English

not defined

no text concepts found

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4 Dynamic response analysis of a wind-train-bridge system with wind barriers T. Zhang1, 2, W. W. Guo1, H. Xia1, Q. P. Hou3,Y. Tian1 School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China 2 Institute of Road and Bridge Engineering, Dalian Maritime University, Dalian 116026, China 3 Research Institute of National Defense Communication, Shijiazhuang Tiedao University, Shijiazhuang 050043, China email: [email protected], [email protected], [email protected], [email protected], [email protected] 1 ABSTRACT: An analysis framework for dynamic interaction of train-bridge system with wind barriers under wind load is proposed based on the theory of bridge wind engineering and structure dynamics. The wind forces acting on bridge, wind barriers and train vehicles include steady forces induced by mean wind and unsteady forces induced by fluctuating wind. The detailed calculation formula of unsteady aerodynamic forces on bridge and vehicles are derived according to wind vibration theory. The bridge is described by modal superposition method based on three-dimensional finite element model, and the vehicle is modeled by a multi-rigid-body system connected with a series of springs and dampers. The effect of wind barriers on the dynamic responses of the train-bridge system is analyzed. By taking a continuous beam bridge on a high-speed railway as an illustrating example, the dynamic responses of the bridge and the train vehicles subjected to strong wind are calculated in the time domain when a train vehicle passing the bridge with or without wind barriers and the running safety indices of vehicles are evaluated. KEY WORDS: Wind-train-bridge coupled system; Wind field; Wind barriers; Dynamic response. 1 INTRODUCTION Wind disaster is one of the main natural disasters that affect the operation safety of railways. In the northwest region of China, railway lines pass through some strong wind areas, where the train service is often interrupted by the wind, and the train overturning accidents occasionally occur [1,2], so effective measures should be taken to ensure the operation of running trains. The experience of Japan's Shinkansen shows that wind barriers can obviously decrease the number of train stops under strong wind [3, 4], so it is necessary and effective to set wind barriers on bridges in strong wind field to ensure the running safety and stability of trains. The high-speed railway line from Lanzhou to Urumqi in west China under construction passes through the strong wind area where the instantaneous wind velocity is up to 60 m/s. When the high-speed train runs on the bridges in this area, the wind force may induce the train to overturn or derail as well as the bridge to vibrate intensely. The dynamic response of the train will decrease under the protection of wind barrier, while the aerodynamic force on the bridge will increase to exacerbate the bridge vibration when the wind barrier is installed on the bridge. Therefore, the running safety of the train on the bridge under wind load and the influence of the wind barrier on the bridge and the train become a vital factor that should be analyzed in the design of railway bridges. With the new built Lanzhou-Xinjiang railway as a research background, this paper research the dynamic problem of a high-speed train running through a continuous beam bridge with and without wind barrier subjected to turbulent wind. A rational analysis framework is established to estimate the vibration response of the coupled vehicle-bridges system in the wind field, so as to evaluate the dynamic performance of the bridge and the running safety of high-speed trains. At the same time, the windbreak structures are studied to improve the running safety of the train, to reduce the interruption on bridge operation when strong wind occurs. 