Abstract
Spherical steel bearings play an important role in the normal longitudinal expansion of the main girders of long-span bridges. With the increase in service time, the wear damage will deteriorate the longitudinal slip performance of spherical steel bearings. In this research, a method of real-time quantitative evaluation on longitudinal slip performance is proposed through monitoring data analysis, correlation analysis, damage evaluation analysis, and experiment data analysis. Monitoring data analysis shows that the temperature field has a good linear relationship with longitudinal displacement. Correlation analysis shows that this relationship is well described by a time-varying multiple linear regression model. Furthermore, bearing friction is used as an index for real-time quantitative evaluation, and a large value of bearing friction indicates serious damage. An evaluation model considering the influence of temperature field and bearing frictions is proposed. The time-varying values of bearing frictions are calculated through Kalman filtering analysis. Experimental results show that the maximum evaluation error of this method is less than 5%, verifying that the proposed method is feasible for real-time quantitative evaluation on the longitudinal slip performance of bridge bearings.
1. Introduction
Spherical steel bearing is one kind of important force-bearing component for long-span bridge structures, which transfers vertical loads from superstructure to substructure and satisfies the longitudinal expansion performance of main girders as well [1–3]. A spherical steel bearing is mainly composed of a top support plate, PTFE plate (significant deformation occurring in PTFE), steel ball core, bottom support plate, and bolts. With the increase in service time, the wear damage of the PTFE plate will deteriorate the longitudinal slip performance of bearings. As shown in Figure 1, the severe wear of the PTFE plate of a suspension bridge, located in Jiangyin, increased the friction force, finally caused the PTFE plate to peel off [4]. What is more, the bearing damage always results in considerable costs and traffic interruption [5–7].
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Structural temperature has a significant effect on the longitudinal displacement of bridge girders [8–15]. For example, Xia et al. [15] carried out field monitoring and numerical analysis of the temperature behavior of the Tsing Ma Suspension Bridge, and the results show that the time histories of the temperature field and its effect have very similar changing trends. Further research results show that temperature has a linear correlation with longitudinal displacement [13–15]. Using long-term monitoring data, Wang et al. [16] built the mathematical model of the linear correlation between longitudinal displacement and temperature field. The wear damage of the PTFE plate will change the linear correlation. Hence, an abnormal correlation indicates bearing damage. For example, Webb et al. [17] detected the partial damage of the flyover’s pier bearings through an analysis of the correlation between pier displacement and uniform temperature.
However, the temperature field of continuous steel truss bridges is very complicated [18, 19]. There are distinct temperature differences in steel truss girders, which may have a serious impact on the longitudinal displacements of main girders [20]. In the current study, the effect of temperature differences in steel truss girders has not been thoroughly considered. In addition, continuous truss bridges usually have many bearings, and it is difficult to localize and quantify the damaged bearings in real-time. In the current study, a period of monitoring data is necessary to mathematically model the correlation between temperature and longitudinal displacement [15–17], which is not appropriate for real-time analysis.
Hence, the aim of this paper is to achieve real-time quantitative evaluation on the longitudinal slip performance of bridge bearings using the monitoring data of temperature and longitudinal displacements. In order to achieve this goal, this paper thoroughly studied the influence of complex temperature fields and bearing frictions on longitudinal displacements of bridge bearings through monitoring data analysis, correlation analysis, damage evaluation analysis, and experiment data analysis. Finally, a time-varying multiple linear regression model containing the influence of temperature field and bearing frictions is proposed for real-time quantitative evaluation on longitudinal slip performance of bridge bearings, which is verified by an experiment test.
2. Monitoring Data Analysis
2.1. Description of the Bridge and the Monitoring System
The bridge in this study is a continuous steel truss arch bridge with a total length of 1,272 m, as shown in Figure 2(a). There are seven sets of bearings, namely 1#–7#, which are movable except 4#. The upper and downstream sides of the six movable bearings are, respectively, installed with a displacement sensor, a total of 12 displacement sensors, respectively, represented by Di,u and Di,d, i = 1, 2, …, 6. All the displacement sensors are the magnetostrictive displacement sensor, the sampling frequency of which is 1 Hz. As shown in Figures 2(b) and 2(c), six temperature sensors are installed on the top chord, bottom chord, and bridge deck chord of this bridge, which are denoted by T, containing T1–T6. All temperature sensors are the fiber bragg grating, the sampling frequency of which is also 1 Hz.
