# Bearing Damage Detection of a Bridge under the Uncertain Conditions Based on the Bayesian Framework and Matrix Perturbation Method

Authors: Baojun Li; Yongzhi Lei; Dongming Zhou; Zhiheng Deng; Yuhou Yang; Minshui Huang
Journal: Shock and Vibration
 (2021)

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## Abstract

The bearing of a bridge, as a critical component, is important in the force transformation of the superstructure; however, due to the service condition and repeated impact load, the bearing is prone to be damaged but difficult to detect the damage; the present research has few studies that focused on the damage detection of the structural bearing. Meanwhile, practical engineering is always surrounded by variational environmental conditions, and sometimes, the element and bearing damage both exist in the structure. Thus, these uncertain conditions all cause inaccurate damage identification results using the vibration-based damage detection method. In order to detect the damage of the structural bearing and improve the precision, firstly, the structural dynamic characteristic equation considering uncertain conditions has been deduced; then, a damage detection framework constructed by the Bayesian theory and perturbation method has been developed in this article; a numerical example of an 8-span concrete continuous beam and a practical example of I-40 steel-concrete composite bridge are utilized to validate the feasibility of the proposed method, and single type and two types of damage cases are studied. The outcomes demonstrate that the damage of structural elements and bearings can be detected with high accuracy. The proposed method is of great applicability and good potential.


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## Body

## 1. Introduction

The bridge structure, as an important tool to connect the two sides of a river or valley, often suffers from heavy traffic flows [1] and serious impact loads. Due to the practical service situation of the bridge, the bridge bearing function, as a critical connector, can transfer the loads coming from bridge decks to bridge piers. The rubber bearings are exposed to the air, which are easy to be aged by the environmental factors [2]. At the same time, under the repeated load, the bearing has been very vulnerable further, which means some diseases, such as separation, shear deformation, bulging deformation, and transverse crack, have often emerged in this component.When the diseases have occurred in the bridge bearings, the transmission of the force in the whole bridge will be changed; even, it will damage the deck and pier in some extreme cases. Thus, the destruction of the bearings will cause a potential risk to the service period of the bridge. Although numerous researchers pay attention to structural health monitoring (SHM), most ideas have focused on the structural element rather than the structural bearings [3–8], which means the research field of this critical component is blank, and it is necessary to put forward some solutions to detect the damage of bridge bearings.Vibration-based structural damage identification methods have been obtained and widely used in various civil engineering examples [9, 10], which provide a possible solution to the bearing damage detection. Chen et al. [11] investigated the feasibility and sensitivity of bearing damage identification using bridge vibration modes and the radial basis function neural network; the results indicate that modal information can reflect the bearing damage clearly. Ni et al. [12] adopted correlation and sensitivity analysis to put forward a damage identification indicator of the bridge pot rubber bearing and then combined with the support vector machine to form a safety-level discriminant model to assess the health state of the bridge pot rubber bearing. However, because the influence severities of modal parameters to structural elements and bearings are discrepant and nonlinear, Huang et al. [13] conducted the modal parameter sensitivity analysis on the damage of structural elements and bridge bearings and deduced the sensitivity coefficients of eigenvalue to the damage factors of structural elements and bearings; then, the regularization method and chaotic moth-flame-invasive weed optimization were used to identify the structural element and bearing damage simultaneously. The outcomes indicate that the accuracy still desires to be improved. Meanwhile, when the structural element and bearing damage have emerged in the structure simultaneously, how to detect the two types of damage accurately? It needs more thinking.On the contrary, due to the tough service conditions of bridge structures, some environmental impacts, such as environmental temperature variations, traffic flow vibration, and climate and weather change, are always acting in structures [14]. All these factors can be considered as uncertainties of the damage identification process, which set the obstacle to identify damage precisely. Aiming to this hardship, Bayesian theory has been introduced in the field of SHM. For example, Ding et al. [15] adopted Bayesian inference and sparse regularization technology to modify the traditional objective function of damage identification; the numerical example and experimental studies show that the robustness is enhanced under the condition of significant uncertainty effect. Hou et al. [16] developed the spare Bayesian learning framework to simultaneously consider the uncertainties and varying temperature conditions; an experimental frame study demonstrated that the method is effective in locating and quantifying structural damage. Wang et al. [17] focused on different uncertainties of each mode in the sparse Bayesian method for damage identification, and variational Bayesian inference was developed; then, the high accuracy of the proposed method was validated by a laboratory-tested frame. Furthermore, the iterative process of Bayesian damage identification method needs to construct the finite element model repetitively, which means the computation is very time consuming. For this problem, one of the solutions is to develop the surrogate model, such as the kriging model [18], artificial neural network, [19] and perturbation model [20].Regarding the present research, there are few studies that have focused on the bearing damage identification, even less to consider the uncertainties of bearing damage detection; thus, in this paper, a bearing damage detection method with good potential has been developed to identify the damage of structural elements and bearings simultaneously; compared to the previous research [13], the proposed one shows high accuracy. The main significance can be summarized as follows: (1) the structural dynamic characteristic equation considering uncertain conditions has been deduced, and then based on the resulting equation, the influences of uncertain conditions on the modal parameters of the simply supported beam have been investigated, which implies the uncertainties should be considered adequately in the damage identification; (2) the Bayesian theory and perturbation method are combined, which is successfully used to identify the damage of structural elements and bearings simultaneously under the influences of uncertainties; (3) only the first several natural frequencies and modal shapes are needed to carry out the proposed damage detection method, which shows good potential and broad applicability.The specific research can be pointed as follows: first of all, based on the structural dynamics and finite element method, the structural dynamic characteristic equation considering bearing damage and environmental temperature variations, as well as element damage, has been obtained, the bearing damage is simulated by introducing the boundary stiffness matrix, and the temperature variations are transformed into elastic modulus decrease. Then, based on the dynamic characteristic equation, a simply supported beam is adopted to evaluate the influences of uncertainties to the modal parameters. After that, Bayesian theory is adopted, and combined with the structural dynamic characteristic equation considering uncertain conditions, the damage detection framework of structural elements and bearings has been formed; additionally, the perturbation method is introduced to avoid the repetitive construction of the finite element model. Finally, a numerical 8-span concrete continuous beam structure and a practical engineering example of I-40 steel-concrete composite bridge are utilized to validate the feasibility and applicability of the proposed framework. The damage detection results indicate that both single type of damage case (only element damage or bearing damage) and two types of damage case (both element and bearing damage) can be identified; furthermore, the influences of uncertainties have been overcome.


## 2. The Evaluation of the Influences of Uncertain Conditions on Modal Parameters

### 2.1. Structural Dynamic Characteristic Equation considering Uncertain Conditions

In order to assess the impacts on the measured modal parameters caused by uncertain conditions, in this section, as shown in Figure1, the simply supported beam with spring support in each end has been introduced to conduct the research. The main function of bridge bearings is to transfer the load in the superstructure to the foundation; when the damage has occurred in this component, the stiffness of the bearing will be changed; thus, the bearings can be simplified and simulated as the simple spring models with the spring constants [21–25]. For the length of each element is 0.25 m, its properties of material, such as Young’s modulus, mass density, inertia moment, and cross-sectional area, are 3.0 × 1010 Pa, 2360 kg/m3, 8.64 × 10−5 m4, and 0.072 m2, respectively; the spring constants of each end are 106 N/m.Figure 1
The simply supported beam.Three factors have been investigated in this section, namely, structural element damage, boundary condition change (bearing damage), and environmental temperature variations. Assuming that the occurrence of damage does not cause the mass change, the structural element damage is simulated by the element stiffness reduction, which can be written as follows [26–29]:(1)Ks=∑i=1nele1−θikie,0≤θi≤1,where Ks means the global stiffness matrix of a structure; nele denotes the total number of structural elements; and kie and θi are i-th element stiffness matrix and i-th stiffness reduction factor, respectively.Meanwhile, the boundary condition change, namely, the damage of bearings, can be represented as the boundary condition matrix, and it can be defined as follows [13, 30]:(2)UxRxUyRyUzRzUxRxUyRyUzRzABCDEFGHSymIJKL,where the diagonal elements on the matrix represent the boundary stiffness in every degree of freedom (DOF); other elements can be described as the synergistic effects. The matrix can be utilized in the 2D finite element model by reducing the unconcerned DOFs. Thus, the damage in the structural bearings can also be described as the similar form as the structural element damage:(3)Kbc=∑j=1nbc1−αjkjbc,0≤αj≤1,where Kbc means the global stiffness matrix of a structural boundary system; nbc denotes the total number of bearings; and kjbc and αj are the j-th bearing stiffness matrix and i-th bearing stiffness reduction factor, respectively.In addition, the environmental temperature variations have also been a serious impact on a civil structure, which can cause the fluctuation of the measured modal parameters [31, 32]. In this paper, the temperature variation is transformed into the reduction of Young’s modulus, which can be calculated as follows [33]:(4)ET=E0⋅1−αE⋅T−T0,where ET and E0 represent Young’s modulus of the material at temperatures T and T0, respectively, and αE stands for the reduction coefficient under variational temperature, which can be determined as 0.3% for the concrete material [14].Thus, based on the aforementioned uncertain conditions and according to the structural dynamic theory and finite element method, the structural undamped dynamic characteristic equation considering uncertain conditions can be written as follows:(5)KsT,θ+Kbcα−λiMs+Mbcφi=0,where KsT,θ stands for the global stiffness matrix of structural elements with the environmental temperature of T and the structural damage severity of θ; Kbcα means the global stiffness matrix of boundary systems with the bearing damage severity of α; Ms and Mbc denote the global mass matrix of structural elements and boundary systems, respectively; and λi and φi are the i-th eigenvalue and eigenvector, respectively. Moreover, it needs to be pointed out that the effects of environmental temperature variations on the boundary systems have been ignored.


