MECHANICAL PROPERTIES AND FLEXURAL APPLICATIONS OF BASALT
FIBER REINFORCED POLYMER (BFRP) BARS
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Sudeep Adhikari
December, 2009
MECHANICAL PROPERTIES AND FLEXURAL APPLICATIONS OF BASALT
FIBER REINFORCED POLYMER (BFRP) BARS
Sudeep Adhikari
Thesis
Approved: Accepted:
______Advisor Department Chair Dr. Anil Patnaik Dr. Wieslaw K. Binienda
______Committee Member Dean of the College Dr. Craig Menzemer Dr. George K. Haritos
______Committee Member Dean of the Graduate School Dr. Kallol Sett Dr. George R. Newkome
______Date
ii ABSTRACT
The Fiber Reinforced polymer (FRP) composites have always found extensive applications in defense and aerospace industries since the period of its inception.
Nowadays, they are finding their way to more novel applications in relation to civil- engineering structures. They are characterized by various beneficial features such as high strength-to-weight ratio, lower specific weight and excellent corrosion and fatigue resistance. These properties have rendered the application of FRP materials as the internal as well as external reinforcement in civil-engineering structures to be a very fertile field of study.
This thesis presents the results of an experimental and analytical investigation on the mechanical properties and their application as the internal reinforcement of the new type of FRP composite material called Basalt Fiber Reinforced Plastic (BFRP) material. The primary objective of the research was the determination of the important mechanical properties of the BFRP bars and their applicability as internal reinforcement in reinforced concrete beams. For the mechanical properties of the BFRP bars, BFRP bars of three different sizes were considered. The provided bars were of nominal size of
3mm, 5mm and 7mm with the volume fraction of 44%, 52% and 41% respectively. For the study of bond-strength of the BFRP bars, four pull-out cylinder tests were conducted for each size.
iii For the study of the flexural-application, fifteen different beams were studied, two of them being the steel-reinforced beams. The study on the mechanical properties
revealed the tensile-strength, rupture-strain and the longitudinal modulus of elasticity of the BFRP bars. From the pull-out cylinder test, a bond-slip model for BFRP bars was
proposed. From the beam tests, it was observed that the ACI 440 can predict the moment-
strength of BFRP reinforced beams with reasonable accuracy. It was also observed that
the original Branson’s equation provides the lower bound and the ACI 440 provides the
upper-bound for the load-deflection curve for BFRP reinforced beam. A relation for the
effective moment of inertia for BFRP beams was proposed for load-deflection analysis.
iv ACKNOWLEDGEMENTS
My heartiest thanks to Dr. Anil Patnaik, my advisor, for her help and guidance in every step of my research. He has been a constant source of great inspiration for me to do my research in the field of fiber reinforced plastic (FRP) bars. I am also very thankful to my committee members, Dr. Kallol Sett and Dr. Craig Menzemer for their invaluable suggestions and corrections.
Also, I would like to thank my family and friends for their continued love and support.
v TABLE OF CONTENTS
Page
LIST OF TABLES ...... xi
LIST OF FIGURES ...... xiii
CHAPTER
I. INTRODUCTION ...... 1
1.1 Research Significance ...... 2
1.2 Research Objective ...... 3
1.3 Research Methodology ...... 4
1.4 Thesis Outline ...... 5
II. LITERATURE REVIEW FOR THE MECHANICAL PROPERTIES OF BFRP BARS ...... 6
2.1 Basic Introduction and the Applicability of FRPs ...... 7
2.1.1 Structural Issues ...... 7
2.1.2 Alkalinity of Concrete...... 8
2.1.3 Unfavorable Environmental Factors ...... 8
2.1.4 Functional Issues ...... 9
2.2 Micro-Mechanics of FRP Material ...... 16
2.3 Discussion on the Test-Methods for Longitudinal Tensile Properties...... 23
vi III. MATERIALS ...... 36
3.1 BFRP Bars ...... 36
3.2 Anchorage Tubes...... 39
3.3 Grouting Material-Epoxy and Sand ...... 40
IV TEST PROCEDURES ...... 42
4.1Preparation of Wooden Framework ...... 45
