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Angle of twist calculator

Find the angle of twist of a solid or hollow shaft in radians and degrees. Or work backwards from a twist limit to the torque, the length, the shear modulus or the diameter. The working is shown step by step.

Units: torque in N·m, power in kW, speed in rev/min, diameters in mm, length in m, shear modulus in GPa and stress in MPa. This is a learning and checking tool: check your own working and your course’s conventions.

Find

Shaft

SectionOuter diameter D (mm)Inner diameter d (mm)Length L (m)Shear modulus G (GPa)

Load

Load given asTorque T (N·m)

Optional

Stress at radius r (mm)Allowable shear stress (MPa)Allowable twist (degrees)

Significant figures

Result

7.51 × 10⁵ mm⁴Polar second moment of area J31.9 MPaMaximum shear stress25.6 MPaShear stress at the inner surface0.0819 radAngle of twist4.69°Angle of twist in degrees2.35° per mTwist per metre9770 N·m/radTorsional stiffness32.8 JStrain energy stored

The torque twists the shaft. The shear stress grows in proportion to the distance from the axis, so the surface carries the most and the material near the axis carries very little. The twist in the sketch is exaggerated.

Send the stress to Mohr’s circle

A shaft surface element is in pure shear. Mohr’s circle gives the principal stresses at 45° to the axis.

Surface: 31.9 MPa pure shear

Working

Shaft and load
Hollow circular shaft: outer diameter D_o = 60 mm, inner diameter D_i = 48 mm. Length L = 2 m, shear modulus G = 26 GPa.
Torque T = 800 N·m.
Polar second moment of area
J = π (D_o⁴ − D_i⁴) / 32 = π × (60⁴ − 48⁴) / 32 = 7.51 × 10⁵ mm⁴.
Shear stress
The shear stress rises in proportion to the radius: τ = T r / J. T = 800 N·m = 8.00 × 10⁵ N·mm.
At the outer surface, r = 30 mm: τ_max = 8.00 × 10⁵ × 30 / 7.51 × 10⁵ = 31.9 MPa.
At the inner surface, r = 24 mm: τ = 25.6 MPa.
The polar section modulus is Z_p = J / r_o = 25000 mm³, so τ_max = T / Z_p.
Angle of twist
θ = T L / (G J). With T in N·m, L in m, G in GPa and J in mm⁴, θ = 1000 T L / (G J) in radians.
θ = 1000 × 800 × 2 / (26 × 7.51 × 10⁵) = 0.0819 rad = 4.69°, which is 2.35° per metre.
Strain, stiffness and energy
Maximum shear strain γ = τ_max / G = 31.9 / 26000 = 0.00123 (1.23 milliradians). It also equals r_o θ / L.
Torsional stiffness k = T / θ = G J / L = 9770 N·m/rad.
Strain energy stored U = ½ T θ = 32.8 J.

Practise torsion questions

Press the button for a question with random numbers. Work it out on paper first, then type your answer.

How the angle of twist calculator works

The calculator uses the standard formulas for the torsion of circular shafts, and shows every step. The steps are:

Choose what to find: the angle of twist, or the torque, length, shear modulus or diameter for a known angle.
Find the polar second moment of area J of the section.
Use θ = T L / (G J) with the units converted (T in N·m, L in m, G in GPa, J in mm⁴).
Check the shear stress τ = T r / J at the torque you have found.
Compare the twist and the stress with your limits.

Assumptions

The shaft is circular, solid or hollow, with a constant section and a constant torque along its length. The material is linear elastic and the twist is small. The stress is the torsional shear stress only: stress concentrations at keyways and shoulders, bending, axial load and fatigue are not included. The direction of the torque does not change the size of the stress or the twist.

Worked example

A solid steel shaft 40 mm in diameter and 1.2 m long must not twist by more than 1.5° under load. G = 80 GPa. Find the largest torque it can carry, and the shear stress at that torque.

