Research on Fluid Viscous Damper Parameters of Cable-Stayed Bridge in Northwest China

To optimize the aseismic performance of nonlinear fluid viscous dampers (FVD) of cable-stayed bridge in the highly seismic zone, Xigu Yellow River Bridge in northwest China is taken as an example. Nonlinear time-history analysis method is used to research on the relation among the internal forces, displacements, and damping parameters of the 650 tonnage FVD.Themethod of getting the minimum of binary functions is used to obtain the optimal parameters of FVD. Also, the 1 : 1 full-scale FVDmodel is made and used in the constitutive relation test. Then the test result of the damping parameters can be got by normal equation method. The optimized method to obtain the damping parameters is further verified. The results indicate that seismic response in key positions of the cable-stayed bridge can be reduced by installing longitudinal nonlinear FVD between the towers and girders if choosing reasonable damping parameters C and ξ. The optimal damping parameters can be calculated accurately by the proposed method of optimizing damping parameters of nonlinear FVD, and the constitutive relation test verifies the correctness of the optimization analysis method. Conclusions concerned can be applied to the design of nonlinear FVD for cable-stayed bridges.


Introduction
For cable-stayed bridges, under longitudinal earthquake input, different tower-girder connection forms will have significant influence on the internal force response of main towers and its foundations, also on the displacements of girder end, and on the relative displacements of the girder and the approach bridge [1,2].The problem that the bridge tower internal force will be greatly increased in consolidated system and the problem that the longitudinal displacements of the girder end and the tower top will be enlarged in longitudinal floating system can be overcome by the damping system installed between the towers and girders [3,4].Therefore, increasing numbers of dampers are favored and used in aseismic system for cable-stayed bridges.However, the aseismic performances of the damper are mainly determined by the designed values of damping parameters C and .
At present, there are few references about the parameter selection of fluid viscous dampers (FVD) in cable-stayed bridges.Researches have shown that choosing the reasonable damping parameters of FVD can reduce the seismic response of key positions of cable-stayed bridges [5][6][7].But the references about the selection of damping parameters mainly rely on subjective experience, which lacks experiment basis for selecting damping parameters [8,9].Therefore, the method, based on the constitutive relation test upon the FVD model, which is used to get the optimal value of the damping parameters of nonlinear FVD [10,11], turns out to be useful for the design of FVD in cable-stayed bridges.
To improve the aseismic performance of the cable-stayed bridge, this paper presents a versatile procedure for the design of nonlinear FVD in cable-stayed bridges.Firstly, the relation among the internal forces, displacements, and damping parameters of the FVD under the action of earthquake is studied.The method of getting the minimum of binary functions is used to obtain the optimal parameters of FVD.Secondly, the results of the methodology adopted to obtain the optimized damping parameters have been confirmed by a constitutive relation test conducted on a full-scale model of the FVD.Seismic response in key positions of the cable-stayed bridge can be reduced by installing longitudinal nonlinear FVD if choosing reasonable damping parameters.

Project Overview
Xigu Yellow River Bridge in northwest China, a key project in Lianhuo national trunk line of expressway, is a cablestayed bridge with the longest span and the tallest tower over Yellow River in northwest China.As is shown in Figure 1, the main bridge is a cable-stayed bridge with double tower, double cable plane, and steel-concrete composite beam, of which the span is (67 + 110 + 360 + 110 + 67) m.Four 650 tonnage dampers are installed between the towers and girders.Reinforced concrete rhombus tower is used and the south tower is 151 meters and the north tower is 147 meters.Steel I-beam and concrete composite girders are used in the main beam, and the composite girder is 2.83 meters high in the center of the steel girder and 3.06 meters in the center of the bridge.Low relaxed galvanized parallel steel tendon, with the 7 mm diameter and tensile strength not less than 1,770 MPa, is used in the stay cables and the longest cable is 187.068meters weighing 17 tons.The seismic peak acceleration is 0.2 g and the predominant period of seismic response spectrum is 0.40 s in the bridge site, which is in eight-degree earthquake zone.
Based on the design and construction data, three-dimension element method is used to establish the finite element analytical model [12,13] of Xigu Yellow River Bridge (Figure 2), in which the single girder mechanical model is utilized in the girder.Tension-only spatial truss element is used in stay cables and Ernst formula is utilized to rectify the elasticity modulus of the cables to verify the sagging effect [14,15].Spatial girder element is utilized to simulate the girders, the towers, and the piers.Damper element is used to simulate FVD [16,17].At the same time, the dead load effect on the structural stiffness is taken into consideration.The masterslave relation is used to simulate the girder joints, the lifting points of the stay cables, and the anchorage zone of the stay cables [18,19].The 6 × 6 coupled spring is adopted to simulate the mutual effect between the pile and soil in both pile foundations of the towers and the piers.The acceleration time-history curve with exceeding probability to be 10 percent in the future 50 years, which is provided from seismological bureau of Gansu Province, is taken as the earthquake load for Xigu Yellow River Bridge [20,21].

