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(28.4)
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when A < 0
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Figure 28.2 shows some experimental points from tests on gray cast iron. Example 3. A 1 4-in-diameter ASTM No. 40 cast iron pin with Sut = 40 kpsi and Suc = 125 kpsi is subjected to an axial compressive load of 800 lb and a torsional moment of 100 lb in. Estimate the factor of safety. Solution. The axial stress is x = The surface shear stress is xy = The principal stresses are A,B = x + y 2 16.3 2 x y 2 16.3 2
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2 2 2 + xy
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F 800 = = 16.3 kpsi A (0.25)2/4
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16T 16(100) = = 32.6 kpsi d 3 (0.25)3
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+ (32.6)2 = 25.45, 41.25 kpsi
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B 41.25 = = 1.64 A 25.45
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The rupture line is the 3 4 locus, and the factor of safety is = SucSut 1 (1 + r)Sut + Suc A
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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.
STRENGTH UNDER STATIC CIRCUMSTANCES 28.9
STRENGTH UNDER STATIC CIRCUMSTANCES
FIGURE 28.2 Experimental data from tests of gray cast iron subjected to biaxial stresses. The data were adjusted to correspond to Sut = 32 kpsi and Suc = 105 kpsi. Superposed on the plot are graphs of the maximum-normal-stress theory, the Coulomb-Mohr theory, and the modified Mohr theory. (Adapted from J. E. Shigley and L. D. Mitchell, Mechanical Engineering Design, 4th ed., McGrawHill, 1983, with permission.)
( 125)(40) = 1.30 [(1 1.64)(40) 125](25.45)
28.3 STRESS CONCENTRATION
Geometric discontinuities increase the stress level beyond the nominal stresses, and the elementary stress equations are inadequate estimators. The geometric discontinuity is sometimes called a stress raiser, and the domains of departure from the elementary equation are called the regions of stress concentration. The multiplier applied to the nominal stress to estimate the peak stress is called the stressconcentration factor, denoted by Kt or Kts, and is defined as Kt = max o Kts = max o (28.5)
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STRENGTH UNDER STATIC CIRCUMSTANCES 28.10
LOAD CAPABILITY CONSIDERATIONS
respectively. These factors depend solely on part geometry and manner of loading and are independent of the material. Methods for determining stress-concentration factors include theory of elasticity, photoelasticity, numerical methods including finite elements, gridding, brittle lacquers, brittle models, and strain-gauging techniques. Peterson [28.1] has been responsible for many useful charts. Some charts representing common geometries and loadings are included as Figs. 28.3 through 28.17. The user of any such charts is cautioned to use the nominal stress equation upon which the chart is based. When the region of stress concentration is small compared to the section resisting the static loading, localized yielding in ductile materials limits the peak stress to the approximate level of the yield strength. The load is carried without gross plastic distortion. The stress concentration does no damage (strain strengthening occurs), and it can be ignored. No stress-concentration factor is applied to the stress. For lowductility materials, such as the heat-treated and case-hardened steels, the full geometric stress-concentration factor is applied unless notch-sensitivity information to the contrary is available. This notch-sensitivity equation is K = 1 + qs(Kt 1) (28.6)
where K = the actual stress-concentration factor for static loading and qs = an index of sensitivity of the material in static loading determined by test. The value of qs for hardened steels is approximately 0.15 (if untempered, 0.25). For cast irons, which have internal discontinuities as severe as the notch, qs approaches zero and the full value of Kt is rarely applied. Kurajian and West [28.3] have derived stress-concentration factors for hollow stepped shafts. They develop an equivalent solid stepped shaft and then use the usual charts (Figs. 28.10 and 28.11) to find Kt. The formulas are
FIGURE 28.3 Bar in tension or simple compression with a transverse hole. o = F/A, where A = (w d)t, and t = thickness. (From Peterson [28.2].)
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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