Shaft and load
Solid circular shaft: diameter D = 40 mm. Length L = 1.2 m, shear modulus G = 80 GPa.
Torque T = 438.649 N·m.
Torque from the angle of twist
Rearrange θ = T L / (G J) for the torque: T = θ G J / L.
θ = 1.5° = 0.0262 rad, J = 2.51 × 10⁵ mm⁴, G = 80 GPa, L = 1.2 m.
With these units T = θ G J / (1000 L) = 0.0262 × 80 × 2.51 × 10⁵ / (1000 × 1.2) = 439 N·m.
Polar second moment of area
J = π D⁴ / 32 = π × 40⁴ / 32 = 2.51 × 10⁵ mm⁴.
Shear stress
The shear stress rises in proportion to the radius: τ = T r / J. T = 439 N·m = 4.39 × 10⁵ N·mm.
At the outer surface, r = 20 mm: τ_max = 4.39 × 10⁵ × 20 / 2.51 × 10⁵ = 34.9 MPa.
The polar section modulus is Z_p = J / r_o = 12600 mm³, so τ_max = T / Z_p.
Angle of twist
θ = T L / (G J). With T in N·m, L in m, G in GPa and J in mm⁴, θ = 1000 T L / (G J) in radians.
θ = 1000 × 439 × 1.2 / (80 × 2.51 × 10⁵) = 0.0262 rad = 1.50°, which is 1.25° per metre.
Strain, stiffness and energy
Maximum shear strain γ = τ_max / G = 34.9 / 80000 = 4.36 × 10⁻⁴ (0.436 milliradians). It also equals r_o θ / L.
Torsional stiffness k = T / θ = G J / L = 16800 N·m/rad.
Strain energy stored U = ½ T θ = 5.74 J.

Twist of common solid shafts

Solid shaft diameter (mm)	J (mm⁴)	Max shear stress (MPa)	Twist per metre (mrad)	Twist per metre (°)
30	79500	189	157	9.01
40	2.51 × 10⁵	79.6	49.7	2.85
50	6.14 × 10⁵	40.7	20.4	1.17
60	1.27 × 10⁶	23.6	9.82	0.563
80	4.02 × 10⁶	9.95	3.11	0.178

Each row is the calculator’s answer for T = 1000 N·m, L = 1 m and G = 80 GPa. Doubling the diameter cuts the twist by a factor of 16.

Solid and hollow shafts compared

The same outer diameter and torque (1000 N·m, G = 80 GPa), with different bores.

Shaft, D = 50 mm	J (mm⁴)	Z_p (mm³)	Max shear stress (MPa)	Twist per metre (°)	Mass compared with solid
Solid	6.14 × 10⁵	24500	40.7	1.17	100%
Hollow, d = 20 mm	5.98 × 10⁵	23900	41.8	1.2	84%
Hollow, d = 30 mm	5.34 × 10⁵	21400	46.8	1.34	64%
Hollow, d = 40 mm	3.62 × 10⁵	14500	69	1.98	36%

Typical shear modulus values

Material	Typical shear modulus G (GPa)
Steel	80
Aluminium	26
Brass	39
Copper	45
Titanium	41

These are typical values for estimates. Use the value from your data book for real work.

Which formulas does this use?

Torsional shear stress in a solid shaft, Twist of a solid circular shaft, Strain energy stored in a twisted shaft, Shaft power from torque and speed, Shear modulus from Young’s modulus and Poisson’s ratio, Second moment of area, rectangle, Hollow rectangular section second moment of area.

Related calculators

Shaft torsion calculator, Shaft diameter calculator, Polar moment of inertia calculator, the Mohr’s circle calculator for the stresses on a shaft surface element, and the beam calculator for bending.

Common questions

What is the angle of twist formula?

θ = T L / (G J), where θ is in radians, T is the torque, L is the length, G is the shear modulus and J is the polar second moment of area. For a solid shaft J = π D⁴ / 32. The calculator also gives the angle in degrees and the twist per metre.

How do I find the torque for a given angle of twist?

Rearrange to T = θ G J / L, with θ in radians. Choose “Torque from the angle of twist” in the Find box, enter the angle in degrees and the calculator does the conversion and shows the working.

Can I find the length, shear modulus or diameter instead?

Yes. The Find box also solves for the length, the shear modulus G or the diameter from a known angle of twist. That lets you size a shaft to a twist limit.

Is the angle of twist in degrees or radians?

The formula gives radians. The calculator shows both, and the twist per metre of length, which is the usual way to quote a stiffness limit. The limit itself comes from the application, a design code or your course.

What assumptions does this calculator make?

It assumes a circular shaft, solid or hollow, of constant section, with a constant torque along its length, linear elastic material and small twists. It ignores stress concentrations at keyways, shoulders and holes, combined bending or axial loads, and fatigue. This is a learning and checking tool, so check the result against your own working and your design code.

Does it work for non-circular shafts?

No. The formulas τ = T r / J and θ = T L / (G J) apply to circular sections, where plane sections stay plane when twisted. Rectangular and other non-circular bars warp, and need a torsion constant and a different stress formula.

Is the calculator free, and does it keep my numbers?

It is free to use, with no sign-up, no adverts and no personal tracking. The numbers you type stay in your browser.