Calculator · Structural Mechanics
Torsional Shear Stress Calculator
Shear stress in a circular shaft twisted about its own axis. The stress is zero on the axis and grows linearly with radius, so the largest value is always at the outer surface. A hollow shaft removes the lightly stressed core and is stiffer per unit mass.
Inputs
Positive by the right-hand rule about the shaft axis.
Where in the wall to report the stress.
Results
| Polar second moment J | 613590 mm⁴ |
|---|---|
| Outer radius c | 25 mm |
| Applied torque T | 1200 N·m |
| Maximum shear τmax = T c / J | 48.892 MPa |
| Shear at ρ = 25 mm | 48.892 MPa |
Assumptions
- · Circular prismatic shaft; plane sections remain plane and radii remain straight.
- · Linear-elastic material and pure torsion (no warping restraint).
Open in the Lab
The Lab carries this shaft over as a single step and lets you add section changes and further torques along the length.
Technical reference
τ(ρ) = T ρ / J
τmax = T c / J
J = π D⁴ / 32 (solid)
J = π (D⁴ − d⁴) / 32 (hollow)
- T
- internal torque carried by the section[N·m]
- J
- polar second moment of area of a circular section[m⁴]
- ρ
- radial distance from the shaft axis[m]
- c
- outer radius[m]
- τ
- shear stress on the cross-section[Pa]
Circular sections only. For a circle the Saint-Venant torsion constant happens to equal the polar second moment of area; for any other shape the section warps and this equality fails, so these relations must not be applied to a rectangular, open or thin-walled non-circular member.
The stress varies linearly from zero on the axis to its maximum at the outer surface. Removing the core costs little strength, which is why hollow shafts are used.
Linear-elastic, pure torsion, no warping restraint. Stress concentrations at a shoulder, keyway, hole or fillet are not included and can raise the local stress well above the value reported here.
The torque used here is the internal torque carried by the section. When several torques act along a shaft, the internal torque changes from one span to the next; the Torsion Lab resolves that automatically.