Construction Monitoring of Cable-stayed Bridges Based on Gray Prediction Model

The construction process of cable-stayed bridges is complex and has many influencing parameters. Construction monitoring plays an important role in the construction process to ensure structural safety and meet design requirements. Gray system theory is applied to analyze and predict structure deformation and cable tension for construction monitoring, which regards the cable-stayed bridge under construction as an interferential gray system with a physical prototype and analyzes the random process as a gray process. The gray prediction model is established by using and evaluating girder and cable tension as two control inputs of the system in the construction process of cable-stayed bridges. The girder and cable tension of subsequent construction stages are predicted, adjusted, and evaluated by using feedback information obtained from measuring and rectifying the gray prediction model to control and adjust the bridge configuration and cable tension effectively. Results show that the gray prediction model has good prediction precision, which can control the structural configuration in the ideal state and meet the design requirements.


INTRODUCTION
A cable-stayed bridge is composed of an inclined cables, tower, and girder and has the characteristics of large crossover ability, lightweight structure, large clearance height, and good earthquake resistance.Therefore, the cablestayed bridge has been extensively adopted as a beautiful style of modern bridge structures.The girder configuration and internal structural force in the completed stage have a close relationship with the construction method because the cable-stayed bridge is a super high-order static structure.The construction process is not only complex (e.g., tensioning the cables, moving the hanging basket, and pouring the concrete) but also has many influencing parameters, such as the material elastic modulus, weight of the structure, and construction loads.Generally, these parameters are assumed ideal based on the design code at the design stage.In the construction process, errors between the actual and theoretical values are difficult to avoid because of various factors.If the errors are not adjusted and controlled in time, then they accumulate and change the theoretical configuration of the girder and endanger structural safety.The closure error was approximately 17 cm when the Pasco-Kennewick cable-stayed bridge in Washington was completed because the construction errors were neglected.When the bridge was completed, it did not meet the ideal state and design requirements [1].Thus, the parameter errors should be adjusted and controlled to ensure that the bridge can meet the design requirements and ideal state.More attention should be focused on construction monitoring of long-span cable-stayed bridges.
*Address correspondence to this author at the School of Highway, Bridge Engineering Chang'an University, Xi'an, Shanxi, 710064, P.R. China; Tel: +86 29 82338970; E-mail: iBridge2014@163.comSome methods, such as Kalman's theory, gray prediction theory model, and least squares method, have been used to identify and predict parameter errors in the construction monitoring of bridge structures.Li Yan-pei and Zhao Yinru successfully applied Kalman's theory based on random process control to the construction monitoring of cablestayed bridges [2][3].The gray prediction model considered the characteristics of construction that can control the bridge configuration and cable tension of cable-stayed bridges [1,4,5].Cable tension is a key factor in determining the girder configuration and internal force distribution.When the girder configuration deviates from the ideal state, it can be adjusted by increasing or decreasing the cable force.Shang Xin adopted gray theory to adjust the evaluation of a cable-stayed bridge based on cable safety [6].Chen Changsong integrated the minimum square method, gray prediction theory model, and influence matrix adjustment method to control the concrete cable-stayed bridge to achieve the optimum mechanical state [7].When more complicated parameters are considered in the construction of bridges, the backpropagation neural network method is capable of computing an extensive range of parameters.Zhou Jiagang and Li Qiao attempted to use the backpropagation neural network method on cable-stayed bridge construction control and obtained a good agreement with the design requirements [8,9].
In the construction process, the mechanical behaviors are changed by the parameter errors.These errors show strong uncertainty, that is, gray characteristics.In this study, we applied the gray prediction model to analyze the parameter errors and predict deck elevation and initial cable tension in the cable-stayed bridge construction process.

