9 Contact Stresses

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CONTACT STRESSES
and figures are included for use in solving various Hertzian problems. Rolling contact
problems, contact stresses with friction, and the fatigue behavior of bodies subjected
to repeated applications of contact loading are briefly described.
9.1 NOTATION
The units for most of the definitions are given in parentheses, using L for length and
F for force.
a Semimajor axis of contact ellipse (L)
A, B Coefficients in equation for locus of contacting points initially separated
by the same distance (L −1 )
Ac Contact area (L 2 )
b Semiminor axis of contact ellipse (L)
d Rigid distance of approach of contacting bodies (L); also total elastic
deformation at origin
E Modulus of elasticity (F/L 2 )
f Friction coefficient
F Force (F)
k Ratio of major to minor axis of contact ellipse, = b/a
p Pressure (F/L 2 )
q Line-distributed load (F/L)
R and R ′ Minimum and maximum radii of curvature for contacting surfaces (L)
zs Distance below center of contact ellipse where maximum shear stress
occurs (L)
θ Angle between planes containing principal radii of curvature for contacting
bodies
ν Poisson’s ratio
σc Maximum compressive stress (F/L 2 )
σys Yield strength in tension (F/L 2 )
τmax Maximum shear stress (F/L 2 )
1, 2 Subscripts designating bodies 1 and 2
Geometric Characteristics of Surfaces Consider a surface F(x, y, z) = 0
(Fig. 9-1). At any point on the surface, the normal to the surface is grad(F). Leta
plane pass through the length of the surface normal at point O, creating a normal
section. The intersection of this plane with the surface is a curve in the normal section
of the surface at point O. An infinite number of normal sections may be taken
through any point on the surface. The following theorem holds [9.1]: At any point
of a surface, two normal sections exist for which the radii of curvature are a minimum
and a maximum; the planes that each contain one of these normal sections are