Damage detection in bridges under moving loads based on subspace projection residuals

Abstract

This paper proposes a data-driven method using subspace projection residual of the responses to identify the damage locations in bridges subjected to moving loads. In this method, a moving window with a certain length determined by the sampling frequency and the fundamental frequency of the measured responses is used to cut out the acceleration responses of the bridge subjected to a moving vehicle. The characteristic subspaces of the windowed signals are subsequently extracted to calculate the local damage index using the subspace projection residual. When the window moves to the damage location, the orthogonality between the active subspace of the damaged state and the null subspace of the healthy state is invalid, which leads to a relatively large projection residual that can be used to localize the damage. To improve the reliability of the proposed approach, a one-side upper confidence limit is introduced. A simply supported beam bridge subjected to a moving mass is simulated to verify the effectiveness of the proposed method. Numerical results indicate that the proposed approach can accurately localize the single and multiple damages, even when the responses are smeared with a significant noise. Experimental tests conducted on a steel beam bridge model also demonstrate the performance and accuracy of the proposed approach. The results demonstrate that the proposed method can localize the damage even with a small number of sensors, indicating the method has a good and promising performance for practical engineering applications.

Keywords

data-driven structural damage localization subspace projection residual moving loads limited number of sensors

Introduction

Structural health monitoring (SHM) becomes more and more important since modern society depends heavily on structural systems like tunnels, buildings, and bridges. SHM aims to monitor and assess the performances of structures in service considering material aging and deterioration, which ensures the designed operational life and reduces maintenance and repair cost. Structural damage detection based on measured vibration data is one of the most significant research topics in the field of SHM (Carden and Fanning, 2004; Yan et al., 2007; Wei and Qiao, 2011). The basic idea is that the dynamic characteristics in relation to structural physical parameters, for example, mass, stiffness, and damping, will alter when the structures suffer damages caused by various external excitations, such as earthquakes and vehicle loadings, and material deterioration. Vibration-based methods can be generally classified as model-based methods and data-driven methods. Model-based methods require establishing an accurate finite element model (FEM) of the structure, and update the current state of the structure through extensive modification and calibration (Alkayem et al., 2018). The model-based methods have been widely studied (Brownjohn et al., 2001; Wu and Li, 2006; Behmanesh and Moaveni, 2015). However, owning to the complexity of civil engineering structures, it is very difficult and time consuming to establish an accurate model in practical applications.

The data-driven methods, on the other hand, detect damage directly from the measured response data of structures (e.g., displacement, acceleration, and strain) for determining whether the structural state is normal. Common data-driven methods include wavelet transform methods, time series methods, and principal component analysis (PCA). The wavelet method can decompose the response signal to identify small changes in the signals (Peng and Chu, 2004; Zhu et al., 2009). Hong et al. (2002) demonstrated the effectiveness of using wavelet transform to detect damage of a beam by means of the estimated Lipschitz exponent. Wu and Wang (2011) detected the damage of a cracked cantilever aluminum beam with the spatial wavelet transform. The time series method performs time series modeling and analysis on structural dynamic responses and realizes damage detection through characteristic changes of model coefficients (Nair et al., 2006; Carden et al., 2008; Sadhu and Hazra, 2013). Jayawardhana et al. (2016) presented an index using the auto-regressive (AR) model coefficients and detected the damage of an experimental steel beam with the Fisher criterion of the computed index. Zhang (2007) proposed a damage feature extraction technique based on time series analysis combining AR model and auto-regressive with exogenous inputs prediction model, to detect the damage of a three-span continuous girder bridge with reasonable damage severity. PCA methods decompose the dynamic responses into the orthogonal space and detect whether the structure is abnormal by the changes in signal characteristics in the main orthogonal direction (Bellino et al., 2010; Torres-Arredondo et al., 2014). Hajrya and Mechbal (2013) developed a robust damage detection method based on PCA, which relies on an original index derived from the projection of the separation matrix. Tibaduiza et al. (2016) utilized the analysis of the residual data matrix to determine the presence of damages based on PCA. Nie et al. (2020) proposed a novel bridge condition monitoring method through the fixed moving principal component analysis, which was successfully applied to both experimental beam bridge and a long span suspension bridge in service.

Similar as the PCA method, the subspace-based method as a data-driven method, performs proper orthogonal decomposition on the Hankel matrix constructed by the structural dynamic responses to obtain the principal components, and then detect the structural damage by using the difference in the subspace information including the modal information behind the principal components (Overschee and Moor, 1996; Juang and Phan, 2001). Among the subspace-based methods, subspace projection residual is commonly used. Basseville et al. (2000) systematically proposed that subspace-based methods for eigenstructure identification can be turned into fault detection methods by using a subspace-based residual function and a chi-square test. Basseville et al. (2004) afterwards detected the possible damaged state of a numerical structure by the residual and a chi-square test, and proposed a damage localization solution by introducing aggregated sensitivities of the modes and mode shapes with respect to structural parameters of FEM. However, there are no experimental tests for further verifications. On the basis of this existing study, Döhler et al. (2014) took the changes in the ambient excitation covariance into account and developed a new robust statistical test to identify the whole condition of structures, which was validated with three numerical examples. Yan and Golinval (2006) proposed multiple indicators to measure the residuals of orthogonal characteristic subspaces and accurately identified the damaged state of an experimental aircraft model. However, most of the methods based on subspace projection residual only concentrate on detecting the occurrence of damage, which is only the first step of SHM. According to Farrar et al. (1994), the complete procedure of SHM generally includes damage existence detection, damage localization, damage severity identification, and the remaining life estimation of structures. These four steps are usually addressed sequentially. Once the damage happens, it is critical to localize it promptly, which paves way to maintain structures in time.

