CHAPTER 3 FINITE ELEMENT MODELLING AND METHODOLOGY...
Transcript of CHAPTER 3 FINITE ELEMENT MODELLING AND METHODOLOGY...
61
CHAPTER 3
FINITE ELEMENT MODELLING AND METHODOLOGY
3.1 General The analysis of stress and deformation of the loading of simple geometric
structures can usually be accomplished by closed-form techniques. As the structures
become more complex, the analyst is forced to use approximations of closed-form
solutions, experimentation, or numerical methods. There are a great many numerical
techniques used in engineering applications for which digital computers are very
useful. In the field of structural analysis, the numerical techniques generally employ a
method which discretizes the continuum of the structural system into a finite
collection of points (or nodes)/ elements called finite elements. The most popular
technique used currently is the finite element method (FEM). Other methods some of
which FEM is based upon include trial functions via variational methods and
weighted residuals the finite difference method (FDM), structural analogues, and the
boundary element method (BEM).
3.1.1 The Finite Difference Method
In the field of structural analysis, one of the earliest procedures for the
numerical solutions of the governing differential equations of stressed continuous
solid bodies was the finite difference method. In the finite difference approximation
of differential equations the derivatives in the equations are replaced by difference
quotients of the values of the dependent variables at discrete mesh points of the
domain. After the equations are replaced by difference quotients of the values of the
dependent variables at discrete mesh points of the domain. After imposing the
appropriate boundary conditions on the structure, the discrete equations are solved
62
obtaining the values of the variables at mesh points. The technique has many
disadvantages, including inaccuracies of the derivatives of the approximated solution,
difficulties in imposing boundary conditions along curved boundaries, difficulties in
accurately representing complex geometric domains, and the inability to utilize non-
uniform and non-rectangular meshes.
3.1.2 The Boundary Element method
The boundary element method developed more recently than FEM, transforms
the governing differential equations and boundary conditions into integral equations,
which are converted to contain surface integrals. Because only surface integrals
remain, surface elements are used to perform the required integrations. This is the
main advantage of BEM over FEM, which require three-dimensional elements
throughout the volumetric domain. Boundary elements for a general three-
dimensional solid are quadrilateral or triangular surface elements covering the surface
area of the component. For two-dimensional and axisymmetric problems, only line
elements tracing the outline of the component are necessary.
Although BEM offers some modelling advantages over FEM, the latter can
analyse more types of engineering applications and is much more firmly entrenched in
today’s computer-aided-design (CAD) environment. Development of engineering
applications of BEM is proceeding however, and more will be seen of the method in
the future.
3.2 Finite Element Method
3.2.1 General
The finite element method (FEM) is a numerical procedure for analysing
structures and continuum. The problem may concern to obtain approximate solutions
to solid mechanics heat transfer and fluid mechanics problem. Basic ideas of the finite
63
element originated from advances in aircraft structural analysis. In 1941 Hrenikoff
presented a solution of elasticity problems using frame work method. Couroint, in
1943 used piecewise polynomial interpolation over triangular sub regions to model
torsional problems. A book by Argyris in 1955 on energy theorems and Matrix
methods laid a foundation for further developments in finite element studies. Turner
et al. derived stiffness matrices for truss, beam & other elements & presented in 1956.
The term finite element method was first coined and used by Clough in 1960. The
first book on finite elements by Zienkiewicz and Chung was published 1967.
The stress analysis in field of Civil, Mechanical and Aerospace
engineering, Naval architecture, Off shore engineering and Nuclear engineering is
invariably complex and for may of the problems it is extremely difficult and tedious
to obtain analytical solutions. In these solutions, engineers usually resort to numerical
methods to solve problems. With the advent of computers, one of the most powerful
techniques that have been developed in the realm of engineering analysis is the finite
element method.
The finite element method can be used for analysis of structures/solids of
complex shapes and complicated boundary conditions. Its applications range from
deformation and stress analysis of automotive, air craft, building and bridge structures
to field analysis of heat flux, fluid flow, magnetic flux, seepage and other flow
problems. With the advances in computer technology and CAD systems, complex
problems can be modelled with relative ease; several configurations can be tested on
computer before the first prototype is built.
64
3.2.2 Basic Concept
In the finite element method of analysis a complex region defining a
continuum is discretized into simple geometric shapes called finite elements. The
material properties and the governing relationships are considered over these elements
and expressed in terms of unknown values at elements corners. An assembly process
duly considering the loading and constraints results in a set of equations. Solution of
these equations gives the approximate behaviour of the continuum.
