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CHARACTERISTICS OF
STRUCTURAL LIGHTWEIGHT CONCRETE
UNDER SHORT-TERM COMPRESSION
MYAT MARLAR HLAING
B.Eng. (Civil), YTU; M.Sc. (Civil Eng.), NUS
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2013
Dedicated to my parents – Mabel & Maurice
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Acknowledgements
The author thanks to all the Professors, Staffs and Students who make her research study successful: Prof. Wee T. H. for giving the chance to conduct this research program; Prof. Mansur M. A. for his instructions during first year of the program; Prof. Zhang M. H. and Prof. Tam C. T. for their suggestions and comments; Dr. Kennan, Dr. Kong, Dr. Kyaw, and Kum for their fruitful discussions; All the laboratory officers at Structural/Concrete Lab especially Mr. Choo, Mr. Koh, Mr. Stanley and Mr. Kamsan for their professional assistance in three phases of experimental program; Former undergraduate student named Ms. Hiew for preparing the specimen as well as participating in testing in one phase of experimental program; National University of Singapore for granting the research scholarship Myat Marlar Hlaing
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Table of Contents
Page No.
CHAPTER 1
Introduction
1.1 Background Knowledge 1
1.2 Objectives 3
1.3 Scope 3
1.4 Significance of the Study 4
1.5 Backbone of the Thesis 4
CHAPTER 2
Plain Concrete
2.1 Introduction 7
2.2 Factors Governing the Sudden Failure 7
2.2.1 Brittle Nature of Concrete 8
2.2.2 Machine-Specimen Interaction 8
2.2.3 Unstable Control System 9
2.3 Literature Review: Modified Test Methods 10
2.3.1 Specimen Loaded in Parallel with Hollow Steel Tube 10
2.3.2 Specimen Loaded in Parallel with Steel Columns 11
2.3.3 Circumferential Strain Control Mode 12
2.3.4 Combination of Axial and Circumferential Strains as
Feedback Signal 13
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2.3.5 Linear Combination of Stress and Strain as Feedback
Signal 14
2.3.6 Linear Combination of Force and Axial Deformation as
Feedback Signal 15
2.4 Snap-Back Behavior and Snap-Through Behavior 16
2.5 Summary on Modified Test Methods 17
2.6 Review the Use of Plain Concrete Characteristic in Analysis 18
2.6.1 Experimental Behavior 18
2.6.2 Predicted Behaviors 19
2.6.3 Literature Review: Plain Concrete Models
Model Suggested by CEB-FIP Code 19
Model Suggested by EC2 Code 21
Codes of Practice 22
2.6.4 Section Analysis: Using Plain Concrete Characteristics 23
2.6.5 Comparative Study: Experimental Behavior and Predicted
Behavior 28
2.7 Summary 28
CHAPTER 3
Experimental Program
3.1 Introduction 35
3.2 Phase 1: Confined Concrete
3.2.1 Concrete Mixtures 35
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3.2.2 Type of Lateral Reinforcement 36
3.2.3 Yield Strength of Lateral Reinforcement 37
3.2.4 Specimen Preparations 37
3.2.5 Test Set-up and Instrumentation 38
3.2.6 Criterion Used in Testing 39
3.2.7 Methodology to Refine the Raw Experimental Data
Correcting the Initial Region of Stress-Strain Curve 40
Correcting the Posture of Stress-Strain Curve 41
3.3 Phase 2: Fiber-Reinforced Concrete
3.3.1 Concrete 42
3.3.2 Specimen Preparations 43
3.3.3 Test Set-up 44
3.4 Phase 3: Spiral-Reinforced Column
3.4.1 Specimen 45
3.4.2 Concrete 45
3.4.3 Specimen preparations
Weakening the Monitoring Zone 46
Strengthening the End Zones 47
Providing the Concrete Cover 47
Controlling the External Eccentricity 47
3.4.4 Test Set-up and Instrumentation 48
3.5 Summary 49
CHAPTER 4
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Confined Concrete: Structural Response
4.1 Introduction 61
4.2 Definition of Compressive Strength Level 63
4.3 Designation of Specimens 63
4.4 Scope and Objective 64
4.5 Observations and Discussion
4.5.1 Response of spiral-reinforced concrete 64
Deformation Capacity 64
Compressive Strength 66
Concrete Strain at Compressive Strength 67
Modulus of Elasticity of Plain Concrete 68
Modulus of Elasticity of Confined Concrete 69
Unloading Manner 69
Failure Mode 69
4.5.2 Mathematical Expressions 70
4.5.3 Influencing variables 70
Pitch of Spiral Reinforcement 70
Diameter of Spiral Wire 71
Pitch of Spiral Reinforcement vs. Diameter of Spiral
Wire 71
Amount of Lateral Reinforcement 71
Compressive Strength of Plain Lightweight Concrete 72
4.6 Summary 72
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CHAPTER 5
Confined Concrete: Stress-Strain Characteristic
5.1 Introduction 85
5.2 Stress Distribution of Concrete in Compression Zone 86
5.3 Designation of Specimens 86
5.4 Scope and Objective 86
5.5 Literature Review: Existing Stress-Strain Models 87
5.6 Suggested Model 88
5.7 Performance of Suggested Model
5.7.1 Using Experimental Data Recorded in Present Study 91
5.7.2 Using Experimental Data Reported by Other Researchers 91
5.8 Applicability of Recommended Model in Analysis
5.8.1 Analysis of Beam Section 93
5.8.2 Analysis of Column 99
5.9 Summary 101
CHAPTER 6
Fiber-Reinforced Concrete
6.1 Introduction 113
6.2 Scope and Significance 115
6.3 Observations and Discussion 115
6.3.1 Deformation Capacity 115
6.3.2 Compressive Strength 117
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6.3.3 Concrete Strain at Compressive Strength Level 118
6.3.4 Modulus of Elasticity 118
6.3.5 Splitting Tensile Strength 118
6.3.6 Oven-Dried Unit weight 119
6.3.7 Failure in Uniaxial Compression 119
6.3.8 Failure in Splitting Tension 119
6.4 Literature Review: Existing Models for Fiber-Reinforced Concrete120
6.5 Suggested Model: Fiber-Reinforced Concrete 120
6.6 Confining System: Combination of Lateral Reinforcement and
Fiber 123
6.6.1 Literature Review: Existing Models 124
6.6.2 Proposed Model 125
6.6.3 Performance of Proposed Model
Assumptions 127
Characteristic of Core Concrete of Columns 129
Performance 131
6.7 Summary 132
CHAPTER 7
Spiral-Reinforced Column
7.1 Introduction 143
7.2 Scope and Objective 144
7.3 Observations and Discussion
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7.3.1 Deformation Capacity 144
7.3.2 Unloading Manner 145
7.3.3 Ultimate Load Capacity 146
7.3.4 Compressive Strength of Concrete in Column 147
7.3.5 Failure mode 147
7.4 Predicted Behaviors of Columns 149
7.5 Summary 151
CHAPTER 8
Conclusions
8.1 Overview 161
8.2 Conclusion Remarks 162
8.2.1 Plain Concrete 162
8.2.2 Confined Concrete: Structural Response 163
8.2.3 Confined Concrete: Stress-Strain Characteristic 164
8.2.4 Fiber-Reinforced Concrete 165
8.2.5 Spiral-Reinforced Column 166
8.3 Recommendation for Further Study 166
Appendix A: List of Symbols 168
Appendix B: Software Program – Section Analysis Using Plain Concrete
Characteristics 171
Appendix C: Software Program – Sectional Analysis Using Lateral-Reinforced
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Concrete Characteristics 179
Appendix D: Stress-Strain Models Proposed in Present Study 199
Appendix E: Literature Review: Parameter Equations and Stress-Strain Models 205
Appendix F: Technical Papers 221
References 222
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List of Table Captions
Page No.
Table 2.1: Literature review on controlling the sudden failure of brittle materials 30 Table 3.1: Concrete mix design of spiral-reinforced lightweight concrete specimens 50 Table 3.2: Spiral-reinforced lightweight concrete specimens 50 Table 3.3: Concrete mix design of fiber-reinforced lightweight concrete specimens 51 Table 3.4: Spiral-reinforced lightweight concrete columns 51 Table 3.5: Concrete mix design of spiral-reinforced lightweight concrete columns 51 Table 4.1: Varying the pitch of spiral reinforcement 74 Table 4.2: Varying the diameter of spiral wire 74 Table 4.3: Varying both the pitch of spiral reinforcement and diameter of spiral wire 74 Table 4.4: Varying the compressive strength of plain lightweight concrete 74 Table 4.5: Varying the specimen size 75 Table 4.6: Validation of Eq. 4.3 75 Table 4.7: Validation of Eq. 4.5 75 Table 4.8: Experimental values of Ec 76 Table 5.1: Confined lightweight concrete specimens reported by other researchers 103 Table 5.2: Reinforced lightweight concrete beams reported by Lim (2007) 103 Table 5.3: Moment capacities of flexural beams 103 Table 5.4: Reinforced lightweight concrete columns reported by Basset and Uzumeri (1986) 104
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Page No. Table 6.1: Fiber-reinforced lightweight aggregate concrete 134 Table 7.1: Varying the fiber dosage 153 Table 7.2: Varying the diameter of spiral wire 153 Table 7.3: Varying the pitch of spiral reinforcement 153 Table 7.4: Details of reinforcement used in columns 153 Table 7.5: Ultimate load capacity of columns 154
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List of Figure Captions
Page No.
Figure 2.1: Load vs. deformation curves of brittle material: (a) snap-back behaviour; (b) snap-through behaviour 31 Figure 2.2: Modified test method reported by Wang et al. (1978) 32
Figure 2.3: Load vs. strain curves of hollow steel tube, of concrete specimen, and of combined concrete specimen and hollow steel tube 32 Figure 2.4: Modified test method reported by Dahl (1992): (a) elevation view; (b) plan view 33 Figure 2.5: Predicted stress-strain curves of plain lightweight aggregate concrete with compressive strength of 38.35 MPa 33 Figure 2.6: Predicted behaviours of beam section (beam 12) when using the stress-strain characteristics of plain concrete 34 Figure 3.1: Spiral reinforcement used in reinforced concrete specimen 52 Figure 3.2: Plan view of core concrete with different types of lateral reinforcement (Park 1975): (a) spiral reinforcement; (b) rectilinear tie 52 Figure 3.3: Elevation view of reinforcing system with different types of lateral reinforcement: (a) spiral reinforcement; (b) discrete lateral reinforcement 53 Figure 3.4: Preparation of straight steel wires for direct tension test 53 Figure 3.5: Test set-up and instrumentation for direct tension test 54 Figure 3.6: Stress-strain properties of lateral reinforcing wires in uniaxial tension 54 Figure 3.7: Test setup and instrumentations for uniaxial compression test 55 Figure 3.8: Appearance of spiral-reinforced lightweight concrete specimens after tested under uniaxial compression: (a) specimens with different pitch of spiral reinforcement; (b) specimens with different diameter of spiral wire 55 Figure 3.9: Correction of initial region, and posture of the stress-strain curve obtained from transducer reading 56
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Page No. Figure 3.10: Spiral-reinforced lightweight concrete column: (a) elevation view; (b) section A-A 57 Figure 3.11a: Round and hooked-end short steel fibers 57 Figure 3.11: Plan view of three spiral-reinforcing cages with different number of longitudinal reinforcement 58 Figure 3.12: Elevation view of three reinforcing cages: Installation of strain gages on longitudinal reinforcement and lateral reinforcement 58 Figure 3.13: Elevation view of three columns: Strain gages are installed in middle zone and fiber-reinforced polymer sheets are wrapped around end zones 59 Figure 3.14: Test set-up and instrumentation for column 59 Figure 3.15: Elevation view of three columns during preparation stage: Weakening the middle zone of columns 60 Figure 3.16: Appearance of three columns with different pitch of spiral reinforcement after testing 60 Figure 4.1: Stress vs. strain curve of spiral-reinforced concrete 77 Figure 4.2: Stress-strain curve of concrete: (a) a curve with descending portion: (b) a curve without descending portion 77 Figure 4.3: Deformation capacity of lateral-reinforced lightweight concrete with regards to pitch of spiral reinforcement, diameter of spiral wire, cylindrical compressive strength of plain lightweight concrete, and specimen size 78 Figure 4.4: Elevation view of spiral reinforcing system with different pitch of spiral reinforcement 83 Figure 4.5: Comparison between computed and experimental values of fco 83 Figure 4.6: Comparison between computed and experimental values of Ec 84 Figure 5.1: Schematic diagram showing the stress-strain characteristics of plain concrete and of lateral-reinforced concrete 105
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Page No. Figure 5.2: (a) cross-sectional view of flexural member; (b) strain distribution diagram; (c) stress distribution diagram of core concrete; (d) stress distribution diagram of cover concrete 105 Figure 5.3: Comparison between experimental stress-strain characteristic of confined lightweight concrete and predicted characteristics obtained from existing models 106 Figure 5.4: Comparison between experimental stress-strain characteristic of confined lightweight concrete and predicted characteristics obtained from suggested models 108 Figure 5.5: Comparison between experimental stress-strain characteristic of confined lightweight concrete reported by other researchers and predicted characteristics obtained from suggested models 110 Figure 5.6: Dimensions of lightweight concrete beams reported by Lim (2007) 111 Figure 5.6a: Predicted behaviours of flexural beam sections are compared with corresponding experimental behaviours 111 Figure 5.7: Dimensions of lightweight concrete columns reported by Basset and Uzumeri (1986) 111 Figure 5.7a: Predicted behaviours of axially loaded columns are compared with corresponding experimental behaviours 112 Figure 6.1: Deformation capacity of fiber-reinforced lightweight aggregate concrete with regards to fiber dosage: (a) group B specimens, (b) group C specimens 135 Figure 6.2: Appearance of plain lightweight concrete specimens after testing in uniaxial compression 136 Figure 6.3: Appearance of fiber-reinforced lightweight concrete specimens after testing in uniaxial compression: (a) group B specimens with different fiber dosage; (b) group C specimens with different fiber dosage 136 Figure 6.4: Appearance of fiber-reinforced lightweight concrete specimens after testing in splitting tension 137
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Page No. Figure 6.5: Comparison between predicted stress-strain characteristics obtained from reported models in literature and experimental characteristics of fiber-reinforced lightweight concrete 138 Figure 6.6: Comparison between predicted stress-strain characteristics obtained from suggested model and experimental characteristics of fiber-reinforced lightweight concrete 139 Figure 6.7: Comparison between predicted stress-strain characteristics obtained from reported models in literature and experimental characteristics of core concrete of reinforced lightweight concrete columns incorporating short steel fiber 140 Figure 6.8: Simplified bi-linear stress-strain relationship of longitudinal reinforcement (using Es and fy provided in mill certificate) 141 Figure 6.9: Load capacity of cover concrete of column is accounted until the column stress reaches a strength level of compressive strength of fiber-reinforced concrete 141 Figure 6.10: Simplified stress-strain relationships of cover concrete of columns with different dosage of fiber 141 Figure 6.11: Comparison between predicted stress-strain characteristics obtained from proposed model and experimental characteristics of core concrete of reinforced lightweight concrete columns incorporating short steel fiber 142 Figure 7.1: Deformation capacity of spiral-reinforced lightweight concrete columns with regards to (a) diameter of spiral wire, (b) pitch of spiral reinforcement, (c) number of longitudinal reinforcement, and (d) fiber dosage 155 Figure 7.2: Load vs. strain relationship of reinforced lightweight concrete column under uniform axial compression 157 Figure 7.3: Failure mode of columns: (a) sudden failure of plain unreinforced column; (b) shear failure of spiral-reinforced column with low amount of lateral reinforcement; (c) spalling failure of spiral-reinforced column with adequate amount of lateral reinforcement; (d) bulging failure of spiral-reinforced column incorporating short steel fibres 158 Figure 7.4: Rupture of lateral reinforcement over longitudinal reinforcement 159 Figure 7.5: Predicted behaviours of reinforced lightweight concrete columns are compared with corresponding experimental behaviours 159
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Summary
Characteristics of structural lightweight concrete under short-term loading are not well
understood due to limited information available in current literature. Insufficient
understanding considerably affects the analysis and design of lightweight concrete
structural members. In addition, design procedures and code specifications for the
members can not be further improved. It is, therefore, obvious that the insufficient
understanding holds back the extensive usage of lightweight concrete in structural
applications, though lightweight concrete possesses many unique properties.
Since insufficient understanding is the main motivation of the study, the study aims to
provide additional understanding on characteristics of structural lightweight concrete.
The study thus observes the characteristics by conducting a series of experimental
program. Lightweight concrete is produced by using expanded clay lightweight coarse
aggregate. Types of the concrete considered are plain (unreinforced) concrete, confined
concrete, and fiber-reinforced concrete. In confined concrete, spiral reinforcement is used
to confine the core concrete effectively. Discrete short steel fibers are used for the same
purpose in fiber-reinforced concrete.
The interested characteristics of the concrete include deformation capacity, compressive
strength, strain at compressive strength, modulus of elasticity, stress-strain characteristic,
splitting tensile strength, unit weight, and failure manner. These characteristics are
observed by varying the variables, namely, pitch of lateral-reinforcement, diameter of
lateral-reinforcing wire, compressive strength of plain lightweight aggregate concrete,
size of specimen, and fiber dosage. Additional understanding of the characteristics, which
xvii
is gained from the study, would be beneficial in analysis and design of structural
lightweight concrete members.
This study also proposes the stress-strain models to predict the stress-strain
characteristics of confined concrete, of fiber-reinforced concrete, and of the concrete
confined by both lateral reinforcement and fiber. The models are applicable for
lightweight aggregate concrete with plain concrete compressive strength ranging between
38 MPa and 58 MPa, with volumetric ratio of spiral reinforcement between 1.7% and
6.8%, and with volume fraction of short steel fiber addition between 0.5% and 1.0%.
In addition, this study presents the experimental behaviors of spiral-reinforced
lightweight concrete column, with and without fiber addition, under uniform axial
compression.
Overall, this study would contribute towards the development of design procedures and
code specifications for structural lightweight concrete members. This study is, therefore,
valuable to those dealing with structural lightweight concrete.
1
CHAPTER 1
Introduction
1.1 Background Knowledge
The definition for structural lightweight aggregate concrete varies from region to region,
and over time. For example, fib Task Group 8.1 (2000a) defines the structural lightweight
aggregate concrete as a concrete with an oven-dry density between 800 kg/ m3 and 2200
kg/m3. On the other hand, ACI 213R (2003) defines it as a concrete with a minimum 28-
day compressive strength of 17 MPa, with an equilibrium density between 1120 and 1920
kg/m3, and made with lightweight coarse aggregate having a maximum bulk density of
880 kg/m3. EC2 (2004), in fact, defines it as a concrete with a density of not more than
2200 kg/m3, and containing lightweight aggregates with a particle density of less than
2000 kg/m3.
Lightweight aggregate concrete is regarded as an efficient construction material for
structural applications due to its distinct properties such as high strength to weight ratio,
reduced inertia force, buoyancy, internal curing, fire resistance, and durability (Berra and
Ferrara 1990; Hoff 1990; ACI 213 R 2003). The high strength to weight ratio significantly
reduces the dead weight of structural members which governs the design of foundation
system, as well as the ground treatment procedures. Furthermore, the high ratio enables
the application for long-span construction technology especially in reinforced concrete
bridges. Alternatively, the buoyancy provides economical solution for offshore platform
2
construction. For these reasons, lightweight aggregate concrete is well known for
providing cost-effective and flexible structural design. Worldwide applications of
lightweight aggregate concrete are reported by fib Task Group 8.1 (2000 b), and ACI
213R (2003).
Nonetheless, lightweight aggregate concrete is a more brittle material compared to normal
weight concrete having the same compressive strength. Because of its brittle property, the
lightweight aggregate concrete tends to fail in a sudden manner. This sudden failure
considerably holds back the widespread usage of lightweight aggregate concrete in
construction industry. Favourably, this sudden failure is overcome by using either
effective lateral reinforcement, or using short steel fiber as a concrete constituent, or
combination of both lateral reinforcement and short steel fiber.
It is noteworthy that stress-strain characteristic of concrete in compression is a
combination of concrete properties, namely modulus of elasticity, compressive strength,
strain at compressive strength, and deformation capacity. For this reason, stress-strain
characteristic alone is able to represent the concrete material property in sectional analysis
and design of structural members. It should be noted that the stress-strain characteristic of
lightweight aggregate concrete is not well understood due to its limited information
available in current literature.
The insufficient understanding considerably affects the analysis of structural lightweight
concrete members, thereby hindering from achieving the safe and cost-effective structural
design. As a result, design procedures and code specifications for the members can not be
further improved. It is obvious that the insufficient understanding holds back the extensive
usage of lightweight aggregate concrete in construction industry. Since the insufficient
3
understanding is a main motivation of the study, this study aims to provide the additional
understanding on the characteristics of structural lightweight aggregate concrete.
1.2 Objectives
The main objectives of the present study are as follows:
a. to capture the experimental characteristics of confined lightweight aggregate
concrete, and of fiber reinforced lightweight aggregate concrete;
b. to propose a stress-strain model of lightweight aggregate concrete which is
applicable in analysis and design of structural lightweight concrete members.
1.3 Scope
In order to achieve the above mentioned objectives, this study focuses on the following
scope:
a. explore the structural response of confined lightweight aggregate concrete to short-
term compression by conducting a series of experimental program;
b. derive a stress-strain model that is capable of predicting the stress-strain
characteristic of confined lightweight aggregate concrete under compression;
c. illustrates the use of the model in structural analysis;
d. observe the characteristics of fiber-reinforced lightweight aggregate concrete under
short-term loading;
e. formulate a model to predict the stress-strain characteristic of fiber-reinforced
lightweight aggregate concrete;
4
f. develop another stress-strain model to generate the stress-strain characteristic of
lightweight aggregate concrete confined by a combination of lateral reinforcement
and short steel fiber;
g. observe the experimental behaviors of spiral-reinforced lightweight concrete
column, with and without steel fiber addition, under uniform axial compression.
1.4 Significance of the Study
a. This study adds to the understanding of response of confined lightweight concrete
subjected to short-term compression. Additional understanding gained from the
study would be beneficial in structural analysis of lightweight concrete members;
b. This study also adds to the knowledge on characteristics of fiber-reinforced
lightweight aggregate concrete subjected to short-term compression. Additional
understanding would be of benefit in approaches for design and analysis of
structural members with such concrete;
c. The proposed stress-strain models would contribute towards the development of
design procedures and code specifications of structural lightweight concrete
members.
1.5 Backbone of the Thesis
First of all, Chapter 1 highlights the background knowledge related to the present study.
Then, it reveals the objectives, scope, and significance of the study. Thereafter, it briefly
describes the thesis which is valuable information to those dealing with structural
lightweight aggregate concrete.
5
Chapter 2 is about the complete stress-strain characteristic of plain lightweight aggregate
concrete. It is noteworthy that several researchers have modified the conventional test
method in order to control the sudden failure of brittle materials under compression. The
chapter reviews the concept behind each modified test method, as well as addresses the
advantages and possible disadvantages of using each method in practice. Furthermore, it
reviews the use of plain lightweight concrete characteristic in structural analysis.
Chapter 3 discusses the details of experimental program. For the ease of discussion, the
experimental program is divided into 3 phases. The test results obtained from this
experimental program are used in discussion of the following chapters.
Chapter 4 is about the response of confined lightweight aggregate concrete to short-term
compression. The response includes deformation capacity, compressive strength, concrete
strain at compressive strength, modulus of elasticity, unloading manner, and failure mode.
The response is explored by varying the variables, namely pitch of spiral reinforcement,
diameter of spiral wire, compressive strength of plain lightweight aggregate concrete, and
specimen size.
Chapter 5 derives a stress-strain model to predict the stress-strain characteristic of
confined lightweight aggregate concrete. This chapter also illustrates the use of the model
in section analysis.
Chapter 6 observes the characteristics of fiber-reinforced lightweight aggregate concrete
subjected to short-term compression. The characteristics are deformation capacity,
compressive strength, strain at compressive strength, modulus of elasticity, splitting
tensile strength, and failure mode. These characteristics are observed by varying the
compressive strength of plain lightweight aggregate concrete, and fiber dosage.
6
Moreover, Chapter 6 formulates a stress-strain model to estimate the stress-strain
characteristic of fiber-reinforced lightweight aggregate concrete. It also proposes another
stress-strain model to generate the stress-strain characteristic of lightweight aggregate
concrete confined by a combination of lateral reinforcement and short steel fiber.
Chapter 7 presents the experimental behaviors of spiral-reinforced lightweight concrete
column, with and without steel fiber addition, under uniform axial compression. These
experimental behaviors of the columns are also used in evaluating the performance of
proposed stress-strain models.
Finally, Chapter 8 outlines the overview of the whole research program. Then, it draws the
conclusions which are valuable information to those dealing with structural lightweight
concrete. It also includes the recommendation section for further research study.
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CHAPTER 2
Plain Concrete
2.1 Introduction
In analysis and design of structural concrete members, complete stress-strain curve of
plain concrete in compression is used to represent the concrete material property
(Popovics 1973; Wang et al. 1978). The complete curve is, in fact, obtained from
experimental testing. In the case of plain lightweight aggregate concrete, it is difficult to
capture the complete curve experimentally. Just beyond the peak compressive strength
level, lightweight aggregate concrete tends to fail suddenly, hindering from obtaining the
complete curve.
To control the sudden failure of brittle materials under compression, several researchers
have modified the conventional test method, and reported the modified test methods in
literature. This chapter reviews the concept behind the modified test methods, as well as
addresses the advantages and possible disadvantages of using each method in practice.
Then, it discusses the nature of stress-strain curve obtained from these methods. In
addition, this chapter reviews the use of plain lightweight concrete characteristic as core
concrete characteristic in structural analysis.
2.2 Factors Governing the Sudden Failure
8
Lightweight aggregate concrete draws a lot of attention as a structural material due to its
marked properties such as high strength to weight ratio, buoyancy, and fire resistance
(Hoff 1990; Berra and Ferrara 1990; ACI 213R 2003). However, lightweight aggregate
concrete is also known as a brittle material which tends to fail suddenly during unloading.
Factors governing the sudden failure are believed to be the brittle nature of lightweight
aggregate concrete, machine-specimen interaction, and unstable control system.
2.2.1 Brittle Nature of Concrete
When lightweight aggregate concrete is loaded under uniaxial compression, it is subjected
to not only vertical compressive stress but also lateral tensile stress. This tensile stress
causes formation and propagation of micro-cracking in coarse aggregate, interfacial
transition zone (ITZ), and mortar matrix. In lightweight aggregate concrete, coarse
aggregate is the weakest component of heterogeneous concrete system, when compared to
ITZ and mortar matrix (Gao et al. 1997; Faust 1997).
When the applied tensile stress exceeds the tensile strength of lightweight coarse
aggregate, cracking inside the coarse aggregate becomes unstable, extends to mortar
matrix and then, connects with existing cracking in mortar matrix. As a result, failure
plane passes through the coarse aggregate, providing smooth fracture surface (Basset and
Uzumeri 1986; Faust 1997; Walraven 2000). Such smooth fracture surface leads to sudden
failure of lightweight aggregate concrete (Zhang and Gjorv 1991; Faust 1997).
2.2.2 Machine-Specimen Interaction
9
When a concrete specimen is loaded under compression in testing frame, not only the
specimen but also the testing frame itself deforms (Chin 1996). When the loading platen
of testing frame is moving downward and pressing down the specimen, stiffness of the
specimen resists back the downward movement of the loading platen. This phenomenon
causes the elastic deformation in the testing frame. Since the elastic deformation increases
with increasing loading, strain energy is accumulated with time in testing frame.
When the specimen starts to unloading due to severe cracking, the stiffness of the
specimen is considerably decreased. As a result, the testing frame begins to release its
stored strain energy to the specimen. For brittle specimen such as lightweight aggregate
concrete, this released energy complicates the failure manner of the specimen.
2.2.3 Unstable Control System
When a monotonic rate of control mode such as constant rate of axial displacement is used
in testing, axial deformation of specimen is expected to increase with time (Eq. 2.1).
When load-deformation curve of specimen displays either snap-back or snap-through
behavior during unloading (Fig. 2.1), axial deformation of specimen is no longer
increasing with time. In this situation, the left-hand side of Eq. 2.1 either decreases or
remains constant, while the right-hand side of Eq. 2.1 is always increasing. Needless to
say, equilibrium condition of the equation (Eq. 2.1) is no longer satisfied. This incident
makes the control system unstable (Okubo 1985), resulting in sudden failure of specimen.
axial deformation of specimen = A × time (2.1)
where A = constant rate of axial displacement
10
2.3 Literature Review: Modified Test Methods
To control the sudden failure of brittle materials under compression, several researchers
have modified the conventional test method. Modifications include loading a specimen in
parallel with either a hollow steel tube (Wang et al. 1978), or four steel columns (Dahl
1992), and establishing the stable feedback signals (Shah et al. 1981, Okubo and
Nishimatsu 1985, Glavind and Stang 1991, Jansen and Shah 1993, Jansen et al. 1995,
Faust 1997). Modified test methods reported in earlier literature are thoroughly reviewed
as below.
2.3.1 Specimen Loaded in Parallel with Hollow Steel Tube
Wang et al. (1978) have proposed a modified test method in which a concrete specimen is
placed inside a hollow steel tube, and then, loaded together with the tube (Fig. 2.2). This
testing method ensures that combined load carrying capacity of concrete specimen and
steel tube always increases throughout the testing (Fig. 2.3). In this context, the machine is
unable to sense the unloading of concrete specimen. Thus, the machine never releases
back the accumulated strain energy to concrete specimen until the end of testing. In this
way, machine-specimen interaction is controlled.
As the steel tube is case-hardened, its stress-strain curve is linearly elastic up to the strain
value of 0.006. Thickness of the steel tube is designed to ensure the condition that the
combined load capacity of concrete specimen and steel tube always increases throughout
the testing, though unloading of concrete specimen occurs in testing. Concrete specimen is
capped to achieve the good leveling with steel tube such that concrete specimen and steel
tube share the applied loading from the start of testing.
11
However, in this test method, deformation of capping material is included in measured
deformation data for concrete specimen. As a result, deformation data of concrete
specimen obtained from this test method is inaccurate. In addition, the presence of the
steel tube obstructs the observation for failure manner of concrete specimen during testing.
2.3.2 Specimen Loaded in Parallel with Steel Columns
Dahl (1992) has modified the testing machine by increasing the stiffness of test rig (Fig.
2.4). The necessary stiffness is obtained from four steel columns which are placed around
and loaded together with concrete specimen. To ensure the good alignment between
concrete specimen and steel columns, concrete specimen is placed on a massive steel
cylinder which is also used as a dynamometer measuring the load capacity of concrete
specimen throughout the testing.
In this test setup, it is reported that top and bottom loading platens are also deformed
under loading, which makes the steel columns follow the angular deformations of loading
platens. As a result, the columns become buckled. Under this condition, a certain amount
of deviations in strain data is detected within each column, as well as among four
columns. Since these strain data of steel columns are used to determine the axial strain of
concrete specimen, the axial strain data of concrete specimen obtained from this test
method is inaccurate.
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2.3.3 Circumferential Strain Control Mode
When subjected to uniaxial compression, concrete specimen deforms not just in axial
direction but also in lateral direction. Since this lateral deformation always increases with
time, it is reported that circumferential strain of specimen can be used as a control mode.
