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

Authors: Gangnian Xu; Youzhi Wang; Shimin Wang; Quan Yuan
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 is(1)X0=X0F1,X0F2,···,X0Fk,···,X0FnIf the cable force difference between the kth and the k-1th tensioning phases is ΔFk, then(2)ΔFk=Fk-Fk-1≠const,k=2,3,···,nwhere ΔF1=1.Therefore, a one-time accumulated generation formula of the original sequence is(3)X1F1=X0F1X1Fk=X1Fk-1+ΔFkX0Fk,k=2,3,···,nAccording to formula (3), formula (1) can be rewritten as an unequal interval time-varying monotonically increasing sequence X(1):(4)X1=X1F1,X1F2,···,X1Fk,···,X1FnThen, the differential equation of grey model can be established by the first-order generation module X(1):(5)dX1dF+aX1=uwhere a is the developing coefficient; u is the grey activity.The solution to the differential equation of the unequal interval sequence is(6)X^1Fk=X1F1-uae-aFk-F1+uaSince X(1)(F1) and a and u are constants, if q=X(1)(F1)-u/a, then formula (6) can be transformed into(7)X^1Fk=qe-aFk-F1-1+X1F1According to formula (3), perform a cumulative reduction and the fitting value can be obtained as(8)X^0F1=X^1F1X^0Fk=X^1Fk-X^1Fk-1ΔFk,k=2,3,···,n


### 2.2. Solution to Grey Parameters

To make the fitting valueX^(1)(Fk) infinitely close to the original value X(1)(Fk), the grey parameters a and q were solved by the relationship between the fitting value and the accumulated data:(9)X1F2=X^1F2=qe-aF2-F1-1+X1F1X1F3=X^1F3=qe-aF3-F1-1+X1F1⋮X1Fn=X^1Fn=qe-aFn-F1-1+X1F1The grey parameters a and q are both constants. Formula (9) contains n-1 equations; any two of these equations could be constructed as simultaneous equation. For example, the ith and jth equations are constructed as follows:(10)X1Fi=qe-aFi-F1-1+X1F1X1Fj=qe-aFj-F1-1+X1F1Solving (10), the value of grey parameter can be gotten and denoted as aij; then(11)aij=1Fj-Filn⁡Fj-F1Fi-F1·X1Fi-X1F1X1Fj-X1F1where aij represents the grey parameter value obtained by the ith and jth equations.Construct simultaneous equation by any two equations in formula (9); (n-1)(n-2)/2 grey parameters could be solved out and the average value of them is(12)a-=2n-1n-2∑i=2n-1∑j=i+1naij=2n-1n-2∑i=2n-1∑j=i+1nln⁡Fj-F1/Fi-F1·X1Fi-X1F1/X1Fj-X1F1Fj-FiLiterature [14] takes the average value a- as the final grey parameter a of the model and substitutes it into (9); then qi can be gotten, and the average value of them is(13)q-=1n-1∑i=2nqi=1n-1∑i=2nX1Fi-X1F1e-aFi-F1-1where qi presents the grey parameter obtained by the ith equation.Takeq- as the final grey parameter q of the model, substitute a and q into formula (7), and perform inverse accumulated generating operation (IAGO). The resultant fitting value X^(0)(Fk) of the original sequence is(14)X^0F1=X1F1X^0Fk=qe-aFk-e-aFk-1e-aF1ΔFk,k=2,3,···,nThen, the prediction value in the next tension stage is(15)X^0Fk+1=X^1Fk+1-X^1FkΔFk+1=qe-aFk+1-e-aFke-aF1ΔFk+1


## 2.1. Deformation Prediction Model of SCS-Enhanced Box Girder

Suppose the original sequence of box girder deformation is(1)X0=X0F1,X0F2,···,X0Fk,···,X0FnIf the cable force difference between the kth and the k-1th tensioning phases is ΔFk, then(2)ΔFk=Fk-Fk-1≠const,k=2,3,···,nwhere ΔF1=1.Therefore, a one-time accumulated generation formula of the original sequence is(3)X1F1=X0F1X1Fk=X1Fk-1+ΔFkX0Fk,k=2,3,···,nAccording to formula (3), formula (1) can be rewritten as an unequal interval time-varying monotonically increasing sequence X(1):(4)X1=X1F1,X1F2,···,X1Fk,···,X1FnThen, the differential equation of grey model can be established by the first-order generation module X(1):(5)dX1dF+aX1=uwhere a is the developing coefficient; u is the grey activity.The solution to the differential equation of the unequal interval sequence is(6)X^1Fk=X1F1-uae-aFk-F1+uaSince X(1)(F1) and a and u are constants, if q=X(1)(F1)-u/a, then formula (6) can be transformed into(7)X^1Fk=qe-aFk-F1-1+X1F1According to formula (3), perform a cumulative reduction and the fitting value can be obtained as(8)X^0F1=X^1F1X^0Fk=X^1Fk-X^1Fk-1ΔFk,k=2,3,···,n


