In this tutorial we examine stresses induced when a cylindrical component, such as a machine shaft, is subjected to torsion generated by a torque load. This analysis is valid only for elastic strains (i.e. complies with Hooke's Law). We consider in turn shear and normal stresses.

### Shear stresses in torsion

Consider the cylindrical section with radius **R** and length **L** shown below.

The rear face of the section is fixed and an anti-clockwise torque **T** is applied to the fromt face. There is an opposing reaction torque on the fixed rear face.

We consider an "elemental tube" inside the cylinder at radius **r** and width **dr**. Under the action of T the element twists and experiences shear strain **γ**. Correspondingly, the front face of the element twists through angle **θ**.

The diagram below shows the shear stress **τ** on the end face varies linearlly with radius **r**. **τ _{max}** is at the outer surface of the cylinder.

In a previous tutorial we established a relation between bending moment **M** and normal stress **σ** for bending. We now do likewise of torque **T** and shear stress **τ**.

By equating the torque induced on the face of the cylinder by the shear stress to the applied torque it is shown that:

where **J** is the **second polar moment of area** about the central axis of a cylinder. Units of J are m^{4}.

*Digression* (Note that the above integral is not the general definition of J, which is the double integral:

) *to continue*:

It can also be shown that the above relation applies to a hollow cylindrical shaft. For a hollow cylinder:

### Normal stresses in torsion

Consider in the diagram below a square element on the outer surface of a cylinder such that the top and bottom sides of the element are parallel to the axis of the cylinder. The cylinder is sbject to a torsional (torque) load.

Stresses **τ** on the sides of this element τ_{xy} and τ_{yx} are pure shear (no normal stresses). This state is represented by the blue τ axis forming the base diameter of Mohr's circle (previous tutorial) shown below with τ_{xy} = τ_{yx} being the maximum shear stress.

Now consider the element rotated 45° such that a line d^{/}b^{/} is parallel with the axis of the cylinder. This state is represented by the red σ axis forming the base diameter of Mohr's circle (remembering that rotation of the base diameter of Mohr's circle is twice the rotation of the stress element) which shows that σ_{1} and σ_{2} are principal stresses equal in magnitude to τ_{max}. Tensile stress σ_{1} acts normal to faces b^{/}c^{/} and a^{/}d^{/} and compressive stress σ_{2} normal to faces a^{/}b^{/} and d^{/}c^{/}.

A consequence of the principal stresses being on these planes is that a component subject to torsion and made from a brittle material will generally fail in tension at an angle of 45° to the longitudinal axis.

Next: Buckling of columns

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