Article 1

A new approach for correlating Paris parameters with fatigue 2

life of aircraft fuselage panels 3

Thiago Arnaud A. Oliveira ¹†, Gilberto Gomes ¹†; Welington Vital¹†, Ramon Silva¹†* and Cynthia M. Oliveira2†*. 4

5
¹Department of Civil and Environmental Engineering, Graduation Program in Structural Engineering of the 6
University of Brasília, Darcy Ribeiro Campus, Brasilia, Brazil. 7
2Department of Civil Engineering of the University Center UDF 8
* Correspondence: ramon.silva@unb.br. 9
† These authors contributed equally to this work 10
11
12

Abstract: This work presents a numerical technique that establishes a relationship between the C 13
and m parameters of the Paris Law and the number of fatigue life cycles. This relationship is partic- 14
ularly important for determining the material's ability to withstand a specified number of cycles in 15
an aircraft fuselage design. The methodology is based on a multiscale problem and comprises two 16
stages: the macro model, which focuses on internal stresses and critical point location, and the micro 17
model, which considers the critical fatigue life cycle number (n) across a "grid" of C and m parame- 18
ters, resulting in the optimal curve N(C,m). To validate the adopted methodology, the academic 19
program BemCracker2D was utilized to calculate the internal stress fields, simulate cracks, and es- 20
timate fatigue life. Three case studies were conducted, and the results indicated that the range of C 21
and m values obtained depends on the configuration of the macro model, the physical parameters 22
of the material, and the defined number of cycles in the design. Ultimately, this computational tech- 23
nique allows for generalization to any fuselage damage analysis model, providing C and m Paris 24
parameter data to mitigate fatigue damage. 25

Keywords: Damage Tolerance; Fatigue; Aircraft fuselage; DBEM; Multiscale analysis. 26

27

28

29

1. Introduction 30
31
The evaluation of Fracture Mechanics (FM) parameters, such as Stress Intensity Factor (SIF), 32

number of loading cycles, and stress and displacement fields in the aircraft fuselage, is challenging 33

due to the complex nature of panel details, including supports, shear clips, and rivets. However, 34

it is crucial to understand the damage process, especially under dynamic loads. Designers are con- 35

tinuously seeking efficient and dependable simulation methods that can provide precise average 36

data for these parameters in order to prevent damage processes and accidents. In this context, 37

automation plays a vital role in conducting multiple analyses for parametric studies, ultimately 38

leading to design optimization [1]. 39

Buildings 2022, 12, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/buildings

Buildings 2022, 12, x FOR PEER REVIEW 2 of 29

To address fracture problems, various numerical methods based on domains, such as the Finite 40

Element Method (FEM), Extended Finite Element Method (XFEM), and Generalized Finite Ele- 41

ment Method (GFEM), as well as boundary-based methods like the Boundary Element Method 42

(BEM) and Dual Boundary Element Method (DBEM), are gaining popularity. Among these meth- 43

ods, DBEM offers several advantages. It simplifies the modeling of the crack area, directly calcu- 44

lates the Stress Intensity Factor (SIF), reduces execution times, and accurately simulates crack 45

growth [2-4]. By discretizing only the boundaries of a solid, DBEM enables the analysis of thou- 46

sands of simulations necessary for probabilistic studies. Furthermore, DBEM can be utilized to 47

study defects, predict fatigue behavior, analyze the damage process, evaluate multiple site dam- 48

ages, and assess reliability [5-9]. 49

There are several numerical simulation programs for structural modeling and analysis, with 50

Abaqus and ANSYS being two of the most popular options for Finite Element Method (FEM). 51

Additionally, FRANC 3D is a numerical simulation program specifically designed for Boundary 52

Element Method analysis (BEM). In addition to these, there are notable home-made software tools 53

such as BemLab and BemCracker2D, which have been prominently used in several scientific 54

works, including references [10-13]. For the technique developed in this work, the use of these 55

programs is crucial, starting from the modeling stage, through processing, and finally for result 56

analysis. The objective of developing this tool is to automate the entire process of defining Paris 57

parameters, thereby assisting in material selection. 58

This work aims to introduce a computational tool that utilizes the Boundary Element Method 59

(BEM) to assess two-dimensional aircraft fuselage models. The tool generates an optimized func- 60

tion for the Paris parameters, which can support a specified number of fatigue life cycles deter- 61

mined by the designer. The process involves the designer creating a macro model and specifying 62

the desired number of cycles. The automated tool then processes the model, evaluates the C and 63

m Paris parameters required to withstand the specified number of cycles, and provides an opti- 64

mized function accordingly. In essence, this paper serves as a continuation of the previous works 65

conducted in references [11-13]. It builds upon the research conducted in those works to develop 66

and present an advanced computational tool that enables the evaluation and optimization of Paris 67

parameters for aircraft fuselage models using BEM. 68

This paper is structured as follows: Section 2 briefly reviews the Evolving Research Trajectory: 69

Insights and innovations over time; Section 3 provides an overview of the current state of the art 70

regarding the fatigue history of aircraft fuselage panels; Section 4 outlines the methodology and 71

the procedure employed in this study. It describes the techniques and tools used for the evaluation 72

and optimization of fatigue parameters in aircraft fuselage models; Section 5 presents the results 73

and discussion obtained from the analysis of three case studies; Finally, Section 6 offers the con- 74

clusions drawn from the results and the technique presented in this paper. 75

76

2. Evolving research trajectory 77

Crack propagation and its implications on material stability have long been subjects of intense 78

scientific investigation. Over the course of a series of interconnected studies, our research journey 79
Buildings 2022, 12, x FOR PEER REVIEW 3 of 29

has traversed through distinct phases, each building upon the insights of its predecessor. In this 80

article, we present a comprehensive overview of this progressive exploration, culminating in our 81

current work that aspires to bridge theoretical understanding with practical applications in aero- 82

space engineering. 83

The foundational stone of our inquiry was laid in the first phase [11], where we challenged the 84

conventional viewpoint that defined instability solely by crack size. Through rigorous analysis and 85

inventive perspectives, we uncovered a pivotal shift — compliance, rather than crack size, 86

emerged as a paramount indicator. We observed that compliance surges towards infinity upon 87

reaching a critical value of 3C, with the number of cycles at this juncture serving as the decisive 88

parameter. A novel analytical calculation for the stress field augmented these revelations. How- 89

ever, the complexity introduced by evaluating thousands of combinations involving random var- 90

iables necessitated a cautious interpretation. 91

Expanding our scope, the second phase saw our investigation extend to various materials [12]. 92