### 2.2. The Evaluation of the Factors

First of all, the factors have been studied to evaluate the influences on the modal parameters (e.g., natural frequencies and vertical modal shapes).


#### 2.2.1. Structural Element Damage

To select one structural element randomly and introducing damage severity that ranges from 0 to 100%, the increasing step size is determined as 1% to calculate the corresponding natural frequencies and modal shapes. Then, the first four-order natural frequencies are extracted and illustrated in Figure2, and the max nodal displacement of each vertical modal shape is obtained and plotted in Figure 3. As shown in Figure 2, the damage occurring in the structural element will cause the decline of the frequencies, especially for the first- and fourth-order frequencies; the falling ranges are larger than others. And from the perspective of the vertical modal shapes, the element damage will cause the max nodal displacements of the first three-order modal shapes to enhance with different extents; meanwhile, the damage also causes some effects on the fourth modal shapes; however, the trend shows ascending first, then declines, and rises finally.Figure 2
The (a) 1st, (b) 2nd, (c) 3rd, and (d) 4th change curves of frequencies under different damage severities in the structural element and bearing.
(a)(b)(c)(d)Figure 3
The (a) 1st, (b) 2nd, (c) 3rd, and (d) 4th change curves of max nodal displacement of each vertical modal shape under different damage severities in the structural element and bearing.
(a)(b)(c)(d)


#### 2.2.2. Structural Bearing Damage

To select the structural bearing in the left end, adopting the way of stiffness reduction to simulate different damage severities at the bearing. Then, the dynamic characteristics of the beam with the corresponding bearing damage are obtained, which can be seen in Figures2 and 3. From Figure 2, it can be seen that the bearing damage has some negative effects on the natural frequencies, especially for the first two-order frequencies, and the variations are more obvious than others. In terms of the max nodal displacements of each modal shape, for the third and fourth modal shapes, there is a positive correlation between the bearing damage and the max nodal displacement. However, the trend shows a decrease first and then a rise in the first modal shape, and for the second, the trend is converse, namely, increasing at first and then falling.


#### 2.2.3. Environmental Temperature Variations

Assuming that the reference environmental temperature is 20°C, the temperature change ranges between 20 and 60°C, and the increasing step size is determined as 0.4°C. To calculate the corresponding natural frequencies and modal shapes, the results are shown in Figures4 and 5. As shown in Figure 4, the first two-order frequencies have shown a slight increase under the rising temperature, and when they achieve some specific values, they remain stable. However, the third and fourth frequencies have shown a sustained increase with the temperature variation. In addition, as shown in Figure 5, the rising temperature will lead to the decline of the max nodal displacements in the first and fourth modal shapes, reversely, an increase in the second modal shape. And the third modal shape will slightly go up and then fall steadily at a specific value.Figure 4
The (a) 1st, 2nd and (b) 3rd, 4th change curves of natural frequencies under different environmental temperatures.
(a)(b)Figure 5
The change curves of max nodal displacement of each vertical modal shape under different environmental temperatures.


### 2.3. The Summary of Evaluation

From the aforementioned analyses, one can conclude that the three factors will cause the fluctuation on the modal parameters to different extents, and the effects are uncertain which are difficult to propose a linear relationship. Meanwhile, although only the single factor rather than compound factors has been studied, the synergistic effects of multiple factors can be predicted to be more serious. As far as the practical situation is concerned, civil engineering structures are always surrounded by severe environmental conditions; this situation means the uncertain conditions are the main error origins of measured modal parameters; furthermore, they threaten the accuracy of the method of bearing damage detection based on modal parameters. Thereby, the uncertainties should be considered in the procedure of bearing damage detection.


## 2.1. Structural Dynamic Characteristic Equation considering Uncertain Conditions

In order to assess the impacts on the measured modal parameters caused by uncertain conditions, in this section, as shown in Figure1, the simply supported beam with spring support in each end has been introduced to conduct the research. The main function of bridge bearings is to transfer the load in the superstructure to the foundation; when the damage has occurred in this component, the stiffness of the bearing will be changed; thus, the bearings can be simplified and simulated as the simple spring models with the spring constants [21–25]. For the length of each element is 0.25 m, its properties of material, such as Young’s modulus, mass density, inertia moment, and cross-sectional area, are 3.0 × 1010 Pa, 2360 kg/m3, 8.64 × 10−5 m4, and 0.072 m2, respectively; the spring constants of each end are 106 N/m.Figure 1
The simply supported beam.Three factors have been investigated in this section, namely, structural element damage, boundary condition change (bearing damage), and environmental temperature variations. Assuming that the occurrence of damage does not cause the mass change, the structural element damage is simulated by the element stiffness reduction, which can be written as follows [26–29]:(1)Ks=∑i=1nele1−θikie,0≤θi≤1,where Ks means the global stiffness matrix of a structure; nele denotes the total number of structural elements; and kie and θi are i-th element stiffness matrix and i-th stiffness reduction factor, respectively.Meanwhile, the boundary condition change, namely, the damage of bearings, can be represented as the boundary condition matrix, and it can be defined as follows [13, 30]:(2)UxRxUyRyUzRzUxRxUyRyUzRzABCDEFGHSymIJKL,where the diagonal elements on the matrix represent the boundary stiffness in every degree of freedom (DOF); other elements can be described as the synergistic effects. The matrix can be utilized in the 2D finite element model by reducing the unconcerned DOFs. Thus, the damage in the structural bearings can also be described as the similar form as the structural element damage:(3)Kbc=∑j=1nbc1−αjkjbc,0≤αj≤1,where Kbc means the global stiffness matrix of a structural boundary system; nbc denotes the total number of bearings; and kjbc and αj are the j-th bearing stiffness matrix and i-th bearing stiffness reduction factor, respectively.In addition, the environmental temperature variations have also been a serious impact on a civil structure, which can cause the fluctuation of the measured modal parameters [31, 32]. In this paper, the temperature variation is transformed into the reduction of Young’s modulus, which can be calculated as follows [33]:(4)ET=E0⋅1−αE⋅T−T0,where ET and E0 represent Young’s modulus of the material at temperatures T and T0, respectively, and αE stands for the reduction coefficient under variational temperature, which can be determined as 0.3% for the concrete material [14].Thus, based on the aforementioned uncertain conditions and according to the structural dynamic theory and finite element method, the structural undamped dynamic characteristic equation considering uncertain conditions can be written as follows:(5)KsT,θ+Kbcα−λiMs+Mbcφi=0,where KsT,θ stands for the global stiffness matrix of structural elements with the environmental temperature of T and the structural damage severity of θ; Kbcα means the global stiffness matrix of boundary systems with the bearing damage severity of α; Ms and Mbc denote the global mass matrix of structural elements and boundary systems, respectively; and λi and φi are the i-th eigenvalue and eigenvector, respectively. Moreover, it needs to be pointed out that the effects of environmental temperature variations on the boundary systems have been ignored.


## 2.2. The Evaluation of the Factors

First of all, the factors have been studied to evaluate the influences on the modal parameters (e.g., natural frequencies and vertical modal shapes).


### 2.2.1. Structural Element Damage

To select one structural element randomly and introducing damage severity that ranges from 0 to 100%, the increasing step size is determined as 1% to calculate the corresponding natural frequencies and modal shapes. Then, the first four-order natural frequencies are extracted and illustrated in Figure2, and the max nodal displacement of each vertical modal shape is obtained and plotted in Figure 3. As shown in Figure 2, the damage occurring in the structural element will cause the decline of the frequencies, especially for the first- and fourth-order frequencies; the falling ranges are larger than others. And from the perspective of the vertical modal shapes, the element damage will cause the max nodal displacements of the first three-order modal shapes to enhance with different extents; meanwhile, the damage also causes some effects on the fourth modal shapes; however, the trend shows ascending first, then declines, and rises finally.Figure 2
The (a) 1st, (b) 2nd, (c) 3rd, and (d) 4th change curves of frequencies under different damage severities in the structural element and bearing.
(a)(b)(c)(d)Figure 3
The (a) 1st, (b) 2nd, (c) 3rd, and (d) 4th change curves of max nodal displacement of each vertical modal shape under different damage severities in the structural element and bearing.
(a)(b)(c)(d)


### 2.2.2. Structural Bearing Damage

To select the structural bearing in the left end, adopting the way of stiffness reduction to simulate different damage severities at the bearing. Then, the dynamic characteristics of the beam with the corresponding bearing damage are obtained, which can be seen in Figures2 and 3. From Figure 2, it can be seen that the bearing damage has some negative effects on the natural frequencies, especially for the first two-order frequencies, and the variations are more obvious than others. In terms of the max nodal displacements of each modal shape, for the third and fourth modal shapes, there is a positive correlation between the bearing damage and the max nodal displacement. However, the trend shows a decrease first and then a rise in the first modal shape, and for the second, the trend is converse, namely, increasing at first and then falling.


