Admissible values for horizontal displacements in accordance with RCDF [16], where h is the height of space between galleries under study.
Abstract
This chapter shows the framework used to obtain data with which the artificial neural network (ANN) was developed. It describes its geometry, properties of the material, sections of structural elements, and loads used. Then, the numerical model of the framework under study is developed in structural analysis using SAP2000® software in order to obtain its modal parameters. In addition, a program made in MATLAB® is shown, from which data with and without damage to the framework under study were obtained, and with which the ANN was developed. Data from the numerical model were used to corroborate data obtained with MATLAB®. The neural model used in this work to detect structural damage is described. Data on damage were obtained simulating a plastic hinge in various elements of a test framework, varying the position of the hinge. The above resulted in obtaining various damage conditions for the same framework, which data thus obtained were used to develop the network. Damage conditions were hierarchized based on their fundamental periods in order to know where is more damage, depending on location of the hinge within the framework. Upon completion of the research, we have concluded that the methodology implemented to detect structural damage is rather simple. It was carried out in four steps.
Keywords
 damage detection
 neural network
 failure condition
 framework
 structural analysis
1. Introduction
Throughout their useful life, structures accumulate gradual damage as time passes. Such damage is caused by actions and natural phenomena such as earthquakes, winds, and explosions, among others. Therefore, and taking into account the functioning and the safety of structures, it is of paramount importance to detect damage, to follow up and monitor structures in order to know their physical conditions, and to increase safety and structural reliability. If damage is detected in a structure early, its evolution may be observed regarding magnitude and size, meaning it can be treated properly. Nowadays, there are various methods used to detect structural damage, all of which have advantages and disadvantages. There is no consensus that one method is the best. Therefore, for those who are not specialists on this topic, reading some specialized references would be useful [1, 2, 3]. In this article, an artificial neural network (ANN) is developed based on MATLAB’s toolboxes [4]. Also, a program with source code was developed using MATLAB^{®} [5], with which data were obtained (condensed matrices and modal parameters (frequencies, periods, and vibration modes) to train the network and, finally, to obtain an ANN capable to detect structural damage.
Damage has been defined as loss of stiffness. In order to identify such loss, plastic hinges in structural elements of the structure under study were simulated to obtain the condensed stiffness matrix with and without damage. Then, dynamic parameters were calculated, as well as their dynamic response with and without damage, which were used to develop the ANN. In this work, stiffness matrices with and without damage, that is, location of damage, are known.
2. Background and approach to the problem
For those devoted to analyze and design structures, it is very important to follow them up in order to know their conditions, specifically their physical conditions, in order to detect if there is any damage. Such followup on what has happened provides certainty regarding good structural safety for the structure and its inhabitants.
Some damage detection methods are visual and experimental, such as acoustic methods and methods based on magnetic, radiographic, and thermal fields. These experimental methods required access to the part of the structure where damage is located. Due to this limitation, these experimental methods detect damage in visible parts of the structure. The need to develop damage detection methods that may be applied to complex structures has led to develop methods that acknowledge changes in modal properties of structures. Every damage detection method using changes in modal properties of the structure is based on modal parameters (fundamental frequencies, vibration modes, and structure’s vibration period), which are functions of physical properties of the structure (stiffness, mass, and damping). Therefore, changes in physical properties of structures such as stiffness cause changes in modal properties.
In civil engineering, studies have been developed to detect structural damage. They have applied ANNs with good results. However, these are recent developments in the area, and for this reason, researchers consider them significant for data development.
