Mechanical

Shaft Diameter Calculator

Size a solid or hollow round shaft from transmitted torque, allowable shear stress, and an optional safety factor.

Formula d = (16T / (pi tau_d))^(1/3)Reviewed Sep 8, 2026

Shaft diameter sizing starts with the torsion equation for a circular shaft. This shaft diameter calculator estimates the minimum outside diameter for a solid or hollow round shaft carrying torque. Enter torque directly, or enter power and RPM so the calculator can first compute torque. The result is a strength-based starting diameter, not a complete shaft design with bending, fatigue, keys, shoulders, bearings, or deflection checks.

Calculation Bench

Input Mode

Shaft Shape

01Transmitted Torque

T · Twisting moment carried by the shaft

02Allowable Shear Stress

τ_allow · Material allowable before safety factor

03Safety Factor

N_s · Use 1 if stress is already derated

Solution

Enter torque, allowable shear stress, and shaft shape to size the shaft.

d = (16T / (pi tau_d))^(1/3)

Formula Sheet

d=\left(\dfrac{16T}{\pi\tau_d}\right)^{1/3}

d_o=\left(\dfrac{16T}{\pi\tau_d(1-k^4)}\right)^{1/3}

T=\dfrac{P}{\omega},\quad \omega=\dfrac{2\pi N}{60}

TTorque
PPower
NShaft Speed
tau_allowAllowable Shear Stress
tau_dDesign Shear Stress
d_oOutside Diameter
d_iInside Diameter
kBore Ratio

Common Metric Stock Diameters Used for Rounding

Range	Common stock diameters
Small shafts	6, 8, 10, 12, 15, 16, 20, 25 mm
General machine shafts	30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80 mm
Large shafts	90, 100, 110, 120, 125, 140, 150, 160, 180, 200 mm
Heavy shafts	225, 250, 280, 300, 350, 400, 450, 500 mm

Variables & Units

Symbol	Variable	Description	Common Units
T	Torque	Transmitted twisting moment on the shaft.	N·m, lb·ft, lb·in
P	Power	Mechanical power transmitted by the shaft.	W, kW, hp
N	Shaft Speed	Rotational speed used to derive torque from power.	rpm, rad/s
tau_allow	Allowable Shear Stress	Permitted torsional shear stress before the entered safety factor is applied.	MPa, ksi
tau_d	Design Shear Stress	Allowable shear stress divided by the safety factor.	MPa, ksi
d_o	Outside Diameter	Minimum outside diameter required by the torsion check.	mm, in
d_i	Inside Diameter	Hollow-shaft inside diameter, equal to k times outside diameter.	mm, in
k	Bore Ratio	Ratio of inside diameter to outside diameter for a hollow shaft.

How to Use This Calculator

01Choose the input mode: direct torque, or power plus shaft speed.
02Choose solid shaft or hollow shaft. For hollow shafts, enter the bore ratio k = inner diameter / outer diameter.
03Enter allowable shear stress for the shaft material. If your stress value is not already derated, enter a safety factor.
04If torque comes from motor power, use the lowest normal operating RPM for the worst-case torque screen; low speed at the same power means higher torque.
05Read the theoretical minimum diameter and the next larger standard stock diameter.
06For real machinery, recheck bending, fatigue, stress concentrations, keyways, shoulders, bearing seats, deflection, and applicable design standards before finalizing the shaft.

How the Formula Works

For a circular shaft in torsion, maximum shear stress is tau = T c / J. A solid round shaft has J = pi d^4 / 32 and c = d / 2, so the surface stress becomes tau = 16T / (pi d^3).

Solving that equation for diameter gives d = (16T / (pi tau_d))^(1/3), where tau_d is the design allowable shear stress. If a safety factor is entered, tau_d = tau_allow / N.

A hollow round shaft uses the same torsion equation but with J = pi(do^4 - di^4) / 32. Writing k = di/do gives do = (16T / (pi tau_d (1 - k^4)))^(1/3).

When power and speed are entered, the calculator first derives torque from T = P / omega, with omega = 2 pi N / 60. Lower RPM at the same power means higher torque, and therefore a larger required shaft diameter.

Worked Example 01

Solid shaft carrying 100 N·m

Known

Torque: 100 N·m
Allowable shear stress: 45 MPa
Safety factor: 1

Formula

d = (16T / (pi tau_d))^(1/3)

Substitution

d = (16 x 100 / (pi x 45,000,000))^(1/3)

Result

d = 22.45 mm, so the next common stock size is 25 mm

The torsion-only diameter is about 22.45 mm. In practice the shaft would be rounded up and then checked again for bending, fatigue, keyways, and deflection.

