# 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.

### Torsion constant

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  