Torsion

Moments vs Torque

Torques act in the longitudinal axis (along the length of a member) whereas a moment acts in other directions

Torsion of Circular Sections

Radi

In this document outer radius is referred to as r whereas some arbitrary radius is referred to as rho ($\rho$)

T = torque l = length p = arbitrary radius G = modulus of rigitity r = radius at outer surface J = second polar moment of radius

Torque Vectors only exist for circular sections

Assumptions

The equations below are only applicable for:

• Pure torque
• No discontinuities
• Materials obey Hooke’s law
• Adjacent cross sections originally plane and parallel remain plane and parallel
• Radial lines remain straight
  • Depends on axisymmetry, so does not hold true for noncircular cross sections.
    • Consequently, only applicable for round cross sections

Important Formulas

Circular Sections

Angle of twist:

Torque

Note that this formula can also be used to calculate Torque!

$\theta =\frac{Tl}{GJ}\left[rad\right]$

Hint

This formula is used for finding G experimentally. You can apply known T to a bar with a known length of L and a known twist.

Shear Stress:
$\tau =\frac{T\rho }{J}$ ${\tau }_{max}=\frac{Tr}{J}$
For solid round sections:
$J=\frac{\pi d^4}{32}$
For solid hollow sections:
$J=\frac{\pi }{32}\left(d_o^4-d_i^4\right)$
Shear Strain The shear strain on the surface of a circular bar (shaft)
$\gamma =\frac{r\theta }{L}$
If you want to measure this gamma at any other radius then:
$\gamma =\frac{\rho \theta }{L}$
Since tau and rho have a linear relationship, then tau/rho is constant

Multiple Torques

If you have a bar subject to multiple torques, then you need to apply method of sections to find the interval torques of each section.

Torsion of Rectangular Sections

Maximum shear stress occurs at the middle of the longest side, b
Shear stress is 0 at the corners

${\tau }_{\max }=\frac{T}{\alpha b{c}^2}\approx \frac{T}{b{c}^2}\left(3+\frac{1.8}{\frac{b}{c}}\right)$
Angle of Twist:
$\theta =\frac{Tl}{\beta b{c}^{3}G}$
Where the values of alpha and beta come from a table.

For a Circular Section:

${\varphi }_{AB}={\frac{T_{AB}{L}_{AB}}{GJ}}_{}$

Power

Torsion can produce Power through torque

Examples

See Goodnotes

General Strategy for Most Questions:

1. Cut sections through shafts and perform static equilibrium analyses to find torque loadings
2. Apply elastic torsion formulas to find min and max stress on a shaft
3. Given allowable shearing stress and applied torque, invert the elastic torsion formula to find the required variable (diameter)