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Materials with Rheological Properties
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Materials with Rheological Properties Calculation of Structures
Constantin Cristescu
First published in France in 2006 by Hermes Science/Lavoisier entitled “Calcul de structures de constructions réalisées avec des matériaux à propriétés rhéologiques” First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 6 Fitzroy Square London W1T 5DX UK
John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA
www.iste.co.uk
www.wiley.com
© ISTE Ltd, 2008 © LAVOISER, 2006 The rights of Constantin Cristescu to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Cristescu, Constantin. [Calcul de structures de constructions réalisées avec des matériaux à propriétés rhéologiques. English] Materials with rheological properties: calculation of structures / Constantin Cristescu. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-012-7 1. Building materials--Mathematical models. 2. Building materials--Analysis. 3. Rheology. I. Title. TA404.8.C7513 2008 624.1'8--dc22 200704394 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-012-7 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.
Table of Contents
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Historical background. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Considering the plastic and rheological properties of materials in calculating and designing resistance structures for constructions . . 1.3. The basis of the mathematical model for calculating resistance structures by taking into account the rheological properties of the materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 1
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3
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4
Chapter 2. The Rheological Behavior of Building Materials 2.1. Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Structural steel for construction . . . . . . . . . . . . . . . 2.2.1. Structural steel for metal construction. . . . . . . . . 2.2.2. Reinforcing steel (non-prestressed) . . . . . . . . . . 2.2.3. Reinforcements, steel wire and steel wire products for prestressed concrete . . . . . . . . . . . . . . . . . . . . . 2.3. Concrete. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 9 19 19 22
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23 32
Chapter 3. Composite Resistance Structures with Elements Built from Materials Having Different Rheological Properties . . . . . . . . . . . 3.1. Mathematical model for calculating the behavior of composite resistance structures: introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Mathematical model for calculating the behavior of composite resistance structures. The formulation considering creep. . . . . . . . . . . . 3.2.1. The effects of the long-term actions and loads: overview . . . . . . 3.2.1.1. Composite structures with discrete collaboration . . . . . . . . . 3.2.1.2. Composite structures with continuous collaboration . . . . . . . 3.2.1.3. Composite structures with complex composition . . . . . . . . . 3.2.2. The effect of repeated short-term variable load actions: overview .
45 45 49 49 61 67 80 86
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Materials with Rheological Properties
3.3. Mathematical model for calculating the behavior of composite resistance structures. The formulation considering stress relaxation. . 3.3.1. The effect of long-term actions and loads: overview . . . . . . 3.3.1.1. Composite structures with discrete collaboration . . . . . . 3.3.1.2. Composite structures with continuous collaboration . . . . 3.3.1.3. Composite structures with complex composition . . . . . . 3.3.2. The effect of repeated short-term variable actions and loads: overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Conceptual aspects of the mathematical model of resistance structure behavior according to the rheological properties of the materials from which they are made . . . . . . . . . . . . . . . . .
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95 95 102 106 115
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120
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125
Chapter 4. Applications on Resistance Structures for Constructions . . 4.1. Correction matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. The displacement matrix of the end of a perfectly rigid body due to unit displacements successively applied to the other end of a rigid body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. The reaction matrix of the end of a perfectly rigid body due to unit forces successively applied to the other end of a rigid body . . . . 4.2. Calculation of the composite resistance structures. Formulation according to the creep . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Preliminaries necessary to systematize the calculation of composite structures in the formulation according to the creep . . . 4.2.2. Composite structures with discrete collaboration . . . . . . . . . 4.2.3. Composite structures with continuous collaboration . . . . . . . 4.2.4. Composite structures with complex composition . . . . . . . . . 4.3. The calculation of composite resistance structures. Formulation according to the stress relaxation . . . . . . . . . . . . . . . . 4.3.1. Preliminaries necessary to systematize the calculation of the composite structures in the formulation according to the stress relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Composite structures with discrete collaboration . . . . . . . . . 4.3.3. Composite structures with continuous collaboration . . . . . . . 4.3.4. Composite structures with complex composition . . . . . . . . . Chapter 5. Numerical Application . . . . . . . . . . . . . . . . . . . . . . 5.1. Considerations concerning the validation of the mathematical model proposed for estimation through calculation of the behavior of the resistance structures by considering the rheological properties of the materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The RALUCA computer applications system. . . . . . . . . . . . 5.3. The resistance structure. . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . .
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129 129
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133 136 140 155
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161 165 172 179
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189 191 198 203
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Table of Contents
5.4.1. The first series of experiments . . . . . . . . . . . . . . . . . . . . . . 5.4.1.1. The particular conditions for the analysis of the mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. The second series of experiments . . . . . . . . . . . . . . . . . . . . 5.4.2.1. The particular conditions for the analysis of the mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3. The third series of experiments . . . . . . . . . . . . . . . . . . . . . . 5.4.3.1. The influence of the parameters defining the creep function . . 5.4.3.2. The stresses state in the structure caused by the contraction of the concrete . . . . . . . . . . . . . . . . . . . . . . . 5.4.3.3. The influence of the deformability of the connection elements on the effort’s distribution among the elements of the structure . . . . . Appendix 1. The Initial Stresses and Strains State of the Structures with Continuous Collaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Simply supported beam with uniformly distributed load . . . . . . . . A.2. Simply supported beam loaded with a concentrated force . . . . . . . A.3. Simply supported beam loaded with a concentrated moment at each end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4. Simply supported beam loaded with concentrated forces applied eccentrically, acting on a direction parallel with the axis of the beam . . .
vii
203 204 206 206 211 211 214 217
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223 227 230
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
283
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
287
Appendix 2. Systems of Integral and Integro-differential Equations . 1. Integro-differential equations whose unknown factors are functions of one variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Integro-differential equations whose unknown factors are functions of two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Integro-differential equations whose unknown factors are functions of one or two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1
Introduction
1.1. Historical background During his evolution, man has continuously improved his ability to use resistance and rigidity properties of materials in order to build the constructions he required. The efficiency of the use of materials, their qualities and the technologies used in building the constructions reflect the level of knowledge of the materials’ properties. The idea of using materials having complementary resistance and rigidity properties was necessary at a time when the only available materials were soil (which is compression resistant), ligneous fibers obtained by primitive work with plants (elongation resistant) and wood (having similar resistance and rigidity properties for both elongation and compression), which was accessible only in wooded areas. This statement is supported by the existence, until today, of some very ancient buildings, made of materials having complementary properties of resistance and rigidity. An example of this, for instance, is the ziggurat of Agargouf (Iraq), built by the Babylonians during the 15th century BC, which is made of unfired (sun dried) clay bricks, laid with bitumen mortar reinforced with a network of reed stems every five layers. Later, the Romans used metallic tie rods to take over the pushing stress in constructions with brick or stone vaults. Some other, more recent, achievements are the use of iron and stone beams (end of the 18th century), wood (compressed bars) and steel (stretched bars) lattice girders and cast iron-steel lattice girders (end of the
2
Materials with Rheological Properties
19th century and beginning of the 20th century), as well as wood-aluminum lattice girders (middle of the 20th century). Before listing the achievements of modern times, we have to remind ourselves of a technique whose origin is extremely ancient but which is still widely used, namely the use of mudbricks (unfired, sun dried, bricks made of clay reinforced with straw). Due to its wide diffusion and applications, the most spectacular achievement in using materials having different physico-mechanical properties to manufacture resistance structures for constructions is reinforced concrete, a creation of the 20th century. Reinforced concrete elements include steel bars laid out in the tensioned areas to compensate for the low tension resistance of the concrete. Further achievements in this field (prestressed concrete and composite structures) may be considered, both from a theoretical point of view and for practical applications, such as improvements to reinforced concrete in order to increase its strength and, implicitly, in order to enlarge its application field. Further achievements of the 20th century are reinforced soil constructions, the use of geogrids that take over the tension stress of soil constructions and, lastly, the manufacture of reinforced concrete with dispersed reinforcements. This brief chronological list of achievements in the field of use of materials having different and complementary resistance and rigidity properties shows that progress is only evident in the technological field, in the creation of strong building materials and in their effective use. This implies a thorough knowledge of the resistance and rigidity properties of materials and an appropriate use of this knowledge in designing and building constructions. Until the end of the 18th century, constructions were built using empirical rules, so that the maximum stresses in their constitutive elements had very reduced values compared to the strength capacity of the materials from which they were made. Innovative solutions, resulting from the architect’s intuition, were rare and they were integrated into current practice only if the on-going behavior of the construction was satisfactory. The resulting constructions were massive, very rigid and required large quantities of materials and labor. The small values of the stresses reduced the vulnerability of constructions in their interaction with the foundation ground. High rigidity of the constructions leads to uniform pressures exerted by the foundations on the foundation ground and, implicitly, to quasi-uniform ground settlements, that do not modify the initial stresses and strains state of the constructions.
Introduction
3
The adoption, during the 17th and 18th centuries, of the linear elastic behavior model of building materials, based on investigations carried out on their properties, allowed the development of mathematical models for calculating resistance structures for constructions. The first to be adopted were the simple structures: straight beams, arches, vaults, frameworks having specific configurations. Towards the middle of 20th century general methods of calculating the stresses and strains of any form and configuration of resistance structures were elaborated by generalization and systematization, the general method of stresses and the general method of strains. 1.2. Considering the plastic and rheological properties of materials in calculating and designing resistance structures for constructions Research aiming to evaluate the safety of resistance structures imposed the adoption of behavior models for building materials that are as close as possible to reality, such as the perfect elastoplastic body model (Prandtl) or the hardening elastoplastic body models. The adoption of these behavior models for building materials within the analysis of resistance structures is difficult and involves a huge amount of calculations. This is why, at the beginning, these models served only for the analysis of the strength capacity of the section of the structural components and for the safety analysis of some simple resistance structures. The development, during the second half of the 20th century, of computers able to perform a large amount of operations in a short period of time, allowed the use of these models for the calculation of resistance structures of constructions. This was achieved by the adaptation, in view of the use of models corresponding to the elastoplastic body or the hardening elastoplastic body, of the general methods given by the constructions’ mechanics for the linear elastic body. In this context, we should mention that the results of the calculations obtained by using the elastoplastic behavior models for building materials serve only to evaluate their safety. The stresses generated in the elements of the resistance structures by the actions and loads of normal operation have, for well-dimensioned structures, values well within linear elastic behavior of the materials from which the structures are made. Although the fundamental notions of rheology were developed during the 19th century, the rheological properties of building materials drew the attention of civil engineers after the failure recorded at the beginning of the 20th century in the manufacturing of prestressed concrete.
4
Materials with Rheological Properties
Based on the progress recorded in the knowledge of the rheological properties of materials, whose behavior is always being researched, some calculation models were worked out for specific types of resistance structures, such as prestressed concrete structures and steel-concrete composite structures. Particular effort has been made to develop calculation models for resistance structures having particular configurations that take into account the rheological properties of the foundation ground. However, it should be mentioned that none of the calculation models developed until today have been based on the description of the phenomenon of continuous redistribution of the stresses and strains among the components of the structures, due to the rheological properties of the materials from which they are made. The successes obtained in building resistance structures made from materials having rheological properties are due to satisfactory but simplified assumptions used for the development of the calculation models as well as for the constructive provisions and devices adopted in order to fit the concrete structure to the calculation model. This work includes a general mathematical model describing the behavior of the resistance structures taking into account the rheological properties of the materials from which the constitutive elements are made, including those corresponding to the foundation grounds. The mathematical model we present is based on the principles and equations of viscoelasticity, on the knowledge we currently have about the viscoelastic rheological properties of the building materials, synthesized in their constitutive laws. As for the resistance structures, where we consider the elastoplastic behavior model of the materials, we must also notice the fact that using the proposed model for calculating the resistance structures by taking into account the rheological properties of the materials can be successful only providing the existence of computers and adequate computing software. 1.3. The basis of the mathematical model for calculating resistance structures by taking into account the rheological properties of the materials The mathematical model this work presents describes evolution in time and, taking into account the rheological properties of materials, of the state of stresses and strains of the resistance structures subject to the actions and loads corresponding to normal operation. This model is based on the constitutive laws of the imperfectly viscoelastic bodies.
Introduction
5
The constitutive laws of building materials are presented in the form of general parametric equations, so that the attribution of particular values to the parameters allows the simulation of the behavior of a broad range of materials having viscoelastic rheological properties. We have to mention that the scope of the mathematical model developed thereafter is imposed by the definition domain of the constitutive laws and that this generally includes the definition domain due to the field of stresses generated in resistance structures by the actions and the loads of the normal operation of a construction. We also have to mention that the stresses and strains state generated in the resistance structures by the actions and loads of normal operation, as well as by the phenomenon of redistribution due to the rheological properties, can strongly influence the strength capacity of the structures, and make them vulnerable to accidental actions and loads. Additionally, we have to mention that the parameters defining the constitutive laws of building materials are influenced by many factors, so that, until now, no quantitative correlation between the size values of the factors of influence and the value of the parameters could be established. The values presented in the design standards and/or specialized literature result from the processing of observed values through statistical methods. The values of the parameters of the constitutive laws used in numerical experiments, which are presented in Chapter 5, were taken from Romanian standards for bridge design or from the specialized literature. Equations describing the evolution of the stresses or strains state for a bar (element) which is homogenous from the point of view of its rheological properties are presented using the analogy principle (Volterra), exposed in Chapter 2, and formulated according to the stress relaxation as well as the creep. The assemblage of the structure from its elements (which are homogenous from the point of view of their rheological properties), as well as the construction of the equations expressing this operation, were carried out by applying the conditions of instantaneous elastic equilibrium. At whatever moment, during the existence of the entire structure, its status of elastic equilibrium implies the simultaneous fulfillment of the following conditions: – the static equilibrium condition – the external forces are balanced by the internal forces; – the compatibility condition – the deformed position of the structure is compatible with the existing connections and with the material continuity of its elements.
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Materials with Rheological Properties
The calculation model this work presents was developed assuming that the offset between the deformed position of the structure and its original position falls in the range of small offsets, so that when we apply the static equilibrium conditions, the balanced system of forces refers to the original position of the structure. Analysis of the types of constitutive laws of building materials having rheological properties highlights the fact that these properties do not meet the condition, presented in Chapter 2, that allows the superposition of the effects (Boltzmann) – see the creep and stress relaxation functions of concrete and high resistance prestressed reinforcements. Equations reflecting the instantaneous elastic equilibrium conditions and describing the evolution in time of the stresses and strains state of the structures according to the rheological properties of the materials take into account the restriction imposed by the fact that applying the Boltzmann principle of superposition is fulfilled neither when it is formulated according to the creep, nor when it is formulated according to the stress relaxation (Chapter 3). The validation of the mathematical model developed in this work was ensured by the development of adequate mathematical and numerical methods able to determine the solutions of the integro-differential equation systems describing the evolution in time of the stresses and strains state of the resistance structures and taking into account the rheological properties of the materials from which the structures are made (Chapter 3). This book contains the development of the generalized mathematical model that includes all the types of resistance structures made of materials having rheological properties (Chapter 3), as well as the specific form – corresponding to the different assembly possibilities of the structures – of the integro-differential equation terms describing their evolution (Chapter 4). Lastly, we should also mention that this book includes two appendices whose purpose is to ease the use of the presented mathematical model. In Appendix 2, we describe the methods and solutions for the three types of integro-differential equations corresponding to the different types of resistance structures from the point of view of the mathematical model, also considering the rheological properties of the materials. In this context, we have to emphasize that the method corresponding to the discrete combined action (cooperation) structures (equations containing discrete unknown quantities, depending only on the “time” variable) was employed within
Introduction
7
the framework of the RALUCA computer applications system, which was used to carry out the numerical experiments described in Chapter 5. The appendix includes the solutions of the equations corresponding to the initial phase of the stresses and strains state of some simple structural elements – falling in the category “with continuous collaboration” – that are submitted to actions with frequently used distribution. The purpose of these solutions in the context of this book is obvious if the reader intends to develop software application; the results presented in the appendix can be used in order to validate the software module operating with the elements of the presented type.
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Chapter 2
The Rheological Behavior of Building Materials
2.1. Preamble From the point of view of the effects of the forces on the bodies, we can distinguish three types of physical phenomena: – the behavior of a solid body: when the deformation of the body tends towards a finite shape; – the behavior of a liquid body: when its deformation pace tends towards a finite value, different from zero; – the fracture: when the continuity of the body is destroyed. In the building activity, it is important to identify when the building materials behave like solids. The technological solutions are the result of the knowledge we have about the properties of the materials and about the mathematical models describing their structures. Ancient constructions prove, by the rather reduced diversity of the adapted solutions and by the particularly large quantity of materials used, compared to the level of the loads that they had to support, that the principal source of inspiration for their achievement was practical experiment. The innovations and new solutions that were adopted became, themselves, models only if they proved a capacity for resistance at least equal to that of the old solutions, over one or more generations. At that time, the principal concern of the builders was the quality of the materials and the practical achievement of the pre-established form of the construction through the development of the applications of rigid body Euclidean geometry.
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Materials with Rheological Properties
The economic and social development of humanity and, implicitly, of transportation and constructions led to the search for new solutions to problems for which the traditional methods could not provide an answer. During the Renaissance (the 15th and 16th centuries), the first systematic tests were carried out to determine the strength capacity of the construction elements (Galileo Galilei). Later, during the 17th and 18th centuries, the systematic study of natural laws spread within the framework of the academic societies that appeared in several European countries. Some discoveries in the field of physics led to the formation of fundamental principles for the disciplines dealing with the calculation of constructions. From this period, we should recall the works of Hooke on the relations between stresses and strains, as well as those of Jacob and Daniel Bernoulli, in the field of bending bars, those of Mariotte and of Leonard Euler concerning stresses in straight beams and strain curves of beams and pillars. During the 18th century, engineers started to use theoretical knowledge they had obtained in the framework of academic research for practical applications. The creation of the engineering schools (in France, National School of Bridges and Roadways, 1747) and the first works in the field of technical mechanics were big steps in this direction. It is also necessary to recall the engineering works of Coulomb and Navier (the latter published one of the first works on strength of materials) who largely contributed to the development of theoretical and experimental studies on the analysis of stresses and strains. Based on the theoretical and experimental knowledge that engineers had accumulated, the theory of elasticity emerged and was developed during the 19th century. This theory (using an advanced mathematical apparatus) represents a synthesis that constitutes an abstract mathematical model for the behavior of the material (elastic) body under the action of forces. Mathematicians and physicists such as Poisson, Cauchy, Saint Venant, Lame, Maxwell, Ostrogradski, Stokes and some others contributed to its development. The solution of even the simplest problems of engineering by the methods of the theory of elasticity faces significant difficulties of the purely mathematical type. Therefore, at the same time as the development of the mathematical theory of elasticity – which aims to find solutions to the problems of the mechanics of the deformable body – another theory was developed: the applied theory of elasticity. In this theory, apart from the hypothesis of the material’s elasticity, also true in the mathematical theory of elasticity, some additional assumptions and simplifications are introduced, leading to approximate solutions that are accessible and applicable as practical solutions to a large number of engineering problems. The results of the mathematical theory of elasticity have restricted practical use in engineering; some
The Rheological Behavior of Building Materials
11
of them, which have been altered, became accessible and were applied in the theory of plasticity and in certain chapters of rheology. The theory of plasticity, whose origins date from the 19th century (the experimental works of Tresca, 1868), was developed in its mathematical form during the first half of the 20th century. It establishes the general laws according to which plastic deformations take place and the stress corresponding to all the stages of plastic deformation. Experimental research, corroborated with the application of the theory of plasticity’s theoretical results to the analysis of the structures, highlighted load-bearing capacity reserves and improved the analysis of construction safety. The syntheses elaborated by Prandtl, Baker, Nadai, Socolovski and Hill had an important contribution in the progress of the theory of plasticity. Rheology studies the relationships between stresses and the strains over time. In order to avoid any confusion with the problems of dynamics in the theory of elasticity, we should note that rheology considers the variation of strain properties of materials over time, under the action of a system of constant forces, or in relation to the modification of the stress status under the action of constant imposed movement over time. From this definition, it results that rheology has a wide field of preoccupations, trying to present the properties of materials in the most general situation, as relations (constitutive laws) between the tensor of the tensions, that of strains and that of their derivatives with respect to time, at a given point and at a given moment. Although the basic concepts were established during the 19th century, the interest of builders in rheology emerged at the same time as the extensive use of concrete as a building material and, particularly, manufacturing structural components from prestressed concrete. We also have to mention that, for the civil engineers – particularly for those who are concerned with structures – rheology only offers a few solutions that are applicable to technical projects; it particularly offers qualitative conclusions, although, in the problems of mechanical engineering, where the rheological phenomena take place (high temperature creep or stress relaxation), the solutions found are practical ones. The brief list of the development stages of deformable body mechanics illustrates the fact that the behavior model of materials under the action of forces has been continuously improved, so that it increasingly reflects their real behavior. This situation is due to the improvement of the investigation means and especially to the problems posed by the use of new building materials and technologies. The huge volume of experimental research deployed in building material testing laboratories has enabled the increasingly precise identification of the physico-mechanical
12
Materials with Rheological Properties
properties of materials, their behavior under the action of forces and under various stress conditions. It was also established that for certain stress domains, which are sometimes restricted, certain properties emerge more powerful. It is obvious that the behavior models of materials adopted by branches of mechanical construction used for calculating resistance structures contain fewer components than the reality, but they offer the advantage of allowing more general logical operations. This work proposes a general model for calculating resistance structures, by considering the rheological properties of the materials from which they are made; it thus constitutes an attempt to apply in practice the knowledge and results obtained within the framework of rheology. As in Romania, some excellent works entirely dedicated to the problems of rheology have been published, it is inappropriate to go over its concepts and models here. However, as far as necessary to the establishment of the conditions making the adoption of the principle of superposition acceptable, we will briefly deal with the concepts of creep and stress relaxation and also with the principle of correspondence, which will be used within the framework of the suggested calculation model. The term creep highlights the behavior of the bodies over time and has a double significance: phenomenon and experiment. A creep experiment consists of supervising the variation over the time of the strains of a body under the action of a system of constant forces throughout the entire experiment. If we notice that the strain is a time dependent function then a creep phenomenon has occurred; however, if the strain is time independent then a creep experiment is carried out, but the result is negative. The creep function is defined as the ratio: f (V 0 , t )
H (t ) V0
where: H(t) – the function expressing the strain variation over time; Vo – the constant stress producing the strain.
(2.1)
The Rheological Behavior of Building Materials
13
Elongation
Primary creep Secondary creep Tertiary creep (unstabilized) (unstabilized) (accelerated) x Breaking
d = minimum speed of creep dt
Time t Figure 2.1. The creep of steels at high temperatures
Figure 2.1 graphically represents the general case of a creep experiment (the creep of steels at high temperatures). When loading the sample, the strain increases very quickly, going from zero to an unspecified value (dependant on the applied stress Vo); this is the initial, instantaneous strain, represented by the OA segment. When the load stops increasing, the strain of the samples increases according to a law represented by the curve ABCD. The tangent to this curve in an unspecified point (curve slope) measures the pace of the strain of the body at the moment corresponding to the considered point. In this perspective, we distinguish three different zones on the curve describing the strain law of the samples over time: – the transient creep zone (or primary creep), represented by segment [AB], along which the pace of the strain decreases continuously, from the maximum value on point A (the moment immediately after the end of loading of the sample), to the minimum value, on point B; – the steady state creep zone (or secondary creep), represented by line segment [BC], along which the pace of the strain is constant and minimal (equal to the pace in point B, placed on curve segment AB). The material behaves like a viscous liquid in this zone; – the accelerated creep zone (or tertiary creep), represented by curve segment [CD], where the pace of the strain grows and reaches its maximum at point D, where the creep fracture occurs.
V
3
V 4
V6 V5 V 5 V 4
Materials with Rheological Properties
Elongation [%]
14
V3
V2
V 2 V1 V1
Time [h] Figure 2.2. The variation of creep with stress
Based on the sample material and test conditions, the three zones (transient, steady state and accelerated creep) can be larger or smaller, or missed out altogether. Figure 2.2 represents creep diagrams obtained for steel samples of the same quality, under the same temperature conditions, but under different stresses. We notice that, for reduced stress values, the curve describing the strain’s variation law has a horizontal asymptote (the limit of the strain’s pace tends towards zero) and the strain limit tends towards a finite value and, thus, the sample behaves like a solid body. If, during a creep experiment, at a given moment, we suddenly stop the stress producing the strain, then we notice that, after the instant decrease of the strain with the value corresponding to the elastic strain, this continues to gradually decrease at an increasingly reduced pace, following a curve whose asymptote is horizontal (Figure 2.3).
0
=0 very
curve
0
reco
0 t1 t 2
t3
tk
t
Figure 2.3. Example of recovery curve
The resulting creep curve after the end of the loading is called the recovery curve.
The Rheological Behavior of Building Materials
15
Ordinate Hp of the recovery curve’s asymptote represents the value of permanent strain. If the permanent strain Hp is zero (the recovery curve’s asymptote is the abscissa axis), then the body is perfectly viscoelastic. Just like creep, stress relaxation has two meanings – phenomenon type and experiment type – and highlights the behavior of the bodies over time. A stress relaxation experiment consists of the follow-up of the variation with time of the stress V in a body that has been suddenly deformed, under a constant given strain, Ho, throughout the entire experiment. If the value of the stress varies over time then the stress relaxation phenomenon occurs. If the value of the stress is independent of time then the result of the recovery experiment is negative. The stress relaxation function is defined by the ratio:
r H0 , t
V t H0
(2.2)
where: V(t) – the function expressing the variation of the stress with time; Ho – the strain generating the stress, constant throughout the entire experiment. The diagram represented in Figure 2.4 illustrates schematically the stress relaxation phenomenon in a semi-logarithmic coordinate system.
0
A
Residual tension
Initial load
Type of loading
Figure 2.4. Stress relaxation in semi-logarithmic coordinate system
In the general case of the long-lasting loads, the stress relaxation diagram has the aspect of the letter S. Based on the test conditions (temperature, the stress corresponding to the initial strain), certain zones of the diagram presented in Figure 2.4 become smaller or may be missed out completely.
16
Materials with Rheological Properties
Figure 2.5 presents the stress relaxation diagrams for the same steel sample under various load conditions. We can make the following observations by analyzing these diagrams: – the increase of the temperature – within the limits where this does not cause any change in the internal structure of the sample material – has similar effects with the growth of the stress corresponding to the initial strain; – the stress decreases towards the same final value as for the same initial stress, whatever the test temperature. These observations have high practical importance because they make it possible to obtain the values of the stress loss during long lasting relaxation intervals, by extrapolating the results of the tests carried out during short intervals (1,000 hours) under the same initial stress at high temperatures (100°C or 150°C). We should mention that a creep or stress relaxation diagram is valid only for the material for which the test was performed. We also have to mention that the creep and stress relaxation are manifestations of the same viscoelastic properties of materials, under specific conditions. 130
Initial load % from T
125
Loading temperature [C ] 22 35 50 100
120 115 110
Stress [daN/cm ]
105 100 112
62
108
22 35 50 100
104 100 96
150
92 95
22
90 85 80 75
50
70 60 0,1
52
35 100 50 150 22
1
100 150
43 35
10 100 1000 10000 1million 100000(=115years)
Figure 2.5. The variation of stress relaxation with the load conditions
The Rheological Behavior of Building Materials
17
The principle of superposition (Boltzmann) If quantities of the same nature s1 and s2, called “causes”, both being measurable, produce quantities e1 and e2, also of the same measurable nature, called “effect”, then Boltzmann’s principle of superposition postulates that the effect of the two causes acting simultaneously s1 + s2 is e1 + e2. This principle is valid within the limits imposed by the assumption of the small effects, negligible compared to the unit, so that, from the mathematical point of view, the relationships between cause and effect are described by linear and homogenous equations. The immediate consequence of this principle is the proportionality between cause and effect; to a cause K u s corresponds the effect K u e. In the case of bodies having rheological properties, the superposition of the effects is made by taking into account relations (2.1) or (2.2). Under the action of stresses V1 and V2 applied simultaneously, strain H(V1, V2, t) will have, by taking into account (2.1), the value:
V1 f (V1 , t ) V2 f (V2 , t )
(V1 V2 ) f (V1 V2 , t )
By developing the second member of the equation presented previously and by identifying the corresponding terms, we obtain the validity condition of the superposition principle for bodies having creep:
f (V 1 V 2 , t )
f (V , t )
f (V 2 , t )
f (t )
(2.3)
Proceeding in a similar way, we obtain the condition for bodies having stress relaxation:
r (H 1 H 2 , t )
r (H 1 , t )
r (H 2 , t )
r (t )
(2.4)
Therefore, in the case of bodies having rheological properties, the principle of superposition can be applied only if the creep and/or stress relaxation functions depend only on time (they do not depend of the cause producing the phenomenon). The bodies to which we can apply the principle of superposition are called Boltzmannian or viscoelastic linear bodies. The principle of correspondence In an elastic linear body, under the action of a system of external forces p(x1, x2, x3), stresses V(x1, x2, x3) occur, satisfying the equilibrium conditions for each
18
Materials with Rheological Properties
element of the body. The following relationship (Hooke) exists between the strains and the stresses:
1 V ( x1 , x 2 , x 3 ) E
H ( x1 , x 2 , x 3 )
Let us consider a body made from a viscoelastic material. At moment t = 0, the body is loaded by a system of forces constant in time:
p ( x1 , x2 , x3 , t )
p ( x1 , x2 , x3 )T (t )
where T(t) is the Heaviside step function. The stresses developed in the body will have the value:
V ( x1 , x2 , x3 , t ) V ( x1 , x2 , x3 )T (t ) where V(x1, x2, x3) is the stress corresponding to the case of the elastic linear body. The strains of the viscoelastic body have the value:
H ( x1 , x 2 , x 3 , t ) V ( x1 , x2 , x 3 ) f (t )
(2.5)
From (2.5) it results that at a given moment t z 0, the strains are distributed in the same way in the elastic linear body having the same shape and the modulus of elasticity E is equal to the inverse of the creep function f(t). Relation (2.5) gives the mathematical expression of the principle of correspondence. If a viscoelastic bar is charged with loads that are suddenly applied at the initial moment t = 0 and that remain constant afterwards, the stresses are the same as those of an elastic linear bar of the same shape, submitted to the same loads, whilst the strains and displacements depend on time and derive from those of the elastic problem, by replacing the modulus of elasticity E with the inverse of the creep function f(t),
1 . f (t )
In a similar way, the principle of correspondence has another formulation, equivalent to the first, according to the stress relaxation. If a viscoelastic bar is submitted to displacements in certain points or sections, which are imposed at the initial moment t = 0 and then maintained as constant over time, the strains and displacements are the same as those of the case of the corresponding elastic linear bar, whilst the stresses depends on time and derive from those of the elastic problem by replacing the modulus of elasticity E with the inverse of the stress relaxation function,
1 . r (t )
The Rheological Behavior of Building Materials
19
T. Alfrey and E. Lee stated the principle of correspondence (or principle of analogy) based on the observation that the Laplace image of the constitutive law of the viscoelastic body coincides, from a mathematical structure point of view, with the constitutive law of the elastic linear body (Hooke). V. Volterra highlighted in his works the fact that solving the viscoelastic problems is equivalent to solving the problems of the theory of elasticity. That is why the principle of correspondence or analogy is called “the Volterra principle”. In what follows, we will present the essential properties of building materials, in particular those occurring in the domain of loads to which the resistance structures are subject during normal operation and which, consequently, condition the behavior over the structures over time.
2.2. Structural steel for construction 2.2.1. Structural steel for metal construction Rolled steel products for general use in constructions, included in the STAS 500/2-80 and the STAS 500/3-80 are employed to build the elements for metal constructions or the metal elements composing the concrete-steel structures. The main mechanical, strength and technological properties of these steels are: elasticity, plasticity, brittleness and hardness. They are defined by: their yield point, their breaking strength during elongation and their relative elongation at break (elasticity and plasticity), their cold bending capacity and their fragility (brittleness) and the size of a ball’s indentation on their surface (hardness). Steels are classified based on their breaking strength minimal value, which, expressed in daN/mm2, represents their grade. The mechanical properties, especially the yield point, depend on the thickness of the rolled steel products, large values being obtained for thinner rolled steel products. Table 2.1 shows the main mechanical and technological properties of the most frequently used steels in constructions. The grade and quality class of the steel used to manufacture the elements of a structure are selected according to the structure’s operation conditions, to the manufacturing conditions, to the nature and size of the loads, as well as to the importance of each element for the safety of the structure. For constructions made up of welded elements, having a composed section, the identification of the quality class of the steel can be made based on the danger
20
Materials with Rheological Properties
coefficient and the construction’s temperature of operation, according to the STAS R8548-70 regulations. The creep and the stress relaxation of the steels from which the resistance structures are made have negligible values in the temperatures of their environment, so that the deformations and displacements are considered independent of time. For the purpose of structural analysis, the real shape of the specific curve of the OL37 steel, presented in Figure 2.6a, is transformed in the simplified theoretical diagram in Figure 2.6b. Since the consolidation effect is not considered in the plastic field’s structural analysis, the analysis diagram is reduced to that of a perfect elastoplastic body, presented in Figure 2.6c. The real shape of the specific curve of the slightly alloyed steels OL44 and OL52 presented in Figure 2.7a is reduced to the simplified theoretical curve in Figure 2.7b. For the plastic field’s structural analysis, we use the curve in Figure 2.7c, corresponding to the elastoplastic cold-hardened material. V
V
V
V
VV V
V
0.09 0.25 2.5 H >25 H [%]
V
R V
H >25 H [%]
H
H =8% H [%]
Figure 2.6. Characteristic stress-strain curves for OL37
The shape analysis of the real specific curves of general use steels shows the fact that the steels have practically a linear elastic behavior up to the values (0.8-0.85) Vc. This observation, corroborated by the fact that, by dimensioning, we exclude the possibility of the appearance of plastic deformations under the actions and the loads of normal operation, makes possible the conclusion that steels used for resistance structures behave as a linear elastic body (Hooke); therefore, the constitutive law for the monoaxial stress state has the form:
H
V
1 E
(2.6)
The Rheological Behavior of Building Materials
21
Steel grade (symbol)
Quality class
Product thickness or diameter a(mm)
OL37
1; 1b; 1a
240
230
210
360-340
25
1.0 a
1.5 a
-
-
2 3 4
240 240 240
230 230 230
210 210 210
360-440 360-440 370-440
25 26 26
1.0 a 1.0 a 1.0 a
1.5 a 1.5 a 1.5 a
69 -
69 -
OL44
2 3 4
280 280 280
270 270 270
250 250 250
430-540 430-540 430-540
22 25 25
2.0 a 2.0 a 2.0 a
2.5 a 2.5 a 2.5 a
59 -
59 -
OL52
2 3 4
350 350 350
340 340 340
330 330 330
510-630 510-630 510-630
21 22 22
2.5 a 2.5 a 2.5 a
3.0 a 3.0 a 3.0 a
59 -
59 -
RCA 37
1; 1b; 1a 2 3
240 240 240
360-440 360-440 360-440
25 25 26
2.0 a 2.0 a 2.0 a
69 -
RCB 52
2 3 4
350 350 350
510-610 510-610 510-610
21 22 22
2.5 a 2.5 a 2.5 a
69 -
a<16 16
a>100
-
a>16
a>16
a<16 a>16
Needle diameter Impact Tensile strength Elongation at for coldstrength R KCU 300/2 hardened break A% 2 N/mm 2 bending, at 180º J/cm
Table 2.1. Mechanical characteristics of structural steel used in constructions
V V
B
HH
E
D'
V B
D
H H
V
H
E
V
tg E
V V H
R B
b
tgE V R H RE
H H ,(%)
H ,(%)
a
V
T'
E = tg D
H =8% H ,(%)
c
Figure 2.7. Characteristic stress-strain curves for alloyed steels
The modulus of elasticity (Young’s modulus) for steel, without considering the grade or quality, has the value: E = 210,000 N/mm2
22
Materials with Rheological Properties
The shear modulus of elasticity, also without considering the brand and quality of the steel, has the value: G = 80,000 N/mm2
2.2.2. Reinforcing steel (non-prestressed) The steel for common reinforcements is especially used as round, smooth or periodically profiled, hot-rolled bars. The usual reinforcement (soft, non-prestressed) is used as resistance reinforcement for reinforced concrete structural components, as passive normal (longitudinal) and shear (transversal) reinforcement (stirrup), for the reinforcements of prestressed elements and also for the manufacture of certain types of connection elements (connectors, steel-concrete spirals, inclined bars, loop bars, etc.), for the construction components having a composite concrete-steel structure. The usual reinforcement is made from carbon steel (soft steels) or slightly alloyed steels (semihard steels). The most frequently used reinforcement types are: – OB37 reinforcing bars, round hot-rolled bars of carbon steels with a diameter of 6 to 40 mm; – PC52 reinforcing bars or PC60 reinforcing bars, round periodically profiled hot-rolled bars of slightly alloyed steels with a diameter from 6 to 40 mm; – drawn hot-rolled steel wires with a smooth surface STMB and the diameter of 3 to 10 mm. The mechanical and technological properties of the reinforcing bars are similar to those of structural steel for general purpose metal constructions (see Table 2.2). Just as in the case of structural steel for general purpose metal constructions, under normal operation conditions (temperature, load level), the creep and stress relaxation of steels for common reinforcements (non-prestressed) are negligible. Between the stresses and the strains, we admit, as in the case of steels for general construction use, the constitutive law (2.6) corresponding to the linear elastic body. The modulus of elasticity for common, hot-rolled reinforcements has the value: Ea = 210,000 N/mm2 The modulus of elasticity for the hot-rolled reinforcements STMB has the value: Ea = 180,000 N/mm2
The Rheological Behavior of Building Materials
23
We should mention that the STMB reinforcement is used in the form of networks welded for reinforcing the plates and walls. The use of this type of reinforcement for reinforcing the elements that are designed under fatigue conditions is forbidden.
340 (35)
32...40
330 (34)
> 40
320 (33)
6...40
PC60 STN
Needle diameter
360 (36)
16...28
Bending angle degrees
6...14 PC52
Elongation at fracture A5 % min
6...12 14...40
Breaking strength Rm N/mm2 (kgf/mm2) min
Nominal diameter mm
OB 37
Cold-hardened test
Creep limit Rpoe N/mm2 (kgf/mm2) min
Steel grade (symbol)
Tensile strength
255 (26) 235 (24)
360 (37)
25
180
0.5 d
510 (52)
20
180
3d
430 (42)
590 (60)
16
180
3d
3.00 4.00
510 (50)
610 (60)
6
4.50 5.00
460 (45)
560 (55)
7
5.60 7.10
460 (45)
560 (55)
8
180
d
8.00 10.00
400 (40)
510 (50)
8
180
d
Table 2.2. Mechanical characteristics of reinforcing steel
2.2.3. Reinforcements, steel wire and steel wire products for prestressed concrete High strength prestressed reinforcements are usually made of carbon steel wires whose properties are improved by heat treatments, mechanical cold-hardened or combinations of heat and mechanical treatments. Romania manufactures the following steel wires and drawn steel wire products for prestressed concrete, made of detensioned carbon steel: – smooth steel wire for prestressed concrete (acronym SBP), manufactured in two strength categories: SBP I and SBP II; – stamped steel wire for prestressed concrete (acronym SBPA) manufactured in two strength categories: SBPA I and SBPA II; – strands for prestressed concrete (acronym TBP) produced by wiring 7 high strength drawn steel wires, which, after being wired, are subject to alignment corrections; – stranded wires for prestressed concrete (acronym LBP) produced by braiding two or three high strength wires.
SBP I
SBP II
SBPA I
SBPA II
Maximal
kgf/mm2 N/mm2 kgf/mm2 N/mm2 kgf/mm2 N/mm2 kgf/mm2 N/mm2
%
1.5
215
2,110
245
2,400
182
1,790
150
1,470
1.5
2.0
205
2,010
235
2,310
175
1,720
143
1,400
1.5
2.5
195
1,910
225
2,210
165
1,620
136
1,330
1.5
3.0
190
1,860
218
2,140
160
1,570
133
1,300
1.5
3.7
180
1,770
208
2,040
153
1,500
125
1,230
2
4.0
175
1,720
203
2,000
148
1,450
122
1,200
2
5.0
170
1,670
198
1,940
145
1,420
118
1,160
2
6.0
165
1,620
193
1,890
140
1,370
115
1,130
2
7.0
160
1,570
188
1,840
135
1,320
112
1,100
2
1.5
195
1,910
225
2,210
165
1,620
136
1,330
1
2
190
1,850
220
2,160
160
1,570
130
1,280
1
2.5
180
1,770
210
2,060
153
1,500
125
1,230
1
3
170
1,670
198
1,940
145
1,420
118
1,170
1
5
170
1,670
198
1,940
145
1,420
120
1,180
2
6
165
1,670
198
1,940
145
1,420
120
1,180
2
7
160
1,570
188
1,840
136
1,330
112
1,100
2
5
155
1,520
183
1,800
132
1,290
108
1,060
1.5
6
150
1,470
178
1,750
127
1,240
105
1,030
1.5
7
150
1,470
178
1,750
127
1,240
105
1,030
1.5
Relaxation test Relaxation after 1,000 hours for initial stress equal to 0.7 R min %
Minimal
Elongation at fracture A min %
Minimal breaking strength Rm
Conventional elasticity limit Rp 0.01
Tensile test in accordance with STAS 66 06 – 78 Conventional creep limit Rp 0.2
Nominal diameter d mm
Materials with Rheological Properties
Steel wire category
24
5.5
5.5
5.5
5.5
Table 2.3. Mechanical characteristics of prestressed steel
The high strength steel wires and stranded wires are used to prestress the concrete in the form of independent cords fixed by adherence or in the form of tendons. The tendons are composed of 6 to 48 parallel steel wires having a diameter of 5 or 7 mm. In the same way, strands can be used to compose the tendons. Table 2.3 presents the principal mechanical and technological properties of steel wires for prestressed concrete. Apart from the high strength steel wire products, bars having a periodic profile PC90 and nominal diameter from 10 to 20 mm, made of steel alloy 65SM11 by hot rolling, are used to prestress the concrete.
The Rheological Behavior of Building Materials
25
Contrary to structural steels or reinforcing steel (non-prestressed reinforcements), loaded in normal operation with stresses that do not exceed 50% of their breaking strength, high strength steels for prestressed concrete are loaded with stresses up to 80% of the breaking strength. Therefore, the manifestation forms of the material’s rheological properties, the creep and the stress relaxation, cannot be ignored. Studies carried out until now prove that the value of stress relaxation or creep depends on factors such as the nature of the steel, the mechanical and heat treatments applied, the value of the initial stress Vi and the shape of the specific diagram V H obtained for static loads of short duration. The technical (conventional) limit of the creep is defined as stress Vt under which the strain is practically negligible, by accepting the value of 3% for the initial strain. For steel wires and high strength steel wire products for prestressed concrete, this limit is close to the technical elastic limit. STAS 10111/2-87 (railway and road bridges, concrete, reinforced and prestressed concrete superstructures; design principles) specifies, for the steels used to manufacture prestressed reinforcements, the final values of the stress loss due to the stress relaxation of the prestressed reinforcement, according to the ratio V po R pn between the initial stress value of the reinforcement and the normalized strength of the reinforcement, expressed as a percentage of initial stress value Vpo (Appendix G, Table 5.5 STAS 10111/2-87). These values are well approximated by the relation:
'V rd
V po
'V r f
- (
V po Rpk
0.5) for 0.5 R pk V po R pk
0 for V po 0.5R pk
(2.7)
Table 2.4 presents, for comparison reasons, the final values of the stress loss 'Vpf, expressed as a percentage of the initial stress values Vpo (prescribed by STAS 10111/2-87) and those obtained using relation (2.7) with the following values for parameter X:
X=
0.45 for prestressed reinforcements made of SBP and SBPA steels 0.55 for prestressed reinforcements made of TBP steels
26
Materials with Rheological Properties
The stress relaxation of the prestressed reinforcement 'Vrt, at a given moment t, related to the final value of the stress loss 'Vpf is given in STAS 10111/2-87 by the coefficient:
K rt
'V rt 'V rf
(Appendix G, Table 5.6 STAS 10111/2-87)
ıp0/Rpk Steel type
0
0.6
0.7
0.8
ǻır/ıp0 (%) SBP, SBPA
0
4.5
9.0
14.0
TBP
0
5
10.5
16.5
SBP, SBPA
0
4.5
9.0
13.5
TBP
0
5.5
11.0
16.5
Values STAS 10111/2 Values obtained using relation (2.7)
Table 2.4. The final values of the stress loss 'Vpf, due to the stress relaxation
Table 2.5 presents, for comparison reasons, the values of function Krt prescribed by STAS 10111/2-87 and those obtained using the relation:
Krt
1 AeD t Be E t
(2.8)
with the following values for its parameters: A = 0.7 D = 0.5 B = 0.3 E = 0.003 By taking into account the definition of the stress relaxation function, given in section 2.1, we can build, using relations (2.7) and (2.8), the analytical expression of the stress relaxation function for the high strength steels used for manufacturing prestressed reinforcements:
r (V p 0 , t )
ª § V p0 · E «1 - ¨ n 0.5 ¸ 1 Ae D t Be E t ¨ Rp ¸ «¬ © ¹
º
»» ¼
(2.9)
The Rheological Behavior of Building Materials
27
24
120 (5 days)
1,000 (42 days)
2,150 (90 days)
8,800 (1 year)
10,000 (1.14 years)
100,000 (11 years)
Krt
T hours
100 (4 days)
If we take into account the fact that Vpo = Hpo u E, it becomes obvious from relation (2.9) that the stress relaxation function of high strength prestressed steels does not meet the necessary condition (2.4) allowing us to apply the principle of superposition (Boltzmann).
0.40
0.51
0.53
0.73
0.79
0.89
0.90
0.98
0.017 0.276
0.608
0.647
0.736
0.771
0.90
0.914
0.999
-91.9 -31.10
19.2
22.1
0.82
-2.46
1.12
1.55
0.19
1
STAS 10111/2-87 0.21 relation (2.8) differences (%)
Table 2.5. Time depending values of the function Krt
Concerning the operation conditions of the high strength steel prestressed reinforcement, we can make the following observations, which allow the formulation of the stress relaxation function (2.9) by using the principle of superposition (Boltzmann): – the value of the stress loss in the prestressed high strength reinforcements is situated between 0 and 14-16.5% of the stress value corresponding to the maximum strain (for the steel wire reinforcements and for the strands reinforcements) if and only if the value of the stress, generated by the strain, exceeds 0.5 Rpk. The stress loss occurs particularly in the time interval succeeding the prestress: 40% after 24 hours, 53% after 120 hours, 73% after 1,000 hours, etc.; – the stress loss of the high strength prestressed reinforcement generates the development of strains in the structural component in which they are encased; the developed strains tend to increase the initial strains of the prestressed reinforcement; – the creep occurring in the concrete elements of the structures, generates the development of strains in the high strength steel prestressed reinforcement that tends to decrease their initial value. From all the above observations it results that the strains of the high strength prestressed reinforcement have small variations that can be ignored in the case of the usual structures, mainly generated by actions tending to mutually cancel themselves out. In this case, to calculate the stress loss in the high strength prestressed reinforcements, we can consider acceptable the value corresponding to the maximum value of strains in the operation interval of the structure. For the
28
Materials with Rheological Properties
resistance structures of the usual constructions, this value corresponds to the initial stress V0, thus:
V0
(H 0
(2.10)
In these conditions, relation (2.9), for the stress relaxation function of the high strength prestressed reinforcements, becomes:
r (t )
>
E 1 C 1 Ae D t Be E t
@
(2.9’)
where C is a constant having the value:
C
§ V0
- ¨¨
© Rpk
· 0,5¸¸ ¹
(2.11)
By taking into account the relation between parameters A and B of the stress relaxation function: A+B=1 and if we note: 1–C=K CuA=L CuB=M
(2.12)
the stress relaxation function (2.9’) takes the final form:
r (t )
E K Le D t Me E t
(2.9”)
Relations (2.9’) and (2.9”) of the stress relaxation functions allow us to use the principle of superposition (Boltzmann) and, thus, determine the creep functions for high strength prestressed steels corresponding to the normal operation of the structures in which they are included. The creep function corresponding to the high strength reinforcing steel Taking into account the creep function for the prestressed reinforcement f(t), unknown for the moment, we can write:
H ( t ) V 0 u f (t )
(2.13)
The Rheological Behavior of Building Materials
29
where:
V0
H0 u E
(2.14)
Considering the stress relaxation function r(t), which is known (obtained above), we can write:
V (t ) H 0 r (t ) and, d >V t @ dt
H0
d > r t @ dt
(2.15)
Let us consider a stress relaxation experiment,
H t
H 0 (constant)
(2.16)
Using the stress relaxation function f(t), relation (2.16’), describing the stress relaxation experiment, takes the form: t
d >V W @ f (t W )dW dt 0
V 0 f (t ) ³
(2.16’)
0
Substituting relation (2.15) into (2.16”) and taking into account relation (2.14), we obtain: t
d > r W @ f (t W )dW 0 dW
H 0 Ef ( t ) H 0 ³
H0
After simplification the relation that takes the final form: t
r (0) f (t ) ³ 0
d >r W @ f (t W )dW dW
1
(2.16”)
The integro-differential equation (2.16”), describing the stress relaxation experiment, was obtained by taking into account that:
V0
V (0) H 0 r (0)
Thus, it results that:
r ( 0)
E
30
Materials with Rheological Properties
Also taking into account the commutative property of the convolution product we have: t
d ³0 dW >r W @ f (t W )dW
t
d
³ dW >r t W @ f (W )dW 0
Applying Duhamel’s convolution principle, the Laplace transform of integrodifferential equation (2.16) takes the expression:
p( p )( p )
1 p
(2.17)
where
( p )
[f(t)]
( p )
[r(t)]
are the Laplace transforms of creep function f(t) and stress relaxation function r(t). From operational equation (2.17) results the expression of the Laplace transform of the designed creep function:
1 p ( p )
( p )
(2.18)
2
or, by taking into account the expression of the Laplace transform of the stress relaxation function: f
³ E>K Le
( p)
D t
@
Me E t e pt dt
0
§K L M · ¸ E ¨¨ p D p E p ¸¹ ©
The Laplace transform of the creep function takes the expanded form: p 2 p D E D E 1 E p p 2 p ªD K M E K L º KD E ¬ ¼
( p )
^
(2.18’)
`
If we note:
\ ( p) a1 , 2
p 2 p D E D E ,
1 ® >D K M E K L @ r 2¯
>D K M E K L @
2
4 KD E ½¾, ¿
The Rheological Behavior of Building Materials
31
the polynomial of the numerator and, respectively, the roots of the polynomial of the denominator from relation (2.18), this expression can be written in concise form:
( p )
\ p 1 E p p a1 p a 2
(2.18”)
Expanding as a sum of simple fractions, relation (2.18’) takes the following form:
( p)
1 ª 1 \ 0 1 \ a1 1 \ a2 º « » E ¬« p a1 a2 p a1 a1 a1 a2 p a2 a2 a2 a1 ¼»
or, finally,
( p)
\ a1 \ a 2 1ª 1 1 1 º « » E ¬ pK a1 a1 a 2 p a1 a 2 a 2 a1 p a 2 ¼
(2.18’’’),
where we considered the relations between the roots and the polynomial’s coefficients of the denominator (Viette’s theorem):
a1a 2
KD E
Using the inverse Laplace transform on relation (2.18”), we obtain the desired creep function:
f (t )
1 E
ª1 º \ a1 \ a 2 e a2 t » e a1 t « a 2 a 2 a1 ¬ K a1 a1 a 2 ¼
(2.19)
In relation (2.19) of the creep function,
a1 a2
U q 0 U q 0
so that, by evaluating the coefficients of the exponential polynomial, there results, at the limit, the following values of the creep function: for t = 0, f (0) for t = , f (f)
1 E
1 1 E K
1 E 1 C
32
Materials with Rheological Properties
Relations (2.13) and (2.15) between the stresses and the strains in time – the constitutive laws of the material – contain, implicitly, parameter E, the material modulus of elasticity (Young). The modulus of elasticity of the high strength steels for prestressed reinforcements, as well as all the steels, is constant in time. For calculating the resistance structures, we use the following values of the modulus of elasticity: – 210,000 N/mm2 – for PC90 bars; – 200,000 N/mm2 – for isolated steel wires and steel wire tendons SBP I, SBP II, SBPA I and SBPA II; – 180,000 N/mm2 – for stranded wires (LBP) and strands (TBP).
2.3. Concrete The main quality parameters of concrete are: compressive strength, permeability, resistance to repeated freezing and unfreezing actions, etc. determined for cubic, prismatic or cylindrical samples, under the conditions established by norms and standards. The experimental specification of the modulus of elasticity, of the strains (longitudinal and transversal) and of the Poisson ratio is carried out on prismatic or cylindrical samples. The strain properties of concrete are in strict correlation with its compressive strength. The compressive strength determined under the conditions imposed by the standards in force, defines the class of the concrete (characteristic strength of the concrete in N/mm2, determined after 28 days on cubic samples of 141 mm in length; maximum 5% of the results may fall below this value). The prismatic strength of the concrete is considered when designing the structures because the conditions of the tests on prismatic samples approximate the load of concrete in resistance structures better. Between the concrete characteristic strength, determined on prismatic samples, Rpr, and the characteristic strength, determined on cubic samples, Rb, we obtain the relation: Rpr = (0.87 – 0.002 Rb) Rb Under long-term actions, the concrete strength, Rbd, is about 10% lower than concrete strength Rb determined on cubic samples: Rbd | 0.90 Rb
The Rheological Behavior of Building Materials
33
The strength and strain properties of concrete depend on the concrete’s class, its age at the moment it is loaded and on the conditions of applying the loads. Figure 2.8 represents the dependence between the growth of concrete strains and the rate of its load. V,[daN/cm ]
105 70
16s
5s
30s
60s
in st a nt an eo
us
140
35 0
0
0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 H [%0 ]
Figure 2.8. Dependence between the growth of concrete strains and the rate of its load
The typical shape of the concrete’s characteristic stress-strain curve when loaded with a monoaxial compression at a constant rate growth of the strain is given in Figure 2.9. The theoretical and experimental studies on the concrete’s characteristic stressstrain curve led to the conclusion that, in the process of loading, the viscous and plastic strains overlap the elastic strains (Figure 2.10). V
0.75
H
E
H
R
0.50 0.25 H
H
H [%0] H
Figure 2.9. Characteristic stress-strain curve for concrete loaded with monoaxial compression at a constant rate growth of the strain
34
Materials with Rheological Properties
The elastic strains H1, which are proportional with stress V1, occur according to Hooke’s law, V1 = H1E, and are characterized by the fact that the concrete’s modulus of elasticity Eb and Poisson’s ratio P have constant values under a load of limited, relatively short-term duration.
V
R 1.00
H
H
H
H
H
H
H
H
H
H
0.75 H
0.50 0.25 H
H
H
H [%]
Figure 2.10. The components of the concrete’s deformations
The experiments showed that the value of the modulus of elasticity is a function of the concrete’s quality and its age at the moment of the test, and that it is influenced by the same factors as the compressive strength to which it is closely related. Just as with the strength, the concrete’s modulus of elasticity grows with age and tends asymptotically, at a variable rate (maximum at the beginning and decreasing afterwards), towards a limit value. Table 2.6 presents the variation of the concrete’s modulus of elasticity and, for comparison, the values obtained with relation (2.20) that will be used throughout this work:
E W
Eb
A Be Dt
(2.20)
The values of the parameters in relation (2.20) that will be used in this work are: A = 0.78 B = 0.29 D = -0.0098
The Rheological Behavior of Building Materials
Variation of Eb according to concrete age Ebt Eb
Calculated using the relation Differences (%)
35
Concrete age 7 days
28 days 45 days 90 days 9 months 6 years
0.76
1.00
1.06
1.11
1.18
1.28
0.95
1.00
1.035
1.111
1.249
1.282
25.0
0.00
-2.36
0.09
5.85
0.16
Table 2.6. The values of concrete’s modulus of elasticity depending on concrete age
The viscous strains H2 characterize the behavior of the concrete under a long-term load and are known as creep. As shown in Figure 2.8, the viscous strains also occur in the case of short lasting loads, when the loading time is practically t z 0. Within the interval of Vb = 0 until Vb = KRb, the viscous strain is proportional to the stress V, and it is called linear creep. Above this limit nonlinear creep occurs, which is accompanied by plastic strains H3. Coefficient K represents the relative value of the load level separating the linear creep zone from the nonlinear creep zone. From the physical point of view, condition Vb > Rbd is equivalent to the occurrence of microcracks in the structure of the concrete and, implicitly, of plastic deformations corresponding to them. If the stress in the concrete exceeds the long-term mechanical strength, Rbd, or its static fatigue strength (Figure 2.11, Vb > Rbd), the microcrack phenomenon becomes general and progressively leads to the concrete’s fracture after a time interval whose length depends on the load stage. In the field of nonlinear creep (Figure 2.11, Rbd > Vb > KRbd), the rate of the nonelastic strains (viscous + plastic) decreases in time to a finite limit different from zero. This led to the hypothesis that such a strain state can generate the fracture of the concrete (Rjanitin). This hypothesis was not confirmed by the experiments because of the difficulty in maintaining the load stage at a constant, taking into account the growth of the concrete’s strength in time.
36
Materials with Rheological Properties
Non-elastic strain
H V >R K R < V
Time
t
Figure 2.11. Concrete’s non-elastic strain according to the stress
In the field of linear creep (Figure 2.11, Vb d KRb), the non-elastic strains are viscous (Figure 2.10) and develop with a decreasing variable rate, maximal immediately after the application of the load, tending towards zero as time passes. The curve strains tend towards an asymptote that is parallel to the time axis. In this case, the strains state tends towards a stable in time state. Coefficient K takes values between 0.30 and 0.80, increasing with the class quality of the concrete. For the concretes Bc 7.5 – Bc 45, the coefficient has average values ranging between 0.51 and 0.75, which enables us to affirm that the stresses generated in the concrete from which the resistance structures are made by the normal operation loads fall in the field of linear creep. In the following sections, we will develop the relations allowing us to determine the calculating quantities for the concrete’s creep in the linear field. These quantities are used in the evaluation of the behavior of resistance structures of constructions during their normal operation. The measure of the creep, denoted Ct, represents the creep strain at moment t for a concrete sample loaded by a unitary stress:
Ct
H cl ,t V
(2.21)
The maximum value, corresponding to the damping interval of this strain (which is theoretically infinite), represents the final measure of the creep: Cf =
H cl ,f V
(2.22)
By comparing relations (2.1) and (2.21), it results that the measure of creep Ct represents the variable in time component of the concrete’s creep function.
