TRACK-BRIDGE INTERACTION ON HIGH-SPEED RAILWAYS
© 2009 Taylor & Francis Group, London, UK
Track-Bridge Interaction on High-Speed Railways Editors
Rui Calçada, Raimundo Delgado & António Campos e Matos Department of Civil Engineering, Faculty of Engineering, University of Porto, Portugal
José Maria Goicolea & Felipe Gabaldón Computational Mechanics Group, Department of Mechanics and Structures, E.T.S. Ingenieros de Caminos, Universidad Politécnica de Madrid, Spain
© 2009 Taylor & Francis Group, London, UK
CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2009 Taylor & Francis Group, London, UK Typeset by Charon Tec Ltd (A Macmillan Company), Chennai, India Printed and bound in Great Britain by Antony Rowe (A CPI-group Company), Chippenham, Wiltshire All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publishers. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Published by:
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Library of Congress Cataloging-in-Publication Data Track-bridge interaction on high-speed railways / edited by Rui Calcada . . . [et al.]. p. cm. Includes index. ISBN 978-0-415-45774-3 (hardcover) — ISBN 978-0-203-89539-9 (ebook) 1. High speed trains. 2. Railroad bridges. 3. Bridges—Live loads. I. Calcada, Rui. TF1460.T73 2008 625.1 4—dc22 ISBN: 978-0-415-45774-3 (hbk) ISBN: 978-0-203-89539-9 (ebook)
© 2009 Taylor & Francis Group, London, UK
2008003167
Table of Contents
Preface
VII
List of Authors
1
2
IX
New evolutions for high speed rail line bridge design criteria and corresponding design procedures D. Dutoit Service limit states for railway bridges in new Design Codes IAPF and Eurocodes J.M. Goicolea-Ruigómez
1 7
3 Track-bridge interaction problems in bridge design A.M. Cutillas
19
4
29
Controlling track-structure interaction in seismic conditions S.G. Davis
5 Track-structure interaction and seismic design of the bearings system for some viaducts of Ankara-Istanbul HSRL project F. Millanes Mato & M. Ortega Cornejo
37
6 Track structure interactions for the Taiwan High Speed Rail project D. Fitzwilliam
55
7 Track-bridge interaction – the SNCF experience P. Ramondenc, D. Martin & P. Schmitt
63
8
Some experiences on track-bridge interaction in Japan N. Matsumoto & K. Asanuma
77
9
Numerical methods for the analysis of longitudinal interaction between track and structure M. Cuadrado Sanguino & P. González Requejo
10
Longitudinal track-bridge interaction for load-sequences P. Ruge, D.R. Widarda & C. Birk
11
Structural analysis of high speed rail bridge substructures. Application to three Spanish case studies J.A. Sobrino & J. Murcia
95 109
129
12 The Italian experience: two case studies M.P. Petrangeli
139
13
149
Rail expansion joints – the underestimated track work material? J. Hess
V © 2009 Taylor & Francis Group, London, UK
VI Table of Contents
14
Dynamic aspects of the high-speed railway bridge across the Hollandsch Diep J.T.F.M. Tünnissen
15 Track-structure interaction in long railway bridges A.J. Reis, N.T. Lopes & D. Ribeiro 16 Track-bridge interaction in railway lines: Application to the study of the bridge over the River Moros R. Simões, R. Calçada & R. Delgado
© 2009 Taylor & Francis Group, London, UK
165 185
201
Preface
The construction of high-speed railways comprises a set of demands, from safety aspects to new types of equipment and construction solutions, involving the most recent and sophisticated technologies. Among these, emphasis is given to the railway behaviour where the structural elements are of great relevance. One of the relevant aspects concerns the effects of the track-bridge interaction, which establishes restricted limits to the vibration and deformability of the structure in order to control the acceleration, the stresses and the track deformations, so that the circulation safety is satisfied, while strongly conditioning the structural design solutions for bridges. The ability to address the multiple issues relevant to this process requires expertise and knowhow, which have been recently developed in this field, with repercussions in terms of the European regulations in this domain. The themes included in this book are mainly based on the papers presented at the workshop “TRACK-BRIDGE INTERACTION ON HIGH-SPEED RAILWAYS” organised by the Faculdade de Engenharia da Universidade do Porto (FEUP) and the Escuela Tecnica Superior de Ingenieros de Caminos Canales y Puertos de Madrid (ETSICCyP). This book is included in a set of three books: one with a more general thematic “BRIDGES FOR HIGH-SPEED RAILWAYS” and other with a more focused thematic, such as the present book, “DYNAMICS OF HIGH-SPEED RAILWAY BRIDGES”. The editors would like to thank all those who contributed to this book, in particular our distinguished guest chapters’ authors who heightened, with their knowledge and expertise, to the present interest and quality of the book, the support of the sponsors for the events which originated the materials for this book, and the institutional support of the Faculty of Engineering of the University of Porto and the RAVE – Rede Ferroviária de Alta Velocidade, S.A. We hope this book will be helpful not only to those professionals involved in the design, construction or maintenance of high speed railway systems, but also to researchers and students working in this field.
VII © 2009 Taylor & Francis Group, London, UK
List of Authors
Antonio Martínez Cutillas, UPM and CFCsl – Spain António Reis, IST and GRID – Portugal Carolin Birk, TU Dresden – Germany Daniel Dutoit, SYSTRA – France Daniel Fitzwilliam, TY Lin – USA Daniel Ribeiro, GRID – Portugal Didier Martin, SNCF – France Dina Rubiana Widarda, TU Dresden – Germany Francisco Millanes Mato, UPM and IDEAM – Spain Joep Tünnissen, JTüDEC – The Netherlands José Maria Goicolea- Ruigómez, UPM – Spain Josef Heß, BWG GmbH – Germany Juan A. Sobrino Almunia, Pedelta and UPC – Spain Juan Murcia, UPC – Spain Kiyoshi Asanuma, RTRI – Japan Manuel Cuadrado Sanguino, Fundación Caminos de Hierro – Spain Mario Paolo Petrangeli, Università Roma “La Sapienza” – Italy Miguel Ortega Cornejo, IDEAM – Spain Nobuyuki Matsumoto, RTRI – Japan Nuno Lopes, GRID – Portugal Patrice Schmitt, SNCF – France Pedro González Requejo, Fundación Caminos de Hierro – Spain Peter Ruge, TU Dresden – Germany Philippe Ramondenc, SNCF – France Raimundo Delgado, FEUP – Portugal Romeu Simões, FEUP – Portugal Rui Calçada, FEUP – Portugal Stuart Davis, Mott MacDonald – United Kingdom
IX © 2009 Taylor & Francis Group, London, UK
CHAPTER 1 New evolutions for high speed rail line bridge design criteria and corresponding design procedures D. Dutoit Systra, Paris, France
ABSTRACT: The high speed rail lines bridges have always had specific design criteria. Nevertheless, with the new development of the analysis of rail stresses due to rail structure interaction, some of the initial criteria used in France can be replaced by limitation of the rail stresses, as described for instance in the Eurocode. This can lead to significant savings, especially in highly seismic zones.
1
MAIN SPECIFIC FEATURES OF HSR BRIDGES PROJECTS
1.1 Typefont, typesize and spacing Historically, the developement of the High Speed Lines in France has been done step by step. Based on actual measurements made of stress concentrations in the rail done on real sites, and based on the experience of track stability and safety, rules were set-up to restrain specific features of the supporting structures within empirial limits in order to provide for the track safety. Usually, and as described today in Eurocode, UIC and present SNCF standards, the structures carrying the long welded rails for high speed trains have specific limitations due to 3 sets of phenomena: • Long Welded rail Rail structure interaction ◦ Additional rail stresses brought by R.S.I. – Temperature variation maximum distance between based points – Deck end rotations – Braking & acceleration forces: maximum displacement under braking and acceleration forces ◦ This controls – The location of expansion joint – The girder stiffness – The support stiffnesses (piers, foundations, bearings) • High speed vehicle Structure dynamic response The high speed rail supports vehicles travelling at high speed. This involves the analysis of the structures dynamic response to address the following items ◦ Control of vertical load (impact at resonance) ◦ Control of acceleration at deck level – Track stability: acceleration at deck level – Rail/wheel contact: acceleration at deck level – Rolling stock stability: acceleration at deck level – Passenger comfort: vertical acceleration in the cars 1 © 2009 Taylor & Francis Group, London, UK
2 Track-Bridge Interaction on High-Speed Railways
◦ Limit fatigue stresses. Rules of • Seismic environment High speed track geometry and stresses In the case of seismic areas, there is the need of additional analysis for the safety of the traffic during a potential earthquake. ◦ Problematic: Find the limits of rail deformation and stresses compatible with the rain full speed operation under earthquake ◦ Steps of the analysis – Life safety analysis (full speed operation compatible with which Service earthquake Peak Ground acceleration?) – Risk analysis (which track stresses and deformation criteria combined with other concomitant sources of stresses?) – Translated that into practical and simple High Speed Serviceability earthquake structure design criteria ◦ Consequence on High Speed Operation Train switch off at certain level of earthquake 2
EVOLUTION USING RAIL STRESSES COMPUTATIONS
In order to address the concerns described above, specific design criteria had been developed by several national codes. These specific criteria involved: • A limitation of maximum distance between bridge expansion joints when using a continuous welded rail, in order to limit the additional stress in the rail due to the difference of displacement between the structure and the rail. • A maximum rotation at bridge ends in order to limit the additional stress in the rail due to bridge end displacement and the corresponding force transmitted by the elasticity of the ballast or of the rail supports in the case of slab track and to ensure the stability of the ballast. This may control the deck rigidity. • A maximum displacement of the bridge when the maximum braking and acceleration force is applied: this may control the foundation, pier and bearing design. In the new evolution, instead of controlling the additional stresses in the rail by the above mentioned limitations, a complete analysis of the additional stresses in the rail due to the bridges supporting the track is limited to the followings Ballasted track
• 72 N/mm2 compression (Risk of track buckling in compression) • 92 N/mm2 tension
Slab track 92 N/mm2 tension and compression. In addition, in case of the ballast track, other criteria shall be satisfied in order to ensure the stability of the ballast (relative displacement of the deck under braking and acceleration, maximum relative displacement of the expansion joint between two bridges under live loads, . . .). This calculation is done by computing, on a computer model describing a significant length of the line on each side of the considered structure: • • • • •
The foundations and the corresponding elasticities due to the soil – foundation interaction Pier flexibilities The bearings (fixed, sliding or its elasticity) The bridge superstructure The tracks, with the rail stiffnesses and the elasticities (horizontal) of the support between the rail and the deck (ballast and ties, slab track, elastomeric pads underneath the rail) • The rail expansion joints
© 2009 Taylor & Francis Group, London, UK
New evolutions for high speed rail line bridge design criteria
3
• The environmental conditions (temperature variations, gradients, . . .) • The train characteristics Based on the corresponding analysis, the piers and foundations can be optimized when compared to the conventional HSR criteria (see example in part 4.1.). In addition, it is also possible to identify critical points on the line where there are concentrations of forces on the bearings and design the sub-structures in order to reduce this unfavourable effect (see example in 4.2.). This cannot be done by using the simplified approach (without the rail interaction analysis). This new computerised method is therefore more economical and safer the simplified one. 3
MAIN CONSEQUENCES
These new design procedures can induce a significant saving in the substructures (foundation, piers). These savings may be magnified in seismic areas. Since the loads applied by a given earthquake increase with the substructure rigidity, the additional elasticity of the substructure due to the new HSR service load criteria will also induce a significant saving in the seismic analysis of the structure. 4
EXAMPLES OF RAIL-STRESSES COMPUTATION
The following examples show that the simplified method (no track structure interaction modelled) used to avoid computerised calculation (track structure interaction modelled) is generally too conservative and cannot identify the critical points on the line where very high bearing reactions can occur. In the following examples, we compare the simplified method and the computerised method on a simple case: • • • • • •
Train type UIC 71 Ballasted track Straight track Double track Rail type UIC 60 Succession of 30 m simply supported spans.
4.1
Comparison between the simplified method and the computerised method – optimisation of the pier and foundation
(a) Simplified method: In the case of a succession of simply supported spans, the braking and acceleration forces applied on one span are fully transmitted to the bearings of the span. In the case of a 30 m simple span, the longitudinal braking and acceleration forces are: F = 33 kN/m × 30 m + 20 kN/m × 30 m = 1590 kN
(1)
The bearing reaction under temperature effect is calculated using the formula 8 × L (L is the length of the span). It can be estimated at 8 kN/m × 30 m = 240 kN. The maximum allowable relative displacement under braking and acceleration forces between two decks is δ = 5 mm. Therefore, the minimum stiffness of the pier and foundation is: K=
F = 318000 kN/m δ
Each pier and foundation shall have a stiffness higher than 318000 kN/m.
© 2009 Taylor & Francis Group, London, UK
(2)
4 Track-Bridge Interaction on High-Speed Railways
(b) Computerised method (including rail structure interaction): 25 spans are modelled. The pier stiffness is the one calculated using the simplified method (see above). We study here the span located at the center of the computer model. The results of the calculations are the following. 25 Spans (30 m long)
Figure 1.
Scheme of the computer model.
Figure 2.
Stresses in the rails under Temperature, braking/acceleration and live loads.
The stresses in the rail is between −25 MPa and +30 MPa.
Figure 3.
Relative displacement between two decks under braking/acceleration.
© 2009 Taylor & Francis Group, London, UK
New evolutions for high speed rail line bridge design criteria
5
The maximum relative displacement between two decks is below 2 mm.
Figure 4.
Bearing reactions under braking/acceleration and live loads.
The maximum bearing reaction is 1019 kN. This represents only 64% of the value given by the simplified method (1590 kN). It can also be noticed that the bearing reaction under temperature effect is almost zero compared to 240 kN calculated by the simplified method. (c) Analysis of the results: The table below shows the results of the computerised calculations. Table 1.
Tensile stress Compression stress Relative displacement
Results
Allowable limits
Ratio
25 MPa 30 MPa 2 mm
92 MPa 72 MPa 5 mm
27% 42% 40%
In addition, the bearing reaction under temperature, braking/acceleration and live loads is only 56% of the value given by the simplified method. (d) Conclusion: It is therefore possible to optimise the piers and foundation. Additional calculations show that even if the pier stiffness is reduced by more than 2, the safety of the track is still ensured. The bearing reactions calculated by the computerised method are also around half of the value calculated by the simplified method. 4.2
Comparison between the simplified method and the computerised method – Identification of the critical points on the line
Due to the link between adjacent girder created by the track, a force applied on one span is transmitted to the adjacent spans.
© 2009 Taylor & Francis Group, London, UK
6 Track-Bridge Interaction on High-Speed Railways
In some case, where there is a sudden variation of pier stiffness, a bearing reaction may be higher than the one calculated using the simplified method (see 4.1.a). The simplified method is, in these cases, too favourable. The case studied here is the same than the previous one (pier stiffness equal to 318000 kN/m), but one pier has a stiffness much higher than the other (due for example to a sudden variation of the ground level). We study in detail the bearing reactions on this pier under braking and acceleration forces. 25 span (30 m)
Very stiff pier
Figure 5.
Scheme of the computer model.
Figure 6.
Maximum bearing reactions under braking/acceleration and live loads.
The maximum bearing reaction is 2036 kN. Conclusion The maximum bearing reaction is 2036 kN, which is 128% of the value calculated using the simplified method (1590 kN – see 4.1.a). Additional calculations shows that, in case of slab track, it is even more unfavourable. The maximum bearing reaction is then 155% of the value calculated using the simplified method. The new computarised method allows therefore a better identification of the overstressed areas, and allows to make the required changes necessary to have a safer track.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 2 Service limit states for railway bridges in new Design Codes IAPF and Eurocodes J.M. Goicolea-Ruigómez Escuela de Ingenieros de Caminos, Technical University of Madrid, Spain
ABSTRACT: The new and enhanced performance needs of bridges for high-speed railway lines have prompted new requirements for design of structures. These have been studied at national and international level within Europe (ERRI, UIC, Eurocode project teams) and have originated new engineering codes for actions and design requirements. Between these we may cite the Eurocodes EN1991-2 (2003) and EN1990-A1 (2005) and the new Spanish code IAPF (2007). An important feature in these codes is the consideration of service limit states. These limit states are unique to railway bridges and are often the critical features conditioning the design. Among these limits are the maximum of displacements and stresses in the rail related to track-bridge interaction, and the limit of accelerations at the track. It must be stressed that some of these service limit states are indeed ultimate limit states related to safety of rail traffic, and hence of the utmost importance. In this work we shall review these limitations, the methods proposed for calculation, and their relevance for design.
1
INTRODUCTION: NEW DESIGN REQUIREMENTS FOR HIGH SPEED RAILWAY BRIDGES
New high speed railway lines are developing at a fast pace in some European countries. In particular, in Spain the plan for transport infrastructure (PEIT 2005–2020) devotes 78000 Million a to high speed railways out of a total investment of 241000 Million a. Railway bridges for the new high speed lines introduce a number of design requirements which cause significant differences not only with road bridges but also with other railway bridges in conventional freight or passenger lines. A first and obvious requirement arises directly from the higher speed of traffic actions. These not only produce a higher individual effect (measured through the impact factor ), but more importantly for speeds above 200 km/h the risk of resonance appears. As a result dynamic analyses must be carried out in general, and furthermore these considerations must be taken into account in the design of structural characteristics. In particular, some structural types such as short span isostatic bridges have been shown to originate high levels of vibration exceeding the limits for comfort and safety. Furthermore the stricter requirements for the high speed lines (e.g. maximum gradients, minimum radii) and geometrical quality originate line layouts in which more and longer viaducts are necessary. This is particularly important in regions with rugged terrain like the Iberian Peninsula. For instance, the new lines in Spain include a number of viaducts longer than 1000 m, some even reaching 3000 m. The consideration of interaction between bridge and track introduces additional requirements to be met by railway bridges. These stem from limitations of stresses in the rail as well as maximum values of relative displacements in deck joints. The magnitude of the track-bridge interaction effects increases with the continuous length (expansion length LT ) of the deck. As a result, in short 7 © 2009 Taylor & Francis Group, London, UK
8 Track-Bridge Interaction on High-Speed Railways
bridges these requirements are not especially restrictive. However in long viaducts, of the lengths commonly required by high speed line layouts, they prove to be an important restriction and must be considered at early stages of design. One option which has been adopted in some lines is to limit the continuous length of the decks, splitting long viaducts into individual simply supported structures. This option may be overly restrictive. Moreover, it bears the disadvantage expressed above that simply supported beam structures develop generally higher vibration response to traffic loads. Furthermore, the optimization of construction procedures in some cases makes advantageous the progressive launching or pushing of a long continuous deck. The above considerations have led the administration and the infrastructure manager (ADIF) in Spain to allow the construction of long continuous viaducts, exceeding in some cases 1000 m length. The bases for calculation and associated limits are defined in IAPF (2007) and at a European level in EN1991-2 (2003). In the remaining of this paper, in section 2 we shall firstly review some design considerations for high speed railway bridges, with special emphasis on those originating from dynamic behaviour. Following, in section 3 we shall review the methods and requirements for track-bridge interaction in the codes, focusing on the new IAPF (2007) compared to EN1991-2 (2003). Here the serviceability limit state checks regarding deformations of the deck will be discussed critically. The paper finishes with a summary of the main conclusions in section 4.
2
DESIGN CONSIDERATIONS FOR RAILWAY BRIDGES FROM DYNAMIC EFFECTS
Dynamic response of railway bridges is a major factor for design and maintenance, especially in new high speed railway lines. The main concern is the risk of resonance from periodic action of moving train loads. In cases when such risk is relevant (e.g. for speeds above 200 km/h) a dynamic analysis is mandatory. The new engineering codes [EN1991-2 (2003), EN1990-A1 (2005), IAPF (2007), FS. (1997)] take into account these issues and define the conditions under which dynamic analysis must be performed. They provide guidelines for models, types of trains to be considered, and design criteria or limits of acceptance [Goicolea, J.M. (2004)]. Resonance for a train of periodically spaced loads may occur when these are applied sequentially to the fundamental mode of vibration of a bridge and they all occur with the same phase, thus accumulating the vibration energy from the action of each axle. If the train speed is v, the spacing of the loads D and the fundamental frequency f0 , defining the excitation wavelength as λ = v/f0 , the condition for critical resonant speeds may be expressed as [EN1991-2 (2003)]: D , i = 1, . . . 4. (1) i Although the basic dynamic phenomena due to moving loads are known since long (e.g. see the book by Fryba, L. (1972)) it has not been until recently that resonance phenomena in bridges from high speed traffic have been well understood and practical methods of analysis developed [ERRI D214. (2002), Domínguez J. (2001)]. From a technical point of view a number of methods for dynamic analysis are available for engineering practice. Briefly, the available methods are: a) dynamic analysis with time integration and moving loads; b) dynamic analysis with time integration and bridge-vehicle interaction; and c) dynamic envelopes based on train signature methods. Rather than discussing these methods here, for which a complete description is given elsewhere [Domínguez, J. (2001), Goicolea, J.M. (2004)] we shall focus on the relevance of dynamic effects for structural designs. In railway bridge design often the most restrictive conditions in practice are the Serviceability Limit States (SLS) [EN1990-A1 (2005), Nasarre J. (2004)] (maximum acceleration, rotations and deflections, etc.). Accelerations must be independently obtained in the dynamic analysis. In the example shown in Figure 1, for a short span simply supported bridge, both maximum displacements and accelerations are obtained independently and checked against their nominal (LM71) or limit λ=
© 2009 Taylor & Francis Group, London, UK
Service limit states for railway bridges in new Design Codes IAPF and Eurocodes
9
values respectively. It is clearly seen that for a resonant train speed the deflection limits are above the LM71 nominal values, resulting in an impact factor greater than unity. A more severe effect is the accelerations which surpass by far the limits, thus invalidating this (purely theoretical) design. Further details may be seen in Goicolea, J.M. (2004). These results have been obtained both with moving loads and with bridge-vehicle interaction, showing that the gain of considering this latter and more advanced model, albeit significant in this case, still yields a non acceptable value. In order to consider the resonant velocities for dynamic calculations, these must be performed generally for all possible speeds. The results may be presented as envelopes of resulting magnitudes in these velocity sweeps. Following we present a typical set of such calculations showing the fact that generally resonance may be much larger for short span bridges. In this representative example, Figure 2 shows the normalised displacement response envelopes obtained for ICE2 train in a velocity sweep between 120 and 420 km/h at intervals of 5 km/h. Calculations are performed for three different bridges, from short to moderate lengths (20 m, 30 m and 40 m). The maximum response obtained for the short length bridge is many times larger that the other. The physical reason is that for bridges longer than coach length at any given time several axles or bogies will be on the bridge with different phases, thus cancelling effects and impeding a clear resonance. We also remark that for lower speeds in all three cases the response is approximately 2.5 times lower than that of the much heavier nominal train LM71. Resonance increases this response by a factor of 5, thus surpassing by a factor of 2 the LM71 response. A significant reduction of vibration is obtained in short span bridges under resonance by using interaction models. This may be explained considering that part of the energy from the vibration is transmitted from the bridge to the vehicles. However, only a modest reduction is obtained for non-resonant speeds. Further, in longer spans or in continuous deck bridges the advantage gained by employing interaction models will generally be very small. This is exemplified in Figure 2, showing results of sweeps of dynamic calculations for the three said bridges of different spans. As a consequence it is not generally considered necessary to perform dynamic analysis with interaction for project or design purposes. The above results are not merely theoretical considerations. It has been seen in practice that they reflect accurately the vibrations taking effect in real high speed railway bridges. To show this we comment some experimental results on an existing high speed bridge. Figure 3 shows the measured resonant response in the bridge over the river Tajo in the Madrid-Sevilla HS line. The bridge consists of a sequence of simply supported isostatic decks with spans of 38 m. The dynamic amplification in this case is noticeable. In spite of this, design responses keep within required limits. However, it is clear that the dynamic performance could be improved by a different structural design. Another well-known issue is the fact that dynamic effects in indeterminate structures, especially continuous deck beams, are generally much lower than isostatic structures [Domínguez, J. (2001)]. The vibration of simply supported beams is dominated clearly by the first mode, and moreover only the loads on the span under consideration excite the motion at a given instant. This makes much more likely a resonant phenomenon, whenever condition (1) is met. On the contrary, the vibration of continuous beams includes significant contributions of a number of modes, and loads on other spans excite the motion of the span under consideration. As a result, the algebraic sum of the effects tends to cancel to a large extent. We show a practical example of this effect in a student project for the Arroyo del Salado viaduct [Sanz, B. (2005)] on the Córdoba-Málaga High Speed Line, with a total length of 900 m and 30 spans of 30 m each. The section is a prestressed concrete box deck, and the proposed solution was a continuous beam deck cast in-situ. The comparison of this solution with a corresponding simply supported multiple span viaduct is shown in Figure 4, where it may be seen the much better performance in terms of dynamic response of the continuous beam deck. Finally, we discuss the consideration of different high speed train types. The existing trains in Europe are defined in EN1991-2 (2003), IAPF (2007), and may be classified into conventional (ICE, ETR-Y, VIRGIN), articulated (THALYS, AVE, EUROSTAR) and regular (TALGO). Variations of these trains which satisfy interoperability criteria have been shown to covered by the dynamic
© 2009 Taylor & Francis Group, London, UK
10 Track-Bridge Interaction on High-Speed Railways
Figure 1. Calculations for simply supported bridge from ERRI D214. (2002) catalogue (L = 15 m, f0 = 5 Hz, ρ = 15000 kg/m, δLM71 = 11 mm), with TALGO AV2 train, for non-resonant (360 km/h, top) and resonant (236.5 km/h, bottom) speeds, considering dynamic analysis with moving loads and with train-bridge interaction. Note that the response at the higher speed (360 km/h) is considerably smaller than for the critical speed of 236.5 km/h. The graphs at left show displacements, comparing with the quasi-static response of the real train and the LM71 model, and those at right accelerations, compared with the limit of 3.5 m/s2 [EN1990-A1 (2005)].
Figure 2. Normalised envelope of dynamic effects (displacement) for ICE2 high-speed train between 120 and 420 km/h on simply supported bridges of different spans (L = 20 m, f0 = 4 Hz, ρ = 20000 kg/m, δLM71 = 11.79 mm, L = 30 m, f0 = 3 Hz, ρ = 25000 kg/m, δLM71 = 15.07 mm and L = 40 m, f0 = 3 Hz, ρ = 30000 kg/m, δLM71 = 11.81 mm). Dashed lines represent analysis with moving loads, solid lines with symbols models with interaction. Damping is ζ = 2% in all cases.
© 2009 Taylor & Francis Group, London, UK
Service limit states for railway bridges in new Design Codes IAPF and Eurocodes
11
Figure 3. Time history of displacements at centre of span in viaduct over Tajo river in HS line Madrid-Sevilla. Simply supported deck with span 38 m, damping ratio ζ = 1.65%. Left graph shows measurements [MFOM (1996)], right graph analytical calculations [Domínguez, J. (2001)]. Horizontal scale is time (s), vertical scale displacements (mm).
Figure 4. Summary of dynamic analysis envelopes with universal HS trains HSLM showing maximum accelerations in the deck. The graph on the left corresponds to the proposed design as continuous beam, which satisfies the requirement for accelerations amax < 3.5 m/s2 . The right graph corresponds to a simple supported bridge with the same deck section; in this case the requirement for maximum accelerations is not fulfilled for high speeds.
effects of the High Speed Load Model (HSLM), a set of universal fictitious trains proposed by ERRI D214. (2002). The use of this new load model is highly recommended for all new railway lines, and incorporated into codes EN1991-2 (2003) and IAPF (2007). More importantly, consideration of HSLM model is mandatory for interoperable lines following the European Technical Specifications for Interoperability (TSI) in high speed lines [EC (2002)]. A useful way to compare the action of different trains and to evaluate the performance of HSLM as an envelope is to employ the so-called dynamic train signature models. These develop the response as a combination of harmonic series, and establish an upper bound of this sum, avoiding a direct dynamic analysis by time integration. Their basic description may be found in [ERRI D214. (2002)]. They furnish an analytical evaluation of an upper bound for the dynamic response of a given bridge. The result is expressed as a function of the dynamic signature of the train G(λ). This function depends only on the distribution of the train axle loads. Each train has its own dynamic signature, which is independent of the characteristics of the bridge. The above expressions have been applied in Figure 5 to represent the envelope of all real existing HS trains in Europe, together with the envelope of HSLM.
© 2009 Taylor & Francis Group, London, UK
12 Track-Bridge Interaction on High-Speed Railways
Figure 5. Envelope of dynamic signatures for European HS trains, together with the envelope of signatures for High Speed Load Model HSLM-A, showing the adequacy of this load model for dynamic analysis.
3 3.1
TRACK-BRIDGE INTERACTION IN CODES IAPF (2007), EN1991-2 (2003) Nature of phenomenon and effects to be evaluated
Track-bridge interaction originates from the fact that longitudinal forces in long welded rail are transmitted both by the structure and the rail to the fixed points at piers or abutments. Furthermore, at joints in the deck there may be structural deformations which could modify the geometry of the track and thus endanger the safety of traffic. For short bridges this issue is not critical and in fact given certain conditions the calculation of the nonlinear models described below may be avoided. However, as has been said above, in high speed lines bridges and viaducts of substantial length are common and hence the issue of track-bridge interaction becomes a critical issue. The basic interpretation and methods agreed internationally are contained in the leaflet by UIC (1999), which summarises the results by ERRI subcommittee D213. Both the Spanish code IAPF (2007) and the Eurocode EN1991-2 (2003) follow generally the recommendations of the said UIC leaflet. They both contain a section describing specifically the objectives of this evaluation, the actions and models to consider and the design requirements. In what follows we describe in summary the main principles, which are common between both codes and, wherever appropriate, underline and comment specifically the differences or additions. In both codes it is stated that consideration of track-bridge interaction is necessary in order to evaluate the following effects: – Forces transmitted to piers and abutments from combined actions of structure and track; – Rail stresses due to variable actions, in particular thermal actions, braking and acceleration longitudinal forces and vertical traffic loads; – Relative movements and deformations at the ends of the deck due to the above variable actions.
3.2
Models to employ in calculation
Several types of structures may be considered from the point of view of track-bridge interaction: a) single deck bridges, be this with one isostatic span or with a multiple span continuous beam, with a fixed bearing at one end; b) continuous beams with multiple spans with a fixed bearing at
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Figure 6. Model to be considered for track-bridge interaction, in a simple case with one deck. The figure shows a deck with one fixed point and two sliding supports, nonlinear “generalised springs” which model the longitudinal interaction between track and deck, and an optional rail expansion device at one end (figure translated from IAPF (2007)).
an intermediate point of the bridge; and c) multiple isostatic spans with fixed bearings ate the end of each span. The general type of model to be considered is depicted schematically in Figure 6, for case a) above. This model considers the track and the deck (both considered deformable elastically), the piers, and the abutments which may also be flexible. A key aspect in the model is the proper consideration of the interaction forces between rail and deck, in the figure represented through generalised springs, which as we shall see below are of nonlinear nature. Finally, in the figure a rail expansion device which would signify a longitudinal discontinuity for the rail is also shown. A characteristic value of these models is the so-called expansion length LT . In the example shown it would be simply the length of the deck between the fixed support on one abutment and the free-sliding joint on the other abutment. The greater the value of LT the greater interaction effects will be introduced at the free sliding joint. When expansion lengths are large the rail stresses may be reduced by the introduction of rail expansion devices. In such case, the horizontal deck forces would be transferred integrally to the fixed bearing, alleviating the effects on the rail. However, rail expansion devices are generally undesirable from the point of view of track engineering and maintenance. Expansion lengths of the order of 100 m may generally be accommodated without resorting to rail expansion devices. Expansion lengths of the order of 300 m to 400 m will very probably necessitate at least one rail expansion device. Expansion lengths greater that this may necessitate at least two expansion devices, say with a fixed point at the center, or other structural solutions. The above mentioned nonlinear generalised springs are defined with bilinear laws, of the type shown in Figure 7. The first branch represents an elastic behaviour, whereas after a given displacement u0 the sliding limit is attained and the constant resistance k is developed. The Eurocode EN1991-2 (2003) leaves the values of (u0 , k) open, to be defined in national annex or other project specifications. The code IAPF (2007) defines values for u0 between 0.5 and 2.0 mm, and for k between 20 and 60 kN/m, depending on the type of track, vertical load etc. The structural model described may be developed within a discretised computer model of the structure with nonlinear material capabilities, such as finite element or other numerical programs. This program must have the capability to solve the resulting set of nonlinear algebraic equations, generally using an iterative procedure by Newton-type iterations until convergence is reached. An essential characteristic of nonlinear models is that superposition of actions is not valid; hence for each calculation the complete set of actions must be applied in the correct sequence to the model. In particular for this case, for each scenario selected the thermal actions would be applied first and then the vertical and longitudinal traffic actions.
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14 Track-Bridge Interaction on High-Speed Railways
Figure 7. Force-displacement interaction law between track and deck. The parameter u0 defines the maximum relative displacement at which sliding starts, with a plasticity or friction-type resistance defined as k. Particular values of u0 and k are defined in IAPF (2007) for different types of tracks and situations. (figure translated from IAPF (2007)).
For short expansion lengths LT both codes allow simplified procedures for calculation. In particular, for LT ≤ 40 m in EN1991-2 (2003) or LT ≤ 60 m (steel) – 90 m (concrete) in IAPF (2007) it may be considered that rail expansion devices are not needed, without a full justification by the nonlinear models above described. For somewhat longer expansion lengths, LT ≤ 110 m, the code IAPF (2007) refers to the simplified procedures defined in UIC (1999) based on charts for evaluating the interaction. Furthermore, the Eurocode EN1991-2 (2003) allows the simplification, for evaluating forces in rails and bearings, of combining linearly the effects of the different actions. As has been said before, strictly speaking this linear combination is not valid; however for computation of forces in general a conservative result will be obtained. This is not generally the case for calculation of deformations, which may be underestimated using this simplification. In the Spanish code IAPF (2007) this simplification is not considered. 3.3
Design criteria
The maximum additional stresses in the rails from the variable actions (thermal and traffic loads) are limited to 72 MPa (compression) or 92 MPa (tension). It is understood that these stresses would apply on top of the existing stresses in the long welded rail, which amount to approx. 105 MPa for a maximum temperature increment T = 50◦ C. Regarding the deformation of the deck, it is required to limit the relative movements at the end of the deck in sliding joints (e.g. between end of the deck and abutment). The following requirements are defined, all related to the said relative movements: – The horizontal movement from braking and acceleration forces must be δB ≤ 5 mm. (called δ2 in IAPF (2007)). Figure 8 shows a schematic representation of this movement. – The horizontal movement from vertical traffic loads must be δH ≤ 5 mm. This movement originates mainly from bending, which produces horizontal movement at points eccentric from the neutral axis. In IAPF (2007) this movement, which is called δ3 , is more precisely defined to be computed not only from the bending caused by vertical traffic loads but also from eccentric horizontal longitudinal loads (i.e. braking or acceleration acting on the rail surface), which also introduce bending moments in the deck. Figure 9 shows a schematic representation of this movement. – The vertical movement from bending and other effects must be δV ≤ 2 mm. This limitation holds for lines with train speeds above 160 km/h which is the case for high speed.
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Figure 8. Maximum longitudinal relative displacement from braking or acceleration actions (δ2 in IAPF (2007) or δB in EN1991-2 (2003)) between two ends at a joint (figure from IAPF (2007)).
Figure 9. Maximum horizontal relative displacement from bending due to vertical or eccentric horizontal actions (δ3 in IAPF (2007) or δH in EN1991-2 (2003)) between two ends at a joint (figure from IAPF (2007)).
In the Spanish code IAPF (2007) again this limit (called δ4 ) is more precisely defined not as vertical but as normal to the rail within a vertical plane. Figure 10 shows a schematic representation of this movement. As may be seen there could be a noticeable difference between the normal movement which actually alters the track geometry in a track with gradient, and the vertical movement which would be zero in this case. For a long viaduct this difference may be critical. Furthermore to the above requirements, the following limit is defined in IAPF (2007), but not in the Eurocode EN1991-2 (2003): – The relative movement between rail and deck (or between rail and abutment platform) must be δ1 ≤ 4 mm under the actions for acceleration and braking. This requirement may also be found in UIC (1999). The above design requirements both for stresses in the rail as well as for movements at the ends of the deck represent Serviceability Limit States (SLS) for the structure. However, in this case the importance of these limit states is paramount, as they represent Ultimate Limit States (ULS) for the railway traffic. It must be clearly understood by any structural engineer that these design criteria
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16 Track-Bridge Interaction on High-Speed Railways
Figure 10. Maximum relative normal displacement in vertical plane from variable actions (δ4 in IAPF (2007) or δV in EN1991-2 (2003)) between two ends at a joint (figure from IAPF (2007)).
are often the critical requirements for railway bridges, contrary to the case for road bridges. This has been clearly set out in the paper by Nasarre J. (2004).
4
CONCLUSIONS
Considering the above, we summarise the following final remarks: – In high speed railway lines it is common to be faced with bridges or viaducts of considerable length, for which special consideration needs to be made at early stages of design to track-bridge interaction effects. – The reduction of dynamic effects is more favourable for continuous beams and for long spans; these factor again favour the consideration of long decks with potential problems for track-bridge interaction. – The proper consideration of track-bridge interaction requires a nonlinear structural model, which requires careful elaboration and checking on the part of adequately skilled structural engineers. Simplifications to this model must be carefully justified and only employed when clearly conservative. – Both the Eurocode EN1991-2 (2003) and the new Spanish code IAPF (2007) contain similar sets of recommendations for the models and design requirements. These criteria originate from the report UIC (1999). REFERENCES Domínguez, J. (2001). Dinámica de puentes de ferrocarril para alta velocidad: métodos de cálculo y estudio de la resonancia. Tesis Doctoral. Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos de Madrid (UPM), 2001. Publicada por la Asociación Nacional de Constructores Independientes (ANCI). EC (2002). TSI-HS2002: Technical specification for Interoperability relating to the infrastructure subsystem of the trans-European high-speed rail system. Commission decision 2002/732/EC 20 may 2002, 2002. Official journal of the European Communities OJ L 12/9/2002. EN1991-2 (2003). European Committee for Standardization. EN1991-2: EUROCODE 1 – Actions on structures, Part 2: Traffic loads on bridges. European Union, 2003. EN1990-A1 (2005). European Committee for Standardization. {EN1990-A1}: EUROCODE 0 – Basis of Structural Design, Amendment A1: Annex {A2}, Application for bridges. European Union, 2005. ERRI D214 (2002). Utilisation de convois universels pour le dimensionnement dynamique de ponts-rails. Synthèse des résultats du D214.2 (Rapport final). European Rail Research Institute (ERRI). ERRI D214 RP9 (1998). European Railway Research Institute subcommitee D214. Design of Railway Bridges for Speed up to 350 km/h; Dynamic loading effects including resonance. Final report, November 1998. FS (1997). Ferrovie dello Stato; Sovraccarichi per il calcolo dei ponti ferroviari.
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Fryba, L. (1972). Vibration of solids and structures under moving loads. Academia, Noordhoff, 1972. Goicolea, J.M. (2004). Dynamic loads in new engineering codes for railway bridges in Europe and Spain. Workshop on Bridges for High Speed Railways, Porto 3–4 June 2004, Faculty of Engineering, University of Porto. IAPF (2007). Instrucción de acciones a considerar en el proyecto de puentes de ferrocarril. Ministerio de Fomento de España, Dirección General de Ferrocarriles. Oct 2007. MFOM (1996). Dirección general de ferrocarriles y transporte∼por carretera. Viaducto sobre el río tajo, línea ave Madrid-Sevilla, ensayos dinámicos. Technical report, Ministerio de Fomento de España, 1996. Nasarre, J. (2004). Estados límite de servicio en relación con los puentes de ferrocarril. In A. Campos R. Delgado, R. Calçada, editor, Bridges for High-Speed Railways, pages 237–250. Civil Engineering Dept., Faculty of Engineering of the University of Porto, 2004. Sanz, B. (2005). Proyecto de viaducto para el ferrocarril de alta velocidad sobre el arroyo del salado. Master thesis, Escuela de Ingenieros de Caminos, UPM, 2005. UIC (1999). Code UIC 774-3R Interaction voie/ouvrages d’art, Recommandations pour les calculs. UIC, Union Internationale des Chemins de Fer, Feb1999. UIC (2006). Leaflet UIC 776-1R: Charges a prendre en consideration dans le calcul des ponts-rails. UIC, Union Internationale des Chemins de Fer, 2006. 5 ed, Aug 2006.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 3 Track-bridge interaction problems in bridge design A.M. Cutillas Technical University of Madrid & Carlos Fernández Casado S.L., Madrid, Spain
ABSTRACT: Track-bridge interaction problems have a main role in the design of bridges, especially at conception stage of long viaducts with high or short piers in high speed railways lines. The presentation will focus on the main aspects of track bridge interaction aspects to be taken into account in the design of these bridges: • Bridge displacement limitations at track level • Railway expansion joint needs Some examples of recent bridges which have been designed in High speed railway lines in Spain will be shown. A special attention will be paid to the Viaduct over the Guadalete river. It is a 3221.70 m long viaduct in which the aforementioned problems were determinant in the conception and design of the bridge.
1
TRACK-DECK INTERACTION PROBLEMS
Railway bridges, in general and high speed railway bridges in particular have to resist important horizontal loads.The horizontal loads are originated by the climate such as the wind. The wind acts on the whole structure surface exposed, the piers and the deck as well as on the live load itself. These loads produce bending moments of the deck’s vertical axis as well as twisting moments which are added to the combined pier-deck effects produced by the live load. The live loads produces two horizontal loads derived from the accelerations. On the one hand, transverse radial accelerations in curved layouts, which bring about the centrifugal force. On the other hand they bring about longitudinal accelerations produced by braking and traction forces. The stresses due to the centrifugal forces produce the same effects as those derived from the transverse wind. These forces can be very important for even though the radii are great so is the speed (Fig. 1).
Figure 1.
Viaduct 2 in Substretch VIII.
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20 Track-Bridge Interaction on High-Speed Railways
The braking forces considered amount to 2 t/m according to EC-1, with a maximum application length of 300 m. The traction forces, in their turn, equal 3,3 t/m with a maximum loaded length of 30 m. Consequently, for long viaducts (longer than 300 m) the maximum force transmitted to the tracks will amount to 700 t. The force transmitted to the deck, bearings and piers will depend on whether the expansion joints are arranged or not on the rail at one or both ends of the deck, as well as on the total length (Fig. 2). Within the whole made of the track (rail, sleepers, ballast), the deck, the bearings and the piers as deformable elements with different mechanical properties there are force transferences produced by external loads or deformations imposed, which bring about the well known phenomena of track-deck interaction (Fig. 3). When the rail is continuous and the track is supported on an element which is not very deformable such as the ground, its sizing depends on the vertical and horizontal loads transmitted by the railway, and the axial forces as a consequence of the deformations restrained by the temperature so the strength reserve to failure is limited. When the tracks are placed over a structure, the imposed deformations produced by the uniform temperature variations and by the creep and shrinkage phenomena in the deck, produce relative displacements between the track and the deck that as a consequence of friction forces with the ballast, produce horizontal loads over both the track and the deck that may exhaust the resistant capacity of the rail. This is the reason why the arrangement
Figure 2.
Bridge over the Ebro river.
Figure 3.
Bridge over Contreras reservoir.
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Track-bridge interaction problems in bridge design
21
of the continuous rail is limited to concrete structures with a total expansion length not greater than 90 m. With the intention to optimise the design and viaducts construction conditions and track exploitation in continuous viaducts it will be necessary, whenever it is possible, to arrange an expansion joint in one of the abutments. In this way the phenomena of track-deck interaction disappear. Otherwise, the viaducts would have to be sub-divided into smaller lengths with the resulting problems of structural joints, duplication of the bearings and the placing of intermediate stiff and resistant elements in order to resist the horizontal loads in each structure. If all these factors are studied and properly balanced: the resistant problem for horizontal forces and the track-deck interaction problems, different longitudinal configurations of the viaducts can be obtained (Fig. 4): • Long continuous viaducts fixed at one abutment with one big expansion joint on the rail at the opposite one (Figs 2–3, 5). • Long continuous viaducts with a stiff intermediate pier and two small expansion joints on the rail at both abutments (Fig. 1). • Continuous viaducts with a stiff intermediate pier with no expansion joints in the rails (Fig. 6). Most of the continuous viaducts we present here are longitudinally fixed in one abutment and have one expansion joint in the rail in the opposite. Since all these viaducts are incrementally
Figure 4.
Rail expansion joints. Longitudinal configuration.
Figure 5.
Viaduct over Huerva river (L = 1122 m).
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22 Track-Bridge Interaction on High-Speed Railways
Figure 6.
Bridge over Llobregat river in Martorell.
launched it is necessary to locate a segment casting yard on one of the abutments. These yards must have the capacity, among other things, to resist the horizontal forces during the launching as well as the loads produced by the wind and temperatures under rest conditions. These are, therefore, elements able to be adapted to become anchorage elements of the deck under service conditions. This is why most of the viaducts presented here, except the Viaduct 2 of the Sub-stretch VIII (Fig. 1) and Martorell Viaduct (Fig. 6), are anchored in the abutment corresponding to the segment casting yard. The abutment-yard whole is the structure in charge of transmitting to the ground the horizontal forces produced by the braking and traction loads, the longitudinal wind and those produced by friction forces of the elastomeric-teflon devices as a consequence of the deformations due to the temperature, creep and shrinkage. The total horizontal forces, under service conditions, are much greater than those corresponding to the situation during construction for the following reasons: During construction, the reactions in the supports produced by the horizontal forces correspond to the total permanent load and not self weight. The coefficient of friction of the teflon supports considered under service conditions amounts to 5%, while during construction it amounts to 2.5%. Actually, the value of 2% is hardly reached if special greases are used to reduce this coefficient. Under service conditions we must take into account the braking and traction forces for viaducts longer than 300 m reach a maximum value of 700 T. In spite of the effectiveness of this structural disposition, we must bear in mind the fact that the viaduct length imposes certain limits: on the one hand, the availability of expansion devices able to admit great movements. At present, there are devices with the movement capacity of up to 1200 mm, which establishes a length limit between 1200 and 1300 m for prestressed concrete viaducts. On the other hand there is a limit, though higher than the previous one, which depends on the capacity of the deck to admit the prestressing force that would counteract these horizontal forces. The concept of the Viaduct 2 of the sub-stretch VIII, mentioned before responds to the need to reduce the movements in the expansion joint and to place the fixed point by a delta shaped pier on a small hill in the valley. This disposition allows us to double the length of the viaducts by placing two small expansion joints in the rail. In MartorellViaduct and intermediateV shaped pier was designed. In order to resist the horizontal forces and to reduce the longitudinal movements in this section, the pier is founded over slurry-walls with a big longitudinal stiffness. (Fig. 6). The deck has a total length of 202 m, the expansion length is 101 m (greater than 90 m). In order to avoid the expansion joints at the abutments a complete track-deck interaction analysis was performed. The deck and rails were modelled connected with
© 2009 Taylor & Francis Group, London, UK
Track-bridge interaction problems in bridge design
Figure 7.