In view of the issues above, the motivation behind this study is to develop a novel data-driven method using subspace projection residual in conjunction with damage sensitive feature (DSF) to localize the structural damage. In this paper, a moving window with a fixed length is firstly defined by a certain principle. When a moving vehicle passes through the bridge, the acceleration response data of the bridge is continuously collected and cut out by the moving window. The subspace information of the windowed data under the current state (i.e., possibly damaged state) and that under the healthy state are simultaneously acquired in order to calculate the real time DSF. When there is no damage in the local bridge section, the active subspace determined by the dominant dynamic information under the current state and the null subspace determined by the secondary dynamic information or noise under the healthy state are nearly completely orthogonal. It means that the projection residuals of them are very small or even close to zero. When damage occurs, large residuals will appear in the projection of the characteristic subspaces under two structural states, which in turn serve as DSF to indicate the existence of the damage. Numerical simulations performed on a beam bridge subjected to a moving mass and experiments performed on a laboratory steel beam bridge model under the moving vehicle are conducted to demonstrate the effectiveness and performance of the proposed approach.

The rest of the paper is organized as follows: dynamic response analysis of a beam bridge subjected to a moving vehicle is introduced in the Dynamic Response Analysis of a Beam Bridge Subjected to a Moving Vehicle section. Methodology about the extraction of the subspaces and DSF is presented in the Methodology section. The Numerical Simulations section gives the numerical simulations to verify the feasibility of the proposed method, while the experimental results are shown in the Experimental Verifications section to further demonstrate the effectiveness and performance of the proposed approach. Finally, the conclusions are summarized in the Conclusions section.

Dynamic response analysis of a beam bridge subjected to a moving vehicle

The coupling vibration of a moving vehicle-bridge system is assumed as a moving mass travels along the beam bridge. The simplified model is shown in Figure 1. When neglecting the damping, rotary inertia, and shearing force effects, the equation of motion for an Euler–Bernoulli beam of length L subjected to a travelling mass M_v with a velocity of v is expressed as (Nikkhoo et al., 2007; Dehestani et al., 2009; Nie et al., 2019)

m (x) \frac{\partial^{2} w (x, t)}{\partial t^{2}} + E I \frac{\partial^{4} w (x, t)}{\partial x^{4}} = M_{v} (g - \frac{\partial^{2} w (x, t)}{\partial t^{2}} - 2 v \frac{\partial^{2} w (x, t)}{\partial x \partial t} - v^{2} \frac{\partial^{2} w (x, t)}{\partial x^{2}}) δ (x - v t)

(1)

where EI is the bending stiffness, m is the mass per unit length of the beam, g is the gravitational acceleration,

δ (x - v t)

is Dirac delta function that represents the location of the moving vehicle, w(x, t) denotes the displacement of the beam, where x and t represent the location of the vehicle on the beam and the corresponding time instant, respectively. Employing the modal superposition method, the displacement

w

can be written in the form of generalized coordinates as

w (x, t) = \sum_{i = 1}^{n} ϕ_{i} (x) q_{i} (t)

(2)

where

q_{i} (t)

is time dependent modal response,

ϕ_{i} (x)

is the ith orthogonal mode shape of the beam, n is the total number of considered modes. Substituting equation (2) into equation (1), and then multiplying both sides by

ϕ_{k} (x)

, using the orthogonality condition of Eigen functions yields

{\overset{..}{q}}_{i} + ω_{i}^{2} q_{i} = \frac{M_{v}}{m (x)} ϕ_{i} (x) [g - \sum_{k = 1}^{n} \overset{..}{q_{k}} ϕ_{k} (x) - 2 v \sum_{k = 1}^{n} \dot{q_{k}} {ϕ^{'}}_{k} (x) - v^{2} \sum_{k = 1}^{n} q_{k} {ϕ^{″}}_{k} (x)]

(3)

where

ω_{i}^{2} = D ϕ_{i} (x) / m (x)

D = E I \frac{\partial^{4}}{\partial x^{4}}

Figure 1.

The simplified beam bridge model subjected to a moving vehicle.

When a damage is introduced in the beam at x₁, dividing the beam into two segments as shown in Figure 1, the discontinuity condition at the damaged location is expressed as (Pala and Reis, 2013)

{w^{'}}_{2} (x_{1}^{+}, t) - {w^{'}}_{1} (x_{1}^{-}, t) = c_{θ} h {w^{″}}_{2} (x_{1}^{+}, t)

(4)

where

w_{1}

and

w_{2}

are the vertical displacements of left and right segments, respectively,

x_{1}^{-}

and

x_{1}^{+}

denote the locations immediately after and before the damage location, respectively, h is the height of the beam,

c_{θ}

is a function of the damage ratio γ (γ =a/h, where a is the damage depth) and it depends on the cross-section geometry of the beam. For a beam bridge with a rectangular cross section,

c_{θ}

is expressed as (Ariaei et al., 2009, 2010)

c_{θ} = 2 {(γ / 1 - γ)}^{2} (5.93 - 19.69 γ + 37.14 γ^{2} - 35.84 γ^{3} + 13.12 γ^{4}) .

(5)

The responses of intact and damaged beam, which are used in the damage detection procedure, can be calculated using the Runge–Kutta integration method.

Methodology

Subspace analysis

The subspace-based methods generally extract the principal components orthogonal to each other through proper orthogonal decomposition (POD) of the Hankel matrix to identify structural modal parameters. Identifying structural damage is conducted through the changes of modal parameters (De Boe et al., 2003; Nguyen and Golinval, 2010). It is noted that the proposed approach in this study detects the damage by the relative changes between characteristic subspaces under the damaged and undamaged states directly, without identifying structural modal parameters. To achieve this goal, the data-driven Hankel matrix of the dynamic responses needs to be constructed firstly. In general, an original dataset matrix X with n variables and m samples can be given as

X = [\begin{matrix} x_{11} & x_{12} & ... & x_{1 m} \\ x_{21} & x_{22} & ... & x_{2 m} \\ ... & ... & ... & ... \\ x_{n 1} & x_{n 2} & ... & x_{n m} \end{matrix}] = [\begin{matrix} x_{1} & x_{2} & \dots & x_{m} \end{matrix}]

(6)

where n denotes the considered sensor number. The matrix X is normalized as a matrix with zero mean and unity covariance for eliminating the influence caused by different magnitudes of variables in X . The data-driven Hankel matrix is defined as

