A 3D simplified model for non-linear stability analysis of the continuous welded rail track.

Dosa, Adam ; Ungureanu, Valentin-Vasile ; Botis, Marius Florin 等

1. INTRODUCTION

At the present day it is impossible to talk about high-speed railway without taking into account the necessity of joints elimination. Because the rail joints have gaps, impacts occur when a railway wheel encounters these discontinuities. These large impact forces may cause damages to wheel, track and vehicle. A modern solution to solve this problem is to eliminate these joints, i.e. to make continuous welded rail (CWR) track, using the aluminothermic welding method or flash butt welding method (Herman, 2000).

The welding of rail should be realized only into prescribed temperature range for decrease the risk of rail track buckling in a hot summer and the risk of rail breaking in cold winter (Van, 1997). Nevertheless during hot summers several hundreds of track buckling occur worldwide and they cause major damage. The number of CWR buckling and the costs of damages and repairs are increasing each year (Van, 1997).

Because the physical model experiments in CWR track buckling are very expensive, the numerical simulations are more convenient (Ungureanu, 2007; Van, 1997).

In this context, at the Civil Engineering Faculty from the Transilvania University of Brasov, Romania, was developed a software for the track stability analysis using a three dimensional simplified non-linear discrete model with thermal and vehicle loads (Dosa & Ungureanu, 2007; Ungureanu, 2007), model called SCFJ (Stabilitatea Caii Fara Joante = Stability of CWR).

2. THE PARAMETERS OF TRACK AND LOADS

The physical and geometrical parameters of the CWR track introduced in the SCFJ model are (Ungureanu, 2007):

1) The longitudinal resistance of the ballast presents a linear or bi-linear displacement-force curve and it is introduced into the model by spring elements which are describing the material nonlinear behaviour of the ballast and of the fastenings for longitudinal displacements.

2) The transversal resistance of the ballast presents a linear, bi-linear or tri-linear displacement-force curve and it is modelled by spring elements which are describing the material nonlinear behaviour of the ballast and of the fastenings for transversal displacements. In all cases the elasto-plastic model includes softening. This kind of ballast behaviour has been measured for consolidated ballast.

3) The torsional resistance of the fastenings is introduced into the model by linear or tri-linear torsional springs. In the case of loaded rail, the behaviour of the three above mentioned resistances can be corrected by taking into account the vertical force acting on each sleeper.

4) The vertical stiffness of the track is modelled by spring elements with linear behaviour.

5) The rail is modelled by beam elements having the following geometrical and physical characteristics:

* area A of the cross section and second order moment about the vertical and horizontal axes [I.sub.z] and [I.sub.y], respectively.

* Young modulus E and thermal expansion coefficient [alpha] of the material.

6) The misalignments of the rail can be described by two types of curves: a complete or a half cosine wave having the total length [lambda] and the amplitude [delta] (fig. 1).

7) The load is introduced into the model by two pairs of vertical forces corresponding to a vehicle with two bogies. The model allows parameterizations of the centre spacing between the bogies, of the spacing between the axles in a bogie and of the axle loads.

8) The length of the model is an input of the program. At the end of the model special infinite boundary elements are introduced--equivalent with the theoretical infinite rail (Van, 1997). These elements are reducing the length of the model and hence the computational effort is smaller. Further reduction can result by using symmetric half structure.

3. THE NUMERICAL ALGORITHM

The numerical algorithm has two phases, since in a simplified manner, the horizontal and vertical behaviour of the CWR track are considered independently (Ungureanu, 2007).

The computational model for vertical loadings is linear elastic consisting of a beam on elastic springs. The nodes of the structure are considered at the sleepers. Each node has two degrees of freedom: the vertical translation w and the rotation [[theta].sub.y]. The system of equations of equilibrium is:

Ka = F (1)

where K is the stiffness matrix of the structure and results by assembling the stiffness matrices k of the beams and the vertical stiffness of the sleepers; a is the displacement vector of the nodes of the structure; F is the vector of forces at the nodes of the structure, which (in this case) results by assembling the vectors [f.sub.0] of the forces on the beams.

The stiffness matrix [k.sub.(4x4)] of a beam is given by:

k = [B.sup.T][k.sup.d] B (2)

where [B.sub.(2x4)] is a transformation vector, which links the vector of displacements of the beam and the reduced vector of displacements of the beam, and [k.sup.d.sub.(2x2)] is the reduced stiffness matrix of the beam reported only to the reduced set of nodal displacements and forces.

[FIGURE 1 OMITTED]

The stiffness matrices and the load vectors of the beams are assembled by the relations (3):

[K.sub.ind,ind] = [K.sub.ind,ind] + k, [F.sub.ind] = [F.sub.ind] + [f.sub.0] (3)

and the vertical stiffness of the sleepers is assembled with the help of the equation (4):

[K.sub.jnd,jnd] = [K.sub.jnd,jnd] + [R.sub.z]L (4)

The constraints of the structure are introduced by setting to zero the displacements of the supports. The free displacements of the nodes result by solving the system of linear equations:

[a.sub.id] = [([K.sub.id,id]).sup.-1][F.sub.id] (5)

Using the vertical displacements w, the vertical force on each sleeper can be computed and then the transversal, longitudinal and torsional resistances are corrected taking into account the forces Q on each sleeper.

