Mechanical Behavior of Reinforced Concrete Beams with … · 2019. 2. 24. · Mechanical Behavior...
Transcript of Mechanical Behavior of Reinforced Concrete Beams with … · 2019. 2. 24. · Mechanical Behavior...
Mechanical Behavior of Reinforced Concrete Beams with Embedded Steel Trusses using Non-
Linear FEM
والمدمجة المسلحة الخرسانیة للكمرات المیكانیكي السلوك دراسة
المحددة العناصر طریقة باستخدام فولاذیة جمالونات مع
By
Ragheb Ibrahim Salim
Supervised by
A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Civil Engineering – Design and Rehabilitation of
Structures
December/2018
Dr. Mohammed Arafa
Dr. Mamoun Al‐Qedra
زةــغ – ةــلامیــــــة الإســـــــــامعـالج
عمادة البحث العلمي والدراسات العلیا
الھنـــــدســــــــــــــةة ــــــــــــــــــــلیـك
قســـــــــــم الھنـدســــــة المدنیـــــــــة
برنامج تصمیـــم وتأھیـــل المنشـــآت
The Islamic University–Gaza
Deanship of Research and Graduate Studies
Faculty of Engineering
Civil Engineering Department
Design and Rehabilitation of Structures
Mechanical Behavior of Reinforced Concrete Beams with
Embedded Steel Trusses using Non-Linear FEM
Declaration
I understand the nature of plagiarism, and I am aware of the University’s policy on this.
The work provided in this thesis, unless otherwise referenced, is the researcher's own
work, and has not been submitted by others elsewhere for any other degree or
qualification.
:Student's name راغب إبراهيم سليم اسم الطالب:
:Signature التوقيع:
17/02/2019 التاريخ:
Date:
I
Abstract
Strengthening of reinforced concrete (RC) beams using embedded steel trusses is a
novel technique to enhance shear and flexural behavior of reinforced concrete beams.
This technique has advantages of being constructed rapidly and easily. Enhancing shear
performance of RC beams using embedded steel trusses is quite unknown within the
engineering community.
The main purpose of this research is to study the mechanical behavior of reinforced
concrete beams with embedded steel trusses using non-Linear FEM and investigate the
effect of different parameters on the behavior of these beams.
The behavior of RC beams was simulated using finite element method. The analyses
were conducted using the finite element computer program ABAQUS. From the
analyses the load-deflection relationships until failure, failure modes and crack patterns
were obtained and compared to the experimental results. The FEM results agreed well
with the experiments regarding failure mode and load capacity.
The validation models were used to investigate the influence of the shear span-to-depth
ratios, shear reinforcement, different shape of embedded truss and longitudinal
reinforcement. To verify the numerical results, a reference analytical model is
employed to calculate the ultimate shear strength of RC control beams and RC beams
with embedded steel trusses at different (a/d) ratios. The numerical results showed well
agreement with the analytical results.
The analysis indicates that the shear capacity of reinforced concrete beams using
embedded steel trusses is inversely dependent on the shear span-to-depth ratio. It also
shows that the shear reinforcement (stirrups) has almost small effect on shear capacity
of reinforcement concrete beams with embedded steel trusses.
The numerical results further demonstrate that longitudinal reinforcement have
significant effect on the shear capacity of reinforced concrete beams using embedded
steel trusses. It is also shows that reinforcement concrete beam using embedded steel
truss with diagonal steel angles in critical shear span only have almost the same failure
load and shear behavior of reinforcement concrete beam using embedded steel truss
with diagonal steel angles in full span.
II
لملخصا
المدمجة ھي تقنیة جدیدة الفولاذیة إن تقویة الكمرات الخرسانیة المسلحة باستخدام الجمالونات
. نفیذتتمیز ھذه التقنیة بسرعة وسھولة الت. لتحسین قوى القص والعزوم للكمرات الخرسانیة المسلحة
المدمجة ةالفولاذیإن تقنیة تحسین قوى القص للكمرات الخراسانیة المسلحة باستخدام الجمالونات
.معروفة لدى كثیر من المھندسین الإنشائیین غیر
ستخدام الأطروحة إلى دراسة السلوك المیكانیكي للكمرات الخرسانیة المسلحة با تھدف ھذه
مجة من خلال طریقة العناصر المحددة، وكذلك دراسة تأثیر بعض الجمالونات الفولاذیة المد
.العوامل على سلوك ھذه الكمرات
مج من خلال برنا تم محاكاة سلوك الكمرات الخرسانیة المسلحة باستخدام نظریة العناصر المحددة
)ABAQUS( وتم التأكد من دقة النموذج عن طریق مقارنة نتائج التحلیل مع النتائج المخبریة .
لدراسات سابقة من خلال مقارنة الحمل الذي یحدث عنده فشل العینة وكذلك شكل الفشل للعینة.
ة الكمرات الخراسانیة المسلحفاعلیة ) على a/dبعد التأكد من دقة النموذج، ثم دراسة تأثیر نسبة (
باستخدام الجمالونات الفولاذیة المدمجة. وكذلك تم دراسة تأثیر حدید التسلیح الطولي والعرضي
تم ولتأكید النتائج وتأثیر استخدام نماذج أخرى من الجمالونات المدمجة على فاعلیة ھذه التقنیة.
یل لدراسات السابقة، أظھرت نتائج التحلحساب قوى القص باستخدام النموذج الریاضي الموثق في ا
باستخدام نظریة العناصر المحددة توافق جید مع نتائج النموذج الریاضي.
ما ، حیث أنھ كلفي ھذه التقنیة) لھا تأثیر عكسي على تحمل قوى القص a/dأن نسبة ( بینت النتائج
ھلذلك أن الحدید العرضي ) قلت قوة تحمل الكمرة لقوى القص. وأثبتت النتائج كa/dزادت نسبة (
. على تحمل قوى القص، أما الحدید الطولي فلھ تأثیر كبیر قلیل تأثیر
الزوایاأن استخدام تجربة نماذج أخرى من الجمالونات المدمجة، فقد بینت النتائج خلالومن
المدمجة في منطقة تمركز قوى القص فقط، لھ تقریبا نفس التأثیر الفولاذیة المائلة في الجمالونات
الفولاذیة على كامل الجمالون. الزوایاالناتج عن استخدام
III
Dedication
To my father
To my mother
To my brothers
To my sisters
My wife,
And my daughter
For their endless support
IV
Acknowledgment
First and foremost, I would like to thank my supervisors, Dr. Mohammed Arafa and
Dr. Mamoun Alqedra, without whom this study would not be accomplished. Their
limitless support, encouragement, and valuable suggestions have guided me throughout
the duration of this thesis.
My deepest thanks to my father Mr. Ibrahim M. Salim, for his encouragement and
limitless support.
Finally, I would like to express my thanks to my wife for her emotional support and
understanding during all this time and my lovely kid Dema.
V
Table of Contents
Abstract ....................................................................................................................... I
Abstract in Arabic ......................................................................................................... II
Dedication .................................................................................................................. III
Acknowledgment ........................................................................................................ IV
Table of Contents ....................................................................................................... V
List of Tables ............................................................................................................ VIII
List of Figures ............................................................................................................. IX
List of abbreviations .................................................................................................... XI
Chapter 1 Introduction ................................................................................................... 1
1.1 General ............................................................................................................... 2
1.2 Problem Statement ............................................................................................... 3
1.3 Research Aim and Objectives ................................................................................ 3
1.4 Methodology ....................................................................................................... 4
1.5 Layout of the thesis .............................................................................................. 4
Chapter 2 Literature Review ....................................................................................... 5
2.1 Introduction ........................................................................................................ 6
2.2 Modes of failure of RC beams ............................................................................... 6
2.3 Shear behavior of RC beams ................................................................................. 7
2.3.1 Behavior of beams without web reinforcement .................................................. 7
2.3.1.1 Internal forces in a beam without stirrups ....................................................... 9
2.3.1.2 Factors affecting the shear strength of beams without web reinforcement ........... 9
2.3.2 Behavior of beams with web reinforcement..................................................... 11
2.4 Shear strengthening of RC beams ........................................................................ 11
2.4.1 Shear strengthening of RC beams with FRP Composites .................................. 11
2.4.2 Shear strengthening of RC beams using high strength concrete .......................... 12
2.4.3 Shear strengthening of RC beams using prestressed concrete ............................ 13
2.4.4 Shear strengthening of RC beams using high-strength steel ............................... 14
2.5 Shear strengthening of RC beams using embedded steel truss .................................. 14
Chapter 3 Mechanical Behavior and Finite Element Modeling of Materials................. 18
3.1 Introduction ...................................................................................................... 19
3.2 Crack models for concrete ................................................................................... 19
3.2.1 Discrete crack model .................................................................................... 19
3.2.2 Smeared crack models .................................................................................. 20
3.2.3 Concrete Damage Plasticity .......................................................................... 20
VI
3.3 FE modeling of reinforcement ............................................................................. 22
3.4 Modeling of the embedded truss - concrete interface .............................................. 22
3.5 Element Types ................................................................................................... 23
3.5.1 Concrete ..................................................................................................... 23
3.5.2 Reinforcement ............................................................................................. 23
3.5.3 Embedded truss elements .............................................................................. 23
3.6 Material Properties ............................................................................................. 23
3.6.1 Concrete ..................................................................................................... 23
3.6.2 Reinforcement ............................................................................................. 25
3.7 Geometry .......................................................................................................... 25
3.8 Meshing............................................................................................................ 28
3.9 Number of Load Increments ................................................................................ 29
3.10 Description of the adopted study ........................................................................ 29
3.11 Summary ........................................................................................................ 32
Chapter 4 Verification of Finite Element Models and Parametric Study ............................. 33
4.1 Introduction ...................................................................................................... 34
4.2 RC beam with conventional reinforcement ............................................................ 34
4.3 RC beam with flat plate steel embedded truss ........................................................ 36
4.4 RC beam with steel angle embedded truss ............................................................. 37
4.5 Parametric Study ............................................................................................... 40
4.6 Effect of Shear Span-To-Depth Ratios (a/d) .......................................................... 40
4.6.1Failure loads and Load-Deflection Response .................................................... 40
4.6.1.1 Von Mises Stress of embedded steel truss at failure load ................................ 42
4.6.1.2 Comparison between numerical and analytical models ................................... 42
4.6.2 Crack pattern and failure modes .................................................................... 44
4.6.2.1 RC beams with conventional reinforcement .................................................. 44
4.6.2.2 RC beams with embedded steel truss ........................................................... 46
4.7 Effect of shear reinforcement .............................................................................. 49
4.7.1 Failure loads and Load-Deflection Response ................................................... 49
4.7.2 Crack pattern and failure modes .................................................................... 51
4.8 Effect of longitudinal reinforcement ..................................................................... 53
4.8.1 Failure loads and Load-Deflection Response ................................................... 53
4.8.2 Crack pattern and failure modes .................................................................... 55
4.9 Effect of shape of the embedded steel truss ........................................................... 55
4.9.1 Failure loads and Load-Deflection Response ................................................... 56
4.9.2 Crack pattern and failure modes .................................................................... 57
VII
4.10 Summary ........................................................................................................ 57
Chapter 5 Conclusion and Recommendations............................................................. 59
5.1 Introduction ...................................................................................................... 60
5.2 Conclusion ........................................................................................................ 60
5.3 Recommendations .............................................................................................. 61
The Reference List ...................................................................................................... 63
VIII
