratio r); by applying formula [15.117] for calculating the modal damping, it is found that it is equal to the average of the damping [o (that of the structure) and [ (that of the link of the pendulous mass). Since the field of application aimed at is that of high rise buildings, it can be assumed that the angular frequencies : , Z1 and Z2 belong to the constant pseudovelocity branch of the elastic design spectrum. Taking the normal rule of dependence of the spectral ordinates in an inverse ratio to the square root of the damping, this pseudo-velocity V can be expressed as: V
Vo
2[ o [0 [
[18.60]
where the result established above for the damping (equal to ([o +[)/2) of the building equipped with its pendulous mass has been used. To calculate the displacement ' for this excitation at constant pseudo-velocity equation [15.100] is used, which in this case is not an upper limit result but a simple equality, since the two modes can be considered as being in phase because of the small difference in their frequencies. The deformation energy Ed is the sum of the two equal terms each one at k'²/2 (as this quantity, which represents the deformation energy of the spring of the building, is also that of the linking spring of the pendular mass); therefore, according to [15.100] and [18.60] we have: Ed = k'² =
1 mV ² 2
mV o2
[o [o [
[18.61]
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from which for displacement ': '=
Vo :
[o [o [
[18.62]
In this equation, the factor Vo/ : against the square root is only displacement 'o which would have the building without its pendulous mass; this leads to the simple formula: ' 'o
[o [o [
[18.63]
which shows that the dynamic shock absorber system tuned in frequency reduces the response by 30% if the damping [ of the pendulous mass-structure link is equal to [o, and by 50% if [ = 3[o (i.e. 15% by taking the usual value of 5% for [o, which is relatively easy to obtain). It is therefore a significant reduction for which the price to pay (in terms of cost and constraints of installation) is not exorbitant (for a building of 20 levels occupying an area of 1,000 m2 on the ground and weighing 20,000 tons, a pendulous mass of 200 tons, i.e., one-hundredth of that of the building, can be represented by a steel cube having a side of around 3 m). The installation conditions of the pendulous mass should take into account the strong amplitude of the displacements of the mass (10 times that of the building for a mass ratio of 1/100, which can correspond to a horizontal displacement of about one meter). In this section we have limited ourselves to the few pieces of information on earthquake-resistant devices other than sandwiched elastomer layer supports or those complementary to these. As indicated earlier it concerns a field where an expanding activity is deployed with a juxtaposition of serious propositions based on analytical or experimental proofs along with some hare-brained ideas thrown around with a rough outline of validation and feasibility [BET 93]. To conclude, a certain number of general observations can be made on the conditions to be fulfilled by the special earthquake-resistant devices to find their place in routine construction work: – the simplicity of the mode of operation of the devices is an important factor for their acceptance by project managers. The complex diagrams involving the use of different mechanisms and their successive intervention discourages a priori anybody who is aware of the extreme variability in the occurrence of the seismic aggression on the one hand and of the reliability of an elaborate system after a long period of inactivity preceding an earthquake on the other; this mainly concerns the use of certain “seismic fuses” in the form of dowel pins. When they are blown they trigger a shift from one behavior mode into another. Such a design can be accepted if there is only one fuse and its action cannot be “short circuited” by an unforeseen routing
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of forces but becomes very dubious if there are several fuses and if the system assumes that they will blow simultaneously; – controlling the effects of aging is also an important factor for project managers since it regulates to some extent the volume of investment when it is a question of preparing the facilities required for inspecting the devices in use (Figure 18.5 and the “cave” around the supports) and their possible replacement during the period of installation; – the economic record of the solutions using these devices (manufacture, implementation, in-service inspection) should be assessed not just comparing with the cost of “conventional” competitive solutions but also considering the possible differences in the performance. At the end of section 18.3.1 it has been indicated that the sandwiched elastomer layer supports currently have a lot of difficulty penetrating the current construction market, probably due to the lack of sufficient information regarding the better investment protection that they offer compared to the usual anti-seismic approach; – attention should be drawn to the fact that the action of certain devices can involve precise knowledge of some specific aspects of seismic movements which are not important when using conventional techniques; they are for example, low frequency components for the systems based on frequency shift (mainly in the case where there is a fear of the killer pulse effect; see sections 4.1.3 and 17.2.4), the displacement limits for the stroke of the hydraulic shock absorbers or the influence of the aftershocks for mechanisms consisting of irreversible changes. Considering the fragmented knowledge on such subjects, a very safe approach seems necessary. This approach can lead to an increase in the safety margins in the design and adopting structural details (like thrust block or additional support systems) when the seismic action exceeds the value taken in the calculation. 18.3.3. Active control
The active control systems correspond to mechanisms whose action is carried out in real time, during a seismic movement, to counteract the dynamic effects inducted in the system by it. The presence of this external control, which in practice is almost always carried out using computer controlled jacks, distinguishes them from passive control systems as the earthquake-resistant devices (mainly the frequency tuned dynamic shock absorber) studied in section 4.3.2 are often called, whose actions are triggered “automatically”. The practical developments of active controls which are still in the experimental stage concern the additional masses situated at a height (top of the building or center of gravity) or on bracing elements. The development of steering software requires work based on the theory of automatic control, for acquiring data (number and
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location of recording points) and simulating structural behavior (considering notably the nonlinearity influence) as well as for compensating for the delays due to the inertia of the mechanical actions. It can be shown [YU 00] that the optimum for an active system acting at the level of the roof involves reproducing an absorbing boundary condition (sections 5.3.2 and 16.1.2) which avoids sending the ascending seismic wave transmitted to the building by the ground movement back down. The doubts raised at the end of section 18.3.2 concerning the reliability of earthquake-resistant devices against the complex action mode are especially applicable to active control at least in the minds of most of the professionals of “ordinary” seismic engineering. However, it should be recognized that the most immediate objections have answers which force us to contemplate their genuineness, for example: – the dependence with respect to a source of energy for the control of the jacks. The problem of availability of this energy during an earthquake can be solved by using reservoirs under pressure, which provide sufficient autonomy (several minutes) even in the case of loss of external sources; – the exceptional character of the seismic loading is not a problem a priori because the system works permanently under the action of the wind; it is possible to control its functioning at all times; – the questions on the capacity of the system to react properly to seismic movements having unusual characteristics are groundless, more so where the software to control the jacks depends on a real time analysis of the seismic movement and the response of the structure and not on preset movements. In spite of these rebuttals it cannot be imagined, at least in the near future, that the earthquake-resistant safety of a building or industrial equipment will depend entirely on the active control system. For the few buildings where such systems have been installed on an experimental basis, the declared objective is mainly to reduce the discomfort to occupants in the case of strong winds. 18.4. Earthquake engineering practices for special risk
Section 18.2.1 dealt with the differences in approach in seismic prevention based on whether the buildings or concerned structures belonged to the normal or special risk category according to the terminology formalized in France by the decree of 14 May 1991.
