dutta roy 2002 cas

Review Article

A critical review on idealization and modelingfor interaction among soilfoundationstructure system

Sekhar Chandra Dutta *, Rana Roy

Department of Applied Mechanics, Bengal Engineering College (Deemed University), Howrah 711 103, West Bengal, India

Received 19 June 2001; accepted 5 April 2002

Abstract

The interaction among structures, their foundations and the soil medium below the foundations alter the actual

behaviour of the structure considerably than what is obtained from the consideration of the structure alone. Thus, a

reasonably accurate model for the soilfoundationstructure interaction system with computational validity, eciency

and accuracy is needed in improved design of important structures. The present study makes an attempt to gather the

possible alternative models available in the literature for this purpose. Emphasis has been given on the physical

modeling of the soil media, since it appears that the modeling of the structure is rather straightforward. The strengths

and limitations of the models described in a single paper may be of help to the civil engineers to choose a suitable one

for their study and design.

2002 Elsevier Science Ltd. All rights reserved.

Keywords: Soilstructure interaction; Modeling; Winkler; Continuum; Elasto-plastic; Nonlinear; Viscoelastic; Finite-element; Seismic

1. Introduction

The response of any system comprising more than

one component is always interdependent. For instance, a

beam supported by three columns with isolated footing

may be considered (Fig. 1). Due to the higher concen-

tration of the load over the central support, soil below it

tends to settle more. On the other hand, the framing

action induced by the beam will cause a load transfer to

the end column as soon as the central column tends

to settle more. Hence, the force quantities and the set-

tlement at the nally adjusted condition can only be

obtained through interactive analysis of the soilstruc-

turefoundation system. This explains the importance of

considering soilstructure interaction.

The three-dimensional frame in superstructure, its

foundation and the soil, on which it rests, together con-

stitute a complete system. With the dierential settlement

among various parts of the structure, both the axial

forces and the moments in the structural members may

change. The amount of redistribution of loads depends

upon the rigidity of the structure and the load-settlement

characteristics of soil. The considerable inuence of the

structural rigidity on the same has been qualitatively

explained in the literature [1] long back. Subsequently,

several studies have been conducted to estimate the eect

of this factor. A critical scrutiny of such studies has been

presented in the literature [2] modeling the soilfounda-

tionstructure system in a number of alternate ap-

proaches. Generally, it may be intuitively expected that

the use of a rigorous model representing the real system

more closely from the viewpoint of mechanics will lead to

better results. But the uncertainty in the determination of

the input parameters involved with such systems may

sometimes reverse such anticipation. Thus, to choose a

detailed model, one should also be careful about the

extent of accuracy with which the parameters involved

with the model can be evaluated. In the present study, an

attempt has been made to scrutinize the various ap-

proaches of modeling the soilstructurefoundation

Computers and Structures 80 (2002) 15791594

www.elsevier.com/locate/compstruc

*Corresponding author. Fax: +91-33-668-2916.

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

0045-7949/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0045-7949 (02 )00115-3

system and also compare the same highlighting their

rigor and suitability for solving practical engineering

problems with desired accuracy.

2. Soilstructure interaction under static loading

Numerous studies [310] have been made on the ef-

fect of soilstructure interaction under static loading.

These studies have considered the eect in a very sim-

plied manner and demonstrated that the force quanti-

ties are revised due to such interaction. A limited

number of studies [6,9,1114] have been conducted

on soilstructure interaction eect considering three-

dimensional space frames. The studies clearly indicated

that a two-dimensional plane frame analysis might

substantially overestimate or underestimate the actual

interaction eect in a space frame. From these studies, it

becomes obvious that the consideration of the interac-

tion eect signicantly alters the design force quantities.

These studies, may be quantitatively approximate, but

clearly emphasize the need for studying the soilstruc-

ture interaction to estimate the realistic force quantities

in the structural members, accounting for their three-

dimensional behaviour.

3. Soilstructure interaction under dynamic loading

Structures are generally assumed to be xed at their

bases in the process of analysis and design under dy-

namic loading. But the consideration of actual support

exibility [15,16] reduces the overall stiness of the

structure and increases the period of the system. Con-

siderable change in spectral acceleration with natural

period is observed from the response spectrum curve.

Thus the change in natural period may alter the seismic

response of any structure considerably. In addition to

this, soil medium imparts damping due to its inherent

characteristics. The issues of increasing the natural pe-

riod and involvement of high damping in soil due to

soilstructure interaction in building structures are also

discussed in some of the studies [17,18]. Moreover, the

relationship between the periods of vibration of struc-

ture and that of supporting soil is profoundly important

regarding the seismic response of the structure. The

demolition of a part of a factory in 1970 earthquake

at Gediz, Turkey; destruction of buildings at Carcas

earthquake (1967) raised the importance of this issue

[19]. These show that the soilstructure interaction

should be accounted for in the analysis of dynamic be-

haviour of structures, in practice. Hence, soilstructure

interaction under dynamic loads is an important aspect

to predict the overall structural response.

4. Model of structurefoundationsoil interacting system

It appears from the foregoing discussion that a com-

pletely misleading behaviour may be obtained unless

the interactive study of the soilstructurefoundation is

conducted. It is generally observed that the modeling of

the superstructure and foundation are rather simpler and

straightforward than that of the soil medium under-

neath. Yet, a lack of simple but reasonably accurate

model of some common structures is often come across.

Hence, the present paper puts forward some idealization

technique for buildings as well as water tanks, which is a

representative inverted pendulum type structure.

However, soil is having very complex characteristics,

since it is heterogeneous, anisotropic and nonlinear in

forcedisplacement characteristics. The presence of uc-

tuation of water table further adds to its complexity. Soil

can be modeled in a number of ways with various levels

of rigor. Hence, the major focus of the present article is

concentrated on soil modeling. However, a guideline in-

dicating an optimum compromise between rigor and ac-

curacy is needed to be furnished with brief details of the

models. Such a literature may help the designers to choose

a suitable model depending on the requirement. This ob-

jective is attempted to be fullled in the present work.

5. Idealization of structure

5.1. Buildings

In the most generalized form, superstructure of the

building frames may be idealized as three-dimensional

Fig. 1. Redistribution of loads in a frame due to soilstructure

interaction.

1580 S.C. Dutta, R. Roy / Computers and Structures 80 (2002) 15791594

space frame using two noded beam elements. The eect

of inll walls may be accounted for by imposing the

loads of the walls on to the beams on which they rest.

Plate element of suitable dimension may be added to

mimic the behaviour of slabs. This idealization appears

to be adequate for analyzing the building frame under

static gravity loading. But under lateral loading, inll

wall imparts considerable lateral stiness to the struc-

ture, since, then it behaves like a compressive strut.

Hence, under lateral loading the eect of the same must

be incorporated as specied in the literature [2023]. It is

well known that if the load coming onto the structure be

such that the stress in the reinforced concrete member of

the building exceeds the yield strength, then under a few

cycles of such loading, the stiness and strength of

concrete members will be degraded. This hysteresis be-

haviour is attempted to be modeled in the literature

[24,25] with various level of rigor. The details of many

such models are available elsewhere [26]. A suitable

model can be picked up depending on the accuracy re-

quired and computational facility available. However,

such degrading eect in stiness and strength seems to

be relatively lesser for buildings made up of steel.

5.2. Water tank

Elevated water tank with frame or shaft-type staging

may be conveniently modeled using any standard nite

element software for the sake of analysis under static

loading. But the performance of such elevated tank be-

comes crucial during earthquake. Then top portion of

the water in the container undergoes sloshing vibration

with a period generally much higher than the container

and the staging, while the remaining portion of water

moves with container, under a lateral ground shaking.

Thus, the system essentially becomes a two-mass model.

Details of such idealization are available in the literature

[27,28]. But during the torsional vibration of the tank,

almost entire amount of water is conceived to vibrate in

sloshing with a period considerably larger than the tor-

sional period of the structure including container and

staging. Details of such modeling are presented in the

literature in an elegant form [29].

The foundation system generally adopted may be

modeled using suitable rectangular or circular plate el-

ements. The strip or grid foundations may be modeled

using well-established theory of beams on elastic foun-

dation. For water tanks and cooling towers, circular or

annular rafts are generally used.

6. Modeling of soil media

The search for a physically close and mathematically

simple model to represent the soil-media in the soil

structure interaction problem shows two basic classical

approaches, viz., Winklerian approach and Continuum

approach. At the foundation-supporting soil interface,

contact pressure distribution is the important parameter.

The variation of pressure distribution depends on the

foundation behaviour (viz., rigid or exible: two extreme

situations) and nature of soil deposit (clay or sand etc.).

Since the philosophy of foundation design is to spread

the load of the structure on to the soil, ideal foundation

modeling is that wherein the distribution of contact

pressure [1] is simulated in a more realistic manner.

