# Damage Identification of Ancient Timber Structure Based on Autocorrelation Function

Authors: Yanfang Hou; Weibing Hu; Xin Wang; Tingting Hou; Congli Sun
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

A damage location method for the autocorrelation peak value change rate based on the vibration response of a random vibration structure is established. To calculate the autocorrelation function of the vibration response of each measurement point, we transformed the maximum values into an autocorrelation peak vector. Under a good condition, the autocorrelation peak vector has a fixed shape; hence, it can be used as a basis for structural damage identification. The two adjacent measurement points with the largest change corresponding to the two nodes of the damage unit and the damage location are determined to calculate the change rate of the autocorrelation peak values between damaged and intact structures. When the degree of damage is 5%, the autocorrelation peak value change rate of the acceleration response on the two nodes of the damage unit is significantly greater than that of the other points, which can accurately determine the damage location, indicating that the damage location index constructed has good damage sensitivity. The damage location index can determine a single damage, as well as a double damage. The antinoise capability of the damage location index gradually improves with an increase in the degree of damage. At 45% degree of damage and signal-to-noise ratio (SNR) of 0 dB, the damage location index can still accurately determine the damage location, which has good antinoise interference capability. The Xi’an Bell Tower is used as a case study, and the feasibility of this method is verified, which provides a new method for the study of damage location of ancient timber structures.


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## Body

## 1. Introduction

Chinese ancient structures, precious cultural heritage, and important parts of human civilization objectively and faithfully reflect the development and features of history, preserving many historical materials, which cannot be recorded in a written language. Therefore, they are known as the “three-dimensional textbooks” and “ready-made museum.” Weather-beaten ancient structures are more valuable not only because they have survived thousands of years, but they are also the remains of history. Once they are destroyed, they can never be recovered.Chinese ancient structures, which are mainly made of wood, can adjust measurements to local conditions and local materials, taking wooden technology to the extreme. For example, the main hall of the Nanchan Temple in Mount Wutai has an extremely strict distribution of wooden components. There are no redundant structures or bracket sets between the columns. In terms of structure, it has reached the state of “to add a point is redundant, to subtract a point is to lack” [1]. Numerous mortise-tenon joints have been found at the Hemudu site in Yuyao, Zhejiang, which reflects the long history of wood structure technology in China. However, ease of burning and deterioration of the wood have led to the collapse and disappearance of Chinese ancient wooden structures.Consequently, studies have been initiated on damage identification methods of ancient wooden structures by scholars worldwide. Wang and Yang [2] applied a vibration response sensitivity damage identification method to identify three different damage states of beams and columns of ancient Tibetan wooden structures. The results show that this method can accurately identify damage to the beams and columns of the ancient Tibetan wooden structure. Wang et al. [3–6] carried out a finite-element simulation of ancient wooden structures under random excitation using wavelet packet energy spectrum and wavelet packet curvature difference method to identify and locate damage to wooden structures. Fan [7] established a finite-element model of a wooden frame and adopted a damage identification method combining the wavelet packet analysis method and stochastic resonance theory to evaluate the damage location and degree of damage of the structure. Choi [8] adopted a model-based damage detection method to identify the locations of common damage defects in wood and estimate the degree of damage. Hu and Afzal [9] proposed a new statistical algorithm, which was validated by the numerical simulation of different degrees of damage and damage locations of wooden beams.Many scholars have studied damage identification methods based on correlation functions. Mohammad used the cross-correlation damage index to determine the damage locations in the length of continuous deck bridges, and a comparison of the obtained analytical results with the experimental results showed that the proposed methods can determine the damage locations with appropriate accuracy for different damage scenarios [10]. Morgantini and Betti presented a structural damage assessment method that relies on a cross-correlation-based damage-sensitive feature, which is extracted from the dynamic response measured at different locations within a structure [11]. Li and Huang proposed an approach to identify structural states from eigenvectors constructed using cross-correlation functions. The method was first applied to damage detection of a numerical model of a jacket platform in the Bohai Sea [12]. Li et al. carried out an experimental study of structural damage detection by applying the cross-correlation function amplitude vector (CorV) of dynamic strain under varying environmental temperatures; the study verified that, under white noise excitation, CorV is only related to natural frequencies, mode shapes, and damping ratios of structures. A laboratory experimental investigation based on an end-fixed steel beam showed that CorV can be used for structural damage detection [13]. Lin and Fajri proposed a structural health monitoring (SHM) system based on detrended fluctuation analysis (DFA) and detrended cross-correlation analysis (DCCA); the performance of the SHM system was verified using a four-layer numerical model [14]. Huo et al. combined cross-correlation function amplitude (CCFA) with a support vector machine (SVM) for damage detection of a bone structure [15]. Zhang et al. [16–18] proposed a new approach to locate damage using the autocorrelation function of vibration response signals and performed detailed investigation of the damage index based on the autocorrelation function to detect the damage of a structure under white noise excitation.At present, studies on damage identification methods for ancient wooden structures mainly focused on the components of the structures [19–21]. However, very few studies have investigated the overall building structures. In addition, existing damage identification methods are often affected by measurement noise and have low antinoise capabilities. In this study, the Xi’an Bell Tower was used as a case study, and a finite-element model of the upper part of the bell tower was established. The damage identification and location index of the autocorrelation peak change rate were constructed based on the principle and property of the autocorrelation function. The two adjacent test points with the largest change rate of the autocorrelation peak value before and after the damage correspond to the two nodes of the damage element, and the damage location of the structure was realized. The damage sensitivity and antinoise capability of the positioning index were analyzed, which provides a new method for damage identification and positioning of ancient wooden structures.


