Damage detection of a bridge model based on operational dynamic strain measurements

Abstract

Vibration data analysis reserves a significant place in structural health monitoring practice since it can be employed for certain engineering problems such as locating damage and assessing damage severity by means of vibration signature interpretation. In this study, a damage detection study is carried out in light of vibration data analysis methods on a four-span bridge model by exploiting strain measurements. The concept of strain operating deflection shape along with an extraction method based on frequency and spatial domain decomposition is proposed. The damage-induced state of the bridge model is compared with the baseline state to detect and locate the damage. The result of strain analysis is also compared with acceleration analysis by simultaneous measurements of strain and acceleration. The power spectrum matrix of responses is estimated using digital signal processing principles. Besides the operating deflection shape analysis, operational modal analysis and correlation analysis of strain modes are conducted to identify and locate the damage.

Keywords

damage detection operating deflection shape operational modal analysis strain measurement and analysis strain modes

Introduction

Structural health monitoring (SHM) may be defined as the assessment and tracking of different structures for conditions such as damage or various anomalies under operational loads. These measurements are directly related to structure’s safety, durability, serviceability, and reliability.

A broad variety of techniques have been used to detect damage on different structures ranging from aerospace to mechanical and civil engineering. A large quantity of studies deal with the interpretation of vibration data using time or frequency domain algorithms. The main idea is to monitor the change of sensitivity in the data between different states. A general and informative review about those methodologies can be found in the literature. Besides vibration measurements and findings, strain measurements reserve an important place not only on finding the localized damages but also on their vicinity and severity (Catbas and Aktan, 2002). A large number of researchers studied the damaged or damage-induced structures by means of strain measurements in conjunction with the vibration data. In most of these scientific research studies, it is very common to use statistical algorithms to have a better understanding about the damage on the structure (Carden and Brownjohn, 2008; Gul and Catbas, 2009, 2011; Omenzetter and Brownjohn, 2006; Zhang, 2007; Zheng and Mita, 2007). Modal strain energy and modal curvature are both classical assessment techniques for damages in structures (Pal and Banerjee, 2015; Yan et al., 2010, 2012). There are also investigations on damage detection using wavelet transforms and also on the application of the same method to operational modal analysis cases (Hester and Gonzalez, 2012). Another study possesses new improved damage identification techniques (Li et al., 2013).

Both strain and acceleration measurements, nowadays, are commonly utilized. However, the former is more difficult to employ than the latter considering the troubles and difficulties in the installation of the metal-foil strain gauges (Figure 1(a)), which, in most cases, are the typical strain measurement sensors. Besides installation problems, another important drawback of the strain gauges is their low capacity if high-frequency measurements are to be made. In recent years, new technological developments on strain sensors have been promising and accelerated the interest in strain measurements. The optical fiber Bragg gratings (FBG; Figure 1(b)) and piezoelectric strain sensors (Figure 1(c)) have been gaining increasing attention in the engineering fields. The FBG sensors are small sized, light weighted, distance independent, have high precision, and can be used to measure very high strain (Malekzadeh and Catbas, 2012). The piezoelectric strain sensors, on the other hand, have a high-frequency measurement range and are compatible with the popular signal conditioners and data acquisition systems with their outstanding advantage of reusability making them even more attractive. Laflamme et al. (2012) used a capacitance sensor built with a soft, stretchable dielectric polymer with attached stretchable metal film electrodes to detect the changes in strain. Wireless sensing technique has also been developed to reduce the instrumentation time and system cost. With the adoption of radiofrequency identification (RFID) for passive wireless sensing, a battery-free strain sensor (Xu and Huang, 2012; Yi et al., 2013) is designed for the strain measurement.

Figure 1.

Some types of strain sensors: (a) metal-foil strain gauge, (b) FBG, and (c) piezoelectric strain sensor.

Operating deflection shape (ODS) is an effective way to investigate how a machine or structure moves during its operation at a specific frequency or a moment in time (Richardson, 1997; Schwarz and Richardson, 1999). Both time domain and frequency domain data can be utilized for the ODS analysis. Time domain ODS is based on the multi-channel time history waveforms acquired spatially from a machine or structure. It shows the vibration motion of a machine or a structure in a certain time interval just like a recorder. The structure’s overall motion as well as the motion of a particular part relative to another can clearly be illustrated. Locations of excessive vibration are easily identified. The frequency domain ODS is based on frequency domain data such as linear spectrums, cross power spectrums, frequency response functions (FRFs), and ODS FRFs (Schwarz and Richardson, 1999). It shows the behavior of a structure at a single frequency, helping the observer find whether or not a resonance frequency is being excited.

