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CHAPTER 8

FINITE ELEMENT ANALYSIS

8.0 GENERAL

The Finite Element Method (FEM) is a numerical analysis for obtaining approximate solutions to

a wide variety of engineering problems. This has developed simultaneously with the increasing

use of high-speed electronic digital computers and with the growing emphasis on numerical

methods for engineering analysis. Although originally developed to study stresses in complex

airframe structures, it has been extended and applied to the broad field of continuum mechanics.

Because of its diversity and flexibility as analysis tool, it is receiving much attention in

engineering field and in industry.

8.1. GENERAL DESCRIPTION OF THE METHOD

The basic concept behind FEM is that a body or structure is divided into smaller elements of

finite dimensions called ‘finite elements’. The original structure is then considered as an

assemblage of these elements at a finite number of joints called ‘nodes’.

The properties of the elements are formulated and combined to obtain the solution for the entire

structure. The shape functions are chosen to approximate the variation of displacement within an

element in terms of displacement at the nodes of the element. The strains and stresses within an

element will also be expressed in terms of the nodal displacement. The principle of virtual

displacement is used to derive the equations of equilibrium for the element and the nodal

displacement will be the unknowns in the equations.

The boundary conditions are imposed and the equations of equilibrium are solved for the nodal

displacement. From the values of the nodal displacement for each element, the stresses and

strains are evaluated using the element properties.

Thus instead of solving the problem for the entire structure in one operation, in this Finite

Element Method attention is mainly devoted to the formulation of properties of the constituent

elements.

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8.2. ADVANTAGES OF FEM

The Finite Element Method has many advantages of its own. Some of them are given below.

Various types of boundary conditions are automatically handled in the

formulation. They are systematically enforced just before the solution, for the

nodal values of the field variables are obtained.

Material anisotropy and in homogeneity can be treated without much difficulty,

Any type of loading can be handled,

Higher order elements may be implemented with relative ease.

The method can efficiently be applied to cater irregular geometry

Spacing of nodes need not follow a pattern or rule

8.3. DISADVANTAGES OF FEM

The Finite Element Method has its own limitations. They are listed below.

There are many types of problems where some other method of analysis may

prove more efficient than the FEM,

There are some trouble spots such as “Aspect Ratio” (ratio of longer to smaller

dimensions), which may affect the final results,

The cost involved in the solution of the problem is high.

8.4. APPLICATIONS OF FEM

Applications of Finite Element Method divide into three categories, depending on the nature of

the problem to be solved. In the first category are the problems known as equilibrium problems

or time independent problems. The majority of applications of Finite Element Method fall into

this category.

For the solution for equilibrium problems in the solid mechanics area we need to find the

displacement distribution and stress distribution for a given mechanical or thermal loading.

Similarly for the solution of equilibrium problems in Fluid mechanics we need to find pressure,

velocity, temperature and density distributions under steady state conditions.

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In the second category are the Eigen value problems of solid and Fluid mechanics. These are

steady state problems whose solution often requires the determination of natural frequencies and

modes of vibration of solids and fluids. Examples of Eigen value problems involving both Solid

mechanics (eg. elasticity, plasticity, static's and dynamics) and Fluid mechanics (viscous and in

viscid) appear in Civil Engineering when the interaction of lakes and dams is considered and in

Aerospace Engineering, when the sloshing of liquid fuels in flexible tanks is involved. Another

class of Eigen value problems includes the stability of structures and the stability of laminar

flow.

In the third category is the multitude of time dependent or problems of continuum mechanics.

This category is composed of the problems that result when time dimension is added to the

problems of the first two categories.

The range of possible applications of the finite Element Method extends to all Engineering

disciplines but Civil, Mechanical and Aerospace engineers are the most frequent users of the

method. In addition to the structural analysis other areas of applications include Heat transfer, lid

mechanics, Electromagnetism, Biomechanics, Geomechanics and acoustics. The method finds

acceptance in multidisciplinary problems where there is coupling between heat transfer and

displacements as well as aero plasticity where there is a strong coupling between external flow

and the distortion of wing.

