158 APPLIED MECHANICS 
153. Thick Hollow Cylinders.—In Art. 90, p. 76, it was shown that 
if a thin cylindrical shell of diameter d and thickness ¢ be subjected to an 
. internal pressure of intensity 
p, the material of the shell 
is subjected to a tensile stress 
of intensity f, given by the 
equation pd =2tf. Inserting 
twice the radius 7 instead of 
the diameter d, then pr=#/. dr 
It is evident that so long as <u 
the thin shell remains circular 
the relation pr =#f will hold 
if the pressure p be transferred 
to he oats: of the shell, Pra, 32. 
but the stress f will become compressive instead of tensile. The assump- 
tion made in proving this relation was, that the stress is uniformly distri- 
buted over the longitudinal section of the shell. This assumption is 
justified for a thin shell, but it cannot be used in the case of a thick 
hollow cylinder. 
Let a thick hollow cylinder (Fig. 222) have an internal radius 7, and 
an external radius 7,, and let there be an internal pressure of intensity 
p, and an external pressure of intensity p,. 
In what follows a thrust or compression will be considered as posi- 
tive, and a pull or tension as negative. It will be convenient to suppose 
that the cylinder is in compression, the external pressure being greater 
than the internal pressure, but the results obtained will be of general 
application, the formule making the stress negative when the internal 
pressure is greater than the external pressure. 
Consider a portion of the cylinder of unit length, and take an inter- 
mediate indefinitely thin ring of it of internal radius r and thickness dr, 
Let the internal radial pressure on this ring be p, and the external 
pressure p + dp. 
Considering the equilibrium of this ring, if the external pressure 
pt+dp acted alone, the stress f produced would be given by the 
equation (p+dp)(r +dr)=/fdr, and if the internal pressure acted alone, 
then pr=fdr. Hence, when both pressures act at the same time 
(p+dp)(r+dr)-—pr=fdr, which reduces to pdr+rdp=fdr, which is 
one relation between p, f, and 7. 
Another relation involving p and f is found from a consideration of 
the strains produced by p and / in the direction of the axis of the 
cylinder. The pressure p will produce a strain in the direction of the 
thickness of the ring equal to p/E, and a strain in the direction of its 
axis equal toa p/E. The stress f will produce a strain in the material of 
the ring, in the direction in which it acts, equal to //E, and a strain in 
the direction of the axis equal too/f/E. Hence the total strain in the 
direction of the axis due to p and f is op/E+o//E, and this strain will 
be uniform throughout, because it is reasonable to suppose that plane 
sections perpendicular to the axis will remain plane. If, therefore, 
o p/K+o7/E is constant, p+/ is constant. Let p+/f=2a. 
From the equation p+/= 2a, f= 2a —p; substituting this value of fin 
the equation pdr+rdp=/fdr, it follows that 2pdr+rdp=2adr. Multi- 
