# Residual Strength Prediction of Aluminum Panels with Multiple Site Damage Using Artificial Neural Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{P}) and the root mean square percentage error (RMSE

_{P}). The ANN optimization covered three of the most commonly used learning algorithms, 12 different activation functions and up to 30 nodes in the hidden layer. Based on this optimization, the Bayesian Regularization (BR) learning algorithm, the Elliot symmetric sigmoid activation function, and seven hidden nodes are used in the ANN model. Our results show that the ANN is able to predict the residual strength for all the different materials and panel configurations with a mean absolute error of about 3.82%. The results also show that the ANN predictions are generally accurate for all the different materials and panel configurations. The obtained ANN predictions are also compared with the residual strength predictions of the best available fracture mechanics semi-analytical and computational models. The comparison shows that the ANN results are of comparable accuracy and even give more accurate results in many cases.

## 2. Background

#### 2.1. Residual Strength of Panels with MSD

_{C}). The fracture toughness is a material property, but for thin sheets, it is also slightly dependent on the thickness, grain orientation, and crack length. Another theory that can be used for predicting the residual strength is the Net Section Yielding (NSY), and it is more applicable to ductile materials. However, experiments have shown that neither of the LEFM nor NSY theories is able to accurately predict the residual strength of panels with MSD for neither ductile nor brittle materials [13].

_{LU}), is found as:

- σ
_{y}: The yield strength of the material. - L: Length of the ligament between the lead crack and MSD crack.
- a: Lead crack half-length.
- $\ell $: MSD crack half-length.
- ${\mathsf{\beta}}_{a}$: The overall SIF correction factor for the lead crack tip.
- ${\mathsf{\beta}}_{\ell}$: The overal SIF correction factor for the adjacent MSD crack tip.

#### 2.2. Applications of ANN in Fracture Mechanics

## 3. Experimental Data

#### 3.1. Unstiffened Panels

#### 3.2. Stiffened Panels

_{stf}) which are also made of aluminum and fixed at the front and back sides of the panel using bolts, as seen in the figure. For the first stiffened panel configuration (one-bay panels, from S-1 to S-21), three different sets of stiffeners were used in the experiments, where each set has a different cross-sectional area, as can be seen in the table.

#### 3.3. Lap-Joint Panels

## 4. ANN Modeling Procedure

_{P}value, it is calculated by averaging 40 MAE

_{P}values of the 40 different random combinations used in the simulation trials.

#### 4.1. Ann Inputs and Structure

_{stf}), it will have a value for the stiffened panels only, while its magnitude will be zero for the unstiffened and lap-joint panels. As a matter of fact, the geometric input parameters being used here are carefully chosen based on the researchers’ experience with this type of fracture mechanics problems and in order to account for the experimental data, which are obtained from different sources. For the material input parameters, two material properties that have significance in fracture mechanics problems are used; these are the yield strength and fracture toughness. In addition to these two material properties, a material identification number is used to designate each of the three different materials being included here. Identification numbers 1, 2, and 3 are assigned to the 2024-T3, 2524-T3, and 7075-T6 aluminum alloys, respectively. A preliminary sensitivity analysis is performed to evaluate the significance of each of the ANN inputs and to see how it affects the residual strength. The sensitivity analysis is done by calculating the Pearson linear correlation coefficient between each of the inputs and the residual strength [56]. Although the sensitivity analysis has shown that the material identification number has a minor effect on residual strength as compared to the yield strength and fracture toughness, however, the material identification number is still included as one of the inputs mainly to designate the materials in case other materials are to be added later. For thin aluminum sheets, the yield strength and fracture toughness values depend on the material type, condition, and grain orientation, as well as sheet thickness. It should be noted here that sheet thickness is not included in the input parameters since both the yield strength and fracture toughness depend on the thickness; therefore, the thickness effect is already accounted for indirectly through these two material properties. Additionally, the fact that the residual strength being used here is the failure stress value rather than the failure load value makes the inclusion of the thickness among the ANN inputs unnecessary. The last input parameter is the panel configuration identification number, which is used to designate each of the different test panel configurations. The experimental data used in this investigation included four distinct test panel configurations. Therefore, identification numbers from 1 to 4 are assigned to distinguish the different configurations where 1: unstiffened panel, 2: one-bay stiffened panel, 3: two-bay stiffened panel with broken stiffener, and 4: lap-joint panel.

#### 4.2. ANN Optimization

_{P}and RMSE

_{P}. Although the MAE

_{P}is more meaningful for reflecting the level of the error, the two metrics are used to get a better understanding of the performance, since the RMSE

_{P}reflects the closeness of the errors to the mean value. The MAE

_{P}and RMSE

_{P}are calculated here as:

_{P}value for the 40 different random datasets combinations. The configuration of the ANN model is optimized in terms of the following:

- (1)
- The adopted back-propagation learning algorithm (${l}_{a}$) used to optimally define the ANN’s internal parameters (i.e., weights and biases). Basically, the weights and biases of the ANN are initially set randomly and then updated iteratively by calculating the error on the training outputs and distributing it back to the ANN layers.
- (2)
- The hidden nodes activation function ($f$) used to process the ANN’s inputs.
- (3)
- The number of nodes ($H$) in the hidden layer.

- Three different possible learning algorithms; ${l}_{a}$= Bayesian Regularization (BR), Levenberg– Marquardt (LM), and Scaled Conjugate Gradient (SCG).
- Twelve different possible activation functions; $f$= ‘logsig’, ‘tansig’, ‘purelin’, ‘tribas’, ‘radbas’, ‘elliotsig’, ‘hardlims’, ‘hardlim’, ‘poslin’, ‘radbasn’, ‘satlin’, ‘satlins’.
- Up to 30 possible number of hidden nodes; $H$= [1–30].

## 5. Results and Discussion

#### 5.1. Training Datasets Selection

_{P}by about 0.2%. Accordingly, this partially randomized training dataset’s selection approach is adapted in here. It should be mentioned here that the MAE

_{P}improvement that results from using the partially randomized approach seems to be small (only 0.2%) because the MAE

_{P}values are averaged for 40 different random combinations (as mentioned previously). However, for some particular data points, the error in the ANN residual strength predictions can be noticeable if the ANN is extrapolating out of the range used for training. It should also be indicated that this approach, which is being followed here to avoid the extrapolation for the ANN predictions, is consistent with the findings reported by Mortazavi and Ince [45] about the poor extrapolation ability of ANN.

