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

Authors: Mehdi Nikoo; Łukasz Sadowski; Faezehossadat Khademi; Mohammad Nikoo
Journal: Applied Computational Intelligence and Soft Computing
 (2017)

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## Abstract

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.


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## Body

## 1. 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 [3–8].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 one-dimensional (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.


## 2. 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 (Figure1). 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].Figure 1
Structural model of (a) a one-dimensional network [9] and (b) a two-dimensional network [14].
(a)
(b)


## 3. 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]: (1)D=δmδu+βδu·Py∫dEh,whereδm is the maximum response of deformation under seismic load.Py is the calculated yield strength.δu is the ultimate deformation under uniform loading.∫dE is the hysteric absorbed energy.β 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.Table 1
Typical damage range in concrete reinforcement frames [13].
Number
Park & Ang index value
Type of damage
1
D
≤
0.1
Without any damage or minor cracking
2
0.1
≤
D
≤
0.25
Minor damage/cracking across the structure
3
0.25
≤
D
≤
0.4
Moderate damage, severe cracking
4
0.4
≤
D
≤
1
Sever damage, crushing of concrete and reinforcement protrusion
5
D
≥
1
Structural collapse


## 4. 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 Table2.Table 2
Data associated with reinforced concrete frames with shear walls [2, 16].
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/m2
Live load at roof
175 kg/m2
Dead load at stories
500 kg/m2
Live load at stories
200 kg/m2
Earthquake danger
Areas with a relatively high risk
Ratio of steel (ρ) in building columns
0.015
≤
ρ
≤
0.035
Twenty-eight day resistance of the concrete cylindrical sample
f
′
c
=
240 kg/cm2
Flow stress in steel
f
y
=
3000 kg/cm2
Shear wall
At the middle bayOne 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.Table 3
Seismic characteristics used in the study [2, 16].
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
Parkfield-Fault 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. ArrayThe 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. Table4 represents the statistical characteristics of the parameters.Table 4
Selected statistical characteristics of the parameters.
Row
Parameter name
Unit
Parameter type
Minimum
Maximum
Average
Standard deviation
1
Peak ground acceleration (PGA)
m/s2
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/s2
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


## 5. Experimental Results

### 5.1. Performance Evaluation of the SOFM

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. Table5 shows the characteristics of the selected SOFM models.Table 5
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
5
∗
5
TanhAxon
Momentum
Square
2
SOFM 2
2
8_4
6
∗
6
LinearTanhAxon
Step
Line
3
SOFM 3
3
4_4_4
7
∗
7
LinearAxon
QuickProp
DiamondTable6 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 R2 for prediction of the Park and Ang damage index for training, validation, and testing.Table 6
The optimized structure of SOFM models in training, validation, and testing.
Number
Model
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
SOFM1
0.9216
y
=
0.9229
x
+
0.0128
0.9330
y
=
0.9102
x
+
0.0133
0.9213
y
=
0.9353
x
+
0.0118
2
SOFM2
0.7590
y
=
0.7178
x
+
0.0483
0.6703
y
=
0.7542
x
+
0.049
0.8250
y
=
0.9174
x
+
0.0232
3
SOFM3
0.6321
y
=
0.6561
x
+
0.0579
0.5323
y
=
0.7059
x
+
0.0713
0.6864
y
=
0.8166
x
+
0.0435Table 7
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
0.070Comparison of the Park and Ang damage index and calculated data for training, validation, and testing for each of the laboratory samples is presented in Figure2.Figure 2
Comparison of the Park and Ang damage index and calculated data for (a) training, (b) validation, and (c) testing.
(a)
(b)
(c)The obtained values of correlation coefficientR2 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.Figure 3
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 (Figure4).Figure 4
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 Figure5.Figure 5
The impact of distances and weights of the neighborhood in a 5 × 5 structure in the SOFM1 model.


