Equations of Equilibrium

EQUATIONS OF EQUILIBRIUM From the Principle of Least Work.—The conditions for the equilibrium of an elastic solid deformed by given forces may be obtained from an earlier form of the principle of least work. Assuming, for simplicity, that the body is deformed only by sur face tractions and that the displacements of points on the surface are prescribed, the internal displacements are supposed to be of such a nature that the total strain energy of the body is a mini mum when the energy associated with the actual deformation is compared with the energy of a fictitious deformation consistent with the same surface displacements.

Giving arbitrary small increments to the component displace ments u,v,w of a point (x,y,z), and assuming that the strains are so small that their expressions (I) may be used in the expression for the total energy, the Eulerian equations of the Calculus of Variations give in this case -=o ax ay az M=cYx+aYy+aY2=o, ax ay az N—+aZy+aZZ =o.

ax az These will be regarded as the differential equations for the equilib rium of an element, of the body, which occupies the position (x,y,z).

The first equation gives f f fL dxdy dz = o, where the integration extends over an arbitrary portion of the body. Transforming this volume integral into a surface integral by the usual method, the resulting equation may be written in the form Transforming the surface integrals into volume integrals, it is found that the three equations are satisfied in virtue of the dif ferential equations of equilibrium and the three equations = = X X y = which were introduced to complete the notation.

With the present definition of stress, the stress components across an area perpendicular to the axis of x are while those across an area perpendicular to the axis of y are The equation = is interpreted as a relation between complementary shearing stresses. The usual convention in interpreting X„ is that it is a stress exerted by the part of the body towards which the normal is drawn. It is clear from the definition that the component stress in the direction (l', m', n') across a plane at right angles to (1, in, n) is The state of stress at a point (x,y,z) is thus completely specified by the six components of stress.

Body Forces.

When an elastic solid is acted upon by a body force like gravity, which acts right through the body, the equa tions of equilibrium take a more general form, and are of type L+ pX = o, where X is the x-component of the body force per unit mass. When the body is in motion the force of inertia may be treated as an additional body force, and the equations of motion are of type z L+pX 2 These equations are particularly useful for a study of the tions of solid bodies and the propagation of waves through them.

It was shown by Poisson that there are two distinct velocities with which waves can be propagated without change of type through a homogeneous isotropic elastic solid. The so-called irrotational waves (or waves of compression) travel with velocity (X+ 2,u) l p, while the equivoluminal waves (or waves of shear) travel with velocity V / VP. Lord Rayleigh (John William Strutt, 3rd baron) discovered that a type of surface wave, compounded in a rather complex way from waves of the two types, can travel with a velocity slightly less than that of the equivoluminal waves. The vibrations of elastic solids. are of great interest in the theory of sound and in the design of engineering structures.

Simple Distributions of Stress.

Examples will now be given to indicate very briefly some of the steps which must be taken in the solution of problems in the theory of elasticity. If T is a constant, the equations are satisfied by the stress com ponents In order that this distribution of stress may be possible it is necessary, however, that there should be a distribution of dis placements (u, v, w) corresponding to the distribution of strain. In the present case it is easily seen that the expressions, satisfy requirements, but generally it is necessary to make use of the conditions of compatibility which lead to a set of differ ential equations for the stresses. In this example the lateral con traction —b is a times the longitudinal extension a, and so a phys ical meaning is found for Poisson's ratio. The state of stress may be supposed to exist in a cylindrical rod whose axis and generators are parallel to the x-axis. The curved surface of the rod is then free from surface tractions, but the end surface must be regarded as acted upon by uniformly distributed normal tractions.

If p is a constant, the stress-components If l,m,n are the direction cosines of the normal to a surface element of the boundary of the solid, the components of stress across this element are — l p, — m p, — np; the stress is thus a uni form normal pressure. It is easily seen that displacements cor responding to the strains can be found, and so the stress distribu tion can exist in a body subjected to a constant hydrostatic pres sure p. The case in which p varies with position and there are body forces is easily treated. This example indicates clearly why k is called the modulus of compression.

