Problem 1 (30 points) A prismatic shaft is made of Material-1 which has shear modulus G. The cross section of the shaft is an equilateral triangle. A torque M is applied and the shaft deforms in torsion with an angle of twist per unit length a. The equations describing the 3 sides of the triangle are: x-3y+2a = 0 (side 1) x + 3y+2 = 0 (side 2) x-a=0 (side 3) y - side 1 side 3 x = -2a side 2 x = a (a) Provide a Prandtl's stress function & that satisfies both the governing equation and the boundary condition. (b) Determine the shear stress distribution x (x,y) and yz(x, y) . (c) Determine the torsional rigidity D of this triangular shaft. Mz D = (d) Determine the warping function x(x, y).