PSAD Civil Engineering Licensure Examination (CELE) Practice Test

This is about the polar moment of inertia for a hollow circular shaft, which determines how resistant the cross-section is to twisting under torque. For torsion, J depends on r^4 across the area, so a hollow shaft is like subtracting the material that’s missing (the inner hole) from a solid disk of the same outer diameter.

The polar moment for a solid disk of diameter D is π/32 times D^4. The inner hole, with diameter d, would have contributed π/32 times d^4 if present as material, so you subtract that. Doing so gives J = π/32 (D^4 − d^4). This correctly accounts for the outer material minus the missing inner material.

The alternatives don’t fit because they either ignore the hole (using D^4 alone), treat the hole as if it contributed positively (using a sum D^4 + d^4), or focus only on the hole’s size (using d^4 alone). The subtraction form matches how the geometry reduces the torsional rigidity.