2000 a Simple Soil-structure Interaction Model. Kocak, Mengi

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A simple soilstructure interaction model

Suleyman Kocak a,*, Yalcin Mengi b

a Department of Civil Engineering, Faculty of Engineering and Architecture, Cukurova University, Adana 01330, Turkeyb Department of Engineering Sciences, Middle East Technical University, Ankara 06531, Turkey

Received 8 October 1998; received in revised form 13 December 1999; accepted 22 December 1999

Abstract

A simple three-dimensional soilstructure interaction (SSI) model is proposed. First, a model is developed for a

layered soil medium. In that model, the layered soil medium is divided into thin layers and each thin layer is represented

by a parametric model. The parameters of this model are determined, in terms of the thickness and elastic properties of

the sublayer, by matching, in frequencywave number space, the actual dynamic stiness matrices of the sublayer when

the sublayer is thin and subjected to plane strain and out-of-plane deformations with those predicted by the parametric

model developed in this study. Then, by adding the structure to soil model a three-dimensional nite element model is

established for the soilstructure system. For the oors and footings of the structure, rigid diaphragm model is em-

ployed. Based on the proposed model, a general computer software is developed. Though the model accommodates

both the static and dynamic interaction eects, the program is developed presently for static case only and will be

extended to dynamic case in a future study. To assess the proposed SSI model, the model is applied to four examples,

which are chosen to be static so that they can be analyzed by the developed program. The results are compared with

those obtained by other methods. It is found that the proposed model can be used reliably in SSI analysis, and ac-commodates not only the interaction between soil and structure; but, also the interaction between footings. 2000

Elsevier Science Inc. All rights reserved.

Keywords: Layered soil medium; Parametric model; Dynamic stiness matrix; Frequency; Wave number; Rigid

diaphragm; Soilstructure interaction; Footingfooting interaction

1. Introduction

As is well known, there are two main methods which are being used in the soilstructure in-

teraction (SSI) analysis, namely, substructure and direct methods [17]. Also it is known that, due

to unbounded nature of soil medium, the computational size of these methods is very large. Forthis reason it is important to establish some simple SSI models which reduce the computational

cost of analysis; but at the same time, accommodate the essential features of interaction. One of

such models is cone model [810]. In that model, the eects of waves in soil are simulated by

modelling the soil by elastic conic bars.

In this study, a new simple model is proposed for three-dimensional SSI analysis. In this model,

it is assumed that the soil has a layered structure and is elastic. Dissipative forces in soil are taken

www.elsevier.nl/locate/apm

Applied Mathematical Modelling 24 (2000) 607635

* Corresponding author.

E-mail address: [email protected] (S. Kocak).

0307-904X/00/$ - see front matter 2000 Elsevier Science Inc. All rights reserved.PII: S 0 3 0 7 - 9 0 4 X ( 0 0 ) 0 0 0 0 6 - 8


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into account through the use of hysteretic damping. If desired, nonlinear eects in soil may be

accounted for by employing equivalent linear method [4]. In the proposed model, the layered soil

medium is subdivided into thin layers and each sublayer is represented by a parametric model.

Parametric model is composed of some distributed springs and ctitious horizontal elementsconnecting the upper and lower ends of the springs. For vertical motion of the soil, distributed

springs are vertical and ctitious horizontal element is a membrane. On the other hand, for

horizontal motion of soil, springs are horizontal and the ctitious horizontal element is a plate

under plane state of stress.

The parametric model thus developed contains three parameters for vertical case, which are

spring constant, membrane force and a mass parameter; and four parameters for horizontal case,

which are spring constant, two parameters belonging to ctitious plate and a mass parameter.

These parameters are related to the thickness and elastic properties of the thin sublayer by

matching, in frequencywave number space, the dynamic stiness matrices of the sublayer de-

termined from the model equations and the exact theory. The actual dynamic stiness matrices of

a thin layer are taken from [11].

Three-dimensional SSI model is obtained by adding the structure to the soil model discussed

above and using the nite element method. It is assumed that structure is composed of vertical

columns, horizontal beams, oors and footings. For the oors and footings of the structure, therigid diaphragm model is used [12]. The use of rigid diaphragm model for footings facilitates the

connection of the structure to soil. Three-dimensional model thus developed accommodates both

soilstructure and footingfooting interactions. The structure might have single or continuous or

mat foundation(s). The proposed model can be used in static or dynamic or earthquake inter-

action analysis. In the case of earthquake analysis, it is assumed that seismic environment is

generated by vertically propagating waves and the layered soil medium rests on a rigid bedrock,

which, in turn, implies that rigid bedrock motion is known. In fact, if the control point is in the

site of structure the rigid bedrock motion can be determined through the use of deconvolution.

On the other hand, if the control point is at outcrop the rigid bedrock motion would be given bycontrol point motion.

The three-dimensional SSI model discussed above has a form suitable for developing a general

purpose computer software. In fact such a computer software is developed in this study in

FORTRAN 77 for static interaction analysis. The extension of the program to dynamic case will

be done in a future study.

The proposed three-dimensional interaction model is assessed by applying it to four example

problems. These problems are chosen to be static so that they can be analyzed by the developed

computer software. In the example problems:

1. a soil layer subjected to a line load,

2. impedance coecients of a strip footing,

3. soil reaction distribution for a rigid strip footing which is under the inuence of a line load,

and

4. three-dimensional SSI analysis of a one-storey building on a soil layer are considered.

Comparisons of the results obtained from the model with those from other methods indicate that

the model can be used reliably in static SSI analysis.

The study is organized as follows. In Section 2, we dene, in general terms, the SSI problem to

be modelled. In Section 3, we propose a model for a layered soil medium. In Section 4, we present

a procedure for determining the model parameters. Section 5 contains a nite element model for

three-dimensional SSI analysis, which is based on the proposed soil model. The connection of thestructure to soil is also discussed in this section. Example problems are given in Section 6. In the

nal section, Section 7, we state some conclusions.

