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Shaft torsion calculator
Enter the torque, or the power and speed, the shaft size and the material. Get the shear stress, the polar second moment of area, the angle of twist and the torsional stiffness for a solid or hollow shaft, with every step of the working shown.
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.
Shaft
Load
Optional
Result
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.
Working
- Shaft and loadSolid circular shaft: diameter D = 50 mm. Length L = 1.5 m, shear modulus G = 80 GPa.
Torque T = 1200 N·m. - Polar second moment of areaJ = π D4 / 32 = π × 504 / 32 = 6.14 × 105 mm⁴.
- Shear stressThe shear stress rises in proportion to the radius: τ = T r / J. T = 1200 N·m = 1.20 × 106 N·mm.
At the outer surface, r = 25 mm: τmax = 1.20 × 106 × 25 / 6.14 × 105 = 48.9 MPa.
The polar section modulus is Zp = J / ro = 24500 mm³, so τmax = T / Zp. - Angle of twistθ = 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 × 1200 × 1.5 / (80 × 6.14 × 105) = 0.0367 rad = 2.10°, which is 1.40° per metre. - Strain, stiffness and energyMaximum shear strain γ = τmax / G = 48.9 / 80000 = 6.11 × 10−4 (0.611 milliradians). It also equals ro θ / L.
Torsional stiffness k = T / θ = G J / L = 32700 N·m/rad.
Strain energy stored U = ½ T θ = 22.0 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 shaft torsion calculator works
The calculator uses the standard formulas for the torsion of circular shafts, and shows every step. The steps are:
- Find the torque. Use the value you have, or convert power and speed with T = P / ω.
- Find the polar second moment of area J of the section: π D⁴ / 32 for a solid shaft and π (D⁴ − d⁴) / 32 for a hollow one.
- Find the shear stress from τ = T r / J. It is largest at the outer surface, r = D / 2.
- Find the angle of twist from θ = T L / (G J), then the torsional stiffness T / θ and the strain energy ½ T θ.
- Compare the stress and the twist with your limits to get a factor of safety.
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 hollow steel drive shaft has an outer diameter of 80 mm and an inner diameter of 60 mm. It is 2.4 m long, G = 80 GPa, and it transmits 120 kW at 1500 rev/min. Find the torque, the maximum shear stress and the angle of twist.
- Shaft and loadHollow circular shaft: outer diameter Do = 80 mm, inner diameter Di = 60 mm. Length L = 2.4 m, shear modulus G = 80 GPa.
Power P = 120 kW at N = 1500 rev/min. - Torque from powerThe angular speed is ω = 2π N / 60 = 2π × 1500 / 60 = 157 rad/s.
Power = torque × angular speed, so T = P / ω = 120 × 1000 / 157 = 764 N·m. - Polar second moment of areaJ = π (Do4 − Di4) / 32 = π × (804 − 604) / 32 = 2.75 × 106 mm⁴.
- Shear stressThe shear stress rises in proportion to the radius: τ = T r / J. T = 764 N·m = 7.64 × 105 N·mm.
At the outer surface, r = 40 mm: τmax = 7.64 × 105 × 40 / 2.75 × 106 = 11.1 MPa.
At the inner surface, r = 30 mm: τ = 8.34 MPa.
The polar section modulus is Zp = J / ro = 68700 mm³, so τmax = T / Zp. - Angle of twistθ = 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 × 764 × 2.4 / (80 × 2.75 × 106) = 0.00834 rad = 0.478°, which is 0.199° per metre. - Strain, stiffness and energyMaximum shear strain γ = τmax / G = 11.1 / 80000 = 1.39 × 10−4 (0.139 milliradians). It also equals ro θ / L.
Torsional stiffness k = T / θ = G J / L = 91600 N·m/rad.
Strain energy stored U = ½ T θ = 3.18 J.
Solid and hollow shafts compared
The same outer diameter and torque (1000 N·m, G = 80 GPa), with different bores. Removing the core saves much more mass than it costs in stiffness.
| Shaft, D = 50 mm | J (mm⁴) | Zp (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% |
Power, speed and torque
The torque a shaft carries follows from the power and the speed. These values come from the calculator.
| Power (kW) | Speed (rev/min) | Torque T = 9549 P / N (N·m) |
|---|---|---|
| 5 | 500 | 95.5 |
| 10 | 1000 | 95.5 |
| 22 | 1450 | 145 |
| 50 | 1500 | 318 |
| 75 | 1000 | 716 |
| 100 | 3000 | 318 |
| 250 | 1500 | 1590 |
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
Angle of twist 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 formula for shear stress in a shaft under torsion?
τ = T r / J, where T is the torque, r is the distance from the axis and J is the polar second moment of area. The stress is zero on the axis and largest at the outer surface, where τmax = T D / (2 J). For a solid shaft this becomes τmax = 16 T / (π D³).
How do you calculate the angle of twist of a shaft?
θ = T L / (G J) in radians, where L is the length, G is the shear modulus and J is the polar second moment of area. Multiply by 180 / π to get degrees. A higher G or a larger J gives less twist.
How do I find the torque from power and speed?
Power equals torque times angular speed, P = T ω, with ω = 2π N / 60 for N in revolutions per minute. So T = 9549 P / N in N·m for P in kW. For example, 75 kW at 1000 rev/min gives about 716 N·m.
Why is a hollow shaft more efficient than a solid one?
The shear stress is highest at the surface and falls to zero at the axis, so the material near the centre carries little of the torque. A hollow shaft with the same outer diameter keeps most of the strength and stiffness with much less mass. The table on this page shows the numbers.
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.