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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Dynamic Infinite Elements for Soil-Structure Interaction Analysis in a Layered Soil Medium C. B. Yun1*, J. M. Kim2 1 Department of Civil & Environmental Engineering, Korea Advanced Institute of Science and Technology,

Daejon 305-701, Korea 2 Department of Civil & Environmental Engineering, Chonnam National University, Yeosu, Chonnam 550-749,

Korea Email: ycb@kaist.ac.kr Abstract: This paper presents dynamic infinite element formulations which have been developed for soil-structure interaction analysis both in frequency and in time domains by the present authors and our colleagues during the past twenty years. Axisymmetric, 2D and 3D layered half-space soil media were considered in the developments. The displacement shape functions of the infinite elements were established using approximate expressions of analytical solutions in frequency domain to represent the characteristics of multiple waves propagating into the unbounded outer domain of the media. The shape functions were determined in terms of the excitation frequency as well as the spatial and material characteristics of the far-field soil region. Thereby the element mass and stiffness matrices become frequency dependent. As far as time domain analysis, the shape functions were further simplified to obtain closed-form frequency-dependent mass and stiffness matrices, which can analytically be transformed into time domain terms by a continuous Fourier transform. The proposed infinite elements were verified using benchmark examples, which showed that the present formulations are very effective for the soil-structure interaction analysis either in frequency or in time domain. Example applications to actual soil-structure interaction problems are also given to demonstrate the capability and versatility of the present methodology.

Key Words: finite element, Infinite element, soil-structure interaction, frequency domain analysis, time domain analysis INTRODUCTION

When an engineer analyzes a structure supported or surrounded by soil medium subjected to dynamic loads, it is used to consider the flexibility of the soil (kinematic interaction) and the energy radiation into outer far-field region (inertial interaction). These effects are noticeable particularly for stiff and/or massive structures resting on relatively soft soil. Thus the interaction phenomena have to be considered in dynamic analysis often called ‘soil-structure interaction’ (SSI) analysis. On the other hand, because the soil domain is practically much larger than the other dimension of the problem such as the size of the structure, it can be regarded as infinite. In order to model the effect of wave radiation into the infinite domain on structural behaviors, several kinds of modeling techniques have been developed including viscous boundary, transmitting boundary, boundary elements, infinite elements, and system identification method [1]. The infinite element method is one of the popular schemes, since it is a mean of extending the well-known finite element method to problems which extend to infinity, at the expense of requiring special shape functions and integration schemes [2-39]. In the infinite element method, the interior region is modeled with finite elements, while the exterior region is represented by infinite elements. Unlike the finite element solution, the behavior in the exterior highly depends on the problem considered, for example the solution of elastostatics problem is non periodic while that of elastodynamic periodic. Thus the shape functions of infinite element for approximating field variables were usually derived using analytical solutions in the exterior region. This may be the reason why lots of infinite element formulations are available and still under development. The idea of the infinite element was originally introduced by Ungless [2]; Bettess published the first paper on infinite element in mid 1970's [3]; and later Astley [4] and Zienkiewicz et al. [5] respectively proposed the wave

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envelope type and the mapped type infinite elements. Owing to these earliest works, the method then became popular and has successfully been extended to various fields. There is now a considerable literature on the topic of infinite elements extended to over 300 papers. Among the research papers, we note Chow & Smith [6], Beer & Meek [7], Lynn & Hadid [8], Rajapakse & Karasudhi [9], and Koh & Lee [10] for elastostatics; Schrefler & Simoni [11] and Karpurapu [12] for consolidation analysis; Honjo & Pokharel [13] and Zhao & Valliappan [14] for seepage flow; Bettess & Zienkiewicz [15], Lau & Ji [16], Park et al. [17, 18], and Astley et al. [19] for the Helmholtz equation including hydrodynamics and acoustics; and other fields [20, 21]. Most of the formulations are reviewed in Bettess’ monograph [22], and more recently Astley presented a review of formulations for scalar wave problems [23]. Among the infinite elements, the static type is judged to be most successful, so that it is now available in some commercial finite element programs. Infinite elements for scalar wave problems governed by the Helmholtz equation have also been developed well enough to apply for industry. However, relatively limited work has been done for elastic wave propagation problems in layered media, such as the SSI analysis in which multiple wave components generated by vibrating structures coupled with neighboring soil travel simultaneously into the unbounded layered half-space medium. Medina carried out pioneering works with Penzien and Taylor in this field [24, 25]. Rajapakse and Karasudhi [26] suggested an infinite element which is capable of propagating multiple waves in a homogeneous half-space. While favorable results were obtained using these elements, the accuracy of the methods was found to deteriorate due to incompatible nature of the shape functions along interfaces with adjacent finite and infinite elements. Yang & Yun [27] and Karasudhi & Liu [28] eliminated this limitation by introducing extra unknowns and effective integration scheme on the infinite formulation. At this juncture it is noteworthy that only ‘decay’ type infinite elements have been proposed for the elastic wave propagation problem, while ‘mapped’ type ones are mostly developed for the scalar wave problem. This is because the ‘decay’ type can easily deal with multiple wave components in the infinite element formulation. Thus the infinite element for the elastodynamic problem necessitates integration over infinite interval, as follows:

0( ) ikxI G x e dx

∞−= ∫ (1)

where 1i ≡ − and k is wavenumber which is in general complex-valued and represents attenuation and frequency of displacement in space. Various schemes including the Gauss- Laguerre quadrature, Newton-Cotes formula and analytical integration can be employed for computation of the integral [22]. In the aforementioned infinite elements for the SSI analysis, the interior near-field consists of a upper cylindrical region and a hemisphere underneath. The exterior far-field of the cylindrical near-field is then modeled by horizontal infinite elements, while the outer domain of the hemispherical near-field is represented by radial infinite elements. Introducing the hemispherical near-field, the body waves which have hemispherical wavefronts can easily and exactly be formulated in the outer hemispherical region. Another way of modeling strategy is proposed by Chow & Smith [6] and Zhang & Zhao [29]. This approach is more natural and economical than the previous method, and its effectiveness is noticeable especially for wider near-field domain. However, in their formulations, the number of wave components included in an infinite element was restricted to one. Yun et al. [30] further developed to include any number of multiple waves in infinite element formulation for the layered half-space. In addition, we note work done by Yang et al. [31] who successfully solved dynamic problems in 3D layered half-space subjected to moving loads. In spite of these progresses, three dimensional infinite element technique for the SSI analysis has left us with a challenging topic, which studied by Zhao & Valliappan [32], Park [33], and recently Seo et al. [34]. It would also be interesting to note some recent works dealing with the wave propagation in two-phase soil media, in which interactions between solid skeleton and pore water are taken into account by considering compressibility and flow of the pore water [35-38]. All the elastodynamic infinite elements aforesaid have dealt exclusively with time-harmonic loading. For the sake of transient analysis, an indirect approach utilizing a special infinite element formulation resulting in closed-form frequency-dependent mass and stiffness matrices in conjunction with a continuous Fourier transform turned out to be very effective [39-41, 33]. Unfortunately a direct time domain elastodynamic infinite element whose shape functions may describe transient behavior has not been developed. This paper presents dynamic infinite element formulations which have been developed for the dynamic SSI analysis both in frequency and in time domains by the present authors and our colleagues during the past twenty years. Axisymmetric, 2D and 3D layered half-space media were considered in the developments. Most of efforts have been devoted to establish wave functions of the infinite elements for those dimensions by using approximate expressions of analytical solutions in frequency domain. They represent the characteristics of multiple waves propagating into the outer domain of multi-layered soil. The shape functions are then constructed based on the shape functions of adjacent finite elements as well as the wave functions which reflect the spatial and material properties of the far-field

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soil. For the sake of time domain analysis, a special infinite element formulation whose element matrices can analytically be transformed into time domain ones by utilizing a continuous Fourier transform. The proposed infinite elements were verified using various benchmark examples, which showed that the present formulations are very effective for SSI analysis either in frequency or in time domain. Example applications to actual SSI problems, including the Hualien LSST project [42-45] and a liquid storage tank on a compliant ground [46], were also given to demonstrate the capability and versatility of the present methodology. DYNAMIC INFINITE ELEMENT FORMULATIONS 1. Governing Equations The harmonic motion of a multi-layered exterior region EΩ in a SSI system can be represented by the Navier's equation as:

( ) EGG Ω=+×∇×∇−⋅∇∇+ in2 2 0uuu ρωλ for a homogeneous linear elastic solid (2)

where the displacement field is defined as u x( ; )ω ωei t ; ω is circular frequency; x denotes location vector; λ and G are Lamé's constants; and ρ is the mass density. Since the problem, except for special cases with a bedrock, remains unsolved, approximate solutions of the free-field problem have been utilized for the shape functions of infinite elements. 2. Geometric Mapping The structure and near-field region, in this study, are modeled by utilizing the standard finite elements, and the exterior is discretized by the proposed horizontal, vertical and corner infinite elements for 2D and axisymmetric bodies; i.e., HIE, VIE and CIE as in Fig. 1(a), while HIE, HCIE, VIE, VCIE, and VHCIE for a 3D body as in Fig. 1(b). The cylindrical coordinate system ( zr ,,θ ) and Cartesian coordinate system ( zx, ) are respectively chosen for the axisymmetric and 2D bodies, while Cartesian coordinate system ( zyx ,, ) for 3D problem. The Fourier series expansion method in the θ–direction is employed for the axisymmetric problem under 3D loads. The global and natural coordinate systems and a typical mesh for finite element incorporating the infinite elements are shown in Fig. 1. In the 2D plane as shown in Fig. 1(a), geometric mappings of the infinite elements from the local coordinates to the global are defined for the HIE, CIE and VIE, as follows:

⎪⎪

⎭

⎪⎪

⎬

⎫

ζ−=η=

ζ−=ξ+=

η=ξ+=

∑

∑

=

=

VIEfor,)(

CIEfor),1(

HIEfor)(),1(

01

00

10

zzxLx

zzxx

zLzxx

N

jjj

N

jjj

in 2D plane (3)

where N is the number of nodes for horizontal and vertical infinite elements; )(ηjL denotes Lagrange’s interpolation function associated with node j; ),( 00 zx are the coordinates at the corner node of the near-field; and

),0[, ∞∈ζξ . The mapping in the 3D space is also defined in the same way [34].