### 2.2.3. Environmental Temperature Variations

Assuming that the reference environmental temperature is 20°C, the temperature change ranges between 20 and 60°C, and the increasing step size is determined as 0.4°C. To calculate the corresponding natural frequencies and modal shapes, the results are shown in Figures4 and 5. As shown in Figure 4, the first two-order frequencies have shown a slight increase under the rising temperature, and when they achieve some specific values, they remain stable. However, the third and fourth frequencies have shown a sustained increase with the temperature variation. In addition, as shown in Figure 5, the rising temperature will lead to the decline of the max nodal displacements in the first and fourth modal shapes, reversely, an increase in the second modal shape. And the third modal shape will slightly go up and then fall steadily at a specific value.Figure 4
The (a) 1st, 2nd and (b) 3rd, 4th change curves of natural frequencies under different environmental temperatures.
(a)(b)Figure 5
The change curves of max nodal displacement of each vertical modal shape under different environmental temperatures.


## 2.2.1. Structural Element Damage

To select one structural element randomly and introducing damage severity that ranges from 0 to 100%, the increasing step size is determined as 1% to calculate the corresponding natural frequencies and modal shapes. Then, the first four-order natural frequencies are extracted and illustrated in Figure2, and the max nodal displacement of each vertical modal shape is obtained and plotted in Figure 3. As shown in Figure 2, the damage occurring in the structural element will cause the decline of the frequencies, especially for the first- and fourth-order frequencies; the falling ranges are larger than others. And from the perspective of the vertical modal shapes, the element damage will cause the max nodal displacements of the first three-order modal shapes to enhance with different extents; meanwhile, the damage also causes some effects on the fourth modal shapes; however, the trend shows ascending first, then declines, and rises finally.Figure 2
The (a) 1st, (b) 2nd, (c) 3rd, and (d) 4th change curves of frequencies under different damage severities in the structural element and bearing.
(a)(b)(c)(d)Figure 3
The (a) 1st, (b) 2nd, (c) 3rd, and (d) 4th change curves of max nodal displacement of each vertical modal shape under different damage severities in the structural element and bearing.
(a)(b)(c)(d)


## 2.2.2. Structural Bearing Damage

To select the structural bearing in the left end, adopting the way of stiffness reduction to simulate different damage severities at the bearing. Then, the dynamic characteristics of the beam with the corresponding bearing damage are obtained, which can be seen in Figures2 and 3. From Figure 2, it can be seen that the bearing damage has some negative effects on the natural frequencies, especially for the first two-order frequencies, and the variations are more obvious than others. In terms of the max nodal displacements of each modal shape, for the third and fourth modal shapes, there is a positive correlation between the bearing damage and the max nodal displacement. However, the trend shows a decrease first and then a rise in the first modal shape, and for the second, the trend is converse, namely, increasing at first and then falling.


## 2.2.3. Environmental Temperature Variations

Assuming that the reference environmental temperature is 20°C, the temperature change ranges between 20 and 60°C, and the increasing step size is determined as 0.4°C. To calculate the corresponding natural frequencies and modal shapes, the results are shown in Figures4 and 5. As shown in Figure 4, the first two-order frequencies have shown a slight increase under the rising temperature, and when they achieve some specific values, they remain stable. However, the third and fourth frequencies have shown a sustained increase with the temperature variation. In addition, as shown in Figure 5, the rising temperature will lead to the decline of the max nodal displacements in the first and fourth modal shapes, reversely, an increase in the second modal shape. And the third modal shape will slightly go up and then fall steadily at a specific value.Figure 4
The (a) 1st, 2nd and (b) 3rd, 4th change curves of natural frequencies under different environmental temperatures.
(a)(b)Figure 5
The change curves of max nodal displacement of each vertical modal shape under different environmental temperatures.


## 2.3. The Summary of Evaluation

From the aforementioned analyses, one can conclude that the three factors will cause the fluctuation on the modal parameters to different extents, and the effects are uncertain which are difficult to propose a linear relationship. Meanwhile, although only the single factor rather than compound factors has been studied, the synergistic effects of multiple factors can be predicted to be more serious. As far as the practical situation is concerned, civil engineering structures are always surrounded by severe environmental conditions; this situation means the uncertain conditions are the main error origins of measured modal parameters; furthermore, they threaten the accuracy of the method of bearing damage detection based on modal parameters. Thereby, the uncertainties should be considered in the procedure of bearing damage detection.


## 3. Bayesian Theory for Damage Detection

### 3.1. The Basic Theory of Bayesian Damage Detection

Bayesian framework is a useful tool to consider the uncertainty from the probabilistic view. In the framework, the posterior probability distribution of the concerned variable can be expressed as follows:(6)Pβ|D,M˜=Pβ|M˜PD|β,M˜PD|M˜,where M˜ stands for the parameterized model; D represents the measured data, e.g., natural frequencies and modal shapes; β means the concerned variable; PD|β,M˜ and Pβ|M˜ are the likelihood function and the prior probability density function (PDF), respectively; and PD|M˜ is a normalizing constant that is unrelated to the concerned variable.Because the selected class of the model has an essential function on the success of model updating, furthermore, on the damage detection [34], assuming that the mass remains constant when the structure is damaged, the linear model considering multiple factors is adopted, which can be written as follows:(7)K=KsT,θ+Kbcα,(8)KsT,θ=∑i=1nele1−θikiT,0≤θi≤1,(9)Kbcα=∑j=1nbc1−αjkjbc,0≤αj≤1,where kiT means the element stiffness matrix corresponding to the temperature of T.To consider the measurement and modelling errors and assuming that both errors follow the Gaussian distribution, the errors of each modal parameter can be evaluated as follows [35]:(10)εf=fi−fiβ∼N0,covfi−1,(11)εφ=φi−φiβ∼N0,covφi−1,where fiβ and φiβ are the i-th analytical natural frequency and modal shape, respectively; fi and φi are the i-th experimental natural frequency and modal shape; β means the concerned variable, and it can be defined as β=θ,α to detect the damage of structural elements and bearings, or to be set as a single variable, such as β=θ for the damage of structural elements or β=α for the damage of structural bearings; and covfi−1 and covφi−1 can be used to measure the uncertainty levels. Based on equations (10) and (11), the likelihood function of measured modal parameters can be deduced as follows:(12)Pfi|β=covfi2π12exp−fi−fiβ22covfi,(13)Pφi|β=covφi2π12exp−φi−φiβTφi−φβ2covφi.Due to the independence of each modal test and assuming each order modal parameter is independent, there is no correlation between the modes. Thus, according to the firstnm-order measured natural frequencies and modal shapes, the final form of the likelihood functions of β can be obtained as follows:(14)Pf|β=∏i=1nmPfi|β=∏i=1nmcovfi2π12exp−12∑i=1nmfi−fiβ2covfi,(15)Pφ|β=∏i=1nmPφi|β=∏i=1nmcovφi2π12exp−12∑i=1nmφi−φiβTφi−φβcovφi.Then, to submit equations (14) and (15) into equation (6), the posterior probability density function of β can be expressed as follows:(16)Pβ|D,M˜=c−1Pf|βPφ|β=c−1∏i=1nmcovfi2π1/2∏i=1nmcovφi2π1/2exp−12∑i=1nmfi−fiβ2covfi+φi−φiβTφi−φβcovφi,where c−1 means a constant number. According to the maximum a posteriori (MAP) principle, optimal β can be obtained by maximizing the posterior probability density function, namely, minimizing the exponential part of equation (16). However, due to the high dimensions of β, it is difficult to calculate; in order to solve this problem, the Monte Carlo Markov Chain (MCMC) method is adopted in this article.


### 3.2. The Metropolis–Hastings Sampling Method

The Metropolis–Hastings (HM) sampling method, as a popular way to obtain Markov chain, has been widely used in the MCMC simulation [36]. The main steps of the HM sampling method can be summarized as follows:(1)
To generate the initial samplext=0 randomly, then t = t+1, and generate candidate sample xt according to the proposal distribution γ.(2)
To calculate the ratio of two PDFsη based on the equation as follows:(17)η=Pxt−1γxt|xt−1Pxtγxt−1|xt=Pxt−1Pxt.(3)
To judge the candidate sample referring to the Metropolis principle, namely, ifη≥1, then xt=xt,t=t+1, to generate the candidate sample again. When η<1, to generate a random number μ, then to judge whether η is greater than or equal to μ, if yes, the candidate sample sample is accepted; but, if η<μ, the candidate sample is rejected, xt=xt−1,t=t+1. After the judgment, generate the candidate sample again.(4)
Whent⟶+∞, the Markov chain has been formed.