From this viewpoint, both the fundamental approaches

have some characteristic limitations. However, the me-

chanical behaviour of subsoil appears to be utterly er-

ratic and complex and it seems to be impossible to

establish any mathematical law that would conform to

actual observation. In this context, simplicity of models,

many a time, becomes a prime consideration and they

often yield reasonable results. Attempts have been made

to improve upon these models by some suitable modi-

cations to simulate the behaviour of soil more closely

from physical standpoint. In the recent years, a number

of studies have been conducted in the area of soil

structure interaction modeling the underlying soil in

numerous sophisticated ways. Details of these modelings

are depicted below in brief.

6.1. Winkler model

Winklers idealization represents the soil medium as a

system of identical but mutually independent, closely

spaced, discrete, linearly elastic springs [30]. According

to this idealization, deformation of foundation due to

applied load is conned to loaded regions only. Fig. 2

shows the physical representation of the Winkler foun-

dation. The pressuredeection relation at any point is

given by

p kw 1

where p is the pressure, k is the coecient of subgradereaction or subgrade modulus, and w is the deection.

A number of studies [3136] (only a few among many

others) in the area of soilstructure interaction have

been conducted on the basis of Winkler hypothesis for

its simplicity. The fundamental problem with the use of

this model is to determine the stiness of elastic springs

used to replace the soil below foundation. The problem

Fig. 2. Winkler foundation [29].

S.C. Dutta, R. Roy / Computers and Structures 80 (2002) 15791594 1581

becomes two-fold since the numerical value of the co-

ecient of subgrade reaction not only depends on the

nature of the subgrade, but also on the dimensions of

the loaded area as well. Since the subgrade stiness is the

only parameter in the Winkler model to idealize the

physical behaviour of the subgrade, care must be taken

to determine it numerically to use in a practical problem.

Hence, several methods proposed to estimate the mod-

ulus of subgrade reaction are also included in the present

work.

Modulus of subgrade reaction or the coecient of

subgrade reaction k is the ratio between the pressure p atany given point of the surface of contact and the set-

tlement y produced by the load at that point:

k p=y 2

The value of subgrade modulus may be obtained in the

following alternative approaches:

(a) Plate load test [3739],

(b) Consolidation test [40,41],

(c) Triaxial test [34,42] and

(d) CBR test [41,4345].

Following some suitable method mentioned to esti-

mate k, a reasonable value of subgrade modulus, theonly parameter to idalize soil stiness, may be obtained.

In the absence of suitable test data, representative values

for the same may be chosen following the guideline

presented in the literature [37]. However, the basic lim-

itations of Winkler hypothesis lies in the fact that this

model cannot account for the dispersion of the load over

a gradually increasing inuence area with increase in

depth. Moreover, it considers linear stressstrain be-

haviour of soil. The most serious demerit of Winkler

model is the one pertaining to the independence of the

springs. So the eect of the externally applied load gets

localized to the subgrade only to the point of its appli-

cation. This implies no cohesive bond exists among the

particles comprising soil medium. Hence, several at-

tempts have been made to develop modied models to

overcome these bottlenecks. These are discussed later in

the present paper.

6.2. Elastic continuum model

This is a conceptual approach of physical represen-

tation of the innite soil media. Soil mass basically

constitutes of discrete particles compacted by some in-

tergranular forces. The problems commonly dealt in soil

mechanics involve boundary distances and loaded areas,

very large compared to the size of the individual soil

grains. Hence, in eect, the body composed of discrete

molecules gets transformed into a statistical macro-

scopic equivalent amenable to mathematical analysis.

Thus, it appears very reasonable to invoke to the theory

of continuum mechanics for idealizing the soil media

[46].

The genesis of continuum representation for the soil

media is perhaps from the research work of Boussinesq

[47] to analyze the problem of a semi-innite, homoge-

neous, isotropic, linear elastic solid subjected to a con-

centrated force acting normal to the plane boundary,

using the theory of elasticity. In this case, some contin-

uous function is assumed to represent the behaviour of

soil medium. In fact, later on it has been concluded that

the nature of supporting elastic medium of any type can

best be described by the deection line of its surface

under a unit concentrated load [48]. In the continuum

idealization, generally soil is assumed to be semi-innite

and isotropic for the sake of simplicity. However, the

eect of soil layering and anisotropy may be conve-

niently accounted for in the analysis [46].

This approach provides much more information on

the stresses and deformations within soil mass than

Winkler model. It has also the important advantage of

simplicity of the input parameters, viz., modulus of

elasticity and Poissons ratio. Solutions for some prac-

tical problems idealizing the soil media as elastic con-

tinuum are available for few limited cases [49,50].

However, this idealization of a semi-innite elastic

continuum leads to many-fold intricacies from mathe-

matical viewpoint [51]. This severely limits the applica-

tion of this model in practice. One of the major

drawbacks of the elastic continuum approach is inac-

curacy in reactions calculated at the peripheries of the

foundation. It has also been found that, for soil in re-

ality, the surface displacements away from the loaded

region decreased more rapidly than what is predicted by

this approach [53]. Thus, this idealization is not only

computationally dicult to exercise but often fails to

represent the physical behaviour of soil very closely, too.

6.3. Improved foundation models

In order to take care of the shortcomings of both the

basic approaches, viz., Winklers model and Continuum

model, some modied foundation models have been

proposed in the literature. These modications have

generally been suggested following two alternate ap-

proaches. In the rst approach, the Winkler foundation

is modied to introduce continuity through interaction

amongst the spring elements by some structural ele-

ments. In the second approach, continuum model is

simplied to obtain a more realistic picture in terms of

expected displacement and/or stresses. These improved

foundation models are briey described below.

6.3.1. Improved versions of winkler model

6.3.1.1. Filonenko-borodich foundation. Fig. 3 shows the

physical representation of FilonenkoBorodich foun-


dation model [54]. As per this model, the connectivity of

the individual Winkler springs is achieved through a thin

elastic membrane subjected to a constant tension T. This

membrane is attached at the top ends of the springs.

Thus, the response of the model is mathematically ex-

pressed as follows.

p kw Tr2w; for rectangular or circular foundation

kw T d2wdx2

; for strip foundation

3

where, r2 Laplace operator o2ox2 o2

oy2; T tensileforce.

Hence, the interaction of the spring elements is char-

acterized by the intensity of the tension T in the mem-

brane. An essentially same foundation model consisting

of heavy liquid with surface tension is also suggested in

the literature [55].

6.3.1.2. Hetenyis foundation. This model suggested in

the literature [31] can be regarded as a fair compromise

between two extreme approaches (viz., Winkler foun-

dation and isotropic continuum). In this model, the in-

teraction among the discrete springs is accomplished by

incorporating an elastic beam or an elastic plate, which

undergoes exural deformation only, as shown in Fig. 4.

Thus the pressuredeection relationship becomes

p kw Dr4w 4

where,

D flexural rigidity of the elastic plate Eph3p=121 lp2;

p is the pressure at the interface of the plate and thesprings; Ep and lp are Youngs modulus and Poissonsratio of plate material; hp is the thickness of the plateand

r4 o4

ox4 o

4

oy4 2 o

4

ox2oy2

Thus, it is seen that the exural rigidity of embedded

beam or plate characterizes the interaction between the

spring elements of the Winkler model. Detailed de-

scriptions of this model as well as some numerical ex-

amples are available in the literature [31,56].

6.3.1.3. Pasternak foundation. In this model, existence of

shear interaction among the spring elements is assumed

which is accomplished by connecting the ends of the

springs to a beam or plate that only undergoes trans-

verse shear deformation [Fig. 5]. The loaddeection

relationship is obtained by considering the vertical equi-

librium of a shear layer. The pressuredeection rela-

tionship is given by

p kw Gr2w 5

where, G is the shear modulus of the shear layer.Thus the continuity in this model is characterized by

the consideration of the shear layer. A comparison of

this model with that of FilonenkoBorodich implies

their physical equivalency (T has been replaced by

G). A detailed formulation and the basis of the de-

velopment of the model have been discussed elsewhere

[57]. Analytical solutions for plates on Pasternak-type

foundations with a brief of the derivation of the model

have been reported in the literature [51,58].

6.3.1.4. Generalized foundation. In this foundation

model, it is assumed that at each point of contact mo-

ment is proportional to the angle of rotation in addition

to the Winklers hypothesis [5962]. This can be ana-

lytically described as follows.

p kw

and

mn k1 dwdn

6

where, mn is the moment in direction, n; n is the directionat any point in the plane of the foundation; and k, k1 areproportionality factors.

The assumption made on the proportionality in this

model is relatively arbitrary [51]. However, a physical

signicance, of the same has also been demonstrated in

the same literature [51].

6.3.1.5. Kerr foundation. A shear layer is introduced in

the Winkler foundation and the spring constants above

and below this layer is assumed to be dierent as per this

formulation [52]. Fig. 6 shows the physical representa-

tion of this mechanical model. The governing dierential

equation for this model may be expressed as follows.Fig. 4. Hetenyi foundation [30].

