# Structural Damage Identification Method of Girder Bridges Based on Multilevel Data Fusion Theory

Authors: Chang-Sheng Xiang; Hai-Long Liu; Yu Zhou; Chen-Yu Liu; Li-Xian Wang
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The single index based on modal strain energy and modal curvature is proved to be effective for damage localization, while its recognition accuracy is not satisfied and can not directly reflect damage degree. Therefore, multilevel data fusion methods are proposed here with three new indexes of modal strain energy dissipation rate (DR), change rate of cross-model modal strain energy (CR), and difference of modal curvature ratio (RD). Firstly, first-level, second-level, and third-level data fusion methods are deduced based on Bayes theory, weighted average criterion, and BP neural network, respectively. The first three order modes data of each index are fused in the first level, and their results are further fused between indexes in the second level, which will be applied for damage position judgement; moreover, results of the second level as input are last fused in the third level in order to predict structural damage degree. Secondly, a simply supported beam and a three-span continuous girder models are simulated to verify effectiveness of multilevel data fusion indexes. It can be concluded that the recognition results agree well with man-made damage cases whatever the positions or degrees are. Finally, a test study on a simply supported steel beam with different damage forms is carried out. The results show that the proposed multilevel data fusion methods have good abilities of sensitivity and anti-interference, are fault-tolerant, have robustness, and would provide certain effective experience for actual damage identification.


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## Body

## 1. Introduction

When bridge structures are in service [1], due to influence of uncertain factors such as environment and vehicle load, structural stiffness and material properties will be degraded to a varying degree, which would affect normal use of structure and pose a threat to driving safety. Therefore, it is particularly important to monitor health of structure, to determine damage position and its extent in time, and to repair defect or to strengthen structure as soon as possible.Presently, damage identification methods based on dynamic fingerprint are most commonly used, which include natural frequency [2], curvature mode [3–5], flexibility matrix [6, 7], and modal strain energy [8, 9]. Ji-Ling et al. [10] applied primary frequency change of local dynamic response to identify local damage of truss structure. Chang-Sheng et al. [11] proposed a damage identification method based on modal curvature utility information entropy by utilizing advantage of modal curvature sensitive to damage, to realize damage localization and quantitative analysis of beam structure. Zi-Chuan et al. [12] used the method of modal strain energy change rate to identify damage of L-shape pipeline. However, the methods above all have their own limitations; their recognition accuracy will also vary with influence of factors such as index selection, test error, and environmental noise. Meanwhile, their recognition results are also changeful when different modal data are adopted for identification, especially for damage quantitative analysis, in which the error obtained by only single indicator will be still large; if it is applied to actual project, it would certainly lead to fault choices of subsequent repair measures.Data fusion is a method where multisource information is comprehensively considered, which has good fault tolerance and reliability [13]. Combining data fusion technology with structural damage identification can effectively improve the accuracy of damage diagnosis. Through redundant or complementary information obtained from multiple damage features, data fusion is carried out according to the fusion algorithm, and then the best synergistic result is obtained, which improves the accuracy of damage identification [14, 15]. Data fusion methods commonly include weighted average criterion [16], Bayes fusion principle [17–19], D-S evidence matrix [20, 21], BP neural network [22–24], and support vector machine (SVM) [25]. Grande and Imbimbo [26] proposed a damage identification method based on the combination of modal strain energy index and D-S evidence criterion, which significantly improved identification efficiency, but the recognition ability of this method depends on the sensibility of vibration modes to damage scenarios. Ji-Lin et al. [27] proposed a method to use D-S evidence criterion to fuse damage identification results of multiple finite element models that consider environmental variation, which has high identification accuracy and practical applicability. Jun et al. [28] found that the damage identification method based on Bayes data fusion can more accurately and effectively determine the location of single damage and multiple damage in beam structures. Xue-Yan and Li-Chang [13] used Bayes data fusion theory to fuse multigroup damage vectors measured under various test environment and multiple sensors, which could improve damage identification accuracy. Bin et al. [29] conducted two-stage information fusion method for multiposition damage identification in a truss frame by using D-S evidence matrix and weighted average criterion and proved that it has better noise immunity.However, the existing data fusion methods still have shortcomings. For example, when using the Bayesian fusion method for damage identification, the prior probability of damage to each element of the structure is not easy to obtain [30]. Due to the uncertainty and subjectivity of prior data, D-S evidence criterion usually faces the problem of evidence conflict [31, 32]. The extraction of feature parameters easily affects the learning time and reasoning performance of neural network fusion system [33].In this paper, modal strain energy dissipation rate (DR), change rate of cross-model modal strain energy (CR), and difference of modal curvature ratio (RD) are selected as basic indexes, to construct multilevel fusion model for structural damage localization and quantitative analysis based on Bayes theorem, weighted average criterion, and BP neural network.


## 2. Data Fusion Method

### 2.1. Fusion Rule

#### 2.1.1. Bayes Theorem

Bayes theorem is one of common theories used in multiobjective information fusion decision. Its principle will constantly update prior probability through last observation data for acquiring posterior probability.Equation of Bayes is known as(1)PAm|B=PB|AmPAm∑j=1mPB|AjPAj,where PAm, PAm|B are prior probability of event Am and its posterior probability when event B occurs.When data fusion theory is applied in research area of structural damage identification, it can be express that there aren information sources (or n evaluation indicators) S1, S2, …, Sn, and m mutual independent recognized targets (or m elements) A1, A2, …, Am, so according to equation (1), PAm|B can be expressed as(2)PAm|S1,…,Sn=PS|AmPAm∑j=1mPS|AjPAj=PS1,…,Sn|AmPAm∑j=1mPS1,…,Sn|AjPAj.While prior probability of target needs to be provided in advance, the damage position of structure usually cannot be foreseen. Usually, damage probability at mid-span is largest and will be smaller at support; therefore, we apply displacement influence line for calculating prior probability [34, 35].As a case study, a simply supported beam is established as shown in Figure1, and based on virtual work principle, structure displacement can be expressed as the following equation:(3)Δ=∫M¯MPEIds.Figure 1
Simply supported beam model.The bending moment at any cross sectionx is(4)Mpx=Fpl−x¯lx0≤x<x¯,Fpl−xlx¯x¯≤x<l.According to equations (3) and (4), displacement influence line then can be derived as(5)φu=−16EIl−slFpx3−16EIFpxll−s3+16EIFpxll−s0≤x<s,−16EIFpsll−x3−16EIl−xlFps3+16EIFpsll−ss≤x<l.Whens=l/2, equation (5) can be easily transformed as follows:(6)φu=−112EIFpx3+116EIFpxl20≤x<l2−112EIFpl−x3+116EIFpl2l−xl2≤x<l.In equation (6), mid-span displacement influence line consists of two continuous and derivable piecewise functions, and it can also be drawn as shown in Figure 2.Figure 2
Displacement influence line of mid-span.From Figure2, the damage prior probability of any interval [a, b] is calculated by its area as the following equation:(7)PAa,b=∫abφudx∫0lφudx.If interval [a, b] is at mid-span, the area of mid-span element should be integrated in two parts, left side and right side of mid-span.


#### 2.1.2. Weighted Average Criterion

The principle of weighted mean method [16] is to assign each index with different weight factor, calculating their weighted mean value, to reach the target that sum of square of Euclidean distance between weighted mean value and each index recognition value is smallest.Mean value of each index is defined asS¯j,(8)S¯j=∑i=1nSijn,where Sij is ith indicator value of jth element, and S¯j is the mean of each index.Weight factorwi of each index is(9)wi=1/ci∑i=1n1/cici≠0,where ci=S¯j−Sij=∑j=1mS¯j−Sij2 is Euclidean distance between each index recognition value and their mean value.So, the weighted mean value can be obtained.(10)DF=∑i=1nwiSi.


#### 2.1.3. BP Neural Network

The backpropagation (BP) neural network first obtains an uncertain reasoning method through a specific learning algorithm and then uses this mechanism to carry out information fusion and relearning through continuous update of network weights. The BP neural network is composed of input layer, output layer, and one or more hidden layers as shown in Figure3, which includes m, n and p neurons in input, output, and hidden layer respectively; here, w is weight value, and h is hidden layer output.Figure 3
BP neural network.When BP neural network is applied to structural damage identification [36, 37], the measured structural features, structural features in the known normal state, and labeled damage feature database are input into neural network for matching. Then, measured features are classified by the self-learning of neural network. Finally, the damage classification results are output.


### 2.2. Index Selection for Fusion

#### 2.2.1. Modal Strain Energy Dissipation Rate (DR)

The index based on modal strain energy can well recognize local damage of structure, and so does the modal strain energy dissipation rate (DR) [38]. Modal strain energy before and after damage is known as(11)MSEiju=ΦiuΤKjΦiu,(12)MSEijd=ΦidΤKjΦid,where Kj is stiffness matrix of element j; Φiu and Φid represent ith modal displacement vector of undamaged and damaged elements, respectively.Considering the damage process of the structure as the dissipation process of modal strain energy of structural element, the DR ofjth element of structure can be expressed as(13)DRij=MSEijd−MSEijuMSEijd−MSEiju+MSEiju,where MSEijd−MSEiju is the change of modal strain energy of jth element of damaged structure.


#### 2.2.2. Change Rate of Cross-Model Modal Strain Energy (CR)

The cross-model modal strain energy (CMMSE) proposed in paper [9] can effectively improve noise immunity of index. Therefore, its change rate (CR) can also be one of damage fusion indexes.The CMMSE before and after damaged can be calculated by equations (11) and (12), respectively, shown as(14)CMMSEiju=ΦiuΤKjΦiu,CMMSEijd=ΦiuΤKjΦid.So, index CR can be expressed as(15)CRij=CMMSEijd−CMMSEijuCMMSEiju.