In this way, equilibrium condition of Eq. 2.2 is maintained throughout the testing.
circumferential strain of specimen = B × time (2.2)
where B = constant rate of circumferential strain
Circumferential strain of concrete specimen can be measured with strain gage. However,
usefulness of the strain gage is limited in testing. Just before the peak compressive
strength level of concrete, cracking usually forms on specimen surface. Due to this
cracking, strain gage becomes detached from the specimen, which terminates the testing
suddenly and prematurely. Thereby, complete stress-strain curve of concrete specimen can
not be obtained by using the strain gage.
Circumferential strain of concrete specimen can also be measured by using either piano
wire (Shah et al. 1981), or link chain with roller (Jansen et al. 1995). These wire and chain
are wrapped around the mid height of concrete specimen where the largest lateral
deformation is expected to occur.
However, using the circumferential strain of concrete specimen as a control mode seems
to be an unreliable control method. The reasons are as follows. Circumferential
deformation of concrete specimen is less pronounced than its axial deformation in initial
part of testing. Hence, it is difficult to detect the circumferential signal in initial part of
13
testing. Moreover, failure mode of plain concrete is mainly due to spalling. Spalling of
concrete pieces from specimen would send the false signal to the control system as if the
specimen is contracting. In addition, the wire and chain would induce the confining effect
to the specimen, which disturbs the actual response of the specimen to loading. On the
other hand, movement of wire and chain produces the friction which induces the strain in
wire and chain. This induced strain interrupts the detection of true circumferential strain of
concrete specimen. For these four reasons, using the circumferential strain of concrete
specimen as a control mode seems to be an unreliable test method.
2.3.4 Combination of Axial and Circumferential Strains as Feedback Signal
Glavind and Stang (1991) have proposed a test method using a combination of axial strain
and circumferential strain of concrete specimen as feedback signal. Up to the peak load
level, axial strain alone is used as the feedback signal (Eq. 2.3), since axial strain is more
pronounced than circumferential strain in initial part of testing. Starting from the peak
load level to the end of testing, a combination of axial strain and circumferential strain is
used as feedback signal (Eq. 2.4), since the axial strain is not always increasing beyond
the peak load level.
From the start of testing to the peak load level,
axial strain of specimen = D × time (2.3)
where D = constant rate of axial strain
14
From the peak load level to the end of testing,
(weight 1 × circumferential strain of specimen) + (weight 2 × axial strain of specimen) = E
× time (2.4)
where E = constant rate of strain
Similar limitations mentioned in Section 2.3.3 except the difficulty in obtaining the signal
at the start of testing seem to occur in this method. The weights of the two strains, which
are constant values, make sure that equilibrium condition of Eq. 2.4 is satisfied. However,
these weights may depend on stiffness of specimen, of testing machine, and relative
stiffness between specimen and testing machine. For this reason, no constant values can be
defined for the two weights to satisfy all types of material and of testing machine. In order
to determine these two weights for each type of material, a lot of trial tests are required by
varying different combinations of the weights until the complete material response is
captured experimentally. Therefore, this kind of modified test method is inconvenient to
use in practice.
2.3.5 Linear Combination of Stress and Strain as Feedback Signal
Okubo and Nishimatsu (1985) have proposed a control method using a linear combination
of stress and strain as feedback signal to test the brittle rocks in compression. The equation
used in the control method is as follows:
timeCE
stressstrain
' (2.5)
15
where C = a constant; E’ = fixed modulus value
To maintain the equilibrium condition of Eq. 2.5, the value of E’ must be larger than slope
of ascending branch of stress-strain curve, but smaller than that of descending branch. In
this way, stable control method is attained.
Nevertheless, the value of E’ seems to be different from material to material, as different
materials vary widely in their response to loading. To determine the value of E’ for each
type of material, a lot of trial tests are required until the complete curve is captured.
Hence, no constant value can be defined for E’ to satisfy all types of material and of
testing machine. Therefore, it is obvious that this kind of modified test method is not
practical.
2.3.6 Linear Combination of Force and Axial Deformation as Feedback
Signal
Jansen and Shah (1993) have also proposed the feedback signal using a linear combination
of force and axial deformation. The feedback signal is described in Eq. 2.6.
signalfeedbackk
forcendeformatio
o
_
(2.6)
To achieve the stable signal, the value of k0 must be between the slope of ascending
branch at 40% of peak strength and smallest tangent of descending branch. For the same
purpose, α must be between 0 and 1. Both k0 and α largely depend on the stiffness of
16
specimen, of testing machine, and relative stiffness between specimen and machine. Thus,
no constant values can be defined for k0 and α to satisfy all types of material, and of testing
machine. As mentioned before in Sections 2.3.4 and 2.3.5, a lot of trial tests are required
to determine these values for each type of material. Therefore, this kind of test method is
not practical.
Faust (1997) has proposed a test method using a linear combination of force and
deformation as feedback signal. The feedback signal is described in Eq. 2.7.
signalfeedbackK
forcendeformatio _ (2.7)
To maintain the equilibrium condition of Eq. 2.7, the value of K is chosen between the
slopes of ascending branch and descending branch of load-deformation curve. In this way,
stable feedback signal is obtained. As mentioned earlier in Sections 2.3.4 and 2.3.5, the
value of K depends on types of material, and of testing machine. Therefore, no constant
value can be defined for K to satisfy all types of material, and of testing machine. A lot of
trail tests are thus required to determine the value of K for each type of material.
Therefore, this test method is also not practical.
2.4 Snap-Back Behavior and Snap-Through Behavior
Provided that axial deformation of a concrete specimen decreases in unloading region
(Fig. 2,1a), post-peak behavior of load-deformation curve of the specimen can be defined
as the snap-back behavior. Provided that axial deformation of a concrete specimen
17
remains constant in unloading region (Fig. 2b), post-peak behavior of load-deformation
curve of the specimen can be defined as the snap-through behavior. In fact, the snap-back
behavior and snap-through behavior are only the curve behaviors which are formed due to
localized failure of tested specimen and elastic recovery of undamaged portions of the
specimen during unloading (Jansen 1993; Faust 1997).
It is found that load-deformation curve obtained from the modified test methods show
snap-back behavior and snap-through behavior. Therefore, it is questionable whether the
curve obtained from these modified test methods represents the true property of tested
specimen in compression.
2.5 Summary on Modified Test Methods
Though all the modified test methods reported in earlier literature are logically sound, they
do have certain limitations. Some of the methods are not practical, while the others are
unreliable. Test method shall be able to provide the complete stress-strain curve of
material without having to conduct a lot of trial tests. Moreover, test method shall not
obstruct the observation for failure manner of specimen during testing. Furthermore,
stability of control mode shall be insensitive to concrete spalling. In addition, test method
shall not disturb the actual response of specimen to loading.
Above all, complete stress-strain curve of tested specimen obtained from the modified test
methods seems to be only the curve behavior that is formed due to localized failure of
tested specimen and elastic recovery of undamaged portions of the specimen during
unloading. Thus, it raises the question of whether the curve obtained from the modified
18
test methods represents the true property of tested specimen in compression. Based on the
above reasoning, none of the modified test methods is applied in the study.
2.6 Review the Use of Plain Concrete Characteristic in Analysis
In structural members, core concrete is strengthened by reinforcement. Since core concrete
is reinforced concrete, it is important to use the confined concrete characteristic in
structural analysis and design.
Using the plain concrete characteristic instead would considerably underestimate the
ultimate behaviors of members in analysis. This would result in overly conservative
design which is uneconomical. In this section, consequences of using the plain concrete
characteristic in analysis are discussed by comparing predicted behaviors of flexural beam
section with corresponding experimental behavior.
2.6.1 Experimental Behavior
Experimental behavior of flexural beam section is obtained from earlier literature (Lim
2007). Dimensions of the beam are 150 mm in width, 300 mm in depth, and 2800 mm in
length between simply supported ends. The beam, which is loaded under two point
loading, is constructed with lightweight aggregate concrete. The concrete is produced by
using expanded clay lightweight coarse aggregate. Details of the reinforcement used in the
beam (beam 12) are summarized in Table 5.1.
19
2.6.2 Predicted Behaviors
Predicted behaviors of flexural beam section are obtained by performing the sectional
analysis. Similar analysis is reported by Bresler (1971), Wang (1977), Fafitis and Shah
(1985), and Chin (1996). The analysis procedures are discussed at length in Section 2.6.4.
Since the analysis involves iteration, repetition and integration procedures, an in-house
computer program is developed and used to save the computing time. Details of the
computer program are given in Appendix B.
In sectional analysis, stress-strain characteristic of plain lightweight aggregate concrete is
used to represent the core concrete characteristic. This stress-strain characteristic is
generated by using two different stress-strain models of plain lightweight aggregate
concrete separately suggested by CEB-FIP Code (CEB-FIP Model Code 90 1993; fib Task
Group 8.1 2000), and EC2 Code (BS EN 1992-1-1 2004). Each stress-strain model is
thoroughly reviewed in Section 2.6.3.
2.6.3 Literature Review: Plain Concrete Models
Model Suggested by CEB-FIP Code
CEB-FIP code reports a stress-strain model for plain lightweight aggregate concrete under
short term loading (CEB-FIP Model Code 90 1993; fib Task Group 8.1 2000). The model,
which is shown in Eq. 2.8, is for lightweight aggregate concrete with cylindrical
compressive strength ranging between 12 MPa and 80 MPa, and with oven-dried density
ranging between 800 kg/m3 and 2200 kg/m3. The model contains only the ascending
20
branch of stress-strain relationship. The negative sign refers to the compressive stress.
Stress-strain relationship generated by the model is shown in Fig. 2.5.
cm
cl
c
cl
Eci
cl
c
cl
c
cl
Eci
c f
E
E
E
E
21
2
(2.8)
where c = concrete stress at any point on stress-strain curve of plain lightweight
aggregate concrete (MPa)
c = concrete strain corresponding to c
Eci
cmcl E
kf
cl = concrete strain corresponding to peak strength level; negative sign shows
compressive strain
k = 1.3 for lightweight aggregate concrete with lightweight aggregate as
coarse aggregate
cmf = cylindrical compressive strength of plain lightweight aggregate concrete at
an age of 28 days (MPa)
3
1
cmo
cmcoci f
fEE
coE = 2.15×104 MPa
cmof = 10 MPa
2
2200
E
21
= oven dry density of lightweight aggregate concrete (kg/m3)
cl
cmcl
fE
clE = secant modulus between the origin and peak point of stress-strain curve
Model Suggested by EC2 Code
EC2 code presents a total of three models to generate the stress-strain relationship of plain
lightweight aggregate concrete (BS EN 1992-1-1:2004). Among them, the first model is
for non-linear structural analysis, while the rest two models are for structural design. The
later models thus incorporate the coefficients and partial safety factors.
Since the study focuses on the stress-strain relationship for structural analysis without
incorporating any coefficients and partial safety factors, only the first model for non-linear
analysis is of interest (Eq. 2.9). The first model is suggested in EC2 code for lightweight
aggregate concrete with density of not more than 2200 kg/m3, with artificial or natural
lightweight coarse aggregate having a particle density of less than 2000 kg/m3, and with
mean cylindrical concrete strength ranging between 17 MPa and 88 MPa. As shown in
Fig. 2.5, the model generates the ascending branch of stress-strain relationship.
21
2
k
k
flcm
c (2.9)
where c = concrete stress at any point on stress-strain curve of plain lightweight
aggregate concrete (MPa)
lcmf = mean cylindrical strength of lightweight aggregate concrete (MPa)
22
1lc
c
c = concrete strain corresponding to c
1lc = concrete strain corresponding to peak strength level
1lc = Elci
lcm
E
kf
k = 1.1 for lightweight aggregate concrete with sand as fine aggregate
2
2200
E
= oven dried density of lightweight aggregate concrete (kg/m3)
Ecmlcm EE (GPa)
Assume that cmllci EE
3.0
1022
cm
cm
fE
Codes of Practice
The models suggested by CEB-FIP code and EC2 code generate the stress-strain
relationship up to peak compressive strength level. Parameters controlling the shape of the
stress-strain curve are compressive strength, concrete strain at compressive strength,
modulus of elasticity, oven dry density of concrete, and type of aggregate. All these
parameters are covered in the two mentioned models. Type of aggregate is accounted by
CEB-FIP Code and EC2 Code as a constant factor of 1.3 and 1.1 respectively.
23
It is noted that ACI 213R-03 does not suggest any models for stress-strain relationship of
plain lightweight aggregate concrete, while BS 8110-2:1985 suggests a stress-strain model
for structural design. Since the model is to be used in design, it incorporates the coefficient
as well as partial safety factor. As mentioned before, this study focuses on the stress-strain
relationship for structural analysis without incorporating any coefficients and factors.
Thus, the model suggested by BS 8110-2:1985 is not taken into account in the present
study.
2.6.4 Sectional Analysis: Using Plain Concrete Characteristics
Sectional analysis is performed to predict the moment-curvature relationship of member
section at mid-span. In the analysis, stress-strain characteristic of plain lightweight
aggregate concrete is used to represent the core concrete characteristic. This stress-strain
characteristic is generated by using the stress-strain models reported by CEB-FIP Code
and EC2 Code (Section 2.6.3). For longitudinal reinforcement, idealized bi-linear stress-
strain relationships are used.
The analysis is based on three principles of mechanics, namely equilibrium condition,
compatibility, and stress-strain relationship. The following assumptions are used in the
analysis: (1) plane section remains plane after bending, ensuring the linear strain
distribution over the effective depth of beam section; (2) there is a perfect compatibility
between concrete and reinforcement, resulting in identical strain value for concrete and
reinforcement located together at any level; (3) stress-strain relationship of longitudinal
reinforcement in compression is identical to that in tension; (4) tensile strength of
concrete, area displaced by longitudinal reinforcement in compression zone of member
24
section, and tension stiffening effect are negligible; and (5) the section is assumed as the
uncracked section.
Geometry of beam section, configuration of reinforcement, stress-strain characteristics of
plain concrete, and of longitudinal reinforcement are accounted in the analysis. Vertical
distance between neutral axis and extreme compression fiber of the beam is defined as the
depth of neutral axis (dNA). Distance along the depth of neutral axis is shown by the
symbol ‘u’.
The following procedures are used in the sectional analysis.
Step 1: specify a small value of concrete strain at extreme compression fiber of beam (εce)
Step 2: assume a trial depth of neutral axis (dNA)
Step 3: find the relationship between strain and distance along depth of neutral axis (ε-u
relationship)
NA
ce
dx
(2.11)
Step 4: determine the relationship between concrete stress and distance along depth of
neutral axis (σ-u relationship) by using σ-ε relationship of plain concrete and ε-u
relationship. This σ-u relationship is the stress distribution diagram drawn over the
compression zone of beam section.
Step 5: compute the area bounded by σ-u curve of concrete, u-axis and the line u = dNA by
using the numerical integration method. The area represents the compressive strength of
concrete per unit width of beam section (A). The area is divided into 5 small segments to
perform the integration procedure.
25
Step 6: calculate the compressive force of concrete (Cp)
Cp = A × width of beam section (2.12)
Step 7: determine the location of compressive force of concrete by using the moment area
method which takes the moment of the stress distribution diagram about the neutral axis
Step 8: compute the tensile steel strain (εts) by using εce, dNA, vertical distance between
neutral axis and center of longitudinal reinforcement in tension zone, and triangular rule.
Similarly, compute the compressive steel strain (εcs) by using εce, dNA, vertical distance
between neutral axis and center of longitudinal reinforcement in compression zone, and
triangular rule.
Step 9: determine the tensile steel stress (fts) by using stress-strain relationship of
longitudinal reinforcement in tension zone and corresponding εts . Likewise, determine the
compression steel stress (fcs) by using stress-strain relationship of longitudinal
reinforcement in compression zone and corresponding εcs.
Step 10: calculate the tensile force of longitudinal reinforcement (Ts), and compressive
force of longitudinal reinforcement (Cs)
Ts = fts × Ats (2.13)
Cs = fcs × Acs (2.14)
where Ats = total cross-sectional area of longitudinal reinforcement in tension zone
Acs = total cross-sectional area of longitudinal reinforcement in compression zone
26
Step 11: check the accuracy of trial depth of neutral axis (dNA) that is assumed in step 2 by
using the force equilibrium condition
Cp + Cs + Ts = 0 (2.15)
Step 12: if the force equilibrium condition is unsatisfied, adjust the trial depth of neutral
axis as shown below, and repeat the iterative procedure from step 2 to step 11 until the
force equilibrium condition becomes satisfied
Adjusting the depth of neutral axis,
if total
stotal
C
TC > 0.1, decrease the trial depth of neutral axis assumed in step 2
if s
totals
T
CT < 0.1, increase the trial depth of neutral axis assumed in step 2
where totalC = Cp + Cs (2.16)
0.1 = a constant used in adjusting the trial depth of neutral axis
Step 13: when the force equilibrium condition is satisfied, find the external moment (Mext)
by using the moment equilibrium condition
Mext = Mint (2.17)
Mext = (C × armcon) + (Cs × armcs) + (Ts × armts) (2.18)
where Mext = external applied moment
Mint = internal resisting moment
27
armcon = vertical distance between neutral axis and compressive strength of
concrete
armcs = vertical distance between neutral axis and compressive strength of
longitudinal reinforcement
armts = vertical distance between neutral axis and tensile strength of longitudinal
reinforcement
Step 14: compute the corresponding curvature at the mid-span of member
NA
ce
d
(2.19)
where = curvature of member section at mid-span
εce = latest specified concrete strain at extreme compression fiber of member
dNA = adjusted depth of neutral axis
Step 15: define the next value for εce by adding an incremental strain value to the previous
value of εce
εce(next value) = εce(previous value) + Δεce (2.20)
where εce(next value) = concrete strain at extreme compression fiber of beam for the next
stage of loading
εce(previous value) = value previously specified in step 14
Δεce = incremental strain value
28
Step 16: repeat the procedure from step 2 to step 15 to find another set of moment-
curvature data until the latest value of εce reaches a desired strain value. In this way,
predicted moment-curvature data sets of beam section at different stages of εce are
obtained.
2.6.5 Comparative Study: Experimental Behavior and Predicted Behavior
As expected, comparative study reveals that predicted behaviors underestimate the
corresponding experimental behavior (Fig. 2.6). Since the core concrete of structural
members is reinforced concrete, it is necessary to use the confined concrete characteristic
in analysis. Using the plain concrete characteristic instead would considerably
underestimate the ultimate behaviors of members. Such underestimation would mislead
the Structural Engineer to increase either the compressive strength of concrete, or size of
members, or amount of reinforcement unnecessarily, resulting in too conservative design.
To avoid such uneconomical design, it is important to use the confined concrete
characteristic in analysis and design.
2.7 Summary
a. Though modified test methods reported in earlier literature are logically sound,
they do have certain limitations. Some of them are not practical, while the others
are unreliable.
b. More importantly, complete stress-strain curve obtained from modified test
methods seem to be only the curve behavior that is formed due to localized failure
of tested specimen, as well as elastic recovery of undamaged portions of specimen
29
during unloading. It is, thus, questionable whether the curve represents the true
property of tested specimen in compression. Based on the above reasoning, none
of the modified test methods is applied in the present study.
c. In addition, core concrete of structural member is strengthened by reinforcement.
Hence, only reinforced concrete characteristic is able to agree the core concrete
characteristic. In other words, plain concrete characteristic is unable to match the
core concrete characteristic. Using the plain concrete characteristic in structural
analysis would only underestimate the ultimate behaviors of members, resulting in
too conservative design.
After realizing the above three factors, this study ceases observing the stress-strain
characteristic of plain lightweight concrete. Instead, this study focuses on the
characteristic of confined lightweight concrete, and of fiber-reinforced lightweight
concrete.
30
Table 2.1: Literature review on controlling the sudden failure of brittle materials
Researcher Year Type of coarse
aggregate Type of concrete
compressive strength of concrete (MPa)
Methodology
Modification
Objective
Wang et al. 1978
dolomitic limestone
normal weight
concrete 21 to 77
load a specimen in parallel with a hollow steel tube
modify the stiffness of
testing machine
to control the machine-specimen
interaction
expanded shale lightweight
concrete 21 to 56
Dahl 1992 Granite normal weight
concrete 10 to 110
load a specimen in parallel with
4 steel columns
Shah et al. 1981 crushed
limestone
normal weight
concrete 70 to 90
use the circumferential strain as
a feedback control variable
modify the testing system
to achieve the stable control
system
Jansen et al. 1995 pea gravel normal weight
concrete 35 to 103
use the circumferential strain as
a feedback control variable
Glavind and Stang 1991 Danish marine
deposits
normal weight
concrete 40
use a combination of axial and circumferential strains as a feedback control variable
Okubo and Nishimatsu
1985 _ brittle rocks _
use a combination of stress and strain as a control variable
Faust 1997 NA lightweight
concrete 47
use a combination of force and deformation as a control
variable
Jansen and Shah 1993 river pea gravel normal weight
concrete 110
use a combination of force and deformation as a control
variable
Note: NA = not available
31
(a)
deformationlo
ad
snap-back behavior
(b)
deformation
load
snap-through behavior
Figure 2.1: Load vs. deformation curves of brittle material: (a) snap-back behavior; (b) snap-through behavior
32
Figure 2.2: Modified test method reported by Wang et al. (1978)
strain
load
hollow steel tube
concrete specimen
combination of concrete specimen
and hollow steel tube
Figure 2.3: Load vs. strain curves of hollow steel tube, of concrete specimen, and of combined concrete specimen and hollow steel tube
33
(a) (b)
Figure 2.4: Modified test method reported by Dahl (1992): (a) elevation view; (b) plan view
0
15
30
45
0 0.001 0.002 0.003
strain
stre
ss (
MP
a)
CEB-FIP (predicted)
EC2 (predicted)
Figure 2.5: Predicted stress-strain curves of plain lightweight aggregate concrete with
compressive strength of 38.35 MPa
34
beam 12
0
20
40
60
80
0 0.02 0.04 0.06 0.08 0.1
curvature (rad/m)
mom
ent
(KN
-m)
experimental (Lim 2007)predicted (CEB-FIP)predicted (EC2)
Figure 2.6: Predicted behaviors of beam section (beam 12) when using the stress-strain characteristics of plain concrete
35
CHAPTER 3
Experimental Program
3.1 Introduction
This chapter discusses the experimental program conducted in the present study. The
experimental program is classified into 3 different phases: phase 1, phase 2, and phase 3
for confined concrete, fiber-reinforced concrete, and spiral-reinforced column
respectively. The main objective of experimental observations is to better understand the
characteristics of the said concretes, and behaviors of the column.
3.2 Phase 1: Confined Concrete
3.2.1 Concrete Mixtures
Lightweight aggregate concrete is produced by using ordinary Portland cement, natural
sand, tap water, and expanded clay lightweight coarse aggregate. Sizes of round and
irregular-shaped lightweight coarse aggregate are between 5 mm and 14 mm. Due to
porous nature of lightweight coarse aggregate, the aggregate absorbs some of the mixing
water during concrete mixing. This results in partial loss of mixing water, affecting the
concrete workability. To avoid this partial loss of mixing water, lightweight coarse
aggregate is pre-soaked in water tank for about 24 hours. Thereafter, free surface water
on the coarse aggregate is drained for about 1 hour, before concrete batching.
Concrete mixtures are designed in accordance with ACI 211.2-98 (1998). Firstly, a
number of trial mixtures are produced to check their 7-day compressive strengths, since
36
the 7-day strength reaches about 90% of 28-day strength (Hoff 1992a). Then, three
different mix designs (Table 3.1) are chosen to produce the concrete with cylindrical
plain compressive strengths (f’c) of 38 MPa, 49 MPa, and 58 MPa respectively.
To increase the compressive strength of concrete, silica fume which is about 7% by
weight of cementitious material is added in the concrete mixture producing the f’c of 58
MPa. Likewise, to maintain the desired workability of concrete mixture, superplasticizer
is added in concrete mixtures except the one producing the f’c of 38 MPa. Slumps of
concrete mixtures, which are measured according to BS EN 12350-2:2000 (2000), are
between 65 mm and 70 mm. Oven-dried densities of concrete, which are measured
according to BS EN 12390-7:2000 (2004), are between 1720 kg/m3 and 1820 kg/m3. The
out-line to out-line horizontal distance of lateral reinforcement is assumed as the diameter
of core concrete.
3.2.2 Type of Lateral Reinforcement
Spiral reinforcement is used as lateral reinforcement due to promising confining effect
(Fig. 3.1). Spiral reinforcement not only provides uniform confining pressure to core
concrete (Fig. 3.2a), but also confines the core concrete continuously along the
longitudinal axis (Fig. 3.3a) (Park and Paulay 1975; Sheikh and Uzumeri 1982; Razvi
and Saatcioglu 1994).
Nominal diameters of spiral wire are 4 mm, 5 mm, 6 mm and 8 mm of which yield
strengths are 1245 MPa, 1457 MPa, 1675 MPa and 1457 MPa respectively. To observe
the stress-strain properties of wires, straight wires which were never spiraled before are
tested in uniaxial tension (Figs. 3.4 and 3.5). The properties are shown in Fig. 3.6. Pitch
37
of spiral reinforcement is varied as 12 mm, 14 mm, 17 mm, 18 mm, 24 mm, 32 mm, 40
mm and 50 mm. Details of the reinforcement used in specimens are also given in Table
3.2.
3.2.3 Yield Strength of Lateral Reinforcement
Since lightweight concrete is a brittle material, a large quantity of lateral reinforcement is
required to maintain the desired ductility in lightweight concrete members. Using the
large quantity, however, congest the reinforcing system in members (Mangat and Azari
1985). Congestion of reinforcement would cause the problems such as difficulties in
fabricating the reinforcement, and in consolidating the concrete mixture.
Muguruma and Watanabe (1990), Nishiyama et al. (1993), Cusson and Paultre (1994),
Razvi and Saatcioglu (1994) and Foster et al. (1998) reported that using high yield
strength of lateral reinforcement is beneficial in increasing the ductile property of lateral-
reinforced normal weight concrete. As such, with high yield strength of lateral
reinforcement, only a moderate amount of lateral reinforcement is required. In this way,
problems related to the congestion of reinforcement can be avoided, while desired ductile
behaviour is maintained in members. For this reason, lateral reinforcement with high
yield strength is used in this phase of the study (Table 3.2).
3.2.4 Specimen Preparations
A drum mixer is used to mix the concrete constituents properly. To obtain the adequate
concrete compaction, freshly cast specimens are vibrated on a vibrating table. At this
stage, careful attention is paid not to over-vibrate the specimens to prevent the concrete
38
segregation. Specimens are covered with wet burlap cloth and plastic sheet for about 24
hours. Then, specimens are removed from the moulds, and placed in a fog room where
the temperature is set at 23oC, and relative humidity is controlled at 100% until they
reach the age of 28 days. Before being tested, specimens are air-cured in laboratory air
for about 7 days to reduce the effect of pore water pressure, and to achieve the
consistency in test results (Attard and Setunge 1996).
Uneven loaded surface of specimen leads to external eccentricity in testing. To avoid this
eccentricity, loaded surface of each cylindrical specimen is grinded in grinding machine
until it becomes smooth and even. The specimens are, for example, named as 57-4-12
where the first number ‘57’ stands for the compressive strength of plain (unreinforced)
concrete, the second number ‘4’ refers to the diameter of spiral wire, and the last one ‘12’
represents the pitch of spiral reinforcement.
In observing the characteristics of lateral-reinforced concrete, single specimen is mostly
tested for each study variable. Reliability of test results is then randomly checked by
testing two identical specimens in some study variables. In the case of plain concrete, 3
identical specimens are prepared and tested for each study variable in order to check the
reliability of test results. Since specimen preparation and testing are done with care,
identical specimens always show similar tested results and profiles. Therefore, the typical
test result and profile of each study variable are used in analyzing the data.
3.2.5 Test Setup and Instrumentation
Response of lateral-reinforced lightweight aggregate concrete to short-term compression
is explored by testing the cylindrical specimens of 100 x 200 mm and 150 x 300 mm
39
under uniform axial compression at MTS 1000 KN UTM testing machine, and at Denison
3000 KN testing machine. Around the mid-height of each specimen, two strain gages of 5
mm in length are fixed on spiral reinforcement to measure the tensile strain developed in
reinforcement. Furthermore, two other strain gages which are 60 mm in length are fixed
vertically in middle portion of each specimen (Fig. 3.7). These gages record axial strain
of specimen until cracking forms on specimen surface.
Besides these strain gages, a total of six transducers are installed to record axial
deformation (shortening) of specimen throughout the testing (Fig. 3.7). The tests are
conducted in displacement controlled mode of 0.1 mm/min. Any of the modified test
methods reviewed in Section 2.3 are not applied in this study. Specimens are tested up to
strain value of 0.03 which is about 10 times of maximum strain of plain lightweight
concrete mentioned in BS EN 1992-1-1:2004.
Appearance of lateral-reinforced specimens after testing is shown in Fig. 3.8. Plain
lightweight concrete specimens (control specimens) are also tested under the same testing
conditions to investigate their response to short-term loading.
3.2.6 Criterion Used in Testing
In general, experimental stress-strain curves are captured until the rupture of spiral
reinforcement. Spiral reinforcement, however, does not rupture in this study. Hence, the
curves are captured until a certain level of strain value is reached. As shown in Fig. 4.3,
most of the curves are captured until the strain value of at least 6εco is reached. As for the
curves in Fig. 4.3h, the curves end before reaching the strain value of 4εco. The reason is
that testing of specimen L38-6-17 has to be terminated earlier due to limitation of the
40
capacity of MTS testing machine used. To be consistent in comparison, the curve of
specimen 38-4-12 is thereby stopped displaying at similar strain value in Fig. 4.3h.
Nonetheless, these two curves in Fig. 4.3h still provide adequate information on
deformation capacity of specimens.
3.2.7 Methodology to Refine the Raw Experimental Data
Correcting the Initial Region of Stress-Strain Curve
It is well known that initial stress-strain response of concrete to compression is linear.
This linear response is clearly seen in stress-strain curve obtained from strain gage
reading (Fig. 3.9b). Since strain gage is usually fixed in middle portion of specimen,
strain gage reading is believed to be free from influence of end zone effect (Mansur et al.
1995). Hence, stress-strain curve obtained from strain gage reading can be regarded as
the true response of concrete to compression.
It is interesting to note that initial region of the curve obtained from transducer reading is
not linear (Fig. 3.9a); the initial region displays some extra deformation in a curvy
manner. Similar observation is reported by Choi et al (1996). This non-linear region
probably arises from imperfect contact among loaded surfaces of specimen, of bearing
plate, and of loading platen of testing machine at the start of testing.
Since this curvy manner is not the true initial response of concrete, it is necessary to
correct the initial region of stress-strain curve obtained from transducer reading. In this
study, correction is done by determining a new origin which is the intersection point
between horizontal strain value axis and tangent line drawn down from linear elastic
region of the curve. Similar correction is carried out by Choi et al (1996). Thereafter,
41
correct initial response is obtained by connecting the new origin with linear elastic region
of the curve. Fig. 3.9b shows the nature of stress-strain curve after correcting the curvy
region.
Correcting the Posture of Stress-Strain Curve
As mentioned in previous section, strain gage reading is free from influence of end zone
effect of specimen. Hence, stress-strain curve obtained from strain gage reading can be
regarded as the true response of concrete to compression. However, usefulness of the
strain gage is limited in compression testing. Just before the compressive strength level of
specimen, cracking usually forms on specimen surface. Due to this cracking, strain gage
becomes detached from specimen, and stops recording the further strain data. For this
reason, strain gage reading does not cover the post-peak strain data.
