
Research Article


Assessing the Load Size Effect in the Soil (Under Single Foundation) Using Finite Element Method 

F. Farrokhzad
and
A.J. Choobbasti



ABSTRACT

In this research the analysis of foundation is efficiently accomplished by using the pure displacementbased finite element method; it should be noted that the effect of the concrete column dimension in the connection place of column to foundation is examined with the aid of finite element method. It is possible to interpret the finite element method as a method of substitute loadings or load cases in the same boundary situation. To meet this objective, single foundations with square shape are analyzed for more than 50 different point loads at the center of the slab. The structural layer is used to define slab geometry, load and boundary conditions which are related to analysis only. At the time of analysis, the objectbased model created by the authors converts into a finite element model, called the analysis model. The finite element mesh used in the analysis is a rectangular mesh, based on a maximum acceptable element size. The stress and displacements in a soil mass due to applied loading are considered in this paper. The authors found that the single footing settlement and stress of soil depend not only on physical and mechanical properties of base soils but also on applied load intensities and the size of the column connected to single foundation, as well as on footing and column rigidity, shape and dimensions. Based on the obtained results, it can be stated that by increasing in the load size (column size) the value of maximum stress and deformation in the soil of under foundation will increase.





Received: December 05, 2010;
Accepted: January 28, 2011;
Published: March 25, 2011


INTRODUCTION
The design of foundations involves the use of many different combinations of
structural elements and foundation types which in turn vary to perform a wide
variety of functions (Trivedi and Sud, 2005). Strip
footings and pad bases are used to deliver and spread superstructure loads over
a suitable area at foundation (formation) level. The foundation is required
to be stiff enough to distribute the loadings onto the substrata in a uniform
manner (Reznik, 1998).
The selection of the appropriate foundation solution is the most important
part of the design process and most difficult to define. Calculation usually
involves analyzing from certain parameters, the forces and stresses involved
in a particular structural element (Atkinson and Han, 2001).
Structural design is the process of exploiting engineering knowledge in an attempt
to select the most suitable and economic structure. Foundation design should
therefore be carried out using a careful blend of geology, soil mechanics, theory
of structures, design of materials, experience, engineering judgment, logic
and downtoearth engineering. Foundation design like other structural design
requires a good sound basic approach in order to achieve a truly successful
result (Zolfaghari and Hajabbasi, 2008).
Recently, extensive studies have been done on the behavior of footing and the
new graphs and several geotechnical researchers have examined formulas. Zhao
and Wang (2008) utilized a finite difference code FLAC to study bearing
capacity factor N_{γ} for ring footings in cohesionless, frictional
and ponderable soil. The present soil model employs MohrCoulomb yield criterion
and associative flow rule. Kumar and Bhattacharya (2010)
have been studied about the bearing capacity of interfering multiple strip footings
by using lower bound finite elements limit analysis. The ultimate bearing capacity
of a number of multiple strip footings, identically spaced and equally loaded
to failure at the same time is computed by using the lower bound limit analysis
in combination with finite elements. They mentioned that the failure load for
a footing in the group becomes always greater than that of a single isolated
footing. Lee et al. (2005) assessed the bearing
capacity of circular footings under surcharge using statedependent finite element
analysis. In their study, estimation of the bearing capacity of footings resting
on a soil mass acted upon by a surcharge is investigated and related to the
cone resistance qc. Nonlinear finite element analyses based on a statedependent
stressstrain model were performed to obtain the loadsettlement responses of
axially loaded circular footings. Various values of relative density, lateral
earth pressure ratio, depth of embedment and footing diameter were considered
in the analyses. Based on the finite element results, loadsettlement curves
were obtained and used to determine the unit limit bearing capacity of the footings
in terms of the cone resistance qc. Values of the unit bearing capacity for
different values of surcharge were in a narrow range while considerable variation
in unit bearing capacity was observed with relative density D_{R}. Algin
(2007) researched on practical formula for dimensioning a rectangular footing.
The relationship among the flexural formulae and the vertical and rotational
equilibrium of forces is correlated analytically to develop a simple and unique
