Development of an efficient finite element model for the dynamic analysis of the train-bridge interaction

Transcription

1 Development of an effiient finite element model for the dynami analysis of the train-bridge interation S. Neves, A. Azevedo & R. Calçada Faulty of Engineering, University of Porto, Porto, Portugal ABSTRACT: The design of high-speed railway bridges omprises a set of demands, from safety and servieability aspets, to new types of equipment and onstrution solutions. In order to perform an aurate and realisti evaluation of the orresponding dynami behavior, adequate analysis tools that take into aount the omplexity of the train-bridge system are required. These omputational tools must be based on effiient algorithms to allow for the ompletion of detailed dynami analyses in a reasonable amount of time. The lassial methods of analysis may be unsatisfatory in the evaluation of the dynami effets of the train-bridge system and fully assessment of the strutural safety, trak safety and passenger omfort. A diret and versatile tehnique for the simulation of the train-bridge interation was implemented in the FEMIX ode, whih is a general purpose finite element omputer program. The presented ase study is an appliation of the proposed formulation, whih proved to be very aurate and effiient. 1 INTRODUCTION The dynami behavior of railway bridges arrying high-speed trains an be analyzed with or without the onsideration of the vehile's own struture. The simulation of the train-bridge system requires several independent submeshes and the onsideration of ontat onditions that represent their interation. The formulation of the ontat between nodal points of the vehile and internal points of a finite element is briefly desribed in this paper. Dynami equilibrium equations in non presribed degrees of freedom, in ontat degrees of freedom and in presribed degrees of freedom are separately developed. Contat ompatibility equations between points of the vehile and internal points of a finite element are also separately developed. All these equations onstitute a single system of linear equations involving displaements, ontat fores and reations as unknowns. After the solution of this system of linear equations the displaements, veloities and aelerations at the urrent time step an be alulated and a new time step is started. This heterogeneous system of linear equations an be effiiently solved by means of the onsideration of several submatries with speifi harateristis. These tehniques have been implemented in FEMIX 4.0, whih is a general purpose finite element omputer program (Azevedo et al, 2003). The new formulation is applied to the analysis of the dynami behavior of the São Lourenço bridge, whih is a bowstring arh bridge. The bridge is loated in the North Line of the Portuguese railway system, in a setion that was reently upgraded to allow the passage at greater speeds of the Alfa pendular train. The numerial model was validated with the results of the ambient vibration test and those measured during several passages of the Alfa pendular train. The dynami response of the train-bridge system is alulated for train speeds exeeding the allowable speed for that setion of the North Line. The orresponding results provide preise indiations on the strutural safety, trak safety and passenger omfort.

2 2 HHT METHOD WITH TRAIN-BRIDGE INTERACTION A simple example is used to introdue the types of degrees of freedom that are onsidered in the formulation of the vehile-struture interation in the ontext of a time step of the Hilber-Hughes-Taylor method (see Fig. 1). On the right, a simply supported beam with two spans (B1 and B2) is subjeted to the ontat of a vehile, shown on the left. The struture of the vehile is also omposed of two beams (B3 and B4). Nodes 7, 8 and 9 are internal points of the beam B1. The loation of these nodes may hange between time steps, depending on the position of the vehile. Eventual gaps between both strutures (g i ) an be easily onsidered in the ompatibility equations, as will be shown later g i B3 B4 B1 B2 Figure 1. Vehile and struture: beams and nodal points. In eah nodal point two degrees of freedom are onsidered (vertial displaement and rotation). Figure 2 shows the generalized displaements in nodal points (1 to 12), the generalized displaements of the ontat points of the struture (13, 14 and 15), the interation fores in the vehile (X 7, X 9 and X 11 ) and the interation fores in the struture (Y 13, Y 14 and Y 15 ). The interation only involves the translational degrees of freedom X 7 X 9 X 11 Y 13 Y 14 Y 15 Figure 2. Vehile and struture: degrees of freedom and interations fores. The following lassifiation of the degrees of freedom is onsidered: F free; X interation (vehile); P presribed; Y interation (struture). This lassifiation is used later in this setion. In the ontext of the Hilber-Hughes-Taylor method (HHT), the dynami equilibrium equation that involves the degrees of freedom in nodal points (1 to 12) is the following M u p p p ( 1 + α) C u& α C u& + ( 1+ α) K u α K u = ( + α) F αf & (1) + 1 In this equation M is the mass matrix, C is the damping matrix, K is the stiffness matrix, F are the applied generalized fores, u are the generalized displaements and α is the main parameter of the HHT method. When α = 0 the HHT method redues to the Newmark method, and for other values of the parameter α, numerial energy dissipation is introdued in the higher modes. The supersript indiates the urrent time step (t + Δt) and the supersript p indiates the previous one (t). Aording to Figure 2 and to the lassifiation indiated above, the F type degrees of freedom are the following: 2, 4, 6, 8, 10 and 12. The X type degrees of freedom orrespond to the