### 3.3. Matrix Perturbation Method

Due to the iterative calculation of Bayesian damage detection, it is noticeable that the computational cost of the calculation of analytical modal parameters in the updating process is very expensive. Meanwhile, the accurate finite element model of the structure is very demanding in every iteration. These requirements make the detection process face a series of challenges. Thereby, researchers proposed the surrogate model to replace the complicated finite element model, which can provide good computation efficiency in the detection process [20].In this paper, the matrix perturbation method is introduced, according to equation (5), and to consider the variations in the structural global stiffness and mass matrix, the 1st-order perturbation formulae for frequencies and modal shapes can be given directly as follows:(18)λip1=φi0TK1−λi0M1φi0,φip1=∑k=1,≠inmφk0TK1−λi0M1φi0λi0−λk0φk0−12φi0TM1φi0φi0.And the 2nd-order perturbation formulae can be expressed as follows:(19)λip2=φi0TK1φip1−λi0M1φip1−λip1M0φip1−λip1M1φi0,φip2=∑k=1,≠inmφk0TK1φip1−λi0M1φip1−λi1M0φip1−λi1M1φi0λi0−λk0φk0,−12φk0TM1φip1+φip1TM0φip1+φip1TM1φi0φi0,where λip1 and λip2 represent the 1st- and 2nd-order perturbation values of the i-th eigenvalue, respectively; φip1 and φip2 stand for the 1st- and 2nd-order perturbation values of the i-th eigenvector, respectively; λi0 and φi0 denote the i-th original eigenvalue and eigenvector, respectively; K1 and M1 mean the 1st-order perturbation values of the stiffness and mass matrices, respectively; and M0 is the original structural mass matrix. Therefore, the i-th eigenvalue and eigenvector after the perturbation can be calculated as follows:(20)λ^i=λi0+ελip1+ε2λip2+Οε3,φ^i=φi0+εφip1+ε2φip2+Οε3,where ε means the perturbation coefficient. The perturbation method only needs the modal parameters of the intact structure to calculate the modal parameters after the physical properties change, which can avoid the repetitive finite element modelling and cut down the computational cost.


### 3.4. Damage Identification Method

The main steps of the proposed damage identification method can be drawn as follows:(1)
Firstly, to construct the finite element model of the structure considering the environmental temperature variations and structural element and bearing damage, as well as random noise, and then its stiffness and mass matrices, eigenvalues, and eigenvectors are obtained(2)
Using stiffness and mass matrices, eigenvalues, and eigenvectors to establish the perturbation model of the structure(3)
Taking the perturbation model and Bayesian framework to detect the damage of structural elements and bearingsThe flowchart is shown in Figure6.Figure 6
The flowchart of the damage identification method.


## 3.1. The Basic Theory of Bayesian Damage Detection

Bayesian framework is a useful tool to consider the uncertainty from the probabilistic view. In the framework, the posterior probability distribution of the concerned variable can be expressed as follows:(6)Pβ|D,M˜=Pβ|M˜PD|β,M˜PD|M˜,where M˜ stands for the parameterized model; D represents the measured data, e.g., natural frequencies and modal shapes; β means the concerned variable; PD|β,M˜ and Pβ|M˜ are the likelihood function and the prior probability density function (PDF), respectively; and PD|M˜ is a normalizing constant that is unrelated to the concerned variable.Because the selected class of the model has an essential function on the success of model updating, furthermore, on the damage detection [34], assuming that the mass remains constant when the structure is damaged, the linear model considering multiple factors is adopted, which can be written as follows:(7)K=KsT,θ+Kbcα,(8)KsT,θ=∑i=1nele1−θikiT,0≤θi≤1,(9)Kbcα=∑j=1nbc1−αjkjbc,0≤αj≤1,where kiT means the element stiffness matrix corresponding to the temperature of T.To consider the measurement and modelling errors and assuming that both errors follow the Gaussian distribution, the errors of each modal parameter can be evaluated as follows [35]:(10)εf=fi−fiβ∼N0,covfi−1,(11)εφ=φi−φiβ∼N0,covφi−1,where fiβ and φiβ are the i-th analytical natural frequency and modal shape, respectively; fi and φi are the i-th experimental natural frequency and modal shape; β means the concerned variable, and it can be defined as β=θ,α to detect the damage of structural elements and bearings, or to be set as a single variable, such as β=θ for the damage of structural elements or β=α for the damage of structural bearings; and covfi−1 and covφi−1 can be used to measure the uncertainty levels. Based on equations (10) and (11), the likelihood function of measured modal parameters can be deduced as follows:(12)Pfi|β=covfi2π12exp−fi−fiβ22covfi,(13)Pφi|β=covφi2π12exp−φi−φiβTφi−φβ2covφi.Due to the independence of each modal test and assuming each order modal parameter is independent, there is no correlation between the modes. Thus, according to the firstnm-order measured natural frequencies and modal shapes, the final form of the likelihood functions of β can be obtained as follows:(14)Pf|β=∏i=1nmPfi|β=∏i=1nmcovfi2π12exp−12∑i=1nmfi−fiβ2covfi,(15)Pφ|β=∏i=1nmPφi|β=∏i=1nmcovφi2π12exp−12∑i=1nmφi−φiβTφi−φβcovφi.Then, to submit equations (14) and (15) into equation (6), the posterior probability density function of β can be expressed as follows:(16)Pβ|D,M˜=c−1Pf|βPφ|β=c−1∏i=1nmcovfi2π1/2∏i=1nmcovφi2π1/2exp−12∑i=1nmfi−fiβ2covfi+φi−φiβTφi−φβcovφi,where c−1 means a constant number. According to the maximum a posteriori (MAP) principle, optimal β can be obtained by maximizing the posterior probability density function, namely, minimizing the exponential part of equation (16). However, due to the high dimensions of β, it is difficult to calculate; in order to solve this problem, the Monte Carlo Markov Chain (MCMC) method is adopted in this article.


## 3.2. The Metropolis–Hastings Sampling Method

The Metropolis–Hastings (HM) sampling method, as a popular way to obtain Markov chain, has been widely used in the MCMC simulation [36]. The main steps of the HM sampling method can be summarized as follows:(1)
To generate the initial samplext=0 randomly, then t = t+1, and generate candidate sample xt according to the proposal distribution γ.(2)
To calculate the ratio of two PDFsη based on the equation as follows:(17)η=Pxt−1γxt|xt−1Pxtγxt−1|xt=Pxt−1Pxt.(3)
To judge the candidate sample referring to the Metropolis principle, namely, ifη≥1, then xt=xt,t=t+1, to generate the candidate sample again. When η<1, to generate a random number μ, then to judge whether η is greater than or equal to μ, if yes, the candidate sample sample is accepted; but, if η<μ, the candidate sample is rejected, xt=xt−1,t=t+1. After the judgment, generate the candidate sample again.(4)
Whent⟶+∞, the Markov chain has been formed.


## 3.3. Matrix Perturbation Method

Due to the iterative calculation of Bayesian damage detection, it is noticeable that the computational cost of the calculation of analytical modal parameters in the updating process is very expensive. Meanwhile, the accurate finite element model of the structure is very demanding in every iteration. These requirements make the detection process face a series of challenges. Thereby, researchers proposed the surrogate model to replace the complicated finite element model, which can provide good computation efficiency in the detection process [20].In this paper, the matrix perturbation method is introduced, according to equation (5), and to consider the variations in the structural global stiffness and mass matrix, the 1st-order perturbation formulae for frequencies and modal shapes can be given directly as follows:(18)λip1=φi0TK1−λi0M1φi0,φip1=∑k=1,≠inmφk0TK1−λi0M1φi0λi0−λk0φk0−12φi0TM1φi0φi0.And the 2nd-order perturbation formulae can be expressed as follows:(19)λip2=φi0TK1φip1−λi0M1φip1−λip1M0φip1−λip1M1φi0,φip2=∑k=1,≠inmφk0TK1φip1−λi0M1φip1−λi1M0φip1−λi1M1φi0λi0−λk0φk0,−12φk0TM1φip1+φip1TM0φip1+φip1TM1φi0φi0,where λip1 and λip2 represent the 1st- and 2nd-order perturbation values of the i-th eigenvalue, respectively; φip1 and φip2 stand for the 1st- and 2nd-order perturbation values of the i-th eigenvector, respectively; λi0 and φi0 denote the i-th original eigenvalue and eigenvector, respectively; K1 and M1 mean the 1st-order perturbation values of the stiffness and mass matrices, respectively; and M0 is the original structural mass matrix. Therefore, the i-th eigenvalue and eigenvector after the perturbation can be calculated as follows:(20)λ^i=λi0+ελip1+ε2λip2+Οε3,φ^i=φi0+εφip1+ε2φip2+Οε3,where ε means the perturbation coefficient. The perturbation method only needs the modal parameters of the intact structure to calculate the modal parameters after the physical properties change, which can avoid the repetitive finite element modelling and cut down the computational cost.


## 3.4. Damage Identification Method

The main steps of the proposed damage identification method can be drawn as follows:(1)
Firstly, to construct the finite element model of the structure considering the environmental temperature variations and structural element and bearing damage, as well as random noise, and then its stiffness and mass matrices, eigenvalues, and eigenvectors are obtained(2)
Using stiffness and mass matrices, eigenvalues, and eigenvectors to establish the perturbation model of the structure(3)
Taking the perturbation model and Bayesian framework to detect the damage of structural elements and bearingsThe flowchart is shown in Figure6.Figure 6
The flowchart of the damage identification method.


