the role of transmitting boundaries in modeling dynamic soil-structure interaction problems

8/11/2019 The Role of Transmitting Boundaries in Modeling Dynamic Soil-Structure Interaction Problems

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International Journal of Engineering and Technology Volume 2 No. 2, February, 2012

ISSN: 2049-3444 2012IJET Publications UK. All rights reserved. 236

The Role of Transmitting Boundaries in Modeling Dynamic Soil-Structure

Interaction ProblemsMohammed Y. Fattah

*, T. Schanz

**, Shatha H. Dawood

***

* Building and Construction Eng. Dept., University of Technology, Baghdad, Iraq

** Ruhr-Universitt Bochum, Germany***Former graduate student, Building and Construction Eng. Dept., University of Technology, Baghdad, Iraq

ABSTRACT

The objective of this paper is to investigate the validity of transmitting boundaries in dynamic analysis of soil-structureinteraction problems. As a case study, the proposed Baghdad metro line is considered. The information about the dimensionsand the material properties of the concrete tunnel and surrounding soil were obtained from a previous study. A parametric

study is carried out to investigate the effect of several parameters including the peak value of the horizontal component ofearthquake displacement records and the frequency of the dynamic load. The computer program (Mod-MIXDYN) is used for

the analysis. The numerical results are analyzed for three conditions; finite boundaries (traditional boundaries), infiniteboundaries modelled by infinite elements (5-node mapped infinite element) presented by Selvadurai and Karpurapu ,1988),and infinite boundaries modelled by dashpot elements (viscous boundaries).

It was found that the transmitting boundary absorbs most of the incident energy. The distinct reflections observed in the "fixedboundaries" case disappear in the "transmitted boundaries" case. This is true for both cases of using viscous boundaries ormapped infinite elements. The type and location of the dynamic load are two controlling factors in deciding the importance ofusing infinite boundaries. It was found that the results are greatly affected when earthquake is applied as a base motion or a

pressure load is applied at the surface ground. The horizontal displacement on the free surface increases significantly whenmodeling infinite boundaries at all values of peak displacement of the earthquake. It increases about (7.0) times the value ofusing finite elements only.

Keywords:Dynamic, soil-structure interaction, infinite element, tunnel.

1. INTRODUCTION

Many different approaches have been proposed for solving

interaction problems considering infinite media by using amechanism which irreversibly transfers energy from thenear field to the far field.

2. THE NON-REFLECTING VISCOUSBOUNDARIES TECHNIQUE

This concept was developed by Lysmer and Kuhlemeyerin 1969. It has been widely used for various dynamic soil-structure interaction problems (Genes and Kocak, 2002).

It is also often referred to as the viscous boundary, sinceits implementation involves placing viscous dashpots atthe boundary. The viscous boundary rests on the well-known observation that the stresses associated with thepassage of plane waves at a station are proportional to thevelocities there.

For example, assuming that the waves in the two-

dimensional case (Figure 1) impinge at a right angle on theartificial boundary, the exact transmitting boundarycondition is formulated as:

0 uVp (1)

0 vVs (2)

where:

: normal stress, : shear stress,

: mass density,

Vp: dilatation (compression) wave velocity =

)2( G ,

Vs: shear wave velocity = G ,

G : shear modulus )1(2E ,

)21)(1(

E

:Poisson's ratio, E: Yong's modulus

u, v: are the normal and tangential displacements.

In discretized form, it is customary to lump the distributeddampers described by Equations 1 and 2, which results in


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International Journal of Engineering and Technology (IJET)Volume 2 No. 2, February, 2012


each node in dampers with coefficients Cn and Ct in thenormal and the tangential directions.

Cn= AVp (3)

Ct= AVs (4)

where:

A: is the applicable area.

Figure 1. Lumped viscous damper (Wolf, 1988)Viscous boundary method appears to treat both dilatational

and shear waves with sufficient accuracy in manyapplications. The viscous forces, or dashpots, do notdepend upon the frequencies of the transmitted waves. Itcan absorb both harmonic and non harmonic waves(Wong, 2004). Viscous boundaries are local in space andtime; therefore this technique is suitable for transient

analysis (Wolf, 1988).

It is commonly believed that one drawback to the viscousboundary is its inability to transmit Rayleigh waves aseffectively as it does the body wave. The use of viscousboundary for problems which involve Rayleigh wavesshould not necessarily be ruled out (Cohen, 1980).

With "Thin Layer" description, , an approachwas developed by Kausel et al. (1975) for plane or axialsymmetric layers on a rigid ground in the frequencydomain which works with exact expressions in thehorizontal directions, and the accuracy of which

corresponds to finite element method in regards of thevertical direction (Rastandi, 2003). In this method, a layer

medium over a rigid base is discretized in relatively thinlayers with respect to the wave length.Thin Layer Element

Method (TLEM) is an example of substructure approachand it is normally formulated in the frequency domain.Though time domain formulation of (TLEM) can becarried out, it was used mainly in the dynamic analysis oflayered soil and was not applied to embedded structures(Jamaa, and Shiojiri, 2000).