608 S. Kocak, Y. Mengi / Appl. Math. Modelling 24 (2000) 607635


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2. General denition of the problem

We consider the soilstructure system shown in Fig. 1. It is assumed that the structure is

composed of vertical columns, horizontal beams, oors and footings. The structure might havesingle or continuous or mat foundation(s). The soil layer, which is assumed to have a layered

structure, rests on a rigid bedrock. To simplify the interaction analysis, the soil material is taken

as a linear elastic material and dissipative forces in soil are accounted for through the use of

Fig. 1. Soilstructure system (a) system (b) interaction forces.

S. Kocak, Y. Mengi / Appl. Math. Modelling 24 (2000) 607635 609


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hysteretic damping. The nonlinear eects in soil, if desired, may be included approximately in the

analyses by using linear equivalent method [4]. The soilstructure system shown in Fig. 1 is re-

ferred to an xi-rectangular i 1 3 frame in which the (x1x2) plane is parallel to the plane of

layering.The structure is under the inuence of some static and dynamic forces and the rigid bedrock is

subjected to an earthquake ground excitation. In the case of earthquake analysis, it is assumed

that seismic environment is generated by vertically propagating waves. This assumption facilitates

the determination of the earthquake displacement ug at the rigid bedrock (see Fig. 1). In fact, if

the control point is at outcrop, the rigid bedrock motion is equal to control point displacement.

On the other hand, if the control point is in the site of the structure, the rigid bedrock motion can

be determined in terms of control point motion by using deconvolution. For example, for a single

elastic, homogeneous soil layer of depth d, the deconvolution formula would be

uFg uFc

cosad1

which relates the rigid bedrock displacement ug to the control point displacement uc in Fouriertransform space. Here, the superscript F designates Fourier transform; a xac the wave numberin vertical direction, where x is the frequency, and c is given by

c cs

laq

pshear wave velocity when the vertical waves are S wavesY

cp

k 2laqp

dilatational wave velocity when the vertical waves are P wavesY

VX

where l is the shear modulus, k 2mla1 2m the Lames constant, m Poissons ratio, and q isthe mass density.

It may be noted that when the vertical waves generating the earthquake motion are S waves,

then the base displacement ug would be horizontal; on the other hand, when they are Pwaves, ug

would be vertical. The dissipative forces in soil can be taken into account by making thereplacement, in Fourier transform space,

l 3 l1 2izsY 2

where zs is the hysteretic damping and i is the imaginary number.

We now start developing the model for the SSI problem stated above.

3. The soil model

For developing this model, the soil part of the soilstructure system in Fig. 1 is divided into

thin layers as shown in Fig. 2. In the gure, the number of sublayers are designated by N; sub-

layers and interfaces are numbered downwards; the depth, mass density, shear modulus andPoisson's ratio of the ith sublayer are designated respectively by hiY qiY li and mi and the layeredsoil medium is referred to an xi-rectangular i 1 3 frame in which the (x1x2) plane is parallelto the plane of layering and x3 axis is directed downwards.

In the proposed model, each thin layer is represented by distributed springs, and two hori-zontal ctitious elements having the same properties which connect the upper and lower ends of

the springs. For vertical motion of the soil, distributed springs are vertical and the ctitious

horizontal element is a membrane (see Fig. 3). Here, one can obtain the Winkler model [13] by

ignoring the ctitious horizontal element, implying that we disregard the interaction between

neighboring springs. The unit cell shown in the gure represents the ith thin layer and contains

three parameters, which are spring constant Ki (dened per unit horizontal area), tension in the



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membrane Ci (dened per unit length) and mass parameter mi (dened per unit horizontal area)

and ui3 and Fi

3 designate, for the ith interface, the vertical displacement and force (per unit area),

respectively.

The model proposed for horizontal motion of the soil is shown in Fig. 4. In this case distributed

springs are horizontal (in x1 and x2 directions) and ctitious horizontal element is a plate under

plane state of stress. To simplify the drawing only the springs in x1 direction are shown in the

gure. The unit cell representing the ith thin layer in this case contains four parameters, which are

Fig. 3. The model for vertical case.

Fig. 2. Representation of soil medium in terms of thin layers.



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spring constant Ki (dened per unit horizontal area), two parameters belonging to ctitious plate

Ci, Si (dened per unit length) and mass parameter mi (dened per unit horizontal area); and

ui ui1Y ui2 and Fi F

i1YF

i2 designate, for the ith interface, the horizontal displacement and

forces (per unit area), respectively. Here, we note that the parameters (Ki, Ci) in this case are

dierent from those of vertical case.

Here, it should be noted that for the proposed model vertical and horizontal motions are

decoupled, which decreases the computational load of the model. However, adding superstructure

on the top of the soil model couples the vertical and horizontal motions of soil, due to use of rigiddiaphragm model for structure foundation. This will be explained in more detail in Section 5.

We now develop the equations of a unit cell of the soil model.

3.1. Equations of unit cell for vertical resistance of soil model

A unit cell of the model for vertical soil resistance is shown in Fig. 5, where K, C, m are the

parameters of that cell. The upper and lower ends of the cell are numbered, respectively, as 1 and

Fig. 4. The model for horizontal case.

Fig. 5. A unit cell for vertical case.



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2 in the gure. The vertical displacement and force of the end ii 1Y 2 are designated by u13YF1

3 and u23YF

23 , respectively.

When we consider the free-body diagram of an innitesimal membrane element in Fig. 5 and

write equilibrium equations (in the sense of D'Alembert principle) we obtain

F13

F23

PR

QS K 1 1

1 1

PR

QS u13

u23

PR

QS Cr2 u

13

u23

PR

QS m u

13

u23

PR

QS 3

which relates the unit cell forces to unit cell displacements. Here dot denotes dierentiation with

respect to time t and r2 is Laplacian operator dened by

r2 o

2

ox21

o2

ox22X 4

Unit cell equation, Eq. (3), may be written in Fourier transform space as

F13

F23

PR

QS K mx2 K

K K mx2

PR

QS u13

u23

PR

QS Cr2 u13

u23

PR

QSY 5

where Fi3 and ui3 i 1Y 2 stand to Fourier transforms of force and displacements. To simplify the

notation, superscript F is not used in Eq. (5) and will not be used hereafter.