Fig. 3. FilonenkoBorodich foundation [52].


1 k2k1

p G

k1r2p k2w Gr2w 7

where, k1 is the spring constant of the rst layer; k2 is thespring constant of the second layer; w is the deection ofthe rst layer.

6.3.1.6. Beam column analogy model. The classical

problem of beams on elastic foundation (Fig. 7) is at-

tempted to be solved in a literature [63] with a new

subgrade model. The nal form of the governing dif-

ferential equation for combined beam-subgrade behav-

iour is obtained as follows:

EbIbd4wxdx4

Cp2 d2wxdx2

Cp1wx qx 8

where, EbIb is the exural stiness of the beam (assumedconstant); wx is the beam settlement qx is the appliedload; Cp1 and Cp2 are constants.For an isotropic, homogeneous layer underlain by a

rigid base, the values of the above constants may be

chosen as Cp1 E=H and Cp2 GH=2 where E is theYoungs modulus of soil, G is the shear modulus of soil,H is the depth to the assumed rigid base.

The above equation is analogous to a beam-column

under constant axial tension of magnitude Cp2, which is

supported on transverse independent springs of sti-

nesses Cp1. Thus, it appears that the continuity amongthe individual Winkler springs is achieved by the pa-

rameter Cp2. However, the modeling of foundation be-comes incorrect due to introduction of a ctitious shear

force [63] while the modeling, as a whole is a signicant

improvement over Winklers hypothesis as a subgrade

model.

6.3.1.7. New continuous winkler model. It has been ob-

served that, to model the continuity in the soil medium,

generally some other structural element is introduced.

But in this model, instead of discrete Winkler springs,

springs are intermeshed so that the interconnection is

automatically achieved [64]. A schematic representation

of the model is shown in Fig. 8. Physically, intercon-

nection among Winkler springs connected to the foun-

dation beam or plate is achieved by some other spring by

virtue of their axial stiness, which are not directly at-

tached to the foundation. Details of the model with

some case studies on beam, plate, hyper shell raft, etc.

resting on elastic foundation are presented in the liter-

ature [64]. The excellence of this model lies in its ability

to account for the eect of the soil outside the bound-

aries of the structure in the modeling.

6.3.2. Improved versions of continuum model

6.3.2.1. Vlasov foundation. Starting from continuum

idealization this foundation model has been developed

using variational principle [65,66]. This model imposes

certain restrictions upon the possible deformations of an

elastic layer. As per this model,

(i) The vertical displacement wx; z wx. hz, suchthat h0 1 and hH 0. This function hz de-scribes the variation of displacement in vertical di-

rection.

(ii) The horizontal displacement ux; z is assumed to bezero everywhere in the soil.

The function hz may be assumed to be linearly de-creasing with depth for a classical foundation of nite

thicknessH. Hence, in this case, hz 1 z=H. For the

Fig. 7. Beamcolumn analogy model to classical beams on

elastic foundation [62]. Fig. 8. Intermeshed Winkler spring model [63].

Fig. 6. Kerr foundation [50].

Fig. 5. Pasternak foundation [55].


foundation resting on relatively thick (or innite thickness)

elastic layer, the choice may be hz sinhcH z=sinhcH , where c is a coecient depending on the elasticproperties of the foundation dening the rate of decrease

of displacements with depth. Then using the principle of

virtual work, response function for this model is ob-

tained and reported in the literature [67] as

p kw 2t d2wdx2

9

where

k E01 m02Z H0

dhdz

2dz; t E0

41 m0Z H0

h2 dz;

E0 E1 m2

and m0 m1 m; E and m are soil constants.

6.3.2.2. Reissner foundation. As per this model, pressure

deection relationship at the interface between founda-

tion slab and subgrade is obtained by the intrusion of a

foundation layer below the slab. This is based on the

following assumptions:

(i) in plane stresses throughout the foundation layer are

negligibly small, and

(ii) horizontal displacements at the upper and lower sur-

faces of the foundation layer are zero.

The pressuredeection relationship is given by

C1w C2r2w p C24C1

r2p 10

where w is the displacement of the foundation surface, pis a distributed lateral load acting on the foundation

surface; C1 E=H ; C2 HG=3; E, G are the elasticconstants of foundation material and H is the thicknessof the foundation layer.

The term H 2G=E in Eq. (10), known as dierentialshear stiness, oers the possibility of obtaining closer

agreement with actual behaviour [68]. This model also

retains the mathematical simplicity of Winkler models.

The classical problem of innitely long rigid strip resting

on Ressiner model and supporting a central line load is

studied in details [69]. It was observed from various

studies that this model predicts higher stress in struc-

tures.

In addition to the above-mentioned models, a few

more improved foundation models have also been pro-

posed in the literature [7075].

7. Applicability of the models

Little evidence is available in the present time to

verify the computational accuracy of the various models

studied to represent the soil medium in soilstructure

interaction analysis. Moreover, it is also dicult to de-

cide the physical quantity, precision of which may in-

dicate the accuracy of the whole computational process.

The dierent idealizations of the soil-media may be

compared with respect to the ways the mechanics of the

problem is treated. However, the model idealizing the

system more rigorously from physical perspective may

deviate more in predicting the behaviour. This may so

happen generally due to the possible uncertainties in the

determination of the parameters involved, number of

which is generally greater in more physically accurate

model. Another matter of considerable interest in the

idealization is to select a model easy to apply.

The various foundation models discussed herein uti-

lize a number of parameters to represent the behaviour

of the soil. Thus, the determination of the parameters

that constitute the model is the basic requirement.

Modulus of subgrade reaction can be conveniently de-

termined from plate load test [37,76]. The values so

obtained can be easily modied for the actual footing.

The other parameters may be obtained from rigid stump

test [74,7780].

Studies have been reported in the area of soilstruc-

ture interaction replacing the soil in a number of dif-

ferent ways. Out of all the models available, Winkler

foundation utilizes only a single parameter. This can be

very conveniently determined and suitably modied for

actual foundation size, shape, etc. to employ in actual

analysis [37,76]. The fundamental limitation of Winkler

idealization lies with the independent behaviour of the

soil springs. Since the degree of continuity of the struc-

ture is suciently higher than the soil media, this ap-

proximation may not be far from reality [38]. Moreover,

a comparison of Winkler solution for a beam on elastic

foundation shows reasonable agreement with classical

solution [31] and the nite-dierence solution [81]. The

most noteworthy series of tests on continuous beams

reported in the literature [82], also corroborate the

ndings obtained through Winkler idealization [81].

Since it is very dicult to arrive at an accurate value for

Youngs Modulus of soil, which is an essential param-

eter in elastic continuum idealization; the approach of

using subgrade modulus nds more appreciation [39].

Further, the validity of Winklers assumptions has been

strongly established for Gibson type soil medium, where

shear modulus of soil varies linearly with depth [83]. It is

also recognized in the literature [84] that even large error

in the assessment of the values of the subgrade modulus

inuences the response of the superstructure quite in-

signicantly. The present practice in design oces gen-

erally adopts a xed base consideration for structural


analysis and design. In this context, the Winkler model,

though oversimplied, seems adequate and suitable for

computational purpose for its reasonable performance

and simplicity.

8. Advanced modeling

In the previous section, pros and cons of classical

modeling of soil media has been discussed in brief. This

section addresses towards the formulation and applica-

bility of some more rened models. The merits and de-

merits of such idealization to analyze the interaction

behaviour have also been reviewed. But before going to

such details, it is worthwhile to present a brief scrutiny

of the complex characteristics of actual soil behaviour,

which is attempted to be modeled.

8.1. Behaviour of soil media

The mechanical behaviour of soil media is so com-

plex that a mathematical simulation of the same is al-

ways a mammoth task to the engineers. Soil is basically

composed of particulate materials. The behaviour of

soil, mainly the stressstraintime property, inuences

the soilstructure interaction phenomenon.

Physically, when a load is applied on the soil mass

(not completely saturated), the soil particles tend to at-

tain such a structural conguration that their poten-

tial energy will be a minimum and hence stability is

achieved. Up to a certain stress level, strain imparted to

the soil mass in this process is elastic and then it may

enter the plastic range depending on the magnitude of

the applied load. This deformation is followed by a

mostly viscoplastic deformation (dominant for ne-

grained soil) due to viscous intergranular behaviour that

implies strain with passage of time. This deformation

occurs by the expulsion of the pore uid and simulta-

neous transfer of excess pore pressure to the solid soil

grains. Hence, the rate of such strain approaches a small

value after a long time. The strain caused by the ex-

pulsion of water from the soil mass is identically equal to

the strain of the soil skeleton. This is because soil skel-

eton is an aggregate of mineral particles, which together

with bound water constitutes the soil mass. This process

is known as primary consolidation. However, after pri-

mary consolidation of the soil structure, continues to

adjust to the load for some additional time and sec-

ondary compression occurs approximately following a

logarithmic function of time [39]. But it is to be noted

that the settlement of any representative soil specimen

may come to an end beforehand if the range of elasticity

of soil is sucient compared to the applied load. Then

the strain will not be a function of time. But for such a

fully saturated soil sample, strain will always be the

function of time, since the external load will rst be

shared by the pore uid under such condition and then

viscoelastic settlement will occur. It has been observed

that the hardening of soil due to consolidation and the

thixotropic processes must be taken into analysis as it

causes manifold increase in the cohesion and angle of

internal friction of soil. Thus well-selected rheological

models in conjunction with the model to represent the

phenomenomenologacal behaviour may oer some use-

ful means to study the interactive system. Attempts have

been extended in the same direction in the following

subsections.