Another way of treating the artificial boundariesmight be by introducing infinite-elements for modeling the

far-field in soil-structure interaction problems. Theseelements were proposed by Bettess (1977) to obtain

solutions to static and steady state dynamic problems.Since then, many authors, including Medina and Penzien

(1982), Yang and Yun (1992) and Yerli et al. (1998) havesuccessfully extended the infinite element formulation to

model wave propagation problems in infinite media(Genes, and Kocak , 2002).

2.1Boundary Element Method (BEM)

It is a semi-numerical method and requires obtaining thefundamental solutions pertaining to the region, providedthat the radiation condition at infinity is satisfied (Genesand Kocak, 2002). In BEM, the differential equation istransformed to a boundary integral equation (Weber,1994). It has advantages because only surfacediscretization of the body is required which decrease the

spatial dimension by one (Genes and Kocak, 2002) whichmakes this method suitable for solving problems involvingonly homogeneous half space.

2.2Multi-Region Boundary Element Method

It is a different BEM approach proposed by Tanrikulu et

al. (2001) who solved two-and three-dimensional layeredtype problems and underground structure problems underharmonic and transient loading. BEM decreases thecomputational mesh significantly. However, it induces anun-symmetric equation system, and its use is limited whennon-linearity is of concern (Genes and Kocak, 2002).

2.3Hybird Numerical Schemes

They combine the FEM for the structure or the near-fieldwith another numerical or analytical method for the far-field. The most widely used Hybrid model is the Coupled

Finite Element and Boundary Element Method (FEBEM)both in time and frequency domain. Two and three-dimensional problems have been solved by FEBEM forexample. This is evident in the works of Von Estroff andKausel (1989), Von Estroff and Firuziaan (2000), Yu et al.(2001). Coupled (FEBEM) analysis exploits theadvantages of the FEM and the BEM and forms arelatively small-size computational mesh, yet requiring theconversion of the system of equations into either FE or BE

form. For these reasons, FE based formulations find wider

application and they are still preferred for manyengineering problems (Genes and Kocak, 2002).

A Perfectly Matched Layer (PML)is an absorbing layermodel for linear wave equations problems that absorb,almost perfectly, propagating waves from all non-tangential angles-of incidence and of all non-zerofrequencies (Basu and Chopra, 2003). Utilizing insights

obtained with electromagnetics PMLs was first introducedby Chew et al. (1997) and Teixeira and Chew (1999).

Basu and Chopra (2003) developed displacementbased(unsplit field) PMLs for anti-plane and plane-strain motion

in visco-elastic media, for time harmonic as well astransient wave motion. These PMLs have been


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Anderson used a term of form 1/(1+r) (r is the radialdirection) in three dimensional elasticity applications.

Medina and Penzin (1982) adopted the same approach.The first explicit stated mapping was made by Beer and

Meek (1981), who used a mapping which included theterms 0:

2 ( j+ ) for < 0 (1)

2 ( j+ ) /(1- ) for > 0 (2)

They split the mapping into two parts, that from = -1 to

= 0 and from = 0 to = 1. The procedure used wasfairly complicated.

Okabe (1983) gave various possible shape functions forinfinite domains, based on what he called "Thegeneralized Lagrange family for the cube".

The form in which Zienkiewicz mapping was originallygiven was simplified and systematized by Marques and

Owen in 1984, who worked out and tabulated the mappingfunction for a large range of commonly used infiniteelements (Bettess, 1992).

4.3Selvadurai and Karpurapu MappedInfinite Elements

In 1986, Selvadurai and Karpurapu have extended the

scope of mapped type infinite elements to analyze theresponse of two-phase media such as saturated soils. The

technique based on conventional shape functions andGauss-Legendre numerical integration was used, thusmaking these elements suitable for implementation ingeneral purpose finite element programs. This kind of

elements was successfully employed for modeling manygeomechanics problems (Karpurapu, 1988). Figure 3

illustrates the composite infinite elements extended toinfinity in direction and Ttable 1 reports their mapping

functions, derivatives and shape functions.

In this study, the derivation of the mapping functions forthe infinite elements will be carried out for two differentcases:

case (a) : Infinite element extended to infinity in thenegative direction.

case (b) : Infinite element extended to infinity in thenegative direction.

Figure 4 presents the composite infinite element, whileTables 2 and 3 report the mapping functions and thederivatives for cases a and b.

