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(36.51) 2z b
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(36.52)
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(36.53)
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where the maximum pressure occurs at the origin of the coordinate system in the contact zone and is po = 2F bl (36.54)
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where l = the length of the contact zone measured parallel to the cylinder axes, and b = the half width. Equations (36.51) to (36.53) give the principal stresses. These equations are plotted in Fig. 36.13. The corresponding shear stresses can be found from a Mohr s circle; they are plotted in Fig. 36.14. Note that the maximum is either xz or yz depending on the depth below the contact surface. The half width b depends on the geometry of the contacting cylinders. The following cases arise most frequently: Two cylinders of equal diameter and of the same material have a half width of b= 2Fd 1 2 l E
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(36.55)
For two cylinders of unequal diameter and unlike materials, the half width is b=
2 2 2F d1d2 1 1 1 2 + l d1 + d2 E1 E2 1/2
(36.56)
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STRESS 36.23
STRESS
FIGURE 36.13 Magnitude of the principal stresses on the z axis below the surface as a function of the maximum pressure for contacting cylinders. Based on a Poisson s ratio of 0.30.
FIGURE 36.14 Magnitude of the three shear stresses computed from Fig. 36.13.
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STRESS 36.24
CLASSICAL STRESS AND DEFORMATION ANALYSIS
For a cylinder of diameter d in contact with a flat plate of unlike material, the result is b=
2 2 2Fd 1 1 1 2 + l E1 E2 1/2
(36.57)
The half width for a cylinder of diameter d1 pressing against a cylindrical socket of diameter d2 of unlike material is b=
2 2 2F d1d2 1 1 1 2 + l d2 d1 E1 E2 1/2
(36.58)
REFERENCES
36.1 F. R. Shanley, Strength of Materials, McGraw-Hill, New York, 1957, p. 509. 36.2 W. C. Young, Roark s Formulas for Stress and Strain, 6th ed., McGraw-Hill, 1989, p. 348 359. 36.3 J. L. Lubkin, Contact Problems, in W. Flugge (ed.), Handbook of Engineering Mechanics, McGraw-Hill, New York, 1962, pp. 42-10 to 42-12.
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Source: STANDARD HANDBOOK OF MACHINE DESIGN
DEFLECTION
Joseph E. Shigley
Professor Emeritus The University of Michigan Ann Arbor, Michigan
Charles R. Mischke, Ph.D., P.E.
Professor Emeritus of Mechanical Engineering Iowa State University Ames, Iowa
37.1 37.2 37.3 37.4
STIFFNESS OR SPRING RATE / 37.2 DEFLECTION DUE TO BENDING / 37.3 PROPERTIES OF BEAMS / 37.3 ANALYSIS OF FRAMES / 37.3
GLOSSARY OF SYMBOLS
a A b C D, d E F G I J k K M N Dimension Area Dimension Constant Diameter Young s modulus Force Shear modulus Second moment of area Second polar moment of area Spring rate Constant Length Moment Number
37.1 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.
DEFLECTION 37.2
CLASSICAL STRESS AND DEFORMATION ANALYSIS
q Q R T U V w W x y
Unit load Fictitious force Support reaction Torque Strain energy Shear force Unit weight Total weight Coordinate Coordinate Deflection Slope, torsional deflection
37.1 STIFFNESS OR SPRING RATE
The spring rate (also called stiffness or scale) of a body or ensemble of bodies is defined as the partial derivative of force (torque) with respect to colinear displacement (rotation). For a helical tension or compression spring, F= where d 4Gy 8D3N thus k= d 4G F = y 8D3N (37.1)
D = mean coil diameter d = wire diameter N = number of active turns
In a round bar subject to torsion, T= GJ thus k= T GJ = (37.2)
and the tensile force in an elongating bar of any cross section is F= AE thus k= F AE = (37.3)
If k is constant, as in these cases, then displacement is said to be linear with respect to force (torque). For contacting bodies with all four radii of curvature finite, the approach of the bodies is proportional to load to the two-thirds power, making the spring rate proportional to load to the one-third power. In hydrodynamic film bearings, the partial derivative would be evaluated numerically by dividing a small change in load by the displacement in the direction of the load.
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