CIV-07

Circular shaft torsion

Shear τ = T r / J and twist θ = T L /(G J).

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TorsionCoulomb / NavierEN 1993-1-1

Governing equation

\tau=\dfrac{T r}{J},\quad J=\dfrac{\pi d^4}{32}

where

T: Torque (kN·m)
d: Diameter (mm)
L: Length (m)
G: Shear modulus (GPa)
\tau: Max shear (MPa)
\theta: Twist (°)

Lecture brief

Historical brief

From Euler’s 1744 elastica and Navier’s beam theory to Mohr’s circle and transformed-section RC, structural mechanics grew as a closed-form craft before finite elements. The lab keeps those governing lines for buckling, flexure, joints and influence. This sheet (CIV-07 — Circular shaft torsion) is the form associated with Coulomb / Navier · EN 1993-1-1. Working symbols: $T$ , $d$ , $L$ , $G$ $\rightarrow$ $\tau$ , $\theta$ . For circular sections Saint-Venant torsion reduces to a linear stress field; J = π d⁴/32 is the polar moment.

Purpose

Purpose: compute

\tau

\theta

from

T

d

L

G

in Structural & civil via

\tau=\dfrac{T r}{J},\quad J=\dfrac{\pi d^4}{32}

Shear τ = T r / J and twist θ = T L /(G J). Use it when a real structural & civil question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

T = 12.000\,\mathrm{kN·m}

d = 80.000\,\mathrm{mm}

L = 2.000\,\mathrm{m}

G = 80.000\,\mathrm{GPa}

, the governing relation

\tau=\dfrac{T r}{J},\quad J=\dfrac{\pi d^4}{32}

yields

\tau = 119.366\,\mathrm{MPa}

\theta = 4.274\,\mathrm{^{\circ}}

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

Calculator

Inputs

Torque T12 kN·mDiameter d80 mmLength L2 mShear modulus G80 GPa

Outputs

Max shear \tau119.366 MPa
Twist \theta4.274 °

Reading speed

Watch on YouTube

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Narration of this film

Solid circular shaft, elastic, uniform T.

For circular sections Saint-Venant torsion reduces to a linear stress field; J = π d⁴/32 is the polar moment.

Reading speed

Watch on YouTube