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7/27/2019 456814236Cover & Table of Contents - Analysis of Structures (an Introduction Including Numerical Methods)
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ANALYSIS OFSTRUCTURES
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ANALYSIS OF
STRUCTURESAN INTRODUCTION INCLUDING
NUMERICAL METHODS
Joe G. Eisley
Anthony M. Waas
College of Engineering
University of Michigan, USA
A John Wiley & Sons, Ltd., Publication
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This edition first published 2011
C 2011 John Wiley & Sons, Ltd
Registered office
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom
For details of our global editorial offices, for customer services and for information about how to apply for
permission to reuse the copyright material in this book please see our website at www.wiley.com.
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Designs and Patents Act 1988.
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, except as permitted by the UK
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Library of Congress Cataloging-in-Publication Data
Eisley, Joe G.
Analysis of structures : an introduction including numerical methods / Joe G. Eisley, Anthony M. Waas.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-97762-0 (cloth)
1. Structural analysis (Engineering)Mathematics. 2. Numerical analysis. I. Waas, Anthony M. II. Title.
TA646.W33 2011
624.171dc22
2011009723
A catalogue record for this book is available from the British Library.
Print ISBN: 9780470977620
E-PDF ISBN: 9781119993285
O-book ISBN: 9781119993278
E-Pub ISBN: 9781119993544
Mobi ISBN: 9781119993551
Typeset in 9/11pt Times by Aptara Inc., New Delhi, India
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We would like to dedicate this book to our families.
To Marilyn, Paul and Susan
Joe
To Dayamal, Dayani, Shehara and Michael
Tony
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Contents
About the Authors xiii
Preface xv
1 Forces and Moments 1
1.1 Introduction 1
1.2 Units 1
1.3 Forces in Mechanics of Materials 3
1.4 Concentrated Forces 4
1.5 Moment of a Concentrated Force 9
1.6 Distributed ForcesForce and Moment Resultants 19
1.7 Internal Forces and StressesStress Resultants 27
1.8 Restraint Forces and Restraint Force Resultants 32
1.9 Summary and Conclusions 33
2 Static Equilibrium 35
2.1 Introduction 35
2.2 Free Body Diagrams 35
2.3 EquilibriumConcentrated Forces 38
2.3.1 Two Force Members and Pin Jointed Trusses 38
2.3.2 Slender Rigid Bars 44
2.3.3 Pulleys and Cables 49
2.3.4 Springs 52
2.4 EquilibriumDistributed Forces 55
2.5 Equilibrium in Three Dimensions 592.6 EquilibriumInternal Forces and Stresses 62
2.6.1 Equilibrium of Internal Forces in Three Dimensions 65
2.6.2 Equilibrium in Two DimensionsPlane Stress 69
2.6.3 Equilibrium in One DimensionUniaxial Stress 70
2.7 Summary and Conclusions 70
3 Displacement, Strain, and Material Properties 71
3.1 Introduction 71
3.2 Displacement and Strain 71
3.2.1 Displacement 72
3.2.2 Strain 723.3 Compatibility 76
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viii Contents
3.4 Linear Material Properties 77
3.4.1 Hookes Law in One DimensionTension 77
3.4.2 Poissons Ratio 81
3.4.3 Hookes Law in One DimensionShear in Isotropic Materials 82
3.4.4 Hookes Law in Two Dimensions for Isotropic Materials 833.4.5 Generalized Hookes Law for Isotropic Materials 84
3.5 Some Simple Solutions for Stress, Strain, and Displacement 85
3.6 Thermal Strain 89
3.7 Engineering Materials 90
3.8 Fiber Reinforced Composite Laminates 90
3.8.1 Hookes Law in Two Dimensions for a FRP Lamina 91
3.8.2 Properties of Unidirectional Lamina 94
3.9 Plan for the Following Chapters 96
3.10 Summary and Conclusions 98
4 Classical Analysis of the Axially Loaded Slender Bar 99
4.1 Introduction 99
4.2 Solutions from the Theory of Elasticity 99
4.3 Derivation and Solution of the Governing Equations 109
4.4 The Statically Determinate Case 116
4.5 The Statically Indeterminate Case 129
4.6 Variable Cross Sections 136