H = [\frac{\begin{matrix} x_{1} & x_{2} & ... & x_{m - 2 r + 1} \\ x_{2} & x_{3} & ... & x_{m - 2 r} \\ ... & ... & ... & ... \\ x_{r} & x_{r + 1} & ... & x_{m - r} \end{matrix}}{\begin{matrix} x_{r + 1} & x_{r + 2} & ... & x_{m - r + 1} \\ x_{r + 2} & x_{r + 3} & ... & x_{m - r + 2} \\ ... & ... & ... & ... \\ x_{2 r} & x_{2 r + 1} & ... & x_{m} \end{matrix}}] = (\frac{X_{p}}{X_{f}})

(7)

where 2r is the user-defined number of row blocks. Referring to literature (Yan and Golinval, 2006; Nguyen and Golinval, 2010), the parameter 2r is set as 10 in this paper because it is sufficient to construct the Hankel matrix while reducing computational cost. The Hankel matrix is split into a “past” part X _p and a “future” part X _f. The core idea is to retain the past information to predict the future information, which allows to take temporal correlations between measurements into account when current data depends on past data (Yan and Golinval, 2006; Nguyen and Golinval, 2010).

The singular value decomposition (SVD) is an effective method to extract the principal components of a matrix. By applying the SVD to the Hankel matrix, we have

H = U S V^{T}

(8)

where U is an orthonormal matrix with each column representing a principal component, V is the time amplitudes of the principal components (De Boe et al., 2003). The principal diagonal of the matrix S gives the singular values sorted by the descending order, which represent the importance of each principal component. The principal components corresponding to the large singular values contain the main dynamic information, and the principal components corresponding to the remaining singular values generally represent the secondary dynamic information and noise. Therefore, U can be divided into two parts, named the active subspace U ₁ and the null subspace U ₂ (Yan and Golinval, 2006), when using those important principal components. Equation (8) can be written as

H = [\begin{matrix} U_{1} & U_{2} \end{matrix}] [\begin{matrix} S_{1} & 0 \\ 0 & S_{2} \end{matrix}] {[\begin{matrix} V_{1} & V_{2} \end{matrix}]}^{T} .

(9)

The contribution ratio of the principal component can be used to accurately reflect the modal participation rate in the energy sense (Ma et al., 2019). The number of principal components j of the active subspace U ₁ can be determined by the cumulative contribution ratio (CCR) of the principal components, which yields

C C R = \frac{\sum_{i = 1}^{j} λ_{i}}{\sum_{i = 1}^{k} λ_{i}}, k = 2 r n

(10)

where

λ_{i}

is the ith principal diagonal element of S . Generally, the CCR of principal components with a value of 90% can be used as a benchmark to select the number of principal components of the active subspace U₁ (Yan and Golinval, 2006; Nguyen and Golinval, 2010).

Damage sensitive feature

In this study, the main goal is to locate the damage of the beam bridge. A moving window is defined to cut out the measured responses, and the characteristic subspaces of the windowed dataset are extracted subsequently. Sliding the window, the ith observed dataset is expressed as

X_{n \times l}^{i} = [\begin{matrix} x_{i, 1} & x_{i + 1,1} & ... & x_{i + l - 1,1} \\ x_{i, 2} & x_{i + 1,2} & ... & x_{i + l - 1,2} \\ ... & ... & ... & ... \\ x_{i, n} & x_{i + 1, n} & ... & x_{i + l - 1, n} \end{matrix}] = [\begin{matrix} x_{i,} & x_{i + 1,} & \dots & x_{i + l - 1,} \end{matrix}]

(11)

where l is the length of the window, i is the ordinal number of the moving window. The corresponding data-driven Hankel matrix is expressed as

H^{i} = [\frac{\begin{matrix} x_{i} & x_{i + 1} & ... & x_{i + l - 2 r} \\ x_{i + 1} & x_{i + 2} & ... & x_{i + l - 2 r + 1} \\ ... & ... & ... & ... \\ x_{i + r - 1} & x_{i + r} & ... & x_{i + l - r - 1} \end{matrix}}{\begin{matrix} x_{i + r} & x_{i + r + 1} & ... & x_{i + l - r} \\ x_{i + r + 1} & x_{i + r + 2} & ... & x_{i + l - r + 1} \\ ... & ... & ... & ... \\ x_{i + 2 r - 1} & x_{i + 2 r} & ... & x_{i + l - 1} \end{matrix}}] .

(12)

The ith data-driven Hankel matrix in equation (12) is analyzed by using SVD, which can be expressed as

H^{i} = [\begin{matrix} U_{1}^{i} & U_{2}^{i} \end{matrix}] [\begin{matrix} S_{1}^{i} & 0 \\ 0 & S_{2}^{i} \end{matrix}] {[\begin{matrix} V_{1}^{i} & V_{2}^{i} \end{matrix}]}^{T}

(13)

where

U_{1}^{i}

U_{2}^{i}

are the ith active subspace and the ith null subspace, respectively.

However, it is necessary to select a value of the window length l which is the key parameter of the proposed method. It should be sufficiently large so that the windowed dataset can capture most of the vibrational information of the responses, but a too large value l will lead to a decrease in the identification resolution of the damage. The length of the moving window in this study is determined as

l = 2 \frac{f_{s}}{f_{1}}

(14)

where f_s is the sampling frequency, f₁ is the fundamental frequency of the responses. The determination of l is based on the Nyquist–Shannon sampling theorem (Shannon, 1949; Romo-Cárdenas et al., 2018), that is, the sampling frequency should be larger than or equal to twice of the largest frequency of the dynamic system. However, the sampling frequency of the measured responses of the beam is set before the dynamic tests. Therefore, in this study, the window length l should be larger than or equal to the total sampling points in two fundamental periods of the responses. If l is less than this value, the dataset in the window will lose the dynamic information of the component corresponding to the fundamental frequency which is generally the most important component in structure vibration responses of a bridge subjected to a moving mass.

When the moving mass passes through the bridge, the acceleration responses of the bridge are collected and segmented by the moving window. The fundamental frequency of the undamaged beam is directly used to compute the length of the used window in all the damage scenarios, since a crack damage will only lead to a small frequency decrease (Nie et al., 2012). A total of m-l+1 windows can be constructed, where m is the total length of the signals. The ith active subspace $U_{1}^{i}$ and the ith null subspace $U_{2}^{i}$ can be extracted by following the above mentioned steps. According to the properties of the principal components, $U_{1}^{i}$ is orthogonal to $U_{2}^{i}$ , that is, the projection between them is a zero matrix, which can be expressed as

U_{1}^{i^{T}} U_{2}^{i} = 0 .

