(3) uniform tension at right angles to the radius vector of
amount
where g is the value of gravity at the surface. The corresponding
strains consist of
(1) uniform contraction of all lines of the body of amount
(2) radial extension of amount
(3) extension in any direction at right angles to the radius
vector of amount
where k is the modulus of compression. The volume is diminished
by the fraction gpajbk of itself. The
parts of the radii vectores within
the sphere r = a{(3 - <r)/(3+3cr)}1/2 are contracted, and the parts
without this sphere are extended. The application of the above
results to the state of the interior of the earth is restricted by the
circumstance that, unless the modulus of compression is much
greater than that of any known material, the stresses and strains
expressed above would, in a sphere of the size of the earth, greatly
exceed the elastic limits.
25. In a spherical shell of homogeneous isotropic material, of
internal radius rx and external radius r0, subjected to pressure pn
on the outer surface, and px on the inner surface, the stress at any
point distant r from the centre consists of
(1) uniform tension in all directions of amount ^>1?r'13- r^0^->
o “ i
/v) — 'fl 7*^
(2) radial pressure of amount —
—
3
Tq r r*
(3) tension in all directions at right angles to the radius vector
of amount
3/°ri
y» 3
, Vi-Pn r'0J1
3 _- rrx°3 ar3 #
The corresponding strains consist of
(1) uniform extension of all lines of the body of amount
-iw
3& TV -r-i3
1
(2) radial contraction of amount ~
^
(3) extension in all directions at right angles to the radius
vector of amount
_1_ P-Po
r0W
4/a r03 - rx r3 ’
where p. is the modulus of rigidity of the material, =^E/(l + o-).
The volume included between
the7* ^two surfaces of the body is
<2^ 3 7)
increased by the fraction/ 1 3—2-^of itself, and the volume
"(To ~ri)
within the inner surface is increased by the fraction
3
3(pi-7?n) 3U 3 +1 pxr-??0r30
4/x r0 - rx
&(r0 - rj )
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
_pp_
, J_ VV
r03 - ri3 ^ 4/x rx ) ’
the greatest tension is zthe transverse tension at the inner surface,
and its amount is p(r0 + rx)l(r(? - r^).
26. 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 § 25 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
Pr-P<?*
r2_~2 >
2/,. 2
(2) radial pressure of amount Pl-W) VT;
(3) hoop tension numerically equal to this radial pressure.
The corresponding strains consist of
(1) uniform extension of all lines of the material at right angles
to the axis of amount
l-o- pyr? -pnT<?
E V-rx2 ’
(2) radial contraction of amount
1+q- P-i-Pn2 TyV
E V-rx r2 ’
(3) extension along the circular filaments numerically equal to
this radial contraction,
SYSTEMS
(4) uniform contraction of the longitudinal filaments of amount
2
2°' Pr -Wq
E 7y - rx2
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
E W-rx2 J
the greatest tension is the hoop tension at the inner surface, and
its amount is p(r<? + r-?)l(r^ - ry ).
27. The results just obtained 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 / the radius of the common
surface. To obtain the stress at any point we superpose the
Y2 ^
system consisting of radial pressure jp-b
3 and hoop tension
o
r- r0 -rx
2 r2
p'A o
upon a system which, for the outer cylinder, consists
a* 2 — ^.2
/pf<l 2 i m2
P-tx
m
and
hoop
tension
P-5
-5 - r wof radial pressure
r* ry
^'2
2
and, for the inner cylinder consists of radial pressure P-^
r 2 r2 + ry The hoop tension at the inner
and hoop tension - P—
surface is less than it would be for a tube of equal thickness
without initial stress in the ratio
2/2
1-?
1.
p rtf + rtf r’2 -1 k
This shows how the strength of the tube is increased by the initial
stress.
28. The problem of determining the distribution of
stress and strain in a circular cylinder, rotating about its
axis, has not yet been completely solved, but 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 21, and of
density p, rotates about its axis with angular velocity w, 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 |w2pa2(3 + cr),
(2) radial pressure of amount |w2pr2(3+ <r),
(3) pressure along the circular filaments of amount |w2pr2(l + 3<r),
(4) uniform tension in all directions at right angles to the axis
of amount ^co2p(Z2 - 322)<r(l + <r)/(l - cr).
The corresponding strains may be expressed as
(1) uniform extension of all filaments at right angles to the axis,
of amount
1
~E~ |wV2(3 + cr),
(2) radial contraction of amount
l-*3 o 2,
—farpr
(3) contraction along the circular filaments of amount
l-ff2, 2 o
—jr- $w2pr“,
(4) extension of all filaments at right angles to the axis of
amount
^ h a>2p(Z2 — 3^2)cr(l +cr),
(5) contraction of the filaments normal to the plane of the disk
of amount
^uPpa2 (3 + cr) - iW2pr2(l + cr) +
u2p(l2 - 3s2)cr
The greatest extension is the circumferential extension near the
centre, and its amount is
(3 + cr)(l -cr) 0 2 cr(l + cr) „ 2
8E
u> + -g^ or pi .
The longitudinal contraction is required to make the plane faces