numerical investigation on the effects of embedding pvc …
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NUMERICAL INVESTIGATION ON THE EFFECTS OF
EMBEDDING PVC PIPES IN REINFORCED CONCRETE BEAMS
FINAL YEAR PROJECT SUBMITTED TO KAMPALA
INTERNATIONAL UNIVERSITY IN PARTIAL FULFILLMENT OF THE
AWARD OF THE DEGREE
Bachelor of Science in Civil Engineering.
BY
AHEEBWA HUSSIEN
1153-03104-00788
SCHOOL OF ENGINEERING AND APPLIED SCIENCES
DEPARTMENT OF CIVIL ENGINEERING
AUGUST 2019
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DECLARATION
I AHEEBWA HUSSIEN hereby declare that the work in this thesis is my own except for
the quotation and summaries which have been acknowledged. The thesis has not been
accepted neither has it ever been submitted for any award of a degree in civil
engineering in any institution.
Signature……………………………………………………………………………
NAME: AHEEBWA HUSSIEN
REG NO BSCE 1153 03104 00788
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APPROVAL
I humbly declare that this final year report has been prepared under supervision.
MR. WAFULA PETER
Signature………………………………………
Date ……………………………………………..
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ACKNOWLEDGEMENT
I thank God for all that long that he has kept me moving through the Engineering
journey that started in 2015 and have enabled me to accomplish the entire course.
Am mush grate full to my family (parents, sister, brother and my loving aunt Akiiki)
for their patience and support in overcoming numerous obstacles I have been facing
through the entire course and this thesis as well.
I would like to thank my supervisor MR Wafula. P for the feedbacks and assistance
rendered always whenever I need help. lastly I thank my friends that really
Encouraged during this study (we had made a web in the library), hadn’t it been a
combined effort throughout I would not have proceeded.
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ABSTRACT
Performance of beams with pvc pipes embedded at the center or with an eccentric
pipes needs design consideration of influencing variables such as the pipe size and the
location of the pipe being embedded. This was performed analytically where six RC
beams of dimension (300x450 and span of 3500) were modeled in Abaqus 6.10-3 of
which one was a solid beam and the 5 had varying pipe sizes and location of the pipe
as designated (B-25.4, B-50.8mm,B-76.2mm,B-101.2mmE50 and 127mmE
100mm).The models were assigned correctly with their material properties both
elastically and plastically of which CDP(concrete damaged plasticity) and other
damage parameters from Abaqus manual were incorporated which helped in defining
the failure mechanisms in concrete. The beam models were treated as simply
supported beam and loaded with a concentrated load(500KN) in addition meshing was
done and the concrete elements were assigned with an element type C3D8R
hexagonal elements and meshed with 50x50mm mesh size and for reinforcements
were meshed with global size of 100mm. The analysis was submitted and results
(stresses, strain, load-deflection ductility index and crack patterns) of beams were
compared to solid beam. The results indicated an increase in stresses in concrete and
bottom reinforcement as the pipe size increases and reduction in loading capacity and
ductility index as the pipe size and location varied. Therefore, the size and location of
the pipes in beams affects the strength, stiffness and ductility of the RC beams.
TABLE OF CONTENTS
DECLARATION .............................................................................................. i
v
APPROVAL ................................................................................................... ii
ACKNOWLEDGEMENT .................................................................................. iii
ABSTRACT .................................................................................................. iv
TABLE OF CONTENTS .................................................................................. iv
LIST OF FIGURES ....................................................................................... vii
LIST OF TABLES .......................................................................................... ix
CHAPTER ONE: INTRODUCTION ............................................................ 1
1.1 Background ............................................................................................ 1
1.2 PROBLEM STATEMENT ............................................................................ 2
1.3 MAIN OBJECTIVES .................................................................................. 3
1.4 SPECIFIC OBJECTIVES ............................................................................ 3
1.5 RESEARCH SIGNIFICANCE: ...................................................................... 3
CHAPTER TWO: LITERATURE REVIEW .................................................. 4
2.1 COMPARISION BETWEEN SOLID AND HOLLOW REINFORCED CONCRETE
BEAMS ......................................................................................................... 4
2.2 REINFORCEMENT DETAILING IN HOLLOW SECTIONS ............................... 6
2.3 FINITE ELEMENT ANALYSIS ..................................................................... 6
2.3.1 FEM Solution Process Procedures ........................................................... 7
2.3.2 Basic Theory ........................................................................................ 7
2.3.3 Element types ...................................................................................... 8
2.4 DEVELOPING MATERIALS MODELS AND FAILURE BEHAVIORS IN MODELS 10
2.4.1 Concrete Plastic Damage Model. .......................................................... 10
2.4.2 Damage Parameters ........................................................................... 11
2.5 FAILURE MODES ................................................................................... 11
2.5.1 Fracture ............................................................................................ 11
2.5.2 Flexural shear cracking ....................................................................... 12
2.5.3 Splitting failure ................................................................................... 13
2.5.4 Web crushing ..................................................................................... 14
2.5.6 STRESS –STRAIN BLOCK OF RECTANGULAR HOLLOW SECTION ............ 16
2.5.7 CONFINEMENT OF REINFORCED CONCRTE COLUMN ............................ 17
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2.5.9 CRACK PROPAGATION ........................................................................ 19
2.5.9.1 SMEARED CRACK MODELS ............................................................... 20
2.6 SOLUTION METHOD ............................................................................. 21
CHAPTER THREE: METHODOLOGY ...................................................... 21
3.0 Introduction ......................................................................................... 21
3.1 Data collection Establishing study parameters ......................................... 22
3.2 Pre process modeling of the structural models in FEM using Abaqus .......... 26
3.2.1 Flowchart of the pre-process modeling and post process analysis in
Abaqus ...................................................................................................... 26
3.3 MODELING OF PARTS ........................................................................... 26
3.5 ASSIGNING MATERIAL PROPERTIES TO PARTS ....................................... 27
3.5.1 Concrete Beam models ....................................................................... 27
3.5.2 REINFORCEMENTS ............................................................................. 29
3.6 ASSEMBLING THE MODLED PARTS ......................................................... 31
3.8 MESHING / DISCRETISATION OF THE MODELS ....................................... 35
3.9 SUBMITION OF THE JOB/STATIC ANALYSIS. .......................................... 37
CHAPTER FOUR: RESULTS AND DISCUSSION ..................................... 38
4.1. STRESS AND STRAIN DISTRIBUTION .................................................... 38
4.4.1 Stress distribution in reinforcements .................................................... 41
CHAPTER FIVE : CONCLUSIONS AND RECCOMMENDATION................ 54
5.1 Conclusion ............................................................................................ 54
5.2 RECOMMENDATIONS ............................................................................ 55
REFERENCES .............................................................................................. 57
APPENDEX 1: FORMULAES .......................................................................... 60
APPENDIX 2: MODIFIED STRESS- STRAIN CURVES UNDER COMPPRESSION
AND TENSION ............................................................................................ 61
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LIST OF FIGURES
Figure 1: Bar element (Hartsuijker & Welleman, 2007) .......................................... 8
Figure 2: Different types of elements used in finite element computations (Malm,
2014) ................................................................................................................ 9
Figure 3: Hourglass modes for 4-noded and 8-noded elements (Malm, 2014) .......... 9
Figure 4Stress-strain relationship for brittle and ductile materials (Class Connection,
2014) .............................................................................................................. 11
Figure 5: Failure propagation of rapture to crack initiation in concrete regarding
tensile capacity (Malm, 2014) ............................................................................ 12
Figure 6: Shear failures source (zacoeb 2003) ..................................................... 13
Figure 7: Splitting progress of the section where splitting cracks have formed in a)
and finally propagated in c) (Engström, et al., 2000) ........................................... 13
Figure 8: Shear force, along an reinforcement bar resisting tension forces, s in
the concrete (Engström, et al., 2000) ................................................................. 14
Figure 9: Bond-stress relationship of ribbed reinforcement bar in confined and
confined section (fib, 2010) ............................................................................... 14
Figure 10: General interaction diagram, (Al-Nuaimi, et al., 2010) .......................... 16
Figure 11Reinforced concrete column strains and stresses,( suprabowo 1996) ....... 16
Figure 12: (Papanikolaou & Kappos, 2009) ......................................................... 17
Figure 13: Confined concrete within a cross-section (Paultre & Légeron, 2008) ...... 18
Figure 14: Stress and strain distribution for unconfined and confined concrete
(Paultre & ........................................................................................................ 19
Figure 15: Definition of different crack formulations for finite elements and definition
of fracture energy (Model Code, 2010). .............................................................. 20
Figure 16: Full Newton-Raphson method (Cervenka & Jendele, 2013) ................... 21
Figure 17: Flow chart of preprocess modelling .................................................... 26
