Two- and Three-dimensional Modelling of Half-Space and Train Track Embankment under Dynamic Loading

UNCORRECTED PROOF

Two- and three-dimensional modelling of half-space and train-trackembankment under dynamic loading

M. Adama,*, G. P¯anzb, G. Schmidc

aDepartment of Civil Engineering, Faculty of Engineering (Shoubra), Zagazig University, El-Hosamy Street, Kaha, El-Kalubia, EgyptbComputational Structural Dynamics, Ruhr University, Bochum, Germany

cDepartment of Civil Engineering, Ruhr University, Bochum, Germany

Abstract

Two modelling approaches for the analyses of half-space and train-track embankment on half-space subjected to dynamic loads are

presented and compared. A three-dimensional (3D) modelling approach is performed by a coupled Boundary Element±Boundary Element

method (BE±BE) and a two-dimensional (2D) one by a coupled Boundary Element±Finite Element method (BE±FE). Both approaches

employ time domain algorithms. The comparison between the results of the presented approaches points out whether a problem can be

treated as a 2D or as a 3D case. As an application, a parametric study of the wave propagation problem in a train-track embankment with an

underlying half-space is presented. q 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Two- and three-dimensional responses; Half-space; Train-track embankment; Wave propagation; BE±BE coupling; BE±FE coupling

1. Introduction

Ground vibrations caused by railway traf®c have

increased considerably during the last three decades leading

to several environmental consequences. Heavy-axle trains

that are running at low speeds on ground surface lines give

rise to surface wave propagation to nearby buildings. The

main component of such vibrations lies in the frequency

range of about 4±50 Hz [1,2]. The waves propagating at

the low end of this frequency range may excite the natural

vertical modes of vibration of the nearby buildings. Vibra-

tions in the frequency range from about 50 to 200 Hz, which

are mainly associated with trains running in tunnels, excite

the bending resonance of walls and ¯oors in buildings which

then radiate sound directly into their interior. Although the

high levels of vibration are associated with heavy-axle

freight traf®c, passenger traf®c and light rail systems can

also cause such type of vibrations and unacceptable levels of

rumbling noise.

Recently, with the expansion of high-speed train

networks, concern has been expressed about the effects of

moving loads on the track, the embankment and the nearby

structures. In some locations with soft ground conditions,

very high levels of displacement have been observed [3]. In

addition, some theoretical work has pointed out a substantial

increase of the vibration level at the line side due to trains

passing with speed, which exceeds the Rayleigh wave speed

of the ground [4]. It is thus important to study these phenom-

ena in order to present engineering solutions for reducing

their impact on the built environment. Since there are a

number of effects of vibration covering surface propagation

in different frequency ranges, it is not surprising that a range

of different modelling approaches are used.

In order to predict the response at a line-side location or at

the foundation of a building, a model for the wave propaga-

tion through the ground must be implemented. Due to inher-

ent limitations, standard numerical techniques for dynamic

analysis are not well suited to consider essential response

controlling factors adequately. For instance, the FE methods

are ideal for problems with a complex geometry and inho-

mogeneous and/or nonlinear materials in ®nite domains. But

when applied to in®nite domains, arti®cial boundaries have

to be used that cause wave re¯ection leading to loss of

accuracy. Many researchers [5±9] suggested some formula-

tions and special types of in®nite elements to handle such

problems. Boundary Element methods are superior in deal-

ing with in®nite domains [10] but they are most ef®cient for

linear problems, without signi®cant enlargement in the

computational effort using sub-domain techniques [11]. To

use the advantages and avoid the limitations of the

two methods, Zienkiewicz et al. [12] proposed a hybrid

Soil Dynamics and Earthquake Engineering 00 (2000) 000±000

SDEE2487

0267-7261/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved.

PII: S0267-7261(00)00068-3

www.elsevier.com/locate/soildyn

* Corresponding author. Fax: 120-13-601-510.

E-mail addresses: [email protected] (M. Adam), p¯[email protected]

bi.ruhr-uni-bochum.de (G. P¯anz), [email protected]

(G. Schmid).

