Fuzzy clustering of time-series model to damage identification of structures

Abstract

Time-series methods have been popularly used for damage identification of civil structure because of its output-only and non-model approach. Since the existence of structural damage is usually vague and not focussed on any particular time point, the switches in damage patterns from one time state to another are necessary to be treated in a fuzzy way. This article develops a damage identification method based on the fuzzy clustering of time-series model. The changes of model coefficients of time-series model are proposed to indicate the undamaged and damaged states by the fuzzy c-means clustering algorithm. The residual errors of time-series model are used to identify the damage location and damage severity. The proposed method is applied to an experimental segment lining and a numerical study of a practical bridge. The results verify that the proposed method is accurate and efficient to detect the structural damage location and severity. Since the computational process of time-series model and fuzzy clustering require low computational cost, the proposed data-based damage identification method is applicable to the online structural health monitoring system of large-scale civil structures.

Keywords

auto-regressive model damage identification fuzzy clustering structural health monitoring time series

Introduction

The structural damage identification methods are widely classified into the model-based methods and data-based methods (Cavadas et al., 2013; Chen et al., 2017; Chen and Xia, 2017; Doebling et al., 1998; Mottershead et al., 2011; Wang et al., 2015, 2016). Model-based methods usually establish a finite element (FE) model of a structure. The change of physical parameters, which is identified by comparing the analytical data and the measured data from the real structure, is used to indicate the structural damage (Li et al., 2015; Ren and Chen, 2010). To accurately present the practical structures, an FE model of a practical structure usually includes a large number of elements, degrees of freedom (DOF) and structural parameters. Dynamic analysis and damage identification based on a large model usually takes up considerable computation storage and is time-consuming. Many strategies have been developed to improve the computational efficiency of model-based methods, such as substructuring methods (Weng et al., 2009; Zhu et al., 2013), model condensation methods (Weng et al., 2014, 2017; Xia and Lin, 2004) and so on. These simplifications on FE model are difficult to release computational load essentially, and the model-based approaches are inherently computationally inefficient.

Data-based approaches identify structural damage by comparing the damage-sensitive parameters of the time history of vibration responses such as the extended Kalman filtering method (Seibold and Weinert, 1996), neural network (Lin et al., 2017; Figueiredo et al., 2011), corresponding Fourier transform (Samman, 1994) and wavelet transform (Giovanni and Paolo, 1997). A crucial point for data-based method is to extract the damage-sensitive features from the time history of vibration responses.

Time-series models especially the auto-regressive (AR) models are widely developed to extract damage-sensitive features, since the AR model coefficients or residuals contain plentiful information including structural natural frequencies, modal shapes and damping ratios (Mosavi et al., 2012; Nair et al., 2006). The statistical time-series modelling set up a hypothetical mathematical model to best describe a time history (Rezaiee-Pajand et al., 2018). Once an appropriate model is fitted to the data, the anomaly from the fitted model can be used for detecting structural damage (Brockwell and Davis, 2002). Sohn et al. (2000) used AR model coefficients as damage-sensitive features to discriminate the undamaged and damaged condition of a bridge column. Sohn and Farrar (2001) built a time-series model for the statistical pattern recognition. A two-stage prediction model (combining AR and AR with exogenous inputs techniques) is constructed for damage detection, and the residual error is defined as the damage-sensitive feature, an 8 DOF mass-spring system was used to investigate the applicability of this approach. Carden and Brownjohn (2008) used the AR moving average model coefficients to distinguish different conditions of a structure. Gul and Catbas (2011) employed the experimental data from different test structures and damage cases to examine a statistical pattern recognition approach for damage assessment. Jayawardhana et al. (2015) proposed the AR model coefficients to detect the damage of a steel beam, and the Fisher information criterion of the computed damage-sensitive feature is used to locate the damage. Zheng and Mita (2009) used a pre-whitening filter to remove the correlations in excitations, and the Itakura distance and the cepstral distance of AR model are used as damage indicators. Lynch et al. (2004) and Rao and Ratnam (2012) used the relative standard deviations of the residual errors between the damaged and undamaged AR with exogenous input models as damage-sensitive feature. Datteo et al. (2017) applied the AR models and principal component analysis to the long-time monitoring of a practical stadium. Ma et al. (2018) proposed the statistical models and multi-scale principal component analysis for the structural response recovery.

