NONLINEAR MECHANICS OF STRUCTURES

Nonlinear Mechanics of Structures

MICHAL KLEIBER AND CZESLA W 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 P olish 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~ !XX: 1(l1(JJlfJ7S!.J4..fJ:fJ-I

e-flBN-I3:978-94-(X9.(l57-1

Published by PWN- Polish Scientific Publishers, Miodowa 10, 00-251 Warszawa, Poland in co-edition with Kluwer Acadcmic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands

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for all other countries: Kluwcr ~ic Publisher, P.O. Box 322, 3300 AH Dordrecht. The Netherlands

Copyright © by PWN-Polish Scientific Publishers, Warszawa 1991

Softeovcr reprint of the hardcover I st edi tion 1991

All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or other­wise, without the prior written permission of the publisher.

Contents

Introduction . . . . . . . . . . . . . . . Conventions and notations . . . . . . . . .

PART I: THEORETICAL FOUNDATIONS

1 General notions and principles Purpose of the chapter . 1.1 Introductory concepts

1.1.1 Space and time 1.1.2 Motion. 1.1.3 Mass . ... . 1.1.4 Forces . . . . 1.1.5 Temperature, energy and entropy .

1.1.6 Heat supply. . . . .

1.2 Fundamental principles . . . . . . 1.2.1 Balance of momentum . . . . . 1.2.2 Balance of moment of momentum 1.2.3 Balance of energy . 1.2.4 Dissipation principle

1.3 Basic fields . . . . . . 1.3.1 Displacements . . . 1.3.2 Strains and strain rates 1.3.3 Stresses and stress rates. 1.3.4 Heat fluxes . . . . . . 1.3.5 Matrix notation for the basic fields.

1.4 Resulting relations . . . 1.4.1 Equation of motion . 1.4.2 Energy equation . . . 1.4.3 Dissipation inequality 1.4.4 Matrix form of the resulting equations

Summary

Problems

XI XIV

3 3 3 3

5 6 8 9

10 10 10 11 11

12 12 12 14 16 16

19 19 21 22 23

24

25

v

Contents

2 Materials, loadings and constraints

Purpose of the chapter . . . . . 2.1 Materials . . . . . . . . .

2.1.1 Real materials and ideal materials 2.1.2 Thermo-visco-elastic materials. 2.1.3 Linear thermo-elastic materials 2.1.4 Thermo-elastic-plastic materials 2.1.5 Matrix form of constitutive relations

2.2 Loadings and heat supply . . . . . . 2.2.1 External agents . . . . . . . . . 2.2.2 Body loadings and internal heat supply 2.2.3 Boundary loadings . . 2.2.4 Boundary heat supply. . . . . .

2.3 Constraints 2.3.1 Boundary and internal constraints 2.3.2 Reactions. . . . . 2.3.3 Bilateral constraints . . . . . . 2.3.4 Constraint functions . . . . . . 2.3.5 Constraints for stresses and heat fluxes

Summary .......... . Problems .......... .

3 Formulation of engineering theories

Purpose of the chapter . . . . .

3.1 General form of governing relations . 3.1.1 Finite variational formulation . . 3.1.2 Incremental variational formulation 3.1.3 Variational equations for stresses and heat fluxes. 3.1.4 Calculation of reactions ..... . 3.1.5 Matrix form of variational equations.

3.2 Method of constraints: finite formulation 3.2.1 Foundations ..... 3.2.2 Shell and plate theories. 3.2.3 Rod and beam theories . 3.2.4 Plane problems . . . .

3.3 Method of constraints: incremental formulation 3.3.1 General equations 3.3.2 Shells and rods 3.3.3 Plane problems

VI

27 27 27 28 29 30 34

37 37 38 38 39

39 39 42 44 46 47

49 50

52

52 52 55 58 59 60 61 61 65 71 74

77 77 78 79

3.4 Method of constraints: discrete description of elements structures . . . . . . .

3.4.1 Discretization concept . 3.4.2 Finite formulation . . . 3.4.3 Incremental formulation 3.4.4 Shell and plate elements 3.4.5 Rod and beam elements 3.4.6 Special cases and matrix notation 3.4.7 Interaction between elements . .

3.5 Evaluation and correction of solutions . 3.5.1 Residual reactions . . . 3.5.2 Evaluation of solutions . . . . . 3.5.3 Correction of solutions . . . . . 3.5.4 On adaptive solution refinements

Summary

Problems

4 Extensions and specifications of the general theory

Contents

and 80 80 80 86 87 89 91 96

97 97 99

101 103

103

104

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 Problems

5 Numerical algorithms and software concepts Purpose of the chapter . . 5.1 Introductory comments . . . . . . . 5.2 Nonlinear quasi-statics . ..... . 5.3 Integration of elastic-plastic constitutive law 5.4 Initial and linearized buckling . . . . . . 5.5 Nonlinear dynamics . . . . . . . . . . . 5.6 Dynamic stability under non-periodic loads 5.7 Nonlinear heat transfer . 5.8 Development of software Summary Problems .. .... . . .

PART II: SELECTED APPLICATIONS

6 Trusses, frames, lattice-type shells

Purpose of the chapter . . . . . 6.1 Trusses . . . . . . . . . .

205 206

212 212 213 225 229 232 237 242 244 246 246

251 251

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 ...... .