The Rheological Behavior of Building Materials
37
In the practice of using the reinforced, prestressed concrete, the creep characteristic, denoted It, is almost exclusively used in order to estimate the behavior of the structures. The creep characteristic represents the ratio between the creep strain Hc1,t at given moment t and the elastic strain He, corresponding to the moment of the load: It =
H cl ,t He
(2.23)
For t = f, corresponding to the (theoretical) strain damping interval, we obtain the (maximum) finite value of the creep characteristic: I=
H cl ,f He
(2.24)
Between the measure of the creep Ct and the creep characteristic It the following relation obviously exists: It = Ct Eb
(2.25)
The creep characteristic of the concrete is determined using a relation having the form: It =I0 K1 K2 …Kn
(2.26)
where I0, K1, K2,…, Kn are functions experimentally determined by taking into account the component materials, the composition and the casting of the concrete, the way the concrete is maintained under load, the value and the type of the loads, the shape and the dimensions of the concrete element, the age of the concrete at its loading and the duration of the load. The number of the functions used as factors in relation (2.26), their symbol, their significance and their field of values are different from one country to another, but three of them have, practically, the same significance: – I0 – the basic (standard) creep characteristic; – K1 – the function that takes into account the influence of the concrete’s age at its loading over the final value of the creep; – K2 – the function that takes into account the variation of the creep with time; an increasing function that asymptotically tends towards the value 1.
38
Materials with Rheological Properties
The other aspects that influence the creep of the concrete and implicitly its characteristic are taken into account, individually or as a whole, using the other coefficients occurring in relation (2.26). The factors occurring in relation (2.26) are given as tables and diagrams in the design specifications, because the attempts to find an analytical expression for the creep characteristic whose parameters could have a physical significance corresponding to the factors has not led, until now, to acceptable results. Our specifications for designing concrete and prestressed concrete elements for bridge, railway and road structures (STAS 10111/2-87) stipulate the evaluation of the creep characteristic using the relation: Ib = KT KR Kb If
(2.27)
where: If – represents the basic (standard) final creep characteristic normalized according to the environment where the construction element is placed and the consistency of the concrete during its casting; Kb – coefficient having values depending on the minimum cross-section of the construction element; KR – function that expresses the variation with time of the final creep characteristic, having values according to the age of the concrete at the moment of its loading (Table 2.7);
KR
Concrete age (days)
7
21
28
45
60
90
180
>360
Values STAS 10111/2-87
-
-
1.00
0.90
0.85
0.75
0.60
0.60
Values obtained using relation (2.29)
1.149
1.045
1.00
0.91
0.848
0.758
0.641
0.6027
-
-
0.00
1.11
-0.23
1.07
-1.38
0.00
Differences (%)
Table 2.7. The values of the function KR depending on the age of the concrete
KT – increasing function expressing the development in time of the creep strain (Table 2.8), having values in the range [0,1]. The maximum value 1 corresponds to the damping interval of the creep phenomenon.
The Rheological Behavior of Building Materials
KT
t – t0 days
0
Values STAS 10111/2-87
-
2
10
20
30
45
60
90
180
39
360 >1100
0.075 0.170 0.240 0.280 0.340 0.390 0.460 0.600 0.76
1,00
Values obtained using 0.00 0.016 0.077 0.147 0.211 0.298 0.375 0.502 0.741 0.923 0.999 relation (2.30) Differences (%)
0,00 -78.7 -54.7 -38.7 -24.6 -12.35 -3.85 9.13
23.5
21.4
0.00
Table 2.8. The values of the function KT depending on time
The function represented in the table for STAS 10111/2-87, is not monotonic. It results that, according to STAS 10111/2-87, the final creep characteristic I for a construction element and under given conditions is dependent on the concrete’s age W at the moment of its load (via function KR):
I W KR (W )Kb If
(2.28)
Substituting relation (2.28) into (2.26), we obtain:
I W , t K T t I W
(2.26’)
The functions KR(W) and KT(t), given by STAS 10111/2-87, are pretty well approximated by the relations: K R*(W) = 0.60 + 0.61e-0.015t
(2.29)
K T*(t) = 1 – 0.4e-0.011t – 0.6e-0.006t
(2.30)
Tables 2.7 and 2.8 present, for comparison, the values given in STAS 10111/287 for the functions KR and KT, as well as those obtained using relations (2.29) and (2.30), which approximate them. From all above it results that the constitutive law of the concrete has the form:
H t
V0 E
>1 (K b K R If )K t (t)@
By noting: C = Kb KR If
(2.31)
40
Materials with Rheological Properties
and by considering the generic form of the interpolating polynomial (the regression function) that approximates the development in time of the concrete creep strain:
KT t
1 Ae D t Be E t
(2.32)
whose coefficients satisfy the relation A+B=1
(2.33)
there results the creep function of the concrete:
>
1 1 C 1 Ae D t Be E t E
f (t )
@
(2.34)
which, by using the notations: 1+C=K CuA=L CuB=M
(2.35)
takes the form:
f (t )
1 K Le D t Me E t E
(2.34’)
The creep function of the concrete allows us to determine the concrete stress relaxation using an operation similar to the one that was used in order to determine the stress relaxation function of the prestressed high strength steel reinforcements. The stress relaxation function of the concrete By considering the stress relaxation function of the concrete r(t), unknown for the moment, we can write: V(t) = H0 r(t)
(2.36)
where:
H0
V0 E
(2.37)
The Rheological Behavior of Building Materials
41
We can also write, by considering the creep function f(t) as known (as previously built):
H t
V 0 f t
d >H t @ dt
V0
(2.38)
d > f t @ dt
By using the mathematical expression of the stress relaxation experiment (2.16”): t
d > r W @ f t W dW 0 dt
r 0 f t ³
1
and, respectively, relation (2.17) of its Laplace transform, we obtain relation (2.17) of the Laplace transform of the integro-differential equation corresponding to the stress relaxation experiment:
p( p )( p )
1 p
from which there results immediately the Laplace transform of the stress relaxation function:
1 ( p ) p2
( p )
(2.39)
where
( p )
[f(t)]
( p )
[r(t)]
are the Laplace transforms of the creep f(t) and, respectively, of the stress relaxation r(t) functions of the concrete. By introducing the expression of the Laplace transform of the creep function into relation (2.39):
p
f
³ E K Le 0
1
D t
Me E t e p t dt
L M 1§ K ¨¨ E© p D p E
· ¸ p ¸¹
(2.40)
42
Materials with Rheological Properties
we obtain the expression of the Laplace transform of the concrete’s relaxation function:
p
E
p 2 p D E DE
(2.41)
p^ p 2 p>D K M E K L @ KDE `
and, respectively, its expression in the form of a sum of simple fractions:
p
§ 1 M a1 \ a 2 1 1 E ¨ ¨ K p a a a p a a 2 a 2 a1 p a 2 1 1 2 1 ©
· ¸ ¸ ¹
(2.41’)
In relation (2.41’), we used the notations:
\ p a1, 2
p 2 p D E DE ,
^
1 >D K M E K L @ r 2
>D K M E K L @2 4 KD E
`
Using the inverse Laplace transform on relation (2.41’), we obtain the desired stress relaxation function:
r t
§1 \ a1 \ a2 a t E¨ e a1 t e2 ¨ K a1 a2 a1 a a a 2 2 1 ©
· ¸¸ ¹
(2.42)
From condition (2.33) and considering notations (2.35), it results, K – M > 0, K – L > 0, and thus: a1 = (– U + q) < 0 a2 = (– U – q) < 0 so that, to the limit, the stress relaxation function of the concrete takes the following values: – for t = 0, r(0) = E; – for t o f, r(f) =
E K
E . 1 C
The Rheological Behavior of Building Materials
43
In the case of the long-lasting cyclic loads, the elastic strain is coupled with a permanent and slowly increasing strain of the type of the creep, having a damping tendency after a number of cycles. The tests showed that the ratio Vmin/Vmax and the frequency of the cycles have a negligible influence on the value of the creep strains, because of the cyclic longlasting loads, and that these strains can be approximated by the same function as those corresponding to the creep generated by the static loads, VD = Vmax having the equivalent duration: T=
N f
where: N – the number of load cycles; f – the frequency of load cycles.
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Chapter 3
Composite Resistance Structures with Elements Built from Materials Having Different Rheological Properties
3.1. Mathematical model for calculating the behavior of composite resistance structures: introduction The rheological properties of the materials from which the elements composing the resistance structures are made, as well as those of the environment with which they are interacting, cause the modification over time of their initial stresses and strains state. The resistance structures are dimensioned for a normal operation during the entire designed working life. To satisfy the above requirement, it is necessary that, in any point of the structures, the stresses and strains generated by the actions and loads to which they are exposed during their operation do not exceed the limit values prescribed by the standards and norms. In this context, the necessity of knowing the values that characterize the structure’s stresses and strains state is obvious, for their operation under satisfactory safety and comfort conditions, as well as for the manufacture of efficient structures from the point of view of material and energy consumption. In what follows, we present a mathematical model for the evaluation of the stresses and strains state of the composite resistance structures with elements built from materials having different rheological properties. We have to specify that this model also includes the interaction of the structures with the environment they are in contact with, under the conditions of the adoption of an adequate model for the structure-environment contact.
46
Materials with Rheological Properties
First, it is necessary to expound the main deformability properties of the materials from which the elements of the resistance structures are made or with which they are interacting and the functions used to model the deformability properties of the materials. Strain H is proportional to stress V and changes according to the age of the material W at the moment when the load generating the stress was applied and also the time t – W elapsed since loading (we consider the creep phenomenon):
H t
V >1 K1 W K 2 t W @ E W
(3.1)
or
H t H 0 K0 W >1 K1 W K 2 t W @ Stress V is proportional to strain H and changes according to the age of the material W at the moment when the load generating the stress was applied and also the time t – W elapsed since loading (we consider the stress relaxation phenomenon):
V t H 0 E W >1 K1 W K 2 t W @ or
V t V 0 K0 W >1 K1 W K 2 t W @
(3.1’)
where: E(W) – the expression describing the variation of the material’s modulus of deformation according to its age W at the moment when the actions and the loads generating the stresses state were applied. The constant E0 is the material’s reference modulus of deformation (the value of the modulus of deformation corresponding to the moment when K0(t) = 1). K1(W) – the function describing the variation of the material’s characteristic of deformability (delayed deformation) according to its age W at the moment when the actions and the loads were applied. K2(t – W) – the function describing the accomplishment of the delayed deformation of material under the action of a constant load according to the time t – W elapsed since the load was applied. The function equates to zero (minimum value, K2(0) = 0) when t – W = 0 and tends to the value 1 (maximum value) when tof (K2(f) = 1).
Composite Resistance Structures
47
If we ignore the rheological properties of the materials (K0(W) = 1 and K1(W) = 0), strain H is proportional to stress V and does not vary over time (the frequent case of steel usage):
H
V E0
(3.1”)
By adopting certain values and certain appropriate expressions for the reference modulus of deformation E0 and, respectively, for functions K0(W), K1(W) and K2(t), relations (3.1) and (3.1’) can be used to estimate the stresses and the strains at any moment of time t due to the actions and the loads applied at the moment W and generating a stresses state in the element. The relations can be used for concrete, steel and high strength prestressed steel elements, as well as the foundation ground. We formulate the following hypotheses: a) During the analysis of the resistance structures’ stresses and strains state by considering the rheological properties of the materials from which their elements are made and the environment with which they come into contact, we admit the principle of superposition (Boltzmann) only for the actions and loads that are simultaneously applied. b) The redistribution of the interior stresses in a resistance structure, due to the rheological properties of the materials from which its elements are made and the environment coming into contact with the structure, is done so that in every crosssection and at any moment the conditions of equilibrium and of compatibility between the strains and the connections (the instantaneous elastic equilibrium condition) are simultaneously satisfied. c) The quantities defining the strains of a correctly dimensioned resistance structure under external loads and actions are functions of time and tend towards a finite value. The variation rates of these quantities reach their maximum value at the moment subsequent to the external loads and actions application (t – W = 0) and tend towards zero over time (to f). Considering the rheological properties of the materials described in relation (3.1) – the creep phenomenon – the strain expression Ds(T) in section s of a structural element, homogenous from the point of view of its behavior in time, under a known stress, variable over time, S(t), has the form:
D S T D S0 W K 0 W >1 K1 W K 2 t W @ T
³ W
dD S0 t K 0 t >1 K1 t K 2 T t @dt dt
(3.2)
48
Materials with Rheological Properties
where the symbols represent: W – the age of the material at the moment when the actions and the loads generating stress S(t) were applied; T – the particular moment of time for which the value of the strain is calculated; t – the time variable; D s0 (W) – the element’s fictitious initial strain in section s, due to value S(W) – at the moment t = W of the variable stress S(t) (calculated by taking into account the constant modulus of deformation, equal to its reference value E0); D s0 (W)K0(W) – the initial strain of the element in section s at the moment t = W, thus corresponding to value S(W) of the variable stress; d D s0 (t ) dt – the basic (infinitesimal) growth, at the moment t, W d t d T, of the dt
fictitious strain in section s of the element under the variable stress S(t), due to the stress variation dS (t ) dt (calculated by taking into account the constant modulus of dt
deformation, equal to its reference value E0);
dD s0 (t ) K (t)dt – the growth, at moment t, W d t d T, of the initial strain in section s 0 dt of the element under the variable stress S(t), due to the stress variation dS (t ) dt . dt
If Ds(T) (the strain of the element in section s at the moment T, W d T d T) is known, then the strain of the element in the respective section at the moment T, Ds(T) takes the form:
D s t D s T D s0 W K 0 W K1 W ª¬ K 2 T W K dD s0 t
T
³
dt
T
dD s0 t
T
³
dt
W
K 0 t ª¬1 K1 t K 2 T t º¼ dt
2
T T º¼ (3.3)
K 0 t K1 t ª¬ K 2 T t K 2 T t º¼ dt
In a similar way, taking into account the rheological properties of material described in relation (3.1’) – the stress relaxation phenomenon – the expression of the reaction Rs(T) in a section s of a structural element, homogenous from the point of view of its behavior in time, under a known strain, variable over time, D(t), has the form: Rs T T
³ W
Rs0 W K 0 W ª¬1 K1 W K 2 T W º¼
dRs t dt
K 0 t ª¬1 K1 t K 2 T t º¼ dt
(3.2’)
Composite Resistance Structures
49
In equation (3.2’), we have used the following notations: W – the age of the material at the moment when the actions and the loads generating strain D(t) were applied; T – the particular moment of time for which the value of the reaction is calculated; t – the time variable; Rs0 (W) – the fictitious reaction of the element in section s, due to the value D(W) – at the moment t = W of the variable strain D(t) (calculated by taking into account the constant modulus of deformation, equal to its reference value E0); Rs0 (W)K0(W) – the initial reaction of the element in section s at the moment t = W, thus corresponding to the value D(W) of the imposed variable deformation; dRs (t ) dt – the basic (infinitesimal) growth, at the moment t, W d t d T, of the dt
fictitious reaction in section s of the element under the imposed variable deformation D(t), due to the deformation variation dD (t ) (calculated by taking into account the dt
constant modulus of deformation, equal to its reference value E0); dRs (t ) K (t)dt – the growth, at the moment t, W d t d T, of the initial reaction, in 0 dt
section s of the element under the imposed variable deformation D(t), due to the deformation variation dD (t ) dt . dt
In the same way, if we know the value Rs(T) (the reaction of the element in the section s at the moment T, W d T d T), the reaction of the element in the respective section at the moment T, Rs(T) takes the form:
Rs T
Rs T Rs W K 0 W K1 W ª¬ K 2 T T K 2 T W º¼ T
³
dRs t dt
W
T
³ T
dRs t dt
K 0 t K1 t ¬ª K 2 T t K 2 T t ¼º dt
(3.3’)
K 0 t ª¬1 K1 t K 2 T t º¼ dt
3.2. Mathematical model for calculating the behavior of composite resistance structures. The formulation considering creep 3.2.1. The effects of the long-term actions and loads: overview We consider a resistance structure composed of elements made of materials having different rheological properties, interior and/or exterior statically indeterminate.
50
Materials with Rheological Properties
Let m be the number of different material types – from the point of view of the rheological properties – used to manufacture the structure’s elements, also including the environment interacting with the structure. We build a basic system of the structure, so that each of its components has a homogenous section from the point of view of the rheological properties. The relative displacement Dij(T) of the structure in the direction of connection force i, due to the stress of the structure’s basic system areas having the rheological properties j by the connection forces and the external actions and loadings, is expressed by the relation:
D ij T D ij0 W K 0 j W j >1 K1 j W j K 2 j T W @ T
³ T
dD ij0 t K 0 j t W j 1 K1 j t W j K 2 j T t dt dt
>
(3.4)
@
From the condition of compatibility between the deformations and the connections of the structure, we obtain:
¦D W K W >1 K W K T W @ m
0 ij
0j
j
1
j
2j
j 1
dDij0 t ¦³ K0 j t W j 1 K1 j t W j K2 j T t dt 0, (i 1,2,3,!, n) dt j 1W m T
>
(3.5)
@
If we know the stresses and strains state at moment T, W d T d T, then by considering relation (3.3), the condition of compatibility between the deformations and the connections of the structure can be expressed as follows:
¦ ^D T D W K W K W >K T W K T W @` m
0 ij
ij
0j
j
j
j
2j
2j
j 1
m dD t ° ij ¦ ® ³ K 0 j t W j 1 K1 j t W j K 2 j T t dt j 1° ¯T dt T
T
³ W
dD ij0 t dt
0
>
@
½° K 0 j t W j K 2 j T t K 2 j T t dt ¾ °¿
>
@
(3.5’)
0, (i 1,2,3, ! , n)
In relations (3.4), (3.5) and (3.5’), we have used the following notations: D ij (T ) – the deformation of the structure at the moment T, W < T < T, in the connection force i direction, due to the stresses generated from the external actions
Composite Resistance Structures
51
and loads, applied at the moment W, and to the corresponding connection forces in the structure’s basic system areas having the rheological properties j; D ij0 (W ) – the fictitious initial deformation of the structure, at the moment t = W, in the connection force i direction, due to the stresses generated by the external actions and loads, applied at the moment W and by the corresponding connection forces in the structure’s basic system areas having the rheological properties j. The value is calculated by taking into account the reference value, E0j, of the sections’ material modulus of deformation;
dD ij0 (t )
dt – the growth of the fictitious deformation of the structure in the dt connection force i direction, at the moment t, W d t d T, due to the stresses generated by the external actions and loads, applied at the moment W, and by the corresponding connection forces in the structure’s basic system areas having the rheological properties j. The value is calculated by taking into account the reference value, E0j, of the sections’ material modulus of deformation; W – the reference age of the structure at the moment when the external actions and the loads were applied; Wj – the age of the material having rheological properties j, composing the structure, at the moment when the external actions and the loads were applied. In equations (3.2), (3.3), (3.5), (3.5’), as well as in those following, unless otherwise specified, the time parameters and variables (t, W, T, T, Wj, etc.) have the same origin. If we take into account that:
D ij0 t E ij t ' ij
(3.6)
where: Eij(t) – the fictitious deformation of the structure in the connection force i direction, at the moment t, W d t d T, due to the stresses generated by the connection forces corresponding to the actions and loads, applied at the moment W in the structure’s basic system areas having the rheological properties j. The value is calculated by taking into account the reference value, E0j, of the sections’ material modulus of deformation; 'ij – the fictitious deformation of the structure on the connection force i direction, due to the stresses generated in the structure’s basic system areas having the rheological properties j by applying the – constant over time – external actions and loads at the moment W. The value is calculated by taking into account the reference value, E0j, of the areas’ material modulus of deformation.
52
Materials with Rheological Properties
It results that:
dD ij0 t
dE ij t
dt
dt
(3.7)
Using relation (3.7) in relations (3.5) and (3.5’), we obtain:
¦D W K W >1 K W K T W @ m
0 ij
j 1
0j
j
1j
j
2j
(3.8)
dEij t ¦³ K0 j t W j 1 Kj t W j K2 j T t dt 0, (i 1,2,3,!, n) dt j 1W m T
>
@
¦^D T D W K W K W >K T W K T W @` m
0 ij
ij
0j
j
1j
j
2j
2j
j 1
m T dE t ij ¦®³ K0 j t W j 1 K1j t W j K2 j T t dt dt j 1 ¯T T dEij t ³ K0 j t W j K1j t W j K2 j T t K2 j T t dt` 0, (i 1,2,3,!, n) W dt
>
@
>
@
By making T = W, and respectively, t = T, in equations (3.8), we obtain:
¦ D W K W m
0 ij
0j
j
0, (i 1,2,3, ! , n)
j 1
respectively
(3.9)
m
¦ D T ij
0, (i 1,2,3, ! , n)
j 1
Using relations (3.9) in equations (3.8), we obtain:
¦ D W K W K W K T t m
0 ij
0j
j 1
m T
¦³ j 1 W
(i
j
1j
d E ij t K 0 j t W dt
1, 2 ,3, ! , n )
2j
j
j
>1 K t W K T t @dt 1j
j
2j
0 , (3.8’)
Composite Resistance Structures
53
¦ D W K W K W >K T W K T W @ m
0 ij
0j
j
1j
j
2j
2j
j 1
m T d E t ij K 0 j t W j 1 K 1 j t W j K 2 j T t dt ¦ ®³ dt j 1 ¯T T d E ij t ½ K 0 j t W j K 1 j t W j K 2 j T t K 2 j T t dt ¾ ³ dt ¿ W
>
@
>
(i
@
0,
1, 2 ,3, ! , n ) (3.8”)
The solutions of the integro-differential equations system (3.8’) and (3.8”) are functions describing the variation of the connection forces values; identifying these functions allows us to evaluate the structure’s stresses and strains state at every moment of its life. Solving the integro-differential equations system (3.8’) and (3.8”) using the traditional methods of the functional analysis that we have employed in Chapter 2 is inoperative from the practical point of view for systems having more than three unknown functions. If we take into account that Eij(t) are linear functions of the connection functions Xi(t), by performing the substitution:
dX i t dt
xi t , (i 1,2,3, ! , n)
(3.10)
the system of integro-differential equations (3.8’) takes the form: m
n T
¦¦ ³ x t N T , t dt F T ijk
k
i
0, (i 1,2,3,!, n)
(3.11)
j 1 k 1W
where:
>
@
N ijk T , t aijk K 0 j t W j 1 K1 j t W j K 2 j T t Fi T
¦D W K W K W K T W m
0 ij
0j
j
1j
j
(3.12)
2j
j 1
Expressions (3.11) compose a system of Voltera integral equations of the first type.
54
Materials with Rheological Properties
The function K2(T – t) expressing the evolution of the deformation of a structure section, homogenous from the point of view of the rheological properties of the material it is made of, over the time elapsed since the external loads application, is at least once differentiable with respect to T. This implies the differentiability with respect to T of the kernels Nijk(T,t) and of the free terms Fi(T). Additionally, considering the properties of the function K2 and relations (3.12) for the kernels and the free terms of the system (3.11), it results that Nijk(T, T) z 0 and Fi(W) = 0. Applying the Volterra-Roux theorem to the system of integral equations (3.11), if the kernels Nijk(T, t) and the free terms Fi(T) of the equations are functions once differentiable with respect to T in the interval (W, t) and if Nijk(T, T) z 0 and Fi(W) = 0, then the system of Voltera integral equations of the first kind (3.11) admits a unique solution, bounded and continuous in this interval. Indeed, if we differentiate with respect to T of the equations from system (3.11), we obtain, after systematizations: n T
n
¦ C x t ¦ ³ x t K T , t dt F T ik
k
k
k 1
ik
i
0, (i 1,2,3, !, n)
(3.13)
k 1W
where:
º w ªm «¦ N ijk T , t » wt ¬j 1 ¼
K ik T , t Fi T
(3.14)
d Fi T dt
Solving, with respect to xk(T), the system of integral equations (3.13) takes the following form: n
T
k 1
W
xi T ¦ O ³ xk t -ik T , t dt Fi T 0, (i 1,2,3, !, n)
(3.15)
which allows the construction of the system solution by successive approximations. Let
xi T xi 0 T O xi1 T O2 xi 2 T ... Ok xik T !, (i 1,2,3, !, n) (3.16) be the solution of system (3.15), where xi0(T), xi1(T), ..., xik(T) are the different successive approximations of the system solution, approximations that must be
Composite Resistance Structures
55
determined. By substituting solution (3.16) into system (3.15) and by identifying the terms, we immediately obtain the following relations:
xi 0 T
F (T ) p T
xi1 T ¦ ³ -ik T , t xi 0 T dt k 1W
(3.17)
p T
xi 2 T ¦ ³ -ik T , t xi1 T dt k 1W
# m T
xik T ¦ ³ -ik T , t xik 1 T dt k 1W
The solution Xi(T) of the integro-differential equations system (3.8) is obtained by integrating the differential equations (3.10), and by imposing the condition that the obtained solution satisfies the equilibrium of the structure and the compatibility between the deformations and the connections of the structure in the initial phase (at t – W = 0), immediately after the loading of the structure, but before the redistribution of the strains due to the rheological properties of the materials:
X i T
t
³W x T dt D , (i i
i
1,2,3,..., n)
(3.18)
This approach serves to demonstrate the existence and the uniqueness of the solution and allows, based on its structure, the following observation. Solution (3.18) of the integro-differential equations system (3.8) is a system of continuous functions, indefinitely differentiable, having continuous derivatives of any order in the interval (W, T), which is obvious if we take into account relations (3.16) and (3.17). This observation indicates a new method for solving the integro-differential equations system (3.8), i.e., by using the Taylor series. Indeed, if Xi(t) are continuous functions, having continuous derivatives in the interval (W, T), up to the order (k + 1) inclusive, then these functions can be expanded according to the powers of the difference (T – T) at any point inside the interval: X i T
T T
k
T T dX i T T T d 2 X i T ! 1! 2! dt dt 2 d k X i T Rk T , (i 1,2,3, ! , n) dt k
X i T K!
2
(3.19)
56
Materials with Rheological Properties
By successive differentiations of any integro-differential equation in relation (3.8’) or (3.8”), we obtain:
dK 2 j T W dE ij T K 0 j T W j D ij0 W K 0 j W j K 1 j W j dT dT 1¯
m
¦® j
T
³
dE ij t dt
W
dK 2 j T t ½ K 0 j t W j K 1 j t W j dt ¾ dT ¿
(3.20)
0, (i 1,2,3,! , n)
° d 2 E ij T dE T ª dK T W j K 0 j T W j ij « 0 j K 0 j T W j K1 j T W j dT dT ¬ dT 1¯
m
¦ ®° j
dK 2 j T T º d 2 K 2 j T W 0 D K W K W ij 0 j j j 1j » dT dT 2 ¼ T dE t d 2 K 2 j T t °½ dt ¾ 0, (i 1,2,3, !, n) ³ ij K 0 j t W j K1 j t W j dt dT 2 °¿ W
d 3 E ij T
m
¦
dT
j 1
K 0 j T W j
d 2 E ij T ª dK 0 j T W j K 0 j T W j K1 j T W j «2 dT ¬ dT
dK 2 j T T º dE ij T ° d 2 K 0 j T W j d 2 K 2 j T T 2 K 0 j T W j K1 j T W j dT ® » 2 dT dT °¯ dT dT 2 ¼
dK1 j T W j º dK 2 j T T ½ ª dK 0 j T W j « K1 j T W j K 0 j T W j ¾ » dT dT dT ¬ ¼ ¿
D W K 0 j W j K 1 j W j 0 ij
T
³ W
d 3 K 2 j T W dT 3
dE ij t d 3 K 2 j T t K 0 j t W j K 1 j t W j dt dt dT 3
0, i 1, 2, 3, ! , n
# m
¦
d k E ij T dT
j 1
T
³ W
K 0 j T W j ! D W K 0 j W j K 1 j W j 0 ij
dE ij t d k K 2 j T t K 0 j t W j K 1 j t W j dt dt dt
d k K 2 j T W dT k
!
0, (i 1,2,3, ! , n)
Composite Resistance Structures
57
For T = W, the integro-differential equation systems (3.8’) and (3.20) are reduced to the following linear algebraic equation systems, having as unknown factors the values in the point W of the unknown functions xi(t), and respectively, those of their 2 d 3 xi t d k xi t : derivatives dxi t , d xi t , ,!, ,! 2 3 dt dt k dt dt
¦ E W K W ¦ ' K W m
m
ij
0j
j
ij
j 1
0j
j
0, (i 1,2,3,!, n)
j 1
m dK W W dEij W K 0 j W j ¦D ij0 W K 0 j W j K1 j W j 2 j dT dT 1 j 1
m
¦ j
0, (i 1,2,3,!, n)
m d 2 Eij W dEij W ª dK0 j W j K W ® ¦ ¦ 0j j « 2 dT j 1 j 1 ¯ dT ¬ dT m
dK W W º d 2 K 2 j W W ½° 0 K 0 j W j K1 j W j 2 j K K D W W W ¾ ij 0j j 1j j » dT dT °¿ ¼ (i 1,2,3,!, n) m
¦
d 3 E ij W dT 3
j 1
m
¦ j 1
0, (3.21)
d 2 E ij W ª dK 0 j W j dK W W º 2 K 0 j W j K1 j W j 2 j « » 2 dT ¬ dT dT ¼
dE ij W ° d 2 E ij W j d 2 K 2 j W W 2 K K W W ® j j j j 0 1 dT 2 °¯ dT 2 dT 2 dK1 j W j º dK 2 j W W ½ ª dK 0 j W j « K1 j W j K 0 j W j ¾ dT »¼ dT ¬ dT ¿
D ijo W K 0 j W j K1 j W j
d 3 K 2 j W W dT 2
0, (i 1,2,3, ! , n)
# m d k E ij W d k K 2 j W W 0 W D W W W K ! K K ¦ ¦ 0j 0j 1j j ij j j dT k dT k j 1 j 1 m
0, (i 1,2,3,!, n)
Indeed, if Eij(t) are linear functions of xi(t), it results that: d E ij t d 2 E ij t d 3 E ij t d k E ij t , , ,! , ,! of any order of the 2 3 dt dt dt dt k 2 3 function Eij(t) are linear functions, respectively, of dxk t , d xk 2 t , d xk 3 t ,! , dt dt dt d p xk t ,! ; dt p
– the derivatives
58
Materials with Rheological Properties
– the coefficient matrices of the functions Eij(t) and of their any order derivatives are identical. Solving the linear algebraic equation systems (3.21), respectively finding the 2 values in W of the unknown factors xk W , dxk W , d xk W , etc., is carried out by dt dt 2 solving successively the respective linear equation systems. The first of systems (3.21) can be solved independently; the others only after finding the solutions of the previous systems, with the help of which the free terms of the system to be solved can be built. If T = T then the integro-differential equations (3.20) are reduced to the following linear algebraic equation systems, having as unknown factors the values in 2 3 4 T of the derivatives dxk t , d xk t , d xk t , d xk t ,!, of the unknown function dt dt 2 dt 3 dt 4 Xk(t): m dE ij T dK T W K 0 j T W j ¦ D ij0 W K 0 j W j K1 j W j 2 j dT dT 1 j 1
^
m
¦ j
dE t dK T t ½ ³ ij K 0 j t W j K1 j t W j 2 j ¾ 0, (i 1,2,3, ! , n) dt dT ¿ W T
(3.22)
m d 2 E ij T dE ij T ª dK 0 j T W j K 0 j T W j K1 j T W j K 0 j T W j ¦ ® ¦ « 2 dT dT j 1 j 1 ¯ dT ¬ m
dK 2 j T T º d 2 K 2 j T W 0 D W W W K K 0j 1j ij j j » dT dT 2 ¼ T dE ij t d 2 K 2 j T t ½° ³ K 0 j t W j K1 j t W j dt ¾ 0, (i 1,2,3,!, n) dT dT 2 °¿ W
m d 3 E ij T d 2 E ij T ª dK 0 j T W j K K 0 j T W j K 1 j T W j T W ¦ j j 0 «2 dT 2 ¬ dT dT 3 j 1
m
¦ j 1
dK 2 j T T º d E ij T ° d 2 K 0 j T W j d 2 K 2 j T T 2 K K T W T W ® 0 1 j j j j » dT dT °¯ dT 2 dT 2 ¼
Composite Resistance Structures
ª dK 0 j T W « dT ¬«
j
K 1 j T W j K 0 j T W
D ij0 W K 0 j W j K 1 j W
T
³
W
dE ij t K 0 j t W dt
j
j
d 3 K 2 j T W dT 3
K t W 1j
j
j
dK 1 j T W
j
59
º dK T T ½°
dT
» ¼»
dt
0 , (i
2j
dT
¾ °¿
d 3 K 2 j T t dT 3
1,2,3, ! , n)
#
m d k E ij T d k K 2 j T W 0 #¦ K 0 j T W j ¦ ! D ij W K 0 j W j K 1 j W j dT k dT k j 1 j 1 m
T
³ W
dE ij t d k K 2 j T t K 0 j t W j K 1 j t W j dt dt dT k
0 , (i 1,2,3, ! , n)
Concerning the expansion in Taylor series of the functions xk around the value W of the variable time and, respectively, finding the values in W of the functions xk and 2 3 of their derivatives dxk , d xk , d xk , etc., we have to add: dt dt 2 dt 3 – the values in T of the functions xk(t) and their expressions (3.19) in the interval (W, T) are known; – the values of the integrals from the right member of systems (3.22) are calculated by substituting the function xk(t) in form (3.19) under the integral and by splitting up, if necessary, the integration interval. Before applying the exposed models on real structures, we should make the following observations concerning the use of the Taylor series for solving the integro-differential equations system (3.8): – the coefficients matrices of equation systems (3.21) and (3.22) are identical to the coefficients matrix of equations system (3.9), a non-singular matrix that express simultaneously the equilibrium of the forces and the compatibility of the deformations with the connections of the structure in the initial phase. The right member expressions of these equation systems also show that not all of the free terms can be zero. It results that equation systems (3.21) and (3.22) admit non-trivial unique solutions, which demonstrates the assertion made before concerning the possibility of expanding in Taylor series the solution of the integro-differential equation system (3.8).
60
Materials with Rheological Properties
– The number of terms in expansion (3.19) depends on the precision required by the evaluation of the functions xk(t) and, implicitly, of the step (t – T). The maximum error due to the use of only (k + 1) terms in the respective expansion is, in the form of Lagrange’s remainder: Rik t
t T k 1 k 1 !
d k 1 xi [ dt k 1
,
T d[ dt ,
(i 1, 2, 3,! , n)
(3.23)
– Finding the solutions of the integro-differential equations system (3.8) by using Taylor’s series expansion is applicable to structures of whatever complexity. Analytical functions that approximate the deformability of the materials, as well as the experimental functions can be used to find the solution. We can set an interesting meaning to the systems of differential equations (3.21) and (3.22) and to their solutions, opening a new path, with a broader applicability field, for the elaboration of the functions that approximate the solutions of the integro-differential equations (3.8’) and (3.8”). Consider the differential equation systems of the order k from (3.21) or (3.22). The differential equation systems from (3.21) or (3.22) that precede those of order k – thus of an order lower than k – together with the initial and contour conditions expressed by the first equations system of (3.21) or, respectively, the values at the time T of the connection forces, according to the case of the considered differential equations system is included in (3.21) or in (3.22), constitute the k additional conditions that the differential equations system solution must fulfill (the Cauchy problem). The differential equation systems of order k from (3.21) or (3.22) approximate the solutions of the integro-differential equation systems (3.8’) or respectively (3.8”) around the value W and respectively T of the variable time. In this context, we have to mention that the equations system from (3.21) and those from (3.22), are independent, being obtained by the successive differentiation of the integro-differential equations (3.8’) and, respectively, (3.8”), around the value W and respectively T of the variable time. In the above presentation, no restriction was imposed on the way the unknown forces – a function of time – xi act. These forces are due to the given external actions and loads and to the interactions of the structure component sections made of materials having different rheological properties. The connection forces xi can be concentrated (discrete), having a fixed position relative to the structure, each of them being a function only of time and/or distributed along the axis of the structure according to a law that has to be determined, thus, being functions of two variables: the section coordinate (position) and time.
Composite Resistance Structures
61
Indeed, by considering the behavior and, implicitly, the implementation of the collaboration between the different areas, having different rheological properties, composing the section of a structural element, we distinguish the following basic types of composite structures: – composite structures with discrete collaboration to which correspond concentrated (discrete) connection forces, variable over time; – composite structures with continuous collaboration to which correspond distributed connection forces, functions of two variables: the section’s coordinate (position) and the time. Each one of these basic types of composite structures corresponds to specific developments of the terms of the equations describing their behavior over time. These developments are presented in the following two sections of this chapter. The knowledge of these developments is also necessary in order to study the composite complex structures. Solving these structures forces us to adopt simultaneously discrete (concentrated) and distributed connection forces. The final part of this chapter will present the composite complex structures. 3.2.1.1. Composite structures with discrete collaboration Even if we consider only the rheological properties of the foundation ground, this case includes the large majority of resistance structures for constructions. However, composite structures whose elements, made out of materials with different rheological properties, are connected by discreetly distributed devices (Figure 3.1) are generally studied under this name: metal plain web or box beams in collaboration with reinforced concrete slabs through rigid shear connection elements (connectors), metal railings beams having compressed reinforced concrete bedplate, etc. For the statically indeterminate structures, initial compression stresses are introduced – through known processes of prestressing using the high strength prestressed reinforcements, through uneven supports, etc. – that cancel the tensile stresses generated by the loads and the actions of the normal operation in certain areas of the reinforced concrete elements composing the structure, and avoid their cracking. The following additional hypotheses are accepted when calculating the composite structures with discrete collaboration: – on the contact surfaces between the structure’s sections having different rheological properties – if such areas exist – no interaction occurs; the collaboration between them is assured only by the shear connection elements (connectors); – the shear connection elements (connectors) allow, in the sections where they are placed, relative displacements between the sections having different rheological
62
Materials with Rheological Properties
properties connected by them. The displacements are elastic, proportional to the stress of the connection elements (connectors), and depend on connection elements of their constructive composition type. Indeed, after the destruction of the adherence between the concrete and the steel, a relative slip between the concrete and steel occurs in the contact surface, under the action of the operation loads, in the sections where the connection elements are placed. The tests revealed the existence of a residual component of this slip, which is negligible compared to the elastic one, preponderant in the case of the correctly dimensioned shear connection elements. We consider a composite discrete collaboration resistance structure and its basic system. By taking into account the known expression from the construction statics – the general method of stresses – of the basic system relative displacement in the connection force i direction, the displacement, due to the stresses inside the structure’s elements having rheological properties j, can be written in the form:
D ij0 W
n
¦G
ijk
xk ' ij
ijk
xk (i 1,2,3,!, n), ( j 1,2,3,!, m)
k 1
E ij W
n
¦G
(3.24)
k 1
where
G ijk ' ij
S p 1 S p 1 ½° °S p 1 1 1 m m ds n n ds qis qks ds ¾ ®³ ¦ is ks is ks ³ ³ p 1 ° S p E0 j I s °¿ S p E0 j As S p G0 j As ¯ S p 1 S p 1 r S p 1 ½° 1 1 ° mis M s ds ³ nis N s ds ³ q is Qs ds ¾ (3.25) ®³ ¦ E 0 j As G0 j As p 1 ° S p E0 j I s °¿ Sp Sp ¯ r
OBSERVATION.– On the integration intervals where the elements of the structure have properties different from j, the value of the integrals from (3.25) is zero. In relation (3.25), we have noted: – mis, mks, nis, nks, qis, qks – bending moments, axial forces and, respectively, shear forces in a section placed between the consecutive sections Sp – Sp + 1 that separate areas of the elements of the structure’s basic system, which are homogenous from the rheological properties point of view, due to its load with the unit stress X1 = 1, and, respectively, Xk = 1;
Composite Resistance Structures
63
– Ms, Ns, Qs – bending moments, axial forces and, respectively, shear forces between sections Sp – Sp + 1, due to the external loads and actions on the basic system;
–
S z2 As
³I
As
2 s
b2
ds – numerical coefficient that depends on the form of the cross-
section; – Is, As – axial inertia moment and, respectively, the area of the current section s, included in the integration interval Sp – Sp + 1; – E0j, G0j – axial and, respectively, shear modulus of elasticity for the material having rheological properties j; – b – the number of the integration intervals (areas of the structure, which are homogenous from the rheological properties point of view). ELEVATION O0O1O2
Ok
Rigid connectors
Reinforced concrete slab
Steel structure
BASIC SYSTEM O0O1O2
Ok
On
Xk X k+1 X k+2
X 3n X 3n+1 X 3n+2
a. Upper deck railway or road bridge consists of lattice steel girder in collaboration with reinforced concrete slab (O -shear connections spacing)
ELEVATION O0O1O2
Shear connectors
Ok
CROSS SECTION Reinforced Shear concrete slab connectors
Reinforced concrete slab
CROSS SECTION Reinforced Shear concrete slab connectors
Structure
b. Boxed composite steel and concrete structure in collaboration with reinforced concrete slab for bridges superstructure
BASIC SYSTEM
O0O1O2
Ok
X 3k X 3k+1 X 3k+2
ELEVATION
HOMOGENEOUS REINFORCED CONCRETE SECTION
HOMOGENEOUS STEEL SECTION
X3
X5
BASIC SYSTEM X7
X9 8
X1
X2
X 10
X11
c. Homogeneous reinforced concrete external statically indeterminate structure (or steel structure if rheological properties of the foundation soil are considered)
Figure 3.1. Composite structures with discrete collaboration
64
Materials with Rheological Properties
By substituting relation (3.24) into the integro-differential equation systems (3.8’) and (3.8”), we obtain:
º ª n ¦ «¦ G ijk xk W ' ij »K 0 j W j K1 j W j K 2 j T W j 1 ¬k 1 ¼ m
m T
¦ ³ G ijk j 1W
>
@
dxk K 0 j t W j 1 K1 j t W j K 2 j T t dt dt
(3.26)
0,
(i 1,2,3, ! , n) ª
m
n
¦ «¬¦ G j 1
>
k 1
@
º x W ' ij »K 0 j W j K1 j W j K 2 j T W K 2 j T W ¼
ijk k
m T n dx ¦ ®³ ¦ Gijk k K 0 j t W j 1 K1 j t W j K 2 j T t dt dt j 1 ¯T k 1
>
T
n
³ ¦ Gijk W k 1
@
(3.27)
½ dxk K 0 j t W j K1 j t W j K 2 j T t K 2 j T W dt ¾ 0, dt ¿
>
@
(i 1,2,3,!, n) By using the matrix representation, integro-differential equations (3.26) and (3.27) take the form:
¦ >G x W ' @K W K W K T W m
j
j
0j
j
1j
j
2j
j 1
m T
(3.26’)
>
@
d ¦ ³ G j >x t @K 0 j t W j 1 K1 j t W j K 2 j T t dt dt j 1W
0
¦>G x W ' @K W K W >K T W K T W @ m
j
j
0j
j
1j
2j
j
2j
j 1
T d ¦®³ G j >x t @K0 j t W j 1 K1 j t W j K2 j T t dt dt j 1 ¯T m
T
³G j W
>
@
(3.27’)
½ d >x t @K0 j t W j K1 j t W j K2 j T t K2 j T t dt¾ 0 dt ¿
>
@
Composite Resistance Structures
65
We will use the systems of linear algebraic equations (3.21) to determine the dxi W d 2 xi W
values xi(W), , , etc., which are necessary to build the functions xi(t), dt dt 2 W < t, by expanding in Taylor series around the value W of the time. By considering the relations from (3.24), we obtain the following matrix representation:
G W x W ¦ ' j K 0 j W j 0 m
j 1
(3.28)
m dK W W d 0 G W >x W @ ¦ G j x W ' j K 0 j W j K1 j W j 2 j dT dT j 1 m dK W dK W W º d2 d >x W @ª« 0 j j K 0 j W j K1 j W j 2 j G W 2 >x W @ ¦ ^G j » dT dT dT j 1 ¬ dT ¼
>
@
d 2 K W W ½ ª d >x W @ ' j º» K 0 j W j K1 j W j 2 j 2 °¾ 0 «G j dT °¿ ¬ dT ¼
G W
m dK W W º ª dK W d3 d2 > @ >x W @«2 0 j j K 0 j W j K 1 j W j 2 j x W G » ¦ j 3 2 dT dT dT dT j 1 ¬ ¼
G j
d 2 K W d 2 K W W d >x W @°® 0 j2 j 2 K 0 j W j K 1 j W j 2 j 2 dT dT °¯ dT
dK 1 j W j º dK 2 j W W ½ ª dK 0 j W j « K 1 j W j K 0 j W j ¾ » dt ¼ dT ¬ dT ¿
>
@
G j x W ' j K 0 j W j K 1 j W j #
G W
d 3 K 2 j W W dT 3
0
m d k K 2 j W W dk > @ ! ' W G W W W x x K K ¦ 0j 1j j j j j dT k dT k j 1
>
@
0
If we follow the construction of the functions xi(t) by expansion in Taylor series around the value T > W of the time variable (assuming that the values xi(T) of the dxi T d 2 xi T d 3 xi T
unknown factors at the time T are known), the values , , , dt dt 2 dt 3 etc., are determined by successively solving the linear algebraic equations (3.22). In matrix representation, by considering (3.24), they have the form:
66
Materials with Rheological Properties m dK T W d >x T @ ¦ ® G j x W ' j K 0 j W j K1 j W j 2 j dt dT j 1¯
>
G T
@
dK 2 j T t ½ d ³ G j >x t @K 0 j t W j K1 j t W j dt ¾ dt dT ¿ W
(3.29)
T
G T
0
m dK 2 j T T º ª dK T W j d2 >x T @ ¦ ®G j d >x T @« 0 j K 0 j T W j K 1 j T W j » 2 dt dT dT dt j 1¯ ¬ ¼
d 2 K 2 j T W G j x W ' j K 0 j W j K 1 j W j dT 2 T d 2 K 2 j T t ½° d dt ¾ 0 ³ G j >x t @K 0 j t W j K 1 j t W j dt dT 2 °¿ W 3 2 m dK dK 2 j T T º T W ª d d j 0j G T 3 >x T @ ¦ G j 2 >x T @ «2 K 0 j T W j K 1, j T W j »
>
@
dt
Gj
dt
j 1
d dt
d K 0 j T W 2
>x T @°® °¯
ª dK 0 j T W « dT «¬
>
j
@
dT
K 1 j T W
G j x W G j K 0 j W T
³G j W
d dt
j
j
2
j
dT
«¬
dT
2 K 0 j T W
K T W 0j
K W 1j
j
K T W 1j
j
dK 1 j T W
j
j
>x t @K 0 j t W j K 1 j t W j
dT 3
dt
2j
» »¼
dT
dT 3 d 3 K 2 j T t
dT 2 dT
>
@
dk >x T @ " G j x T ' j K 0 j W dt k
T
³G W
j
d >x t @K 0 j t W dt
j
K t W 1j
j
j
¾ °¿
0
#
G T
º dK T T ½°
d 3 K 2 j T W
j
d K 2 j T T
»¼
2
K W d 1j
j
d k K 2 j T t dT
k
k
dt
K 2 j T W dT
k
0
In equations (3.28) and (3.29), we have noted: Gj, 'j – the matrix of the coefficients Gijk, and, respectively, the free terms vector 'ij, calculated using relations (3.25); x t , x W , x T – the vectors of the connection forces xk and those of their k k derivatives d >x t @, d >x W @, d >x T @ at the time W, and, respectively, T. dt dt k dt k
Composite Resistance Structures
67
In addition, in equations (3.28) and (3.29), we have considered the following relations:
G W
m
¦G
j
j 1
G T
K 0 j W j (3.30)
m
¦G
j
K 0 j T
j 1
The elements of matrices Gi and vectors 'j are calculated by considering the undeformed lengths of the elements composing the structure. 3.2.1.2. Composite structures with continuous collaboration The composite structures with continuous collaboration are those structures that are calculated by accepting the hypothesis that along the contact surface between the areas having different rheological properties, composing the sections of the structure, the connection elements are continuously distributed (Figure 3.2) and they allow relative displacements of the sections, proportional to the stress. From the calculus hypothesis point of view, the composite structures with continuous collaboration can be considered as the limit case of the composite structures with discrete collaboration, when the distance between the connection elements tends towards zero. Displacements E ij ( r , t ) and D 0 ij ( r ,W ) on the connection force i direction in the structure’s section r, due to the stresses generated in structure’s basic system areas having the rheological properties j, by the unknown connection forces xk(s,t), and, respectively, by the unknown connection forces xk(s,t) and by the external actions and loads applied to the structure at the moment W, can be written in the form:
E ij r , t
n
l
¦ ³K r , s x s, t ds ijk
k
k 1 0
D ij0 r ,W
n
(3.31)
l
¦ ³K r , s x s,W ds ' r ijk
k
ij
k 1 0
In relations (3.31), we have noted: xi(s,t) – the function expressing the value of the connection force in the i direction, at the moment t, in the current section s of the composite structure, corresponding to the external actions and loads applied to the structure at the moment W;
68
Materials with Rheological Properties
Kijk(r.s) – the function expressing the initial fictitious specific displacement of the structure’s basic system on the direction of the connection force i, in the current section r, due to the stresses generated in the structure’s basic system areas having rheological properties j, by a unit force acting in the connection force k direction in the section s; 'ij(r) – the function expressing the initial fictitious specific displacement of the structure’s basic system, in the connection force i direction, in the current section r, due to the stresses generated by the external actions and loads in the structure’s basic system areas having the rheological properties j. We consider a differential element of the basic system of a composite structure with continuous collaboration. We note: X ijk (r ) – the initial fictitious specific displacement of the structure’s basic system in the connection force i direction, in section r, due to the stresses generated in the structure’s basic system areas having the rheological properties j by the unit force acting in the direction of the connection force k in section r; ' ij (r ) – the initial fictitious specific displacement of the structure’s basic system on the connection force i direction, in section r, due to the stresses generated in the structure’s basic system areas having the rheological properties j by the external actions and loads; Oik (r ) – the displacement of the structure’s basic system in the connection force i direction, in section r, due to the stresses generated in the connection elements by a unit force applied in the section r in the connection force k direction. To the specific displacement Oik (r ) there corresponds the specific rigidity cik (r ) of the connection elements, so that, between these sizes the following relation exists:
Oik (r ) 1 / cik (r ) The rigidity cik (r ) varies with the type and with the placement mode of the shear connection elements; it is experimentally determined. By analyzing, on the structure’s basic system, the interaction between the stresses and the strains in a section r of the structure, with the stresses and the displacements of the other sections, we can make the following observation: – the stresses and strains state in any unspecified section of the structure is in reciprocal interaction only with the adjacent neighboring sections and only due to the variation, according to the section coordinates, of the connection forces generating the stresses in the connectors (connection elements). The coupling of the equations describing the stresses and strains state in one unspecified section with the
Composite Resistance Structures
69
adjacent neighboring sections is therefore carried out only by the sizes expressing the variation, in the respective section, of the unknown forces generating stresses in the connection elements (connectors); – the evaluation of displacements X ijk (r ) and ' ij (r ) will be carried out by using the lengths of the structure’s basic system not deformed elements. OBSERVATION.– The deformability of the connection elements (connectors) is not taken into account when we evaluate displacements X ijk (r ) and ' ij (r ) . The displacements due to the deformability of the connection elements are evaluated separately. OBSERVATION.– In what follows, we consider a section of differential constant length dl, in a composite structure. After having removed the additional connections, whatever area having rheological properties j and an unspecified orientation relative to an arbitrary coordinates system, will have the undeformed length:
dl ' dl 1
1 N j r
E j W A j r
§ dy r · § dz r · 1 ¨ ¸ ¨ ¸ dl © dl ¹ © dl ¹ 2
2
or
dl ' dl H j r dl where we have noted: Nj(r) – the axial force that stresses the area with the rheological properties j, in the structure’s section r, due to the actions and loads applied to the structure before the moment W;
dy (r ) , dz (r ) – the derivatives of the axis equation for the area having the dl dl rheological properties j, in the structure’s section r.
70
Materials with Rheological Properties ELEVATION Reinforced concrete slab
Elastic shear connectors (headed studs)
CROSS SECTION Elastic shear connectors Reinforced conctrete slab (headed studs) Steel structure
x
X1(x) X2(x) X3(x)
Steel structure BASIC SYSTEM
a. Composite steel and concrete structure with countinuous colaboration carried out with dens distributed elastic shear connections (headed studs).
ELEVATION Reinforced concrete slab
Elastic shear connectors steel spirals
CROSS SECTION Reinforced Elastic shear connectors conctrete slab steel spirals Steel girders
Steel girders BASIC SYSTEM
b. Composite steel and concrete structure with countinuous collaboration carried out with elastic shear connections made of steel spirals.
x
X1(x) X2(x) X3(x) CROSS SECTION
LONGITUDINAL SECTION
Reinforced concrete beam Foudation soil c. Continuous reinforced beam which span continuously on a medium with or without rheological properties
BASIC SYSTEM X1(x) x X2(x) X3(x)
Figure 3.2. Composite structures with continuous collaboration assured by elastic shear connection elements
By taking into account the previous observation and by using the Dirac delta function G(r,s), we can write:
K ijk r , s G r , s -ijk r Therefore, by considering the property of the Dirac delta function, relations (3.31) take the form:
E ij r , t
n
¦- r x r , t ijk
k
k 1
E il r , t
w ¦ O r n
2
k 1
D ij0 r ,W
x k r , t
(3.31’)
w r2
ik
n
¦- r x r ,W ' r ijk
k 1
k
ij
Composite Resistance Structures
71
0 if s z r ½ ¾ has the property ¯z 0 if s r ¿
OBSERVATION.– The Dirac delta function G(r,s) = ® s
³
that, in an interval (a, b), r ( a, b ) : (r , s)ds 1 . a
If we substitute relation (3.31’) in (3.8’) and (3.8”) and by considering the fact that the connection elements can have only elastic deformations (therefore, for the material of the connection elements K0(W) = 1, K1(W) = 1, K2(t) = 0), we obtain:
ª º ¦ «¬¦ - r x r ,W ' r »¼K W K W K T W m
n
j 1
k 1
ijk
m T
k
ij
n
¦ ³ ¦ -ijk r j 1 W k 1 n
T
k 1
W
¦ Oik r ³
w xk r , t dt
w xk r , t dt w 2 rw t
0j
j
1j
j
2j
>
@
K 0 j t W j 1 K1 j t W j K 2 j T t dt
(3.32)
3
0, (i 1,2,3, ! , n)
ª º ¦ «¬¦- r x r, t ) ' r »¼K W K W >K T W K T W @ m
n
j 1
k 1
ij
m T
k
n
¦ ³ ¦-ijk r j 1T k 1 m T
n
ij
m
j
1j
j
2j
2j
w xk r , t K 0 j t W j >1 K1 j t W j K 2 j T t @dt wt
¦ ³ ¦- ijk r j 1W k 1
0j
w xk r , t K 0 j t W j K1 j t W j >K 2 j T t K 2 j T t @dt wt
w 3 xk r , t dt 0, (i 1,2,3,!, n) 2 T w rw t T
¦ Oik r ³ k 1
OBSERVATION.– By hypothesis, the connection elements allow relative displacements between the structure’s sections that they connect, proportional to the stresses loading them. In the structure’s section r, the stresses inside the shear connectors are generated by the variation wxk/wr of the connection forces and they are equal to it, so that the displacement on the connection force xi direction due to the connection force xK has the value:
G lik r O ik
w xk r , t wr
72
Materials with Rheological Properties
The variation of this displacement on the length dr of the considered element will thus have the value:
w >G lik r @dr wr
Oik
w 2 x r , t dr w r2
By using the matrix representation, equations (3.32) take the form:
¦ >- r x r,W ' r @K W K W K T W m
j
j
0j
j
1j
j
2j
j 1
m T
¦ ³ - j r j 1W
T
O r ³ W
w >x r , t @K 0 j t W j >1 K1 j t W j K 2 j T t @dt wt
w3 >x r , t @dt 0 w 2 rw t
(3.33)
¦ >- r x r,W ' r @K W K W >K T W K T W @ m
j
j
0j
j
1j
j
2j
2j
j 1
m T
¦ ³ - j r j 1T
m T
¦ ³ - j r j 1W
w >x r , t @K 0 j t W j >1 K1 j t W j K 2 j T t @dt wt
w >x r , t @K 0 j t W j K1 j t W j >K 2 j T t K 2 j T t @dt wt
w3 >x r , t @dt 0 w 2 rw t T T
O r ³
The obtained equation systems (3.32) and (3.33) describe the evolution of the stresses and strains state of a composite structure with continuous collaboration. If we substitute relation (3.31’) in (3.21) and by using the matrix representation, we obtain the following partial differential equation systems: m w2 0 2 > x r , t @ - r , W x r , W ¦ ' j r K 0 j W j w r j 1 w3 w O r 2 >x r ,W @ - r ,W >x r ,W @ wr wt wT m dK W W ¦ >- j r x r ,W ' j r @K 0 j W j K1 j W j 2 j 0
O r
j 1
dT
(3.34)
Composite Resistance Structures
73
w4 w2 x r r W W , , > @ >x r ,W @ w r 2w t 2 wT 2 m dK W W º ª dK W w ¦ - j r >x r ,W @« 0 j j K 0 j W j K1 j W j 2 j » dT wT j 1 ¬ dT ¼
O r
>
d 2 K 2 j W W
@
¦ - j r x r , W ' j r K 0 j W j K1 j W j m
j 1
dT 2
0
w3 w5 > @ >x r , W @ , , W W x r r wT3 w r 2w t 3 m dK W W º ª dK W w2 >x r , W @«2 0 j j K 0 j W j K 1 j W j 2 j ¦ - j r » 2 dT dT wT j 1 ¬ ¼
O r
m
¦ - j r j 1
d 2 K W d 2 K W W w >x r , W @°® 0 j2 j 2 K 0 j W j K 1 j W j 2 j 2 wT dT °¯ dT
dK 1 j W j º dK 2 j W W ½ ª dK 0 j W j « K 1 j W j K 0 j W j ¾ » dT ¼ dT ¬ dT ¿ 3 m d K 2 j W W 0 ¦ - j r x r , W ' j r K 0 j W j K 1 j W j dT 3 j 1
>
@
#
O r
w k 2 wk W W , , x r r > @ >x r ,W @ " w r 2w t k wT k
"
m d k K 2 j W W ¦ - j r x r , W ' j r K 0 j W j K 1 j W j 0 dT k j 1
>
@
The first differential equations system of (3.34) can be reduced to a system of first order ordinary differential equations (if we consider the fact that, for t = W, it describe the structure’s initial stresses and strains state, which does not vary with the time). Together with the boundary conditions (the Cauchy problem):
'( s0 )
O ( s 0 ) u ( s 0 ,W )
(3.35)
74
Materials with Rheological Properties
this system allows the determination of unknown connection forces at the moment t = W (in the initial phase). Given the system of partial differential equations of order k in (3.34); the equation systems of (3.34) which precede it, together with the boundary conditions (3.35), represent k additional conditions which must be fulfilled by the solution of the considered system and constitute the Cauchy problem attached to this system. In these conditions, the solution of the system of partial differential equations of order k approximates, for toWW, the solution of the first integral equations system from (3.33). The precision of the approximation increases with the order k of the system of partial differential equations and decreases when the interval t – W grows. The systems of partial differential equations that are obtained by making the substitution (3.31) in (3.22) describe the structure’s stresses and strains state at the moment ș, ș > W. As demonstrated before, the use of the equations obtained in this way assumes the stresses and strains state is known and, implicitly, the connection forces at the moment t = W – in the initial phase – at the moment ș, W > 0, as well as the functions describing the evolution of the stresses and strains state and those of the connection forces in the time interval W – ș. In matrix representation, they have the form:
O r
w3 >x r , T @ - r , T w >x r , T @ 2 w r wT wT
m dK T W ¦ - j r x r ,W ' j r K 0 j W j K1 j W j 2 j dT j 1
>
@
m T
¦ ³ - j r j 1W
O r
(3.36)
dK T t w >x r , t @K 0 j t W j K1 j t W j 2 j dt wt dT
w4 w2 T T x r , r , > @ >x r ,T @ w r 2w t 2 wT2
m
¦- j r j 1
m T
dK 2 j T T º ª dK T W j w K 0 j T W j K1 j T W j >x r ,T @« 0 j » wT dT dT ¬ ¼
¦ ³ - j r j 1W
0
d 2 K T t w >x r , t @K 0 j t W j K1 j t W j 2 j 2 dt wt dT
0
Composite Resistance Structures
75
w5 w3 T T , , x r r > @ >x r , T @ w r 2w T 3 wT 3 m dK 2 j T T º ª dK T W j w2 >x r , T @«2 0 j ¦ - j r K 0 j T W j K 1 j T W j » 2 dT dT wT j 1 ¬ ¼
O r
m
¦ - j r j 1
d 2 K T W d 2 K T T w >x r , T @°® 0 j 2 j 2 K 0 j T W j K 1 j T W j 2 j 2 wT dT dT °¯
dK 1 j T W j º dK 2 j T T ª dK 0 j T W j « K 1 j T W j K 0 j T W j » dT dT dT ¬ ¼ m d 3 K 2 j T T ¦ - j r x r , W ' j r K 0 j W j K 1 j W j dT 3 j 1
>
m T
@
¦ ³ - j r j 1W
d 3 K T t w >x r , t @K 0 j t W j K 2 j t W j 2 j 3 dt wt dT
0
# O r
w k 2 wk > @ >x r , T @ " T T x r , r , w r 2w T k wT k
>
@
¦ - j r x r , W ' j r K 0 j W j K 1 j W j m
j 1
m T
¦ ³ - j r j 1W
d k K 2 j T T dT k
d k K T t w >x r , t @K 0 j t W j K 1 j t W j 2 j k dt 0 wt dT
In equations (3.34) and (3.36), we have noted:
- r ,W
¦- r K W m
j
0j
j
j 1
- r ,T
¦- r K T W m
j
0j
j
j 1
The equations obtained so far in this section have been established in the hypothesis (considered at the beginning) that, between the different areas of the structure, having different rheological properties, along the contact surfaces between them, may appear relative displacements proportional to the stresses loading the connection elements.
