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.