(20 points) An isotropic, linearly elastic, prismatic rod with an equilateral triangle cross- section is subjected...

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(20 points) An isotropic, linearly elastic, prismatic rod with an equilateral triangle cross- section is subjected to torsion. The origin of the coordinate system is at the centroid of the cross-section, and the following Prandtl stress function is suggested: (x, y) = c(x + 3y 2a/3) (x 3y 2a/3) (x + a/3). I - (a) Verify that lateral boundary conditions and compatibility are satisfied with an appro- priate choice of the constant c. (b) Determine the shear stresses, r(x, y) and zy(x, y), in terms of the applied axial torque T, shear modulus G, and geometry of the cross-section. (c) Find the maximum shear stress Tmax and its location(s) in the cross-section. (d) Plot the distribution of zy along the x-axis. (20 points) An isotropic, linearly elastic, prismatic rod with an equilateral triangle cross- section is subjected to torsion. The origin of the coordinate system is at the centroid of the cross-section, and the following Prandtl stress function is suggested: (x, y) = c(x + 3y 2a/3) (x 3y 2a/3) (x + a/3). I - (a) Verify that lateral boundary conditions and compatibility are satisfied with an appro- priate choice of the constant c. (b) Determine the shear stresses, r(x, y) and zy(x, y), in terms of the applied axial torque T, shear modulus G, and geometry of the cross-section. (c) Find the maximum shear stress Tmax and its location(s) in the cross-section. (d) Plot the distribution of zy along the x-axis.