Influence of the nonlinear behaviour of ballast on the dynamics of simply supported railway bridges

. In this work, the vertical motion of a simply supported railway bridge which is subjected to the circulation of high speed trains was studied. A system consisting of two-layer beam was considered to model the dynamics of the bridge structure. The upper beam represents the rails with the sleepers and the lower beam the bridge deck. These two beams are coupled through distributed nonlinear springs that model the ballast action. The characteristics of these elements were identified from experimental measurements performed on real rail track. Considering the circulation of high speed train at given velocity, the influence of the nonlinear stiffness of the ballasted track on the response of the bridge system was analyzed. This was achieved by using the Galerkin method and the Runge-Kutta scheme to solve numerically the nonlinear partial differential equations governing the motion of the two beams. It was found that the nonlinear behaviour of the ballast affects notably the dynamics of the bridge, especially when the ballast stiffness is low. The proposed modelling enables to get more understanding regarding the vertical dynamics of ballasted track bridge in high speed line.


Introduction
The design of high-speed railway bridges requires studying their dynamic response. These structures may experience in fact severe transient accelerations under the passage of high speed trains. This happens at resonance which occurs for specific train speeds that depend on the general bridge characteristics: mass per unit length, bending stiffness, span length, damping as well as the configuration of circulating train. It was observed that simply supported ballasted track bridges can undergo high vertical acceleration levels during the passage of trains at velocities exceeding 200 / km h . Adverse effects may then result such as ballast destabilization, passenger discomfort, structural fatigue, loss of contact between wheel and rail, and even derailment [1][2][3][4][5].
In the actual design practice of railway bridges, the track is assumed to act as an additional mass and its stiffness is ignored. This hypothesis is made in order to get an evaluation on the safety side. However, in reality, track stiffness is not infinite and affects the bridge dynamics differently than by its mass. Improved modelling is then needed to predict realistically the vibration levels of ballasted track bridges under the circulation of high speed trains.
A large number of published papers have focused on different aspects of the complex coupled dynamics of vehicle-track-bridge-foundation system [6][7]. Authors have derived various models either under full 3D kinematics or by assuming 2D approximation of the problem. They have considered rigid, linear or nonlinear ballast deformation.
In the context of linear based investigations, Biondi et al. [8] proposed a 2D train-track-bridge model where the rail and the bridge have been represented as two beams coupled through a viscoelastic connection layer. By using the substructuring approach, they succeeded in extracting the dynamic response of the vehicles, rails and bridge. Regueiro et al. [3] investigated numerically the influence of ballasted track models on the dynamic response of medium span railway viaducts. They have considered 2D and 3D models for the track and two models for the loading. The authors noted that 2D models can be used for non-skewed viaducts. By comparing the predicted accelerations to field measurements, they proved that including the track superstructure has a favourable effect on the maximum acceleration. However, the authors have pointed out that the computations and measurements made do not reflect the resonance situation which needs further investigation. Zhai et al. [4] presented a review on the general problem of train-track-bridge interaction. This covers both the development of numerical models and experimental tests. The authors have argued that the consideration of track structure stiffness makes more practical analysis of the system vibration. Martinez-Rodrigo et al. [9] introduced a 2D numerical model that was fitted based on information from experimental tests. They concluded that the proposed approach enables to take into account the effects of the track components and the ballast coupling on the dynamic response of real bridges. They noted that high dispersion exists in the track parameters as given in literature even for similar track infrastructures. They mentioned that the rail-pad stiffness seems to have the most effect on the bridge maximum acceleration at resonance. Malveiro et al. [10] presented the calibration of a dynamic model for a railway viaduct by means of experimental data. Lou et al. [11] proposed a coupled numerical model to analyse the dynamic problem of train-track-bridge interaction system. The authors concluded that the double-layer track model, with sleepers considered, affects the natural frequencies of bridge in comparison with the single-layer track model where sleepers are ignored. The doublelayer track model was found to be more accurate.
Contrary to the above mentioned studies where the ballast stiffness was stated linear, it was recognized in literature that the nonlinear stiffness of the track is important to consider. It was found that it can play a positive role in the design of railway bridges [12][13]. Based on experimental testing performed on a single ballasted track railway bridge, Rebelo et al. [2] proved that there exists an important nonlinear behaviour related to the variation of the natural frequency with the amplitude of vibration. By using the continuous wavelet transform Ülker-Kaustell and Karoumi [14] demonstrated that, for an observed range of acceleration amplitudes, the natural frequency decreases and the damping ratio increases with increasing vibration amplitudes. The same conclusions were drawn by Fink and Mahr [15]. The influence of variations affecting resonance response characteristics in the case of simply supported railway bridge has been also studied by Ülker-Kaustell and Karoumi [6]. These authors have stated that these variations induce decreased critical speeds and resonant amplitudes. Based on experimental tests, Dahlberg [16] showed that the ballast in railway track has a highly nonlinear behaviour. He suggested that ballast foundation can be modelled by adding a cubic nonlinear stiffness term.
The nonlinear behaviour of ballasted track was also investigated by Ansari et al. [17] and Iwnicky [18]. But, this was performed for rails on foundation. The aim of this work is to analyze the influence of the nonlinear behaviour of the ballast on the vertical dynamic response of a railway bridge as function of the loading speed. The modelling is based on a 2D approach. It is assumed that the bridge and the rails behave as a system of two coupled beams which are connected continuously through vertical nonlinear springs and linear dampers which simulate together the ballast effect. The beams are assumed elastic and obeying to the Euler-Bernoulli theory. Torsion and the dynamics along the longitudinal direction are not studied.
The remainder of this paper is organized as follows. The partial differential equations of the two-layer beam system are derived at first. Then, the Galerkin method is employed for spatial discretization of these equations, before applying the Runge-Kutta method to calculate the time history of the dynamic response. A case of study is considered after that to analyse the effect of ballast nonlinearity on bridge dynamics as function of the train speed.  The upper beam represents the rails with the added mass of sleepers, whereas the lower one represents the bridge structure with added mass of the ballast. Both beams are assumed to be parallel and simply supported, with identical length L .