PRINCIPLES OF THE GRAY PREDICTION MODEL
Gray system theory, an analytical system based on gray relation, processes and manages the original series of data through gray processing and gray generation by using the existing information and the original data [10].Gray differential equations are established as the main body of the gray prediction model GM (1,1), which can predict future changes in the system.This predictive control method allows for the application of gray theory in bridge construction control technology.The method regards the gray dynamic model GM(1,1) as a prediction model.Then, the method is optimized and rectified accordingly.All of the factors that affect the status of bridge construction are eliminated by gray theory and reflected in the target vector concentration, thereby allowing the completion of the prediction process by using the target vector.
The predictive control method predicts the structure state before and after each construction stage, with full consideration of the various factors that affect the construction process and the construction goal for each stage to ensure that construction proceeds as intended.An error inevitably occurs between the predicted and actual states.Thus, specific errors that affected the construction goal are considered in the subsequent stage.The aforementioned rules are followed until the construction is finished and the structure state fulfills the design.
The construction of long-span bridges is a multivariable, high-level, time-varying, and complex process.Although building an accurate model through this process is difficult, predictive control is an effective program to solve the difficulties in building the model and controlling the system complexity.In this study, gray system theory is adopted to predict and analyze the construction process of a cablestayed bridge.
The cable-stayed bridge and its various loads are regarded as a system.Although considerable information is available about the system, further information remains unknown.Therefore, this system is a gray technology system that is affected by noise and has a physical prototype.
In gray system theory, a dynamic model with a group of differential equations is developed, which is called gray differential model GM (1,1).The gray differential model GM(1,1) is a prediction function model with a differential equation expressed as follows: where ( ) x is a cumulative value of the raw data series ( ) 0 x and calculated as follows: Thus, the prediction model is generated by using the data series ( ) x .The accuracy of the model can be tested by residual size, posteriori, and rolling inspection.If the accuracy of the model cannot meet the requirements, then the model is rectified by the residuals.Then, feedback correction of the model is implemented through model metabolism.
Only a few data (as few as four) are needed to distinguish and build a GM model.The equation is a gray differential model, called GM(1,1), as it includes only one variable ( ) x , where a and b are the coefficients.In gray system theory terms, a denotes a developing coefficient and b denotes the gray input.( ) ( ) x k is a gray derivative that maximizes the information density of a given series to be modeled.

CONSTRUCTION PROCESS OF THE CABLE-STAYED BRIDGE
Upon completion of the piers, pylons, and other associated works in the construction process, steel girder erection and cable installation become typical repetitive cycles, as shown in Fig. (1).The lifting crane, which is mounted on the last erected segment, hooks up the segment from the transportation barge.This segment is lifted to the design level and is secured to the temporary support frame as a temporary fixing.This segment is welded to the preceding segment.Then, the inclined cable is installed on the segment.The cable is first anchored to the tower and the deck.After anchorage work is completed, the cable is stressed by a heavy jack until the designed length is achieved.One back-span cable and one mid-span cable should be stressed simultaneously to balance the bending force acting on the tower.The typical erection cycle is repeated until the last closure segment, which is operated with a distinct method.

CONSTRUCTION MONITORING OF CABLE-STAYED BRIDGES
For cable-stayed bridges, when various factors are effectively controlled during the construction process, the alignment and internal force of the girder can be adjusted only by initializing the heights of girders and the tension forces of cables.Therefore, the initial heights of girders and the tension forces of cables are two control inputs of the system.Given that a change in initial girder height does not affect the internal structural force, the alignment can be adjusted using the initial height without changing the internal structural force.As a result, the completed stage alignment need not be changed for the design alignment target.Internal structural forces are caused by the difference between various parameters of construction and the design values.Only the tension forces of cables adjusted to allow the internal force distribution to meet the design expectations.Therefore, the ultimate goal of the control method is to adjust the girder alignment in the completed stage and ensure that the distribution of internal structural forces meets the design requirements.

(1) Adjustment of Cable Tension
The increment of vertical displacement of the cantilever end girder and several adjacent segments during construction should be consistent with the design value by confirming the initial tension forces of cables in the succeeding phase, in which ( 1) r T k + segment is tensioned and ( 1) k + segment is welded.The cable to be tensioned at the cantilever end is denoted as the first i number, and the corresponding cantilever end node is the i node.The girder segment constructed before the i number cable is tensioned and denoted as the i number segment.The initial tension force of the cable is ( 1) r T k + when the ( 1) k + segment is welded, and it is expressed as follows: ( 1) ( 1) ( 1), where ( 1) k β + is the tension adjustment factor of the ( 1) k + number cable.

① Prediction of the vertical displacement increment at the cantilever end before and after the cable is tensioned
The adjustment value of the ( 1) k + number cable tension force is determined based on the definition series 1 x .The design value is ( 1) d T k + , and the vertical displacement increment design value is 1 ( 1) d k ∆ + .Thus, the corresponding model output value 1 ( 1) m k ∆ + is expressed as follows: ( 1). ( 1)

② Prediction of the Vertical Displacement Increment at the Cantilever end Before and After the Girder Segment is Lifted
Based on the definition series 1 y , the vertical displacement increment of the design value for the cantilever + is expressed as follows: .

Lifting the Girder Segment
If girder elevation is determined during construction, then deformation of the crane must be predicted accordingly.
1 ( ) z i is defined as the ratio between the actual deformation value of the hanging basket and the theoretical value when the i number girder segment is lifted.The prediction model GM(1,1) is set up for 1 ( ) z i , and the crane deformation value is 1 ( 1) + .As a result, the model output value is expressed as follows: .
④ Confirmation of Segment Elevation at ( 1)  k + Stage ( 1) l H k + Girder elevation at the ( 1) k + number stage is calculated as follows: ( 1) ( 1), where ( 1) d H k + is the design elevation value of the cantilever end after the ( 1) k + number cable is tensioned.Consequently, once the cable tension is determined at the ( 1) k + stage, the predicted values of 1 ( 1) and 1 ( 1) m k ′′ ∆ + can be obtained from past and present data.Then, the unique elevation value can be identified.

