Analysis of Stress

When a long rod of uniform cross-section is stretched by means of a suspended weight, a practically uniform tensile stress will act on horizontal surfaces, and vertical surfaces will be free from stress. The material is then said to be subjected to "simple longi tudinal stress." When a short pillar or block is compressed by opposite forces applied at its ends, we have, to a somewhat less close approximation, a state of simple longitudinal compressive stress.

Compound Stress.

If the rod or block is subjected in ad dition to forces acting on its sides, a more complex state of stress is presented. We may regard it as a combination of two or more simple longitudinal stresses, and we describe it as a compound stress. In fig. I, the first block (A) is subjected to simple longi tudinal stress (tensile) in the di rection ox; block B is sub jected to simple longitudinal compressive stress in the perpendic ular direction oy. When the two systems of applied force are combined, as in block C, we have a state of compound stress made up of two simple longitudinal components. In block D, three simple longitudinal components go to make up the state of stress.

Principal Stresses.-6.

That part of the theory of elasticity which is called analysis of stress deals with the combination of simple stresses and, conversely, with the resolution of compound stresses into their "simple" components. The most important theorem in the subject may be stated as follows :—At any point in a material, however complicated may be the state of stress at that point, three planes can be found, each perpendicular to the other two, which have the property that the stresses transmitted across them are purely normal. These planes are termed principal planes of stress for the point considered, and the corresponding stresses I, are termed principal stresses.

We may imagine that a very small rectangular block of mate , rial, containing the point in question, has its faces parallel to the three principal planes. The theorem states that only normal stresses will act upon those faces, as shown in case D of fig. 1, which accordingly represents the most general state of compound stress that can occur. The three normal stresses pi, p2, P3 will in general all be different, and one or more may assume negative values (representing compressive stresses, as explained on p. 51).

Stress Equations of Motion or Equilibrium.-7.

The di rections of the principal stresses will in general vary from point to point, and cannot be determined until we have calculated the state of stress. For this purpose we form, in the first place, the equations of motion or of equilibrium for a small rectangular block of the material having edges parallel to three fixed axes Ox, Oy, Oz : hence, imagining the dimensions of the block to be made indefinitely small, we derive three equations of which the follow ing is typical : In this equation, x, y and z are the components of a point re ferred to the fixed axes; is the normal component of the stress on a plane through (x,y,z) which is perpendicular to Ox; and X, are the tangential components, parallel to Ox, of the stresses on planes through (x,y,z) which are perpendicular to Oy and Oz respectively. X is the "body force" per unit mass (due, e.g., to

gravitation or electrical attractions), p the density of the ma terial, and the acceleration in the direction Ox, at the point (x, y, z).

Two-Dimensional Stress-Systems.-8.

The derivation of these equations, and a proof of the theorem stated in § 6, will be found in the article ELASTICITY. We shall here confine attention to the special case in which there is no stress on planes perpendicular to Oz.

Fig. 2 shows a rectangular block of dimensions bx, by, I, in the directions of Ox, 0y, Oz respectively. The point A has co-ordinates x, y, and the stress on the face AD has a normal component which we denote (as above) by similarly, the stress on the face AB has a normal component which we denote by Y,. The tangen tial stress on AB has a component in the direction of Ox which we denote (as above) by and similarly, the tangential stress on Mohr's Circle Diagram for Compound Stress.—II. If, in fig. 3, the rectangular faces are principal planes of stress, X, will be zero. Writing and P2, in (4), for Y, and respectively, we have for this case: It is clear that p and q will be given, in terms of p2, by a circular diagram constructed as shown in fig. 4. If CA is drawn, at an angle 20 to OCN, to meet the circle at A, the co-ordinates ON, AN of A will represent p and q respectively.

Again, it is clear that the stress on AB (fig. 3) will not be af fected by the addition of a third principal stress acting on the triangular faces of the prism; so, in the general case, the circle BAX still gives the stresses on planes which are to the direction of In the same way, if OE (fig. 4) represents points on a circle having EB as diameter will relate the normal and tangential components of stress for all planes parallel to and points on a circle having EX as diameter will relate these components for all planes parallel to P2.

On planes which are inclined to all three of the principal planes, the stresses, in the general case, will depend upon all three of P2, p3. But it may be proved that points taken, in fig. 4, to relate the normal and tangential components of stress on such planes will in all cases lie within the shaded area of the diagram. Thus, if the normal stress on a plane is specified by ON, the intensity of the tangential stress lies between limits given by AN and A'N.

Case of Two Equal and Opposite Principal If, in equations (6), we make p, equal and opposite to the stress on planes which are equally inclined to the principal planes of stress (20 =9o°) will be given by p=0, q= pi. (7) Hence we see that a state of stress represented by equal and opposite principal stresses of intensity p and -p is equivalent to a state of simple shearing stress, of intensity p, on planes inclined at 45° to the principal planes of stress.