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a) Show that phi = C [ (3)1/2 a (y2 + x2 ) - 2 y3 + 6 y x2 + A] is a possible stress function by verifying that it satisfies Poisson's equation for an equilateral triangular cross section with side length a. Evaluate the constant C in terms of the shear modulus G and the rate of twist alpha.
b) Show that phi given in part (a) is indeed the stress function for the equilateral triangular section below by verifying that it also satisfies the appropriate boundary condition. Find the constant A.
c) A steel bar with a triangular section has Young’s modulus E = 210 GPa, Poisson’s ratio nu = 0.33 and side a = 5 cm. The length of the bar is L = 1m, and the total angle of twist is alpha = 0.5 radians. Use your results from parts (a) and (b) above to evaluate the torsional constant (J=T/(Gtheta)) for the bar and to calculate the maximum shear stresses occurring in the cross section.
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