4.2 Preparation of Specimens ...... 46
4.3 Experimental Program ...... 51
V RESULTS AND ANALYSIS ...... 54
5.1 Research Objective ...... 55
5.2 3 mm Nominal Diameter BFRP Bars ...... 55
5.3 5 mm Nominal Diameter BFRP Bars ...... 58
5.4 7 mm Nominal Diameter BFRP Bars ...... 60
VI CONCLUSIONS AND RECOMMENDATION ...... 68
VII LITERATURE REVIEW FOR THE BOND-CHARACTERISTICS OF FRPs ...... 71
7.1 Basic Introduction to Bond ...... 71
7.2 Basic Mechanics of Bond-Stress ...... 73
7.3 Bond in the Context of FRP Bars ...... 75
7.4 Difference in Bond-Behavior of Steel and FRPs ...... 76
7.5 Bond-Behavior of Steel...... 78
7.6 Mode of Transfer of Bond in FRPs...... 79
7.7 Constitutive Relation for Bond-Slip Mechanism for FRPs ...... 85
7.7.1 Local Bond-Slip Relationship ...... 87
vii 7.8 Analytical Models of Bond-Slip Relationship ...... 89
7.9 Experimental Methods and Analysis ...... 97
VIII MATERIAL AND MIXES ...... 107
8.1 Materials ...... 107
8.1.1 Basalt FRP Bars ...... 108
8.1.2 Anchorage Tubes ...... 110
8.1.3 PVC conduits, Hard Rubbers and Steel Plates...... 111
8.1.4 Grouting Material-Epoxy and Sand ...... 113
8.1.5 Coarse Aggregate ...... 113
8.1.6 Fine Aggregate ...... 113
8.1.7 Water ...... 114
IX TEST PROCEDURES ...... 115
9.1 Preparation of Specimens ...... 117
9.1.1 Preparation of Wooden Frameworks ...... 117
9.1.2 Casting of Pull-Out Cylinders for Compressive Strength Test ...... 119
9.1.3 Grouting Procedures ...... 123
9.2 Experimental Program ...... 124
9.2.1 Pull-out Test of the Specimens ...... 125
X RESULTS AND ANALYSIS ...... 133
10.1 Research Objective ...... 133
10.2 Bond-slip Analysis of Basalt FRP Bars ...... 134
10.2.1 Basalt FRP Bar with Nominal Diameter 3mm ...... 134
10.2.2 Basalt FRP Bar with Nominal Diameter 5mm ...... 136
viii 10.2.3 Basalt FRP Bar with Nominal Diameter 7mm ...... 138
10.2.4 Bond-slip Modeling of Basalt FRP bars ...... 142
10.3 Stress-Strain Analysis for Basalt FRP Bars from Pull-Out Test Data ...... 146
XI CONCLUSIONS AND RECOMMENDATION ...... 150
XII LITERATURE REVIEW FOR STUDY OF BFRP REINFORCED BEAM ...... 152
12.1 General Introduction to Beam Element ...... 152
12.2 Reinforced Concrete Beams ...... 156
12.3 FRP Reinforced Concrete Beams ...... 161
12.3.1 Stress-Strain Behavior of FRPs ...... 162
12.3.2 Stiffness of the FRP Bar ...... 165
12.4 Deflection Analysis of Reinforced Concrete Beams ...... 168
12.5 Deflection of FRP Reinforced Beams...... 171
12.6 Branson’s Equation in the Context of FRP Beams ...... 176
XIII MATERIALS AND MIXES ...... 189
13.1 Materials ...... 189
13.1.1 Basalt FRP Bars ...... 190
13.1.2 Coarse Aggregate ...... 192
13.1.3 Fine Aggregate ...... 192
13.1.4 Water ...... 192
13.1.5 Steel Stirrups ...... 192
13.1.6 Steel Bars ...... 193
13.1.7 Strain-Gages ...... 193
XIV TEST PROCEDURES ...... 194
ix 14.1 Preparation of Specimen ...... 197
14.1.1 Preparation of Formworks ...... 198
14.1.2 Preparation of Cages and Strain Gages ...... 198
14.1.3 Casting of Beams and Cylinders ...... 204
14.2 Experimental Program ...... 208
XV RESULTS AND ANALYSIS ...... 215
15.1 Research Objective ...... 216
15.1.1 Moment Strength of the BFRP Reinforced Beams ...... 216
15.1.2 ACI 440.0R-06 Method ...... 217
15.1.3 Strain-Compatibility Method ...... 217
15.2 Moment Strength of the BFRP Reinforced Beams ...... 219
15.2.1 Cracking Moment for BFRP Reinforced Beams ...... 226
15.2.2 Moment Strength of the Steel Reinforced Control Beams ...... 230
15.3 Load-Deflection Analysis for the BFRP Reinforced Beams ...... 233
XVI. CRACK-MAP ANALYSIS OF THE BFRP REINFORCED BEAMS ...... 238
16.1 Crack maps for the BFRP Reinforced Beams...... 241
XVII CONCLUSION AND RECOMMENDATION ...... 245
17.1 Moment strength of the BFRP Reinforced Beams ...... 245
17.2 Moment Strength of the Steel Beams ...... 246
17.3 Cracking Moments of the BFRP Reinforced Beams ...... 247
17.4 Load-Deflection Analysis of the BFRP Reinforced Beams ...... 247
REFERENCES ...... 249
APPENDICES ...... 252
x APPENDIX A. LIST OF NOTATIONS ...... 253
APPENDIX B. PLOTS, DATA, AND SAMPLE CALCULATION ...... 255
