Buckling of bars, plates, and shells - · PDF fileContents xi Problem Set 3.3.3 279 3.3.4 Buckling of SimplySupported Rectangular Plates under Combined In-PlaneBendingand Compression

BUCKLING OF

BARS, PLATES,AND SHELLS

ROBERT M. JONES

Professor Emeritus of Engineering Science and Mechanics

Virginia Polytechnic Institute and State UniversityBiacksburg, Virginia 24061-0219

Bull Ridge Publishing

Biacksburg, VirginiaUnited States of America

copyright © 2006 by Bull Ridge Publishingall rights reserved

CONTENTS

PREFACE xix

1 INTRODUCTION TO BUCKLING 1

1.1 FUNDAMENTAL DEFINITIONS OF

MECHANICAL SYSTEMS BEHAVIOR 2

1.1.1 Stability 2

1.1.2 Instability 3

1.1.3 Buckling Load 5

1.1.4 Maximum Load 9

1.1.5 General Load-Deformation Behavior including Buckling 10

1.2 BOOK CONCENTRATION AND PHILOSOPHY 10

1.2.1 Introduction 10

1.2.2 Buckling of Bars 12

1.2.3 Buckling of Plates 14

1.2.4 Buckling of Shells 15

1.2.5 Summary 17

1.3 CLASSIFICATION OF TYPES OF

BUCKLING AND POSTBUCKLING BEHAVIOR 18


1.3.2 General Nature of Linear Eigenvalue Analysis 18

1.3.3 Types of Postbuckling Behavior 20

1.3.4 Initial Geometric Imperfections Effect on Postbuckling Behavior 22

1.3.5 Degree of Stability in Stable Postbuckling Behavior 24

1.3.6 Summary 25

1.4 IMPORTANCE OF EXPERIMENTAL RESULTS

IN DEVELOPMENT OF THEORIES 26


1.4.2 Comparison of Theoretical and Experimental Results 26

1.4.3 Experimental Results are Used

More than 'Just' to Validate Theory 28

1.4.4 Experimental Results are

the Very Basis of all Theoretical Models 30

1.4.5 Summary 32

1.5 SOME IMPORTANT DISTINCTIONS IN ENGINEERING ANALYSIS 33


1.5.2 Presumption versus Assumption 33

1.5.3 Presumption versus Restriction 40

1.5.4 Misinterpretations and their Consequences 42

1.5.5 Summary 44

1.6 BOOK ROAD MAP 46

CHAPTER 1 REFERENCES 47

vil

viii Contents

2 BUCKLING OF BARS 49

2.1

2.2

INTRODUCTION

EULER'S COLUMN EQUATION

49

50

2.2.1 Buckling of an Axially Compressed Pinned-Pinned Bar 50

2.2.2 Buckling of an Axially Compressed Fixed-Pinned Bar 56

2.2.3 Buckling of an Axially Compressed Fixed-Fixed Bar 57

2.2.4 Buckling of an Axially Compressed Fixed-Free Bar 58

2.2.5 Buckling of an Axially Compressed Pinned-Free Bar

with Elastically Restrained Ends 59

2.2.6 Buckling of an Axially Compressed Pinned-Free Bar

with an Elastic Translational Spring at the Free End 60

2.2.7 Buckling of an Axially Compressed Pinned-Pinned Bar

with a Mid-Span Transverse Spring 62

2.2.8 Summary 66

Problem Set 2.2 66

2.3 AN ALTERNATIVE BAR BUCKLING DIFFERENTIAL EQUATION.

67

2.3.1 Derivation of an Alternative Differential Equation 67

2.3.2 Buckling of an Axially Compressed Pinned-Pinned Bar 69

2.3.3 Buckling of an Axially Compressed Fixed-Pinned Bar 70

2.3.4 Buckling of an Axially Compressed Fixed-Fixed Bar 71

2.3.5 Buckling of an Axially Compressed Fixed-Free Bar 72

2.3.6 Buckling of an Axially Compressed Fixed-Guided Bar 73

2.3.7 Buckling of an Axially Compressed Pinned-Guided Bar 75

2.3.8 The Effective Length Conceptto Account for Boundary Conditions 76

2.3.9 Summary 76

Problem Set 2.3 78

2.4 LARGE DEFLECTIONS OF BUCKLED BARS —

THE PROBLEM OF THE ELAST1CA 79