Track-deck interaction model.
Figure 8.
Martorell bridge. Axial forces on the rail.
23
non linear springs corresponding to the mechanical behaviour of the ballast. The main conclusions of the analysis were: The stress increment in the rail due temperature actions was 42 MPa. (Fig. 8) The maximum stress increment in the rail, in compression, due to temperature and brakingtraction forces was 77.6 MPa. The longitudinal displacement in the deck due to braking-traction forces was 4.27 mm and 5.2 mm taking into account the foundation influence. The braking force transferred to de deck is 80% of the total force applied at the rail level.
2 2.1
VIADUCT OVER THE GUADALETE RIVER IN SPAIN Structural concept. General Description
The Viaduct over the Guadalete river is located in the railway line between Sevilla and Cádiz. It has a total length of 3221.70 m. It is located in a circular alignment of 2200 m of radius. The average height of the different piers is 10 m. The viaduct crosses twice the Guadalete River and the roads CA 2011 and CA 9023. A bridge which fulfils all the functional and structural requirements was designed from the beginning taken into account the appearance suitable adapted to the landscape which is a very flat fertile valley. (Fig. 9) One of the most specific features of the bridge is the extremely length and the foundation conditions. The soil has more than 25 m deep very soft layer. These soil conditions are inadequate for resisting horizontal forces due to braking and traction railway loads.
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24 Track-Bridge Interaction on High-Speed Railways
Figure 9.
Figure 10.
Viaduct over the Guadalete river.
Viaduct over Guadalete river. Typical cross section.
The length of the viaduct asks for a typological study taken into account track-deck interaction problems due to imposed deformations, horizontal loads and to the position of the rail and deck expansion joints along the viaduct. In this study different structural, constructional and environmental problems were considered (Fig. 11). From the structural point of view, the strength to resist the horizontal forces due to braking and seismic loads should be compatible with the flexibility to reduce as much as possible the stresses due to thermal and long term deformations. It was decided to avoid any expansion joint on the rails in order to improve as much as possible the exploitation of the railway. A maximum length of 200 m between structural deck expansion joints was limited in order to avoid any overstressing on the rails due to track-deck interaction problems. In order to fulfill all the environmental requirements, the supports over the river beds were reduced as much as possible. A precast solution was designed to allow industrialized construction procedures suited to a very long viaduct (Fig. 12). The final solution adopted is a twin precast box girder 2.20 m deep and located below each railway axis, 2.15 m apart. The total deck width is 13.0 m (Fig. 10). This deck allows to span the length of 30 m as a simply supported structure along most of the length of the bridge and to span 49 m with the help of two additional precast arches. This concept allows with a very repetitive structure to cross the longest spans due to the presence of Guadalete river beds and the road CA
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Track-bridge interaction problems in bridge design
Figure 11.
Viaduct over Guadalete river. Simply supported girders.
Figure 12.
Viaduct over Guadalete river. Precast arches.
25
2011. With these criteria the viaduct is split in 7 stretches with the lengths and spans shown in Table 1. (Fig. 13) The use of continuous precast arches allows to balance the horizontal forces due to permanent loads on the intermediate supports creating a well suited and new structure (Fig. 14). 2.2 Track deck interaction problems During the design conception the main idea of a jointless rail bridge has been present in order to reduce maintenance problems. Two main aspects for track-deck interaction problems were studied: 1. The structural scheme to resist the horizontal forces due to braking and traction railway loads. 2. The maximum displacements due to thermal and long term deformations to avoid overstresses on the rails produced by track-deck interaction problems. By these reasons the piers under simply supported girders are able to resist the horizontal loads from each span and the expansion length of 30 m is far from producing any overstressing on the
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26 Track-Bridge Interaction on High-Speed Railways
Table 1. Viaduct over Guadalete river. Span distribution. Stretch
Length (m)
1 2
823.50 207.00
3 4
810.00 207.00
5 6
547.20 207.00
7 Total
N◦ Spans
420.00
27 2 3 27 2 3 18 2 3 14
3221.70
101
Figure 13.
Viaduct over Guadalete river. Span distribution.
Figure 14.
Viaduct over Guadalete river. Precast arches.
Span (m) 30.50 30.00 49.00 30.00 30.00 49.00 30.40 30.00 49.00 30.00
rails. For the greater spans the continuous precast arches are able to resist properly the horizontal loads with an expansion length to avoid any overstressing problems on the rails. Preliminary analysis to control track-deck interaction problems were performed: the expansion lengths limitation were fulfil so the maximum displacement due to braking loads should be limited to 5 mm at deck level on the structural deck expansion joints. These displacements were obtained taken into account the flexibility of the soil-foundation structure using the stiffness matrix of the whole. It was assumed that all the braking loads are transmitted to the continuous deck which can be considered as an upper limit of the total load because of the continuity of the rails (Fig. 15).
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Track-bridge interaction problems in bridge design
Figure 15.
Displacement control under railway braking loads.
Figure 16.
Track-deck interaction structural model.
Figure 17.
Braking loads. Axial forces on the rail with soil structure-interaction model.
27
A detailed model for the track interaction problems was made in order to confirm the main assumptions done in the preliminary analysis. The continuous precast arches were modelled with plane bar elements. Four additional simply supported spans are added to avoid any perturbation on the expansion joints results. The rails were included in the model as an additional structure. The connection between the rail and the deck has been done with perfectly elastoplastic elements which represent the track-deck behaviour, as it has been mentioned above. The maximum forces and stiffnesses vary according the loaded or unloaded track situation (Fig. 16). The different load cases were done in two different models: 1. Built-in model, in which the arches are completed built in the foundation 2. Model with soil-structure interaction: in which the soil-foundation stiffness matrix has been included. Three load cases have been considered on each model: 1. Case 1: Railway braking load at rail level. 2. Case 2: Deck increment of temperature −20◦ C. 3. Case 3: Deck increment of temperature +20◦ C. The summary of the main results is as follows: 1. For the braking loads the soil structure stiffness has a relevant importance in the total displacements and on the overstressing in the rails. (Figs 17–18) 2. For the thermal actions soil-structure interaction stiffness has no influence in the results. (Figs 19–20) 3. The increment of stresses due to the combination of thermal actions and braking forces are lower than the limits 72 MPa in compression and 92 MPa in tension.
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28 Track-Bridge Interaction on High-Speed Railways
Figure 18.
Braking loads. Axial forces on the rail built-in model.
Figure 19.
Thermal action +20◦ C on the deck. Axial forces on the rail with soil structure-interaction model.
Figure 20.
Thermal action +20◦ C on the deck. Axial forces on the rail in built in model.
REFERENCES ENV 1991-3:1995. Eurocode 1. Basis of design and actions on structures. Part 3. Traffic loads on bridges. Manterola, J., Astiz, M.A., Martínez, A. Puentes de ferrocarril de alta velocidad Revista de Obras Públicas n◦ 3386 pp. 43–77 Abril (1999). Manterola Armisén, J.; Martínez Cutillas, A. Prestressed concrete railways bridges Workshop Bridges for high speed railways. Oporto (2004).
© 2009 Taylor & Francis Group, London, UK
CHAPTER 4 Controlling track-structure interaction in seismic conditions S.G. Davis Mott MacDonald, United Kingdom
ABSTRACT: In seismic regions, a major constraint affecting the design of high speed rail viaducts is the track-structure interaction during the operation of the trains, and in particular when the train must remain safe at high speeds under the service design earthquake. The paper describes studies undertaken to establish design criteria and the analysis methods to be used for the design of high speed rail viaducts. These studies comprised non-linear analysis of the track and viaduct structure system under the actions of temperature variation, traction and braking, and multiple input nonlinear time history analysis for the response under earthquake loading. A series of parametric studies were undertaken and the conclusions from this work used to form the basis of a design specification.
1
TRACK-STRUCTURE INTERACTION IN NORMAL OPERATION
Track-structure interaction is the transfer of the traction and braking longitudinal loads between the track and the viaduct deck through the ballast. Ballast behaviour under these actions is non-linear and load and temperature dependent. Typically, viaducts for high speed railways are a succession of simply supported spans such that no rail movement joints are needed. However it is essential that the viaduct is designed to limit the relative displacements between decks to 5mm so that the rails are not overstressed or the ballast destabilised under the traction and braking loads. The fundamental principles of the analysis method to calculate the relative displacements are given in the Eurocode (UIC Code 774-3R), but use of these requires a finite element analysis modelling the elasto-plastic friction behaviour between the track and the bridge deck. As part of Mott MacDonald’s role in the Taiwan High Speed Rail project, studies were carried out, aimed at providing simplified methodology for use at the detailed design stage. Existing in-house software was adapted to analyse models comprising elastic deck and track members connected by non-linear ballast members as shown in Figure 1. The method and results were validated, and the process was automated to
Deck Node
Track 1 Node
Ballast Member
Track 1 Member Deck Member
Track 2 Member 1m
Figure 1.
1m
1m
Track-structure interaction analysis model.
29 © 2009 Taylor & Francis Group, London, UK
1m
30 Track-Bridge Interaction on High-Speed Railways
analyse as many structures as possible in the time available. A parametric study was then conducted to determine the key factors affecting track-structure performance, and to develop the simplified method of analysis. Nearly 800 non-linear models, each with 25 viaduct spans, were created and analysed. Analysis was carried out in two stages. In the first stage, seasonal temperature expansion or contraction was applied to the viaduct with the train absent. Loads in the ballast members were extracted from this model and used to provide the starting position for the second stage analysis with the vertical load of the train and traction/braking loading. The ballast stiffnesses used, and examples of the viaduct models analysed are shown in Figure 2. Models with pier heights from 8 m to 25 m were analysed. All the piers had the same cross section and the same foundation stiffness, and by converting pier heights to equivalent substructure stiffnesses, it was possible to express the geometric relationship between the taller piers and the standard piers in a dimensionless form. By determining the stiffness ratio of the taller piers to the standard piers at which the 5 mm limit on relative displacement between decks was reached, a series of design charts were produced as shown in Figure 3.
Ballast Stiffness
(Loaded) Rectangular valley
V-shaped valley
(Unloaded)
Sloped step valley Displacement
Figure 2.
Ballast stiffness and viaduct models.
0
Stiffness Ratio of Feature Stiffnesses, R/S
2 0.2
4 0.3
6
0.4
n increasing →
1 0.1
10
0.5 0.6 0.7 0.8
Rectangular Valley Design Lines 0.9 1 30
40
50
60
70
80
90
100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300
Standard Pier Stiffness (MN/m)
Figure 3.
Example of design chart.
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Controlling track-structure interaction in seismic conditions
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31
TRACK-STRUCTURE INTERACTION UNDER THE SERVICE EARTHQUAKE
High speed rail systems in seismic zones are commonly designed against two levels of design seismic event – the service earthquake and the ultimate (or no collapse) earthquake. The aim is to ensure safe running during low return period events and the avoidance of collapse during events with a suitably low chance of occurrence during the design life of the system. The Taiwan High Speed Rail project, located in one of the most seismically active areas in the world, adopted this approach. The viaducts are designed to resist without collapse seismic forces arising from design peak ground accelerations up to 0.4 g, and to remain fully serviceable under a PGA of one third of this. To ensure that trains on the viaduct can brake safely to a stop from full design speed during service event earthquakes, track displacements must remain within the allowable values with the structure subjected to dead load, temperature variation, train braking load, and seismic excitation. In the main, track-structure interaction is concerned with relative longitudinal movements of adjacent viaduct spans at the structure’s movement joints (usually over supports) and the requirement is to prevent buckling, fracture or excessive displacement of the rails. Relative transverse movement of adjacent spans is usually restrained by seismic shear keys or buffers on the piers that allow only minimal movements at joints (of the order of 2–3 mm). However, examples exist where the designer has chosen a form of construction comprising a series of contiguous portal frames, and this form of articulation permits relative transverse movements that are several times larger than normal. Clearly, the magnitude of these movements also needs to be controlled in order to ensure running safety in an earthquake. For long structures, spatial variations in seismic ground motions can be significant even for uniform site conditions due to complex mechanisms such as wave passage, scattering, and attenuation effects. Hence non-linear time history analysis was adopted, and three orthogonal displacement time histories were generated at each support location to match the seismic design spectra and statistical spatial coherency functions. Since the track displacement is a function of both vertical load and loading history, a series of three-dimensional non-linear finite element analyses were performed simulating the loading sequence. Dead load was applied to the structure first without the presence of rail or ballast elements. Rail and ballast elements were then “birthed” and temperature variation was applied to the girders. Following ballast property modifications to account for train vertical loading, train braking loads were imposed on the structure and the structure was subjected to earthquake ground displacements. Relative movements at the track centrelines and the forces in the rails were recovered for assessment of track/structure performance, the key parameter being a limit on relative displacement between decks of 25 mm. The computer program ADINA was used. Figures 4 and 5 show the finite element modelling to represent the viaduct. Beam elements represent the bridge deck, columns and rails; contact surfaces with friction simulate ptfe sliding bearings at the free ends of the box girders at the connection to the column; non-linear plastic truss elements simulate the ballast in the longitudinal direction; and rigid links and stiff springs the interconnection of these components. The pile cap is modelled as a rigid system between the base of each column and the soil, with the mass lumped at the pile cap’s centre. The soil-structure interaction matrix forms the interface between the underside of the pile cap and the ground such that the supporting soil nodes are the boundaries of the model and represent the free-field ground in the vicinity of each foundation. The free-field ground motions are applied to the boundaries of the model, across the soil structure interaction matrices, and these displacements subject the structure to the seismic excitation. The free-field ground is assumed to move with a displacement history that is unaffected by the presence of the structure. Two viaduct models were analysed, each consisting of 25 identical spans, 30m long as shown in Figure 6. The uniform model represents a typical viaduct crossing flat terrain with uniform pier heights of 10 m and a pier cross section 2.7 m square, and the valley model a crossing over a gentle valley with pier heights varying from 10 m to 18 m. For each configuration, the 25 spans represent part of a viaduct that could be many kilometres in length. The 750 m length of the viaduct model is
© 2009 Taylor & Francis Group, London, UK
32 Track-Bridge Interaction on High-Speed Railways Rail Deck girder
Pier
Pile cap
Figure 4.
Elevation showing modelling of pier and deck members.
Centroid of Deck Deck Node Node-to-Node Contact with Friction Column Member
Bearing Node Rigid Transverse Connection Pile/Soil Damping Matrix Structure Node
Ballast Node Longitudinal Ballast Stiffness Deck Node Rigid Transverse Connection
12x12 Pile Stiffness Matrix u(t) - Displacement Time History Input
Mass of Pile Cap
Ground Node
Rigid Link
Figure 5.
Modeling of pier and deck members.
able to accommodate the 300 m long train with the train located sufficiently far from the boundaries of the model, such that boundary effects are negligible. The static and dynamic loads were imposed on the structure in the following sequence. First, a non-linear static analysis was performed for dead load and temperature through an incremental load step solution procedure using the full Newton method for equilibrium iterations. The temperature rise was applied to the deck concrete box girders only. Next, longitudinal and vertical loads were applied statically to one of the two tracks to simulate the train braking at
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Controlling track-structure interaction in seismic conditions
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Uniform Model
Valley Model
Figure 6.
Viaduct models for seismic analysis.
the onset of the seismic event. The vertical load is the weight of the train and the braking force is a uniformly distributed longitudinal load over a 300 m long segment of the viaduct between Piers 7 and 17. Finally, a multiple support input non-linear dynamic analysis was performed for 28 seconds of seismic load using a step-by-step direct time integration method. The Newmark method, an implicit finite difference solution scheme, was employed to solve the problem in the time domain with time increments of 0.01 second. For each time step, the full Newton method was used for the equilibrium iterations. The initial studies showed that with the proposed viaduct configuration and structural geometries, except for sections of viaduct with tall relatively slender piers, track structure performance would be within the 25 mm limit for relative displacement between decks. This relative displacement is the aggregated effect at the track centreline of longitudinal movement due to braking and seismic motions, beam end rotation under vertical load, and angular bending in plan due to transverse seismic motions. However the information obtained was insufficient to develop usable criteria for simplifying the design process. In subsequent studies, a series of simplified analyses were conducted to examine the sensitivity of the structural response to various simplifying assumptions. Simpler modelling of the coupling between decks through the track ballast system was seen as particularly desirable, and hence a number of linear response spectrum and time history analyses were undertaken. Results from single versus multiple support excitations, linear versus non-linear ballast and bearing behaviour, and use of response spectrum versus time history analyses on the response of both models were studied in detail. Comparisons of the responses of the valley model using response spectrum, linear and non-linear time history analysis are presented in Figure 7. Generally, the responses obtained from linear time history analysis (both multi- and singlesupport excitation) and the response spectrum analysis (constraint nodes method) are equivalent, whereas those from response spectrum analysis (absolute sum method) are almost always overpredicted by a large margin. The maximum relative displacement from the nonlinear analysis is about 20 mm, compared with 65–70 mm from the linear analyses. Thus it appears that there is considerable benefit to be obtained from modelling the nonlinear behaviour of the ballast in a track structure interaction analysis for earthquake loading. The results from the parametric studies described above provided confidence that outline designs prepared using simplified methods are feasible and could be developed to meet detailed contract requirements. Designers should nevertheless be required to carry out full track-structure analyses for the service level earthquake case for all sections of viaduct, during the detailed design phase. It was clear from the studies that nonlinear time history analysis would be required in order to obtain sufficient accuracy in the predictions of relative displacements of adjacent spans. However, it was considered that there was a relatively small difference between applying a synchronous time history excitation input to each pier and including wave passage effects by applying time delayed displacements as inputs to each pier. Following another study, it was concluded that the additional effect of spatial variation could be adequately represented by adding a relative displacement that is a direct function of span length. This displacement modifies (increases) the value of displacement obtained by applying synchronous time history input. The value is around 2.5 mm for 30 m spans.
© 2009 Taylor & Francis Group, London, UK
34 Track-Bridge Interaction on High-Speed Railways
160
Response Spectrum Constraint Nodes
Time History Single Support Excitation
Response Spectrum Absolute Sum
Time History Nonlinear Model
Time History Multi-Support Excitation
Relative Displacement (mm)
140 120 100 80 60 40 20 0 3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Pier No.
Figure 7.
3
Comparison of relative displacements.
OBSERVATIONS DURING IMPLEMENTATION
The majority of the designers for the THSR Project carried out the track-structure interaction analysis solely on the basis of the non-linear time history method, including the coupling effects of the track. However, several of the designers also presented results of analyses using linear response spectrum methods without track. In addition, the authors’ firm, as Independent Checking Engineer, carried out a number of check analyses using a variety of methods. It has therefore been possible to make some comparisons between the results. Using the linear response spectrum method, disregarding the track, the relative longitudinal displacements at joints under earthquake are generally in the range 15 to 40 mm, but in some cases as high as 60 mm. Using the non-linear time history method, including the track, the relative displacements under earthquake were always found to reduce to values comfortably within the specification limit. Indeed, the maximum displacement calculated by any of the designers or by the Independent Checking Engineer for the standard viaducts is around 12 mm (worst case with highly non-uniform stiffnesses with adjacent piers). Typical values with reasonably consistent pier top stiffnesses are around 5 mm. These results are in good agreement with the pre-project work described in Sections 1 and 2 of this paper. The THSR Design Specification limits rail stresses to 167 MPa in tension and 147 MPa in compression, values that are considered to provide an acceptable margin of safety against either fracture or buckling. The rail stress can only be reliably checked by means of a non-linear time history analysis, with the non-linear connections between the rails and the structure properly modelled. The designers were provided with bilinear (elastic/plastic) stiffness relationships for the trackwork for use in their models, as shown in Figure 2. The designers have generally reported rail stresses from their analyses that are comfortably within the specification limits – commonly up to a maximum of 100 MPa. The checks carried out by the Independent Checking Engineer have revealed similar results.
© 2009 Taylor & Francis Group, London, UK
Controlling track-structure interaction in seismic conditions
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35
CONCLUSIONS
From pre-project studies, confirmed by reviews and checks of the actual designs for a major high speed rail project, it is clear that unless the appropriate track-structure interaction analysis methods are adopted the displacements at structure joints can be over-estimated, and that this could lead to uneconomic design. It is essential to perform a track-structure interaction analysis based on non-linear modelling incorporating the coupling between the track and the viaduct decks, and using appropriate acceleration time histories as ground motion input. The stiffness of the foundation must be included in the model, although the response of the structure itself often dominates the displacements at deck level, particularly for high piers. A key requirement is to model as accurately as possible the non-linear interface between the track and the structure, in order to derive peak rail stresses in addition to the relative displacements between viaduct spans at structure movement joints. Using the above approach, it has been found that the relative longitudinal displacements between viaduct spans remain comfortably within the limit specified. Similarly, peak rail stresses are generally well below the limits required to provide a margin of safety against either fracture or buckling. Typically, for “service” level PGAs of up to 0.13 g, peak relative displacements are of the order of 12 mm and peak rail stresses in the range 80–100 MPa. Nevertheless, a minimum level of structure stiffness is necessary to achieve compliance with the requirements and large variations in stiffness between adjacent piers are to be avoided.
ACKNOWLEDGEMENTS The permission of the Taiwan High Speed Rail Corporation (THSRC) to publish technical information on this project is gratefully acknowledged. The Independent Checking Engineer for the THSR Project was IREG (International Railway Engineering Group), a Joint Venture of DE-Consult, Electrowatt Engineering, Mott MacDonald and SYSTRA. REFERENCES “Track-Structure Interaction on the Taiwan High Speed Rail Viaducts”, Place, J.D., Davis, S.G. and Barron, M. Paper presented at the IABSE Symposium ‘Structures for High Speed Railway Transportation’, Antwerp, August 2003. “Bridges for High Speed Railways – Special Considerations”, Davis, S.G. and Wilkins, A.G. Paper presented at the Institution of Civil Engineers Conference ‘Current and Future Trends in Bridge Design, Construction and Maintenance’, Kuala Lumpur, October 2005.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 5 Track-structure interaction and seismic design of the bearings system for some viaducts of Ankara-Istanbul HSRL project F. Millanes Mato Universidad Politécnica de Madrid & IDEAM S.A., Madrid, Spain
M. Ortega Cornejo IDEAM S.A., Madrid, Spain
ABSTRACT: This paper describes the specific problems concerning the design and dimensioning of the bearing systems for some viaducts of the Ankara-Istanbul HSRL. These structures, with total lengths of up to 2232 m, consist of a series of simply supported 34-meter-long spans, whose cross-section is constituted by a deck with multiple precast I-beams, transversely braced at their ends. Because of the area’s high seismic activity, with ground accelerations of up to 0,21·g, the structure’s design turned out to be highly conditioned by the global seismic structural response. In addition to the classical deck-piers-foundation interaction, it proved essential to take into account the problems dealing with the rail-structure interaction as well as the bearings’ design, strength and ductility. These considerations led to technically innovative dispositions for the aforesaid bearings.
1
INTRODUCTION
As part of the High Speed Railway Line Ankara-Istanbul Rehabilitation Project, IDEAM, S.A. has collaborated with Eng. Manuel Alpañés OHL’s Technical Office Director, in the execution project for the structures of Viaduct 1 and 4 with 1032,80 m and 2232,45 m length respectively, built by the Contractors OHL/Alarko/G&O Joint Venture. Dr. Eng. Juan Carlos Lancha of OHL’s Technical Office has also collaborated with IDEAM, S.A. in the multimodal seismic analysis of the Viaducts. The new High Speed Railway Line has been projected for a speed of 250 km/h. According to the client, the deck of both Viaducts should consist of precast prestressed double-T deck beams placed in contact with each other (to eliminate the need for slabs with permanent shuttering) and cast in place concrete slab. In order to fulfil precasting and transportation requirements, the maximum span should not be greater than 35 m. 2
DESCRIPTION OF THE VIADUCTS
Viaduct 4 is the longest viaduct of the Ankara-Istanbul High Speed Railway Line with 2.232,45 m long, and the deck consists of 66 simply-supported spans distributed as follows in the viaduct (Figs 1 and 2): – First span with 23,40 meters between the axis of bearings at abutment and axis of pier 1. – 64 typical spans with 34,00 meters between centrelines of adjacent piers. – Last span with 33,05 meters between the centreline of pier 65 and the axis of bearings at abutment 2. 37 © 2009 Taylor & Francis Group, London, UK
38 Track-Bridge Interaction on High-Speed Railways
Figure 1.
Figure 3.
View of Viaduct 4 during construction.
Figure 2.
Lateral View of Viaduct 4 concluded.
Viaduct cross section.
According to the client, the bridge-deck cross-section is made up of 12 precast prestressed beams 1,95 m deep and 0,93 m wide, abutting to each other (Fig. 3). Just 0,03 meters are kept free between adjacent beams to allow for tolerances in the execution. Cast in place concrete compression slab is laid over the beams, with thickness varying from 0,25 meters to 0,34 meters to obtain transversal slope required. The width of the deck remains constant at 11,50 m. Longitudinal displacements of the spans will be restrained in one end and allowed at the other one. Appropriate design of the bearings system in the deck is carried out according to this condition as described later. In these conditions expansion joints at the rails over any of the abutments will not be required, in view of the supports layout and mentioned spans. The track-structure interaction analysis confirms this assumption. Lateral displacements are restrained at both ends of each span with lateral bearings between the deck and the pier or abutment. A simple transverse bracing between the beams is designed to achieve with transverse requirements. All the abutments are of the closed type, which allows for an easy construction of transition structures, as required in High Speed Railway systems.
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction and seismic design of the bearings system
Figure 4.
39
Front and lateral view of a pier.
Piers consist of transverse frames with two hollow prismatic shaped columns, 2,20 m wide and 3,40 m long, connected at the top with transverse cap supporting the beams. This upper element also includes lateral restraints for the deck (Figs. 4 and 5). The high of the piers of Viaduct 4 varies from 7,40 m to 18,03 m. The foundations of the piers have been designed with 16 piles of 1,20 m of diameter. Viaduct 1 is 1.032,80 m long, and the deck consists of 31 simply-supported spans distributed as follows in the viaduct (Fig. 6): – First span with 23,40 meters between the axis of bearings at abutment and axis of pier 1. – 29 typical spans with 34,00 meters between centrelines of adjacent piers. – Last span with 23,40 meters between the axis of pier 30 and the axis of bearings at abutment 2. The main difference between Viaduct 1 and 4 is the geometry of the piers. The piers of Viaduct 1 consist of transverse frames with three cylindrical shaped columns (Fig. 7), 2,00 m diameter, connected at the top with transverse cap supporting the beams. This upper element also includes lateral restraints for the deck, the same way as it happened on Viaduct 4. The high of the piers of Viaduct 1 are remarkable lower than Viaduct 4, and varies from 4,24 to 7,84 m high. 3
DESIGN CRITERIA TRACK-STRUCTURE INTERACTION AND SEISMIC CRITERIA
In the case of a bridge with statically determinate short spans, the thermal effects, shrinkage, creep and bending do not introduce important displacements or stresses in the rails. However, from the point of view of the track-structure interaction, “fixed points” in the deck will be needed to ensure that braking loads are transferred correctly. The creation of such “fixed
© 2009 Taylor & Francis Group, London, UK
40 Track-Bridge Interaction on High-Speed Railways
Figure 5.
View of one pier of Viaduct 4.
Figure 6.
View of Viaduct 1 during construction.
points” means that supports must be fixed in one direction and that the flexibility of piers and foundation must be limited. For this reason the use of specific bearings is required, fixed in the longitudinal direction at one end of the beams and free to slide at the other. A specific analysis for this structure has been developed according to Eurocode 1, part 2 [1], so as to guarantee that stiffness requirements for the substructure are fulfilled according to track structure interaction, and no excessive displacements or stresses appear on the rails.
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction and seismic design of the bearings system
Figure 7.
41
Front and lateral view of one pier of Viaduct 1.
Due to the high level of seismic activity in the area, seismic criteria are especially relevant for the design of the structure. In general seismic design runs counter to provisions for track-structure interaction. In fact, fixed points and rigid piers are required to assure suitable behaviour of the structure with regard to braking/start loads. This increases the seismic loads on the piers and foundations. 3.1
Seismic design
According to the specifications transmitted by the client, the calculations have been carried out for an acceleration of the ground of 0,21·g for Viaduct 4 and 0,25·g for Viaduct 1. In the event of such an important earthquake, failure of the structure (Ultimate Limit State) must be prevented. The basic standards applied in the calculations of the Viaducts have been the Structural Eurocodes, and according to the client requirements, seismic loads to be used in the elastic analysis for seismic effects have been obtained from the elastic seismic response spectrum in AASHTO “Standard specifications for highway bridges” [2]. All the calculations after seismic analysis have been done according to Eurocodes, including plastic capacity analysis to obtain forces for the design of bearings and foundations. Design details for reinforcement steel and others have also fulfilled requirements from Eurocodes for seismic design. As above mentioned, the elastic horizontal response spectrum used in the design is obtained from AASHTO. The soil profile type is classified as ST IV, which is on the safety side. In these conditions Figure 8 shows elastic horizontal spectrum considered in the calculations. With regard to vertical component of seismic action, AASHTO neglects this component in the design. As AASHTO is the standard considered for the elastic response spectrum, no vertical effects have been considered in the design. In addition, the same basic conclusion can be obtained according to Eurocodes. They state the effects of vertical seismic component on the substructure need only be investigated when the piers are subjected to high bending stresses due to permanent actions on the deck [3] or in the exceptional case of bridges located within 5 km from an active semi tectonic fault according to [4], which is not our case. As a result, vertical component of seismic actions need not to be considered on the substructure.
© 2009 Taylor & Francis Group, London, UK
42 Track-Bridge Interaction on High-Speed Railways
Just for stability of bearings or uplift forces in the prestressed deck some checking could be necessary according to Eurocodes, which have been the only incidence in the design from vertical seismic action. Seismic design forces for individual members of the bridge have been determined by dividing the elastic forces obtained from the spectrum of Figure 8 by the appropriate “behaviour factor” q. This factor has the same significance than the Response Modification Factor R in AASHTO. According to the height of the piers in the longitudinal direction for the shorter pier of Viaduct 4 q = 2, 98 ≈ 3. As q = 2,98 is obtained for the shortest pier we used q = 3 for all the piers in longitudinal direction for Viaduct 4. This value is coincident with R = 3 established by AASHTO for single columns. So both standards have been simultaneously fulfilled in longitudinal direction. Meanwhile, in the transverse direction, in Viaduct 4, for piers with more than 12 m high q = 3, and for piers with less than 12 m high (7,40 m minimum), q = 2,238. As AASHTO establishes R = 5 for multiple column bent piers it is clear that ductile factors for transverse direction according to Eurocodes are on the safety side from Response Modification Factors from AASHTO. Bearings and forces transmitted to foundations have been obtained by a plastic capacity analysis including appropriate overstrength factors according to Eurocodes, as required by the client. Equivalent linear analysis has been adopted for all elements, including ductile components as piers. For the estimation of element stiffness, stiffness values corresponding to uncracked cross-section, or effective values after cracking, can be used. As an overestimation of the stiffness leads to results which are on the safety side regarding the seismic actions, uncracked stiffness have been used in the calculations of seismic forces. This assumption is not true in the case of displacements, and calculations with effective stiffness after cracking is made in the case of ductile components (piers) so as to obtain safety values for displacements. Rigid deck behaviour is considered, taking into account that the deformations of the deck are negligible compared to the displacements of the substructure elements. Piers and abutments are assumed to be fixed to the soil foundation when evaluating seismic forces, and no soil-structure effects has been considered. However, flexibility of the foundation has
Figure 8.
Horizontal spectrum for seismic actions.
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction and seismic design of the bearings system
43
been considered according to geotechnical parameters of the ground included in the geotechnical report to obtain seismic displacements. Mean values of permanent masses (self-weight of deck, piers and abutments, and dead loads) have been used. Null quasi-permanent values of the masses corresponding to variable actions have been considered, in the event of normal traffic intensity on the line, according to Eurocodes (ENV 1998-2:1994 [3] and prEN1998-2. Draft April 2004 [4]). The elastic forces and displacements shall be determined independently along two perpendicular axes, longitudinal, and transversal. As there are no interactions between decks in the longitudinal direction (due to the fixed-sliding bearings system) (Fig. 9), a single degree of freedom model is appropriate for modelling the behaviour of the structure in longitudinal direction. A single mode spectral analysis has been used. The equivalent static seismic forces from deck on abutments and piers in the longitudinal direction are derived from the inertia forces corresponding to the fundamental natural period of the structure (abutment or pier and one span of the deck) in that direction, using the relevant ordinate of the response spectrum. Fi = MiSd(Ti)
(1)
Where: – Mi is the effective mass attributed to the abutment or the pier “i”, which corresponds to the mass of one span. – Ti is the fundamental period of the same element (abutment or pier). From the safety side inertia forces from the own mass of piers have been considered with the same seismic spectral acceleration. However, inertia forces from the own mass of the abutments is expected not to be significantly amplified and seismic ground acceleration have been considered. As there are few interactions between decks in the transverse direction, a single degree of freedom model could be appropriate enough for modelling the behaviour of the structure in that direction. A single mode spectral analysis also has been used in the transverse direction. The same way, the equivalent static seismic forces from deck on abutments and piers in the transverse direction are derived from the inertia forces corresponding to the fundamental natural period of the structure (abutment or pier and one span of the deck) in that direction, using the relevant ordinate of the response spectrum. As it was made for the longitudinal analysis, from the safety side, inertia forces from the own mass of piers have been considered with the same seismic spectral acceleration. However, inertia forces from the own mass of the abutments is expected not to be significantly amplified and seismic ground acceleration have also been considered. Conclusions from the transversal single degree of freedom model are confirmed by additional multimodal analysis of the structure. Transverse seismic forces from multimodal analysis confirmed the values obtained with the single degree of freedom model as described later. Interaction between adjacent decks from out of phase transverse displacements of piers is also obtained with multimodal analysis, which has been used for the design of bearings.
Figure 9.
Longitudinal bearing system with fixed-sliding bearing devices.
© 2009 Taylor & Francis Group, London, UK
44 Track-Bridge Interaction on High-Speed Railways
The design forces for piers have been calculated by dividing the elastic forces, obtained as explained before, by the appropriate behaviour factor q. It has been done a Plastic Capacity analysis according to Eurocodes to obtain the design forces for bearings and foundations of piers, and the smaller of these results or the elastic forces have been considered in the design.
3.2
Multimodal seismic analysis
The response of the structure to a transverse seismic action has been carried out, as described before, considering a system of one degree of freedom for each pier and abutment. Additional calculations have been made to verify the results previously obtained. The whole deck has been modelled with elastic supports (transversal displacement and vertical rotations) equivalent to the pier-foundation (abutment-foundation) system. The calculations have been carried out by mean of a SpectralAnalysis, with Modal Superposition. The results for each mode are obtained and, later, combined to obtain the total response of the structure to a transverse seismic action. It’s important to remark that, because of the typology of the bridge, the obtained modes are very close in frequencies, so, the method chosen to combine the results for each mode, is very important to get a good accuracy in the calculations. In this way, the selected method is the CQC method defined by Wilson, der Kiureghian and Bayo, because is the most accurate method to consider the correlation between modes, very significant in this train bridge because the frequencies are very close, and the cross terms of the quadratic combination, very significant and with a sign that can be plus or minus. 3.2.1 Transverse reactions The transversal reactions obtained in each element, pier or abutments are shown in Figure 10 in comparison with the previously obtained with the systems of one degree of freedom. It has been made a parametric analysis considering the vertical rotation free in both ends of each span, or taking into account an elastic support in the origin of the span in the vertical rotation, equivalent to the system pier-foundation, and two more studies considering the elastic constant multiplied and divided by 10. Figure 10 shows that the results from those models of the whole deck are similar between them and similar to the previously obtained with the one degree of freedom system, with no significant variations. 3.2.2 Longitudinal displacements and forces on bearings The longitudinal displacements and forces on bearings due to the action of a transverse seismic excitation have also been analysed. The displacements and forces have been obtained through two models of the whole deck. The first one considering the vertical rotation free in both ends of each span, and the second one, considering an elastic support in the vertical rotation, equivalent to the system pierfoundation. Analyzing Viaduct 4, with the first model, considering the vertical rotation free, in most of the piers the longitudinal displacements of the bearings due to the transversal seismic action are less than 1 mm (in thirty four piers), between 1 mm and 2 mm in fourteen piers, between 2 mm and 3 mm in four piers and more than 3 mm in thirteen piers. The maximum value is 6,51 mm, in pier number sixty one. Because of the pin connections on piers, the maximum displacements in both ends of each span are almost the same. In the second model, with elastic connection of rotation around the vertical axis on pier, in the line with the guided bearings (beginning of each span), the maximum longitudinal displacement in one bearing, with no cracked torsional inertia considered in piers, reduces to 4,92 mm.
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction and seismic design of the bearings system
Figure 10.
Transversal reactions on piers obtained from the multimodal analysis.
Figure 11.
Longitudinal forces in the 4 central bearings due to a vertical moment.
45
To evaluate with more accuracy the efforts in the 4 longitudinally fixed bearings (Fig. 12), we took the vertical moment reaction obtained from the second model. This moment is equivalent to a longitudinal distribution of forces with the following results (Fig. 11): – in twenty nine piers and in the abutment AB2 the maximum force in one bearing reached a value greater than 3000 kN.
© 2009 Taylor & Francis Group, London, UK
46 Track-Bridge Interaction on High-Speed Railways
Figure 12.
Bearings of one typical span.
– in eighteen of these piers and in the abutment AB1 this value is greater than 4000 kN – in eleven piers this value is greater than 5000 KN. The maximum value obtained is 13270 kN, in pier number twenty three. – In thirty six piers this force is less than 300 t. These results are obtained with no cracked torsional inertia in piers. Additional calculations have been made considering the possibility of torsional cracking of the elements of the piers. This way, we have considered a torsional cracked inertia of the piers equal to the total inertia divided by three. The results are similar in the case of transverse reactions but the longitudinal reaction in the external fixed bearing changes significantly, until near fifty per cent. The displacements in this case are between the obtained in the first model, with pin connections, and the obtained with the second model, with elastic support of the rotation around the vertical axis, with total torsional inertia of the elements of the piers. After these studies, due to the transverse seismic action relevant longitudinal forces on bearings are developed. However, these forces are associated to minor longitudinal displacements of bearings to vanish the forces. It is obviously preferable for design purposes to vanish these forces with minor displacements under seismic actions than withdrawing these forces with rigid systems under seismic action. As a result, a minimum ductility was required in the design of the bearings under seismic actions. According to the results of the studies, a capability of displacement about ±10 mm was enough to vanish from the safety side all longitudinal forces on bearings from transverse seismic action. So it was required in the design of the bearings ±10 mm of displacement without developing longitudinal forces than that obtained from the just longitudinal seismic action. In this way, longitudinal effects from transverse seismic action can be neglected in the analysis of bearings. Admissible displacements in the bearings guarantee vanishing these effects without increasing forces on bearings higher than that required for longitudinal seismic actions. Obviously, these requirements for seismic actions have to be simultaneously fulfilled with rigid response of the bearings under service actions.
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction and seismic design of the bearings system
Figure 13.
Bearings of one typical span.
Figure 14.
Detail of cast in place concrete blocks between the inferior heads of the beams.
4
47
CONSIDERATIONS ABOUT THE BEARINGS SYSTEM
The serviceability conditions required from the bearings system are shown in the project’s plans, and present some specific requirements in its response to the important horizontal loads acting on the bridge, chiefly braking and seismic actions. The project’s specifications can be met by means of a certain variety of solutions, each one of them employing different technologies.
4.1
Specific requirements regarding the bearings system
Each span of the deck is supported on 24 vertical bearing devices, placed at the ends of each of the 12 beams of the deck. The 4 central bearings located at the beginning of the span (right side of the pier), must withstand the horizontal longitudinal forces acting on the deck. The rest of the supports (12 placed at the end of the span (left side of the pier) and 8 of the bearings located at the beginning of the span (right side of the pier) do not absorb horizontal loads, so they are completely free, longitudinally and transversally (Figs. 12, 13, and 16). Vertical bearings characteristics of Viaduct 4 are show in Figure 17.
© 2009 Taylor & Francis Group, London, UK
48 Track-Bridge Interaction on High-Speed Railways
Figure 15.
Frontal view of cast in place concrete blocks between the inferior heads of the beams.
Figure 16.
Detail of one vertical bearing.
The horizontal transverse forces are resisted by means of 4 lateral bearings placed at the corners of the pier’s head. (Figs 18 and 19) Figure 13 shows the cross section of the deck in the support line. In both ends, the inferior heads of the beams are joined together at the same level of the lateral bearings with cast in place concrete blocks (Figs. 14 and 15), for transmitting the horizontal reaction of the deck to the lateral bearings of the piers. The project allows for the design of the 24 bearings employing either neoprene or POT-type devices. If neoprene bearings are used, their disposition would be as follows (Fig. 12): – 12 elastomeric sliding bearings at the end of each span (left side of the pier). – 4 elastomeric unidirectional bearings, free in the transverse direction, located at the beginning of the 4 central beams (right side of the piers) in order to withstand the longitudinal horizontal forces. – 8 elastomeric bearings placed at the beginning of the span (4 at each side of the deck). In the case of using POT-type bearings, their arrangement would result in: – 20 free Pot bearings (Fig. 21), 12 of them located at the end of each span (left side of the pier) (Fig. 22), and 8 at the beginning of the span (4 at each side of the deck). – 4 unidirectional POT bearings, free in the transverse direction, placed at the beginning of the 4 central beams. (Fig. 23) However, both options are mutually exclusive. It is not possible to use elastomeric bearings and POT-type devices simultaneously, neither in the same line of supports nor in different lines. The difference in the stiffness of these devices renders them incompatible in the bridge, irrespective of their location.
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction and seismic design of the bearings system
Figure 17.
Viaduct’s 4 table with vertical bearings characteristics.
Figure 18.
Detail of one lateral bearing.
Figure 19.
View of one lateral bearing.
© 2009 Taylor & Francis Group, London, UK
49
50 Track-Bridge Interaction on High-Speed Railways
Figure 20.
Viaduct’s 4 table with horizontal bearings characteristics.
Figure 21.
View of one free POT-bearing.
On the other hand, the lateral bearings must be elastomeric, standard type at the beginning of the span (right side of the pier), and sliding type for those at the end (left side of the pier). (Figs 18, 19 and 20) In addition to absorbing the vertical and horizontal forces as well as the design displacements, there are two basic conditions to be observed in the bearing’s design. a) Distribution of the horizontal longitudinal force among the 4 transversely oriented central bearings. The horizontal longitudinal forces must be withstood by the set of the 4 central transversely oriented bearings. The force criterion is established according to the total force to be resisted by the 4 bearings, instead of that resisted by each individual element. It is the provider’s responsibility to appropriately distribute that total force among the 4 bearings according to the system’s own characteristics. Along these lines, it is necessary to observe EN 1337-1:2000 [5] in order to account for the possible undesirable incidence on each support’s loading state caused by an adverse distribution of the execution tolerances and gaps between the bearing device’s elements responsible for transmitting the horizontal actions.
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Track-structure interaction and seismic design of the bearings system
Figure 22.
View of the free sliding POT-bearing.
51
Figure 23. View of the four central unidirectional POT-bearings.
In the case of using elements of high stiffness and little ductility in order to transmit the horizontal forces, as it is usual in POT-type bearings and in the guiding slots of oriented neoprene bearings, the gaps and execution tolerances, even within the range admitted by the legislation, may very significantly alter the actions distribution in the central supports with respect to a hypothetic uniform share. The system’s inability to offer a ductile enough response, and therefore to regularise the loads on the supports, may lead to failure without previous warning. For this reason, and since the acceptable actions distribution on the bearings depends basically on the chosen system’s intrinsic characteristics, it proved advisable, in the project stage, to specify in the plans the total force to be resisted by the system, leaving the proper forces distribution to the provider’s know-how according to the eventually implemented system’s features. b) Stiffness in service conditions and displacement capability under sustained load caused by the seismic action. The 4 central bearings which transmit the horizontal longitudinal forces must display a rigid response under service conditions (braking and start forces), but it is necessary to allow for a ±10 mm displacement capability (with respect to the initial position) under the seismic action, without exceeding the maximum seismic longitudinal force shown in the plans. The bearing’s displacement limitation under maximum load is necessary to dissipate/release the effects caused by an offset in the pier’s transverse displacement when subjected to transverse seism. The vertical axis torsion thus induced on the deck, added to the torsional stiffness of the frames that constitute the piers, would introduce great longitudinal forces in the oriented bearings if they were rigid, as it is required under serviceability conditions. This is why it is necessary to relax this rigid response under service loads when it comes to evaluating the seismic response, letting the bearings absorb the rotation around the vertical axis by developing admissible stresses. A longitudinal displacement of ±10 mm without exceeding the maximum longitudinal seismic action (as specified in the project) is enough to eradicate the incidence of transverse piers offset under transverse seismic action. This displacement capability under seismic action leads to a very favorable consequence in the design stage: the distribution of the longitudinal seismic forces can be assumed as uniform for the 4 bearings, since the system has the displacement capability to regularise the forces acting on each of them. The aforesaid concern about the distribution of the horizontal longitudinal forces, and the incidence of the bearing’s execution tolerances and gaps on that share, is therefore limited to service loads and not to seismic actions (with slightly higher values). 4.2
Different proposals of the bearing system made by the contractor
Focusing on those proposals which try to meet the project’s specifications, there were 4 different alternatives that could resolve the bearing system.
