Critical Review of Soil-Structure Interaction Modeling

Review Article
A critical review on idealization and modeling
for interaction among soil–foundation–structure 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 soil–foundation–structure interaction system with computational validity, efficiency
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.
Ó 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Soil–structure 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 finally adjusted condition can only be
obtained through interactive analysis of the soil–struc-
ture–foundation system. This explains the importance of
considering soil–structure interaction.
The three-dimensional frame in superstructure, its
foundation and the soil, on which it rests, together con-
stitute a complete system. With the differential 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 influence 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 effect
of this factor. A critical scrutiny of such studies has been
presented in the literature [2] modeling the soil–founda-
tion–structure 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 soil–structure–foundation
Computers and Structures 80 (2002) 1579–1594
www.elsevier.com/locate/compstruc
*
Corresponding author. Fax: +91-33-668-2916.
E-mail address: scdind@netscape.net (S.C. Dutta).
0045-7949/02/$ - see front matter Ó 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. Soil–structure interaction under static loading
Numerous studies [3–10] have been made on the ef-
fect of soil–structure interaction under static loading.
These studies have considered the effect in a very sim-
plified manner and demonstrated that the force quanti-
ties are revised due to such interaction. A limited
number of studies [6,9,11–14] have been conducted
on soil–structure interaction effect considering three-
dimensional space frames. The studies clearly indicated
that a two-dimensional plane frame analysis might
substantially overestimate or underestimate the actual
interaction effect in a space frame. From these studies, it
becomes obvious that the consideration of the interac-
tion effect significantly alters the design force quantities.
These studies, may be quantitatively approximate, but
clearly emphasize the need for studying the soil–struc-
ture interaction to estimate the realistic force quantities
in the structural members, accounting for their three-
dimensional behaviour.
3. Soil–structure interaction under dynamic loading
Structures are generally assumed to be fixed at their
bases in the process of analysis and design under dy-
namic loading. But the consideration of actual support
flexibility [15,16] reduces the overall stiffness 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
soil–structure 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 soil–structure interaction
should be accounted for in the analysis of dynamic be-
haviour of structures, in practice. Hence, soil–structure
interaction under dynamic loads is an important aspect
to predict the overall structural response.
4. Model of structure–foundation–soil interacting system
It appears from the foregoing discussion that a com-
pletely misleading behaviour may be obtained unless
the interactive study of the soil–structure–foundation 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
force–displacement characteristics. The presence of fluc-
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 fulfilled 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 soil–structure
interaction.
1580 S.C. Dutta, R. Roy / Computers and Structures 80 (2002) 1579–1594

space frame using two noded beam elements. The effect
of infill 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, infill
wall imparts considerable lateral stiffness to the struc-
ture, since, then it behaves like a compressive strut.
Hence, under lateral loading the effect of the same must
be incorporated as specified in the literature [20–23]. 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 stiffness 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 effect in stiffness 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 finite
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 flexible: 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-
fications 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
Winkler’s 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 confined to loaded regions only. Fig. 2
shows the physical representation of the Winkler foun-
dation. The pressure–deflection relation at any point is
given by
p ¼ kw ð1Þ
where p is the pressure, k is the coefficient of subgrade
reaction or subgrade modulus, and w is the deflection.
A number of studies [31–36] (only a few among many
others) in the area of soil–structure 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 stiffness 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) 1579–1594 1581