8.5. ANSYS

8.5.1 General

The ANSYS computer program is a general purpose Finite Element Modelling Package for

numerically solving a variety of mechanical problems. These problems include static and

dynamic structural analysis (both linear and non linear), steady state and transient heat transfer

problems, mode-frequency and buckling analyses, acoustic and electromagnetic problems and

various types of field and coupled-field applications. The program contains many special

features which allow nonlinearities or secondary effects to be included in the solution such as

plasticity, large strain, hyper elasticity, creep, swelling, large deflections, contact, stress

stiffening, temperature dependency, material anisotropy and radiation.

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As ANSYS has been developed, other special capabilities, such as sub structuring, sub

modelling, random vibration, kinetostatics, kinetodynamics, free convection fluid analysis,

acoustics, magnetics, piezoelectrics, coupled-field analysis and design optimization have been

added to the program. These capabilities contribute further to making ANSYS a multi-purpose

analysis tool for varied engineering disciplines.

8.5.2 Program Overview

The ANSYS element library contains more than sixty elements for static and dynamic analyses,

over twenty for heat transfer analyses and numerous magnetic field and special purpose

elements. This variety of elements allows the ANSYS program to analyze two and three

dimensional frame structures, piping systems, two dimensional plane and axis symmetric solids,

flat plates, axis symmetric and three dimensional shells and non-linear problems including

contact, interface and cables.

The program is divided into many processors where each processor has a particular job to

perform.

1. Pre Processor: This builds the model.

2. Solution Processor: This is for assigning loads, constrains and finally to get Finite

element solution.

3. General Post Processor: This is for further processing and viewing the results over the

entire model at specific time points.

4. Time History Post Processor: Reviews results at specific points in the model as a function

of time.

5. Topological optimization: Execute several topological optimization iterations.

6. ROM Tool:

7. Design optimization: This improves an initial design.

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8. Probabilistic Design: This accounts for the inaccuracies and uncertainties influencing the

outcome of an analysis by the use of a random input variable

9. Radiation Matrix: This calculates radiation view factors and generates radiation matrix

for thermal analysis.

10. Run Time Statistics: Predicts CPU time, wave front requirements etc for an analysis.

11. Session Editor: Allows the user to modify or save commands issued since the last

RESUME or SAVE command.

A Graphical User Interface (GUI) is available throughout the program, to guide new users

through the learning process and provide more experienced users with multiple windows, pull-

down menus, dialogue boxes, tool bars and on-line documentation.

8.6. NON-LINEAR ANALYSIS

8.6.1 Introduction

In a FE formulation, we assume that the displacements of the FE assemblage are infinitesimally

small and that the material is linearly elastic. In addition, it was also assumed that the nature of

boundary conditions remain unchanged during the application of load on the FE assemblage.

With these assumptions, the FE equilibrium equations derived were for static analysis.

K U = R

These equations correspond to a linear analysis of a structural problem, because, U is a linear

function of the applied load vector “R”. If the loads are “αR” instead of “R”, where α is a

constant, the corresponding displacements are “αU”.

The linearity of a response prediction rests on the assumptions, which have been entered in the

equilibrium equations. The fact that the displacements must be small has entered to the

evaluation of matrix [K] and the load vector [R]. The strain-displacement matrix [B] of each

element was assumed to be constant and independent of the element displacements. The

assumption of a linear material is implied in the use of constant stress-strain matrix [C].

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But, the real life problems don’t follow this statement. All the materials are not linearly elastic.

The boundary conditions change during the application of the load. The displacements and

strains may not be small. When we experience all the above problems in analysis, we should

consider non-linearity. The generalized definition for nonlinear problems is given below.

“The problems which do not develop from one stage to another in a single smooth series of

stages are called non-linear problems”.

8.6.2 Causes of Non-Linear Behaviour

The non-linear behaviour in a structure is caused by any of the following reasons.

Geometric Non-linearity

Material Non-linearity

Changing Boundary Conditions

8.6.3 Geometric Non-Linearity

If a structure experiences large deformations, then its changing geometric configurations can

cause the structure to respond non-linearly. The geometric non-linearity can be further classified

into two types.

Large deformation, large strain.

Large Deformation, but small strain.

8.6.4. Material Non-Linearity

The Hooke’s Law states that, within elastic limit, the stress is directly proportional to strain. All

the linear elastic materials should obey the Hooke’s law strictly. But, all the materials are not

fully elastic and beyond the elastic limit, the stress is not proportional to strain. When these kinds

of materials are to be analyzed, material non-linearity should be considered.