8.2. Elasto-plastic idealization

In the soilstructure interaction analysis, nonlinear

behaviour of soil mass is often modeled in the form of an

elasto-plastic element. Up to a certain stress level, de-

formation occurs linearly and proportional to the ap-

plied stress. This behaviour may be represented by ideal

reversible spring. A Hookean spring element is the best

suitable representation for the same. The perfectly plas-

tic deformation of the soil mass can be well represented

with the help of a Coulomb unit [85]. But when an elastic

element (Hookean Spring) is connected in series with a

plastic element, a new schematic system known as St.

Venants unit is formed. Use of such a single element

generally shows an abrupt transition of soil from elastic

to plastic state. Instead, the use of a large number of St.

Venants units in parallel (Fig. 9) represents the elasto-

plastic behaviour of soil more accurately. Use of a

number of springs helps to facilitate the simulation of

the gradual transition of soil strain from elastic to plastic

zone. The following expression may be used in terms of

strain moduli for elstic and plastic strains (eep), respec-tively;

eep MerMp log ruru r 11

where Me is the elastic strain modulus of soil; Mp is theplastic strain modulus of soil and ru is the ultimate loadthat soil can sustain.

Conceptually, the above mechanical model may ap-

pear to be useful enough. But problems occur in view of

the choice of the parameters as well as the proper ad-

justment of such springs at the base of the structure. At

Fig. 9. St. Venant elasto-plastic unit in parallel [84].


this sequence, the use of more recently developed elasto-

plastic soil models [86] are invoked.

As per this idealization, dierent convenient forms of

the various yield criteria of soils such as Tresca yield

criterion, Von Mises yield criterion, MohrCoulomb

yield criterion, DruckerPrager yield criterion, etc.

[46,87] may be suitably chosen in the modeling. A ow

rule to describe the post-yielding behaviour may be

adopted following deformation theory or, the incre-

mental or ow theory [88]. In the deformation theory,

the plastic strains are uniquely dened by the state of

stress, whereas in the incremental theory the plastic

strains depend upon a combination of factors, such as

increments of stress and strain and the state of stress.

For general elasticplastic behaviour, the incremental

theory of plasticity is often employed for its generality.

The constitutive modeling of soil to be adopted in the

analysis may be developed using either of the incre-

mental method, iterative method, initial strain method

and initial stress method. Detailed formulations of the

same with their suitability in application have been de-

picted in the literature [88,89]. The major advantage of

such formulation is that it permits the computer coding

of the yield function and the ow rule in the general

form and necessitates only the specication of the con-

stants involved that may be conveniently obtained. Re-

cently, an elasto-plastic model for unsaturated soil in

three-dimensional stresses has been developed in the

literature [90].

Attempt has been made to investigate the interactive

behaviour using elasticperfectly plastic behaviour of

subsoil for a plane frame-combined footing-soil system

[91,92]. The eect for the inuence of strain-hardening

characteristics of soil in the elasto-plastic soilstructure

interaction of framed structures has also been under-

taken [93]. However, the use of this model is not very

popular because, in spite of the mathematical intricacies

involved, it does not yield reasonable performance to

predict the interactive behaviour.

8.3. Nonlinear idealization

The stressstrain behaviour of soil is virtually non-

linear. The solution of nonlinear problems is normally

achieved by one of the three basic techniques: incre-

mental procedure, iterative procedure and mixed pro-

cedure. Mathematical formulations of these techniques

have been presented in the literature in considerable

details [88]. The major advantage of the incremental

procedure is its generality to use in analyzing almost all

types of nonlinear behaviour, barring some work-soft-

ening materials; but it is time-consuming. On the con-

trary, the iterative scheme works faster and may be

utilized in bi-modular and work-softening materials,

where incremental method fails. However, the iterative

method fails to assure convergence to the exact solution

and cannot be suitably applied to dynamic problems as

well as the materials having path-dependent behaviour.

To minimize the disadvantages of each, incremental it-

erative or mixed technique is recommended that com-

bine the advantages of the both.

However, the outputs from any numerical or ana-

lytical technique are acceptable only to the extent that

the constitutive relation of the material is accurate.

Nonlinear stressstrain relationship may be represented

either with discrete values in tabular form (obtained

from laboratory test results) where interpolation is made

for intermediate values or in the functional form.

Mathematical spline functions can provide a satisfactory

functional representation of stressstrain curves and of

the tangent moduli computed as the rst derivative of

the curves [88]. The most popular functional approach

to describe the same is to characterize the soil with hy-

perbolic relationship [94,95]. But the inadequacy of the

model has been clearly shown in the literature [96].

Another mathematical model accounting for the soil

nonlinearity has been proposed in the literature [97]. In

the recent time, a nonlinear elastic model to simulate

stressstrain relationships over a wide range of strains

has been advanced [98]. So, when the load on the soil

from the superstructure does not become so high that

plastic strain occurs in the soil mass, this model can be

suitably employed.

Study has been made on the interactive behaviour on

simplied structural models with nonlinear soil behav-

iour [99]. A rigorous computational method accounting

for nonlinear load-settlement characteristics of consoli-

dation was validated from model tests and was reported

in the literature [11,100,101]. Two studies used this

scheme and showed that dierential settlement may

cause a many-fold increase in the axial force and mo-

ment of the corner columns [12,14]. Recently, a rigorous

computational scheme accounting for the three-dimen-

sional behaviour of the structure as well as the nonlinear

consolidation characteristics of clayey soil has been de-

veloped by the authors [102]. Perhaps this three-dimen-

sional structural representation considering nonlinear

soil behaviour is a reasonably accurate representation of

the interactive system. Yet a scrutiny of the existing

literature reveals that only a few studies have considered

the same.

8.4. Viscoelastic idealization

The real deformation characteristics of soil media

(particularly ne-grained) under the application of any

load are always time-dependent to some extent de-

pending on the permeability of soil media. Loading

applied to saturated layers of clay, at the rst instance,

causes an increase in pressure in the pore water of soil.

With time, the pore water pressure will dissipate re-

sulting in progressive increase of eective stress in soil


skeleton. This leads to time-dependent settlement of

foundation. There are numerous instances of rheological

processes in the foundations leading to large and non-

uniform settlement. Considerable displacement of re-

taining walls from their original position and instability

of slopes and embankments [103] are two classical ex-

amples apart from the usual time-dependent settlement

of footings of building frames. Similar observations

have been reported elsewhere [104]. Hence, a general

approach governing the deformation of soil with time

considering the rheological process at the micro level is

necessary. Various models are available to describe the

rheological properties of clayey soil such as mechanical

model, theory of hereditary creep, engineering theory of

creep, theory of plastic ow and molecular theory of

ow [103]. Details of these models are available else-

where [105]. However, for the sake of completeness,

brief accounts of some of such models are described

below.

The mechanical models represent the rheological

properties of the soil skeleton by a combination of elas-

tic, viscous and plastic elements. These models are gen-

erally formed by a combination of spring and dashpot

in series (e.g., Maxwell model; shown in Fig. 10) or in

parallel (e.g., Kelvin model; shown in Fig. 11). A de-

tailed discussion of these models with their physical

interpretation has been furnished in the literature [105].

In the Shvedov model, the elastic element is connected in

series with the viscous element and then in parallel with

the St. Venants plastic element [106].

These mechanical models predict the shear strain

more accurately. Hence, attempts have been made to

develop models that account for the process of consol-

idation also. They describe the mechanism of transmis-

sion of load to the soil skeleton and water. Extensive

research eorts [107113] have been made to idealize the

one-dimensional consolidation characteristics of soil as

viscoelastic model. This gives an insight to the secondary

consolidation phenomenon as well. The various pa-

rameters involved in these mechanical models may be

suitably determined following the treatise on the same

[103]. A recent review [114] concluded that no such

model is available that can suitably describe the time-

dependent behaviour for the soil at any stress level. A

new such model is also proposed in the same literature

[114].

A three-dimensional viscoelastic nite element for-

mulation for studying the interactive behaviour of space

frame considering the stressstrain versus time response

of supporting soil media has been made to observe the

importance of such detailed modeling [115]. Observation

of the results obtained shows that the time-independent

analysis may often lead to estimates which is needed to

be accounted in the design for safety. Hence, to arrive

at the complementary recommendations for the design

of structures resting on consolidating soil, viscoelastic

idealization of the same is desired to be considered.