xi LIST OF TABLES
Table Page
1: Tensile Test Results for 3mm Basalt Bars ...... 56
2: Tensile Test Results for 5mm Basalt Bars ...... 59
3: Tensile Test Results for 7mm Basalt Bars ...... 62
4: Guaranteed Rupture-Strains ...... 68
5: Guaranteed Tensile-Strengths ...... 69
6: Guaranteed Modulus of Elasticity ...... 69
7: Details for the Pull-Out Including the Cylinders for Compressive Strength Test ...... 139
8: Summary of the Test Results from the Pull-Out Test ...... 141
9: Summary of Modulus of Elasticity from Pull-Out Test ...... 149
10: Details of the Reinforcement Ratios for the Beams ...... 197
11: Details of Strain-Gages Used in Beams ...... 202
12: Compressive Strength Details for the Beam-Tests ...... 221
13: Equivalent Beam Properties Based on the Guaranteed Properties of the BFRP Bars ...... 222
14: Equivalent Beam Properties Based on Average Properties of the BFRP Bars ...... 223
15: Moment-Strengths of BFRP Reinforced Beams by Different Methods ...... 224
16: Cracking-Moment Comparison for the BFRP Reinforced Beams ...... 227
17: Cracking-Load Comparison for BFRP Beams ...... 229
xii 18: Moment-Strengths Comparison of the Steel-Beams ...... 232
xiii LIST OF FIGURES
Figure Page
1: Gripping-Systems as per ASTM D 1936 for the Tensile-Test ...... 27
2: Different Grip-Arrangements by Various Researchers ...... 28
3: Typical Basalt FRP bar Used for the Tensile Strength Test ...... 38
4: Steel Tube and Steel Cap for the Anchorage ...... 40
5: Structural Epoxy Used for Grouting ...... 41
6: Schematic for the Tensile-Test Set-up for the Basalt Bar ...... 43
7: Longitudinal Details of the Tensile-Test Specimen ...... 44
8:Cross-Sectional Details of the Tensile-Test Specimen ...... 44
9: Schematic of the Framework Arrangement ...... 45
10: Actual Arrangement of Specimens on the Framework ...... 46
11: Structural Epoxy Used for Grouting ...... 47
12: Arrangement for Mixing the Epoxy...... 48
13: Epoxy and the Sand Mixture ...... 49
14: Arrangement to Pour Epoxy Inside the Tube ...... 50
15: Tubes Filled with Epoxy on One Side ...... 51
16: Arrangement of Tensile-Test Specimen in Universal Testing Machine...... 52
17: Positioning of Strain-Gage on the Specimen ...... 53
xiv 18: Typical Failure mode for 3 mm Basalt Bar...... 56
19: Typical Stress-Strain Curve for 3 mm Basalt Bar ...... 57
20: Typical Failure Mode for 5 mm Bar ...... 58
21: Typical Stress-Strain Curve for 5 mm Basalt Bar ...... 60
22: Tensile Rupture Failure of 7 mm Bar ...... 61
23: Debonding of the Basalt Bar with the Anchors ...... 61
24: Typical Stress-Strain Curve for 7 mm Basalt Bar ...... 63
25: A Specimen with Imperfection as a Dent ...... 65
26: Cross-Section of the Tested Specimens ...... 66
27: Longitudinal Section of the 7mm specimen after Testing ...... 67
28: Local Reduction in Bond Strength at the Anchor Ends ...... 67
29: Typical Basalt FRP Bar Used for the Pull-Out Test ...... 109
30: Anchor for the Pull-Out Test ...... 111
31: PVC Conduits for the Pull-Out Cylinders at Both Edges ...... 112
32: Steel Plate and the Hard Rubber ...... 112
33: Cross-Section and Elevation of the Pull-Out Specimen ...... 115
34: PVC Conduits at the Edges ...... 116
35: Wooden Framework ...... 118
36: Horizontal Arrangement to Hold the Specimens ...... 119
37: Arrangement for Specimen Casting ...... 120
38: Cylinders for Compressive Strength ...... 121
39: Concrete Cylinders ...... 122
40: Cylinders on Horizontal Platform ...... 123
xv 41: Test Set-up for the Pull-Out Test ...... 125
42: Specimen Positioning in UTM ...... 126
43: Arrangement for Dial-gauge Set-Up ...... 127
44: Grip for the Steel Tube ...... 128
45: Specimen Set-up in the Machine ...... 129
46: Pull-Out Failure ...... 130
47: Tensile Failure ...... 130
48: Slip Failure of the Specimen ...... 131
49: Amount of Slip in One Specimen ...... 131
50: All of the Tested Pull-Out Specimens ...... 132
51: Two Specimens with Different Failure Modes ...... 132
52: Specimen C6 after Tension Failure ...... 135
53: Specimen with Same Size but Different Failure Modes ...... 137