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Seismic Engineering
This aspect will not be discussed in section 18.4 which shall simply give a brief description of some of the practices used for structures such as nuclear power plants, industrial plants which deal with dangerous products in large quantities and large dams. It is important to note the use of the word “practices” in the title. In fact, the “strict” approach which characterizes the special risk (see section 18.2.1) has given rise to texts describing technical procedures (such as the American Regulatory Guides or the French fundamental rules of safety in the nuclear field) which have in reality a very weak regulatory status (that of a ministerial circular in France). This leaves the user, at least theoretically, with the possibility of applying another procedure provided its validity is justified. In addition, the draft for these documents often does not specify the details and can lead to misinterpretation. As a result it is necessary to talk of “practices”, i.e. the know-how of the jurisprudence of enforcement of a rule (see section 7.1.2 for seismic hazard in French nuclear power plants) rather than the strict enforcement of a precise formal text. All attempts to give a higher status (law, decree, order) to documents regulating seismic prevention for special risk have given rise to texts which although laying down certain general principles still remain very vague regarding the practical modalities of enforcement (section 18.4.2 on the order of 10 May 1993). In what follows, emphasis will be on French practices, but also those already in force in other countries mainly in the USA and Japan will also be mentioned because they have often served as references. Nuclear facilities (section 18.4.1), chemical factories (section 18.4.2) and dams (section 18.4.3) will be studied successively. 18.4.1. Nuclear power plants and facilities
Earthquake engineering practices for nuclear power plants have been greatly influenced by the practices in force in the countries of origin of different reactor systems. Thus the plants depending on the light water system (pressurized or boiling) which are by far the most numerous in the global nuclear park, have adopted, in many of the countries using this type of reactor for producing electrical energy, the principles and rules of earthquake-resistant design developed in the USA in the 1970s. In certain aspects such as determining the seismic hazard, many of these countries including France have defined specific national practices which do not reflect the American influence. However, if the concrete realities (i.e. the spectrum actually taken into account for designing, the calculation methods and verification criteria) are examined, they definitely have a common origin. That is why in France all active nuclear plants have been designed either with the USNRC spectrum
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(Figure 9.5) or with a spectrum derived from recordings in California by using the elastic analysis mainly inspired by American engineering practices. Japanese practices are significantly different (mainly in the classification of buildings and equipment depending on their importance for safety and the continuity with normal risk that was mentioned in section 18.2.1) and so are the practices in eastern countries (where the nuclear plants designed during the time of the USSR rarely took into account the seismic risk and in the cases where earthquake-resistant measures have been taken, they have resulted mainly from the use of seismic coefficients). In terms of principles, the American approach retains two levels of seismic aggression for designing facilities. The highest is indicated as SSE (Safe Shut Down Earthquake) which indicates clearly that this level refers only to the safety objective (stopping the reactor and maintaining the safe shutdown condition after the earthquake). The link with the regional or local seismic hazard should correspond to a very low probability of exceedance which is, however, not clearly specified (the practice in force in the USA for determining hazard at that time, i.e. in the 1970s, was mainly deterministic). This principle of taking into account the “maximum” tremor just for nuclear safety (and not for continuing operations) involves the study and analysis of all the facilities to draw an exhaustive list of structures and materials which go towards maintaining safety either directly or indirectly (when their destruction or damage could jeopardize the proper functioning of the elements directly instrumental in safety). This demand for a “safety analysis” specific to the facility under study is certainly one of the major contributions of the nuclear industry to seismic prevention. If the notion of SSE is unambiguous and has been accepted (often with other acronymns, such as SMS for “séisme majoré de sûreté” in France which means “increased earthquake for safety”) by all the countries, the situation is not the same for the second level, or OBE (Operating Basis Earthquake), of the American approach. OBE shows that this level which is lower than SSE (reduction factor with respect to the SSE, taken initially equal to ½) is aimed at maintaining the functioning of the nuclear plant after the earthquake; the practical applications of its definition have given rise to a certain number of questions and difficulties: – does maintaining the functional capacities concern all the parts of the installation (including those such as the AC turbo generators or cooling towers which have no safety role) or only the elements which contribute to safety during operation (which are greater in number than those necessary for a safe shutdown, already aimed at by SSE)? In the strict sense of an operational earthquake it is the first option that should be retained but the second is equally justifiable as it enables quick resumption of work (under nominal safety conditions after possible repairs of the non-classified elements which would have been damaged);
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Seismic Engineering
– continuity in the supply of energy by a power producer even during an earthquake can be imposed through regulations by the government. However, this does not mean that each nuclear plant run by this producer will be bound by these regulations; it is for the producer to define the actions to be taken to ensure such a mission depending on the structure of his network and his power resources considering the fact that an earthquake, even a violent one, will affect only one part of the region serviced by them. Until now, it has been the responsibility of the owner to take the correct decision to ensure the functioning of an electrical power plant after an earthquake (which involves certain technical problems still to be solved such as reducing the vulnerability of ceramic insulators; see section 12.2.6 and Figure 12.64); – the significance of OBE has given way to different interpretations which continue to feed the controversies. Initially this level of seismic aggression was supposed to represent events with a relatively high probability of occurrence (in any case higher than that associated with SSE) during the operating life of the plant; the undisputable logic of this idea was destroyed by the imposition of the value ½ for the ratio between OBE and SSE; it is obvious that this ratio depends on the tectonic context of the region of the site and that the choice of ½ makes sense only in the highly active zones (in the intraplate zones of moderate seismic activity, i.e. in a large area of Europe and also in the eastern USA this ratio should be clearly lower); – this ratio between OBE and SSE has been further aggravated by the fact that by taking ½ it is the OBE that becomes the seismic event that governs the design of mechanical equipment to a large extent (specifically the nuclear reactor itself, that is the most important element from a safety point of view). This paradoxical situation arises from the hypothesis on damping (which is lower for OBE than for SSE; see Table 15.2) and the stricter justification criteria in OBE because of which equipment verified for OBE fulfills ipso facto the verifications with respect to the SSE, which is for that matter an earthquake twice as strong. It was therefore very difficult for the American authorities to question the choice of the value of ½ which would mean accepting a reduction in safety margins. The de facto desertion of the nuclear program in the USA, at least regarding new plants, has avoided the obligation of “clearing the air” but it can be observed that some of the last American nuclear plants have in fact been authorized with an OBE/SSE ratio of only 1/3 (which practically eliminates the importance of the OBE in the designing). Apart from the USA, the countries which have continued to construct nuclear plants using light water reactors have followed different practices when it comes to the two levels of earthquake; the subject is delicate and it is necessary to distinguish between the officially proclaimed doctrine of safety and the actually followed design practices. In France, the SMS level determined by the procedures of RFS 1.2.c (section 7.1.2) corresponds, according to this document, to “the most aggressive earthquakes to be retained for the evaluation of the seismic hazard while designing
Technical Aspects of Prevention
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an installation”. These installations should be “constructed for the seismic stress envelopes of those induced by the movements associated with the SMS”, in such a manner that “the continuance of the important safety functions such as the shutdown of the cooling and containment of radioactive products during and/or after plausible earthquakes which can affect the concerned installation site, can be ensured”. Comparing the French SMS with the American SSE is therefore totally legitimate. However, in the case of earthquakes of an intensity lower than the SMS, the RFS I.2.c simply states that “the SMSs can be preceded or followed by quakes capable of reaching the level of SMHV” (i.e. the maximum historically plausible earthquake whose macroseismic intensity level is lower by one degree on the MSK scale than that of the SMS; section 7.1.2). There being no further precision either in this RFS or in any other, on the seismological significance attributed to these earthquakes (events that can happen during the operating period or aftershocks of a SMS), or on the practical design rules to be associated with them, this type of declaration mainly contributes to confuse people. As a matter of fact, the dimensioning is carried out with respect to SDD (Séisme de Dimensionnement – design earthquake, whose spectrum surrounds that of the SMS) for all buildings and equipment having a safety function and also with DSD (Demi Séisme de Dimensionnement – half design earthquake) for most mechanical materials. Furthermore, in the qualification procedures of electrical materials on shaking tables (section 18.1.2), the trials at the SDD level are followed by several trials at the DSD level. These design practices indicate more a continuity of “the American heritage” than the assertion of a specific French approach. Currently there is a change in the importance given to the lesser intensity earthquake which could become an inspection earthquake. When its level is exceeded by an actual earthquake the nuclear plant will have to be stopped in order to go ahead with the inspection of important safety elements. The responsibility for choosing this level of earthquake falls on the owner; if he chooses a weak level, he runs the risk of not only stopping the plant frequently but also of not being able to restart it rapidly because he has to prove that the safety functions have not been impaired; it is therefore in the interest of the owner to retain a sufficiently higher level in order to get authorization to restart after a reasonably strong earthquake without incurring additional investment. This approach is undoubtedly more realistic than the one devolved initially on the OBE because the respective responsibilities of the safety authorities and the owner are clearly spelled out. The principle of two levels is retained in the guides published by the IAEA (International Agency for Atomic Energy), with the deliberate choice of neutral acronymns (S1 and S2) so as not to revive the old quarrels related to the OBE. These documents are interesting as they are revised periodically and this enables them to consider the “prescribed analysis of past experiences” but are hardly directional
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because their acceptance by all the member states of this agency implies compromising on certain delicate points [COL 91a]. The Japanese approach as indicated earlier is quite different from the American approach and is interesting due to its elaborate nature of seismic classification. According to the importance of their role in maintaining safety conditions, buildings and equipment are classified into four categories A, B, C and As; the elements classified as A, B, C are verified only for an S1 level earthquake (which is closer to an SSE than to an OBE, but determining this level is governed by specific Japanese rules), with decreasing severity criteria when moving from A to C and which are mainly based on using seismic coefficients. This enables a continuity with the normal risk (which can be identified with the category C) to be established by clearly bringing out the increase of safety factors for nuclear installations (which corresponds to a multiplication by 3 for category A). The elements classified as As whose safety function is the most important are subjected to verification at the level S2 (which is a “marginal” earthquake whose ratio with S1 varies from 4/3 to 3/2 depending on the site) through dynamic calculations possibly taking into account certain nonlinear behavior (even though in practice this possibility seems to have been seldom used in the studies related to designing). The Japanese approach seems more logical than the one adopted in the USA where there is total separation between the special and normal risk. It is regrettable that the American approach is being followed by most countries including France. One of the reasons for this separation is certainly due to the fact that in the 1970s, when the American practices were defined, the earthquake-resistant codes applicable to the normal risk were still, including in the USA, based on the use of seismic coefficients without explicitly taking into account the ductile deformation capacities of the structures (section 8.1.2 and Table 8.1), following empirical formulae. It would have seemed safe for the potentially dangerous structures like nuclear power plants to follow a completely different and more secure path based on intensive calculations in the conditions where it is best controlled, i.e., in the field of linear behavior of materials. Thus we have just ended up in a set of design practices which forms a complex chain whose different links mobilize actions belonging to different disciplines. Figure 18.19 is a (simplified) flow diagram of the seismic studies for PWR plants (pressurized water reactors) which were built in France. There are a number of stages to be crossed before arriving at the final stage (dimensioning and rating); amongst these steps those which present perceptible uncertainties are: – seismo-tectonic studies (identifying seismogenic structures and characterizing source zones);
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– calculation of the design spectrum in the free field (dispersion of the correlations enabling the calculation of the spectral ordinate, influence of the depth of hypocenters, site effects); – modeling of the buildings and ground (soil-structure interaction, representation of the damping); – calculation of the floor spectra (definition of the accelerograms tuned to the ground spectrum, smoothing and enveloping procedures); – modeling of the equipment (damping value, representation of the support conditions); – design criteria (effective margin with respect to failure). In general those in charge of one of the actions in this chain solve the uncertainty problem at their own level by referring to documents such as the Regulatory Guides or the fundamental rules of safety or by abiding by practices that are not formalized but recognized as acceptable by the profession. The uncertainties are not made explicit but often prudent choices are preferred which can however vary from country to country. For example, as indicated in the notes of Table 15.2, the damping considered for the piping is weaker (by a factor of 4 or 6) in Japan than in the USA. This difference cannot be attributed to technology implemented in the two countries. Considering the number of steps to arrive at the design values, the independent management of the uncertainties at every step and the general tendency to be on the safe side, a high degree of conservatism is normally observed at the end of the chain. It is this “piling up of safety factors”, along with the almost exclusive practice of elastic calculation and the utilization of the static equilibrium criteria of the forces, which basically ensures the seismic safety of nuclear power plants and it is not, as is commonly believed, the only choice of designing seismic movements. It has been seen in section 18.1.1 that, for piping, these design practices ended up with a considerable margin (factor of about ten) with respect to the risk of destruction under the effect of the forces of inertia.