The hysteretic damping can be introduced into the unit cell equations by making the

replacement

K 3 K1 2izs 6

in Eq. (5).

3.2. Equations of unit cell for horizontal resistance of soil model

A unit cell of the model for this case is shown in Fig. 6, where K, C, Sand m are the parameters

of the cell. In that gure (u1, F1) and (u2, F2) denote, respectively, the displacements and forces of

the upper and lower ends of the cell. Here, we use the notation

ui ui1Y ui2Y F

i Fi1YFi

2 i 1Y 2 7

for unit cell displacement and forces. It may be noted that, in Eq. (7), superscript designates the

end of unit cell and subscript designates the direction of displacement or force.

Fig. 6. A unit cell for horizontal case.



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The lower and upper ends of the distributed horizontal springs are connected by ctitious

plates which are under plane state of stress. When we consider the free-body diagram of inni-

tesimal plate element in Fig. 6 and write equilibrium equations (in the sense of D'Alembert

principle) we obtain

F1i

F2i

PR

QS K 1 1

1 1

PR

QS u1i

u2i

PR

QS C

2r2

u1i

u2i

PR

QS Soi D

1

D2

PR

QS m u

1i

u2i

PR

QS i 1Y 2 8

which relates horizontal forces and displacements of unit cell. Here, oi oaoXi (partial derivativewith respect to xi) and D

k is dilatation at kth end of unit cell dened by

Dk o1uk1 o2u

k2 k 1Y 2 9

An important point is now in order. For the horizontal resistance of the soil model, we assume

that the ctitious horizontal element connecting the springs is a plate which is under the inuence

of plane state of stress. Thus, the deformation of this plate would involve two independent elasticconstants, for example elasticity modulus and Poissons ratio. This explains why Eq. (8) contains

two independent parameters C and S, of course, in addition to the spring constant K and mass

parameter m.

Unit cell equation, Eq. (8), may be written in Fourier transform space as

F1i

F2i

PR

QS K mx

2 K

K K mx2

PR

QS u1i

u2i

PR

QS C

2r2

u1i

u2i

PR

QS Soi D

1

D2

PR

QS i 1Y 2X 10

The hysteretic damping ratio can be introduced into the unit cell equations by using the cor-

respondence principle in Eq. (6).

4. Determination of model parameters

Unit cell Eqs. (5) and (10) contain, respectively, three and four parameters, which are

for vertical case X KvY CvY mvY

for horizontal case X KhY ChY SY mhY11

where the subscripts v and h are used, respectively, for vertical and horizontal cases. In this

section, the parameters of the proposed model will be determined, in terms of the thickness h,

mass density q, and elastic properties l and m of the sublayer (see Fig. 7) by matching, in fre-

quencywave number space, the dynamic stiness matrices of a thin sublayer (subjected to plane

strain and out-of-plane deformations), determined from the model equations and exact theory.

Here it may be noted that plane strain and out-of-plane deformations in soil are produced,

respectively, by P, SV and SH waves.

The actual dynamic stiness equations can be found in [11] for a thin sublayer undergoing

plane strain and out-of-plane deformations. These equations for plane strain and out-of-plane

cases can be written in the common form

F D UY 12

where FY U and D are, respectively, force, displacement vectors and dynamic stiness matrixwhich are dened by, for plane strain and out-of-plane cases,



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for plane strain case:

F F11 iF1

3 F2

1 iF2

3

TY u u11 iu

13 u

21 iu

23

T13

and

D GBkAk2 x2MY 14

where k is being wave number in x1 direction (see Fig. 7) and

G

1

h

l 0 l 0

0 A 0 A

l 0 l 0

0 A 0 A

PTTTR QUUUSY

B 1

2

0 E 0 P

E 0 P 0

0 P 0 E

P 0 E 0

PTTTR

QUUUSY

A h

6

2A 0 A 0

0 2l 0 l

A 0 2A 0

0 l 0 2l

PTTTRQUUUSY

M qh

6

2 0 1 0

0 2 0 1

1 0 2 0

0 1 0 2

PTTTR

QUUUSY

15

Fig. 7. A typical thin layer.



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in which

A 21 m

1 2mlY P

1

1 2mlY E

4m 1

1 2mlY 16

for out-of-plane case:

F F12 F2

2

Tu u12 u

22

T17

and

D GAk2 x2MY 18

in which

Gl

h

1 1

1 1 !Y A lh

6

2 1

1 2 !Y M qh

6

2 1

1 2 !X 19

In Eq. (14), B denes the coupling between horizontal and vertical motion of the sublayer for

plane strain case. In this study, it is assumed that the free-eld system is subjected to verticalwaves and thus, the coupling described above can be assumed to be small. In view of this

assumption, Eq. (14) can be written as

D GAk2 x2M 20

when B is ignored.

To obtain the stiness equations of the proposed model for plane strain and out-of-plane cases,

we rst note that model displacements are dependent only on x1 and t, i.e.,

uik uikx1Y t i 1Y 2Y k 1 3X 21

When the unit cell equations, Eqs. (3) and (8), are written in Fourier transform space (with respect

to time t and x1) in view of Eq. (21), one can obtain the dynamic stiness matrix of the model

again in the form of Eq. (20), but, in this case G, A, M matrices are dened by

for plane strain case:

G

Kh 0 Kh 00 Kv 0 Kv

Kh 0 Kh 00 Kv 0 Kv

PTTR

QUUSY

A

Cha2 S 0 0 00 Cv 0 0

0 0 Cha2 S 00 0 0 Cv

PTTR

QUUSY 22

M

mh 0 0 0

0 mh 0 0

0 0 mh 0

0 0 0 mh

PTTR

QUUSY



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for out-of-plane case:

GKh Kh

Kh Kh !Y A Cha2 0

0 Cha2 !Y M mh 0

0 mh !X 23The model parameters may be obtained by matching the G, A and M matrices given in

Eqs. (15) and (19) and Eqs. (22) and (23) which belong, respectively, to the exact theory and

model. If o-diagonal terms in A and M matrices in Eqs. (15) and (19) are lumped on the diagonal

terms, the model parameters may be found as

Kh l

hY Ch lhY S

lh

21 2mY mh

qh

2Y

Kv 21 ml

1 2mhY Cv

lh

2Y mv

qh

2Y

24

where mh and mv represent the mass values of the thin layer lumped at its upper and lower ends.