Rumelhart and his research team rediscovered the backpropagation learning algorithm (backpropagation). From 1986, the picture was encouraging with respect to the research and development of artificial neural networks [6, 7]. Wu et al. used a backpropagation neural network to identify damage in a threestored building, modeled as a shear building in two dimensions, when being subject to earthquake excitation [8]. Damage was modeled, reducing stiffness of elements from 50 to 75%. Neural network was used to identify acceleration records and damage level of every element based on Fourier transform. The first 200 points in the fast Fourier transform (FFT) (0–20 Hz) were used as input in the network. A 200node input layer and a 10node hidden layer were selected as architecture of the network, and 42 training cases were used to train the network. In the first test, only acceleration data from the last floor were used, and the neural network was capable of identifying only the damage from the third floor with some exactness. The second network used acceleration data from two levels as input. This network was capable of diagnosing damage on the first and third levels with up to 25% approximation, but it was still incapable of predicting, with much precision, damage on the second level. The method is based on full knowledge of the acceleration record of two from three degrees of freedom.
Chen and Kim used a backpropagation neural network as a data processing technique for a metallic structure of a tridimensional bridge instrumented with accelerometers and strain gauges [9]. They studied feasibility to identify and locate structural damage using MATLAB’s Neural Network Toolbox^{®}. Vibration signals were obtained from a series of experiments on the scale structure. In their study, they developed four networks: the first one to identify damage using acceleration records, the second one to locate damage using data from strain gauges, the third one using signals simulated through analysis of finite elements and signals from the accelerometer, and the fourth one using acceleration records obtained from analysis of the finite elements model.
Marwala shows a technique based on using a set of neural networks with modal properties (frequencies and vibration modes) and the wavelet transform simultaneously in order to identify damage in structures [10]. The proposed method was tested with simulated data from a threedegreesoffreedom springmass system. The result showed that performance of the method is not influenced by noise in data. Finally, the method was used to identify damage in 10 cylindrical shells. The set was capable to identify damage better than the three methods used separately.
In this work, artificial neural networks are applied to detect structural damage in steel buildings. An artificial neural network is designed to detect damage caused by actions present in the useful life of structures. In order to determine performance of the neural network, a structure is analyzed in various single and multiple damage scenarios. Damage is considered as loss of stiffness [11, 12]. The ANN is compared to data obtained in the numerical model and the program developed in MATLAB^{®}. Learning of the developed neural network is based on epochs. The main idea here is that the network evolves for a period of time (called epoch), with an example sequence, under which it must render proper response. Upon completion of an epoch, the network is restarted in order that the new starting condition of such does not depend on the final condition of the previous epoch. Variation of weights is made upon completion of each epoch, and, as for the multilayer case, it is carried out by lots, that is, all the increases of patternsepoch are calculated in an incremental manner, which is the one most used for recurrent networks.
As for hypotheses and objectives, it may be considered that side displacements were taken into account in order to make static condensation to horizontal displacements, that is, a stiff floor framework was considered. Only beams are axially stiff, and columns are completely flexible. Another hypothesis is that nodes would be absolutely stiff. In order to obtain data with and without damage (global stiffness matrices, condensed stiffness matrices, modal parameters, and dynamic response), plastic hinges (damage) were simulated on structural elements, based on the hypothesis that these hinges were near to the node, after the structure was subject to stress causing one or more elements to surpass their yield stress.
The objective is, throughout, to detect structural damage in steel buildings applying an ANN to monitor the structure in order to know and obtain proper knowledge of the physical conditions of such to be able to make reinforcement or rehabilitation decisions.
3. Method
A threelevel framework is modeled in the structural analysis program SAP2000^{®} [13]. From this modal parameter (frequencies, periods, and mode shapes) of the framework was obtained with and without damage. A program was developed in MATLAB^{®} [5], from which the global and condensed stiffness matrices were obtained at horizontal degrees of freedom, with and without damage. Modal parameters and dynamic response with and without damage were calculated, which served for the development of the ANN, and a failure condition was considered to define a serious damage condition.
