INTRODUCTION
At first Terzaghi (1943) studied change in the properties
of soil behavior beneath and around the foundation (Terzaghi
(1943); Meyerhof and Hanna, 1963). It is based ultimate
bearing capacity theory for strip footing. This theory is explained by failure
mechanism and width footing. Later on, Meyerhof and Hansen
(1963) expanded Terzaghi theory. They considered other parameters to complete
Terzaghi equation. More studies about bearing capacity are concentrated on the
third coefficient bearing capacity (Nγ) and failure mechanism is assumed
general (Fig. 1), which is explained by Terzaghi
(1943) and Bowles (1982).
Recently, Zhu et al. (2001) showed that bearing
capacity and Terzaghi failure wedge position are related to width footing. According
to studies, this researches for strip footing, while width increased from 0.1
to 10 m, ultimate bearing capacity calculated, increased from 620 to 12300 kPa
and for circle footing ultimate bearing capacity increased from 290 kPa to 6730
kPa and wedge angel decreases from 57.9° to 52.2°.

Fig. 1: 
Terzaghi failure mechanism 
In later study performed
by Junhwan et al. (2005), using finite elements
method (FEM) on bearing capacity it was concluded that there are many uncertainties
in the Terzaghi theory. More challenges concentrated on real development and
explanation of bearing capacity parameters.
Another important subject, not considered in footing ultimate bearing
capacity, is the interaction effects between earth and footing. In this
paper first the different interaction types are both defined and classified
and then considering the interaction, the behavior and role of earth under
the small and large foundations are examined. Position of failure mechanism
is also observed.
INTERACTION CLASSIFICATION
The interaction between the sections of structures is one of the most important
and the most complex issue in engineering which is currently of particular interest
to the researchers, designers and structural engineers; This is due to lack
of information and insufficient knowledge in this area and the fact that this
interaction has not yet been included in valid codes practice, generally do
not take account of it. But due to the existing developments and more technological
abilities as well as the owner’s desire, studying and analyzing interaction
is a must. To achieve this goal, the applicable solutions should be introduced
which are in the hand of structural and geotechnical engineers. In general,
three types (Zienkiewicz, 1991; Zienkiewicz
and Taylor, 2000) of interactions including the geotechnical materials having
high ductility, steel and concrete can be defined and studied:
Interaction in the form of compatibility in different material interface
and contact surface (Compatibility equations contact): In this mode,
the different material should be related to each other defining interface
elements in contact surfaces. This element provides the compatibility
and continuity conditions between the two continuums. This element contains
material properties of the two media and can stand the stresses up to
a certain limit, besides, stress transits between the two media continuously.
After the stress reaches its final limit in the said element, the lateral
elements are allowed to act independently causing segregation, i.e., in
this stage the compatibility equations are not met.
Geometrical interaction (Large deformations and slipping on the contact
surface): The displacement of match points of the two neighboring
media is extremely noticeable and huge; so, it is likely that part of
a medium is transferred and moved to another medium. Thus, we will have
failure in some part of a medium. The stressstrain relations are not
applicable to the boundary points of the two media and at the point of
interface the geometrical behavior of the materials should be studied
as nonlinear.
Behavioral interaction (Discontinuity of the material medium):
In this case, it is possible to say that although the materials are constant
and uniform, the behavioral properties are different at different points
of the ambient material; this kind of interaction is conceptual and is
only considered in overall modeling of the foundation system and the effective
part of the earth (domain). However, in the interaction of earth and structure,
beside structural system, the nonbearing members such as walls and fillers
and should be considered as well. This results in the complexity of the
analytical model and the modeling method.
Currently, most models are finite elements or boundary elements that
their application will be commenced or the earth is considered as a set
of springs with linear stiffness which has basic differences with the
real state. Calculating the deformations, earth behavior should be taken
in to account. In small deflections, the deflection effect on the finishing
or partitions or fillers become important, while under the major deformations,
especially during the earthquake, the location of the filler walls is
due to mass distribution and creating rigidity and interfering with the
stiffness matrix. Namely, in large deformations of the earth, the limit
and the definition of the structure is changed owing to the entrance of
nonstructural elements to the structure. Here, new elements are also
