Abstract
This paper aims to achieve the accurate prediction of box girder deformation in SCS-enhanced PSC continuous box girder bridges. To this end, the authors proposed a box girder deformation prediction model based on unequal interval grey model (UIGM) and residual composite correction (RCC). Firstly, the tension forces of cables and the deformation data of box girder were regarded as the time sequence and the original unequal interval sequence, respectively, and the UIGM was constructed to predict the box girder deformation. Secondly, the sine function was constructed on the average features of the residual sequence waveform, and the periodic sequence function was generated by harmonic transform. The two functions were employed to extract the implicit periodic information and overcome the limitation of the UIGM (i.e., the UIGM only generates the change rate of a single index). Finally, the Markov chain method was adopted to treat the random fluctuations and further enhance the prediction accuracy and adaptability of the model. Then, the proposed UIGM-RCC was applied to predict the box girder deformation of Dongming Huanghe River Highway Bridge and forecast the settlement of a building mentioned in previous research. The results show that the proposed model can reflect the exact periodicity and random fluctuations of box girder deformation in SCS-enhanced PSC continuous box girder bridges. The research findings provide a meaningful reference for improving deformation prediction accuracy in SCS-enhanced structures.
1. Introduction
Prestressed-concrete (PSC) continuous box girder bridges often suffer from midspan lag and box girder cracking [1, 2]. These problems can be resolved by enhancing the bridges with a stay cable system (SCS) [3]. However, it is immensely difficult to predict the deformation of the SCS-enhanced box girder because of the fuzzy, stochastic, and indeterminant mechanical form and deformation mechanism of the bridges under natural and anthropic factors [4]. Considering the limits of existing theories and methods, the most practical way to elevate prediction accuracy lies in the estimation of girder deformation based on monitoring data [5–7].
In view of the indeterminacy, poor information, and high cost of sample data, many scholars have successfully introduced the grey system theory to solve engineering problems [8–10]. Nevertheless, the common grey prediction model only applies to strictly equal interval sequences, adding to the difficulty of engineering application. Taking the SCS as an example, the cable force control often relies on the force difference of unequal interval cables, with the aim to make tensioning more flexible. Moreover, the grey system theory may have low prediction accuracy and increased error, owing to the weak periodicity and strong random fluctuations of the box girder’s deformation sequence [11].
Based on the box girder deformation data of an SCS-enhanced PSC continuous box girder bridge, this paper discloses the relationship between the deformation pattern and the grey theory and proposes a deformation prediction model combining the unequal interval grey model (UIGM) and the residual composite correction (RCC). On one hand, the UIGM was adopted to solve the indeterminacy, poor information, and high cost of the sample data. On the other hand, the RCC was employed to enhance the low accuracy of the UIGM and the adaptability of the whole prediction model. Specifically, the sine function was constructed on the average features of the residual sequence waveform; the periodic sequence function was generated by harmonic transform. The two functions were employed to correct the residual sequence, which is random and alternatively positive and negative. Finally, the prediction results were compared with the data of the previous research. The comparison shows that the proposed prediction model has high accuracy and strong adaptability. This research provides a new control method for box girder deformation on SCS-enhanced bridges and boasts great theoretical and practical potential.
2. The Building and Solving of Unequal Interval Grey Model
Based on unequal interval grey theory [13], the cable force in tensioning phase was regarded as the original time sequence, and the vertical deformation of box girder recorded at the measuring points was deemed as the original unequal interval sequence. Then, the authors created the deformation prediction model of SCS-enhanced box girder in tensioning phase. The monitoring data on box girder deformation were processed by the sequential operators to disclose the change pattern, generate data sequences with certain regularity, and establish the corresponding differential equation, laying the basis for quantitative prediction of change.
2.1. Deformation Prediction Model of SCS-Enhanced Box Girder
Suppose the original sequence of box girder deformation isIf the cable force difference between the th and the th tensioning phases is , thenwhere .
Therefore, a one-time accumulated generation formula of the original sequence isAccording to formula (3), formula (1) can be rewritten as an unequal interval time-varying monotonically increasing sequence :Then, the differential equation of grey model can be established by the first-order generation module :where is the developing coefficient; is the grey activity.
The solution to the differential equation of the unequal interval sequence isSince and and are constants, if , then formula (6) can be transformed intoAccording to formula (3), perform a cumulative reduction and the fitting value can be obtained as
2.2. Solution to Grey Parameters
To make the fitting value infinitely close to the original value , the grey parameters and were solved by the relationship between the fitting value and the accumulated data:The grey parameters and are both constants. Formula (9) contains equations; any two of these equations could be constructed as simultaneous equation. For example, the th and th equations are constructed as follows:Solving (10), the value of grey parameter can be gotten and denoted as ; thenwhere represents the grey parameter value obtained by the th and th equations.
Construct simultaneous equation by any two equations in formula (9); grey parameters could be solved out and the average value of them isLiterature [14] takes the average value as the final grey parameter of the model and substitutes it into (9); then can be gotten, and the average value of them iswhere presents the grey parameter obtained by the th equation.
Take as the final grey parameter of the model, substitute and into formula (7), and perform inverse accumulated generating operation (IAGO). The resultant fitting value of the original sequence isThen, the prediction value in the next tension stage is
3. The Residual Composite Correction (RCC)
3.1. The Sine and the Periodic Sequence Function Residual Correction (SPSFRC)
After the girder deformation trend was fitted and predicted by the UIGM, the residual sequence was alternatively positive and negative, showing an unobvious periodicity. For better fitting and prediction accuracy, the sine function, constructed on the average features of the residual sequence waveform, and the periodic sequence function, generated by harmonic transform, were employed to modify the residual sequence [15]. The two functions are collectively referred to as the SPSFRC. Coupled with the UIGM, the SPSFRC can extract the implicit periodic information and overcome the limitation of the UIGM (i.e., the UIGM only generates the change rate of a single index). The SPSFRC is established in the following steps.
Step 1. Build the residual sequence .
Step 2. Fit by sine function , constructed on the average features of the residual sequence waveform.where ; is the average value of residual sequence wave time span; its optimal value follows the principle of the minimum quadratic sum of the fitting values residual for sine function; that is, and .
Step 3. Fit by periodic sequence function generated by harmonic variation:where .
Step 4. Introduce the weight coefficient () to combine the weights with formulas (17) and (18) and obtain the residual fitting value .Also, the optimal value of and follows the principle of the minimum quadratic sum of the fitting values residual by SPSFRC; that is, and .
Step 5. Calculate the prediction value modified by the SPSFRC.The box girder deformation of the th tensioning phase can be predicted based on the data of the previous tensioning phases. Then, the residual composite correction value of the tensioning phase is .
3.2. Markov Chain Residual Correction
The prediction based on the Markov chain [16] mainly derives the possible future state of a system based on the current state and the change trend of system variables. This approach is particularly suitable for problems with strong randomness. Under the influence of various factors, the box girder deformation has a stochastic growth rate and growth probability. Therefore, the Markov chain-based prediction can enhance the random features of the box girder deformation sequence and improve the prediction results. Suppose that the random system is in state at time . Then, its state at time has nothing to do with the state before time :The actual sequence is rationally divided into several states . Then, the state transition probability at step iswhere is the number of states transferred from to by steps in the system sequence; is the amount of raw data in state of the sequence.
Define the state transition matrix at step aswhere .
The division of the state interval is generally combined with the mean value and standard deviation of the data sequence [17]. The centre of each interval, that is, the average value of the two endpoints, is recorded as the state centre . For Markov chain-based residual prediction, the residual sequence was divided into states . transition probability vectors (the rows of a matrix) are needed to consider the transition processes in steps. The probability of residual state at the target moment is the sum of the vectors. The elements of each vector are denoted as . If there are a total of intervals of state , then the residual correction prediction value of the target moment, that is, the next moment, iswhere is the state weight; is the state interval centre.
After Markov chain-based prediction, the prediction value of the UIGM-RCC is corrected as
4. Case Study
4.1. Overview
The main bridge of Dongming Huanghe River Highway Bridge is a PSC continuous box girder bridge. The 990 m long bridge has a total of nine spans within the lengths: 75m-120m-120m-120m-120m-120m-120m-120m-75m. To overcome girder cracking and midspan sag, the main bridge was reinforced by the SCS, marking the first project of its kind in China. The bridge towers, the main beam, and the cables are connected as follows. The cables are fixed onto the top of bridge towers via steel anchor boxes and sliding cable saddles. The trimmer beams, laid laterally beneath the box girder, are attached to the bottom of the box girder by base plate [3]. The force on the stay cables is transmitted to the main beam through the trimmer beams and base plates. The designed force of long cable is 2,700 kN and that of short cable is 2,100 kN. During the construction, the cables were tensioned symmetrically by 8 lifting jacks. Thus, the cables become shorter and shorter from the middle towers (61# and 62#) to the side towers (58# and 65#). To capture the box girder deformation during the tensioning, 114 vertical displacement measuring points were placed upstream and downstream of the bridge. In the actual project, the cable tensioning phases are divided into 7 stages, and the final tension values of long and short cables are 90% of the design cable values of the bridge, which are 2,430 kN and 1,890 kN, respectively. This paper elaborates and studies according to the tension force values of short cables, namely, 630 kN, 1,050 kN, 1,155 kN, 1,365 kN, 1,575 kN, 1,785 kN, and 1,890 kN. The drawing of Dongming Huanghe River Highway Bridge strengthened by the stay cable system is shown in Figure 1.
4.2. Example Analyzing
To verify the performance of the proposed prediction model, the deformation values of the box girder at C21 measuring point in the 1st~7th tensioning phases were selected for modelling analysis (Table 1). The values of the first 6 tensioning phases were taken as the raw data, and the deformation value of the 7th tensioning phase was considered as the target of prediction.
(1) The fitting values and preliminary prediction values were obtained by the UIGM, and then the residual sequence of the fitting values was figured out.
Table 2 contains the fitting values of the first 6 tensioning phases. It can be seen that the UIGM has a large error in fitting, but the error is gradually decreasing. For the 7th tensioning phase, the predicted value of UIGM is 40.7586 mm, which is quite different from measuring value, and the relative error is 13.7019%.
(2) Calculate the optimal composite parameters of the SPSFRC, the fitting values of the residual sequence, and the prediction values after one correction.
First, the sine function was used to modify , and the optimal parameter was determined by the quadratic sum of the fitting values. Let and let the step length be 1. The optimal result was obtained at . Figure 2 shows the results of the first 300 iterations. In Figure 2, the triangle represents the corresponding point of the optimal result.
After that, the residual sequence was modified by the periodic sequence function and sine function generated by harmonic transform. The optimal parameters and were still determined by the quadratic sum of the fitting values. Suppose that with a step length of 0.01 and with a step length of 1. In this case, the cumulative residual of the fitting values was minimal () at , , and . Figure 3 shows the results of the first 300 iterations. In Figure 3, the triangle represents the corresponding point of the optimal result.
There are 6 pieces of data in the residual sequence of the first 6 tensioning phases. The fitting values modified by the SPSFRC areThe 7th tensioning phase, that is, 1,890 kN, was substituted into the SPSFRC, putting the correction value at 1.0441. Thus, the prediction value after one correction isThe above analysis shows that the SPSFRC improved the prediction accuracy by lowering the relative error to 11.49%.
(3) Divide the residual sequence into different state intervals according to the Markov chain, solve the state transition matrix of the residual sequence, and figure out the Markov residual prediction value and the final prediction value .
The residual sequence was divided into 3 state intervals based on the mean value and standard deviation of the residual sequence: [-1.9,-0.6], [-0.6,+0.6], and [+0.6,+1.9], corresponding to states , , and . These states should cover the entire range of the residual sequence. The range of the 3 states is [−3.71, 0.02], [0.02, 3.46], and [3.46, 7.18], respectively. The corresponding states are shown in Table 3.
The residual value of the 7th tensioning phase was predicted based on the transitions of the first 6 tensioning phases. The transition contains a total of 3 steps, and the transition probabilities are listed in Table 4.
Solved by , the state weights are and , and the state centres are , and . Then, the Markov chain-based residual prediction value of the 7th tensioning phase isThus, the final deformation of the box girder of the 7th tensioning phase isThe fitting accuracy of the proposed UIGM-RCC was compared with the UIGM model and the measured values (Table 5). It is clear that the proposed model outshines the UIGM model in fitting accuracy. It can be seen from the prediction result of the 7th deformation value that the relative errors of UIGM-RCC and UIGM are 0.2243% and 13.7019%, respectively. The results show that the fitting and predicted values of the proposed model in this paper are close to the measuring values.
In the same way, the deformation values of the 8th~12th times (i.e., 2,100 kN, 2,205 kN, 2,310 kN, 2,415 kN, and 2,520 kN tension forces) are extrapolated. Table 6 contains the prediction values obtained by UIGM and UIGM-RCC. For intuitiveness, the data in Tables 5 and 6 were converted into graphs (Figure 4).
For further verification of the fitting effect, prediction accuracy, and adaptability, the box girder deformation values of other measuring points in the first 6 tensioning phases were selected as the original data (Table 7) to predict the deformation in the 7th tensioning phase. denotes distance from center line of 57# pier in Table 7.
The optimal parameter values and prediction results are shown in Tables 8 and 9, respectively.
As shown in Table 9, the UIGM-RCC achieved a small deviation between the prediction values and the measured values. The mean relative error was only 1.11%, far below the 8.71% of the UIGM. Figure 5 compares the box girder deformation predicted by the two models. Note that the absolute deviation of predicted values from the measured values is magnified 5 times.
Table 10 shows the 8th~12th times’ prediction results of the deformation obtained by UIGM-RCC. The results show that the prediction value of the measuring point C21 is the largest when the cable tension is drawn to the design tension value of the bridge, and its value is 48.1295 mm. When the cable tension is drawn to the 1.2 times’ design tension value of the bridge, the measuring point C21 is raised by 3.0109 mm compared with the span in the 7th tensioning phase, and the prediction deformation is over 50 mm.
In addition, the UIGM-RCC and the UIGM were also applied to calculate the settlements of the building in literatures [11, 12] in the 9th and 10th phases according to the measured data of the first eight phases. Tables 11 and 12 compare the fitting, predicted results, and relative error of UIGM: literature [11], literature [12], and our model. Through the comparison of results, it is learned that the UIGM-RCC has much higher prediction accuracy.
Note that the absolute deviation of calculated values from the measured values is magnified 3 times. In Figure 6, it can be seen that the results of the UIGM-RCC agree well with the measured values.
According to the measured data of the first eight phases, the UIGM-RCC is used to predict the settlement in the 160th~210th days in this paper, as shown in Table 13. The results show that the settlement is over 5 mm in the 174th day and the settlement is 5.6431 mm in the 210th day.
5. Conclusion
The monitoring and prediction of box girder deformation are essential to SCS-enhanced PSC continuous box girder bridges. The effective and timely prediction can guarantee the smoothness and intelligent control of tensioning. However, the box girder deformation features periodicity and random fluctuations under the joint action of environmental and anthropic factors, making it difficult to predict the deformation with common models.
The UIGM can partially overcome the difficulty in deformation prediction. However, this model faces a large deviation between the prediction value and the measured value (mean relative error: 8.71%). In light of this, the prediction results of the UIGM were firstly modified by the SPSFRC and then corrected by the Markov chain. After the two corrections, the mean relative error was reduced to 1.11%. In this way, the author created a new method for long-term deformation prediction of SCS-enhanced box girders. The proposed model was applied to predict the settlement of a building. The mean relative error of the prediction results was only 2.29%, far lower than that of the UIGM and other models. This further validates the prediction accuracy of the UIGM-RCC and expands the application scope of the model.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of the paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 11372165).