Torsional Stress in Shaft: Formula & Worked Example
The torsional stress in a shaft formula is τ = T · r / J, where τ is the shear stress (MPa), T is the applied torque (N·mm), r is the distance from the shaft centre to the outer surface (mm), and J is the polar moment of inertia (mm⁴). Maximum torsional shear stress always occurs at the outer radius. This formula applies to both solid and hollow circular cross-sections and is the starting point for any shaft torque stress calculation example.
Worked Example: Solid Shaft Under Torque
Consider a solid steel drive shaft with a diameter of 50 mm transmitting a torque of 1 500 N·m. The goal is to find the maximum torsional shear stress and compare it against a design limit.
This is a representative scenario found in conveyor drives, gearbox output shafts, and pump couplings — exactly the type of calculation that needs to be verified before finalising a shaft diameter. If you need to cross-check torque values upstream, the Motor Torque Calculation guide walks through deriving T from power and speed.
Given Data
- Shaft diameter: d = 50 mm
- Applied torque: T = 1 500 N·m = 1 500 000 N·mm
- Material: Medium carbon steel, yield strength Sy = 350 MPa
- Design safety factor: n = 2
Step 1 — Calculate Polar Moment of Inertia (J)
For a solid circular shaft: J = π · d⁴ / 32
J = π × (50)⁴ / 32 = π × 6 250 000 / 32 = 613 592 mm⁴ (≈ 6.136 × 10⁵ mm⁴)
Step 2 — Identify the Outer Radius (r)
r = d / 2 = 50 / 2 = 25 mm
Step 3 — Apply the Torsion Formula
τ = T · r / J = 1 500 000 × 25 / 613 592 = 61.1 MPa
Step 4 — Check Against Allowable Shear Stress
Allowable shear stress: τallow = Sy / (2 × n) = 350 / (2 × 2) = 87.5 MPa
Since 61.1 MPa < 87.5 MPa, the shaft diameter is acceptable under pure torsion. A combined loading check (bending + torsion) would be the logical next step in a full shaft design.
Torsion Formula and Variables
The governing equation for torsional shear stress comes directly from the theory of elasticity for circular cross-sections, codified in ASME B106.1M — Design of Transmission Shafting. The table below defines every symbol used in the torsion formula.
| Symbol | Quantity | SI Unit | Notes |
|---|---|---|---|
| τ | Torsional shear stress | MPa (N/mm²) | Maximum at outer radius |
| T | Applied torque (twisting moment) | N·mm | Convert N·m × 1000 to get N·mm |
| r | Radial distance from centre to point | mm | Use r = d/2 for maximum stress |
| J | Polar moment of inertia | mm⁴ | Depends on cross-section geometry |
| d | Shaft outer diameter | mm | Nominal machined diameter |
| do | Hollow shaft outer diameter | mm | Used in hollow shaft J formula |
| di | Hollow shaft inner diameter | mm | Bore diameter |
| G | Shear modulus (modulus of rigidity) | GPa | ~80 GPa for steel; used for angle of twist |
| L | Shaft length | mm | Used for angle of twist: φ = TL / GJ |
| φ | Angle of twist | radians | φ = TL / GJ |
| Sy | Material yield strength | MPa | From material datasheet or standard |
| n | Safety factor | — | Typically 1.5–3 for rotating machinery |
Polar Moment of Inertia: Solid vs Hollow Shaft
The polar moment of inertia J quantifies a cross-section's resistance to torsion. A larger J means lower stress for the same torque — which is why shaft diameter has such a powerful effect on torsional capacity (stress scales with d³ for a solid shaft when r and J are both expressed in terms of d).
| Cross-Section | J Formula | Notes |
|---|---|---|
| Solid circular shaft | J = π · d⁴ / 32 | Most common in drive shafts |
| Hollow circular shaft | J = π · (do⁴ − di⁴) / 32 | Better strength-to-weight ratio |
Hollow shafts are preferred in aerospace and automotive applications where weight reduction is critical. For the same outer diameter, removing material near the centre reduces J by a smaller proportion than it reduces mass, making hollowing an efficient trade-off. This intersects directly with press and shrink fit design when hollow shafts must interface with hubs — see the Press Fit vs Shrink Fit guide for those interface calculations.
ASME B106.1M references the polar section modulus Zp = J / r as a convenient shorthand, allowing the torsion formula to be written as τ = T / Zp. For a solid shaft, Zp = π · d³ / 16, which simplifies hand calculations when only the maximum surface stress is needed.
Step-by-Step Calculation Method
Follow this procedure for any shaft torque stress calculation, whether solid or hollow, for any consistent unit system.
- Determine the applied torque T. Calculate from power and rotational speed: T (N·mm) = (Power in W × 1 000) / (2π × n / 60), where n is speed in rpm. Alternatively, derive from gear ratios using the Gear Ratio Calculation guide. Convert the final value to N·mm before proceeding.