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Figure 18.19. Simplified flow diagram of the studies for the earthquake-resistant designing of the French nuclear power plants using pressurized water reactors (PWR)
Given the existing knowledge on seismic movements and the dynamic response of structures it should be possible to improve nuclear power plant design, which was done mainly based on the knowledge of the 1970s, significantly. The improvements could involve [BET 90b]: – adopting a more physics-based process using the existing analysis of past experience for defining behavior patterns and numerical models especially concerning the real mode of damage and design criteria; – explaining the uncertainties at every stage so as to ensure coherence in the choice of safety factors (which can be incongruous because of the present practice where different groups take decisions independently) and to assess as objectively as possible the residual risk at the end of the dimensioning chain:
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– introducing a certain dose of probabilism in the approach so that the uncertainties mentioned in the previous section can be handled rationally; – looking for simplifications in most of the calculating methods with a view to facilitate the quality control of the designs and to identify, as in the Japanese approach, the level of prevention obtained compared to that required by the regulation applicable to normal risk structures. Developing a new methodology along these guidelines would probably be just a lip service considering the volume of work to be put in and the present situation of the nuclear industry in most of the countries where it is the activities connected to the reanalysis of existing installations that keep the nuclear analysts busy more than the development of new projects. Under these conditions, as mentioned earlier, revising design practices cannot become a priority more so when their present state is considered satisfactory from the point of view of safety. We shall discuss in section 18.5.1 the seismic diagnosis of existing installations, for which the application of “design type” calculations produces very pessimistic results. 18.4.2. Chemical, oil and gas plants
In France, the order of 10 May 1993 “fixing the earthquake-resistant rules applicable to installations subjected to the legislation on classified installations” has proceeded based on a political will to bring these installations (which belong mainly to the chemical, oil and gas industry) in line with nuclear power plants in connection with the prevention of risks associated with earthquakes. The text of this order is restricted to putting forth a few principles (study of seismic hazard based on the notions of SMHV and SMS (see section 7.1.2), the need for a safety analysis of the installation, justification by the owner of the maintenance of the safety functions in case of an earthquake) which correspond to those of the nuclear industry; on two aspects it deviates from its model; it involves the possibility of using a standard spectrum (instead of a spectrum resulting from a special study of the hazard) for the sites situated in zones O and Ia of the zoning applicable to the normal risk (Figure 7.3) and the possible recourse to encroachment of the plastic range for certain justifications provided there is compatibility with the safety function of the element under consideration. This order posed and continues to pose a certain number of application problems as it has no backing from any established industrial practice (nor substantiated by the existence of technical documents which can be used as guides) unlike for the nuclear power plants. That is why a circular from the Ministry of Environment, published on 27 May 1994, has tried to provide details on the methods to be used to determine the hazard,
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establish the safety functions and carry out the explanatory calculations for concerned users (industrial and administrative). For safety analysis it also contains a classification of the behavior demands in the following categories: – stability, imposed on the elements for which it is sufficient to prevent the risk of collapse, or certain portions falling, to avoid damage to equipment or adjacent structures; – integrity, imposed on the elements which should maintain certain passive functions (for example, the watertight integrity of a wall); – operational capacity, for the mechanical elements through which fluid flows. These elements should have a guaranteed deformation limit in order to ensure maintenance of the liquid rate of flow and, generally, prevent any obstacle in the fulfillment of the safety functions; – usability, imposed on the elements which should maintain certain active functions (for example the closing of a valve). However, such a classification will have some practical value only if it spells out some acceptability criteria (in terms of stresses and/or deformation) which are themselves linked to the calculation methods. On these two grounds the circular still remains quite a qualitative text: for example, when it states that “the approach using behavior coefficients is effective when the stability, integrity and even the operational capacity are looked into” without mentioning the order of magnitude of the acceptable values for these behavior coefficients. The conditions for a “reasonable” implementation (i.e. a satisfactory one from the point of view of public safety as well as of the thoroughness of the technique used) of a text that avoids any “commitment” on a quantitative level depends naturally on the progressive planning of a rule of “good practice”. Unlike the French approach of being in line with nuclear installations, other countries seem to follow a different path when it comes to applying seismic prevention rules to petrochemical installations. Since 1980, Japan has an earthquake-resistant design code for factories producing pressurized gaseous products (including phosgene, nitrogen dioxide, chlorine and fluoride). This code is based on the use of seismic coefficients and thus belongs to the “same family” as the codes applicable to buildings and structures at normal risk. The USA, which has recently (around 1990) taken into account the problem of potentially dangerous installations (other than nuclear), has defined the prevention measures concerning them in the latest earthquake-resistant codes which should merge into the International Building Code (IBC).
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These installations would be considered as a part of the intermediate category between buildings and “ordinary” structures (that can be considered as normal risk in France) and the structures termed “sensitive” (such as nuclear). At the technical level the values of the behavior coefficients (varying in general in the range 2–3) have been proposed; these can rely upon documents published earlier by professional associations such as ASME (American Society of Mechanical Engineers) or the API (American Petroleum Institute). 18.4.3. Dams
Amongst civil engineering structures that can fall under the special risk category, dams belong to a category for which there is abundant analysis of past seismic experience. This is due to their number (according to the world register of the International Commission on Large Dams, nearly 13,000 dams of a height more than 15 meters have been commissioned since 1900), to the duration of their operating period (the oldest date back to more than a century) and to the fact that most of them are constructed in earthquake prone mountainous regions. This analysis of past experience is generally good because we have yet to come across any instance of a large dam being completely destroyed (releasing all the retained water downstream) under the action of an earthquake even though certain cases occured in the epicentral zone of seismic activities of high magnitude producing very violent movements. Figures 18.20, 18.21 and 18.22 show three dams which have gone through major tremors without any rupture (but with minor damage). However, a few examples of complete destruction of low height dams constructed with mining waste fillings are known. Ultimately it is the type of dam and the special conditions of their site, mainly their foundations, which determine the potential risks related to earthquakes; these are now systematically taken into account for important structures.
