7/29/2019 co03_2

1/7

Analysis of High-speed Train Related GroundVibrations by a Hybrid Method

TORBJRN EKEVIDNILS-ERIK WIBERG

Paper presented at IABSE Symposium "Structures for high-speed railway transportation", Antwerp,

Belgium, August 27-29, 2003

Department of Structural Engineering and MechanicsComputational MechanicsCHALMERS UNIVERSITY OF TECHNOLOGYGteborg, Sweden, 2003

7/29/2019 co03_2

2/7

Analysis of high-speed train related ground vibrations by a hybrid method

Torbjrn EKEVIDPh. D.Chalmers University ofTechnology, GteborgSWEDEN

Torbjrn Ekevid, born 1973,received his Ph. D. in StructuralMechanics from ChalmersUniversity of Technology.

Nils-Erik WIBERGProfessorChalmers University ofTechnology, GteborgSWEDEN

Nils-Erik Wiberg, born 1938,received his Ph. D. in StructuralMechanics from ChalmersUniversity of Technology.

Summary

On actual railway lines, particular large vibrations amplitudes have been noticed at sites with softclay deposits as high-speed trains have begun to operate on the lines. To understand thephenomenon and to propose countermeasures to limit the effects of the propagating waves, detaileddynamic analyses are necessary. A displacement based finite element method is not very suitable fordynamic analysis of this type of problem since the method not fulfils the radiation condition. Wepropose a hybrid approach combining conventional finite element techniques and the ScaledBoundary Finite Element Method (SBFEM). A number of advantages can be identified; detailedmodelling of track components including nonlinear effects is possible; generated waves aretransmitted through the boundary and further the size of the model can substantially be reduced.

Keywords: Wave propagation, High-speed trains, Railway mechanics, Hybrid method.

1. IntroductionIn Sweden, the research dealing with wave propagation phenomena related to high-speed trains wasinitiated by field observations of extreme vibration amplitudes at Ledsgrd[1] south of Gteborg.The phenomenon of high-speed train vibrations has later also been observed at other locations inSweden but also at locations in Holland and Germany where the high-speed trains operate at muchhigher velocities. However, compared to the vibrations observed in Ledsgrd, the amplitudes atthose locations are comparably small.

Vibrations and propagating waves are generated by railway traffic even at low train speeds sinceimperfections at the train wheel surface and at the steel rail cause variations in the contact forcesbetween the rail and the wheel, which in addition to the quasi-static deflection of the ground

induces vibrations (propagating waves) originating at the contact point. These high-frequent wavesbut also the transmitted sound are normally what people recognise as disturbances from railwaytraffic. However, as the speed of the train reaches or exceeds the critical wave velocity of thecompound track-ground structure, the response dramatically changes. Although no imperfectionsare present the moving loads generate propagating shock waves. Theoretically, if no damping waspresent in the ground and the materials were perfectly homogenous, there exist points wheredisplacements and stresses become infinitely large. However, since such conditions do not exist inreality, the displacement and stress amplitudes are limited but high amplitudes could be expected incomparison to normal conditions at subsonic speed. Solutions of an idealised model consisting of amoving point load on elastic half-space have been studied by example Frba[2]. In Ekevid[3] a moredetailed problem description is given.

2. The hybrid methodThe coupling between the conventional finite element domain and the unbounded domain

7/29/2019 co03_2

3/7

approximated by SBFEM [4]-[5] are based on dynamic equilibrium. In the program used for thesimulations, the approach based on Lagrange multipliers for handling the constraint equations hasbeen adopted. The approach gives a very clear and consistent implementation of both linear andnon-linear constraint equations. However, compared to the conventional approach where theconstraint equations are condensed in advance, the major drawback is the enlarged system of

equations. The dynamic equilibrium for the FE-domain could be expressed as:( )ext a u i + + + =F F F F R 0 (1)

= 0 (2)where extF are the external forces and , ,a u F F F are forces from inertia, internal and constraintequations defined as

:

t

Ta

u ext

d

d d d

= =

+= =

F u u F

u

F F u b u t

&&

% %

%

% % % %

(3)

Further, iR are interaction forces from the unbounded domain at the interface region, see Fig. 1,obtained from

( ) ( )i

t

i

i

0

t d d

= R M u&&%

(4)

wherei

is the surface representing the unbounded domain and M the unit impulse responsematrix computed by the SBFEM. Moreover, represents a set of algebraic equations either fromconstraint equations or boundary conditions.

x

zy

Scaled Boundary FEMFEMi

R

Fig. 1. Interaction forces in the interfaceregion from the unbounded domain.