## 2. Basic Theory of Autocorrelation Function [22–24]

### 2.1. Correlation and Autocorrelation Functions

Correlation is an important measure of the relationship between two or more random variables. Suppose thatx(t) and y(t) are two waveforms corresponding to two different signals in a random vibration process. To measure the similarity between the two waveforms, one can be moved relative to the other. When the two waveforms reach a certain degree of similarity, it can be said that the two random variables x(t) and y(t) are related.For a random vibration processx(t), its autocorrelation indicates the similarity between signal x(t) and the delay signal x(t + τ). In other words, the autocorrelation function describes the correlation degree of the random signal x(t) at any two different times t and t + τ, which is defined as the average value of the product of x(t) and x(t + τ), namely, E[x(t)x(t + τ)]. If the waveform x(t) is divided into N equally spaced discrete values x1, x2,…, xN, then(1)Extxt+τ=1N∑i=1Nxitxit+τ.In the above equation,τ is the moving value of the time coordinate, not time, and it can be either positive or negative.If the waveform is continuous, it should be discretized, the number of sampling points isN, and the time interval is Δt, then(2)N−1=TΔt.If the value of Δt is small, or N ⟶ ∞, dt can be used to represent Δt, then equation (2) becomes(3)1N=ΔtT+Δt=dtT.By substituting equation (3) into (1) and changing the sum into an integral, we obtain(4)ϕxxτ=limT⟶∞1T∫0Txtxt+τdt.In the equation,T is the time history of the selected waveform and τ is the time difference between the two groups of vibration data, which is called the lag.The size ofφ(τ) was quantified to measure the similarity between the two groups of vibration data. φ(τ) is related not only to the characteristics of the signal, but also to the time interval τ. If φ(τ) reaches the maximum value at τ = τi, it means that they are the most similar, that is, the most relevant degree.


### 2.2. Autocorrelation Peak Vector

According to the concept of the relevant function, the smaller the time intervalτ, the closer the relationship between x(t + τ) and x(t), and the greater the φxx(τ) value. As the value of τ increases, x(t + τ) and x(t) can reach a state of irrelevance, where φxx(τ) tends to a minimum, or zero. Therefore, the autocorrelation function has the following properties:(1)
Whenτ is very large, the autocorrelation function of a random variable with an average value of 0 approaches 0, namely,(5)limτ⟶0ϕxxτ=ϕxx±∞=0.This shows that there is no correlation betweenx(t + τ) and x(t).(2)
Whenτ = 0, the autocorrelation function of a random variable with an average value of zero is maximized and is equal to the mean square value of the signal, namely,(6)ϕxx0=Ex2t=Dx.(3)
The autocorrelation functionφxx(τ) will never be greater than its initial value when τ takes any value, namely,(7)ϕxxτ≤ϕxx0.This is because whenτ = 0, x(t + τ) is related to x(t). The correlation between two states of a random process cannot be greater than that between the state at the same point and itself; that is, the signal is strongly correlated with itself, and the correlation is the largest. Hence, the maximum number of φxx(τ) is φxx(0).For a structure with random vibration, if the dynamic response (acceleration, velocity, displacement, etc.) ofN measurement points is simultaneously measured, then the maximum value of the autocorrelation function of the dynamic response of each measurement point is φxx(0) = δii. All the maximum values constitute the autocorrelation peak vector, denoted as(8)A=δ11,δ22,…,δnn.When an engineering structure is damaged, the dynamic characteristics and dynamic response of the structure will change. The autocorrelation peak is calculated based on the dynamic response of the structure. Therefore, when the structure is damaged, the autocorrelation peak also changes. The degree of change of each element value before and after the damage can be measured by the peak value change rate as follows:(9)Δii=δiid−δiiuδiiu×100%,where δiiu is the peak value of the autocorrelation function of the dynamic response of each measurement point of the intact structure, δiid is the peak value of the autocorrelation function of the dynamic response at each measurement point of the damaged structure, and Δii is the rate of change of the dynamic response of the autocorrelation peak at each measurement point before and after structural damage.The greater the change rate of the autocorrelation peak value of the dynamic response, the greater the degree of damage of the structure. Therefore, the autocorrelation peak change rate can be used to construct the damage identification and location index to identify structural damage.The maximum value of the peak change rate is normalized as follows:(10)Δii∗=ΔiimaxΔii.