76
Materials with Rheological Properties
In the case of these structures, the hypothesis that the plane sections normal on the axis of the structure before the deformation remain plane and normal on the axis of the structure after its deformation (Bernoulli) is not achieved. Yet there are situations where at least two of the areas (but not all of them) of the composite structure with continuous collaboration (areas having different rheological properties) interact so that on the surface separating them no relative displacement can appear. This case is taken into account by introducing into the respective equations a specific rigidity of the connection elements corresponding to a zero displacement (cik(r) o f Oik(r) = 1/cik(r) = 0). Therefore, the equations established so far in this chapter allow the determination of the evolution in time of the stresses and strains state and of the connection forces that correspond to them in the composite structures with continuous collaboration, to which, at least on a separation surface between two areas having different rheological properties, the collaboration is ensured by shear connectors that allow relative displacements proportional to the stresses loading them. If the connection elements do not allow relative displacements (slips) (Figure 3.3) on any of the separation surfaces between different areas of a composite structure with continuous collaboration, having different rheological properties – so that the hypothesis of Bernoulli is respected – equations (3.33) describing the evolution of the stresses and strains state and, implicitly, of the connection forces corresponding to them, take the form:
¦ >- r x r , W ' r @K W K W K T W m
j
j
0j
j
1j
j
2j
(3.33’)
j 1
w ¦ ³ - j r >x r , t @K 0 j t W j >1 K 1 j t W j K 2 j T t @dt wt j 1 W m T
0
¦ >- r x r , W ' r @K W K W >K T W K T W @ m
j
j
0j
j
1j
j
2j
2j
j 1
m T
¦ ³ - j r j 1T m T
¦ ³ - j r j 1 W
w >x r , t @K 0 j t W j >1 K1 j t W j K 2 j T t @dt wt w >x r , t @K 0 j t W j K1 j t W j >K 2 j T t K 2 j T t @dt wt
0
Composite Resistance Structures
77
The evolution of the stresses state in an unspecified section of the structure is independent of the stresses state in its other sections, which means that the coupling 2 term O (r ) w >x r , t @ is canceled. wr 2 Indeed, equations (3.33’) are obtained from (3.33) by taking into account the fact that the absence of relative displacements between the different areas of a composite structure, having different properties, implies infinitely rigid connection elements, thus: 1/cij(r) = 0, (i, j = 1,2,3,...,n) or, in other words, O(r) { 0 CROSS SECTION
ELEVATION Reinforced concrete for grouting
Prestressed prefab beams BASIC SYSTEM
Prestressed tendons
Prestressed concrete beams Prestressing tendons a. Composite prestressed prefab concrete and reinforced concrete structure with continuous collaboration ensured by the adherence.
x (x) x (x) x (x) x (x) x (x) x (x) ELEVATION
CROSS SECTION Grouting reinforced concrete (cast in the second stage) Prestressed steel girders
BASIC SYSTEM
Rienforced concrete cast in first stage b. Composite - preflex girders made of prestressed steel girders (by jacking at supports) encased in reinforced concrete with continuous collaboration ensured by the adherence.
Figure 3.3. Composite structures with continuous collaboration assured by adherence (finite rigid collaboration elements)
78
Materials with Rheological Properties
In the same way, from (3.34) and from (3.36), we obtain: - r , W x r , W ¦ ' j r K 0 j W j 0 m
(3.34’)
j 1
m dK W W w >x r ,W @ ¦ >- j r x r ,W ' j r @K 0 j W j K1 j W j 2 j 0 dT wT j 1 m dK W W º ª dK W w2 w - r ,W 2 >x r ,W @ ¦- j r >x r ,W @« 0 j j K 0 j W j 2 j » wT dT wT j 1 ¬ dT ¼
- r ,W
m d 2 K 2 j W W ¦ - j r x r ,W ' j r K 0 j W j K1 j W j 0 dT 2 j 1 m dK W W º ª dK W w3 w2 - r ,W 3 >x r ,W @ ¦- j r 2 >x r ,W @«2 0 j j K 0 j W j K1 j W j 2 j » dT dT T w wT j 1 ¬ ¼
>
m
¦- j r j 1
@
d 2 K W d 2 K W W w >x r ,W @°® 0 j2 j 2K 0 j W j K1 j W j 2 j 2 dT wT °¯ dT
dK1 j W j º dK 2 j W W ½ ª dK 0 j W j K1 j W j K 0 j W j « ¾ » dT ¼ dT ¿ ¬ dT m d 3 K 2 j W W 0 ¦ - j r x r ,W ' j r K 0 j W j K1 j W j dT 3 j 1
>
@
#
- r ,W
wk >x r ,W @ " wTk
"
m d k K 2 j W W ¦ - j r x r ,W ' j r K 0 j W j K1 j W j dT k j 1
>
@
0
and, respectively: - r ,T
m dK T W w >x r ,T @ ¦ >- j r x r ,T ' j r @K 0 j W j K1 j W j 2 j wT dT j 1
(3.36)
dK T t w >x r , t @K 0 j t W j K1 j t W j 2 j dt 0 dT wt j 1W m dK 2 j T T º ª dK 0 j T W j w2 w - r ,T 2 >x r ,T @ ¦ - j r >x r ,W @« K 0 j T W j K1 j T W j » wt dT dT wT j 1 ¬ ¼ m T
¦ ³ - j r
>
@
¦ - j r x r ,W ' j r K 0 j W j K1 j W j m
j 1
m T
¦ ³ - j r j 1W
d 2 K 2 j T W dT 2
d 2 K T t w >x r , t @K 0 j t W j K1 j t W j 2 j 2 dt wt dT
0
Composite Resistance Structures - r ,T
79
2 m dK 2 j T T º ª dK T W j w3 >x r ,T @ ¦- j r w 2 >x r,T @«2 0 j K 0 j T W j K1 j T W j » 3 dt dT wT wT j 1 ¬ ¼
m
¦ - j r j 1
d 2 K T W d 2 K T T w >x r,T @°® 0 j 2 j 2 K 0 j T W j K1 j T W j 2 j 2 . dT wT dT °¯
dK1 j T W j º dK 2 j T T ª dK 0 j T W j K1 j T W j K 0 j T W j « » dT dT dT ¼ ¬ m d 3 K 2 j T W j ¦ - j r x r ,W ' j r K 0 j W j K1 j W j dT 3 j 1
>
m T
¦ ³ - j r j 1W
@
d 3 K T t w >x r , t @K 0 j t W j K1 j t W j 2 j 3 dt 0 wt dT
#
- r ,T
wk >x r ,T @ " wTk
"
>
@
¦ - j r x r ,W ' j r K 0 j W j K1 j W j m
j 1
m T
¦ ³ - j r j 1W
d k K 2 j T W dT k
d k K T t w >x r , t @K 0 j t W j K1 j t W j 2 j k dt 0 wt dT
The equation systems of (3.34’) and (3.36’) allow the approximation, with the required accuracy degree, of the solutions of the integro-differential equation systems (3.33’) around the values W and, respectively, T, of the variable time. The integro-differential equation systems describe the evolution of the stresses and strains state of the composite structures with continuous collaboration, whose behavior imposes the adoption of the hypothesis that in all the points of the contact surfaces between the different areas of the structures, having different rheological properties, their relative displacements are zero. In this class of structures where the adherence between the areas having different rheological properties is maintained during the entire lifetime of the structure, are included: – composite preflex beams; – prestressed concrete (after the accomplishment of the adherence between the prestressed reinforcement and the concrete); – reinforced concrete composite structures (precompressed concrete, etc.).
80
Materials with Rheological Properties
3.2.1.3. Composite structures with complex composition As agreed in section 3.2, under this denomination we will analyze the composite structures made of materials having different rheological properties (among them, we can also include the foundation ground) for which, in order to describe the evolution in time of their stresses and strains state, it is necessary to adopt, simultaneously, discrete connection forces as well as the distributed connection forces. The discrete forces are variable only with time whilst the distributed connection forces are variable with the coordinate of the section and with time. We consider a composite structure with complex composition and its basic system. By taking into account (3.24) and (3.31’), displacements Eij, El and D0 take the following form in matrix representation:
>E r , t @ ij
>E l r , t @
>D
0
r ,W @
ªl º ªG j º K j r y r , t dr » « ³ « T » x t « 0 » ¬«K j r ¼» «¬ - j r y r , t »¼ I ª º 2 « » w « O r 2 > y r , t @» wr ¬« ¼»
(3.37)
ªl º ª Gj º K j r y r , W dr » ª ' j º « ³ «K T r » x W « 0 » « D j r » ¬ ¼ ¬ j ¼ r y r , W ¬« j ¼»
In (3.37), we have noted: x(t) – the vector of the discrete connection forces xi(t); y(r,t) – the vector of the distributed connection forces, yi(r,t); Kj(r) – the matrix of the functions Kijk(r) representing the displacement of the structure’s basic system on the discrete connection force xi(t) direction, due to the stresses generated in the structure’s basic system areas having the rheological properties j by a force applied in the section r of the basic system on the distributed connection force yk(r,t) direction;
Composite Resistance Structures ELEVATION Prestressed tendons
Shear connectors densely distributed
Reinforced concrete slab
Steel structure BASIC SYSTEM Y1(5) Y2(5) Y3(5)
O0 O1 O2
Ok
Y1 (s) Y4 (s) Y2 (s) Y5 (s) Y3 (s) Y6 (s)
x
ELEVATION Prestressed tendons
Reinforced concrete slab
CROSS SECTION Reinforced Dense elastic shear concrete slab connectors
Steel structure a. Complex structure. The collaboration between the concrete slab and the upper flangeof the steel structure is ensured by the dense elastic shear connectors. In tension area the concrete slab is precompressed.
CROSS SECTION Prestressed Rigid shear tendons connectors Reinforced concrete slab
Steel structure Y1 (s) Y2 (s) Y3 (s)
DETAIL "A" X3n+1 ds X3n+2 X3n+3
BASIC SYSTEM O0 O1
X 3k X 3k+1 X 3k+2
Ok
On
"A"
ELEVATION
On
81
Plain web girders
b. Composite complex structure consists of steel plain web girders and concrete slab The collaboration between elements is ensured by the shear connectors having high capacity to transmit the forces and disposed at great spacing (discrete collaboration). In tensile areas the slab is precompressed.(continuous collaboration by the adherence).
CROSS SECTION Precompressed concrete structure Prestressed tendons
BASIC SYSTEM X
X1
c. External statically indeterminate precompressed concrete structure.
X4 X
Y1(s) Y2(s) Y3(s)
X5
Figure 3.4. Composite structures with complex composition
Dj(r) – the vector of the functions Dij(r) representing the displacement of the basic system of the structure in section r on the distributed connection force yi(r.t) direction, due to the stresses generated in the structure’s basic system areas having the rheological properties j by the external loads and actions; ) – the zero vector, with dimensions equal to the vector x(t). Matrices Gj, Qj(r), O(r), as well as vector 'j, have the significance established in the previous sections.
82
Materials with Rheological Properties
By substituting (3.37) in (3.8’) and in (3.8”), we obtain the integro-differential equation systems describing the evolution of the stresses and strains state of the composite structures with complex composition. The first has as initial phase moment W of the external loads and actions application; the other has (as initial phase) a moment T, T > W, of the structure’s lifetime for which the stresses and strains state is known, as well as the function describing its evolution in the interval (T – W). Below, we will use only the compact expression in the matrix representation form of the equations, so that the integro-differential equation systems that we are talking about will have the following form: ªl º °° ª G j º « ³ - j r y r ,W dr » ª ' j º °½ « » W x » ¾ K 0 j W j K1 j W j K 2 j T W ® T «0 »« ° ¬«- j r ¼» « - r y r ,W » «¬ D j r »¼ °¿ ¼ j °¯ ¬
m
¦
j 1
ªl º½ w K j r ª¬ y r , t º¼ dr » ° °ª G º « ³ j d ° wt » ¾° K t W u » ª¬ x t º¼ « 0 ³ ®« T j « »° 0 j dt « » w 0 ° ¬K j r ¼ « » ª º r y r , t ¼ ° j ° wt ¬ ¬ ¼¿ ¯ l
T w3 u ª1 K1 j t W j K 2 j T t º dt ³ O r ¬ª y r , t ¼º dt ¬ ¼ w r 2w t W
(3.38)
0
and, respectively: m
¦
j 1
½ ªl º °ª G j º « ¦K j r y r ,W dr » ª ' j º ° » x W « 0 « » ®« T ¾K 0 j W j K1 j W j u » ° «¬K j r »¼ « - j r y r ,W » ¬« D j r ¼» ° ¬ ¼ ¯ ¿
ªl º½ w °ª G º « ³ K j r ª¬ y r , t º¼ dr » ° j d ° wt »°u » ¬ª x t ¼º « 0 u ¬ª K 2 j T W K 2 j T W ¼º ³ ® « T « » ¾° w T ° «¬K j r »¼ d t « - j r ª¬ y r , t º¼ » ° ° wt ¬ ¼¿ ¯ T
T ª G j º d ° » ª¬ x t º¼ u K 0 j t W j ª1 K1 j t W j K 2 j T t º d t ³ ® « T ¬ ¼ ° «¬K j r »¼ d t W¯
ªl º½ w « ³ K j r ¬ª y r , t ¼º dr » ° wt » ° K t W K t W ª K T t K T t º dt «0 1j 2j j j ¬ 2j ¼ « » ¾° 0 j w « - j r ª¬ y r , t º¼ » ° wt ¬ ¼¿
M ª º « » 3 ³« w »d t ª º O r y r t , T« ¬ ¼» w r 2w t ¬ ¼ T
0
(3.39)
Composite Resistance Structures
83
By considering relations (3.37) and by noting:
w >E j r , t @ wt k k
º ªl wk > y r , t @dr » K r « j k ³ ª Gj º d w t » «K T r » k >x t @ « 0 » « wk ¬ j ¼ dt « - j r wt k > y r , t @ » ¼ ¬ k
(3.40)
I ª º wk k 2 « » E > @ r , t w l « O r 2 k > y r , t @» wt k wr wt ¬ ¼ the integro-differential equation systems (3.38) and (3.39) take the form:
¦ ^>D r ,W @K W K W K T W m
0
0j
j
1j
j
2j
j 1
T
³ W
T
³ W
w >E r , t @K 0 j W j >1 K1 j W j K 2 j T t @d t` wt j
(3.38’)
w >E l r , t @d t 0 wt
and, respectively:
¦ ^>D r ,W @K W K W >K T W K T W @ m
0
0j
j
1j
j
2j
2j
j 1
T
³ T
T
w >E j r, t @K 0 j t W j >1 K1 j t W j K 2 j T t @d t wt
³
w >E l r , t @K 0 j t W j K1 j t W j >K 2 j T t K 2 j T t @d t` wt
³
w >E l r , t @d t wt
W
(3.39’)
0
Using this compact expression in the following and by considering (3.37) and (3.40), equations (3.21) and (3.22) will have, in the case of the composite structures with complex composition, the following form:
84
Materials with Rheological Properties
>E l r ,W @ ¦ >D 0j r , W @K 0 j W j m
0
j 1
m w >E l r , W @ ¦ ® w >E r , W @K 0 j W j wt j 1 ¯ wt
>
@
D 0j r , W K 0 j W j K 1 j W j
d K 2 j W W ½ ¾ 0 dt ¿
m w 2 w2 , E W > @ r ® 2 >E j r , W @K 0 j W j ¦ l wt 2 j 1 ¯ wt
d K 2 j W W º ª d K 0 j W j w > E j r , W @« K 0 j W j K 1 j W j » wt dT ¬ dT ¼
d 2 K 2 j W W ½° D 0j r , W K 0 j W j K 1 j W j ¾ 0 dT 2 °¿
>
@
m w3 w3 >E j r,W @K 0 j W j > @ r E W , ¦ l 3 wt 3 j 1 wt
d K 2 j W W º ª d K 0 j W j w2 > @ r K K E W W W , 2 « » j j j j j 0 1 dT dT wt 2 ¬ ¼
° d 2 K 0 j W j d 2 K 2 j W W w >E j r , W @® 2 K 0 j W j K 1 j W j 2 wt dT 2 °¯ d T d K 1 j W j º d 2 K 2 j W W ½° ª d K 0 j W j K 1 j W j K 0 j W j « ¾ » dT ¼ dT 2 °¿ ¬ dT d 3 K 2 j W W D 0j r , W K 0 j W j K 1 j W j dT 3
>
@
0
# k m wk >E l r ,W @ ¦ w k >E j r ,W @K 0 j W j " k wt j 1 wt
"
>
@
d k K 2 j W W
D r , W K 0 j W j K 1 j W j 0 j
dT k
0
(3.41)
Composite Resistance Structures
85
and, respectively: m w w ª¬ El r ,T º¼ ¦ ® ª E j r , T º K 0 j T W j ¬ ¼ wt j 1¯w t
ªD 0j r ,W º K0 j W j K1 j W j ¬ ¼ T
³ W
d K 2 j T W dT
w ª E r , t ¼º K 0 j t W j K1 j t W wt ¬ j
d K 2 j T t °½ ¾ dT °¿
0
ª d K0 j T W j 2 m °w « K 0 j T W j K1 j T W j ¬ª El r , T ¼º ¦ ® 2 ¬ª E j r ,W ¼º « dT wt j 1°w t ¬ ¯
w2
2
d K 2 j T T º d 2 K 2 j T W 0 » ªD j r ,W º K 0 j W j K1 j W j ¼ dT dT 2 »¼ ¬ T w d 2 K 2 j T t ½° ª E j r , t º K 0 j t W j K1 j t W j dt¾ 0 ³ ¬ ¼ dT °¿ W wt
w3 w t3
wt
2
ªE ¬
ª d K0 j T W
j
r , T º¼ «« 2
w ª ° d K0 j T W E r , T º¼ ® wt ¬ j dT 2 °
ª d K0 j T W « « dT ¬
j
K
W
2K
0j
0j
w ª E r , t º¼ K 0 j t W wt¬ j
j
T W j K 1 j T W j
d 3 K 2 j T t dT 3
K1 j t W j
j
T W j K 1 j T W j
T W j K 0 j T W j
1j
T
j
K
¯
ªD 0j r , W º K 0 j W j K 1 j W j ¬ ¼ ³
j
dT
¬
2
j 1
2
w3 ª E j r , T º¼ K 0 j T W w t3 ¬
m
ª¬ E l r , T º¼ ¦
w
d K2 j T W
j
dT
º» » ¼
d K 2 j T T º » » dT ¼
d 2 K 2 j T T dT 2
d K 2 j T T ½° ¾ dT °¿
d 3 K 2 j T t dT 3
0
#
wk wt
k
ª¬ El r , T º¼
m
¦
j 1
wk ª E j r ,T º¼ K 0 j T W j " w tk ¬
"
ªD 0j r ,W º K 0 j W j K1 j W j ¬ ¼ T
³ W
d k K 2 j T W dT k
d k K 2 j T t w ª dt E j r , t º¼ K 0 j t W j K1 j t W j ¬ wt dT 3
0
(3.42)
86
Materials with Rheological Properties
As mentioned, the equation systems (3.42) are used when knowing the structure’s stresses and strains state at the moments W and T, as well as the functions describing its evolution in the time interval of W – T . In the case of equation systems (3.41), the first of them and the boundary conditions of the (3.35) type allow the determination of the structure’s initial stresses and strains state. The solution of the differential equation systems of order k from (3.41) or, according to the case, from (3.42), to which we attach, as additional conditions that must be satisfied by the solution (the Cauchy problem), the differential equation systems preceding them and the initial and the contour conditions represents an approximation of the solution of the integro-differential equations system (3.38) or, respectively, (3.39). The class of the composite structures with complex composition includes: – all the bar, pillar and composite girder structures, with continuous collaboration, statically indeterminate on the exterior, independent of the interaction with the environment supporting them; – frame structures, whatever materials they are made of, if they are continuously supported and if we consider the interaction with the environment (supporting them); – structures on externally statically indeterminate frames, made of combinations of composite elements with continuous collaboration and composite elements with discrete collaboration (for instance, steel pillars encased in concrete and composite concrete steel beams with discrete collaboration, etc.).
3.2.2. The effect of repeated short-term variable load actions: overview We consider as known the dynamic response of the structures made of materials having rheological properties for each of the short-term, repeated variable actions and loads (to which they are submitted); therefore, we know the variation of the stresses and the strains in all the points of the structures only during their dynamic response. We mention that the dynamic response of the structures can be experimentally determined, using empirical relations or, in simple particular cases, can be calculated, by using dynamics methods. In the following, we will treat the evolution in time of the stresses and strains states generated by the dynamic response of the structures made of materials having rheological properties submitted to variable short-term actions and loads. Thus, it is obvious in what follows, that the variation of the stresses and strains is due only to the rheological properties of the materials, so that the variation speed of the stresses
Composite Resistance Structures
87
and strains is low enough to admit that the inertial forces they generate are practically zero. The structure’s stresses and strains state at moment T, when submitted to a variable action or load for a term 't = T – W (where W is the beginning moment and T the moment when action ceases) has the expression: m
T
¦³ W j 1
>
@
>
@
d E j t ' j t K 0 j t W j 1 K1 j T W j K 2 j T t dt dt
0; (3.43)
By splitting up the integration interval and by taking into account that outside the time interval (W – T):
Q t { 0, ' j t
' j Q t , *
we obtain:
T d E j t ' j t K 0 j t W j 1 K1 j t W j K1 j T t dt ®³ ¦ j 1 ¯ W dt (3.43’) T dE j t ½ ³ K 0 j t W j 1 K1 j t W j K 2 j T t dt ¾ 0 dt ¿ T m
>
@
>
@
>
@
We have to specify that the effect of the variation of the real actions and loads on the structures can be described through linear combinations of the effect on the structure of certain actions and loads imposing a simple variation of the structure (Figure 3.5); this implies the necessity of studying the structure’s behavior only in this last situation.
V
E
W
't
X
t
Figure 3.5. The approximation of a short-term variable load with a short-term constant load
88
Materials with Rheological Properties
In the same way, we have to admit, for the moment without justifying, that the behavior, at the moment T, of the structure submitted to a short-term variable load 'ti = T – Wi, T > T, is sufficiently well described by a short-term constant action or load whose effect is defined by the time interval 'ti and the average value of the considered effect Emed. The time interval of loading and unloading is short compared to the total duration of the load. We note:
I j T
T
³W
>
@
>
@
d E j t ' j t K 0 j t W j 1 K1 j t W j K 2 j T t dt. dt
In equations (3.43) and (3.43’), 'j(t) can be expressed as:
' j t '*j Q t where:
'*j – has the same significance as the one previously used and corresponds to a constant reference (unit) load; Q(t) – factor describing the variation in time of the temporary load acting on the structure. Based on the previously mentioned hypothesis as well as on the linear relationship between the structure’s instantaneous answer and the actions and loads the structure bears, we can write:
d Q t t oW dt
Q; d E t dt
E med
(3.44)
t oW
d d Q t W t T 0; E t W t T 0; dt dt d d Q t t oT Q; E t E med . dt dt t oT
E t W t T
E med
const.
Composite Resistance Structures
89
By considering Gt the duration of the loading and, respectively, of the unloading, )j (T) can be expressed: W Gt
I j T
³W
>
@
>
@
d E j t ' j t K 0 j t W j 1 K1 j t W j K 2 j T t dt dt
T Gt
d >E t ' t @K t W >1 K t W K T W @dt ³ W G dt j
j
0j
j
1j
j
2j
j
t
T
d >E t ' t @K t W >1 K t W K T t @dt ³ dt T G j
j
0j
j
1j
j
2j
t
By taking into account relations (3.44) and the fact that the duration Gt of the loading and unloading respectively is short compared to the duration of the load, and by considering the relations:
W Gt
W
Gt o 0 T Gt T Gt o 0, we obtain:
>E W ' W @K W W >1 K W W K T W @ >E T ' T @K T W >1 K T W K T T @
I j T
j
j
T
³ W
0j
j
dE j t dt
0j
j
1j
j
1j
j
>
2j
j
2j
j
(3.45)
@
K 0 j t W j 1 K1 j t W j K 2 j T t dt
The short-term of the loading and unloading makes it possible to use the relations obtained before, in the following way:
¦ >E W ' W @K W W
0
¦ >E T ' T @K T W
0
m
j
j
0j
j
j 1
m
j
j 1
j
0j
j
(3.46)
90
Materials with Rheological Properties
By substituting expression (3.45) of function )j (T) into (3.43’) and by considering relations (3.46), we obtain the final expression of the equations describing the stresses and strains state under the short-term variable action or load:
¦ >E W ' W @K W W K W W K T W >E T ' T @K T W K T W K T T m
j
j
0j
j
1j
j
2j
j 1
j
T
³ W
j
0j
j
1j
j
(3.47)
2j
dE j t K 0 j t W j 1 K 1 j t W j K 2 j T t dt dt
>
@
0
The form of equation (3.47) allows us to observe that, under the conditions imposed by the hypothesis regarding the accomplishment of the variable short-term action or load and those regarding the response of the structure, the effect at the moment T on the structure is equivalent to that due to the permanent actions or loads, equal in terms of intensity and of contrary direction, corresponding to the effect Emed, applied at the moments T and W of the structure loading and structure unloading respectively. If we know the structure’s stresses and strains state at the moment T, due to a finite number k – 1 of short-term variable loads and/or actions ('ti = T – Wi, Wi < Ti), so that Wi < Ti d T, i = 1, 2, 3, ... k – 1, the stresses and strains state at the moment T > 0 is described by the equation: m
T
j 1
¯W1
° d ¦ ®°³ dt >E t ' t @K t W K t W >K T t K T t @dt T
³ T
j
j
0j
j
>
1j
j
2j
2j
>
@
d ½ E j t ' j t @K 0 j t W j 1 K1 j t W j K 2 j T t dt ¾ dt ¿
(3.48)
0
We note:
Iij T
T
d ³W dt >E t ' t @K t W K t W >K T t K T t @dt , j
i
(i 1, 2, 3, ! , k 1)
j
0j
j
1j
j
2j
2j
Composite Resistance Structures
91
By taking into account the considerations that led to relation (3.45), it results that:
>E W '
W i @K oj W i W j K1 j W i W j >K1 j T W i K 2 j T W i @ >E j T j ' j T i @K 0 j T i W j K1 j T i W j >K 0 j T T i K 2 j T T i @ T dE j t ³ K 0 j t W j K 1 j t W j >K 2 j T t K 2 j T t @dt ,
Iij T
j
i
j
(3.49)
dt
Wi
(i 1,2,3, ! k 1)
If, after the moment T, but in the time interval T – T, the structure is submitted to a number n – k + 1 of short-term variable actions and loads ('ti = T – Wi, i = k, k + 1, ..., n – 1, n), that are also accomplished under the conditions previously imposed, the integro-differential equation describing the structure’s stresses and strains state will be obtained by substituting (3.49) in (3.48) and by considering relation (3.47):
¦ ¦ ^>E W ' W @K W W K W W >K T W K T W @ >E T ' T @K T W K T W >K T T K T T @ k 1
m
i 1
j 1
j
j
T
³
i
j
j
dE ji t dt
Wi
j
i
0j
i
0j
i
i
j
1j
j
1j
i
i
2j
j
2j
j
2j
i
2j
i
>
@
K 0 j t W j K1 j t W j K 2 j T t K 2 j T t dt`
i
i
(3.50)
¦ ^>E W ' W @K W W K W W K T W >E T ' T @K T W K T W K T T n
m
i k
j 1
¦
j
j
T
³
i
dE ji t
Ti
dt
m Wk
¦ ³ j 1 T
i
j
i
0j
0j
i
j
>
i
1j
j
i
1j
j
i
j
2j
2j
j
i
@
K 0 j t W j 1 K1 j t W j K 2 j T t dt`
dE j t dt
i
j
>
@
K 0 j t W j 1 K1 j t W j K 2 j T t dt 0
where T d Wk. Form (3.50) of the integro-differential equation system describing the behavior of the structures made of elements having rheological viscoelastic properties, enables, by particularizations, a very interesting observation.
92
Materials with Rheological Properties
Indeed, if we suppose that the materials from which the structure is made behave like a perfectly viscoelastic body, then: K0 j t
K 0 j (constant)
K1 j t
K1 j (constant)
(3.51)
By considering relations (3.44) and (3.51), the integro-differential equations system (3.50) takes the form:
¦ ¦ ^>E W ' W @K k 1
m
i 1
j 1
j
T
³
dE j t dt
Wi
Wi
i
0j
>
½° K 0 j K 1 j K 2 j T t K 2 j T t dt ¾ °¿
>
j
j
@
j
j 1
dE j t dt
m Wk
¦³ j 1 T
@
K 1 j K 2 j T W i K 2 j T W i K 2 j T T i K 2 j T T i
m
i k
T
j
¦ ^>E W ' W @K
n
¦ ³
i
j
0j
(3.52)
>
@
K 1 j K 2 j T W j K 2 j T T j
>
@
K 0 j 1 K 1 j K 2 j T t dt`
dE j t dt
>
@
K 0 j 1 K 1 j K 2 j T t dt
0
For a finite value, 't = T – W,
K 2 j t
K 2 j t 't tof
1 tof
In reality, there is a (minimum) finite value TL – 'T (equal to or greater than the duration of the phenomenon of stress redistribution in the structure, due to the rheological properties of the materials) so that the following relation is valid:
K 2 j T L | K 2 j T L 'T | 1 Therefore, we can determine a (maximum) value TL < T, so that, for an established T (T < T), which is sufficiently large compared to TL, the following relations exist:
K 2 j T T i
K 2 j T T i
K 2 j T T L
K 2 j T W i 1
K 2 j T W i 1
K 2 j T T L 1
(3.53)
Composite Resistance Structures
93
for any pair of values Wi and Ti that fulfill the condition:
W i T i d T L ,i 1,2,3,...k 1 By considering relation (3.44), it results that: k 1 T i
¦³
dE j t
>
@
K 0 j K1 j K 2 j T t K 2 j T t dt
dt
i 1 Wi
0,
so that we can write: k 1 T
¦³
dE j t dt
i 1 Wi
>
dE j t
k 1 T
¦³
dt
i 1 Tj
@
K 0 j K1 j K 2 j T t K 2 j T t dt
>
(3.54)
@
K 0 j K1 j K 2 j T t K 2 j T t dt
0
By substituting (3.54) in (3.52) and by considering (3.53), the integro-differential equation system (3.52) takes the form:
¦ ¦ ^>E W ' W @K k 1
m
k L 1
j 1
j
i
j
T
K 2 j T T i @ ³
dE j t dt
TL
j
i k
dt
m Wk
¦ ³ j 1 T
>
j
j
i
>
@
K 0 j K 1 j K 2 j T t K 2 j T t dt`
0j
>
@
K 1 j K 2 j T W i K 2 j T T i
j 1
dE j t
Wj
>
K 1 j K 2 j T W i K 2 j T W i K 2 j T T i
m
¦ T
0j
¦ ^>E W ' W @K
n
³
i
(3.52’)
@
K 0 j 1 K 1 j K 2 j T t dt`
dE j t dt
>
@
K 0 j 1 K 1 j K 2 j T t dt
0
These results allow the following observations and conclusions: – in the case of the structures made of materials having properties of perfectly viscoelastic bodies, the stresses and strains state at the moment T, due to short-term variable actions and loads, is influenced only by the (short-term) variable actions and loads occurring in a time interval TL – T whose duration depends on the speed of
94
Materials with Rheological Properties
the stress redistribution phenomenon’s consumption, respectively, on the form of the functions K2j(t). The short-term variable actions and loads prior to the lower end TL of the interval TL – T do not have any influence on the stresses and strains state at the moment T. The functions describing the evolution in time of the structure’s stresses and strains state, due to only one short-term variable action or load, are independent of the application moment of the action or load; – in the case of the real structures, made of materials behaving like imperfect viscoelastic bodies until the age of 3-5 years, only the short-term variable actions and loads stressing the structure during this period influence the stresses and strains state along the entire lifetime of the structure. When the materials reach their maturity (3-5 years) – which makes them behave like perfectly viscoelastic bodies – the effect of the short-term variable actions and loads is considered as corresponding to this situation (see the previous section). Therefore, it is necessary to solve the integro-differential equation system (3.50) only for the short-term variable actions and loads borne by the structure until the maturation of the materials composing it, by following the order of accomplishment of the actions and by using the values by which they are determined. We obtain in this way the stresses and strains state of the structure due to these actions and loads during the entire operation period of the structure. In this case, the superposition of the effects is implicit. For short-term variable actions and loads that stress the structure after the maturation of the materials, we can solve integro-differential equation system (3.50) for each type of action or load. This way, we obtain the functions describing the variation of the stresses and strains state for the entire duration of the structure’s operation and for each type of action or load. The stresses and strains state of the structures up to a certain moment is obtained by totaling up the values which characterize, for the chosen moment, the effect of the permanent actions and loads (long-term) and short-term variable actions and loads supported by the structure until that moment. The values that characterize the effects of the short-term variable actions and loads applied after the maturation of the materials from which the structure is made can be determined by considering the fact that the functions describing the variation in time of their effects is independent of the action or load application moment. In this way, we can obtain the extreme values characterizing the stresses and strains state of the structure. These results serve to protect the structure against the loss of local and/or general stability, exceeding the stresses corresponding to the linear elastic behavior of the materials or, depending on the case, of those over which the nonlinear creep develops, as well as against the development of deformations unacceptable for the normal operation of the structure.
Composite Resistance Structures
95
The variations compared to the average value (the cycle of loading and unloading compared to the average value) of the stresses generated by short-term variable actions and loads in the elements composing the resistance structures are accomplished in short time intervals compared to the duration of the average effort action, Emed, considered in the calculus. Therefore, they have a sufficiently short absolute duration to allow the assertion that, in this time interval, the variation in time of parameters K0j(t) and K1j(t) is zero. Therefore, we can consider that the variation of the stresses compared to the average value Emed, independently of their accomplishment moment, acts on a structure made of materials behaving like perfectly viscoelastic bodies and, subsequently, their stress tends to cancel out in time. This tendency is accentuated by the variation of the stresses in both directions compared to the average value of the stress Emed justifying the hypothesis concerning the substitution in the calculus of the short-term variable actions and loads by constant actions and loads having the same duration. The obtained results show that in order to determine the history of the stresses necessary for calculating fatigue, we must use the first equation (3.46) for each short-term variable action or load. The obtained results enable us to affirm that, for calculating fatigue, the decrease of the rigidity of the concrete component of the composite steel and concrete structures is not justified.
3.3. Mathematical model for calculating the behavior of composite resistance structures. The formulation considering stress relaxation 3.3.1. The effect of long-term actions and loads: overview We consider the interior and/or exterior statically indeterminate composite resistance structure, with elements made of m materials having different rheological properties, referred to in section 3.2.1. However, in this case, the basic system is obtained by blocking displacements of the structure’s nodes. In the following, we consider as structure’s nodes the zones where two or several component elements are concurrent, as well as the connection zones, continuous or discrete, between the areas made of materials having different rheological properties. The reaction Rij(T) that occurs between blocking element i due to the stresses generated in the structure’s elements having rheological properties j by the external
96
Materials with Rheological Properties
actions and loads and by the displacements of the structure’s nodes (rotations and translations), determined on the basic system, will have the following expression (see equation (3.2’)):
Rij T T
³
>
@
Rij W K 0 j W j 1 K1 j W j K 2 j T W j
d Rij t dt
W
>
(3.55)
@
K 0 j t W j 1 K1 j t W j K 2 j T t d t
Considering the static equilibrium conditions, it results that:
¦ R W K W >1 K W K T W @ m
ij
0j
j 1
m T
¦³ j 1W
j
1j
d Rij t dt
j
2j
j
(3.56)
>
@
K 0 j t W j 1 K1 j t W j K 2 j T t d t 0, (i 1,2,3, !, n)
By knowing the structure’s stresses and strains state at the moment T (W < T
¦ ^R T R W K W K W >K T W K T W @` m
ij
j 1
m T
¦³ j 1W
m W
¦³ j 1T
ij
0j
j
1j
j
2j
j
2j
j
dRij t K 0 j t W j K1 j t W j K 2 j T t K 2 j T t d t dt
>
@
dRij t K 0 j t W j 1 K1 j t W j K 2 j T t d t dt
>
@
(3.56’)
0,
(i 1,2,3,......, n) In expressions (3.55), (3.56) and (3.56’) we have denoted: Rij(W) – the fictitious reaction, in the blocking direction i, due to the stresses generated by the external actions and loads applied at the moment W and to the displacements of the structure’s nodes corresponding to these in the structure’s basic system areas having rheological properties j. This value is calculated by considering the reference value Eoj of the material’s modulus of deformation; Rij(T) – the reaction, in the blocking direction i, at the moment T (W < T < T), due to the stresses generated by the external actions and loads applied at the moment W and to the displacements of the structure’s nodes corresponding to these in the structure’s basic system areas having rheological properties j;
Composite Resistance Structures
97
dRij (t )
dt – the fictitious reaction increase, at the moment t (W < t < T), in the dt blocking direction i, due to the stresses generated by the external actions and loads applied at the moment W and by the displacements of the structure’s nodes corresponding to these, in the structure’s basic system areas having the rheological properties j. The value is calculated by considering the reference value Eoj of the material’s modulus of deformation. Considering that:
Rij t Uij t ij
(3.57)
where: Uij(t) – the fictitious reaction in the blocking direction i at the moment t ( W < t < T), due to the stresses generated by displacements of the structure’s nodes, corresponding to the actions and loads applied at the moment W, in the structure’s basic system areas having rheological properties j. This value is calculated by considering the reference value Eoj of the material’s modulus of deformation; ij – the fictitious reaction in the blocking direction i, due to the stresses generated by the external actions and loads (constant in time and applied at the moment W) in the structure’s basic system areas having rheological properties j, calculated on the basic system by considering the reference value Eoj of the material’s modulus of deformation. It results that:
dRij t
dUij t
dt
dt
By making T = W, and, respectively, T = T, in equations (3.56) and, (3.56’) respectively, we obtain:
¦ R W K W m
ij
0j
j
0, (i 1,2,3,!, n)
(3.58)
j 1
and, respectively: m
¦ R T ij
j 1
0, (i 1,2,3,!, n)
(3.59)
98
Materials with Rheological Properties
By substituting relation (3.58) in (3.56’), it results that:
¦ R W K W K W K T W m
ij
0j
j 1
m T
¦³
j
1j
d U ij t
2j
>
@
K 0 j t W j 1 K1 j t W j K 2 j T t d t
dt
j 1W
j
(3.60)
0,
(i 1,2,3,......., n) and, respectively:
¦ R W K W K W >K T W K T W @ m
0j
ij
j 1
m T
¦³ j 1W
m T
¦³
d U ij t dt d U ij t
j 1T
1j
j
dt
j
2j
2j
>
@
K 0 j t W j K1 j t W j K 2 j T t K 2 j T t d t (3.60’)
>
@
K 0 j t W j 1 K1 j t W j K 2 j T t d t
0,
(i 1,2,3,........., n) The solutions of integro-differential equation systems (3.60) and (3.60’) are functions of time and describe the variation of the values of the structure’s node displacements (rotations and translations); using these functions, we can estimate the evolution in time of the stresses and strains state in any point of the structure. Although integro-differential equation systems (3.60) and (3.60’) have a different physical significance, they have the same mathematical expression as integro-differential equation systems (3.8’) and (3.8”). Therefore, the statements concerning the possibilities and the integration procedures, made in section 3.2.1, for systems (3.8’) and (3.8”), are also valid for systems (3.60) and (3.60’) and they will not be discussed further. Using successive differentiations of integro-differential equation system (3.60), we obtain:
d K 2 j T W d U ij T K 0 j T W j Rij W K 0 j W j K1 j W j dT dT 1¯
m
¦® j
T
³ W
d U ij t dt
d K 2 j T t ½ K 0 j t W j K1 j t W j d t ¾ 0, dT ¿
(i 1,2,3,!, n)
(3.61)
Composite Resistance Structures
d U ij T ª d K 0 j T W j K 0 j T W j K1 j T W j u « dT ¬ dT dT 1 d K 2 j T T º d 2 K 2 j T W R K K u W W W ij j j 0j 1j » dT dT 2 ¼ d 2 U ij T
m
¦ j
99
2
T
³
K 0 j T W j
d U ij t
W
dt
K 0 j t W j K1 j t W j
d 3 U ij T
m
¦
dT 3
j 1
u
K 0 j T W j
d 2 K 2 j T t dT 2
dt
0,(i 1,2,3,!, n)
d 2 U ij T ª d K 0 j T W j K 0 j T W j K 1 j T W ju «2 dT dT 2 ¬
d K 2 j T T º d U ij T ° d 2 K 0 j T W j 2 K 0 j T W j K1 j T W j u ® » dT d T °¯ dT 2 ¼
d K 2 j T T ª d K 0 j T W j d K 1 j T W j º « K 1 j T W j K 0 j T W j »u 2 dT dT dT ¬ ¼ 3 d K 2 j T T ½ d K 2 j T W u ¾ Rij W K 0 j W j K 1 j W j dT dT 3 ¿
u
T
³ W
d U ij t dt
K 0 j t W j K 1 j t W j
d 3 K 2 j T t dT 3
dt
0,
(i 1,2,3,....., n) # m
¦
d k U ij T dT k
j 1
K 0 j T W j "
" Rij W K 0 j W j K 1 j W j T
³ W
d U ij t dt
d k K 2 j T W dT k
d k K 2 j T t K 0 j t W j K 1 j t W j dt dT k
0,
(i 1,2,3, ! , n) By making T = W in integro-differential equation systems (3.56) and (3.61) and by considering (3.57), we obtain the following linear algebraic equation systems, having as unknown factors the values, in W, of the functions describing the evolution
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Materials with Rheological Properties
in time of the displacements (rotations and translations) of the structure’s nodes and of their derivatives of order 1, 2, 3, ..., k, etc.:
¦ U W K W ¦ W K W m
m
ij
j 1 m
¦
d U ij W dT
j 1 m
¦
0j
ij
0j
0, (i 1,2,3,!, n)
j
j 1
d K 2 j W W
K 0 j W j ¦ Rij W K 0 j W j m
dT
j 1
d 2 U ij W dT 2
j 1
j
¦
d 3 U ij W dT 3
j 1
0, (i 1,2,3,!, n)
m d U W d K d K 2 j W W º ª ° ij 0 j W j K 0 j W j ¦ ® K 0 j W j K 1 j W j » « dT j 1° ¼ ¯ dT ¬ dT
d 2 K 2 j W W ½° Rij W K 0 j W j K 1 j W j ¾ dT 2 °¿ m
(3.62)
K 0 j W j ¦ m
j 1
0, (i 1,2,3, ! , n)
d 2 U ij W ª d K 0 j W j d K 2 j W W º K 0 j W j K 1 j W j «2 » 2 dT dT dT ¼ ¬«
d 2 K 2 j W W d U ij W ° d K 0 j W j 2 W W K K ® 2 1 j j j j d T °¯ d T 2 dT 2 d K 1 j W j º d K 2 j W W ½ ª d K 0 j W j « K 1 j W j K 0 j W j ¾ » dT ¼ dT ¬ dT ¿ 2
d 3 K 2 j W W Rij W K 0 j W j K 1 j W j dT 3 # m
¦ j 1
d k U ij W dT k
K 0 j W j ¦ m
j 1
d k U ij W d T k 1
0, (i
1,2,3, ! , n)
^"`
d k K 2 j W W Rij W K 0 j W j K 1 j W j dT k
0, (i 1,2,3,! , n)
Similarly, using the successive differentiations of integro-differential equation system (3.60’), for t = T, we obtain the following linear algebraic equation systems, having as unknown factors the values in T of the functions describing the evolution in time of the displacement of the structure’s nodes and of their derivatives of order 1, 2, 3, ...k, etc.
Composite Resistance Structures
101
m d K 2 j T W ª K 0 j T W j ¦ « Rij W K 0 j W j K 1 j W j dT j 1 dT j 1¬ (3.63) T d U ij t d K 2 j T t º K 0 j t W j K 1 j t W j ³ d t » 0, (i 1,2,3, ! , n) dt dT ¼ W m d 2 U T m d U ij T ª d K 0 j T W j ij K 0 j T W j K 1 j T W j u K 0 j T W j ¦ ® ¦ « 2 dT ¬ dT dT j 1 j 1¯ m
¦
d U ij
d K 2 j T T º d 2 K 2 j T W R K K W W W » 0j 1j ij j j dT dT 2 ¼ T d U ij t d 2 K 2 j T t ½° ³ K 0 j t W j K 1 j t W j d t ¾ 0, (i 1,2,3,! , n) dt dT 2 °¿ W 2 m d 3 U T m d U ij T ª d K 0 j T W j ij K0 j T W j ¦ K 0 j T W j K1 j T W j u ¦ «2 3 dT dT dT 2 ¬ j 1 j 1 u
u u u
d K 2 j T T ½ d 3 K 2 j T W W W W R K K ¾ ij j j 0j 1j dT dT 3 ¿
W
m
d 2 K 2 j T T ª d K 0 j T W j d K 2 j T W j º K1 j T W j K 0 j T W j « »u 2 dT dT dT ¬ ¼
T
¦
d K 2 j T T º d U ij T ° d 2 K 0 j T W j 2K 0 j T W j K1 j T W j u ® » dT d T °¯ dT 2 ¼
³ #
d U ij t dt
d k U ij T dT k
j 1
d 3 K 2 j T t K 0 j t W j K1 j t W j dt dT 3
K 0 j T W j ¦ m
j 1
0, (i 1,2,3,!, n)
d k U ij T d T k 1
^"`
d k K 2 j T W Rij W K 0 j W j K1 j W j dT k T
³ W
d U ij t dt
K 0 j t W j K1 j t W j
d k K 2 j T t dT k
dt
0, (i 1,2,3,!, n)
The values in W and, respectively, in T of the functions and of their derivatives of order 1, 2, 3, ..., k serve, as shown in section 3.2.1, to produce the Taylor series that approximates the solutions of integro-differential equation systems (3.60) and, respectively, (3.60’), around their values in W and, respectively, in T.
102
Materials with Rheological Properties
In the following, we are going to show the specific developments of the equation terms describing the behavior in time of the basic types of composite structures, as well as for the general case of composite structures with complex composition. The classification criteria of composite structures, their properties and the properties of the materials and elements from which they are made, as well as the hypothesis concerning their behavior, are listed in sections 3.2.2, 3.2.3 and 3.24, and will not be discussed further. 3.3.1.1. Composite structures with discrete collaboration In the case of the formulation, according to the stress relaxation of the mathematical model of the behavior of composite resistance structures with discrete collaboration described in section 3.2.2, the basic system is obtained by blocking the displacements (rotations, translations) of the structures’ nodes. The reaction in blocking element i due to the stresses inside the elements of the structure made of materials having rheological properties j has the expression: Rij W
m
¦r
Z k W ij
m
Z k t , (i 1,2,3,!, n), ( j 1,2,3,!, m)
ij k
(3.64)
k 1
U ij t
¦r
ij k
k 1
In relations (3.64), we have denoted: rijk – the unit reaction in blocking element i, due to the stresses generated in the elements made of materials having rheological properties j by a unit displacement in blocking direction k; ij – the reaction in blocking element i, due to the stresses generated in the elements made of materials having rheological properties j by the external actions and loads applied to the structure’s basic system; =k(t) – the displacement (rotation or translation) of the structure in blocking direction k. By substituting relation (3.64) in (3.60) and (3.60’), the integro-differential equation systems describing the behavior of the composite structures with discrete collaboration will have the following form in the matrix representation:
¦ >r Z W @K W K W K T W m
j
j
0j
j
1j
2j
j
(3.65)
j 1
m T
¦ ³ rj j 1 W
>
@
d >Z W @K 0 j t W j 1 K 1 j t W j K 2 j T t d t dt
0
Composite Resistance Structures
103
and, respectively:
¦ >r Z W @K W K W >K T W K T W @4 m
j
j
0j
j
1j
j
2j
2j
j 1
m T
¦ ³ rj j 1W
m T
¦ ³ rj j 1T
>
@
d >Z t @K 0 j t W j K1 j t W j K 2 j T t K 2 j T t d t dt
>
@
d >Z t @K 0 j t W j 1 K1 j t W j K 2 j T t d t dt
0
Determining values =(W) of the functions describing the node displacement 2 evolution of the structure in time and of their derivatives d=(W ) , d =(W ) , etc. for dt dt 2 the moment W, necessary to build the Taylor series that approximates, around the value t = W of the functions =(t) argument, can be carried out by solving successively, the algebraic linear equation systems, obtained by substituting, relation (3.64) in (3.62), which takes the following form in the matrix representation: m
r W Z W ¦ j K0 j W j j 1
r W
r W
(3.66)
0
m d K2 j W W d Z W ¦ rj Z W j K0 j W j K1 j W j dT dT j 1
>
>
@
@
m ª d K 0 j W j d 2 K 2 j W W º d2 ° d > @ > @ Z r Z K K W W W W « » ® ¦ 0j 1j j j j dt 2 dT 2 j 1° ¬« d T ¼» ¯ dt
>
@
d 2 K 2 j W W ½° ¾ 0 dT 2 °¿ d K 2 j W W º ª d K 0 j W j d2 r j 2 >Z W @«2 K 0 j W j K 1 j W j » d T dT dt ¼ ¬
r j Z W j K 0 j W j K 1 j W j r W
m d3 >Z W @ ¦ 3 dT j 1
rj
d 2 K W d 2 K W W d >Z W @°® 0 j2 j 2 K 0 j W j K1 j W j 2 j 2 dT dT °¯ d T
d K 1 j W j º d K 2 j W W ½ ª d K 0 j W j K 1 j W j K 0 j W j « ¾ » dT ¼ dT ¿ ¬ dT d 3 K 2 j W W r j Z W j K 0 j W j K 1 j W j dT 3
>
#
0
@
0
104
Materials with Rheological Properties
r W
k 1 m dk >Z >W @@ ¦ d k 1 >Z W @^"` k dT j 1 dT
d k K 2 j W W r j Z W j K 0 j W j K 1 j W j dT k
>
@
0
Algebraic linear equation systems (3.67), obtained by the substitution of (3.64) 2 3 in (3.63), serve to determine the values dZ (T ) , d Z (T ) , d Z (T ) , etc. of the dt dt 2 dt 3 functions =(t) derivatives in T, necessary, together with the known values =(T), to build the Taylor series that approximate the solution of the second integrodifferential equation system (3.65) around the value T of the argument. We also use the matrix representation to write linear algebraic equation system (3.67):
r T
m d K T W d >Z T @ ¦ ® r j W j K 0 j W j K1 j ( W j 2 j dT dT j 1¯
>
T
³ rj W
r T
@
d K T t ½ d >Z t @K 0 j t W j K1 j t W j 2 j dt¾ dt dT ¿
0
m ª d K T W j d2 K 0 j T W j K 1 j T W j u >Z T @ ¦ °®r j d >Z T @« 0 j 2 dT dT j 1° ¬ ¯ dt
d 2 K 2 j T W d K 2 j T T º W W W r Z K K j j j j » 0j 1j dT dT 2 ¼ T d 2 K 2 j T t ½° d ³ r j >Z t @K 0 j t W j K 1 j t W j dt¾ 0 dt dT 2 °¿ W u
(3.67)
>
@
Composite Resistance Structures
r T u
105
2 m ª d K T W j d3 >Z T @ ¦ r j d 2 >Z T @«2 0 j K 0 j T W j K 1 j T W j u 3 dT dT dt j 1 ¬
° d 2 K 0 j T W j d K 2 j T T º d T r Z > @ 2 K 0 j T W j u ® » j dT dt dT 2 °¯ ¼
u K 1 j T W j
d 2 K 2 j T T ª d K 0 j T W j « K 1 j T W j dT dT 2 ¬
d K 1 j T W j º d K 2 j T T K 0 j T W j » dT dT ¼
d 3 K 2 j T W r j Z W j K 0 j W j K 1 j W j dT 3
>
T
³ rj W
@
d 3 K T t d >Z t @K 0 j t W j K1 j t W j 2 j 3 d t dt dT
0
#
r T
k 1 m dk >Z T @ ¦ r j d k 1 >Z T @^"` k dT dT j 1
d k K 2 j T W r j Z W j K 0 j W j K1 j W j dT k
>
T
³ rj W
@
d k K T t d >Z t @K 0 j t W j K1 j t W j 2 j dt dt d Tk
0
In equation systems (3.65), (3.66) and (3.67), we have denoted: rj, j – the (square) matrix of the rijk coefficients and, respectively, the free terms Rij vector; =(t), =(W), =(T) – the vectors of the values of the functions and their derivatives d >Z t @, d >Z W @ that describe the evolution in time of displacements (rotations, dt dt translations) of the structure’s nodes and of their derivatives, at the moments t, W, and T respectively.
106
Materials with Rheological Properties
In equation systems (3.56) and (3.67) we took into account the existence of the following relations:
r W
¦ r K W m
0j
j
j
(3.68)
k 1
r T
¦ r K T W m
0j
j
j
j 1
3.3.1.2. Composite structures with continuous collaboration The composition and the properties of the composite structures with continuous collaboration were described in section 3.2.3. Below we will present the developments of the terms of the general equations obtained within the framework of section 3.3.1, which are specific to these structures. Reactions Uij(r,t) and ij (r ,W ) in the blocking element from section r of the displacement in direction i, due to the stresses generated in the elements having rheological properties j of the basic system’s structure by the unknown displacements =k(s,t) – for Uij(r,t) – and, respectively, by the unknown displacements =k(s,t) together with the external actions and loads applied to the structure at the moment W – for ij (r ,W ) – can be written as follows:
U ij r , t
l
n
¦ ³ [ r , s Z s, t d s ij k
k
(3.69)
k 1 0
ij r ,W
n
l
¦ ³ [ r , s Z s,W d s R r ij k
k
ij
k 1 0
In relations (3.69), we have denoted: =k(s,t) – the function expressing the value at moment t of the composite structure displacement in the current section s in direction k, due to the external actions and loads applied to the structure at moment W; [i,j,k(r,s) – the function expressing the value of the reaction in the current section r in direction i due to the stresses generated in the structure’s elements having rheological properties j by a unit displacement applied to the basic system of the structure, in the current section s, in direction k; Rij(r) – the function expressing the value of the reaction in the current section r, in direction i, due to the stresses generated by the external actions and loads in the elements of the structure having rheological properties j, determined on the basic system of the structure.