Clustering is a process to classify data into different classes or clusters. The objects in the same cluster are similar, while the objects between different clusters are distinct. Therefore, it is helpful to discriminate the parameters of a structure from undamaged and damaged states. Kalpakis et al. (2001) used the linear predictive coding cepstrum of time series to cluster AR integrated moving average time series. Shumway (2003) used the time-varying spectra for clustering non-stationary time series. A survey of time-series clustering methods was made by Liao (2005).

The switches in patterns of civil structure from one state to another state are usually vague, and the structural damage does not appear at a specific time point. The fuzzy clustering method, for example, the fuzzy c-means clustering method, can naturally treat the problem with a fuzzy membership to each cluster (Bezdek, 1981). A fuzzy c-means clustering method was first developed in information science for the analysis of the disordered large data and was employed by Samuel et al. (2007) to quantify the damage-sensitive feature in an unsupervised learning mode. Maharaj and D’Urso (2011) applied a fuzzy clustering approach to the cepstral analysis and verified that the fuzzy clustering method has greater sensitivity and adaptivity for clustering time series than the non-fuzzy clustering methods.

This article proposed the fuzzy clustering of time-series model for the damage identification of large-scale structures. The model coefficients of time-series AR model are used to indicate the undamaged state to the damaged state by the fuzzy c-means clustering algorithm. The damage location and damage severity are identified by the residual errors of time-series AR model. The proposed method is applied to an experimental segment lining and a numerical study of a practical bridge to prove its effectiveness and accuracy in damage identification.

Damage-sensitive feature from AR model

The dynamic properties of a structure are affected by the structural damages, resulting in a change in the statistical characteristics of the measured time histories. The damage detection can be performed by fitting AR model to vibration signals of undamaged and damaged structures. The vibration data are necessary to be standardized (or normalized) before fitted to the AR models, since the loading conditions (different magnitudes and directions of loads) and/or environmental conditions might influence the data.

The vibration data from sensor i is partitioned into n different windows x_i,j(t), where i denotes the sensor number and j (j = 1, 2,…, n) denotes the jth window of data. The normalized signal y_i,j(t) is obtained as

y_{i, j} (t) = \frac{x_{i, j} (t) - μ_{i, j}}{σ_{i, j}}

(1)

where $μ_{i, j}$ and $σ_{i, j}$ are the mean and standard deviation of x_i,j(t).

Let y_i,j(t) and z_i,j(t) represent the normalized time-series windows in undamaged states and damaged states, AR models are fitted for y_i,j(t) and z_i,j(t) as

y_{i, j} (t) = \sum_{k = 1}^{p} ϕ_{k} y_{i, j} (t - k) + e (t)

(2)

z_{i, j} (t) = \sum_{k = 1}^{p} φ_{k} z_{i, j} (t - k) + τ (t)

(3)

where y_i,j(t) and z_i,j(t) are the time history at time t; $y_{i, j} (t - k)$ and $z_{i, j} (t - k)$ represent the kth previous points. $ϕ_{k}$ and $φ_{k}$ are the AR model coefficients, k = 1, 2, 3,…, p. e(t) and $τ (t)$ are the residual errors of the model with zero mean and constant variance. p is the order for the AR model. The order p can be determined based on Akaike’s information theoretic criterion (Akaike information criterion (AIC)) or Akaike’s final prediction error (FPE) (Akaike, 1974).

The AR coefficients of all n windows for sensor i in undamaged and damaged states are written as

Φ_{i}^{u} = [\begin{matrix} ϕ_{1, 1} & ϕ_{1, 2} & \dots & ϕ_{1, p} \\ ϕ_{2, 1} & ϕ_{2, 2} & \dots & ϕ_{2, p} \\ ⋮ & ⋮ & \dots & ⋮ \\ ϕ_{n, 1} & ϕ_{n, 2} & \dots & ϕ_{n, p} \end{matrix}]

(4)

Φ_{i}^{d} = [\begin{matrix} φ_{1, 1} & φ_{1, 2} & \dots & φ_{1, p} \\ φ_{2, 1} & φ_{2, 2} & \dots & φ_{2, p} \\ ⋮ & ⋮ & \dots & ⋮ \\ φ_{n, 1} & φ_{n, 2} & \dots & φ_{n, p} \end{matrix}]

(5)

After fitting the AR model for time series, the coefficient matrix for the sensor i from the undamaged state $(Φ_{i}^{u})$ and damaged state $(Φ_{i}^{d})$ are estimated for clustering analysis. The coefficient matrix $[Φ_{i}^{u}; Φ_{i}^{d}]$ will be used as the cluster samples to identify the damage in the next section. For clarity, $[Φ_{i}^{u}; Φ_{i}^{d}]$ is represented by $Φ$ in the following chapters.

