/sin [alpha]' cos [alpha]' sin [alpha] cos [alpha] \
C ( ------------------------- - ------------------------ ),
\ r' r /
and the flexural couple is
/cos^2 [alpha]' cos^2 [alpha]\
A ( -------------- - ------------ ).
\ r' r /
The wrench (R, K) along the axis by which the spring can be held in
the form ([alpha]', r') is given by the equations
sin [alpha]' /cos^2 [alpha]' cos^2 [alpha]\
R = A ------------ ( -------------- - ------------- ) -
r' \ r' r /
cos [alpha]' /sin [alpha]' cos [alpha]' sin [alpha] cos [alpha]\
C ------------- ( ------------------------- - ----------------------- ),
r' \ r' r /
/cos^2 [alpha]' cos^2 [alpha]\
K = A cos [alpha]' ( -------------- - ------------- ) +
\ r' r /
/sin [alpha]' cos [alpha]' sin [alpha] cos [alpha]\
C sin [alpha]' ( ------------------------- - ----------------------- ).
\ r' r /
When the spring is slightly extended by an axial force F, = -R, and
there is no couple, so that K vanishes, and [alpha]', r' differ very
little from [alpha], r, it follows from these equations that the axial
elongation, [delta]x, is connected with the axial length x and the
force F by the equation
E[pi]c^4 sin [alpha] [delta]x
F = -------- ------------------------- --------,
4r^2 1 + [sigma] cos^2 [alpha] x
and that the loaded end is rotated about the axis of the helix through
a small angle
4[sigma]Fxr cos [alpha]
-----------------------
E[pi]c^4
the sense of the rotation being such that the spring becomes more
tightly coiled.
66. A horizontal pointer attached to a vertical spiral spring would be
made to rotate by loading the spring, and the angle through which it
turns might be used to measure the load, at any rate, when the load is
not too great; but a much more sensitive contrivance is the twisted
strip devised by W.E. Ayrton and J. Perry. A very thin, narrow
rectangular strip of metal is given a permanent twist about its
longitudinal middle line, and a pointer is attached to it at right
angles to this line. When the strip is subjected to longitudinal tension
the pointer rotates through a considerable angle. G.H. Bryan (_Phil.
Mag._, December 1890) has succeeded in constructing a theory of the
action of the strip, according to which it is regarded as a strip of
_plating_ in the form of a right helicoid, which, after extension of the
middle line, becomes a portion of a slightly different helicoid; on
account of the thinness of the strip, the change of curvature of the
surface is considerable, even when the extension is small, and the
pointer turns with the generators of the helicoid.
If b stands for the breadth and t for the thickness of the strip, and
[tau] for the permanent twist, the approximate formula for the angle
[theta] through which the strip is untwisted on the application of a
load W was found to be
Wb[tau](1 + [sigma])
[theta] = ---------------------------------------.
/ (1 + [sigma]) b^4[tau]^2\
2Et^3 ( 1 + ------------- - ---------- )
\ 30 t^2 /
The quantity b[tau] which occurs in the formula is the total twist in
a length of the strip equal to its breadth, and this will generally be
very small; if it is small of the same order as t/b, or a higher
order, the formula becomes 1/2Wb[tau](1+[sigma])/Et^3, with sufficient
approximation, and this result appears to be in agreement with
observations of the behaviour of such strips.