Soil Dynamics and Earthquake Engineering ± Model 5 ± Ref style 3 ± AUTOPAGINATION 2 01-09-2000 16:45 SD0761 ks

Alden

UNCORRECTED PROOF

BE±FE method for complex geometry, heterogeneous and

nonlinear problems. Then, the coupled BE±FE method was

used for transient response investigation of ¯exible two-

dimensional (2D) strip foundations, layered half-space and

2D trenches under external dynamic loads and seismic wave

excitations [13±15]. Different 2D models are used to study

the propagation in layered soil and vibration due to train

passage in tunnels [16±18]. The dynamic responses of

three-dimensional (3D) surface foundations and the 3D

soil railway-track interaction problem are investigated by

a coupled BE±FE approach [19,20]. The ®rst application

of the coupled technique to inelastic problems was

performed by Pavlatos and Beskos [21] and recently, Abou-

seeda and Dakoulas [22] applied it to the nonlinear dynamic

earth dam±foundation interaction.

To investigate the wave propagation phenomena due to a

high speed moving load, 3D models have to be used.

Yoshioka et al. [23]present the in¯uence of masses and

axle arrangement on vibration amplitudes for different

train velocities. The peak displacement amplitudes of a

half-space surface arising from a passing rectangular load

are given in Ref. [24]as a function of load velocity. Jones et

al. [25]and Lefeuve-Mesgouez [26] present extensive

studies of moving oscillating loads on layered ground and

Hanazato et al. [27]investigate the effect of pavement

structures on the reduction of ground vibration. Chang et

al. [28]made a summary of former models for analysis

of the soil±train-track embankment system, called

ªGEOTRACKº and a recent wider survey on the modelling

of ground-borne vibration from railways is presented by

Petyt and Jones[2].

In this paper, two time domain modelling approaches to

investigate the dynamic response and the wave propagation

in the train-track embankment and the underlying half-space

are demonstrated and compared. A 3D model by means of a

coupled Boundary Element±Boundary Element (BE±BE)

method and a 2D one that employs the coupled Boundary

Element±Finite Element (BE±FE) method are presented.

The radiation conditions are ful®lled in both approaches

and no absorbing boundary conditions have to be used.

First, the validity of each algorithm is demonstrated by

comparison with results from literature. Second, in order

to determine the applicability and the limitation of 2D solu-

tions, the responses of a half-space due to vertical and hori-

zontal impulse loads are obtained by both methods and

compared. Last, as a main application, a parametric study

of the wave propagation through a train-track embankment

and along the surface of the underlying half-space is

performed using both approaches.

2. 2 Analytical and numerical approaches

2.1. Boundary integral formulation

The differential equation of LameÂ-Navier describes the

displacement ®eld of a linear-elastic continuum that is

subjected to dynamic loads. Under the assumption of homo-

geneous initial conditions and vanishing body forces this

differential equation can be transformed to the following

boundary integral equation [29]

cikuk�j; t� �ZGx

Zt

0up

ik�x; t; j; t� ti�x; t�dt dGx

2ZGx

Zt

0tpik�x; t; j; t� ui�x; t�dt dGx; �1�

where tpik and up

ik are the fundamental solutions (see Ref.

[30]) for traction and displacement at the ®eld point x at

time t, caused by a Dirac-load acting at the boundary point jat time t . The fundamental solutions depend on the material

parameters of the continuum, i.e. the velocities of the long-

itudinal and the transverse waves, cp and cs, respectively and

the mass density r .

The ui and the ti represent the boundary values for the

displacements and the traction, respectively. The integration

is carried out over the boundary G with respect to x as

indicated using Gx. The matrix cik includes the integral-

free terms, which depend on the geometry in the vicinity

of the source point j.

Generally, for each point on the boundary G either the

displacements or the traction are known, and Eq. (1) is used

to evaluate the unknown boundary values.

2.2. 3D BEM

For numerical solution the boundary integral equation Eq.

(1) is discretized in time and space and solved. The time

domain is subdivided into equal intervals where a constant,

linear or quadratic time interpolation function is used to

represent the continuous time history of traction and displa-

cements. Then, the time integration of Eq. (1) for the bound-

ary values can be carried out analytically (see Ref. [31])

leading to functions that depend on space variables only.