Damage detection by fuzzy clustering method

In this part, the fuzzy c-means algorithm will be developed to construct the statistical modelling of AR coefficients for damage identification. In fuzzy c-means clustering, the membership of a data-point in a cluster is a fuzzy decision. A data-point is considered to be a member of every cluster with a given possibility membership value that ranges from 0 to 1.

The fuzzy c-means clustering minimizes the total fuzzy dissimilarity within each cluster in terms of the objective function

\min : J_{m} = \sum_{a = 1}^{N} \sum_{b = 1}^{C} u_{a, b}^{m} {‖ Φ_{a} - c_{b} ‖}^{2}

(6)

where $Φ_{a}$ stands for the ath observation sample, that is, the ath row of the coefficients matrix $[Φ_{i}^{u}; Φ_{i}^{d}]$ ; c_b is the centroid of the bth cluster; N and C are the total amount of the observation samples and clusters, respectively. $u_{a, b}^{m}$ is the membership degree of the ath sample to the bth cluster with the constraints

\sum_{b = 1}^{C} u_{a, b}^{m} = 1, u_{a, b}^{m} > 0

(7)

m is the fuzziness parameter and normally set to 2 (Maharaj and D’Urso, 2011; Yang et al., 2008).

To solve the optimal problem, the Lagrange method is employed here by taking into account the membership degree constraints (equation (7); Bezdek and Pal, 1992), and the solution is obtained by the following iterative formulas:

Choose the cluster numbers C, the fuzzy parameter m and initialize the membership degrees matrix $u_{a, b}^{0}$ .

Update the centroids $c_{b}^{(q + 1)}$ by equation (8), where the superscript (q) denotes the qth iteration

c_{b}^{(q + 1)} = \frac{\sum_{a = 1}^{N} u_{a, b}^{(q)} Φ_{a}}{\sum_{a = 1}^{N} u_{a, b}^{(q)}}

(8)

Update the membership degrees $u_{a, b}^{(q + 1)}$ by means of equation (9)

u_{a, b}^{(q)} = \frac{1}{\sum_{k = 1}^{C} {(\frac{‖ Φ_{a} - c_{b}^{(q + 1)} ‖}{‖ Φ_{a} - c_{k} ‖})}^{\frac{2}{m - 1}}}

(9)

Steps 2 and 3 are repeated until $| u_{a, b}^{(q + 1)} - u_{a, b}^{(q)} | < ε$ . Afterwards, the optimized u and c are achieved for clustering.

For the damage-detection procedure, the C and m are set to 2, and the membership degrees are randomly initialized with the constraints (equation (7)). In consequence, the membership degrees u for all the coefficients samples and centroid of the two clusters are calculated by the fuzzy c-means clustering method. The Euclidean distance l between the centroid of the two clusters is defined as

l = \sqrt{\sum_{k = 1}^{p} {(c_{1}^{k} - c_{2}^{k})}^{2}}

(10)

where $c_{1}^{k}$ and $c_{2}^{k}$ stand for the kth dimension of the centroid of cluster c₁ and c₂, respectively. l increases with the severity of the damage, which can be used to indicate the damage severity.

An index r is defined with the membership degree u as

r = \frac{{\bar{u}}^{d}}{{\bar{u}}^{u}}

(11)

where ${\bar{u}}^{u}$ is the average membership degree of coefficients samples $Φ_{i}^{u}$ to cluster c₁ or c₂, ${\bar{u}}^{d}$ is the average membership degree of coefficients samples $Φ_{i}^{d}$ to the same cluster. Two threshold values – upper and lower thresholds – are calculated using the statistical analysis of the r index when the structures are undamaged

Th = μ r \pm 3 σ r

(12)

where $μ r$ and $σ r$ stand for the mean and the standard deviation of the r indexes, respectively. In statistics, the threshold of $\pm 3 σ r$ includes 99.7% of the undamaged data, so the outlier can be used to detect the damage.

