BUCKLING OF BARS, PLATES, AND SHELLS. Virginia Polytechnic Institute and State University Biacksburg, Virginia

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1 BUCKLING OF BARS, PLATES, AND SHELLS ROBERT M. JONES Science and Mechanics Professor Emeritus of Engineering Virginia Polytechnic Institute and State University Biacksburg, Virginia Bull Ridge Publishing Biacksburg, Virginia United States of America copyright 2006 by Bull Ridge Publishing all rights reserved

2 CONTENTS PREFACE xix 1 INTRODUCTION TO BUCKLING FUNDAMENTAL DEFINITIONS OF MECHANICAL SYSTEMS BEHAVIOR Stability Instability Buckling Load Maximum Load General Load-Deformation Behavior including Buckling BOOK CONCENTRATION AND PHILOSOPHY Introduction Buckling of Bars Buckling of Plates Buckling of Shells Summary CLASSIFICATION OF TYPES OF BUCKLING AND POSTBUCKLING BEHAVIOR Introduction General Nature of Linear Eigenvalue Analysis Types of Postbuckling Behavior Initial Geometric Imperfections Effect on Postbuckling Behavior Degree of Stability in Stable Postbuckling Behavior Summary IMPORTANCE OF EXPERIMENTAL RESULTS IN DEVELOPMENT OF THEORIES Introduction Comparison of Theoretical and Experimental Results Experimental Results are Used More than 'Just' to Validate Theory Experimental Results are the Very Basis of all Theoretical Models Summary SOME IMPORTANT DISTINCTIONS IN ENGINEERING ANALYSIS Introduction Presumption versus Assumption Presumption versus Restriction Misinterpretations and their Consequences Summary BOOK ROAD MAP 46 CHAPTER 1 REFERENCES 47 vil

3 viii Contents 2 BUCKLING OF BARS INTRODUCTION EULER'S COLUMN EQUATION Buckling of an Axially Compressed Pinned-Pinned Bar Buckling of an Axially Compressed Fixed-Pinned Bar Buckling of an Axially Compressed Fixed-Fixed Bar Buckling of an Axially Compressed Fixed-Free Bar Buckling of an Axially Compressed Pinned-Free Bar with Elastically Restrained Ends Buckling of an Axially Compressed Pinned-Free Bar with an Elastic Translational Spring at the Free End Buckling of an Axially Compressed Pinned-Pinned Bar with a Mid-Span Transverse Spring Summary 66 Problem Set AN ALTERNATIVE BAR BUCKLING DIFFERENTIAL EQUATION Derivation of an Alternative Differential Equation Buckling of an Axially Compressed Pinned-Pinned Bar Buckling of an Axially Compressed Fixed-Pinned Bar Buckling of an Axially Compressed Fixed-Fixed Bar Buckling of an Axially Compressed Fixed-Free Bar Buckling of an Axially Compressed Fixed-Guided Bar Buckling of an Axially Compressed Pinned-Guided Bar The Effective Length Concept to Account for Boundary Conditions Summary 76 Problem Set LARGE DEFLECTIONS OF BUCKLED BARS THE PROBLEM OF THE ELAST1CA Introduction Exact Load-Deflection Behavior Approximate Load-Deflection Behavior Just After Buckling Other Large-Deflection Buckling Problems Practical Significance of the Euler Buckling Load BUCKLING OF BARS BY ENERGY PRINCIPLES Basic Energy Principles Application of Basic Energy Principles to Stability of a Spring-Supported Bar Buckling Criteria Derivation of Governing Differential Equation and Boundary Conditions for Bar Buckling by Energy Principles Rigorous Derivation of the Differential Equation and Boundary Conditions for Bar Buckling by Energy Principles Equilibrium Analysis Buckling from a Straight Prebuckled Shape Buckling from a Non-Straight Prebuckled Shape 115 Problem Set Summary 118