List of Tables Table (3.1): Material properties of steel........................................................................ 31
Table (3.2): Material properties of concrete ………………………………………………. 31
Table (3.3): Descriptions of tested specimens……………………………………………... 31
Table (4.1): Ultimate failure loads for RC beams with conventional reinforcement and RC beams with embedded steel truss …………………………………………………………………………………….. 41
Table (4.2): Comparison between the calculated and numerical ultimate load carrying capacity of RC beams ............................................................................................................. 43
Table (4.3): Comparison of ultimate loads carrying capacity of HSTC beams with and without shear reinforcement (stirrups)................................................................................................ 50
IX
List of Figures Figure (3.1): Drucker–Prager boundary surface (Drucker and Prager 1952) ........................ 20
Figure (3.2): Concrete damage plasticity deviatoric plane (Systèmes 2014) ........................ 21
Figure (3.3): Compression hardening relationship for RC beam models 3 .......................... 24
Figure (3.4): Tension stiffening (displacement) for RC beam models ................................ 25
Figure (3.5): 3-D View of the RC beam modeled in ABAQUS ........................................ 26
Figure (3.6): 3-D View of the embedded conventional steel reinforcement modeled in
ABAQUS .................................................................................................................. 26
Figure (3.7): 3-D View of the embedded flat plate steel truss modeled in ABAQUS ........... 27
Figure (3.8): 3-D View of the embedded steel angle truss modeled in ABAQUS ................ 27
Figure (3.9): 3-D View of the concrete meshed model of RC beam. ................................. 28
Figure (3.10): Meshed model of the embedded flat plate steel truss. .................................. 28
Figure (3.11): Meshed model of the embedded steel angle truss. ....................................... 29
Figure (3.12): Profile and cross section detail of SRCB-1 and SRCB-2 (Zhang, Fu et al. 2016)
................................................................................................................................. 30
Figure(3.13): Profile and cross section detail of SRCB-3 (Zhang, Fu et al. 2016) ................ 30
Figure (3.14) : Profile and cross section detail of SRCB-4 (Zhang, Fu et al. 2016) .............. 31
Figure (4.1): Deflected shape for RC beam with conventional reinforcement. .................... 34
Figure (4.2) : load-deflection curve RC beam with conventional reinforcement .................. 35
Figure (4.3): crack pattern at failure for RC beam with conventional reinforcement ............ 35
Figure (4.4): Deflected shape for RC beam with flat plate steel embedded truss.................. 36
Figure (4.5): load-deflection curve for RC beam with flat plate embedded truss ................. 36
Figure (4.6): crack pattern at failure for RC beam with flat plate steel embedded truss ........ 37
Figure (4.7): Deflected shape for RC beam with steel angle embedded truss ...................... 37
Figure (4.8): load-deflection curve for RC beam with steel angle embedded truss ............... 38
Figure (4.9): Strain curve (strain gauge 1) of steel truss rod. ............................................. 38
Figure (4.10): Strain curve (strain gauge 2) of steel truss rod. ........................................... 39
Figure (4.11): crack pattern at failure for RC beam with steel angle embedded truss ........... 39
Figure (4.12): Shear span to depth ratio (a/d) .................................................................. 40
Figure (4.13): load-deflection relationship for RC beams with embedded steel truss at
different (a/d) ............................................................................................................. 41
Figure (4.14): Von Mises Stress of embedded steel truss at failure load ............................. 42
Figure (4.15): crack pattern at failure for RC beam with conventional reinforcement at (a/d) =
1 ............................................................................................................................... 44
Figure (4.16): crack pattern at failure for RC beam with conventional reinforcement at (a/d) =
1.25........................................................................................................................... 44
Figure (4.17): crack pattern at failure for RC beam with conventional reinforcement at (a/d) =
1.5 ............................................................................................................................ 45
Figure (4.18): crack pattern at failure for RC beam with conventional reinforcement at (a/d) =
1.75........................................................................................................................... 45
Figure (4.19): crack pattern at failure for RC beam with conventional reinforcement at (a/d) =
2 ............................................................................................................................... 45
Figure (4.20): crack pattern at failure for RC beam with conventional reinforcement at (a/d) =
2.25........................................................................................................................... 46
Figure (4.21): crack pattern at failure for RC beam with conventional reinforcement at (a/d) =
2.5 ............................................................................................................................ 46
Figure (4.22): crack pattern at failure for RC beam with embedded steel truss at (a/d)
= 1.25 ........................................................................................................................ 47
X
Figure(4.23): crack pattern at failure for RC beam with embedded steel truss at (a/d) =
1.5 ............................................................................................................................ 47
Figure (4.24): crack pattern at failure for RC beam with embedded steel truss at (a/d)
= 1.75 ........................................................................................................................ 47
Figure (4.25): crack pattern at failure for RC beam with embedded steel truss at (a/d) = 2 ... 48
Figure (4.26): crack pattern at failure for RC beam with embedded steel truss at (a/d)
= 2.25 ........................................................................................................................ 48
Figure (4.27): crack pattern at failure for RC beam with embedded steel truss at (a/d) =
1 ............................................................................................................................... 48
Figure (4.28): crack pattern at failure for RC beam with embedded steel truss at (a/d)
= 2.5 ......................................................................................................................... 49
Figure (4.29): load-deflection relationship for RC beams using embedded steel angle truss
without stirrups .......................................................................................................... 50
Figure (4.30): crack pattern at failure for the beam using embedded steel truss without stirrups
at (a/d) =1 .................................................................................................................. 51
Figure (4.31): crack pattern at failure for the beam using embedded steel truss without stirrups
at (a/d) =1.25 .............................................................................................................. 51
Figure (4.32): crack pattern at failure for the beam using embedded steel truss without stirrups
at (a/d) =1.5 ............................................................................................................... 52
Figure (4.33): crack pattern at failure for the beam using embedded steel truss without stirrups
at (a/d) =1.75 .............................................................................................................. 52
Figure (4.34): crack pattern at failure for the beam using embedded steel truss without stirrups
at (a/d) =2 .................................................................................................................. 52
Figure (4.35): crack pattern at failure for the beam using embedded steel truss without stirrups
at (a/d)=2.25 .............................................................................................................. 53
Figure (4.36): crack pattern at failure for the beam using embedded steel truss without stirrups
at (a/d) =2.5 ............................................................................................................... 53
Figure (4.37): load-deflection relationship for RC beam using embedded steel truss with
reduction in the long. reinf. Ratio and RC beam with conventional reinf. ........................... 54
Figure (4.38): load-deflection relationship for RC beams using embedded steel truss with and
without reduction in the long. reinf. ratio ....................................................................... 54
Figure (4.39): crack pattern at failure for RC beams using embedded steel angle truss with
reduction in the longitudinal reinforcement ratio ............................................................. 55
Figure (4.40): embedded steel truss with diagonal steel angles in critical shear span only .... 55
Figure (4.41): load-deflection relationship for RC beam using embedded steel truss with
diagonal steel angles in critical shear span only and RC beam with conventional reinf. ....... 56
Figure (4.42): load-deflection relationship for RC beam using embedded steel truss and RC
beam using embedded steel truss with diagonal steel angles in critical shear span only ........ 56
Figure (4.43): pattern of crack at failure for RC beam using embedded steel truss with
diagonal steel angles in critical shear span only .............................................................. 57
XI
List of abbreviations RC: Reinforced Concrete.
GFRP: Glass Fiber Reinforced Polymer.
FRP: Fiber Reinforced Polymer.
NSM: Near Surface Mounted.
CFRP: Carbon Fiber Reinforced Polymer.
HSRC: High Strength Reinforced Concrete.
HSTCB: Hybrid Steel Trussed Concrete Beam.
CDP: Concrete Damage Plasticity.
1
Chapter 1 Introduction
2
Chapter 1
Introduction
1.1 General
When principal tensile stresses within the shear region of a reinforced concrete beam
exceed the tensile strength of concrete, diagonal cracks develop in the beam, eventually
causing failure. The brittle nature of concrete causes the collapse to occur shortly after
the formation of the first crack, (Lim and Oh 1999). So, the shear failure pattern of
reinforced concrete beam is more critical and unsafe than the flexural failure pattern of
the same beam. Thus, in order to enhance the shear capacity of concrete beams, the
improvement of the brittle and poor performance of concrete in tension has been
proposed and studied in the last few decades.
Many researches have been conducted to enhance the shear strength of reinforced
concrete beams through using pre-stressed concrete, high strength concrete, steel fiber
concrete, ultra-high performance concrete, and high-strength steel. Nevertheless, these
enhancing measures need complex construction technology and special materials.
Another technique to enhance shear and flexure strength of reinforced concrete beams
is to adopt prefabricated steel trusses embedded in cast-in-place concrete beams, which
has advantages of being constructed rapidly and easily,(Zhang, Fu et al. 2016).
Shear strengthening of reinforced concrete beams using embedded steel trusses
technique is quite unknown within the engineering community. Moreover, it has been
used for three decades in Italy. Through this technique, the steel trusses can bear their
own weight and the weight of slab and fresh concrete without any provisional support
during a first assembly stage. Then, when the concrete become hard, the embedded
trusses can collaborate with the cast in place concrete. Although, interest for this
technique is growing up in many countries because of its advantages, there are not
specific regulations for this technique in American or European codes.
While experimental methods of investigation are extremely useful in obtaining
information about the mechanical behavior of reinforced concrete beams using
embedded steel trusses, the use of numerical models helps in developing a good
understanding of the behavior at lower cost.
3
In this research, non‐linear finite element analysis models for strengthening of
reinforced concrete beams with embedded steel trusses was presented. A finite element
software package ABAQUS was utilized to study the mechanical behavior of reinforced
concrete beams of small shear span-depth ratio with embedded steel trusses. The effect
of the following parameters on the behavior of strengthened beams namely, shear-span
to depth ratios, shear reinforcement at different shear-span to depth ratios, truss shape,
and longitudinal reinforcement were obtained.
1.2 Problem Statement
Shear strengthening of reinforced concrete beams using embedded steel trusses
technique has been used for three decades in Italy; in addition, the interest of this
technique is growing up in many countries. However, there are not specific regulations
for this technique in American and European codes and this technique is quite unknown
in engineering community, (Monaco 2016). Therefore, this study is to address this
problem and study the mechanical behavior of reinforced concrete beams with
embedded steel trusses and the influences of the shear span-to-depth ratios, shear
reinforcement, different shape of embedded truss and longitudinal on the shear behavior
of these beams.
Because of the high cost and large equipment required for full-scale tests, and small-
scale tests; development of a reliable analytical software model is desirable. The finite
element method computer software, ABAQUS, will be used to produce a model, which
closely resembles the experimental available data.
1.3 Research Aim and Objectives
The main aim of this research is to study the mechanical behavior of reinforced concrete
beams with embedded steel trusses using non-Linear FEM. The following are the
objectives of this study:
1. Determine the most capable methods that can be used for modeling and simulation
of reinforced concrete beams with embedded steel trusses.
2. Develop a 3D model reinforced concrete beams with embedded profile steel trusses
in ABAQUS software.
