Basic Principles of Plate Theory

1. Preliminaries.- 1.0 Motivation.- 1.1 Vectors-algebra.- 1.2 Vectors-calculus.- 1.3 Matrices.- 1.4 Statics-equilibrium.- 1.5 Summation convention and index notation.- 1.6 Elements of beam theory.- 1.7 Conclusions.- 2. Statics and Kinematics of Plate Bending.- 2.0 Introduction.- 2.1 The stress resultants.- 2.2 Principal values.- 2.3 The moment circle.- 2.4 Equilibrium equations-rectangular coordinates.- 2.5 Plate bending kinematics-rectangular coordinates.- 2.6 Equilibrium equations-polar coordinates-radial symmetry.- 2.7 Plate bending kinematics-polar coordinates-radial symmetry.- 2.8 Conclusions.- 3. Elastic Plates.- 3.0 Introduction.- 3.1 Elastic theory of plate bending-moment/curvature relations.- 3.2 Elastic theory of plate bending-governing equation.- 3.3 Circular plates-radial symmetry.- 3.4 Some simple solutions for circular plates.- 3.5 Simple solutions for problems in rectangular coordinates.- 3.6 Further separation of variable features-rectangular plates.- 3.7 Solution by finite differences.- 3.8 Some other aspects of plate theory.- 3.9 Stability of plates.- 3.10 Conclusions.- 4. Plastic Plates.- 4.0 Introduction.- A. Solid metal plates.- 4.1 Yield criteria.- 4.2 The bound theorems.- 4.3 The normality rule.- 4.4 Circular plates-square yield locus.- 4.5 Circular plates-Tresca yield locus.- 4.6 Plates of other shapes-square and regular shapes.- B. Reinforced concrete plates.- 4.7 Yield line theory-I. Fundamentals.- 4.8 Yield line theory-II. Further isotropic examples.- 4.9 Yield line theory-III. Orthotropic problems.- 4.10 Hillerborg strip theory.- 4.11 Conclusions.- 5. Optimal Plates.- 5.0 Introduction.- 5.1 Problem formulation.- 5.2 Constant curvature surfaces and principal directions.- 5.3 Basic results-corners.- 5.4 Some complete results.- 5.5 Moment volumes.- 5.6 Some theory.- 5.7 Conclusions.- 5.8 Exercises.- 6. Bibliography and Exercises.- 6.0 Bibliography.- 6.1 Exercises.- Appendix Geometry of Surfaces.- A.0 The need for geometry.- A.1 Geometry of a plane curve-curvature.- A.2 Length measurement on a surface-first fundamental form.- A.3 The normal to a surface.- A.4 Normal curvature-second fundamental form.- A.5 The derivatives of n-the Weingarten equations.- A.6 Directions on a surface.- A.7 The principal curvatures.- A.8 Principal directions.- A.9 Curvature and twist along the coordinate lines.- A.10 The curvature matrix.- A.11 The curvature circle.- A.12 Continuity requirements.- A.13 Special surfaces.- A.14 Summary-the geometrical quantities required for the construction of a plate theory.