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Numerical Modeling of Dynamic Soil-Structure Interaction during

Earthquakes

Delong Zuo

Abstract

The developing history of numerical modeling of dynamic soil-structure

interaction was reviewed, with emphasis given to the parallel development of

the direct method and the substructure method. Formulation of a model

developed with the substructure method was discussed in detail. Future

development in the modeling of soil structure interaction was anticipated and

possible challenges were suggested.

Key Words: Soil-Structure Interaction, Direct Method, Substructure Method,

Time Domain Analysis, Frequency Domain Analysis

Introduction

During the last couple of decades, it has been well recognized that the soil on

which a structure is constructed may interact dynamically with the structure

during earthquakes, especially when the soil is relatively soft and the structure is

stiff. This kind of dynamic soil-structure interaction can sometimes modify

significantly the stresses and deflections of the whole structural system from the

values that could have been developed if the structure were constructed on a

rigid foundation. Two important characteristics that distinguish the dynamic

soil-structure interaction system from other general dynamic structural systems

are the unbounded nature and the nonlinearity of the soil medium. Generally,

when establishing numerical dynamic soil-structure interaction models, the

following problems should be taken into account:

1. radiation of dynamic energy into the unbounded soil;

2. the hysteretic nature of soil damping;

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3. separation of the soil from the structure;

4. possibility of soil Liquefaction under seismic loads; and

5. other inherent nonlinearities of the soil and the structure.

However, due to the complexity of dynamic soil-structure interaction,

numerical modeling of this phenomenon still remains a challenge. There still

exist many difficulties to cover in one model all the problems listed above.

Current models usually stress one or several of these problems.

Developing History and Classification of Numerical Models

Modeling and analysis of dynamic soil-structure interaction during

earthquakes initiated with the Finite Element Method in the 1960’s. They have

gone through various stages, but always in two distinct directions, that is, the

substructure method and the direct method, depending on the modeling method

for the soil around the structure. In the substructure method, the soil-structure

system is divided into two substructures: a structure that may include a portion

of nonlinear soil adjacent to it and the unbounded soil. The unbounded soil

region is usually represented by an impedance matrix, which may be attached to

the dynamic stiffness matrix of the structure. In the direct method, the structure

and the soil adjacent to it are modeled directly.

For a long time, modeling of dynamic soil structure-interaction was carried out

in the frequency domain, which restricted the analysis of the soil-structure

system to be linear. Nonlinearity of the soil was taken into account only in an

approximate manner through equivalent linear analysis procedure in which

dynamic soil parameters were adjusted in accordance with the peak or the

average strain during iterative solutions of the system in frequency domain,

whereas the structure had to be assumed to be linear. [1] To address this problem,

the direct method went into the time domain, using well-established procedure

of structural dynamics. But, at this stage, the direct method still could not model

the energy radiation effect, whereas the substructure method, which remained in

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the frequency domain, could model this phenomenon very well. In response,

there began in the direct method the development of “transmitting boundaries”,

such as the early “viscous boundary”(Fig. 1a) proposed by Lysmer, J. etc.[2] and

then the various kinds of “consistent boundaries”(Fig. 1b). The general purpose

of these ”transmitting boundaries” is to avoid the reflection of waves emanating

from the structure and the adjacent soil. Some more recent “transmitting

boundaries” are frequency dependent and made the direct method enter the

frequency domain again and ready to model the hysteretic nature of soil

damping.

On the other side, the substructure method incorporated with it the rapid

developing Boundary Element Method, with which theoretically any type of

geometrical boundary conditions can be simulated in both frequency domain

and time domain by discrete modeling of the soil-structure interface. At this

stage, the analysis of the unbounded soil region was carried out by discretized

Green’s Function,[3] whereas the soil adjacent to the structure was treated as a

part of the structure with Finite Element Modeling.

More recent research of dynamic soil-structure modeling tends to be

concentrated in the time domain, not only because the problem of nonlinearity

can be better simulated in the time domain than in the frequency domain, but

also that “ the typical structural analyst is not accustomed to working in the

frequency domain; his natural approach is to consider the sequence of

(a) (b)

Fig. 1 Schematic Representation of Viscous Boundary and Consistent Boundary[1]

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developments from one time to the next—that is to apply the time domain

concept”.[4] At the same time, the concept of discretizing the soil-structure

interface has been strengthened. Fig. 2 shows a typical model of the direct

method and a typical model of the substructure method.[1] Both models consist of

an irregular soil zone and a regular soil zone divided by a so-called “Interaction

Horizon”, through which the seismic input motion is applied. The counter part

of the “Interaction Horizon” in the direct method and the substructure method

are the “transmitting boundaries” and the “irregular-regular soil interface”

respectively. In both methods, the irregular soil zone is modeled with Finite

Element Method and material and geometric nonlinearities are considered,

whereas the regular soil zone is usually taken as linear. The main difference

between this two models is that the transmitting boundaries of the direct method

should be uncoupled both in time and space, which leads to approximate

expressions for the boundaries, while in the substructure method, the dynamic

stiffness matrix of the unbounded regular soil zone can be obtained rigorously in

the frequency domain using convolution theorem of Fourier Transform. The

location of the interaction horizon in both methods is very important. Research

has been performed in this field to provide references.[5]

