MecMovies 4.0 : M6.21: Maximum Torque for Composite Shaft

Diagram shows an aluminum shaft A B with a support at A, marked 1 of length 4,000 millimeters and a brass shaft B C with a support at C, marked 2 of length 3,000 millimeters with a flange in between at B. The shafts are centered on the x axis. 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 flange $B$ 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

$\frac{T_{1} L_{1}}{J_{1} G_{1}} + \frac{T_{2} L_{2}}{J_{2} G_{2}} = 0$

to obtain the following:

$\frac{τ_{1} L_{1}}{c_{1} G_{1}} + \frac{τ_{2} L_{2}}{c_{2} G_{2}} = 0$