Table Of ContentNONLINEAR MECHANICS OF STRUCTURES
Nonlinear Mechanics
of Structures
MICHAL KLEIBER AND CZESLAW WOZNIAK
Institute of Fundamental Technological Research,
Polish Academy of Sciences, Warsaw
KLUWER ACADEMIC PUBLISHERS
DORDRECHT/ BOSTON/LONDON
PWN - POLISH SCIENTIFIC PUBLISHERS
WARSAW
Library of Congress Cataloglng-in-Publicatlon Data
Kleiber, Michal.
[Nieliniowa mechanika konstrukcji. English]
Nonlinear mechanics of structures/Michal Kleiber and Czeslaw
Wozniak; [translated by the authors from the Polish edition].
p. cm.-
Translation of: Nieliniowa mechanika konstrukcji,
Includes bibliographical references.
flBN-l3: ~lfM747-8
I. Mechanics Applied. 2. Noolinear mechanics I. Wotoiak,
Czeslaw. II. Tille.
TA350, K65313 1990
620. l- dc20 90-4290
flBN-l3: ~01().(j747~ e-flBN-I3:978-94-(X9.(l57-1
!XX: 1(l1(JJlfJ7S!.J4..fJ:fJ-I
Published by PWN- Polish Scientific Publishers,
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Copyright by PWN-Polish Scientific Publishers, Warszawa 1991
