MecMovies 4.0 : M6.2: Torsion Theory for Circular Sections

Four points j k m n are marked on a grid system. J k m n forms a square of side d x. After torsion is applied the points k and m move to points k prime m prime. The length of each side remains d x.

A similar argument could be developed for any interior surface located at a radial distance $ρ$ from the shaft centerline. The shear strain $γ$ on an interior surface at $ρ$ is expressed as:

$γ = \frac{ρϕ}{L}$

We can combine...

$γ = \frac{ρϕ}{L} and γ_{\max} = \frac{cϕ}{L}$

by solving for $ϕ / L$ ...

$\frac{ϕ}{L} = \frac{γ}{ρ} = \frac{γ_{\max}}{c}$

to give...

$γ = \frac{ρ}{c} γ_{\max}$

This equation shows that shear strains in a circular shaft vary linearly with the radial distance $ρ$ from the centerline. The shear strain is zero at the centerline and increases linearly until it reaches a maximum value on the outermost surface of the shaft.