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

Question

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, [latex]D[/latex] = 2, [latex]d[/latex] = 1.8, and [latex]r[/latex] = 0.25. The material has [latex]S_{ut}[/latex] = 100 kpsi.

Accepted Answer

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

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

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

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

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

[latex] q=\frac{1}{1+\frac{\sqrt{a}}{\sqrt{r}}} [/latex] (6.13)

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

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

[latex] K_f=1+q\left(K_t-1\right) [/latex] (6.11b)

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

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

Table 6-6 Neuber's Constant for Steels
[latex] S _{ ut }( ksi ) [/latex]	[latex]\sqrt{a}\left(\operatorname{in}^{0.5}\right) [/latex]
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
[latex] S _{ ut }( kpsi ) [/latex]	[latex]\sqrt{a}\left(\operatorname{in}^{0.5}\right) [/latex]
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
[latex] S _{ ut }( kpsi ) [/latex]	[latex]\sqrt{a}\left(\operatorname{in}^{0.5}\right) [/latex]
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

Question 6.3: Determining Fatigue Stress-Concentration Factors Problem A r...

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