Only the transducers are able to capture both pre-peak and post-peak deformation data.
However, the pre-peak portion obtained from transducer reading is found to be different
from that obtained from strain gage reading (Fig. 3.9b). The reason is that transducer
reading includes not only the specimen deformation but also other extra deformation.
This extra deformation includes (1) deformation at the interfaces between two different
materials which are concrete specimen and steel bearing plate, (2) deformation of bearing
plate, and (3) deformation of steel platform on which specimen is placed and loaded.
Since the curve obtained from strain-gage reading is regarded as the true response of
concrete to compression, it is necessary to correct the posture of the curve obtained from
transducer reading. Correction is done as shown in Eq. 3.1 by using the correction factor
suggested by Mansur et al. (1995). After the correction, pre-peak portion of the stress-
42
strain curve obtained from transducer reading becomes almost identical to that obtained
from strain gage reading (Fig. 3.9c).
fEE sgtp
tpactual
11 (3.1)
where actual = actual concrete strain at corresponding stress level of f
tp = concrete strain measured by transducers at corresponding stress level of f
Etp = initial tangent modulus of plain concrete based on the stress-strain curve
derived from transducers
Esg = initial tangent modulus of plain concrete based on the stress-strain curve
derived from strain gauges.
3.3 Phase 2: Fiber-Reinforced Concrete
3.3.1 Concrete
In producing the fiber-reinforced lightweight aggregate concrete, round and hooked-end
steel fibers (Figure 3.11a) are manually distributed inside concrete mixer as the last
concrete constituent. Hooked-end steel fibers are used, since they develop the good
bonding with surrounding mortar matrix (Bentur and Mindness 2007). Durability wise,
Bentur and Mindess 2007 reported that corrosion occurs in particular fibers located near
the outer surface of concrete. More importantly, such corrosion does not affect the
integrity, strength, toughness, and abrasion resistance of concrete (Bentur and Mindess
2007).
43
The fibers used in the study are 30 mm in length, 0.375 mm in diameter; equivalent
aspect ratio is about 80. Volume fractions of fiber addition are 0.5%, 0.75%, 1%, and
1.3% by volume of concrete (Table 3.3). Thus, reinforcing index which is a product of
volume fraction and aspect ratio of fiber varies from 0.4 to 1.04.
In this phase, a total of 12 different concrete mixtures (Table 3.3) are produced. Among
them, 3 mixtures are plain concrete mixtures (control mixtures), while the rest are fiber-
reinforced concrete mixtures. Target compressive strengths of plain concrete (f’c) are 41
MPa, 55 MPa, and 59 MPa.
3.3.2 Specimen Preparation
In each concrete mixture, 6 cylindrical specimens of 100×200 mm and 6 cube specimens
of 100 mm are prepared. To obtain the adequate concrete compaction, freshly cast
specimens are vibrated on a vibrating table. During the first 24 hours from casting,
specimens are cured under wet burlap cloth and plastic sheet. Specimens are then
removed from the moulds, and moist cured in a fog room where the temperature is set at
23’C, and relative humidity is controlled at 100%.
After 28 days of moist curing, specimens are air dried in laboratory air for about 7 days to
reduce the effect of pore water pressure and to obtain the consistency in test results
(Attard and Setunge 1996). Furthermore, loaded surface of cylindrical specimens is
grinded to obtain the smooth and level surface. Such grinding also removes the parts of
the fibers that protrude from the surface.
Based on the f’c, specimens are classified into three different groups: A, B, and C for the
specimens with f’c of 41 MPa, 55 MPa, and 59 MPa respectively (Table 3.3). Specimens
44
are, for example, named as B-0.5 where B represents the group name, and 0.5 stands for
the volume fraction of fiber added in concrete. Variability of test results is checked by
testing 3 identical specimens for each study variable. Since specimen preparation and
testing are done with care, identical specimens always exhibit similar test results and
profiles. Therefore, the typical test result and profile of each study variable are used in
analyzing the data.
3.3.3 Test Set-up
To capture the load-strain relationship of fiber-reinforced concrete, cylindrical specimens
are tested under uniform axial compression at MTS 1000 KN testing machine (Fig. 3.7).
Specimens are tested up to strain value of 0.023 which is about 7 times of maximum
strain of plain lightweight concrete mentioned in BS EN 1992-1-1:2004. Methodology
used in refining the raw experimental data is discussed at length in Section 3.2.6.
Splitting tensile strength of the concrete is determined by applying the compressive force
along the length of cylindrical specimens (ASTM C 496/C 496 M).
Oven-dried density of the concrete is measured on 100 mm cube specimens. After the
moist curing, cube specimens are placed in an oven of which temperature is set at 105’C.
Weight of each specimen is recorded in every 24 hours. When a specimen shows a
weight difference which is less than 0.2% in adjacent recordings, the lastly measured
weight is regarded as the oven-dried weight of the specimen (BS EN 12390-7). Then,
oven-dried density of the specimen is readily obtained by dividing the oven-dried weight
with specimen volume.
45
Careful attention is paid to be consistent in specimen preparation, casting, curing, and
testing procedures.
3.4 Phase 3: Spiral-Reinforced Column
3.4.1 Specimen
To capture the experimental load-strain relationship of lightweight concrete column, a
total of eleven circular columns are tested under uniform axial compression; seven of
which are spiral-reinforced columns, three of which are spiral-reinforced columns
containing short steel fibers, while one of which is plain (unreinforced) column.
Dimensions of the columns are 900 mm in height and 250 mm in diameter (Fig. 3.10).
Based on the slenderness ratio, the columns are classified as short columns.
Thickness of cover concrete in reinforced column is 20 mm. Diameter of core concrete in
reinforced column, which is measured from outer-line to outer-line horizontal distance of
spiral reinforcement, is 210 mm (Fig. 3.10). Pitch of spiral reinforcement is equally
spaced throughout the height of column.
Nominal diameters of spiral wire are 6 mm, 8 mm, and 10 mm of which yield strengths
are 617 MPa, 580 MPa, and 532 MPa respectively. Pitch of spiral reinforcement is varied
as 50 mm, 75 mm, and 100 mm. Longitudinal reinforcing bars are evenly distributed in
each column. Total numbers of longitudinal reinforcing bars used in the columns are 4, 6,
and 8 (Fig. 3.11). Nominal diameter, yield strength, and elastic modulus of longitudinal
reinforcing bars are 16 mm, 537 MPa, and 179,663 MPa respectively. Details of
reinforcement used in the columns are also summarized in Table 3.4.
46
3.4.2 Concrete
Lightweight aggregate concrete is produced by using ordinary Portland cement, natural
sand, tap water, and expanded clay lightweight coarse aggregate. The concrete mix
design is shown in Table 3.5. Maximum size of coarse aggregate is 15 mm. Coarse
aggregate is pre-soaked in water for such purpose, as described in Section 3.2.1.
Cylindrical compressive strength of plain (unreinforced) lightweight concrete, which is
used in the columns, is 45.8 MPa.
In fiber-reinforced concrete mixtures, round and hook-end steel fibers are added inside
concrete mixer as the last concrete constituent. These short steel fibers are 30 mm in
length and 0.375 mm in diameter, providing the aspect ratio of 80. Volume fraction of
fibers added in the columns are 0.5%, 0.75%, and 1% by volume of concrete used in
corresponding columns. Cylindrical compressive strengths of fiber-reinforced concrete
used in the columns are 50.62 MPa, 51.8 MPa, and 54.44 MPa for fibre dosages of 0.5%,
0.75%, and 1% respectively.
A superplasticizing admixture is added in fiber-reinforced concrete mixtures to obtain the
desired concrete workability. To obtain the adequate concrete compaction, poker vibrator
is employed.
3.4.3 Specimen Preparation
Weakening the Monitoring Zone
During testing, the middle zone of each column is monitored by sensors such as strain
gages and transducers (Figs. 3.12, 3.13, 3.14). To weaken this monitoring zone, cross-
sectional diameter of each column is narrowed down by 10 mm over the column length
47
of 200 mm (Figs. 3.10 and 3.15). Weakening the zone ensures that column would fail in
the monitoring zone (Fig. 3.16) where the sensors are installed (Foster and Attard 1999).
In this way, the sensors are able to capture the true axial deformation of column
throughout the testing.
Strengthening the End Zones
To prevent the premature end zone failure at the interfaces between two different
materials which are concrete column and steel plate, the two ends of each column are
strengthened by wrapping two layers of fibre-reinforced polymer sheets over the length
of 200 mm (Figs. 3.10 and 3.13). After wrapping the fibre-reinforced polymer sheets,
columns are left to dry the epoxy resin in laboratory environment for about 5 days.
Providing the Concrete Cover
A concrete cover of 5 mm thick is provided between the end of longitudinal bars and top
loaded surface of each column in order to avoid the direct loading being applied on
longitudinal bars. Similarly, a concrete cover is also kept between the end of longitudinal
bars and bottom surface of each column. Due to the presence of fibre-reinforced polymer
sheets at the ends of column (Figs. 3.10 and 3.13), these concrete covers do not induce
any weakening effect in column.
Controlling the External Eccentricity
To eliminate or mitigate the external eccentricity in testing, special attention is paid to
ensure that the columns are loaded under uniform axial compression. To obtain the flat
48
and level surfaces in each column, Plaster-Of-Paris (POP) is applied over the top and
bottom surfaces of column. Then, a small amount of compression loading is applied to
column before the setting (hardening) of POP, and is maintained throughout the setting
process of POP. Moreover, at the beginning of testing, preloading is done up to 100 KN
in order to detect any external eccentricity. The position of column in testing frame is
adjusted until no external eccentricity is detected in preloading.
3.4.4 Test Setup and Instrumentation
The columns are tested under uniform axial compression at servo-controlled testing
machine (Fig. 3.14). The tests are conducted in the displacement-controlled mode.
Instead of using monotonic rate of axial displacement, different rates of axial
displacement ranging from 0.05 mm/min to 0.2 mm/min are used. The rate of axial
displacement is reduced from just before the peak load level to immediate post-peak load
levels in order to control the sudden failure of the columns. Similar slow rate of loading
is also reported by Hadi (2005).
In each column, three strain gages of 5 mm in length are fixed on longitudinal reinforcing
bars to record the compression strain developed in the bars. Likewise, another three strain
gages of 5 mm in length are fixed on spiral reinforcement to measure the tension strain
developed in the spiral (Fig. 3.12). Furthermore, three strain gages of 60 mm in length are
fixed vertically at mid-height of each column (Fig. 3.13), which is about 120 degrees
apart from each other around the circumference of column, to measure the vertical strain
data of column until the cracking forms on cover concrete.
49
Moreover, four transducers of 50 mm in length are mounted in middle zone of column to
measure the axial deformation (shortening) of the zone (Fig. 3.14). All the wires of the
sensors are connected to the data logger system to capture the corresponding data
throughout the testing.
3.5 Summary
This chapter presents the details of experimental program conducted in the study. The test
results of the experimental program are used in the discussion of the following chapters.
50
Table 3.1: Concrete mix design of spiral-reinforced lightweight concrete specimens
f'c
(MPa) cement (kg/m3)
silica fume
(kg/m3)
w/c (or)
w/cm
water (kg/m3)
sand (kg/m3)
coarse aggregate (kg/m3)
38 360 0 0.5 180 765 607 49 460 0 0.35 161 731 607 58 493 37 0.28 148 686 607
Table 3.2: Spiral-reinforced lightweight concrete specimens
specimen name
spiral reinforcement
f'c (MPa)
dimension of
cylinder (mm)
d (mm)
s (mm)
ρs (%)
fy (MPa)
38-4-12 4 12 4.55 1245 38 100×200 38-4-14 4 14 3.90 1245 38 100×200 38-4-32 4 32 1.70 1245 38 100×200 38-4-50 4 50 1.09 1245 38 100×200 38-5-14 5 14 6.09 1457 38 100×200 38-8-24 8 24 9.10 1457 38 100×200 49-4-14 4 14 3.90 1245 49 100×200 49-5-14 5 14 6.09 1457 49 100×200 57-4-12 4 12 4.55 1245 58 100×200 57-4-14 4 14 3.90 1245 58 100×200 57-4-40 4 40 1.36 1245 58 100×200 57-5-14 5 14 6.09 1457 58 100×200 57-6-18 6 18 6.82 1675 58 100×200 57-8-31 8 31 7.04 1457 58 100×200
L38-6-17 6 17 4.52 1675 37 150×300 L57-6-17 6 17 4.52 1675 56 150×300
Note: d = diameter of spiral wire; s = pitch of spiral reinforcement; ρs = volumetric ratio of spiral reinforcement which is determined by Eq. 4.2; fy = yield strength of spiral reinforcement; f’c is cylindrical compressive strength of plain lightweight aggregate concrete.
51
Table 3.3: Concrete mix design of fiber-reinforced lightweight concrete specimens
specimen name
f'c
(MPa) cement (kg/m3)
silica fume
(kg/m3)
w/c (or)
w/cm
water (kg/m3)
sand (kg/m3)
coarse aggregate (kg/m3)
fiber dosage
(%)
A-0 41 360 0 0.5 180 765 607 0 A-1 41 360 0 0.5 180 765 607 1 B-0 55 493 37 0.28 148 686 607 0
B-0.5 55 493 37 0.28 148 686 607 0.5 B-0.75 55 493 37 0.28 148 686 607 0.75
B-1 55 493 37 0.28 148 686 607 1 B-1.3 55 493 37 0.28 148 686 607 1.3 C-0 59 493 37 0.25 132 711 607 0
C-0.5 59 493 37 0.25 132 711 607 0.5 C-0.75 59 493 37 0.25 132 711 607 0.75
C-1 59 493 37 0.25 132 711 607 1 C-1.3 59 493 37 0.25 132 711 607 1.3
Table 3.4: Spiral-reinforced lightweight concrete columns
no. column name
f'c (MPa)
longitudinal reinforcement
lateral reinforcement
fiber (%)
d (mm)
fy (MPa)
Es (103xMPa)
no. of bar
reinforce-ment ratio
(%)
d
(mm) fy
(MPa) s
(mm)
I 6-6-50 45.8 16 523 180 6 2.45 6 617 50 0
II 6-8-50 45.8 16 523 180 6 2.45 8 580 50 0
III 6-10-50 45.8 16 523 180 6 2.45 10 532 50 0 IV 4-10-75 45.8 16 523 180 4 1.63 10 532 75 0 V 6-10-75 45.8 16 523 180 6 2.45 10 532 75 0 VI 8-10-75 45.8 16 523 180 8 3.27 10 532 75 0 VII 6-10-100 45.8 16 523 180 6 2.45 10 532 100 0 VIII 6-10-75 45.8 16 523 180 6 2.45 10 532 75 0.5 IX 6-10-75 45.8 16 523 180 6 2.45 10 532 75 0.75 X 6-10-75 45.8 16 523 180 6 2.45 10 532 75 1 XI plain 45.8 0 0 0 0 0 0 0 0 0
Note: Reinforcement ratio is calculated by dividing the total cross-sectional area of longitudinal reinforcement with cross-sectional area of column. Table 3.5: Concrete mix design of spiral-reinforced lightweight concrete columns
column no. f'c
(MPa) cement (kg/m3)
w/c (or)
w/cm
water (kg/m3)
sand (kg/m3)
coarse aggregate (kg/m3)
fiber dosage
(%)
I to VII, & XI
45.8 460 0.35 161 731 607 0
VIII 45.8 460 0.35 161 731 607 0.5 IX 45.8 460 0.35 161 731 607 0.75 X 45.8 460 0.35 161 731 607 1
52
Figure 3.1: Spiral reinforcement used in reinforced concrete specimen
(a) (b) Figure 3.2: Plan view of core concrete with different types of lateral reinforcement (Park 1975): (a) spiral reinforcement; (b) rectilinear tie
effectively confine area of core concrete
53
(a) (b)
Figure 3.3: Elevation view of reinforcing system with different types of lateral reinforcement: (a) spiral reinforcement; (b) discrete lateral reinforcement
Figure 3.4: Preparation of straight steel wires for direct tension test
spiral reinforcement
discrete lateral reinforcement
longitudinal reinforcement
54
Figure 3.5: Test setup and instrumentation for direct tension test
0
400
800
1200
1600
2000
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
strain
str
ess
(MP
a)
4 mm diameter bar
5 mm diameter bar
6 mm diameter bar
8 mm diameter bar
Figure 3.6: Stress-strain properties of lateral reinforcing wires in uniaxial tension
55
Figure 3.7: Test setup and instrumentation for uniaxial compression test
(a) (b)
Figure 3.8: Appearance of spiral-reinforced lightweight concrete specimens after testing under uniaxial compression: (a) specimens with different pitch of spiral reinforcement;
(b) specimens with different diameter of spiral wire
loading platen
spherical seating platen
strain gage
transducer stand
bottom bearing plate
top bearing plate
transducers
56
(a) initial region of the curve obtained from transducer reading
0
40
80
120
0 0.01 0.02 0.03 0.04strain
stre
ss (
MP
a)
(b) after correcting the initial region of the curve obtained
from transucer reading
0
40
80
120
0 0.01 0.02 0.03 0.04
strain
stre
ss (
MP
a)
transducer reading
strain gage reading
(c) after correcting the posture of the curve obtained from transducer reading
0
40
80
120
0 0.01 0.02 0.03 0.04
strain
stre
ss (
MP
a)
transducer reading
strain gage reading
Figure 3.9: Correction of the initial region and posture of the stress-strain curve obtained from transducer reading
curvy manner
57
(a) elevation view (b) section A-A
Figure 3.10: Spiral-reinforced lightweight concrete column
Figure 3.11a: Round and hooked-end short steel fibers
250 mm
260 mm
350 mm
900 mm A A
210 mm
250 mm
Φ 16 mm
200 mm
200 mm
350 mm
200 mm
monitoring zone of 400 mm
strengthening the end zone with FRP sheets
strengthening the end zone with FRP sheets
58
Figure 3.11: Plan view of three reinforcing cages with different numbers of longitudinal reinforcement
Figure 3.12: Elevation view of three reinforcing cages: Installation of strain gages on longitudinal reinforcement and lateral reinforcement
59
Figure 3.13: Elevation view of three columns: strain gages are installed in middle zones and fiber-reinforced polymer sheets are wrapped in end zones
Figure 3.14: Test set-up and instrumentation for column
60
Figure 3.15: Elevation view of three columns during preparation stage: Weakening the middle zones of columns
Figure 3.16: Appearance of three columns with different pitch of spiral reinforcement after testing
61
CHAPTER 4
Confined Concrete: Structural Response
4.1 Introduction
Deformation capacity of structural concrete member is an ability to maintain the load
carrying capacity when the member is undergoing large degree of inelastic deformation
due to overloading. Under overloading, a member with low deformation capacity would
fail suddenly, while that with high deformation capacity would fail in a gradual manner.
Gradual failure would enable the member to redistribute its moment and loading to
surrounding members, which would mitigate the progressive collapse of the whole
structure (Bresler 1971; Mander et al. 1988; Bjerkeli et al. 1990; Hoff 1990). Moreover,
gradual failure would provide a lot of warning signs, such as member bending, concrete
cracking, and concrete spalling, which alert the occupants in the structure to evacuate. In
addition, gradual failure would provide adequate time for evacuation. It is, therefore, no
doubt that deformation capacity of members plays an important role in saving the human
lives as well as property (McCormac 1998).
Deformation capacity of structural member is closely linked to that of lateral-reinforced
concrete (Sheikh and Uzumeri 1982). Unfortunately, deformation capacity of lateral-
reinforced lightweight concrete is not fully understood due to limited information
available in current literature. Such insufficient understanding largely affects the
prediction of member capacity, resulting in either unsafe or uneconomical design. This
62
study thus observes the deformation capacity of confined lightweight concrete under
uniform axial compression.
In the observation, deformability parameter which is measured on stress versus strain
curve is used. Deformability parameter is defined as the ratio between compressive stress
of spiral-reinforced concrete at strain value of 3εco and compressive strength of spiral-
reinforced concrete (Eq. 4.1 and Fig. 4.1). The parameter which simply quantifies the
deformation capacity serves in comparing the deformation capacity of specimens.
deformability parameter = cof
f3 (4.1)
where f3 = compressive stress of spiral-reinforced concrete at strain value of 3εco (MPa)
fco = compressive strength of spiral-reinforced concrete (MPa)
εco = concrete strain at fco
On the other hand, ultimate load capacity of member is closely related to compressive
strength of confined concrete. Compressive strength of confined lightweight concrete is
not well documented in current literature. Lack of proper documentation affects the
prediction of ultimate load capacity of lightweight concrete members, leading to too
conservative design. Therefore, this study also explores the compressive strength of
confined lightweight concrete under uniform axial compression.
As shown in Table 3.2, small scale specimens are tested in this phase of the study.
Therefore, volumetric ratio of lateral reinforcement, which can be applied in any
specimens regardless of specimen size, is used herein. Volumetric ratio of spiral
63
reinforcement (ρs) is defined as the ratio between volume of spiral reinforcement in one
loop, and volume of core concrete within one pitch in length (Eq. 4.2).
sdD
A
c
ss
4 (4.2)
where As = cross-sectional area of spiral wire
Dc = diameter of core concrete
d = diameter of spiral wire
s = pitch of spiral reinforcement which is the center-line to center-line distance
between adjacent spiral wires
4.2 Definition of Compressive Strength Level
In this study, compressive strength of confined concrete is defined based on the shape of
stress-strain curve. For the stress-strain curve that clearly shows the descending portion
beyond the peak strength level (Fig. 4.2a), the peak strength level is regarded as the
compressive strength of lateral-reinforced concrete. It is also common that some stress-
strain curves showing strain hardening characteristic do not have any descending portion
(Fig. 4.2b); the direction of these curves dramatically changes at a certain strength level.
In this case, the curve direction changing point is assumed as the compressive strength of
confined concrete.
4.3 Designation of Specimens
64
Lateral-reinforced concrete specimens are named as 38-4-32 where the first number ‘38’
stands for the compressive strength of plain lightweight aggregate concrete, the second
number ‘4’ represents the diameter of spiral wire, while the last one ‘32’ refers to the
pitch of spiral reinforcement.
4.4 Scope and Objective
Response of spiral-reinforced lightweight concrete to short-term compression is explored
by conducting an experimental program. The response includes deformation capacity,
compressive strength, concrete strain at compressive strength, modulus of elasticity, and
failure mode. The response is explored by varying the pitch of spiral reinforcement,
diameter of spiral wire, compressive strength of plain concrete, and specimen size. The
purpose of the study is to gain better understanding on structural response of confined
lightweight concrete which is closely linked to behaviors of lightweight concrete
members. Understanding of the concrete response gained from the study would be
beneficial in analysis and design of structural lightweight concrete members.
4.5 Observations and Discussion
4.5.1 Response of Spiral-reinforced Concrete
Deformation Capacity
In observing the deformation capacity, stress history and strain history of each specimen
are normalized against its compressive strength and concrete strain at compressive
strength respectively (axial stress/fco & axial strain/εco) (Fig. 4.3). Since normalized
stress-strain curves offset the difference in compressive strength and concrete strain at
65
compressive strength among specimens with different variables, these curves offer
effective comparison for deformation capacity (Luc 1991, Hsu & Hsu 1994a, Nataraja et
al. 1999). Strain hardening characteristic in post-peak region of normalized curve
indicates level of deformation capacity. Apart from normalized curves, deformability
parameter (Eq. 4.1) is also used in observation. High value of deformability parameter
indicates high level of deformation capacity.
Test results show that deformation capacity of spiral-reinforced concrete noticeably
increases with reducing pitch of spiral reinforcement (Figs. 4.3a and 4.3b; Table 4.1).
Reducing the pitch would increase the effectively confined area of core concrete along
the longitudinal axis (Fig. 4.4) (Park and Paulay 1975), which in turn increases the
deformation capacity.
Furthermore, deformation capacity increases with an increase in diameter of spiral wire
(Figs. 4.3c, 4.3d, and 4.3e; Table 4.2). Increasing the diameter of spiral wire would
increase the stiffness of spiral reinforcement, which in turn reduces the radial expansion
of spiral reinforcement. Reducing the radial expansion of spiral reinforcement provides
better confining effect to core concrete. Therefore, increasing the diameter of spiral wire
increases the deformation capacity.
Deformation capacity of spiral-reinforced concrete also depends on compressive strength
of plain lightweight concrete (f’c), decreasing with an increase in f’c (Figs. 4.3f and 4.3g;
Table 4.4). It is known that deformation capacity of plain lightweight concrete decreases
with an increase in f’c (Zhang and Gjorv 1991; Faust 1997). Hence, this characteristic of
plain lightweight concrete is still pronounced in spiral-reinforced lightweight concrete.
66
Deformation capacity is found to be unaffected by changing the size of cylindrical
specimen from 100×200 mm to 150×300 mm (Figs. 4.3h and 4.3i; Table 4.5).
Compressive Strength
For given f’c and diameter of spiral wire, compressive strength of spiral-reinforced
concrete (fco) becomes increased with reducing pitch of spiral reinforcement (Table 4.1).
Reducing the pitch would increase the effectively confined area of core concrete along
the longitudinal axis (Fig. 4.4) (Park and Paulay 1975), which in turn increases the fco.
For given f’c and pitch of spiral reinforcement, fco increases with an increase in diameter
of spiral wire (Table 4.2). Increasing the diameter of spiral wire would increase the
stiffness of spiral reinforcement, which in turn reduces the radial expansion of spiral
reinforcement. Reducing the radial expansion of spiral reinforcement provides better
confining effect to core concrete. Thus, increasing the diameter of spiral wire increases
the fco.
Since fco depends on pitch of spiral reinforcement, diameter of spiral wire, and f’c, fco can
be expressed as a function of these variables as shown in Eq. 4.3. Experimental data
obtained from present study serves as the database in formulating Eq. 4.3.
'433 cco fs
df (MPa) (4.3)
where d = diameter of spiral wire (mm)
s = pitch of spiral reinforcement (mm)
f’c = cylindrical compressive strength of plain lightweight concrete (MPa)
67
Eq. 4.3 is verified by showing the consistency between computed data obtained from Eq.
4.3 and experimentally measured data. Besides the experimental data recorded in present
study, experimental data reported by Shah et al. (1983), Sudo et al. (1993), and Campione
and Mendola (2004) are used in comparison. However, these experimental data reported
by other researchers are not used in formulating Eq. 4.3.
Similarly, apart from Eq. 4.3, equations reported by Mander et al. (1988), Hsu and Hsu
(1994), Mansur et al. (1997), fib Task Group 8.1 (2000), EC2 Code (2004) are used in
generating the computed data. It should be mentioned here that the equations reported by
Mander et al. (1988), Hsu and Hsu (1994), Mansur et al. (1997) are originally developed
for lateral-reinforced normal weight concrete.
As displayed in Fig. 4.5 and Tables 4.6, computed data obtained from Eq. 4.3 agree well
with all the experimental data. This agreement shows the applicability of Eq. 4.3.
Therefore, Eq. 4.3 is applicable for the concrete with compressive strength of plain
lightweight concrete ranging between 38 MPa and 58 MPa, and with volumetric ratio of
spiral reinforcement between 1.1% and 6.8%.
Concrete Strain at Compressive Strength (εco)
εco is an important parameter in defining the complete stress-strain characteristic of
confined lightweight concrete, and in estimating the ultimate behaviors of flexural beams
and of columns under combined axial loading and bending moment. Test results show
that εco largely depends on fco. This study, thus, suggests an empirical equation of εco as a
function of fco (Eq. 4.4).
68
)104(0018.0 5coco f (4.4)
where fco = compressive strength of spiral-reinforced lightweight concrete (MPa).
Modulus of Elasticity of Plain Concrete (Ec)
Ec represents the elastic response of plain concrete to compression. Experimental values
of Ec are measured according to ASTM C469 – 02 (2006). Mathematical expression for
Ec is formulated as a function of f’c (Eq. 4.5). The equation is then verified by comparing
computed data obtained from the equation with experimental data recorded in this study
(Fig. 4.6).
Apart from the Eq. 4.5, equations suggested by CEB-FIP Code (2000), EC2 Code (2004),
and BS Standard (2008) are carefully assessed. The equation suggested by CEB-FIP
Code is valid for lightweight aggregate concrete with oven-dried density between 800
kg/m3 and 2200 kg/m3, while the equation by EC2 Code is for the concrete with
cylindrical concrete strength between 17 MPa and 88 MPa, and the equation by BS
Standard is for the concrete with cube concrete strength between 20 MPa and 60 MPa.
The equation suggested by ACI 213R-03 (2003) is excluded in the assessment. The
reason is that the valid range of cylindrical concrete strength of the equation, which is
between 21 MPa and 35 MPa, is outside the considered range of the present study (38
MP and 58 MPa).
As shown in Fig. 4.6, Eq. 4.5 is capable of generating the computed data that are in good
agreement with experimental data. This agreement clearly shows that Eq. 4.5 is
applicable for plain lightweight concrete with f’c ranging between 38 MPa and 58 MPa.
69
'16018000 cc fE (MPa) (4.5)
where f’c = cylindrical compressive strength of plain lightweight concrete (MPa)
Modulus of Elasticity of Confined Concrete
Test results indicate that modulus of elasticity of spiral-reinforced lightweight concrete
(Eco) is unstable. This unstableness may arise from the presence of air and water
entrapped under reinforcement. These entrapped air and water create weak spots inside
concrete, which in turns affects the Eco. Experimental value of Eco is measured between
two points: one point corresponds to 50 μ strain while the other point corresponds to 40%
of compressive strength in pre-peak region (ASTM C469-02 2006). At the latter point,
lateral reinforcement is not fully activated yet. In other words, presence of lateral
reinforcement around specimen can be neglected in determining Eco. Thus, it can be
assumed that Eco is independent of any variables of lateral reinforcement. Since Eco
largely depends on f’c, Eco can be shown as a function of f’c alone, and can be assumed to
be equivalent to modulus of elasticity of plain lightweight concrete (Ec).
Unloading Manner
During unloading, plain concrete specimens always show the sudden drop in load
carrying capacity. In contrast, lightly reinforced specimens such as specimens 38-4-32,
38-4-50, and 57-4-40 (Table 3.2) show the gradual drop in load carrying capacity (Figs.
4.3a and 4.3b). As adequately reinforced specimens are concerned (Table 3.2), they even
show the strain hardening characteristic (Fig. 4.3).
70
Failure Mode
Failure mode of plain concrete specimens is sudden failure, while that of spiral-
reinforced specimens is mainly due to spalling. Concrete pieces start spalling from outer
layer of spiral-reinforced specimens, when stress-strain curve reaches immediate post-
peak region. Appearance of specimens after testing is displayed in Fig. 3.8.
4.5.2 Mathematical Expressions
Mathematical expressions for fco, εco, and Ec (Eqs. 4.3 through 4.5) are determined by
performing the regression analysis. The analysis is, in fact, performed by using statistical
analysis software. Adequacy of the fit equations to corresponding experimental data is
checked with coefficient of determination, and residual analysis. In addition, simple
forms of equations are intentionally chosen to ensure that the equations are user-friendly.
4.5.3 Influencing Variables
The variables considered in this section are pitch of spiral reinforcement, diameter of
spiral wire, amount of spiral reinforcement, and compressive strength of plain lightweight
aggregate concrete.
