Shaft torsion and angle of twist, explained

Twist a round shaft and two separate questions appear: will it break (stress) and will it wind up too far (stiffness). Torsion theory answers both with one geometric property — the polar moment of inertia J — and two short formulas:

τ = T·r / J    θ = T·L / (G·J)

T is the torque, r the outer radius, L the loaded length and G the shear modulus (≈79 GPa for steel). For round sections:

J = π·d⁴/32 (solid)   J = π·(D⁴ − d⁴)/32 (hollow)

The fourth power is the whole story of shaft design. Ten percent more diameter is 46% more J — which works out to 25% less stress (∝1/d³) and 32% less twist (∝1/d⁴) at the same torque. Diameter is never a rounding decision in torsion.

Worked example — 50 mm solid steel shaft

A 50 mm solid shaft carries T = 1,000 N·m over L = 1 m (steel, G = 79 GPa, allowable shear 80 MPa):

J = π·50⁴/32 = 613,592 mm⁴

τ = 1,000,000 × 25 / 613,592 = 40.7 MPa

θ = 1,000,000 × 1,000 / (79,000 × 613,592) = 0.0206 rad = 1.18°

Stress sits at half the allowable (safety factor 1.96; the shaft could carry 1,964 N·m before reaching 80 MPa) — but the twist is already past the ~1°/m general screening band, and a precision drive would reject it outright. That split verdict is the normal one: long shafts are twist-limited, not strength-limited. The shaft torsion calculator runs all four numbers — stress, twist, capacity and safety factor — from the same inputs.

Why hollow shafts win

Shear stress varies linearly from zero at the axis to maximum at the surface — the core carries almost nothing. Bore a 30 mm hole through the same 50 mm shaft:

J = π·(50⁴ − 30⁴)/32 = 534,070 mm⁴ — 87% of solid

at 64% of the weight. Stress rises only to 46.8 MPa and twist to 1.36°. Scale that logic up and you get every propeller shaft and driveshaft made: a larger, thinner tube beats a smaller solid bar on every torsional metric per kilogram. The practical ceiling is wall buckling and the connections at the ends — which is where the keyway and spline capacity pages take over.

Common mistakes

• Using I instead of J. Bending uses the second moment I (π·d⁴/64); torsion uses the polar moment J (π·d⁴/32) — exactly double for round sections. Mixing them halves or doubles everything.
• Round-shaft formulas on non-circular sections. Rectangles and open profiles (slotted tubes, angles) twist by a different mechanism and are drastically softer — these formulas only hold for closed round sections.
• Forgetting stress concentrations. τ = T·r/J is the smooth-shaft stress; keyways, cross-holes and shoulder fillets multiply it locally (a standard keyway roughly doubles it for fatigue purposes).
• Checking strength but not twist. The worked example passes stress with margin and still twists past common limits — always read both outputs.

Adjacent checks: the shaft stress calculator combines torsion with bending for the real loading case, the critical speed calculator covers the whirl limit that catches long thin shafts, and the torque-power-RPM calculator turns the motor nameplate into the T these formulas need.

Frequently asked questions

What is the angle of twist formula?

θ = T·L / (G·J): torque times length over shear modulus times polar moment of inertia, in radians (×180/π for degrees). For a 50 mm solid steel shaft (G = 79 GPa, J = 613,592 mm⁴) carrying 1,000 N·m over 1 m, θ = 1.18°.

What is the shaft torsion stress formula?

τ = T·r / J — maximum shear at the outer surface, where r is the outer radius and J the polar moment of inertia (π·d⁴/32 solid, π·(D⁴−d⁴)/32 hollow). Stress is zero at the center, which is why hollow shafts are so efficient.

How much twist is acceptable in a shaft?

Commonly quoted screening limits run from roughly 0.25°/m for precision machine drives to about 1°/m for general transmission shafting — but the real limit is the application: backlash-sensitive positioning wants far less, a torque-only shaft can tolerate more. Check both stress and twist; long shafts usually hit the twist limit first.

Why are drive shafts hollow?

Torsion loads the outer fibres: stress and stiffness both scale with r⁴ terms, while weight scales with r². Boring a 30 mm hole through a 50 mm shaft keeps 87% of the torsional stiffness at 64% of the weight — the inner material was barely working.