3. Validate the numerical model with respect to the experimental database available in
the literature.
4
4. Study the effect of selected parameters such as: shear span-depth ratio, shear
reinforcement at different shear-span to depth ratios, shape of truss and longitudinal
reinforcement.
1.4 Methodology
To achieve the stated objectives of this research, the following tasks will be
conducted:
1. Review of available literature for the finite element modeling and experimental
works related to the subject.
2. Utilize the finite element software ABAQUS to develop a 3-D non-linear model of
reinforced concrete beams of small shear span-depth ratio with embedded steel trusses.
3. Validate the developed model by means of experimental and numerical results from
a previous studies.
4. Evaluate the sensitivity of critical modeling parameters.
5. Draw conclusions and suggest recommendations.
1.5 Layout of the thesis
The contents of the chapters are presented below to give an overview of the structure
of the master thesis. This thesis consists of five chapters.
Chapter 1 discusses the plan of this thesis.
Chapter 2 presents more details on the literature review of modes of failure in RC
beams, as well as on the shear behavior in RC beams and its background. It also presents
many different techniques to increase the shear strength of reinforced concrete beams.
Chapter 3 presents a finite element analysis of reinforced concrete beams with
embedded steel trusses using ABAQUS based on the experimental results.
Chapter 4 presents a verification of ABAQUS finite element models and the effect
of the following parameters on the behavior of verified strengthened beams: shear-span
to depth ratios, shear reinforcement, truss shape, and longitudinal reinforcement.
Chapter 5 summarizes the main conclusions, and the overall findings of this project
with recommendations for further actions to be taken.
5
Chapter 2
Literature Review
6
Chapter 2
Literature Review
2.1 Introduction
In this chapter, modes of failure in RC beams are initially reviewed. Subsequently a
study of shear behavior in RC beams and its background has been discussed. Finally,
many different techniques to increase the shear strength of reinforced concrete beams
have been presented.
The research, performed throughout this project involves the use of prefabricated steel
trusses embedded in cast-in-place concrete beams that is a novel technique to enhance
shear and flexure strength of reinforced concrete beams. There has been limited
research work performed using of prefabricated steel trusses embedded in cast-in-place
concrete beams as a shear strengthening technique and hence, only a small number of
publications are available for reference work in this regard. The few literatures
pertaining to use of this technique as a shear strengthening technique which been
published till date has been reviewed in this chapter.
2.2 Modes of failure of RC beams
Several types of failure modes may occur when RC beams are subjected to either
uniformly distributed load or a concentrated load. There are four possible modes of
failure: anchorage failure, crushing failure, flexural failure and shear failure as
explained below:
Anchorage failure: for RC beam, the tensile force in the reinforcement at the end of
the beam must be transferred to the surrounding concrete by bond action between the
two materials; this anchorage requires a certain transmission length. When cracks
develop closer to support, the transmission length gets shorter, and then the beam fails
due to in-sufficient anchorage capacity.
Crushing failure: in compression zone, when the stresses caused by the increasing
load exceed the compressive strength of concrete, crushing of concrete may lead to
brittle failure of the beam. In the tensile zone, if the stresses in the reinforcement exceed
the yielding strength of the reinforcement before crushing failure happens, the beam
could fail for bending.
7
Flexural failure: the first crack would form in the region of maximum bending
moment, where the maximum principal stress attains the material tensile strength. As
the amount of reinforcement increased, several cracks are formed along the beam. In
early stages, these cracks are approximately normal to the beam axis and, as cracking
progresses, they grow in the presence of combined normal and shear stress as mixed-
mode flexural shear cracks.
Shear failure: this mode of failure is discussed in detail in the following subsection.
2.3 Shear behavior of RC beams
The study of shear behavior in concrete structures has been going on since a century
and the foundations of knowledge on shear were provided by Mörch in 1909. Until the
year 1955, researchers were of the view that shear was a simple problem to deal with.
Afterwards, researchers realized that shear in concrete beams cannot be designed as
traditionally as it was done earlier. It is only since the last six decades, researchers have
been focusing their work to develop a common and an efficient consensus on design
for shear which could be internationally acceptable. As a result, many theories have
been developed to explain the shear behavior in beams and to estimate its shear
capacity.
It is well known that shear failures are inherently more dangerous than flexural failures,
since shear failures normally exhibit fewer significant signs of distress and warnings
than flexural failures. The determination of shear strength of RC members is based on
several assumptions, all of which are not yet proved to be correct. It is important to
realize that there is a considerable disagreement in the research community about the
factors that most influence shear capacity.
The behavior of RC beams under shear may be categorized into two types; these types
of behavior are briefly discussed in the following subsections.
2.3.1 Behavior of beams without web reinforcement
The mechanism of the brittle-type diagonal tensile failure of RC beams with no shear
reinforcement (stirrups) is complex and not yet fully understood. The behavior of beams
failing in shear may vary widely, depending on the (a/d) ratio (shear span to effective
depth ratio) and the amount of web reinforcement. (Kani 1966)
8
According to Kani (1966) the shear spans can be divided into three types: short, slender,
and very slender shear spans. The term deep beam is also used to describe beams with
short shear spans.
Very short shear spans, with (a/d) from 0 to 1, develop inclined cracks joining the load
and the support. These cracks, in effect, destroy the horizontal shear flow from the
longitudinal steel to the compression zone, and the behavior changes from beam action
to arch action. Here, the reinforcement serves as the tension ties of a tied arch and has
a uniform tensile force from support to support. The most common mode of failure in
such a beam is an anchorage failure at the ends of the tension tie, (Kani 1966).
Beams with (a/d) ranging from 1 to 2.5 develop inclined cracks and, after some internal
redistribution of forces, carry some additional loads due to arch action. These beams
may fail by splitting failure, bond failure, shear tension, or shear compression failure,
(Kani 1966).
In slender shear spans, those having (a/d) from about 2.5 to about 6, the inclined cracks
disrupt equilibrium to such an extent that the beam fails at the inclined cracking load.
When the load is applied and gradually increased, flexural cracks appear in the mid-
span of the beams, which are more or less vertical in nature. With further increase of
load, inclined shear cracks develop in the beams, at about 1.5d– 2d distance from the
support, which are sometimes called primary shear cracks. The typical cracking in the
slender beams without transverse reinforcement, leading to the failure, involves two
branches. The first branch is the slightly inclined shear crack, with the typical height of
the flexural crack. The second branch of the crack, also called secondary shear crack or
critical crack, initiates from the tip of the first crack at a relatively flatter angle, splitting
the concrete in the compression zone. The failure is by shear compression due to the
crushing of concrete, without ample warning and at comparatively small deflection.
The nominal shear stress at the diagonal tension cracking at the development of the
second branch of inclined crack is taken as the shear capacity of the beam, (Kani 1966).
Very slender beams, with (a/d) greater than about 6, will fail in flexure prior to the
formation of inclined cracks, (Kani 1966).
9
2.3.1.1 Internal forces in a beam without stirrups
According to Fenwick and Pauley (1968) The factors assumed to be carrying shear
force in cracked concrete to the supports when no shear reinforcement are listed below:
1. Shear resistance of uncraked concrete (Vc).
2. Interlocking action of aggregates (Va).
3. Dowel Action of steel reinforcement (Vd).
Before cracking, a reinforced concrete beam acts like a homogeneous beam. After
bending cracks appear, shear displacement occurs along an inclined crack and dowel
action in reinforcements gets mobilized. When the two faces of a bending crack of
moderate width are given a shear displacement relative to each other, a number of
coarse aggregate particles projecting across the crack interlock with each other generate
significant shear resistance. As the applied shear force is increased, the dowel action is
the first to reach the capacity after which a proportionally large shear force is transferred
through aggregate interlock. The aggregate interlock mechanism is probably the next
to fail, necessitating a rapid transfer of a large shear force to the concrete compression
zone, which as a result of this sudden shear transfer, the beam often fails abruptly and
explosively, ( Fenwick and Pauley 1968).
2.3.1.2 Factors affecting the shear strength of beams without web reinforcement
Shear behavior of a beam without shear reinforcement is largely determined by six
factors: the tensile strength of the concrete, the longitudinal reinforcement ratio, the
ratio of shear span to effective depth, size of beam, and the presence of axial forces,
(Wright and MacGregor 2012).
These factors will be discussed below: -
1. Shear span to effective depth ratio. The shear span-to-depth ratio (a/d) is one of
the major parameters that affect the shear strength of reinforcement concrete beams.
All studies showed that shear strength of reinforcement concrete beams decreases with
the increase of the shear span-to-depth ratio (a/d), (Michael 1984, Shah and Ahmad
2008, Alhamad, Al Banna et al. 2017, Hu and Wu 2018). However, (Kani 1966) found
that the shear span-to-depth ratio (a/d) affects the inclined cracking shears and ultimate
shears of portions of members with a/d less than 2. For longer shear spans, (a/d) has
little effect on the inclined cracking shear and can be neglected.
2. Longitudinal reinforcement ratio. According to Lee and Kim (2008) the shear
strength of the RC beams drops significantly if the longitudinal reinforcement ratio
10
decreases below (1.2–1.5) percent. When the steel ratio, is small, flexural cracks extend
higher into the beam and open wider than would be the case for large values of steel
ratio. An increase in crack width causes a decrease in the maximum values of the
components of shear - the aggregate interlock and the dowel action - that are transferred
across the inclined cracks by dowel action or by shear stresses on the crack surfaces.
Similar, (Yu, Che et al. 2011) conducted seven experiments on reinforcement concrete
beams and commented that, there is a decreasing trend of shear strength as the
longitudinal reinforcement ratio decreases. Hamid, Ibrahim et al. (2016) showed that,
the shear capacity of concrete beams longitudinally reinforced with glass fiber-
reinforced polymer (GFRP) bars were affected by high reinforcement ratio of
longitudinal GFRP bars.
3. Size of beam. According to ACI specifications, the shear capacity is proportional to
the depth of the member. T. Shioya and Okada (1990) tested reinforced concrete beams
with depths ranging from 100 to 3000 mm. The results show that the shear stress at
failure decreases when the depth of the member increases. Similarly, Ghannoum
(1998) performed twelve experiments on nominal strength and high strength concrete
beams and commented that, the size effect is very evident in both nominal strength and
high strength concrete series. Furthermore, he found that the shallower specimens were
consistently able to resist higher shear stress than the deeper ones.
4. Compressive strength of concrete. The shear strength is the function of the
compressive strength. Yaseen (2016) showed that, the shear strength increases by
approximately 55 % and 72% when the compressive strength of concrete increased
from 434 MPa to 61 MPa and then to 119 MPa. On the other hand, Kong (1996) showed
that the concrete compressive strength within the range of 60 to 90 MPa had a little
influence on the shear strength of the beams.
5. Tensile strength of concrete. According to Angelakos, Bentz, and Collins (2001),
concrete tensile strength influence the shear failure process. The concrete tensile
strength mainly affects the shear failure modes. Abdul-Zaher, Abdul-Hafez et al.
(2016), showed that the shear strength increased by 11.43% and 28.57% when tensile
strength increased by 12.5 and 31.25 respectively.