Composite Resistance Structures
107
The development of the mathematical model for the behavior of these structures according to the stress relaxation is made based on a diagram. The diagram is obtained by reducing the homogenous sections to their axis (from the point of view of the rheological properties of the materials from which the structures are made) and by replacing the continuous connections between them by equivalent discrete connections, distributed at infinitely small distances (the distance between two adjacent connection elements is ds). We attribute to the elements of the diagram the physical and mechanical characteristics of the sections they replace as well as the respective rheological properties. The basic system of the structure is obtained by blocking the displacements of the nodes of the described diagram. Following the analysis, on the basic system of the structures with continuous collaboration – obtained as shown above – of the interactions between possible displacements of whatever node and the reactions in the blocking elements that prevent the displacements of the other nodes of the basic system, we can make the following observation, which is similar to that of section 3.2.3. The stresses and strains state in whatever unspecified section of the structure is in reciprocal interaction only with the adjacent neighboring sections and only due to the variation, according to the section coordinates, of the nodes’ relative displacement (rotations and translations). The coupling of the equations describing the stresses and strains state in one unspecified section with the stresses and strains state of adjacent neighboring sections is thus carried out only by the sizes expressing the variation, in the respective section, of the unknown deformations generating stresses in the connection elements (connectors). NOTE.– To the variation of the displacement in direction k, w >= k ( r , t ) @ , in the wr current section r of the structure, corresponds the occurrence in the connection elements of the elastic forces and, implicitly, of a variation of the reaction on the blocking from direction i, in the current section r of the structure, whose value is:
rlik r cik r
w >Z k r , t @ wr
The variation of the reaction over the infinitesimal length dr of the structure’s element will have the value:
w >rlik r @d r wr
cik r
w2 >Z k r , t @d r wr 2
108
Materials with Rheological Properties
The evaluation of the reactions, in the basic system nodes, is made by taking into account the stresses existing prior to the actions and loads applied at moment W. By taking into account the previous observation and by using the Dirac delta function G(r,s), we can write:
U ij r , t
m l
¦ ³ r r d sG r , s Z s, t d s ij k
k
k 1 0
U il r ,W
w ¦ ³ c r G r , s n
l
ik
k 1 0
2
Z k s, t ds ds2
Therefore, by considering the properties of the Dirac delta function, relations (3.69) take the following form:
U ij r , t
n
¦ r r Z r , t ij k
k 1
U il r , t
n
¦ cik r
k
w 2 Z k r , t
k 1
ij r ,W
(3.69’)
dr2
n
¦ r r Z r ,W R r ij k
k
ij
k 1
In the above equations, we have denoted: rijk(r) – the function expressing the value of the reaction in the current section r, in direction i, due to the stresses generated in the elements having the rheological properties j, by a unit displacement applied in section r of the structure in direction k, determined on the structure’s basic system; Rij (r) – the function expressing the value of the reaction in the current section r in direction i, due to the stresses generated in the elements having rheological properties j, by the external actions and loads applied to the structure at moment W, determined on the structure’s basic system; cik(r) – the function expressing the value of the reaction in the current section r, in direction i, due to the stresses generated in the connection elements (connectors) by a unit displacement in the current section r of the structure, in direction k, determined on the structure’s basic system. By substituting the relations from (3.69’) in (3.60) and (3.60’) and by taking into account the properties of the connection elements to develop only elastic deformations (K0(W) = 1, K1(t) = 0, K2(t) = 0), it results, in the matrix representation:
Composite Resistance Structures
109
¦ >r r Z r,W R r @K W K W K T W m
j
j
0j
j
1j
2j
j
j 1
m T
¦ ³ r j r j 1W
T
c r ³ W
w >Z r , t @K 0 j t W j >1 K1 j t W j K 2 j T t @d t wt
w3 >Z r , t @d t wr 2w t
(3.70)
0
and, respectively:
¦ >r r Z r,W R r @K W K W >K T W K T W @ m
j
j
0j
j
1j
j
2j
2j
j 1
m T
¦ ³ r j r j 1W
>
w >Z r , t @K 0 j t W j K1 j t W j u wt m T
@
u K 2 j T t K 2 j T t d t ¦ ³ r j r
>
j 1T
@
(3.70’)
w >Z r , t @K 0 j t W j u wt
w3 >Z r , t @d t 2 T wr w t
T
u 1 K1 j t W j K 2 j T t d t c r ³
0
The integro-differential equation systems (3.70) and (3.70’) describe the evolution in time of the stresses and strains state of the composite structures with continuous collaboration. The solutions of these systems are approximated, by integrating the differential equation systems obtained by successive differentiation with respect to time, of one of the two integro-differential equation systems (3.70) or (3.70’), for T = W, and, respectively, for T = T, or – which is equivalent – by substituting relations (3.69’) in (3.62) and, respectively, in (3.63). In this way, we obtain, in the matrix representation, the following differential equations with partial derivative systems: m w2 Z r ,W @ r r ,W Z r ,W ¦ R j r K 0 j W j 0 2 > wr j 1 w3 w >Z r ,W @ c r 2 >Z r ,W @ r r ,W wT wr w T m d K 2 j W W ¦ >r j r Z r ,W R j r @K 0 j W j K1 j W j 0
c r
j 1
dT
(3.71)
110
Materials with Rheological Properties
c r
2 m w4 >Z r ,W @ r r ,W w 2 ¦ ª«rj r w Z r ,W º» u 2 2 wr w T wT wT j 1¬ ¼
ª d K 0 j W j d 2 K 2 j W W º u« K 0 j W j K1 j W j » dT 2 «¬ d T »¼ 2 m d K 2 j W W ¦ r j r Z r ,W R j r K 0 j W j K1 j W j 0 dT 2 j 1
>
c r
@
w5 w3 W W > @ >Z r ,W , , Z r r r wT3 wr 2 w T 3
m
¦ r j r j 1 m
¦ r j r j 1
ª d K 0 j W w2 Z r ,W @«2 2 dT wT «¬
K 0 j W j K1 j W j
d K 2 j W W º » dT ¼
d 2 K W d 2 K W W w >Z r ,W @°® 0 j2 j K 0 j W j K1 j W j 2 j 2 wT dT °¯ d T
d K 1 j W j º d K 2 j W W ½ ª d K 0 j W j « K 1 j W K 0 j W j ¾ » dT ¼ dT ¿ ¬ dT 3 m d K 2 j W W ¦ r j r Z r ,W R j r K 0 j W j K 1 j W j 0 dT 3 j 1
>
@
# c r
w k 2 wr 2w T k
m
¦ j 1
ª º wk Z r r r Z r ,W » " W W , , « k wT ¬ ¼ d k K 2 j W W r j r Z r ,W R j r K 0 j W j K 1 j W j dT k
>
@
0
and, respectively:
c r
w3 >Z r ,W @ r r ,T w Z r ,T 2 wt wr w T
m d K 2 j T W ¦ r j r Z r ,T R j r K 0 j W j K1 j W j dT j 1
>
m T
¦ ³ r j r j 1W
@
d K T t w >Z r , t @K 0 j t W j K1 j t W j 2 j dt wT dT
0
Composite Resistance Structures
c r
111
m w4 w2 w > @ > @ >Z r ,T @u , , , T T T Z r r r Z r r j r ¦ 2 2 2 wT wr w T wT j 1
d K 2 j T T º ª d K 0 j T W j u« K 0 j T W j K1 j T W j » dT dT ¬ ¼ 2 m d K 2 j T W ¦ r j r Z r ,W R j r K 0 j W j K1 j W j dT 2 j 1
>
@
m T
¦ ³ r j r j 1W
c r
d 2 K T t w >Z r , t @K 0 j t W j K1 j t W j 2 j 2 d t 0 wT dT
m w5 w3 w2 > @ > @ >Z r,T @u T T T Z r r r Z r r r , , , ¦ j wr 2w T 3 wT3 wT 2 j 1
d K 2 j T T º ª d K 0 j T W j u «2 K 0 j T W j K1 j T W j » dT dT ¬ ¼ m
¦ r j r j 1
d 2 K T W d 2 K T T w >Z r,T @°® 0 j 2 j 2K 0 j T W j K1 j T W j 2 j (3.71’) wT dT dT °¯
d K1 j T W j º d K 2 j T T ½ ª d K 0 j T W j K1 j T W j K 0 j T W j « ¾ » dT dT dT ¬ ¼ ¿ 3 m d K 2 j T W ¦ r j r Z r ,W R j r K 0 j W j K1 j W j dT 3 j 1W
>
m T
¦ ³ r j r j 1W
@
d 3 K T t w >Z r, t @K 0 j t W j K1 j t W j 2 j dt 0 wT dT 3
#
c r
k w k 2 >Z r ,T @ r r ,T w k >Z r ,T @ " 2 k wr w T wT
m d k K 2 j T W ¦ r j r Z r ,W R j r K 0 j W j K1 j W j dT k j 1
>
m T
@
¦ ³ r j r j 1W
d k K T t w >Z r , t @K 0 j t W j K1 j t W j 2 j k d t wT dT
In equations (3.71) and (3.71’), we have denoted:
r r ,W
¦ r r K W m
j
0j
j
j 1
r r ,T
¦ r r K T W m
j
j 1
0j
j
0
112
Materials with Rheological Properties
For the differential equations with partial derivatives of order k systems, from (3.71) and (3.71’), the differential equation systems preceding them and the boundary conditions are additional conditions that must be fulfilled by the solutions of the respective differential equations; they constitute the Cauchy problem attached to them. Integro-differential equation systems (3.70) and (3.70’) and also differential equation systems with partial derivatives deriving from them (3.71) and (3.71’) were established by considering the hypothesis that the connection elements (connectors) that ensure the collaboration between the section’s areas made of materials having different rheological properties allow relative displacements between them, proportional to the stresses which they transmit, thus satisfying the condition of rigidity: 1 / cij(r) z 0 If the collaboration between the various parts of the structure’s sections is carried out so that no relative displacement between them is allowed along the separation surface (which is equivalent to setting up an infinite rigidity to the connection elements, 1/cij(r) = 0) then it results, as shown in section 2.3.2, that the stresses and strains state in an unspecified section r evolves independently of the stresses and strains state in the other sections of the structure. Therefore, the equations describing the evolution in time of the stresses and strains state in an unspecified section for this type of structure do not contain the coupling term with the stresses and strains state in the adjacent sections like equations (3.70), (3.70’), (3.71) and (3.71’). The integro-differential equations that describe the evolution in time of the stresses and strains state, in an unspecified section r of this type of structure, will have the following form in matrix representation:
¦ >r r Z r ,W R r @K W K W K T W m
j
j
0j
j
1j
j
2j
j 1
m
¦ r j r j 1
w >Z r , t K 0 j t W j @>1 K1 j t W K 2 j T t @dt wT
0
¦ >r r Z r ,W R r @K W K W >K T W K T W @ m
j
j
0j
j
1j
j
2j
2j
j 1
m T
¦ ³ r j r j 1W
>
w >Z r , t @K 0 j t W j K1 j t W j u wT
@
m T
u K 2 j T t K 2 j T t d t ¦ ³ r j r j 1T
w >Z r , t @K 0 j t W j 0 wT
(3.72)
Composite Resistance Structures
113
For an unspecified section r of the structure, the solutions of the first or, respectively, of the second integro-differential equation system (3.72), around the values t = W or, according to the case, t = T, can be approximated by using the Taylor series expansion based on the values at moment W, or, respectively, T, of the unknown functions and their derivatives. These values are obtained by solving successively the respective algebraic linear equation systems (3.73), obtained by making t = W or t = T in the integro-differential equation systems, obtained by the successive differentiation of one of the systems in (3.72), or, equivalently, by making the substitution:
U j r , t r j r Z r , t
j r ,W r r Z r , t R j r in equation systems (3.62):
r r ,W Z r ,W ¦ R j r K 0 j W j 0 m
(3.73)
j 1
m d K W W w >Z r ,W @ ¦ >r j r Z r ,W R j r @K 0 j W j K1 j W j 2 j 0 dT wT j 1 m ª d K W w2 w r r ,W 2 >Z r ,W @ ¦ r j r >Z r ,W @« 0 j j K 0 j W j K1 j W j u wT wT j 1 ¬ dT
r r ,W
d 2 K 2 j W W º m u » ¦ r j r Z r ,W R r K 0 j W j K1 j W j u dT 2 »¼ j 1 2 d K 2 j W W u 0 dT 2 m d K W W º ª d K W w3 w2 r r ,W Z r r r W , > @ >Z r ,W @«2 0 j j K 0 j W j K 1 j W j 2 j ¦ » j 3 2 dT dT wT wT j 1 ¬ ¼
>
m
¦ r j r j 1
@
d 2 K W d 2 K W W w >Z r ,W @°® 0 j2 j 2 K 0 j W j K 1 j W j 2 j 2 wT dT °¯ d T
d K 2 j W j º d K 2 j W W ½ ª d K 0 j W j K 1 j W j K 0 j W j « » ¾ d T ¼» dT ¿ ¬« d T
m d 3 K 2 j W W ¦ r j r Z r , W R r K 0 j W j K 1 j W j 0: dT 3 j 1
>
#
@
114
Materials with Rheological Properties
r r ,W
wk wTk
ª¬ Z r ,W º¼
m
¦ ! " j 1
m
¦ ª¬ r j r Z r ,W R j r º¼ K 0 j W j K1 j W j j 1
d k K 2 j W W
0
dT k
and, respectively, in equation systems (3.63): r r ,W
m dK 2 j T W w ª¬ Z r ,T º¼ ¦ ª¬ r j r Z r ,W R j r º¼ K 0 j W j K1 j W j wT dT j 1
m T
w ª Z r , t º¼ K0 j t W j K1 j t W ¦ ³ rj r w T¬ j 1W r r ,T
dT
0
dt
dT 2
m T
¦ ³ rj r
j 1W
d 2 K 2 j T t w ª¬ Z r , t º¼ K 0 j t W j K1 j t W j dt wT dT 2
m
ª¬ Z r ,T º¼ ¦ r j r
w2
d 2 K 2 j W W dT 2
m
j 1
m T
¦ ³ rj r j 1W
º» dK2 j W W ½°
ª dK 0 j W j dK1 j W j K1 j W j K 0 j W j « « dT dT ¬
¦ ª¬ r j r Z r ,W R j r º¼ K 0 j W j K1 j W j
#
0
ª dK 0 j W j ª¬ Z r , t º¼ « 2 K 0 j W j K1 j W j u « dT w T3 w T2 j 1 ¬ 2 dK 2 j W W º m w ° d K0 j W j ª¬ Z r , t º¼ ® u 2 K 0 j W j K1 j W j u » ¦ rj r dT wT dT 2 °¯ ¼» j 1
w3
u
(3.74)
dK 2 j T T º m » ¦ ª¬ r j r Z r ,W R j r º¼ K 0 j W j K1 j W j u dT »¼ j 1
d 2 K 2 j T W
r r ,T
dK 2 j T t
ª dK 0 j T W j m w ª¬ Z r ,T º¼ ¦ r j r ª¬ Z r , T º¼ « K0 j T W j u « wT dT wT2 j 1 ¬
w2
u K1 j T W j u
» ¼
d 3 K 2 j W W dT 3
dT
d 3 K 2 j T t w ª¬ Z r , t º¼ K 0 j t W j K1 j t W j dt wT dT 3
0
¾ °¿
Composite Resistance Structures
r r ,T
wk
m
ª¬ Z r ,T º¼ ¦ "
wTk
j 1
m
¦ ª¬ r j r Z r ,W R j r º¼ K 0 j W j K1 j W j j 1
m T
¦ ³ rj r j 1W
115
d k K 2 j W W dT k
d k K 2 j T t w ª¬ Z r , t ¼º K 0 j t W j K1 j t W j dt wT dT k
0
The structures falling in this class from the behavior and, respectively, the hypothesis used to elaborate the mathematical model for calculation of the evolution of the stresses and strains state from the point of view of time have already been listed at the end of section 3.2.3. 3.3.1.3. Composite structures with complex composition Based on the definition given in section 3.2.4 for the composite structures with complex composition and by considering relations (3.52) and (3.69’), the reactions Ui, Ul and j can be written in the following form:
U j r , t
ªl º ª rj º q j r [ r , t dr » « ³ «q r » Z t « 0 » ¬ j ¼ ¬« p j r [ r , t ¼»
(3.75)
I º » w2 «c r 2 [ r , t » wT ¼ ¬ ª
U l r , t «
j r ,W
º ªl ª rj º q j r [ r ,W dr » ª R j º « ³ «q r » Z W « 0 » « Pj r » ¼ ¬ j ¼ «¬ p j r [ r ,W »¼ ¬
In relations (3.75), we have denoted: =(t) – the vector of displacements =i(t) of the structure’s nodes (discrete collaboration); [(r,t) – the vector of functions [i(r,t) expressing the value of the displacement on the direction i in the current section r of the structure (continuous collaboration); rj – the matrix of reactions rijk in the direction of discrete blocking element i due to the stresses generated in the structure’s basic system areas having the rheological
116
Materials with Rheological Properties
properties j by a unit displacement applied in the direction of the discrete blocking element k; pj(r) – the matrix of functions Pijk(r) expressing the value of the reaction in section r of the continuous blocking element i direction, due to the stresses generated in the structure’s basic system areas having rheological properties j by a unit displacement applied in the same section r, in the direction of continuous blocking element k; qi(r) – the matrix of functions qijk(r) representing the value of the reaction in discrete blocking element i, due to the stresses generated in the structure’s basic system areas having rheological properties j by a unit displacement applied in section r, in the direction of continuous blocking element k: (kijk(r) = kkji(r); Rj – the vector of values Rij of the reactions in (discrete) blocking element i due to the stresses generated in the structure’s basic system areas having rheological properties j by the external actions and loads; Pj(r) – the vector of functions Pij(r) representing the values of the reaction in section r of continuous blocking element i due to the stresses generated in the structure’s basic system areas having rheological properties j by the external actions and loads; ) – the zero vector having the same dimensions as vector =(t); c(r) – the matrix of functions cik(r) representing the values of the reaction in section r, in the direction of continuous blocking element i, due to the stresses generated by a given unit strain in the connection elements from section r in the direction of continuous blocking element k. By substituting relations (3.75) in (3.60), we obtain the integro-differential equations system describing the evolution in time of the stresses and strains state of the composite structures with complex composition, after moment W of the external actions and loads application: m
¦ j 1
½ ªl º q j r [ r , t dr » ª R j º ° °ª r j º « ³ K W K W K T W ®« T » Z W « 0 » « Pj r » ¾ oj j 1 j j 2 j ¬ ¼ ° °¬ q j r ¼ «¬ p j r [ r , t »¼ ¿ ¯
º½ ªl w °ª r º « ³ q j r >[ r , t @dr » ° d ° j wt » °¾ K 0 j t W j u ³ ®« T » >Z t @ « 0 q r w dt »° « j ¬ ¼ W ° « p j r w t >[ r , t @ » ° °¯ ¼¿ ¬ I ª º T » w3 u 1 K 1 j t W j K 2 j T t dt ³ « >Z r , t @» dt «c r 2 W w r wT ¬ ¼
(3.76)
T
>
@
0
Composite Resistance Structures
117
In the same way, by substituting relations (3.75) in (3.60’), we obtain the integrodifferential equation system describing the evolution in time of the stresses and strains state of the composite structures with complex composition, after moment T, subsequent to moment W of the external actions and loads application (T > W): m
¦ j 1
ªl º °ª r j º « q j r [ r , t dr » ª R j º ½°K W K W u ®« T » Z W « ³0 » « Pj r » ¾° 0 j j 1 j j ¼¿ °¬q j r ¼ [ p r r t , « »¼ ¬ ¬ j ¯
ªl º½ w q j r >[ r , t @dr » ° °ª r º « ³ d ° j wt » °¾ u u K 2 j T W K 2 j T W ³ ®« T » >Z t @ « 0 q r w dt « »° ¼ W °¬ j « p j r w t >[ r , t @ » ° °¯ ¬ ¼¿ u K 0 j t W j K1 j t W j K 2 j T t K 2 j T t dt
>
@
T
>
@
ªl º½ w °ª r º « ³ q j r >[ r , t @dr » °° d ° j wt » ¾ K 0 j t W j u ³ ®« T » Z t « 0 q r ¼ dt w « »° T °¬ j [ p r r , t > @ « j wt »° °¯ ¬ ¼¿ I ª º T » dt 0 w3 u 1 K1 j t W j K 2 j T t dt ³ « «c r 2 >[ r , t @» T wr wt ¬ ¼
(3.76’)
T
>
@
We note:
wk >p j r, t @ wtk
ªl º wk q j r k >[ r , t @dr » « k ³ ª rj º d wt » «q T r » k >Z t @ « 0 « » wk dt ¬ j ¼ « p j r w t k >[ r , t @ » ¬ ¼
I ª º wk k 2 « » w U > @ r , t l «c r 2 k >[ r , t @» wtk wr wt ¬ ¼
(3.77)
118
Materials with Rheological Properties
Using notations (3.75) and (3.77), the integro-differential equations (3.76) and (3.76’), describing the evolution in time of the stresses and strains state of the composite structures with complex composition, take the following form that is more compact: °
T
j 1° ¯
W
m
¦ ® ª¬ R j r ,W º¼ K0 j W j K1 j W j K 2 j T W ³
u ª1 K1 j t W j ¬
w ª U j r , t º¼ K0 j t W j u wt ¬
T
w ª¬ Ul r , t º¼ dt K 2 j T t º dt` ³ ¼ w T t
(3.78)
0
and, respectively: m
^
¦ ª«¬ R j r ,W º»¼ K0 j W j K1 j W j ª¬ K 2 j T W K 2 j T W º¼ j 1
T
w ª º ª º ¬ U j r , t ¼ K 0 j t W j ¬ K 2 j T t K 2 j T t ¼ dt w W t T w ª ª º º ³ ¬ U j r , t ¼ K 0 j t W j ¬1 K1 j t W j K 2 j T t ¼ dt w W t
³
^
T
w ª¬ Ul r , t º¼ dt T wt
³
(3.78’)
`
0
The solutions of integro-differential equation systems (3.78) and (3.78’) are approximated around the value t = IJ of the variable time, by the solution of the differential equation system of order k, which is obtained by substituting (3.75) and (3.77) in (3.62): ª¬ Ul r ,W º¼
m
¦ ª¬ R j r ,W º¼ K0 j W j 0
j 1
m ° w dK 2 j W W °½ w ª Ul r ,W ¼º ¦ ® ¬ª U j r ,W ¼º K 0 j W j ¬ª R j r ,W ¼º K 0 j W j K1 j W j ¾ 0 ¬ wt w t dT j 1° °¿ ¯
° w 2 ª U r ,W º K 0 j W j ¼ 2 ¬ j wt j 1° ¯w t ª dK 0 j W j dK 2 j W W º w ª » U j r ,W º¼ « K 0 j W j K1 j W j ¬ « » wt dT dT ¬ ¼ 2 ½ d K 2 j W W ° ª¬ R j r ,W º¼ K 0 j W j K1 j W j ¾ 0 dT 2 °¿
w2
2
ª¬ Ul r ,W º¼
m
¦®
(3.79)
Composite Resistance Structures
119
3 2 m ª dK W w3 >U l r , W @ ¦ w 3 >U j r, W @K 0 j W j w 2 >U j r , W @«2 0 j j K 0 j W j K 1 j W j u 3 dT wt wt j 1 wt ¬
u
dK 2 j W W dT
° d 2 K 0 j W j d 2 K 2 j W W w K K U j r , W @® W W 2 0j 1j j j 2 wt dT 2 °¯ dT
dK 1 j W j º dK 2 j W W ½ ª dK 0 j W j K 1 j W j K 0 j W j « ¾ » dT ¼ dT ¬ dT ¿
>
@
R j W K 0 j W j K 1 j W j
d 3 K 2 j W W dT 3
0
# k m wk >U l r ,W @ ¦ w k >U j r ,W @K 0 j W j " k wt j 1 wt
d k K 2 j W W R j W K 0 j W j K1 j W j dT k
>
@
0
and, respectively, around the value t = T of the variable time, by the solution of the differential equations system of order k, obtained by substituting (3.75) and (3.77) in (3.63): m dK T W w >U l r ,T @ ¦ ® w >U j r , T @K 0 j T W j >R j r ,W @K 0 j W j K1 j W j 2 j wt w t dT j 1¯
T
³ W
dK 2 j T t º w > U j r , t @K 0 j t W j K 1 j t W j dt » wt dT ¼
0
2 m w2 >U l r ,T @ ¦ ® w 2 >U j r ,T @K 0 j T W j w >U j r ,T @u 2 wt wt j 1 ¯w t
dK 2 j T T º ª dK 0 j T W j u« K 0 j T W j K1 j T W j » dT dT ¬ ¼ d 2 K 2 j T W R j r ,W K 0 j W j K1 j W j dT 2 T d 2 K 2 j T t ½° w U j r , t K 0 j t W j K1 j t W j ³ dt ¾ 0 dT 2 °¿ W wt
>
@
>
@
(3.79’)
120
Materials with Rheological Properties m w3 w3 w2 > @ >U j r ,T @u > @ r r K U T U T T W , , ¦ 0j l j j 3 w t3 wt2 j 1 wt
dK 2 j T T º ª dK 0 j T W j u «2 K 0 j T W j K 1 j T W j » dT dT ¬ ¼ 2 ° d K 0 j T W j d 2 K 2 j T T w >U r , T @® K K T W T W 2 0j 1j j j wt dT 2 dT 2 °¯ dK 1 j T W j º dK 2 j T T ½ ª dK 0 j T W j K 1 j T W j K 0 j T W j « ¾ » dT dT dT ¿ ¬ ¼ T
³ W
d 3 K 2 j T t w > dt U j r , t @K 0 j t W j K 1 j t W j wt dT 3
0
# k m wk >U l r ,T @ ¦ w k >U j r ,T @K 0 j T W j " k wt j 1 wt
d k K 2 j T W U j r , T K 0 j W j K 1 j W j dT k T d k K 2 j T t w ³ dt U j r , t K 0 j t W j K 1 j t W j dT k W wt
>
@
>
@
0
As shown in sections 3.2.3 and 3.3.3, the differential equation systems preceding that of order k from (3.79) or (3.79’), together with the boundary conditions, or, respectively, with the initial conditions, are additional conditions that the solutions of the equation systems of order k from (3.69) and (3.79’) must fulfill. They constitute the Cauchy problem attached to them.
3.3.2. The effect of repeated short-term variable actions and loads: overview As with the case of the formulation according to the creep, we mention that, in what follows, the dynamic response of the structure to short-term variable actions and loads is known (determined by analytical, experimental or empirical methods). Therefore, the stresses in the elements of the structure can be determined by increasing the stresses and strains corresponding to the static response of the structure under the respective variable actions and loads, multiplied by the dynamic coefficient corresponding to the given situation. The stresses and strains state at moment T of a structure submitted to short-term variable actions or loads, acting in the time interval 't = T – W, where W is the beginning moment, and T is the moment where the action ceases, has the expression:
Composite Resistance Structures T
m
¦³ W j 1
>
@
>
@
d U j t j t K 0 j t W j 1 K1 j t W j K 2 j T W j dt dt
121
0 (3.80)
By splitting up the integration interval and by considering that outside the interval of time (W – T),
Q t { 0, j t j Q t , t 0 W T T we obtain: T
d ¦ ®³ dt >U t t @K t W >1 K t W K T W @dt m
j 1
¯W
T
³ T
j
dU j t dt
0j
j
j
1j
j
2j
½ K 0 j t W j 1 K1 j t W j K 2 j T W j dt ¾ ¿
>
@
j
(3.80’)
0
Below, we admit as valid the hypothesis made in the case of the formulation according to the creep (section 3.2.2): – the effect of the variation of the real actions and loads on the structures can be described by linear combinations of the effects on the structures of certain actions and loads that imply a simple variation of the effects, which implies the necessity of studying the behavior of the structures based only on this last hypothesis; – the behavior at moment T of the structures submitted to a variable short-term action or load, 'ti = Ti – Wi, T > Ti, is well approximated by a short-term constant action or load, whose effect is defined by the interval of time 'ti and by the average value of the considered effect. The time of loading and unloading is short compared to the duration of the load. In the integro-differential equations (3.80) and (3.80’), we have considered that j t can have the form:
j t j Q t where:
* j – have the same significance as j used previously and correspond to a
constant, long-term reference (unit) action or load; Q(t) – the factor describing the variation in time of the short-term action or load applied to the structure.
122
Materials with Rheological Properties
By taking into account relations (3.44) and by employing the reasoning and mathematical developments of the formulation according to the creep of the equations describing the behavior of the structures submitted to short-term variable actions and/or loads (section 3.2.2), as well as the relations:
¦ >U W ' W @K W W
0
¦ >U T ' T @K T W
0
m
j
j
0j
j
(3.81)
j 1 m
j
j
0j
j
j 1
that represent the adaptation of relation (3.58) at moment W of the short-term variable action or load application and, respectively, at moment T of its end, equations (3.80) and (3.80’) take the final expression:
¦ >U W W @K W W K W W K T W >U T T @K T W K T W K T T m
j
0j
j
1j
j
2j
j
j 1
j
T
³ W
dU j t dt
0j
j
K 0 j t W
j
1j
j
j
2j
>1 K t W K T t @dt 1j
j
2j
(3.82)
0
As we could expect, the matrix integro-differential equation (3.82) that describes, according to the stress relaxation, the behavior at moment T of a composite structure submitted to a short-term variable load confirms the observation that the effect at moment T of a short-term variable action or load is equivalent to that of some permanent actions or loads equal in intensity and in the opposite direction, corresponding to the average effect Emed, applied to moments W and T corresponding to the action or load application and, respectively, to the ending moment of the action (unloading of the structure). We suppose the stresses and strains state of the structure at moment T to be known, due to a finite number, k – 1, of short-term variable actions and loads ('ti = Ti – Wi, T < Ti), as well as Wi < T1 < T, i = 1, 2, 3, ..., k – 1. The stresses and strains state of the structure at moment T (T > T) is described by the equation: m
T
j 1
¯W1
° d ¦ ®°³ dt >U t t @K t W K t W >K T t K T t @dt T
³ T
j
>
j
0j
1j
j
>
j
0j
2j
@
d ½ U j t j t @K 0 j t W j 1 K1 j t W j K 2 j T t dt ¾ 0 dt ¿
(3.82’)
Composite Resistance Structures
123
If, after moment T, in the interval T – T, the structure is submitted to a number n – k + 1 of short-term variable actions and loads, by using the same reasoning and the same mathematical developments as in section 3.2.2, the integro-differential equation (3.82) describing the behavior of the structure at moment T, due to all the n short-term variable actions and loads, takes the form:
¦ ¦ ^>U W W @K W W K W W >K T W K T W @ >U T T @K T W K T W K T T K T T k 1
m
i 1
j 1
j
i
j
T
³
j
j
j
dU j t
i
0j
i
1j
j
1j
j
i
2j
j
i
i
2j
i
2j
2j
i
>
i
i
@
K 0 j t W j K 1 j t W j K 1 j T t K 2 j T t dt`
dt
Wi
0j
i
(3.83)
¦ ^>U W W @K W W K W W K T W >U T T @K T W K T W K T T n
m
i k
j 1
¦
j
j
T
³
j
j
dU j t
dt
j 1 T
0j
i
0j
i
i
j
i
1j
1j
j
j
j
i
2j
j
2j
i
j
>
dU j t
m Wk
¦ ³
j
@
K 0 j t W j 1 K 1 j t W j K 2 j T t dt`
dt
Wi
j
>
@
K 0 j t W j 1 K 1 j t W j K 2 j T t dt
0
where the condition T d Wk is respected. If we consider the structure made of materials having the properties of a perfectly viscoelastic body, which supposes: K0 j t
(3.84)
K 0 j (constant)
K1 j t
K1 j (constant)
then the integro-differential equation (3.83) takes the form:
¦ ¦ ^>U W W @K k 1
m
i 1`
j 1
j
T
³
dt
0j
>
j
dt
m Wk
¦ ³ j 1 T
>
@
i
j
j
0j
>
>
@
K 1 j K 2 k T T i K 2 j T W i
@
K 0 j 1 K 1 j K 2 j T t dt`
dU j t dt
(3.85)
½ K 0 j K 1 j K 2 j T t K 2 j T t dt ¾ ¿
j 1
dU j t
@
K 1 j K 2 j T W i K 2 j T W i K 2 j T T j K 2 j T T i
m
i k
Wi
i
¦ ^>U W W @K
¦ T
j
dU j t
Wi
n
³
i
>
@
K 0 j 1 K 1 j K 2 j T t dt
0
124
Materials with Rheological Properties
For a finite value 't = T – W
K 0 j t
K 2 j t 't t of
1 tof
Practically, there is a (minimal) finite value TL – 'T, at least equal to the duration of the realization of the phenomenon of stresses redistribution in the structure due to the rheological properties of the materials, so that the following relation is valid:
K 2 j T L | K 2 j T L 'T | 1 Therefore, it results that for fixed T (T < T), sufficiently large, there is a maximum value TL (TL < T) that makes possible the relations:
K 2 j T T i
K 2 j T W i 1
K 2 j T T i
K 2 j T W i 1
K2 j T T L
K2 j T T i
(3.86)
1
for any pair of values Wi, Tj meeting the condition:
W i T i d T L , i 1,2,!, k 1. Under these conditions,
dU j t
k 1 T
¦³
dt
i 1 Wi
k 1 T
¦³
>
dU j t dt
j 1 TL
@
K 0 j K1 j K 2 j T t K 2 j T t dt
>
@
K 0 j K1 j K 2 j T t K 2 j T t dt
and integro-differential equation (3.85) takes the form:
¦ ¦ ^>U W W @K k 1
m
i L 1
j 1
j
T
³ T
dU j t dt
L
n
¦ i k
T
³ Wi
i
j
i
0j
>
K 1 j K 2 j T W i K 2 j T W i K 2 j T T i K 2 j T T L @
>
¦ ^>U W W @K m
j
i
j
i
0j
>
dt
@
K 1 j K 2 j T T i K 2 j T W i
j 1
dU j t
(3.85’)
@
K 0 j K 1 j K 2 j T t K 2 j T t dt `
m ½° dU j t K 0 j 1 K 1 j K 2 j T t dt ¾ ¦ ³ K 1 j 1 K 1 j K 2 j T t dt dt °¿ j 1 T
>
@
Wk
>
@
0
Composite Resistance Structures
125
The obtained results and the structure of the integro-differential equations (3.83), (3.85) and (3.85’) confirm the observations and the conclusions presented at the end of section 3.2.2.
3.4. Conceptual aspects of the mathematical model of resistance structure behavior according to the rheological properties of the materials from which they are made The equilibrium established in the deformed position, corresponding to the actions and loads applied to a resistance structure, is characterized by the simultaneous achievement of two conditions: – the static equilibrium condition: the totality of the external and internal forces is in equilibrium; – the compatibility condition: the deformed position of the structure and of the elements composing the structure respects all the existing connections, including the material continuity of its elements. These conditions, used in all the fields of the deformable body mechanics, also constitute, in the case of the study of composite structures, the necessary instrument for the research concerning its behavior according to the time. The application of these conditions is made by considering the character of the structure’s deformation. As a preliminary, we establish if the conditions are fulfilled for the 1st order calculation or if it is necessary to use the 2nd order calculation. Due to the reduced deformability of the materials composing the resistance structures for constructions, in the majority of the cases, the deformed position of the structure is very close to the initial one. Keeping the deformations of the structures in the domain of the small displacements allows the simplification of the calculation by writing the equilibrium equations relating the system of forces to the undeformed position of the structure. The stresses state in a composite structure, interior and/or exterior statically indeterminate, can be defined under the conditions of the acceptance of the validity of the hypothesis formulated previously, according to the external actions and loads and to a number of independent parameters y1, y2, y3, …, yn in the form: E(s,t) = e1(s)y1(t) + e2(s)y2(t) + ... + en(s)yn(t) + Ep(s)
(3.87)
We have denoted: Ep(s) – the function expressing the value of the stress in section s of the structure due only to the external actions and loads – thus considering parameters y1(t), y2(t), y2(t), ..., yn(t) as zero;
126
Materials with Rheological Properties
ei(s) – the function of influence corresponding to the considered parameters. Therefore, the stress at moment t, in section s – function E(s,t) – results from a linear combination of the function expressing the value of the stress in the given section. The last function is due to the external loads and actions; the others come from the successive loading of the structure with: y1(t) = 1, y2(t) = 1, y3(t) = 1, ..., yn(t) = 1 It results that it is necessary to be able to establish functions Ep(s) and independent linear influence functions e1(s), e2(s), ..., en(s), as well as the condition equations from which result parameters y1, y2, y3, ..., yn. These aims are carried out by the simultaneous consideration of the two instantaneous elastic equilibrium conditions. Indeed, if we consider as unknown factors the additional connection forces, the static equilibrium is satisfied indifferently of the values set to the unknown factors, because the number of available equations is equal to the minimal necessary number of connection forces. If we consider as unknown the displacements of the nodes of the structure (rotations and translations), the compatibility condition is satisfied whatever the values of these unknown factors, because we respect the connections and the continuity of the structure’s elements. Therefore, it results that the achievement of one of the two conditions leads to an infinity of solutions for the stresses state of the structure. The identification of the correct solution, corresponding to the real stresses and strains state in the structure, is made by the simultaneous use of the two conditions. Once functions Ep(s) and e1(s), e2(s), e3(s), ..., en(s) are determined, according to the condition corresponding to the nature of the chosen parameters, we determine the values of the parameters using the equations obtained by using the other condition. So, in the case of using the parameters stresses (connection forces) and the condition of compatibility and by using the 1st version of the principle of the correspondence, we obtain the system of integro-differential equations (3.8’) and (3.8”), the formulation according to the creep. In the case of using the displacement parameters, from the static equilibrium condition, by using the 2nd version of the principle of the correspondence, we obtain the integro-differential equation systems (3.60) and (3.60’), the formulation according to the stress relaxation. The functions of influence e1(s), e2(s), ..., en(s), corresponding to parameters y1, y2, ..., yn, must satisfy the condition of being linear and independent between them and each of them must meet the specific condition of the nature of the parameters. This is also carried out by the extension of the basic system concept in the case of
Composite Resistance Structures
127
composite structures. The basic system is conventional and it is obtained from the considered structure, by canceling out all the parameters. When the parameters are connection forces, their cancellation is equivalent to the suppression of the respective additional connections of the structure. As the additional connections represent only a part of the connections of the structure, their choice is arbitrary. It results that, for the same structure, several basic systems are possible. The number of unknown factors is equal to the number of additional connections. When the parameters are displacements, their cancellation is equivalent to the introduction of fictitious connections in the direction of the displacement corresponding to it, so that we arrive at a system of complete connections. It results, therefore, that the basic system is unique. The number of unknown factors is equal to the number of introduced additional connections. The significance of diagrams Ep(s), e1(s), e2(s), ..., en(s), is obtained by adopting particular values for the selected parameters. Effectively, if we admit that all the parameters of equation (3.87) are zero, we obtain: E(s) = Ep(s) Therefore, Ep(s) is the function expressing the variation, determined on the basic system, of the value of the stress due to the external actions and loads according to the position of the section. In a similar way, if we remove the external actions and loads stressing the structure and if we cancel out all the selected parameters, except one, to which we set the unit value, we obtain: E(s) = ei(s) It results that ei(s) is the function describing the variation of the value of the stress generated, on the basic system, by the action of the unit load unit, yi(t) = 1. From all this, we obtain the analogy between solving the composite structures and solving the resistance structures studied by the statics of the constructions. Effectively, if in equations (3.8’), (3.8”), (3.60) and (3.60’), we substitute, according to the case, the functions of creep or stress relaxation by the constant values E and respectively 1/E, we obtain the specific equations for the general displacement method of the statics of the constructions. It thus results that the statics of the constructions is a particular case of the mathematical model of the behavior of composite structures whose elements are made of materials with different rheological properties.
128
Materials with Rheological Properties
In this context, we must mention that, similarly, when solving the resistance structures by using the general method of displacements from statics of the constructions, the unique basic system obtained by adapting the formulation according to the stress relaxation of the mathematical model for the behavior of the composite structures, allows us to obtain some benefits concerning the organization and the systematization of the calculation for the formulation of algorithms used for the development of computer programs. In this respect, we must mention that only the unique basic system, specific to the mathematical model of the behavior of the composite structures in the formulation according to the stress relaxation, offers the premises necessary to the entire automation of the calculation of the composite structures. In this section, dealing with the conceptual aspects of solving the composite structures, we consider it opportune to mention that, above, we have deliberately renounced presenting the operational calculation methods as usable means for determining the solutions of the integro-differential equation systems (3.8’), (3.8”) and (3.60), (3.60’) describing the behavior of the composite structures according to the creep and, respectively, to the stress relaxation. The use, in Chapter 2, of the Laplace transform in order to determine the solutions in the simple cases of integrodifferential equations (2.16”), describing a stress relaxation experiment, suggests the difficulties which we confront in the case of the complex integro-differential equation systems, such as those obtained in this work. We have considered that the operational calculation methods are unfit for the development of a general method for solving the complex integro-differential equation systems such as those used in the description of the behavior of resistance structures by considering the rheological properties of the materials from which they are made. Operational calculation methods can be successfully used to solve the particular cases of composite structures leading to simple integro-differential equation systems.
Chapter 4
Applications on Resistance Structures for Constructions
4.1. Correction matrix “Modeling” resistance structures for calculation imposes the introduction of perfectly rigid parts (infinite rigidity) that help us to consider the effect of the eccentric fixings and/or, according to the case, that of the rigidity’s increase in the fixing zone of the components within the structure. This approach to the problem allows us to obtain a relatively simple mathematical model that reflects with enough precision the real behavior of the structure without a significant increase in calculation effort and, particularly, without implying an increase in the computational load. The increase in computational load would lead, in the case of numerical applications, to solutions that are less precise because of the accumulated precision errors (such errors increase with the number of the equations in the system). We also have to mention that this approach to the problem of eccentric fixings and to the increasing rigidity of the resistance structures’ components in the nodes fixing zone allows the analysis of the neglected phenomena (local slips and crushing due to concentrations of stresses, etc.) effect on the obtained solutions. The abovementioned approach also allows the analysis of the approximation due to introducing the perfectly rigid, fictitious connections, by varying their length and/or by varying the sizes that characterize the elastic properties of the materials from which the structure is made.
130
Materials with Rheological Properties
The correction matrices which will be determined below allow us to obtain the flexibility matrix, or, according to the case, the rigidity matrix, for the structural components that can be modeled as a cranked beam (having angles and bends) whose ends are perfectly rigid (Figure 4.1). To obtain the flexibility matrix or rigidity matrix we have to use the flexibility matrix or, respectively, the rigidity matrix of the bar located between the infinitely rigid parts. It is important to mention that there is no restriction concerning the shape of the bar. The correction matrices are the same whatever the shape of the bar’s axis (straight bars, plane curved bars, space (cranked) curved bars, etc.). y
y
xk
K x z
I z
y
xi
z Figure 4.1. Cranked bar with perfect (infinite) rigid ends
exi, exk – projection on axis X of the perfectly rigid part of the bar’s end i and k respectively; ey – projection on axis Y of the perfectly rigid parts from the bar’s ends; ez – projection on axis Z of the perfectly rigid parts from the bar’s ends; Values exi, exk, ey, ez are given in the axis system Oxyz, specific to the bar.
4.1.1. The displacement matrix of the end of a perfectly rigid body due to unit displacements successively applied to the other end of a rigid body In Figure 4.2, the perfectly rigid body is represented by the cranked bar I, M, N, K, fixed at point i. By displacing the rigid body through translation in the X direction with the unit value, 'xi = 1, we obtain the following displacements in point k: 'xk = (ex + 1) ex
Applications on Resistance Structures for Constructions
131
'yk = ey ey 'zk = ez ez Txk = 0 Tyk = 0 Tzk = 0. y x
N
K(e x ,e y ,e z ) x
z
y
I(0,0,0)
z
M
Figure 4.2. Cranked bar with a perfect (infinite) rigid end
In a similar way, for the translations in the Y and Z directions, we obtain: for 'yi = 1:
'xk = ex ex 'yk = (ey + 1) ey 'zk = ez ez Txk = 0 Tyk = 0 Tzk = 0
for 'zi = 1:
'xk = ex ex 'yk = ey ey 'zk = (ez + 1) ez Txk = 0 Tyk = 0 Tzk = 0
By applying successively finite rotations Tx, Ty, Tz around axes Ox, Oy and Oz respectively (which pass through point i), we obtain, for point k, the following displacements: for Tx:
for Ty:
'xk = ex ex 'yk = ez sin Tx ey (cos Tx 1) 'zk = (cos Tx 1) ey sin Tx Txk = Tx Tyx = 0 Tzk = 0 'xk = ex (cos Tx 1) ez sin Ty 'yk = ey ey
132
Materials with Rheological Properties
'zk = ex sin Ty + ez (cos Ty 1) Txk = 0 Tyk = 0 Tzk = 0 'xk = ex (cos Tz 1) ey sin Tz 'yk ex sin Tz ey (cos Tz 1) 'zk = ez ez Txk = 0 Tyk = 0 Tzk = Tz
for Tz:
for the small angles T: cos T | 1 sin T | T By using the values of the trigonometric functions for small angles in the relations for the displacements in point k of the rigid body, due to the displacements in point i and then by considering unit rotation angles T, we obtain the required displacement matrix:
>D@
ª1 «0 « «0 « «0 «0 « ¬«0
0 0
0
ez
1 0
ez
0
0 1 ey
ex
0 0 0 0
1 0
0 1
0 0
0
0
ey º e x »» 0 » » 0 » 0 » » 1 ¼»
(4.1)
4.1.2. The reaction matrix of the end of a perfectly rigid body due to unit forces successively applied to the other end of a rigid body Consider the rigid body represented in Figure 4.2. By applying successively, in point k, unit forces in all the possible directions in space, we obtain in fixed point i the following reactions: for Fx
1,
for Fy
1,
Rx = 1 Ry = 0 Rz = 0 Rx = 0 Ry = 1
Mx = 0 My ez Mz = ey Mx = ez My = 0
Applications on Resistance Structures for Constructions
for Fz for Mx for My for Mx
Rz = 0 Rx = 0 Ry = 0 Rz = 1 1, Rx = 0 Ry = 0 Rz = 0 1, Rx = 0 Ry = 0 Rz = 0 1, Rx = 0 Ry = 0 Rz = 0
1,
133
Mz ez Mx ey My = ex Mz = 0 Mx = 1 My = 0 Mz = 0 Mx = 0 My = 1 Mz = 0 Mx = 0 My = 0 Mz = 1
By joining together the above results, we obtain the matrix of the reactions in fixed point i of a perfect rigid body when we successively apply unit forces in point k of the rigid body.
>E @
ª 1 « 0 « « 0 « « 0 « e z « «¬ e y
0 1 0 ez 0 ex
0 0 1 ey ex 0
0 0 0 1 0 0
0 0 0 0 1 0
0º 0»» 0» » 0» 0» » 1»¼
(4.2)
4.2. Calculation of the composite resistance structures. Formulation according to the creep 4.2.1. Preliminaries necessary to systematize the calculation of composite structures in the formulation according to the creep The traditional form of the condition equations, in the matrix representation, for solving the statically indeterminate resistance structures by using the general stress method is:
G x 's
0
(4.3)
134
Materials with Rheological Properties
In order to systematize matrix G of the unit displacements and vector 's of the free terms, we start from the general expression of the displacements:
'
sT F b Q U s
(4.4)
In the above relation, we have denoted: ' – the displacement vector for the structure; the number of elements in the vector being equal to the sum of the number of the statically indeterminate unknown factors Xi and of the number of displacements that we want to determine; s – the transformation matrix whose element ski is a vector containing the stresses of the end of the bar k due to unit load in direction i acting on the structure’s basic system. The number of columns of the matrix is equal to the number of elements of vector ';
F – the flexibility matrix of the structure’s bars; this is a square diagonal matrix whose elements, distributed on the leading diagonal, consist of (square) flexibility matrices of the structure’s bars; b – the transformation matrix whose element bki is represented by a vector including the stresses of the current end of bar k, due to a unit load in direction i acting on the structure’s basic system; Q – the vector of the external action and load components on the bar’s ends and of the statically indeterminate unknown factors; Us – the displacement vector of the ends of the bars produced in the direction of the stresses in the end by the action of the loads applied to the bars, calculated on the structure’s basic system. By separating displacements '0, which we want to determine, from displacements 'x, in the direction of the statically indeterminate unknown factors, which serves to determine them, equation (4.4) takes the form:
ª' 0 º «""» « » ¬ 'x ¼
T ª s0T º ª P º ª s0 º « » « » F >b # b @ « ""»» «""»G s P 0 x «"" » « T «¬ bxT »¼ ¬ X ¼ «¬ bx »¼
>@
(4.5)
In relation (4.5), we have denoted: '0 – the vector of the structure displacements to be determined; 'x – the vector of the structure displacements in the directions of the statically indeterminate unknown factors; s0 – the transformation matrix whose element s0ki is a vector containing stresses in the end of bar k due to an auxiliary unit-load applied in the point and in the
Applications on Resistance Structures for Constructions
135
direction we want to determine the structure’s displacement and acting on the structure’s basic system; b0 – the transformation matrix whose element b0ki is a vector including the stresses in the end of bar k due to a unit load on the external load Pi (effectively applied) direction acting on the structure’s basic system; bx – the transformation matrix whose element bxki is the vector including the stresses in the end of bar k due to a unit load in the direction of the connection force (the statically indeterminate unknown factor) Xi acting on the structure’s basic system; P – the vector of the external actions and loads (effectively applied); X – the vector of the statically indeterminate unknown factors; Gs – the matrix whose element Gski, corresponding to bar k directly under the load Pi, is a vector including the displacements of the bar’s end due to the respective load considered equal to the unit (Pi = 1). By extracting from relation (4.5) the displacements in the direction of the statically indeterminate unknown factors and by considering the compatibility condition 'x = 0, we obtain:
>b
T x
@ >
@
F bx X bxT F b0 bxT G s P
0
(4.6)
By comparing the traditional form (4.3) of the compatibility equation with equation (4.6), there results:
G
>b
T x
F bx
@
(4.7)
From equation (4.5), we can obtain, after solving equation (4.6), vector '0 of the structure’s displacements which we want to determine:
'0
>S F @ >b P b X @ s G P T 0
0
x
T 0
s
(4.6’)
By considering that certain sizes composing equations (3.8’) and (3.8”) and describing the stresses and strains state’s evolution in time of the composite structures, and the sizes composing equation (4.3) have the same significance, there results the possibility of adapting the results of this section to systematize the calculation of the composite structures.
136
Materials with Rheological Properties
4.2.2. Composite structures with discrete collaboration By adapting relations (4.7) to the calculation of the composite structures with discrete collaboration, integro-differential equations (3.26’) and (3.27’) take the form:
>
@`
ªm T º «¦ bxj K 0 j W j K1 j W j K 2 j T W » F bx X F b0 G s P ¬j1 ¼ T m ½ dX ³ ®¦ bxjT K 0 j t W j 1 K1 j t W j K 2 j T t ¾ F bx dt dt W ¯j 1 ¿
^
>
@
(4.8)
0
and, respectively:
>
@`
m T ½ ®¦ bxj K 0 j W j K1 j W j K 2 j T W K 2 j T W ¾ F bx X F b0 G s P ¯j 1 ¿ T (4.8’) ½ m T dX dt ³ ®¦ bxj K 0 j t W j 1 K1 j t W j K 2 j T t ¾ F bx dt T ¯j 1 ¿ T m ½ dX dt 0 ³ ®¦ bxjT K 0 j t W j K1 j t W j K 2 j T t K 2 j T t ¾ Fbx dt W ¯j 1 ¿
@^
>
>
@
>
@
In integro-differential equations (4.8) and (4.8’), we have denoted: m – the number of materials having rheological properties different from the structure; bxj – transformation matrix bx, corresponding to the material having rheological properties j; element bxjki, corresponding to bar k and the statically indeterminate unknown factor Xi, has non-zero values only if the bar has rheological properties j. Between matrices bx and bxi there is the relation:
bx
m
¦b
xj
j 1
From expressions (4.8) and (4.8’) and, respectively, (4.9) and (4.10), it results that the estimation of the stresses and strains state’s evolution in time for a composite structure with discrete collaboration is possible if we know the functions describing the creep of the materials from which the structure’s elements are made and which are involved in the calculation of matrices bxki(W), bxki(t) and bxki(T). The construction and significance of matrices bxj, bx, b0, F, s0, ds, which we mentioned at the beginning of this section, are those known from the statics of constructions.
Applications on Resistance Structures for Constructions
137
APPLICATION 1.– We need to determine matrices bxj, F, bx, b0, ds necessary for the calculation of the stresses and strains state’s evolution in time for the structure in Figure 4.3a. Apart from the geometric elements given in Figure 4.3a, we also know: – the characteristics of the concrete part of the beam section: Eb, Ab, Ib; – the characteristics of the metal part of the beam section: E, A, I; – the elastic characteristics of the connection elements: 1/c. y
q
cgcs
y
y
a)
cgss Ok
y
b)
z
On
x x i+1 xi+2 Ok
1 xi+1=1
m
x i+1=1 1
1 c)
Ok
n
m
Ok
1
ey 1
xi+2=1
i=3(k-1)+1
m
t
xi+2=1
xi =1
q=1
d)
Mq
1 M k-1
Mk
Key: a) scheme of calculus, geometric elements and cross-section of the structure b) basic system and the unknown factors c) diagrams of internal forces on the basic system due to unit unknown factors d) diagrams of internal forces on the basic system due to action of loads on structure Figure 4.3. Composite structures with discrete collaboration
SOLUTION.– The structure is (3 x n) times statically indeterminate and the selected basic system is that presented in Figure 4.3b. The unit diagrams corresponding to the load with the statically indeterminate unknown factors xi = 1, xi + 1 = 1 and xi + 2 = 1 (from the panel k) are given in Figure 4.3c, and the diagram corresponding to load q, effectively applied, in Figure 4.3d.
138
Materials with Rheological Properties
The resistance structure is plane, loaded horizontally. The flexibility matrices of the bars are:
Mj
Mk
j
k
O
f
ªO 0 « EA « O « 0 3 EI « « 0 O «¬ 6 EI
º » O » » 6 EI » O » 3EI »¼
f
ª O « EA « « 0 « « « 0 ¬
º » O2 »» 2 EI » O3 » 3EI »¼
Fixed at both ends bar
Mk N j
k
O
k
Tk
Fixed-hinged bar
Mk N j
k T k
O
k
f
0
O EI
O2
2 EI
ª º «0 0 0 » «0 0 0 » « 1» «0 0 » c¼ ¬
Shear connector bar (cantilever) This results in the flexibility matrix of the structure’s bars:
F
ª0 « 0 « 1 « « c « « « « « « « « « « « « « « « « « « « « «¬
O EA
O 3EI
O
6 EI
O 6 EI
O
3EI
0 0 1 c
O Eb Ab
O Eb I b
O2
2 Eb I b
0
º » » » » » » » » » » » » » » » » » » » » O2 » 2 Eb Ib » O3 » » 3Eb I b »¼
0
Applications on Resistance Structures for Constructions
Transformation matrices bxj and bx: bx1
bx
ª 0 « 0 « « 0 « « 1 « e y « « ey « 0 « « 0 « 0 « « 0 « « 0 «¬ 0
bx2
bx3
0 0º ª 0 0 0»» «« 0 0 0» « 1 » « 0 0» « 0 0 0» « 0 » « 0 0» « 0 0 0» « 0 » « 0 0» « 0 0 0» « 1 » « 0 0» « 0 » « 1 0» « 0 0 1»¼ «¬ 0
0 º ª0 0 »» ««0 0 0 » «0 » « 0 0 » «0 1 0 » «0 » « 1 O » «0 0 0 » «0 » « 0 0 » «0 0 0 » «0 » « 0 0 » «1 » « 0 0 » «0 0 0 »¼ «¬0 0 0
bx
0 0º 0 0»» 0 0» » 0 0» 0 0» » 0 0» 0 0» » 0 0» 0 0» » 0 0» » 0 0» 0 0»¼
ª 0 « 0 « « 1 « « 1 « e y « « ey « 0 « « 0 « 1 « « 1 « « 0 «¬ 0
The displacements of the bar’s end under a uniform unit load:
Tj
T k
'l
0
O3 24 EI q=1
O
O
2
8 1 1
from which results: T
Us
ª O3 O3 º 0 # # « 24 EI 24 EI » q ¬ ¼
0 º 0 »» 0 0 » » 0 0 » 1 0 » » 1 O» 0 0 » » 0 0 » 0 0 » » 0 0 » » 1 0 » 0 1 »¼ 0 0
139
140
Materials with Rheological Properties
b0
ª 0 º « 0 » « » « 0 » « » « 0 » « M k 1 » « » « M k » « 0 » « » « 0 » « 0 » « » « 0 » « » « 0 » «¬ 0 »¼
Gs
ª « « « « « « O3 « 3 « O « « « « « « « « «¬
0 º » 0 » » 0 » 0 » 24 EI » » 24 EI 0» » 0 » 0 » » 0 » » 0 » 0 » »¼ 0
P
q
4.2.3. Composite structures with continuous collaboration Just like the composite structures with discrete collaboration, we can systematize the calculation of the stresses and strains state’s evolution in time of the composite structures with continuous collaboration. We start from the relation:
Hx ½ °J ° ° xy ° °°J xz °° ® ¾ °J yz ° ° wy ° ° ° °¯ wz °¿
ª1 EA º Nx ½ « »°T ° J y GA « »° y ° « » °° Tz °° J z GA « »® ¾ 1 GI t « » °M x ° « » °M y ° 1 EI y « »° ° 1 EI z ¼» °¯ M z °¿ ¬«
(4.9)
or
H
f Q
In (4.9), we present the deformability matrix f of a spatial element. In the case of plane problems and when we ignore certain effects (for example, the shearing force, the torsion moment, etc.), the dimensions of matrix b and the dimensions of matrix f are proportionally reduced.
