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Torsion Calculator

Plan your structural engineering project with our free torsion calculator. Get precise measurements, material lists, and budgets.

Reviewed by Abdullah, Technical Content Specialist

Reviewed by Abdullah, Technical Content Specialist

Formula

tau_max = Tc/J | theta = TL/(GJ) | J = pi*R^4/2 (solid)

Maximum shear stress equals the applied torque times the outer radius divided by the polar moment of inertia. The angle of twist equals the torque times the length divided by the product of shear modulus and polar moment of inertia. For solid circular sections, J equals pi times the radius to the fourth power divided by 2.

Worked Examples

Example 1: Solid Steel Shaft

Problem:Find the maximum shear stress and angle of twist for a solid 100mm diameter steel shaft, 2m long, under 5000 N-m torque. G = 80 GPa.

Solution:J = pi * 50^4 / 2 = 9,817,477 mm4\ntau_max = T*c/J = 5,000,000 * 50 / 9,817,477 = 25.46 MPa\ntheta = TL/(GJ) = 5,000,000*2000/(80,000*9,817,477) = 0.0127 rad = 0.73 deg

Result:tau_max = 25.46 MPa, angle of twist = 0.73 degrees

Example 2: Hollow Shaft Comparison

Problem:Same conditions but with a hollow shaft (100mm outer, 60mm inner diameter).

Solution:J = pi*(50^4 - 30^4)/2 = 8,545,132 mm4\ntau_max = 5,000,000 * 50 / 8,545,132 = 29.27 MPa\ntheta = 5,000,000*2000/(80,000*8,545,132) = 0.0146 rad = 0.84 deg

Result:tau_max = 29.27 MPa, twist = 0.84 deg (13% more stress, 36% less weight)

Frequently Asked Questions

What is torsion in structural and mechanical engineering?

Torsion is the twisting of a structural member when it is loaded by torques (moments) that produce rotation about the longitudinal axis. The resulting shear stresses vary linearly from zero at the center to a maximum at the outer surface for circular sections. Torsion is common in shafts, beams loaded eccentrically, and spandrel beams in concrete frames where floor loads apply twisting moments.

What is the torsion section modulus Zp?

The torsion section modulus Zp (also called the polar section modulus) equals the polar moment of inertia J divided by the outer radius c. It relates the applied torque directly to the maximum shear stress through the formula tau_max = T/Zp. This is analogous to the flexural section modulus S that relates bending moment to bending stress. A larger Zp means a lower maximum shear stress for a given torque.

References

Reviewed by Abdullah, Technical Content Specialist · Editorial policy