## Q. 8.5

Review Example Analysis 4. Suppose we have a 2 in. diameter shaft with a ½ in. square key. Suppose the key length is 1½ in. Suppose further that the maximum shear stress $τ_{max}$ of the key material is 30,000 psi. Determine the maximum shaft moment $T_{max}$ which can be transmitted through the joint.

## Verified Solution

From Eq. (b) of the solution of Example Analysis 34 we have the relation between the shaft moment T and the shear stress τ:

$τ=\frac{T}{ra\ell } or T=ra\ellτ$    (a)

where as before r, a, and$\ell$ are the shaft radius, key side width, and key length, respectively.
By substituting the values of the given data into Eq. (a) we obtain the maximum shaft moment
$T_{max}$ as:

$T_{max} = (1)(0.5)(1.5)(30)(10)^{3} = 22,500$ in lb

or

$T_{max} = 1875$ ft lb     (b)

Comment

The result of Eq. (b) is a theoretical maximum. The shear stress maximum is an ideal material value and the formula of Eq. (a) is developed assuming ideal geometry. Therefore, for actual designs, we should limit the shaft moment transmission to only a fraction of the value given by Eq. (b).