International Journal of Mechanical Engineering and ... · COMPUTATION OF HARMONIC...

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 Othman

1, 2, Abdessalem Chamekh

1, 3

1Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University,

P.O. Box 80248. Jeddah 21589, Saudi Arabia 2LUNAM Université, Ecole Centrale de Nantes, GeM, UMR CNRS 6183, BP 92101,

44321 Nantes cedex 3, France 3LGM, 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

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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 , , be a reference for the 3D space. Without loss of generality, we assume that

the soil is defined by the volume Ω M , , ; 0 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:

, , , , , , , for any , , ! " (1)

where, is the elasto-dynamic stress tensor due to the displacement field , , , , is the soil

density and , , , is the acceleration field.

The above equation is valid in the domain ". On the top surface Γ# M , , ; 0, the

following boundary conditions hold:

, , , 0 $0, if ' 0 # sin2+,# - -, if . 0 /, (2)

0 0 00 0 00 0

,# is the frequency of the external load and the stress # is such as

0 , , , 0 12 1. (3)

1 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 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 5, 6, of reference 7, 8 , . It is worth noting that the

displacement field and the stress tensor are independent of the angle 6. Moreover, the displacement

in the direction 8 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. m<=, 220 MPa and 0.35, respectively. Hence, the pressure wave speed and certainly the

shear wave speed are lower than 400 m. s<>. 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. s<> 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 forceIt 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 It is widely known that finite element results are sensitive to choice of mesh size. Actually,


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

It is widely known that finite element results are sensitive to choice of mesh size. Actually,


, 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 forceIn 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 In this section, we are interested in the region away from the concentrated force. Namely, we


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


In this section, we are interested in the region away from the concentrated force. Namely, we


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

On one hand, Section 3.1 yields 0.05m as the best mesh size around the concentrated force.


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


On one hand, Section 3.1 yields 0.05m as the best mesh size around the concentrated force.


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.

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