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CHAPTER VIII

ANALYTICAL MODELLING

NON LINEAR FINITE ELEMENT ANALYSIS OF RPC BEAMS AND

COLUMNS USING ANSYS SOFTWARE

8.0 INTRODUCTION

The nonlinear response of RC structures can be computed using the

finite element method (FEM). This analytical method, gives the

interaction of different nonlinear effects on RC structures. The success

of analytical simulation is in selecting suitable elements, proper

material models and in selecting proper solution method. The FEM is

well suited modeling composite material with material models. The

various finite element software packages available are ATENA,

ABAQUS, Hypermesh, Nastran, ANSYS etc. Amongst the available finite

element package for the non-linear analysis ANSYS (Analysis System),

an efficient finite element package is used for of the present study.

This chapter discusses the procedure for developing analysis model in

ANSYS v11.0 & the procedure for nonlinear analysis of Reactive Powder

Concrete structural components is discussed. This chapter discusses

the models and elements used in the present analysis of ANSYS. The

graphical user interface in ANSYS provides an efficient and powerful

environment for solving many anchoring problems. ANSYS enables

virtual testing of structures using computers, which is the present

trend in the research and development world. Concrete is represented

as solid brick elements; the reinforcement provided by fibre is

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simulated by bar elements. All the necessary steps to create these

models are explained in detail and the steps taken to generate the

analytical load-deformation response of the beam are discussed. The

results from the finite element model are compared with the

experimental results by load deformation plots and cracking patterns.

8.1 DESIGN DETAILS OF BEAM AND COLUMN

The beams designed for Finite Element Model (FEM) in ANSYS 11.0

took up the experimental study for the analytical study. The design

details of the beam are shown in Fig. 8.4.1. The same beam is modeled

in ANSYS using the following procedure

The columns designed for the experimental study was taken up for the

analytical study by FEM in ANSYS 11.0. The design details of the

column were shown in Fig.8.4.2. The same column is modeled in

ANSYS using the following procedure.

To create the finite element model in ANSYS there are multiple

tasks that are to be completed for the model to run properly. Models

can be created using command prompt line input or the Graphical User

Interface (GUI). For this model, the GUI was utilized to create the

model. This section describes the different tasks and entries into used

to create the FE calibration model.

Three basic steps involved in ANSYS include:

Preprocessing:

Building FEM model

Geometry Construction

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Mesh Generation (right element type!)

Application of Boundary and load conditions

Solving:

Submitting the model to ANSYS solver

Post processing:

Checking and evaluating results

Presentation of results- Stress/Strain contour plot, Load deflection

plots etc.

8.2 ELEMENT TYPE USED IN THE MODEL

Concrete generally exhibits large number of micro cracks, especially,

at the interface between coarse aggregates and mortar, even before it is

subjected to any load. The presence of these micro cracks has a great

effect on the mechanical behavior of concrete, since their propagation

during loading contributes to the nonlinear behavior at low stress levels

and causes volume expansion near failure. Some micro cracks may

develop during loading because of the difference in stiffness between

aggregates and mortar. Since the aggregate-mortar interface has a

significantly lower tensile strength than mortar, it constitutes the

weakest link in the composite system. This is the primary reason for

the low tensile strength of concrete. The response of a structure under

load depends largely on the stress-strain relation of the constituent

materials and the magnitude of stress. The stress-strain relation in

compression is of primary interest because mostly for compression

members are cast using concrete. The actual behavior of concrete

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should be simulated using the chosen element type. For the present

type of model solid 65 elements was chosen. The element types for this

model are shown. The Solid65 element was used to model the concrete.

This element has eight nodes with three degrees of freedom at each

node – translations in the nodal x, y, and z directions. This element is

capable of plastic deformation, cracking in three orthogonal directions,

and crushing. A schematic representation of the element is shown in

Fig 8.1.

Fig 8.1 Solid 65 Elements in ANSYS

The element has eight nodes having three degrees of freedom at

each node: translations in the nodal x, y, and z directions. Up to three

different rebar specifications may be defined. The solid capability may

be used to model the concrete while the rebar capability is available for

modeling reinforcement behavior.

Fibre reinforcement is modeled through Link 8. Link 8 is a

uniaxial tension-compression element with three degrees of freedom at

each node: translations in the nodal x, y, and z directions as shown in

Fig. 8.2.

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Fig. 8.2 Link 8 Element in ANSYS

8.3 COMBIN 14 ELEMENT

COMBIN14 has longitudinal or torsional in 1-D,2-D, or 3-D

applications(Fig.8.3). The longitudinal spring-damper option is a

uniaxial tension-compression element without three degrees of freedom

at each node x, y, and z directions. No bending or torsion is considered.