2.4.2 Exact Load-Deflection Behavior 80

2.4.3 Approximate Load-Deflection Behavior

Just After Buckling 85

2.4.4 Other Large-Deflection Buckling Problems 90

2.4.5 Practical Significance of the Euler Buckling Load 91

2.5 BUCKLING OF BARS BY ENERGY PRINCIPLES 92

2.5.1 Basic Energy Principles 92

2.5.2 Application of Basic Energy Principlesto Stability of a Spring-Supported Bar 95

2.5.3 Buckling Criteria 97

2.5.4 Derivation of Governing Differential Equation and

Boundary Conditions for Bar Buckling by Energy Principles 100

2.5.5 Rigorous Derivation of the Differential Equation and

Boundary Conditions for Bar Buckling by Energy Principles 106

2.5.5.1 Equilibrium Analysis 109

2.5.5.2 Buckling from a Straight Prebuckled Shape 111

2.5.5.3 Buckling from a Non-Straight Prebuckled Shape 115

Problem Set 2.5.5 117

2.5.6 Summary 118

Contents ix

2.6 APPROXIMATE BUCKLING LOADS BY ENERGY METHODS 119


2.6.2 The Rayleigh Method 119

2.6.2.1 Strain Energy Integrals and Buckling Criteria 120

2.6.2.2 Boundary Conditions 123

2.6.2.3 Buckling of an Axially Compressed Pinned-Pinned Bar 127

2.6.2.4 Buckling of an Axially Compressed Fixed-Free Bar 130

2.6.2.5 Summary 134


2.6.3 The Rayleigh-Ritz Method 135

2.6.4 Buckling of a Bar under Its Own Weight 138


2.6.5 Buckling of Bars with Step Changes in Cross Section 140


2.6.6 Buckling of an Axially Compressed Pinned-Pinned Bar

with a Mid-Span Transverse Spring 142


2.6.7 Summary 144

2.7 EFFECT OF TRANSVERSE SHEAR STIFFNESS 145


2.7.2 Differential Equation Approach to Effect of Transverse Shearon Deflection of an End-Loaded Cantilever Beam 145

2.7.3 Energy Methods Approach to Effect of Transverse Shearon Deflection of an End-Loaded Cantilever Beam 149

2.7.4 Differential Equation Approach to Effect of Transverse Shear

on Buckling of an Axially Compressed Fixed-Free Bar 152

2.7.5 Energy Methods Approach to Effect of Transverse Shearon Buckling of an Axially Compressed Pinned-Pinned Bar 154

2.7.6 Importance of Transverse Shear Stiffness on Buckling of Bars 155

Problem Set 2.7 155

2.8 EFFECT OF INITIAL GEOMETRIC IMPERFECTIONS 156

Problem Set 2.8 160

2.9 EFFECT OF RESTRAINED THERMAL EXPANSION 161


2.9.2 Origins and Conditions of Thermal Stress and Thermal Strain 161

2.9.3 Fundamental Thermoelastic Relations 165

2.9.4 Bar Support Conditions 171

2.9.5 Thermal Buckling of an Axially Restrained Pinned-Pinned Bar

subjected to a Uniform Temperature Change 174

2.9.6 Thermal Behavior of an Axially Restrained Bar subjected to

a Linear Through-the-Thickness Temperature Change 178

2.9.7 Summary 184

Problem Set 2.9 186

2.10 PLASTIC BUCKLING OF BARS 187


2.10.2 Reduced-Modulus Theory 189

2.10.3 Tangent-Modulus Theory 191

2.10.4 Transcendental Plastic Buckling Equation 191

2.10.5 Solution Strategy 193

2.10.6 Representation of the Nonlinear Stress-Strain Curve 196

2.10.7 Numerical Results 198

X Contents

2.10.8 Comparison of Reduced-Modulus Theory with

Tangent-Modulus Theory and Measured Buckling Loads 199

2.10.9 Summary 200

Problem Set 2.10 201

2.11 ASPECTS OF DESIGN OF BARS AGAINST BUCKLING 202


2.11.2 Analysis versus Design 202

2.11.3 End and Lateral Support Conditions 207

