Dynamic analysis of rail track for high speed trains. 2D approach

In the framework of an ongoing national research project involving the University of Minho, the National Laboratory of Civil Engineering and the New University of Lisbon, different commercial FEM codes (DIANA, PLAXIS and ANSYS) were tested to model the dynamic performance of a high speed train track. Initially a plain strain model is considered in a robin test using experimental data from a well documented instrumented standard ballast rail track under the passage of a HST at 314 km/h. Numerically predicted results are presented and assessed, and comparison is made between the different codes and also the experimental data. model to calculate the force time history under the sleeper; this force is then introduced in a plain strain model of the tunnel cross section. This type of approach could be interesting to develop for rail-track-foundation modelling. Anyway, the reliability of all these numerical models depends largely on the accuracy of the input data and the choice of an appropriate underlying theory and can be evaluated through comparison with results from experiments and theoretical analysis. In this respect the results presented in this paper are a first contribution of the projectfor this assessment. 2 MODELLING OF DYNAMIC PERFORMANCE OF RAIL TRACK UNDER HIGH SPEED MOVING LOADS 2.1 Background In the process of modeling and design, material models are an important component. Associated with the material models, it is necessary to take into account the tests needed to obtain their parameters. This chain of operations must be always borne in mind at the design stage in order to ensure a good planning of the tests needed for the models. This process will be completed by choosing the performance models and relevant design criteria (Gomes Correia, 2001, 2005). For a more practical application, the complexity of all the process is divided in three levels: routine design, advanced design and research based design. An outline of these levels of design were reported in COST 337, action related with pavements and summarized by Gomes Correia (2001). Each process should be object of verification, calibration and validation. Verification is intended to determine whether the operational tools correctly represent the conceptual model that has been formulated. This process is carried out at the model development stage. Calibration refers to the mathematical process by which the differences between observed and predicted results are reduced to a minimum. In this process parameters or coefficients are chosen to ensure that the predicted responses are as close as possible to the observed responses. The final process is validation that ensures the accuracy of the design method. This is generally done using historical input data and by comparing the predicted performance of the model to the observed performance. Based in this general framework, some particular aspects are developed hereafter in relation with the use of different numerical tools that are available for be used at routine and advanced modeling and design. Lord (1999) emphasized the empirical rules still used at the construction and design levels of rail track. He also noticed the importance to consider dynamic aspects in design. Rail track design is probably one of the most complex soil-structure interaction problems to analyse. The various elements in design process comprise (Lord, 1999): (1) multi-axle loading varying in magnitude and frequency; (2) deformable rails attached to deformable sleepers with flexible fixings, with sleeper spacings which can be varied; (3) properties and thickness of ballast, sub-ballast, prepared subgrade (if adopted); (4) properties of underlying soil subgrade layers. At routine design level, several railway track models are operational and some commercially available. The most popular and simplest model for rail track design represents the rail as a beam, with concentrated wheel loads, supported by an elastic foundation. The stiffness of elastic foundation incorporates the sleepers, ballast, sub-ballast and subgrade, but it is not possible to distinguish between the contribution of the sleeper and underlying layers. This simplified approach has been used to establish dimensionless diagrams in order to quickly assess the maximum track reactions both for a single axle load and a double axle load when the track parameters are changed (Skoglund, 2002). More sophisticated models have been developed which represent the rails and sleepers as beams resting on a multiple layer system (as in pavements) comprised of the ballast, sub-ballast and subgrade. These models include the commercial programs ILLITRACK, GEOTRACK and KENTRACK cited by Lord (1999). In these models incorporating