© 2009 Taylor & Francis Group, London, UK
52 Track-Bridge Interaction on High-Speed Railways
Figure 24.
View of one bearing device with pre-compressed neoprene.
– Those based on sacrifice shear locks which materialise the transition from a rigid response (service) to the allowance for a ±10 mm displacement under seismic action. – Those which attempt to materialise a hinge allowing the relative rotation of the deck with respect to the piers, essential in order to release the effects caused by the pier’s transverse offset when subjected to transverse seism. – Those using dissipative antiseismic devices also implementing impact transmission functions, which provide with the necessary stiffness under service conditions. – Those implementing pre-compressed neoprene in order to achieve the described type of response. (Fig. 24) The latter alternative with pre-compressed rubber consists of a transversely-guided Pot-type bearing sandwiched between two laminated bar-precompressed rubber bearings, and the whole set placed in a metal seat anchored to the substructure. The bars used to pre-compress the rubber are left in the bearing working as a guide for the system, as well as preventing decompression. However, these bars fail to prevent the rubber from overloading in case of seism. With the aforesaid disposition, and until the longitudinal load exceeds the rubber’s precompression force, the bearing shows basically a rigid response, since the applied load is diverted by decompressing the bars, barely causing any appreciable deformations. Once the load is greater than the precompressing force, it is directly transmitted to the laminated rubber, whose compressibility has been previously gauged to admit a maximum relative displacement of ±10 mm when the applied load attains the peak value of the seismic action. To that end the POT rests on a sliding surface in the lower seat.
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Track-structure interaction and seismic design of the bearings system
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This bearings system ensures an adequate response to the two basic, previously described, requirements. On the one hand, the suggested system adequately displays a rigid response under service conditions and, on the other, allows for a relative displacement of ±10 mm when subjected to the maximum design seism. This kind of bearings can be assumed to distribute longitudinal loads uniformly among the four central members. Even assuming the existence of a non-favourable execution tolerances distribution in those elements responsible for transmitting longitudinal actions – which would lead to an overload in one bearing with respect to the others–, the system’s response is guaranteed. The load would be directed towards the most penalised bearing –that with the worst tolerance condition – until it reached the rubber precompression force. But after that, it would suffice with a displacement within the POT guaranteed tolerance range – millimetric – to cause the adjacent bearings to be put in load until balancing the applied load, without further complication in the system’s behaviour. That is to say, the system’s flexibility necessary to regularise the incidence of execution tolerances on the POT guides is guaranteed, neither hampering the system’s global response, nor causing a risk of overloading which might lead to premature failure under service loads. Overall, this bearings alternative provides an adequate response to the design requirements. It also presents some additional convenient features, such as: – Absence of fusible elements on which the stiffness change between the service and seism situations depends. – It does not contain fatigue-sensitive elements, which might either unleash premature failure or require over dimensioning to avoid it. – Easily inspection, maintenance, and replacement, even periodic or after an important seism. The precompressed rubber solution allows us to satisfactorily meet the design requirements, therefore being the eventually chosen alternative.
REFERENCES [1] [2] [3] [4]
Eurocode 1, part 2: Traffic loads on Bridges. (EN-1991-2:2003) American Association of State Highway and Transportation officials. (AASHTO-1994) Eurocode 8: Design provisions for earthquake resistance – Part 2: Bridges. (ENV 1998-2:1994) Eurocode 8: Design of structures for earthquake resistance – Part 2: Bridges. (prEN 1998-2. April 2004 (Draft)) [5] Structural Bearings. Part 1: General design rules. (EN 1337-1:2000)
© 2009 Taylor & Francis Group, London, UK
CHAPTER 6 Track structure interactions for the Taiwan High Speed Rail project D. Fitzwilliam TY Lin International, San Diego, CA, USA
ABSTRACT: This paper describes the analyses for one of the special structures of the Taiwan High Speed Rail project. These analyses addressed the rail-structure interaction of a ballast track portion of the alignment subjected to a train braking load and a seismic load. Results for a response spectrum analysis and a nonlinear time history analysis are presented. Advantages and limitations of the two methods are discussed.
1
INTRODUCTION
The Taiwan High Speed Rail project consists of over 26 kilometres of viaducts and bridges, crossing Taiwan from north to south for nearly the entire length of the island. Structures on the project were analyzed for two types of earthquakes: Type 1 (severe) and Type II (moderate). Repairable damage in the inelastic range is expected for a Type I earthquake based on capacity design. The structure is designed for safe operation at maximum speed with no yielding under a Type II earthquake. Relative gap displacements across structure movement joints and rail stresses were computed for the rail-structure interaction analysis, using dynamic time history analysis and response spectrum analysis. For Type II earthquakes, the dynamic analysis included dead weight of the structure and train, braking, temperature expansion or contraction, and ground motion applied at the foundations.
2
BACKGROUND
With the faster speeds and higher capabilities of computers, many civil engineering methods need to be updated. Many methods that are being used today to analyze and design structures were developed years ago. Computer technology has progressed much faster than the fields in which they are used. It is the engineers’ responsibility to look at older methods of analysis and determine if they were limited by the computer technology of the time. One method that may be improved by using advanced computers is designing bridges for dynamic loadings. Two methods are used for modelling structures subject to seismic excitations. The most common method is response spectrum analysis. A more demanding analysis method is time history analysis. Response spectrum analysis is useful for models with a linear elastic response to the loads being applied to it. When nonlinear behaviour is expected, response spectrum analysis cannot provide a direct solution without a careful iteration procedure. Time history analysis is able to directly solve in either linear or nonlinear domains as long as the nonlinear properties of the structure can be adequately modelled. Currently, many engineers separate structures into two main parts: the superstructure, which is often linear; and a foundation, which is likely to be nonlinear. In the case of this project, the nonlinear portion is the ballast connecting the rails to the deck. These two portions are then modelled separately. The first step is to apply the earthquake or other dynamic loading to the superstructure, which is modelled on fixed supports at the base of each pier. The maximum reactions at the base are then found and applied to a separate model of the substructure. 55 © 2009 Taylor & Francis Group, London, UK
56 Track-Bridge Interaction on High-Speed Railways
The displacements of the nonlinear substructures are found and used to compute an equivalent stiffness to replace the fixed supports of the superstructure model. The dynamic loading is again applied to the modified structure and new base reactions are found. The process is repeated until it converges on reactions and displacements that correspond in both the superstructure model and the substructure model. The internal forces in the two models are then used in the design and further analysis of the bridge. The procedure is similar whether the nonlinear portion is in the foundation or in any other element of the structure. In the design of rail bridges, consideration must be given to the method in which the rails are connected to the deck. If connection of the rails is done either with direct fixation or by using ballast, forces applied by trains to the rails are transmitted to the structure through elements, which are likely to respond with nonlinear behaviour. Similarly, forces transmitted from the structure to the rails, such as those resulting from seismic movements, must also pass through these nonlinear elements.
3
PROJECT DESCRIPTION
The Taiwan High Speed Rail project consists of over 26 kilometres of viaducts and bridges, crossing the Erh Jen and Ah Kung Tien rivers and includes a 420-meter long, four-track structure through Taiwan Station. Spans vary in length from 15 to 45 meters. The typical superstructure width for the twin track box girder is 13 meters. Tracks centers are typically placed at a distance of 2.25 meters to either side of the center line of the box girder (Fig. 1). The box girder superstructure is constructed on twin wall columns (Photo 1) in a high seismic region using a travelling false work system that enables casting one complete span at a time. This paper describes the structure on Contract 295, which consists of 27 km of elevated viaduct and one station. The typical structure is composed of cast-in-place concrete post-tensioned box girders built monolithic with twin pier walls. Continuous structures are used in track switching
Photo 1.
View of the box girder superstructure.
Figure 1.
Box girder and track configuration.
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Track structure interactions for the Taiwan High Speed Rail project
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areas, the longest being 150 m. On these structures, the end piers provide support through bearings to reduce the demands on the pier walls and foundation. These piers also have shear keys and lock up devices, which engage under earthquake and braking loads.
3.1 Analysis scope The analyses discussed in this paper were done for design unit DU-33-4 (Fig. 2). This design unit is located at the point where the Chiaotou workshop connection lines merges with the HSR mainline. It consists of side-by-side twin connected boxes joined with a CIP slab. The rail structure interaction model is a combination of two design units. The first is a 201-meter long structure consisting of three single span units and a three span continuous structure. Split wall piers support the single span units. The split wall piers have monolithic connections to the box girder. An expansion joint is provided at the centerline of each pier. Bearings with a transverse and longitudinal shear key (allowing plan rotation) are included at the top of these piers. The interior supports to the continuous structure are provided by portals frames, which are spanning the Tien Pao stream, at a skew to the deck. The round columns at piers four and seven are tied together transversely with a portal beam similar to the beam between five and six. The second design unit is a 300-meter long structure, which consists of two single unit spans followed by a seven span continuous structure and then two more single unit spans. The interior supports to the continuous structure are provided by a combination of bearing and rigid piers. The spans for these two design units vary from 17 to 37 meters. Additional 30-meter boundary spans were added to both ends of the analysis models.
3.2 Analysis considerations and results To accurately model the behavior of the structure, many details have to be carefully considered. Among these details are expansion joints, nonlinear ballast springs, rails and additional mass from trains. The details of these elements and their effects on the results are described in the following sections.
Figure 2.
Analysis model showing three single span units and one three span continuous unit.
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58 Track-Bridge Interaction on High-Speed Railways
Normal Ballast SpringCurves
Earthquake Ballast Spring Curves 80 Loaded curve
60 40
Unloaded curve
20 0 0
1
2
3
disp (mm)
4 5 Load state after train + braking
Figure 3. Nonlinear ballast spring behavior including temperature loads.
Stiffness (kN/m/track)
Stiffness (kN/m/track)
70 Loaded curve
60 50 40
Expected curve
Load state after temperature
30
Unloaded curve
20 10 0 0
1
2 3 disp (mm)
4
5
Figure 4. Nonlinear ballast spring behavior including braking loads.
3.2.1 Expansion Joints The single span units are supported by sliding bearings at one end and fixed at the opposite end by a shear key. A lock-up device accompanies the moving type connection. This indicates a bearing connection, where transverse and vertical displacements are constrained, but longitudinal displacements are free for “normal” operational loads, excluding braking and traction. At each bearing, all box girder rotations are free. The pair of bearings transmits the torsional moment in the box girder into the column in the form of a force couple. The bearings include lock-up devices, which are active during high frequency loading such as earthquake, and braking loads. These devices restrain the longitudinal displacements during high relative velocity motions. Two lock-up devices are used at each pier. They are 1.9 meters left and right of the centerline of the box, and aligned beneath the box girder bottom soffit at the height of the top of tabletop piers. A fixed connection is located at the opposite end of each span with a moving connection. This indicates a pair of bearings (which transmit vertical loads only) along with a combination longitudinal/transverse shear key. This connection allows plan rotation of the box girders relative to the pier, and transverse rotation of the box girder (i.e. simple span). All other degrees of freedom are constrained to the pier. To simulate the rail continuity at the ends of the structure, rail springs were modeled with an equivalent stiffness to the structural stiffness of the rail in adjacent spans. 3.2.2 Ballast modeling The longitudinal connection between the ballast and the box girder is through a nonlinear ballast truss element. The longitudinal force-displacement relation of the ballast is shown in Figures 3 and 4. It is considered to be an elastic-perfectly plastic material. The nonlinear truss provides this behavior by connecting the mass of the ballast to the box girder. For the Type II earthquake time history analysis the nonlinear representation of the ballast is explicitly modeled. The appropriate rail tracks were loaded by the design train. The ballast associated with the loaded tracks was assigned the higher yield force of 60 kN/m, the unloaded tracks were assigned the lower 20 kN/m yield force. Figure 5 shows the force in the rails with the weight of one train placed aver the expansion joint. Figure 6 shows the resulting rail force with the additional braking force applied for the train in the same location. 3.2.3 Rails Rails were included explicitly in the Type II time history analyses. A rail similar to the standard AREA 57 kg/m was assumed with a cross-sectional area of 72.65 mm2 . 3.2.4 Train masses Train nodes were defined at the center of gravity of the train, and attached to the ballast node using a stiff element. For the time history analysis, including breaking, one uniform train mass of 60 kN/m
© 2009 Taylor & Francis Group, London, UK
Track structure interactions for the Taiwan High Speed Rail project
Figure 5.
Forces in rails loaded with the weight of one train.
Figure 6.
Forces in rails loaded with the weight of one train and braking force.
59
was included on the entire length of the structure, at one set of train nodes. Also, one train engine was assumed centered. 3.2.5 Piers The Piers are modeled with cracked section properties for the track-structure interaction analysis, with section inertias reduced to 70 % of their gross values. 3.3
Dynamic loading
For the dynamic, a displacement time history was applied at the base of each of the pier foundations. The displacement time histories were computed via double integration of the acceleration time histories. Acceleration time histories were provided, which included coherence effects due to wave scattering and wave passage effects. However, the motions were only defined for 15 stations, evenly spaced at 30-meter intervals. This was not adequate for models with up to 37 stations and variable foundation spacing. Previous experience indicated that the coherence effects are negligible compared to the wave passage effects. Therefore, a single time history was used for input to the model, with wave passage effects accounted for by applying a delay for each pier. The wave speed was 2.5 km/s. The wave started at the one end of the model and proceeded directly along the
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60 Track-Bridge Interaction on High-Speed Railways
Figure 7.
Maximum tension forces in rails during time history analysis of Earthquake Type II.
length of the model (i.e., the angle between the direction of travel of the wave front and the structure centerline is zero.)For the Type II rail-structure time history analysis, the ballast truss elements must be preloaded by the dead load of the train and structure with the unloaded properties. After that, the loaded properties must be used for those tracks that carry trains during the analysis. The following procedure was used to accomplish this. First, the dead load and the train load was applied using the unloaded ballast properties. The loading condition of each ballast truss element was examined, and a new nonlinear material was generated with the same condition at the end of the dead load as at the first point (assuming the response was elastic). The loaded properties started from that point (Fig. 5). The new nonlinear properties were input into the model, and the dynamic time history analysis was started again with the dead load and train load included. Spring properties used in the nonlinear models can be seen in Figures 3 and 4. A time step of 0.01 seconds is used, so frequencies up to about 50 Hertz are accounted for in the analysis. Damping as specified to be 5 percent for all vibration modes. In a time history analysis, Rayleigh damping is used whereby two particular frequencies can be damped 5 percent exactly, but other modes will have different damping. An assumption of 5 percent damping for periods of 0.1 and 2.0 seconds was deemed appropriate to approximate a result of overall 5 percent damping. For horizontal modes, no significant modes are below 0.1 seconds, so the model is slightly under damped. For the vertical direction, most of the modes occur in the region of 0.1 and 0.2 seconds, so the damping is approximately 5 percent.
4
CONCLUSIONS
The forces shown in Figure 7 indicate that the rail reaches a maximum tension at an expansion joint. The shape of the tension diagram indicates the spans on opposite sides of the expansion joint are moving toward one another and the ballast spring elements are in the nonlinear range. The diagram shows the force increasing from zero to 2200 kN, a length of just over 35 m. This corresponds to the 60 kN/m limit for the ballast springs shown in Figures 3 and 4. The diagrams also indicate that the extent of the slipping in the ballast is confined to approximately one span at the time step shown. In order to model the same behavior in a response spectrum analysis, the location, state and number of nonlinear elements must be known before running the analysis. This is at best a guess in the initial system and the effects of the initial guess can greatly influence the results of the analysis. In
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this way, the designer is tempted to produce the desired results by adjusting the initial guess until the result is within the allowable range. The response spectrum analysis was performed as a preliminary indicator to evaluate if the more sophisticated and complex time history analyses were required. The results of the comparison show that the nonlinear nature of the rail-structure interaction makes time history analysis indispensable whenever there is concern about the stresses and deformations of the rail or box girders. With computer capacities and processor speeds of today, the time history analysis should no longer be avoided in favor of a response spectrum analysis when nonlinear behavior is expected. REFERENCE Taiwan High Speed Rail Corporation, Taiwan High Speed Rail Design Specifications, Vol. 9. 21 January, 1999.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 7 Track–bridge interaction – the SNCF experience P. Ramondenc, D. Martin & P. Schmitt SNCF Engineering Direction, Bridge Department, Paris, France
ABSTRACT: The communication deals with the SNCF experience of track-bridge interaction. Following items will be successively treated: – long welded rails: conception, design and current effects on platform – bridges as singular points in the current platform – behaviour of the attach from the track on the bridge deck – dimensioning of structures including track-bridge interaction, preliminary approach and f.e.m. calculation – associated equipment (joints and dilatation devices) – conception of structure in order to manage the phenomenon – examples of good practices in terms of general design for structures submitted to track-bridge interaction. 1
LONG WELDED RAILS
1.1
Generalities
In France, since 30 or 40 years old, all new tracks have been designed with long welded rails. This evolution was necessary to: – improve the comfort of the passengers – increase the speed of the trains (>120 km/h). Nevertheless, long welded rails have only been enabled by recent technical progresses, such as: – the availability of very high strength steels (fe > 900 MPa) and the progress in welding techniques – the development of modern and reliable track equipments: concrete sleepers, modern connection between rails and sleepers, ballast, . . .
Figure 1.
Connection between the rail and sleeper.
It has also been necessary to better understand the behaviour of the track, especially in terms of lateral resistance of the ballast, track stability and influence of temperature. 63 © 2009 Taylor & Francis Group, London, UK
64 Track-Bridge Interaction on High-Speed Railways
1.2
Main characteristics
First, it appears that when a track is equipped with long welded rails, all sections do not behave on the same way. In the neighbourhood of sections where the rails are cut, one can observe a relative slipping between the track and the platform when conditions (temperature, load) change. This zone is called “end breathing zone”. It works as a transition zone for the compression state in the rails, as detailed in the figure below: Track compression (kN)
End breathing track lengths < 150 m
< 150 m
E.A.␣.⌬Tr x
Total length of the LWR
Figure 2.
End breathing lengths.
In this figure following definitions apply: E = Elasticity modulus for rails (210 000 MPa) A = Section area of track α = Thermal dilatation coefficient for steel (1 × 10−5 ◦ K−1 to 1.2 × 10−5 ◦ K−1 ) Tr = Temperature variation in rails
Figure 3.
Special section at the end of the tunnel.
Along the tracks, other special sections may appear where additional effects may have influence on the track: – Each interruption of the LWR (switches, . . .) – bridges (= deformable platform . . .), – end of tunnels (discontinuity of exposure . . .) In some cases, where stresses may be too high, rails have to be cut and special Track Dilatation Devices have to be implemented.
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65
LONG WELDED RAILS AND BRIDGES
2.1
Generalities
When LWR cross bridges: – loads that are applied to the track (train. . .) are (partially) transmitted to the deck. – imposed deformations of the bridge induce additional stresses in rails. That is the reason why engineers speak about interaction. F
αF
Figure 4.
Transmission of loads from track to the bridge.
To guaranty security, it must be controlled that the total longitudinal force in the track (self behaviour + influence of the bridges), including second order effects will not lead to lateral buckling of the rails, or excessive slipping in the ballast. As a simplification, the global longitudinal resistance is assumed to be divided in two parts: – one part is reserved to current track behaviour – the other part is available for additional effects due to the bridge
Figure 5.
Division of the track resistance in 2 parts.
This approach represents a simplification of the reality, in order to be able to study the bridge effects apart from the current behaviour. It is a conservative approach.
2.2
Bridge additional effects
Three main mechanical phenomena have to be considered: (1) Thermal actions on bridges (uniform temperature variations, gradient); (2) Braking and acceleration forces (horizontal traffic actions); (3) Vertical loads effects (vertical traffic actions) Long term actions may usually be neglected (especially effects of creep and shrinkage, because of track maintenance and long term rearrangement of ballast). It is also assumed that only longitudinal forces and displacements in rails are taken into account.
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66 Track-Bridge Interaction on High-Speed Railways
2.2.1 Temperature variations Assuming a variation of the temperature in the bridge T, it leads to a variation of normal force in the track, according to following equations: F = Ku1 =
f (x)dx = −F2 − F1 ;
u2 − u1 = α LT
(1)
As instance, for a simply supported span, the general shape of the longitudinal force in the rails is:
Frail
F1
F2
f(x)
⌬T F K
u2
u1
Figure 6.
Temperature effects on a simply supported span.
2.2.2 Braking or acceleration Assuming a longitudinal effort Fb transmitted by the train to the track, one part is transmitted to the bridge, so that the variation of the normal forces in the rail verify following equations: F = Ku =
f (x)dx = −Fb − F2 − F1
(2)
Frail F1
f(x)
Fb
F2
F K u
Figure 7.
Braking/acceleration effects on a simply supported span.
© 2009 Taylor & Francis Group, London, UK
u
Track-bridge interaction – the SNCF experience
67
As instance, for a simply supported span, the general shape of the the longitudinal force in the rails is:
Frail
Fv F1
F2
f(x)
θ
h
K
Figure 8.
Vertical load effects on a simply supported span.
2.2.3 Vertical loads Assuming a vertical effort Fv transmitted by the train to the track, the normal forces in the rail vary, due to the rotation of the section at the abutments. As instance, for a simply supported span, the general shape of the longitudinal force in the rails is: Usually, due to deck translation, F2 < F1 .
3
TRACK/BRIDGE INTERACTION – F.E.M. ANALYSIS
To approach the track/bridge interaction a f.e.m. calculation may in many cases be performed. Main characteristics of the model are following: – bridge + pears + foundations are represented by beam/plates elements – rails are introduced in the model by additional beams (with at least 100 m at each abutment) – the connection between rail and deck is described by non linear elements (representing the sleepers and the ballast) Rail
platform
Figure 9.
3.1
bridge
Modelling of the connection between track and deck.
Behaviour low for the track – deck connexion
The behaviour for the connexion between the track and the deck is described by a force density f(x) that is a non-linear function of the relative displacement. u(x) = displacement of rail v(x) = displacement of support f (x) = density of the connexion effort between track and support
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68 Track-Bridge Interaction on High-Speed Railways
u(x)
Rail
f(x) f(x)
f
v(x) Support (bridge/platform)
u(x)-v(x)
0
Figure 10.
Behaviour for the connexion between the track and the deck.
The resulting force in the track becomes: Nt =
f (u(x) − v(x))dx
(3)
Nevertheless, to simplify the analysis, the function f (x) is usually assumed to be of bilinear type. Two different domains can be identified: linear elastic domain and pure friction domain. f
f loaded track
loaded track
60 kN/m
60 kN/m unloaded track
40 kN/m unloaded track
20 kN/m
2 mm
0
u(x)-v(x)
0
ballasted track
Figure 11.
0,5 mm
u(x)-v(x)
unballasted track
Bilinear functions for ballasted and unballasted tracks.
3.2 Analytical calculation model To perform the calculation, the program must enable the implementation of the bilinear behaviour function f (u(x) − v(x)). Many programs may be adequate nowadays. It remains possible in most cases to perform the computation with a simple approach, requiring only successive elastic linear analysis, as the non-linear effects remain simple. The principle of a simple approach (iterative elastic calculation) may be the following, in instance for the application of the temperature effects. Rail
platformt
Figure 12.
K
bridge
Elastic computation model for a simplified approach.
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Step 1: application of a fraction α1 T on the elastic model, until “plastification” of the first connexion element i1 . The contact force in each (elastic) contact element is fi,1 : fi,1 = 20 kN/m (unloaded ballasted track) Boarder of the elastic domain
elastic domain
Element i1
Figure 13.
Computation model for the 1st step.
Step 2: suppression of the connexion on the element i1 , and application of a fraction α2 T on the modified elastic model, until “plastification” of the second connexion element i2 . The contact force in each (elastic) contact element is fi,2 : fi1,2 = 0 kN/m fi2,1 + fi2,2 = 20 kN/m (unloaded ballasted track) Boarder of the elastic domain
Element i1
Element i2
Figure 14.
Computation model for the 2nd step.
Next steps: the same processus is n-time repeated until: nj=1 αj = 1.0 n The final force in each contact element is: fi = j=1 fi,j The distribution of effort Nt in the rails can than be established, as well as: – The relative displacement u(x) − v(x) between track and support, – The efforts in the structure . . . – Braking and acceleration loads have to be taken from EN 1991–2 (UIC load model and if relevant, SW0/SW2). They are directly applied to the track beam. – Temperature variation of the deck must be conform to EN 1991–5 and is applied to the bridge deck. It supposes that at medium temperature, stresses in the track disappear (That requires specific attention about welding conditions). This method leads to relevant results as soon as the discretisation of the structure remains precise enough. As a consequence a great number of steps is usually necessary. A few precisions can be given for the application of the loads: The structural effects of the 3 loads are finally combined according to the following rules: – For structural effects in the bridge: application of factors (EN 1990-A2) – For stresses in rails: rough superposition (i = 1) (conservative)
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70 Track-Bridge Interaction on High-Speed Railways
3.3
Standards
Following standards are applicable: – UIC 774-3 leaflet “Track/bridge interaction – recommendations for calculations” – EN 1991-2 (section 6, §6.5.4.5), inspired from the UIC leaflet Acceptability criteria are the following: – Under braking and acceleration, total bridge displacement δB < 5 mm – Under vertical loads, total bridge displacement at joint δH < 8 mm, when no considering the track/bridge interaction (10 mm either) (In case of succession of independent decks, δB and δH are the relative displacements between two consecutive decks) Additional stresses in rails σ < 72 MPa in compression; σ < 92 MPa in traction (for ballasted track equipped with rails type UIC 60 . . .)
3.4
Example of application
The method as been applied to the “viaduc de la Savoureuse”, which is under construction on the new high speed line “TGV Rhin-Rhône”.
66 m
45 m
Figure 15.
Viaduc de la Savoureuse (under construction).
A f.e.m. calculation has been performed with the software Ansys. The different load cases are applied and the results are given in term of additional stresses in the rails. For instance, with the temperature load, the following distribution of stresses appears: An alternative computation has also been performed, with the use of an Excel sheet. A virtual model of the bridge, including 5 spans has been considered (it represents a total length of 375 m). The deformation (traction/compression and flexion) of each deck is also neglected, but horizontal stiffness of each pear is properly evaluated. The whole bridge is divided in 20 cm long elements, for the evaluation of track/bridge interaction The stresses in the rails can on this model also be evaluated, for all load cases. Finally, the difference between Excel computation and Ansys analysis is less than 5%.
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Track-bridge interaction – the SNCF experience
Figure 16.
F.e.m. calculation model.
Figure 17.
Stresses in the rails due to temperaure effects.
abscisses des joints
77
abscisses des appuis
77
Figure 18.
143
132
209
198
275
264
Virtual model of the bridge for simplified analysis.
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341
330
407
396
451
71
72 Track-Bridge Interaction on High-Speed Railways
Temperature +35˚
40
Charges verticales
35 30
30
25
10 0
0
66
132
198
264
330
396
462
-10 -20
528
contrainte rails (MPa)
stresses rails (MPa)
20 20 15 10 5 0 0
66
132
198
264
330
396
462
528
-5 -30
-10 -15
-40
abscisse (m)
Temperature ∆T = 35ºK
Figure 19.
3.5
abscisse (m)
Braking/acceleration on central spam
Stress distribution under several load cases.
Simplified method for small bridges
The track/bridge interaction leads to additional horizontal effects that are assumed to be proportional to the dilatation length: F = k · L L
Figure 20.
L1
L2
Definition of dilatation length for several cases.
EN 1991-2 allows this method while L < 40 m. The French National Annex allows extending the domain of validity of this method to dilatable length L < 100 m, but with additional requirements about track dilatation devices (TDD) (in conformity with the former SNCF code). Other simplified methods: – EN 1991-2 proposes an alternative method in informative appendix G. This method has been invalidated by the French National Annex. – UIC 774-3 leaflet proposes rules, formulae and graphs for monolithic bridge decks and succession of monolithic bridge decks. They can be used for pre-calculation purposes. 4
GENERAL CONCEPTION – ADDITIONAL REMARKS
Standards give all necessary indications to calculate the effect of track/bridge interaction, but it is not enough to perform a valuable design of the bridges. If the additional stresses in rails exceed codes limits, or if the total dilatable length is greater than 90 m (concrete and composite bridges) or 60 m (steel bridges), LWR must be cut and Track Dilatation Devices (TTD) must be installed. Nevertheless, LWR must be as long as possible (for maintenance), and bridge engineers have to take this recommendation into account. 4.1 Track dilatation devices Practical considerations apply for the use of Track Dilatation devices. – TDD are long (about 30 m) and need to be installed on a stable platform. – In plan, the track must be straight or with a constant curvature (circle) to install TDD.
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– For TDD, the track can not be simultaneously curved in plan and in elevation. – The maximum dilatable lengths consistent with the greatest track dilatation devices in France are about 450 m.
Figure 21.
Track dilatation device.
In addition to Track Dilatation Devices it is necessary to put special expansion joints, that are able to retain ballast.
Figure 22.
4.2
Special expansion joints to retain ballast.
Solution for very long continuous bridges
For very long continuous bridges, the following conception is used: the Track Dilatation Devices are installed of “inert spans” (= single span with fixed bearings). Each inert span is equipped with a double Track Dilatation Device. TDD
TDD
L < 900 m
~40 m
fixed bearing mobile bearing retaining ballast expansion joint
Figure 23.
Solution for very long continuous bridges.
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TDD
L < 900 m
74 Track-Bridge Interaction on High-Speed Railways
This solution has been applied on several viaducts like “Haute Colme” and “Avignon”.
Figure 24.
“Viaduc de Haute Colme” – 1829 m (composite structure).
Figure 25.
“Viaducs d’Avignon” – 2 × 1500 m (concrete structures).
4.3
Solution for very long bridges without TDD
4.3.1 Simply supported decks It remains possible to design the bridge with only simply supported spans, with an individual length smaller than 90 m (for concrete or composite structures). Usually the economic span length is 30 to 40 m. In this case, no Track Dilatation Devices is needed.
L < 90 m
Figure 26.
Solution with simply supported spans.
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Track-bridge interaction – the SNCF experience
75
4.3.2 Repeated continuous decks An other solution consists in building succession of short continuous decks (2 or 3 spans – maximal length 90 m). This solution is economically interesting.
L < 90 m
Figure 27.
Solution with repeated continuous spans.
4.4 Additional remarks In terms of conception and construction of high-speed railways, layout, geometry and maintainability of track and security equipments are of great importance. They induce particular requirements for bridge design. Usually, bridges must be adapted to the track and not the opposite. Only track experts are competent to validate global bridge/track design, according to the local conditions (curvature, switches, maintenance . . .) Standards (EN 1991-2) give general specifications for the evaluation of track/bridge interaction, especially to perform non-linear calculations. Extensive computation models provide results that must be carefully checked. In most cases simple analytical calculation are relevant to provide good estimations of the effects.
5
CONCLUSIONS
The experience of the SNCF in track-bridge interaction has begun with the construction of the new high-speed lines (1975–1980). Nowadays this part of the design of bridges is well known. It remains a field where solutions must be found in relation with track experts and people in charge with the maintenance. The temptation is high to design systems that are complex and expensive in terms of maintenance. REFERENCES ERRI D 213, European Rail Research Institute, 1999. Etude générale de la répartition des efforts longitudinaux sur les ponts-rails, SNCF, 1990. Etude de l’interaction voie-ouvrages d’art dans le cas d’ouvrages longs constitués d’ouvrages unitaires, SNCF, 1993. UIC 774-3 leaflet “Track/bridge interaction – recommendations for calculations”. EN1991-2: Eurocode 1 “Actions on structures – Part 2: Traffic loads on bridges”.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 8 Some experiences on track-bridge interaction in Japan N. Matsumoto & K. Asanuma Railway Technical Research Institute, Tokyo, Japan
ABSTRACT: Some experiences and studies on the subject of track-bridge interaction in Japan are presented. As for the problems in ordinary conditions, studies on the influence of the temperature change between CWR and bridge girders, experiences on the vibration isolation using a newly developed track system, and experiences on the rail extension and contraction at edges of a long span bridge are introduced. As for the problems in seismic conditions, studies and experiences on the vehicle’s running safety on structures, and displacement limit criteria of track and structures are introduced with some explanation on a vehicle/track/structure interaction analysis program.
1
INTRODUCTION
Various kinds of problems are related to the track-bridge interaction. In this paper, some experiences related to them in Japan are briefly introduced. In the daily circumstance, one of the most popular problems may be the longitudinal force induced from continuously welded rail (CWR) to a bridge due to the temperature change. A brief history on the theoretical studies conducted in Japan is introduced. Next popular problem may be the rail expansion and contraction at the edges of a long span bridge due to its deflection. From 1970s, long span road-rail bridges had been constructed to connect Honshu (Main Island) with Shikoku Island of Japan. The problem was studied to design adequate expansion joint devices at that time. Outlines of the bridge, the expansion joint device and the running test which was conducted after the completion of the bridge are introduced. Many severe earthquakes attack Japan. Therefore, not only the seismic strength of railway structure but also the running safety of railway vehicle on the structure during an earthquake motion is our great concern. A vehicle/track/structure dynamic interaction analysis program to analyze the running safety of vehicle on structure subjected to earthquake motion is briefly introduced. An idea of structure to increase running safety is also introduced. A design standard, which was published lately, to specify displacement limits of structure, is introduced lastly.
2
CWR ON BRIDGE
2.1 Theoretical studies and design guideline The primary studies on CWR was started in the late 1930s in Japan. However, the first extensive theoretical study on the influence of the temperature change between CWR and bridge girders was conducted by Fukazawa & Onishi (1962) in order to apply CWR to the Tokaido Shinkansen line. They assumeF: (1) the resistance force per unit length in the longitudinal direction to railcreepage is constant; (2) the expansion and contraction of the girder occur freely; (3) the relative displacement between the rail and the girder induces uniform longitudinal forces into rail through the fastening devices on the girder; and (4) the neutral temperature and the temperature change of rail and girder are the same. Then, they studied using several dimensions of single span girders and multiple span girders. 77 © 2009 Taylor & Francis Group, London, UK
78 Track-Bridge Interaction on High-Speed Railways
Figure 1.
A diagram to indicate CWR applicable span length (Fukazawa & Onishi (1962)).
Figure 2.
Action of longitudinal resistance force and axial force distribution (Miyai (1976)).
In the study of a single span simply supported girder which exists in a confined section of CWR, they set the maximum axial force in rail as shown in Equation 1, and some limit values for rail fracture strength, track buckling strength and gap opening at rail fracture. As a result, they derived a design diagram indicating the CWR applicable span length as shown in Figure 1 and concluded that it would be practically possible to place CWR of 50 kg rail on 50 to 75 m span length girder without expansion joints, depending on the magnitude of resistance force adjacent to the girder r0 . PM = Pt + 0.5rl
(1)
where Pt = axial force in a rail due to temperature change (Pt = E · A · β · t); E =Young’s modulus; A = section area of a rail; t = temperature change; l = span length of a girder; r = resistance force per unit length for a rail on girder; β = coefficient of expansion of rail. As for the multiple span girders, Fukazawa & Onishi recommended to use the FF-MM type support arrangement to relax the axial force distribution of rail in each span. Here, the FF-MM means that fixed supports of two adjacent simply supported girders are placed on a same pier and movable supports of next two adjacent girders are also placed on another same pier. After the work of Fukazawa & Onishi, Miyai (1976) studied analytically the behavior of CWR on multiple span girders using a computer program in order to expand the applicability of the FM-FM type support arrangement, which is favorable for the seismic design of bridge piers. He adopted the same assumption of which Fukazawa & Onishi used, and modeled the axial force distribution of rail as shown in Figure 2. In his computation, the position to produce the same rail and girder expansion is sought iteratively first to draw the axial force distribution diagram. Then,
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79
Figure 3.
Maximum axial force of rail when expansion joints are placed at bridge ends (Miyai (1976)).
Figure 4.
Maximum span length of girders when they exist in a confined section of CWR (Miyai (1976)).
the maximum axial force of rail, the displacement of rail at its end and the potential rail fracture position to make the maximum gap opening are computed. In the case of a bridge composed of multiple span simply supported girders, which have a constant span length, the FM–FM type support arrangement and expansion joints at the bridge ends, Miyai obtained some diagrams to show the characteristics of the maximum axial force of rail (see Figure 3), the displacement of rail at its end and the maximum gap opening of rail. As the result, he derived that the maximum axial force Pmax would not excess Pt + r · l/4 and concluded that the practical maximum CWR length would be approximately 100 m on 10 girders when 60 kg rail was used and some values were assumed as follows: r = 5 kN/m, Pmax = 1 MN, the rail displacement limit y0 = 100 mm, and the rail gap opening limit D = 70 mm. On the other hand, in the case that a bridge composed of multiple span girders exists in a confined section of CWR, it had been thought that the axial force of rail was accumulated according to an increase of the number of girders. However, he found that the axial force of rail would not be accumulated in the middle region of the bridge because the position which produces the same rail and girder expansion shifts toward the center of a girder in the region. He concluded that CWR could be used on FM–FM supported multiple span girders as far as each span length is shorter than 50 m under the condition of r = r0 = 5 kN/m and Pmax = 1 MN. 2.2
Design guideline
Terashima & Hiraoka (1977) summarized the result of Miyai and proposed a design guideline for the arrangement of girder supports and the position of expansion joints (EJ) as shown in Table 1
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80 Track-Bridge Interaction on High-Speed Railways
Table 1. Arrangement of support and position of EJ for bridges exist in CWR section. Total length of Bridge L (m)
Maximum span length of girder l (m)
L < 100
– l ≤ 10 10 < l < 50
No restriction No restriction FF-MM FM-FM
EJ is unnecessary EJ is unnecessary EJ is unnecessary Within about 100 m from the end of breathing section of bridge.
L ≥ 100
50 ≤ l < 100
FF-MM FM
EJ is unnecessary Within about 100 m from the end of breathing section of bridge. Gap opening shall be checked.
l ≥ 100
FF-MM & FM-FM
Near the end of breathing section of the max span girder. Axial force and gap opening shall be checked.
Figure 5.
Arrangement of support
Position of EJ
Rail fastening devices for CWR on girder.
assuming the resistance force per unit length for a rail on girder r is less than 5 kN/m. The value has been adopted in the Design Standard for Railway Structures (Steel and Composite Structures) (1992). Then, the Design Standard for Railway Structures (Steel and Composite Structures) specifies the characteristic value of longitudinal force due to CWR, when designing bridges, as 10 kN/m for a track. And, the upper limit value of the axial load for a track is specified as 2 MN. 2.3
Rail fastening devices
The resistance force per unit length for a rail on girder is designed as 5 kN/m, 2.5 kN/m or 0 in practice. In order to control the resistance force, rail fastening devices for non-ballasted track as shown in Figure 5 are used on bridges in Japan. The rail fastening device shown in Figure 5(a) has the resistance force r of 5 kN/m and the one in Figure 5(b) has no resistance force. They are combined to produce the designed resistance force. The rail pad for these rail fastening devices is placed beneath the baseplate not to make excess friction between the rail and the baseplate. 2.4
New non-ballasted track which mitigate axial force of CWR on bridge
The slab track as shown in Figure 6 is popularly used in Japan as a non-ballasted track. This type of track, however, can not be installed striding over the ends of girders because the structure of the slab track is reinforced concrete and the slab plate requires a homogeneous plane support using
© 2009 Taylor & Francis Group, London, UK
Some experiences on track-bridge interaction in Japan
Figure 6.
Slab track.
Figure 7.
Floating ladder track.
Figure 8.
Analysis on behavior of CWR on Floating ladder track.
81
cement asphalt mortar. On the other hand, a new non-ballasted track, named “Floating ladder track,” shown in Figure 7 which was developed by Wakui, Matsumoto & Inoue (1996), can stride over the ends of girders because it is composed of prestressed concrete longitudinal beam and can slide on the concrete bed. Then, it can mitigate the maximum axial force in a rail. Asanuma et al. (2002) analyzed the characteristics of axial force of rail when the floating ladder track was used striding over the ends of girders. Figure 8(a) shows an example of the analytical models with the
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82 Track-Bridge Interaction on High-Speed Railways
FM-FM supported multiple span girders (5 girders of 50 m span length) and Figure 8(b) shows the computed axial forces of rail by the temperature change of 40◦ C. They found that the floating ladder track can reduce the maximum axial force by approximately 5 percents comparing with a direct fastened track in this case. The floating ladder track has already been used on some bridges of conventional revenue lines (1067 mm-gauged lines) in Japan. 3
TRACK STRUCTURE AND STRUCTURE BORNE NOISE
It is a very important item in designing a non-ballasted track structure whether it can reduce noise and vibration of structure or not. In order to reduce the vibration of structure effectively, the following measures can be thought: (1) increase of rigidity of rail, (2) smoothening of top surface of rail, (3) lowering stiffness of rail fastener, (4) decrease of rigidity of intermediate mass of track, (5) increase of intermediate mass of track, (6) improvement of supporting elasticity for intermediate mass of track. Some non-ballasted track structures as shown in Figure 9 have been used in Japan to reduce the vibration of structure underneath. The floating ladder track is also a very effective measure to reduce the vibration of structure and then the structure borne noise. Okuda et al. (2003) measured the vibration acceleration of concrete viaduct. Figure 10(a) shows a comparison of the vertical acceleration vibration on concrete
Figure 9.
Examples of non-ballasted tracks to decrease vibration of structure.
Figure 10.
Vibration reduction effect of floating ladder sleeper for structure.
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track-bed between the floating ladder track and the elastically supported embedded crosstie track, when an identical passenger-car train of 16 ton axle load running at a speed of 80 km/h. According to the frequency analysis of the vibration, as shown in Figure 10(b), the floating ladder track can reduce the vibration by approximately 20 dB over the elastically supported embedded crosstie track. Therefore, the floating ladder track is a solution to the question of how best to support the rail from the viewpoint of reducing structure borne noise. In other words, the floating ladder track reduces the vibration by approximately 20dB in the audible range to virtually eliminate the structure borne noise, which is the source of trouble with viaducts and bridges.
4 4.1
DEFLECTION OF LONG SPAN BRIDGE Honsyu Shikoku connecting bridge
In order to connect Honsyu (Main Island) to Shikoku Island of Japan, the Seto-Ohashi Bridges were constructed and opened in April 1988. The total length of railway linking Honshu and Shikoku is 32.4 km, highway being 37.5 km, of which 13.1 km with the strait part. The six main long span bridges as shown in Table 2 were constructed. Figure 11(a) shows a photograph of South and North Bisan-Seto Bridges. Figure 11(b) shows a typical cross section of the bridges. They are road-rail bridges. They have a four-lane highway provided on the upper deck of the stiffening girder, and a conventional (1067 mm-gauged) line and Shinkansen line (which will be installed in future) on the lower deck.
Table 2. Main Bridges of Seto-Ohashi Bridges.
Figure 11.
Name of bridge
Structural type
Span (m)
Shimotsui-Seto Bridge Hitsuishijima Bridge Iwagurojima Bridge Yoshima Bridge North Bisan-Seto Bridge South Bisan-Seto Bridge
Suspension Cable stayed Cable stayed Truss Suspension Suspension
230 + 940 + 230 185 + 420 + 185 185 + 420 + 185 175 + 245 + 165 274 + 990 + 274 274 + 1100 + 274
South and North Bisan-Seto Bridges.
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84 Track-Bridge Interaction on High-Speed Railways
Figure 12.
Deflection of suspension bridge.
Figure 13.
Type 1500 transition girder system.
4.2 Technical development for train running safety on suspension bridge The suspension bridge is a flexible structure and shows, in general, a larger deflection comparing with the other type of structures. At the stage of a schematic design for South Bisan-Seto Bridge, which has the center span length of 1100 m and a two-hinge stiffening truss girder, it was computed that the bridge would deflect 5 m in the vertical direction, as schematically shown in Figure 12, and 8 m in the horizontal at the span center due to the train and automobile loads, the effect of temperature change and the wind load. It was also computed that the expansion of approximately 1.5 m would occur at the end of the stiffening truss girder with the angular rotation of approximately 30/1000 in both the vertical and horizontal direction. We had no experience in running a high speed train on the structure showing such large deflections and deformations. Therefore, extensive studies were conducted experimentally using an actual test track and theoretically using computer simulation programs. Itoh et al. (1972) tested the behavior of railway vehicle running through an actual test track on 5 span girders which could control their angler rotations at ends of girders. Matsuura & Wakui (1979) studied the allowable angler rotation using a simulation program that could compute a 3D multi-body vehicle model run on locus with a specified angular rotation. As the result of studies, a structural system as shown in Figure 13 to avoid the concentration of angular rotation at the bridge end was developed. The system has a transition girder of 15 m span length to divide the place where the angular rotation occurs into two and can restrict the maximum angular rotation within 7/1000 which is required to maintain the running safety and riding comfort criteria for 160 km/h running (Ishiguro & Matsuura (1988)). The expansion and contraction of rail due to the deflection of suspension bridge was also a problem. New devices were required to be developed to overcome the problem. The expansion and contraction stroke was computed summing up the rail breathing of ±200 mm due to temperature
© 2009 Taylor & Francis Group, London, UK
Some experiences on track-bridge interaction in Japan
Figure 14.
Type 1500 expansion joint device.
Figure 15.
Loading test train on bridge.
85
Figure 16. Measured deflection and breathing displacement.
change, ±200 mm due to train loading on double track and ±100 mm due to automobile loading. Then, the type 1500 expansion joint device was developed through some tests conducted in the Sanyo Shinkansen line (Hashimoto et al. (1975)) and installed on the suspension bridge. After the completion of the construction of South Bisan-Seto Bridge, a running test using a train composed of 10 locomotives (each weight: 100 t) and 3 freight wagons as shown in Figure 15 and observed the performance of the transition girder and the expansion joint device. It was observed that the maximum measured deflection and angular rotation of stiffening girders were almost equal to those of computed values and the maximum breathing stroke of rail at the expansion joint device as shown in Figure 16 was approximately 80 percent of the computed value (Iwata et al (1988)).