3.2.3 Basic steps in the Finite Element Method
The following are the steps adopted for analyzing a structural engineering
problem by the finite element method.
1. Discretization of the domain
The continuum is divided into a number of finite elements by imaginary lines
or surfaces. The interconnected elements may have different sizes and shapes. The
choice of the simple elements or higher order element straight or curved, it’s shape,
refinement are to be decided before the mathematics formulation starts.
2. Identification of variables
The elements are assumed to be connected at their intersecting points referred
to as nodal points. At each node, generalized displacements are the unknown degrees
of freedom. They are dependent on the problem at hand. For example in a plane stress
problem the unknowns are two linear translations at each nodal point.
3. Choice of approximating functions
Once the variables and local coordinate system have been chosen. The next
step is the choice of displacement function. In fact it is the displacement function that
is the starting point of the mathematical analysis. This function represents the
variation of the displacements within the element. The function can be approximated
65
in a number of ways. The displacement function may be approximated in the form of
a linear function or a higher order function. The shape of element or the geometry
may also be approximated. The co ordinates of corner nodes define the element shape
accurately if the element is actually made of straight line or plates.
4. Formation of the element stiffness matrix
After the continuum is discritised with desired element shapes, the element
stiffness matrix is formulated. This can be done in a number of ways. Basically it is a
minimization procedure whatever may be the approach adopted. For certain elements,
the form involves a great deal of sophistication. With the exception of a few simple
elements, the element stiffness matrix for majority of elements is not available in
explicit form. As such they require numerical integration for their evaluation.
The geometry of the element is defined in reference to a global frame. In many
problems such as those of rectangular plates, the global and local axis systems are
coincident and for them no further calculation is needed at the element level beyond
computation of element stiffness matrix in local coordinates. Coordinates
transformation must be done for all elements where in is needed.
5. Formulation of the overall stiffness matrix
After the element stiffness matrices in the global coordinates are formed, they
are assembled to form the overall stiffness matrix. The assembly is done through the
nodes, which are common to adjacent elements. At the nodes, the continuity of the
displacement function and possibly their derivatives are established. The overall
stiffness matrix is symmetric and banded.
6. Incorporation of boundary conditions
The boundary restraint conditions are to be imposed in the stiffness matrix.
There are various techniques available to satisfy the boundary conditions. In some of
66
these approaches, the size of the stiffness matrix may be reduced or condensed in its
final form. To ease the computer programming aspect and to elegantly incorporate the
boundary conditions, the size of the overall stiffness matrix is kept the same.
7. Formulation of element load matrix
The loading forms an essential parameter in many structural engineering
problems. The loading inside the element is transferred at the nodal points and
consistent element load matrix is formed. Sometimes, based on the typicality of
problem, the load matrix may be simplified.
8. Formation of the overall load matrix
Like the overall stiffness matrix, the element loading matrices are assembled
to form the overall loading matrix. This matrix has one column per loading case and it
is either a column vector or a rectangular matrix depending on the number of loading
conditions.
9. Solution of simultaneous conditions
All the equations required for the solution of the problem are now developed.
In the displacement method, the unknowns are the nodal displacements. The gauss
elimination and Cholesky’s factorization are the most commonly used procedures for
the solution of simultaneous equations. These methods are well suited to a small or
moderate number of equations. For large sized problems, a frontal technique is one of
the methods of obtaining solution. For systems of large order, Gauss-Seidel or Jacobi
iterations are more suited.
10. Calculation of stress or stress-resultants
In the previous step, nodal displacements are calculated and these values are
utilized for the calculation of stresses or stress-resultants. This may be done for all
elements of the continuum or it may be limited only to some predetermined elements.
67
Results may be obtained by graphical means. It may be desirable to plot the
contour of the deformed shape of the continuum. The contour of the principal stresses
may be one of the sought after items for certain category of problems.
3.2.4 Advantages
The main advantage of finite element analysis can be put in one sentence. The
physical problems which were so far intractable and complex for any closed bound
solution can now be analyzed by this method. The advantages in relation to the
complexity of the problem are stated below.
Finite element method is fast, reliable and accurate.
It can analyse any structure with complex loading and boundary conditions
i.e. the method can be efficiently applied to cater irregular geometry and
can handle any type of loading.
It can analyse structures with different material properties i.e. material
anisotropy and non homogeneity can be catered without much difficulty.
This method is easily amenable to computer programming.
It can analyse structures having variable thickness.
