30120140504021 2

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 4, April (2014), pp. 169-182 © IAEME
169
COMPUTATION OF HARMONIC GREEN’S-FUNCTIONS OF A
HOMOGENEOUS SOIL USING AN AXISYMMETRIC FINITE ELEMENT
METHOD
Adel Shaukath1
, Ramzi Othman1, 2
, Abdessalem Chamekh1, 3
1
Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University,
P.O. Box 80248. Jeddah 21589, Saudi Arabia
2
LUNAM Université, Ecole Centrale de Nantes, GeM, UMR CNRS 6183, BP 92101,
44321 Nantes cedex 3, France
3
LGM, Ecole Nationale d'ingénieurs de Monastir, Avenue Ibn El Jazzar, 5019, Munastîr, Tunisia
ABSTRACT
Ground-borne vibrations are disturbing to human beings. In order to model and reduce these
vibrations, the calculation of the harmonic Green’s-functions of the soil is highly required. In the
past, the problem was approached by an analytical methodology. For the first time, this paper
proposes to compute these Green’s-functions by using an axisymmetric finite element approach.
Careful attention was paid to the convergence of results regarding the mesh size. The new proposed
solution was applied to calculate the harmonic Green’s-functions of a homogeneous half-plane soil.
The new methodology has a great potential to be used in the modelling of railway and automobile
traffic induced vibrations.
Keywords: Axisymmetry, Explicit Scheme, Green’s-Functions, Homogenous Soil, Railway-Induced
Vibration
I. INTRODUCTION
When trains move, vibrations are caused because of the uneven nature of railway tracks,
misalignment of wheels of the trains or discontinuities of the wheels and tracks [1].For similar
reasons, vibrations are also generated by other terrestrial vehicle traffic. Earthquakes can also initiate
vibrations. These vibrations propagate through the soil to the foundations of neighbouring buildings
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 5, Issue 4, April (2014), pp. 169-182
© IAEME: www.iaeme.com/ijmet.asp
Journal Impact Factor (2014): 7.5377 (Calculated by GISI)
www.jifactor.com
IJMET
© I A E M E

170
and structures, causing disturbances, noise pollution and even damage [2, 3]. Moreover, they can
also have unwelcome effects on the health of occupants and they can disturb the work vibration-
sensitive equipment. In order to prevent these undesirable effects, ground-borne vibrations have to be
reduced or isolated as much as possible.
The formulation of a suitable model is difficult because of the number of parameters involved
in describing the underground environment. For modelling of vibrations due to the underground train
traffic, the tunnel structure and the surrounding soil are assumed much stiffer than the track [4]. Due
to this assumption, the railway induced vibrations problem is approached in two steps. Firstly, the
wheel-rail-(track) interaction problem is separately solved to derive the dynamic axle forces. These
forces are considered as an input external force for the second problem that deals with the track-
(tunnel)-soil interaction problem. These problems aims at predicting the vibration propagation in the
underground environment.
Various models are designed in order to predict these vibrations: The numerical solutions
obtained earlier were time-consuming because of the slow processing speeds of computers.
Analytical approaches were preferred. However, with the help of increasing processing speeds of
computers, the calculation time can be reduced admirably. Now, numerical models are highly
preferred in the modelling of underground railway-induced vibrations, mainly by using
finite/infinite-element models [5] and coupled finite element–boundary element (FE–BE) models
[6, 7]. Even though two-dimensional numerical models offer computation times that are much
shorter than three-dimensional models [8], they are unable to account for wave propagation along the
track. In addition, they can not accurately simulate the radiation damping of the soil [6]. On the other
hand, 3-D numerical models are highly expensive in terms of computation requirements. The current
tendency is towards the use of numerical methods that exploit the invariance in structure(track and
tunnel) longitudinal axis direction (2.5D finite/infinite element or FE–BE models). These techniques
are effective when considering very long structures partially or totally embedded in soil [8, 9]. They
indeed take into account the soil-structure coupling which is of great importance [10, 11]. The
structure is modelled by techniques of structural dynamics that can be deterministic or probabilistic
[12]. The soil is regarded as a layered half-space, which is made of horizontal elastic homogenous
layers that rest on a homogenous elastic half-space. The Green’s-function [13] or the fundamental
solutions [14] can be used to describe the dynamic characteristics of a layered half-space.
The fundamental solutions are based on the computation of integrals that require the
introduction of a fictitious boundary, across which the radiation conditions have to be satisfied [15].
The Green’s functions are approached in two different ways: approximate and analytical solutions.
An example of approximate methods is the thin-layer method [16]. The analytical methods give
solutions that are expressed in terms integrals that have infinite or semi-infinite integration path
[17-20].However, the evaluation of these integrals is mostly possible after a tedious mathematical
work.
To the best of the authors’ knowledge, no one has used the finite element method to calculate
the Green’s functions of a homogenous or a layered soil. We think that the computational time cost
has discouraged researchers to look in this direction. However, the increasing performance of
computers nowadays may invert this tendency. The aim of this work is to explore the possibility of
computing the soil harmonic Green’s function by using the finite element method. To the best of the
authors’ knowledge, this has never been undertaken before.
II. METHODOLOGY
2.1. Problem statement
We would like to use Finite Element capability and Green’s function approximation to model
the multi-layered soil. Green’s functions are calculated using the displacements caused by a