- Record shaft geometry. For a solid shaft, note the outer diameter d. For a hollow shaft, note both do and di. If keyways or splines are present, use the reduced effective cross-section diameter per ASME B106.1M.
- Calculate the polar moment of inertia J. Solid: J = π · d⁴ / 32. Hollow: J = π · (do⁴ − di⁴) / 32. Work in mm throughout to keep units consistent.
- Identify the radius r for maximum stress. For maximum shear stress at the outer surface, r = do / 2. For stress at a specific radius (for example, the inner bore of a hollow shaft), substitute that radius instead.
- Apply the torsion formula. τ = T · r / J. The result is in MPa when T is in N·mm, r in mm, and J in mm⁴.
- Calculate the allowable shear stress. Use τallow = Sy / (2 × n) for ductile materials under the maximum shear stress theory (Tresca criterion). The distortion energy theory (von Mises) gives τallow = Sy / (√3 × n), which is slightly less conservative.
- Compare and decide. If τ < τallow, the shaft is adequate in torsion. If τ ≥ τallow, increase d or select a higher-strength material. Where bending moments also act, combine stresses using the equivalent torque method: Te = √(M² + T²).
- Check angle of twist if stiffness matters. φ = T · L / (G · J). For precision machinery, φ is often limited to 0.25°–0.5° per metre of shaft length as a practical stiffness criterion.
Common Mistakes in Torsion Calculations
Even experienced engineers make unit and geometry errors in torsion problems. The following issues appear repeatedly in design reviews and are worth a deliberate check before signing off.
- Using diameter instead of radius for r. The formula uses the radial distance r, not the full diameter. Substituting d instead of d/2 doubles the calculated stress and leads to unnecessary oversizing — or, if used backwards in sizing, a shaft that is critically undersized.
- Ignoring stress concentration factors (Kt). Keyways, holes, shoulders, and grooves raise local stress significantly. ASME B106.1M and published engineering reference data provide stress concentration factors that must multiply τ at the feature location.
- Applying the formula to non-circular sections. The Coulomb torsion formula τ = T · r / J is valid only for circular cross-sections. Square, rectangular, or I-section shafts require different methods based on Saint-Venant's torsion theory.
- Neglecting combined loading. A shaft in a real drive system carries bending from gear or pulley forces alongside torsion. Treating them separately and checking each alone can give a false sense of safety. Always compute the equivalent torque or use principal stress analysis when both are present.
- Using nominal diameter for a hollow shaft with thick walls. For hollow shafts, using only do in a solid-shaft formula significantly overestimates J and underestimates stress. Calculate J correctly with both diameters.
- Overlooking fatigue under cyclic torque. For rotating shafts under fluctuating loads, the static shear stress limit is insufficient. Apply the ASME fatigue design criteria or an appropriate endurance limit correction for the material and surface condition.
For shaft-to-hub interfaces that sit alongside these stress considerations, the H7/p6 Press Fit Interference Calculation article covers the contact pressure and hoop stress side of the problem — directly complementing torsional design when torque is transferred through an interference fit rather than a key.
Frequently Asked Questions
What is the torsional stress formula for a shaft?
The torsional shear stress formula is τ = T · r / J, where T is the applied torque in N·mm, r is the outer radius in mm, and J is the polar moment of inertia in mm⁴. Maximum shear stress occurs at the outer surface of the shaft where r is greatest.
How do you calculate the polar moment of inertia for a solid shaft?
For a solid circular shaft, J = π · d⁴ / 32, where d is the shaft diameter in mm. For a hollow shaft, J = π · (do⁴ − di⁴) / 32, where do is the outer diameter and di is the inner bore diameter. Both values must be in the same unit — millimetres are recommended for SI work.
What is the difference between torsional shear stress and bending stress in a shaft?
Torsional shear stress acts circumferentially and is caused by an applied torque. Bending stress is a normal (tensile or compressive) stress caused by transverse loads or overhanging moments. When both act together, principal stress theory or the equivalent torque method Te = √(M² + T²) must be used to find the critical stress state.
What is an acceptable shear stress limit for a steel shaft?
A common design rule for ductile steel under static loading limits shear stress to τallow = Sy / (2 × n), where n is the safety factor (typically 1.5–3 depending on loading certainty). ASME B106.1M provides detailed guidance for transmission shafting. Always consult the applicable standard rather than relying on a generic rule for critical applications.
Does the torsion formula apply to hollow shafts?
Yes. The formula τ = T · r / J applies equally to hollow circular shafts. Only the polar moment of inertia changes: J = π · (do⁴ − di⁴) / 32. Hollow shafts are mass-efficient because the low-stress material near the neutral axis is removed, reducing weight with a relatively small loss in torsional capacity.