The torsional spring-damper option is a purely rotational element with

three degrees of freedom at each node: rotations about the nodal x, y

and z axes. No bending or axial loads are considered.

Fig. 8.3 COMBIN 14 Element in ANSYS

8.4 REAL CONSTANT

Real constant Set 1 is used for the Solid65 element to define the

geometrical parameters of embedded with fibres. A value of zero was

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entered for all real constants for solid65. Real Constant set 4 and 5 are

defined for COMBIN 14 element and Link8 (Fig.8.4). Value for spring

constant 114.78 for COMBIN 14 and for Link 8, the bilinear stress –

strain for fibres were entered as per Fig.8.8a.

Fig. 8.4 Real Constant Values For Various Elements Types

8.5 MATERIAL PROPERTIES

Parameters needed to define the material models were obtained

from experimental study. Some of the parameters were obtained from

the literature. As seen in Fig 8.5, there are multiple parts of the

material model for each element. Concrete Material Model Number 1

refers to the Solid65 element. The Solid65 element requires linear

isotropic and multilinear isotropic material properties to properly model

concrete. The multilinear isotropic material uses the von Mises failure

criterion along with the Willam and Warnke (1974) model to define the

failure of the concrete. Ex is the modulus of elasticity of the concrete

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(E), and PRXY is the Poisson’s ratio (ν). The material properties given in

the present model is shown Table 8.1.

Fig. 8.5 Material Property given to SOLID65

Table 8.1 Material Property Given for the Calibration Model

The compressive uniaxial stress-strain relationship for the

concrete model was obtained by idealizing the stress strain curve

obtained from the experimental study. The multilinear curve is used to

help with convergence of the nonlinear solution. A typical idealized

multilinear stress strain curve for RPC is shown in (Fig. 8.6).

Material property RPC concrete

EX (MPa) 39 GPa to 48.5 GPa

Poissons ratio 0.23

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Fig. 8.6 A Typical Stress Strain Curve For RPC with 2% 13mm fibre for Non

Linear Analysis

8.6 Failure surface models of concrete

The model is capable of predicting the failure of concrete materials.

Both cracking and crushing failure modes are to be accounted for. The

two input strength parameters i.e., ultimate uniaxial tensile and

compressive strengths are needed to define a failure surface for the

concrete. Willam and Warnke (1974) developed a widely used model for

the triaxial failure surface of unconfined plain concrete. The failure

surface in principal stress-space is shown in Fig 8.7a&b. The

mathematical model considers a sextant of the principal stress space

because the stress components are ordered according to σ1≥σ2≥σ3.

These stress components are the major principal stresses.

The failure surface is separated into hydrostatic (change in volume)

and deviatory (change in shape) sections as shown in Fig. 8.7b. The

hydrostatic section forms a meridianal plane which contains the

equisectrix σ1 =σ 2 =σ 3 as an axis of revolution (see Fig. 8.7b). The

deviatory section in Fig.8.7a&b lies in a plane normal to the equisectrix

(dashed line in Fig. 8.7b).

0

20

40

60

80

100

120

140

160

180

0 0.005 0.01 0.015 0.02

Co

mp

ress

ive

str

ess

(MP

a)

Strain

1%13mm 2%13mm

182

Fig. 8.7 Fig. 8.7b

Fig. 8.7a & 8.7b Failure Surface of Plain Concrete Under Triaxial Conditions

(Willam and Warnke 1974)

The Willam and Warnke (1974) Fig.8.7 mathematical model of the

failure surface for the Concrete has the following advantages:

1. Close fit of experimental data in the operating range;

2. Simple identification of model parameters from standard test

data;

3. Smoothness(e.g. continuous surface with continuously varying

tangent planes);

4. Convexity (e.g. monotonically curved surface without inflection

points).

Based on the above criteria, a constitutive model for the concrete

suitable for FEA Implementation of the Willam and Warnke material

model in ANSYS requires that nine different constants be defined.