(15)

Assuming that $U_{1, c u r r e n t}^{i}$ is the ith active subspace under the current state (i.e., possibly damaged state) and $U_{2, h e a l t h y}^{i}$ is the ith null subspace under the healthy state, respectively. The residual of the former projection on the latter can be given as

Δ_{i} = U_{1, c u r r e n t}^{i^{T}} U_{2, h e a l t h y}^{i} (i = 1, \dots, m - l + 1)

(16)

If there is no damage in the current state, $Δ_{i}$ is almost a zero matrix. It means that the projection residual between the active subspace under the current state and the null subspace under the healthy state is almost zero. It should be mentioned that even if there is no damage in the current state, $Δ_{i}$ will not be strictly equal to a zero matrix, since it is affected by uncertainties and measurement noise (Yan and Golinval, 2006). However, when the current state is damaged, the orthogonality between the active subspace under the current state and the null subspace under the healthy state will be invalid to a certain extent, which leads to a relatively large projection residual. In order to measure the residual matrix $Δ_{i}$ , a damage sensitive feature, namely, subspace projection residual (SPR), is defined as

S P R_{i} = {‖ v e c (Δ_{i}) ‖}_{2}^{2}

(17)

where vec(.) denotes the column stacking operator, and ||.||₂ denotes the L2 norm operator. The index is the square of the norm.

By analyzing the windowed subspace information and calculating the damage index SPR, the local singularity of SPR value with the vehicle passing through the bridge could be used to indicate the damage location.

Flowchart of the proposed algorithm

The overall methodology followed in this paper is shown in Figure 2, which comprises of four key parts: windowing, construction, extraction, and identification. For understanding the detailed process, the basic steps of the proposed approach are described as follows:

(i) Windowing: The length of the moving window l is determined based on the sampling frequency and the fundamental frequency of the measured accelerations. The responses of both the current state (possibly damaged) and healthy state are segmented to obtain the ith windowed responses matrix $X_{n \times l, c u r r e n t}^{i}$ and $X_{n \times l, h e a l t h y}^{i}$ , respectively.

(ii) Construction: The matrix X is normalized to have zero mean and unity covariance, and the data-driven Hankel matrices $H_{c u r r e n t}^{i}$ and $H_{h e a l t h y}^{i}$ are constructed, respectively.

(iii) Extraction: SVD is applied to these two Hankel matrices to extract the active subspace $U_{1, c u r r e n t}^{i}$ under the current state and the null subspace $U_{2, h e a l t h y}^{i}$ under the healthy state, respectively. The projection residual $Δ_{i}$ is obtained subsequently to check the orthogonality.

(iv) Identification: The damage sensitive feature SPR is calculated to determine whether it exceeds the threshold. If SPR is larger than the threshold, the local structural damage is located by multiplying the windowed time domain responses and the speed of moving vehicle. Otherwise, let i=i+1, and the iteration continues. Sliding the moving window to obtain the (i+1)th windowed responses matrices, the same procedure in the above steps is followed to analyze the whole time domain responses.

Figure 2.

Flowchart of the proposed approach.

Numerical simulations

Numerical model

To demonstrate the effectiveness and feasibility of using the proposed approach for locating the damage, a simply supported beam as shown in Figure 1 is employed as an example for numerical studies. The dimension of the beam is defined as follows: the height, width, and length of the beam are set as h=0.1 m, b=0.2 m, and L=10 m, respectively. The Young’s modulus, density, and the sampling frequency are set as E=200 GPa, ρ=7850 kg/m³ and f_s=500 Hz, respectively. The moving mass with a weight of 200 kg is used to simulate a passing vehicle with various velocities travelling on the beam model. The Rayleigh damping coefficients in this finite element calculation are α=0.001 and β=0.0001, respectively. In this simulation, the considered modal number n in equation (2) is selected to be three, since the first three modes are sufficient to capture the dominating vibration of the bridge in the vehicle-bridge systems (Zhu et al., 2018; Li et al., 2019). Seven measurement points are set at each one-eighth length of the beam bridge to collect the acceleration responses, which will be used in the subsequent damage localization.

Single damage detection

Scenarios of different damage severities

The cracks with the depth ratio $γ$ equal to 10%, 20%, 30%, 40%, and 50%, respectively, located at the position of L_d=0.3 (L_d=x₁/L is the relative distance from the left support of the beam), are assumed as five damage scenarios with different damage severities by deleting elements in the numerical simulations for single damage detection. The velocity of the moving mass here is set as 0.2 m/s.

Following the procedure as shown in Figure 2, the windowed damage index can be calculated. The length of moving step is the sampling interval. In order to determine the window length with equation (14), the fundamental frequency of the bridge should be identified first. The acceleration responses and the corresponding Fourier spectrum at mid-span from undamaged, 30% damaged and 50% damaged states are shown in Figure 3. It can be observed that it is difficult to identify the damage reliably by utilizing the change in responses and natural frequencies. As shown in Figure 3(b), the fundamental frequency f₁ of the response is 2.213 Hz. Thus, the window length l is obtained as 451 (l=2*500/2.213).

Figure 3.

Dynamic response at mid-span from different damage scenarios: (a) acceleration responses and (b) Fourier spectrum.