54: Specimen C7 at the Start of the Loading ...... 138
55: Specimen C9 with Considerable Slip during Loading ...... 140
56: Specimen C4 (5mm) Load-Slip Curves ...... 144
57: specimen C3 (5mm) Load-Slip curves ...... 144
58: Specimen C11 (7mm) Load-Slip Curves ...... 145
59: Specimen C10 (7mm) Load-Slip Curve ...... 145
60: Specimen C9 (7mm) Load-Slip Curves ...... 146
61: Typical Stress-Strain Curve for 7mm BFRP Bar from Pull-Out Test ...... 147
62: Typical Stress-Strain Curve for 5mm BFRP bar from Pull-Out Test ...... 148
63: Typical Stress-Strain Curve for 3mm BFRP bar from Pull-Out Test ...... 148
xvi 64: Comparison of the Member and Bare Response ...... 173
65: Effect of Reinforcement Ratio on the Deflection ...... 179
66: BFRP Bars Used in Beams ...... 191
67: Steel Stirrups Used for Shear Reinforcement ...... 193
68:Cross-Section Details of the Beams...... 195
69: Reinforcement Cages ...... 199
70: Formworks and Preparation of Cages ...... 200
71: Students Learning to Fix the Strain Gages on the Basalt bar ...... 203
72: Students working on the Strain-Gages...... 203
73: Finishing of the Top Surface of the Beams ...... 205
74: Concrete Works for the Beams ...... 206
75: Application of Strain-Gage on the Concrete Surface ...... 207
76: Clamping the Strain-Gage on the Surface of the Beam ...... 208
77: Schematic of the Test Set-up for the Beam Test ...... 209
78: Actual Arrangement for Four-Point Bending ...... 210
79: Arrangement at Supports ...... 211
80: Position of Beam in UT Machine and MTS Machine ...... 212
81: Arrangement for the Load-Application ...... 213
82: LVDT Positioning for the Deflection Data ...... 214
83: Moment-Strength Comparison for BFRP Beams ...... 225
84: Ultimate Load Comparison for BFRP Reinforced Beams...... 226
85: Cracking-Moment Comparison for the BFRP Reinforced Beams ...... 228
86: Graphical Comparison of Cracking-Load for BFRP Beams ...... 229
xvii 87: Stress-Strain Curve for No-3 Bar...... 231
88: Stress-Strain Curve for No-4 Steel Bar...... 232
89: Graphical-Comparison of the Moment-Strengths of the Steel Beams ...... 233
90: Typical Load-Deflection Curves for BFRP Reinforced Beam ...... 236
91: Typical Load-Deflection Curves for BFRP Reinforced Beam ...... 237
92: Crack-Map for BFRP Reinforced Beams B-1 and B-2 ...... 241
93: Crack-Map for BFRP Reinforced Beams B-3 and B-4 ...... 242
94: Crack-Map for BFRP Reinforced Beams B-5 and B-6 ...... 242
95: Crack-Map for BFRP Reinforced Beams B-7, B-8 and B-9 ...... 243
96: Crack-Map for BFRP Reinforced Beams 2B-1and 2B-2 ...... 243
97: Crack-Map for BFRP Reinforced Beams 2B-3 and 2B-4 ...... 244
98: Crack-Map for BFRP Reinforced Beams 2B-5 and 2B-6 ...... 244
xviii CHAPTER I
INTRODUCTION
The last 100 years had been the most eventful epoch in the human history in regard to the major and groundbreaking paradigm-shifts in the world-views, technology and material-science during the period. If the development of quantum mechanics and relativity can be regarded as the major shift in our material world-view, it won’t be an over exaggeration if we consider the introduction of composite materials to be the paradigm-shift in the field of materials. The past 100 years represents the evolution of
FRP composite materials in the similar way iron and then steel characterized 19th century
(L.Hollaway, 1993). Apart from their multifarious application, particularly in the performance-based sectors such as defense and aerospace, FRP composite materials are gradually finding their way to establish themselves to be a very good alternative material for the civil-engineering structural applications. Over the last 20 years the FRP composite materials have developed into economically and structurally viable construction material for buildings and bridges (Bank, 2006). They have proved themselves to be a very ingenious substitution to the conventional metallic materials even in civil engineering application, where both strength and stiffness play pivotal role. FRP materials are characterized by low specific weight, higher strength and electromagnetic transparency.