6. Axial forces. Bhide and Collins (1989) researched the effect of axial tension on the
shear behavior of reinforcement concrete beams, he found that the axial tension
increases the inclined crack width and reduces the aggregate interlock, and hence, the
shear strength provided by the concrete is reduced. Shaaban (2004) conducted an
11
experimental and analytical investigation on the shear behavior of high strength fiber
reinforced concrete beams and found that, increasing the axial compression stress level
to 0.2 led to an increase in the first crack load, ultimate load by 24% and 10%, a
reduction in the deflection by (19-30%).
2.3.2 Behavior of beams with web reinforcement
The purpose of web reinforcement is to ensure that the full flexural capacity can be
developed. Prior to inclined crack, the strain in the stirrups is equal to the corresponding
strain of the concrete. Because concrete cracks at a very small strain, the stress in the
stirrups prior to inclined cracking will not exceed the compressive strength of the
concrete. Thus, stirrups do not prevent inclined cracks from forming; they work after
the cracks have formed, (Wright and MacGregor 2012).
After the first inclined crack, redistribution of shear stresses occurs, with some parts of
the shear being carried by the concrete and the rest by the stirrups, Vs. Further loading
will result in the shear stirrups carrying increasing shear, with the concrete contribution
remaining constant. The presence of shear reinforcements restricts the growth of
diagonal cracks and reduces their penetration into the compression zone. This leaves
more uncracked concrete in the compression zone for resisting the combined action of
shear and flexure. The stirrups also counteract the widening of cracks, making available
significant interface shear between the cracks. They also provide some measure of
restraint against the splitting of concrete along the longitudinal reinforcement, also
increasing the dowel action. With further loading and opening of cracks, the interface
shear decreases, forcing of (shear resistance of uncracked concrete and dowel action)
to increase at an accelerated rate the stirrups also start to yield. Soon, the failure of the
beam follows either by splitting (dowel) failure or by compression zone failure due to
the combined shear and compression, (Subramanian 2013).
2.4 Shear strengthening of RC beams
Various techniques have been used to increase shear strength for reinforced concrete
beams in the past decades. These techniques will be discussed below.
2.4.1 Shear strengthening of RC beams with Fiber Reinforced Polymer (FRP)
Composites
Thanasis (1998) studied the shear performance of strengthening of reinforced concrete
beams using epoxy - bonded composite materials in the form of laminates or fabrics
12
through experimentation research. The test results indicate that using epoxy - bonded
composite materials appears to be a highly effective technique. The experimental
results further demonstrated that, the effectiveness of FRP increases as the fibers’
direction becomes closer to the perpendicular to the diagonal crack.
Diagana et al. (2002) studied the shear performance of strengthening reinforced
concrete beams using bonded carbon fiber through experimentation research. The test
results indicate the shear capacity of strengthened beam affected by the applied
composite fabric area, the spacing between the steel stirrups, and the longitudinal steel
bars diameter of reinforced beam.
Al-Mahaidi et al. (2006) studied the bond characteristic between carbon fiber reinforced
polymer (CFRP) and concrete though numerical research. Twelve shear-lap specimens
were modeled using a combination of smeared and discrete cracks to investigate their
ultimate loads, and crack patterns. The numerical results indicated that, transverse
cracks have a significant influence on the bond stress distributions.
Rahal and Rumaih (2011) studied the shear performance of reinforced concrete T-
beams strengthened in shear using near surface mounted (NSM) carbon fiber reinforced
polymer (CFRP) bars and conventional steel reinforcing bars through experimentation
research. Four large scale reinforced concrete T-beams were tested to investigate their
structural performance and ultimate shear strength. Comparing with the common
reinforced concrete T-beams, the test results indicate that using of CFRP increased the
ultimate shear capacity 37% - 92%, reduced the width of the diagonal cracks, and
improve the flexural ductility. The experimental results further demonstrated that
orienting the NSM bars at 45°and extending their anchorage into the flange concrete
improved the efficiency of strengthening.
2.4.2 Shear strengthening of RC beams using high strength concrete
Ashour and Faisal (1992) presented test results of eighteen high strength reinforced
concrete beams without stirrups which were tested to investigate their flexural and shear
strength and behavior under loading. The results indicated that addition of fibers
increased the ultimate shear strength and improved flexural ductility, depending upon
the shear-span/depth ratio and transformed the mode of failure into a more ductile one.
13
Cladera and Marí (2005) studied the shear performance of reinforced concrete beams
that had a compressive strength ranged from 50 to 87 MPa through experimental
research. Eighteen reinforced concrete beams with and without shear reinforcement
were tested to investigate their structural behavior and ultimate shear strength. The test
results indicated, for beams without web reinforcement, the failure shear strength
increased as the concrete compressive strength increased in condition of beams without
web reinforcement. Fragile response for high-strength concrete beams with stirrups is
less than similar beams without web reinforcement.
Perera and Hiroshi (2013) studied the shear performance reinforced high-strength
concrete beams without shear reinforcement though experimental research. The
concrete beams which had compressive strength more than 100 MPa were tested to
investigate their structural performance and ultimate shear strength. It is found that
shear strength started to decrease due to the smooth fracture surface and brittleness.
Saeed and Abubaker (2016) studied the shear performance of high strength reinforced
concrete (HSRC) beams without stirrups through experimental research. Twelve
specimens were tested to investigate their structural performance and ultimate shear
strength. The test results indicated that, the ultimate shear strength of high strength
reinforced concrete (HSRC) beams without stirrups didn't increase significantly when
each of compressive strength and (a/d) ratio increased.
2.4.3 Shear strengthening of RC beams using prestressed concrete
Mikata, Inoue et al. (2001) studied the effects of prestress level and the distribution of
prestress over the section on shear capacity used the test results of prestressed concrete
beams without shear reinforcement and prestressed reinforced concrete beams. Test
results showed that the measured shear cracking load increases as the amount of
introduced prestress increase and when the stress distribution over the section changes
from triangular to rectangular, and the measured inclination of diagonal cracks
decreases with increasing the introduced prestress and decreases when the stress
distribution over the section is changed from triangular distribution to rectangular
distribution.
Hou, Nakamura et al. (2017) studied the shear resistance mechanism of reinforced
concrete and prestressed concrete tapered beams without stirrups. Three series of seven
beams with different parameters (a/d) ratios and prestress levels were tested. The results
14
indicated that when the prestress level increased, the inclination of the shear stress flow
decreased while the effective depth of the critical section became larger, and the shear
capacity of prestressed concrete tapered slender beam without stirrups became higher
than that of prestressed concrete constant depth beam without stirrups because of the
larger critical section.
2.4.4 Shear strengthening of RC beams using high-strength steel
Hassan and Paul (2008) studied the effects of high-strength steel on shear performance
using test results of concrete beams reinforced with either conventional- or high-
strength steel and tested up to failure. Six large-size beams were tested to study the
effect of the shear span depth ratio, concrete compressive strength, as well as the type
and the amount of longitudinal steel reinforcement. The experimental results showed
that using high-strength steel affected significantly the shear behavior of the concrete
beams. The experimental results further demonstrated that, using high-strength steel
improve the mode of failure to shear compression failure.
Lee and Sang-Woo (2011) studied the effects of high-strength stirrups shear capacity
using the test results of thirty-two simply supported RC beams. The results indicated all
the beams with stirrups with a yield strength ≤ 700 MPa (101,500 psi) failed after
reaching their yield strains, unrelatedly to the compressive strength of the concrete,
whereas the shear failure mode of the beams with a yield strength > 700 MPa (101,500
psi) was affected by the compressive strength of the concrete.
2.5 Shear strengthening of RC beams using embedded steel truss
Strengthening of reinforced concrete beam using embedded steel truss is a novel
technique to enhance shear and flexural behavior of reinforced concrete beam. In this
technique the steel truss is embedded in concrete core. The truss is usually made up of
steel angles that represent the bottom and the top chords, a system of steel angles or
steel plates welded in order to form the diagonals of the truss. The web elements work
as struts and ties, which can be inclined or according to different geometries.
The RC beams with embedded steel trusses represent a structural solution to improve
the shear performances and ultimate shear strength for RC beams with small shear span
to depth ratios, also frequently introduced within seismic frame structure.
15
The main advantages in their use are higher construction speed with the minimum site
labor and economical convenience. There has been limited research work performed
using of prefabricated steel trusses embedded in cast-in-place concrete beams as a shear
strengthening technique and hence, only a small number of publications are available
for reference work in this regard. The few literatures pertaining to use of this technique
as a shear strengthening technique, which been published until date, will be discussed
below.
Tesser and Scotta (2013) studied the shear and flexural behavior of composite steel
truss and concrete beams with inferior precast concrete base. Twenty-four beam
specimens with different depth were tested to investigate their shear and flexural
behavior. The results of flexural test indicated that, all the beams suffered yielding of
the steel bottom chord. The results of shear test showed that, all the beams were affected
by inclined cracks in the portion between the load application point closer to the support
and the support itself. The experimental results further demonstrated that, the specimens
which are characterized by shear span to depth ratio lower than 4.6, failed in shear.
Tullini and Minghini (2013) studied the bending moment capacity and shear connection
strength of composite beams with concrete – encased steel truss through finite element
formulation based on Newark's classical models. Simply supported beams subjected to
uniformly distributed load are considered. Test results indicated that, for medium spans,
the bending strength turns out to be proportional to the beam span length and the
ultimate conditions are in fluenced by the shear connections. In the other region,
including medium up to long spans, the bending strength is constant with the beam span
length and the ultimate conditions are essentially governed by the concrete com-
pressive strength at midspan. The numerical results demonstrate that, in the presence of
a brittle shear connection, the ultimate limit state is governed by the ultimate slip at the
supports. Whereas, in the presence of a ductile shear connection, the beam ductility is
limited by premature compression failure of concrete.
Colajanni, La Mendola et al. (2014) studied the transformation mechanism of stress in
hybrid steel trussed-concrete beams through experimentation research. Push-out tests
were used to investigate the tensile strength of beam specimens. The results of push-
out tests indicated, the specimens exhibit almost brittle failures due to the collapse of
the concrete in tension with the steel tensile web rebar that developed large inelastic
deformations, mainly concentrated at the ends close to the bottom steel plate.
16
Monti and Petrone (2015) developed shear capacity equations for composite steel truss
beam. These equations were obtained from developed mechanics-based shear models.
In the study two different equations were proposed: the first one was derived from an
analytical method that considered the contribution of the concrete whereas the second
equation obtained from a simplified method which does not consider the contribution
of the concrete. The results of both quotations showed significant agreement with
experimental and numerical results.
Campione, Colajanni et al. (2016) developed a calculation method for the pridiction of
the shear resistance of the hyprid steel trussed concrete beams. These beams are
prefabricated steel truss embedded within a concrete cast in palce. The analytical model
was developed based on the results of previous experimental three-point bending test.
Furthermore the analytical results were supported by the results of previous finit
element modeling.
Zhang, Fu et al. (2016) studied the effects of shear embedded steel trusses on shear
performance of reinforced concrete beams using the test results of five beam specimens
characterized by small shear span to depth ratio. The experimental results showed that,
using steel angle truss adding horizontal reinforcement increased the ultimate shear
strength compared with the common reinforced concrete beams. The experimental
results further confirmed that embedding the steel trusses in reinforced concrete beams
is indeed a promising novel technique that can significantly improve the structural
behavior of reinforced concrete beams in shear failure.
