Analysis of Stress

ANALYSIS OF STRESS The Notion of Stress.-4. Let us imagine, in the first place, that a heavy cube (A) of the structureless material rests with its bottom face horizontal and in contact with a similar cube (B). To maintain equilibrium, an action must be exerted at the sur face of contact, whereby B pushes upwards against the weight of A, whilst A pushes down on B with equal force. Mutual action of this kind is termed a stress: when, as in this example, its direction is at right angles to the surface at which it acts, it is termed "normal stress." Action of the same kind (although different in amount) must evidently be exerted at any horizontal surface which divides either cube into two parts : the material below the surface must push upwards on that above, and the material above the surface must push downwards on that below. So the idea of stress can be extended from mutual actions between two bodies to mutual actions between different parts of the same body; the reality of the surface across which a stress is transmitted is a matter of indifference.

Again, we may conceive the total action across a surface to be made up of contributions from every part of that surface. In this way we arrive at the notion of superficial intensity of stress. If a portion of area S makes a contribution P to the total action, then the ratio P/S measures the average intensity of stress on that area. If S (and therefore P) is indefinitely small, so that the area is effectively concentrated at a point, the average intensity of stress may be identified with the actual local intensity at that point.

We employ the symbol p to denote this local intensity of stress. If equal contributions to the total action are made by equal areas, p will have a constant value over the surface considered ; in such cases we say that the intensity of stress is uniform, or that the stress is "uniformly distributed." More generally, p will vary from point to point.

Normal and Tangential Stress.

The action at a given sur face will not necessarily be directed at right angles to that surface. Let us imagine that the cube A, in the case already considered, is subjected to a horizontal t orce which tends to slide it off B, and that sliding is resisted by friction at the surface of contact.

Evidently there is a mutual action between A and B, at this sur face, of a kind to which we have given the term stress; and fur ther, action of the same sense and total amount must be exerted across any imaginary horizontal surface, lying below the point of application of F, which divides A into two parts. It would not be legitimate to say that this last action has its origin in friction ; but we see the necessity for the concept of stresses, which may or may not be "uniformly distributed," having directions parallel to the surfaces at which they act. Such stresses are termed tangential, or "shearing" stresses.

Resolution of Stress.-5.

In the general case, the action at a surface may have any inclination to that surface. Thus a stress, like a force, may be resolved into components having any three specified directions ; and conversely normal and tangential stresses may be superposed, or "compounded." Simple Longitudinal Stress.—Normal stress of the type al ready considered will be brought into existence when uniform pressures (that is, distributed normal forces of uniform intensity, acting inwards) are applied to the top and bottom faces of a cube. If the directions of the applied forces are reversed (fig. IA), the stress across a horizontal surface will have the same direction and magnitude as before, but it will now be opposite in sense. We term it a tensile stress, because the applied forces tend to stretch the cube : stress of the former type is termed compressive.

Thus we see that normal stress may be of two kinds,—viz., tensile or compressive,—which differ only in respect of sense; compressive stress may be regarded as tensile stress of negative intensity. This convention will be adopted in what follows: we shall use the symbol p to denote tensile stress, and we shall rep resent compressive stresses by giving a negative sign to p. Tan gential stress will frequently be represented by the symbol q.