171
concentrated load of unit magnitude, applied on the soil layer. Initially, we assume the soil to be
homogeneous, for the sake of simplification. However, the approach that will be presented in Section
2.2 can extended to the case of a layered soil, with almost no extra effort.
Let ൫݁Ԧ௫,݁Ԧ௬, ݁Ԧ௭൯ be a reference for the 3D space. Without loss of generality, we assume that
the soil is defined by the volume ൌ ሼMሺ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ሻ; ‫ݖ‬ ൑ 0ሽ and that the force is applied at the origin
(Figure 1). Moreover, we are dealing in this work only with harmonic Green’s functions. This means
that the concentrated load is harmonic with respect to time. As a first step, the concentrated load is
assumed to be in the vertical direction. Green’s functions due to horizontal forces are out-of-the
scope of the work.
Neglecting body forces, the elasto-dynamic equations in the homogeneous soil are written as:
݀ଓ‫ݒ‬ሬሬሬሬሬሬԦ ൬ߪ൫‫ݑ‬ሬԦሺ‫,ݐ‬ ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ሻ൯൰ ൌ ߩ ܽԦሺ‫,ݐ‬ ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ሻ, for any ሺ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ሻ ‫א‬ ߗ (1)
where, ߪ is the elasto-dynamic stress tensor due to the displacement field ‫ݑ‬ሬԦሺ‫,ݐ‬ ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ሻ, ߩ is the soil
density and ܽԦሺ‫,ݐ‬ ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ሻ is the acceleration field.
The above equation is valid in the domain ߗ. On the top surface Γ଴ ൌ ሼMሺ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ሻ; ‫ݖ‬ ൌ 0ሽ, the
following boundary conditions hold:
ߪ௭௭ሺ‫,ݐ‬ ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ ൌ 0ሻ ൌ ฬ
0, if ‫ݐ‬ ൏ 0
ߪ଴ sinሺ2ߨ݂଴ ‫ݐ‬ሻ ߜሺ‫ݔ‬ሻ ߜሺ‫ݕ‬ሻ, if ‫ݐ‬ ൒ 0
, (2)
ߪ ൌ
0 0 0
0 0 0
0 0 ߪ௭௭
݂଴ is the frequency of the external load and the stress ߪ଴ is such as
‫׬‬ ߪ௭௭ሺ‫,ݐ‬ ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ ൌ 0ሻ ݀ܵௌ
ൌ 1. (3)
ܵ is a surface of Γ଴ that should include the origin, where the load is applied.
The above problem is invariant when any rotation around the axis ݁Ԧ௭ is applied. Therefore,
the 3D model can be simplified to the axisymmetric model depicted in Figure 2. It is then better
represented in cylindrical coordinates ሺ‫,ݎ‬ ߆, ‫ݖ‬ሻ of reference ሺ݁Ԧ௥, ݁Ԧ௵, ݁Ԧ௭ሻ. It is worth noting that the
displacement field and the stress tensor are independent of the angle ߆. Moreover, the displacement
in the direction ݁Ԧ௵ is zero.
2.2. Finite element approach
We numerically solve the problem represented in Figure 2 by the use of the finite element
method. The main challenge is to overcome the difficulty due to the infinite volume of the soil.
However, only a finite part of the soil can be modelled with finite element method. Thus, fictitious
boundaries have to be assumed. Moreover, additional assumptions are considered to reduce the size
of the numerical problem.

