The paper presents the use of a selforganizing 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.
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 [
In the selforganizing feature map (SOFM), cells are organized in various sensual areas with regular and significant computational maps [
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 [
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.
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 [
In the SOFM, processor units are placed within the nodes of a onedimensional or twodimensional network (Figure
Structural model of (a) a onedimensional network [
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. [
Typical damage range in concrete reinforcement frames [
Number  Park & Ang index value  Type of damage 

1 

Without any damage or minor cracking 
2 

Minor damage/cracking across the structure 
3 

Moderate damage, severe cracking 
4 

Sever damage, crushing of concrete and reinforcement protrusion 
5 

Structural collapse 
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
Data associated with reinforced concrete frames with shear walls [
Frame  Special reinforced concrete with a shear wall 

Height of each story  3.2 m 
Bay of each frame  5 m 
Dead load at roof  600 kg/m^{2} 
Live load at roof  175 kg/m^{2} 
Dead load at stories  500 kg/m^{2} 
Live load at stories  200 kg/m^{2} 
Earthquake danger  Areas with a relatively high risk 
Ratio of steel ( 

Twentyeight day resistance of the concrete cylindrical sample 

Flow stress in steel 

Shear wall  At the middle bay 
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 [
Seismic characteristics used in the study [
Number  Peak ground acceleration (PGA)  Earthquake name  Station 

1  0.254  Imperial Valley 1979  Chihuahua 
2  0.27  Imperial Valley 1979  Chihuahua 
3  0.231  Northridge 1994  Hollywood Storage 
4  0.145  San Fernando 1971  Lake Hughes #1 
5  0.210  San Fernando 1971  Hollywood Stor Lot 
6  0.134  Superstition Hills 1987  Wildlife LiquefactionArrey 
7  0.134  Superstition Hills 1987  Wildlife iquefaction Arrey 
8  0.119  Superstition Hills 1987  Salton Sea Wildlife Refuge 
9  0.186  Superstition Hills 1987  Plaster City 
10  0.247  Superstition Hills 1987  Calipatria Fire Station 
11  0.135  Landers 1992  Barstow 
12  0.385  Cape Mendocino 1992  Rio Dell Overpass 
13  0.549  Cape Mendocino 1992  Rio Dell Overpass 
14  0.164  Coalinga 1983  ParkfieldFault Zone 3 
15  0.126  Whittier Narrows 1987  Beverly Hills 
16  0.239  Northridge, 1994  LA, Baldwin Hills 
17  0.143  Imperial Valley, 1979  El Centro Array #12 
18  0.240  Loma Prieta, 1989  Anderson Dam Downstream 
19  0.247  Loma Prieta, 1989  Anderson Dam Downstream 
20  0.159  Loma Prieta, 1989  Agnews State Hospital 
21  0.244  Loma Prieta, 1989  Anderson Dam Downstream 
22  0.179  Loma Prieta, 1989  Coyote Lake Dam Downstream 
23  0.309  Imperial Valley, 1979  Cucapah 
24  0.207  Loma Prieta, 1989  Sunnyvale Colton Ave 
25  0.117  Imperial Valley, 1979  El Centro Array #13 
26  0.074  Imperial Valley, 1979  Westmoreland Fire Station 
27  0.209  Loma Prieta, 1989  Sunnyvale Colton Ave 
28  0.139  Imperial Valley, 1979  El Centro Array #13 
29  0.110  Imperial Valley, 1979  Westmoreland Fire Station 
30  0.269  Loma Prieta, 1989  Hollister Diff. Array 
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
Selected statistical characteristics of the parameters.
Row  Parameter name  Unit  Parameter type  Minimum  Maximum  Average  Standard deviation 

1  Peak ground acceleration (PGA)  m/s^{2}  Input  0.074  0.549  0.207  0.095 
2  Input time of the earthquake to a structure  s  Input  0.005  0.020  0.011  0.004 
3  Time  s  Input  21.880  39.990  36.187  5.896 
4  Frequency  Hz  Input  0.025  0.046  0.029  0.006 
5  Input acceleration to the building (Acc)  m/s^{2}  Input  0.100  1.500  0.795  0.406 
6  Displacement  mm  Input  11.025  1023.293  222.576  208.891 
7  Park & Ang damage index  —  Output  0.008  0.823  0.153  0.138 
Three Kohonen ANNs (Square, Line, and Diamond) were employed in this research for the SOFM. From 412 sets of data, 70% (288 sets) were used for training, 15% (62 sets) were used for validation, and 15% (62 sets) were used for testing of the ANN. Different stimulation functions, including LinearTanhAxon, LinearAxon, and TanhAxon, were used. Table
The characteristics of the selected SOFM models.
Row  Model name  Number of inputs  Number of outputs  Number of hidden layers  Number of nods in the hidden layer  Column and Row in the network  Training algorithm  Transfer function  Neighborhood shape 

