## La

(For a flat plate R2 is infinite, for a cylinder in a cylindrical bearing Ri is negative). Stress conditions at the surface on the load axis are then:

az =<re = —po oy = —po ax= —2vpo (along cylinder length) The maximum shear stress is:

tmax = 0.295po - 0.3po occurring at a depth beneath the surface of 0.786 b and on planes at 45° to the load axis.

In cases such as gears, bearings, cams, etc. which (as will be discussed later) can be likened to the contact of parallel cylinders, this shear stress will reduce gradually to zero as the rolling load passes the point in question and rise again to its maximum value as the next load contact is made. However, this will not be the greatest reversal of shear stress since there is another shear stress on planes parallel and perpendicular to the load axes known as the "alternating" or "reversing" shear stress, at a depth of 0.5 b and offset from the load axis by 0.85 b, which has a maximum value of 0.256 po which changes from positive to negative as the load moves across contact.

The maximum shear stress on 45° planes thus varies between zero and 0.3 po (approx) with an alternating component of 0.15 po about a mean of 0.15 po. The maximum alternating shear stress, however, has an alternating component of 0.256 po about a mean of zero -see Fig. 10.4. The latter is therefore considerably more significant from a fatigue viewpoint.

N.B.: The above formulae assume the length of the cylinders to be very large in comparison with their radii. For short cylinders and/or cylinder/plate contacts with widths less than six times the contact area (or plate thickness less than six times the depth of the maximum shear stress) actual stresses can be significantly greater than those estimated by the given equations.

### 10.1.3. Combined normal and tangential loading

In normal contact conditions between contacting cylinders, gears, cams, etc. friction will be present reacting the sliding (or tendency to slide) of the mating surfaces. This will affect the stresses which are set up and it is usual in such cases to take the usual relationship between normal and tangential forces in the presence of friction viz. F = fiR or q = p.po where q is the tangential pressure distribution, assumed to be of the same form as that of the normal pressure. Smith and Liu(20) have shown that with such an assumption:

(a) A shear stress now exists on the surface at the contact point introducing principal stresses which are different from ax, crv and crz of the normal loading case.

(b) The maximum shear stress may exist either at the surface or beneath it depending on whether p. is greater than or less than 1/9 respectively.

(c) The stress range in the y direction is increased by almost 90% on the normal loading value and there is also a reversal of sign. A useful summary of stress distributions in graphical form is given by Lipson and Juvinal<21).

10.1.4. Special case 2 - Contacting spheres

For contacting spheres, eqns. (10.9) and (10.8) become Maximum compressive stress (normal to surface)

with a maximum value of oc = — l.SP/na oc = — l.SP/na

Contact dimensions (circular)

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