**A. Fatigue damage growth due to fuselage pressurization**

Fatigue crack growth can be modeled in a number of ways. Beden et al. [30] provided an extensive review of crack growth models. Mohanty et al. [31] used an exponential model to model fatigue crack growth. Scarf [32] advocated the use of simple models, when the objective was to demonstrate the predictability of crack growth. In this chapter, a simple Paris-Erdogan model [33] is considered to describe the crack growth behavior. However, other advanced models can also be used.

Damage in the fuselage skin of an airplane is modeled as a through-the-thickness center crack in an infinite plate. The life of an airplane can be viewed as consisting of damage growth cycles, interspersed with inspection and repair. The cycles of pressure difference between the interior and the exterior of the cabin during each flight is instrumental in fatigue damage growth. The crack growth behavior is modeled using the Paris-Erdogan model, which gives the rate of damage size

growth as a function of half damage size ð Þ *a* , pressure differential ð Þ *p* , thickness of fuselage skin ð Þ*t* , fuselage radius ð Þ*r* , and Paris-Erdogan model parameters, *C* and *m*:

$$\frac{da}{dx} = \mathcal{C}(\Delta K)^m \tag{1}$$

where the range of stress intensity factor is approximated with the stress *Δσ* ¼ *pr=t* as

$$
\Delta K = \Delta \sigma \sqrt{\pi a} \tag{2}
$$

The following critical crack size can cause failure of the panel and is approximated as

$$
\sqrt{a\_{cr}} = \frac{K\_{IC}}{\Delta \sigma \sqrt{\pi}} \tag{3}
$$

the left and right corresponds to a value of *C*. For a given value of *C*, there are only certain possible values of the slope, m. **Figure 5** plots those permissible ranges of slop ð Þ *m* , for a given value of intercept ð Þ *C* . It can be seen from **Figure 5** that the slope and log ð Þ *C* are negatively correlated; the correlation coefficient is found to

*Advantages of Condition-Based Maintenance over Scheduled Maintenance Using Structural…*

Kim et al. [35], Packman et al. [36], Berens and Hovey [37], Madsen et al. [38], Mori and Ellingwood [39], and Chung et al. [40] have modeled the damage detection probability as a function of damage size. In this chapter, the inspection of

In scheduled maintenance and in SHM-based maintenance assessment, the detection probability can be modeled using the Palmberg Equation [41] given by

> *a ah <sup>β</sup>*

*<sup>β</sup>* (4)

<sup>1</sup> <sup>þ</sup> *<sup>a</sup> ah*

The direct integration procedure is a method used to compute the probability of an output variable with random input variables. In general, Monte Carlo simulation

fuselage skins for damage is modeled using the Palmberg equation.

*Pd*ð Þ¼ *a*

The expression gives the probability of detecting damage with size 2*a*. In Eq. (4), ah is the half damage size corresponding to 50% probability of detection, and *β* is the randomness parameter. Parameter *ah* represents average capability of the inspection method, while *β* represents the variability in the process. Different values of the parameter, *ah* and *β*, are considered to model the inspection for scheduled maintenance and also for SHM-based maintenance assessment. **Table 4** shows the parameters used in the damage growth model, as well as the inspection

be about �0.8.

**Figure 5.**

model.

**41**

**C. Direct integration procedure**

**B. Inspection model**

*Possible region of Paris-Erdogan model parameters.*

*DOI: http://dx.doi.org/10.5772/intechopen.83614*

where *KIC* is the fracture toughness of an infinite plate with a through-the-thickness center crack loaded in the Mode-I direction.

In the above damage growth process, the following uncertainty is considered: uncertainty in the Paris-Erdogan model parameters, pressure differential, and initial crack size. The damage size after N flight cycles depends on the aforementioned parameters and is also uncertain. The values of uncertain parameters are tabulated in **Table 4**.

It is approximated that all fuselage skins are made of aluminum alloy 2024-T3 with dimensions of 57<sup>0</sup> � 57<sup>0</sup> � 0*:*063 17 ð *:*4 m � 17*:*4 m � 1*:*6 mm). Newmann et al. (Pg 113, **Figure 3**) [34] showed the experimental data plot between the damage growth rate and the intercept and slope, respectively, of the region corresponding to stable damage growth. As the region of the stable damage growth can be bounded by a parallelogram, the estimates of the bounds of the parameters, *C and m*, are obtained from **Figure 3** of Newmann et al. [34].

For a given value of intercept *C*, there is only a range of slope ð Þ *m* permissible in the estimated parallelogram. To parameterize the bounds, the left and right edges of the parallelogram were discretized by uniformly distributed points. Each point on


#### **Table 4.**

*Parameters for crack growth and inspection.*

*Advantages of Condition-Based Maintenance over Scheduled Maintenance Using Structural… DOI: http://dx.doi.org/10.5772/intechopen.83614*

**Figure 5.** *Possible region of Paris-Erdogan model parameters.*

the left and right corresponds to a value of *C*. For a given value of *C*, there are only certain possible values of the slope, m. **Figure 5** plots those permissible ranges of slop ð Þ *m* , for a given value of intercept ð Þ *C* . It can be seen from **Figure 5** that the slope and log ð Þ *C* are negatively correlated; the correlation coefficient is found to be about �0.8.
