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This fits into the category of n = 2 with a stress function eqn. (8.109);

Using eqns. (8.103) the stresses can be written:

orr = -(2A2 + 6B2/r4 + 4D2/r2)cos 20 om = (2A2 + 6B2/r4 + 12C2r2)cos2<9 I*, = (2A2 - 6B2/r4 + 6C2r2 - 2D2/r2)sin26>, The four constants are found such that arr and r^ satisfy the boundary conditions: at r = a, arr = x^ = 0

From these, at r — R —> oo, orr = — cos 29, x^ = —— sin 29

Thus:

The sum of the stresses given by eqns. (8.120) and (8.121) is that proposed by Kirsch. At the edge of the hole arr and r^ should be zero and this can be verified by substituting r = a into these equations.

The distribution of ogg round the hole, i.e. r = a, is obtained by combining eqns. (8.120) and (8.121):

The stress concentration factor (S.C.F) defined as Peak stress/Average stress, gives an S.C.F. = 3 for this case.

The distribution across the plate from point A is:

This is shown in Fig. 8.38(b), which indicates the rapid way in which age approaches axx as r increases. Although the solution is based on the fact that R » a, it can be shown that even when R = 4a, that is the width of the plate is four times the diameter of the hole, the error in the S.C.F. is less than 6%.

Using the stress distribution derived for this case it is possible, using superposition, to obtain S.C.F. values for a range of other stress fields where the circular hole is present, see problem No. 8.52 for solution at the end of this chapter.

A similar, though more complicated, analysis can be carried out for an elliptical hole of major diameter 2a across the plate and minor diameter 2b in the stress direction. In this case the S.C.F. = 1 + 2a/b (see also §8.3). Note that for the circular hole a — b, and the S.C.F. = 3, as above.

8.27.9. Other useful solutions of the biharmonic equation

(a) Concentrated line load across a plate

The way in which an elastic medium responds to a concentrated line of force is the final illustrative example to be presented in this section. In practice it is neither possible to apply a genuine line load nor possible for the plate to sustain a load without local plastic deformation. However, despite these local perturbations in the immediate region of the load, the rest of the plate behaves in an elastic manner which can be adequately represented by the governing equations obtained earlier. It is thus possible to use the techniques developed above to analyse the concentrated load problem.

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