## 4. Example Studies

### 4.1. Numerical Example

#### 4.1.1. Example Introduction and Damage Cases

In this section, as shown in Figure7, an 8-span concrete continuous beam structure with 9 vertical bearings and 48 elements has been used to verify the proposed bearing damage detection method. The beam structure is with Young’s modulus of 3.0 × 1010 Pa, the mass density of 2360 kg/m3, Poisson’s ratio of 0.15, the inertia moment of 8.64 × 10−5 m4, and the cross-sectional area of 0.072 m2; and for each element, its length is 0.25 m. In addition, the structural element is modelled by the 6-DOF beam element, and the bearing is simulated by the spring element of 6.0 × 106 N/m.Figure 7
The 8-span continuous beam structure.There are seven damage cases that are preset to validate the feasibility of the proposed damage detection method, which are listed in Table1. In the practical situation, sometimes, the damage occurs in the elements and bearings simultaneously; thus, the preset cases can not only study the single type of damage, e.g., structural element or bearing, but also consider the multiple types of damage. As listed in Table 1, case 1 stands for the situations of element damage that has occurred in the two ends; cases 2–4 are utilized to consider the damage emerged in elements and bearings at the same spatial location simultaneously; and cases 5–7 mean the damage of the element and bearing rising in different spatial locations, respectively. Moreover, the environmental factors such as temperature variation and random noise effects are both considered. The temperature variation can be simulated according to equation (4), and the random noise can be simulated as follows [37]:(21)fik,φik=fi,φi⋅1+σrand,where fi,φi and fik,φik mean the i-th noise-free and contaminated natural frequencies and modal shapes, respectively; σ stands for the noise pollution level [38–40]; and rand is a random number that ranges from −1 to 1.Table 1
The damage cases of the continuous beam structure.
Noise level (%)Temperature variation (reference temperature is 0 (°C))CaseDamage severity @ element numberDamage severity @ bearing numberDetection goal320150% @ 1/E30% @ 48250% @ 130% @ 1#E3E4B and E550% @ 2630% @ 1#B6E7B and ENote: E and B denote the element damage and bearing damage, respectively.


#### 4.1.2. Damage Identification and Results’ Discussion

Aiming to every damage case, the modal test number is 50, and the first five-order natural frequencies and modal shapes are adopted; and the iterative number is determined as 5000 for case 1 and 10,000 for cases 2–7. Meanwhile, another method in the paper [13] is also adopted to make a comparison, namely, the identification method using optimization algorithm and sensitivity analysis (OASA), which can make a comparison with the proposed method. As far as OASA is concerned, herein, some parameters of CMF-IWO algorithm except for the iterative number and population size are the same as in [13]. The iterative number is defined as 500 for case 1 and 1000 for cases 2–7, and the population size is defined as 100 for each damage case; for every damage case, to run the procedure for 7 times, the average identified results are extracted. The Markov chains and histograms of the proposed method are extracted and shown in Figures 8–14, and the damage detection results of two methods are illustrated in Table 2.Figure 8
The Markov chains and histograms of damage case 1.Figure 9
The Markov chains and histograms of damage case 2.Figure 10
The Markov chains and histograms of damage case 3.Figure 11
The Markov chains and histograms of damage case 4.Figure 12
The Markov chains and histograms of damage case 5.Figure 13
The Markov chains and histograms of damage case 6.Figure 14
The Markov chains and histograms of damage case 7.Table 2
The damage identification results of the continuous beam structure.
CasePreset damage severityIdentified damage severity (Bayesian method)Identified damage severity (OASA method)150% @ 146.96% @ 133.53% @ 130% @ 4827.54% @ 4846.51% @ 48230% @ 1#36.46% @ 1#29.83% @ 1#350% @ 148.74% @ 149.96% @ 1430% @ 1#27.80% @ 1#26.95% @ 1#50% @ 150.68% @ 139.62% @ 1530% @ 1#31.45% @ 1#29.61% @ 1#650% @ 2646.68% @ 2649.91% @ 26730% @ 1#28.01% @ 1#39.91% @ 1#50% @ 2654.44% @ 2664.93% @ 26As shown in Figure8, the proposed method shows a good ability to detect the damage of structural elements, the max error is 3.04%, and two Markov chains of case 1 always remain nonstationary; there is no smooth segment in them; and the histograms illustrate that the mean values of element and bearing damage parameters obey the preset values.And in Figures9 and 10, although there is another damage situation that has some negative effects on the detection process, the bearing and element damage can be measured independently. And the max errors are 6.46% that emerge in case 2. Considering the practical situation of bridge engineering, the finding means that the proposed method is possible to apply in the practical application of detecting the bearings or structural elements individually.Meanwhile, Figure11 shows that the method can detect the damage of the bearing and structural element simultaneously. However, the Markov chain of the structural element indicates some fluctuations, especially in the early stage. This situation represents that researchers should pay more attention to study the scenario similar to case 4 in the future.In terms of Figures12 and 13, the same conclusion as Figure 8 can be drawn. And the max error that occurred in case 6 is 3.32%.From the perspective of Figure14, the Markov chain of bearing damage parameter is very stable, but another one fluctuates intensely, which can be owed to the selection of the variance of assuming proposal distribution. Thus, how to determine a suitable variance is a key factor to identify the damage of the element accurately when bearing and element damage exist simultaneously.Furthermore, from Table2, although the OASA method can accurately tackle the single structural element damage cases or single bearing damage cases, e.g., cases 2-3 and 5-6, however, regarding the multiple-damage case, namely, structural element and bearing damage, compared to the proposed method, the identified results of the OASA method are worse. For case 1, the detected damage extents of the structural element and bearing have been exchanged, which is owed to the negative effect of uncertainties or the poor global optimization ability of the optimization algorithm. And for cases 4 and 7, the identified damage severities are not consistent with the preset values.To summarize, from the aforementioned analyses and discussions, one can conclude that based on the proposed method, it has good feasibility to detect the structural bearing damage and structural element damage independently and even the two types of damage simultaneously. At the same time, the proposed method can overcome the distraction of uncertainties such as temperature variations and random noise, which shows good potential in the practical application.


### 4.2. Engineering Example

#### 4.2.1. The Introduction of the Example

In order to further verify the practicability of the proposed method, in this section, a practical engineering of I-40 steel-concrete bridge has been exploited. This bridge is sited in Albuquerque, New Mexico, USA, and the related studies have become the classical cases in the field of structural health monitoring. A vibration test was carried out to collect the dynamic characteristics [41], and the obtained data have been widely used to verify the diverse damage detection approach. Figure 15 illustrates the overall view and experimental setup of the bridge.Figure 15
I-40 bridge: (a) overall layout; (b) experimental setup.
(a)(b)In the middle span of the north plate girder of the bridge, there are four damage cases which have been introduced, which can be seen in Figure16. After the introduction, the modal test was conducted for every damage case, and for D-1 to D-4, the corresponding ambient temperatures and approximate damage severities were recorded, which are (15.5°C, 5%), (28.9°C, 10%), (26.1°C, 32%), and (20.0°C, 92%), respectively.Figure 16
Damage cases of the I-40 bridge.Referring to the report of the modal experiment and collected modal characteristics, the finite element model has been programmed using MATLAB [22, 32, 42], which is described in Figure 17. In this model, the concrete deck and the web of the plate girder are both simulated by the 4-node shell element, and other components, such as the flanges of the girder, stringers, floor beams, and concrete piers, are modelled based on the 2-node beam element. And the bridge bearings are tackled with a spring element; for each direction, the spring constants are kx = 3.17 × 106 N/m, ky = 1.26 × 106 N/m, and kz = 4.29 × 107 N/m, respectively. In addition, the detailed material properties can be referred to the paper [42]. After obtaining the finite element model, the natural frequencies and modal assurance criterion (MAC) are utilized to measure the correlation, which are listed in Table 3.Figure 17
The finite element model of the I-40 bridge.Table 3
Analytical and measured natural frequencies and MAC.
ModeMeasured (Hz)Analytical (Hz)Error (%)MAC12.48282.48210.030.994922.95933.00161.410.984133.49913.41762.380.990544.07914.03651.060.970854.16684.03693.220.971864.63104.65610.540.9709From the data listed in Table3, it can be seen that the max error between analytical natural frequencies and measured ones is merely 3.22%; meanwhile, the first six MAC values are all larger than 0.97. The comparison indicates that the model shows a good reality to reflect the dynamic characteristics of the actual engineering structure, which can be adopted to implement the subsequent damage detection procedure.


