Stresses in Beams, Plates, and Shells, Third Edition

Preface xiiiAcknowledgments xviiList of Symbols xixPart 1 *Fundamentals1 Basic Concepts 31.1 Introduction 31.2 Methods of Analysis 41.3 Conditions of Equilibrium 51.4 Stress Defined 71.4.1 Components of Stress 81.4.2 Sign Convention 91.5 Internal-Force Resultants 91.6 Differential Equations of Equilibrium 121.7 Transformation of Stress 141.7.1 Mohr's Circle for Stress 161.8 Strain Defined 181.9 Components of Strain 201.9.1 Conditions of Compatibility 211.10 Large Strains 221.11 Transformation of Strain 231.12 Engineering Materials 241.12.1 Stress-Strain Diagrams 251.13 Hooke's Law, Poisson's Ratio 261.14 Rational Design Procedure 301.15 Factor of Safety 311.16 Problem Formulation and Solutions 321.16.1 Significant Digits 331.16.2 Computational Tools 33References 34Problems 342 Stresses in Simple Structural Members 412.1 Introduction 412.2 Types of Structures 422.3 Axially Loaded Members 452.4 Stress Concentration Factors 482.5 Torsion of Circular Bars 492.5.1 Shear Stress 502.5.2 Angle of Twist 512.6 Stresses in Beams 522.6.1 Normal Stress 532.6.2 Shear Stress 542.6.3 Shear Flow 552.7 Deflection of Beams by Integration 562.8 Beam Deflections by Superposition 612.9 Thin-Walled Pressure Vessels 632.10 Yield and Fracture Criteria 652.10.1 Maximum Principal Stress Theory 652.10.2 Coulomb-Mohr Theory 662.10.3 Maximum Shear Stress Theory 672.10.4 Maximum Distortion Energy Theory 682.10.5 A Typical Case of Combined Loadings 682.11 Strain Energy 712.12 CastigUano's Theorem 732.12.1 Statically Indeterminate Structures 76References77Problems 77Part II Plates3 Elements of Plate-Bending Theory 873.1 Introduction 873.2 Historical Development of Plate and Shell Theory 883.3 General Behavior of Plates 893.4 Strain-Curvature Relations 913.4.1 Mohrs Circle of Curjvature 933.5 Stresses and Stress Resultants 943.6 Equations for Transformation of Moment 973.7 Variation of Stress within a Plate 983.8 The Governing Equation for Deflection of Plates 1013.8.1 Reduction of Plate-Bending Problem to That of Deflection of a Membrane 1023.9 Boundary Conditions 1033.10 Exact Theory of Plates 1063.11 Methods for Solution of Plate Deflections 1093.12 Strain Energy of Plates 1163.13 Energy Methods in Theory of Plates 1173.13.1 The Principle of Virtual Work 1173.13.2 The Principle of Minimum Potential Energy 1183.13.3 The Ritz Method 1193.14 *Natural Frequencies of Plates by the Energy Method 119References 121Problems 1224 Circular Plates 1274.1 Introduction 1274.2 Basic Relations in Polar Coordinates 1274.3 The Axisymmetrical Bending 1324.4 Equations of Equilibrium for AxisymmetricallyLoaded Circular Plates 1334.5 Uniformly Loaded Circular Plates 1354.6 *Effect of Shear on the Plate Deflection 1394.7 Local Stresses at the Point of Application of a Concentrated Load 1404.8 Circular Plates under a Concentrated Load at the Center 1414.8.1 A Short Catalog of Solutions 1444.9 Annular Plates with Simply Supported Outer Edges 1444.10 Deflection and Stress by Superposition 1524.10.1 Design Tables for Annular Plates 1524.11 The Ritz Method Applied to Bending of Circular Plates 1554.12 Asymmetrical Bending of Circular Plates 1614.13 *Deflection by the Reciprocity Theorem 163References 164Problems 1655 Rectangular Plates 1715.1 Introduction 1715.2 Navier's Solution for Simply Supported Rectangular Plates 1715.3 Simply Supported Rectangular Plates under Various Loadings 1745.4 Lévy's Solution for Rectangular Plates 1805.4.1 Simply Supported Rectangular Plate underUniform Loading 1835.5 Lévy's Method Applied to Rectangular Plates under Nonuniform Loading 1915.6 Rectangular Plates under ^Distributed Edge