multiple layer systems, the design criterion is identical as for pavements, keeping vertical strain or vertical stress at the top of subgrade soil below a determined limit. This criterion is an indirect verification of limited permanent settlements at the top of the system, having the same drawbacks as mentioned for pavements. Gomes Correia & Lacasse (2005) summarized some values of allowed permanent deformations for rail track adopted in some countries. To overcome the drawbacks of the previous models, two directions of advanced modelling are identified. The first category of models aim at improving the theory of beam resting on continuous medium by introducing a spring-dashpot to better simulate a multiple layer system. Furthermore, the model was also improved by introducing a moving load at constant speed and also an axial beam force (Koft and Adam, 2005). Figure 1 is a sketch of the model. This model is able to determine dynamic response in different rail track systems due to a load moving with constant speed. A drawback of the model is that it is limited to beams with finite length and consequently only steady state solutions can be provided. Figure 1. Sketch of a flexible beam resting on continuous spring-dashpot elements loaded by a moving single load (Koft and Adam. 2005). The second group of advanced rail track models use FEM and hybrid methods. The hybrid methods couple FEM and multi-layer systems (Aubry et al., 1999; Madshus, 2001). The trackembankment system is modelled by FEM and the layered ground through discrete Green’s functions (Kaynia et al., 2000). The software developed is called VibTrain. The models referred by Aubry et al. (1999) and Madshus (2001) use frequency domain analysis having the drawback to require linear behaviour of materials. However, solutions in the time domain also exist (Hall, 2000 – mentioned by Madshus, 2001). This last family of models incorporating track-embankment-ground is very powerful as it simulates behaviour at all speeds from low up to the critical speed. As referred by Madshus (2001), these models need further validation by field monitoring. Dynamic materials characterization should be strongly encouraged, as well as dynamic field observations, mainly for ballasts. In this paper the second group of advanced models was tested for a case study. These first results only address calculations done in plain strain conditions (2D). This project also intends to put into operation a numerical model where, by incorporating the global behaviour of the whole of the railway platform and the supporting soil, will serve to quantify advantages and disadvantages, of the methods used at present in the maintenance of ballast platforms. It will be also possible to quantify and predict the consequences that can have, on this type of structures, the increase of the circulation velocity and the axle load in high speed trains. 2.2 Application of different commercial software for a case study 2.2.1 Presentation of the case study The experimental data used to calibrate de models is obtained from the literature (Degrande & Schillemans, 2001) with material parameters summarized in Table1. These data correspond to vibration measurements made during the passage of a Thalys (high speed train – HST) at 314 km/h on a track between Brussels and Paris, more precisely near Ath, 55 km south of Brussels. The geometry and load characteristics of the HST are presented in Figure 2. The HST track is a classical ballast track with continuously welded UIC 60 rails fixed with a Pandroll E2039 rail fixing system on precast, prestressed concrete monoblock sleepers of length l = 2.5m, width b = 0.285 m, height h = 0.205m (under the rail) and mass 300 kg. Flexible rail pads with thickness t = 0.01m and a static stiffness of about 100 MN/m, for a load varying between 15 and 90 kN, are under the rail. The track is supported by ballast and sub-ballast layers, a capping layer and the supporting soil. Table 1. Geometry and material parameters of the HST track (after Degrande & Schillemans, 2001) Element Parameter Value Sleeper Poisson 0.2 * Young Modulus 30 GPa * Mass density 2054 Kg/m Rail/sleeper interface Thickness 0.01 m Stiffness 100 MN/m Rail (UIC60) Area 76.84 cm Inertia Ix 3055 cm Inertia Iz 512.9 cm Tortional inertia 100 cm * Volumic Weight 7800 Kg/m Poisson 0.3 * Young Modulus 210 GPa * Ballast (25/50) Stiffness 0.3 m Mass density 1800 Kg/m * Poisson 0.1 * Young Modulus 200 MPa * Damping 0.01 * Sub-ballast (0/32) Stiffness 0.2 m Mass density 2200 Kg/m * Poisson 0.2 * Young Modulus 300 MPa * Damping 0.01 * Capping layer (0/80 a 0/120) Stiffness 0.5 m Mass density 2200 Kg/m * Poisson 0.2 * Young Modulus 400 MPa * Damping 0.01 * Soil 1 Compression wave velocity 187 m/s Mass density 1850 Kg/m Stiffness 1.4 