3 "supports" of the separated vehile struture, being the following: 7, 9 and 11. The P type degrees of freedom are the main strutural supports 1, 3 and 5. The Y type degrees of freedom 13, 14 and 15 onsist on the internal displaements of beam B1 at the ontat points. Aording to Azevedo et al (2007), the dynami response an be obtained by solving the following system of linear equations K ( ) FF K FX 1+ α d FX u F F F ( ) ( ) ( ) ( ) K XF K XX 1+ α I XX u X = F X α 1 α I 1 α f X g XF XX It is possible to demonstrate that in Equation (2) the oeffiient matrix of the system of linear equations is symmetri. XX X X (2) 3 APPLICATION TO THE DYNAMIC ANALYSIS OF A BOWSTRING ARCH BRIDGE 3.1 Dynami model of the bridge The São Lourenço bridge is loated at the km of the North Line of the Portuguese railway system, in a reently upgraded setion for the passage of the Alfa pendular train at a maximum speed of 220 km/h. The bridge is a bowstring arh omposed of two half-deks spanning 42 m with a single trak in eah (REFER, 2003). Figure 3 shows a view of the bridge. Eah dek onsists of a 40 m thik prestressed onrete slab suspended by two lateral arhes (see Fig. 4). The arhes are linked in the upper part by transversal girders with a retangular hollow setion and diagonals in double angles that assure the braing of the arhes. Figure 3. São Lourenço bridge and Alfa pendular train. The bridge was modeled with 2D beam elements. The influene of the dek-trak omposite effet in the dynami behavior of railway bridges, due to the transmission of shear stresses between the dek and the rails at the ballasted layer, was treated by Ribeiro et al (2007) and onsidered in the urrent finite element model (see Fig. 4). Figure 4. FEM model of the São Lourenço bridge.

4 The first three vertial mode shapes of the bridge (1V, 2V and 3V) are represented in Figure 5. 1 st vertial mode (1V): f = 4.25 Hz 2 nd vertial mode (2V): f = 6.20 Hz 3 rd vertial mode (3V): f = 9.94 Hz Figure 5. Mode shapes and orresponding frequenies. 3.2 Dynami model of the Alfa pendular train The Alfa pendular train is omposed of six arriages. Eah arriage has two bogies with two axles eah. The total weight of the omposition is t, tare weight, and t for a regular loading. The maximum weight per axle is 14.4 t and the total length of the omposition is m. The dynami model of the train is shown in Figure 6 and onsists of rigid bodies simulating the vehile box (mass M ) and the bogies (mass M b ), springs with stiffness K p (or K s ) and dashpots with damping p (or s ) simulating the primary (or seondary) suspensions, springs with stiffness K h simulating the wheel-rail ontat and masses M e simulating the axles and wheels. The train is modeled with 2D beam elements (see Fig. 6). (1) M (3) (5) (2) (4) (6) M b M e s, k s p, k p k h 2,70 2,70 2,70 2,70 2,70 19,00 25,90 19,00 6,90 25,90 19,00 27,65 Figure 6. Dynami model of the Alfa pendular train.