#### 5.2. Optimum ANN Configuration

_{P}value (averaged over the 40 different random combinations). For identifying the best of the three learning algorithms (SCG, LM, BR), an ANN is developed using the training dataset based on each algorithm, and the validation dataset is used to determine the optimum configuration for each. Table 6 reports the best ANN configurations obtained for each learning algorithm in terms of the hidden nodes activation function (${f}_{opt}$) and the number of hidden nodes (${H}_{opt}$). The table shows that the BR learning algorithm outperforms the LM and SCG learning algorithms significantly in terms of all performance metrics: MAE

_{P}, RMSE

_{P}, and the coefficient of determination (R

^{2}). From the table, it can also be seen that the BR learning algorithm gives the best performance using the Elliot symmetric sigmoid (elliotsig) activation function along with 30 hidden nodes (based on the MAE

_{P}value of the validation datasets). To further clarify the effect of the number of hidden nodes on the ANN performance, the evolution of the MAE

_{P}versus the considered numbers of hidden nodes for each learning algorithm (using the optimum activation function) is shown in Figure 5. The figure shows that the BR learning algorithm continuously outperforms the two other learning algorithms for any number of hidden nodes. The optimum number of hidden nodes (${H}_{opt}$), indicated by the asterisk in Figure 5, for each learning algorithm are those that minimize the MAE

_{P}value using the validation datasets. The figure shows that for the BR algorithm, the optimum number of hidden nodes is 30, which corresponds to the lowest MAE

_{P}value. By carefully inspecting the curve, it can be seen that the MAE

_{P}value dropped rabidly at the beginning (at seven hidden nodes), and it remained relatively steady afterwards. Based on that, it can be seen that taking the number of hidden nodes to be seven would probably be good enough, and it will not make much difference in the residual strength prediction accuracy. As a matter of fact, some researchers use some rules of thumb for choosing the number of hidden nodes to be used in ANN models. One of the most commonly used rules of thump suggests that the number of hidden nodes should be somewhere in between the number of input nodes and the number of output nodes (i.e., between one and nine hidden nodes for our case) [62]. Based on this rule of thumb, taking the number of hidden nodes to be seven seems to be more reasonable, although our optimization results indicate that the 30 hidden nodes gives slightly lower MAE

_{P}value (3.38% for 30 nodes vs. 3.43% for seven nodes). To further investigate that, the residual strength prediction performance of the seven hidden nodes and the 30 hidden nodes ANNs is also compared using the testing datasets (instead of the validation datasets that are used in the optimization). The compression based on the testing datasets (averages for the 40 random dataset combinations) shows that the seven hidden nodes gives slightly more accurate residual strength predictions where the MAE

_{P}values using the seven and 30 hidden nodes are 3.82% and 3.92%, respectively. Consequently, the number of hidden nodes is taken to be seven, especially that such a small number of hidden nodes makes the ANN less complex and reduces the computational effort. Therefore, in summary, the ANN model that is being adopted in this study uses the “BR” learning algorithm, the “elliotsig” activation function, and seven nodes in the hidden layer. All the residual strength predictions presented in the succeeding sections are obtained using this ANN configuration.

#### 5.3. ANN Residual Strength Predictions

_{P}= 2.35%, and the testing dataset that gives the worst performance has an MAE

_{P}= 6.35%, while the average MAE

_{P}for the 40 testing datasets is 3.82%. This in fact shows the importance of using the cross-validation technique that makes the results more credible, since it eliminates the variations associated with the selection of the data points used in the testing dataset. It should be stressed here that the overall performance metrics reported in this paper are calculated by averaging over the 40 testing dataset combinations.

^{2}) values for the training, validation, and testing datasets are also shown in the figure. It should be noticed here that the coefficient of determination for the training dataset is clearly higher than those for the validation and training datasets. This is in fact quite expected, since the training data are already "seen" by the ANN (since they are used for training); therefore, the ANN can predict the residual strength for the training dataset more accurately that the "unseen" validation and testing datasets. The overall coefficient of determination for all the 147 data points shown in Figure 9 is 99.46%, which is another indicator of the goodness of the ANN predictions.

_{P}and RMSE

_{P}values for the three different materials and for the three general panel configurations, as well as the totals for all materials and panel configurations. From the table, it is evident that the ANN predictions are generally accurate for all the deferent materials. However, it also can be seen that there are some relatively small differences in the residual strength predictions error levels between the different materials and panel configurations. The largest error is observed for the Al 7075-T6 material (MAE

_{P}= 8.2%). The relatively high error associated with this material is rather expected, since the number of data points used for training the ANN is smaller than that for other materials (only eight data points are used for training). On the other hand, the best ANN performance is observed for the lap-joint panels where the MAE

_{P}is 1.81%. This error value is lower than that observed for the unstiffened panels of the same Al 2024-T3 material (MAE

_{P}= 4.61%), even though the number for panels used for ANN training for the unstiffened panels is larger than that for the lap-joint panels. In fact, this difference can be attributed to the fact that the unstiffened panels data have more variation where they include different thicknesses, material conditions, and grain orientations; while on the other hand, the lap-joint panels are all of the same thickness and grain orientation. Therefore, it is quite normal that the ANN is able to give better predictions for the lap-joint panels, since there is no variation in their material properties. It is probably worth mentioning here that using the actual values of the material properties (obtained by testing samples of the same sheet material) might slightly improve the ANN prediction accuracy. However, as mentioned previously, adopting the standard handbook values of the material properties is much more convenient for engineering use. By comparing the values of the two performance metrics given in the table, it can be seen that the RMSE

_{P}values are consistently higher than the MAE

_{P}values, but the difference is generally not very high. This difference between the RMSE

_{P}and the MAE

_{P}values comes from the variability of the error values for the individual predictions, and the fact that the difference is not very high indicates that the variability is not that significant. Finally, the table also shows the overall error for all the testing datasets where the MAE

_{P}is equal to 3.82%. For the same group of testing datasets, the average error for the residual strength predictions obtained using the semi-analytical models (Equation (2) to Equation (4)) is 4.1%. Comparing the overall error values of the ANN and semi-analytical models predictions shows that both have the same level of accuracy. However, obtaining residual strength predictions using the ANN is easier, and the fact that a single ANN model is used for all materials and panel configurations makes it even more convenient.