These functions can not be integrated analytically for an

arbitrary boundary geometry. Therefore, the boundary G is

divided into constant or quadratic isoparametric quadrilat-

eral boundary elements with one or eight nodes per element,

respectively. After discretization, Eq. (1) can be written in

the following form:

cikui�j; tN�1XLl�1

XNm�1

ZG l

T �N2m11�ik �x; j�dG lx

�l�u�m�i

�XLl�1

XNm�1

ZG l

U�N2m11�ik �x; j�dGlx

�l�t�m�i ; �2�

where Tik and Uik are the traction and displacement kernels,

respectively, resulting from the fundamental solution's

temporal integration. The outer summation in Eq. (2) is

carried out over the total number of elements L, and the

inner one is carried out over the number of time steps N.

Gaussian quadrature is used to evaluate the integrals over

M. Adam et al. / Soil Dynamics and Earthquake Engineering 00 (2000) 000±0002


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UNCORRECTED PROOFthe boundary Gl The quadrature scheme has to be modi®ed

for N� mto account for the singularities of T 1ik and U1

ik when

the distance r between the source point j and the ®eld point

x approaches zero [32]. In this case, the traction kernel has a

strong singularity �O�1=r 2�� and the corresponding integral

exists in a principal value sense only. The displacement

kernel has a weak singularity �O�1=r��; therefore its integral

is regular. However, particular techniques are required in

both cases to obtain a higher accuracy with a lower number

of Gaussian points. In this work, a transformation to trian-

gular polar coordinates approach is used [33]. The quadratic

elements are divided into triangles and subsequently each

triangle is mapped onto a quadratic reference element in

such a way that the singular point of the triangle becomes

one side of the reference element, thus reducing the order of

the singularity by one, which is suf®cient for the displace-

ment kernels. The strongly singular traction kernels are trea-

ted with an additional Taylor-series expansion in

combination with a subtraction of the singular terms [34±

37]. This expansion and subtraction allows for the analytical

integration of the strong singularity, while the regular and

weakly singular parts are integrated numerically.

After integration, Eq. (2) can be written in matrix notation

as:

U1tN � T1uN 1 EN; EN �

XNm�2

TmuN2m11 2 UmtN2m11

�3�where Um and Tm are the coef®cient matrices of the system

at time mDt. For the current time step N, all traction vectors

tm, m� 1 to N, and previous displacement vectors um, m� 1

to N 2 1, are known.

2.2.1. BEM±BEM coupling in time domain

In order to consider problems containing several domains

with different material properties, a substructure method is

applied. The interfaces of two adjacent regions are coupled

under the condition that equilibrium and compatibility are

maintained. In general, two possibilities exist to perform the

BEM±BEM coupling in time domain, the ®rst approach

using elements of the same size with different time steps

for both domains. The second approach uses identical time

step for both domains but different element sizes [38]. In

this work the second possibility is employed. To perform the

coupling, traction conditions were formulated in an integral

sense, i.e. the sum of the integral of the surface traction over

the two common surfaces of each element pair to be coupled

has to vanish. For the displacements linear interpolation was

used to formulate the coupling conditions, which allows to

couple domains where nodes of both regions do not have to

coincide at the interface. This also allows to couple domains

with different shape interpolation functions.

2.2.2. Numerical improvements

An algorithm to save computational time and memory for

3D BE calculations was developed and implemented. It is

based on the translation property of the fundamental solu-

tions and can be used for all 3D problems whose geometry

has an axis such that all cross-sections perpendicular to this

axis are identical (see Fig. 1 for an example). A geometry

like this can be assumed for the analysis of many problems

related to traf®c loads, like vibration caused by trains, planes

on the runway or trucks on a road. The algorithm can be

applied for both moving and stationary loads. A uniform

element mesh is required for the algorithm.