Figure 4(a) shows the original curve of damage index SPR for 50% damaged scenario, and the triangle on the horizontal axis shows the damage location. It can be observed that many peaks interfere with the results as shown in Figure 4(a). Hence, a Moving Filter Function (MFF) (Nie et al., 2017) is utilized to smooth the results to eliminate the fluctuations but retain the damage information. In the following studies, all the cases under different damage scenarios are smoothed by MFF as

\bar{a} (i) = \frac{1}{δ} \sum_{j = i - \frac{δ}{2}}^{j = i + \frac{δ}{2}} a (i + j)

(18)

where a and i are the signal to be smoothed and the time point, respectively. δ is the span of the moving function that can be defined as

δ = T_{δ} f_{s}

(19)

where T_δ is the time span of the dynamic component to be smoothed. Taking T_δ equal to 1/f_s, the smoothed result of 50% damage scenario is shown in Figure 4(b). The smoothed curve has an obvious peak at the position L_d = 0.3, which is the introduced damage location in the simulation. However, there are still many relatively small peaks that may be caused by signal changes appear in other locations. In order to improve the effectiveness of the proposed approach, a one-side upper confidence limit (UCL) (Han et al., 2005; Ren et al., 2008; Li et al., 2014) is introduced to define a threshold for a more reliable damage detection. It is expressed as

T^{α} = μ + Z_{α} σ

(20)

where

T^{α}

is the confidence limit value,

μ

and

σ

are the mean value and standard deviation of the obtained damage indices, respectively,

Z_{α}

is the value of a standard normal distribution with zero mean and unit variance such that the cumulative probability is

100 (1 - α)

α

is selected to be 1% in this study, which means that 99% of the SPR curve are under this confidence limit, which can be considered as a threshold value for the possible abnormality in structural health monitoring. Since the threshold is based on the statistical properties of the calculated damage indices from measured responses, the damage index values which are larger than the threshold value can be used to indicate the damage locations. Figure 5 shows the results under the other four different scenarios with damage severity varying from 10% to 40%. Only the SPR values at the position L_d = 0.3 are higher than the threshold in all cases, indicating that the damage can be localized confidently and accurately. Moreover, the value of SPR increases with the increasing damage severity, which leads the SPR value to be capable of indicating the damage severity to some extent.

Figure 4.

Damage index SPR of 50% damaged beam: (a) original curve and (b) smoothed curve.

Figure 5.

Damage detection results with different damage degrees at Ld = 0.3 when velocity. is 0.2 m/s: (a) 10%, (b) 20%, (c) 30%, and (d) 40%.

Scenarios of different damage locations

In order to evaluate the efficacy of the proposed method to detect the damage at different locations, scenarios with damage at the position of L_d = 0.5 and different severities are also studied. The structural physical parameters are the same as those mentioned in the above Scenarios of Different Damage Severities section except the damage location. As shown in Figure 6, all the curves of different damage severities have obvious peaks at L_d = 0.5, which is the introduced damage location in the numerical model. The results indicate that the proposed approach can well localize the damage at different locations.

Figure 6.

Damage detection results with different damage degrees at Ld = 0.5 when velocity is 0.2 m/s: (a) 10%, (b) 20%, (c) 30%, (d) 40%, and (e) 50%.

Effect of velocity of the moving vehicle

The effect of the moving vehicle velocity on the identification accuracy is investigated in this section. Two different velocities, namely, 0.5 m/s and 0.8 m/s, are considered, while the damage location is set at the position of L_d = 0.3.

As is shown in Figures 7 and 8, the curves for the cases with a moving vehicle velocity of 0.5 m/s and 0.8 m/s have obvious peaks that are higher than the threshold at the location L_d = 0.3, indicating the damage can be reliably located. The detection results of these two scenarios with higher speeds of 0.5 m/s and 0.8 m/s have less peaks that may disturb detection accuracy than those in the scenarios with a low moving speed. This is because when the moving mass velocity is high, the amount of the windowed segments is less than that in the scenarios with low velocities, which leads to a decrease in vibration information.

Figure 7.

Damage detection results with different damage degrees at Ld = 0.3 when velocity is 0.5 m/s: (a) 10%, (b) 20%, (c) 30%, (d) 40%, and (e) 50%.

Figure 8.

Damage detection results with different damage degrees at Ld = 0.3 when velocity is 0.8 m/s: (a) 10%, (b) 20%, (c) 30%, (d) 40%, and (e) 50%.

Multiple damage detection

In this section, damage detection of multiple damage scenarios is performed. Two damages at L_d = 0.3 and L_d = 0.6 are introduced in the model, and the velocity of the moving mass is set as 0.2 m/s. Five multiple damage scenarios, namely, MD1, MD2, MD3, MD4, and MD5 are defined, as listed in Table 1. The results are shown in Figure 9. As shown, there are peaks whose values exceed the threshold lines at the position of imposed damages, indicating the multiple damage locations can be detected. For case MD1 (as shown in Figure 9(a)), two damages are localized accurately, and the peak at L_d = 0.3 is higher than that at L_d = 0.6, since the damage at the position L_d = 0.3 is severer. For case MD2 (as shown in Figure 9(b)), the peak at L_d = 0.6 becomes more obvious due to the increased damage level. It is noted that two imposed damages can be well identified. As shown in Figure 9(c–e), with the damage severity at L_d = 0.6 increasing, the corresponding peak becomes higher while less prominent at L_d = 0.3, indicating that only the damage at L_d = 0.6 is detected. In terms of multiple damage detection, the performance of the proposed method depends on the damage severities, and the severer one can be always confidently identified, although there is also a relatively high peak around the damage location with a lower damage severity.

Table 1.

Damage degrees for multiple damage scenarios.

Damage location	Damage scenarios
Damage location	MD1, %	MD2, %	MD3, %	MD4, %	MD5, %
L_d = 0.3	20	20	20	20	20
L_d = 0.6	10	20	30	40	50

Figure 9.

Damage detection results under different multiple damage scenarios: (a) MD1, (b) MD2, (c) MD3, (d) MD4, and (e) MD5.

Robustness to noise

The responses smeared with different levels of Gaussian noise are used for damage detection in this section, in order to study the robustness of the proposed approach. The noise is added to the original signal, and the Signal to Noise Ratio (SNR) which is used to quantify the noise level can be defined as (Elliott, 2000)

S N R = 10 \lg \frac{P_{x}}{P_{N}}

(21)

where

P_{x}

and

P_{N}

are the power of the original signal and measurement noise, respectively. Two SNRs, namely, 40 dB and 25 dB, are considered here. The velocity of the moving mass is set as 0.2 m/s, while the damage is imposed at L_d = 0.5. Figure 10 shows the results using acceleration responses contaminated with 40 dB noise. It can be observed that the damage location is identified clearly under damage scenarios with different severities, which indicates that the proposed method is capable of locating the single damage in a noisy environment.

