Mechanics of Materials

Polar Moment of Inertia

The section property that sets a round shaft’s resistance to twisting, J = πd⁴/32 for a solid shaft.

The polar moment of inertia J = ∫r² dA measures how a cross-section’s area is spread about its centre. In torsion it plays the role I plays in bending: shear stress τ = Tr/J and angle of twist φ = TL/(GJ).

Because J grows with the fourth power of diameter, a modest increase in diameter makes a shaft much stiffer and stronger in torsion. A hollow shaft, J = π(d_o⁴ − d_i⁴)/32, keeps most of the strength of a solid one at a fraction of the weight, because the removed core carried little stress.

How much of the FE Civil exam is Mechanics of Materials?