Figure 3. Composite infinite elements in direction (Karpurapu, 1988)

Table 1. Mapping and shape functions for composite mapped infinite element (Karpurapu, 1988)

a. Mapping functions and derivatives

ode, i

2-noded

SuperparametricMi

5-noded

Serendipity isoparametricMi

5-nodedMi/

5-nodedMi/

1 )1()1( )1()1( 2)1()1( )1()21(

2 )1()1( )1()1( 2)1()1( )1()21(

3 )1(2)1)(1( )1()1(2 2 22 )1()1(2 )1(4

4 )1(2)1)(1( )1(2)1)(1( 2)1()1( )1(2)1(

5 )1(2)1)(1( 2)1()1( )1(2)1(

b. Shape Functions


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Node, i 2-nodedSuperparametric Ni

5-nodedSerendipity isoparametric Ni

1 4)1)(1( 4)1)(1(4)1)(1(4)1)(1( 22

2 4)1)(1( 4)1)(1(4)1)(1(4)1)(1( 22

3 2)1)(1( 2

4 2)1)(1( 2

5 2)1)(1( 2

Table 2: Mapping functions and their derivatives of the 5-noded serendipity mapped infinite element

adopted in this study for case (a)

Figure 4. The composite infinite elements adopted in this study for cases (a) and (b)

Table 3: Mapping functions and their derivatives of the 5-noded serendipity mapped infinite elements adopted in

this study for case (b).

Node, i

i i

5-nodedserendipity Mi case b

5-nodedMi/

5-nodedMi/

1 1 1 )1()1( 2)1()1( )1()21(

2 -1 1 )1()1( 2)1()1( )1()21(

3 0 1 )1()1(2 2 22 )1()1(2 )1(4

4 -1 -1 )1(2)1)(1( 2)1()1( )1(2)1(

5 1 -1 )1(2)1)(1( 2)1()1( )1(2)1(

In this paper, the 5-nodeed serendipity mapped infinite element has also been added to the finite element models

Node, ii i

5-nodedserendipity Mi case a.

5-nodedMi/

5-nodedMi/

1 1 1 )1()1( 2)1()1( )1()21(

2 1 -1 )1()1( 2

)1()1(

)1()21(

3 1 0 )1()1(2 2 22 )1()1(2 )1(4

4 -1 -1 )1(2)1)(1( 2)1()1( )1(2)1(

5 -1 1 )1(2)1)(1( 2)1()1( )1(2)1(


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of the computer program (Mod-MIXDYN).

4.4Computer Program

The computer program called MIXDYN (Owen andHinton, 1980) was modified to (Mod-MIXDYN) by

implementing the mapped infinite elements to it; i.e., 5-noded coding of the mapped infinite element extendedfrom that presented by Selvadurai and Karpurapu in 1988.The program Mod-MIXDYN is coded in Fortran languageand implemented on a Pentium-IV personal computer.

5. Description of the Case Study

The proposed Baghdad metro line consists of two routes of32 km long and 36 stations designed as fortified shelters,as a first stage. The tunnel of reinforced concrete has acircular cross section with 5.0 m inner diameter (Diam.)and a 0.45 m lining thickness. The reinforcement details

and the method of construction were not known. A typicalgeological section for a specified position which is

characterized by a tunnel axis depth of 18 m is shown inFigure 5. This figure also illustrates the soil properties of

the three layers that characterize the investigated section.The properties of the materials involved in the problemsare detailed in Table 5.

5.1Modeling and Boundary Conditions

Case 1A finite element mesh with traditional boundaries (Figure6) is adopted to model the Baghdad metro line section.

Isoparametric quadrilateral eight-nodded elements are usedto model the concrete tunnel and the surrounding soil. Theboundary side nodes are restrained laterally (only vertical

displacements are possible), while the lower end nodes ofthe entire boundary mesh are restrained vertically (only

horizontal displacements are possible).

Figure 5. Geological profile of the investigated Baghdad

metro line section.

Case 2:

The 5-node mapped infinite elements presented by

Selvadurai and Karpurapu (1988) extends to infinity to theright with the other two mapped infinite elements (wheninfinite direction extends to the left and down), derived inthis study, are added to the mesh of case 1 as shown inFigure 7.

Case 3:A finite element mesh (as in case 1) with dashpot elementsis considered to model the viscous boundaries as shown in

Figure 8

Figure 6. Case 1 - Finite element mesh with traditional boundaries.


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Table 4 summarizes the general information about thefinite, infinite and dashpot elements meshes for cases 1, 2

and 3. The material properties for the problem are shownin table 5.

Table 4. General information about the mesh adopted for the three different cases.

Type of information

Case (1)

Finiteelements

only

Case (2)Finite and infinite

elements

Case (3)Finite and dashpot

elements

No. of nodes 544 581 618

No. of finite elements 160 160 160

No. of infinite elements - 20 -

No. of dashpot elements - - 50

No. of restrained nodes 37 37 50

Analysis type Plane strain, Large displacement

Damping parameters 0.00075

0

Figure 7. Case 2 - The finite element mesh with infinite boundary elements

Table 5: Material properties

PropertyConcrete

lining

Soil layers

(1)

Silty clay

(2)

Sand to siltysand

(3)

