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Determine the value of the constants a, b, c and e and hence show that the stresses are:

8.50 (C). A cantilever of unit width length L and depth 2a is loaded by a linearly distributed load as shown in Fig. 8.51, such that the load at distance x is qx per unit length. Proceeding from the sixth order polynomial derive the 25 constants using the boundary conditions, overall equilibrium and the biharmonic equation. Show that the stresses are:

Examine the state of stress at the free end (x = 0) and comment on the discrepancy of the shear stress. Compare the shear stress obtained from elementary theory, for L/2a = 10, with the more rigorous approach with the additional terms.

8.51 (C). Determine if the expression <t> = (cos 30)/r is a permissible Airy stress function, that is, make sure it satisfies the biharmonic equation. Determine the radial and shear stresses (arr and t^) and sketch these on the periphery of a circle of radius a.

2 ~ 6 -, o>r = -t cos 0(3 — 5 cos^ 9), r^ = —y cos^ #sin0. rs r

8.52 (C). The stress concentration factor due to a small circular hole in a flat plate subject to tension (or compression) in one direction is three. By supeiposition of the Kirsch solutions determine the stress concentration factors due to a hole in a flat plate subject to (a) pure shear, (b) two-dimensional hydrostatic tension. Show that the same result for case (b) can be obtained by considering the Lamé solution for a thick cylinder under external tension when the outside radius tends to infinity. [(a) 4; (b) 2.]

8.53 (C). Show that <f> - Cr2(a - 0 + sin 0 cos 9 — tan a cos 9) is a permissible Airy stress function and derive expressions for the corresponding stresses arr, nm and

These expressions may be used to solve the problem of a tapered cantilever beam of thickness carrying a uniformly distributed load q/unit length as shown in Fig. 8.52.

Show that the derived stresses satisfy every boundary condition along the edges 0 = 0° and 9 = a. Obtain a value for the constant C in terms of q and a and thus show that:

Compare this value with the longitudinal bending stress at 9 = 0° obtained from the simple bending theory when a = 5° and a = 30°. What is the percentage error when using simple bending?

C = --, —0.2% and —7.6% (simple bending is lower)

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