Determination of Damage in Reinforced Concrete Frames with Shear Walls Using Self-Organizing Feature Map

. The paper presents the use of a self-organizing feature map (SOFM) for determining damage in reinforced concrete frames with shear walls. For this purpose, a concrete frame with a shear wall was subjected to nonlinear dynamic analysis. The SOFM was optimized using the genetic algorithm (GA) in order to determine the number of layers, number of nodes in the hidden layer, transfer function type, and learning algorithm. The obtained model was compared with linear regression (LR) and nonlinear regression (NonLR) models and also the radial basis function (RBF) of a neural network. It was concluded that the SOFM, when optimized with the GA, has more strength, flexibility, and accuracy.


Introduction
Damage to concrete structures mainly occurs because of inadequate management, incorrect maintenance, overloading, exposure to chemical components, climatic factors, and also extra loads such as earthquakes [1]. As mentioned by Nikoo et al. [2], earthquakes are the most devastating of these factors. Their destruction mechanism can cause extraordinary damage to a structure. In the last few years, much attention has been given to the use of artificial neural networks (ANNs) for solving various civil engineering problems [3][4][5][6][7][8].
In the self-organizing feature map (SOFM), cells are organized in various sensual areas with regular and significant computational maps [6,9]. As described by Kohonen [9], processor units are placed within the nodes of a onedimensional (or more) network and are regulated in a competitive learning process [9,10]. Therefore, the SOFM can be seen as a topographic map for input models, in which the units' locations correspond to the inherent features of the input models. Competitive learning is applied in such networks, and in each step the units compete in order to be activated. At the end of the initial step of this competition only one unit wins and its weights are changed differently when compared to the weights of other units. This kind of learning is called unsupervised learning [6]. In previous papers, the SOFM was described in more detail [6,10].
In the genetic algorithm (GA), chromosomes with high competence have a higher chance of repeating in the selected population of the replication process. The basic operators of the GA are reproduction, crossover, and mutation [11]. The GA ends when certain criteria, such as a certain number of generations or the average standard deviation performance of individuals, are fulfilled [12].
The main objective of this study was to evaluate the abilities of the SOFM in determining damage in reinforced concrete frames with shear walls. The SOFM was optimized using the GA in order to determine the number of layers, number of nodes in the hidden layer, transfer function type, and learning algorithm. The obtained model was compared with linear regression (LR), nonlinear regression (NonLR), and the radial basis function (RBF) of a neural network.   Figure 1: Structural model of (a) a one-dimensional network [9] and (b) a two-dimensional network [14].

A Short Description of the Self-Organizing Feature Map (SOFM)
In the SOFM, the competitive learning method is used for training and is based on specific characteristics of a developed human brain. The cells in the human brain are organized in various sensual areas with regular and significant computational maps [6,9]. In the SOFM, processor units are placed within the nodes of a one-dimensional or two-dimensional network ( Figure 1). These units are regulated in a competitive learning process and compared to the input models [9,10]. Therefore, the SOFM can be seen as a topographic map for the input models, in which the units' locations correspond to the inherent features of the input models. Competitive learning is applied in such networks, and in each step the units compete in order to be activated. At the end of the initial step of this competition, only one unit wins and its weights are changed differently when compared to the weights of other units. This kind of learning is called unsupervised learning [6].

The Park and Ang Damage Index
One of the most useful methods proposed for quantifying the calculation of damage in concrete structures is the Park and Ang model. As mentioned by Valles et al. [13], it is defined as follows [15]: where is the maximum response of deformation under seismic load.
is the calculated yield strength. is the ultimate deformation under uniform loading. ∫ is the hysteric absorbed energy. is the resistance reduction parameter according to hysteric energy. The Park and Ang index value is between 0 and 1. The damage range is shown in Table 1.

Experimental Setup
To determine the distribution function for the Park and Ang damage index, a concrete frame with a shear wall was selected. Lateral loading of the mentioned structure was then applied. In the next step, the structure was designed. The data associated with reinforced concrete frames with shear walls is listed in Table 2.
One of the main parameters influencing the input energy of structures is the earthquake accelerogram applied in seismic analysis. The extent of input energy applied to the structure is more dependent on input mapping than its structural characteristics [2]. In this research, thirty earthquakes were used for nonlinear dynamic analysis, as listed in Table 3.
After the analysis, the overall Park and Ang damage index was extracted using version 4.0 of IDARC 2D software.
The input parameters in this research include the following: peak ground acceleration (PGA); input time of the earthquake to a structure; time; frequency; input acceleration to the building (Acc); and also displacement. The output parameter is the Park and Ang damage index. Table 4 represents the statistical characteristics of the parameters.

