One way of calculating the torsional constant is by using the Prandtl Stress Function (another is by using warping functions). Unfortunately this is not straightforward, which is why the values (usually approximate) for common shapes are tabulated. ![]() Which leaves the problem of how to calculate $J_T$. When a member does not have circular symmetry then we can expect that it will warp under torsion and therefore $J_T \neq I_P$. The case of a circular rod under torsion is special because of circular symmetry, which means that it does not warp and it's cross section does not change under torsion. The polar moment of inertia on the other hand, is a measure of the resistance of a cross section to torsion with invariant cross section and no significant warping. Where $T$ is the applied torque, $L$ is the length of the member, $G$ is modulus of elasticity in shear, and $J_T$ is the torsional constant. The torsion constant $J_T$ relates the angle of twist to applied torque via the equation:
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