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## 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.

## 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