Worked Example 02

10 kW at 955 rpm

Known

Power: 10 kW
Speed: 955 rpm
Allowable shear stress: 45 MPa

Formula

T = P / omega, omega = 2 pi N / 60

Substitution

omega = 2 pi x 955 / 60 ≈ 100.01 rad/s; T = 10,000 / 100.01 ≈ 99.99 N·m

Result

d ≈ 22.45 mm for a solid shaft

Power is not enough by itself because torque depends on speed. At roughly 955 rpm, 10 kW corresponds to about 100 N·m, giving the same diameter as the direct-torque example.

Worked Example 03

Hollow shaft with k = 0.60

Known

Torque: 100 N·m
Allowable shear stress: 45 MPa
Bore ratio: 0.60

Formula

do = (16T / (pi tau_d (1 - k^4)))^(1/3)

Substitution

do = (16 x 100 / (pi x 45,000,000 x (1 - 0.60^4)))^(1/3)

Result

do ≈ 23.5 mm and di ≈ 14.1 mm

Removing material from the center reduces polar moment, so the outside diameter must increase. Hollow shafts can still be efficient, but wall thickness, buckling, manufacturing, and fatigue details need separate checks.

Applications

01Early sizing of solid transmission shafts under torque
02Comparing solid and hollow shaft outside diameters for the same torque
03Converting motor power and RPM into a shaft torque before sizing
04Teaching the cube-root relationship between torque, stress, and shaft diameter

Assumptions

01The shaft is circular, prismatic, and loaded in pure torsion at the checked section.
02Allowable shear stress is appropriate for the material, loading condition, and design method.
03Safety factor is applied by dividing allowable shear stress before solving diameter.
04Power/RPM mode assumes steady transmitted mechanical power and constant shaft speed.

Where This Model Stops

01Does not include bending moment, combined stress, shock factors, fatigue, stress concentrations, keyways, splines, shoulders, grooves, holes, or press fits.
02Does not apply ASME shaft design shock/fatigue factors or modified Goodman/Soderberg fatigue checks.
03Does not check angle of twist, lateral deflection, critical speed, bearing spacing, coupling fit, or manufacturing tolerances.
04Does not choose a material or allowable stress for you; material properties and safety factors must come from the project standard or engineer.
05Round up the result and recheck the final stock diameter with the real geometry, loads, and governing machine-design standard.

References

[1]
Torsion: Summary of key formulas
TU Delft OpenCourseWare
Reference for tau = Tr/J and angle-of-twist equations for circular shafts.
[2]
Shaft Size Calculator
Omni Calculator
Cross-check reference for solid and hollow shaft torsion diameter relationships.
[3]
Shaft Diameter Calculator
MachineCalcs
Reference for pure torsion sizing and the distinction between torsion-only and combined bending/torsion shaft sizing.

Frequently Asked Questions

What is the shaft diameter formula for pure torsion?

For a solid circular shaft, d = (16T / (pi tau_d))^(1/3), where T is torque and tau_d is the design allowable shear stress. For a hollow shaft, include the factor (1 - k^4), where k = inner diameter / outer diameter.

Can this calculator size a shaft with bending?

No. This tool is intentionally limited to pure torsion. Real shafts often carry bending from gears, pulleys, overhung loads, and bearing reactions, so combined stress and fatigue checks are usually required before final design.

Should safety factor multiply torque or divide stress?

This calculator divides allowable shear stress by the safety factor. For the same result, you can think of it as increasing the required torque by that factor before solving, because diameter varies with the cube root.

Why does doubling torque not double shaft diameter?

For pure torsion, diameter is proportional to the cube root of torque. Doubling torque increases required diameter by 2^(1/3), or about 26%, not 100%.

Is the next standard diameter the final shaft size?

No. It is only a practical rounding aid. The final diameter must account for stock availability, machining allowances, keyways, stress raisers, bearing fits, fatigue, twist, and the applicable machine-design standard.

Why can bending control a shaft even when torsion looks safe?

Gears, pulleys, sprockets, and overhung loads create bending moments between bearings. Those bending stresses combine with torsion and can govern fatigue long before a pure-torsion diameter check looks critical.

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