Damage identification by residual errors

After the undamaged and damaged states are discriminated by the fuzzy clustering of AR coefficients, the residual errors of the AR model are used to identify the location and severity of damage. The damage-sensitive feature dsf is defined to estimate the correlation of residual errors from two sensors as

ds f_{i} = \frac{| var (e_{i}^{d}) var (e_{i + 1}^{u}) - var (e_{i + 1}^{d}) var (e_{i}^{u}) |}{var (e_{i}^{d}) var (e_{i + 1}^{u})}

(13)

where $e_{i}^{u}$ and $e_{i}^{d}$ are the residual errors of time series in the undamaged state and damaged state fitting the AR model (equation (2)), respectively. dsf denotes the difference of residual errors between the sensors i and i+1. If the structure is damaged, dsf will be increased at the damaged locations. In consequence, dsf can be used to identify damage location and severity.

Experiment on segmental lining

The subject of the experiment is the full-scale two-rings reinforced concrete segmental lining shown in Figure 1(a). The outside diameter of the segmental lining is 4000 mm, the inside diameter is 3500 mm as shown in Figure 1(b). A full ring consists of six segments, including a top segment (F segment), two contiguous segments (L1 and L2 segment), and three standard segments (B1, B2 and B3 segment). The segments are connected with 12 high-strength bolts in radial direction and 14 high-strength bolts in longitudinal direction. The segmental lining is excited by a hammer in the radial direction on the outside surface of the Ring 1 segmental lining, corresponding to the measurement point 1 (MP1) shown in Figure 1(a). A series of hammer hits were recorded in each test configuration. Four DH187E accelerometers with the sensitivity of 50 mV/g were placed at four measurement points (MP1, MP2, MP3 and MP4 in Figure 2), which were numbered with sensors 1–4 to measure the responses of the B2 and B3 segments in the direction perpendicular to the surface of the lining. The sampling frequency is 1000 Hz.

Figure 1.

Full-scale segmental lining: (a) test site view and (b) cross-section (unit: mm).

Figure 2.

Sensor location and impact point.

The segmental lining was tested in four cases as Table 1. The damage is simulated by loosening Bolt 23 (BL23, see Figure 1(b)) that connects B2 and B3 segments. Case 1 is the undamaged state and Cases 2–4 are the damaged states. The typical acceleration time history of MP1 and MP2 is shown in Figure 3. The time history in all four cases is fitted to the AR model. The optimal model order for AR model is selected as 35, according to the converged FPE value at 0.52 in Figure 4 (Akaike, 1974). Afterwards, the AR model coefficients and residual errors of each test are calculated. The proposed method is conducted for the coefficient matrix from the undamaged state $(Φ_{i}^{u})$ and damaged state $(Φ_{i}^{d})$ with the cluster number set to 2. The result of the index r (equation (11)) and membership degrees to the two clusters c₁ and c₂ are shown at Table 2. The thresholds are calculated as 0.8503 and 1.1531.

Table 1.

Description of test cases.

Case	Condition	Description
1	Undamaged (U)	Bolt 23 was not loosened
2	Damaged (D1)	Bolt 23 was loosened 180°
3	Damaged (D2)	Bolt 23 was loosened 270°
4	Damaged (D3)	Bolt 23 was removed

Figure 3.

Typical acceleration time history of MP1 and MP2: (a) MP1 and (b) MP2.

Figure 4.

FPE criterion.

Table 2.

The average of the membership degree to cluster c₁ and c_2.

Condition	c ₁			c ₂
Condition	U	D1/D2/D3	r	U	D1/D2/D3	r
Case 1	0.5352	0.6022	1.09	0.4648	0.3978	0.86
Case 2	0.8820	0.1809	0.21	0.1180	0.8191	6.94
Case 3	0.8275	0.2247	0.27	0.1725	0.7753	4.49
Case 4	0.8904	0.2291	0.26	0.1096	0.7709	7.03

In Case 1, because the membership degrees of the coefficients samples from the undamaged and test states are similar and all indices r are within the threshold, the coefficients samples are not able to be divided into two clusters. It means the structure is undamaged. In Cases 2–4, the average membership degrees of the coefficients samples from undamaged and damaged states are distinctly discriminated into two clusters with all the indices r exceeding the threshold, which means the structure is damaged in Cases 2–4. Therefore, the proposed method can be used to identify the damage of the structure. In particular, the average membership degree values in Case 1 and Case 4 was compared in Figure 5. In Figure 5(a) and (b), the membership degrees of the coefficients samples from the undamaged states are very similar and are not able to be discriminated. It means that the test samples cannot be divided into two clusters according to the fuzzy c-means clustering method, and there is no damage to the structure. In Figure 5(c) and (d), the membership degrees of test samples from undamaged states and damaged case (D3) are quite different, and they are classified into two clusters. This is because the damage happened on the structure.