#### 2.2.3. Difference of Modal Curvature Ratio (RD)

Based on method of central difference, modal curvature of structure can be approximately calculated as follows:(16)Φik'′=Φik+1−2Φik+Φik−1h2,where Φik'′, Φik represent modal curvature and displacement of kth node of ith mode, respectively. h is distance between adjacent nodes.The ratio ofkth node modal curvature to all nodes is defined as R, and its values before and after damaged can be acquired by the following equations:(17)Riku=Φik′′u∑k=1nΦik′′u,Rikd=Φik′′d∑k=1nΦik'′d.Here, we introduce a new index namedRD, which is the difference of R damaged and undamaged as equation (18) shown as follows:(18)RDik=Rikd−Riku.


### 2.3. Procedure of Multilevel Fusion

#### 2.3.1. First-Level Fusion Method Based on Bayes Theorem

The first-level fusion flow chart of simply supported beam is shown in Figure4; it is a kind of fusion within each index itself, respectively, by index values of the first-three order modes based on Bayes fusion theorem. The detailed steps are as follows:(1)
Calculating first-three order modal information of structure by finite element software, and then according to three indicesDR, CR, and RD mentioned above, their values can be achieved from equations (13), (15), and (18) by MATLAB software.(2)
Calculating probability distribution vectorsz1,z2,z3 of first-three order modes of each index, respectively.(3)
Fusing first-three order modal probability assignmentsz1,z2,z3 of each index, and obtaining first-level fusion indexes DR1, CR1, and RD1 for damage location judgement. The greater the fusion probability is, the greater the element damage possibility would be.Figure 4
First-level fusion method based on Bayes theorem.


#### 2.3.2. Second-Level Fusion Method Based on Weighted Average Criterion

Second-level fusion process is shown in Figure5, and the concrete steps are as follows:(1)
According to the first-level fusion recognition method, calculating values of DR1, CR1, and RD1 for every element, respectively, and substituting them into equation (8), so that we can obtain mean S¯.(2)
SubstitutingS¯ into equation (9), we can obtain a set of weight factors w1,w2 and w3 for indexes DR1, CR1, and RD1 respectively.(3)
Substituting DR1, CR1, RD1, and weight factors w1,w2,w3 into equation (10), second-level fusion index DF will be obtained, which can be applied for damage location identification, and also as input for quantitative analysis in the third-level fusion.Figure 5
Second-level fusion method based on weighted average criterion.


#### 2.3.3. Third-Level Fusion Method Based on BP Neural Network

Third-level fusion process is shown in Figure6, and the concrete steps are as follows:(1)
Establishing various damage cases, and according to equation (10), the second-level fusion results DF of all damage cases are calculated.(2)
The second-level fusion results DF of all cases are divided into two parts: training set and prediction set.(3)
The training set DF is input into the neural network for training, and the network parameters are continuously adjusted through the learning of training set until network output is close to expected output, and the training is stopped.(4)
Input prediction set DF into the trained neural network for damage degree recognition, and finally output the residual stiffness DD3 of damaged unit.Figure 6
Third-level fusion method based on BP neural network.


## 2.1. Fusion Rule

### 2.1.1. Bayes Theorem

Bayes theorem is one of common theories used in multiobjective information fusion decision. Its principle will constantly update prior probability through last observation data for acquiring posterior probability.Equation of Bayes is known as(1)PAm|B=PB|AmPAm∑j=1mPB|AjPAj,where PAm, PAm|B are prior probability of event Am and its posterior probability when event B occurs.When data fusion theory is applied in research area of structural damage identification, it can be express that there aren information sources (or n evaluation indicators) S1, S2, …, Sn, and m mutual independent recognized targets (or m elements) A1, A2, …, Am, so according to equation (1), PAm|B can be expressed as(2)PAm|S1,…,Sn=PS|AmPAm∑j=1mPS|AjPAj=PS1,…,Sn|AmPAm∑j=1mPS1,…,Sn|AjPAj.While prior probability of target needs to be provided in advance, the damage position of structure usually cannot be foreseen. Usually, damage probability at mid-span is largest and will be smaller at support; therefore, we apply displacement influence line for calculating prior probability [34, 35].As a case study, a simply supported beam is established as shown in Figure1, and based on virtual work principle, structure displacement can be expressed as the following equation:(3)Δ=∫M¯MPEIds.Figure 1
Simply supported beam model.The bending moment at any cross sectionx is(4)Mpx=Fpl−x¯lx0≤x<x¯,Fpl−xlx¯x¯≤x<l.According to equations (3) and (4), displacement influence line then can be derived as(5)φu=−16EIl−slFpx3−16EIFpxll−s3+16EIFpxll−s0≤x<s,−16EIFpsll−x3−16EIl−xlFps3+16EIFpsll−ss≤x<l.Whens=l/2, equation (5) can be easily transformed as follows:(6)φu=−112EIFpx3+116EIFpxl20≤x<l2−112EIFpl−x3+116EIFpl2l−xl2≤x<l.In equation (6), mid-span displacement influence line consists of two continuous and derivable piecewise functions, and it can also be drawn as shown in Figure 2.Figure 2
Displacement influence line of mid-span.From Figure2, the damage prior probability of any interval [a, b] is calculated by its area as the following equation:(7)PAa,b=∫abφudx∫0lφudx.If interval [a, b] is at mid-span, the area of mid-span element should be integrated in two parts, left side and right side of mid-span.


### 2.1.2. Weighted Average Criterion

The principle of weighted mean method [16] is to assign each index with different weight factor, calculating their weighted mean value, to reach the target that sum of square of Euclidean distance between weighted mean value and each index recognition value is smallest.Mean value of each index is defined asS¯j,(8)S¯j=∑i=1nSijn,where Sij is ith indicator value of jth element, and S¯j is the mean of each index.Weight factorwi of each index is(9)wi=1/ci∑i=1n1/cici≠0,where ci=S¯j−Sij=∑j=1mS¯j−Sij2 is Euclidean distance between each index recognition value and their mean value.So, the weighted mean value can be obtained.(10)DF=∑i=1nwiSi.


### 2.1.3. BP Neural Network

The backpropagation (BP) neural network first obtains an uncertain reasoning method through a specific learning algorithm and then uses this mechanism to carry out information fusion and relearning through continuous update of network weights. The BP neural network is composed of input layer, output layer, and one or more hidden layers as shown in Figure3, which includes m, n and p neurons in input, output, and hidden layer respectively; here, w is weight value, and h is hidden layer output.Figure 3
BP neural network.When BP neural network is applied to structural damage identification [36, 37], the measured structural features, structural features in the known normal state, and labeled damage feature database are input into neural network for matching. Then, measured features are classified by the self-learning of neural network. Finally, the damage classification results are output.


## 2.1.1. Bayes Theorem

Bayes theorem is one of common theories used in multiobjective information fusion decision. Its principle will constantly update prior probability through last observation data for acquiring posterior probability.Equation of Bayes is known as(1)PAm|B=PB|AmPAm∑j=1mPB|AjPAj,where PAm, PAm|B are prior probability of event Am and its posterior probability when event B occurs.When data fusion theory is applied in research area of structural damage identification, it can be express that there aren information sources (or n evaluation indicators) S1, S2, …, Sn, and m mutual independent recognized targets (or m elements) A1, A2, …, Am, so according to equation (1), PAm|B can be expressed as(2)PAm|S1,…,Sn=PS|AmPAm∑j=1mPS|AjPAj=PS1,…,Sn|AmPAm∑j=1mPS1,…,Sn|AjPAj.While prior probability of target needs to be provided in advance, the damage position of structure usually cannot be foreseen. Usually, damage probability at mid-span is largest and will be smaller at support; therefore, we apply displacement influence line for calculating prior probability [34, 35].As a case study, a simply supported beam is established as shown in Figure1, and based on virtual work principle, structure displacement can be expressed as the following equation:(3)Δ=∫M¯MPEIds.Figure 1
Simply supported beam model.The bending moment at any cross sectionx is(4)Mpx=Fpl−x¯lx0≤x<x¯,Fpl−xlx¯x¯≤x<l.According to equations (3) and (4), displacement influence line then can be derived as(5)φu=−16EIl−slFpx3−16EIFpxll−s3+16EIFpxll−s0≤x<s,−16EIFpsll−x3−16EIl−xlFps3+16EIFpsll−ss≤x<l.Whens=l/2, equation (5) can be easily transformed as follows:(6)φu=−112EIFpx3+116EIFpxl20≤x<l2−112EIFpl−x3+116EIFpl2l−xl2≤x<l.In equation (6), mid-span displacement influence line consists of two continuous and derivable piecewise functions, and it can also be drawn as shown in Figure 2.Figure 2
Displacement influence line of mid-span.From Figure2, the damage prior probability of any interval [a, b] is calculated by its area as the following equation:(7)PAa,b=∫abφudx∫0lφudx.If interval [a, b] is at mid-span, the area of mid-span element should be integrated in two parts, left side and right side of mid-span.


## 2.1.2. Weighted Average Criterion

The principle of weighted mean method [16] is to assign each index with different weight factor, calculating their weighted mean value, to reach the target that sum of square of Euclidean distance between weighted mean value and each index recognition value is smallest.Mean value of each index is defined asS¯j,(8)S¯j=∑i=1nSijn,where Sij is ith indicator value of jth element, and S¯j is the mean of each index.Weight factorwi of each index is(9)wi=1/ci∑i=1n1/cici≠0,where ci=S¯j−Sij=∑j=1mS¯j−Sij2 is Euclidean distance between each index recognition value and their mean value.So, the weighted mean value can be obtained.(10)DF=∑i=1nwiSi.