Figure 18: Modeled parts of the Beam ................................................................ 27
Figure 19: Material Assignment .......................................................................... 28
Figure 20summary of plastic parameters ............................................................. 28
Figure 21: Tabulated values of the compressive behavior and tensile behavior ....... 29
Figure 22: Material Assignment in steel .............................................................. 31
Figure 23: Typical figure showing the Assembling of different parts ...................... 31
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Figure 24: Beam models, Assembled and Assigned with material properties. ......... 32
Figure 25: Static step being defined ................................................................... 33
Figure 26: Typical beam showing support conditions ........................................... 34
Figure 27: Static load being applied. ................................................................... 35
Figure 28: Meshed beam models ........................................................................ 36
Figure 29: Stress distribution in concrete ............................................................ 39
Figure 30: Variation of stresses in concrete for all models .................................... 40
Figure 31: Stress distribution in reinforcement .................................................... 42
Figure 32: Chart of stress variation in models ...................................................... 43
Figure 33: Strains in concrete models ................................................................. 46
Figure 34: Strain distribution in steel .................................................................. 47
Figure 35: Load – midspan deflection ................................................................. 48
Figure 36: Crack patterns in concrete models ...................................................... 53
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LIST OF TABLES
Table 1: List of Data collected ............................................................................ 23
Table 2: compressive behaviors (ABAQUS MANUAL 2008) .................................... 24
Table 3: Tension stiffening model and (ABAQUS/CEA manual 2008) ..................... 24
Table 4: model parameters characterizing it performance under compound stresses
(Abaqus manual 2008) ...................................................................................... 25
Table 5: stress distribution of beams in concrete ................................................. 39
Table 6: stresses in bottom layer reinforcements for all modeled beams ................ 43
Table 7: summaries of strain maximum values in concrete and bottom
reinforcements ................................................................................................. 44
Table 8: Ductility index and ultimate loads .......................................................... 49
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LIST OF ACRONYMS NSC Normal strength concrete
HSC High strength concrete
PVC Polyvinylchloride
EC Eurocode standards
BS British standards
RC Reinforced concrete
FEM finite element method
FEA finite element analysis
DOF Degree of freedom
CDP concrete Damaged plasticity
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CHAPTER ONE: INTRODUCTION
1.1 Background
Hollow concrete sections have been often used during beam and column designs in
most of countries, particularly for very tall bridges in seismic areas including California,
New Zealand, japan, Italy, china and Malaysia etc. for reducing the mass and therefore
minimizing the self-weight contribution to the inertia model of vibration during an
earth quake. A similar leaf is borrowed in to construction industries where PVC pipes
are embedded in to concrete structural element to allow access for services such as
drains and electric wiring. the provision of such openings /embedding pvc pipes in the
structural element may result in to loss of strength, stiffness, ductility and hence
significant structural damage may be sustained, if the provision of such concerns are
not considered adequately during the design stage (k-s son et al 2005).
When using a hollow column instead of a solid column, the shear flow becomes closer
to a thin walled tube and little investigation has been done for evaluating the shear
strength for such members (Shin, et al., 2013). In Ec2, there is no specific shear
formula for hollow reinforced columns, which are based on empiric formulas from
testing of solid sections consisting of normal strength concrete. The shear resisting
mechanism in the BS8110 is based on the area from the webs where full scale testing
of hollow columns has shown to be more dependent on the gross cross section of the
column, especially in large cross sections. In design of columns in high-rise structures,
the amount of ductility is significant in order to prevent a brittle failure of the column
when subjected to large lateral forces and the BS8110 does not specifically define a
ductility factor. Several other research groups have addressed this issue and
formulated shear formulas based on the ductility factor in order be able to calculate
the full response and preferred failure mode of the column when failing (Priestley, et
al., 2002). A research group has recently developed a shear strength formula,
specifically designed for ductile hollow rectangular reinforced concrete sections which
take into account the gross cross section of the hollow section when calculating the
shear strength of the section (Shin, et al., 2013).
According to SNn03-2847,2007 it is mentioned that channels and pipes and hooks
planted in both columns and beams shall not occupy more than 4% the cross section
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area for the section required for the strength and fire protection of which this does
not satisfy the behaviors or how the components responds due to embedment.
Plastic deformation capacity and energy dissipation of reinforced concrete hollow
beams and columns may not be adequate to ensure effective confinement of the
thinner web causing the drop of member’s ductility and shear strength. For safety of
RC members, these behaviors shall critically be investigated. Last decades many
researches have been conducted on the behaviors of hollow reinforced structural
elements as though there is PVC embedded longitudinally through the elements and
their experimental tests seemed to be agreeing with the analytical analysis using Non
-linear finite element programs (FEM) Since it allows one to perform a full scale
analysis for structural elements and saves time compared to experimental methods
which necessitate to have a testing rig and the machines which are quite expensive.
In addition, element program, the variables that are not measured in the field when
doing the experiment can be known in more detail and complete. The practice of
embedding PVC has been controversial due to lack of understanding and limited
guidelines by the building codes on the structural response of such structural members
under various loading combination. This gives urge to me to investigate the behaviors
/effects of embedding pipes in to the structural members and which I hope
understanding the structural response for such scenario towards the structural
elements will help the fill the void that has been there.
1.2 PROBLEM STATEMENT
In countries like Uganda, openings and holes are often provided in reinforced concrete
beams to all access for services such as drains and electric wirings, however plastic
deformation capacity and the energy dissipation of reinforced concrete hollow or
embedded with pvc pipes may not be adequate to ensure effective confinement of the
thinner web because of drop in strength, stiffness and ductility thus leading to partial
or full collapse of buildings.
Therefore, from the above hypothetic observation there may be reduced strength,
stiffness and ductility that could lead to catastrophic failure. Understanding the
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behavior of concrete member embedded with PVC is important in developing the safe
design procedures.
1.3 MAIN OBJECTIVES
To study the strength, stiffness, ductility and crack patterns of the hollow structural
members under loading and compare with solid one.
1.4 SPECIFIC OBJECTIVES
1) To establish parameters, develop material model based on studies found in
literatures in addition to existing code (BS in conjunction with EC) and Abaqus
manual.
2) To Model structural models in non-linear FEM using Abaqus CEA 6.1
3) To analyze the numerical modeled models.
4) To compare and validate of the results found analytically.
1.5 RESEARCH SIGNIFICANCE:
In design and construction of reinforced concrete members, the ductility failure should
be prevented. The presence of PVC pipes as hollow parts in reinforced concrete
columns and beams decreases strength, stiffness and ductility of the structural
members thus sustain damages which may lead to catastrophic failures. Therefore,
understanding the behavior of RC beams and columns with embedded PVC pipes is
important for developing safe design procedures for designers.
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CHAPTER TWO: LITERATURE REVIEW
2.1 COMPARISION BETWEEN SOLID AND HOLLOW REINFORCED
CONCRETE BEAMS
According to Zacoeb (2003), the experimental research about the Flexural
Capacity of Reinforced Concrete Short Column with Variations Hole which performed
with the eccentric load of 60 mm and 100 mm. The ratios of holes were 0%, 3.63%,
7.95% and 12.50% and 20.28% with cross-sectional sections of 150 mm x 150 mm.
The result from the experimental research shown that the hollow column with hole
ratio up to 20,28% and 60 mm load eccentricity shown a decrease of maximum
capacity and stiffness experienced to 11,90% and 53,88% relative to solid column.
Whereas column with 100 mm load eccentricity, did not indicate a decrease in
maximum load capacity and relative stiffness to solid columns. The maximum increase
of ductility experienced by hollow column with hole ratio of 20.28% and 60 mm load
eccentricity was 16.38% relative to solid column. The models of solid and hollow
column collapse with 60 mm load eccentricity, shows the pattern of compression
failures. While the hollow columns with hole ratios up to 20.28% and 100 mm load
eccentricities, did not show any increase in ductility relative to solid column. Cracks in
solid or hollow columns generally exhibit a fracture pattern that is in the same direction
as the tensile reinforcement at the tension and spalling side of the tap. The model of
solid and hollow column collapse with 100 mm load eccentricity shows the same
pattern of tension failures.
Ali said Alnuaimi et al 2005 presented a comparison study on the test results of seven
hollow and seven solid reinforced concrete beams. All beams were designed to resist
combined load of bending, torsion and shear. Every pair (hollow and solid) were
designed for the same load combination and received similar reinforcement. The
beams were 300x 300mm cross-section and 3800mm long and the internal hollow
core for the hollow were 200x 200mm creating a peripheral wall of thickness 50mm.
The main variable studied were the ratio of bending to torsion which varied between
0.19 and 2.62 and the ratio in the web of the shear stress due shear force also varied
between 0.59 and 6.84.