The computational model in the horizontal plane is a straight or curved beam on elastic supports with misalignments (fig. 1). The nodes of the structure are considered at the sleepers. At each node are introduced longitudinal, transversal and rotational spring elements which are modelling the sleepers. The infinite boundary elements at the ends of the model have equivalent characteristics (Young modulus and thermal expansion coefficient) in order to replace the theoretical infinite rail (Van, 1997). The loading of the model is an increase of the temperature in the rail. The characteristics of the beams and of the springs correspond to the two rails of the track panel.

A node has three degrees of freedom: two linear displacements in the horizontal plane, u and v and the rotation [[theta].sub.z] around the vertical axis. In the analysis of the structure the goal is to obtain the displacement-temperature curve. The problem is solved through an updated Lagrangean formulation by a displacement control based incremental process. The behaviour of the system is determined as a sequence of increments of the state parameters (forces and displacements). In the current increment j, characterized by a small control displacement [delta][v.sub.cj], the nonlinear behaviour of the system can be approximated by linear relations:

[a.sub.j+i] = [a.sub.j] + [delta][a.sub.j], [delta][F.sub.j] = [K.sub.j][delta][a.sub.j] (6)

In the above equation [a.sub.j] is the displacement vector in the current configuration, [delta][a.sub.j] is the increment of the displacements, [delta][F.sub.j] is the incremental load vector and [K.sub.j] is the incremental (tangent) stiffness matrix of the structure.

From equations (6), results the incremental scheme:

[delta][a.sub.j] = [([K.sub.j]).sup.-1][delta][F.sub.j], [a.sub.j+1] = [a.sub.j] + [delta][a.sub.j] (7)

For better performances of the incremental algorithm, Heun's midpoint rule has been adopted (Felippa, 2007). The algorithm uses a displacement control technique. The vector [delta][F.sub.j] is not explicitly expressed and the displacement increment [delta][a.sub.j] results from an unknown variation of the temperature, which produces a known increase of the control displacement.

The tangent stiffness matrix [K.sub.j] in the j-th increment depends on the parameters of the system in the current step. The stiffness matrices [k.sub.t(6x6)] of the beam elements are:

[k.sub.t] = EA/L x [r.sup.T]r + [B.sup.T]([k.sup.d] + [k.sup.d.sub.G])B + [N.sub.j] /L x [z.sup.T]z (8)

Matrices [k.sup.d] and [k.sup.d.sub.G] are the material and geometric stiffness matrices, respectively. They are expressed with the reduced set of displacements which produce deformations and they are not containing the rigid body displacements of the beam. This reduced form of the stiffness matrices needs less computational effort and speeds up significantly the computation. Equation (8) introduces the non-linear effect of the axial force [N.sub.j]. The complete tangent stiffness matrix in the updated lagrangean formulation has two more terms corresponding to the variation of the length of the beam in bending and to the effect of the shear force (Dosa & Ungureanu, 2007). Since in the current cases the structure is divided in a sufficient number of beams, the errors are very small, when neglecting these two terms. In a comparative study using the complete tangent stiffness matrix and equation (8) the differences between the resulting limit temperatures were only at the fifth digit (Dosa & Ungureanu, 2007).

4. CONCLUSION

In the free market conditions, but also because of the global heating phenomenon, which generates many problems on the operation and maintenance of the CWR track in a safe and profitable manner, it is very useful to develop one numerical model for the CWR track buckling simulation on purpose to the evaluations of the safety degree against of CWR track buckling (Dosa & Ungureanu, 2007; Ungureanu, 2007).

SCFJ model is different from the others, because it includes a correction of the torsional resistance of the fastenings, which takes into account the vertical loads of the vehicle. The computations are simplified by decoupling the horizontal and the vertical behaviour of the CWR track, and by including the possibility of use of a symmetric half of the structure (Dosa & Ungureanu, 2007). The model was validated by comparative analyses with similar models and with the results found in the literature. The agreement with these results is very good. In addition, the SCFJ program has shorter running times than similar programs (Ungureanu, 2007).

The SCFJ model opens perspectives for future researches, about:

* the development of the SCFJ model to allow quantification of the influence of the different neutral temperatures of rails and the influence of different wear of the rails;

* the possibility to take into consideration different fixing temperatures of the rails along the track;

* the possibility to analyse the CWR track buckling in dynamic conditions.

5. REFERENCES

Dosa, A. & Ungureanu, V.V. (2007). Discrete model for the stability of continuous welded rail, In: Intersections/Intersecfii, Vol.4, No.1, 2007, "Transportation Infrastructure Engineering", pp 25-34, ISSN 1582-3024

Felippa, C. (2007). Purely Incremental Methods: Load Control, Available from: http://www.colorado.edu/engineering/CAS/ courses.d/NFEM.d/NFEM.Ch17.d/NFEM.Ch17.pdf Accessed: 2009-06-13

Herman, A. (2000). Calea fara joante--teorie si aplicatii (Continuous welded rail track--theory and applications), Editura Mirton, ISBN 973-585-164-4, Timisoara, Romania

Ungureanu, V.V. (2007). Cercetari privind simularea pierderii stabilitatii caii fara joante (Researches about simulation of continuous welded rail buckling), Teza de doctorat, Universitatea "Transilvania" din Brasov, Brasov, Romania

Van, M.A. (1997). Stability of Continuous Welded Rail Track, PhD Thesis, Delft University Press, ISBN: 90-407-1485-1, Delft, Netherlands