5 3.3 Experimental validation of the numerial model In order to validate the numerial model an experimental ampaign was undertaken, whih inluded an ambient vibration test and several measurements of passages of the Alfa pendular train (Ribeiro et al, 2007). The ambient vibration test was used to identify the dynami properties of the struture, namely its natural frequenies, mode shapes and damping oeffiients. The vertial aelerations at some points of the dek were measured during several passages of the Alfa pendular train. A good agreement between the natural frequenies obtained with the numerial model and the experimental values ould be observed (Azevedo et al, 2008). 3.4 Dynami response of the train-bridge system Several dynami analyses were performed for speeds ranging between 140 an 420 km/h, with a 5 km/h step, onsisting of 57 simulated passages. The dynami analyses were performed using the HHT method with onstant average aeleration, i.e., with α = 0, β = 1/4 and γ = 1/2 (Hughes, 2000), and a time step equal to s. Rayleigh damping is adopted in the present study. The damping ratios of modes 1V and 3V were obtained in the experimental ampaign, and are equal to 1.4% and 2.4%, respetively. Figure 7 shows the maximum vertial aeleration at the span quarter-point and mid-point for the passage of the Alfa pendular train at speeds ranging between 140 and 420 km/h Aeleration (m/s 2 ) Quarter point Speed (km/h) 2.0 Aeleration (m/s 2 ) Mid point Speed (km/h) Figure 7. Maximum vertial aeleration at quarter-point and mid-point. Resonane peaks an be observed in Figure 7 for train speeds of 395 km/h, at the span quarter-point, and for train speeds of 195 km/h, 230 km/h, 295 km/h and 310 km/h, at mid-point. The resonane phenomena may our for trains with regularly spaed axles, traveling at ritial speeds defined in EN (2003). v d n j (i,j) =, i =1,2,3,4,...,1 2,1 3,1 4,... i res (3)

6 where d is the harateristi length and n j is the jth natural frequeny of the bridge. The harateristi length is a distane (many times repeated) between the vehiles or axles that at some ritial speeds may indue a periodi exitation of the bridge and ause the resonane phenomenon. The harateristi length of the Alfa pendular train is 25.9 m (see Fig. 6). Aording to Figure 7 and to Equation (3), the ritial speeds are: i) taking into onsideration the span quarter-point and an exitation of the struture with a frequeny whih is equal to the mode 1V frequeny (v res = /1 3.6 = 396 km/h 395 km/h); ii) onsidering the mid-point and an exitation with a frequeny whih is equal to 1/2 and 1/3 of the frequeny of mode 2V (v res = /2 3.6 = 289 km/h 295 km/h and v res = /3 3.6 = 193 km/h 195 km/h) and 1/3 and 1/4 of the frequeny of mode 3V (v res = /3 3.6 = 309 km/h 310 km/h and v res = /4 3.6 = 232 km/h 230 km/h). The onsideration of the train-bridge interation in the dynami analysis is partiularly useful sine it allows for an aurate evaluation of the vibrations of the rail ars, and, onsequently, for an assessment of the riding omfort of the passengers. Figure 8 shows the time histories of the vertial aeleration in the first and last ar bodies at a speed of 395 km/h, for whih resonane of mode 1V ours. 1.9 Aeleration (m/s 2 ) Fisrt ar body Last ar body Time (s) Figure 8. Vertial aeleration in the first ar body and last ar body. Figure 9 shows the maximum vertial aeleration in the first and last ar bodies of the train for speeds ranging between 140 and 420 km/h. The maximum level of aeleration ours in the last ar body for speeds that orrespond to the resonane of the bridge. 2.5 Aeleration (m/s 2 ) First ar body Last ar body Speed (km/h) Figure 9. Maximum vertial aeleration in the first ar body and last ar body. The dynami analyses with train-bridge interation were performed for 57 different speeds, whih orresponded to a total exeution time of 2624 s ( 44 min) in a 2.13 GHz personal omputer. The bridge submesh has 1860 non presribed degrees of freedom and the train submesh