Figure 5.

The membership degree to the cluster: (a) c1 of Case 1, (b) c2 of Case 1, (c) c1 of Case 4 and (d) c2 of Case 4.

Figure 6 compares the randomly selected model coefficients from the undamaged state (Case 1) and D3 state (Case 4). In Figure 6(a) and (b), the 5th, 6th, 20th and 23rd model coefficients of the damaged state are far away from those of the undamaged state, whereas the 11th, 12th, 15th and 17th model coefficients (Figure 6(c) and (d)) are very near. Although the model coefficients are not separated obviously in Figure 6(c) and (d), the fuzzy c-means clustering analysis is advantageous to make the best separation of model coefficients to detect the structural damage.

Figure 6.

Model coefficients of Case 4: (a) $ϕ_{5}$ and $ϕ_{6}$ , (b) $ϕ_{20}$ and $ϕ_{23}$ , (c) $ϕ_{15}$ and $ϕ_{17}$ and (d) $ϕ_{12}$ and $ϕ_{11}$ .

Afterwards, the damage index dsf is calculated to locate the damage by equation (13). The dsf values between two sensors are calculated and displayed in Figure 7. The sensor pairs are divided into two groups, group A contains sensor pairs 1-2, 1-4, 2-3 and 3-4 which are coming from two sides of the segments, group B contains sensor pairs 1-3 and 2-4 which are coming from only one side of the segments. The dsf of group A are greater than group B, which means the damage exists on the location related to sensor pairs of group A. A common feature of those sensor pairs is that all those sensor pairs are located on the two sides of contiguous segments connected by the high-strength bolts. It is obvious that the loosening of the bolts will weaken the connection between the two segments, which influences the dsf value of the sensor pairs of group A coming from two sides of contiguous segments rather than the others (group B) coming from one side; because the other sensor pairs come from only one side of the segments which is insusceptible to the bolts. Actually, the dsf values of the sensor pairs of group B are very small. Therefore, the damage-sensitive index dsf accurately identifies the bolt loosening of Bolt 23, and its value increases with the increase of damage severity.

Figure 7.

dsf of sensor pairs: (a) Case 3 and (b) Case 4.

Numerical study to Jun Shan Yangtze bridge

In this section, the proposed damage identification method is applied to a numerical study of Jun Shan Yangtze Bridge to evaluate its effectiveness in large structures. The Jun Shan Yangtze Bridge is a cable-stayed bridge located in Wuhan, China. The bridge is 4881.178 m in length and 33.5 m in width. It is modelled by 611 nodes, 758 elements with 3634 DOF in Figure 8. A total of 19 uniformly distributed nodes in the bridge are selected as the measurement points. The bending rigidity of two elements (denoted with ‘D1’ and ‘D2’ in Figure 8) in the mid-span of the girder are reduced to simulate the damage states in Table 3. D1 is located between measurement points 8 and 9, and D2 is situated between points 14 and 15. Case 1 stands for the undamaged state. In Case 2, the bending rigidity of D1 is reduced by 10%, 20%, 30% and 40%, while other elements remain unchanged. In Case 3, the bending rigidity of D2 is reduced by 10%, 20%, 30% and 40%, respectively. A Gaussian white noise excitation is applied to the mid-span of the bridge in Z direction. The sampling frequency is set to 100 Hz, and the time last 50 s. A typical time history of the excitation is shown in Figure 9.

Figure 8.

Model of the Jun Shan Yangtze Bridge.

Table 3.

The stiffness reduction in the elements of the bridge.

Case	Case 1	Case 2	Case 3
Stiffness reduction	No reduction	D1 (10%, 20%, 30%, 40%)	D2 (10%, 20%, 30%, 40%)

Figure 9.

A typical time history of the excitation.

The response time series consist of 5000 points and are separated into 19 windows. Each window contains 500 points, and half of them are overlapped with the former window. The AR model order is selected as 10 following the FPE criterion as shown in Figure 10. The total sample consists of the model coefficients from the undamaged and damaged structural model and is used for damage identification according to the proposed method with the cluster number set to 2. The result of membership degrees to the two clusters c₁ and c₂ in Case 2 are shown in Table 4. The thresholds are calculated as 0.8376 and 1.1653.

Figure 10.

FPE criterion.

Table 4.

The average membership degree to cluster c₁ and c_2.

