MecMovies 4.0 : M6.22: Maximum Torque Applied to Coaxial Shaft

Diagram shows a shaft A B with aluminum segment marked 1 of length 400 millimeters inside which is a brass shaft marked 2. Both the shafts are centered on the central axis x. A torque T is applied at B.

In this problem, the allowable shear stresses in shafts $(1)$ and $(2)$ are given, and we must compute the maximum external torque that can be applied to free end $B$ without exceeding either one of those limits.

The relationship between torque and shear stress is:

$τ = \frac{Tc}{J}$

Using this relationship, write expressions for $T_{1}$ and $T_{2}$ :

$T_{1} = \frac{τ_{1} J_{1}}{c_{1}} and T_{2} = \frac{τ_{2} J_{2}}{c_{2}}$

Substitute these expressions into the compatibility equation

$\frac{T_{1} L_{1}}{J_{1} G_{1}} = \frac{T_{2} L_{2}}{J_{2} G_{2}}$

to obtain the following:

$\frac{τ_{1} L_{1}}{c_{1} G_{1}} = \frac{τ_{2} L_{2}}{c_{2} G_{2}}$