Question 8.4: Square keys are commonly used in shaft/hub connections as re......
Square keys are commonly used in shaft/hub connections as represented in Figure a. When a moment T is applied to the shaft, the key is subjected to shear forces. We can keep the shear forces from creating excess shear stresses by either: (i) increasing the key cross-section area; or (ii) increasing the length of the key. Suppose a shaft with radius r is transmitting a moment T through the key to the hub. Let the key cross-section side be: a and length be: \ell as in Figure b. Determine an expression for the shear stress τ in the key in terms of the shaft moment and the shaft and key dimensions.
Envision a free-body diagram of say the upper half of the key, housed in the hub. Figure c depicts the equivalent shearing force components V on the half key.
By inspection and comparison of Figures b and c we immediately see that V is:
V=\frac{T}{r} (a)
The shear stress τ is simply the shear force V divided by the sheared stress area A. Therefore, τ is:
τ=\frac{V}{A}= \frac{\left\lgroup\frac{T}{r} \right\rgroup }{\left(a\ell\right) }
or simply
τ=\frac{T}{ra\ell} (b)