These 9 constants are

1. Shear transfer coefficients for an open crack;

2. Shear transfer coefficients for a closed crack;

3. Uniaxial tensile cracking stress;

4. Uniaxial crushing stress (positive);

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5. Biaxial crushing stress (positive);

6. Ambient hydrostatic stress state for use with constants 7 and 8;

7. Biaxial crushing stress (positive) under the ambient hydrostatic

stress state(constant 6);

8. Uniaxial crushing stress (positive) under the ambient hydrostatic

stress state(constant 6);

9. Stiffness multiplier for cracked tensile condition.

Typical shear transfer coefficients range from 0.0 to 1.0, with 0.0

representing a smooth crack (complete loss of shear transfer) and 1.0

representing a rough crack (no loss of shear transfer). Convergence

problems occur when the shear transfer coefficient for the open crack

drop below 0.2. No deviation of the response occurs with the change of

the coefficient. Therefore, the coefficient for the open crack was set to

0.65 .The uniaxial tensile cracking stress is based upon the modulus of

rupture. For the present model, the uniaxial tensile cracking stress was

given as varies between 7MPa to 12 MPa for RPC concrete with various

dosages of steel fibre (Fig.8.11).

Numerous general purpose computer programs are available for the

analysis of reinforced concrete structures. However, modeling the effect

of fibres on concrete, fibre bond/slip and the bridging effects across has

still not taken into account in FEM analysis in SFRC structures in any

of these programs. Padmarajaiah.S.K. et al., (2002)62 developed a

model for finite element assessment of flexural strength of prestressed

concrete beams with fibre reinforcement.

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In Finite element studies using ANSYS to simulate the effect of steel

fibres in a concrete matrix its behavior has been decomposed into two

components. Firstly, the multiaxial stress state in concrete failure

surface and the stress-strain properties. Secondly, the fibres along the

beam length have also been modeled as truss elements explicitly in

order to capture the crack propagation resistance through bridging

action. Tension stiffening and bond slip between concrete and fibre

reinforcement have been considered in the model using Linear springs.

All the flexure critical beams having fibre over the full depth or partial

depth are observed to have failed in flexure with fibre pull-out across

the cracks, rather than through yielding of the fibre. In order to

simulate the effect of steel fibres in a concrete matrix, its behavior has

been decomposed into two components. The multiaxial state of stress

in concrete due to the presence of fibre has been simulated by

modifying the failure surface of concrete and a typical stress strain is

shown in Fig.8.6. The bridging action of fibres resisting crack

propagation has been modeled using three-dimensional LINK8 (truss)

elements explicitly. Material Model Number 4 refers to the Link8

element. The Link8 element is being used for all the steel fibre

reinforcement in the concrete. The model requires the modulus of

elasticity of steel Es as 200GPa and Poisson’s ratio (0.3). The fraction of

the entire volume of the fibre present along the entire longitudinal axis

of the longitudinal beam has been modeled explicitly, in the flexure

zone. In the case of beams containing fibres, were modeled only over

half the depth in the flexure zone. (Fibres in shear were ignored) The

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effect of tension stiffening and bond-slip at the interface between these

fibre elements and concrete elements has also been simulated using

COMBIN 14 (linear springs) elements with appropriate properties to

capture the effects of bond, bond-slip and peel off.

8.7 Modelling the Flexure and Compression Specimen

The beam was modeled as a volume. The zero values for the Z

coordinates coincide with the center of the cross-section for the

concrete beam. To obtain good results from the Solid65 element, a

mapped mesh is used. Selection of element size is an important factor

in the finite element analysis of concrete structures. It has been

reported by Padmarajaiah,et.al.,(2002), that the smallest element

dimension in an FE model is controlled by the size of coarse aggregate

used. The mesh size used for the study of angle section in flexure and

compression is 10mm x 10mm. The compression member with various

heights 600mm, 400mm, 300mm and 200mm were simulated in Ansys

using SOLID 65, LINK 8 and COMBIN 14(Fig.8.8).

The command ‘merge items’ merges separate entities that have the

same location. These items will then be merged into single entities.

Caution must be taken when merging entities in a model that has

already been meshed because the order in which merging occurs is

significant. Merging key points before nodes can result in some of the

nodes becoming “orphaned”; that is, the nodes lose their association

with the solid model. The orphaned nodes can cause certain operations

(such as boundary condition transfers, surface load transfers, and so

on) to fail. Care must be taken to always merge in the order in which

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the entities appear. All precautions were taken to ensure that

everything was merged in the proper order. Also, the lowest number

was retained during merging.