## 2.2. Solution to Grey Parameters

To make the fitting valueX^(1)(Fk) infinitely close to the original value X(1)(Fk), the grey parameters a and q were solved by the relationship between the fitting value and the accumulated data:(9)X1F2=X^1F2=qe-aF2-F1-1+X1F1X1F3=X^1F3=qe-aF3-F1-1+X1F1⋮X1Fn=X^1Fn=qe-aFn-F1-1+X1F1The grey parameters a and q are both constants. Formula (9) contains n-1 equations; any two of these equations could be constructed as simultaneous equation. For example, the ith and jth equations are constructed as follows:(10)X1Fi=qe-aFi-F1-1+X1F1X1Fj=qe-aFj-F1-1+X1F1Solving (10), the value of grey parameter can be gotten and denoted as aij; then(11)aij=1Fj-Filn⁡Fj-F1Fi-F1·X1Fi-X1F1X1Fj-X1F1where aij represents the grey parameter value obtained by the ith and jth equations.Construct simultaneous equation by any two equations in formula (9); (n-1)(n-2)/2 grey parameters could be solved out and the average value of them is(12)a-=2n-1n-2∑i=2n-1∑j=i+1naij=2n-1n-2∑i=2n-1∑j=i+1nln⁡Fj-F1/Fi-F1·X1Fi-X1F1/X1Fj-X1F1Fj-FiLiterature [14] takes the average value a- as the final grey parameter a of the model and substitutes it into (9); then qi can be gotten, and the average value of them is(13)q-=1n-1∑i=2nqi=1n-1∑i=2nX1Fi-X1F1e-aFi-F1-1where qi presents the grey parameter obtained by the ith equation.Takeq- as the final grey parameter q of the model, substitute a and q into formula (7), and perform inverse accumulated generating operation (IAGO). The resultant fitting value X^(0)(Fk) of the original sequence is(14)X^0F1=X1F1X^0Fk=qe-aFk-e-aFk-1e-aF1ΔFk,k=2,3,···,nThen, the prediction value in the next tension stage is(15)X^0Fk+1=X^1Fk+1-X^1FkΔFk+1=qe-aFk+1-e-aFke-aF1ΔFk+1


## 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 sequenceE(0)(Fk).(16)E0Fk=X0Fk-X^0Fk,k=1,2,···,nStep 2.
FitE(0)(Fk) by sine function E^1(0)(Fk), constructed on the average features of the residual sequence waveform.(17)E^10Fk=Asin⁡2πFk+T/2Twhere A=1/n∑i=1nE(0)(Fk); T is the average value of residual sequence E(0)(Fk) wave time span; its optimal value follows the principle of the minimum quadratic sum e1min of the fitting values residual for sine function; that is, e1min=min⁡e11,e12,···,e1i,···,e1n and e1i=∑k=1n[E1(0)(Fk)-E^1i(0)(Fk)]2.Step 3.
FitE(0)(Fk) by periodic sequence function E^2(0)(Fk) generated by harmonic variation:(18)E^20Fk=a0+∑i=1maicos⁡ωiFk+bisin⁡ωiFkwhere ωi=2iπ/Fn,i=1,2,⋯,m,a0=1/n∑k=1nE(0)(Fk),ai=2/n∑k=1nE0Fkcos⁡(ωiFk),bi=2/n∑k=1nE0Fksin⁡(ωiFk).Step 4.
Introduce the weight coefficientξ (0≤ξ≤1) to combine the weights with formulas (17) and (18) and obtain the residual fitting value E^(0)(Fk).(19)E^0F1=E0F1E^0Fk=1-ξE^1Fk+ξE^2Fk,k=2,3,···,nAlso, the optimal value of ξ and m follows the principle of the minimum quadratic sum e2min of the fitting values residual by SPSFRC; that is, e2min=min⁡e21,e22,···,e2i,···,e2n and e2i=∑k=1n[E(0)(Fk)-E^i(0)(Fk)]2.Step 5.
Calculate the prediction value modified by the SPSFRC.(20)X^10F1=X^0F1X^10Fk=X^0Fk+E^0Fk,k=2,3,···,nThe box girder deformation of the k+1th tensioning phase can be predicted based on the data of the previous k tensioning phases. Then, the residual composite correction value of the Fk+1 tensioning phase is X^1(0)(Fk+1).


