Fundamentals of Structural Mechanics

484 Pages·2013·6 MB·English

Fundamentals of
Structural Mechanics
Second Edition
Fundamentals of
Structural Mechanics
Second Edition
Keith D. Hjelmstad
University of Illinois at Urbana-Champaign
Urbana-Champaign, Illinois
^ Springer
ISBN 0-387-23330-X
elSBN 0-387-23331-8
©2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
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Printed in the United States of America. (BS/DH)
9 87 6 5 4 3 21
springeronline.com
To the memory of
Juan Carlos Simo
(1952-1994)
who taught me
the joy of mechanics
and
To my family
Kara, David, Kirsten, and Annika
who taught me
the mechanics of joy
Contents
Preface xi
1 Vectors and Tensors 1
The Geometry of Three-dimensional Space 2
Vectors 3
Tensors 11
Vector and Tensor Calculus 33
Integral Theorems 45
Additional Reading 48
Problems 49
2 The Geometry of Deformation 57
Uniaxial Stretch and Strain 58
The Deformation Map 62
The Stretch of a Curve 65
The Deformation Gradient 67
Strain in Three-dimensional Bodies 68
Examples 69
Characterization of Shearing Deformation 74
The Physical Significance of the Components of C 77
Strain in Terms of Displacement 78
Principal Stretches of the Deformation 79
Change of Volume and Area 84
Time-dependent motion 91
Additional Reading 93
Problems 94
viii Contents
3 The Transmission of Force 103
The Traction Vector and the Stress Tensor 103
Normal and Shearing Components of the Traction 109
Principal Values of the Stress Tensor 110
Differential Equations of Equilibrium 112
Examples 115
Alternative Representations of Stress 118
Additional Reading 124
Problems 125
4 Elastic Constitutive Theory 131
Isotropy 138
Definitions of Elastic Moduli 141
Elastic Constitutive Equations for Large Strains 145
Limits to Elasticity 148
Additional Reading 150
Problems 151
5 Boundary Value Problems in Elasticity 159
Boundary Value Problems of Linear Elasticity 160
A Little Boundary Value Problem (LBVP) 165
Work and Virtual Work 167
The Principle of Virtual Work for the LBVP 169
Essential and Natural Boundary Conditions 181
The Principle of Virtual Work for 3D Linear Solids 182
Finite Deformation Version of the Principle of Vu*tual Work 186
Qosure 188
Additional Reading 189
Problems 190
6 The Ritz Method of Approximation 193
The Ritz Approxunation for the Little Boundary Value Problem .. 194
Orthogonal Ritz Functions 207
The Finite Element Approximation 216
The Ritz Method for Two- and Three-dimensional Problems .... 226
Additional Reading 233
Problems 234
7 The Lmear Theory of Beams 241
Equations of Equilibrium 243
The Kinematic Hypothesis 249
Constitutive Relations for Stress Resultants 252
Contents ix
Boundary Conditions 256
The Limitations of Beam Theory 257
The Principle of Virtual Work for Beams 262
The Planar Beam 266
The BemouUi-Euler Beam 273
Structural Analysis 278
Additional Reading 282
Problems 283
8 The Linear Theory of Plates 293
Equations of Equilibrium 295
The Kinematic Hypothesis 300
Constitutive Equations for Resultants 304
Boundary Conditions 308
The Limitations of Plate Theory 310
The Principle of Virtual Work for Plates 311
The Kirchhoff-Love Plate Equations 314
Additional Reading 323
Problems 324
9 Energy Principles and Static Stability 327
Wtual Work and Energy Functionals 330
Energy Principles 341
Static Stability and the Energy Criterion 345
Additional Reading 352
Problems 353
10 Fundamental Concepts m Static Stability 359
Bifurcation of Geometrically Perfect Systems 361
The Effect of Imperfections 369
The Role of Linearized Buckling Analysis 375
Systems with Multiple Degrees of Freedom 378
Additional Reading 384
Problems 385
11 The Planar Buckling of Beams 389
Derivation of the Nonlinear Planar Beam Theory 390
A Model Problem: Euler's Elastica 2>91
The General Linearized Buckling Theory 408
Ritz and the Linearized Eigenvalue Problem 415
Additional Reading 421
Problems 423
X Contents
12 Numerical Computation for Nonlinear Problems 431
Newton's Method 433
Tracing the Equilibrium Path of a Discrete System 438
The Program NEWTON 444
Newton's Method and Wtual Work 446
The Program ELASTICA 452
The Fully Nonlinear Planar Beam 454
The Program NONLINEARBEAM 462
Summary 469
Additional Reading 469
Problems 470
Index 473
Preface
The last few decades have witnessed a dramatic increase in the application
of numerical computation to problems in solid and structural mechanics. The
burgeoning of computational mechanics opened a pedagogical gap between
traditional courses in elementary strength of materials and the finite element
method that classical courses on advanced strength of materials and elasticity
do not adequately fill. In the past, our ability to formulate theory exceeded our
ability to compute. In those days, solid mechanics was for virtuosos. With the
advent of the finite element method, our ability to compute has surpassed our
ability to formulate theory. As a result, continuum mechanics is no longer the
province of the specialist.