Combined with the structure’s three-dimensional (3D) geometry and visual animation tightly, the ODS method is widely applied in practice to solve various vibration problems, by offering a simpler, more straightforward approach for fault detection. Pai and Jin (2000) developed a boundary effect detection method to extract damage-induced boundary-layer effects by measuring ODSs using a scanning laser vibrometer. Hence, damages such as surface slots and stiffened sections can be located using a sliding-window least-squares curve-fitting technique to decompose an ODS into central- and boundary-layer solutions. The method was then improved to locate different types of damages (Pai and Young, 2001). Ganeriwala et al. (2009) used ODS to detect the unbalance in a rotating equipment. The change in ODS was proposed as an early warning indicator of unbalance in rotating components. Sundaresan et al. (1999) used a scanning laser vibrometer to compute and then compare the ODSs of healthy and damaged turbine blades. Gul and Catbas (2008) identified modal parameters using ODS and modal ODS were combined with assumed modal scaling factors for damage identification and localization purposes. Changes in curvature of the ODSs were used to locate the damage. Zhang et al. (2013) proposed a new damage detection algorithm called global filtering method for beam- and plate-like structures based on ODS curvature. A vehicle was used to move along a line on damaged structure as an exciter. The ODS curvature was then constructed from dynamic responses of the vehicle and not from the structure. Based on wavelet decomposition, the experimental ODS curvature was made smoother to be used as the baseline. Then, the effect of damage could be detected through comparison of the ODS curvature to the filtered one. The method only requires the information from damaged structures. Recently, a new concept called residual ODS (Asnaashari and Sinha, 2013, 2014) has been defined for beam-like structures. Based on the ODS analysis, the method maps the deflection of cracked structures at the exciting frequency and its higher harmonics to identify the location of the crack.

ODSs are extracted from measurement data taken from 2 or more degrees-of-freedom (DOFs) of the machine or structure. The data are typically a measure of surface motion and can be in displacement, velocity, or acceleration units. In this article, the strain measurement data are utilized for the ODS analysis, and a new method based on the frequency and spatial domain decomposition (Wang et al., 2005) is proposed to obtain the frequency domain strain ODSs, which allows to investigate the structural dynamic behaviors at a specific frequency clearly. The theoretical background of extraction of the ODS is first presented. Then, the strain ODS analysis is applied to detect the simulated damage of a four-span bridge model from two different states, namely, baseline and damage induced. To further locate the position of the damage, the acceleration and strain modes are identified from the operational responses, and the correlation analysis is conducted as well. The respective performance of simultaneously acquired strain and acceleration data from the bridge model are carefully compared in the ODS analysis and operational modal analysis.

Theoretical background

A well-known method to obtain the ODS from response only or operating data is to compute the so-called ODS FRF. An ODS FRF is formed by replacing the magnitude of the cross spectrum between a response and the reference responses with the magnitude of the auto spectrum of the response. Mathematically, this is equivalent to

ODS FR F_{ij} (ω) = \sqrt{P_{ii} (ω)} \frac{P_{ij} (ω)}{| P_{ij} (ω) |}

(1)

where $P_{ij}$ means the cross spectrum between response i and j, and $P_{ii}$ represents the auto spectrum of response i. Then, the ODS FRF values at a specified frequency, which are taken from all the calculated ODS FRFs, make up the ODS vector. In this article, a new type of ODS is proposed. It is defined as the singular vector of the response spectrum matrix. The response spectrum matrix consists of all the auto spectrums and cross spectrums between m responses

[G_{yy} (ω)] = [\begin{matrix} P_{11} (ω) & P_{12} (ω) & \dots & P_{1 m} (ω) \\ P_{21} (ω) & P_{22} (ω) & \dots & P_{2 m} (ω) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ P_{m 1} (ω) & P_{m 2} (ω) & \dots & P_{mm} (ω) \end{matrix}]

(2)

The response spectrum matrix can easily be proved to be a Hermitian matrix. Taking the singular value decomposition (SVD) of the response spectrum matrix gives

