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Shaft diameter calculator

Find the smallest solid or hollow shaft diameter for a given torque, or for a given power and speed, from an allowable shear stress and an optional twist limit. The working is shown, with the next whole millimetre.

Units: torque in N·m, power in kW, speed in rev/min, diameters in mm, length in m, shear modulus in GPa and stress in MPa. This is a learning and checking tool: check your own working and your course’s conventions.

Shaft and limits

Load

Optional twist limit

Result

716 N·mTorque from the power45.0 mmOuter diameter for strength46 mmRounded up: diameter37.5 MPaShear stress at the rounded size

Working

  1. Design basis
    Power 75 kW at 1000 rev/min.
    Torque T = 716 N·m. Allowable shear stress 40 MPa. Solid shaft.
  2. Torque from power
    ω = 2π N / 60 = 105 rad/s, so T = P / ω = 75 × 1000 / 105 = 716 N·m.
  3. Diameter needed for strength
    For a solid shaft τ = 16 T / (π D3), and for a hollow shaft τ = 16 T Do / (π (Do4 − Di4)) = 16 T / (π Do3 (1 − k4)).
    Rearranging, Do = (16 T / (π τallow (1 − k4)))1/3 = (16 × 7.16 × 105 / (π × 40 × 1.00))1/3 = 45.0 mm.
  4. Choose the size
    The larger of the diameters governs. Rounding up to the next whole millimetre gives Do = 46 mm.
  5. Check the rounded size
    At Do = 46 mm the shear stress is 37.5 MPa, inside the limit.

Practise shaft sizing questions

Press the button for a question with random numbers. Work it out on paper first, then type your answer.

How the shaft diameter calculator works

The calculator uses the standard formulas for the torsion of circular shafts, and shows every step. The steps are:

  1. Find the torque. Use the value you have, or convert power and speed with T = P / ω.
  2. Rearrange τ = T r / J for the diameter that gives exactly the allowable shear stress.
  3. If you give a twist limit, rearrange θ = T L / (G J) for the diameter that gives exactly that twist.
  4. Take the larger of the two diameters, and round up to the next whole millimetre.
  5. Check the stress and twist at the rounded size.

Assumptions

The shaft is circular, solid or hollow, with a constant section and a constant torque along its length. The material is linear elastic and the twist is small. The stress is the torsional shear stress only: stress concentrations at keyways and shoulders, bending, axial load and fatigue are not included. The direction of the torque does not change the size of the stress or the twist.

Worked example

A hollow shaft with an inner to outer diameter ratio of 0.6 carries a torque of 2500 N·m. The allowable shear stress is 50 MPa, and the twist must not exceed 1° over 2 m. G = 80 GPa. Find the outer and inner diameters, to the next whole millimetre.

  1. Design basis
    Torque T = 2500 N·m. Allowable shear stress 50 MPa. Allowable twist 1° over 2 m, G = 80 GPa. Hollow shaft with Di / Do = 0.6.
  2. Diameter needed for strength
    For a solid shaft τ = 16 T / (π D3), and for a hollow shaft τ = 16 T Do / (π (Do4 − Di4)) = 16 T / (π Do3 (1 − k4)).
    Rearranging, Do = (16 T / (π τallow (1 − k4)))1/3 = (16 × 2.50 × 106 / (π × 50 × 0.870))1/3 = 66.4 mm.
  3. Diameter needed for stiffness
    θ = T L / (G J) with J = π Do4 (1 − k4) / 32 gives Do = (32 × 1000 T L / (π G θallow (1 − k4)))1/4.
    θallow = 1° = 0.0175 rad, so Do = 80.5 mm.
  4. Choose the size
    The larger of the diameters governs: stiffness governs here. Rounding up to the next whole millimetre gives Do = 81 mm and Di = 48.6 mm.
  5. Check the rounded size
    At Do = 81 mm the shear stress is 27.5 MPa, inside the limit and the twist is 0.974°.
  6. Compared with a solid shaft
    A solid shaft for the same torque and stress needs D = 63.4 mm. The hollow shaft carries the same torque with 29.8% less material, because the material near the axis is lightly stressed.

Shaft diameter for common torques

Torque (N·m)Solid shaft diameter (mm)Rounded up (mm)Hollow shaft outer diameter, d / D = 0.6 (mm)
10023.42424.5
25031.73233.2
50039.94041.8
100050.35152.7
250068.36971.5
5000868790.1

Each row is the calculator’s answer for an allowable shear stress of 40 MPa, from strength alone.

Power, speed and torque

If you know the power and the speed, find the torque first.

Power (kW)Speed (rev/min)Torque T = 9549 P / N (N·m)
550095.5
10100095.5
221450145
501500318
751000716
1003000318
25015001590

Which formulas does this use?

Torsional shear stress in a solid shaft, Twist of a solid circular shaft, Strain energy stored in a twisted shaft, Shaft power from torque and speed, Shear modulus from Young’s modulus and Poisson’s ratio, Second moment of area, rectangle, Hollow rectangular section second moment of area.

Related calculators

Shaft torsion calculator, Angle of twist calculator, Polar moment of inertia calculator, the Mohr’s circle calculator for the stresses on a shaft surface element, and the beam calculator for bending.

Common questions

How do you find the diameter of a shaft for a given torque?

Rearrange τ = 16 T / (π D³) for a solid shaft to get D = (16 T / (π τallow))^(1/3). Enter the torque and the allowable shear stress, and the calculator shows the working and rounds up to the next millimetre.

How do I find the shaft diameter from power and rpm?

Convert to torque first with T = 9549 P / N, for P in kW and N in rev/min, then use the strength formula. The calculator does both steps for you.

What allowable shear stress should I use?

It depends on the material, the factor of safety and the code or course you follow. A common start is the shear yield strength of the material, about half the tensile yield strength by the Tresca criterion, divided by a factor of safety. Use the value your problem or standard gives.

Should strength or stiffness decide the diameter?

Check both. The calculator finds the diameter for the stress limit and, if you give a twist limit, for the twist limit too. The larger one governs, and long or precise shafts are often governed by twist.

What assumptions does this calculator make?

It assumes a circular shaft, solid or hollow, of constant section, with a constant torque along its length, linear elastic material and small twists. It ignores stress concentrations at keyways, shoulders and holes, combined bending or axial loads, and fatigue. This is a learning and checking tool, so check the result against your own working and your design code.

Is the calculator free, and does it keep my numbers?

It is free to use, with no sign-up, no adverts and no personal tracking. The numbers you type stay in your browser.