Reliability Analysis of Numerical Simulation of Sea?Crossing Bridge Deformation Under Wave Forces

LUAN Yuanzhong，YU Jian*，HU Junwei，DONG Yue，GUI Weizhen，JI Zhaolei

1.College of Geodesy and Geomatics，Shandong University of Science and Technology，Qingdao 266590，P.R.China；2.Linyi Huibaoling Iron Ore Limited Company，Linyi 277700，P.R.China

Abstract: To study on the numerical simulation calculation reliability of sea-crossing bridge under complex wave forces，the paper applied GPS deformation monitoring and numerical simulation calculation by researching Qingdao Jiaozhou Bay Sea-Crossing Bridge. The db3 wavelet three-layer decomposition was used on the horizontal movement of the sea-crossing bridge and the wind speed of the waves to analyze their correlation. The complex wave forces value of Qingdao Jiaozhou Bay Sea-Crossing Bridge was loaded on FLAC3D software successfully to make numerical simulation calculation of bridge deformation. Since the accuracy of the GPS deformation monitoring reaches millimeter level，it was used to monitor the exact value of the bridge deformation to judge the reliability of numerical simulation. The relative errors of displacement in X，Y and Z directions were between 33% and 41% through comparison. It could be seen that the numerical simulation error was relatively large，which was mainly due to various environmental factors and the deviation of applied wave forces. However，numerical simulation generally reflects the deformation law of the sea-crossing bridge under complex wave forces，providing an effectively technical support for the safe operation assessment of the sea-crossing bridge.

Key words：sea-crossing bridge；GPS monitoring；three-dimensional deformation；numerical simulation of wave forces；relative error

0 Introduction

In recent years，China has successively built several sea-crossing bridge traffic facilities，such as the Qingdao Jiaozhou Bay Sea-Crossing Bridge，Hangzhou Bay Sea-Crossing Bridge， Shanghai Donghai Bridge，etc. In the complex ocean motion environment［1-3］，sea-crossing bridge needs to bear the effects of tidal phenomena and seasonal ocean di?sasters，such as hurricanes，and other phenome?na［4-6］as well as the effects of vehicles. Therefore，it is necessary to comprehensively study the effects of the wave forces on the bridge pier and effectively simulate the deformation of the bridge，which en?sures the safe operation and deformation warning of the bridge［7-10］.

The traditional monitoring method of bridge vertical displacement is mainly second-order or third-order leveling. Three-dimensional deformation of bridge is mainly monitored by GPS，geo-robots，three-dimensional laser scanner and DinSAR （or PS-DinSAR） measurement technology［11］. These monitoring tools are applied in the deformation mon?itoring of the sea-crossing bridge，and the measur?ing accuracy meets the requirements of bridge moni?toring. Over the years，a lot of in-depth researches on wave loads have been conducted by scholars at home and abroad and there are many breakthroughs in the researches. McIver studied the wave forces on single pile and pile group under the action of irregu?lar waves［12］. Mostafa et al. used the dynamic p?y curve to simulate the interaction of pile-soil，and cal?culated the dynamic response of pile group under wave load through finite element method［13］. Wilson et al.calculated the displacement of the pile body un?der the action of wave forces by using the dynamic stiffness matrix of the lateral vibration of the contin?uous system of the single pile offshore platform［14］.It can be seen that relative researches have been car?ried out about the deformation monitoring of the seacrossing bridge and simulating the pier deformation under the effect of wave loads. However，some as?pects have been barely discussed on evaluating the accuracy of FLAC3Dnumerical simulation of bridge deformation and analyzing the main factors affecting the numerical simulation［15］，based on the observa?tion of GPS deformation.

Therefore，based on the research background of Qingdao Jiaozhou Bay Sea-Crossing Bridge，this paper studied the GPS deformation monitoring and numerical simulation deformation calculation and discussed the real-time relationship between the wave forces and the dynamic deformation of bridge.The quantitative expressions of the wave forces of the pier were determined and loaded on the bridge dynamically in real-time. The difference and relative error between the simulated data of bridge deforma?tion and the measured data of GPS were calculated，so as to evaluate the reliability of the numerical sim?ulation of bridge deformation.

