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(36.42)
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where the subscripts i and o refer to the inner and outer radii, respectively, and the subscripts t and l refer to the tangential (circumferential) and longitudinal directions. Radial stresses of lesser magnitude will also exist, although not at the inner or outer surfaces. If the tubing of Fig. 36.10b is thin, then the inner and outer stresses are equal, although opposite, and are lo = to = ( T)E 2(1 ) (36.43) ( T)E li = ti = 2(1 ) at points not too close to the tube ends.
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36.6 CONTACT STRESSES
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When two elastic bodies having curved surfaces are pressed against each other, the initial point or line of contact changes into area contact, because of the deformation, and a three-dimensional state of stress is induced in both bodies. The shape of the contact area was originally deduced by Hertz, who assumed that the curvature of the
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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STRESS 36.20
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CLASSICAL STRESS AND DEFORMATION ANALYSIS
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two bodies could be approximated by second-degree surfaces. For such bodies, the contact area was found to be an ellipse. Reference [36.3] contains a comprehensive bibliography. As indicated in Fig. 36.11, there are four special cases in which the contact area is a circle. For these four cases, the maximum pressure occurs at the center of the contact area and is po = 3F 2 a2 (36.44)
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where a = the radius of the contact area and F = the normal force pressing the two bodies together. In Fig. 36.11, the x and y axes are in the plane of the contact area and the z axis is normal to this plane. The maximum stresses occur on this axis, they are principal stresses, and their values for all four cases in Fig. 36.11 are
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FIGURE 36.11 Contacting bodies having a circular contact area. (a) Two spheres; (b) sphere and plate; (c) sphere and spherical socket; (d) crossed cylinders of equal diameters.
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STRESS 36.21
STRESS
x = y = po po 1 + z2/a2
z 1 1 tan 1 (1 + ) a z/a 2(1 + z2/a2)
(36.45)
z =
(36.46)
These equations are plotted in Fig. 36.12 together with the two shear stresses xz and yz. Note that xy = 0 because x = y.
FIGURE 36.12 Magnitude of the stress components on the z axis below the surface as a function of the maximum pressure. Note that the two shear-stress components are maximum slightly below the surface. The chart is based on a Poisson s ratio of 0.30.
The radii a of the contact circles depend on the geometry of the contacting bodies. For two spheres, each having the same diameter d, or for two crossed cylinders, each having the diameter d, and in each case with like materials, the radius is a= 3Fd 1 2 8 E
(36.47)
where and E are the elastic constants. For two spheres of unlike materials having diameters d1 and d2, the radius is a=
2 1 1 1 2 3F d1d2 2 + E1 E2 8 d1 + d2 1/3
(36.48)
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STRESS 36.22
CLASSICAL STRESS AND DEFORMATION ANALYSIS
For a sphere of diameter d and a flat plate of unlike materials, the radius is a=
2 2 3Fd 1 1 1 2 + E1 E2 8 1/3
(36.49)
For a sphere of diameter d1 and a spherical socket of diameter d2 of unlike materials, the radius is a=
2 2 1 1 1 2 3F d1d2 + E1 E2 8 d2 d1 1/3
(36.50)
Contacting cylinders with parallel axes subjected to a normal force have a rectangular contact area. We specify an xy plane coincident with the contact area with the x axis parallel to the cylinder axes. Then, using a right-handed coordinate system, the stresses along the z axis are maximum and are x = 2 po 1+ z2 b2
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