# 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.

#### Angular twist

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

$$\varphi=\frac{T\cdot l}{G\cdot J_T}$$

$J_T$ is the torsion constant. It is equal to the polar moment of inertia $I_z$ 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 and maximum stress are given in the table below.

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

$$G=\frac E{2\left(1+v\right)}{\approx}0.38E \textsf{ (For most metals)}$$

#### Rotational stiffness

The rotational stiffness of a torsionally loaded beam is:

$$K_z=\frac{T}{\varphi}=\frac{G\cdot J_T}{l}$$

#### Maximum shear stress

For a torsionally loaded beam with a circular cross-section, the maximum shear stress can be calculated with:
$$\tau_{max}=\frac{Tr}{J_T}$$

For non-circular cross-sections the equations in the table below can be used.

Cross section ConditionTorsion constant JTMaximum shear stress
$$J_T=I_Z=\frac{\pi}{2}r^4$$$$\tau_{max}=\frac{2T_{max}}{{\pi}r^3}$$
$$J_T=I_Z=\frac{\pi}{2}(r_o^4-r_i^4)$$$$\tau_{max}=\frac{2T_{max}r_o}{\pi\left(r_o^4-r_i^4\right)}$$
$$h=w$$$$J_T=\frac{9}{64}w^4$$$$\tau_{max}=\frac{4.808T}{w^3}$$
$$h\geq\ w$$$$J_T=\frac{1}{16}hw^3\left(\frac{16}{3}-3.36\frac{w}{h}\left(1-\frac{w^4}{12h^4}\right)\right)$$
$$\approx0.33\cdot\ hw^3-0.21\cdot\ w^4+0.017\cdot\left(\frac{w^2}{h}\right)^4$$
$$\tau_{max}=\frac{3T}{hw^2}\left(1+0.6095\frac{w}{h}+0.8865\left(\frac{w}{h}\right)^2-1.8023\left(\frac{w}{h}\right)^3+0.9100\left(\frac{w}{h}\right)^4\right)$$
For thin walled structures$$J_t=\frac{2t^2\left(w-t\right)^2\left(h-t\right)^2}{\left(w+h\right)t-2t^2}$$$$\tau_{average}=\frac{T}{2t(w-t)(h-t)}$$

Note: stress is nearly uniform if t is small. There will be higher stresses at inner corners unless fillets of fairly large radius are provided.

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