3.2.5 Disadvantages
One should not form the idea that the F.E.M is the most efficient for the
analysis of any type of structural engineering or physical problem. There are many
types of problems where some other method of analysis may prove efficient than the
F.E.M.
Main disadvantages of this method are the cost involved in the solution of
the problem. (simpler computer methods such as finite strip or other same
analytic methods for the vibration and stability analysis of simpler
68
structure will lead to more economic solution, but these methods will work
within their own limitations and will not be as versatile as the F.E.M)
It is difficult to model all problems accurately and the results obtained are
approximate.
The result depends upon the number of elements used in the analysis.
Data preparation is tedious and time consuming.
3.2.6 Limitations of finite element method
In whatever sophisticated manner the problem might be formulated and
solved, it has been done so within the frame work of its assumptions using
proper engineering judgment. However, one has to be exercised in
interpreting the results.
Not that all conceivable existing complicated problems have been solved
by the finite element method. There are still some which has remained
intractable till today.
Due to the requirement of large computer memory and time, computer
programs based on the FEM can be run only in high speed digital
computers. For some problems, there may be considerable amount of input
data. Errors may creep up in their preparation and results thus obtained
may also appear to be acceptable which indicates deceptive state of affairs.
It is always desirable to make a visual check of the input data as described
in the next section.
3.3 General Procedure
In the finite element method, the structure under consideration is divided into
smaller zones, called as elements. The elements are assumed are to be connected to
each other at certain points called nodes. It is at the nodes that we compute the
69
displacements. Thus a body having degrees of freedom equal to two or three times the
number of nodes approximates the body with infinite number of degrees of freedom.
With the increase in number of nodes, a better solution (closer to the exact) is
obtained. The displacements at any point within an element are related to the
displacements at the nodes by a set of functions, the most common form being a
polynomial. The polynomials chosen should be such that they give continuity within
the elements and the displacements must be compatible between adjacent elements.
The element stiffness matrices and the element load vectors are formulated and
assembled to get global vectors for the entire structure. After applying the required
boundary conditions, the formulated equations are solved to obtain the nodal
displacements. From the displacement field within the element, strains can be
calculated. From the strains, using the stress- strain relations, stresses can be
calculated. The following sections present the formulation of stiffness matrix in detail.
3.4 Three dimensional elements
In the present work three dimensional elements have been used for modelling
SIFCON slabs. Accordingly details of 3 D elements are provided in sections below.
3.4.1 Basic Finite Element Relationships
The basic steps is the derivation of the element stiffness matrix, which relate
the nodal displacement vector d , to the nodal force vector f are described below.
Considering a body subjected to a set of external forces, the displacement
vector at any point within the element eU is given by
ee dNU
Where, [N] is the matrix of shape functions, {d} e the column vector of nodal
displacements.
70
The strain at any point can be determined by differentiating the displacement vector
as ee UL
Where, [L] is the matrix of differential operator. In expanded form, the strain vector
can be expressed as:
=
zu
xw
yw
zv
xv
yu
zwyvxu
zx
yz
xy
z
y
x
Substituting equation displacement into strain gives:
ee dB
Where, [B] is strain-nodal displacements matrix given by:
NLB
The stress vector can be determined by using the appropriate stress-strain relationship
as: ee D
Where, [D] is the constitutive matrix and e is:
TzxyzxyZZyyxxe
From the above equations, the stress-nodal displacement relationship can be
expressed as: ee dBD
For writing the force-displacement relationship, the principal of virtual displacements
are used. If any arbitrary virtual nodal displacement, {d} e, is imposed, the external
work, Wext., will be equal to the internal work Wint.
71
Wext = Wint
In which
eext feT
dW
and
dVW e
Te int
Where, {f} e is the nodal force vector. Substituting equation strain into Wint, we get:
v
eTT
e dVBdW ...int
From equation stress and Wint,
v
eTT
e ddVBDBdW .....int
and equation Wext = Wint can be written as:
efeT
d = v
eT ddVBDB
eT
d .....
or
v
eT
e ddVBDBf .....
Letting:
v
Te dVBDBK .....
Then
eee dKf .
Where, [K]e is the element stiffness matrix. Thus, the overall stiffness matrix can be
obtained by: n v
Te dVBDBK ....
72
The total external force vector {f} is then:
dKf
Where, {d} is the unknown nodal point displacements vector.