## 2.1.3. BP Neural Network

The backpropagation (BP) neural network first obtains an uncertain reasoning method through a specific learning algorithm and then uses this mechanism to carry out information fusion and relearning through continuous update of network weights. The BP neural network is composed of input layer, output layer, and one or more hidden layers as shown in Figure3, which includes m, n and p neurons in input, output, and hidden layer respectively; here, w is weight value, and h is hidden layer output.Figure 3
BP neural network.When BP neural network is applied to structural damage identification [36, 37], the measured structural features, structural features in the known normal state, and labeled damage feature database are input into neural network for matching. Then, measured features are classified by the self-learning of neural network. Finally, the damage classification results are output.


## 2.2. Index Selection for Fusion

### 2.2.1. Modal Strain Energy Dissipation Rate (DR)

The index based on modal strain energy can well recognize local damage of structure, and so does the modal strain energy dissipation rate (DR) [38]. Modal strain energy before and after damage is known as(11)MSEiju=ΦiuΤKjΦiu,(12)MSEijd=ΦidΤKjΦid,where Kj is stiffness matrix of element j; Φiu and Φid represent ith modal displacement vector of undamaged and damaged elements, respectively.Considering the damage process of the structure as the dissipation process of modal strain energy of structural element, the DR ofjth element of structure can be expressed as(13)DRij=MSEijd−MSEijuMSEijd−MSEiju+MSEiju,where MSEijd−MSEiju is the change of modal strain energy of jth element of damaged structure.


### 2.2.2. Change Rate of Cross-Model Modal Strain Energy (CR)

The cross-model modal strain energy (CMMSE) proposed in paper [9] can effectively improve noise immunity of index. Therefore, its change rate (CR) can also be one of damage fusion indexes.The CMMSE before and after damaged can be calculated by equations (11) and (12), respectively, shown as(14)CMMSEiju=ΦiuΤKjΦiu,CMMSEijd=ΦiuΤKjΦid.So, index CR can be expressed as(15)CRij=CMMSEijd−CMMSEijuCMMSEiju.


### 2.2.3. Difference of Modal Curvature Ratio (RD)

Based on method of central difference, modal curvature of structure can be approximately calculated as follows:(16)Φik'′=Φik+1−2Φik+Φik−1h2,where Φik'′, Φik represent modal curvature and displacement of kth node of ith mode, respectively. h is distance between adjacent nodes.The ratio ofkth node modal curvature to all nodes is defined as R, and its values before and after damaged can be acquired by the following equations:(17)Riku=Φik′′u∑k=1nΦik′′u,Rikd=Φik′′d∑k=1nΦik'′d.Here, we introduce a new index namedRD, which is the difference of R damaged and undamaged as equation (18) shown as follows:(18)RDik=Rikd−Riku.


## 2.2.1. Modal Strain Energy Dissipation Rate (DR)

The index based on modal strain energy can well recognize local damage of structure, and so does the modal strain energy dissipation rate (DR) [38]. Modal strain energy before and after damage is known as(11)MSEiju=ΦiuΤKjΦiu,(12)MSEijd=ΦidΤKjΦid,where Kj is stiffness matrix of element j; Φiu and Φid represent ith modal displacement vector of undamaged and damaged elements, respectively.Considering the damage process of the structure as the dissipation process of modal strain energy of structural element, the DR ofjth element of structure can be expressed as(13)DRij=MSEijd−MSEijuMSEijd−MSEiju+MSEiju,where MSEijd−MSEiju is the change of modal strain energy of jth element of damaged structure.


## 2.2.2. Change Rate of Cross-Model Modal Strain Energy (CR)

The cross-model modal strain energy (CMMSE) proposed in paper [9] can effectively improve noise immunity of index. Therefore, its change rate (CR) can also be one of damage fusion indexes.The CMMSE before and after damaged can be calculated by equations (11) and (12), respectively, shown as(14)CMMSEiju=ΦiuΤKjΦiu,CMMSEijd=ΦiuΤKjΦid.So, index CR can be expressed as(15)CRij=CMMSEijd−CMMSEijuCMMSEiju.


## 2.2.3. Difference of Modal Curvature Ratio (RD)

Based on method of central difference, modal curvature of structure can be approximately calculated as follows:(16)Φik'′=Φik+1−2Φik+Φik−1h2,where Φik'′, Φik represent modal curvature and displacement of kth node of ith mode, respectively. h is distance between adjacent nodes.The ratio ofkth node modal curvature to all nodes is defined as R, and its values before and after damaged can be acquired by the following equations:(17)Riku=Φik′′u∑k=1nΦik′′u,Rikd=Φik′′d∑k=1nΦik'′d.Here, we introduce a new index namedRD, which is the difference of R damaged and undamaged as equation (18) shown as follows:(18)RDik=Rikd−Riku.


## 2.3. Procedure of Multilevel Fusion

### 2.3.1. First-Level Fusion Method Based on Bayes Theorem

The first-level fusion flow chart of simply supported beam is shown in Figure4; it is a kind of fusion within each index itself, respectively, by index values of the first-three order modes based on Bayes fusion theorem. The detailed steps are as follows:(1)
Calculating first-three order modal information of structure by finite element software, and then according to three indicesDR, CR, and RD mentioned above, their values can be achieved from equations (13), (15), and (18) by MATLAB software.(2)
Calculating probability distribution vectorsz1,z2,z3 of first-three order modes of each index, respectively.(3)
Fusing first-three order modal probability assignmentsz1,z2,z3 of each index, and obtaining first-level fusion indexes DR1, CR1, and RD1 for damage location judgement. The greater the fusion probability is, the greater the element damage possibility would be.Figure 4
First-level fusion method based on Bayes theorem.


### 2.3.2. Second-Level Fusion Method Based on Weighted Average Criterion

Second-level fusion process is shown in Figure5, and the concrete steps are as follows:(1)
According to the first-level fusion recognition method, calculating values of DR1, CR1, and RD1 for every element, respectively, and substituting them into equation (8), so that we can obtain mean S¯.(2)
SubstitutingS¯ into equation (9), we can obtain a set of weight factors w1,w2 and w3 for indexes DR1, CR1, and RD1 respectively.(3)
Substituting DR1, CR1, RD1, and weight factors w1,w2,w3 into equation (10), second-level fusion index DF will be obtained, which can be applied for damage location identification, and also as input for quantitative analysis in the third-level fusion.Figure 5
Second-level fusion method based on weighted average criterion.


### 2.3.3. Third-Level Fusion Method Based on BP Neural Network

Third-level fusion process is shown in Figure6, and the concrete steps are as follows:(1)
Establishing various damage cases, and according to equation (10), the second-level fusion results DF of all damage cases are calculated.(2)
The second-level fusion results DF of all cases are divided into two parts: training set and prediction set.(3)
The training set DF is input into the neural network for training, and the network parameters are continuously adjusted through the learning of training set until network output is close to expected output, and the training is stopped.(4)
Input prediction set DF into the trained neural network for damage degree recognition, and finally output the residual stiffness DD3 of damaged unit.Figure 6
Third-level fusion method based on BP neural network.


## 2.3.1. First-Level Fusion Method Based on Bayes Theorem

The first-level fusion flow chart of simply supported beam is shown in Figure4; it is a kind of fusion within each index itself, respectively, by index values of the first-three order modes based on Bayes fusion theorem. The detailed steps are as follows:(1)
Calculating first-three order modal information of structure by finite element software, and then according to three indicesDR, CR, and RD mentioned above, their values can be achieved from equations (13), (15), and (18) by MATLAB software.(2)
Calculating probability distribution vectorsz1,z2,z3 of first-three order modes of each index, respectively.(3)
Fusing first-three order modal probability assignmentsz1,z2,z3 of each index, and obtaining first-level fusion indexes DR1, CR1, and RD1 for damage location judgement. The greater the fusion probability is, the greater the element damage possibility would be.Figure 4
First-level fusion method based on Bayes theorem.


## 2.3.2. Second-Level Fusion Method Based on Weighted Average Criterion

Second-level fusion process is shown in Figure5, and the concrete steps are as follows:(1)
According to the first-level fusion recognition method, calculating values of DR1, CR1, and RD1 for every element, respectively, and substituting them into equation (8), so that we can obtain mean S¯.(2)
SubstitutingS¯ into equation (9), we can obtain a set of weight factors w1,w2 and w3 for indexes DR1, CR1, and RD1 respectively.(3)
Substituting DR1, CR1, RD1, and weight factors w1,w2,w3 into equation (10), second-level fusion index DF will be obtained, which can be applied for damage location identification, and also as input for quantitative analysis in the third-level fusion.Figure 5
Second-level fusion method based on weighted average criterion.


## 2.3.3. Third-Level Fusion Method Based on BP Neural Network

Third-level fusion process is shown in Figure6, and the concrete steps are as follows:(1)
Establishing various damage cases, and according to equation (10), the second-level fusion results DF of all damage cases are calculated.(2)
The second-level fusion results DF of all cases are divided into two parts: training set and prediction set.(3)
The training set DF is input into the neural network for training, and the network parameters are continuously adjusted through the learning of training set until network output is close to expected output, and the training is stopped.(4)
Input prediction set DF into the trained neural network for damage degree recognition, and finally output the residual stiffness DD3 of damaged unit.Figure 6
Third-level fusion method based on BP neural network.