### 5.2. Comparison of Selected SOFM Models

#### 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 (2)y=fx1,x2,…⟶y=a0+a1x1+a2x2+⋯,where y is the dependent variable; x1, x2, etc. are the independent variables; and a1, a2, a3, 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)Y=-0.206-0.0246X1+0.449X2+0.00350X3+2.08X4+0.7635X5+0.000500X6LR10.174-0.0894X1+0.74X2-0.00278X3-4.71X4+0.28218X5LR20.572-0.1014X1+1.63X2-0.00589X3-7.12X4LR3.In the above equation, Y is the damage to the entire frame, X1 is the PGA variable, X2 is the input time variable, X3 is the time variable, X4 is the frequency variable, X5 is the acceleration variable, and X6 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.Table 8
The structure of LR models in training, validation, and testing.
Number
Model
Type
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
LR 1
Linear regression (LR)
0.8924
y
=
0.8847
x
+
0.017
0.9098
y
=
0.9715
x
+
0.0053
0.8925
y
=
0.9319
x
+
0.0123
2
LR 2
0.6883
y
=
0.6627
x
+
0.0486
0.6693
y
=
0.826
x
+
0.035
0.7946
y
=
0.845
x
+
0.032
3
LR 3
0.0096
y
=
0.0098
x
+
0.1509
0.0062
y
=
0.0131
x
+
0.1494
0.0241
y
=
0.0182
x
+
0.1515Table 9
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
0.118Figure 6
Comparison of the Park and Ang damage index obtained using LR and calculated data for (a) training, (b) validation, and (c) testing.
(a)
(b)
(c)In the LR1 model, the values ofR2 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.


#### 5.2.2. Nonlinear Regression (NonLR)

In nonlinear regression (NonLR), the PARK_ANG parameter (y) is a dependent value and displacement (x) is an independent value. The best NonLR model, which was more coordinated with damage data, was obtained via (4)Y=0.5404∗exp⁡-x-922.95982NonLR10.977∗sin⁡0.0006597∗x+0.01211NonLR2.The analysis results of the selected NonLR models are presented in Table 10 and the results obtained from the statistical indices are presented in Table 11. Figure 7 shows the results arising from different NonLR models.Table 10
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
0.050Table 11
The structure of NonLR models in training, validation, and testing.
Number
Model
Type
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
NonLR 1
Nonlinear regression(NonLR)
0.7351
y
=
-
9.5762
x
+
4.4271
0.8014
y
=
-
13.122
x
+
4.85
0.7925
y
=
-
11.403
x
+
4.5353
2
NonLR 2
0.8781
y
=
0.8708
x
+
0.0203
0.9174
y
=
0.9506
x
+
0.0083
0.8272
y
=
0.878
x
+
0.0191Figure 7
Comparison of the Park and Ang damage index obtained by NonLR and calculated data for (a) training, (b) validation, and (c) testing.
(a)
(b)
(c)


#### 5.2.3. 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 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 Table12. 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.Table 12
Structure of different models for assessing the Park & Ang damage.
Number
Model
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
SOFM1
0.9216
y
=
0.9229
x
+
0.0128
0.933
y
=
0.9102
x
+
0.0133
0.9213
y
=
0.9353
x
+
0.0118
2
LR1
0.8924
y
=
0.8847
x
+
0.017
0.9098
y
=
0.9715
x
+
0.0053
0.8925
y
=
0.9319
x
+
0.0123
3
RBF
0.8054
y
=
0.8151
x
+
0.0238
0.8304
y
=
0.9509
x
+
0.0187
0.8789
y
=
0.9784
x
+
0.0122
4
NonLR2
0.8781
y
=
0.8708
x
+
0.0203
0.9174
y
=
0.9506
x
+
0.0083
0.8272
y
=
0.878
x
+
0.0191Table 13
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
0.050Figure 8
Comparison of the obtained results for (a) validation and (b) testing.
(a)
(b)When consideringR2 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.


## 5.1. Performance Evaluation of the SOFM

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. Table5 shows the characteristics of the selected SOFM models.Table 5
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
5
∗
5
TanhAxon
Momentum
Square
2
SOFM 2
2
8_4
6
∗
6
LinearTanhAxon
Step
Line
3
SOFM 3
3
4_4_4
7
∗
7
LinearAxon
QuickProp
DiamondTable6 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 R2 for prediction of the Park and Ang damage index for training, validation, and testing.Table 6
The optimized structure of SOFM models in training, validation, and testing.
Number
Model
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
SOFM1
0.9216
y
=
0.9229
x
+
0.0128
0.9330
y
=
0.9102
x
+
0.0133
0.9213
y
=
0.9353
x
+
0.0118
2
SOFM2
0.7590
y
=
0.7178
x
+
0.0483
0.6703
y
=
0.7542
x
+
0.049
0.8250
y
=
0.9174
x
+
0.0232
3
SOFM3
0.6321
y
=
0.6561
x
+
0.0579
0.5323
y
=
0.7059
x
+
0.0713
0.6864
y
=
0.8166
x
+
0.0435Table 7
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
0.070Comparison of the Park and Ang damage index and calculated data for training, validation, and testing for each of the laboratory samples is presented in Figure2.Figure 2
Comparison of the Park and Ang damage index and calculated data for (a) training, (b) validation, and (c) testing.
(a)
(b)
(c)The obtained values of correlation coefficientR2 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.Figure 3
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 (Figure4).Figure 4
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 Figure5.Figure 5
The impact of distances and weights of the neighborhood in a 5 × 5 structure in the SOFM1 model.