1 These features have made the FRP composite material more attractive for the field of civil-engineering, owing to their innate capability to meet more functional requirements along with the stringent structural demands. The simplicity associated with the application of FRP composite renders them very attractive in civil-engineering application, especially when dead-weight, space or time constraints are existent
(V.M.Karbhari, 2007).
1.1 Research Significance
Many structures and bridges in the United States as in the world are facing severe problem of deterioration leading to functional inadequacy due to the deleterious environmental factors, increased load, ageing and corrosion. Reinforced concrete bridges, especially in the USA, have suffered from chronic corrosion through deicing. From the span of 1987 to 1993, 200,000 tones of epoxy coated steel rebars was used in bridge decks but significant percent of it has degraded (L.Hollaway, 1993). This fact gives us some indication of the big financial burden associated with the replacement of the existing structures with the new materials and also at the same time, justifies the dire necessity of the introduction of alternative materials in the civil-engineering structural application. These alternative materials are also sought to provide the novel functional benefits.
This situation has necessitated the comprehensive and extensive research on the properties and the applicability as the structural materials of such kind of materials. The identification of the pertinent mechanical properties and the test of their applicability 2 thus, constitute the major objective of the research which further signifies its significance.
Another significance of this research is the goal to develop the sufficient background for the further study of long-term behavior of the particular material. The study of the fundamental properties of any new material can be regarded as the prerequisite for any detailed study of that material. The properties identified from the short-term experiments might influence the design of long-term experiments (related with long-term behavior such as creep and fatigue), thereby reducing the overall cost of any specific study and further modification (C.M.R.Dunn, 1991).
1.2 Research Objective
The primary objective of the research is the identification of the mechanical properties of Basalt Fiber Reinforced Plastic (BFRP) bars of three different sizes and the study of their application as the internal reinforcement in the beam element. The identification of the mechanical properties thus ensues the determination of the guaranteed tensile strength, guaranteed rupture strain and guaranteed modulus of elasticity (longitudinal) of the BFRP bars of three different sizes. It also included the study of the bond-behavior of the BFRP bar with the concrete-matrix. The bond-test was aimed to determine the relative bond-strengths of the different sizes of BFRP bars and to check the adequacy of the provided surface deformation to develop the required bond- stress. From the Load-slip data, it was aimed to develop a constitutive bond-slip relationship for the BFRP bar and concrete.
3 Another important objective was the study of the applicability of the BFRP bars as the internal reinforcement in the reinforced concrete beam structures. The research was undertaken to identify the sufficiency of the current standards to predict the various structural properties. Since BFRP bars are new materials, it is very imperative to know the adequacy and limitation of the current standards in regard to this new kind of material. One of the objectives of the beam-test is also the study of the load-deflection behavior of the BFRP reinforced beams. A secondary objective of this research was the
determination of various properties of BFRP bars such that they can be retrieved for the
study of long-term behavior and other durability related properties of the BFRP bars.
1.3. Research Methodology
This study includes the determination of the mechanical properties of BFRP
bars and their application in reinforced concrete beam. For the determination of tensile-
strength and modulus of elasticity, the relevant test-method conforming to ASTM standards was followed. For the study of bond behavior, a test method was finalized based on the extensive literature review of the similar kind of researches and tests. For the determination of bond-slip constitutive model, various analytical models were studied first. The load-slip data was analyzed and on the basis of its conformation to the various models, a constitutive bond-slip model was proposed. All the BFRP reinforced beams were tested under four-point bending and the moment-strengths were predicted based on
ACI rectangular stress-block model and the strain-compatibility method using parabolic
stress-strain curve for concrete.
4 For the load deflection analysis of BFRP reinforced beams, three different relations for the effective moment of inertia for the cracked section were considered.
Three analytical load-deflection curves were plotted and compared to the experimental curve. From the relative comparison of the three different methods and the actual load deflection curves, a modified relation for the effective moment of inertia for the cracked
BFRP section was proposed.
1.4. Thesis Outline
The thesis can be viewed to be comprised of three different major sections. Each section is organized as a sub-thesis in itself. Tensile characterization of the BFRP bars which constitute the first section is divided into five chapters which includes the literature-review for the particular subject, materials and mixes, test-procedures, results and analysis and finally the conclusion and recommendations. Study of the bond- behavior of the BFRP bars constitutes the second section and is divided into five different sections which include literature-review followed by materials, experimental program, results and analysis and conclusion and recommendations. Study of the flexural behavior of the BFRP reinforced beams which constitutes the third section is divided into six chapters which includes literature review followed by materials , experimental program , results and analysis and conclusions and recommendations.
