Torsion Calculator
Plan your structural engineering project with our free torsion calculator. Get precise measurements, material lists, and budgets.
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-mm 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.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
Can I use the results for professional or academic purposes?
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.
How accurate are the results from Torsion Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
Does Torsion Calculator work offline?
Once the page is loaded, the calculation logic runs entirely in your browser. If you have already opened the page, most calculators will continue to work even if your internet connection is lost, since no server requests are needed for computation.