It is further evident
that the internal stresses will obey a definite but very simple law,
namely, there will be in the hollow cylinder a layer whose radius is
sqrt(R r0), in which the stress is _nil_; from this layer the
stresses increase toward the external and the internal radii of the
cylinder, where they attain a maximum, being in compression in the
internal layers and in tension in the external ones.
The internal pressures corresponding to these stresses may be found by
means of very simple calculations. The expression for this purpose,
reduced to its most convenient form, is as follows:
R / R \ / r0 \
p_x = T -------- ( --- - 1 ) ( 1 - ----- ) (4)
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 abscissae, since their value is the same throughout all the layers
of metal and equal to the elastic limit, 3,000 atmospheres.
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