The algorithm represents an extension of the algorithm

developed by Neidhart [38] for a plane geometry. As the

in¯uence of a given source point on a given ®eld element

depends on their relative position and their orientation with

respect to the global coordinate system only, the entry in the

in¯uence matrices for source point c1 and ®eld element F1

will be identical to the entry for source point c2 and element

F2 (see Fig. 1). Therefore, only the source points located

within the two rows on either end of the geometry (see

grey-shaded rows in Fig. 1) have to be considered to obtain

all matrix entries. For all other combinations of source

points and ®eld elements the result of the integration are

identical to the result of the source point and the ®eld

element obtained by a translation of the source point parallel

to the axis of the geometry to one of the two reference rows

M. Adam et al. / Soil Dynamics and Earthquake Engineering 00 (2000) 000±000 3

Fig. 1. Reduction of computational time and memory in 3D-BEM based on

the translation property of the fundamental solution.

Fig. 2. 2D BE±FE domains.


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UNCORRECTED PROOF

at the end of the geometry and a translation of the ®eld

element with the same translation vector.

Using this scheme has the following advantages: The

in¯uence matrices can be stored as rectangular matrices

where the number of rows corresponds to the number of

source points in the two reference rows multiplied by the

degrees of freedom of each node. The number of

columns corresponds to the total number of degrees of

freedom of the discretized domain. Therefore, the

computational effort increases only linearly with the

length of the cylinder. Furthermore, for a constant

number of source points in the reference rows the larger

the problem the higher the advantage of the proposed

algorithm compared to the standard one, which calcu-

lates and stores all entries. The algorithm works for

constant space interpolation as well as for elements

with order space interpolation functions.

Considering the example of a train-track embankment

discretized with 50 (space constant) elements in the direc-

tion of the axis of the geometry and ®ve elements in each

row perpendicular to the axis of the geometry, using the

standard scheme each in¯uence matrix consists of 5 £ 10 £3 rows and columns, i.e. each matrix has 22 500 entries.

Using the proposed scheme, only 5 £ 2 £ 3 rows of the

matrix have to be calculated and saved, whereas the number

of columns is equal to the number of columns in the stan-

dard scheme. This results in 450 entries per matrix, only 2%

of the number of entries for the standard scheme. As a result

the computational time to assemble the matrices for each

time step and the memory required to save each matrix is

reduced by a factor of 50 for this example. If a sparse matrix

scheme is employed together with the standard scheme, the

number of 22 500 can be reduced, since the zero entries are

not stored. However, the memory savings using a sparse

matrix scheme in time domain BEM are small compared

to the proposed method. For the time required to assemble

the in¯uence matrices no savings can be obtained compared

to the standard scheme.

Together with the substructure method the proposed algo-

rithm can reduce the computational cost considerably. If the


Fig. 3. Surface discretization for 3D BEM model.

Fig. 4. Vertical displacements due to vertical load (a) presented 3D BE method's results (b) other numerical and analytical results (after Neidhart [38] and

Pekeris[39]) (P, S and R denote the arrival time of the corresponding wave).


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UNCORRECTED PROOF

geometry of the problem consists of several domains (e.g.

train-track embankment, half-space and foundation), the

proposed algorithm can be used for those domains where

the geometry requirements Ð as stated above Ð hold (e.g.

train-track embankment and half-space), while the other

domains (e.g. foundation) have to be treated with the stan-

dard scheme.

2.3. 2D coupled BE±FE method

In the following, a general 2D BE±FE coupling formula-

tion is presented where the problem in hand is divided into

two parts as shown in Fig. 2. The uniform half-space domain

C reduces to a plane region and is modelled by boundary

elements. The non-homogeneous part V, that can be an

embankment, a dam or a structure will be modelled by ®nite

elements. The compatibility and equilibrium conditions

have to be satis®ed along the common interface line

between the two parts Gi. It should be noted that the integral

representation in Eq. (1) holds on the same form for both 3D

and 2D problems but reduces to a line integral and the

subscripts i, k take the values one and two only, for the

2D case.