## 2.1. Correlation and Autocorrelation Functions

Correlation is an important measure of the relationship between two or more random variables. Suppose thatx(t) and y(t) are two waveforms corresponding to two different signals in a random vibration process. To measure the similarity between the two waveforms, one can be moved relative to the other. When the two waveforms reach a certain degree of similarity, it can be said that the two random variables x(t) and y(t) are related.For a random vibration processx(t), its autocorrelation indicates the similarity between signal x(t) and the delay signal x(t + τ). In other words, the autocorrelation function describes the correlation degree of the random signal x(t) at any two different times t and t + τ, which is defined as the average value of the product of x(t) and x(t + τ), namely, E[x(t)x(t + τ)]. If the waveform x(t) is divided into N equally spaced discrete values x1, x2,…, xN, then(1)Extxt+τ=1N∑i=1Nxitxit+τ.In the above equation,τ is the moving value of the time coordinate, not time, and it can be either positive or negative.If the waveform is continuous, it should be discretized, the number of sampling points isN, and the time interval is Δt, then(2)N−1=TΔt.If the value of Δt is small, or N ⟶ ∞, dt can be used to represent Δt, then equation (2) becomes(3)1N=ΔtT+Δt=dtT.By substituting equation (3) into (1) and changing the sum into an integral, we obtain(4)ϕxxτ=limT⟶∞1T∫0Txtxt+τdt.In the equation,T is the time history of the selected waveform and τ is the time difference between the two groups of vibration data, which is called the lag.The size ofφ(τ) was quantified to measure the similarity between the two groups of vibration data. φ(τ) is related not only to the characteristics of the signal, but also to the time interval τ. If φ(τ) reaches the maximum value at τ = τi, it means that they are the most similar, that is, the most relevant degree.


## 2.2. Autocorrelation Peak Vector

According to the concept of the relevant function, the smaller the time intervalτ, the closer the relationship between x(t + τ) and x(t), and the greater the φxx(τ) value. As the value of τ increases, x(t + τ) and x(t) can reach a state of irrelevance, where φxx(τ) tends to a minimum, or zero. Therefore, the autocorrelation function has the following properties:(1)
Whenτ is very large, the autocorrelation function of a random variable with an average value of 0 approaches 0, namely,(5)limτ⟶0ϕxxτ=ϕxx±∞=0.This shows that there is no correlation betweenx(t + τ) and x(t).(2)
Whenτ = 0, the autocorrelation function of a random variable with an average value of zero is maximized and is equal to the mean square value of the signal, namely,(6)ϕxx0=Ex2t=Dx.(3)
The autocorrelation functionφxx(τ) will never be greater than its initial value when τ takes any value, namely,(7)ϕxxτ≤ϕxx0.This is because whenτ = 0, x(t + τ) is related to x(t). The correlation between two states of a random process cannot be greater than that between the state at the same point and itself; that is, the signal is strongly correlated with itself, and the correlation is the largest. Hence, the maximum number of φxx(τ) is φxx(0).For a structure with random vibration, if the dynamic response (acceleration, velocity, displacement, etc.) ofN measurement points is simultaneously measured, then the maximum value of the autocorrelation function of the dynamic response of each measurement point is φxx(0) = δii. All the maximum values constitute the autocorrelation peak vector, denoted as(8)A=δ11,δ22,…,δnn.When an engineering structure is damaged, the dynamic characteristics and dynamic response of the structure will change. The autocorrelation peak is calculated based on the dynamic response of the structure. Therefore, when the structure is damaged, the autocorrelation peak also changes. The degree of change of each element value before and after the damage can be measured by the peak value change rate as follows:(9)Δii=δiid−δiiuδiiu×100%,where δiiu is the peak value of the autocorrelation function of the dynamic response of each measurement point of the intact structure, δiid is the peak value of the autocorrelation function of the dynamic response at each measurement point of the damaged structure, and Δii is the rate of change of the dynamic response of the autocorrelation peak at each measurement point before and after structural damage.The greater the change rate of the autocorrelation peak value of the dynamic response, the greater the degree of damage of the structure. Therefore, the autocorrelation peak change rate can be used to construct the damage identification and location index to identify structural damage.The maximum value of the peak change rate is normalized as follows:(10)Δii∗=ΔiimaxΔii.


## 3. Overview and Establishment of Finite-Element Model of the Xi’an Bell Tower [25, 26]

### 3.1. Overview of the Xi’an Bell Tower

Built during the Hongwu period of the Ming Dynasty (1384), the Xi’an Bell Tower (see Figure1) is one of the largest, grandest and best-preserved Ming architectures in China. It is located at the intersection of the East, West, South, and North four main streets in the city center of Xi’an; it is the landmark building of Xi’an, known as the “Pearl of the Ancient City.” The bell tower covers an area of approximately 1,377.6 m2 and is composed of three parts: the platform, tower, and roof. The lower platform is a cuboid with an underside length of 35.5 m and a height of 8.6 m. The interior of the high platform is composed of plain packed earth layer compaction and there are arch holes in the middle of all sides with a height and width of 6 m and 6 m, respectively. The main body of the building is a 27.4 m two-story wooden structure with triple eaves. The plane is a square, surrounded by cloisters. The bell tower is a timber frame structure with four central columns and 12 surrounding columns. The lower end of the columns is placed on a rammed earth foundation through the plinth, and the upper end extends to the roof.Figure 1
The Xi’an Bell Tower.The Xi’an Bell Tower is similar to the frame structure, whose wall does not bear weight but only plays the role of enclosure and separation of the space. The columns are connected by beams with a high span ratio through mortise and tenon joints, which together with the wall form an antilateral force system, enhancing the integrity of the bell tower structure. The plane of the bell tower structure is symmetrical, and the distribution of mass and stiffness is even. This arrangement not only prevents stress concentration due to the excessive internal force of the corner column but also makes the dynamic characteristics of all directions on the building plane roughly consistent; therefore, the torsion effect of the structure caused by the deviation between the mass center and the stiff center is reduced.In this study, the finite-element model of the upper frame was established based on the structural characteristics of the Xi’an Bell Tower. A damage identification method based on the autocorrelation function of random vibration structures is proposed; a damage identification and location index of the autocorrelation peak change rate was constructed and the damage location of the structure was realized. This resulted in a new method for damage identification and positioning of ancient wooden structures.


### 3.2. Establishment of Finite-Element Model

The timber frame structure of the upper part of the bell tower is composed of 12 pillars C-1 on the periphery and four pillars C-2 at the center, as well as timber beams B-1, B-2, and B-3. A schematic diagram of the bell tower is shown in Figure2. The section size and material parameters of each member are listed in Table 1.Figure 2
Diagram of the Xi’an Bell Tower [27].Table 1
Section size of beam and column [27].
Column componentC-1C-2—Cross-sectional diameter (mm)500700—Beam componentB-1B-2B-3Cross-sectional size (mm × mm)300 × 700300 × 800200 × 300The parameters of the wood material are based on the measured data in reference [28], as listed in Table 2.Table 2
Material parameters [28].
MaterialElasticity moduli (MPa)Density (kg/m−3)Poisson’s ratioTimber8307.34100.25One notable feature of ancient Chinese wooden structures is that beams and columns are connected by mortise and tenon joints, unlike steel and reinforced concrete structures. In a mortise and tenon joint, the tenon has a certain length, and there is a certain gap between the mortise and tenon. Therefore, the connection is neither rigid nor hinge, but a semirigid connection [29–33]. The energy dissipation caused by friction and sliding between the mortise and tenon is an important contributor to the good seismic performance of ancient wooden structures [34]. Han studied and demonstrated the stiffness of the mortise and tenon joints of wooden structures and obtained the stiffness value, as presented in Table 3 [35].Table 3
Stiffness value of mortise and tenon joints.
NameKx = Kz (N/m)Ky (N/m)RX = RY = RZ (N·m)Side span1.71 × 1072.07683 × 1086.244 × 108A finite-element software program, ANSYS, was used to build the wooden frame model. The coupling function of the joints and the stiffness of the spring element were used to simulate the semirigid characteristic of the mortise and tenon joints. The mortise and tenon joint was realized to establish several overlapping finite-element nodes at the connecting position of the beam and column, representing the ends of each beam and column of the mortise and tenon joint. The spring element was applied only at the position where the beam and column are connected, and no spring element was applied between the posts. The spring element connects the column end node ① with beam end nodes ②, ③, ④, and ⑤. Six spring elements were used to simulate the end nodes of the column and beam, and the spring stiffness parameters of each degree-of-freedom direction of the beam and column were set to realize the mortise and tenon joint, as shown in Figure3.Figure 3
Mortise-tenon joint between beam and column.Three-dimensional elastic elements were used to simulate the wooden beams and columns. The roof mass was applied to the top of the column in the form of concentrated mass to consider the influence of the inertia force of the roof mass, and the mass element was used to simulate a large roof. The finite-element model of the timber frame structure in the upper part of the bell tower is shown in Figure4.Figure 4
Wooden structure of the Xi’an Bell Tower.


## 3.1. Overview of the Xi’an Bell Tower

Built during the Hongwu period of the Ming Dynasty (1384), the Xi’an Bell Tower (see Figure1) is one of the largest, grandest and best-preserved Ming architectures in China. It is located at the intersection of the East, West, South, and North four main streets in the city center of Xi’an; it is the landmark building of Xi’an, known as the “Pearl of the Ancient City.” The bell tower covers an area of approximately 1,377.6 m2 and is composed of three parts: the platform, tower, and roof. The lower platform is a cuboid with an underside length of 35.5 m and a height of 8.6 m. The interior of the high platform is composed of plain packed earth layer compaction and there are arch holes in the middle of all sides with a height and width of 6 m and 6 m, respectively. The main body of the building is a 27.4 m two-story wooden structure with triple eaves. The plane is a square, surrounded by cloisters. The bell tower is a timber frame structure with four central columns and 12 surrounding columns. The lower end of the columns is placed on a rammed earth foundation through the plinth, and the upper end extends to the roof.Figure 1
The Xi’an Bell Tower.The Xi’an Bell Tower is similar to the frame structure, whose wall does not bear weight but only plays the role of enclosure and separation of the space. The columns are connected by beams with a high span ratio through mortise and tenon joints, which together with the wall form an antilateral force system, enhancing the integrity of the bell tower structure. The plane of the bell tower structure is symmetrical, and the distribution of mass and stiffness is even. This arrangement not only prevents stress concentration due to the excessive internal force of the corner column but also makes the dynamic characteristics of all directions on the building plane roughly consistent; therefore, the torsion effect of the structure caused by the deviation between the mass center and the stiff center is reduced.In this study, the finite-element model of the upper frame was established based on the structural characteristics of the Xi’an Bell Tower. A damage identification method based on the autocorrelation function of random vibration structures is proposed; a damage identification and location index of the autocorrelation peak change rate was constructed and the damage location of the structure was realized. This resulted in a new method for damage identification and positioning of ancient wooden structures.


## 3.2. Establishment of Finite-Element Model

The timber frame structure of the upper part of the bell tower is composed of 12 pillars C-1 on the periphery and four pillars C-2 at the center, as well as timber beams B-1, B-2, and B-3. A schematic diagram of the bell tower is shown in Figure2. The section size and material parameters of each member are listed in Table 1.Figure 2
Diagram of the Xi’an Bell Tower [27].Table 1
Section size of beam and column [27].
Column componentC-1C-2—Cross-sectional diameter (mm)500700—Beam componentB-1B-2B-3Cross-sectional size (mm × mm)300 × 700300 × 800200 × 300The parameters of the wood material are based on the measured data in reference [28], as listed in Table 2.Table 2
Material parameters [28].
MaterialElasticity moduli (MPa)Density (kg/m−3)Poisson’s ratioTimber8307.34100.25One notable feature of ancient Chinese wooden structures is that beams and columns are connected by mortise and tenon joints, unlike steel and reinforced concrete structures. In a mortise and tenon joint, the tenon has a certain length, and there is a certain gap between the mortise and tenon. Therefore, the connection is neither rigid nor hinge, but a semirigid connection [29–33]. The energy dissipation caused by friction and sliding between the mortise and tenon is an important contributor to the good seismic performance of ancient wooden structures [34]. Han studied and demonstrated the stiffness of the mortise and tenon joints of wooden structures and obtained the stiffness value, as presented in Table 3 [35].Table 3
Stiffness value of mortise and tenon joints.
NameKx = Kz (N/m)Ky (N/m)RX = RY = RZ (N·m)Side span1.71 × 1072.07683 × 1086.244 × 108A finite-element software program, ANSYS, was used to build the wooden frame model. The coupling function of the joints and the stiffness of the spring element were used to simulate the semirigid characteristic of the mortise and tenon joints. The mortise and tenon joint was realized to establish several overlapping finite-element nodes at the connecting position of the beam and column, representing the ends of each beam and column of the mortise and tenon joint. The spring element was applied only at the position where the beam and column are connected, and no spring element was applied between the posts. The spring element connects the column end node ① with beam end nodes ②, ③, ④, and ⑤. Six spring elements were used to simulate the end nodes of the column and beam, and the spring stiffness parameters of each degree-of-freedom direction of the beam and column were set to realize the mortise and tenon joint, as shown in Figure3.Figure 3
Mortise-tenon joint between beam and column.Three-dimensional elastic elements were used to simulate the wooden beams and columns. The roof mass was applied to the top of the column in the form of concentrated mass to consider the influence of the inertia force of the roof mass, and the mass element was used to simulate a large roof. The finite-element model of the timber frame structure in the upper part of the bell tower is shown in Figure4.Figure 4
Wooden structure of the Xi’an Bell Tower.


## 4. Damage Localization and Identification of the Xi’an Bell Tower

### 4.1. Intrinsic Form of Autocorrelation Peak Vector

The central spans of the first and second floors of the timber frame of the bell tower, namely, beams I and II in Figure4, were selected for analysis. The number of elements of beam I is 230 to 235 and the corresponding number of nodes is 800 to 806, whereas the number of elements of beam II is 454 to 459 and the corresponding number of nodes is 1,048 to 1,054, as shown in Figure 5.Figure 5
Number of nodes and elements of beams I and II.The Xi’an Bell Tower is located at the intersection of the East, West, South, and North streets, and the surrounding traffic environment is complicated. Lines 2 and 6 of the Xi’an Subway meet at the bottom of the Bell Tower. The Bell Tower Station of Line 2 is approximately 60 m south of the bell tower. To analyze the impact of subway operation on the bell tower, Zhao [26] calculated the acting load of Xi’an Subway Line 2 and Line 6 during single line operation using the empirical analysis model of train load and obtained the load values under different running speeds.WhenV1 = 20 km/h, F(t) = 70,000 + 31.95 sin 3.49 t + 91.3 sin 17.44 t + 292.1 sin 69.8 t.WhenV2 = 40 km/h, F(t) = 70,000 + 127.8 sin 6.98 t + 365.2 sin 34.89 t + 1168.5 sin 139.56 t.V1 and V2 were applied to plinth node 501 as random excitation sample 1 and random excitation sample 2, respectively. The excitation time is t = 12 s and the time step is Δt = 0.01 s.TheX-direction horizontal acceleration responses of each node on beams I and beam II were extracted. Taking node 803 of beam I and node 1048 of beam II as examples, the autocorrelation value of node acceleration was calculated, as shown in Figure 6.Figure 6
Normalized autocorrelation values of (a) node 803 and (b) node 1048.
(a)(b)It can be observed from the diagram that the autocorrelation function of the acceleration response of nodes 803 and 1048 is largest whenτ = 0 because the signal correlated with itself and the degree of correlation is highest.Taking beam I as an example, the autocorrelation function of each node on the beam was calculated at subway loads ofV1 = 20 km/h and V2 = 40 km/h. The maximum value was transformed into an autocorrelation peak vector, and the peak value vectors of the autocorrelation functions of each node of the wooden frame intact structure under random excitations 1 and 2 were recorded as A1 and A2, respectively, as shown in Figure 7.Figure 7
Inherent form of the intact structures.It can be observed from the figure that the peak value curves of the two autocorrelation functions under two different random excitations are almost identical. This shows that, for a random vibration structure, the autocorrelation peak vector of the dynamic response of each measurement point has a fixed shape when the structure is intact. Therefore, the change in the peak value of the autocorrelation function of the dynamic response of the random vibration structure can be used to identify the damage to the structure.