## 3. Damage Recognition Results of Simply Supported Beam

### 3.1. Simple Supported Beam Model

A concrete simply supported beam model is established by using ANSYS 17.0 as shown in Figure7. Beam length is 6.00 m, cross section dimension (width × height) is 0.20 m × 0.12 m, elastic modulus E = 30 GPa, Poisson’s ratio ε = 0.17, and density ρ = 2500 kg/m3. Beam189 element model is selected, and beam structure is divided into 25 elements, 26 nodes.Figure 7
Model of simply supported beam.The structural stiffness is declined by reducing element elastic modulusE; to simulate structure damage, in total, 20 damage cases are considered here as shown in Table 1.Table 1
Damage cases of simply supported beam.
Damage caseDamage element no.Damage degree (%)135231036104620562569157925812109122010123011181512182013221014221515315910163159151712151820181220182519615121518152063012201825


#### 3.1.1. Modal Analysis

In order to reduce error for damage identification, the first-three modal shapes are used for analysis, so the first-three order frequencies of undamaged and damaged cases are calculated firstly by ANSYS 17.0 software, shown in Table2. When beam is damaged, the natural frequency of structure will also change, and the modal shape cloud diagram of nondamage case is shown in Figure 8.Table 2
Modal frequencies of damage cases.
CaseFrequency1st order mode2nd order mode3rd order modeUndamaged case5.242521.04847.652Case 15.241421.03347.586Case 25.240221.01547.514Case 35.233020.95947.493⋮⋮⋮⋮Figure 8
Modal shape cloud diagram of nondamage case. (a) 1st order mode. (b) 2nd order mode. (c) 3rd order mode.
(a)(b)(c)


#### 3.1.2. Prior Probability

The displacement influence line is generated when unit load moves on beam, and the prior probabilityPAm of every element can be calculated by equation (7), as shown in Table 3.Table 3
Prior probabilities of elements.
Element no.Prior probability10.00383590420.01145856730.0189337640.026163250.03304857660.03949158470.0453939280.0506572890.05518336100.058873856110.061630507120.06335488130.063949312……Note. Prior probabilities of the other 12 elements are the same due to symmetry.


### 3.2. Damage Localization Analysis of Single Index

First, we use three original indicators (DR, CR, and RD) to analyze structural damage separately, and the results of damage cases are shown in Figure9.Figure 9
The identification results of single index for single (a), two (b), and three (c) damage cases. (a) Case 1. (b) Case 15. (c) Case 20.
(a)(b)(c)


#### 3.2.1. Single Damage Identification

From Figure9(a), it is obvious that maximum peaks appear at element No.3 in 1st order modal curves of DR, CR, and RD, which correspond to the damage assumed in case 1, so they all can search damage location. However, in 2nd and 3rd order modal curves, not only element No. 3, there would appear other peaks at health elements, which cannot be ignored. For example, in the 3rd order modal curve of RD (blue line), the values of undamaged elements No. 8 and No. 9 are very large, so it may lead misjudgment.


#### 3.2.2. Two Damage Cases Identification

In Figure9(b), for case 15, from 1st and 3rd order modal curves of DR and CR, peaks at damage element No. 3 and No. 9 can be searched, while their 2nd order modal curve would be not significant because other peaks appeared at elements No. 12 and No. 14. In RD modal curves, it would only be effective for 1st order mode to lock damage location.


#### 3.2.3. Three Damage Cases Identification

In Figure9(c), both methods of DR and CR can identify damage position by searching peaks at damaged elements No. 6, No. 12, and No. 18. But in the curves of RD, values of undamaged elements No. 5 and No. 7 will exceed those of damaged elements No. 12 and No. 18, so it is not easy to distinguish which element is damaged or not.From the above analysis, when multiple damage occurs, the ability of single index and its accuracy are both significantly declined for damage recognition, and its probability of misjudgment would be increased.


### 3.3. Damage Localization Analysis of First-Level Fusion Based on Bayes Theory

Based on Bayes fusion theory, by fusing damage probabilities of the first three order modes of DR, CR, and RD, respectively, three first-level fusion indexes named DR1, CR1, and RD1 will be obtained, which can be applied for damage location identification according to the flow chart in Figure 4.


#### 3.3.1. Two Damage Cases Identification

In Figure10, the peaks position of two damage cases for cases 15, 16, 17, and 18 can be easily found, which indicates that DR1, CR1, and RD1 can be qualified for damage localization. But from vertical coordinate, the peak values of damage probabilities are irregular and cannot be in accordance with damage set in cases; therefore, the damage degree will not be directly reflected in fusion results.Figure 10
Fusion recognition results of two damage cases for DR1 (a), CR1 (b), and RD1 (c).
(a)(b)(c)


#### 3.3.2. Multiple Damage Identification

According to the results of case 20 shown in Figure11, indexes DR1, CR1, and RD1 can locate damage position by peak searching, and the greater the damage degree is, the higher and more obvious the peaks are; for example, damage degree of element No. 6 is biggest (30%), so its peak is highest.Figure 11
Fusion recognition results of multiple damage (case 20).In summary, compared with single index, after Bayes first-level fusion, damage localization abilities of DR1, CR1, and RD1 are significantly improved, which can reduce the impact of health elements and increase the accuracy of damage identification. Comparing their peak values with each other, the order of their ability is DR1≥CR1≥RD1, so DR1 and CR1 are preferred for structural damage localization in actual engineering application.


### 3.4. Damage Localization Analysis of Second-Level Fusion Based on Weighted Average Criterion

According to the flow chart in Figure5, the second-level fusion index will be obtained by applying weighted average criterion. The recognition results are as follows.In Figure12, some single damage cases are selected for analysis, and damage elements can be clearly found by searching maximum peaks from the curve. It should be noted that health element values are very small, so we consider that second-level fusion index can weaken “proximity effect” of RD1 in first-level fusion. For multiple damage cases (Figure 13), second-level fusion index also has the same identification ability.Figure 12
Recognition results of second-level fusion for single damage.Figure 13
Recognition results of second-level fusion for multi-damage.


### 3.5. Damage Quantitative Analysis of Third-Level Fusion Based on BP Neural Network

From the above, we know that whatever the single index is, first-level or second-level fusion, they all cannot easily determine the structural damage degree. Therefore, a third-level fusion model based on BP neural network is established here for further quantitative analysis.A three-layer BP neural network is adopted for training here, which takes second-level fusion results as input and takes structural residual stiffness as output (DD3). In the training process, maximum iterative steps are 100, and precision is 10−5. Lastly, the output data can be easily processed to list absolute error (AE), relative percentage error (RPE), mean absolute error (MAE), and mean relative percent error (MRPE), which can be used as output evaluation indices. The detail calculation is as follows:(19)AE=Vkt−V^ktRPE=Vkt−V^ktVkt×100%MAE=1M∑i=1MVkt−V^ktMRPE=1M∑i=1MVkt−V^ktVkt×100%,where M is the number of test samples; Vkt is real structural residual stiffness; V^kt is output residual stiffness of BP neural network.


#### 3.5.1. Taking Cases 4, 8, and 14 as Prediction Set

According to equation (10), the secondary fusion results DF of all damage conditions are calculated, and the DF values of cases 4, 8, and 14 are used as prediction set, and the DF values of other cases are used as the training set. The training set is input into the BP neural network for training. After the accuracy meets the requirements, the prediction set is input into BP neural network for damage degree prediction, and the residual structural stiffness DD3 is output. At the same time, according to equation (2), DR1, CR1, and RD1 of all cases are calculated, respectively, as control group, and they are also divided into training set and prediction set and input into BP neural network, and output DR2, CR2, and RD2. The prediction results of damage degree of prediction set are shown in Table 4.Table 4
Damage quantitative analysis comparation for DD3, DR2, CR2, and RD2.
IndexCaseC4C8C14Ideal outputActual outputAERPE (%)Ideal outputActual outputAERPE (%)Ideal outputActual outputAERPE (%)DD310.800.82610.02613.26750.90.92440.02442.70690.850.84080.00921.080220.77840.02162.69820.87730.02272.51980.87590.02593.045530.80880.00881.09380.88540.01461.61960.86090.01091.2850DR210.81930.01932.41590.83390.06617.34160.90120.05126.028120.78320.01682.09580.84720.05285.86260.89910.04915.779030.77120.02883.60030.82180.07828.69150.92480.07488.7986CR210.75460.04545.67680.93030.03033.36840.89550.04555.355320.76730.03274.09310.84120.05886.52980.87100.02102.465030.75350.04655.81390.87760.02242.49430.86370.01371.6137DR210.83180.03183.97550.93310.03313.68250.90990.05997.049720.81420.01421.77590.86710.03293.65960.90640.05646.630430.83100.03103.87930.85150.04855.39300.90360.05366.3116For further evaluating recognition effect of DD3, a detailed comparation is analyzed for RPE, MAE. and MRPE of DD3, DR2, CR2, and RD2. In column of RPE, the maximum value of DD3 is 3.27%, which is 4.93% lower than those of others by 8.80%. For MAE, DD3 result for case 4 can be simply calculated, and its value is 0.0188, less than that of DR2, CR2, and RD2 (0.0216, 0.0415, and 0.0257); MRPEs of cases 4, 8, and 14 for DD3 are 2.35%, 2.28%, and 1.80%, respectively, while those of DR2, CR2, and RD2 are 2.71%, 7.30%, and 6.87%; 5.19%, 4.13%, and 2.99%; and 3.21%, 4.25%, and 6.66%, respectively. It is obvious that mean relative percent error of index DD3 is significantly lower than that of DR2, CR2, and RD2.


#### 3.5.2. Taking Single Case from 1 to 14 as Prediction Set, Respectively

Predicted results of 14 groups will be output following the above steps; specifically, their AE results are drawn as a histogram into Figure14 for comparison. Maximum AE value of DD3 (black) is 0.014, while the other DR2 (red), CR2 (blue), and RD2 (green) are all beyond 0.04, and also their MAE value is still larger than that of DD3.Figure 14
Comparison of absolute error of damage degree.Overall, the damage degree actual output of DD3 is closer to ideal output, which means that the third-level fusion method is preferred for structural damage degree identification.