5. A nite element model for three-dimensional SSI analysis

In this section, the structure will be added to soil foundation through the use of nite element

method and the SSI model thus developed will be described briey. In the development of this SSI

model we do the following assumptions:

1. For the soil we use the model proposed in Section 3.

2. The structure is composed of vertical columns, horizontal beams, oors and footings. The

structure might have single or continuous or mat foundation(s).

3. We use rigid diaphragm model for oors and footings [12]. The use of rigid diaphragm model

for footings facilitates the connection of structure to soil foundation.

4. The foundation of the structure is perfectly bonded to soil foundation.The nite element model is based on writing stiness and mass matrices for soil, beam, column

and plate elements and combining them by coding technique, in view of rigid diaphragm as-

sumption, so that the equilibrium equations (in the sense of D Alembert principle) and com-

patibility conditions at the nodes are satised. A special care will be given to the connection of

structure to soil foundation.

As the equations of beam, column and plate elements are well established and the use of rigid

diaphragm model in the formulation of three-dimensional response of structures is well known in

literature, in what follows we develop only, based on the proposed model in Section 3, the stiness

and mass matrices for soil elements and emphasize the procedure by which the soil foundation

and structure can be connected.

Since the equations of the proposed model are uncoupled for vertical and horizontal cases, wepresent the stiness and mass matrices of soil elements for the aforementioned cases separately.

However, it should be noted that when the soil elements are combined with the structure the

vertical and horizontal degrees of freedoms (DOFs) of soil elements would become coupled due to

the presence of the structure foundation which is modelled by using rigid diaphragm assumption.

5.1. The soil element for vertical case

A three-dimensional element associated with a unit cell of the soil model undergoing vertical

displacements is shown in Fig. 8. As the soil model eliminates the dimension in vertical direction,

the element in the gure should be taken as dimensionless in that direction. The ctitious vertical



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dimension is used in the gure to dierentiate the upper and lower ends of the cell. The horizontal

faces of the elements are taken as quadrilateral and the corners of the element are numbered as

shown in the gure.In the formulation we use bilinear interpolation functions for both geometry and vertical

displacement distributions ui3x1Yx2 i 1Y 2 over the upper and lower faces of the element.When we use virtual work principle for the unit cell equations given in Eq. (5) and follow the

procedure used in conventional nite element formulation, we obtain the dynamic stiness matrix

as, in Fourier transform space,

D ~K ~C x2M ~K

~K ~K ~C x2M

!25

for the three-dimensional soil element. In Eq. (25) the rst and second rows represent, respec-

tively, the force DOFs of the upper and lower faces, and the rst and second columns represent

the displacement DOFs of the upper and lower faces. Consequently, the o-diagonal matricesdescribe the coupling between the upper and lower faces of the element. The matrices~KY ~C and M in Eq. (25) are dened as

~K Kv

A

N NT dAY

~C Cv

A

o1N o2N o1N

T

o2NT

4 5dAY

M mv

A

N NT dAY

26

Fig. 8. Three-dimensional soil element (vertical case).



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where A is the area of the horizontal face of the element and Nis a vector of the dimension 4 1containing bilinear interpolation functions. The elastic and dynamic eects are both included in

the dynamic stiness matrix in Eq. (25). The hysteretic damping can be introduced into the model

through the use of Eq. (6).

5.2. The soil element for horizontal case

A three-dimensional element associated with a unit cell of the model undergoing horizontal

displacements is shown in Fig. 9. As stated for vertical case, ctitious vertical dimension is used to

dierentiate upper and lower ends of the cell. The horizontal faces of the element are taken as

quadrilateral and the corners of the element are numbered as shown in the gure.

As in vertical case, here again we use bilinear interpolation functions for both geometry and

horizontal displacement distributions ui1x1Yx2Y ui2x1Yx2Y i 1Y 2) over the upper and lower

faces of the element. When we use virtual work principle for the unit cell equations given in

Eq. (10) and follow the procedure used in conventional nite element formulation, we obtain

the dynamic stiness matrix as, in Fourier transform space,

D A1 A2 B

BT A1 A2

PR

QS 27

for the three-dimensional soil element. In Eq. (27) the rst and second rows represent, respec-

tively, the force DOFs of the upper and lower faces, and the rst and second columns represent

the displacement DOFs of the upper and lower faces. The matrices A1Y A2 and D in Eq. (27) aredened as

Fig. 9. Three-dimensional soil element (horizontal case).



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A1

~K Cha2 ~C11Cha2 ~C22 S ~C11

S~C21

S~C12~

K Cha2~

C11Cha2 ~C22 S~C11

PTTTR

QUUUSY 28

A2 x2M 0

0 x2M

!Y

B BT ~K 0

0 ~K

!Y

in which

~

K Kh

ANN

T

dAY

~Cij

A

oiNojNT dA iYj 1Y 2Y

M mh

A

N NT dAX

29

In Eq. (28), the rst row and column of the matrices represent the DOFs in x1 and the second row

and column represent the DOFs in x2 direction. The hysteretic damping again can be introduced

into the model through the use of Eq. (6).

5.3. The connection of the structure to soil

In this section, the connection of the structure to soil foundation is discussed very briey. In

the SSI system considered in this study, the foundation of the structure is assumed to be per-

fectly bonded to soil foundation which implies that the structure and soil, at the points of

contact area, would move together. To satisfy this continuity condition between the structure

and soil foundation, the equations of soil elements should be modied when they have nodes

over the contact area. For vertical case, the modied equations can be obtained by introducing

rotational DOFs about x1 and x2 axes, in addition to vertical DOF, for the nodes lying over

contact area. On the other hand, for horizontal case, the modied equations can be derived by

imposing rigid diaphragm constraints, for horizontal DOFs, to the nodes lying over the contact

area.

