**3. Methodology**

Initially, by submitting the macro model to the normal load P and shear load Q, the stress fields on the plate are obtained from the analysis of the continuum mechanics regarding the Eqs. (3), as shown in **Figure 2**.

Thus, analytically, the stress fields in a body in the elastic regime submitted to normal and shear stress, as shown in **Figure 2**, are represented in Eqs. (3):

$$
\sigma\_{\mathbf{x}} = -\frac{2\mathbf{z}}{\pi} \int\_{-\mathbf{a}}^{\mathbf{b}} \frac{\mathbf{p}(\mathbf{s})(\mathbf{x}-\mathbf{s})^2}{\left((\mathbf{x}-\mathbf{s})^2 + \mathbf{y}^2\right)^2} \mathrm{d}\mathbf{s} - \frac{2}{\pi} \int\_{-\mathbf{a}}^{\mathbf{b}} \frac{\mathbf{q}(\mathbf{s})(\mathbf{x}-\mathbf{s})^3}{\left((\mathbf{x}-\mathbf{s})^2 + \mathbf{y}\right)^2} \mathrm{d}\mathbf{s}
$$

$$
\sigma\_{\mathbf{y}} = -\frac{2\mathbf{z}^3}{\pi} \int\_{-\mathbf{a}}^{\mathbf{b}} \frac{\mathbf{p}(\mathbf{s})}{\left((\mathbf{x}-\mathbf{s})^2 + \mathbf{y}^2\right)^2} \mathrm{d}\mathbf{s} - \frac{2\mathbf{y}^2}{\pi} \int\_{-\mathbf{a}}^{\mathbf{b}} \frac{\mathbf{q}(\mathbf{s})(\mathbf{x}-\mathbf{s})}{\left((\mathbf{x}-\mathbf{s})^2 + \mathbf{y}^2\right)^2} \mathrm{d}\mathbf{s}
$$

$$
\tau\_{\mathbf{x}\mathbf{y}} = -\frac{2\mathbf{z}^2}{\pi} \int\_{-\mathbf{a}}^{\mathbf{b}} \frac{p(\mathbf{s})(\mathbf{x}-\mathbf{s})}{\left((\mathbf{x}-\mathbf{s})^2 + \mathbf{y}^2\right)^2} \mathrm{d}\mathbf{s} - \frac{2\mathbf{y}}{\pi} \int\_{-\mathbf{a}}^{\mathbf{b}} \frac{q(\mathbf{s})(\mathbf{x}-\mathbf{s})^2}{\left((\mathbf{x}-\mathbf{s})^2 + \mathbf{y}^2\right)^2} \mathrm{d}\mathbf{s} \tag{3}
$$

With the stress field obtained in the macro model, analyzes are performed in the microelement. The microelement is represented by a square region of unitary side with a load on the right and top edges, and supported on the left and bottom edges, with a central hole and two 45° inclined cracks representing initial defects in the piece, as shown in **Figure 3**. Stresses σx, σy, and τ from the stress field at an internal point of the macro analysis, applied at the load ratio R = 0.5.

#### **Figure 2.**

*Global–local analysis: Macro model with loadings and dimensions; and detail of the stress in the microelement (units in cm).*

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

#### **Figure 3.**

*Macro model and positions for initial defect analysis (microelements) and detail of the microelement for the microanalysis (units in cm).*

**Figure 4.** *Points of the number of cycles* versus *compliance ratio at each increment. (a) Top edge. (b) Right edge.*

The microanalysis was performed in three different positions of the macro element. Position 1 considers the microelement in the center of the plate with coordinates (0.10); position 2 considers this element at the origin of the system axis (0,0); and position 3, at the limit of the application of the external request (8.0), such positions are illustrated in **Figure 3**.

When performing the analysis on the microelement, BemCracker2D brings, as a result, the number of cycles and the displacement of the boundary mesh for each crack propagation increment. With these results, the compliance is calculated from the average of the displacements of each edge and the respective stress on the considered edge (right or top), thus obtaining the points to form the curve of the number of cycles *versus* compliance, as shown in **Figure 4**.

Each point (from left to right) represents an increase in crack size. Initial compliance is on the abscissa axis. As the cracks spread, the plate loses rigidity, exponentially increasing the compliance *versus* cycles ratio. From these points, a spline curve is fitted as seen in **Figure 5**. With the curves, the number of cycles is obtained, which corresponds to the compliance C (Cx and Cy) two times initial compliance 2C (2Cx and 2Cy), and three times initial compliance 3C (3Cx and 3Cy).

The results of the number of cycles were evaluated for the four crack tips of the microelement as shown in **Figure 6**, where C1 T1 means Crack 1 Tip 1; C2 T2, Crack 2 Tip 2, and so on, with the lowest number of cycles being considered to achieve the results for the 2C and 3C.

