(1) uniform extension of all lines of the material at right angles to
the axis of amount
1 - [sigma] p1r1^2 - p0r0^2
----------- ---------------,
E r0^2 - r1^2
(2) radial contraction of amount
1 + [sigma] p1 - p0 r0^2 r1^2
----------- ----------- ---------,
E r0^2 - r1^2 r^2
(3) extension along the circular filaments numerically equal to this
radial contraction,
(4) uniform contraction of the longitudinal filaments of amount
2[sigma] p1r1^2 - p0r0^2
-------- ---------------.
E r0^2 - r1^2
For a shell subject only to internal pressure p the greatest extension
is the circumferential extension at the inner surface, and its amount
is
p /r0^2 + r1^2 \
-- ( ----------- + [sigma] );
E \r0^2 - r1^2 /
the greatest tension is the hoop tension at the inner surface, and its
amount is p(r0^2 + r1^2)/(r0^2 - r1^2).
78. When the ends of the tube, instead of being free, are closed by
disks, so that the tube becomes a closed cylindrical vessel, the
longitudinal extension is determined by the condition that the
resultant longitudinal tension in the walls balances the resultant
normal pressure on either end. This condition gives the value of the
extension of the longitudinal filaments as
(p1r1^2 - p0r0^2)/3k(r0^2 - r1^2),
where k is the modulus of compression of the material. The result may
be applied to the experimental determination of k, by measuring the
increase of length of a tube subjected to internal pressure (A.
Mallock, _Proc. R. Soc. London_, lxxiv., 1904, and C. Chree, _ibid._).
79. The results obtained in S 77 have been applied to gun construction;
we may consider that one cylinder is heated so as to slip over another
upon which it shrinks by cooling, so that the two form a single body in
a condition of initial stress.
We take P as the measure of the pressure between the two, and p for
the pressure within the inner cylinder by which the system is
afterwards strained, and denote by r' the radius of the common
surface. To obtain the stress at any point we superpose the
r1^2 r0^2 - r^2
system consisting of radial pressure p ---- ----------- and hoop tension
r^2 r0^2 - r1^2
r1^2 r0^2 + r^2
p ---- ----------- upon a system which, for the outer cylinder,
r^2 r0^2 - r1^2
r'^2 r0^2 - r^2
consists of radial pressure P ---- -----------
r^2 r0^2 - r'^2
r'^2 r0^2 + r^2
and hoop tension P ---- -----------, and for the inner cylinder consists
r^2 r0^2 - r'^2
r'^2 r^2 - r1^2 r'^2 r^2 + r1^2
of radial pressure P ---- ----------- and hoop tension P ---- -----------.
r^2 r'^2 - r1^2 r^2 r'^2 - r1^2
The hoop tension at the inner surface is less than it would be for a
tube of equal thickness without initial stress in the ratio
P 2r'^2 r0^2 + r1^2
1 - -- ----------- ----------- : 1.
p r0^2 + r1^2 r'^2 - r1^2
This shows how the strength of the tube is increased by the initial
stress. When the initial stress is produced by tightly wound wire, a
similar gain of strength accrues.
80. In the problem of determining the distribution of stress and strain
in a circular cylinder, rotating about its axis, simple solutions have
been obtained which are sufficiently exact for the two special cases of
a thin disk and a long shaft.
Suppose that a circular disk of radius a and thickness 2l, and of
density [rho], rotates about its axis with angular velocity [omega],
and consider the following systems of superposed stresses at any point
distant r from the axis and z from the middle plane:
(1) uniform tension in all directions at right angles to the axis of
amount (1/8)[omega]^2[rho]a^2(3 + [sigma]),
(2) radial pressure of amount (1/8)[omega]^2[rho]r^2(3 + [sigma]),
(3) pressure along the circular filaments of amount
(1/8)[omega]^2[rho]r^2(1 + 3[sigma]),
(4) uniform tension in all directions at right angles to the axis of
amount (1/6)[omega]^2[rho](l^2 - 3z^2)[sigma](1 + [sigma])/(1 -
[sigma]).