η

( )freeb ωr

2-D plane strainfinite elements

HIE

Vertically Incident Plane Body Waves

Structures

Inhomogeneous Near-Field Soil

Horizontally Layered Far-Field Soil

Homogeneous Half-Space

Equivalent earthquake

force on bΓ

bΓ

bΓ

HIE

CIE

VIE

ζ

ξ

ηξ

ζ

(a) 2D or axisymmetric plane (b) 3D space

Figure 1: Modeling of soil-structure systems with multi-layered half-space by FE-IE method

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Table 1 Approximate wave functions (see Fig. 1 to refer local coordinate systems)

Horizontal layer Underlying half-space

Body waves Surface waves Body waves Surface waves

2D (2) ( )mH kr ikxe− (2) ( )mH kr | |ikx ze e μ− −

Axi.

Exact wave

functions 3D

(2) ( )mh kR (2) ( )mH kr (2) ( )mh kR (2) | |( ) zmH kr e−μ

2D 0(0.25 )ikxe− + ξ 0ikxe− ξ 0(0.25 )ikxe− + ζ ,

0 0(0.25 ) (0.25 )ikx ikxe e− + ξ − + ζ 0ikxe e− ξ −μζ

Axi. 0(0.75 )ikre− + ξ 0(0.25 )ikre− + ξ (0.75 )e ik− + ζ , 0 0(0.75 ) (0.75 )e eikr ikr− + ξ − + ζ

0(0.25 )ikre e− + ξ −μζ Approximate wave

functions

3D

(0.75 )jx jikxe−θ + ξ , (0.75 )j

y jikye−θ + η , (0.75 ) (0.75 )j j

x j y jikx ikye e−θ + ξ −θ + η

(0.25 )jx jikxe−θ + ξ , (0.25 )j

y jikye−θ + η , (0.25 ) (0.25 )j j

x j y jikx ikye e−θ + ξ −θ + η

(0.75 )jz ike−θ + ζ

(0.75 ) (0.75 )j jx j zikx ike e−θ + ξ −θ + ζ ,

(0.75 ) (0.75 ) (0.75 )j j jx j y j zikx iky ike e e−θ + ξ −θ + η −θ + ζ

e−μζ , (0.25 )j

x jikxe e−θ + ξ −μζ , (0.25 ) (0.25 )j j

x j y jikx ikye e e−θ + ξ −θ + η −μζ

3. Formulations for Frequency Domain Analysis The displacement field of an infinite element in the 2D plane,

);( ωxu , can be approximated as:

∑∑= =

ωω=ωN

j

M

mjmjmN

1 1)();();( pxxu or ( ; ) ( ; ) ( )pω = ω ωu x N x p (4)

where N is the number of nodes or the number of wave functions depending on the type of infinite element; M is the number of wave functions included in the formulation; and p jm( )ω is a unknown vector in the generalized coordinate associated with N jm . The shape function N jm is expressed in the 2D plane as:

( ; ) ( ) ( ; )xjm j mN L fω = η ξ ωx for HIE; ( ; ) ( ; ) ( ; )x z

jm m mN f fω = ξ ω ζ ωx for CIE; and ( ; ) ( ) ( ; )zjm j mN L fω = η ζ ωx

for VIE. In the shape functions, xmf and z

mf are wave functions in x- and z-axis respectively which can be obtained from the approximate expressions of analytical solution as given in Table 1. All the wave functions have unit value at FE-IE interface, i.e., ξ = 0 or ζ = 0. In Table 1, wavenumber k associated with the body waves varies with layer and is a function of frequency. On the other hand, the values of k and μ associated with surface waves are in general nonlinear function of frequency, which is called ‘dispersion’. In this study, the dispersion curves are determined by solving transcendental eigen equations prior to the SSI analysis. For the purpose of assembling element matrices, it is required to express the displacement field in terms of shape functions associated with nodal displacements, displacements along sides of the infinite element and internal displacements. Hence, Eq. (4) is rewritten as:

( ; ) ( ; ) ( )qω ω ω=u x N x q (5)

in which ],,[);( bsdq NNNxN =ω ; TTTT bsdq ,,= ; d is nodal displacement vector; while s and b are

amplitude vectors of side and bubble modes, respectively. Typical shape functions for HIE and CIE are shown in Fig. 2 and Fig. 3. Thus, a transformation matrix between two generalized coordinates, )(ωp in Eq. (4) and )(ωq in Eq. (5), can be introduced as:

)()( ω=ω qTp pq , pqpq TxNxN );();( ω=ω (6)

where pqT is the constant transformation matrix which is defined for each type of infinite elements [30].