## 5.2. Comparison of Selected SOFM Models

### 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 (2)y=fx1,x2,…⟶y=a0+a1x1+a2x2+⋯,where y is the dependent variable; x1, x2, etc. are the independent variables; and a1, a2, a3, 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)Y=-0.206-0.0246X1+0.449X2+0.00350X3+2.08X4+0.7635X5+0.000500X6LR10.174-0.0894X1+0.74X2-0.00278X3-4.71X4+0.28218X5LR20.572-0.1014X1+1.63X2-0.00589X3-7.12X4LR3.In the above equation, Y is the damage to the entire frame, X1 is the PGA variable, X2 is the input time variable, X3 is the time variable, X4 is the frequency variable, X5 is the acceleration variable, and X6 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.Table 8
The structure of LR models in training, validation, and testing.
Number
Model
Type
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
LR 1
Linear regression (LR)
0.8924
y
=
0.8847
x
+
0.017
0.9098
y
=
0.9715
x
+
0.0053
0.8925
y
=
0.9319
x
+
0.0123
2
LR 2
0.6883
y
=
0.6627
x
+
0.0486
0.6693
y
=
0.826
x
+
0.035
0.7946
y
=
0.845
x
+
0.032
3
LR 3
0.0096
y
=
0.0098
x
+
0.1509
0.0062
y
=
0.0131
x
+
0.1494
0.0241
y
=
0.0182
x
+
0.1515Table 9
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
0.118Figure 6
Comparison of the Park and Ang damage index obtained using LR and calculated data for (a) training, (b) validation, and (c) testing.
(a)
(b)
(c)In the LR1 model, the values ofR2 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.


### 5.2.2. Nonlinear Regression (NonLR)

In nonlinear regression (NonLR), the PARK_ANG parameter (y) is a dependent value and displacement (x) is an independent value. The best NonLR model, which was more coordinated with damage data, was obtained via (4)Y=0.5404∗exp⁡-x-922.95982NonLR10.977∗sin⁡0.0006597∗x+0.01211NonLR2.The analysis results of the selected NonLR models are presented in Table 10 and the results obtained from the statistical indices are presented in Table 11. Figure 7 shows the results arising from different NonLR models.Table 10
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
0.050Table 11
The structure of NonLR models in training, validation, and testing.
Number
Model
Type
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
NonLR 1
Nonlinear regression(NonLR)
0.7351
y
=
-
9.5762
x
+
4.4271
0.8014
y
=
-
13.122
x
+
4.85
0.7925
y
=
-
11.403
x
+
4.5353
2
NonLR 2
0.8781
y
=
0.8708
x
+
0.0203
0.9174
y
=
0.9506
x
+
0.0083
0.8272
y
=
0.878
x
+
0.0191Figure 7
Comparison of the Park and Ang damage index obtained by NonLR and calculated data for (a) training, (b) validation, and (c) testing.
(a)
(b)
(c)


### 5.2.3. 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 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 Table12. 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.Table 12
Structure of different models for assessing the Park & Ang damage.
Number
Model
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
SOFM1
0.9216
y
=
0.9229
x
+
0.0128
0.933
y
=
0.9102
x
+
0.0133
0.9213
y
=
0.9353
x
+
0.0118
2
LR1
0.8924
y
=
0.8847
x
+
0.017
0.9098
y
=
0.9715
x
+
0.0053
0.8925
y
=
0.9319
x
+
0.0123
3
RBF
0.8054
y
=
0.8151
x
+
0.0238
0.8304
y
=
0.9509
x
+
0.0187
0.8789
y
=
0.9784
x
+
0.0122
4
NonLR2
0.8781
y
=
0.8708
x
+
0.0203
0.9174
y
=
0.9506
x
+
0.0083
0.8272
y
=
0.878
x
+
0.0191Table 13
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
0.050Figure 8
Comparison of the obtained results for (a) validation and (b) testing.
(a)
(b)When consideringR2 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.


