C3.1 Torsion Formula

Torsion is basically the stress due to torque. Many structures experience torque (e.g. torque wrench, car shaft, etc) and therefore it is important to quantify the stress caused by torque to help us design safe structures.

Torsion applies shear rather than normal stress, as seen in the illustration below:

torque causing shear in structures rather than normal stress

C3.1 Torsion Formula

Torsion is basically the stress due to torque. Many structures experience torque (e.g. torque wrench, car shaft, etc) and therefore it is important to quantify the stress caused by torque to help us design safe structures.

Torsion applies shear rather than normal stress, as seen in the illustration below:

torque causing shear in structures rather than normal stress

Torsion formula

The torsional shear stress can be calculated using the following formula:

Torsion formula and torsion shear stress distribution along radius

Note:

T is the internal torque at the region of interest, as a result of external torque loadings applied to the member (units: Nm)
r is the radius of the point where we are calculating the shear stress (units: m or mm)
J is the polar moment of inertia for the cross-section (units: m⁴ or mm⁴)

Notice that the higher the radius r, the higher the torsional shear stress. Therefore at r_max, we have τ_max. We usually denote r_max as c:

Maximum torsional shear stress formula

Polar moment of inertia, J

This variable basically measures the resistance to torsional loading. It is a function of the geometry (not mass); the larger the cross-section, the bigger the polar moment of inertia.

We mostly deal with solid or hollow circular cross-sections:

Polar moment of inertia for solid and hollow circular cross-sections

Sign convention

We use the right-hand rule as our positive sign convention. First we define an axis direction, then all torque directions are determined according to the axis and the right-hand rule:

Right-hand rule for torque

Let’s look at an example now.

Torsion formula

The torsional shear stress can be calculated using the following formula:

Torsion formula and torsion shear stress distribution along radius

Note:

T is the internal torque at the region of interest, as a result of external torque loadings applied to the member (units: Nm)
r is the radius of the point where we are calculating the shear stress (units: m or mm)
J is the polar moment of inertia for the cross-section (units: m⁴ or mm⁴)

Notice that the higher the radius r, the higher the torsional shear stress. Therefore at r_max, we have τ_max. We usually denote r_max as c:

Maximum torsional shear stress formula

Polar moment of inertia, J

This variable basically measures the resistance to torsional loading. It is a function of the geometry (not mass); the larger the cross-section, the bigger the polar moment of inertia.

We mostly deal with solid or hollow circular cross-sections:

Polar moment of inertia for solid and hollow circular cross-sections

Sign convention

We use the right-hand rule as our positive sign convention. First we define an axis direction, then all torque directions are determined according to the axis and the right-hand rule:

Right-hand rule for torque

Let’s look at an example now.