At the present stage, the well-established Newmark scheme has been used for the time steppingprocedure of the governing ordinary differential equations (1) subjected to the constraint equations(2). By introducing approximations and corresponding test functions, linearisation of (1) and (2)with respect to the primal unknowns

n 1+a%

andn 1+

%, corrections ia

%, i

%are computed from

( ) ( ) ( )( )

* ii T ida

i id

=

RaM

R 0

%

%

(5)

Here, d is the Jacobian of the constraint equations (2) and ( )*i

M is the equivalent mass matrixdefined as

7/29/2019 co03_2

4/7

( ) ( )*2

i itn

n 1

tt

2

= + +M M M K (6)

where M and ( )i

tK are conventional mass and tangent stiffness matrices and

1

M is the first

discretizised unit impulse response matrix. The right-hand side of (5) represents the residual forces

and violation of the constraint equations according to.

( ) ( )ii ext a u

a n 1 n 1 n 1 n 1

+ + + += + +R F F F F (7)

( )i

i n 1

2

n

2

t

+=

R (8)

As the corrections ia%

and i%

have been computed accelerations, velocities, displacements andLagrange multipliers could be are updated from

;

;

i 1 i i i 1 i i

n 1 n 1 n 1 n 12

i 1 i i i 1 i inn 1 n 1 n n 1 n 1

tt 2

+ ++ + + +

+ ++ + + +

= + = +

= + = +

a a a

v v a u u a

% % % % % %

% % % % % %

(9)

The equations (5) are for intermediate problem sizes solved by a direct solver. For larger problemsan iterative solver is necessary and in Ekevid[6] an efficient iterative solver based on a specialpreconditioner of the indefinite system (5) and multigrid technique is proposed.

3. Numerical simulations of high-speed train induced vibrations

To study the propagating waves generated by high-speed trains, a model recovering the conditionsat Ledsgrd has been used. The result also demonstrates the performance of the proposed hybridmethod.

3.1 The computational modelThe model of the railway section (total length 80 m), see Fig. 2, consists of rail, sleeper, ballast,subballast, dry crust and the in-situ clay material.

X

Y

Z

B

A

C

Fig. 2. Model used for simulation of vibrations

caused byhigh-speed railway traffic.

Clay3

Clay2

Clay1

Dry crust

Clay2

Clay1

Dry crustB2B1

Superelement

Fig. 3. Material layer definition.

Foundations and cantilever beams for the electrical power supply system have also been included tothe model. In order to reduce the size of the model, a symmetry plane is assumed along the tracksection such that just one half has to be considered, see Fig. 2. The material in the model is assumed

7/29/2019 co03_2

5/7

to be elastic isotropic and the parameters have either been estimated from field measurementsconducted by J&W[1], from laboratory tests made by NGI[1] or taken from Hall[7]. More advancedconstitutive laws could have been used for the track and its vicinity since finite elements are used tomodel this part. However, the usage of advanced material models becomes meaningless if thematerial models and the corresponding parameters are not estimated from material tests and for the

moment, no such tests have been conducted and evaluated for the Ledsgrd case. Fig. 3 shows thematerial layers in the subground and the embankment.

F1F3

F2F3F3F3F3F3 F4

F5

15 187 187 187 107

3

(m)

F = 181 kN1 F3 = 122 kN

F2 = 180 kN F4 = 117 kN

F5 = 160 kN

Fig. 4. Amplitudes and distances between

axle loads of a X2000 train.

To reduce the model to a reasonable size and avoid reflections at the boundary, SBFEM are usedalong some of the boundaries. The SBFEM discretization is in the example restricted to 191 nodesand 159 elements, see Fig. 5. Fig. 3shows the geometrical extension of the material layers for scaledboundary discretization due to the position of the similarity centre. For further informationregarding the definition of similarity centre, see Ekevid[8].

Fig. 5. Scaled boundary finite element

discretization.

Fig. 6. Conventional finite element

discretization.

The SBFEM discretization is treated as a superelement and can be reused repeatedly to reduce thecomputational effort when examining the effect of different train speeds or modifications of thetrack structure. The analysis procedure for constructing the SBFEM superelement requires a fixedtime step. As a consequence, adaptation of the time step size is not possible using the hybridmethod.

3.2 Result obtained in the analysis

The FE-discretization shown in Fig. 6 consists of 14584 nodes and 9654 elements; mostly 8-nodebrick elements but also some beam elements. To couple the rail to the sleepers, the sleepers to theballast layer, the cantilever beams to the foundations etc., the facility to automatically generate

7/29/2019 co03_2

6/7

constraint equations in the FEM90 program by giving candidates for the coupling in node sets hasbeen used. The model involves approximately 50000 unknowns and the model is analysed for 3.5 swhere the time step size is kept fixed equal to 0.004 s. A direct solver has been used to solve thesystem of equations. Two speeds, 50 and 70 m/s, of the train are considered in the analysis.