The governing equation of motion of the ®nite element

domain due to a time dependent applied load P(t) can be

expressed and partitioned for the current time step N as:

Moo Moi

Mio Mii

" #�uo

�ui

( )N

1Coo Coi

Cio Cii

" #_uo

_ui

( )N

1Koo Koi

Kio Kii

" #

�uo

ui

( )N

�Po

Pi

( )N

(4)

where M, C and K are the mass, damping and stiffness

matrices, respectively. The vectors contain the nodal accel-

erations, velocities and displacements, respectively. The

variables on the interface Gi are denoted by the subscript i

and the other variables are denoted by the subscript o. Simi-

larly, using the subscript i for the interface nodes and s for

the surface nodes of the boundary element domain, Eq. (3)

can be written as:

U 1ss U1

si

U1is U1

ii

" #ts

ti

( )N

� T1ss T1

si

T1is T1

ii

" #us

ui

( )N

1Es

Ei

( )N

: �5�

Employing the traction-free condition on the surface, and

solving for the interface node variables, the nodal traction

along the interface can be expressed as:

{ti}N � �Kb�{ui}

N 2 {Ebu}N; �6�

where [Kb] and {Ebu} are the results from the necessary


Fig. 5. Horizontal displacements in the same direction of an applied horizontal load (a) presented 3D BE method's results (b) other numerical and analytical

results (after Neidhart [38] and Triantafyllidis[40]) (P, S and R denote the arrival time of the corresponding wave).

Fig. 6. Half-space under surface vertical strip load impulse.


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UNCORRECTED PROOF


Fig. 7. Vertical responses calculated at points A, B, and C (a) presented methods' results (3D BE and 2D BE±FE) (b) others' results (after Abouseeda et al.

[47]).

Fig. 8. Surface discretizations of half-space for comparison with 2D (a) Discretizations for different ratios of L/B (b) Section A±A.


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UNCORRECTED PROOF

matrix and vector operations. The principle of virtual work

is used employing interpolation functions over each bound-

ary element to transform the traction in Eq. (6) into the

nodal force vector {Pbb}:

{Pbb}N � �Kbb�{ui}N 2 {Fb}N

; �7�

where [Kbb] and {Fb} represent [Kb] and {Ebu} after trans-

formation, respectively. Then, the principle of weighted

residual is applied along the interface to obtain the coupled

equation of motion as:

Koo Koi

Kio Kii 1 Kbb

" #p uo

ui

( )N

�Po

Pbb

( )Np

�8�

where the superscript p denotes the resulting coupled terms

including the contributions of the boundary element domain

as well as the mass and damping matrices of the ®nite

element domain.

3. Numerical examples and discussions

3.1. Validations

The 3D response of an elastic half-space to a Heaviside

type point load is obtained by a 3D Boundary Element

Method and compared with other numerical and analytical

solutions. Coupling is not required here, since the half-space

is treated as one domain. The discretized model is shown in

Fig. 3 where the black square indicates the location where

the load is applied at t� 0 and the black circle indicates the

location where the displacements are evaluated. The results

of the present method are shown in Figs. 4(a) and 5(a) while

Figs. 4(b) and 5(b) display the numerical results of Neidhart

[38] as well as the analytical solutions of Pekeris [39] and

Triantafyllidis [40]. The horizontal axis displays dimension-

less time, where r is the distance between the loaded point

and the observation point. P, S and R indicate the arrival of

the longitudinal, the transverse and the Rayleigh

waves, respectively. The vertical axis displays dimension-

less displacements where Ai is the force applied in the


Fig. 9. Responses to vertical load at a distance of 1 m from the line load (a) vertical (b) horizontal.



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UNCORRECTED PROOF

i-direction. The parameters uzz and uyy denote the value of

the static solution of the corresponding Boussinesq-

problem, which is given by

uzz � uzGpr

Az

� v 2 1

2� 20:375 �9�

for the vertical response to a vertical loading and by

uyy �uyGpr

Ay

� 1 2 v

2� 0:375 �10�

for the horizontal response to a horizontal loading.

Different meshes A, B, C and D are used to investigate the

in¯uence of discretization, where mesh D represents the

largest mesh and mesh A is the smallest one as shown in

Fig. 3. The overall behavior of the presented results is in

very good agreement with the other solutions as shown in

Fig. 4 for vertical response due to vertical load and Fig. 5 for

horizontal response due to horizontal load. As a result of the

®nite size of the boundary elements and the approximation

of a point load to a surface area load, the numerical solution

can only approximate the abrupt change at the singularity at

the arrival of the Rayleigh-wave front. After the Rayleigh

wave has passed, the numerical and the analytical solutions

converge to the corresponding static solution given in Eqs.

