The experimental bridge is a typical steel box girder bridge. The total length of the bridge is 172.5 m, and its width is 10.25 m. It has pin bearings at fixed ends and pin-roller bearings at movable sides. One of the movable bearings was found damaged with partial unseating, which is a typical issue of these types of bearings due to earthquakes (Dicleli and Bruneau, 1995; Kim et al., 2006; Santo et al., 2012; Ellenberg et al., 2016). The first span of the bridge from the south side, bound for incoming traffic, was the target span for the response measurements since the malfunctioned bearing had been discovered there. Figure 2 shows the site condition of the damaged bearing where the roller support was partially moved out, and additional damage was prevented with temporary retrofitting, done with a wooden stopper on both sides. This arrangement restricted the longitudinal movement of the bearing and imparted the sliding malfunction (Lu et al., 2017; Chu et al., 2022; Sun et al., 2022). The condition of the neighboring bearing was found to be satisfactory by visual inspection and thus chosen for comparison purposes.

In order to investigate the performance of both bearings in similar conditions, an identical test vehicle weight of 1.78 ton was driven closer to each bearing in each lane to input the same effects on both bearings. For clarity, the location of both bearings is shown in Figure 3. The malfunctioned bearing is located at the end of the slow lane, and the bearing in good condition is positioned at the end of the fast lane. Since the movable side of the bridge with pin-roller bearings is the only end that has both standard and malfunctioning bearings, further discussion will, therefore, be purely pertinent to this side of the bridge span, unless stated otherwise. This span was chosen for investigation because it was found to have both malfunctioned and standard bearings located side by side. The favorable situation was exploited to develop a rational approach.

Microelectromechanical Systems (MEMS) accelerometers and laser displacement sensors (Gangone et al., 2011; Xiao et al., 2017; Whelan et al., 2008) were installed on the structure to record the response of both malfunctioned and good condition bearings. The Complementary Metal Oxide Semiconductor (CMOS) Multi-Function Analog Laser Sensor of the IL-300 series from KEYENCE corporation of America was used with a sampling frequency of 100 Hz. The M-A351AU triaxial MEMS wireless accelerometer of EPSON Japan recorded the acceleration response of the structure with a sampling frequency of 1,000 Hz. The wireless accelerometer is an IoT-based sensor designed explicitly for analyzing vibrations and monitoring the structural health of a system. It is a low-power device that uses Wi-Fi to communicate and has a built-in 2.4 GHz antenna that allows it to transmit data wirelessly over a maximum range of 200 m. These sensors measured outputs as voltage values and then converted them to the standard displacement unit (mm) and acceleration (m/sec²). Objectively, a longer measurement duration was chosen, which allowed for recording enough information on imposed actions. A test vehicle was used to collect the response of the structure, and the data was captured using a data acquisition system connected to the laser sensors and accelerometer sensors. The data was continuously recorded and stored in the data acquisition system during the 6-h test period. Figure 4A shows the typical installation of laser displacement sensors using special stands, and Figure 4B displays the placement of wireless sensors with firmly placed double-sided tape. The reference axis of these sensors is also shown in their respective figures with additional information on sensor placement.

For purposeful sensor placement, two laser displacement sensors were installed at each bearing location. These sensors were affixed to a steel component near the bearing using strong magnetic support, as depicted in Figure 4A. The horizontal laser sensors were positioned at the sliding component of the bearing to capture longitudinal displacement and assess sliding performance. Similarly, vertical laser sensors were oriented towards the main girders, approximately 80 mm away from the bearing, to measure the vertical displacement of the girder. This vertical displacement is indicative of the rotational performance of the bearings. The rotational performance of the girder and bearing components is estimated by computing the arctangent function of the girder’s vertical response, divided by the distance between the bearing and the specific measurement point on the girder. Furthermore, wireless accelerometers were installed using double taps with strong adhesive characteristics to ensure proper placement on the girders and piers, as illustrated in Figure 4B. The acceleration response of the near-bearing girder and pier was recorded to analyze the vibration characteristics of these components. This approach allows for the development of an efficient bearing performance evaluation methodology, circumventing the cumbersome laser displacement sensor placement procedures. Additionally, it bypasses the potential errors or uncertainties associated with numerical integration for displacement calculations. The experimental results are discussed in subsequent sections.