Monaco (2016) studied the shear performance of hybrid steel trussed concrete beams
(HSTCB) through finite element method. Numerical simulation of experimental three-
point bending test was developed for the aim. The numerical results indicated that,
detailing model with cohesive interaction is more accurate than simplified finite
element model with perfect bond hypothesis.
Kareemi, Petrone et al. (2016) studied the shear performance of composite beams made
of prefabricated steel trusses encase in structural concrete through experimentation
research. Eight beam specimens subjected to uniformly distributed load were tested to
investigate their ultimate shear strength. The experimental results demonstrate that
composite truss beams have an effective performance in shear.
17
Colajanni, La Mendola et al. (2017) investigated the mechanical response of hybrid
steel trussed-concrete beams under shear through experimentation research. Two series
of beams subjected to positive and negative moment respectively were tested to
investigate their ultimate shear strength and structural performance. The results
indicated that in almost all cases fragile shear failure occurred mainly because of the
crisis of the compressed concrete strut involved in the mechanism.
Ballarini, La Mendola et al. (2017) used finite element method to study the failure
behavior of hybrid steel trussed-concrete beams under three-point bending test. The
numerical model was compared with previous experimental data, and showed well
agreement to the experimental results. The numerical results indicated that, the small-
size beam exhibited shear failure, while the large-size beam experienced flexural
failure.
Colajanni, La Mendola et al. (2018)studied the failure mode and the stress transfer
mechanism of semi precast hybrid trussed-concrete beams through experimentation
research. Six beams specimens subjected to four-point bending with deferent (a/d)
ratios were tested. The experimental results showed that the flexural failure load and
the connection failure load are almost coincident for (a/d) ratios in the range 3.6 < (a/d)
< 4.8.
18
Chapter 3
Mechanical Behavior and
Finite Element Modeling
of Materials
19
Chapter 3
Mechanical Behavior and Finite Element Modeling of Materials
3.1 Introduction
Finite element method is a numerical procedure for finding estimated solutions to
boundary element problems for partial differential equations, which incorporates for
connecting numerous simple element equation over a large domain. Furthermore, it
subdivides the whole problem into smaller and simple parts called finite elements.
Nowadays, finite element analysis is an important tool for studying the structural
response of concrete. Experiments are the most accurate method to study the behavior
of reinforced concrete members, but it is expensive and cannot be counted on to give
all the information needed.
In recent years many finite element analysis packages has been developed, one of these
packages is ABAQUS\CAE, which is used in this study. ABAQUS/CAE is a software
tool for analysis a wide variety of finite element models, is divided in to different
modules: part, property, assembly, step, interaction, mesh, job and visualization for
defining geometry and material properties, applying boundary conditions and load,
generating mesh and analysis the model.(Systèmes 2014)
In this chapter, a finite element analysis of reinforced concrete beams using embedded
steel trusses has been performed to develop models based on the experimental results.
The models were created using ABAQUS/CAE. Three categories of beam are
considered in this research and the test data is taken from the experiment conducted by
Zhang, Fu et al. (2016).
3.2 Crack models for concrete
According to Bazant and Planas (1998), concrete cracking may be modelled using
discrete crack models, smeared crack models or concrete damage plasticity models. The
following sections present further details of these models.
3.2.1 Discrete crack model
In this approach, the crack is treated as a geometrical entity. This method implies a
continuous change in nodal connectivity, which consider serious drawback of the
approach, due to the nature of the finite element displacement method. The other
drawback of this approach that, the crack is constrained to follow a predefined path
20
along the element edges. It means that, to decide where and how the cracks may arise
a great amount of work is required, (ACI 2004).
3.2.2 Smeared crack models
The smeared crack concept is the counterpart of the discrete crack concept. In this
approach a cracked solid is imagined to be a continuum. Accordingly, the cracked
material can be described with a stress-strain relation. As this means that, the smeared
crack approach is a more attractive procedure than the discrete crack approach, because
it does not impose restrictions with respect to the orientation of the crack planes.
According to Rots and Blaauwendraad (1989), the smeared crack models can be
subdivided into fixed and rotating smeared crack. In the first kind of model, the
orientation of the crack is fixed during the entire computational process, whereas in the
second kind of model the orientation of the crack can rotate with the axes of principal
strain. The smeared crack model is most effective for use with concrete shell elements,
so that will not be used for this model.
3.2.3 Concrete Damage Plasticity (CDP)
According to the hypothesis of Drucker and Prager (1952), failure is determined by
non-dilatational strain energy and the boundary surface. The shape of energy and the
boundary surface in the stress space assumed to be the shape of a cone. The advantage
of this assumption is no complication in numerical applications due to surface
smoothness.
1 Figure (3.1): Drucker–Prager boundary surface (Drucker and Prager 1952)
21
Concrete damage plasticity model available in ABAQUS software is a modification of
Drucker–Prager hypothesis. According to the modification, the failure surface is
governed by parametric Kc, (Systèmes 2014).
Parametric Kc is a ratio of distances between the hydrostatic axis and the compression
and the tension meridian respectively in the deviatoric cross section. This ratio is always
higher than 0.5 and less than 1. The CDP model recommends to assume Kc = 2/3,
(Systèmes 2014).
Experimental results show that, the shapes of the plane’s meridians are curved. On the
other hand, CDP model assumes the plastic potential surface in the meridional plane is
formed of a hyperbola, so that the shape is adjusted through eccentricity. Parameter
eccentricity can be calculated as a ratio of tensile strength to compressive strength
(Kmiecik and Kamiński 2011). The CDP model recommends to assume eccentricity =
0.1, (Systèmes 2014).
Another parameter describing the state of the material is a ratio of the strength in the
biaxial state to the strength in the uniaxial state. The recommended value by Systèmes
(2014) for this ratio is 1.12. The last parameter describing the performance of concrete
under compound stress is dilation angle. In modeling dilation angle = 31 degree
recommended by (Systèmes 2014).
Figure (3.22): Concrete damage plasticity deviatoric plane (Systèmes 2014)
22
In this study, concrete damaged plasticity method was used to simulate the concrete
behavior. This method was used by (Salh 2014, Yosef Nezhad Arya 2015), and showed
well agreement with experimental results.
3.3 FE modeling of reinforcement
There are three approaches to model steel reinforcement in finite element models for
reinforcement concrete: the discrete model, the smeared model, and the embedded
model, (Tavárez 2001).
The reinforcement in the discrete model is connected to the nodes of the concrete mesh.
Therefore, the concrete will share same nodes and the steel meshed and the concrete
and the reinforcement will occupy the same regions. This approach suffers from the
drawback that the location of reinforcement restricts the mesh of the concrete and the
steel volume is not deducted from the volume of concrete. The smeared model assumes
that the reinforcement is uniformly distributed in the concrete element in defined region
of the finite element mesh.
In the embedded model the displacement of the reinforcement steel is compatible with
the surrounding concrete element. In this approach the restriction of the concrete mesh
is overcome because the stiffness of the reinforcement steel is evaluated separately from
the concrete. The embedded model, models reinforced steel efficiently specially, when
the reinforcement is complex, but it increases the run time and computational cost due
to the increasing of the number of nodes and the degree of freedom.
In this study, the embedded method has been used to model the reinforcement in the
reinforced concrete beams. (Salh 2014, Yosef Nezhad Arya 2015) applied this method
to model the reinforced steel and gave acceptable results compared to the experimental
evidence.
3.4 Modeling of the embedded truss - concrete interface
There are two models which deal with the case of reinforced concrete beams with
embedded steel truss. The first one provides a simplified modeling of the concrete-truss
elements interaction by tying the nodes of the two surfaces in contact; the second one
provides the insertion of cohesive contact elements between the truss elements and the
surrounding concrete. The simplified approach, which is used in this study is very easy
to apply in practice and generates a constraint between the nodes and the meshes of the
23
elements that rigidly connected each other. (Monaco 2016, Ballarini, La Mendola et al.
2017) used this method to model the reinforced steel and gave acceptable results
compared to the experimental evidence.
3.5 Element Types ABAQUS provides an extensive element library that helps efficiently to solve different
problems. Each element in ABAQUS has a unique name, such as B22, B23H, SC6R,
or C3D8R.
3.5.1 Concrete
To model the concrete in 3D models, an eight node linear brick, reduced integration
element which called C3D8R was used. This element has 8-nodes with three degree of
freedom at each node. These elements can predict cracking, plastic deformation and
crushing.
3.5.2 Reinforcement
The reinforcement bars were modelled by using 3D beam element. A 2-node beam
element in three-dimensional with linear interpolation formulations which called B31
was used. The steel bars were embedded into the concrete element, so that no interface
element was needed and perfect bond between concrete and reinforcement was
assumed.
3.5.3 Embedded truss elements
To model the elements of the embedded truss in 3D models, the simplest element type
C3D8R was used to develop favorite convergence and to reduce the computational
times. Monaco (2016) used this element to model truss elements, and showed well
agreement with experimental results.
3.6 Material Properties
3.6.1 Concrete
The ABAQUS program requires the uniaxial stress-strain relationship of concrete in
tension and compression in order to be able to model concrete. In this study for concrete
under compression a relationship proposed by Kabaila, Saenz et al. (1964) was adopted
as shown in equation (3.1). See Figure (3.3) for compression hardening relationship for
RC beam models.
(3.1)
24
In which δ is the compressive stress and δp is the experimentally determined maximum
compressive stress (41 MPa), ε is the strain and εp is the corresponding strain (0.0035),
and α is an experimentally determined coefficient representing the elastic modulus of
the concrete.
Figure (3.3): Compression hardening relationship for RC beam models 3
For concrete under uniaxial tension, the tension softening curve of Hordijk (1991) was
used as follows:
Where fc' = 41 MPa, ft = 4 MPa, wt and wcr are respectively the crack opening
displacement and the crack opening displacement at the complete release of stress, GF
is the required fracture energy, and C1=3 and C2=6.93 are constants obtained from
tensile test of concrete. In equation (3.4) da is the maximum aggregate size and it was
assumed to be 20 mm. See Figure (3.4) for tension stiffening (displacement) for RC
beam models.
0
5
10
15
20
25
30
35
40
45
0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004
Stre
ss M
pa
Strain
Compression Hardening
(3.3)
(3.4)
(3.2)
25
3.6.2 Reinforcement
An elastic-plastic constitutive relationship with strain hardening is assumed for
reinforcing steel and embedded steel truss. This model generally yields acceptable
results for the response prediction of RC members, (Neale, Ebead et al. 2005, Khan,
Al-Osta et al. 2017). According to Bǎzant (1982), the Poisson’s ratio was assumed to
be 0.3, and the elastic modulus was assumed to be 200 GPa.
3.7 Geometry
Three models are built to simulate the beam specimens as show in Figures (3.5) to (3.8).