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Figure 18.20. Lower Crystal Springs dam. This 47 m high arch-gravity type dam survived the major San Francisco earthquake (18 April 1906) without damage even though it is situated about 400 m from the San Adreas Fault. On the right, its structure in blocks with meshing is shown
Figure 18.21. Sefidrud dam situated less than a kilometer from the fault responsible for the Manjil earthquake (Iran) of 2 June 1990 (magnitude 7.7); this buttress dam suffered some damage (look at the crest) but its stability was not impaired
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Figure 18.22. Pacoima dam near Los Angeles
The 125 m high arch dam at Pacoima has become a legend in the seismographic history of strong movements by providing an acceleration of 1.2 g for the horizontal components during the San Fernando earthquake (9 February 1971). This record remained untouched for a long time. This value was exceeded during the Northridge earthquake (17 January 1994) where a seismograph placed on the crest of the structure recorded peaks of 2.3 g in horizontal and 1.7 g in vertical. The similar minor damages observed in the two cases consisted mainly of cracks at the concretestone contact on the left bank and some rock falls [COL 94b]. The methods of seismic analysis of dams have been the subject of advice documents published by the ICOLD (International Commission on Large Dams); they vary a lot depending on the type of structure and the level of complexity of the models. These can be very simple (at least for the preliminary project studies) in cases where plane two dimensional drawings are sufficient, like the gravity-dams made of concrete or embankments of earth fillings whose crest is rectilinear and the profiles are constant over a large portion of the length. This profile being closer to a triangle the eigenmodes of vibrations can be easily calculated, at least approximately, using the established analytical formulae for a triangular beam working under shear force. Thus, in the case of an homogenous material, eigenfrequencies fn can be found depending on the height H and velocity c of the
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shear waves: gn =
]n c 2S H
[18.64]
where ]n are the zeros of the Bessel function of the first kind and of order O (]1 = 2.405; ]2 = 5.520, ]3 = 8.654, etc.). By applying Rayleigh’s approximation ([9.84]), by taking a linear profile for the deformed shape of the first mode, f1 becomes [POS 85]: g1 =
6 c 2S H
i.e., a result close to that of [18.64] (because 2.405).
[18.65]
6
2.449 as compared with ]1 =
In the case where it is accepted that the material modulus increases proportionally with the square root of the depth (measured from the crest), which is a more realistic hypothesis than that of the homogenous material for earthfill dams we get for the eignfrequencies [GAZ 80]: gn =
n c 3H
[18.66]
by taking c as the value of the velocity of waves at the base of the dam. Expressions [18.64] to [18.66], which do not bring the slopes of the dam faces downstream or upstream into the picture, provide simple ways of estimating the accelerations sustained by the structure based on the spectrum that represents the seismic action. These simple methods are sufficient to study the stability of the structure considering the effect of water (horizontal thrust due to the retention and vertical force due to pressure acting on the base) but they should be substantiated with more elaborate analysis when it comes to assessing the liquefaction risk of the earth fill (which has caused several cases of rupture mentioned earlier). As indicated in section 17.1.1, earth fill dams are one of the rare cases where actual nonlinear calculations using accelerograms are carried out in order to check the design. Dams with curved cress (arches or curved gravity dams) and those which close narrow and deeply embanked valleys depend on the principle of three dimensional models whose complexity cannot be compared with that of the methods just discussed. The representation of the dynamic effects of the retained water (which in the model diagram is often treated using analytical formulae such as that by
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Westergaard [POS 85]), requires a joint calculation for fluid-structure in which the effects of compressibility of water cannot be ignored. This leads to very “specific” models that can be handled only by a few specialists. The above considerations give a brief idea of the specific nature, diversity and different degrees of complexity of seismic studies related to dams. It appears almost impossible even undesirable to regulate the contents of this study apart from fixing the hazard level and the general objectives of safety. 18.5. Seismic diagnosis and reinforcement of the existing framework
The earthquake-resistant codes for normal risk and the practices in force for special risk that have been discussed in sections 18.2 and 18.4 respectively, apply to new constructions. In principle they are not intended for the assessment of the vulnerability of the existing framework, not even for defining the reinforcement measures which would help reduce it. However, these themes appear more and more as the main stakes of seismic prevention. In fact, the recent implementation of the codes even in highly exposed regions and the low rate of renewal of housing stock (about 2% per year, i.e. an average life of 50 years for constructions in most developed countries) are such that the seismic risk remains significant for a great number of buildings and structures constructed without special precautions against this risk. Recent earthquakes such as the ones in Northridge and Kǀbe which affected regions (California and Japan) considered as references in the matter of seismic prevention have shown that the codes of the 1970s can be insufficient mainly from the point of view of structural details (see section 12.2.2). It may seem strange that the awareness regarding the overriding importance of the existing framework is only a relatively recent preoccupation considering the number of papers on the themes of vulnerability and rehabilitation in the World Conferences on Earthquake Engineering; Table 18.9 summarizes the evolution of this number for the last seven World Conferences [DAV 97]:
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Conference WCEE
Number of papers “vulnerability-rehabilitation ”
on Percentage of the total
New Delhi 1977
10
1.6%
Istanbul 1980
13
1.8%
San Francisco 1984
23
2.8%
Tokyo Kyoto 1988
29
2.9%
Madrid 1992
50
4.2%
Acapulco 1996
104 127
7.6% 8.5%
Auckland 2000
Table. 18.9. Papers on vulnerability-rehabilitation at the WCEE
This late manifestation of interest by the seismic engineering community may be due to the following reasons: – the topic is technically more difficult than that of the new constructions because it involves adapting to diverse realities that are often elusive; – it often requires close collaboration between engineers and representatives of other disciplines (urban planners, architects, architects of historical monuments, sociologists) with whom they are not used to working on new projects; – given the possible economic impact and the difficulty of having a communication policy on such a sensitive topic, the incentive given by governments to pursue this type of study has often indicated a lot of conservatism. 18.5.1. The different aspects of seismic diagnosis
Historically, the notion of seismic diagnosis appeared first as a need for public safety after a damaging earthquake. It involves deciding rapidly which amongst those buildings that have been affected by the tremors are too damaged to be reoccupied given the risk of aftershocks. This type of diagnosis is generally a brief report considering the limited time within which it should be carried out and was formalized for the first time in California through procedures which are now accepted in many seismic regions. The inspected buildings generally have colored posters (green, yellow and red) stuck on them with the following code: – a green poster signifies that a rapid evaluation did not indicate any structural damage; the occupants are allowed to resettle in the building;
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– a yellow poster signifies that the building is damaged but is not in danger for the time being; however the risk of collapse is not excluded in case of powerful aftershocks; the occupants can go in at their own risk to recover their personal belongings but in no way can they live there; – a red poster indicates immediate danger and forbids any one from entering even for a short while and staying in the vicinity of the building. Carrying out rapid inspections to determine these posters and checking the ban on reinstallation which can affect several thousands of buildings imply the excellent organization and cooperation of a great number of qualified people. For the Northridge earthquake on 17 January 1994, the county of Los Angeles had mobilized its own inspectors (around 900) and the Corps of Engineers from the US army (about 300 inspectors). In addition several engineers belonging to professional associations (such as the SEAOC, Structural Engineers Association of California, or the ASCE, American Society of Civil Engineers) or to the inspection services of other cities of the USA volunteered their help. Three weeks after the earthquake more than 60,000 buildings had been inspected out of which 2,061 had red posters and 8,232 had yellow ones [COL 94a]. The inspection teams, each consisting of two people, use a knowledge-based system [COL 94a] which makes a check-list for each building inspected. Strategic buildings such as emergency operation control centers or fire stations are inspected on a priority basis according to specific procedures. Once the emergency situation is over, decisions have to be taken regarding the damaged buildings (repair, or partial or total demolition) based on a more elaborate diagnosis than the one described above. It is necessary to examine certain parts of the structure (foundations, bracing elements hidden by the esthetic covers or insulations) which are not directly accessible and thus escape rapid diagnostic procedures. In section 12.2.3 it was observed that during the Northridge earthquake the problem of welded joints giving way in a great number of metallic structures affected buildings which otherwise looked intact. There is an interaction between the investigation level necessary for the diagnosis and the type of decision to be taken for the concerned building. When damages are recognizably too heavy to warrant repairs the “emergency” diagnosis carried out after the earthquake need not be fine-tuned and the decision to demolish stands. However, for less critical damage with only certain parts severely damaged there is a choice between repairs to bring it back to the “original state” and reinforcement by replacing the old damaged parts with new ones. In the second option the structural drawing of the building should be redesigned taking into account the bracing or weight supporting pattern of the new structure and additional investigations may be limited to the parts belonging to the old structure which play a