Figure 10.

Damage detection results for different damage degrees using 40 dB noise-contaminated responses: (a) 10%, (b) 20%, (c) 30%, (d) 40%, and (e) 50%.

When the noise level is expanded to 25 dB, as shown in Figure 11(a), false detection occurs at around L_d = 0.8 when the damage level is 10% while the actual damage location can be well detected. However, with the increasing of damage degree, the damage location can be localized confidently and accurately without any false detection, as shown in Figure 11(b–e). The results indicate that the proposed approach has a good robustness performance on damage localization using the noisy measurement responses.

Figure 11.

Damage detection results for different damage degrees using 25 dB noise-contaminated responses: (a) 10%, (b) 20%, (c) 30%, (d) 40%, and (e) 50%.

Experimental verifications

Experimental setup

To further verify the applicability and efficacy of utilizing the proposed method for damage localization, dynamic tests on a steel beam bridge model subjected to a moving vehicle are conducted, and the measured acceleration responses are subsequently used to detect the damage. Details about the experimental setup are shown in Figure 12. As shown in Figure 12(a), the model has three spans, but only the middle span with the length of 6m was instrumented and tested in this experiment. To avoid the error caused by signals selection, the acceleration responses of the bridge started to be collected when the trolley enters the main bridge and stopped when it leaves. The two side spans built with wood board were served as the approach and ending bridges to ensure that the moving vehicle passes through the middle span with a constant speed. The tested beam model is a hollow rectangular steel tube with the cross-section dimension of 200 mm (width) × 100 mm (height) × 3 mm (thickness), and the edge with 200 mm width is treated as the bridge deck. Both ends of the model are supported with a steel angle and bonded together with a strong metal adhesive.

Figure 12.

Experimental setup of bridge model subjected to a moving vehicle: (a) tested bridge model, (b) the trolley guide rails and the vehicle model, (c) accelerometers, (d) direct-current motor, and (e) DH5922 N dynamic data acquisition system.

To facilitate the control of vehicle movement, two steel angles are used as trolley guided rails bonded on the bridge deck using the strong metal adhesive with the angle placed upward. The surface of the wheel contacting to the rail is an inverted triangular groove, as shown in Figure 12(b), so that the moving direction of the trolley can be well controlled by the guide rails. Two masses of the trolley including the additional static weight of 10.5 kg and 20.5 kg are studied in this experimental tests. The traction power is provided by a direct-current (DC) motor, as shown in Figure 12(d), and the different speeds of the moving vehicle are realized by adjusting the gearbox which controls the traction speed. Dynamic tests with two moving velocities, namely, 0.25 m/s and 0.5 m/s are conducted. Integrated Circuits Piezoelectric (ICP) accelerometers as shown in Figure 12(c) and DH5922 N dynamic data acquisition system as shown in Figure 12(e) are used to collect the acceleration responses of the bridge model. The sampling frequency is set as 500 Hz. The schematic experimental setup and sensor locations are illustrated in Figure 13. Seven accelerometers are installed on the bottom of the bridge, dividing the bridge into eight segments with equal intervals. S1, S2, …, S7 are used to label the seven sensors, respectively, with an increasing order in labeled number from the left to right end of the bridge model.

Figure 13.

The schematic diagram of the experiment and distribution of the measurement points.

In this experimental study, the damage was created by using an electrical grinding wheel cutter to introduce a minor crack, as shown in Figure 14, at specific positions of the testing model measured from the left side. Dynamic tests under the undamaged scenario and four different damage scenarios including two single damage (SD) scenarios and two multiple damage (MD) scenarios are conducted. The damage scenarios in this experimental studies are listed in Table 2. In single damage scenario 1 (SD1), a crack over the entire width of the deck penetrates through the thickness of the bottom flange with a depth of 3 mm, at the location 0.68 L. In SD2, a deeper crack is extended from SD1 to a depth of 5 mm into the web. In order to further demonstrate the performance of this method under multiple damage scenarios, another damage is introduced at the position of 0.38 L of the total model length from the left side. In multiple damage scenario 1 (MD1), two cracks with the depth of 4 mm and 7 mm are introduced at the position of L_d = 0.38 and L_d = 0.68, respectively. In MD2, two cracks with the same depth of 7 mm are introduced at two damage positions.

Figure 14.

The imposed damage in the model: (a) elevational view of the damage, (b) crack at the bottom of the bridge model, and (c) electrical grinding wheel cutter.

Table 2.

Damage scenarios in the experimental studies.

Damage location	Depth of the introduced crack
Damage location	SD1	SD2	MD1, mm	MD2, mm
L_d = 0.38	—	—	4	7
L_d = 0.68	3mm	5mm	7	7

Results of single damage detection

Figure 15 shows the measured responses of sensor No. 4 measured from different damage scenarios and the corresponding Fourier spectrum when the vehicle velocity is 0.25 m/s and the vehicle mass is 10.5 kg. It is noticed that no significant difference is observed between the undamaged and damaged states, since the damage of the introduced local minor crack does not prominently affect the natural frequencies of the structure, which means that it is difficult to utilize the change in natural frequencies to identify the damage reliably under these circumstances. However, the information on vibration frequencies of the bridge model is important in the proposed method because it is one of the crucial parameter in the determination of the length of the moving window. The Fourier spectrum indicates that the responses consist of three significant components with natural frequencies of 10.4 Hz, 33.2 Hz and 77.8 Hz, respectively, and the frequency component at 77.8 Hz dominates the vehicle-bridge coupled vibration in this case. Substituting the fundamental frequency f₁ =10.4 Hz into equation (14), the length of moving window l is calculated as 97 (l = 2 × 500/10.4).

Figure 15.

Responses measured by S4 and the corresponding Fourier spectrum in different damage scenarios with a moving velocity of 0.25 m/s and the mass of 10.5 kg: (a) acceleration responses and (b) the corresponding Fourier spectrum.

