456814236cover & table of contents - analysis of structures (an introduction including numerical...

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

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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


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


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


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


5.6 Analysis and Design of Pin-jointed Trusses 202


6 Torsion 213


6.2 Torsional Displacement, Strain, and Stress 213



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


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


7.4 The Statically Determinate Case 271

7.5 Work and EnergyCastiglianos Second Theorem 278

7.6 The Statically Indeterminate Case 281



7.9 Shear Stress in Non Rectangular Cross SectionsThin Walled Cross Sections 302

7.10 Design of Beams 309

7.11 Large Displacements 313


8 A General Method (FEM) for the Bending of Beams 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



9 More about Stress and Strain, and Material Properties 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


10 Combined Loadings on Slender BarsThin Walled Cross Sections 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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x Contents

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


11 Work and Energy MethodsVirtual Work 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


12 Structural Analysis in Two and Three Dimensions 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


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


14.2 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


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


16 Evolution in the (Intelligent) Design and Analysis of Structural Members 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


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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xii Contents

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

456814236cover & table of contents - analysis of structures (an introduction including numerical...

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