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

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Printed in the United States of America. (BS/DH) 987 654321 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. Fundamentals of Structural Mechanics xiii Chapters 9 through 11 concern structural stability. Chapter 9 explores the concept of energy principles, observing that if an energy functional exists we can deduce it from a virtual-work functional by a theorem of Vainberg. The relationship between virtual work and energy provides an opportunity for fur­ ther exploration of the calculus of variations. This chapter ends with the ob­ servation that one can use an energy criterion to explore the stability of static equilibrium if the system possesses an energy functional. Chapter 10 gives a brief, illustrative invitation to static stability theory. Through the examination of some simple discrete systems, we encounter many of the interesting phe­ nomena associated with nonlinear systems. We distinguish limit points from bifurcation points, explore the effects of imperfections, and examine the role of linearized buckling analysis. Chapter 11 extends the ideas of chapter 10 to continuous systems, applying the machinery developed in chapter 9 to nonlin­ ear planar beam theory. Structural stability problems create a strong need for a general approach to nonlinear computations. Chapter 12 provides an introduction to nonlinear computations in mechanics. Newton's method serves as the unifying frame­ work for organizing the nonlinear computations. The arc-length method is of­ fered as a general strategy for numerically tracing equilibrium paths of nonlin­ ear mechanical systems. We illuminate the curve-tracing algorithm by numerically solving Euler's elastica and subsequently apply the algorithm to the solution of the fully nonlinear beam problem. Computer programs are pres­ ented at each level of the development to help cement the understanding of the algorithms.

Juan C. Simo, to whose memory this book is dedicated, was my closest and dearest friend. We were graduate students at the University of California at Berkeley in the early 1980s. We spent countless hours in the coffee shops near campus discussing mechanics. I learned to appreciate mechanics by watching his deep and clear insight flow from his pen onto his "pad yellow," as he called it in his inimitable Spanish accent. In his hands, the equations of mechanics came to life. Juan's love for mechanics, his tireless pursuit of knowledge, and his gift for developing and expressing theory made an indelible mark on me. His influence is clearly written on these pages. Juan Simo passed away on September 26, 1994, at the age of 42, after an eight-month battle with cancer. In his short career, Juan made tremendous con­ tributions to the field of computational mechanics, many of them in the area of nonlinear structural mechanics. Unfortunately, a classical education in structural mechanics leaves the student ill-equipped to appreciate Juan's con­ tributions (not to mention the contributions of many others). The approach I have taken in this book was inspired by the hope of narrowing the gap between classical structural mechanics and some of the modem innovations in the field. xiv Preface I owe a great debt to Juan that I can repay only by passing on what he taught me to the next generation of scholars and engineers. I hope this book defrays some of that debt. The first edition of the book was bom as a brief set of class notes for my course Applied Structural Mechanics at the University of Illinois. I am in­ debted to Bill Hall, Narbey Khachaturian, and Arthur Robinson for enabling the teaching opportunity that led to this book. I have loved every minute of the 24 times I have taught this course. I am indebted to the many students who have taken my course, first for inspiring me to write the book and then for gamely trying to learn from it. I appreciate the help of my former students and piost- docs—Parvis and Bijan Banan (a.k.a. "the bros"), Jiwon Kim, Ertugrul Taciro- glu, Eric Williamson, and Ken Zuo—^for their assistance with the first edition and later the completion of the solutions to all of the problems in that edition. I also appreciate the support of my colleagues—especially Bob Dodds, Dennis Parsons, and Glaucio Paulino—^for believing in the course enough to make it a cornerstone of our graduate curriculum in structural engineering at Illinois. This second edition of the book is informed by nearly a decade of using the first edition in my class. I have refined the story and added some important top­ ics. I have tightened up some of the things that were a little loose and loosened a few that were a bit tight. I even rewrote the computer programs in MATLAB. I have expanded the number of examples in the text and I have augmented the problems at the ends of the chapters—^tapping into my extensive collection of problems that have grown from my proclivity to facilitate the learning of me­ chanics through a diet of fortnightly "quizzes." The revisions for the second edition were largely made during the fall semester of 2003. The students in that class endured last minute delivery of the new chapters and did yeoman's work in tracking down typographical errors. I am especially appreciative of my own research assistants—Steve Ball, Kristine Cochran, Ghadir Haikal, Kalyanaba- bu Nakshatrala, and Arun Prakash—^for proofreading the text and making suggestions for its improvement. Special thanks to Kalyan for providing a tidy proof of Vainberg's theorem. Finally, I am grateful to my wife, Kara, and my children, David, Kirsten, and Aimika for being cheerful and supportive while the book robbed them of my time and attention. While it was Juan Simo who taught me the joy of mechan­ ics, my family has taught me the mechanics of joy. Keith D. Hjelmstad