5 5.1
VEHICLE, TRACK & BRIDGE INTERACTION AT EARTHQUAKE Background
Earthquakes occur frequently in Japan. Some are severe enough to damage tracks and structures. According to the past experiences, track damages which affect the train operation are classified as follows: (1) track deformation caused by the damage of roadbed, (2) track deformation caused by destruction or large deformation of infrastructures, (3) angular rotation or gap between adjacent structures, (4) settlement of abutment back soil, (5) buckling of track without damages of roadbed or structures. Moreover, without any track damages or structure failures, the railway vehicle may have a chance to derail when it is subjected to a severe transverse oscillation caused by earthquake.
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86 Track-Bridge Interaction on High-Speed Railways
So, the running safety of vehicle during an earthquake is one of the most important problems in design consideration. Studies on the running safety of train on structures subjected to earthquake motions started in the late 1960s in Japan when the practical construction investigation of the Honsyu Shikoku Connecting Bridges was started. Nishioka (1969), Nishioka & Hashimoto (1980) studied the response behavior of railway vehicles subjected to earthquake motions using 3D multi-body vehicle model. After their studies, few studies had done until the Hyogo-ken Nanbu Earthquake (Kobe Earthquake) occurred on January 1995 which caused catastrophic damages not only to building and highway structures but also to railway structures. After the earthquake, extensive studies on the seismic design of structures including strengthening methods were conducted. Consequently, the evaluation process of structural safety and ductility of members subjected to severe earthquakes was reflected in the Design Standard for Railway Structures (Seismic Design) published in 1999. As for the running safety of train at earthquake, Matsuura (1998) studied using 3D multi-body vehicle model and Matsumoto et al. (2003) studied using a vehicle/structure dynamic interaction model with 3D multi-body vehicle model and 3D FEM. The outputs were reflected in the Design Standard for Railway Structures (Deflection Limits) published in 2006.
5.2
Running safety analysis model
Matsumoto et al. (2003 & 2007) studied the running safety of train on structure at earthquake a vehicle/structure dynamic interaction analysis program named DIASTARS (Dynamic Interaction Analysis for Shinkansen Train And Railway Structures). The schematics of mechanical models of DIASTARS are shown in Figure 17(a). The vehicle model consists of a body, two trucks and four wheelsets. These elements are modeled by rigid bodies being connected to each other with non-linear springs and dampers as shown in Figure 17(b). The vehicle model moves at a constant speed. Then, the number of degrees of freedom for one vehicle is 31. The contact forces, such as the creep force and the flange pressure depending on the profiles of rail head and wheel tread are taken into consideration. The structures are modeled by the combination of finite elements. Interaction forces are transferred through running lines defined in advance on structure model. The equations of motions of vehicle and structure are coupled to solve the dynamic interaction. When the running safety of vehicle subjected to earthquake motion is analyzed, it is very important to take into consideration not only the non-linear characteristics of springs and dampers in vehicle but also the jumping and contact behaviors of wheel from rail surface.
Figure 17.
Schematics of DIASTARS.
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Figure 18 shows the schematics of wheel/rail contact model. Although it is ideal to trace the wheel/rail contact point using a 3D contact model, it is time consuming to use the model in every iteration computation. In the DIASTARS, wheel/rail contact point is traced using a 2D contact model assuming the following things: (1) the yaw and roll displacements are taken into consideration as the motion of vehicle, but drift of contact point due to these displacements is ignored; (2) the creep force is produced in the tangential direction at contact point; (3) the contact point and contact angle α are obtained depending on the relative displacement between wheel and rail using some pre-computed contact curves for wheel and rail surface profiles. This interaction analysis model was verified using semi-vehicle model placed on a shaking table as shown in Figure 19 which is consist of an actual track, an actual bogie and a half carbody model (Miyamoto et al. (2004), Matsumoto et al. (2007)). Figure 20 shows an example of experiment and analysis response results when the vehicle model was oscillated transversely by 5 cycles of sinusoidal displacement of 0.5 Hz. Figure 21 shows threshold displacement curves to produce 3 mm uplift of wheel. From these comparisons, it was found that the analysis model could predict the behavior of railway vehicle subjected to earthquake motions. Several analyses have been conducted on the running safety of train on structures subjected to an earthquake motion. Figure 22 shows a schematic model to evaluate the influence of non-linear response of structure. The hysteresis model of pier is a basic tri-linear type model (the Masing model) as shown in Figure 23. Seismic waves are induced to bases of piers in the transverse direction perpendicular to the traveling direction of train. The structure is excited with the seismic motions of the same phase. The damping ratio of structure was set as 0.05. The train speed was set as 300 km/h. Figure 24 shows the influence of the non-linear response of structures that have different seismic horizontal coefficient Kh (=(yield strength)/(self weight of structure)). It was found that the running safety threshold ground acceleration would increase when the structure respond non-linearly.
Figure 18.
Wheel and rail contact model.
Figure 19.
Semi-vehicle model used in shaking table test.
© 2009 Taylor & Francis Group, London, UK
160
6
Load (kN)
Disp. (mm)
88 Track-Bridge Interaction on High-Speed Railways
4 2 0
80 40 0
-2 0
2
4
6
8
10
12
14
16
0
Time (sec) Experiment
6 4 2 0
4
6
8
10
12
14
16
10
12
14
16
Time (sec) Experiment
120 80 40 0
-2 0
2
4
6
8
10
12
Time (sec) Analysis
14
16
0
2
4
Figure 20.
Comparison of wheel responses.
Figure 21.
Comparison of running safety threshold curve.
Figure 22.
Schematic of analysis model.
6
8
Time (sec) Analysis
A
(a) Vertical displacement of wheel
5.3
2
160
Load (kN)
Disp. (mm)
120
(b) Wheel load
Figure 23.
Hysteresis model.
Structure to make higher running safety during earthquake
From above-mentioned vehicle/structure dynamic interaction analyses, it was obtained that the running safety of train on structure subjected to earthquake motions would be improved when the natural period of structure became shorter and it would be recommendable to make it shorter than 0.5 sec. Natural periods of typical railway viaducts in Japan are estimated as 0.6 to 1.1 sec. In order to make it shorter than 0.5 sec, viaduct piers have to be stiffened twice to four times as stiff as conventional ones. It may be possible to adopt the wall-type pier to satisfy the requirement, but it increases the overall weight of structure and can not effectively provide damping. Then, to actualize shorter natural period effectively, Matsumoto et al. (2003) proposed a new viaduct system using lightweight tracks, prefabricated track-supporting girders, which can make the weight of
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Ground acceleration (gal)
800 700
Non-linear structure Kh=0.3 Non-linear structure Kh=0.4
600 500 400 300 200 100 0 0.0
Linear structure Ground Acc. to make Kh=0.3 structure yield 0.5
1.0
1.5
2.0
2.5
Natural period of structure (sec)
Figure 24.
Running safety threshold curves considering non-linear response of structure.
Figure 25.
Overview of viaduct and structure of damper and brace.
superstructure halve compared with conventional viaducts, and rigid-frame piers with braces and damper as shown in Figure 25(a). This system can increase the lateral stiffness quite easily using a small section of steel brace. The structure of damper is very simple. Details of damper and brace are shown in Figure 25(b). The energy assumption caused by plastic shear deformation of web steel plate is used as damping. In order to avoid total buckling and warping, flanges and stiffeners are attached to the web plate. A shaking table test was conducted with a 1/2.5 scaled model of rigid-frame RC pier with steel damper and brace to confirm its seismic performance. The brace is anchored to the footing or mid-columns. Relations between input accelerations and response displacements of specimens are shown in Figure 26. In the figure, the results of RC frame without damper and brace are also shown. It was confirmed that the displacements are well restrained compared with the case of RC frame without damper and brace. On the other hand, it can be suggested from Figure 23 that the seismic isolation structure may increase the running safety during an earthquake because the threshold line of non-linear response increases in the long period region. It is, however, necessary to take into consideration the influence of angular rotation and alignment irregularity because they will have a large differential displacement between adjacent structures that have long natural periods. Moreover, the influence of the existence of track on the response of structures shall be studied. Ikeda et al. (2005) conducted shaking table test as shown in Figure 27 and found that longitudinal response of bridge on seismic isolation support would restricted approximately 20% because of the existence of track.
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90 Track-Bridge Interaction on High-Speed Railways
Figure 26.
Relation between maximum input acceleration and maximum displacement of structures.
Rail
Oscillating direction (longitudinal)
Rail
Rail fastener Seismic isolated bridge girder
Figure 27.
5.4
Abutment
Shaking table test on seismic isolation structure with non-ballast track.
Design standard
“Design Standard for Railway Structures and Commentary (Displacement Limits)” was published in February 2006. The purpose of this design standard is to prescribe appropriate verification methods for displacement limits of structures based on the running safety and riding comfort of a train in ordinary conditions, running safety during an earthquake (in a seismic condition) and damage to tracks. Design provisions associated with the displacement and/or deformation limits of structures determined from the viewpoint of the running safety, riding comfort of train and track damages used to be prescribed as parts of the respective design standards for steel structures, concrete structures and seismic design. The new displacement limit standard, on the other hand, was formulated into an independent and comprehensive design standard. The main characteristics of the displacement limit standard are (1) adoption of the performance-based design method, (2) review of provisions relating to runability in ordinary conditions, and (3) review of provisions relating to running safety in a seismic condition. As for the ordinary conditions, limit values associated with the displacement of structures, deflection of girders and differential displacement on track surfaces which occur between structure ends (angular rotation and alignment irregularity) are prescribed. The applicable maximum train speed is 360 km/h. Limit values that took into consideration the combination of the deflection of girder and the alignment irregularity which occurs due to the deformation of elastic bearing are also prescribed.
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Table 3. Design limit values of girder deflection determined by running safety in ordinary conditions (for Shinkansen). Span length Lb (m) Number of spans
Maximum speed (km/h)
Single span
260 300 360
Multiple span
260 300 360
Lb ≤ 60 m
Lb ≥ 70 m
Lb /700 Lb /900 Lb /1100 Lb /1200 Lb /1500 Lb /1900
Lb /1400 Lb /1700 Lb /2000
Table 4. Design limit values of girder deflection determined by riding comfort (for Shinkansen). Span length Lb (m) Maximum speed (km/h)
Lb ≤ 20 m
30 m
40 m
50 m
Single span
260 300 360
Lb /2200 Lb /2800 Lb /3500
Lb /1700 Lb /2000 Lb /3000
Lb /1200 Lb /1700 Lb /2200
Lb /1000 Lb /1300 Lb /1100 Lb /1800 Lb /1500
Multiple span
260 300 360
Lb /2200 Lb /2800 Lb /3500
Number of spans
Lb /2800
Lb ≥ 60 m
Lb /1700 Lb /2000 Lb /2200
Table 5. Design limit values of angular rotation on track surfaces determined by running safety in ordinary conditions (for Shinkansen). Vertical direction (θ/1000)
Transverse direction (θ/1000)
Maximum speed (km/h)
Parallel shift
Folding
Parallel shift
Folding
210 260 300 360
4.0 3.0 2.5 2.0
4.0 3.0 2.5 2.0
2.0 1.5 1.0 1.0
2.0 2.0 1.0 1.0
Tables 3 and 4 show the deflection limits determined by running safety and riding comfort, respectively. Tables 5 and 6 show the angular rotation limits as well. Table 7 shows the limit values determined by track damage criteria. The fundamental idea of structural design for seismic conditions in the displacement limit standard is to reduce the possibility of derailment during an earthquake by adopting structures that have higher running safety against an earthquake as much as possible. On the other hand, it has become apparent from recent research that it is sometimes difficult to ensure running safety merely by structural measures if earthquake motion exceeds a certain level and induced from the transverse direction. Even in such cases, earthquake motion does not necessarily lead to injury of passengers. However, considering the influence of high-speed running on damage, especially for Shinkansen structures, appropriate and effective risk reducing measures must be devised from the viewpoint of
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92 Track-Bridge Interaction on High-Speed Railways
Table 6. Design limit values of angular rotation on track surfaces determined by riding comfort in ordinary conditions (for Shinkansen). Vertical direction (θ/1000)
Transverse direction (θ/1000)
Maximum speed (km/h)
Parallel shift
Folding
Parallel shift
Folding
210 260 300 360
4.0 3.0 2.5 2.5
4.0 3.5 3.0 2.5
2.5 2.0 1.5 1.0
2.5 2.0 1.5 1.0
Table 7. Design limit values for angular rotation and alignment irregularities determined by track damage criteria in ordinary conditions. Angular rotation (θ/1000) Folding/Parallel shift
Alignment irregularity δ (mm)
Displacement direction
Track type
50 N Rail
60 kg Rail
50 N Rail
60 kg Rail
Vertical direction
Slab track Ballast track
3.5 6.0
3.0 7.0
3.0 3.0
2.0 2.0
Horizontal direction
Slab track Ballast track
4.0 5.5
4.0 5.5
2.0 2.0
2.0 2.0
both the hard and soft aspects of the entire railway system against severe earthquake motion (of an intensity exceeding Level-1 earthquake motion). These include quick slowdown of the train by use of an earthquake early detection system or limiting disasters after derailment by installing devices for preventing large deviation of the train from the track. The practical verification method for the transverse vibration displacement associated with running safety was determined taking spectrum intensity SI as a verification index and Level-1 earthquake motion as a scale gauge for the sake of verification as shown in Figure 28. Limit values were reviewed and determined taking into consideration: (1) the results of running safety analysis using more than ten earthquake motions with different spectrum characteristics and (2) the influence of angular rotation and alignment irregularities between structure ends. It is also recommended in design to minimize the differential displacement of track surfaces in a seismic condition as much as possible. The design limit values for the differential displacement of track surfaces in a seismic condition are prescribed as shown in Tables 8 and 9.
6
CONCLUSIONS
Some experiences related to the track-bridge interaction in Japan were briefly introduced. A brief history on the theoretical studies on CWR placed on bridge conducted in Japan was introduced. A design guideline on the position of EJ depending on the arrangement of fixed and moveable bridge supports and a new non-ballasted track which may mitigate the axial force of rail were introduced. A special expansion joint was used on long span bridges connecting Honshu (Main Island) and Shikoku Island of Japan. Outlines of the bridge, the expansion joint device and the running test which was conducted after the completion of the bridge are introduced.
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Figure 28. Limit value SI L of displacement associated with running safety in a seismic condition and response values SI according to Level-1 earthquake motion. Table 8. Design limit values for angular rotation and alignment irregularities determined by running safety in a seismic condition. Angular rotation (θ/1000)
Direction Lateral
Maximum speed (km/h) 130 160 210 260 300 360
Parallel shift Lb = 10 m
Lb = 30 m
Folding
Alignment irregularity (mm)
3.5 3.0 2.5 2.0
8.0 6.0 4.0 3.5 3.0 2.0
14 12 10 8 7 6
7.0 6.0 5.5 5.0 4.5 4.0
Table 9. Design limit values for angular rotation and alignment irregularities determined by track damage criteria in a seismic condition. Angular rotation (θ/1000) Folding/Parallel shift
Alignment irregularity δ (mm)
Displacement direction
Track type
50 N Rail
60 kg Rail
50 N Rail
60 kg Rail
Vertical direction
Slab track Ballast track
5.0 7.5
3.5 6.5
4.5 3.5
3.5 4.0
Horizontal direction
Slab track Ballast track
6.0 8.0
6.0 8.0
2.0 2.0
2.0 2.0
A vehicle/structure dynamic interaction analysis program to analyze the running safety of vehicle on structure subjected to earthquake motion is briefly introduced. An idea of structure to increase running safety was also introduced. A design standard, which was published lately, to specify displacement limits of structure, was introduced.
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94 Track-Bridge Interaction on High-Speed Railways
REFERENCES Asanuma, K., Matsumoto, N. & Wakui, H. 2002. Behavior of Newly Developed Floating LadderTrack Bridging Structure Boundaries Subjected to Temperature Changes, Proceedings of Railway Mechanics, No.6, Japan Society of Civil Engineers: 7–11 (in Japanese). Fukazawa, Y. & Onishi, A. 1962. A Very Long Rail Laid on Bridge, Quarterly Report of RTRI, Vol.3, No.1: 26–31. Hashimoto, K., Miyoshi, K. & Fujita, R. 1975. Running Vibration Test of Series 961 Shinkansen Test Vehicle, Railway Technical Research Pre-Report, No.75–11, RTRI. (in Japanese). Ikeda, M., Toyooka, A., Murata, K., Yanagawa, H., Kataoka, H. & Iemura, H. 2005. The Effects of the Track Structures on the Seismic Behavior of Isolation System Bridges, RTRI Report, Vol.19, No.3: 23–28. (in Japanese). Ishiguro,Y. & Matsuura, A. 1988. Constructions of Honshu-Shikoku Bridges and Seto-Ohashi Bridge, Japanese Railway Engineering, No.106: 19–24. Itoh, F., Okuda, T., Hirata, G., Arai S. & Fujita R. 1972. Runnning Safety Test of Car against Derailment over an Angular Bend, Quarterly Report of RTRI, Vol.13, No.2: 70–76. Iwata, J., Miura, S. & Ando, K. 1988. Several Track Test Conducted at Opening of Seto Bridge Line, Journal of Japan Railway Engineering Association, Vol.31, No.12: 18–21. (in Japanese). Matsumoto, N., Wakui, H., Sogabe, M., Okano, M., Ohuchi, H. & Tanabe, M. 2003. Design Concept and Practice Ensure Running Safety at Earthquake, IABSE Sympoium Report-Antwerp 2003, Structures for High Speed Railway Transportation, Vol.87, International Association for Bridge and Structural Engineering: 46–47 on book and ANT129 on CD. Matsumoto, N., Tanabe, M. Wakui, H. & Sogabe, M. 2007. A Dynamic Interaction Analysis Model for Railway Vehicles and Structures Which Takes Into Account Non-linear Response, Journal of Structural Mechanics and Earthquake Engineering, Vol.63, No.3, Japan Society of Civil Engineers: 533–551. Matsuura, A. & Wakui, H. 1979. Allowable Bent-Angle of Long-Spanned Suspension Bridges Determined by Running Property of Railway Cars, Quarterly Report of RTRI, Vol.20, No.3: 106–111. Matsuura, A. 1998. Simulation for Analyzing Direct Derailment Limit of Running Vehicle on Oscillating Tracks, Journal of Structural Mechanics and Earthquake Engineering, Vol.15, No.1, JSCE: 63–72. Miyai, T. 1976. Theoretical Study of the Long Welded Rail Continuously Laid on Equally Spanned Bridges, Railway Technical Research Report, No.991. (in Japanese). Miyamoto T., Matsumoto, N., Sogabe, M., Shimomura, T., Nishiyama, Y. & Matsuo, M. 2004. Railway Vehicle Dynamic Behavior against Large-Amplitude Track Vibration, Quarterly Report of RTRI, Vol.45, No.3: 111–115. Nishioka, T. 1969. A Theoretical Study on the Behavior of Railway Vehicles Considering Vibration of Track, Journal of the Japan Society of Civil Engineer, No.172, Japan Society of Civil Engineers: 43–57. (in Japanese). Nishioka, T. & Hashimoto, S. 1980. Running Stability of Two-Axle Freight Car on Bridges at Lateral Earthquake Shock, Journal of the Japan Society of Civil Engineers, No.296, Japan Society of Civil Engineers: 61–72. (in Japanese). Okuda, H., Asanuma, K., Matsumoto, N. & Wakui, H. 2003. Environmental Performance Improvement of Railway Structural System, IABSE Sympoium Report-Antwerp 2003, Structures for High Speed Railway Transportation, Vol.87, International Association for Bridge and Structural Engineering: 242–243. Railway Technical Research Institute (ed.) 1992. Design Standard for Railway Structures and Commentary (Steel and Composite Structures). Tokyo: Maruzen. (in Japanese). Railway Technical Research Institute (ed.) 1999. Design Standard for Railway Structures and Commentary (Seismic Design). Tokyo: Maruzen. (in Japanese). Railway Technical Research Institute (ed.) 2006. Design Standard for Railway Structures and Commentary (Deflection Limits). Tokyo: Maruzen. (in Japanese). Terashima, M. & Hiraoka, S. 1997. Support Arrangement of Bridges to be Installed CWR, Structure Design Documents, No.51, Japan Railway Civil Engineering Association: 2–6. Wakui, H., Matsumoto, N. & Inoue, H. 1996. Ladder Sleeper and NewTrack Structures Development, Quarterly Report of RTRI, Vol.37, No.3: 110–111.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 9 Numerical methods for the analysis of longitudinal interaction between track and structure M. Cuadrado Sanguino & P. González Requejo Fundación Caminos de Hierro, Madrid, Spain
ABSTRACT: This paper analyzes the numerical methods used to assess the phenomenon of longitudinal track-structure interaction in railway bridges, focusing on the key issue of combination of actions. The different computing methods are presented as examples of application for viaduct design.
1
INTRODUCTION
In railway bridges, the variations of temperature, the strains due to deck shrinkage and creep and the action of railway loads can generate additional stress on continuous rails and thus induce relative displacements between track and deck or subgrade. The value of such additional stress, measured as the increment with respect to stress on the rail at a point distant enough from the bridge, may exceed the maximum limit values according to current standards, particularly where large expansion lengths can be expected (continuous decks or long-span isostatic bridges). On the other hand, the relative displacements between track and deck or subgrade can also modify the conditions of track grid stability. Therefore an analysis is needed of the interaction between rail and structure focusing on two main aspects: – Determination of the additional stress on rail and the eventual need of installing one or more expansion devices; – Determination of the absolute displacements of deck and relative displacements between track and deck or subgrade, the latter shall be limited in order to avoid track grid general instability due to ballast layer deconsolidation. 2 2.1
COMPUTING METHOD Preliminary considerations
The UIC Leaflet 774-3 [12] and Eurocode in 1991-2:2003 [4] include the basic methodology for the analysis of track-deck interaction and describe the actions to be considered and the limit values to be complied with as regards both stress on the rail and displacements. These standards are based upon prior research on this phenomenon [11]. Besides the theoretical approach [5][6], the standards describe a method for the analysis of interaction based upon numerical methods that idealize the behavior of all the elements and actions taking part in the phenomenon, computing stresses and displacements. Furthermore, the UIC Leaflet 774-3 [12] states that the models used shall be validated before being actually used through the analysis of test cases. The computing model presented in this paper and used for the case study of chapter 4 has been validated with the relevant test cases [2]. 95 © 2009 Taylor & Francis Group, London, UK
96 Track-Bridge Interaction on High-Speed Railways
Figure 1.
2.2
Schematic of the model.
Computing model
The numerical model (see diagram in Fig. 1) is made up of various types of finite elements: beam element (idealization of rails and bridge deck), non linear spring element (idealization of the interaction between rail and track base, either the subgrade or the deck proper, and propagated through the fastening-sleeper-ballast system,) and linear spring type element (idealization of the behavior of deck bearings, i.e., in each case the bearing-abutment or bearing-pier system). This model will take into account the different actions to be considered for interaction analysis (variations of temperature in different elements, strains on the deck due to shrinkage and creep of concrete, as well as horizontal or vertical overloading), and will allow computing the generated additional stress and displacements. This is a two dimensional model that can be specifically developed, or as it is the case in this paper, based upon an off-the-shelf FEM package. The description of the elements used for the idealization of each one of the components of the infrastructure is detailed below. 2.2.1 Rails The rails are idealized by means of beam-type elements, with two nodes and three degrees of freedom per node (longitudinal and vertical displacements and rotation in the vertical plane). As a simplification, these elements are located in a horizontal line although not at their actual position but at the upper height of the deck, which has no impact on the results (the diagram in Figure 1 shows different heights for the sake of clarity). 2.2.2 Subgrade The subgrade is considered with infinite stiffness and idealized with connecting nodes located at the same height of the rail where any node displacement is prevented. 2.2.3 Deck The deck has been idealized using two types of elements, both beam-type elements with two nodes and three degrees of freedom per node: – horizontal elements, located at the height of the center of gravity of the deck section, having the same parameters of bending and longitudinal stiffness as the said section; – vertical elements, embedded onto horizontal elements, with a very large bending and longitudinal stiffness; these elements connect the intermediate fiber of the deck with the bearings and with nodes on the upper face of the deck, the latter being located at the same height of the nodes of the rail.
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Resistance per track unit length (kN)
70 60 50 Loaded track Unloaded track (good maintenance) Unloaded track (fair maintenance)
40 30 20 10 0 0
2
4
6
8
10
12
Displacement (mm)
Figure 2. Resistance of track to relative displacements with respect to bearing per unit length (UIC 54 or UIC 60 rails).
2.2.4 Interaction between rails and base of track (deck or subgrade) In the vertical direction, and for the purposes of the interaction analysis, a rigid union between rail and deck or subgrade is considered. Therefore, in the model the connecting nodes of the rail and the connecting nodes of the subgrade or upper face of deck cannot have relative vertical displacement. In the longitudinal direction, the connection between the track and its base or supporting structure (either the deck or the subgrade) is characterized by the track resistance to relative displacements with respect to the base. This resistance has two components: the resistance to rail displacement with respect to the sleeper and the resistance of the sleeper to displacement with respect to the ballast. The resistance increases with the displacement until becoming practically constant. Therefore a simplified approach can establish a bilinear relationship between force and overall displacement, taking into account both components, thus characterized by the displacement uo , and the maximum resistance k (see Fig. 2). The following characteristics are adopted for ballasted track [1] [4] [12]: k = 12 kN per unit length of track, for unloaded track with fair maintenance level; k = 20 kN per unit length of track, for unloaded track with good maintenance level; k = 60 kN unit length of track, for loaded track; uo = 2 mm in all cases 2.2.5 Deck bearings Two types of bearings can be considered: – Fixed, with a finite stiffness: they are idealized by restricting the vertical displacement of the corresponding connection point, and the association of a horizontal elastic spring with a constant depending on the characteristics of the bearing-abutment system (K1 in Fig. 1) or bearing-pier system (K2 in Fig. 1). – Sliding: they are idealized by restricting the vertical displacement of the corresponding connection point and allowing the horizontal displacement. The friction is usually disregarded.
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98 Track-Bridge Interaction on High-Speed Railways
2.3 Actions to be considered The actions that can produce interaction effects are all those which lead to the generation of relative displacements between track and deck or subgrade. These actions are described below. 2.3.1 Variations of temperature The standards ([4] [12]) include among the actions to be considered two actions relative to variations of temperature, the uniform variations in rail and deck and the thermal gradient in deck and piers. Generally speaking, the effects of the thermal gradient will be disregarded in the interaction analysis. Without an expansion device, the variation of temperature in the rail (Tc ) does not produce relative displacements between rail and deck or subgrade, thus the only variation of temperature to be considered is the change in temperature of the deck (Tt ). For the interaction analysis, without considering expansion devices, the stresses in the rail due to variation of temperature of the deck are considered as “additional stress”, to be added to the stresses eventually due to the variation of temperature of the confined rail (σc = αc · Tc · Ec ). When an expansion device has been fitted in the area being assessed, the variation of temperature of the deck and the variation of temperature of the rail shall be taken into account. 2.3.2 Horizontal forces The forces of braking and traction defined in the standards [4] shall be considered. In the case of dual track viaducts, the action of braking in one track will be combined with the traction in the other track. Several positions of a train shall be assessed in order to find the worst-case situation for each scenario. 2.3.3 Vertical forces The models of vertical loads defined in the standards [4] shall be considered. As a simplification of the UIC 71 load model a train of loads 300 m long and a uniform load of 80 kN/m can be used (like in the examples of the UIC Leaflet 774-3 [12]). As regards the dynamic factor, the UIC Leaflet 774-3 [12] only states explicitly that the longitudinal displacement of the upper face of the deck due to bending shall be checked. On the other hand the Eurocode 1991-2 [4] states that the dynamic effects can be disregarded in the interaction analysis. 2.3.4 Shrinkage and creep The standards [4] [12] state among other actions to be taken into account the phenomena of shrinkage and creep in the deck, although without an explicit statement on how these have to be considered for the calculation. The overall strains of the deck due to shrinkage and creep can reach significant values (around ε ≈ 10−3 , equivalent to a strain due to a temperature decrease of the deck of 100◦ C). Although rapidly decreasing with time, they should be taken into account in all cases, as they generate stresses in the rail and relative displacements between rail and deck [7] [8] [9]. The strain curve due to the combined effects of shrinkage and creep can be computed according to the procedures stated in standards on concrete, in the Spanish case, EHE [10], and represented as a reduction of equivalent temperature (Fig. 3). Once the time elapsed between deck concreting and rail soldering has been established, the values of this curve, allow finding the maximum variation of temperature equivalent to the strain due to shrinkage and creep at infinite time (Tt,ret,flu ), which shall always be a negative value. 2.4
Combination of actions
The combination of actions to be considered will depend on each of the verifications to be made (on stresses or displacements). However, it shall be pointed out that the linear combination of stresses and strains obtained by separate calculation of each elementary action (temperature,
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T equivalent for Shrinkage & Creep Time (years)
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
-10 -20
T (ºC)
-30 -40 -50
DTeq = erf/a
-60 -70 -80 -90
Figure 3.
Example of curve of temperature reduction equivalent to the strain due to shrinkage and creep.
braking/traction, bending, shrinkage and creep) is not correct as the model of rail–deck interaction is itself non linear, due to the behavior of the union between track and deck or subgrade. On the other hand, this law is also dependent on the track being loaded or unloaded, and therefore it depends on the position of the train of loads, which further complicates a “step by step” non linear calculation, which would be a more realistic approach.
2.4.1 Simplified method of superposition The UIC Leaflet 774-3 [12] admits as a simplification the linear combination of the results obtained in the separate analysis of each elementary action. However, several considerations have to be made as regards this simplification. If including the effect of shrinkage and creep this simplification is excessively conservative [3]. The method presented in this paper is more realistic than the linear superposition of states. It is based upon the assessment of the worst-case situations for the rail and the most precise calculation possible of the stresses in rail with in such situations although considering the concomitance of the temperature variations arising from strains due to shrinkage and creep (using a variation of temperature equivalent to the sum of both) and not their linear superposition. Consequently, without any expansion device and considering only the variations of temperature of the deck, the worst-case assumptions will be as follows: H1) Tt = maximum value and Tt,ret,flu = 0 H2) Tt = minimum value and Tt,ret,flu = minimum value (always negative) On the other hand, considering expansion devices, the variations of temperature of the rail and deck are both taken into account and thus the worst-case situations will be more than two as previously described. Therefore it shall be necessary to find several combinations of variations of deck temperature, including eventually the equivalent variations of temperature from the strain due to shrinkage and creep, and variations of temperature of rail in order to fully assess all worst-case assumptions. Considering traffic actions, if train braking and/or traction are produced on the bridge proper, once the horizontal forces are working the strain due to vertical bending has already occurred. Therefore it seems that the proper method should consider two calculation steps. The first step shall consider vertical loads (at their definitive position on the bridge) and then computing the bending strain. The second step, where the model has already experienced strain due to vertical bending,
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100 Track-Bridge Interaction on High-Speed Railways
F
N = 60 kN/m
60 kN/m
increasing N
FN N =0
F0
2 mm
Figure 4. load.
u
Resistance of track against relative displacements with respect to bearing, as a function of vertical
shall include the horizontal forces of braking/traction. This calculation is quite straightforward as the characteristics of the non linear elements remain unchanged during the two calculation steps. 2.4.2 “Step by step” calculation method As a prerequisite of the “step by step” calculation, it is necessary to define a law for the longitudinal behavior of the fastening-ballast system as a function of the vertical load. Then based upon the laws previously established for loaded and unloaded track, a law like the one shown in Figure 4 could be defined where the maximum longitudinal load is a function of the vertical load. The value of maximum force is obtained with the following expression: FN = F0 + µN F0 = 12 kN/m F80 = 60 kN/m µ = 0.60 with ballast properly maintained: F0 = 20 kN/m F80 = 60 kN/m µ = 0.75 Once this law has been defined, a full analysis is possible considering the application of applied strains and subsequently the forces of the train moving along the model track in successive steps (vertical and eventually horizontal loads). with ballast poorly maintained:
2.4.3 Comparison of results A comparative assessment of the two calculation methods previously described has been made with several of the test cases stated in the UIC Leaflet 774-3 [12]. Table 1 next summarizes the results of the calculations carried out using the two methods as well as a comparative analysis. This comparison shows that the linear superposition of applied strains and traffic loads generates conservative results as regards both stresses in rail and reactions in bearing. The corresponding error is usually small (lower than 10% on average and never greater than 25% in the cases analyzed). Figure 5 shows the results of stresses in rail for the test case E1-3 of the UIC Leaflet 774-3 [12]. This is a single-span bridge 60 m long, with a restricted bearing in abutment 1 and a sliding support in abutment 2. The actions considered are: increase of deck temperature +30◦ C; vertical traffic load of 80 kN/m; braking load of 20 kN/m. The calculation has been carried out for a train running from abutment 1 to abutment 2, where the train head is located over abutment 2, and assuming it will brake in that position.
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Table 1. Comparison of results with test cases according to UIC Leaflet 774-3 [12]. Case
Effect
Simplified (1)
Step-by-step (2)
Error 1-(1)/(2)
C 1-3
Max. Comp. (MPa) Rel. Displ. (m) Abs. Displ. (m) Reaction (kN) Max. Comp. (MPa) Rel. Displ. (m) Abs. Displ. (m) Reaction (kN)
−56.5 −1.26E−02 1.78E−02 956.0 −48.7 −1.63E−03 2.85E−03 1840.3
−48.1 −1.37E−02 1.84E−02 802.3 −46.3 −2.25E−03 3.52E−03 1703.5
−17% 8% 3% −19% −5% 28% 19% −8%
Max. Comp. (MPa) Rel. Displ. (m) Abs. Displ. (m) Reaction (kN) Max. Comp. (MPa) Rel. Displ. (m) Abs. Displ. (m) Reaction (kN)
−45.7 −1.05E−02 1.45E−02 538.4 −25.9 −8.23E−04 −1.89E−03 921.7
−44.4 −1.07E−02 1.47E−02 526.4 −25.7 −8.60E−04 −1.69E−03 919.0
−3% 2% 1% −2% −1% 4% −12% 0%
Max. Comp. (MPa) Rel. Displ. (m) Abs. Displ. (m) Reaction (kN) Max. Comp. (MPa) Rel. Displ. (m) Abs. Displ. (m) Reaction (kN)
−66.0 −1.70E−02 2.32E−02 977.8 −56.2 −1.89E−03 3.33E−03 2380.8
−55.4 −1.75E−02 2.36E−02 783.4 −54.1 −2.00E−03 4.03E−03 2179.2
−19% 3% 2% −25% −4% 6% 17% −9%
Max. Comp. (MPa) Rel. Displ. (m) Abs. Displ. (m) Reaction (kN) Max. Comp. (MPa) Rel. Displ. (m) Abs. Displ. (m) Reaction (kN)
−57.5 −1.46E−02 1.98E−02 631.9 −32.9 −8.63E−04 −2.60E−03 1287.9
−53.0 −1.47E−02 2.03E−02 581.6 −32.9 −9.10E−04 −2.30E−03 1270.7
−8% 1% 2% −9% 0% 5% −13% −1%
C 4-6
D 1-3
D 4-6
E 1-3
E 4-6
F 1-3
F 4-6
On the other hand, in the calculation of displacements, both absolute of deck and relative between track and subgrade or deck, the linear superposition is not conservative. As in the previous case the error is not significant, lower than 10% on average and never greater than 28% in the cases analyzed, although great care must be taken in the analysis of the results as regards displacements whenever carrying out this type of simplified calculations. Figure 6 shows the results of displacements of rail and deck in the same case E1-3. In most of the cases the simplified method should be sufficient. Nevertheless some bridges can be labeled as “threshold” structures, for which after the analysis has been made using the simplified method, the stresses in rail without expansion device are slightly over the permissible levels. After a simplified analysis the actual worst-case situations will have been detected for the rail, and a “step by step” calculation could then be executed for such situations. This will allow to check how conservative was the first calculation as regards stresses, and then get more realistic values of displacements. Chapter 4 describes an example of calculation for a viaduct with these features.
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102 Track-Bridge Interaction on High-Speed Railways
Figure 5. Comparison of stresses in rail with simplified and “step by step” methods. Case E1-3 UIC leaflet 774-3 [12].
3 3.1
VERIFICATION Stresses in rail
3.1.1 Verification of stresses in continuous rail In the case of a track with UIC 60 rail, concrete sleepers, ballast layer thickness greater than 30 cm and radii greater than 1500 m, the standards [4] [12] state that the permissible additional stress due to track-deck interaction is 72 MPa in compression and 92 MPa in traction. It is assumed that this increment is computed with respect to the stress in rail at a sufficient distance from the bridge. 3.1.2 Verification of stresses in rail with expansion device In this case the calculation shall not determine the additional stress but the stresses in rail due to all longitudinal effects (temperature effects on deck and rail, braking/traction, bending, shrinkage and creep). The stresses obtained will be compared to the sum of the permissible limit additional stress and the confining stress of rail corresponding to a variation of temperature of the rail (the maximum variation stated in the standards of ±50◦ C can be considered, generating a stress of 126 MPa in the UIC 60 rail, with α = 1.2 × 10−5 ◦ C−1 ). 3.2
Displacements
The relative displacements between track and deck are restricted in order to avoid ballast deconsolidation. The limitation of deck displacements ensures compliance with limiting stresses in the rail.
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Numerical methods for the analysis of longitudinal interaction
103
Figure 6. Comparison of displacements of rail and deck with simplified and “step by step” methods. Case E1-3 UIC leaflet 774-3 [12].
The verification on displacements stated by the standards [4] [12] are the following: – Under loads of braking/traction, the relative displacement between track and deck or subgrade shall be lower than 4 mm. – Under loads of braking/traction, the horizontal displacement of deck in abutment or between adjacent decks shall be lower than 5 mm without expansion device or 30 mm with expansion device. – Under traffic vertical loads of braking and/or traction and thermal gradient, the strain of the deck due to bending shall not generate horizontal displacements of the upper side of the deck in abutments or between adjacent decks greater than 8 mm. – Under traffic vertical loads, with their corresponding dynamic factor, and temperature actions, the strain of the deck due to bending shall not generate vertical displacements of the upper side of the deck in abutments or between adjacent decks greater than 3 mm for tracks with a project speed V lower than 160 km/h, or 2 mm when V can be greater than 160 km/h. – With continuous rail, the horizontal displacements of the deck in abutment or between adjacent decks due to daytime variation of temperature shall not be greater than a value in the range 10 mm to 15 mm (to be defined by each administration; in the Spanish case: 13 mm for a maximum daytime variation of temperature of 5◦ C [1]). Otherwise an expansion device shall be fitted. 3.3
Calculation of the maximum displacement of the expansion device
Whenever an expansion device has to be installed, it shall be dimensioned depending on its maximum displacement, thus selecting the appropriate type of device (expansion devices are available with a maximum displacement of up to 1200 mm).
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104 Track-Bridge Interaction on High-Speed Railways
In this case the displacements of the rail in the device due to a variation of temperature of deck and rail (Tt ; Tc ) and due to shrinkage and creep shall be taken into account, while the displacements due to braking/traction or bending can be disregarded. 4
EXAMPLE OF CALCULATION
An example of calculation is presented next, carried out with the simplified superposition method described in 2.4.1 and the “step by step” calculation method described in 2.4.2. 4.1
Description of viaduct
This is a continuous bridge of four spans with lengths of 20.0 and 35.0 m, and an overall length of 110.0 m. The deck is made up of prestressed precast concrete (HP-55) u-beams with an edge of 2.15 m, and a reinforced concrete slab HA-30 0.35 m thick. The slab includes a longitudinal joint in the axis of the bridge, thus structurally there are two independent decks, one for each track. All the bearings of the deck are neoprene-teflon sliding bearings. In bearing E-1 the deck is anchored to the abutment by means of prestressing strands. Figure 7 shows a schematic diagram of the viaduct. 4.2
Model of calculation
The characteristics of the different elements of the calculation model are as follows: • Rails: four rails are idealized, two per track: Cross-section = 2 × 0.007686 m2 ; Inertia = 2 × 3.055 × 10−5 m4 ; E = 200,000 MPa; ν = 0.30; α = 1.2 × 10−5◦ C−1 • Deck: The parameters are: Cross-section = 3.39 m2 , bending inertia = 2.77 m4 , edge = 2.50 m, Height of center of gravity = 1.59 m E = 32.325 MPa; ν = 0.20; α = 1.0 × 10−5◦ C−1 • Bearings of the deck: The sliding bearings are idealized without friction, allowing the horizontal displacement of the corresponding node. The anchoring of deck to bearing E-1 is idealized by means of an elastic spring with a constant equal to the stiffness of the anchoring. This stiffness has been estimated, assuming an abutment with infinite stiffness, as the stiffness of the compressed neoprene interfacing deck and abutment: kn =
E n An en
where: kn : stiffness of anchoring En : Young’s modulus of neoprene = 420 MPa An : total cross-section of neoprene = 2 × 0.50 × 0.30 = 0.30 m2 en : net thickness of neoprene = 0.021 m thus: Kn ≈ 6,000,000 kN/m 110 20
Figure 7.
35
Schematic diagram of the viaduct.
© 2009 Taylor & Francis Group, London, UK
35
20
Numerical methods for the analysis of longitudinal interaction
105
4.2.1 Actions taken into account Variation of temperature: The maximum value of variation of deck temperature has been considered: Tt = ±35◦ C Braking/traction: As there are two independent decks, one for each track, only braking forces have been considered. The values of the Spanish standards [1] have been used: Braking force Qf = α · q f · L f where: α = 1.21 qf = 20 kN/m Lf = 300 m Bending: As a simplification of the model of loads UIC 71, a train of loads has been used with a length of 300 m and a uniform load of 80 kN/m multiplied by the coefficient of classification α = 1.21, with further multiplication where applicable by a dynamic factor: = 1.07. Shrinkage and creep: The following value has been used for the reduction of the equivalent deck temperature with strains due to shrinkage and creep from rail soldering instant till infinite time: Tt,ret,flu = −23◦ C. 4.2.2 Combination of actions As described in 2.4, with no expansion device, the worst-case assumptions would be two: H1) Tt = maximum value and Tt,ret,flu = 0 H2) Tt = minimum value and Tt,ret,flu = minimum value (always negative) These two situations shall be combined with the bending and braking actions, according to the two methods described in this paper (simplified and “step by step”). 4.3
Results
The first calculation has been carried out with the simplified method, where the deck undergoes two variations of temperature (concomitant with a variation of temperature of the rail of 50◦ C) and the vertical and braking forces. In the case of traffic overloads four positions of the train have been considered. Figure 8 shows the results of overall stresses in rail. As can be seen there is a maximum of compression on the unrestricted abutment at 202.7 MPa. Therefore the limit values of the standards are slightly exceeded (72 MPa added to 126 MPa of confinement stress: 198 MPa). For the simplified calculation the worst-case scenario has been identified as occurring for the reduction of temperature of deck combined with the equivalent temperature of shrinkage and creep and the train head located over abutment 2 in the instant of braking (Fig. 9). For this situation a “step by step” calculation has been carried out. In the first step of calculation, the variations of temperature of rail and deck are entered. In the following steps the vertical loads are entered, advancing from the embankment adjacent to abutment 1 till the definitive position shown in Figure 9. In this position the horizontal loads of braking are entered in the last step of calculation. Figure 10 shows the comparison of results for stresses in rail using both methods, for vertical and horizontal forces and the sum of all actions. As can be seen the “step by step” method implies a reduction of maximum compressions that become lower than the limit values of the standards.
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106 Track-Bridge Interaction on High-Speed Railways
Figure 8. Envelopes of stresses in rail due to variations of temperature, bending and braking, obtained with the simplified method.
Figure 9.
Actions in worst-case situation for stresses in rail.
The verification of displacements are detailed next. Verifications 2 to 5 have been made for the situation analyzed with the “step by step” method (the sign criteria for absolute displacements adopted is [+] for direction from abutment 1 to abutment 2 and [−] for direction from abutment 2 to abutment 1, the sign criteria for relative displacements is [+] for receding and [−] for approaching): 1. Displacement of end of deck (absolute in abutment or relative between adjacent decks) due to daytime variation of temperature (±5◦ C): • In end of deck over E2: ±4.9 mm < 13.0 mm 2. Relative displacement between track and bearing (deck or subgrade) against braking/traction: • Maximum (E-2): ±2.1 mm < 4.0 mm 3. Displacement of end of deck (absolute in abutment or relative between adjacent decks) against braking/traction: • Maximum (E-2): ±1.9 mm < 5.0 mm 4. Horizontal displacement of upper face of deck (absolute in abutment or relative between adjacent decks) due to deck bending at deck end without expansion device: • Maximum (E-1): 0.2 mm < 8.0 mm.
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Numerical methods for the analysis of longitudinal interaction
107
BENDING
5
Stresses in rail (MPa)
Step-by-step method Simplified method
0
-5
20
BRAKING Stresses in rail (MPa)
Step-by-step method Simplified method
10
0
-10
-20
0
-20
COMBINATION:DT, BENDING AND BRAKING Stresses in rail (MPa)
-40
Step-by-step method Simplified method Limit of compression
-60 -80 - 100 -120 -140 -160 -180 -200 -220
Figure 10.
Comparison of results for stresses in rail.
5. Vertical displacement of deck (absolute in abutment or relative between adjacent decks) against deck bending (this value depends on the rotation of deck in bearing and the separation between axis of bearing and end of deck): • separation between axis of bearing in abutments and end of deck: 1.0 m • maximum rotation in abutments (E1): 0.5 × 10−3 rad • maximum vertical displacement in abutments: 0.5 mm < 2.0 mm As can be seen, all limit values are easily complied with. Furthermore, the earlier comment on stresses would not justify in this case the installation of an expansion device.
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108 Track-Bridge Interaction on High-Speed Railways
5
CONCLUSIONS
The numerical methods of the analysis of longitudinal track-structure interaction have been presented, where two different scenarios of combination of actions have been considered: – simplified method of combination; – “step by step” calculation method. Both methods have been used with a set of test cases described in UIC Leaflet 774-3 [12]. The analysis of the results allows to characterize the simplified method as conservative for the calculation of stresses in rail and reactions in bearings, but not conservative for displacements of rail and deck. In most of the cases the simplified method will be sufficient, although “threshold” cases can occur where a thorough analysis of a specific situation could be needed using a “step by step” calculation.