What an engineer needs to know about mechanics has been forever changed
by our capacity to compute. This book attempts to capitalize on the pedagogi
cal opportunities implicit in this shift of perspective. It now seems more ap
propriate to focus on fundamental principles and formulations than on classical
solution techniques.
The term structural mechanics probably means different things to different
people. To me it brings to mind the specialized theories of beams, plates, and
shells that provide the building blocks of common structures (if it involves
bending moment then it is probably structural mechanics). Structural elements
are often slender, so structural stability is also a key part of structural mechan
ics. This book covers the fundamentals of structural mechanics. The treatment
here is guided and confined by the strong philosophical framework of continu
um mechanics and is given wings to fly by the powerful tools of numerical
analysis.
xii Preface
In essence, this book is an introduction to computational structural mechan
ics. The emphasis on computation has both practical and pedagogical roots.
The computational methods developed here are representative of the methods
prevalent in the modem tools of the trade. As such, the lessons in computation
are practical. An equally important outcome of the computational framework
is the great pedagogical boost that the student can get from the notion that most
problems are amenable to the numerical methods advocated herein. A theory
is ever-so-much more interesting if you really believe you can crunch numbers
with it. This optimistic outlook is a pedagogical boon to learning mechanics
and the mathematics that goes along with it.
This book is by no means a comprehensive treatment of structural mechan
ics. It is a simple template to help the novice learn how to think about structural
mechanics and how to express those thoughts in the language of mathematics.
The book is meant to be a preamble to further study on a variety of topics from
continuum mechanics to finite element methods. The book is aimed at ad
vanced undergraduates and first-year graduate students in any of the mechani
cal sciences (e.g., civil, mechanical, and aerospace engineering).
The book starts with a brief account of the algebra and calculus of vectors
and tensors (chapter 1). One of the main goals of the first chapter is to introduce
some requisite mathematics and to establish notation that is used throughout
the book. The next three chapters lay down the fundamental principles of con
tinuum mechanics, including the geometric aspects of deformation and motion
(chapter 2), the laws governing the transmission of force (chapter 3), and ele
ments of constitutive theory (chapter 4).
Chapters 5 and 6 concern boundary value problems in elasticity and their
solution. We introduce the classical (strong form) and the variational (weak
form) of the governing differential equations. Many of the ideas are motivated
with the one-dimensional ''little boundary value problem'' The Ritz method
is offered as a general approach to numerical computations, based upon the
principle of virtual work. Although we do not pursue it in detail, we show how
the Ritz method can be specialized to form the popular and powerful/z«te ele
ment method. The Ritz method provides a natural tool for all of the structural
mechanics computations needed for the rest of the book.
Chapters 7 and 8 cover the linear theories of beams and plates, respectively.
These structural mechanics theories are developed within the context of three-
dimensional continuum mechanics with the dual benefit of lendmg a deeper
understanding of beams and plates and, at the same time, of providing two rele
vant applications of the general equations of continuum mechanics presented
in the first part of the book. The classical constrained theories of beams (Ber-
nouUi-Euler) and plates (Kirchhoff-Love) are examined in detail. Each theory
is cast both as a classical boundary value problem and as a variational problem.