4 Contents ix 2.6 APPROXIMATE BUCKLING LOADS BY ENERGY METHODS Introduction The Rayleigh Method Strain Energy Integrals and Buckling Criteria Boundary Conditions Buckling of an Axially Compressed Pinned-Pinned Bar Buckling of an Axially Compressed Fixed-Free Bar Summary 134 Problem Set The Rayleigh-Ritz Method Buckling of a Bar under Its Own Weight 138 Problem Set Buckling of Bars with Step Changes in Cross Section 140 Problem Set Buckling of an Axially Compressed Pinned-Pinned Bar with a Mid-Span Transverse Spring 142 Problem Set Summary EFFECT OF TRANSVERSE SHEAR STIFFNESS Introduction Differential Equation Approach to Effect of Transverse Shear on Deflection of an End-Loaded Cantilever Beam Energy Methods Approach to Effect of Transverse Shear on Deflection of an End-Loaded Cantilever Beam Differential Equation Approach to Effect of Transverse Shear on Buckling of an Axially Compressed Fixed-Free Bar Energy Methods Approach to Effect of Transverse Shear on Buckling of an Axially Compressed Pinned-Pinned Bar Importance of Transverse Shear Stiffness on Buckling of Bars 155 Problem Set EFFECT OF INITIAL GEOMETRIC IMPERFECTIONS 156 Problem Set EFFECT OF RESTRAINED THERMAL EXPANSION Introduction Origins and Conditions of Thermal Stress and Thermal Strain Fundamental Thermoelastic Relations Bar Support Conditions Thermal Buckling of an Axially Restrained Pinned-Pinned Bar subjected to a Uniform Temperature Change Thermal Behavior of an Axially Restrained Bar subjected to a Linear Through-the-Thickness Temperature Change Summary 184 Problem Set PLASTIC BUCKLING OF BARS Introduction Reduced-Modulus Theory Tangent-Modulus Theory Transcendental Plastic Buckling Equation Solution Strategy Representation of the Nonlinear Stress-Strain Curve Numerical Results 198

5 X Contents Comparison of Reduced-Modulus Theory with Tangent-Modulus Theory and Measured Buckling Loads Summary 200 Problem Set ASPECTS OF DESIGN OF BARS AGAINST BUCKLING Introduction Analysis versus Design End and Lateral Support Conditions Material Properties Effect of Creep Earthquakes Environmental Effects for Fiber-Reinforced Composite Materials Bar Cross-Sectional Shape Design Codes and Their Development Summary Remarks on Design BUCKLING OF BARS SUMMARY 221 CHAPTER 2 REFERENCES, BUCKLING OF RECTANGULAR PLATES INTRODUCTION CONSISTENT DERVIVATION OF EQUILIBRIUM EQUATIONS, BUCKLING EQUATIONS, AND BOUNDARY CONDITIONS Variation of a Double Integral with Three Dependent Variables First Variation of a Double Integral Equilibrium Differential Equations and Boundary Conditions for von Karman Plate Theory First Variation of the Second Variation of a Double Integral Buckling Differential Equations and Energy Expressions for a Rectangular Plate with In-Plane Loads Summary CLASSICAL BUCKLING THEORY SOLUTIONS Introduction Buckling of Simply SupportedJ-tectangular Plates under Uniform Compression Nx The Buckling Differential Equation Approach The Strain Energy Integral Approach Results and Discussion 267 Problem Set Buckling of Simply Supported Rectangular Plates under Uniform Biaxial Loading Nx and Ny Basic Solution Manner of Biaxial Load Introduction General Nature qfjhe Buckling Results Results for Both Nx and Ny Compressive_ Results for Tensile Ny ancf Compressive Nx Behavior under Large Tensile Loads Nonproportional Loading Effect of Plate Aspect Ratio Summary Remarks on Buckling under Biaxial Loading 279