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2.2. Temperature Features
Due to the interference of the external environment of the bridge site, the monitoring data will appear abnormal phenomena, such as “spike,” “jump,” and “drift.” As shown in Figure 3(a), the daily time–history curve of structure temperature presents distinct daily characteristics but appears at the “spike” point at about 9:45 am. In addition, from 15:30 to 20:30 pm, there are many “spikes” in the temperature curve. If the abnormal data are not processed, the accuracy of the analysis will be affected. The time window smoothing method is mostly used to solve this problem [21, 22]. The length of the time window is depended on the data scale and timeliness, which is 10 min in this paper, i.e., the raw data will be averaged every 10 min. The daily time–history curve of T becomes smooth after 10-min averaged in Figure 3(a). The structure temperature data after processing is named TP, containing TP1–TP6, which is used for correlation analysis and correlation model construction below. The daily time-history of structure temperature on the 13th day is shown in Figure 3(b); we can see that the daily time–history curves are similar to a sine wave, and there is a temperature gradient (TG) in the main beam.
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Temperature is mainly affected by solar radiation. T1 and T2 are located on the top chord, which is significantly affected by sunlight, while T5 and T6 are located on the bottom plate of the bridge, which are not exposed to sunlight. Therefore, the TG is investigated, which is denoted by TPi,j (namely, TPi,j = TPi − TPj). TG contains TG between two sides of a single chord (TGS) and TG between different chord members (TGM) in this paper. The average values are used to represent the uniform temperature TPu (namely, TPu = (TP1 + TP2)/2), TPm (namely, TPm = (TP3 + TP4)/2), and TPd (namely, TPd = (TP5 + TP6)/2). The maximum and minimum values of temperature differences are shown in Table 1. It can be seen that there are significant TGs in this bridge; the maximum and minimum of TGS are 5.76 and −11.81°C, respectively. The maximum and minimum of TGM are 11.16 and −7.66°C, respectively. Hence, both TGS and TGM should be taken into consideration in the analysis of the temperature effect on the longitudinal displacement.
2.3. Displacement Features
The longitudinal displacement of each set of bearings is measured by two displacement sensors; the average value of two measured displacements is calculated to represent the displacement of each set of bearings in this paper, which are denoted as D1–D6. In the raw data, the longitudinal displacement is caused by many factors, such as vehicle, train, wind loads, etc. Figure 4 shows the extraction process of temperature-induced displacement (TID). Wavelet decomposition is usually used for filtering the raw data to get the TID, which is the low-frequency information [23]. Then, the displacements after filtering are averaged every ten minutes to get the same length of temperature data; the displacement data after processing is named DP, containing DP1–DP6, which is used for correlation analysis and correlation model construction below.
The time–history curve of TP1, TP3, and DP1 is shown in Figure 5; it can be seen that the time–history curve of TP1, TP3, and DP1 has very similar changing trends and presents distinct seasonal characteristics during the sampling period. In addition, according to the previous analysis, there is significant TG in these steel truss bridges, which should be taken into consideration.
3. Correlation Modeling Analysis
In this paper, the correlation coefficient R, shown in Equation (1), is introduced to describe the correlation between TP and DP, which is between 0 and 1, and the larger R is, the stronger the correlation between variables.where Rn,m denotes the correlation coefficient between DPn and TPm, DPn,i denotes the ith value of DPn, TPm,i denotes the ith value of TPm, and denotes the average value of DPn and TPm, respectively, N denotes the total amount of data. In this paper, n = 1, 2, … 6 and m = 1, 2, …, 6.
According to Equation (1), the daily correlation coefficient between DP and TP can be calculated. Figure 6 is the scatter diagram of the daily correlation coefficient of TP1, TP3, TP5 versus DP1; we can see that there is a significant positive correlation between the TID and temperature, and the correlation coefficient R is mainly 0.85–1.0. Hence, a linear regression model is appropriately used to describe this linear correlation. To consider the temperature variables between chords, TP1–TP6 are used temperature variables in this linear regression model.
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The model is a multiple regression equation as follows:where DPj(t) denotes the regression value of the DPj at time t, j = 1, 2, …, M. TPi(t) denotes TPi at time t, i = 1, 2, …, N. γi, j denotes the linear constant coefficient between TPi(t) and DPj(t), and Cj is the constant term. In this paper, M = 6 and N = 6.
