Shaft Torsion Calculator: Shear Stress and Angle of Twist

Shear stress and angle of twist for a solid circular shaft carrying a torque.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

A shaft carrying torque T is loaded in shear. The shear stress is zero on the axis and largest at the surface: τ = 16 T / (π d³) for a solid round shaft. The angle of twist over a length L is θ = T L / (G J), where J is the polar second moment of area and G the shear modulus.

If you add the yield strength, the calculator compares the surface shear stress with the shear yield strength (taken as the tensile yield divided by the square root of 3, the von Mises value) and shows the smallest diameter or largest torque that reaches your target factor of safety.

Formulas used

  • \tau_{max} = \frac{16 T}{\pi d^3}
  • J = \frac{\pi d^4}{32}
  • \theta = \frac{T L}{G J}

Worked example

With the default numbers in the calculator above:

  1. Polar second moment of area: 251327 mm⁴
  2. Maximum shear stress (at the surface): 39.8 MPa
  3. Angle of twist: 1.44°
  4. Factor of safety against yield in shear: 3.63
  5. Smallest shaft diameter that reaches FoS 1.5: 29.8 mm
  6. Largest torque at FoS 1.5: 1209 N m

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

How do I get the torque from motor power?

Torque equals power divided by angular speed: T = P / ω, with power in watts and speed in radians per second.

Does a keyway change the answer?

Yes. A keyway raises the local stress, and rotating shafts fail by fatigue at stresses well below yield. This calculator does not include them.

Is the angle of twist a limit?

Often. For a long shaft or an accurate drive the twist angle can govern before the stress does.

Related calculators

Further reading: Hibbeler, Mechanics of Materials; Gere and Goodno, Mechanics of Materials; Shigley, Mechanical Engineering Design.

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