1 Bridge Deformation Monitoring

The action of sea breeze accompanied by waves on bridges is very complex，and its forces are divided into three directions：Vertical，horizontal and torsional directions［16］. A bridge deformation monitoring network has been established at a sec?tion of the Qingdao Sea-crossing Bridge，as shown in Fig.1. Q1，Q2，Q3，Q4，Q5and Q6are GPS moni?toring points arranged for the bridge deck. The length of the bridge section studied is 610 m，and the width is 190 m. Q2and Q5are located in the per?pendicular bisector of the connecting line between the two piers，40 cm away from the edge. Q1，Q2and Q3stand 300 m apart and their connecting line is parallel to the passing direction of the bridge. Q4，Q5and Q6are the same as above，and all points are dis?tributed equidistantly on both sides of the bridge deck. Continuously Operating Reference Station was used as the base station to collect the three-di?mensional deformation data of the bridge to avoid the conversion error between coordinate systems.WGS-84 coordinate system was used to calculate the three-dimensional deformation value of the bridge. Fig.1 shows two groups of piers of the bridge：D1and D2. Anemometers were installed at the position corresponding to the bridge beam side of Q1，Q2，and Q3to collect the wind speed value of the monitoring part in real time.

The paper collected nearly 10 000 monitoring data of point Q2in 54 days from May 1 to 23 June，2015. After data processing analysis，the horizontal displacement and the wind speed of the bridge were obtained. The curve graph according to the data is shown in Fig.2.

Fig.1 Bridge monitoring point diagram layout

Fig.2 Curve of wind speed and bridge horizontal displace?ment in X direction

2 Wavelet Analysis of Correlation Between Wind Speed and Hori?zontal Displacement Values

The horizontal displacement in X direction（perpendicular to the bridge deck direction）and the wind speed data were measured at point Q2for 54 d.It can be seen from Fig.2 that there is no obvious correlation between the wind speed and the displace?ment of the bridge，and the trend of data change is not regular. Wavelet decomposition is performed on the measured wind speed and the horizontal displace?ment values of the bridge.

Db3 wavelet，coif2 wavelet and Dmey wavelet were applied to deal with the noise processing of the measured deformation data. After comparing the root mean square error（RMSE）and signal to noise ratio（SNR），Dmey wavelet has the least RMSE and the largest SNR. The mean square error of each layer and SNR were compared through db3 wavelet decomposition from layer 1 to layer 5，which indi?cated that the db3 wavelet three-layer decomposi?tion had the best denoising effect.

RMSE of the variance between the original da?ta and the denoised data is expressed as

where n is the size of the signal，f (n) the original signal，andthe denoised signal. The smaller the RMSE is，the higher the similarity between the denoised signal and the original signal is，and the better the effect is.

SNR is a commonly used index to measure the noise in data，so it is often used to measure the de?noising effect. Its unit is decibel，and it is expressed as

In the paper，the RMSE of wind speed is 0.858 9. The RMSE of horizontal displacement is 1.624 7. The SNR of wind speed is 10.268 9. The SNR of horizontal displacement is 9.334 1.

Db3 wavelet was chosen for multi-scale decom?position. There are two purposes of wavelet analy?sis，one is to denoise the observation values，and the other is to explore the correlation between the horizontal displacement values of bridge and the wind speed.The db3 wavelet three-layer decomposi?tion of wind speed and the details of db3 wavelet de?composition are shown in Fig.3 and Fig.4，respec?tively. The db3 wavelet three-layer decomposition of horizontal displacement values and the details of db3 wavelet decomposition are illustrated in Fig.5 and Fig.6，respectively.

Fig.3 Db3 wavelet three-layer decomposition of wind speed

Fig.4 Details of db3 wavelet decomposition of wind speed

Fig.5 Db3 wavelet three-layer decomposition of hori?zontal displacement

Fig.6 Details of db3 wavelet decomposition of hori?zontal displacement

According to the correlation coefficient formu?la，Matlab software is utilized to calculate the corre?lation between the wind speed data after wavelet de?noising and the horizontal displacement of bridge af?ter wavelet denoising，shown as

where vXis the wind speed after denoising，uYthe horizontal displacement in X direction after denois?ing，Cov(vX，uY) the covariance of vXand uY，Var[vX] the variance of vX，and Var[uY] the vari?ance of uY.