By dissecting the growth rates of cracks across diverse metallic alloys, we unveiled correlations 93

between fracture properties and these propagation rates. Our efforts bore fruit with the identifica- 94

tion of the lower and upper limits of instability curves, adding a quantifiable dimension to our 95

explorations. Moreover, this phase's conclusions were constrained by the specificity of materials 96

under examination. 97

In our third phase [13], the focus shifted to numerical simulations. We endeavored to bridge 98

theory and practice, validating its predictions against empirical data. This step forward, coupled 99

with the direct calculation of stress fields via the Boundary Element Method (BEM), fortified our 100

analytical arsenal. Although initially constrained to a specific model, this phase paved the way for 101

future endeavors. 102

103

2.1. Current Endeavors 104

Building upon the insights accrued through this evolutionary journey, our current work 105

emerges as a synthesis of cumulative knowledge. By reintroducing certain parameters as ran- 106

dom variables, we aim to identify the most critical combinations yielding worst-case scenarios. 107

Our approach transcends specific models, achieving a level of generalization essential for real- 108

world applicability. It is important to acknowledge the limitations that underpinned our prior 109

research, meticulously considered as we strive to strike a balance between flexibility and pre- 110

cision. 111

In this respect, and in a summarized way, we delve into the intricacies of each phase, re- 112

vealing the contributions, limitations and innovations of our research trajectory, as seen in the 113

chart below. 114

[11] [12] [13] [Current]

Buildings 2022, 12, x FOR PEER REVIEW 4 of 29

 Compliance is as-  Discovery of the mini-  First attempt to relate  Reintroduction of R,

sessed as the defining mum and maximum Paris' C,m with fa- L1, L2 as random vari-
variable for instability limits of instability tigue life using this ables, allowing identi-
instead of crack size. curves. method. fication of the combi-
 Discovery that com-  Direct calculation of nation yielding the
pliance already tends stress field through worst-case scenario.
to infinity upon the BEM.  Application now be-
reaching the value of  Limited application comes fully general-
3C, with the number solely to the first ized for various mod-
of cycles at this point model [11] for valida- els, as per the design-
being critical. tion of the stress er's requirements.
 Analytical calculation field.
of the stress field.  The numerical stress
 Evaluation of a thou- field enables generali-
sand combinations zation to any case.
with random varia-  Evaluation con-
bles P, Q, R, L1, L2, C, strained to a single
m. combination of R, L1,
 Discovery of the criti- L2, as assessment of
cal position for that singular values of this
specific case. parameter does not
reveal the worst-case
scenario.
115

Finally, in the present phase, the research incorporates the accumulated insights and pro- 116

gresses significantly. The reintroduction of random variables R, L1, and L2 allowed for the 117

identification of the combination yielding the worst case, offering a more precise view of indi- 118

vidual influences. The approach was meticulously designed for complete generalizability, 119

adapting to designers' needs in diverse scenarios and models. Nevertheless, the limitations of 120

the previous approach were carefully contemplated, and the new model aims to strike a balance 121

between flexibility and precision. This stage represents a pivotal stride toward practical ap- 122

plicability and potential contributions to the industry. 123

124

3. Literature review 125

In recent papers, there is a series of fatigue studies on structural elements. Among them, 126

the works [14-17] stand out. Ma [14] examined fatigue life prediction in automobile compo- 127

nents. In this work, multiaxial random fatigue damage was adopted to predict the fatigue life 128

of half-shaft and the results show that the prediction method is reliable and meets the service 129

life and safety requirements. Zhang [15] presented a power exponential fatigue equivalent 130

damage model capable of describing the residual strength degradation of materials to improve 131

the fatigue life predictions, considering that when the loading sequence of fatigue loads 132
Buildings 2022, 12, x FOR PEER REVIEW 5 of 29

changes, the fatigue cumulative damage prediction tends to present a significant error. Liu [16] 133

improved the accuracy of parameter prediction for small-sample data, considering the exist- 134

ence of error in samples, the error circle was introduced to analyze original samples. The author 135

discovered that the S-N curve obtained by the error circle method is more reliable; the S-N 136

curve of the Bootstrap method is more reliable than that of the Maximum Likelihood Estimation 137

(MLE) method. Li [17] improved fatigue life analysis method for optimal design of electric mul- 138

tiple units (EMU) gear, which aims at defects of traditional Miner fatigue cumulative damage 139

theory. The results show that it is more corresponded to engineering practice by using the im- 140

proved fatigue life analysis method than the traditional method. 141

Similarly, to approximate the Paris law to the material used in aircraft fuselage, Breitbarth 142

[18] investigated fatigue crack growth in sheets of aluminum alloy AA2024-T3 under high- 143

stress conditions. In this experiment, high-stress intensity factors cause plastic zone sizes that 144

extend up to approximately 100 mm from the crack tip. The da/dN-ΔK data obtained in this 145

study provide crucial information about the fatigue crack growth and damage tolerance of very 146

long cracks under high-stress conditions in thin lightweight structures. 147

Still applying these concepts, Toor [19], [20] discussed the requirements for designing a fail- 148

safe fuselage structure for aircraft. It highlights the importance of light weight and high oper- 149

ating stresses for an efficient structural component that must perform its intended function, 150

have a long service life, and be produced at a reasonable cost. 151

Regarding the damage tolerance, Sayar [21] presented a two‐stage fatigue life evaluation 152

of a stiffened aluminium aircraft fuselage panel with a bulging circumferential crack and a bro- 153

ken stringer. In this work, the authors concluded that bulging of the skin due to the internal 154

pressure can have significant effect on the stress intensity factor, resulting in fast crack propa- 155

gation after the stringer is completely broken. Bakuckas Jr. [22] showed the potential for ad- 156

vanced fuselage panels with varying emerging metallic structures technologies (EMST) to have 157

improved fatigue and damage-tolerance performance compared to panels constructed using 158

conventional materials and fabrication processes. Abdi [23] described a new analysis approach 159

for evaluating the durability and damage tolerance of exterior aircraft attachment installations, 160

which involves considering multiple crack interactions. The analysis was used to evaluate the 161

fatigue crack initiation and propagation in the fuselage skin and doublers made of wrought 162

aluminum alloys. The results showed that the fatigue damage state in the components at the 163

designed operational life will not exceed the static safety requirements, and therefore, the FAA 164

accepted the damage tolerance analysis. 165

About the use of computational analysis, Carta [24] validated a numerical method of anal- 166

ysis for predicting the damage tolerance of reinforced panels found in aircraft fuselage. The 167

study uses a fracture mechanics approach with several models simulated with the finite ele- 168

ment solver ABAQUS to determine fatigue crack growth rates. The results showed that differ- 169

ent solutions for improving the damage tolerance of aircraft reinforced panels can be tested 170

virtually before performing experiments. Proppe [25] presented a probabilistic framework for 171

computing the failure probability of aircraft structural elements under the concept of damage 172
Buildings 2022, 12, x FOR PEER REVIEW 6 of 29

tolerance, which requires the aircraft to have sufficient residual strength in the presence of dam- 173

age during service inspections. The problem of multi-site damage (MSD) is considered, and 174

uncertainties in crack initiation, crack growth, yield stress, and fracture toughness are described 175

by random variables. The finite element alternating method (FEAM) was used for crack growth 176

calculations, and importance sampling is employed to obtain the probability of failure due to 177