6 Contents xi Problem Set Buckling of Simply Supported Rectangular Plates under Combined In-Plane Bending and Compression 279 Problem Set Buckling of Simply Supported Rectangular Plates under In-Plane Shear Nxv 285 Problem Set Buckling of Uniformly Compressed Rectangular Plates Simply Supported along Two Opposite Loaded Edges with Various Boundary Conditions along Two Unloaded Edges One Edge Simply Supported and the Other Edge Free One Edge Clamped and the Other Edge Free 297 Problem Set Physical Restraints against Buckling Examples for a Uniaxially Loaded Plate Examples for a Biaxially Loaded Plate Summary Remarks for Physical Restraint 300 Problem Set Summary of Classical Buckling Results POSTBUCKLING BEHAVIOR OF SIMPLY SUPPORTED RECTANGULAR PLATES UNDER UNIFORM COMPRESSION Nx EFFECT OF INITIAL GEOMETRIC IMPERFECTIONS Introduction Analysis and Experimental Results Consequences of Nonlinear Load-Deflection Behavior Summary EFFECT OF MULTIPLE FIBER-REINFORCED LAYERS Introduction Governing Equations for Buckling of Laminated Plates Basic Restrictions, Assumptions, and Consequences Equilibrium Equations for Laminated Plates Buckling Equations for Laminated Plates Solution Techniques Buckling of Simply Supported Laminated Plates under In-Plane Loading Specially Orthotropic Laminated Plates Symmetric Angle-Ply Laminated Plates Antisymmetric Cross-Ply Laminated Plates Antisymmetric Angle-Ply Laminated Plates General Remarks on Effects of Stiffnesses Tow-Placed, Variable-Stiffness Plates Summary 350 Problem Set EFFECT OF RESTRAINED THERMAL EXPANSION Introduction Fundamental Thermoelastic Relations In-Plane Loading and Restraint Conditions What In-Plane Boundary Conditions can Result in Thermal and/or Mechanical Buckling? Load versus Displacement in the Laboratory Loading and Restraint Conditions in Engineering Practice 367

7 Contents Elastic In-Plane Edge Restraint Thermal Buckling without In-Plane Edge Restraint In-Plane Displacement Notation In-Plane Boundary Condition Notation Compatibility of Boundary Conditions at Corners and on Adjacent Edges Boundary Support and Thermal Loading Conditions Leading to Non-Classical Problems Summary Equivalent Mechanical Load Concept Buckling under a Uniform Temperature Change of a Uniaxially In-Plane Restrained Simply Supported Plate Buckling under a Uniform Temperature Change of a Biaxially in-plane Restrained Simply Supported Plate Buckling of a Partially In-Plane Restrained Plate under Uniform Temperature Change and Mechanical Load Introduction Manner of Load Introduction Buckling Differential Equation and Solution Results and Discussion Behavior for Heating and Tensile Nx Behavior for Heating and Compressive Nx Behavior for Cooling and Compressive Nx Nonproportional Loading Effect of Poisson's Ratio Effect of the Plate Aspect Ratio Material-Property-Related Limitations Summary of Thermal and Mechanical Buckling Summary and Observations on Plate Thermal Buckling Other Topics in Thermal Buckling of Plates Various Edge Support Conditions Nonuniform Heating Time-Dependent Temperature Distributions Limitations of the Linear Elastic Material Model Temperature-Dependent Material Properties Nonlinear Stress-Stress Behavior Analogy between Temperature and Moisture Effects Concluding Remarks 407 Problem Set EFFECT OF PLASTIC DEFORMATION Introduction Derivation of Buckling Criterion Fundamental Relations in Jp Deformation Theory Variations of Strains during Buckling Variations of Stresses during Buckling Variations of Forces and Moments during Buckling Buckling Criterion Solution Strategy Numerical Results Comparison of Theoretical and Experimental Results Summary 421 Problem Set

