Introduction to the Theory of Shells

Preface

I. Preludes

The Lama Problem

A Derivation of Beam Theory

II. The Theory of Surfaces

The First Fundamental Form

Curvature and the Second Fundamental Form

The Gauss-Codazzi Conditions and the Fundamental Form

The Surface of Revolution

Some Terminology for Surfaces

III. The Construction of a Shell Theory

The Basic Assumptions

Shell Coordinates

Strain-Displacement Relations

Stress Resultants and Strain Energy

Equations of Equilibrium

Simplifications of the Strain Energy Functional and the Stress-Strain Relations

The Kirchoff Boundary Conditions

Appendix IIIA. Verification of Equation (56)

Appendix IIIB. Strain-Displacement Relations in Curvilinear Coordinates

Appendix IIIC. Verification of Equation (67)

Appendix IIID. Alternate Derivation of the Equilibrium Equations

Appendix IIIE. Strain Parameter Values for Rigid Body Motions

IV. Membrane Shells

General Formulation of Membrane Theory

Shells of Revolution with Straight Generators

Some Examples of Axisymmetric Shells of Revolution

Appendix IVA. Stresses in a Pressurized Oval Cylinder

Appendix IVB. Stresses in an Ogival Dome

V. The Bending of Circular Cylinders

Basic Relations and Simplifications

The Donnell, Sanders and Flügge Equations

The Axisymmetric, Semi-Infinite Cylinder

Decay Lengths and Edge Effects

The Donnell Equation and Some of Its Solutions for Asymmetric Deformation

Appendix VA. Cylinders with Variable Wall Thickness

Appendix VB. Influence Coefficients for the Axisymmetric Cylinder

Appendix VC. The Maxwell-Betti Theorem

VI. Shells of Revolution

General Formulation and Uncoupling Procedures

The Reissner-Meissner Theory of Axisymmetric Shells of Revolution

The Geckeler Approximation for Steep Shells

The Reissner Theory for Shallow Shells

Appendix VIA. The Shell Mating Problem

VII. Axisymmetric Vibrations of Circular Cylinders

Free Vibrations—Frequencies, Mode Shapes, Orthogonality

Forced Vibrations—Normal Modes Solution

Forced Vibrations—Williams' Method for Time-Dependent Boundary Conditions

Bibliography

Index