172
The ground-borne vibrations typically involve frequencies in the range of 0 to 100Hz [1]. In
this work, the numerical analysis is carried out assuming a harmonic force of frequency 50Hz, which
corresponds to middle of the frequency range of interest. This leads to a periodicity of 20 ms. The
numerical simulation is undertaken during 0.24s. This is higher than 10 periods.
In the literature review, the vibration problems are approached by using a modal analysis
method. This assumes the computation of structure modes. Subsequently, the displacement is written
as a combination of these modes with appropriate weights. Actually, this solution cannot be
considered here, because the modes highly depend on the fictitious boundaries. Changing these
boundaries will lead to different modes, and consequently, to a different displacement solution. In
this work, we propose to solve the elasto-dynamic problem in the time domain by an explicit
integration scheme. To the best of the author’s knowledge, this is the first time that the explicit
integration scheme is being used to compute Green’s functions of homogeneous/layered soil. Solving
the problem in the time domain is a significant advantage, since the effects of fictitious boundaries
can be cancelled, if we should use an appropriate size of the finite soil part that will be included in
the numerical model.
In this work, we assume a soil of density, Young’s modulus, and Poisson’s ratio equal to
2500kg. mିଷ
, 220 MPa and 0.35, respectively. Hence, the pressure wave speed and certainly the
shear wave speed are lower than 400 m. sିଵ
. Consequently, a bounded domain of the soil of
100m*100m is modelled (Figure 3). The model size is taken as 100m2
, since wave speeds are lower
than 400 m. sିଵ
and the computation time is 0.25 s. Thus the waves generated by the concentred
force propagate along a distance shorter than 100 m in 0.24s. In this case, the fictitious boundary
conditions that are created at 100m away from the concentrated load have no effect on the wave
propagation. The displacement field that is computed in this framework is then independent of the
fictitious boundaries.
Axisymmetric model is selected, as it takes into account the 3D nature of the soil.
Meanwhile, it implies significantly reduced model size and consequently, calculation time. A
concentrated force of one Newton is applied in the z-direction at the origin, since, by definition,
Green’s function is the response of a structure, to a concentrated force of one Newton.
Due to the concentrated force applied, it is difficult to determine the optimum mesh size close
to the loading point, and far away from it, as most of the displacement is caused within a region of
1m from the load. The 100m*100m model is divided into two parts, one with a finer mesh, and the
other with larger meshes of various sizes. The mesh size influence is studied in two steps. First we
have studied the best mesh size on the narrow region around the applied load of size 1m*1m.
Second, we have looked for the best mesh on the remaining part.
In the first step, only the region around the concentred force is modelled. Different mesh
sizes are considered: 0.2, 0.05, 0.01m. Subsequently, the displacements, obtained by the different
meshes, are compared at several points in order to decide the optimum mesh. In terms of the second
step, we considered the whole soil part, i.e., 100m*100m. We consider a mesh size of 0.25m for the
1m*1m region around the load. For the second larger part, we apply various mesh sizes of 5 m, 2m,
1 m, 0.5 m, 0.35 m and 0.2 m, in order to get the optimum mesh size. Likewise, the displacements
obtained by the different meshes are compared at several points.
The results are stored with a step time of 1 ms in order to have 20 points in a period of 20ms.
The finite element model is undertaken using the explicit scheme of Abaqus.