8 Contents xiii 3.9 ASPECTS OF DESIGN OF PLATES AGAINST BUCKLING Introduction Steps in the Structural Design Process Design of an Isotropic Metal Plate Design of an Isotropic Metal Plate Against Failure Caused by High Stresses Design of an Isotropic Metal Plate Against Failure by Buckling Design of a Stiffened Isotropic Metal Plate Against Failure by Buckling Design of a Laminated Composite Plate Design of a Laminated Composite Plate Stresses 431 Against Failure Caused by High Design of a Laminated Composite Plate Against Failure by Buckling Design of a Stiffened Laminated Composite Plate Against Failure by Buckling Genetic Algorithm Approach to Laminate Design Summary of Design of a Laminated Composite Plate Summary of Design of Plates Against Buckling BUCKLING OF PLATES SUMMARY General Problems Review of Experimental Work Consequences of Stable Postbuckling Behavior Boundary Conditions in Experiments Extensions of Present Coverage Buckling of Plates of Nonlinear Composite Materials Thermal Buckling of Cross-Ply Composite Plates 453 CHAPTER 3 REFERENCES BUCKLING OF CIRCULAR CYLINDRICAL SHELLS INTRODUCTION DERIVATION OF EQUILIBRIUM AND BUCKLING EQUATIONS PLUS ASSOCIATED BOUNDARY CONDITIONS Equilibrium Differential Equations for Moderately Large Deflections Buckling Differential Equations and Energy Expressions for Combinations of Lateral Pressure, Axial Compression, and In-Surface Shear 478 Problem Set CLASSICAL BUCKLING THEORY SOLUTIONS Introduction Batdorf's Classical Results Axial Compression Lateral Pressure Hydrostatic Pressure Comparison of Results for Axial Compression, Lateral Pressure, and Hydrostatic Pressure Torsion Use of Batdorf's Modified Donnell Equation for Axially Compressed Circular Cylindrical Shells

9 xiv Contents with Clamped Edge Buckling Boundary Condition C Use of All Three Buckling Differential Equations for Axially Compressed Circular Cylindrical Shells with Clamped Edge Buckling Boundary Condition C Summary for Classical Buckling Results 517 Problem Set POSTBUCKLING BEHAVIOR 519 Problem Set EFFECT OF PREBUCKLING DEFORMATIONS ON BUCKLING UNDER AXIAL COMPRESSION 529 Problem Set EFFECT OF INITIAL GEOMETRIC IMPERFECTIONS EFFECT OF ECCENTRIC STIFFENERS Introduction Derivation of Buckling Criterion Variations of Stresses and Strains during Buckling Variations of Forces and Moments during Buckling Buckling Differential Equations Buckling Criterion Numerical and Experimental Results The Stiffener Eccentricity Effect Summary 554 Problem Set EFFECT OF MULTIPLE FIBER-REINFORCED LAYERS Introduction Derivation of Buckling Criterion Orthotropic Stress-Strain Relations Variations of Stresses and Strains during Buckling Variations of Forces and Moments during Buckling Buckling Differential Equations Buckling Criterion Numerical Example General k-z Results for Antisymmetric Cross-Ply Laminates Axial Compression Lateral Pressure Summary for Antisymmetric Cross-Ply Laminates Results for Unsymmetric Cross-Ply Laminates Axial Compression Lateral Pressure Summary for Unsymmetrically Laminated Shells Summary for Laminated Shells 575 Problem Set EFFECT OF RESTRAINED THERMAL EXPANSION Introduction Fundamental Thermoelastic Relations In-Surface Loading and Restraint Conditions What In-Surface Boundary Conditions can Result in Thermal and/or Mechanical Buckling of a Shell? Load Versus Displacement Application in the Laboratory Loading and Restraint Conditions