Pitch of Spiral Reinforcement
Influence of the pitch on characteristics of spiral-reinforced lightweight concrete is
observed by varying the pitch, while all other variables are kept constant. Test results
show that reducing the pitch considerably increases both deformation capacity and
compressive strength of spiral-reinforced concrete (Table 4.1). As mentioned earlier,
71
reducing the pitch would increase the effectively confined area of core concrete along the
longitudinal axis (Fig. 4.4) (Park and Paulay 1975), which in turn increases the
deformation capacity and compressive strength.
Diameter of Spiral Wire
Effect of the diameter of spiral wire is investigated by varying the diameter, while the rest
of the variables are kept constant. As displayed in Table 4.2, increasing the diameter
increases both deformation capacity and compressive strength of spiral-reinforced
concrete. As mentioned before, increasing the diameter of spiral wire would increase
stiffness of spiral reinforcement, which in turn reduces radial expansion of spiral
reinforcement. Reducing the radial expansion of spiral reinforcement provides better
confining effect to core concrete. Therefore, increasing the diameter of spiral wire
increases the deformation capacity and compressive strength of spiral-reinforced
concrete.
Pitch of Spiral Reinforcement vs. Diameter of Spiral Wire
When the pitch and diameter are varied concurrently, the pitch is the one that governs the
characteristics of lightweight concrete. As shown in Table 4.3, reducing the pitch
improves the deformation capacity as well as compressive strength, though diameter of
wire is also reduced at the same time. This phenomenon indicates that the pitch
outweighs the diameter in influencing the characteristics of lightweight concrete.
Amount of Lateral Reinforcement
72
When either the pitch of spiral reinforcement or diameter of spiral wire is kept constant,
increasing the amount of spiral reinforcement (ρs) increases the deformation capacity as
well as compressive strength of spiral-reinforced lightweight concrete (Tables 4.1 and
4.2).
However, varying the amount of spiral reinforcement becomes ineffective, when the
pitch and diameter are varied concurrently (Table 4.3). In this context, the pitch of spiral
reinforcement governs the deformation capacity and compressive strength.
Compressive Strength of Plain Lightweight Concrete (f’c)
As shown in Table 4.4, increasing the f’c decreases the deformation capacity of spiral-
reinforced lightweight concrete.
4.6 Summary
a. Deformation capacity of spiral-reinforced lightweight concrete becomes
decreased with an increase in compressive strength of plain lightweight concrete.
Nevertheless, the deformation capacity can be increased by reducing the pitch of
spiral reinforcement, and by increasing the diameter of spiral wire.
b. Compressive strength of spiral-reinforced lightweight concrete also depends on
pitch of spiral reinforcement and diameter of spiral wire, increasing with a
decrease in the pitch, and with an increase in the diameter.
c. Pitch of spiral reinforcement outweighs diameter of spiral wire in influencing the
characteristics of spiral-reinforced lightweight concrete.
73
d. Modulus of elasticity of spiral-reinforced lightweight concrete is unstable due to
the presence of air and water entrapped under reinforcement. This study assumes
that modulus of elasticity of spiral-reinforced lightweight concrete is equivalent to
that of plain lightweight concrete.
e. Compressive strength of spiral-reinforced lightweight concrete, concrete strain at
compressive strength, and modulus of elasticity of plain lightweight concrete are
essential parameters in analysis and design of lightweight concrete structural
members. These parameters can be determined by using the suggested equations.
The equations are applicable for the concrete with compressive strength of plain
lightweight concrete ranging between 38 MPa and 58 MPa, and with volumetric
ratio of spiral reinforcement between 1.1% and 6.8%.
f. Failure mode of spiral-reinforced lightweight concrete is mainly due to spalling.
Understanding of the concrete response gained from the study would be beneficial in
analysis and design of lightweight concrete structural members.
74
Table 4.1: Varying the pitch of spiral reinforcement
compared specimens
s (mm)
ρs (%)
deformability parameter
fco (MPa)
εco
38-4-12 12 4.55 1.22 48.94 0.0042 38-4-14 14 3.90 1.17 47.12 0.0041 38-4-50 50 1.09 0.67 39.19 0.0028 57-4-12 12 4.55 1.11 71.55 0.0047 57-4-14 14 3.90 1.08 64.07 0.0042 57-4-40 40 1.37 0.59 62.29 nil
Note: Some specimens do not have strain value due to early damage of strain gauge before reaching the compressive strength level. Table 4.2: Varying the diameter of spiral wire
compared specimens
d (mm)
ρs (%)
deformability parameter
fco (MPa)
εco
38-4-14 4 3.90 1.17 47.12 0.0041 38-5-14 5 6.16 1.23 50.06 0.0042 49-4-14 4 3.90 1.05 59.53 nil 49-5-14 5 6.16 1.18 66.05 nil 57-4-14 4 3.90 1.08 64.07 0.0042 57-5-14 5 6.16 1.16 64.54 nil
Note: Some specimens do not have strain value due to early damage of strain gauge before reaching the compressive strength level. Table 4.3: Varying both the pitch of spiral reinforcement and diameter of spiral wire
compared specimens
s (mm)
d (mm)
ρs (%)
f'c (MPa)
fy (MPa)
deformability parameter
fco (MPa)
εco
38-5-14 14 5 6.16 38 1457 1.23 50.06 0.0042 38-8-24 24 8 9.52 38 1457 1.06 46.92 0.0040 57-5-14 14 5 6.16 58 1457 1.08 64.54 nil 57-8-31 31 8 7.31 58 1457 0.99 56.11 nil
Note: Some specimens do not have strain value due to early damage of strain gauge before reaching the compressive strength level. Table 4.4: Varying the compressive strength of plain lightweight concrete
compared specimens
f'c (MPa)
deformability parameter
fco (MPa)
εco
38-4-14 38 1.17 47.12 0.0041 49-4-14 49 1.06 59.53 nil 57-4-14 58 1.08 64.07 0.0042 38-5-14 38 1.23 50.06 0.0042 49-5-14 49 1.18 66.05 nil 57-5-14 58 1.16 64.54 nil
Note: Some specimens do not have strain value due to early damage of strain gauge before reaching the compressive strength level.
75
Table 4.5: Varying the specimen size
compared specimens
size of cylinder
deformability parameter
fco (MPa)
εco
38-4-12 100×200 1.22 48.94 0.0042 L38-6-17 150×300 1.23 55.65 nil 57-4-12 100×200 1.11 71.55 0.0047
L57-6-17 150×300 1.13 74.86 nil Note: Some specimens do not have strain value due to early damage of strain gauge before reaching the compressive strength level. Table 4.6: Validation of Eq. 4.3
using the equation reported by
mean standard deviation
Mander et al 1.99 0.71 Hsu & Hsu 1.72 0.60
Mansur et al 1.33 0.51 EC2 1.21 0.35
CEB-FIP 1.21 0.34 present study (Eq. 3) 0.93 0.12
Note: mean is the mean of the ratios between computed and experimental values of fco. Mean value close to 1 indicates good agreement between computed and experimental values. Standard deviation value close to 0 indicates low degree of scatter of the ratios away from mean value. Table 4.7: Validation of Eq. 4.5
using the equation reported by
mean standard deviation
CEB-FIP 0.93 0.035 EC2 0.91 0.036 BS 0.75 0.040
present study (Eq. 5) 1.00 0.037 Note: mean is the mean of the ratios between computed and experimental values of Ec. Mean value close to 1 indicates good agreement between computed and experimental value. Standard deviation value close to 0 indicates low degree of scatter of the ratios away from mean value.
76
Table 4.8: Experimental values of Ec
f'c (MPa)
Ec (experimental values, MPa)
Ec (computed
values from Eq. 4.5, MPa)
30.23 22000 22900 35.27 20600 23600 37.10 23100 24000 38.78 23400 24200 38.92 25500 24200 45.80 25300 25300 49.32 26900 25900 49.40 26900 25900 50.70 26900 26100 51.04 25700 26200 51.79 27000 26300 54.57 27800 26700 54.72 26800 26800 55.50 26300 26900 56.03 27400 27000 57.48 27000 27200 57.66 27000 27200 58.08 27200 27300 58.79 27000 27400
77
0
10
20
30
40
0 0.005 0.01 0.015 0.02
axial strain
axia
l str
ess
(MP
a)
Figure 4.1: Stress vs. strain curve of spiral-reinforced concrete
(a) a curve with descending portion
0
15
30
45
0 0.005 0.01 0.015 0.02
axial strain
axia
l str
ess
(MP
a)
descending portion
ascending portion
peak strength level
Figure 4.2: Stress-strain curve of concrete (Continued)
fco
f3
εco 3εco
78
(b) a curve without descending portion
0
35
70
0 0.005 0.01 0.015 0.02
axial strain
axia
l str
ess
(MP
a)
ascending portion
another ascending portion heading towards different direction
curve direction turning point
Figure 4.2: Stress-strain curve of concrete: (a) a curve with descending portion; (b) a curve without descending portion
(a) varying the pitch of spiral reinforcement
0
0.5
1
1.5
0 1 2 3 4 5 6 7
normalized strain
norm
aliz
ed s
tres
s
38-4-12
38-4-14
38-4-32
38-4-50
Figure 4.3: Deformation capacity of spiral-reinforced lightweight concrete (Continued)
(axial strain/εco)
(axi
al s
tres
s/f c
o)
79
(b) varying the pitch of spiral reinforcement
0
0.25
0.5
0.75
1
1.25
0 1 2 3 4 5 6 7
normalized strain
norm
aliz
ed s
tres
s
57-4-12
57-4-14
57-4-40
(c) varying the diameter of spiral wire
0
0.5
1
1.5
0 1 2 3 4 5 6 7
normalized strain
norm
aliz
ed s
tres
s
38-5-14
38-4-14
Figure 4.3: Deformation capacity of spiral-reinforced lightweight concrete (Continued)
(axial strain/εco)
(axial strain/εco)
(axi
al s
tres
s/f c
o)
(axi
al s
tres
s/f c
o)
80
(d) varying the diameter of spiral wire
0
0.5
1
1.5
0 1 2 3 4 5 6
normalized strain
norm
aliz
ed s
tres
s
49-5-14
49-4-14
(e) varying the diameter of spiral wire
0
0.5
1
1.5
0 1 2 3 4 5 6 7normalized strain
norm
aliz
ed s
tres
s
57-5-1457-4-14
Figure 4.3: Deformation capacity of spiral-reinforced lightweight concrete (Continued)
(axial strain/εco)
(axial strain/εco)
(axi
al s
tres
s/f c
o)
(axi
al s
tres
s/f c
o)
81
(f) varying the compressive strength of plain concrete used in specimens
0
0.5
1
1.5
0 1 2 3 4 5 6 7
normalized strain
norm
aliz
ed s
tres
s
38-4-1449-4-1457-4-14
(g) varying the compressive strength of plain concrete used in specimens
0
0.5
1
1.5
0 1 2 3 4 5 6
normalized strain
norm
aliz
ed s
tres
s
38-5-1449-5-1457-5-14
Figure 4.3: Deformation capacity of spiral-reinforced lightweight concrete (Continued)
(axial strain/εco)
(axial strain/εco)
(axi
al s
tres
s/f c
o)
(axi
al s
tres
s/f c
o)
82
(h) varying the size of specimens
0
0.5
1
1.5
0 1 2 3 4
normalized strain
norm
aliz
ed s
tres
s
L38-6-17
38-4-12
(i) varying the size of specimens
0
0.5
1
1.5
0 1 2 3 4 5 6 7
normalized strain
norm
aliz
ed s
tres
s
L57-6-1757-4-12
Figure 4.3: Deformation capacity of spiral-reinforced lightweight concrete with regards to pitch of spiral reinforcement, diameter of spiral wire, cylindrical compressive strength of plain lightweight concrete, and specimen size Note: Specimens are, for example, named as 57-4-12 where ‘57’ refers to the compressive strength of plain lightweight concrete, ‘4’ means the diameter of spiral wire, ‘12’ denotes the pitch of spiral reinforcement. Size of specimens is 100×200 mm, except the two specimens named L38-6-17, and L57-6-17 of which dimensions are 150×300 mm.
(axial strain/εco)
(axial strain/εco)
(axi
al s
tres
s/f c
o)
(axi
al s
tres
s/f c
o)
83
Figure 4.4: Elevation view of spiral reinforcing system with different pitch of spiral reinforcement
0
50
100
150
200
0 50 100 150 200
experimental f co (MPa)
com
pu
ted
f co
(MP
a)
Mander et alHsu & HsuMansur et alEC2CEB-FIPpresent study (Eq. 3)line of equality
Figure 4.5: Comparison between computed and experimental values of fco
pitch
pitch
effectively confined area of core concrete
84
15000
20000
25000
30000
15000 20000 25000 30000
experimental E c (MPa)
com
pu
ted
Ec
(M
Pa
)
CEB-FIP
EC2
BS
present study (Eq. 5)
line of equality
Figure 4.6: Comparison between computed and experimental values of Ec
85
CHAPTER 5
Confined Concrete: Stress-Strain Characteristic
5.1 Introduction
Stress-strain characteristic of confined concrete is somewhat different from that of plain
concrete. Generally, the former shows higher compressive strength, and higher
deformation capacity (Fig. 5.1) (Ahmad and Mallare 1994; El-Dash and Almad 1995). In
structural members, core concrete is strengthened by lateral reinforcement such that
stress-strain characteristic of confined concrete is required to represent the core concrete
characteristic in structural analysis and design.
On the other hand, using the stress-strain characteristic of plain concrete instead would
considerably underestimate the ultimate behaviors of members (Section 2.6.5). Such
underestimation would mislead the Structural Engineer to increase either the compressive
strength of concrete, or size of members, or amount of reinforcement unnecessarily,
resulting in uneconomical design. Therefore, it is essential to use the stress-strain
characteristic of confined concrete to represent the core concrete characteristic in analysis
and design.
Unfortunately, stress-strain characteristic of confined lightweight aggregate concrete is
not well understood due to limited information available in current literature. Insufficient
understanding of the characteristic largely affects the accuracy of structural analysis,
resulting in either uneconomical or unsafe design. This study thus fills the knowledge gap
by suggesting a stress-strain model to predict the stress-strain characteristic.
86
5.2 Stress Distribution of Concrete in Compression Zone
Stress distribution diagram of concrete drawn in compression zone of flexural member
section is, in fact, the relationship between concrete stress (σ) and distance along the
depth of neutral axis (u) (Fig. 5.2). This stress-distance (σ-u) relationship is readily
converted from stress-strain relationship of concrete by using linear strain distribution of
concrete. Since the strain distribution is linear, the shape of the stress distribution
diagram happens to be identical to that of the stress-strain diagram.
5.3 Designation of Specimens
As mentioned in Section 4.3, lateral-reinforced concrete specimens are, for example,
named as 38-4-32 where the first number ‘38’ stands for the compressive strength of
plain lightweight concrete, the second number ‘4’ represents the diameter of spiral wire,
while the last one ‘32’ refers to the pitch of spiral reinforcement.
5.4 Scope and Objective
This chapter reviews the existing stress-strain models for confined normal weight
concrete. Then, it derives a new stress-strain model for confined lightweight aggregate
concrete. Thereafter, it verifies the validity of the new model, as well as demonstrates the
usefulness of the model in applications. The objective of the study is to recommend a
reliable stress-strain model that is capable of predicting the stress-strain characteristic of
confined lightweight aggregate concrete. Applying the recommended model in analysis
and design of structural lightweight concrete members, accurate analysis and optimum
design would be achieved.
87
5.5 Literature Review: Existing Stress-Strain Models
Stress-strain model for confined lightweight aggregate concrete is scarce in current
literature due to insufficient study on its stress-strain characteristic. However, a number
of stress-strain models for confined normal weight concrete are reported by various
researchers (Ahmad and Mallare 1994; El-Dash and Ahmad 1995). Since the response of
normal weight concrete to loading is somewhat different from that of lightweight
aggregate concrete (Faust 1997), it is interesting to check whether the existing models for
confined normal weight concrete are suitable for confined lightweight aggregate
concrete.
In this section, stress-strain models for confined normal weight concrete reported by
Mander et al. (1988), Hsu and Hsu (1994), and Mansur et al. (1997) are reviewed. These
models are selected due to the following reasons: (1) they are able to generate the
complete stress-strain characteristic for confined normal weight concrete; (2) they are
convenient for routine use, since computing procedures involved are simple; (3)
parameters used in the models are easily determined. Performance of these models is
observed by comparing predicted stress-strain characteristics obtained from the models
with experimental characteristic of confined lightweight aggregate concrete recorded in
the present study.
As shown in Fig. 5.3, these models significantly overestimate the stress-strain
characteristics for confined lightweight aggregate concrete. Hence, using these normal
weight concrete models in analysis of lightweight concrete members would affect the
analysis and design. This finding serves as a motivation to develop a new model for
confined lightweight aggregate concrete.
88
5.6 Suggested Stress-Strain Model
Apart from costly design operation which involves preliminary design procedure,
physical modeling and actual design procedure, most of the design operations solely
depend on predicted behaviors of members obtained from structural analysis. In this
context, accuracy of the prediction is important in producing the optimum design. To
achieve the acceptable accuracy in prediction, it is essential to use the reliable stress-
strain model of lateral-reinforced concrete in analysis.
This section suggests a stress-strain model to predict the stress-strain characteristic of
confined lightweight aggregate concrete. The model is formulated based on experimental
stress-strain characteristic recorded in present study. The model consists of two
mathematical expressions: one is to generate the pre-peak portion of the stress-strain
curve, while the other is to yield the post-peak portion.
The mathematical expression for the pre-peak portion (Eq. 5.1) is modified based on
Mander et al.’s model (1988).
c
coc
coc
cof
f
1
for co 0 (5.1)
The mathematical expression for the post-peak portion (Eq. 5.2) is developed by using
statistical analysis.
2 zyxf for co (5.2)
89
f and ε are the concrete stress and corresponding strain at any point on stress-strain curve.
fco and εco are the compressive strength of confined lightweight aggregate concrete and
concrete strain at compressive strength level (Fig. 5.1). βc is a parameter controlling the
shape of the pre-peak portion. x, y, z are the parameters influencing the shape of post-
peak portion. Empirical equations for fco and εco are formulated in Section 4.5.1, and
reported in Eq. 4.3 and Eq. 4.4 respectively. βc, x, y and z are determined by using Eqs.
5.3 through 5.6.
(5.3)
3
2'200
4040068
s
d
fx
c
(5.4)
d
sfy c 50092300 ' (5.5)
2
3' 70007.020400
d
sfz c (5.6)
In the above equations, d refers to the diameter of spiral wire, s represents the pitch of
spiral reinforcement which is center-line to center-line distance between adjacent
reinforcement, fc’ is the cylindrical compressive strength of plain lightweight aggregate
concrete (Fig. 5.1), and Ec is the elastic modulus of plain lightweight aggregate concrete.
Empirical expression for Ec is formulated in Section 4.5.1, and stated in Eq. 4.5. In this
study, elastic response of confined lightweight aggregate concrete (Eco) is assumed to be
cco
coc
E
f
1
1
90
similar to that of plain lightweight aggregate concrete (Ec). The discussion with regards
to this assumption is provided in Section 4.5.1.
As shown in Eqs. 4.3 through 4.5, and Eqs. 5.3 through 5.6, the parameters used in the
model largely depend on the variables f’c, d, and s. These variables are the known
variables at analysis and design stages. Using these variables in the model reflects their
influence on the stress-strain characteristic.
Empirical equations for x, y and z (Eqs. 5.4 through 5.6) are formulated by performing
regression analysis. The analysis is, in fact, performed by using statistical analysis
software. Adequacy of the fit equations to experimental data is determined with
coefficient of determination, and residual analysis. Moreover, simple forms of equations
are intentionally used to ensure that the equations are user-friendly.
In integrating the area under complete stress-strain curve, continuous transition between
the pre-peak portion and post-peak portion is crucial. This continuous transition is
ensured by using the equal signs in concrete strain ranges of the pre-peak equation (Eq.
5.1) and post-peak equation (Eq. 5.2).
In the model, core concrete and lateral reinforcement are assumed as a single composite
material. Hence, lateral expansion of core concrete, hoop tensile stress developed in
lateral reinforcement, and interaction between core concrete and lateral reinforcement are
ignored in deriving the model. By using Eqs. 5.1 through 5.6, and Eqs. 4.3 through 4.5,
stress-strain characteristic of confined lightweight aggregate concrete is readily
generated.
5.7 Performance of Suggested Model
91
5.7.1 Using Experimental Data Recorded in Present Study
Performance of suggested stress-strain model is evaluated by comparing predicted stress-
strain characteristics obtained from the model with experimental characteristics recorded
in the present study. Details of confined concrete specimens are shown in Table 3.2.
Information regarding the specimen preparation, test set-up and instrumentation are
discussed in Section 3.2.
As shown in Fig. 5.4, predicted characteristics agree well with experimental ones up to
strain value of 0.03 except specimens 38-4-50 and 57-4-40. These specimens have low
volumetric ratio of lateral reinforcement of 1.09% and 1.36% respectively (Table 3.2). It
is, therefore, concluded that the recommended model is applicable for lateral-reinforced
lightweight aggregate concrete with compressive strength of plain lightweight aggregate
concrete ranging between 38 MPa and 58 MPa, and with volumetric ratio of lateral
reinforcement between 1.7% and 6.8%.
5.7.2 Using Experimental Data Reported by Other Researchers
A stress-strain model is supposed to satisfy not only the experimental data used in
deriving the model, but also any relevant experimental data that are independent from
model derivation procedures. In this section, performance of suggested stress-strain
model is examined by comparing predicted stress-strain characteristics obtained from
suggested model with experimental characteristics of confined lightweight aggregate
concrete reported by Shah et al. (1983), and Sudo et al. (1993).
Experimental data considered in this section are not used in deriving the recommended
model. Summary of specimens’ specifications are given in Table 5.1. Shah et al. (1983)
92
tested the cylindrical specimens that have dimensions of 75×150 mm, compressive
strengths of plain lightweight aggregate concrete (f’c) of 37 MPa and 41 MPa, volumetric
lateral reinforcement ratio (ρs) of 1.89%, and yield strength of lateral reinforcement (fy) of
1117 MPa. They used expanded shale (Materialite and Solite) as lightweight coarse
aggregate. Their two other specimens that have ρs of 0.29% are excluded in this study due
to low value of ρs which is outside the specified range of the model (between 1.09% and
6.82%).
Sudo et al. (1993) tested the cylindrical specimens that have dimensions of 150 mm ×
300 mm, f’c of 49 MPa and 66.7 MPa, ρs between 1.74% and 5.23%, and fy of 571 MPa.
Details of all these specimens are also shown in Table 5.1. Experimental data of 4
specimens that have f’c of 21 MPa reported by Campione and Mendola (2004) are
excluded in this study due to low value of f’c which is outside the specified range of the
model (between 38 MPa and 58 MPa).
Fig. 5.5 reveals the comparison between predicted and experimental stress-strain
characteristics. As shown in Figs. 5.5a and 5.5b, the model provides slightly stiffer pre-
peak portions. The reason may be that predicted characteristics for specimens I and II
represent the curves that have already been modified by correction factor of which role in
refining the raw experimental data is discussed in Section 3.2.4. On the contrary,
experimental characteristics of specimens I and II might not have been modified by the
correction factor, resulting in slightly stiffer predicted curves. Moreover, the model
somewhat overestimates the compressive strength of these two specimens, and post-peak
characteristic of specimen II. Nevertheless, the model is able to predict the overall trend
of the characteristics satisfactorily.
93
Regarding the specimens III through V, the model could predict the entire characteristics
well. The model also predicts well for specimens VI through VIII, although slightly
underestimating the characteristics. The reason for such underestimation is that the
specimens use high compressive strength of plain lightweight concrete of 66.7 MPa
which is outside the specified range of the model (between 38 MPa and 58 MPa).
In summary, the model is derived without using the experimental data considered in this
section. Nonetheless, the model could predict the overall trend of the curves
satisfactorily. Therefore, the model can be used for confined lightweight aggregate
concrete that is made with lightweight coarse aggregate having similar particle density
and strength investigated in this study, and with different yield strengths of lateral
reinforcement of 571 MPa and 1117 MPa.
5.8 Applicability of Recommended Model in Analysis
5.8.1 Analysis of Beam Section
Applicability of recommended stress-strain model in sectional analysis is examined by
comparing predicted and experimental moment-curvature relationships of flexural beam
sections at mid-span. Similar comparison is carried out in the work of Chin (1996) for
high-strength normal weight concrete. Experimental data of beam sections are obtained
from earlier literature (Lim 2007). Dimensions of the beams are 150 mm in width, 300
mm in depth, and 2800 mm in length between simply supported ends. Specifications of
the beams which are tested under two-point loading are given in Table 5.2.
Predicted data of beam sections are obtained from sectional analysis in which suggested
stress-strain model is used to generate the stress-strain characteristic of core concrete of
94
beams. For longitudinal reinforcement, simplified bi-linear stress-strain relationships are
used. For cover concrete, stress-strain relationship of plain lightweight concrete is used.
Sectional analysis is based on three principles of mechanics, namely equilibrium
condition, compatibility, and stress-strain relationship. The following assumptions are
used in the analysis: (1) plane section remains plane after bending, ensuring the linear
strain distribution of both concrete and reinforcement over the effective depth of member
section; (2) there is a perfect compatibility between concrete and reinforcement, resulting
in identical strain value for concrete and reinforcement located together at any level; (3)
stress-strain relationship of longitudinal reinforcement in compression is identical to that
in tension; (4) tensile strength of concrete, area displaced by longitudinal reinforcement
in compression zone of member, and tension stiffening effect are negligible; and (5) the
section is assumed as the uncracked section.
Behaviors of member section depend on geometry of section, configuration of
reinforcement, stress-strain characteristics of core concrete, of cover concrete, and of
longitudinal reinforcement. Thereby, these factors are accounted in the analysis. As
shown in Fig. 5.2, vertical distance between neutral axis and extreme compression fiber
of beam is defined as the depth of neutral axis (dNA). Distance along the depth of neutral
axis is shown by the symbol ‘u’. Over the member cross-section, horizontal distance
between outer layers of shear reinforcement is defined as the horizontal width of core
concrete (hcore). Likewise, vertical distance between outer layer of shear reinforcement in
compression zone and neutral axis is defined as the vertical region of core concrete
(Vcore).
The procedures used in sectional analysis are as follows:
95
Step 1: specify a small value of concrete strain at extreme compression fiber of beam (εce)
Step 2: assume a trial depth of neutral axis (dNA)
Step 3: compute the vertical region of core concrete (Vcore)
Vcore = dNA - thickness of cover concrete (5.7)
Step 4: find the relationship between strain and distance along depth of neutral axis (ε-u
relationship)
NA
ce
du
(5.8)
Step 5: determine the relationship between stress and distance along depth of neutral axis
(σ-u relationship) of core concrete by using σ-ε relationship of core concrete, and ε-u
relationship. This σ-u relationship of core concrete is the stress distribution diagram
drawn over the compression zone of member section.
Step 6: compute the area bounded by σ-u curve of core concrete, u-axis and the line u =
Vcore by using numerical integration method. This area represents the compressive
strength of core concrete per unit width of core concrete (Acore). This area can be
classified into two parts: area under pre-peak portion of σ-u curve, and area under post-
peak portion. Area under each portion is divided into 5 segments in integrating.
Step 7: calculate the compressive force of core concrete (Ccore)
Ccore = Acore × hcore (5.9)
Step 8: determine the location of compressive force of core concrete by using the moment
area method which takes the moment of the stress distribution diagram about the neutral
axis.
96
Step 9: similarly, determine the σ-u relationship of cover concrete by using σ-ε
relationship of cover concrete and ε-u relationship.
Step 10: since σ-ε relationship of cover concrete is linear, area under σ-u curve of cover
concrete can be easily calculated. The area represents the compressive strength of cover
concrete per unit width of cover concrete (Acover).
Step 11: calculate the compressive force of cover concrete (Ccover)
Ccover = Acover × width of cover concrete (5.10)
Step 12: compute the tensile steel strain (εts) by using εce, dNA, vertical distance between
neutral axis and center of longitudinal reinforcement in tension zone, and triangular rule.
Similarly, compute the compressive steel strain (εcs) by using εce, dNA, vertical distance
between neutral axis and center of longitudinal reinforcement in compression zone, and
triangular rule.
Step 13: determine the tensile steel stress (fts) by using σ-ε relationship of longitudinal
reinforcement in tension zone, and corresponding εts. Likewise, determine the
compression steel stress (fcs) by using σ-ε relationship of longitudinal reinforcement in
compression zone, and corresponding εcs.
Step 14: calculate the tensile force of reinforcement (Ts), and compressive force of
reinforcement (Cs)
Ts = fts × Ats (5.11)
Cs = fcs × Acs (5.12)
97
where Ats = total cross-sectional area of longitudinal reinforcement in tension zone
Acs = total cross-sectional area of longitudinal reinforcement in compression zone
Step 15: check the accuracy of the trial depth of neutral axis (dNA) that is assumed in step
2 by using the force equilibrium condition
Ccore + Ccover + Cs + Ts = 0 (5.13)
Step 16: if the force equilibrium condition is not satisfied, adjust the trial depth of neutral
axis as shown below, and repeat the iterative procedure from step 2 to step 15 until the
force equilibrium condition becomes satisfied.
Adjusting the depth of neutral axis,
if total
stotal
C
TC > 0.1, decrease the trial depth of neutral axis assumed in step (2)
if s
totals
T
CT < 0.1, increase the trial depth of neutral axis assumed in step 2
where totalC = Ccore + Ccover + Cs
0.1 = a constant used in adjusting the trial depth of neutral axis
Step 17: when the force equilibrium condition is satisfied, find the external moment
(Mext) by using the moment equilibrium condition
Mext = Mint (5.14)
98
Mext = (Ccore × armcore) + (Ccover × armcover) + (Cs × armcs) + (Ts × armts) (5.15)
where Mext = external applied moment
Mint = internal moment capacity
armcore = vertical distance between neutral axis and compressive strength of core
concrete
armcover = vertical distance between neutral axis and compressive strength of cover
concrete
armcs = vertical distance between neutral axis and compressive strength of
reinforcement
armts = vertical distance between neutral axis and tensile strength of
reinforcement
Step 18: compute the corresponding curvature at the mid-span of member
NA
ce
d
(5.16)
where = curvature of member section at mid-span
εce = latest specified concrete strain at extreme compression fiber of member
dNA = adjusted depth of neutral axis
In this way, moment-curvature data set of member section at specified stage of εce is
obtained.
Step 19: thereafter, define the next value of εce by adding an incremental strain value to
the previous value of εce
99
εce(next value) = εce(previous value) + ∆εce (5.17)
where εce(next value) = concrete strain at extreme compression fiber of beam for next
stage of loading
εce(previous value) = value previously specified in step 18
∆εce = incremental strain value
Step 20: repeat the procedure from step 2 to step 19 to find another set of moment-
curvature data until the latest specified value of εce reaches the desired value. In this way,
predicted moment-curvature data sets of member section at different stages of εce are
obtained.
Sectional analysis provides predicted moment-curvature relationships of beam sections.
Since the analysis involves iteration, repetition, and integration procedures, an in-house
computer program is developed and used to save computing time. Details of the computer
program are shown in Appendix C.
Fig. 5.6a displays the comparison between predicted and experimental moment-curvature
relationships. Furthermore, predicted moment capacities are also compared with
experimental ones in Table 5.3. Fig. 5.6 and Table 5.4 clearly show that prediction agrees
well with experimental results, confirming the applicability of suggested stress-strain
model in analysis.