Stiffness reduction (%)	c ₁			c ₂
Stiffness reduction (%)	Undamaged	Damaged	r	Undamaged	Damaged	r
10	0.3650	0.3302	0.90	0.6350	0.6698	1.05
20	0.8286	0.6530	0.79	0.1714	0.3470	2.02
30	0.9163	0.4413	0.48	0.0837	0.5587	6.68
40	0.9879	0.0366	0.04	0.0121	0.9634	7.96

It is seen in Table 4 that the membership degrees u of the coefficients samples from the undamaged and damaged states are similar with all indices r within the threshold, when minor damage happens on the structure, for example, the stiffness reduction is 10%. With the increase of the damage severity, the discrepancy of the membership degrees of the coefficients samples from the undamaged and damaged states increases, and all the indices r exceed the threshold. The coefficients samples from the undamaged and damaged states are divided into two different clusters according to the fuzzy c-means clustering method, which means that the proposed method can be used to detect the structural damage.

Figure 11 shows the result of the membership degrees of the measured point 8 in Case 2 with 10% stiffness reduction. The membership degrees from undamaged and damaged states are separated slightly, which indicates a minor damage to the bridge according to the fuzzy c-means clustering method. Figure 12 shows the membership degrees of Case 2 with 40% stiffness reduction. For the data of undamaged state, the membership degrees to cluster c₁ in undamaged state are greater than 0.95 with the average value of 0.991. The average membership degree to cluster c₂ in damaged state is around 0.009. For the damaged state, the average values are 0.052 and 0.948 for cluster c₁ and c₂, respectively. Therefore, the coefficients from undamaged and damaged state are separated into two different clusters clearly, and the proposed method is capable of identifying the local damage. Figure 13 shows the distances between the centroids of two clusters, and the distance of centroids grows with the increase of damage severity.

Figure 11.

Membership degree to the cluster with 10% stiffness reduction: (a) c1 and (b) c2.

Figure 12.

Membership degree to the cluster with 40% stiffness reduction: (a) c1 and (b) c2.

Figure 13.

Distances between the centroids of two clusters in Case 2.

Figure 14 shows the damage index dsf (equation (13)) for locating the damage. The dsf₈ is much larger than that of the other measured points in all damaged states. This indicates that the damage is close to measurement point 8. The proposed method is capable of identifying the damage location D1 accurately. In addition, dsf grows gradually with the increase of damage severity of the bridge. Therefore, the proposed indices can be used to identify the damage location and severity accurately.

Figure 14.

dsf of sensor pairs in Case 2: (a) 10%, (b) 20%, (c) 30% and (d) 40%.

Without losing generality, another random damage location D2 is selected to illustrate the effectiveness of the proposed method. Identically, a largest value of dsf is seen in the 14th measurement point as Figure 15, which is close to the damage location D2. If the dsf₁₄ in different damage severity is displayed together in Figure 16, dsf₁₄ grows with the increase of damage severity. In consequence, the proposed method is effective to locate the local damages and identify the damage severity for large civil structures.

Figure 15.

dsf of sensor pairs in Case 3: (a) 10%, (b) 20%, (c) 30% and (d) 40%.

Figure 16.

dsf of MP14 in different damage severity in Case 3.

Practical structures are subject to harsh service environment, where noise is present in the measurements. The robustness of the proposed methods needs to be discussed for practical applications. Taking Case 2 for instance, a Gaussian white noise is added to the structural response with signal-to-noise-ratio (SNR) levels = 26 dB. SNR is defined as (Yang and Nagarajaiah, 2014)

SNR = 20 \log_{10} \frac{RMS (signal)}{RMS (noise)}

(14)

The thresholds are calculated as 0.8668 and 1.1347. Table 5 shows the result of the average membership degree and r index in the noise condition. It is seen that the coefficients samples from undamaged and damaged states cannot be distinguished with all the r within the threshold in minor damage condition. When major damage happens on the structure, for example, with 20% stiffness reduction, the average membership degree of these two states belong to different clusters, and all the r index exceed the threshold, which shows the ability of the proposed method to detect the damage. Figure 17 shows damage location of Case 2 with noise, the 8th point has the biggest dsf value in all damage states, which means that the proposed method can exactly locate the damage under the measurement noise. In addition, dsf₈ again grows gradually with the increase of damage severity.

Table 5.

The average membership degree to cluster c1 and c2 in the noise condition.

