E10.11 Deriving indices for torsion (Figure E10.8). A shaft of length L, loaded in torsion, must support a specified torque 7* without failing and be as cheap as possible. Section shape is a variable and "failure" again means the first onset of plasticity. Derive the material index. The table summarizes the requirements. Function Constraints Objective Free variables Cheap shaft Specified failure torque T* Length L specified Minimize material cost C Choice of material Section shape and scale Torque T, Angle of twist e T.0 Figure E10.8 Area A