4.7 Thermal Stress and Strain in an Axially Loaded Bar 142
4.8 Shearing Stress in an Axially Loaded Bar 143
4.9 Design of Axially Loaded Bars 145
4.10 Analysis and Design of Pin Jointed Trusses 1494.11 Work and EnergyCastiglianos Second Theorem 153
4.12 Summary and Conclusions 162
5 A General Method for the Axially Loaded Slender Bar 165
5.1 Introduction 165
5.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix 165
5.3 The Assembled Global Equations and Their Solution 169
5.4 A General MethodDistributed Applied Loads 182
5.5 Variable Cross Sections 196
5.6 Analysis and Design of Pin-jointed Trusses 202
5.7 Summary and Conclusions 211
6 Torsion 213
6.1 Introduction 213
6.2 Torsional Displacement, Strain, and Stress 213
6.3 Derivation and Solution of the Governing Equations 216
6.4 Solutions from the Theory of Elasticity 225
6.5 Torsional Stress in Thin Walled Cross Sections 229
6.6 Work and EnergyTorsional Stiffness in a Thin Walled Tube 231
6.7 Torsional Stress and Stiffness in Multicell Sections 239
6.8 Torsional Stress and Displacement in Thin Walled Open Sections 242
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Contents ix
6.9 A General (Finite Element) Method 245
6.10 Continuously Variable Cross Sections 254
6.11 Summary and Conclusions 255
7 Classical Analysis of the Bending of Beams 2577.1 Introduction 257
7.2 Area PropertiesSign Conventions 257
7.2.1 Area Properties 257
7.2.2 Sign Conventions 259
7.3 Derivation and Solution of the Governing Equations 260
7.4 The Statically Determinate Case 271
7.5 Work and EnergyCastiglianos Second Theorem 278
7.6 The Statically Indeterminate Case 281
7.7 Solutions from the Theory of Elasticity 290
7.8 Variable Cross Sections 300
7.9 Shear Stress in Non Rectangular Cross SectionsThin Walled Cross Sections 302
7.10 Design of Beams 309
7.11 Large Displacements 313
7.12 Summary and Conclusions 314
8 A General Method (FEM) for the Bending of Beams 315
8.1 Introduction 315
8.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix 315
8.3 The Global Equations and their Solution 320
8.4 Distributed Loads in FEM 327
8.5 Variable Cross Sections 341
8.6 Summary and Conclusions 345
9 More about Stress and Strain, and Material Properties 347
9.1 Introduction 347
9.2 Transformation of Stress in Two Dimensions 347
9.3 Principal Axes and Principal Stresses in Two Dimensions 350
9.4 Transformation of Strain in Two Dimensions 354
9.5 Strain Rosettes 356
9.6 Stress Transformation and Principal Stresses in Three Dimensions 358
9.7 Allowable and Ultimate Stress, and Factors of Safety 361
9.8 Fatigue 363
9.9 Creep 3649.10 Orthotropic MaterialsComposites 365
9.11 Summary and Conclusions 366
10 Combined Loadings on Slender BarsThin Walled Cross Sections 367
10.1 Introduction 367
10.2 Review and Summary of Slender Bar Equations 367
10.2.1 Axial Loading 367
10.2.2 Torsional Loading 369
10.2.3 Bending in One Plane 370
10.3 Axial and Torsional Loads 372
10.4 Axial and Bending Loads2D Frames 375
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10.5 Bending in Two Planes 384
10.5.1 When I yz is Equal to Zero 384
10.5.2 When I yz is Not Equal to Zero 386
10.6 Bending and Torsion in Thin Walled Open SectionsShear Center 393
10.7 Bending and Torsion in Thin Walled Closed SectionsShear Center 39910.8 Stiffened Thin Walled Beams 405
10.9 Summary and Conclusions 416
11 Work and Energy MethodsVirtual Work 417
11.1 Introduction 417
11.2 Introduction to the Principle of Virtual Work 417
11.3 Static Analysis of Slender Bars by Virtual Work 421
11.3.1 Axially Loading 421
11.3.2 Torsional Loading 426
11.3.3 Beams in Bending 427
11.3.4 Combined Axial, Torsional, and Bending Behavior 43011.4 Static Analysis of 3D and 2D Solids by Virtual Work 430
11.5 The Element Stiffness Matrix for Plane Stress 433
11.6 The Element Stiffness Matrix for 3D Solids 436
11.7 Summary and Conclusions 437
12 Structural Analysis in Two and Three Dimensions 439
12.1 Introduction 439
12.2 The Governing Equations in Two DimensionsPlane Stress 440
12.3 Finite Elements and the Stiffness Matrix for Plane Stress 445