## 6. Concluding Remarks

_{P}) is 3.82%, which puts it at the same accuracy level (and even better for many configurations) of the best available semi-analytical and computational approaches. The main conclusions of this study can be summarized in the following points:

- Proper selection of the input parameters for representing the different materials and the geometric and panel configurations is essential for obtaining good results using ANN. For instance, two fracture-related material properties (yield strength and fracture toughness) and a designation number are used as inputs to account for the three different materials being used here. The material properties also "indirectly" account for sheet thickness, material condition, and grain orientation, and this eliminates the need for some inputs. In addition, using the standard handbook values of the material properties (instead of the actual properties obtained from testing) makes the ANN modeling approach being used here simpler and more convenient for engineering use.
- Optimizing the ANN model is essential for obtaining high-accuracy predictions. The ANN optimization carried out here showed that there could be quite a significant difference in the prediction accuracy when using different learning algorithms, hidden node activation functions, and numbers of hidden nodes. In this investigation, the best ANN performance was obtained using the Bayesian Regularization learning algorithm with the Elliot symmetric sigmoid activation function and seven hidden nodes.
- In order to avoid bias and get more reliable results, it is essential to implement a randomized selection procedure for the data points used in the training, validation, and testing datasets; also, the randomized selection needs to be repeated several times (cross-validation). Additionally, it is beneficial to include some fixed, manually selected data points that cover the upper and lower limit values of the different inputs within the training group. This will avoid the extrapolation in the ANN predictions and thus improve the accuracy.
- There are some differences (relatively small yet noticeable) in the average residual strength prediction error values between the different materials and panel configurations. These differences in the error values can be attributed to two factors: (i) the different number of data points available for training the ANN for the different materials and configurations and (ii) the amount of variation in the different inputs parameters within the different materials and configurations. For instance, the highest error is observed for the 7075-T6 material (MAE
_{P}= 8.2%) since only eight data points are used for training. In fact, eight data points is a very small number, and it is definitely not sufficient for training an ANN; however, the error level is still reasonable, since this material is not used alone for developing an ANN model but rather among a larger group of data points that share many common geometric and configuration inputs, although the materials are different. On the other hand, the best accuracy (MAE_{P}= 1.81%) is observed for the lap-joint panels, since they all share the same sheet thickness, material condition, and grain orientation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Swift, T. Widespread fatigue damage monitoring: Issues and concerns. In Proceedings of the 5th International Conference on Structural Airworthiness of New and Ageing Aircraft, Hamburg, Germany, 16–18 June 1993; pp. 113–150. [Google Scholar]
- Broek, D. The Effects of Multi-Site Damage on The Arrest Capability of Aircraft Fuselage Structures; FractuREsearch Report No. TR9302; Foster Miller Inc.: Waltham, MA, USA, 1993. [Google Scholar]
- Schijve, J. Multiple-site damage in aircraft fuselage structures. Fatigue Fract. Eng. Mater. Struct.
**1995**, 18, 329–344. [Google Scholar] [CrossRef] - Dawicke, D.S.; Newman, J.C. Evaluation of Various Fracture Parameters for Predictions of Residual Strength in Sheets with Multi-Site Damage. In Proceedings of the 1st Joint NASA/FAA/DOD Conference on Aging Aircraft, Ogden, UT, USA, 8–10 July 1997. [Google Scholar]
- Ingram, J.E.; Kwon, Y.S.; Duffie, K.J.; Irby, W.D. Residual strength analysis of skin splices with multiple site damage. In Proceedings of the 2nd Joint NASA/FAA/DOD Conference on Aging Aircraft, Williamsburg, VA, USA, 31 August–3 September 1998; Langley Research Center: Hampton, VA, USA, 1999; pp. 427–436. [Google Scholar]
- Kuang, J.H.; Chen, C.K. The failure of ligaments due to multiple-site damage using interactions of dugdale-type cracks. Fatigue Fract. Eng. Mater. Struct.
**1998**, 21, 1147–1156. [Google Scholar] [CrossRef] - Thomson, D.; Hoadley, D.; McHatton, J. Load Tests of Flat and Curved Panels with Multiple Cracks; Foster-Miller Draft Final Report to the FAA Technical Center; Foster Miller Inc.: Waltham, MA, USA, 1993. [Google Scholar]
- Dewit, R.; Fields, R.J.; Mordfin, L.; Low, S.R.; Harne, D. Fracture Behavior of Large-Scale Thin-Sheet Aluminum Alloy. In 1995 National Fracture Symposium; American Society for Testing and Materials: West Conshohocken, PA, USA, 1995. [Google Scholar]
- Smith, B.L.; Saville, P.A.; Mouak, A.; Myose, R.Y. Strength of 2024-T3 aluminum panels with multiple site damage. J. Aircr.
**2000**, 37, 325–331. [Google Scholar] [CrossRef] - Smith, B.L.; Hijazi, A.L.; Haque, A.K.M.; Myose, R.Y. Strength of stiffened 2024-T3 aluminum panels with multiple site damage. J. Aircr.
**2001**, 38, 764–768. [Google Scholar] [CrossRef] - Hijazi, A.L.; Smith, B.L.; Lacy, T.E. Linkup Strength of 2024-T3 Bolted Lap Joint Panels with Multiple Site Damage. J. Aircr.
**2004**, 41, 359–364. [Google Scholar] [CrossRef] - Hijazi, A.L.; Lacy, T.E.; Smith, B.L. Comparison of residual strength estimates for bolted lap-joint panels. J. Aircr.
**2004**, 41, 657–664. [Google Scholar] [CrossRef] - Smith, B.L.; Hijazi, A.L.; Myose, R.Y. Strength of 7075-T6 and 2024-T3 aluminum panels with multiple-site damage. J. Aircr.
**2002**, 39, 354–358. [Google Scholar] [CrossRef] - Smith, B.L.; Flores, T.L.; Hijazi, A.L. Link-up strength of 2524-T3 and 2024-T3 aluminum panels with multiple site damage. J. Aircr.
**2005**, 42, 535–541. [Google Scholar] [CrossRef] - Hijazi, A.L. Residual Strength of Thin-Sheet Aluminum Panels with Multiple Site Damage. Ph.D. Thesis, Wichita State University, Wichita, KS, USA, 2002. [Google Scholar]
- Labeas, G.; Diamantakos, J. Analytical prediction of crack coalesce in Multiple Site Damaged structures. Int. J. Fract.
**2005**, 134, 161–174. [Google Scholar] [CrossRef] - Pidaparti, R.M.V.; Palakal, M.J.; Rahman, Z.A. Simulation of structural integrity predictions for panels with multiple site damage. Adv. Eng. Softw.
**2000**, 31, 127–135. [Google Scholar] [CrossRef] - Dawicke, D.S.; Newman, J.C. Residual strength predictions for multiple site damage cracking using a three-dimensional finite element analysis and a CTOA criterion. In Fatigue and Fracture Mechanics; ASTM International: West Conshohocken, PA, USA, 1999; Volume 29. [Google Scholar]
- Xu, W.; Wang, H.; Wu, X.; Zhang, X.; Bai, G.; Huang, X. A novel method for residual strength prediction for sheets with multiple site damage: Methodology and experimental validation. Int. J. Solids Struct.
**2014**, 51, 551–565. [Google Scholar] [CrossRef] [Green Version] - Wu, N.; Xie, L.; Zhao, F.; Chen, B. Residual strength assessment to panels with multiple site damage by method of system reliability. In Proceedings of the 9th International Conference on Reliability Maintainability and Safety IEEE, Guiyang, China, 12–15 June 2011; pp. 74–79. [Google Scholar]
- Pidaparti, R.M.; Jayanti, S.; Palakal, M.J. Residual strength and corrosion rate predictions of aging aircraft panels: Neural network study. J. Aircr.
**2002**, 39, 175–180. [Google Scholar] [CrossRef] - Pidaparti, R. Aircraft structural integrity assessment through computational intelligence techniques. Struct. Durab. Health Monit.
**2006**, 2, 131–148. [Google Scholar] - Haykin, S. Neural Networks: A Comprehensive Foundation; Prentice-Hall Inc.: Upper Saddle River, NJ, USA, 2007. [Google Scholar]
- He, W.; Yan, Z.; Sun, Y.; Ou, Y.; Sun, C. Neural-learning-based control for a constrained robotic manipulator with flexible joints. IEEE Trans. Neural Netw. Learn. Syst.