The methods used to extract information about the frequency content and similarity between two signals are CPSD and cross-correlation. CPSD and Cross-correlation provide information about the similarities and dissimilarities in the frequency content of vibration signals. CPSD is a frequency domain analysis technique used to examine the relationship between two vibration signals. The CPSD is obtained by multiplying each signal’s Fourier transform and multiplying their complex conjugates. Mathematically, the CPSD can be expressed using Eq. 1: S x y f = F x t .   F y t * (1)

Here, S x y f represents the CPSD between signals x(t) and y(t) at frequency f, F x t and F{y(t)} correspond to the Fourier Transforms of signals x(t) and y(t) respectively and “∗” denotes the complex conjugate operator. The Fourier Transform operator F{x(t)} is defined in Eq. 2 as below: F x t = ∫ − ∞ ∞ x t e − 2 j π f t d t (2) where x(t) represents the time-domain signal, t denotes time, j is the imaginary unit, and f denotes frequency. The expression in Eq. 1 provides information about how the spectral content of one signal relates to the other signal across various frequencies. CPSD is employed to analyze girders’ vertical acceleration response and piers’ longitudinal acceleration response for both healthy and damaged bearing locations. Whereas cross-correlation is a time domain analysis technique that measures the similarity between two signals as a function of time lag. It represents the impulse response between the girders’ vertical acceleration response and the piers’ longitudinal acceleration response. It provides information about the time lag and strength of the correlation between these two signals. In this case, the inverse fast Fourier transform (IFFT) is performed on the CPSD signals to obtain the results in the time domain. Mathematically, the cross-correlation function, CC(t), is calculated using Eq. 3 as follows: C C t = F − 1 S x y f (3) where F − 1 denotes the Inverse Fourier Transform operator, and Sxy(f) represents the CPSD between the signals x(t) and y(t) at frequency f.

The Inverse Fourier Transform operator, F − 1 , is defined in Eq. 4 as: F − 1 X f = ∫ − ∞ ∞ X f e 2 j π f t d f (4) where X(f) represents the frequency-domain signal, f denotes frequency, and t represents time. The utility of this function has been discussed in Section 3.2.

This study further proposes the CS as a quantitative estimation approach for assessing bearing performance in bridge systems. The mathematical expression for CS estimation is provided below in Eq. 5. C S = ∑ C C t (5) where CS represents the condition score, the sum is taken over all the correlation values. The absolute value (|cross-correlation|) is calculated for each correlation value, ensuring a positive outcome. The square root (√) is then applied to each absolute correlation value. Finally, the square root is taken of the sum of the square roots of the absolute correlation values, resulting in the CS.

To demonstrate the application of the proposed CS analysis, a bridge model previously investigated (Farrar et al., 2000) was utilized to assess a wide range of bearing conditions. The Abaqus software (Abaqus Analysis user’s guide, 2013) was used to develop a detailed Finite Element Model (FEM) of the bridge, which includes a concrete deck supported by six steel beams. The bridge has a width of 7.3 m and a length of 15.2 m. The bridge pier measures 2.5 m in height and 1 m in width. The steel beams in the model are W30 × 116 wide-flange beams, and four evenly spaced cross braces with C12 × 25 channel sections are positioned along the span’s length. Figure 5 provides various perspectives of the bridge model and its meshed geometry. The material properties for concrete and steel used in the study are presented in Table 1.

The bridge model used in this study incorporates essential boundary conditions, with one end supported by sliding bearings and the other end supported by pin bearings. A tie constraint was implemented to establish the interconnection between the bridge deck slab, main girder, and cross braces. For the finite element analysis, the bridge structure was discretized using 3D solid elements with a refined mesh employed at the locations of contact interaction.

A contact interaction with friction formulation was defined to simulate the sliding bearing performance. This formulation allowed for the simulation of the sliding behavior of the bearing with different coefficients of friction. A range of coefficient of friction values was used, assuming that a higher coefficient of friction indicates a malfunctioned bearing. The contact interaction was assigned to the corresponding surfaces of the bridge pier and main girder, incorporating friction to account for the sliding bearing in the analysis.

To replicate the effects of traffic, the recorded traffic acceleration was applied as a loading condition at the bottom of the main girder near the pier. The longitudinal acceleration response of the pier was then estimated from the analysis to calculate the CS and establish the relationship between the CS and the coefficient of friction. By employing this numerical modeling approach, a comprehensive investigation of the relationship between the CS and the coefficient of friction becomes possible, offering valuable insights into the bearing performance of the bridge.

