Fatigue Life of Instrument Panel with a Crack A long plate of an instrument is of width 2w and thickness t. The panel is subjected to an axial tensile load that varies from Pmin to Pmax with a complete cycle every 15 s. Before loading, on inspection, a central transverse crack of length 2a is

Question

Fatigue Life of Instrument Panel with a Crack

A long plate of an instrument is of width [latex]2w[/latex] and thickness [latex]t[/latex]. The panel is subjected to an axial tensile load that varies from [latex]P_{\min}[/latex] to [latex]P_{\max}[/latex] with a complete cycle every 15 s. Before loading, on inspection, a central transverse crack of length [latex]2a[/latex] is detected on the plate. Estimate the expected life.

Given: [latex]a[/latex] = 0.3 in., [latex]t[/latex] = 0.8 in., [latex]w[/latex] = 2 in., [latex]P_{\max}[/latex] = [latex]2P_{\min}[/latex] = 144 kips.

Assumption: The plate is made of an AISI 4340 tempered steel.

Accepted Answer

See Tables 6.1, 6.2, and 7.5.

The material and geometric properties of the panel are

[latex] \begin{array}{l} A=3.6 imes 10^{-10}, \quad n=3, \quad K_c=53.7 ksi \sqrt{\operatorname{in}_{.}} \quad S_y=218 ksi , \ \lambda=1.02, \quad ext { for } a / w=0.15 \quad ext { (Case A of Table } 6.1) \end{array} [/latex]

Note that the values of [latex]a[/latex] and t satisfy Table 6.2. The largest and smallest normal stresses are

[latex] \sigma_{\max }=\frac{P_{\max }}{2 w t}=\frac{144}{2(2)(0.8)}=45 ksi , \quad \sigma_{\min }=22.5 ksi [/latex]

The cyclical stress range is then Δσ = 45 − 22.5 = 22.5 ksi.

The final crack length at fracture, from Equation 7.39, is found to be

[latex] a_f=\frac{1}{\pi}\left(\frac{K_c}{\lambda \sigma_{\max }} ight)^2=\frac{1}{\pi}\left(\frac{53.7}{1.02 imes 45} ight)^2=0.436 in . [/latex]

Substituting the numerical values, Equation 7.41 results in

[latex] N=\frac{a_f^{1-n / 2}-a^{1-n / 2}}{A\left(1-\frac{n}{2} ight)(1.77 \lambda \Delta \sigma)^n} [/latex] (7.41)

[latex] N=\frac{0.436^{-0.5}-0.3^{-0.5}}{3.6\left(10^{-10} ight)(-0.5)[(1.77)(1.02)(22.5)]^3}=25,800 ext { cycles } [/latex]

With a period of 15 s, approximate fatigue life [latex]L[/latex] is

[latex] L=\frac{25,800(15)}{60(60)}=107.5 h [/latex]

Table 6.1 Geometry Factors λ for Some Initial Crack Shapes
Case A. Tension of a long plate with a central crack	[latex]a/w[/latex] 0.1 0.2 0.3 0.4 0.5 0.6	[latex]\lambda[/latex] 1.01 1.03 1.06 1.11 1.19 1.30
Case B. Tension of a long plate with an edge crack	[latex]a/w[/latex] 0 ([latex] w \longrightarrow \infty [/latex]) 0.2 0.4 0.5	[latex]\lambda[/latex] 1.12 1.37 2.11 2.83
Case C. Tension of a long plate with double-edge cracks	[latex]a/w[/latex] 0 ([latex] w \longrightarrow \infty [/latex]) 0.2 0.4 0.5 0.6	[latex]\lambda[/latex] 1.12 1.12 1.14 1.15 1.22
Case D. Pure bending of a beam with an edge crack	[latex]a/w[/latex] 0.1 0.2 0.3 0.4 0.5 0.6	[latex]\lambda[/latex] 1.02 1.06 1.16 1.32 1.62 2.10

Table 6.2 Yield Strength [latex]S_{y}[/latex] and Fracture Toughness [latex]K_{c}[/latex] for Some Engineering Materials
	[latex]S_{y}[/latex]		[latex]K_{c}[/latex]		Minimum Values of [latex]a[/latex] and [latex]t[/latex]
Metals	(MPa)	(ksi)	[latex]\sqrt{m} [/latex]	[latex]\sqrt{in.} [/latex]	(mm)	(in.)
Steel AISI4340	1503	(218)	59	(53.7)	3.9	(0.15)
Stainless steel AISI 403	690	(100)	77	(70.1)	31.1	(1.22)
Aluminum 2024-T851 7075-T7351	444 392	(64.4) (56.9)	23 31	(20.9) (28.2)	6.7 15.6	(0.26) (0.61)
Titanium Ti-6A1–4V Ti-6a1–6V	798 1149	(116) (1671)	111 66	(101) (60.1)	48.4 8.2	(1.91) (0.32)

Table 7.5 Paris Equation Parameters for Various Steels
	A
Steel	SI Units	U.S. Units	[latex]n[/latex]
Ferritic–pearlitic	[latex]6.90 imes 10^{−12}[/latex]	[latex]3.60 imes 10^{−10}[/latex]	3.00
Martensitic	[latex]1.35 imes 10^{−10}[/latex]	[latex]6.60 imes 10^{−9}[/latex]	2.25
Austenitic stainless	[latex]5.60 imes 10^{−12}[/latex]	[latex]3.00 imes 10^{−10}[/latex]	3.25
Source: Based on Barsom, J.M. and Rolfe, S.T., Fracture and Fatigue Control in Structures, 3rd ed., Oxford, U.K., Butterworth Heinemann, 1987.

Question 7.7: Fatigue Life of Instrument Panel with a Crack A long plate o...

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