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35.1 M. G. Fontana and N. D. Greene, Corrosion Engineering, 2d ed., McGraw-Hill, New York, 1978. 35.2 E. Rabald, Corrosion Guide, 2d ed., rev., Elsevier Scientific Publishing, Amsterdam, 1968. 35.3 R. J. Fabian and J. A. Vaccari (eds.), How Materials Stand Up to Corrosion and Chemical Attack, Materials Engineering, vol. 73, no. 2, February 1971, p. 36.
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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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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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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
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Joseph E. Shigley
Professor Emeritus The University of Michigan Ann Arbor, Michigan
36.1 DEFINITIONS AND NOTATION / 36.3 36.2 TRIAXIAL STRESS / 36.5 36.3 STRESS-STRAIN RELATIONS / 36.6 36.4 FLEXURE / 36.12 36.5 STRESSES DUE TO TEMPERATURE / 36.16 36.6 CONTACT STRESSES / 36.19 REFERENCES / 36.24
36.1 DEFINITIONS AND NOTATION
The general two-dimensional stress element in Fig. 36.1a shows two normal stresses x and y , both positive, and two shear stresses xy and yx, positive also. The element is in static equilibrium, and hence xy = yx. The stress state depicted by the figure is called plane or biaxial stress.
FIGURE 36.1 Notation for two-dimensional stress. (From Applied Mechanics of Materials, by Joseph E. Shigley. Copyright 1976 by McGraw-Hill, Inc. Used with permission of the McGraw-Hill Book Company.)
36.3 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.
STRESS 36.4
CLASSICAL STRESS AND DEFORMATION ANALYSIS
Figure 36.1b shows an element face whose normal makes an angle to the x axis. It can be shown that the stress components and acting on this face are given by the equations = x + y x y + cos 2 + xy sin 2 2 2 x y sin 2 + xy cos 2 2 (36.1)
(36.2)
It can be shown that when the angle is varied in Eq. (36.1), the normal stress has two extreme values. These are called the principal stresses, and they are given by the equation 1, 2 = x + y 2 x y 2
2 2 + xy 1/2
(36.3)
The corresponding values of are called the principal directions. These directions can be obtained from 2 = tan 1 2 xy x y (36.4)
The shear stresses are always zero when the element is aligned in the principal directions. It also turns out that the shear stress in Eq. (36.2) has two extreme values. These and the angles at which they occur may be found from 1, 2 = x y 2
2 2 + xy 1/2
(36.5)
2 = tan 1
x y 2 xy
(36.6)
The two normal stresses are equal when the element is aligned in the directions given by Eq. (36.6). The act of referring stress components to another reference system is called transformation of stress. Such transformations are easier to visualize, and to solve, using a Mohr s circle diagram. In Fig. 36.2 we create a coordinate system with normal stresses plotted as the ordinates. On the abscissa, tensile (positive) normal stresses are plotted to the right of the origin O, and compression (negative) normal stresses are plotted to the left. The sign convention for shear stresses is that clockwise (cw) shear stresses are plotted above the abscissa and counterclockwise (ccw) shear stresses are plotted below. The stress state of Fig. 36.1a is shown on the diagram in Fig. 36.2. Points A and C represent x and y, respectively, and point E is midway between them. Distance AB is xy and distance CD is yx. The circle of radius ED is Mohr s circle. This circle passes through the principal stresses at F and G and through the extremes of the shear stresses at H and I. It is important to observe that an extreme of the shear stress may not be the same as the maximum.
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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