For 3D infinite element formulation, the wave functions shown in Table 1 are established in conjunction with local coordinate systems depicted in Fig. 1. All the procedures are similar to those as for the 2D formulation given above, although the shape functions and the transformation are much more complicated as the dimension increase. Typical shape functions are shown in Fig. 4, and Fig. 5 demonstrate how the shape functions for the 3D infinite elements describe a displacement field.

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The element stiffness and mass matrices of the infinite element can be computed in a way similar to the finite element method, as follows:

( )e

T eqq q q d

Ωω = Ω∫K B DB , ( )

e

T eqq q q d

Ωω = ρ Ω∫M N N (7)

Thus the dynamic stiffness matrix of infinite element, )(ωqqS , is also constructed in the same manner of the finite element method as:

2( ) (1 2 ) ( ) ( )qq qq qqiω = + ξ ω − ω ωS K M (8)

in which ξ is the hysteretic damping ratio of the infinite element.

An effective integration procedure is proposed by Yang & Yun [27] by devising wave-by-wave integration procedure and by modifying the Gauss-Laguerre quadrature, and later elaborated by Yun et al. [30] by employing the transformations as:

K T K Tqq pqT

pp pq= , M T M Tqq pqT

pp pq= (9)

where Tpq is the transformation matrix defined in Eq. (6). It should be noted that computations of the matrices K pp and M pp are much easier than those of qqK and qqM , since shape function pN involves single wave

component only while qN contains multiple wave functions. The integration including single wave functions in the infinite direction may be carried out by using the Gauss-Laguerre quadrature as [27]:

k

N

k

ksx WsxG

sdxexG ∑∫

=

−∞≅

int

10

)(1)( (10)

where s is in general a complex number; xk and Wk are the Gauss-Laguerre integration point and weight factor, respectively; and N int is the number of required integration points. In general, this integration scheme gives good results for non-oscillating functions of G x( ), while it may yield erroneous results for oscillating functions. In this study, the five-point Gauss-Laguerre rule has been used for all numerical examples, i.e., Nint = 5 .

The equation of motion for the SSI system including impedance matrix of the far-field soil, )(ωgbbS , can be written as:

⎭⎬⎫

⎩⎨⎧

ωω

=⎭⎬⎫

⎩⎨⎧

ωω

⎥⎦

⎤⎢⎣

⎡ω+ωω

ωω)()(

)()(

)()()()()(

b

s

b

sgbbbbbs

sbss

FF

UU

SSSSS

(11)

where )(ωU and )(ωF are respectively displacement and force vectors, and )(ωS is the dynamic stiffness matrix in frequency domain. Subscript b stands for the nodes along the interface between the near- and far-field and s for those of the structure and the near-field soil region.

+1

0-1

η

ξ

+

Re{ }NS,12 (ξ,η)

(b) Side mode

-1 0

+1 η

ξ

Re{N11(ξ,η)}

-1 0

+1 η

ξ

Re{N21(ξ,η)}

(a) Nodal modes

+1

0-1

η

ξ

Re{ }Nb,22(ξ,η)

(c) Bubble mode

Figure 2: Example shape functions of a 3-node horizontal infinite element: Real parts only (Axisymmetric bodies)

— 185 —

Re{ ( , )}N21 ζ ξ

(0,0)

ζ

ξ Re{ ( , )}N22 ζ ξ

(0,0)

ζ

ξ

Figure 3: Typical shape functions for a corner infinite element: Real parts only (Axisymmetric bodies)

η ξ

ζ

η ξ

ζ

η ξ

ζ

(a) Nodal mode (b) Edge mode (c) Face mode

Figure 4: Typical shape functions for a 3D horizontal infinite element: Real parts only (3D body) XY

Z

FE

STR

HIEHIE

HCIE

YX

Z

STR

HCIE

HIE

HIE

VIE

FE

VCIE

VCIE

(a) Plan view (b) 3D view

Figure 5: An example of displacement field constructed using 3D infinite elements (3D body)

4. Formulations for Time Domain Analysis For the SSI analysis in frequency domain, the dynamic stiffness matrices of the infinite elements shall be calculated at each frequency step in accordance with wavenumbers which are nonlinear function of the frequency. The frequency-dependency of the dynamic stiffness is a distinct characteristic of the SSI system. However, it makes the time domain analysis of the system be extremely difficult or inefficient compared with the frequency domain analysis [1].