If stress-strain relations are linear or non-linear but elastic, there is a unique relation exists

between stress and strain. But, if there are plastic strains, the stress-strain relation is path

dependent, not unique. A given state of stress can be produced by many different straining

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procedures. In addition, different materials require different material theories. The essential

computational problem of material non-linearity is that equilibrium equations must be written

using material properties that depend on the strains, but the strains are not known in advance.

Hence, based on material models only, the solution algorithms can be developed.

The material non-linear problems can be classified further into the following types.

Non-linear Elastic

Non-linear Plastic

Non-linear Hyper-elastic

Non-linear Hypo-elastic

Non-linear Creep

Non-linear Visco-plastic

Table 8.1 provides a brief overview of the major classes of material selection.

Table 8.1 Different Types of Material

S.No. Material Model Characteristics Examples

1 Elastic (Linear or

Non-linear)

Stress is a function of strain

only. Same stress path on

unloading as on loading.

Almost all the materials

provided the stresses are small

enough. Steel, Cast Iron, Glass,

Rock, Wood before yielding or

fracture.

2 Hyper-Elastic Stress is calculated from strain

energy functional, W Rubber like materials

3 Hypo-Elastic Stress increments are calculated

from strain increments. Concrete models

4 Elasto Plastic Linear elastic behaviour until

yield.

Metals, Soils, Rocks when

subjected to high stresses.

5 Creep

Time effect of increasing strain

under constant load or

decreasing stress under constant

deformation.

Metals at high temperature.

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8.7 FINITE ELEMENT MODELLING OF COMPOSITE BEAMS

8.7.1 General

Modeling is one of the important features in Finite Element Analysis. It takes around 40% to

60% of the total solution time. Improper modeling of the structures leads to the unexpected

errors in the solution. So, proper care should be taken for modeling the structures to avoid errors.

A good idealization of the geometry reduces the running time of the solution considerably. A

three dimensional structure can be easily analyzed by considering it as a two dimensional

structure without any variation in results. So, creative thinking in idealizing and meshing of the

structure helps not only in considerable reduction of time but also in less memory of the system.

8.7.2 Modelling of Composite Beams

Finite Element Modeling of Composite beams in ANSYS consist of three stages, which are

explained below.

Selection of element type

Assigning material properties

Modeling and meshing the geometry

8.7.3 Element Types

Selection of proper element types is another important criterion in Finite Element Analysis. For

composite beams, the C-R steel sheet trough and braces were modeled by using SHELL 63

element in ANSYS. The Concrete portion was modeled by using a special element developed

particularly for Concrete by ANSYS, SOLID 65 element. The stud connectors and reinforcement

were modeled by using LINK 8 Element. SOLID 45 elements were used to model the steel plates

provided at support and loading points. All the elements used in the analysis are explained below

briefly.

8.7.4 SHELL 63

SHELL 63 is used to model the thin walled structures effectively. This element has four nodes

with six degrees of freedom at each node namely three translational degrees of freedoms and

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three rotational degrees of freedom. This element has stress stiffening and large deformation

effects. It is possible to give four different thicknesses on four corners of this element. The

thicknesses can be given in the real constant option.This element is illustrated in Fig. 8.1

Fig. 8.1 SHELL 63 Element

8.7.5 SOLID 65

SOLID 65 elements are used to model reinforced concrete problems or reinforced composite

materials, such as fiber glass. This element has eight nodes, with each node having three

translational degrees of freedom in the nodal X, Y & Z directions as shown in 8.2. The element

may be used to analyze cracking in tension and crushing in compression. The element itself is

used to analyze problems with or without reinforced bars. Upto three rebar specifications may be

defined. The rebar facility can be removed by assigning the volume ratio as zero.

Fig. 8.2 SOLID 65 Element

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8.7.6 LINK 8

LINK 8 is a spar, which may be used in a variety of engineering applications. Depending upon

the applications, the element may be thought of as a truss element, a cable element, a reinforcing

bar and a bolt. The three-dimensional spar element is having two nodes and each node having

three translational degrees of freedom. This element is capable of plasticity, creep, swelling and

stress stiffening effects. The cross sectional area can be given as the real constant. This element

is shown in Fig.8.3.