Sand

Modulus of elasticity, E (MPa) 20000 80 180 150

Poissons ratio, 0.15 0.333 0.263 0.284

Friction angle o - 30o 40o 37o

Cohesion, c (MPa) - 0.2 0.15 0Density , (kg/m

3) 2400 1950 1970 2000


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6. LOADING

Tunnels may be subjected to different types of dynamicloads such as earthquakes, impact loads resulting fromtraffic accidents if they serve as metro lines or traffictunnels, external blast loads coming from direct or indirect

hits and the internal blast loads resulting from theexplosions of demolitions charges caused by sabotage

works. In this study, two types of loads are considered:

6.1Time-History Displacement

The earthquake load is applied as base displacementmotion. The earthquake record of El-Centro in California,which hit on 18

th May 1940 has been used. It is

characterized by a s magnitude equal to 6.7 on Richterscale and a peak acceleration of 0.35g. This earthquakewas scaled to get three different values of peak

accelerations: 0.045, 0.065 and 0.13g, which were giventransformed to peak displacements Ao: 0.035, 0.05 and 0.1m by means of the fast Fourier transform. The ground

motion displacement time-history of El-Centro earthquakeis shown in Figure 9.

6.2Surface Pressure Load

A sinusoidal pressure load P(t) acting at the free surface

above the tunnel was adopted.

It can be represented by the following equation:

P(t) = 380 * sin(4.18879**t) kN/m2

for 0 < t =< 1.5 sec.

= 0.0kN/m2, for 1.5 < t < 3.0 sec.

where: = Angular frequency (rad./sec.).

f = Cyclic frequency (Hz). = /2

T = period (sec./rad.). = 1/f

Figure 8. Case 3Finite element mesh with viscous boundaries


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Figure 9. The ground motion displacement time-history of El-Centro earthquake

Three different frequencies f: 0.16, 0.24, 0.32 Hz ( = 1.0,

1.5, 2.0 rad./sec.) were considered for the acting sinusoidal

pressure load, as shown in Figure 10.

(a)

(b)

(c)

Figure 10: Pressure load - time history for the three values of .

7. PARAMETRIC STUDY

7.1Effect of Peak Displacement

The horizontal component of El-Centro earthquake wasscaled to have three different values of peak displacement

to study its effect on the

behavior of the tunnel and soil.

Figures 11 and 12 show the deflected shape of axes a-aand b-b (shown in Figure 8) respectively at time when the

peak displacement occurs along them for both cases (1 and3), at three different values of peak displacement (Ao =0.035m, 0.05m and 0.1m).

It can be noticed that the horizontal displacement alongaxis a-a increases when modeling infinite boundaries in

Peak value =0.035 m @2.04 sec.

-0.045

-0.030

-0.015

0.000

0.015

0.030

0.045

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time (sec.)

Gr

ounddisplacement(m)

= 1.0 rad. / sec.

-570

-380

-190

0

190

380

570

0 0.5 1 1.5 2 2.5 3

Time (sec.)

P(t)(kN

/m

2)

= 1.5 rad. / sec.

-570

-380

-190

0

190

380

570

0 0.5 1 1.5 2 2.5 3

Time (sec.)

P(t)(

kN/m

2)

= 2.0 rad. / sec.

-570

-380

-190

0

190

380

570

0 0.5 1 1.5 2 2.5 3

Time (sec.)

P(t)(kN/m

2)


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case 3, this increase becomes larger near the groundsurface about (7.0) times the value of case 1 at equal times.

On the other hand, the peak value of vertical displacementdecreases along axis b-b to about 0.2 the value of case 1 at

about the same time.

Figures 13 and 14 show the time history of horizontal

displacement at node B (shown in Figure 8), for both cases1 and 3 respectively for the three different values of (Ao).

It can be concluded from these figures that the horizontaldisplacement on the tunnels boundary decayssignificantly when using viscous boundaries (case 3) for

all the considered peak displacements. This trend may beattributed to the absorption of the reflected waves made by

using viscous boundaries which is faster than do theinfinite elements. Figures 15 to 20 show the normal

stresses xandytime- history, and the shear stress xyatnode B (shown in Figure 8) for the three different values

of the peak horizontal displacement Ao for both cases 1and 3.

A summary of the comparison between the peak values ofdisplacements and stresses predicted at the selectedlocations are listed in Table 6 below.

Table 6. Comparison of results

(a) Comparison between case 3 and case 1 for peak values of displacements and their times for axes a-a , b-b and nodes B and F at

three different values of the parameter (Ao).

Case(3)/Case(1)

Location Displacement Ao (m) Displ. ratio Time ratio

axis a-a Ux

0.035 1.00 1.00

0.05 1.00 1.00

0.1 1.00 1.00

axis b-b Uy

0.035 -0.21 0.97

0.05 0.21 0.97

0.1 0.21 0.97

Node BUx

0.035 -1.10 0.43

0.05 -1.10 0.43

0.1 -1.10 0.43

Node F Ux0.035 -0.54 0.730.05 -0.54 0.73

0.1 -0.54 0.73

(b) Comparison between case 3 and case 1 peak values of normal stresses x, yand shear stress xy at node B for three differentvalues of parameter (Ao).

