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2 2 vm = ( Am + Bm Am Bm)1/2 =
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3 16T d 3
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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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STRENGTH UNDER DYNAMIC CONDITIONS
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FIGURE 29.19 The principal stresses due to the steady stresses Am and Bm appear on the distortion-energy ellipse as point D. The transform to equivalent distortion energy in tension is point E, which becomes the abscissa of point P. The principal stresses due to stress amplitude Aa and Ba appear as point D ; the transform is E , which becomes the ordinate of point P.
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FIGURE 29.20 (a) A shaft subjected to a steady torque T and completely reversed flexure due to bending moment M; (b) the mean-stress element and the stress-amplitude element.
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29.30 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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STRENGTH UNDER DYNAMIC CONDITIONS 29.31
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FIGURE 29.21 Designer s fatigue diagram for geared shaft showing load line of slope r = 2M/( 3T ), the operating point P, using the Goodman failure locus, and the designer s line reflecting a design factor of n.
For the amplitude-stress element, x,max = Aa = 32M d 3 32M d 3 x,min = Ba = 0 32M d 3
and the corresponding von Mises stress is
2 2 va = ( Aa + Ba Aa Ba)1/2 =
32M d 3
If this is an element of a geared shaft, then M and T are proportional and the locus of possible points is a radial line from the origin with a slope of r= va 32M = vm d 3 d3 2M = 3T 3 16T (29.29)
This is called the load line. If data on failures have been collected and converted to von Mises components and a Goodman line is an adequate representation of the failure locus, then for the designer s line in Fig. 29.21, va vm 1 + = Se Su n (29.30)
where n = design factor. Substituting for va and vm and solving for d, we obtain d= 32n M 3T + Se 2Su
(29.31)
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STRENGTH UNDER DYNAMIC CONDITIONS 29.32
LOAD CAPABILITY CONSIDERATIONS
Data points representing failure are plotted embracing a significant stress commitment, and the plotted load line represents the same belief. It is appropriate that equations such as Eq. (29.31) be labeled with two adjectives: (1) significant stress and (2) failure locus. For example, Eq. (29.31) could be called a distortion-energy Goodman equation. For the case where moments, torques, and thrusts contribute both steady and alternating components of stress, then the distortion-energy Goodman equation for the critical location is 16 d 3Se where 2Ma + Pa d 4
2 2 + 3T a 1/2
16 d 3Su
2Mm +
Pmd 4
2 + 3T m
1 = 0 (29.32) n
n= Ma = Mm = Ta = Tm = Pa = Pm = Se = Su = d=
design factor component of bending moment causing flexural stress amplitude component of bending moment causing flexural stress, steady component of torque causing shear-stress amplitude component of torque causing shear stress, steady component of axial thrust causing tensile-stress amplitude component of axial thrust causing tensile stress, steady local fatigue strength of shaft material ultimate local tensile strength local shaft diameter
Since the equation cannot be solved for d explicitly, numerical methods are used.
29.8 SURFACE FATIGUE
When cylinders are in line contact, sustained by a force F, a flattened rectangular zone exists in which the pressure distribution is elliptical. The half width of the contact zone b is b= 2F [1 2]/E1 + [1 2]/E2 1 2 (1/d1) + (1/d2) (29.33)
The largest stress in magnitude is compressive and exists on the z axis. As a pressure, its magnitude is pmax = 2F b (29.34)
Along the z axis, the orthogonal stresses are [29.6] x = 2pmax y = pmax z = 2 1+ z b
z b 1+ z b
(29.35) 2 z b (29.36) (29.37)
1 1 + (z/b)2
pmax (z/b)2
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