To explain the connection more explicitly, we refer to Fig. 10 showing the nite element meshof a soil foundation referred to an x1x2x3 global coordinate system. In the gure, the shaded

regions represent the contact areas of the footings of the superstructure; Mrepresents the master

point of rigid diaphragm model used for each contact area; D1Y D2 and Dh represent, respectively,the displacements in x1, x2 (horizontal) directions and rotation about x3 (vertical) axis. The

horizontal displacements (d1Y d2) of an arbitrary point in contact area are related to master pointdisplacements (D1YD2YDh) by, in view of rigid diaphragm assumption,

d1 D1 x2 xM2 DhY

d2 D2 x1 xM1 DhY

30



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where (xM1 YxM2 ) and (x1Yx2) denote, respectively, the horizontal coordinates of the master point M

and arbitrary point. We now discuss the connection procedure for horizontal and vertical cases

separately.

(a) Horizontal case: Horizontal displacements of the nodes lying over the contact area are

related to the horizontal displacements and rotation (about vertical axis) of the master point by

Eqs. (30). By using these constraints in Eq. (27), the modied equations can be obtained, for

horizontal case, for the soil elements having nodes over the contact area. It should be noted that

these modications should be done not only for the element A which lies completely in the

contact area; but, also for the neighboring elements B and C which have nodes in that area (see

Fig. 10).

(b) Vertical case: As stated previously, the connection for this case could be done by intro-

ducing rotational DOFs about x1 and x2 axes for the nodes lying over the contact area (see

Fig. 10), which dictates that we modify Eq. (25) for the elements having nodes on that area. To

simplify the formulation and the computer implementation, this modication is done in the

present study not only for the aforementioned elements; but, also for the elements outside the

contact area.

The modied form of the equations for horizontal and vertical cases are not given here not to

lengthen the paper.

Fig. 10. Modelling the footings by rigid diaphragms.



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5.4. Information about the computer program

A computer program is developed, based on the proposed model, for three-dimensional SSI

analysis. The program is written in FORTRAN 77. Though the proposed model accommodatesboth static and dynamic eects, presently the computer program is developed for static interaction

analysis and the extension of the computer program to dynamic case will be done in a future

study.

The system stiness matrix is formed as a skyline matrix which is stored in out-of-core by

blocks and the solution is obtained by LDU decomposition algorithm [14]. The numbering of the

DOFs of the nodes is done automatically by the program so that the computational load of the

solution of the system equations is minimum.

The program has some modules working sequentially. The input le has some blocks con-

taining information about system, coordinates, layers, etc., similar to those of SAP90 [15]. The

output les have a structure easy to interpret.

In the program, only the upper surface of soil is discretized. The nite element meshes of the

layers underneath are generated by projecting the nodes of the top surface on horizontal layersand numbered by the program automatically. With this generation facility the nite element mesh

for the whole domain is assembled by the program, copying the nite element mesh of the top soil

layer like slices of meshes for as many of sublayers as dened. Therefore, user do not have to mesh

the layers underneath the top of soil resulting minimum data preparation eort. Once the mesh on

the top of soil is dened the other nodes of the sublayers are generated by the program provided

that the number of layers is dened by the user. However, for the denition of superstructure the

complete nite element mesh has to be entered by the user.

The computer software can be obtained, together with its manual, from the authors upon

request.

6. Example problems

In this section, with the object of assessing the model we analyze some example problems, and

compare the results obtained from the model with those from other methods. The example

problems considered in this section are:

1. a soil layer subjected to a line load,

2. impedance coecients of a strip footing,

3. soil reaction distribution for a rigid strip footing which is under the inuence of a line load,

4. three-dimensional SSI analysis of a one-storey building on a soil layer.

Problems 1, 2 and 3 are two-dimensional and involve only the soil foundation without super-structure on it. As the mesh of the top surface in these problems is two-dimensional and has a

simple form, to save space, it is not given in the paper. However, for Problem 4, which is three-

dimensional and contains a superstructure, all the details regarding the mesh used in the analysis

are given. As mentioned before, program discretizes the horizontal layers of soil model auto-

matically.

6.1. Problem 1: a soil layer subjected to a line load

We consider an elastic layer resting on a rigid bedrock and subjected to a vertical or hori-

zontal line load. The vertical loading case is shown in Fig. 11. The layer has a depth of H and is

referred to an x1x2x3 global coordinate system. The vertical line load P extends to innity in



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out-of-plane direction (along x2 axis). In horizontal loading case, the line load P acts in x1direction. The shear modulus and Poisson's ratio of the layer are designated by l and m in the

gure.The analyses of the problems stated above are performed by the computer program developed

in this study and the results are compared with the ones obtained from boundary element method

(BEM) [1620], in terms of nondimensional (ND) variables, in Figs. 12 and 13 (for the Poisson

ratio m 0.2). It may be noted that BEM is a semi-analytical method whose accuracy is muchbetter than FEM, in particular, for dynamic problems and for problems with the solution domain

extending to innity in some directions. The accuracy of the boundary element results in this

problem is insured by increasing the number of boundary elements employed in the analysis until

the convergence of the results is achieved. In the model analyses, the model parameters are de-

termined through the use of Eq. (24). In Figs. 12 and 13, ND vertical and horizontal displace-

ments (u3) and u1 are dened by

u3Y u1 lP

u3Y u1 31

and

x1 x1

a32

in which a represents a characteristic length. The results are obtained when

H a

and are shown in the gures only for x16 0 due to the symmetry of the problem.

Fig. 11. An elastic layer subjected to a vertical line load.



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The model prediction for the vertical displacement distribution of soil surface for vertical line

load problem is obtained by taking the number of sublayers as NS 1Y 2Y 4Y 10 and compared withthe BEM results in Fig. 12. From the gure it may be observed that when the number of sublayers

(NS) increases the model prediction approaches that of BEM. Fig. 13 gives the same comparison

for horizontal line load problem, which shows that the observation made for vertical case holds

also for this case.