Case(3)/Case(1)

Stresses Ao (m) Stress ratio Time ratio

x

0.035 0.83 1.87

0.05 0.83 1.870.1 0.84 1.87

y

0.035 -0.92 1.84

0.05 -0.92 1.84

0.1 -0.92 1.84

xy

0.035 0.85 0.09

0.05 0.85 0.09

0.1 0.85 0.09

7.2Effect of the Frequency of the SurfacePressure Load

By applying sinusoidal pressure load P(t), the effect of the

angular frequency () is studied. The pressure load isapplied at the surface.


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Figures 21 and 22 show the deflected shape of the axes a-aand b-b respectively, at the time when the peak

displacement occurs along them for the three cases 1, 2

and 3 at three different values of angular frequency (=1.0, 1.5, 2.0 rad./sec.).

The deflected shape of axis a-a has a harmonic (sinusoidal)shape for the three cases. The deflected shape of axis a-a

for case 2 is close to that of case 1 when = 1.0 rad./sec.,

while it is close to that of case 3 when = 2.0 rad./sec.

It is clear that the presence of viscous boundaries leads to

inverse the deflected shape of the axis b-b when = 2.0rad./sec., while the peak value of the vertical displacement

along axis b-b for case 3 is close to that in case 2 when = 2.0 rad./sec. but at opposite direction.

The effect of the surface pressure load frequency () onthe vertical displacement at node A is shown in Figures 23

and the horizontal displacement at node F is shown inFigure 24 for cases 1, 2 and 3. The peak value of thevertical displacement at node A increases as the frequencyincreases for cases 1 and 2 while it decreases for case 3.

The vertical and horizontal displacements at nodes A and

F respectively in case 3 decays rapidly compared to cases1 and 2.

The presence of viscous boundary in case 3 causes anincrease of the peak value of vertical displacements atnode A in the amount of about 2.1 times the value of case

1 when (= 1.0 rad./sec), while it decreases in amount of

about 0.89 times the value for case 1 when ( = 2.0rad./sec). On the other hand , in case 3 the verticaldisplacement at node A reaches the peak value at timegreater of about 1.8 times than the time measured for case

1 when (= 1.0 rad./sec).

Figures 25 to 30 trace the time-history of the normal

stresses x , yandshear stress xyat nodesB and D forthe three different values of frequency for cases 1 and 3,respectively.

A summary of the comparison between the peak values ofdisplacements and stresses of Figures 21 to 30 is listed inTable 7.

Table 7. Comparison between the results

(a) A comparison between cases 2, 3 and case 1 peak values of displacements and their times for axes a-a , b-b and nodes A and F at

three different values of (

).

Case(2)/Case(1) Case(3)/Case(1)

Location Displacementw

rad./sec.Displ.ratio Time ratio Displ. ratio

Timeratio

axis a-a Ux

1 -0.51 0.29 1.35 1.08

1.5 0.49 0.93 1.38 1.56

2 0.57 3.48 1.18 1.39

axis b-b Uy

1 0.77 0.80 1.05 1.22

1.5 0.87 4.50 0.95 0.23

2 -0.95 2.55 -0.80 0.40

Node A Uy

1 0.42 0.50 2.09 1.78

1.5 0.70 0.89 1.31 1.15

2 -0.99 1.90 -0.89 0.55

Node F Ux

1 0.52 0.81 1.35 1.08

1.5 0.49 0.93 1.38 1.56

2 0.57 3.48 1.18 1.39

(b) A comparison between case 3 and case 1 peak values of normal stresses x, yand shear stress xy for nodes B and D at three

different values of

.

Case(3)/Case(1)

Node Stresses

rad./secStress ratio Time ratio

B x

1 -0.52 3.20

1.5 0.55 0.20


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2 -0.45 0.41

D x

1 0.11 0.35

1.5 0.13 0.37

2 -0.13 0.20

B y

1 -0.97 3.14

1.5 -5.20 0.202 -0.91 0.43

D y

1 -0.49 3.20

1.5 0.85 0.77

2 -0.42 0.40

B xy

1 2.04 0.99

1.5 -1.74 0.35

2 1.52 1.67

D xy

1 -2.15 3.00

1.5 -2.07 0.35

2 1.96 1.67

8. CONCLUSIONS

A dynamic finite-element analysis is carried out for a soil-structure interaction problem considering transmitting

boundaries. Two types of boundaries are considered inaddition to the traditional boundaries: viscous boundaries

and mapped infinite elements. The following conclusionsare drawn

i. The transmitting boundary absorbs most of theincident energy. The distinct reflections observed for

the "fixed boundaries" disappear by using "transmittedboundaries". This is true for both cases of using

viscous boundaries or mapped infinite elements.

ii. The type and location of the dynamic load representtwo controlling factors in deciding the importance ofusing infinite boundaries. It was found that the resultspresent significant differences when earthquake isapplied as a base motion or a pressure load is appliedat the surface ground.

iii. The horizontal displacement on the free surface

increases significantly when modeling infinite

boundaries for all values of the earthquake peakdisplacement. It increases about 7.0 times the valueobtained by using finite elements only.

iv. The peak value of the vertical displacement at nodesA, B, E and F (located at the tunnels crown and sidewalls, and at the surface above the tunnel and at thesurface 6.5 m away from tunnels centre respectively)

increases with the frequency of the surface pressureload for both cases 1 and 2 (traditional boundaries andmapped infinite elements respectively) while itdecreases for case 3 (viscous boundaries).