Fig.8.8. (a) & (b) Rheological representation of a FRC element by

Padmarajaiah.S.K., and Ananth Ramasamy(2002)

8.8 LOADING AND BOUNDARY CONDITIONS FOR BEAM AND

COLUMN

Displacement boundary conditions are needed to constrain the

model to get a unique solution. To ensure that the model acts the same

way as the experimental beam, boundary conditions need to be applied

at where the supports and loadings exist. Loading applied was applied

at loading point. Since it is a quarter beam model, at one end of the

beam support, Uy is restrained to ensure roller support conditions and

other end is restrained against x direction ensuring the symmetry

boundary conditions along the longitudinal section. Similarly, along the

z direction all the nodes are constrained ensuring symmetry boundary

condition along cross section. The loading was applied on the nodes at

one-third point. The range of load applied for flexure was between 10kN

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to 25 kN for various dosages of RPC’s. The loading was applied at a

distance of 167mm from the support for span to depth ratio 7.5 for

flexure.

The loads range from 100kN to 170 kN for RPC compression

members. Similarly, the loading was applied at the centroid for the

compression members. The bottom nodes are restrained in the

longitudinal direction.(Fig.8.9 a & b)

Fig.8.9 (a)&(b) Loading Conditions in Flexure and Compression specimen model

The finite element model for this analysis is a simple beam under

transverse loading. For the purposes of this model, the Static analysis

type is utilized. The Solution Controls command dictates the use of a

linear or non-linear solution for the finite element model. In the

particular case considered in this thesis, the analysis is small

displacement and static type. The time at the end of the load step refers

to the ending load per load step. The commands used to control the

solver and output is shown in Table 8.2 & 8.3.

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Table 8.2 Commands used for the Nonlinear Algorithm

The commands used for the nonlinear algorithm and convergence

criteria are shown in Table 8.3. All values for the nonlinear algorithm

are set to defaults.

Table 8.3 Nonlinear Algorithm and Convergence Criteria

Parameters

8.9 Techniques for Nonlinear 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 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 ANSYS program

(ANSYS v.11) 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, 8.10 shows the

189

use of the Newton-Raphson approach in a single degree of freedom

nonlinear analysis.

Fig. 8.10 Newton-Raphson iterative solution (2 load increments) (ANSYS v11.0)

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 (ANSYS v11.0). In this study, for the reinforced concrete

solid elements, convergence criteria were based on force and

displacement, and the ANSYS program initially selected the

convergence tolerance limits. It was found that convergence of solutions

for the models was difficult to achieve due to the nonlinear behavior of

reinforced concrete. Therefore, the convergence tolerance limits were

increased to a maximum of 5 times the default tolerance limits (0.1%

for force checking and 1% for displacement checking) in order to obtain

convergence of the solutions.

190

8.10 LOAD STEPPING AND FAILURE DEFINITION FOR FE MODELS

For the nonlinear analysis, automatic time stepping in the ANSYS

program predicts and controls load step sizes. Based on the previous

solution history and the physics of the models, if the convergence

behavior is smooth, automatic time stepping will increase the load

increment up to a selected maximum load step size. If the convergence

behavior is abrupt, automatic time stepping will bisect the load

increment until it is equal to a selected minimum load step size. The

maximum and minimum load step sizes are required for the automatic

time stepping.

8.11 BEHAVIOUR OF CRACKED CONCRETE

The nonlinear response of concrete is often dominated by

progressive cracking which results in localized failure. So it is

important to study the behaviour of concrete at the cracked zone.

8.11.1Description of a Cracked Section

The structural member has cracked at discrete locations where the

concrete tensile strength is exceeded. At the cracked section all tension

is carried by the steel fibre reinforcement. Tensile stresses are,

however, present in the concrete between the cracks, since some

tension is transferred from steel fibre to concrete through bond. The

magnitude and distribution of bond stresses between the cracks

determines the distribution of tensile stresses in the concrete and the

reinforcing steel fibre between the cracks. Additional cracks can form

between the initial cracks, if the tensile stress exceeds the concrete

tensile strength between previously formed cracks. The final cracking

191

state is reached when a tensile force of sufficient magnitude to form an

additional crack between two existing cracks can no longer be

transferred by bond from steel fibre to concrete.

Primary cracks formation starts when the concrete reaches its

tensile strength. The size, orientation and placement of the steel fibre

controls the extent and number of cracks at the primary cracks. The

concrete stress drops to zero and the steel fibre carries the entire

tensile force. The concrete between the cracks, however, still carries

some tensile stress, which decreases with increasing load magnitude.