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role in the new drawing. If the first option, i.e., repairs to restore the “original state” is chosen then there should be a more in-depth diagnosis. The above observations refer to the immediate or delayed post-seismic diagnosis which is not a part of the preventive measures but which has played and still plays an important role in formalizing the analysis of past experiences. The preventive diagnosis developed considerably in the recent years (see Table 18.9) consists of two different aspects depending on whether it concerns a particular building or the whole of a zone or city. Typical examples of specific diagnosis are given below: – priceless historical heritage monuments requiring major reinforcement work to protect them from seismic threat; – strategic installations for the management of a seismic crisis (emergency relief coordination centers, fire or police stations, hospitals, etc.) corresponding to the normal risk category D in France (see section 18.2.1), but too old to have been designed following earthquake-resistant norms; – installations belonging to the special risk category which were constructed at a time when the earthquake engineering precautions were not imposed through regulations or involved practices that are now considered obsolete. The specific diagnosis, whether carried out with a view to prevent or to decide repairs necessary after an earthquake, aims at providing an assessment of the seismic aggression level beyond which there is no guarantee that the structure will hold. Unlike the immediate post-seismic diagnosis which follows a qualitative approach based on the analysis of past experience, the specific diagnosis should count on the quantitative elements; these result from calculations on models of varying complexity and whose implementation often poses a certain number of difficulties [DAV 97] concerning the following points: – information gathering regarding the building and the soil. This information includes documents prepared at the time of construction (geotechnical reports, implementation plans, calculation notes) but which may not be available easily and additional investigation results intended to verify the conformity of the building, in its current state, with the original drawings and to fill up certain lacunae in the data through in situ measures or on the collected samples; the delicate points in this information collection often include knowledge about the ground in contact with the foundations, detail of the reinforcement in the reinforced concrete parts and state of the joints (which if sealed can cause significant interactions between adjacent structures); – assessment of the degree of participation of non-structural elements in the functioning of the bracing system (section 12.3.1 and 15.1.4),
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– the choice of behavior coefficients which, in principle, cannot fall in line with those of the earthquake-resistant codes applicable to the designing of the new constructions because the structural details of the existing building do not conform to the recommendations of these codes (as indicated in section 9.3.1 the calculation with forces reduced due to the use of a behavior coefficient higher than one will be valid only if the structural details required by the codes are respected). These difficulties confirm that the critical limit of seismic aggression for an existing building cannot be estimated based only on an approach through calculation and should include expert conclusions drawn from the analysis of past experience. These conclusions could correspond to the identification of fragile elements (by not respecting the appropriate structural details or due to an incorrect design such as using short columns (see section 12.2.2) or connecting parts weaker than the elements that they connect (see section 12.2.3), to the at least partial reference to the notion of macroseismic intensity to place the seismic action taken into account in the calculation with respect to the results obtained from it or to the reasoning through analogy. In France, special risk installations have given rise to purely “computational” diagnoses using linear models of calculation directly inspired by those which correspond to the design practice used in the new structures. Several installations from 1960–1980 have used this analysis which has always led to very low levels of the acceleration spectrum range, of about 0.05 g, beyond which the behavior will no longer be acceptable, at least according to the criteria used in designing. Such a result is hardly credible because these levels of acceleration cannot produce movements that can destroy (Table 14.4) the industrial structures which are generally constructed in a better way compared to average housing constructions (whose observed behavior is at the base of the intensity scales; see section 14.1.3). The consequences of this blind submission to the requirements of a calculating mode unsuitable for diagnosis are the following [COL 95b]: – the almost systematic negative result of this diagnosis makes it difficult to find from amongst these installations those that present a real risk; – in order to maintain an acceptable level of safety the owner is pushed to go for more and more complicated methods of modeling and calculations supposed to identify the available margin in the designing of his buildings; the debate on the details of methods often becomes sterile and loses all link with reality and safety issues; – defining the corrective measures (eliminating fragile elements, reinforcing certain parts, improving the bracing drawing and the links between elements) which would be necessary in a certain number of cases is not made easy by the
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presentation of the results of the calculation which corresponds to a designing logic. For example, for reinforced concrete structures a certain practice favors the presentation of reinforcement maps corresponding to the equilibrium of the forces for the calculation spectrum of the site; the simple fact that with respect to these maps the real reinforcement of the structure is insufficient does not necessarily help to decide the modifications to be carried out to the building. The seismic diagnosis of a particular structure is thus a specialty which requires a lot of experience and good general earthquake engineering knowledge; considering the diversity of cases encountered in practice, it is just not possible to reduce the diagnosis to the application of the drawings based on pre-established calculations in a different context; the field of historical monuments, most of which are huge structures of stonework constructed using techniques that are no longer practiced, now involves a level of expertise that has been developed mainly in the most earthquake prone countries of the Mediterranean basin. As opposed to the special diagnosis which was just discussed briefly, general diagnosis which is commonly known as a vulnerability study, concerns a set of structures having a common construction site (it could be a block i.e. a series of buildings constructed side by side without spacing, a section of a city or even an entire city) or structural characteristics linked to their function (for example schools built under the same reference plan). The aim is to evaluate the damages which could arise after an earthquake of a given level and their socio-economic effects. This type of study is often associated with defining the priorities in terms of reinforcement and investment in the field of civil safety. Most of the methodologies proposed for these vulnerability studies consist of the following stages [DAV 97]: – determining the seismic hazard based on a microzonation study (see section 7.2.3) aimed at identifying the zones affected by the appearance of a site effect or induced phenomena (liquefaction, landslides). The importance of this study depends on the scale adopted (block, section of a city, city) for the evaluation of the vulnerability of an urbanized area; when it involves the vulnerability of a set of similar buildings constructed on different sites (schools mentioned above), the principle of determination of the seismic hazard based on the general provisions of earthquake-resistant codes can be adhered to; – inventory of the constructions found in the zone in question and classification of these following a typological approach; – establishing vulnerability curves for each of the families of the typological classification. These curves will give the variation of a damage index following a parameter characterizing the level of seismic aggression (zero period acceleration of
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a spectrum or macroseismic intensity) and they can be drawn for different levels of probability of occurrence of the damage; – evaluation of the damage based on the level of earthquake at habitations of the zone studied and the vulnerability curves which characterize them; – evaluation of the consequences of these damages according to the ultimate purposes which can be very different from the vulnerability study: assessment of the human and economic losses, functionality of the emergency services (fire service, police, research and rescue teams in collapsed buildings, emergency medical assistance) and hospitals, importance of the measures to be taken for providing the homeless with temporary accommodation and items of vital need (water, food supplies, heating, etc.). Drawing the vulnerability curves which represent the quantification tool for this type of study along with the hazard derived from the microzonation, can be carried out, in the simplest cases where the seismic aggression is characterized by its intensity, based on the description of the damage associated with the different degrees of the scale used. This method is unsuitable for buildings that depart too much from the classification adopted in the scale (such as classes A, B and C of the MSK scale which are insufficient for most modern constructions, as seen in section 14.1.3). It is then necessary to use calculations for tracing these vulnerability curves. Certain documents [COL 87] and [COL 92c] have suggested sample curves based on a combination of calculations and expert conclusions, for most of the buildings encountered commonly; two examples of such curves are given in Figure 18.23 [DAV 97].
Figure 18.23. Vulnerability curves drawn on an EDF diagram (Expected Damage Factor in ordinate) – I (Intensity according to the Mercalli scale modified in abscissa) for light wooden structures (on the left) and non-reinforced stonework (on the right); according to [DAV 97]
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Other approaches can be used for the determination and graphic representation of the vulnerability curves. For example, to characterize the seismic aggression level using the zero period acceleration A of a response spectrum, the relation between the damage factor E (represented as the ratio between the cost of repairs and that of the construction of the building) and A using a series of curves each one corresponding to a value of the vulnerability index IV [DAV 97] can be expressed. This index whose numerical values are conventionally limited to the interval 0–100 is arrived at by calculating a weighted sum of the grades attributed to a certain number of criteria characterizing the type of building (some of which are the nature of the materials used, structural drawing, number of floors, regularity, age of the structure, etc.) For a given value of Iv, the relation between E and A is determined by calculations that enable the assessment of the damage (as the push-over method described in section 17.2.4). Figure 18.24 gives an example of these relations [DAV 97]: This figure shows that relations E-A have been schematized by tri-linear diagrams (no damage below a value AO and total damage above a value A1, with a linear variation between these two values, which obviously depend on the vulnerability index IV). This willingness to schematize shows the rudimentary nature of these evaluations which could be masked by taking recourse to sufficiently elaborate calculations. In section 17.2.4 it was observed that the push over method in its present form could lead to an underestimation of damage at the higher levels of the buildings (Figure 17.11) and was not well adapted to irregular structures sensitive to torsion.
