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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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
187
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