(9) and (10), respectively. To achieve a better agreement

with the analytical solution a ®ner mesh with more elements

and smaller time steps would be required. However, since

this section is only for validation no convergence studies are

conducted at this point.

The two presented methods (3D BEM and 2D BE±FE

coupling) are now applied to obtain the response of an elas-

tic half-space under discontinuous boundary stress distribu-

tion as shown in Fig. 6. This test example was used for

comparison by many authors using different methods

[22,41±43]. A vertical impulse traction q � 68:95MPa is

applied on a strip of width b � 152:4 m: The elastic half-

space has a Young's modulus E � 17240 MPa; mass

density r � 3150 kg=m3 and a Poisson's ratio n � 0.25.

For the analysis using the 3D Boundary Element approach,

a length of about 3000 m is discretized in the x-direction and

a width of about 1000 m in the y-direction.



Fig. 12. Responses to horizontal load at a distance of 1 m from the line load (a) vertical (b) horizontal.


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UNCORRECTED PROOF

The vertical response is calculated at points A, B, and C

as shown in Fig. 6. The results of the present methods are

given in Fig. 7(a), while the results of the other authors are

scanned from Ref. [22]and displayed in Fig. 7(b). We

assumed the positive direction downward opposite to

other results, which makes no difference on the results. It

can be seen from Fig. 7(a) that for the 3D computation

results are obtained that are almost identical to those of

the 2D computation. The results demonstrate clearly the

good agreement of the presented methods with the

published solutions at all three points.

3.2. 2D and 3D responses of half-space

Plane-strain conditions and 2D solutions are assumed for

problems that have in®nite length in the third direction. In

order to ®nd whether 2D analysis of problems with ®nite

length in the third direction yields a good approximation to

the 3D solution, a problem of half-space under a line load

(see Fig. 8) is investigated using both presented approaches.

The amplitude of the line load is 1 kN/m for the time t

between 0 and 0.03 s and zero otherwise.

In the 3D model, a certain width B is discretized in

the y-direction and different stretches of length L are

discretized in the x-direction as shown in Fig. 8(a).

Unless stated otherwise, the length of the line load is

assumed to be equal to the discretized length. In the 2D

modelling, the same width B is discretized in the y-

direction. The line load is applied in the x-direction in

the middle of the discretized width B.

The case of vertical load is analyzed and the response at

three points of evaluation (see Fig. 8(a)) with different

distances from the line load along the width B are obtained.

The vertical and horizontal displacements at distances of 1,

3 and 5 m are shown in Figs. 9±11, respectively. The solid

lines display the 2D results, the dashed and the dotted lines

give the 3D ones. The comparison is performed in terms of

the L/B ratio. It can be seen that as the ratio L/B increases,

the 3D solutions approach the 2D ones. A very good match-

ing is obtained for L=B � 2:7 and higher for the vertical

response and a good matching for the maximum value is

achieved when L=B � 1:4: Concerning the horizontal

response, the best matching is obtained when L=B � 2:7

and no better matching can be obtained even for a very





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UNCORRECTED PROOF

high ratio of L/B. The same behavior is concluded at other

distances, which are not shown here.

For applied horizontal load, the responses at distances of

1, 3 and 5 m are shown in Figs. 12±14 respectively. In these

cases the matching between results of the two methods is

obtained for a higher L/B ratio than in the previous case,

especially for the vertical responses. This may be due to the

wave scattering by the truncated surface edges, which is

stronger in the case of horizontal load.

In addition another 3D calculation is performed, where

the discretized length in the x-direction is larger than the line

load length, which is denoted by L � 5:4B�p� in Fig. 8(a).

The results of this case are compared with the results of the

3D calculation of the similar case with shorter discretization

in the x-direction and shown in Fig. 15. The obtained iden-

tical results re¯ect that further increasing the discretized

area in the third direction has no effect on the result and it

is enough to discretize the loaded area only.

Based on these results, we can state that a problem can be

treated as a 2D case if the length to width ratio is about three

or more. It can be well approximated by a 2D solution when

this ratio is not less than 1.5. For smaller ratios, the 2D

model is not appropriate and the problem must be treated

as a 3D case.