Applications on Resistance Structures for Constructions
141
By adapting matrices bx and b0, defined in the previous section, to the composite structures with continuous collaboration, and by making the substitution:
- j r E xjT r (r ) E x (r ) ' j r E xjT r (r ) E x (r )
O r E OT r (r ) E x (r )
integro-differential equations (3.33) take the form:
¦ >E r r E r X r ,W E r r E r P@K W K W K T W m
T xj
T xj
x
0j
0
j
1j
j
2j
j 1
m T
>
¦ ³ E xjT r r E x r @ j 1W
w X r , t K 0 j t W j >1 K1 j t W j K 2 j T t @dt wt
(4.10)
w3 X r , t dt 0 2 T wr wt
>
T
E OT r r E O r @ ³
and, respectively:
¦ >E r r E r X r,W r E r P@K W K W >K T W K T W @ m
T xj
0
x
0j
j
1j
j
2j
2j
j 1
m T
>
w X r , t K 0 j t W j >1 K1 j t W j K 2 j T t @dt wt
>
w X r , t K 0 j t W j K1 j t W j >K 2 j T t K 2 j T t @dt wt
¦ ³ E xjT r r E x r @ j 1T
m T
¦ ³ E xjT r r E x r @ j 1W
>
(4.10’)
w3 X r , t dt 0 2 T wr wt T
E OT r r E O r @ ³
In integro-differential equations (4.10) and (4.10’), we have denoted: m – the number of parts composing the structure’s section, that are homogenous from the point of view of the rheological behavior of the material from which they are made; ȕxj(r) – the equilibrium matrix ȕx(r) corresponding to the material having rheological properties j; the element ȕxjxi(r), corresponding to part k of the structure’s cross-section (of coordinate r) and to the unknown factor Xi(r), takes non-zero values only if the structure is made of a material having rheological properties j; ȕO(r) – the equilibrium matrix similar to matrix ȕx(r) corresponding to the shear connection elements (connectors); element ȕxki(r) corresponding to component k of
142
Materials with Rheological Properties
the section of the structure and to the unknown factor Xi takes non-zero values only if element k is a connector and, obviously, only if Xi, can cause its deformation;
(r) – the flexibility matrix of the elements composing the section of the structure, which is a diagonal (square) matrix, whose elements, distributed on the leading diagonal, are the deformability matrices fii(r) of each part of the structure’s section, homogenous from the point of view of their rheological properties. Between matrices ȕx(r) and ȕxj(r), there is the relation:
E x r
m
¦ E r xj
j 1
If we take into account representation (4.10) and (4.10’) of the integrodifferential equations (3.33), the differential equation systems (3.34), (3.35) that serve to approximate the solutions of equations (3.33) can be written as follows: O
w2 X r , W E xT0 r ,W r E x r ,W X r ,W > r E 0 r H s @ Q w r2
^
O
`
0
w3 w X r ,W E xT0 r ,W r E x r ,W X r ,W 2 wT w r wT
E xT11 r , W ^ r E x r ,W X r ,W > r E 0 r H s @Q` 0
w4 w2 T X r W E r W r E r W X r , W , , , x0 x w r 2w T 2 wT 2 w X r ,W E xT22 r ,W r E x r ,W wT
O
E xT21 r ,W ^ r E x r ,W X r ,W > r E 0 r H s @Q` 0
O
w5 X r ,W E xT0 r ,W r E x r ,W X r ,W 2 3 w r wT
E xT33 r ,W r E x r ,W
w2 w X r ,W E xT32 r ,W r E x r ,W X r ,W 2 wT wT
E xT31 r ,W ^ r E x r ,W X r ,W > r E 0 r H s @Q` 0 #
O
w k 2 wk X r , W E xT0 r , W r E x r , W k X r , W " 2 w r w Tk wT
E xkT 1 r , W ^ r E x r , W X r , W > r E 0 r H s @Q` 0
(4.11)
Applications on Resistance Structures for Constructions
143
respectively, for the situation where we know the stresses and strains state at moment T > W. T w3 w X r ,W E xT0 r ,T r E xw T r ,T X r ,T ³ E xT11 r , t r E x r u 2 wT w r wT W w u X r , t dt E xT10 r ,T ^ r E x r X r ,T > r E 0 r H s @Q` 0 wT
O
w4 w2 T , W E , T E X r r r r X r , T x0 x w r 2w T 2 wT 2 w E xT22 r , T r E x r X r , T wT
O
T
³ E xT21 r , t r E x r W
(4.12)
w X r , t dt wT
E xT20 r , T ^ r E x r X r ,W > r E 0 r H s @Q` 0
w3 w5 T , , X r r r r X r , T W E T E x0 x wT3 w r 2w T 3
O
E xT33 r ,T r E x r
w2 X r ,T wT 2
E xT32 r ,T r E x r
T w w X r ,T ³ E xT31 r , t r E x r X r , t dt wt wT W
E xT30 r ,T ^ r E x r X r ,W > r E 0 r H s @Q` 0 # O
wk w k 2 X r , W E xT0 r , T r E x r X r , T " 2 k wTk w r wT T
³ E xkT 1 r , t r E x r W
w X r , t dt wt
E xkT 0 r , T ^ r E x r X r , W > r E 0 r H s @Q ` 0 In equations (4.11) and (4.12), we have denoted:
O
E OT r r E O r ;
E x 0 r ,W
¦ E r K W ; m
xj
j 1
0j
j
144
Materials with Rheological Properties
E x11 r ,W
dK W W ¦ E r K W K W dT ; m
2j
xj
0j
j
j 1
E x 22 r ,W
1j
K W K W dK W W º; » dT
ª dK W ¦ E r « dT m
0j
xj
j 1
¬
j
2j
j
0j
1j
j
¼
d K W W ; ¦ E r K W K W dT dK W W º ª dK W r ,W ¦ E r «2 K W K W »; dT dT ¬ ¼ d K W d K W W r ,W ¦ E r °® 2 K W K W dT ° dT
E x 21 r ,W
2
m
2j
0j
xj
1j
j
j
2
j 1
E x 33
j
m
0j
j
2j
xj
0j
j
1j
j
j 1
E x 32
2
m
2
0j
xj
j 1
j
2j
0j
2
¯
j
1j
j
2
dK W º dK W W ½ ª dK W « 0 j j K1 j W j K 0 j W j 1 j j » 2 j ¾; dT ¼ dT ¬ dT ¿ m d 3 K 2 j W W ; E x 31 r ,W ¦ E xj r K 0 j W j K1 j W j dT 3 j 1 #
E xk1 r ,W
¦ E xj r K 0 j W j K1 j W j m
d k K 2 j W W dT k
j 1
E x 0 r ,T
;
¦ E r K T W ; m
0j
xj
j
j 1
E x11 r , t
dK T t ¦ E r K t W K t W dT ; m
2j
xj
0j
j
1j
j
j 1
dK T W ¦ E r K W K W dT ; dK T T º ª dK T W r ,T ¦ E r « K T W K T W »; dT dT
E x10 r ,T
m
2j
0j
xj
1j
j
j
j 1
E x 22
m
0j
xj
j 1
E x 21 r , t
2j
j
0j
¬
d ¦ E r K t W K t W m
0j
xj
j
1j
1j
K 2 j T t dT 2
d 2 K 2 j T T
¦ E xj r K 0 j W j K1 j W j m
j 1
E x 33 r ,T
2
j
j 1
E x 20 r , T
j
dT 2
j
¼
;
;
dK T T º ª dK T W ¦ E r «2 dT K T W K T W dT »; m
0j
xj
j 1
¬
2j
j
0j
j
1j
j
¼
Applications on Resistance Structures for Constructions
E x 32 r , T
d 2 K 0 j T W j
° ¦ E r ®° m
xj
j 1
dT
¯
2
145
d 2 K 2 j T T 2 K 0 j T W j K1 j T W j dT 2
dK1 j T W j º dK 2 j T T ½ ª dK 0 j T W j « K 1 j Y W j K 0 j T W j ¾; » dT dT dT ¬ ¼ ¿
E x 31 r , t
d ¦ E r K t W K t W m
0j
xj
1j
j
j 1
E x 30 r ,T
j
dT 3
;
d 3 K 2 j T T ; dT 3
j 1
d k K 2 j T t
¦ E r K t W K t W m
0j
xj
1j
j
j
j 1
E xk 0 r , T
K 2 j T t
¦ E xj r K 0 j W j K1 j W j m
#
E xk1 r , t
3
m
¦ E r K W K xj
j
1
0j
j
1j
dT k
;
d k K 2 j T T W j ; dT k
As shown in section 3.2.3, equations (3.33), (3.34), (3.36), as well as their systematized form (4.10), (4.10’), (4.11) and (4.12), given in this section, describe the stresses and strains state’s evolution in time in composite structures with continuous collaboration, where the connection elements (connectors) allow elastic relative displacements between the elements of the structure’s section. In the case of perfect collaboration, the collaboration between the elements of the structure’s sections is performed without the possibility of relative displacements. Therefore, the stresses and strains state’s evolution in time is described by equations (3.33’), (3.34’), (3.36’) or, in the systematized form, by equations (4.10), (4.11) and (4.12), where we will adopt value 0 for matrix O. APPLICATION 2.– We need to determine matrices ȕO(r), ȕxj(r), ȕ0(r), (r) necessary for the calculation of the stresses and strains state’s evolution in time of the composite structure with continuous collaboration from Figure 4.4. We consider that the structure has a constant section and uniformly distributed shear connection elements (connectors). The physico-mechanical characteristics of the elements composing the structure’s section are: – metal part of the section: E, G, A, I; – concrete part I of the section: E1, G1, A1, I1;
146
Materials with Rheological Properties
– concrete part II of the section: E2, G2, A2, I2; The elastic characteristics of the connectors are 1/c1, 1/c2. NOTE.– According to the calculation hypothesis, on the undashed sections of the stress diagrams, the respective stresses do not produce deformations.
2 1
1
cgcs II cgcs I cgss
q
2
y
z
Figure 4.4. Composite structures with continuous collaboration; scheme of calculus, geometric elements and basic system
The composite structure with continuous collaboration is considered, at the limit, as a composite structure with discrete collaboration where the distance between the connectors tends towards zero. This way of considering the composite structure with continuous collaboration allows us to choose a basic system similar to that used in Application 1 for the analysis of the composite structure with discrete collaboration (Figure 4.5a). Figures 4.5b and 4.5c illustrate, on a large scale, the action mode of connection forces Xi and, respectively, of the external actions on the elements composing the structure’s section in one of its current sections x. By ignoring the contribution of small infinities of a higher order than the deformation of the components of the structure’s section, we have only retained, for the construction of matrices E, the presented stress diagrams. The flexibility matrix of the structure’s elements When analyzing a plane structure, loaded horizontally, we retain only the elements corresponding to the respective plane problem in the generic matrix given in relation (4.12). Also, in the case of the connection elements (connectors) (in the basic system from Figure 4.5a, the vertical elements), the matrix is adapted according to their deformability properties.
Applications on Resistance Structures for Constructions 6
5
a)
x4 x1
x3
b)
1
2
x2
1
n
1
1
1
e1
t
x1=1
m
1 1
t
x2=1
1 1
m
x3=1
147
148
Materials with Rheological Properties
1 n
1
x4=1
t
1
1
m
e2 1 t
1
x5=1
1 m
1 c)
x6=1
M
Mx
dM M+ dx dx
Q=1
T Tx
dT T+ dx dx
Key: a) basic system and the unknown factors b) diagrams of internal forces on the basic system due to unit unknown factors c) diagrams of internal forces on the basic system due Q = 1 Figure 4.5. Composite structures with continuous collaboration
Applications on Resistance Structures for Constructions
149
We obtain: – for the metal part of the section:
fm
0 0 º ª1 EA « 0 J m GA 0 »» « «¬ 0 0 1 EI »¼
– for the concrete part I of the section:
f b1
0 0 º ª1 E1 A1 « 0 0 »» J 1 G1 A1 « «¬ 0 0 1 E1I1 »¼
– for the concrete part II of the section:
fb2
0 0 º ª1 E2 A2 « 0 J 2 G2 A2 0 »» « «¬ 0 0 1 E2 I 2 »¼
– for the shear connectors between the metal part of the section and the concrete part I of the section, and, respectively, between the concrete parts I and II of the section:
f c1
ª0 0 «0 1 c 1 « ¬«0 0
0º 0»» f c 2 0»¼
ª0 0 «0 1 c 2 « ¬«0 0
0º 0»» 0»¼
The flexibility matrix of the elements of the section of the composite structure with continuous collaboration is obtained by assembling the flexibility matrices of the elements composing the section and by considering the direction of the structure’s path indicated in Figure 4.5a.
ª fb2 « « « « « « ¬
f c1 f b1 fc2
º » » » » » f m »¼
150
Materials with Rheological Properties
Equilibrium matrices ȕxj, ȕx and ȕ0 are built by “picking” the stresses that are borne in the elements composing the section of the composite structure when we successively set the unit value to the unknown connection forces (the stress diagrams of Figure 4.5b) and respectively when we set the unit value to the external actions (Figure 4.5c). The same succession of elements (the structure’s traverse) is respected when building matrices E, as that used in the composition of the flexibility matrix of the structure’s elements: 0 0 1 0 0º 0 0 0 1 0»» 0 0 0 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» E xb 2 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 0¼»
E xb1
ª0 «0 « «0 « «0 «0 « «0 «0 « «0 «0 « «0 « «0 «0 « «0 «0 « ¬«0
Ex
¦E
m
j 1
xj
ª0 «0 « «0 « «0 «0 « «0 «1 « «0 «0 « «0 « «0 «0 « «0 «0 « ¬«0
0 0 0 0 0º 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 0 0 0» E xm3 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 0¼»
ª0 0 0 0 0 0º «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 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 0 1 0 0 » « 0 1 0 0 1 0 » » « ¬« e1 0 1 e2 0 1»¼
Applications on Resistance Structures for Constructions
Ex
ª0 0 0 1 0 0º «0 0 0 0 1 0» » « «0 0 0 0 0 1» » « «0 0 0 0 0 0» «0 0 0 0 0 0» » « «0 0 0 0 0 0» «1 0 0 0 0 0» » « 0 0 0 0 »E O «0 1 «0 0 1 0 0 0» « » «0 0 0 0 0 0» « » «0 0 0 0 0 0» «0 0 0 0 0 0» « » « 1 0 0 1 0 0 » « 0 1 0 0 1 0 » » « ¬« e1 0 1 e2 0 1¼»
ª0 «0 « «0 « «0 «0 « «0 «0 « «0 «0 « «0 « «1 «0 « «0 «0 « ¬«0
0 0 0 0 0º 0 0 0 0 0»» 0 0 0 0 0» » 0 0 0 0 0» 0 0 1 0 0» » 0 0 0 0 0» 0 0 0 0 0» » 0 0 0 0 0» E 0 0 0 0 0 0» » 0 0 0 0 0» » 0 0 1 0 0» 0 0 0 0 0» » 0 0 0 0 0» 0 0 0 0 0» » 0 0 0 0 0¼»
ª0 «0 « «0 « «0 «0 « «0 «0 « «0 «0 « «0 « «0 «0 « «0 «T « ¬« M
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 T
0 0
0 0 M
151
0 0º 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 0¼»
A special mention must be made concerning the consideration of the shear connectors’ contribution to the development of certain deformations; these exist 2 only if w x1 z 0 and is proportional with this size. Therefore, in the composition of ws 2 matrices bxj, bx and b0, the elements corresponding to the shear connectors elements are zero. When composing matrix EO, we set to the elements corresponding to the connection elements the values given in the stress diagrams (Figure 4.5b). We set the value zero to the other elements of the matrix. APPLICATION 3.– We need to determine matrices (r), Exj(r), Ex(r), EO(r) and E0(r), necessary for the calculation of the stresses and strains state’s evolution in time for the composite structure with continuous collaboration from Figure 4.6. We consider that the structure has a constant section and that the connection elements (connectors) are uniformly distributed. The physico-mechanical characteristics of the elements comprising the structure’s section are: – metal part of the section: E, G, A, I; – concrete part of the section: Eb, Gb, Ab, Ib; – the elastic characteristics of the shear connectors: 1/c.
152
Materials with Rheological Properties
In Figure 4.7a, we present the basic system of the structure; Figures 4.7b and 4.7c illustrate the unit diagrams necessary for the composition of matrices E.
y
cgcs cgss z
Figure 4.6. Composite structures with continuous collaboration; scheme of calculus, geometric elements and basic system
The flexibility matrix of the elements of the structure’s section: – for the metal part of the section:
fm
ª1 EA º « » J m GA « » «¬ 1 EI »¼
– for the concrete part of the section:
fb
ª1 Eb Ab « « «¬
J b Gb I b
º » » 1 Eb I b »¼
– for the connection elements (connectors):
fO
ª0 º « 1c » « » «¬ 0»¼
The flexibility matrix of the structure’s elements is obtained by assembling the matrices of the parts entering in the composition of the structure’s section by considering the sense of the structure’s traverse indicated in Figure 4.7a.
Applications on Resistance Structures for Constructions
ª fb « « « ¬
fO
º » » f m »¼
Equilibrium matrices Exj(r), Ex(r), EO(r), and E0:
E xb1
EO
ª1 «0 « «0 « «0 «0 « «0 «0 « «0 «0 ¬
0 0º 1 0»» 0 1» » 0 0» 0 0» E xb2 » 0 0» 0 0» » 0 0» 0 0»¼
ª0 0 «0 0 « «0 0 « «0 0 «0 1 c « «0 0 «0 0 « «0 0 «0 0 ¬
0º 0»» 0» » 0» 0» E0 » 0» 0» » 0» 0»¼
ª0 0 0º «0 0 0» « » «0 0 0» « » «0 0 0» « 0 0 0 »Ex « » «0 0 0» « 1 0 0 » « » « 0 1 0 » « e 0 1» ¬ ¼ ª0º «0» « » «0» « » «0» «0» « » «0» «0» « » «T » «M » ¬ ¼
m
¦E j 1
xj
ª1 0 0º «0 1 0» « » «0 0 1» « » «0 0 0» «0 0 0» « » «0 0 0» « 1 0 0 » « » « 0 1 0 » « e 0 1» ¬ ¼
153
154
Materials with Rheological Properties x3
a)
x1 x2
b)
1 n
1
1
1
e
t
x1=1
m
1 1
t
x2=1
m
x3=1
1 1 c) T
T T+dT dxdx Q=1 M
M
M+dMdx dx
Key: a) basic system and the unknown factors b) diagrams of internal forces on the basic system due to unit unknown factors c) diagrams of internal forces on the basic system due external loads (Q = 1) Figure 4.7. Composite structures with continuous collaboration
Applications on Resistance Structures for Constructions
155
4.2.4. Composite structures with complex composition The systematization of the calculation of the stresses and strains state’s evolution in time of composite structures with complex composition is made by using the results obtained previously for the calculation of the composite structures with discrete collaboration and for the calculation of the composite structures with continuous collaboration. By using the notations from sections 4.2.2 and 4.2.3, and by considering the significance of the respective sizes, relations (3.37) can be written as follows:
>E r, t @ j
>E O r , t @
>D
0 j
r ,W @
ªl T º ª E xjT F bx º « ³ E yi E xyY r , t dr » « T » X t « 0 » ¬« E yj E xy ¼» «¬ E yjT E yY r , t »¼ 0 ª º « T » w2 « E O E O w r 2 >Y r , t @» ¬ ¼
(4.13)
> >
ªl T º T ª bxjT F bx º « ³ E yj E xyY r ,W dr » ª bxj F b0 G s W X « T » «0 » « E E H 0 s ¬« E yj E xy ¼» «¬ E yjT E yY r ,W »¼ ¬« yj
@º» Q @¼»
and integro-differential equations (3.38) and (3.39) become: m
¦ j 1
T ªl T º °ª E xj Fbx º « ³ E yj E xyY r ,W dr » » X W « 0 ®« T » °«¬ E yij E xy »¼ «¬ E yjT E yY r ,W »¼ ¯
> >
ª bT Fb G s « xjT 0 «¬ E yj E 0 H s
@º» Q½°K W K W K @»¼ ¾°¿ 0j
j
1j
j
2j
T W
ªl T º½ w E yj E xy >Y r , t @dr » ° ° ª bT F b º « ³ d ° wt » °¾ u ³ ®« xjT x » > X t @ « 0 w dt E E « »° T « yj xj ¼» W °¬ « E yj E y w t >Y r , t @ » ° °¯ ¬ ¼¿ T
>
@
u K 0 j t W j 1 K1 j t W j K 2 j T t dt 0 T ª º 3 ³ « E T E w >Y r , t @» dt O O » W « w r 2w t ¼ ¬
0
(4.14)
156
Materials with Rheological Properties
and, respectively:
T ½ ªl T º T °ª bxj F bx º « ³ E yj E xyY r ,W dr » ª bxj F b0 G s º ° » X W « 0 ®« T ¦ » « E T E H » Q ¾ u j 1 » ° T 0 s ¼ °¬« E xj E xy ¼» «¬ E yj E yY r ,W »¼ ¬« yj ¿ ¯ u K 0 j W j K1 j W j K 2 j T W K 2 j T W
> >
m
>
@ @
@
(4.15)
ª T º½ w E yj E xy >Y r , t @dr » ° ° ª bT F b º « ³ d ° wt » °¾ u ³ ®« Txj x » >X t @ « 0 w « »° T T °« ¬ E yj E xy »¼ dt « E yj E y w t >Y r , t @ » ° °¯ ¬ ¼¿ u K 0 j t W j 1 K1 j t W j K 2 j T t dt l
T
>
@
º½ ªl T w Y r , t dr > @ E E ° ª bT F b º » °° « yj xy ³ d ° wt »¾ u ³ ®« xjT x » > X t @ « 0 w »° « T « E yj E xj ¼» dt W °¬ « E yj E y w t >Y r , t @ » ° °¯ ¼¿ ¬ T
>
@
u K 0 j t W j K1 j t W j K 2 j T t K 2 j T t dt 0 T ª º 3 ³ « E T E w >Y r ,W @» dt O O » T « w r 2w t ¬ ¼
0
By substituting (4.13) in the equation systems (3.14) and (3.42) whose solutions approximate the solution of the integro-differential equation systems (4.14) and (4.15) and by considering that:
wk >E r , t @ w tk j
ªl T º wk E E « ³0 yj xy w t k >Y r , t @dr » ª b F bx º d k »; « » k >X t @ « « » wk T «¬ E E xy »¼ dt « E yj E y w t k >Y r , t @ » ¬ ¼ T xj T yj
0 ª º wk k 2 « » w > @ r t , E T « E O E O w r 2w t k >Y r , t @» w tk O ¬
(4.16)
¼
we obtain the form of the differential equations which approximate (around values W or, depending on the case, T of the variable time) the solutions of the integro-
Applications on Resistance Structures for Constructions
157
differential equation systems (4.14) and (4.15) respectively specific to the composite structures with complex composition. To save space, we will not give the results of this substitution, but we have to make the following observation concerning the determination of the solutions for the integro-differential equation systems (4.14) and (4.15). Therefore, apart from the approximation of the solutions of the integrodifferential equation systems (4.14) and (4.15) by Taylor series or by the numerical integration of the differential equation systems obtained by the substitution that we have mentioned above, we can simultaneously use the two processes. The two processes are the construction of the Taylor series corresponding to the unknown discrete functions and the numerical integration of the unknown distributed functions, which corresponds to partitioning the structure into substructures. In relations (4.13), (4.16) and, implicitly, in equations (4.14), (4.15), we have denoted: Exy – the transformation (equilibrium) matrix – whose element Exymm(r) represents the effort in element m, in section r of the basic system – due to a unit force applied in the direction of discrete connection force Xn. APPLICATION 4.– We need to determine matrices bxj, F, bx, b0, Gx, , Ey, Exy, E0, necessary for the calculation of the stresses and strains state’s evolution in time for the composite structure with complex composition in Figure 4.5. We consider that the structure has a constant section and that the connection elements (connectors) are uniformly distributed. The structure (the composite beam with continuous collaboration having elastic connectors) is simply supported at one end and fixed at the other end. The supports can settle and they have the same rheological properties. The geometric elements of the structure are given in Figure 4.8; the physicomechanical characteristics of their components are: – the rigidity of the supports:
U1
^U v1 ` and U 2
Uv2 ½ ® ¾ ¯ UT 2 ¿
– the metal section E, G, A, I; – the concrete section Eb, Gb, Ab, Ib; – the elastic characteristics of the shear connectors: 1/c.
158
Materials with Rheological Properties y
y1 y2
cgcs
cgss z U 1 UQ1
x1
x2 UQ2 U UX2 2
Figure 4.8. Composite structures with continuous collaboration; scheme of calculus, geometric elements and basic system
The flexibility matrix of the elements of the structure: – for the elements of the structure that are directly loaded by the discrete unknown factors: - the simple support:
f1
ª 1 º « » ¬ U v1 ¼
- the fixed support
f2
ª 1 «U « v2 « 0 «¬
º 0 » » 1 » UT 2 »¼
- the metal part of the beam:
f3
ª 1 « 3EI « 1 « ¬ 6 EI
1 º 6 EI » 1 » » 3EI ¼
– for the elements of the structure directly loaded by the distributed unknown factors: - the metal beam:
I1
ª 1 « EA « « 0 ¬
º 0» 1» » EI ¼
Applications on Resistance Structures for Constructions
159
– the concrete slab:
ª 1 «E I « b b « 0 «¬
I2
º 0 » » 1 » Eb I b »¼
– the shear connectors: ĭ2 = [1/c]. y(x) 2
x1
y1(x) 1 1 x 1=1
r
1 e l ex l
m
n
1
1
e l e
1
e 1
t
y(x) =1 1
m
m
y(x) =1 2
1 M
r M
1 l x2=1
M+dM dx dx
Q=1
x l
m
Q=1 R1
R1
Figure 4.9. Composite structures with continuous collaboration; basic system and the unknown factors, diagrams of internal forces on the basic system due to unit unknown factors and external load on structure
160
Materials with Rheological Properties
The flexibility matrices of the structure’s elements:
F
ª f1 « « «¬
f2
º » » f 3 »¼
ªI1 º « » I 2 « » «¬ I3 »¼
The transformation (equilibrium) matrices: – for the discrete unknown factors:
bx1
ª e l 1 l º «el 1 l »» « « e 1 »;bx 2 « » 0 0 » « «¬ 0 0 »¼
ª0 0 º «0 0 » « » «0 0 »; E xy « » «0 0 » «¬e 1»¼
ª 0 «ex l « « 0 « « 0 «¬ 0
0º x l »» 0 »;bx » 0» 0 »¼
3
¦b
xi
i 1
ª e l 1 l º «el 1 l »» « « e 1 » « » 0 0 » « «¬ e 1 »¼
– for the distributed unknown factors:
E y1
ª1 «e « «0 « «0 «¬0
0º 1»» 0»; E y 2 » 0» 0»¼
ª0 0º «0 0» » « « 1 0 »; E y » « « 0 1» «¬ 0 0 »¼
2
¦E
i
i 1
ª1 0º «e 1 »» « « 1 0 »; E O » « « 0 1» «¬ 0 0 »¼
ª0 «0 « «0 « «0 «¬1
0º 0»» 0» » 0» 0»¼
– for the external loads (a load Q):
b0
ª R1 º «R » « 2» « 0 »;G s « » «0» «¬ 0 »¼
ª0º «0» « » « 0 »; E 0 « » « Qi » «Q j » ¬ ¼
ª 0 º «M » « x» « 0 »;H s « » « 0 » «¬ 0 »¼
ª0 º «0 » « » «0 » « » «0 » «¬0»¼
The flexibility matrices of the structure’s elements, as well as the transformation matrices, are built and have the same significance as in the case of the composite structures with discrete collaboration and, respectively, with continuous collaboration. An additional observation must be made concerning matrix Exy: this matrix is built on the same principles as matrix E0, but considering the elements with continuous collaboration of the structure loaded by the external forces Xi (discrete unknown factors).
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161
4.3. The calculation of composite resistance structures. Formulation according to the stress relaxation 4.3.1. Preliminaries necessary to systematize the calculation of the composite structures in the formulation according to the stress relaxation As in the case of the formulation according to the creep, the formulation according to the stress relaxation of the composite structures’ calculation can be systematized by using knowledge of the statics of constructions, namely of the general displacement method. The condition equations, in the general displacement method, express the equilibrium of the structure’s nodes and have the matrix representation: RxZ+P=0
(4.17)
where we have denoted: R – the (global) rigidity matrix of the entire structure; Z – the vector of the unknown factors (displacements), formed by the succession of vectors Zk, and which includes the “unknown” displacements (generalized displacements – translations or rotations) of node k; P – the vector of the equivalent effects (reduced to the nodes) of the external actions and loads, formed by the succession of vectors Pk, includes all the components of the effect of the external actions and loads acting on the elements of the structure that are concurrent in node k; The composition of rigidity matrix R of the structure and vectors Z and P is visible in the relation which is at the origin of relation (4.17) and which expresses the equilibrium of the structure’s nodes. The resultant stresses from the ends of all the bars, which meet in a node must be equivalent to the resultant of the effect of the external loads reduced to that node:
¦E i
k
Pk
k
where:
E k i
rkk Z k rkm Z m
(4.18)
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Materials with Rheological Properties
From (4.17) and (4.18) it results that the rigidity matrix of the entire structure is composed as follows: – the sub-matrix:
¦r
Rkk
kk
– the sub-matrix:
Rkm
¦r
(4.19)
km
The values of the matrices’ elements and vectors from relations (4.17), (4.18) and (4.19) are given in the structure’s general system of axes. Between these values and the values expressed in the bar’s system are the following relations:
Ek
A0 E k
Zk
A0 Z k
r i
A T r i A
(4.19’)
where we have used: symbols without bars for the values expressed in the general system of axes and symbols with bars for the values expressed in the bar’s system of axes. In relation (4.19), we have denoted:
A0
ªO xx « «O yx « O zx ¬
O xy O yy O zy
O xz º » O yz » O zz »¼
(4.20)
the vector-based charge matrix (rotation) from the structure’s general system of axes to the bar’s system of axes. The line elements of this matrix are cosine directors of each axis
A
X , Y , Z , determined in the structure’s general system of axes. ª A0 «0 « «0 « ¬0
0 A0 0 0
0 0 A0 0
0º 0 »» 0» » A0 ¼
(4.20’)
Applications on Resistance Structures for Constructions
163
If we take into account the above considerations, the total stresses in the end of a bar are:
Ek ½ [ k ½ ª r kk ® ¾ ® ¾ « ¯ Em ¿ ¯[ m ¿ ¬r mk
ª A0 « r km º « 0 » r mm ¼ « 0 « ¬0
0
0
A0
0
0
A0
0
0
0º 0 »» Z k ½ ® ¾ 0 » ¯Z m ¿ » A0 ¼
(4.21)
where we have denoted [(i) the vector of the effect of external actions and loads on bar i obtained by reducing external actions and loads to the bar nodes. Before moving towards the calculation of the composite structures, it is necessary to make a comment concerning the rigidity matrix of a specific element of the composite structures, namely the shear connection element (connector). By hypothesis, the connection elements allow relative displacements of the elements they connect, displacements that are carried out by shearing the elements in a section placed in their separation plan. The force transmitted by the shear connectors is proportional to the relative displacement of the section of the connection elements which are adjacent to the shearing sections. If we consider the elastic constant c of a shear connector (c – the force that causes a relative unit displacement of the sections adjacent to the shearing section of the shear connectors), the rigidity matrix is:
rO
0 ª R « 0 c « « 0 0 « 0 « 0 « 0 0 « « 0 caz « R 0 « c « 0 « 0 0 « « 0 0 « 0 0 « «¬ 0 cbz
0 0
0 0
0 0
0 ca z
R 0
0 c
0 0
0 0
0 0
c 0
0
ca y 0
0 0
0 0
0 0
c 0
0
cby 0
ca y
0
ca 2y
0
0
0
0 0
0 0
0 0
caz2 0
0 R
caz 0
ca y 0 0
caz 0
0 0
c 0
0 c
0 0
0 0
0 0
caz bz
0
cbz
0 0 0 c 0 ca y 0 0 cby 0 ca y by 0
0
0
0 cby 0
0 ca y by 0 0
0 0
0 0
0
0
0
cby 0 0 cby2
0 º cb z »» 0 » » 0 » 0 » » cazbz » 0 » » cbz » 0 » » 0 » » 0 » cbz2 »¼
164
Materials with Rheological Properties
Values R and of the elements of the rigidity matrix of the connection elements, expressing their rigidity to an axial load and to torsion (around the x axis), do not depend on the single deformability parameter c, assigned by hypothesis to the connection elements, and, therefore, they are not specified. However, by hypothesis, we admit that, in these directions, the relative displacements between the connected elements of the structure are zero:
'X i 'X j
0
'T xi 'T xj
0
Therefore: – at bar end i:
Rxi M xi
'X i 'X j 0
'T xi 'T xj 0
– at bar end j:
Rxj M xj
'X i 'X j 0
'T xi 'T xj 0
This result is obtained for any finite value of elements R and . Therefore, by assigning value 0 (zero) to the respective elements, we introduce errors neither in the evaluation of the structure’s rigidity, nor in the calculation of its stresses and strains state (if, by writing the equations of condition, we take into account the hypothesis that we have mentioned).
Applications on Resistance Structures for Constructions
165
We obtain:
rO
0 ª0 «0 c « «0 0 « 0 «0 «0 0 « 0 ca « z «0 0 « «0 c «0 0 « «0 0 « 0 «0 «¬0 cbz
0 0 c 0
0 0 0 0
ca y 0 0
0 0 0
0 c 0 cby
0 0 0
0
0 0 ca y 0 ca y2
0 ca z 0 0 0 caz2
0 0
0
0 ca y 0 0 ca y by 0 0
caz 0 0 0 caz bz
0 0 0 c 0 0 0 0 0 0 0 caz 0 0 0 0 0
0 0 c 0 ca y
c 0 0
0 0 0 cbz
0 0 0 0
0 0 cby 0
0 0
0 ca y by 0 0 0 0
0 c 0
0 0 0 cby 0 0
cby 0
cby2 0
0 0
0 º cbz »» 0 » » 0 » (4.22) 0 » » caz bz » 0 » » cbz » 0 » » 0 » » 0 » cbz2 »¼
In this way, the possibility of rigid body-like displacement of connectors (at the same time as the deformation of the entire structure) is not prevented.
4.3.2. Composite structures with discrete collaboration The significance of the sizes composing equations (3.53) and (3.53’):
¦ ^>R Z W P @K W K W K T W m
j
j
0j
j
1j
j
2j
j 1
T
³ Rj W
>
@
d >Z W @K 0 j W j 1 K1 j t W j K 2 j T t dt` 0 dt
and, respectively:
¦ ^>R Z W P @K W K W >K T W K T W @ m
j
j
0j
j
1j
j
2j
2j
j 1
T
³ Rj W
T
³ Rj T
>
@
d >Z t @K 0 j t W j K1 j t W j K 2 j T t K 2 j T t dt dt
>
@
d >Z t @K 0 j t W j 1 K1 j t W j K 2 j T t dt` 0, dt
166
Materials with Rheological Properties
as well as the significance of the sizes from equations (4.17) indicates, by analogy, the way to systematize the composition of the matrices and the vectors defining the systems of integro-differential equations (3.53) and (3.53’). The systems of integrodifferential equations (3.53) and (3.53’) model the evolution in time of the stresses and strains state of the composite structures and, implicitly, of the differential equation systems (3.54) and (3.55), whose solutions approximate the solutions from the discussion about the integro-differential equation systems. Therefore, by considering (4.18), there results:
* jkk
¦r
jkk
– the sub-matrix placed on the leading diagonal of partial rigidity
matrix *j (corresponding to the elements of the structure made from materials having rheological properties j); the sum of the sub-matrices corresponding to node k of all the structure’s elements having rheological properties j and an end in node k;
* jkm
¦r
jkm
– the sub-matrix of matrix *j placed outside the leading
diagonal (on the row corresponding to node k and on the column corresponding to node m) corresponding to the element of the structure having properties j and connecting nodes k and m; P*jk – the vector composing vector Pj that includes the resultant components of the stresses in node k obtained by reducing to the node the external actions and loads that stress the elements having rheological properties j that join this node; Z*k – the vector including node k displacement components. In the composition of matrices j and vectors P and Z, the supports will be considered as nodes of the structure. In this context, we must note the composition mode of the partial rigidity matrix of the structure, corresponding to the supports that can settle. We consider the support having rheological properties j, j (1, 2, ..., m), placed in node k of the structure. The rigidity matrix of the support that can settle from node k has the expression (for structures in space):
rjkk
ª U1 « « « « « « « ¬«
U2 U3
º » » » » K 0 j t 1 K1 j t K 2 j t » » » U 6 ¼»
>
U4 U5
@
(4.23)
Applications on Resistance Structures for Constructions
167
where Ui, i (1, 2,..., 6), has values corresponding to the physico-mechanical characteristics of the support (according to the dimensions, nature and quality of material, etc.). For the directions on which the displacement of the node is free (it is not restrained by the existence of the support), the corresponding elements Ui, i (1, 2,... 6) have the value zero. The value zero is also assigned to elements Ui for the directions on which the support does not allow displacements (it is absolutely rigid – see the observation in the previous section, concerning the shear connection elements). Partial rigidity matrix j, corresponding to the supports having rheological properties j, is composed of sub-matrices:
jkk
r jkk
distributed on the leading diagonal, corresponding to the respective support nodes. The sign of the elements from partial rigidity matrix j, corresponding to the supports that can settle, is obtained from the equilibrium of the respective nodes and signifies a reduction of the structure’s rigidity, which corresponds to the reality. Matrices Rj and, respectively, vectors Pj and Z entering in the composition of integro-differential equation systems (3.53) and (3.53’) are obtained by using and rearranging matrices Rj and, respectively, vectors P* and Z* (which also implies a reduction of the dimensions) as follows: – we add rows and then columns corresponding to the unknown factors that are equal by hypothesis (rotations and displacements – in the axial direction of the connection elements – of the nodes bound by the connection elements). The result of each of these operations is registered in one of two rows and, respectively, one of the columns; the others (the row and the column) are eliminated. In this way, the dimension of the matrices is reduced with the number of eliminated rows and columns. This operation implies the addition of the elements of the Pj* vectors corresponding to the rows that are added in matrix Rj*; the result is registered in the place corresponding to the retained columns of Rj and the other element (the even element) is eliminated. In vector Z*, we will operate only the elimination of the element corresponding to that which is also eliminated from the Pj* vectors; – we eliminate the rows and columns corresponding to the displacements that are completely restrained by the supports (supports that cannot settle), according to the case. APPLICATION 5.– We need to determine the rigidity matrices r of the elements of the structure in Figure 4.10 and the table of the unknown factors which are equal by
168
Materials with Rheological Properties
hypothesis; the table is necessary for the rearrangement of matrices Rj* and vectors Pj* and Z*. The geometric elements of the structure are given in Figure 4.10a. y
cgcs cgss
z O
O
5O
a. Composite structure with discrete collaboration 1
3
2
4
5
7
9
6
8
10 11
7 8
9 10
b. Basic system ec=a+b=e
O
c. Basic system
1 2
ec=0
3 4
O 5O
5 6 O
11
O 5O
Figure 4.10. Composite structure with discrete collaboration; basic systems
The physico-mechanical characteristics of the structure’s components are: – the rigidity of the supports (identical rheological properties):
U1
0 ½ ° ° ® U v1 ¾; °0 ° ¯ ¿
U11
UV 11 ½ ° ° ® U H 11 ¾; °U ° ¯ T 11 ¿
– the metal section (constant) E, G, A, I; – the concrete section (constant) Eb, Gb, Ab, Ib; – the elastic characteristic of the shear connectors (identical, distributed at distance O), c; – the rigidity matrices of the elements of the structure (the plane structure) directly solicited by the discrete unknown factors:
Applications on Resistance Structures for Constructions
- the simple support (node 2):
rr 22
ªUr 2 « 0 « «¬ 0
0 0º 0 0»» 0 0»¼
- the fixed support (node 11):
rr11
ª UV 11 « 0 « «¬ 0
0 º 0 »» UT 11 »¼
0
U 0 H 11
- the metal beam element (of equal length, O):
A0m
rm
>I @ ; ª EA « O « « 0 « « 0 « « EA « « O « 0 « « « 0 ¬
>I@ ;
T 0
12 EI
O3
6 EI
4 EI
O2
O
0
0
12 EI
6 EI
O3
O2
EA
2 EI
O2
O
0
12 EI
O3
6 EI
0
O2
EA
6 EI
º » 6 EI » 2 » O » 2 EI » O » » 0 » » 6 EI » O2 » 4 EI » » O ¼
0
O 0
O2
6 EI
0
0
O
12 EI
0
O3
6 EI
0
O2
- the current concrete element (of equal length, O):
A0b
rb
>I@ ; ª Eb Ab « O « « 0 « « 0 « « Eb Ab « O « « 0 « « « 0 ¬
>I@ ;
T
0 12 Eb I b
O3
6 Eb I b
O2
Eb Ab
0
O
0
12 Eb I b
O3
4 Eb I b
0
6 Eb I b
0
0
Eb Ab
0
12 Eb I b
6 Eb I b
0
12 Eb I b
6 Eb I b
2 Eb I b
0
6 Eb I b
6 Eb I b
0
O2
O3
O2
O
O2 O
O
O2
O3
O2
º » 6 Eb I b » 2 » O » 2 Eb I b » O » » 0 » » 6 Eb I b » O2 » 4 Eb I b » O »¼ 0
169
170
Materials with Rheological Properties
- the concrete element having a particular 9 – 11 form (of equal length, O):
A0b
>I@ ; ª1 «0 « «0 « «0 «0 « ¬«0
T
0 0 0 0 0º 1 0 0 0 0»» 0 1 0 0 0» » 0 0 1 0 e» 0 0 0 1 0» » 0 0 0 0 1¼»
This results in:
rb 9 11
ª E b Ab « O « « 0 « « 0 « « « E b Ab « O « 0 « « « eE b Ab « O ¬
T T rb T
0 12 E b I b
O3
6Eb I b
0
6Eb I b
O2
4 Eb I b
O2
O
0
0
12 E b I b
6Eb I b
O3
O2
6Eb I b
O2
2 Eb I b
O
– the shear connectors:
A0c
ª0 1 0 º «1 0 0» » « «¬0 0 1»¼
Therefore:
rc
ª c « 0 « « cb « « c « 0 « «¬ ca
0 0
cb 0
0 cb 2 0 0
cb 0
0
cb
c 0 ca º 0 0 0 »» cb 0 cab » » c 0 ca » 0 0 0 » » ca 0 ca 2 »¼
E b Ab
0
O
0 0 E b Ab
O
0 eE b Ab
O
12 E B I b
O3
6Eb I b
O2 0 12 E b I b
O3
6Eb I b
O2
º » » 6Eb I b » 2 » O » 2 Eb I b » O » eE b I b » » O » 6Eb I b » O2 » 2 e E b Ab 4 E b I b » » O O ¼
eE b Ab
O
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The vector of the loads, P*: For a uniform load q distributed on a doubly fixed beam i – j having the span O:
Vi i
1 qO 2
Vj j
1 2 qO 12
we obtain:
P1
½ ° 0 ° °° 1 °° qO ¾;Pk ® ° 2 ° ° 1 qO2 ° °¯ 12 °¿
0½ ° ° ®qO ¾,k °0° ¯ ¿
3,5,7,9;Pk
0½ ° ° ®0¾,k °0° ¯ ¿
2,4,6,8;P10
½ ° 0 ° °° 1 °° ® qO ¾ ° 2 ° ° 1 qO2 ° °¯12 °¿
The equal unknown factors by hypothesis: – the displacements of node i are:
'i
'X i ½ ° ° ® 'Yi ¾ °T ° ¯ Zi ¿
Z 3 i1 1 ½ ° ° ®Z 3 i 1 2 ¾ °Z ° ¯ 3 i 1 3 ¿
Zi
By hypothesis:
'Yi
'Yi 1;
T Zi T Zi 1, i 1,3,5,7,9 which results in:
Z 3i1 Z 3i
Z 3i 2 ; Z 3i 3 , i 1,3,5,7,9
In Figure 4.10c, we present a particular alternative to the structure’s basic system: the nodes that are bound by the connection elements (connectors) (1, 2; 3, 4; 5, 6; 7, 8; 9, 10) merge from the geometrical point of view, and are separated by the
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Materials with Rheological Properties
surface between the concrete and the higher base of the metal beam. The rigidity matrices of the steel and concrete elements are obtained using the relations:
rm
TmT rm Tm ;
rb
TbT rb Tb ;
rc
rc a
0,b
0
The other calculation elements are identical. This alternative has the advantage of being more suggestive with regard to the way the hypothesis of the equality of the vertical displacements (axial direction of the connection elements) and rotation of the nodes bound by the connection elements influence the behavior of the real structure. Indeed, the nodes that form a pair (the nodes bound by connection elements: 1, 2; 3, 4; 5, 6; 7, 8; 9, 10) cannot carry out an independent vertical rotation and displacement (in the axial direction of the connection elements) and, thus, for these displacements, each of these pairs behaves as a unique node. This fact is reflected in the mathematical model by the addition operations of the columns and the rows corresponding to respective displacements, which are made by rearranging matrices Rj, as well as vectors Pj and Z.
4.3.3. Composite structures with continuous collaboration The systematization of the composition of the matrices is made by taking into account the general considerations included in section 4.3.1, as well as the note on the basic system of Figure 4.10c (Application 5) from the previous section. The matrices in question are those which play a role in the integro-differential equation systems (3.58) and (3.58’):
¦ ^>R r Z r ,W P @K W K W K T W m
j
j
0j
j
1j
j
2j
j 1
T
³ R j W W
T
Rc r ³ W
w >Z r ,W @K 0 j t W j >1 K1 j t W j K 2 j t W j @dt` wt
w3 >Z r , t @dt w r 2w t
0
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173
and, respectively:
¦ ^>R W Z r ,W P @K W K W >K T W K T W @ m
j
j
0j
j
1j
j
2j
2j
j 1
T
³ R j r W
W
³ R j r T
w >Z r , t @K 0 j t W j K1 j t W j >K 2 j T t K 2 j T t @dt wt w >Z r , t @K 0 j t W j >1 K1 j t W j K 2 j T t @dt` wt
w3 >Z r , t @dt 0 w r 2w t T T
Rc r ³
and, implicitly, of those involved in the composition of differential equation systems (3.59) and (3.59’), whose solutions approximate the solutions of integro-differential equations (3.58) and (3.58’) describing the stresses and strains state’s evolution in time of the composite structures with continuous collaboration. The response of a differential element, homogenous from the point of view of its elastic and rheological properties, when we impose on it a field of specific deformations, is illustrated by the expression:
Ex ½ °E ° ° y° °° E z °° ® ¾ °M x ° °M y ° ° ° °¯ M z °¿
ª EA « GA « Fy « « « « « « « ¬«
GA
Fz
GI t GI y
º » Hx ½ » °J ° » ° xy ° » °°J xz °° »® ¾ » ° Ix ° » °Z y ° » ° ° » ¯°Z z ¿° EI ¼»
or
E
r H
(4.24)
where r is the rigidity matrix of the differential element in its own system of axes (the abscissa is directed in the element’s length direction, the other axes being the central and principal axes of the element’s section).
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Materials with Rheological Properties
If the connection of the element to the nodes of the structure is eccentric, we compose the correction (transformation) matrix T, corresponding to the real situation and we determine the rigidity matrix corresponding to the situation:
r T T r T
(4.24’)
In the general system of coordinates of the structure, the rigidity matrix of the element is determined using the known expression:
r
AT r A
By mentioning that, in the case of the composite structures with continuous collaboration, the rigidity of the connection elements (connectors) c (KN/m2) represents the ratio between the reaction of the connection elements to an imposed unit length deformation and the length on which the connection elements are distributed, thus:
c
lim O o0
R G
O
1
Considering that in the case of the basic system having merged pairs of nodes (separated by the surface that separates the two materials bound by connection elements), the length of the connection elements is zero and, thus, the segments a and b in Figure 4.11 and, respectively, in relation (4.22) are also zero, we obtain:
rc
ªc «0 « «0 « «0 «0 « «0 « c « «0 «0 « «0 « «0 «¬ 0
0 0 0 0 0 c 0 0 0 0 0º 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 0 0 0» » 0 0 0 0 0 0 0 0 0 0 0» 0 0 0 0 0 c 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 0 0 0 0 0 0 0 0» 0 0 0 0 0 0 0 0 0 0 0»¼
(4.22’)
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175
We have anticipated the configuration of the basic system above. For reasons related to the control of the calculation and to the simplification of the expressions, we prefer, in what follows, basic system having pairs of merged nodes. By “pairs of (merged) nodes”, we understand zones made of materials having different rheological properties, bound by connection elements belonging to the separation surface between them. Therefore, we evaluate separately the contribution of each component to the rigidity of the structure. To re-establish the real operation of the structure we have to total up the partial rigidity (the contribution of each element of the section considered separately) and to impose the compatibility conditions of the deformations (to reorganize and transform the rigidity matrix of the structure by taking into account the equal values of certain deformations). We can observe that the contribution of the connection elements (connectors) to the rigidity (deformability) and, implicitly, to the distribution of the stresses among the elements of the structure, involves only the second order variation of the deformations that cause their shearing. Therefore, the partial matrix corresponding to the connection elements cannot be added to those corresponding to the elements that make up the section of the structure. From this it results that:
jkk
r jkk – the sub-matrix distributed on the leading diagonal of the rigidity
R*j,
matrix corresponding to the element made of materials having rheological properties j, thus assembling the sub-matrix corresponding to node k of the structure’s element having an end in k and rheological properties j;
ckk
rckk – the sub-matrix distributed on the leading diagonal of the rigidity matrix R*c, corresponding to the connection elements (connectors), assembling the sub-matrix corresponding to the node k of the rigidity matrix rc of the connection elements that have an end in the node k; ckm
rckm – the sub-matrix of matrix R*c, placed outside the leading diagonal,
on the row corresponding to node k and on the column corresponding to node m, formed by the sub-matrix corresponding to the connection element (connector) that connects nodes k and m;
Pk r – the vector whose elements represent the components of the resultant effects of the external actions and loads in node k (section r);
Z k r , t – the vector whose elements are unknown (generalized) displacements of nodes k.
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Materials with Rheological Properties
Matrices Rj, Rc and vectors P and Z are obtained by the transformation (described above) of matrices R*j, R*c and, respectively, of vectors P* and Z*, by considering the accepted hypothesis concerning the behavior of the composite structures with continuous collaboration. APPLICATION 6.– We need to determine rigidity matrices Rj, Rc and vectors P and Z corresponding to the structure in Figure 4.11. We consider that the structure has a constant section over all its length and the geometric elements are given. The physico-mechanical characteristics of the elements composing the structure’s section are: – concrete section 1: Eb1, Gb1, Ab1, Ib1; – concrete section 2: Eb2, Gb2, Ab2, Ib2; – metal section: E, G, A, I; – elastic characteristic of the connection elements (connectors) (constant over the entire length of the structure): c12, c23. The rigidity matrices of the elements of the structure (the plane structure): – concrete section 1 (ey = e1, A = [I]):
ª Eb1 Ab1 « 0 ¬
Tb1
ª1 e1 º «0 1 »; ¬ ¼
rb1
rb1
TbT1 rb1Tb1
ª Eb1 Ab1 «e E A ¬ 1 b1 b1
0 º ; Eb1 I b1 »¼
e1 Eb1 Ab1 Eb1 e22 Ab 2 I b 2
º »; ¼
z
1 2
1
1 2 3
2
2
cgcs 1 cgss 2 cgss
1
y
3 2 x
Figure 4.11. Composite structure with continuous collaboration; cross-section of the structure and basic system
1
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177
– concrete section 2 (ey = e2, A = [I]):
Tb 2
ª1 e 2 º «0 1 »; ¬ ¼
rb 2
TbT2 rb 2Tb 2
rb 2
ª E b 2 Ab 2 « 0 ¬
ª E b 2 Ab 2 «e E A ¬ 2 be b 2
º ; E b 2 I b 2 »¼ 0
e2 E b 2 Ab 2
E b 2 e Ab 2 I b 2 2 2
º »; ¼
– metal section (ey = 0 , A = [I]):
ª1 0º «0 1» ; r ¬ ¼
T
ª EA 0 º « 0 EI » ; T ¬ ¼
ª EA 0 º « 0 EI » ; ¬ ¼
TT r T
– shear connectors. If we adopt, for this particular case, the rigidity matrix of the connection elements obtained within the framework of Application 5, we obtain the following expressions for the rigidity matrices of the connection elements between parts (bodies) 1–2 and 2–3:
rc12
c12 º , c12 »¼
ª c12 « c ¬ 12
ª c23 « c ¬ 23
rc 23
c23 º c23 »¼
This results in: – rigidity matrices, R*j:
R
ªrb1 0 0º « 0 0 0»; R 2 « » «¬ 0 0 0»¼
Rc
ª c12 « 0 « « c12 « « 0 « 0 « ¬« 0
1
0 0
c12 0
0 c12 c23 0 0 0 0
c23 0
ª0 0 «0 r b2 « «¬0 0 0 0
0º 0»»; R3 0»¼ 0 0
0 c23 0 0 0 0
c23 0
0º 0»» 0» » 0» 0» » 0¼»
ª0 0 0º «0 0 0»; « » «¬0 0 r »¼
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Materials with Rheological Properties
– the vector of the loads, P*:
P
>K x , x , K x , x , K x , x @ 1
1
2
2
3
3
T
;
– the vector of the unknown factors, Z*:
Z
>H x , T x , H x , T x , H x , T x @ 1
1
2
2
3
3
T
;
– the unknown factors having, by the hypothesis, equal values:
T1 x T 2 x T 3 x T x ; Transforming matrices R*j, R*c and vectors P* and Z* by taking into account the equal values of unknown factors ș(x), we obtain:
R1
R3
P
º » »; R 2 » 0 0 0 » 0 0 Eb1 e12 Ab1 I b1 ¼ 0 0º c12 ª c12 » « 0 0» c c c ; Rc « 12 12 23 « 0 EA 0 » c23 » « 0 EI ¼ 0 0 ¬
ª Eb1 Ab1 « 0 « « 0 « ¬e1 Eb1 Ab1
ª0 «0 « «0 « ¬0
0 0 0 0
0 0 0 0
e1 Eb1 Ab1 0
>K1 x ,K 2 x ,K3 x ,M 0 x @ ;
0 ª0 «0 E A b2 b2 « «0 0 « ¬0 e2 Eb 2 Ab 2
0 c23 c23 0
0 0 0
0 e2 E b 2 Ab 2
0
0 Eb 2 e Ab 2 I b 2 2 2
º » »; » » ¼
0º 0»» ; 0» » 0¼
T
where:
K1 x K2 x K3 x N 0 x and, respectively:
1 x 2 x 3 x M 0 x where: N0(x) and M0(x) are the values of the sectional stresses (the axial force and, respectively, the bending moment in section x) determined on the external statically determined structure; K(x) and F(x), (i = 1, 2, 3) are determined, for the initial phase, by using the formulation of the problem according to the creep.
Applications on Resistance Structures for Constructions
179
>[1 x ,[ 2 x ,[3 x ,T x @T ;
Z
If, between parts (bodies) i and k of the section, the collaboration is perfect:
d2 >[i x [k x @ 0 dx 2 cik o f , and d2 >[i x [ x x @ 0 dx 2
cik
This result is obtained by substituting into the matrix: Rc, cik = 0 and it constitutes the manner of systematic realization of the rigidity matrix Rc* and, respectively, Rc of the structure. 4.3.4. Composite structures with complex composition
The significance of the matrices and vectors’ elements involved in the composition of the integro-differential equation systems that describe the evolution of the stresses and strains state of the composite structures with complex composition (see section 3.3.4), on one hand, and define the integro-differential equation systems (3.53), (3.53’) and (3.58), (3.58’) that describe the behavior in time of the composite structures with discrete collaboration and, respectively, with continuous collaboration, on the other hand and the results obtained in sections 4.3.2 and 4.3.3 concerning the systematized composition of the matrices and vectors, allow us to write equations (3.64) (3.64’) in the following form: m
¦ j 1
½ ªl º °ª R j º «³ j s [ s,W ds» Pj ½°K W K W K T W Z W ®« T » «0 » ®q j s ¾¾ 0 j j 1 j j 2 j ¯ ¿° °¬ j s ¼ , s s U [ W « j ¬ ¼» ¯ ¿
ªl º½ w j s >[ s, t @ds» ° °ª R º « ³ d ° j wt » °¾K 0 j t W j u ³ ®« T » >Z t @ « 0 s w dT « »° ¼ W °¬ j « U j s w t >[ s, t @ » ° °¯ ¬ ¼¿ T
I º Tª »dt 0 w3 u 1 K1 j t W j K 2 j T t dt ³ « « Uc s 2 >[ s, t @» W wr wt ¬ ¼
>
@
(4.25)
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Materials with Rheological Properties
and, respectively: ½ º ªl j s [ s,W ds » Pj ½° °ª R j º « ³ u ®« T » Z W « 0 ¦ » ®q j s ¾¾ j 1 °¬ j s ¼ ¯ ¿° s U » « j ¼ ¬ ¿ ¯ u K 0 j W j K1 j W j K 2 j T W K 2 j T W m
>
@
º½ ª w j s >[ s, t @ds » ° °ª R º « ³ d ° j wt » °¾ K 0 j t W j K1 j t W j u ³ ®« T » >Z t @ « 0 s w dt »° « j ¼ W °¬ « U j s w t >[ s, t @ » ° °¯ ¼¿ ¬ º½ ªl w T ° « ³ j s >[ s, t @ds » °° °ª R j º d t w »¾ u u K 2 j T t K 2 j T t dt ³ ®« T » >Z t @ « 0 s ¼ dt w »° « T °¬ j « U j s w t >[ s, t @ » ° ¼¿ ¬ ¯° l
T
>
@
I º T ª » dt w3 u K 0 j t W j 1 K1 j t W j K 2 j T t dt ³ « s 2 >[ s, t @» U « c T wr wt ¬ ¼
>
@
(4.25’) 0
In relations (4.25) and (4.25’), we have denoted:
j – the rigidity matrix corresponding to the discrete unknown factors, thus to the structure’s elements with rheological properties j and finite dimensions, homogenous from the point of view of the rheological properties (see section 4.3.2); Pj – the vector composed from the Pjk vectors, whose elements are the components corresponding to the discrete unknown factors’ directions of the resultant stresses obtained by reducing to node k the effects of the external actions and loads on the structure’s elements having rheological properties j and finite dimensions (see section 4.3.2); Z(t) – the vector whose components are vectors including the components of the unknown factors of the discrete displacements of the structure (see section 4.3.2); Uj(s) – the rigidity matrix corresponding to the distributed unknown factors, to the section’s parts having rheological properties j, of infinitesimal length, homogenous from the point of view of the rheological properties (see section 4.3.3); Uc(s) – the rigidity matrix corresponding to the shear connectors and respectively to the distributed unknown factors composing the resistance structure’s elements (composite structure with continuous collaboration) (see section 4.3.3); qj(s) – the vector having as elements the vectors of the functions whose values in section s represent the components, in the direction of the distributed unknown
Applications on Resistance Structures for Constructions
181
factors of the stresses in the structure’s section element having rheological properties j due to the external actions and loads (see section 4.3.3); [(s,t) – the vector whose elements are vectors of the functions whose values in section s represent the components of the distributed unknown factors (see section 4.3.3);
j(s) – the matrix (of the influence functions) whose elements jki(s), distributed on the row corresponding to the discrete unknown factor i, in the column corresponding to the distributed unknown factor k, are functions whose values in section s represent the reaction along the direction of the discrete unknown factor i due to a unit displacement in the direction of the distributed unknown factor k, applied in section s. Obviously, we consider only the values corresponding to the section parts having rheological properties j, homogenous from the point of view of the rheological properties. Matrices Rj, Pj and Z(t), corresponding to the discrete unknown factors, are obtained as described in section 4.3.2, using only the rigidity characteristics of the finite size elements, homogenous from the point of view of the rheological properties, entering in the composition of the structure. Also, matrices Uj(s), Uc(s), qj(s), [(s,t), corresponding to the distributed unknown factors, thus to the elements of the composite structure with continuous collaboration of the composite structure with complex composition, are obtained as described in section 4.3.3. Matrices j(s), involved only in the analysis of the composite structure with complex composition are obtained by carrying out, on the initial matrices j*(s), the operations corresponding to the transformation of matrices Rj*, Pj*, Z*(t) and Uj*(s), pc*(s), q*(s), [*(s,t), into matrices Rj, Pj, Z(t) and, respectively, Uj (s), pc(s), q(s), [(s,t). In this context, we notice that matrices q*(s) and j*(s) are obtained by solving the respective cases of loads of the composite structures with continuous collaboration (that constitute elements of the composite structure with complex collaboration), by using the formulation of the problem according to the creep. APPLICATION 7.– We need to determine matrices Rj, P, Z, Uj(s), Uc(s), q(s), j(s) and [(s,t), corresponding to the structure in Figure 4.8 from Application 4. Additionally, we consider that the upper flange (concrete) of the composite (sub)structure with continuous collaboration is anchored to the support by a connector having the characteristic of rigidity c (Figure 4.12).