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976
ISSN 0976 – 6359(Online), Volume 5, Issue
Fig 1: Schematic representation of the 3D problem
Fig 2: Schematic representation of the axisymmetric model
6359(Online), Volume 5, Issue 4, April (2014), pp. 169-182 © IAEME
173
Schematic representation of the 3D problem
Schematic representation of the axisymmetric model

Fig 3: Schematic representation of the bounded axisymmetric model
III. RESULTS
3.1. Mesh size around the concentrated force
It is widely known that finite element results are sensitive to choice of mesh size. Actually,
using coarse mesh decreases computation time cost. However, this increases the stiffness of structure
and distributes concentrated forces on a larger surface. On
appropriate to represent the physical behaviour.
computation cost.
In this section, we are interested in determining
1m) around the concentrated force that gives the best trade
good representation of the real mechanical behaviour.
A 1m * 1m region around the origin is meshed using three mesh sizes
Figure 4 shows an overview of the magnitude of displacement in this region as obtained by
mesh and 0.05-m mesh. The displacement is clearly concentrated around the applied force. Reducing
the mesh size, reduces the region where the displacement is significant.
achieve mesh convergence right under the concentrated force. However, this region is less than 0.15
m * 0.2 m (Figure 4(b)). It is even more narrow if 0.1
to obtain a suitable mesh that can give accurate results with non
In order to confirm this, the horizontal and/or vertical displacement
are represented in figure 6. The three nodes are selected half way from the origin. Nodes 1, 2 and 3
are (0.5 m, 0 m), (0.5 m, -0.5 m) and (0, 0.5 m), respectively. Figure 6(a) depicts the horizontal
displacement at node 1 computed the three mesh sizes: 0.2, 0.05 m and 0.01 m. The 0.2
displacement is slightly higher than 0.05
displacements are almost superimposed. This last conclusion is also observed in figures 6(b), 6(c),
6(d) and 6(e) which depict the vertical displacement of no
2, the vertical displacement of node 2 and the vertical displacement of node3, respectively.
Therefore, we can state that mesh convergence is obtained for mesh sizes lower than 0.05 m.
174
Schematic representation of the bounded axisymmetric model
concentrated force
and distributes concentrated forces on a larger surface. On the other hand, using finer meshes is more
appropriate to represent the physical behaviour. However, it can be highly expensive in terms of
In this section, we are interested in determining the optimal mesh size in a small region (1m *
m) around the concentrated force that gives the best trade-off between low computation cost and
good representation of the real mechanical behaviour.
A 1m * 1m region around the origin is meshed using three mesh sizes: 0.2, 0.05 and 0.01
the magnitude of displacement in this region as obtained by
The displacement is clearly concentrated around the applied force. Reducing
the mesh size, reduces the region where the displacement is significant. Actually, it will be hard to
m * 0.2 m (Figure 4(b)). It is even more narrow if 0.1-m mesh is used. Outside, it should be possible
t can give accurate results with non-prohibitive computation cost.
In order to confirm this, the horizontal and/or vertical displacements at three nodes (Figure 5)
The three nodes are selected half way from the origin. Nodes 1, 2 and 3
0.5 m) and (0, 0.5 m), respectively. Figure 6(a) depicts the horizontal
displacement at node 1 computed the three mesh sizes: 0.2, 0.05 m and 0.01 m. The 0.2
slightly higher than 0.05-m and 0.02-m mesh displacements. The
6(d) and 6(e) which depict the vertical displacement of node 1, the horizontal displacement of node
Schematic representation of the bounded axisymmetric model
, using finer meshes is more
can be highly expensive in terms of
the optimal mesh size in a small region (1m *
off between low computation cost and
: 0.2, 0.05 and 0.01 m.
the magnitude of displacement in this region as obtained by 0.2-m
The displacement is clearly concentrated around the applied force. Reducing
ally, it will be hard to
m mesh is used. Outside, it should be possible
prohibitive computation cost.
at three nodes (Figure 5)
The three nodes are selected half way from the origin. Nodes 1, 2 and 3
0.5 m) and (0, 0.5 m), respectively. Figure 6(a) depicts the horizontal
displacement at node 1 computed the three mesh sizes: 0.2, 0.05 m and 0.01 m. The 0.2-m mesh
. These two last
de 1, the horizontal displacement of node