If we let the wave function to be as ( ; ) exp( )mf ξ ω = −κ ξ , then the exponent of the wave function becomes implicit functions of the frequency. Therefore the dynamic stiffness matrix for the infinite domain can be only obtained numerically at each frequency point. To derive the dynamic stiffness matrix analytically, the exponent can be further approximated as linear function of frequency as:

( ) ( )m ma i Cκ ω ≅ + ω (12)

where the newly introduced positive constant, a , which is related to the geometric attenuation, is taken to be the same for all wave components. The value of a in Eq. (12) is determined by an error minimization procedure [39]. Owing to the simplification, the element mass and stiffness matrices are expressed analytically in terms of frequency and constant matrices as:

0 12

1 1( )( )qq a i a i

ω = ++ ω + ω

M M M , 0 1 21( ) ( )qq a i

a iω = + + ω +

+ ωK K K K (13)

— 186 —

where 0M , 1M , 0K , 1K and 2K are complex-valued constant matrices. Finally, assembling the mass and stiffness matrices of the infinite elements, the impedance matrix of the far-field soil region, ( )g

bb ωS , can be obtained as:

0 1 2 32

1 1( )( )

g g g g gbb i

a i a iω = + ω + +

+ ω + ωS S S S S (14)

where 0gS , 1

gS , 2gS and 3

gS are complex-valued constant matrices.

Since the impedance matrix of the far-field region is obtained as an analytical function of frequency, Eq. (11) can be analytically transformed into time domain by taking Fourier transform as [40]:

( )01 2 30

( )( ) ( ) ( )( ) ( ) ( ) ( ) { ( ) } ( )

sss sb s s ss sb stgg g g a t

bs bb b b bs bb b b b

tt t tt t t t t e d− −τ

⎧ ⎫⎡ ⎤ ⎧ ⎫ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎡ ⎤ ⎪ ⎪+ + =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎢ ⎥ + − + − τ τ τ⎣ ⎦⎣ ⎦ ⎩ ⎭ ⎩ ⎭ ⎣ ⎦ ⎩ ⎭ ⎪ ⎪⎩ ⎭∫fM M u u K K u0 0

M M u u K K S u0 S f S S u&& &

&& & (15)

where )(tu and ( )tf are displacement and force vectors in time domain respectively; g0S and g

1S represent

contributions of the exterior region to the static stiffness and viscous damping matrices; and g2S and g

3S are related to the lingering responses on the interface between the near- and far-field. The SSI system may be analyzed in time domain by using Eq. (15). Furthermore, in case that the nonlinear behavior of the structure or the near-field soil is significant, the nonlinear effect may easily be included in Eq. (15) and the response can effectively be analyzed in time domain. 5. Earthquake Response Analysis The earthquake responses can be obtained by solving the wave radiation problem with equivalent earthquake force ( )free

b ωr applied along interface bΓ as shown in Fig. 1(a). The force

vector can be calculated using the free-field motions (displacement freebu and traction free

bt ) and the impedance matrix of the far-field regardless of the properties of near-field as [47]:

( ) ( ) ( ) ( )free g free freeb bb b bω = ω ω − ωr S u At (16)

where A is a constant transformation matrix. When a nonlinear analysis is carried out by changing its shear modulus and damping ratio which are equivalent to a reference strain in the region, the dynamic stiffness matrix for the near-field soil shall be modified at each iteration step. However, the earthquake force vector can be assumed unchanged during the iteration process, if the far-field region was selected so far from the structure that nonlinear behavior of soil may be negligible in the region.

For the purpose of time domain earthquake response analysis, the equivalent earthquake input force, ( )freeb ωr , is

firstly computed along the interface in the frequency domain. Then, the earthquake forces on the interface in the time domain, ( ) ( )free

b bt t=f r , are computed through the discrete Fourier transform technique. Finally, the earthquake responses are computed by solving Eq. (15), in which force vector ( )s tf is equal to zero [41]. VERIFICATION EXAMPLES 1. A Rigid Strip Foundation on an Elastic Layer with Rigid Bedrock (2D IE) A rigid strip foundation on an elastic soil with rigid bedrock is modeled as in Fig. 6. Two Rayleigh waves and two body waves are used in the infinite element formulation. The vertical compliance of the foundation was obtained using the numerical and closed-form frequency-dependent infinite elements, and compared with each other along with reference solution by Tassoulas [48] in Fig. 6. Very good agreements have been found between them. 2. A Rigid Circular Disk on Layered Hals-Space Medium (Axisymmetric IE) Horizontal impedance has been obtained for the case that the depth of the upper horizontal layer (h) is R0 . Both the horizontal layer and the half-space are assumed to be elastic, homogeneous and isotropic but having different soil properties; i.e., shear wave velocities (C Cs s1 2 0 8/ .= ), densities (ρ ρ1 2 0 85/ .= ) and Poisson's ratio (ν ν1 2 0 25= = . ). Two body waves and three Rayleigh waves have been also used for infinite elements. The effect of the horizontal dimension of the near-field has been also investigated using three different meshes shown in Fig. 6; i.e., r R0 012= . , 15 0. R and 2 0R . The results shown in Fig. 7 indicate that reasonable solution can be obtained if the horizontal dimension of the near-field is taken to be larger than 15 0. R . The results are also found to be in good agreement with the analytical solutions obtained by Luco [49].

