Calculation of Displacements

Abstract

So far, we have restricted attention to the calculation of stresses and to problems in which the boundary conditions are stated in terms of tractions or force resultants, but there are many problems in which displacements are also of interest. For example, we may wish to find the deflection of the rectangular beams considered in Chapter 5, or calculate the stress concentration factor due to a rigid circular inclusion in an elastic matrix, for which a displacement boundary condition is implied at the bonded interface.

If the stress components are known, the strains can be written down from the stress-strain relations (1.77) and these in turn can be expressed in terms of displacement gradients through (1.51). The problem is therefore reduced to the integration of these gradients to recover the displacement components.

The method is most easily demonstrated by examples, of which we shall give two — one in rectangular and one in polar coördinates.