The correlation coefficient of wind speed data after wavelet denoising and bridge deformation data after denoising is -0.31. The significance of magni?tude of the correlation coefficient is currently incon?sistent in the statistical field，but it is generally con?sidered as follows. The correlation coefficients be?tween 0.00 — ± 0.30，between ± 0.30 — ± 0.50，between ± 0.50 — ± 0.80 and between ± 0.80 —± 1.00 belong to micro correlation，real correla?tion，significant correlation and high correlation，re?spectively. Accordingly，it can be seen that the wind speed and the horizontal displacement in X di?rection of bridge are not closely related，and the ef?fect of wind speed does not serve as the main factor for the bridge deformation in a certain range.

By comparing and analyzing Figs.3 and 4 with Figs.5 and 6，it can be seen that by changing wind direction，horizontal displacement varies in X direc?tion，the magnitude of the horizontal displacement changes with the wind speed，and the former lags behind the latter.

Therefore，it is quite essential to monitor the wind speed and its change，which can reflect the de?formation of the bridge. Although the specific rela?tionship between wind speed and horizontal displace?ment in X direction is not identified，results indicate that only keeping the wind speed in a certain range can guarantee the safety of bridge.

3 Calculation of Wave Forces

The wave forces is composed of drag force k2u|u"|and wave acceleration inertial force k1u"，cal?culated by the Morison equation

Pier’s wave force is calculated as an isolated building because of small size，and the ratio of the width of the wavefront to the wavelength is D / L ＜0.2. The average high tide level is 3.41 m and the wave height is 1.49 m，corresponding to effective wave period T=60 s and wavelength 52.8 m.

The wave force of each pile in a group is differ?ent from that of a single pile，which is mainly caused by the phase difference and the flow field in?terference between the piles in the group［17］. In this paper，only the pile group coefficient caused by phase difference is considered，which is defined as

According to the relevant regulations of the sea?port hydrological code，when calculating the total wave force on each pile at the same time，it should be multiplied by the group pile coefficient K，which is 0.866 according to the aforementioned formula.

According to the Morison equation，the pro?gram is compiled in Matlab language to calculate the wave forces of particles on water surface and the time-history curve is shown in Fig.7.

Fig.7 Time-history curves of wave force

4 Equation of Wave Particle Mo?tion

Applying the theory of linear wave superposi?tion，the vertical displacement of sea wave can be decomposed into an infinite superposition of simple waves with frequency=wn，amplitude=anand ini?tial phase=εn. In this way，the formula of discrete and continuous vertical displacement fluctuation can be obtained as：

Discrete vertical displacement fluctuation is

Continuous vertical displacement fluctuation is

According to the law of wave particle motion，the horizontal velocity and acceleration of the parti?cles can be expressed as：

Horizontal velocity of the particles is

Horizontal acceleration of the particles is

where ε，εnare the random variables；Gζ(w)is the spectrum；λ the wavelength；k the wave number，T the period and w the circular frequency.

5 Numerical Simulation of Bridge Deformation

5.1 Model establishment

The studied section of Jiaozhou Bay Bridge is a pile group foundation with 24 root piles in total.Each pile has a length of 70 m，a diameter of 2.5 m，and a pile spacing of 6.25 m. The cushion cap is a round chamfered octagon with a length of 42 m，a width of 23.25 m and a thickness of 6 m，as shown in Fig.8. The modeling method is to build a single pile model first，and then build a group pile mod?el［18］.

Fig.8 Top view of pile group foundation

Fig.9 Rendering of pile group foundation model

Radcylinder is chosen to build the soil mass around single piles，and then cylinder is selected to build single piles. During the process of modeling，mirror reflect is used to obtain the single pile model.Since the force acting on the pile body is axisymmet?ric，the 1/4 model of the actual shape is selected for simulation. The basic unit of pile group is single pile and the commands such as reflect，copy and merge should be used in the modeling of pile group. The model is shown in Fig.9.

5.2 Numerical simulation

The Coulomb shear model is adopted for the contact surface of the model. As pile group and cap are both linear elastic bodies，elastic material is used in simulation. The plastic material is selected for the soil around the pile and Mohr Coulomb con?stitutive model is used. Numerical simulation param?eters are shown in Table 1.