m Shear wave velocity 100 m/s Poisson 0.3 Damping 0.03 Soil 2 Compression wave velocity 249 m/s Mass density 1850 Kg/m Stiffness 1.9 m Shear wave velocity 133 m/s Poisson 0.3 Damping 0.03 Soil 3 Compression wave velocity 423 m/s Mass density 1850 Kg/m Stiffness Infinite Shear wave velocity 226 m/s Poisson 0.3 Damping 0.03 * Adopted values Figure 2 shows the configuration of the Thalys HST referred by Degrande & Lombaert (2000), consisting of 2 locomotives and 8 carriages; the total length of the train is equal to 200.18 m. The locomotives are supported by 2 bogies and have 4 axles. The carriages next to the locomotives share one bogie with the neighbouring carriage, while the 6 other carriages share both bogies with neighbouring carriages. The total number of bogies equals 13 and, consequently, the number of axles on the train is 26. The carriage length Lt, the distance Lb between bogies, the axle distance La and the total axle mass Mt of all carriages are summarized. Figure 2. Geometry and load characteristics of the Thalys HST (Degrande & Lombaert, 2000) The location of the measurement points (accelerometers) used for this work is presented in Figure 3 and the results are shown hereafter together with numerical predictions. Figure 3. Location of the accelerometers 2.2.2 Simulation of Thalys HST moving at 314 km/h The loads to be considered in the calculations are those that derive from the passing of the train. The loading history depends, naturaly, on the train speed. Because the calculations are to be done in plain strain conditions places some difficulties in the definition of the loading model arise because any point load in a plain strain model corresponds to a distributed load in the treedimensional model. Therefore, it is necessary to consider some simplifications and assumptions and, in the interpretation of the results, to limit its validity to the distances reported in Gutowski & Dym (1976). The maximum length of the Thalys train is LT=196.7 m (maximum distance between extremity axles) so, the results obtained in 2D model considering a linear load can be considered as valid for distances: m L d T 62 7 . 196 ≈ = = π π (1) Although the loads transmitted by the axle are discrete, the stiffness of the structural elements of the railroad superstructure provides some distribution of the load. For circulation speeds – V – lower than the critical speed Vcr the loading in each point due to the passage of a axle follows approximately the distribution presented in Figure 4. The exact shape of the curve depends on the load speed, on the response of the railroad superstructure and its foundation, being that for higher speeds (but still inferior to the critical one) the curve tends to be thinner. When the speed approaches the critical speed, the curve loses the symmetry and the maximum value occurs after the load. Figure 4. Load due to a axle at sub-critical speed A possible approach to define the load distribution can be considered admitting a distribution adjusted to the sleeper spacing as is considered in the Japanese regulations. In accordance with this document, a changeable part between 40 and 60% of the load is distributed to the adjacent sleepers. A simplified way to establish the load distribution consists of using the solution of the Winkler beam for the movement of a load. In accordance with this simplified model, the quasi static response in displacement is given by: ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = − L s L s e kL Q s w L s sin cos 2 / (2) where Q if the applied load, k the reaction module of the foundation, L the characteristic length of the beam and s the coordinate in a moving referential. It seems reasonable to admit that, for speeds inferior to the critical one, the distribution of load underneath each axle will follow an analogous distribution: ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = − L s L s e L F s F L s e sin cos 2 / (3) In the previous equation F(s) represents the distribution of the force due to each axle as a function of force Fe correspondent to the axle. The value of characteristic length L can be adjusted to obtain a certain amount of axle load at the point s=0 (underneath the axle). Transformation between static referential “x” (in global coordinates) and the moving referential “s” is obtained through: ( ) t V x L s 0 1 − = (4) where V0 represents the train speed and t the time. Thus, admitting that for s=0 one has 60% of the axle load, L=0.831 will be obtained. The load distribution corresponding to each axle is presented in Figure 5 for a unitary load. Figure 5. Load distribution to a unitary axle load The effect of the train can now be obtained considering the overlapping of all the axles, in accordance with the load distribution of the Thalys train: