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}

$T \in \left[10, 2000\right]$

where

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

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

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

$T \in \left[10, 2000\right]$

$T = 10$ $\tau = 1.587$

Along the curve

②

Domain of the variable — blue

T_{\min} = 10,\quad T_{\max} = 2000

$T \in \left[10, 2000\right]$

$T = 10$ $T = 10$

Along the curve

③

Response curve — green

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

$T \in \left[10, 2000\right]$

$T = 10$ $\tau = 1.587$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\tau}{\mathrm{d}T}

$T \in \left[10, 2000\right]$

$T = 22.52$ $\tau^{\prime} = 0.159$

Along the curve

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

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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