### 3.6. Noise Immunity Analysis

#### 3.6.1. Noise Simulation Method

In actual working condition, due to external environment, test equipment error, and manual operation irregularities, the noise will be always existing; it would affect measured data so that there is often a certain degree difference between real values. Therefore, in order to better verify effectiveness of proposed methods, certain level noise is added into modal information when simulation, for analyzing their antinoise performance.Usually, the way to add noise [39] is applied through the following equation:(20)Φijn=Φij+Φmax,i×randn×δ,where Φij and Φijn represent vibration shape before and after noise addition, respectively; Φmax,i is maximum value of ith order vibration shape; randn represents Gaussian white noise, where the mean value and standard deviation are 0 and 1, respectively; δ is noise level.


#### 3.6.2. Noise Immunity Analysis of the First-Level Fusion

For noise immunity comparative analysis between single index and first-level fusion index, 5% and 10% level noise are added to damage cases 15 and 20, respectively. Figures15 and 16 show antinoise analysis results of two and three damage cases, respectively. In (a) and (b), with increase of damage elements or noise level, damage probability and amount of health elements are also increased, which would lead to ambiguity for actual damage judgement. But after first-level data fusion based on Bayes theory, in (c), damage probabilities of health elements are significantly reduced, which makes peaks at damage position more prominent and can be easily and accurately found. So, first-level fusion index can improve recognition accuracy and has strong robustness than single index.Figure 15
Antinoise analysis results of case 15. (a) 5% noise level of single index. (b) 10% noise level of single index. (c) 5%, 10% noise level of first-level fusion index.
(a)(b)(c)Figure 16
Antinoise analysis results of case 20. (a) 5% noise level of single index. (b) 10% noise level of single index. (c) 5%, 10% noise level of first-level fusion index.
(a)(b)(c)


#### 3.6.3. Noise Immunity Analysis of the Second-Level Fusion

Taking damage cases 1, 15, and 20 as examples with 5% and 10% noise, respectively, second-level fusion results will be calculated and drawn into histogram for comparation in Figure17, in which peaks would be easily searched at damaged elements whatever the single, double, or three damage cases are, which means that the second-level fusion index has strong ability to eliminate noise interference.Figure 17
Antinoise analysis results of the second-level fusion index.


## 3.1. Simple Supported Beam Model

A concrete simply supported beam model is established by using ANSYS 17.0 as shown in Figure7. Beam length is 6.00 m, cross section dimension (width × height) is 0.20 m × 0.12 m, elastic modulus E = 30 GPa, Poisson’s ratio ε = 0.17, and density ρ = 2500 kg/m3. Beam189 element model is selected, and beam structure is divided into 25 elements, 26 nodes.Figure 7
Model of simply supported beam.The structural stiffness is declined by reducing element elastic modulusE; to simulate structure damage, in total, 20 damage cases are considered here as shown in Table 1.Table 1
Damage cases of simply supported beam.
Damage caseDamage element no.Damage degree (%)135231036104620562569157925812109122010123011181512182013221014221515315910163159151712151820181220182519615121518152063012201825


### 3.1.1. Modal Analysis

In order to reduce error for damage identification, the first-three modal shapes are used for analysis, so the first-three order frequencies of undamaged and damaged cases are calculated firstly by ANSYS 17.0 software, shown in Table2. When beam is damaged, the natural frequency of structure will also change, and the modal shape cloud diagram of nondamage case is shown in Figure 8.Table 2
Modal frequencies of damage cases.
CaseFrequency1st order mode2nd order mode3rd order modeUndamaged case5.242521.04847.652Case 15.241421.03347.586Case 25.240221.01547.514Case 35.233020.95947.493⋮⋮⋮⋮Figure 8
Modal shape cloud diagram of nondamage case. (a) 1st order mode. (b) 2nd order mode. (c) 3rd order mode.
(a)(b)(c)


### 3.1.2. Prior Probability

The displacement influence line is generated when unit load moves on beam, and the prior probabilityPAm of every element can be calculated by equation (7), as shown in Table 3.Table 3
Prior probabilities of elements.
Element no.Prior probability10.00383590420.01145856730.0189337640.026163250.03304857660.03949158470.0453939280.0506572890.05518336100.058873856110.061630507120.06335488130.063949312……Note. Prior probabilities of the other 12 elements are the same due to symmetry.


## 3.1.1. Modal Analysis

In order to reduce error for damage identification, the first-three modal shapes are used for analysis, so the first-three order frequencies of undamaged and damaged cases are calculated firstly by ANSYS 17.0 software, shown in Table2. When beam is damaged, the natural frequency of structure will also change, and the modal shape cloud diagram of nondamage case is shown in Figure 8.Table 2
Modal frequencies of damage cases.
CaseFrequency1st order mode2nd order mode3rd order modeUndamaged case5.242521.04847.652Case 15.241421.03347.586Case 25.240221.01547.514Case 35.233020.95947.493⋮⋮⋮⋮Figure 8
Modal shape cloud diagram of nondamage case. (a) 1st order mode. (b) 2nd order mode. (c) 3rd order mode.
(a)(b)(c)


## 3.1.2. Prior Probability

The displacement influence line is generated when unit load moves on beam, and the prior probabilityPAm of every element can be calculated by equation (7), as shown in Table 3.Table 3
Prior probabilities of elements.
Element no.Prior probability10.00383590420.01145856730.0189337640.026163250.03304857660.03949158470.0453939280.0506572890.05518336100.058873856110.061630507120.06335488130.063949312……Note. Prior probabilities of the other 12 elements are the same due to symmetry.


## 3.2. Damage Localization Analysis of Single Index

First, we use three original indicators (DR, CR, and RD) to analyze structural damage separately, and the results of damage cases are shown in Figure9.Figure 9
The identification results of single index for single (a), two (b), and three (c) damage cases. (a) Case 1. (b) Case 15. (c) Case 20.
(a)(b)(c)


### 3.2.1. Single Damage Identification

From Figure9(a), it is obvious that maximum peaks appear at element No.3 in 1st order modal curves of DR, CR, and RD, which correspond to the damage assumed in case 1, so they all can search damage location. However, in 2nd and 3rd order modal curves, not only element No. 3, there would appear other peaks at health elements, which cannot be ignored. For example, in the 3rd order modal curve of RD (blue line), the values of undamaged elements No. 8 and No. 9 are very large, so it may lead misjudgment.


### 3.2.2. Two Damage Cases Identification

In Figure9(b), for case 15, from 1st and 3rd order modal curves of DR and CR, peaks at damage element No. 3 and No. 9 can be searched, while their 2nd order modal curve would be not significant because other peaks appeared at elements No. 12 and No. 14. In RD modal curves, it would only be effective for 1st order mode to lock damage location.


### 3.2.3. Three Damage Cases Identification

In Figure9(c), both methods of DR and CR can identify damage position by searching peaks at damaged elements No. 6, No. 12, and No. 18. But in the curves of RD, values of undamaged elements No. 5 and No. 7 will exceed those of damaged elements No. 12 and No. 18, so it is not easy to distinguish which element is damaged or not.From the above analysis, when multiple damage occurs, the ability of single index and its accuracy are both significantly declined for damage recognition, and its probability of misjudgment would be increased.


## 3.2.1. Single Damage Identification

From Figure9(a), it is obvious that maximum peaks appear at element No.3 in 1st order modal curves of DR, CR, and RD, which correspond to the damage assumed in case 1, so they all can search damage location. However, in 2nd and 3rd order modal curves, not only element No. 3, there would appear other peaks at health elements, which cannot be ignored. For example, in the 3rd order modal curve of RD (blue line), the values of undamaged elements No. 8 and No. 9 are very large, so it may lead misjudgment.


## 3.2.2. Two Damage Cases Identification

In Figure9(b), for case 15, from 1st and 3rd order modal curves of DR and CR, peaks at damage element No. 3 and No. 9 can be searched, while their 2nd order modal curve would be not significant because other peaks appeared at elements No. 12 and No. 14. In RD modal curves, it would only be effective for 1st order mode to lock damage location.


## 3.2.3. Three Damage Cases Identification

In Figure9(c), both methods of DR and CR can identify damage position by searching peaks at damaged elements No. 6, No. 12, and No. 18. But in the curves of RD, values of undamaged elements No. 5 and No. 7 will exceed those of damaged elements No. 12 and No. 18, so it is not easy to distinguish which element is damaged or not.From the above analysis, when multiple damage occurs, the ability of single index and its accuracy are both significantly declined for damage recognition, and its probability of misjudgment would be increased.


## 3.3. Damage Localization Analysis of First-Level Fusion Based on Bayes Theory

Based on Bayes fusion theory, by fusing damage probabilities of the first three order modes of DR, CR, and RD, respectively, three first-level fusion indexes named DR1, CR1, and RD1 will be obtained, which can be applied for damage location identification according to the flow chart in Figure 4.


### 3.3.1. Two Damage Cases Identification

In Figure10, the peaks position of two damage cases for cases 15, 16, 17, and 18 can be easily found, which indicates that DR1, CR1, and RD1 can be qualified for damage localization. But from vertical coordinate, the peak values of damage probabilities are irregular and cannot be in accordance with damage set in cases; therefore, the damage degree will not be directly reflected in fusion results.Figure 10
Fusion recognition results of two damage cases for DR1 (a), CR1 (b), and RD1 (c).
(a)(b)(c)


### 3.3.2. Multiple Damage Identification

According to the results of case 20 shown in Figure11, indexes DR1, CR1, and RD1 can locate damage position by peak searching, and the greater the damage degree is, the higher and more obvious the peaks are; for example, damage degree of element No. 6 is biggest (30%), so its peak is highest.Figure 11
Fusion recognition results of multiple damage (case 20).In summary, compared with single index, after Bayes first-level fusion, damage localization abilities of DR1, CR1, and RD1 are significantly improved, which can reduce the impact of health elements and increase the accuracy of damage identification. Comparing their peak values with each other, the order of their ability is DR1≥CR1≥RD1, so DR1 and CR1 are preferred for structural damage localization in actual engineering application.