2 2.1 NUMERICAL SIMULATION OF WIND LOAD Wind velocity field An autoregressive model can be used to simulate the wind velocity field which is essentially stochastic time series. m related turbulent wind time histories v(X, Y, Z, t) = [v1(x1,y1, z1,t) v2(x2, y2, z2, t) …vm (xm , ym, zm, t)] can be generated by p v (X,Y,Z,t ) fk v (X,Y,Z,t k Dt ) N (t ) (1) k 1 where X [x1 x2 xm ]T , Y [y1 y2 ym ]T , Z [z1 z2 zm ]T , ( xi , yi , z i ) is the coordinate the i-th point, i=1,2,3,…,m; p is the order of AR model; t is the time interval of wind field simulation; k is the autoregressive coefficient m m square matrix of AR model, k=1,2,…, p; and N(t) is a zero-mean independent stochastic process with given variance. For convenience, v(X, Y, Z, t) is written as v(t). Based on the assumption of wind field simulation and the characteristics of auto-correlation function expressed by R( j Dt ) E[v (t ) v T (t j Dt )] (2) R ( j Dt ) R ( j Dt ) (3) the relationship between the correlation function R ( j t ) and the autoregressive coefficient k is given by p R( j t ) k R[( j k ) t ] N (t ) v T (t j t ) k 1 (j 1,2,, p ) (4) 1147 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 p R (0) R ( k t ) R k N (5) Dse (t ) k 1 where RN is the covariance matrix. Using Equation (4), it finally gets the autoregressive coefficient matrix k, and then substituting k into Equation (5), RN can be solved. When the coefficient matrix k and the covariance matrix RN of AR model are calculated, the fluctuating wind velocity time series can be easily generated using Equation (1). 2.2 1 Lst = u 2CL ( ) B 2 1 Dst = u 2CD ( ) D 2 1 2 M st = u CM ( ) B 2 2 (6-a) (6-b) (6-c) where the subscript st represents the static forces; u is the mean wind velocity; CL(α), CD(α) and CM(α) are, respectively, the non-dimensional lift, drag and moment coefficients, which are determined by the structure size and wind attack angle α, whose values can be measured from wind tunnel tests of section model; B and D are, respectively, the width and height of the bridge deck segment. The buffeting forces per unit length are commonly expressed in terms of quasi-steady model as follows [5]: 1 2 D u (t ) u B 2 CD ( ) (7-a) 2 u B u (t ) D w( t ) 1 [CL ( ) CD ( )] Lbf (t ) u 2 B 2CL ( ) (7-b) u B u 2 u (t ) w( t ) 1 CM ( ) M bf (t ) u 2 B 2 2CM ( ) (7-c) u u 2 Dbf (t ) where the subscript bf represents the buffeting force; CD dCD d , CL dCL d , CM dCM d ; u(t) and w(t) are the lateral and vertical components of the fluctuating wind velocity, respectively. The self-excited forces per unit length, i.e. lift Lse(t), drag Dse(t), and moment Mse(t) are commonly described utilizing flutter derivatives in frequency domain as follows [6]: Lse (t ) h B 1 2 u B KH1* ( K ) b KH 2* ( K ) b K 2 H 3* ( K )b u u 2 K 2 H 4* ( K ) 1148 K 2 P4* ( K ) M se (t ) hb p p KH 5* ( K ) b K 2 H 6* ( K ) b B u B (8-a) pb h h KP5* ( K ) b K 2 P6* ( K ) b B u B (8-b) h B 1 2 2 * u B KA1 ( K ) b KA2* ( K ) b K 2 A3* ( K )b u u 2 + K 2 A4* ( K ) Wind load acting on the bridge The wind forces acting on the bridge include the static force caused by mean wind, the buffeting force caused by fluctuating wind, and the self-exciting force caused by the interaction between the wind and bridge motions. Each component contains forces from three directions of drag force, lift force and moment. The lift force Lst, drag force Dst and moment Mst per unit length caused by mean wind can be calculated according to the classical airfoil theory: p B 1 2 * u B KP1 ( K ) b KP2* ( K ) b K 2 P3* ( K )b u u 2 hb p p KA5* ( K ) b K 2 A6* ( K ) b B u B (8-c) where the subscript se represents the self-excited wind forces; H i* , Pi * and Ai* (i=1,…,6) are frequency dependent flutter derivatives from wind tunnel tests; K B u is reduced frequency; is the circular frequency of vibration; hb, pb, and b are vertical, lateral, and torsional displacement of bridge, respectively. 2.3 Wind load acting on the train Similar to the bridge, wind forces acting on a train in crosswind field can be divided into two parts, i.e. steady aerodynamic forces induced by the mean wind velocity component of natural wind and unsteady aerodynamic forces induced by the fluctuating wind velocity component [7, 8]. The wind forces acting on the bogie and wheel-sets of the vehicle are neglected because of their small windward area, thus only wind forces acting on the car-body are taken into account, which mainly refer to side force FS, lift FL and rolling moment M with respect to the mass center of the car body. The wind forces acting on the car-body can be given [9]: w 1 1 2uu FS = AVR2 CFS ( )+ AVR2 CFS ( ) 2 CFS ( ) (9-a) VR u 2 2 w 1 1 2uu