### 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 X is in state in at time t. Then, its state at time t+1 has nothing to do with the state before time t:(21)PXn+1=in+1∣X0=i0,X1=i1,···,Xn=inThe actual sequence is rationally divided into several states S1,S2,···,Sn. Then, the state transition probability at step f is(22)Pikf=Nikf∑k=1nNikfwhere Nik(f) is the number of states transferred from Si to Sk by f steps in the system sequence; ∑k=1nNik(f) is the amount of raw data in state Si of the sequence.Define the state transition matrix at stepf as(23)Pf=P11fP12f···P1nfP21fP22f···P2nf⋮⋮⋮Pn1fPn2f···Pnnfwhere ∑k=1nPik(f)=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 χi. For Markov chain-based residual prediction, the residual sequence E(0)(Fk) was divided into n states S. r transition probability vectors (the rows of a matrix) are needed to consider the transition processes in r steps. The probability of residual state at the target moment is the sum of the r vectors. The elements of each vector are denoted as η1,η2,⋯,ηn. If there are a total of n intervals of state S, then the residual correction prediction value of the target moment, that is, the next moment, is(24)MFk+1=λ1χ1+λ2χ2+···+λnχn(25)λi=ηiη1+η2+···+ηnwhere λi is the state weight; χi is the state interval centre.After Markov chain-based prediction, the prediction value of the UIGM-RCC is corrected as(26)X^20Fk+1=X^10Fk+1+MFk+1


## 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 sequenceE(0)(Fk).(16)E0Fk=X0Fk-X^0Fk,k=1,2,···,nStep 2.
FitE(0)(Fk) by sine function E^1(0)(Fk), constructed on the average features of the residual sequence waveform.(17)E^10Fk=Asin⁡2πFk+T/2Twhere A=1/n∑i=1nE(0)(Fk); T is the average value of residual sequence E(0)(Fk) wave time span; its optimal value follows the principle of the minimum quadratic sum e1min of the fitting values residual for sine function; that is, e1min=min⁡e11,e12,···,e1i,···,e1n and e1i=∑k=1n[E1(0)(Fk)-E^1i(0)(Fk)]2.Step 3.
FitE(0)(Fk) by periodic sequence function E^2(0)(Fk) generated by harmonic variation:(18)E^20Fk=a0+∑i=1maicos⁡ωiFk+bisin⁡ωiFkwhere ωi=2iπ/Fn,i=1,2,⋯,m,a0=1/n∑k=1nE(0)(Fk),ai=2/n∑k=1nE0Fkcos⁡(ωiFk),bi=2/n∑k=1nE0Fksin⁡(ωiFk).Step 4.
Introduce the weight coefficientξ (0≤ξ≤1) to combine the weights with formulas (17) and (18) and obtain the residual fitting value E^(0)(Fk).(19)E^0F1=E0F1E^0Fk=1-ξE^1Fk+ξE^2Fk,k=2,3,···,nAlso, the optimal value of ξ and m follows the principle of the minimum quadratic sum e2min of the fitting values residual by SPSFRC; that is, e2min=min⁡e21,e22,···,e2i,···,e2n and e2i=∑k=1n[E(0)(Fk)-E^i(0)(Fk)]2.Step 5.
Calculate the prediction value modified by the SPSFRC.(20)X^10F1=X^0F1X^10Fk=X^0Fk+E^0Fk,k=2,3,···,nThe box girder deformation of the k+1th tensioning phase can be predicted based on the data of the previous k tensioning phases. Then, the residual composite correction value of the Fk+1 tensioning phase is X^1(0)(Fk+1).


## 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 X is in state in at time t. Then, its state at time t+1 has nothing to do with the state before time t:(21)PXn+1=in+1∣X0=i0,X1=i1,···,Xn=inThe actual sequence is rationally divided into several states S1,S2,···,Sn. Then, the state transition probability at step f is(22)Pikf=Nikf∑k=1nNikfwhere Nik(f) is the number of states transferred from Si to Sk by f steps in the system sequence; ∑k=1nNik(f) is the amount of raw data in state Si of the sequence.Define the state transition matrix at stepf as(23)Pf=P11fP12f···P1nfP21fP22f···P2nf⋮⋮⋮Pn1fPn2f···Pnnfwhere ∑k=1nPik(f)=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 χi. For Markov chain-based residual prediction, the residual sequence E(0)(Fk) was divided into n states S. r transition probability vectors (the rows of a matrix) are needed to consider the transition processes in r steps. The probability of residual state at the target moment is the sum of the r vectors. The elements of each vector are denoted as η1,η2,⋯,ηn. If there are a total of n intervals of state S, then the residual correction prediction value of the target moment, that is, the next moment, is(24)MFk+1=λ1χ1+λ2χ2+···+λnχn(25)λi=ηiη1+η2+···+ηnwhere λi is the state weight; χi is the state interval centre.After Markov chain-based prediction, the prediction value of the UIGM-RCC is corrected as(26)X^20Fk+1=X^10Fk+1+MFk+1