Data with damage were obtained by simulating a plastic hinge in various elements of the framework. The position of the hinges varied to obtain various damage conditions for the same framework. Data obtained were used to develop the network, and an ANN was obtained to detect structural damage. Regarding how damage conditions are determined, it must be considered that, in the moderate seismresistant design of reinforced concrete frameworks, there is the fundamental philosophy to dimension and detail the elements of concrete frame in order that a plastic strong columnweak beam mechanism may be developed [14]. In addition, due to the complexity of the problem and for a quick and easy obtaining of data to train the ANN and for simplicity purposes, it was proposed that hinges or damages on the framework appear on the ends of structural elements (beams or columns). As for the question: Are all the damage conditions considered as possible? The answer is yes, since forming hinges in concrete frameworks usually takes place on the ends of their structural elements (beams or columns) [15]. It must be remembered that the ANN shown is in the development phase and still may not be compared to a real building. The ANN was trained for a threestory frame.
Damage detection was carried out in four steps: first, extraction of modal parameters and condensed matrix; second, establishing failure condition for a serious damage condition; third, treatment of modal data to be used in the development of the ANN; and fourth, detection of damage to be carried out with the ANN.
3.1 Static condensation
In order to carry out static condensation, the side stiffness matrix
3.2 Obtaining modal parameters
To obtain modal parameters, the diagonal mass matrix is
where
To obtain the dynamic response where there is movement on the base, the secondorder linear differential equation
The distribution of these forces in height is defined by vector
The contribution of the ith mode to
4. Failure condition
The network will identify if there is damage or not. If there is any, it will consider it a failure condition, in order to determine if the structure has failed or not. Therefore, failure condition is defined as a limit service condition, that is, the presence of displacements affecting proper functioning of the building (Figure 2) not damaging its capacity to bear loads. In accordance with Article 209 of the Rules for Buildings in Mexico City [16, 17], admissible displacements are as shown in Table 1.
4.1 Presentation of a 2D framework
The flat framework of Figure 3 was analyzed considering that beams are infinitely stiff regarding the columns and that the columns are fixed on the base. Stiffness of columns and beams and the spaces between galleries are in SI units. All the elements are made of steel with a I W14X311 section (see Table 2).
For the dynamic analysis of the framework, a program was developed in MATLAB^{®}. With it condensed stiffness matrices and modal data (periods, frequencies, and modal shapes) were obtained. To obtain structural response, the framework was placed under stress by the seismic acceleration record in accordance with the department of communication and transportation for the seism that occurred in 1985 and a 5% damping. Data for analysis are as follows (see Table 3).
The geometry of the structure is described in Table 4. Column five of this table shows
Loads applied on the framework are shown in Figure 4. Magnitude of loads was fixed at 25 ton in order that, when carrying out the seismic analysis, the vibration periods would fall on the constant part of the design spectrum of the NTC, Construction Rules for Mexico City.
4.2 2D numerical model
The framework shown in Figure 3 was modeled with a structural analysis software, with the same geometry and properties of material, in order to obtain modal data (periods, frequencies, and modal shapes), as well as structural response for a I W14X311 section. The numerical model used was carried out to compare data obtained from such with those obtained with the program in MATLAB^{®} in order to calibrate the numerical model.
4.3 Program in MATLAB^{®} to simulate 2D framework
A program was developed in MATLAB^{®}, with which the framework was analyzed, obtaining data with and without damage for the development of the ANN. For calculation of data with damage, the total stiffness matrix
The program must be supplied with the geometry and properties of material, number of floors, damping, degrees of freedom to be condensed, NGLC, displaced degrees of freedom, NGLD, mass per floor, and accelerogram. The stiffness method is used to obtain the total stiffness matrix
5. Application of the neural network to detect structural damage
Let us see three cases of application of the ANN which were created. A threelevel framework was analyzed, in which plastic hinges on the ends of its elements were simulated. Plastic hinges varied in position to obtain various damage conditions. In addition, to simplify the problem, the
5.1 Case 1: test of the ANN
For this example, the periods of every damage case were used to carry out the test of the ANN. From the above framework, a condition without damage was obtained, as well as 15 damage conditions. Table 5 shows the elements damaged for every condition, as well as the location of damage. It should be mentioned that data were obtained with the program developed in MATLAB^{®}. Structural analysis software SAP2000^{®} was used only to verify that data obtained in the program created in MATLAB^{®} were correct. Figure 6 shows the structural response of Case 1 obtained with SAP2000^{®} and the program created in MATLAB^{®}, which were used to verify that results obtained with the program developed using MATLAB^{®} were correct and, therefore, to train the ANN.