Figure 16 shows the single damage detection results of scenarios with different vehicle masses when the velocity is 0.25 m/s. As shown in Figure 16(a), when the mass is 10.5 kg, it can be observed that the damage can be well localized with the peak at the position of L_d = 0.68, although there are some fluctuations at other locations. In order to improve the robustness of the proposed approach, a one-side upper confidence limit as used in numerical studies is introduced. The parameter $α$ in equation (20) is selected to be 1%. In SD1, the peak higher than the threshold can be used to indicate the damage, but there is a minor false detection at the position of L_d = 0.6. With the increase of damage degree, in SD2, the damage at L_d = 0.68 is identified confidently by the obvious peak at the corresponding location. As described in the numerical simulation, the SPR value increases as the damage severity increases. Another case with the vehicle mass of 20.5 kg is also applied in the test to investigate the sensitivity of the proposed method to the vehicle mass. As shown in Figure 16(b), the threshold values of two single damage conditions are 6.295 and 7.194, respectively. Only the SPR values near the position of L_d = 0.68 are higher than the threshold values in the two damage scenarios, indicating that the damage can be reliably localized under these two scenarios. In general, the SPR values at L_d = 0.68 from scenarios with a moving mass weight of 20.5 kg are larger than those from scenarios with 10.5 kg. It is because the increase of trolley mass will increase the relative vibration amplitude of the introduced damage location. When the moving vehicle passes through the damage location, the difference in subspace projection between the healthy and damaged state also increases. Furthermore, increasing the vehicle mass more significantly excites the structure, leading to the increase in the signal to noise ratio, therefore making the damage more confidently detected.

Figure 16.

Single damage detection results with different masses when the vehicle velocity is 0.25 m/s: (a) 10.5 kg and (b) 20.5 kg.

Measured responses from scenarios with the vehicle speed of 0.5 m/s are also utilized for damage localization. Figure 17(a) shows the results when the mass is 10.5 kg. It can be observed that there is a peak higher than the threshold in each scenario near the true location L_d = 0.68. For the scenario with the mass of 20.5 kg, as shown in Figure 17(b), the damage location can be well identified by peaks at L_d = 0.68 under two damage conditions, and the SPR value of SD2 is larger than that of SD1, which is consistent with the previous observation.

Figure 17.

Single damage detection results with different masses when the vehicle velocity is 0.5 m/s: (a) 10.5 kg and (b) 20.5 kg.

Results of multiple damage detection

Multiple damage detection is studied in this experiment with the vehicle velocity of 0.25 m/s. When the mass is 10.5 kg, as shown in Figure 18(a), the relatively larger damage at L_d = 0.68 can be accurately identified. However, another peak exceeding the threshold appears at about L_d = 0.42, which deviates from the actual position of the other damage at L_d = 0.38. It is because the vehicle has a wheelbase, which makes the model not strictly equal to a concentrated mass. When the front of the vehicle model reaches the location L_d = 0.38, the bridge model is not well excited yet. That is, the vehicle model has just entered into the bridge and the bridge is not well excited, which makes the vibration information of the bridge insufficient. These issues cause the identification accuracy of a damage near the vehicle entrance end to be less accurate than that of a damage far away from the entrance end under this damage scenario. In MD2, as shown in Figure 18(b), with the increase of damage degree, SPR peak values increase accordingly. It should be noted that there is still a deviation in the damage detection at L_d = 0.38, with the same explanation described above, even though the damage at L_d = 0.68 can be better identified.

Figure 18.

Multiple damage detection results under different scenarios when the vehicle velocity is 0.25 m/s and vehicle mass is 10.5 kg: (a) MD1 and (b) MD2.

Figure 19 shows the multiple damage detection results of scenarios with the vehicle velocity of 0.25 m/s and the mass of 20.5 kg. It can be found that the identification deviation of the first damage at L_d = 0.38 is greatly improved with the increase of the vehicle mass because a heavier vehicle can better excite the bridge model. In MD1, as shown in Figure 19(a), two peaks exceeding the introduced threshold at the positions of introduced damages indicate the locations of damages accurately, while the SPR value corresponding to the severer damage at the position of L_d = 0.68 is larger than that of the less severe one at L_d = 0.38. As shown in Figure 19(a), in MD2, the SPR value of the peak at L_d = 0.38 becomes larger due to the increased damage level, and two damages are well localized.

Figure 19.

Multiple damage detection results under different scenarios when the vehicle velocity is 0.25 m/s and vehicle mass is 20.5 kg: (a) MD1 and (b) MD2.

Discussions

The above results demonstrate that the proposed method is capable of locating the structural damage simulated with the dynamic responses measured from the sensors installed along the bridge model. Yet in practice, it is difficult and expensive to arrange and maintain a large number of sensors, and it is also time consuming and costly to analyze the massive data for SHM. This section investigates the feasibility of the proposed method with a less number of sensors for damage localization. The same experimental data presented above are used. Data from three sensors are used for locating a single damage in this section. The three cases with different sensor combinations are discussed here, as listed in Table 3. It is noted that the sensor numbers are shown in Figure 13.

Table 3.

Sensor combinations in the experimental studies with a less number of sensors.

Case no	Sensor combinations
1	S5, S6, S7
2	S1, S6, S7
3	S2, S3, S4

First, data from three sensors S5, S6, and S7 near the damage are used. Figure 20(a) shows the results of scenarios with the vehicle velocity of 0.25 m/s and the mass of 10.5 kg using these three sensors. In SD1, although there is a minor false identification at around L_d = 0.6, the significant peak exceeding the threshold at the position of L_d = 0.68 indicates that the damage can be localized by data from three sensors. As the damage severity increases, in SD2, the damage is confidently identified by the combination of SPR peak value and the introduced threshold, and the amplitude becomes larger at L_d = 0.68. When the mass weight is increased to 20.5 kg, as shown in Figure 20(b), it can be observed that under these two scenarios, there are obvious peaks over the thresholds at the position of the imposed damage, which further verifies the feasibility of the proposed approach using less number of sensors.

Figure 20.

Single damage detection results with different masses when the vehicle velocity is 0.25 m/s using sensors S5, S6 and S7: (a) 10.5 kg and (b) 20.5 kg.

