COMPOUND STRAINS AND STRESSES 159 
plying both sides by r, 2prdr+r°dp=d(pr*)=2ardr. Integrating 
pr =2afrdr = 2a 5 3 +c=ar*+c, where c is a constant of integration. 
When r=r,, p=p,,_ therefore Py =ari+e. 
” T= Vos P=Pro : Pali =ar?+c. 
iin thew equations a = 2X3 P2 “ang cao ied 
Zs rar er 
From the equations pr?=ar?+c, and f=2a-—yp, and the values of 
~ @and c just determined, 
f= Pi, Pari _ (ry — Por irs ’ 
r§-Tri (73 = ri)r 
Putting r=r,, f becomes /,, the stress at the inside surface, 
fa teri— rot +rt) 
t3-7r} 
Putting r=r, , f becomes f,, the stress at the outside surface, 
fea PhO td = pyri 
3-1} 
_ The most common cases in practice are those in which p, is so small 
_ compared with p, that the former may be neglected, or p, = 0. 
—p,(r?2 +72 — 2p r 
If p,=0 then f, = =e Te, fy= 7 at, 
and fa seri (rare), 
r3—ri r2 
~ the minus sign in front in each case See i the stress is tensile. 
The tensile stress J, has the value, — £ ri + a9 hanes it is easy to deduce 
"3 
2 _ (ate ae Lime, n/ (Bees w Pt) 1, 
mt J Ji-VA nd 5 Ji-Py 
_ where t is equal to 7,—7r,, the theives of the cylinder. It is evident 
_ that for all possible values of ¢, /, must be greater than 7. 
The foregoing formule are Known as Lamé’s formule. 
Exercises IXb. 
Notr.—The answers given at the end of the book to Exercises 5, 6, and 8 on crane 
hooks were obtained on the assumption that the moment of resistance of a curved 
bar to bending is the same as that for a straight bar. The answers are therefore 
only approximate. See the latter part of Art. 150, p. 155. 
1, The crank shaft shown in Fig. 223 is subjected to a thrust P along 
its axis. Determine the magnitude of P, in tons, 
when the maximum tensile stress produced by it a 
in the crank pin is 4 tons per square inch. Lak gee 
What will then be the maximum compressive © i fe? ie 
Stress in the pin? pe): 
2. A link (Fig. 224) of rectangular section, af 
8 inches deep and 2 inches wide, is subjected . 
to tension by a load P, the line of action of which ’ 
is to the axis of the central part of the p+ —+—|+] p< 
+ at a distance 2 above it. When «=} 
P, if the maximum tensile stress in 
6 tons per square inch. Find also Fig. 223. 