**Figure 5.** *Number of cycles* versus *compliance curves. (a) Upper edge. (b) Right edge.*

**Figure 6.** *Crack tips.*

Structural aircraft fuselage commonly uses the 2024 aluminum alloy as a base material due to its high ability to withstand damage, good mechanical strength, and corrosion resistance [43, 44]. Thus, the material considered for the fuselage panel was aluminum alloy 2024-T3, yield and limit strengths were 338 MPa and 476 MPa, respectively, Young's modulus and Poisson's coefficient of 74 GPa and 0.33, respectively, and fracture toughness (Kc) of 34 MPa√m [45]. The Paris Law's parameters C = 7.20e-11 and m = 3.52 [46] were considered with the fatigue load ratio being R = 0.5.

The proposed analysis considered the effects on fatigue life by changing the values of the following variables: external loadings P (normal) and Q (shear) the macro analysis; contained in the Paris Law C and m; and initial defects R (hole radius), L1 (upper crack) and L2 (lower crack), from the microanalysis.

The following is an example of the technique, considering the values of the variables represented in **Table 1**:

P and Q are the normal and shear stresses, respectively, C and m are the Paris constants, r the radius of the central hole, L1 and L2 the size of the upper and lower cracks, respectively. For these values, we obtain the stress fields of the macro analysis illustrated in **Figure 7**.

Position 1.

**Table 2** shows the resulting stresses considering the microelement at position 1 of **Figure 3**.

**Figure 8(a)** shows the 10 crack propagation increments, while **Figure 8(b)** shows the deformed microelement after all the increments, highlighting two details: *A Probabilistic Approach in Fuselage Damage Analysis* via *Boundary Element Method DOI: http://dx.doi.org/10.5772/intechopen.98982*


### **Table 1.**

*Values of case study 1 variables.*

**Figure 7.** *Calculated stress field.*


#### **Table 2.**

*Stress results at position 1.*

#### **Figure 8.**

*Analysis for the microelement in position 1.*

1.Since normal σ<sup>y</sup> stress has a magnitude higher than the σ<sup>x</sup> (**Table 2**), strains were much higher in y to x, deformation still occurs contrary to the direction of applied stress (tensile and σ<sup>x</sup> deformation in the negative direction of the xaxis) due to Poisson effect. This implies disregarding this negative compliance for the analysis. Therefore, **Figure 9** considers only the analysis for the upper border.

#### **Figure 9.**

*Number of cycles x compliance for the micro element in position 1.*


#### **Table 3.**

*Stress results at position 2.*

2.The analysis was able to detect the loss of local stiffness near the crack zones. At the upper edge, the microelement near the upper crack had a much greater displacement than at the other points; the same occurring on the lateral edge close to the lower crack.

The smallest number of cycles to reach 2Cy and 3Cy, we have N(2C) = 1.3635e +04 and N(3C) = 1.3811e+04, respectively.

Position 2.

**Table 3** shows the resulting stress field considering the microelement at position 2 of **Figure 3**.

**Figure 10(a)** shows the crack increments. This analysis also resulted in ten propagation increments. **Figure 10(b)** shows the deformed microelement after all the increments. As a result, it can be seen that as the stresses have similar magnitudes (σ<sup>x</sup> = 349.00 MPa and σ<sup>y</sup> = 360.47 MPa), the deformations in the element have a certain symmetry. Thus, the two edges that have compliance, should be assessed the lowest number of cycles for each crack tip resulting in the first value of 2C and 3C. The result points N(2C) = 1.8601e+03 and N(3C) = 1.8636e+03, as shown in **Figure 11**.

Position 3.

**Table 4** shows the resulting stress field considering the microelement at position 3 of **Figure 3**.

**Figure 12(a)** illustrates the crack increments. This analysis also resulted in 10 propagation increments. Then, **Figure 12(b)** represents the deformed microelement after all the increments. As a result, it is noticed that as σ<sup>x</sup> has a magnitude

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

**Figure 10.**

*Analysis for the microelement at position 2.*

#### **Figure 11.**

*Number of cycles x compliance for the microelement at position 3.*


#### **Table 4.**

*Stress field (MPa) at position 3.*

much greater than σy, the element has more prominent deformation on the right edge. In this analysis, the two edges also show compliance, and the lowest number of cycles for each crack tip that results in the first value of 2Cx and 3Cx must be evaluated. The result points N(2C) = 1.2467e+03 and N(3C) = 1.2521e+03, as shown in **Figure 13**.

**Figure 12.**

*Analysis for the microelement at position 3.*

**Figure 13.**

*Number of cycles* versus *compliance for the microelement at position 3.*