Softeovcr reprint of the hardcover I st edition 1991
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Contents
Introduction . . . . . . . . . . . . . . . XI
Conventions and notations . . . . . . . . . XIV
PART I: THEORETICAL FOUNDATIONS
1 General notions and principles
Purpose of the chapter . 3
1.1 Introductory concepts 3
1.1.1 Space and time 3
1.1.2 Motion. 3
1.1.3 Mass . ... . 5
1.1.4 Forces . . . . 6
1.1.5 Temperature, energy and entropy. 8
1.1.6 Heat supply. . . . . 9
1.2 Fundamental principles . . . . . . 10
1.2.1 Balance of momentum . . . . . 10
1.2.2 Balance of moment of momentum 10
1.2.3 Balance of energy . 11
1.2.4 Dissipation principle 11
1.3 Basic fields . . . . . . 12
1.3.1 Displacements . . . 12
1.3.2 Strains and strain rates 12
1.3.3 Stresses and stress rates. 14
1.3.4 Heat fluxes . . . . . . 16
1.3.5 Matrix notation for the basic fields. 16
1.4 Resulting relations. . . 19
1.4.1 Equation of motion . 19
1.4.2 Energy equation . . . 21
1.4.3 Dissipation inequality 22
1.4.4 Matrix form of the resulting equations 23
Summary 24
Problems 25
v
Contents
2 Materials, loadings and constraints
Purpose of the chapter . . . . . 27
2.1 Materials . . . . . . . . . 27
2.1.1 Real materials and ideal materials 27
2.1.2 Thermo-visco-elastic materials. 28
2.1.3 Linear thermo-elastic materials 29
2.1.4 Thermo-elastic-plastic materials 30
2.1.5 Matrix form of constitutive relations 34
2.2 Loadings and heat supply . . . . . . 37
2.2.1 External agents . . . . . . . . . 37
2.2.2 Body loadings and internal heat supply 38
2.2.3 Boundary loadings . . 38
2.2.4 Boundary heat supply. . . . . . 39
2.3 Constraints 39
2.3.1 Boundary and internal constraints 39
2.3.2 Reactions. . . . . 42
2.3.3 Bilateral constraints . . . . . . 44
2.3.4 Constraint functions . . . . . . 46
2.3.5 Constraints for stresses and heat fluxes 47
Summary .......... . 49
Problems .......... . 50
3 Formulation of engineering theories
Purpose of the chapter . . . . . 52
3.1 General form of governing relations . 52
3.1.1 Finite variational formulation . . 52
3.1.2 Incremental variational formulation 55
3.1.3 Variational equations for stresses and heat fluxes. 58
3.1.4 Calculation of reactions ..... . 59
3.1.5 Matrix form of variational equations. 60
3.2 Method of constraints: finite formulation 61
3.2.1 Foundations ..... 61
3.2.2 Shell and plate theories. 65
3.2.3 Rod and beam theories . 71
3.2.4 Plane problems . . . . 74
3.3 Method of constraints: incremental formulation 77
3.3.1 General equations 77
3.3.2 Shells and rods 78
3.3.3 Plane problems 79
VI
Contents
3.4 Method of constraints: discrete description of elements and
structures . . . . . . . 80
3.4.1 Discretization concept . 80
3.4.2 Finite formulation . . . 80
3.4.3 Incremental formulation 86
3.4.4 Shell and plate elements 87
3.4.5 Rod and beam elements 89
3.4.6 Special cases and matrix notation 91
3.4.7 Interaction between elements . . 96
3.5 Evaluation and correction of solutions . 97
3.5.1 Residual reactions . . . 97
3.5.2 Evaluation of solutions . . . . . 99
3.5.3 Correction of solutions . . . . . 101
3.5.4 On adaptive solution refinements 103
Summary 103
Problems 104
4 Extensions and specifications of the general theory
Purpose of the chapter . . . . . . . . . . . . . . . . . . 106
4.1 Domain discretization . . . . . . . . . . . . . . . . . 107
4.2 Solving differential equations by weighted residual methods 109
4.3 Assembly of element matrices and vectors 113
4.4 Elastic pin-jointed element in space 118
4.5 Elastic frame element in space 123
4.6 Elastic lattice-type structures 131
4.6.1 Discrete models . . . 131
4.6.2 Continuous models 141
4.7 Engineering shell theories 147
4.7.1 Foundations 147
4.7.2 General form of equations of motion. 151
4.7.3 Special theories . . . . . . . . . . 158
4.7.4 Example: axisymmetric deformations of shells 162
4.8 Thin shell finite element for nonlinear axisymmetric analysis 164
4.9 Linear strain triangular element for nonlinear plane stress analysis 182
4.10 Constitutive matrices for thermo-elastic-plastic materials .. 193
4.11 On inelastic analysis under non-proportionally varying loads. 197
4.11.1 Elastic-plastic structures subjected to mechanical loads 197
4.11.2 Thermo-elastic-plastic structures subjected to mechanical
and thermal loads . . . . . . . . . . . . . . . . .. 203
vn
Contents
Summary 205
Problems 206
5 Numerical algorithms and software concepts
Purpose of the chapter . . 212
5.1 Introductory comments . . . . . . . 212
5.2 Nonlinear quasi-statics . ..... . 213
5.3 Integration of elastic-plastic constitutive law 225
5.4 Initial and linearized buckling . . . . . . 229
5.5 Nonlinear dynamics . . . . . . . . . . . 232
5.6 Dynamic stability under non-periodic loads 237
5.7 Nonlinear heat transfer . 242
5.8 Development of software 244
Summary 246
Problems .. .... . . . 246
PART II: SELECTED APPLICATIONS
6 Trusses, frames, lattice-type shells
Purpose of the chapter . . . . . 251
6.1 Trusses . . . . . . . . . . 251
6.1.1 Nonlinear effects in truss analysis - a model problem 251
6.1.2 Accuracy of computations and further examples . . . 261
6.1.3 Elastic-plastic truss under non-proportionally varying loads 272
6.2 Frames . . . . . . . . . . . . . . . . . . . . . . 277
6.2.1 Elastic-plastic constitutive matrices for beam elements 277
6.2.2 Limit state conditions . . . . . . . . . . . . . 283
6.2.3 Nonlinear analysis of curved beams . . . . . . . . 290
6.2.4 Examples of numerical frame analysis up to collapse . 291
6.2.5 Elastic-plastic beam under non-proportionally varying loads. 298
6.2.6 Elastic buckling of plane grid . . . . . . . . . . . 305
6.2.7 Approximate large displacement analysis of frames using
buckling mode superposition . . . . . . 306
6.3 Lattice-type structures . . . . . . . . . . . 312
6.3.1 Lattice shells -linear and "second-order" theories. 312
6.3.2 Buckling of elastic grid . . . . . . . . . . . . . 318
6.3.3 Perfectly-plastic lattice-type plates - discrete model 322
6.3.4 Limit load of polar grids - continuous model . . . 339
6.3.5 Minimum weight design of ideally plastic rectangular dense grid 351
Summary 360
Problems 360
VIII
Contents
7 Thin plates loaded in-plane
Purpose of the chapter . . . . . . . . . . . 363
7.1. Elastic-plastic bending of a cantilever beam 363
7.2 Limit analysis of perforated plates . . .. 369
7.3 Elastic-plastic analysis of a cantilever under non-proportionally