Figure 18.24. Relation between the damage factor E and the acceleration of calibration A of the ground spectrum, for different values of the vulnerability index IV (according to [DAV 97])
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On the other hand, the tri-linear schematization, in trying to simplify the calculation of the damage becomes suitable for the sensitivity studies of the variations of the parameters (such studies are always necessary for judging the reliability of the results obtained). Vulnerability studies are generally carried out for a seismic hazard level corresponding to that of the earthquake-resistant codes or to the repeatability scenarios of a famous historical earthquake; as examples of this second possibility the study carried out for Lisbon can be mentioned (simulation of the big earthquake of 1 November 1755 in case it reoccurred today [OLI 96]), or the study related to the last highly damaging earthquake that struck France (Rognes-Lambesc 11 June 1909); in the latter case the simulation carried out in 1982 provided the following estimates [LAM 97]: Simulation 1982 Number of deaths Number of those wounded Direct costs
400 to 970
46
1,850 to 5,650
250
4,660 million Francs
1,500 to 2,250 million Francs (value in 1982)
400 to 500 million Francs
Not possible to estimate due to lack of data
(repair and reconstruction) Indirect costs (economic impact)
Real earthquake (1909)
This type of simulation, even though its results are tainted with uncertainties, enables us to assess the impact that a new occurrence of these catastrophes could have at the human and the economic level. It can be observed that the vulnerability seems much higher in 1982 than in 1909 which is a frequently observed tendency in this type of study mainly because the increase of urbanization was not accompanied by a significant improvement in the safety of current constructions. The methodologies of the vulnerability studies can also be used for purposes other than the assessment of damage suffered by a set of existing buildings. The damage-acceleration relations of the type shown in Figure 18.24 can be applied to future constructions designed and built following the recommendations of an earthquake-resistant code with a view to assessing the level of protection chosen by the code: adequate, permissive or overdemanding, considering the seismicity of the region under study.
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It can be assumed that a structure designed according to an earthquake-resistant norm has a tri-linear vulnerability curve (Figure 18.24) corresponding to a low value of the vulnerability index IV and that the nominal acceleration for the application of this norm is included between A0 and A1 (by being normally closer to A0, than to A1, even if the objective officially proclaimed by the codes is that structures do not collapse). In this hypothesis a simple calculation helps in determining the cost of the seismic damages to be foreseen for a homogenous group of buildings having this vulnerability curve and constructed in a zone of diffused seismicity (section 7.2.5); this cost estimate provides an element of judgment in choosing the nominal acceleration. The following hypotheses are made: – the seismicity of the region is described by an untruncated law by GutenbergRichter, i.e., the annual number N (M) of earthquakes of magnitude equal to or higher than M is given by (section 6.11): N (M) = 10a-bM
[18.67]
a and b being two constants (b close to 1); – in conformity with the hypothesis of diffused seismicity, the depth h, of the hypocenters has a uniform distribution between a lower limit h1 and an upper limit h2; – the law of attenuation of acceleration A has the following form: A = C eDM/R
[18.68]
R being the focal distance, C and D the constants; the “normal” value of D is ¼ Ln10 = 0.576 [5.36]; – the buildings in the diffuse seismicity zone that are assumed to be identical and distributed uniformly (at the rate of n per unit area) have a tri-linear vulnerability law, i.e., the cost cd of the damages caused by an earthquake producing an acceleration A is given by: cd = 0 for 0 d A d Ao
A Ao cd = co A A for Ao d A d A1 1 0 cd = co for A1 d A co being the construction cost of the building.
[18.69]
Technical Aspects of Prevention
923
By taking the case of an earthquake of magnitude M whose hypocenter is at a depth h, the focal distances Ro and R1 corresponding to the values Ao and A1 of the acceleration become equal to: Ro =
C DM e ; R1 Ao
C DM e A1
Ao Ro A1
[18.70]
There are three hypothetical cases to be foreseen to determine the damage caused by the earthquake (Figure 18.25).
Figure 18.25. The three possible cases of damage caused by an earthquake whose hypocenter is at a depth h
1) Ro < h acceleration A at the surface is lower than Ao everywhere and there is no damage. 2) R1 < h < Ro acceleration A is more or equal to Ao (but lower than A1) inside a circle with center at the epicenter and of radius R02 h² ; the total cost Cd of the damages is therefore, according to [18.69] and [18.68]: Cd
³
R02 h ²
0
ª Ro º nco Ao « 1» 2S rdr A1 Ao ¬ r ² h² ¼
[18.71]
r being the epicentral distance. For Cd, the elementary calculation for this integral gives: Cd = S nco
Ao 2 Ro h A1 Ao
[18.72]
924
Seismic Engineering
3) h < R1 the acceleration is higher than A1 inside the circle of radius and
included
between
AO
and
A1
in
the
ring
domain
R12 h ²
represented
by R h² d r d R h² ; 2 1
2 0
the total cost Cd of the damage is:
Cd = nco S R12 h ² ³
R02 h ² R12 h ²
nco
º Ao ª Ro 1» 2S rdr « A1 Ao ¬ r ² h ² ¼
[18.73]
i.e., after a basic calculation: §A · Cd = S n co ¨ o R02 h² ¸ A © 1 ¹
[18.74]
From expressions [18.72] and [18.74] of Cd it is possible to calculate the cost Cd,a (h) of the damage caused every year (on an average) by all the earthquakes put together occurring at depth h, irrespective of their magnitude, which however has to be higher than the limit MO represented by: Ao =
C D Mo e h
[18.75]
in order for there to be damages; the limit M1 is also defined as: A1 =
C D M1 e h
[18.76]
By using equation [6.18] for the distribution density of earthquakes according to their magnitude and equations [18.72] and [18.74] of Cd, which are valid respectively for Mo d M d M1 and for M t M1, for Cd,a (h), the following equation is arrived at: Cd,a (h) = (b Ln 10) 10a S nco x ª M1 bM Ao º f §A · Ro h ²dM ³ 10 bM ¨ o R02 h ² ¸ dM » « ³M 10 M 1 A1 Ao © A1 ¹ ¬« o ¼»
[18.77]
Technical Aspects of Prevention
and for the total annual cost Cd, distribution along the depth: h2
Cd, a, t = ³ Cd , a (h) h1
a, t
925
of damage, considering the hypothesis of equal
dh h2 h1
[18.78]
By replacing in [18.77] RO by its expression [18.70] with respect to M, the integrals whose calculations are basic are obtained and finally using [18.78] the following is obtained: Cd, a, t =
ª Ao 1J § A ·1J º h3J h3J 2 «§¨ ·¸ » 1 ¨ 1 ¸ C C h h A A J 1 J 2 J 3 « » 1 0 ¬© ¹ © ¹ ¼ 2 1 2S nco 10
a
C
[18.79]
with: J=
b
D
[18.80]
Ln10
This parameter J whose “normal” value corresponding to b = 1 and D = ¼ Ln 10, is 4, has already been seen in section 7.2.5 ([7.43]) where it is included in addition to factor E of geometrical enhancement, which has been taken here as equal to 1 [18.68]. With J = 4, formula [18.79] gives: Cd, a, t =
S 3
4
nco 10
a
C A12 A1 Ao A02 h1h2 A13 A03
[18.81]
To this cost of damage, that the application of an earthquake-resistant code aims to minimize but cannot eliminate, the cost of protection should be added, i.e. the famous “earthquake resistance supplementary cost” (see the introduction to Part 7); a plausible hypothesis consists of proposing an expression of the following form for this supplementary cost cp: G
§A · cp = co ¨ o ¸ ¨ Ap ¸ © ¹
[18.82]
Ap being a reference acceleration and G an exponent which is probably higher than one because the supplementary cost increases rapidly with the expectations of the protection; with for example G = 2, AO = 2m/s² and Ap = 10 m/s², Cp represents 4% of the construction cost Co of the building which is the order of magnitude (probably a little overestimated, refer to the introduction to Part 7). By multiplying by the total
926
Seismic Engineering
number of buildings in the region under study and by the annual renewal rate r of the housing stock, the total annual earthquake resistance supplementary cost Cp, a, t comes to: G
§A · Cp, a, t = n 6t r co ¨ o ¸ ¨ Ap ¸ © ¹
[18.83]
6t being the total area of the region. By introducing the notations: k=
A1 ;K A0
2S 10a 1 k 1J h13J h23J J 1 J 2 J 3 k 1 h2 h1
[18.84]
and taking the sum of Cd, a, t and Cp, a, t as the annual cost Cs, a, t of the seismic risk in the region, we obtain: G ª § C ·J §A · º Cs, a, t = n co « K ¨ ¸ r ¦t ¨ o ¸ » ¨ Ap ¸ » « © Ao ¹ © ¹ ¼ ¬
[18.85]
This expression of Cs, a, t shows that if Ao is made to vary by taking a constant value for the ratio k = A1/AO, there exists an optimum value A0* for which the cost of the seismic risk is minimal; canceling the derivative of Cs, a, t with respect to Ao gives for A0* : 1
A0* Ap
ª J K º J G « » ¬ G r ¦t ¼
J
§ C · J G ¨¨ ¸¸ © Ap ¹
[18.86]
By taking the values J = 4 and G = 2 and by using expression [18.84] for K, A0* becomes: A0* Ap
ª 2S 1 k k ² 10a º 3 « » k r ¦ t h1h2 ¼ ¬ 3
1/ 6
§ C · ¨¨ ¸¸ © Ap ¹
2/3
[18.87]
and for the ratio of Cs , a ,t to its minimum value Cs*, a ,t obtained for Ao = A0* : Cs , a , t Cs*, a ,t
4
1 § A0* · 2 § Ao · ¨ ¸ ¨ ¸ 3 © Ao ¹ 3 © A0* ¹
2
[18.88]
Technical Aspects of Prevention
927
It is observed that the exponent 1/6 of equation [18.87] is such that the influence on A0* of the choice of the values of most of the parameters (k, r, h1, h2) is not very significant; for example, if h1 is divided by 2, A0* increases only by 12%. The most significant parameter is the constant C of the law of attenuation [18.68] which is raised to the power of 2/3, in the same way as the reference acceleration Ap. With the values: k = 3; r = 0.02; C = 1m²/s² ([7.20]); 6t = 40,000 km²; h1 = 5 km; h2 = 20 km; AP = 10 m/s² For a zone of moderate seismicity, A0* (a = 4 i.e. a centennial earthquake of magnitude 6) is: A0*
1.53m / s ²
[18.89]
and in a zone of strong seismicity (a = 6 i.e. a centennial earthquake of magnitude 8): A0*
3.29m / s ²
[18.90]
Formula [18.88] enables us to study the variation of the cost with respect to Ao; Table 18.10 shows that the cost increases only relatively slowly for Ao higher than A0* , whereas the increase is very fast for values lower than A0* . In other words, over sizing is cheaper than under sizing.