involving in which their interfering rate depends on their geometry, material
and adherence (link) to the main structure. This change in the stiffness
matrix as wall as the interference of the new elements are changed with
respect to the rate of deformation and failure of the old elements which
can be considered and estimated using failure mechanic. To compute the
stresses and deformations, the earth is usually modeled as a continuous
medium (semi infinity). In this study, efforts have been made considering
large deformations and interaction of foundation type (21) and part of
the earth beneath and around it is modeled and foundation dimension variation
effect on the earth behaviors under and around the foundations is studied
and analyzed.
ANALYSIS METHOD OF EARTH AND FOOTING INTERACTION
There are generally two methods of analysis for analyzing earth and large footings
interaction (Owen and Hinton, 1980; Zienkiewicz
and Taylor, 2000; Khazaie and Amirshahkarami, 2007):
Bearing analysis using the limiting equilibrium method (failure analysis
and determining the safety factor): In order to study and determine the
safety factor regarding the behavior of the earth beneath the large foundations,
the limiting state analysis should be conducted first. As for the limiting analysis
method, it is usually necessary to define and have the failure mechanism, i.e.,
define and illustrate surface areas on which the material (medium) slips and
fails. When loading the foundation, some segments (areas) are formed in the
earth beneath and around the foundation including the foundation itself and
these segments slide on each other. The abovementioned failure mechanism has
been defined by Terzaghi (1943) and is only true for the
small foundations (spread and strap footing. The bearing capacity relations
have been presented based on it; however, it seems that is not the case for
foundations with large dimensions (Mat foundation).
Deformation analysis: This analysis is very complicated and difficult.
Here, the quality of the footing behavior on the ground under the distributed
load and effective parameters on the layers behavior and the earth elements
are presented. What is very important is distinguishing and identifying
behavioral differences between layers and the soil beneath the foundation
depth in the earth under the foundation. Columns of soil located exactly
in the middle and under the foundation there are different behaviors of
soil columns around the footing. From analytical point of view, such behavior
is only possible when numerical methods are used; that is, the base of
analysis theory of deformations.
COMPUTATIONAL MODEL
The proposed model: In this study to predict and interpret the results
obtained from ground behavior analysis beneath large and small foundations,
softened continuous loops method of a conceptual model has been used to adjust
the concentration of deformations (localization) and to interpret more real
behavior of materials (Zienkiewicz and Taylor, 2000; Amirshahkarami,
19782007).
According to this model in the method of analysis based on deformations
due to the uniformly assumption of chains behavior as well as uniformity
of the load rate in them, the failures are similar to each other and total
deformation equals sum of the deformations in all loops. To justify the
concentration of deformations in Fig. 2 it is assumed
that from behavioral point of view, five loops are not similar to each
other and one of the loops for instance loop No. 3 is weaker than the
other loops. Before the behavior of loop No. 3 gets loops, the behavior
of the system as well as the behavior of all loops is similar and from
the moment loop No. 3 reaches loose state, the overall system acts in
the feeling state caused by loop No. 3. In this mode, loop No. 3 is under
load whereas loops No. 1, 2, 4 and 5 bear unloading mode and shrink (Fig.
3).

Fig. 2: 
Schematic diagram of softened continuous loops 

Fig. 3: 
Schematic curve of softened continuous loops behavior 
Here, total strain of the system with respect to the strain in loop No.
3 which is in the feeling mode and also deducting, the amount of shrinkage
in other loops, the following equation is obtained:
If shrinkage rate in loops No.1, 2, 4 and 5 are equal, (then) we have:
It can be observed that by applying loading, one loop starts noticeable deforming
and ultimately fails while others tend to return to their initial form, namely,
the deformations are concentrated on one loop and the rate of deformation in
other loops is decreased, this phenomenon is know as deformation concentration
(Localization).
Modeling: Modeling and finite analysis of elements have been performed
in ANSYS software version 8.1. In this study in order to geometrically idealize
according to Fig. 4, consider three types of interaction between
foundation and ground. Constant media including the footing and part of the
soil located under and around the footing which is effective in bearing and
deformation (settlement) and is large enough (about 4B at sides and beneath
the large footing) have been taken into account a twodimensional medium; Such
that the boundaries of the media under study are far away enough from the point
the footing is located and stress as well as deformations at those points are
insignificant and negligible (Katsutoshi et al., 1998,
2000).