5.8.2 Analysis of Column
100
Applicability of suggested stress-strain model in column analysis is also evaluated by
comparing predicted and experimental behaviours of reinforced lightweight concrete
columns subjected to uniform axial compression. Behaviours of axially loaded columns
are best described by load-strain relationship. Experimental behaviours are obtained from
earlier literature (Basset and Uzumeri 1986). Dimensions of the columns are 305 by 305
mm in cross section, and 1956 mm in height. Specifications of the columns are given in
Table 5.4.
Predicted behaviors are obtained from column analysis in which suggested stress-strain
model is used to generate the stress-strain characteristic of core concrete of the columns.
Diameter of core concrete is assumed as the outside-to-outside horizontal distance of
lateral reinforcement. Stress-strain relationship of longitudinal reinforcing bars in tension
and compression are assumed to be identical. Simplified bi-linear stress-strain diagrams
are assumed as the stress-strain relationships of the bars. At any stage of axial
deformation of column, axial strain developed in core concrete and longitudinal
reinforcement are assumed to be identical.
In predicting the overall load capacity of column, components of load carried by core
concrete and longitudinal reinforcement are separately calculated, and then added
together. It may be noticed that component of load carried by the cover concrete of
column (column shell) is ignored in the analysis. In this study, peak load capacity and
post-peak behavior of column are of main concern. Since load capacity of cover concrete
is negligible at and beyond the peak load level, component of load carried by cover
concrete is ignored in analysis. This assumption is consistent with experimental
observation reported by Nishiyama et al. (1993).
101
Fig. 5.7a displays the comparison between predicted behaviors and experimental
behaviors of the columns. The slight underestimation obtained in column 10 (Fig. 5.7a) is
due to low compressive strength of plain concrete (34 MPa) which is outside the
specified range of the model (38 MPa to 58 MPa). Overall, predicted behaviors of the
columns are in good agreement with corresponding experimental behaviors. This
agreement demonstrates the usefulness of suggested stress-strain model in column
analysis.
5.9 Summary
a. Stress-strain characteristic of confined lightweight aggregate concrete is not well
understood due to limited information available in current literature. Insufficient
understanding of the characteristic largely affects the analysis and design of
structural lightweight concrete members.
b. A number of stress-strain models are reported in earlier literature to estimate the
stress-strain characteristic of confined normal weight concrete. However, these
models are unsuitable for confined lightweight aggregate concrete. This study
thus suggests a stress-strain model to predict the stress-strain characteristic of
confined lightweight aggregate concrete.
c. The model is applicable for confined lightweight aggregate concrete with
compressive strength of plain lightweight concrete ranging between 38 MPa and
58 MPa, and with volumetric ratio of lateral reinforcement between 1.7% and
6.8%. The model is also applicable for the concrete made with lightweight coarse
aggregate having similar particle density and strength investigated in this study,
102
and with yield strength of lateral reinforcement ranging between 540 MPa and
1680 MPa.
103
Table 5.1: Confined lightweight concrete specimens reported by other researchers
researchers f'c
(MPa)
type of lightweight
coarse aggregate
spiral reinforcement
dimension of axially
loaded concrete cylinder
(mm)
specimen name
fy (MPa)
ρs (%)
Shah et al (1983)
37 expanded shale
1117 1.89 75 × 150
I 41 1117 1.89 II
Sudo et al (1993)
49 artificial
lightweight coarse
aggregate *
571 1.74
150 × 300
III 49 571 2.61 IV 49 571 5.23 V
66.7 571 1.74 VI 66.7 571 2.61 VII 66.7 571 5.23 VIII
Note: specific gravity of 1.63 and 1.68
Table 5.2: Reinforced lightweight concrete beams reported by Lim (2007)
beam name
f'c (MPa)
longitudinal reinforcement in tension zone of beam
longitudinal reinforcement in compression zone of beam
shear reinforcement
no. of bar
d (mm)
fy (MPa)
Es (GPa)
no. of bar
d (mm)
fy (MPa)
Es (GPa)
d
(mm) s
(mm) fy
(MPa) Es
(GPa)
8 44.42 2 16 512 183 2 10 540 185 10 130 540 185 11 44.42 2 20 532 178 2 10 540 185 10 130 540 185 12 38.35 4 13 542 187 2 10 540 185 10 130 540 185 13 38.35 4 16 512 183 2 10 540 185 10 130 540 185
Note: cs = center-line to center-line distance between adjacent shear reinforcement; '
cf = cylindrical
compressive strength of plain lightweight aggregate concrete; d = diameter of reinforcement; Es = modulus of elasticity of reinforcement;
yf = yield strength of reinforcement.
Table 5.3: Moment capacities of flexural beams
beam name
ultimate moment capacity (KN-m)
experimental value
(Lim 2007)
predicted value
ratio of experimental
value to predicted
value 8 53.6 49.93 1.07
11 85.1 77.43 1.1 12 66.8 62.46 1.07 13 88.8 79.78 1.11
104
Table 5.4: Reinforced lightweight concrete columns reported by Basset and Uzumeri (1986)
column name
f'c (MPa)
longitudinal reinforcement
lateral reinforcement
no. of bar
d (mm)
fy (MPa)
d (mm)
s (mm)
fy (MPa)
10 34 4 30 491 7.94 76.2 533 15 37.2 4 20 418 7.94 76.2 533
105
0
0
axial strain
axia
l str
ess
f' c
f co
lateral-reinforced concrete
plain concrete
ε o ε co
Figure 5.1: Schematic diagram showing the stress-strain characteristics of plain concrete
and of lateral-reinforced concrete C Vcore deff neutral axis T (a) (b) (c) (d)
Figure 5.2: (a) cross-sectional view of flexural member; (b) strain distribution diagram; (c) stress distribution diagram of core concrete; (d) stress distribution diagram of cover
concrete
dNA
εst
εc
εsc
ø
hcore
u u
106
(a) 100 by 200 mm specimen: 38-4-12
0
30
60
90
120
0 0.005 0.01 0.015 0.02 0.025 0.03
axial strain
axia
l str
ess
(Mpa
)
Mander et al. (predicted)Hsu & Hsu (predicted)Mansur et al. (predicted)experimental (present study)
(b) 100 by 200 mm specimen: 38-4-32
0
20
40
60
80
0 0.005 0.01 0.015 0.02 0.025 0.03
axial strain
axia
l str
ess
(MP
a)
Mander et al. (predicted)Hsu & Hsu (predicted)Mansur et al. (predicted)experimental (present study)
Figure 5.3: Comparison between experimental stress-strain characteristic of confined lightweight concrete and predicted characteristics obtained from existing models (Cont’d)
107
(c) 100 by 200 mm specimen: 57-6-18
0
50
100
150
200
0 0.005 0.01 0.015 0.02 0.025 0.03
axial strain
axia
l str
ess
(MP
a)
Mander et al. (predicted)Hsu & Hsu (predicted)Mansur et al. (predicted)experimental (present study)
(d) 150 by 300 mm specimen: L38-6-17
0
35
70
105
140
0 0.005 0.01 0.015 0.02 0.025
axial strain
axia
l str
ess
(MP
a)
Mander et al. (predicted)Hsu & Hsu (predicted)Mansur et al. (predicted)experimental (present study)
Figure 5.3: Comparison between experimental stress-strain characteristic of confined lightweight concrete and predicted characteristics obtained from existing models
108
(a) 38-4-12
0
35
70
0 0.01 0.02 0.03
(b) 38-4-14
0
35
70
0 0.01 0.02 0.03
(c) 38-4-32
0
15
30
45
0 0.01 0.02 0.03
(d) 38-4-50
0
20
40
0 0.01 0.02
(e) 38-5-14
0
40
80
0 0.01 0.02 0.03 0.04
(f) 49-4-14
0
40
80
0 0.01 0.02 0.03 0.04
(g) 49-5-14
0
45
90
0 0.01 0.02 0.03 0.04
(h) 57-4-12
0
45
90
0 0.01 0.02 0.03 axial strain
experimental predicted (suggested model)
Figure 5.4: Comparison between experimental stress-strain characteristics of confined lightweight concrete and predicted characteristics obtained from suggested model
(Cont’d)
axia
l str
ess
(MP
a)
109
(i) 57-4-14
0
40
80
0 0.01 0.02 0.03
(j) 57-4-40
0
35
70
0 0.005 0.01 0.015 0.02
(k) 57-5-14
0
45
90
0 0.01 0.02 0.03
(l) 57-6-18
0
50
100
0 0.01 0.02 0.03
(m) L38-6-17
0
40
80
0 0.01 0.02
(n) L57-6-17
0
50
100
0 0.01 0.02 axial strain
experimental predicted (suggested model)
Figure 5.4: Comparison between experimental stress-strain characteristics of confined lightweight concrete and predicted characteristics obtained from suggested model
axia
l str
ess
(MP
a)
110
(a) specimen I
0
15
30
45
0 0.005 0.01 0.015
(b) specimen II
0
15
30
45
0 0.005 0.01 0.015
(c) specimen III
0
20
40
60
0 0.005 0.01 0.015 0.02 0.025
(d) specimen IV
0
35
70
0 0.005 0.01 0.015 0.02
(e) specimen V
0
40
80
0 0.01 0.02 0.03
(f) specimen VI
0
40
80
0 0.005 0.01 0.015 0.02
(g) specimen VII
0
45
90
0 0.005 0.01 0.015 0.02
(h) specimen VIII
0
50
100
0 0.005 0.01 0.015 0.02 0.025 axial strain experimental (Shah et al.) experimental (Sudo et al.) predicted (suggested model)
Figure 5.5: Comparison between experimental stress-strain characteristics of confined lightweight concrete reported by other researchers and predicted characteristics obtained
from suggested model
axia
l str
ess
(MP
a)
111
Figure 5.6: Dimension of lightweight concrete beams reported by Lim (2007)
(a) beam 8
0
30
60
0 0.04 0.08
(b) beam 11
0
45
90
0 0.035 0.07
(c) beam 12
0
40
80
0 0.05 0.1
(d) beam 13
0
50
100
0 0.03 0.06 curvature (rad/m)
experimental (Lim 2007) _____ predicted (present study)
Figure 5.6a: Predicted behaviors of flexural beam sections are compared with corresponding experimental behaviors
mom
ent c
apac
ity
(KN
/m)
112
Figure 5.7: Dimension of lightweight concrete columns reported by Basset and Uzumeri (1986)
(a) column 10
0
1500
3000
4500
0 0.004 0.008 0.012
(b) column 15
0
1500
3000
4500
0 0.005 0.01
axial strain
experimental (Basset and Uzumeri 1986) _____ predicted (present study)
Figure 5.7a: Predicted behaviors of axially loaded columns are compared with corresponding experimental behaviors
load
cap
acit
y (K
N)
113
CHAPTER 6
Fiber-Reinforced Concrete
6.1 Introduction
Due to segregation, plastic shrinkage and drying shrinkage of concrete, concrete suffers
from cracking even before subjected to any external loading (Hsu et al. 1963; Samaha
and Hover 1992). External loading then accelerates the extension of existing cracking,
while inducing new cracking. In lightweight aggregate concrete, coarse aggregate is the
weakest component of heterogeneous concrete system, when compared to mortar matrix,
and interfacial transition zone (Gao et al. 1997; Faust 1997). With increasing external
loading, cracking inside lightweight coarse aggregates becomes unstable, extends to
mortar matrix, and then, connects with existing cracking in mortar matrix. As a result,
failure plane passes through the coarse aggregate, which provides smooth fracture surface
(Faust 1997). This smooth fracture surface leads to brittle failure (sudden failure) of
lightweight aggregate concrete (Zhang and Gjorv 1991; Faust 1997).
The brittleness obviously holds back the widespread usage of lightweight aggregate
concrete in structural applications, though this concrete possesses many distinct
properties such as high strength to weight ratio, buoyancy, internal curing, and fire
resistance. Favorably, the brittleness appears to be overcome by using short steel fibers as
a concrete constituent. Fibers are able to carry some of the tensile stresses developed in
mortar matrix, and to transfer these stresses from less stable part of mortar matrix to more
stable part (Yazici et al. 2007). In this way, fibers restrain the unstable propagation of
114
cracking in concrete, and control the brittle failure of concrete (Fanella and Naaman
1985).
Over the last four decades, researchers have paid attention to characteristics of fiber-
reinforced lightweight aggregate concrete (Ritchie and Al-Kayyali 1975). However, the
characteristics are not well understood due to wide variety in type of lightweight
aggregate, and of fiber (Gao et al. 1997; Kayali et al. 2003). Insufficient understanding of
the characteristics largely affects the analysis and design of structural members
constructed with fiber-reinforced lightweight aggregate concrete.
In this study, characteristics of fiber-reinforced lightweight aggregate concrete under
short-term loading are observed experimentally. An improved understanding of the
characteristics gained from the present study would be beneficial in structural analysis
and design.
In this study, compressive strength enhancement of concrete due to fiber addition (∆fc) is
determined by subtracting the compressive strength of plain lightweight concrete (f’c)
from that of fiber-reinforced lightweight concrete (ffiber) (Eq. 6.1). Similarly, tensile
strength enhancement of concrete due to fiber addition (∆fst) is determined by subtracting
the splitting tensile strength of plain lightweight concrete (fst(plain)) from that of fiber-
reinforced lightweight concrete (fst) (Eq. 6.2).
'cfiberc fff (MPa) (6.1)
)( plainststst fff (MPa) (6.2)
115
6.2 Scope and Significance
This chapter explores the characteristics of fiber-reinforced lightweight aggregate
concrete under short-term loading. An experimental program is designed and
implemented to better understand the characteristics, namely, deformation capacity,
compressive strength, strain at compressive strength, modulus of elasticity, splitting
tensile strength, and failure mode. These characteristics are explored by varying the
compressive strength of plain lightweight aggregate concrete, and fiber dosage.
Furthermore, this chapter formulates an analytical stress-strain model to predict the
reliable stress-strain characteristic of fiber-reinforced lightweight aggregate concrete.
Additionally, this chapter proposes another stress-strain model to generate the stress-
strain characteristic of lightweight aggregate concrete confined by a combination of
lateral reinforcement and short steel fibers. Overall, this chapter enriches the knowledge
on the characteristics of fiber-reinforced lightweight aggregate concrete. Profound
understanding of the characteristics would be beneficial in analysis and design of
structural members constructed with fiber-reinforced lightweight aggregate concrete.
6.3 Observations and Discussion
Each discussion is based on typical test result of three identical specimens belonging to
the same concrete batch, and going through the same preparation, and testing procedures.
6.3.1 Deformation Capacity
Post-peak behavior of concrete stress-strain curve indicates deformation capacity of
concrete. In observing the deformation capacity of concrete specimens with different
116
variables, it is advantageous to use normalized stress-strain curves, since these curves
offset the difference in compressive strength, and concrete strains at compressive strength
among specimens. Stress history and strain history of each specimen are normalized
against its compressive strength and concrete strain at compressive strength respectively
(Fig. 6.1).
Test results show that deformation capacity of fiber-reinforced lightweight aggregate
concrete depends on fiber dosage. In general, deformation capacity becomes increased
with an increase in fiber dosage. However, overusing the fibers would reduce back the
deformation capacity. Overusing the fibers leads to non-uniform fiber distribution in
concrete mixture, which results in inadequate concrete compaction. As a consequence,
undesirable air is entrapped under fibers, and concrete becomes less homogeneous. This
phenomenon considerably affects the deformation capacity of concrete. Therefore, care
should be taken not to exceed the optimum fiber dosage beyond which the deformation
capacity would be affected.
Fig. 6.1 clearly shows that optimum fiber dosage of group C specimens is 0.5%, while
that of group B specimens is 1%. Since the target compressive strength of plain concrete
for group C specimens is higher than that of group B specimens, water content used in
concrete mixtures producing group C specimens is lower. As a result, fluidity of the
concrete mixtures producing group C specimens is lower. Since it is more difficult to mix
the fibers well in concrete mixtures with low fluidity, optimum fiber dosage of group C
specimens is lower than that of group B specimens.
117
6.3.2 Compressive Strength
Compressive strength of fiber-reinforced lightweight aggregate concrete (ffiber) generally
increases with an increase in fiber content (Table 6.1). In group B specimens, strength
enhancement due to fiber addition (∆fc) varies from 13% to 32% for fiber dosage ranging
between 0.5% and 1.3%. For the same range of fiber dosage, group C specimens exhibit
the strength enhancement (∆fc) from 9% to 20%.
It is noted that strength enhancement in group C specimens is rather low. As mentioned
in Section 6.3.1, it is more difficult to mix the fibers well in concrete mixtures producing
group C specimens. Non-uniform distribution of fibers in concrete mixtures affects
concrete integrity of group C specimens. As a result, strength enhancement in group C
specimens is rather low.
The main purpose of using short steel fibers is to enhance the deformation capacity of
concrete. Compressive strength of concrete, on the other hand, can be effectively
increased by other means such as adjusting the water to cement ratio in concrete mix
design, and using efficient admixtures. However, strength enhancement due to fiber
addition should not be ignored. Acknowledging the strength enhancement would be
beneficial in optimizing the structural design.
Since the strength enhancement depends on f’c and RI, the ffibre can be expressed as a
function of these two variables. To estimate the ffibre in structural analysis and design, an
empirical equation (Eq. 6.3) is formulated based on experimental data recorded in this
study.
fcfibre Vff 120012.14.6 ' (MPa) (6.3)
118
f’c is the compressive strength of plain lightweight aggregate concrete (MPa). Vf is the
volume fraction of fiber. Eq. 6.3 can be used for fiber-reinforced lightweight aggregate
concrete with f’c between 40 MPa and 60 MPa, and with Vf ranging between 0.5% and
1.3%.
6.3.3 Concrete Strain at Compressive Strength Level (εfibre)
εfibre is an important parameter in estimating the ultimate behaviors of flexural members,
and of column under combined axial loading and bending moment. As shown in Table
6.1, experimental results of εfibre provide the mean value of 0.0029. This study thus
suggests the εfibre value to be 0.0029, regardless of concrete compressive strength, and of
fiber dosage added in the concrete.
6.3.4 Modulus of Elasticity
Modulus of elasticity of fiber-reinforced lightweight aggregate concrete (Efibre) represents
the elastic stress-strain response of the concrete to compression. Efibre is measured
according to ASTM C 469 – 02 (2006). Test results show that Efibre generally increases
with an increase in fiber dosage (Table 6.1).
6.3.5 Splitting Tensile Strength
Splitting tensile strength of fiber-reinforced lightweight aggregate concrete (fst) is used in
structural analysis and design to determine the shear strength of concrete, and
development length of reinforcement (ASTM C 496/C 496M; Hoff 1992 b). In this study,
fst is observed with regards to fiber dosage. It is found that fst increases with an increase in
119
fiber dosage. Strength enhancement due to fiber addition varies from 121 % to 233 % for
fiber dosage between 0.5 % and 1.3 % (Table 6.1). Hence, the benefit of adding the fiber
in lightweight aggregate concrete is more pronounced in fst compared to ffiber.
6.3.6 Oven-Dried Unit Weight
Unit weight of fiber-reinforced lightweight concrete naturally increases with an increase
in fiber dosage. However, only a slight increase is detected. The unit weight is found to
fall between 1769 kg/m3 and 1877 kg/m3 (Table 6.1). Therefore, high strength to weight
ratio, which is a desirable property of lightweight aggregate concrete, is still maintained
in fiber-reinforced lightweight aggregate concrete.
6.3.7 Failure in Uniaxial Compression
As failure mode of plain concrete specimen is concerned, visible cracking starts forming
with low cracking noise just before the peak strength level of concrete. During unloading,
concrete pieces heavily fall down from the specimen, which is later followed by sudden
failure. At the end of testing, hour-glass shaped specimen is usually obtained (Fig. 6.2).
On the contrary, fiber-reinforced concrete specimen usually fails in a gradual manner.
During unloading, fiber-reinforced specimen becomes bulging in lateral directions,
without showing excessive concrete spalling (Fig. 6.3).
6.3.8 Failure in Splitting Tension
120
At the end of testing, plain concrete specimen suddenly fails into two separate halves.
Unlike the plain concrete, fiber-reinforced concrete specimen merely shows some
cracking at failure (Fig. 6.4).
6.4 Literature Review: Existing Models for Fiber-Reinforced Concrete
Hsu and Hsu (1994) and Mansur et al. (1999) have reported the mathematical stress-
strain models for fiber-reinforced normal weight concrete. The models, which are two of
the most referenced models in literature, are convenient for routine use. However, it is
uncertain whether the models are also valid for fiber-reinforced lightweight aggregate
concrete.
In this section, the models are reviewed by comparing predicted stress-strain
characteristics obtained from the models with experimental characteristics of fiber-
reinforced lightweight concrete recorded in present study. As shown in Fig. 6.5, predicted
characteristics disagree with experimental ones. Hsu’s model underestimates the post-
peak behavior of the curves, while Mansur et al.’s model overestimates the post-peak
behavior. The models are originally derived for normal weight concrete such that they are
unsuitable for lightweight aggregate concrete. This finding serves as a motivation to
develop a new model that is capable of predicting the stress-strain characteristic for fiber-
reinforced lightweight aggregate concrete.
6.5 Suggested Model: Fiber-Reinforced Concrete
In this section, a stress-strain model is suggested to predict the stress-strain characteristic
of fiber-reinforced lightweight aggregate concrete. Experimental stress-strain curve of the
121
concrete can be classified into three different portions: pre-peak, post-peak, and tail
portions. The model is thus made up of three mathematical expressions; each expression
represents each portion of the curve accordingly. The model is, in fact, a refined version
of Hsu and Hsu’s model (1994) that clearly defines the tail portion of stress-strain curve.
Mathematical expression for pre-peak portion of stress-strain curve is shown in Eq. 6.4.
fibre
fibrefibre
fibrefibrefibref
f
2
12
2
for fibre 0 (6.4)
Mathematical expression for post-peak portion is shown in Eq. 6.5.
baf for tfibre (6.5)
Mathematical expression for tail portion is shown in Eq. 6.6.
7.0
exp6.0fibre
t
fibretfibre kff
for t (6.6)
where
cfibre
fibrefibre
E
f
1
1 (6.7)
122
fc
Vf
a 4005400
200' (6.8)
fc
Vf
b 31340010
32700'
6
(6.9)
fc
t Vfk
003.024.306.0
' (6.10)
3
2'500
2.5003.0 f
ct V
f (6.11)
f and ε are the concrete stress and corresponding concrete strain at any point on stress-
strain curve. ffibre and εfibre are the compressive strength of fiber-reinforced lightweight
aggregate concrete and corresponding concrete strain. βfibre is a parameter controlling the
shape of pre-peak portion. Ec is the modulus of elasticity of plain lightweight aggregate
concrete. a and b are the parameters controlling the shape of post-peak portion. εt is the
concrete strain corresponding to 60% of compressive strength on post-peak portion. kt is
a parameter controlling the shape of tail portion.
For simplification, Ec instead of Efibre is used in the model. Ec is estimated by using the
empirical equation suggested in Eq. 4.5. This study assumes that tail portion of the curve
starts from 60% of compressive strength on post-peak portion of the curve.
The model contains eight parameters, namely ffibre, εfibre, βfibre, a , b , kt, εt, and Ec. These
parameters are computed by using Eq. 6.3, Eqs. 6.7 through 6.11, and Eq. 4.5. These
equations are formulated by performing the regression analysis of experimental data
recorded in present study. The analysis is, in fact, performed by using statistical analysis
software. Adequacy of the fit equations to the experimental data is determined with
123
coefficient of determination, and residual analysis. More importantly, simple forms of
equations are intentionally chosen to ensure that the equations are user-friendly.
Variables considered in the equations are f’c and RI which are the known variables at
structural analysis and design stages. Using these variables in the model clearly reflects
their influences on stress-strain characteristic.
Continuous transition among the pre-peak, post-peak, and tail portions of the stress-strain
curve is the key in integrating the area under the curve correctly. This continuous
transition is ensured by using the equal signs in concrete strain ranges of the equations
(Eqs. 6.4 through 6.6) generating the pre-peak, post-peak, and tail portions.
This study also checks the validity of the model by comparing predicted stress-strain
characteristics obtained from the model with experimental characteristics. As shown in
Fig. 6.6, predicted characteristics agree quite well with experimental ones up to strain
value of 0.023. This agreement shows the validity of suggested model. The model is
effective, and straightforward. The model is applicable for fiber-reinforced lightweight
aggregate concrete with f’c ranging between 40 MPa and 60 MPa, and Vf between 0.5%
and 1.0%.
6.6 Confining System: Combination of Lateral Reinforcement and Fiber
In structural members, core concrete is always confined by lateral reinforcement. To
represent the characteristic of core concrete of fiber-reinforced lightweight concrete
members, stress-strain characteristic of lightweight concrete confined by a combination
of lateral reinforcement and fiber is required. However, the stress-strain characteristic is
not fully understood due to limited information available in current literature. Such
124
insufficient understanding directly affects the analysis and design of members
constructed with fiber-reinforced lightweight aggregate concrete. This study thus
proposes a stress-strain model to predict the stress-strain characteristic of lightweight
aggregate concrete confined by a combination of lateral reinforcement and short steel
fiber.
6.6.1 Literature Review: Existing Models
Hsu and Hsu (1994) and Mansur et al. (1997) have reported the mathematical stress-
strain models for normal weight concrete confined by conventional lateral reinforcement
in conjunction with short steel fiber. The models, which are two of the most referenced
models in literature, are convenient to use routinely. However, it is uncertain whether the
models are also valid for confined lightweight aggregate concrete.
In this section, performance of the models is reviewed by comparing predicted stress-
strain characteristics obtained from the models with experimental characteristics of core
concrete of reinforced lightweight concrete columns incorporating short steel fiber.
Details of the columns are described in Section 3.4 and Table 3.4. Experimental stress-
strain characteristics of core concrete of the columns are filtered from experimental load-
strain relationships of the columns. Details of the filtering procedures are discussed at
length in Section 6.6.3. Comparative study reveals that predicted characteristics disagree
with experimental ones (Fig. 6.7). This disagreement indicates that the models are unable
to generate the stress-strain characteristics for confined lightweight aggregate concrete.
Since the models are originally derived for confined normal weight concrete, they are
unsuitable for confined lightweight aggregate concrete.
125
As shown in Figure 6.7, Hsu and Hsu’s model overestimates the peak strength, strain at
peak strength, and post-peak behavior of the curves. On the other hand, Mansur et al.’s
model underestimates the peak strength, while it overestimates the strain at peak strength,
and post-peak behavior of the curves. Therefore, it is necessary to develop a new model
that is capable of generating the stress-strain characteristic for lightweight aggregate
concrete confined by a combination of lateral reinforcement and short steel fiber.
6.6.2 Proposed Model
To predict the stress-strain characteristic of lightweight aggregate concrete confined by a
combination of lateral reinforcement and short steel fiber, an analytical stress-strain
model is proposed herein. The model consists of two mathematical expressions: one is for
pre-peak portion of the stress-strain curve, while the other is for post-peak portion.
Mathematical expression for pre-peak portion, which is a refined version of Mander et
al.’s model (1988), is described in Eq. 6.12.
c
peakc
peakc
peakf
f
1
for peak 0 (6.12)
Mathematical expression to generate the post-peak portion is described in Eq. 6.13.
2 rqpf for peak (6.13)
126
f and ε are the concrete stress and corresponding strain at any point on stress-strain curve.
fpeak is the compressive strength of lightweight aggregate concrete confined by a
combination of lateral reinforcement and fiber. εpeak is the concrete strain at fpeak. βc, p, q,
and r are the parameters controlling the shape of stress-strain curve. These parameters are
estimated by using Eqs. 6.14 through 6.19.
s
dVff fcpeak 4311759 ' (MPa) (6.14)
)104(0018.0 5peakpeak f (6.15)
cpeak
peakc
E
f
1
1 (6.16)
3
2'200
)120012.14.6(
4040068
s
d
Vfp
fc
(6.17)
d
sVfq fc 50010700102300 ' (6.18)
2
3' 700)120012.14.6(07.020400
d
sVfr fc (6.19)
In the above equations, d stands for the diameter of spiral wire, s represents the pitch of
spiral reinforcement, and f’c refers to the cylindrical compressive strength of plain
127
lightweight aggregate concrete. Vf is the volume fraction of fiber. For simplification,
elastic modulus of plain lightweight aggregate concrete (Ec) is used in the model. To
estimate the value of Ec, the empirical equation suggested in Eq. 4.5 is used.
The proposed model is user-friendly. The model contains four variables, namely,
diameter, pitch, f’c and Vf . These variables are the known variables at analysis and design
stages. Using these variables in the model reflects their influences on the stress-strain
characteristic.
Continuity between pre-peak portion and post-peak portion of stress-strain curve is the
key in integrating the area under the curve properly. This continuity is ensured by using
the equal signs in concrete strain ranges of pre-peak equation (Eq. 6.12) and of post-peak
equation (Eq. 6.13).
In this study, core concrete and lateral reinforcement are assumed as a single composite
material. Hence, lateral expansion of core concrete, hoop tensile stress developed in
lateral reinforcement, and interaction between core concrete and lateral reinforcement are
disregarded in deriving the model. In fact, the model is formulated without using any
experimental data. Instead, the model is formulated by linking the models reported in
other parts of the research program (Sections 5.6 and 6.5). With Eqs. 6.12 through 6.19,
and Eq. 4.5, stress-strain characteristic of lightweight aggregate concrete confined by a
combination of lateral reinforcement and short steel fibres can be readily predicted.
6.6.3 Performance of Proposed Model
Assumptions
128
Nominal diameter of longitudinal reinforcement is used in column analysis. Stress-strain
properties of reinforcement in tension and compression are assumed to be identical. 0.2%
offset method is used in determining the yield strength of reinforcement. Simplified bi-
linear stress-strain diagram shown in Fig. 6.8 is assumed as the stress-strain property of
reinforcement. At any stage of axial deformation of a column, axial strains developed in
cover concrete, core concrete, and longitudinal reinforcement are assumed to be identical.
Cover concrete of column is assumed to carry the load until the column stress reaches a
stress level that is equivalent to compressive strength of fiber-reinforced lightweight
aggregate concrete (ffibre). As shown in Fig. 6.9, ffibre is lower than peak strength of
column. When the column stress reaches the stress level equivalent to ffibre, visible
cracking is formed on cover concrete. Since such cracking affects the integrity of cover
concrete, contribution of cover concrete is ignored after the formation of cracking. In
other words, component of load carried by cover concrete is accounted until the column
stress reaches the stress level equivalent to ffibre. This assumption agrees with
experimental observation recorded in present study.
For simplification, stress-strain relationship of cover concrete is assumed to be linear
(Fig. 6.10). The linear diagram is drawn by using the modulus of elasticity (Efiber) and
ffiber. Values of Efiber and ffiber are determined on 100 by 200 mm cylindrical fiber-
reinforced concrete specimen under uniaxial compression.
Presence of fibers in concrete delays the spalling of cover concrete (Hadi 2007). In
addition, this study observes that extent of cover concrete spalling is low in columns
incorporating fibers. Thus, diameter of core concrete of columns incorporating fibers is
129
assumed as the outer-line to outer-line horizontal distance of lateral reinforcement which
is 210 mm.
Characteristic of Core Concrete of Column
Experimental load-strain relationships of columns are obtained by testing three spiral-
reinforced lightweight concrete columns incorporating short steel fiber under uniaxial
compression. Dimensions of the columns are 250 mm in diameter and 900 mm in height.