928
Seismic Engineering *
AO/ A 0
Cs , a ,t / Cs*, a ,t
0.2
208.36
0.4
13.18
0.6
2.81
0.8
1.24
1.0
1.00
1.2
1.12
1.4
1.39
1.6
1.76
1.8
2.19
2.0
2.69
Table 18.10. Variation of the cost of the seismic risk with respect to the acceleration AO of the vulnerability curve (the optimum value corresponds to A0 A0 )
The above exercise is obviously too schematic and simple to represent a real situation; mainly the tendencies indicated for the ratio of 2 of the nominal accelerations between a zone of strong seismicity and a zone of moderate seismicity, or the identification of the law of attenuation as the most sensitive issue in the studies of the hazard are however coherent with what is observed in practice when the national regulations are compared or when the summary of sensitivity studies is carried out for the evaluation of the seismic hazard. 18.5.2. Rehabilitation and reinforcement
Seismic rehabilitation of existing buildings and structures is a general term which can mean carrying out simple repairs of the damage caused by an earthquake as well as reinforcement of the structure to improve its performance. In both cases it is necessary to carry out a detailed diagnosis not only to define clearly the work to be executed but also to direct the strategy of rehabilitation. A simple repair to restore the original state is applicable only to cases where the damages are relatively minor, i.e., the behavior of the structure as a whole can be considered satisfactory. When the damages are major and the volume of repair work is high it is generally possible to bring in some significant improvement to the structural details without significantly increasing the budget. Of course, the views of insurance companies towards these “enhancement” repairs are crucial. A typical
Technical Aspects of Prevention
929
example is that of chimneys which in conventional masonry constructions are often delicate elements and not very well maintained (see section 12.2.6) and represent a significant human risk, even for an earthquake of moderate intensity (VII degree on the MSK scale). Repairing a vulnerable chimney should always make it more resistant and this can generally be done through chaining. When the focus is on reinforcement, the often debated question is the level of seismic performance to be achieved after the work. Several answers are possible depending on the function of the concerned structure, the administrative framework in which the work to be done is placed and the available financial means: – the level aimed at is the same as for new constructions irrespective of the investment amount necessary; this will be the case for a historical monument of great heritage value or for a building designed initially using earthquake-resistant norms but which has undergone modifications in its structure or function (making it move from class C to class D in the French regulation; see Table 18.5); – the level aimed at is lower than for a new construction either because the budget does not allow improvements of the same level or because this lower level is deemed sufficient considering the short duration of the residual life of the installation; – a particular level of protection is not openly aimed at but a decision can be taken to include improvements in certain architectural or structural details to reduce seismic vulnerability while carrying out some other work not necessarily for seismic prevention (for example, renovation of an old section of a city). As indicated in the introduction to Part 7, the earthquake-resistant reinforcement for one building and a fortiori for a group of constructions can lead to heavy expenditure. Currently, it is thus not prescribed by general regulations (which would involve taking a decision regarding the answer to the previous question on the level of protection), in countries that have a seismic prevention policy. Only certain isolated operations can be mentioned regarding certain quarters or sections of cities as well as systematic programs of reinforcement for certain types of buildings and structures (for example, schools and bridges in California). These programs are often stretched over several years because of financial constraints; defining the priorities is therefore essential for carrying out these programs and has constituted till now the main repercussions of the vulnerability studies described in section 18.5.1. From a practical point of view, earthquake-resistant reinforcement is not currently codified by any norms (even though Eurocode 8 has devoted one of its sections to it) but has been discussed in a certain number of documents and guides in different countries; in the USA, the big project launched by FEMA (Federal Emergency Management Agency) in 1989 produced a set of texts ([ROJ 96] and
930
Seismic Engineering
[COL 92c]) giving techniques to be used and identifying the need for research and development. Reinforcement procedures are varied and can be implemented in different ways considering the architectural requirements on the external aspect or the minimization of disturbances for installations which must continue to function during the work. This diversity added to the fact that every example is practically a special case explains why the standardization of reinforcement is hardly possible in practice and may even not be desirable. However, the commonly used techniques can be classified into two main categories [DAV 97]. The first category corresponds to the creation of a complementary structure (in reinforced concrete or using a metallic frame) which can be a partial or total substitute to the existing structure for the bracing function and possibly for the weight-supporting function. This complementary structure includes, in addition to the elements that resist the seismic forces, those which can transmit them to unaffected parts of the existing structure and to the foundations and also those which contribute to a good transfer of forces in the bracing system (such as the floors and the roofing which must act as diaphragms). Therefore, it may be necessary to reinforce certain elements of the old structure, mainly the foundations. The elements of the complementary structure that we frequently come across are: – concrete shear walls which are very efficient for reinforcing structures using cross beams; – St Andrew’s crosses made of steel girders which in certain cases can be placed outside the existing structure; – prestressed cables, whose fixing is often less cumbersome than that of the two previous systems because their connections with the existing structure are limited to its nodes. The second category of reinforcement techniques regroups all the improvement procedures of the elements of the existing structure while still retaining its structural pattern for the balance of forces. For structures made of stone/brickwork the conventional method of reinforcing by chaining or using tie rods is practiced along with the addition of plastic sheets reinforced with carbon fibers or the prestressing of load bearing panels. For structures with cross beams the reinforcement of the concrete columns by lining either with metallic sheaths or with resin consisting of carbon fibers is the most common method. For metallic cross beams whose beam to column connection presents a risk of rupture of the welded parts (see notes in section 12.2.3 on the identification of this problem after the Northbridge earthquake), an efficient procedure consists simply of reinforcing the connection by
Technical Aspects of Prevention
931
welding “hips” in its angles and reestablishing the normal “strong columns-weak beams” hierarchy according to which the plastic hinges should appear first in the beams rather than in the columns, and in any case, before the damage to the connections [YU 00b]. The technique of metallic lining was implemented in a systematic manner in California for the reinforced concrete piers of bridges and viaducts; this program of reinforcement was triggered after the Whittier Narrows (1 October 1987) and particularly the Loma Prieta (17 October 1989) earthquakes which damaged a number of bridges. The Northridge earthquake (17 January 1994) enabled verification of the efficiency of this technique. The 115 bridges of the county of Los Angeles which were reinforced before this earthquake all resisted whereas the bridges which were not reinforced suffered some damage or even collapsed [COL 94a]; the role of lining is essentially to ensure the confinement of the concrete of the pier by averting the mode of destruction shown in Figure 12.18. Finally, we shall conclude this brief presentation on rehabilitation and reinforcement by mentioning the possibility of protecting an existing building by constructing it on special earthquake-resistant supports (see section 18.3); this method has been used for the town hall of Salt Lake City (Utah).