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Materials with Rheological Properties
b
1
2
a
3 Figure 4.12. Composite structures with complex composition; basic system
The matrices and vectors corresponding to the distributed unknown factors: – the concrete section: ey = e; E = Eb; A = Ab; I = Ib; A0b = [I];
Tb
ª1 eº «0 1»; ¬ ¼
rb
A0Tb TbT rb Tb A0b
ª Eb Ab «eE A ¬ b b
º Eb e Ab I b »¼ eEb Ab
2
– the steel section: ey = e; E = E0; A = A0; I = I0; A00 = [I];
T0
ª1 0º «0 1»;r0 ¬ ¼
T A00 T0T r0 T0 A00
ª E0 A0 « 0 ¬
– the elastic connectors:
rcb 0
ªc «0 « « c « ¬0
0 c 0º 0 0 0»» ; 0 c 0» » 0 0 0¼
– the vector of the loads:
q s
>nb s ,mb s ,n0 s ,m0 s @T ;
– the vector of the unknown factors:
[ s, t
>H b s, t ,Tb s, t ,H 0 s, t ,T 0 s, t @T ;
0 º ; E0 I 0 »¼
Applications on Resistance Structures for Constructions
– the rigidity matrices
ª Eb Ab «eE A « b b « 0 « ¬ 0
U b s
U cb 0 s
ªc « « « c « ¬0
U j s : eEb Ab
Eb eAb I b 0 0
0 0º 0 0»» ; 0 0» » 0 0¼
U 0 s
ª0 «0 « «0 « ¬0
0
0
0
0
0 E0 A0 0
0
0 º 0 »» ; 0 » » E0 I 0 ¼
0 c 0º 0 0 0»» ; 0 c 0» » 0 0 0¼
Using the hypothesis: Tb(s,t) = T0(s,t) = T(s,t) We obtain:
ª Eb Ab 0 0 U b s «« 0 «¬eEb Ab 0 ª c c U cb 0 s «« c c «¬ 0 0
º »; 0 » Eb eAb I b »¼
eEb Ab
0º 0»» 0»¼
>nb s ,n0 s ,m s @T
q s
[ s, t
where m s
U 0 s
0 ª0 «0 E A 0 0 « «¬0 0
0 º 0 »»; E0 I 0 »¼
mb s mo s
>H b s, t ,H 0 s, t ,T s, t @T
– the matrices and vectors corresponding to the discrete unknown factors: - the (simple) support of node 1:
r1
rv1
- the (fixed) support of node 2:
r2
rv1 0
0 rv 2
0
0
183
0 0 ; rT 2
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Materials with Rheological Properties
- the connector of nodes 2–3:
0 0 k
ªk « 0 « « 0 « « k « 0 « «¬ 0
rk 23
0 0 0 0
0 0
0 0
k
0 0 0 0
0 0
0 0º 0 0»» 0 0» »; 0 0» 0 0» » 0 0»¼
- the vector of the free terms:
>V1,H 2 ,V2 ,2 ,H 3 ,V3 ,3 @T ;
P
- the vector of the discrete unknown factors:
>'Y1 t , 'X 2 t , 'Y2 t ,T 2 t , 'X 3 t , 'Y3 t ,T3 t @T ;
Z t
- the rigidity matrices R*j:
1
R
ª rV 1 « 0 « « 0 « « 0 « 0 « « 0 « 0 ¬
0
0
0
rH 2 0 0
0
0
rV 2 0
0 rT 2
0
0
0
0
0
0
0
0
0
0 0 0º 0 0 0»» 0 0 0» » 0 0 0 », 0 0 0» » 0 0 0» 0 0 0»¼
Rk
ª0 0 «0 k « «0 0 « «0 0 «0 k « «0 0 «0 0 ¬
0 0
0 0
k
0 0
0
0 0
0
By hypothesis:
'Y2
'Y3
'Y and T 2
T3 T
We obtain:
R1
P
ª rV 1 « 0 « « 0 « « 0 «¬ 0
0
0
0
rH 2 0
0 0
0 0
0 0
0 rV 2 0 0
>V1,H 2 ,H 3 ,V ,M @
T
0 º 0 »» 0 »; » 0 » rT 2 »¼
where
Rk 23
ª0 0 «0 k « «0 k « «0 0 «¬0 0
0 0 k 0 k 0 0 0 0
0
0 0 k 0 0 0 0 0 0
0
º » » » » » »¼
0 0º 0 0»» 0 0» » 0 0»; 0 0» » 0 0» 0 0»¼
Applications on Resistance Structures for Constructions
V
V2 V3 and M
Z t
185
2 3 ;
>'Y1,'X 2 ,'X 3 ,'Y ,T @T ;
– the matrices of the influence functions j(s). Just as with the elements of the free terms’ vectors qj*(s), Pj* or, respectively, qj(s) and Pj, the elements of the matrices j*(s) and, respectively, j(s), are determined by the evaluation, in the initial phase, of the stresses state generated in the given structure by the successive loading of the structure’s basic system by unit displacements in the direction of the discrete unknown factors, using the formulation of the problem according to the creep. As we have mentioned in the application statement, by comparison with the structure described in Application 4 (Figure 4.8), the present structure (Figure 4.12) has a higher degree of complexity, due to the deformable connection of the concrete plate of the beam with the fixed support. Under these conditions, by using the same basic system as in Application 4 (Figure 4.9), the flexibility matrices of the structure’s elements are: – for the elements of the structure directly loaded by the discrete unknown factors: - the (simple) support:
f1
ª1º « r »; ¬ V1 ¼
- the (fixed) support:
f2
ª 1 « « rH 2 « 0 « « « 0 «¬
º 0 » » 0 »; » 1 » » rT 2 »¼
0 1 rV 2 0
- the metal part of the beam:
f3
ª 1 « 3E I 0 0 « « 1 «¬ 6 E0 I 0
1 º 6 E0 I 0 » » 1 » 3E0 I 0 »¼
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Materials with Rheological Properties
- the connection element of the concrete flange of the beam with support:
f4
ª1º «¬ k »¼
– for the elements of the structure directly loaded by the distributed unknown factors: - the metal beam:
I1
ª 1 «E A « 0 0 « 0 «¬
º 0 » » 1 » E0 I 0 »¼
- the (concrete) flange of the beam:
I2
ª 1 «E A « b b « 0 «¬
º 0 » » 1 » E b I b »¼
- the uniformly distributed connection elements (connectors):
I3
ª1 º «¬ c »¼
It results that the flexibility matrices of the structure’s elements:
F
ª f1 «0 « «0 « ¬0
0
0
f2
0
0
f3
0
0
0º 0 »» ; 0» » f4 ¼
º ªI1 »; « I2 « » «¬ I3 »¼
Applications on Resistance Structures for Constructions
– the transformation (balance) matrices: - the matrices corresponding to the discrete unknown factors, - the metal beam:
I1
ª 1 «E A « 0 0 « 0 «¬
º 0 » » 1 » E0 I 0 »¼
- the beam’s (concrete) flange:
I2
ª 1 «E A « b b « 0 «¬
º 0 » » 1 » Eb I b »¼
- the uniformly distributed connection elements (connectors):
I3
ª1 º «¬ c »¼
It results that the flexibility matrices of the structure’s elements:
F
ª f1 «0 « «0 « ¬0
0
0
f2 0 0
0 f3 0
0º 0 »» ; 0» » f4 ¼
ªI1 º « »; « I2 » «¬ I3 »¼
– the transformation (balance) matrices: - the matrices corresponding to the discrete unknown factors:
bx1
ª e « l « 1 « e « « l « e « 0 « « 0 « 0 ¬
1º » l 0» 1» » l »;b 1» x 2 0» » 0» 0 »¼
ª0 «0 « «0 « «0 «0 « «e «0 ¬
0º 0»» 0» » 0»;bx 3 0» » 1» 0»¼
ª0 «0 « «0 « «0 «0 « «0 «1 ¬
0º 0»» 0» » 0»;bx 0» » 0» 0»¼
¦b
xi
ª e « l « 1 « e « « l « e « 0 « « e « 1 ¬
1º » l 0» 1» » l » 1» 0» » 1» 0 »¼
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Materials with Rheological Properties
- the matrices corresponding to the distributed unknown factors:
ª1 «e « «0 « «0 «¬0
E y1
0º 1»» 0»;E y 2 » 0» 0»¼
ª0 0º «0 0» « » « 1 0 »;E y « » « 0 1» «¬ 0 0 »¼
ª1 0º «e 1» « » « 1 0 »;E O « » « 0 1» «¬ 0 0 »¼
ª0 «0 « «0 « «0 «¬1
0º 0»» 0»;E xy » 0» 0»¼
ª0 « ex «l «0 « «0 «0 ¬
0º x» l» 0 »; » 0» 0 »¼
- the matrices corresponding to the external actions and loads:
b0
ªV1 º «0» « » «V2 » « » « 0 »;G s «0» « » «0» «0» ¬ ¼
ª0º «0» « » «0» « » « 0 »; E 0 «T1 » « » «T 2 » «0» ¬ ¼
ª 0 º « M x » « 0 » « 0 »;H s « » « 0 » «¬ 0 »¼
ª0º «0» « » «0» « » «0» «¬0»¼
- the matrices corresponding to the loading of the structure by unit displacements, in the direction of the discrete unknown factors (of the basic system corresponding to the formulation of the problem according to the stress relaxation):
ª bxjT F b0 G s 1 yj 0 Hs
m
¦ «E E «¬ j
º Q K W »»¼ 0j
j
ª «1 « «0 « «0 «0 ¬
eº l» 1» » 1 l» 0 0» 0 0 »¼ e
Chapter 5
Numerical Application
5.1. Considerations concerning the validation of the mathematical model proposed for estimation through calculation of the behavior of the resistance structures by considering the rheological properties of the materials
The mathematical model of a physical phenomenon is accepted if, and only if, the values of the sizes (entered into and determined by the calculation) describing its evolution are confirmed by the observations carried out on the respective phenomenon in the natural world and/or reproduced in the laboratory. In this case, the observations carried out on the resistance structures for constructions submitted to long-term actions and loads revealed the redistribution of the stresses and strains phenomenon in time, according to the properties of the materials from which the elements of the structures are made and/or with which they are interacting. These observations have also imposed the study of the rheological properties of the building materials and foundation ground. The analysis of the experimental results carried out until now has allowed us to determine the constitutive laws of common building materials, on one hand and, on the other, the values of the parameters involved in the respective equations. Also, the in situ observations of the operating resistance structures, and research studies performed in laboratories, have allowed the development of empirical methods good enough to consider, when dimensioning, the effects of the rheological properties specific to every type of structure: prestressed concrete, reinforced concrete, composite steel and concrete structures, etc.
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The development of a general and precise mathematical model able to estimate, through calculation, the behavior of the resistance structures for constructions also taking into account the effects of the rheological properties of the materials, such as the one we are proposing in this work, represents, without any doubt, obvious progress. Indeed, this development fits in a broader sphere of concerns aiming to approach the physical nonlinear calculation of the structures while designing them assisted by the computer. As we already pointed out, the mathematical model of a physical phenomenon is poorer than the phenomenon itself. Therefore, in the case of integrating the mathematical models of certain “basic” physical phenomena in the mathematical model of a complex physical phenomenon, the neglected aspects, respectively the parameters that are ignored in the description of the evolution of the “basic” physical phenomena, can have a determining influence in the evolution of the complex phenomenon integrating them. From this observation results the need to confirm the proposed mathematical model. The validation of the mathematical model able to estimate, through calculation, the evolution of the stresses and strains state for the structures composed from elements having different rheological properties is essential, particularly if we consider the behavior in time of the resistance structures, their resistance, their stability and functionality throughout their entire operation. In order to carry out the huge computational load assumed by the use of the presented mathematical model, we worked out the algorithms, as well as a computer applications system capable of carrying out the calculation. The RALUCA computer applications system, developed based on the mathematical model proposed for the estimation by calculation of the behavior of the resistance structures by taking into account the rheological properties of the materials from which their elements are made allows us to perform, in a reasonable time interval, a large amount of numerical experiments and can be used with various purposes: – to validate the proposed mathematical model by comparing the results obtained through calculation with the observations made in situ on the real structures, or with the results obtained by using some simple calculation models, such as those accepted in the design specifications or presented in the specialized literature; – to check and/or to evaluate the projects for new constructions as well as for existing constructions; – to perform numerical experiments, aiming to evaluate the influence of the variation of parameters defining the rheological properties of materials on the resistance structures’ operational safety (resistance, stability, functionality).
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Certainly, the use of the proposed calculation model and the RALUCA computer applications system in the last two alternatives presented above can be considered only when the operations that validate the model, envisaged by the first alternative, gave satisfactory results. In this case, the use of the proposed calculation model imposes in designing the structures that, in order to fit in the existent calculation models, expensive or unaesthetic solutions, or those which harm the behavior of the structure in its operation are necessary. From the mathematical point of view, the RALUCA computer applications system is based on the formulation, according to the creep, of the equations describing the evolution of the stresses and strains state of the discrete collaboration and/or perfect continuous collaboration structures, as well as of the structures with complex composition, whose collaboration between the constitutive elements (made from materials having different rheological properties) is discrete and/or perfectly continuous. The modules allowing the solution of the discrete collaboration structures, externally statically determined, are operational, so that it is possible to carry out numerical experiments in order to validate the calculation model. When we chose the discrete collaboration structures in order to carry out the numerical experiments to validate the proposed calculation model, we kept in mind the existence of a great number of resistance structures implemented using solutions of this category, namely, composite steel and concrete structures. A large number of thorough observations and measures exist for composite steel and concrete structures, as well as a rich specialized literature confirming their behavior during operation, according to the specifications of the design and implementation standards. 5.2. The RALUCA computer applications system
The evolution in time of the stresses and strains state of the composite structures with discrete collaboration is described by integro-differential equation systems (4.8) and (4.8’). Among the possible methods for the construction of the solutions of the equation systems in section 3.2.1, we have chosen the Taylor series expansion method, approximating the statically indeterminate unknown functions. This method is presented in detail in Appendix 2. In this section, we will present only a few specific aspects of usage of the mathematical-numerical method of estimating, through calculation, the composite structures’ behavior, in general, and of estimating those with discrete collaboration, in particular.
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Integro-differential equation system (4.8) describes the behavior of composite structures with discrete collaboration submitted to actions and loads applied at moment W when the structure operates like a composite structure, although the implementation of the resistance structure is not yet completed. Integro-differential equation system (4.8’) describes the behavior of a composite structure with discrete collaboration at an unspecified moment ș, subsequent to the initial moment W (ș > W), submitted to forces equal to the known values at moment ș of the statically indeterminate unknown functions and supposing as known the analytical expressions of these functions in the time interval W – ș. It results that integro-differential equation system (4.8’) allows us to evaluate the behavior of the structure during its entire operation time, by considering the stages of its implementation and the supported loads. Therefore, it results that the use of the two integro-differential equation systems is necessary for the evaluation of the behavior of composite structures in general and for composite structures with discrete collaboration in particular. Integro-differential equation system (4.8) provides the first values of the statically indeterminate unknown functions and their derivatives of any necessary order, as well as, implicitly, their analytical expressions (approximations) at whatever moment și > W. In this way, it is possible to use integro-differential equation system (4.8’) for the evaluation of the structure’s behavior in the interval of time the structure is implemented and, later on, during the operation of the structure. In this context, it should be mentioned that the moments și dividing the stages of the structure’s implementation and operation of the structure must meet the following conditions: – ensure the evaluation of the structure’s behavior with the necessary precision; – correspond to the moments of the structure’s (technological) implementation and, thus, to the realization of the loads in the normal operation. The stresses and strains state and, respectively, the values of the unknown functions, can be simultaneously evaluated only for the loads that are applied simultaneously. For the loads applied at different moments, the stresses and strains state is evaluated separately and then, the values are superposed (they are added algebraically). The RALUCA computer applications system is developed based on the mathematical model presented before and has a modular structure, thus presenting several advantages: – it allows us to test (in all the possible situations) the correct operation of each module thoroughly and debug easily in the case of faults occurring during the tests;
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– it allows the introduction of modules for pre-processing of the input data and for post-processing of the results, so that data is introduced and, respectively, extracted in the format used by the current design, which facilitates access to the applications system’s usage. The currently operational modules of the RALUCA computer applications system allow us to calculate the evolution in time of the stresses and strains state for externally statically determined composite structures with discrete collaboration. In this context, we shall specify that by adding the corresponding modules, the applications system will be able to solve composite resistance structures with perfect continuous collaboration, both externally statically determined and externally statically indeterminate, as well as resistance structures with complex composition externally statically indeterminate, made up from composite structures with discrete and/or continuous perfect collaboration. In the following, we will briefly present the operational modules of the RALUCA computer applications system. RALUCA – the main program (module), calls the specialized modules of the computer applications system, according to the needs. The main operations directly performed or directed by the main module are: – processing of the problem’s input data and, then, processing and organization in the format imposed by its final use (within the framework of the computer applications system); – setting up the integration subintervals of the differential equation systems, according to the moments when the components of the structure become active and to the moments when the external actions and loads are applied, in order to obtain, with a minimum computing effort, the required precision in the evaluation of the statically indeterminate unknown functions. The integration interval of approximately 2,500 days, configured in the program, is sufficient compared to the approximately 1,600 days, resulting from the numerical experiments and which represent the extinction time of the phenomenon of redistribution of the stresses among the structure’s elements, due to the rheological properties of the materials from which they are made. This interval is evaluated in the specialized literature as between 3 and 6 years (1,100 – 2,200 days). Numerical experiments revealed the influence of the number of subintervals and particularly the major influence of the law of variation of the size of integration subintervals on the convergence and, respectively, on the precision of the values obtained for the statically indeterminate unknown functions. In the program an exponential law of variation of the size of integration subintervals was adopted,
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Materials with Rheological Properties
physically justified by the attenuation, in time, of the redistribution speed of the stresses among the elements of the composite structure. The size of the integration interval (2,500 days), the law of variation of the size of integration subintervals, their number (approximately 200) and, implicitly, their size, have resulted from the analysis of the accuracy of the results of the numerical experiments carried out on several structures. The tests that were carried out considered integration intervals of 5,000, 4,000 and 2,500 days, different laws of variation of the size of integration subintervals and a number of integration subintervals between 1,000 and 50. We have to mention that, the program controls the reach of the extinction level of the phenomenon of redistribution of the stresses among the structure’s elements and if after 2,500 days this level has not been reached the program will inform the user: – it calls the modules MFSVTL, VTLCUB, SEIDTLI and, according to the case, CSFII K or TLIAAC; – it processes and displays the obtained results so that the user can consult them. MFSVTL – the flexibility matrix of the structure; the vectors of the free terms – calculates and retains the elements of the structure’s flexibility matrices corresponding to each particular test and to each type of material so that they can be used in later calculations. In addition, it calculates and retains the elements of the vectors of the free terms corresponding to each load and to each type of material. By load, we understand the totality of the external actions and forces that are simultaneously applied. For judicious use of the computer memory and in order to reduce the computing effort, we calculate and retain only the elements of the leading diagonals of the flexibility matrices and the range containing non-zero elements, which is (situated) above the diagonal. In addition, for the same purposes, calculation of both the elements of the flexibility matrices and the elements of the vectors of the free terms is performed by successively calculating and using the flexibility matrices of the elements of the structure, the matrices corresponding to the equilibrium vectors of the statically indeterminate unknown functions and, respectively, to the external actions and loads. This module calls the VENECS and MEFLEX modules. VENECX – the equilibrium vector for the statically indeterminate unknown factor Xi. It calculates the elements of the equilibrium vector corresponding to the
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statically unspecified unknown functions Xi and arranges them to easily calculate the product with flexibility matrices of the elements of the structure (not assembled, see the assertion associated with the description of the MFSVTL module, concerning judicious memory usage). MEFLEX – the matrices of the product of the equilibrium vector with the flexibility matrices of the elements of the structure. It performs and retains the product between the equilibrium vector corresponding to the statically indeterminate unknown functions Xi and the flexibility matrices of the elements of the structure. VTLCUB – the vectors of the free terms of the concrete’s contraction or expansion. It calculates the elements of the free vectors corresponding to the action of contraction or expansion of concrete for all the integration subintervals. The respective vectors are stored in the file of the free terms, at the address corresponding to the integration subintervals and, respectively, to the equation system for which they were calculated. SEFIIK – the state of initial stresses due to load k. It calculates the initial values (at the moment of load k’s application) of the statically indeterminate unknown functions and it also calculates and retains in the file of the free terms, the contribution of the initial values of the statically indeterminate unknown functions to the free terms’ value. – It calculates the correction factors (of the structure’s flexibility matrices and the vectors of the free terms) corresponding to each material type and to the moment of load k’s application and, respectively, to the initial moment of the integration subinterval. – It assembles, by superposing the corrected structure’s partial flexibility matrices corresponding to each material type, the structure’s flexibility matrix corresponding to the moment of load k’s application. – It assembles the vectors of the free terms similarly to the flexibility matrix. – It obtains the initial values of the statically indeterminate unknown functions. – It determines, for every subinterval of integration subsequent to the application of load k, the contribution of the initial stresses state to the evolution in time of this; thus, it determines, based on the stresses generated by load k and the values of the statically indeterminate unknown factors, the corrected values of the free terms. – It stores the result in the file of the free terms. This module calls the modules RMETRS, TRIANG, RVTLET, SOLSIS, EXVECT, VIFDIF and starts calculating the evolution in time of the structure’s stresses and strains state.
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Materials with Rheological Properties
RMETRS – the rearrangement of the flexibility matrices according to the number of active components corresponding to the technological stage of implementation of the structure. The rearrangement is done by the elimination from the structure’s flexibility matrices corresponding to the elements of the finite structure, of the lines and columns corresponding to the elements that are inactive during the subinterval of integration. TRIANG – it prepares for solving the linear equation system, by decomposing the matrix of the coefficients in the product of two triangular matrices. For judicious use of the computer memory, the module works with a band matrix. RUTLET – the rearrangement of the free terms’ vectors according to the number of active components corresponding to the technological stage of structure’s implementation. The rows corresponding to the elements of the structure that are inactive during the subinterval of integration are eliminated from the vectors of the free terms corresponding to the elements of the finite structure. SOLSIS – solution of the linear equations system. It determines the solutions of the linear equations system by taking into account the fact that the triangular matrices, resulted from the structure’s flexibility matrix, are stored in the form of band matrices. EXVECT – the expansion of the vector of the unknown factors. The vector of the solutions for the system of linear equations corresponding to the number of structure’s components active at a given moment, is expanded to correspond to the final solution of the structure, by introducing zero elements (rows) into the positions corresponding to the structure’s inactive components at that moment. VIFDIF – the vector of the loads due to the differentiated creep. It calculates the contribution to the structure’s stresses and strains state evolution in time of the different development modes of the creep deformations of the structure’s components. TLIAAC – the free terms corresponding to the loads applied prior to the establishment of the collaboration. In the case of resistance structures that are implemented in stages, some of the components become active (cooperate within the structure) when a part of the loads already acts on the structure. During the redistribution of the stresses due to the rheological properties of the materials, the structure’s components, which become active after the moment of application of the loads, take part in accepting the stresses generated by the loads. – The correction factors of the free terms’ vectors corresponding to each material type and to the initial moment of the integration subinterval are calculated.
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– We determine, for all the integration subintervals subsequent to the moment of the establishment of the collaboration, the contribution of the initial stresses and strains state at its evolution in time. The result is stored in the file of the free terms. Therefore, the calculation of the stresses and strains state’s evolution in time is started generated by the stress redistribution process among the components of the structure. EIDTLI – the solutions of the integro-differential equations; the calculation of free integral terms. It determines the values of the derivates of order 1 ÷ 8 of the statically indeterminate unknown functions corresponding to integration subinterval i, that are stored in the file of the results, just as the already known values of these functions for the initial moment of subinterval i. Using these sizes, it determines the values of the statically indeterminate unknown functions at the end of subinterval i (that coincides to the initial moment of the following subinterval, i + 1), as well as the influence of the variation of the statically indeterminate unknown functions from integration subinterval i on the evolution of the stresses and strains state in the subsequent subintervals. The calculated values are superposed on the corresponding values, which already exist in the file of the free terms. – The correction factors of the structure’s flexibility matrices and the elements of the matrix from the right member of the equations system, corresponding to each material type and, respectively, to the initial moment of integration subinterval i, are calculated. – The corrected structure’s partial flexibility matrices corresponding to each material type, the structure’s flexibility matrix corresponding to the initial moment of integration subinterval i are assembled by superposition. – The values of the first derivative of the statically indeterminate unknown functions at the initial moment of integration subinterval i are calculated. – The values of the derivatives of order 2 ÷ 8 of the statically indeterminate unknown functions at the initial moment of integration subinterval i are calculated. – The values are stored in the results file, at the initial moment of integration subinterval i, of the statically indeterminate unknown functions and their derivatives of order 1 ÷ 8. – The values of the statically indeterminate unknown functions are calculated at the end of integration subintervals i, that constitutes their values at the initial moment of the following integration subinterval (i + 1). – The influence of the variations of the statically indeterminate unknown functions is calculated from the integration subinterval i on the subsequent evolution of the structure’s stresses and strains state. – The corrected values of the free terms are stored in the file of the free terms.
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This module calls the modules RMETRS, TRIANG, RVTLEX, EXVECT and VIFDIF, presented above. 5.3. The resistance structure
The resistance structure on which numerical experiments for the validation of the proposed calculation model were made was selected in order to meet the following requirements: – it allows, by adopting particular values for certain parameters, the verification of the adopted numerical methods by comparing the calculated results with the results obtained by exact methods of calculation; – it allows, by adopting particular values for certain parameters, the verification, by comparing the obtained results using the adopted mathematical model with the results obtained for the same values of the parameters using simpler, approximate calculation methods, authorized in designs or recommended in specialized literature; – it is a structure currently used, with a known operation behavior in accordance with the behavior estimated by the simplified calculation model used for its design; – it allows us to follow and interpret the results obtained by the use of the proposed calculation model. The resistance structure chosen to carry out the numerical experiments for the validation of the calculation model developed in this work is the standard bridge floor, a composite steel and concrete structure for railway bridges, having a span of 35.2 meters. In Figure 5.1 the principal geometrical elements of the structure are given, as well as its mode of composition. The bridge floor is a boxed structure (Figure 5.1b), composed of a metal profile open on the upper part (reversed ʌ), assembled by welding steel sheets and strips, and a reinforced concrete slab, laid out on the upper part, closing the box. Inside, the metal part of the structure is equipped with transversal rigidity frames and with longitudinal rigidity elements in the form of ribs made of steel strips welded to the webs of the structure’s metal skeleton, in the form of welded T profile, applied by welding on the sheet which forms the bottom flange of the bridge. At the upper part, each edge of the webs of the metal skeleton of the structure is strengthened using a strip steel which also supports the connectors and contributes to taking over the loads acting on the structure in the implementation stages as well as during use. The reinforced concrete slab that forms the upper flange of the structure also has the role of assuming and transmitting the local effects of the loads that stress it directly (the weight of the waterproof layer, the ballast bed, the track, the convoys, etc.).
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A
a) Lateral view
A
b) Cross section c) Detail A
Figure 5.1. Composite structure with discrete collaboration for a railway bridge
The connection elements (connectors) ensuring the collaboration (the take over and the transmission of interaction forces) between the metal and concrete part of the structure, have the form given in Figure 5.1c. They are welded on the upper surfaces of the upper flange of the structure’s metal skeleton and, respectively, they are included in the mass of the reinforced concrete slab that forms the upper flange of the structure. The distribution of the shear connectors along the length of the structure is given in Figure 5.1a. Steel section E=210,000 N/mm2
Concrete section E=37,500 N/mm2
Central zone
Towards the supports
Section area (cm2)
1,972
1,450
17,400
The ordinate of the center of gravity “yg” (cm)
61.75
78.18
26.05
Table 5.1. Mechanical characteristics of the structure’s cross-section
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Materials with Rheological Properties
NOTE.– The ordinates of the centers of gravity are measured based on the lower surface of the respective element. As shown in Figure 5.1a and b, the bridge floor’s section varies only by the thickness of steel sheet that constitutes its bottom flange; in the central zone of the deck, the sheet is 30 mm thick and, on its side zones, towards the supports, it is reduced to 15 mm. To implement the deck, the design has recommended the use of OL37EP steel for the metal skeleton of the structure and, respectively, a concrete corresponding to the class quality Bc25 for its reinforced slab. According to the quality of the materials recommended by the design for the manufacturing of the deck and the dimensions of the structure (see Figure 5.1), it results in the characteristics of calculation given in Table 5.1. For the shear connectors, we have determined the flexibility constant c = 1.965u10-5. In the same way, according to the conditions of implementation and the environment (where the structures of this type are placed), we have adopted the following values for the contraction and final creep of the concrete: Hct = 35u10-5 I = 3.0 The deck was modeled for calculation, like a Vierendeel beam, having the bottom flange composed of the metal component of the structure and that upper one of its concrete slab (each one having the characteristics of rigidity corresponding to relevant sections), laid out like bars in the centers of gravity of the two components of the deck. The uprights are bars of infinite rigidity for bending and axial forces equal to that of the connectors for the shear forces and having the same distribution on the structure as that of the connectors.
Numerical Application
xk
201
xj
e
xi
Om
i=3(m-1)+1 j=i+1 k=i+2 panel m-basic system
Figure 5.2. Structure’s basic system (panel m)
In Figure 5.2, the basic system adopted in the calculation model is presented, generically, through a fragment, namely an unspecified panel m. The length of the panel and the characteristics of the elements of the basic system are those corresponding to the structure between the consecutive connectors m – 1, m. The stresses and strains state depends on the structure’s sequence of implementation. Figure 5.3 presents the sequence of implementation of the deck, the loads corresponding to each stage and, respectively, the stresses they generate. The technology, illustrated in Figure 5.3, is adapted to the efficient use of the structure’s components. Indeed, the temporary support of the deck’s metal component, as shown in Figure 5.3a, constitutes an inexpensive method of prestressing the metal component, by using the deck’s weight itself, the weight of the slab’s formwork, the weight of the reinforcement and that of the concrete recently grouted for the implementation of the slab. All these weights are included in distributed load q1 (Figure 5.3a). The adjustment of the stresses due to prestressing is carried out by variation of the distance between the temporary supports or, which amounts to the same thing, by the variation of the length of the consoles. The stresses generated in the structure’s metal component in this situation of support and load, are “frozen” there during the hardening of the concrete of the slab. The actions and loads subsequent to the hardening of the concrete of the slab (generally, 28 days) are taken over by the composite structure, and the stresses produced by them in the metallic component of the deck are superposed on those due to the prestressing.
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Materials with Rheological Properties q1
a
R
P=R
R
q2
P=R
b
q3
c
Figure 5.3. Loads and their effects on the structure according to its sequence of construction
The situation that we obtain after setting the deck on its final supports, after the removal of the temporary ones and after the removal of the slab – operations that are carried out after hardening of concrete – is illustrated in Figure 5.3b. The distributed load q2 adds together the weight of the formwork of the slab and the weight of the water lost by the concrete (approx. 1.0 KN/m3) by evaporation; the concentrated forces P compensate for the cancellation of the reactions corresponding to the temporary supports.
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The permanent loads imposed by fitting the deck for normal operation (waterproof layer, ballast bed, track etc., which, totaled up, give the distributed load denoted by q3) are carried out at various times; the loads corresponding to them are illustrated in Figure 5.3c. 5.4. Numerical experiments
The numerical experiments carried out and presented below must reveal the fidelity of the description, by the mathematical model developed in this work, of the behavior of the composite resistance structures whose elements have different rheological properties and, thus, they constitute the support of the validation of this model. To validate the obtained results and, consequently, the proposed mathematical model, the numerical experiments were organized in the following way: – a first series of experiments, has the limited purpose of checking the accuracy of the results, thus the adequacy of the mathematical and numerical methods used for the solution of integro-differential equation systems (3.8’) and (3.8”). This checking is more necessary under the conditions of use of the computer, where we cannot follow all the calculation stages. – the second series of experiments, aims to check the veracity of the obtained results using the proposed model, comparing them with those obtained for the same structure and under the same conditions, but using the methods prescribed in the design standards and/or recommended in specialized literature. – the third and final series of experiments, aims, on one hand, to check the concordance between the numerical values obtained by the use of the proposed calculation model and those obtained by observations on real structures or, if possible, on the intuitive model of the phenomenon and, on the other hand, but along the same lines, to explain the mismatches between the results analyzed in the second series of experiments. We will present below the numerical experiments we have selected from each series because of their significance. The structure on which the numerical experiments were made, its characteristics and its calculation parameters are those presented in section 5.3; the structure of the creep function is that of section 2.3. In all the described cases, the particular values of the parameters and the loads corresponding to the experiment will be presented. 5.4.1. The first series of experiments
This series includes the experiments carried out to check each module of the RALUCA computer applications system, as well as those using different
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combinations and configurations, including the (final) work configuration. These experiments confirmed the correct operation of the modules under all conditions. They implicitly confirmed the adequacy of the mathematical method adopted for the solution of the integro-differential equation systems (3.8”), as well as of the numerical methods that we have used. Among all the numerical experiments of this series, we choose to present hereafter the one that is relevant for the discussed problem and which, from the mathematical and numerical analysis point of view, validates the results obtained within the framework of the other experiments. 5.4.1.1. The particular conditions for the analysis of the mathematical model The structure is studied under the following particular conditions. We also assign to the shear connectors (connectors ensuring the concrete-steel collaboration) an infinite rigidity to the shearing forces – which is equivalent to assigning a zero value to the flexibility coefficient of the connectors: c = 0. The value of the concrete’s modulus of elasticity is constant in time, equal to the reference modulus of elasticity, which is equivalent to assigning the unit value to the K0 function, thus: K0(t) = 1. The creep of the concrete does not manifest itself, which corresponds to assigning an identically zero value to the function which describes the variation in time of the final creep’s characteristic K1, thus: K1(t) = 0. The structure is stressed by the action corresponding to the development in time of shrinkage of concrete starting from the zero value, at the moment when the concrete is grouted, and tending (increasing), in time, towards the maximum value, the final value of the shrinkage Hct. The law according to which the shrinkage develops is the same as the one corresponding to the development of the creep, thus: t = 0, t o f,
H ct K2(t) = 0 H ct K 2(t) = H
In Figure 5.4, we present the evolution in time of the axial force’s value that develops in the concrete component of the section, in the central panel of the structure. The maximum value of the axial force developing in the concrete is: F * = –1778.40 KN
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and it is reached after approx. 1,800 days. Calculated by statistical construction methods under the announced particular conditions, the axial force developing in the concrete component of the structure, in the central panel, due to its shortening and corresponding to the shrinkage of concrete in the respective panel ('O = Hct u O, where O is the length of the panel), has the value F = 1,850.92 KN. The relative deviation of the F* value, obtained with the RALUCA computer applications system, compared with the value F calculated by statistical construction methods, is:
F F* u 100 = 3.91% F which imposes the conclusion that the mathematical method ensures the required convergence towards the solution of integro-differential equation systems (3.8’) and (3.8”), and the numerical methods used are adapted with a purpose, namely determining the values of the statically indeterminate unknown functions, with whatever level of precision. 0
50 100 150 200 250 300 350 400 450 500 550 600 650 700 T(days)
-200 -400 -600 -800 -1000 -1200 -1400 -1600 -1778,4 -1800 -2000 N(kN) Variation of the axial effort in reinforced concrete slab due to shrinkage, in absence of the creep. Figure 5.4. Axial effort in structure’s reinforced slab
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Materials with Rheological Properties
5.4.2. The second series of experiments
This series is represented by a unique example. 5.4.2.1. The particular conditions for the analysis of the mathematical model We analyze the evolution in time of the structure’s stresses and strains state, by taking into account the following conditions. The characteristics and the calculation parameters of the composite structure’s elements correspond to the situation described in section 5.3. The structure is submitted to the stresses of the following actions and loads: – the shrinkage of concrete; – external loads having the values and the implementation moments given in Figure 5.3. Figure 5.5 illustrates the evolution in time of the stresses state in the mid-span section of the deck, calculated by using the mathematical model developed in this work. For comparison, Figure 5.6 presents the evolution in time of the stresses state, in the same section of the deck, but the calculation is made by using the traditional model of the transformed sections. NOTE.– The method of the transformed section consists of the transformation of the structure’s (non-homogenous) concrete-steel composite section into homogenous steel sections by replacing its concrete component with an equivalent steel component. The transformation is made by dividing the mechanical characteristics of the concrete part of the section by the equivalence coefficient, which takes the following values: E nc = n = o for short-term static loads; Eb
nc = kn =
Eo 1 Mcl for the final phase, in the case of long-term loads; Eb
nc = k’n =
Eo 1 M cl for the moment t, 0 d t d f, in the case of long-term static Eb
loads; where:
M cl
M cl 1 kt and it expresses the variation in time of the characteristic of the
creep of the concrete.
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The notations used in the present note are those used in the tables of design standards for the calculation of composite steel and concrete structures and in the specialized literature. From the analysis of the shapes of the presented stresses diagrams, we obtain the perfect analogy between the obtained results using the two calculation methods. The comparison of the values of the stresses corresponding to the stages of implementation and operation of the structure imposes the following observations: – the values of the stresses in the different stages of implementation of the structure (Figure 5.5a, b, c and Figure 5.6a, b, c) and the initial value, corresponding to the beginning of its operation (Figures 5.5d and 5.6d), which corresponds to the moment t = 35 days, calculated using the proposed mathematical model and, respectively, the model of the transformed sections, are practically identical. The differences are explained by accepting the simplifying assumption of the plane sections (Bernoulli) (which supposes, implicitly, the perfect continuous collaboration between the concrete and steel components of the structure) within the framework of the calculation model of the transformed sections. We can also verify that, in the short interval of time until the beginning of the operation of the structure (in the analyzed case, 35 days), the effect of the concrete’s creep on the structure’s stresses state, considered by the two calculation models used, cannot lead to significant differences.
367.75 t=0
282.766 t=28+ t
302.846 t=28+ t
a
b
c
309.834
720.05
t=35+ t
t=35+ t
d
e
-39.372 -25.126
-68.519 -35.488 -30.216
869.673 t
8
-3.497 -34.071 -35.572 -147.24 -1.227 150.371 -16.980 100.685 -16.788 -134.548
f
Stresses state evolution in central panel of the structure taking account of time variation of the concrete modulus of elasticity, k 0=k0(t) and respectively, of this one creep coefficient, k 1=k1(t). a-before removing of the temporary supports, at concrete casting; b-before removing of the temporary supports; c-after removing of the temporary supports and striking of the slab; d-before application of the uniformly distributed load q3; e-after the application of the uniformly distributed load q3; f-in operation state, under permanent loads action, after extinction of the stress redistribution phenomena due to materials rheological properties.
Figure 5.5. Stress state evolution in the structure’s central panel
-1101.50
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Materials with Rheological Properties
If we compare the diagrams of stresses presented in Figures 5.5f and 5.6d, we can verify that, by using the mathematical model elaborated in this work, we obtain in the final phase – i.e. at the end of the redistribution process, due to the creep, of the stresses among the structure’s elements – lower values, in the concrete component, and higher values in the steel component than the values obtained using the calculation model of the transformed sections. The maximum differences between the corresponding stresses determined by the use of the two calculation methods remain, however, within reasonable limits, 'Vbmax = –0.926 N/mm2 for the concrete component (approx. 4.52% of the concrete’s resistance used for calculation), and respectively, 'Vomax = 4.427 N/mm2 for the steel component of the structure (approx. 1.925% of the steel creep stress). Therefore, we can note that designing composite steel and concrete structures, by using the calculation model of the transformed sections, while being sufficient for the concrete component, remains slightly insufficient for the whole structure, because it implies the over stress of the structure’s steel component. The value of the axial force that develops in the concrete component of the structure, obtained by using one of the calculation models gives an overall image of the dependence of the structure’s resistance capacity value on the adopted calculation model for estimating the stresses state. Indeed, starting from the relation: M = EoIoMo + EbIbMb + N u e we obtain: M § EoIoMo + N u e while observing that: EbI bMb << EoIoMo, where Mb | Mo = M (the specific rotation in one of the sections of the structure). If our purpose is not to over stress the metal component of the structure, it is necessary that EoIoMo = constant. Therefore, it results that: 'M = 'N u e. Consequently, it results that in the structure’s resistance economy, the contribution of the concrete slab is significant only through the axial force it develops, so that the variation of the axial force implies the proportional variation of the structure’s resistance capacity. Therefore, we estimate globally the influence of the error in the evaluation of the axial force developed in the concrete component, caused by the use of the calculation model, on the structure’s resistance capacity.
Numerical Application
-147.24 182.23
a
301.22
708.77
t=28+' t
t=35+' t
825.40 t
b
c
d
-32.01
-817.72
8
367.24 t=0
-53.91 -34.53 -549.45
-32.63 -25.36
209
Stress state evolution in structure steel section, computed with transformed cross section method; the modulus of elasticity and the creep coefficient are those corresponding to the moment of the load aplication, constant in time. a-before removing of the temporary supports, ar concrete casting; b-before removing of the temporary supports; c-before application of the uniformly distributed load q ; 3 d-in operation state, under permanent loads action, after extinction of the stress redistribution phenomena due to materials rheological properties. Figure 5.6. Stress state evolution in the structure’s central panel calculated using the transformed cross-section method
Figure 5.7 shows the evolution in time of the axial force that develops in the concrete component of the structure, calculated using the method we have presented in this work (thus, using the RALUCA computer applications system); it results that this axial force tends, in its final phase, towards the value N = 5,986.9 KN. By using the calculation model of the transformed sections for the evaluation of the axial force corresponding to the same situation, we obtain the value N = 7,453.8 KN.
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Materials with Rheological Properties
N(kN) 10000 9500 9000 8500 8000 7500 7000 6500 6000 5986.8 5500 5062.1 5000 4880.4 4500 4000 3500 3000 2500 2000 1500 1000 500 T(days) -35.11 0 50 400 450 500 550 600 650 700 100 150 200 250 300 350 -500 35 -1000 9919.3
Figure 5.7. Evolution of the axial force developed in concrete slab taking into account the structure’s sequence of construction
By using the two methods, the variation of the percentage deviation in the evaluation of the structure’s resistance capacity has the value: Nb Nb* u e u 100 M max
10.385%
Even if they are small in absolute value, the differences between the values of the stresses in the final phase, calculated by using the two models, have significant values compared to those obtained in the initial phase. These differences are due to the increase in time of the value of concrete’s modulus of elasticity and, at the same time, the decrease of the value of concrete’s creep characteristic. These two phenomena and the consequences that they imply on the distribution of the stresses in the elements of the structure will be discussed below.
Numerical Application
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5.4.3. The third series of experiments
This series is represented by 14 experiments that will be analyzed in two groups, each containing four experiments, and another group, of six experiments. The first two groups of this series reveal the influence of the configuration assigned to the creep function on the distribution of the stresses among the elements of the structures, due to the loading of the structure with external forces and, respectively, to the shrinkage of the concrete slab. The other group of experiments reveals the influence of the configuration assigned to the creep function, but also to the flexibility assigned to the shear connectors on the distribution of the stresses in the elements of the structure and, particularly, on the connection elements themselves. For reasons of concision and expressivity, the evolution in time of the stresses state is represented by its synthetic indicator, namely the axial force that develops in the concrete component of the structure, in one of its central panels. In addition, in order to facilitate the analysis of the obtained results and to draw in this way some relevant conclusions, in all the numerical experiments presented below, we considered the same values that characterize, in the initial phase, the elastic and rheological properties of the materials the structure is made of. The (qualitative and quantitative) analysis of the mode the rheological properties of materials influence the evolution of the stresses state in the structure is done by comparing the results obtained through numerical experiments considering particular configurations of the creep function, adapted to reveal the influence of each one of these properties, as well as their totality. 5.4.3.1. The influence of the parameters defining the creep function The influence of the rheological properties of the concrete on the evolution of the stresses state of the structure submitted to the actions of the external forces (in the analyzed case, a uniformly distributed load q = 79.60 KN/ml) is illustrated in the diagrams presented in Figure 5.8. The analysis of the qualitative and quantitative influence of the rheological properties of the concrete on the evolution of the stresses state developed in the elements of the structure is made by comparing the results of the numerical experiments. This is done by taking into account, successively, the hypothesis concerning the rheological properties of the concrete that are adequate to this purpose and concretized in the adoption of the corresponding particular configurations of the creep function. As a reference base, we consider the values of the axial force that develops in the concrete component of the structure, presented in Figure 5.8a, obtained in the numerical experiments carried out by observing the assumption that both the
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Materials with Rheological Properties
modulus of elasticity and the creep characteristic of the concrete are constant in time. The final value towards which the axial force from the concrete component of the structure tends is N = 3,859 KN. The following values of the respective functions from the concrete’s creep function correspond to the assumptions made in these experiments on the rheological properties of the concrete: K0 = 1.0 and K1 = 1.0 The influence of the (increasing) variation in time of the concrete’s modulus of elasticity on the evolution of the structure’s stresses state is illustrated in Figure 5.8b. In this experiment, we considered as constant the characteristic of the concrete’s creep; the value of this characteristic corresponds to the age of the concrete at the moment of the external force’s application. The configuration of the creep function corresponds in this case to that resulting when we adopt the following values for functions K0 and K1: K0 = K0(t) and, respectively, K1 = 1.0 In this case, the final value towards which the axial force tends is N = 3,824.3 KN. This value of the axial force, lower than its reference value by ǻN1 = 35.3 KN, allows us to conclude that the (increasing) variation in time of the concrete’s modulus of elasticity implies the decrease of the resistance capacity of the structure submitted to the actions of the permanent loads. The influence of the (decreasing) variation in time of the concrete’s creep characteristic on the evolution of the structure’s stresses state is illustrated in Figure 5.8c. In this experiment, we have considered that the concrete’s modulus of elasticity is constant in time and has the reference value that corresponds to its class quality. Corresponding to these assumptions, the creep function takes the configuration resulting from the adoption of the following particular values for the functions K0 and K1: K0 = 1.0 and, respectively, K1 = K1(t). Determined under these conditions, the axial force takes, in the final phase, the value N = 3,803.3 KN, thus a value that is lower than the reference value by 'N2 = 56.3 KN. This allows the assertion that the (decreasing) variation in time of the concrete’s creep characteristic also implies the decrease of the resistance capacity of the structure submitted to the action of permanent external loads.
Numerical Application
213
The evolution of the stresses state in the structure, calculated for the real hypothesis, which is that of simultaneous, increasing variation of the concrete’s modulus of elasticity value and decreasing variation of the creep characteristic is represented in Figure 5.8d. The creep function, corresponding to this hypothesis, results by the replacement, in its general expression, of the expressions adopted for the functions K0 and K1, thus: K0 = K0(t) and K1 = K1(t). N(kN) 5000 4800 4600 4400 K0=1.0 4200 K1=1.0 4000 3859.5 3800 3600 3400 0 2850 100 150 200 250 300 350 400 450 500 550 600 650 700 T(days) N(kN) 4880.8
5000 4800 4600 4400 K =K (t) 4200 K =1.0 4000 3824.3 3800 3600 3400 0 2850 100 150 200 250 300 350 400 450 500 550 600 650 700 T(days)
a
4880.6
b
N(kN) 5000 4800 4600 4400 K0=1.0 4200 K1=K1(t) 4000 3803.3 3800 3600 3400 0 2850 100 150 200 250 300 350 400 450 500 550 600 650 700 T(days) 4880.8
c
N(kN) 5000 4800 4600 4400 K0=K0(t) 4200 K1=K1(t) 4000 3771.7 3800 3600 3400 0 2850 100 150 200 250 300 350 400 450 500 550 600 650 700 T(days) 4880.6
d
Figure 5.8. Evolution of the axial force developed in concrete slab due to uniformly distributed load Q = 79.60 kN/m on the structure, taking into account the configuration of the concrete creep function
214
Materials with Rheological Properties
The axial force, calculated with this hypothesis, tends, in the final phase, towards the value N = 3,771.7 KN, and is 'N3 = 87.9 KN lower than the reference value, and also lower than the values obtained in the two previous experiments. We can, thus, affirm that, in the case of composite resistance structures submitted to the actions of continuous (external) loads, the decrease of their resistance capacity is amplified by the simultaneous existence, in conformity with the reality, of the phenomena of (increasing) variation in time of the modulus of elasticity and (decreasing) concrete’s creep characteristic. The accumulation of the effects of these two phenomena is done, as expected, according to a nonlinear law. In the case of the stresses due to permanent loads, the addition of the two phenomena leads to a slight drop of the effects considered separately. Indeed: 'N1 + 'N2 > 'N3 We must add that these two phenomena, the increasing variation of the modulus of elasticity and, respectively, the decreasing variation of the concrete’s creep characteristic constitute two of the sources of the concrete’s behavior as imperfect viscoelastic body (viscoelastic body whose asymptote of the recovery curve is a straight line parallel with the abscissa and its ordinate is equal to the permanent strain, Hp > 0). Moreover, we should emphasize the fact that the stresses state in the initial phase, generated by the external loads, is independent of the configuration of the creep function and depends only on the values, in the moment of application of the loads, of the parameters characterizing the elastic properties of the materials from which the structure’s elements are made. The stresses state, identical in the initial phase in the four experiments that we have described above and illustrated in Figure 5.8, confirms this assertion. 5.4.3.2. The stresses state in the structure caused by the contraction of the concrete The evolution of the stresses state, in one of the central panels of the analyzed structure, due to the development in time of the contraction of the concrete (Hc = 35 u 10-5) is represented in Figure 5.9. Just as in the case of the previous group of numerical experiments, the evolution of the stresses state is synthetically represented by the evolution in time of the axial force that develops in the concrete component of the structure, determined for the configuration of the creep function adequate for the hypothesis made on the variation of the modulus of elasticity and of the concrete’s creep characteristic; the hypothesis used for the respective experiment.
Numerical Application
215
For the configuration of the creep function, corresponding to the hypothesis that the modulus of elasticity, as well as the concrete’s creep characteristic, are constant in time, the evolution of the axial force that develops in the concrete component of the structure, is illustrated in Figure 5.9a. This force, which tends towards the value N = –1,387.6 KN in the final phase, represents the reference value within the framework of the analysis of the results of this group of numerical experiments. In Figure 5.9b the evolution of the stresses state is represented, calculated for the configuration of the creep function, corresponding to the hypothesis of the (increasing) variation in time of the value of the modulus of elasticity and a constant value of the concrete’s creep characteristic. The axial force tends, in this hypothesis, towards the final value N = –1,423.5 KN, with N1 = 35.9 KN lower than the reference value obtained in the previous numerical experiments. Therefore, it results that the increasing variation of the concrete’s modulus of elasticity has as an effect, also in the case of the stresses generated by the shrinkage of concrete, a decrease of the structure’s resistance capacity. Figure 5.9c presents the evolution of the stresses state for the configuration of the creep function corresponding to the hypothesis that the value of the concrete’s modulus of elasticity remains constant in time, and the value of its creep characteristic varies in a decreasing way in time. Determined under these conditions, the axial force that develops in the concrete component of the structure tends, in the final phase, towards the value N = –1,444.0 KN, with 'N2 = 56.4 KN lower than the reference value. Therefore, we can affirm that the decreasing variation of the concrete’s creep characteristic value leads, according to the reference hypothesis, to the “consumption” of a fraction of the structure’s resistance capacity. The diagram presented in Figure 5.9d illustrates the variation of the stresses state in the structure, for the configuration of the creep function, corresponding to the real hypothesis that the value of the modulus of elasticity, as well as the value of the concrete’s creep characteristic, is variable, increasing and respectively decreasing in time. The final value towards which tends, in this case, towards the axial force is N = –1,551.4 KN, with 'N3 = 63.8 KN lower than the reference value. The structure’s resistance capacity, calculated in this hypothesis, is lower than the one corresponding to the reference hypothesis, as well as those corresponding to the previous hypotheses; in this case, the accumulation of the effects is carried out according to a nonlinear law. However, the accumulation of the two phenomena leads, in the case of the stresses generated by the development in time of the shrinkage of concrete, to a severe amplification of the effects considered separately. Indeed: 'N1 + 'N2 << 'N3.
216
Materials with Rheological Properties
To remove any ambiguity, it is necessary to emphasize the positive effect of the creep phenomenon on the evolution of the stresses state, generated in the structure by the development in time of the shrinkage of concrete. This assertion becomes obvious if we compare the values in the final phase of the axial stresses developing in the concrete component of the structure, calculated in the numerical experiments based on the hypothesis supposing the presence of the creep phenomenon (this corresponds to the attribution of non-zero values to creep function k1 and to the representation in Figure 5.9), with the value resulting from the numerical experiments based on the hypothesis that the creep phenomenon does not appear (this corresponds to the attribution of a zero value to function k1 and to the representation in Figure 5.4). 0 -200 -400 -600 -800 -1000 -1200 -1387.6-1400 -1600
50 100 150 200 250 300 350 400 450 500 550 600 650 700 T(days)
K0=1.0 K1=1.0
N(kN) 0 -200 -400 -600 -800 -1000 -1200 -1400 -1423.5 -1600
a
50 100 150 200 250 300 350 400 450 500 550 600 650 700 T(days)
K0=K0(t) K1=1.0
b
N(kN) 0 -200 -400 -600 -800 -1000 -1200 -1400 -1444 -1600
50 100 150 200 250 300 350 400 450 500 550 600 650 700 T(days)
K0=1.0 K1=K1(t)
c
N(kN) 0 -200 -400 -600 -800 -1000 -1200 -1400 -1551.4 -1600
50 100 150 200 250 300 350 400 450 500 550 600 650 700 T(days)
K0=K0(t) K1=K1(t)
d
N(kN)
Figure 5.9. Variation of the axial effort in the concrete slab due to developing in time of the concrete shrinkage, taking into account the configuration of the creep function
Numerical Application
217
5.4.3.3. The influence of the deformability of the connection elements on the effort’s distribution among the elements of the structure The final group of experiments from this series has the objective of revealing the way the concrete’s creep phenomenon and the deformability of the shear connectors influence the distribution of the stresses among the elements of the structure. We have considered that it would be more suggestive to represent the shearing forces that develop in the connectors, in the diagrams illustrating the results of these final six experiments. The pertinence of this statement is obvious if we consider the fact that the shearing forces loading the shear connectors is equal to the difference between the axial forces that develop in the concrete components of the structure, in the panels adjacent to the connectors. Alternatively, which comes back to the same, the axial force in the concrete component of the structure in a given panel is equal to the sum of the shear forces that develop in all the connectors located on the one or the other side of the respective panel. Illustrated by the shearing forces developed in the connection elements and corroborated by the results of the previous experiments, the numerical experiments we will present below offer an image on the behavior of the entire structure. We must mention, moreover, that the numerical experiments of this group were carried out on the hypothesis, corresponding to the real situation, of the increasing variation of the modulus of elasticity and, respectively, the decreasing value of the concrete’s creep characteristic. In the diagrams illustrating the numerical experiments of this group, the evolution of the shearing forces that develop in the connection elements, is represented, for each connector, by two line segments perpendicular to the abscissa in the point corresponding to the position of the connection element on the structure and having a length which is proportional to the values of the shearing forces developed in the connector. A continuous line represents the initial phase shear force while a dashed line represents the final phase shear force. The results of the numerical experiments carried out for the given structure, submitted to the action of the uniformly distributed external load: Q = 79.60 KN/ml and represented in Figure 5.10 and 5.11, corroborated by those obtained in the numerical experiments presented in the previous group, highlight the evolution of the shearing forces that develop in the connection elements. This evolution starts from the maximum value, that occurs at the moment of the application of the load (in the initial phase), towards a minimum value, realized when the stress redistribution phenomenon among the elements of the structure is over (1,500 – 1,800 days).
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Materials with Rheological Properties
The correlation, mentioned above, between the shearing forces and the axial forces that develop in the connection elements and, respectively, in the components of the structure, implies the conclusion that their evolution is described by continuous, monotone decreasing functions, that tend asymptotically towards a limit value, which is their value in the final phase. In the initial phase, as well as in the final phase, the shearing forces that develop in the connection elements are distributed according to a curve close to that obtained by using the expression that serves to determine the value of the slip taken over by a connector in the method of the transformed sections:
Li
T u Sz u Oi Iz
In the expression of the slip presented above, the significance of the symbols is that known from the resistance of materials. F 500 400 300 200 100 0 a-taking into account rigid shear connectors
F
X
500 400 300 200 100 0 b-taking into account elastic shear connectors
Key: initial stage values final stage values
Figure 5.10. Shear forces developed in the shear connectors
X
Numerical Application
219
The deviations from this distribution law of the shearing forces that develop in the connection elements are generated by the rigidity discontinuities of the structure and/or by discontinuities in the arrangement of the connectors on the structure. Indeed, in Figure 5.10, which illustrates the distribution of the shearing forces that develop in the connectors, calculated on the real structure (which presents discontinuities of both types), the deviations of the mentioned law take place in the end sections, the section having the abscissa 8.20 m, where the section of the structure changes, and also the section having the abscissa 11.20 m, where the distance between the connectors is increased in order to allow a compound joint implementation. For comparison, in Figure 5.11 is illustrated the distribution of the slip forces that develop in the connectors, calculated for a structure having a constant section over its entire length and having the same number of connectors, but uniformly distributed on the structure. In this case, we only notice the discontinuity at the ends of the structure. The influence of the connectors’ flexibility on the distribution of the stresses among the elements of the structure will be analyzed based on the results of the four illustrated numerical experiments – two, in Figure 5.10, where the results obtained for the real structure (having a rigidity discontinuity and whose connectors are distributed to non-uniform distances) are shown and two, in Figure 5.11, for the fictitious structure (having a constant rigidity and whose connectors are uniformly distributed over the entire length of the structure). The two structures are submitted to the action of a uniformly distributed load (Q = 79.60 KN/ml). The most unfavorable situations of stress for the structure are those corresponding to the initial and the final phase. This statement is the consequence of the fact that the shearing forces and the axial forces that develop in the connectors and, respectively, in the elements of the structure, evolve according to a law that corresponds to a continuous, monotone, decreasing function whose extreme values are at the ends of its definition interval.
220
Materials with Rheological Properties F 500 400 300 200 100 0 F
a-taking into account rigid shear connectors
X
500 400 300 200 100 0 b-taking into account flexible shear connectors
X
Key: initial stage values final stage values
Figure 5.11. Shear stresses distribution in shear connectors in a constant section structure and uniformly repartition connectors due to uniformly distributed load Q = 79.60 kN/m
The values of the shear stresses that develop in the connectors under the conditions of the real structures are lower than those determined, by calculation, in the hypothesis of the connectors having infinite rigidity. The axial forces that develop in the components of the structure, in its most stressed section, are slightly influenced by the flexibility of the connectors. The sum of the absolute values of the shear stresses that develop in the connection elements is influenced very little by their flexibility.
