52a 352b 352c 125 ch03qxd 122002 720 am page

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Unformatted text preview: oncentration factor, defined as the ratio of the maximum stress at the hole to the nominal stress o is therefore k = 3 o/ o = 3. To depict the variation of r1r, /22 and 1r, /22 over the distance from the origin, dimensionless stresses are plotted against the dimensionless radius in Fig. 3.14. The shearing stress r 1r, /22 = 0. At a distance of twice the diameter of the hole, that is, r = 4a, we obtain L 1.037 o and r L 0.088 o. Similarly, at a distance r = 9a, we have L 1.006 o and r L 0.018 o, as is observed in the figure. Thus, simple tension prevails at a distance of approximately nine radii; the hole has a local effect on the distribution of stress. This is a verification of SaintVenant’s principle. The results expressed by Eqs. (3.52) are applied, together with the method of superposition, to the case of biaxial loading. Distributions of maximum stress 1r, /22, obtained in this way (Prob. 3.36), are given in Fig. 3.15. Such conditions of stress concentration occur in a thin-walled spherical pressure vessel with a small circular hole (Fig. 3.15a) and in the torsion of a thin-walled circular tube with a small circular hole (Fig. 3.15b). 126 Chapter 3 Two-Dimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 127 FIGURE 3.15. Tangential stress distribution for = ; /2 in the plate with the circular hole subject to biaxial stresses: (a) uniform tension; (b) pure shear. FIGURE 3.16. Elliptical hole in a plate under uniaxial tension. It is noted that a similar stress concentration is caused by a small elliptical hole in a thin, large plate (Fig. 3.16). It can be shown that the maximum tensile stress at the ends of the major axis of the hole is given by max = o ¢1 + 2 ≤ b a (3.53) Clearly, the stress increases with the ratio b/a. In the limit as a : 0, the ellipse becomes a crack of length 2b, and a very high stress concentration is produced; material will yield plastically around the ends of the crack or the crack will propagate. To prevent such spreading, holes may be drilled at the ends of the crack to effectively increase the radii to correspond to a smaller b/a. Thus, a high stress concentration is replaced by a relatively smaller one. 3.12 NEUBER’S DIAGRAM Several geometries of practical importance, given in Table 3.2, were the subject of stress concentration determination by Neuber on the basis of mathematical analysis, as in the preceding example. Neuber’s diagram (a nomograph), which is used with the table for determining the stress concentration factor k for the configurations shown, is plotted in Fig. 3.17. In applying Neuber’s diagram, the first step is the calculation of the values of 2h/a and 2b/a. Given a value of 2b/a, we proceed vertically upward to cut the appropriate curve designated by the number found in column 5 of the table, then horizontally to the left to the ordinate axis. This point is then connected by a straight line to a point on the left abscissa representing 2h/a, according to either scale e or f as indicated in column 4 of 3.12 Neuber’s Diagram 127 ch03.qxd 12/20/02 7:20 AM Page 128 TABLE 3.2 Type of change of section A Type of loading P 2bt 3M 2b2t Tension Bending B P bt 6M b 2t Tension Bending C Nominal stress Scale for 2h/a Curve for k f 1 f 2 f 3 f 4 P 2bt f 5 Bending D Tension 3Mh 2t1c3 - h32 e 5 f 6 f 7 e 8 e 9 Tension Bending Direct shear Torsional shear P b2 4M b3 1.23V b2 2T b3 the table. The value of k is read off on the circular scale at a point located on a normal from the origin. [The values of (theoretical) stress concentration factors obtained from Neuber’s nomograph agree satisfactorily with those found by the photoelastic method.] Consider, for example, the case of a member with a single notch (Fig. B in the table), and assume that it is subjected to axial tension P only. For given a = 7.925 mm, h = 44.450 mm, and b = 266.700 mm, 2h/a = 2.37 and 2b/a = 5.80. Table 3.2 indicates that scale f and curve 3 are applicable. Then, as just described, the stress concentration factor is found to be k = 3.25. The path followed is denoted by the broken lines in the diagram. The nominal stress P/bt, multiplied by k, yields maximum theoretical stress, found at the root of the notch. EXAMPLE 3.5 A circular shaft with a circumferential circular groove (notch) is subjected to axial force P, bending moment M, and torque T (Fig. D of Table 3.2). Determine the maximum principal stress. 128 Chapter 3 Two-Dimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 129 FIGURE 3.17. Neuber’s nomograph. ¢ ≤+ C2 Solution For the loading described, the principal stresses occur at a point at the root of the notch which, from Eq. (1.16), are given by 2 = 1,2 x 2 ; x 2 xy , 3 =0 (a) where x and xy represent the normal and shear stresses in the reduced cross section of the shaft, respectively. We have x = ka My P + kb , A I xy = kt xy = kt Tr J or x = ka P 4M + kb 3 , 2 b b 2T b3 (b) Here ka, kb, and kt denote the stress concentration factors for axial force, bending moment, and torque, respectiv...
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