7 - Displacements of elastic shells stressed according to the membrane hypothesis

Summary

Introduction

We are now in a position to bring together the work of chapters 2, 4 and 6 in order to calculate the displacement of elastic shells which carry applied load, according to the membrane hypothesis, by direct-stress resultants only. In the case of a shell which is statically determinate according to the membrane hypothesis the procedure is straightforward, and consists of the same three steps which are used in the calculation of distortion of other kinds of statically determinate structure:

1. (i) Given the shell and its loading, and appropriate edge support conditions, use the equilibrium equations to find the direct-stress resultants, as in chapter 4.

2. (ii) Given the elastic properties of the material of which the shell is made (E, v) and the thickness of the shell, use Hooke's law (chapter 2) to determine the surface strains in the shell.

3. (iii) Solve the strain–displacement equations, as in chapter 6, together with the appropriate boundary conditions, to determine the displacement of the shell.

Most of the problems which we shall investigate in the following chapters will involve interaction between stretching and bending effects in shell structures. It may seem odd therefore to wish to perform the sequence of calculations listed above, since in practice the membrane hypothesis will rarely be valid. And indeed, most of the results which will be obtained in the present chapter will reappear later as special cases of more general analyses, incorporating bending effects, which will be performed in subsequent chapters.