© bara are made of the same material, and that they 
Pe eee 
7 
SIMPLE STRAINS AND STRESSES 77 
From the analogy between this case and that of a thin cylindrical shell 
under fluid pressure (Art. 90) it may be concluded that R, the resultant 
of the centrifugal forces gq... . on one half of the 
hoop, is equal to dq = 24awv*/g, and equating this to if 
theresistanceof the hoop tobursting, 2a/,= 24awv?/g, T 
therefore f,= 12wv?/g, where /; is the stress (in lbs. ’ H 
‘ ’ 
per square inch) due to centrifugal force. a 
The foregoing result may be demonstrated in = 
another way. Consider a small portion of the hoop 7 
(ig. 91) subtending an angle @ at the centre. 
is in equilibrium under the action of the r) 
centrifugal force F and the tensions TT. The ee 
weight of the portion under consideration is Jawd8, y 
and F = 12aw@v?/g. From the triangle of forces Fic. 91. 
F =T9, since @ is a very small angle. Also T=fa, 
therefore TO =f,a0 = 12aw6v?/g, and f,= 12wv?/g. 
93. Cottered Joints.—TFig. 92 shows two bars of diameter d joined 
yl together with their axes in the same straight line. The upper bar is 
enlarged at its lower end to form a socket, which fits over the enlarged 
upper end of the lower bar. A cotter passes through the two as shown. 
It will be assumed in what follows that the two 
are Ss core ge Eanes Phen Fe ‘ Y 
or the parts o whey , (1) , > 
Z $12 ea St, Verh 7d". Me 
The weakest cross section of the part of diameter 
d, is at the cotter hole, where the area of the cross 
section is very nearly ai —d,t, and therefore 
T=(Ta}- at) 4 eee: 
The weakest part of the socket is the cross section 
at the cotter hole, where the area is 
q(D? - 43) -(D-a,)t, 
LWWWWWE IS 
therefore T={F(D?- a})-(b-a,ye\y, eg ee 
The cotter will shear at two sections, therefore 
T=20tf, . Faw Ts (4) 
The bearing area of the cotter on the lower bar is d,t, therefore 
sO a i cae ae aaa (5) 
The bearing area of the cotter on the socket is (D, —d,)¢, therefore 
CaM tied) f+ 8. s:- » (6) 
_ Assuming T and the stresses to be known, the foregoing six equations 
are sufficient for determining the dimensions d, d,, D, D,, 6, and ¢. 