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It was found out that the concrete core participates in the beams behaviors and
strength cannot be ignored when combined load of bending, shear and torsion are
present. It participation depends partly on the ratio of shear -stress due to torsion to
the shear. All solid beams cracked and failed at higher load than their counterpart
hollow beams and the smaller the ratio of torsion to bending the larger the difference
in failure loads between hollow and solid beams and the longitudinal steel yields while
the transverse steel experienced strain values.
Dr- Thaar saud (2015) presented a study of six reinforced concrete moderate deep
beams with embedded pvc pipes. The tests studied the effects of installation of PVC
pipes on the behavior of reinforced moderate deep beams. The test perimeters were
the diameter and the location of the pipes. The beams 1000mm length, 150mm width
and 300mm depth. One of the beams was constructed without pipe as ac control and
the remaining five had embedded pipes. Five pipe diameters were used 25.4, 40.5,
50.8, 76.2 and 101.6mm and these pipes were inserted longitudinally either at the
center of the beam or near the tension reinforcement. The beams were simply
supported tested under central concentrated load until failure.
The results indicated that, the pipe diameter less than 1
3 of the beam width had limited
effect on the capacity and rigidity of the beam and for the larger pipes the ultimate
strength of the beam decreased between 16.7%- 33% and the beam stiffness
decreased 10.3%- 297%.
Dr Nabeel Hassan Ali (2005) presented a study which was devoted to investigate the
behavior and ultimate strength of reinforced concrete short columns with different
shapes of transverse openings. The experimental part includes investigation of the
ultimate strength of tested columns Variables considered in the test program include
different shapes of openings with the same opening ratio of 0.133. In the numerical
part, the columns have been analyzed using nonlinear finite element model. The finite
element analysis has been carried out to analyze the tested columns to determine the
stresses at the longitudinal bars and the stresses in concrete at the sections of
openings. An increase in the ultimate strength of about 2.06% is noticed when the
single opening having 20mm diameter is replaced by two symmetrical openings
of10mm diameter each. Also, a decrease in the ultimate strength of about 2.88% and
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5.97% is noticed when the single circular opening of 20 mm diameter is replaced by
20x20 mm square opening or20x40 mm rectangular opening respectively.
Abhay (2014) performed the study of structural behavior of Hollow (Box- type) and
Solid reinforced concrete members in the RCC framed building under Seismic load
using ETABS software. He concluded that maximum node displacement of hollow
members given by ETABS is less as compared to solid members. 20% to 27%
reduction in the storied overturning moment due to hollow members in RCC framed
building was observed. Storied shear for RCC framed building having hollow members
is decreased by 27% as compared to solid members. 74.1687 ton of M30 concrete is
saved by using hollow (Box-type) members in RCC framed building so it leads to
economical.
2.2 REINFORCEMENT DETAILING IN HOLLOW SECTIONS
Reinforcement detailing in hollow sections are more complicated than solid sections,
correct spacing and constructability of these sections need to be properly considered
and crucial to ensure a ductile behavior of the structure if extreme loads are applied
such as strong earthquakes, (Subramanian, 2011). Especially important is the
arrangement of reinforcement within the plastic hinge region where the transverse
reinforcement should be designed to avoid shear failure, splitting failure in anchorage
zones, prevent buckling of longitudinal bars and to effectively confine the concrete in
order to ensure a ductile behavior when failing, (Paultre &Légeron, 2008).
2.3 FINITE ELEMENT ANALYSIS
FEM is a powerful tool commonly used for analyzing a broad range of engineering
problems in different environments. Fem is employed extensively in the analysis of
solids, structures, heat transfers and fluids. Buyukkaagozi (2010). For the accurate
assessment of the inelastic behavior characteristics of hollow structural components
or any form of structural member with a sophisticated geometry, three -dimension
finite element analysis and reliable constitutive modeling are required (Maekawa et al,
2001)
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A typical finite element on a software system requires the following
i. Nodal point spatial locations (geometry)
ii. Elements connecting the nodal points
iii. Mass properties
iv. Boundary conditions or restraints
v. Loading or forcing function details
vi. Analysis options
Because FEM is a discretization method, the number of degrees of freedom of a FEM
model is necessarily finite. They are collected in a column vector called u. This vector
is generally called the DOF vector or state vector. The term nodal displacement vector
for u is reserved to mechanical applications.
2.3.1 FEM Solution Process Procedures
i. Divide structure into pieces (elements with nodes) (discretization/meshing)
ii. Connect (assemble) the elements at the nodes to form an approximate system
of equations for the whole structure (forming element matrices)
iii. Solve the system of equations involving unknown quantities at the nodes (e.g.,
displacements)
iv. Calculate desired quantities (e.g., strains and stresses) at selected elements
2.3.2 Basic Theory
In a finite element analysis, elements are idealized in order to represent the behavior
of a certain structure. The structure being analyzed is divided into elements with
material properties and certain geometry. The geometry of the elements depends on
which type of elements that are used. Generally, the structure may be divided into
rectangular and triangular elements with a certain number of nodes in order to
represent the behavior of the structure. The nodes of the elements have a certain
degrees of freedom (DOF), which allows rotation and displacement in any wanted
direction. A 2D bar element with two nodes could represent a steel rod with two DOF,
where rotation and displacement perpendicular to the bar axis is prohibited. This
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means the bar is only allowed to displace in its own plane and forces may only vary
within its own direction, (Hartsuijker & Welleman, 2007).
Figure 1: Bar element (Hartsuijker & Welleman, 2007)
The displacements [d] within this element may be computed according as below,
where
[k] Represents the stiffness of the structure and the force [f] that is applied at the
node.
[d]=[𝑘]−1 [f]
The choice of element type depends on what type of structure, component or material
that should be represented. In general, any type of geometric shape may be modeled
with the tetra elements and it allows for a great variety of imposed loads such as heat
transfer, dynamic loads and nonlinear material models. In FE-analysis it is also possible
to simulate nonlinear fracture mechanics. The brick elements can be used to model
any rectangular geometry and computational time may be decreased when using this
kind of elements.
2.3.3 Element types
When using FEM software there are several different types of elements that can be
used when modeling geometry. The most common types of elements are; truss-,
beam-, shell- and solid elements. The truss element only has three DOF at each node,
i.e only translations and no bending in each node. Both the beam- and the shell
element have six DOF at each node and can interpret both translations and bending
at the nodes. The solid element can only represent translations at each node, i.e. it
has only three DOF. In structural analysis there are different element classes that have
these different properties, in some examples of the different element classes are
presented (Malm, 2014).
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Figure 2: Different types of elements used in finite element computations
(Malm, 2014)
Each element can have a different amount of nodes and integration points.
Displacements and rotations are calculated at the nodes of the element. Based on the
displacements, the strain in the element can be calculated at the integration points,
also known as Gauss points, of each element. Based on the strain, the corresponding
stresses are calculated at the integration points. In 4-noded and 8-noded elements,
the number of integration points depends on the type of integration, it can either be
a full integration or a reduced integration. The integration points for a 4-noded and 8-
noded element are show with full integration and reduced integration respectively for
both element types.
Figure 3: Hourglass modes for 4-noded and 8-noded elements (Malm,
2014)
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2.4 DEVELOPING MATERIALS MODELS AND FAILURE BEHAVIORS IN
MODELS
Reinforced concrete is a complicated material to be modelled within finite element
packages. A proper material model in finite element model should inevitably be
capable of representing both elastic and plastic behavior of concrete in compression
and tension. The complete compressive behavior should include both elastic and
inelastic behavior OF concrete including strain softening regimes. Simulation of proper
behavior under tension should include tension softening, tension stiffening and local
bond effects in reinforced concrete elements. Therefore, the development of a finite
element model (FEM) may need intensive material testing to incorporate into the
material model in any of the finite element [FE] packages available. There are quite
large numbers of numerical material models available in the literature with potential
to develop complete stress-strain curves of concrete for compression and tension
separately based on experiment results. However, these methods are not directly
applicable with the input format required for the finite element packages (Dassault
Systèmes Simulia Corp. [SIMULIA], 2008).
2.4.1 Concrete Plastic Damage Model.
As discussed in the before, the structural behavior of RC structures is highly complex,
because the joint operation of concrete and steel. Concrete behavior is brittle, but,
under stress reversal, tensile cracks might close, then broken parts being reassembled.