Details of reinforcement used in the columns (columns VIII, IX, X) are summarized in
Table 3.4. Information regarding the specimen preparations, test set-up, and
instrumentations are provided in Sections 3.4.3 and 3.4.4. Elevation and cross-section
details of the columns are shown in Fig. 3.10.
Experimental stress-strain characteristic of core concrete of column can be filtered from
experimental load-strain relationship of column (Fafitis and Shah 1985; Nagashima et al.
1992; Nishiyama et al. 1993; Pessiki 2001). In filtering procedure, load-strain
relationship of column, stress-strain relationships of longitudinal reinforcement and of
cover concrete, and cross-sectional areas of longitudinal reinforcement and of cover
concrete are used. The detail procedure is discussed as below.
At a given axial deformation of column, stress developed in longitudinal reinforcement is
calculated by using experimentally measured strain, and stress-strain relationship of
reinforcement. With this stress value and total cross-sectional area of reinforcement,
component of load carried by longitudinal reinforcement (Ps) is computed. Likewise, at a
given axial deformation of column, stress developed in cover concrete (shell of column)
130
is calculated by using experimentally measured strain, and stress-strain relationship of
concrete containing fibers only. With the stress developed in and cross-sectional area of
cover concrete, component of load carried by cover concrete (Pcover) is computed.
Until the column stress reaches a stress level that is equivalent to ffiber, overall load
capacity of column (Ptotal) can be estimated by summing the components of load carried
by longitudinal reinforcement (Ps), cover concrete (Pcover) and core concrete (Pcore) (Eq.
6.20). In this context, Pcore is computed by subtracting both Ps and Pcover from Ptotal. After
the column stress passes the stress level of ffiber, Pcover is ignored (Eq. 6.21). In this
situation, Pcore is calculated by subtracting Ps alone from Ptotal. Thereafter, stress
developed in core concrete is computed by dividing Pcore with cross-sectional area of core
concrete. In this way, experimental stress-strain characteristic of core concrete of column
is readily obtained.
Until the column stress reaches the stress level equivalent to ffiber,
ercorestotal PPPP cov (6.20)
After the column stress passes the stress level equivalent to ffiber,
corestotal PPP (6.21)
In reality, the withdrawal of Pcover from Ptotal is expected to be in a less sudden fashion,
since the spalling of fiber-reinforced cover concrete is less sudden than that of plain cover
131
concrete. However, for simplicity, the contribution of Pcover is withdrawn suddenly in
analysis, once the column stress reaches ffiber. Sudden withdrawal of Pcover causes sudden
increase in Pcore. It is reflected in the upper part of the pre-peak portion of experimental
curves displayed in Fig. 6.7 and Fig. 6.11.
Performance
Performance of proposed stress-strain model (Section 6.6.2) is examined by comparing
predicted stress-strain characteristics obtained from the model with experimental stress-
strain characteristics of core concrete of columns tested in present study. It is interesting
to note that predicted characteristics represent the stress-strain characteristics of lateral-
reinforced cylindrical specimens that are 100 mm in diameter and 200 mm in height. On
the contrary, experimental characteristics stand for the stress-strain characteristics of
columns’ core concrete of which dimensions are 210 mm in diameter and 900 mm in
height. Although the scales of these two specimens are different, scale-effect is ignored in
comparative study. In addition, the model is derived without using any experimental data
of the columns. Nevertheless, favorable agreement between predicted characteristics and
experimental ones is observed as shown in Fig. 6.11. This agreement shows the validity
of proposed stress-strain model (Section 6.6.2).
In conclusion, the proposed model (Section 6.6.2) is capable of defining the stress-strain
characteristic for the concrete that has compressive strength of plain lightweight
aggregate concrete (f’c) of 45.8 MPa, volumetric ratio of lateral reinforcement (ρs) of
2.09%, and fiber dosage (Vf) between 0.5% and 1.0%. To bring out the valid ranges of
these variables for the model, further experimental investigation is required.
132
6.7 Summary
a. Deformation capacity of fiber-reinforced lightweight concrete increases with an
increase in fiber dosage. However, fiber dosage should not exceed an optimum
amount. Beyond the optimum amount, further increase in fiber dosage would
reduce back the deformation capacity.
b. Though both the compressive strength and splitting tensile strength of fiber-
reinforced lightweight concrete increase with an increase in fiber dosage, benefit
of using fiber is more pronounced in splitting tensile strength.
c. Unit weight of the concrete naturally increases with increasing fiber dosage.
Nonetheless, only a slight increase is detected. Therefore, high strength to weight
ratio, which is a desirable property of lightweight aggregate concrete, is still
maintained in fiber-reinforced lightweight aggregate concrete.
d. Presence of short steel fibers in lightweight concrete has an influential effect on
modulus of elasticity of concrete, and concrete strain at compressive strength.
Modulus of elasticity generally increases with increasing fiber dosage. Similarly,
a slight increase in the concrete strain is detected with increasing fiber dosage.
e. This chapter suggests a stress-strain model to predict the stress-strain
characteristic of fiber-reinforced lightweight aggregate concrete. The model is
applicable for the concrete with f’c ranging between 40 MPa and 60 MPa, and
with Vf between 0.5% and 1.0%.
f. Moreover, this chapter proposes another stress-strain model to predict the stress-
strain characteristic of lightweight aggregate concrete confined by a combination
of lateral reinforcement and short steel fiber. The model is capable of predicting
133
the stress-strain characteristic for the concrete that has f’c of 45.8 MPa, ρs of
2.09%, and Vf between 0.5% and 1.0%. The valid ranges of these variables, which
can be used in the model, are still open to investigate.
g. On the whole, this chapter enriches the knowledge on the characteristics of fiber-
reinforced lightweight aggregate concrete under short-term loading. An improved
understanding of the characteristics gained from the present study would be
beneficial in analysis and design of structural members constructed with fiber-
reinforced lightweight aggregate concrete.
134
Table 6.1: Fiber-reinforced lightweight aggregate concrete
specimen name
f'c (MPa)
Vf (%)
ffiber (MPa)
Δfc (%)
ω (kg/m3)
εfiber Efiber
(103xMPa) fst(plain) (MPa)
fst (MPa)
Δfst (%)
A-1 41 1 51.92 26 1769 0.0029 23.2 3.17 7.57 138
B-0.5 55 0.5 62.15 13 1778 0.0030 23.0 3.70 8.21 121
B-0.75 55 0.75 62.57 13 1778 0.0032 27.3 3.70 9.85 166
B-1 55 1 67.77 23 1830 0.0027 27.8 3.70 9.12 146
B-1.3 55 1.3 72.97 32 1836 0.0032 28.3 3.70 11.18 202
C-0.5 59 0.5 68.41 15 1834 0.0028 29.1 3.47 8.44 143
C-0.75 59 0.75 64.73 9 1850 0.0026 27.2 3.47 9.06 161
C-1 59 1 71.13 20 1841 0.0029 28.3 3.47 8.89 156
C-1.3 59 1.3 70.91 20 1877 0.0028 29.9 3.47 11.58 233
Note: f’c = cylindrical compressive strength of plain lightweight concrete, Vf= volume fraction of fiber,
ffiber = compressive strength of fiber-reinforced concrete, Δfc = compressive strength enhancement due to
fiber addition = 100'
'
c
cfiber
f
ff; ω = oven-dried unit weight of fiber-reinforced concrete, εfibre =
concrete strain at ffiber, Efiber = elastic modulus of fiber-reinforced concrete, fst(plain) = splitting tensile strength
of plain concrete, fst = splitting tensile strength of fiber-reinforced concrete, Δfst = splitting tensile strength
enhancement due to fiber addition = 100)(
)(
plainst
plainstst
f
ff .
135
(a)
0
0.5
1
0 2 4 6 8
normalized strain
norm
aliz
ed s
tres
s
B-0.5
B-0.75
B-1
B-1.3
(b)
0
0.5
1
0 2 4 6 8
normalized strain
norm
aliz
ed s
tres
s C-0.5
C-1
C-1.3
Figure 6.1: Deformation capacity of fiber-reinforced lightweight aggregate concrete with regards to fiber dosage: (a) group B specimens; (b) group C specimens
136
Figure 6.2: Appearance of plain lightweight concrete specimens after testing in uniaxial compression
(a)
Figure 6.3: Appearance of fiber-reinforced lightweight concrete specimens after testing in uniaxial compression (Cont’d)
B
137
(b)
Figure 6.3: Appearance of fiber-reinforced lightweight concrete specimens after testing in uniaxial compression: (a) group B specimens with different fiber dosage; (b) group C
specimens with different fiber dosage.
Figure 6.4: Appearance of fiber-reinforced lightweight concrete specimens after testing in splitting tension
C
138
(a) B-0.5
0
35
70
0 0.01 0.02 0.03
strain
stre
ss (
MP
a)
experimental
predicted (Hsu's model)
predicted (Mansur's model)
(b) C-0.5
0
40
80
0 0.01 0.02 0.03
strain
stre
ss (
MP
a)
experimental
predicted (Hsu's model)
predicted (Mansur's model)
Figure 6.5: Comparison between predicted stress-strain characteristics obtained from reported models in literature and experimental characteristics of fiber-reinforced
lightweight concrete
139
(a) B-0.5
0
35
70
0 0.01 0.02
experimentalpredicted
(b) B-0.75
0
35
70
0 0.01 0.02
experimentalpredicted
(c) B-1
0
35
70
0 0.01 0.02
experimentalpredicted
(d) B-1.3
0
40
80
0 0.01 0.02
experimentalpredicted
(e) C-0.5
0
35
70
0 0.01 0.02
experimentalpredicted
(f) C-1
0
40
80
0 0.01 0.02
experimental
predicted
axial strain
Figure 6.6: Comparison between predicted stress-strain characteristics obtained from suggested model and experimental characteristics of fiber-reinforced lightweight concrete
axia
l str
ess
(MP
a)
140
(a) column VIII: 0.5% fiber dosage
0
20
40
60
80
0 0.01 0.02 0.03strain
stre
ss (
MP
a)experimentalpredicted (Hsu & Hsu's model)predicted (Mansur et al.'s model)
(b) column IX: 0.75% fiber dosage
0
20
40
60
80
0 0.01 0.02 0.03strain
stre
ss (
MP
a)
experimentalpredicted (Hsu & Hsu's model)predicted (Mansur et al.'s model)
(c) column X: 1% fiber dosage
0
20
40
60
80
0 0.01 0.02 0.03strain
stre
ss (
MP
a)
experimentalpredicted (Hsu & Hsu's model)predicted (Mansur et al.'s model)
Figure 6.7: Comparison between predicted stress-strain characteristics obtained from reported models in literature and experimental characteristics of core concrete of
reinforced lightweight concrete columns incorporating short steel fiber
141
0
200
400
600
0 0.01 0.02 0.03strain
stre
ss (
MP
a)
Figure 6.8: Simplified bi-linear stress-strain relationship of longitudinal reinforcement (using Es and fy provided in the mill certificate)
0
25
50
75
0 0.01 0.02 0.03strain
stre
ss (
MP
a)
stress-strain behavior ofa column under uniformaxial compression
a strength level equivalentto compressive strengthof fiber-reinforcedconcrete
peak strength of column
Figure 6.9: Load capacity of cover concrete of column is accounted until the column stress reaches a strength level of compressive strength of fiber-reinforced concrete.
0
20
40
60
0 0.001 0.002 0.003 strain
stre
ss (
MP
a)
0.5% fiber dosage
0.75% fiber dosage
1% fiber dosage
Figure 6.10: Simplified stress-strain relationships of cover concrete of columns with different dosages of fiber
Φ 16 mm
142
(a) column VIII: 0.5% fiber dosage
0
20
40
60
0 0.01 0.02 0.03
strain
stre
ss (
MP
a)experimental
predicted (suggested model)
(b) column IX: 0.75% fiber dosage
0
25
50
75
0 0.01 0.02 0.03strain
stre
ss (
MP
a)
experimental
predicted (suggested model)
(c) column X: 1% fiber dosage
0
25
50
75
0 0.01 0.02 0.03
strain
stre
ss (
MP
a)
experimental
predicted (suggested model)
Figure 6.11: Comparison between predicted stress-strain characteristics obtained from proposed model and experimental characteristics of core concrete of reinforced
lightweight concrete columns incorporating short steel fiber.
143
CHAPTER 7
Spiral-Reinforced Column
7.1 Introduction
In current literature, a limited number of studies have been reported on ultimate behaviors
of reinforced lightweight concrete column under compression. As a result, it is difficult to
estimate the behaviors with reliable accuracy. On the other hand, design procedures and
code specifications for the column have not been well established yet (Hoff 1990).
Since column supports many parts of the whole structure, column failure leads to
progressive failure of the structure (McCormac 1998). For this reason, deformation
capacity of column is much more concerned compared to that of other types of structural
member. This study thus explores the deformation capacity of the column by conducting
the experimental program.
Though the main objective of using the lateral reinforcing system in column is to enhance
the deformation capacity, load capacity enhancement shall not be ignored.
Acknowledging the load capacity enhancement in analysis and design would noticeably
reduce the size of column. Reducing the size would bring the benefits such as (1)
reducing the cost of materials used in column, (2) increasing the rentable floor area, and
(3) reducing the dead load of column. Hence, this study also investigates the load
capacity enhancement in spiral-reinforced lightweight concrete column.
Previous research on reinforced normal weight concrete column shows that using the
short steel fibers in column could improve the deformation capacity as well as ultimate
144
load capacity of column (Foster and Attard 1999). To ensure whether the similar trend
also exists in reinforced lightweight concrete column, this study investigates the influence
of short steel fiber on ultimate behaviors of spiral-reinforced lightweight concrete
column.
7.2 Scope and Objective
This chapter explores the experimental behaviors of spiral-reinforced lightweight
concrete column under uniform axial compression. The behaviors include deformation
capacity, unloading manner, ultimate load capacity, compressive strength of concrete in
column, and failure manner. These behaviors are explored by varying the variables,
namely diameter of spiral wire, pitch of spiral reinforcement, amount of longitudinal
reinforcement, and amount of short steel fiber. Primary purpose of this chapter is to
explore the experimental behaviors of the column under compression. And, secondary
objective is to demonstrate the prediction for ultimate behaviors of the column.
7.3 Observations and Discussion
7.3.1 Deformation Capacity
In observing the deformation capacity of columns, normalized load-strain curves are used
to balance out the difference in ultimate load and concrete strain at ultimate load level
among columns with different variables. Load history and strain history of each column
are normalized against its ultimate load capacity and concrete strain at ultimate load level
respectively.
145
For a given diameter of spiral wire, reducing the pitch of spiral reinforcement increases
the deformation capacity of the column (Fig. 7.1a). Reducing the pitch would increase the
effectively confined area of core concrete along the longitudinal axis (Fig. 4.4) (Park and
Paulay 1975), which in turn increases the deformation capacity.
For a given pitch of spiral reinforcement, increasing the diameter of spiral wire improves
the deformation capacity of the column (Fig. 7.1b). Increasing the diameter of spiral wire
would enhance the stiffness of spiral reinforcement, which in turn reduces the radial
expansion of spiral reinforcement. Reducing the radial expansion of spiral reinforcement
provides better confining effect to core concrete of column. Thereby, increasing the
diameter of spiral wire improves the deformation capacity.
Test results show that varying the number of longitudinal reinforcement from 4 to 8 does
not affect the deformation capacity of the column (Fig. 7.1c). Similarly, short steel fiber
has little influence on the deformation capacity (Fig. 7.1d). The optimum fiber dosage
that can improve the deformation capacity is only 0.75%. Beyond the optimum dosage,
deformation capacity would be reduced back with increasing fiber dosage.
7.3.2 Unloading Manner
To obtain the desired deformation capacity in column, spiral reinforcement is supposed to
yield around the ultimate load level of column. Test results show that spiral
reinforcement yields only in post-peak unloading region which is far below the ultimate
load level (Fig. 7.2). It shows that lateral reinforcement is not fully active in immediate
post-peak region.
146
Neville (1997) reported that interfacial transition zone is improved in lightweight
aggregate concrete. Moreover, restraining effect of lightweight coarse aggregate on
surrounding mortar matrix is low due to small difference between elastic stiffness of
lightweight coarse aggregate and that of mortar matrix (Bremner and Holm 1986; Neville
1997). As a result, amount of microcracking is low in lightweight aggregate concrete,
causing low degree of lateral expansion of the concrete (Foster et al. 1998).
Lateral reinforcement is, in fact, the passive confinement which requires activation in
order to confine the core concrete effectively. Since this activation is obtained from
lateral expansion of core concrete, low lateral expansion of lightweight aggregate
concrete delays the activation process. As a result, lateral reinforcement is not fully active
and is unable to confine the core concrete effectively in immediate post-peak region. It
appears to be the reason why load carrying capacity of reinforced lightweight concrete
column drop suddenly in immediate post-peak region.
7.3.3 Ultimate Load Capacity
Increasing the fiber dosage enhances the ultimate load capacity of the column (Table
7.1). The ultimate load capacity is also enhanced by increasing the diameter of spiral wire
(Table 7.2), and pitch of spiral reinforcement (Table 7.3).
As mentioned before, increasing the diameter of spiral wire would increase the stiffness
of spiral reinforcement, which in turn reduces the radial expansion of spiral
reinforcement. Reducing the radial expansion of spiral reinforcement provides better
confining effect to core concrete of column. Thus, increasing the diameter of spiral wire
enhances the ultimate load capacity of the column.
147
Increasing the pitch reduces the congestion of reinforcement, which reduces the
formation of plane of separation between cover concrete and core concrete of column.
Since cover concrete is less interrupted by the plane, it seems to contribute more load
capacity to the column. As a result, ultimate load capacity of the column increases with
an increase in pitch of spiral reinforcement.
7.3.4 Compressive Strength of Concrete in Column
The ratio between compressive strength of concrete in column and that in cylinder is used
in order to determine the strength reduction factor in column due to size effect, concrete
segregation, and to determine the strength gain in column due to lateral reinforcement
(Basset and Uzumeri 1986). In this study, compressive strength of concrete in axially
loaded column is obtained by testing the plain (unreinforced) column under uniform axial
compression, while that of 100 by 200 mm plain concrete cylinder is obtained by testing
the cylinder under uniaxial compression. The ratio between the above two strength is
found to be 0.96. This finding is in line with the observation reported by Basset and
Uzumeri (1986), though only one plain column is tested in the study.
7.3.5 Failure Mode
As shown in Fig. 7.3a, plain column (column XI) fails suddenly into two separate pieces,
when the load-strain curve reaches the immediate post-peak unloading region. Since
lightweight coarse aggregate is the weakest component of heterogeneous concrete
system, failure plane passes through the coarse aggregate (Section 2.2.1). This
148
phenomenon provides smooth fracture surface, resulting in sudden failure of plain
column.
In this study, minimum required amount of spiral reinforcement suggested in ACI 318
(2005) Clause 10.9.3 is used as a guideline, though the amount is meant for reinforced
normal weight concrete column. In reinforced column with low amount of lateral
reinforcement (column I), shear failure is observed (Fig. 7.3b). Amount of lateral
reinforcement provided in the column is much lower than minimum required amount
suggested by ACI-318 (2005) (Table 7.4). Moreover, small diameter of spiral wire is
used in the column, which offers low stiffness in spiral reinforcement. Due to inadequate
amount of and low stiffness of lateral reinforcement, lateral reinforcement is unable to
confine the core concrete of the column effectively, resulting in shear failure.
To avoid this undesirable shear failure, it is required to use the adequate amount of and
adequate stiffness of lateral reinforcement. This would ensure that available deformation
capacity in column exceeds demanding capacity during overloading.
In reinforced columns with adequate amount of lateral reinforcement (column II to
column VI), spalling failure is observed (Fig. 7.3c). At and beyond the ultimate load level
of each column, concrete pieces fall down from cover concrete of column, and then
continue to fall down from outer layer of core concrete of column. Due to differential
shrinkage between cover concrete and core concrete, as well as outward buckling of
longitudinal reinforcement, high tensile stress is developed at cover-core interface (Foster
et al. 1998; Foster 2001). On the other hand, bond strength between cover concrete and
core concrete is low due to the plane of separation which is formed by closely spaced
149
reinforcement (ACI 441R-96). It seems that these high tensile stress and low bond
strength are responsible for the spalling of concrete pieces.
When short steel fiber are added in spiral-reinforced columns (column VIII to column X),
heavy spalling of concrete pieces does not occur in the columns. Instead, the columns
become bulging in outward radial direction (Fig. 7.3d).
All the columns are tested until the rupture of spiral reinforcement. As shown in Fig. 7.4,
the rupture always occurs over longitudinal reinforcement. Similar observation is
reported by Hadi (2005). Location of the rupture emphasizes the fact that spiral
reinforcement is subjected to tensile stress not only from lateral expansion of core
concrete but also from outward buckling of longitudinal reinforcement.
7.4 Predicted Behaviors of Column
This section demonstrates the prediction for ultimate behaviors of spiral-reinforced
lightweight concrete column under uniform axial compression. Predicted behaviors are
obtained from column analysis in which recommended stress-strain model (Section 5.6)
is used to generate the stress-strain characteristic of core concrete of the column.
Diameter of core concrete is assumed as the center-line to center-line horizontal distance
of lateral reinforcement, since concrete pieces fall down from cover concrete (shell of
column), and then from outer layer of core concrete during testing.
In the analysis, nominal diameter of longitudinal reinforcement is used. Stress-strain
properties of the reinforcement in tension and compression are assumed to be identical.
0.2% offset method is used in determining the yield strength of the reinforcement.
Simplified bi-linear stress-strain diagram shown in Fig. 6.8 is assumed as the stress-strain
150
property of the reinforcement. At any stage of axial deformation of a column, axial
strains developed in cover concrete, core concrete and longitudinal reinforcement are
assumed to be identical.
In predicting the overall load capacity of column, components of load carried by core
concrete and longitudinal reinforcement are separately calculated, and then added
together. It may be noticed that component of load carried by cover concrete is ignored in
the analysis. Experimental observation reports that cracking forms on cover concrete
before load carrying capacity of column reaches the ultimate load level. As the cracking
affects the integrity of cover concrete, load capacity of cover concrete is negligible at and
beyond the ultimate load level. Since the ultimate load level and post-peak behavior are
of main concern in this study, component of load carried by cover concrete is ignored in
the analysis. The assumption is also consistent with experimental observation reported by
Nishiyama et al. (1993).
This section also examines the accuracy of the prediction by comparing predicted
behaviors with corresponding experimental behaviors recorded in this study. As
displayed in Fig. 7.5, predicted behaviors are in close agreement with experimental
behaviors, though slightly overestimating the post-peak behavior. This slight
overestimation may arise from buckling of longitudinal reinforcement which reduces the
load capacity of longitudinal reinforcement. For simplification, such reduction in load
capacity is not taken into account in the analysis. In addition, buckling of longitudinal
reinforcement tends to interrupt the confining mechanism of lateral reinforcement by
pushing away the lateral reinforcement in outward radial direction. In this situation,
lateral reinforcement could not confine the core concrete effectively, which reduces the
151
load capacity of core concrete. To simplify the analysis procedures, such reduction in
load capacity is also not accounted in the analysis. Ignoring the reduced load capacity of
longitudinal reinforcement and of core concrete seems to be the reason for slightly
overestimating the post-peak behavior.
Table 7.5 also compares the predicted and experimental ultimate load capacities of the
columns. As shown in Table 7.5, the predicted values agree with experimental values.
Overall, this section contributes towards the prediction of the ultimate behaviors of
spiral-reinforced lightweight concrete column under compression.
7.5 Summary
a. Reducing the pitch of spiral reinforcement increases the deformation capacity of
spiral-reinforced lightweight concrete column, but decreases the ultimate load
capacity of the column.
b. Increasing the diameter of spiral wire enhances both the deformation capacity and
ultimate load capacity of the column.
c. Varying the number of longitudinal bars from 4 to 8 does not affect the
deformation capacity.
d. Increasing the dosage of short steel fiber increases the ultimate load capacity of
the column, though it has little influence on the deformation capacity.
e. Test results show that lateral reinforcement yields only in the post-peak region
which is far below the ultimate load level of the column. As a result, lateral
reinforcement is unable to confine the core concrete of the column effectively in
152
immediate post-peak region. As a result, load carrying capacity of the column
drops suddenly after the peak load level.
f. For the concrete used in this study, the ratio between compressive strength of
concrete used in plain column and that used in 100 by 200 mm concrete cylinder
is found to be 0.96.
g. Regarding the failure mode of the column, a total of four different failure modes
are detected: sudden failure, shear failure, spalling failure, and bulging failure for
plain (unreinforced) column, reinforced column with low amount of lateral
reinforcement, reinforced column with adequate amount of lateral reinforcement,
and reinforced column incorporating short steel fiber respectively.
In summary, this chapter, which presents the experimental behaviors of lightweight
concrete column, contributes towards the prediction of the ultimate behavior of spiral-
reinforced lightweight concrete column under compression.
153
Table 7.1: Varying the fiber dosage
column number
column name
fiber dosage
(%)
ultimate load capacity of
column (KN)
V 6-10-75 0 2406 VIII 6-10-75 0.5 2572 IX 6-10-75 0.75 2846 X 6-10-75 1 2884
Table 7.2: Varying the diameter of spiral wire
column number
column name
diameter of spiral
wire (mm)
ultimate load capacity of
column (KN)
I 6-6-50 6 1872 II 6-8-50 8 2146 III 6-10-50 10 2309
Table 7.3: Varying the pitch of spiral reinforcement
column number
column name
pitch of spiral reinforcement
(mm)
ultimate load
capacity of column
(KN) III 6-10-50 50 2309 V 6-10-75 75 2406
VII 6-10-100 100 2422
Table 7.4: Details of reinforcement used in columns
column no.
column name
spiral reinforcement
longitudinal reinforcement
fiber dosage
(%)
ρs(req) (%)
ρs (%)
ρs (%)
I 6-6-50 1.39 1.10 2.45 0 II 6-8-50 1.48 1.99 2.45 0 III 6-10-50 1.62 3.14 2.45 0 IV 4-10-75 1.62 2.09 1.63 0
V 6-10-75 1.62 2.09 2.45 0
VI 8-10-75 1.62 2.09 3.27 0 VII 6-10-100 1.62 1.57 2.45 0 VIII 6-10-75 1.62 2.09 2.45 0.5 IX 6-10-75 1.62 2.09 2.45 0.75 X 6-10-75 1.62 2.09 2.45 1 XI plain 0 0 0 0
154
Note: ρs(req), ρs, and ρ stated in Table 7.4 are determined by using Eqs. 7.3, 4.3, and 7.4 respectively.
y
c
c
greqs f
f
A
A '
)( 145.0
(7.3)
g
s
A
A (7.4)
where Ac = cross-sectional area of core concrete of column; f’c = cylindrical compressive
strength of plain concrete used in column; Ag = gross cross-sectional area of column;
ρs(req) = minimum required amount of spiral reinforcement (ACI 318- 05); As = total
cross-sectional area of longitudinal reinforcement; ρs = amount of spiral reinforcement
provided in column; fy = yield strength of lateral reinforcement ≤ 689 M Pa.
Table 7.5: Ultimate load capacity of columns
column no.
column name
ultimate load capacity (KN)
experimental value
predicted value
ratio of experimental
value to predicted
value I 6-6-50 1872 2206 0.85 II 6-8-50 2146 2264 0.95 III 6-10-50 2309 2321 0.99 IV 4-10-75 2254 2034 1.11 V 6-10-75 2406 2224 1.08 VI 8-10-75 2616 2415 1.08 VII 6-10-100 2422 2178 1.11
155
(a)
0
0.25
0.5
0.75
1
0 2 4 6 8 10 12 14
normalized strain
norm
aliz
ed lo
ad
pitch 50 mm
pitch 75 mm
pitch 100 mm
(b)
0
0.25
0.5
0.75
1
0 2 4 6 8 10 12
normalized strain
norm
aliz
ed lo
ad
dia: of spiral 6 mm
dia: of spiral 8 mm
dia: of spiral 10 mm
Figure 7.1: Deformation capacity of spiral-reinforced lightweight concrete columns
(Continued)
156
(c)
0
0.25
0.5
0.75
1
0 1 2 3 4 5 6 7 8
normalized strain
norm
aliz
ed lo
ad
4 longitudinal bars
8 longitudinal bars
(d)
0
0.25
0.5
0.75
1
0 1 2 3 4 5 6
normalized strain
norm
aliz
ed lo
ad
0.5% fiber dosage0.75% fiber dosage1% fiber dosage
Figure 7.1: Deformation capacity of spiral-reinforced lightweight concrete columns with regards to (a) diameter of spiral wire, (b) pitch of spiral reinforcement, (c) number of
longitudinal reinforcement, and (d) fiber dosage
157
0
500
1000
1500
2000
2500
3000
0 0.005 0.01 0.015 0.02 0.025 0.03
axial strain
load
car
ryin
g ca
paci
ty (
KN
) ultimate load level
point of spiral yielding
point of spiral rupture
Figure 7.2: Load vs. strain relationship of reinforced column under uniform axial
compression
158
(a) (b)
(c) (d)
Figure 7.3: Failure mode of columns: (a) sudden failure of plain unreinforced column; (b) shear failure of spiral-reinforced column with low amount of lateral reinforcement; (c)
spalling failure of spiral-reinforced column with adequate amount of lateral reinforcement; (d) bulging failure of spiral-reinforced column incorporating short steel
fibers
159
Figure 7.4: Rupture of lateral reinforcement over longitudinal reinforcement
(a) column I
0
1200
2400
0 0.01 0.02
(b) column II
0
1200
2400
0 0.012 0.024
(c) column III
0
1200
2400
0 0.012 0.024
(d) column IV
0
1200
2400
0 0.017 0.034
strain experimental _____ predicted
Figure 7.5: Predicted behaviors of reinforced lightweight concrete columns are compared
with corresponding experimental behaviors. (Continued)
load
(K
N)
160
(e) column V
0
1300
2600
0 0.015 0.03
(f) column VI
0
1400
2800
0 0.012 0.024
(g) column VII
0
1200
2400
0 0.012 0.024
strain experimental _____ predicted
Figure 7.5: Predicted behaviors of reinforced lightweight concrete columns are compared with corresponding experimental behaviors.
load
(K
N)
161
CHAPTER 8
Conclusions
8.1 Overview
This study explores the characteristics of structural lightweight aggregate concrete
subjected to short-term compression. The concrete is produced by using expanded clay
lightweight coarse aggregate. Types of the concrete considered are plain (unreinforced)
concrete, confined concrete, and fiber-reinforced concrete. In confined concrete, spiral
reinforcement is used to confine the core concrete effectively. Discrete short steel fibers
are used for the same purpose in fiber-reinforced concrete.
The concrete characteristics include deformation capacity, compressive strength, strain at
compressive strength level, modulus of elasticity, splitting tensile strength, oven-dry unit
weight, and failure manner. These characteristics are explored experimentally by varying
the compressive strength of plain concrete, diameter of spiral wire, pitch of spiral
reinforcement, size of specimen, and fiber dosage.
This study also suggests analytical stress-strain models for confined concrete, fiber-
reinforced concrete, and the concrete confined by a combination of lateral reinforcement
and fiber. The models are applicable for lightweight aggregate concrete with plain
concrete strength ranging between 38 MPa and 58 MPa, with volumetric ratio of spiral
reinforcement between 1.7% and 6.8%, and with volume fraction of fiber addition
between 0.5% and 1.0%.
