**4. Probabilistic analysis**

The values obtained previously refer to a deterministic analysis. A probabilistic analysis using Monte Carlo (MC) sampling performed a thousand simulations using BEM varying the values of P, Q, C, m, R, L1, and L2. The statistical parameters for those variables used for the MC simulation are listed in **Table 5**.

The results are presented from **Figures 14**–**16**. **Figure 14** refers to the results for microelement in Position 1. It can be seen that the instability of the microelement occurs for load cycles varying between 10<sup>3</sup> and 10<sup>6</sup> . **Figure 15** refers to the results


*A Probabilistic Approach in Fuselage Damage Analysis* via *Boundary Element Method DOI: http://dx.doi.org/10.5772/intechopen.98982*

#### **Table 5.**

*Mean and standard deviation values for each variable.*

**Figure 14.** *Monte Carlo simulations of compliance* versus *number of cycles in position 1.*

for the microelement in Position 2. It can be seen that the instability of the microelement occurs for load cycles varying between 10<sup>2</sup> and 10<sup>5</sup> . **Figure 16** refers to the results for Position 3. It can be seen that the instability of the microelement occurs for small cycles of load ranging between 101 and 10<sup>5</sup> .

It can be noted that when the compliance reaches the value of three times the initial compliance (3C), the element already becomes unstable and tends to increase infinitely as can be seen in **Figures 16**–**18**. **Table 6** shows the minimum number of cycles *N* that leads the microelement to reach 3C, thus occurring instability. The smallest *N*-value is the worst analysis case, which is the most conservative case. For Position 1 the element becomes unstable in 1.113e+03 cycles. For Position 2, instability occurs in 1.12e+02 cycles, in this case, there is a sudden reduction of load cycles regarding the previous case (ten times less than the element in Position 1). In position 3, the instability occurs in 4.7e+01 cycles, being the worst situation in the case of occurrence of initial defects in the panel. **Figure 17** shows the superposition of the maximum and minimum limits of all curves in each position.

**Figure 15.** *Monte Carlo simulations of compliance* versus *number of cycles in position 2.*

**Figure 16.** *Monte Carlo simulations of compliance* versus *number of cycles in position 3.*

**Figure 18** shows the Absolute Frequency (AF) of the incidences of compliance values corresponding to the initial compliance C, 2C, and 3C. **Figure 19(a)**–**(c)**, for the microelement, show the positions 1, 2, and 3, respectively. Note the increase in the dispersion of the value distribution due to higher standard deviation observed for increasing compliance levels.

*A Probabilistic Approach in Fuselage Damage Analysis* via *Boundary Element Method DOI: http://dx.doi.org/10.5772/intechopen.98982*

**Figure 17.** *Compliance* versus *number of cycle limits in positions 1, 2, and 3.*

#### **Figure 18.**

*Absolute frequency of the incidences of compliances values corresponding to the initial compliance C, 2C, and 3C for the three positions.*


#### **Table 6.**

*Minimum number of cycles that lead the microelement to instability.*

**Figure 19** shows the probability density function *f N*ð Þ <sup>3</sup>*<sup>c</sup>* to the micro element damage tolerance analysis for the N cycles at 3C. It is noticed that the element in position 3 has a marked curvature and consequently a lesser variability of damage tolerance, that is, a selection range of *N* which is limited to 3C is limited to 0.4e+05. The element in position 2 has a variation of *N* that reaches 3C up to 0.7e+05.