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


### 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 (Table1). 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.Table 1
The deformation values of C21 in different tensioning phases.
Tensioning phase Tensioning force/kN Deformation/mm 1 630 25.30 2 1,050 31.33 3 1,155 33.27 4 1,365 37.15 5 1,575 42.78 6 1,785 45.18 7 1,890 47.23(1) The fitting valuesX^(0)(Fk) and preliminary prediction values X^(0)(Fk+1) were obtained by the UIGM, and then the residual sequence E(0)(Fk) of the fitting values was figured out.Table2 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%.Table 2
The fitting values by UIGM.
Tensioning force/kN Measured values/mm Fitting values/mm Residual/mm Relative error/% 630 25.30 25.3000 0.0000 0.0000 1,050 31.33 32.4076 -1.0776 -3.4395 1,155 33.27 34.4138 -1.1438 -3.4379 1,365 37.15 35.6871 1.4629 3.9378 1,575 42.78 37.4548 5.3252 12.4479 1,785 45.18 39.3101 5.8699 12.9923(2) Calculate the optimal composite parameters of the SPSFRC, the fitting values of the residual sequence, and the prediction valuesX^1(0)(Fk+1) after one correction.First, the sine function was used to modifyE(0)(Fk), and the optimal parameter Tbest was determined by the quadratic sum e1 of the fitting values. Let T∈[1,10,000] and let the step length be 1. The optimal result e1min=27.2228mm2 was obtained at T=114s. Figure 2 shows the results of the first 300 iterations. In Figure 2, the triangle represents the corresponding point of the optimal result.Figure 2
Relationship between parameterT and quadratic sum e1 of the fitting values.After that, the residual sequence was modified by the periodic sequence function and sine function generated by harmonic transform. The optimal parametersξbest and mbest were still determined by the quadratic sum e2 of the fitting values. Suppose that ξ∈[0,1] with a step length of 0.01 and m∈[1,10,000] with a step length of 1. In this case, the cumulative residual of the fitting values was minimal (e2min=3.0365mm2) at T=114s, ξ=0.17, and m=10. Figure 3 shows the results of the first 300 iterations. In Figure 3, the triangle represents the corresponding point of the optimal result.Figure 3
Relationship between parametersξ and m and quadratic sum e2 of the fitting values at Tbest=114s.There are 6 pieces of data in the residual sequence of the first 6 tensioning phases. The fitting values modified by the SPSFRC are(27)E^0Fk=E^0F1,E^0F2,E^0F3,E^0F4,E^0F5,E^0F6=0.0000,-2.3696,-2.2550,1.5893,5.5899,5.6544The 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(28)X^10F7=X^0F7+E^0F7=40.7586+1.0441=41.8027The 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 matrixP(f) of the residual sequence, and figure out the Markov residual prediction value M(Fk+1) and the final prediction value X^2(0)(Fk+1).The residual sequence was divided into 3 state intervals based on the mean valueμ and standard deviation σ of the residual sequence: [μ-1.9σ,μ-0.6σ], [μ-0.6σ,μ+0.6σ], and [μ+0.6σ,μ+1.9σ], corresponding to states S1, S2, and S3. 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.(29)P1=23130000001,P2=131313001000,P3=01323000000.Table 3
States of residual sequence.
Tensioning force/kN Residual State 630 0.0000 S 1 1,050 -1.7078 S 1 1,155 -1.1438 S 1 1,365 1.4629 S 2 1,575 5.3252 S 3 1,785 5.8699 S 3The 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 Table4.Table 4
State transition probabilities.
Transition step State Transition probability S 1 S 2 S 3 1 (1,785 kN) S 3 0 0 1 2 (1,575 kN) S 3 0 0 0 3 (1,365 kN) S 2 0 0 0  Total probabilityλ 0 0 1Solved byη, the state weights are λ1=λ2=0 and λ3=1, and the state centres are χ1=-1.8425,χ2=1.7394, and χ3=5.3214. Then, the Markov chain-based residual prediction value of the 7th tensioning phase is(30)MF7=λ1·χ1+λ2·χ2+λ3·χ3=0×-1.8425+0×1.7394+1×5.3214=5.3214Thus, the final deformation of the box girder of the 7th tensioning phase is(31)X^20F7=X^10F7+MF7=41.8027+5.3214=47.1241The 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.Table 5
Comparison of the fitting values by the different prediction models.
Tensioning phase Tensioning force/kN Measured values/mm UIGM UIGM-RCC Fitting values/mm Relative error/% Fitting values/mm Relative errors/% 1 630 25.30 25.3000 0.0000 25.3000 0.0000 2 1,050 31.33 32.4076 -3.4394 30.0380 4.1238 3 1,155 33.27 34.4138 -3.4380 32.1588 3.3400 4 1,365 37.15 35.6871 3.9377 37.2764 -0.3404 5 1,575 42.78 37.4548 12.4478 43.0448 -0.6189 6 1,785 45.18 39.3101 12.9923 44.9645 0.4769In 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. Table6 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).Table 6
Comparison of the predicted values by the different prediction models.
