Page 79 - Tribology in Machine Design

66 Tribology in machine design

square of the distance OQ from the point of load application. This is an
indication of the rate at which stresses die out. The deflection of the surface
at a radial distance r is inversely proportional to r, and hence, is a hyperbola
asymptotic to axes OR and OZ. At the origin, the stresses and deflections
theoretically become infinite, and one must imagine the material near 0 cut
out, say, by a small hemispherical surface to which are applied distributed
forces that are statistically equivalent to the concentrated force P. Such a
surface is obtained by the yielding of the material.
An analogous case is that of concentrated loading along a line of length /
(case 2). Here, the force is P./l per unit length of the line. The result is a
normal stress directed through the origin and inversely proportional to the
first power of distance to the load, not fading out as rapidly. Again, the
stress approaches infinite values near the load. Yielding, followed by work-
hardening, may limit the damage. Stresses in a knife or wedge, which might
be used to apply the foregoing load, are given under case 3. The solution for
case 2 is obtained when 2y.=n, or when the wedge becomes a plane.
In the deflection equation of case 1, we may substitute for the force P, an
expression that is the product of a pressure p, and an elemental area, such as
the shaded area in Fig. 3.1. This gives a deflection at any point, M, on the
Figure 3.1 surface at a distance r = s away from the element, namely

where v is the Poisson ratio. The total deflection at M is the superposition
or integration over the loaded area of all the elemental deflections, namely

2
where 77 is an elastic constant (1 —v )/£. If the pressure is considered
uniform, as from a fluid, and the loaded area is a circle, the resulting
deflections, in terms of elliptic integrals, are given by two equations, one for
M outside the circle and one for M inside the circle. The deflections at the
centre are given under case 4 of Table 3.1. The stresses are also obtained by
a superposition of elemental stresses for point loading. Shear stress is at a
maximum below the surface.
If a rod in the form of a punch, die or structural column is pressed against
the surface of a relatively soft material, i.e. one with a modulus of elasticity
much less than that of the rod, the rod may be considered rigid, and the
distribution of deflection is initially known. For a circular section, with
deflection w constant over the circle, the results are listed in case 5. The
pressure p is least at the centre, where it is 0.5p avg, and it is infinite at the
edges. The resultant yielding at the edges is local and has little effect on the
general distribution of pressure. For a given total load, the deflection is
inversely proportional to the radius of the circle.