Fig. 13. Horizontal displacement distribution on soil surface (horizontal line load).

Fig. 12. Vertical displacement distribution on soil surface (vertical line load).



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6.2. Problem 2: impedance coecients of a strip footing

We consider a rigid strip footing of width Bperfectly bonded to an elastic layer of depth H(see

Fig. 14). The layer rests on a rigid bedrock. The rigid strip footing (which extends to innity inout-of-plane direction) is subjected to static horizontal and vertical forces F1Y F3 and moment Mat its center as shown in the gure. The strip footing system is referred to an x1x3 coordinate

system in which x1 axis is horizontal and x3 is directed downwards. The thickness of the footing

will be ignored in the analysis.

The static impedance relation for the rigid strip footing would be in the form

F1F3M

PR

QS SHH 0 SHM0 SVV 0

SMH 0 SMM

PR

QS D1D3

a

PR

QSY 33

where SHH, SVV and SMM represent, respectively, the static horizontal, vertical and rocking im-

pedance coecients and SHM SMH designates the coupling impedance coecient between thehorizontal and rocking motions of the footing; a, D1 and D3 are, respectively, the rotation, and thehorizontal and vertical translations of the rigid footing. The coupling impedance coecient SHMwill be disregarded in the analysis as its value is small compared to those of the other impedance

coecients. The vertical and horizontal impedance coecients in Eq. (33) are computed by the

proposed model by subdividing the soil layer into thin layers and the variations of these coe-

cients with the layer depth are compared with those obtained from ``BEM'' in Figs. 15 and 16. Inthe model analyses (carried out by the computer program developed in this study) the rigidity of

footing is accounted for by choosing large values for its bending and axial rigidities. In the gures,

the ND depth is given by

H H

B34

and ND impedance coecients are dened by

SHHY SVV SHH

lY

SVV

l

Y 35

where l is shear modulus of soil layer.

Fig. 14. Strip footing problem.



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The model results are obtained when each layer of depth B is subdivided into NS 20 sub-layers and parameters are determined through the use of Eq. (24) and Poisson s ratio is taken to

be m 0X2. Figs. 15 and 16 show, respectively, the variation of vertical and horizontal impedancecoecients with the depth of soil layer. From the gures one can see that model results are fairly

close to those of BEM.

Fig. 15. Variation of vertical impedance with depth.

Fig. 16. Variation of horizontal impedance with depth.



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6.3. Problem 3: soil response under a rigid strip footing

As in Problem 2, here we consider a rigid strip footing of width Bwhich lies on the top surface

of an elastic layer of depth H(see Fig. 17). The base of the layer is rigid. The system is referred toa rectangular xi-frame in which the (x1x2) plane coincides with the top surface of the layer and x3axis is directed downwards. The strip footing is subjected, along its center line, a vertical uniform

line load P as shown in the gure. In the analysis, we ignore the thickness of the footing.

The object in this problem is to determine the soil reaction underneath the footing by using the

proposed model and compare it with that obtained by the BEM. We do the computations in terms

of ND variables by choosing

H H

B 4

and Poissons ratio m 0X2. The model parameters are determined through the use of Eqs. (24). Inthe model analyses (carried out by the computer program developed in this study) the rigidity of

footing is accounted for by assigning a large value for its bending rigidity. The soil reactiondistribution is obtained by the model through the subdivision of the soil layer into

NS 1Y 2Y 8Y 32Y 64 36

sublayers and is compared with BEM results in Fig. 18. In the gure, ND horizontal coordinate x1is given by

x1 x1

B37

Fig. 17. Rigid strip footing.



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and ND soil reaction "p is dened by

p pB

P38

in which p is the soil pressure acting on the footing. The gure shows that as the number of

subdivisions NS, used in approximating the layer response, increases the model prediction for thesoil reaction distribution approaches that obtained by BEM, and that the model is capable to

describe the singularities in the soil pressure along the edges of the rigid strip footing.

6.4. Problem 4: one-storey building supported by soil layer

We consider a one-storey building having the dimensions, and material and sectional prop-

erties shown in Fig. 19. The one-storey building has an elastic oor surrounded with beams, which

stands, at its corners on columns supported by rigid square footings. The soil layer overlies a rigid

bedrock. The soilstructure system is referred to a global X1X2X3 coordinate system as shown in

the gure, where X1X2 coincides with the top surface of the layer and X3 axis is directed upwards.

The object in this problem is to analyze the soilstructure system, described above, under the

inuence of some loadings by the developed computer program (which is based on the proposedmodel) and compare the results with those of SAP90 [15].

SAP90 is a widely used structural analysis program and it can accommodate discrete springs on

any point of structure. Using this program one can model soil with discrete springs as in Winkler

Model. For the purpose of comparing our program (based on the proposed model) with a

software which is practically used by engineers, this problem is chosen.

In the analysis by the proposed model, the soil layer is divided into NS 4 sublayers whichimplies that each thin layer has a depth of

h H

NS 0X5 mX 39

Fig. 18. Soil response distribution under strip footing.



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Fig. 19. Soilstructure system considered in Problem 4.



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The model parameters can be obtained through the use of Eqs. (24), in view of soil properties

given in Fig. 19 and Eq. (39), as

Kv 53333X3 kNam3

Y Cv 2500 kNamYKh 20000 kNam

3Y Ch 5000 kNamY S 4166X67 kNamX40

In the analyses performed by the model, the oor and footings of the system are modelled as rigid

diaphragms. The network used in the analysis by the proposed model is shown in Fig. 20.

