A notched plate made of the AISI 4340 (aircraft quality) steel of Table 14.1 has an elastic stress concentration factor of kt = 2.80. If the nominal stress is cycled between Smax = 750 and Smin = 50 MPa, how many cycles can be applied before fatigue cracking is expected?

Question

A notched plate made of the AISI 4340 (aircraft quality) steel of Table 14.1 has an elastic stress concentration factor of [latex]k_{t} = 2.80[/latex]. If the nominal stress is cycled between [latex]S_{max} = 750 ext{ and } S_{min} = 50[/latex] MPa, how many cycles can be applied before fatigue cracking is expected?

Accepted Answer

The identical situation has already been analyzed in Ex. 13.4 to estimate the local notch stresses and strains. In particular, the cyclic stress–strain curve and Neuber’s rule were used, as in Eqs. 14.51 and 14.52,

[latex]\varepsilon_{\max }=\frac{\sigma_{\max }}{E}+\left(\frac{\sigma_{\max }}{H^{\prime}} ight)^{1 / n^{\prime}}, \quad \frac{\left(k_t S_{\max } ight)^2}{E}=\sigma_{\max } \varepsilon_{\max }[/latex] (14.51)

[latex]\varepsilon_a=\frac{\sigma_a}{E}+\left(\frac{\sigma_a}{H^{\prime}} ight)^{1 / n^{\prime}}, \quad \frac{\left(k_t S_a ight)^2}{E}=\sigma_a \varepsilon_a[/latex] (14.52)

to obtain both the maximums and amplitudes of stress and strain. The resulting values are

[latex]\sigma_{\max }=972 MPa , \quad \quad \varepsilon_{\max }=0.02192[/latex]

[latex]\sigma_a=755 MPa , \quad \quad \varepsilon_a=0.00615[/latex]

Note that the steps followed in Ex. 13.4 correspond to (1) and (2) in Fig. 14.15. Plotting the stress–strain response as in step (3) gives the result shown previously as Fig. E13.4. Also, the mean stress during cycling is

[latex]\sigma_m=\sigma_{\max }-\sigma_a=972-755=217 MPa[/latex]

where cycle-dependent relaxation is assumed not to occur.

The life can now be estimated. If the Morrow mean stress method is chosen, [latex]ε_{a} ext{ and } σ_{m}[/latex] are needed, and we substitute material constants from Table 14.1 into Eq. 14.20

[latex]\varepsilon_a=\frac{\sigma_f^{\prime}}{E}\left(2 N^* ight)^b+\varepsilon_f^{\prime}\left(2 N^* ight)^c[/latex] (14.20)

to obtain

[latex]\varepsilon_a=\frac{1758}{207,000}\left(2 N^* ight)^{-0.0977}+2.12\left(2 N^* ight)^{-0.774}[/latex]

Substituting the value we found for [latex]ε_{a}[/latex] and solving numerically for N* gives

[latex]N^*=3011 ext { cycles }[/latex]

The life [latex]N_{f}[/latex] with the [latex]σ_{m}[/latex] effect included is then obtained from Eq. 14.23, for which [latex]N_{m i}^*=N^*[/latex].

[latex]N_f=N_{m i}^*\left(1-\frac{\sigma_m}{\sigma_f^{\prime}} ight)^{-1 / b}=3011\left(1-\frac{217}{1758} ight)^{1 / 0.0977}=781 ext { cycles }[/latex]

We could also choose a different mean stress method, such as the SWT parameter, in which case [latex]\varepsilon_a ext { and } \sigma_{\max }[/latex] are employed with Eq. 14.30,

[latex]\sigma_{\max } \varepsilon_a=\frac{\left(\sigma_f^{\prime} ight)^2}{E}\left(2 N_f ight)^{2 b}+\sigma_f^{\prime} \varepsilon_f^{\prime}\left(2 N_f ight)^{b+c}[/latex] (14.30)