10 Contents xv in Engineering Practice Elastic In-Surface Edge Restraint Thermal Buckling Without In-Surface Edge Restraint Equivalent Mechanical Load Concept Buckling of an Axially Restrained Circular Cylindrical Shell under a Uniform Temperature Change Prebuckling Equilibrium Solution Buckling Solution Thermal Buckling Results Correlation with Measured Buckling Temperature Change Buckling of an Axially Restrained Circular Cylindrical Shell under a Uniform Temperature Change and Lateral Pressure Prebuckling Equilibrium Solution Buckling Solution Thermal Buckling Results Summary and Other Topics in Thermal Buckling of Shells 616 Problem Set EFFECT OF PLASTIC DEFORMATION Introduction Derivation of Buckling Criterion for Single-Layered Shells Fundamental Relations in J2 Deformation Theory Variations of Strains during Buckling Variations of Stresses during Buckling Variations of Forces and Moments during Buckling Buckling Criterion Solution of Buckling Criterion for Single-Layered Shells Numerical Results for Single-Layered Shells Single-Layered Circular Cylindrical Shell under Hydrostatic Pressure Stiffened Large-Diameter Booster-Interstage Shell under Biaxial Loading Derivation of Buckling Criterion for Multilayered Shells Introduction Fundamental Relations in J2 Deformation Theory Variations of Strains during Buckling Variations of Stresses during Buckling Variations of Forces and Moments during Buckling Buckling Criterion Solution of Buckling Criterion for Multilayered Shells Introduction Determination of the Yield Load Determination of Layer and Stiffener Stresses, Strains, and Material Properties at an Estimated Buckling Load Calculation of Absolute Minimum Buckling Load at an Estimated Buckling Load Comparison of Estimated Buckling Load and Absolute Minimum Calculated Buckling Load Numerical Results for Multilayered Shells Summary 652 Problem Set

11 xvi Contents 4.11 ASPECTS OF DESIGN OF SHELLS AGAINST BUCKLING Introduction The Importance of Experiments in the Development of Shell Buckling Analysis and Design Knockdown Approach to Buckling-Critical Shell Design Review of the NASA Shell Design Monographs Enhanced Shell Analysis Capabilities Sophisticated Experimental Measurements Integration of Sophisticated Analyis and Sophisticated Experiments in the Design Process Summary and What's to Come High-Fidelity Analysis and Design of Buckling-Critical Shells High-Fidelity Measurements and Experimental Procedures Development of High-Fidelity Analyses High-Fidelity Design Procedure Design of Buckling-Critical Stiffened Isotropic Metal Shells Design of Buckling-Critical Laminated Composite Shells Summary Remarks on Shell Design Against Buckling BUCKLING OF SHELLS SUMMARY 685 CHAPTER 4 REFERENCES SUMMARY AND OTHER TOPICS INTRODUCTION DESIGN-RELATED PERSPECTIVE ON CONTRAST BETWEEN BAR, PLATE, AND SHELL BUCKLING BEHAVIOR General Response Design Considerations and Factor of Safety Comparison with Experimental Results OTHER CLASSICAL BUCKLING TOPICS History of Buckling Comments on Theoretical Derivations in this Book Other Structural Elements and Structures Solution Procedures for Buckling Problems Other Structural Configurations Sandwich Construction Laminated Fiber-Reinforced Composite Structures ADVANCED BUCKLING TOPICS Introduction Ziegler's Classification of Stability Problems Nonconservative Systems Stochastic Excitation Dynamic Buckling Chaos ON THE TRANSITION TO COMPUTATIONAL APPROACHES How to Learn Complex Buckling Behavior Simple Analytical Approaches versus Numerical Approaches Computational Tools Available Replacement of Measured Behavior by Computer Simulation Designing Structures 721

12 Contents xvii 5.6 THE INCORRECT BENCHMARK PANEL BUCKLING SOLUTION AND COMPUTER PROBLEM-SOLVING LESSONS LEARNED Introduction General Characteristics of the Panel Buckling Problem Characteristics of the Incorrect Benchmark Solution Characteristics of the Correct Panel Buckling Solution Common Difficulties in Engineering Problem Solving Summary SUMMARY REMARKS 735 CHAPTER 5 REFERENCES 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 C.4 STRESS-STRAIN RELATIONS FOR PLANE STRESS IN AN ORTHOTROPIC MATERIAL 761 C.5 STRESS-STRAIN RELATIONS FOR 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

13 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

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