CONTENTS Preface Xlll 1. Basic Equations of Theory of Elasticity 1 1.1Introduction 1 1.2Theory of Stress 1 1.2.1Concept of stress 1 7.2.2Stress components3 1.2.3Equilibrium equations5 7.2.4Stresses on an oblique plane 7 1.3Theory of Strain 9 1.3.1Deformation 9 1.3.2Strain and strain-displacement relations 10 1.3.3Strain compatibility 13 1.4Generalised Hooke's Law (Stress-strain relations) 14 1.5Differential Equations 20 1.6Summary 22 2. Basic Equations of Thin Plate Theory 23 2.1Introduction 23 2.2Assumptions 25 2.3Slopes and Curvatures of a Bent Plate 26 2.4Strain-Curvature Relations 29 2.5Moment-Curvature Relations 31 2.6Equilibrium Equations 33 2.6.7Rectangular plate 33 2.6.2Circular plate 39 2.7Boundary Conditions42 2.7.1Rectangular plate 42 2.7.2Circular plate47 2.8Summary of Basic Equations 49 2.8.1Basic equations in Cartesian coordinate system 49 2.8.2Basic equations in polar coordinate system 50 2.9Summary52 Bending of Isotropic Rectangular Plates 53 3.1Introduction 53 3.2Pure Bending and Cylindrical Bending of Rectangular Plates 53 3.3Navier Solution for an All-round Simply Supported Rectangular Plate 57 3.4Levy Solution for Rectangular Plates 68 3.4.1Rectangular plate with all edges simply supported and subjected to a uniformly distributed load of intensity q 74 3.4.2Rectangular plate with one pair of edges simply supported and the other pair fixed and subjected to a uniformly distributed load of intensity q 79 3.4.3Rectangular plate with two edges simply supported while the third edge is fixed and the fourth edge is free. The plate is subjected to a uniformly distributed load of intensity q 83 3.4.4All-round simply supported rectangular plate subjected to a uniformly distributed load of intensity q, with the origin at the centre of the plate 86 3.4.5 All-round simply supported rectangular plate subjected to end moments 88 3.5Method of Superposition for the Analysis of Rectangular Plates with Arbitrary Boundary Conditions 91 3.5.1 Rectangular plate with one pair of edges simply supported and the other pair fixed 91 3.5.2 Rectangular plate with all the four edges fixed and subjected to a uniformly distributed load 94 3.6 Bending Analysis of Continuous Rectangular Plates 101 3.7Rectangular Plate Subjected to Patch Load 106 3.7.7Approach 1 106 3.7.2Approach 2 111 3.8Semi-Infinite Rectangular Plate Subjected to Uniform Load 114 3.9Infinitely Long Simply Supported Rectangular Plate 117 3.10Summary 119 Problems 120 Bending of Orthotropic Rectangular Plates 125 4.1Introduction 125 4.2The Navier Solution for Orthotropic Plates 128 4.3The Levy Solution for Orthotropic Plates ,130 4.3.1An all-round simply supported rectangular Orthotropic plate subjected to a uniformly distributed load 136 4.4Semi-infinite Orthotropic Rectangular Plate 139 4.5Bending of an Infinitely Long Orthotropic Plate Subjected to a Line Load 141 4.6Summary 146 Problems 146 Bending of Circular Plates 147 5.1Circular Plates Subjected to an Arbitrary Load 147 5.2Symmetric Bending of Circular Plates 152 5.2.7Simply supported solid circular plate subjected to a uniformly distributed load of intensity q 153 5.2.2Simply supported solid circular plate subjected to an end moment 157 5.2.3A solid circular plate fixed or clamped along the boundary and subjected to a uniformly distributed load of intensity q 158 5.2.4Annular plate simply supported on the outer periphery and free in the inner periphery and subjected to a uniformly distributed load of intensity q 159 5.2.5A simply supported solid circular plate subjected to a partially distributed load 161 5.2.6A clamped solid circular plate subjected to a concentrated load at the centre166 5.3Circular Plate Subjected to Asymmetric Load 168 5.3.1 Simply supported solid circular plate subjected to asymmetric load 168 5.3.2Annular plate fixed at the inner periphery and free at the outer periphery, subjected to linearly varying asymmetric load 171 5.3.3Clamped solid