REFERENCES [1] Comisión redactora IAPF. 2006. Borrador de Instrucción de acciones a considerar en el proyecto de puentes de ferrocarril. [2] Cuadrado, M. & González, P. 2000. Interacción entre carril continuo y tablero de puente. Calibrado del modelo de cálculo según la Ficha UIC 774-3. [3] Cuadrado, M. & González, P. 2004. Consideración de las deformaciones por retracción y fluencia en el estudio del fenómeno de interacción vía -tablero en el proyecto de puentes ferroviarios. Revista de Obras Públicas. [4] Eurocódigo EN 1991-2. 2003. Acciones en estructuras. Parte 2: Cargas de tráfico en puentes. [5] Fryba, L. 1985. Thermal interaction of long welded rails with railway bridges. Rail International. 3, pp. 5–24. [6] Fryba, L. 1997. Continuous welded rail on railway bridges. World Congress on Railway Research. Firenze. [7] González, P., Cuadrado, M., Nasarre, J. & Romo, E. 2002. Alta velocidad: fenómeno de interacción vía-tablero en puentes. Revista de Obras Públicas. [8] González, P., Cuadrado, M. & Romo, E. IABSE. 2002. Consideración del fenómeno de interacción víatablero en el proyecto de puentes ferroviarios. Congreso puentes de ferrocarril: Proyecto construcción y conservación. Madrid. [9] González, P. & Cuadrado, M. 2000. Apoyo a la Dirección de los Proyectos de la Línea de alta velocidad ente Córdoba y Málaga. Plataforma. Informe sobre trabajos específicos adicionales: Interacción entre carriles continuos y tableros de viaductos ferroviarios o plataforma. Metodología y criterios de dimensionamiento de aparatos de dilatación de vía. [10] Ministerio de Fomento. 1999. EHE. Instrucción de Hormigón Estructural. [11] Ramondenc, P. Track/Bridge interaction. 1997. World Congress on Railway Research. Firenze. [12] Union Internationale des Chemins de fer. 1999. Fiche UIC 774-3R. Interaction voie-ouvrages d’art. Recommendations pour les calculs.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 10 Longitudinal track-bridge interaction for load-sequences P. Ruge, D.R. Widarda & C. Birk Lehrstuhl Dynamik der Tragwerke, Fakultät Bauingenieurwesen, TU Dresden, Germany
ABSTRACT: European codes recommend an independent nonlinear treatment of three significant loading cases in order to calculate longitudinal forces in continuously welded rails on bridgedecks: Temperature change along one year, braking/accelerating and finally bending of the bridge deck. The final rail-stresses are obtained by summing up the different values found independently. This is against the nonlinear character of the problem. An additional effect concerns the change of the longitudinal coupling stiffness between railway and bridge deck when a train passes the bridge. This contribution focuses on a true nonlinear combination of all loading cases and on the influence of the longitudinal stiffness change. As this change occurs in a rather short time, mass-acceleration forces have to be considered in order to assess their influence compared with a pure statical analysis.
1
INTRODUCTION
In general, rails as major part of any railway construction rest on the subgrade. Thus the stresses σ in continuously welded rails due to temperature change T are directly and totally related to T with σ = αET . However, if the track is supported by a flexible structure like a bridge, the homogeneous situation is disturbed and the track-bridge interaction has to be analyzed. In former times track and the supporting structure had been treated and analysed separately. However, this approach cannot be used if the rails are continuously welded and expansion devices have to be considered. In this case the track-bridge interaction cannot be neglected in order to ensure stability, safety and serviceability. In this context, longitudinal loads are of special importance. These are caused by braking, uniform temperature change of the supporting structure or bending of the bridge deck. In References [Ruge et al., 2004, Ruge et al., 2005b, Ruge et al., 2005a] it has been demonstrated that each of the loads can cause remarkable longitudinal rail stresses. Apart from the above situations, an additional loading case is identified and treated in this paper which is not mentioned in codes so fare. It is shown that additional longitudinal forces are caused by the sudden change of the longitudinal coupling between track and bridge by a passing train. A detailed discussion and mathematical description of this loading case is given in section 3, numerical results are presented in section 5 in order to show the importance of this effect. The investigation of longitudinal loads and their influence on the forces in continuously welded rails on bridgedecks has been much discussed for the last 20 years. As a result the European norm EN 1991–2 [EN9, 2003] provides information on loads on bridges, design methods and approaches today. Together with national appendixes like DIN FB 101 in Germany a common set of rules is available which will harmonize national norms throughout Europe. According to Reference [DIN, 2003] different longitudinal loading cases are analysed separately considering a nonlinear stiffness law of the ballast. Such separate treatments of the loading cases braking and temperature change can be found in References [Ruge et al., 2004, Ruge et al., 2005b, Ruge et al., 2005a]. A comprehensive treatment of longitudinal effects in general is presented in the books [Fryba, 1996] chapter 14 and [Esveld, 2001] chapter 7. The maximum longitudinal stresses 109 © 2009 Taylor & Francis Group, London, UK
110 Track-Bridge Interaction on High-Speed Railways
follow from a subsequent summation of the results corresponding to different loading cases. This approximate approach completely neglects the influence of preceding events on the current load process. In fact, a more realistic combination of loading cases is required to obtain correct results. Then the sequence of events is of importance and the history of a load process has to be taken into account. For this purpose, a truly nonlinear track-bridge interaction model resulting from a correct combination of loading cases is developed in this paper. The nonlinear problem is solved by a sequence of linear steps with incremental displacements; thus the overall system-stiffness depends on the load-deformation history. In statics, the stepwise linearized differential equations can be solved exactly and enables one to establish a corresponding exact stiffness formulation. In dynamics, the exact dynamical stiffness matrix is frequency-dependant and thus not available in the time-domain in a straightforward manner. Therefore, in dynamics, the problem is solved by means of a variational formulation using a quadratic interpolation for the rail-displacements in space and a linear one in time. 2
TRACK-BRIDGE INTERACTION
A typical track-bridge system is shown in Figure 1. A rail of longitudinal stiffness EA is resting on a bridge structure of length L. On the left-hand side abutment (point A) the horizontal bridge movement is restrained by an elastic support modelled as a spring with stiffness kA [N/m]. On the right-hand side abutment (point B) the horizontal √ displacement of the structure is enabled by sliding bearing. Two springs with a stiffness k0 = k1 = EAcu [N/m] represent the track resting on the adjacent dams. Such springs of stiffness k0 , k1 can only be used in regions where the rail-subsoil interaction is elastic during the whole loading-history. Bridge and rail are coupled through the ballast or fastening system in case of a ballasted or rigid track, respectively. This is modelled as a continuous elastic bond of distributed stiffness c[N/m2 ]. The value of the latter parameter depends on whether the track is loaded or not: cu N/m2 for unloaded track, (1) 2 cl N/m for loaded track. The above elastic coupling between bridge and rail can be visualized as a coupling shearelement subject to a displacement difference uD as shown in Figure 1(b). uD = uR − uB uD = sign(uR − uB )˜u
Figure 1.
Longitudinal track-bridge interaction.
© 2009 Taylor & Francis Group, London, UK
for |uR − uB | < u˜ , for |uR − uB | ≥ u˜ .
(2)
Longitudinal track-bridge interaction for load-sequences
111
uR longitudinal displacement of the rail, uB longitudinal displacement of the upper surface of the bridge. One important characteristic of the track-bridge system is that the above deformation of the coupling element is limited to a critical value u˜ . Below this limit value, a linear elastic relationship between the displacement difference uD and the longitudinal restoring force q shown in Figure 1 is valid. Unloaded track: q = −cu uD , Loaded track: q = −cl uD .
(3)
Here, a positive force +q is assumed to act on the rail if the displacement of the former is smaller than that of the bridge; consequently (−q) acts onto the bridge. Above ±˜u the rail is slipping relative to the ballast or concrete strip. The corresponding nonlinear stiffness law for the ballasted and rigid track is shown in Figure 2. The latter illustration indicates that a situation where the rail slips relative to the coupling element corresponds to a constant longitudinal restoring force q˜ with q = q˜ = −c · sign(uR − uB )˜u
for
|uR − uB | ≥ u˜ .
(4)
However, Equation 4 is valid for a virgin track-bridge system, which has not undergone any previous deformation only. For a sequence of loading cases the situation is more complicated, as will be explained in the following. 2.1
Influence of the load history
Consider a track-bridge system which has experienced one single longitudinal loading process, such as temperature change. The displacement difference (uR − uB ) obtained for this first loading case is shown in Figure 3. The relative deformation uD follows immediately. Based on this result the rail can be divided into regions where the elastic coupling to the bridge is retained and where the rail slips relative to the coupling element. In the following, these two situations will be referred to as ‘elastic’ and ‘plastic’, respectively. Depending on the state of deformation of the coupling element there is a remaining deformation capacity k available for a subsequent loading case to proceed elastically. This is illustrated in
Figure 2.
Nonlinear stiffness law for rails UIC 60 and sleepers B 70 W.
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112 Track-Bridge Interaction on High-Speed Railways
Figure 3. Deformation situation after a preceding loading case. (a) displacement difference (uR − uB ), (b) deformation uD of the coupling element and (c) corresponding available capacity for a subsequent elastic deformation.
Figure 4 for the case of a positive relative deformation 0 < uD < +˜u. Here, the terms kr and kl denote the capacity for proceeding elastically corresponding to a possible positive and negative additional deformation zD of the coupling element, respectively. The above available capacity can be calculated using Equations (5) and (6).
© 2009 Taylor & Francis Group, London, UK
kr = u˜ − uD ,
(5)
kl = −˜u − uD .
(6)
Longitudinal track-bridge interaction for load-sequences
Figure 4.
113
Definition of deformation capacity.
The following limiting relationships apply: −2˜u ≤ kl ≤ 0,
0 ≤ kr ≤ 2˜u.
(7)
At the same time, the sum of the absolute values of the positive and negative capacity is constant. |kl | + kr = 2˜u.
(8)
In Figure 3 it can be seen that the negative and positive available deformation capacity enclose a band of width 2˜u. A subsequent loading case proceeds elastically if the additional relative deformation zD of the coupling element, zD = zR − zB , (9) zR additional longitudinal displacement of the rail, zB additional longitudinal displacement of the upper surface of the bridge, lies within this band. Otherwise, the rail slips relative to the coupling element. kl ≤ zD ≤ kr zD < kl , zD ≤ kr
→ →
elastic plastic
(10)
If a subsequent loading case proceeds plastically, a longitudinal restoring force q as given in Equations (11) and (12) acts on the rail. q = −kr · c
for
zR − zB ≥ 0,
(11)
q = |kl | · c
for
zR − zB < 0.
(12)
In case of a positive additional deformation of the coupling element, the positive capacity kr is used to determine q, otherwise kl is relevant. Using Equations (5) and (6), formulations (11) and (12) can be written as: q = (uD − u˜ )c
for
zR − zB ≥ 0,
(13)
q = (uD + u˜ )c
for
zR − zB < 0.
(14)
Equations (11)–(14) will be used for a derivation of stiffness formulations in the following sections.
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114 Track-Bridge Interaction on High-Speed Railways
3
EXACT STIFFNESS FORMULATIONS FOR LONGITUDINAL STATIC LOADS
In reality the loading cases • • • •
uniform temperature change of the supporting structure sudden change of ballast stiffness when the train passes the bridge bending of the supporting structure braking/accelerating
follow each other with elastic/plastic regions changing. In any case, the actual relative deformation uD of the coupling element between rail and bridge due to an arbitrary number of preceding load processes is assumed to be known. The special case of a virgin structure is included in the following formulation with uD = 0. At this point it is beneficial to consider a typical nonlinear equation with coefficients C1 ; C3 . The solution u for a sequence of 2 loads f1 followed by f2 can be found by first solving (15) with f1 for u1 C1 u1 + C3 u13 = f1 → solution u1 .
(15)
Afterwards u12 due to the sequence f2 following f1 can be found by increments z added to u1 . u12 = u1 + z → C1 (u1 + z) + C3 (u1 + z)3 = f1 + f2 (C1 + 3C3 u12 ) z + 3C3 u1 z 2 + C3 z 3 = f2 + (f1 − C1 u1 − C3 u13 )
(16)
0
This equation is characterized by modified coefficients which depend on u1 and thus on the load/deformation history. These considerations can be transferred to the nonlinear track-bridge system. As explained in section 2.1, the response of the latter to a subsequent load strongly depends on the loading history. Consequently, the total displacement uR,total of the rail is the sum of the displacements due to preceding loading cases and the additional deformation zR due to the current load. uR,total = uR + zR , u B = uB + z B .
(17)
The displacement uB of the bridge (assumed to be rigid) is treated in a similar manner. In order to derive a stiffness formulation, Kz = r,
(18)
for the additional displacements z a typical continuous track-bridge system as shown in Figure 1(a) is analysed. Here, it is distinguished between an elastic coupling of rail and bridge and a plastic situation where the rail slips relative to the structure. 3.1
Elastic coupling
Each of the 4 loading situations is governed by a differential equation for zR (x) and the discrete value zB . EAzR − c(zR − zB ) = q(uR (x), uB ). The deformation-history is hidden in the right side q(x).
© 2009 Taylor & Francis Group, London, UK
(19)
Longitudinal track-bridge interaction for load-sequences
115
The homogeneous solution zRh (x) can be normalized with respect to z0 = zRh (x = 0), z1 = zRh (x = l) and zB . zRh (x) = pT h(x), zRh (x = 0) = z0 , zRh (x = l) = z1 ; ⎡ ⎤ ⎡ ⎤ sinh γ(l − x) z0 ⎢ sinh γx ⎥ 1 ⎢z ⎥ . p = ⎣ 1 ⎦, h(x) = ⎣ zB −sinh γ(l − x) − sinh γx + sinh γl ⎦ sinh γl 1 0
(20)
So fare, the addition longitudinal forces L0h and L1h at the end points due to the homogeneous solution are known from L = EAzR . ⎤ −coth γl ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ √ ⎢ sinh γl ⎥ T = EAzRh (x = 0) = p cEA⎢ ⎥, γl ⎥ ⎢ ⎥ ⎢ tanh ⎣ 2 ⎦ 0 ⎡
L0h
⎡
L1h
1 ⎢ − sinh γl ⎢ ⎢ coth γl √ ⎢ T = EAzRh (x = l) = p cEA⎢ γl ⎢ ⎢ −tanh ⎣ 2 0
(21)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
(22)
The last position p4 = 1 in p is provided with respect to the additional solution zRq due to the right-hand side q with zRq (x = 0) = 0 and zRq (x = l) = 0 in order to keep the normalization undisturbed. The analysis is significantly simplified and applicable for all 4 loading cases if the right hand side is interpolated in a linear manner. x x + q1 . q(x) = q0 1 − l l
(23)
Then the last position h4 (x) in h simply changes from h4 = 0 to h4 (x) =
−1 [q0 sinh γ(l − x)] + q1 sinh γx] c sinh γl
(24)
and L0 , L1 are completed accordingly: zR = zRh + zRq ,
⎡
⎤ 0 √ 0 ⎥ cEA ⎢ ⎢0 ⎥ L0 = L0h + pT , ⎢ ⎥ c ⎣ 1 1 1 ⎦ − q1 − q0 coth γl − γl sinh γl γl
© 2009 Taylor & Francis Group, London, UK
(25)
116 Track-Bridge Interaction on High-Speed Railways
⎡
⎤ 0 √ 0 ⎥ cEA ⎢ ⎢0 ⎥ L1 = L1h + pT ⎢ ⎥. c ⎣ 1 ⎦ 1 1 − − q1 coth γl − q0 sinh γl γl γl
(26)
Evaluating the equilibrium of forces at the points 0, 1, A and for the whole system, Figure 1, x=0: L0 = k0 z0 x = L : −L1 = k1 z1 A: LA = kA zA overall : L A = L1 − L 0
(27)
and taking the longitudinal forces L0 , L1 from Equations (21)–(27), ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ z0 k0 0 0 L0 (27) : ⎣−L1⎦ = Kk z; Kk = ⎣0 k1 0 ⎦, z = ⎣z1⎦, LA zA 0 0 kA ⎡ ⎤ L0 ⎦ = −Ksc z + r, −L1 (21) − (22) : ⎣ LA = L1 − L0 results in a stiffness formulation to describe the elastic behaviour of a continuous track-bridge system due to a linear load. (Ksc + Kk )z = r,
⎡
−1 sinh γl
⎤
γl 2⎥ ⎥ γl ⎥ T = Ksc , Ksc coth γl − tanh ⎥ 2⎥ ⎥ γl ⎦ ∗ 2 tanh 2 ⎤ ⎡ 1 1 1 ⎢ coth γl − γl γl − sinh γl ⎥ ⎥ ⎢ 1⎢ 1 1 ⎥ 1 ⎥ q0 , γ 2 = c . ⎢ r = ⎢ − coth γl − γ ⎢ γl EA sinh γl γl ⎥ ⎥ q1 ⎣ γl ⎦ γl − tanh − tanh 2 2 ⎢ coth γl ⎢ √ ⎢ = cEA⎢ ⎢ ∗ ⎢ ⎣ ∗
3.2
(28)
− tanh
(29)
Plastic coupling
In this situation, the elastic coupling between rail and bridge is ineffective and the rail as well as the bridge are loaded by the internal longitudinal restoring forces qR , which are restricted to qR = ±2˜uc. In addition, there can be an external load p(x) from braking, acting directly onto the rail. Here, too, it is advantageous to describe the right-hand side in a linear manner.
© 2009 Taylor & Francis Group, London, UK
x q(x) = (qR1 − qR0 ) + qR0 , l
(30)
x +(p1 − p0 ) + p0 . l
(31)
Longitudinal track-bridge interaction for load-sequences
117
Thus, the governing differential equation EAzR = −q
(32)
is solved by zR = pT h(x) ⎡
1−
⎤
x l
⎢ ⎥ ⎡ ⎤ ⎢ ⎥ z0 x ⎢ ⎥ ⎥, p = ⎣ z1 ⎦, h = ⎢ ⎢ ⎥ l ⎢ 3 ⎥ 1 3 ⎣ q0 x ⎦ q x 1 − 3x2 + 2lx + lx − 6EA l 6EA l q0 = qR0 + p0 , q1 = qR1 + p1 .
(33)
The longitudinal forces L0 , L1 (L > 0: tension)
EA 1 L0 =− −L1 l −1
q0 l 2 q1 l 1 −1 z0 + + 1 z1 6 1 6 2
(34)
are taken to establish the element stiffness matrix K and the corresponding right-hand side: [Kz]Element = rElement ⎡ EA ⎣ 1 −1 K = l 0
⎤ ⎡ ⎤ −1 0 z0 1 0⎦, z = ⎣ z1 ⎦, zA 0 0 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2 1 2 p0 l ⎣ ⎦ p1 l ⎣ 1 ⎦ qR0 l ⎣ ⎦ qR1 l ⎣ ⎦ 1 + 2 + 1 + 2 + r = 6 6 −3 6 0 6 0 −3
(35) (36)
(37)
The element stiffness is completed by the external stiffnesses k0 , k1 and kA , shown in Figure 1. ⎡
K = KElement
k0 +⎣ 0 0
0 k1 0
⎤ 0 0 ⎦. kA
(38)
More details concerning the static analysis are given in German written papers [Ruge et al., 2004, Ruge et al., 2005b, Ruge et al., 2005a] and a summarizing report [Ruge and Birk, 2006]. The new aspect, treated in this paper, is devoted to the question, if the change in the couplingstiffness causes significant dynamical forces. Then the differential equation for the rail has to be complemented by mass-accelerations with the mass density ρ [kg/m] of the track. −EAzR + c(zR − zB ) + ρ¨zR = q(x, t).
(39)
Here the exact solutions zR (x, t), zB (t) can be formulated but in the frequency-domain zR (x, t) = zˆR (x)eiωt , zB (t) = zˆB eiωt ; however the excitation from coupling-change is highly transient. Therefore we decided to establish a fully discretized formulation: quadratic in the space domain (x), linear (Newmark) in the time domain (t).
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118 Track-Bridge Interaction on High-Speed Railways
4
FINITE ELEMENT FORMULATION IN DYNAMICS
The discretization in the space domain results in a system of differential equations in the time domain. The dynamic analysis caused by change of coupling-stiffness follows statical loadings; thus additional displacements have to be considered which are denoted by z, M(t)¨z(t) + D(t)˙z(t) + K(t)z(t) = Q(t, u)
(40)
where mass matrix M, damping matrix D, stiffness matrix K and load vector Q are generated by evaluating Hamilton’s principle. t1 t1 L(z, z˙ , t)dt + δzT Q∗ dt = 0. (41) δH = δ t0
t0
∗
Here Q represents nonpotential forces and L = T − V − W is Lagrange’s function with kinetic energy, potential energy and work of external force, respectively. ∂V ∂W d ∂T − Q∗ + = . dt ∂˙z ∂z ∂z
(42)
The kinetic energy of rails and bridge girder, 1 1 mR (x, t)[˙zR (x, t)]2 dx + mB z˙B (t)2 , TRB = 2 2 T T d ∂TRB z¨ R (t) x mR h hdx 0 = MRB (t)¨z(t) = , dt ∂˙z 0 mB z¨B (t)
(43) (44)
generates the element mass matrix:
MRB
⎡ 2ρAl ⎢ 15 ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ sym ⎣
ρAl 15 8ρAl 15
ρAl 30 ρAl 15 2ρAl 15
−
⎤ 0
⎥ ⎥ ⎥ 0 ⎥ ⎥. ⎥ ⎥ 0 ⎥ ⎦
(45)
mB
Here z¨ R is the acceleration vector of the rail, z¨B the acceleration of the bridge, h is the column of shape functions and mR = ρA [kg/m] is the mass density of the rail with cross section A. The additional displacements zR of the rail are approximated by quadratic shape functions with corresponding displacements p in the nodes as shown in Figure 5. zR (x) = pT h, 1 hT = 2 [(l 2 − 3lx + 2x2 )(4lx − 4x2 )(−lx + 2x2 )], l
4.1
⎡ ⎤ z0 p = ⎣ z1 ⎦ z2
(46)
Stiffness and damping matrix
The elastic potential VR of the rail generates the elastic stiffness-matrix K R of a single element: 1 2 (47) VR = ER AR [zR (x, t)] dx, 2 x
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Longitudinal track-bridge interaction for load-sequences
l/2
119
l/2
z2
z1
z0 x
Figure 5.
Quadratic interpolation for additional rail-displacements
∂VR T h h dx [zR (t)], = KR z(t) = ER AR ∂z x ⎡ ⎤ −8 1 ER AR ⎣ 7 16 −8 ⎦. KR = 3l sym 7
(48)
(49)
An additional stiffness-part follows from the elastic coupling between rail and bridge. The corresponding elastic potential: Vc =
1 2
x
c(x, t)[zR (x, t) − zB (t)]2 dx,
(50)
∂Vc = Kc (t)z(t) ∂z T (x)dx − x c(x, t)h(x)dx zR (t) x c(x, t)h(x)h = ZB (t) − x c(x, t)hT (x)dx x c(x, t)dx
(51)
gives the stiffness matrix due to the coupling interface: ⎡ Kc =
4
2 16
cl ⎢ ⎢ ⎢ 30 ⎣sym
−1 2 4
⎤ −5 −20⎥ ⎥ ⎥. −5⎦ 30
(52)
Damping of the system is assumed to be proportional to the mass and stiffness with corresponding factors cK and cM . (53) D = ck K + cM M. The values cK and cM depend on the first 2 angular eigenfrequencies of the undamped system: cK = 2D(w1 + w2 ),
cM = cK w1 w2 .
D is the classical damping coefficient which is up to D ≈0.03 for concrete structures. 4.2
Load due to change of ballast stiffness
The quantity c(x, t) represents the change of coupling stiffness; either from unloaded to loaded when the train comes or from loaded to unloaded when it leaves the bridge. The load vector due to this change can be deduced from the work of external forces.
∂W = r(t) = − ∂z
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c (x, t) uD h(x)dx . − x c(x, t)uD dx
x
(54)
120 Track-Bridge Interaction on High-Speed Railways
The load vector depends on the displacements uB and uR from the previous loads as described in Equation (2) and in Figure 3(b). The quadratic interpolation uR = hT uR for the previous displacements describes any history; either elastic t)h(x)hT (x)dx − x c(x, t)h(x)dx uR (t) ∂W x c(x, = r(t) = − , (55) ∂z uB (t) − x c(x, t)hT (x)dx x c(x, t)dx ⎡ ⎤⎡ ⎤ 4 2 −1 −5 u0 c(x, t)l ⎢ 2 16 2 −20⎥⎢ u1 ⎥ , (56) r =− ⎣−1 2 4 −5⎦⎣ u2 ⎦ 30 −5 −20 −5 30 uB or plastic with |u0 − uB | = |u1 − uB | = |u2 − uB | = u˜ : ⎡ ⎤ 1 c(x, t)l ⎢ 4⎥ · sign(uD )˜u⎣ ⎦. r=− 1 6 −6
(57)
The dynamical treatment of this load can be realized by means of a sudden change of the couplingforces like an impulse which acts in a time instant. However, the train needs some time to affect the whole bridge to be in the loaded condition and thus a continuous change of the ballast stiffness in the time domain is more realistic. In order to facilitate the analysis, the change of the coupling stiffness is assumed to be constant along the bridge. It is represented by means of a continuous function in time during a certain interval. The loaded situation starts from the unloaded value (cu ) and increases until cl within T seconds as formulated in Equation (58) and shown in Figure 6. ⎧ t t ⎨ cu 1 − + cl , t ≤ T; (58) ca (t) = T T ⎩ cl, t > T. The time t = 0 starts when the first wheel of the train enters the bridge and ends with t = T when the last wheel leaves it. In every single time-step corresponding to the actual time-instant tj the actual loaded coupling stiffness ca is treated as a constant value with the final maximum value cl : c(x, tj ) = ca (tj ) − cu .
Figure 6.
Continuous change of coupling stiffness in time.
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(59)
Longitudinal track-bridge interaction for load-sequences
4.3
121
Load due to actual slipping
If the actual situation in the coupling element is characterized by slipping, an internal longitudinal force q acts on the rail (+q) as well as on the bridge (−q). However, only a part of this force is caused by the actual differences zD defined by the additional displacements zR and zB . The contributions uD from the previous loads have to be subtracted. The difference zD is defined in the same manner as uD in Equation (2). q(x, t) = c(x, tj ) · (uD − sign(zD )˜u).
(60)
Here c(x, tj ) is the actual coupling stiffness and u˜ denotes the critical displacement of the coupling element which is a constant value independent with respect to c(x, tj ). The interface-force due to actual slipping follows from applying Equation (55): r(t) =
x
c(x, t)h(x)hT (x)dx − x c(x, t)hT (x)dx
− x c(x, t)h(x)dx uR (t) c(x, t)˜uh(x)dx . − x uB (t) − x c(x, t)˜udx x c(x, t)dx
(61)
The numerical form follows from the quadratic interpolation of the previous displacements; the right-hand side for an elastic history, ⎡
4 ca l ⎢ ⎢ 2 r= 30 ⎣−1 −5
2 16 2 −20
−1 2 4 −5
⎤⎡ ⎤ ⎡ ⎤ 1 −5 u0 ⎢ u1 ⎥ ca l ⎢ 4⎥ −20⎥ ⎥ ⎥⎢ ⎥ − · sign(zD )˜u⎢ ⎣ 1⎦, −5⎦⎣ u2 ⎦ 6 −6 30 uB
(62)
can be specialized for a plastic history with |uR − uB | = u˜ : ⎡
⎡ ⎤ ⎤ 1 1 ca l ⎢ 4⎥ ca l ⎢ 4⎥ r= · sign(uD )˜u⎣ ⎦ − · sign(zD )˜u⎣ ⎦. 1 1 6 6 −6 −6
(63)
The minimum r = 0 of this force in Equation (63) occurs if uD and zD are equally oriented. The maximum occurs when they are opposite to each other. 5 5.1
EXAMPLES Simply supported bridge
A simply supported bridge of 60 m length is taken as an example for track-bridge interaction. In a first step the track bridge interaction is analyzed in a pure static manner. A subsequent dynamical analysis shows the differences between the static and the dynamic approach. The system’s data are given in Table 1. Eurocode 1, part 2 as well as the german version in [DIN, 2003] for design of railway bridges allow the linear summation of loading cases, although some loads cause nonlinear reactions of the coupling interface. Figure 7 shows the elastic and plastic regions of the interface due to warming up (δT ), bending (w/l) ˆ and braking (p). Plastic regions caused by the first loading influence the following deformations and consequently there are significant differences between a truly nonlinear treatment of the sequence of loadings and a pure summation of the results found independently. Figure 8(a) demonstrates the longitudinal stresses along the rail due to warming up, bending and braking found independently but paying attention to nonlinear slipping. Included in the same figure is the linear combination of these stresses. Comparing the linear combination of separated
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122 Track-Bridge Interaction on High-Speed Railways
Table 1. Geometry, material properties and loading data for simply supported bridge system. Length of the bridge Stiffness of the elastic support Cross section area of bridge Mass density of bridge Longitudinal rail stiffness Cross section area of rail Stiffness of unloaded track Stiffness of loaded track Critical elastic relative deformation Temperature Thermal expansion coefficient of concrete bridge Braking force Distance from neutral axis to upper surface Distance from neutral axis to lower surface Mass density of rail Velocity of train Damping value
L = 60 m kA = 6 × 108 N/m AB = 3.5070 m2 ρB = 2400 kg/m3 EAR = 3.23 × 109 N AR = 15372 mm2 cu = 6.0 × 107 N/m2 cl = 12.0 × 107 N/m2 u˜ = 0.0005 m t = 30◦ C α = 1.0 × 10−5 p = 20000 N/m hu = 1.21 m hl = 4.79 m ρR = 7900 kg/m3 vtrain = 60 m/sec D = 0.01
Figure 7. Elastic and plastic regions due to warming up T = +30K, bending w/L ˆ = 1/2500 and braking p = +20000 N /m.
(a) Each loading case are analyzed separately and added linearly.
Figure 8.
(b) Loading case is analyzed sequentially with correct combination.
Longitudinal rail stresses due to warming up T = +30K, bending and braking p = +20000 N/m.
loads with the sequential analysis of loading cases as shown in Figure 8(b), the latter gives smaller values for both; maximum compression and tension. Table 2 and Table 3 show the numerical values for both situations. The sequential loading analysis gives 27% less maximum compression than the linear combination. As the maximum value of each loading case does not appear at the same point, the maximum value of the summation is not equal to the summation of maximum stresses of each loading case.
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123
Table 2. Maximum longitudinal rail stresses due to temperature change, bending and braking obtained separately and added linearly. σmin [N/mm2 ]
Load T = +30 K p = +20000 N/m Bending
(1) (2) (3)
(1) + (2) + (3)
σmax [N/mm2 ]
−66.506 −19.276 −34.051
58.624 19.276 49.914
−119.833
76.383
Table 3. Maximum longitudinal rail stresses due to a correct sequentially combination of the loading cases temperature change, bending and braking. σmin [N/mm2 ]
σmax [N/mm2 ]
T = +30 K Bending after (1) p = +20000 N/m after (1) + (3)
−66.506 −15.322 −7.514
58.624 53.174 9.527
(1) + (2) + (3)
−87.472
69.895
Load (1) (2) (3)
Figure 9.
Longitudinal stress due to change of coupling stiffness after temperature.
Finally, the system’s answer to a change of the coupling stiffness is studied by comparing 3 different situations. • Pure static analysis • Sudden, impact-like change with dynamics • Continuous change within a time-period T with dynamics As previous load before stiffness-change warming up is taken because typically this load causes plastic intervals in the coupling interface. Figure 9 shows the resulting longitudinal stresses at the movable support for all of these situations. The time-interval is taken to T = 1 sec. Obviously all curves tend to the static level. The sudden change results in oscillating stresses with initial peaks whereas the continuous smooth change generates a monotonic decreasing curve. It should be noted that the minimum stresses occur at the moveable support.
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124 Track-Bridge Interaction on High-Speed Railways
Table 4. Maximum longitudinal rail stress due temperature change followed by change of coupling stiffness. Load (1) (2) (3) (4)
Figure 10.
Tl Tu → Cstatic Tu → Cdyn,sudden Tu → Cdyn,continuous
σmin [N/mm2 ]
σmax [N/mm2 ]
−66.506 −46.611 −51.719 −46.620
58.624 46.956 51.715 46.961
System plot for double-track bridge system of Altmühltal bridge.
Table 5. Geometry, material properties and loading data for Altmühltal track-bridge system. Length of the bridge Cross section area of the bridge-girder Stiffness of the elastic support Modulus elasticity of rail Rigid track Cross section area of rail 2AUIC60 Stiffness of unloaded track Stiffness of loaded tack Critical elastic relative deformation u = ur − uB Ballasted track Cross section area of rail 2AS54 Stiffness of unloaded track Stiffness of loaded track Critical elastic relative deformation u = ur − uB Temperature changing Thermal expansion coefficient of the bridge Damping value
L = 79 m AB = 9.914467 m2 kA = 0.5 × 108 N/m ER = 21 × 1010 N/mm2 = ARr = 15372 mm2 cur = 6.0 × 107 N/m2 clr = 12.0 × 107 N/m2 u˜ r = 0.0005 m = ARb = 13896 mm2 cr = 1.0 × 107 N/m2 clb = 3.0 × 107 N/m2 u˜ b = 0.002 m t = 30◦ C α = 1.0 × 10−5 D = 0.01
Table 4 shows the maximum values of compression and tension in the rail. The notation T indicates the loading case ‘warming up’. The subscript l for T denotes that the results where calculated for a loaded track. The unloaded situation is indicated by Tu and only this situation is followed by the load-case ’change of coupling stiffness’ with c. 5.2
Double-track bridge
The next example is devoted to an existing bridge upon the river Altmühl on the high-speed line from Nürnberg to Ingolstadt in Germany as shown in Figure 10. This railway bridge is not typical because two different types of tracks run along the bridge; a rigid one and a ballasted track. Each track has it’s individual nonlinear stiffness law as shown in Figure 2. The system’s data are described in Table 5. As both tracks are coupled by means of the bridgedeck, loading of one track will affect the other one. Here, results are presented in Table 6 for one train passing the bridge; either on the rigid track or on the ballasted one. Consequently change of coupling stiffness occurs only on one track but influences the other track remaining unloaded.
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Longitudinal track-bridge interaction for load-sequences
125
Table 6. Maximum longitudinal stress of double track bridge due to change of coupling stiffness after temperature case. Ballasted track is loaded
Rigid track is loaded
Ballasted track
Rigid track
Ballasted track
Rigid track
Loading case
σmin N/mm2
σmax N/mm2
σmin N/mm2
σmax N/mm2
σmin N/mm2
σmax N/mm2
σmin N/mm2
σmax N/mm2
(1) (2) (3) (4)
−21.974 −31.448 −50.378 −31.473
29.077 45.125 58.918 45.139
−33.981 −34.061 −34.085 −34.023
41.685 41.666 41.707 41.683
−21.974 −22.011 −22.051 −22.013
29.077 29.064 29.077 29.077
−33.982 −36.350 −40.408 −36.355
41.685 48.292 52.693 48.295
Tu Tu → Cstatic Tu → Cdyn,sud Tu → Cdyn,cont
The first row dedicated to Tu shows the stresses due to ‘warming up’ without any load from passing trains. Therefore the stresses appear twice. The maximum compression occurs at the right moveable support whereas the maximum tension is located about 33 up to 38 m from the left-end of the bridge. Figures 11(a) and 11(b) show the longitudinal stresses in both tracks due to a train passing along the ballasted track. Thus change of coupling stiffness happens only there. The compression stress in the loaded track increases continuously (Fig. 11(a)) or oscillates (Fig. 11(b)). But finally, both curves tend to the same final stress-level which is more or less the same as for the pure statical analysis without including mass-accelerations. The coupling-effect onto the neighboring rigid track is neglectable as can be seen from 11(a) and 11(b). Similar results occur when the train passes the bridge along the rigid track as is shown in 11(c) and 11(d). 5.3
Repetitive loading
This example deals with the influence of multiple passing-cycles on the rail-stresses. Results are presented for the same double track bridge as treated in section 5.2. Here, ‘warming up’ is followed by a pair of changes of coupling stiffness; from unloaded to loaded and reverse from loaded to unloaded. This first cycle is followed by two more. Figure 12(a) shows the compression stress in that rail which is not passed by the train. Starting with temperature change at point 1 where the interface-coupling is in slip condition, point 2 shows the stress due to change of stiffness from unloaded to loaded and point 3 the stress from the way back from loaded to unloaded condition. Two more cycles, 3-4-5 and 5-6-7, take place in a common rather narrow stress-band with remaining stresses after each cycle not exceeding the stresses from ‘warming up’. Corresponding results for the loaded track due to train-passing are shown in Figure 12(b). Starting with temperature change (point 1) the stress becomes larger due to change of stiffness from unloaded to loaded (point 2). On the way back to unloaded interface the coupling element deforms into the opposite direction; thus the stress in point 3 becomes smaller. Further cycles, 3-4-5, 5-6-7 are restricted to a narrow band. After some cycles the stresses in the unloaded rail are lower than after ‘warming up’. 6
CONCLUSION
Modern railway tracks are characterized by continuously welded rails and by avoiding expansion devices at the ends of railway-bridges. Thus a careful treatment of the track-bridge interaction is of significant importance as is confirmed by European codes. There a nonlinear coupling stiffness in the track-bridge interface is recommended; however without regard to load-sequences or any deformation history.
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126 Track-Bridge Interaction on High-Speed Railways
Figure 11. Longitudinal stress of rail at the right-end of the bridge due change of coupling stiffness after temperature case with dynamic analysis.
Figure 12.
Stress of rail at the right-end of the bridge due to repetitive train-passings on rigid track.
In this paper it has been shown that regard to the history can reduce the maximum rail-stresses such, that expansion devices can be avoided. The importance of the loading case ‘change of coupling stiffness’ due to train-passing has been clarified by treating three different scenarios; two characterized by dynamics and the third one by statics. Results, achieved so fare, indicate similar rail-stresses: either found statically or dynamically with a continuous change-over of the coupling stiffness from unloaded-to-loaded situation or vice versa.
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These rail-stresses due to ‘change of coupling stiffness’ should not be neglected as is shown by means of one example in the paper. On the other hand it has been found, that multiple loadcycles from train-passing after temperature-increase in summer has happened typically proceed elastically in interface-regions where temperature caused a plastic behaviour.
ACKNOWLEDGEMENTS The cooperation on track-bridge interaction with Deutsche Bahn AG, represented by Dr. Lamine Bagayoko and Dipl.-Ing Gerd Schmälzlin is highly appreciated. REFERENCES EN9 (2003). Eurocode 1, Part 2 (EN 1991–2). Actions on Structures; Traffic loads on bridges. European Committee for Standardization (CEN), Brussels. DIN (2003). DIN-Fachbericht 101. Einwirkungen auf Brücken. Berlin. Esveld, C. (2001). Modern Railway Track. Zaltbommel: MRT Production. Fryba, L. (1996). Dynamics of railway bridges. London: Thomas Telford. Ruge, P. and Birk, C. (2006). Longitudinal forces in continuously welded rails on bridgedecks due to nonlinear track-bridge interaction. Computer & Structures, 85:458–475. Ruge, P., Birk, C., Muncke, M., and Schmälzlin, G. (2005a). Schienenlängskräfte auf Brücken bei nichtlinearer überlagerung der Lastfälle Temperatur, Tragwerksbiegung, Bremsen. Bautechnik, 82:818–825. Ruge, P., Schmälzlin, G., and Trinks, C. (2005b). Schienenlängskräfte auf Brücken infolge Biegung. Bautechnik, 82:69–80. Ruge, P., Trinks, C., Muncke, M., and Schmälzlin, G. (2004). Längskraftbeanspruchung von durchgehend geschweißten Schienen auf Brücken für Lastkombinationen. Bautechnik, 81:537–548.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 11 Structural analysis of high speed rail bridge substructures. Application to three Spanish case studies Juan A. Sobrino Pedelta & University of Catalonia, Barcelona, Spain
Juan Murcia University of Catalonia, Barcelona, Spain
ABSTRACT: This paper describes the track-bridge interaction phenomenon in bridges for high speed railways, proposing a methodology for the structural analysis of the substructure taking into account the following aspects: horizontal loads and imposed displacements due to the longterm behaviour of concrete and temperature using a step by step analysis and, if required, the construction process, behaviour of ballast using a non-linear links between the track and the superstructure and behaviour of POT bearings, interaction between foundation and ground conditions. The methodology fulfils the specifications of Eurocode 1 and consist of modelling the whole structure (track-bridge) with a 2D or 3D model using linear bar elements and non-linear springs and the analysis of the different short-time and long-time scenarios. To illustrate the methodology, three case studies of concrete viaducts of the HSR line Madrid-Barcelona-French Border are presented. The results are compared with those obtained from the simplified design methods.
1
INTRODUCTION
After the conclusion of the first Spanish High Speed Railway Line in 1992, connecting Madrid and Seville, the strategic railway infrastructure plan developed by the Spanish Ministry of Public Works as a part of the objective of the European Union (EU) of developing a Trans-European High-Speed Rail System, includes the construction of more than 4000 km of HSR in a period of fifteen years [1]. The directives for the rail system interoperability constitute an impelling element for the railway sector, as new lines, trains and equipment within the EU countries should be either built or renovated. As a result of the complexity of the Spanish geography, approximately a 10% of the railway system consists of bridges and tunnels. Construction of railway bridges for high speed lines represents a significant cost of the network. Due to the important magnitude of vertical and horizontal loads, the design of these bridges requires a judicious selection of the structural configurations and erection procedure. 2 2.1
DESIGN CRITERIA OF HSR BRIDGES IN SPAIN Design Codes
The HSR lines in Spain are developed by the Ministry of Public Works. The technical specifications required by the owner for the design of these bridges are as follows: • Loads should be according to the Spanish Code for Railway Bridges [2] [3] but a check is also required to fulfil the specifications of Eurocode 1.2 (Traffic Loads on Bridges) and the Annex 2 of Eurocode 1, specifying additional Serviceability Limit States [4] [5]. 129 © 2009 Taylor & Francis Group, London, UK
130 Track-Bridge Interaction on High-Speed Railways
• Design of structural elements should be carried out according to the Spanish Codes for concrete structures or the recommendations for the design of composite and steel road bridges. 2.2
Specific relevant aspects for the design of HSR Bridges
Internal forces due to railway traffic loads are 2 to 2.5 times larger than those induced by road traffic loads. Dead loads of two ballasted tracks, including all bridge finishes, weights 120 kN/m and the effect of ballast should be incremented for the design about 30% to take into account possible increments during the life of the bridge. Horizontal loads originated by railway traffic (braking and traction, nosing force, wind, trackstructure interaction and centrifugal forces) are also much bigger than similar effects in road bridges. As an example, maximum braking and traction force in a HSR standard 300 m viaduct is 7000 kN. The equivalent force in a similar road bridge is 850 kN. Loads induced by centrifugal forces in railway bridges could also range from 300% to 1500% of the ones caused by the same force in road bridges. Apart from the heavy loads considered for the design of HSR bridges, there are some specific Serviceability Limit States (SLS) to be verified in this type of bridges that could be summarized as follows: • Verification of vibrations for traffic safety, limiting the maximum vertical peak deck acceleration induced by real trains (for instance, the recommended value for a ballasted track is 3.5 m/s2 ). • Verification of deck twist and vertical deformations of the deck for traffic safety. • Verification of the maximum vertical deflection for passenger comfort, depending on the train speed and span length. • Verification of track, limiting rail stresses due to combined response of the structure and track to variable actions, limiting the longitudinal displacements induced by traction and braking (for instance to only 5 mm for welded rails without rail expansion devices or with only one expansion device at one end of the deck), etc. Due to these significant loads and the very strict Serviceability Limit States to be fulfilled, structural elements are much stiffer than in road bridges and for this reason, the optimization of materials and, in particular, the selection of a judicious structural system and the deck’s slenderness is basic to obtain economical solutions. 3
TRACK-DECK INTERACTION
The design basis of the track-bridge interaction phenomenon is established in the Eurocode 1 [4] [5] and in the UIC 774-3 leaflet [6]. The interaction is taken into account in the numerical model by means of non linear springs that reproduce the horizontal interaction of the system rail + fastenings + sleepers and the deck (Fig. 1). Non linearity is modelled by a bilinear horizontal force per unit rail length – relative displacement law, as shown in Figure 2. In the structural analysis, the existence of rail expansion devices, the type and the loading and maintenance state of track (affecting the value of the k parameter of bilinear law) and the substructure stiffness have also to be considered. It is necessary to include the substructure in the track-bridge interaction model as long as it can affect significantly the global behaviour of the bridge [7]. The situations in which the interaction appears are those where a relative horizontal displacement between the track and the deck occurs. The UIC 774-3 leaflet specifies the main actions to be taken into account: the changes in temperature in the rail and in the deck, the horizontal forces due to braking and acceleration and the bending of the deck caused by vertical traffic loads (bending generates horizontal movements at the end of the deck due to its rotation). The values of the above mentioned actions are defined in the EC-1 [4] [5] and in the UIC 774-3. The linear combination of the results obtained from the independent calculation of each action (temperature, braking/acceleration, bending) is not valid since the problem is not linear. A rigorous study demands to carry out a stepby-step non linear analysis (called complete calculation in the UIC 774-3), including all the actions
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Structural analysis of high speed rail bridge substructures
Figure 1.
131
Elements to be considered in the analysis of track-bridge interaction.
Longitudinal stiffness of the ballasted bed F (KN/ml) k=0
Loaded track
60
k=60 KN/m2 40
k=0 20
k=20 KN/m
Unloaded track
2
0 0
Figure 2.