Stiffness reduction (%)	c ₁			c ₂
Stiffness reduction (%)	Undamaged	Damaged	r	Undamaged	Damaged	r
10	0.5062	0.4634	0.92	0.4938	0.5366	1.09
20	0.5777	0.3715	0.64	0.4223	0.6285	1.49
30	0.9505	0.2740	0.29	0.0495	0.7260	14.67
40	0.9785	0.0556	0.06	0.0215	0.9444	43.93

Figure 17.

dsf of sensor pairs in Case 2 with noise: (a) 10%, (b) 20%, (c) 30% and (d) 40%.

Conclusion

This article proposes a data-based damage-detection method based on the fuzzy clustering time-series model. The AR model coefficient is used as features for the fuzzy clustering to detect the existence of damage. The residual error of AR model between adjacent sensors is used to identify the damage location and severity. The proposed method is verified by an experiment on a segmental lining experiment and a numerical study on Jun Shan Yangtze Bridge. Using the fuzzy c-means clustering method, the model coefficients from the undamaged and damaged states are classified into two clusters efficiently. The dsf index from residual error of AR model is larger at the sensors near the damage, and the dsf index grows with the increase of damage severity. The proposed method is effective for fast damage identification of civil structures.

Footnotes

Acknowledgements

The author would like to acknowledge the support provided by the Basic Research Program of China (Grant No.: 2016YFC0802002), the National Natural Science Foundation of China (NSFC, contract no.: 51778258, 51629801 and 51578260) and the Fundamental Research Funds of the Central Universities (HUST, Grants No.: 2016JCTD113, 2014TS130 and 2015MS064).

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: Authors’ have received funding from Basic Research Program of China (Award ID 2016YFC0802002) and National Natural Science Foundation of China (Institution ID : 10.13039/501100001809 and Award ID 51778258, 51629801 and 51578260).

References

Akaike

(1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control 19(6): 716–723.

Bezdek

(1981) Pattern Recognition with Fuzzy Objective Function Algorithm. New York: Plenum Press.

Bezdek

Pal

(1992) Fuzzy Models for Pattern Recognition: Methods That Search for Structures in Data. New York: IEEE Press.

Brockwell

Davis

(2002.) Introduction to Time Series and Forecasting. New York: Taylor & Francis.

Carden

Brownjohn

(2008) ARMA modelled time-series classification for structural health monitoring of civil infrastructure. Mechanical Systems and Signal Processing 22(2): 295–314.

Cavadas

Smith

IFC

Figueiras

(2013) Damage detection using data-driven methods applied to moving-load responses. Mechanical Systems and Signal Processing 39(1–2): 409–425.

Chen

Xia

(2017) Advanced technologies in disaster prevention and mitigation. Advances in Structural Engineering 20(8): 1141–1142.

Chen

Weng

Zhi

et al . (2017) Response control of a large transmission-tower line system under seismic excitations by using friction dampers. Advances in Structural Engineering 20(8): 1155–1173.

Datteo

Lucà

Busca

(2017) Statistical pattern recognition approach for long-time monitoring of the G.Meazza stadium by means of AR models and PCA. Engineering Structures 153: 317–333.

10.

Doebling

Farrar

Prime

(1998) A summary review of vibration-based damage identification methods. Shock and Vibration Digest 30(2): 91–105.

11.

Figueiredo

Park

Farrar

et al . (2011) Machine learning algorithms for damage detection under operational and environmental variability. Structural Health Monitoring 10(6): 559–572.

12.

Giovanni

Paolo

(1997) Wavelet analysis of structures: statics, dynamic and damage identification. Meccanica 32: 223–230.

13.

Gul

Catbas

(2011) Structural health monitoring and damage assessment using a novel time series analysis methodology with sensor clustering. Journal of Sound and Vibration 330(6): 1196–1210.

14.

Jayawardhana

Zhu

Liyanapathirana

et al . (2015) Statistical damage sensitive feature for structural damage detection using AR model coefficients. Advances in Structural Engineering 18(10): 1551–1562.

15.

Kalpakis

Gada

Puttagunta

(2001) Distance measures for effective clustering of ARIMA time-series. In: IEEE international conference on data mining, San Jose, CA, 29 November–2 December, pp. 273–280. New York: IEEE.

16.

Chen

Xie

et al . (2015) A comparative study on frequency sensitivity of a transmission tower. Journal of Sensors 2015: 610416.

17.

Liao

(2005) Clustering of time series data – a survey. Pattern Recognition 38(11): 1857–1874.

18.