8 Plate and shell problems .

Purpose of the chapter . . . . . . . . . . . . . . 8.1 Thin elastic-plastic axisymmetric shells. . . . . . 8.2 Elastic-viscoplastic analysis of axisymmetric shells . 8.3 Buckling of elastic-plastic spatial plate assemblies Summary ......... . .... .

9 Heat conduction and thermal stress problems

384

385 385 405 410 430

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 "engineer­ing" 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 specify­ing the constraints we are able to systematically derive classical and non­classical 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 compat­ible 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-elastic­plastic 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

Introduction

The solution methods for the discretized problems are described in Chapter 5. Two basic sections in this chapter deal with numerical algorithms for static and dynamic nonlinear problems, respectively. The algorithms are based on repetitive solutions of sets of linear algebraic equations describing the instantaneous equilibrium of the discretized body and are additionally endowed with some Newton-Raphson iterative corrections to improve the accuracy of the results. Part I of the book is completed by a short discussion of software concepts which are typically employed while developing computer programs.

The contents of Part I of the book is made more specific in Part II, in which we consider different discretized models for beams, plates and shells. A number of effective finite elements is described in detail and test calculations are given to illustrate their performance in modelling different aspects of nonlinear structural behaviour. Trusses, frames, grids and shell-like structures made of regularly distributed beams are dealt with in Chapter 6. Chapter 7 is devoted to the analysis of thin plates loaded in-plane whereas Chapter 8 concentrates on shell-type problems. A short discussion of heat conduction and thermal stress problems given in Chapter 9 completes the book.

Problems for independent studies are provided at the end of most chapters so that students can check their understanding of the subjects. At places, problems serve also the purpose of extending the scope of the book into areas which could not have been covered for the sake of compactness of the presenta­tion.

The course, as outlined here, is designed for a two-semester sequence at the graduate level. On the whole, it is believed that the treatment used in this book is in harmony with current trends toward a more fundamental approach in engineering education.

Conventions and notations

Throughout the book we shall use such mathematical objects as scalars, vectors, scalar and vector functions, vector and tensor components, coordinates of points, etc. We shall also perform certain operations on these objects such as scalar or vector products. We assume that the reader knows the elementary vector and tensor calculus both in the absolute and the component form. As far as possible we shall use the following conventions:

1. Sub- and superscripts i, j, k, I, m, n run over the sequence 1, 2, 3 and are related to a cartesian coordinate system (so-called spatial coordinate system) in the physical space identified with the Euclidean 3-space R3 of spatial points. Sub- and superscripts ex, fl, y, b run also over the sequence 1,2,3 (unless a different range is clearly stated) and are related to a coordinate system introduced exclusively in a region D occupied by a body in a certain reference configuration (so-called material coordinate system). Ranges of any other indices appearing in the text will always be explained prior to their use.

2. Summation convention holds with respect to the indices ex, fl, ... and i,j, ... , provided they are repeated twice in a given expression and situated in it on the upper and lower level, respectively. Ricci symbol and Kronecker delta are denoted by 8 iik, bii respectively (position of i, j, k is immaterial).

3. Spatial points are denoted by z, Z E R3, and their (spatial) coordinates by Zk; points of D are denoted by X, XED, and their (material) coordinates by Xa (instead of k or ex we may be using any other pertinent Latin or Greek superscript, respectively). The symbols i and t stand for time instants with t used to indicate a specific rather than an arbitrary time.

4. Scalars are represented mostly by small light-faced Greek italics: e, ex, n. Scalar-valued functions are denoted by e( . ), ex( . ), n( .) etc.; domains of these functions are specified in the text.

5. Vectors and vector-valued functions are represented by small bold

face Latin or Greek Roman letters; examples: f, X, i(·), i ( . ), respectively; their components are represented by small face Latin or Greek italics : fk, X", jk( .), XA( . ).

XIV

Conventions and notations

6. Second order tensors and tensor-valued functions are represented by A A

capital bold face Latin letters: T, Sand T('), S( . ); their components are

denoted by Tii, S"(3 and rU(.), S"(3( · ), respectively. 7. Higher order tensors and tensor functions are represented by

A A

German capitals: <£, i> and <£(.), i>('), respectively; their components are indicated by light-faced Roman letters: Ciikl, H"(3""(') etc. Symbols C['], H['] stand for linear mappings of the form Ciikl Dkl> H"(3".,S)ld, respectively.

8. Comma denotes partial differentiation with respect to the spatial

coordinates (example: r.i == :£T) or to the material coordinates (example:

T.t1. == :;rx). Dot denotes substantial differentiation with respect to the time

coordinate (hence g = ~; provided g = g(X, 1) and X = (xrx) are the

material coordinates). The gradient with respect to the material coordinates

X = (X") is denoted by V (example: Vn = (n.tn.2n.3), n.rx == :;'"

where n = n(X, i)). 9. The component forms of products a' b, a x b, Tn, SE are akbb 8kl ma,b'",

Tkln!. S"(3 E(3)1, respectively. 10. Symbols tr A, det A stand for trace and determinant of an arbitrary

second order tensor A. Symbols AT, A-I stand for the transpose and the inverse of A, respectively, 1 is a unit tensor.

11. Sets of points or vectors are represented by capital light face Greek letters (examples: D, rJ)); sets of mappings are denoted by a light face script (examples: ry .cr\.

xv