— 187 —

∞

∞

∞

∞

b

H=2b

0 .0 1.0 2.0 3.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Re

Tassoulas

Numerical IEAnalytical IE

0 .0 1.0 2.0 3.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-Im

(a) FE & IE mesh (b) Compliance of vertical motion

Figure 6: Vertical compliance of a rigid strip foundation on a layer with fixed bedrock (2D analysis) CL R0

h

CL R0CL R0

kHH

cHH

h

R0

0r

Present

Luco

study⎫⎬⎭

r R0 0 12/ .=r R0 0 15/ .=r R0 0 2 0/ .=

0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 60.2

0.6

1.0

1.4

a RCS

00

1= ω a R

CS0

0

1= ω

Figure 7: Horizontal impedances of a rigid disk on a layered half-space using three different finite

element meshes for the near-field (h R/ .0 1 0= ) (Axisymmetric analysis)

0 1 2 3ao=ωB/cs

0

0.5

1

kV

cV

This study

Chow(1987)

hBL

0 1 2 3

ao=ωB/cs

0

0.5

1kV

cV

(a) Vertical impedance (h/B=4) (b) Vertical impedance (h/B=6)

Figure 8: Vertical impedance of rectangle footing on a soil layer with bedrock (L/B=2, ν=0.33, ξ =0.05) (3D analysis)

a b cs0 = ω /

— 188 —

3. A Rigid Rectangular Disk on Layered Half-Space (3D IE) A rigid rectangular footing on a horizontal soil layer with an underlying rigid rock is analyzed. The impedance function of the rectangular footing is expressed as

o o( ) = ( +i )(1 2 )staz z dk a k k a c i+ ξ , where

stazk is the vertical static stiffness. HIE and HCIE are used for the horizontal

exterior region of the soil layer, while the boundary condition along the interface with the bed rock is taken as fixed. Two cases with different depth ratios ( 4h B= and 6B ) are considered. Fig. 8 shows the results for the rectangular footing along with the results by Chow [50]. The present results are found to be in good agreement with the reference values. 4. A Strip Foundation on a Two-Phase Medium (2D Body) As a demonstrative extension of the present methodology, we present results of forced vibration of a strip foundation on a two-phase soil layer with fixed bedrock, as in Fig. 9. In the two-phase layers, the wave propagation mechanism is much more complicated than that in single-phase soil, e.g., three body wave components exist including P1-, P2- and S-waves. However, the infinite element formulation essentially remains the same and straightforward [37]. Fig. 9 shows the accuracy of the present method, and indicates that multiple wave components shall be considered in the analysis even though other researchers [35, 36, 38] obtained reasonable results by considering one or two wave components only.

Q=389.41MPa, E=100MPa α=1, ν=0.3 n=0.4, k=0.02 m/s ρ=2.0Mg/m3, ρf=1.0Mg/m3

15m

B=5m

x0

HIE HIE HIE HIE HIE HIE

20MPa

0 10 20 30 40 50 60 70 80Frequency (rad/sec)

0.0

1.0

2.0

3.0

4.0

Ampl

itude

Amplitude of Footing Displacement (u_z)

(x0/B = 11)

0 10 20 30 4 0 5 0 60X-coordinate (m )

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

Am

plitu

de

Real (u_z) at ground surf a c e ( ω = 2 0 r a d / s e c)

x 0 / B =

1 1

Figure 9: Frequency response of a 2D strip foundation on a two-phase soil layer upon fixed bedrock

(Exact: ) (FE only: ◇ ) (FE-IE : ◆= including 1 wave, ■= 2 waves, ●= 3 waves)

5. A Rigid Circular Disk on Flexible Hollow Caisson (Axisymmetric IE and 3D IE) A rigid circular footing on a RC caisson shown in Fig. 10 is analyzed using the FE-IE approach. The axisymmetric and 3D infinite elements are employed for modeling the far-field of soil. The material properties are given in the figure. The impedance functions are normalized by 0RGS for the horizontal and the vertical components, while they are normalized by 3

0RGS for the rocking and the torsional components. The results are shown in Fig. 10, and compared with those based on the indirect boundary element method by Chen & Penzien [51]. Excellent agreements have been achieved using both types of axisymmetric and 3D infinite elements. APPLICATION EXAMPLES

1. Simulation of Forced Vibration Test & Earthquake Responses (Axisymmetric Body) This example presents the result of an international cooperative research on the post-correlation analysis of forced vibration tests and the prediction of earthquake responses of a large-scale seismic test (LSST) structure. Through the system identification technique, the properties of the soil layers are revised so that the best correlation in the responses is obtained as shown Fig. 11. The revised values are shown in Table 2 [41]. Utilizing the revised soil properties as the initial linear value, the seismic responses are predicted for an earthquake using the equivalent linearlization technique. It has been found that the predicted responses by the equivalent nonlinear procedure are in excellent agreement with the observed responses as shown in Fig. 12.

