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# Ch5sol - d0/2 1 d/2 x 11 32 44 yield elastic yield 1 b...

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yield elastic yield cylinder α Shape factor of 1.5 1.5 1.0 1 2 3 d /2 d /2 x 3 4 b 2 5 6 c c t: constant 1 0′ 1″ 11″ 2 4 3 9 7 8 y .28″ 34.20″ 6 1 5 C x 5.1 Fig. P5-1 In order to minimize the potential stress concentration at the reentrant corners at the bottom of a rectangular keyway, the sharp corners are smoothed out by a circular hole. Show that the Prandtl stress function (Sokolnikoff 1956) leads to the solution of the circular shaft with a circular keyway, shown in Fig. P5-1. Determine the constant m and the expressions of the stresses, and on the boundaries . If , show that the ratio of the maximum shear stresses that are developed in is approximately 2 to 1. Solution Because of the nature of the problem, it would be convenient to transform the coordinate system from the Cartesian to the cylindrical coordinate system. The relationships between variables in the two systems are Hence, . For , (a) 1

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To get the higher order derivative with respect to x , it is only necessary to repeat the above operation; hence (b) In the same manner, (c) Adding these together, one obtains (d) From Eq. (5.1.15) (e) 2
(f) Substituting in the Poisson equation gives where is the rate of twisting. Hence, (g) (h) (i) On the boundary , . Substituting r into equations (h) and (i) yields 3

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(j) (k) (l) (m) On the boundary , . Substituting r into equations (h) and (i) yields (n) (o) (p) 4
(q) For , Hence, the ratio is 5.2. Three rods with solid cross sections, square, equilateral triangle, and circle, have equal cross-sectional areas and are subjected to equal twisting moments (Saada 1974). Compute the maximum shearing stresses developed and St. Venant torsional

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Ch5sol - d0/2 1 d/2 x 11 32 44 yield elastic yield 1 b...

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