FL = AVR2 CFL ( )+ AVR2 CFL ( ) 2 CFL ( ) (9-b) VR u 2 2 M w 1 1 2uu ( ) AVR2 HCMv ( )+ AVR2 H CMv ( ) 2 CMv V u 2 2 R (9-c) where VR2 VT2 u 2 , VR is the wind velocity relative to the train, VT is the train speed; A and H are the reference area and height of the vehicle, respectively; Ci ( ) and Ci( ) ( i FS , FL , M v ) are the side force, lift force and moment coefficients of the vehicle and their first order derivatives at the wind attack =0 . In the right side of Equation (9), the first term represents the steady aerodynamic force, while the other two terms represent the unsteady, respectively. It can be seen from Equation (9) that the fluctuating components of wind velocity field u and w should be obtained in order to decide the unsteady aerodynamic forces. In addition, the aerodynamic coefficients of the vehicle and their first order derivations also should be provided. Of course, the turbulent wind velocities can be simulated at a series of points along a longitudinal line passing through the mass center of the car-body by the method in Section 2.1. Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Modeling of train In this paper, a 4-axle vehicle with two suspension systems is taken as an example to demonstrate the modeling of the vehicle. To simplify the analysis but with enough accuracy, the five assumptions which are described in detail in Ref. [10] are used in the modeling of the vehicle subsystem. Both the car-body and each bogie have five degrees-offreedom, including the floating, the lateral movement, the rolling, the yawing, and the pitching. Each wheel-set has three degrees-of-freedom, including the floating, the lateral movement, and the rolling with respect to its mass center. As a result, the total degrees-of-freedom of the vehicle are 27. Then the vehicle and the bridge are associated with the supposed wheel-rail relation and take the track irregularity as the excitation source. 3.2 Modeling of bridge The bridge is composed of girders, piers, abutments, deck system and track system. Due to the complexity of coupled components, suppose the girder and the track have no relative deformation and the elastic deformation of the track system is also neglected. Thus based on the finite element fundamentals, the bridge is discretized as a three-dimensional finite element model. By applying the modal comprehension analysis technique where the generalized coordinates of bridge vibration modes are solved rather than the motion equations of the bridge directly, the total number of the degrees-of-freedom of the system is significantly reduced and the coupled equations of motion are efficiently solved. Detailed formulations can be found in Ref. [11]. 3.3 Wind-train-bridge system considering wind barriers Based on the previous model of the vehicle, bridge and wind field, the wind-train-bridge dynamic interaction analysis model is derived considering the effect of the wind barrier. Since the stiffness of wind barrier on is small relative to the bridge, and only the stand column of the wind barrier is connected with the bridge, it is not considered that the wind barrier influence on the bridge modes. So only the effect of the wind barrier on the wind forces acting on the bridge and the train is considered to calculate the dynamic response of the system. The motion equations of train-bridge coupling system under wind load can be expressed as follows: M vv 0 Cvv 0 X v + M bb X b Cbv K vv K bv Cvb X v Cbb X b K vb X v Fv0 Fvst +Fvust K bb X b Fb0 F bst +F bbf +F bse acceleration, velocity and displacement vectors of the train and the generalized coordinate vector of the bridge, its first order derivation, its second order derivation, respectively; Fv0 and Fb0 are the force vectors due to the train-bridge interaction through the track and wheels under the moving train, respectively; F bst , F bbf , and F bse are the modal static wind force vector, buffeting force vector and the self-excited force vector of the bridge, respectively; Fvst and Fvust are the steady force vector and unsteady force vector of the vehicle. Note that different aerodynamic coefficients are adopted with or without wind barriers when calculating the aerodynamic forces on the bridge and the train. CASE STUDY 4 4.1 System input data During the design of the Lanzhou-Xinjiang high-speed railway, the wind-train-bridge calculation program was used to calculate the dynamic response of the bridge and the running safety indices of the train traveling on the bridge with or without wind barriers when the wind is normal to the motion of the vehicle at the level of the vehicle mass center. In the wind prone region of the Lanzhou-Xinjiang railway, the whole bridge is installed with the single-side 4 m height wind barrier, as shown in Figure 1. 