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Unformatted text preview: ely. These factors are determined from curves 6, 7, and 9 in Fig. 3.17. Thus, given a set of shaft dimensions and the loading, formulas (a) and (b) lead to the value of the
maximum principal stress 1.
In addition, note that a shear force V may also act on the shaft (as in
Fig. D of Table 3.2). For slender members, however, this shear contributes very little to the deflection (Sec. 5.4) and to the maximum stress.
3.13 CONTACT STRESSES Application of a load over a small area of contact results in unusually high stresses.
Situations of this nature are found on a microscopic scale whenever force is transmitted through bodies in contact. There are important practical cases when the
3.13 Contact Stresses 129 ch03.qxd 12/20/02 7:20 AM Page 130 geometry of the contacting bodies results in large stresses, disregarding the stresses
associated with the asperities found on any nominally smooth surface. The Hertz
problem relates to the stresses owing to the contact of a sphere on a plane, a sphere
on a sphere, a cylinder on a cylinder, and the like. The practical implications with respect to ball and roller bearings, locomotive wheels, valve tappets, and numerous
machine components are apparent.
Consider, in this regard, the contact without deformation of two bodies having
spherical surfaces of radii r1 and r2, in the vicinity of contact. If now a collinear pair
of forces P acts to press the bodies together, as in Fig. 3.18, deformation will occur,
and the point of contact O will be replaced by a small area of contact. A common
tangent plane and common normal axis are denoted Ox and Oy, respectively. The
first steps taken toward the solution of this problem are the determination of the
size and shape of the contact area as well as the distribution of normal pressure acting on the area. The stresses and deformations resulting from the interfacial pressure are then evaluated.
The following assumptions are generally made in the solution of the contact
1. The contacting bodies are isotropic and elastic.
2. The contact areas are essentially flat and small relative to the radii of curvature
of the undeformed bodies in the vicinity of the interface.
3. The contacting bodies are perfectly smooth, and therefore only normal pressures
need be taken into account.
The foregoing set of assumptions enables an elastic analysis to be conducted.
Without going into the derivations, we shall, in the following paragraphs, introduce
some of the results.* It is important to note that, in all instances, the contact pressure varies from zero at the side of the contact area to a maximum value c at its
FIGURE 3.18. Spherical surfaces of two bodies compressed
by forces P. *A summary and complete list of references dealing with contact stress problems are
given by Refs. 3.12 through 3.16. 130 Chapter 3 Two-Dimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 131 Two Spherical Surfaces in Contact
Because of forces P (Fig. 3.18), the contact pressure is distributed over a small circle
of radius a given by
a = 0.88 B P1E1 + E22r1r2
E1E21r1 + r22 1/3 (3.54) where E1 and E2 ( r1 and r2 ) are the respective moduli of elasticity (radii) of the
spheres. The force P causing the contact pressure acts in the direction of the normal
axis, perpendicular to the tangent plane passing through the contact area. The
maximum contact pressure is found to be
c = 1.5 P
a2 (3.55) This is the maximum principal stress owing to the fact that, at the center of the
contact area, material is compressed not only in the normal direction but also in
the lateral directions. The relationship between the force of contact P, and the relative displacement of the centers of the two elastic spheres, owing to local deformation, is
= 0.77 B P ¢
≤ ¢ + ≤R
E2 1/3 2 (3.56) In the special case of a sphere of radius r contacting a body of the same material but having a flat surface (Fig. 3.19a), substitution of r1 = r, r2 = q , and
E1 = E2 = E into Eqs. (3.54) through (3.56) leads to
a = 0.88 ¢
1/3 PE 2
= 0.62 ¢ 2 ≤ ,
1/3 c P2
= 1.54 ¢ 2 ≤
2E r 1/3 (3.57) For the case of a sphere in a spherical seat of the same material (Fig. 3.19b) substituting r2 = - r2 and E1 = E2 = E in Eqs. (3.54) through (3.56), we obtain
a = 0.88 B
= 1.54 B 2Pr1r2
E1r2 - r12
1/3 P21r2 - r12
2E 2r1r2 R 1/3 c = 0.62 B PE 2 ¢ r2 - r1 2
2r1r2 1/3 (3.58) FIGURE 3.19. Contact load: (a) in sphere on
a plane; (b) in ball in a spherical seat. 3.13 Contact Stresses 131 ch03.qxd 12/20/02 7:20 AM Page 132 FIGURE 3.20. Contact load: (a) in two cylindrical rollers; (b) in
cylinder on a plane. Two Parallel Cylindrical Rollers
Here the contact area is a narrow rectangle of width 2b and length L (Fig. 3.20a).
The maximum contact pressure is given by
c b=B where = 2P
L1r1 + r22
1 (3.59) + 1E2 2
2 ≤R 1/2 (3.60) In this expression, Ei1 i2 and ri, with i = 1, 2, are the moduli of elasticity (Poisson’s
ratio) of the two rollers and the...
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This note was uploaded on 01/09/2011 for the course ME 201 taught by Professor Yok during the Spring '08 term at Boğaziçi University.
- Spring '08