Fig 4: Overview of the magnitude of displacement: (a)
Fig 5: Definition of nodes 1, 2 and 3 for the 1m * 1m region
175
agnitude of displacement: (a) 0.2-meter mesh, and (b) 0.05
Definition of nodes 1, 2 and 3 for the 1m * 1m region
meter mesh, and (b) 0.05-meter mesh

176
Fig 6: Vertical and/or horizontal displacements at three nodes obtained with three different meshes

3.2. Mesh size away from the concentrated force
In this section, we are interested in the region away from the concentrated force. Namely, we
are studied the total model, except the 1m*1m region that is already investigated in Section 3.1. Six
nodes are selected for the mesh convergence study (Figure 7); more precisely, Node 1 (0 ,
Node 2 (25m , -25m), Node 3 (25m , 0), Node 4 (0 ,
0). The vertical displacement in these points is determined using six different mesh size, na
2m, 1m, 0.5m, 0.35m and 0.2m. They are depicted in Figure 8. For all nodes, displacements obtained
by 0.35 and 0.2m-mesh size are very close. The displacements obtained by 0.5m
slightly different. The signal periodicity is almost th
However, the amplitude of oscillations is slightly different.
1m, 2m and 5m-mesh size are quite different from results obtained by the former three meshes. We
conclude that mesh convergence is obtained with 0.35m size.
Fig 7: Nodes studied for mesh convergence
177
from the concentrated force
convergence study (Figure 7); more precisely, Node 1 (0 ,
25m), Node 3 (25m , 0), Node 4 (0 , -50 m), Node 5 (50m , -50m) and Node 6 (50m,
0). The vertical displacement in these points is determined using six different mesh size, na
mesh size are very close. The displacements obtained by 0.5m
slightly different. The signal periodicity is almost the same as obtained by 0.35 and 0.2m
However, the amplitude of oscillations is slightly different. On the opposite the displacements by
mesh size are quite different from results obtained by the former three meshes. We
mesh convergence is obtained with 0.35m size.
Nodes studied for mesh convergence
convergence study (Figure 7); more precisely, Node 1 (0 , -25 m),
50m) and Node 6 (50m,
0). The vertical displacement in these points is determined using six different mesh size, namely, 5m,
mesh size are very close. The displacements obtained by 0.5m-mesh size are
e same as obtained by 0.35 and 0.2m-mesh size.
On the opposite the displacements by
mesh size are quite different from results obtained by the former three meshes. We

Fig 8: Vertical displacements at six nodes obtained with different meshes: (a) Node 1, (b) Node 2,
(c) Node 3, (d) Node 4, (e) Nod
178
Vertical displacements at six nodes obtained with different meshes: (a) Node 1, (b) Node 2,
(c) Node 3, (d) Node 4, (e) Nod3e 5, and (f) Node 6
Vertical displacements at six nodes obtained with different meshes: (a) Node 1, (b) Node 2,

