MCPCalc

Shaft Torsion Calculator

Torsional shear stress, angle of twist, and polar moment J for solid and hollow circular shafts, from torque or from power and rpm, with an allowable-stress sizing check.

Scratchpad (not saved)

Outside diameter of the shaft.

Bore diameter. Used for hollow shafts only; must be smaller than the outer diameter.

Unit for the value entered above.

Length over which the torque acts.

Unit for the value entered above.

Used when Torque Input is direct.

Unit for the value entered above.

Used when torque is computed from power and speed.

Unit for the value entered above.

rpm

Shaft speed in revolutions per minute. Used with power.

Used only when the material is set to Custom.

Unit for the value entered above.

Enter 0 to skip the sizing check.

Unit for the value entered above.

What This Calculator Does

A shaft that transmits torque is loaded in pure shear. This calculator takes the shaft geometry and either a torque or a power and speed, then returns the polar moment $J$, the maximum shear stress at the outer surface, and the angle of twist over the shaft length. If you enter an allowable shear stress, it also checks the shaft against it and gives the smallest outer diameter that meets it for the same bore ratio. Solid and hollow circular shafts are supported, in metric or imperial units. Torque from a motor is computed as $T = P/\omega$.

It combines Shaft Type, Outer Diameter Do, Inner Diameter Di, Diameter Unit to estimate Maximum shear stress τmax, Angle of twist θ, Polar moment J.

Formula & Method

Polar moment of area: Jsolid=πD432,Jhollow=π(Do4−Di4)32J_{solid} = \frac{\pi D^4}{32}, \qquad J_{hollow} = \frac{\pi\left(D_o^4 - D_i^4\right)}{32} Maximum shear stress, at the outer surface: τmax=T (Do/2)J\tau_{max} = \frac{T\,(D_o/2)}{J} Angle of twist over length LL: θ=TLGJ\theta = \frac{TL}{GJ} Torque from power and speed: T=Pω=P2πn/60T = \frac{P}{\omega} = \frac{P}{2\pi n/60} where TT is torque, DoD_o and DiD_i are the outer and inner diameters, GG is the shear modulus, LL is the length, PP is power, and nn is speed in rpm. Minimum outer diameter for an allowable stress τall\tau_{all} with bore ratio k=Di/Dok = D_i/D_o: Do=[16 Tπ τall(1−k4)]1/3D_o = \left[\frac{16\,T}{\pi\,\tau_{all}\left(1-k^4\right)}\right]^{1/3}

Notation used in the formulas: RR = Maximum shear stress τmax; x1x_{1} = Shaft Type; x2x_{2} = Outer Diameter Do; x3x_{3} = Inner Diameter Di; x4x_{4} = Diameter Unit; x5x_{5} = Shaft Length L; x6x_{6} = Length Unit.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it to size a drive shaft, check twist in a long line shaft, or compare a solid shaft with a hollow one of the same outer diameter. Because $J$ scales with $D^4$, a small increase in diameter buys a large reduction in both stress and twist.

Worked Examples

Solid steel shaft, 50 mm, 1000 N·m (metric)

Given: D = 50 mm, T = 1000 N·m, L = 1 m, G = 79.3 GPa Method: solid circular section, so J = πD⁴/32, then τ = T(D/2)/J and θ = TL/(GJ). J = π(0.050)⁴/32 = 6.136×10⁻⁷ m⁴ τmax = 1000 × 0.025 / 6.136×10⁻⁷ = 40.74 MPa θ = 1000 × 1 / (79.3×10⁹ × 6.136×10⁻⁷) = 0.02055 rad = 1.178° For an allowable stress of 40 MPa the minimum solid diameter is [16×1000/(π×40×10⁶)]^(1/3) = 50.3 mm, so this shaft is 0.3 mm short of that limit.

Torque from motor power, then stress (imperial)

Given: 20 hp at 1750 rpm, solid steel shaft, D = 2 in, L = 36 in, G = 11.5×10⁶ psi Method: find torque from power first, T = 63,025 × hp / rpm in lbf·in. T = 63,025 × 20 / 1750 = 720.3 lbf·in J = π(2)⁴/32 = 1.571 in⁴ τmax = 720.3 × 1 / 1.571 = 458 psi θ = 720.3 × 36 / (11.5×10⁶ × 1.571) = 0.001435 rad = 0.082° Both are small. A 2 in shaft is oversized for this load unless there are keyways, shock loads, or bending on top of the torque.

