Rotational Stiffness Calculator — Angular Stiffness & Torsional Rigidity
Compute the rotational (angular) stiffness of a component — either directly from applied torque and measured angle of twist, or from a shaft's material and geometry using the Coulomb torsion formula k_r = G·J/L.
Calculation method
N·m
°
Torque required to produce one radian of angular deflection
- 1
Convert angle to radians
5 × π ÷ 180 = 0.087266 - 2
Rotational stiffness k_r = T ÷ θ
100 ÷ 0.087266 = 1,145.92
How does this calculator work?
Rotational stiffness k_r = T/θ is the torque needed per radian of twist. For a solid shaft it equals G·J/L where G is the shear modulus, J = π·d⁴/32 is the polar moment of inertia, and L is the shaft length. Enter either torque + angle or shaft properties to compute k_r in N·m/rad.
Formula
How this is calculated
Rotational stiffness (also called torsional stiffness or angular stiffness) describes how much torque a structure needs per unit of angular deflection: k_r = T/θ, where T is the applied torque in newton-metres and θ is the angle of twist in radians. The higher the stiffness, the more torque is needed to rotate a member by a given angle. It is the rotational analogue of the familiar linear spring constant k = F/x.
For a solid cylindrical shaft made of an isotropic elastic material, the stiffness can be predicted from geometry and material alone using the Coulomb torsion formula: k_r = G·J/L. Here G is the shear modulus of the material (80 GPa for structural steel, about 26 GPa for aluminium, 0.001 GPa for rubber), J = π·d⁴/32 is the polar moment of inertia of the circular cross-section (in m⁴), and L is the shaft length (m). Shorter, wider, and stiffer shafts are harder to twist.
This calculator plots torque against twist angle for a linear (Hookean) regime, where the slope of the line equals k_r. The model assumes small angles within the elastic range and a uniform, solid circular cross-section; hollow shafts, non-circular profiles, temperature effects and plastic deformation are not covered here.
Frequently asked questions
Rotational stiffness (k_r = T/θ) relates torque to twist angle about the longitudinal axis. Bending stiffness (EI/L) relates bending moment to curvature about a transverse axis. They use different material constants (G for shear, E for bending) and different second moments of area (polar J vs. planar I).
The SI unit is newton-metres per radian (N·m/rad). Because radians are dimensionless, this is equivalent to N·m, but the per-radian convention makes the stiffness interpretation clear. Some datasheets quote N·m/degree — divide N·m/rad by 57.296 to convert.
The shaft-geometry mode uses J = π·d⁴/32 for a solid cross-section. For a hollow shaft, replace J with π·(d_o⁴ − d_i⁴)/32. You can compute that J independently and use the direct mode (set torque and the resulting angle) to get k_r without geometry constraints.
Also known as
TG we-Calculate Editorial Team. (2026). Rotational Stiffness Calculator — Angular Stiffness & Torsional Rigidity [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/rotational-stiffness-calculator
TG we-Calculate Editorial Team. "Rotational Stiffness Calculator — Angular Stiffness & Torsional Rigidity." TG we-Calculate. 2026. https://we-calculate.com/calculator/rotational-stiffness-calculator.
TG we-Calculate Editorial Team, "Rotational Stiffness Calculator — Angular Stiffness & Torsional Rigidity," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/rotational-stiffness-calculator
@misc{wecalculate_rotational_stiffness_calculator, title = {Rotational Stiffness Calculator — Angular Stiffness & Torsional Rigidity}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/rotational-stiffness-calculator}}, year = {2026}, note = {TG we-Calculate} }
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