### 4.2. Single Damage Location Analysis

According to the intrinsic morphological characteristics of an intact structure, when the structure is damaged, the peak curve of the autocorrelation function fluctuates, reducing the correlation between the autocorrelation peak curves of the damaged structure and intact structure.Structural damage can be simulated by reducing the stiffness of the elements. The specific method is as follows: two types of materials with different elasticity moduli are defined, the intact value and the one with a certain percentage reduction, corresponding to the intact and the damaged structures, respectively. Damage simulation of the structure can be realized by replacing a certain element of the beam with a reduced elastic modulus. Select beam I as the research object, when each element of 230 to 235 on the beam is damaged by 10%, the rate of change of the autocorrelation peak value of the acceleration response of each node between the damaged structure and the intact structure is shown in Figure8.Figure 8
The peak rate of change between the damaged structure and the intact structure.
(a)(b)(c)(d)(e)(f)It can be observed from the figure that when damage occurs to a certain element on the beam, the peak rate of change of the two nodes corresponding to the element is the largest. For instance, when unit 230 is damaged, the peak rate of change of nodes 800 and 801 at both ends of the element before and after the damage is the maximum; when unit 231 is damaged, the peak rate of change of nodes 801 and 802 of the unit is the maximum. The corresponding relation is also satisfied when other units are damaged separately. Therefore, the damage location on the structure can be determined by the maximum absolute value of the autocorrelation peak rate of change of the dynamic response of each measurement point before and after the damage, that is, the area between two adjacent nodes that has the largest change rate of the autocorrelation peak value corresponds to the damage location of the structure; therefore, the damage location of the structure is realized.


### 4.3. Double Damage Location Analysis

The middle span beam II of the second floor of the wooden frame was added as the research object. The number of elements of the beam is 454 to 459, and the corresponding number of nodes is 1,048 to 1,054. Beams I and II were set to be damaged simultaneously to analyze the effect of the rate of change of the autocorrelation peak value on double damage identification and location of the structure. The different working conditions of the double damage are listed in Table4.Table 4
Double damage conditions.
Damage conditionsCondition 1Condition 2Condition 3Condition 4Damage elements230/459231/456232/456232/458Corresponding nodes800, 801/1053, 1054801, 802/1050, 1051802, 803/1050, 1051802, 803/1052, 1053The peak rate of the autocorrelation of the acceleration response before and after damage under different damage conditions is shown in Figure9.Figure 9
The peak rate of change under different damage conditions. (a) Elements 230 and 459 damaged simultaneously by 10%, (b) elements 231 and 456 damaged simultaneously by 10%, (c) elements 232 and 456 damaged simultaneously by 10%, and (d) elements 232 and 458 damaged simultaneously by 10%.
(a)(b)(c)(d)It can be observed from the figure that when beams I and II are damaged at the same time, the damage index of the autocorrelation peak rate of the dynamic response before and after damage can also be used to accurately locate the damage and meet the damage location requirements. Therefore, the autocorrelation peak rate of change damage localization index is also effective for double damage.


### 4.4. Antinoise Interference Capability of Damage Index

Taking element 232 of beam I as an example to simulate the influence of noise on the damage location index in practical engineering, the degrees of damage are 5%, 10%, 20%, 30%, 35%, 40%, and 45%. White Gaussian noise with different noise levels was added to the nodal acceleration. The antinoise interference capability of the damage index under different degrees of damage was calculated, and the results are shown in Figure10.Figure 10
Antinoise capability of the damage index under different degrees of damage. Element 232 damaged by (a) 5%, (b) 10%, (c) 20%, (d) 30%, (e) 35%, (f) 40%, (g) 45%, and (h) 50%.
(a)(b)(c)(d)(e)(f)(g)(h)It can be observed from Figure10(a) that when element 232 is damaged by 5% (in the case of no noise), the two adjacent nodes with the highest autocorrelation peak change rate before and after structural damage are 802 and 803, which are also the two nodes at both ends of unit 232. This shows that the damage localization index constructed by the autocorrelation peak change rate can accurately identify early damage to the structure and has good capability in terms of damage identification and localization. At a signal-to-noise ratio (SNR) of 30 dB, the damage index can still accurately locate the structure and is not affected by measurement noise. However, when SNR = 20 dB, node 806 has the maximum peak change rate and is affected by noise, which does not meet the requirements of the structural damage location.When the unit damage is 10% and SNR = 20 dB, the two adjacent nodes with the highest peak change rates are 802 and 803, indicating that the damage location index is not affected by measurement noise, and the damage location can still be accurately identified. It can be observed from Figures10(a) to 10(g) that the antinoise interference capability of the damage location index gradually improved with an increase in the degree of damage. When the unit damage is 45%, it can still meet the requirements of the damage location under the condition of SNR = 0 dB, and accurate positioning is also possible. However, when the degree of damage is 50%, the antinoise capability of the damage location index remains stable at SNR = 0 dB, indicating that the antinoise capability of the damage location index of the autocorrelation peak change rate tends to be stable at 50% degree of damage.Therefore, for structures subjected to random vibration, the change rate of the peak value of the autocorrelation function at each measurement point before and after damage was used as the damage location index owing to its good antinoise interference capability.