Fig. 8.3 LINK 8 Element

8.7.7 SOLID 45

SOLID 45 is a three-dimensional brick element used to model isotropic solid problems. It has

eight nodes, with each node having three translational degrees of freedom in the nodal X, Y & Z

directions. This element may be used to analyze the large deflection, large strain, plasticity and

creep problems. It has no real constants. This element is illustrated in Fig. 11.4

Fig. 8.4 SOLID 45 Element

UX

UZ

UY

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8.8 MATERIAL PROPERTIES

From the experiments conducted, the following values are found out and listed.

Compressive strength of concrete cubes

Yield stress of C-R steel sheet

Yield stress of reinforcing bars.

Coupon tests are done in order to determine the yield stress and modulus of elasticity of C-R

sheet and reinforcing bars. The values tabulated below are used for calculating the important

properties required for specifying material non-linearity.

8.9 NON-LINEAR MATERIAL MODEL FOR CONCRETE

The challenging task in modeling the COMPOSITE beams is the development of the behaviour

of concrete. Concrete is purely non-linear material and it has different behaviour in compression

and tension. The tensile strength of concrete is typically 8% to 15% of the compressive strength.

Fig. 8.5 shows the typical stress-strain curve for normal weight concrete.

In compression, the stress-strain curve of concrete is linearly elastic up to about 30% of the

maximum compressive strength. Above this point, the stress increases gradually up to the

maximum compressive strength, and then descends into a softening region, and eventually

crushing failure occurs at an ultimate strain εcu. In tension, the stress-strain curve for concrete is

approximately linearly elastic up to the maximum tensile strength. After this point, the concrete

cracks and the strength decreases gradually to zero.

ANSYS has its own non-linear material model for concrete. Its reinforced concrete model

consists of a material model to predict the failure of brittle materials, applied to a three-

dimensional solid element in which reinforcing bars may be included. The material is capable of

cracking in tension and crushing in compression. It can also undergo plastic deformation and

creep. Three different uniaxial materials, capable of tension and compression only, may be used

as a smeared reinforcement, each one in any direction. Plastic behaviour and creep can be

considered in the reinforcing bars too. For plain cement concrete model, the reinforcing bars can

be removed.

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Fig. 8.5 Typical Uniaxial Compressive and Tensile

Stress-Strain Relationship of Concrete

As per the ANSYS concrete model, two shear transfer coefficients, one for open cracks and other

for closed ones, are used to consider the amount of shear transferred from one end of the crack to

other.

Following are the input data required to create the material model for concrete in ANSYS.

Elastic Modulus, (Ec)

Poisson’s Ratio, (ν)

Ultimate Uniaxial compressive strength, (fc’)

Ultimate Uniaxial tensile strength, (ft)

Shear transfer coefficient for opened crack, (β0)

Shear transfer coefficient for closed crack, (βc)

ACI equations are used to find out the input values. Finally, the values are converted to SI units

and taken as input. The equations are as follows,

Ec = 5000 √ fc’

ft = 0.7√ fc’

Where, Ec, fc’ and ft are in MPa.

Peak Compressive Stress

Strain at Max.Compressive Stress, ε0

Compression Softening

fc’

fc

Tension

εcu

E0

ε

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y/4 y/4 y/4 y/4

fc

’

0.3fc

’

o 1

y = o- 1

Poisson’s ratio for concrete is assumed to be 0.2 for all the beams. Damien Kachlakev et.al.

Conducted numerous investigations on full-scale beams and they found out the shear transfer

coefficient for opened crack as 0.2 and for closed crack as 1.

Even though the above parameters are enough for the ANSYS non-linear concrete model, it is

better to keep a stress-strain curve of concrete as a backbone for achieving accuracy in results.

Fig. 8.6 Simplified Compressive Uniaxial Stress-Strain Curve for Concrete

The stress-strain curve for concrete can be constructed by using the Desayi and Krishnan

equations. Multi-linear kinematic behaviour is assumed for the stress-strain relationship of

concrete, which is shown in Fig. 8.6. It is assumed that the curve is linear up to 0.3 fc’.

Therefore, the elastic stress-strain relation is enough for finding out the strain value.

ε1 = fc’1/ Ec = (0.3 fc’)/ Ec ---------------------------- (1)

The Ultimate strain can be found out from the following formula.

