144 APPLIED MECHANICS 
of the bending moment B and twisting moment T. T, is called the 
equivalent twisting moment. 
In using the formula T,=B+ ,/B?+T?= ree” it must be remem- 
bered that 7 is not a shear stress, but a tensile stress. This is a point 
which is frequently misunderstood, and it may be well to restate the case 
as follows:—When a shaft of 
diameter d is subjected to a pure A— as B A ao B 
twisting momentT, then T = ne sf, 4 ¢ ais | = 1 
where f is the shear stress on the ! f if Pq . iP 
faces AB, BC, CD, and DA } I 4 i 
(Fig. 193) of a square element ei Esa r, a 
of the skin of the shaft, AB Fic, 193. Fic. 194. 
being parallel to the axis, but 
f is also the tensile stress on AC. If a bending moment B 
is added, this shifts the plane of tensile stress to EF (Fig. 
194), and increases its value from f to 7, the value of 7 being 
(B+ /B?+T?}. Hence it may be said that in the formulae 
T= itt and T, = ee jf and r are both tensile stresses. 
It is easy to show that if B, is a bending moment which will produce 
the same maximum normal stress as a bending moment B and a twisting 
moment T acting together, then B,=4B+4,/B?+T% In applying this 
to a shaft, B, must be equated to x50 , where f is the maximum normal 
stress, 
145. Maximum Shear Stress due to Combined Twisting and 
Bending. — Let ABCD 
(Fig. 195) be an in- 
definitely small square N 
prism of the material + " 
of a shaft of diameter P- 
d which is subjected a yj 
to a bending moment + "2 
B and a twisting mo- D 
ment T, the face ABCD ; 
being on the surface of wifes ae 
the shaft in the neighbourhood of the greatest bending stress. AB 
is parallel to the axis of the shaft. The faces AD, CB, AB, and 
CD are subjected to shear stress f= — The faces AD and CB are 
T 
also subjected. to bending stress. In Fig. 195 the bending stress is a 
\32B 
tensile stress p=—-. These stresses produce a pure normal stress 7 
A— — ——! B 
et ow 
<= «+ + + 
on planes parallel to\CJ, and a pure normal stress 7, on planes parallel 
to DK, which is perpendicular to CJ, By Art. 143, putting 7=0, 
1 =4p+h Jp? +47? and r,=4p-4 Jp + 47% 
