Torsional Stiffness Calculator

Torsional stiffness (k) is a measure of a structural member's resistance to angular (twisting) deformation when subjected to a torque. It is the rotational analog of the linear spring constant, defined as the ratio of applied torque to the resulting angle of twist. A stiffer member requires more torque to produce the same twist angle. See also our Torsional Constant (K) — Torsional Constant.

There are two common approaches. The experimental formula is k = T / φ, where T is the applied torque (N·m) and φ is the measured angle of twist (rad). The geometric formula is k = GJ / L, where G is the shear modulus of the material, J is the polar moment of inertia of the cross-section, and L is the length of the beam.

Torsional stiffness is expressed as torque per unit angle. In SI units, this is N·m/rad. In imperial units, common expressions are lbf·ft/rad or lbf·in/rad. Since radians are dimensionless, the unit simplifies to N·m or lbf·ft in some contexts, but the /rad notation is retained to clarify it is a rotational stiffness.

The polar moment of inertia (J) describes how a cross-section's area is distributed relative to its centroidal axis. For a solid circular shaft of radius r, J = πr⁴/2. For a hollow shaft with outer radius R and inner radius r, J = π(R⁴ − r⁴)/2. It directly controls how resistant a section is to torsional deformation. You might also find our use the Principal Stress Calculator useful.

For a torsional spring, you can use the experimental formula k = T / φ, applying a known torque T and measuring the resulting angular deflection φ. This works because torsional spring stiffness shares the same units and definition as structural torsional stiffness (torque divided by angle).

Use the geometric formula (k = GJ / L) when you know the material properties and beam geometry but haven't physically tested the member. It only applies to straight, prismatic beams. Use the experimental formula (k = T / φ) when you have measured torque and twist values, or when dealing with non-standard shapes like torsional springs.

Yes. In the geometric formula k = GJ / L, the shear modulus G is a material property. Steel has G ≈ 80 GPa, aluminum ≈ 26 GPa, and titanium ≈ 41 GPa. A stiffer material (higher G) results in higher torsional stiffness for the same geometry, meaning less twist under the same torque.

Torsional stiffness is inversely proportional to beam length (k = GJ / L). Doubling the length of a beam halves its torsional stiffness, meaning it will twist twice as much under the same torque. Shorter, stockier members are therefore significantly stiffer in torsion than long, slender ones.