5
CHAPTER II
LITERATURE REVIEW FOR THE MECHANICAL PROPERTIES OF BFRP BARS
This section will provide the comprehensive literature review done in relation to the determination of mechanical properties of Basalt FRP bars. Considering the nature of materials being used for the manufacture of the FRP reinforcing bars, basalt can be regarded to be an entirely new type of material. This requires the detailed study of the past researches and investigations done on the prevalent FRP materials such as carbon, glass and aramid. Since the structural behavior of all the FRP materials are expected to be similar in their nature, the study of the researches done on the available materials can be very insightful for the understanding of the structural behavior of the material considered for our investigation. This is also intended to provide us the in formations regarding the test-methods and the analysis procedures that may be required for the subsequent applications of the determined properties.
6 2.1. Basic Introduction and the Applicability of FRPs
We all know that Reinforced concrete has established itself as a dominant construction methodology in the modern day construction industry for a long time due to some very good elastic properties. The high modulus of elasticity renders steel very good
ductility which subsequently enhances its post-yielding behavior to a significant degree.
The ability of steel to work in both elastic and plastic regime allows the steel to
work efficiently even after undergoing yielding and hence facilitates the redistribution of
load-effects such as moments. However, the method of using the reinforced-concrete as a
composite material with steel embedded in the concrete-matrix can generate multifarious issues which can’t be underestimated from the structural and functional view-point. They can be addressed under the following topics:
2.1.1. Structural Issues
There are various issues related with the application of reinforced-concrete owing to their propensity to generate various issues when used in civil engineering structures. They have been addressed below.
7 2.1.2 Alkalinity of Concrete
Concrete is a heterogeneous mixture of aggregate, cement and water. Aggregate is generally the chemically inert component of the concrete. Cement is a siliceous compound which primarily consists of 50% tricalcium silicate, 25% dicalcium silicate,
10% tricalcium aluminate, 10% tetra calcium aluminoeferrite and 5% gypsum or hydrated calcium sulphate. When water is added, the cement-components are hydrated and result in various chemical compounds which finally define the characteristics of the resulting concrete such as strength and setting time. Concrete can quickly attains a PH of about 12.4 or 12.5 due to the development of a saturated solution of calcium hydroxide such that the PH in concrete can vary from 15 to 12.5 (Concrete Const., 2004).
One of the most expensive and deleterious effect of the inherent alkalinity of the concrete is due to the chemical stripping of the chlorine compound from an intermixed latex due to the high PH of concrete, resulting in a process called Dehydrohalogination.
This severely contributes for the corrosion of the steel (Concrete Const, 2004). Corrosion of steel embedded in concrete by chloride ions has been identified as one of the major reason for the gradual deterioration of the concrete (B.Benmokrane, 1996).
2.1.3 Unfavorable Environmental Factors
Various structures such as marine structures, bridges, tunnels, sea walls, coastal structures, chemical and wastewater treatment plants, parking garages and retaining walls are subjected to highly aggressive and hostile environments. A combination of these
8 environmental factors can initiate corrosion of steel. For the last few decades, the
premature degradation of the structures due to the corrosion of steel has been a major
concern (B.Tighiouart, 1999). The untimely deterioration of reinforcement steel due to
corrosion has been observed in various instances. The corrosion of steel results in its
increase in volume which causes excessive tensile stress in the concrete thus resulting in
spalling (B.W.Wambeke, 2006). This clearly gives us some idea of the devastating
influence that this factor can engender on the whole structure. The corrosion of steel can
result in the deterioration of concrete and ultimately the loss of serviceability of the
structure (ACI 440.1R-06).
2.1.4. Functional Issues
Regardless of the structural issues, due to its huge area of influence, civil-
engineering is bound to seek some new construction materials pertinent to different
functional criteria. It is worthwhile and very relevant to quote M.Mashima and
K.Iwamoto on this topic.
“there is a great interest for the civil-engineers to apply new material for practical construction works. For example, continuous fiber reinforcement is manufactured by binding a highly strong fiber within resin. This reinforcement, a FRP rod could substitute rebar and or PC tendon.”
The above excerpt greatly clarifies the need for the new kinds of civil-
engineering materials in relation to various functional needs and requirements. FRP bars
as a substitute for the steel as a reinforcement material in the reinforced concrete
9 structures seem to perfectly serve the purpose. FRP bars can provide the following
distinct advantages over the conventional steel reinforcement:
2.1.4.1 Higher Strength/Stiffness to Weight Ratio
Owing to their low specific gravity, the FRP materials possess a considerable
higher strength-to-weight and modulus-to-weight ratios in compared to metallic
materials. For instance, the specific gravity of most types of steel is nearly equal to 8. On
the other hand, most of the FRP materials have specific gravity values less than two.
Along with the higher strength of the FRP material, this property can be very useful and advantageous depending on the nature of the structure. This feature of the FRP material
can serve both structural and functional issues pertinent to the particular structure.