This drop in concrete tensile stress with increasing load is associated

with the breakdown of bond between steel fibre and concrete. At this

stage a secondary system of internal cracks, called bond cracks,

develops around the steel fibre, which begins to slip relative to the

surrounding concrete. Since cracking is the major source of material

nonlinearity in the serviceability range of concrete structures, realistic

cracking models need to be developed in order to accurately predict the

load-deformation behavior of concrete members. The selection of a

cracking model depends on the purpose of the finite element analysis. If

overall load deflection behavior is of primary interest, without much

concern for crack patterns and estimation of local stresses, the

"smeared" crack model is probably the best choice. If detailed local

behavior is of interest, the adoption of a "discrete" crack model might

be necessary. Unless special connecting elements and double nodes are

introduced in the finite element discretization of the structure, the well

established smeared crack model results in perfect bond between steel

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and concrete, because of the inherent continuity of the displacement

field.

8.11.2 Modelling of Crack in Concrete

The process of crack formation can be divided into three stages. The

uncracked stage is before the limiting tensile strength is reached. The

crack formation takes place in the process zone of a potential crack

with decreasing tensile stress on crack face due to crack bridging effect.

Finally, after a complete release of the stress, the crack opening

continues without the stress. The tension failure of concrete is

characterized by a gradual growth of cracks, which join together and

eventually disconnect larger parts of the structure. It is usually

assumed that cracking formation is a brittle process and that the

strength in tension loading direction abruptly goes to zero after such

cracks have formed.

The discrete approach is physically attractive but this approach

suffers from few drawbacks, such as, it employs a continuous change

in nodal connectivity, which does not fit in the nature of finite element

displacement method; the crack is considered to follow a predefined

path along the element edges and excessive computational efforts are

required. The second approach is the smeared crack approach. In this

approach, the cracks are assumed to be smeared out in a continuous

fashion. Within the smeared concept, two options are available for

crack models: the fixed crack model and the rotated crack model. In

both models, the crack is formed when the principal stress exceeds the

tensile strength. It is assumed that the cracks are uniformly distributed

193

within the material volume. The element includes a smeared crack idea

for handling cracking in tension zones and a plasticity algorithm to

account for the possibility of concrete crushing in compression zones.

Each element has eight integration points at which cracking and

crushing checks are performed (Fig.8.11).

Fig. 8.11 Gaussian Integration Points in solid 65 and Model of Crack in ANSYS

The element behaves in a linear elastic manner until either of the

specified tensile or compressive strengths is exceeded. Cracking (or

crushing) of an element is initiated once one of the element principal

stresses, at an element integration point, exceeds the specified tensile

or compressive concrete strength. The formation of a crack is achieved

by the modification of the stress-strain relationships of the element to

introduce a plane of weakness in the principal stress direction.

8.12 RESULTS &DISCUSSIONS OF FE ANALYSIS OF RPC BEAMS

The RPC beam modeled in ANSYS 11.0 is compared with the

experimental results. Typical RPC beam modeled in Ansys is shown in

Figs.8.12 &8.13. The loading applied was 10kN to 22 kN at a distance

of 116mm from the support.

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8.12.1 Behaviour at Different Load Stages-RPC Beams

The nonlinear analysis was procedure adopted same as that

described in the previous section. The load deformation response

obtained and the crack pattern of the beams was shown in Fig. 8.17

and 8.19.

Table 8.4 gives the typical Load-deflection response obtained from

the tests along with the FE results for RPC beams under pure bending.

From the load deflection response (Fig.8.14 a-d), it is clear that the

initial portion of the load deflection curve is in close agreement with the

experimental findings. Addition of fibres increased the cracking and

ultimate strength and reduces the deformational characteristics. As

seen from the load-deflection curves the FEM load response prediction

is close to the experimental results in the working load range. However,

as the load reached the peak it is seen that the FEM results are stiffer

than the corresponding test results. An examination of the Table 8.4

reveals that the cracking values for these beams were almost the same

for various fibre contents. This may be due to the inability to account

for the actual heterogeneity existing in the test beam is not simulated

in the analytical model as the same property is assigned to all concrete

elements for various fibre contents. The present analytical model

predicts well the behavior of the beam similar to experimental beam. In

the pre peak regime, flexural cracks development in the experiment is

quite smooth whereas in the numerical solution curve it is flat and

“sudden”. This is because the ANSYS cracking option does not include

properly the tensile stress relaxation. That fact does not generally affect

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the solution since the tensile steel capacity is available; therefore, the

sudden stress drop at the cracking points explains the discrepancy

between the two curves at the beginning of nonlinear process.

