Question 7.7: Fatigue Life of Instrument Panel with a Crack A long plate o...
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 P_{\min} to P_{\max} with a complete cycle every 15 s. Before loading, on inspection, a central transverse crack of length 2a is detected on the plate. Estimate the expected life.
Given: a = 0.3 in., t = 0.8 in., w = 2 in., P_{\max} = 2P_{\min} = 144 kips.
Assumption: The plate is made of an AISI 4340 tempered steel.
See Tables 6.1, 6.2, and 7.5.
The material and geometric properties of the panel are
\begin{array}{l} A=3.6 \times 10^{-10}, \quad n=3, \quad K_c=53.7 ksi \sqrt{\operatorname{in}_{.}} \quad S_y=218 ksi , \\ \lambda=1.02, \quad \text { for } a / w=0.15 \quad \text { (Case A of Table } 6.1) \end{array}
Note that the values of a and t satisfy Table 6.2. The largest and smallest normal stresses are
\sigma_{\max }=\frac{P_{\max }}{2 w t}=\frac{144}{2(2)(0.8)}=45 ksi , \quad \sigma_{\min }=22.5 ksi
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
a_f=\frac{1}{\pi}\left(\frac{K_c}{\lambda \sigma_{\max }}\right)^2=\frac{1}{\pi}\left(\frac{53.7}{1.02 \times 45}\right)^2=0.436 in .
Substituting the numerical values, Equation 7.41 results in
N=\frac{a_f^{1-n / 2}-a^{1-n / 2}}{A\left(1-\frac{n}{2}\right)(1.77 \lambda \Delta \sigma)^n} (7.41)
N=\frac{0.436^{-0.5}-0.3^{-0.5}}{3.6\left(10^{-10}\right)(-0.5)[(1.77)(1.02)(22.5)]^3}=25,800 \text { cycles }
With a period of 15 s, approximate fatigue life L is
L=\frac{25,800(15)}{60(60)}=107.5 h
| Table 6.1 Geometry Factors λ for Some Initial Crack Shapes |
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| Case A. Tension of a long plate with a central crack
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a/w 0.1 0.2 0.3 0.4 0.5 0.6 |
\lambda 1.01 1.03 1.06 1.11 1.19 1.30 |
| Case B. Tension of a long plate with an edge crack
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a/w 0 ( w \longrightarrow \infty ) 0.2 0.4 0.5 |
\lambda 1.12 1.37 2.11 2.83 |
| Case C. Tension of a long plate with double-edge cracks
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a/w 0 ( w \longrightarrow \infty ) 0.2 0.4 0.5 0.6 |
\lambda 1.12 1.12 1.14 1.15 1.22 |
| Case D. Pure bending of a beam with an edge crack
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a/w 0.1 0.2 0.3 0.4 0.5 0.6 |
\lambda 1.02 1.06 1.16 1.32 1.62 2.10 |
| Table 6.2 Yield Strength S_{y} and Fracture Toughness K_{c} for Some Engineering Materials |
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| S_{y} | K_{c} | Minimum Values of a and t |
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| Metals | (MPa) | (ksi) | \sqrt{m} | \sqrt{in.} | (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 |
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| A | |||
| Steel | SI Units | U.S. Units | n |
| Ferritic–pearlitic | 6.90 \times 10^{−12} | 3.60 \times 10^{−10} | 3.00 |
| Martensitic | 1.35 \times 10^{−10} | 6.60 \times 10^{−9} | 2.25 |
| Austenitic stainless | 5.60 \times 10^{−12} | 3.00 \times 10^{−10} | 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. | |||