Polar moment of inertia

The polar moment of inertia J is the second moment of a cross-section’s area about an axis normal to its plane, J = integral of r^2 dA, equal to I_x + I_y for the same origin. It sets torsional stiffness: the twist rate of a closed thin-walled section is theta’ = T / (G * J), with T the torque and G the shear modulus. For hull-girder torsion the relevant property is the torsion constant of the open or closed box: open sections, such as container ships with wide deck openings, have low torsional stiffness and warp under quartering seas, which is why warping stress enters their strength assessment. Units are length to the fourth power.