## 4.1. Intrinsic Form of Autocorrelation Peak Vector

The central spans of the first and second floors of the timber frame of the bell tower, namely, beams I and II in Figure4, were selected for analysis. The number of elements of beam I is 230 to 235 and the corresponding number of nodes is 800 to 806, whereas the number of elements of beam II is 454 to 459 and the corresponding number of nodes is 1,048 to 1,054, as shown in Figure 5.Figure 5
Number of nodes and elements of beams I and II.The Xi’an Bell Tower is located at the intersection of the East, West, South, and North streets, and the surrounding traffic environment is complicated. Lines 2 and 6 of the Xi’an Subway meet at the bottom of the Bell Tower. The Bell Tower Station of Line 2 is approximately 60 m south of the bell tower. To analyze the impact of subway operation on the bell tower, Zhao [26] calculated the acting load of Xi’an Subway Line 2 and Line 6 during single line operation using the empirical analysis model of train load and obtained the load values under different running speeds.WhenV1 = 20 km/h, F(t) = 70,000 + 31.95 sin 3.49 t + 91.3 sin 17.44 t + 292.1 sin 69.8 t.WhenV2 = 40 km/h, F(t) = 70,000 + 127.8 sin 6.98 t + 365.2 sin 34.89 t + 1168.5 sin 139.56 t.V1 and V2 were applied to plinth node 501 as random excitation sample 1 and random excitation sample 2, respectively. The excitation time is t = 12 s and the time step is Δt = 0.01 s.TheX-direction horizontal acceleration responses of each node on beams I and beam II were extracted. Taking node 803 of beam I and node 1048 of beam II as examples, the autocorrelation value of node acceleration was calculated, as shown in Figure 6.Figure 6
Normalized autocorrelation values of (a) node 803 and (b) node 1048.
(a)(b)It can be observed from the diagram that the autocorrelation function of the acceleration response of nodes 803 and 1048 is largest whenτ = 0 because the signal correlated with itself and the degree of correlation is highest.Taking beam I as an example, the autocorrelation function of each node on the beam was calculated at subway loads ofV1 = 20 km/h and V2 = 40 km/h. The maximum value was transformed into an autocorrelation peak vector, and the peak value vectors of the autocorrelation functions of each node of the wooden frame intact structure under random excitations 1 and 2 were recorded as A1 and A2, respectively, as shown in Figure 7.Figure 7
Inherent form of the intact structures.It can be observed from the figure that the peak value curves of the two autocorrelation functions under two different random excitations are almost identical. This shows that, for a random vibration structure, the autocorrelation peak vector of the dynamic response of each measurement point has a fixed shape when the structure is intact. Therefore, the change in the peak value of the autocorrelation function of the dynamic response of the random vibration structure can be used to identify the damage to the structure.


## 4.2. Single Damage Location Analysis

According to the intrinsic morphological characteristics of an intact structure, when the structure is damaged, the peak curve of the autocorrelation function fluctuates, reducing the correlation between the autocorrelation peak curves of the damaged structure and intact structure.Structural damage can be simulated by reducing the stiffness of the elements. The specific method is as follows: two types of materials with different elasticity moduli are defined, the intact value and the one with a certain percentage reduction, corresponding to the intact and the damaged structures, respectively. Damage simulation of the structure can be realized by replacing a certain element of the beam with a reduced elastic modulus. Select beam I as the research object, when each element of 230 to 235 on the beam is damaged by 10%, the rate of change of the autocorrelation peak value of the acceleration response of each node between the damaged structure and the intact structure is shown in Figure8.Figure 8
The peak rate of change between the damaged structure and the intact structure.
(a)(b)(c)(d)(e)(f)It can be observed from the figure that when damage occurs to a certain element on the beam, the peak rate of change of the two nodes corresponding to the element is the largest. For instance, when unit 230 is damaged, the peak rate of change of nodes 800 and 801 at both ends of the element before and after the damage is the maximum; when unit 231 is damaged, the peak rate of change of nodes 801 and 802 of the unit is the maximum. The corresponding relation is also satisfied when other units are damaged separately. Therefore, the damage location on the structure can be determined by the maximum absolute value of the autocorrelation peak rate of change of the dynamic response of each measurement point before and after the damage, that is, the area between two adjacent nodes that has the largest change rate of the autocorrelation peak value corresponds to the damage location of the structure; therefore, the damage location of the structure is realized.