## 3.3.1. Two Damage Cases Identification

In Figure10, the peaks position of two damage cases for cases 15, 16, 17, and 18 can be easily found, which indicates that DR1, CR1, and RD1 can be qualified for damage localization. But from vertical coordinate, the peak values of damage probabilities are irregular and cannot be in accordance with damage set in cases; therefore, the damage degree will not be directly reflected in fusion results.Figure 10
Fusion recognition results of two damage cases for DR1 (a), CR1 (b), and RD1 (c).
(a)(b)(c)


## 3.3.2. Multiple Damage Identification

According to the results of case 20 shown in Figure11, indexes DR1, CR1, and RD1 can locate damage position by peak searching, and the greater the damage degree is, the higher and more obvious the peaks are; for example, damage degree of element No. 6 is biggest (30%), so its peak is highest.Figure 11
Fusion recognition results of multiple damage (case 20).In summary, compared with single index, after Bayes first-level fusion, damage localization abilities of DR1, CR1, and RD1 are significantly improved, which can reduce the impact of health elements and increase the accuracy of damage identification. Comparing their peak values with each other, the order of their ability is DR1≥CR1≥RD1, so DR1 and CR1 are preferred for structural damage localization in actual engineering application.


## 3.4. Damage Localization Analysis of Second-Level Fusion Based on Weighted Average Criterion

According to the flow chart in Figure5, the second-level fusion index will be obtained by applying weighted average criterion. The recognition results are as follows.In Figure12, some single damage cases are selected for analysis, and damage elements can be clearly found by searching maximum peaks from the curve. It should be noted that health element values are very small, so we consider that second-level fusion index can weaken “proximity effect” of RD1 in first-level fusion. For multiple damage cases (Figure 13), second-level fusion index also has the same identification ability.Figure 12
Recognition results of second-level fusion for single damage.Figure 13
Recognition results of second-level fusion for multi-damage.


## 3.5. Damage Quantitative Analysis of Third-Level Fusion Based on BP Neural Network

From the above, we know that whatever the single index is, first-level or second-level fusion, they all cannot easily determine the structural damage degree. Therefore, a third-level fusion model based on BP neural network is established here for further quantitative analysis.A three-layer BP neural network is adopted for training here, which takes second-level fusion results as input and takes structural residual stiffness as output (DD3). In the training process, maximum iterative steps are 100, and precision is 10−5. Lastly, the output data can be easily processed to list absolute error (AE), relative percentage error (RPE), mean absolute error (MAE), and mean relative percent error (MRPE), which can be used as output evaluation indices. The detail calculation is as follows:(19)AE=Vkt−V^ktRPE=Vkt−V^ktVkt×100%MAE=1M∑i=1MVkt−V^ktMRPE=1M∑i=1MVkt−V^ktVkt×100%,where M is the number of test samples; Vkt is real structural residual stiffness; V^kt is output residual stiffness of BP neural network.


### 3.5.1. Taking Cases 4, 8, and 14 as Prediction Set

According to equation (10), the secondary fusion results DF of all damage conditions are calculated, and the DF values of cases 4, 8, and 14 are used as prediction set, and the DF values of other cases are used as the training set. The training set is input into the BP neural network for training. After the accuracy meets the requirements, the prediction set is input into BP neural network for damage degree prediction, and the residual structural stiffness DD3 is output. At the same time, according to equation (2), DR1, CR1, and RD1 of all cases are calculated, respectively, as control group, and they are also divided into training set and prediction set and input into BP neural network, and output DR2, CR2, and RD2. The prediction results of damage degree of prediction set are shown in Table 4.Table 4
Damage quantitative analysis comparation for DD3, DR2, CR2, and RD2.
IndexCaseC4C8C14Ideal outputActual outputAERPE (%)Ideal outputActual outputAERPE (%)Ideal outputActual outputAERPE (%)DD310.800.82610.02613.26750.90.92440.02442.70690.850.84080.00921.080220.77840.02162.69820.87730.02272.51980.87590.02593.045530.80880.00881.09380.88540.01461.61960.86090.01091.2850DR210.81930.01932.41590.83390.06617.34160.90120.05126.028120.78320.01682.09580.84720.05285.86260.89910.04915.779030.77120.02883.60030.82180.07828.69150.92480.07488.7986CR210.75460.04545.67680.93030.03033.36840.89550.04555.355320.76730.03274.09310.84120.05886.52980.87100.02102.465030.75350.04655.81390.87760.02242.49430.86370.01371.6137DR210.83180.03183.97550.93310.03313.68250.90990.05997.049720.81420.01421.77590.86710.03293.65960.90640.05646.630430.83100.03103.87930.85150.04855.39300.90360.05366.3116For further evaluating recognition effect of DD3, a detailed comparation is analyzed for RPE, MAE. and MRPE of DD3, DR2, CR2, and RD2. In column of RPE, the maximum value of DD3 is 3.27%, which is 4.93% lower than those of others by 8.80%. For MAE, DD3 result for case 4 can be simply calculated, and its value is 0.0188, less than that of DR2, CR2, and RD2 (0.0216, 0.0415, and 0.0257); MRPEs of cases 4, 8, and 14 for DD3 are 2.35%, 2.28%, and 1.80%, respectively, while those of DR2, CR2, and RD2 are 2.71%, 7.30%, and 6.87%; 5.19%, 4.13%, and 2.99%; and 3.21%, 4.25%, and 6.66%, respectively. It is obvious that mean relative percent error of index DD3 is significantly lower than that of DR2, CR2, and RD2.


### 3.5.2. Taking Single Case from 1 to 14 as Prediction Set, Respectively

Predicted results of 14 groups will be output following the above steps; specifically, their AE results are drawn as a histogram into Figure14 for comparison. Maximum AE value of DD3 (black) is 0.014, while the other DR2 (red), CR2 (blue), and RD2 (green) are all beyond 0.04, and also their MAE value is still larger than that of DD3.Figure 14
Comparison of absolute error of damage degree.Overall, the damage degree actual output of DD3 is closer to ideal output, which means that the third-level fusion method is preferred for structural damage degree identification.


## 3.5.1. Taking Cases 4, 8, and 14 as Prediction Set

According to equation (10), the secondary fusion results DF of all damage conditions are calculated, and the DF values of cases 4, 8, and 14 are used as prediction set, and the DF values of other cases are used as the training set. The training set is input into the BP neural network for training. After the accuracy meets the requirements, the prediction set is input into BP neural network for damage degree prediction, and the residual structural stiffness DD3 is output. At the same time, according to equation (2), DR1, CR1, and RD1 of all cases are calculated, respectively, as control group, and they are also divided into training set and prediction set and input into BP neural network, and output DR2, CR2, and RD2. The prediction results of damage degree of prediction set are shown in Table 4.Table 4
Damage quantitative analysis comparation for DD3, DR2, CR2, and RD2.
IndexCaseC4C8C14Ideal outputActual outputAERPE (%)Ideal outputActual outputAERPE (%)Ideal outputActual outputAERPE (%)DD310.800.82610.02613.26750.90.92440.02442.70690.850.84080.00921.080220.77840.02162.69820.87730.02272.51980.87590.02593.045530.80880.00881.09380.88540.01461.61960.86090.01091.2850DR210.81930.01932.41590.83390.06617.34160.90120.05126.028120.78320.01682.09580.84720.05285.86260.89910.04915.779030.77120.02883.60030.82180.07828.69150.92480.07488.7986CR210.75460.04545.67680.93030.03033.36840.89550.04555.355320.76730.03274.09310.84120.05886.52980.87100.02102.465030.75350.04655.81390.87760.02242.49430.86370.01371.6137DR210.83180.03183.97550.93310.03313.68250.90990.05997.049720.81420.01421.77590.86710.03293.65960.90640.05646.630430.83100.03103.87930.85150.04855.39300.90360.05366.3116For further evaluating recognition effect of DD3, a detailed comparation is analyzed for RPE, MAE. and MRPE of DD3, DR2, CR2, and RD2. In column of RPE, the maximum value of DD3 is 3.27%, which is 4.93% lower than those of others by 8.80%. For MAE, DD3 result for case 4 can be simply calculated, and its value is 0.0188, less than that of DR2, CR2, and RD2 (0.0216, 0.0415, and 0.0257); MRPEs of cases 4, 8, and 14 for DD3 are 2.35%, 2.28%, and 1.80%, respectively, while those of DR2, CR2, and RD2 are 2.71%, 7.30%, and 6.87%; 5.19%, 4.13%, and 2.99%; and 3.21%, 4.25%, and 6.66%, respectively. It is obvious that mean relative percent error of index DD3 is significantly lower than that of DR2, CR2, and RD2.


## 3.5.2. Taking Single Case from 1 to 14 as Prediction Set, Respectively

Predicted results of 14 groups will be output following the above steps; specifically, their AE results are drawn as a histogram into Figure14 for comparison. Maximum AE value of DD3 (black) is 0.014, while the other DR2 (red), CR2 (blue), and RD2 (green) are all beyond 0.04, and also their MAE value is still larger than that of DD3.Figure 14
Comparison of absolute error of damage degree.Overall, the damage degree actual output of DD3 is closer to ideal output, which means that the third-level fusion method is preferred for structural damage degree identification.


## 3.6. Noise Immunity Analysis

### 3.6.1. Noise Simulation Method

In actual working condition, due to external environment, test equipment error, and manual operation irregularities, the noise will be always existing; it would affect measured data so that there is often a certain degree difference between real values. Therefore, in order to better verify effectiveness of proposed methods, certain level noise is added into modal information when simulation, for analyzing their antinoise performance.Usually, the way to add noise [39] is applied through the following equation:(20)Φijn=Φij+Φmax,i×randn×δ,where Φij and Φijn represent vibration shape before and after noise addition, respectively; Φmax,i is maximum value of ith order vibration shape; randn represents Gaussian white noise, where the mean value and standard deviation are 0 and 1, respectively; δ is noise level.