The response of the bearings under the passage of the test vehicle that was driven closer to either bearing is presented in Figure 6. From the figure of longitudinal displacement response, the sliding malfunction of the retrofitted bearing can be identified as consistent with the in-situ condition. This bearing exhibited negligibly small longitudinal displacements even though the test vehicle passed closer to it (see Figures 6A, B). Figures 6C, D display the vertical displacement response of the girders at a malfunctioned bearing location. Both downward and upward displacements can be observed in contrast to their counterparts, which exhibited mostly downward displacements. Sliding malfunction of the bearing altered the rotational response of the girder and resulted in greater vertical movements. Usually, under standard vehicle passage, the sliding and rotation phenomenon simultaneously takes place at the bearing location. Nevertheless, in case of sliding malfunctioning, the only mechanism left behind to compensate for the imposed actions is rotation, which is reflected in terms of more vertical movements of the girder. Likewise, the girder at a malfunctioned bearing revealed more considerable rotational demands and supported this conclusion, even though the test vehicle was not passing closer to the malfunctioned bearing (Figure 6D).

An overview of the typical acceleration response recorded at the bearing vicinity under traffic loading is presented in Figure 7. From the overall response, it is difficult to identify the various characteristics of response and hard to understand the different mechanisms. Displacement and rotation estimates from acceleration response under the test vehicle are expected to infer the equivalent information described in the laser displacement sensor response discussions. Additionally, the aforementioned technique involves the use of a test vehicle drive by approach. It requires the absence of any other vehicle on the bridge, which is sometimes burdensome to realize, and it is onerous to avoid the influence of other traffic. Therefore, keeping in view the concerns associated with the drive by approach, the ambient acceleration response of bearing locality under any arbitrary traffic was further utilized and presented in the following discussion.

Ideally, bridge bearings should receive the same input excitations when their corresponding bridge girder midspans have similar acceleration responses (Yamamura and Tanaka, 1990; Tsai et al., 2007; Zhu et al., 2014). In view of this notion, the acceleration response of the girder midspans was also acquired by placing sensors in each lane, as shown in Figure 3. Figure 7 shows the almost identical mid span acceleration response of both girders obtained with in-depth exploration. Additionally, the corresponding response of both bearings with observable discrepancies is presented in Figure 8, and details are discussed further. Figure 8 shows the combined plots of the acceleration response of the girder in the vertical direction and the acceleration response of the pier in the longitudinal direction at both malfunctioned and good condition bearing locations. Response of only stated directions was considered since other response information may not be significant. For instance, a pier in the vertical direction is appreciably rigid and will not exhibit any noteworthy excitations. Response in the transverse direction also may not be of interest as no major force is acting under normal operations, and these bearings are not intended to accommodate movements in the lateral direction. Response of the girder in either the vertical or longitudinal direction at the bearing vicinity could be considered. The vertical direction response was selected because of its direct relation with bearing malfunction, as illustrated next.

It can be seen from the plots that the acceleration response of the girder in the vertical direction at the location of the healthy bearing is larger than the vertical response of the girder at the malfunctioned bearing location (see Figures 8A, B). The conceivable reason for this phenomenon is the increased rotational demand of the girder, as per the previous discussion. The frequent change in the vertical displacement direction of the girder caused the comparatively reduced acceleration response (Roberts et al., 2015; Martínez et al., 2017; Xia et al., 2017). Furthermore, increased bearing fixity attracted a relatively larger share of input actions at the malfunctioned bearing location, which was reflected in terms of the enlarged acceleration response of the pier in a longitudinal direction. The current approach to identifying malfunctioning bearings has limitations, such as the difficulty of finding similar inputs to bearings and the influence of multiple vehicles on the bridge. However, using CPSD and cross-correlation, we can develop a more systematic approach to analyze the near bearing vertical acceleration response of girders and the longitudinal acceleration response of piers. This will allow us to better understand the performance of bearings and identify any malfunctions.

The preceding discussion emphasized the importance of analyzing the near bearing vertical acceleration response of girders and the longitudinal acceleration response of piers to understand the performance of bearings. In order to develop a more comprehensive approach, the mutual characteristics of these response parameters using CPSD and Cross-correlation were considered in the context of bearing malfunctioning. Figure 9A shows the CPSD plots of these signals normalized by their corresponding maximum amplitudes. This frequency domain plot could reveal the dissimilarity of these signals, which might need to be clarified. The cross-correlation function is shown in Figure 9B.