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[20]Pissanetsky, S., A Simple Infinite Element, Compel,3, No. 2, Boole Press, Dublin, pp107-114, 1984.

[21]Rastandi, I. J., Modelization of Dynamic Soil-Structure Interaction Using Integral Transform-FiniteElement Coupling, Technician University Munchen,2003.

[22]Ross, M., Modeling Methods for Silent Boundaries

in Infinite Media, ASEN 5519-006, Fluid-StructureInteraction, Aerospace Engineering Sciences-

University of Colorado, Feb., 2004.

[23]Selvadurai, A. P. S. and Gopal, K. R., ConsolidationAnalysis of Screw Plate Test, Proc. 39th Canadian

Geotech. Conf., Ottawa. 167-178, 1986.

[24]Tanrikulu, A. H., Yerli, H. R., and Tanrikulu, A. K.,Application of the Multiregion Boundary Element

Method to Dynamic Soil-Structure InteractionAnalysis., Computers and Geotechnics, Vol. 28,

2001.

[25]Teixeira, F. L. and Chew, W. C., Unified Analysis ofPerfectly Matched Layers using Differential Forms.,Microwave and Optical Technology Letters, 20(2),1999.

[26]Von Estorff, O., and Kausel, E., Coupling ofboundary and finite elements for soil-structure

interaction problems., Earthquake Engineering andStructural Dynamics, 18, 1989.

[27]Von Estorff, O., and Firuziaan, M., CoupledBEM/FEM Approach for Non-linear Soil-StructureInteraction.,Eng Anal Boundary Elements, 24, 2000.

[28]Weber, B., Rational Transmitting Boundaries forTime-Domain Analysis of Dam-Reservoir

Interaction, Institute for Construction, Zurich, 1994.

[29]Wolf, J. P., "Soil Structure-Introduction Analysis inTime Domain", Prentice Hall, 1988.

[30]Wolf, J. P. and Song, C., The Scaled BoundaryFinite-Element Method-a Primer Derivations,Computers and Structures., 78, 191-210, 2000.

[31]Wong, C. W., Sesmic Behavior of Micropiles,

Washington state University, 2004.

[32]Yang, S. and Yun, C., Axisymmetric InfiniteElements for Soil-Structure Interaction Analysis.,Eng. Struct., Vol. 14(6), 1992.

[33]Yerli, H. R., Temel, B. and Kiral, E., Transient

Infinite Elements for 2D Soil-Structure InteractionAnalysis, Journal of Geotechnical and Geo-

environmental Engineering, ASCE, Paper No. 13807Vol. 124. No. 10, October 1998.

[34]Yu, G., Mansur, W. J., Carrer, J. A. M., and Lie, S.T., A More Stable Scheme for BEM/FEM CouplingApplied to Two-Dimensional Elastodynamics.,Computers and Structures, 79, 2001.


14/23



a a

b b

c

c

Figure 11. Deflected shape of axis a-a at time of peak

value of horizontal displacement for three different values

of (Ao)cases (1 and 3)

Figure 12. Deflected shape of axis b-b at time of peak

value of vertical displacement for three different values of

(Ao)cases (1 and 3)

Ao = 0.035 m

0

5

10

15

20

25

30

-0.1 -0.075 -0.05 -0.025 0 0.025

Horizontal Displacement (m)

VerticalDistance

(m

)

Ux-Case1 Ux-Case3

Ao = 0.035 m

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 5 10 15 20 25

Horizontal Distance (m)

VerticalDisplacement(m

)

Uy-Case1 Uy -Case3

Ao = 0.05 m

0

5

10

15

20

25

30

-0.1 -0.075 -0.05 -0.025 0 0.025


VerticalDistance

(m)

Ux-Case1 Ux-Case3

Ao = 0.05 m

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 5 10 15 20 25


VerticalDisplacement(m)

Uy-Case1 Uy -Case3

Ao = 0.1 m

0

5

10

15

20

25

30

-0.1 -0.075 -0.05 -0.025 0 0.025


VerticalDistance

(m)

Ux-Case1 Ux-Case3

Ao = 0.1 m

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 5 10 15 20 25


V

erticalDisplacement(m)

Uy-Case1 Uy -Case3


15/23



a a

b b

c c

Figure 13. Horizontal displacement - time history at node

B for three different values of (Ao) - case (1).


B for three different values of (Ao) - case (3).

