**B. Inspection model**

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*:

*dx* <sup>¼</sup> *<sup>C</sup>*ð Þ *<sup>Δ</sup><sup>K</sup> <sup>m</sup>* (1)

*<sup>π</sup><sup>a</sup>* <sup>p</sup> (2)

*<sup>π</sup>* <sup>p</sup> (3)

*m*p

log10(5E-10)]

*da*

*pr=t* as

ated as

in **Table 4**.

**Table 4.**

**40**

*Parameters for crack growth and inspection.*

where the range of stress intensity factor is approximated with the stress *Δσ* ¼

ffiffiffiffiffiffi *acr* <sup>p</sup> <sup>¼</sup> *KIC*

center crack loaded in the Mode-I direction.

*Reliability and Maintenance - An Overview of Cases*

obtained from **Figure 3** of Newmann et al. [34].

*<sup>Δ</sup><sup>K</sup>* <sup>¼</sup> *<sup>Δ</sup><sup>σ</sup>* ffiffiffiffiffi

where *KIC* is the fracture toughness of an infinite plate with a through-the-thickness

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

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

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

**Parameter Type Value** Initial crack size ð Þ *a*<sup>0</sup> Random LN(0.2, 0.07)mm Pressure ð Þ *p* Random LN(0.06, 0.003)MPa Radius of fuselage ð Þ*r* Deterministic 2 m (76.5 in) Thickness of fuselage skin ð Þ*t* Deterministic 1.6 mm (0.063 in) Mode-I fracture toughness ð Þ *KIC* Deterministic 36.58*MPa* ffiffiffiffi

Paris-Erdogan law constant ð Þ *C* Random U[log10(5E-11),

Paris-Erdogan law exponent ð Þ *m* Random U[3, 4.3] Palmberg parameter for scheduled maintenance ð Þ *ah*�*man* Deterministic 12.7 mm (0.5 in) Palmberg parameter for scheduled maintenance *βman* ð Þ Deterministic 0.5 Palmberg parameter for SHM based inspection ð Þ *ah*�*shm* Deterministic 5 mm (0.2 in) Palmberg parameter for SHM based inspection ð Þ *βshm* Deterministic 5.0

The following critical crack size can cause failure of the panel and is approxim-

*Δσ* ffiffiffi

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 fuselage skins for damage is modeled using the Palmberg equation.

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

$$P\_d(a) = \frac{\left(\frac{a}{a\_h}\right)^{\beta}}{\mathbf{1} + \left(\frac{a}{a\_h}\right)^{\beta}}\tag{4}$$

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 model.
