UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering,Department of Civil Engineering Mechanics and MaterialsFall 2004 Professor: S. GovindjeeHW #12: CE231 / MSE2111. Consider the tapered beam shown below.(a) Determine the expression for A in the stress function φ = Arθ cos θ in terms of H andthe angle α.(b) Plot the bending and shear stresses at x = L/2 for L = 1 and L = 10. On the samefigures, plot the Bernoulli-Euler parabolic solution for shear in a cantilever and the bendingstresses in a cantilever; (see Popov or any other strength of materials text). Assume H = 1for the plots.αxyHL1thickness = 12. For the curved beam shown below:(a) Determine the stress field by starting with the stress functionΦ = (Ar ln(r) + Br3+ C/r) sin(θ)(b) Set-up the linear equations that are used to find A, B, and C in terms of a, b, and P .(c) At θ = π/2 plot the normal (bending) stress on the section and compare it to theBernoulli-Euler solution. Make plots for (a, b) = (0.5, 2) and (a, b) = (8.5, 10). AssumeP = 1 for the plots.(d) Comment on the physical implication of σrrin the curved beam. Relate it to the situationin straight beam.1abPthickness = 13. A circular plane strain disk made of a linear elastic isotropic material is subjected to aradial pressure, p(θ) = ˆσ cos(2θ), on its perimeter.(a) Find the maximum hoop stress σθθ.(b) Determine the hoop strain