2.1.4.2 Electromagnetic Transparency
This property can be particularly advantageous in some specific cases and can be identified as the functional merit of the FRP material in comparison to the conventional metallic reinforcement. In the cases of hospitals and research labs, where electromagnetic transparency is required to allow the smooth functioning of various electronic devices, FRP reinforcement can serve as a very right alternative.
10 2.1.4.3 Higher Corrosion Resistance
This property of FRP materials can serve both the structural and functional merit to the respective structure. The higher corrosion resistance of the FRP material can provide the structural engineer with the flexibility to improvise the various structural responses and to finalize the optimum arrangement. From ACI 2008, it can be seen that
the crack-width allowance for the FRP reinforced structures is higher compared to the
corresponding allowance for the steel-reinforced structures. This can be evidently
attributed to the better corrosion resistance of FRP materials. This will further provide the
designer more room to limit the deflection criteria for the structure. Thus, it can be
inferred that this merit of the FRP material can act synergistically with some of the
structural response of the structure which might allow the structural engineer to come up
with better and reasonable design combinations.
2.1.4.4 Higher Fatigue Resistance
Most of FRP materials are characterized by the higher resistance against the
fatigue loading in compare to their metallic counterparts. As already mentioned, FRP
materials are anisotropic materials such that the structural properties varies with the
direction. The higher fatigue resistance may be due to the inherent inhomogeniety and
anisotropy in the microstructure of the FRP material. The inhomogeniety in the
microstructure can provide mechanisms for high energy absorption on microscopic scale
(M.M.Schwartz, 1997), thus, possibly, increasing the fatigue resistance of the FRP
11 materials. This feature can be sufficiently exploited as a very good structural novelty
associated with the FRP materials.
2.1.4.5 Tailorable Properties
The anisotropic nature of the FRP material can provide a very good opportunity
to tailor its various physical, mechanical and structural properties as per the
circumstantial requirement of the structure. For instance, the fibers can be selectively
provided in the direction of higher stresses or provided to increase the stiffness in the
desired direction. The fibers can be selectively pretreated to produce the situation of zero
coefficient of thermal expansion. (M.M.Schwartz, 1997). This characteristic of FRP
material can be strikingly significant to meet the various structural and functional
requirements under stringent conditions.
2.1.4.6 Inherent Damping
The FRP materials are characterized by very good inherent damping property.
As already been mentioned, due to the anisotropy and inhomogeniety in their
microstructure, FRP materials can develop a mechanism for high energy absorption on
microscopic scale similar to the yielding phenomenon exhibited by the metallic materials
capable of undergoing plastic deformation. This can result in better absorption of vibrational energy and hence the reduction in the noise level and the vibration transferred to the adjacent structures (M.M.Schwartz, 1997).
12 2.1.4.7 Lower Coefficient of Thermal Expansion
The FRP materials generally have lower coefficient of thermal expansion
compared to the metallic reinforcing materials. The coefficient of thermal expansion can be generally attributed to the change in the intermolecular bond between the atoms. The coefficient of thermal expansion of a FRP material is dominated by fiber in the longitudinal direction and by the polymer matrix in transverse direction. The molecular organization of the fiber which is generally the combination of various compounds and the polymer matrix which is generally a polymer consisting of a combination of large molecules, renders them more molecular stability compared to the isotropic metallic
materials, thus resulting in lower coefficient of thermal expansion. Hence, the FRP
materials possess more dimensional stability over a wide range of temperature. This property can render FRP materials more functional advantages over their metallic counterparts.
In spite of all these very good structural and functional superiority over the traditional metallic materials, FRP materials have their own deficiencies that are worth discussing here. One of the major demerits is the very high cost of the raw materials required and the cost associated with the fabrication. Owing to their antistrophic microstructure, they possess a directional dependency in their structural properties. The
longitudinal properties are dominated by the fiber whereas the transverse properties are
dominated by the polymer matrix. For example, the strength of the carbon fiber
perpendicular to the fiber axis is 10 times less than the strength parallel to the
13 longitudinal axis. (M.M.Schwartz, 1997). This can be attributed to the weakness associated with the matrix.
Another problem associated with the FRP material is the possibility of the environmental degradation of the polymer matrix. Various polymer matrix composites are susceptible to moisture absorption from the ambience which may result in the dimensional variation which eventually can give rise to the generation of adverse internal stress. One of the environmental factors affecting the performance of FRP material is their susceptibility to Ultra-Violet radiation exposure. This may results in the scission or alteration of polymer molecules. There may be loss in the bond strength in the fiber- matrix interface resulting from the ingression of fluids into the material. From the past studies, it is known that the moisture may result in the acceleration in the static fatigue in glass fibers. Similarly, Kevlar 49 fibers exhibit reduction in their tensile strength and modulus in response to the moisture absorption (M.M.Schwartz, 1997).