Lin

Nie

(2017) Structural damage detection with automatic feature-extraction through deep learning. Computer-Aided Civil and Infrastructure Engineering 32(12): 1025–1046.

19.

Lynch

Sundararajan

Law

et al . (2004) Embedding damage detection algorithms in a wireless sensing unit for operational power efficiency. Smart Materials and Structures 13(4): 800–810.

20.

Hao

et al . (2018) Structural response recovery based on improved multi-scale principal component analysis considering sensor performance degradation. Advances in Structural Engineering 21(2): 241–255.

21.

Maharaj

D’Urso

(2011) Fuzzy clustering of time series in frequency domain. Information Sciences 181(7): 1187–1211.

22.

Mosavi

Dickey

Seracino

et al . (2012) Identifying damage locations under ambient vibrations utilizing vector autoregressive models and Mahalanobis distance. Mechanical Systems and Signal Processing 26(1): 254–267.

23.

Mottershead

Link

Friswell

(2011) The sensitivity method in finite element model updating: a tutorial. Mechanical Systems and Signal Processing 25(7): 2275–2296.

24.

Nair

Kiremidjian

Law

(2006) Time series based damage detection and localization algorithm with application to the ASCE benchmark structure. Journal of Sound and Vibration 291(1–2): 349–368.

25.

Rao

Ratnam

(2012) Health monitoring of welded structures using statistical process control. Mechanical Systems and Signal Processing 27: 683–695.

26.

Ren

Chen

(2010) Finite element model updating in structural dynamics by using the response surface method. Engineering Structures 32(8): 2455–2465.

27.

Rezaiee-Pajand

Entezami

Shariatmadar

(2018) An iterative order determination method for time-series modeling in structural health monitoring. Advances in Structural Engineering 21(2): 300–314.

28.

Samman

(1994) Vibration testing for nondestructive evaluation of bridges. Journal of Structural Engineering 120(1): 269–300.

29.

Samuel

Júnior

Junior

(2007) Damage detection in a benchmark structure using AR-ARX models and statistical pattern recognition. Journal of the Brazilian Society of Mechanical Sciences and Engineering 29(2): 174–184.

30.

Seibold

Weinert

(1996) A Time domain method for the localization of cracks in rotors. Journal of Sound and Vibration 195(1): 57–73.

31.

Shumway

(2003) Time-frequency clustering discriminant analysis. Statistic and Probability Letters 63(3): 307–344.

32.

Sohn

Farrar

(2001) Damage diagnosis using time series analysis of vibration signals. Smart Materials and Structures 10(3): 446–451.

33.

Sohn

Czarnecki

Farrar

(2000) Structural health monitoring using statistical process control. Journal of Structural Engineering 126(11): 1356–1363.

34.

Wang

Zhu

(2016) A new three-dimensional electromechanical impedance model for an embedded dual-PZT transducer. Smart Materials and Structures 25(7): 075002.

35.

Wang

Zhang

Zhu

(2015) Embedded electromechanical impedance and strain sensors for health monitoring of a concrete bridge. Shock and Vibration 2015: 821395.

36.

Weng

Tian

Zhu

et al . (2017) Dynamic condensation approach to calculation of structural responses and response sensitivities. Mechanical Systems and Signal Processing 88: 302–317.

37.

Weng

Xia

et al . (2009) Improved substructuring method for eigensolutions of large-scale structures. Journal of Sound and Vibration 323(3–5): 718–736.

38.

Weng

Zhu

et al . (2014) Dynamic condensation approach to the calculation of eigensensitivity. Computers and Structures 132: 55–64.

39.

Xia

Lin

(2004) A new iterative order reduction (IOR) method for eigensolutions of large structures. International Journal for Numerical Methods in Engineering 59(1): 153–172.

40.

Yang

Hsieh

(2008) Alpha-cut implemented fuzzy clustering algorithms and switching regressions. IEEE Transactions on Systems, Man, and Cybernetics, Part B 38(3): 588–603.

41.

Yang

Nagarajaiah

(2014) Structural damage identification via a combination of blind feature extraction and sparse representation classification. Mechanical System and Signal Processing 45(3): 1–23.

42.

Zheng

Mita

(2009) Localized damage detection of structures subject to multiple ambient excitations using two distance measures for autoregressive models. Structural Health Monitoring 8(3): 207–216.

43.

Zhu

Mao

Weng

(2013) Calculation of dynamic response sensitivity to substructural damage identification under moving load. Advances in Structural Engineering 16(9): 1621–1631.