[G_{yy} (ω)] = \sum_{r = 1}^{m} σ_{r} v_{r} v_{r}^{H} \approx \sum_{r = 1}^{N} σ_{r} v_{r} v_{r}^{H}

(3)

where $σ_{r}$ and $v_{r}$ are, respectively, the singular value and singular vector, and N is the number of dominant singular values. In Wang et al. (2005), the mode indicator function (MIF) is defined as the singular values computed from the response spectrum matrix at each spectral line. The peaks detected in the MIF plot indicate the existence of modes, and the frequencies corresponding to those peaks are known as the modal frequencies. In the case that the dimension of the response spectrum matrix is larger than the dominant modes at a specific frequency, the singular vector becomes related to the mode shape. Around each peak, the singular vectors calculated from the adjacent spectral lines do not change much. Therefore, we can define the singular vectors as a new type of frequency domain ODS. Due to its derivation from the frequency and spatial domain decomposition (FSDD), it is named after FSDD ODS. One outstanding advantage of FSDD ODS is that, all the ODSs related to the dominant singular values at a specific frequency can separately be displayed. It helps investigator go deeply into the structure’s dynamic signatures.

The response spectrum matrix in equations (2) and (3) is computed from either displacement or acceleration responses. The same analogy can also be adapted to strain responses. According to the modal superposition theorem, the displacement vector can be expressed by the modal coordinates and mode shapes

{u} = \sum_{r = 1}^{N} q_{r} {φ_{r}}

(4)

where ${u}$ is the displacement vector, $q_{r}$ is the rth modal coordinate, and ${φ_{r}}$ is the rth displacement mode vector. The strain is the partial derivative of the displacement. As an example, the normal strain in x-direction may be described as

{ε_{x}} = \frac{\partial {u}}{\partial x} = \frac{\partial}{\partial x} (\sum_{r = 1}^{N} q_{r} {φ_{r}}) = \sum_{r = 1}^{N} q_{r} {φ_{r}^{ε}}

(5)

where ${φ_{r}^{ε}} = ((\partial {φ_{r}}) / \partial x)$ represents the strain mode vector. In the steady state of the system, the modal coordinate, a function of frequency and time, can be expressed as

q_{r} = \frac{{φ_{r}}^{T} {F} e^{j ω t}}{k_{r} - ω^{2} m_{r} + j ω c_{r}}

(6)

where $k_{r}$ , $m_{r}$ , and $c_{r}$ are the rth modal stiffness, mass, and damping terms, respectively. ${F}$ is the force vector. By substituting equation (6) into equation (5), the strain frequency response function (SFRF) can be obtained as

[H_{ε}] = \sum_{r = 1}^{N} \frac{{φ_{r}^{ε}} {φ_{r}}^{T}}{k_{r} - ω^{2} m_{r} + j ω c_{r}}

(7)

Equation (7) can also be formulated in the form of partial fractions of poles and residues

[H_{ε}] = \sum_{r = 1}^{N} (\frac{[R_{r}]}{j ω - λ_{r}} + \frac{{[R_{r}]}^{*}}{j ω - λ_{r}^{*}})

(8)

where $λ_{r}$ is the pole and $[R_{r}]$ is the residual matrix

[R_{r}] = \frac{{φ_{r}^{ε}} {φ_{r}}^{T}}{a_{r}}

(9)

$a_{r}$ is the modal participation factor which indicates the contribution of rth mode to the vibration response of the system. The strain response spectrum matrix can be expressed as

[G_{ε ε}] = [H_{ε}]^{*} [G_{FF}] [H_{ε}]^{T}

(10)

where $[G_{FF}]$ is the excitation force spectrum matrix, which can be simplified to a constant real diagonal matrix $[C]$ over a specified frequency range with flat spectrums. Broadband and random excitations such as the traffic on a bridge or a wind blowing against a building, as well as impulsive forces in nature, can all be assumed to have a flat spectrum approximately. By substituting equation (8) into equation (10), the strain response spectrum matrix $[G_{ε ε}]$ can be written as the summation of four terms

[G_{ε ε}] = \sum_{r = 1}^{N} (\frac{[A_{r}]}{j ω - λ_{r}} + \frac{{[A_{r}]}^{H}}{- j ω - λ_{r}^{*}} + \frac{{[A_{r}]}^{*}}{j ω - λ_{r}^{*}} + \frac{{[A_{r}]}^{T}}{- j ω - λ_{r}})