0,002
u (m)
Longitudinal resistance of the track respect to longitudinal displacement.
and the loading conditions of the track (affecting the non-linear springs representing the effect of the track-deck interaction). However, the UIC 774-3 admits the linear combination of the results to check the additional stresses in the rail. For some specific cases, a simplified methodology using diagrams and tables is provided to obtain the stresses in the rail and the reactions at the supports. On the other hand, the deformation of the concrete deck due to creep and shrinkage can be as important as for the temperature changes and should also be considered [8]. In [9] a method to combine this action with the rest of actions is proposed in order to obtain the stress envelope for the rail. The temperature changes and the creep and shrinkage are considered together in the same load case, as an equivalent temperature change of the deck. The worst situation for the rail would happen in one of these cases: a) maximum increase of temperature in the early ages (without creep and shrinkage deformation), and b) maximum concomitant decrease of temperature and maximum creep and shrinkage deformation. The braking/acceleration and bending results would be added afterwards.
4
METHODOLOGY FOR THE STRUCTURAL ANALYSIS OF HSR BRIDGES
4.1 Analysis model The analysis model used to obtain the longitudinal forces distribution among the viaduct includes the track, the deck, the substructure and their interaction non-linear effects (fastenings/ballasted
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132 Track-Bridge Interaction on High-Speed Railways
Figure 3.
Force-displacement cycle for a sliding POT bearing.
bed and bearings). The rails and the deck are modelled with beam elements in their centre of gravity. The supports are modelled in the same way, taking also into account the loss of stiffness due to the long-term shrinkage. Geometric and material non linearities have not been considered; which would turn out to be suitable for very high supports. The foundations are represented as equivalent stiffness for horizontal movement and rotation in the support base. The track system (rail + fastenings + sleppers + ballast) and deck interaction is represented by non-linear springs with a bilinear law according to Figure 2. POT bearings, fixed, guided or sliding, are the most commonly used type of bearings in the Spanish high-speed railway bridges. They have two different friction coefficients (static and dynamic) should be considered. This friction coefficient µ varies from 0.5% to 5%. The behaviour of the POT for horizontal movements can be represented by a spring with a Coulomb’s friction law as shown in Figure 3. The model of the whole set in 2 dimensions is shown in Figure 4. In case there is no rail expansion device in the end of the viaduct, it is necessary to include the adjacent platform (300 m behind the abutment according to the UIC 774-3 leaflet [6]).
4.2 Analysis type The type of analysis that is going to be carried out is the so called complete calculation [7] [8]. It consists in a step-by-step analysis for each different action and static arrangement.
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Structural analysis of high speed rail bridge substructures
Figure 4.
4.3
133
Continuous-deck bridge model.
Step-by-step load combinations
The steps considered in the analysis are listed below: 1. Construction of the structure (under self-weight -D1- and prestressing if existing). 2. Phase between the conclusion of the bridge and the welding of the rails, taking into account the short-term creep and shrinkage deformation of the deck (evaluated after 3 months). 3. Introduction of the imposed dead load (D2) of the superstructure (in steps 1 and 2 the loads act in the deck and the substructure. Only after welding the rails the track-bridge interaction starts to act. Rails and the non-linear springs representing the track-deck interaction have to be included. 4. Final creep and shrinkage deformation (considering two cases: just after welding the rails and, eventually, the long-term situation). 5. Change of temperature (positive or negative) of the deck. Change of temperature of the rails is only considered when an expansion device exists at the end of the viaduct. 6. Vertical and horizontal (braking/acceleration) loads due to traffic according to EC-1. 7. Beginning/final of the process. 8. Steps 1, 2 and 3 are always considered in the same way. However, in each one of the steps 4, 5 and 6 a pair of hypotheses exist with respect to the operating action. It means having 8 different load combinations for the substructure, if the rails are continuous. If an expansion device is placed near the bridge it would also be necessary to consider the change of temperature of the track (step 5), which represents16 different load combinations. The results will enable to obtain the force envelopes of the supports. In the next section, this methodology is used to analyse three Spanish high-speed railway viaducts. The results are compared with the obtained by means of simplified calculation procedures. 5
CASE STUDIES
The proposed methodology has been applied in three viaducts of the HSR Madrid-Barcelona, with commonly used static arrangements. As a first estimation, longitudinal forces can be evaluated to be equal to the friction force generated when the bearing slides under vertical permanent loads: µND1 +D2 . On the other hand, at the fixed point (in general placed in one of the abutments) the longitudinal force is evaluated as the sum of the external horizontal forces due to braking, acceleration, etc. (Fext ) and the compensation of the forces of friction generated in the rest of the supports ( µND1 +D2 ). Therefore the bearing are
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134 Track-Bridge Interaction on High-Speed Railways
usually expected to slide independently from the support stiffness. It is also assumed that moving supports are unable to generate additional friction under vertical traffic load. These assumptions omit the real effect of track-bridge interaction and substructure stiffness. In the following lines, the results obtained with the methodology proposed in this paper are compared to these reference values. 5.1
Selles Viaduct
Selles Viaduct is a continuous post-tensioned concrete bridge with an overall length of 167 m (Fig. 5). The decks has 5 spans of 28 + 3 × 37 + 28 m. The typical cross-section (Fig. 6) is a box girder 3.3 m depth (L/11.2). It has a fixed support in one of the abutments and longitudinally sliding (guided or fiexed in the transverse direction) POT bearings at the rest of the supports. The pier height varies from 10 to 12 m (Fig. 7). A numerical model, following the criteria described in paragraph 4, has been carried out. According to the obtained results (Table 1), the piers take a significant part of the external longitudinal forces (braking/acceleration). It is due to the fact that the bearing slides under traffic loading, reaching the value µND1 +D2 +q . However, achieving this maximum depends on the stiffness of the
Figure 5.
View of the bridge.
Figure 6.
Typical cross-section of Selles Viaduct.
Figure 7.
Elevation of Selles viaduct.
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Structural analysis of high speed rail bridge substructures
135
substructure. For high pier stiffness and/or low sliding friction coefficient, sliding is needed to make movements compatible (case µ = 0.01). For very slender piers or high sliding friction coefficient (case µ = 0.05) , the compatibility can be achieved by column deformation without bearing sliding. Therefore the value is below µND1 +D2 +q as happens for the second scenario. The force at the fixed point is lower than expected (Table 2) because piers take braking and acceleration forces and also a part of these external forces are transferred to the embankments through the rails by means of the track-bridge interaction. On the other hand, the continuity of the rails can introduce over-stresses in the bridge system as bridge deformations are constrained. 5.2 Avernó Viaduct This viaduct consists on 14 spans having a total length of 810 m (maximum span length 60 m) (Fig. 8). The distribution of spans is 45 + 12 × 60 + 45 m. Having small curvature it has been nevertheless analyzed as a straight bridge for our purpose. The deck is a post-tensioned concrete box girder (Fig. 9) with constant depth of 4 m (H = L/15). The fixed support is in one of the abutments. Some piers reach a maximum height of 40 m (Fig. 10). Similar conclusions can be drawn from the analysis of the Avernó viaduct (Tables 1 and 2). Piers can take a significant part of the horizontal traffic load. Even though the piers are higher than in
Table 1. Longitudinal force at the top of piers 1 and 4 in Selles River and Averno viaducts respectively. Viaduct
µ
%µND1 +D2
% µND1 +D2 +q
Selles River Avernó
0,05 0,01 0,05 0,01
125% 148% 125% 148%
85% 100% 84% 100%
Table 2. Longitudinal force in top of fixed abutment in Selles River and Averno viaducts. Viaduct
µ
%Fext + µND1 +D2
%Fext + µND1 +D2 +q
% FH UIC simplified
Río Selles Avernó
0,05 0,01 0,05 0,01
67% 108% 59% 53%
55% 102% 53% 50%
59% 98% – –
Figure 8.
View of Avernó bridge.
© 2009 Taylor & Francis Group, London, UK
Figure 9.
Typical cross-section of Selles viaduct.
136 Track-Bridge Interaction on High-Speed Railways
Figure 10.
Elevation of Avernó viaduct.
Figure 11.
View of Anguera bridge.
Figure 12. viaduct.
Typical cross-section of Anguera
Selles bridge, they are not slender enough to prevent sliding of the bearings for low sliding friction coefficient. The force in the abutment is lower than the maximum friction admissible load due to the capacity of the rest of the supports to take some horizontal load. 5.3 Anguera Viaduct The long viaduct over Anguera River consists of 28 simply supported spans of 34 m (Figs 11 and 12). The deck is formed by pre-cast prestressed concrete U girders with 2.45 m depth and a 30 cm in situ slab. Piers do not exceed 10 m of height, being very stiff. Every span has a fixed support and the movable one in each side, therefore each pier has a fixed bearing and a sliding one. Longitudinal forces for this structural arrangement are usually evaluated similarly as for continuous viaducts, but considering each span independent from the other. This criterion obviously doesn’t consider accurately the track-bridge interaction. On the other hand, for similar span lengths, the friction forces generated in the sliding bearings of contiguous spans are equivalent and are compensated at the pier. According to the analysis the simplified criteria (par. 5) is clearly conservative since it does not consider the redistribution of forces through the continuity of the rails (Tables 3 and 4). Moreover, this continuity of the track allows to transferring part of these forces to the adjacent platform.
6
CONCLUSIONS
Track-bridge interaction has a main role in the global behaviour of the HSR bridges in the distribution of the longitudinal forces and therefore has to be taken into account in the structural analysis of the substructure. A methodology to obtain longitudinal forces in supports has been presented in this paper. It has been applied to three Spanish HSR viaducts and compared with the commonly
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137
Table 3. Longitudinal force in top of column no 7 in Anguera viaduct. µ
%FH simplified
0,05 0,01
44% 50%
Table 4. Longitudinal force in top of fixed abutment in Anguera viaduct. µ 0,05 0,01
% FH , simplified UIC 71% 58%
simplified calculation procedures. Differences are shown between both methods. From the comparison it is shown the relevance of considering track-bridge interaction, sliding support behaviour and support stiffness in the structural analysis. Further developments should include geometric nonlinearity of high supports and variable ballast stiffness and sliding friction coefficient.
REFERENCES [1] [2] [3] [4] [5] [6] [7]
“Strategic infrastructures and transport plan”, Ministry of public works, 2005. “Code IAP – Actions on railway bridges”, Ministry of public works, 1972 (in Spanish). “Code RPX – Composite bridge Code”, Ministry of public works, 1995 (in Spanish). EN 1991-2 “Eurocode 1: Actions on structures. Part 2 Traffic loads on bridges”. EN 1990 PrAnnex A2 “Eurocode: Basis of design. Annex 2: Application for bridges”, 2002. UIC, Leaflet-774-3 Track/bridge interaction Reccomendations for calculations, 2nd edition. 2003. Manterola, J.; Astiz, M.A.; Martínez, A. Puentes de Ferrocarril de Alta Velocidad, Revista de Obras Públicas No 3386, abril 2000 (in Spanish). [8] González Requejo, P; et al. Alta velocidad: El fenómeno de interacción vía- tablero en puentes. Revista de Obras Públicas No 3418., Febrero 2002 (in Spanish). [9] Cuadrado Sanguino, M.; González Requejo, P. Consideración de las deformaciones por retracción y fluencia en el estudio del fenómeno de interacción vía-tablero en el proyecto de puentes ferroviarios. Revista de Obras Públicas, No 3446, Julio-Agosto 2004 (in Spanish).
© 2009 Taylor & Francis Group, London, UK
CHAPTER 12 The Italian experience: two case studies M.P. Petrangeli University “La Sapienza”, Department of Civil Engineering, Rome, Italy
ABSTRACT: The construction of the first Italian HS Railways, linking Rome to Florence, was completed at the end of the eighty when the bridge crossing the Arno river was opened to the trains running at 250 km/h. This p.c. bridge has a 230 m continuous deck rigidly connected to the piers, an expansion length of 115 m so resulting for the thermal effects. In spite of the absence of joints in the ballasted track, its behaviour in about 20 years of service has been excellent. The cable stayed bridge over the PO river is the second example illustrated in the paper. It has been completed in 2006 and will be operated at the speed of 300 km/h. The 192 m main span and the two 104 m side spans made Rail Expansion Devices necessary at both the ends of the bridge: it will be only exception on the whole Italian HS Railway network.
1
INTRODUCTION
The construction of the Italian HS Railway network begun with the 236 km line linking Rome to Florence at a maximum speed of 250 km/h, first in Europe at the time. Its construction started in 1970 and 193 km were put in service since 1977 but it was completed only on 1989 because of the limited budget. After a stop lasted more than ten years, new lines have been built and the final situation is illustrated in Figure 1. All the 1400 km Italian HS Railway network, once completed, will have the Continuous Welded Rails (CWR) with the only exception of the cable stayed bridge over the PO river; for this reason simply supported spans are utilized in almost all the viaducts of the new lines. Some trough arches and continuous girders are the few exceptions, all respecting the expansion lengths suggested in the annexe G of Eurocode 1 (EN 1991-2), i.e. 90 m for concrete and composite decks and 60 m for steel structures. The only bridge not respecting these limits is the one crossing the Arno river, designed before the mentioned EC was issued, and therefore strictly monitored during its life.
2 2.1
THE BRIDGE OVER THE ARNO RIVER Short description of the structure
The bridge crosses the Arno (Fig. 2) in a section where the river is about 160 m wide because of a dam located not far from the crossing; the height of the central piers is 55 m above the water level. It is a prestressed concrete long viaducts with 43 simply supported 35 m spans and the central part constituted by a four spans (45-70-70-45 m) frame, the total length between the structural discontinuities so resulting 230 m (Petrangeli 1991). 139 © 2009 Taylor & Francis Group, London, UK
140 Track-Bridge Interaction on High-Speed Railways
Figure 1.
Italian HS Railway network.
Figure 2.
View of the completed bridge.
The Italian Railway (FS) asked to have the major spans deck composed by two separate box girders, one for each track. This choice was justified by security reasons and by the will of an easy maintenance. The three central piers are rigidly connected to both the girders that are, on the contrary, simply supported over the two piers of transition to the approach viaducts (Fig. 3). The three central piers reach the bedrock through very large caissons (22 m diameter) so they can reasonably be considered fixed at the base. The bridge is located in a seismic area with a PGA of 0,25 g and has been designed, on request by FS, to stand up also if a span should collapse because of an extreme event.
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The Italian experience: two case studies
Figure 3.
2.2
141
Longitudinal and transversal sections of the bridge.
Combined track/structure system
The thermal fixed point can be located at the central pier because of the symmetry of the structure. Since both the simply supported spans of the approach viaducts have the fixed bearings facing the main spans, the expansion length results to be 115 m (Fig. 3). The stiffness of substructure per track per m of deck, having supposed the piers fixed at the base because of the very stiff foundations adopted, is K = 2,4 E 3 kN/m
(1)
and the horizontal displacement of the upper deck edge due to end rotation is less than 1 mm. According to Eurocode 1 and assuming α = 10E-6
T = 35◦ C
k20 /k60 = 20/60 kN/m
(2)
the maximum permissible expansion Length should be, in the case of a single bridge deck: LTP = ∼90 m < 115 m
(3)
In spite of this discrepancy no disorder has been surveyed in the ballasted track in about 20 years of service. The interaction between track and structure was analyzed in the case of seismic actions too. Non linear dynamic analysis were carried out both ignoring the presence of the rails (the ballasted track is taken into account only as dead load) or considering the combined track/structure system. The EL CENTRO NS accelerogram scaled to 0,25 g was assumed as input for the calculation. The working presence of the track leads to a significant reduction of the bending moments in the piers, up to 40%, as well as in the movement of the deck joint: +/−10 cm instead of +/−15 cm (Figs 4a, 4b). Of course large stresses in the rails due to this action arise. In this example they reach a maximum value of 150 N/mm2 and therefore the capability of the rails to absorb them depends on the stress level due to the temperature effects at the moment of the earthquake.
3 3.1
THE CABLE STAYED BRIDGE OVER THE PO RIVER Short description of the structure
The new railway linking Bologna to Milan crosses the Po near Piacenza in a section where the river is usually about 350 m wide, up to 1 km between the main embankments. The bridge is 1200 m long, 400 m devoted to cross the ordinary riverbed, an obliquity of 22◦ resulting between the tracks
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142 Track-Bridge Interaction on High-Speed Railways
Figure 4a. Bending moment diagrams under seismic actions measured at central pier footing. Left diagram: bridge + tracks; right diagram: bridge only.
Figure 4b. Bending moment diagrams under seismic actions measured at central pier top. Left diagram: bridge + tracks; right diagram: bridge only.
and the river (Petrangeli 2006). Two approach viaducts, respectively 6 and 4 km long, complete this work, the most important of the whole line. Three types of structure are present in the crossing in addition to the standard approach viaducts already mentioned: (i) the cable stayed bridge, (ii) 12 simply supported decks on the right bank and (iii) two continuous p.c. box girders necessary to overpass the main embankments (Fig. 5). The decks are subdivided in such a way that Rail Expansion Devices (RED) are necessary to keep the expansion length within the allowable limits. This will be the only exception along the HS Railway Italian network, all equipped with CWR. The relevant part of the crossing has a 192 m central span and two 104 m long side spans. The deck is a p.c. continuous box girder with the fixed point at one tower, sliding bearings at the second tower and at the transition piers. Expansion lengths of 296 and 104 m derived from this arrangement of the bearings, joints in the rails so being required. The height of the cross section is constant and equal to 4,5 m (L/42,7) along the central span; it varies and decreases to 3,70 m in the side spans, in order to fit the height of the other decks. The towers are 60 m high from the footing, 51 m from the deck. The top of the towers, where the stays are anchored, is a steel-concrete composite structure. The stays are made of 55 to 91 zinc-coated, singularly greased and sheathed 0,6 super strands (Della Vedova et al. 2006). The foundation of each tower has the footing (shaped to reduce the drag force) supported by 28 piles, 2 m diameter and 65 m long. Dynamic analysis: three different trains (ETR 500, TGV, ICE) have been considered for the dynamic analysis considering the dynamic behavior of the vehicle as well as the irregularities of the track.
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The Italian experience: two case studies
Figure 5.
143
General view of the whole bridge.
Seismic analysis have been carried out in the elastic range according to the Italian Railway Specifications and the European Code 8. Because of the low seismicity, these actions did not design the bridge but few sections in the upper part of the towers, while they were relevant for the bearings and the joints.
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144 Track-Bridge Interaction on High-Speed Railways
Figure 6.
Construction of the main deck.
All the decks have been built by cantilever method (Fig. 6) with cast in situ segments, with the exception of the 13 simply supported spans made by precast beams. A number of physical tests have been executed to assess the theoretical assumption, the most outstanding being: (i) test on a half scale model reproducing a segment of the deck with a stay anchorage, carried on in the yard; (ii) fatigue test on a full scale model of the steel box embedded in the upper part of the tower to anchor the stays and (iii) fatigue tests on three stays (composed by 55, 73 and 91 0,6” strands) complete of anchorage. Both fatigue tests were carried out in the EC Joint Research Centre of ISPRA (Petrangeli & Polastri 2004, Petrangeli 2006). Due to the importance of the bridge, a large number of sensors have been permanently placed on it. The monitored quantities are: loads on the piles, stress and temperature in the most representative sections of the deck and the towers as well, forces transmitted by a number of stays and bearings, geometrical data like the angular rotation of towers and the deflection of the decks and, finally, the scour near the piers in the riverbed. All the data will be collected inside the cable-stayed deck and from there automatically transmitted to a remote office located in Bologna that will manage all the bridges of the line.
3.2
Combined track/structure system
Two RED at both ends of the cable stayed bridge have been initially designed. The actions due to temperature have been assumed as follows: – Temperature variation of the deck +/− 15◦ C – Temperature variation of the rail +30◦ C/−40◦ C – Maximum difference in temperature between rail and deck: the two previous actions must be considered separately in case of concrete bridge, says the Italian Code. It means the maximum difference in temperature between rail and deck coincide with the absolute temperature variation of the rail (+30◦ C/−40◦ C). Loads Model 71 and SW2 multiplied by α = 1,1 as requested by the Italian National Code (Ferrovie dello Stato 1995) have been considered while breaking and traction forces have been assumed as per EC 1-2.
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The Italian experience: two case studies
Figure 7.
145
Sap2000 model, detail of the deck-tower junction.
The LM 71 has been supposed to run both from Milan to Bologna and vice-versa (SW2 running in the opposite direction) because of the lack of symmetry of the bridge. Also the relative position of the two trains has been chosen as to maximize the studied effect. The plastic shear resistances of the track for the unloaded/loaded conditions have been assumed respectively 20/60 KN/m. The pile foundations have been considered as elastic springs with two limit deformability: – zero, in calculating the maximum force transmitted to the fixed bearing; – corresponding to the maximum scour allowed in service (the piles have been supposed to be free for 8 m depth) and to the lower bound of the soil characteristic, in calculating the maximum displacements. The detail of the mesh at the junction between the deck and the towers is shown in Figure 7. The 1D elements representing the rail were made 4,5 m long, instead of the maximum value of 2,0 m imposed by FS, to fit the length of the deck segments (the bridge was built by cantilever method). A sensitivity analysis showed the variations were negligible. Figure 8 shows the most meaningful results. Both the maximum track-structure relative displacements (3,5 mm < 5 mm) and the maximum stress in the rail (109 N/mm2 in compression and 104 N/mm2 in tension) satisfy the imposed limits.
3.3
Rail expansion devices
Large movements must be allowed under the seismic actions, up to +/−235 mm according to the FS Specifications (Ferrovie dello Stato 1996). A RED allowing these displacements resulted, at construction stage, to be complicated and difficult to maintain; each one was therefore split into two minor RED and the final disposition was as illustrated in Figure 9. With this configuration all the quantities resulted to be smaller than those calculated in service with only two REDs as previously illustrated. Relative displacements and stress in the rails are therefore far below the allowable limits. All the equipment used has been supplied by the General Contractor CEPAV 1 headed by ENJ.
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146 Track-Bridge Interaction on High-Speed Railways
Figure 8. Sap2000 model and results of structure analysis 1) Longitudinal force on fixed bearing; 2) compression tension on track; 3) traction tension on track; 4) track-structure relative displacement.
Figure 9.
4
RED final disposition.
CONCLUSIONS
Some considerations can be derived from the two case studies illustrated. The 20 years oldArno bridge, also if operated at a speed now considered not very high (250 km/h), has the CWR with 115 m expansion length and no problems have been registered in these years. It means that with very stiff foundations and a framed structures probably the usual limits can be slightly exceeded.
© 2009 Taylor & Francis Group, London, UK
The Italian experience: two case studies
Figure 10.
147
View of the Po bridge nearing its completion.
The cable stayed PO bridge put in evidence a different problem to solve in seismic regions, that is the performance requested to the RED during a strong earthquake. No damages were accepted in this case for the PO bridge, very expensive devices so being necessary. This request should be deeply discussed since the seismic actions, in other bridges of the same line equipped with the CWR, if unfavourably combined with the thermal actions, can bring the rail to collapse. REFERENCES Della Vedova M., Evangelista L., Petrangeli M.P., Prevedini C. 2006. “The Stays of the Cable-stayed Bridge over Po River: Design, Testing and Technological Choices”. ID 14-24 (Vol. 2 – pagg 434–435) – Session 14 Reinforcing and prestressing materials and systems – 2◦ FIB Congress – Naples, 5–8 June 2006 EN1991-2: Actions on structures- Part 2: Traffic loads on bridges EN 1991-2 Ferrovie dello Stato 1995. Sovraccarichi per il calcolo dei ponti ferroviari – Istruzioni per la progettazione, esecuzione e il collaudo Istruzione n.1/SC/PS-OM/2298 del 2.06.95 agg.13.01.97 Ferrovie dello Stato 1996. Istruzioni tecniche per manufatti sotto binario da costruire in zona sismica. Istruzione 44B del 14.11.96 Petrangeli, M.P. 1991. A railway bridge over the Arno river near Arezzo for the Rome-Florence high speed line, Industria Italiana del Cemento N. 657 Luglio-Agosto 1991 Petrangeli, M.P. & Polastri, A. 2004 Prove a fatica su un concio di torre del nuovo ponte strallato sul Po Costruzioni Metalliche N. 4 Luglio-Agosto 2004 Petrangeli, M.P. 2006. Cable Stayed Bridge over river Po. Special issue on Italian concrete structures for the Second fib Congress – Naples, June 5–8 2006 of Industria Italiana del Cemento 820
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CHAPTER 13 Rail expansion joints – the underestimated track work material? J. Hess BWG GmbH & Co.KG, Germany
ABSTRACT: Rail Expansion Joints REJ are some time underestimated during the planning and design phase of bridge projects. For operators and maintainers they are than later on the root cause of operating and maintenance troubles. As track work components do, they have to carry and guide wheels. In addition, REJ compensate the length variations of rails and have to allow compensation of longitudinal movements of rails caused by movements of engineering structures such as bridges and viaducts. However in reality this means in details the compensation of the length variations of rails, bridging of the gap of the structures, accommodation of vertical and horizontal rotations of ends of structures and adjustment of the different track resilience at both sides of the RMJ. In some case studies and outlooks it will be shown how REJ have been incorporated into engineering structures built in the past or in the planning stage.
1
GENERAL
The volume of substances changes according to the ambient temperature. In physics, this behavior is characterized by the coefficient of linear expansion. The coefficient of linear expansion is different for every substance. One way of dealing with this phenomenon is demonstrated in the formerly practiced track installation with rail joint gaps. These gaps between the rails provided them with sufficient space to expand or contract, respectively, depending on the temperature. In modern types of permanent way, rails are continuously welded. The longitudinal changes associated with temperature variation have to be adjusted by the rail itself as well as the rail fastenings, including sleepers and ballast, via means of tensile and compressive strength. This can only be achieved successfully if the permanent way and the foundation show approximately the same expansion behavior. However, there are areas such as bridge constructions where the different lengthwise movements between bridge construction and permanent way would result in the bridge being damaged by the permanent way. In this case, the lengthwise movement of the permanent way has to be separated from the bridge construction by the respective devices. This is the field of application of the rail expansion joint. Rail expansion joints (REJ) are devices which permits longitudinal relative movement of two adjacent rails, while maintaining correct guidance and support. These longitudinal movements may be required in interrupted CWR (continuously welded rails), bridge movement or a combination of both. The maximum admitted movement of two tracks against each other is called “Expansion capacity C”. It is evaluated in dependence of the geometrical frame conditions, as can be seen in Figure 1 with: (1) C = Dmax − Dmin The mean position is the position where the expansion capacity and the relative displacement of rails are half way, and the bearers are in nominal position. The actual position shall be checked by reference points. 149 © 2009 Taylor & Francis Group, London, UK
150 Track-Bridge Interaction on High-Speed Railways
Figure 1.
Minimum, maximum and mean position of rail expansion joints (REJ).
The mean position is used for design and acceptance. The reference points are also used for installation. This longitudinal movements influenced by – Temperature – Creeping and shrinking of the concrete – Braking and acceleration are calculated during planning of the bearing structure. Dependent on the guidelines for the maximum construction gap, the requirements of the bridge mounts and the maximum admitted distances between the interpolation points the kind of REJ for each construction and its position in the construction has to be fixed. From the various possible constructions of REJ, for high-speed traffic only constructions without interruption of the running edge are used, as for example on high-speed tracks in Europe (Germany, Spain, Netherlands) or Asia (Taiwan). Basically, two modes of action can be differentiated: • REJ with fixed stock rails and movable switch rails (Fig. 2) • REJ with fixed switch rails and movable stock rails (Fig. 3) In rail expansion joints with movable switch rails there is an inevitable excess width of the track gauge in the area of the wheel transition zone due to the relative movement of the switch rail. This wide to gauge is a point of unsteadiness and leads especially in high-speed traffic to a discontinuous run of the vehicles and thus to a reduced comfort and an enhanced abrasion. In rail expansion joints with fixed switch rails and longitudinal moving stock rails there is no widening of the gauge.
© 2009 Taylor & Francis Group, London, UK
Rail expansion joints – The underestimated track work material?
Figure 2.
REJ with fixed stock rails and movable switch rails.
Figure 3.
REJ with fixed switch rails and movable stock rails.
2
151
DESIGN AND FUNCTION OF RAIL EXPANSION JOINTS – REJ
The design of rail expansion joints resembles those of switches. Therefore, their large components also consist of switch and stock rails. Two different construction principles are employed in order to compensate for linear expansion. In case of the conventional construction principle the stock rail is mounted firmly in longitudinal direction while the switch rail in closed position to the stock rail is longitudinally movable. Since the stock rail features a circular geometry behind the switch rail, the track gauge at the point of the switch rail changes with every displacement of the mobile switch. Thus, the lengthwise movement of the switch rail along the stock rail changes the track gauge. This causes railways to be uncomfortable and leads to increased wear. In order to solve this problem, BWG (member of the VAE Group) developed a new generation of rail expansion joints using a new design principle that transfers the lengthwise movement of the
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152 Track-Bridge Interaction on High-Speed Railways
switch rail to the stock rail. In this case, the horizontal geometry of the stock rail corresponds to an elastic bending line that ensures a constant track gauge in any expansion position and results in reduced wear and higher traveling comfort. In order to provide the longitudinally fixed switch with some freedom of movement (approx. ±30 mm), the switch and stock rails are aligned vertically in a defined system via a special rail tie. In this connection, the stock rail is subject to a lower restraining force than the switch rail. Additionally, the restrain force generated at the switch rail and the stock rails differ due to different friction coefficients of the synthetic rail pads. 2.1
Design principle
Rail expansion joints are used in bridge constructions in order to compensate the different changes in length between the track system and the bridge, resulting from brake forces, temperature forces as well as creeping and shrinkage of concrete. The following rail expansion joint systems can here be used, dependent on the length of the expansion joints and the track system (slab track or ballasted track): • SAV-B 300: expansion length up to ±150 mm • SAV-B 600: expansion length up to ±300 mm • SAV-B 1200: expansion length up to ±600 mm The application, the choice as well as the details of completion are accordant to the specific conditions of the construction. Also the conditions regarding the rotation angle and the offsets on the track system end have to be taken into account. With the rail expansion joints the longitudinal forces resulting from braking forces, temperature forces as well as creeping and shrinking of the concrete, are uncoupled from the track. In the expansion joint the rails are separated in a way to provide a flatly inclined rail joint without interruption of the running edge. The rail joint consists of a longitudinal movable stock rail from the rail profile UIC 60 E1 A1. Figure 4 shows the composition and the way of function in a scheme. On the junction bridge deck/abutment, the switch part as a “fixed part” can be located above the abutment resp. behind it as well as on the bridge deck, on the junction bridge deck/bridge deck (above piers) the fixed part (switch rail) is located on the comparatively inflexible part of the bridge or on the side of the fixed bearing respectively. With the rail fastening system, the switch rail is braced more tightly than the stock rail. Thus, the longitudinal forces affecting the section of tracks are adjusted. The movable stock rail is pushed in the shape of an arch behind the mostly immobile switch rail, thus the track gauge stays accurate up to the pointed area of the switch rail.
Figure 4.
Schematical illustration of the way of function.
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153
The rail head of the switch rail shows geometry like the arching of a stock rail under its permanent weight. The horizontal (elastical) bent stock rail produces a homogenous knife edge load on the back of the switch rail, which prevents a diverge between switch rail and stock rail. To achieve the required fixpoint switch rail and stock rail are fixed vertically by the rail fastening system, whereas the stock rail is fixed with less power. Additionally, different restrain of the switch rail and stock rails are created by inserting synthetic pads with different friction coefficients. To bypass the gap different constructions are used in dependence of the length of the expansion joint, which are adapted to the moment of inertia, the admitted bending and the railbase stress of the rail track in a way, that the admitted values and the admitted distance of the bases of max. 0.65 m can be maintained: • up to 300 mm (±150 mm): • up to 600 mm (±300 mm):
Standard rails or filled section rails at bridge joint. Use of a steel sleeper, which is fixed on the side carrier enabling a longitudinal movement. It is fixed with a crossbar control system in the movement groove. • up to 1200 mm (±600 mm): Use of two steel sleepers fixed on the side carriers enabling a longitudinal movement. A crossbar control mechanism keeps the steel sleepers in a homogenous distance in the movement groove. The single components of the expansion joints are shown in the lay-outs and the sectional drawings. This construction makes a homogenous bedding and elasticity of the track system possible to a large extent. The area of the bridge expansion joint differs in dependence of the expansion length. The movement of the bridge produces a differing sleeper distance at the sleepers of the bridge expansion joints. It has to be made sure, that the maximum railbase stress and arching is still not exceeded. For a expansion length up to ±150 mm (SAV 300), the expansion joint will be bridged with a larger moment of inertia, dependent on the profile of the standard rail profile (Vignol) or the profile of the filled section rails. If the expansion joints are larger, e.g. ±300 mm (SAV 600) or ±600 mm (SAV 1200), supporting carriers are used above the joint. They consist of two steel profiles, which are located laterally of the driving rail on the sleepers. The carriers are fixed immovably with the sleepers on the fixed side of the joint and can balance the longitudinal movements of the bridge on the movable side of the joint. Lengthwise movable steel sleepers are on the carriers as support of the driving rail, depending on the maximum width of the bridge joint. They are centered in the bridge joint with an arm- or crossbar control mechanism installed ahead. Elastic ribbed base plates enable especially in case of larger bridge joints a more homogenous arching line over the entire construction length of the rail expansion joints. Due to the special characteristic of the elastic system of the bedding of ribbed base plates with its flat load deflection curve dynamic parts of the wheel load on the sleeper are reduced significantly. The sleepers are decoupled dynamically from the track and stay more evenly in the ballast due to the reduced alternation of loads and reduced dynamic load parts as with inflexible bedding of the plates. Additionally the elastic ribbed plates cushion the trackway with a base stiffness of 17.5 kN/mm and reduced the impact sound. To increase the reduction of the longitudinal forces over the sleeper into the ballast, the sleepers are connected on both sides at the head with connecting irons. This connection is only disrupted at the bridge joint. In the slab track system an elastic bedding is obligatory, this is done with the same rail bases, probably with an amended base stiffness. With the uniformity of the track gauge, the favorable conductance of longitudinal forces and the homogenous track subsidence due to the elastically bedded track rail, the intervals of maintenance of the rail expansion joints in this delicate area are influenced positively. Rail expansion joints for bridges with a wide span can additionally provided with extra base plates. These contain an elastic element, which adjusts the vertical arching line (rotation angle) of the bridge. The expansion joints can be installed on fixed as well as elastic bases (ERL).
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154 Track-Bridge Interaction on High-Speed Railways
Figure 5.
Rail expansion joint SAV-B 300 (without crossbar control mechanism).
Figure 6.
Rail expansion joint SAV-B 600 (with single crossbar control mechanism).
The drawings (Figs. 5–7) show the bridging of the bridge joint with the admitted rail expansion joint device with and without crossbar control system and with up to two steel sleepers. In dependency on the specific frame conditions of the construction and possible special constructions of the construction joint (e.g. installation of transition slab or bridging constructions) also in larger expansion lengths models without the shown crossbar control mechanism or steel sleeper constructions are possible.
2.2
Rails
The stock rails are manufactured from profile UIC 60 E1 according to UIC-Code 861-2 with a quality of 900 A according to UIC-Code 860 HT 350. The switch rails are made of the asymmetric profile UIC 60 E1 A1 with a quality of 900 A according to UIC-Code 860 HT 350.
2.3
Gauge
The gauge of the rail expansion joint amounts to 1435–1437 mm.
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Rail expansion joints – The underestimated track work material?
Figure 7.
2.4
155
Rail expansion joint SAV-B 1200 (with double crossbar control mechanism).
Components
2.4.1 Bridging girder A steel profile with cross sections and specifications indicated in Fig. 9 is installed as bridging girder. It is fixed at the side of the rigid bearing; for the steel sleepers and at the movable side, longitudinal displacement is realized by respective fastenings with low creep resistance. 2.4.2 Crossbar control mechanism Depending on the expansion lengths and the specific requirements from bridge movements, intermediate sleepers in the bridge gap may be necessary to comply with maximum admissible base plate distances. They are centered in the bridge gap by crossbar constructions or they are installed at constant distances (a ≤ 0.65 m). If due to particular constructive measures at the bridge structure (e.g., installation of a transition plate construction) the compliance with the maximum base plate distances is ensured, steel sleepers and crossbar construction do not have to be arranged. 2.4.3 Rail fastening In the area of the stock rail elastic ribbed base plates ERL of BWG are installed with a base plate stiffness of 17.5 kN/mm (upon customer request also other base plate stiffnesses like 22.5 kN/mm). Rails (inclination 1:40 or 1:20) are fastened according to rail expansion joint drawings by tension clamps Skl 3, Skl (B) 12, Skl (B) 15 or clamping plates. To ensure the required creep resistances, elastic inserts with the respective properties (elasticity, friction coefficient) are applied. To buffer higher lifting forces (12 kN < Zu ≤ 27 kN), the bases plates ERL (mod.) modified for transitions on bridge structures can be used. Thanks to the application of two springs with stiffness of 1.125 kN/mm, they feature upwards and downwards static base plate stiffness of approx. 20 kN/mm. To contain the springs, caps Ka 4 are replaced by caps Ka 10. The fastening of the ribbed base plate (bolting of cap and vulcanized bushing) is identical to the standard base plate ERL. The horizontal base plate stiffness is not unaffected by the modification of the upper elastomer spring and therefore remains unchanged. 2.4.4 Sleepers In the area of the rail expansion joint, turnout sleepers (e.g., turnout sleepers of BWG for ballast and slab track or according to the type of slab track, e.g., sleepers PVT-M of Pfleiderer AG/Rail. One for slab track type RHEDA 2000) are applied. The standard sleeper spacing in the area of the rail expansion joint amounts to 0.60 m.
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156 Track-Bridge Interaction on High-Speed Railways
2.4.5 References for planning The rail expansion joints to be applied are determined based on calculations on stability of the building regarding the determined linear expansions, the end tangent rotation angle and the offsets at the building gaps. The stability of the bridging construction or the base plate forces resulting from the end tangent rotation angle, bearing plate offset and wheel load at the building gaps are to be analyzed. Modification of standard rail expansion joints (e.g. omission of the crossbar construction also at bigger expansion lengths) may be applied for special constructions of the building gap (e.g., transition bridge girder). 2.4.6 Details for delivery and assembly Rail expansion joints are factory-preassembled onto sleepers and delivered in neutral position (adjusting measure a = 0). In this case, the center mark at the stock rail is located at the point of the blade. If delivered without sleepers, respective transport protections are installed. The rail expansion joints are to be adjusted when they are installed and/or before they are welded to the adjoining track. The adjustment measure (displacement of the point of the blade against the center mark on the stock rail) depends on the expected alterations in length from crawling and contracting, on the current temperature of the structure as well as on the calculated dislocations from braking and accelerating.
3 3.1
SPECIAL SUPPORT POINTS BWG FOR TRANSITION AREAS ON BRIDGES Development
On transition areas such as abutment – bridge deck and bridge deck – bridge deck on piers, the deformation of the civil structure (expansion, contraction, deflection) will excite lifting forces in the rail support points directly adjacent to the bridge gap. These forces arise from the rotation angle and offsets resulting from the deformation of the bridge deck as shown in Figure 8. The lifting forces are to be calculated in accordance with the applicable regulations for railway bridges. On bridges with very different length and overhangs or very large overhangs, the forces of the rail fastener can exceed the limit of the standard rail fastening systems. In cooperation with DB AG, BWG enhanced and modified the common turnout support point “Elastic Ribbed Plate Support – (ERL)” in such a way as to allow to accept uplift forces up to approx. 27 kN (standard rail support point loarv-300-1: 12 kN) while ensuring the same upwards and downwards rail fastening elasticity. Authorizations for trial operations in network of the German railway DB AG exists for these designs. Such support points have been installed on various bridges in the high-speed lines Hanover–Berlin, Cologne–Rhine/Main and Nuremberg–Ingolstadt as well as bridges in the region of Berlin and the High speed line in Taiwan.
Figure 8.
Schematic drawing of torsion and offset.
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3.2
157
Design and function of the special support points
The special support point was evolved from the elastic ribbed base plate support for turnouts. In this case, a downwards support point stiffness of approx. 18 kN/mm deflection and an upward support point stiffness of approx. 150 kN/mm is achieved by elastomer springs vulcanized to the base plate. In order to make it possible to accept uplift forces with respective elasticity and necessary motion range, the support point was modified such as to feature an upwards and downwards support point stiffness of approx. 20 kN/mm. For this purpose, the hard top elastomer springs were replaced by 2 coil springs each with a stiffness of 1125 N/mm, which are inserted into the associated cap. The design of the special support point and of the special cap for the coil springs is shown in Figure 9. The characteristic lines of elastomer and coil springs are harmonized so that there will be a support point stiffness of approx. 20 kN/mm when the support point moves upwards on a path of 1.5 mm and when it moves downwards on a path of 4.5 mm. This support point stiffness results from the interaction of the lower suspension spring of approx. 15 kN/mm and the upper pre-tension spring of approx. 5 kN/mm. The characteristic lines for the special support point are shown in Figure 10.
Figure 9.
Figure 10.
Special support point ERL (mod.) for transition area of bridges.
Characteristic lines for the special elastic fastening support point.
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158 Track-Bridge Interaction on High-Speed Railways
This design ensures a virtually even bearing and elasticity of the permanent way (track stiffness). 3.3
Components
3.3.1 Rail fastenings Rail fastening is realized in accordance with the drawings of the special support points with tension clamps Skl B 12. Other clamps or clamping plates may be used. With regard to ensuring the required creep resistance, elastic inserts are used which are provided with the corresponding characteristics (elasticity, coefficient of friction). The fastening of the ribbed plate (bolted connection of the cap and the bushing/adjustment cone) is identical to the one of the standard support point ERL. The horizontal support point stiffness is not affected by the modifications of the upper elastomer spring and remains therefore unchanged. 3.3.2 Sleepers Special support points in the area of the bridge joint may be installed on sleepers (e.g. sleepers PVT-M from Pfleiderer AG/Rail. One for slab track type Rheda 2000, turnout sleepers of BWG) or on blocks, arranged on the ends of civil structures. The maximum sleeper or support point spacing in the area of the bridge joint is 650 mm. Al larger expansion paths, the special support points will be installed in combination with rail expansion joints of BWG type or, if required, in combination with customized solutions (e.g. adjustment plates as track transition constructions). 3.3.3 Information regarding calculation and design With regard to defining the arrangement of the special support points to be used, apply calculations for stability of the construction concerning length expansion (temperature, braking/acceleration, creeping and shrinkage) and resulting rotational angles as well as the offsets at the bridge joints (Fig. 11). Proof must be established with regard to stability of the bridging construction respectively the support point forces resulting from the rotational angle, offset and wheel load at the bridge joints. Special support points can also be combined with special constructions of the bridge joint (e.g. transition slab, standard expansion joints or modifications of standard expansion joints). Adjustments to other axle loads and base plate stiffnesses are possible on demand. 3.3.4 Information regarding delivery and assembly On request, special support points are also available pre-assembled on sleepers or separately with assembly on site. During installation, installation instructions for elastic ribbed plate support points must be observed.
4 4.1
SPECIAL CONSTRUCTIONS OF RAIL EXPANSION JOINTS Great end tangent rotation angle and offset
At the application of slab track, bearing torsions due to a deformation of the superstructure plays a dominating role because longitudinal movements and twists transform into lifting forces or bending strains, respectively, that have to be accommodated by the rail base plates and the construction of slab track in the transition area. The deformations of the structure occurring at the transitions thrust bearing – super-structure and the resulting forces at the structure gaps are depicted schematically in Figure 11. To determine these forces, for bridges respective calculations are carried out, the so called proof of serviceability, as evidence for the compatibility of the system and the adherence to admissible forces and deformations for the overall system, rails and components of the slab track. In particular cases, these calculations lead to the conclusion that special solutions of transition construction are necessary because movements, twists and forces are only admissible within the defined limits in order to guarantee the serviceability and sustainability of the construction.
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Rail expansion joints – The underestimated track work material?
Figure 11.
159
Evidences at superstructure ends for superstructures with slab track.
4.2 Adjusting plates as transitional trackway constructions Principally, adjusting plates are small bridges in steel, concrete or composite construction that overstretch the bridge gap and support the permanent way. They are movable in track direction and are rigidly supported in transverse direction. Adjusting plates have several functions: – The vertical offset is translated into a longitudinal inclination over the length of the adjusting plate. The plates thus reduce the effect of the vertical offset at the bridge gap and the connected lifting forces at the rail fastenings (Fig. 12). – If the adjusting plate is applied at the joint between two bridge superstructures, the end tangent rotation angle occurring between the two superstructure ends is divided in smaller angles so
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160 Track-Bridge Interaction on High-Speed Railways
that the rail base plate forces from the end twist of the superstructures are reduced considerably (Fig. 13). – Great movements of the joint are divided to smaller joint displacements between the adjusting plates so that the admissible rail base plate distances are adhered to (Fig. 14). The application of the adjusting plates generally provides evidence for the lifting forces, if need be, in combination with the aforementioned special base plates. Transition slab A is situated above the bridge gap, its length depends on the position of the bridge supports. Transition slab are installed next to transition slab A if the overall dilatation at the movable bridge gap is that big that by a distribution over two gaps the admissible base plate distance cannot be adhered to. At the gaps of the transition slabs, locking constructions ensure that the gaps do not open beyond the maximum value of the rail base plate distance.
Figure 12.
Transition slab at transition abutment – superstructure.
Figure 13.
Transition slab at transition superstructure – superstructure (above piers).
Figure 14.
Transition slabs for the compensation of great longitudinal movements.
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5
161
EXAMPLES OF REALIZATION:
5.1
Bridge Hollandsch Diep, HSL Zuid (The Netherlands)
BWG together with the customer (Rheda 2000 vof/infraspeed) and the constructional steelwork company (STOG, Munich) modified rail expansion joint SAV-B 1200 (Fig. 15) with maximum a expansion length of ±600 mm so that both movements and forces can be distributed and accommodated in an optimum way. The double crossbar construction was replaced by a bridging girder that distributes longitudinal movements and twists to two gaps. The principle is a small bridge placed on the bridge as shown in Figure 16. The particularities of this approx. 1,200 m long bridge – – – –
great bridge length and high expansion length due to temperature changes steel construction with great span and connected deflection/end tangent rotation angle cantilever at the overhang of the bridge and offset at the expansion joint to be expected long-term settlements under operation in the area of the thrust bearing and the ramp
lead to deformation values and forces that cannot be accommodated by standard constructions. With view to the twists, the supports of the bridging girders (small bridge) are arranged optimally above the supports of the steel bridge or the ramp respectively. The gaps be-side the bridging girder accommodate the lateral movements of the bridge. The design of the modified rail expansion joint is composed of the following parts: – Rail expansion joint SA 1200 with stock rails extended by approx. 5 m, overall length approx. 21.05 m, without crossbar design,
Figure 15.