2.11.4 Material Properties 211

2.11.4.1 Effect of Creep 211

2.11.4.2 Earthquakes 212

2.11.4.3 Environmental Effects for

Fiber-Reinforced Composite Materials 213

2.11.5 Bar Cross-Sectional Shape 214

2.11.6 Design Codes and Their Development 215

2.11.7 Summary Remarks on Design 221

2.12 BUCKLING OF BARS SUMMARY 221

CHAPTER 2 REFERENCES,

223

3 BUCKLING OF RECTANGULAR PLATES 227

3.1 INTRODUCTION 227

3.2 CONSISTENT DERVIVATION OF EQUILIBRIUM EQUATIONS,BUCKLING EQUATIONS, AND BOUNDARY CONDITIONS 230

3.2.1 Variation of a Double Integral with Three Dependent Variables 230

3.2.2 First Variation of a Double Integral 233

3.2.3 Equilibrium Differential Equations and Boundary Conditions

for von Karman Plate Theory 240

3.2.4 First Variation of the Second Variation of a Double Integral 249

3.2.5 Buckling Differential Equations and Energy Expressionsfor a Rectangular Plate with In-Plane Loads 255

3.2.6 Summary 262

3.3 CLASSICAL BUCKLING THEORY SOLUTIONS 264


3.3.2 Buckling of Simply SupportedJ-tectangular Plates

under Uniform Compression Nx 264

3.3.2.1 The Buckling Differential Equation Approach 264

3.3.2.2 The Strain Energy Integral Approach 266

3.3.2.3 Results and Discussion 267


3.3.3 Buckling of Simply Supported Rectangular Plates

under Uniform Biaxial Loading Nx and Ny 269

3.3.3.1 Basic Solution 269

3.3.3.2 Manner of Biaxial Load Introduction 270

3.3.3.3 General Nature qfjhe Buckling Results 271

3.3.3.4 Results for Both Nx and Ny Compressive_ 2733.3.3.5 Results for Tensile Ny ancf Compressive Nx 274

3.3.3.6 Behavior under Large Tensile Loads 274

3.3.3.7 Nonproportional Loading 275

3.3.3.8 Effect of Plate Aspect Ratio 277

3.3.3.9 Summary Remarks on Buckling under Biaxial Loading 279

Contents xi



under Combined In-Plane Bending and Compression 279



under In-Plane Shear Nxv 285


3.3.6 Buckling of Uniformly Compressed Rectangular Plates

Simply Supported along Two Opposite Loaded Edges with

Various Boundary Conditions along Two Unloaded Edges 292

3.3.6.1 One Edge Simply Supportedand the Other Edge Free 294

3.3.6.2 One Edge Clamped and the Other Edge Free 297


3.3.7 Physical Restraints against Buckling 298

3.3.7.1 Examples for a Uniaxially Loaded Plate 299

3.3.7.2 Examples for a Biaxially Loaded Plate 300

3.3.7.3 Summary Remarks for Physical Restraint 300


3.3.8 Summary of Classical Buckling Results 301

3.4 POSTBUCKLING BEHAVIOR OF

SIMPLY SUPPORTED RECTANGULAR PLATES

UNDER UNIFORM COMPRESSION Nx 302



3.5.2 Analysis and Experimental Results 308

3.5.3 Consequences of Nonlinear Load-Deflection Behavior 315

3.5.4 Summary 319

3.6 EFFECT OF MULTIPLE FIBER-REINFORCED LAYERS 320


3.6.2 Governing Equations for Buckling of Laminated Plates 322

3.6.2.1 Basic Restrictions, Assumptions, and Consequences 322

3.6.2.2 Equilibrium Equations for Laminated Plates 325

3.6.2.3 Buckling Equations for Laminated Plates 329

3.6.2.4 Solution Techniques 332

3.6.3 Buckling of Simply Supported Laminated Plates

under In-Plane Loading 333

3.6.3.1 Specially Orthotropic Laminated Plates 335

3.6.3.2 Symmetric Angle-Ply Laminated Plates 337

3.6.3.3 Antisymmetric Cross-Ply Laminated Plates 339