Conversely, steel behavior is ductile, with extremely rare fractures, and broken parts
cannot be reunited. Therefore, concrete behavior can be better described with damage
models, whereas plasticity models better represent steel behavior. Nevertheless, since
steel brings additional ductility, the behavior of concrete belonging to reinforced
concrete can be even better described with models that combine damage and
plasticity. These models are particularly well suited for reproducing failure modes that
are based on tensile cracking and compression crushing. Modifications for two stress-
strain curves under compression and tension are suggested to be used with the
damaged plasticity model in ABAQUS because is capable of representing the formation
of cracks and post-cracking behavior of reinforced concrete elements. Therefore,
development of a proper damage simulation model using the concrete damaged
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plasticity model will be useful for the analysis of reinforced concrete structures under
any loading combinations including both static and dynamic loading (“Abaqus Analysis
User Manual – Abaqus Version 6.8” [Abaqus Manual], 2008).The modified two stress
– strains are shown in appendix 2
2.4.2 Damage Parameters
The tensile damage parameter, dt is defined as the ratio of the cracking strain to the
total strain. Similarly, the compressive damage parameter, dc is defined as the ratio
between the inelastic strain and total strain. If damage parameters are not specified,
the model behaves as a plasticity model.
2.5 FAILURE MODES
2.5.1 Fracture
Fracture is one of the most important concepts in structural engineering. Basically,
fracture can describe as one single body that is being separated into pieces by an
imposed stress. There are principally two different fracture modes, ductile and brittle.
The main difference between the two modes is the amount of plastic deformation that
the material endures before fracture occurs. Ductile materials such as steel undergo
larger plastic deformations while brittle materials such as concrete show no or little
plastic deformations before fracture occurs.
Figure 4: Stress-strain relationship for brittle and ductile materials (Class
Connection, 2014)
In concrete, initiation and propagation of cracks are vital to in order to determine the
type of fracture and how the crack propagates through the material gives a good
insight into the mode of fracture. In ductile materials, the crack propagates slowly
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and contributes to large plastic deformations. Usually the crack will not extend without
an increase in stress. When there is a brittle fracture, cracks spread very rapidly with
no or little plastic deformations. The cracks will continue to propagate and grow once
they are initiated in a brittle material. Another important characteristic of crack
propagation is how the crack is advancing through the material. In HSC, cracks tend
to propagate through the aggregates due to the high compression forces which cause
a more brittle failure compared to regular strength grades when the crack travels
around the aggregate stones which will lead to a more ductile behavior (Bailey, 1997).
Figure 5: Failure propagation of rapture to crack initiation in concrete
regarding tensile capacity (Malm, 2014)
For several reasons, a ductile fracture behavior is preferred in design. This is because
brittle failures occur very rapidly, which can lead to catastrophically consequences
without any warning. Ductile materials plastically deform slowly and the problem can
be corrected before the structure collapses. Because of the larger plastic
deformations, more strain energy is needed to cause a ductile fracture, which will lead
to a more forgiving failure (Bailey, 1997).
2.5.2 Flexural shear cracking
When a column in a moment resisting frame is subjected to high lateral loads in a
seismic event or high wind loads, a preferred failure mode would be controlled flexural
crack failure. The flexural cracks are initiated from the base of the column face
propagated along the height of the column, The transverse reinforcement in the
cracked regions transfers the shear force and resists the cracks from widening. The
inclination of the crack angle has been seen to be initiated at 45° and decline towards
30° as yielding of reinforcement is progressing. The inclination also depends on the
ratio between transverse and longitudinal reinforcement. The flexural cracks are
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initiated in the tensile region and when they reach the compressive zone, they are
closed by the compressive forces. For columns with low shear span ratios, h/d<2, the
flexural cracks could propagate thorough the width of the column from corner to
corner and would could cause a premature diagonal tension failure
Figure 6: Shear failures source (zacoeb 2003)
2.5.3 Splitting failure
Splitting is generally a local problem, caused by bond-slip between the longitudinal
reinforcement and concrete when subjected to tensile forces,
Figure 7: Splitting progress of the section where splitting cracks have
formed in a) and finally propagated in c) (Engström, et al., 2000)
The ribbed reinforcement bars resist the applied tension stress in the concrete, where
a splitting failure is initiated by micro-cracks forming around the ribs of the reinforcing
bar. As the slip of the longitudinal reinforcement bar is continued, the cracks between
the ribs of the longitudinal bars will also continue to propagate until the cracks have
formed across the whole section and the final rapture will occur (fib, 2010).
14
Figure 8: Shear force, along an reinforcement bar resisting tension
forces, s in the concrete (Engström, et al., 2000)
In columns, the slip failure is generally at the ends of the anchorage zones around
the ribbed reinforcing bars. The splitting failure surface will occur along the face of
the concrete cover and is influenced by the confining action of the column in order to
prevent splitting. The confining reinforcement will prevent the longitudinal bars from
slipping and cracks to propagate between the ribbed bars up to a certain amount of
confinement, until the failure is governed by pull-out failure.
Figure 9: Bond-stress relationship of ribbed reinforcement bar in confined
and confined section (fib, 2010)
Pull out failure is when the stresses become too large and the ribs cannot resist the
tension stresses, which will lead to a continued slip until final failure occurs. The
longitudinal bars could fracture the end of the concrete section (Engström, et al.,
2000).
2.5.4 Web crushing
If high compression forces are present, diagonal web crushing may occur in hollow
columns. The web crushing is dependent on the inclination of the diagonal
compressive struts. The diagonal web crushing is also dependent on the center of
gravity of the section where a deeper section will increase the capacity in order to
resist web crushing (Priestley, et al., 2002)
15
2.5.5 INTERACTION DIAGRAM
When a column or pier is subjected to a moment and an axial load, it has to be
designed with the aid of an interaction diagram according to the ACI 318. In an
interaction diagram the strain in the maximum compressive fiber is assumed to be
equal to the maximum usable strain of concrete, εcu=0.003. The strain relationship
between the strain in the reinforcement and the concrete is assumed to be directly
proportional to the distance from the neutral axis. Furthermore, the tensile strength
of the concrete should be neglected when performing axial and flexural calculations
of reinforced concrete when dealing with interaction diagrams. The interaction
diagram displays the relationship between axial load, Pn and moment, Mn for a given
reinforcement amount, As. The moment is given by the relationship Pn·e=Mn. This is
useful since even if there is no moment present, the axial load is never in practice
centric. This is because there are always imperfections in the structure and columns
are never perfectly straight, which results in a moment that corresponds to the axial
load times an eccentricity. The interaction diagram will then display all different
capacity combinations of axial loads and moments for a given reinforcement ratio. If
the ultimate loads are within the interaction curve then there is enough reinforcement
and the section will not fail with the current loads. Most often it is desirable to design
the cross-section so that the ultimate loads are within the tension controlled section
of the interaction diagram, which will provide a ductile failure. This can be achieved
by letting the reinforcement strain εs>εy=εcu, which implicate that the steel
reinforcement will yield before the concrete reaches its compressive strength.
However, in large columns this can be difficult to achieve because of the large axial
forces that are present, which will give a compression-controlled failure. Due to this it
can be wise to have some safety margins to the compression-controlled surface of the
interaction diagram, since this kind of failure is very abrupt and can cause a collapse
of the structure (Ansell, et al., 2012).
16
Figure 10: General interaction diagram, (Al-Nuaimi, et al., 2010)
2.5.6 STRESS –STRAIN BLOCK OF RECTANGULAR HOLLOW SECTION
The concrete stress distribution is approximated to be uniformly rectangular and with
a magnitude of 0.85fc’, which should be distributed over a portion of the compressive
zone that is equal to a=β*X. Where X is the distance from the top compressive fiber
to the neutral axis and β is a reduction factor for X depending on the concrete class,
( suprabowo 1996)
Figure 11: Reinforced concrete column strains and stresses,( suprabowo
1996)
The Equation for balancing of forces can be seen as follow:
𝑝𝑛 = 𝑐𝑐 − 𝑐𝑐𝑝 + 𝑐𝑠 − 𝑇𝑠
17
𝑐𝑐 = 0.85𝑓′𝑐 . 𝑏. 𝑎 (Concrete compressive force)
𝑐𝑐𝑝 = 0.85𝑓′𝑐𝑅2(𝜃4 − 𝑠𝑖𝑛𝜃4. 𝑐𝑜𝑠𝜃4) (The reduction due to the
hole)
𝑐𝑠 = 𝐴′𝑆 . 𝑓′
𝑠 (steel compression force)
𝑇𝑠 = 𝐴𝑆 . 𝑓𝑠 (steel stress force)
Moment balance equation can be seen as follows:
𝑚𝑛 = 𝑝𝑛. 𝑒
𝑚𝑛 = 𝑚𝑐𝑐 − 𝑚𝑐𝑝 + 𝑚𝑐𝑠 + 𝑚𝑇𝑠 Moment balance equation
𝑚𝑐𝑐 = 0.85𝑓′𝑐 . 𝑏 𝑎 (𝑦 −𝑎
2) Moment of concrete
𝑚𝑐𝑝 = 0.57𝑓𝑐′. 𝑅3𝑠𝑖𝑛3𝜃4 Moment reduction due the
hole
𝑚𝑐𝑠 = 𝐴′𝑆. 𝑓′
𝑠(𝑦 − 𝑑) Moment of compression
steel
𝑚𝑐𝑠 = 𝐴′𝑆. 𝑓′
𝑠(𝑦 − 𝑑) Moment in tension steel
2.5.7 CONFINEMENT OF REINFORCED CONCRTE COLUMN
When a reinforced concrete column is subjected to compression forces, the concrete
will transfer forces in its lateral direction due to the Poisson effect. The Poisson effect
is the volumetric expansion of concrete and when the expansion is restrained by
transverse reinforcement in the perimeter, the concrete core will be confined, (Razvi
&Saatcioglu, 1999).
