## Q. 8.3

Two circular, equal diameter, transmission shafts are to be coupled end-to-end as in Figure a. Suppose the coupling joint is to consist of a collar and pins as illustrated in Figure b. Let the shafts each have radius R and let the pin radii be r, as in Figure c. Let an axial moment (or “torque”) T be transmitted by the shafts through the coupling joint. Assuming ideal geometry, determine the stress exerted on the pins.   ## Verified Solution

From the end view in Figure b, we see that during the torque transmission the shaft and collar will exert shear forces at the ends of the pins as in Figure d. Assuming ideal geometry, and then considering a moment balance about the shaft axis, we see that the shear force magnitude V is related to the transmission torque T by the simple expression:

$2RV=T or V=\frac{T}{2R}$   (a)

Finally, the shear stress τ on the pin is simply the shearing force divided by the pin cross-section area A. That is:

$τ=\frac{V}{A}=\frac{T}{2\pi r^{2}R }$    (b)

Comment

Equation (b) presents a design formula for the pin radius r given a transmission torque, shaft radius, and shear stress strength. Since in practice the coupling geometry is likely to be less than ideal, a safer design formula is obtained by simply assuming all the shear force concentration at one end of the pin. Under this assumption Eq. (b) is replaced by:

$τ=\frac{T}{2\pi r^{2}R }$      or       $r=\left[\frac{T}{\pi r τ} \right] ^{1/2}$ 