Table 1 Parameters of model materials

Considering the influence of the superstructure on the pier，the upper gravity load is first applied on the cap［19］. What is usually loaded in FLAC3Dsimu?lation is the stress，and the stress on the surface of the cap should be obtained by simple calculation，where the stress loaded is the load carried per unit area. Key monitoring points are set in the command flow. During the process of numerical simulation，the sinking of the foundation piles of the following locations ought to be monitored respectively：In the middle of the pile cap，at the top of the side pile cap along the bridge and at the top of the lateral side pile cap.

The displacement changes at different positions of piles are recorded. The center pile in the pile group foundation is selected as the analysis object，and the displacement changes are recorded at differ?ent positions on the upper，middle and lower of the central foundation pile. The vertical displacement nephogram of pile group under the gravity load of superstructure is shown in Fig.10，and that of pier is shown in Fig.11.

Fig.10 Nephogram of vertical displacement

Fig.11 Nephogram of vertical displacement of pier

On the basis of the gravity load of the super?structure，the wave force load and wind load are ap?plied. The measured wind speed of the bridge sec?tion is low，which affects less on the bridge defor?mation. Besides，this paper focuses on the effect of the wave force，which gives rise to the bridge defor?mation. Therefore，the wind load and wave force load are considered as a comprehensive action.According to the action law，i.e.，wave force timehistory curve shown in Fig.7 and combined with the measured wind speed value，the vertical and hori?zontal displacement nephograms of wave force load and wind load continuously applied to the pier at a certain time are shown in Fig.12 and Fig.13.

Fig.12 Nephogram of vertical displacement of pier

Fig.13 Nephogram of horizontal displacement of pier

The 3-D displacement values of pier of eight equally spaced discrete moments under the action of periodic wave force is shown in Table 2.

Table 2 Numerical simulation of deformation at different moments

6 Comparison of GPS Observed Values and Numerical Simulation Values

Comparing the above three-dimensional dis?placement values of the bridge pier and GPS defor?mation observed values in the corresponding time，the displacement differences are shown in Table 3.

It can be seen from Table 3 that the average rel?ative error of displacement difference between GPS deformation observation and numerical simulation is 33% in X direction，41% in Y direction and 35% in Z direction. The displacement of the bridge pier does not represent the displacement of the bridge.This is only an approximate numerical method. In fact，the displacement of the bridge is caused by such environmental factors as wind，wave，current and so on. At the same time，there are errors in ap?plying wave forces. The results of the numerical simulation basically reflect variation law of the threedimensional displacement of the pier，which pro?vides a more reliable technical means for the safety assessment of the bridge in response to the hurri?cane and other marine disasters.

Table 3 Comparison of 3?D displacement between measurement and simulation values at different moments

7 Conclusions

Based on the GPS deformation monitoring of Qingdao Jiaozhou Bay Sea-Crossing Bridge，this pa?per calculates the wave forces on the pier and stud?ies the numerical simulation of the bridge deforma?tion under the dynamic loading of the wave forces.The conclusions are as follows：

（1）The db3 wavelet three-layer decomposition was performed respectively on the GPS measured values，the horizontal displacement values of bridge and the corresponding wind speed values. The re?sults indicate that the deformation of the sea-cross?ing bridge is related to the wind speed，and the dis?placement is proportional to the wind speed. With the change of wind direction，horizontal displace?ment varies in X direction，and the change of the dis?placement lags behind that of the wind speed.

（2）According to the wave motion parameters and Morison equation，the pile coefficient caused by the phase difference was introduced to obtain the time-history curve of the wave forces of the pier wa?ter surface particles.

（3）According to the mechanical parameters of the bridge pier，FLAC3Dnumerical simulation soft?ware was used to make the numerical simulation cal?culation of the dynamic loading of bridge pier in realtime wave forces after the gravity load of the super?structure considered. So the real-time dynamic de?formation of sea-crossing bridge was obtained.

（4）Through the comparative analysis between the measured values and the numerical simulation deformation values of the sea-crossing bridge，the relative error of displacement in X direction，Y di?rection and vertical direction is between 33% and 41%，indicating that the error of the numerical sim?ulation is relatively large. In addition to other envi?ronmental factors，the real-time dynamic application of wave force is one of the main factors affecting the accuracy of numerical simulation. Therefore，it is of vital importance to calculate the wave forces accu?rately.

The numerical simulation，however，basically reflects the deformation law of the sea-crossing bridge，which provides the technical support for its safety assessment.