162
This study also investigates the experimental behaviors of spiral-reinforced lightweight
concrete column, with and without steel fiber addition, under uniform axial compression.
These experimental behaviors of the columns are also used in evaluating the performance
of suggested stress-strain models.
8.2 Conclusion Remarks
This study continues the line of the work of Bresler (1971), Wang (1977), Fafitis and
Shah (1985), Mander et al. (1988), Hsu and Hsu (1994 a; 1994 b), and Chin (1996). For
the ease of discussion, the study is subdivided into five phases: plain concrete, confined
concrete: structural response, confined concrete: stress-strain characteristic, fiber-
reinforced concrete, and spiral-reinforced column. These five phases complements each
other in contributing the knowledge that is valuable to those dealing with structural
lightweight aggregate concrete.
8.2.1 Plain Concrete
a. It is difficult to capture the complete stress-strain curve of plain lightweight
aggregate concrete in compression. The reason is mainly due to sudden failure of
specimen during unloading. To avoid this sudden failure, several researchers have
modified the conventional test method. Though modified test methods are
logically sound, they do have certain limitations. Some of them are not practical,
while others are unreliable.
b. Moreover, the complete curve obtained from modified test methods seems to be
the result of localized failure of specimen, as well as elastic recovery of
163
undamaged portions of specimen during unloading. It is, thus, questionable
whether the curve represents the true property of specimen in compression. Based
on the above reasoning, none of the modified test methods is applied in the
present study.
c. In addition, core concrete of structural member is strengthened by reinforcement.
Hence, only reinforced concrete characteristic is able to agree the core concrete
characteristic. In other words, plain concrete characteristic is unable to match the
core concrete characteristic. Using the plain concrete characteristic in structural
analysis would underestimate the ultimate behaviors of members, resulting in too
conservative design.
After understanding the above 3 factors, this study ceases observing the characteristics of
plain lightweight concrete. Instead, this study focuses on the characteristics of confined
lightweight concrete and fiber-reinforced lightweight concrete.
8.2.2 Confined Concrete: Structural Response
Response of confined lightweight aggregate concrete to compression is not well
understood due to limited information available in current literature. Such insufficient
understanding largely affects the structural analysis and design. This study thus explores
the response to short-term compression. The response includes deformation capacity,
compressive strength, concrete strain at compressive strength, modulus of elasticity,
unloading manner, and failure manner.
This study also formulates the mathematical equations for essential parameters namely
compressive strength of spiral-reinforced lightweight aggregate concrete, concrete strain
164
at compressive strength, and modulus of elasticity of plain lightweight aggregate
concrete. The equations are applicable for lightweight aggregate concrete with plain
compressive strength ranging between 38 MPa and 58 MPa, with volumetric ratio of
spiral reinforcement between 1.1% and 6.8%. Better understanding gained from the study
would be beneficial in analysis and design of structural lightweight concrete members.
8.2.3 Confined Concrete: Stress-Strain Characteristic
Stress-strain characteristic of confined lightweight aggregate concrete is yet to be well
understood. Such limitation hinders from achieving the safe and cost-effective structural
design.
In this study, a stress-strain model, which is capable of generating the complete stress-
strain characteristic of confined lightweight aggregate concrete, is developed based on the
recorded experimental data. The applicability of the model is checked by comparing the
generated characteristic of the model with experimental characteristic recorded in this
study, as well as those reported in literature by other researchers.
Moreover, the use of the model in structural analysis is illustrated by applying the model
in section analysis, generating the predicted behavior of structural members, and
comparing that predicted behavior with experimental behavior of structural members
such as reinforced lightweight concrete beams and columns. Experimental data of the
beams are obtained from the literature, while those of the columns are obtained from the
present study.
The model is applicable for lightweight aggregate concrete with plain compressive
strength ranging between 38 MPa and 58 MPa, volumetic ratio of spiral reinforcement
165
between 1.7% and 6.8%, concrete made with lightweight coarse aggregate having similar
particle density and strength investigated in this study, and with yield strength of lateral
reinforcement ranging between 540 MPa and 1680 MPa.
8.2.4 Fiber-Reinforced Concrete
It is known that adding short steel fibers in normal weight concrete is another way to
enhance the deformation capacity of the concrete. This study thus explores the
deformation capacity of lightweight concrete with regard to addition of short steel fibers.
Moreover, compressive strength, concrete strain at compressive strength, modulus of
elasticity, splitting tensile strength, and concrete unit weight are also explored.
This study then suggests a mathematical stress-strain model which is capable of
generating the complete stress-strain characteristic of fiber-reinforced lightweight
aggregate concrete. The model is applicable for fiber-reinforced lightweight aggregate
concrete with plain compressive strength ranging between 40 MPa and 60 MPa, and with
volume fraction of fiber addition up to 1.0%.
In addition, this study presents another stress-strain model for lightweight aggregate
concrete confined by a combination of lateral reinforcement and short steel fiber. The
model is capable of generating the stress-strain characteristic for the concrete that has
compressive strength of plain lightweight aggregate concrete of 45.8 MPa, volumetric
ratio of lateral reinforcement of 2%, and volume fraction of short steel fiber addition up
to 1.0%. The ranges of these variables, which can be used in the latter model, are still
open to investigate.
166
8.2.5 Spiral-Reinforced Column
Since characteristics of concrete are closely related to behaviors of column constructed
with the concrete, this study aims to observe the experimental behaviors of spiral-
reinforced lightweight concrete columns under uniform axial compression. The behaviors
are observed with respect to the pitch and diameter of spiral reinforcement, amount of
longitudinal reinforcement, dosage of short steel fiber addition, size of specimen, and
failure manner.
The experimental data of the columns are utilized in demonstrating the usefulness of the
proposed stress-strain model of lateral-reinforced lightweight concrete. In addition, the
experimental data are used in checking the applicability of the proposed stress-strain
model for lightweight concrete confined by a combination of lateral reinforcement and
short steel fiber.
8.3 Recommendation for Further Study
The followings are recommended for further research study:
o The stress-strain models recommended in the present study are limited by the
extent of influencing variables. For this reason, it would be favorable to further
refine the models to cover much wider ranges of the variables.
o ACI-318 Clause 10.9.3 suggests the minimum required amount of spiral
reinforcement for reinforced normal weight concrete column under compression.
The suggested amount ensures that normal weight concrete column would exhibit
adequate ductile behaviour during overloading. It is yet to be observed whether
167
the suggested amount is also applicable to the design of reinforced lightweight
concrete column.
o It is of interest to explore the ultimate behaviors of reinforced lightweight
concrete column with different shape of cross-section, with different
configuration of lateral reinforcement, and subjected to different loading
conditions.
168
Appendix A
List of Symbols
βfibre = a parameter used in stress-strain model for fibre-reinforced lightweight aggregate concrete to control the shape of pre-peak portion of stress-strain curve kt = a parameter used in stress-strain model for fibre-reinforced lightweight aggregate concrete to control the shape of tail portion of stress-strain curve βc = a parameter used in stress-strain model for lightweight concrete confined by a combination of lateral reinforcement and short steel fibres to control the shape of pre-peak portion of stress-strain curve RI = a reinforcing index which is a product of multiplying the aspect ratio and volume fraction of fibre
actual = actual concrete strain at corresponding stress level of f
sc = center-to-center horizontal distance between adjacent shear reinforcements in flexural member fco = compressive strength of confined lightweight aggregate concrete Acore = compressive strength of core concrete per unit width of core concrete region ffibre = compressive strength of fibre-reinforced lightweight aggregate concrete fpeak = compressive strength of lightweight aggregate concrete confined by a combination of lateral reinforcement and short steel fibres εce = concrete strain at extreme compression fiber of flexural member εfibre = concrete strain corresponding to ffibre
εpeak = concrete strain corresponding to fpeak
tp = concrete strain measured by transducers at corresponding stress level of f
εco = confined concrete strain corresponding to fco εt = concrete strain corresponding to 60% of compressive strength of fibre-
169
reinforced lightweight aggregate concrete on post-peak region Ac = cross-sectional area of core concrete of column As = cross-sectional area of spiral wire as = cross-sectional area of spiral wire f’
c = cylindrical compressive strength of plain lightweight aggregate concrete dNA = depth of neutral axis Dc = diameter of confined core concrete dc = diameter of longitudinal reinforcement in compression zone of flexural member section dt = diameter of longitudinal reinforcement in tension zone of flexural member section dcol = diameter of longitudinal reinforcement used in column ds = diameter of shear reinforcement in flexural member d = diameter of spiral wire Es = elastic modulus of reinforcement Ec = elastic modulus of plain lightweight aggregate concrete Ag = gross cross-sectional area of column Esg = initial tangent modulus of plain concrete based on the stress-strain curve derived from strain gauges Etp = initial tangent modulus of plain concrete based on the stress-strain curve derived from transducers Pcore = load capacity contributed by confined core concrete of column Ptotal = load carrying capacity of column Pcover = load capacity contributed by cover concrete of column Ps = load capacity contributed by longitudinal reinforcement βc = parameter controlling the shape of the pre-peak portion of stress-strain curve
170
x,y,z = parameters influencing the shape of the post-peak portion of stress-strain curve
ba, = parameters used in stress-strain model for fibre-reinforced lightweight concrete to control the shape of post-peak portion of stress-strain curve p, q, r = parameters used in stress-strain model for lightweight concrete confined by a combination of lateral reinforcement and short steel fibres to control the shape of post-peak portion of stress-strain curve s = pitch of spiral reinforcement ρs(req) = required volumetric ratio of spiral reinforcement according to ACI 318-05 fst = splitting tensile strength of fibre-reinforced lightweight aggregate concrete ε = strain corresponding to f εo = strain corresponding to f’
c f = stress at any considered point on stress-strain curve As = total cross-sectional area of longitudinal bars ρs = volumetric ratio of spiral reinforcement used in specimen fy = yield strength of reinforcement
171
Appendix B
Software Program – Sectional Analysis Using Plain Concrete Characteristics % ce is the concrete strain at extreme compression fiber of beam
% NA is the assumed depth of neutral axis
% Ec is the modulus of elasticity of lightweight aggregate concrete
% edepth is the effective depth of beam
% dsc is the diameter of longitudinal rebars located in compression zone of beam section
% dst is the diameter of longitudinal rebars located in tension zone of beam section
% f0 is the compressive strength of plain lightweight aggregate concrete
% s0 is the concrete strain corresponding to f0
% sc0 is the steel strain corresponding to yield strength of longitudinal rebars located in compression zone
% st0 is the steel strain corresponding to yield strength of longitudinal rebars located in tension zone
% l is a variable controlling the repetition procedure in the program
% M is the moment of beam section at corresponding stage of l
% cur is the curvature of beam section at corresponding stage of l
% s is the concrete stress at corresponding stage of l
% fce is the concrete strain at extreme compression fiber of beam at corresponding stage of l
% fNA is the depth of neutral axis at corresponding stage of l
% Core is the compressive strength of concrete
% Coremoment is the internal moment capacity contributed by concrete
% A, B, and C are the parameters used in first derivative of the mathematical equation generating the
relationship between lightweight concrete stress and distance along depth of neutral axis
% Tsl is the tensile strength of longitudinal rebars located at lower layer of the configuration
% Tsu is the tensile strength of longitudinal rebars located at upper layer of the configuration
% Tsl_moment is the internal moment capacity contributed by longitudinal rebars located at lower layer of
the configuration
% Tsu_moment is the internal moment capacity contributed by longitudinal rebars located at upper layer of
the configuration
% Tstot is the total tensile strength of longitudinal rebars
% Ctot is the total compressive strength of beam section including compressive strength of concrete, and of
longitudinal rebars in compression zone
% Cs is the compressive strength of longitudinal rebars in compression zone
% Cs_moment is the internal moment capacity contributed by longitudinal rebars in compression zone
172
% h is the width of each segment under the curve showing the relationship between concrete stress and
distance along depth of neutral axis
% a_arm is the lever arm of corresponding segment
% f is the first derivative of the mathematical equation generating the relationship between lightweight
concrete stress and distance along depth of neutral axis
% sarea is the area of corresponding segment under the curve showing the relationship between concrete
stress and distance along depth of neutral axis
% sc is the compressive strain of longitudinal rebars in compression zone
% fsc is the compressive stress of longitudinal rebars in compression zone
% stl is the tensile strain of longitudinal rebars located at lower layer of the configuration
% fstl is the tensile stress of longitudinal rebars located at lower layer of the configuration
% Tsl_moment is the internal moment capacity contributed by longitudinal rebars located at lower layer of
the configuration
% Tsu_moment is the internal moment capacity contributed by longitudinal rebars located at upper layer of
the configuration
% pi is the value representing 3.14159
B.1 Using the Model Suggested by CEB-FIP Code ce=0.0001;
NA=84;
slope=ce/NA;
Ec=33653;
edepth=244.5;
dsc=10;
dst=13;
f0=38.35;
s0=0.002245;
sc0=0.002919;
st0=0.002898;
l=1;
M(1)=0;
cur(1)=0;
s(1)=0;
fce(1)=0;
fNA(1)=0;
while ce<0.02
sc=ce*(NA-35)/NA;
173
stl=ce*(edepth+12.5+6.5-NA)/NA;
stu=ce*(edepth-12.5-6.5-NA)/NA;
Core=0;
Coremoment=0;
A=22205*slope;
B=3.3*10^6*slope^2;
C=-311.86*slope;
Tsl=0;
Tsu=0;
Tsl_moment=0;
Tsu_moment=0;
Tstot=0;
Ctot=0;
Cs=0;
Cs_moment=0;
if ce<=s0
h=NA/5;
a_arm(1)=-0.5*h;
for i=1:1:5
f=@(i)((((1+(C*h*(i-1)))*(A-(2*B*h*(i-1))))-(((A*h*(i-1))-(B*(h^2)*((i-1)^2)))*C))/((1+(C*h*(i-1)))^2));
s(i+1)=s(i)+(h*f(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*h*150*10^-3;
Core=Core+sarea(i+1);
a_arm(i+1)=a_arm(i)+h;
Coremoment=Coremoment+(sarea(i+1)*a_arm(i+1));
end
else ce>s0
h=(s0*NA/ce)/5;
a_arm(1)=-0.5*h;
for i=1:1:5
f=@(i)((((1+(C*h*(i-1)))*(A-(2*B*h*(i-1))))-(((A*h*(i-1))-(B*(h^2)*((i-1)^2)))*C))/((1+(C*h*(i-1)))^2));
s(i+1)=s(i)+(h*f(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*h*150*10^-3;
Core=Core+sarea(i+1);
a_arm(i+1)=a_arm(i)+h;
Coremoment=Coremoment+(sarea(i+1)*a_arm(i+1));
end
174
end
if sc<sc0
fsc=540*sc/0.002919;
else fsc=540;
end
Cs=fsc*pi*dsc^2*2*10^-3/4;
Cs_moment=Cs*(NA-35);
if stl<st0
fstl=542*stl/0.002898;
else fstl=542;
end
Tsl=fstl*pi*dst^2*2*10^-3/4;
Tsl_moment=Tsl*(edepth+12.5+6.5-NA);
if stu<st0
fstu=542*stu/0.002898;
else fstu=542;
end
Tsu=fstu*pi*dst^2*2*10^-3/4;
Tsu_moment=Tsu*(edepth-12.5-6.5-NA);
Tstot=Tsl+Tsu;
Ctot=Core+Cs;
if Ctot>Tstot
ii=(Ctot-Tstot)/Ctot;
if ii<=0.05
M(l)=(Coremoment+Cs_moment+Tsl_moment+Tsu_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else ii>0.05
ce=ce;
NA=NA-1;
slope=ce/NA;
l=l;
175
end
else Tstot>Ctot
jj=(Tstot-Ctot)/Tstot;
if jj<=0.05
M(l)=(Coremoment+Cs_moment+Tsl_moment+Tsu_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else jj>0.05
ce=ce;
NA=NA+1;
slope=ce/NA;
l=l;
end
end
end
>> M’
>> cur’
>> fce’
>> fNA’
B.2 Using the Model Suggested by EC2 Code ce=0.0001;
NA=84;
slope=ce/NA;
Ec=21725;
edepth=244.5;
dsc=10;
dst=13;
f0=38.35;
s0=0.00294;
sc0=0.002919;
st0=0.002898;
176
l=1;
M(1)=0;
cur(1)=0;
s(1)=0;
fce(1)=0;
fNA(1)=0;
while ce<0.02
sc=ce*(NA-35)/NA;
stl=ce*(edepth+12.5+6.5-NA)/NA;
stu=ce*(edepth-12.5-6.5-NA)/NA;
Core=0;
Coremoment=0;
A=-306.12*slope;
B=14348*slope;
C=4.44*10^6*slope^2;
Tsl=0;
Tsu=0;
Tsl_moment=0;
Tsu_moment=0;
Tstot=0;
Ctot=0;
Cs=0;
Cs_moment=0;
if ce<=s0
h=NA/5;
a_arm(1)=-0.5*h;
for i=1:1:5
f=@(i)((((1+(A*h*(i-1)))*(B-(2*h*(i-1))))-(((B*h*(i-1))-(C*(h^2)*((i-1)^2)))*A))/((1+(A*h*(i-1)))^2));
s(i+1)=s(i)+(h*f(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*h*150*10^-3;
Core=Core+sarea(i+1);
a_arm(i+1)=a_arm(i)+h;
Coremoment=Coremoment+(sarea(i+1)*a_arm(i+1));
end
else ce>s0
h=(s0*NA/ce)/5;
a_arm(1)=-0.5*h;
177
for i=1:1:5
f=@(i)((((1+(A*h*(i-1)))*(B-(2*h*(i-1))))-(((B*h*(i-1))-(C*(h^2)*((i-1)^2)))*A))/((1+(A*h*(i-1)))^2));
s(i+1)=s(i)+(h*f(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*h*150*10^-3;
Core=Core+sarea(i+1);
a_arm(i+1)=a_arm(i)+h;
Coremoment=Coremoment+(sarea(i+1)*a_arm(i+1));
end
end
if sc<sc0
fsc=540*sc/0.002919;
else fsc=540;
end
Cs=fsc*pi*dsc^2*2*10^-3/4;
Cs_moment=Cs*(NA-35);
if stl<st0
fstl=542*stl/0.002898;
else fstl=542;
end
Tsl=fstl*pi*dst^2*2*10^-3/4;
Tsl_moment=Tsl*(edepth+12.5+6.5-NA);
if stu<st0
fstu=542*stu/0.002898;
else fstu=542;
end
Tsu=fstu*pi*dst^2*2*10^-3/4;
Tsu_moment=Tsu*(edepth-12.5-6.5-NA);
Tstot=Tsl+Tsu;
Ctot=Core+Cs;
if Ctot>Tstot
ii=(Ctot-Tstot)/Ctot;
if ii<=0.05
M(l)=(Coremoment+Cs_moment+Tsl_moment+Tsu_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
178
NA=NA;
slope=ce/NA;
l=l+1;
else ii>0.05
ce=ce;
NA=NA-1;
slope=ce/NA;
l=l;
end
else Tstot>Ctot
jj=(Tstot-Ctot)/Tstot;
if jj<=0.05
M(l)=(Coremoment+Cs_moment+Tsl_moment+Tsu_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else jj>0.05
ce=ce;
NA=NA+1;
slope=ce/NA;
l=l;
end
end
end
>> M’
>> cur’
>> fce’
>> fNA’
179
Appendix C
Software Program – Sectional Analysis Using Lateral-Reinforced Concrete Characteristics % ce is the concrete strain at extreme compression fiber of beam
% NA is the assumed depth of neutral axis
% pstress is the compressive strength of lateral-reinforced lightweight aggregate concrete
% pstrain is the concrete strain corresponding to the compressive strength of lateral-reinforced lightweight
aggregate concrete
% Ec is the modulus of elasticity of lightweight aggregate concrete
% edepth is the effective depth of beam
% dsc is the diameter of longitudinal rebars located in compression zone of beam section
% dst is the diameter of longitudinal rebars located in tension zone of beam section
% beta is the parameter used in the stress-strain model of lateral-reinforced lightweight aggregate concrete
% f0 is the compressive strength of plain lightweight aggregate concrete
% s0 is the concrete strain corresponding to f0
% sc0 is the steel strain corresponding to yield strength of longitudinal rebars located in compression zone
% st0 is the steel strain corresponding to yield strength of longitudinal rebars located in tension zone
% l is a variable controlling the repetition procedure in the program
% pit is the pitch of spiral reinforcement
% dia is the diameter of spiral wire
% x, y, and z are the parameters used in first derivative of the mathematical equation generating the
relationship between lateral-reinforced lightweight concrete stress in post-peak region and distance along
depth of neutral axis
% M is the moment of beam section at corresponding stage of l
% cur is the curvature of beam section at corresponding stage of l
% s is the concrete stress in pre-peak region at corresponding stage of l
% fce is the concrete strain at extreme compression fiber of beam at corresponding stage of l
% fNA is the depth of neutral axis at corresponding stage of l
% c is the concrete strain corresponding to the end of confined core concrete in compression zone of beam
section
% st is the tensile strain of longitudinal rebars
% Core is the compressive strength of concrete
% Coremoment is the internal moment capacity contributed by concrete
% A, B, and C are the parameters used in first derivative of the mathematical equation generating the
relationship between confined lightweight concrete stress and distance along the depth of neutral axis
180
% dpeak is the distance under pre-peak portion of the curve showing the relationship between concrete
stress and distance along depth of neutral axis
% d is the distance between neutral axis and the end of confined core concrete in compression zone
% k is the width of each segment under pre-peak portion of the curve showing the relationship between
confined concrete stress and distance along depth of neutral axis
% t is the width of each segment under post-peak portion of the curve showing the relationship between
confined concrete stress and distance along depth of neutral axis
% a_arm_a is the lever arm of corresponding segment under pre-peak portion of the curve showing the
relationship between confined concrete stress and distance along depth of neutral axis
% a_arm_d is the lever arm of corresponding segment under post-peak portion of the curve showing the
relationship between confined concrete stress and distance along depth of neutral axis
% A_tri is the compressive strength of cover concrete at the sides of core concrete region in compression
zone
% A_trape is the compressive strength of cover concrete at the top of core concrete region in compression
zone
% f is the first derivative of the mathematical equation generating the relationship between confined
lightweight concrete stress in pre-peak region and distance along depth of neutral axis, when the variable h
is used
% fanother is the first derivative of the mathematical equation generating the relationship between confined
lightweight concrete stress in pre-peak region and distance along depth of neutral axis, when the k variable
is used
% g is the first derivative of the mathematical equation generating the relationship between confined
lightweight concrete stress in post-peak region and distance along depth of neutral axis
% ds is the concrete stress in post-peak region at corresponding stage of l
% Tsl is the tensile strength of longitudinal rebars located at lower layer of the configuration
% Tsu is the tensile strength of longitudinal rebars located at upper layer of the configuration
% Tsl_moment is the internal moment capacity contributed by longitudinal rebars located at lower layer of
the configuration
% Tsu_moment is the internal moment capacity contributed by longitudinal rebars located at upper layer of
the configuration
% Tstot is the total tensile strength of longitudinal rebars
% Ctot is the total compressive strength of beam section including compressive strength of concrete, and of
longitudinal rebars in compression zone
% Cs is the compressive strength of longitudinal rebars in compression zone
% Cs_moment is the internal moment capacity contributed by longitudinal rebars in compression zone
% h is the width of each segment under the curve showing the relationship between concrete stress and
distance along depth of neutral axis
181
% a_arm is the lever arm of corresponding segment
% sarea is the area of corresponding segment in the pre-peak region of the curve showing the relationship
between concrete stress and distance along depth of neutral axis
% darea is the area of corresponding segment in post-peak region of the curve showing the relationship
between concrete stress and distance along depth of neutral axis
% fc_at_c is the cover concrete stress at the end of core concrete region
% fc_at_ce is the cover concrete stress at the extreme compression fiber of beam
% fst is the tensile stress of longitudinal rebars
% Ts is the tensile strength of longitudinal rebars
% Ts_moment is the internal moment capacity contributed by longitudinal rebars
% sc is the compressive strain of longitudinal rebars in compression zone
% fsc is the compressive stress of longitudinal rebars in compression zone
% stl is the tensile strain of longitudinal rebars located at lower layer of the configuration
% fstl is the tensile stress of longitudinal rebars located at lower layer of the configuration
% Tsl_moment is the internal moment capacity contributed by longitudinal rebars located at lower layer of
the configuration
% Tsu_moment is the internal moment capacity contributed by longitudinal rebars located at upper layer of
the configuration
% pi is the value representing 3.14159
C.1 Beam 8 ce=0.0001;
NA=84;
slope=ce/NA;
pstrain=0.0036;
pstress=43.9;
Ec=25751.76;
edepth=262;
dsc=10;
dst=16;
beta=1/(1-(pstress/(pstrain*Ec)));
B=beta-1;
f0=44.42;
s0=0.0023;
sc0=0.002919;
st0=0.00279;
l=1;
pit=130;
182
dia=10;
y=2317.3+(8.9*f0)-(480.6*pit/dia);
z=-20412.4-(0.069*(f0^3))+(677.6*(pit/dia)^2);
M(1)=0;
cur(1)=0;
s(1)=0;
fce(1)=0;
fNA(1)=0;
while ce<0.02
c=ce*(NA-20)/NA;
sc=ce*(NA-35)/NA;
st=ce*(edepth-NA)/NA;
Core=0;
Coremoment=0;
h=(NA-20)/5;
a_arm(1)=-0.5*h;
A=pstress*beta*slope/pstrain;
C=slope^beta/pstrain^beta;
dpeak=pstrain/slope;
d=c/slope;
k=dpeak/5;
t=(d-dpeak)/5;
a_arm_a(1)=-0.5*k;
Ts=0;
Ts_moment=0;
Ctot=0;
a_arm_d(1)=dpeak-(0.5*t);
A_tri=0;
A_trape=0;
Cover=0;
Covermoment=0;
Cs=0;
Cs_moment=0;
if d<=dpeak
for i=1:1:5
f=@(i)(((B+(C*h^beta*(i-1)^beta))*A-(A*h*(i-1)*C*beta*h^B*(i-1)^B))/(B+(C*h^beta*(i-1)^beta))^2);