Similar conclusions have been made elsewhere [116].

Thus it appears that modeling the foundation soil, as

viscoelastic medium may be more appropriate.

8.5. Finite element modeling

The widespread availability of powerful computers

has brought about a sea change in the computational

aspect recently. Since the scope of numerical methods is

incomparably wider than that of analytical methods, the

use of general-purpose nite element method has at-

tained a sudden spurt to study the complex interactive

Fig. 10. Maxwell model [104].

Fig. 11. Kelvin model [104].


behaviour. The method is so general that it is possible

to model many complex conditions with a high degree

of realism, including nonlinear stressstrain behaviour,

non-homogeneous material conditions, changes in ge-

ometry and so on. However, care must be taken about

the possibilities of inaccuracy arising out of numerical

limitations while interpreting the results [88]. Neverthe-

less, this seems to be the most powerful and versatile

tool for solving soilstructure interaction problem.

The method is a special extended form of matrix

analysis based on variational approach, where the whole

continual is discretized into a nite number of elements

connected at dierent nodal points. Displacements func-

tions, i.e., the displacement within the element is

not known and hence to be judiciously assumed. Thus

knowing the stiness matrix for each element, overall

stiness matrix may be determined. Hence, from the

global loading conditions and boundary conditions

nodal unknowns may be generated.

The general principles and use of this method is well

documented in the literature [88,89]. A nite element

procedure for the general problem of three-dimensional

soilstructure interaction involving nonlinearities due to

material behaviour, geometrical changes and interface

behaviour is also presented in the literature [117]. The

viscoelastic behaviour of soil may also be conveniently

modeled in this method. Such a suitable scheme has been

presented in considerable details in the literature [118].

Discontinuous behaviour may occur at the interface of

soil and structure. Several studies [119123] have been

made to develop interface elements, use of which is

proved to be useful to take care of this discontinuity.

The stiness matrix for the interface element has been

explicitly presented in the literature [124]. In view of its

generality, the present paper recommends the use of the

same to study the soilstructure interaction behaviour at

least for important structure, if possible.

9. Dynamic soilstructure interaction

The consideration of soil-exibility increases the pe-

riod of vibration. Hence, a considerably dierent re-

sponse from that of reality may be obtained if this eect

is not considered. Such observations have been made by

the authors while analyzing the building with isolated

footing [125]. There are two currently used procedures

for analyzing seismic vibration of structures incorpo-

rating the eect of soilstructure interaction: (1) Elastic

half space theory [126], (2) Lumped mass or lumped

parameter method [39]. The strengths and limitations of

the available methods have been discussed in details in

the literature [127,128]. However, on the basis of an

extensive literature survey, it is suggested elsewhere

[39,128] that the lumped mass approach is more reliable

and substantially more general than the other alternative

procedures. Hence, the present paper recommends the

use of the same and provides a brief outline here. As per

this method, three translational and three rotational

springs are attached along three mutually perpendicular

axes and three rotational degrees of freedom about the

same axes below each of the foundation of the structure.

The stinesses of these springs for arbitrary shaped

footings (except annular one) resting on homogeneous

elastic half-space have been suggested in the literature

[129]. Conceptual background to develop such stiness

functions has been presented in the literature [130]. The

expressions for these spring stinesses, the shape factors

and the factors accounting for the depth of embedment

involved to compute the same have been suggested after

an extensive literature survey, study based on boundary

element method and experimental verication [129].

Dynamic stiness for machine foundations resting on

layered soil systems has been discussed elsewhere [131].

An analytical method to estimate the stiness of the

foundations embedded into the stratum over rigid rock

corresponding to dierent stress distribution below the

foundation has been elegantly presented in the literature

[132]. This study highlights on the sensitivity of the stress

distribution below the foundation in the estimation of

the dynamic stiness of the underlying soil media. The

stiness of annular footings has been derived in some

other literature [133135]. It has been observed that the

stinesses of the springs are dependent on the frequency

of the forcing function, more strongly if the foundation

is long and on saturated clay [136,138]. In fact, the in-

ertia force exerted by a time varying force imparts a

frequency dependent behaviour, which seems to be more

conveniently incorporated in stiness in the equivalent

sense. Thus the dependence of the stiness of equivalent

springs representing the deformable behaviour of soil is

due to the incorporation of the inuence that frequency

exerts on inertia, though purely stiness properties are

frequency independent. This frequency dependence is

suggested to be incorporated by multiplying the equiv-

alent spring stinesses by a frequency dependent factor.

This factor is plotted as a function of a non-dimensional

parameter a0 where a0 xB=Vs [129]. Here, x is thefrequency of the forcing function, B is the half of thelateral dimension of the footing and Vs is the shear wavevelocity in the soil medium. But in an earthquake mo-

tion, a large spectrum of waves with wide ranges of

frequencies participates together. Hence, it is dicult to

consider any frequency dependent multiplier to compute

dynamic stiness and damping coecient as is suggested

in the literature [129]. In fact, other literatures [138,139]

have not recommended the use of such multiplication

factors perhaps due to the same reason. However, the

critical situations that may occur due to the consider-

ation of these frequency dependent factors have been

studied for buildings on grid foundation in a very lim-

ited form [140]. The study reveals that the eect of


frequency dependent soil-exibility on the behaviour of

overall structural system may be higher than what is

obtained from the frequency independent behaviour,

only to a limited extent. The additional damping eect

imparted by the soil to the overall system may also be

conveniently accounted for in this method of analysis

[39,129]. However, it is agreed that at certain complex

sites, nite element idealization of elastic half space de-

noting soil below foundation may prove useful. An

outline of such procedure has been given in the literature

in a very lucid manner [141]. Nevertheless, it is believed

in the recent time that nite element method is not ca-

pable of idealizing the innite soil medium properly.

Hence it is suggested in the literature [136144] to model

the innite soil media using boundary element method

and the nite structure with nite element method.

These two dierent means of idelizations may be suit-

ably matched at the interface through equilibrium and

compatibility conditions. An extremely ecient scheme

for the analysis of soilstructure interaction system

using coupling model of nite elements, boundary ele-

ments, innite elements and innite boundary elements

has been elegantly presented in the literature [145] re-

cently. This scheme will be of ample help in case of

layered soil also. The eect of soilstructure interaction

on vibrating pile foundation can be studied following

the analytical formulation or numerical modeling of

vibrating beams partially embedded in a Winkler foun-

dation, presented in well-accepted literature [146,147].

The discussion on the sensitivity of the nite element

models for the same provided in such literature [146] can

be of help in nite element modeling for pile vibration

problem with varying degrees of renement depending

on the required level of accuracy.

10. Conclusions

The review of the current state-of-the art of the

modeling of soil as applied in the soilstructure inter-

action analysis leads to the following broad conclusions.

(1) To accurately estimate the design force quantities,

the eect of soilstructure interaction is needed to

be considered under the inuence of both static

and dynamic loading. To obtain the same, realistic

yet simplied modeling of the soilstructurefoun-

dation system is obligatory.

(2) Winkler hypothesis, despite its obvious limitations,

yields reasonable performance and it is very easy

to exercise. So for practical purpose, this idealization

should, at least, be employed, instead of carrying out

an analysis with xed base idealization of structures.

(3) The consolidation phenomenon of clayey soil

follows a nonlinear stresssettlement relationship.

Hence, to achieve a more realistic analysis of the

soilstructure interaction behaviour involving clayey

soil, nonlinear modeling of soil is desired. To per-

form such an analysis, incremental iterative tech-

nique appears to be the most suitable and general

one.

(4) The clayey soil having low permeability possesses

time-dependent behaviour under sustained loading.

In such time-dependent process of soilstructure in-

teraction, critical condition may occur at any time

during the process in some situation. Under such cir-

cumstances, modeling the soil as viscoelastic me-

dium can only provide the crucial input for design.

(5) Modeling the system through discretization into a

number of elements and assembling the same using

the concept of nite element method has proved to

be a very useful method, which should be employed

for studying the eect of soilstructure interaction

with rigor. In fact, the technique becomes useful to

incorporate the eect of material nonlinearity, non-

homogeneity and anisotropy of the supporting soil-

medium if needed to be accounted due to the case

specic nature of any particular problem.

(6) The eect of soilstructure interaction on dynamic

behaviour of structure may conveniently be ana-

lyzed using Lumped parameter approach. However,

resort to the nite element modeling may be taken

for the important structure where more rigorous

analysis is necessary.

(7) The paper may help to arrive at a suitable method of

analysis by properly weighing the strength and limi-

tation of the same against the particular characteris-

tics and need of the problem at hand. The further

details of a method may be obtained from picking

the right reference from the exhaustive list presented

in the paper.