ε 0 = 2 fc’/ Ec ---------------------------- (2)

The total strain in the non-linear region is calculated and corresponding stresses for the strains

are found out by using the following formula.

fc(2 , 3 & 4) = (Ec ε)/(1+ (ε / ε 0)2) ---------------------------- (3)

The above input values are given as material properties for concrete to define the non-linearity.

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8.10. Failure Criteria for Concrete

ANSYS non-linear concrete model is based on William-Warnke failure criteria. As per the

William-Warnke failure criteria, at least two strength parameters are needed to define the failure

surface of concrete. Once the failure is surpassed, concrete cracks if any principal stresses are

tensile while crushing occurs if all the principal stresses are compressive. Tensile failure consists

of a maximum tensile stress criterion. Unless plastic deformation is taken into account, the

material behaviour is linearly elastic until failure.

When the failure surface is reached, stresses in that direction have a sudden drop to zero,

provided there is no strain softening neither in compression nor in tension. This indicates that the

descending portion in strain-strain curve of concrete is not considered in ANSYS non-linear

concrete model.

Fig. 8.7 3-D Failure Surface for Concrete

A three-dimensional failure surface for concrete is shown in Fig. 8.7. The most significant non-

zero principal stresses are in the x and y directions respectively. Three failure surfaces are shown

as the projections on the σxp-σyp plane. The mode of failure is the function of the sign of σZP

(principal stress in Z direction). For example, if σxp and σyp, both are negative (compressive) and

σZP is slightly positive (tensile), cracking would be predicted in a direction perpendicular to σZP.

However, if σZP is zero or slightly negative, the material is considered as crushed.

σZP >0(cracking)

σZP= 0(crushing)

Cracking

Cracking Cracking

σxp

fc

’

fc’

σyp

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8.11 NON-LINEAR MATERIAL MODEL FOR STEEL

Steel reinforcement in the experimental beams is constructed with typical Fe 415 Grade steel

reinforcing bars. Properties like young’s modulus and yield stress, for the steel reinforcement

used in this FEM study are found out by conducting the required tests on the sample specimens.

The steel for the finite element models is assumed to be an elastic-perfectly plastic material and

identical in tension and compression. Poisson’s ratio of 0.3 is used for the steel reinforcement in

this study Fig. 8.8 shows the stress-strain relationship used in this study.

Fig 8.8 Stress strain curve for steel

8.12 MODELLING THE GEOMETRIC SHAPE

The dimensions of the full-size beams are 150 mm x 230 mm x 2300 mm. By taking advantage

of the symmetry of the beams; a quarter of the full beam is modeled. In the ideal case of

Composite beams, the bond between the concrete and steel plays a major role. There will always

be a slip at the steel concrete interface. However, in this study, perfect bond between materials is

assumed. By using the Merge option in ANSYS, the coinciding nodes of the C-R sheet and

concrete are shared and thus composite action is assured.

8.13 FINITE ELEMENT DISCRETIZATION

The first step in finite element analysis after the creation of the model is meshing. In other

words, the model is divided into a number of finite elements, and after loading, stress and strain

are calculated at integration points of these small elements. An important step in finite element

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modeling is the selection of the mesh density. A convergence of results is obtained when an

adequate number of elements are used in a model. The accuracy of the results is directly

proportional to the number of elements chosen. But if the number of elements goes beyond a

limit, the running time to get a solution becomes more and Convergence problems also arise.

Thus there are an optimum number of elements using which we get reliable and accurate results.

The Fig. 8.9 shows the meshed model of Composite beams.

Fig. 8.9 Trough plate, braces, studs & reinforcements

8.14 ANALYSIS OF COMPOSITE BEAMS

8.14.1 General

The Composite beams are analysed both in the linear stage and non-linear stage. In the

serviceability stage, deflections are found out. For the analysis, one-fourth of the full-scale model

is used due to the symmetry of the beams.

8.14.2. Loading and Boundary Conditions

The full scale models are tested in two point loading. Due to the symmetric nature of

COMPOSITE beam one-fourth model is analysed. A steel plate of 10mm thick and

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100mmx75mm cross section is provided at the support to avoid the concentration of stresses.