becomes two-fold since the numerical value of the co-
efficient 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 stiffness 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 coefficient of
subgrade reaction k is the ratio between the pressure p at
any given point of the surface of contact and the set-
tlement y produced by the load at that point:
k ¼ p=y ð2Þ
The value of subgrade modulus may be obtained in the
following alternative approaches:
(a) Plate load test [37–39],
(b) Consolidation test [40,41],
(c) Triaxial test [34,42] and
(d) CBR test [41,43–45].
Following some suitable method mentioned to esti-
mate k, a reasonable value of subgrade modulus, the
only parameter to idalize soil stiffness, 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 influence area with increase in
depth. Moreover, it considers linear stress–strain be-
haviour of soil. The most serious demerit of Winkler
model is the one pertaining to the independence of the
springs. So the effect 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 modified 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 infinite 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 effect, 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-infinite, 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 deflection line of its surface
under a unit concentrated load [48]. In the continuum
idealization, generally soil is assumed to be semi-infinite
and isotropic for the sake of simplicity. However, the
effect 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 Poisson’s 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-infinite 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 difficult 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., Winkler’s model and Continuum
model, some modified foundation models have been
proposed in the literature. These modifications have
generally been suggested following two alternate ap-
proaches. In the first approach, the Winkler foundation
is modified to introduce continuity through interaction
amongst the spring elements by some structural ele-
ments. In the second approach, continuum model is
simplified to obtain a more realistic picture in terms of
expected displacement and/or stresses. These improved
foundation models are briefly described below.
6.3.1. Improved versions of winkler model
6.3.1.1. Filonenko-borodich foundation. Fig. 3 shows the
physical representation of Filonenko–Borodich 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 ¼ kw Tr2
w; for rectangular or circular foundation
¼ kw T
d2
w
dx2
; for strip foundation
ð3Þ
where, r2
Laplace operator o2
ox2 þ o2
oy2; T ¼ tensile
force.
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. Hetenyi’s 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 flexural deformation only, as shown in Fig. 4.
Thus the pressure–deflection relationship becomes
p ¼ kw þ Dr4
w ð4Þ
where,
D ¼ flexural rigidity of the elastic plate
¼ ðEph3
pÞ=ð12ð1 lpÞ2
Þ;
p is the pressure at the interface of the plate and the
springs; Ep and lp are Young’s modulus and Poisson’s
ratio of plate material; hp is the thickness of the plate
and
r4

o4
ox4
þ
o4
oy4
þ 2
o4
ox2oy2
Thus, it is seen that the flexural 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 load–deflection
relationship is obtained by considering the vertical equi-
librium of a shear layer. The pressure–deflection rela-
tionship is given by
p ¼ kw Gr2
w ð5Þ
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 Filonenko–Borodich 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 Winkler’s hypothesis [59–62]. This can be ana-
lytically described as follows.
p ¼ kw
and
mn ¼ k1
dw
dn
ð6Þ
where, mn is the moment in direction, n; n is the direction
at any point in the plane of the foundation; and k, k1 are
proportionality factors.
The assumption made on the proportionality in this
model is relatively arbitrary [51]. However, a physical
significance, 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 different as per this
formulation [52]. Fig. 6 shows the physical representa-
tion of this mechanical model. The governing differential
equation for this model may be expressed as follows.
Fig. 4. Hetenyi foundation [30].
Fig. 3. Filonenko–Borodich foundation [52].