3.3. Final results
On one hand, Section 3.1 yields 0.05m as the best mesh size around the concentrated force.
On the other hand, Section 3.2 gives 0.35m as the best mesh size for the remaining part of the model.
These mesh size are chosen to calculated the
Figure 9 depicts the horizontal and vertical displacement for four points along the horizontal free
surface. Likewise, Figure 10 represents
along the vertical axe of symmetry. The two
displacement starts for the nearest points first. Moreover, the displacements vanish for points that
highly distant from the concentrated force.
Fig 9: Propagation along the free horizontal surface: (a) Horizontal, and (b) Vertical displacements
179
These mesh size are chosen to calculated the Green’s functions due to a 50-Hz harmonic force.
represents the horizontal and vertical displacements for four points
vertical axe of symmetry. The two figures show clearly wave propagation effects since the
highly distant from the concentrated force.
n along the free horizontal surface: (a) Horizontal, and (b) Vertical displacements
Hz harmonic force.
the horizontal and vertical displacements for four points
wave propagation effects since the
n along the free horizontal surface: (a) Horizontal, and (b) Vertical displacements

Fig 10: Propagation along the vertical axe of symmetry: (a) Horizontal, and (b) Vertical
IV. CONCLUSIONS
In this work, the axi-symmetric finite element method was applied for the first to
computation of the harmonic Green’s
easily extendable to layered half plane soils. Without loose of generality,
assuming a concentrated 50Hz harmonic force applied in the origin.
order to eliminate boundary effects. The soil is split in two parts. The first is around the concentrated
force where mesh convergence is obtained by 0.05m mesh size. The remaining second part, is
optimally meshed with 0.35 m elements.
showed a significant wave propagation effects. The methodology developed in this work has a great
potential mainly in the modelling of railway and automobile traffic induced vibrations.
should follow to extend this work to layered soils, to predict the response of harmonic waves and to
solve soil-structure interaction problems.
180
Propagation along the vertical axe of symmetry: (a) Horizontal, and (b) Vertical
displacements
symmetric finite element method was applied for the first to
Green’s functions of a homogeneous half plane soil. This approach is
easily extendable to layered half plane soils. Without loose of generality, this study was undertaken
assuming a concentrated 50Hz harmonic force applied in the origin. The model size is chosen in
is obtained by 0.05m mesh size. The remaining second part, is
optimally meshed with 0.35 m elements. The displacements obtained by the optimal mesh sizes
potential mainly in the modelling of railway and automobile traffic induced vibrations.
structure interaction problems.
Propagation along the vertical axe of symmetry: (a) Horizontal, and (b) Vertical
symmetric finite element method was applied for the first to
functions of a homogeneous half plane soil. This approach is
this study was undertaken
The model size is chosen in
is obtained by 0.05m mesh size. The remaining second part, is
The displacements obtained by the optimal mesh sizes
potential mainly in the modelling of railway and automobile traffic induced vibrations. Further works

