Such stresses are termed injurious
or detrimental stresses. With t0 negative, the strength of the
cylinder increases with the numerical value of t0, and those stresses
which cause compression in the layers nearest to the bore of the
cylinder and tension in the outer layers are termed beneficial or
useful stresses.
[Footnote 1: Lame holds that in a homogeneous tube subjected to
the action of two pressures, external and internal, the
difference between the tension and the compression developed at
any point of the thickness of the tube is a constant quantity,
and that the sum of these two stresses is inversely proportional
to the square of the radius of the layer under consideration. Let
r0, R, and r_x be the respective radii, p0, p?, and p_x the
corresponding pressures, and T0, T?, and T_x, the tensions, then
we have:
T0 - p0 = T_x - p_x (1)
(T0 + p0) r0 squared = (T_x + p_x) (r_x) squared (2)
T_x - p_x = T? - p? (3)
(T_x + p_x)(r_x) squared = (T? + P?)R squared (4)
if the radii are known and p and p? be given, then deducing from
the above equations the values T0 and T?, and also the variable
pressure p_x, we determine--
p0 r0 squared(R squared + (r_x) squared) - p? R squared((r_x) squared + r0 squared)
T_x = ------------------------------------------
(R squared + r0 squared) (r_x) squared
This is the formula of Lame, from which, making p? = 0, we obtain
the expression in the text.
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