Table 8.4 shows the typical finite element results comparison with

the test results at five stages of loading for the selected beam. The first

stage was taken before the crack initiation (≈20% peak load), the

second stage at initiation of first flexure crack, the third and fourth

stage at a working load level taken to be the peak load/1.5 and the last

stage at the peak load. From the results, it is observed for all the

beams, the load and deflection before crack and at first crack in the

analysis were very much in agreement with the experimental values. At

working load level and at the peak load level the values of load obtained

from the FEM were close to the experimental results. However, the

deflection obtained from FEM was less than those in the test at working

load level, at the peak. One possible reason for the lower deflection may

be because linear springs were used to simulate the bond slip where as

the behavior may be highly nonlinear at these load levels. The ratio of

load obtained in FE analysis to experimental loads ranged from 0.68 to

1.2 for 6mm fibres, 1.08 to 0.8 times for 13mm fibres and 1.08 to 1.2

times for the hybrid fibres i.e., Combination of 6mm and 13mm fibres.

The Bending stress distribution across the beam cross section shows

the increase in tensile stress in the tensile zone compared with the

other High Performance Concrete. The Neutral axis lies at 23.7mm from

the bottom of the beam. The theoretical calculations also confirm the

neutral axis but the bending stresses are one fourth of the theoretical

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bending stress in compression as well as in tension (Fig.8.15. and

Fig8.16)

Fig. 8.12 Fig. 8.13 Fig. 8.12 Details of Beam

Fig. 8.13 3D beam modelled in ANSYS

Fig.8.14 (a) Fig. 8.14(b)

Fig.8.14(c) Fig.8.14(d)

Fig.8.14(a-c) Flexural Stress-Strain curves for different dosages of fibres

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5

Load

(kN

)

Deflection (mm)

Angle flexure 1f6 Angle Flexure ansys 1f6

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4

Load

(kN

)

Deflection (mm)

Angle Flexure 2f6 Angle Flexure ansys 2f6

0 2 4 6 8

10 12 14 16 18

0 0.1 0.2 0.3 0.4 0.5 0.6

Load

(kN

)

deflection (mm)

2%13mm Experimenral

Ansys

0 2 4 6 8

10 12 14 16 18

0 0.1 0.2 0.3 0.4 0.5 0.6

Load

(kN

)

deflection (mm)

1%6mm+1%13mm Experimenral

Ansys

197

Fig.8.15 Typical Bending Stress distribution across the cross section of the

beam for 2% 6mm fibre and 2% 13mm

Fig.8.16 Theoretical Bending Stress distribution across the cross section of the beam for 2% 6mm fibre and 2% 13mm

198

Table 8.4 Comparison of FEM results at various stages for the RPC

beams under Flexure.

Specimen id

Stages

FEM Expt Specimen

id

FEM Expt.

Load

(kN)

δ(mm) Loa

d (kN)

δ

(mm)

Load

(kN)

δ

(mm)

Load

(kN)

δ

(mm)

1f6 1 1.2 0.28 0 0 2f6 1.2 0.062 2 0.1

2 2.4 0.336 2 0.12 2.4 0.124 4 0.21

3 4.2 0.392 4 0.21 4.2 0.218 6 0.36

4 6.9 0.448 6 0.3 6.9 0.357 8 0.53

5 9.45 0.504 8 0.38 9.45 0.489 10 0.9

3f6 1 1.6 0.092 2 0.28 1f13 1.6 0.092 2 0.26

2 3.2 0.183 4 0.52 3.2 0.183 4 0.36

3 4.8 0.274 6 0.73 4.8 0.274 6 0.47

4 6.4 0.366 7 0.88 6.4 0.366 8 0.69

5 9.6 0.458 8 0.99 9.6 0.458 9 0.74

2f13 1 3 0.165

2

0.11 1f61

f13

2.1 0.11 2 0.15

2 4.5 0.247 5 0.3 4.74 0.24 4 0.25

3 6 0.33 8 0.41 7.68 0.39 6 0.35

4 7.5 0.413 12 0.76 9.78 0.49 8 0.43

5 14.32 0.579 18 1.13 10.7 0.54 11.5 0.58

1f62f13 1 2.05 0.09 2 0.04

2 4.18 0.19 4 0.1

3 6.31 0.29 8 0.23

4 10.8 0.51 14 0.48

5 14.36 0.68 18 0.71

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Fig.8.17 a-b Crack Pattern at the 46thand 51st (12kN) load Step

Fig.8.17c Fig.8.18

Fig.8.4.17c Crack Pattern at the failure stage

Fig.8.18 Bending Stresses for 2% 13mm 55th(12kN) load Step

8.12.2 Initial Flexural Crack and Formation of Diagonal Crack

The cracking pattern(s) in the beam was obtained using the

Crack/Crushing plot option in ANSYS (v11.0). Vector Mode plots must

be turned on to view the cracking in the model. Crack pattern of a

typical flexure beam of fibre dosage 3%6mm is discussed in the

following paragraph.