In the analysis by SAP90, we put springs at the lower ends A, B, C and D of the columns (see

Fig. 19) to describe the inuence of the soil on the response of the structure. Each of the column

ends A, B, C and D have six springs associated with its three translational and three rotational

DOFs. In the present analysis these spring constants are determined through the use of BEM by

considering a square rigid footing of side 1 m overlying on an elastic layer of depth 2 m. Their

computed values are:

Kx Ky 37800 kNam translational springs in X1 and X2 directions Y

Kz 28400 kNam translational spring in X3 directionY

ax ay 6200 kN marad rotational springs about X1 and X2 axesY

az 4780 kN marad rotational spring about X3 axisX

41

It may be noted that, in computing these spring constants, the interactions between the footings

are ignored.

In this problem, we analyze the soilstructure system under consideration for three dierent

loadings. They are:

Loading 1: A 10 kN horizontal concentrated force in X1 direction is applied to the master point

of the oor which is chosen to be at the centroid of the oor (joint number 527 in Fig. 20).Loading 2: A 100 kN vertical concentrated force is applied to the joint 442 in Fig. 20 (which

acts on the column A in Fig. 19).

Loading 3: A 100 kN vertical concentrated force is applied to the joint 462, and at the same

time, a 10 kN horizontal concentrated force in X1 direction and a 7.5 kN m moment about X3 axis

are applied to the master point of the oor (joint number 527 in Fig. 20).

6.4.1. Loading 1As stated previously, this loading involves a 10 kN horizontal concentrated force in X1 di-

rection applied to the master point of the oor (joint 527 in Fig. 20 which is at the centroid of the

oor).

In this loading, we study only the rotations (about vertical axis, X3) of the footings produced

by the aforementioned applied horizontal force.SAP90 results yield no rotations for the footings, which may be anticipated in view of the

symmetry of the structure and of the fact that we ignore the interaction between the footings in

SAP90 analysis. On the other hand, as the proposed model permits the footings to interact with

each other, it predicts the rotations of the footings as sketched in Fig. 21 showing that the front

(A, B) and rear (C, D) footings rotate in opposite directions of the amount

h3 0X2872 105 radX 42



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Fig. 20. Finite element network used in Problem 4.



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To check the order of the amount of rotation in Eq. (42), we approximate the layer by a half-

space and consider the top surface on which four horizontal loads (in X1 direction) of 2.5 kN are

acting at the points A, B, C, D (see Fig. 19). The rotation at one of these points can be obtained

by computing, through the use of Cerrutis solution [21], the rotation at that point produced by

the horizontal forces at the other points and superposing the results. The rotational distortion of

the footings obtained using this procedure is

h3 0X2990 105 rad 43

which is slightly higher than that in Eq. (42) (which was found by the model). This small dierence

may be expected in view of the fact that the value in Eq. (43) is obtained for a half-space while the

supporting soil medium in our soilstructure system is a layer of thickness 2 m.

6.4.2. Loading 2A 100 kN concentrated vertical force is applied at the joint 442 in Fig. 20 (on the column A).

We analyze this system by the computer program developed in this study and SAP90 and

compare the results in Tables 1 and 2, where E designates the relative error between SAP90 and

model results dened by

E jVmodel VSAPj

MaxjVSAPjY 44

Fig. 21. Rotational distortion of footings.



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where Vmodel and VSAP are element forces computed, respectively, by model and SAP90, and

MaxjVSAPj is the absolutely maximum value of VSAP obtained by considering the whole system. Astudy of Tables 1 and 2 indicates that the results obtained by SAP90 and model do not dier

much, implying that the interaction between the footings may be ignored with respect to the

element forces for the problem under consideration.

6.4.3. Loading 3In this loading case, a 100 kN vertical concentrated force is applied to the joint 462, and at the

same time, a 10 kN horizontal concentrated force in X1 direction and a 7.5 kN m moment about

X3 axis are applied to the master point of the oor (joint number 527 in Fig. 20).

The comparison of some column and beam element forces obtained by the model and SAP90

are given in Tables 3 and 4. Studying the relative error E, we see that, for the reason stated for

Loading 2, the two results do not dier much for this loading too.

Table 3

Column element forces (Loading 3)

Element # Node # Moment

(kN m)

Shear

(kN)

Normal

(kN)

Torsional moment

(kN m)

SAP Model E

(%)

SAP Model E

(%)

SAP Model E

(%)

SAP Model E

(%)

1 133 )1.25 )0.08 8 1.94 2.77 11 )54.6 )55.4 1 0.12 0.10 16

442 7.07 8.38 9

2 141 )7.59 )9.81 15 )7.61 )8.20 8 )20.6 )20.0 1 0.12 0.10 16

450 )15.2 )14.8 3

3 301 )2.15 )2.26 1 )0.73 )0.64 1 )15.3 )15.4 1 0.12 0.10 16

514 )0.04 )0.35 2

4 309 )4.05 )5.20 8 )3.60 )3.93 4 )9.93 )9.26 1 0.12 0.10 16

522 )6.75 )6.60 1

Table 1

Column element forces (Loading 2)

Ele me nt # Node # Moment (kN m) Shear (kN) Normal (kN)

SAP Model E (%) SAP Model E (%) SAP Model E (%)1 133 )0.14 )0.36 10 )0.71 )0.41 42 )96.4 )97.4 1

442 2.26 1.58 30

2 141 )1.07 )0.89 8 0.71 0.52 27 )2.77 )1.93 1

450 )1.04 )0.68 16

3 301 0.14 0.20 2 )0.70 )0.63 10 )2.77 )1.93 1

514 2.24 1.70 24

4 309 )1.07 )0.87 9 0.70 0.51 27 1.96 1.39 1

522 )1.02 )0.67 16

Table 2

Beam element forces (Loading 2)

Element # Node # SAP Model E (%)1 442 2.25 1.57 30

8 450 2.26 1.69 25

9 514 )1.14 )0.74 18

16 522 )1.14 )0.75 17



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7. Conclusions

In the study, rst a model is proposed for layered soil foundations; then, based on this soil

foundation model, a nite element model is proposed for the three-dimensional SSI analysis.

Now, in what follows, we repeat some important points which are already stated in the previous

sections and draw some conclusions from the ndings of the study:

(a) In the study, the model parameters are obtained through the match of dynamic stiness

matrices of a sublayer as determined from the exact theory and the model. Alternatively, it is also

possible to determine the soil model parameters via optimization, which involves selection of a

base problem and tting, through optimization, the model and exact responses for that base

problem. This alternative approach may be the subject of a future study.