s(i+1)=s(i)+(h*f(i));
183
sarea(i+1)=(s(i)+s(i+1))*0.5*h*110*10^-3;
Core=Core+sarea(i+1);
a_arm(i+1)=a_arm(i)+h;
Coremoment=Coremoment+(sarea(i+1)*a_arm(i+1));
end
else d>dpeak
for i=1:1:5
fanother=@(i)(((B+(C*k^beta*(i-1)^beta))*A-(A*k*(i-1)*C*beta*k^B*(i-1)^B))/(B+(C*k^beta*(i-
1)^beta))^2);
s(i+1)=s(i)+(k*fanother(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*k*110*10^-3;
Core=Core+sarea(i+1);
a_arm_a(i+1)=a_arm_a(i)+k;
Coremoment=Coremoment+(sarea(i+1)*a_arm_a(i+1));
end
ds(1)=s(6);
for j=1:1:5
g=@(j)((y*slope)+(2*z*slope^2*(dpeak+((j+1)*t))));
ds(j+1)=ds(j)+(t*g(j));
dsarea(j+1)=(ds(j)+ds(j+1))*0.5*t*110*10^-3;
Core=Core+dsarea(j+1);
a_arm_d(j+1)=a_arm_d(j)+t;
Coremoment=Coremoment+(dsarea(j+1)*a_arm_d(j+1));
end
end
if ce<=s0
fc_at_c=f0*c/s0;
fc_at_ce=f0*ce/s0;
A_tri=0.5*(NA-20)*fc_at_c*2*20*10^-3;
A_trape=(fc_at_c+fc_at_ce)*0.5*20*150*10^-3;
Cover=A_tri+A_trape;
Covermoment=(A_tri*2*(NA-20)/3)+(A_trape*(NA-10));
elseif ce>s0 & c<s0
fc_at_s0=f0;
fc_at_c=f0*c/s0;
a=(NA*s0/ce)-(NA-20);
A_tri=0.5*(NA-20)*fc_at_c*2*20*10^-3;
184
A_trape=(fc_at_c+fc_at_s0)*0.5*a*150*10^-3;
Cover=A_tri+A_trape;
Covermoment=(A_tri*2*(NA-20)/3)+(A_trape*((0.5*a)+NA-20));
else c>=s0
fc_at_s0=f0;
b=NA*s0/ce;
A_tri=0.5*b*fc_at_s0*2*20*10^-3;
Cover=A_tri;
Covermoment=A_tri*2*b/3;
end
if sc<sc0
fsc=540*sc/0.002919;
else fsc=540;
end
Cs=fsc*pi*dsc^2*2*10^-3/4;
Cs_moment=Cs*(NA-35);
if st<st0
fst=512*st/0.00279;
else fst=512;
end
Ts=fst*pi*dst^2*2*10^-3/4;
Ts_moment=Ts*(edepth-NA);
Ctot=Core+Cover+Cs;
if Ctot>Ts
ii=(Ctot-Ts)/Ctot;
if ii<=0.05
M(l)=(Coremoment+Covermoment+Cs_moment+Ts_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else ii>0.05
ce=ce;
NA=NA-1;
185
slope=ce/NA;
l=l;
end
else Ts>Ctot
jj=(Ts-Ctot)/Ts;
if jj<=0.05
M(l)=(Coremoment+Covermoment+Cs_moment+Ts_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else jj>0.05
ce=ce;
NA=NA+1;
slope=ce/NA;
l=l;
end
end
end
>> M’
>> cur’
>> fce’
>> fNA’
C.2 Beam 11 ce=0.0001;
NA=84;
slope=ce/NA;
pstrain=0.0036;
pstress=43.9;
Ec=25751.76;
edepth=260;
dsc=10;
dst=20;
186
beta=1/(1-(pstress/(pstrain*Ec)));
B=beta-1;
f0=44.42;
s0=0.0023;
sc0=0.002919;
st0=0.002989;
l=1;
pit=130;
dia=10;
y=2317.3+(8.9*f0)-(480.6*pit/dia);
z=-20412.4-(0.069*(f0^3))+(677.6*(pit/dia)^2);
M(1)=0;
cur(1)=0;
s(1)=0;
fce(1)=0;
fNA(1)=0;
while ce<0.02
c=ce*(NA-20)/NA;
sc=ce*(NA-35)/NA;
st=ce*(edepth-NA)/NA;
Core=0;
Coremoment=0;
h=(NA-20)/5;
a_arm(1)=-0.5*h;
A=pstress*beta*slope/pstrain;
C=slope^beta/pstrain^beta;
dpeak=pstrain/slope;
d=c/slope;
k=dpeak/5;
t=(d-dpeak)/5;
a_arm_a(1)=-0.5*k;
Ts=0;
Ts_moment=0;
Ctot=0;
a_arm_d(1)=dpeak-(0.5*t);
A_tri=0;
A_trape=0;
187
Cover=0;
Covermoment=0;
Cs=0;
Cs_moment=0;
if d<=dpeak
for i=1:1:5
f=@(i)(((B+(C*h^beta*(i-1)^beta))*A-(A*h*(i-1)*C*beta*h^B*(i-1)^B))/(B+(C*h^beta*(i-1)^beta))^2);
s(i+1)=s(i)+(h*f(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*h*110*10^-3;
Core=Core+sarea(i+1);
a_arm(i+1)=a_arm(i)+h;
Coremoment=Coremoment+(sarea(i+1)*a_arm(i+1));
end
else d>dpeak
for i=1:1:5
fanother=@(i)(((B+(C*k^beta*(i-1)^beta))*A-(A*k*(i-1)*C*beta*k^B*(i-1)^B))/(B+(C*k^beta*(i-
1)^beta))^2);
s(i+1)=s(i)+(k*fanother(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*k*110*10^-3;
Core=Core+sarea(i+1);
a_arm_a(i+1)=a_arm_a(i)+k;
Coremoment=Coremoment+(sarea(i+1)*a_arm_a(i+1));
end
ds(1)=s(6);
for j=1:1:5
g=@(j)((y*slope)+(2*z*slope^2*(dpeak+((j+1)*t))));
ds(j+1)=ds(j)+(t*g(j));
dsarea(j+1)=(ds(j)+ds(j+1))*0.5*t*110*10^-3;
Core=Core+dsarea(j+1);
a_arm_d(j+1)=a_arm_d(j)+t;
Coremoment=Coremoment+(dsarea(j+1)*a_arm_d(j+1));
end
end
if ce<=s0
fc_at_c=f0*c/s0;
fc_at_ce=f0*ce/s0;
A_tri=0.5*(NA-20)*fc_at_c*2*20*10^-3;
188
A_trape=(fc_at_c+fc_at_ce)*0.5*20*150*10^-3;
Cover=A_tri+A_trape;
Covermoment=(A_tri*2*(NA-20)/3)+(A_trape*(NA-10));
elseif ce>s0 & c<s0
fc_at_s0=f0;
fc_at_c=f0*c/s0;
a=(NA*s0/ce)-(NA-20);
A_tri=0.5*(NA-20)*fc_at_c*2*20*10^-3;
A_trape=(fc_at_c+fc_at_s0)*0.5*a*150*10^-3;
Cover=A_tri+A_trape;
Covermoment=(A_tri*2*(NA-20)/3)+(A_trape*((0.5*a)+NA-20));
else c>=s0
fc_at_s0=f0;
b=NA*s0/ce;
A_tri=0.5*b*fc_at_s0*2*20*10^-3;
Cover=A_tri;
Covermoment=A_tri*2*b/3;
end
if sc<sc0
fsc=540*sc/0.002919;
else fsc=540;
end
Cs=fsc*pi*dsc^2*2*10^-3/4;
Cs_moment=Cs*(NA-35);
if st<st0
fst=532*st/0.002989;
else fst=532;
end
Ts=fst*pi*dst^2*2*10^-3/4;
Ts_moment=Ts*(edepth-NA);
Ctot=Core+Cover+Cs;
if Ctot>Ts
ii=(Ctot-Ts)/Ctot;
if ii<=0.05
M(l)=(Coremoment+Covermoment+Cs_moment+Ts_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
189
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else ii>0.05
ce=ce;
NA=NA-1;
slope=ce/NA;
l=l;
end
else Ts>Ctot
jj=(Ts-Ctot)/Ts;
if jj<=0.05
M(l)=(Coremoment+Covermoment+Cs_moment+Ts_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else jj>0.05
ce=ce;
NA=NA+1;
slope=ce/NA;
l=l;
end
end
end
>> M’
>> cur’
>> fce’
>> fNA’
190
C.3 Beam 12 ce=0.0001;
NA=84;
slope=ce/NA;
pstrain=0.002781;
pstress=37.96;
Ec=23431.52;
edepth=244.5;
dsc=10;
dst=13;
beta=1/(1-(pstress/(pstrain*Ec)));
B=beta-1;
f0=38.35;
s0=0.0023;
sc0=0.002919;
st0=0.002898;
l=1;
pit=130;
dia=10;
y=2317.3+(8.9*f0)-(480.6*pit/dia);
z=-20412.4-(0.069*(f0^3))+(677.6*(pit/dia)^2);
M(1)=0;
cur(1)=0;
s(1)=0;
fce(1)=0;
fNA(1)=0;
while ce<0.02
c=ce*(NA-20)/NA;
sc=ce*(NA-35)/NA;
stl=ce*(edepth+12.5+6.5-NA)/NA;
stu=ce*(edepth-12.5-6.5-NA)/NA;
Core=0;
Coremoment=0;
h=(NA-20)/5;
a_arm(1)=-0.5*h;
A=pstress*beta*slope/pstrain;
C=slope^beta/pstrain^beta;
191
dpeak=pstrain/slope;
d=c/slope;
k=dpeak/5;
t=(d-dpeak)/5;
a_arm_a(1)=-0.5*k;
Tsl=0;
Tsu=0;
Tsl_moment=0;
Tsu_moment=0;
Tstot=0;
Ctot=0;
a_arm_d(1)=dpeak-(0.5*t);
A_tri=0;
A_trape=0;
Cover=0;
Covermoment=0;
Cs=0;
Cs_moment=0;
if d<=dpeak
for i=1:1:5
f=@(i)(((B+(C*h^beta*(i-1)^beta))*A-(A*h*(i-1)*C*beta*h^B*(i-1)^B))/(B+(C*h^beta*(i-1)^beta))^2);
s(i+1)=s(i)+(h*f(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*h*110*10^-3;
Core=Core+sarea(i+1);
a_arm(i+1)=a_arm(i)+h;
Coremoment=Coremoment+(sarea(i+1)*a_arm(i+1));
end
else d>dpeak
for i=1:1:5
fanother=@(i)(((B+(C*k^beta*(i-1)^beta))*A-(A*k*(i-1)*C*beta*k^B*(i-1)^B))/(B+(C*k^beta*(i-
1)^beta))^2);
s(i+1)=s(i)+(k*fanother(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*k*110*10^-3;
Core=Core+sarea(i+1);
a_arm_a(i+1)=a_arm_a(i)+k;
Coremoment=Coremoment+(sarea(i+1)*a_arm_a(i+1));
end
192
ds(1)=s(6);
for j=1:1:5
g=@(j)((y*slope)+(2*z*slope^2*(dpeak+((j+1)*t))));
ds(j+1)=ds(j)+(t*g(j));
dsarea(j+1)=(ds(j)+ds(j+1))*0.5*t*110*10^-3;
Core=Core+dsarea(j+1);
a_arm_d(j+1)=a_arm_d(j)+t;
Coremoment=Coremoment+(dsarea(j+1)*a_arm_d(j+1));
end
end
if ce<=s0
fc_at_c=f0*c/s0;
fc_at_ce=f0*ce/s0;
A_tri=0.5*(NA-20)*fc_at_c*2*20*10^-3;
A_trape=(fc_at_c+fc_at_ce)*0.5*20*150*10^-3;
Cover=A_tri+A_trape;
Covermoment=(A_tri*2*(NA-20)/3)+(A_trape*(NA-10));
elseif ce>s0 & c<s0
fc_at_s0=f0;
fc_at_c=f0*c/s0;
a=(NA*s0/ce)-(NA-20);
A_tri=0.5*(NA-20)*fc_at_c*2*20*10^-3;
A_trape=(fc_at_c+fc_at_s0)*0.5*a*150*10^-3;
Cover=A_tri+A_trape;
Covermoment=(A_tri*2*(NA-20)/3)+(A_trape*((0.5*a)+NA-20));
else c>=s0
fc_at_s0=f0;
b=NA*s0/ce;
A_tri=0.5*b*fc_at_s0*2*20*10^-3;
Cover=A_tri;
Covermoment=A_tri*2*b/3;
end
if sc<sc0
fsc=540*sc/0.002919;
else fsc=540;
end
Cs=fsc*pi*dsc^2*2*10^-3/4;
193
Cs_moment=Cs*(NA-35);
if stl<st0
fstl=542*stl/0.002898;
else fstl=542;
end
Tsl=fstl*pi*dst^2*2*10^-3/4;
Tsl_moment=Tsl*(edepth+12.5+6.5-NA);
if stu<st0
fstu=542*stu/0.002898;
else fstu=542;
end
Tsu=fstu*pi*dst^2*2*10^-3/4;
Tsu_moment=Tsu*(edepth-12.5-6.5-NA);
Tstot=Tsl+Tsu;
Ctot=Core+Cover+Cs;
if Ctot>Tstot
ii=(Ctot-Tstot)/Ctot;
if ii<=0.05
M(l)=(Coremoment+Covermoment+Cs_moment+Tsl_moment+Tsu_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else ii>0.05
ce=ce;
NA=NA-1;
slope=ce/NA;
l=l;
end
else Tstot>Ctot
jj=(Tstot-Ctot)/Tstot;
if jj<=0.05
M(l)=(Coremoment+Covermoment+Cs_moment+Tsl_moment+Tsu_moment)*10^-3;
cur(l)=ce/NA;
194
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else jj>0.05
ce=ce;
NA=NA+1;
slope=ce/NA;
l=l;
end
end
end
>> M’
>> cur’
>> fce’
>> fNA
C.4 Beam 13 ce=0.0001;
NA=84;
slope=ce/NA;
pstrain=0.002781;
pstress=37.96;
Ec=23431.52;
edepth=241.5;
dsc=10;
dst=16;
beta=1/(1-(pstress/(pstrain*Ec)));
B=beta-1;
f0=38.35;
s0=0.0023;
sc0=0.002919;
st0=0.00279;
>> l=1;
pit=130;
195
dia=10;
y=2317.3+(8.9*f0)-(480.6*pit/dia);
z=-20412.4-(0.069*(f0^3))+(677.6*(pit/dia)^2);
M(1)=0;
cur(1)=0;
s(1)=0;
fce(1)=0;
fNA(1)=0;
while ce<0.02
c=ce*(NA-20)/NA;
sc=ce*(NA-35)/NA;
stl=ce*(edepth+12.5+8-NA)/NA;
stu=ce*(edepth-12.5-8-NA)/NA;
Core=0;
Coremoment=0;
h=(NA-20)/5;
a_arm(1)=-0.5*h;
A=pstress*beta*slope/pstrain;
C=slope^beta/pstrain^beta;
dpeak=pstrain/slope;
d=c/slope;
k=dpeak/5;
t=(d-dpeak)/5;
a_arm_a(1)=-0.5*k;
Tsl=0;
Tsu=0;
Tsl_moment=0;
Tsu_moment=0;
Tstot=0;
Ctot=0;
a_arm_d(1)=dpeak-(0.5*t);
A_tri=0;
A_trape=0;
Cover=0;
Covermoment=0;
Cs=0;
Cs_moment=0;
196
if d<=dpeak
for i=1:1:5
f=@(i)(((B+(C*h^beta*(i-1)^beta))*A-(A*h*(i-1)*C*beta*h^B*(i-1)^B))/(B+(C*h^beta*(i-1)^beta))^2);
s(i+1)=s(i)+(h*f(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*h*110*10^-3;
Core=Core+sarea(i+1);
a_arm(i+1)=a_arm(i)+h;
Coremoment=Coremoment+(sarea(i+1)*a_arm(i+1));
end
else d>dpeak
for i=1:1:5
fanother=@(i)(((B+(C*k^beta*(i-1)^beta))*A-(A*k*(i-1)*C*beta*k^B*(i-1)^B))/(B+(C*k^beta*(i-
1)^beta))^2);
s(i+1)=s(i)+(k*fanother(i));
sarea(i+1)=(s(i)+s(i+1))*0.5*k*110*10^-3;
Core=Core+sarea(i+1);
a_arm_a(i+1)=a_arm_a(i)+k;
Coremoment=Coremoment+(sarea(i+1)*a_arm_a(i+1));
end
ds(1)=s(6);
for j=1:1:5
g=@(j)((y*slope)+(2*z*slope^2*(dpeak+((j+1)*t))));
ds(j+1)=ds(j)+(t*g(j));
dsarea(j+1)=(ds(j)+ds(j+1))*0.5*t*110*10^-3;
Core=Core+dsarea(j+1);
a_arm_d(j+1)=a_arm_d(j)+t;
Coremoment=Coremoment+(dsarea(j+1)*a_arm_d(j+1));
end
end
if ce<=s0
fc_at_c=f0*c/s0;
fc_at_ce=f0*ce/s0;
A_tri=0.5*(NA-20)*fc_at_c*2*20*10^-3;
A_trape=(fc_at_c+fc_at_ce)*0.5*20*150*10^-3;
Cover=A_tri+A_trape;
Covermoment=(A_tri*2*(NA-20)/3)+(A_trape*(NA-10));
elseif ce>s0 & c<s0
197
fc_at_s0=f0;
fc_at_c=f0*c/s0;
a=(NA*s0/ce)-(NA-20);
A_tri=0.5*(NA-20)*fc_at_c*2*20*10^-3;
A_trape=(fc_at_c+fc_at_s0)*0.5*a*150*10^-3;
Cover=A_tri+A_trape;
Covermoment=(A_tri*2*(NA-20)/3)+(A_trape*((0.5*a)+NA-20));
else c>=s0
fc_at_s0=f0;
b=NA*s0/ce;
A_tri=0.5*b*fc_at_s0*2*20*10^-3;
Cover=A_tri;
Covermoment=A_tri*2*b/3;
end
if sc<sc0
fsc=540*sc/0.002919;
else fsc=540;
end
Cs=fsc*pi*dsc^2*2*10^-3/4;
Cs_moment=Cs*(NA-35);
if stl<st0
fstl=512*stl/0.00279;
else fstl=512;
end
Tsl=fstl*pi*dst^2*2*10^-3/4;
Tsl_moment=Tsl*(edepth+12.5+8-NA);
if stu<st0
fstu=512*stu/0.00279;
else fstu=512;
end
Tsu=fstu*pi*dst^2*2*10^-3/4;
Tsu_moment=Tsu*(edepth-12.5-8-NA);
Tstot=Tsl+Tsu;
Ctot=Core+Cover+Cs;
if Ctot>Tstot
ii=(Ctot-Tstot)/Ctot;
if ii<=0.05
198
M(l)=(Coremoment+Covermoment+Cs_moment+Tsl_moment+Tsu_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else ii>0.05
ce=ce;
NA=NA-1;
slope=ce/NA;
l=l;
end
else Tstot>Ctot
jj=(Tstot-Ctot)/Tstot;
if jj<=0.05
M(l)=(Coremoment+Covermoment+Cs_moment+Tsl_moment+Tsu_moment)*10^-3;
cur(l)=ce/NA;
fce(l)=ce;
fNA(l)=NA;
ce=ce+0.0002;
NA=NA;
slope=ce/NA;
l=l+1;
else jj>0.05
ce=ce;
NA=NA+1;
slope=ce/NA;
l=l;
end
end
end
>> M’
>> cur’
>> fce’
>> fNA’
199
Appendix D
Stress-Strain Models Proposed in Present Study
D.1 Stress-Strain Model of Confined Lightweight Aggregate Concrete
The mathematical expression for the pre-peak portion of confined lightweight aggregate
concrete is as follows:
c
coc
coc
cof
f
1
for co 0 (D.1)
The mathematical expression for the post-peak portion of confined lightweight aggregate
concrete is as follows:
2 zyxf for co (D.2)
'433 cco fs
df (MPa)
)104(0018.0 5coco f
'16018000 cc fE (MPa)
cco
coc
E
f
1
1
200
3
2'200
4040068
s
d
fx
c
d
sfy c 50092300 '
2
3' 70007.020400
d
sfz c
f and ε are the concrete stress and corresponding strain at any point on stress-strain curve.
fco and εco are the compressive strength of confined lightweight aggregate concrete and
concrete strain at compressive strength level. βc is a parameter controlling the shape of
the pre-peak portion. x, y, z are the parameters influencing the shape of post-peak
portion.
In the above equations, d refers to the diameter of spiral wire (mm), s represents the pitch
of spiral reinforcement (mm), fc’ is the cylindrical compressive strength of plain
lightweight aggregate concrete (MPa), and Ec is the elastic modulus of plain lightweight
aggregate concrete (MPa)
201
D.2 Stress-Strain Model of Fiber-Reinforced Lightweight Aggregate
Concrete
The stress-strain curve of fiber-reinforced lightweight aggregate concrete can be
classified into three portions: pre-peak, post-peak, and tail portions. The model is thus
made up of three mathematical expressions; each expression represents each portion of
the curve accordingly.
The mathematical expression for the pre-peak portion of the stress-strain curve is as
follows:
fibre
fibrefibre
fibrefibrefibref
f
2
12
2
for fibre 0 (D.3)
The mathematical expression for the post-peak portion is as follows:
baf for tfibre (D.4)
The mathematical expression for the tail portion is as follows:
7.0
exp6.0fibre
t
fibretfibre kff
for t (D.5)
202
fcfibre Vff 120012.14.6 ' (MPa)
0029.0co
'16018000 cc fE (MPa)
cfibre
fibrefibre
E
f
1
1
fc
Vf
a 4005400
200'
fc
Vf
b 31340010
32700'
6
fc
t Vfk
003.024.306.0
'
3
2'500
2.5003.0 f
ct V
f
f and ε are the concrete stress and corresponding concrete strain at any point on stress-
strain curve. ffibre and εfibre are the compressive strength of fiber-reinforced lightweight
aggregate concrete and corresponding concrete strain. f’c is the compressive strength of
plain lightweight aggregate concrete (MPa). Vf is the volume fraction of short steel fiber.
βfibre is a parameter controlling the shape of pre-peak portion. Ec is the modulus of
elasticity of plain lightweight aggregate concrete. a and b are the parameters controlling
the shape of post-peak portion. εt is the concrete strain corresponding to 60% of
compressive strength on post-peak portion. kt is a parameter controlling the shape of tail
portion.
203
D.3 Stress-Strain Model of Lightweight Aggregate Concrete Confined
by a Combination of Lateral Reinforcement and Short Steel Fiber
The model consists of two mathematical expressions: one is for pre-peak portion of the
stress-strain curve, while the other is for post-peak portion.
Mathematical expression for pre-peak portion is as follows.
c
peakc
peakc
peakf
f
1
for peak 0 (D.6)
Mathematical expression to generate the post-peak portion is as follows:
2 rqpf for peak (D.7)
s
dVff fcpeak 4311759 ' (MPa)
)104(0018.0 5peakpeak f
'16018000 cc fE (MPa)
cpeak
peakc
E
f
1
1
204
3
2'200
)120012.14.6(
4040068
s
d
Vfp
fc
d
sVfq fc 50010700102300 '
2
3' 700)120012.14.6(07.020400
d
sVfr fc
f and ε are the concrete stress and corresponding strain at any point on stress-strain curve.
fpeak is the compressive strength of lightweight aggregate concrete confined by a
combination of lateral reinforcement and fiber. εpeak is the concrete strain at fpeak. Ec is the
elastic modulus of plain lightweight aggregate concrete. βc, p, q, and r are the parameters
controlling the shape of stress-strain curve.
In the above equations, d stands for the diameter of spiral wire (mm), s represents the
pitch of spiral reinforcement (mm), and f’c refers to the cylindrical compressive strength
of plain lightweight aggregate concrete (MPa). Vf is the volume fraction of fiber.
205
Appendix E
Literature Review: Parameter Equations and Stress-Strain Models
E.1 Compressive Strength of Confined Concrete
E.1.1 Mander et al. (1988)
Mander et al. (1998) reported the following equation for compressive strength of
confined normal weight concrete.
)294.71254.2254.1('
'''''
co
lcolcocc f
fffff (MPa)
where 'cof = unconfined concrete compressive strength
yhsel fkf 2
1'
cc
se
d
s
k
1
21
'
for circular spiral as transverse reinforcement
yhf = yield strength of transverse reinforcement
s = volumetric ratio of transverse reinforcement
's = clear spacing between spiral reinforcement
sd = diameter of spiral reinforcement
cc = ratio of area of longitudinal reinforcement to area of core of section
206
E.1.2 Hsu & Hsu (1994)
Hsu & Hsu (1994) reported the following equation for compressive strength of confined
high strength normal weight concrete with unconfined compressive strength more than 69
MPa.
'' 73.214 ccc ff (kip/in2)
where 'ccf = peak stress of confined high strength normal weight concrete
'cf = compressive strength of unconfined high strength concrete
= volumetric ratio of lateral reinforcement = DS
As4
sA = cross-sectional area of circular hoop (in2)
D = outside diameter of circular hoop (in)
S = tie spacing (in)
E.1.3 Mansur et al. (1997)
Mansur et al. (1997) reported the following equation for compressive strength of
confined high strength normal weight concrete with unconfined compressive strength
ranging from 60 MPa to 120 MPa, with yield strength of transverse tie of 493 MPa, and
with volumetric ratio of lateral tie between 0 to 4.51%.
23.1"
6.01
o
ys
o
o
f
f
f
f (MPa)
where "of = strength of confined high strength normal weight concrete
207
of = strength of unconfined high strength normal weight concrete
yf = yield strength of transverse tie
s = volumetric ratio of lateral tie
E.1.4 CEB-FIP Code (2000)
The Code reported the following equation for compressive strength of confined
lightweight aggregate concrete.
cccccc f
ff 2* 1.11
(MPa)
where *ccf = compressive strength of confined lightweight aggregate concrete
ccf = compressive strength of plain lightweight aggregate concrete
2 = lateral compressive stress due to confinement
E.1.5 EC2 Code (2004)
The Code reported that the following equation of compressive strength of confined
lightweight aggregate concrete may be used when more precise data are not available.
However, without having the precise data from experiment, 2 , which is a parameter of
the equation, would be unknown.
lcklckclck f
kff 2
, 1
(MPa)
208
where k = 1.1 for lightweight aggregate concrete with sand as fine aggregate
lckf = characteristic concrete strength of lightweight aggregate concrete
clckf , = compressive strength of confined concrete
2 = effective lateral compressive strength at ultimate limit state due to
confinement
E.2 Modulus of Elasticity of Plain Concrete
E.2.1 CEB-FIP Code (2000)
The Code reported the following equation for modulus of elasticity of plain lightweight
concrete. It mentioned that lightweight concrete has a lower modulus of elasticity than
normal weight concrete, and recommended the further research.
2'
)( 220069003320
cLWACc fE (MPa)
where 'cf = compressive strength of lightweight aggregate concrete
= density of lightweight aggregate concrete
E.2.2 EC2 Code (2004)
The Code recommends the following equation for modulus of elasticity of lightweight
aggregate concrete with cylinder compressive strength of concrete ranging between 17
MPa and 88 MPa. The Code states that tests shall be carried out in order to determine the
modulus of elasticity values when accurate data are required.
209
23.0
22001022
lcmlcm
fE (MPa)
where = oven-dry density of lightweight aggregate concrete
lcmE = secant modulus of lightweight aggregate concrete
lcmf = cylinder compressive strength of concrete
E.2.3 ACI 213R (2003)
ACI 213R stated that the modulus of elasticity of lightweight aggregate concrete varies
between ½ and ¾ of modulus of elasticity of normal weight concrete of the same
strength. ACI 213R recommended the following equation for lightweight aggregate
concrete with density between 1440 kg/m3 and 2480 kg/m3, with concrete compressive
strength between 21 MPa and 35 MPa.
'5.1 043.0 ccc fwE (MPa)
where cE = modulus of elasticity of lightweight aggregate concrete
cw = density of lightweight aggregate concrete
'cf = compressive strength of lightweight aggregate concrete
E.2.4 BS8110 (2008)
BS recommended the following equation for lightweight aggregate concrete with
characteristic cube strength between 20 MPa and 60 MPa.
210
2
)( 24002.0
cuoLWACc fKE (GPa)
where )(LWACcE = modulus of elasticity of lightweight aggregate concrete
cuf = characteristic cube strength at 28 days
oK = a constant closely related to the modulus of elasticity of aggregate
= density of lightweight aggregate concrete (kg/m3)
E.3 Stress-Strain Model of Confined Normal Weight Concrete
E.3.1 Mander et al. (1988)
Mander et al. (1998) suggested the following stress-strain model for confined normal
weight concrete.
r
ccc xr
xrff
1
'
where cc
cx
151
'
'
co
cccocc f
f
co = unconfined concrete strain which is assumed as 0.002
'cof = unconfined concrete strength
'ccf = compressive strength of confined normal weight concrete
c = concrete longitudinal compressive concrete strain
211
secEE
Er
c
c
'5000 coc fE
cc
ccfE
'
sec
)294.71254.2254.1('
'''''
co
lcolcocc f
fffff
'cof = unconfined concrete compressive strength
yhsel fkf 2
1'
cc
se
d
s
k
1
21
'
for circular spiral as transverse reinforcement
yhf = yield strength of transverse reinforcement
s = volumetric ratio of transverse reinforcement
's = clear spacing between spiral reinforcement
sd = diameter of spiral reinforcement
cc = ratio of area of longitudinal reinforcement to area of core of section
E.3.2 Hsu & Hsu (1994)
Hsu & Hsu (1994) suggested the following stress-strain model for confined high strength
normal weight concrete with compressive strength of unconfined concrete more than 69
MPa.
212
n
c
c
xn
xn
1
where '
cc
c
f
f
oc
x
2.06.1exp c
)(1
1'
ito
c
E
f
for 1
DS
As4
n = 1
'' 73.214 ccc ff
ooc 18134.0
3'5 10114.2109.8 co f
3'2 1028312.3102431.1 cit fE
'ccf = peak stress of confined high strength normal weight concrete
'cf = compressive strength of unconfined high strength concrete
= volumetric ratio of lateral reinforcement = DS
As4
sA = cross-sectional area of circular hoop (in2)
D = outside diameter of circular hoop (in)
213
S = tie spacing (in)
itE = slope at the origin of stress-strain curve or initial tangent modulus
E.3.3 Mansur et al. (1997)
Mansur et al. (1997) reported the following stress-strain model for confined high strength
normal weight concrete with compressive strength of unconfined concrete ranging from
60 to 120 MPa, and with volumetric ratio of lateral tie ranging from 0 to 4.51%.
For the ascending portion of stress-strain curve of confined high strength normal weight
concrete,
"
""
1o
ooff
where
ito
o
E
f"
"
1
1
8.0"
6.21
o
ys
o
o
f
f
23.1"
6.01
o
ys
o
o
f
f
f
f
3
1
10184 oit fE (MPa)
214
For the descending portion of the stress-strain curve of confined normal weight concrete,
2
"1
"1
"
1
k
o
oo
k
k
ff
where
o
ys
f
fk
77.21
17.019.22
o
ys
f
fk
E.4 Stress-Strain Model of Fiber Reinforced Normal Weight Concrete
E.4.1 Hsu & Hsu (1994)
The model is suggested for fiber-reinforced high strength normal weight concrete with
plain compressive strength more than 10,000 psi (70 MPa), with volume fraction of hook
end steel fiber ranging from 0 to 1%. The aspect ratio of the fiber used in the study is
reported as 60.
For the ascending and descending portions of stress-strain curve,
nxn
xn
1 for dxx 0
where '
c
c
f
f
o
x
215
ito
c
E
f
'
1
1
for 1
For tailed portion,
addd xxk exp for xxd
where 6.0d
7.0dk
8.0a
CA
fc
3'
501.8717.1 3 fVA
742.226.0 fVC
11 ' Cfa co
22 ' CfaE cit
Table showing the value of a1, C1, a2 and C2
Vf (%) a1 C1 a2 C2 0.5 0.00014 0.00184 43.66 3629.240.75 0.00012 0.00217 35.51 3792.86
1 0.00018 0.00165 33.77 3792.59
216
E.4.2 Mansur et al. (1999)
Mansur et al. (1999) reported the stress-strain model of fiber reinforced high strength
normal weight concrete with plain compressive strength ranging from 70 to 120 MPa,
and with volume fraction of hook-ended steel fibers up to 1.5%. The dimensions of the
fibers are reported as 30 mm in length and 0.5 mm in diameter.
For the ascending portion of stress-strain curve,
o
ooff
1
where
ito
o
E
f
1
1
35.000000072.00005.0 of
o flV
3
1
40010300 ofit fVE (MPa)
For the descending portion of stress-strain curve,
2
11
1
k
o
oo
k
k
ff
217
where
5.20.3
1 5.2150
lV
fk f
o
1.13.1
2 11.0150
lV
fk f
o
Vf = volume fraction of hook-ended steel fiber
l = length of hook-ended steel fiber
= diameter of hook-ended steel fiber
E.5 Stress-Strain Model of Normal Weight Concrete Confined by a
Combination of Lateral Reinforcement and Short Steel Fiber
E.5.1 Hsu & Hsu (1994)
The model is suggested for fiber-reinforced high strength normal weight concrete with
plain compressive strength more than 10,00 psi (70 MPa), volume fraction of hook-end
steel fiber ranging from 0 to 1%, and volumetric ratio of tie reinforcement ranging from
0.38% to 1.15%. The aspect ratio of hook-end steel fibers is reported as 60.
For the ascending and descending portions of stress-strain curve,
cn
c
c
xn
xn
1
where cc
c
f
f'
oc
x
218
ito
c
E
f
'
1
1
for 1
ac k exp
1n
volumetric ratio of confinement
'' , ccc ff = compressive strength of unconfined and confined concrete
oco , = strain at peak stress of unconfined and confined concrete
c , = material parameter of unconfined and confined concrete
fV = fiber volume fraction
Table showing the equations for parameters f’cc and εoc
Vf(%) f'cc, ksi εoc, in./in
0.50 197.95ρ+f'c 0.2252ρ+εo
0.75 186.76ρ+f'c 0.2322ρ+εo
1.00 190.47ρ+f'c 0.2360ρ+εo
Table showing the values of β, k, a
Vf(%) β k a 0.50 5.11 5.70 0.44 0.75 4.71 3.30 0.33 1.00 4.10 1.70 0..2
E.5.2 Mansur et al. (1997)
Mansur et al. (1997) reported the stress-strain model of confined fiber reinforced high
strength normal weight concrete with plain compressive strength ranging from 60 to 120
MPa, with volumetric ratio of lateral tie ranging from 0 to 4.51%, and with volume
219
fraction of hook-ended steel fibers of 1%. The dimensions of the fibers are reported as 30
mm in length and 0.5 mm in diameter.
For the ascending portion of stress-strain curve,
"
""
1o
ooff
where
ito
o
E
f"
"
1
1
2"
2.621
o
ys
o
o
f
f
23.1"
63.111
o
ys
o
o
f
f
f
f
3
1
9704 oit fE (MPa)
For the descending portion of stress-strain curve,
2
"1
"1
"
1
k
o
oo
k
k
ff
220
where 12.033.31
o
ys
f
fk
35.062.12
o
ys
f
fk
221
Appendix F
Technical Papers
The outcomes of the present study have been reported in the following technical papers:
F.1 Journal Papers
(1) Myat M. H., Wee T. H., and Tamilselvan T. “Response of spiral-reinforced lightweight concrete to short-term loading.” Journal of Materials in Civil Engineering, ASCE, Dec 2010, 22, No. 12, 1295-1303.
F.2 Conference and Symposium Papers
(1) Myat M. H. and Wee T. H. Behaviours of spiral reinforced lightweight aggregate concrete columns. Proc., 32nd Conference on Our World in Concrete & Structures, CI-Premier, Singapore, 2007, 335-341.
(2) Myat M. H. and Wee T. H. Properties of fibre reinforced lightweight aggregate
concrete. Recent Advances in Concrete Technology. DEStech Publications, Inc., Washington, D.C., USA, 2007, 679-686.
(3) Myat M. H. and Wee T. H. Controlling the sudden failure of materials under
compression. Proc., 20th KKCNN Symposium on Civil Engineering, Jeju, Korea, 2007, 45-49.
(4) Myat M. H. and Wee T. H. Compressive stress-strain model for confined
lightweight concrete subjected to short-term loading. Proc., CSCE 2009 Annual Conference, Newfoundland, Canada, 2009.
222
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