In addition, the network used is of the backpropagation type, which uses the error backpropagation algorithm or generalized delta rule (Figure 7). The network is provided with the following data to be trained: error = 0.001 and maximum number of epochs = 10,000. From the analysis of the framework, condensed stiffness matrices
5.1.1 Result
Results obtained where as follows:
It may be seen that the maximum displacements obtained with SAP2000^{®} are almost equal to those obtained with MATLAB^{®}, as well as the periods.
Inputs are the weight and period matrices and the objective. The output is a modified weight matrix and a calculated matrix close to the objective matrix. This diagram describes the error backpropagation algorithm or generalized delta rule explained above.
In tables in Annex I, it can be seen that the ANN works properly with a small error. In order to reduce the error obtained by the network, the ANN may be provided with a smaller error than that proposed, to reduce the difference between the required input and output. Here, the error of the network is 0.00099999 and the number of epochs required is 5976. Table 6 for weights shows learning of the network.
Since the error of the network is lower than the error proposed, training is completed. In addition, the number of epochs required for training was 5976 of the 10,000 that were proposed at the beginning. As was previously mentioned, if the output calculated by the network is closer to the objective matrix given to it, a smaller error must be proposed. Table 6 also shows the average error percentage between the objective and calculated matrices, which is acceptable. It must be considered that the error is a percentage measured between periods to be calculated by the ANN and the ones actually calculated.
5.2 Case 2: damage detection
In this application, condensed stiffness matrices were used to carry out all the training of the ANN, different from Case 1, where the periods were used for test purposes and to verify that the ANN was working properly. Table 7 shows 48 damage conditions and which bar is damaged for every case.
5.3 Training of the network
To train the network, there are 48 condensed stiffness matrices, from which one belongs to the framework without damage matrix zero, and the remaining 47 are matrices with damage. Matrix number 47 was selected as the objective matrix, leaving it fixed for every training case. Matrix number 47 belongs to a framework with damage (Figure 8).
Figure 3 framework selected to train the ANN. Damage may be observed in bars 7 and 3. Framework number 47 was selected since it is known that an articulated column from the base of the framework is more unfavorable than an intermediate one, though numerically it was found that damage is higher if the hinge is in an intermediate column (Figure 9). Case 42 shown may be seen in Figure 8.
The input data to train the network were (i) condensed matrices for every framework; (ii) the weight matrix, created from the correlation of two matrices, the stiffness matrix containing the principal diagonal of each condensed matrix and the period matrix; and (iii) bias matrix, started with a unit matrix. The training began with condensed matrix number zero, the matrix without damage, and matrices were given in increasing order until matrix number 47. For the weight’s matrix, the starting matrix was the one obtained in the correlation; the matrix obtained upon completion of training for each case became the input weight matrix for the following case and so on, until reaching matrix number 47. The same was carried out with the bias matrix. It should be mentioned that stiffness matrices were multiplied by a factor that reduced values to decimal numbers, because the network used only accepts values lower than the unit since they have a sigmoid activation function. In addition, training was supervised for every case. The maximum number of epochs (training cycles) proposed was 30,000, while the error calculated by the network should be lower than 0.001, and the
5.3.1 Results
Table 8 shows the data proposed for training of Case Zero (Damage Zero) and data obtained by the ANN upon completion of training.
It can be observed that the 30,000 epochs proposed were required to train the ANN. In addition, it may be seen that the error calculated by the ANN is higher than that proposed as objective error. This error was left out, since it is not significant. The error calculated by the ANN could be reduced by increasing the number of epochs, but then training would be longer. Therefore, it was decided to leave these results out and, as it was already mentioned, the input matrix for the ANN, in this case the condensed matrix zero (Damage Zero), the objective matrix that was the condensed matrix of Case 47 (case with damage), and the output matrix calculated by the ANN. The ANN searches for a weight matrix to make the input matrix adjust to the objective matrix, which tells us how efficient its learning is. As may be seen in Table 9, the output matrix is almost equal to the objective matrix, with a 16% error. This procedure was made for the 48 cases considered, obtaining 48 matrices almost equal to the objective matrix (Case 47).