The displacements, u = 211y, v = o, w = o, corresponding to a simple shear are possible in connection with this distribution of strain, and the reason for calling ,u the modulus of shear or rigidity of the material is apparent. It is sometimes convenient to resolve a given distribution of stress into a uniform normal pressure, for which 3p= — (X. and a residual distribu tion of shearing stresses which would by themselves produce no volume change. This gives a general definition of pressure.

The Thin Plate.

The deformation of a thin plate bounded by plane surfaces z= ±h may in some cases be discussed with the aid of a theory of generalized plane stress, due to Louis Napoleon George Filon. It is assumed in this theory that is everywhere zero, and that the faces z = ±h are free from stress. Average values of u, X,., etc., are defined by means of equations such as 2h u= -h the bar over a symbol denoting that an average has been taken. Since and vanish at z = - - h, it is found that a ' ax + ay . = o, ax + ay = o;and these equations may be satisfied, in the same way as in the earlier theory of Sir George Airy, by writing .11,= 32x — 32X , — — a y2 a x2 Xyaxay The differential equation for x may be found by first observing that The elimination of u and v now gives Airy's differential equation, 04x (34X ax ay ay The quantities X„ may be regarded as generalized corn- ponents of stress, and au ' a v ' + ay au as generalized com ponents of strain. At each point of the plate there will be prin cipal axes of generalized strain, and these may also be regarded as principal axes of generalized stress. If these are taken as axes of co-ordinates Xy = o, and become the principal stresses which will be denoted by the symbols P and Q.

Photoelastic Methods.

Many deductions from the theory have been confirmed by the photoelastic methods of research based upon the discovery, made by Sir David Brewster in 1816, that, when a piece of glass is loaded and viewed in polarized light under suitable conditions, it shows brilliant colour effects due to the fact that the glass has changed its optical properties and has become doubly refracting. Brewster suggested that the stress distribution in masonry bridges might be investigated by construct ing glass models, and after many years the suggestion was fol lowed up by Augustin Mesnager in Paris. By means of the ap paratus devised by E. G. Coker it is possible to measure, to an accuracy of about 2%, the stress distribution, under any system of stress loads, in any body which can be represented by a plate model of transparent material stressed in its own plane.

Tests are frequently made with models cut from transparent celluloid, but a new transparent material, "phenolite," which ap pears to be superior to celluloid, has recently been developed in Tokyo by Z. Tuzi. The model is illuminated by plane polarized light, and an image of it projected on the screen through an analyser whose plane of polarization is perpendicular to that of the polar izer. All points at which the principal stresses are either parallel to or at right-angles to the plane of polarization are then readily determined, and, by rotating both polarizer and analyser simul taneously, the directions of the principal stresses can be found at every point of the specimen. The difference P — Q can be found by the colour shown at the point in question when the model is illuminated by circularly polarized light. If either P or Q is zero, as in simple tension or compression, the intensity of stress may be derived from a colour scale, such as that given in the following table, or may be derived by comparison with a simple tension member of the same material subjected to a known stress. Usually the maximum stresses occur at the inner and outer edges of the material, and when these surfaces are not directly loaded the stress across a line at right angles to the boundary can be read off at once.

of Materials (1908) ; H. Lamb, Statics (1912) ; J. Prescott, Applied Elasticity (1924) ; S. Timoshenko and J. M. Lessells, Applied Elasticity (1926) ; J. Case, Strength of Materials (1925) ; E. G. Coker's apparatus is described in booklets issued by Adam Hilger, Ltd., con taining a bibliography on the subject of photoelasticity. The theory of elastic stability is discussed in R. V. Southwell, "Memoir," Phil. Trans. Roy. Soc., A. (1913) . (H. BN.)