Input Motion

Regular Soil Zone

Transmitting Boundary (Interaction Horizon)

Irregular Soil Zone

Irregular Soil Zone

Interaction Horizon Regular

Soil Zone

Input Motion

(a) (b)

Fig. 2 Models of the Direct Method and the Substructure Method[1]

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It should be pointed out that numerical modeling of dynamic soil-structure

interaction is still in its course of development. There are still no standard

numerical models available. The various current models are no longer restricted

only in the time or the frequency domain alone. Techniques used to establish

numerical models are not restricted to be finite element method or boundary

element method. On the contrary, all these are always incorporated with one and

another,[6][7] and some new analysis techniques have been introduced into the

problem, such as the infinite element method.[8][9] To demonstrate the

formulation of numerical model of dynamic soil-structure interaction, a model

developed in the 1980’s is discussed in detail in the next section, because many

later models, though all have their different features, are based on this typical

model.

Example Numerical Model of Soil-Structure Interaction

Fig. 3 shows a model developed in the 1980’s, which used the substructure

method and was formulated in the time domain. In the model, the line joining

the nodes whose variables have subscript b separates the unbounded soil region

and the region consists of the structure and its adjacent soil (Variables of this

region have subscript s). The structure and the adjacent soil in this model are

modeled nonlinearly, whereas the unbounded soil region is treated as linear.

b s

Structure Adjacent Soil

Unbounded Soil

Fig. 3 A typical Model of Substructure Method

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The free field (the virgin soil before the construction of the structure, denoted

by superscript f) in the model can be treated as the superposition of the

excavated soil (denoted by superscript g) and the excavation (denoted by

superscript e)(Fig. 4). Although the soil adjacent to the structure will exhibit

nonlinear behavior in the interaction analysis, the calculation is based on the

free-field motion determined for linear soil. The time domain dynamic stiffness

matrix )]([ tS fbb of the free field contains the forces needed to give the free field soil

unit-impulse displacements. It is given as the inverse Fourier-Transform of the

dynamic stiffness matrix )]([ ωfbbS in the frequency domain:

ωωπ

ω deStS tifbb

fbb )]([

2

1)]([ ∫

∞

∞−= (1)

When the excavation is taken into account, the time domain dynamic stiffness

matrix )]([ tS gbb of the excavated soil can also be acquired by a formula similar to

(1), with the superscripts being changed to g.

The expression of the dynamic stiffness matrix in the frequency domain

( )]([ ωgbbS ) can be determined using several different methods, including the

boundary element method,[1] the infinite element method,[8] and the Green’s

Function.[3]

f g

b f

bbf

b Su gbb

gb Su

e

ebbS

Fig. 4 Reference Soil System

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With the dynamic stiffness given, the basic equation of motion in the time

domain can be formulated as:

−=

−+

+

∫

∫

t gb

gbb

t tb

gbbb

s

tb

ts

bbbs

sbss

dutS

dutStP

tP

tu

tu

MM

MM

0

0

})()]{([

}0{

})()]{([

}0{

})({

})({

})({

})({

][][

][][

τττ

τττ&&&&

(2)

In the equation, )(tu t denotes the vector of the total displacement;

)(tu gb represents the so-called scattering motion; [Μ] is the Mass matrix and {Ρ} is

the vector of the nonlinear internal forces of the system consisted of the structure

and the adjacent soil. The interaction forces acting on the boundary nodes are

equal to the convolution integral of the dynamic stiffness matrix )]([ tS gbb and the

displacement relative to the ground.

ττττ duutStR gb

tb

t gbbb }))({})()]({([})({

0−−= ∫ (3)

Material damping of the soil is very important to the analysis of the whole

dynamic soil-structure interaction system. It has been discovered that different

damping models used in the numerical soil-structure interaction models usually

bring very different results.[7] Since identification of the soil damping is still a

difficult problem, linear visco-elastic damping models, such as the Voigt Model

(Fig 5a) and the Three-Parameter Kelvin model, (Fig. 5b) are generally adopted.

In these two models, material damping is introduced by using a complex

material constant E*.

(a) Viogt Model (b) Three-Parameter Kelvin Model

Fig. 5 Linear Visco-Elastic Damping Models

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In the Voigt Model:

)21(* iEE ζω+= (4)

In the Three-Parameter Kelvin Model:

i

iEE

ζωζω

++

=1

21* (5)

In both models, E is the Young’s medullas of the unbounded soil, and ζ is the

damping ratio. Material damping of the soil is not introduced into the soil-

structure interaction system implicitly, but by replacing the Young’s medullas E

in the dynamic soil stiffness matrix with E*.