3.3. Transient response of a train-track embankment on

half-space

As an application, the responses of half-space to station-

ary train-track loading are investigated. Fig. 16 shows the

dimensions and the con®guration of the considered train-

track embankment. The 0.8 m thick embankment layer is

assumed to be resting on a uniform half-space. The train-

track load is represented by two concentrated line loads in

the form of an impulse with unit amplitude (1 kN/m) that

lasts for 0.02 s. The time history and the frequency content

of such a load are given in Fig. 17(a) and (b), respectively.

The duration of the unit amplitude of 0.02 s is chosen in

such a way that the frequency content of the load covers the

typical range of frequencies of vibration as they are caused

by heavy-axle trains. Different values for the shear wave

velocity vs are assumed for ®ve cases as given in Table 1.

The mass density r � 2000 kg/m3 and Poisson's ratio

n � 1/3 are assumed to be identical for the embankment


Fig. 15. Responses to vertical load at a distance of 1 and 5 m (3D approach) (a) vertical (b) horizontal.

Fig. 16. Train-track embankment on half-space.


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UNCORRECTED PROOF

and the half-space in all cases. Case 1 is the reference case,

which allows for comparison with the half-space results,

since the material properties of embankment and half-

space are identical. Case 2 is an example of a harder half-

space supporting the embankment, while in Case 3 the

half-space is softer than the embankment. Case 4 represents

an extreme case of supporting half-space while Case 5 is

close to the case of a rigid track. To ®nd out the wave

propagation characteristics in the embankment layer and

through the half-space, the responses at four different loca-

tions, A, B, C, and D as shown in Fig. 16 are obtained.

The ®rst, the second and the ®fth case are analyzed by

both presented approaches (3D BE±BE coupling and 2D

BE±FE coupling), where the 3D analysis is performed

using the algorithm presented in Section 2.2.2. In the x-

direction a length of 48 m and in the y-direction a width

of 16 m (half-space) was discretized. For Case 5 the half-

space surface was modelled using 4800 space constant

elements with three degrees of freedom each, while the

train-track embankment was modelled using 1320 space

constant elements. After coupling, this results in a system

with 18 360 degrees of freedom, which was solved for 48

time steps of 0:00 �2 s duration. This problem was solved on a

PC with 1 Gb of random access memory; while using the

standard scheme this amount of memory would not be suf®-

cient to save the in¯uence matrices required for only one

time step of the total time history.

The results of the 2D and the 3D analysis are compared in

Figs. 18±20, for Cases 1, 2 and 5, respectively. The good

agreement between the results of both methods can be

observed in all three ®gures. The kinks that occur in Case

1 (see Fig. 18(a)) for the vertical displacement at point A at

t� 0.005 and 0.025 s are due to the wave scattering at the

edges of the embankment (see Fig. 16). Besides the kinks

the results presented in Fig. 18(a) are qualitatively very

similar to the half-space results given in Fig. 9(a). This

can be expected since the embankment and the underlying

half-space possess the same material properties and there-

fore the only differences between Case 1 and the half-space

case considered in Section 3.2 are the geometry of the


Fig. 17. Applied line load (a) time history (b) frequency content.

Fig. 18. Comparison of 2D and 3D responses at two different locations for Case 1 (a) vertical (b) horizontal.

Table 1

Shear-wave velocities for different cases of analysis in meters per second

[m/s]

Case 1 Case 2 Case 3 Case 4 Case 5

Embankment 250 250 250 100 500

Half-space 250 400 150 600 250


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UNCORRECTED PROOF

embankment and the fact that the embankment is subjected

to two line loads while the half-space in Section 3.2 is

subjected to one line load only.

Looking at Fig. 18(a) the kinks obtained from the 2D

analysis are sharper than those of the 3D analysis, since a

smaller time step was used in the 2D analysis. The agree-

ment for the maximum amplitudes is good. The deviations

for time t . 0.06 s for the horizontal response at point C (see

Figs. 18(b), 19(b), 20(b)) are due to the fact that point C is

very close to the border of the discretized area for the 3D




Fig. 21. Responses at four locations for different shear-wave velocities (a) Case 1 (b) Case 2.


