(1) uniform extension of all lines of the body of amount
1 p1r1^3 - p0r0^3
-- ---------------,
3k r0^3 - r1^3
1 p1 - p0 r0^3 r1^3
(2) radial contraction of amount ----- ----------- ---------,
2[mu] r0^3 - r1^3 r^3
(3) extension in all directions at right angles to the radius vector
of amount
1 p1 - p0 r0^3 r1^3
----- ----------- ---------,
4[mu] r0^3 - r1^3 r^3
where [mu] is the modulus of rigidity of the material, = 1/2E/(1 +
[sigma]). The volume included between the two surfaces of the body is
increased
p1r1^3 - p0r0^3
by the fraction --------------- of itself, and the volume within the
k(r0^3 - r1^3)
inner surface is increased by the fraction
3(p1 - p0) r0^3 p1r1^3 - p0r0^3
---------- ----------- + ---------------
4[mu] r0^3 - r1^3 k(r0^3 - r1^3)
of itself. For a shell subject only to internal pressure p the
greatest extension is the extension at right angles to the radius at
the inner surface, and its amount is
pr1^3 / 1 1 r0^3 \
----------- ( -- + ----- ---- );
r0^3 - r1^3 \3k 4[mu] r1^3 /
the greatest tension is the transverse tension at the inner surface,
and its amount is p(1/2 r0^3 + r1^3)/(r0^3 - r1^3).
77. In the problem of a cylindrical shell under pressure a
complication may arise from the effects of the ends; but when the ends
are free from stress the solution is very simple. With notation
similar to that in S 76 it can be shown that the stress at a distance
r from the axis consists of
(1) uniform tension in all directions at right angles to the axis of
amount
p1r1^2 - p0r0^2
---------------,
r0^2 - r1^2
p1 - p0 r0^2 r1^2
(2) radial pressure of amount ----------- ---------,
r0^2 - r1^2 r^2
(3) hoop tension numerically equal to this radial pressure.