expression to determine the required minimum footing area under full compression.
The practice of estimating the size of footing in structural design usually
employs the conventional iterative process initiated with the educated guess
of designers.
The stability of a structure depends upon the stability of the supporting soil
(Leucci, 2006). Two important factors that are to be
considered are: (1)the foundation must be stable against shear failure of supporting
soil. (2)the foundation must not settle beyond a tolerable limit to avoid damage
to the structure.
The finite element method is a technique for solution of mathematical problems
governed by systems of partial differential equations (Bathe
and Zhang, 2004). It can produce close approximate solutions to problems
with highly complex geometries, material behaviors and boundaries which would
result in highly complex field variations in the solution variables (Mukherjee
et al., 2000). The method accomplishes this by subdividing the solution
space into many pieces (the finite elements) sufficiently small that the variations
in the solution variables can be well approximated within each by very simple
functions (Liu et al., 2001). Implementation
of the method numerically on modern digital computers enables highly accurate
solutions with extremely large numbers of small elements. All of the governing
equations are then solved on all of the elements and the elemental solutions
are assembled into the solution for the whole subject to compatibility and continuity
requirements (Narra, 2009).
The finite element method is a numerical procedure for obtaining solution to many of the problems encountered in engineering analysis . In this research the authors used Finite element method for utilizing discrete elements to obtain the joint displacement and under foundation soil pressure for different load size. In order to assess the effect of column dimension in the connection place to single foundation, different columns and single foundations are modeled. MATERIALS AND METHODS
This research, carrying out in Babol University of Technology (Iran) from Dec
2009 to Sep 2010, models are objectbased and consist of point, line and area
objects to which assignments are made to define structural members, such as
slabs, columns and supports, as well as to define loads (Curley
et al., 2011). In this project Area objects are used to model slabs,
soil supports and surface loads. Point objects are used for column supports
and concentrated loads.
Properties are assigned to each object to define the structural behavior of
that object in the model (Handayani and Prawito, 2008).
Some properties, such as slab properties that contain both material and section
definitions, are named entities that must be specified before assigning them
to objects (Choobbasti et al., 2009).
Supports may be assigned to point, line and area objects and similar to properties.
The most difficult part of a settlement analysis is the evaluation of the modulus
of elasticity E_{S} that would conform to the soil condition in the
field. There are two methods by which E_{S} can be evaluated (Kouzer
and Kumar, 2010). They are laboratory method (triaxial tests) and field
method (plate load tests, standard penetration test, static cone penetration
test,…). Slab property data and subgrade modulus are shown in Table
1.
There are several reasons for the popularity of finite element methods. Large
code segments can be implemented for a wide class of problems (Demkowicz,
2000). The software can handle complex geometry. Little or no software changes
are needed when boundary conditions change, domain shapes change, or coefficients
vary (Hiller and Bathe, 2003). A typical finite element
software framework contains a preprocessing module to define the problem geometry
and data; a processing module to assemble and solve the finite element system
(Brenner and Scott, 2002); and a post processing module
to output the solution and calculate additional quantities of interests. A finite
element analysis of any physical problem requires that a mesh of finite elements
be generated. Because the generation of finite element meshes is a fundamental
step and can require significant human and computational effort (Mushari
AlNaeem, 2008). Figure 1 shows typical finite element
meshes (or finite element assemblages) modeling of the foundation. In mentioned
case the finite elements are used to represent the volume of the system. The
finite elements are connected at the nodal points located at the corners and
along the sides and in the faces of the elements (Hughes
and Wells 2005). However, nodal points can also be located within the volume
of an element (Liu, 2002b). An important feature is that
the finite elements do not overlap geometrically but together fill the complete
volume of the solid. In finite element based model, a twoway slab of arbitrary
shape is meshed into isotropic thin plate bending elements, using the area objects.
Those elements are three to fournode elements with one vertical and two rotational
degrees of freedom at each node which capture outofplane bending behavior
(Liu, 2002a).
Table 1: 
Slab property data and subgrade modulus 

