Basic Principles of Plate Theory

Adding another volume, even if only a slim one, to the technical books already published requires some justification. Mine is, firstly, that plate theory is not well represented in the available elementary texts, and secondly that no existing text adequately covers modern applications. The present account is intended to be elementary (though this is a relative term) while still providing stimulation and worthwhile experience for the reader. Special features of interest will I hope be the treatment of geometry of surfaces and the attempts around the end of the work to speculate a little. The detailed treatment of geometry of surfaces has been placed in an appendix where it can readily be referred to by the reader. My interest in plate theory extends back many years to the energetic and stimulating discussions with my supervisor, Professor R. W. Tiffen, at Birkbeck College, London, and a debt to him remains. Interest was rekindled for me by Dr R. E. Melchers when I supervised him in Cambridge some ten years ago, and more recently my stay at Strathclyde University and encouragement and stimulation in the Civil Engineering Department led me to undertake the present work. The typescript was prepared by Ms Catherine Drummond and I thank her warmly for this and other assistance, always cheerfully offered. My thanks also to the publishers and the referees for useful comments and advice. P.G.L.

Table of Contents:
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.- B. Reinforced concrete plates.- 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.5The 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.