The dynamic properties of the framework were used to graph the period and the frequency, as well as to graphically observe the damage to the framework. Figure 10 shows the case with Damage Zero, case with Damage 47, case with intermediate damage (Case 22), and the case with serious damage (Case 4) of 48 cases considered. These graphs show something already known, that is, that frequency is a function of stiffness and that period is a function of frequency. If stiffness changes, dynamic properties of the framework will also change. If stiffness decreases, frequency also decreases; therefore, the period increases. This shows that the structure lost stiffness (damage) and, therefore, became more flexible, thus increasing the vibration time (period). Therefore, the vibration period of Damage Zero framework is lower than the vibration period of the framework with serious damage (Case 42) (Figure 10). It should also be noted that mass was constant in every case. Graphs have been used to define an epsilon value (Table 10) both for the period and the frequency, respectively. With this parameter, through the ANN, the case of damage may be specified, as well as its range.
The damage conditions could also be placed hierarchically, based on their fundamental periods, in order to know where there is more damage, depending on the location of the hinge within the framework. It was observed that, when hinges were placed on the higher corners of the framework, damage was the same, regardless of the hinge being in the column or the girder. The same happened when the hinges were placed on the corners of the lower part of the framework. It should be highlighted that the damage is the same whether hinges are in the higher or lower corners of the framework. In other cases, the same did not happen. An important observed aspect is that cases where there was higher damage are those where hinges were inside the framework, that is, in intermediate elements. Case 47 was expected to be a serious damage case, where elements 7 and 9 of the frameworks are articulated. However, the serious damage case is Case 42, where hinges are in elements 1 and 7 of the frameworks.
5.4 Case 3: damage detection
In this example, the ANN was trained with modal matrices of every damage case. As it has already been mentioned, there are 48 different cases, from which one is the case without damage. For Case Zero, its condensed stiffness matrix
There are also 47 cases of damage, through which the same information can be known.
Here the subscript shows the case of damage, and the superscript shows the number of levels. Thus,
6. Conclusions and discussion
According to Test 1, the ANN works properly calculating the objective matrix in an efficient manner, with an average 5% error. This may be lower if the number of training epochs of the ANN increases.
From Test 2 it can be concluded that, if training cases increase, the number of epochs required for learning of the ANN is higher. This is because input data of the ANN is larger, and therefore, more time is required for training. Also, from Test 2 it is concluded that the ANN has a 16% error when calculating the objective matrix, which corresponds to the increase of training cases. However, if the number of training epochs increases, the error of the ANN will decrease because the ANN will have more learning time.
Cases of serious damage are those where hinges are in intermediate elements, and cases where damage is of the same magnitude are those where hinges are in the higher and lower corners of the framework.
A hinge column on the base is more unfavorable than a hinge column in higher floors, but numerically, damage is higher when there are hinge elements in higher floors.
It may be interesting to see the damage level and period relation. How is period and damage related, the level of damage and period elongation is consistent. How much different damage levels coincide with the same period value and vice versa. But these points of interest may not be addressed in this work, since there was no parameter to measure the damage level on the framework. Damage is only found with changes in modal parameters, as well as with the increase of the vibration period.
Upon completion of the research, we have concluded that the methodology implemented to detect structural damage is rather simple. It was carried out in four steps: (i) extraction of modal parameters and condensed matrix, (ii) establishing failure condition for a serious damage condition, (iii) treatment of modal data to be used in the development of the ANN, and (iv) detection of damage to be carried out with the artificial neural network.
Acknowledgments
This article and its corresponding research were carried out, in part, with the research projects IPN SIP20181253 and IPN SIP20196119. The authors thank the reviewers for carefully reading the paper and for their constructive comments and suggestions which have improved this paper.
Annex I
Constructiong of an artificial neural network for detection of structural damage (Tables I1–I4).
Principal stiffness kg/m  