12.4 Thin Flat PlatesClassical Analysis 452
12.5 Thin Flat PlatesFEM Analysis 45512.6 Shell Structures 459
12.7 Stiffened Shell Structures 466
12.8 Three Dimensional StructuresClassical and FEM Analysis 470
12.9 Summary and Conclusions 477
13 Analysis of Thin Laminated Composite Material Structures 479
13.1 Introduction to Classical Lamination Theory 479
13.2 Strain Displacement Equations for Laminates 480
13.3 Stress-Strain Relations for a Single Lamina 482
13.4 Stress Resultants for Laminates 486
13.5 CLT Constitutive Description 489
13.6 Determining Laminae Stress/Strains 492
13.7 Laminated Plates Subject to Transverse Loads 493
13.8 Summary and Conclusion 498
14 Buckling 499
14.1 Introduction 499
14.2 The Equations for a Beam with Combined Lateral and Axial Loading 499
14.3 Buckling of a Column 504
14.4 The Beam Column 512
14.5 The Finite Element Method for Bending and Buckling 515
14.6 Buckling of Frames 524
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Contents xi
14.7 Buckling of Thin Plates and Other Structures 524
14.8 Summary and Conclusions 527
15 Structural Dynamics 529
15.1 Introduction 52915.2 Dynamics of Mass/Spring Systems 529
15.2.1 Free Motion 529
15.2.2 Forced MotionResonance 540
15.2.3 Forced MotionResponse 547
15.3 Axial Vibration of a Slender Bar 548
15.3.1 Solutions Based on the Differential Equation 548
15.3.2 Solutions Based on FEM 560
15.4 Torsional Vibration 567
15.4.1 Torsional Mass/Spring Systems 567
15.4.2 Distributed Torsional Systems 568
15.5 Vibration of Beams in Bending 56915.5.1 Solutions of the Differential Equation 569
15.5.2 Solutions Based on FEM 574
15.6 The Finite Element Method for all Elastic Structures 577
15.7 Addition of Damping 577
15.8 Summary and Conclusions 582
16 Evolution in the (Intelligent) Design and Analysis of Structural Members 583
16.1 Introduction 583
16.2 Evolution of a Truss Member 584
16.2.1 Step 1. Slender Bar Analysis 584
16.2.2 Step 2. Rectangular BarPlane Stress FEM 58516.2.3 Step 3. Rectangular Bar with Pin HolesPlane Stress Analysis 586
16.2.4 Step 4. Rectangular Bar with Pin HolesSolid Body Analysis 587
16.2.5 Step 5. Add Material Around the HoleSolid Element Analysis 588
16.2.6 Step 6. Bosses AddedSolid Element Analysis 590
16.2.7 Step 7. Reducing the WeightSolid Element Analysis 591
16.2.8 Step 8. Buckling Analysis 592
16.3 Evolution of a Plate with a HolePlane Stress 592
16.4 Materials in Design 594
16.5 Summary and Conclusions 594
A Matrix Definitions and Operations 595
A.1 Introduction 595
A.2 Matrix Definitions 595
A.3 Matrix Algebra 597
A.4 Partitioned Matrices 598
A.5 Differentiating and Integrating a Matrix 598
A.6 Summary of Useful Matrix Relations 599
B Area Properties of Cross Sections 601
B.1 Introduction 601
B.2 Centroids of Cross Sections 601
B.3 Area Moments and Product of Inertia 603
B.4 Properties of Common Cross Sections 609
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C Solving Sets of Linear Algebraic Equations with Mathematica 611
C.1 Introduction 611
C.2 Systems of Linear Algebraic Equations 611
C.3 Solving Numerical Equations in Mathematica 611
C.4 Solving Symbolic Equations in Mathematica 612C.5 Matrix Multiplication 613
D Orthogonality of Normal Modes 615
D.1 Introduction 615
D.2 Proof of Orthogonality for Discrete Systems 615
D.3 Proof of Orthogonality for Continuous Systems 616
References 617
Index 619
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About the Authors
Joe G. Eisley received degrees from St. Louis University, BS (1951), and the California Institute of
Technology, MS (1952), PhD (1956), all in the field of aeronautical engineering. He served on the faculty
of the Department of Aerospace Engineering from 1956 to 1998 and retired as Emeritus Professor of
Aerospace Engineering in 1998. His primary field of teaching and research has been in structural analysis
with an emphasis on the dynamics of structures. He also taught courses in space systems design and
computer aided design. After retirement he has continued some part time work in teaching and consulting.