Another important factor that affects the analysis of this numerical model is

the constitutive model of the soil adjacent to the structure. In the 1980’s, when

this model was established, the nonlinear elastic soil constitutive laws were

generally adopted.

Future Prospects and Recommendations

Because of the reasons stated ahead, future numerical models of dynamic soil

structure interaction are likely to continue to be concentrated in the time domain.

And direct method may become more attractive to researchers and engineers

because it can consider the nonlinearity of the unbounded soil as long as

sufficiently accurate transmitting boundaries can be developed.[4] However,

resulting transmitting boundaries may not be as simple as attaching certain

masses, springs and dashpots directly to the interaction horizon and may come

up with sophisticated coupling both in time and space.[10] The substructure

method, on the other hand, with its clear concept and rigorous formulation, will

also continue to develop.

Although current research are focused on such places as the interaction

horizon, the unbounded soil and so on, future research may pay more attention

to the soil adjacent to the structure, since it is this region that is in consistent

dynamic contact with the vibrating structure. Such behavior of the soil as its

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separation from the structure foundation may change the behavior of the whole

system drastically. To address this problem, new forms of finite elements have

been developed to model the soil-structure interface, such as the Goodman

Contact Element and the Thin Layer Element. It can be foreseen that more work

will be done in this area.

One other important phenomenon associated with soil-structure interaction is

the liquefaction of the soil adjacent to the structure. On the one hand, it is very

hard to judge when and where exactly soil liquefaction will occur during soil-

structure interaction in earthquakes. On the other hand, once the soil liquefies, it

will become so soft that the supporting force of the structure will be totally lost,

which is a very difficult situation to model. There has been some research done

to address the problem of pure soil liquefaction, but little work has been done to

introduce this issue into the numerical model of dynamic soil-structure

interaction. Apparently, there should be and will be more research carried out in

this field.

In one word, since numerical modeling of dynamic soil structure is still at its

developing stage, future models will consider more affecting factors. And only

this way, can more accurate results be achieved.

Conclusion

Dynamic soil-structure interaction is an important part that should be

included in the seismic analysis when the structures are stiff and the soil is soft.

Numerical models using the substructure method are very rigorously formulated

but may not be able to consider some of the nonlinear properties of the system,

whereas the models using the direct methods can take the nonlinearities of the

unbounded soil into consideration but approximate transmitting boundaries

have to be included. In both methods, damping from the soil is very important.

Developments of both methods are now concentrated in the time domain and

may continue to be in the future. More study should be carried out to understand

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and model the phenomena of soil-structure separation and soil liquefaction

better. Currently, if nonlinearity of the far field soil is not an important issue in

the analysis, the substructure method is more practical; otherwise, the direct

method is more desirable.

References:

[1] M. Nuray Aydinoglu, “Development of Analytical Techniques in Soil-structure

Interaction”, Developments in Dynamic Soil-Structure Interaction, Kluwer

Academic Publishers, 1992

[2] lysmer, J., Kuhlemeyer, R.L. “Finite Model for Infinite Media”, Journal of

Engineering Mechanics Division, ASCE, V 95, pp. 377-392, 1969

[3] Wolf, J. P. and Oberbhuber, P., “Non-linear Soil-structure Interaction Analysis

Using Green’s Function of Soil in the Time Domain”, Earthquake Engineering and

Structural Dynamics, v 13 n 2, pp. 213-223 , Mar-Apr 1985

[4] Wolf, J. P., “Soil-structure Interaction in Time Domain”, Prentice-Hall,

Englewood Cliffs, N.J., 1988

[5] Wolf, J.P. “a Comparison of Time-Domain Transmitting Boundaries”, Earthquake

Engineering and Structural Dynamics, V 14, pp.655-673, 1986

[6] Bernal, Dionisio and Youssef, Akram, “A Hybrid Time Frequency Domain

Formaulation for Nonlinear Soil-structure Interaction”, Earthquake Engineering and

Structural Dynamics, v27, pp.673-685, 1998

[7] Yazdchi, M., Khalili, N. and Valliappan, S. “Dynamic Soil-structure Interaction

Analysis Via Coupled Finite-Element-Boundary-Element Method”, Soil Dynamics and

Earthquake Engineering, v18, pp.499-517, 1999

[8] Doo-Kie Kim and Chung-Bang Yun, “Time Domain Soil-structure Interaction

Analysis in Two-Dimensional Medium Based on Analytical Frequency-dependent

Infinite Elements”, International Journal of Numerical Methods in Engineering, v

47, pp.1241-1261, 2000

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[9] Xiong Zhang, J.L. Wegner and J.B. Haddow, “Three Dimensional Dynamic Soil-

structure Interaction Analysis in the Time Domain”, Earthquake Engineering and

Structural Dynamics, v 28, pp. 1501-1524, 1999

[10] Wolf, J. P. “Consistent Lumped-Parameter Models for Unbounded Soil: Frequency

Independent Stiffness, Damping and Mass Matrix”, Earthquake Engineering and

Structural Dynamics, v 20, pp. 33-42, 1991