The initial flexural cracks formed in the constant bending moment

region (at the load of 9.9N for 3% 6mm) as reckoned from the link in

the initial segment of the load deformation curve. As load increases, the

depth of the flexural cracks in the constant bending moment region

200

increased and more cracks appeared adjoining the existing cracks.

However, with further increase of load the cracks stopped growing in

length and additional cracks consisting of flexural cracks formed in the

constant bending moment region.

(a) Crack pattern at load of 2450N (b) Crack pattern at load step 2477N

(c)Crack pattern at load step 2800N (d) Deformed beam with cracks

Fig. 8.19 (a-d) Crack pattern at Various Stages 2% 13mm-RPCC BEAM

Fig. 8.20. Typical Crack pattern -RPC BEAM

After the load stage of 41 the cracking increases in the flexural zone.

Many adjoining diagonal cracks extend from the loading point towards

the bottom nodes and crushing of concrete occurs below the loading

plate. This is followed dropping of load and failure on a brittle fashion

201

with little increase in deflection beyond this stage in tests (Fig.8.17)

However in case of finite element analysis because of withholding the

crushing option, the deflection starts further increasing and crushing of

concrete in the constant bending region at the failure load of 11200N.

Therefore, large number of flexure crack develops at the constant

moment region. The beam no longer can support additional load as

indicated by an insurmountable convergence failure and the analysis is

interrupted. The crack patterns of beam in pure bending are shown in

Figs. 8.17 & 8.19.

8.12.3 Behaviour of Compression Members At different Load

Stages

The behaviour of compression members with various aspect ratios

such as 2.5, 3.75, 5 and 7.5 are compared in the Tables 8.5-8.8. The

predicted values from FE analysis mostly correlates well with the

experimental results. The stress-strain curves show the slopes are the

same at the pre-peak loads. Fig.8.21 gives the stress – strain curves for

RPC specimens with various fibre contents for an aspect ratio of 7.5.

The bar charts (Fig.8.24) shows the compressive stresses of RPC

specimens with various aspect ratios. The figures indicate the results

are well correlating with the experimental results. The Fig.8.22 shows

the progress of compressive stress at various load stages. The

Maximum stress distribution along the diagonal of the specimen

coincides with the failure pattern of the compressive specimen during

experiments (Fig.8.23). The propagation of the critical diagonal crack

202

provokes growth of concrete plastic strain and relevent material

softening.

The peak loads values vary 10 to 25% more than experimental

values in all the cases. The present analytical model predicts well the

behavior of the compression specimen similar to the experimental

specimen. In pre peak regime, the maximum stresses are formed along

the diagonals indicating the buckling of the specimen. The failed

specimen (Fig.8.23) confirms the orientation of failure as that of the

predicted stresses (Fig.8.22).

Fig.8.21 Compressive stress-strain curves for different dosages of fibres for

aspect ratio 7.5.

0

20

40

60

80

100

120

140

0 2000 4000 6000 8000 10000 12000 14000 16000

Com

pre

ssiv

e S

tress (

MPa)

Compressive strain

Height of angle 600mm

2f6H600 3f6H600 1f13H600 2f13H600 1f61f13H600 1f62f13H600 1f6H600 ansys2f6-600 ansys-3f6-600 ansys - 1f13-600 ansys-2f13 ansys-1f6+1f13-600 ansys 1f6+2f13-600 Ansys 2f6-600

1f6 2f6 3f6 1f13 1f61f13 2f13 1f62f13

203

Fig.8.22(a) Fig.8.22(b)

Fig.8.22( c) Fig 8.23

Fig 8.22 Compressive Stress Pattern at different Load step. Fig 8.23 Typical Diaagonal tensile failure of RPC Specimen under compression.