MSD. Kennedy [26] developed a computational technique to predict failure loads in composite 178

structures with through-the-thickness cracks. The discrete crack model with a finite element 179

program was used to simulate damage growth and predict failure over a range of crack sizes. 180

The technique was applied to two laminates and a composite aircraft fuselage, and the results 181

showed good agreement with experimental tests. 182

Madhavi [27] investigated the damage tolerance design of a transport aircraft fuselage 183

structure, which is subjected to high internal pressurization during each take-off and landing 184

cycle leading to metal fatigue. The study focused on the stress intensity factor for a longitudinal 185

crack under pressurization load and investigates crack initiation, growth, fast fracture, and 186

crack arrest features in the stiffened panel. The analysis was performed using the MSC NAS- 187

TRAN solver and pre-processed using MSC PATRAN software in order to prevent further 188

crack propagation. 189

In the upcoming section, the methodology of the proposed technique for damage tolerance 190

in aircraft fuselage is presented that used remote sensing. This methodology has been devel- 191

oped based on the extensive analysis conducted to understand the state of the art and specific 192

contributions in this field. 193

194

4. Material and method 195

In this section, a computational technique for optimizing the fatigue life of aircraft fuselage 196

parts using compliance is presented. This technique builds upon the previous works published 197

in [11-13] and introduces several innovations: 198

1. The initial defects (R, L1, and L2) are once again considered as random variables, with 199

the same values as presented in the first paper [11]. By evaluating thousands of interactions 200

with different defect sizes, the damage tolerance can be assessed for the worst-case scenario 201

of initial defects; 202

2. The C and m Paris parameters are optimized to identify the values that can withstand 203

the specified number of cycles defined in the design; 204

3. The technique creates an optimized function that aids in material selection for aircraft 205

fuselage design. 206

To accomplish this, the developed technique utilizes an algorithm, named BLBC_Algor, that 207

integrates the BemLab and BemCracker2D software tools, whose idea is the following: 208

• The designer creates a macro model in BemLab and defines the desired number of cycles 209

(n*). 210
Buildings 2022, 12, x FOR PEER REVIEW 7 of 29

• BemCracker2D computes the internal stress field in the macro analysis and identifies 211

the stress peak before reaching yielding, allowing for elastic analysis. 212

• The algorithm positions the micro element at this stress peak and calculates the number 213

of cycles required for compliance to reach 3C (N3C). 214

• Considering a range of initial defects in the micro model (R, L1, L2), the algorithm de- 215

termines the values of the material's physical parameters (C and m) that result in the mini- 216

mum number of cycles (N3C) reaching the user-defined value (n*). 217

• Finally, an objective function is derived, which establishes the relationship between the 218

physical parameters of the material (C and m) that ensure the desired number of cycles for 219

the entire plate, ensuring structural integrity. 220

BLBC_Algor 221

1. Input: The designer provides the macro model and specifies the desired number of cycles for 222

fatigue life (n*). 223

2. Random Generation of Initial Defects: The algorithm generates random values for the initial 224

defect parameters (R, L1, and L2) based on their predefined range. 225

3. BEM Analysis: The Bemcracker2D software is utilized to perform the boundary element 226

analysis, computing the internal stresses field. 227

4. Crack Simulation and Fatigue Life Calculation: The BemCracker2D software is employed 228

to simulate crack growth and estimate the fatigue life based on the evaluated initial defects. 229

5. Optimization of Paris Parameters: An optimization algorithm is employed to iteratively 230

adjust the C and m values in the Paris Law until the specified number of cycles is achieved. 231

6. Creation of Optimized Function: The resulting optimized Paris parameters are used to create 232

a function that assists in material selection for the aircraft fuselage design. 233

By following this algorithmic process, the technique enables the optimization of fatigue life 234

for aircraft fuselage parts, considering random initial defects, and provides valuable insights 235

for material selection in the design process. 236

The objective function holds significant relevance for aircraft designers, as it allows them to 237

determine the appropriate material for a specified number of cycles. For instance, if a designer 238

aims to achieve instability at n*=104 cycles, the optimization process will reveal the specific set 239

of physical parameters (C and m) that the material must possess in order to minimize N3C and 240

reach the desired number of cycles outlined in the project (n*). By utilizing the objective func- 241

tion, designers can make informed decisions about material selection, ensuring the structural 242

integrity and safety of the aircraft over the defined lifespan. 243

244

5. Material and method 245

Buildings 2022, 12, x FOR PEER REVIEW 8 of 29

Here is an example model that demonstrates the computational technique described in 246

BLBC_Algor algorithm. The macro model used in this example has already been validated in 247

previous works [11], [13]. The novelty introduced in this paper begins from step 4 onwards. 248

The technique is divided into two analyses: the macro analysis, which involves designing 249

the model to identify the location of the stress peak (steps 1 and 3), and the micro analysis, 250

which simulates initial damage to determine the critical number of fatigue life cycles (steps 2 251

and 4). Finally, in steps 5 and 6, an optimized function is developed to relate the Paris parame- 252

ters to fatigue life. For the sake of simplicity, the steps 1 and 3 can be found in [13] and will not 253

be shown. 254

According to step 2 of the algorithm, the initial defects of the fuselage (R, L1 and L2) were 255

treated as random variables, with values matching those initially defined in [11] and statistical 256

properties represented in Table 1. These variables followed a lognormal distribution. A total of 257

one thousand combinations of these variables were analyzed. The section 4.2 of the paper will 258

present the results of the analysis, specifically the "Minimum N(C,m) curve." This curve illus- 259

trates the combination of C and m values that yield the minimum number of cycles (N3C) re- 260

quired to achieve the user-defined value (n*). By examining this curve, designers will gain in- 261

sights into the optimized values of C and m that can ensure the desired fatigue life for the 262

fuselage structure. 263

Table 1. Statistics of the variables R, L1, and L2. 264

Random Mean Standard Coefficient of

variable deviation variation
R (cm) 0.1 2.0e-02 0.23
L1 (cm) 0.1 1.9e-02 0.22
L2 (cm) 0.1 1.9e-02 0.22
265