**2018**, 29, 5993–6003. [Google Scholar] [CrossRef] [PubMed] - Al-Dahidi, S.; Ayadi, O.; Alrbai, M.; Adeeb, J. Ensemble approach of optimized artificial neural networks for solar photovoltaic power prediction. IEEE Access
**2019**, 7, 81741–81758. [Google Scholar] [CrossRef] - Cao, J.; Fang, Z.; Qu, G.; Sun, H.; Zhang, D. An accurate traffic classification model based on support vector machines. Int. J. Netw. Manag.
**2017**, 27, e1962. [Google Scholar] [CrossRef] - Mehdy, M.M.; Ng, P.Y.; Shair, E.F.; Saleh, N.I.; Gomes, C. Artificial neural networks in image processing for early detection of breast cancer. Comput. Math. Methods Med.
**2017**, 2017, 2610628. [Google Scholar] [CrossRef] [Green Version] - Abueidda, D.W.; Koric, S.; Sobh, N.A. Topology optimization of 2D structures with nonlinearities using deep learning. Comput. Struct.
**2020**, 237, 106283. [Google Scholar] [CrossRef] - Abueidda, D.W.; Koric, S.; Sobh, N.A.; Sehitoglu, H. Deep learning for plasticity and thermo-viscoplasticity. Int. J. Plast.
**2020**, 136, 102852. [Google Scholar] [CrossRef] - Szklarek, K.; Gajewski, J. Optimisation of the Thin-Walled Composite Structures in Terms of Critical Buckling Force. Materials
**2020**, 13, 3881. [Google Scholar] [CrossRef] - Altarazi, S.; Ammouri, M.; Hijazi, A. Artificial neural network modeling to evaluate polyvinylchloride composites’ properties. Comput. Mater Sci.
**2018**, 153, 1–9. [Google Scholar] [CrossRef] - Shokry, A.; Gowid, S.; Kharmanda, G.; Mahdi, E. Constitutive models for the prediction of the hot deformation behavior of the 10% Cr steel alloy. Materials
**2019**, 12, 2873. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dagli, C.H. (Ed.) Artificial Neural Networks for Intelligent Manufacturing; Springer Science & Business Media: Dordrecht, The Netherlands, 2012. [Google Scholar]
- Nasiri, S.; Khosravani, M.R.; Weinberg, K. Fracture mechanics and mechanical fault detection by artificial intelligence methods: A review. Eng. Fail. Anal.
**2017**, 81, 270–293. [Google Scholar] [CrossRef] - Balcıoğlu, H.E.; Seçkin, A.Ç.; Aktaş, M. Failure load prediction of adhesively bonded pultruded composites using artificial neural network. J. Compos. Mater.
**2016**, 50, 3267–3281. [Google Scholar] [CrossRef] - Hakim, S.J.S.; Razak, H.A. Structural damage detection of steel bridge girder using artificial neural networks and finite element models. Steel Compos. Struct.
**2013**, 14, 367–377. [Google Scholar] [CrossRef] [Green Version] - Janssens, O.; Slavkovikj, V.; Vervisch, B.; Stockman, K.; Loccufier, M.; Verstockt, S.; Van Hoecke, S. Convolutional neural network based fault detection for rotating machinery. J. Sound Vib.
**2016**, 377, 331–345. [Google Scholar] [CrossRef] - Shu, J.; Zhang, Z.; Gonzalez, I.; Karoumi, R. The application of a damage detection method using Artificial Neural Network and train-induced vibrations on a simplified railway bridge model. Eng. Struct.
**2013**, 52, 408–421. [Google Scholar] [CrossRef] [Green Version] - Feng, S.; Zhou, H.; Dong, H. Using deep neural network with small dataset to predict material defects. Mater. Des.
**2019**, 162, 300–310. [Google Scholar] [CrossRef] - Nechval, K.N.; Nechval, N.A.; Bausova, I.; Skiltere, D.; Strelchonok, V.F. Prediction of fatigue crack growth process via artificial neural network technique. Int. J. Comput.
**2006**, 5, 21–32. [Google Scholar] - Gajewski, J.; Sadowski, T. Sensitivity analysis of crack propagation in pavement bituminous layered structures using a hybrid system integrating Artificial Neural Networks and Finite Element Method. Comput. Mater. Sci.
**2014**, 82, 114–117. [Google Scholar] [CrossRef] - Lee, J.A.; Almond, D.P.; Harris, B. The use of neural networks for the prediction of fatigue lives of composite materials. Compos. Part A Appl. Sci. Manuf.
**1999**, 30, 1159–1169. [Google Scholar] [CrossRef] - Hamdia, K.M.; Lahmer, T.; Nguyen-Thoi, T.; Rabczuk, T. Predicting the fracture toughness of PNCs: A stochastic approach based on ANN and ANFIS. Comput. Mater. Sci.
**2015**, 102, 304–313. [Google Scholar] [CrossRef] - Mohanty, J.R.; Verma, B.B.; Ray, P.K.; Parhi, D.K. Application of artificial neural network for fatigue life prediction under interspersed mode-I spike overload. J. Test. Eval.
**2010**, 38, 177–187. [Google Scholar] - Mortazavi, S.M.; Ince, A. An artificial neural network modeling approach for short and long fatigue crack propagation. Comput. Mater. Sci.
**2020**, 185, 109962. [Google Scholar] [CrossRef] - Seibi, A.; Al-Alawi, S.M. Prediction of fracture toughness using artificial neural networks (ANNs). Eng. Fract. Mech.
**1997**, 56, 311–319. [Google Scholar] [CrossRef] - Ince, R. Prediction of fracture parameters of concrete by artificial neural networks. Eng. Fract. Mech.
**2004**, 71, 2143–2159. [Google Scholar] [CrossRef] - Pidaparti, R.M.V.; Palakal, M.J. Neural network approach to fatigue-crack-growth predictions under aircraft spectrum loadings. J. Aircr.
**1995**, 32, 825–831. [Google Scholar] [CrossRef] - Pidaparti, R.M.V.; Palakal, M.J. Fatigue crack growth predictions in aging aircraft panels using optimization neural network. AIAA J.
**1998**, 36, 1300–1304. [Google Scholar] [CrossRef] - Spear, A.D.; Priest, A.R.; Veilleux, M.G.; Ingraffea, A.R.; Hochhalter, J.D. Surrogate modeling of high-fidelity fracture simulations for real-time residual strength predictions. AIAA J.
**2011**, 49, 2770–2782. [Google Scholar] [CrossRef] [Green Version] - Candelieri, A.; Sormani, R.; Arosio, G.; Giordani, I.; Archetti, F. Assessing structural health of helicopter fuselage panels through artificial neural networks hierarchies. Int. J. Reliab. Saf.
**2013**, 7, 216–234. [Google Scholar] [CrossRef] - Anderson, T.L. Fracture Mechanics: Fundamentals and Applications; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Handbook, M. Metallic Materials and Elements for Aerospace Vehicle Structures; Military Handbook No. MIL-HDBK-5H, Section 5; Defense Standardization Program: Riverside, OH, USA, 1998. [Google Scholar]
- Dawicke, D.S.; Newman, J.C., Jr.; Starnes, J.H., Jr.; Rose, C.A.; Young, R.D.; Seshadri, B.R. Residual strength analysis methodology: Laboratory coupons to structural components. In Proceedings of the 3rd Joint NASA/FAA/DOD Conference on Aging Aircraft; NASA Langley Technical Report Server, Hampton, VA, USA; 2000. [Google Scholar]
- Bendat, J.S.; Piersol, A.G. Random Data: Analysis and Measurement Procedures; John Wiley & Sons: Hoboken, NJ, USA, 2011; Volume 729. [Google Scholar]
- Pearson’s Correlation Coefficient. In Encyclopedia of Public Health; Kirch, W. (Ed.) Springer: Dordrecht, The Netherlands, 2008. [Google Scholar] [CrossRef]
- Hagan, M.T.; Demuth, H.B.; Beale, M.H. Neural Network ToolboxTM 6 User’s Guide; MathWorks: Natick, MA, USA, 2015. [Google Scholar]
- Baghirli, O. Comparison of Lavenberg-Marquardt, Scaled Conjugate Gradient and Bayesian Regularization Backpropagation Algorithms for Multistep Ahead Wind Speed Forecasting Using Multilayer Perceptron Feedforward Neural Network. Master’s Thesis, Uppsala University, Uppsala, Sweden, 2015. [Google Scholar]
- Peters, S.O.; Sinecen, M.; Gallagher, G.R.; Pebworth, L.A.; Hatfield, J.S.; Kizilkaya, K. Comparison of linear model and artificial neural network using antler beam diameter and beam length of white-tailed deer (Odocoileus virginianus). J. Anim. Sci.
**2016**, 94, 823–824. [Google Scholar] [CrossRef] - Al-Dahidi, S.; Ayadi, O.; Adeeb, J.; Louzazni, M. Assessment of Artificial Neural Networks Learning Algorithms and Training Datasets for Solar Photovoltaic Power Production Prediction. Front. Energy Res.
**2019**, 7, 130. [Google Scholar] [CrossRef] [Green Version] - Arora, M.; Ashraf, F.; Saxena, V.; Mahendru, G.; Kaushik, M.; Shubham, P. A Neural Network-Based Comparative Analysis of BR, LM, and SCG Algorithms for the Detection of Particulate Matter. In Advances in Interdisciplinary Engineering; Springer: Singapore, 2019; pp. 619–634. [Google Scholar]
- Heaton, J. Deep Learning and Neural Networks; Artificial Intelligence for Humans; Heaton Research: St. Louis, MO, USA, 2015; Volume 3. [Google Scholar]

