What is the polar moment of inertia for a solid shaft?

For a solid circular shaft of diameter d, the polar moment of inertia J = pi*d^4/32. For a hollow shaft with outer diameter D and inner diameter d, J = pi*(D^4 - d^4)/32.

What is the angle of twist formula?

The angle of twist theta (in radians) = T*L/(G*J), where T is applied torque (N*m), L is shaft length (m), G is shear modulus (Pa), and J is polar moment of inertia (m^4). Convert to degrees by multiplying by 180/pi.

What is a typical allowable shear stress for a steel shaft?

For carbon steel shafts, allowable shear stress typically ranges from 40 to 70 MPa for general power transmission. High strength steels may allow 80 to 120 MPa. A safety factor of 2 to 3 is commonly applied against the material yield shear strength (approximately 0.577 times tensile yield strength per von Mises criterion).

What is the difference between a solid and hollow shaft under torsion?

A hollow shaft has a higher strength-to-weight ratio than a solid shaft of equal outer diameter. Material near the centre of a solid shaft contributes little to torsional strength (stress is zero at the centre axis), so removing it via a hollow bore saves weight with minimal strength reduction. Hollow shafts are preferred for weight-critical applications like vehicle driveshafts.

Shaft Torsion & Angle of Twist Calculator – Solid & Hollow Shafts | CalcEngines

Mechanical Engineering Tools · CalcEngines

Shaft Torsion & Angle of Twist Calculator

Calculate maximum shear stress and angle of twist for solid or hollow circular shafts. Uses the torsion formula T/J = τ/r = Gθ/L.

τ = Tr/J θ = TL/GJ Solid & hollow 10 materials Safety factor check SI & Imperial

Shaft Cross-Section

Quick Presets

Shaft Dimensions

Outer Diameter D

Outside diameter of shaft

Inner Diameter d

Enable hollow shaft above

Shaft Length L

Length between fixed ends

Loading & Material

Applied Torque T

Applied torsional moment

Material G = 80 GPa · τ_allow = 60 MPa

Shear Modulus G

GPa

Auto-filled from material

Allowable Shear Stress τ_allow

MPa

Auto-filled · override if needed

Allowable Twist Angle

°/m

Typical limit: 0.5°–1.0° per metre

Imperial

Key Results

Enter shaft dimensions, torque, and material above, then click Calculate.

Cross-Section & Geometry

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Angle of Twist Results

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All Stress & Stiffness Values

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Formula Reference

Quantity	Formula	Variables
Polar moment J (solid)	J = πD⁴ / 32	D = outer diameter
Polar moment J (hollow)	J = π(D⁴ − d⁴) / 32	D = outer, d = inner diameter
Max shear stress τ	τ = T × r / J	r = D/2, T in N·m, J in m⁴
Angle of twist θ	θ = TL / (GJ)	L in m, G in Pa, result in radians
Torsional rigidity	GJ	Units: N·m²
Section modulus Z_p	Z_p = J / r = 2J / D	Z_p = πD³/16 (solid)
Torsion equation	T/J = τ/r = Gθ/L	Full torsion formula
Twist per unit length	θ/L = T / (GJ)	Result in rad/m → convert ×180/π for °/m

Core torsion relation: T/J = τ/r = Gθ/L · Shear stress is maximum at the outer surface · Zero at the neutral axis (centre of solid shaft)

Common Questions

What is the torsion formula for a shaft? +

The torsion formula is T/J = τ/r = Gθ/L. From this: maximum shear stress τ = Tr/J, and angle of twist θ = TL/(GJ), where T is torque (N·m), J is polar moment of inertia (m⁴), r is outer radius (m), G is shear modulus (Pa), L is length (m).

How do I calculate the polar moment of inertia? +

For a solid shaft: J = πD⁴/32. For a hollow shaft: J = π(D⁴ − d⁴)/32, where D is the outer diameter and d is the inner diameter. Note that J scales with diameter to the fourth power — doubling the diameter increases J by 16 times.

What is a typical allowable twist angle for a shaft? +

A common design rule is to limit twist to 0.5° to 1.0° per metre of shaft length for power transmission shafts. Precision shafts (machine tool spindles, instrument drives) may require 0.1°/m or less. Flexible couplings can tolerate more. Enter your allowable twist rate above to check compliance.

Why is a hollow shaft more efficient than a solid shaft? +

Material at the centre of a solid shaft carries very little shear stress (stress is zero at the axis and maximum at the surface). Removing that central material via a bore saves significant weight with minimal strength reduction. A hollow shaft with d/D = 0.6 retains about 87% of the solid shaft’s torsional strength at only 64% of its weight.

What is torsional rigidity and why does it matter? +

Torsional rigidity is the product GJ (N·m²). It quantifies a shaft’s resistance to twisting — a higher GJ means less angular deflection for the same applied torque. It depends on both the material (G) and the cross-section geometry (J). Angle of twist θ = TL/(GJ), so doubling GJ halves the twist.