#### 4.2.2. Damage Identification and Results’ Discussion

Based on the aforementioned actual damage cases and the finite element model, there are several simulated bearing damage cases that have been introduced to combine the actual structural element damage cases to verify the proposed bearing damage detection method. For each damage case, the first five natural frequencies and modal shapes are adopted to detect the damage of the structural element and bearing. The iterative number is defined as 20,000, and the initial value is 1 for every damage case. The damage cases and corresponding results can be seen in Table4.Table 4
Damage cases and identification results of the I-40 bridge.
Ambient temperature (°C)CaseDamage severity @ element numberDamage severity @ bearing numberDetection goalIdentification results15.5D-1(a)5% @ 12/E5.12% @ 12D-1(b)5% @ 1230% @ 1#B29.94% @ 1#D-1(c)5% @ 1230% @ 1#E and B5.25% @ 1231.89% @ 1#D-1(d)5% @ 1230% @ 1#50% @ 2#E and B5.21% @ 1230.10% @ 1#49.48% @ 2#28.9D-2(a)10% @ 12/E10.1% @ 12D-2(b)10% @ 1230% @ 1#B29.99% @ 1#D-2(c)10% @ 1230% @ 1#E and B10.09% @ 1229.99% @ 1#D-2(d)10% @ 1230% @ 1#50% @ 2#E and B9.85% @ 1230.61% @ 1#49.52% @ 2#26.1D-3(a)32% @ 12/E32.28% @ 12D-3(b)32% @ 1230% @ 1#B30.05% @ 1#D-3(c)32% @ 1230% @ 1#E and B31.04% @ 1226.80% @ 1#D-3(d)32% @ 1230% @ 1#50% @ 2#E and B31.56% @ 1230.95%@ 1#49.51% @ 2#20.0D-4(a)92% @ 12/E91.77% @ 12D-4(b)92% @ 1230% @ 1#B30.07% @ 1#D-4(c)92% @ 1230% @ 1#E and B92.88% @ 1226.93% @ 1#D-4(d)92% @ 1230% @ 1#50% @ 2#E and B92.20% @ 1221.38% @ 1#54.84% @ 2#Note: E and B denote the element damage and bearing damage, respectively.As shown in Table4, the proposed method can not only identify the single-type damage situation (e.g., only element damage or bearing damage) but also assess the multiple-type damage scenario accurately. In case D-1, the max identified error of the structural element is 0.21%, and for the bearing, it is 1.89%; for case D-2, the max errors have emerged in D-2(d), which are 0.15% and 0.61% for the element and bearing, respectively. And for cases D-3 and D-4, the accuracies of damage detection have declined to some extent, namely, the max errors are 3.20% (D-3(c), element damage) and 4.84% (D-3(d), bearing damage), respectively.In conclusion, although there are some negative influences of ambient temperature variations and environmental noise, the damage detection results are of great accuracy, which indicates the proposed method has good robustness. The uncertainties in the damage detection process can be overcome using the Bayesian framework; meanwhile, the damage of the structural element and bearing can be detected simultaneously, which is very useful in the practical application. The proposed damage detection framework shows good potential for the bearing damage detection.


## 4.1. Numerical Example

### 4.1.1. Example Introduction and Damage Cases

In this section, as shown in Figure7, an 8-span concrete continuous beam structure with 9 vertical bearings and 48 elements has been used to verify the proposed bearing damage detection method. The beam structure is with Young’s modulus of 3.0 × 1010 Pa, the mass density of 2360 kg/m3, Poisson’s ratio of 0.15, the inertia moment of 8.64 × 10−5 m4, and the cross-sectional area of 0.072 m2; and for each element, its length is 0.25 m. In addition, the structural element is modelled by the 6-DOF beam element, and the bearing is simulated by the spring element of 6.0 × 106 N/m.Figure 7
The 8-span continuous beam structure.There are seven damage cases that are preset to validate the feasibility of the proposed damage detection method, which are listed in Table1. In the practical situation, sometimes, the damage occurs in the elements and bearings simultaneously; thus, the preset cases can not only study the single type of damage, e.g., structural element or bearing, but also consider the multiple types of damage. As listed in Table 1, case 1 stands for the situations of element damage that has occurred in the two ends; cases 2–4 are utilized to consider the damage emerged in elements and bearings at the same spatial location simultaneously; and cases 5–7 mean the damage of the element and bearing rising in different spatial locations, respectively. Moreover, the environmental factors such as temperature variation and random noise effects are both considered. The temperature variation can be simulated according to equation (4), and the random noise can be simulated as follows [37]:(21)fik,φik=fi,φi⋅1+σrand,where fi,φi and fik,φik mean the i-th noise-free and contaminated natural frequencies and modal shapes, respectively; σ stands for the noise pollution level [38–40]; and rand is a random number that ranges from −1 to 1.Table 1
The damage cases of the continuous beam structure.
Noise level (%)Temperature variation (reference temperature is 0 (°C))CaseDamage severity @ element numberDamage severity @ bearing numberDetection goal320150% @ 1/E30% @ 48250% @ 130% @ 1#E3E4B and E550% @ 2630% @ 1#B6E7B and ENote: E and B denote the element damage and bearing damage, respectively.


### 4.1.2. Damage Identification and Results’ Discussion

Aiming to every damage case, the modal test number is 50, and the first five-order natural frequencies and modal shapes are adopted; and the iterative number is determined as 5000 for case 1 and 10,000 for cases 2–7. Meanwhile, another method in the paper [13] is also adopted to make a comparison, namely, the identification method using optimization algorithm and sensitivity analysis (OASA), which can make a comparison with the proposed method. As far as OASA is concerned, herein, some parameters of CMF-IWO algorithm except for the iterative number and population size are the same as in [13]. The iterative number is defined as 500 for case 1 and 1000 for cases 2–7, and the population size is defined as 100 for each damage case; for every damage case, to run the procedure for 7 times, the average identified results are extracted. The Markov chains and histograms of the proposed method are extracted and shown in Figures 8–14, and the damage detection results of two methods are illustrated in Table 2.Figure 8
The Markov chains and histograms of damage case 1.Figure 9
The Markov chains and histograms of damage case 2.Figure 10
The Markov chains and histograms of damage case 3.Figure 11
The Markov chains and histograms of damage case 4.Figure 12
The Markov chains and histograms of damage case 5.Figure 13
The Markov chains and histograms of damage case 6.Figure 14
The Markov chains and histograms of damage case 7.Table 2
The damage identification results of the continuous beam structure.
CasePreset damage severityIdentified damage severity (Bayesian method)Identified damage severity (OASA method)150% @ 146.96% @ 133.53% @ 130% @ 4827.54% @ 4846.51% @ 48230% @ 1#36.46% @ 1#29.83% @ 1#350% @ 148.74% @ 149.96% @ 1430% @ 1#27.80% @ 1#26.95% @ 1#50% @ 150.68% @ 139.62% @ 1530% @ 1#31.45% @ 1#29.61% @ 1#650% @ 2646.68% @ 2649.91% @ 26730% @ 1#28.01% @ 1#39.91% @ 1#50% @ 2654.44% @ 2664.93% @ 26As shown in Figure8, the proposed method shows a good ability to detect the damage of structural elements, the max error is 3.04%, and two Markov chains of case 1 always remain nonstationary; there is no smooth segment in them; and the histograms illustrate that the mean values of element and bearing damage parameters obey the preset values.And in Figures9 and 10, although there is another damage situation that has some negative effects on the detection process, the bearing and element damage can be measured independently. And the max errors are 6.46% that emerge in case 2. Considering the practical situation of bridge engineering, the finding means that the proposed method is possible to apply in the practical application of detecting the bearings or structural elements individually.Meanwhile, Figure11 shows that the method can detect the damage of the bearing and structural element simultaneously. However, the Markov chain of the structural element indicates some fluctuations, especially in the early stage. This situation represents that researchers should pay more attention to study the scenario similar to case 4 in the future.In terms of Figures12 and 13, the same conclusion as Figure 8 can be drawn. And the max error that occurred in case 6 is 3.32%.From the perspective of Figure14, the Markov chain of bearing damage parameter is very stable, but another one fluctuates intensely, which can be owed to the selection of the variance of assuming proposal distribution. Thus, how to determine a suitable variance is a key factor to identify the damage of the element accurately when bearing and element damage exist simultaneously.Furthermore, from Table2, although the OASA method can accurately tackle the single structural element damage cases or single bearing damage cases, e.g., cases 2-3 and 5-6, however, regarding the multiple-damage case, namely, structural element and bearing damage, compared to the proposed method, the identified results of the OASA method are worse. For case 1, the detected damage extents of the structural element and bearing have been exchanged, which is owed to the negative effect of uncertainties or the poor global optimization ability of the optimization algorithm. And for cases 4 and 7, the identified damage severities are not consistent with the preset values.To summarize, from the aforementioned analyses and discussions, one can conclude that based on the proposed method, it has good feasibility to detect the structural bearing damage and structural element damage independently and even the two types of damage simultaneously. At the same time, the proposed method can overcome the distraction of uncertainties such as temperature variations and random noise, which shows good potential in the practical application.