The concrete part and truss part are done as 3D deformable solid elements, but the
reinforcement steel part is done as a two-node linear 3D truss element, then all parts are
merged together in the assembly module. The bond between the embedded truss
elements and the surrounding concrete was considered as perfect bond. Therefore, the
embedded option was used in defining the truss elements inside the host element which
was the concrete beam.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.05 0.1 0.15 0.2 0.25
Stre
ss M
pa
Displacement (mm)
TensionStiffening
4 Figure (3.4): Tension stiffening (displacement) for RC beam models
26
Figure (3.5): 3-D View of the RC beam modeled in ABAQUS 5
Figure (3.6): 3-D View of the embedded conventional steel reinforcement modeled in ABAQUS6
27
Figure (3.7): 3-D View of the embedded flat plate steel truss modeled in ABAQUS7
Figure (3.8): 3-D View of the embedded steel angle truss modeled in ABAQUS8
28
3.8 Meshing
Study of the mesh convergence is very important to determine a suitable mesh density.
ABAQUS/CAE provides different meshing techniques. In this study, for plain concrete
structured mesh is selected, and for the embedded truss multiple meshed are selected.
The mesh element for concrete and embedded truss is three-dimensional solid, which
called C3D8R. A 2-node beam element in three-dimensional with linear interpolation
formulations B31 is used for the rebar elements. According to Systèmes (2014) the best
size of the mesh range from 10% to 15% of the depth, so 40 mm element size had been
used in this model as shown in Figures (3.9) to (3.11).
Figure (3.9): 3-D View of the concrete meshed model of RC beam. 9
Figure (3.10): Meshed model of the embedded flat plate steel truss. 10
29
3.9 Number of Load Increments
The number of load increment may have a major effect on the accuracy of the results.
The structural response at each increment can be calculated by ABAQUS through
dividing the applied load into smaller increments. The increments characteristics are
specified in Step Module. ABAQUS analysis may miss key events in the response if
the increment is too large, such as the beginning of yielding in a member. However,
selecting a smaller step-size also leads to a larger computational expense, due to more
increments being calculated. The large number of increments also leads to larger post-
process files and accompanying post-process times. For the ABAQUS analyses
developed for this study, load increment size was chosen by trial and error to get an
accurate solution with reasonable computation time.
3.10 Description of the adopted study
To verify the finite element models, the test data from the experimental program
conducted by Zhang, Fu et al. (2016) was used. In this experiment, five full-scale beam
specimens were tested, named SRCB-1 to 5. These specimens (1800 mm long and 300
mm high) were designed according to ACI318-14. The specimens had a rectangular
cross section of 200 mm x 300 mm. The specimens had three steel bars with 22 mm
11 Figure (3.11): Meshed model of the embedded steel angle truss.
30
diameter in the bottom and two steel bars with 16 mm diameter in the top of the beam
for longitudinal reinforcement. The stirrups consisted of 8 mm diameter bars spaced at
150 mm were used for shear reinforcement in SRCB-1 and SRCB-3-5 and spaced at 75
mm in SRCB-2.
SRCB-1 and SRCB-2 were the control beams, Figure (3.12). SRCB-3 was a reinforced
concrete beam with embedded longitudinal angle steel and vertical flat steel as the
strengthening steel skeletons, Figure (3.13). SRCB-4 and SRCB-5 were the reinforced
concrete beams with an embedded longitudinal angle steel and diagonal angle steel as
the strengthening steel skeletons, but SRCB-5 had additional horizontal web
reinforcements, Figure (3.14). The material properties are listed in Table (3.1) and (3.2),
and the outline of the tested specimens is listed in Table (3.3).
Figure (3.12): Profile and cross section detail of SRCB-1 and SRCB-2 (Zhang, Fu et al. 2016)
13 Figure(3.13): Profile and cross section detail of SRCB-3 (Zhang, Fu et al. 2016)
31
Type of steel
Diameter,
thickness, (mm)
Yield strength
(MPa)
Ultimate strength
(MPa)
Modulus of
elasticity (GPa)
Reinforcement ɸ 8 363 465 210
Reinforcement ɸ 12 405 522 200
Reinforcement ɸ 16 378 472 200
Reinforcement ɸ22 393 557 200
Flat steel 30×4 266 363 200
Steel angle 40×40×4 345 519 200
Steel angle 30×30×3 348 522 200
Test specimen SRCB-1 SRCB-2 SRCB-3 SRCB-4 SRCB-5
Shear span to depth ratio (a/d) 1.4 1.4 1.5 1.5 1.5
Type of loading
Two Point
Loads
Two Point
Loads
Two Point
Loads
Two Point
Loads
Two Point
Loads
Figure (3.14) : Profile and cross section detail of SRCB-4 (Zhang, Fu et al. 2016)
Table (3.1): Material properties of steel (Zhang, Fu et al. 2016)
Table (3.2): Material properties of concrete (Zhang, Fu et al. 2016)
Table (3.3): Descriptions of tested specimens (Zhang, Fu et al. 2016)
32
3.11 Summary
This chapter provided an overview of the modeling techniques that are used to analyze
the mechanical behavior of RC beams with embedded steel truss. The analyses were
conducted using the finite element computer program ABAQUS.
The concrete was modelled using a plastic damage model. The embedded steel
reinforcement model has been used to simulate the reinforced steel and embedded truss.
This method of modelling the steel reinforcement solves the mesh restriction problem
that appears in discrete and smeared modelling of reinforcement, by evaluation of
stiffness of reinforcement elements separately from the concrete elements. This method
provides a perfect bond between the host element (concrete) and the slave element (steel
rebar and embedded truss). Moreover, in this method, the displacement of steel bars
and embedded truss will be compatible with the displacement of the surrounding
concrete elements.
A brief discussion was provided to issues related to the selection of mesh size and load
increment size that effected the accuracy of the solution.
Test specimen
SRCB-1 SRCB-2 SRCB-3 SRCB-4 SRCB-5
Compressive strength (MPa)
41.54 41.73 44.11 40.41 42.36
Modulus of elasticity (GPa)
34.11 34.10 34.56 33.72 34.13
33
Chapter 4 Verification of Finite Element Models and
Parametric Study
34
Chapter 4 Verification of Finite Element Models
and Parametric Study
4.1 Introduction
In this chapter, the developed finite element models were verified by comparing results
obtained from the FE analysis with results obtained from the adopted experimental
tests. The verification process was based on the following criteria: load – mid span
deflection curves, strain curves, and loads and deflection at failure. Having the finite
element model validated, a parametric study was performed using the verified model to
evaluate the effect of the following parameters on the behavior of strengthened beams:
shear-span to depth ratios, shear reinforcement, truss shape, and longitudinal
reinforcement.
4.2 RC beam with conventional reinforcement
Results of RC beams models with conventional reinforcement showed that, the load at
failure is 350 kN and the corresponding deflection is 5.6 mm. The deflected shape at
failure for the beam is shown in Figure (4.1)
A comparison of the load-deflection response between the FEM and the test results for
the RC beam with conventional reinforcement is shown in Figure (4.2). The failure load
predicted by FE model is 3% higher than that obtained from the test results, the mid-
span deflection at failure which predicted by FE analysis is 7% less than the mid-span
deflection obtained from the experimental results.
Figure (4.1): Deflected shape for RC beam with conventional reinforcement. 15
35
.
The crack pattern at failure load is shown in Figure (4.3). When applied loads increase,
diagonal tensile cracks appear. Then continued to develop and propagate toward the
loading point until the failure. Zhang, Fu et al. (2016) mentioned that the mode of
failure of RC beam with conventional reinforcement is shear tension failure, which
agrees very well with that obtained from the finite element model at the failure load as
shown in Figure (4.3).
Figure (4.3): crack pattern at failure for RC beam with conventional reinforcement 17
Figure (4.2) : load-deflection curve RC beam with conventional reinforcement 16
0
50
100
150
200
250
300
350
400
0 1 2 3 4 5 6 7 8
Load
(kN
)
Deflection (mm)
FE results
EXP results
36
4.3 RC beam with flat plate steel embedded truss
Results of RC beams models with flat plate steel embedded truss indicated that, the load
at failure is 465 kN and the corresponding deflection is 4.6 mm. The deflected shape at
failure for the beam is shown in Figure (4.4)
A comparison of the load-deflection response between the FEM and the test results for
RC beam with flat plate embedded truss is shown in Figure (4.5). The failure load
predicted by FE model is 4% less than that obtained from the test results, the mid-span
deflection at failure which predicted by FE analysis is 6% less than the mid-span
deflection obtained from the experimental results.
Figure (4.4): Deflected shape for RC beam with flat plate steel embedded truss 18
0 1 2 3 4 5 6 7
0
100
200
300
400
500
600
Deflection (mm)
Load
(kN
)
FE results
EXP results
Figure (4.5): load-deflection curve for RC beam with flat plate embedded truss 19
37
The crack pattern at failure load is shown in Figure (4.6). The beam failed in the shear-
flexural mode. It is obvious that the first cracks appear near the mid span and continued
to develop toward to the mid height of the beam. These cracks were followed by
diagonal shear cracks near the support. Diagonal shear cracks continued to develop and
propagate toward the loading point until the failure. Zhang, Fu et al. (2016) mentioned
that the mode of failure of RC beam with flat plate steel embedded truss is shear-
flexural failure, which agrees very well with that obtained from the finite element model
at the failure load as shown in figure (4.6).
4.4 RC beam with steel angle embedded truss
Results of RC beams models with steel angle embedded truss revealed that, the load at
failure is 510 kN and the corresponding deflection is 5.2 mm. The deflected shape at
failure for the beam is shown in Figure (4.7)
Figure (4.6): crack pattern at failure for RC beam with flat plate steel embedded truss 20
Figure (4.7): Deflected shape for RC beam with steel angle embedded truss 21
38
A comparison of the load-deflection response between the FEM and the test results for
RC beam with steel angle truss is shown in Figure (4.8). The failure load predicted by
FE analysis is 2% higher than the failure load obtained from the experimental results,
the mid-span deflection at failure which predicted by FE analysis is 8% less than the
mid-span deflection obtained from the experimental results.
As shown in Figure (3.14) in chapter (3) for RC beam with steel angle embedded truss,
two strain gauges are installed on the rod of steel truss, the comparison between strain
curves (strain gauge1 and strain gauge 2) are shown in Figures (4.9) and (4.10). These
figures indicated that the strain curves obtained from the finite element analysis agrees
well with the experimental data for the RC beam reinforced with embedded steel truss.
0 500 1000 1500 2000 2500 3000
0
100
200
300
400
500
600
700
Microstrain
Load
s (k
N)
FE results
Exp results
Figure (4.9): Strain curve (strain gauge 1) of steel truss rod. 23
Figure (4.8): load-deflection curve for RC beam with steel angle embedded truss 22
0
100
200
300
400
500
600
700
0 2 4 6 8 10 12 14
Load
(kN
)
Deflection (mm)
FE Resluts
EXP Results
39
The crack pattern at failure load is shown in Figure (4.11). When applied loads increase,
flexural cracks appear near the mid span and continue to develop toward to the mid
height of the beam. Then, diagonal shear cracks appear and continue to develop and
propagate toward the loading point until the failure. Zhang, Fu et al. (2016) mentioned
that the mode of failure of RC beam with steel angle embedded truss is shear-flexural
failure, which agrees very well with that obtained from the finite element model at the
failure load as shown in figure (4.11).