**Figure 2.**The stiffened panels’ configurations: (

**a**) one-bay panel, (

**b**) two-bay panel with broken middle stiffener.

**Figure 5.**MAE

_{P}evolution vs. the number of hidden neurons based on the validation dataset for the three learning algorithms (averages for the 40 random combinations of the validation dataset).

**Figure 6.**The unstiffened panels experimental residual strength values along with the predictions obtained by the ANN and the semi-analytical models (WSU2, MLU2524, and MLU7075) [9,13,14] for one of the testing datasets (

**top**), together with the residual strength prediction errors (

**bottom**). The shown ANN predictions are for the best of the 40 random combinations.

**Figure 7.**The stiffened panels experimental residual strength values along with the predictions obtained by the ANN and the semi-analytical model (WSU2) [10] for one of the testing datasets (

**top**), together with the residual strength prediction errors (

**bottom**). The shown ANN predictions are for the best of the 40 random combinations.

**Figure 8.**The lap-joint panels experimental residual strength values along with the predictions obtained by the ANN and the semi-analytical model (WSU2) [11] and FEM simulation [12] for one of the testing datasets (

**top**), together with the residual strength prediction errors (

**bottom**). The shown ANN predictions are for the best of the 40 random combinations.

**Figure 9.**Correlation of ANN predictions with experimental residual strength values for all panels (the full 147 data points used for training, validation, and testing). The shown ANN predictions are for the best of the 40 random combinations.

Panel ID | Source | Mat. Cond. | Grain Orient. | σ_{y}MPa | K_{C}MPa.m ^{1/2} | t mm | W mm | a mm | ℓ mm | L mm | σ_{Exp}MPa |
---|---|---|---|---|---|---|---|---|---|---|---|

U-1 | WSU | Clad | L-T | 310.3 | 114.3 | 1.6 | 610 | 93.35 | 4.45 | 3.81 | 79.84 |

U-2 | WSU | Clad | L-T | 310.3 | 114.3 | 1.6 | 610 | 90.81 | 4.45 | 6.35 | 97.15 |

U-3 | WSU | Clad | L-T | 310.3 | 114.3 | 1.6 | 610 | 88.27 | 4.45 | 8.89 | 112.18 |

U-4 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 84.46 | 8.26 | 8.89 | 94.25 |

U-5 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 83.19 | 6.99 | 11.43 | 110.04 |

U-6 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 81.92 | 5.72 | 13.97 | 120.04 |

U-7 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 80.65 | 4.45 | 16.51 | 132.52 |

U-8 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 118.75 | 4.45 | 3.81 | 67.57 |

U-9 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 116.21 | 4.45 | 6.35 | 83.36 |

U-10 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 113.67 | 4.45 | 8.89 | 94.94 |

U-11 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 109.86 | 8.26 | 8.89 | 82.33 |

U-12 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 108.59 | 6.99 | 11.43 | 97.15 |

U-13 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 107.32 | 5.72 | 13.97 | 105.7 |

U-14 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 106.05 | 4.45 | 16.51 | 119.77 |

U-15 | WSU | Clad | L-T | 310.3 | 114.3 | 1.6 | 610 | 144.15 | 4.45 | 3.81 | 59.02 |