The results of scenarios with the vehicle velocity of 0.5 m/s and the mass of 10.5 kg is shown in Figure 21(a). Peaks exceeding the threshold near the position of L_d = 0.68 in both scenarios indicate the damage is correctly identified. For the scenario with 20.5 kg, as shown in Figure 21(b), although false identifications occur at the left end due to the end effect, damage at L_d = 0.68 is correctly identified.

Figure 21.

Single damage detection results with different masses when the vehicle velocity is 0.5 m/s using sensors S5, S6 and S7: (a) 10.5 kg and (b) 20.5 kg.

To better demonstrate the effectiveness of the proposed method when a small number of sensors are used, the sensor S5 close to the damage is replaced with the sensor S1 far away from the damage in Case 2, that is, the sensor combination S1, S6, S7 is used. With the vehicle velocity of 0.25 m/s and the mass of 10.5 kg, as shown in Figure 22(a), under two different operational conditions, the SPR index is larger than the threshold at L_d = 0.68, indicating the damage can be correctly located. The detection results with the moving mass of 20.5 kg are shown in Figure 22(b). In SD1, although there are several peaks, the damage location can still be identified using the defined damage index and the introduced threshold. In SD2, as the degree of damage increases, a peak that clearly exceeds the threshold near L_d = 0.68 indicating the reliable identification.

Figure 22.

Single damage detection results with different masses when the vehicle velocity is 0.25 m/s using sensors S1, S6 and S7: (a) 10.5 kg and (b) 20.5 kg.

When the vehicle velocity and the mass are 0. 5 m/s and 10.5 kg, respectively, as shown in Figure 23(a), in SD1, though there is a peak at the position of L_d = 0.68, the damage cannot be localized by the combination of the peak and threshold. In SD2, the SPR value near L_d = 0.68 slightly exceeds the threshold value, which can be utilized to localize the damage but with less confident. When the speed is increased to 0.5 m/s, the performance of the method becomes less promising in damage localization. This is because when the speed increases, the total number of data samples are reduced under the same sampling frequency, resulting in more reduction of vibration information. However, this problem can be solved by increasing the mass of the vehicle. As shown in Figure 23(b), when the mass is 20.5 kg, except for the misjudgment caused by the end effect at the left end, there is a peak at the position of L_d = 0.68 which indicates the existence of damage.

Figure 23.

Single damage detection results with different masses when the vehicle velocity is 0.5 m/s using sensors S1, S6 and S7: (a) 10.5 kg and (b) 20.5 kg.

To further investigate the influence of sensor location on damage localization, the sensor combination with sensors S2, S3, and S4 placed far away from the damage is used in Case 3. Figure 24 shows the results of scenarios with different vehicle masses when the velocity is 0.25 m/s. As shown in Figure 24(a), when the mass is 10.5 kg, though there is a minor false identification at the position of L_d = 0.8, the peak at L_d = 0.68 exceeding the threshold can still indicates the damage. As the severity of damage increases, in SD2, the damage can be confidently localized by the SPR index. When the mass is 20.5 kg, as shown in Figure 24(b), the performance of damage detection is also enhanced. Under two different damage conditions, the damage at L_d = 0.68 are well identified. These observations indicate that a larger moving mass introduces higher structural vibration responses and more significant vehicle-bridge interaction, leading to better identification accuracy.

Figure 24.

Single damage detection results with different masses when the vehicle velocity is 0.25 m/s using sensors S2, S3 and S4: (a) 10.5 kg and (b) 20.5 kg.

As shown in Figure 25(a), when the moving speed is 0.5 m/s and the mass is 10.5 kg, as mentioned above, the detection performance is weakened. As the mass increases to 20.5 kg, as shown in Figure 25(b), it is observed that the damage at L_d = 0.68 can be identified by the combination of the SPR value and the introduced threshold under both damage conditions. The detection results obtained from the proposed method with different sensor combinations demonstrate the effectiveness of the proposed method using a small number of sensors. When the moving vehicle has a relatively low velocity and a relatively large mass, the proposed approach can identify the damage well in both damage conditions. In terms of the required minimum sensor number and the optimal locations of sensors, systematic study is necessary to determine the optimal number and location of sensors, which will be conducted in the future study.

Figure 25.

Single damage detection results with different masses when the vehicle velocity is 0.5 m/s using sensors S2, S3 and S4: (a) 10.5 kg and (b) 20.5 kg.

Conclusions

This paper proposes a novel data-driven method based on subspace projection residual to localize the damage in beam bridges subjected to moving loads. A moving window with a certain length determined by the sampling frequency and the fundamental frequency of the measured responses is used to cut out the signals. The characteristic subspaces of the windowed responses are subsequently extracted to calculate the local damage index for damage localization. When there is damage in the local bridge section, the orthogonality condition between the active subspace projections determined by the dominant dynamic information under the current state and the null subspace determined by the secondary dynamic information or noise under the healthy state is invalid, which leads to the residual index SPR larger than the introduced threshold for damage detection. In order to verify the feasibility and reliability of the proposed method, a simply supported beam bridge subjected to a moving mass is considered in numerical study. Numerical results indicate that the proposed approach can accurately localize the single and multiple damages even when the responses are smeared with a significant noise. Experimental verifications on a laboratory steel beam bridge model further demonstrate the efficacy and accuracy of the propose approach to identify the locations of single and multiple damages. Discussions on the performance and effectiveness of using a small number sensors are presented. The results demonstrate that the proposed method can be used to localize the damage of beam bridges with a small number of sensors. However, it is also observed that the proposed approach may give false identifications. A larger mass moving at a slower speed leads to more reliable damage identifications. Further research will be conducted in the future to quantify the sensitivities of the proposed method in localizing small damages with respect to the number of required sensors, locations of the sensors, the optimized vehicle and bridge mass ratios and vehicle speed, and apply the method for practical applications to identify the conditions of real civil engineering structures.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the projects in key areas of Guangdong Province (No. 2019B111106001); KEY Laboratory of Robotics and Intelligent Equipment of Guangdong Regular Institutions of Higher Education (No.2017KSYS009); and the DGUT innovation center of robotics and intelligent equipment (No.KCYCXPT2017006).

ORCID iDs

Zhenhua Nie

Jun Li

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