67. _Thin Plate under Pressure._--The theory of the deformation of
plates, whether plane or curved, is very intricate, partly because of
the complexity of the kinematical relations involved. We shall here
indicate the nature of the effects produced in a thin plane plate, of
isotropic material, which is slightly bent by pressure. This theory
should have an application to the stress produced in a ship's plates. In
the problem of the cylinder under internal pressure (S 77 below) the
most important stress is the circumferential tension, counteracting the
tendency of the circular filaments to expand under the pressure; but in
the problem of a plane plate some of the filaments parallel to the plane
of the plate are extended and others are contracted, so that the
tensions and pressures along them give rise to resultant couples but not
always to resultant forces. Whatever forces are applied to bend the
plate, these couples are always expressible, at least approximately in
terms of the principal curvatures produced in the surface which, before
strain, was the middle plane of the plate. The simplest case is that of
a rectangular plate, bent by a distribution of couples applied to its
edges, so that the middle surface becomes a cylinder of large radius R;
the requisite couple per unit of length of the straight edges is of
amount C/R, where C is a certain constant; and the requisite couple per
unit of length of the circular edges is of amount C[sigma]/R, the latter
being required to resist the tendency to anticlastic curvature (cf. S
47). If normal sections of the plate are supposed drawn through the
generators and circular sections of the cylinder, the action of the
neighbouring portions on any portion so bounded involves flexural
couples of the above amounts. When the plate is bent in any manner, the
curvature produced at each section of the middle surface may be regarded
as arising from the superposition of two cylindrical curvatures; and the
flexural couples across normal sections through the lines of curvature,
estimated per unit of length of those lines, are C(1/R1 + [sigma]/R2)
and C(1/R2 + [sigma]/R1), where R1 and R2 are the principal radii of
curvature. The value of C for a plate of small thickness 2h is
(2/3)Eh^3/(1 - [sigma]^2). Exactly as in the problem of the beam (SS 48,
56), the action between neighbouring portions of the plate generally
involves shearing stresses across normal sections as well as flexural
couples; and the resultants of these stresses are determined by the
conditions that, with the flexural couples, they balance the forces
applied to bend the plate.
[Illustration: FIG. 28.]
68. To express this theory analytically, let the middle plane of the
plate in the unstrained position be taken as the plane of (x, y), and
let normal sections at right angles to the axes of x and y be drawn
through any point. After strain let w be the displacement of this
point in the direction perpendicular to the plane, marked p in fig.
28. If the axes of x and y were parallel to the lines of curvature at
the point, the flexural couple acting across the section normal to x
(or y) would have the axis of y (or x) for its axis; but when the
lines of curvature are inclined to the axes of co-ordinates, the
flexural couple across a section normal to either axis has a component
about that axis as well as a component about the perpendicular axis.
Consider an element ABCD of the section at right angles to the axis of
x, contained between two lines near together and perpendicular to the
middle plane. The action of the portion of the plate to the right upon
the portion to the left, across the element, gives rise to a couple
about the middle line (y) of amount, estimated per unit of length of
that line, equal to
/dP^2w dP^2w \
C ( ----- + [sigma]----- ), = G1,
\dPx^2 dPy^2 /
say, and to a couple, similarly estimated, about the normal (x) of
amount
dP^2w
-C(1-[sigma]) ------, = H,
dPxdPy
say. The corresponding couples on an element of a section at right
angles to the axis of y, estimated per unit of length of the axis of
x, are of amounts
/dP^2w dP^2w\
-C( ----- + [sigma]----- ), = G2
\dPy^2 dPx^2/
say, and -H. The resultant S1 of the shearing stresses on the element
ABCD, estimated as before, is given by the equation
dPG1 dPH
S1 = ---- - ---
dPx dPy
(cf. S 57), and the corresponding resultant S2 for an element
perpendicular to the axis of y is given by the equation
dPH dPG2
S2= - --- - ----.
dPx dPy
If the plate is bent by a pressure p per unit of area, the equation of
equilibrium is
dPS1 dPS2
---- + ---- = p, or, in terms of w,
dPx dPy
dP^4w dP^4w dP^4w p
----- + ----- + 2---------- = --.
dPx^4 dPy^4 dPx^2dPy^2 C
This equation, together with the special conditions at the rim,
suffices for the determination of w, and then all the quantities here
introduced are determined. Further, the most important of the
stress-components are those which act across elements of normal
sections: the tension in direction x, at a distance z from the middle
plane measured in the direction of p, is of amount
3Cz /dP^2w dP^2w\
- ---- ( ----- + [sigma]----- ),
2h^3 \dPx^2 dPy^2/
and there is a corresponding tension in direction y; the shearing
stress consisting of traction parallel to y on planes x = const., and
traction parallel to x on planes y = const., is of amount
3C(1 - [sigma])z dP^2w
---------------- ------;
2h^3 dPxdPy
these tensions and shearing stresses are equivalent to two principal
tensions, in the directions of the lines of curvature of the surface
into which the middle plane is bent, and they give rise to the
flexural couples.