U-16 | WSU | Clad | L-T | 310.3 | 114.3 | 1.6 | 610 | 141.61 | 4.45 | 6.35 | 73.98 |

U-17 | WSU | Clad | L-T | 310.3 | 114.3 | 1.6 | 610 | 139.07 | 4.45 | 8.89 | 83.71 |

U-18 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 135.26 | 8.26 | 8.89 | 71.23 |

U-19 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 133.99 | 6.99 | 11.43 | 83.50 |

U-20 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 132.72 | 5.72 | 13.97 | 93.91 |

U-21 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 131.45 | 4.45 | 16.51 | 108.73 |

U-22 | WSU | Clad | T-L | 275.8 | 109.9 | 1.6 | 610 | 160.66 | 8.26 | 8.89 | 71.23 |

U-23 | NIST | Bare | L-T | 324.1 | 111.0 | 1.02 | 2286 | 254 | 6.35 | 6.35 | 61.50 |

U-24 | NIST | Bare | L-T | 324.1 | 111.0 | 1.02 | 2286 | 177.8 | 5.08 | 7.62 | 84.12 |

U-25 | NIST | Bare | L-T | 324.1 | 111.0 | 1.02 | 2286 | 71.12 | 7.62 | 10.16 | 137.9 |

U-26 | NIST | Bare | L-T | 324.1 | 111.0 | 1.02 | 2286 | 195.58 | 5.08 | 15.24 | 97.91 |

U-27 | NIST | Bare | L-T | 324.1 | 111.0 | 1.02 | 2286 | 241.3 | 6.35 | 19.05 | 88.95 |

U-28 | NIST | Bare | L-T | 324.1 | 111.0 | 1.02 | 2286 | 96.52 | 7.62 | 22.86 | 161.34 |

U-29 | NIST | Bare | L-T | 324.1 | 111.0 | 1.02 | 2286 | 273.05 | 6.35 | 25.4 | 91.29 |

U-30 | NIST | Bare | L-T | 324.1 | 111.0 | 1.02 | 2286 | 127 | 5.08 | 33.02 | 151.69 |

U-31 | FM * | Clad | T-L | 268.9 | 113.2 | 1.02 | 508 | 101.6 | 3.81 | 8.89 | 97.43 |

U-32 | FM * | Clad | T-L | 268.9 | 113.2 | 1.02 | 508 | 96.52 | 6.35 | 11.43 | 99.98 |

U-33 | FM * | Clad | T-L | 268.9 | 113.2 | 1.02 | 508 | 40.64 | 10.16 | 12.7 | 144.8 |

U-34 | FM * | Clad | T-L | 268.9 | 113.2 | 1.02 | 508 | 63.5 | 12.7 | 12.7 | 106.05 |

U-35 | FM * | Clad | T-L | 268.9 | 113.2 | 1.02 | 508 | 93.98 | 6.35 | 13.97 | 110.32 |

U-36 | FM * | Clad | T-L | 268.9 | 113.2 | 1.02 | 508 | 40.64 | 6.35 | 16.51 | 171.55 |

U-37 | FM * | Clad | T-L | 268.9 | 113.2 | 1.02 | 508 | 91.44 | 6.35 | 16.51 | 118.94 |

U-38 | FM * | Clad | T-L | 268.9 | 113.2 | 1.02 | 508 | 76.2 | 6.35 | 31.75 | 155.14 |

U-39 | FM * | Clad | T-L | 268.9 | 113.2 | 1.02 | 508 | 38.1 | 12.7 | 38.1 | 194.78 |

U-40 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 90 | 7.5 | 8 | 106.83 |

U-41 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 90 | 7.5 | 12 | 120.83 |

U-42 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 90 | 7.5 | 18 | 132 |

U-43 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 111 | 7.5 | 10 | 103 |

U-44 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 111 | 7.5 | 15 | 107.83 |

U-45 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 113 | 7.5 | 15 | 113.33 |

U-46 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 113 | 7.5 | 20 | 119.67 |

U-47 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 136 | 7.5 | 20 | 99 |

U-48 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 138 | 7.5 | 30 | 110.67 |

U-49 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 143 | 7.5 | 20 | 100 |

U-50 | SJTU * | Clad | L-T | 303.4 | 117.6 | 1 | 600 | 148 | 7.5 | 30 | 106.67 |

**Table 2.**Experimental data for the unstiffened (Clad) 2524-T3 panel configurations [14].

Panel ID | Grain Orient. | σ_{y}MPa | K_{C}MPa.m ^{1/2} | t mm | W mm | a mm | ℓ mm | L mm | σ_{Exp}MPa |
---|---|---|---|---|---|---|---|---|---|