Figure 12: (Papanikolaou & Kappos, 2009)
When the concrete expansion is restrained, tensile pressure is applied in the parameter
reinforcement which will create an inward radial pressure acting on the concrete core
and thus
18
When the concrete is degraded in an unconfined section, the Poisson ratio will increase
from 0.2to 0.5 due to the increased cracking and crushing of the material. Depending
on the confinement effectiveness of the section, the lateral expansion that is restrained
will increase the ductility and thus enhancing the concrete strength of the section since
the damage propagation is prevented due to the confining effect (Imran &
Pantazopoulou, 2001) (Model Code, 2010).
The inward radial pressure created by the confining action will develop an arching
effect between the transverse reinforcement layer as well as within the section, Figure
2-10 (Paultre &Légeron,2008).
Figure 13: Confined concrete within a cross-section (Paultre & Légeron,
2008)
When designing hollow sections, the confined concrete section becomes more like a
closed boxed wall section where an encircling hoop in each separate wall with
intermediate cross-ties confine the concrete. The theory of the stress-strain
relationship will therefore become the same for a solid column except that each wall
acts as a separate confining section (Mander, et al.1988).
In the confinement model that Mander et al. proposed in 1988, the maximum stress
in the confined regions of the section is determined from the maximum strain when
the cross-ties fracture, which takes into account the strain-hardening behavior of the
steel when it is yielding. This criterion has been derived by (Mander, et al., 1988)
19
which is an energy balance between the confined strain increase in the concrete and
the maximum yield strength in cross-ties(Paultre & Légeron, 2008).
Figure 14: Stress and strain distribution for unconfined and confined
concrete (Paultre &
Légeron, 2008)
2.5.9 CRACK PROPAGATION
Fracture energy
The formation of a new crack is called crack initiation and the amount of energy, Gf,
which is required to fully open a crack, is called fracture energy, The fracture energy
depends on the tensile capacity of the member, the crack width and the shape of the
softening behavior in the concrete (Leckie & Dal Bello, 2009).
There are currently two approaches used in numerical models when calculating crack
propagation, the smeared crack approach and the discrete crack model, in the discrete
crack model, a crack is formed between the interfaces of two elements. When the
crack is formed, the boundary conditions of the element change, which will require
modifying the meshing of the element at the crack-tip in the crack initiation stage. In
the smeared crack approach, a crack band within a length of the concrete element is
propagated and modeled as isotropic stiffness degradation in the stiffness matrix (fib,
2010) (Hofstetter & Meschke, 2011).
20
Figure 15: Definition of different crack formulations for finite elements
and definition of fracture energy (Model Code, 2010).
2.5.9.1 SMEARED CRACK MODELS
Smeared crack models use the fracture energy to compute the total strain required in
order to
compute the critical crack width. In one-dimensional crack models, the strain
composes of two different relations, the elastic strain εe, which is derived from Hooke’s
law and the additional
strain due to crack initiation εc, Equation 5-6. Before the tensile capacity is reached
within the linear elastic domain, no deformations will occur and the crack growth will
return to zero strain.
When the tensile capacity of the section is reached, micro cracks will develop until the
critical cracking width is reached and the section no longer can carry tensile forces in
the cracked region
(Hofstetter & Meschke, 2011).
𝜀 = 𝜀𝑒 + 𝜀𝑐
The smeared crack model then uses the strain εc to calculate the propagation of the
crack until the critical crack width, wc is found,The crack at any point may be evaluated
as the strains times the crack band width, Lt in the element.
ω= 𝜀*𝑙𝑡
There are two different types of crack models, a fixed crack model and a rotating
cracking model. The fixed crack model locks the direction of the crack in the crack
initiation phase and the crack is formed in that direction. There is a possibility for
cracks to form in the direction of the plane that may be approximated by taking into
account a factor β, which correlates with the shear modulus G. Using a fixed value of
β, could introduce errors since stresses will be able to be transferred through the
cracks, even though the crack is wide open. Therefore, β should be set to a low value,
and if a constant value is used that should be approximately 0.01. There is also a
possibility to vary the value of β, which should decrease to zero as the crack is fully
formed. The rotating crack model allows the plane of the crack to change direction
and is assumed to remain perpendicular to the direction of principal strain. This allows
21
new cracks to be initiated that are not parallel to the original crack initiation (Hofstetter
& Meschke, 2011).
2.6 SOLUTION METHOD
Newton-Raphson (N-R) Method
The Newton-Raphson method describes a way of generating the force, P versus
displacements, u curve, whose shape is not known at the outset. The initial
displacement is imagined to be equal
to zero, then a load is applied and the corresponding displacement is searched. The
initial tangent stiffness, k0 is then derived and a displacement is calculated. But this
displacement does not correspond to the given load increment, so a new tangent
stiffness is derived from the current deformation. Then this tangent stiffness spawns
a new displacement. This continues until the tangent stiffness converges with the
load-deformation curve, and then the load is increased to P2 and the steps above is
repeated until the total load is applied and the corresponding deformation is
calculated.
Figure 16: Full Newton-Raphson method (Cervenka & Jendele, 2013)
CHAPTER THREE: METHODOLOGY
3.0 Introduction
This chapter consists of set strategies that were adapted during the investigation of
the effects of embedding PVC pipes in structural members and it includes; Establishing
modeling parameters based on the existing codes (EC2 and BS8110), Abaqus user
22
manuals and the literature reviews. that was put as the input during the pre-
processing modeling of structural models in non- linear FEM using Abaqus6.1
General Flow chart showing process to be followed to achieve the objectives
3.1 Data collection Establishing study parameters
Table1 Summarizes Geometric properties, reinforcement details and material
characteristics that was adapted for the development of FEM beam models, their
behavior for linear and Non-linear behavior of concrete and steel (modulus of
elasticity, poisons ratio, density yielding of steel strain ɛst=0.0022 and concrete strain
ɛc=0.0035 (BS8110). Stating failure mechanism in concrete for concrete using
concrete plasticity damages which brings in the non linearity of concrete for example
defining concrete crushing under compressive stresses and cracking under tensile
stresses under compressive behaviors and tension behaviors respectively. The values
of damage parameters ranges from 0-1 which defines the stiffness degradation in
concrete as shown in table 1.
DATA COLLECTION PRE- PROCESS
modeling of specimen
in Non –FEM using
ABAQUS
POST - PROCESS
Analysis of the
numerically modeled
specimen
VALIDATION OF
RESULTS
23
Table 1: List of Data collected
Notations
B-solid concrete beam
B-25.4mm beam with PVC pipe size 25.4mm
B-50.8mm beam with pipe size 50.8mm
B-76.2mm beam with pipe size 76.2mm
B-101.6E 50mm beam with eccentric pipe of 101.6mm at 50mm
B-127E 100mm beam with eccentric pipe of 127mm at 100mm
Beam models
designations
Geometric
properties
Reinforcement
details
Material properties
B-Solid
B-25.4mm
B-50.8mm
B-76.2mm
B-101.6E50mm
B-127E100mm
300x450mm
Span 3500m
2T 16mm TOP
3T 25mm BOTTOM
Link- 8mm@200mm
Concrete
Concrete grade 30
Strength =30mpa
Poison ratio 0.2
Modulus of elasticity=32Gpa
Ultimate strain concrete ɛc =0.0035
steel
strength of Steel –main 460mpa
strength of Links -250mpa
Poison ratio 0.3
Modulus of elasticity =200Gpa
Ultimate strain steel ɛst =0.0022
24
FINITE ELEMENT MATERIAL MODEL development
Concrete damaged plasticity
Development of material models for the concrete beams models based on the
experiments derived using the experimentally verified numerical method by Hsu and
Hsu (1994) was adapted from the abaqus manual 2008. This was used to relatively
simulate the real concrete behaviors under tension and compression and which
represents the formation of cracks and post cracking behaviors of reinforced concrete.
