The distributions of these stresses are shown on Fig. 8.33. Of particular note is the nonlinear distribution of the am stress. This predicts a higher inner fibre stress than the simple bending (a = My/I) theory.

8.27.7. Case 4. Asymmetric case n — 1. Shear loading of a circular arc cantilever beam

To illustrate this form of stress function the curved beam is again selected; however, in this case the loading is a shear loading as shown in Fig. 8.34.

As previously the beam is of narrow rectangular cross-section and unit width. Under the shear loading P the bending moment at any cross-section is proportional to sin 0 and, therefore it is reasonable to assume that the circumferential stress <rm would also be associated with sin0. This points to the case n = 1 and a stress function given in eqn. (8.108).

i.e. <(> = (A^r3 +Bt/r + C,r + D,rlnr)sin6 (8.113)

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

0 0

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