3.6.3.4 Antisymmetric Angle-Ply Laminated Plates 342

3.6.4 General Remarks on Effects of Stiffnesses 346

3.6.5 Tow-Placed, Variable-Stiffness Plates 349

3.6.6 Summary 350

Problem Set 3.6 351




3.7.3 In-Plane Loading and Restraint Conditions 365

3.7.3.1 What In-Plane Boundary Conditions can Result in

Thermal and/or Mechanical Buckling? 365

3.7.3.1.1 Load versus Displacement in the Laboratory 366

3.7.3.1.2 Loading and Restraint Conditions

in Engineering Practice 367

Contents

3.7.3.1.3 Elastic In-Plane Edge Restraint 368

3.7.3.1.4 Thermal Buckling without

In-Plane Edge Restraint 369

3.7.3.2 In-Plane Displacement Notation 370

3.7.3.3 In-Plane Boundary Condition Notation 370

3.7.3.4 Compatibility of Boundary Conditions

at Corners and on Adjacent Edges 372

3.7.3.5 Boundary Support and Thermal Loading Conditions

Leading to Non-Classical Problems 376

3.7.3.6 Summary 376

3.7.4 Equivalent Mechanical Load Concept 377

3.7.5 Buckling under a Uniform Temperature Changeof a Uniaxially In-Plane Restrained Simply Supported Plate 379

3.7.6 Buckling under a Uniform Temperature Changeof a Biaxially in-Plane Restrained Simply Supported Plate 384

3.7.7 Buckling of a Partially In-Plane Restrained Plate

under Uniform Temperature Change and Mechanical Load 387

3.7.7.1 Introduction 387

3.7.7.2 Manner of Load Introduction 387

3.7.7.3 Buckling Differential Equation and Solution 389

3.7.7.4 Results and Discussion 391

3.7.7.4.1 Behavior for Heating and Tensile Nx 392

3.7.7.4.2 Behavior for Heating and Compressive Nx 392

3.7.7.4.3 Behavior for Cooling and Compressive Nx 393

3.7.7.4.4 Nonproportional Loading 394

3.7.7.4.5 Effect of Poisson's Ratio 394

3.7.7.4.6 Effect of the Plate Aspect Ratio 395

3.7.7.4.7 Material-Property-Related Limitations 395

3.7.7.5 Summary of Thermal and Mechanical Buckling 395

3.7.8 Summary and Observations on Plate Thermal Buckling 396

3.7.9 Other Topics in Thermal Buckling of Plates 400

3.7.9.1 Various Edge Support Conditions 400

3.7.9.2 Nonuniform Heating 401

3.7.9.3 Time-Dependent Temperature Distributions 403

3.7.9.4 Limitations of the Linear Elastic Material Model 404

3.7.9.5 Temperature-Dependent Material Properties 404

3.7.9.6 Nonlinear Stress-Stress Behavior 405

3.7.9.7 Analogy between Temperature and Moisture Effects 406

3.7.9.8 Concluding Remarks 407

Problem Set 3.7 407

3.8 EFFECT OF PLASTIC DEFORMATION 408


3.8.2 Derivation of Buckling Criterion 409

3.8.2.1 Fundamental Relations in Jp Deformation Theory 409

3.8.2.2 Variations of Strains during Buckling 411

3.8.2.3 Variations of Stresses during Buckling 411

3.8.2.4 Variations of Forces and Moments during Buckling 413

3.8.2.5 Buckling Criterion 414

3.8.3 Solution Strategy 415

3.8.4 Numerical Results 417

3.8.5 Comparison of Theoretical and Experimental Results 419

3.8.6 Summary 421

Problem Set 3.8 421

Contents xiii

3.9 ASPECTS OF DESIGN OF PLATES AGAINST BUCKLING 422


3.9.2 Steps in the Structural Design Process 422

3.9.3 Design of an Isotropic Metal Plate 425

3.9.3.1 Design of an Isotropic Metal Plate

Against Failure Caused by High Stresses 425

3.9.3.2 Design of an Isotropic Metal Plate

Against Failure by Buckling 426

