chap3_0130473928

Qxd 122002 720 am page 129 figure 317 neubers

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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 problem: 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 center. 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 R 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 1 121 1 + = 0.77 B P ¢ ≤ ¢ + ≤R r1 r2 E1 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 2Pr a = 0.88 ¢ ≤, E 1/3 PE 2 = 0.62 ¢ 2 ≤ , 4r 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 R, E1r2 - r12 1/3 P21r2 - r12 2E 2r1r2 R 1/3 c = 0.62 B PE 2 ¢ r2 - r1 2 ≤R 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 bL 4Pr1r2 1¢ L1r1 + r22 E1 2 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.

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