7 mm., Fig. 2, has been traced,
after calculating, by means of the usual well known formulae, the amount
of pressure exerted by the hoop on the tube, as well as the stresses and
pressures inside the tube and the hoop, before and after firing. A
comparison of these curves with those on Fig. 1 will show the difference
between the internal stresses in a homogeneous and in a built-up
cylinder. In the case of the hooped gun, the stresses in the layers
before firing, both in the tube and in the hoop, diminish in intensity
from the inside of the bore outward; but this decrease is comparatively
small. In the first place, the layer in which the stresses are = 0 when
the gun is in a state of rest does not exist. Secondly, under the
pressure produced by the discharge, all the layers do not acquire
simultaneously a strain equal to the elastic limit. Only two of them,
situated on the internal radii of the tube and hoop, reach such a
stress; whence it follows that a cylinder so constructed possesses less
resistance than one which is homogeneous and at the same time endowed
with ideally perfect useful initial stresses. The work done by the
forces acting on a homogeneous cylinder is represented by the area _a b
c d_, and in a built-up cylinder by the two areas _a' b' c' d'_ and _a"
b" c" d"_.
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