(11)

[A_{r}] = \sum_{k = 1}^{N} (\frac{{[R_{k}]}^{*} [C] {[R_{r}]}^{T}}{- λ_{k}^{*} - λ_{r}} + \frac{[R_{k}] [C] {[R_{r}]}^{T}}{- λ_{k} - λ_{r}})

(12)

If the tested structure does not have heavy damping, equation (12) can be simplified

[A_{r}] \approx \frac{{φ_{r}}^{H} [C] {φ_{r}}}{a_{r}^{*} a_{r}} {φ_{r}^{ε}}^{*} {φ_{r}^{ε}}^{T} = d_{r} {φ_{r}^{ε}}^{*} {φ_{r}^{ε}}^{T}

(13)

where $d_{r}$ is the real scalar. In the vicinity of the modal peak, the last two terms of equation (11) can be ignored. So

{[G_{ε ε}]}_{ω \to ω_{r}}^{T} \approx \sum_{r = 1}^{N} (\frac{d_{r} {φ_{r}^{ε}} {φ_{r}^{ε}}^{H}}{j ω - λ_{r}} + \frac{d_{r} {φ_{r}^{ε}} {φ_{r}^{ε}}^{H}}{- j ω - λ_{r}^{*}}) = [ψ_{ε}] diag (2 Re (\frac{d_{r}}{j ω - λ_{r}})) [ψ_{ε}]^{H}

(14)

Equation (14) has the same expression as the SVD of the strain response spectrum matrix. Therefore, the strain FSDD ODS can also be obtained from the frequency and spatial domain decomposition of the strain response spectrum matrix. Moreover, the similar procedures as those in the FSDD literature (Wang et al., 2005) can be applied to obtain the modal frequencies and damping ratios from the strain response spectrum matrix.

The four-span bridge model

The model seen in Figure 2 is a four-span bridge-type structure located at the University of Central Florida. This structure allows the users to change the boundary conditions and also the type of connections between its individual elements, thus allowing different damage states to be induced (Figure 2(b)). To provide connection between girders and deck, sets of four 1/4 in bolts and plates are utilized. Through this configuration, the girders are connected to one another by means of only the deck. The stiffness of the structure can be changed by removing or loosening bolts at different locations. The two approach spans of the structure have 1220 mm length each (Figure 3). The two inner spans are 3048 mm long and are instrumented with sensors. Two high-speed steel (HSS) 25 × 25 × 3 girders, separated 610 mm from each other, support a 3.2-mm-thick and 1220-mm-wide steel deck. In all, 16 accelerometers and 20 strain gauges are placed on the two inner spans along the girders to measure the acceleration in the vertical direction and the surface strain in the axial direction of the girders. To exert the bridge randomly, two toy trucks are mobilized back and forth along the lane between the girders. The acceleration and strain responses are simultaneously acquired by a National Instrument data acquisition system, which consists of two eight-channel SXCI-1531 ICP accelerometer conditioning modules and three eight-channel SXCI-1520 modules with the accessory SCXI-1314 terminal block for strain gauges.

Figure 2.

(a) The four-span bridge, (b) convertible boundary conditions, (c) and bolts linking girder and deck.

Figure 3.

The schematic and sensor placement of the four-span bridge.

Baseline state measurement and analysis

For the baseline state, the response signals are acquired in the original state of the bridge (simply supported) at a sampling frequency of 250 Hz and a sampling period of 500 s. Figure 4 shows the measured acceleration and strain responses of two close measurement positions. It can be seen that the two types of responses look very different, and in both responses an approximate periodicity due to the back and forth movement of the toy trucks can be found. By applying a de-trend procedure in the MATLAB environment, the pre-strain found in the strain responses is removed. Then, the acceleration and strain responses are divided into segments to apply the Hanning window and the fast Fourier transform (FFT) subsequently. Finally, the acceleration and strain response matrices are obtained by the calculation of auto and cross power spectrums from the responses. In Figure 5, the auto power spectrums of the sample responses from Figure 4 are plotted in the first 20 Hz frequency band. The modal peaks in the two spectrum curves are similar. However, the magnitude of the strain spectrum reduces quickly in the frequency band that is higher than 10 Hz, which is caused by the limited performance of strain gauges in the high-frequency band.

Figure 4.

Typical acceleration and corresponding strain responses.