In Equation (2), a long period of monitoring temperature and displacement data is needed to determine the values of γi, j, so Equation (2) is not suitable for real-time analysis. Real-time analysis helps to identify the bearing damage in time, so Equation (2) is modified into a time-varying linear regression equation as follows:where γi, j(t) denotes the time-varying linear regression coefficient of at time t.
However, the time histories of TP1–TP6 have similar changing trends, which are not suitable to be treated as independent variables. Principal components analysis is used to extract the principal components of the temperature field, which are independent. The principal components of temperature are sorted according to the contribution rate, and the greater the contribution rate, the more original information contained in the principal component. As shown in Figure 7(a), the six temperature principal components, denoted as P1–P6, are ranked in descending order of contribution rate. We can see that the cumulative contribution rate of P1, P2, and P3 is 99.41%, and the trends of the time–history curves of them are different, as shown in Figure 7(b). The principal components are independent of each other, which can be treated as independent variables. Hence, Equation (3) is changed as follows:
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Kalman filtering method is used to determine the values of γi, j(t). In this method, a state equation is given as follows:where Ψj(t) is a column vector of containing time-varying parameters γi, j(t), i.e.,Ψj(t) = (γ1,j(t) γ2,j(t) … γQ, j(t)]T. Ωj is an iteration matrix. Vj(t) is a column vector of containing white noises. By virtual of Equations (4) and (5), the values of γi, j(t) are determined using the Kalman filtering method. For example, the values of P1, P2, P3, and DP1 of the bridge are substituted into Equation (4), and then the values of γ1,1(t), γ1,2(t), and γ1,3(t) are determined, as shown in Figure 8(a). It can be seen that the changing trends of γ1,1(t), γ1,2(t), and γ1,3(t) are stationary with time after a quick convergence. The changing trends help to identify the bearing damage. If the bearing is in good health, the time-varying coefficient tends to be flat; otherwise, the damage is occurring. Hence, Figure 8(a) indicates that the bearing 1# is in good health during the monitoring period. The result is correct because the model is established based on the data of the bridge in the early stage of operation.
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In order to verify the accuracy of Equation (4), the P1(t), P2(t), P3(t), γ1,1(t), γ1,2(t), and γ1,3(t) are substituted into Equation (4) to obtain the simulated DP1,s(t). Then, the simulated and monitoring displacements are compared, as shown in Figure 8(b). It can be seen that the simulated values are very close to the monitoring values, indicating that it is appropriate to use Equation (4) for a description of the correlation. Figures 8(c) and 8(d) show the time-history curves of calculated and measured values at the early and late sampling periods, respectively; we can see that as the amount of data increases, the simulated values become more accurate.
4. Damage Evaluation Analysis
As mentioned above, the temperature field of bridge girders has an effect on bearing displacements. The wear damage inside the bearings will cause friction, existing between the steel ball core and the PTFE plate, which prevents the bearings from moving, as illustrated in Figure 9. Therefore, when bearing wear occurs, the longitudinal displacement is also affected by the friction, which is modeled by an equation as follows:where DPj(t) denotes the total displacement of the jth bearing at time t. DPT,j(t) denotes the displacement caused by structure temperature at time t, and DPf,j(t) denotes the displacement caused by friction at time t. DPj, T(t) has a good linear correlation with structure temperature, which is expressed as Equation (7a), and DPj, f(t) is expressed as Equation (7b).where fj(t) denotes the friction of the jth bearing, δj denotes the jth longitudinal flexibility coefficient.
When the sliding friction of the bridge bearings increases, it indicates that the sliding plates inside the bearings are worn. Therefore, the wear condition of bearings can be evaluated in real time by monitoring the change of bearing friction force in real time. The specific calculation method of bearing friction is described as follows:
(1) The temperature field and displacement monitoring data are substituted into Equations (6) and (7), so the displacement residual Rj(t) includes the friction force of the bearings as follows:where ej(t) is a stationary sequence with a mean value of zero caused by model fitting error. Combined Equation (6), Rj(t) (j = 1, 2, ⋯, M) can be furthermore represent as follows:
(2) Rj(t)−Rj−1(t) represented by ΔRjj−1(t), is calculated to reduce repetitive frictional terms as follows:where ϑjj−1(t) is the difference of model fitting error ej(t) (ϑjj−1(t) = ej(t)-ej−1(t)).