(b) As noted in Problem 4 of Section 6, the model proposed in the study for SSI analysis

accommodates not only the interaction between the structure and soil; but, also the interaction

between the footings. This makes the proposed model general and suitable for the analysis of

structures having multiple footings or a mat foundation, etc.(c) In the proposed model, we use the rigid diaphragm assumption for both the oors and

footings, each footing having a separate master point. The use of the rigid diaphragm assumption

for footings facilitates the connection of the structure to the soil foundation.

(d) The model accommodates the layered structure of the soil foundation through the repre-

sentation of sublayers by unit cells each having dierent model parameters.

(e) The results of Loading 2 in Problem 4 of Section 6 indicate that the interaction between the

footings may be ignored, with respect to the element forces, for the one-storey structure analyzed

in that problem. We think that this result is due to the symmetry of the one-storey structure. We

expect that the footingfooting interaction eects may become important for a general non-

symmetric structure, in particular, in dynamic analysis. We plan to study this aspect of interaction

analysis in future.

(f) As stated previously, SAP90 can be used in SSI analysis only when the structure has rigid

footings, not continuous beam or mat foundations. Under these conditions, in order to use SAP90

in the interaction analysis, we rst determine the spring constants for each of the footings using

``BEM''; then, attaching these springs to the base levels of the columns, we analyze the structure.

We note that this kind of analysis ignores the footingfooting interaction eects. In view of the

arguments given above, we can summarize the shortcomings of using SAP90 in SSI analysis as:

1. it can be used only if the structure has rigid footings,

2. it requires a separate analysis by ``BEM'', and

3. it ignores the footingfooting interaction eects.

The model proposed in this study eliminates all these shortcomings.

Table 4

Beam element forces (Loading 3)

Element # Node # Moment (kN m) Torsional moment (kN m)

SAP Model E (%) SAP Model E (%)1 442 )1.14 )2.48 7 5.85 5.89 1

4 445 17.95 17.53 2 )2.12 )2.16 1

8 450 12.79 12.34 2 2.70 2.71 0

9 514 2.47 2.15 2 )1.69 )1.76 1

11 516 4.60 4.39 1 )0.41 )0.45 1

16 522 4.55 4.46 1 )2.0 )1.96 1

20 469 16.68 16.40 2 2.31 2.33 1

27 477 )4.14 )4.04 1 )0.56 )0.58 1



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(g) The model is proposed for the three-dimensional dynamic SSI analysis; but, the computer

software, as stated above, is developed for static case. The program will be extended to dynamic

case in a future study.

References

[1] J.P. Wolf, Dynamic SoilStructure Interaction, Prentice-Hall, Englewood Clis, NJ, 1985.

[2] J.P. Wolf, SoilStructure Interaction Analysis in Time Domain, Prentice-Hall, Englewood Clis, NJ, 1988.

[3] R.W. Clough, J. Penzien, Dynamics of Structures, second ed., McGraw-Hill, Tokyo, 1993.

[4] J. Lysmer, T. Udaka, C. Tsai, H.B. Seed, FLUSH: a computer program for approximate 3D dynamic analysis of

soilstructure problems, Report of Earthquake Engineering Research Center, University of California, Berkeley,

Report No. EERC75-30, 1975.

[5] S. Gupta, T.W. Lin, J. Penzien, C.S. Yeh, Hybrid modelling of soilstructure interaction, Report of Earthquake

Engineering Research Center, University of California, Berkeley, Report No UCB/EERC-80/09, 1980.

[6] A. Gomez-Masso, J. Lysmer, J.-C. Chen, H.B. Seed, Soil structure interaction in dierent seismic environments,

Report of Earthquake Engineering Research Center, University of California, Berkeley, Report No UCB/EERC-

79/18, 1979.

[4] J.A. Gutierrez, Substructure method for earthquake analysis of structure-soil interaction, Report of Earthquake

Engineering Research Center, University of California, Berkeley, Report No EERC 76-9, 1976.

[8] J.W. Meek, J.P. Wolf, Cone models for homogenous soil, part I, J. Geotech. Eng. 118 (1992) 667.

[9] J.W. Meek, J.P. Wolf, Cone models for homogenous soil, part II, J. Geotech. Eng. 118 (1992) 686.

[10] J.P. Wolf, W. Meek, Cone models for a soil layer on a exible rock half-space, Earthquake Eng. Struct. Dyn. 22

(1993) 185.

[11] E. Kausel, J.M. Roesset, Stiness matrices for layered soils, Bull. Seismol. Soc. Am. 71 (1981) 1743.

[12] A. Ghali, A.M. Neville, Structural Analysis, Chapman and Hall, London, 1978.

[13] E. Winkler, Die Lehre von Der Elastizitaet und Festigkeit, Dominiqus, Prag, 1867.

[14] G. Dhatt, G. Touzot, G. Cantin, The Finite Element Method Displayed, Wiley, Canada, 1985.

[15] E.L. Wilson, A. Habibullah, SAP90 Structural Analysis Programs, Computers and Structures, Berkeley, CA,

1984.

[16] Y. Mengi, A.H. Tanrikulu, A.K. Tanrikulu, Boundary Element Method for Elastic Media: An Introduction,METU Publications, Ankara, 1994.

[17] P.K. Banerjee, The Boundary Element Method in Engineering, McGraw-Hill, London, 1994.

[18] G.D. Manolis, D.E. Beskos, Boundary Element Method in Elastodynamics, Unwin Hyman, London, 1987.

[19] A.A. Becker, The Boundary Element Method in Engineering: A Complete Course, McGraw-Hill, London, 1992.

[20] J. Trevelyan, Boundary Elements for Engineers: Theory and Applications, Computational Mechanics Publica-

tions, Southampton-Boston, 1994.

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2000 a Simple Soil-structure Interaction Model. Kocak, Mengi