5.1. N(C,m) curve 266

By utilizing the combination of R, L1, and L2 values specified in Table 2, the technique in- 267

volves varying C and m in a grid form within the domain of C = [5e-11, 10e-11] and m 268

= [2.5, 3.0]. This grid-based approach allows for the systematic exploration of different combina- 269

tions of C and m values. 270

Table 2. Example of R, L1, and L2 combination. 271

Random Value
variable
R (cm) 0.167
L1 (cm) 0.131
L2 (cm) 0.087
272
Buildings 2022, 12, x FOR PEER REVIEW 9 of 29

As a result, the technique generates a set of data points that establish the relationship between 273

C and m values and the corresponding number of cycles. These data points are graphically rep- 274

resented in Figure 1 (a), providing a visual representation of how different values of C and m 275

impact the fatigue life of the fuselage structure. By examining this figure, designers can gain val- 276

uable insights into the optimal ranges of C and m that can achieve the desired number of cycles 277

for the given combination of R, L1, and L2 values. 278

Continuing with Figure 1 (a), the next step involves interpolating the data points to obtain a 279

surface that represents the relationship between C, m, and the number of cycles. This interpolated 280

surface is depicted in Figure 1 (b), providing a visual representation of how C and m values im- 281

pact the fatigue life of the fuselage structure. 282

Figure 1. a) Points of the number of cycles for each combination (C,m); b) N(C, m) surface. 283

284

To find the values of C and m that result in the minimum number of cycles required by the 285

project, one must identify the intersection between the relationship curve and the surface of the 286

desired number of cycles. For example, considering the project's specified number of cycles as 287

1e+04, the intersection is indicated on the red line in Figure 2 (a). This intersection represents the 288

values of C and m that satisfy the fatigue life requirement. Figure 2 (b) displays the curve on the 289

C x m plane, indicating the specific values of C and m that the material must possess to support 290

the requested number of cycles defined by the user. This curve is also represented in Figure 5 as 291

the C and m graph, supporting 10,000 cycles. These visualizations aid in identifying the optimal 292

values of C and m for achieving the desired fatigue life of the fuselage structure. 293

294
Buildings 2022, 12, x FOR PEER REVIEW 10 of 29

Figure 2. Intersection between N(C,m) and the design number of cycles (n*). 295

296

5.2. Minimum N(C,m) curve 297

Indeed, it is observed that each combination of initial defects (R, L1, L2) leads to a distinct 298

N(C,m) curve. This characteristic is illustrated in Figure 3, which showcases three examples of 299

combinations (C1, C2, C3) from Table 3. Each combination generates a unique N(C,m) curve, re- 300

flecting the influence of the initial defect parameters on the fatigue life of the fuselage structure. 301

The variability in the N(C,m) curves emphasizes the importance of considering different combi- 302

nations of initial defects in the analysis. It highlights how different defect patterns can impact the 303

relationship between C, m, and the number of cycles, underscoring the need for comprehensive 304

and probabilistic studies to assess the tolerance to damage for various scenarios. 305

C1
C2

C3

306
Figure 3. N(C,m) curves for combinations 1, 2, 3. 307

308

Table 3. Values of R, L1, L2 to combinations 1, 2, 3. 309

Random Combination 1 Combination 3 Combination 3

variable (blue curve) (red curve) (purple curve)
R (cm) 0.088 0.073 0.086
L1 (cm) 0.111 0.081 0.093
L2 (cm) 0.130 0.105 0.119
310

These three curves intersect the desired number of cycles for the design (n*=104) at different 311

points, as depicted in Figure 4. The worst-case scenario is represented by the lower curve, as the 312

higher curves can endure a greater number of cycles for the same combination of C and m com- 313

pared to the lower curve. Consequently, adopting the worst-case scenario is favorable for safety, 314

as it corresponds to the lowest N(C,m) curve among the thousand combinations analyzed. By 315

considering the worst-case scenario, designers prioritize safety and ensure that the chosen com- 316

bination of C and m values will support the desired number of cycles, even in the most challeng- 317

ing conditions represented by the lower N(C,m) curve. This approach accounts for potential var- 318

iations in initial defect patterns and guarantees the structural integrity and reliability of the fuse- 319

lage under fatigue loading. 320

Buildings 2022, 12, x FOR PEER REVIEW 11 of 29

n*
n*

321

Figure 4. Different N(C,m) curves intersections to the number of cycles of project (n*). 322

323

5.3. m(C) curve 324

The lower N(C,m) curve from the thousand combinations is indeed the minimum N(C,m) 325

curve. The intersection of this minimum curve with the desired number of cycles (n*), represented 326

by the black surface, results in the m(C) curve, as depicted in Figure 5. It is important to note that 327

the intersection between the minimum N(C,m) curve and the black surface (representing the de- 328

sired number of cycles) corresponds to the optimal values of C and m that satisfy the fatigue life 329

requirement. Any other intersections between different N(C,m) curves and the black surface will 330

be above the m(C) curve. 331

332

Figure 5. Relationship m(C) that results in the design number of cycles 104. 333

334

By focusing on the intersection of the minimum N(C,m) curve and the desired number of 335

cycles, designers can identify the specific values of C and m that the material needs to possess in 336

order to support the requested number of cycles defined by the user. This ensures the selection of 337

appropriate material properties for achieving the desired fatigue life of the fuselage structure. 338

339

340
Buildings 2022, 12, x FOR PEER REVIEW 12 of 29

5.4. Varying the design number of cycles (n*) 341

Additionally, if the designer wishes to consider a different design number of cycles (n*), the 342

black surface in Figure 2 will shift upwards or downwards, altering the intersection point with 343

the N(C,m) curve. Consequently, this leads to the movement of the red curve m(C) within the C 344

x m plane, as demonstrated in Figure 6. The vertical movement of the black surface corresponds 345

to a change in the desired number of cycles, and this adjustment influences the selection of the 346

optimal values of C and m. As a result, the intersection between the N(C,m) curve and the black 347

surface will shift accordingly, affecting the position and shape of the red curve m(C) within the C 348

x m plane. Finally, Figure 6 provides a visualization of how the red curve m(C) changes in re- 349

sponse to modifications in the design number of cycles (n*). This understanding enables designers 350

to assess the impact of different fatigue life requirements on the optimal material properties (C 351

and m) for the fuselage structure. 352

353

354

Figure 6. Relationship m(C) that results in the design number of cycles 0.5x104, 0.75x104 e 355

