MEC-01

Elastic shaft twist

θ = T L /(G J) for a circular shaft.

Reading speed

ShaftsShigleyISO 4156

Governing equation

\theta=\dfrac{T L}{G J},\quad J=\dfrac{\pi d^4}{32}

$d \in \left[8, 120\right]$

where

T: Torque (N·m)
L: Length (m)
d: Diameter (mm)
G: Shear modulus (GPa)
\theta: Twist angle (°)

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

\theta=\dfrac{T L}{G J},\quad J=\dfrac{\pi d^4}{32}

$d \in \left[8, 120\right]$

$d = 8$ $\theta = 356.2$

Along the curve

②

Domain of the variable — blue

d_{\min} = 8,\quad d_{\max} = 120

$d \in \left[8, 120\right]$

$d = 8$ $d = 8$

Along the curve

③

Response curve — green

\theta=\dfrac{T L}{G J},\quad J=\dfrac{\pi d^4}{32}

$d \in \left[8, 120\right]$

$d = 8$ $\theta = 356.2$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\theta}{\mathrm{d}d}

$d \in \left[8, 120\right]$

$d = 8.704$ $\theta^{\prime} = -120.7$

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-01 — Elastic shaft twist) is the form associated with Shigley · ISO 4156. Working symbols: $T$ , $L$ , $d$ , $G$ $\rightarrow$ $\theta$ . Integrating γ = r θ/L with Hooke τ = G γ and τ = T r / J recovers the torsion formula of Coulomb and Navier.

Purpose

Purpose: compute

\theta

from

T

L

d

G

in Mechanical via

\theta=\dfrac{T L}{G J},\quad J=\dfrac{\pi d^4}{32}

θ = T L /(G J) for a circular shaft. 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 = 250.000\,\mathrm{N·m}

L = 0.800\,\mathrm{m}

d = 30.000\,\mathrm{mm}

G = 80.000\,\mathrm{GPa}

, the governing relation

\theta=\dfrac{T L}{G J},\quad J=\dfrac{\pi d^4}{32}

yields

\theta = 1.801\,\mathrm{^{\circ}}

. Solid circular steel shaft. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Torque T250 N·mLength L0.8 mDiameter d30 mmShear modulus G80 GPa

Outputs

Twist angle \theta1.801 °

Reading speed

Watch on YouTube

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Narration of this film

Solid circular steel shaft.

Integrating γ = r θ/L with Hooke τ = G γ and τ = T r / J recovers the torsion formula of Coulomb and Navier.

Reading speed

Watch on YouTube