181
REFERENCES
[1] G. Degrande, D. Clouteau, R. Othman, M. Arnst, H. Chebli, R. Klein, P. Chatterjee and B.
Janssens, “A numerical model for ground-borne vibrations from underground railway traffic
based on a periodic finite element–boundary element formulation” J. Sound Vib., vol. 293
(3-5), pp. 645-666, 2006.
[2] H. Chebli, R. Othman, D. Clouteau, M. Arnst and G. Degrande, “3D periodic BE–FE model
for various transportation structures interacting with soil,” Comput. Geotech. Vol. 35,
pp. 22–32, 2008.
[3] S. Jones and H. Hunt, “Predicting surface vibration from underground railways through
inhomogeneous soil”, J. Sound Vib., vol. 331, (3-5), pp. 2055-2069, 2012.
[4] L. Fryba, “Vibrations of solids and structures under moving loads”, Thomas Telford Ed.,
1999.
[5] Y. B. Yang and H. H. Hung, “Soil vibrations caused by underground moving trains”,
J. Geotech. Geoenviron. Eng. vol. 134, 1633–1644, 2008.
[6] S. Gupta, M. F. M. Hussein, G. Degrande, H. E. M. Hunt and D. Clouteau, “A comparison of
two numerical models for the prediction of vibrations from underground railway traffic”, Soil
Dyn. Earthquake Eng. 27, pp. 608–624, 2007.
[7] X. Sheng, C. J. C. Jones and D. J. Thompson, “Modelling ground vibration from railways
using wavenumber finite- and boundary-element methods”, Proc. Roy. Soc. A – Math. Phys.
Eng. Sci. vol. 461, pp. 2043–2070, 2005.
[8] L. Andersen and C. J. C. Jones, “Coupled boundary and finite element analysis of vibration
from railway tunnels-a comparison of two- and three-dimensional models”, J. Sound Vib.,
vol. 293, pp. 611–625, 2006.
[9] D. Clouteau D. Aubry and M.L. Elhabre “Periodic BEM and FEM-BEM coupling:
applications to seismic behaviour of very long structures” Computational Mechanics vol. 25,
pp. 567-577, 2000.
[10] H. Chebli, R. Othman and D. Clouteau “Response of periodic structures due to moving
loads”, C. R. Mecanique vol. 334, pp. 347–352, 2006.
[11] H. Chebli, D. Clouteau and L. Schmitt, “Dynamic response of high-speed ballasted railway
tracks: 3D periodic model and in situ measurements”, Soil Dyn. Earthquake Eng. vol. 28,
pp. 118–131, 2008.
[12] M. Arnst, D. Clouteau, H. Chebli, R. Othman and G. Degrande, “A non-parametric
probabilistic model for ground-borne vibrations in buildings”, Prob. Eng. Mech. vol. 21,
pp.18–34, 2006.
[13] A. Strukelj, T. Plibersek and A. Umek, “Evaluation of Green’s-function for vertical point-
load excitation applied to the surface of a layered semi-infinite elastic medium”, Arch Appl
Mech, vol. 76, pp.465–479, 2006.
[14] J. Miklowitz, “The Theory of Elastic Waves and Waveguides”. North-Holland, Amsterdam,
1980.
[15] M. Premrov and I. Spacapan, “Solving exterior problems of wave propagation based on an
iterative variation of local DtN operators”, Appl. Math. Model, vol. 28, pp. 291–304, 2004.
[16] E. Kausel, “An explicit solution for the Green’s-function for dynamic loads in layered media”
Research Report R81–13, Massachusetts Institute of Technology, 1981.
[17] A. V. Vostroukhov, S. N. Verichev, A. W. M. Kok and C. Esveld, “Steady-state response of a
stratified half-space subjected to a horizontal arbitrary buried uniform load applied at a
circular area”, Soil Dyn. Earthq. Eng. vol. 24, pp. 449–459, 2004.
[18] R. Y. S. Pak and B. B. Guzina, “Three-dimensional Green’s-functions for a multilayered
half-space in displacement potentials”, J. Eng. Mech. vol. 128(4), pp. 449–461, 2002.

182
[19] C. D. Wang, C. S. Tzeng, E. Pan, and J. J. Liao, “Displacements and stresses due to a vertical
point load in an inhomogeneous transversely isotropic half-space”, Int. J. Rock. Mech. Min.
Sci. vol. 40, pp. 667–685, 2003.
[20] T. Kobayashi and F. Sasaki, “Evaluation of Green’s-function on semi-infinite elastic
medium. KICT Report, No. 86, Kajima Research Institute, Kajima Corporation, 1991.

30120140504021 2

Recommended

Recommended

More Related Content

What's hot

What's hot (18)

Viewers also liked

Viewers also liked (9)

Similar to 30120140504021 2

Similar to 30120140504021 2 (20)

More from IAEME Publication

More from IAEME Publication (20)

Recently uploaded

Recently uploaded (20)

30120140504021 2