MecMovies 4.0 : M6.19: Shear Stresses in Coaxial Shafts

Diagram shows a shaft A B with an aluminum segment of length 1,000 millimeters and a brass shaft in the inside. Both the shafts are centered on the central axis x. A torque T is applied at B. The radial and longitudinal lines are shown on the shaft. The deformation of the shaft is shown with a counterclockwise segment of angle phi subscript 1 = phi subscript 2 = phi subscript B, angle of twist.

Substitute the torque-twist relationships...

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

...into the geometry of deformation relationship...

$ϕ_{1} = ϕ_{2}$

...to obtain the compatibility equation:

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

This equation is based upon the deformations that will occur in the composite shaft, expressed in terms of the unknown internal torques $T_{1}$ and $T_{2}$