If R and r0 be the external and internal radii of the cylinder, and
if we suppose the external pressure _nil_, then, if the pressure inside
the bore be P0, the stress on the radius r_x is determined by the
following expression deduced from the well-known fundamental formulae of
Lame:[1]
r0 squared R squared + (r_x) squared
T = P0 ------- . -------------
R squared-r0 squared (r_x) squared
From which we see that T is a maximum when r_x = r0, i.e., for
the layer immediately next to the bore of the cylinder. Calling t0
the internal stress in this layer, and T0 the stress resulting from
the action inside the bore of the pressure P0, and allowing that the
sum of both these quantities must not exceed the elastic limit U of the
material, we have--T0 = U - t0. And for this value of T0, the
corresponding pressure inside the bore will be
R squared - r0 squared
P = (U - t0) ----------.
R squared + r0 squared
This pressure increases with the term (U - t0). With t0 positive,
i.e., when the internal stresses in the thickness of the hollow
cylinder are such that the metal of the layers nearest to the bore is
in a state of tension and that of the outer layers in a state of
compression, then the cylinder will have the least strength when t0
has the greatest numerical value.
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