Figure (4.11): crack pattern at failure for RC beam with steel angle embedded truss 25
0 500 1000 1500 2000 2500 3000
0
100
200
300
400
500
600
Microstrain
Load
s (k
N)
FE results
Exp results
Figure (4.10): Strain curve (strain gauge 2) of steel truss rod.24
40
4.5 Parametric Study
As seen in the previous section, the developed FE models for the RC beam with
conventional reinforcement, RC beam with flat plate embedded truss and RC beam with
steel angle truss embedded truss are able to predict the failure loads and the failure
modes very closely to what were observed in the experiments. According to
experimental and finite element analysis results, the steel angle truss is the optimum
layout with respect to the plate steel truss when embedded in concrete, therefore, a
parametric study was performed to evaluate the effect of different parameters on RC
beam with steel angle embedded truss as explained in the following sections.
4.6 Effect of Shear Span-To-Depth Ratios (a/d)
To evaluate the effect of (a/d) ratio on the behavior of reinforced concrete beam with
embedded steel truss, fourteen beams model with different (a/d) ranged from 1 to 2.5
were analyzed using the verified FE model. The different (a/d) ratios were attained by
changing the distance between the loading points as shown in Figure (4.12).
4.6.1Failure loads and Load-Deflection Response
Table (4.1) shows the ultimate failure load for RC beams with conventional
reinforcement and RC beams with embedded steel truss. A comparison between two
beams typologies shows that, all RC beams with embedded steel truss showed a
significant increase in failure loads. For all RC with conventional reinforcement with
Figure (4.12): Shear span to depth ratio (a/d) 26
41
different (a/d) ratios, a fixed reinforced steel content of 2.12 was used, while the steel
content of 3.25 was used in all RC beams with embedded steel trusses.
Table (4.1): Ultimate failure loads for RC beams with conventional reinforcement and RC beams with embedded steel truss 4
(a/d) RC beams with
conventional reinforcement (kN)
RC beams with embedded steel
truss (kN) Percentage increased (%)
1 480 780 62.5%
1.25 420 650 54.7%
1.5 365 550 50.6%
1.75 300 475 58.3%
2 270 400 48.1%
2.25 235 365 55.3%
2.5 217 340 56.6%
It is clear form table (4.1) that, the ultimate failure load decreases with the increase in
the (a/d) ratios. The increase in ultimate failure load is much noticeable for lower (a/d)
ratios. The increase in the failure load for RC beams with conventional reinforcement
and RC beams with embedded steel truss is respectively about 129% and 121% when
the (a/d) ratios decreased from 2.5 to 1. It can be concluded that, the rate of increasing
in the ultimate load is almost constant with the increase in the (a/d) ratios for RC beams
with conventional reinforcement and RC beams with embedded steel truss.
0
100
200
300
400
500
600
700
800
900
0 1 2 3 4 5 6 7
Load
s (k
N)
Deflection (mm)
(a/d)=1
(a/d)=1.25
(a/d)=1.5
(a/d)=1.75
(a/d)=2
(a/d)=2.25
(a/d)=2.5
Figure (4.13): load-deflection relationship for RC beams with embedded steel truss at different (a/d) 27
42
Figure (4.13) shows the deflection of RC beams with embedded steel truss at different
(a/d) ratios. It can be seen that increasing of the (a/d) ratios increases the beam
maximum mid-span deflection. RC beams with embedded steel truss at (a/d) ratio = 1,
shows maximum load at failure and maximum central deflection value.
4.6.1.1 Von Mises Stress of embedded steel truss at failure load
Von Mises Stress was used to describe the distribution of stress, and color in each mesh
showed the stress value. The stress increase is obtained when color turns from blue to
red, and the von Mises Stress values can be obtained. The Von Mises Stress of
embedded steel truss was shown in Figure (4.14). The maximum Von Mises Stress, 348
MPa appears in the bottom of the central region of the embedded steel truss, around the
supports and around the applying loads. The stress increased from 217 MPa to 295 MPa
at the diagonal steel angles.
4.6.1.2 Comparison between numerical and analytical models
To verify the numerical results, the ultimate shear strength of RC beams with
conventional reinforcement and RC beams with embedded steel trusses was calculated
using equations (4.1) to (4.5) which were derived by Zhang, Fu et al. (2016).
V� = ∑ �������∑ �������
�������
�−
∑ �������
���
� =��±√������
��
� = �∑ ����
��� ��
����� − (����)�
� = ���
� ∑ �������
�−
(∑ �������∑ ��������∑ ���)∑ �������
����
����
����
����
(4.1)
(4.2)
(4.3)
(4.4)
Figure (4.14): Von Mises Stress of embedded steel truss at failure load 28
43
� =�∑ �������∑ ��������∑ ����
�������
���� �
�
����
Where ∑Tsi is the total ultimate tensile forces of longitudinal reinforcement and angle
steel; ∑Tvi is the total ultimate tensile forces of vertical stirrups and vertical component
of steel angle; hsi are the distances from the top fibers of the beam to the centroidal
position of longitudinal reinforcements and angle steel; dvi are the distances from the
loading point to the centroidal position of vertical stirrups and vertical component of
angle steel; a is the shear span of the beam; b is the width of the beam; Vu is the ultimate
shear-flexural strength of the reinforcement concrete beam; υ is the softened coefficient
of compressive strength of concrete; and x is the equivalent compressive depth of the
beam in the shear compression zone.
Comparison between the calculated and numerical ultimate load carrying capacity of
RC beams are presented in Table (4.2). It is clear that, the numerical results agree well
as compared with the analytical results.
Table (4.2): Comparison between the calculated and numerical ultimate load carrying capacity of RC beams 5
a/d
FEM results Calculated results
Control Beams, (kN)
Beams with embedded truss
reinf. (kN)
Control Beams, (kN)
Beams with embedded truss
reinf. (kN)
1 480 780 475.856 745.504
1.25 420 650 406.451 612.906
1.5 365 550 354.116 522.201
1.75 300 475 285.426 448.535
2 270 400 251.076 391.775
2.25 235 365 223.902 347.024
2.5 217 340 201.906 311.018
(4.5)
44
4.6.2 Crack pattern and failure modes
The following sections explained the crack pattern and the modes of failure for RC
beams with conventional reinforcement and RC beams with embedded steel angle truss
at deferent (a/d) ratios ranged from 1 to 2.5.
4.6.2.1 RC beams with conventional reinforcement
For the beams with conventional reinforcement and having (a/d) ratios between 1 and
2 the crack pattern is similar as shown in Figure 4.15 to 4.19. The beams failed in shear
failure mode. For these beams diagonal shear cracks appears near to supports when the
applied load increases. As the load increased, the diagonal cracks developed further up
to the critical failure of the beams. For beams with (a/d) = 2, few
hairline flexural cracks were observed.
Figure (4.15): crack pattern at failure for RC beam with conventional reinforcement at (a/d) = 1 29
Figure (4.16): crack pattern at failure for RC beam with conventional reinforcement at (a/d) = 1.25 30
45
Figure (4.17): crack pattern at failure for RC beam with conventional reinforcement at (a/d) = 1.5 31
Figure (4.18): crack pattern at failure for RC beam with conventional reinforcement at (a/d) = 1.75 32
Figure (4.19): crack pattern at failure for RC beam with conventional reinforcement at (a/d) = 2 33
46
For the beams with (a/d) ratios 2.25 and 2.5, the beams failed in shear - flexural and
flexural modes respectively as shown in Figure 4.20 and 4.21. For thes beams vertical
cracks appear in the mid span region between the two point loads when the applied load
increases. As the load increased, the flexural cracks developed further up to the critical
failure of the beams. Flexural failure of these two beams may be occurred due to the
distance between the two loading points, this distance is less than one third clear span,
according to ASTM (2010) this test can be considered as flexural test.
.
4.6.2.2 RC beams with embedded steel truss
For the RC beams with embedded steel truss and having (a/d) ratios between 1.25 and
2.25, flexural cracks were formed in the bottom of the central region of the beam and
continued to develop and elongate toward the mid height of the beams. Then diagonal
shear cracks began to form in the critical shear span regions followed by the appearance
Figure (4.20): crack pattern at failure for RC beam with conventional reinforcement at (a/d) = 2.25 34
Figure (4.21): crack pattern at failure for RC beam with conventional reinforcement at (a/d) = 2.5 35
47
of the main diagonal cracks. Flexural cracks kept their length and width until failure.
This beams failed in shear-flexural mode as shown in Figure 4.22 to 4.26
Figure (4.22): crack pattern at failure for RC beam with embedded steel truss at (a/d) = 1.25 36
Figure(4.23): crack pattern at failure for RC beam with embedded steel truss at (a/d) = 1.5 37
Figure (4.24): crack pattern at failure for RC beam with embedded steel truss at (a/d) = 1.7538
48
For the beam with (a/d) ratio = 1, the RC beam failed in shear mode as shown in Figure
4.27. It is clear that from the Figure, when applied loads increase few hairline vertical
cracks appears. As the load increased diagonal shear cracks appears near to supports
and developed further up to the critical failure of the beam.
Figure (4.25): crack pattern at failure for RC beam with embedded steel truss at (a/d) = 239
Figure (4.26): crack pattern at failure for RC beam with embedded steel truss at (a/d) = 2.25 40
Figure (4.27): crack pattern at failure for RC beam with embedded steel truss at (a/d) = 1 41
49
For the beam with (a/d) ratio = 2.5, the RC beam failed in flexural mode as shown in
Figure 4.28. It is clear that from the Figure when applied loads increase flexural cracks
appears. As the load increased the flexural cracks developed further up to the critical
failure of the beam. Hairline diagonal cracks were observed before failure.
4.7 Effect of shear reinforcement
RC beams with embedded steel truss had stirrups consisted of 8 mm diameter bars
spaced at 150 mm as shown in Figure (3.6) in chapter (3). To evaluate the effect of
shear reinforcement (stirrups) on the behavior of RC beam with embedded steel truss,
seven reinforced concrete beams using embedded steel truss without stirrups model
with different (a/d) ratios ranged from 1 to 2.5 were analyzed using the verified FE
model.
4.7.1 Failure loads and Load-Deflection Response
Table (4.3) shows the comparison of ultimate loads carrying capacity of RC beams
using embedded steel truss with and without shear reinforcement (stirrups). As seen
from the table, web reinforcements (stirrups) have a small effect on the ultimate loads
carrying capacity of RC beams using embedded steel truss.
Figure (4.28): crack pattern at failure for RC beam with embedded steel truss at (a/d) = 2.5 42
50
Table (4.3): Comparison of ultimate loads carrying capacity of HSTC beams with and without shear reinforcement (stirrups) 6
a/d RC beams using
embedded steel truss with stirrups, (kN)
RC beams using embedded steel truss without stirrups, (kN)
% of Reduction
1 780 720 7.7
1.25 650 585 10.0
1.5 550 498 9.5
1.75 475 425 10.5
2 400 380 5.0
2.25 365 342 6.3
2.5 340 310 7.3
From the Table (4.3) it is clear that, for (a/d) ratios between 1 and 2 the failure load
decrease significantly when (a/d) ratio decrease. For (a/d) ratios between 2 and 2.5 the
reduction rate in failure load is minimal. Figure (4.29) shows a variation of the failure
loads with different (a/d) ratios predicted by the FE models. It is cleared from the plot
that the failure loads decreases with the increase in the (a/d) ratios. The increase in
failure loads is much perceptible for lower (a/d) ratios.