400 3.1 DYNAMIC MODEL OF WIND-TRAIN-BRIDGE SYSTEM WITH WIND BARRIERS 30 1340/2 1340/2 305 3 Figure 1. Single-side wind barriers on the bridge (Unit: cm) The bridge concerned is composed of 5-span continuous PC beams with the length of 40+3×64+40 m, whose finite element model is shown in Figure 2. The natural vibration characteristics of the bridge are analyzed to obtain the frequency and the vibration modes. The range of the first 60 order natural vibration frequency is 0.437 Hz~32.01 Hz. (10) where: the subscripts v and b represent the train and bridge, respectively; Mvv, Cvv and Kvv are the mass, damping and stiffness matrices of the train, respectively; Mbb, Cbb and Kbb are the mass, damping and stiffness matrices of the bridge, respectively; Kvb and Kbv , Cvb and Cbv are the stiffness and damping matrices due to the interaction between the bridge , X and X , X , X are the and the train; X v , X b v v b b Figure 2. Finite element model of the continuous beam bridge The train in the case study is the ICE train in Germany, composed of 2×(3M+1T), where M represents the motor-car and T the trailer-car. The height and width of the car-body are 3.5 m and 2.7 m. The average static axle loads are 160 kN for a motor-car and 146 kN for a trailer car. The other parameters of the ICE train can be found in Ref. [5]. All the vehicles are the same in a train and the wind is normal to the motion of the 1149 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Table 1 Aerodynamic coefficient CM Bridge 1.09 0.47 0.15 Vehicle 1.37 0.052 0.82 Bridge 2.03 -0.13 0.24 0.057 0.13 Vehicle 0.33 0.094 0.21 4.2 0.74 -0.40 --- 2 15 10 5 0 -5 -10 -15 -- Bridge response Shown in Figure 3 are the maximum displacements of the bridge with and without wind barriers under different mean wind velocities when the train speed is 200 km/h. 0 2 acceleration [cm/s ] 0.10 0.05 0.00 5 10 15 20 25 30 35 mean wind velocity [m/s] 5 10 2 acceleration [cm/s ] vertical displacement [cm] without wind barrier with wind barrier 0.25 0.24 0.23 0.22 5 10 15 20 25 30 mean wind velocity [m/s] 35 Figure 3. Maximum displacement of bridge vs. wind velocity at VT=200 km/h 1150 15 time [s] 20 30 25 30 u=0 m/s 5 0 -5 -10 5 10 0.27 0 25 10 0 0.26 15 20 time [s] (a) Lateral acceleration 0.15 0 10 u=30 m/s without wind barrier with wind barrier 0.20 5 15 10 5 0 -5 -10 -15 0.25 lateral displacement [cm] u=0 m/s 0 -- 2 With 4m wind barriers CM CL acceleration [cm/s ] Without wind barriers CL CD Item It can be seen that wind force has obvious influence on the lateral displacements of the bridge, which increase with the wind velocity significantly, while it has relatively little effect on the vertical displacements. At the same time, the lateral displacements of the bridge with wind barrier are larger than those without wind barrier, while the vertical displacements of the bridge with wind velocity are the opposite. For the bridge without wind barrier, Figure 4 shows the time histories of lateral and vertical accelerations of the bridge when the train runs at 200 km/h on the third span whether or not considering the wind forces. acceleration [cm/s ] vehicle at the level of the vehicle mass center. The train runs on a straight track at a constant speed. The track irregularities are generated by harmonic synthetic method on the basis of the German PSD functions of rail irregularities for high-speed railways, whose detailed expressions can be found in Ref. [5]. The length of the simulated data is 2000 m with the maximum amplitude being 4.20 mm in the lateral direction, 5.80 mm in the vertical direction and 0.002 rad in the torsional direction. Then the wind velocity time series are simulated according to the given PSD functions in Ref. [12] adopting the method in Section 2.1. In the addition, the aerodynamic coefficient of the bridge and the vehicle with respect to wind angle at the zero wind angle of attack also should be given to calculate the wind forces by the wind tunnel test, listed in Table 1. 