Numerical Application
221
X
0 -100 -200 -300 -400
-1300 -1400 -1500 F
a-taking into account infinite rigid shear connectors X
0 -100 -200 -300 -400 -500 -600 -700 -800 -900 F
b-taking into account flexible shear connectors
Figure 5.12. Shear stresses distribution in shear connectors due to concrete shrinkage
The flexibility of the connectors has a spectacular influence on the distribution of the stresses that develop in the connectors, and, implicitly, on the axial stresses that develop in the components of the structure, in the case of the action due to the development in time of the shrinkage of concrete. The numerical experiment carried out by assigning an infinite rigidity to the connection elements (c = 0), highlights the fact that all the stress is transmitted to the elements of the structure by the connection elements placed at its ends. The stresses of the other connectors being zero or, in the case of those placed in the sections where there are discontinuities of the structure’s rigidity or of the connectors’ distribution, without practical significance (see Figure 5.12a). It results that the axial force that develops in the elements of the structure is practically constant over its entire length.
222
Materials with Rheological Properties
The numerical experiment carried out on the hypothesis corresponding to the real situation, that of the connection elements having a finite rigidity (c z 0), whose results are presented in Figure 5.12b, highlights the fact that the shear stresses that develop in the connection elements take non-zero values for all of them, but, in their large majority, without practical significance. The first 4–5 connectors, placed at the ends of the structure, are exceptions because the effort determined in the previous experiment (corresponding to the hypothesis of the infinitely rigid connectors) is practically distributed among them. In this experiment, we have highlighted the non-uniformities in the distribution of the shear stresses that develop in the connection elements, generated by the structure’s rigidity discontinuities and by distribution discontinuities of the connectors on the structure. The most strongly stressed connection elements are placed at the ends of the structure, but the value of the shearing force that develops in these connection elements is about 56.4% of the value of the shearing force determined on the (previous) hypothesis of the infinitely rigid connectors. Just as in the case of a structure submitted to the action of external forces, in the case of the action due to the development in time of the concrete’s contraction, the sum of the absolute values of the shear stresses that develop in the connection elements is slightly influenced by their flexibility. At the end of the presentation of the numerical experiments carried out for the validation of the calculation model developed in this work, we must emphasize the fact that the results obtained and presented in this chapter are in total agreement with the observations and measurements taken on the real structures of this type. Moreover, we can affirm that some of the presented results and conclusions contribute to the theoretical foundation of some of the instructions included in the design standards described on an empirical basis, such as those concerning the distribution on the connection elements of the slip stresses generated by the load of the structure by external forces, or that concerning the resumption of the stresses generated by the shrinkage of concrete only by the connectors located at the ends of the structure. The final conclusion which emerges after the presentation of the results of the numerical experiments is that the mathematical model for the estimation by calculation of the stresses and strains state evolution in the resistance structures considering the rheological properties of the materials from which they are made, that are worked out and developed in this work, accurately reflects the behavior of the real structures and, consequently, it can be put into practice.
Appendix 1
The Initial Stresses and Strains State of the Structures with Continuous Collaboration
We determine the analytical solutions of the differential equations modeling the behavior of a composite structure with continuous collaboration at the moment of the application of the load. Being worked out for a particular, simple structure, and for some of the most common loading cases, the analytical solutions have a reduced practical importance. We have considered them useful for verifying the algorithms and, respectively, the calculation programs, which will be developed for the numerical integration of the differential equations that will be used to determine the solutions of the integrodifferential equations describing the behavior in time of the structures made out of materials having rheological properties. As mentioned at the end of Appendix 2, apart from their practical interest, the results presented below have important theoretical interest, because they indicate the operating process of the composite structures with continuous collaboration. We will use the first differential equation of (4.14), in the following form: O
>
@
>
@
d2 x r ,W E xT0 r ,W r E x r ,W x r ,W E xT0 r E0 r E xT0 H c Q d r2
0 (A01)
224
Materials with Rheological Properties
y cgcs cgss z
The first two terms of the differential equation are dependent only on the mechanical characteristics of the elements composing the structure, the last being, also, dependent on the actions and the loads applied to it. If we consider only the deformations produced by the axial force and by the bending moment for the concrete and steel parts of the section and only the slip deformations of the connectors, it results that:
fm
ª 1 « EA « « 0 ¬
ª 1 «E A « b b « 0 «¬
º 0 » ;f 1 » b » EI ¼
º 0 » »; f 1 » O Eb I b »¼
ª1 «c «0 ¬
º 0» , 0»¼
The flexibility matrix of the elements of the structure takes the form:
ª 1 «E A « b b « « « « « « « « « « ¬
1 Eb I b
1 c
0 1 EA
º » » » » » » » » » » 1 » » EI ¼
Appendix 1
225
The equilibrium matrices Ex, EO, E0 have the following configuration:
ª1 «0 « «0 « «0 «0 « ¬«0
Ex
1
0º 1»» 0» »;E x 0» 2 0» » 0¼»
ª0 0º «0 0» « » «0 0» « »;E O «0 0» « 1 0 » « » ¬« e 1»¼
ª0 «0 « «1 « «0 «0 « ¬«0
0º 0»» 0» »;E 0 0» 0» » 0¼»
ª 0 º « 0 » « » « 0 » « » « 0 » « 0 » « » ¬« M r ¼»
Thus, by taking into account the relations: – K02 Ł 1 for steel; and – K01 = K01(W) for concrete, the terms of equation (A01) take the following expressions: ª1 º « c 0» « 0 0» ¬ ¼
E OT r E O
O
2
E xT r , W
¦ E r K W
0
j 1
E x r
T xj
2
¦ E r j 1
xj
0j
0 0 0 1 e º ª K 01 W ; « 0 K 01 W 0 0 0 1»¼ ¬
0º ª1 «0 1 »» « «0 0» » « 0» «0 « 1 0 » » « ¬« e 1¼»
If we note:
1 E
K01 W Eb
it results that:
E xT r ,W r 0
ª 1 « E A b « « 0 «¬
0 1 E Ib
0 0
1 EA
0 0
0
e º EI » » 1 » EI »¼
226
Materials with Rheological Properties
and, respectively:
E xT r ,W r E x r 0
E xT r ,W r E 0 r 0
ª 1 1 e2 « « E Ab EA EI e « «¬ EI ª e º « EI » « M r » « » ¬ EI ¼
º » » 1 1 » E I b EI »¼
e EI
Finally, differential equation system (A01) takes the form:
1 w 2 X1 § 1 e2 · e eM r 1 ¸¸ X 1 ¨¨ X2 ° 2 EI EI © E Ab EA EI ¹ ° c wr ® ° e X §¨ 1 1 ·¸ X M r 0. 2 ° EI 1 ¨ E I EI ¸¹ EI b © ¯
0 (A02)
From the second equation of the system (A02), it results that:
E Ib M r eX1 . E I b EI
X2
(A03)
If we introduce (A03) into the first equation (A02), it results in the following, non-homogenous differential equation of order 2:
· 1 d 2 X1 § 1 1 e2 1 ¨ ¸¸ X 1 eM r 0 (A04) 2
¨ c dr E I b EI © E Ab EA E I b EI ¹
If we note:
:2
§ 1 · e2 1 ¸ c ¨¨ ¸ © E Ab EA E I b EI ¹
Appendix 1
227
and:
K
ce E I b EI
differential equation (A04) takes the form:
d 2 X1 :2 X1 dr 2
KM r
(A05)
The general solution of the non-homogenous differential equation (A05):
X 1 r
X 1 r X 1 r
(A06)
where X 1 (r) is the solution of the homogenous equation associated with equation (A05) and X1*(r) is a particular solution of non-homogenous equation (A05). The solution of the homogenous equation associated with equation (A05) is:
X 1 r C1 e :r C 2 e :r .
(A07)
The form of the particular solution X*1 of the non-homogenous equation (A05) resembles the free term of the equation and will be determined for the load case which will be studied below. A.1. Simply supported beam with uniformly distributed load q
M(x)
x
228
Materials with Rheological Properties
The equation of the moment diagram (the free term of the differential equation) is:
M r
1 qx l x 2
(A08)
The particular solution of the differential equation will have the form:
X 1 x
Ax 2 Bx C
thus:
>
@
d X 1 x 2 Ax B dx d2 X 1 x 2 A. 2 dx
>
(A09)
@
If we introduce (A08) and (A09) into (A05), we obtain:
2 A : 2 Ax 2 Bx C
1 1 Kqlx Kqx 2 2 2
(A05’)
and, by identification, we obtain:
1 Kqx 2 o A 2
: 2 Ax 2
1 Kqlx oB 2
: 2 Bx 2 A : 2C
0oC
1 Kq , 2: 2 1 Kql , 2: 2
1 Kq . :4
Thus,
X 1 x
1 1 §1 · Kq¨ : 2 x 2 : 2lx 1¸. 4 : 2 ©2 ¹
If we substitute (A07) and (A10) in (A06) and if we note:
P
1 Kq :4
(A10)
Appendix 1
229
the general solution (A06) takes the form:
1 §1 · X 1 x C1e :x C2 e :x P¨ : 2 x 2 : 2lx 1¸ 2 ©2 ¹
(A06’)
The integration constants C1 and C2 are determined from the boundary conditions. Therefore: – for x
0oX 1 0 0oC1 C2 P
– for x
lo X 1 l 0oC1e :l C2 e :l P 0
0
from which it results that:
C1 C2
1 e :l e :l e :l e :l 1 P :l e e :l
P
The solution of equation system (A02), corresponding to the load situation given in (A08), takes the expression: ª 2 K 1 § sh:x sh: l x ·º ¨1 ¸» ° X 1 r 2 qx l x «1 2 2 sh:l x l x : : © ¹¼ ¬ ° ®
eK ª E Ib 1 2 § sh:x sh: l x ·º ½ ° X r qx l x ®1 2 «1 2 ¨1 ¸» ¾
° 2 sh:l E I b EI 2 ¹¼ ¿ ¯ : ¬ : x l x © ¯
or, finally: ª § §l · ·º ¨ ch:¨ x ¸ ¸» ° « 1 2 K 2 © ¹ ¸ »; ¨1 ° X r qx l x «1 2 ¸» 1 ° 1 « : x l x ¨ :2 2 ch :l ¸» ¨ ° « 2 ¹¼ © ¬ ° ® ª § §l · ·º ½ ° ¨ ch:¨ x ¸ ¸» ° ° « ° E Ib 1 2 2 ° eK « © ¹ ¸» ° ¨1 qx l x ®1 2 1 2 ° X 2 r
¸» ¾ ¨ 1 « 2 E I EI x l x : : ° b ° ch l : ¸» ° ¨ « ° °¯ 2 ¹¼ °¿ © ¬ ¯
(A11)
230
Materials with Rheological Properties
A.2. Simply supported beam loaded with a concentrated force
Pb ; l Pa ; Rk l Pab . M max l Ri
i
j
P
k
x
The diagram of the bending moments has the following equation: – on section i-j:
M x
Pb x l
(A12)
– on section j-k:
M x
Pa l x l
(A13)
The particular solution of non-homogenous differential equation (A05) corresponding to the two sections of the beam: – for section i-j:
X 1 x
>
Ax B;
@
d X 1 x A; dx d2 X 1 x 0. dx 2
>
@
(A14)
Appendix 1
231
If we substitute (A12) and (A14) in (A05), we obtain:
: 2 Ax B K
Pb x l
and after identification, it results that:
KPb ; B=0; : 2l KPb X 1 x 2 x ; :l
A
and, respectively:
X 1s x C1e:x C2e :x
KPb x,0 d x d a : 2l
(A15)
– for section j-k:
X 1 x
>
Ax B;
@
d X 1 x A; dx d2 X 1 x 0. dx 2
>
(A16)
@
If we substitute (A13) and (A16) in (A05), it results that:
: 2 Ax B K
Pa l x l
and after identification, it results that:
A B
KPa ; : 2l KPa 2 ; :
232
Materials with Rheological Properties
and, respectively:
KPa x l : 2l
X 1 x
(A17)
From the boundary conditions, thus for x = 0 and x = 1, we obtain:
C1 C 2 C3 e
:l
0;
C 4 e :l
0,
thus:
C2
C1;
C4
C3e 2 :l .
(A18)
From the conditions:
X 1s a
X 1d a ;
d X 1s r dr
d X 1d r dr
r a
r a
;
and by considering (A18), it results that:
C1 e :a e :a C3 e :a e: l b ° ® :a :a :a : l b °C1 e e C3 e e ¯
0; KP . :3
(A19)
If we keep in mind that:
e:a e: l b e
O:
e
: l b
e
,
e:l e :b e:b ; e
:l
:b
e
:b
equation system (A19) takes the form:
C1sh:a C3e:l sh:b 0; ° ® KP . :l °C1ch:a C3e ch:b 2: 3 ¯
(A19’)
Appendix 1
233
If we introduce into (A15) and (A17) the values of coefficients C1 and C3 obtained by solving system (A19) and by taking into account (A18), it results that: – for 0 d x d a
X 1s r
X 2s r
KPbx § l sh:b sh:x · ¨1 ¸; :bx sh:l ¹ l: 2 ©
E Ib Pbx ª eK § l sh:b sh:x ·º 1 ¨1 ¸.
l «¬ : 2 © :bx sh:l ¹»¼ E I b EI
(A20)
– for a d x d 1
X 1d r X 2d r
KPa l x § l sh:a sh: l x · ; ¨¨1 ¸ :a l x sh:l ¸¹ l: 2 ©
E Ib Pa l x ª eK «1 2
E I b EI l ¬ :
§ l sh:a sh: l x ·º ¨¨1 ¸» :a l x sh:l ¸¹¼ ©
A.3. Simply supported beam loaded with a concentrated moment at each end
Mi Ri i Mi
Ri
M j Mi
Mj j R j Mj
;
l Mi M j ; Rj l x· x § M r M i ¨1 ¸ M j l¹ l ©
x M i M i M j . l
234
Materials with Rheological Properties
X 1 r
Ax B;
>
@
d X 1 r A; dx d2 X 1 r 0. dx 2
>
@
X 1 r
K :2
(A21)
§ Mi M j · ¨¨ x M i ¸¸ l © ¹
(A22)
If we substitute (A22) and (A21) in (A05) and after identification, it results that: X 1 r
K :2
§ Mi M j · ¨¨ x M i ¸¸ l © ¹
thus:
X 1 r C1e :x C 2 e :x
K :2
§ Mi M j · ¨¨ x M i ¸¸ l © ¹
(A23)
Using the boundary conditions,
x 0 l
X 1 0 0
x l l
X 1 l 0
we determine the values of constants C1 and C2, so that, finally, we obtain:
X 1 r
K :2
M Mj § ¨¨ M i i l ©
· x ¸¸ ¹
§ M sh: l x M j sh:x · l ¨1 ¸; i ¨ M l x M x ¸ : sh l i j © ¹ M Mj § ¨¨ M i i l ©
· x ¸¸ ¹
X 2 r
E Ib E I b EI
ª eK «1 2 «¬ :
§ M sh: l x M j sh:x ·º l ¨1 ¸» i ¨ M l x M x ¸» sh:l i j © ¹¼
(A24)
Appendix 1
235
A.4. Simply supported beam loaded with concentrated forces applied eccentrically, acting on a direction parallel with the axis of the beam
e j Pj ei Pi
; l ei Pi e j Pj ; l e j Pj ei Pi
Rj
M r
(A25)
x ei Pi ;
l
i
Mi
Pj
j
Vi
j R j
Vi i e iPi
ejPj N
N(r)=
Pi, if the fixed support is placed as in the figure (on the left) Pj, if the fixed support is placed on the right
If we take into account that, in this case, the axial force is independent of the coordinates and that:
ei Pi
M i;
e j Pj
M j,
it results that:
M r M i N r N
M j Mi l
x; (A25’)
236
Materials with Rheological Properties
In this case, vector E0 takes the form:
E0
0 ½ ° 0 ° ° ° °° 0 °° ¾ ® ° 0 ° ° N ° ° ° ¯° M ¿°
so that the elements of the product vector following values:
E r ,W r E 0 r T x0
^E r ,W r E r ` T x
0
will take the
N r eM r ½ ° EA EI ° . ® M r ¾ ° ° EI ¿ ¯
In this case, differential equation system (A01) takes the form: 1 w 2 X1 § 1 e2 · e N eM 1 ¸X1 ¨ X2 ° 2 ¸ ¨ E A c EA EI EI EA EI x w b ° ¹ © ® § 1 e M 1 · ° ¸ ¨ ° EI X 1 ¨ E I EI ¸ X 2 EI 0 b ¹ © ¯
0
(A26)
If we eliminate the unknown function X2 from the equations of system (A26), we obtain:
w 2 X1 : 2 X 1 ON KM 2 wx where we have noted:
§ 1 e2 · 1 ¸ c¨¨ ¸ EA EI E A b ¹ © 1 O c ; EA e K c . E I b EI
:2
0,
(A27)
Appendix 1
237
It is obvious that the particular solution of non-homogenous differential equation (A27) will be the sum of the particular solutions of the non-homogenous equations:
w 2 X1 : 2 X 1 ON wx 2
0,
and, respectively:
w 2 X1 : 2 X 1 KM wx 2
(A28)
0
The last equation (A28) corresponds to the loading case we have treated in section A.3 above. We search for a particular solution X1** corresponding to the first equation (A28) in the form of a constant, X1** = K, which, substituted into the first equation (A28), allows us to write the particular solution:
X 1
O :2
N
so that, if we consider the particular solution X1*, obtained in section A.3 above, for the second equation (A28), we can write the general solution of the differential equation (A27) in the form:
X1
C1e :x C 2 e :x
1 :2
ª § Mi M j « K ¨¨ M i l «¬ ©
From the boundary conditions:
1 °°C1 C 2 : 2 KM i ON i ; ® °C e :l C e :l 1 KM ON , 2 j j °¯ 1 :2
º · x ¸¸ ON » »¼ ¹
(A29)
238
Materials with Rheological Properties
we obtain the values for constants C1 and C1, so that, finally, the unknown functions take the form:
X1
M Mj 1 ª § 2 « K ¨¨ M i i : ¬ © l
° º ° · :2 x ¸¸ ON » ®1 ¹ ¼ ° K §¨ M M i M j ¨ i °¯ l ©
· x ¸¸ ON ¹
(A30)
½ 1 1 ª º ª º «i : 2 KM i ON » sh: l x « j : 2 KM j ON » sh:x °° ¼ ¬ ¼ ¬ ¾; sh:l ° °¿ X2
M Mj E Ib § ¨Mi i l E I b EI ¨©
·° · e § e: 2 OlN ¸®1 x ¸¸ 1 2 ¨ K ¨ ¸ M i l x M j x ¹°¯ K M i l x M j x OlN : © ¹
>
1 1 ª º ª «¬i : 2 KMi ON »¼ sh: l x «¬j : 2 KM j ON sh:l
@
º»¼ sh:x ½°° ¾ ° °¿
Relations (A30) are valid if:
i d N ; j d N, and act directly on the concrete part of the section. If they act on the metal part, in relations (A30) we introduce the values: i j 0 . It should be emphasized that the above results were obtained using the hypothesis that the materials composing the structure are subjected to stresses in the elastic linear field, so that the use without restrictions of the effects superposition principle (Boltzmann) is allowed. The analytical expressions of the axial forces X1(r) generated in the concrete part of the section of coordinates “r” of the analyzed composite structure with continuous collaboration allow the following comment. If we take into account that, by using the method of the transformed sections, axial effort Nb in the concrete part of the section of coordinate “r ” of the composite structure has the value:
Nb
³V
Ab
b
dA
Appendix 1
239
and, if we consider that:
Vb
M r y; IT
Sb
³ ydA,
Ab
we obtain:
Sb M r IT
N b r
(A31)
where we have denoted: IT – the inertial axial moment of the entire (transformed) section. Therefore, the diagram of the axial force Nb has the aspect of the diagram of bending moments, from which it is obtained by the multiplication by the factor S b . IT In what follows, if we take into account that, for a section of coordinates “r”,
K :2
Sb IT
it results that the expression of the axial effort X1(r) in the concrete part of the current section “r”, obtained by the integration of differential equation (A01) in different loading situations, has the following generic expression:
X 1 r
N b r 1 f r
(A32)
in the definition field r (0,1) of the function X1(r), f(r) d 1. It results that the diagram of the axial force X1(r) is obtained from the diagram of the axial force Nb(r) obtained by the method of the transformed sections multiplied by the correction factor (1 – f(r)) d 1 and, thus, X1(r) d Nb(r), for 0 d r d 1. Therefore, the axial force determined by the integration of differential equation (A01) has a smaller value than the one determined by the traditional method of the transformed sections and confirms the results obtained for the initial phase by the integration of the differential equation describing the behavior of the composite structures mentioned in Chapter 5.
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Appendix 2
Systems of Integral and Integro-differential Equations
Method for solving the systems of integral and integro-differential equations used in the analysis of resistance structures
Although we have presented throughout this work basic indications concerning the methods of integration for integral and integro-differential equations that represent the mathematical model of the behavior of composite resistance structures, we consider that the work would be incomplete without Appendix 2. It provides the reader with instruments that will help him to use the proposed mathematical model. We have considered necessary and useful the detailed analysis of the original methods elaborated by the author with the purpose of integrating the systems of integro-differential equations used throughout this work since these methods can be regarded as the most adequate for solving the analyzed problem. We also have to mention that the systems of integro-differential equations of the type used in this work can be easily transformed into systems of integral equations by performing a basic substitution. The three integration methods for the systems of integral or integro-differential equations correspond to those obtained by the construction of the mathematical models for the types of composite resistance structures analyzed (with discrete collaboration, with continuous collaboration and with complex collaboration). They are identical in their broad outlines and are based on the same hypothesis:
242
Materials with Rheological Properties
– the systems of integro-differential equations that govern the modeled process are known both for the case when the process is described from its initiation at the moment W (t t W) and for the case when the description begins at an unspecified moment ș ( t t ș, ș (W ,T)) from process development; – the value of the unknown functions ^x(t)` is known for a given moment t (t (W, T)) from the analysis interval as it follows: - the values of the unknown functions ^x(W)` at the moment of process initiation W, - the values of the unknown functions ^x(ș)` at the moment ș (W < ș < T) situated in the interval of process development; – the unknown functions ^x(t)` are known in the time interval W t t t ș, between the moment W of the process initiation and the moment ș, when the systems of integro-differential equations are used to describe the development of the process starting from an unspecified moment ș (W < ș < T) of the development interval. 1. Integro-differential equations whose unknown factors are functions of one variable
We consider the system of integro-differential equations which models the behavior of the composite structures with discrete collaboration, corresponding to the case when the process is described from its initiation at the moment W;
¦ >G x W ' @K W K W K T W m
j
j
0j
j
1j
j
2j
j 1
m T
>
@
d ¦ ³ G j >x t @K 0 j t W j 1 K 1 j t W j K 2 j T t dt dt j 1W
(App01)
0
whose solutions must satisfy the initial conditions imposed by the system of equations:
G W x W ¦ ' j K 0 j W j 0 m
j 1
(App02)
Appendix 2
243
We consider the system of integro-differential equations which modulates the behavior of the composite structures with discrete collaboration, corresponding to the case when the description of the process begins at an unspecified moment ș (t t ș, ș (W,T)) from process development interval:
¦ >G x W ' @K W K W >K T W K T W @ m
j
j
0j
j
1j
j
2j
2j
j 1
m T d ¦ ®³ G j >x t @K 0 j t W j 1 K1 j t W j K 2 j T t dt dt j 1 ¯T
>
T
³G j W
(App01’)
@
>
@
d >x t @K 0 j t W j K1 j t W j K 2 j T t K 2 j T t dt` 0 dt
and values ^x(ș)` of unknown functions ^x(t)` at moment ș. Obviously, if functions ^x(t)` represent the solution of system of integrodifferential equations (App01) then they verify both system (App01) and the system (App01’) that describe the same process and vice versa. It is admitted (for the case of the mathematical model describing the behavior of the composite structures, the demonstration was given in section 3.2.1) that unknown functions ^x(t)` are continuous and indefinitely differentiable. Consequently, functions ^x(t)` and their derivatives verify, for any values of argument t (W,T), the integro-differential equations of systems (App01) and (App01’), and respectively, of their derivatives of any order. It should be emphasized that we did not impose any restriction on the size of intervals (W,T) and (ș, T) where the systems of integro-differential equations (App01) and (App01’) operate. Therefore, we can affirm that systems of integrodifferential equations (App01) and (App01’) and their derivatives of any order are also true when T o W and, respectively, T o ș. In these conditions, system of equations (App02) expressing the initial conditions (for t = W) imposed on unknown functions ^x(t)` together with the equations system obtained by successive differentiation of system of integrodifferential equations (App01) and by carrying out T = W, form a linear algebraic equation system, that has the following expression in matrix representation:
244
Materials with Rheological Properties
G W x W ¦ ' j K 0 j W j 0 m
j 1
m dK 2 j W W d G W >x W @ ¦ G j x W ' j K 0 j W j K 1 j W j dT dT j 1
>
G W
@
m dK W W º ª dK W d2 >x W @ ¦ ^G j d >x W @« 0 j j K 0 j W j K 1 j W j 2 j » 2 dT dT dT j 1 ¬ dT ¼
d 2 K W W ½ d ª >x W @ ' j º» K 0 j W j K 1 j W j 2 j 2 °¾ «G j dT ¬ dT ¼ °¿
G W
(App03)
0
0
2 m dK W dK W W º d3 >x W @ ¦ G j d 2 >x W @ª«2 0 j j K 0 j W j K1 j W j 2 j » 3 dT dT dT dT j 1 ¬ ¼
Gj
d 2 K W d 2 K W W d >x W @°® 0 j2 j 2 K 0 j W j K1 j W j 2 j 2 dT dT °¯ dT
dK 1 j W j º dK 2 j W W ½ ª dK 0 j W j K 1 j W j K 0 j W j « ¾ » dt ¼ dT ¬ dT ¿
>
@
G j x W ' j K 0 j W j K 1 j W j
# G W
d 3 K 2 j W W dT 3
0
m d k K 2 j W W dk > @ x x K K " ' W G W W W ¦ j j j j 0j 1j dT k dT k j 1
>
@
0
Solved successively, matrix equations (App03) give the values of unknown 2 k functions ^x(t)` and values ® d x W ½¾, ® d x W ½¾,!, ® d x W ½¾ of their derivatives 2 k ¯ dT ¿ ¯ dT ¿ ¯ dT ¿ at moment W of the process initiation. Therefore, we can determine the values of the unknown functions ^x(t)` at moment ș = W + 't, W < ș < T, by using the Taylor series expansion of functions ^x(t)`. 2 k ½ ½ ^x T1 ` ^x W ` 1 ® dx W ½¾'t1 1 ® d x W2 ¾ 't1 2 ...... 1 ® d x kW ¾ 't1 k (App04)
1! ¯ dT ¿
2! ¯ dT
¿
k! ¯ dT
¿
The values ^x(ș1)`, at moment ș1 = W + 't1, of unknown functions ^x(t)` constitute the initial values for the following calculation stage. The size of step 't1 is imposed by the precision required for the determination of the values of the unknown functions. Generally, this size is small in comparison to
Appendix 2
245
the duration of the process described by equation system (App01). In order to determine the values of functions ^x(t)` in any moment of the development interval of the modeled process, we impose the carry over of the calculation process by using the equation systems obtained by the successive differentiation of the integrodifferential equation systems (App01’). By successively differentiating equation (App01’) and by carrying out T = ș, we obtain a succession of linear algebraic equations:
G T
m dK T W d >x T @ ¦ ^G j x W ' j K 0 j W j K1 j W j 2 j dt dT j 1
>
@
dK T t ½ d ³ G j >x t @K 0 j t W j K1 j t W j 2 j dt ¾ dt dT ¿ W
(App03’)
T
G T
0
m dK 2 j T T º ª dK T W j d2 K 0 j T W j K1 j T W j >x T @ ¦ ®G j d >x T @« 0 j » 2 dt dT dT dt j 1¯ ¬ ¼
d 2 K 2 j T W G j x W ' j K 0 j W j K1 j W j dT 2 T d 2 K 2 j T t ½° d ³ G j >x t @K 0 j t W j K1 j t W j dt ¾ 0 dt dT 2 °¿ W 3 2 m dK 2 j T T º ª dK T W j d d G T 3 >x T @ ¦ G j 2 >x T @«2 0 j K 0 j T W j K1, j T W j » dt dT dT dt j 1 ¬ ¼
>
Gj
@
d 2 K T W d 2 K T T d >x T @°® 0 j 2 j 2 K 0 j T W j K1 j T W j 2 j 2 dt dT dT °¯
dK1 j T W j º dK 2 j T T ½ ª dK 0 j T W j K1 j T W j K 0 j T W j « ¾ » dT dT dT ¿ ¬ ¼ 3 d K 2 j T W G j x W G j K 0 j W j K1 j W j dT 3
>
T
@
³G j W
d 3 K T t d >x t @K 0 j t W j K1 j t W j 2 j 3 dt dt dT
0
#
G T
d k K 2 j T W dk > @ T G T W W " ' x x K K j j j j 0j 1j dt k dT k
T
³G j W
>
@
d k K T t d >x t @K 0 j t W j K1 j t W j 2 j k dt dt dT
0
246
Materials with Rheological Properties
The successive solving of the linear algebraic equations (App03’) gives the values, at the unspecified moment și, W < și < T, of the derivatives dk ^x(t )`, k 1,2,3,!, n of the unknown functions. dt k The initial values ^x(și)` of the unknown functions corresponding to calculation stage “i” are the end values obtained in the previous calculation stage (i – 1). The values of the integrals contained by equations (A03’) will also be determined using the Taylor series expansion that approximates functions ^x (t)` in the stages preceding the calculation stage. By using the Taylor series expansion of unknown functions ^x(t)`, we obtain their final value in calculation stage i:
^x T i 1 ` ^x T i ` 1 ® dx T i ½¾ ' t i 1! ¯
dt
¿
1 d 2 x T i ½ 1 d k x T i ½ 2 k ® ¾ ' t i ..... ® ¾ ' t i 2 k ! ¯ dt k ¿ 2! ¯ dt ¿
(App04’)
By systematization and employing the symbols and relations used in our work in section 4.2.2, the system of matrix equations (App03) takes, if we use the first eight successive derivatives of integro-differential equation system (App01), the following configuration: X ½ ªA º ° dX ° « » ° ° A « » ° dt ° »° . ° « A »° . ° « A »° « ° »® . ¾ « A »° . ° « A »° « ° »° . ° « A »° « ° A » ° 8. ° « d X » « A¼ ° 8 ° ¬ ¯ dt ¿
ª B11 « . « « . « « . « . « « B61 «B « 71 « B81 «B ¬ 91
B 22 B33 . . . . . B92
B93
.
.
.
.
.
1 ½ º° ° »° X ° » ° dX ° » ° dt ° »° . ° »° ° »® . ¾ »° . ° »° ° »° . ° »° ° » ° 7. ° d X » B99 ¼ ° 7 ° ¯ dt ¿
(App05)
Appendix 2
247
In a similar way, and also by using the first (eight) successive derivatives of integro-differential equations system (App01), matrix equation system (App03’) takes the form: dX ½ º° ªA ° » ° dt. ° « A »° « ° »° . ° « A »° « ° A »° . ° « ® » . ¾ « A ° »° « A »° . ° « ° » . ° « A »° 8 ° « A»¼ ° d X ° ¬« °¯ dt 8 °¿
ª C1 «C « 2 «C 3 « « . « . « « . « . « ¬«C8
B33 B43
B44
B53 B63
B55 B66
B73
B77
B83 B93
B88 B94
B95
B96
B97
B98
1 ½ º ° dX ° »° ° » ° dt ° »° . ° »° . ° ° »° »® . ¾ »° . ° ° »° »° . ° »° d 7 X ° B99 ¼» °° 7 °° ¯ dt ¿
In matrix equation systems (App05), (App05’), we have denoted: A
¦D K j
0J
j
B11
2¦ E xT F j E 0 K 0 j j
B21
dK 2 j
2¦ E xT F j E 0T1
dT
j
dK 2 j
B22
D jT1
B31
2¦ E xT F j E 0T1
dT d 2K2 j dT 2
j
B32
¦D T
j 1
j
B33
dT 2
dK · § dK 0 j T1 2 j ¸¸ dT ¹ © dT
¦ D ¨¨ j
j
B41
d 2K2 j
2¦ E xT F j E 0T1
d 3K2 j
j
B42
¦ D jT1 j
B43
j
dT 3 2
T2
dK 2 j
2T1
dT © dT dK · § dK ¦j D j ¨¨ 2 dT0 j T1 dT2 j ¸¸ © ¹ j
B44
d 3K2 j
§ d 2 K0 j
¦ D ¨¨
dT 3
d 2K2 j · ¸ dT 2 ¸¹
(App05’)
248
Materials with Rheological Properties
B51
d 4K2 j
2¦ E xT F j E 0T1
dT 4
j
B52
d 4K2 j
¦D T
j 1
dT 4
j
B61
2¦ E xT F j E 0T1
d 5K2 j
j
B62
¦D T
d K2 j
j 1
j
B64
dT 5
5
dT 5
§ d 3K0 j
¦ D ¨¨ 4 j
3T3
dK 2 j
8T2
d 2K2 j
3 dT dT 2 © dT § d 2K d 2K dK ¦j D j ¨¨ 6 dT 0 j 3T2 dT2 j 4T1 dT 22 j © dK · § dK ¦j D j ¨¨ 4 dT0 j T1 dT2 j ¸¸ © ¹ 6 d K 2j 2¦ E xT F j E 0T1 dT 6 j
6T1
j
B65 B66
B71 B72
¦D T
B74 B75 B76
B82
dT 6
§ d 5K0 j dK 2 j d 2K2 j d 3K2 j d 4K2 j d 5K2 j ¨ D T T T T T 5 10 10 5 ¦j j ¨ dT 5 5 dT 1 2 3 4 dT 2 dT 3 dT 4 dT 5 © § d 4 K0 j d 4 K2 j d 3K2 j dK 2 j d 2K2 j ¨ 5 4 15 20 10 T T D T T ¦j j ¨ dT 4 1 2 4 3 dT 4 dT 3 dT 2 dT © § d 3K dK d 2K d 3K · ¦j D j ¨¨10 dT 30 j 6T3 dT2 j 15T2 dT 22 j 10T1 dT 32 j ¸¸ © ¹ §
¦ D ¨¨10 j
d 2 K0 j
4T2
dK 2 j
5T1
dT 2 dT © dK · § dK ¦j D j ¨¨ 5 dT0 j T1 dT2 j ¸¸ B81 © ¹ d 7 K2 j ¦j D jT1 dT 7 j
B77
· ¸ ¸ ¹
d 6K2 j
j 1
j
B73
d 3K2 j · ¸ dT 3 ¸¹
d 2K2 j · ¸ dT 2 ¸¹
2¦ E xT F j E 0T1 j
d 7 K2 j dT 7
· ¸ ¸ ¹
· ¸ ¸ ¹
Appendix 2
B83
§ d 6K0 j dK 2 j d 2K2 j d 3K2 j ¨ D T 6 T 15 T ¦j j ¨ dT 6 6 dT 5 4 dT 2 dT 3 ©
20T3
B84
d 4K2 j dT
§ d 5K0 j
j
B87
dT 5
©
5
15T1
dT 4
§ d 4K0 j D ¦j j ¨¨15 dT 4 © § d 3K0 j ¦j D j ¨¨ 20 dT 3 10T3 © § d 2 K0 j D ¦ j j ¨¨15 dT 2 5T2 © dK 2 j § dK 0 j T1 dT dT ©
¦ D ¨¨ 6
B88
j
j
B91
¦ D jT1
§ d 7 K0 j
j
35T3 B94
B95
© dT
d 5K2 j dT 5
7
T7
d 3K2 j · ¸ dT 3 ¸¹
d 2K2 j · ¸ dT 2 ¸¹
· ¸¸ ¹
dT 5
21T1
§ d 5K0 j D ¦j j ¨¨ 21 dT 5 © d 4K2 j dT
4
dK 2 j dT
7T6
d 2K2 j dT
2
21T5
d 3K2 j dT
3
35T4
d 4K2 j dT 4
d 7K2 j · ¸ dT 6 dT 7 ¸¹ dK 2 j d 2K2 j d 3K2 j d 4K2 j 6T6 35T 84 115 T T 4 3 dT dT 2 dT 3 dT 4
21T2
d 5K2 j
105T2
dT
6T1
dT 2
15T1
dT 8
§ d 6 K0 j D ¦j j ¨¨ 7 dT 6 ©
70T2
dK 2 j
d 2K2 j
dT 8
¦ D ¨¨ j
dT
24T2
d 8K2 j
j
B93
dK 2 j
d 8K2 j
2¦ E xT F j E 0T1 j
B02
6T1
d 5K2 j · ¸ dT 5 ¸¹ dK 2 j d 2K2 j d 3K2 j d 4K2 j T T 45 20 10T4 36T3 2 1 dT dT 2 dT 3 dT 4
d 4K2 j
40T2
B86
d 6K2 j · ¸ dT dT 6 ¸¹ dK 2 j d 2K2 j d 3K2 j 5T5 24T4 45T3 2 dT dT dT 3 d 5K2 j
15T2
4
¦ D ¨¨ 6 j
B85
249
d 6K2 j
7T1
d 6 K2 j · ¸ dT 6 ¸¹ dK 2 j d 2K2 j d 3K2 j 126 T 15T5 70T4 3 dT dT 2 dT 3
35T1
d 5K2 j · ¸ dT 5 ¸¹
· ¸ ¸ ¹
250
Materials with Rheological Properties
B96
§ d 4K0 j dK 2 j d 2K2 j ¨ D T T 35 20 70 ¦j j ¨ dT 4 4 3 dT dT 2 ©
84T2
B97
d 3K2 j 3
dT
§
¦ D ¨¨ 35
35T1
d 3K0 j
j
d 4K2 j · ¸ dT 4 ¸¹
15T3
dK 2 j
35T2
dT dT 3 dT 2 © § d 2K0 j dK 2 j d 2K2 j · ¨ ¸ D T T 21 6 7 ¦j j ¨ dT 2 2 1 dT dT 2 ¸¹ © dK · § dK ¦j D j ¨¨ 7 dT0 j T1 dT2 j ¸¸ ¹ © T w X dK 2 j B21 ¦ ³ D j T dt w T 1 dT j W j
B98 B99
C1
T
C2
B31 ¦ ³ D j j W
B41 ¦ ³ D j j W
T
C4
B51 ¦ ³ D j j W
T
C5
B61 ¦ ³ D j j W
T
C6
B71 ¦ ³ D j j W
T
C7
B81 ¦ ³ D j j W
T
C8
B91 ¦ ³ D j
Dj
E xT r E x
j W
T1 T2 T3
2
w X d K2 j T w T 1 dT 2 dt
T
C3
d 2K2 j
3
w X d K2 j T dt w T 1 dT 3 4
w X d K2 j T dt w T 1 dT 4 5
w X d K2 j T dt w T 1 dT 5 6
w X d K2 j T1 dt wT dT 6 7
w X d K2 j T1 dt wT dT 7 8
w X d K2 j T1 dt wT dT 8
K 0 j K1 j dK 0 j
K1 j K 0 j
dT d 2K0 j dT 2
K1 j 2
dK1 j
dT dK 0 j dK 1 j dT
dT
K0 j
d 2 K1 j dT 2
21T1
d 3K2 j · ¸ dT 3 ¸¹
Appendix 2
251
§ d 2 K 0 j dK 1 j dK 0 j d 2 K 1 j · d 3 K1 j ¨ ¸ K0 j 3 ¨ dT 2 dT dT dT 2 ¸¹ dT 3 K 1 j dT 3 © d 4 K0 j d 3 K 0 j dK1 j d 2 K 0 j d 2 K1 j dK 0 j d 3 K1 j T5 K 4 6 4 1j dT 4 dT 3 dT dT 2 dT 2 dT dT 3 d 4 K1 j K0 j dT 4 5 § d 3 K 0 j d 2 K1 j d 2 K 0 j d 3 K1 j · d K0 j d 4 K 0 j dK 1 j ¸ ¨ T6 K 5 10 1j ¨ dT 3 dT 2 dT 5 dT 4 dT dT 2 dT 3 ¸¹ © dK 0 j d 4 K1 j d 5 K1 j 5 K0 j 4 dT dT dT 5 5 6 d K0 j d K 0 j dK1 j d 4 K 0 j d 2 K1 j T7 K 6 15 1j dT 6 dT 5 dT dT 4 dT 2 d 3 K 0 j d 3 K1 j d 2 K 0 j d 4 K1 j dK 0 j d 5 K1 j d 6 K1 j 20 15 6 K0 j 3 3 2 4 5 dT dT dT dT dT dT dT 6 T4
d 3K0 j
In order to simplify and to make the expression more suggestive, we have denoted:
½ dk ® k x t ¾ ¿T ¯ dT
W
½ dk ® k x W ¾ ¿ ¯ dT
We can stipulate that the solutions of integro-differential equation systems (App01’), (App01) corresponding to the unknown functions of one variable are approximated by successively solving linear algebraic equation systems (App03), (App03’) or their condensed expressions (App05), (App05’). 2. Integro-differential equations whose unknown factors are functions of two variables
Given the integro-differential equation system describing the behavior of the composite structures with continuous collaboration starting from moment W of process initiation:
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Materials with Rheological Properties
¦ >- r x r,W ' r @K W K W K T W m
j
0j
j
j
1j
j
2j
j 1
m T
¦ ³ - j r j 1W
T
O r ³ W
w >x r , t @K 0 j t W j >1 K1 j t W j K 2 j T t @dt (App06) wt
w3 >x r , t @dt 0 w 2 rw t
whose solutions must satisfy (in the moment of process initiation) the initial conditions imposed by the equation system:
O r
m w2 > @ x r , t r , W x r , W ' j r K 0 j W j 0 ¦ w r2 j 1
(App07)
and the integro-differential equation system describing the behavior of those structures from moment ș, W < ș < T, of process development interval:
¦ >- r x r,W ' r @K W K W >K T W K T W @ m
j
j
0j
j
1j
j
2j
2j
j 1
m T
¦ ³ - j r j 1T
m T
¦ ³ - j r j 1W
w >x r , t @K 0 j t W j >1 K1 j t W j K 2 j T t @dt wt
(App06’)
w >x r , t @K 0 j t W j K1 j t W j >K 2 j T t K 2 j T t @dt wt
w3 >x r , t @dt 0 2 T w rw t T
O r ³
and the system of values ^x(r, ș)` of unknown functions ^x(r, t)` at moment ș, W < ș < T, which constitute the initial conditions that the solutions of equation system (App06’) must satisfy. By hypothesis, integro-differential equations (App06), (App06’) have a common solution (they describe the same process). Unknown functions ^x(r, t)` are continuous and indefinitely differentiable according to time. Therefore, they verify for each possible value of arguments r and t (falling, thus, in their definition domain), the integro-differential equations (App06), (App06’) as well as their derivative of any order.
Appendix 2
253
By successively differentiating integro-differential equation system (App06), for T = W, we obtain the following systems of differential equations with partial derivatives: m w2 > @ x r , W r , W x r , W ' j r K 0 j W j 0 ¦ w r2 j 1 w3 w O r 2 >x r ,W @ - r ,W >x r ,W @ wT w r wT m dK 2 j W W ¦ >- j r x r ,W ' j r @K 0 j W j K1 j W j 0
O r
(App08)
dT
j 1
2 m w4 >x r ,W @ - r ,W w 2 >x r ,W @ ¦- j r w >x r ,W @ 2 2 wT w r wT wT j 1
O r
dK 2 j W W º ª dK 0 j W j « K 0 j W j K1 j W j » dT ¼ ¬ dT m d 2 K1 j W W ¦ - j r x r ,W ' j r K 0 j W j K1 j W j 0 dT 2 j 1
>
@
#
w k 2 wk W W x r r , , > @ >x r ,W @ " wr 2wT k wT k
O r
>
@
¦ - j r x r ,W ' j r K 0 j W j K1 j W j m
j 1
d k K 2 j W W
0
dT k
By proceeding in a similar way with integro-differential equation system (App06’), for T = ș, we obtain:
O r
w3 wr 2 dT
>x r ,T @ - r ,T w >x r ,T @
>
wT
@
¦ - j r x r ,W ' j r K 0 j W j K1 j W j m
j 1
m T
¦ ³ - j r j 1W
dK 2 j T W j dT
(App08’)
dK T t w >x r , t @K 0 j t W j K1 j t W j 2 j dt wt dT
0
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Materials with Rheological Properties
O r
m w w2 w4 > @ > @ , , , T - j r >x r ,T @u x r r x r T T ¦ 2 2 2 wT wT wr wT j 1
dK 2 j T T º ª dK 0 j T W j u« K 0 j T W j K 1 j T W j » dT dT ¼ ¬ m d 2 K 2 j T W j ¦ - j r x r ,W ' j r K 0 j W j K 1 j W j dT 2 j 1
>
@
d 2 K 2 j T t w ¦ ³ - j r >x r , t @K 0 j t W j K 1 j t W j dt wt dT 2 j 1W m T
0
#
O r
k w k 2 >x r ,T @ - r ,T w k >x r ,T @ " k 2 wr wT wT
>
@
¦ - j r x r ,W ' j r K 0 j W j K1 j W j m
j 1
m T
¦ ³ - j r j 1W
d k K 2 j T W j dT k
d k K T t w >x r , t @K 0 j t W j K1 j t W j 1 j k dt wt dT
0
By setting t = W and by making the substitution:
w >x r , T @T W x1 1 r ,W ; wT w2 >x r , T @T W x2 2 r ,W ; wT 2 # wk >x r , T @T wT k
W
x k k r , W
equation system (App07) expressing the initial conditions that must be satisfied by the solutions of integro-differential equation system (App06) and by system of equation with partial derivatives (App08) change into the following system of ordinary linear differential equations of order 2:
d2 x r , W - r , W x r ,W r 0; dr d2 O r 2 x 1 r , W - r , W x 1 r , W 1 r dr O r
(App09)
0;
Appendix 2
O r
255
d 2 2 x r , W - r , W x 2 r , W 2 r 0; dr 2
# O r
d 2 k x r ,W - r ,W x k r ,W k r 0, dr 2
where we have noted: m
¦ ' r K
r
j
j 1
W ; j
dK1 j W W
m
¦ >- r x r ,W ' r @K W K j dT ; dK W W º ª dK W r ¦ °®- r x r ,W « K W K W » dT dT °
1 r 2
0j
j
0j
j
j 1
0j
1
¯
1j
0j
j
1j
j
¬
>
@
- j r x r ,W ' j r K 0 j W j K1 W j
k r
1j
j 1 m
#
j
1j
j
¼
d K1 j W W ½° ¾; dT 2 °¿ 2
° d k K 0 j W W ½° k r x r r K K W ' W W " , ® ¾ ¦ 0j 1 j j j j dT k °¿ j 1° ¯
>
m
@
By solving successively differential equation systems (App09), we determine the value, at moment W, of unknown functions ^x(r,t)` and their derivatives ^x (1)(r,t)`, ^x (2)(r,t)`, ..., ^x(k)(r,t)`. If we set, this time, the coordinate of section r = ri, we can write, by using the expansion in Taylor series, the value of unknown functions ^x(r,t)` in the chosen section ri, at moment ș1 = W + 't1:
^x ri , T1 ` ^x ri , W ` "
't1 k!
k
't 't1 1 x ri , W 1 1! 2!
^
^x r ,W `; i k
i
`
2
^x r ,W ` 2
i
(App10)
1,2,3, ! , n
The size of step 't1 is limited by the precision required for the determination of the values of the unknown functions. Therefore, it is necessary to use equations (App08’) after having set the time t = ș, to determine the values of the unknown
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Materials with Rheological Properties
functions for any value in the domain of existence of the independent variables. By making the substitution:
w >x r , T @T T x1 1 r ,T : wT w2 >x r , T @T T x2 2 r ,T ; 2 wT # wk >x r , T @T T xk k r ,T wT k The time being fixed, t = ș, the differential equations with partial derivatives (App08’) change into the following system of ordinary linear differential equations of order 2:
d 2 1 x r ,T - r ,T x 1 r ,T F1 r ,T 0; dr 2 d2 O r 2 x 2 r , T - r , T x 2 r , T F2 r , T 0; dr # d2 O r 2 x k r ,T - r ,T x k r ,T Fk r ,T 0 dr O r
(App09’)
where the following notations have been used: F1 r , T
¦ ®>- r x r ,W ' r @K W K W m
j 1
¯
j
j
0j
T
³ - j r x 1 r , t K 0 j t W j K 1 t W j W
F2 r , T
j 1
>
ª ° ¦ ®°- r x r , T « m
¯
1
j
¬
j
dK 0 j T W j dT
@
³ - j r x 1 r , t K 0 j t W j K 1 j t W j W
#
dK 2 j T W j
j
dK 1 j T t
- j r K r ,W ' j r K 0 j W j K 1 j W j T
1j
dT
dT
½ dt ¾; ¿
K 0 j T W j K 1 j T W j
d 2 K 2 j T W j dT 2
d 2 K 2 j T t dT
2
½° dt ¾; °¿
dK 2 j T T º » dT ¼
Appendix 2
Fk r , T
257
m
¦ ^" j 1
"
>
@
- j r x r ,W ' j r K 0 j W j K 1 j W j T
d k K 2 j T W j
³ - j r x 1 r , t K 0 j t W j K 1 j t W j W
dT k
d k K 2 j T t dT k
dt `.
By successively solving system of differential equations (App09’), we determine the value, at moment ș, W < ș, of derivatives ^x(1)(r,ș)`, ^x(2)(r,ș)`, ^x(3)(r,ș)`, ..., ^x(k)(r,ș)` of the unknown functions. By setting variable r = ri, we write, using the Taylor series expansion, the value of unknown functions ^x(r,t)` at moment și + 1 = ș + 'ti, in the selected ri section.
^x ri ,T i1 ` ^x ri ,T i ` 'ti ^x1 1 ri ,T i ` 'ti
2
't k i ^x k r ,T `; i i
k!
1!
i
2!
^x r ,T ` ... 2
i
i
(App10’)
1,2,3,!, n
By systematizing, differential equation systems (App09) and (App09’) take, if we use the first eight successive derivatives of equations (App06) and (App06’), the following condensed form, similar to that obtained by solving the integrodifferential equation systems having unknown functions of only one variable:
ª O « « « « « « « « « « « « ¬
O O O O O O O
º » » » » » d2 »u 2 » dr » » » » O »¼
X r ,W ½ ° X 1 r , W ° ° ° ° X 2 r , W ° ° ° 3 ° X r , W ° ° ° 4 ® X r , W ¾ ° X 5 r , W ° ° ° 6 ° X r , W ° ° ° 7 ° X r , W ° °¯ X 8 r , W °¿
258
Materials with Rheological Properties
ªD º X r ,W ½ « » ° X 1 r ,W ° D ° « » ° « » ° X 2 r ,W ° D ° « » ° 3 D (App10) « » ° X r ,W ° » u °® X 4 r ,W °¾ « D « » ° 5 ° D « » ° X r ,W ° « » ° X 6 r ,W ° D « » ° 7 ° D « » ° X r ,W ° « D »¼ ¯° X 8 r ,W ¿° ¬ 1 ª E11 º ½ «E » ° ° « 21 E 22 » ° X r ,W ° « . » ° X 1 r ,W ° E 33 « » ° 2 ° E 44 « . » ° X r ,W ° « . » u °® X 3 r ,W °¾ E 55 « » ° 4 ° E 66 « . » ° X r ,W ° « . » ° X 5 r ,W ° E 77 « » ° 6 ° E 87 E 88 . . . . « E 81 E 82 » ° X r ,W ° 7 «E » ¬ 91 E 92 E 93 E 94 E 95 E 96 E 97 E 98 E 99 ¼ °¯ X r ,W °¿ and, respectively:
ª O « O « « O « O « « O « O « « « «¬
º » » » » 2 »u d » dr 2 » » » O » O »¼
X 1 r , T ½ ° ° 2 ° X r , T ° ° X 3 r , T ° ° ° 4 ° X r , T ° ¾ ® 5 ° X r , T ° ° X 6 r , T ° ° ° 7 ° X r , T ° ° X 8 r , T ° ¿ ¯
Appendix 2
ªD º X 1 r , T ½ ° « » ° 2 D « » ° X r , T ° « » ° X 3 r , T ° D ° « » ° 4 D ° X r , T ° « » u « » ® X 5 r , T ¾ D ° « » ° 6 D « » ° X r , T ° « » ° X 7 r , T ° D ° « » ° D »¼ °¯ X 8 r , T °¿ «¬ 1 ª9 1 º ½ «9 E » ° X 1 r ,T ° 33 « 2 » ° ° «9 3 E 43 E 44 » ° X 2 r , T ° « » ° 3 ° . . E 55 «9 4 » u ° X r , T ° «9 5 » ® X 4 r , T ¾ . . E 66 ° « » ° 5 . . E 77 «9 6 » ° X r ,T ° «9 E » ° X 6 r , T ° E 84 E 88 83 ° « 7 » ° 7 «¬9 8 E 93 E 94 E 95 E 96 E 97 E 98 E 99 »¼ °¯ X r , T °¿
259
(App10’)
The expressions of the elements from the matrices of the systems of the integrodifferential equations (App10) and (App10’) are similar to those obtained for the systems of integro-differential equations with unknown functions of only one variable (App05), (App05’). They are determined by using the relations and symbols in section 4.2.3. We should emphasize that the solutions of integro-differential equation systems (App06), (App06’) corresponding to the unknown functions that depend on two variables, are approximated by successively solving the differential linear equation systems of order 2 presented in condensed form in (App10) and (App10’).
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Materials with Rheological Properties
3. Integro-differential equations whose unknown factors are functions of one or two variables
Given the system of integro-differential equations with unknown functions ^x( t)` of only one variable and ^y(r,t)` of two variables: m
¦ j 1
½ ªl º - j r y r ,W dr » ª ' j º ° °ª G j º « ³ K W K W K T W ®« T » x W « 0 » « D j r » ¾ 0 j j 1 j j 2 j ¬ ¼° °¬- j r ¼ r y r , W « » j ¬ ¼ ¿ ¯
ªl º½ w °°ª G j º d « ³K j r > y r , t @dr » °° wt » ¾ K 0 j t W j u ³ ®« T » >x t @ « 0 K r ¼ dt w « »° W °¬ j > @ r y r , t «¬ j wt »¼ ¿° ¯° T
>
@
T
u 1 K1 j t W j K 2 j T t ³ O r W
(App11)
w3 > y r , t @dt 0 w r 2w t
whose solutions must satisfy the initial conditions imposed by the equation system: ªl º 0 ª º m ª Gj º K r y3 r ,W dr » 2 « » ¦ « T » x W « ³ j d K W 0 «0 » 0j j « O r 2 > y r ,W @» j 1 ¬K j r ¼ r y r , W dr ¬ ¼ 3 ¬« j ¼»
(App12)
Equation system (App11) describes a process from moment W of its initiation. Given: m
¦ j 1
½ º ª l K j r y r ,W dr » ª ' j º ° °ª G j º ¦ « x K W K W u W ®« T » » «¬ D j r »¼ ¾° 0 j j 1 j j « 0 °¬K j r ¼ r y r , W j ¼» ¬« ¯ ¿
(App11’)
Appendix 2
261
º½ ªl w °°ª G º d « ³K j r > y r , t @dr » °° j wt »¾ u u K 2 j T W K 2 j T W ³ ®« T » >x t @ « 0 K r ¼ d t w « »° T °¬ j r y r t > @ , «¬ j wt »¼ °¿ °¯
>
@
T
T °ª G j º d u K 0 j t W j 1 K1 j t W j K 2 j T t d t ³ ®« T » >x t @ W ° ¯¬K j r ¼ d t
>
@
º½ ªl w « ³K j r > y r , t @dr » °° wt » ¾ K 0 j t W j K1 j t W j K 2 j T t K 2 j T t dt «0 w »° « > @ r y r , t »¼ ¿° «¬ j wt
>
T ª 0 º »dt w3 ³« > @ r y r , t O » T « wr 2wt ¬ ¼
@
0
the system of integro-differential equations describing the same process from moment ș, ș (W ,T) of the process development and the system of values ^x(ș)` and ^y(r,ș)` of unknown functions ^x(t)` and, respectively, ^y(r,t)`, at moment ș. Functions ^x(t)` and ^y(r,t)` obviously verify both equation systems (App11), (App11’) and are, as we established, continuous and indefinitely differentiable over the entire definition domain. Since there are no restrictions on the intervals (W, T) and, respectively, (ș, T), where equation systems (App11) and (App11’) operate, functions ^x(t)` and ^y(r,t)` must verify the equation systems and their derivatives of any order. This is also true when To W and, respectively, To ș, or, at the limit, for T = W or, according to the case, T = ș. Equation system (App12) expressing the initial conditions (for t = W) imposed on functions ^x(t)` and ^y(r,t)` together with the systems of equations obtained by the successive differentiation of equation system (App11) and by making T = W, form a differential equation system of order 2, which has the following expression, in matrix representation: 0 ª º m « »¦ d2 « O r dr 2 > y r ,W @» j 1 ¬ ¼
º½ ªl °ª G j º « ³K j r y3 r ,W dr » ° K W 0 x W ®« T » »¾ 0 j j «0 °¯¬K j r ¼ «¬ - j r y3 r ,W »¼ °¿
262
Materials with Rheological Properties
0 ª º m « »¦ d 2 1 , O W r y r « » j1 dr 2 ¬ ¼
>
@
ªl º½ 1 °ª G j º 1 « ³ K j r y r , W dr » ° W x ®« T » «0 » ¾ K 0 j W j °¯¬K j r ¼ «¬ - j r y 1 r , W »¼ °¿
½ ªl º dK 2 j W W K r y r , W dr » ª ' j º ° °ª G j º ®« T » x W « ³0 j « 0 K W K W « » D j r » ¾ 0 j j 1 j j K r dt j ¼ ¬ ¼° °¬ , W r y r « » j ¬ ¼ ¯ ¿ º½ ªl 0 ª º m °ª G º K j r y 2 r , W dr » ° j 2 « 2 ³ « » ¦ ®« T » x W 0 d 2 » ¾ K 0 j W j « « O r 2 y r , W » j 1 °¬K j r ¼ dr ¬ ¼ «¬ - j r y 2 r , W »¼ °¿ ¯
>
@
º½ ªl 1 dK 2 j W W º °ª G j º « K r y r , W dr » °ª dK 0 j W j ®« T » x 1 W « ³0 j K 0 j W j K 1 j W j » » ¾« dT r K dT j 1 ¼ ¼ °¬ °¬ » « , r y r W j ¼¿ ¬ ¯ ½ º ªl d 2 K 2 j W W K r y r , W dr » ª ' j º ° °ª G j º « ®« T » x W « ³0 j K 0 j W j K 1 j W j ¾ » » D j r « dT 2 ¬ ¼° °¬K j r ¼ r y r W , » « j ¼ ¬ ¯ ¿ l ª º½ 0 ª º m °ª G º K r y 3 r , W dr » ° « » ¦ ®« T j » x 3 W « ³0 j d 2 3 « » ¾K 0 j W j « O r 2 y r , W » j 1 °¬K j r ¼ dr ¬ ¼ «¬ - j r y 3 r , W »¼ °¿ ¯
>
0
@
ªl º½ 2 dK 2 j W W º °ª G j º 2 « ³ K j r y r , W dr » °ª dK 0 j W j K 0 j W j K 1 j W j ®« T » x W « 0 ¾«2 » » K r dT dT 2 ¼ ¼ °¬ °¬ j » « W , r y r j ¬ ¼¿ ¯ ªl º½ 2 1 d 2 K 2 j W W °ª G j º 1 « K r y r , W dr » °° d K 0 j W j ®« T » x W « ³0 j 2 W W K K ¾ ® j j 0j 1j » ° dT 2 dT 2 1 °¯ °¬K j r ¼ « » W , r y r j ¬ ¼¿ ¯ dK 1 j W j º d 2 K 1 j W W ª dK 0 j W j « K 1 j W j K 0 j W j » dT ¼ dT 2 ¬ dT ½ ªl º d 3 K 2 j W W K r y r , W dr » ª ' j º ° °ª G j º ®« T » x W « ³0 j « 0 ¾ K 0 j W j K 1 j W j » « » dT 3 ¬ D j r ¼ ° °¬K j r ¼ , W r y r « » j ¬ ¼ ¯ ¿ #
ªl º½ k 0 ª º m °ª G º j «³K j r y r ,W dr » ° 2 k « » d W x k ¦ ®« T » «0 » ¾K 0 j W j ! « O r 2 y r ,W » j 1 °¬K j r ¼ dr ¬ ¼ «¬ - j r y k r ,W »¼ °¿ ¯
>
@
½ ªl º d k K 2 j W W K r y r ,W dr » ª ' j º ° °ª G j º ®« T » x W «³0 j « K 0 j W j K1 j W j ¾ » « » D j r dT k ¼° °¬K j r ¼ «¬ - j r y r ,W »¼ ¬ ¯ ¿
(App13)
0
Appendix 2
263
Solved successively, the systems of differential equations composing equation system (App13) give the values ^x(W)`, ^x(1)(W)`, ^x(2)(W)`, ..., ^x(k)(W)` and ^y(r,W)`, ^y(1)(r,W)`, ..., ^y(k)(r,W)` of unknown functions ^x(t)` and ^y(r,t)` and of their derivatives according to time, at the initial moment W. By using the expansion in Taylor series, and by considering fixed the variable r in the case of functions ^y(r,t)`, we can determine the values of the unknown functions at a moment ș1 = W + ' t1, ș1 > W:
^x T1 ` ^x W ` 't1 ^x 1 W ` 't1
2
1!
