146 APPLIED MECHANICS 
146. Stresses in a Cranked Shaft.—A cranked shaft is a good 
example of a structure subjected to both bending and twisting, and 
particular attention should be directed to the fact that the crank pins 
are generally subjected to twisting as well as the shaft itself. The forces 
acting on a cranked shaft usually vary in magnitude and direction as the 
shaft revolves, and each part of the shaft must be designed to withstand 
the greatest straining action which may come upon it. 
A simple example will serve to indicate how the stresses in a cranked 
shaft are determined. Fig. 197 shows a cranked shaft turning in bear- 
Fig. 197. Fria. 198. 
ings at Aand E. The shaft has two crank pins B and D, the axes of 
which are in the same plane as AK, the axis of the shaft. The parts 
A, B, C, D, and E are each 3} inches in diameter. Fig. 198 shows how 
the shaft is loaded when the cranks are in a vertical position. There is 
a pure torque on the left-hand end of the shaft, a force P of 4800 Ibs. 
on the crank pin B, and a force Q of 6000 lbs. on the crank pin D; the 
lines of action of P and Q are perpendicular to the plane containing the 
axes of the crank pins. It is required to find the maximum stresses in 
the pins B and D, and in the shaft at C. 
Imagine the shaft produced to the points F and H directly opposite 
to the centres of the crank pins B and D respectively. The equilibrium 
of the shaft will not be affected by applying at F forces P, and P, acting 
in opposite directions and each parallel and equal to P. Nor will the 
equilibrium be disturbed by applying at H forces Q, and Q, each 
equal and parallel to Q, as shown. 
P and P, being equal and parallel forces acting in opposite directions 
form a couple, and since a couple can only have a turning effect, there — 
can be no pressure on the bearings due to these forces. The forces Q 
and Q, also form a couple. The reactions on the shaft at the bearings 
at A and E must therefore be due to the forces P, and Q,. Taking 
moments about A, R, is found to be 3120 Ibs., and taking moments 
about E, R, is found to be 1920 Ibs. | 
Consider . the straining actions on the crank pin D. The -. 
force to the right of D is R,, and this produces a bending moment 
= 3120 x 8 = 24,960 inch-lbs., anda twisting moment = 3120 x 6 = 18,720 | 
inch-lbs. Using the Rankine formula, the equivalent twisting moment 
at D due to these is 
24960+ ,/24960? + 18720? = 56,160 inch-lbs. 
If f is the maximum stress in the pin D, then 782 3f = 56,160, from 
which f= 6671 Ibs. per square inch. 
Consider next the straining actions on the shaft at C. Taking the 
