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Chapter 6

Q. 6.3

Determining Fatigue Stress-Concentration Factors

Problem    A rectangular, stepped bar similar to that shown in Figure 4-36 (p. 190) is to be loaded in bending. Determine the fatigue stress-concentration factor for the given dimensions.

Given    Using the nomenclature in Figure 4-36, D = 2, d = 1.8, and r = 0.25. The material has S_{ut} = 100 kpsi.

F4-36

Step-by-Step

Verified Solution

1    The geometric stress-concentration factor K_{t} is found from the equation in Figure 4-36:

K_t=A\left(\frac{r}{d}\right)^b          (a)

where A and b are given in the same figure as a function of the D/d ratio, which is 2 / 1.8 = 1.111. For this ratio, A = 1.014 7 and b = –0.217 9, giving

K_t=1.0147\left(\frac{0.25}{1.8}\right)^{-0.2179}=1.56          (b)

2    The notch sensitivity q of the material can be found by using the Neuber factor \sqrt{a} from Figure 6-35 and Tables 6-6 to 6-8 in combination with equation 6.13 (p. 345), or by reading q directly from Figure 6-36. We will do the former. The Neuber factor from Table 6-6 for S_{ut} = 100 kpsi is 0.062. Note that this is the square root of a:

q=\frac{1}{1+\frac{\sqrt{a}}{\sqrt{r}}}        (6.13)

q=\frac{1}{1+\frac{\sqrt{a}}{\sqrt{r}}}=\frac{1}{1+\frac{0.062}{\sqrt{0.25}}}=0.89          (c)

3    The fatigue stress-concentration factor can now be found from equation 6.11b (p. 343):

K_f=1+q\left(K_t-1\right)         (6.11b)

K_f=1+q\left(K_t-1\right)=1+0.89(1.56-1)=1.50          (d)

4    The files EX06-03 are on the CD-ROM.

 

Table 6-6
Neuber’s Constant for Steels
S _{ ut }( ksi ) \sqrt{a}\left(\operatorname{in}^{0.5}\right)
50 0.130
55 0.118
60 0.108
70 0.093
80 0.080
90 0.070
100 0.062
110 0.055
120 0.049
130 0.044
140 0.039
160 0.031
180 0.024
200 0.018
220 0.013
240 0.009

 

Table 6-7
Neuber’s Constant for Annealed Aluminum
S _{ ut }( kpsi ) \sqrt{a}\left(\operatorname{in}^{0.5}\right)
10 0.500
15 0.341
20 0.264
25 0.217
30 0.180
35 0.152
40 0.126
45 0.111

 

Table 6-8
Neuber’s Constant for Hardened Aluminum
S _{ ut }( kpsi ) \sqrt{a}\left(\operatorname{in}^{0.5}\right)
15 0.475
2 0.380
30 0.278
40 0.219
50 0.186
60 0.162
70 0.144
80 0.131
90 0.122
F6-35
F6-36.1
F6-36.2