69. In the special example of a circular plate, of radius a, supported
at the rim, and held bent by a uniform pressure p, the value of w at a
point distant r from the axis is
1 p /5 + [sigma] \
-- -- (a^2 - r^2) ( ----------- a^2 - r^2),
64 C \1 + [sigma] /
and the most important of the stress components is the radial tension,
of which the amount at any point is (3/32)(3 + [sigma])pz(a^2 - r)/h^3;
the maximum radial tension is about (1/3)(a/h)^2p, and, when the
thickness is small compared with the diameter, this is a large
multiple of p.
70. _General Theorems._--Passing now from these questions of flexure and
torsion, we consider some results that can be deduced from the general
equations of equilibrium of an elastic solid body.
The form of the general expression for the potential energy (S 27)
stored up in the strained body leads, by a general property of quadratic
functions, to a reciprocal theorem relating to the effects produced in
the body by two different systems of forces, viz.: The whole work done
by the forces of the first system, acting over the displacements
produced by the forces of the second system, is equal to the whole work
done by the forces of the second system, acting over the displacements
produced by the forces of the first system. By a suitable choice of the
second system of forces, the average values of the component stresses
and strains produced by given forces, considered as constituting the
first system, can be obtained, even when the distribution of the stress
and strain cannot be determined.
[Illustration: FIG. 29.]
Taking for example the problem presented by an isotropic body of any
form[4] pressed between two parallel planes distant l apart (fig. 29),
and denoting the resultant pressure by p, we find that the diminution
of volume -[delta]v is given by the equation
-[delta]v = lp/3k,
where k is the modulus of compression, equal to (1/3)E/(1 - 2[sigma]).
Again, take the problem of the changes produced in a heavy body by
different ways of supporting it; when the body is suspended from one
or more points in a horizontal plane its volume is increased by
[delta]v = Wh/3k,
where W is the weight of the body, and h the depth of its centre of
gravity below the plane; when the body is supported by upward
vertical pressures at one or more points in a horizontal plane the
volume is diminished by
-[delta]v = Wh'/3k,
where h' is the height of the centre of gravity above the plane; if
the body is a cylinder, of length l and section A, standing with its
base on a smooth horizontal plane, its length is shortened by an
amount
-[delta]l = Wl/2EA;
if the same cylinder lies on the plane with its generators horizontal,
its length is increased by an amount
[delta]l = [sigma]Wh'/EA.
71. In recent years important results have been found by considering the
effects produced in an elastic solid by forces applied at isolated
points.
Taking the case of a single force F applied at a point in the
interior, we may show that the stress at a distance r from the point
consists of
(1) a radial pressure of amount
2 - [sigma] F cos [theta]
----------- ----- -----------,
1 - [sigma] 4[pi] r^2
(2) tension in all directions at right angles to the radius of amount
1 - 2[sigma] F cos [theta]
-------------- -------------,
2(1 - [sigma]) 4[pi]r^2
(3) shearing stress consisting of traction acting along the radius
dr on the surface of the cone [theta] = const. and traction acting
along the meridian d[theta] on the surface of the sphere r = const. of
amount
1 - 2[sigma] F sin [theta]
-------------- ----- -----------,
2(1 - [sigma]) 4[pi] r^2
where [theta] is the angle between the radius vector r and the line of
action of F. The line marked T in fig. 30 shows the direction of the
tangential traction on the spherical surface.
[Illustration: FIG. 30.]
Thus the lines of stress are in and perpendicular to the meridian
plane, and the direction of one of those in the meridian plane is
inclined to the radius vector r at an angle
/2 - 4[sigma] \
1/2tan^(-1) ( ------------ tan [theta] ).
