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FIGURE 28.16 Grooved round bar in bending. o = Mc/I, where c = d/2 and I = d4/64. (From Peterson [28.2].)
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28.4.1 Stress Intensities In Fig. 28.18a, suppose the length of the tensile specimen is large compared to the width 2b. Also, let the crack, of length 2a, be centrally located.Then a stress-intensity factor K can be defined by the relation K0 = ( a)1/2 (28.8)
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where = average tensile stress. The units of K0 are kpsi in1/2 or, in SI, MPa m1/2.
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FIGURE 28.17 Grooved round bar in torsion. o = Tc/J, where c = d/2 and J = d4/32. (From Peterson [28.2].)
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FIGURE 28.18 Typical crack occurrences. (a) Bar in tension with interior crack; (b) bar in tension with edge crack; (c) flexural member of rectangular cross section with edge crack; (d) pressurized cylinder with radial edge crack parallel to cylinder axis.
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Since the actual value of K for other geometries depends on the loading too, it is convenient to write Eq. (28.8) in the form KI = C ( a)1/2 where C= KI K0 (28.10) (28.9)
Values of this ratio for some typical geometries and loadings are given in Figs. 28.19 and 28.20. Note that Fig. 28.18 must be used to identify the curves on these charts. Additional data on stress-intensity factors can be found in Refs. [28.5], [28.6], and [28.7]. The Roman numeral I used as a subscript in Eq. (28.9) refers to the deformation mode. Two other modes of fracture not shown in Fig. 28.18 are in-plane and out-ofplane shear modes, and these are designated by the Roman numerals II and III. These are not considered here (see Ref. [28.4], p. 262).
28.4.2 Fracture Toughness When the stress of Eq. (28.9) reaches a certain critical value, crack growth begins, and the equation then gives the critical-stress-intensity factor KIc.This is also called the
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STRENGTH UNDER STATIC CIRCUMSTANCES 28.19
STRENGTH UNDER STATIC CIRCUMSTANCES
FIGURE 28.19 Stress-intensity charts for cracks shown in Fig. 28.18a and c. Letters A and B identify the ends of the crack shown in Fig. 28.18a. Values of /h > 2 will produce curves closer to the curve for pure bending.
fracture toughness. Since it is analogous to strength, we can define the design factor as n= Kc K (28.11)
Some typical values of Kc are given in Table 28.1. For other materials, see Ref. [28.8].
28.5 NONFERROUS METALS
Designing for static loads with aluminum alloys is not much different from designing for the steels. Aluminum alloys, both cast and wrought, have strengths in tension and compression that are about equal. The yield strengths in shear vary from about 55 to 65 percent of the tensile yield strengths, and so the octahedral shear theory of failure is valid. The corrosion resistance (see Chap. 35), workability, and weldability obtainable from some of the alloys make this a very versatile material for design.And the extrusion capability means that a very large number of wrought shapes are available.
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STRENGTH UNDER STATIC CIRCUMSTANCES 28.20
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FIGURE 28.20 Stress-intensity chart for cracks shown in Figs. 28.18b and d. The curve h/b = has bending constraints acting on the member.
However, these alloys do have a temperature problem, as shown by the curves of strength versus temperature in Fig. 28.21. Other aluminum alloys will exhibit a similar characteristic. Alloying elements used with copper as the base element include zinc, lead, tin, aluminum, silicon, manganese, phosphorus, and beryllium. Hundreds of variations in the percentages used are possible, and consequently, the various copper alloys may have widely differing properties. The primary consideration in selecting a copper alloy may be the machinability, ductility, hardness, temperature properties, or corrosion resistance. Strength is seldom the primary consideration. Because of these variations in properties, it is probably a good idea to consult the manufacturer concerning new applications until a backlog of experience can be obtained.
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