Parameter Analysis of Nonlinear FVD
FVD consist of (Figure 3) cylinder, piston, hydraulic valve, piston rod, and silicone oil [22].Under the earthquake load, the movement of the structure pushes the piston and the cylinder to produce relative displacement, and the reciprocating motion of the piston drives the flow of silicone oil.The friction between the molecules and that between the surface of the cylinder and the fluid generates heat so that the seismic energy can be converted into heat energy and the damping effect can be realized [23,24].FVD are common shock absorbers in bridge engineering, and the fundamental principle of damping property is [25][26][27][28] where  is the damping force,  is the damping coefficient,  is the relative velocity of dampers, and  is the velocity index.In particular, when the velocity index is equal to 1, the damping force is proportional to the relative velocity and the damper turns out to be a linear one.
As can be seen from ( 1), the response of the damper to the structure is different because of the different damping parameters of FVD.However, most of researches mainly rely on subjective experience to select damping parameters and lack a unified method to select the parameters quantitatively [29][30][31][32].Therefore, it is necessary to do sensitivity analysis of damping parameters  and  to discuss the variation of the structural response with the change of parameters, which is useful for the parameter design of the FVD.
In bridge engineering, the value of velocity index  is usually in the range of 0.2∼2.0, and the damping coefficient  is usually in the range of 1000∼20000 kN×(s/m)  .Considering the FVD of Xigu Yellow River Bridge and the finite element analysis model, the damping coefficient  is set as 5000, 7500, 10000, 15000, and 20000 kN×(s/m)  , respectively, and the velocity index  is set as 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7, respectively.According to the trend of the calculation results of controlled internal force in the south tower bottom, it is easier to obtain the appropriate damping parameters when  is set as 2500, 5000, 7500, 10000, and 15000 kN×(s/m)  , respectively.So there are a total of 30 conditions for parameter analysis.

Parameter Analysis of Controlled Internal Force.
Because the cable-stayed bridge is a symmetrical structure, the south tower is taken as an example.The finite element model shown in Figure 2 is used to do parameter analysis in the 30 working conditions.The actions applied to the bridge include only earthquake acceleration at the bottom of each pier, which are provided from seismological bureau of Gansu Province of China and have 2% of exceeding probability of 50 years.Nonlinear time-history analysis method is used to research on the relation among the internal forces, displacements, and damping parameters.
As is shown in Figure 4, because of different damping coefficient C, the numerical values of longitudinal shear force in the tower bottom will change with velocity index .And it is not hard to figure out that when the damping  coefficient  is in the range of 5,000∼10,000 kN×(s/m)  and the velocity index  is in the range of 0.4∼0.7, the seismic shear force in tower bottom is at a low level.Similarly, because of different damping coefficient C, the bending moment in tower bottom will change with velocity index .And it is easy to find that when the damping coefficient  is in the range of 7,500∼10,000 kN×(s/m)  and the velocity index  is in the range of 0.4∼0.5 or when damping coefficient  is equal to 15,000 kN×(s/m)  and velocity index  is equal to 0.7, the bending moment in tower bottom is at a low level.Therefore, from the perspective of reducing the seismic response in tower bottom, the optimal effect can be achieved when damping coefficient  is in the range of 7,500∼10,000 kN×(s/m)  and velocity index  is in the range of 0.4∼0.5.
As is shown in Figure 5, for the foundation shear force, when damping coefficient  is less than 10,000 kN×(s/m)  , the foundation shear force response decreases with  decreasing, and when damping coefficient  is greater than 10,000, the foundation shear force response decreases first and then increases with  increasing.As for foundation bending moment, with the increase of  and the decrease of , the foundation bending moment response gets smaller and smaller.Therefore, from the perspective of reducing the internal force in tower bottom foundation, the optimal effect can be achieved when damping coefficient  is in the range of 7,500∼15,000 kN×(s/m)  and damping constant  is in the range of 0.3∼0.5.

Parameter Analysis of Controlled Displacement.
As is shown in Figure 6(a), the girder end displacements decrease with the increase of damping coefficient .When damping coefficient  is greater than 10,000, the influence of  on the girder end displacement is weakened, and the increase of  has little influence on the girder end displacements.Therefore, from the perspective of reducing the girder end displacements, the optimal effect can be reached when damping coefficient  is in the range of 10,000∼20,000 kN×(s/m)  and velocity index  is in the range of 0.3∼0.5.
As is shown in Figure 6(b), when damping coefficient  is less than 10,000 kN×(s/m)  , the relative displacements between the south main bridge and the approach bridge increase with  increasing, but when damping coefficient  is greater than 10,000 kN×(s/m)  , the relative displacements decrease first and then increase with  increasing.However, from the perspective of reducing the displacements, the optimal effect can be achieved when damping coefficient  is in the range of 10,000∼20,000 kN×(s/m)  and velocity index  is in the range of 0.3∼0.5.
Based on the above analysis, the seismic response in the tower bottom, the tower foundation, and the girder can be reduced effectively by installing longitudinal nonlinear FVD between the towers and girders when the velocity index  is in the range of 0.4∼0.5 and the damping coefficient  is about 10,000 kN×(s/m)  .

Optimization of Damping Parameters.
As is shown in Sections 3.1 and 3.2, the damping parameters of the FVD can be preliminarily determined qualitatively by simulating and analyzing the finite element model in multiple conditions.However, when determining the optimal damping parameter in many researches [33][34][35][36], the results of multiple situations are simply put to take the common intersection.This method lacks quantitative analysis of multiple conditions.It is obvious that the internal forces or displacements of key positions are binary function of damping parameters  and .Thus, least square method can be used to perform the surface fitting of the relation among the internal forces or the displacements of key positions with damping parameters  and .And then the problem is transformed to solve the extreme values of the nonlinear binary function.The least squares fitting is realized by the LSQCURVEFIT command in MATLAB, and the surface fitting results are as Table 1.
The coefficient of determination  2 is where  is the sample number, ŷ is the corresponding value of regression equation,  is the sample average, and   is the sample corresponding value.The coefficient of determination  2 reflects the goodness of fitting, and its range is [0, 1].The closer the  2 is to 1, the better the regression equation is fitted.
To obtain the optimal damping parameters, the results of surface fitting in Table 1 are arranged: (3)  The constraint condition is as follows:  ∈ [2000,20000] and  ∈ [0.2, 0.7].The damping parameters  and  can be got as long as the minimal values of (2) with constraint condition are solved.Take the average value of the three sets of damping parameters as the optimal damping parameters:  = 10871 kN×(s/m)  and  = 0.415.The above-mentioned method can be applied to obtain the optimal damping parameters when to design the nonlinear FVD in bridges.

Model Making and Experiment
To further verify the optimization method of damping parameters in Section 3.3, in laboratory for training and research in earthquake engineering and seismology at EUCENTRE (Pavia, Italy), the damper tester showed in Figure 7 is applied to do the constitutive relation test on the 1 : 1 full-scale damper model of Xigu Yellow River Bridge in Lanzhou.The installation for testing is shown in Figure 8, and the main parameters of the damper model are shown in Table 2. Make the damper tester control the piston rod to move at the speed of 3 mm/s, 30 mm/s, 60 mm/s, and 80 mm/s successively so that the constitutive relation test can be conducted.Then the relation between the damping force and the relative movement velocity of the piston pod can be tested.Conduct 3 complete axial displacement cyclic loadings in each set.Namely, the displacement is controlled from 0 to the  half of the positive maximum range and then to the half of the negative maximum range, and back to 0 at last.In the first set of test, the relative displacement of the controlled piston pod is shown in Figure 9, and the time-history curve of the FVD relative velocity is shown in Figure 10, and the time-history curve of the damping force is shown in Figure 11.
As is shown from the test results in Figures 10 and 11, when the relative displacement of the damper piston pod is  a sine curve, the time-history curve of the velocity and the time-history curve of the damping force can be seen as cosine curve approximately.And the damping force changes greatly when the velocity of the damper is close to 0. The damping forces of the FVD in four sets tests with the relative velocities are shown in Figure 12.Obviously, the test value curve of the damping force is consistent with the theoretical value curve.Applying a logarithmic law at each member of (1), the following expression is obtained: When the damping force  and velocity  are already known, (4) represents a linear problem in the parameters log  and .To solve the least squares problem, the normal equations method can be used. [ So the value of log  and  can be calculated with Substituting the damping forces and velocities tested in the four sets of tests into (6), the damping parameters  = 10496 kN×(s/m)  and  = 0.427 can be got.The damping parameters of the test are consistent with that of the optimal method, which further verifies the method of optimizing damping parameters.

Conclusion
Taking Xigu Yellow River Bridge in Lanzhou where the seismic intensity is eight degrees as an example, the method of combining finite element simulation and the 1 : 1 full-scale FVD model test is adopted to discuss the reasonable FVD parameters of a cable-stayed bridge in highly seismic zone.The following conclusions can be reached: (1) The FVD are longitudinally installed between the towers and girders in cable-stayed bridge, whose damping parameters have great significance on the seismic response.
After the simulation analysis in 30 conditions by using the finite element software, the reasonable damping parameters can be preliminarily determined.For the nonlinear FVD of Xigu Yellow River Bridge, the seismic response in the tower bottom, tower foundation, and the girder can be effectively reduced when the velocity index  is in the range of 0.4∼0.5 and the damping coefficient  is about 10000 kN×(s/m)  .
(2) When nonlinear FVD are used in the design of bridges, least square method can be applied to conduct the surface fitting on the relation among the internal forces, displacements, and damping parameters  and .So the problem is transformed to solve the extreme values of the nonlinear constraint binary function.The optimal values of damping parameters can be got explicitly by the method, which is useful for the design of nonlinear FVD installed in cable-stayed bridges.
(3) Through the constitutive relation test in the 1 : 1 fullscale FVD model of Xigu Yellow River Bridge in Lanzhou, the results show that when the relative displacement of the damper piston pod is a sine curve, the time-history curve of the velocity and the time-history curve of the damping force can be seen as cosine curve approximately.And the damping force changes greatly when the velocity of the damper is close to 0.
(4) Through the comparison between the finite element simulation analysis and the full-scale FVD model test, the test value curve of the damping force keeps consistent with the theoretical value curve.The damping parameters  = 10496 kN×(s/m)  and  = 0.427 are got, which agrees with the result of the finite element analysis.The method of optimizing damping parameters is further verified.

Figure 2 :
Figure 2: Finite element model of Xigu Yellow River Bridge.

Figure 4 :Figure 5 :
Figure 4: Change of controlled internal force in the south tower bottom with damping parameters.

Figure 6 :
Figure 6: Change of controlled displacements with damping parameters.

Figure 10 :Figure 11 :
Figure 10: Velocity time-history curve of the damper piston pod.

Figure 12 :
Figure 12: Damping force changes with the velocity.

Table 2 :
Main parameters of the FVD model.