\5 - 4[sigma] /
The corresponding displacement at any point is compounded of a radial
displacement of amount
1 + [sigma] F cos [theta]
-------------- ------ -----------
2(1 - [sigma]) 4[pi]E r
and a displacement parallel to the line of action of F of amount
(3 - 4[sigma])(1 + [sigma]) F 1
--------------------------- ------ --.
2(1 - [sigma]) 4[pi]E r
The effects of forces applied at different points and in different
directions can be obtained by summation, and the effect of
continuously distributed forces can be obtained by integration.
72. The stress system considered in S 71 is equivalent, on the plane
through the origin at right angles to the line of action of F, to a
resultant pressure of magnitude 1/2F at the origin and a radial traction
of amount
1 - 2[sigma] F
-------------- --------,
2(1 - [sigma]) 4[pi]r^2
and, by the application of this system of tractions to a solid bounded
by a plane, the displacement just described would be produced. There is
also another stress system for a solid so bounded which is equivalent,
on the same plane, to a resultant pressure at the origin, and a radial
traction proportional to 1/r^2, but these are in the ratio 2[pi]:r^(-2),
instead of being in the ratio 4[pi](1 - [sigma]) : (1 - 2[sigma])r^(-2).
[Illustration: FIG. 31.]
The second stress system (see fig. 31) consists of:
(1) radial pressure F'r^(-2),
(2) tension in the meridian plane across the radius vector of amount
F'r^(-2) cos [theta] /(1 + cos [theta]),
(3) tension across the meridian plane of amount
F'r^(-2)/(l + cos [theta]),
(4) shearing stress as in S 71 of amount
F'r^(-2) sin [theta]/(1 + cos [theta]),
and the stress across the plane boundary consists of a resultant
pressure of magnitude 2[pi]F' and a radial traction of amount
F'r^(-2). If then we superpose the component stresses of the last
section multiplied by 4(1 - [sigma])W/F, and the component stresses
here written down multiplied by -(1 - 2[sigma])W/2[pi]F', the stress
on the plane boundary will reduce to a single pressure W at the
origin. We shall thus obtain the stress system at any point due to
such a force applied at one point of the boundary.
In the stress system thus arrived at the traction across any plane
parallel to the boundary is directed away from the place where W is
supported, and its amount is 3W cos^2[theta]/2[pi]r^2. The
corresponding displacement consists of
(1) a horizontal displacement radially outwards from the vertical
through the origin of amount
W(1 + [sigma]) sin [theta] / 1 - 2[sigma] \
-------------------------- ( cos [theta] - --------------- ),
2[pi]Er \ 1 + cos [theta]/
(2) a vertical displacement downwards of amount
W(1 + [sigma])
-------------- {2(1 - [sigma]) + cos^2[theta]}.
2[pi]Er
The effects produced by a system of loads on a solid bounded by a
plane can be deduced.
The results for a solid body bounded by an infinite plane may be
interpreted as giving the local effects of forces applied to a small
part of the surface of a body. The results show that pressure is
transmitted into a body from the boundary in such a way that the
traction at a point on a section parallel to the boundary is the same at
all points of any sphere which touches the boundary at the point of
pressure, and that its amount at any point is inversely proportional to
the square of the radius of this sphere, while its direction is that of
a line drawn from the point of pressure to the point at which the
traction is estimated. The transmission of force through a solid body
indicated by this result was strikingly demonstrated in an attempt that
was made to measure the lunar deflexion of gravity; it was found that
the weight of the observer on the floor of the laboratory produced a
disturbance of the instrument sufficient to disguise completely the
effect which the instrument had been designed to measure (see G.H.
Darwin, _The Tides and Kindred Phenomena in the Solar System_, London,
1898).
73. There is a corresponding theory of two-dimensional systems, that is
to say, systems in which either the displacement is parallel to a fixed
plane, or there is no traction across any plane of a system of parallel
planes. This theory shows that, when pressure is applied at a point of
the edge of a plate in any direction in the plane of the plate, the
stress developed in the plate consists exclusively of radial pressure
across any circle having the point of pressure as centre, and the
magnitude of this pressure is the same at all points of any circle which
touches the edge at the point of pressure, and its amount at any point
is inversely proportional to the radius of this circle. This result
leads to a number of interesting solutions of problems relating to plane
systems; among these may be mentioned the problem of a circular plate
strained by any forces applied at its edge.
