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 −C (∂2w∂y2 + σ ∂2w∂x2), = G2 say, and −H. The resultant
S1 of the shearing stresses on the element ABCD, estimated as
before, is given by the equation S1 = ∂G1∂x − ∂H∂y (cf. § 57), and the
corresponding resultant S2 for an element perpendicular to the
axis of y is given by the equation S2= −∂H∂x − ∂G2∂y. If the plate
is bent by a pressure p per unit of area, the equation of equilibrium
is ∂S1∂x + ∂S2∂y = p, or, in terms of w,
| ∂4w | + | ∂4w | + 2 | ∂4w | = | p | . |
| ∂x4 | ∂y4 | ∂x2∂y2 | 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 −3Cz2h3 (∂2w∂x3 + σ(∂2w∂y2), 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 − σ)z2h3 ∂2w∂x∂y; 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 | (a2 − r 2) ( | 5 + σ | a2 − r 2 ), |
| 64 C | 1 + σ |
and the most important of the stress components is the radial tension, of which the amount at any point is 332(3 + σ) pz (a2 − r)/h3; the maximum radial tension is about 13(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.
 |
| Fig. 29. |
The form of the general expression for the potential energy (§ 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.
Taking for example the problem presented by an isotropic body of any form[1] pressed between two parallel planes distant l apart (fig. 29), and denoting the resultant pressure by p, we find that the diminution of volume -δv is given by the equation
where k is the modulus of compression, equal to 13E / (1 − 2σ). 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
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
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
if the same cylinder lies on the plane with its generators horizontal, its length is increased by an amount
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 − σ | F | cos θ | , | ||
| 1 − σ | 4π | r 2 |
(2) tension in all directions at right angles to the radius of amount
| 1 − 2σ | F | cos θ | , | ||
| 2(1 − σ) | 4π | r 2 |
(3) shearing stress consisting of traction acting along the radius dr on the surface of the cone θ = const. and traction acting along the meridian dθ on the surface of the sphere r = const. of amount
| 1 − 2σ | F | sin θ | , | ||
| 2(1 − σ) | 4π | r 2 |
where θ 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.
 |
| Fig. 30. |
 |
| Fig. 31. |
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
| 12 tan−1 ( | 2 − 4σ | tan θ ). |
| 5 − 4σ |
The corresponding displacement at any point is compounded of a radial displacement of amount
| 1 + σ | F | cos θ | ||
| 2(1 − σ) | 4πE | r |
and a displacement parallel to the line of action of F of amount
| (3 − 4σ) (1 + σ) | F | 1 | . | ||
| 2(1 − σ) | 4πE | 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 § 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 12F at the origin and a radical traction of amount 1 − 2σ2(1 − σ) F4πr 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 2π : r −2, instead of being in the ratio 4π(1 − σ) : (1 − 2σ)r −2.
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
(3) tension across the meridian plane of amount
(4) shearing stress as in § 71 of amount
and the stress across the plane boundary consists of a resultant
pressure of magnitude 2πF′ and a radial traction of amount F′r −2. If- ↑ The line joining the points of contact must be normal to the planes.
