Torsional Stiffness Calculator — Shaft Twist & Shear Stress
Find the torsional stiffness (k = GJ/L) of a solid circular shaft and, from an applied torque, the angle of twist and maximum shear stress. Select a preset material or enter a custom shear modulus.
Shaft material
GPa
mm
mm
N·m
Torque required per radian of twist: k = G × J / L
- 1
Polar moment J = π × d⁴ ÷ 32
π × 50⁴ ÷ 32 = 613,592.3J in mm⁴; scales with the fourth power of diameter. - 2
Torsional stiffness k = G × J ÷ L
80 × 613,592.3 ÷ 500 = 98,174.77G in GPa, J in mm⁴, L in mm — the units cancel to give N·m/rad.
How does this calculator work?
Torsional stiffness k = GJ/L (N·m/rad), where J = πd⁴/32 for a solid circular shaft. Angle of twist φ = TL/(GJ) and max shear stress τ = Tr/J. Enter shaft diameter, length, and material shear modulus to get k; add the applied torque for φ and τ.
Formula
How this is calculated
Torsional stiffness k (N·m/rad) is the torque needed to twist a shaft through one radian. It combines three quantities: the material's shear modulus G (resistance of the material to shear deformation), the polar moment of inertia J (how the cross-sectional area is distributed), and the shaft length L. A stiffer material (higher G), fatter shaft (higher J ∝ d⁴) or shorter length all increase k.
From k and an applied torque T, the angle of twist follows directly from φ = T/k = TL/(GJ). This is the rotation one end experiences relative to the other. For a solid circular shaft J = πd⁴/32, so doubling the shaft diameter reduces twist by a factor of 16. The maximum shear stress occurs at the outer surface: τ_max = T·r/J, where r = d/2. Shear stress should stay below the material's shear yield strength (roughly 0.577 × tensile yield for ductile metals).
Shear modulus values used here are representative values at approximately 20 °C (steel ~80 GPa, aluminium alloys ~25–26 GPa, copper ~48 GPa, titanium alloys ~41–44 GPa) and can vary with alloy grade and temperature. For critical engineering calculations, verify G against the specific material datasheet.
Frequently asked questions
The torsional constant J is a purely geometric property of the cross-section (mm⁴ or m⁴). Torsional stiffness k = GJ/L also incorporates the material (G) and length (L), giving units of N·m/rad — it is the actual spring rate of the shaft in rotation.
The angle of twist φ = TL/(GJ) is directly proportional to length. Doubling the shaft length doubles the twist under the same torque, because the same strain is distributed over twice the material.
The shaft begins to deform plastically — it will not spring back to its original angle when the torque is removed, and may fracture at higher stresses. Design practice typically limits τ_max to 40–60% of the shear yield strength to maintain a safety margin.
Also known as
TG we-Calculate Editorial Team. (2026). Torsional Stiffness Calculator — Shaft Twist & Shear Stress [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/torsional-stiffness-calculator
TG we-Calculate Editorial Team. "Torsional Stiffness Calculator — Shaft Twist & Shear Stress." TG we-Calculate. 2026. https://we-calculate.com/calculator/torsional-stiffness-calculator.
TG we-Calculate Editorial Team, "Torsional Stiffness Calculator — Shaft Twist & Shear Stress," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/torsional-stiffness-calculator
@misc{wecalculate_torsional_stiffness_calculator, title = {Torsional Stiffness Calculator — Shaft Twist & Shear Stress}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/torsional-stiffness-calculator}}, year = {2026}, note = {TG we-Calculate} }
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