Materials and method 2.1 Modelling the ballasted track bridge
The assumption of the two beams having the same finite length constitutes a simplification with respect to the real rails conditions, as these can be considered to expand infinitely in both sides. In fact, there exists a transition zone at both ends of the bridge where the track is supported differently by soil, abutment and the bridge structure. The representation of boundary conditions according to the finite length assumption may not be always sufficient. However, to capture the real outcome of real boundary conditions in the zone of bridge ends, 3D modeling is needed. This is because complex phenomena take place there such stress concentration, and the Euler-Bernoulli beam theory is not sufficient.
The axial deformations are neglected and no explicit slip in the horizontal direction is considered. The influence of the track irregularities is neglected and the passing trains are introduced by means of moving loads model. These assumptions regarding track-bridge vertical kinematics enable to focus specifically on the interaction in the vertical direction resulting from the nonlinear stiffness of the ballast. Yang et al. [19] have used a synthetic beam model similar to that one given in Figure 1, but they have used linear elements between the track and bridge beams.
In the following, the vertical displacement of the rails beam and the bridge are denoted as ( , ) r w x t and ( , ) s w x t where the subscripts 'r' and 's' refer to the rails and bridge structure, respectively. t is the time and x is the spatial coordinate along the axis of the beams with the origin taken at the left extremity. The continuous layer connecting the two beams is characterized by the linear elastic stiffness l k , nonlinear elastic stiffness nl k and the damping coefficient w c . The resulting vertical force per unit length is taken under the following form: In Eq. (1), the quantity r s w w w = − designates the relative displacement between rails and bridge structure.
The nonlinear constitutive part of the stiffness, namely the cubic term in Eq. (1), is conform to the dependency that was introduced by Dahlberg [ [20], the partial differential equations governing the transverse motion of the doublelayer beam system can be expressed as follows: where i EI and i m are the bending stiffness and the mass per unit length of the ith beam, respectively, with i r = or s for the rails beam or the bridge beam.
According to the recommendations of the EN 1991-2 [1], the damping ratio of the ballasted track and the bridge structure are attributed the same value. The damping depends on the span and the bridge type (filler beam, pre-stressed concrete, reinforced concrete…). The damping coefficients which are denoted r c and s c in Eq. (2) can then be calculated from the assigned damping ratio by using the mass and bending frequencies of the rails beam and the bridge beam. For steel and composite bridges having length exceeding 30m , the Eurocode stipulations recommend: It should be mentioned that additional damping resulting from the longitudinal slip of the ballast should be added to the previous damping. It was demonstrated that this damping is nonlinear and depends on the amplitude of vibration [21]. However, a linear approximation of this damping is made here. Considering the damping value given in [22], the global coefficient of damping is set equal to 0 07 The forcing term v F in Eq. (2) is the railway excitation which is simulated by means of constant moving loads. From literature, consideration of the more complete train-bridge interaction leads in general to a reduction in the vertical acceleration of the bridge at resonance. The dynamic load is assumed to be applied vertically at the centerline of the rails beam. It is taken under the following general form [6,23] where k F is the force exerted by the kth load and acting at the rails beam point: s w x t are identical for the two beams. They are expressed as:

Numerical solution of the nonlinear equations
The system of Eq. (2) is nonlinear. The Galerkin method is employed in order to perform spatial discretization into a system of ordinary differential equations. This consists at first in selecting a set of pertinent trial functions. These can be for example the modal functions of a linear elastic beam satisfying the boundary conditions in Eq. (4). Then, the vertical displacements are expanded as explicit series in terms of these basic functions [24]: In Eqs. (6.1) and (6.2), the quantities 4  . Considering the situation of forced vibrations, the initial conditions are all set to zero such that: The time interval used for the calculation of the system forced response corresponds to the duration of train passage which depends on the actual train speed. Considering similar orders of truncation of the two series = = Nmq Nmu N , Eqs. (6.1) and (6.2) yield the following system of equations: where ( ) Eq. (8) is a nonlinear system of 2N coupled second order ordinary differential equations. Each equation contains nonlinear terms which are due to the cubic ballast stiffness nonlinearity. The restriction of motion to the first bending mode is not generally sufficient. This was recognized by Ding et al. [12] who have analyzed the convergence of the Galerkin method for the vibration of a single finite length Euler-Bernoulli beam resting on a nonlinear foundation and subjected to a moving concentrated force. The authors have found that the convergence of the Galerkin truncation depends on the intervening parameters and large truncation terms may be needed in some cases. Here, the retained number of modes is fixed by reference to the EN 1991-2 [1] stipulating the need to cover the frequency band ranging from 0 to 30Hz , or for the superior limit 1.5 times the frequency of the first mode if this exceeds the bound of 30Hz .
By taking into consideration the initial conditions given in Eq. (7) and for given geometrical characteristics of the double-layer beam system, the generalized coordinates ( ) In the following, the nonlinear model will be compared to the linear model that can be retrieved from Eq. (8) by dropping the nonlinear terms.

Results and discussion
In this section, the Skidträsk bridge [6,14] is modeled as a simply supported two-layer beam. This bridge has a medium length of 36 L m = . The real bridge is a steelconcrete composite structure which presents a small horizontal skew, but skewness was not included in the present modeling. The objective is to assess in this particular case the effect of ballast nonlinear behavior as it can be predicted by Eq. (8). Comparison is performed between the predicted results based on the present proposed model and those of the linear model.
One of the most important issues to deal with in the context of nonlinear model as given by Eq. (8), is how to identify the mechanical characteristics of the ballast elements in terms of the spring stiffness and damping defined in Eq. (1), namely the constants: , , l nl w k k c . In this work, reference is made to the experimental work performed by Dahlberg [16] to fix the parameters of the nonlinear ballast model and to [22] to estimate damping value. Dahlberg has identified the nonlinear forcedisplacement characteristics from measurements performed on a newly built track in Sweden which is supported by a rigid foundation. It was done under the circulation of the Swedish high speed train X2000 with the speed 198 / c km h = . The author has noted that the ballasted track presents highly nonlinear behavior with hardening characteristics. Besides, he pointed out that a linearized track model can only be used for one single axle load.
The track parameters used in the following are recalled in Table 1. The flexural rigidity of the upper beam is due to that of two UIC-60 rails. The mass per unit length is taken as that of the track superstructure which includes the rails and the sleepers [25]. They are assumed to remain constant along the railway track bridge. The Swedish Steel Arrow train [6] will be used as a railway excitation to the coupled track-bridge structure. The train consists of 24 wagons with each of them having a length of 13.9m . Axle loads of the locomotives and wagons are given respectively by 19.5 tons and 22.5 tons . Train speeds investigated in the following are ranging from 50 / km h to 300 km h . The other geometrical characteristics of this train are given in [6].
Considering the linear model of the bridge, the resonance occurs when the train crosses the bridge at critical speeds which are given according to [23,26] where w D is the wagon length, which is equal to 13.9m for the Steel Arrow train and n f is the nth natural frequency of the bridge.
As for the bridge considered here, the first mode frequency is 1 3.855 f Hz = , one can predict the following three critical speeds belonging to the considered interval of speeds:  From Fig. 2, one can verify that the three linear resonant peaks predicted according to the theoretical values given in Eq. (11) appear in the displacement curves. However, other local peaks are present in the bridge acceleration curves of Fig. 3. This occurs for both the linear and nonlinear models with the amplitudes of the new peaks of acceleration not exceeding those associated to the theoretical peaks. Fig. 2 shows that for the bridge displacement, the two models predict the same response. However, for the rails, the nonlinear model predicts reduced level of relative displacement. This can be understood from the fact that the nonlinear model presents a strengthening effect in comparison to the linear model.