Deformation Forecast of Main Girder Enhanced by Stay Cable System with Unequal Interval Grey Model and Residual Composite Correction

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 functionwas 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 buildingmentioned in previous research.The results show that the proposedmodel 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.


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][6][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][9][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 SCSenhanced 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.

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.
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 X(0) (  ) of the original sequence is Then, the prediction value in the next tension stage is

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.
Also, the optimal value of  and  follows the principle of the minimum quadratic sum 2 min of the fitting values residual by SPSFRC; that is, Step 5. Calculate the prediction value modified by the SPS-FRC.
The box girder deformation of the  + 1th tensioning phase can be predicted based on the data of the previous  tensioning phases.Then, the residual composite correction value of the  +1 tensioning phase is X(0) 1 ( +1 ).

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  + 1 has nothing to do with the state before time : The actual sequence is rationally divided into several states  1 ,  2 , ⋅ ⋅ ⋅,   .Then, the state transition probability at step  is where 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  as where ∑  =1  ()  = 1.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  (0) (  ) 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  1 ,  2 , ⋅ ⋅ ⋅ ,   .If there are a total of  intervals of state , then the residual correction prediction value of the target moment, that is, the next moment, is where   is the state weight;   is the state interval centre.After Markov chain-based prediction, the prediction value of the UIGM-RCC is corrected as 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.

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 X(0) (  ) and preliminary prediction values X(0) ( +1 ) were obtained by the UIGM, and then the residual sequence  (0) (  ) 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 40.7586mm, 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 X(0) 1 ( +1 ) after one correction.First, the sine function was used to modify  (0) (  ), and the optimal parameter  best was determined by the quadratic sum 1 of the fitting values.Let  ∈ [1, 10, 000] and let the step length be 1.The optimal result 1 min = 27.2228mm 2 was obtained at  = 114s.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  best and  best were still determined by the quadratic sum 2 of the fitting values.Suppose that  ∈ [0, 1] with a step length of 0.01  and  ∈ [1, 10, 000] with a step length of 1.In this case, the cumulative residual of the fitting values was minimal (2 min = 3.0365mm 2 ) at  = 114s,  = 0.17, and  = 10. 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 are  The 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 is The 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 ( +1 ) and the final prediction value X(0) 2 ( +1 ).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  1 ,  2 , and  3 .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  1 =  2 = 0 and  3 = 1, and the state centres are  1 = −1.8425, 2 = 1.7394, and  3 = 5.3214.Then, the Markov chain-based residual prediction value of the 7th tensioning phase is Thus, the final deformation of the box girder of the 7th tensioning phase is The 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.

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

Figure 1 :
Figure 1: Drawing of Dongming Huanghe River Highway Bridge strengthened by the stay cable system.

2 )Figure 2 :
Figure 2: Relationship between parameter  and quadratic sum 1 of the fitting values.

Figure 3 :
Figure 3: Relationship between parameters  and  and quadratic sum 2 of the fitting values at  best = 114s.

Figure 4 :
Figure 4: Comparison of the different model results.

Figure 5 :
Figure 5: Prediction results of measuring points.

Table 1 :
The deformation values of C21 in different tensioning phases.

Table 2 :
The fitting values by UIGM.

Table 3 :
States of residual sequence.

Table 5 :
Comparison of the fitting values by the different prediction models.

Table 6 :
Comparison of the predicted values by the different prediction models.

Table 7 :
The measured values of C14∼C23 measuring points.

Table 8 :
Optimal parameters of the different measuring points.

Table 11 :
Fitting values of building settlement.Fitting values/mm Relative errors/% Fitting values/mm Relative error/% Fitting values/mm Relative error/% Fitting values/mm Relative error/%