2524-1 | L-T | 310.3 | 204.4 | 1.6 | 610 | 93.35 | 4.45 | 3.81 | 102.11 |

2524-2 | L-T | 310.3 | 204.4 | 1.6 | 610 | 90.81 | 4.45 | 6.35 | 124.18 |

2524-3 | L-T | 310.3 | 204.4 | 1.6 | 610 | 88.27 | 4.45 | 8.89 | 139.69 |

2524-4 | T-L | 275.8 | 180.2 | 1.6 | 610 | 84.46 | 8.26 | 8.89 | 125.42 |

2524-5 | T-L | 275.8 | 180.2 | 1.6 | 610 | 83.19 | 6.99 | 11.43 | 144.17 |

2524-6 | T-L | 275.8 | 180.2 | 1.6 | 610 | 81.92 | 5.72 | 13.97 | 156.31 |

2524-7 | T-L | 275.8 | 180.2 | 1.6 | 610 | 80.65 | 4.45 | 16.51 | 172.44 |

2524-8 | T-L | 275.8 | 180.2 | 1.6 | 610 | 118.75 | 4.45 | 3.81 | 89.01 |

2524-9 | T-L | 275.8 | 180.2 | 1.6 | 610 | 116.21 | 4.45 | 6.35 | 107.08 |

2524-10 | T-L | 275.8 | 180.2 | 1.6 | 610 | 113.67 | 4.45 | 8.89 | 120.94 |

2524-11 | T-L | 275.8 | 180.2 | 1.6 | 610 | 109.86 | 8.26 | 8.89 | 108.67 |

2524-12 | T-L | 275.8 | 180.2 | 1.6 | 610 | 108.59 | 6.99 | 11.43 | 124.32 |

2524-13 | T-L | 275.8 | 180.2 | 1.6 | 610 | 107.32 | 5.72 | 13.97 | 136.73 |

2524-14 | T-L | 275.8 | 180.2 | 1.6 | 610 | 106.05 | 4.45 | 16.51 | 150.1 |

2524-15 | L-T | 310.3 | 204.4 | 1.6 | 610 | 144.15 | 4.45 | 3.81 | 77.5 |

2524-16 | L-T | 310.3 | 204.4 | 1.6 | 610 | 141.61 | 4.45 | 6.35 | 93.77 |

2524-17 | L-T | 310.3 | 204.4 | 1.6 | 610 | 139.07 | 4.45 | 8.89 | 103.15 |

2524-18 | T-L | 275.8 | 180.2 | 1.6 | 610 | 135.26 | 8.26 | 8.89 | 92.32 |

2524-19 | T-L | 275.8 | 180.2 | 1.6 | 610 | 133.99 | 6.99 | 11.43 | 107.63 |

2524-20 | T-L | 275.8 | 180.2 | 1.6 | 610 | 132.72 | 5.72 | 13.97 | 116.94 |

2524-21 | T-L | 275.8 | 180.2 | 1.6 | 610 | 131.45 | 4.45 | 16.51 | 129.21 |

2524-22 | T-L | 275.8 | 180.2 | 1.6 | 610 | 160.66 | 8.26 | 8.89 | 80.67 |

**Table 3.**Experimental data for the unstiffened (Bare) 7075-T6 panel configurations [13].

Panel ID | Grain Orient. | σ_{y}MPa | K_{C}MPa.m ^{1/2} | t mm | W mm | a mm | ℓ mm | L mm | σ_{Exp}MPa |
---|---|---|---|---|---|---|---|---|---|

7075-1 | T-L | 468.9 | 76.9 | 1.8 | 610 | 84.46 | 6.99 | 10.16 | 100.6 |

7075-2 | T-L | 468.9 | 76.9 | 1.8 | 610 | 109.86 | 8.26 | 8.89 | 81.02 |

7075-3 | T-L | 468.9 | 76.9 | 1.8 | 610 | 108.59 | 6.99 | 11.43 | 94.25 |

7075-4 | T-L | 468.9 | 76.9 | 1.8 | 610 | 108.59 | 5.72 | 12.7 | 96.94 |

7075-5 | T-L | 468.9 | 76.9 | 1.8 | 610 | 107.32 | 4.45 | 15.24 | 105.36 |

7075-6 | T-L | 468.9 | 76.9 | 1.8 | 610 | 133.99 | 5.72 | 12.7 | 81.22 |

7075-7 | T-L | 468.9 | 76.9 | 1.8 | 610 | 132.72 | 5.72 | 13.97 | 90.46 |

7075-8 | T-L | 468.9 | 76.9 | 1.8 | 610 | 135.26 | 6.99 | 10.16 | 75.22 |

7075-9 | T-L | 468.9 | 76.9 | 1.8 | 610 | 158.12 | 5.72 | 13.97 | 77.29 |

7075-10 | T-L | 468.9 | 76.9 | 1.8 | 610 | 191.14 | 8.26 | 3.81 | 32.61 |

7075-11 | T-L | 468.9 | 76.9 | 1.8 | 610 | 189.87 | 8.26 | 5.08 | 41.23 |

7075-12 | T-L | 468.9 | 76.9 | 1.8 | 610 | 188.60 | 8.26 | 6.35 | 46.68 |

**Table 4.**Experimental data for the stiffened (Clad) 2024-T3 panel configurations [10].

Panel ID | Stiff. Config. | Grain Orient. | σ_{y}MPa | K_{C}MPa.m ^{1/2} | A_{stf}mm ^{2} | t mm | W mm | a mm | ℓ mm | L mm | σ_{Exp}MPa |
---|---|---|---|---|---|---|---|---|---|---|---|

S-1 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 118.75 | 4.45 | 3.81 | 73.09 |

S-2 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 116.21 | 4.45 | 6.35 | 86.95 |

S-3 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 113.67 | 4.45 | 8.89 | 99.91 |

S-4 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 108.59 | 6.99 | 11.43 | 95.43 |

S-5 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 107.32 | 5.72 | 13.97 | 110.66 |

S-6 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 106.05 | 4.45 | 16.51 | 122.18 |

S-7 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 144.15 | 4.45 | 3.81 | 74.67 |

S-8 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 141.61 | 4.45 | 6.35 | 88.67 |

S-9 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 139.07 | 4.45 | 8.89 | 97.08 |

S-10 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 133.99 | 6.99 | 11.43 | 96.25 |

S-11 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 132.72 | 5.72 | 13.97 | 112.66 |

S-12 | One-Bay | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 131.45 | 4.45 | 16.51 | 127.83 |

S-13 | One-Bay | L-T | 310.3 | 114.3 | 161.3 | 1.6 | 610 | 113.67 | 4.45 | 8.89 | 107.08 |

S-14 | One-Bay | L-T | 310.3 | 114.3 | 151.6 | 1.6 | 610 | 108.59 | 6.99 | 11.43 | 108.6 |

S-15 | One-Bay | L-T | 310.3 | 114.3 | 151.6 | 1.6 | 610 | 107.32 | 5.72 | 13.97 | 119.9 |

S-16 | One-Bay | L-T | 310.3 | 114.3 | 151.6 | 1.6 | 610 | 106.05 | 4.45 | 16.51 | 130.8 |

S-17 | One-Bay | L-T | 310.3 | 114.3 | 161.3 | 1.6 | 610 | 144.15 | 4.45 | 3.81 | 81.5 |

S-18 | One-Bay | L-T | 310.3 | 114.3 | 151.6 | 1.6 | 610 | 133.99 | 6.99 | 11.43 | 114.04 |

S-19 | One-Bay | L-T | 310.3 | 114.3 | 161.3 | 1.6 | 610 | 132.72 | 5.72 | 13.97 | 120.32 |

S-20 | One-Bay | L-T | 310.3 | 114.3 | 161.3 | 1.6 | 610 | 131.45 | 4.45 | 16.51 | 139.35 |

S-21 | One-Bay | L-T | 303.4 | 117.6 | 105 | 1.02 | 610 | 81.92 | 5.72 | 13.97 | 130.73 |

S-22 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 107.32 | 5.72 | 13.97 | 80.53 |

S-23 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 108.59 | 6.99 | 11.43 | 70.88 |

S-24 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 109.86 | 8.26 | 8.89 | 58.68 |

S-25 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 132.72 | 5.72 | 13.97 | 75.85 |

S-26 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 133.99 | 6.99 | 11.43 | 67.85 |

S-27 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 135.26 | 8.26 | 8.89 | 56.75 |

S-28 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 158.12 | 5.72 | 13.97 | 72.26 |

S-29 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 159.39 | 6.99 | 11.43 | 63.92 |

S-30 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 160.66 | 8.26 | 8.89 | 54.75 |

S-31 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 183.52 | 5.72 | 13.97 | 68.74 |

S-32 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 184.79 | 6.99 | 11.43 | 60.95 |