 Fig. 1: 
(a) 3D view of structural layer in detail and (b) the bilinear
rectangular elements of the finite element 
 Fig. 2: 
(a) 3D view of foundation and spring which simulate the subsoil
and (b) boundary condition 
The finite element mesh used in this analysis is a rectangular mesh based on
a maximum acceptable element size. However, extra mesh lines are introduced
at all locations of objects and object boundaries. Additional mesh lines can
be introduced at specified locations by adding gridlines. Natural coordinate
systems can be defined for 2D elements. They are more convenient system for
both analytical and numerical integration. Elements produced by the quad tree
techniques have good geometric shapes near boundaries in this research (Farrokhzad
et al., 2010).
After the mesh has been generated, subdividing all the area objects that have been assigned slab properties creates slab elements. Support properties are lumped into discrete springs and are assigned to finite element mesh points (Fig. 2).
Support properties of all point objects are assigned directly to the corresponding
finite element mesh point. Similarly, support properties of all area objects
are applied to all the mesh points that exist within the area and on the boundaries
of the area, based on the tributary area associated with the mesh point. Point
loads of all point objects are assigned directly to the corresponding finite
element mesh point. Similarly, surface loads of all area objects are applied
to all the mesh points that exist within the area and on the boundaries of the
area based on the tributary area associated with the mesh point. All of the
internal meshing and assigning performed by the program does not alter the number
or size of the objects which allows revisions and modifications to be executed
at the object level.
RESULTS AND DISCUSSION
The accuracy of the finite element analysis results, measured on the exact
solution of the mathematical model, highly depends on the use of an appropriate
mesh and this holds true in particular when coarse meshes need be used to reduce
the computer time employed for complex analyses. Hence, effective mesh generation
procedures are most important (Braess, 2002).
The analysis model consists of joints, frame elements, plate elements and springs in contrast to the point, line and area objects in the userdefined objectbased model. In addition, object meshing is performed with control over the maximum mesh dimension. It should be noted that maximum mesh dimension was selected 25 cm. Each slab element is an isotropic, thick platebending element. This research applies an iterative procedure using the original stiffness and corrective load vectors to obtain the notension results. The concept of stress is closely associated with the concept of a continuum. Thus, when we speak of the stress acting at a point, we envision the forces against the sides of an infinitesimally small cube, which is composed of some homogeneous material.
Results from the theory of elasticity are often used to compute the stresses
induced within soil masses by externally applied loads (Li,
2004). The assumption of this theory is that stress is proportional to strain.
Most of the useful solutions from this theory also assume that soil is homogeneous
and isotropic. Soil seldom if ever exactly fulfills and often seriously violates,
these assumptions. Yet the soil engineer has little choice but to use the results
of this theory together with engineering judgment (Ghosh
and Sharma, 2010).
The processing module would need more information when adaptivity is performed.
It, for example, would need a link to the geometric information in order to
refine elements along a curved boundary (Piedrahita and
Montana, 2007). Even without adaptivity, the processing software may want
access to geometric information when using elements with curved edges or faces.
If the finite element basis were known at the preprocessing stage, space could
be reserved for edge and interior nodes or for a symbolic factorization of the
resulting algebraic system.
Figure 3 shows the results of different analysis in assessing the relation of load size and deformation of under foundation soil. It is obvious that increasing the dimension of column cause to increase in deformation or settlement of soil. The soil is modeled as springs under the foundation by increasing the dimension of the column more springs are affected and more deformation would occur. In this research load shape was assumed square and load dimension varied between 30 to 60 cm, 25 and 50 ton were selected as column loads and dimension of the single foundation was assumed 2 m by 2 m in surface and 0.6 m in depth. Figure 4 shows the variation of stress in the soil against the load size. Based on above results and the direct relation of stress and deformation, as it is obvious in figure, the maximum stress will increase when the size of column become larger.
A critical review on idealization and modeling for interaction among soilfoundationstructure
system has been done by Dutt and Roy (2002). The interaction
among structures, their foundations and the soil medium below the foundations
alter the actual behavior of the structure considerably than what is obtained
from the consideration of the structure alone .
 Fig. 3: 
The relation of load size and deformation (settlement of
soil), (a) applied load: 25 ton and (b) applied load: 50 ton 
 Fig. 4: 
The relation of load size and stress in soil, (a) applied
load: 25 ton and (b) applied load: 50 ton 
The mentioned study makes an attempt to gather the possible alternative models
available in the literature for this purpose. Emphasis had been given on the
physical modeling of the soil media, since it appears that the modeling of the
structure is rather straightforward. In another research, by means of a semianalytical
finite element approach an embedded circular footing under monotonic horizontal
and moment loading is studied (Bouzid and Vermeer, 2007).
In a nonhomogeneous soil whose shear modulus is characterized by a power law
variation with depth, horizontal, rocking and coupled modes of displacement,
expressed in terms of influence factors are thoroughly examined. The same effect
of interface conditions on the soil normal stresses developed beneath the embedded
footing for the case of loading applied at the footing top. By using small scale
model tests, the interference effect on the vertical loaddeformation behavior
of a number of equally spaced strip footings, placed on the surface of dry sand
was investigated by Kumar and Bhoi (2008). The interference
effect becomes prominent with increase in soil friction angle. In contrast to
an increase in the bearing capacity, with decrease in spacing of footings, an
increase in the footing settlement associated with the ultimate state of shear
failure was observed. The present experimental observations were similar to
the result of this paper by the available theory based on finite element method.
In the study of Lee et al. (2005) estimation
of the bearing capacity of footings resting on a soil mass acted upon by a surcharge
is investigated. Nonlinear finite element analyses based on a statedependent
stressstrain model were performed to obtain the loadsettlement responses of
axially loaded circular footings. It can be stated that the results of above
study and the research which is done in Babol University of technology, show
same thing in different way.
CONCLUSION Foundation design is one of the most challenging aspects of engineering and no twofoundation conditions are the same. In this study, the authors have proposed analysis based on finite element method for a nonlinear elastic problem in order to asses the load size effect (column dimension) in connection place to foundation on behavior of subsoil. The numerical results coincide with the analysis very well. The aim here was to link the behavior and modeling of foundations and soil to prior knowledge. The settlement of the footing is recorded against the size of the applied load or the dimension of column. The shape of the curve obtained by analysis depends generally on the size and shape of the footing, the column shape and the composition of the supporting soil. It can be noted that the applied load intensities and the size of the column connected to single foundation are the effective parameters on footing settlement and stress of soil. It can be concluded that increasing the dimension of column cause to increase in deformation or settlement of soil. The soil was modeled as springs under the foundation, by increasing the dimension of the column more springs are affected and more deformation would occur and finally it was obtained that the maximum stress will increase when the size of column become larger.

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