circular plate subjected to an eccentric concentrated load 173 5.3.4A solid circular plate subjected to a uniform load and supported on K equidistant rigid discrete supports 177 5.4Summary 182 Problems 182 6.Approximate Methods 186 6.1 Introduction 186 6.2Principles of Virtual Work and Minimum Potential Energy 186 6.3Ritz Method 190 6.3.1All-round simply supported rectangular plate subjected to a uniformly distributed load 191 6.3.2All-round clamped rectangular plate subjected to a uniformly distributed load 193 6.3.3Rectangular plate simply supported on two opposite edges, clamped and free on the third and fourth edges, and subjected to a uniformly distributed load. 195 6.4Galerkin Method 197 6.4.1All-round simply supported plate subjected to a uniform load 199 6.4.2All-round clamped plate subjected to a uniform load 200 6.4.3Analysis of rectangular plates with various types of edge conditions using beam functions 203 6.5Summary 210 Problems 210 7.Numerical Methods 212 7.1Introduction 212 7.2Finite Difference Method 213 7.2.7Rectangular plate 218 7.2.2Boundary conditions 223 7.2JLoad distribution 227 7.2.4All-round simply supported square plate subjected to uniformly distributed load q 227 7.2.4.1All-round clamped square plate subjected to uniform load 232 7.2.5Plates of different geometrical shapes 234 7.2.5.1Bending analysis of an all-round simply supported skew plate with each side a, and subjected to a uniform lateral load 239 7.2.6Improved finite difference method 241 7.3Finite Element Method 242 7.3.1Rectangular plate 243 7.3.1.1General concepts 243 7J.7.2Rectangular element 250 7.3.1.3All-round clamped square plate subjected to a uniform load 255 7.5.2Circular plates subjected to axisymmetric load 260 7.4Summary 265 Problems 265 8. Shear Deformation Theories 267 8.1Introduction 267 8.2First Order Shear Deformation Plate Theory (FSDPT) 269 8.2.1Rectangular plate 269 8.2.1.1All-round simply supported rectangular plate subjected to an arbitrary lateral load of intensity q 275 5.2.2Circular plate 279 8.2.2.1Bending analysis of a clamped solid circular plate subjected to a uniformly distributed load of intensity q based on FSDPT . 282 8.3Higher Order Shear Deformation Plate Theory (HSDPT) 284 8.3.1Bending analysis of an all-round simply supported rectangular plate subjected to an arbitrary lateral load 290 8.4Shear Deformation Theory including Normal Strain 292 8.5Summary 293 9. Bending Analysis of Laminated Composite Plates 295 9.1Introduction 295 9.2Terminology and Assumptions 295 9.3Equilibrium Equations 297 9.4Strain-Displacement Relations 300 9.5Stress-Strain Relations 302 9.5.7Force-Displacement Relations 305 9.6Governing Differential Equations 307 9.7Determination of Strains and Stresses 310 9.8Boundary Conditions 313 9.9Lamination Configuration Types 314 9.10Bending Analysis of Symmetric Laminated Plates 317 9.10.1All-round simply supported symmetric cross ply laminated plate subjected to a uniformly distributed load 317 9.11 Bending Analysis of Antisymmetric Laminated Plate 324 9.11.1 All-round simply supported antisymmetric cross ply laminated plate 324 9.12Cylindrical Bending of Laminated Plates 330 9.13Summary 334 Problems 335 10. Analysis of Thick Plates 336 10.1Introduction 336 10.2Three-dimensional Solution for a Rectangular Orthotropic Plate 337 10.3Three-dimensional Solution for an Isotropic Plate 345 10.4Analysis of Thick Rectangular Plates using the Method of Initial Functions352 10.4 JMethod of initial functions 352 10.4.2Symmetrically loaded plate (Extension problem) 357 10.4.3Plate loaded antisymmetrically with respect to middle plane (Bending problem) 361 10.4.4Application to an all-round simply supported plate 363 10.4.5Approximate theories 367 10.5Cylindrical Bending of Thick Plates 368 70.5.7Orthotropic plate 368 10.5.1.1Simply supported long rectangular plate 370 70.5.2Isotropic plate 374 10.6Summary 377 Appendix 378 References 401 Index 404