## 4.3. Double Damage Location Analysis

The middle span beam II of the second floor of the wooden frame was added as the research object. The number of elements of the beam is 454 to 459, and the corresponding number of nodes is 1,048 to 1,054. Beams I and II were set to be damaged simultaneously to analyze the effect of the rate of change of the autocorrelation peak value on double damage identification and location of the structure. The different working conditions of the double damage are listed in Table4.Table 4
Double damage conditions.
Damage conditionsCondition 1Condition 2Condition 3Condition 4Damage elements230/459231/456232/456232/458Corresponding nodes800, 801/1053, 1054801, 802/1050, 1051802, 803/1050, 1051802, 803/1052, 1053The peak rate of the autocorrelation of the acceleration response before and after damage under different damage conditions is shown in Figure9.Figure 9
The peak rate of change under different damage conditions. (a) Elements 230 and 459 damaged simultaneously by 10%, (b) elements 231 and 456 damaged simultaneously by 10%, (c) elements 232 and 456 damaged simultaneously by 10%, and (d) elements 232 and 458 damaged simultaneously by 10%.
(a)(b)(c)(d)It can be observed from the figure that when beams I and II are damaged at the same time, the damage index of the autocorrelation peak rate of the dynamic response before and after damage can also be used to accurately locate the damage and meet the damage location requirements. Therefore, the autocorrelation peak rate of change damage localization index is also effective for double damage.


## 4.4. Antinoise Interference Capability of Damage Index

Taking element 232 of beam I as an example to simulate the influence of noise on the damage location index in practical engineering, the degrees of damage are 5%, 10%, 20%, 30%, 35%, 40%, and 45%. White Gaussian noise with different noise levels was added to the nodal acceleration. The antinoise interference capability of the damage index under different degrees of damage was calculated, and the results are shown in Figure10.Figure 10
Antinoise capability of the damage index under different degrees of damage. Element 232 damaged by (a) 5%, (b) 10%, (c) 20%, (d) 30%, (e) 35%, (f) 40%, (g) 45%, and (h) 50%.
(a)(b)(c)(d)(e)(f)(g)(h)It can be observed from Figure10(a) that when element 232 is damaged by 5% (in the case of no noise), the two adjacent nodes with the highest autocorrelation peak change rate before and after structural damage are 802 and 803, which are also the two nodes at both ends of unit 232. This shows that the damage localization index constructed by the autocorrelation peak change rate can accurately identify early damage to the structure and has good capability in terms of damage identification and localization. At a signal-to-noise ratio (SNR) of 30 dB, the damage index can still accurately locate the structure and is not affected by measurement noise. However, when SNR = 20 dB, node 806 has the maximum peak change rate and is affected by noise, which does not meet the requirements of the structural damage location.When the unit damage is 10% and SNR = 20 dB, the two adjacent nodes with the highest peak change rates are 802 and 803, indicating that the damage location index is not affected by measurement noise, and the damage location can still be accurately identified. It can be observed from Figures10(a) to 10(g) that the antinoise interference capability of the damage location index gradually improved with an increase in the degree of damage. When the unit damage is 45%, it can still meet the requirements of the damage location under the condition of SNR = 0 dB, and accurate positioning is also possible. However, when the degree of damage is 50%, the antinoise capability of the damage location index remains stable at SNR = 0 dB, indicating that the antinoise capability of the damage location index of the autocorrelation peak change rate tends to be stable at 50% degree of damage.Therefore, for structures subjected to random vibration, the change rate of the peak value of the autocorrelation function at each measurement point before and after damage was used as the damage location index owing to its good antinoise interference capability.


## 5. Conclusion

In this study, the peak change rate of the autocorrelation function of the dynamic response before and after damage was used as the damage index to locate the damage to the Xi’an Bell Tower by reducing the stiffness of each element to simulate the damage to the structure. The results show that(1)
The peak value vector of the autocorrelation function of the dynamic response of the intact structure under random vibration has a fixed shape, which can be used as the basis for structural damage identification and location.(2)
The rate of change of the peak value of the autocorrelation function of the acceleration dynamic response to each measurement point before and after the structural damage was calculated. The two adjacent measurement points with the highest degree of change corresponded to two nodes at both ends of the damage unit, indicating that using the rate of change of the autocorrelation peak value as the damage location index could accurately determine the damage location of the structure.(3)
The antinoise capability of the damage location index gradually improved with an increase in the degree of damage. When the degree of damage of the structure is 45%, the damage location index can still accurately determine the damage location of the structure if the SNR is 0 dB, and the damage location index is not affected by measurement noise. At 50% degree of damage, the antinoise capability of the positioning index tended to be stable.Therefore, the autocorrelation peak change rate of the dynamic response before and after damage was used as the damage location index, which can accurately determine the damage location of the structure and has good antinoise capability. However, one limitation of the method proposed in this study is that it requires many measurement points; therefore, optimizing the location and number of measurement points is a future research direction.



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*Source: 6683666-2021-07-07.xml*