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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 thinwalled spherical pressure vessel with a small
circular hole (Fig. 3.15a) and in the torsion of a thinwalled circular tube with a
small circular hole (Fig. 3.15b).
126 Chapter 3 TwoDimensional 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 TwoDimensional 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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