Tensioning phase Tensioning force/kN Predicted values/mm UIGM UIGM-RCC 7 1,890 40.7586 47.1241 8 2,100 41.7640 48.1295 9 2,205 42.2790 48.6445 10 2,310 42.8025 49.1680 11 2,415 43.3345 49.7000 12 2,520 43.8754 50.2409Figure 4
Comparison of the different model results.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 (Table7) to predict the deformation in the 7th tensioning phase. L denotes distance from center line of 57# pier in Table 7.Table 7
The measured values of C14~C23 measuring points.
Measuring points L/m Measured values/mm 630 kN 1,050 kN 1,155 kN 1,365 kN 1,575 kN 1,785 kN 1,890 kN C14 585.00 7.61 9.47 9.82 9.99 10.50 10.38 10.41 C15 615.00 18.98 23.55 24.76 27.17 30.49 32.11 34.15 C16 645.00 7.03 7.42 8.16 8.53 9.06 9.25 9.35 C17 705.00 8.04 12.30 12.64 13.33 14.02 16.45 17.01 C18 735.00 16.48 24.67 25.34 28.77 31.48 33.20 34.25 C19 765.00 4.18 6.01 8.05 9.68 10.03 11.45 11.45 C20 825.00 11.92 14.90 15.06 15.37 16.73 17.88 18.01 C21 855.00 25.30 31.33 33.27 37.15 42.78 45.18 47.23 C22 885.00 11.05 15.81 16.29 17.88 19.45 21.57 21.60 C23 952.50 6.09 7.14 7.83 8.06 8.24 8.26 8.45The optimal parameter values and prediction results are shown in Tables8 and 9, respectively.Table 8
Optimal parameters of the different measuring points.
Measuring points ξ T/s m Measuring points ξ T/s m C14 0.35 204 7 C19 0.33 204 7 C15 0.17 114 10 C20 0.17 114 14 C16 0.33 204 7 C21 0.17 114 10 C17 0.26 107 7 C22 0.17 114 12 C18 0.35 203 7 C23 0.07 203 7Table 9
Prediction values of C14~C23 measuring points.
Measuring points L/m Measured values/mm UIGM UIGM-RCC Predicted values/mm Relative error/% Predicted values/mm Relative error/% C14 585.00 10.41 10.2650 1.3929 10.3759 0.3276 C15 615.00 34.15 29.2886 14.2354 33.2063 2.7634 C16 645.00 9.35 8.9535 4.2406 9.3561 -0.0652 C17 705.00 17.01 14.7196 13.4650 16.9213 0.5215 C18 735.00 34.25 30.4025 11.2336 33.2873 2.8108 C19 765.00 11.45 10.8515 5.2271 11.2477 1.7668 C20 825.00 18.01 16.2766 9.6247 17.8585 0.8412 C21 855.00 47.23 40.7586 13.7019 47.1241 0.2242 C22 885.00 21.60 19.0744 11.6926 21.2667 1.5431 C23 952.50 8.45 8.2588 2.2627 8.4285 0.2544As shown in Table9, 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.Figure 5
Prediction results of measuring points.Table10 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.Table 10
Prediction results of C14~C23 measuring points.
Measuring points Prediction results/mm 2,100 kN 2,205 kN 2,310 kN 2,415 kN 2,520 kN C14 10.4508 10.4885 10.5264 10.5645 10.6027 C15 33.8028 34.1071 34.4155 34.7281 35.0449 C16 9.5084 9.5859 9.6642 9.7434 9.8236 C17 16.5406 16.6357 16.7317 16.8284 16.9261 C18 33.8811 34.1838 34.4904 34.8010 35.1156 C19 11.8436 12.1578 12.4834 12.8208 13.1706 C20 17.9959 18.0653 18.1350 18.2051 18.2756 C21 48.1295 48.6445 49.1680 49.7000 50.2409 C22 21.6033 21.7746 21.9479 22.1232 22.3005 C23 8.5352 8.5892 8.6437 8.6986 8.7540In 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.Table 11
Fitting values of building settlement.
Time/d Measured values/mm UIGM Literature [11] model Literature [12] model UIGM-RCC Fitting values/mm Relative errors/% Fitting values/mm Relative error/% Fitting values/mm Relative error/% Fitting values/mm Relative error/% 0 1.11 1.1100 0.0000 1.3000 -17.1171 1.5400 -38.7387 1.1100 0.0000 15 1.45 1.7211 -18.6966 1.4900 -2.7586 1.7300 -19.3103 1.3030 10.1379 31 1.93 1.9025 1.4249 1.7400 9.8446 1.9600 -1.5544 2.0797 -7.7565 43 2.08 2.0823 -0.1106 1.9500 6.2500 2.1400 -2.8846 2.0535 1.2740 57 2.52 2.2209 11.8690 2.2200 11.9048 2.3900 5.1587 2.4648 2.1905 71 2.92 2.4326 16.6918 2.5400 13.0137 2.6600 8.9041 2.8516 2.3425 84 3.20 2.7052 15.4625 2.8700 10.3125 2.9300 8.4375 3.2343 -1.0719 127 3.96 3.2512 17.8990 4.2900 -8.3333 4.0800 -3.0303 4.0055 -1.1490Table 12
Predicted values of building settlements.
Time/d Measured values/mm UIGM Literature [11] model Literature [12] model UIGM-RCC Predicted values/mm Relative error/% Predicted values/mm Relative error/% Predicted values/mm Relative error/% Predicted values/mm Relative error/% 140 4.38 3.8847 11.3082 4.8500 -10.7306 4.5000 -2.7397 4.4436 -1.4521 154 4.68 4.0684 13.0684 5.5400 -18.3761 5.0100 -7.0513 4.5335 3.1303Note that the absolute deviation of calculated values from the measured values is magnified 3 times. In Figure6, it can be seen that the results of the UIGM-RCC agree well with the measured values.Figure 6
Comparison of predicted results by the different models.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 Table13. 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.Table 13
Prediction results of the 160th~210th days by UIGM-RCC.
Time/d Predicted values/mm 160 4.8577 174 5.0603 185 5.2285 198 5.4383 210 5.6431


