MEC-09

Circular shaft torsion

τ = T r / J, θ = T L / (G J). Elastic circular shafts.

Reading speed

StrengthSaint-Venant

Governing equation

\tau=\dfrac{T r}{J}

where

T: Torque (N·m)
r: Radius (mm)
J: Polar inertia (mm^4)
\tau: Shear stress (MPa)

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-09 — Circular shaft torsion) is the form associated with Saint-Venant. Working symbols: $T$ , $r$ , $J$ $\rightarrow$ $\tau$ . J = π d⁴/32. Plane sections warp not in the circular case.

Purpose

Purpose: compute

\tau

from

T

r

J

in Mechanical via

\tau=\dfrac{T r}{J}

τ = T r / J, θ = T L / (G J). Elastic circular shafts. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

T = 200.000\,\mathrm{N·m}

r = 20.000\,\mathrm{mm}

J = 126000.000\,\mathrm{mm^4}

, the governing relation

\tau=\dfrac{T r}{J}

yields

\tau = 31.75\,\mathrm{MPa}

. A shaft, a torque, a helical fibre. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Torque T200 N·mRadius r20 mmPolar inertia J126000 mm^4

Outputs

Shear stress \tau31.75 MPa

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A shaft, a torque, a helical fibre.

J = π d⁴/32. Plane sections warp not in the circular case.

Reading speed

Watch on YouTube