— 189 —

Hei tω

R0= 4 m

1 m

12 m

1 m

Caisson

0.6 m

Mei tω

Tei tωVei tω

Ec Es/ = 500ρc ρs/ = 1.5νc νs= 0.25=ξs= 0.02ξc= 0.0

HIE

HIE

HIE

HIE

HIE

HIE

HIE

HIE

HIE

HIE

CIEVIEVIEVIEVIEVIE

cVV

kVV

0.0 0.5 1.0 1.5 2.0 2.5 3.0-40

-20

0

20

40

60

80

a0 = R0ωVS

kTT

cTT

0.0 0.5 1.0 1.5 2.0 2.5 3.00

10

20

30

40

50

60a0 = R0ω

VS

Chen and PenzienPresent study

cHH

kHH

0.0 0.5 1.0 1.5 2.0 2.5 3.010

20

30

40

kMM

cMM

0.0 0.5 1.0 1.5 2.0 2.5 3.40

80

120

160

200

240

280

a0 = R0ωVS

a0 = R0ωVS

cHM

kHM

0.0 0.5 1.0 1.5 2.0 2.5 3.0

20

40

60

80

100

a0 = R0ωVS

(a) Using axisymmetric FE-IE

HIE

VIE

VCIE

HCIE

HIEHIE

0 1 2 3

ao=ωB/cs

10

15

20

25

30

kHH

cHH

Chen & Penzien(1986)

This study

0 1 2 3

ao=ωB/cs

40

80

120

160

200

240

280

kMM

cMM

(b) Using 3D FE-IE Figure 10: Impedances of a rigid disk on a hollow caisson embedded in homogeneous half-space

H11

H7GL

GL -2 m

GL -5.15 m

GL -12.15 m

Sand-1

Gravel-3

Gravel-4

Sand-2

Backfill-1

Backfill-2

Gravel-1

V3

H15

V2 V1

Excitation

Measurements

Gravel-2

30

250 200 265 150 541

110150

315 Base Slab

1363

0.5 m

∞ ∞

∞

V9138 30 326 110

C.L.

H12

Shell-1

Shell-2

Shell-3

1163

Roof Slab

∞

SB-1

SB-2

SG-2 SG-1

0 5 10 15 20

Frequency (Hz)

0.0

2.0

4.0

6.0

8.0

Ampl

itude

(mm

/MN

) H15 Test (D1)

Calculated

Figure 11: Simulated and measured forced vibration test (D1-direction)

— 190 —

0 1 1 0 100F r eque n c y ( H z )

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

Spec

tral A

ccel

erat

ion

(g)

(a) H15RecordedNonlinearLinear

Spec

tral A

ccel

erat

ion

(g)

0 1 1 0 100Fr e q u e n c y ( H z )

0.0

0.2

0.4

0.6

0.8

(b) H7

Figure 12: Response spectra of simulated and measured accelerations in structure (D1-direction)

Table 2 Identified values of the parameters for from the Hualien LSST structure using FVT data, and equivalent linear properties (Note: Values in parentheses are hysteretic damping ratios in percent.)

Parameters (VS in m/sec, E in GPa)

FVT simulation & Linear Earthquake Response Analysis

Nonlinear Earthquake Response Analysis (May 1, 1995)

Sand-1, Sand-2 (VS) 133 (2.0), 231 (2.0) 113 (4.7), 199 (4.5) Backfill-1, SB-1 (VS) 270 (2.0) 244 (2.4), 207 (5.3) Backfill-2, SB-2 (VS) 325 (2.0) 257 (4.8), 234 (6.4) Gravel-1, SG-1 (VS) 308 (2.0) 262 (5.0), 248 (6.0) Gravel-2, SG-2 (VS) 281 (2.0) 214 (7.0), 203 (7.9)

Gravel-3, Gravel-4 (VS) 333 (2.0), 388 (2.0) 211 (7.2), 340 (4.4) Roof & Base (E) 28.2 (2.0) 28.2 (2.0)

Shell (E): Upper, Middle, Lower 19.7 (2.0), 21.3 (2.0), 21.8 (2.0) 19.7 (2.0), 21.3 (2.0), 21.8 (2.0)

2. Time Domain SSI Analysis For verification of the proposed time-domain SSI analysis procedure, earthquake response analysis of a multi-layered free field half-space shown in Fig. 13 is carried out. The near-field soil region is discretized with plane strain finite elements and the remaining far-field soil region is modeled by the analytical frequency-dependent infinite elements [39]. The properties of the soil layers are shown in Table 2. A horizontal acceleration record is used as the input control motion on the ground surface, which is the NS-component of an earthquake measured at Hualien, Taiwan on January 20, 1994. The peak ground acceleration is 0.0318g, and the time history is shown in Fig. 13. The acceleration histories are compared with those of the free field analysis which are obtained based on the frequency domain method. The results in Fig. 13 show excellent agreements.