3.4.2 Eight-Noded Solid Element
In NISA, 3-D concrete solid is used for the 3-D modelling of solids with or
without fibres. The solid is capable of cracking in tension and crushing in
compression. In concrete applications, for example, the capability of the solid
element may be used to model the concrete, while the rebar capability is available for
modelling fibre behaviour. The element is defined by eight nodes having three
degrees of freedom at each node: translations of the nodes in x, y and z-directions.
The most important aspect of this element is the treatment of nonlinear
material properties. The concrete is capable of cracking, crushing, plastic
deformation, and creep. An eight noded hexahedral solid element is incorporated in
this study to simulate the behaviour of SIFCON. The element is defined by eight
nodes and by the isotropic material properties. The geometry, node locations, and the
coordinate system for this element are shown in Fig. 3.1.
Fig. 3.1 Solid – 3 D Concrete Element
The element is bounded by six quadrilateral faces and has eight nodes. The geometry of
the element is described by the Cartesian coordinates ( ) of the eight nodes.
Each node i has three degrees of freedom .
73
The nodal degrees of freedom vector q is represented as
),,,( 111 wvuq T
)
The corresponding vector of nodal forces is
T= {Fx1, Fy1, Fz1, Fx2, Fy2, Fz2 ….Fx8, Fy8, Fz8)
The displacement components (u, v, w) at any point (X, Y, Z) within the HEXA 8
element have to be interpolated in terms of the nodal degrees of freedom.
The displacement can be expressed as
{u} = [N] {q}
Where, [N] is the matrix of shape functions,
821
821
821
000000000000000000
NNNNNN
NNNN
{q} = the column vector of nodal displacements i.e.(Xi, Yi, and Zi are
displacement components of node i.
For the interpolation functions Ni (i = 1, --, 8), it is now a standard practice to use the
concept of a parent element in the covariant natural coordinate system ξ, η, ζ. Fig.
3.1(a) shows an eight noded element with node numbering and the natural co-
ordinates.
Fig. 3.1(a) 8-node hexahedron and the natural coordinate’s ξ, η, μ
74
The parent element is a bi-unit cube with nodes located at it eight corners. Any
variable Φ is approximated over the parent element domain using the tri-linear
polynomial:
Φ=α1+ α2ξ+ α3η+ α4ζ+ α5ξη+ α6ηζ+ α7ξζ+ α8ζξη
The polynomial coefficients (i =1,--, 8), in terms of the nodal values of Φ namely, Φ i
(i=1,--, 8). The result is
Φ=
Where
Natural coordinates of nodes of the parent element
Node i 1 -1 -1 -1 2 +1 -1 -1 3 -1 +1 -1 4 +1 +1 -1 5 -1 -1 +1 6 +1 -1 +1 7 -1 +1 +1 8 +1 +1 +1
These are the shape functions for the HEXA 8 element and are listed
below for the HEXA 8 elements.
75
These functions have the property of
at all other nodes
8
11
iiN
The relation between the Cartesian coordinates and the natural coordinates to
represent the parent element to the shape of the HEXA 8 element is
Where ( denote the Cartesian coordinate of the node i.
The geometry of the HEXA 8 element is thereby represented by the shape functions
(i=1,--,8) and the nodal coordinates ( (i=1,--,8). The centroid of the
element is located at ξ=η=ζ=0; the six faces of the element are identified by ξ=+
1,η=+1,ζ=+1
The assumed displacement functions will satisfy the continuity requirements at
interfaces between elements.
76
Element strain matrix
can be written as {ε}= {B} {q}
Where
[B]=
Element stress
The stresses σ at any point with in the HEXA 8 element are evaluated using
{σ}= [D] [B] {q}-[D] {
Where the initial strains due to thermal expansion are
and denotes the temperature change at node i.
Element stiffness matrix
The stiffness matrix [k] for the HEXA 8 element is evaluated using
eT dVBDBk
77
Where edV = dddJdet and J is the (3 x 3) Jacobian Matrix
which is defined as the matrix connecting the derivatives in local and global
coordinate systems. The integration in the above equation is performed numerically
using Gauss Quadrature.
The integration can now be performed over the parent element domain using
dddJBDBk T det1
1
1
1
1
1
We note here that both B and J are involved functions of ξ, η, ζ.
Consistent nodal force vector: body force
The nodal force vector fb due to body force b is evaluated using
e
Tb dVbNf
The integration can be performed over the parent element domain using
dddJbNfT
b det1
1
1
1
1
1
Consistent nodal force vector: Initial strain
The nodal force vector finit due to initial strain εinit is evaluated using
einitTinit dVDBf
The integration can be performed over the parent element domain using
dddJDBf initT
init det1
1
1
1
1
1
3.5 Finite Element Program used in this work
NISA is a proprietary engineering analysis program developed and marketed
by Cranes Software International Limited (Bangalore), which is used for FE
modelling in the present work.