1x104. 356

357

Finally, there exists an objective function that establishes the relationship between the physi- 358

cal parameters of the material (C and m) and the safe number of cycles for the entire plate. This 359

objective function is represented by the m(C) curve, which is obtained through a polynomial 360

curve fitting process. The order of the polynomial curve is optimized using the Bayesian Infor- 361

mation Criterion (BIC) that is a statistical measure that helps determine the most appropriate 362

model by selecting the one that yields the lowest BIC value, which is given below, where 𝑛𝑛 is the 363

�
number of points, 𝜎𝜎𝑒𝑒 is the error variance, and 𝑘𝑘 represents the polynomial degree:
2 364

�𝑒𝑒2 � + 𝑘𝑘 ln (𝑛𝑛)
𝐵𝐵𝐵𝐵𝐵𝐵 = 𝑛𝑛 ln�𝜎𝜎 (1)
365

In this study, it was observed that a polynomial degree up to 5 was sufficient to accurately 366

represent the intersection curve. The BIC was utilized to identify the minimum BIC value along 367

with the corresponding polynomial degree. The BIC compares different polynomial models with 368

varying degrees to determine the optimal order for the polynomial curve. For each case study, 369

the minimum BIC value was identified by considering polynomial degrees ranging from 1 to 5. 370
Buildings 2022, 12, x FOR PEER REVIEW 13 of 29

The corresponding polynomial, with the minimum BIC value and the selected degree, was chosen 371

to represent the curve m(C) accurately. 372

This approach ensures that the curve m(C) is represented by a polynomial equation of appro- 373

priate degree, striking a balance between the model complexity and its ability to capture the rela- 374

tionship between C and m accurately. Figure 7 illustrates that, in this specific example, the lowest 375

BIC value is achieved with a polynomial degree of 5. 376

-1100

-1150

-1200

-1250

-1300

-1350

-1400
2 3 4 5

377

Figure 7. Polynomial order that optimizes the BIC. 378

379

As a result, the m(C) curve presented in Figure 7 can be accurately represented as (2). This 380

equation will effectively represent the relationship between the physical parameters C and m for 381

achieving the desired number of cycles n*. 382

𝑚𝑚(𝐶𝐶) = − 1.17𝑥𝑥1051 ∗ 𝐶𝐶 5 + 4.10𝑥𝑥1041 ∗ 𝐶𝐶 4 − 5.71𝑥𝑥1031 ∗ 𝐶𝐶 3 + 4.01𝑥𝑥1021 ∗ 𝐶𝐶 2

(2)
− 1.48𝑥𝑥1011 ∗ 𝐶𝐶 + 5.02
383

6. Results and discussions 384

385
To provide a demonstration of the technique's results, three case studies are presented. In 386

each case study, a design engineer defines a Macro model by specifying different combinations 387

of external loads and displacement constraint arrangements. The computational technique then 388

processes the model and optimizes its fatigue life by employing the proposed methodology. By 389

utilizing the technique, the design engineer can assess and enhance the fatigue life of the Macro 390

model, taking into account the specified external loads and displacement constraints. The com- 391

putational tool automates the process and provides optimized solutions for achieving the desired 392

fatigue life for each case study. 393

394

6.1. CASE STUDY 1 395

The Case Study 1 is presented in Figure 8. This model represents a fuselage piece subjected to 396

normal (P) and shear (Q) external loads with values shown in Table 4 and with displacement 397

restraint of degree 1 in the direction perpendicular to the other nodes. 398

Buildings 2022, 12, x FOR PEER REVIEW 14 of 29

Table 1. Case Study 1 variable values.

Macro analysis variables
P1 (MPa) 60.47
Q1 (MPa) 42.78
P2 (MPa) 43.58
Q2 (MPa) 90.70

Figure 1. Case Study 1 Macro model.

399

From the initial model, BemCracker2D calculates the internal stress fields of the macro model, 400

as shown in Figures 9, 10 and 11, with the dimensions of the model highlighted on the x-y axes in 401

meters. 402

403

Figure 9. Sigma x stress field in Case Study 1 (MPa). 404

405

406
Figure 10. Sigma y stress field in Case Study 1 (MPa). 407
Buildings 2022, 12, x FOR PEER REVIEW 15 of 29

408
Figure 11. Shear stress field in Case Study 1. 409

410

Using the stress field, the critical stress location is analyzed using the von Mises criterion be- 411

fore reaching yielding, as shown in Figure 12. 412

413

Figure 12. Stress peak location in Case Study 1. 414

415

By identifying the location of the stress peak, the method positions the micro-element at this 416

point and applies the internal stresses from this point, as shown in Figure 13. At this point, the 417

values of the internal stresses are represented in Table 5. 418

419

Von Mises
0 200

180

160
-5
140

120
-10
100

80
-15
60

40

-20 20
-20 -15 -10 -5 0 5 10 15 20

Figure 13. Positioning the Micro element in Case Study 1. 420

421

422
Buildings 2022, 12, x FOR PEER REVIEW 16 of 29

Table 5. Stress values in Case Study 1. 423

σ x (MPa) 43.58
σ y (MPa) 168.81
τ xy (MPa) 90.70
424

The values of C and m of the Paris Law are varied, and the resulting number of fatigue cycles 425

is calculated for each combination (C,m), as shown in Figure 14. 426

427

Figure 14. Points of the number of cycles for each combination (C,m) in Case Study 1. 428

429

The interpolation of these points results in the surface that relates the number of cycles to each 430

combination (C,m), generating the function N(C,m), as shown in Figure 15. 431

432

Figure 15. N(C,m) surface in Case Study 1. 433

434

Then, the method positions the number of project cycles (n*) defined by the designer, such 435

that the intersection of N(C,m) with n* results in the combination of C and m from the Paris Law 436

for the required number of cycles in the project, as shown in Figure 16. 437
Buildings 2022, 12, x FOR PEER REVIEW 17 of 29

438
Figure 16. Intersection of N(C,m) relation from Case Study 1 with the design number of cy- 439

cles (n*=104). 440

441

Thus, Figure 17 shows the combination of C and m from Paris Law for the number of cycles 442

of 104. 443

444

Figure 17. Relationship m(C) that results in the design number of cycles 10 .
4 445