The two numerical stress-strain relationship that is tension stiffening relationship and
compressive stress –strain relationships are listed in table2 and table 3 respectively
and their plots in appendex2. Inaddition other parameters that characterizes model
performance under compound stress (uniaxial state ) that are a modification of
drucker- pranger strength hypothesis defines surface failures of the model under
stresses are listed in table 4.
Table 2: compressive behaviors (ABAQUS MANUAL 2008)
Table 3: Tension stiffening model and (ABAQUS/CEA manual 2008)
COMPRESSIVE BEHAVIOR
Stress(mpa) Inelastic strain
15 0
20.1978 7.47E-05
30.00061 9.88E-05
40.30378 0.000154
50.00769 0.000762
40.23609 0.002558
20.23609 0.005675
5.257557 0.011733
COMPRESSIVE DAMAGE
Inelastic strain dc
0 0
7.47E-05 0
9.88E-05 0
0.000154 0
0.000762 0
0.002558 0.195402
0.005675 0.596382
0.011733 0.894862
25
Table 4: model parameters characterizing it performance under compound
stresses (Abaqus manual 2008)
Parameter name Value
Dilatation angle 38
Eccentricity 1
fbo/fco 1.12
K 0.67
Viscosity parameter 0.1
TENSION BEHAVIOR
stresses(mpa) cracking strain damage(dt)
1.99893 0 0
2.842 3.33E-05 0
1.86981 0.00016 0.406411
0.862723 0.00028 0.69638
0.226254 0.000685 0.920389
0.056576 0.001087 0.980093
26
3.2 Pre process modeling of the structural models in FEM using Abaqus
3.2.1 Flowchart of the pre-process modeling and post process analysis in
Abaqus
Figure 17: Flow chart of preprocess modelling
3.3 MODELING OF PARTS
The FE concrete models of dimension 450x300mm with span length 3500mm,
reinforcements (2T16 on top and 3T25 at the bottom and link of 8mm diameter) were
modeled in ABAQUS to constitute up the whole reinforced concrete beam that is suite
for the static loading during the application of the load.in addition the loading plate
was also modeled (300x300x10mm) and for continuum elements were 3D deformable
models with solid shape that is of extrusion type, reinforcements (main bars and
stirrups) were 3D deformable wire shape and planar type. This procedure was
repeated for different models that have pvc pipes B-25.4mm, B-50.8mm, B-76.2mm,
3D Modeling of parts in ABAQUS Assigning material properties for models
Modulus of elasticity Poison ratio concrete,0.2and 0.3
steel Strength of concrete and steel
Assembling the models
Static step
Loading and defining boundary
conditions Loading increments
of 0.1-0.01 0f the total load
Meshing /discretization of the specimens
Submission of job and execution. Post analysis study
Results Stress – strain distribution in the specimen
Load –deflection relationship load carrying
capacity of modeled models
Examining the stiffness and ductility of the solid
models and compare with hollow models
Simulation of modes of failures and crack patterns
in the models
27
B-101.6E50mm, B-127E100mm as shown in the figure17and figure 18 after modelling,
material assignment and assembling of different parts to come up with a concrete
beam.
Figure 18: Modeled parts of the Beam
3.5 ASSIGNING MATERIAL PROPERTIES TO PARTS
3.5.1 Concrete Beam models
The FE concrete models were modeled in ABAQUS and then assigned with general
material properties which includes density 2400E-6kg/mm3, Elastic modulus 32Gpa,
poisons ration of 0.2 this is shown below. Units used were (kg, N,mm,mpa).
28
Figure 19: Material Assignment
For Non-linearity of concrete and simulation of the failure mechanism which is cracking
under tensile stresses and crushing under compressive stresses, a plastic module was
introduced under which the concrete damaged plasticity (CDP) option was considered
with different parameters entered into tabular form and this is shown in table 2,3 and
concrete damage includes yield stresses, inelastic strains and damage parameters
under compression behaviors while as the yield stresses, cracking strains and damage
parameters under tension behaviors and the behavior of such parameters are shown
figure 20 and 21.
Figure 20: summary of plastic parameters
29
Figure 21: Tabulated values of the compressive behavior and tensile
behavior
3.5.2 REINFORCEMENTS
The steel bar used in the modeled reinforced beams were 3T25mm at the bottom,
2T16mm at the top with stirrup of size 8mm diameter spaced at 200mm. the bars
were assumed to have strength of 460mpa with an elastic modulus of 210Gpa. The
steel reinforcements were assigned with a poisons ratio of 0.3. the reinforcement
plasticity behaviors were assumed to first yield at strength 0f 250mpa with zero strain,
second yield at 360 with 0.001 strain and the last yield at 460mpa with 0.0022. in
addition full bond contact between the steel reinforcement and concrete was
presumed by applying the embedded option that connects the reinforcement elements
30
(reinforcements are defined as embedded) to the concrete elements (defined as the
host) the material properties are listed below in figure 22.
31
Figure 22: Material Assignment in steel
3.6 ASSEMBLING THE MODLED PARTS
After assignment of material properties to the rightful modeled parts, parts were assembled
to form a composite element (a reinforced concrete section) as shown below in figure 23
Figure 23: Typical figure showing the Assembling of different parts
32
Figure 24: Beam models, Assembled and Assigned with material
properties.
33
STATIC STEP
General static step was created, that incorporates full Newton Raphson solution
method that describes the way generating force (p) relates to the displacement (u)
this was intended to provide a controlled rate of loading on to the model. the analysis
was given a static time of 1ms and Nlgeom (Non-linear geometry) was included with
specified dissipated energy fraction of 0.0002. The static step was optioned to an
automatic increment step with a maximum number of 200 increments, initial
increment size of 0.1 with a minimum of 1E-006 and maximum of 1. this is shown in
figure 25.
Figure 25: Static step being defined
34
3.7 LOADING AND DEFINING BOUNDARY CONDITIONS
BOUNDARY CONDITIONS.
All concrete models were treated as simply supported beams with supports defined as
pinned at one end which means in simple terms the support is constrained in the x
and y axis and a roller at another end which constrained in the y axis to allow
movement in the x axis and this shown in figure 26.
Figure 26: Typical beam showing support conditions
LOADING.
Loading of the concrete models was done through the steel loading plate that was
created initially during the modeling of different parts that constitute the beam to
transfer the loading to the finite modeled beams under a controlled rate of load
application that was defined in the static step. The steel plate was located at the
midspan of the beam based on the reference point created (RP= 1750, 465,150 which
is x,y,z respectively). The load applied on the loading plate was a concentrated load
(point) of magnitude 500KN to all the models and was applied in a static steps from
0.1 to 0.01 of the total load as shown in figure 27.
35
Figure 27: Static load being applied.
3.8 MESHING / DISCRETISATION OF THE MODELS
The concrete beam and the loading plate were assigned with an element type 3Dstress
that has a linear geometry order with C3D8R hexagonal elements dominating which
has eight (8) node linear brick (continuum solids) that reduces on the integration with
the application of hourglass control.
Meshing was done to all the models that is to say the concrete beam, the loading plate
and the reinforcements (main bars and stirrup).in finite analysis the accuracy of the
results element sizes is a function of the element size used (meshing size) where the
finner the messing the more the accurate results with more converging eugine values
though this slows down the static analysis since it will require more time to get the
job completed. And the course meshing has some variation in its results because it
generates few equations of the system and requires less time. In this thesis a justified
meshing size was adapted 50x50mm for the concrete models and 75x75mm size for
the steel loading plate for the master and slave interaction that was defined for
transfer of load from the plate elements to the beam elements this is shown in
figure28.
36
Figure 28: Meshed beam models
37
3.9 SUBMITION OF THE JOB/STATIC ANALYSIS.
From the static step that was created before the application of boundary condition
and the loading on to the models. Submission of the models for the static analysis was
done using a solution method called Newton Raphson’s method that sufficiently proves
to simulate how testing machines apply loads in steps describes a way of generating
the force, P versus displacements, u curve, whose shape is not known at the outset.
The initial displacement is imagined to be equal to zero, then a load is applied and the
corresponding displacement is searched. The analysis was complete when all the step
frame defined had converged because of sufficient maximum increment size 1 and the
minimum of 1E-005.
38
CHAPTER FOUR: RESULTS AND DISCUSSION
4.1. STRESS AND STRAIN DISTRIBUTION
The distribution of stresses for the FE models (B-solid, B-25.4mm, B-50.8mm, B-
76.2mm, B-101.6E50mm and B-127E100mm) in concrete are shown in figure 29 and
summarized in table5.