## 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 (2)y=fx1,x2,…⟶y=a0+a1x1+a2x2+⋯,where y is the dependent variable; x1, x2, etc. are the independent variables; and a1, a2, a3, 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)Y=-0.206-0.0246X1+0.449X2+0.00350X3+2.08X4+0.7635X5+0.000500X6LR10.174-0.0894X1+0.74X2-0.00278X3-4.71X4+0.28218X5LR20.572-0.1014X1+1.63X2-0.00589X3-7.12X4LR3.In the above equation, Y is the damage to the entire frame, X1 is the PGA variable, X2 is the input time variable, X3 is the time variable, X4 is the frequency variable, X5 is the acceleration variable, and X6 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.Table 8
The structure of LR models in training, validation, and testing.
Number
Model
Type
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
LR 1
Linear regression (LR)
0.8924
y
=
0.8847
x
+
0.017
0.9098
y
=
0.9715
x
+
0.0053
0.8925
y
=
0.9319
x
+
0.0123
2
LR 2
0.6883
y
=
0.6627
x
+
0.0486
0.6693
y
=
0.826
x
+
0.035
0.7946
y
=
0.845
x
+
0.032
3
LR 3
0.0096
y
=
0.0098
x
+
0.1509
0.0062
y
=
0.0131
x
+
0.1494
0.0241
y
=
0.0182
x
+
0.1515Table 9
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
0.118Figure 6
Comparison of the Park and Ang damage index obtained using LR and calculated data for (a) training, (b) validation, and (c) testing.
(a)
(b)
(c)In the LR1 model, the values ofR2 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.


## 5.2.2. Nonlinear Regression (NonLR)

In nonlinear regression (NonLR), the PARK_ANG parameter (y) is a dependent value and displacement (x) is an independent value. The best NonLR model, which was more coordinated with damage data, was obtained via (4)Y=0.5404∗exp⁡-x-922.95982NonLR10.977∗sin⁡0.0006597∗x+0.01211NonLR2.The analysis results of the selected NonLR models are presented in Table 10 and the results obtained from the statistical indices are presented in Table 11. Figure 7 shows the results arising from different NonLR models.Table 10
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
0.050Table 11
The structure of NonLR models in training, validation, and testing.
Number
Model
Type
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
NonLR 1
Nonlinear regression(NonLR)
0.7351
y
=
-
9.5762
x
+
4.4271
0.8014
y
=
-
13.122
x
+
4.85
0.7925
y
=
-
11.403
x
+
4.5353
2
NonLR 2
0.8781
y
=
0.8708
x
+
0.0203
0.9174
y
=
0.9506
x
+
0.0083
0.8272
y
=
0.878
x
+
0.0191Figure 7
Comparison of the Park and Ang damage index obtained by NonLR and calculated data for (a) training, (b) validation, and (c) testing.
(a)
(b)
(c)


## 5.2.3. 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 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 Table12. 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.Table 12
Structure of different models for assessing the Park & Ang damage.
Number
Model
Training
Validation
Testing
R
2
Equation
R
2
Equation
R
2
Equation
1
SOFM1
0.9216
y
=
0.9229
x
+
0.0128
0.933
y
=
0.9102
x
+
0.0133
0.9213
y
=
0.9353
x
+
0.0118
2
LR1
0.8924
y
=
0.8847
x
+
0.017
0.9098
y
=
0.9715
x
+
0.0053
0.8925
y
=
0.9319
x
+
0.0123
3
RBF
0.8054
y
=
0.8151
x
+
0.0238
0.8304
y
=
0.9509
x
+
0.0187
0.8789
y
=
0.9784
x
+
0.0122
4
NonLR2
0.8781
y
=
0.8708
x
+
0.0203
0.9174
y
=
0.9506
x
+
0.0083
0.8272
y
=
0.878
x
+
0.0191Table 13
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
0.050Figure 8
Comparison of the obtained results for (a) validation and (b) testing.
(a)
(b)When consideringR2 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.


## 6. 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.



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