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29.6 STRENGTH AT CRITICAL LOCATIONS
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The critical locations of strength-limited designs can be identified as regions in which load-induced stresses peak as a result of distribution of bending moment and/or changes in geometry. Since the strength at the critical location in the geometry and at condition of use is required, it is often necessary to reflect the manufacturing process in this estimation. For heat-treatable steels under static loading, an estimate of yield or proof strength is required, and under fatigue loading, an estimate of the ultimate strength of the endurance limit is needed for an adequacy assessment. For the design process, strength as a function of intensity of treatment is required. In Chap. 33, the quantitative estimation methods of Crafts and Lamont and of Grossmann and Field for heat-treatable steels are useful. For cold work and cold heading, the methods of Datsko and Borden give useful estimates. Consider an eyebolt of cold-formed 1045 steel hot-rolled quarter-inch-diameter rod in the geometry of Fig. 29.14. The properties of the hot-rolled bar are Sy = 60 kpsi Su = 92.5 kpsi m = 0.14 f = 0.58 o = 140 kpsi
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FIGURE 29.14 Geometry of a cold-formed eyebolt.
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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 29.23
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STRENGTH UNDER DYNAMIC CONDITIONS
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At section AA on the inside surface, the true strain is estimated as i = 1 2d ln 1 + 2 D = 1 2(0.25) ln 1 + 2 1
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= | 0.203| = 0.203 The yield strength of the surface material at this location is estimated as, from Table 33.1, quo = 0.203 = 0.1015 1+1 qyo = 0.1015 = 0.0844 1 + 2(0.1015)
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(Sy)tLc = o ( qyo)m = 140(0.0844)0.14 = 99 kpsi The ultimate strength at this location is estimated as, from Table 33.1, (Su)tLc = (Su)o exp ( quo) = 92.5 exp (0.1015) = 102.4 kpsi Both the yield strength and the ultimate strength have increased. They are nominally equal because the true strain of 0.203 exceeds the true strain of 0.14 which occurs at ultimate load. The yield strength has increased by 65 percent and the ultimate strength has increased by 11 percent at this location. The strength at the inside and outside surface at section BB has not changed appreciably. The changes at the sections above BB are improvements in accord with the local geometry. For dynamic strength, the endurance limits have changed in proportion to the changes in ultimate strength. At section AA the R. R. Moore endurance limit is estimated to be Se = Su 102.4 = = 51.2 kpsi 2 2
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an improvement of 11 percent. Since the strengths vary with position and stresses vary with position also, a check is in order to see if section AA or section BB is critical in a tensile loading of the eyebolt. The increase in yield strength and endurance limit due to cold work, while present, may not be helpful. Consider the strip spring formed from bar stock to the geometry of Fig. 29.15. Just to the right of the radius the original properties prevail, and the bending moment is only slightly less than to the left of section AA. In this case, the increased strength at the critical location is not really exploitable.
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FIGURE 29.15 A latching spring cold formed from 3 4-in-wide No. 12 gauge strip.
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STRENGTH UNDER DYNAMIC CONDITIONS 29.24
LOAD CAPABILITY CONSIDERATIONS
FIGURE 29.16 Logic flowchart for estimation of localized ultimate strength or endurance limit for heattreated steels.
For parts that are heat-treated by quenching and tempering, the methods and procedures are given in Fig. 29.16 (see Chap. 33). If a shaft has been designed, an adequacy assessment is required. An estimate of the strength at a location where the shaft steps in Fig. 29.17 from 1 to 1.125 in is necessary. The specifications include the material to be 4140 steel quenched in still oil with mild part agitation and tempered for 2 hours at 1000 F. The material earmarked for manufacture has a ladle analysis of
C Percent Multiplier 0.40 0.207 Mn 0.83 3.87 P 0.012 S 0.009 Si 0.26 1.18 Ni 0.11 1.04 Cr 0.94 3.04 Mo 0.21 1.65
The experience is that a grain size of 71 2 can be maintained with this material and heat treatment. The multipliers are determined by the methods of Chap. 33. The ideal critical diameter is estimated as DI = 0.207(3.87)(1.18)(1.04)(3.04)(1.65) = 4.93 in The factors are D = 5.3, B = 10, and f = 0.34. The addition factors are AMn = 2.1 ASi = 1.1 ANi = 0.03 ACr = 4.9 AMo = 3.78 A = 11.91
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