📋

Walk-through

How to Use This Calculator

4 steps

1

Choose a solid or hollow shaft

Pick Solid or Hollow (tube) from the shaft-type dropdown. A solid shaft needs a single diameter; a hollow shaft needs both an outer and an inner diameter, since torsion loads only the circular cross-section of material that actually exists.

2

Enter the shaft dimensions in millimetres

Fill in the diameter (or outer and inner diameter for a tube) in millimetres. The calculator uses these to compute the polar moment of inertia (J) for the cross-section.

3

Enter the torque, length, and shear modulus

Enter the applied torque in newton-metres, the shaft length in metres, and the material's shear modulus in gigapascals (steel is about 79–81 GPa, aluminum about 26 GPa). The calculator updates instantly.

4

Read the result — switch tabs to change the headline value

The result card always shows all three quantities. Use the Shear Stress, Angle of Twist, or Polar Moment tab to pick which one is promoted to the large headline number; the other two stay visible in the detail line below it.

⚡

Reference

Formula & Methodology

3 formulas

Torsional shear stress

τ = Tr / J

The shear stress τ at the outer surface of a circular shaft under torque T equals T times the outer radius r, divided by the polar moment of inertia J. It is maximum at the outer surface and zero at the center of the shaft.

Angle of twist

θ = TL / (GJ)

The angle of twist θ (in radians) over a shaft of length L equals the torque T times L, divided by the product of the material's shear modulus G and the polar moment of inertia J. The calculator also converts θ to degrees for readability.

Polar moment of inertia (circular section)

J = πd⁴/32 (solid); J = π(D⁴−d⁴)/32 (hollow)

J measures how a circular cross-section's area is distributed around its center. For a solid shaft of diameter d, J = πd⁴/32. For a hollow shaft (tube) with outer diameter D and inner diameter d, subtract the inner circle's contribution: J = π(D⁴−d⁴)/32.

📖

Glossary

Key Terms Explained

7 terms

Torsion The twisting of an object caused by an applied torque about its longitudinal axis. Torsion produces shear stress that varies from zero at the center of a circular shaft to a maximum at its outer surface.

Angle of twist The angular rotation (θ) of one end of a shaft relative to the other, produced by an applied torque over the shaft's length. Measured in radians in the underlying formula, commonly reported in degrees.

Polar moment of inertia (J) A geometric property of a cross-section that measures its resistance to torsion, analogous to how the ordinary moment of inertia measures resistance to bending. Larger J means less shear stress and less twist for the same applied torque.

Torsional shear stress The shear stress produced within a shaft by an applied torque, denoted τ. It is maximum at the outer surface of the shaft (τ = Tr/J) and decreases linearly to zero at the central axis.

Shaft A rotating or twisted structural member, typically circular in cross-section, that transmits torque — for example a drive shaft, axle, or torsion bar.

Shear modulus (G) Also called the modulus of rigidity, a material property relating shear stress to shear strain, analogous to Young's modulus for normal stress and strain. Stiffer materials (higher G) twist less under the same torque.

Torque A rotational force, equal to a force applied at a distance from an axis of rotation (force × lever arm). Measured in newton-metres (N·m); torque is the load that causes torsion in a shaft.

👥

Scenarios

Real-World Examples

3 worked examples

🔩

Machine designer

Solid steel drive shaft

Shaft type Solid Diameter (d) 50 mm Torque (T) 500 N·m Length (L) 1 m Shear modulus (G) 79 GPa

J = π(50)⁴/32 ≈ 613,592 mm⁴. τ = Tr/J = (500,000 N·mm × 25 mm) / 613,592 mm⁴ ≈ 20.37 MPa — well below the roughly 150–200 MPa shear-yield range of common structural steels, so this shaft has comfortable margin at this load.

⚙️

Mechanical engineer

Longer, thinner shaft — angle of twist

Shaft type Solid Diameter (d) 40 mm Torque (T) 300 N·m Length (L) 1.5 m Shear modulus (G) 79 GPa

J = π(40)⁴/32 ≈ 251,327 mm⁴. θ = TL/GJ ≈ 0.0227 rad ≈ 1.30° of twist over the 1.5 m length. Reducing the diameter from 50 mm to 40 mm shrinks J by a factor of (40/50)⁴ ≈ 0.41, which is why a modest diameter drop produces a much larger twist for a similar torque and length.