Fig. 4: 
Geometric modeling and domain characteristics of materials 
The constant grillage is independent from the footing dimensions
in the studied media in order to avoid the effects of grillage change which
is later followed in all large and small foundation models. The concrete footing
and the soil with plane 82 elements have been modeled. This element is in the
form of an eightnode and twodimensional having three transitional degrees
of freedom in each node and is able to take elastoplastic behavior and large
deformations in the plane strain state. To define the interaction between footing
and ground, the 2node contact element (contact 12) is used. This 2node contact
element; i.e., a node to node (contact), contains three transitional degrees
of freedom in each node and has the ability of the initial distance. Pressure
resistant in the axial direction and Coulomb frictional shear in the tangential
direction are among the characteristics (features) of this element. The tangential
and axial stiffness come into action when the initial distance is closed, the
axial and tangential stiffness are defined based on stiffness of the contact
surface and tangential resistance is also considered in the Coulomb frictional
state applying the friction factor of the contact surface. It is worth mentioning
that the grillage in footing and soil beneath footing have been conducted in
a good manner so that for the contact elements, the corresponding nodes lie
adjacent to each other.
Table 1: 
Mechanical properties and Characteristics of materials
of the model 

^{1}: Young’s modulus soil, ^{2}:
Poisson’s ratio soil, ^{3}: Internal frictional angel
soil, ^{4}: Cohesion soil, ^{5}: Unit weight soil,
^{6}: Sticking stiffness contact element, ^{7}: Normal
stiffness contact element and ^{8}: Coefficient of friction
contact element 
In this study, the concrete behavioral model has been considered elastically
and for soil, the DruckerPrager behavioral model has been used which takes
into account the material behavior elastoplastically (Khazaie
and Amirshahkarami, 2007). The mechanical properties and characteristics
of the model materials have been shown in Table 1.
Boundary conditions and loading: With respect to the fact that
the geometrical model has defined twodimensional and is in the form of
plane strain, the vertical boundaries have been limited such that the
displacement is only possible in the vertical direction without friction
and there is no horizontal transition. The horizontal boundary (bed) has
been bounded such that it can only move horizontally. Loading on the large
and small foundations have been carried out with uniform stress and small
steps. The models then have been analyzed and compared with each other.
In order to analyze non linearly, the full NewtonRaphson step by step
method applied and when the solution is converged in each substep, the
outputs are stored in the nodes.
Analysis of the results with a view to the proposed model: As
shown in Fig. 5a and b under the
small footings, the ground beneath and around the footing both bear large
deformation (failure) together. This is fully noted when studying the
differential stresses and strains state in the ground under and around
the small foundation which is clear from Fig. 5c and
d, that complies with the Terzaghi theory. In large
foundations, which can be observed from Fig. 6a and
b, the earth deformations are mainly bound to the ground
beneath the footing and to some extent to the punching or plastic deformation
concentration mode takes place in the boundary ground between beneath
and around the footing, Fig. 6d. This kind of failure
mechanism is different from the failure mechanism proposed by Terzaghi
on small foundations (strip footing) and the existing bearing capacity
relations which have been obtained regarding the limit analysis (failure
mechanism), are not valid for large footings.





Fig. 5: 
Small footing (B = 2 m) under uniformly distributed
load (a) Deformed Mesh, (b) Vertical Deformation cantor, (c) Von Misses
equivalent plastic stress (deviatory), (d) Von Misses equivalent plastic
strain (deviatory), (e) Principle stress vectors and (f): Principle
plastic strain vectors 