(3) The friction force of each bearing can be calculated by transforming Equation (10) as follows:
Then the vector set of fj(t) is denoted by F(t), the vector set of δj is denoted by Φ, and the vector set of ΔRjj−1(t)−ϑjj−1(t) is denoted byω. Equation (11) is simplified as follows:
The daily maximum values of F(t) are selected, denoted by Fv(t), during the sampling period (Fv(t) = (f1,v(t) f2,v(t) … fM,v(t))T). Then, the fitting curves of Fv(t), represented by Mj(t), can be obtained based on Fv(t). Finally, the degraded bearings can be identified and evaluated according to the trend of Mj(t). For example, when Mj(t) shows an increasing trend, it means that bearing j is in a degraded state. Otherwise, bearing j is healthy.
5. Experiment Test Analysis
5.1. Description of Experiment Design
The experiment model is a two-span continuous beam bridge with three bearings, as shown in Figure 10. Bearing 1# is fixed on the test bench, and Bearings 2# and 3# are movable in the longitudinal direction, which are simulated by pulleys. The bridge girder is simulated by Springs 1 and 2 (δ1 = 4.17 mm/N, δ2 = 2.63 mm/N). The friction forces of bearings are simulated by Springs 3 and 4, i.e., the compression of each spring produces a force f1(t) or f2(t) that prevents the bearing from moving. Because it is difficult to simulate temperature action Pi(t), an actuator is used to apply a horizontal load A(t) on the one end of the girder to simulate the influence of Pi(t) on displacement. This is feasible because A(t) also has a linear correlation with D(t). A(t) is measured by a high-precision pressure sensor, which is located between the beam end and the actuator. Red marks are pasted at the bottom of the bearings, as shown in Figure 11. When the bearings move, a camera will track the displacements Di(t) of the centroids of red mark points using the image recognition algorithm [24]. Under this experiment condition, Equations (11) and (12) are written as follows:
5.2. Experimental Result
In this experiment, A(t) is loaded in 14 steps produced by an actuator, and the values of D1(t) and D2(t) are measured by the camera. The values in each step are shown in Table 2. Then, by substituting A(t), D1(t), and D2(t) into Equations (6) and (7), the values of f1(t) and f2(t) are determined using the proposed method, as shown in Table 2. In addition, the actual friction values and f1a(t) of Bearings 1 and 2 are calculated according to Hooke’s law by Equation (14a), as shown in Table 2. The relative errors E1 and E2 are calculated by Equation (14b), as shown in Table 1.
where ki is the stiffness coefficient of the Spring 3 or 4 (k1 = 0.24N/mm, k2 = 0.38N/mm), i = 1,2. It can be seen from Table 2 that fi(t) is close to fia(t), and the relative error is less than 5%, verifying that the proposed method is feasible for real-time quantitative evaluation on bearing damage of continuous beam bridges.
6. Conclusion
In this research, a method of real-time quantitative evaluation on the longitudinal slip performance of spherical steel bearings is proposed through monitoring data analysis, correlation modeling analysis, damage evaluation analysis, and experiment test analysis. The main conclusions are drawn as follows:(1)The longitudinal displacement of the bearing exhibits a strong linear correlation with the monitored temperature. Since temperature and displacement change at every moment, a constant coefficient model cannot dynamically describe the real-time the longitudinal slip performance of the bearings. Then, a multiple linear regression model with time-varying coefficients is proposed, and the time-varying coefficients in the model are solved by the Kalman filter method. This model describes the longitudinal displacement of the bearing accurately and can help to identify bearing damage.(2)Considering the influence of temperature field and bearing frictions, a time-varying multiple linear regression model is proposed for real-time quantitative evaluation of the longitudinal slip performance of spherical steel bearings. In this model, time-varying bearing friction is used as an evaluation index, which is used for real-time quantitative evaluation. Experimental analysis shows that the maximum evaluation error is less than 5%, verifying that the proposed method is feasible for real-time quantitative evaluation.(3)There are significant vertical and transversal temperature differences between truss members in steel truss bridges. A suggestion is that significant vertical and transversal temperature differences should be taken into consideration in the analysis of temperature effects.
Data Availability
Data are available on request from the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors gratefully acknowledge the National Natural Science Foundation of China (grant number 51908545) and the 2021 Science and Technology R&D Project (Science and Technology Project of State Grid Jiangsu Electric Power Co., Ltd.) (no. J2021022). This work was supported by the Innovative and Entrepreneurial Talent Plan of Jiangsu Province of China and the Natural Science Foundation of Jiangsu Province of China (grant number BK20180652).