78
A Numerically Integrated element for Systems Analysis (NISA) is a general
purpose finite element program to analyse a wide spectrum of problems encountered
in engineering mechanics.
The analysis comprises of three steps:
1. Pre-processing
2. Processing
3. Post-processing
It involves providing required data and instructions. The steps that are often
used in working with the program are as follows:
Geometric Modelling - Here the domain boundaries are plotted by using grids
and lines.
Discretization -At this stage the domain is divided into finite elements. The
elements can be formed as 4 to 12 node quadrilateral, or a 3 to 6 node triangle
depending on the order of the element.
Material Properties - The material properties are identified by giving material
identification numbers to elements. For each material the properties are given
in an existing tabular form provided in the package.
Boundary Conditions - Each element has three degrees of freedom per node
displacements (UX, UY, and UZ) and rotations (RX, RY, and RZ). Based on
the problem the boundary conditions are applied by specifying displacement
and rotation values.
Loads – It involves specifying external loads through applied forces and
pressure data. The forces are applied at nodal points and pressures are applied
on the element faces.
79
Package based Information – It includes saving of the required files for
processing. These files have all the information regarding the given problem.
Output file for the given data is processed. The results can be viewed by
opening the output file. The various results that can be seen in NISA are Stress
contours, deformations, animations and graphs for the above results. Plots can be
drawn in post-processing; it also prepares the data in a suitable format to import data
to other graphing packages.
3.6 MODELLING AND VALIDATION DETAILS
3.6.1 Development of Finite Element Model
Development of a finite element model for predicting the load deflection
response of SIFCON slab elements involve various stages which are addressed in the
following sections.
3. 6.1.1. Methodology
The load – deflection response of the slab has been modelled by the finite
element method. To study the effect of volume fraction of fibre in modelled slab and
load deflection response, two categories of slab, viz., slab with all fixed edges and all
simply supported edges are considered. The basic properties of SIFCON for
modelling used in this study have been obtained from experiments described in
Chapter 4. For each category experimental values for young’s modulus, E, yield
stress, y Poisson’s ratio, for different volume fraction of fibre has been used
from Table 4.10 in this validatation problem. For the study of non-linear response of
slab beyond the elastic limit stress –strain data from experimental work has been used
i.e. multilinear stress strain data to converge the nonlinear solution from Table 4.11.
The region of the domain and boundary conditions of the modelled slab for the both
80
the categories are as shown in Fig.3.2 and Fig.3.3. The details of test program are
presented in Table 3.1.
.
Fig. 3.2 Finite element discretization for the SIFCON slab fixed on all its edges
Fig. 3.3 Finite element discretization for the SIFCON slab Simply supported on all its edges
Table 3.1 Details of test program
S.No. Slab Type Slab Designation Percent
volume of fibres
1.
SIFCON slab with all four
edges fixed
SIF0S4F-8 8
2 SIFCON slab with all four
edges simply supported
SIF4S0F-8 8
81
3.6.1.2 Finite element analysis
The SIFCON slab is modelled using 3D solid elements. The developed model
can simulate all the possible support and load conditions that usually occur in normal
building slabs. To estimate the optimal number of elements in each direction of the
slab, studies have been conducted which give a constant deflection with increasing
number of elements. These elements have three degrees of freedom per node. To
represent the support condition, proper boundary conditions have been used. When
the support is fixed, all the degrees of freedom (Ux, Uy, Uz,) have been restrained and
for simple support case only vertical degrees of freedom (Uz) have been restrained.
Load is applied on the surface of the elements as pressure load which is the ultimate
load of the panel obtained from the experimental study.
The slab selected for analysis has a dimension of 600 mm x 600mm, and a
thickness of 50mm. It is discritised using solid element having a size of 30. A uniform
pressure of 0.192N/mm2, which is close to the ultimate load of the slab (SIF0S4F)
obtained from experiments conducted by (Ramana, 2006), is applied on the surface of
the elements. A load of 0.0761N/mm2 has been applied on the simply supported slab.