39
Figure 29: Stress distribution in concrete
Table 5: stress distribution of beams in concrete
BEAM MODELS STRESSES IN CONCRETE(mpa) INCREASE(%)
B-Solid 2.560 0.0
B-25.4mm 5.382 110.2
B-50.8mm 5.448 112.8
B-76.2mm 8.536 233.4
B-101.6E50mm 9.467 269.8
B-127E100mm 12.500 388.3
The maximum principal stress values in concrete for all modeled beams are listed in
table5The amount of stresses was increased to 110.2%, 112.8%, 233.4%,269.8%,
388.3% for beams B-25.4mm, B-50.8mm, B-76.2mm, B-101.6E50mm, B-127E100mm
respectively as shown in figure 29 and compared with the control solid beam. This
40
clearly shows that embedding pvc pipes at the center of the beams creates a big
margin value of stresses that ranges from 2-3 times of the normal stresses in the solid
model and in models whose pipes were placed with eccentricity from the center (B-
101.6E50mm and B-127E100mm) the value ranges from 4-6times the principle
stresses in solid beam. The increase in stresses at the web sides of the beams clearly
shows how weak concrete is in tension compared to compression and a mere damage
in the tension zone makes the beams to stress at its extreme fibers leading failures
and collapse of structural members.
Figure 30: Variation of stresses in concrete for all models
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
400.0
450.0
0
2
4
6
8
10
12
14
B-Solid B-25.4mm B-50.8mm B-76.2mm B-101.6E50mm
B-127E100mm
BEAM MODELS
INC
REA
SE %
STR
ESSE
S (M
PA
)
STRESSES IN CONCRETE
STRESSES IN CONCRETE(mpa) INCREASE(%)
41
4.1.1 Stress distribution in reinforcements
The distribution of principal stresses for the FE models (B-solid, B-25.4mm, B-50.8mm,
B-76.2mm, B-101.6E50mm and B-127E100mm) in reinforcements are shown in figure
31.
42
Figure 31: Stress distribution in reinforcement
The maximum stress values in bottom reinforcement (T25mm) for all modeled beams
are listed in table 6 The amount of stresses was increased to 14.7%,17.9% ,24.9%,
25.7%,34.1% for beams B-25.4mm, B-50.8mm,B-76.2mm,B-101.6E50mm,B-
127E100mm respectively as compared with the control solid beam. As discussed
earlier concrete behaviors is brittle as result of stress increase and conversely the
behavior of steel is ductile and rarely exhibits fractures under preferable condition
which means that steel enables concrete to behave in a ductile manner (this could be
43
a small value) when they are not stressed to drawn line nearing their yielding strength
and embedding pipes in the core or at an eccentricity below the core of the beam
weakens more the tension zone of the beam and of which concrete exhibits weakness
in tension thus stressing the tension reinforcements forcing them to yield when
concrete has already crushed. Therefor this can be strongly a firmed that embedding
pvc pipes for service utilities and use of hollow pre-stressed member increases on the
stresses at the bottom and leads to a decrease in the carrying capacities because of
the reduction in the gross cross-sectional area of the beam that would be in position
to resist the stresses that reach the bottom reinforcements thus leading to failure and
collapse of structural members.
Table 6: stresses in bottom layer reinforcements for all modeled beams
Figure 32: Chart of stress variation in models
RESULTS FOR STRAIN DISTRIBUTION IN THE MODELS
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
400.0
B-Solid B-25.4mm B-50.8mm B-76.2mm B-101.6E50mm B-127E100mm
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
Stre
ss(m
pa)
beam models
incr
eas
ed
%stresses in reinforcement at the bottom
STRESSES AT BOTTOM BARS(mpa) INCREASE(%)
BEAM MODELS STRESSES AT BOTTOM
BARS(mpa)
INCREASE(%)
B-Solid 260.2 0.0
B-25.4mm 298.4 14.7
B-50.8mm 306.7 17.9
B-76.2mm 325.0 24.9
B-101.6E50mm 327.0 25.7
B-127E100mm 348.9 34.1
44
Table 7 Presents strain distribution in the models whereby PEEQT represents
maximum strain values in concrete and PEEQ represents maximum strain values in
the bottom reinforcements as shown in plots of strain in models in figure 33 The
maximum values of strain in reinforcements and concrete are compared with the
initially set value which is the yield strain of steel(ES=0.0022) and maximum strain in
concrete beyond which concrete crushes (Ec=0.0035).
All reinforcement in the models exhibited maximum strain value that was greater than
the yield strain value (0.0022) this clearly means that all the tension reinforcements
yielded before the concrete precedes with cracking and thus had to fail in a ductile
manner though the ductility level decreased as the pipe diameter embedded in to the
beams increased and their location lowered from the center of the beam to bottom
reinforcements.
The solid beam experienced a high strain values that was greater than the limit strain
values and also the strain value in steel exceeded 0.0022 which is preferred section
failure (tension controlled section) that gives warning during its failure time and it
exhibits prominent cracks in the beams unlike B-25,4mm, B-50.8mm, B-76.2, B-
101.6mmE50mm, B-127E100mm.
Table 7: summaries of strain maximum values in concrete and bottom
reinforcements
BEAM MODELS PEEQT (concrete) PEEQ(reinforcement)
B-Solid 0.00780 0.0094
B-25.4mm 0.00210 0.0044
B-50.8mm 0.00149 0.0068
B-76.2mm 0.00185 0.0052
B-101.6E50mm 0.00165 0.0045
B-127E100mm 0.00130 0.0034
45
46
Figure 33: Strains in concrete models
47
Figure 34: Strain distribution in steel
LOAD –DEFLECTION BEHAVIOR
Figure 35 shows the load –deflection response for the modeled beams. Generally, all
the beams behavior from the initial to the first cracking load of 100-150KN was linear
which means that their stiffness or slope was constant until the first cracking and then
changed to a nonlinear form with varying slope then a substantial increase in
deflection with small increase in load occurred till reaching its ultimate load. A solid
beam exhibited a high ultimate load of 457KN with a high flexural capacity which is
seen clearly as the area under the load-deflection curve, undergoing large deflection.
48
The increase in percentage of the reduction of the ultimate loading capacities of the
beams as compared with control solid beam varied from 9%,21%,23%,28% and 34%
for B-24.5, B-50.8, B-76.2, B-101E50mm and B-127E100mm respectively. The
location of the pipes has a significant effect on the ultimate load of beams. For beams
B-101mmE50mm and B-127mmE100mm -diameter eccentric pipe, the ultimate load
reduced by 28% and 34% and this reduction was the larger than that of beam B-24.5,
B-50.8, B-76.2mm diameter pipe installed at its center. This is due to the installation
pipe near the tension reinforcement in beam this can be seen that the increasing pipe
sizes embedded in the beams lead to the increase in the reduction of the ultimate
loading capacity of the beam and as well as the reduction in stiffness This is the result
of a decrease in moment of inertia of beams cross sections due to the presence of
pipes. This is shown in table8
Figure 35: Load – midspan deflection
0
50000
100000
150000
200000
250000
300000
350000
400000
450000
500000
0 10 20 30 40 50 60 70
LOA
D P
(N)
MIDSPAN DEFLECTION(mm)
LOAD -DEFLECTION CURVE
B-101.6mmE50mm
B-127E100mm
B-76.2mm
B-SOLID
B-25.4mm
B-50.8mm
49
DEFLECTION DUCTILITY INDEX
A summary of deflection ductility index, midspan deflection at ultimate load, deflection at the
yield load, % increase in reduction of the ductility index, ultimate load and % increase in the
reduction ultimate load or load carrying capacities are presented in table 8
The deflection ductility index was calculated using the equation below
𝜇∆ =∆𝑈
∆𝑦
Where 𝜇∆ is the deflection ductility index, ∆𝑈 is the midspan deflection at ultimate load and
∆𝑦is the deflection at the yield load.
Table 8: Ductility index and ultimate loads
From table 8 the solid beam experienced a ductility index of 4.4 and for other beams
experienced 3.1,1.53,1.44,1.35 and 1.2 which shows an increasing percentage in the
reduction of ductility index expressed as 30%,65%,67%,69% and 73% for beams B-
24.5, B-50.8, B-76.2, B-101E50mm and B-127E100mm respectively in comparison
with the solid beam.
BEAM
MODELS
MIDSPAN.
