Every time a motor spins a pump, a gearbox drives a wheel, or a wind turbine feeds a generator, a shaft is quietly carrying the torque between them. That shaft looks like a plain cylinder, but along its length the material is working hard: each cross-section is being twisted relative to its neighbour, and the surface is in pure shear. Push the torque too high and the shaft yields, twists permanently, or fails outright.
The good news is that for a circular shaft the stress and the twist follow from two compact formulas. This article explains where they come from, works a full numerical example, and flags the assumptions that quietly break if you are not careful.
Why this calculation matters
Torsion is the defining load case for most rotating machinery. Drive shafts, propeller shafts, axles, gearbox input shafts, agitator spindles, and screw conveyors all exist to transmit torque, and the question the engineer must answer is always the same: is the shaft strong enough, and is it stiff enough?
Strength and stiffness are two separate checks. The strength check compares the maximum shear stress against the material's allowable shear stress — exceed it and the shaft yields. The stiffness check looks at the angle of twist; a shaft can be perfectly safe from yielding yet twist so much that gears mesh poorly, timing drifts, or torsional vibration becomes a problem. A precision indexing shaft might be governed entirely by twist, while a heavy-duty drive shaft is governed by stress. You need both numbers to size a shaft with confidence.
The core method
When a torque T is applied to a circular shaft, the cross-sections rotate relative to one another but stay flat and circular — this is the key simplifying fact for circular sections. The shear stress at any radius r grows linearly from zero at the centre to a maximum at the outer surface:
tau = T * r / J
J is the polar second moment of area, a purely geometric quantity that measures how the cross-section's material is distributed about its axis. For a solid circular shaft of diameter d:
J = pi * d^4 / 32
The maximum shear stress occurs at the outer surface, where r equals the shaft radius:
tau_max = T * (d/2) / J
The angle of twist over a length L follows from the same analysis. It depends on the torque, the length, the geometry through J, and the material through its shear modulus G:
theta = T * L / (G * J)
Theta comes out in radians. Two features are worth noticing. Stress and twist both scale inversely with J, and J scales with the fourth power of diameter — so a modest increase in shaft diameter is a powerful way to cut both stress and twist. And because the inner material carries so little stress, hollow shafts are remarkably efficient: removing the lightly loaded core saves weight while keeping most of the strength.
A worked example
Take a solid circular steel shaft of diameter d = 40 mm transmitting a torque T = 500 N.m. The shear modulus of steel is G = 79 GPa, and we want the maximum shear stress plus the twist over a length L = 1 m.
Step 1 — polar second moment of area.
J = pi * d^4 / 32
J = pi * (0.040)^4 / 32
J = 2.513e-7 m^4
Step 2 — maximum shear stress at the outer surface.
The outer radius is r = d/2 = 0.020 m.
tau_max = T * r / J
tau_max = 500 * 0.020 / 2.513e-7
tau_max = 39.8 MPa
So the surface of the shaft sees about 39.8 MPa of shear stress. For typical structural steel, with an allowable shear stress well above this, the shaft is comfortably safe against yielding.
Step 3 — angle of twist over 1 metre.
theta = T * L / (G * J)
theta = 500 * 1 / (79e9 * 2.513e-7)
theta = 0.0252 rad
Converting to degrees, 0.0252 rad multiplied by 180/pi gives about 1.44 degrees. A common rule of thumb for power-transmission shafts is to keep twist below roughly one degree per metre, so this shaft is slightly above that guideline — a hint that, if twist mattered for this application, a small diameter increase would be worth considering even though the stress check passes easily.
Common mistakes
Confusing polar moment with the rectangular second moment. For bending you use I = pi d^4 / 64; for torsion you use J = pi d^4 / 32. They differ by exactly a factor of two for a solid circle. Grabbing the wrong one halves or doubles your answer.
Forgetting to halve the diameter. The stress formula uses radius, not diameter. Plugging d straight into tau = T r / J doubles the computed stress. Always convert d to r = d/2 for the surface stress.
Leaving the twist angle in radians when a spec is in degrees. The formula theta = T L / (G J) returns radians. Drawings and tolerances are almost always in degrees. Multiply by 180/pi before comparing against a limit.
Applying circular-shaft theory to non-circular sections. The clean linear stress distribution relies on cross-sections staying plane. Square, rectangular, and open sections warp out of plane under torque and need different treatment — their torsional constant is not pi d^4 / 32, and the peak stress sits at a different location.
Ignoring stress raisers. The formulas give the nominal stress in a smooth shaft. Keyways, shoulders, splines, and cross-holes concentrate stress locally, sometimes by a factor of two or more. The nominal value is a baseline, not the peak the material actually feels.
Try the interactive NovaSolver calculator
Running these formulas once is straightforward, but real shaft design means juggling power, speed, section choice, and material at the same time. The Shaft Torsion Design Calculator on NovaSolver handles that: enter power and RPM and it derives the torque, then pick a cross-section — solid circle, hollow circle, solid square, or rectangle — and a material to get the maximum shear stress, the twist angle, the required diameter, the safety factor, and the critical speed, all updating in real time.
Related calculators
- Torsional Vibration Analysis — when the shaft drives inertias and you need natural frequencies of the multi-DOF system, not just static stress.
- Torsional Pendulum Simulator — to build intuition for how torsional stiffness and inertia set the period of oscillation.
- Gear Tooth Stress Calculator — for the next link in the drivetrain, where the torque you computed turns into tooth bending and contact stress.
You can browse the full set in the mechanical engineering tools hub.
Closing note
Shaft torsion is one of the cleanest design calculations in mechanical engineering: a geometric term J, a stress formula, and a twist formula, and you have both checks you need. Keep the fourth-power dependence on diameter in mind — it is your most effective lever for cutting stress and twist alike — and remember that strength and stiffness are separate questions with separate limits. Use radius where the formula asks for radius, convert twist to degrees before comparing it to a spec, and treat the smooth-shaft result as the starting point you then adjust for keyways and shoulders.
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