446

Finally, the curve equation is obtained through a polynomial regression with optimal degree 447

defined by the BIC method. In this case, the polynomial with the lowest BIC was of degree 5, as 448

shown in Figure 18. 449

450
Buildings 2022, 12, x FOR PEER REVIEW 18 of 29

-350

-400

-450

-500

-550

-600

-650

-700

-750

-800
2 2.5 3 3.5 4 4.5 5

451

Figure 18. Degree of the polynomial that optimizes the BIC. 452

453

As a result, the m(C) curve presented in Figure 17 can be accurately represented as: 454

𝑚𝑚(𝐶𝐶) = − 4.26𝑥𝑥1050 ∗ 𝐶𝐶 5 + 1.44𝑥𝑥1041 ∗ 𝐶𝐶 4 − 1.97𝑥𝑥1031 ∗ 𝐶𝐶 3 + 1.40𝑥𝑥1021 ∗ 𝐶𝐶 2

(3)
− 6.09𝑥𝑥1010 ∗ 𝐶𝐶 + 4.63
455

For Case Study 1, the combination of C and m from Paris Law for the number of cycles of 104 456

is represented in Equation (3). Therefore, the computational technique provides the designer of 457

Case Study 1 the physical parameters of the material that can withstand the required number of 458

cycles in the project. 459

460

6.2. CASE STUDY 2 461

Case Study 2 presents a similar model to Case Study 1 but adds normal (P3) and shear (Q3) 462

stresses, as shown in Figure 19. Again, the values of each stress are presented in Table 6. 463

Table 6. Case Study 2 variable values.

Macro analysis variables
P1 (MPa) 60.47
Q1 (MPa) 42.78
P2 (MPa) 43.58
Q2 (MPa) 90.70
P3 (MPa) 70.22
Q3 (MPa) 30.63
Figure 19. Case Study 2 Macro model.

464

Based on the initial model, BemCracker calculates the internal stress fields of the macro 465

model, as shown in Figures 20, 21 and 22, with the dimensions of the model highlighted in the x- 466

y axes in meters. 467

Buildings 2022, 12, x FOR PEER REVIEW 19 of 29

468
Figure 20. Sigma x stress field in Case Study 2 (MPa). 469

470

471
Figure 21. Sigma y stress field in Case Study 2 (MPa). 472

473

Figure 22. Shear stress field in Case Study 1. 474

475

With the stress field, the critical stress location is analyzed using the von Mises criterion before 476

reaching yielding, as shown in Figure 23. 477

Buildings 2022, 12, x FOR PEER REVIEW 20 of 29

478

Figure 23. Stress peak location in Case Study 2. 479

480

Upon identifying the location of the stress peak, the method positions the microelement at 481

this peak and applies the internal stresses at this point, as shown in Figure 24. At this point, the 482

values of the internal stresses are represented in Table 7. 483

Von Mises
0 200

180

160
-5
140

120
-10
100

80
-15
60

40

-20 20
-20 -15 -10 -5 0 5 10 15 20

484

Figure 24. Positioning the Micro element in Case Study 2. 485

486

Table 7. Stress values in Case Study 2. 487

σ x (MPa) 43.58
σ y (MPa) 152.86
τ xy (MPa) 90.70
488

It can be noticed that the stress values are similar to those of Case Study 1, since the critical 489

point was the same, the only difference being the shear stress. Therefore, the values of C and m 490

of the Paris Law are varied, and the resulting fatigue cycle count is calculated for each combina- 491

tion (C, m), as shown in Figure 25. 492

493
Buildings 2022, 12, x FOR PEER REVIEW 21 of 29

494

Figure 25. Points of the number of cycles for each combination (C,m) in Case Study 2. 495

496

497

498

Figure 26. Intersection of N(C,m) relation from Case Study 2 with the design number of cy- 499

cles (n*=104). 500

501

Figure 27 shows the combination of C and m for the number of cycles of 104. 502

503

Figure 27. - Relationship m(C) resulting in the design number of cycles 104. 504

505

Finally, the equation of the curve is obtained through a polynomial regression with optimal 506

degree determined using BIC method. In this case, the polynomial with the lowest BIC was of 507

degree 5, and the combination of C and m from Paris Law for the number of cycles of 104 is 508

represented by Equation (4). 509

Buildings 2022, 12, x FOR PEER REVIEW 22 of 29

-300

-400

-500

-600

-700

-800

-900

510
2 2.5 3 3.5 4 4.5 5

511
50 5 41 4 31 3 21 2
𝑚𝑚(𝐶𝐶) = − 3.52𝑥𝑥10 ∗ 𝐶𝐶 + 1.24𝑥𝑥10 ∗ 𝐶𝐶 − 1.76𝑥𝑥10 ∗ 𝐶𝐶 + 1.31𝑥𝑥10 ∗ 𝐶𝐶
10
(4)
− 5.98𝑥𝑥10 ∗ 𝐶𝐶 + 4.76
512

For Case Study 2, the combination of C and m from Paris Law for the number of cycles of 104 513

is represented in Equation (4). Therefore, the computational technique provides the designer of 514

Case Study 2 the physical parameters of the material that can withstand the required number of 515

cycles in the project. 516

517

6.3. CASE STUDY 3 518

Case Study 3 presents a model with unbalanced stresses on each boundary, free edges, and 519

second-degree displacement constraints at some nodes on the left boundary, as shown in Figure 520

28. Once again, the values of each loading are presented in Table 8. 521

Table 8. Case Study 3 variable values.

Macro analysis variables
P1 (MPa) 3.97
Q1 (MPa) 18.83
P2 (MPa) 5.54
Q2 (MPa) 9.96
P3 (MPa) 7.18
Q3 (MPa) 12.40

Figure 28. Case Study 3 Macro model.

522

Figure 29 illustrates the internal normal (σ x and σ y ) and shear (τ xy ) stress fields. 523
Buildings 2022, 12, x FOR PEER REVIEW 23 of 29

Figure 29. Sigma x stress field in Case Study 2 (MPa). 524

525

526

From the stress field in Figure 29, the critical stress location is analyzed using the von Mises 527

criterion before reaching yielding, as shown in Figure 30. 528

Buildings 2022, 12, x FOR PEER REVIEW 24 of 29

529

Figure 30. Stress peak location in Case Study 3. 530

531

Upon identifying the location of the stress peak, the method positions the micro-element at 532

this point and applies the internal stresses at this point, as shown in Figure 31. The values of the 533

internal stresses at this point are represented in Table 9. 534

Von Mises
0 400

-0.1

350

-0.2

300
-0.3

-0.4
250

-0.5

200
-0.6

-0.7
150

-0.8

100 Table 2. Stress values in Case Study 3.

-0.9

-1
0 0.1 0.2
50
σ x (MPa) 280.21
σ y (MPa) 92.46
Figure 2. Positioning the Micro element in τ xy (MPa) 216.75

Case Study 3.