1

þ
k2
k1

p ¼
G
k1
r2
p þ k2w Gr2
w ð7Þ
where, k1 is the spring constant of the first layer; k2 is the
spring constant of the second layer; w is the deflection of
the first 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 final form of the governing dif-
ferential equation for combined beam-subgrade behav-
iour is obtained as follows:
EbIb
d4
wðxÞ
dx4
Cp2
d2
wðxÞ
dx2
þ Cp1wðxÞ ¼ qðxÞ ð8Þ
where, EbIb is the flexural stiffness of the beam (assumed
constant); wðxÞ is the beam settlement qðxÞ is the applied
load; 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 ¼ E=H and Cp2 ¼ GH=2 where E is the
Young’s 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 stiff-
nesses Cp1. Thus, it appears that the continuity among
the individual Winkler springs is achieved by the pa-
rameter Cp2. However, the modeling of foundation be-
comes incorrect due to introduction of a fictitious shear
force [63] while the modeling, as a whole is a significant
improvement over Winkler’s 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 stiffness, 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 effect 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 wðx; zÞ ¼ wðxÞ. hðzÞ, such
that hð0Þ ¼ 1 and hðHÞ ¼ 0. This function hðzÞ de-
scribes the variation of displacement in vertical di-
rection.
(ii) The horizontal displacement uðx; zÞ is assumed to be
zero everywhere in the soil.
The function hðzÞ may be assumed to be linearly de-
creasing with depth for a classical foundation of finite
thickness H. Hence, in this case, hðzÞ ¼ 1 ðz=HÞ. For the
Fig. 7. Beam–column 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 infinite thickness)
elastic layer, the choice may be hðzÞ ¼ ðsinh½cðH zÞ =
sinh½cH Þ, where c is a coefficient depending on the elastic
properties of the foundation defining 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 ¼ kw 2t
d2
w
dx2
ð9Þ
where
k ¼
E0
ð1 m0Þ2
Z H
0
dh
dz
2
dz; t ¼
E0
4ð1 m0Þ
Z H
0
h2
dz;
E0 ¼
E
ð1 mÞ2
and m0 ¼ mð1 mÞ; E and m are soil constants.
6.3.2.2. Reissner foundation. As per this model, pressure–
deflection 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 pressure–deflection relationship is given by
C1w C2r2
w ¼ p
C2
4C1
r2
p ð10Þ
where w is the displacement of the foundation surface, p
is a distributed lateral load acting on the foundation
surface; C1 ¼ E=H; C2 ¼ HG=3; E, G are the elastic
constants of foundation material and H is the thickness
of the foundation layer.
The term H2
G=E in Eq. (10), known as differential
shear stiffness, offers the possibility of obtaining closer
agreement with actual behaviour [68]. This model also
retains the mathematical simplicity of Winkler models.
The classical problem of infinitely 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 [70–75].
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 soil–structure
interaction analysis. Moreover, it is also difficult to de-
cide the physical quantity, precision of which may in-
dicate the accuracy of the whole computational process.
The different 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 modified for the actual footing.
The other parameters may be obtained from rigid stump
test [74,77–80].
Studies have been reported in the area of soil–struc-
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 modified 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 sufficiently 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 finite-difference solution [81]. The
most noteworthy series of tests on continuous beams
reported in the literature [82], also corroborate the
findings obtained through Winkler idealization [81].
Since it is very difficult to arrive at an accurate value for
Young’s Modulus of soil, which is an essential param-
eter in elastic continuum idealization; the approach of
using subgrade modulus finds more appreciation [39].
Further, the validity of Winkler’s 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
influences the response of the superstructure quite in-
significantly. The present practice in design offices gen-
erally adopts a fixed base consideration for structural

analysis and design. In this context, the Winkler model,
though oversimplified, 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 refined 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 stress–strain–time property, influences
the soil–structure 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 configuration 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 fine-
grained soil) due to viscous intergranular behaviour that
implies strain with passage of time. This deformation
occurs by the expulsion of the pore fluid 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 sufficient 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 first be
shared by the pore fluid 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 offer 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 soil–structure 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.
Venant’s 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.
Venant’s 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 ¼ Mer þ Mp log
ru
ru r
ð11Þ
where Me is the elastic strain modulus of soil; Mp is the
plastic strain modulus of soil and ru is the ultimate load
that 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, different convenient forms of
the various yield criteria of soils such as Tresca yield
criterion, Von Mises yield criterion, Mohr–Coulomb
yield criterion, Drucker–Prager yield criterion, etc.
[46,87] may be suitably chosen in the modeling. A flow
rule to describe the post-yielding behaviour may be
adopted following deformation theory or, the incre-
mental or flow theory [88]. In the deformation theory,
the plastic strains are uniquely defined 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 elastic–plastic 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 flow rule in the general
form and necessitates only the specification 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 elastic–perfectly plastic behaviour of
subsoil for a plane frame-combined footing-soil system
[91,92]. The effect for the influence of strain-hardening
characteristics of soil in the elasto-plastic soil–structure
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 stress–strain 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 stress–strain 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 stress–strain curves and of
the tangent moduli computed as the first 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
stress–strain 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
simplified 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 differential 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 fine-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 first 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 effective 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 flow and molecular theory of
flow [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. Venant’s 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 efforts [107–113] 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 finite element for-
mulation for studying the interactive behaviour of space
frame considering the stress–strain 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 finite 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 stress–strain 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 soil–structure interaction problem.
The method is a special extended form of matrix
analysis based on variational approach, where the whole
continual is discretized into a finite number of elements
connected at different nodal points. Displacements func-
tions, i.e., the displacement within the element is
not known and hence to be judiciously assumed. Thus
knowing the stiffness matrix for each element, overall
stiffness 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 finite element
procedure for the general problem of three-dimensional
soil–structure 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 [119–123] have been
made to develop interface elements, use of which is
proved to be useful to take care of this discontinuity.
The stiffness 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 soil–structure interaction behaviour at
least for important structure, if possible.
9. Dynamic soil–structure interaction
The consideration of soil-flexibility increases the pe-
riod of vibration. Hence, a considerably different re-
sponse from that of reality may be obtained if this effect
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 effect of soil–structure 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 stiffnesses 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 stiffness
functions has been presented in the literature [130]. The
expressions for these spring stiffnesses, 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 verification [129].
Dynamic stiffness for machine foundations resting on
layered soil systems has been discussed elsewhere [131].
An analytical method to estimate the stiffness of the
foundations embedded into the stratum over rigid rock
corresponding to different 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 stiffness of the underlying soil media. The
stiffness of annular footings has been derived in some
other literature [133–135]. It has been observed that the
stiffnesses 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 stiffness in the equivalent
sense. Thus the dependence of the stiffness of equivalent
springs representing the deformable behaviour of soil is
due to the incorporation of the influence that frequency
exerts on inertia, though purely stiffness properties are
frequency independent. This frequency dependence is
suggested to be incorporated by multiplying the equiv-
alent spring stiffnesses by a frequency dependent factor.
This factor is plotted as a function of a non-dimensional
parameter a0 where a0 ¼ xB=Vs [129]. Here, x is the
frequency of the forcing function, B is the half of the
lateral dimension of the footing and Vs is the shear wave
velocity 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 difficult to
consider any frequency dependent multiplier to compute
dynamic stiffness and damping coefficient 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 effect of