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Index
A
B
acceleration 60 relative 556 vertical 555 zero period 384, 698 accelerogram natural 765 synthetic 388, 765 accident post-seismic 580 seismogenous 319 AFPS90 recommendations 656 aftershock 45 amplification of the soil layers 266 analysis modal 433, 607 safety 901, 907, 908 spectral modal 394, 401, 722 angle incidence 136, 137 of refraction 126, 127 anti-nodes of vibration 405 archeoseismicity 288 aseismic soil creep 481 asperities 232, 233 attenuation depth 256 geometric 136 inelastic 139 of seismic waves 135 azimuth 31, 186, 201
basemat uplift 442, 793 benchmark 768 bracing elements 395, 528, 627 Brune’s formula 81 C calculation response spectra 388 transient seismic 433 Cauchy-Schwarz inequality 38, 681 caving 49, 50, 72, 148 center of gravity 88 character “broad band” 268 “narrow band” 268 coefficient behavior 418, 501 ductility 370, 423, 424 energy loss 139 friction 499, 778 Poisson 115, 240 seismic 365 static 802 column vector of the degrees of freedom 400 of the direction of excitation 400 component horizontal 178 vertical 178 constructive measures 423
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Seismic Engineering
continental collision 15, 18, 19, 25 continental drift 9 convection current 7, 14 convergence zone 15 correlation intensity-motion parameter 595 Wells and Coppersmith 69, 82 cratons 10 criteria deformation 374 stress 374 cumulative effect of charge cycles 443 cut-off frequency 385 D damping critical 377 internal 368, 714, 720 radiative 717 reduced 140, 377 viscous 257, 619 deconvolution 251 deformation energy 132, 896 degrees of freedom of translation 400, 730 dephasing of motion 127 design capacity 427 deterministic method 197, 317 development on the basis of eigenmodes 614, 637 diagnosis post-seismic 916 seismic 913 diaphragm 395, 930 differential displacement 487 digital model 213, 232, 237 displacement drift 383, 441 distance epicentral 101, 120 focal 101, 121 distortion reference 242, 263 strain 240 Doppler effect 46, 203 ductility coefficient 370, 423, 424 Duhamel’s integral 378, 408, 657
E earthquake artificial 49 characteristic 305 deep 23 induced 50 intermediate 23, 25 interpolate 22, 33, 65 intraplate 25, 46, 65 non-tectonic 47 slow 46 superficial 23, 97 tectonic 5 volcanic 48 eigenfrequency 377 eigenmodes of vibration 492, 911 eigenperiod 376, 401, 418, elastic design spectra 427 domain 366, 437 rebound model 56 response spectrum 362, 372, 767 energy balance 61 flow 133 equation Euler’s 829 Fokker-Planck 776, 780 equivalent static force 181, 370, 863 Eurocode 8 279 European Macroseismic Scale 582 extension mechanism 486 F fault continental transform 15, 25, 76 listric 32 North Anatolian 19, 37 reverse 30, 67, 473 San Andreas 19 transform 13, 19, 24 field far 163 free 178 near 163, 168 finite differences 711, 722, 753
Index floor spectra 392 forces of horizontal inertia 365 forecast long-term 312 short-term 309 foreshock 46 foundations compensated 512 deep 512 Fourier series 435, 725 frequency basic 392 dominant 211, 220, 261 resonant 142 friction Coulomb 768, 778, 780, 822 dissymmetric 801, 803, 807 symmetric 800 fundamental mode 405, 621, 645 G generalized Brownian motion 445 geometric attenuation 136 ground displacement 493, 760 spectra 392 H H/V method 270 hazard residual 279 seismic 275 tectonic 343 uniform 325, 351 hazard map 329 historical seismicity 283 horizontal shear force at ground level 368 Hypocenter 38-42, 94 I-J Impedance ratio 124, 214, 253 index of refraction 125 inelastic attenuation 139 instrumental seismicity 294
955
intensity Arias 99, 170, 186 epicentral 319 macroseismic 581 International Building Code 908 International Commission on Large Dams 51, 909, 911 intraplate area 19, 23 isoseismal line 202, 589 isostasy 9 K-L killer pulse 185, 234, 667, 815, 821 kinetic energy 61, 786 landslide undersea 154 lateral resistance 361, 395, 519, 523, 564 lateral spreading 145, 481, 485, 490 law Aki scale 64 attenuation 186 average 86, 196 direct 86 Green 155 Gutenberg and Richter 69, 224, 227 Hooke 58 attenuation laws of intensity 598, 600 Kovesligethy-Sponheuer 601 of refraction 122, 128, inverse 86 lineament 282 linear elastic behavior 372, 395, 709 liquefaction 143, 573 lithosphere 56 localized dampers 434 logarithmic decrement 377 M macroseismic radius 592 epicenter 38, 39, 285 magnitude as per the surface wave 101 as per the volume wave 101 definition given by Japan Meteorological Agency 101
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Seismic Engineering
main shock 74, 238, 561 mantle 7 matrix damping 400, 408, 619, 723, 774 mass 400, 620, 637, 774 rigidity 608, 679, 732, 774 stiffness 400, 637 Maximum Credible Earthquake (MCE) 315, 320 maximum historically probable earthquake (SMHV) 318 microseismicity 29 microzoning 330 mid-oceanic ridges 12, 14, 15, 24 modulus shear 58, 240, 406, 705, 868 Young’s 115, 394, 412, 563, 635, 685, 697, 742, 796 Mohorovicic discontinuity 116, 125 monochromatic excitation 219 motion fault 45, 197, 201, 331 vertical 39, 48, 181, 184 N natural dam 150, 153, 825 neotectonics 282 noise unfiltered white 446 white 386, 445 background 269, 270, 331 non-damped eigenmodes 401, 638 O-P obduction 17 operating basis earthquake (OBE) 624, 901 opposition of phase 131 orogeny 9 paleoliquefaction 144, 294, 297 paleoseismicity 33, 288 partial collapse 513, 526, 527, 533, 583 participation factors 407, 410, 614, 638 Plan for Prevention of Risks (PPR) 333 plastification 418-420, 427 plate subducted 17, 24, 27, 33, 67, 72
subducting 24 plate tectonics 9, 15 Poisson’s law of distribution 299 polarity 13, 39, 185 post elastic behavior 362, 853 post-seismic fire 578 prediction of earthquakes 307 of seismic motion 186 probabilistic method 317, 329 propagation velocity 8, 45, 308 pulsation 174, 215, 225, 799 Q quadratic combination 218 cumulation 185 quadri-logarithmic diagram 382, 383 quarter wave resonator 217, 225 R Rayleigh’s method 652 Regulatory Guide 624, 625 relation Karnik 595, 598 Levret-Albaret 598 Mohammadioun 598 response spectrum of elastic oscillators 375, 386 return period 279, 296 rigid modes of response 428 risk normal 331, 629, 858, 859 seismic 107, 277, 580 special 327, 418, 859, 899 rock fall 72 Rules PS 69/82 368 S safe maximum earthquake (SMS) 318, 901 safe shut down earthquake (SSE) 624, 901 scale intensity 55, 546 JMA 586, 587, 588 Mercalli 582
Index MSK 318, 582, 583 Richter 93, sea floor spreading 12 sea quake 158 second order effects 431 seiche 158 seismic action 359 luminescence 160 moment 55, 60 source 55, 232 Seismic Engineering Design Earthquake (SEDE) 316 Seismic Safety Evaluation Earthquake (SSEE) 316 seismograph 50, 73, 163 seismological epicenter 38, 285 segmentation of the fault 232 signal coherence 491 simple oscillator 375, 408 size classification of the building 368 soil/structure interaction 703 spectral ordinate 389, 392, 905 spectrum design 418, 430, 440 displacement response 451 Kanaï-Tajimi 455 USNRC 184, 389 square root of the sum of the squares 411 standard deviation 195, 231 static equivalent 374 horizontal force 365 stick model 399, 674 stress drop 57, 60, 63, 81 strike-slip left-lateral 31, 37 right-lateral 31, 37
strong vibratory motions 165, 223 structural design spectrum 425 subduction 16, 310 T-U theorem by Kausel 709, 712 Matsushima 681 Newmark 372, 422, 885 torsion oscillations 494 transitory differential displacements 113 tsunami 154-158 V-Z vertical inertial effect 553 vibration nodes 405 Wadati-Benioff surface 24 wave Rayleigh 117, 118 compression 39, 562 incident 123, 134, 215, 220 longitudinal 115 Love 117, 118 reflected 123, 127, 138 refracted 123, 126, 127 seismic 121, 132 shear 221, 760 surface 211 transverse 115, 117 number 119 train 248 wavefront 136, 711 Winkler spring 746 yamatsunami 152 zone of seismic activity 368
957
