MechClarityMechanical Engineering Calculators

Shafts · Elastic torsion of circular sections

Shaft Torsional Stress / Angle of Twist Calculator

Mechanical Engineering Calculators for nominal torsional shear stress, elastic angle of twist, polar moment, section modulus, and stiffness of solid or hollow circular shafts.

Reference calculator #020

Enter circular shaft geometry and elastic data

Inputs stay in your browser. Values are normalized to canonical units before calculation.

Choose a solid round shaft or a concentric hollow round shaft.

Enter the constant torque magnitude acting through the evaluated segment.

Use the constant outside diameter over the entered shaft length.

This value is ignored for a solid shaft; use d = 0 for a clean shareable state.

Enter the length over which the elastic angle of twist is required.

Verify G for the actual material condition and temperature. The default is illustrative, not a material selection.

Display nominal maximum shear stress in Pa, MPa, psi, or ksi.

Display elastic twist in radians, degrees, or arcminutes.

Calculated output

Results

shaft-torsional-stress-angle-of-twist/1.0.0
Nominal maximum shear stress τ_max39.788736 MPa
Elastic angle of twist θ
1.437406 °
Torsional stiffness k_t
19930.263794 N·m/rad
Polar second moment of area J
251327.412 mm⁴
Polar section modulus Z_p
12566.371 mm³
Inner-to-outer diameter ratio d / D
0%

Valid nominal elastic circular-shaft torsion result

Circular shaft torsion diagramA circular shaft under equal and opposite torque, with a cross-section showing outer and inner diameters and labels for nominal shear stress and elastic twist.TTL = 1000 mmApplied torque: 500 N·mEntered G = 79.3 GPaD = 40 mmd = 0 mmSolid circular sectionτ_max = 39.788736 MPaθ = 1.437406 °k_t = 19930.263794 N·m/rad
Nominal linear-elastic torsion of a uniform circular shaft. Stress concentrations, combined loading, fatigue, yield, and allowable-twist selection are not evaluated.
Scope and assumptions
  • The shaft is straight, prismatic, and circular with constant outer and inner diameters over the entered length.
  • The torque magnitude is constant over the evaluated shaft segment and the material is homogeneous, isotropic, and linear elastic at the entered shear modulus.
  • The calculation uses Saint-Venant torsion for solid or hollow circular shafts and reports nominal outer-surface shear stress.
  • The entered shear modulus is verified for the actual material condition and operating temperature; no material database or default-material claim is applied.
  • Keyways, splines, cross-holes, shoulders, grooves, fillets, fits, surface condition, residual stress, and other stress concentrations are excluded.
  • Yield, fatigue, combined bending and axial load, shock, reversal, critical speed, stability, vibration, deflection limits, and shaft selection are excluded.

Calculation engine: shaft-torsional-stress-angle-of-twist/1.0.0

Circular-shaft torsion formulas

For a concentric hollow circular shaft, the polar second moment of area is:

J = π(D⁴ − d⁴) / 32

Set d = 0 for a solid shaft. The nominal outer-surface shear stress and elastic angle of twist are:

τ_max = T(D/2) / J     θ = TL / (JG)

The calculator also reports torsional stiffness k_t = GJ/L and polar section modulus Z_p = J/(D/2).

Symbol Meaning Canonical calculation unit
T Applied torque magnitude N·m
D Outer diameter m inside the formula
d Concentric inner diameter m inside the formula
L Uniform shaft length m inside the formula
G Entered shear modulus Pa
J Polar second moment of area m⁴, displayed in mm⁴
τ_max Nominal maximum shear stress Pa
θ Elastic angle of twist rad

Worked solid-shaft example

Use the default inputs: T = 500 N·m, D = 40 mm, d = 0, L = 1,000 mm, and an illustrative entered G = 79.3 GPa.

  1. Convert geometry to metres: D = 0.04 m and L = 1 m.
  2. Calculate J = π(0.04)⁴/32 = 2.513274 × 10⁻⁷ m⁴, or 251,327.412 mm⁴.
  3. Calculate τ_max = 500(0.02)/J = 39.788736 MPa.
  4. Calculate θ = 500(1)/(J × 79.3 × 10⁹) = 0.025087 rad = 1.437406°.
  5. Calculate k_t = GJ/L = 19,930.263794 N·m/rad.

The example demonstrates arithmetic only. The entered G is not a material recommendation, and the result is not compared with any allowable stress or allowable twist.

Solid versus hollow circular shafts

For the same outer diameter and torque, increasing the concentric bore reduces J and Z_p, which increases nominal maximum stress and twist. The exact circular-tube equation is used; no thin-wall approximation is substituted.

When “solid circular shaft” is selected, the engine uses d = 0. A nonzero inner-diameter field is ignored and reported as a warning so a shared URL cannot silently imply hollow geometry. When “hollow circular shaft” is selected, d must be positive and smaller than D.

Enter and verify shear modulus

Shear modulus affects elastic twist and stiffness but not the nominal stress from Tρ/J. Verify G from current material data for the actual alloy, temper, heat treatment, temperature, and condition. Do not infer yield strength, endurance strength, or allowable twist from G.

The unit engine converts Pa, MPa, GPa, psi, and ksi through canonical pascals. Display rounding occurs only after calculation.

Engineering scope and limitations

This calculator applies to one straight, uniform, solid or concentric hollow circular shaft under constant torque magnitude in linear-elastic Saint-Venant torsion. It excludes:

  • noncircular, open, thin-walled multicell, composite, anisotropic, tapered, stepped, or curved shafts;
  • keyways, splines, cross-holes, shoulders, grooves, threads, press fits, welds, fillets, residual stress, and surface finish effects;
  • yielding, plastic torsion, fatigue, mean and alternating stress, stress-concentration factors, notch sensitivity, and safety factors;
  • combined bending, axial, transverse, or thermal loads; torque shock, reversal, start-up, braking, or measured load spectra;
  • lateral deflection, critical speed, whirl, torsional vibration, resonance, damping, couplings, bearings, alignment, and system compliance;
  • material selection, allowable stress, allowable twist, dimensional tolerances, inspection acceptance, manufacturing feasibility, certification, or final engineering approval.

Use verified material data, the governing design code or company method, and qualified engineering review when shaft failure or excessive twist can affect safety, reliability, or adjacent components.

Frequently asked questions

What is the torsional shear stress formula for a solid circular shaft?

For a solid circular shaft, J = πD⁴/32 and the nominal maximum surface shear stress is τ_max = T(D/2)/J, which simplifies to 16T/(πD³).

How is a hollow shaft different in this calculator?

A hollow shaft uses J = π(D⁴-d⁴)/32. The maximum nominal shear stress still occurs at the outer surface, while angle of twist is calculated from TL/(JG).

Does the shear modulus affect torsional stress?

Not in this linear circular-shaft stress equation. G affects angle of twist and torsional stiffness, while T and section geometry set the nominal stress.

Does this calculator include a keyway or shoulder stress concentration?

No. It reports nominal stress for a uniform circular section. Apply an appropriate verified stress-concentration and fatigue method separately for keyways, splines, shoulders, cross-holes, grooves, or other discontinuities.

Can the result approve a shaft diameter or material?

No. The calculator does not compare stress with yield or fatigue allowables, choose a material, set an allowable twist, or evaluate combined loading, dynamics, critical speed, manufacturing details, or safety factors.

References and review status

Reviewed . References support the circular-shaft elastic torsion equations and unit conversions; they do not imply shaft approval, material verification, or standards conformity.