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


## 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 (Table1). 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.Table 1
The deformation values of C21 in different tensioning phases.
Tensioning phase Tensioning force/kN Deformation/mm 1 630 25.30 2 1,050 31.33 3 1,155 33.27 4 1,365 37.15 5 1,575 42.78 6 1,785 45.18 7 1,890 47.23(1) The fitting valuesX^(0)(Fk) and preliminary prediction values X^(0)(Fk+1) were obtained by the UIGM, and then the residual sequence E(0)(Fk) of the fitting values was figured out.Table2 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%.Table 2
The fitting values by UIGM.
Tensioning force/kN Measured values/mm Fitting values/mm Residual/mm Relative error/% 630 25.30 25.3000 0.0000 0.0000 1,050 31.33 32.4076 -1.0776 -3.4395 1,155 33.27 34.4138 -1.1438 -3.4379 1,365 37.15 35.6871 1.4629 3.9378 1,575 42.78 37.4548 5.3252 12.4479 1,785 45.18 39.3101 5.8699 12.9923(2) Calculate the optimal composite parameters of the SPSFRC, the fitting values of the residual sequence, and the prediction valuesX^1(0)(Fk+1) after one correction.First, the sine function was used to modifyE(0)(Fk), and the optimal parameter Tbest was determined by the quadratic sum e1 of the fitting values. Let T∈[1,10,000] and let the step length be 1. The optimal result e1min=27.2228mm2 was obtained at T=114s. Figure 2 shows the results of the first 300 iterations. In Figure 2, the triangle represents the corresponding point of the optimal result.Figure 2
Relationship between parameterT and quadratic sum e1 of the fitting values.After that, the residual sequence was modified by the periodic sequence function and sine function generated by harmonic transform. The optimal parametersξbest and mbest were still determined by the quadratic sum e2 of the fitting values. Suppose that ξ∈[0,1] with a step length of 0.01 and m∈[1,10,000] with a step length of 1. In this case, the cumulative residual of the fitting values was minimal (e2min=3.0365mm2) at T=114s, ξ=0.17, and m=10. Figure 3 shows the results of the first 300 iterations. In Figure 3, the triangle represents the corresponding point of the optimal result.Figure 3
Relationship between parametersξ and m and quadratic sum e2 of the fitting values at Tbest=114s.There are 6 pieces of data in the residual sequence of the first 6 tensioning phases. The fitting values modified by the SPSFRC are(27)E^0Fk=E^0F1,E^0F2,E^0F3,E^0F4,E^0F5,E^0F6=0.0000,-2.3696,-2.2550,1.5893,5.5899,5.6544The 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(28)X^10F7=X^0F7+E^0F7=40.7586+1.0441=41.8027The 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 matrixP(f) of the residual sequence, and figure out the Markov residual prediction value M(Fk+1) and the final prediction value X^2(0)(Fk+1).The residual sequence was divided into 3 state intervals based on the mean valueμ and standard deviation σ of the residual sequence: [μ-1.9σ,μ-0.6σ], [μ-0.6σ,μ+0.6σ], and [μ+0.6σ,μ+1.9σ], corresponding to states S1, S2, and S3. 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.(29)P1=23130000001,P2=131313001000,P3=01323000000.Table 3
States of residual sequence.