Distinct information on bearing performances could be observed in terms of differences in amplitude, phase, and even total response duration. The malfunctioned bearing seems to have very stiff behavior and smaller excitations along with lesser excitation duration (about three times smaller) when compared to its counterpart. The disparity in energy dissipation mechanisms was found to be the leading cause of the discrepancy between the responses. Suitable condition bearings respond to incoming actions through a combination of sliding and rotation, leading to a longer response duration. Whereas a malfunctioned bearing uses the rotational ability alone to act against the imposed actions that have relatively limited capability to respond, and henceforth, it transfers the energy to the underlying pier that could also be seen from its larger longitudinal response.

The proposed CS methodology is next employed herein to demonstrate its superior functionality over the conventional techniques. While traditional methods rely on subjective evaluations, leading to inconsistencies, the CS considers parameters such as amplitude and response duration from cross-correlation of accelerations to provide a comprehensive evaluation metric. By combining these parameters, a holistic assessment of the bearing condition is obtained. The CS enhances the objectivity and standardization of evaluations, aiding in maintenance decisions and ensuring long-term bridge integrity. The obtained CS values for healthy and malfunctioned bearings were found to be 1.682 and 1.311, respectively. The noticeable difference in these values indicates contrasting conditions between the two types of bearings. The CS value is expected to be larger for healthy and properly functioning bearings, indicating a higher structural integrity and performance level. The lower CS value associated with the malfunctioned bearings suggests a compromised condition with reduced capabilities to withstand external forces and vibrations. This finding highlights the importance of the CS as a quantitative measure for distinguishing between healthy and malfunctioned bearings, providing valuable insights for maintenance and decision-making processes in bridge asset management. The subsequent section expounds on applying the proposed approach through the utilization of numerical modeling techniques.

The bridge model prescribed in Section 2.5 was simulated with varying conditions of bearing degradation, including both healthy and malfunctioning states. To establish a correlation between the CS and the coefficient of friction, different models with distinct friction coefficients were employed. The aim was to investigate the relationship between the CS and the health of the bearings. A lower coefficient of friction indicated a healthier bearing, while a higher coefficient pointed towards a malfunctioning bearing.

During the analysis, parameters such as the amplitude and response duration of cross correlation of acceleration response were considered to calculate the CS values for each bearing condition. This quantitative assessment allowed for direct comparisons among different conditions, enabling an objective evaluation of bearing performance. The findings, presented in Figure 10A, clearly demonstrate the relationship between the CS and the coefficient of friction. The graph illustrates how variations in the coefficient of friction correspond to changes in the CS.

It was observed that a lower coefficient of friction correlated with a higher CS, indicating a healthier bearing. Conversely, a higher coefficient of friction was associated with a lower CS, indicating a malfunctioning bearing. Within the range of 0–0.5 coefficient of friction, the CS exhibited a direct decrease. However, beyond a coefficient of friction of 0.5, the acceleration response became minimal, resulting in an insignificant change in the CS, which reached a nearly constant value.

It should be noted that the current methodology for CS assessment relies on acceleration response data. However, for bearings that are severely malfunctioning, estimating the CS may be challenging due to the smaller amplitude of the acceleration response. In such cases, the damage is often visually apparent and can be readily identified. It is crucial to address significant damage and implement preventive maintenance measures to mitigate the effects of severe damage. The current methodology is recommended for identifying damage in bearings that may not be visually apparent and serves as a valuable tool in maintenance and decision-making processes.

The research presented in this study demonstrates the practical application of the proposed CS estimation method through numerical modeling, providing a convenient approach for assessing bearing conditions. The obtained CS values offer valuable quantitative indicators that bridge owners and engineers can use to prioritize maintenance actions and make informed decisions regarding repair or replacement strategies. Additionally, establishing a database of CS-coefficient of friction relationships using numerical modeling enables estimating the current state of bearing function, enhancing the evaluation of bearing conditions and improving our understanding of bearing performance for proactive maintenance and the long-term durability and safety of bridge structures.

The comparison of the response of bearings with different health conditions paved the way for the applicability of the presented method. The damage identification methods based on comparative assessment of damaged and undamaged cases have already been used in past research. Typically, the artificial damage states had been induced in the structure for devising the damage identification methodologies (Li et al., 2021). On the contrary, the propitious availability of damaged and undamaged bearings at the target bridge site made it possible to devise a more realistic response-based bearing damage identification scheme.

