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174
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
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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
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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.
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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
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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
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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.