3.9.3.3 Design of a Stiffened Isotropic Metal Plate


3.9.4 Design of a Laminated Composite Plate 431

3.9.4.1 Design of a Laminated Composite Plate

Against Failure Caused by High Stresses 431

3.9.4.2 Design of a Laminated Composite Plate


3.9.4.3 Design of a Stiffened Laminated Composite Plate


3.9.4.4 Genetic Algorithm Approach to Laminate Design 442

3.9.4.5 Summary of Design of a Laminated Composite Plate 447

3.9.5 Summary of Design of Plates Against Buckling 447

3.10 BUCKLING OF PLATES SUMMARY 448

3.10.1 General Problems 448

3.10.2 Review of Experimental Work 449

3.10.2.1 Consequences of Stable Postbuckling Behavior 449

3.10.2.2 Boundary Conditions in Experiments 450

3.10.3 Extensions of Present Coverage 451

3.10.3.1 Buckling of Plates of Nonlinear Composite Materials 451

3.10.3.2 Thermal Buckling of Cross-Ply Composite Plates 453


4 BUCKLING OF CIRCULAR CYLINDRICAL SHELLS 463


4.2 DERIVATION OF EQUILIBRIUM AND BUCKLING EQUATIONS

PLUS ASSOCIATED BOUNDARY CONDITIONS 467

4.2.1 Equilibrium Differential Equations for

Moderately Large Deflections 467

4.2.2 Buckling Differential Equations and Energy Expressionsfor Combinations of Lateral Pressure, Axial Compression,and In-Surface Shear 478

Problem Set 4.2 489

4.3 CLASSICAL BUCKLING THEORY SOLUTIONS 490


4.3.2 Batdorf's Classical Results 491

4.3.2.1 Axial Compression 491

4.3.2.2 Lateral Pressure 498

4.3.2.3 Hydrostatic Pressure 501

4.3.2.4 Comparison of Results for Axial Compression,Lateral Pressure, and Hydrostatic Pressure 502

4.3.2.5 Torsion 504

4.3.3 Use of Batdorf's Modified Donnell Equation for

Axially Compressed Circular Cylindrical Shells

xiv Contents

with Clamped Edge Buckling Boundary Condition C3 508

4.3.4 Use of All Three Buckling Differential Equations for

Axially Compressed Circular Cylindrical Shells

with Clamped Edge Buckling Boundary Condition C1 512

4.3.5 Summary for Classical Buckling Results 517

Problem Set 4.3 518

4.4 POSTBUCKLING BEHAVIOR 519

Problem Set 4.4 528

4.5 EFFECT OF PREBUCKLING DEFORMATIONS ONBUCKLING UNDER AXIAL COMPRESSION 529

Problem Set 4.5 533


4.7 EFFECT OF ECCENTRIC STIFFENERS 540



4.7.2.1 Variations of Stresses and Strains during Buckling 543


4.7.2.3 Buckling Differential Equations 545


4.7.3 Numerical and Experimental Results 547

4.7.4 The Stiffener Eccentricity Effect 550

4.7.5 Summary 554

Problem Set 4.7 554

4.8 EFFECT OF MULTIPLE FIBER-REINFORCED LAYERS 555



4.8.2.1 Orthotropic Stress-Strain Relations 557

4.8.2.2 Variations of Stresses and Strains during Buckling 558

4.8.2.3 Variations of Forces and Moments during Buckling 5594.8.2.4 Buckling Differential Equations 560


4.8.3 Numerical Example 562

4.8.4 General k-Z Results for Antisymmetric Cross-Ply Laminates 564



4.8.4.3 Summary for Antisymmetric Cross-Ply Laminates 569

4.8.5 Results for Unsymmetric Cross-Ply Laminates 570



4.8.5.3 Summary for Unsymmetrically Laminated Shells 5744.8.6 Summary for Laminated Shells 575

Problem Set 4.8 576




4.9.3 In-Surface Loading and Restraint Conditions 5914.9.3.1 What In-Surface Boundary Conditions can Result in

Thermal and/or Mechanical Buckling of a Shell? 5914.9.3.2 Load Versus Displacement Application