535

The Figure 32 shows the values of C and m from the Paris Law and the result of fatigue cycle 536

number for each combination (C,m). 537

538
Buildings 2022, 12, x FOR PEER REVIEW 25 of 29

539

Figure 32. Points of the number of cycles for each combination (C,m) in Case Study 3. 540

541

The intersection of N(C,m) with n* results in the combination of C and m of the Paris Law for 542

the required number of cycles in the design, as shown in Figure 33. 543

544
Figure 33. Intersection of N(C,m) relation from Case Study 3 with the design number of cy- 545

cles (n*=104). 546

547

Figure 34 shows the combination of Paris C and m for the number of cycles of 104. 548

549

Figure 34. Relationship m(C) resulting in the design number of cycles 104. 550
Buildings 2022, 12, x FOR PEER REVIEW 26 of 29

Finally, the curve equation is obtained through a polynomial regression with optimal degree 551

defined by the BIC method. In this case, the polynomial with the lowest BIC was of degree 4, as 552

shown in Figure 35. 553

-50

-100

-150

-200

-250

-300

-350
2 2.5 3 3.5 4 4.5 5

554

Figure 35. Degree of the polynomial that optimizes the BIC. 555

556

𝑚𝑚(𝐶𝐶) = 6.93𝑥𝑥1043 ∗ 𝐶𝐶 4 − 3.76𝑥𝑥1033 ∗ 𝐶𝐶 3 + 7.51𝑥𝑥1022 ∗ 𝐶𝐶 2 − 6.80𝑥𝑥1011 ∗ 𝐶𝐶 + 5.59 (5)

557

For Case Study 3, the combination of C and m from Paris Law for the number of cycles of 104 558

is represented in Equation (5). Therefore, the computational technique provides the designer of 559

Case Study 3 the physical parameters of the material that can withstand the required number of 560

cycles in the project. 561

562

7. Final remarks 563

564

In summary, the methodology presented in this study offers a valuable alternative to the con- 565

ventional damage tolerance analysis approach, which typically revolves around critical crack size. 566

Instead, this methodology focuses on evaluating compliance as a key factor for assessing instabil- 567

ity. 568

The utilization of the Boundary Element Method (BEM) was crucial in the development of 569

this technique. BEM's flexibility allowed for the evaluation of stress peak locations and compli- 570

ance at the edges of micro-analysis elements. This innovation has led to the establishment of a 571

meaningful relationship between the Paris constants and the concept of damage tolerance, exem- 572

plified by the curve that correlates the Paris constants with the desired number of cycles. 573

The results clearly demonstrate that the N(C,m) curve is influenced by several factors, includ- 574

ing model configuration, physical material parameters, and the specific number of cycles defined 575

in the project. Consequently, for each model, there exists a range of C and m values that can fulfill 576

the project requirements. 577

Moreover, the automation of the technique and the utilization of the BemLab and BemCracker 578

computational programs allow for the generalization of the approach to encompass any fuselage 579
Buildings 2022, 12, x FOR PEER REVIEW 27 of 29

damage analysis model. The case studies presented in this study have showcased the effective- 580

ness of this methodology, yielding valuable parametric data for the Paris constants and ensuring 581

damage tolerance while preventing the structure from reaching a critical limit state. 582

In conclusion, this methodology offers a novel and comprehensive approach to damage tol- 583

erance analysis, departing from traditional crack size-based evaluations. It provides valuable in- 584

sights for optimizing the fatigue life of aircraft fuselage structures, ultimately enhancing their 585

safety and reliability. 586

587

Author Contributions: Regarding conceptualization, methodology, using of software, validation, 588

formal analysis, investigation, resources, data curation, writing—original draft, preparation, writ- 589

ing—review and editing, visualization, supervision, project administration, and funding ac-qui- 590

sition for this article, all authors have provided the same contribution. All authors have read and 591

agreed to the published version of the manuscript. 592

Funding: This research received no external funding. 593

Institutional Review Board Statement: Not applicable. 594

Informed Consent Statement: Not applicable. 595

Data Availability Statement: All data are contained within the article. 596

597

Acknowledgment 598
599

The authors are grateful to the Brazilian Coordination for the Improvement of Higher Education 600

(CAPES) for the supporting funds for this research. The authors also thank the Graduate 601

Programme in Structural Engineering and Civil Construction in the Department of Civil and 602

Environmental Engineering at the University of Brasilia, Brazil. 603

604
605
606
References 607
1. J. Ju and X. You, “Dynamic fracture analysis technique of aircraft fuselage containing damage subjected to 608

blast,” Math Comput Model, vol. 58, no. 3–4, pp. 627–633, Aug. 2013, doi: 10.1016/j.mcm.2011.10.044. 609

2. A. Portela, M. H. Aliabadi, and D. P. Rooke, “The dual boundary element method: Effective implementation for 610

crack problems,” Int J Numer Methods Eng, vol. 33, no. 6, pp. 1269–1287, Apr. 1992, doi: 10.1002/nme.1620330611. 611

3. A. Portela, M. H. Aliabadi, and D. P. Rooke, “Dual boundary element analysis of cracked plates: singularity sub- 612

traction technique,” Int J Fract, vol. 55, no. 1, pp. 17–28, May 1992, doi: 10.1007/BF00018030. 613

4. G. E. Blandford, A. R. Ingraffea, and J. A. Liggett, “Two-dimensional stress intensity factor computations using 614

the boundary element method,” Int J Numer Methods Eng, vol. 17, no. 3, pp. 387–404, Mar. 1981, doi: 615

10.1002/nme.1620170308. 616

5. R. Citarella, “MSD crack propagation by DBEM on a repaired aeronautic panel,” Advances in Engineering Soft- 617

ware, vol. 42, no. 10, pp. 887–901, Oct. 2011, doi: 10.1016/j.advengsoft.2011.02.014. 618

6. R. Citarella, P. Carlone, R. Sepe, and M. Lepore, “DBEM crack propagation in friction stir welded aluminum 619
Buildings 2022, 12, x FOR PEER REVIEW 28 of 29

joints,” Advances in Engineering Software, vol. 101, pp. 50–59, Nov. 2016, doi: 10.1016/j.advengsoft.2015.12.002. 620