15 10 5 0 -5 -10 -15 15 20 time [s] 25 30 u=30 m/s 0 5 10 15 20 time [s] 25 30 (b) Vertical acceleration Figure 4. Mid-span acceleration time histories of the third span of the bridge at VT=200 km/h Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Table 2 Maximum accelerations of the bridge at the third span 15 20 25 30 without wind barrier with wind barrier offload factor 0.40 0.30 0.20 0.10 0.00 0 5 10 15 20 25 30 mean wind velocity [m/s] 35 0.50 without wind barrier with wind barrier 0.40 0.30 0.20 11.90 12.10 11.90 12.00 12.30 0.10 11.80 12.00 11.90 11.50 12.80 0 5 8.10 8.40 8.62 7.54 9.68 Vehicle response The maximum car-body accelerations as the train runs at 200 km/h are listed in Table 3. Table 3 Maximum car-body accelerations of the train Mean wind velocity 5 10 15 20 25 30 /(m/s) Without 41.14 43.07 52.32 60.60 82.55 110.39 Lateral/ wind barriers (cm/s2) With wind 41.22 40.65 41.30 43.03 46.56 48.31 barriers Without 59.53 59.51 59.57 59.57 58.94 59.24 Vertical/ wind barriers 2 (cm/s ) With wind 59.49 59.45 59.58 59.59 58.34 61.09 barriers It can be seen from the table that the lateral car-body accelerations increase obviously with the wind velocity without installing wind barriers, while increase slowly with wind barriers. When the mean wind velocity is 30 m/s, the maximum lateral car-body accelerations are 48.31 cm/s2 and 110.39 cm/s2 with and without wind barriers, respectively. However, the vertical car-body acceleration has no big change with the increase of wind velocity. The running safety indices of the train such as offload factors, derailment factors, overturning factors and lateral 10 15 20 25 30 mean wind velocity [m/s] 35 2.50 7.08 8.04 8.14 8.02 8.67 overturning factor 4.3 10 0.50 without wind barrier with wind barrier 2.00 1.50 1.00 0.50 0.00 0 lateral force of wheel-set [kN] Mean wind velocity 5 /(m/s) Without 11.60 Lateral/ wind barriers 2 (cm/s ) With wind 11.80 barriers Without 8.18 Vertical/ wind barriers 2 (cm/s ) With wind 8.19 barriers forces of the wheel-set are shown in Figure 5, as the mean wind velocity from 0 m/s to 30 m/s at VT=200 km/h. derailment factor The time histories show that the accelerations reach the maximum value when the train travels on the bridge, while the wind force has slight effect, because the accelerations are mainly caused by the track irregularity and train excitation. The acceleration decreases rapidly after the train leaves the span without wind action, while it obviously has turbulent characteristic with wind action. The maximum lateral and vertical accelerations of the third span are 11.8 cm/s2 and 8.18 cm/s2 without wind, and they are 12.3 cm/s2 and 9.68 cm/s2 with the mean wind velocity 30 m/s, respectively, indicating that the wind force has no great influence on the bridge acceleration. Furthermore, the bridge acceleration is also calculated with wind barriers. The results of the mid-span maximum accelerations of the third span of the bridge are listed in Table 2 when the train runs at 200 km/h. It can be found that the lateral and vertical accelerations of the bridge changed little with the wind velocity, and the lateral accelerations are larger than the vertical ones, indicating that the wind barriers have little effect on the maximum acceleration of the bridge. 5 50 10 15 20 25 30 mean wind velocity [m/s] 35 without wind barrier with wind barrier 45 40 35 30 0 5 10 15 20 25 30 mean wind velocity [m/s] 35 Figure 5. Running safety indices of the train 1151 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 As can be seen, the safety indices of the train without wind barriers are much larger than those with wind barriers, especially when the wind velocity is high. These indices increase rapidly with the wind velocity, and the greater the wind velocity is, the faster the increase growth, particularly for offload factors and overturning factors. The growth amplitude is larger without wind barriers than that with wind barriers. Without wind barriers, the maximum overturning factor is up to 1.25 when the mean wind velocity reaches 25 m/s, while it is 0.32 with wind barriers. Moreover, these safety indices change slowly with the wind velocity when wind barriers are installed. Therefore, the wind barrier can effectively reduce the dynamic response of the train and improve the running safety. 