Shear Modulus and Torque Reference

MaterialShear modulus G
Carbon and alloy steel79.3 GPa (11.5×10⁶ psi)
Aluminum 606126 GPa (3.8×10⁶ psi)
Titanium (Ti-6Al-4V)44 GPa (6.4×10⁶ psi)

Torque delivered by common motor ratings at 1500 rpm is $T = 9549\,P_{kW}/n$ in N·m. These are the values the calculator produces from power and speed:

Motor power750 rpm1500 rpm3000 rpm
5.5 kW70.0 N·m35.0 N·m17.5 N·m
15 kW191.0 N·m95.5 N·m47.7 N·m
30 kW382.0 N·m191.0 N·m95.5 N·m
75 kW954.9 N·m477.5 N·m238.7 N·m

Shear modulus varies with alloy and temper. Use the datasheet value for a specific grade when it matters.

Reference

The torsion formulas for circular shafts follow Coulomb's analysis and are derived in Beer and Johnston, Mechanics of Materials, and in Shigley's Mechanical Engineering Design.

Common Mistakes

  • Using the torsion formula on a non-circular shaft. Square, rectangular, and keyed sections warp under torque, and $\tau = Tr/J$ does not apply.
  • Entering the diameter where the radius belongs, or the reverse. Every $D$ field here is a diameter.
  • Mixing torque units. 1 lbf·ft is 12 lbf·in. The unit selector converts for you, so set it to match the number you typed.
  • Forgetting that power and torque are tied together by speed. Halving the rpm doubles the torque for the same power, so a slow shaft is loaded far more heavily than a fast one.
  • Reading a low nominal stress as safe. Keyways, shoulders, and press fits raise local stress well above $\tau_{max}$ from this formula.
  • Using the hollow-shaft formula with an inner diameter that is not smaller than the outer diameter.

When Not To Use This Calculator

  • Non-circular cross sections, such as square or splined shafts.
  • Stress concentrations at keyways, shoulders, holes, and fillets. Apply a stress concentration factor or analyze the feature directly.
  • Shafts that also carry bending or axial load, which need a combined-stress check such as von Mises.
  • Yielding, fatigue, and fluctuating torque. This calculator gives the elastic, static stress only.
  • Thin-walled open sections, which twist very differently from closed circular tubes.

Frequently Asked Questions

How do I calculate the angle of twist of a shaft?
Use θ = TL/(GJ), with torque T, length L, shear modulus G, and polar moment J. For a solid circular shaft J = πD⁴/32, so the angle falls with the fourth power of diameter. The result is in radians. Multiply by 180/π for degrees.
What is the polar moment of inertia J and how is it different from the moment of inertia of a beam?
J is the polar second moment of area, J = π(Do⁴ − Di⁴)/32 for a circular shaft. It measures a circular section's resistance to twisting. The moment of inertia I used in beam bending measures resistance to bending. For a circular section J = 2I.
How do I find shaft torque from power and rpm?
Torque is power divided by angular speed: T = P/ω, with ω = 2πn/60 in rad/s. In SI, T in N·m equals 9549 × kW / rpm. In US units, T in lbf·in equals 63,025 × hp / rpm.
Is a hollow shaft stronger than a solid shaft?
Per unit of weight, yes. Material near the axis carries little shear stress, so removing it lowers weight much more than it lowers torsional stiffness. A hollow shaft with the same outer diameter is somewhat weaker in absolute terms than a solid one, but its strength-to-weight ratio is higher.
What shear modulus G should I use for steel?
79.3 GPa (11.5×10⁶ psi) is a standard value for carbon and alloy steels. Stainless steels run slightly lower, around 77 GPa. Aluminum alloys are near 26 GPa. Use the datasheet value for the specific material when accuracy matters.

Inputs Used

  • Shaft Type: Used directly in the calculation.
  • Outer Diameter Do: Outside diameter of the shaft.
  • Inner Diameter Di: Bore diameter. Used for hollow shafts only; must be smaller than the outer diameter.
  • Diameter Unit: Unit for the value entered above.
  • Shaft Length L: Length over which the torque acts.
  • Length Unit: Unit for the value entered above.
  • Torque Input: Used directly in the calculation.
  • Torque T: Used when Torque Input is direct.
  • Torque Unit: Unit for the value entered above.
  • Power P: Used when torque is computed from power and speed.
  • Power Unit: Unit for the value entered above.
  • Speed n: Shaft speed in revolutions per minute. Used with power.
  • Material (Shear Modulus G): Used directly in the calculation.
  • Custom Shear Modulus G: Used only when the material is set to Custom.
  • Custom G Unit: Unit for the value entered above.
  • Allowable Shear Stress (optional): Enter 0 to skip the sizing check.
  • Allowable Stress Unit: Unit for the value entered above.

See Also

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