## 4.1.1. Example Introduction and Damage Cases

In this section, as shown in Figure7, an 8-span concrete continuous beam structure with 9 vertical bearings and 48 elements has been used to verify the proposed bearing damage detection method. The beam structure is with Young’s modulus of 3.0 × 1010 Pa, the mass density of 2360 kg/m3, Poisson’s ratio of 0.15, the inertia moment of 8.64 × 10−5 m4, and the cross-sectional area of 0.072 m2; and for each element, its length is 0.25 m. In addition, the structural element is modelled by the 6-DOF beam element, and the bearing is simulated by the spring element of 6.0 × 106 N/m.Figure 7
The 8-span continuous beam structure.There are seven damage cases that are preset to validate the feasibility of the proposed damage detection method, which are listed in Table1. In the practical situation, sometimes, the damage occurs in the elements and bearings simultaneously; thus, the preset cases can not only study the single type of damage, e.g., structural element or bearing, but also consider the multiple types of damage. As listed in Table 1, case 1 stands for the situations of element damage that has occurred in the two ends; cases 2–4 are utilized to consider the damage emerged in elements and bearings at the same spatial location simultaneously; and cases 5–7 mean the damage of the element and bearing rising in different spatial locations, respectively. Moreover, the environmental factors such as temperature variation and random noise effects are both considered. The temperature variation can be simulated according to equation (4), and the random noise can be simulated as follows [37]:(21)fik,φik=fi,φi⋅1+σrand,where fi,φi and fik,φik mean the i-th noise-free and contaminated natural frequencies and modal shapes, respectively; σ stands for the noise pollution level [38–40]; and rand is a random number that ranges from −1 to 1.Table 1
The damage cases of the continuous beam structure.
Noise level (%)Temperature variation (reference temperature is 0 (°C))CaseDamage severity @ element numberDamage severity @ bearing numberDetection goal320150% @ 1/E30% @ 48250% @ 130% @ 1#E3E4B and E550% @ 2630% @ 1#B6E7B and ENote: E and B denote the element damage and bearing damage, respectively.


## 4.1.2. Damage Identification and Results’ Discussion

Aiming to every damage case, the modal test number is 50, and the first five-order natural frequencies and modal shapes are adopted; and the iterative number is determined as 5000 for case 1 and 10,000 for cases 2–7. Meanwhile, another method in the paper [13] is also adopted to make a comparison, namely, the identification method using optimization algorithm and sensitivity analysis (OASA), which can make a comparison with the proposed method. As far as OASA is concerned, herein, some parameters of CMF-IWO algorithm except for the iterative number and population size are the same as in [13]. The iterative number is defined as 500 for case 1 and 1000 for cases 2–7, and the population size is defined as 100 for each damage case; for every damage case, to run the procedure for 7 times, the average identified results are extracted. The Markov chains and histograms of the proposed method are extracted and shown in Figures 8–14, and the damage detection results of two methods are illustrated in Table 2.Figure 8
The Markov chains and histograms of damage case 1.Figure 9
The Markov chains and histograms of damage case 2.Figure 10
The Markov chains and histograms of damage case 3.Figure 11
The Markov chains and histograms of damage case 4.Figure 12
The Markov chains and histograms of damage case 5.Figure 13
The Markov chains and histograms of damage case 6.Figure 14
The Markov chains and histograms of damage case 7.Table 2
The damage identification results of the continuous beam structure.
CasePreset damage severityIdentified damage severity (Bayesian method)Identified damage severity (OASA method)150% @ 146.96% @ 133.53% @ 130% @ 4827.54% @ 4846.51% @ 48230% @ 1#36.46% @ 1#29.83% @ 1#350% @ 148.74% @ 149.96% @ 1430% @ 1#27.80% @ 1#26.95% @ 1#50% @ 150.68% @ 139.62% @ 1530% @ 1#31.45% @ 1#29.61% @ 1#650% @ 2646.68% @ 2649.91% @ 26730% @ 1#28.01% @ 1#39.91% @ 1#50% @ 2654.44% @ 2664.93% @ 26As shown in Figure8, the proposed method shows a good ability to detect the damage of structural elements, the max error is 3.04%, and two Markov chains of case 1 always remain nonstationary; there is no smooth segment in them; and the histograms illustrate that the mean values of element and bearing damage parameters obey the preset values.And in Figures9 and 10, although there is another damage situation that has some negative effects on the detection process, the bearing and element damage can be measured independently. And the max errors are 6.46% that emerge in case 2. Considering the practical situation of bridge engineering, the finding means that the proposed method is possible to apply in the practical application of detecting the bearings or structural elements individually.Meanwhile, Figure11 shows that the method can detect the damage of the bearing and structural element simultaneously. However, the Markov chain of the structural element indicates some fluctuations, especially in the early stage. This situation represents that researchers should pay more attention to study the scenario similar to case 4 in the future.In terms of Figures12 and 13, the same conclusion as Figure 8 can be drawn. And the max error that occurred in case 6 is 3.32%.From the perspective of Figure14, the Markov chain of bearing damage parameter is very stable, but another one fluctuates intensely, which can be owed to the selection of the variance of assuming proposal distribution. Thus, how to determine a suitable variance is a key factor to identify the damage of the element accurately when bearing and element damage exist simultaneously.Furthermore, from Table2, although the OASA method can accurately tackle the single structural element damage cases or single bearing damage cases, e.g., cases 2-3 and 5-6, however, regarding the multiple-damage case, namely, structural element and bearing damage, compared to the proposed method, the identified results of the OASA method are worse. For case 1, the detected damage extents of the structural element and bearing have been exchanged, which is owed to the negative effect of uncertainties or the poor global optimization ability of the optimization algorithm. And for cases 4 and 7, the identified damage severities are not consistent with the preset values.To summarize, from the aforementioned analyses and discussions, one can conclude that based on the proposed method, it has good feasibility to detect the structural bearing damage and structural element damage independently and even the two types of damage simultaneously. At the same time, the proposed method can overcome the distraction of uncertainties such as temperature variations and random noise, which shows good potential in the practical application.


## 4.2. Engineering Example

### 4.2.1. The Introduction of the Example

In order to further verify the practicability of the proposed method, in this section, a practical engineering of I-40 steel-concrete bridge has been exploited. This bridge is sited in Albuquerque, New Mexico, USA, and the related studies have become the classical cases in the field of structural health monitoring. A vibration test was carried out to collect the dynamic characteristics [41], and the obtained data have been widely used to verify the diverse damage detection approach. Figure 15 illustrates the overall view and experimental setup of the bridge.Figure 15
I-40 bridge: (a) overall layout; (b) experimental setup.
(a)(b)In the middle span of the north plate girder of the bridge, there are four damage cases which have been introduced, which can be seen in Figure16. After the introduction, the modal test was conducted for every damage case, and for D-1 to D-4, the corresponding ambient temperatures and approximate damage severities were recorded, which are (15.5°C, 5%), (28.9°C, 10%), (26.1°C, 32%), and (20.0°C, 92%), respectively.Figure 16
Damage cases of the I-40 bridge.Referring to the report of the modal experiment and collected modal characteristics, the finite element model has been programmed using MATLAB [22, 32, 42], which is described in Figure 17. In this model, the concrete deck and the web of the plate girder are both simulated by the 4-node shell element, and other components, such as the flanges of the girder, stringers, floor beams, and concrete piers, are modelled based on the 2-node beam element. And the bridge bearings are tackled with a spring element; for each direction, the spring constants are kx = 3.17 × 106 N/m, ky = 1.26 × 106 N/m, and kz = 4.29 × 107 N/m, respectively. In addition, the detailed material properties can be referred to the paper [42]. After obtaining the finite element model, the natural frequencies and modal assurance criterion (MAC) are utilized to measure the correlation, which are listed in Table 3.Figure 17
The finite element model of the I-40 bridge.Table 3
Analytical and measured natural frequencies and MAC.
ModeMeasured (Hz)Analytical (Hz)Error (%)MAC12.48282.48210.030.994922.95933.00161.410.984133.49913.41762.380.990544.07914.03651.060.970854.16684.03693.220.971864.63104.65610.540.9709From the data listed in Table3, it can be seen that the max error between analytical natural frequencies and measured ones is merely 3.22%; meanwhile, the first six MAC values are all larger than 0.97. The comparison indicates that the model shows a good reality to reflect the dynamic characteristics of the actual engineering structure, which can be adopted to implement the subsequent damage detection procedure.