Top view of slab track, rail expansion joint and transition slab.
Figure 16.
System transition slab.
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162 Track-Bridge Interaction on High-Speed Railways
– Bridging girder of Co. STOG with 4 base plate lines (direct fastening at the steel plate) and lengthwise movable and rotary bearing above the expansion joint Bridge – Ramp, – 2 short transition plates for lateral guidance of the bridging girder. Lifting forces still exceeding the capacity of standard rail fastenings despite this special design can be accommodated by special base plates also designed by BWG and called Elastic Bridge Support Point – mod. ERL. In relation to the problems that occurred and to the alternatives (fundamental changes to the bearing structures to reduce deflections and foundation measures for a further reduction/prevention of settlements), the depicted and realized design (4 rail expansion joints were installed) represents an economic and sustainable solution. As shown beyond mere static/dynamic aspects safetyrelevant issues have been covered as well, as for example with additional equipment/check rails for derailment protection. Similar to the bridging designs already realized in 1995 at the Havel and Havel-Waterway-Bridge, it can be considered as a standard solution for great longitudinal movements, deflections and if needed in combination with the special base plates for high lifting forces. 5.2
Hongkong, Airport Express, Tsing Ma Bridge, Ma Wan Viaducts, Kap Shui Mun Bridge
The Lantau Fixed Crossing is formed by the world’s longest suspension bridge, the Tsing Ma Bridge (TMB), the Ma Wan Viaducts and the Kap Shui Mun cable-stayed bridge. Parameters to be taken into consideration included a 2.2 m gap, a speed of 140 km/h, and about 198 trains per day on an average. The main problems to solve were the compensation of the length variations of rails, the bridging of the gap, the compensation of the rotations of bridge around the vertical or horizontal axis and the adjustment of the different track resilience at both sides of the RMJ. VAE has developed a solution with a continuous running table and a continuous running edge to ensure a smooth ride. Over the whole length of the RMJ there are no gaps which leave the wheels unguided. The possible 2,2 m gap between the abutment and the bridge is divided into 4 small gaps by means of sledges. The supporting structure for the REJ is designed as a flexible connection between the primary structure and the abutment. The main goal concerning the design of the REJ’s was to introduce a modern technical solution with an optimum of Riding comfort ↓ less wear and noise ↓ longer life cycle and less maintenance Another goal during the design period was to find and implement as many approved trackwork solutions as possible. The design of the supporting structure had to meet the demands imposed upon the track with respect to vertical and horizontal acceleration and riding comfort as well as the requirements mentioned above. VAE has developed a cast tank-shaped expansion sleeve made of high manganese steel in order to accommodate a UIC 60 thick-web stock rail. In principle the design follows the switch design principles for a chamfered switch with straight machining for the foot of the stock rail. The transition from the stock rail to the expansion sleeve or vice versa remains constant over the whole range of movements. For bridging the gaps widths of 1410 mm respectively 1670 mm, VAE finally decided on a solution with 3 sledges. These sledges divide the wide gap into 4 small gaps enabling the adherence to the required maximum base plate distances.
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Rail expansion joints – The underestimated track work material?
Figure 17.
Section switch rail – stock rail.
Figure 18.
Transition slabs with sledges.
163
An equal size of all four spacings is guaranteed by means of a lever connection mounted below the sledges. The connection between the abutment at Tsing Yi and the primary bridge structure is made by a flexible “spline girder” and an “end girder”. The spline girder transforms all rotations around the vertical or horizontal axis into longitudinal movements by forming a smooth sag (concave) or crest curve (convex). The longitudinal movements of each track on the Tsing Ma Bridge (TMB) are also transmitted to the end girders via the spline girders. The end girders are mounted on 4 vertical bearings and are guided by 4 horizontal bearings. These bearings can be adjusted hydraulically during installation and slide upon polished stainless steel bearing plates.
6
CONCLUSION
Rail expansion joints of BWG and VAE are applied in the area of bridge constructions and in normal conditions as well as in geologically instable areas. They are equally suited for ballast and slab track. For Vignol rails, the standard expansion lengths 300 mm, 600 mm and 1200 mm are available. Customized expansion lengths and special designs can be realized upon consultation and in cooperation with the builder. Standard rail expansion joints are designed to be easily integrated in the general planning of the permanent way on bridges and adjoining thrust bearings and soil constructions. They do not require any particular changes to the building design. In case of special designs, as they may be required for great end tangent rotation angle and/or great expansion lengths, an early coordination between bridge planner, planner of the permanent way and producer of the rail expansion joints is absolutely necessary to develop an economic and low-maintenance solution that meets all requirements.
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REFERENCES Bringfried Belter, Peter Ablinger, Josef Hess, Gilbert Waldmann Railway Track at the High Speed Line Cologne – Rhine (Main – Slab Track) Eisenbahn-Ingenieurkalender – EIK – 2002 Systematical Decision Ballasted Track or Slab Track Report to results of the analysis and examination within the scope of the project “Fahrbahnstrategie SMP-T”, published by ifv Bahntechnik (Project team Fahrbahnstrategie, DBSystemtechnik, Strategy Consultancy Fontin & Company, ibt Dr. Ablinger) Special constructions of Rail Expansion Joints Report about development and adaption of Rail Expansion Joint SAV60 – 1200 and Elastic Ribbed Base Plates for the Bridge Hollandsch Deep in the course of High Speed Line HSL Zuid, The Netherlands EI – Eisenbahningenieur (57) 4/2006 Johannes Rohlmann, Josef Hess New standard for turnouts and rail expansion joints for high-speed traffic – installation in a high-speed track in Taiwan Track System Development, Engineering – Construction – Experiences RTR Special - SLAB TRACK, 09/2006 Johannes Rohlmann, Josef Hess New standard for turnouts and rail expansion joints for high speed traffic with large scale operation in Taiwan Eisenbahntechnische Rundschau – ETR, 07+08/2007 Stephan Gerke, Jürgen Haase, Josef Hess Development and construction of turnouts VDV Schriftenreihe Band 26 Gleisbau: Planung – Bau – Vermessung Jürgen Haase, Josef Hess, René Brehm Turnouts in High Speed Applications RTR Special – The German High Speed Rail System, Evolution • Quality • Track construction, 03/2008
© 2009 Taylor & Francis Group, London, UK
CHAPTER 14 Dynamic aspects of the high-speed railway bridge across the Hollandsch Diep J.T.F.M. Tünnissen Joep Tünnissen Dynamic Engineering Consultancy, Veghel, The Netherlands
ABSTRACT: The structures and track system which are part of the Dutch High-Speed Railway connection between Amsterdam and Paris, have all been analyzed for their dynamic behavior with regard to structural integrity and passenger comfort. This paper describes the general approach of these dynamic analyses as performed in the High-Speed Line-Zuid project, and goes into more detail with regard to bridge-track interactions at the bridge across the Hollandsch Diep, one of the most eye-catching and largest structures in this project. Relevant issues are the optimization of the level of passenger comfort by introduction of a pre-camber in the alignment of the track system and the dynamic behavior of the steel transition slabs, which as a special structure in the track system allow for the horizontal expansion and contraction of the bridge.
1
INTRODUCTION
October 2007, the Dutch part of the High-Speed Railway connection between Amsterdam and Paris nears its completion and thereby its goal to reduce the travelling time between both cities to just 3 hours. In the 125 kilometers from Amsterdam, via Rotterdam to the Belgium border, of which 85 kilometers have been destined for high-speed railway, trains will pass 170 different structures, designed to cope with train velocities up to 300 km/h. In the 2nd half of the 90s the HSL-Zuid Project-organization, representing the Dutch State Department of Transportation, developed the visualization and engineering tools for the architectural and structural elements in the High-Speed Line-Zuid project. Due to its magnitude and complexity it was decided to divide the contract into 7 subcontracts. 6 of these subcontracts dealt with the design and construction of the railway structures, each covering a separate part of the track. They were granted to different joint ventures between contractors in the year 2000. The 7th subcontract covered the design and construction of the track system as well as the future maintenance of the entire railway system and was granted to Infraspeed in 2002. The author has been a member of the design team in 4 out of these 7 subcontracts, as an advisor on dynamic aspects and as the responsible party for the dynamic analyses performed on the primary (main structures and track system) and secondary (architectural elements and noise barriers) structures. The Dutch terrain has required many adaptations to the high-speed railway structures and track system as the track encounters highways, rivers, canals, ditches and environmentally important areas on its way. Together with the necessity for piled foundations, due to the soft clay type of soil in the western part of the Netherlands, the track has become a chain of different structures with dilatations generally every 20 to 30 m. Dealing with the design criteria concerning the dynamic behavior of these structures and the implementation of a ballastless track system, has made the HSL-Zuid project a challenging engineering experience. 165 © 2009 Taylor & Francis Group, London, UK
166 Track-Bridge Interaction on High-Speed Railways
2
HIGH-SPEED RAILWAY STRUCTURES IN HSL-ZUID
The natural and man-made obstacles in the path of the track, the need for pile foundations and the ever developing engineering and architectural insight during the design phase, have led to a variety of structures. Some of these structures are highly visible, such as the elevated long-viaducts near Hoofddorp and Bleiswijk (see Fig. 1), and the bridge crossing the Hollandsch Diep, which with a total span of 1192 meters is the longest HSR-bridge between Amsterdam and Paris (see Fig. 2). Others are hardly noticeable, such as the settlement-free slabs, which cover about 33 km of the track and with their typical length of 30 m per slab often provide the link with other types of structures. Except for the train passengers, some major achievements are not visible at all, such as the Ringvaart aqueduct and the tunnels underneath the Green Heart, with a diameter of almost 15 meters, one of the largest drilled tunnel in the world, and the rivers Oude Maas and Dordtsche Kil. Due to the soft clay type of soil in the western part of the Netherlands most of these structures are supported by pile foundations, reaching into the Pleistocene sand bed. Only in the last 3.5 kilometers towards the Belgium border the sand bed reaches the surface and provides a solid foundation. At this location the transition into a ballast track is established in order to connect to the Belgium part of the high-speed railway track. The prize-winning design for the bridge across the Hollandsch Diep by architects Benthem & Crouwel, is a result of a competition held by the HSL-Zuid Project-organization. The optimization
Figure 1.
Long-viaduct (6 km) near Bleiswijk. (Photo: JTüDEC).
Figure 2.
Bridge across the Hollandsch Diep. (Photo: JTüDEC).
© 2009 Taylor & Francis Group, London, UK
Dynamic aspects of the high-speed railway bridge across the Hollandsch Diep
167
of the design and the final construction of the bridge was in the hands of HSL Drechtse Steden, a joint venture of contractors responsible for the track between the city of Barendrecht and the southern end of the bridge. Consisting of steel box-girders and U-shaped sections topped by a concrete deck, the 10 main spans measure 105 m each to coincide with the spans of the old railway bridge located next to it. The V-shaped hammer-pieces with a length of 45 m and a maximum height of 11.40 m are interconnected by 60 m long field members. In the engineering phase minor changes to the columns were introduced in order to obtain a maximum spread of the rubber bearings. The increase of restraint at these locations resulted in an increase of the natural frequency of the bridge which was beneficial to the level of passenger comfort. 3
RHEDA 2000 TRACK SYSTEM
The choice of a ballastless track system is based on economics, ease of maintenance and favorable experiences with this system abroad. However, due to the soft soil conditions and the variety of supporting structures with their inherent need for dilatations, the classical solution of a continuous ballastless track could not be implemented in the HSL-Zuid project without some major adjustments. The task of coming up with a solution was given to Rheda 2000 vof, which led to the introduction of 2 different types of track. One, a continuous slab poured directly onto the structure in areas which are relatively insensitive to settlement such as tunnels and slabs on embankment and two, a jointed slab which is poured on an intermediate layer, consisting of a 4 mm thick polypropylene geo-textile op top of the structures in settlement sensitive areas. The jointed slab is anchored to the structure in pre-designated free-drilling zones by means of HILTI high quality stainless steel dowels with a diameter of 40 mm. This is the most commonly type used throughout the track (67%) for structures such as the settlement-free slabs, viaducts and bridges with dilatations approximately every 15 to 30 meters, but also on the bridge across the Hollandsch Diep which is categorized as being sensitive to settlement/vertical deflection. In the 163 km of Rheda 2000® slab track system, the rails are the only continuous elements connecting adjacent structures. They are held in place by a Vossloh type IOARV 300 rail-fastening system which is provided with a highly elastic intermediate layer. These layers are responsible for the transfer of horizontal and vertical forces into the Pfleiderer type B355 W60M concrete bi-block sleepers. The Rheda 2000® slab track system is completed by casting the prefab concrete sleepers into a reinforced concrete slab on site. The reinforcement consists of lattice trusses which provide stable dimensions and assures the required gauge of the track. The concrete slab grade B35, as used in the HSL-Zuid project, has a standard height, without cant, of 240 mm and a width of 2600 or 2800 mm, depending on the location. The slab is reinforced throughout its entire length for systematic prevention of cracks. See Figure 3. Fastening IOARV 300 Sleeper B355 (c.t.c. 650 mm typical) 3000 2600 Dowel 232
240
Intermediate layer
free drilling zone 500
Settlement-Free Plate
Figure 3.
Rheda 2000 slab track system on settlement-free plate.
© 2009 Taylor & Francis Group, London, UK
168 Track-Bridge Interaction on High-Speed Railways
Figure 4. Rheda 2000 on settlement-free plate. (Photo: JTüDEC)
Figure 5. Steel transition slab on BHD. (Photo: Courtesy of Mr. P. Meijvis, DMC bv, The Netherlands)
Figure 6. Rough positioning of Rheda 2000 with Fassetta. (Photo: Courtesy of Mr. P. Meijvis, DMC bv, The Netherlands)
The interconnecting components of the Rheda 2000®track system, being the stainless steel dowels and the intermediate layer have undergone several tests in order to determine their capacity to withstand fatigue loadings (dowels) and to establish the stiffness characteristics and frictional behavior under static and dynamic loadings (intermediate layer). Dynamic analyses have been performed on the interaction between structures and track system near transitions, especially with regard to the dynamic forces in the rail-fastening system.
© 2009 Taylor & Francis Group, London, UK
Dynamic aspects of the high-speed railway bridge across the Hollandsch Diep
169
Due to the fact that the engineering phase for the structures started 2 years prior to that of the track system, assumptions were made with regard to the system’s mass ranging from 650 to 3750 kg/m per track. Without cant, the mass of the Rheda 2000® slab track with one pair of UIC60 type rails eventually comes down to approximately 1800 kg/m per track.
4
DYNAMIC ANALYSES
4.1
Design Criteria
To aid the design teams of the contractors during the engineering phase of the structures, guidelines have been prepared and provided by the HSL-Zuid Project-organization. These guidelines are based on national and European standards anno 1999, such as ENV 1991-3:1995, supplemented with available research data and experiences with high-speed railways abroad. In guideline HSL600E, titled “Loads and Deformations of Structures”, the criteria with regard to the dynamic behavior of structures are stipulated. These can be summarized as follows: 1. The vertical accelerations in the structure, as calculated in the center of the track, shall not exceed 0.50 g in case of ballastless track, for frequencies ranging from 0 to 20 Hz. This is a way to ensure that the structural accelerations will not negatively affect the passing train (rebound) and is introduced as a safety measure against derailment. It does not mean in any way that acceleration signals outside the mentioned frequency range do not matter or may be ignored in the structural analyses. 2. To account for the dynamic response of a structure a dynamic coefficient φ2 is introduced based on the determinant length (Lφ ) of the structure at hand. The value of this coefficient ranges from 1.00 to 1.67, as applicable for carefully maintained track. By means of dynamic analysis a second coefficient φr has to be established, equal to the quotient of the dynamic and static bending moments at any governing location in the structure. The maximum of these coefficient φ2 and φr is then used as the general multiplier φ applicable to the load models (LM 71, SW/0 & /2) used for static analyses. 3. In order to ensure a proper level of passenger comfort, the vertical acceleration of the train is limited to a maximum of 1.0 m/s2 (classification: very good). However, the natural frequency of the bridge across the Hollandsch Diep lies close to 1.0 Hz, which is near the range of natural rigid car body frequencies of most trains (0.6 to 2.0 Hz). Therefore an additional criterion has been introduced in the form of a weighted level of “incomfort” of harmonic vibration (LIh ) which may not exceed the value of 45 [ms−5/3 ]. The limiting value of 45 is time dependent with a maximum duration of passage of 15 s. Derived from ERRI D190 (“Permissible deflection of steel and composite bridges for velocities V > 160 km/h”, December 1995) this criterion is based on empirical data, as shown in the following figure. Taking into account a weighting factor of 0.40, the formulae for the LIh -value reads: LIh = 43.1
T
1/3 |av (t)|3 dt
≤ 45
(1)
0
where av (t) = the train acceleration; and T = the duration of passage. These (3) design criteria do only indirectly apply to the track system. In the requirements concerning the track system it is stated that the application of the track system shall have no negative effect on the behavior of the supporting structures, or in other words, have no negative impact on the results of the dynamic analyses already performed on these structures. In the matter of passenger comfort, the criterion differs from that of the structures as it more specifically includes the roughness effects of the rails, by means of actual field data. As stipulated in UIC513 (Full Method), the level of comfort is to be determined by means of field measurements
© 2009 Taylor & Francis Group, London, UK
170 Track-Bridge Interaction on High-Speed Railways
100
100% Satisfied 100% Unsatisfied the in-betweens LIh limit value (45)
Passenger (in-) comfort [%]
90 80 70 60 50 40 30 20 10 0 0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
LIh value Figure 7.
Empirical data, source for the LI h comfort criterion as per ERRI D190.
of accelerations at floor and seat level. A typical measurement period is 5 minutes in which data is collected every 5 seconds, resulting in a total of 60 measurements per period. Measurements are to be taken at the maximum operation velocity, 300 km/h in this project, and in all three (x:y:z) axis, resulting in one measurement every 417 m. Data is processed, filtered and weighted into 50% and 95% probability ranges. The required comfort level of maximum 2 is applicable to both seated [NVA ] and standing [NVD ] positions, as per following formulas: Wd 2 Wb Wc b 2 NVA = 4(aW ZP95 ) + 2 (aYA95 ) + (aZA95 ) + 4(aXD95 ) ≤ 2
(2)
Wd Wd Wd d 2 2 2 NVD = 3 16(aW XP50 ) + 4(aYP50 ) + (aZP50 ) + 5(aYP95 ) ≤ 2
(3)
In order to predict the field measurements dynamic analyses have been performed before the actual installation of the Rheda 2000® slab track system, and with the train types specified in paragraph 4.2. Assumptions have been made for the accelerations in the horizontal, lateral and transverse direction based on the ratio (1:2:3) ( source: prof. C. Esveld, TU Delft). By collecting data every 417 m the chance that relevant data on smaller structures are included tends to be small. During the passage of the 1192 m long bridge across the Hollandsch Diep, the governing structure with regard to level of passenger comfort, only 3 field measurements are collected out of the 60 in 1 period. By assuming that structures in front of and following the bridge will show an equal or W better comfort performance, a ratio of the weighted 50% and 95% train accelerations (aW Z50 /aZ95 ) 2 of 1.0 is derived. This leads to a maximum allowable weighted RMS-value of 0.210 m/s . Applying a weighting factor of 0.40 this results in a non-weighted RMS-value of 0.525 m/s2 . Due to the fact that structures before and after the bridge will most probably show better comfort performances, this approach is considered to be on the conservative side. 4.2
Real Train types
The dynamic analyses required by guideline HSL600E have been performed with the, at that moment in time, relevant or real high-speed train types such as the French THALYS and German ICE3M plus ICMAT. Their characteristics, axle loadings and positions, are specified in attachment
© 2009 Taylor & Francis Group, London, UK
Dynamic aspects of the high-speed railway bridge across the Hollandsch Diep
Figure 8.
171
Real Train characteristics THALYS1&2, ICE3M1&2 and ICMAT.
1 of this guideline and are shown in Figure 8. For the THALYS and IC3M trains, single and double configurations are considered, identified by index 1 and 2. The THALYS2 characteristic is equivalent to train type B in Annex F of the European standard EN 1991-2:2003. Hence, the dynamic analyses have been performed with 5 real train types. To obtain the vertical body accelerations from these train types, they have been modeled as SDOF spring-mass systems with a mass equal to the axle-forces divided by the g-force. The spring frequencies range from 0.80 to 1.20 Hz, while the damping ratio is set to be 0.10, both as per guideline. Hence: Mass M = Axle Loading/g-force and Spring Constant K = M (2πf)2 . In general, the structures and track system have been analyzed for the passage of the THALYS and ICE3M trains at velocities ranging from 160 to 360 km/h being 1.2 times the maximum design velocity or line speed of 300 km/h, and from 160 to 264 km/h for the ICMAT train as its maximum permitted vehicle speed lies at 220 km/h. An exception for the upper bound velocity is made for the bridge across the Hollandsch Diep by reducing the factor of 1.2 to 1.1, resulting in a maximum velocity of 330 km/h. In the early stages of engineering the track system characteristics were not yet included in the dynamic analyses. To study the favorable effects of the rails distributing the axle loadings in longitudinal direction, a bell-shaped distribution was taken into account with a maximum spread of 2.50 m. Structures with spans up to 20 m and structures with large cantilevered ends, of which there are several in the HSL-Zuid project, show a reduction in the dynamic response. For structures with spans larger than 20 m the effect of load distribution is found to be negligible.
4.3
Computer program DRS
In 1999, during the tender phase of the structures in the HSL-Zuid project, the need arose for software able to perform dynamic analyses and quick parameter studies. At that moment in time the existing mainframe software packages were found to be too time-consuming with regard to the number of analyses to be performed. Therefore, a new computer program was developed by the author based on software readily available and used for seismic and shock analyses, focused on the queries at hand. This program called DRS, which stands for Dynamic Response of Structures has been validated and was approved for usage in this project by the HSL-Zuid Project-organization. In the 8 years since 1999 the program has been updated continuously and has proven to be a versatile tool in getting insight in the dynamic behavior of structures, track system, architectural elements and noise barriers.
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172 Track-Bridge Interaction on High-Speed Railways
The heart of the program is based on a Householder and QR-algorithm eigenvalue solution and direct numerical integration using the Newmark β – Hilber α method. Depending on the structure’s geometry there is a choice to construct 2-dimensional beam or 3-dimensional grid models, using 1-dimensional elements. The program contains a graphical interface and is able to directly transfer data to MS-Excel for further processing. Special modules are included, dealing with shear deformation, rail level geometry and signal filtering by means of the Fast Fourier Transform method, which is relevant for the criterion with regard to the maximum allowable, filtered structural accelerations. 5 5.1
PASSENGER COMFORT ON THE BRIDGE ACROSS THE HOLLANDSCH DIEP Dynamic model
The structure of the bridge consists of relative torsion stiff structural elements such as steel box-girders at the V-shaped part of the hammer-pieces and steel U-shaped sections topped by a concrete deck. The average load of the real trains THALYS and ICE3M is 21 to 22 kN/m which is about 8% of bridge’s total dead load of 275 kN/m. The centerlines of the tracks are 5.50 m apart and therefore 2.75 m off center with respect to the centerline of the bridge. The center of the tracks coincides with the support locations of the steel U-shaped sections. These considerations have led to the conclusion that for this bridge a 2-dimensional model will be sufficiently accurate. The original design of the bridge assumed a line-support in the center of the columns/piers. By locating the laminated rubber bearing into the four corners of the columns and by introducing an additional horizontal, in longitudinal direction, restriction of motion by means of hydraulic dampers, the natural frequency of the bridge has been increased, which is beneficial to the level of passenger comfort. Dynamic analyses run with these dampers show a reduction of approximately 10 points on the LIh -scale. Structural damping is an important factor in the dynamic analyses. For the bridge structure a damping ratio of 0.010 was applied as per guideline HSL600E. For the rails and rail-fastenings damping ratios of 0.005 and 0.100 respectively were taken into account. 5.2 Vertical rail level geometry considerations The dynamic analyses for the bridge across the Hollandsch Diep were performed twice. The purpose of the first round of analyses (2001) was to establish the structural integrity of the bridge and the level of passenger comfort with regard to the applicable design criteria. Static analyses revealed that temperature fluctuations and creep of the concrete deck will cause a deformation of the vertical alignment of the bridge (deck) and consequently the rail level geometry. As a result the level of passenger comfort will be compromised. This systematic deformation [SD] in great lines follows the bridge’s natural deflection pattern under dead load, causing disturbing frequencies of 0.79 Hz at a train velocity of 300 km/h to 0.87 Hz at 330 km/h (main span = 105.0 m). Not only do these frequencies come close to the bridge’s natural frequency of 1.10 Hz, they also fall within the range of frequencies, 0.80 to 1.20 Hz, to be considered for the real trains. Therefore, resonant effects are likely to occur. The maximum differential systematic deformation was established at 5 mm comprising of 3 mm at mid-span and 2 mm at the center of the hammer-pieces. In the case the deformation at mid-span is upward directed (positive) the most favorable situation [SDR, with R = Reversed] is derived as the train is ‘flattening’ its way across the bridge deck. Unfortunately, the opposite situation may also occur. As a solution to this problem a corrective deflection [CD] or pre-camber of the rail level was suggested by the design team of the bridge structure and incorporated by Rheda 2000 vof. See Figures 10 and 11. This pre-camber is defined as follows: 2πa δ(a) = δini 1 − cos Li where δini = + 1.5 mm and Li = 70.0 m for the end-spans and = 105.0 m between piers and a = the distance along the bridge deck/track.
© 2009 Taylor & Francis Group, London, UK
Dynamic aspects of the high-speed railway bridge across the Hollandsch Diep
Figure 9.
173
DRS 2D-model of the bridge across the Hollandsch Diep. 105
V = 330 km/h V = 300 km/h V = 270 km/h LIh limit value (45)
90
LIh value
75 60 45 30 15 0 -10.0
-7.5
-5.0
-2.5
0.0
2.5
5.0
7.5
10.0
12.5
15.0
(Differential) Systematic Deformation of Bridge Deck/Rails [mm]
Figure 10.
The effect of systematic deformation of the rails on passenger comfort.
A second round of analyses was performed 2 years later in 2003, as part of the scope of Infraspeed, to analyze the possible effects of the Rheda 2000® slab track system on the bridge structure and to establish the final requirements with regard to the vertical rail level geometry. The effect of rail roughness has been incorporated in more detail by combining actual field data for short wavelengths, ranging from 3 to 25 m, with theoretical data for long wavelengths in excess of 25 m (source: prof. C. Esveld, TU Delft). By use of a scaling factor it is possible to alter the contribution of the long wavelengths to get in compliance with the required comfort level (see paragraph 4.1). However, deviations from the starting value of 1.00 will have an impact on the level
© 2009 Taylor & Francis Group, London, UK
174 Track-Bridge Interaction on High-Speed Railways
Abutment
Pier
Pier
Pier
105.0 m (total of 10 spans)
70.0 m
+ 2 mm
- 3 mm Systematic Deformation
47.5 m Abutment
45.0 m Pier
60.0 m Pier
Pier
105.0 m (total of 10 spans)
70.0 m
+ 3 mm
- 2 mm Systematic Deformation (Reversed)
Abutment + 3 mm
Pier 70.0 m
Pier
Pier
105.0 m (total of 10 spans)
Corrective Deflection/Pre-camber
Figure 11.
Systematic deformation and required corrective deflection or pre-camber.
of accuracy required during the final preparation of the vertical rail level and possibly requires a closer monitoring of this level while the track is in operation. Note that during the first dynamic analyses of the bridge, rail roughness effects were assumed to be negligible due to the longs spans of the bridge, which resulted in a roughness factor equal to 1.00 according to formulae: −L2 /100 froughness = 1 + 0.28e φ where Lφ = the span of the bridge with a minimum of 42.5 m (at both ends). In order to comply, the scaling factor for the bridge across the Hollandsch Diep had to be reduced to 0.55 (see Fig. 12). For all other HSL-structures, analyzed for their interaction with the Rheda 2000® track system, scaling factors were found to be within a range of 0.92 to 1.12. This meant that the preparation of the vertical rail level on the bridge would require more attention and accuracy than at other structures, which was to be expected as this bridge structure was and still is the most comfort sensitive structure in the entire HSL-Zuid project. 5.3
Results dynamic analyses
For the 5 real train types used in the dynamic analyses, type ICE3M2 proved to be responsible for the governing results in all of the dynamic criteria considered. For all 4 ICE3M and THALYS trains the maximum results were obtained at a natural train frequency of 0.89 Hz. The effect of rail roughness on the structural acceleration of the bridge deck remains small. At peak values an increase of approximately 10% is found compared to a situation with a perfect alignment of the rail level. See Figure 14. With a maximum value of 0.40 m/s2 or 0.04 g, obtained for the ICE3M2 train at 330 km/h, the criterion not to exceed 0.50 g is easily fulfilled. Maximum results for the THALYS2 and ICMAT trains were established at 0.26 and 0.20 m/s2 at corresponding velocities of 240 and 220 km/h. The impact of train types THALYS1 and ICE3M1 proved to be smaller than for their double-sized companions. Due to the low natural frequency of the bridge structure (1.10 Hz), filtering the structural acceleration signal for frequencies between 0 and 20 Hz did not result in any reduction of these accelerations. With regard to the magnitude of the structural accelerations it could also be concluded that the dynamic coefficient φr , part of the 2nd dynamic criterion (see paragraph 4.1), would not exceed the value of 1.00 and would therefore not become governing. This can be verified considering the following. The distributed load of the real train ICE3M is 21.9 kN/m. During its
© 2009 Taylor & Francis Group, London, UK
Dynamic aspects of the high-speed railway bridge across the Hollandsch Diep
175
10.0 9.0 8.0 7.0
(scaled: 0.55) Long Waves
6.0
Short plus (scaled) Long Waves
Vertical Rail Level [mm]
5.0 4.0 3.0 2.0 1.0 0.0 -1.0
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
180.0
200.0
-2.0 -3.0 -4.0 -5.0 -6.0 -7.0
Location [m]
Figure 12.
Vertical rail level geometry as applicable for the first 245 m on the bridge.
Figure 13.
Graphical display of the bridge and train vertical displacements by program DRS.
© 2009 Taylor & Francis Group, London, UK
220.0
240.0
176 Track-Bridge Interaction on High-Speed Railways
0.450 0.400 0.350
ICE3M2 at 330 km/h [SDCD]
ICE3M2 at 330 km/u [SDCDRGF055]
0.300 0.250 0.200
Bridge deck accelerations [m/s2]
0.150 0.100 0.050 0.000 0.0 -0.050
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
1100.0
-0.100 -0.150 -0.200 -0.250 -0.300 -0.350 -0.400 -0.450
Location [m]
Figure 14.
Governing vertical structural accelerations in the bridge deck.
presence on the bridge the structure shows a response of maximum 0.04 times its dead load which results to 11.0 kN/m (= 0.04 × 275 kN/m). Train load and structural response combined amounts to 32.9 kN/m, which is considerably less than the uniform distribution load of 80.0 kN/m belonging to Load Model 71 which is used in the static analyses. In general terms, the dynamic coefficient φr results to 0.41 (= 32.9/80.0), thus less than 1.00. This approach however is only valid for structures with large spans and a high ratio of dead load versus live load. For smaller structures, such as viaducts the dynamic analyses showed higher φr -values, however hardly ever exceeding 1.67, the upper bound value for coefficient φ2 . Comparing both criteria with regard to the level of passenger comfort, the results show a strong similarity between the LIh - and RMS-values, which is obvious as they are based on the same principles. In the case of the bridge across the Hollandsch Diep the ratio LIh over corresponding RMS-value results to 111, i.e. for a RMS-value of 0.525 m/s2 the corresponding LIh -value = 58.3. See also Figure 15. In the worst case scenario, combining systematic deformation and corrective deflection or precamber [SDCD] a maximum LIh -value of 52 is derived. According to Figure 7, the number of 100% satisfied passengers would reduce from 84% to 78%, while the number of 100% unsatisfied passengers almost remains steady at 10%. Due to the fact that the conditions for which the maximum calculated systematic deformations occur are rare, the level of passenger comfort is only compromised for short periods of time and then only in certain areas of the train as can be seen in Figure 16. Most of these areas coincide with the axle positions of the locomotive sections. As a consequence of these arguments, the increase of the LIh -value from 45 to 52 has been acknowledged by the HSL-Zuid Project-organization. When both approaches for the level of passenger comfort are being compared, it could be concluded that the criteria for the structure are more strict than those applicable to the track system. However, the effects of vertical rail imperfections or rail roughness (short and long wavelengths) have not been taken into account during the first dynamic analyses concerning the structure as
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Dynamic aspects of the high-speed railway bridge across the Hollandsch Diep
177
0.700 75.0 [PA] [RMS]
[CD] [RMS]
[SDCD] [RMS]
[CDRGF055] [RMS]
[SDCDRGF055] [RMS]
Upper Limit RMS-value
[PA] [LIh]
[CD] [LIh]
60.0
0.500
[SDCD] [LIh]
[CDRGF055] [LIh]
55.0
0.450
[SDCDRGF055] [LIh]
Upper Limit LIh-value
50.0
0.600
2
Maximum RMS trainaccelerations [m/s ]
0.550
70.0 65.0
45.0
0.400
40.0
0.350
LIh values
0.650
35.0
0.300
30.0 0.250 25.0 0.200 20.0 0.150
15.0
0.100
10.0
0.050
5.0
0.0 0.000 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0 310.0 320.0 330.0
Train velocity [km/h]
Legend: [PA] [SD] [CD] [RGF055]
Figure 15.
: Perfect Alignment of the rails : Systematic Deformation of the bridge deck due to temperature and creep effects : Corrective Deflection or pre-camber of the rail level : vertical Rail level Geometry with scaling factor of 0.55 for long wavelengths
Maximum RMS train accelerations and LIh -values for governing train type ICE3M2.
0.700 75.0 0.650 ICE3M2 [CD]
ICE3M2 [SDCD]
ICE3M2 [SDRCD]
0.550
THALYS2 [CD]
THALYS2 [SDCD]
THALYS2 [SDRCD]
65.0 60.0
0.500
55.0
0.450
50.0
0.400
45.0 40.0
0.350
35.0
0.300
30.0 0.250 25.0 0.200 20.0 0.150
15.0
0.100
10.0
0.050
5.0
0.000 0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
Axle Position [m]
Figure 16.
Maximum train acceleration for each axle-position types THALYS2 and ICE3M2.
© 2009 Taylor & Francis Group, London, UK
0.0 400.0
LIh value
RMS train acceleration [m/s2]
70.0 0.600
178 Track-Bridge Interaction on High-Speed Railways
they were nullified by the definition of the roughness factor. The RMS-range from 0.470 m/s2 , corresponding with a LIh -value of 52, to 0.525 m/s2 is used to allow for a, however reduced (scaling factor = 0.55), realistic vertical rail level geometry in the Rheda 2000® track system analyses. As a result of the first dynamic analyses performed on the bridge across the Hollandsch Diep an alternative for the LIh -criterion was introduced in the way of a maximum allowable vertical train acceleration of 0.70 m/s2 valid for structures with 3 or more repetitive spans in a row. The value of 0.70 m/s2 coincides with a LIh -value of 45 ms−5/3 and takes out the dependency on the duration of passage in the LIh approach. Another important issue is the impact of the vertical rail level geometry on the axial forces in the rail-fastenings and the bending moments in the rails. In general, the forces in the rail-fastenings are of a higher importance as the fastenings are the weaker link in the system. During a quasi-static passage of train ICE3M the benchmark values are established at 3.0 kN for the tensile (upward) forces and −36.3 kN for the compression (downward) forces in the fastenings. See Figure 17. The THALYS and ICMAT trains show similar results, all in proportion to the maximum axle loading of 170 kN for ICE3M and THALYS and 225 kN for ICMAT (see Fig. 8), and in combination with a center-to-center sleeper distance of 600 mm to 650 mm. Hence, a pair of rail-fastenings or a sleeper location is subjected to approximately 45% of the axle loading. This is a recurring result for all HSL-structures analyzed and shows that the distribution of a single axle loading covers at least 3 but most probably 5 sleeper locations. The systematic deformation [SD] of the vertical rail level has a minor effect which is only noticeable for the compression forces in the rail-fastenings (see Figure 17). More relevant is the rail roughness effect [RGF055] which results in an amplification of both tensile and compression forces. However, the component of the long wavelengths in this signal, has a minor to negligible impact on the results, which seem to be solely dictated by the short wavelengths. A plausible reason for this fact is that the frequency range of these short waves, especially at high train velocities, lies much closer to the natural frequencies of the track system.
40.0 30.0 20.0 10.0 0.0 0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
180.0
200.0
Axial Forces in (2) Fastenings [kN]
-10.0
Benchmark [Tension] [SD] [Tension] [RGF -Factor 0.55] [Tension] Benchmark [Compression] [SD] [Compression] [RGF -Factor 0.55] [Compression]
-20.0 -30.0 -40.0 -50.0 -60.0 -70.0 -80.0 -90.0 -100.0 -110.0 -120.0
Location [m]
Figure 17.
Rail fasting forces in first 240 m of BHD during the passage of train ICE3M(2).
© 2009 Taylor & Francis Group, London, UK
220.0
240.0
Dynamic aspects of the high-speed railway bridge across the Hollandsch Diep
179
In the design of the rail-fastening system, limitations to the allowable tensile forces have been formulated and a minimum dynamic coefficient φ2 of 1.67 has been incorporated. In the worst case scenario, combining rail level situations [SD] and [RFG055], the dynamic coefficient φr for the rail fastenings in compression amounts to 1.62. However, due to the application of a pre-camber in the rail level this value is reduced to 1.50. In those areas where the tensile forces in the fastening systems may exceed the allowable values, special fastenings with a higher capacity for tensile forces have been applied. This is especially valid at the dilatations between the smaller structures in the HSL-track, where differential vertical displacements and rotations are likely to occur. At the bridge across the Hollandsch Diep, these effects have been reduced by the implementation of steel transition slabs to keep the dynamic amplification of the forces and bending moments in rail-fastenings and rails within the required limits. 6
STEEL TRANSITION SLAB
6.1 Transition slab design With a total, non-dilated span of 1192 m the hybrid bridge-structure across the Hollandsch Diep needs to be able to react freely to temperature fluctuations. Therefore areas for expansion and contraction are required at both ends of the bridge at its connection with the approach ramps. Furthermore, the differences in rotations and vertical displacements between the connecting ends of the bridge and the approach ramps, need to be restricted in order to limit the bending moments and forces and the rails and rail-fastenings. To comply with both requirements and ensure the continuity of the track system, 4 so-called transition slabs have been designed and constructed to cover the expansion/contraction areas. Consisting of a steel plate structure with dimensions 2700 × 2380 × 200 mm, they each support 2 rails and 4 pairs of rail-fastenings. The expansion and contraction of the bridge has been determined to be plus and minus 300 mm at both ends. The average distance between the first pair of railfastenings on the transition slab and the first pair of rail-fastenings on either the bridge or the approach ramps is 500 mm. By allowing the distance between these rail-fastenings to vary from 350 mm to 650 mm, together with the application of special fastenings which allow the rails to slide through, the required expansion and contraction can be assured. Each transition slab is resting on 4 vertically adjustable supports, which are fixed in pairs to the approach ramps and bridge ends, and are located directly above the supports/rubber bearings of these structures. PE-elements are used as an intermediate layer between the supports and the steel transition slab, to provide sliding, damping and spring characteristics. The final shape and dimensions of the transition slabs have been the result of the dynamic analyses performed on these structures. See Figures 5 and 18.
2570 [OP] (= [CP] + 300 mm) 2270 [CP] (Center Support Position)
200
2380
232
2550
1970 [IP] (=[CP] - 300 mm) 635
100
500
PLAN VIEW
Figure 18.
650
650
650
1970 [IP] (Inner Support Position) 2270 [CP] (Center Support Position) 2570 [OP] (Outer Support Position) 2770
500
c.t.c. rails 1500
2700
40
500
Approach Ramp
Expansion/Contraction Area
SIDE VIEW
Steel transition slab connecting bridge with approach ramps.
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Bridge HD
180 Track-Bridge Interaction on High-Speed Railways
train body mass train spring and damping characteristics
rail(s)
1500 mm
rail-fastening spring and damping characteristics transition slab mass, flexural and damping characteristics transition slab supports stiffness and damping characteristics
CL track
Figure 19.
Cross section DRS model of transition slab.
Figure 20.
1st and 2nd mode shape of transition slab at 37 and 59 Hz respectively.
Special provisions are taken with regard to the horizontal fixation of the slab, perpendicular to the track and the proper guidance of the slab in longitudinal direction. A steel U-shaped section is bolted down to the slab in the center of track, in line with the concrete derailment provision on the bridge and approach ramps. 6.2
Dynamic model
With program DRS a 3-dimensional grid model has been constructed to incorporate the spring and damping characteristics of the transition slab supports, the steel plate structure of the transition slab, the spring and damping characteristics of 4 pairs of rail-fastenings on the slab plus 10 pairs on both sides of the slab, and 2 rails type UIC60. See Figures 19 and 20. The steel plate structure and rails provide only little damping to the system. A ratio of 0.005 is taken into account. A considerably higher damping ratio of 0.100 is provided by the rail-fastening as they incorporate a highly elastic intermediate layer. Another source of damping lies within the PE-elements which are part of the special supports of the transition slab. Tests, as performed by the Technical University of Munich on the stiffness and damping characteristics of these supports, show that damping ratios ranging from 0.050 to 0.140 may be expected. In the dynamic model and analyses the lower bound value of 0.050 has been assumed. 6.3
Results dynamic analyses
The dynamic analyses performed on the transition slab have led to a change of the original design, which showed a configuration of H and U-shaped beams. The response of this type of slab was
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181
40.0 37.5 35.0
Due to DW of transition slab (30 kN/support) K = 250 MN/m, Dratio = 0.005 K =250 MN/m, Dratio = 0.020
32.5
Upward directed support forces [kN]
30.0 27.5
K =250 MN/m, Dratio = 0.050 K =500 MN/m, Dratio = 0.005 K =500 MN/m, Dratio = 0.050 K =1000 MN/m, Dratio = 0.005
25.0 22.5
K =1000 MN/m, Dratio = 0.020 K =1000 MN/m, Dratio = 0.050
20.0 17.5 15.0 12.5 10.0 7.5 5.0 2.5 0.0 160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
Train velocity [km/h]
Figure 21.
Maximum uplift support forces versus train velocity, during passage of ICE3M(2) train.
such that the filtered, structural accelerations would not comply with the design criteria, while in upward motion the dynamic forces at the supports would overcome the static forces due to dead load, resulting in loss of contact. This is an undesirable effect with regard to the support’s capacity to withstand fatigue. By introducing a design with a higher stiffness-mass ratio the natural frequency of the combined system (rails, fastenings, transition slab and supports) has been increased. In general, the filtered vertical structural accelerations will decrease with an increase of the system’s natural frequency. At the same time, the increase of mass works out positively as a measurement to reduce uplift. From test results the vertical (secant) stiffness of the supports was established which ranged between 380 and 1125 MN/m, depending on the frequency of the applied load and the number of load cycles. The natural frequency of the system largely depends on the support stiffness and ranges from 37 to 54 Hz (1st mode shape, see Fig. 20) within the established range of support stiffness. These results have been obtained for the situation where the supports are located in the outer position (see also Fig. 18), hence when the bridge-structures is fully contracted. This is considered to be the governing situation, creating the largest span and consequently the lowest natural frequency possible. Due to the increase of natural frequency, the filtered vertical accelerations have been reduced to a maximum value of 0.24 g. The unfiltered results however, show values exceeding 1.00 g. Although these values occur at the center of the track and show a decrease towards the supports, the possibility of a temporary uplift, has been thoroughly investigated. In Figure 21, the impact of the support stiffness (selected values: 250, 500 and 1000 MN/m) and damping ratio (selected values: 0.005, 0.020 and 0.050) is shown for the uplift support forces during the passage of the governing train type ICE3M(2). With the increase of the support stiffness, the position of the peak values for the uplift forces shift to higher train velocities, without showing a severe reduction nor increase in value. As to be expected, the support damping provides a considerable reduction of these peak values. Within the stiffness range of 250 to 1000 MN/m, a damping ratio of 0.050 (5% of the critical damping) reduces the uplifting support forces to 50–60%
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182 Track-Bridge Interaction on High-Speed Railways
30.0 20.0
Support Forces [kN]
10.0 0.0 0.000 -10.0
0.500
1.000
1.500
2.000
2.500
3.000
3.500
4.000
4.500
5.000
5.500
-20.0 -30.0 -40.0 -50.0 -60.0 -70.0
Time [s]
Figure 22.
Time-history of support forces during passage of ICE3M(2) at 290 km/h.
of the support forces due to the dead load of the system, which is 30 kN per support. This results in a minimum safety factor against uplift of 1.67. Having adapted the design of the transition slab with the restriction of uplifting forces at the supports in mind has also been beneficial to the requirements with regard to the dynamic coefficients, the level of passenger comfort and the forces in the rail-fastenings as they do not reveal anything out of the ordinary. The dynamic bending moments in the transition slab remain below the static bending moments derived during the passage of static load models LM71 and SW2. This results in a dynamic coefficient φr below unity. Due to its short determinant length [Lφ ] coefficient φ2 is governing with a value of 1.67. The upward dynamic response of the steel plate is compensated by its dead weight. In a conservative approach of the passenger comfort, the governing RMS-value of the vertical train accelerations is established at 0.290 m/s2 , hence below the maximum allowable value of 0.525 m/s2 . The dynamic amplification factor of the forces in the rail fastenings does not exceed 1.40. In Figure 22, the time-history results for the support forces are shown during the passage of train type ICE3M2 at a velocity of 290 km/h, with a support stiffness of 500 MN/m and a damping ratio of 0.050. The characteristics of the train, axle positions and loading effects (2 times 8 wagons with 4 axles each) are clearly recognizable. Out of the ordinary structures, especially small sized structures such as these transition slabs, combining a lot of different functions on a concentrated area, often require more attention than usual and may create more questions than answers. The basic design principle of these transition slabs had already been applied and proven itself before in HSR tracks abroad. New, however were the operational train velocity, the train characteristics and the depth of the dynamic analyses. Coming up with a workable solution proved to be an interesting engineering challenge.