### 3.6.2. Noise Immunity Analysis of the First-Level Fusion

For noise immunity comparative analysis between single index and first-level fusion index, 5% and 10% level noise are added to damage cases 15 and 20, respectively. Figures15 and 16 show antinoise analysis results of two and three damage cases, respectively. In (a) and (b), with increase of damage elements or noise level, damage probability and amount of health elements are also increased, which would lead to ambiguity for actual damage judgement. But after first-level data fusion based on Bayes theory, in (c), damage probabilities of health elements are significantly reduced, which makes peaks at damage position more prominent and can be easily and accurately found. So, first-level fusion index can improve recognition accuracy and has strong robustness than single index.Figure 15
Antinoise analysis results of case 15. (a) 5% noise level of single index. (b) 10% noise level of single index. (c) 5%, 10% noise level of first-level fusion index.
(a)(b)(c)Figure 16
Antinoise analysis results of case 20. (a) 5% noise level of single index. (b) 10% noise level of single index. (c) 5%, 10% noise level of first-level fusion index.
(a)(b)(c)


### 3.6.3. Noise Immunity Analysis of the Second-Level Fusion

Taking damage cases 1, 15, and 20 as examples with 5% and 10% noise, respectively, second-level fusion results will be calculated and drawn into histogram for comparation in Figure17, in which peaks would be easily searched at damaged elements whatever the single, double, or three damage cases are, which means that the second-level fusion index has strong ability to eliminate noise interference.Figure 17
Antinoise analysis results of the second-level fusion index.


## 3.6.1. Noise Simulation Method

In actual working condition, due to external environment, test equipment error, and manual operation irregularities, the noise will be always existing; it would affect measured data so that there is often a certain degree difference between real values. Therefore, in order to better verify effectiveness of proposed methods, certain level noise is added into modal information when simulation, for analyzing their antinoise performance.Usually, the way to add noise [39] is applied through the following equation:(20)Φijn=Φij+Φmax,i×randn×δ,where Φij and Φijn represent vibration shape before and after noise addition, respectively; Φmax,i is maximum value of ith order vibration shape; randn represents Gaussian white noise, where the mean value and standard deviation are 0 and 1, respectively; δ is noise level.


## 3.6.2. Noise Immunity Analysis of the First-Level Fusion

For noise immunity comparative analysis between single index and first-level fusion index, 5% and 10% level noise are added to damage cases 15 and 20, respectively. Figures15 and 16 show antinoise analysis results of two and three damage cases, respectively. In (a) and (b), with increase of damage elements or noise level, damage probability and amount of health elements are also increased, which would lead to ambiguity for actual damage judgement. But after first-level data fusion based on Bayes theory, in (c), damage probabilities of health elements are significantly reduced, which makes peaks at damage position more prominent and can be easily and accurately found. So, first-level fusion index can improve recognition accuracy and has strong robustness than single index.Figure 15
Antinoise analysis results of case 15. (a) 5% noise level of single index. (b) 10% noise level of single index. (c) 5%, 10% noise level of first-level fusion index.
(a)(b)(c)Figure 16
Antinoise analysis results of case 20. (a) 5% noise level of single index. (b) 10% noise level of single index. (c) 5%, 10% noise level of first-level fusion index.
(a)(b)(c)


## 3.6.3. Noise Immunity Analysis of the Second-Level Fusion

Taking damage cases 1, 15, and 20 as examples with 5% and 10% noise, respectively, second-level fusion results will be calculated and drawn into histogram for comparation in Figure17, in which peaks would be easily searched at damaged elements whatever the single, double, or three damage cases are, which means that the second-level fusion index has strong ability to eliminate noise interference.Figure 17
Antinoise analysis results of the second-level fusion index.


## 4. Damage Recognition Results of a Three-Span Continuous Girder Bridge

The damage identification indexes based on multilevel data fusion has a good effect on simply supported beam, while, for actual bridge in complicated loading, practicability of those methods needs to be further verified.


### 4.1. Finite Element Model

A three-span continuous box girder model is established by ANSYS 17.0 software as shown in Figure18, which is a single box and single cell structure with three equal spans in total 90 meters (30 m + 30 m + 30 m). Concrete material is C50, elastic modulus is 3.45 × 107 kN/m2, Poisson’s ratio is 0.2, density is 25 kN/m3, and mass density is 2.549 kN/m3/g. The beam is divided into 90 elements, 91 nodes, and the size of cross section is shown in Figure 19.Figure 18
Three-span continuous box girder.Figure 19
Cross section of box girder (cm).


### 4.2. Damage Case and Modal Shape

The method of stiffness reduction is still applied for simulating damage of continuous girder. Usually, damage near support is not easy to be identified [1], so we particularly set damage elements near support, as shown in Figure 20, and set damage to No. 2, 29, 31, 59, and 61. The damage cases with different degrees with 5% noise level are shown in Table 5. The first-three modal shapes of three-span continuous girder under undamaged case can be calculated and shown in Figure 21.Figure 20
Damage location of three-span continuous girder.Table 5
Damage cases of continuous girder.
Damage caseDamage element no.Damage degree (%)21291059152221531206115Figure 21
Vibration modes for 1st (a), 2nd (b), and 3rd (c) order of three-span continuous girder.
(a)(b)(c)


### 4.3. Damage Localization Analysis of Single Index

Identification curves for DR and CR indices of first-three order are shown in Figure22. Because RD index is poor for continuous beam damage identification, it will not participate in fusion. From the histogram, peaks would appear not only at damaged elements, but also at health elements, so we prove once again that it will cause “false damage” phenomenon and is not easy to identify damage near support.Figure 22
Damage identification results for case21 (a) and case22 (b) of single index.
(a)(b)


### 4.4. Damage Localization Analysis of Multi-Level Fusion

The three-span continuous girder is shown in Figure23. According to equation (7), damage prior probability of each element can be calculated, and the results are shown in Figure 24.Figure 23
Three-span continuous girder model.Figure 24
Damage prior probability of continuous girder.In order to accurately identify damage near support and reduce interference from undamaged elements, the first-level fusion and second-level fusion methods are introduced for three-span continuous girder damage recognition, and their results for cases 21 and 22 are shown in Figure25.Figure 25
Recognition results for continuous girder of multilevel fusion.Compared with single index results, after Bayes first-level fusion, the value of peaks at damage elements is more higher than that at health elements, which indicates that health element will significantly reduce interference to damaged one, and probability of misjudgment would be declined, so that identification accuracy will be improved.Moreover, superposing the two results of first-level fusion DR1 and CR1, we can obtain second-level fusion result DF, from the histogram, values at undamaged elements are almost zero, and peaks at damage elements are prominent. Therefore, second-level fusion index will have strong anti-interference performance and present better recognition effect.


## 4.1. Finite Element Model

A three-span continuous box girder model is established by ANSYS 17.0 software as shown in Figure18, which is a single box and single cell structure with three equal spans in total 90 meters (30 m + 30 m + 30 m). Concrete material is C50, elastic modulus is 3.45 × 107 kN/m2, Poisson’s ratio is 0.2, density is 25 kN/m3, and mass density is 2.549 kN/m3/g. The beam is divided into 90 elements, 91 nodes, and the size of cross section is shown in Figure 19.Figure 18
Three-span continuous box girder.Figure 19
Cross section of box girder (cm).


## 4.2. Damage Case and Modal Shape

The method of stiffness reduction is still applied for simulating damage of continuous girder. Usually, damage near support is not easy to be identified [1], so we particularly set damage elements near support, as shown in Figure 20, and set damage to No. 2, 29, 31, 59, and 61. The damage cases with different degrees with 5% noise level are shown in Table 5. The first-three modal shapes of three-span continuous girder under undamaged case can be calculated and shown in Figure 21.Figure 20
Damage location of three-span continuous girder.Table 5
Damage cases of continuous girder.
Damage caseDamage element no.Damage degree (%)21291059152221531206115Figure 21
Vibration modes for 1st (a), 2nd (b), and 3rd (c) order of three-span continuous girder.
(a)(b)(c)


## 4.3. Damage Localization Analysis of Single Index

Identification curves for DR and CR indices of first-three order are shown in Figure22. Because RD index is poor for continuous beam damage identification, it will not participate in fusion. From the histogram, peaks would appear not only at damaged elements, but also at health elements, so we prove once again that it will cause “false damage” phenomenon and is not easy to identify damage near support.Figure 22
Damage identification results for case21 (a) and case22 (b) of single index.
(a)(b)


## 4.4. Damage Localization Analysis of Multi-Level Fusion

The three-span continuous girder is shown in Figure23. According to equation (7), damage prior probability of each element can be calculated, and the results are shown in Figure 24.Figure 23
Three-span continuous girder model.Figure 24
Damage prior probability of continuous girder.In order to accurately identify damage near support and reduce interference from undamaged elements, the first-level fusion and second-level fusion methods are introduced for three-span continuous girder damage recognition, and their results for cases 21 and 22 are shown in Figure25.Figure 25
Recognition results for continuous girder of multilevel fusion.Compared with single index results, after Bayes first-level fusion, the value of peaks at damage elements is more higher than that at health elements, which indicates that health element will significantly reduce interference to damaged one, and probability of misjudgment would be declined, so that identification accuracy will be improved.Moreover, superposing the two results of first-level fusion DR1 and CR1, we can obtain second-level fusion result DF, from the histogram, values at undamaged elements are almost zero, and peaks at damage elements are prominent. Therefore, second-level fusion index will have strong anti-interference performance and present better recognition effect.