Figure 5.

Typical spectrums of acceleration and strain.

The size of the acceleration response spectrum matrix is 16 × 16, while the size of the strain response spectrum matrix is 20 × 20. By taking the SVD of the response spectrum matrices at each spectral line, the MIFs and the FSDD ODSs can be obtained. Figure 6 shows the first MIF of acceleration responses which consists of the largest singular value at each spectral lines and the corresponding FSDD ODSs at some typical frequencies. The dynamic change of ODSs along the frequency band is clearly observed.

Figure 6.

Acceleration mode indicator function and FSDD ODS in the baseline state.

Figure 7 shows the first MIF of strain responses which consists of the largest singular value at each spectral lines and the corresponding FSDD ODSs at some typical frequencies. Comparing Figure 6 with Figure 7, although the acceleration and strain MIFs have different shapes, they both indicate the same structural modes, that is, the peaks of the two plots are exactly at the same frequencies. Again, it can be found that the strain MIF has relatively lower magnitude than the acceleration MIF in the frequency band higher than 10 Hz. The strain FSDD ODSs along the girder are visualized as vertical displacement in Figure 7. It can be seen that the strain ODS is very different than the acceleration ODS. The color of the strain ODS at the two ends of the bridge model is blue, which means that the magnitudes are close to zeros. It correctly reflects the property of the pin supports located at those positions.

Figure 7.

Strain mode indicator function and FSDD ODS in the baseline state.

Damaged state measurement and analysis

The two boundary conditions of the bridge model at right outer end are converted from simply supported to fixed condition to simulate a damaged state. The converted boundary conditions are close to strain sensor #1 and #11, and to accelerometer sensor #1 and #9. The measurement and analysis are conducted again to extract the FSDD ODSs from the simultaneously obtained 16 acceleration and 20 strain time waveforms in the damaged state.

Figure 8 shows the acceleration MIF and FSDD ODSs in the damaged state. Compared to the baseline state in Figure 6, the modal peaks in the MIF are generally moved toward right direction, due to the enhancement of stiffness by fixing the pin supports. Some slight changes in the FSDD ODSs can also be observed.

Figure 8.

Acceleration mode indicator function and FSDD ODS in the damaged state.

Figure 9 shows the strain MIF and FSDD ODSs in the damaged state. It can be concluded again that the strain MIF indicates the same structural modes as the acceleration MIF. Compared to the baseline state in Figure 7, the FSDD ODSs show a significant change. The color of the strain ODS at the left end stays as blue, while the color turns into red at the right end, which means that the magnitude at this end is not zero any more. It clearly reflects the change in boundary condition at the right end, from simply supported condition to the fixed condition.

Figure 9.

Strain mode indicator function and FSDD ODS in the damaged state.

Modal analysis and modes correlation analysis

To locate the damage position, a full operational modal analysis and correlation analysis are further conducted with the N-Modal analysis software. Four dominant modes are identified from both the acceleration and strain responses with the operational modal identification algorithm. The results are listed in Table 1. It is clear that the modal parameters extracted from different responses are very close. Additionally, the acceleration and strain response parameters belonging to damage state are listed in Table 2. Compared to the baseline state analysis results, the modal frequencies are increased due to the increment in stiffness caused by the fixed pins. Figure 10 shows the variation of the strain modal frequencies for two different states investigated.

Table 1.

The modal parameters identified in the baseline state.

Mode	Frequency			Damping
Mode	Acc. (Hz)	Strain (Hz)	Diff. (%)	Acc. (%)	Strain (%)	Diff. (%)
1	4.52	4.50	−0.44	4.94	4.88	−1.21
2	7.51	7.50	−0.13	4.33	4.25	−1.84
3	8.75	8.72	−0.34	3.94	3.84	−2.53
4	17.01	17.05	0.23	1.36	1.33	−2.20

Table 2.

The modal parameters identified in the damaged state.

Mode	Frequency			Damping
Mode	Acc. (Hz)	Strain (Hz)	Diff. (%)	Acc. (%)	Strain (%)	Diff. (%)
1	5.37	5.34	−0.56	5.69	5.79	1.76
2	8.41	8.44	0.36	3.16	3.15	−0.32
3	10.14	10.14	0	3.44	3.47	−0.87
4	17.62	17.61	−0.06	1.44	1.42	−1.39

Figure 10.