2!
^x W ` ... 'kt! ^x W ` k
2
^y r,T1 ` ^y r,W ` 't1 ^y 1 r,W ` 't1
2
1!
2!
k
1
(App14)
^y r,W ` ... 'kt! ^y r,W `. k
2
1
k
The size of step 't1 is limited by the precision imposed by the determination of the solutions. Therefore, it is necessary to continue the calculation process by considering the system of initial values ^x(ș1)` and ^y(r,ș1)` for the solutions determined by using the equation systems obtained by the successive differentiation of the system of integro-differential equations (App11’) and by making T = ș. 0 ª º « » d 2 1 r y r , O T « » dr 2 ¬ ¼
>
@
ªl º½ 1 °ª G j º 1 « ³ K j r y r , T dr » ° x T ®« T » «0 » ¾ K 0 j T W j 1 °¬K j r ¼ « »¼ °¿ r y r , T ¬ j ¯
½ ªl º dK 2 j T W K r y r ,W dr » ª ' j º ° °ª G j º K W K W « ®« T » x W « ³0 j « » D j r » ¾ 0 j j 1 j j dT ¬ ¼ ° °¬K j r ¼ r y r , W « » j ¬ ¼ ¯ ¿ ªl º½ 1 °ª G j º 1 « K r y r , t dr » ° ³ ®« T » x t « ³0 j » ¾ K 0 j t W j K1 j t W K r ¼ 1 W °¬ j « »¼ °¿ r y r t , j ¬ ¯ T
0 ª º m « »¦ d 2 2 « O r 2 y r , T » j 1 dr ¬ ¼
>
@
dK dT T t 2j
0
ªl º½ 2 °ª G j º 2 « ³ K j r y r , T dr » ° ®« T » x T « 0 » ¾ K 0 j T W j °¬K j r ¼ «¬ - j r y 2 r , T »¼ °¿ ¯
ªl º½ 1 dK 2 j T T º °ª G j º « K r y r , T dr » °ª dK 0 j T W j ®« T » x 1 T « ³0 j K 0 j T W j K 1 j T W j » » ¾« dT dT 1 ¬ ¼ °¬K j r ¼ ° « » T r y r , j ¬ ¼¿ ¯ ½ ªl º d 2 K 1 j T W K r y r ,W dr » ª ' j º ° °ª G j º ®« T » x W « ³0 j « K 0 j W j K 1 j W j ¾ » « » D j r dT 2 ¼° °¬K j r ¼ «¬ - j r y r ,W »¼ ¬ ¯ ¿ ªl º½ 1 d 2 K 2 j T t °ª G j º 1 « K r y r , t dr » ° ³ ®« T » x t « ³0 j dt ¾ K 0 j t W j K 1 j t W j » r K dT 2 1 ¼ W °¬ j ° « » r y r , t j ¬ ¼ ¯ ¿ T
0
264
Materials with Rheological Properties
ªl º½ 3 (App13’) ° ª G j º 3 « ³ K j r y r , T dr » ° x T ®« T » ¾ K 0 j T W j 0 « » °¬K j r ¼ «¬ - j r y 3 r , T »¼ °¿ ¯ ªl º½ 2 dK 2 j T T º °ª G j º 2 « K r y r , T dr » °ª dK 0 j T W j K 0 j T W K 1 j T W ®« T » x T « ³0 j ¾«2 » » r K dT dT 2 ¼ °¬ j ¼ °¬ « » , r y r T j ¬ ¼ ¯ ¿ º½ 2 ªl 1 d 2 K 2 j T T °ª G j º 1 « K r y r , T dr » °° d K 0 j T W j 2 K K ®« T » x T « ³0 j T W T W ® ¾ 0 1 j j j j » ° dT 2 dT 2 1 °¯ °¬K j r ¼ » « , r y r T j ¼¿ ¬ ¯ dK 1 j T W j º dK 1 j T T °½ ª dK 0 j T W j K 1 j T W j K 0 j T W j « ¾ » dT dT dT °¿ ¬ ¼ l ª º d 3 K 2 j T W K r y r ,W dr » ª ' j º ½° °ª G j º K 0 j W j K 1 j W j ®« T » x W « ³0 j « ¾ » « » D j r ° dT 32 ¬ ¼¿ °¬K j r ¼ , r y r W « » j ¬ ¼ ¯
0 ª º m « »¦ d 2 3 « O r 2 y r , T » j 1 dr ¬ ¼
>
@
ªl º½ 1 d 3 K 2 j T T °ª G j j º 1 « K r y r , t dr » ° W W K t K t dt ³ ®« T » x t « ³0 j ¾ 1j j j » 0j K r ¼ dT 3 1 W °¬ j ° « » , r y r t j ¬ ¼¿ ¯ T
0
#
0 º m ª »¦ « d 2 k r O T y r , » j1 « dr 2 ¼ ¬
>
@
º½ ªl k °ª G j º k « ³ K j r y r , T dr » ° ®« T » x T « 0 » ¾K 0 j T W j " °¯¬K j r ¼ «¬ - j r y k r , T »¼ °¿
½ º ªl d k K 1 j T W j K r y r , W dr » ª ' j º ° °ª G j º « ®« T » x W «³0 j ¾K 0 j W j K 1 j W j » » « D j r ¼ ° d kT °¬K j r ¼ «¬ - j r y r , W »¼ ¬ ¯ ¿ º½ ªl 1 d k K 2 j T T °ª G j º 1 «³ K j r y r , t dr » ° ³ ®« T » x t « 0 dt ¾ K 0 j t W j K 1 j t W j » K r ¼ dT k 1 W °¬ j ° » « K r y r , t j ¼ ¬ ¯ ¿ T
0
The successive solving of equation systems (App13’) leads to the values ^x(1)(și)`, ^x (și)`, ..., ^x(k)(ș1)` and ^y(1)(r,W)`, ^y(2)(r,W)`, ..., ^y(k)(r,W)` of the derivatives of the order 1, 2, ..., k according to the time of the unknown functions ^x(t)` and ^y(r,t)` at moment și. Therefore, by considering the initial values ^x(și)` and ^y(r,și)` of functions ^x(t)` and ^y(r,t)` obtained in the previous calculation stage and by using the Taylor series expansion, with the variable r fixed, we can write: (2)
^x T i1 ` ^x T i ` 'ti ^x 1 T i ` 'ti
2
1!
2!
^x T ` .... 'kt! ^x T ` k
2
^y r , T i1 ` ^y r , T i ` 'ti ^y 1 r , T i ` 'ti
2
1!
i
i
2!
(App14’)
k
i
^y r, T ` ... 'kt! ^y r , T ` k
2
i
i
k
i
Appendix 2
265
From what we have discussed up until now, it results that the determination of solutions ^x(t)` and ^y(r,t)` of differential equation systems (App11) is reduced to the integration of differential equation systems (App13) and (App13’). Therefore, we will discuss below the integration of these differential equation systems. In order to simplify matters, we have noted:
>E r , t @ j
>E l r , t @
>D r,W @ D j
ªl º ª G º K j r y r , t dr » « ³ ; «K T r » x t « 0 » ¬ j ¼ r y r t , ¬« j ¼» I ª º « »; w2 « O r 2 > y r , t @» wr ¬ ¼ l ª º ª Gj º K j r y r ,W dr » ª ' j º « ³ ; «K T r » x W « 0 » « D j r » ¬ j ¼ ¬ ¼ W r y r , ¬« j ¼»
(App15)
Therefore, we can write:
wk > E l r , t @ wt k
wk >E r , t @ wt k j
0 ª º w k 2 « »; «¬ O r wr 2wt k > y r , t @»¼ º ªl wk > y r , t @dr » K r k «³ j k ª Gj º d wt » «K T r » k >x t @ « 0 dt » « wk ¬ j ¼ «¬ - j r wt k > y r , t @ »¼
or, by using the already used condensed form:
wk dt
k
wk dt k
> E r , t @ l
>E r, t @ j
0 ª º d 2 k « », «¬ O r dr 2 y r , t »¼ ªl º ª G j º k K j r y k r , t dr » « ³ «K T r » x t « 0 » k ¬ j ¼ «¬ - j r y r , t »¼
(App16)
266
Materials with Rheological Properties
According to the principle of the superposition of the effects, the unknown functions of two variables can be written as follows:
^ y r , t ` >z r , t @ ^ x t ` ^ w r , t `
(App17)
The elements of matrix [z(r, t)] and vector ^w(r,t)` are, for the moment, unknown functions. Obviously, the number of columns of matrix [z(r, t)] is equal to the number of elements of vector ^x(t)`. The number of rows of matrix [z(r, t)] and the number of elements of vector ^w(r,t)` is equal to the number of unknown functions of two variables ^y(r,t)`. The significance of the newly introduced functions does not have any importance for the moment. If we substitute relation (App17) in (App15), it results that:
>El r , t @
>E r, t @ j
0 0 ª º ª º « » x t « »; w2 w2 «¬ O r wr 2 >z r , t @»¼ «¬ O r wr 2 w r , t »¼ l ª º l ½ °³ K j r w r , t dr ° «G j ³ K j r >z r , t @dr » ¾; 0 « » x t ® 0 T ° ° «¬ K j r - j r >z r , t @ »¼ ¯ - j r w r, t ¿
(App18)
l ¯ ¦£¦ l ¦² ¡ G K r z r ,W ¯ dr ° ¦¦¨ K j r w r ,W dr ¦¦¦ £¦¦ ' j ¦²¦ j j ¨ t ¡ ¢ ± ° D j r ,W ¯ ¡ » ¤ ». ° x W ¤¦ 0 ¡¢ °± 0 ¦¦ ¦¥¦ D j r ¦¦¼ ¡ T ° ¦ ¦¦ - r w r ,W ¦¦ ¡¢K j r - j r ¢ z r ,W ¯± dr °± ¥ ¼
If we note, in what follows:
dk ^ x t `; dt k
z k t z k r , t w k r , t A j t
l
³K j r >z r , t @dr 0
Bj r , t
wk ^ z r , t `; wt k wk ^ z r , t `; wt k
- j r > z r , t @
w - j r 0 wt w K j r 0 wt (App19)
A jk t
³K r >z r , t @dr l
k
j
0
k
Bj
r , t
>
- j r z k r , t
@
Appendix 2 l
C j t
³K j r w r, t dr 0
Cj
k
l
t ³K j r w k r , t dr 0
G j r , t
- j r w r , t
G j k r , t
- j r w k r , t
E r , t
w2 > z r , t @ wr 2 w2 O r 2 w r , t wr
E k r , t
O r
F r , t
O r
267
F k r , t
w 2 k z r , t wr 2 w2 O r 2 w k r , t wr
>
@
then, expressions (App15) and (App16) take the form:
ª 0 º ª 0 º « F r , t » « E r , t » x t ; ¬ ¼ ¬ ¼ k 0 ª 0 º ½ ª º w E r , t BN x t ¾; > @ ®« l k « » » k wt ¬ Fk r , t ¼ ¯¬ E r , t ¼ ¿
>El r , t @
>E r, t @
ª C j t º ª G j A j t º «G r , t » «K T r B r , t » x t ; j ¬ j ¼ ¬ j ¼
wk E j r , t wt k
@
j
>
(App20)
k °ª G j A j t º °½ ° C j t ½° ¾ BN k ®« T ® k » x t ¾; °¯G j r , t °¿ °¯¬K j r E j r , t ¼ °¿
ª G j A j W º ' j C j W ½ ¾; « T » x W ® ¬K j r B j r , W ¼ ¯ D j r G j r , W ¿
>D q r ,W @ j
wk D j q r ,W wt k
>
@
0,
where we have noted:
ª Lp t º k p ° ª L t º °½ k p BN k ® « t , » x t ¾ ¦ Ck « p » x °¯ ¬ M t ¼ °¿ p 0 ¬ M t ¼ K! and 0! = 1, thus, Ckp p! k p !
Ck q
Ck k
1
(App21)
268
Materials with Rheological Properties
By considering relations (App15), (App20) and (App21), the systems of differential equations (App13) and (App13’) take the following expression: m ª 0 º ª 0 º « F r ,W » « E r ,W » x W ¦ j 1 ¬ ¼ ¬ ¼
' j C j W ½ ª G j A j W º «K T r B r ,W » x W ® D r G r ,W ¾ K 0 j W j 0 j j j ¼ ¯ j ¬ ¿
ª 0 º ½ m ª 0 º « F r ,W » BN1 ®« E r ,W » x W ¾ ¦ ¼ ¬ 1 ¼ ¯¬ ¿ j1
§ ° C j1 W ½° °ª G j A j W º ½° · ¨® BN1 ®« T » x W ¾ ¸¸ K 0 j W j . ¨ °G j1 r ,W ¾° , K W r B r °¯¬ j °¿ ¹ j ¼ ¿ ©¯
§ ª G j A j W º dK1 j W W ' j C j W ½ · ¨« T x W ® ¾ ¸¸ K 0 j W j K1 j W j ¨ K j r B j r ,W » W D r G r , dt j ¼ ¯ j ¿¹ ©¬
ª 0 º ½ m ª 0 º « F r ,W » BN 2 ®« E r ,W » x W ¾ ¦ ¼ ¬ 2 ¼ ¯¬ ¿ j1
(App22)
0
§ G j A j W º ½° ° C 2 W ½° · ¨ BN °®ª« x W ¾ ® 2j ¾ ¸ K 0 j W j ¨ 2 °¬K Tj r B j r ,W »¼ °¿ °¯G j r ,W °¿ ¸¹ ¯ ©
§ °ª G j A j W º ½° ° C 1 W ½° ·§ dK 0 j W j dK 2 j W W · ¸¸ ¨ BN1 ®« T K 0 j W j K1 j W j x W ¾ ® 1j ¾ ¸¸¨¨ » ¨ dT °¯¬K j r B j r ,W ¼ °¿ °¯G j r ,W °¿ ¹© dT ¹ © § ª G j A j W º d 2 K 2 j W W ' j C j W ½ · ¸ K 0 j W j K1 j ¨« T x W ® ¾ » ¨ K j r B j r ,W ¸ dT 2 ¼ ¯D j r G j r ,W ¿ ¹ ©¬ ½ m ª 0 º ª 0 º « F r ,W » ®« E r ,W » x W ¾ ¦ ¼ ¬ 3 ¼ ¯¬ ¿ j1
0
§ G A W ½ 3 ½· ¨ BN °®ª« j T j º» x W °¾ °® C j W °¾ ¸K W j 3 0j 3 ¨ , r K W , G r W ° °¿ ¸¹ ° ° j ¼ ¯¬ ¿ ¯ j ©
§ °ª G j A j W º ½° ° C 2 W ½° ·§ dK 0 j W j x W ¾ ® 2j ¨ ®« T K 0 j W j K 1 j W ¾ ¸¨ 2 » ¨ °¬K j r B j r ,W ¼ dT °¿ °¯G j r ,W °¿ ¸¹¨© ©¯
dK dT W W ·¸¸ 2j
¹
§ °ª G j A j W º ½° ° C j W ½° ·§ d K 0 j W j d K 2 j W W 2 K 0 j W j K 1 j W j ¨ BN 1 ®« T ¾ ¸¸¨¨ » x W ¾ ® 1 2 ¨ , r B r K W , G r W dT 2 °¿ ¹© dT °¯¬ j °¿ °¯ j j ¼ © dK 1 j W j · dK 2 j W W · § dK 0 j W j ¸ ¸¸ K 1 j W j K 0 j W j ¨¨ dT ¸¹ dT © dT ¹ 1
2
§ ª G j A j W º d 3 K 2 j W W ' j C j W ½ · x W ® ¨« T ¾ ¸¸ K 0 j W j K1 j W j ¨ K j r B j r ,W » dT 3 ¯ D j r G j r ,W ¿ ¹ ¼ ©¬
2
0
#
½ m ª 0 º ª 0 º « F r ,W » ®« E r ,W » x W ¾ ¦ ¼ ¬ k ¼ ¯¬ ¿ j1
k § · ¨ BN °®ª G j A j W º x W ½°¾ °® C j W ½°¾ ¸ K W » 0j k « T j k ¨ , r B r K W °¿ °¯G j r ,W °¿ ¸¹ j ¼ ¯°¬ j ©
§ °ª G j A j W º ½° ¨ BN k 1 ®« T » x W ¾ " ¨ K W W , r r °¯¬ j °¿ j ¼ ©
§ ª G j A j W º d k K 2 j W W ' j C j W ½ · ¸ K 0 j W j K 1 j W j ¨« T x W ® ¾ » ¨ K j r B j r ,W ¸ dT k ¼ ¯ D j r G j r ,W ¿ ¹ ©¬
0
Appendix 2
269
and, respectively: ª 0 º ½ m ª 0 º « F r ,T » BN1 ®« E r ,T » x T ¾ ¦ ¼ ¬ 1 ¼ ¯¬ ¿ j 1
§ G j A j T º ½° ¨ BCN °®ª« » x T ¾ T 1 ¨ r B r K T , °¿ j ¼ ¯°¬ j ©
§ ª G j A j W º C j1 T ½ · ' j C j W ½ · ® ¾ ¸¸ K 0 j T W j ¨¨ « T » x W ® D r G r ,W ¾ ¸¸ u D r r B r T K W , , j j ¯ j1 ¿¹ ¯ j ¿¹ ¼ ©¬ j dK 2 j T W j u K 0 j W j K1 j W j dT T§ °ª G j A j t º ½° ° C 1 t ½° · dK 2 j T t ³ ¨ BN1 ®« T x t ¾ ® 1j dt ¾ ¸¸ K 0 j t W j K1 j t W j » ¨ dT k °¯¬K j r B j r , t ¼ °¿ °¯G j r , t °¿ ¹ W © ª 0 º ½ m ª 0 º « F r ,T » BN 2 ®« E r ,T » x T ¾ ¦ ¼ ¬ 2 ¼ ¯¬ ¿ j1
0
§ G j A j T º ½° ¨ BN °®ª« x T ¾ ¨ 2 °¬K Tj r B j r ,T »¼ °¿ ¯ ©
(App22’)
§ °ª G j A j T º ½° ° c 2 T °½ · ® 2j ¾ ¸ K oj T W j ¨¨ BN1 ®« T » x T ¾ K T r B r , °¯G j r ,T °¿ ¸¹ ° °¿ j ¼ ¯¬ j ©
§ ª G j A j W º d 2 K 2 j T W j ' j C j W ½ · ¸ K 0 j W j K1 j W j x W ® ¨« T ¾ » ¸ ¨ ¬K j r B r ,W ¼ dT 2 ¯D j r G j r ,W ¿ ¹ © T§ °ª G j A j t º ½° ° C j1 t ½° · d 2 K 2 j (T t ) ³ ¨ BN1 ®« T x t ¾ ® 1 dt ¾ ¸¸ K 0 j t W j K1 j t W j » ¨ dT 2 °¯¬K j r B j r , t ¼ °¿ °¯G j r , t °¿ ¹ W ©
ª 0 º ½ m ª 0 º « F r ,T » BN 3 ®« E r ,T » x T ¾ ¦ ¼ ¬ 3 ¼ ¯¬ ¿ j1
0
§ G j A j T º ½° ¨ BN °®ª« x T ¾ ¨ 3 °¬K Tj r B j r ,T »¼ °¿ ¯ ©
§ °ª G j A j T º ½° ° C 2 T ½° · ° C 3 T ½° · ® 3j x T ¾ ® 2j ¾ ¸¸ K 0 j T W j ¨¨ BN 2 ®« T ¾¸ » K j r B j r , T ¼ G j r ,T °¿ ¸¹ °¯G j r ,T °¿ ¹ ° ° ° ¬ ¯ ¯ ¿ © °ª G A j T º ½° dK 2 j T T · § § dK 0 j T W j ¸¸ ¨ BN1 ®« T j K 0 j T W j K1 j T W j ¨¨ 2 » x T ¾ ¨ K T , r B r dT dT °¿ j ¹ © © ¼ ¯°¬ j ° C 1 T °½ ·§ d 2 K 0 j T W j d 2 K 2 j T T ® 1j 2 K 0 j T W j K1 j T W j ¾ ¸¸¨¨ 2 dT dT 2 °¯G j r , T °¿ ¹©
§ dK 0 j T W j dK1 j T W j · dK 2 j T T · ¸ ¸¸ ¨¨ K1 j T W j K 0 j T W j ¸ dT dT dT ¹ © ¹ 3 · § ª G j A j W º W ' C d K ½ j j 2 j T W j ¨« T x W ® ¾ ¸¸ K 0 j W j K1 j W j ¨ K j r B j r ,W » dT 3 ¯ D j r G j r ,W ¿ ¹ ¼ ©¬ T§ °ª G j A j t º ½° ° C j1 t ½° · d 3 K 2 j T t ¸ K 0 j t W j K 1 j t W j ³ ¨ BN1 ®« T dt x t ¾ ® 1 ¾ » ¨ dT 3 °¯¬K j r B j r , t ¼ °¿ °¯G j r , t °¿ ¸¹ W ©
#
0
270
Materials with Rheological Properties
°ª G j A j T º ½° ª 0 º ½ m ª 0 º « F r ,T » BN k ® « E r ,T » x T ¾ ¦ BN k ®«K T r B r ,T » x T ¾ °¿ j ¼ ¬ k ¼ ¼ ¯¬ ¿ j1 ¯°¬ j °ª G j A j T º ½° ° C k T °½ · ® kj ¾ ¸ K 0 j T W j BN k 1 ®« T » x T ¾ " °¯G j r ,T °¿ ¸¹ °¯¬K j r B j r ,T ¼ °¿
§ ª G j A j W º d k K 2 j T W j ' j C j W ½ · ¸ K 0 j W j K1 j W j ¨« T x W ® ¾ » ¨ K j r B j r ,W ¸ dT k ¯ D j r G j r ,W ¿ ¹ ¼ ©¬ T§ °ª G j A j t º ½° ° C 1 t ½° · d k K 2 j T t ¸ K 0 j t W j K1 j t W j ³ ¨ BN1 ®« T x t ¾ ® 1j dt ¾ » ¨ dT k °¯¬K j r E j r , t ¼ °¿ °¯G j r , t °¿ ¸¹ W ©
0
In differential equation systems (App22) and (App22’), we consider ^x(t)` = 0 which implies:
dk ^x t ` { 0 dt k Therefore:
^x W ` 0,^x T ` k
k
0, k 1,2,3,! , n
We obtain the following systems of linear differential equations of order 2, where the unknown functions are ^w(r,t)`:
>F r,W @ ¦ ^D j r G j r,W `K 0 j W j m
0
j 1
dK2 j W W
0 > F r ,W @ ¦ ^G r ,W ` K W ^ D r G r ,W ` K W K W dT dK W W · § dK W ¸¸ >F r ,W @ ¦ ^G r ,W `K W ^G r ,W `¨¨ K W K W dT dT m
1
1
j
0j
j
j
j
0j
j
1j
j
j 1 m
2
2
j
0j
0j
1 j
j
d 2 K 2 j T W j
^D j r G j r ,W `K 0 j W j K 1 j W j
dT 2
1j
j
0j
©
j 1
j
1j
j
¹
(App23)
§ d K 0 j W d K 2 j W W § dK 0 j W j K 1 j W j G j1 r ,W ¨ 2 K 0 j W j K 1 j W j ¨¨ ¨ dT 2 dT 2 © dT ©
^
`
2
dK 1 jW j K 0 j W j dT #
2
2 d 3 K 2 j W W · d K 2 j W W · ¸ ^D j r G j r ,W `K 0 j W j K 1 j W j ¸¸ ¸ dT 2 dT 3 ¹ ¹
0
Appendix 2
271
>Fk r ,W @ ¦ ^G jk r ,W `K 0 j W j ^G jk 1 r ,W ` " m
j 1
^D j r G j r ,W `K 0 j W j K1 j W j
d k K 2 j W W
0
dT k
and, respectively:
>F1 r ,T @ ¦ ^G j1 r ,T `K 0 j T W j ^D j r G j r ,T `K 0 j W j K1 j W j m
j 1
T
^
`
³ G j1 r , t K 0 j t W j K1 j t W j W
dK 2 j T t dT
dt
dK 2 j T W dT
0
>F2 r ,T @ ¦ ^G j2 r ,T `K 0 j T W j ^G j1 r ,T `u m
j 1
dK 2 j T T · § dK1 j T W j ¸¸ u ¨¨ K 0 j T W j K1 j T W j dT dT © ¹ d 2 K 2 j T W ^D j r G j r ,W `K 0 j W j K1 j W j dT 2 T d 2 K 2 j T t ³ G j1 r , t K 0 j t W j K1 j t W j dt 0 dT 2 W
^
(App23’)
`
§ dK 0 j T W j dT ©
>F3 r ,T @ ¦ ^G j3 r ,T `K 0 j T W j ^G j2 r ,T `¨¨ 2 m
j 1
§ d 2 K 0 j T W j dK 2 j T T · ¸¸ G j1 r ,W ¨ K 0 j T W j K1 j T W j 2 K 0 j T W j K1 j T W j ¨ dT dT 2 ¹ ©
^
`
d 2 K 2 j T T § dK 0 j T W j dK 1 j T W j · dK 2 j T T · ¸¸ ¸¸ ¨¨ K 0 j T W j K 0 j T W j 2 dT dT dT dT ¹ ¹ © d 3 K 2 j T W j ^D j r G j r ,W `K 0 j W j K 1 j W j dT 3 T d 3 K 2 j T t ³ G j1 r , t K 0 j t W j K 1 j t W j dt 0 dT 3 W
^
`
#
>Fk r ,T @ ¦ ^G jk r ,T `K 0 j T W j ^G jk 1 r ,T ` " " m
j 1
^D j r G j r ,W `K 0 j W j K1 j W j T
^
`
³ G j1 r , t K 0 j t W j K1 j t W j W
d k K2 j
dT k d k K 2 j T t dT k
dt
0
272
Materials with Rheological Properties
The differential equation systems (App23) allow the determination of the values taken by the functions ^w(r,t)` and their derivatives at the initial moment W. In a similar way, the differential equation system (App23') leads to the determination of the values of the derivatives w 1 r , T , w 2 r , T ,.., w k r , T of the function {w(r,t)` at moment ș.
^w r , W `, ^w 1 r , W `, ^w 2 r , W `,..., ^w k r , W `
^
`^
` ^
`
By introducing the values of functions ^w(r,t)` and their derivatives, obtained by the integration of the systems of differential equations (App23) and (App23’), into the differential equation system (App23) and (App23’), we obtain: m ª 0 º « E r ,W » x W ¦ j 1 ¬ ¼
ª G j A j W º ª' j C j W º «K T r B r ,W » x W « » K 0 j W j 0 0 j ¬ ¼ ¬ j ¼
½ m ª 0 º BN1 ®« x W ¾ ¦ » ¿ j1 ¯¬ E r , W ¼
§ G j A j W º ½° C 1 W ½ · ¨ BN °®ª« x W ¾ ® j ¾ ¸ K 0 j W j » 1 T ¨ °¯¬K j r B j r , W ¼ °¿ ¯ 0 ¿ ¸¹ ©
§ ª G j A j W º dK 2 j W W ' C j W ½ · ¨« T 0 x W ® j ¾ ¸¸ K 0 j W j K1 j W j ¨ ¬K j r B j r , W »¼ 0 dT ¯ ¿¹ © °ª G j A j W º ½° C j2 W ½ · ª 0 º ½ m §¨ BN 2 ®« x BN x W W T ¾ ¾ ¸ K 0 j W j ® ¾® « ¦ » 2 » °¯¬K j r B j r , W ¼ °¿ ¯ 0 ¿ ¸¹ ¯¬ E r , W ¼ ¿ j 1 ¨©
§ °ª G j A j W º ½° C 1 W ½ ·§ dK 0 j W j dK 2 j W W · ¸¸ x W ¾ ® j K 0 j W j K1 j W j ¨ BN1 ®« T ¾ ¸¸¨¨ » ¨ r B r K W , dT dT 0 ° ° j j ¹ ¿ ¹© ¼ ¿ ¯ ¯¬ © § ª G j A j W º d 2 K 2 j W W ' C j W ½ ·¸ x W ® j ¨« T ¾ ¸ K 0 j W j K1 j W j ¨ K j r E j r , W » 0 dT 3 ¿¹ ¯ ¼ ©¬
ª 0 º ½ m BN 3 ®« x W ¾ ¦ » ¯¬ E r , W ¼ ¿ j 1
0
§ ª G Aj W º ½° C j 3 W ½· ¨ BN 3 °®« T j ¾¸ K0 j W j » x W ¾ ® ¨ °¯¬K j r B j r ,W ¼ °¿ ¯ 0 ¿¸¹ ©
(App24)
§ °ª G j A j W º ½° C 2 W ½ ·§ dK 0 j W j dK 2 j W W · ¸¸ K 0 j W j K1 j W j ¨ BN 2 ®« T x W ¾ ® j ¾ ¸¸¨¨ 2 » ¨ dT dT °¯¬K j r B j r ,W ¼ °¿ ¯ 0 ¿ ¹© ¹ ©
§ °ª G j A j W º ½° C j1 W ½ ·§ d 2 K 2 j W d 2 K 2 j W W ¸ ¨ 2 W W W ¨ BN1 ®« T x K K ¾ ¾ ® » j j 0j 1j 3 ¨ dT 2 °¯¬K j r B j r ,W ¼ °¿ ¯ 0 ¿ ¸¹¨© dT © .
§ dK 0 j W j · dK 2 j W W · § ª G j A j W º ¸¸ ¨ « T ¨¨ K1 j W j K 0 j W j ¸¸ » x W ¨ dT ¹ © ¬K j r B j r ,W ¼ © dT ¹ d 3 K 2 j W W ' C j W ½ · ® j ¾ ¸¸ K 0 j W j K1 j W j 0 dT ¯ ¿¹
#
0
Appendix 2 ½ m ª 0 º BN k ®« x W ¾ ¦ » ¿ j1 ¯¬ E r , W ¼
273
§ G j A j W º ½° C k W ½ · ¨ BN °®ª« x W ¾ ® j ¾ ¸ K 0 j W j " » k T ¨ °¯¬K j r B j r ,W ¼ °¿ ¯ 0 ¿ ¸¹ ©
§ ª G j A j W º d k K 2 j W W ' C j r ½ ·¸ x W ® j K 0 j W j K1 j W j ¨« T ¾ » ¨ K j r B j r ,W 0 dT k ¯ ¿ ¸¹ ¼ ©¬
0
and, respectively: ª 0 º ½ m BN1 ®« x T ¾ ¦ » ¯¬ E r ,T ¼ ¿ j1
§ G A j T º ½° C 1 T ½ · ¨ BN °®ª« j x T ¾ ® j ¾ ¸ K 0 j T W j ¨ 1 °¬K Tj r B r ,T »¼ °¿ ¯ 0 ¿ ¸¹ ¯ ©
§ ª G j A j W º dK 2 j T W j ' C j W ½ ·¸ x W ® j ¨« T ¾ ¸ K 0 j W j K1 j W j ¨ K j r B j r ,W » 0 dT ¿¹ ¯ ¼ ©¬
T§ °ª G j A j t º ½° ªC j1 t º · dK T t ¸ K 0 j t W j K1 j t W j 2 j ³ ¨ BN1 ®« T x t ¾ « dt 0 » » ¸ ¨ dT °¯¬K j r B j r , t ¼ °¿ ¬ 0 ¼ ¹ W © °ª G j A j T º ½° C 2 T ½ · ª 0 º ½ m § x T ¾ ® j BN 2 ®« x T ¾ ¦ ¨ BN 2 ®« T ¾ ¸ K 0 j T W j » » ¨ °¯¬K j r B j r , T ¼ °¿ ¯ 0 ¿ ¸¹ ¯¬ E r , T ¼ ¿ j1 ©
§ ½° C 1 T ½ ·§ dK 0 j T W j °ª G j A j T º x T ¾ ® j ¨ BN1 ®« T ¾ ¸¨ » ¨ dT °¿ ¯ 0 ¿ ¸¹¨© °¯¬K j r B j r , T ¼ ©
(App24’)
dK 2 j T T · § ª G j A j W º ¸¸ ¨ « T K 0 j T W j K1 j T W j » x W ¨ dT ¹ © ¬K j r B j r , W ¼
d 2 K 2 j T W j ' C j2 T ½ · ® j ¾ ¸¸ K 0 j W j K1 j W j dT 2 0 ¯ ¿¹
T § °ª G j A j t º ½° C j1 ½ · d 2 K 2 j T t ³ ¨ BN1 ®« T x t ¾ ® ¾ ¸K 0 j t W j K1 j t W j dt » ¨ dT 2 °¯¬K j r B j r , t ¼ °¿ ¯ 0 ¿ ¸¹ W ©
0
274
Materials with Rheological Properties
°ª G j A j T º ½° C 3 T ½ · ª 0 º ½ m BN 3 ®« x T ¾ ¦ BN 2 ®« T x T ¾ ® j ¾ ¸ K 0 j T W j » » °¯¬K j r B j r ,T ¼ °¿ ¯ 0 ¿ ¸¹ ¯¬ E r ,T ¼ ¿ j1 § °ª G j A j T º ½° C 2 T ½ ·§ dK 0 j T W j x T ¾ ® j ¨ BN 2 ®« T ¾ ¸¨ 2 » ¨ dT °¯¬K j r B j r ,T ¼ °¿ ¯ 0 ¿ ¸¹¨© ©
°ª G j A j T º ½° dK 2 j T T · § ¸¸ ¨ BN1 ®« T K 0 j T W j K1 j T W j » x T ¾ ¨ , r B r K T dT °¯¬ j °¿ j ¹ © ¼
d 2 K 2 j T T C 1 T ½ ·§ d 2 K 0 j T W j ® j 2 K 0 j T W j K1 j T W j ¾ ¸¸¨¨ 2 dT dT 2 ¯ 0 ¿ ¹©
dK1 j T T · dK 2 j T T · § dK 0 j T W j ¸¸ ¸¸ K1 j T W j K 0 j T W j ¨¨ dT dT dT ¹ ¹ © 3 § ª G j A j W º d K 2 j T W j ' C j W ½ · x W ® j ¨« T ¾ ¸¸ K 0 j W j K1 j W j ¨ K j r B j r ,W » 0 dT 3 ¯ ¿¹ ¼ ©¬
T § °ª G j A j W º ½° C 1 t ½ · d 3 K 2 j T t ³ BN1 ¨ ®« T x t ¾ ® j ¾ ¸ K 0 j t W j K1 j t W j dt » ¨ °¬K j r B j r , t ¼ dT 3 °¿ ¯ 0 ¿ ¸¹ W ©¯
0
# ª 0 º ½ m BN k ®« x T ¾ ¦ » ¯ ¬ E r ,T ¼ ¿ j1
§ G j A j T º ½° C k T ½ · ¨ BN °®ª« x T ¾ ® j ¾ ¸ K 0 j T W j » T k ¨ °¿ ¯ 0 ¿ ¸¹ ¯°¬K j r B j r ,T ¼ ©
°ª G j A j T º ½° BN k 1 ®« T » x T ¾ " °¯¬K j r B j r ,T ¼ °¿
§ ª G j A j W º d k K 2 j T W j ' j C j W ½ · ¸ x K K W W W ¨« T ® ¾ ¸ 0 j j 1j j ¨ K j r B j r ,W » 0 dT K ¯ ¿¹ ¼ ©¬ T § °ª G j A j t º ½° C j1 t ½ · d k K 2 j T t ¸ K 0 j t W j K1 j t W j x t ¾ ® dt ³ BN1 ¨ ®« T ¾ » ¨ °¬K j r B j r , t ¼ ° ¯ 0 ¿ ¸¹ dT k W ¿ ©¯
0
In the systems of linear differential equations of order 2 (App24) and (App24’), obtained in this manner, the unknown factors are the values, at moment W, and, respectively, ș, of unknown functions ^w(r,t)` and ^z(r,t)` and their derivatives. The systems of linear differential equations (App24) and (App24’) can each be decomposed into two equation systems corresponding to the two rows of each equation system composing them. We obtain a system of linear algebraic equations, and, respectively, a system of linear ordinary differential equations of order 2.
Appendix 2
275
Thus, equation system (App24) is split into the following equation systems: – the system of algebraic equations:
¦ >G m
j 1
j
@
A j X W ^' j C j W ` K 0 j W j 0
¦ BN ^>G m
1
j
j 1
@
^
(App25)
>
`
@
A j W x W ` C j1 W K 0 j W j G j A j W x W ^' j C j W ` u
dK 2 j W W u K 0 j W j K1 j W j dT
0
¦ BN ^>G A W @x W ` ^C W ` K W BN ^>G A W @x W ` ^C W ` dK W W · § dK W ¸¸ >G A W @x W ^' C W ` ¨¨ K W K W m
2
2
j
0j
j
j
j
0j
j
j
1
1 j
j
j 1
2j
0j
dT
©
j
1j
j
d 2 K 2 j W W K 0 j W j K1 j W j dT 2
¦ BN ^>G m
3
j
j 1
@
^
dT
j
¹
j
j
j
0
`
>
@
^
`
A j W x W ` C j3 W K 0 j W j BN 2 ^G j A j W x W ` C j2 W u
dK 2 j W W · § dK 0 j W j ¸¸ BN1 ^G j A j W x W ` C j1 W u u ¨¨ 2 K 0 j W j K1 j W j dT dT ¹ ©
>
@
^
§ d 2 K 2 j W j d 2 K 2 j W W u¨ W W K K 2 0j 1j j j ¨ dT 2 dT 2 ©
dK1 j W j · dK 2 j W W · § dK 0 j W j ¸ ¸ ¨¨ K1 j W j K 0 j W j ¸ dT dT ¸¹ dT © ¹
d 3 K 2 j W W G j A j W x W ^' j C j W ` K 0 j W j K1 j W j dT 3
>
@
0
#
¦ BN ^>G m
k
j 1
j
@
^
`
A j W x W ` C jk W K 0 j W j "
"
>
@
G j A j W x W ^' j C j W ` K 0 j W j K1 j W j
d k K 2 j W W dT k
0
`
276
Materials with Rheological Properties
– the system of differential equations: m
>E e,W @x W ¦ j 1
>K
T j
r B j r ,W @x W K 0 j W j
0
BN1 \ ¢ E r ,W ¯± x W ^ m
BN \ ¢¡K
1
j 1
q K 0 j W j K1 j W j
T j
r B j r ,W ¯±° x W ^ K 0 j W j ¢¡K Tj r B j r ,W ¯±° x W
q
dK 2 j W W
dT m
0
BN 2 \ ¢ E r ,W ¯± x W ^ BN 2 \ ¢¡K Tj r B j r ,W ¯±° x W ^ K 0 j W j 1
j
BN1 \ ¡¢K Tj r B j r ,W ¯±° x W ^ q dK W
dK 2 j W W ¬ 0 j j K W K W q
j j 0j 1j dT dT ® ¡¢K Tj r B j r ,W ¯°± x W
q K 0 j W j K1 j W j
d 2 K 2 j W W
dT 2
(App26)
0
BN 3 \ ¢ E r ,W ¯± x W ^ m
BN3 \ ¡¢K Tj r B j r ,W ¯±° x W ^ K 0 j W j BN 2 \ ¢¡K Tj r B j r ,W ¯±° x W ^ ¸ j 1
dK W W ¬ dK 0 j W j
T ¯ ¸ 2 K 0 j W j K1 j W j 2 j BN1 \¡¢K j r B j r ,W ±° x W ^ ¸ dT dT ®
d 2 K 2 j W
d 2 K 2 j W W
K W K W 2 ¸
j j 0j 1j dT 3 dT 2 dK W
dK1 j W j ¬ dK 2 j W W ¬ 0j j K1 j W j K 0 j W j
® dT dT ® dT ¢¡ K Tj r B j r ,W ±°¯ x W
K 0 j W j K1 j W j
#
d 3 K 2 j W W
dT 3
0
Appendix 2 m
277
BN k \¢ E r ,W ±¯ x W ^ BN k \¢¡ K Tj r B j r ,W ¯±° x W ^ K 0 j W j " j 1
" ¡¢ K Tj r B j r ,W °±¯ x W
K 0 j W j K1 j W j
d k K 2 j W W
0
dT k
In a similar way, equation system (App24’) is split into: – an algebraic equation system:
¦ BN ^>G m
1
j
j 1
>
@
A j T x T ` ^C j1 W ` K 0 j T W j
@
G j A j W x W ^' j C j W ` K 0 j W j K1 j W j T
³ W
dK 2 j T W j
dT dK T t BN1 ^G j A j t x t `C j1 t K 0 j t W j K1 j t W j 2 j dt dT
>
@
¦ BN ^>G A T @x T ` ^C T ` K T W BN ^>G A T @x T ` ^C T ` dK T T · § dK T W ¸¸ ¨¨ K T W K T W m
0
2
2
j
j
j
0j
1
j
j
1 j
j
j 1
0j
j
>
j
1j
j
dT
@
G j A j W X W ^' j C j W ` K 0 j W j K1 j W j T
>
@
^
`
³ BN1 ^G j A j t x t ` C j1 t W
(App25’)
2j
0j
dT
©
d 2 K 2 j T t K 0 j t W j K1 j t W j dt dT 2
0
¹ d 2 K 2 j T W j dT 2
278
Materials with Rheological Properties
¦ BN ^>G A T @x T ` ^C T ` K T W BN ^>G A T @x T ` C T § dK T W ¨ dK T T ·¸ m
3
j
3 j
j
0j
j
j 1
2
j
2
j
j
j ¨2 0 j K 0 j T W j K1 j W j dT © BN1 ^G j A j T x T ` C j1 T
2j
¸ ¹
dT
> @ ^ ` K T W d 2 K T W K T W dT
2 § d 0j K 2 j T T j .¨¨ j j 0j 2 1j dT 2 © dK T T · dK 2 j T T · § dK T W ¸¸ ¸¸ ¨¨ 0 j K1 j T W K 0 j T W j 1 j dT dT dT © ¹ ¹ 2
>
@
G j A j W x W ^' j C j W ` K 0 j W j K1 j W j T
>
@
^
`
^
`
d 3 K 2 j T W j
dT 3 d 3 K 2 j T t
³ BN1 ^G j A j t x t ` C j1 t K 0 j t W j K1 j t W j W
0
dT 3
#
¦ BN ^>G m
k
j
j 1
>
@
>
@
G j A j W x W ^' j C j W ` K 0 j W j K1 j W j T
>
@
^
d k K 2 j T W j
dT k d k K 2 j T t
`
³ BN1 ^G j A j t x t ` C j1 t K 0 j t W j K1 j t W j W
@
A j T x T ` C jk T K 0 j T W j BN k 1 ^G j A j T x T ` ...
dT k
dt
– a differential equation system:
^>
@ `
BN 1 ^>E r , T @x T ` ¦ BN 1 K Tj r B j r , T x T K 0 j T W j m
j 1
>
@
K Tj r B j r ,W x W K 0 j W j K 1 j W j
dK 2 j T W j dT
T dK 2 j T t ³ BN 1 K Tj r B j r , t x t K 0 j t W j K 1 j t W j dt dT W
^>
@ `
0
0
Appendix 2
^>
279
@ `
BN 2 ^>E r , T @x T ` ¦ BN 2 K Tj r B j r , T x T K 0 j T W j m
j 1
@ ` ^> dK T T · § dK T W ¸¸ .¨¨ K T W K T W BN1 K Tj r B j r , T x T 0j
0j
dT
©
>
1j
j
j
@
K Tj r B j r , W x t K 0 j W j K1 j W j T
^>
dT
¹ d K 2 j T W j 2
dT 2
@ `
³ BN1 K Tj r B j r , t x t K 0 j t W j K1 j t W j W
(App26’)
2j
j
^>
d 2 K 2 j T t dT 2
dt
0
@ `
BN 3 ^>E r ,T @x T ` ¦ BN 3 K Tj r B j r ,T x T K 0 j T W j m
j 1
^> @ ` dK T T · § dK T W ¸¸ K T W K T W .¨¨ 2
BN 2 K Tj r B j r , T x T 0j
2j
j
0j
j
dT © BN1 K Tj r B j r , T x T
^>
1j
j
@ `
dT
¹
§ d K 2 j T W d 2 K 2 j T T K K .¨ 2 T W T W 0j 1j j j ¨ dT 3 dT 2 © dK1 j T T · dK 2 j T T · § dK 0 j T W j ¸¸ ¸¸ ¨¨ K1 j T W j K 0 j T W j dT dT dT © ¹ ¹ 3 d K 2 j T W j K Tj r B j r ,W x W K 0 j W j K1 j W j dT 3 T d 3K 2 j T t ³ BN1 K Tj r B j r , t x t K 0 j t W j K1 j t W j dt 0 dT 3 W 2
>
@
^>
@ `
#
^>
@ `
BN k ^>E r , T @x T ` ¦ BN 3 K Tj r B j r , T x T K 0 j T W j m
^>
j 1
@ `
BN k 1 K Tj r B j r , T x T ...
>
@
K Tj r B j r , W x W K 0 j W j K 1 j W j T
^>
@ `
d k K 2 j T W j dT k
³ BN 1 K Tj r B j r , t x t K 0 j t W j K 1 j t W j W
d k K 2 j T t dT k
dt
0
280
Materials with Rheological Properties
In order to determine unknown functions ^z(r,t)`, we consider the differential equation system (App26), or, according to the case, (App26’), and we integrate them by successively assigning unit values to an element of the vector of unknown functions ^x(t)`, and by assigning the zero value to all the other elements. Let p be the number of elements (of the functions) composing vector ^x`. We obtain:
^x` 1
x1 1 ° 1 ° x2 °° x3 1 ® ° . ° . ° 1 °¯ x p
x1 2 ° 2 ° x2 °° x3 2 ® ° . ° .. ° 2 °¯ x p
1½ ° 0° 0°° ¾;^x` 2 ° ° ° 0°¿
0½ ° 1° 0°° ¾;!;^x` p ° ° ° 0°¿
x1 p ° p ° x2 °° x3 p ® ° . ° . ° p °¯ x p
0½ ° 0° 0°° ¾ ° ° ° 1°¿
(App27)
In this way, we successively obtain specific solutions for unknown functions ^z(r,t)`. If we consider the number of elements (functions) composing the vector of unknown functions ^y(r,t)`, it results that, with each integration, we obtain q unknown functions zij(r,t), or, respectively, zij(r, ș). The index j = 1, 2, 3, …, p corresponds to unknown factor xj(t), while the index i = 1, 2, 3, …, q corresponds to unknown factor yi(r,t). We obtain the form of the matrix of functions z(r, t):
>z r , t @
ª z11 «z « 21 « z 31 « « . « . « ¬« z q1
z12 z 22
z13 z 23
z 32
z 33
. .
. .
z q2
z q3
. . z1 p º . . z 2 p »» . . z3 p » » . . . » . . . » » . . z qp ¼»
(App28)
Column k of matrix >z(r,t)@ thus represents the functions, the specific solutions of differential equation systems (App26) or (App26’), corresponding to unknown factors yi(r,t), when we assign the unit value to unknown function xk(t) (xk(t) = 1). If we assign an arbitrary value ck to function xk(t), thus xk(t) = ck, then, obviously, differential equation system (App26) or (App26’) will have the solutions:
z k r ,W ck z k r ,W
Appendix 2
281
or, respectively:
z k r ,T ck z k r ,T Therefore, it results that any linear combination:
[ k r ,W
p
¦ c z r ,W j kj
j 1
or, respectively:
[ k r ,T
p
¦ c z r,T j kj
j 1
is a solution of differential equation system (App26), or, according to the case, (App26’), where the elements of vector ^x(t)`, take the values: xi = ci; i = 1, 2, 3, …, p Solving the systems of differential equations (App26) and (App26’), under the conditions of assigning values (App27) to vectors ^x(W)` or ^x(ș)`, provides, if we use equations system (App26), the values of the elements of matrix >z(r,t)@ and derivatives of the functions composing it at moment W, or, if we use equation system (App26’), the matrices of the derivatives of these functions at the moment ș. By introducing the values of functions >z(r,t)@ and their derivatives at moment W into the equations system (App25), this changes into a system of linear algebraic equations having as unknown factors the values of unknown functions ^x(t)` and their derivatives at moment W. In a similar way, by introducing values >z(r,t)@ and their derivatives at moment ș into equation system (App25’), we obtain the values, at moment ș, of derivatives of unknown functions ^x(t)`. The values of derivatives of unknown functions ^x(și+1)` and ^y(r,și+1)` at moment și+1 = și +'ti are found using relations (App14) and (App14’) by taking into account form (App17) of the unknown functions of the two variables:
^y r , t ` >z r , t @^x t ` ^w r , t ` and the form taken by their derivatives according to the time:
dk ^ y r , t ` dt k
^ y r , t ` k
BN k > z r , t @ ^ x t ` ^ w r , t `
k
282
Materials with Rheological Properties
The values:
^x W ` , ^y r ,W ` >z r ,W @^x W ` ^w r ,W `
(App29)
and, respectively:
^x T ` , ^y r ,T ` >z r ,T @^x T ` ^w r ,T `
(App29’)
are the values of the solutions for equation systems (App22) and, respectively, (App22’) in moments W and ș. Indeed, this assertion is obvious if we consider that these equation systems represent the addition, equation by equation, of the equations from systems (App23) or (Add23’) with those from systems (App24) or (App24’). The solutions of systems (App23) and (App23’) are: ^w (r,t)`, and, respectively: ^w (r,ș)`
(App30)
The solutions of systems (App24) and (App24’) are:
^x W ` ^[ r ,W ` >z r ,W @^x W ` and, respectively:
^x T ` ^[ r ,T ` >z r ,T @^x T `
(App31)
The values of the solutions of differential equation systems (App22) and (App22’), presented in (App29) and (App29’), represent the sum of differential equation systems (App23) or (App23’) given in (App30) with the values of the solutions of equation systems (App24) or (App24’) whose expressions are given in (App31).
Bibliography
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284
Materials with Rheological Properties
Caquot A., Kerisel Y., Tratat de mecanica pamânturilor, Editura tehnică, Bucharest, 1968. Constantinescu A., Rotestein B., Lascu S., Fluajul metalelor, Editura tehnică, Bucharest, 1970. Constantinescu D., Efectele structurilor ale curgerii lente ale betonului, Editura Academiei, Bucharest, 1985. Cristescu C., “Calculul grinzilor încovoiate alcatuite din elemente prefabricate úi monolitizate cu beton armat turnat pe úantier”, in Revista Transporturilor úi telecomunicaĠiilor, no. 4, 1976. Cristescu C., “Structuri de rezistenĠă alcatuite din elemente confecĠionate din materiale cu proprietăĠi reologice diferite”, in Studii úi Cercetări de Mecanică Aplicată, May 1987 (unpublished article) Dicter G. Jr, Metalurgie mecanică, Editura tehnică, Bucharest, 1970. Dodescu G., Toma M., Metode de calcul numeric, Editura didactică úi pedagogică, Bucharest, 1976. Dragos L., Principiile mecanicii mediilor continue, Editura Academiei, Bucharest, 1982. Filonenco-Borodici, Teoria elasticităĠii, Editura tehnică, Bucharest, 1958. Gheorghiu A., Iordanescu M., Prelegeri orientative de sinteză la disciplina Statică, Stabilitate úi Dinamica ConstrucĠiilor, Institutul de ConstrucĠii Bucharest, 1972. Ghiocel D., Lungu D., SiguranĠa construcĠiilor, Institutul de ConstrucĠii, Bucharest, 1973. Haimovici A., EcuaĠii diferenĠiale úi ecuaĠii integrale, Editura didactică úi pedagogică, Bucharest, 1965. Hawranek A., Steinhardt O., Theorie und Berechnung der Stahlbrücken, SpringerVerlag, Berlin, 1958 Herberg W., ConstrucĠii de beton precomprimat, volumes I and II, Editura tehnică, Bucharest, 1959-1961. Iacob C. et al., Matematici clasice úi moderne, volume III, Editura tehnică, Bucharest, 1981.
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286
Materials with Rheological Properties
Smirnov V.-I., Curs de matematici superioare, volumes I-V, Editura tehnică, Bucharest, 1953-1963. Smoleanski M.-L., Tabele de integrale nedefinite, Editura tehnică, Bucharest, 1972. Soare M., Structuri discrete úi structuri continue în mecanica construcĠiilor, Editura Academiei, Bucharest, 1986. Stussi F., Statique appliquée et résistance des matériaux, volume I, Dunod, Paris, 1964. Stussi F., Statique appliquée et résistance des matériaux, volume II, Dunod, Paris, 1967. Sabac I.G., Matematici speciale, volumes I-II, Editura didactică úi pedagogică, Bucharest, 1981-1983. Teodorescu P.-P., Ille V., Teoria elasticităĠii úi introducere în mecanica corpurilor deformabile, volumes I-IV, Dacia, Cluj-Napoca, 1976-1984. Vaicum A., Studiul reologic al corpurilor solide, Editura Academiei, Bucharest, 1978. Vaicum A., Tasarea construcĠiilor, volumes I-II, Editura tehnică, Bucharest, 19881989. *** Stahlbau, Band 1 – Cologne, 1961, Stahlbau GMBH. *** STAS 500/2,3-80: OĠeluri de uz general pentru construcĠii *** STAS 6482/2,3,4-73 Sârme de oĠel úi produse pentru beton precomprimat. *** STAS 438/-80 OĠel beton laminat la cald. *** STAS 438/2-80 Sârmă trasă pentru beton armat. *** STAS 438/3-80 Plase sudate pentru beton armat. *** STAS 10111/2-87 Suprastructuri din beton, beton armat úi beton precomprimat. PrescripĠii de proiectare. *** InstrucĠiuni tehnice pentru calculul úi alcatuirea constructivă a structurilor compuse din oĠel-beton. Indicativ P83-81.
Index
B
E
Bernoulli, 10, 76, 207 Boltzmann, 6, 17, 27, 28, 47, 240
EIDTLI, 197 EXVECT, 196
C
H
Cauchy problem, 60, 73, 74, 86, 112, 120 Compatibility condition, 5, 125, 126, 135 connectors, 22, 61, 62, 68, 69, 71, 76, 107, 108, 112, 141, 145, 146, 149, 151, 152, 157, 159, 163, 165, 168, 170, 171, 174, 175, 176, 177, 180, 182, 186, 187, 198, 199, 200, 201, 204, 211, 217, 219, 220, 221, 222, 226 Correction matrix, 129 Creep, 12 Creep experiment, 12, 13, 14 Creep function, 12, 18, 28, 30, 31, 36, 40, 41, 203, 211, 212, 213, 214, 215, 216
Heaviside step function, 18 Hooke, 10, 18, 19, 20
D Dirac’s delta function, 70, 71, 108 Displacement matrix, 130, 132
I Integro-differential equation, 6, 29, 30, 41, 56, 57, 59, 60, 64, 79, 82, 83, 91, 93, 94, 98, 99, 100, 101, 102, 104, 109, 113, 117, 118, 122, 123, 124, 126, 128, 156, 157, 166, 167, 172, 179, 191, 192, 203, 204, 205, 247, 253, 259, 261
L Laplace transform, 30, 31, 41, 42
M MEFLEX, 195 MFSVTL, 194 Modulus of elasticity, 21
288
Materials with Rheological Properties
P
T
Principle of correspondence, 17 Principle of superposition, 6, 12, 17, 27, 28, 47
Taylor series, 55, 59, 65, 101, 103, 104, 113, 157, 191, 246, 248, 257, 259, 265, 266 TLIAAC, 196 TRIANG, 196
R RALUCA, 7, 190, 191, 192, 193, 203, 205, 209 Reaction matrix, 132 Recovery curve, 14, 15, 214 Reinforcing steel (non-prestressed), 22 Resistance structure, 47, 49, 62, 95, 125, 192, 198 RMETRS, 196 RUTLET, 196
V
S
Young’s modulus, 2
SEFIIK, 195 Shearing force, 140, 222 SOLSIS, 196 STAS 10111/2-87, 25, 26, 27, 38, 39, 288 STAS 500/2-80, 19 STAS 500/3-80, 19 STAS R8548-70, 20 Static equilibrium condition, 5, 96, 125, 126 Stress relaxation, 15 Stress relaxation experiment, 15, 29, 41, 128 Stress relaxation function, 15, 18, 26, 27, 28, 29, 30, 40, 41, 42
VENECX, 194 Viette’s theorem, 31 VIFDIF, 196 Volterra principle, 19 Volterra-Roux theorem, 54 VTLCUB, 195
Y