Sand 1

Sand 2

Gravel 1

Gravel 2Half-space

2.0 m

3.15 m

7.0 m

Vertically Incident P-wave and SV-wave

9-node FE

Boundary for earthquake force input

3-node HIE

3-node VIE

1-node CIE

z

x A2 A3A1

B2 B3B1

C2 C3C1

D2 D3D1

-0.04

0.00

0.04

Acce

lera

tion(

g) (a) Control motion (Free surface)

-0.04

0.00

0.04

Acce

lera

tion(

g) (b) Earthquake analysis (A1)

-0.04

0.00

0.04

Acce

lera

tion(

g) (c) Earthquake analysis (A2)

0 10 20 30Time (sec)

-0.04

0.00

0.04

Acce

lera

tion(

g) (d) Earthquake analysis (A3)

(a) Free field analysis problem (b) FE-IE mesh (c) Time history responses

Figure 13: Earthquake response analysis of a layered soil medium

— 191 —

3. Flexible RC tank on a layered half-space (Axisymmetric Body) In order to demonstrate the soil-structure interaction effect on the member forces of a liquid storage tank, a stress analysis is carried out for a structure depicted in Fig. 14 under various soil conditions. The structure is supported by a horizontal layer with the underlying bedrock. Three values of the shear wave velocity for the horizontal soil layer, i.e., 500m/s, 800m/s, and 5000m/s, are considered in this investigation. The other material properties for the structure and soil regions are given in Fig. 14. The frequency-dependent mass matrix associated with the DOF along wetted interface is included in the equation of motion in the frequency domain, in which the fluid motion is represented by Bessel functions [46]. An acceleration time history with PGA of 0.14g, which is compatible with a design response spectrum for a rock site is simulated for the earthquake input as in Fig. 12. In this analysis, the control acceleration is assigned at the top of bedrock as a horizontal outcrop motion. Thus, seismic motion can be amplified at the ground surface depending on the properties of the horizontal soil layer. Member forces are calculated on the vertical shell for three different soil conditions including both the fluid-structure interaction and soil-structure interaction, and their maximum values are plotted along the height of the structure in Fig. 14. For the purpose of comparison, the maximum member forces are also computed using ANSYS program for the same structure but on a rigid ground. A fully coupled fluid-structure-soil interaction analysis cannot be carried out by ANSYS program. In ANSYS analysis, the input ground acceleration at the fixed base is prepared for each soil condition by carrying out the free-field analysis using SHAKE91 program. Thus, the solution by ANSYS can be considered as the response for the same input motion but excluding the soil-structure interaction effect. Two sets of the results for a rigid soil condition by the present and ANSYS analysis (in Fig. 14) are found in good agreements, which confirms the accuracy of the present analysis. The results for the softer soil conditions indicate that the member forces on the shell reduce considerably as the soil stiffness decreases. This result re-confirms that accurate dynamic analysis of a large liquid storage tank considering the soil-structure interaction may yield cost-effective cross-section for the structure.

Bedrock (ρs=2.5Mg/m3, Vs=5000m/sec νs=0.2, h=2%)

50m H=40m

R=30m

104m

20mCompliant Soil

C.L.

(ρs=2.5Mg/m3, νs=0.2, h=2%)

3m

RC Shell ρc=2.6Mg/m3, Es=30MPa, νc=0.2, h=2%

0.8m

Water

ρw=1.0Mg/m3

0 2000 4000 6000 8000Member force (KN/ m)

0

10

20

30

40

50

Hei

ght f

rom

flui

d bo

ttom

(m)

0 2000 4000 6000 8000Member force (KN/ m)

0

10

20

30

40

50

Hei

ght f

rom

flui

d bo

ttom

(m)

0 2000 4000 6000 8000Member force (KN/ m)

0

10

20

30

40

50

Hei

ght f

rom

flui

d bo

ttom

(m)

(a) Vs=5000m/s (b) Vs=800m/s (c) Vs=500m/s

Figure 14: RC liquid storage tank and maximum member force profiles for a RC liquid storage tank (Nt at 0oθ = , Nz at 0oθ = , and Ntz at 90oθ = )

Nt Nz Ntz

SSI excluded(ANSYS)

Nt Nz Ntz

SSI included (Present Study)

0 1 0 20 30 T i m e ( s e c ) - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2

Acc

eler

atio

n (g

)

— 192 —

CONCLUDING REMARKS This paper presented infinite element formulations for the dynamic SSI analysis of an axisymmetric body subject to 3-D loads, a 2D problem, and a 3D analysis both in frequency and in time domains. The displacement shape functions of the infinite elements were constructed using approximate expressions of analytical solutions in frequency domain to represent the characteristics of multiple waves propagating into the infinite multi-layered soil medium. For the sake of time domain analysis, the dispersive wavenumbers included in shape functions were linearized to obtain closed-form frequency-dependent element matrices, which can analytically be transformed into the time domain terms by a continuous Fourier transform. The proposed elements were verified using several benchmark examples. Comparisons with the results by other studies showed that the present formulations are very effective for the SSI analysis either in frequency or in time domain. Example applications to actual SSI problems were also given to demonstrate the versatility and effectiveness of the present methodology. Acknowledgement This study has been supported by SISTEC (KOSEF/MOST: Grant No. R11-2002-101-03001-0). Their financial supports are greatly acknowledged. Also the second author would like to thank KOSEF (Grant No. R01-2003-000-10635-0) for the financial support to this study.

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