S-33 | Two-Bay * | T-L | 275.8 | 109.9 | 105 | 1.6 | 610 | 186.06 | 8.26 | 8.89 | 51.85 |

S-34 | Two-Bay * | L-T | 310.3 | 114.3 | 105 | 1.6 | 610 | 107.32 | 5.72 | 13.97 | 97.01 |

S-35 | Two-Bay * | L-T | 310.3 | 114.3 | 105 | 1.6 | 610 | 132.72 | 5.72 | 13.97 | 84.26 |

S-36 | Two-Bay * | L-T | 310.3 | 114.3 | 105 | 1.6 | 610 | 158.12 | 5.72 | 13.97 | 82.33 |

**Table 5.**Experimental data for the lap-joint (Clad) 2024-T3 panel configurations [11].

Panel ID | Grain Orient. | σ_{y}MPa | K_{C}MPa.m ^{1/2} | t mm | W mm | a mm | ℓ mm | L mm | σ_{Exp}MPa |
---|---|---|---|---|---|---|---|---|---|

LJ-1 | T-L | 268.9 | 109.9 | 1.42 | 610 | 106.52 | 3.65 | 16.83 | 126.73 |

LJ-2 | T-L | 268.9 | 109.9 | 1.42 | 610 | 106.52 | 4.92 | 15.56 | 115.77 |

LJ-3 | T-L | 268.9 | 109.9 | 1.42 | 610 | 107.79 | 3.65 | 15.56 | 126.73 |

LJ-4 | T-L | 268.9 | 109.9 | 1.42 | 610 | 107.79 | 4.92 | 14.29 | 113.35 |

LJ-5 | T-L | 268.9 | 109.9 | 1.42 | 610 | 107.79 | 6.19 | 13.02 | 106.53 |

LJ-6 | T-L | 268.9 | 109.9 | 1.42 | 610 | 109.06 | 3.65 | 14.29 | 119.84 |

LJ-7 | T-L | 268.9 | 109.9 | 1.42 | 610 | 109.06 | 4.92 | 13.02 | 113.56 |

LJ-8 | T-L | 268.9 | 109.9 | 1.42 | 610 | 109.06 | 6.19 | 11.75 | 105.77 |

LJ-9 | T-L | 268.9 | 109.9 | 1.42 | 610 | 109.06 | 7.46 | 10.48 | 101.49 |

LJ-10 | T-L | 268.9 | 109.9 | 1.42 | 610 | 131.92 | 3.65 | 16.83 | 107.29 |

LJ-11 | T-L | 268.9 | 109.9 | 1.42 | 610 | 131.92 | 4.92 | 15.56 | 102.6 |

LJ-12 | T-L | 268.9 | 109.9 | 1.42 | 610 | 133.19 | 3.65 | 15.56 | 105.91 |

LJ-13 | T-L | 268.9 | 109.9 | 1.42 | 610 | 133.19 | 4.92 | 14.29 | 97.29 |

LJ-14 | T-L | 268.9 | 109.9 | 1.42 | 610 | 133.19 | 6.19 | 13.02 | 90.32 |

LJ-15 | T-L | 268.9 | 109.9 | 1.42 | 610 | 134.46 | 3.65 | 14.29 | 102.18 |

LJ-16 | T-L | 268.9 | 109.9 | 1.42 | 610 | 134.46 | 4.92 | 13.02 | 96.25 |

LJ-17 | T-L | 268.9 | 109.9 | 1.42 | 610 | 134.46 | 6.19 | 11.75 | 88.26 |

LJ-18 | T-L | 268.9 | 109.9 | 1.42 | 610 | 134.46 | 7.46 | 10.48 | 81.91 |

LJ-19 | T-L | 268.9 | 109.9 | 1.42 | 610 | 157.32 | 3.65 | 16.83 | 89.29 |

LJ-20 | T-L | 268.9 | 109.9 | 1.42 | 610 | 157.32 | 4.92 | 15.56 | 86.26 |

LJ-21 | T-L | 268.9 | 109.9 | 1.42 | 610 | 158.59 | 3.65 | 15.56 | 87.84 |

LJ-22 | T-L | 268.9 | 109.9 | 1.42 | 610 | 158.59 | 4.92 | 14.29 | 81.29 |

LJ-23 | T-L | 268.9 | 109.9 | 1.42 | 610 | 158.59 | 6.19 | 13.02 | 75.02 |

LJ-24 | T-L | 268.9 | 109.9 | 1.42 | 610 | 159.86 | 3.65 | 14.29 | 84.46 |

LJ-25 | T-L | 268.9 | 109.9 | 1.42 | 610 | 159.86 | 4.92 | 13.02 | 79.78 |

LJ-26 | T-L | 268.9 | 109.9 | 1.42 | 610 | 159.86 | 6.19 | 11.75 | 74.12 |

LJ-27 | T-L | 268.9 | 109.9 | 1.42 | 610 | 159.86 | 7.46 | 10.48 | 69.23 |

**Table 6.**Optimum artificial neural networks (ANN) configuration (based the validation dataset) for each learning algorithm and its performance metrics (averages for the 40 random combinations of the validation dataset).

SCG | LM | BR | |
---|---|---|---|

${f}_{opt}$ | satlin | logsig | elliotsig |

${H}_{opt}$ | 19 | 19 | 30 |

MAE_{P} [%] | 6.99 | 4.59 | 3.38 |

RSME_{P} [%] | 9.61 | 7.22 | 4.9 |

R^{2} [%] | 85.77 | 91.27 | 96 |

**Table 7.**The ANN predictions overall performance metrics for the different materials and panel configurations (averages for the 40 random combinations of the testing datasets).

Unstif (2024) | Unstif (2524) | Unstif (7075) | Stif | Lab-joint | All | |
---|---|---|---|---|---|---|

MAE [%]_{P} | 4.61 | 2.86 | 8.2 | 3.62 | 1.81 | 3.82 |

RMSE [%]_{P} | 6.36 | 3.38 | 9.96 | 4.49 | 2.21 | 5.88 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hijazi, A.; Al-Dahidi, S.; Altarazi, S.
Residual Strength Prediction of Aluminum Panels with Multiple Site Damage Using Artificial Neural Networks. *Materials* **2020**, *13*, 5216.
https://doi.org/10.3390/ma13225216

**AMA Style**

Hijazi A, Al-Dahidi S, Altarazi S.
Residual Strength Prediction of Aluminum Panels with Multiple Site Damage Using Artificial Neural Networks. *Materials*. 2020; 13(22):5216.
https://doi.org/10.3390/ma13225216

**Chicago/Turabian Style**

Hijazi, Ala, Sameer Al-Dahidi, and Safwan Altarazi.
2020. "Residual Strength Prediction of Aluminum Panels with Multiple Site Damage Using Artificial Neural Networks" *Materials* 13, no. 22: 5216.
https://doi.org/10.3390/ma13225216