Fig. 7 Deformation of the railway section at

train speed 50 m/s.

Fig. 8 Deformation of the railway section at

train speed 70 m/s.

For the sub-critical speed (50 m/s) the deformation patterns are concentrated to the bogies and theresponse of the model could be said to be more or less quasi static. For the trans-critical speed (70m/s), the response changes and waves originating at the position of the bogies, forming a V-shapebehind the position of the bogie are propagating in the ground material. In Fig. 7 and Fig. 8,snapshots of the deformation pattern (deformations are magnified 200 times) for each speed of thetrain are available. The figures show that the responses at the two speeds are substantially different.In order to make the response easier to interpret, the model and result are mirrored in the symmetryplane and the positions of the bogies are visualized by means of arrows.

Time [s]

Disp

lacement

[mm

]

2

0

-2

-8

-6

-4

-10

-120 0.5 3.02.51.5 2.01.0 3.5

Fig. 9 Time history of the deflection of the

railhead at train speed 50 m/s.

Dis

pla

cemen

t[m

m]

Time [s]0 0.5 3.02.51.5 2.01.0 3.5

2

0

-2

-8

-6

-4

-10

-12

Fig. 10 Time history of the deflection of the

railhead at train speed 70 m/s.

Fig. 9a and b show the time-history response of the railhead deflection in point C, see Fig. 2, at thedifferent train speeds. The vibration amplitudes (difference between the maximum and theminimum value during the train passage) are approximately twice as large for the transcritical speed

(70 m/s) compared to the subcritical speed (50 m/s). The difference in vibration amplitude seems tobe even more pronounced if a comparison between the results in Fig. 7 and Fig. 8 is made.

7/29/2019 co03_2

7/7

However, it is likely the principal change of the response that makes propagating waves present atthe transcritical speed contributes to this impression. From Fig. 9 and Fig. 10 it is also possible tonotice the static deflection of the model, which indicates that the SBFEM works properly.

4. Discussion, Conclusions and Acknowledgements

Detailed modelling of railroad components and methods taking the unbounded media into accountare fundamental for accurate and reliable simulations of high-speed train induced vibrations. Asexpected, the response close to critical speed is much higher than for subcritical speed. It shouldalso be noted that if the unsprung masses were taken into account, the response would increase tolarge extent[9].

The SBFEM in combination to conventional FEM results in an approach fulfilling theserequirements. In comparison to BEM, the SBFEM does not require fundamental solutions and sincethe method is displacement-based, coupling to finite elements is straightforward. Since the method,like BEM, is global in both space and time, convolutions and dense matrices are features includedin the solutions procedure. This makes the method expensive from a computational point of viewand limits the applicability. On the other hand the method can handle unbounded domains, which is

necessary when using FEM with artificial boundaries.Financially the work has been supported by Banverket (Swedish National Rail Administration andhas been part of the work within CHARMEC; the national centre of excellence in railwaymechanics research.

5. References

[1] BANVERKET, High speed lines on soft ground: evaluation and analyses of measurementsfrom the West Coast line, Swedish National Rail Administration, Borlnge, 1999.

[2] FRBA L., Vibration of solids and structures under moving loads, Thomas Telford, London1999, 494 pp.

[3] EKEVID T., On computational wave propagation in solids with emphasis on high-speedtrain related ground vibrations, Lic. thesis, Department of Structural Mechanics, ChalmersUniversity of Technology, Gteborg, 2000, 99 pp.

[4] WOLF J.P., The scaled Boundary Finite Element Method, Wiley, Chichester 2003, 378 pp.

[5] SONG C. and WOLF J.P., The scaled boundary finite-element method-alias the infinitesimalfinite-element cell method- for elastodynamics, Comput. Methods Appl. Mech. Engrg., Vol.147, 1997, pp 329-355.

[6] EKEVID T., Computational Solid Wave Propagation Numerical Techniques and IndustrialApplications, Ph. D. thesis, Department of Structural Mechanics, Chalmers University ofTechnology, Gteborg, 2002, 170 pp.

[7] HALL L., Simulations and analyses of train-induced ground vibrations A comparative studyof two- and three dimensional calculations with actual measurements, Ph. D. thesis, Divisionof Soil and Rock Mechanics, Royal Institute of Technology, Stockholm (2000).

[8] EKEVID T. and WIBERG N-E., Wave Propagation Related to High-Speed Train - a ScaledBoundary FE-approach for Unbounded Domains, Comput. Methods Appl. Mech. Engrg, Vol.191, No. 36, pp 3947-3964, 2002.

[9] M.X.D. LI, EKEVID T. and WIBERG N-E., An integrated vehicle-track-ground model forinvestigating the wheel/rail dynamic forces due to high axle loads, To be presented atCM2003, Gothenburg, Sweden, June 10-13, 6pp.