5 CONCLUSIONS This paper studies the dynamic response of a multi-span continuous beam bridge passing through a high-speed train under turbulent winds. There are several cases, including different mean wind velocities, the bridge without and with wind barriers, to obtain the change rule of the dynamic response and the effect of the wind barriers. Some conclusions are summarized as follows: (1) Wind forces have influence on the displacement responses of the bridge. The lateral displacement is very small without wind action but significantly increases with the wind velocity, and the lateral displacement is larger with wind barriers than that without wind barriers. However, the vertical displacement has little change with the wind velocity. Wind force has no great impact on bridge accelerations whether or not wind barriers are installed. (2) For bridge without wind barriers, wind has strong influence on lateral car-body accelerations, and they increase rapidly with the wind velocity, while for the bridge with wind barriers, this influence is very slight, which indicates that the wind barriers can enhance the passengers comfort level. Wind force does not produce significant impact on vertical car-body accelerations whether or not to adopt wind barriers. (3) The offload factor, derailment factor, overturning factor and lateral wheel-set force of the train under wind action increase with wind velocity, and the greater the wind velocity, the faster the increase rate. The increasing amplitude without wind barriers is much larger than that with wind barriers, indicating that the running safety of the train on the bridge can be improved by adopting the proposed wind barriers. (4) The calculated results show that wind barriers lead to reduction of vehicle response, but increase of bridge response, in general, more advantages than disadvantages. ACKNOWLEDGMENTS The research described in this paper is supported by the Major State Basic Research Development Program of China (“973” Program: 2013CB036203), and the Fundamental Research Funds for the Central Universities of China (No.2013YJS054), the Natural Science Foundation of China (51308034). REFERENCES [1] 1152 Q. K. Liu, Y. L Du and F. G. Qiao. Train crosswind and strong wind countermeasure research in Japan, Journal of the China Railway Society, Vol. 30(1), 82-88, 2008. Z.Y. Qian. Strong wind disaster and control countermeasure for northwest China railways, Chinese Railways, Vol. 51(3), 1-4 to 14,2009. [3] T. Noguchi and T. Fujii. Minimizing the effect of natural disasters, Japan Railway & Transport Review, Vol. 23, 52-59, 2000. [4] T. Fujii, T. Maeda and H. Ishida, et al. Wind-induced accidents of train/vehicles and their measures in Japan, Quarterly Report of Railway Technical Research Institute, Vol. 40(1), 50-55, 1999. [5] H. Xia, G. D. Roeck, J. M. Goicolea. Bridge vibration and controls: New Research, Nova Science Publishers, New York, 2011. [6] X. Z. Chen, M. Matsumoto and A. Kareem. Time domain flutter and buffeting response analysis of bridges, Journal of Engineering Mechanics, Vol. 126(1), 7-16, 2000. [7] C. J. Baker. Ground vehicles in high cross winds part 1: steady aerodynamic forces, Journal of Fluids and Structures, Vol. 5(2), 69-90, 1991. [8] C. J. Baker. Ground vehicles in high cross winds part 2: unsteady aerodynamic forces, Journal of Fluids and Structures, Vol. 5(2), 91-111, 1991. [9] T. Zhang, H. Xia and W. W. Guo. Analysis on running safety of train on bridge with wind barriers subjected to cross wind, Wind and Structures, Vol. 17(2), 203-225, 2013. [10] H. Xia and N. Zhang. Dynamic interaction of vehicles and structures, Science Press, Beijing, second edition, 2005. [11] W. W. Guo, Y. L. Xu and H. Xia, et al. Dynamic response of suspension bridge to typhoon and trains II: numerical results, Journal of Structure Engineering, Vol. 133(1), 12-21, 2007. [12] Ministry of Communications of PRC. Wind resistant design specification for highway bridges JTG/T D60-01-2004, China Communications Press, Beijing, 2004. [2]