Angle of Twist Calculator for Solid and Hollow Shafts (θ = TL/GJ)
Find the angle of twist of a solid or hollow circular shaft from torque, length, diameter and material, or the torque for a given twist, with J, shear stress and stiffness.
How it works
θ = T × L ÷ (G × J) (radians) · Solid shaft: J = π D⁴ ÷ 32 · Hollow shaft: J = π (D⁴ − d⁴) ÷ 32 · θ in degrees = θ × 180 ÷ π · Torque for a twist: T = θ G J ÷ L · Max shear stress τ = T (D/2) ÷ J · Torsional stiffness k = G J ÷ L
When a torque twists a circular shaft, each cross-section rotates a little relative to the next. Over a length L the ends rotate by θ = TL ÷ (GJ), where G is the shear modulus of the material and J is the polar moment of inertia of the section, π D⁴ ÷ 32 for a solid shaft and π (D⁴ − d⁴) ÷ 32 for a tube. The result is in radians; the calculator converts it to degrees. Because J grows with the fourth power of the diameter, a shaft only 19% thicker twists half as much, and a tube loses little stiffness by removing its lightly stressed core. Choose “Torque for a twist” to work backwards: the torque that produces an allowed angle is θGJ ÷ L. The calculator also reports the maximum shear stress at the surface (T × radius ÷ J), the torsional stiffness GJ ÷ L, the twist per metre or foot, and a table of the same shaft in other common materials. Inputs can be in N·m, lbf·in or lbf·ft, metres, millimetres, inches or feet, and results in SI or US units. The formula assumes a straight, uniform, circular shaft of one material carrying a constant torque in the elastic range. For a stepped shaft or one with several torques, find the twist of each uniform segment and add them, keeping the sign of each torque. This page finds the twist; the shaft diameter calculator sizes a shaft for strength instead.
Worked example
A solid steel shaft 50 mm in diameter and 1 m long carries 1,000 N·m, with G = 79.3 GPa. J = π × 50⁴ ÷ 32 = 613,592 mm⁴. θ = 1,000,000 N·mm × 1,000 mm ÷ (79,300 MPa × 613,592 mm⁴) = 0.02055 rad = 1.178°. The surface shear stress is 1,000,000 × 25 ÷ 613,592 = 40.74 MPa, the torsional stiffness is 849.2 N·m per degree, and the outer surface moves 0.514 mm around the circumference.
Sources: Omni Calculator: Angle of twist (Hibbeler shear moduli, 100 mm / 3 m example) · The Engineering ToolBox: Torsion of shafts (50 mm steel shaft examples) · The Engineering ToolBox: Modulus of rigidity of common materials · Nylon 6/6 data sheet (tensile modulus 420,000 psi ≈ 2.9 GPa; G ≈ E ÷ 2(1 + ν) with ν ≈ 0.4)
Frequently asked questions
What is the formula for the angle of twist?
θ = TL ÷ (GJ): torque times length divided by shear modulus times polar moment of inertia. With T in N·mm, L in mm, G in MPa and J in mm⁴ (or lbf·in, in, psi and in⁴), θ comes out in radians. Multiply by 180/π for degrees.
How do I find the polar moment of inertia of a shaft?
For a solid round shaft J = π D⁴ ÷ 32. For a hollow shaft J = π (D⁴ − d⁴) ÷ 32, where d is the inside diameter. A 50 mm solid shaft has J = 613,592 mm⁴; a 50 mm tube with a 30 mm bore has 534,071 mm⁴.
How much does a steel shaft twist?
A 100 mm solid shaft 3 m long under 10,000 N·m with G = 80 GPa twists 0.0382 rad, about 2.19°. A 50 mm shaft 1 m long under 1,000 N·m twists about 1.2°.
What is the shear modulus of steel?
About 77–80 GPa (11–11.6 × 10⁶ psi) for carbon and structural steels, roughly 77 GPa for stainless steel. Aluminum alloys are about 24–28 GPa, so an aluminum shaft twists about three times as much as the same steel shaft.
How do I find the twist of a stepped shaft?
Split it into segments where the torque, diameter and material are constant, compute TL ÷ (GJ) for each and add the results with the sign of each internal torque. Run this calculator once per segment.
Related calculators
Results are textbook estimates for uniform circular shafts in the elastic range. Have a qualified engineer check any shaft whose failure could injure people or damage equipment.