Shaft Torsion Calculator
Find torsional shear stress in a circular shaft.
Results update as you type.
How it works
When a shaft transmits torque, it experiences torsional shear stress, and this calculator computes it for a solid circular shaft. You enter the applied torque and the shaft diameter, and it returns the maximum torsional shear stress and the polar moment of inertia. The maximum stress occurs at the outer surface of the shaft and equals the torque times the outer radius divided by the polar moment of inertia, J, which for a solid round shaft is pi times the diameter to the fourth power over 32. This is a fundamental calculation for designing drive shafts, axles, torsion bars, and any rotating component that transmits power, since the shaft must carry the torque without its shear stress exceeding the material's limit. The strong dependence on diameter, to the fourth power in J, means that increasing shaft diameter dramatically reduces stress, which is why a modest size increase greatly strengthens a shaft against torsion. The stress is zero at the center and rises linearly to a maximum at the surface, so the outer material does the most work, which is also why hollow shafts can be nearly as strong as solid ones at much less weight. Engineers compare this maximum shear stress to the material's allowable shear stress, with a safety factor, to size shafts. This calculator handles solid shafts; hollow shafts use a modified J. Enter the torque and diameter to find the torsional shear stress.
τ = T·r / J, where J = πd⁴/32 for a solid shaft
For a solid circular shaft under torque, the maximum shear stress occurs at the surface and equals the torque times the outer radius, divided by the polar moment of inertia J. For a solid shaft, J is pi times the diameter to the fourth power, divided by 32.
Source:Mechanics of materials
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter applied torque and shaft diameter.
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Max torsional shear stress updates instantly as you type — no button to press.
Frequently asked questions
What is torsional shear stress?
Torsional shear stress is the shear stress that develops in a shaft when it is twisted by a torque. It is zero at the center and greatest at the outer surface. For a solid circular shaft it equals the torque times the outer radius divided by the polar moment of inertia, and it must stay below the material's shear strength.
What is the polar moment of inertia?
The polar moment of inertia, J, measures a shaft's resistance to twisting based on how its cross-sectional area is distributed about the central axis. For a solid round shaft, J equals pi times the diameter to the fourth power divided by 32. A larger J means less shear stress and twist for a given torque.
Why does shaft diameter matter so much?
The polar moment of inertia grows with the fourth power of the diameter, so even a small increase in diameter greatly increases J and sharply reduces the torsional shear stress. This means slightly thicker shafts are much stronger against torsion, which is why diameter is the primary design variable for shafts transmitting torque.
Why are hollow shafts efficient?
Because torsional stress is highest at the surface and zero at the center, the material near the axis contributes little to resisting torque. Removing it to make a hollow shaft saves significant weight while retaining most of the strength, which is why hollow shafts are common where weight matters, such as in vehicles and aerospace.
Does this work for hollow shafts?
This calculator uses the solid-shaft formula for the polar moment of inertia. A hollow shaft has a different J, equal to pi over 32 times the outer diameter to the fourth minus the inner diameter to the fourth. For hollow shafts, use that modified J; the stress is still torque times outer radius over J.