7. R. J. Price and J. Trevelyan, “Boundary element simulation of fatigue crack growth in multi-site damage,” Eng 621

Anal Bound Elem, vol. 43, pp. 67–75, Jun. 2014, doi: 10.1016/j.enganabound.2014.03.002. 622

8. L. Morse, Z. S. Khodaei, and M. H. Aliabadi, “Multi-Fidelity Modeling-Based Structural Reliability Analysis 623

with the Boundary Element Method,” Journal of Multiscale Modelling, vol. 08, no. 03n04, p. 1740001, Sep. 2017, doi: 624

10.1142/S1756973717400017. 625

9. X. Huang, M. H. Aliabadi, and Z. S. Khodaei, “Fatigue Crack Growth Reliability Analysis by Stochastic Bound- 626

ary Element Method,” CMES-Computer Modeling in Engineering & Sciences, vol. 102, no. 4, pp. 291–330, 2014, doi: 627

10.3970/cmes.2014.102.291. 628

10. G. Gomes and A. C. O. Miranda, “Analysis of crack growth problems using the object-oriented program bem- 629

cracker2D,” Frattura ed Integrità Strutturale, vol. 12, no. 45, pp. 67–85, Jun. 2018, doi: 10.3221/IGF-ESIS.45.06. 630

11. T. A. A. Oliveira, G. Gomes, and F. Evangelista Jr, “Multiscale aircraft fuselage fatigue analysis by the dual 631

boundary element method,” Eng Anal Bound Elem, vol. 104, pp. 107–119, Jul. 2019, doi: 10.1016/j.enga- 632

nabound.2019.03.032. 633

12. G. Gomes, T. Oliveira, and F. Evangelista Jr, “A Probabilistic Approach in Fuselage Damage Analysis via Bound- 634

ary Element Method,” in Advances in Fatigue and Fracture Testing and Modelling, IntechOpen, 2022. doi: 635

10.5772/intechopen.98982. 636

13. G. Gomes, T. A. A. Oliveira, and A. M. Delgado Neto, “A new methodology to predict damage tolerance based 637

on compliance via global-local analysis,” Frattura ed Integrità Strutturale, vol. 15, no. 58, pp. 211–230, Sep. 2021, doi: 638

10.3221/IGF-ESIS.58.16. 639

14. X. Ma, X. Liu, H. Wang, J. Tong, and X. Yang, “Fatigue Life Prediction of Half-Shaft Using the Strain-Life 640

Method,” Advances in Materials Science and Engineering, vol. 2020, pp. 1–8, Aug. 2020, doi: 10.1155/2020/5129893. 641

15. M. Zhang, G. Hu, X. Liu, and X. Yang, “An improved strength degradation model for fatigue life prediction con- 642

sidering material characteristics,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 43, no. 5, p. 643

275, May 2021, doi: 10.1007/s40430-021-02997-4. 644

16. X. Liu, J. Liu, H. Wang, and X. Yang, “Prediction and evaluation of fatigue life considering material parameters 645

distribution characteristic,” International Journal of Structural Integrity, vol. 13, no. 2, pp. 309–326, Mar. 2022, doi: 646

10.1108/IJSI-11-2021-0118. 647

17. Y.-H. Li, C. Zhang, H. Yin, Y. Cao, and X. Bai, “Modification optimization-based fatigue life analysis and im- 648

provement of EMU gear,” International Journal of Structural Integrity, vol. 12, no. 5, pp. 760–772, Oct. 2021, doi: 649

10.1108/IJSI-07-2021-0072. 650

18. E. Breitbarth, T. Strohmann, and G. Requena, “High‐stress fatigue crack propagation in thin AA2024‐T3 sheet 651

material,” Fatigue Fract Eng Mater Struct, vol. 43, no. 11, pp. 2683–2693, Nov. 2020, doi: 10.1111/ffe.13335. 652

19. P. M. Toor, “On damage tolerance design of fuselage structure (longitudinal cracks),” Eng Fract Mech, vol. 24, 653

no. 6, pp. 915–927, Jan. 1986, doi: 10.1016/0013-7944(86)90276-6. 654

20. P. M. Toor, “On damage tolerance design of fuselage structure—circumferential cracks,” Eng Fract Mech, vol. 26, 655

no. 5, pp. 771–782, Jan. 1987, doi: 10.1016/0013-7944(87)90140-8. 656

21. B. Sayar and A. Kayran, “Two-stage fatigue life evaluation of an aircraft fuselage panel with a bulging circum- 657

ferential crack and a broken stringer,” Fatigue Fract Eng Mater Struct, vol. 37, no. 5, pp. 494–507, May 2014, doi: 658

10.1111/ffe.12127. 659

22. J. G. Bakuckas et al., “Assessment of emerging metallic structures technologies through test and analysis of fuse- 660

lage structure,” SN Appl Sci, vol. 1, no. 11, p. 1521, Nov. 2019, doi: 10.1007/s42452-019-1471-7. 661
Buildings 2022, 12, x FOR PEER REVIEW 29 of 29

23. F. Abdi, Y. Xue, M. Garg, B. Farahmand, J. Housner, and K. Nikbin, “An analysis approach toward FAA certifi- 662

cation for damage tolerance of aircraft components,” The Aeronautical Journal, vol. 118, no. 1200, pp. 181–196, Feb. 2014, 663

doi: 10.1017/S0001924000009064. 664

24. F. Carta and A. Pirondi, “Damage tolerance analysis of aircraft reinforced panels,” Frattura ed Integrità Strut- 665

turale, vol. 5, no. 16, pp. 34–42, Apr. 2011, doi: 10.3221/IGF-ESIS.16.04. 666

25. C. Proppe, “Probabilistic analysis of multi-site damage in aircraft fuselages,” Comput Mech, vol. 30, no. 4, pp. 667

323–329, Mar. 2003, doi: 10.1007/s00466-002-0408-x. 668

26. T. C. Kennedy, M. H. Cho, and M. E. Kassner, “Predicting failure of composite structures containing cracks,” 669

Compos Part A Appl Sci Manuf, vol. 33, no. 4, pp. 583–588, Apr. 2002, doi: 10.1016/S1359-835X(01)00145-2. 670

27. N. Madhavi and R. Saritha, “Two bay crack arrest capability evaluation for metallic fuselage,” IOP Conf Ser Ma- 671

ter Sci Eng, vol. 455, no. 1, p. 012015, Dec. 2018, doi: 10.1088/1757-899X/455/1/012015. 672