Moments 1955.7 Method of Superposition Applied to Bending ofRectangular Plates 1995.8 *The Strip Method 2025.9 *Simply Supported Continuous Rectangular Plates 2065.10 *Rectangular Plates Supported by Intermediate Columns 2095.11 Rectangular Plates on Elastric Foundation 2125.11.1 Simply Supported Plates 2135.11.2 Plates with Arbitrary Boundary Conditions 2135.12 The Ritz Method Applied to Bending of Rectangular Plates 215References 222Problems 2226 Plates of Various Geometrical Fprms 2296.1 Introduction 2296.2 *Method of Images 2296.3 Equilateral Triangular Plate with Simply Supported Edges 2326.3.1 Equilateral Triangtllar Plate under Uniform Moment M0 along its Boundary 2336.3.2 Equilateral Triangular Plate under Uniform Load p0 2346.4 Elliptical Plates 2356.4.1 Uniformly Loaded Elliptic Plate with Clamped Edge 2356.4.2 Uniformly Loaded Elliptic Plate with SimplySupporred Edge 2376.5 Sector-Shaped Plates 2376.6 *Stress Concentration around Holes in a Plate 239References 243Problems 2437 Numerical Methods 2477.1 Introduction 2477.2 Finite Differences 2487.3 Solution of the Finite Difference Equations 2527.3.1 Load Representation 2547.4 *Plates with Curved Boundaries 2647.5 *The Polar Mesh 2687.6 *The Triangular Mesh 2697.7 The Finite Element Method 2727.8 Properties of a Finite Element 2737.8.1 Displacement Matrix 2747.8.2 Strain, Stress, and Elasticity Matrices 2747.9 Formulation of the Finite Element Method 2767.10 Beam Element 2797.10.1 Methods of Assemblage of the [k]e's 2807.11 Triangular Finite Element 2827.11.1 Displacement Function 2837.11.2 The Stiffness Matrix 2857.11.3 External Nodal Forces 2857.12 Rectangular Finite Element 2877.12.1 Displacement Function 2877.12.2 The Stiffness Matrix 2887.12.3 External Nodal Forces 291References 293Problems 2948 Anisotropic Plates 2998.1 Introduction 2998.2 Basic Relationships 3008.3 Determination of Rigidities 3028.4 Rectangular Orthotropic Plates 3038.4.1 Application of Navier's Method 3058.4.2 Application of Lévy's Method 3078.4.3 Application of the Finite Difference Method 3088.5 Elliptic and Circular Orthotropic Plates 3108.6 Deflection by the Energy Method 3118.7 *Plates of Isotropic Multilayers 3158.8 The Finite Element Solution 3168.9 A Typical Layered Orthotropic Plate 3198.10 Laminated Composite Plates 322References 327Problems 3289 Plates under Combined Lateral and In-Plane Loads 3319.1 Introduction 3319.2 Governing Equation for the Deflection Surface 3319.3 Buckling of Plates 3359.4 Application of the Energy Method 3399.5 *The Finite Difference Solution 3459.6 Plates with Small Initial Curvature 3499.7 *Bending to a Cylindrical Surface 351References 355Problems 35510 Large Deflections of Plates 35910.1 Introduction 35910.2 Plate Behavior When Deflections Are Large 36010.3 Comparison of Small- and Large-Deflection Theories 36110.3.1 An Approximate Method for the Circular Plate 36110.3.2 Exact Solution for the Circular Plate Problem 36310.4 General Equations for Large Deflections of Plates 36410.5 Deflections by the Energy Method 36710.6 The Finite Element Solution 37110.6.1 Rectangular Finite Element373References 374Problems 37511 Thermal Stresses in Plates 37711.1 Introduction 37711.2 Stress, Strain, and Displacement Relations 37811.3 Stress Resultants 37911.4 The Governing Differential Equations 38011.5 Simply Supported Rectangular Plate Subject to an Arbitrary Temperature Distribution 38211.6 Simply Supported Rectangular Plate with Temperature Distribution Varying over the Thickness 38311.7 Analogy between Thermal and Isothermal PlateProblems 38511.7.1 Plates with Clamped Edges 38511.7.2 Plates with Simply Supported or Free Edges 38611.8 Axisymmetrically Heated