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and the fatigue locus itself is expressed as Sa Se
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Sm Sy
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(17.28)
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Combining Eqs. (17.19) and (17.20) with (17.28) gives d= 32n K f Ma Se K f Ma Se
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Tm Sy Tm Sy
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(17.29) (17.30)
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1 32 = n d 3
2 1/2
which are called D.E. elliptic or ASME-elliptic equations. On the designer s fatigue diagram, the slope of a radial load line r is given by r= d 3 a 32K f Ma = = m d 3 16 3Tm 2 3 K f Ma Tm (17.31)
The expressions for d and n in Eqs. (17.29) and (17.30) are for a threat from fatigue failure. It is also possible on the first revolution to cause local yielding, which changes straightness and strength and involves now-unpredictable loading. The Langer line, Sa + Sm = Sy, predicts yielding on the first cycle.The point where the elliptic locus and the Langer line intersect is described by 2Se /Sy Sa = Se 1 + (Se /Sy)2 Sm 1 (Se /Sy)2 = Sy 1 + (Se /Sy)2 The critical slope contains this point: rcrit = 2(Se /Sy)2 1 (Se /Sy)2 (17.34) (17.32) (17.33)
If the load line slope r is greater than rcrit, then the threat is from fatigue. If r is less than rcrit, the threat is from yielding. For the Gerber fatigue locus, the intersection with the Langer line is described by Sa = Sm = and rcrit = Sa /Sm. Example 4. At the critical location on a shaft, the bending moment Ma is 2520 in lbf and the torque Tm is 6600 in lbf. The ultimate strength Sut is 80 kpsi, the yield strength Sy is 58 kpsi, and the endurance limit Se is 31.1 kpsi. The stress concentration
2 Sut 2Se Sy 2Se 2 Sut 1 2Se
1 + 1+
2 4(Sut S 2 ) y 2 (S ut /Se 2Sy)2
(17.35) (17.36)
2 4S e (1 Sy /Se) 2 Sut
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SHAFTS 17.17
SHAFTS
factor corrected for notch sensitivity Kf is 1.54. Using an ASME-elliptic fatigue locus, ascertain if the threat is from fatigue or yielding. Solution. From Eq. (17.31), r= From Eq. (17.34), rcrit = 2(31.1/58)2 = 0.807 1 (31.1/58)2 2(1.54)2520 3 6600 = 0.679
Since rcrit > r, the primary threat is from fatigue. Using the Gerber fatigue locus, rcrit = Sa /Sm = 26.18/31.8 = 0.823. For the distortion energy Gerber failure locus, the relation for the strength amplitude Sa is given in Eq. (29.34) and CSa in Eq. (29.35); these quantities are given by Eqs. (29.37) and (29.38), respectively, for the ASME-elliptic failure locus.
17.8 CRITICAL SPEEDS
Critical speeds are associated with uncontrolled large deflections, which occur when inertial loading on a slightly deflected shaft exceeds the restorative ability of the shaft to resist. Shafts must operate well away from such speeds. Rayleigh s equation for the first critical speed of a shaft with transverse inertial loads wi deflected yi from the axis of rotation for simple support is given by Ref. [17.6] as = g wiyi wiy2 i (17.37)
where wi is the inertial load and yi is the lateral deflection due to wi and all other loads. For the shaft itself, wi is the inertial load of a shaft section and yi is the deflection of the center of the shaft section due to all loads. Inclusion of shaft mass when using Eq. (17.37) can be done. Reference [17.7], p. 266, gives the first critical speed of a uniform simply supported shaft as = 2
EI 2 = 2 m
gEI A
(17.38)
Example 5. A steel thick-walled tube with 3-in OD and 2-in ID is used as a shaft, simply supported, with a 48-in span. Estimate the first critical speed (a) by Eq. (17.38) and (b) by Eq. (17.37). Solution. (a) A = (32 22)/4 = 3.927 in2, I = (34 24)/64 = 3.19 in4, w = A = 3.925(0.282) = 1.11 lbf/in. From Eq. (17.38), = 2 482 386(30)(106 )(3.19) = 782.4 rad/s = 7471 r/min 3.927(0.282)
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