varying loads. . . . . . . . . . . . . 376
7.4 In-plane buckling of an elastic-plastic strip . . . . . . 379
7.5 Necking of an elastic-plastic strip . . . . . . . . . . 380
7.6 Extension of a thin rectangular plate with a central hole 383
Summary ...... . 384
8 Plate and shell problems .
Purpose of the chapter . . . . . . . . . . . . . . 385
8.1 Thin elastic-plastic axisymmetric shells. . . . . . 385
8.2 Elastic-viscoplastic analysis of axisymmetric shells . 405
8.3 Buckling of elastic-plastic spatial plate assemblies 410
Summary ......... . .... . 430
9 Heat conduction and thermal stress problems
Purpose of the chapter . . . . . . . . . 431
9.1 Heat flow in flash and friction welding. 431
9.2 Thermally induced stresses in elastic-plastic plate 440
9.3 Elastic-plastic cylindrical shell under thermo-mechanical load 443
9.4 Elastic-plastic truss under nonproportionally varying temperature
and mechanical load 445
Summary . 448
Bibliography. 449
Index . . . 457
IX
Introduction
The aim of this book is to provide a unified presentation of modern mechanics
of structures in a form which is suitable for graduate students as well as for
engineers and scientists working in the field of applied mechanics. Traditionally,
students at technical universities have been taught subjects such as continuum
mechanics, elasticity, plates and shells, frames or finite element techniques
in an entirely separate manner. The authors' teaching experience clearly
suggests that this situation frequently tends to create in students' minds an
incomplete and inconsistent picture of the contemporary structural mechanics.
Thus, it is very common that the fundamental laws of physics appear to students
hardly related to simplified equations of different "technical" theories of
structures, numerical solution techniques are studied independently of the
essence of mechanical models they describe, and so on.
The book is intended to combine in a reasonably connected and unified
manner all these problems starting with the very fundamental postulates
of nonlinear continuum mechanics via different structural models of "engineer
ing" accuracy to numerical solution methods which can effectively be used for
solving boundary-value problems of technological importance.
The authors have tried to restrict the mathematical background required
to that which is normally familiar to a mathematically minded engineering
graduate.
Of course, no single book can fully embrace the broad field of nonlinear
structural mechanics. Holding the book to reasonable length did not permit
to include many interesting, and certainly important issues. In making our
choice we have primarily aimed at logical presentation of the selected material
and its anticipated usefulness in the long run. In particular, it is emphasized
that the book is not intended to provide the reader with a great number of
sophisticated nonlinear theories applicable to the refined analysis of specific
structural configurations. Quite to the contrary, after giving a thorough
continuum mechanics background we consistently develop merely relatively
simple, or conventional, theories and apply them to solve numerous practical
problems of engineering significance. It is our strong belief, however, that the
general framework laid down in the present book will turn out to be a proper
tool to develop more sophisticated theories in the near future.
XI
Introduction
A brief outline of the main portions of the book is as follows. The first
part devoted to theoretical foundations of nonlinear mechanics of structures
starts with the short presentation of the fundamental notions and laws of
nonlinear continuum mechanics. We introduce the notation and get the reader
acquainted with the description of motion of the material continuum, the states
of strain and stress and finally with the conservation laws. The temperature
effects are included into the formulation.
In Chapter 2 we outline the theory of constitutive equations and thoroughly
discuss loadings and boundary constraints typical of structural configurations.
Specific constitutive laws are given for materials known as linear elastic,
visco-elastic and thermo-elastic-plastic.
Chapter 3 is of fundamental significance to the approach consequently
followed in the book since it deals with the key concept of internal constraints
imposed on the kinematics and/or state of stress of the continuum. By specify
ing the constraints we are able to systematically derive classical and non
classical equations governing different "engineering" theories which describe
the nonlinear behaviour of beam-, plate- and shell-type configurations. The
considerations of this chapter are based on a number of variational principles.
Emphasis is placed on demonstrating the power of the principles in deriving
the governing equations in a systematic way. Next, the method of constraints
is used to introduce a discrete description of structural configurations. By
this we mean setting up approximate models of structures described in terms
of algebraic rather than differential equations. An interesting feature of the
approach to discretization as presented in this book is that it is again based
upon the internal constraint concept thus making the discussion fully compat
ible with that given previously in Sections 3.2, 3.3 of this chapter. Apart from
some attractive error estimates offered by the way we look at these problems
this portion of the book (Secs. 3.4, 3.5) essentially discusses the foundations
and applications of the well-known and powerful finite element method in
its displacement and mixed-type versions. Discretization with respect to space
and time variables is discussed and final sets of nonlinear algebraic equations
are derived.
Some useful extensions and specifications of the general theory proposed
in Chapter 3 are dealt with in the next Chapter 4. We consider in it such
problems as the domain discretization, assembly procedures to be used in
summing up the element matrices and vectors, specific forms of typical element
matrices as well as an explicit form of equations describing thermo-elastic
plastic materials. The theoretical background for the so-called shake-down
analysis of structures is also presented to be used later while discussing specific
boundary-value solutions to inelastic problems under variable loads.
XII