Tensioning force/kN Residual State 630 0.0000 S 1 1,050 -1.7078 S 1 1,155 -1.1438 S 1 1,365 1.4629 S 2 1,575 5.3252 S 3 1,785 5.8699 S 3The 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 Table4.Table 4
State transition probabilities.
Transition step State Transition probability S 1 S 2 S 3 1 (1,785 kN) S 3 0 0 1 2 (1,575 kN) S 3 0 0 0 3 (1,365 kN) S 2 0 0 0  Total probabilityλ 0 0 1Solved byη, the state weights are λ1=λ2=0 and λ3=1, and the state centres are χ1=-1.8425,χ2=1.7394, and χ3=5.3214. Then, the Markov chain-based residual prediction value of the 7th tensioning phase is(30)MF7=λ1·χ1+λ2·χ2+λ3·χ3=0×-1.8425+0×1.7394+1×5.3214=5.3214Thus, the final deformation of the box girder of the 7th tensioning phase is(31)X^20F7=X^10F7+MF7=41.8027+5.3214=47.1241The 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.Table 5
Comparison of the fitting values by the different prediction models.
Tensioning phase Tensioning force/kN Measured values/mm UIGM UIGM-RCC Fitting values/mm Relative error/% Fitting values/mm Relative errors/% 1 630 25.30 25.3000 0.0000 25.3000 0.0000 2 1,050 31.33 32.4076 -3.4394 30.0380 4.1238 3 1,155 33.27 34.4138 -3.4380 32.1588 3.3400 4 1,365 37.15 35.6871 3.9377 37.2764 -0.3404 5 1,575 42.78 37.4548 12.4478 43.0448 -0.6189 6 1,785 45.18 39.3101 12.9923 44.9645 0.4769In 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. Table6 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).Table 6
Comparison of the predicted values by the different prediction models.
Tensioning phase Tensioning force/kN Predicted values/mm UIGM UIGM-RCC 7 1,890 40.7586 47.1241 8 2,100 41.7640 48.1295 9 2,205 42.2790 48.6445 10 2,310 42.8025 49.1680 11 2,415 43.3345 49.7000 12 2,520 43.8754 50.2409Figure 4
Comparison of the different model results.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 (Table7) to predict the deformation in the 7th tensioning phase. L denotes distance from center line of 57# pier in Table 7.Table 7
The measured values of C14~C23 measuring points.
Measuring points L/m Measured values/mm 630 kN 1,050 kN 1,155 kN 1,365 kN 1,575 kN 1,785 kN 1,890 kN C14 585.00 7.61 9.47 9.82 9.99 10.50 10.38 10.41 C15 615.00 18.98 23.55 24.76 27.17 30.49 32.11 34.15 C16 645.00 7.03 7.42 8.16 8.53 9.06 9.25 9.35 C17 705.00 8.04 12.30 12.64 13.33 14.02 16.45 17.01 C18 735.00 16.48 24.67 25.34 28.77 31.48 33.20 34.25 C19 765.00 4.18 6.01 8.05 9.68 10.03 11.45 11.45 C20 825.00 11.92 14.90 15.06 15.37 16.73 17.88 18.01 C21 855.00 25.30 31.33 33.27 37.15 42.78 45.18 47.23 C22 885.00 11.05 15.81 16.29 17.88 19.45 21.57 21.60 C23 952.50 6.09 7.14 7.83 8.06 8.24 8.26 8.45The optimal parameter values and prediction results are shown in Tables8 and 9, respectively.Table 8
Optimal parameters of the different measuring points.
Measuring points ξ T/s m Measuring points ξ T/s m C14 0.35 204 7 C19 0.33 204 7 C15 0.17 114 10 C20 0.17 114 14 C16 0.33 204 7 C21 0.17 114 10 C17 0.26 107 7 C22 0.17 114 12 C18 0.35 203 7 C23 0.07 203 7Table 9
Prediction values of C14~C23 measuring points.