APPLICATION OF GM(1,1) MODEL TO THE CONSTRUCTION MONITORING OF CABLE-STAYED BRIDGES (1) Project Overview of Sutong Cable-Stayed Bridge
The Sutong Bridge crosses the Yangtze River approximately 100 km upstream from Shanghai, China and connects the cities of Suzhou and Nantong located on the southern and northern banks, respectively.The bridge is a seven-span double-pylon and double-cable plane steel box girder cable-stayed bridge, with a span arrangement of 100 + 100 + 300 + 1,088 + 300 + 100 + 100 = 2,088 m, as shown in Fig. (2).The Sutong Bridge is the second longest cablestayed bridge in the world, with a record-breaking construction in the history of bridge building.

(2) Engineering Application
Gray theory is applied to the construction control of the cable-stayed bridge.The construction state is as follows: No. 15 cable is tensioned, and the initial height of the No. 16 segment is predicted.At the same time, the initial tension force of the No. 16 cable is determined.

1) Confirmation of 1 ( ) x i , the Vertical Displacement Increment Prediction for the Cantilever End When the No. 10 Cable is Tensioned
The measured series 1 ( ) x i identifies four cables (Nos. 6,7, 8, and 9) that have been tensioned, and the No. 10 cable, which will be tensioned, as a group.The model GM(1,1) is established based on actual and theoretical data.The prediction results are shown in Table 1.

2) Confirmation of 1 ( ) y i , the Vertical Displacement Increment Prediction for the Cantilever end when the No. 10 Girder Segment is Erected
The measured series 1 ( ) y i identifies the adjacent four girder segments, which have been constructed.The model GM(1,1) is established based on actual and theoretical data.The prediction results are shown in Table 2.

3) Confirmation of 1 ( ) z i , Prediction of Crane Deformation
The theoretical crane deformation is 80 mm.From the measured data, the prediction results of crane deformation are as follows: 1.36 for side span 1 (5) z and 1.17 for main span 1 (5) z .

4) Prediction of the Initial Cable Tension for the No. 10 Cable
Based on the determined value of 1 (5) x and the predicted value of 1 (5) y , the adjustment factor 1 (10) β of the tension force of the No. 10 cable and the corresponding cable force are obtained as follows: No. 10 cable of the side span: 1 ( Prior to building the No. 10 girder segment, the initial evaluation of the side span and main span are first analyzed.Subsequently, the predicted initial height is compared with the theoretical height. First, the result (10) 5,992 kN r T = obtained from the previously presented equation is substituted into Formula (8).Thus, the predicted value of the vertical displacement increment of the cantilever end 1 (10) m ∆ is obtained as follows: Second, when the No. 10 segment is set up, the predicted value of the vertical displacement 1 (10) m ′ ∆ can be obtained by using Formula (9), as follows: (5) (10) 0.783 390 305 mm.
Finally, when the No. 10 segment is set up, the predicted value of the hanging basket deformation may be obtained by using Formula (10), as follows: (5) (10) 1.36 80 109 mm.
The design evaluation value of the girder bottom is (10)  As the No. 10 segment of the main span is erected, the predicted value of the initial evaluation 1 (10) H is 77.727 m, which was calculated using the aforementioned method.The theoretical value of the initial evaluation of this segment is 77.698 m.Consequently, the actual initial height should be higher by 2.9 cm.
When gray system theory is used in the construction control of the cable-stayed bridge, positive results are obtained during construction.Generally, the deviation error of the cable force during the completed stage is not greater than 10% and the closure error is only 5 mm.
The gray prediction model GM(1,1) is applied to construction monitoring of the Sutong cable-stayed bridge.This model can control the parameters and predict the errors in the subsequent stage effectively.When the structure is finished, the cable force distribution is smooth and does not involve non-mutation.Furthermore, the tensions in the cables are in an ideal state.The actual configuration of the girder is in good agreement with the theoretical alignment, as shown in Fig. (4).The upstream and downstream cable forces were measured in the field after the Sutong cablestayed bridge was completed [11].A comparison of the calculated and measured results shows that a slight difference exists between them, as shown in Fig. (3).The deviation error of the cable force during the completed stage is not greater than 10%.Furthermore, cable tension stress is in the range of Chinese design standards.The deformation of the girder and the tower is sensitive to the cable force.During construction, the cable forces are usually adjusted to change the internal force as well as the displacement of the girder and the tower.Furthermore, the cable force is a key factor in determining the actual geometric profile of the completed stage.

Table 1 .
Measured and predicted values of 1 ( ) x i .

Table 2 .
Measured and predicted values of 1 ( ) y i .
The theoretical value of the No. 10 girder segment is 73.402 m, whereas the actual value should be lower by 2.6 cm.