Fig. 6: 
Large footing (B = 20 m) under uniformly distributed
load (a) Deformed Mesh, (b) Vertical Deformation cantor, (c) Von Misses
equivalent plastic stress (deviatory), (d) Von Misses equivalent plastic
strain (deviatory), (e) Principle stress vectors and (f) Principle
plastic strain vectors 
The failure mechanism in the soil beneath large footings is observed
internally (a column of soil under the footing undergoes failure) thus
swelling and movement of soil around the foundation is not seen (the passive
zone is not distinguished/found clearly). This mechanism is different
from what Terzaghi has proposed for small footings (limit width) on the
soils with average and high compaction. Increasing lateral pressure (confining)
the ground located under the footing will increase the pressure bearing
in this region comparing regions outside this region (under the footing)
which result in the formation of a column which resistant soil strongly
(with high confining pressure) beneath the footing and it results in increasing
the stress depth of influence and hence settlement would be higher, reasonable
and expectable. According to the conventional failure mechanism, stress
is distributed in the soil beneath and around the footing, so, the stress
penetration depth decreases (formation of stress bubbles). This phenomenon
is highly significant in settling the decline of small footing which is
uniform and under loading.
According to the proposed model in this paper, in order to predict and
interpret the results obtained from numerical analysis, the deformation
concentration phenomenon leads to the stress releasing (unloading) in
parts of the media. When a large footing undergoes load up to a certain
amount of loading, the behavior of the ground beneath and around the footing
is similar and the whole system undergoes deformation (Fig.
7).
Increasing rate of loading, the earth separated to deferent zones: the soil
column beneath the footing enters the plastic behavior range and hence the bearing
capacity of the segment increases which results in concentration and increasing
stress in this region and ultimately deformation and settlement increase. In
the mean time, the behavior of the parts outside beneath the footing (around
the footing) is elastic and under unloading. Therefore it can be inferred that
the ground outside beneath the large footing does not have a significant influence
on the footing bearing and zone of between this parts is in plastic deformation
and shearing (Fig. 8). In the part of earth that located under
the footing σ_{1} (main stress) will be vertical and in the other
parts it will be horizontal and in this parts the earth have a shear fracture
and in the parts that maximum main stress vertical (σ_{1}), soil
is bearing. So, in the parts that we have a horizontal (σ_{1})
the back part of this zone is passive which will be susceptible to breakage,
but in the parts the σ_{1} is vertical, zone is active and bearing,
so if footing have a large zone of the vertical σ_{1}, this footing
have a large bearing. In the small footing this zone is small and this zone
will be progress by increase of footing dimension (Owen and
Hinton, 1980; Zienkiewicz and Taylor, 2000; Khazaie
and Amirshahkarami, 2007).

Fig. 7: 
Stressstrain curve in domain 

Fig. 8: 
Failure mechanism, (a) Large footing and (b)Small footing 
In the large footing due to the increasing the confined zone that located
under the foundation and decreasing deviatory stress in this zone bearing
on the footing can increase. Also this agent cause the increase of reaction
earth in the middle of footing compared to the edges. In region 1 the
reaction of soil is less than the region 2, then by drawing the soil reaction
curve and determining the inclination point we could separate two regions.
Therefore when the earth is put under the vertical pressure the confined
pressure will increase and it can effects to the region 1. Due to this
pressure, region 1 want to move toward the edges or move up, according
to the boundary conditions, none of these moves are possible for these
regions then this will react as confined pressure. The confined pressure
that acts from region 1 to region 2 will cause reduction of deviatory
stress and breakage from shear state in the region 2. In the small footing
region 1 under pressure of region 2 move up and shear fracturing will
occur.
CONCLUSION
Based on the results obtained in this research, the failure mechanism
for small foundations, proposed by Terzaghi and other researchers are
not applicable to design large foundations in which the failure mechanism
occurs regionally within the interface between the earths around and underneath
them and rigid wedge angle will be increased up to 90°^{}.
Also by considering the change of property and distinction between soil
behavior when it is placed under small and large foundations, predicting
suitable tests in study of earth and finally selecting the best improvement
method for earth underneath the foundation regarding foundation dimensions
is a very important subject. On the other hand, the settlement of control
problem taking into account the increase of stress in depth and in large
foundations is more important than small foundations. It’s obvious
that the 3D modeling can offer more accurate results and it’s being
prepared by the author. But the real modeling in lab is proposed by other
researchers.