7 has 192. The total exeution time of eah analysis orresponds to the period between the entry of the first axle in the bridge and the exit of the last axle. 4 ASSESSMENT OF THE DYNAMIC BEHAVIOR Based on the regulations of EN (2003) and EN1990-A2 (2005), the dynami response of the bridge is evaluated in terms of strutural safety, trak safety and passenger omfort. The orresponding results are presented in this setion. 4.1 Strutural safety The alulation of the dynami amplifiation fators, ϕ dyn, resulting from the passage of the train at high speeds, is based on the results of the time history analysis of the train-bridge system. These values of ϕ dyn are orreted by a fator ϕ in order to take into aount for the trak irregularities. Table 1 shows a omparison between the maximum vertial displaements, at the span quarter-point and at mid-point, obtained with a dynami analysis with the Alfa pendular train, and by means of the appliation of the load model LM 71, being these results multiplied by the orresponding dynami amplifiation fator Φ 2. Table 1. Maximum vertial displaements at the span quarter-point and mid-point. Model Quarter-point Mid-point mm mm LM Alfa pendular Trak safety The norm EN1990-A2 (2005) speifies defletion and vibration limits that must be taken into aount in the design of railway bridges. These limits guarantee traffi safety in terms of: i) vertial aeleration of the dek; ii) dek torsion; iii) vertial deformation of the dek; and, iv) horizontal deformation of the dek. When, for example, the vertial aeleration of the dek exeeds its limit value, instability of the ballast may our, or even a loss of ontat between wheel and rail. For the ase of ballasted dek bridges, the peak value of the vertial aeleration of the dek should not exeed 3.5 m/s 2 ( 0.35g). The aeleration limit is violated only at the span quarter-point and for speeds exeeding 310 km/h (see Fig. 7). 4.3 Passenger omfort Sine the passenger omfort depends on the vertial aeleration of the ar body, the norm EN1990-A2 (2005) limits its value to 1.0, 1.3 or 2.0 m/s 2, whih orrespond to a omfort level lassifiation of "very good", "good" or "aeptable". By observing Figure 9 it an be readily onluded that the omfort level at the first ar body is "very good" for any speed inluded in the studied range. For the ase of the last ar body, the omfort level is "very good" for speeds up to 345 km/h. For speeds near 395 km/h (resonane speed) the omfort level remains lose to aeptable. 5 CONCLUSIONS In the present work an integrated model whose aim is the dynami analysis of strutures by the Hilber Hughes Taylor method is proposed. This algorithm treats the interation between moving vehiles and a struture suh as a bridge. This work provides a signifiant improvement rela-

8 tively to the method proposed by Delgado & Cruz (1997) and Calçada (1995), sine the ompatibility between vehile and struture is no longer imposed by an iterative method, but by means of an integrated formulation that onsiders as variables displaements and ontat fores. The governing system of equations omprises dynami equilibrium equations and ompatibility equations. The system of linear equations that arises at eah time step is effiiently solved by Gaussian elimination, by onsidering several submatries and their own harateristis. The proposed formulation is applied to the dynami behavior of a bowstring arh bridge. The dynami response of the train-bridge system was alulated for train speeds only ourring in high-speed lines, allowing for a better understanding of the behavior of the system and its resonane effets, in terms of strutural safety, trak safety and passenger omfort. A signifiant improvement in terms of effiieny ould also be observed, without ompromising the auray of the results. ACKNOWLEDGEMENTS The authors wish to aknowledge the support provided by RAVE in the ontext of the protool between RAVE and FEUP. This paper reports researh developed under finanial support provided by "FCT - Fundação para a Ciênia e Tenologia", Portugal. REFERENCES Azevedo, A., Barros, J.A.O., Sena-Cruz, J.M. & Ventura-Gouveia, A Eduational software for the design of strutures. Proeedings of the III Engineering Luso-Mozambian Congress. Maputo, Mozambique. 2003: (in Portuguese). Azevedo, A., Neves, S. & Calçada, R Dynami analysis of the vehile-struture interation: a diret and effiient omputer implementation. Proeedings of CMNE Congress on Numerial Methods in Engineering and XXVIII CILAMCE - Iberian Latin-Amerian Congress on Computational Methods in Engineering. FEUP. Porto. 2007: 123 (Abstrat) + 16 pages (Paper n published on CD-ROM by APMTAC). Azevedo, A., Neves, S. & Calçada, R Dynami analysis of the train-bridge interation: an aurate and effiient numerial method. Proeedings of EURODYN Conferene of the European Assoiation for Strutural Dynamis. University of Southampton. Southampton, United Kingdom (to be published). Calçada, R Dynami effets on bridges due to high-speed railway traffi. MS Thesis, Faulty of Engineering of the University of Porto. Porto (in Portuguese). Delgado, R. & Santos, S. M Modelling of railway bridge-vehile interation on high-speed traks. Computers and Strutures 63: EN1990-A Appliation for Bridges. European Committee for Standardization (CEN). Brussels. EN Ations on Strutures Part 2: General Ations Traffi loads on bridges. European Committee for Standardization (CEN). Brussels. Hughes, T.J.R The Finite Element Method. New York: Dover Publiations, In. REFER, E.P Rede Ferroviária Naional, Replaement of the S. Lourenço bridge at km 158,662. Design Projet Projet brief. Lisbon. Ribeiro, D., Calçada, R. & Delgado, R Experimental alibration of the numerial model of the S. Lourenço railway bridge. VI Conferene on Steel and Composite Constrution. Porto (in Portuguese).