204

Fig.8.24a&b Compressive Stress Pattern at different aspect ratios

Fig.8.24c&d Compressive Stress Pattern at different aspect ratio

0 20 40 60 80

100 120 140 160 180

Co

mp

ress

ive

Str

ess

(M

Pa)

Ansys Experimental

0

20

40

60

80

100

120

140

160

Co

mp

ress

ive

str

ess

(MP

a)

Ansys Experimental

0

20

40

60

80

100

120

140

Co

mp

ress

ion

Str

ess

(MP

a)

Ansys Experimental

0

20

40

60

80

100

120

140

160

Co

mp

ress

ion

Str

ess

(MP

a) Ansys Experimental

205

Table 8.5 Comparison of Experimental and Ansys Results for L/d ratio 2.5

Specimen id. L/d Ansys peak

stress

peak

strain

Ansys % of increase

(MPa) (MPa) (10-6) (10-6) stress strain

1f6H200 2.5 101.4 103.3 2257 2120 -1.86 -10.58

2f6H200 2.5 114.89 124.6 2585 2200 -7.81 -14.89 3f6H200 2.5 122.36 146.7 2240 2420 -16.57 8.03

1f13H200 2.5 124.18 143.3 2273 2500 -13.34 9.98

2f13H200 2.5 139.05 153.3 2642 3000 -9.31 13.55

1f61f13H200 2.5 120.3 126.3 2058 2210 -4.72 7.38

1f62f13H200 2.5 137.53 156.7 2485 2785 -12.21 12.07

Table 8.6 Comparison of Experimental and FE Results for L/d ratio

3.75

Specimen id. L/d Ansys peak stress

Ansys peak strain

% of increase

(MPa) (MPa) (10-6) (10-6) Stress Strain

1f6H300 3.75 107.5 95.72 2132 2020 12.30 5.54

2f6H300 3.75 108.2 100 2200 2195 8.2 0.22

3f6H300 3.75 90.04 90.31 2100 2335 -0.29 -10.06

1f13H300 3.75 115.91 121.1 2482 2273 -4.25 9.19

2f13H300 3.75 140 130 2320 2550 7.69 -9.01

1f61f13H300 3.75 112.8 118.2 2200 2108 -4.60 4.36

1f62f13H300 3.75 129.51 129.5 2462 2695 0 -8.65

206

Table8.7 Comparison of Experimental and Ansys Results for L/d ratio 5.0

Table 8.8 Comparison of Experimental and Ansys Results for L/d ratio 7.5

Specimen id. L/d

peak stress

Ansys peak strain

Ansys % diff.

(Mpa)

(10-6)

Stress Strain

1f6H600 7.5 87.27 101.29 2114 2200 16.06 4.07

2f6H600 7.5 96.67 112.22 2140 2340 16.08 9.34

3f6H600 7.5 101.6 113.33 2180 2400 11.54 10.09

1f13H600 7.5 109.9 112.2 2186 2300 2.09 5.21

2f13H600 7.5 116.2 128.3 2255 2340 10.41 3.76

1f61f13H600 7.5 97.9 110.23 2084 2360 12.59 13.24

1f62f13H600 7.5 106.7 117.92 2211 2490 10.51 12.62

The failures of the compression specimens were attained by

crushing under the load plate. This was seen in all the crack

patterns which were indicated by the increased concentration of

diagonal crack.

Specimen id. L/

d

Ansys peak

stress

Ansys

Strain

peak

strain

% of

increase

(MPa) (MPa) (10-6) stress strain

1f6H400 5 76.55 88.73 1900 1765 -12.3 7.64

2f6H400 5 100.3 105.9 2557 2342 -5.3 9.18

3f6H400 5 101.4 113.3 2552 2322 -10.5 9.9 1f13H400 5 115.9 109.9 2127 1967 5.45 8.13

2f13H400 5 129.1 121.1 2682 2462 6.61 8.94

1f61f13H400 5 114.9 101.4 2532 2322 13.3 9.04

1f62f13H400 5 115.4 116.3 2585 2695 -0.77 -4.08

207

8.13 CONCLUSIONS OF ANALYTICAL STUDIES OF RPC ANGLE

SECTIONS UNDER FLEXURE AND COMPRESSION

1. Based on the points raised in numerical results and discussion

sections, the following conclusions are drawn from this numerical

research:

2. The 3D ANSYS modeling is able to properly simulate the

nonlinear behavior of RPC in Flexure and Compression.

3. ANSYS 3D concrete element is very good concerning flexural

development but poor concerning the crushing state. It may be

possible to overcome the deficiency employing a certain

multilinear plasticity model available in ANSYS but the lack of

experimental data for material parameters especially RPCs is a

drawback.

4. The concrete finite element model does not consider adequately

tension stiffening, tension softening & bond slip behaviour.

5. The load-deformation characteristics obtained from the finite

element solution was in close agreement with the experimental

results at four critical stages of loading.

6. The crack pattern at both initial and at failure predicted by FEM

was in close agreement with the experiment results, indicating

that the effect of fibres on the concrete strength and ductility and

its bridging effects in arresting crack propagation have been

suitably captured.