These loads are applied on respective slabs incrementally as 100 load steps and
Newton Raphson’s method has been used to facilitate the non-linear analysis for
solution convergence. In non-linear static analysis, a von-mises yield criterion with
elastic piece wise linear hardening curve is used in this slab model analysis. In the
simply supported case, one fourth of the model with symmetric boundary conditions
on the two edges has been considered for the analysis which also represents the full
model analysis. The solid elements representing the slab were 2000 brick elements
and 2646 nodes with three degrees of freedom per node.
3.6.1.3 Non - linear Solution In nonlinear analysis, the total load applied to a finite element model is
divided into a series of load increments called load steps. At the completion of each
82
incremental solution, the stiffness matrix of the model is adjusted to reflect nonlinear
changes in structural stiffness before proceeding to the next load increment. The
NISA program uses Newton-Raphson equilibrium iterations for updating the model
stiffness.
Newton-Raphson equilibrium iterations provide convergence at the end of
each load increment within tolerance limits. Fig. 3.4 shows the use of the Newton-
Raphson approach in a single degree of freedom nonlinear analysis.
Fig. 3.4 Newton-Raphson iterative solution (2 load increments)
Prior to each solution, the Newton-Raphson approach assesses the out-of-
balance load vector, which is the difference between the restoring forces (the loads
corresponding to the element stresses) and the applied loads. Subsequently, the
program carries out a linear solution, using the out-of-balance loads, and checks for
convergence. If convergence criteria are not satisfied, the out-of-balance load vector
is re-evaluated, the stiffness matrix is updated, and a new solution is attained. This
iterative procedure continues until the problem converges. It was found that
convergence of solutions for the models was difficult to achieve due to the nonlinear
behaviour of SIFCON. Therefore, the convergence tolerance limits were increased to
a maximum of 5 times the default tolerance limits (0.5% for force checking and 5%
for displacement checking) in order to obtain convergence of the solutions.
Load
Displacement
Load points
83
3.6.2 Results and Discussions
For validation purpose, finite element analysis has been carried out on FE
modelled SIFCON slab without openings and the results have been compared well
with the measured values presented in Table 3.2 obtained by experimentation
(Ramana, 2006).
Table 3.2 Central deflection values for various square plates with uniform load for 8% percentage volume of fibre fraction
S.No Nomenclature Ultimate load (kN)
Maximum central deflection at ultimate load (mm)
FEA Measured value
1 SIF0S4F-8 69.12 13.11 13.00 2 SIF4S0F-8 27.00 11.61 9.42
Fig. 3.5 shows the comparison of load deflection response of SIFCON slab
restrained on all its edges and SIFCON slab simply supported on all its edges with 8%
volume of fibre fraction as obtained from FE analysis and experimental results.
Fig. 3.5 Comparison of load deflection response of SIFCON slab
The analysis of results shows good agreement between analytical values and
experimental values. Fig. 3.6 illustrates deflections along or across the mid span for
the ultimate pressure pattern for SIF0S4F and SIF4S0F SLABS with 8% volume of
fibre fraction. The results illustrated in the figures show that at the ultimate pressures,
the slabs having fixed boundary edge condition had larger deflection than the slab
84
having simply supported edge boundary condition with 8% volume of fibre fraction.
The deflection of the simply supported slab and fixed slab are not the same. Lot of
difference is there between two values. That difference was due to scale effect. The
maximum value of deflection for the simply supported slab at the mid-span is smaller
than that in the fixed slab. This is because edges of the slab are just resting over the
supports and hence undergone less deformation with low load carrying capacity. In
the case of fixed slab, the edges of the slab are restrained effectively and hence
undergo maximum deformation with maximum load carrying capacity. Whereas, the
maximum value of deflection for the simply supported slab at the mid-span is higher
than that in the fixed slab for the same load.
Variation of deflection along side of the SIF4S0F and SIF0S4F model slabs with 8% volume of fiber fraction
0
2
4
6
8
10
12
14
0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 570 600Side, a(mm)
Def
lect
ion ,
Δ(m
m) SIF4S0F-8%
SIF0S4F-8%
Fig. 3.6 Deflection variation across mid span of slab for fixed edge and simply
supported boundary condition
3.7 Summary
The details of FEM, model used in the present investigation for FE analysis
and modelling and validation details are presented in this chapter. The SIFCON slabs
restrained on all its edges and simply supported on all its edges with 8% volume of
fibre fraction are modelled using 3D solid elements for validation purpose. The
results obtained from FE analysis shown good agreement with experimental values
(Ramana, 2006). This validated model will be used to carry out FE analysis of
SIFCON slabs with or without different type and size openings at different locations.