R + r0 \ r_x / \ r_x /

In order to represent more clearly the distribution of stresses and
pressures in the metal of a homogeneous ideally perfect hollow cylinder,
let us take, as an example, the barrel of a 6 in. gun--153 mm. Let us
suppose T = 3,000 atmospheres; therefore, under the most favorable
conditions, P0 = 1.41 T, or 4,230 atmospheres. From Equation (1) we
determine R = 184.36 mm. With these data were calculated the internal
stresses and the pressures from which the curve represented in Fig. 1 is
constructed. The stresses developed under fire with a pressure in the
bore of 4,230 atmospheres are represented by a line parallel to the axis
of the abscissæ, since their value is the same throughout all the layers
of metal and equal to the elastic limit, 3,000 atmospheres. If, previous
to firing, the metal of the tube were free from any internal stresses,
then the resistance of the tube would be

R² - r²_0
P0 = U ----------- ,
R² + r²_0

or 2,115 atmospheres--that is, one-half that in the ideally perfect
cylinder. From this we perceive the great advantage of developing useful
initial stresses in the metal and of regulating the conditions of
manufacture accordingly. Unless due attention be paid to such
precautions, and injurious stresses be permitted to develop themselves
in the metal, then the resistance of the cylinder will always be less
than 2,115 atmospheres; besides which, when the initial stresses exceed
a certain intensity, the elastic limit will be exceeded, even without
the action of external pressures, so that the bore of the gun will not
be in a condition to withstand any pressure because the tensile stress