Acknowledgements

The support received from a UGC Major Research

Project (no. F.1413/2000 (SR-I)) is gratefully acknowl-

edged. The help rendered by Mr. K. Bhattacharya, a

Graduate student of B.E. College (D.U.) is also sincerely

appreciated.

Appendix A. Introduction to references

Ref. [1] explains the inuence of structural rigidity

apart from soil exibility on the amount of load distri-

butions due to soilstructure interaction. A suitable

iterative method for estimation of the eect of soil

structure interaction is outlined in Ref. [3]. Ref. [14]

provides an idea about the eect of dierential settle-

ment on design force quantities of various building


frames with isolated footings. Remedial measure to re-

duce this eect is also suggested in this literature. Refs.

[20,21] provide the approach for accounting the contri-

bution of the brick walls to the lateral stiness of the

buildings. The detailed information about various im-

portant models, namely FilonenkoBorodich Founda-

tion model, Hetenyis Foundation model, Pasternak

Foundation model, Kerr Foundation model, Beam

Column analogy model, and New Continuum model can

be obtained from Refs. [54], [31,56], [57], [52], [63], [64],

respectively. Refs. [65,66,68] provide the details of two

improved versions of continuum model, namely, Vlasov

Foundation and Reissner Foundation, respectively. Vali-

dated computational scheme of accounting for nonlinear

loadsettlement characteristics of consolidation settle-

ment in framesoil interaction process was reported in

[100,101]. Ref. [114] proposed a model, which can suit-

ably depict the time-dependent behaviour for the soil at

any stress level. Modeling of foundation soil interface

with the help of nite element discretization is explicitly

presented in [124]. Ref. [129] provides the dynamic

stiness as well as damping characteristics of soil me-

dium supporting any arbitrary shaped foundation. In-

cluding the eect of the frequency of the forcing function

in dynamic stiness of soil medium, it becomes a

benchmark literature in the area of dynamic soilstruc-

ture interaction. Modeling required to address the

problem of soilstructure interaction of pile foundation

in vibrating condition nds a detailed treatment in two

pioneering literatures, Refs. [146,147].

References

[1] Taylor DW. Fundamentals of soil mechanics. New York:

John Wiley and Sons; 1964.

[2] Roy R, Dutta SC, Moitra D. Soilstructure interaction in

buildings with isolated and grid foundations: a critical

study on the state of the art with recommendations. The

Bridge and Structural Engineers 2002;31(4):1536.

[3] Chamecki C. Structural rigidity in calculating settlements.

J Soil Mech Found Div ASCE 1956;82(1):119.

[4] Morris D. Interaction of continuous frames and soil media.

J Struct Eng Div ASCE 1966;(5):1343.

[5] Larnach WJ. Computation of settlement of building frame.

Civ Eng Publ Works Rev 1970;65:10404.

[6] Lee IK, Brown PT. Structuresfoundation interaction

analysis. J Struct Eng Div ASCE 1972;(11):241331.

[7] Subbaa KS, Rao H, Sharaadaa BV. Interaction analysis of

frames with beam footing. Proc Indian Geotech Conf

Roorkee 1985;(l):38995.

[8] Allam MM, Subba Rao KS, Subramanya IVV. Frame soil

interaction and winkler model. In: Proceedings of Institu-

tion of Civil Engineers, Part 2. 1991. p. 47794.

[9] Deshmukh AM, Karnarkar SR. Interaction analysis of

plane frame with soil. Proc Indian Geotech Conf Surat

India 1991;1:3236.

[10] Noorzaei J, Godbole PN, Viladkar MN. Nonlinear soil

structure interaction of plane framesa parametric study.

Comput Struct 1993;49(3):5616.

[11] Dutta SC, Bhattacharya G, Moitra D. Eect of soil

structure interaction on building frames. Proc Indian

Geotech Conf 1999;1:1236.

[12] Dutta SC, Maiti A, Moitra D. Eect of soilstructure

interaction on column moment of building frames. J Inst

Eng (India) 1999;(May):17.

[13] Bhattacharya G, Dutta SC, Roy R, Dutta S, Roy TK. A

simple approach for framesoil interaction analysis. In:

IGC: The Millenium Conference, Indian Institute of

Technology, Powai, Mumbai, 2000. p. 11922.

[14] Roy R, Dutta SC. Dierential settlement among isolated

footings of building frames: the problem, its estimation

and possible measures. Int J Appl Mech Eng 2001;6(1):

16586.

[15] Martel RR. Eect of foundation on earthquake motion.

Civ Eng 1940;10(1):710.

[16] Tanabashi R, Ishizaki H. Earthquake damages and elastic

properties of the ground. Bulletin No.4, Disaster Preven-

tion Research Institute, Koyoto University. 1953.

[17] Stewart JP, Fenres GL, Seed RB. Seismic soilstructure

interaction in buildings I: analytical method. J Geotech

Geoenviron Eng Div 1999;125(1):2637.

[18] Stewart JP, Seed RB, Fenres GL. Seismic soilstructure

interaction in buildings II: empirical ndings. J Geotech

Geoenviron Eng Div 1999;125(1):3848.

[19] Dowrick DJ. Earthquake resistant design: a manual for

engineers and architects. New York: John Wiley and Sons

Ltd; 1977.

[20] Smith BS. Lateral stiness of inlled frames. J Struct Eng

Div ASCE 1962;88(6):18399.

[21] Smith BS, Carter C. A method of analysis of inlled

frames. Proc Inst Civil Engrs 1969;44(2):3148.

[22] Mallick DV, Severn RT. The behaviour of inlled frames

under static loading. Proc Inst Civ Engrs 1967;38:63956.

[23] Mallick DV, Severn RT. Dynamic characteristics of inlled

frames. Proc Inst Civ Engrs 1967;39:26187.

[24] Park YJ, Reinhorn AM, Kunnath SK. IDARC: Inelastic

structural analysis of reinforced concrete frame-shear wall

structures. Report No. NCEER-87-0008, National Centre

for Earthquake Research, Bualo, NY, USA. 1987.

[25] Das PK, Dutta SC. Simplied hysteresis models to

recognize progressive damage of R.C. asymmetric build-

ings under seismic torsion. In: Proceedings of the national

symposium of advances in structural dynamics and design

(ASDD), Chennai, January 911, 2001. p. 21927.

[26] Dutta SC. Hysteresis models for nonlinear behaviour of

RC members. Asian J Struct Eng 1995;1(1):3770.

[27] Housner GW. The dynamic behaviour of water tanks. Bull

Seismol Soc Am 1963;53(2):3817.

[28] Newmark NM, Rosenblueth E. Fundamentals of earth-

quake engineering. New Jersy, USA: Prentice-Hall; 1971.

[29] Dutta SC. Seismic torsional behaviour of elevated tanks

for improved code provisions: elastic behaviour. J Inst

Engrs (India) 2000;80:16981.

[30] Winkler E. Die Lehre von der Elasticitaet und Festigkeit

1867; Prag, Dominicus.

[31] Hetenyi M. Beams on elastic foundations. University of

Michigan Press; 1946.


[32] Popov EP. Successive approximation for beams on elastic

foundations. Trans ASCE 1951;116:1083108.

[33] Baker AL. Raft foundations. third ed. London: Concrete

Publications Ltd; 1957.

[34] Vesic AB. Bending on beams resting on isotropic elastic

solid. J Eng Mech Div ASCE 1961;2(87):3553.

[35] Kramrisch F, Rogers P. Simplied design of combined

footings. J Soil Mech Div ASCE 1961;87(5):1944.

[36] Brown CB, Laurent JM, Tilton JR. Beam-plate system on

winkler foundation. J Eng Mech Div ASCE 1977;103(4):

589600.

[37] Terzaghi KV. Evaluation of coecients of subgrade

reaction. Geotechnique 1955;5(4):297326.

[38] Kurian NP. Modern foundations: introductions to ad-

vanced techniques. Tata McGraw-Hill Publishing Com-

pany Limited; 1986.

[39] Bowles JE. Foundation analysis and design. McGraw-Hill

International Editions, 5th ed. Civil Engineering Series;

1996.

[40] Yong RNY. A study of settlement characteristics of model

footings on silt. In: Proceedings of the 1st Pan-American

conference on soil mechanics and foundation engineering

Mexico DF. 1960;3:492513.

[41] Recordon E. Determination of soil characteristics neces-

sary for foundation calculations on elastic soils. Proc 4th

Int Conf Soil Mech Found Eng London 1957;1:4148 [in

French].

[42] Vesic AB. Beams on elastic subgrade and Winklers

hypothesis. In: Proceedings of 5th International Confer-

ence on Soil Mechanics and Foundation Engineering, Paris

1961. p. 84550.

[43] Nascimento V, Simoes A. Relation between CBR and

modulus of strength. Proc 4th Int Conf Soil Mech Found

Eng London 1957;2:1668.

[44] Black WPM. The calculation of laboratory and in-situ

values of california bearing ratio from bearing capacity

data. Geotechnique 1961;11(1):1421.