in the Laboratory 592

4.9.3.3 Loading and Restraint Conditions

Contents xv

in Engineering Practice 593

4.9.3.4 Elastic In-Surface Edge Restraint 594

4.9.3.5 Thermal Buckling Without In-Surface Edge Restraint 595

4.9.4 Equivalent Mechanical Load Concept 596

4.9.5 Buckling of an Axially Restrained Circular Cylindrical Shell

under a Uniform Temperature Change 598

4.9.5.1 Prebuckling Equilibrium Solution 599

4.9.5.2 Buckling Solution 601

4.9.5.3 Thermal Buckling Results 603

4.9.5.4 Correlation with

Measured Buckling Temperature Change 605

4.9.6 Buckling of an Axially Restrained Circular Cylindrical Shell under

a Uniform Temperature Change and Lateral Pressure 607

4.9.6.1 Prebuckling Equilibrium Solution 607

4.9.6.2 Buckling Solution 609

4.9.6.3 Thermal Buckling Results 613

4.9.7 Summary and Other Topics in Thermal Buckling of Shells 616

Problem Set 4.9 618

4.10 EFFECT OF PLASTIC DEFORMATION 619


4.10.2 Derivation of Buckling Criterion for Single-Layered Shells 620

4.10.2.1 Fundamental Relations in J2 Deformation Theory 620





4.10.3 Solution of Buckling Criterion for Single-Layered Shells 626

4.10.4 Numerical Results for Single-Layered Shells 628

4.10.4.1 Single-Layered Circular Cylindrical Shell

under Hydrostatic Pressure 628

4.10.4.2 Stiffened Large-Diameter Booster-Interstage Shell

under Biaxial Loading 630

4.10.5 Derivation of Buckling Criterion for Multilayered Shells 633


4.10.5.2 Fundamental Relations in J2 Deformation Theory 634





4.10.6 Solution of Buckling Criterion for Multilayered Shells 641


4.10.6.2 Determination of the Yield Load 642

4.10.6.3 Determination of Layer and Stiffener Stresses,Strains, and Material Propertiesat an Estimated Buckling Load 644

4.10.6.4 Calculation of Absolute Minimum Buckling Load

at an Estimated Buckling Load 648

4.10.6.5 Comparison of Estimated Buckling Load

and Absolute Minimum Calculated Buckling Load 648

4.10.7 Numerical Results for Multilayered Shells 649

4.10.8 Summary 652

Problem Set 4.10 653

xvi Contents

4.11 ASPECTS OF DESIGN OF SHELLS AGAINST BUCKLING 654


4.11.2 The Importance of Experiments in the Developmentof Shell Buckling Analysis and Design 656

4.11.3 Knockdown Approach to Buckling-Critical Shell Design 657

4.11.4 Review of the NASA Shell Design Monographs 660

4.11.4.1 Enhanced Shell Analysis Capabilities 660

4.11.4.2 Sophisticated Experimental Measurements 661

4.11.4.3 Integration of Sophisticated Analyis and

Sophisticated Experiments in the Design Process 662

4.11.4.4 Summary and What's to Come 662

4.11.5 High-Fidelity Analysis and Design of Buckling-Critical Shells 663

4.11.5.1 High-Fidelity Measurements and

Experimental Procedures 663

4.11.5.2 Development of High-Fidelity Analyses 666

4.11.5.3 High-Fidelity Design Procedure 674

4.11.6 Design of Buckling-Critical Stiffened Isotropic Metal Shells 683

4.11.7 Design of Buckling-Critical Laminated Composite Shells 684

4.11.8 Summary Remarks on Shell Design Against Buckling 684

4.12 BUCKLING OF SHELLS SUMMARY 685


5 SUMMARY AND OTHER TOPICS 699


5.2 DESIGN-RELATED PERSPECTIVE ON CONTRAST

BETWEEN BAR, PLATE, AND SHELL BUCKLING BEHAVIOR 700

5.2.1 General Response 700