frequency dependent soil-flexibility 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 effect
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, finite 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 finite element method is not ca-
pable of idealizing the infinite soil medium properly.
Hence it is suggested in the literature [136–144] to model
the infinite soil media using boundary element method
and the finite structure with finite element method.
These two different means of idelizations may be suit-
ably matched at the interface through equilibrium and
compatibility conditions. An extremely efficient scheme
for the analysis of soil–structure interaction system
using coupling model of finite elements, boundary ele-
ments, infinite elements and infinite 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 effect of soil–structure 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 finite element
models for the same provided in such literature [146] can
be of help in finite element modeling for pile vibration
problem with varying degrees of refinement 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 soil–structure inter-
action analysis leads to the following broad conclusions.
(1) To accurately estimate the design force quantities,
the effect of soil–structure interaction is needed to
be considered under the influence of both static
and dynamic loading. To obtain the same, realistic
yet simplified modeling of the soil–structure–foun-
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 fixed base idealization of structures.
(3) The consolidation phenomenon of clayey soil
follows a nonlinear stress–settlement relationship.
Hence, to achieve a more realistic analysis of the
soil–structure 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 soil–structure 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 finite element method has proved to
be a very useful method, which should be employed
for studying the effect of soil–structure interaction
with rigor. In fact, the technique becomes useful to
incorporate the effect of material nonlinearity, non-
homogeneity and anisotropy of the supporting soil-
medium if needed to be accounted due to the case
specific nature of any particular problem.
(6) The effect of soil–structure interaction on dynamic
behaviour of structure may conveniently be ana-
lyzed using Lumped parameter approach. However,
resort to the finite 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 influence of structural rigidity
apart from soil flexibility on the amount of load distri-
butions due to soil–structure interaction. A suitable
iterative method for estimation of the effect of soil–
structure interaction is outlined in Ref. [3]. Ref. [14]
provides an idea about the effect of differential settle-
ment on design force quantities of various building

frames with isolated footings. Remedial measure to re-
duce this effect is also suggested in this literature. Refs.
[20,21] provide the approach for accounting the contri-
bution of the brick walls to the lateral stiffness of the
buildings. The detailed information about various im-
portant models, namely Filonenko–Borodich Founda-
tion model, Hetenyi’s 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
load–settlement characteristics of consolidation settle-
ment in frame–soil 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 finite element discretization is explicitly
presented in [124]. Ref. [129] provides the dynamic
stiffness as well as damping characteristics of soil me-
dium supporting any arbitrary shaped foundation. In-
cluding the effect of the frequency of the forcing function
in dynamic stiffness of soil medium, it becomes a
benchmark literature in the area of dynamic soil–struc-
ture interaction. Modeling required to address the
problem of soil–structure interaction of pile foundation
in vibrating condition finds a detailed treatment in two
pioneering literatures, Refs. [146,147].
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