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 (Hooke's Law).

### 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 opposong reaction torque on the fixed rear face.

We consider an "elemental tube" inside the cylinder at radius**
r **and width **dr**. Undere 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 shows the shear stress** τ **on the end
face varies linearlly with radius r. ** τ**_{max}
is at the outer surface of the cylinder.

For bending we established a relation between bending moment M
and normal stress σ. 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}.

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

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

### Normal stresses in torsion

Consider 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
load.

Stresses **τ** on the sides of this element are pure
shear (no normal stresse). This state is represented by the
blue **τ** axis of Mohr's circle below which shows that
**τ **on the element** **is the**
**maximum shear stress.

Now consider the element rotated 45° such that the sides of the
element are parallel with lines ac and bd. This state is
represented by the red **σ** axis of Mohr's circle
(remembering that the rotation of the circle diameter is twice
that 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 plane bd
and compressive stress **σ**_{2} normal to
plane ac.

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.