74. The results stated in S 72 have been applied to give an account of
the nature of the actions concerned in the impact of two solid bodies.
The dissipation of energy involved in the impact is neglected, and the
resultant pressure between the bodies at any instant during the impact
is equal to the rate of destruction of momentum of either along the
normal to the plane of contact drawn towards the interior of the other.
It has been shown that in general the bodies come into contact over a
small area bounded by an ellipse, and remain in contact for a time which
varies inversely as the fifth root of the initial relative velocity.
For equal spheres of the same material, with [sigma] = 1/4, impinging
directly with relative velocity v, the patches that come into contact
are circles of radius
/45[pi]\ ^(1/5) /v \ ^(2/5)
( ------ ) ( -- ) r,
\ 256 / \V /
where r is the radius of either, and V the velocity of longitudinal
waves in a thin bar of the material. The duration of the impact is
approximately
/2025[pi]^2\ ^(1/5) r
(2.9432) ( ---------- ) --------------- .
\ 512 / v^(1/5) V^(4/5)
For two steel spheres of the size of the earth impinging with a
velocity of 1 cm. per second the duration of the impact would be about
twenty-seven hours. The fact that the duration of impact is, for
moderate velocities, a considerable multiple of the time taken by a
wave of compression to travel through either of two impinging bodies
has been ascertained experimentally, and constitutes the reason for
the adequacy of the statical theory here described.
75. _Spheres and Cylinders._--Simple results can be found for spherical
and cylindrical bodies strained by radial forces.
For a sphere of radius a, and of homogeneous isotropic material of
density [rho], strained by the mutual gravitation of its parts, the
stress at a distance r from the centre consists of
(1) uniform hydrostatic pressure of amount (1/10)g[rho]a(3 -
[sigma])/(1 - [sigma]),
(2) radial tension of amount (1/10)g[rho](r^2/a)(3 - [sigma])/(1
-[sigma]),
(3) uniform tension at right angles to the radius vector of amount
(1/10)g[rho](r^2/a) (1 + 3[sigma])/(1 - [sigma]),
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
(1/30)k^(-1)g[rho]a(3 - [sigma])/(1 - [sigma]),
(2) radial extension of amount (1/10)k^(-1)g[rho](r^2/a)(1 +
[sigma])/(1 - [sigma]),
(3) extension in any direction at right angles to the radius vector of
amount
(1/30)k^(-1)g[rho](r^2/a) (1 + [sigma])/(1 - [sigma]),
where k is the modulus of compression. The volume is diminished by the
fraction g[rho]a/5k of itself. The parts of the radii vectors within
the sphere r = a{(3 - [sigma])/(3 + 3[sigma])}^(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 involves a
neglect of the caution emphasized in S 40, viz. that the strain
determined by the solution must be small if the solution is to be
accepted. In a body of the size and mass of the earth, and having a
resistance to compression and a rigidity equal to those of steel, the
radial contraction at the centre, as given by the above solution,
would be nearly 1/3, and the radial extension at the surface nearly
1/6, and these fractions can by no means be regarded as "small."
76. In a spherical shell of homogeneous isotropic material, of
internal radius r1 and external radius r0, subjected to pressure p0 on
the outer surface, and p1 on the inner surface, the stress at any
point distant r from the centre consists of
p1r1^3 - p0r0^3
(1) uniform tension in all directions of amount ---------------,
r0^3 - r1^3
p1 - p0 r0^3 r1^3
(2) radial pressure of amount ----------- ---------,
r0^3 - r1^3 r^3
(3) tension in all directions at right angles to the radius vector of
amount
p1 - p0 r0^3 r1^3
1/2 ----------- ---------.
r0^3 - r1^3 r^3