Ao = 0.035 m

-0.02

-0.01

0.00

0.01

0.02

0 5 10 15 20 25 30Time (sec.)

HorizontalDisplacement(m

Ux -Case1

Ao = 0.035 m

-0.02

-0.01

0.00

0.01

0.02

0 5 10 15 20 25 30Time (sec.)


Ux -Case3

Ao = 0.05 m

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0 5 10 15 20 25 30Time (sec.)


Ux -Case1

Ao = 0.05 m

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0 5 10 15 20 25 30Time (sec.)


Ux -Case3

Ao = 0.1 m

-0.06-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 5 10 15 20 25 30Time (sec.)


Ux -Case1

Ao = 0.1 m

-0.06-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 5 10 15 20 25 30Time (sec.)


Ux -Case3


16/23



a a

b b

c c

Figure 15. Normal stressx- time history at node B for

three different values of (Ao) - case (1)


three different values of (Ao) - case (3)

Ao = 0.035 m

-200

-150

-100

-50

0

50

100

150

200

0 5 10 15 20 25 30Time (sec.)

x(kN/

m2)

Sx -Case1

Ao = 0.035 m

-200

-150

-100

-50

0

50

100

150

200

0 5 10 15 20 25 30Time (sec.)

x(kN/

m2)

Sx -Case3

Ao = 0.05 m

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30Time (sec.)

x(kN/m

2)

Sx -Case1

Ao = 0.05 m

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30Time (sec.)

x(kN/m

2)

Sx -Case3

Ao = 0.1 m

-600

-400

-200

0

200

400

600

0 5 10 15 20 25 30Time (sec.)

x(kN/m

2)

Sx -Case1

Ao = 0.1 m

-600

-400

-200

0

200

400

600

0 5 10 15 20 25 30Time (sec.)

x(kN/m

2)

Sx -Case3


17/23



a a

b b

c c

Figure 17. Normal stressy- time history at node B for

three different values of (Ao) - case (1).



Ao = 0.035 m

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30Time (sec.)

y(kN/

m2)

Sy -Case1

Ao = 0.035 m

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30Time (sec.)

y(kN/

m2)

Sy -Case3

Ao = 0.05 m

-3000

-2000

-1000

0

1000

2000

3000

0 5 10 15 20 25 30Time (sec.)

y(kN/m

2)

Sy -Case1

Ao = 0.05 m

-3000

-2000

-1000

0

1000

2000

3000

0 5 10 15 20 25 30Time (sec.)

y(kN/m

2)

Sy -Case3

Ao = 0.1 m

-6000

-4000

-2000

0

2000

4000

6000

0 5 10 15 20 25 30Time (sec.)

y(kN/m

2)

Sy -Case1

Ao = 0.1 m

-6000

-4000

-2000

0

2000

4000

6000

0 5 10 15 20 25 30Time (sec.)

y(kN/m

2)

Sy -Case3


18/23



a a

b b

c c

Figure 19. Shear stressxy- time history at node B for




Ao = 0.035 m

-1000

-500

0

500

1000

0 5 10 15 20 25 30Time (sec.)

xy(kN

/m

2)

Sxy -Case1

Ao = 0.035 m

-1000

-500

0

500

1000

0 5 10 15 20 25 30Time (sec.)

xy(kN

/m

2)

Sxy -Case3

Ao = 0.05 m

-1500

-1000

-500

0

500

1000

1500

0 5 10 15 20 25 30Time (sec.)

xy(kN/m

2)

Sxy -Case1

Ao = 0.05 m

-1500

-1000

-500

0

500

1000

1500

0 5 10 15 20 25 30Time (sec.)

xy(kN/m

2)

Sxy -Case3

Ao = 0.1 m

-3000-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

3000

0 5 10 15 20 25 30Time (sec.)

xy(kN/m

2)

Sxy -Case1

Ao = 0.1 m

-3000-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

3000

0 5 10 15 20 25 30Time (sec.)

xy(kN/m

2)

Sxy -Case3


19/23



a a

b b

c c

Figure 21. Deflected shape of axis a-a at time of peak

value of horizontal displacement for three different values

of ( - cases (1, 2 and 3).

Figure 22. Deflected shape of axis b-b at time of peak

value of vertical displacement for three different values of

() - cases (1, 2 and 3).

= 1.0 rad /sec.0

5

10

15

20

25

30

-0.005 -0.0025 0 0.0025 0.005


VerticalDista

nce

(m)

Ux-Case1 Ux-Case3Ux-Case2

= 1.0 rad /sec.

-0.05

-0.0375

-0.025

-0.0125

0

0 5 10 15 20 25


VerticalDisplace

ment(m)

Uy-Case1 Uy -Case3Uy -Case2

= 1.5 rad /sec.0

5

10

15

20

25

30

-0.005 -0.0025 0 0.0025 0.005


VerticalDistance

(m)


= 1.5 rad /sec.