Moreover, a single line support is placed under the centerline of the steel plate to allow rotation

of the plate. In the quarter model, as the two sides of the beam are continuous, the displacement

in the direction perpendicular to the plane is arrested. The one fourth models with applied load

and boundary conditions are shown in Fig. 8.2.

8.14.3 Application of Loads

The load application in failure analysis is entirely different from the conventional way of

applying loads. In conventional analysis, the results can be obtained for the applied load in the

structure. The structure never fails and it responds to the load applied to the structure. The loads

are known initially and response is predicted for the applied load. But, in failure analysis, the

load that causes failure in the structure is not known. In order to find out the failure load, a

failure criterion is specified for the materials. Then the load is applied on the structure. The

application of load consists of two stages.

Load step

Sub step

The load at which the structure may fail can be assumed arbitrarily. A load of three to four times

greater than the assumed failure load is considered. This heavy load is subdivided into number of

small loads. This small load is called” Load step”. Each load step is solved gradually and the

solution is obtained for each load step. The experimental set up for the analysis of Composite

beams is shown in Fig.8.10.

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Fig. 8.10 One-fourth model after applying boundary conditions

Fig. 8.11 Experimental Test set up

To get accurate results, the load step is further divided into a number of small load increments.

Inside every load step, this small load increment is solved and the solutions are obtained. This

small load increment is called “sub step”. When the displacement gets exceeded due to unstable

model, the solution stops. The total load at this point is the failure load of the structure. The load

steps and sub steps are shown in Fig. 8.12.

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Sub steps

Loadstep1

step1 1

Fig. 8.12 Load Steps and Sub Steps

The ANSYS 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. A force convergence criterion with a tolerance limit of 5% was

adopted for avoiding the divergence problem.

8.15 COMPUTATION RESOURCES

In this thesis work, Pentium IV processor with 1.0GB RAM 9 was used to perform the analysis.

The time taken for modeling and analyzing each COMPOSITE beam was around one to two

hours. Each problem requires around 21MB for database memory.

= Load steps

= Sub Steps

Load step2

1

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Fig. 8.13 Full- Scale model of Composite beam

Fig. 8.14 One-Fourth-model idealization

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8.16 COMPARISON OF RESULTS

8.16.1 Comparison

Load deflection curve comparison between Experimental and ANSYS results are presented

Table 8.2 Load deflection Values of Experiment and ANSYS

T1B1-100mm C/C

S.NO LOAD ( kN) EXP DEF (mm) ANSYS (mm)

1 0.00 0.00 0.00

2 6.71 0.32 0.49

3 13.43 0.92 0.81

4 20.14 1.44 1.29

5 26.86 2.06 1.72

6 33.57 2.90 2.22

7 40.28 3.10 2.62

8 47.00 3.53 3.11

9 53.71 4.00 3.51

10 60.43 4.49 4.02

Fig. 8.15 Load deflection curve Comparison for 100 mm spacing of Bracings

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Table 8.3 Load deflection Values of Experiment and ANSYS

T2B1-150mm C/C

S.NO LOAD EXP DEF (mm) ANSYS (mm)

1 0.00 0.00 0.00

2 6.71 0.29 0.31

3 13.43 0.65 0.54

4 20.14 0.96 0.79

5 26.86 1.40 1.16

6 33.57 2.16 1.66

7 40.28 2.62 2.21

8 47.00 3.27 2.74

9 53.71 3.77 3.36

0.000

10.000

20.000

30.000

40.000

50.000

60.000

0.000 1.000 2.000 3.000 4.000

deflection mm

load

kN EXPERIMENTAL

ANSYS

Fig 8.16 Load deflection curve Comparison for 150 mm spacing of Bracings

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Table 8.4 Ultimate Load and Moment in Beams

BEAM TYPE ULTIMATE LOAD (kN) ULTIMATE MOMENT

(kNm)

T1B1 112.6 39.41

T1B3 130 45.5

T1B4 160 56

T2B1 112.2 39.27

T2B3 112 39.2

T2B4 110 38.5

8.17 CONCLUSIONS

The Load-deflection curves for the beams analysed with ANSYS to that of Experimental

values, ANSYS gives less deflection.

ANSYS always predicts lesser deflection than the experimental work. In the

serviceability stage, the percentage of variation between the two is in the range of 5 -

20%. This is because of neglecting the strain-softening region in the stress-strain curve

for the material model in concrete.