Laser displacement sensor-based assessment could provide a detailed comparative study and differentiate the bridge structure’s bearing malfunction under undamaged and damaged conditions. These findings are consistent with the work of Zhao et al. (Zhao et al., 2015b), which has demonstrated the accuracy and viability of measurements of displacements response and the ability to operate using non-contact measurements for safe and efficient monitoring. However, this method depends on the known performance of healthy/undamaged bearing to compare with. This unavailability of performance data of a healthy bearing could be recorded from similar bridge structures, and analytical and numerical modeling approaches could be utilized otherwise. Further, the application and erection of laser sensors pose practical challenges, including the need for access to a bearing locality, special stands for instrument erection, and sufficient space for data acquisition setup. These requirements may hinder the feasibility and practicality of utilizing laser sensors for displacement measurements in bridge monitoring applications. As an alternative, accelerometers offer a promising solution for displacement estimation, as they do not necessitate direct contact with the bearing locality and can be installed in a more straightforward manner. However, it is essential to consider potential errors or uncertainties associated with displacement calculations based on acceleration data using double integration. Factors such as noise, drift, and integration errors can affect the accuracy of displacement estimates derived from acceleration data, highlighting the importance of robust signal processing techniques and calibration procedures to minimize these sources of error. Therefore, while accelerometers offer advantages in terms of ease of installation, it became imperative to explore alternative approaches that circumvent these limitations while ensuring the reliable bearing health assessment.

In the absence of laser sensor data, CPSD and Cross-correlation of acceleration data collected from the bridge girder and abutment could be utilized for acceleration response signals. The vibration response of a location caused by traffic excitation at a particular location and its proximity in the system could be effectively utilized for bridge-bearing performance evaluation. The CPSD and Cross-correlation relationship between the input signal and the output signal in the system corresponding to measured responses of damaged and undamaged conditions deviate from each other. From this result, the presence of damage in the bearing and the extent of damage greatly influence the vibrational response of the structure and the Cross-correlation of acceleration data in the structural system. The practicality of the proposed correlation-based assessment method has already been explored by different researchers (Yoon et al., 2017; Wang et al., 2021) and could be employed as a potential framework for bridge-bearing damage identification.

Initial measurement data on the target structure revealed that bearing damage significantly impacted the CPSD and cross-correlation. The damage from the original state of the structure alters the magnitude of the cross-correlation of accelerations, and its value is reduced from the undamaged state. The cross-correlation is inversely related to the presence of damage. It could be linked with the extent of the damage if sufficient data on various damage states and vibrational characteristics of the structure are available. The required data sets could be gathered from the numerical modeling of bridge structures with bearings of various damage states to collect data on vibrational characteristics and acceleration response. The experimental findings make it clear that this method needs reference cross-correlation data of undamaged structures for further applications in the current phase.

Furthermore, regarding the specific conditions of the vehicle running over the bridge for condition assessment of bearings, it is essential to consider several factors. Firstly, while the proposed methodology is designed to be less influenced by the presence of multiple vehicles on the bridge, it is recommended to avoid the presence of multiple vehicles during testing. Additionally, maintaining an average speed of 30 km/h during the test is crucial, as driving too slowly or too quickly may compromise the accuracy of the methodology. Therefore, it is recommended that the speed of the test vehicle closely aligns with the specified speed of 30 km/h. Furthermore, for effective bearing performance evaluation, the test vehicle should move closer to the steel bearings under investigation and over the main girder directly supported by that bearing, if feasible. Thus, adherence to these specific test conditions would ensure the reliability and accuracy of the proposed methodology for bearing condition assessment.

While the current study primarily focuses on the identification of individual bearing health status, it is pertinent to note that plate girder bridges commonly feature multiple bridge bearings arranged on the piers. The method proposed in this article relies on the test vehicle passing closer to the bearing or over the main girder directly resting over the bearing. Consequently, this methodology is deemed more suitable for assessing the performance of individual bearings rather than groups of bearings. Future research efforts should aim to explore the applicability of this method for evaluating the health status of groups of bearings in plate girder bridges. Moreover, with the easy installation of acceleration sensors and data acquisition, the proposed approach could be further gradually matured. It should strive to become a dependable damage identifier in bridge bearings based on extensive data set correlations. Figure 10B illustrates the schematic framework of the proposed approach for bearing condition assessment.