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UNCORRECTED PROOF

analysis. The comparison con®rms our ®nding from the

previous analysis Ð 2D and 3D results match well for a

ratio of L/B of about three Ð now also for the case of

substructures with different material properties. As the 2D

and the 3D results are the same the following discussion is

based on the results of the 2D approach.

The time histories of vertical responses at the selected

four locations are shown in Fig. 21(a) and (b), for Case 1

and Case 2, respectively. The time delay and the amplitude

decrease for the maximum response at each location are

clearly depicted for both cases. The slight effect of wave

scattering by the inclined edges of the embankment for Case

1 is shown. In Case 2 a larger effect can be observed due to

the wave scattering by the subsurface between the embank-

ment and the half-space. As the half-space has more rigidity

and higher wave speed in Case 2, the maximum peaks arrive

earlier at locations B, C and D. In addition, the maximum

responses at all locations are reduced to about 50 to 60% of

the responses in Case 1.

The time histories of the responses at the locations A and

C for all cases are shown in Figs. 22 and 23, respectively,

and the maximum response amplitudes along the surface are

given in Fig. 24. Obviously, Case 4, which represents a very

weak embankment layer on a very rigid half-space gives the

highest response on the embankment and the lowest

response along the half-space surface. After the load has

ended, the embankment layer continues to vibrate with its

fundamental natural frequency similar to a single layer on a

base rock (see Fig. 22). Although the half-space surface has

the lowest response, a free vibration is observed at location

C after the load has ended, which is due to waves radiating

from the free vibrating embankment layer (see Fig. 23). In

contrast, Case 2 results in a lower response without free

vibration after the load has ended, although the half-space

is also more rigid than the embankment. Therefore, it can be

stated that the more rigid half-space results in smaller

responses under the condition that the rigidity ratio between

the half-space and the overlying embankment layer is not so

large that the embankment behaves as a layer on a base rock.

When the half-space is less rigid than the embankment, as in

Case 3, higher responses are obtained at the embankment,

and highly ampli®ed responses occur in particular in the


Fig. 22. Responses at point A for all cases (a) vertical (b) horizontal.

Fig. 23. Responses at point C for all cases (a) vertical (b) horizontal.


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UNCORRECTED PROOF

horizontal direction along the half-space surface. The time

of the maximum response is delayed due to the low wave

speed inside the half-space as shown in Figs. 23 and 24. In

Case 5, where a relatively rigid embankment is used,

reduced responses are obtained at the embankment with

slight reduction of the responses along the surface at other

locations.

From the above results, it can be stated that Case 2

presents the best solution for the model under consideration.

It should be mentioned here that, for the design of train-

track embankment, a more detailed parameter study is

required according to the site conditions.

4. Conclusions

Two modelling approaches for analyses of a half-space

subjected to dynamic loads and train-track loads are

presented in time domain. The validity of each approach

is demonstrated by comparison with other results from

literature. Good agreement was found between both meth-

ods for the investigation of half-space, train-track embank-

ment on half-space with the same material properties and

train-track embankment on half-space with different mate-

rial properties.

The comparison between results of the two methods

points out that a problem can be treated as a 2D case if

the length to width ratio L/B is about three or more. For

ratios less than 1.5 the 3D solution is essential.

An algorithm to reduce computational time- and

memory-requirements based on the translation property of

the fundamental solution was presented and its ef®ciency

was demonstrated. This algorithm allows to investigate the

3D analysis of moving loads on a train-track embankment

on a half-space for practical applications.

For the design of a train-track embankment the use of a

soft train-track embankment layer over a much more rigid

half-space should be avoided, but a more detailed case by

case parametric study based on the site conditions is

required.

Acknowledgements

The ®rst author wants to express his thanks to the

Graduiertenkolleg Computational Structural Dynamics for

the support of his research stay at the Ruhr University

Bochum. The second author thanks the German Research

Foundation (DFG) for the ®nancial support through a

scholarship.

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Two- and Three-dimensional Modelling of Half-Space and Train Track Embankment under Dynamic Loading

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Transcript of Two- and Three-dimensional Modelling of Half-Space and Train Track Embankment under Dynamic Loading