Damage type  Level  1  2  3 
SD0 

55047511.35  43986421.33  15198916.05 
D1 

54877708.7  36097371.39  9399138.702 
D2 

54771544.01  35317166.31  9121527.737 
D3 

54771544.01  35317166.31  9121527.737 
D4 

54877708.7  36097371.39  9399138.702 
D5 

52859001.57  30346342.81  9268800.799 
D6 

53635426.09  32269306.5  9737254.051 
D7 

51836691.31  43952306.34  13441604.94 
D8 

51573961.66  43939537.94  13450263.27 
D9 

44998259.72  29296754.08  13808325.01 
D10 

46785291.09  31313923.98  14088832.2 
D11 

38598400.77  34400828.39  14998080.93 
D12 

41125388.66  35987149.14  15086548.41 
D13 

54,908,398  41044048.98  15126241.06 
D14 

54923344.34  40895188.06  15112225.05 
D15 

36667939.84  41919304.71  15169346.77 
Periods  

Damage type  Level  1  2  3 
SD0 

0.299830  0.090200  0.050792 
D1 

0.318255  0.104710  0.052938 
D2 

0.318365  0.105415  0.053246 
D3 

0.318365  0.105415  0.053246 
D4 

0.318255  0.104710  0.052938 
D5 

0.303981  0.104247  0.055981 
D6 

0.303965  0.103230  0.054903 
D7 

0.348754  0.096427  0.051103 
D8 

0.348938  0.096669  0.051148 
D9 

0.323188  0.090897  0.060636 
D10 

0.323013  0.090790  0.058691 
D11 

0.319120  0.092516  0.060238 
D12 

0.318989  0.092179  0.058335 
D13 

0.362777  0.092909  0.051041 
D14 

0.362999  0.093013  0.051068 
D15 

0.311588  0.100941  0.055899 
Weights correlation 


1  0.19607059  0.36461043  −0.75429818 
2  0.47564924  −0.19211267  −0.73301055 
3  0.39640392  −0.89248216  0.149395599 
Periods, input matrix, and target matrix  

Damage type  Level  1  2  3 
SD0 

0.299830  0.090200  0.050792 
D1 

0.318255  0.104710  0.052938 
D2 

0.318365  0.105415  0.053246 
D3 

0.318365  0.105415  0.053246 
D4 

0.318255  0.104710  0.052938 
D5 

0.303981  0.104247  0.055981 
D6 

0.303965  0.103230  0.054903 
D7 

0.348754  0.096427  0.051103 
D8 

0.348938  0.096669  0.051148 
D9 

0.323188  0.090897  0.060636 
D10 

0.323013  0.090790  0.058691 
D11 

0.319120  0.092516  0.060238 
D12 

0.318989  0.092179  0.058335 
D13 

0.362777  0.092909  0.051041 
D14 

0.362999  0.093013  0.051068 
D15 

.0311588  0.100941  0.055899 
Periods, output matrix  

Damage type  Level  1  2  3 
SD0 

0.30661  0.09916  0.05424 
D1 

0.32025  0.09805  0.05403 
D2 

0.32029  0.09801  0.05401 
D3 

0.32029  0.09801  0.05401 
D4 

0.32025  0.09805  0.05403 
D5 

0.30896  0.09837  0.05378 
D6 

0.30914  0.09847  0.05381 
D7 

0.34395  0.09714  0.05503 
D8 

0.34409  0.09713  0.05502 
D9 

0.32224  0.09747  0.05483 
D10 

0.32248  0.09763  0.05481 
D11 

0.31927  0.09763  0.05467 
D12 

0.31953  0.09779  0.05466 
D13 

0.35497  0.09666  0.05549 
D14 

0.35515  0.09665  0.05549 
D15 

0.31459  0.09814  0.05409 
Difference, target matrix, and output matrix  

Damage type  Level  1  2  3 
SD0 

−0.006784  −0.008962  −0.003451 
D1 

−0.001994  0.006660  −0.001092 
D2 

−0.001921  0.007404  −0.000760 
D3 

−0.001921  0.007404  −0.000760 
D4 

−0.001994  0.006660  −0.001092 
D5 

−0.004983  0.005873  0.002200 
D6 

−0.005173  0.004756  0.001091 
D7 

0.004801  −0.3000715  −0.003928 
D8 

0.004845  −0.000458  −0.003877 
D9 

0.000947  −0.006573  0.005807 
D10 

0.000530  −0.006841  0.003883 
D11 

−0.000149  −0.005118  0.005573 
D12 

−0.000541  −0.005614  0.003680 
D13 

0.007803  −0.003753  −0.004452 
D14 

0.007851  −0.003637  −0.004426 
D15 

−0.003003  0.002805  0.001814 
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