# BEAM THEORY: TORSION

## Engineering Fundamentals

### INTRODUCTION

When a straight beam is subjected to an axial moment, each cross section twists around its torsional center. Shear stresses occur within the cross sectional planes of the beam.

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### Angular twist

For a torsionally loaded beam, the angular twist is described by:

is the shear modulus. The relation between the shear modulus and the elastic modulus is defined by the following formula:

(For most metals)

### Rotational stiffness

The rotational stiffness of a torsionally loaded beam is:

For a torsionally loaded beam, the maximum torque load can be calculated with:

is the torsion constant. It is equal to the polar moment of inertia if the cross section is circular.

For non-circular cross sections warping occurs which reduces the effective torsion constant. For these shapes, approximate solutions of the torsion constant are given in the table below.

Cross section

Torsion constant

Cross section

Torsion constant

Cross section

Torsion constant

Cross section

Torsion constant

Cross section

Torsion constant

With h>w

Cross section

Torsion constant