[45] Barata FE. Contribution to a better application and more

correct analysis of bearing plate tests. In: 3rd Pan-

American Conference on Soil Mechanics and Foundation

Engineering, Carcas, Venezuela, Vol. 1. 1967. p. 591612.

[46] Harr ME. Foundation of theoretical soil mechanics. New

York: McGraw-Hill Book Company; 1966 [International

Student Edition].

[47] Boussinesq H. Applications des Potentiels alEtude de

lEquilibre et du Mouvement des Soilds Elastique. Paris:

Gauthier-Villars; 1885.

[48] Fletcher DQ, Hermann LR. Elastic foundation represen-

tation of continuum. J Eng Mech Div ASCE 1971;97(1):

73147.

[49] Biot MA. Bending of innite beams on an elastic foun-

dation. J Appl Mech Trans Am Soc Mech Eng 1937;

59:A17.

[50] Gorbunov-Posadov GI. Beams and plates on elastic base

[in Russian] Moscow, USSR: Stroizdat; 1949.

[51] Kerr AD. Elastic and viscoelastic foundation models.

J Appl Mech Trans ASME 1964;31(4):4918.

[52] Kerr AD. A study of a new foundation model. Acta Mech.

1965;I/2:13547.

[53] Foppl A. Vorlesungen uber Technische Mechanik, fourth

ed. vol. 3. B.B. Teubner, Leipzig, Germany 1909. p. 228.

[54] Filonenko-Borodich MM. Some approximate theories of

elastic foundation Uchenyie Zapiski Moskovskogo Gosu-

darstvennogo Universiteta [in Russian] Mekhanica 1940;

46:318.

[55] Schiel F. Der Schwimmende Balken. Z wandte Math Mech

1942;22:25562.

[56] Hetenyi M. A general solution for the bending of beams on

an elastic foundations of arbitrary continuity. J Appl Phys

1950;21:558.

[57] Pasternak PL. On a new method of analysis of an elastic

foundation by means of two foundation constants [in

Russian]. Gosudarstvennoe Izdatelstvo Literaturi po Stro-

itelstvu i Arkhitekture, Moscow, USSR. 1954.

[58] Wang CM, Iang YX, Wang Q. Axisymmetric buckling of

reddy circular plates on Pasternak foundation. J Eng Mech

ASCE 2001;127(3):2549.

[59] Pasternak PL. Theory of beams on a continuous elastic

rotating and elastically settling foundation [in Russian]

Nauch Isledovatelskaya Konferencia MISI 1937.

[60] Kovalskii BS. Stress analysis of heat exchange apparatus

[in Russian]. Inzhinierny Sbornik, Academy of Science

USSR 1950. p. 6.

[61] Sokolov SN. Circular plate on a generalized foundation

[in Russian]. Inzhinierny Sbornik, Academy of Sciences

USSR. 1952.

[62] Jakovlev JuV. On the analysis of heat exchange apparatus

[in Russian]. Trudi Kharkovskogo Aviatsionnogo Institute

1954;(15).

[63] Horvath JS. Beam-column-analogy model for soilstruc-

ture interaction analysis. J Geotech Eng 1993;119(2):358

64.

[64] Kurian NP, Manojkumar NG. A new continuous model

for soilstructure interaction. J Struct Eng 2001;27(4):269

76.

[65] Vlasov VZ. Structural mechanics of thin walled three

dimensional systems [in Russian] Moscow: Stroizdat; 1949.

[66] Vlasov VZ, Leontiev NN. Beams, plates and shells on an

elastic foundation [in Russian] Moscow, USSR: Fizmatgiz;

1960.

[67] Lecture Notes on Soilstructure-interaction by Sarvesh

Chandra, Department of Civil Engineering, Indian Insti-

tute of Technology, Kanpur, UP.

[68] Ressiner M. A note on deections on plate on a visco-

elastic foundation. J Appl Mech Trans ASME 1958;25(1):

1445.

[69] Rhines WJ. A study of the Ressiner Foundation model. In:

Proceedings of 5th US National Conference of Applied

Mechanics. 1966. p. 319.

[70] Harr ME, Davidson JL, Da-Min HO, Pombo LE,

Ramaswamy SV, Rosner JC. Euler beams on a two-

parameter elastic foundation model. J Geotech Eng Div

ASCE 1969;95(4):93348.

[71] Rao NSKV, Das YC, Ananthakrishnan M. Variational

approach to beams on elastic foundations. J Eng Mech Div

ASCE 1971;97(2):27194.

[72] Nogami T, Lain YC. Two parameter layer model for

analysis of slab on elastic foundation. J Eng Mech Div

ASCE 1987;113(9):127991.

[73] Horath JS. Subgrade models for soilstructure interaction

analysis. Found Engr Curr Principles Pract Proc ASCE

1989;20(2):599612.


[74] Girija Vallabhan CV, Das YC. Modied Vlasov Model for

beams on elastic founadtions. J Geotech Eng 1991;117(6):

95666.

[75] Wieghardt K. Uber den blaken auf nachgiebiger Unter-

large. Z Angew Math Mech Berlin Germany 1922;2:16584.

[76] Teng WC. Foundation and design. New Delhi: Prentice

Hall of India Private Ltd; 1987.

[77] Siva Reddy A, Pranesh MR. Warping stresses in concrete

strip slabs on Hetenyi Foundation. Indian Concr J 1969;

54(5).

[78] Pranesh MR. Warping stresses in concrete slabs on

ground. Ph.D. Thesis submitted to Indian Institute of

Science, Bangalore, October 1970.

[79] Siva Reddy A, Pranesh MR. Warping stresses in slabs on

Ressiner Foundation. J Transp Eng ASCE 1971;97(3):511

26.

[80] Siva Reddy A, Pranesh MR. Warping stresses in concrete

slabs on FilonenkoBorodich model. Indian Concr J 1973;

47(7).

[81] Bowles JE. Analytical and computer methods in geotech-

nical engineering. New York: Tata McGraw-Hill Publish-

ing Company Ltd; 1974.

[82] Vesic AS, Johnson WH. Model studies of beams resting on

a silt subgrade. J Soil Mech Found Div ASCE 1963;89(1):

131.

[83] Gibson RE. Some results concerning displacements and

stresses in a non-homogeneous elastic half-space. Geotech-

nique 1967;17:5867.

[84] Smolira M. Analysis of tall buildings by the force-

displacement method. UK: McGraw-Hill Book Company

Limited; 1975.

[85] Zeevart L. Foundation engineering for dicult sub-soil con-

ditions. New York: Van Nostrand Reinhold Company; 1972.

[86] Noorzaei J. Non-Linear soilstructure interaction in

framed structures. Ph.D. Thesis, Civil Engineering De-

partment, University of Roorkee, Roorkee, India. 1991.

[87] Das BM. Advanced soil mechanics. Civil Engineering

Series. McGraw-Hill International Editions. 1983.

[88] Desai CS, Abel JF. Introduction to the nite element

method: a numerical method for engineering analysis. CBS

Publishers and Distributors; 1987.

[89] Zienkiewicz OC, Taylor RL. The nite element method. 2.

Physics Series. McGraw-Hill International Editions.

[90] Sun DN, Matsuoka H, Yao YP, Ichihara W. An elasto-

plastic model for unsaturated soil in three-dimensional

stresses. Soils Found JGS 2000;40(3):1728.

[91] Viladkar MN, Godbole PN, Noorzaei J. Soilstructure

interaction in plane frames using coupled nite innite

elements. Comput Struct 1991;39(5):53546.

[92] Noorzaei J, Viladkar MN, Godbole PN. Elasto-plastic

analysis for soilstructure interaction in framed structures.

Comput Struct 1995;55(5):797807.

[93] Noorzaei J, Viladkar MN, Godbole PN. Inuence of

Strain hardening on soil-p-structure interaction of framed

structures. Comput Struct 1995;55(5):78995.

[94] Kondner RB. Hyperbolic stressstrain response: cohesive

soil. J Soil Mech Found Eng Div Proc ASCE 1963;89(1):

11543.

[95] Duncan JM, Chang CY. Nonlinear analysis of stress and

strain in soils. J Soil Mech Found Eng Div Proc ASCE

1970;96(5):162953.

[96] Tatsuoka F, Shibuya S. Deformation characteristics of

soils and rocks from elds and laboratory tests. Keynote

Lecture for Session 1. In: Proceedings of 9th Asian regional

conference on SMFE, Bangkok. 1991. p. 2.

[97] Jardine RJ, Potts DM, Fourie AB, Burland JB. Studies of

the inuence of non-linear stressstrain characteristics in

soilstructure interaction. Geotechnique 1986;36(3):377

96.

[98] Goto S, Burland JB, Tatsuoka F. Non-linear soil model

with various strain levels and its application to axisym-

metric excavation problem. Soils Found JGS 1999;39(4):

1119.