Performance Evaluation of the SOFM. Three Kohonen
ANNs (Square, Line, and Diamond) were employed in this  Table 5 shows the characteristics of the selected SOFM models. Table 6 presents the optimized structure of the SOFM models for training, validation and testing. Table 7 presents the statistical results of different models in the SOFM. As can be seen from these tables, the SOFM1 model enjoys the highest values of the correlation coefficient 2 for prediction of the Park and Ang damage index for training, validation, and testing.
Comparison of the Park and Ang damage index and calculated data for training, validation, and testing for each of the laboratory samples is presented in Figure 2.
The obtained values of correlation coefficient 2 for the Park and Ang damage index for the SOFM1 model were 0.9330, 0.9216, and 0.9221 for training, validation, and testing, respectively. Additionally, the values of mean absolute error (MAE), mean square error (MSE), and root mean square error (RMSE) are less than those for the two other models. Considering the above, MSE versus epoch for the SOFM1 model is presented in Figure 3.
Considering the above, the best ANN for adaptation of input data is the SOFM with a 5 × 5 structure (Figure 4).
In addition, the impact of distances and weights of the neighborhood in a 5 × 5 structure in the SOFM1 model is presented in Figure 5.

5.2.1.Linear Regression (LR).
First, linear regression (LR) was used [15]. LR models are based on a data oriented technique, where the collected data is directly associated with each other. The process behind this data is not considered. In a specific       form of LR, data is modeled using linear predictor functions. Unknown model parameters are then estimated from the data [16]. In LR, two or more independent variables have a major effect on the dependent variable shown in equation where is the dependent variable; 1 , 2 , etc. are the independent variables; and 1 , 2 , 3 , etc. are the equation regression coefficients. In this research, various models of LR are investigated using MINITAB software. The best LR model, which was more coordinated with damage data, was obtained using (3) In the above equation, is the damage to the entire frame, 1 is the PGA variable, 2 is the input time variable, 3 is the time variable, 4 is the frequency variable, 5 is the acceleration variable, and 6 is the displacement variable. The analysis results of the three LR models are presented in Table 8 and the results obtained from the statistical indices are presented in Table 9. Figure 6 shows the results arising from different LR models.
In the LR1 model, the values of 2 are equal to 0.8925, 0.9098, and 0.8924 for training, validation, and testing, respectively. The values of MAE, MSE, and RMSE for the LR1 model are lower than those in the other two LR models.

Nonlinear Regression (NonLR).
In nonlinear regression (NonLR), the PARK_ANG parameter ( ) is a dependent value and displacement ( ) is an independent value. The best NonLR model, which was more coordinated with damage data, was obtained via The analysis results of the selected NonLR models are presented in Table 10     indices are presented in Table 11. Figure 7 shows the results arising from different NonLR models.

Summary of the Comparison.
To determine the optimized structure of the radial basis function (RBF) neural network, version 5.0 of NeuroSolutions Software was used. The RBF with a structure of 6-1-4, the TanhAxon training algorithm, and the QuickProp transfer function were selected.
To evaluate the performance of the optimized SOFM1 model in assessing the Park and Ang damage, the obtained     Table 12. Statistical results of different models are presented in Table 13. In Figure 8, a comparison of the obtained results for validation and testing is presented. When considering 2 coefficient, the slope of the straight line and statistical indices in each of the three models, it can be concluded that the accuracy of the SOFM1 model is higher than the accuracy of the other two LR models and also the RBF network in the case of training, validation, and test steps.

Conclusion
In this paper, the self-organizing feature map (SOFM) was used to evaluate damage in reinforced concrete frames with shear walls. For this purpose, a concrete frame with a shear wall was subjected to nonlinear dynamic analysis. The extent of damage to the frame was calculated using the Park and Ang index.
The SOFM was optimized using the genetic algorithm (GA) in order to determine the number of layers, number of nodes in the hidden layer, transfer function type, and learning algorithm. The obtained model was compared with linear regression (LR) and nonlinear regression (NonLR) models and also the radial basis function (RBF) neural network. It can be concluded that the SOFM that is optimized with GA enjoys more strength, flexibility, and accuracy.