1  SOFM 1  6  1  1  6 

TanhAxon  Momentum  Square 
2  SOFM 2  2  8_4 

LinearTanhAxon  Step  Line  
3  SOFM 3  3  4_4_4 

LinearAxon  QuickProp  Diamond 
Table
The optimized structure of SOFM models in training, validation, and testing.
Number  Model  Training  Validation  Testing  


Equation 

Equation 

Equation  
1  SOFM1  0.9216 

0.9330 

0.9213 

2  SOFM2  0.7590 

0.6703 

0.8250 

3  SOFM3  0.6321 

0.5323 

0.6864 

Statistical results of different SOFM models.
Number  Model  MAE  MSE  RMSE  

Training  Validation  Testing  Training  Validation  Testing  Training  Validation  Testing  
1  SOFM1  0.024  0.020  0.024  0.002  0.001  0.001  0.041  0.028  0.033 
2  SOFM2  0.048  0.054  0.042  0.005  0.004  0.003  0.072  0.065  0.052 
3  SOFM3  0.061  0.063  0.056  0.008  0.007  0.005  0.089  0.085  0.070 
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
Comparison of the Park and Ang damage index and calculated data for (a) training, (b) validation, and (c) testing.
The obtained values of correlation coefficient
MSE versus epoch in training and validation of the SOFM1 model.
Considering the above, the best ANN for adaptation of input data is the SOFM with a 5 × 5 structure (Figure
Structure for the adaptation of input data in training and validation of the SOFM1 model.
In addition, the impact of distances and weights of the neighborhood in a 5 × 5 structure in the SOFM1 model is presented in Figure
The impact of distances and weights of the neighborhood in a 5 × 5 structure in the SOFM1 model.
First, linear regression (LR) was used [
The structure of LR models in training, validation, and testing.
Number  Model  Type  Training  Validation  Testing  


Equation 

Equation 

Equation  
1  LR 1  Linear regression (LR)  0.8924 

0.9098 

0.8925 

2  LR 2  0.6883 

0.6693 

0.7946 


3  LR 3  0.0096 

0.0062 

0.0241 

Statistical results of different LR models.
Number  Model  MAE  MSE  RMSE  

Training  Validation  Testing  Training  Validation  Testing  Training  Validation  Testing  
1  LR 1  0.027  0.023  0.026  0.002  0.001  0.002  0.048  0.033  0.039 
2  LR 2  0.050  0.048  0.040  0.007  0.004  0.003  0.082  0.066  0.054 
3  LR 3  0.114  0.089  0.099  0.021  0.012  0.014  0.145  0.111  0.118 
Comparison of the Park and Ang damage index obtained using LR and calculated data for (a) training, (b) validation, and (c) testing.
In the LR1 model, the values of
In nonlinear regression (NonLR), the PARK_ANG parameter (
Statistical results of different NonLR models.
Model  MAE  MSE  RMSE  

Training  NonLR1  2.792  10.840  3.292 
NonLR2  0.030  0.003  0.051  
Validation  NonLR1  3.182  12.835  3.583 
NonLR2  0.021  0.001  0.031  
Testing  NonLR1  2.375  8.372  2.894 
NonLR2  0.031  0.002  0.050 
The structure of NonLR models in training, validation, and testing.
Number  Model  Type  Training  Validation  Testing  


Equation 

Equation 

Equation  
1  NonLR 1  Nonlinear regression 
0.7351 

0.8014 

0.7925 

2  NonLR 2  0.8781 

0.9174 

0.8272 

Comparison of the Park and Ang damage index obtained by NonLR and calculated data for (a) training, (b) validation, and (c) testing.
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 614, 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 results were compared with the results derived from the SOFM, LR models, RBF network, and NonLR. This comparison was conducted in three steps of training, testing, and validation. The obtained results are shown in Table
Structure of different models for assessing the Park & Ang damage.
Number  Model  Training  Validation  Testing  


Equation 

Equation 

Equation  
1  SOFM1  0.9216 

0.933 

0.9213 

2  LR1  0.8924 

0.9098 

0.8925 

3  RBF  0.8054 

0.8304 

0.8789 

4  NonLR2  0.8781 

0.9174 

0.8272 

Statistical results of different models.
Number  Model  MAE  MSE  RMSE  

Training  Validation  Testing  Training  Validation  Testing  Training  Validation  Testing  
1  SOFM1  0.024  0.020  0.024  0.002  0.001  0.001  0.041  0.028  0.033 
2  LR1  0.027  0.023  0.026  0.002  0.001  0.002  0.048  0.033  0.039 
3  RBF  0.043  0.039  0.036  0.004  0.002  0.002  0.064  0.048  0.044 
4  NonLR2  0.030  0.021  0.031  0.003  0.001  0.002  0.051  0.031  0.050 
Comparison of the obtained results for (a) validation and (b) testing.
When considering
In this paper, the selforganizing 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.
The authors declare that there are no conflicts of interests regarding the publication of this paper.