A cantilevered beam is to be made of 7075-T6 aluminum with a uniaxial yield strength of 469 MPa. The beam is 1m long, and is loaded with the following limit loads: a bending moment of 1 kN-m and an axial torque of 2 kN-m (these are applied simultaneously).  Neglect stress concentrations at the wall [due to warping effects, in part b, the maximum torsion stress at the wall will be greater than that expected from simple mechanics of materials, but you may use the mechanics of materials approach here].

a) Suppose a solid circular shaft is to be made of this material.  Determine the minimum diameter required to ensure that no yielding occurs at the limit load?  Use the Von Mises criterion.  Justify your work clearly. As part of your work, indicate clearly the x-y-z coordinate(s) of the critical point used in your analysis, and how this is determined.

b) Suppose a solid square shaft is to be made of this material.  Determine the minimum side length required to ensure that no yielding occurs at the limit load?  Use the Von Mises criterion. Justify your work clearly. As part of your work, indicate clearly the x-y-z coordinate(s) of the critical point used in your analysis, and how this is determined.

c) Compare the two cases and comment.  Which of the two structures will be lighter? Also suggest alternate designs (still using aluminum) that might be lighter still (no calculations needed here, but justify/explain your reasoning).