5.2.2 Design Considerations and Factor of Safety 702

5.2.3 Comparison with Experimental Results 704

5.3 OTHER CLASSICAL BUCKLING TOPICS 705

5.3.1 History of Buckling 705

5.3.2 Comments on Theoretical Derivations in this Book 705

5.3.3 Other Structural Elements and Structures 705

5.3.4 Solution Procedures for Buckling Problems 706

5.3.5 Other Structural Configurations 706

5.3.5.1 Sandwich Construction 706

5.3.5.2 Laminated Fiber-Reinforced Composite Structures 708

5.4 ADVANCED BUCKLING TOPICS 710


5.4.2 Ziegler's Classification of Stability Problems 710

5.4.3 Nonconservative Systems 713

5.4.4 Stochastic Excitation 714

5.4.5 Dynamic Buckling 714

5.4.6 Chaos 714

5.5 ON THE TRANSITION TO COMPUTATIONAL APPROACHES 715

5.5.1 How to Learn Complex Buckling Behavior 715

5.5.2 Simple Analytical Approaches versus Numerical Approaches 717

5.5.3 Computational Tools Available 718

5.5.4 Replacement of Measured Behavior by Computer Simulation 720

5.5.5 Designing Structures 721

Contents xvii

5.6 THE INCORRECT BENCHMARK PANEL BUCKLING SOLUTION

AND COMPUTER PROBLEM-SOLVING LESSONS LEARNED 723


5.6.2 General Characteristics of the Panel Buckling Problem 724

5.6.3 Characteristics of the Incorrect Benchmark Solution 725

5.6.4 Characteristics of the Correct Panel Buckling Solution 728

5.6.5 Common Difficulties in Engineering Problem Solving 733

5.6.6 Summary 734

5.7 SUMMARY REMARKS 735


APPENDIX A: MAXIMA AND MINIMA OF

FUNCTIONS OF A SINGLE VARIABLE 738

APPENDIX A REFERENCE 742

APPENDIX B: ABSOLUTE MINIMUM OF

A FUNCTION OF TWO VARIABLES 743

APPENDIX B REFERENCES 748

APPENDIX C: BEHAVIOR OF FIBER-REINFORCED

LAMINATED COMPOSITE MATERIALS

AND STRUCTURAL ELEMENTS 749

C.1 INTRODUCTION 749

C.2 FIBER-REINFORCED LAMINATED COMPOSITE MATERIALS 750

C.2.1 Introduction 750

C.2.2 Laminae 751

C.2.3 Laminates 752

C.2.4 Advantages 752

C.3 STRESS-STRAIN RELATIONS FOR

ANISOTROPIC AND ORTHOTROPIC MATERIALS 756


PLANE STRESS IN AN ORTHOTROPIC MATERIAL 761


A LAMINA OF ARBITRARY ORIENTATION 763

C.6 CLASSICAL LAMINATION THEORY 767

C.6.1 Lamina Stress-Strain Behavior 768

C.6.2 Stress and Strain Variation in a Laminate 768

C.6.3 Resultant Laminate Forces and Moments 772

C.6.4 Summary 776

C.7 SPECIAL CASES OF LAMINATE STIFFNESSES 779

C.7.1 Single-Layered Configurations 780

C.7.2 Symmetric Laminates 782

C.7.3 Antisymmetric Laminates 789

C.7.4 Unsymmetric Laminates 793

C.7.5 Stacking-Sequence Notation 794

C.7.6 Balanced Laminates 795

xviii Contents

C.7.7 Hybrid Laminates 795

C.8 SUMMARY REMARKS 795

APPENDIX C REFERENCES 796

APPENDIX D: ELEMENTS OF DESIGN PHILOSOPHY 797

D.1 INTRODUCTION 797

D.2 WHAT IS ENGINEERING DESIGN? 798

D.3 PHILOSOPHIES OVER THE AGES

OF HOW TO ENSURE SAFE DESIGNS 801

D.4 THE ROLE OF PAST FAILURES IN

UNDERSTANDING AND IMPROVING DESIGN PRACTICE 802

D.4.1 de Havilland Comet Fatigue Failures 803

D.4.2 Hyatt Regency Hotel Skywalk Failure 804

D.5 FAILURE MODES IN DESIGN 808

D.6 SUMMARY REMARKS 811

APPENDIX D REFERENCES 812

INDEX 813

Documents

Buckling of bars, plates, and shells - · PDF fileContents xi Problem Set 3.3.3 279 3.3.4 Buckling of SimplySupported Rectangular Plates under Combined In-PlaneBendingand Compression