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The distributions of these stresses are shown in Fig. 8.34. They are similar to that for the pure moment application. The simple bending (a = My/I) result is also shown. As in the previous case it is noted that the simple approach underestimates the stresses on the inner fibre.

8.27.8. Case 5-The asymmetric cases n ^ 2-stress concentration at a circular hole in a tension field

The example chosen to illustrate this category concerns the derivation of the stress concentration due to the presence of a circular hole in a tension field. A large number of stress concentrations arise because of geometric discontinuities-such as holes, notches, fillets, etc., and the derivation of the peak stress values, in these cases, is clearly of importance to the stress analyst and the designer.

The distribution of stress round a small circular hole in a flat plate of unit thickness subject to a uniform tension axx, in the x direction was first obtained by Prof. G. Kirsch in 1898 The width of the plate is considered large compared with the diameter of the hole as shown in Fig. 8.35. Using the Saint-Venant's^ principle the small central hole will not affect the

Fig. 8.35. Elements in a stress field some distance from a circular hole.

t G. Kirsch Verein Deutsher Ingenieure (V.D.I.) Zeitschrift, 42 (1898), 797-807. * B. de Saint-Venant, Mem. Acad. Sc. Savants E'trangers, 14 (1855), 233-250.

stress distribution at distances which are large compared with the diameter of the hole-say the width of the plate. Thus on a circle of large radius R the stress in the x direction, on 0 = 0 will be oxx. Beyond the circle one can expect that the stresses are effectively the same as in the plate without the hole.

Thus at an angle 6, equilibrium of the element ABC, at radius r — R, will give orr.AC = UxxBC cos 0, and since, cos 6 — BC/AC orr — Oxx cos2 6,

Note the sign of r^ indicates a direction opposite to that shown on Fig. 8.35. Kirsch noted that the total stress distribution at r = R can be considered in two parts:

(a) a constant radial stress axx/2

(b) a condition varying with 26, that is; orr = — cos 26, t^ = - — sin 26.

The final result is obtained by combining the distributions from (a) and (b). Part (a), shown in Fig. 8.36, can be treated using the Lamé equations; The boundary conditions are:

Using these in the Lamé equation, arr =A+B/r2

Fig. 8.36. A circular plate loaded at the periphery with a uniform tension.

When a these can be modified to A = — and B = ——a

Thus

Parr (b), shown in Fig 8.37 is a new case with normal stresses varying with cos 29 and shear stresses with sin 29.

Fig. 8.37. A circular plate loaded at the periphery with a radial stress = — cos29 (shown above) and a shear ctr ■ f«»/2

Fig. 8.37. A circular plate loaded at the periphery with a radial stress = — cos29 (shown above) and a shear a** ■ -in stress =--sm 20.

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