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

scene 14 of 21

To find the resultant moment for the shaft, we've got to add up all of the elemental moments that exist on the entire cross section.

To do this, we'll integrate $dM$ over the shaft area $A$ .

$\int dM = \int_{A} ρ τ dA$

Previously, we developed a relationship between $τ$ and $τ_{\max}$ :

$τ = \frac{ρ}{c} τ_{\max}$

Substitute this expression for $τ$ into the integral:

$\int dM = \int_{A} \frac{τ_{\max}}{c} ρ^{2} dA$