### 4.2.2. Damage Identification and Results’ Discussion

Based on the aforementioned actual damage cases and the finite element model, there are several simulated bearing damage cases that have been introduced to combine the actual structural element damage cases to verify the proposed bearing damage detection method. For each damage case, the first five natural frequencies and modal shapes are adopted to detect the damage of the structural element and bearing. The iterative number is defined as 20,000, and the initial value is 1 for every damage case. The damage cases and corresponding results can be seen in Table4.Table 4
Damage cases and identification results of the I-40 bridge.
Ambient temperature (°C)CaseDamage severity @ element numberDamage severity @ bearing numberDetection goalIdentification results15.5D-1(a)5% @ 12/E5.12% @ 12D-1(b)5% @ 1230% @ 1#B29.94% @ 1#D-1(c)5% @ 1230% @ 1#E and B5.25% @ 1231.89% @ 1#D-1(d)5% @ 1230% @ 1#50% @ 2#E and B5.21% @ 1230.10% @ 1#49.48% @ 2#28.9D-2(a)10% @ 12/E10.1% @ 12D-2(b)10% @ 1230% @ 1#B29.99% @ 1#D-2(c)10% @ 1230% @ 1#E and B10.09% @ 1229.99% @ 1#D-2(d)10% @ 1230% @ 1#50% @ 2#E and B9.85% @ 1230.61% @ 1#49.52% @ 2#26.1D-3(a)32% @ 12/E32.28% @ 12D-3(b)32% @ 1230% @ 1#B30.05% @ 1#D-3(c)32% @ 1230% @ 1#E and B31.04% @ 1226.80% @ 1#D-3(d)32% @ 1230% @ 1#50% @ 2#E and B31.56% @ 1230.95%@ 1#49.51% @ 2#20.0D-4(a)92% @ 12/E91.77% @ 12D-4(b)92% @ 1230% @ 1#B30.07% @ 1#D-4(c)92% @ 1230% @ 1#E and B92.88% @ 1226.93% @ 1#D-4(d)92% @ 1230% @ 1#50% @ 2#E and B92.20% @ 1221.38% @ 1#54.84% @ 2#Note: E and B denote the element damage and bearing damage, respectively.As shown in Table4, the proposed method can not only identify the single-type damage situation (e.g., only element damage or bearing damage) but also assess the multiple-type damage scenario accurately. In case D-1, the max identified error of the structural element is 0.21%, and for the bearing, it is 1.89%; for case D-2, the max errors have emerged in D-2(d), which are 0.15% and 0.61% for the element and bearing, respectively. And for cases D-3 and D-4, the accuracies of damage detection have declined to some extent, namely, the max errors are 3.20% (D-3(c), element damage) and 4.84% (D-3(d), bearing damage), respectively.In conclusion, although there are some negative influences of ambient temperature variations and environmental noise, the damage detection results are of great accuracy, which indicates the proposed method has good robustness. The uncertainties in the damage detection process can be overcome using the Bayesian framework; meanwhile, the damage of the structural element and bearing can be detected simultaneously, which is very useful in the practical application. The proposed damage detection framework shows good potential for the bearing damage detection.


## 4.2.1. The Introduction of the Example

In order to further verify the practicability of the proposed method, in this section, a practical engineering of I-40 steel-concrete bridge has been exploited. This bridge is sited in Albuquerque, New Mexico, USA, and the related studies have become the classical cases in the field of structural health monitoring. A vibration test was carried out to collect the dynamic characteristics [41], and the obtained data have been widely used to verify the diverse damage detection approach. Figure 15 illustrates the overall view and experimental setup of the bridge.Figure 15
I-40 bridge: (a) overall layout; (b) experimental setup.
(a)(b)In the middle span of the north plate girder of the bridge, there are four damage cases which have been introduced, which can be seen in Figure16. After the introduction, the modal test was conducted for every damage case, and for D-1 to D-4, the corresponding ambient temperatures and approximate damage severities were recorded, which are (15.5°C, 5%), (28.9°C, 10%), (26.1°C, 32%), and (20.0°C, 92%), respectively.Figure 16
Damage cases of the I-40 bridge.Referring to the report of the modal experiment and collected modal characteristics, the finite element model has been programmed using MATLAB [22, 32, 42], which is described in Figure 17. In this model, the concrete deck and the web of the plate girder are both simulated by the 4-node shell element, and other components, such as the flanges of the girder, stringers, floor beams, and concrete piers, are modelled based on the 2-node beam element. And the bridge bearings are tackled with a spring element; for each direction, the spring constants are kx = 3.17 × 106 N/m, ky = 1.26 × 106 N/m, and kz = 4.29 × 107 N/m, respectively. In addition, the detailed material properties can be referred to the paper [42]. After obtaining the finite element model, the natural frequencies and modal assurance criterion (MAC) are utilized to measure the correlation, which are listed in Table 3.Figure 17
The finite element model of the I-40 bridge.Table 3
Analytical and measured natural frequencies and MAC.
ModeMeasured (Hz)Analytical (Hz)Error (%)MAC12.48282.48210.030.994922.95933.00161.410.984133.49913.41762.380.990544.07914.03651.060.970854.16684.03693.220.971864.63104.65610.540.9709From the data listed in Table3, it can be seen that the max error between analytical natural frequencies and measured ones is merely 3.22%; meanwhile, the first six MAC values are all larger than 0.97. The comparison indicates that the model shows a good reality to reflect the dynamic characteristics of the actual engineering structure, which can be adopted to implement the subsequent damage detection procedure.


## 4.2.2. Damage Identification and Results’ Discussion

Based on the aforementioned actual damage cases and the finite element model, there are several simulated bearing damage cases that have been introduced to combine the actual structural element damage cases to verify the proposed bearing damage detection method. For each damage case, the first five natural frequencies and modal shapes are adopted to detect the damage of the structural element and bearing. The iterative number is defined as 20,000, and the initial value is 1 for every damage case. The damage cases and corresponding results can be seen in Table4.Table 4
Damage cases and identification results of the I-40 bridge.
Ambient temperature (°C)CaseDamage severity @ element numberDamage severity @ bearing numberDetection goalIdentification results15.5D-1(a)5% @ 12/E5.12% @ 12D-1(b)5% @ 1230% @ 1#B29.94% @ 1#D-1(c)5% @ 1230% @ 1#E and B5.25% @ 1231.89% @ 1#D-1(d)5% @ 1230% @ 1#50% @ 2#E and B5.21% @ 1230.10% @ 1#49.48% @ 2#28.9D-2(a)10% @ 12/E10.1% @ 12D-2(b)10% @ 1230% @ 1#B29.99% @ 1#D-2(c)10% @ 1230% @ 1#E and B10.09% @ 1229.99% @ 1#D-2(d)10% @ 1230% @ 1#50% @ 2#E and B9.85% @ 1230.61% @ 1#49.52% @ 2#26.1D-3(a)32% @ 12/E32.28% @ 12D-3(b)32% @ 1230% @ 1#B30.05% @ 1#D-3(c)32% @ 1230% @ 1#E and B31.04% @ 1226.80% @ 1#D-3(d)32% @ 1230% @ 1#50% @ 2#E and B31.56% @ 1230.95%@ 1#49.51% @ 2#20.0D-4(a)92% @ 12/E91.77% @ 12D-4(b)92% @ 1230% @ 1#B30.07% @ 1#D-4(c)92% @ 1230% @ 1#E and B92.88% @ 1226.93% @ 1#D-4(d)92% @ 1230% @ 1#50% @ 2#E and B92.20% @ 1221.38% @ 1#54.84% @ 2#Note: E and B denote the element damage and bearing damage, respectively.As shown in Table4, the proposed method can not only identify the single-type damage situation (e.g., only element damage or bearing damage) but also assess the multiple-type damage scenario accurately. In case D-1, the max identified error of the structural element is 0.21%, and for the bearing, it is 1.89%; for case D-2, the max errors have emerged in D-2(d), which are 0.15% and 0.61% for the element and bearing, respectively. And for cases D-3 and D-4, the accuracies of damage detection have declined to some extent, namely, the max errors are 3.20% (D-3(c), element damage) and 4.84% (D-3(d), bearing damage), respectively.In conclusion, although there are some negative influences of ambient temperature variations and environmental noise, the damage detection results are of great accuracy, which indicates the proposed method has good robustness. The uncertainties in the damage detection process can be overcome using the Bayesian framework; meanwhile, the damage of the structural element and bearing can be detected simultaneously, which is very useful in the practical application. The proposed damage detection framework shows good potential for the bearing damage detection.


## 5. Conclusions

In this paper, firstly, based on the structural dynamics and finite element method, the structural dynamic characteristic equation considering uncertain conditions, such as environmental temperature variations, structural element damage, and bearing damage, has been proposed. Then, based on the proposed equation, the influences of the uncertain conditions have been evaluated firstly using a simply supported beam. After that, a damage detection method has been promoted by introducing the Bayesian damage detection theory and matrix perturbation method; in Bayesian damage detection theory, regarding the uncertainties of damage detection, the posterior probability density function of damage parameters is obtained based on the natural frequencies and modal shapes; at the same time, the matrix perturbation method is used to obtain the 2nd-order perturbation formulae of modal parameters, which can save the computational cost. Furthermore, a numerical example of an 8-span concrete continuous beam structure and a practical example of I-40 steel-concrete composite bridge are exploited to validate the effectiveness of the proposed method. There are some conclusions and outlooks that can be drawn as follows:(1)
The uncertain conditions, especially for the synergistic effects of multiple factors, can cause serious fluctuations of structural modal parameters; meanwhile, the fluctuations cannot be evaluated as a linear relationship, which will lead to inaccurate damage detection. Thus, uncertainties should be considered in the process.(2)
The proposed damage detection method is of great robustness and good damage identification accuracy, which is validated in the numerical example and practical engineering. It can identify the damage of the structural elements and bearings simultaneously under the distraction of uncertainties.(3)
The high accurate detection of bearing damage when the structural element exists indicates the proposed method can be applied to practical engineering, which means there is some possibility to develop a useful damage detection approach of structural bearings based on this idea.(4)
However, some extreme damage cases, for instance, most of the bearings and even all bearings of a structure are damaged, have not been studied. In this article, the most serious situation is one element and two bearings have been damaged, which means some further research studies need to be performed in the future.



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*Source: 5576362-2021-05-20.xml*