Figure (4.29): load-deflection relationship for RC beams using embedded steel angle
truss without stirrups 43
0
100
200
300
400
500
600
700
800
900
0 1 2 3 4 5 6 7
Load
s (k
N)
Deflection (mm)
a/d=1
a/d=1.25
a/d =1.5
a/d=1.75
a/d=2
a/d=2.25
a/d=2.5
51
4.7.2 Crack pattern and failure modes
Figure 4.30 to 4.36 shows the crake pattern for RC beams using embedded steel truss
without shear reinforcement. It can be concluded that, RC beams using embedded steel
truss without shear reinforcement had almost the same behavior and failure pattern of
the RC beams with embedded steel truss and with shear reinforcement at different (a/d)
ratios.
Figure (4.30): crack pattern at failure for the beam using embedded steel truss without stirrups at (a/d) =1 44
Figure (4.31): crack pattern at failure for the beam using embedded steel truss without stirrups at (a/d) =1.25 45
52
Figure (4.32): crack pattern at failure for the beam using embedded steel truss without stirrups at (a/d) =1.5 46
Figure (4.34): crack pattern at failure for the beam using embedded steel truss without stirrups at (a/d) =2 48 49
Figure (4.33): crack pattern at failure for the beam using embedded steel truss without stirrups at (a/d) =1.7547
53
4.8 Effect of longitudinal reinforcement
To evaluate the effect of longitudinal reinforcement on the behavior of reinforced
concrete beam with embedded steel truss, longitudinal reinforcement ratio was reduced
in the verified FE model. Three steel bars with 16 mm diameter were used instead of
three steel bars with 22 mm diameter in the bottom of the verified simulated beam to
get the same longitudinal reinforcement ratio of the control beam.
4.8.1 Failure loads and Load-Deflection Response
A comparison of failure load with the decrease in the longitudinal reinforcement ratio
for RC beam using embedded steel angle truss presented graphically in Figure (4.37).
As seen from the figure, RC beam using embedded steel truss with the decrease in the
longitudinal reinforcement ratio did not show a significant increase in failure load
compared to the RC beam with conventional reinforcement.
Figure (4.35): crack pattern at failure for the beam using embedded steel truss without stirrups at (a/d)=2.25 50
Figure (4.36): crack pattern at failure for the beam using embedded steel truss without stirrups at (a/d) =2.5 51
54
It can be seen from the Figure (4.37), the failure load obtained by FEA is 380 kN, and
the mid span deflection at failure is 5.5 mm. Compared to the RC beam with embedded
steel truss, reduction of the longitudinal reinforcement ratio, showed a significant
decrease in failure load. As seen in Figure (4.38) the failure load obtained from FEA is
decreased for 510 kN to 380 kN.
0 1 2 3 4 5 6 7
0
50
100
150
200
250
300
350
400
Defection (mm)
Load
(kN
)
RC beam using embedded steeltruss with reduction in thelong. reinf. ratioRC beam wiith conventionalreinf.
Figure (4.37): load-deflection relationship for RC beam using embedded steel truss with reduction in the long. reinf. Ratio and RC beam with conventional reinf.52
0 1 2 3 4 5 6 7
0
100
200
300
400
500
600
Deflection (mm)
Load
(kN
)
RC beam with embeddedsteel truss without reductionlong. Reinf. ratioRC beam with embeddedsteel truss with reductionlong. Reinf. ratio
Figure (4.38): load-deflection relationship for RC beams using embedded steel truss with and without reduction in the long. reinf. ratio 53
55
4.8.2 Crack pattern and failure modes
Compared to RC beam with conventional reinforcement, embedded steel truss
improves the mode of failure at the same longitudinal reinforcement ratio. As seen in
Figure (4.39) RC beam using embedded steel truss with the reduced in the longitudinal
reinforcement ratio failed in shear flexural mode, whereas RC beam with conventional
reinforcement failed in shear tension mode at the same longitudinal reinforcement ratio.
4.9 Effect of shape of the embedded steel truss
To evaluate the effect of shape of the embedded truss, diagonal steel angles were used
in critical shear span of the verified simulated beam as shown in Figure (4.40).
Figure (4.40): embedded steel truss with diagonal steel angles in critical shear span only 55
Figure (4.39): crack pattern at failure for RC beams using embedded steel angle truss with reduction in the longitudinal reinforcement ratio 54
56
4.9.1 Failure loads and Load-Deflection Response
Compared to RC beam with conventional reinforcement, using embedded steel truss
with diagonal angle steel in critical shear span only, showed a significant increase in
failure load as seen in Figure (4.41). It is clear that, using embedded steel truss with
diagonal angle steel in critical shear span only increased failure load from 350 kN to
465 kN, that mean the failure load increased by 30%.
0 2 4 6 8 10 12 14
0
100
200
300
400
500
600
deflection (mm)
Load
(kN
)
RC beam with embeddedsteel truss (diagonal steelangles in shear span only)RC beam with conventionalreinforcement
Figure (4.41): load-deflection relationship for RC beam using embedded steel truss with diagonal steel angles in critical shear span only and RC beam with conventional reinf. 56
0 2 4 6 8 10 12 14
0
100
200
300
400
500
600
700
Deflection (mm)
Load
(kN
)
RC beam with embeddedsteel truss
RC beam with embeddedstee truss (diagonal steelin shear span only)
Figure (4.42): load-deflection relationship for RC beam using embedded steel truss and RC beam using embedded steel truss with diagonal steel angles in critical shear span only
57
Compared to RC beam with embedded steel truss in full span length, using embedded
steel truss with diagonal angle steel in critical shear span only did not show a significant
decrease in failure load as seen in Figure (4.42). It is clear from the figure that, using
embedded steel truss with diagonal angle steel in critical shear span only decreased
failure load from 510 kN to 465 kN, that mean the failure load decreased by 9% only.
4.9.2 Crack pattern and failure modes
Compared to the RC beam with conventional reinforcement, using embedded steel truss
with diagonal angle steel in critical shear span only, showed an improvement in beam
ductility and mode of failure.
As seen in Figure (4.43) using embedded steel truss with diagonal steel angles in critical
shear span only failed in shear flexural mode, while the RC beam with conventional
reinforcement failed in shear tension mode at the same longitudinal reinforcement ratio.
For RC beams with embedded steel angle truss, if diagonal steel angles were used in
the critical shear span only or in full span, we can get the same mode of failure. In both
cases, the beams failed in shear flexural mode. 4.10 Summary
In this chapter, the developed finite element models were verified by comparing the
results obtained from the FE analysis with results obtained from the adopted
experimental test. The FEM results agree will with the experiments regarding failure
mode and load capacity.
Figure (4.43): pattern of crack at failure for RC beam using embedded steel truss with diagonal steel angles in critical shear span only5 8
58
The validation models were used to investigate the influence of the shear span-to-depth
ratios, shear reinforcement, different shape of embedded truss and longitudinal
reinforcement. To verify the numerical results, a reference analytical model is
employed to calculate the ultimate shear strength of RC control beams and RC beams
with embedded steel trusses at different (a/d) ratios. The numerical results showed well
agree with the analytical results.
The analysis indicated that, using embedded steel truss enhanced the shear capacity and
improved the ductility of the RC beams. The analysis also indicated that the shear
capacity of reinforced concrete beams using embedded steel trusses is inversely
dependent on the shear span-to-depth ratio. It is also showed that the shear
reinforcement (stirrups) has almost small effect on shear capacity of reinforcement
concrete beams with embedded steel trusses.
The numerical results further demonstrated that longitudinal reinforcement have
significant effect on the shear capacity of reinforced concrete beams using embedded
steel trusses. It is also showed that reinforcement concrete beam using embedded steel
truss with diagonal angles steel in critical shear span only have almost the same failure
load and shear behavior of reinforcement concrete beam using embedded steel truss
with diagonal angles steel in full span.
59
Chapter 5
Conclusion and
Recommendations
60
Chapter 5 Conclusions and Recommendations
5.1 Introduction
The thesis investigated a non-linear analysis of RC beams with embedded steel trusses
using FE software ABAQUS. The FE models are initially built for the three simply-
supported beams with embedded steel trusses subjected to two-point load. The results
of the three beams are verified against the experimental results in terms of the load-
deflection response, the failure load and the mode of failure. The models are used to
perform a parametric study of the effect of the following parameters on the behavior of
strengthened beams: shear-span to depth ratios, shear reinforcement at different shear-
span to depth ratios, truss shape, and longitudinal reinforcement. In the following
sections, the important conclusions drawn from this study and research
recommendations are presented.
5.2 Conclusion
1. The ABAQUS FE models are able to analyze reinforcement concrete beams with
embedded steel trusses and predict the failure load and the mode of failure closely
as observed in the experimental tests.
2. The difference between the FEA results and experimental results are within 5%
range of accuracy in terms of failure load prediction while the concrete strain at mid-
span from the FEA is 8% lower than the test results.
3. The failure loads of all reinforcement concrete beams using embedded steel trusses
are higher when compared to RC beams with conventional reinforcement.
4. Using of embedded steel trusses improved reinforcement concrete beam ductility
and mode of failure.
5. Reinforcement concrete beam using embedded steel truss with shear-span to depth
ratio value 1 failed in shear mode. However, using embedded steel truss improves
the ductility of the beam.
6. All reinforced concrete beams using embedded steel trusses with shear-span to depth
ratio between 1.25 and 2.5 failed in shear flexural mode.
7. For different (a/d) ratios, all RC beams using embedded steel angle trusses showed
reduction in deflection compare with control beams.
61
8. The average increase in the ultimate failure loads for all the reinforcement concrete
beams using embedded steel trusses is 55% when compared to RC beam with
conventional reinforcement.
9. In general, the behavior of tested beams influenced by (a/d) ratio. It was found that
the increase of (a/d) ratio from 1 to 2.5 decrease the ultimate failure load for RC
beam with conventional reinforcement and RC beam with embedded steel angle
trusses beams by about 129% and 121% respectively.
10. The reduction rate of ultimate failure load is very small when the shear-span to
depth ratio between 2 and 2.5.
11. All RC beam with conventional reinforcement with (a/d) ratios between 1.25 and
2.5 failed in shear, while RC beams using embedded steel angle trusses at the same
(a/d) ratios failed by shear flexural mode.
12. For RC beams using embedded steel angle trusses with different (a/d) ratios, web
reinforcement have some effect on shear strength, whereas have almost no effect
on failure mode of the beams.
13. Reduced longitudinal reinforcement ratio for reinforcement concrete beams using
embedded steel trusses, decrease significantly the failure load.
14. Failure mode of reinforcement concrete beams using embedded steel trusses do not
affect by reduction of longitudinal reinforcement ratio.
15. Reinforcement concrete beam using embedded steel truss with diagonal steel
angles in critical shear span only have almost the same failure load and shear
behavior of reinforcement concrete beam using embedded steel truss with diagonal
steel angles in full span.
5.3 Recommendations
Based on the findings and conclusions of the current study, the following
recommendations are made for future research:
1. The effect of different factors such as size of the beam, use of different shapes
of embedded trusses, and concrete compressive and tensile strength on the shear
behavior of RC beams should be studied.
2. The performance of high strength concrete beams with embedded steel trusses
should be studied.
62
3. Shear behavior of reinforced concrete slender and deep beams using embedded steel
trusses should be studied.
63
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