Anthony M. Waas is the Felix Pawlowski Collegiate Professor of Aerospace Engineering and Professor
of Mechanical Engineering, and Director, Composite Structures Laboratoryat the University of Michigan.
He received his degrees from Imperial College, Univ. of London, U.K., B.Sc. (first class honors, 1982),
and the California Institute of Technology, MS (1983), PhD (1988) all in Aeronautics. He joined the
University of Michigan in January 1988 as an Assistant Professor, and is currently the Felix Pawlowski
Collegiate Professor. His current teaching and research interests are related to lightweight composite
aerostructures, with a focus on manufacturability and damage tolerance, ceramic matrix compositesfor hot structures, nano-composites, and multi-material structures. Several of his projects have been
funded by numerous US government agencies and industry. In addition, he has been a consultant to several
industries in various capacities. At Michigan, he has served as the Aerospace Engineering Department
Graduate Program Chair (19982002) and the Associate Chairperson of the Department (20032005).
He is currently a member of the Executive Committee of the College of Engineering. He is author or
co-author of more than 175 refereed journal papers, and numerous conference papers and presentations.
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Preface
This textbook is intended to be an introductory text on the mechanics of solids. The authors have targeted
an audience that usually would go on to obtain undergraduate degrees in aerospace and mechanical
engineering. As such, some specialized topics that are of importance to aerospace engineers are given
more coverage. The material presented assumes only a background in introductory physics and calculus.
The presentation departs from standard practice in a fundamental way. Most introductory texts on
this subject take an approach not unlike that adopted by Timoshenko, in his 1930 Strength of Materials
books, that is, by primarily formulating problems in terms of forces. This places an emphasis on statically
determinate solid bodies, that is, those bodies for which the restraint forces and moments, and internal
forces and moments, can be determined completely by the equations of static equilibrium. Displacements
are then introduced in a specialized way, often only at a point, when necessary to solve the few statically
indeterminate problems that are included. Only late in these texts are distributed displacements even
mentioned. Here, we introduce and formulate the equations in terms of distributed displacements from
the beginning. The question of whether the problems are statically determinate or indeterminate becomes
less important. It will appear to some that more time is spent on the slender bar with axial loads than thatparticular structure deserves. The reason is that classical methods of solving the differential equations
and the connection to the rational development of the finite element method can be easily shown with
a minimum of explanation using the axially loaded slender bar. Subsequently, the development and
solution of the equations for more advanced structures is facilitated in later chapters.
Modern advanced analysis of the integrity of solid bodies under external loads is largely displacement
based. Once displacements are known the strains, stresses, strain energies, and restraint reactions are
easily found. Modern analysis solutions methods also are largely carried out using a computer. The
direction of this presentation is first to provide an understanding of the behavior of solid bodies under
load and second to prepare the student for modern advanced courses in which computer based methods
are the norm.
Analysis of Structures: An Introduction Including Numerical Methods is accompanied by a website(www.wiley.com/go/waas) housing exercises and examples that use modern software which generates
color contour plots of deformation and internal stress. It offers invaluable guidance and understanding
to senior level and graduate students studying courses in stress and deformation analysis as part of
aerospace, mechanical and civil engineering degrees as well as to practicing engineers who want to
re-train or re-engineer their set of analysis tools for contemporary stress and deformation analysis of
solids and structures.
We are grateful to Dianyun Zhang, Ph.D candidate in Aerospace Engineering, for her careful reading
of the examples presented.
Corrections, comments, and criticisms are welcomed.
Joe G. EisleyAnthony M. Waas
June 2011
Ann Arbor, Michigan