**C. Direct integration procedure**

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 can be used to calculate the probability, but it requires many samples, and the results have sampling error. In this chapter, the direct integration process is used to compute the probability of having a specific crack size. The damage size distribution is a function of initial crack size, pressure differential, and Paris-Erdogan model parameter ð Þ *C; m* , which are all random:

$$f\_N(a) = h(a\_0, f(p), f(C, m))\tag{5}$$

where *a*0*, f <sup>N</sup>*ð Þ *a ,* and *f p*ð Þ represent the initial crack size, the probability density function of crack size after N cycles, and the pressure differential, respectively. *J C*ð Þ *; m* is the joint probability density of the Paris-Erdogan model parameters ð Þ *C; m* . The probability of crack size being less than *aN* after *N* cycles is the integration of the joint probability density of input parameters over the region that results in a crack size being less than or equal to *aN*, that is,

$$\Pr(a \le a\_N) = \int \dots \int\_R a\_0 J(\mathcal{C}, m) f(p) \mathrm{d}R \tag{6}$$

where *R* represents the region of ð Þ *a*0*;C; m; p* which will give *a*≤*aN*.

Based on preliminary analysis performed by the authors, the effect of random pressure differential was averaged out over a large number of flight cycles. Therefore, the average of the pressure differential is used in the following calculation. Hence, Eq. (6) reduces to be a function of *m* and *C*, as

$$F\_N(40) = \iint\_A J(C, m) \,\mathrm{d}C \mathrm{d}m \tag{7}$$

**Author details**

**43**

Ting Dong, Raphael T. Haftka and Nam H. Kim\* University of Florida, Gainesville, FL, USA

\*Address all correspondence to: nkim@ufl.edu

provided the original work is properly cited.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

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

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

where *A* represents the region of f g *C; m* that would give *aN*≤40*mm* for a given initial crack size, *a*0. The parallelogram in **Figure 6** is the region of all possible combinations of Paris-Erdogan model parameters, f g *C; m* . For the initial crack size, *a*<sup>0</sup> ¼ 1*mm*, cracks in the gray triangular region will grow beyond 40 mm after *N* ¼ 50*;* 000 cycles. If the initial crack size is distributed, then the integrand is evaluated at different values in the range of the initial crack size, and the trapezoidal rule is used to compute the probability at the desired crack size.

**Figure 6.** *Regions of* f g *C; m for N* ¼ 50*;* 000 *and a*<sup>0</sup> ¼ 1*mm.*

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