Circular Plates 388References 391Problems 392Part III Shells12 Membrane Stresses in Shells 39712.1 Introduction 39712.2 Theories and General Behavior of Shells 39712.3 Load Resistance Action of a Shell 39912.4 Geometry of Shells of Revolution 40112.5 Symmetrically Loaded Shells of Revolution 40212.6 Some Typical Cases of Shells: of Revolution 40512.6.1 Spherical Shell 40612.6.2 Conical Shell 40612.6.3 Circular Cylindrical Shell 40812.7 Axially Symmetric Deformation 41712.8 Asymmetrically Loaded Shells of Revolution 41912.9 *Shells of Revolution under Wind Loading 42112.10 Cylindrical Shells of General'shape 42412.11 *Folded Structures 42812.12 *Shells of General Form 42912.13 *Breakdown of Elastic Action in Shells 433References 435Problems 43513 Bending Stresses in Shells 44313.1 Introduction 44313.2 Shell Stress Resultants 44313.3 Force, Moment, and Displacement Relations 44513.4 Compound Stresses in a Shell 44813.5 Strain Energy in the Bending and Stretching of Shells 44813.6 Axisym metrically Loaded Circular Cylindrical Shells 44913.7 A Typical Case of the Axisym metricallyLoaded Cylindrical Shell 45313.8 Shells of Revolution under Axisym metrical Loads 45713.8.1 Conical Shells 45913.8.2 Spherical Shells 45913.8.3 Cylindrical Shells 46013.9 Governing Equations for Axisym metrical Displacements 46013.10 Spherical Shells under Axisym metrical Load 46213.11 Comparison of Bending and Membrane Stresses 46513.12 *Simplified Theory of Spherical Shells underAxisymmetrical Load 46613.13 The Finite Element Representations of Shells of General Shape 47013.14 The Finite Element Solution of Axisym metrically Loaded Shells 471References 474Problems 47514 Applications to Pipes, Tanks, and Pressure Vessels 47714.1 Introduction 47714.2 Pipes Subjected to Edge Forces and Moments 47814.2.1 Long Pipes 47814.2.2 Short Pipes 48014.3 Reinforced Cylinders 48214.3.1 Cylinders with Collars That Prohibit Deflection 48214.3.2 Cylinders with Collars That Resist Deflection 48314.3.3 Cylinders with Closed Ends 48414.4 Cylindrical Tanks 48414.5 Thermal Stresses in Cylinders 48814.5.1 Uniform Temperature Distribution 48814.5.2 Radial Temperature Gradient 48914.6 Thermal Stresses in Compound Cylinders 49114.7 Discontinuity Stresses in Pressure Vessels 49414.8 Cylindrical Vessel with Hemispherical Heads 49514.9 Cylindrical Vessel with Ellipsoidal Heads 49814.10 Cylindrical Vessel with Flat Heads 49914.11 *Design Formulas for Conventional Pressure Vessels 500References 502Problems 50315 Cylindrical Shells under General Loads 50715.1 Introduction 50715.2 Differential Equations of Equilibrium 50815.3 Kinematic Relationships 50915.4 The Governing Equations for Deflections 51215.5 *Approximate Relations 51315.6 A Typical Case of Asymmetrical Loading 51415.7 Curved Circular Panels 51815.8 *A Simple Theory of Bending of Curved Circular Panels 52015.9 *Curved Circular Panels with Ends Simply Supported and Straight Edges Free 52315.10 Inextensional Deformations 52715.11 A Typical Layered Orthotropic Cylindrical Shell 53115.12 Laminated Composite Cylindrical Shells 53515.13 *Symmetrical Buckling under Uniform Axial Pressure 53715.14 Nonsymmetrical Buckling under Uniform Axial Compression 540References 544Problems 544AppendicesA Fourier Series Expansions 547A.1 Single Fourier Series 547A.2 Half-Range Expansions 549A.3 Double Fourier Series 551Reference 552B Tables 553B.l Conversion Factors: SI Units to U.S. Customary Units 553B.2 SI Unit Prefixes 554B.3 Typical Properties for Some Common Materials 555B.4 Properties of Common Areas 577B.5 Beam Deflection and Slopes 558B.6 Restrained Beam Reactions and Deflections 560Answers to Selected Problems 563Index 56