Measuring points L/m Measured values/mm UIGM UIGM-RCC Predicted values/mm Relative error/% Predicted values/mm Relative error/% C14 585.00 10.41 10.2650 1.3929 10.3759 0.3276 C15 615.00 34.15 29.2886 14.2354 33.2063 2.7634 C16 645.00 9.35 8.9535 4.2406 9.3561 -0.0652 C17 705.00 17.01 14.7196 13.4650 16.9213 0.5215 C18 735.00 34.25 30.4025 11.2336 33.2873 2.8108 C19 765.00 11.45 10.8515 5.2271 11.2477 1.7668 C20 825.00 18.01 16.2766 9.6247 17.8585 0.8412 C21 855.00 47.23 40.7586 13.7019 47.1241 0.2242 C22 885.00 21.60 19.0744 11.6926 21.2667 1.5431 C23 952.50 8.45 8.2588 2.2627 8.4285 0.2544As shown in Table9, 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.Figure 5
Prediction results of measuring points.Table10 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.Table 10
Prediction results of C14~C23 measuring points.
Measuring points Prediction results/mm 2,100 kN 2,205 kN 2,310 kN 2,415 kN 2,520 kN C14 10.4508 10.4885 10.5264 10.5645 10.6027 C15 33.8028 34.1071 34.4155 34.7281 35.0449 C16 9.5084 9.5859 9.6642 9.7434 9.8236 C17 16.5406 16.6357 16.7317 16.8284 16.9261 C18 33.8811 34.1838 34.4904 34.8010 35.1156 C19 11.8436 12.1578 12.4834 12.8208 13.1706 C20 17.9959 18.0653 18.1350 18.2051 18.2756 C21 48.1295 48.6445 49.1680 49.7000 50.2409 C22 21.6033 21.7746 21.9479 22.1232 22.3005 C23 8.5352 8.5892 8.6437 8.6986 8.7540In 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.Table 11
Fitting values of building settlement.
Time/d Measured values/mm UIGM Literature [11] model Literature [12] model UIGM-RCC Fitting values/mm Relative errors/% Fitting values/mm Relative error/% Fitting values/mm Relative error/% Fitting values/mm Relative error/% 0 1.11 1.1100 0.0000 1.3000 -17.1171 1.5400 -38.7387 1.1100 0.0000 15 1.45 1.7211 -18.6966 1.4900 -2.7586 1.7300 -19.3103 1.3030 10.1379 31 1.93 1.9025 1.4249 1.7400 9.8446 1.9600 -1.5544 2.0797 -7.7565 43 2.08 2.0823 -0.1106 1.9500 6.2500 2.1400 -2.8846 2.0535 1.2740 57 2.52 2.2209 11.8690 2.2200 11.9048 2.3900 5.1587 2.4648 2.1905 71 2.92 2.4326 16.6918 2.5400 13.0137 2.6600 8.9041 2.8516 2.3425 84 3.20 2.7052 15.4625 2.8700 10.3125 2.9300 8.4375 3.2343 -1.0719 127 3.96 3.2512 17.8990 4.2900 -8.3333 4.0800 -3.0303 4.0055 -1.1490Table 12
Predicted values of building settlements.
Time/d Measured values/mm UIGM Literature [11] model Literature [12] model UIGM-RCC Predicted values/mm Relative error/% Predicted values/mm Relative error/% Predicted values/mm Relative error/% Predicted values/mm Relative error/% 140 4.38 3.8847 11.3082 4.8500 -10.7306 4.5000 -2.7397 4.4436 -1.4521 154 4.68 4.0684 13.0684 5.5400 -18.3761 5.0100 -7.0513 4.5335 3.1303Note that the absolute deviation of calculated values from the measured values is magnified 3 times. In Figure6, it can be seen that the results of the UIGM-RCC agree well with the measured values.Figure 6
Comparison of predicted results by the different models.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 Table13. 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.Table 13
Prediction results of the 160th~210th days by UIGM-RCC.
Time/d Predicted values/mm 160 4.8577 174 5.0603 185 5.2285 198 5.4383 210 5.6431


## 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.



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*Source: 6373086-2018-06-21.xml*