Changes of strain modal frequencies from baseline to damaged state.

Figure 11 shows the acceleration mode shapes and the strain mode shapes in the baseline state. Figure 12 shows the acceleration and strain mode shapes in the damaged state.

Figure 11.

Acceleration and strain mode shapes in the baseline state (DOF numbers marked on the first graph).

Figure 12.

Acceleration and strain mode shapes in the damaged state (DOF numbers marked on the first graph).

Comparing Figure 11 with Figure 12, the difference in strain modes can obviously be seen from baseline to damaged state, while the difference in acceleration modes is not distinctive. Modal assurance criterion (MAC) can be employed to check the correlation between two modal vectors

MAC (φ_{1}, φ_{2}) = \frac{| φ_{1}^{H} φ_{2} |^{2}}{φ_{1}^{H} φ_{1} φ_{2}^{H} φ_{2}}

(15)

The MAC values between each pair of acceleration modes (baseline and damage) are 0.94, 0.85, 0.87, and 0.92, respectively, which are rather high. It proves that the acceleration modes are global features of structures, and they are not sensitive to the local damage. On the other hand, the strain modes change obviously at the right end of the bridge model and the MAC values between mode pairs are 0.77, 0.79, 0.68, and 0.74, respectively. The MAC values will be greatly improved if the elements related to the damaged positions are removed from the mode vectors. In the current damage state, the average MAC value increases from 0.74 to 0.89.

An extension of the MAC, the coordinate modal assurance criterion (COMAC) is also employed to analyze the strain modes correlation. COMAC can be used to identify which measurement DOFs contribute negatively to a low value of MAC, and it should be calculated after pairing of modes and normalization. The COMAC on DOF p over N mode pairs can be expressed as

COMAC (p) = \frac{\sum_{r = 1}^{N} | φ_{1 pr}^{*} φ_{2 pr} |^{2}}{\sum_{r = 1}^{N} | φ_{1 pr} |^{2} \sum_{r = 1}^{N} | φ_{2 pr} |^{2}}

(16)

Figure 13 shows the COMAC bar graphs of acceleration and strain modes. For the acceleration modes, there is no obvious measurement DOF that has low COMAC. Nevertheless, for the strain modes, the COMAC values at measurement DOF 1, 11, and 20 are much lower than others. DOF 1 and 11 are the positions where the boundary conditions were changed from simply supported to fixed state. DOF 20 corresponds to the position of roller, where the magnitude of each mode is almost zero. Therefore, the COMAC calculation at DOF 20 is very readily to be influenced by the measurement noise or identification error.

Figure 13.

COMAC of acceleration modes and strain modes in the baseline and damaged states.

Conclusion

A method based on the frequency and spatial domain decomposition is proposed to estimate strain ODS, which can be considered as a useful tool to investigate the structural dynamic behavior at a specific frequency clearly. By changing the boundary conditions of the four-span bridge model, a local damaged state is simulated. Both the acceleration ODS analysis and the strain ODS analysis are applied to detect the damage. The strain FSDD ODS is found to be better in reflecting the damage.

To further locate the position of the damage, the acceleration modes and strain modes are extracted from acceleration and strain responses, respectively, via an operational modal analysis process. The strain modal frequencies and damping ratios are almost the same with acceleration frequencies and damping ratios, whereas the strain mode shapes are different from the acceleration mode shapes. It should be mentioned that strain ODS reflects the damage of the structure more clearly especially at locations where there is little vibration such as (near) supports. While there is little vibration at supports and acceleration ODS cannot clearly indicate the change at the boundary conditions, strain ODS indicates the change quite clearly. By checking the MAC and COMAC correlation between the baseline strain modes and damaged strain modes, the damaged positions are successfully located.

It seems reliable and reasonable to estimate the dynamic characteristics of low-frequency structures, such as bridges and buildings, from the strain measurements obtained using the conventional foil strain gauges. With the development of strain sensor technology, it is very possible to cover higher frequency band in the strain analysis.

The proposed algorithm and procedure were also used to detect more local and minor damage states, such as loosening some of the bolts which connect the bridge deck to the girders. Neither strain nor acceleration modes could reflect this kind of damage well.

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: The research project described in this article and the co-authors are funded by Florida Department of Transportation (FDOT), Federal Highway Administration of the USA; the Fundamental Research Funds for the Central Universities (NO. NS2013004), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) of PRC.

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