7
CONCLUSION
The structures in the HSL-Zuid project generally show natural frequencies in the range of 3 to 10 Hz. Usually, the governing dynamic criteria for this type of structures are the (filtered) structural accelerations, which depend on span length, type and velocity of passing train and the presence of cantilevered ends. The THALYS train for instance shows governing results for spans between 16 and 20 m, while the ICE3M train shows governing results for spans just over 20 m. Passenger comfort becomes an issue for structures with considerably larger spans, certainly in situations where there are 3 or more similar spans in a row.
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183
Figure 23. Transition slab just hoisted into position. (Photo, courtesy of Mr. P. Meijvis, DMC bv, The Netherlands).
Figure 24. Rheda 2000 slab track system on bridge across the Hollandsch Diep. (Photo, courtesy of Mr. P. Meijvis, DMC bv, The Netherlands)
© 2009 Taylor & Francis Group, London, UK
184 Track-Bridge Interaction on High-Speed Railways
One of the largest structures in the HSL-Zuid project, the bridge across the Hollandsch Diep and one of the smallest structures, the transition slab covering the expansion areas of this bridge, have in common that their dynamic behavior has played a considerable factor in their final design. Considering the many dynamic analyses performed on other HSL- structures, these structures have shown features which were out of the ordinary. The low natural frequency of the bridge across the Hollandsch Diep of 1.10 Hz makes this structure relative insensitive to the dynamic impact of real train loads, resulting in low structural accelerations. However, trains with natural body frequencies ranging between 0.8 to 1.6 Hz are easily exited, herewith compromising the required level of passenger comfort. Several measures were taken to keep the passenger comfort within and acceptable level, of which one has been the application of a pre-camber in the vertical rail level during the installation of the Rheda 2000 slab track system. The bridge-structure and the track system are subject to different comfort criteria. The more severe criterion for the bridge structure allows for a better approach of the rail roughness impact in the design of the track system. However, for the sake of clarity both criteria should merge into one. The transition slab is a structural element with a relative high natural frequency. Sliding over the 4 supports but not vertically fixated the main problem here lies in the possible loss of contact which is not beneficial to the fatigue strength of the supports. By increasing the stiffness over mass ratio and establishing the damping properties of the sliding PE-elements in the supports, the (filtered) structural accelerations of the transition slab and the uplift forces in the supports are kept within acceptable limits. REFERENCES High Speed Line – South project guidelines HSL600E “Loads and Deformations on HS-structures”, 1999. EN 1991-2: Eurocode 1, “Actions on structures – Part 2: Traffic loads on bridges”, 2003. “Dynamics of Railway Bridges”, Ladislav Fryba (ISBN: 0-7277-2044-9), 1996. “Structural Dynamics by Finite Elements”, William Weaver Jr., Paul R. Johnston (ISBN: 0-13-853508-6), 1987. “Finite Element Procedures”, Klaus-Jürgen Bathe (ISBN: 0-13-301458-4), 1996.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 15 Track-structure interaction in long railway bridges A.J. Reis Technical University of Lisbon & GRID SA, Lisbon, Portugal
N.T. Lopes & D. Ribeiro GRID SA, Lisbon, Portugal
ABSTRACT: The concept design for long railway bridges shall take into consideration the need to balance imposed deformations of the deck, due to thermal effects, shrinkage and creep in concrete and composite structures and induced stress effects in the rails. The need to reduce the number of rail expansion devices requires moderate lengths of continuous superstructures and so the introduction of expansion joints in the deck. In seismic zones, the concept design of the bridge shall balance the advantage of short continuous lengths of the superstructure, to reduce track-structure interaction effects, with the inconvenient of transferring the seismic forces to a limited number of piers. The bridge design shall take into consideration a variety of other variable actions inducing stresses in the rails due to longitudinal displacements, namely associated to braking forces and vertical actions. Due to the continuity of the rails on the structural expansion joints, the deformations of the deck induce stresses in the rails that need to be checked. The main design criteria are now specified in Eurocode 1-Part 2. The concept design for two long railway bridges, located in seismic zones, is discussed in the present paper. Prestressed concrete and steel-concrete composite superstructures are considered. To check track-structure interaction, a numerical model based on EC1-Part 2 and UIC 774-3 was developed and results are presented for one of these viaducts.
1
INTRODUCTION
In concept design of long railway bridges (above 400 m) one of the main constraints is the trackstructure interaction, i.e. the concept design of the track itself. The ballasted continuous track is very popular among the track owners, since they can avoid introduction of rail expansion devices, and so, reduce track maintenance costs. On the other hand, the maintenance of ballasted tracks is a very well known problem and track owners have large experience on dealing with it. Although, this concept applied to long viaducts raises several problems regarding the track-structure interaction, namely the effect of structural deformations on rail stresses. This is not a specific problem of the High Speed Railway bridges, so from this point of view, there is no difference between HSR bridges and conventional railway bridges. Generally, regular bridges without special constraints, can be conceived for continuous ballasted track, adopting short spans and as many structural expansion joints as needed, but when other important structural constraints are imposed, such as topographical, geotechnical and seismic constraints, the structural design may prevail. This paper focus particularly on the design of two solutions for long railway bridges in similar environment, but with two different solutions to solve the track-structure interaction problem. The bridges are inserted in the same track stretch of the Portuguese south railway line, linking Lisbon to Algarve, but have slightly different ground conditions, and both are implanted in a seismic area. In this stretch, the design speed is 220 km/h. 185 © 2009 Taylor & Francis Group, London, UK
186 Track-Bridge Interaction on High-Speed Railways
Table 1. General features of the bridges (in meters). Access Viaducts
Length Span distribution Maximum Height Number of rail expansion devices Continuous segments
Figure 1.
São Martinho Viaduct
North
South
1115 35 + 6 × 37.5 + + 19 × 45 22.5 2
1140 17 × 45 + + 10 × 37.5 22.0 2
852 30 × 28.4
260 + 45 + 765 + 45
45 + 720 + 37.5 + 337.5
7 × 113.6 + 56.8
13.5 none
São Martinho Viaduct Cross-Section.
One of the design solutions was adopted in both approach viaducts – North and South – to the new bridge over river Sado, at Alcácer do Sal. The other solution refers to the São Martinho viaduct, a few kilometres towards north from the Sado crossing (Table 1). 2 2.1
BASES OF DESIGN São Martinho Viaduct
São Martinho river is one of the most important tributaries of Sado river’s north bank. The location of the new railway viaduct, with the same name, is not far from the estuary of Sado, and is included in its Natural Reserve, an important sanctuary of birds. The viaduct is now under construction and has a prestressed concrete deck, composed of two main girders connected by the railway platform concrete slab. It is a ballasted double track solution with continuous rail (Figure 1). The crossing of São Martinho valley has required a long viaduct with more than 800 m, as the geotechnical conditions did not allowed an earthfill solution. There were not many constraints to piers implantation, so it was possible to establish a very regular distribution of spans; with 30 consecutive 28,4 m spans and a viaduct total length of 852 m. With 28,4 m spans, it was possible to divide the bridge length in a total of 8 continuous segments, 7 with 4 spans and 113,6 m length, and 1 last (south side) with 2 spans and 56,8 m length (Figure 2).
© 2009 Taylor & Francis Group, London, UK
Figure 2. São Martinho Viaduct. Prestressed Concrete Deck, with 852 m (30 × 28,4 m) total length. Double-beam section with diaphragms at supports. Total width: 13 m, for double track.
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188 Track-Bridge Interaction on High-Speed Railways
Figure 3.
São Martinho Viaduct. Typical 4-span Viaduct Segment.
A significant part of the total length of the viaduct was implanted over very thick alluvial deposits, imposing deep pile foundations (20–30 m). Despite the high deformability of the foundations, it was possible to maintain the continuity of the rail track without any expansion device, mobilizing a total of 4 stiff piers in each viaduct segment, that are responsible to control structural deformations and ensure track safety requirements (Figure 3).
2.2 Approach Viaducts to new Bridge over Sado River The two approach viaducts to the new railway bridge over Sado River have basically the same solution, with steel-concrete composite decks and concrete piers. Like in São Martinho Viaduct, it is a double track ballasted deck. The cross section is composed of two plate girders 2,6 m high, supporting a concrete slab with variable thickness (30–40 cm). For many reasons, but mainly because of the simplicity and low intrusive process, the constructive solution was the incremental launching method, where steel girders present many advantages. This part of the Sado river valley is inside the Natural Reserve of the Sado estuary, and here, as in São Martinho River, there is a large area of alluvial deposits, where all piers should be founded. The incremental launching of steel girder imposes, for economical reasons, 40–50 m spans, and the adopted solution combines few 37.5 m spans with the typical 45 m span. Considering the spans, the piers height (over 20 m) and the poor geotechnical conditions, a fractioned superstructure like São Martinho it wasn’t feasible, therefore, the adopted solution included expansion rail devices, already foreseen in the bridge over Sado, where the structure is continuous along 480 m extension (Figure 4). The viaducts superstructure was divided in two continuous parts with a single simply supported span in the middle, where the rail expansion devices should be implanted. The distribution of continuous superstructure segments is presented in Table 1, and consists in a main long segment of 720 or 765 m length, between the bridge and the simply supported span, and a second long segment with 260 or 337.5 m length, fixed in the abutments. The transition to the bridge has also expansion rail devices (both sides) implanted in a simply supported span. In order to control longitudinal deformations and to face longitudinal horizontal forces, such as braking and acceleration, that for this continuous length totalize 7000 kN, the main continuous segments were fixed not to one single pier but to as much piers as allowed by thermal effects. The concentration of forces in one single very stiff pier was considered uneconomical, as regular piers, with some additional reinforcement, proved to be adequate for that purpose. This solution is more favourable for the foundations design as well. To face the seismic action, controlling the global deformation, some additional piers need to be mobilized, adopting special sliding bearings with dampers, fixing the pier under seismic action. For the second continuous segment, the superstructure is fixed at the abutment (Figure 5).
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction in long railway bridges
Figure 4.
3 3.1
189
Approach viaducts cross-section.
SEISMIC DESIGN São Martinho Viaduct
Seismic design of São Martinho Viaduct has faced a particular problem, as some localized sands occurring in the south part of the alluvial deposits reveal to have high liquefaction potential. The general solution consists on fixing each segment of 113.6 m, on its own piers, including the transition pier, as presented in Figure 3. In the design it was left enough clearance between adjacent segments, to prevent the risk of shock during seismic action. The last three segments, implanted over one layer of sands with high liquefaction potential were connected with dampers, between each other and at the abutment, to reduce the seismic impact to the foundations (see Figure 6). 3.2 Approach viaducts to new Bridge over Sado River The seismic design of the approach viaducts, as previously described, considers two types of force transmission, as the viaducts are divided in two continuous segments. The continuous segment adjacent to the abutments is fixed at one point – the abutment itself. The main continuous segments, with 720 or 765 m long, are continuously fixed for seismic action along a set of 8 piers, 4 of them with special sliding bearings, provided with dampers, that are only fixed under seismic actions (Figure 7). The single supported stretches are fixed at the transition piers.
4 4.1
TRACK-BRIDGE INTERACTION Introduction
Continuous tracks when crossing support discontinuities, such as embankment-bridge transitions, or structural expansion joints, transmit horizontal forces directly applied to the support, which have a stiffness discontinuity, producing stresses concentration in the rails.
© 2009 Taylor & Francis Group, London, UK
Figure 5. Approach viaducts to the new Bridge over Sado River (South). Composite steel-concrete deck, approx. 1100 m total length. Double plated steel girders section, with tubular diaphragms. Total width varying between 13 m and 15.7 m. Double track. © 2009 Taylor & Francis Group, London, UK
Track-structure interaction in long railway bridges
Figure 6.
191
Particular Solution for south part of São Martinho Viaduct. Connections with dampers.
In the same way, where continuous rails restrain the free movement of the bridge deck, deformations of the bridge deck (e.g. due to thermal variations, vertical loading, creep and shrinkage) produce longitudinal forces in the rails and in the fixed bridge bearings. The effects resulting from the combined response of the structure and the track to variable actions shall be taken into account for the design of the bridge superstructure, fixed bearings, the substructure and for checking load effects in the rails. The next specifications, based on section 6.5.4 of EN 1991-2:2003 are valid for conventional ballasted tracks, which is the design case to be discussed.
4.2
Combination of Actions
The following actions shall be taken into account: – Traction and braking forces: For double track bridges the braking force in one track must be considered with the traction forces in the other track. – Thermal effects in combined structure and track system: Temperature variations in the bridge should be taken as TN (uniform temperature variation), with γ and ψ taken as 1,0. – Classified vertical traffic loads (including SW/0 and SW/2 where required). Associated dynamic effects may be neglected. – Other actions such as creep, shrinkage, temperature gradient etc. shall be taken into account for the determination of rotation and associated longitudinal displacement of the end sections of the decks where relevant.
© 2009 Taylor & Francis Group, London, UK
Figure 7.
Approach viaduct (North) seismic design. Dampers and bearings distribution.
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction in long railway bridges
193
When determining the combined response of track and structure to traction and braking forces, the traction and braking forces should not be applied on the adjacent embankment unless a complete analysis is carried out considering the approach, passage over and departure from the bridge of rail traffic on the adjacent embankments to evaluate the most adverse load effects.
4.3 Tensions on the track For rails on the bridge and on the adjacent abutment the permissible additional rail stresses due to the combined response of the structure and track to variable actions should be limited to the following design values: – Compression: 72 N/mm2 ; – Tension: 92 N/mm2 . The above given limiting values are valid for tracks complying with: – UIC 60 rail with a tensile strength of at least 900 N/mm2 ; – Straight track or track radius r ≥ 1500 m; – For ballasted tracks with heavy concrete sleepers with a maximum spacing of 65 cm or equivalent track construction; – For ballasted tracks with at least 30 cm consolidated ballast under the sleepers. In the viaducts to be discussed, all the above criteria are satisfied. When any of the criteria is not satisfied special studies should be carried out or additional measures provided.
4.4
Deformation of the structure
4.4.1 Longitudinal displacement Due to traction and braking δB shall not exceed the following values: – 5 mm for continuous welded rails without rail expansion devices or with a rail expansion device at one end of the deck, – 30 mm for rail expansion devices at both ends of the deck where the ballast is continuous at the ends of the deck, – movements exceeding 30 mm shall only be permitted where the ballast is provided with a movement gap and rail expansion devices provided. Where δB is: – the relative longitudinal displacement between the end of a deck and the adjacent abutment or, – the relative longitudinal displacement between two consecutive decks. Due to vertical traffic actions up to two tracks loaded with load model LM 71 (and where required SW/0) δB shall not exceed the following values: – 8 mm when the combined behaviour of structure and track is taken into account (valid where there is only one or no expansion devices per deck), – 10 mm when the combined behaviour of the structure and track is neglected. Where δB [mm] is: – the longitudinal displacement of the upper surface of the deck at the end of a deck due to deformation of the deck.
© 2009 Taylor & Francis Group, London, UK
194 Track-Bridge Interaction on High-Speed Railways
4.4.2 Vertical displacement The vertical displacement of the upper surface of a deck relative to the adjacent construction (abutment or another deck) δV due to variable actions shall not exceed the following values: – 3 mm for a Maximum Line Speed at the Site of up to 160 km/h, – 2 mm for a Maximum Line Speed at the Site over 160 km/h.
5
TRACK-BRIDGE INTERACTION MODELATION
5.1
Principles
For the determination of load effects in the combined track/structure system a model based upon Figure 8 (from EC1, part 2: EN 1991-2:2003) was used. The longitudinal load/displacement behavior of the track or rail supports may be represented by the relationship shown in Figure 9 (from EC1, part 2: EN 1991-2:2003) with an initial elastic shear resistance [kN/mm of displacement per m of track] and then a plastic shear resistance k [kN/m of track]. The diagrams (4) and (6) were adopted, with the following values (from UIC 774-3): Loaded track (4): k = 60 kN/m, u0 = 2 mm Unloaded track (6): k = 20 kN/m, u0 = 2 mm For the design case to be discussed there will be a change in the future from one way/track to two way/track use. Hence all cases are taken in account. For the calculation of the total longitudinal support reaction FL and in order to compare the global equivalent rail stress with permissible values, the global effect is calculated as follows: FL =
ψ0i Fli
(1)
with: Fli The individual longitudinal support reaction corresponding to the action i; ψ0i For the calculation of load effects in the superstructure, bearings and substructures the combination factors defined in EN 1990 A2 shall be used; ψ0i For the calculation of rail stresses, ψ0i shall be taken as 1.0.
Figure 8.
Model of track-structure system.
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction in long railway bridges
195
When determining the effect of each action, the non-linear behaviour of the track stiffness shown in Figure 9 should be taken into account. The “k” value depends if the track is loaded or unloaded. This means that for each loading, a different computer model has to be calibrated, according to the loaded and unloaded positions of the track. The longitudinal forces in the rails and bearings resulting from each action may be combined using linear superimposition.
5.2
Model Description
The track/structure interaction model developed for the viaduct was a global frame model, with bar elements to simulate: – – – –
deck, piers, piles, track.
The computer model simulates the entire viaduct and also an additional of 300 m (from each side of the viaduct) of track in the ground, above the backfill. All the 7 independent frames were included in the computer model (7 × 4 × 28.4 + 1 × 2 × 28.4 = 852 m), according to the general principles of modelation and the global structural model. The connection between the track and the structure was modelated by elements with nonlinear response (as defined above), and each frame element has 1 m length, in general.
Figure 9.
Variation of longitudinal shear force with longitudinal track displacement for one track.
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196 Track-Bridge Interaction on High-Speed Railways
Figure 10.
Track-structure analysis model, including the backfill of the north abutment and the first 5 spans.
When we have a ballasted track, according to UIC 774-3, we may take: – u0 = 2 mm, – Track-structure plastification force: Loaded track (4): k = 60 kN/m, Unloaded track (6): k = 20 kN/m, The track and the nonlinear modelation were extended to both sides of the structure of the viaduct, to take into account the track on the ground, admitting the soil rigid, as defined in EC1. On Figure 10, one show the analysis model, between the piers P1 and P5, where are modelated: – the foundations, – the track over the deck, – the track over de terrain. In this section, there are already discontinuities between the deck and the abutment (where the structure begins) and over the pier P4 (with doubling of nodes on top), being the track, and his connections, continuous. Four braking positions and six traction positions were modelated, with the purpose of investigating the most unfavorable position for these actions. It was concluded that the most unfavorable effect for the tracks occur where discontinuities in the structure exist, due to differential displacements. There are differential displacements imposed to the structure that the track has to follow, and the load positions try to maximize this effect. The actions in the modelations were (following EC1-part2): i) ii) iii) iv)
Traction loads: Qlak = 33 [kN/m] × La,b [m] ≤ 1000 [kN] Braking loads: Qlbk = 20 [kN/m] × La,b [m] ≤ 6000 [kN] Average temperature in structure: 20 ◦ C with Tcon = −16◦ C e Texp = +22◦ C Vertical loads: Load Model 71
In Figure 11 are shown the traction and braking position forces adopted.
6 ANALYSIS OF THE RESULTS In Table 2 one list the most unfavorable stresses obtained for two critical sections of the track: near the abutment (at the expansion joint) and at the expansion joint between two different frame decks (at the intermediate piers). The safety check of the track is made according to EC1-2 for the maximum additional compression stress in the track, due to this action (Table 3). For the deformation check of the structure, due to traction and braking actions, this type of analysis, based in a track-structure interaction, allows to check the limit defined in EC1-2 of 5 mm. On Table 4 one presents the results for each loading case and also the combined value of the displacement.
© 2009 Taylor & Francis Group, London, UK
Figure 11.
Track-structure interaction model: localization scheme: traction actions (Axx) and breaking actions (Fxx).
© 2009 Taylor & Francis Group, London, UK
198 Track-Bridge Interaction on High-Speed Railways
Table 2. Track verification: Axial force. Action
Abutment Joint kN
Pier Joint kN
Braking Temperature (22◦ C) LM71 – Vert. Total
886 1121 <10 2017
831 1306 <10 2147
Table 3. Track verification. Security Check (4 rails > 550 × 4 = 2200 kN). Security Check
Abutment Joint kN
Pier Joint kN
N Allow. – Comp. 72 MPa Rail security check.
2200 0.917
2200 0.976
Table 4. Deck displacements due to traction and braking actions. f1
f2
f3
f4
a11
a12
a21
Zone Abut. N P04 – N P04 – S P08 – N P08 – S P12 – N P12 – S P16 – N P16 – S P20 – N P20 – S P24 – N P24 – S P28 – N P28 – S Abut. S
7
a22
a31
a41
máx.f
máx.a
d.máx
0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.2 0.8 0.9 0.4 0.3 0.1 0.1 0.0 0.0
0.1 0.1 0.3 0.3 0.8 0.7 0.2 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.1 2.2 2.5 2.4 2.2 2.2 2.6 2.6 2.7 2.7 2.4 2.4 2.4 2.3 0.3 0.2
0.8 0.6 0.3 0.3 0.8 0.8 0.2 0.2 0.8 0.9 0.8 0.7 0.2 0.1 0.1 0.1
2.9 2.8 2.7 2.7 3.0 3.0 2.8 2.7 3.6 3.6 3.2 3.1 2.6 2.4 0.5 0.2
mm 2.1 2.2 2.5 2.4 1.8 1.6 0.4 0.3 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.2 0.9 1.0 2.4 2.4 2.4 2.3 0.3 0.2
0.1 0.1 0.3 0.4 1.6 1.7 2.6 2.6 2.7 2.7 0.7 0.6 0.1 0.1 0.0 0.0
0.0 0.1 0.5 0.6 2.2 2.2 2.6 2.5 2.1 2.0 0.4 0.4 0.1 0.1 0.0 0.0
0.8 0.6 0.2 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.2 0.2 0.8 0.8 0.2 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1
0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.4 0.5 0.8 0.7 0.2 0.1 0.0 0.0
CONCLUSIONS
A discussion on track-structure interaction for long railway bridges was presented. The problem of imposed deformations and seismic actions were discussed taking into consideration the need to “balance” the continuity of the structure with the requirements for minimum track expansive joints. A design case was presented according to the Eurocodes, namely in terms of actions and in terms of structural safety verification. The stress analysis allows concluding that the limit stress on the rail is almost reach out, for compression, being, in general, the piers joints the most unfavorable. Although, the braking force produces higher stresses near the abutment joint, due to absorption of this force in the backfill of the abutments.
© 2009 Taylor & Francis Group, London, UK
Track-structure interaction in long railway bridges
Figure 12.
Track axial force diagram due to temperature (structure heating).
Figure 13.
Track axial force diagram due to braking (maximum stress near the abutments).
Figure 14.
Track axial force diagram due to braking (maximum stress near intermediate pier).
199
Besides, for compatibility of displacements, the temperature generates higher stresses on the track near the piers joint (there is a higher displacement amplitude). The vertical forces, in this model, were not critical, and induce very low stress amplitude. The displacement limit of 5 mm, was also verified with some clearance. In conclusion, the track interaction analysis allowed to check the track safety, in terms of Eurocodes criteria, for a joint distance larger than 90 m, although in this situation, the length of 4 × 28.4 = 113.6 m is almost at the allowable limit for stress verification.
© 2009 Taylor & Francis Group, London, UK
200 Track-Bridge Interaction on High-Speed Railways
ACKNOWLEDGEMENTS Thanks are due to REFER for allowing the publication of some results of specific studies developed for the design of the new stretch of the Portuguese south railway line, near Alcácer do Sal, that includes the New Sado River railway crossing. REFERENCES EN1991-2, Actions on structures – Part 2: General actions – Traffic Loads on Bridges, European Committee for standardization, CEN (2003). UIC Code 774-3-R, “Track/bridge interaction – Recommendations for calculations”, 2nd edition, October 2001, Union International des Chemins de Fer, UIC, 2001.
© 2009 Taylor & Francis Group, London, UK
CHAPTER 16 Track-bridge interaction in railway lines: Application to the study of the bridge over the River Moros R. Simões, R. Calçada & R. Delgado Faculty of Engineering of the University of Porto, Porto, Portugal
ABSTRACT: The present work deals with the phenomenon of track-deck interaction in railway bridges. Following a brief description of the methodology of analysis and the safety checks established in the most recent European regulations on this matter (EN1991-2), an application is presented regarding the study of the bridge over the river Moros, a 476,0 m long continuous deck bridge with a box type of cross section, designed to operate in the Segóvia – Valladolid high-speed railway line.
1
INTRODUCTION
One of the fundamental aspects to take into account in the conception and design of railway bridges refers to the effects resulting from the track-deck interaction. The demands currently faced by modern railway lines have led to the adoption of continuous welded rail (CWR), towards a reduction in maintenance efforts and an increase in the level of comfort of the passengers. When a rail is placed continuously over a bridge or viaduct, both structures have to resist to the loads resulting from temperature variations, from the traction and/or braking of the trains, from creep and shrinkage of the deck, which produce efforts in the rails and forces in the supports of the deck. The effects resulting from the combined response of the bridge and the track to these loads should be taken into account not only in the design of the structural elements, but also in the check of the stress state in the rails and of the deformations of the structure, in order to ensure the stability of the track and thus guarantee the safety of circulation [1, 2, 3, 4]. The procedures related with the determination of these effects were included in the latest version of EN1991-2 [5] which reflects the most recent research carried out in this field. In the present work, after a brief description of the methodology of analysis and the safety checks established in the referred regulation, its application is presented to the study of the bridge over the river Moros, a 476,0 m long continuous deck bridge with a box type of cross section, designed to operate in the Segóvia – Valladolid high-speed railway line. This study involves a sensitivity analysis in relation to the parameters that influence the combined response of the track and the structure, namely the length of the zone of track in embankment and the stiffness of the abutment/foundation group. 2 ANALYSIS METHODOLOGY 2.1
Introduction
In 2001, sheet UIC 774-3R [6] was published, reflecting the works of the expert committee D213 of ERRI on the effects of the track-deck interaction in railway bridges. The recommendations 201 © 2009 Taylor & Francis Group, London, UK
202 Track-Bridge Interaction on High-Speed Railways
on this sheet were mostly entirely transferred to the EN1991 2 standard, under a section entitled “Combined response of structure and track”. In this section, the loads and respective combinations to consider are specified, as well as the methods of structural models of the track-bridge system and the safety criteria to be adopted. 2.2
Loads
According to EN1991-2, for the assessment of the effects of the track-bridge interaction the following loads should be taken into account: (i) vertical train loads; (ii) longitudinal forces resulting from starting or braking of the trains; (iii) uniform or differential temperature changes; (iv) shrinkage and creep of the concrete. The vertical permanent or transient loads to be applied are translated by the LM71 (and SW/0 or SW/2) load model. The longitudinal forces resulting from starting or braking are provided by the following expressions: i) starting forces: QLak = 33 [kN/m] × La,b [m] ≤ 1000 [kN], for load models LM71, SW/0 and SW/2; ii) braking forces: QLbk = 20 [kN/m] × La,b [m] ≤ 6000 [kN], for load models LM71and SW/0; QLbk = 35 [kN/m] × La,b [m], for load model SW/2. The characteristic values of the uniform temperature variation in the track and the deck, according to sheet UIC774-3R and EN1991-2, can be taken equal to ±50◦ C and ±35◦ C, respectively. The characteristic values and the distribution of the differential change in temperature in the crosssection are defined in the pre-standard prEN1991-1-5 [7] as a function of the deck typology and of the upper coating. In what concerns the retraction or creep of the concrete, the effects to be considered will depend on the time lag between the construction of the deck and the installation of the track. 2.3
Structural modelling of the track-bridge system
The study of the track-deck interaction involves the implementation of numerical models that enable to consider the configuration and properties of the structure and the track. The model should be able to adequately represent the supports, the columns, and the foundations. In some cases, it is possible to replace the mentioned elements by a connection of equivalent stiffness to that of the foundation/column/support group (Fig. 1). The elements corresponding to the rails and the deck should be located at the level of the respective centres of gravity. Likewise, the connections corresponding to support devices should be placed at the level of their centres of rotation (Fig. 2).
1
2
3
5
5
4 1 - Track
2 - Longitudinal non-linear spring
5 - Embankment
6 3 - Rail expansion joint (REJ)
4 - Deck
6 - Longitudinal springs reproducing the longitudinal stiffness K of a fixed support
Figure 1. Simplified structural model for the analysis of the effects of the track-deck interaction (adapted from [6]).
© 2009 Taylor & Francis Group, London, UK
Track-bridge interaction in railway lines
203
The longitudinal behaviour of the track-deck connection can be modelled as a spring which behaviour is translated by a load/displacement relation similar to that illustrated in Figure 3, that is, a bi-linear relation consisting of an initial elastic section up to a relative displacement u0 and a section corresponding to the plastification of the connection for a load of value k, given in kN/m per metre of track length. For the case of a ballasted track, u0 is equal to 2 mm and k takes the values of 20 or 60 kN/m per metre of track development, whether the track is unloaded or loaded, respectively. Hence, the effects of the track-deck interaction have to be derived by means of a non-linear analysis of the system. In accordance with EN1991-2, the total effects may be determined by linear superposition of the effects obtained for each of the loads acting individually. The total normal stress in the rails, or the total longitudinal reaction in the supports, can then be obtained by means of the expression: (1) FL = (ψ0i · FLi ) where FLi is the effect of load i and 0i is the combination coefficient relative to load i, which values are defined in EN 1990-A2 [8]. For the calculation of the normal stress in the rails, the combination coefficients should be considered equal to 1.0. 2.4 Track safety check In terms of the checks regarding track safety, EN1991-2 specifies relative limit states to the normal stress in the rails and the displacements of the structure. In what concerns the rails, the stability analyses carried out for UIC60 rails, manufactured with a steel of strength equal to or higher than 900 N/mm2 , with a curvature radius equal to or higher than rail axis Longitudinal non-linear spring
Longitudinal non-linear spring
Und
efo
rma
ble
bar
de
Un
Neutral axis of deck
Z
ble
ma
for
Z
r
ba
Fixed support
X
(a) Longitudinal model view
Y (b) Section in deck support
Longitudinal shear force in track per unit length, k [kN/m/m of track]
Figure 2. Location in height of the elements corresponding to the rails, the deck and the supports (adapted from [6]).
Figure 3.
Resistence of rail in sleeper (Loaded track)
Resistence of sleeper in ballast (Unloaded track)
u0
Relative displacement between rail and top of supporting deck, u [mm]
Load-displacement relation of the track-deck connection (adapted from [6]).
© 2009 Taylor & Francis Group, London, UK
204 Track-Bridge Interaction on High-Speed Railways
1500 m, fixed to concrete sleeper with a spacing of 0.65 m or less, over a consolidated ballast layer with a minimum thickness of 0.30 m, have demonstrated that the normal stress in these elements, for the combined action of diverse loads, should not exceed the following values: a) 72 N/mm2 , in compression; b) 92 N/mm2 , in tension. In terms of the structure displacements, the following limits should not be exceeded: i) due to starting and braking, the relative longitudinal displacement between the end of the deck and the abutment, or the relative longitudinal displacement between two consecutive spans of the deck, δB, should not go above 5 mm, for the case of a continuous welded rail (CWR) without dilation devices or with one dilation device in one of the ends of the deck, and 30 mm, for the case of a deck with dilation joints in the rails at both ends and a continuous ballast layer over those ends; ii) due to vertical loads, the longitudinal displacement of the upper surface of the deck in its end, δH, should not exceed 8 mm when the composite behaviour of the track and the structure is taken into account, and 10 mm when that composite behaviour is ignored; iii) furthermore, the vertical displacement of the upper surface of the deck in relation to the adjacent structural element (abutment or span of the deck), δV, should not exceed 3 mm, for maximum speeds on site of 160 km/h, and 2 mm, for maximum speeds greater than 160 km/h. 3 APPLICATION TO THE STUDY OF THE BRIDGE OVER RIO MOROS 3.1
Description
The bridge over the river Moros, designed to operate in the Segóvia – Valladolid high-speed railway line, is composed by a continuous deck with total length of 476.0 m, consisting of seven intermediate spans of 56.0 m and two end spans of 42.0 m (Fig. 4). Longitudinally, the deck has a fixed support (AF) located over the abutment E1, while the remaining supports are mobile (AM). The track exhibits one dilation device (AD) at the end of the deck adjacent to abutment E2. The deck, of pre-stressed reinforced concrete, presents a box type of cross-section and its geometry is represented in Figure 5, serving as support to two railway tracks (V1 e V2). The track is composed of UIC60 rails placed over concrete sleepers, spaced between them by 0.60 m, positioned on top of a ballast layer with a minimum thickness of 0.35 m under the sleepers. The outline at the bridge site presents a longitudinal gradient of 0.21%, with a plan radius of 11 444 m. 3.2
Modelling
In Figure 6 is illustrated the finite element model developed for the analysis of the track-deck interaction. The deck has been discreetized by means of beam elements with the following characteristics: elasticity modulus E = 35 GPa; section A = 11.19 m2 ; moment of inertia I = 28.82 m4 and thermal dilation linear coefficient αC = 1.0 × 10−5 /◦ C. Each of the tracks was modelled by means of beam elements placed in the alignment of the respective axles. An extension of the track has been modelled corresponding to the length of the
z[m] REJ 890 880 870 860 850
Track 892m
FR
FR
FR
FR
FR
REJ - Rail expansion joint
Figure 4.
FR
FR -Free support FF- Fixed support
Longitudinal view of the bridge over river Moros.
© 2009 Taylor & Francis Group, London, UK
FR Moros River 862m
FR
FR
FF
x[m]
Track-bridge interaction in railway lines
205
0.30 0.30
deck and 300.0 m to the side of the abutment E1 in order to simulate the track over the adjacent embankment (Fig. 6a)). The characteristics of these elements correspond to two UIC 60 rails, that is: elasticity modulus E = 210 GPa; section A = 0.0153 m2 ; moment of inertia I = 6.11 × 10−5 m4 ; thermal linear dilation coefficient α = 1.2 × 10−5 /◦ C. The connection between the elements of the track and of the deck, and between the elements of the deck and the supports was simulated by means by rigid bars (Fig. 6b)). The elements that model the track-deck connection, with non-linear behaviour translated by the relation presented in Figure 3, were placed at a height corresponding to the base of the sleepers. As previously mentioned, the deck is supported by the E1 abutment by means of a fixed support. The stiffness of the elastic support corresponding to the abutment/foundation group was determined based on a model using volume finite elements (Fig. 7) where the elasticity moduli for the foundation soil and the abutment were considered to be Es = 190 MPa and Ec = 25 GPa, respectively, from which k = 660 × 103 kN/m was obtained. In order to assess the influence of the group stiffness, other situations were also analysed corresponding to stiffness values equal to 330 × 103 kN/m (0.5 k), 3300 × 103 kN/m (5 k) and infinite stiffness, that is, deck fixed to the support. The influence of the track length in the embankment zone beyond the E1 abutment was also analysed for the k = 660 × 103 kN/m scenario, and lengths of 100 and 200 m were considered, besides the 300 m.
N.A.
Figure 5.
Cross section of the deck.
z [ m]
1.76 1.76
3.00
2.75
N.A.
0.73
3.00
1.64
N.A.
2.75 1.64 4.50 5.23
3.00
0.10
3.00
0.73
a)
E1
b)
c)
d)
Figure 6. Structural model of the track-deck system: (a) front view; (b) cross section of the support; (c) cross section of the span and (d) detail of the model at the zone of the fixed support.
© 2009 Taylor & Francis Group, London, UK
206 Track-Bridge Interaction on High-Speed Railways
Figure 7.
Finite element model of the abutment-foundation system.
3.3 Analysis and results In the analysis of the track-bridge interaction were considered the longitudinal loads due to starting (A) and braking (F) of the compositions, the vertical loads corresponding to the LM71 load model and the load corresponding to an uniform temperature variation (TU) in the deck of ±35◦ C and in the track of ±50◦ C. Loads resulting from retraction, creep and differential temperature variations were not considered in the present study. For the determination of the effects induced by the LM71 load model, diverse loading configurations were considered in order to obtain the most unfavourable values for each parameter. In what the longitudinal loads are concerned, the most adverse scenario was considered as corresponding to the consideration of braking in track V1 (→) and of traction in track V2 (→). In Figure 8 are presented the results relative to the normal stress on the rails of track V2, which provided the most adverse results, for the different loads and for the different stiffness values considered for the abutment/foundation system. In Table 1 are presented, in turn, the maximum values of the support reactions and the relative longitudinal (δl ) and vertical (δv ) displacements between the ends of the deck and the respective abutment. Combined observation of Figure 8 and Table 1, enables to conclude that: i) the stiffness of the abutment/foundation group has great influence in the values of the normal stress on the rails and of the support reactions; ii) for the LM71 load model, the maximum tensions on the rails occur over the fixed support E1; iii) for an uniform temperature variation loading, the compressions on the rails at the abutment E1 vary according to the stiffness of the abutment/foundation group; iv) for a loading considering the LM71 load model and an uniform temperature variation, the increase of the stiffness of the abutment/foundation group results in an increase of the normal stresses in the rails and of the support reactions; v) for starting and braking loading, the maximum stresses (compression or tension according to the sign of these forces) occur over the fixed support E1;
© 2009 Taylor & Francis Group, London, UK
Track-bridge interaction in railway lines
207
Track T2
Additional rail stresses [MPa]
20 0 -20 -40 -51,74
-60 -80
-66,28 -76,64 -86,33
Fixed support k=3300x10³ kN/m k=660x10³ kN/m k=330x10³ kN/m
-100 -120
-90,83
-126.11
-140
a)
0
100
200
300
400
500
600
x [m] Track T2
20 Fixed support k=3300x10³ kN/m k=660x10³ kN/m k=330x10³ kN/m
Additional rail stresses [MPa]
15 10
15,30 14,28 11,54 9,27
5 0 -5 -10 -15 -20
b)
0
100
200
300
400
500
600
x [m]
Track T2
20 Additional rail stresses [MPa]
10 0 -10 -16,27
-20 -30
-29,89
-40 -50
Fixed support k=3300x10³ kN/m k=660x10³ kN/m k=330x10³ kN/m
-60 -70
-57,23
-77,07
-80
c)
0
100
200
300 x [m]
400
500
600
Figure 8. Normal stress in the rails of track V2 as a function of the stiffness of the abutment/foundation group for the loads: (a) of an uniform temperature variation; (b) of the LM71 load model and (c) of starting and braking.
© 2009 Taylor & Francis Group, London, UK
208 Track-Bridge Interaction on High-Speed Railways
Table 1. Support reactions and longitudinal and vertical displacements of the deck for various loads as a function of the stiffness of the abutment/foundation group. Displacement [mm] Stiffness of the abutment/foundation group [kN/m]
E2 Loads
Support reaction [kN]
δl
δv
Fixed support, K=∞
TU LM71 F(V1)/A(V2)
2713 −471 −6529
−168,47 −0,73 3,04
0 0,21 0
K = 3300 × 103
TU LM71 F(V1)/A(V2)
2614 −439 −6106
−169,50 0,59 5,00
0 0,21 0
K = 660 × 103
TU LM71 F(V1)/A(V2)
2316 −355 −5251
−172,10 −0,22 10,79
0 0,21 0
K = 330 × 103
TU LM71 F(V1)/A(V2)
2038 −285 −4640
−174,54 0,10 16,80
0 0,21 0
Track T2
20
Additional rail stresses [MPa]
0 -20 -40 -51,74
-60 -80
-91,77
300,0m
-100
200,0m -120
-123,29
100,0m
-126.11
-140 0
100
200
300
400
500
600
700
x [m]
Figure 9. Normal stress in the rails due to a loading of uniform temperature variation in the track and deck, for track lengths in the embankment zone of 100, 200 and 300 m.
vi) for this loading scheme, the normal stresses in the rails increase with the decrease of the stiffness of the abutment/foundation group; vii) the relative longitudinal displacements in the moving bearing E2 due to traction and braking loads are strongly influenced by the stiffness of the abutment/foundation group, taken values of 3.04 mm, for k = ∞, and of 16.80 mm, for k = 330 × 103 kN/m; viii) the maximum values of the longitudinal support reactions have occurred for the case corresponding to infinite stiffness of the abutment/foundation group. In Figure 9 are presented the normal stress results relative to track V2, for an uniform temperature variation loading, for track lengths in the embankment zone beyond abutment E1 equal to 100, 200 and 300 m. By looking at the figure it is possible to conclude that, in this case, the minimum length to consider is of about 200 m, from which results an error of approximately 2.2%, taking as reference the compression stress obtained from length 300 m.
© 2009 Taylor & Francis Group, London, UK
Track-bridge interaction in railway lines
Fixed support (K → ∞), Track T2
K=3 300×10 3 kN/m, Track T2 240
∆TU=+15ºC; BT ( → )
120
) Additional rail stresses [MPa]
∆TU=-15ºC; LM71; BT (
160
→
200
126,03
117,06
51,30
80
61,10
40 0 -40
-54,78
-80
-52,62 -108,72
-120
-126,04
-160
200
∆TU=-15ºC; LM71; BT (
160
∆TU=+15ºC; BT ( → )
120
) 126,08
125,56
51,89
80
47,34
40 0 -40
-50,40
-80
-52,60 -113,53
-120
-126,08
-160 -200
-200
-240
-240 0
100
200
300
400
500
600
700
0
100
200
300
400
500
600
700
x [m]
x [m]
K=660×103 kN/m, Track T2
K=660×103 kN/m, Track T2 240
∆TU=-15ºC; LM71; BT (
160
∆TU=+15ºC; BT ( → )
120
) Additional rail stresses [MPa]
→ →
200
126,11
141,21
51,89
80
47,00
40 0 -40
-50,20
-80
-52,60
-120
-132,53
-126,11
-160 -200
200
∆TU=-15ºC; LM71; BT (
160
∆TU=+15ºC; BT ( → )
120
→
240
Additional rail stresses [MPa]
→
Additional rail stresses [MPa]
240
209
) 126,11
141,21
51,89
80
47,00
40 0 -40
-50,20
-80
-52,60
-120
-132,53
-126,11
-160 -200
-240
-240 0
100
200
300
400
500
600
700
0
100
200
300
400
500
600
700
x [m]
x [m]
Figure 10. Maximum and minimum normal stresses in the rails of the V2 track, for the different values of the stiffness of the abutment/foundation group. Table 2. Track safety check in terms of deck displacements. Stiffness of the abutment/foundation group [kN/m]
δB [mm]
∞ 3300 × 103 660 × 103 330 × 103
3.04 5.00 10.79 16.80 5.0 mm
δH [mm]
δV [mm]
2.18 2.19 2.21 2.23
0.24 0.24 0.24 0.24
8.0 mm
2.0 mm
3.4 Track safety check For the track safety check, the combined action of the loads corresponding to an uniform temperature variation (±50◦ C in the track and ±35◦ C in the deck), of traction and braking loads and of the LM71 load model were considered. The combined effects were obtained by means of linear superposition of the effects associated to each isolated load, according to expression (1). In Figure 10 are presented the tension and compression envelopes of the rails in track V2 for the different values of the stiffness of the abutment/foundation groups. The observation of the figure also enables to conclude that the tension (92 + 126 = 218 MPa) and compression (−72−126 = −198 MPa) limits are not exceeded in any of the analysed situations. In Table 2 are presented the maximum values of the relative longitudinal displacement between the end of the deck and the abutment for the braking and starting loads (δB ), of the relative longitudinal displacement of the upper surface of the deck for the LM71 load model (δH ), and of the the relative longitudinal displacement of the upper surface of the deck in relation to the abutment (δV ).
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210 Track-Bridge Interaction on High-Speed Railways
From the inspection of the above table it is possible to conclude that it is only for those cases where the abutment/foundation group is stiffer (k = ∞ e k = 3300 × 103 kN/m) that the displacement δB is inferior to the limit established in EN1991-2. The remaining displacements, δH e δV , stay within the limits for all analysed cases.
4
CONCLUSIONS
In the present paper, the effects resulting from the track-structure interaction have been assessed for the case of the bridge over the river Moros, a continuous deck bridge with a box type of cross section designed for the high-speed railway line between Segóvia and Valladolid. In terms of connections, the deck features a fixed support at one of the ends and moving supports in the other abutment and columns. The track contains a dilation device, next to the abutment with moveable bearings. Analyses were carried out for different values of track length in the embankment zone and of the stiffness of the abutment/foundation group. The obtained results enable to conclude that it will be necessary to model a minimum length of 200 m beyond the embankment zone, and that the results in terms of normal stresses in the rails, of longitudinal support reactions and deck displacements are highly inflenced by the stiffness of the abutment/foundation group. In terms of track safety, it has only been possible to control the relative longitudinal displacement between the end of the deck and the abutment, calculated for the starting and braking loads, for the highest values of the stiffness of the abutment/foundation group, that is, k = ∞ and k = 3300 × 103 kN/m. All other limits have been respected from all the cases considered for the stiffness of the abutment/foundation group.
ACKNOWLEDGEMENTS The authors wish to acknowledge RAVE for the support provided and to Prof. José Maria Goicolea of ETSICCyP in Madrid for the design elements of the bridge over river Moros and for all information on this issue kindly made available.
REFERENCES [1] UIC Code 774-3-R, “Track/bridge interaction – Recommendations for calculations”, 2nd edition, Union Internationale des Chemins de Fer, UIC, 2001. [2] Parra, F., “Proyecto de una puente de alta velocidad”, E.T.S. Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 2005. [3] Simões, R., “Interacção via-estrutura em pontes ferroviárias em linhas de alta velocidade”, Tese de Mestrado, FEUP, Porto, Portugal, 2006 (to be submitted). [4] EN1991-2, Actions on structures – Part 2: General actions – Traffic Loads on Bridges, European Committee for standardization, CEN, 2003. [5] EN1991-1-5, Actions on structures – Part 1-5: General actions – Thermal actions, European Committee for standardization, CEN (2003). [6] EN1990-A2, Basis of Structural Design – Annex A2: Application for bridges (normative), Final PT Draft, European Committee for standardization, CEN (2005).
© 2009 Taylor & Francis Group, London, UK