-0.05

-0.0375

-0.025

-0.0125

0

0 5 10 15 20 25




= 2.0 rad /sec.0

5

10

15

20

25

30

-0.005 -0.0025 0 0.0025 0.005


VerticalDistance

(m)


= 2.0 rad /sec.

-0.05

-0.0375

-0.025

-0.0125

0

0.0125

0.025

0.0375

0.05

0.0625

0 5 10 15 20 25





20/23



a a

b b

c c

Figure 23. Vertical displacement - time history at node A

for three different values of (

) - cases (1, 2 and 3).


F for three different values of (

) - cases (1, 2 and 3).

= 1.0 rad / sec.

-0.02

-0.01

0.00

0.01

0.02

0 0.5 1 1.5 2 2.5 3Time (sec.)

VerticalDisplac

ement(m)

Uy -Case1 Uy -Case3Uy -Case2

= 1.0 rad / sec.

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0 0.5 1 1.5 2 2.5 3Time (sec.)

HorizontalDispla

cement(m

Ux -Case1 Ux -Case3Ux -Case2

= 1.5 rad / sec.

-0.02

-0.01

0.00

0.01

0.02

0 0.5 1 1.5 2 2.5 3Time (sec.)



= 1.5 rad / sec.

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0 0.5 1 1.5 2 2.5 3Time (sec.)



= 2.0 rad / sec.

-0.02

-0.01

0.00

0.01

0.02

0 0.5 1 1.5 2 2.5 3Time (sec.)



= 2.0 rad / sec.

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0 0.5 1 1.5 2 2.5 3Time (sec.)




21/23



a a

b b

c c



) - cases (1 and 3).

Figure 26. Normal stressx- time history at node D for



= 1.0 rad / sec.

-300

-200

-100

0

100

200

300

0 0.5 1 1.5 2 2.5 3Time (sec.)

x(kN/

m2)

Sx -Case1 Sx -Case3

= 1.0 rad / sec.

-75

-60

-45

-30

-15

0

15

30

45

60

75

0 0.5 1 1.5 2 2.5 3Time (sec.)

x(kN/

m2)

Sx -Case1 Sx -Case3

= 1.5 rad / sec.

-300

-200

-100

0

100

200

300

0 0.5 1 1.5 2 2.5 3Time (sec.)

x(kN/m

2)

Sx -Case1 Sx -Case3

= 1.5 rad / sec.

-75

-60

-45

-30

-15

0

15

30

45

60

75

0 0.5 1 1.5 2 2.5 3Time (sec.)

x(kN/m

2)

Sx -Case1 Sx -Case3

= 1.5 rad / sec.

-300

-200

-100

0

100

200

300

0 0.5 1 1.5 2 2.5 3Time (sec.)

x(kN/m

2)

Sx -Case1 Sx -Case3

= 2.0 rad / sec.

-75

-60

-45

-30

-15

0

15

30

45

60

75

0 0.5 1 1.5 2 2.5 3Time (sec.)

x(kN/m

2)

Sx -Case1 Sx -Case3


22/23



a a

b b

c c




Figure 28. Normal stressy - time history at node D for



= 1.0 rad / sec.

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

0 0.5 1 1.5 2 2.5 3Time (sec.)

y(kN/

m2)

Sy -Case1 Sy -Case3

= 1.0 rad / sec.

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3Time (sec.)

y(kN/

m2)

Sy -Case1 Sy -Case3

= 1.5 rad / sec.

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

0 0.5 1 1.5 2 2.5 3Time (sec.)

y(kN/m

2)

Sy -Case1 Sy -Case3

= 1.5 rad / sec.

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3Time (sec.)

y(kN/m

2)

Sy -Case1 Sy -Case3

= 2.0 rad / sec.

-1200-1000

-800

-600

-400

-200

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3Time (sec.)

y(kN/m

2)

Sy -Case1 Sy -Case3

= 2.0 rad / sec.

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5 3Time (sec.)

y(kN/m

2)

Sy -Case1 Sy -Case3


23/23



a a

b b

c c




Figure 30. Shear stressxy- time history at node D for



= 1.0 rad / sec.

-80

-60

-40

-20

0

20

40

60

80

0 0.5 1 1.5 2 2.5 3Time (sec.)

xy(kN

/m

2)

Sxy -Case1 Sxy -Case3

= 1.0 rad / sec.

-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2 2.5 3Time (sec.)

xy(kN

/m

2)


= 1.5 rad / sec.

-80

-60

-40

-20

0

20

40

60

80

0 0.5 1 1.5 2 2.5 3Time (sec.)

xy(kN/m

2)


= 1.5 rad / sec.

-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2 2.5 3Time (sec.)

xy(kN/m

2)


= 2.0 rad / sec.

-80

-60

-40

-20

0

20

40

60

80

0 0.5 1 1.5 2 2.5 3Time (sec.)

xy(kN/m

2)


= 2.0 rad / sec.

-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2 2.5 3Time (sec.)

xy(kN/m

2)


the role of transmitting boundaries in modeling dynamic soil-structure interaction problems

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