## 5. Test Study

### 5.1. Test Overview

A test model of simply supported rectangular steel plate beam is constructed here for further verification. Cross section size is100mm×8mm (width × height), length is 2000 mm, calculated span is 1750 mm, elastic modulus of steel is E=2.0795×108kN/m2, and density is ρ=7698kg/m3, as shown in Figure 26.Figure 26
Simply supported steel beam test mode.Within the calculated span length, 36 white lines are signed on surface of steel girder with equal distance of 5 cm, for accurate positioning acceleration sensors installation. When the test, single-point excitation, and multipoint acceleration collection method are applied, here, six signal sensors are employed to obtain modal information of beam; among them, five sensors are set as a working group with a distance of 5 cm moving on beam, the other one as a base control group, for calculating relative value of working group. The system INV-9812 is used for data collection, and analysis software DASP2003 for signal processing. The sensors layout is shown in Figure27.Figure 27
Simply supported beam sensors layout.
(a)(b)


### 5.2. Damage Pattern and Dynamic Parameter Acquisition

The damage pattern in this experiment is realized by symmetrical cutting method, actual cut form of steel.Beam is shown in Figure28, cutting width is 10 mm, and cutting depth and position are listed in Table 6.Figure 28
Damage form of steel beam. (a) Steel beam damage case. (b) Steel beam notch form.
(a)(b)Table 6
Damage forms of tested steel beam.
Damage formDamage position and cutting depthDamage degree (%)1No damage2300 mm cut 25 mm503500 mm cut 20 mm401250 mm cut 25 mm504450 mm cut 20 mm40875 mm cut 20 mm401300 mm cut 20 mm40The modal parameters and corresponding frequency spectrum curve of simply supported beam under different damage forms can be obtained by processing acceleration signal data with analysis system DASP2003. Some frequency spectrum curve diagrams are shown in Figure29.Figure 29
Frequency spectrum curve of simply supported beam. (a) Form 1. (b) Form 2. (c) Form 4.
(a)(b)(c)According to equation (7), damage prior probability PAm of every element of tested simply supported beam is calculated, and its parabolic diagram is shown in Figure 30.Figure 30
Damage prior probability of tested simply supported beam.


### 5.3. Test Results Analysis

#### 5.3.1. Damage Localization Analysis of Single Index

The results of single index recognition are shown in Figures31 and 32. From the graph, whatever the form is, form 2 or form 4, peaks will not only appear at the actual damaged elements 6, 9, 17, and 26 we set, but also appear at other health elements, so that we cannot distinguish which element will be damaged and which will not.Figure 31
Damage identification results of form 2 for DR (a), CR (b), and RD (c).
(a)(b)(c)Figure 32
Damage identification results of form 4 for DR (a), CR (b), and RD (c).
(a)(b)(c)


#### 5.3.2. Damage Localization Analysis of Multilevel Fusion

Because the above result figures are disorganized, and we will be puzzled to find the actual damaged elements, we need to take effective measures for further processing. The first-level data fusion for forms 2 and 4 and their second-level data fusion are calculated, respectively, and drawn in Figures33 and 34.Figure 33
Recognition results of first-level fusion index for form 2 (a) and form 4 (b).
(a)(b)Figure 34
Recognition results of the second-level fusion index.From Figure33, peaks in histogram, whatever the indexes are, DR1, CR1, or RD1, are all obvious and just happen at damaged elements, while values at other position can be almost ignored. Therefore, the first-level data fusion model is a very nice data processing method and will be qualified for damage identification.In Figure34, in second-level fusion results, whatever the curve of single, double, or three damage cases is, peaks will still be easily locked, which well match the damage forms set before, and the “proximity effect” is also significantly weakened, so the second-level data fusion model is superior method for priority selection.


## 5.1. Test Overview

A test model of simply supported rectangular steel plate beam is constructed here for further verification. Cross section size is100mm×8mm (width × height), length is 2000 mm, calculated span is 1750 mm, elastic modulus of steel is E=2.0795×108kN/m2, and density is ρ=7698kg/m3, as shown in Figure 26.Figure 26
Simply supported steel beam test mode.Within the calculated span length, 36 white lines are signed on surface of steel girder with equal distance of 5 cm, for accurate positioning acceleration sensors installation. When the test, single-point excitation, and multipoint acceleration collection method are applied, here, six signal sensors are employed to obtain modal information of beam; among them, five sensors are set as a working group with a distance of 5 cm moving on beam, the other one as a base control group, for calculating relative value of working group. The system INV-9812 is used for data collection, and analysis software DASP2003 for signal processing. The sensors layout is shown in Figure27.Figure 27
Simply supported beam sensors layout.
(a)(b)


## 5.2. Damage Pattern and Dynamic Parameter Acquisition

The damage pattern in this experiment is realized by symmetrical cutting method, actual cut form of steel.Beam is shown in Figure28, cutting width is 10 mm, and cutting depth and position are listed in Table 6.Figure 28
Damage form of steel beam. (a) Steel beam damage case. (b) Steel beam notch form.
(a)(b)Table 6
Damage forms of tested steel beam.
Damage formDamage position and cutting depthDamage degree (%)1No damage2300 mm cut 25 mm503500 mm cut 20 mm401250 mm cut 25 mm504450 mm cut 20 mm40875 mm cut 20 mm401300 mm cut 20 mm40The modal parameters and corresponding frequency spectrum curve of simply supported beam under different damage forms can be obtained by processing acceleration signal data with analysis system DASP2003. Some frequency spectrum curve diagrams are shown in Figure29.Figure 29
Frequency spectrum curve of simply supported beam. (a) Form 1. (b) Form 2. (c) Form 4.
(a)(b)(c)According to equation (7), damage prior probability PAm of every element of tested simply supported beam is calculated, and its parabolic diagram is shown in Figure 30.Figure 30
Damage prior probability of tested simply supported beam.


## 5.3. Test Results Analysis

### 5.3.1. Damage Localization Analysis of Single Index

The results of single index recognition are shown in Figures31 and 32. From the graph, whatever the form is, form 2 or form 4, peaks will not only appear at the actual damaged elements 6, 9, 17, and 26 we set, but also appear at other health elements, so that we cannot distinguish which element will be damaged and which will not.Figure 31
Damage identification results of form 2 for DR (a), CR (b), and RD (c).
(a)(b)(c)Figure 32
Damage identification results of form 4 for DR (a), CR (b), and RD (c).
(a)(b)(c)


### 5.3.2. Damage Localization Analysis of Multilevel Fusion

Because the above result figures are disorganized, and we will be puzzled to find the actual damaged elements, we need to take effective measures for further processing. The first-level data fusion for forms 2 and 4 and their second-level data fusion are calculated, respectively, and drawn in Figures33 and 34.Figure 33
Recognition results of first-level fusion index for form 2 (a) and form 4 (b).
(a)(b)Figure 34
Recognition results of the second-level fusion index.From Figure33, peaks in histogram, whatever the indexes are, DR1, CR1, or RD1, are all obvious and just happen at damaged elements, while values at other position can be almost ignored. Therefore, the first-level data fusion model is a very nice data processing method and will be qualified for damage identification.In Figure34, in second-level fusion results, whatever the curve of single, double, or three damage cases is, peaks will still be easily locked, which well match the damage forms set before, and the “proximity effect” is also significantly weakened, so the second-level data fusion model is superior method for priority selection.


## 5.3.1. Damage Localization Analysis of Single Index

The results of single index recognition are shown in Figures31 and 32. From the graph, whatever the form is, form 2 or form 4, peaks will not only appear at the actual damaged elements 6, 9, 17, and 26 we set, but also appear at other health elements, so that we cannot distinguish which element will be damaged and which will not.Figure 31
Damage identification results of form 2 for DR (a), CR (b), and RD (c).
(a)(b)(c)Figure 32
Damage identification results of form 4 for DR (a), CR (b), and RD (c).
(a)(b)(c)


## 5.3.2. Damage Localization Analysis of Multilevel Fusion

Because the above result figures are disorganized, and we will be puzzled to find the actual damaged elements, we need to take effective measures for further processing. The first-level data fusion for forms 2 and 4 and their second-level data fusion are calculated, respectively, and drawn in Figures33 and 34.Figure 33
Recognition results of first-level fusion index for form 2 (a) and form 4 (b).
(a)(b)Figure 34
Recognition results of the second-level fusion index.From Figure33, peaks in histogram, whatever the indexes are, DR1, CR1, or RD1, are all obvious and just happen at damaged elements, while values at other position can be almost ignored. Therefore, the first-level data fusion model is a very nice data processing method and will be qualified for damage identification.In Figure34, in second-level fusion results, whatever the curve of single, double, or three damage cases is, peaks will still be easily locked, which well match the damage forms set before, and the “proximity effect” is also significantly weakened, so the second-level data fusion model is superior method for priority selection.


## 6. Conclusion

Based on data fusion theories, this paper takes the modal strain energy dissipation rate (DR), change rate of cross-model strain energy (CR), and difference of modal curvature ratio (RD) as basic indicators to construct multilevel data fusion models for structural damage identification; also the noise immunity and experimental verification are analyzed, and the following conclusions are obtained:(1)
In the process of multilevel data fusion method, both first-level and second-level can locate damage positions at wheresoever mid-span or near support, and whatever the simply supported beam or three-span continuous girder is, the second-level proved that its recognition effectiveness is better for multidamage. The third-level method can directly reflect structural damage degree, and it is more superior for the results of second-level as input for BP neural network training than those of first-level.(2)
The indicators based on multilevel data fusion method reinforced each other due to multisource information superposition, so that they can improve reliability of damage localization and accuracy of damage quantification.(3)
By experimental analysis, the multilevel data fusion method can also accurately identify real damage although the modal data are affected by factors such as site test condition and individual error, which indicate that they have strong robustness and fault tolerance. Those methods need further verification by field test.



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*Source: 9962169-2022-04-16.xml*