DEFLECTION
AT
ULTIMATE
LOAD(mm)
DEFLECTION
AT YIELD
load(mm)
DUCTILITY
INDEX(∆u/∆y)
% increase
in
reduction
of ductility
index
ultimate
load
(KN)
%
increase
in
reduction
of
ultimate
load
B-Solid 66.12 15.03 4.4 0 457 0
B-25.4mm 33.55 10.82 3.1 30 415.9 9
B-50.8mm 25.68 16.78 1.53 65 360 21
B-76.2mm 24.01 16.67 1.44 67 352 23
B-
101.6E50mm 23.98 17.76 1.35 69 329 28
B-
127E100mm 22.45 18.71 1.2 73 302 34
50
The solid control beam(B-solid) exhibited a high ductility index which indicates that
the structural member was capable of undergoing large deformation prior to failure in
other words the steel yields before the crushing of the concrete this clearly defines
how tension controlled section behaves. There was a sharp percentage increase in
reduction of the ductility index the moment the pipe of 25.4mm diameter was
embedded at the center and the increase in doubled for the 50.8mm diameter pvc
pipe while at the center. It also increase to 69% and 73% for B-101mm and B-127mm
diameter for beams with an eccentric pipe at 50mm and 100mm respectively this
shows that the location of the pipes has a significant effect on the ductility of beams
and this because the would be cross-section area to resist the in plane tensile stresses
is replaced by the pipes that makes the bottom reinforcement and concrete susceptible
to rapid strain rate that makes the steel yield after the concrete has crushed or steel
yields and concrete crushes simultaneously thus exhibiting brittle failure modes.
CRACK PATERNS OF AND MODES FAILURES
Crack patterns
At load of 185KN, vertical fine cracks started to develop at the midspan of solid beam
(B-Solid) and at the approach of the supports from the midspan. As the internal load
was increased, few more visual flexural cracks developed at the midspan and the
further ends of the beam surface nearing the supports and comparing the cracking
area (crack intensity) of the solid beam with the modeled beams with pipes embedded
at the center or with an eccentric pipe was less. This is shown in figure 36.
All finite modeled beams with embedded varying pipe sizes experienced diagonal
cracks except for the solid beam. At loads of 70 to 150KN, few fine vertical flexural
cracks started to develop at mid span of these beams, followed by the destruction of
the bond between the reinforcing steel and the surrounding concrete at support. In
beams B-25.4mm and B-76.2mm more diagonal crack developed in region between
the face of support and the point load. As the external load increased, the diagonal
cracks widened and extended to the top compression fibers of the beams. Beam B-
50.8mm experienced a cracking patterns at ultimate strain that reasonably shows a
concentration of splitting cracks at the midspan of the beam yet the cracks were
51
experienced at lower loading step compared to the solid beam and beam B-25.4mm
as shown.
Modes of failures
In general, the failure modes were shear and the flexure with cracks propagated from
the extreme surface of beams and the supports (diagonal cracks). The solid beam
experienced flexural failure (when steel yields) which shows that the beam failed in
flexure which was a ductile one. Flexural failures are preferred to shear because the
beam attained its total flexural capacity at a high ultimate load compared to other
beams (those embedded with pipes) this is clearly observed under load –deflection
curve figure 35 and it presents warning before failure and the time of occurrence can
be predicted in real life of structures.
For the other beams whose core had been replaced with a pipe, it be at the center or
an eccentricity (B-24.5, B-50.8, B-76.2, B-101E50mm and B-127E100mm)
experienced a combination of failure modes which was flexure –shear and diagonal
tensile cracks. In addition, bond failure resulting from splitting of concrete in vertical
direction also occurred extending from the perimeter of pipe to the bottom face of
beam. Therefore the flexure –shear failure of which some exhibits diagonal cracks at
the end of the end of the beam are characterized under brittle failure modes by
crushing of concrete before the bottom reinforcement yields and this type of failure is
very acute because it does not give warning during it occurrence .later beam models
experienced such because of the reduction in flexural capacity and they could not
sustain large deformation as compared to the solid beam this is clearly shown under
load-deflection behavior for the models in figure 35
52
53
Figure 36: Crack patterns in concrete models
54
CHAPTER FIVE : CONCLUSIONS AND RECCOMMENDATION
5.1 Conclusion
Basing on the finite element analysis of the modeled beams with varying pipe size
embedded at the center or with an eccentric pipe, the following conclusions were
observed.
Stresses and strain in concrete and reinforcement
1. The pvc pipes embedded in the concrete beams led to an increase in stresses
for both concrete and reinforcement because of the reduction in the moment
of inertia and the would be cross section area to resist the in plane stresses is
not there and this clearly explains how the core of concrete beams helps in the
confinement process and absorbing the in plane compressive stresses.
Strength /loading capacity and stiffness
2. The installation of PVC pipes in reinforced concrete beams decreases the
strength and rigidity of the beams depending on sizes and locations of these
pipes.
3. The size of the pipe inserted in reinforced concrete beams influences the
capacity of the beams. For the beam with pipe of smaller diameter, the ultimate
strength decreased by 9% with respect to the ultimate strength of the solid
beam. The decrease in capacity reached to 28-34% for the beam with eccentric
pipe of larger diameter.
Ductility, load deflection behavior and failure modes in beams
4. A high ductility index indicates that a structural member is capable of
undergoing large deformation prior to failure and the ductility index ranging
from 3-5 is considered adequate for design of beams especially in areas of
seismic design and redistribution of moment.
55
5. The pipe inserted at the center of the beam decreases the ductility index of
beam more than the beam with an eccentric pipe installed below the center of
beam near the tension reinforcement. The central pipe decreased the ductility
index of the beam maximally by 1.44(67%), while eccentric pipe decreased the
ductility index maximally by 1.2(73%).
6. Solid beam exhibited flexural failure modes which are preferred it attains total
flexural failures at its highest ultimate carrying capacity compared to the beams
which were embedded with pipe at the center or with an eccentricity who
experienced a combination of failure modes that was flexure-shear failure
modes of which most exhibited diagonal cracks. Flexural failure (when
reinforcement yield) modes are preferred compared to shear or a combination
of failure modes because concrete crushing precedes yielding of
reinforcements.
7. For beams , the beam section should be designed as tension controlled section
with minimum steel percentage so as to predict the tension section failure
where the reinforcements yields before the concrete crushes and this can be
seen that before the concrete crushes , the beam can undergo noticeable large
deflection before it reaches its failure point and even the cracks are visible that
gives warning to the occupants and the engineers to vacate and rectify the
failure than other sections (compression controlled and balanced section which
exhibits brittle failure and they give no warning during their occurrence.
5.2 RECOMMENDATIONS
1. FEM software (ABAQUS/CEA) is quite challenging in understanding the concept
of how to prototype something into reality. From engineering back ground
material properties of the object are the one to define the behaviors of the
object of which some are rated linearly (over rating). This is not sufficient in
fem because one needs to define Non-linear material properties and also
defines the failure mechanisms for example for concrete models. Therefore,
concrete behavior can be better described with damage models, whereas
plasticity models better represent steel behavior. Nevertheless, since steel
56
brings additional ductility, the behavior of concrete belonging to reinforced
concrete can be even better described with models that combine damage and
plasticity. These models are particularly well suited for reproducing failure
modes that are based on tensile cracking and compression crushing concrete
which sufficiently provided reliable results this could help users of fem to exploit
more features of this tool of which I believe results my thesis could be in
position to act as a benchmark.
2. There is no need to take any precautions regarding the beam with central pipe
when the diameter of the pipe is about or less than a fifth (1/5) of the beams
width because however much there is reduction in the flexural capacity and
reduction in stiffness but still they exhibited ranged carrying capacities
compared to beams whose pipe size is greater than a fifth of the beam width
and was located at distance below the center of the beams.
3. The most preferable location to install pipe in reinforced concrete beams is at
the center of the beams away from the tension reinforcement in order to avoid
bond failure.
4. For larger diameters of pipes installed either at the center or below the center
of the beams, special design should be undertaken so as not to comprise the
carrying capacity of the beams for example providing two layers of bottom
reinforcement where the first layer could be at concrete cover and another one
at a distance of about 50-75mm from the first layer reinforcement and special
stirrup arrangement this helps the beam to undertake large deformation before
it reaches it failure.
57
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60
APPENDEX 1: FORMULAES
ɛ𝑡~𝑐𝑡 = ɛ𝑡- ɛ𝑜𝑡
𝑒𝑙
ɛ𝑜𝑡𝑒𝑙 = ơ/𝐸𝑜
ɛ𝑡~𝑝𝑙
= ɛ𝑡~𝑐𝑡 -
𝑑𝑡 ᵟ𝑡
(1−𝑑𝑡 )𝐸𝑜
ɛ𝑐~𝑙𝑛 = ɛ0 - ɛ𝑜𝑐
𝑒𝑙
ɛ𝑜𝑐𝑒𝑙 = ᵟ𝑐 /𝐸𝑜
ɛ𝑐~𝑝𝑙
= ɛ𝑐~𝑙𝑛 -
𝑑𝑐 ᵟ𝑐
(1−𝑑𝑐 )𝐸𝑜
61
APPENDIX 2: MODIFIED STRESS- STRAIN CURVES UNDER
COMPPRESSION AND TENSION
Typical compressive stress-strain relationship with damage properties (source
ABAQUS manual 2008)
Typical Tensional stress-strain relationship with damage properties ( source ABAQUS
manual 2008)