giving [latex]N_f=1800[/latex] cycles

Table 14.1 Cyclic Stress–Strain and Strain–Life Constants for Selected Engineering Metals. $^1$
		Tensile Properties				Cyclic σ-ε Curve			Strain–Life Curve
Material	Source	$\sigma_o$	$\sigma_u$	$\tilde{\sigma}_{f B}$	% RA	E	$H^{\prime}$	$n^{\prime}$	$\sigma_f^{\prime}$	b	$\varepsilon_f^{\prime}$	c
(a) Steels
SAE 1015 (normalized)	(8)	228 (33.0)	415 (60.2)	726 (105)	68	207,000 (30,000)	1349 196)	0.282	1020 (148)	-0.138	0.439	−0.513
Man-Ten² (hot rolled)	(7)	332 (46.7)	557 (80.8)	990 (144)	67	203,000 (29,500)	1096 (159)	0.187	1089 (158)	-0.115	0.912	-0.606
RQC-100 (roller Q & T)	(2)	683 (99.0)	758 (110)	1186 (172)	64	200,000 (29,000)	903 (131)	0.0905	938 (136)	−0.0648	1.38	−0.704
SAE 1045 (HR & norm.)	(6)	382 (55.4)	621 (90.1)	985 (143)	51	202,000 (29,400)	1258 (182)	0.208	948 (137)	−0.092	0.260	−0.445
SAE 4142 (As Q, 670 HB)	(1)	1619 (235)	2450 (355)	2580 (375)	6	200,000 (29,000)	2810 (407)	0.040	2550 (370)	−0.0778	0.0032	−0.436
SAE 4142 (Q & T, 560 HB)	(1)	1688 (245)	2240 (325)	2650 (385)	27	207,000 (30,000)	4140 (600)	0.126	3410 (494)	-0.121	0.0732	−0.805
SAE 4142 (Q & T, 450 HB)	(1)	1584 (230)	1757 (255)	1998 (290)	42	207,000 (30,000)	2080 (302)	0.093	1937 (281)	−0.0762	0.706	−0.869
SAE 4142 (Q & T, 380 HB)	(1)	1378 (200)	1413 (205)	1826 (265)	48	207,000 (30,000)	2210 (321)	0.133	2140 (311)	– 0.0944	0.637	−0.761
AISI 4340² (Aircraft Qual.)	(3)	1103 (160)	1172 (170)	1634 (237)	56	207,000 (30,000)	1655 (240)	0.131	1758 (255)	−0.0977	2.12	−0.774
AISI 4340 (409 HB)	(1)	1371 (199)	1468 (213)	1557 (226)	38	200,000 (29,000)	1910 (277)	0.123	1879 (273)	−0.0859	0.640	−0.636
Ausformed H-11 (660 HB)	(1)	2030 (295)	2580 (375)	3170 (460)	33	207,000 (30,000)	3475 (504)	0.059	3810 (553)	−0.0928	0.0743	−0.7144
(b) Other Metals
2024-T351 Al	(1)	379 (55.0)	469 (68.0)	558 (81.0)	25	73,100 (10,600)	662 (96.0)	0.070	927 (134)	−0.113	0.409	−0.713
2024-T4 Al³ (Prestrained)	(4)	303 (44.0)	476 (69.0)	631 (91.5)	35	73,100 (10,600)	738 (107)	0.080	1294 (188)	−0.142	0.327	−0.645
7075-T6 Al	(5)	469 (68.0)	578 (84)	744 (108)	33	71,000 (10,300)	977 (142)	0.106	1466 (213)	−0.143	0.262	−0.619
Ti-6Al-4V (soln. tr. & age)	(1)	1185 (172)	1233 (179)	1717 (249)	41	117,000 (17,000)	1772 (257)	0.106	2030 (295)	−0.104	0.841	−0.688
Inconel X (Ni base, annl.)	(1)	703 (102)	1213 (176)	1309 (190)	20	214,000 (31,000)	1855 (269)	0.120	2255 (327)	−0.117	1.16	−0.749
Notes: ¹The tabulated values either have units of MPa (ksi), or they are dimensionless. ²Test specimens prestrained, except at short lives, also periodically overstrained at long lives. ³For nonprestrained tests, use same constants, except $σ^{\prime}_{f} = 900(131) \text{and} b = −0.102$ . Sources: Data in (1) [Conle 84]; (2) author’s data on the ASTM Committee E9 material; (3) [Dowling 73]; (4) [Dowling 89] and [Topper 70]; (5) [Endo 69] and [Raske 72]; (6) [Leese 85]; (7) [Wetzel 77] pp. 41 and 66; (8) [Keshavan 67] and [Smith 70].

Question 14.3: A notched plate made of the AISI 4340 (aircraft quality) ste...

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