🛠️

Weight-conscious designer

Hollow shaft, same outer diameter

Shaft type Hollow (tube) Outer diameter (D) 50 mm Inner diameter (d) 40 mm Torque (T) 500 N·m Length (L) 1 m Shear modulus (G) 79 GPa

J = π(50⁴−40⁴)/32 ≈ 362,265 mm⁴ — about 59% of the solid 50 mm shaft's J. For the same 500 N·m torque, τ ≈ 34.50 MPa and θ ≈ 1.00° — roughly 1.7× the stress and twist of the solid shaft in example 1, but with far less than 1.7× the material. This is the classic torsion trade-off: hollowing out a shaft removes the least-effective material (near the center, which barely resists torsion), so a hollow shaft is usually the better choice when weight matters and the higher stress and twist stay within limits.

📄

Deep Dive

Understanding Torsion, Torsional Shear Stress, and Angle of Twist

3 chapters · 4 min read

Torsion shows up anywhere a shaft, axle, or torsion bar transmits rotational force — drive shafts, motor shafts, bolted fasteners under tightening torque, and torsion springs. This calculator finds the two numbers engineers check first: the torsional shear stress at the shaft's surface, and how far the shaft twists under load.

How the Torsion Calculator works

The calculator first computes the polar moment of inertia J for the chosen circular cross-section — J = πd⁴/32 for a solid shaft, or J = π(D⁴−d⁴)/32 for a hollow tube, which subtracts the moment of inertia of the missing inner circle from the outer circle's. From J it derives the torsional shear stress τ = Tr/J at the outer surface (r is the outer radius, where stress is highest) and the angle of twist θ = TL/(GJ) over the entered shaft length. Both formulas assume the shaft is a straight, uniform circular section under pure torsion — the standard elastic torsion theory used for round shafts, axles, and torsion bars.

Inputs and what they mean

Diameters are entered in millimetres, torque in newton-metres, shaft length in metres, and shear modulus in gigapascals — the calculator converts internally so shear stress comes out in megapascals (MPa) and the angle of twist in both radians and degrees. Diameter has the largest effect on both results because it enters the polar moment of inertia to the fourth power: halving the diameter multiplies both stress and twist roughly sixteenfold for the same torque. Shear modulus reflects the material — steel (about 79–81 GPa) twists far less than aluminum (about 26 GPa) or plastics under the same load, which is why material choice matters as much as geometry for stiffness-critical shafts.

Limits and edge cases

This calculator covers pure torsion of a straight, uniform circular (solid or hollow) shaft in the elastic range — it does not account for combined bending-and-torsion loading, stress concentrations at keyways, shoulders, or holes, non-circular cross-sections (which require a different, shape-dependent torsion constant), or buckling of thin-walled tubes under torsion. For a hollow shaft, keep the wall reasonably thick relative to the diameter; very thin-walled tubes can buckle locally before reaching the shear stress this formula predicts. Always compare the computed shear stress against the material's shear yield strength (roughly 0.5–0.6 times its tensile yield strength for ductile metals) with an appropriate safety factor, and have a qualified engineer review any load-bearing shaft design.

❓

Questions

Frequently Asked Questions

6 questions

What is the formula for torsional shear stress?+

Torsional shear stress is τ = Tr/J, where T is the applied torque, r is the outer radius of the shaft, and J is the polar moment of inertia of the cross-section. Stress is maximum at the outer surface (r) and zero at the center.

What is the formula for the angle of twist?+

The angle of twist is θ = TL/(GJ) in radians, where T is torque, L is the shaft length, G is the material's shear modulus, and J is the polar moment of inertia. The calculator also converts θ to degrees.

How do you calculate the polar moment of inertia?+

For a solid circular shaft of diameter d, J = πd⁴/32. For a hollow shaft (tube) with outer diameter D and inner diameter d, J = π(D⁴−d⁴)/32 — the outer circle's moment of inertia minus the inner (missing) circle's.

How does a hollow shaft differ from a solid one?+

A hollow shaft has a smaller polar moment of inertia than a solid shaft of the same outer diameter, so it produces more stress and twist for the same torque — but it also uses much less material. Because material near a shaft's center barely contributes to torsional strength, hollow shafts are usually the more material-efficient choice when some increase in stress and twist is acceptable.

What units does this calculator use?+

Diameters are entered in millimetres, torque in newton-metres (N·m), shaft length in metres, and shear modulus in gigapascals (GPa). Results are shown in megapascals (MPa) for shear stress, degrees (and radians) for the angle of twist, and mm⁴ for the polar moment of inertia.

What makes a shaft stiffer against twisting?+

Stiffness against twist comes from a larger polar moment of inertia J (bigger diameter has by far the biggest effect, since J scales with diameter to the fourth power) and a higher shear modulus G (a stiffer material). A shorter shaft also twists less for the same torque, since the angle of twist is directly proportional to length.

Download a one-page PDF of your numbers instantly. Add your email to also get our occasional calculator tips — no spam, unsubscribe anytime. Privacy.

Torsion Calculator

Shaft

Result