[99] Varadrajan A, Ray S. Non-Linear soilstructure interac-

tion analysis of a raft foundation. Indian Geotech Conf

1999:714.

[100] Mandal A, Moitra D, Dutta SC. Soilstructure interaction

on building frame: a small scale model study. Int J Struct

1998;18(2):93109.

[101] Dasgupta S, Dutta SC, Bhattacharaya G. Eect of soil

structure interaction on building frames on isolated foot-

ings. J Struct Eng 1999;26(2):12934.

[102] Roy R, Dutta SC, Moitra D. Behaviour of Grid Founda-

tion Supporting Multistorey Frames: A Study through

Idealized Model. Proceedings of the International Confer-

ence on Mathematical Modelling, University of Roorkee,

January 2931, 2001. p. 4527.

[103] Ter-Martirosyan ZG. Rheological parameters of soils and

design of foundations. Oxford and IBH Pbulishing Com-

pany Limited; 1992.

[104] Noda T, Fernado GSK, Asaoka A. Delayed failure in soft

clay foundations. Soils Found JGS 2000;40(1):8597.

[105] Vyalov SS. Reologicheskie osnovy mekhvnikigruntov

(Rheological Principles of Soil Mechanics). Moskow:

Vysshaya Shkola; 1978.

[106] Caddell RM. Deformation and fracture of solids. Engle-

wood Clis, NJ: Prentice-Hall Inc; 1980.

[107] Taylor DW, Merchant W. A theory of clay consolidation

accounting for secondary compression. J Math Phys

1940;19:167.

[108] Taylor DW. Research on consolidation of clays. Mas-

sachusetts Institute of Technology, Publication no 82,

1942.

[109] Tan TK. Discussion. Proc 4th Int Conf Soil Mech Found

Eng 1957;3:278.

[110] Gibson RE, Lo KY. A theory of consolidation for soils

exhibiting secondary compression. Norwegian Geotechni-

cal Institute, Publication No. 41, 1961.

[111] Schiman RL, Ladd CC, Chan AT. The secondary

consolidation of clay. In: I.U.T.A.M., Symposyum of

Rheological Soil Mechanics, Grenoble 1964. p. 273.

[112] Barden L. Consolidation of clay with non-linear viscosity.

Geotechnique 1965;15(4):34562.

[113] Barden L. Primary and secondary consolidation of clay

and peat. Geotechnique 1968;18:114.

[114] Hashiguchi K, Okayasu T. Time-dependent elasto-plastic

constitutive equation based on the subloading surface model

and its applications to soils. Soils Found JGS 2000;40(4):

1936.

[115] Viladkar MN, Ranjan G, Sharma RP. Soilstructure

interaction in time domain. Comput Struct 1993;46(3):

42942.


[116] Nasri V, Magnan JP. Eect of soil consolidation on space

frame-raft-soil interaction. J Struct Eng 1977;123(11):

152834.

[117] Desai CS, Phan HV, Perumpral JV. Mechanics of three-

dimensional soilstructure interaction. J Eng Mech Div

ASCE 1982;108(5):73147.

[118] Thomas HR, Bendani K. Primary/secondary compression

solution algorithm. J Comput Civ Eng 1988;2(4):38097.

[119] Goodman RE, Taylor RL, Brekkee TL. A model for the

mechanics of jointed rock. J Soil Mech Found Eng ASCE

1968;94(3):63759.

[120] Zienkiewicz OC, Best B, Dullage C, Stagg KG. Analysis of

non-linear problem in rock mechanics with particular

reference to jointed rock systems. In: Proceedings of 2nd

International Congress of the International Society for

Rock Mechanics, 1970. p. 5019.

[121] Ghaboussi J, Wilson E, Isenberg J. Finite element analysis

for rock joints and interfaces. J Soil Mech Found Eng

ASCE 1973;99(10):83348.

[122] Pande GN, Sharma KG. On joint/interface element and

associated problems of numerical ill conditioning. Int

J Anal Meth Geomech 1979;2:293300.

[123] Beer G. An isoparametric joint/interface element for

nite element analysis. Int J Numer Meth Eng 1985;21:

585600.

[124] Viladkar MN, Godbole PN, Noorzaei J. Modelling of

interface for soilstructure interaction studies. Comput

Struct 1994;52(4):76579.

[125] Roy R, Dutta SC, Moitra D. Eect of soilstructure

interaction on dynamic behaviour of building frames on

isolated footings. In: Proceedings of the national sympo-

sium on advances in structural dynamics and design

(ASDD), January 911, 2001, Chennai. p. 57986.

[126] Sung TY. Vibrations in semi-innite solids due to periodic

surface loading. ASTM STP no. 156. 1953. p. 3568.

[127] Seed HB, Lysmer J. Soilstructure interaction analysis for

seismic response. J Geotech Eng Div ASCE 1975;101(5).

[128] Hall JR, Kissenpfennig JF, Rizzo PC. Discussion on the

paper of soilstructure interaction analysis for seismic

response. J Geotech Eng Div ASCE 1976;102(6).

[129] Gazetas G. Formulas and charts for impedances of surface

and embedded foundations. J Geotech Eng Div ASCE

1991;117(9):136381.

[130] Veletsos AS, Vebric B. Elastic response functions for elastic

foundations. J Eng Mech Div ASCE 1974;100(EM2).

[131] Sridharan A, Gandhi NSVVSJ, Suresh S. Stiness coe-

cients of layered soil systems. J Geotech Eng Div ASCE

1990;116(4):60423.

[132] Baidya DK, Sridharan A. Stinesses of the foundations

embedded into elastic stratum. Indian Geotech J 1994;

24(4):35367.

[133] El-Shafee OM, Gould PL. Dynamic axisymmetric soil

model for a exible ring footing. Earthquake Eng Soil

Dynam 1980;8(5):47998.

[134] Tassoulas JL, Kausel E. On the dynamic stiness of

circular ring footings on an elastic stratum. Int J Numer

Anal Meth Geomech 1984;8:41126.

[135] Veletsos AS, Tang Y. Vertical vibration of ring founda-

tions. Earthquake Eng Struct Dynam 1987;15(1):121.

[136] Dobry R, Gazetas G. Dynamic response of arbitrarily

shaped foundations. J Geotech Eng Div ASCE 1986;

112(2):10935.

[137] Dobry R, Gazetas G, Stokoe KH. Dynamic response of

arbitrarily shaped foundations: experimental verications.

J Geotech Eng Div ASCE 1986;112(2):13654.

[138] Parmalee RA, Perelman DS, Lee SL. Seismic response of

multistorey structures on exible foundations. Bull Seismol

Soc Am 1969;3:106170.

[139] Prakash S, Puri VK. Foundation for machines: analysis

and design. New York: John Wiley and Sons; 1988.

[140] Roy R, Dutta SC. Eect of soilstructure interaction on

dynamic behaviour of building frames with grid founda-

tions. In: Proceedings of Struct Eng Conven (Sec 2001), An

International Meet, Roorkee, India, October 2931, 2001.

p. 694703.

[141] Kameswarao NSV. Vibration analysis and foundation

dynamics. Wheeler Publishing; 1998.

[142] Kokkinos FT, Spyrakos CC. Dynamic analysis of exible

strip-foundations in the frequency domain. Comput Struct

1991;39(5):47382.

[143] Wolf JP. Dynamic soilstructure interaction. Prentice-

Hall: Englewood Clis, NJ; 1985.

[144] Spyrakos CC, Patel PN, Beskos DE. Dynamic analysis of

exible embedded foundations: plane strain case. In:

Proceedings of the 3rd international conference on com-

puting methods and experimental measures, Port Carcas,

Greece, 1986.

[145] Zhang C, Xinfeng C, Guanglun WA. coupling model of

FE-BE-IE-IBE for non-linear layered soilstructure inter-

actions. Earthquake Eng Struct Dynam 1999;28:42141.

[146] West RP, Pavlovic MN. Finite-element model sensitivity in

the vibration of partially embedded beams. Int J Numer

Meth Eng 1999;44(4):51733.

[147] West RP, Pavlovic MN. A parametric study of the

clustering of modes in the vibration of partially embedded

beams. Int J Mech Sci 1999;41(5):54759.


A critical review on idealization and modeling for interaction among soil-foundation-structure systemIntroductionSoil-structure interaction under static loadingSoil-structure interaction under dynamic loadingModel of structure-foundation-soil interacting systemIdealization of structureBuildingsWater tank

Modeling of soil mediaWinkler modelElastic continuum modelImproved foundation modelsImproved versions of winkler modelFilonenko-borodich foundationReissner foundationImproved versions of continuum modelVlasov foundationApplicability of the models

Advanced modelingBehaviour of soil mediaElasto-plastic idealizationNonlinear idealizationViscoelastic idealizationFinite element modeling

Dynamic soil-structure interactionConclusionsAcknowledgements

Introduction to references

References

dutta roy 2002 cas

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