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This value of the constructive strain allows estimation of the exponent c from 1 = f N c: f c= log 1/ f log (0.000 365/0.821) = = 0.4788 log Ne log 107
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Now Eq. (29.24) can be written as Sf = 103(0.14) 0.14 exp (0.14)0.8210.14N f 0.4778(0.14) exp (0.821Nf 0.4788) which simplifies to Sf = 151.8N f 0.067 exp (0.821N f 0.4788) Table 29.12 can be constructed. See Ref. [29.8] for notch-sensitivity corrections for low cycle strengths.
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29.7 COMBINED LOADING
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Simple loading is regarded as an influence that results in tension, compression, shear, bending, or torsion, and the stress fields that result are regarded as simple. Combined loading is the application of two or more of these simple loading schemes. The stresses that result from both simple and combined loading are three-dimensional. Applying the adjective combined to stresses is inappropriate. The nature of both yielding and fatigue for ductile materials is best explained by distortion-energy (octahedral shear, Henckey-von Mises) theory. For variable loading, the stress state is plotted on a modified Goodman diagram that has tensile mean stresses as abscissa and
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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 29.28
LOAD CAPABILITY CONSIDERATIONS
TABLE 29.12 Fatigue Strength Ratio S f /Su as a Function of Cycles to Failure for Annealed 4340 Steel
tensile stress amplitude as ordinate. The stress amplitude is that present in a uniform tension that induces the same distortion-energy amplitude (octahedral shear amplitude) as is present in the critical location of the machine part. The steady stress is that stress present in a uniform tension that induces the same steady distortion energy (steady octahedral shear) as is present in the critical location of the machine part.The plotting process involves conversion of the actual stress state to the equivalent uniform tension circumstances. The von Mises axial tensile stress that has the same distortion energy as a general three-dimensional stress field is, in terms of the ordered principal stresses 1, 2, and 3, v = ( 1 2)2 + ( 2 3)2 + ( 3 1)2 2
(29.25)
If one of the principal stresses is zero and the other two are A and B, then
2 2 v = ( A + B A B)1/2
(29.26)
If the axes xy are not principal, then
2 2 v = ( x + y x y + 3 2 )1/2 xy
(29.27)
If the concern is yielding, then yielding begins when the von Mises stress equals the tensile value of Sy. If the concern is fatigue, then failure occurs when the von Mises steady stress and amplitude equal the simple steady tension and amplitude that result in failure. If Eq. (29.26) is equated to a critical value cr, then
2 2 2 A + B A B = cr
Treating the preceding equation as a quadratic in A, we have A = 1 1 B 2 2 (2 cr)2
2 3 B
(29.28)
On a plot in the A B plane, the critical-stress magnitude can be observed at six places, three tensile and three compressive.The locus is an ellipse with the major axis
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.
STRENGTH UNDER DYNAMIC CONDITIONS 29.29
STRENGTH UNDER DYNAMIC CONDITIONS
FIGURE 29.18 The distortion-energy critical-stress ellipse. For any point A, B on the ellipse, the uniaxial tension with the same distortion energy is the positive abscissa intercept cr, 0.
having a unity slope, as depicted in Fig. 29.18. The octahedral stress (von Mises stress) tensile equivalent is the A-axis intercept of the ellipse. For the Goodman diagram, the transformation is to move the point representing the stress condition A, B to the abscissa, while staying on the ellipse. This is done by Eq. (29.26). For three-dimensional stress, the surface is an ellipsoid in the 1 2 3 space, and the transformation is accomplished by Eq. (29.25). Figure 29.19 shows the conversions of the steady-stress condition and the stress-amplitude condition to the respective simpletension equivalents for the purposes of plotting a point representing the equivalent stress state on the modified Goodman diagram. In Fig. 29.20 an element on a shaft sees a steady torque and fully reversed bending stresses. For the steady-stress element, xym = 16T d 3 Am = 16T d 3 Bm = 16T d 3
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