Another issue of major concern is when FRP materials are to be used in the condition of elevated temperatures. The longitudinal strength and modulus of an unidirectional FRP material is unaffected by the increase in the temperature for all practical purpose. However, the transverse and off-axis properties are significantly reduced as the temperatures approach the so-called glass-transition temperature of the polymer matrix. Glass-Transition-Temperature can be defined as the temperature after which the polymer changes from a glasslike state to a state resembling rubber. This can be regarded as the applicable temperature range for all practical purposes. Once this limit is crossed, there can be observed significant drop in the strength and the modulus of the material due to thermal softening. The glass-transition temperature for the polymer
14 matrix in FRP material is relatively low while those of organic fibers such as carbon and
aramid are high. The resulting change in the stiffness properties after crossing this limit may be important as the serviceability design criteria seeks to impose a limitation on the response like deflection of the structure ( L.C.Bank, 1993).
The inherent inhomogeniety and anisotropy of the FRP material also render the
analysis of the structure more difficult and demanding. Anistrophy can be identified as
the most distinguishing property of the FRP materials, hence more attention is demanded
to control this property and its effects on analytical and design methodologies
(L.Hollaway, 1993). This can be regarded as the major demerit associated with the use of
the FRP material in civil-engineering structural purposes. However, from the above
discussion, it is evident that the within the framework of FRP material, owing to the
direct control over the material properties ,the material properties can be modified so as
to conform to the stringent structural requirements. So we can be optimistic of the fact
that with the refinement of the production methods and analysis procedures, it is possible
to obtain a FRP material with the optimum properties conforming to the particular
structural requirement. On the same token, the wide applicability of the FRP material to
meet the particular functional requirement can never be underestimated. For the high-
performance characteristics such as strength and stiffness, FRP materials will always be
the last resort.
15 2.2. Micro-Mechanics of FRP Material
A structural composite can be defined as a heterogeneous material system which
consists of two or more constituents on a macroscopic scale, which when combined, can
yield new emergent properties that are superior to the constituent elements. Thus a fiber reinforced composite polymer consists of high strength and stiffness fibers embedded in a
polymer matrix with relatively less strength and stiffness separated by distinct interfaces
between them. Generally, the structural composite is a two-phase system where the
constituent fiber and the matrix retain their physical and chemical integrity, but at the
same time, enables us to extract some properties which cannot be achieved by the
application of the individual components alone (M.M.Schwartz,1997). The reinforcement
of low stiffness polymer matrix with high strength stiffness fiber uses the plastic flow
under stress of the polymer for the transfer of load to the fiber (L.Hollaway, 1993).
Generally, the fiber is the principal load carrying component whereas the polymer matrix is liable to provide the composite with the desired orientation, to facilitate the load transfer to the fiber and to provide protection to the fiber against the environmental factors associated with elevated temperature and humidity conditions.
The interface between the fiber and the polymer matrix is an anisotropic region characterized by gradation of properties. This is very important in the sense that the transfer of load from matrix to the fiber is critically dependent on the performance of this region (L.Hollaway, 1993). The analytical procedures also presume a perfect bond between the fiber and the matrix such that there exists a perfect strain-compatibility in this region. This region is also very important for the determination of the fracture
16 toughness property and the composite-resistance to aqueous and corrosive environment.
Composite which has weak interface region has lower strength but higher resistance to fracture and the composites with strong interface region has high strength but renders the material more brittle.
The following assumptions are made for the analysis of composite materials.
These assumptions enable the formulation of the mathematical models that are able to predict the properties of the composite material analytically: a. Matrix and fiber are perfectly elastic materials. b. There is a perfect bond between the matrix and the fiber such that there is perfect strain- compatibility at the interface. c. The material closer to the fiber has the same properties as the properties of the material in general d. The fibers are arranged in a regular and repetitive manner
The composite material can be called as monolithic (or single phase), biphase
(two-phase) or multi-phase. The most important physical attributes which differentiates a composite material from the conventional materials is heterogeneity and anisotropy
(L.C.Bank, 1993). Heterogeneity signifies that the composite is constituted by the material with varying properties while anisotropy refers to the directional dependence of the properties of the respective composite. Heterogeneity is a relative concept since it is scale-variant. A material which is homogenous in macroscopic scale may not be homogenous in microscopic scale. Anisotropy is a characteristic which becomes significant while it is considered under the continuum-mechanics concept. If the observable mechanical properties of the given material are variable depending on the
17 direction, then the material response can be considered to be anisotropic. An isotropic
material has infinite planes of symmetry where as an anisotropic material has no plane of
symmetry. Orthtropic materials, which can related to the composite materials is a special case of anisotropy constituted by three mutually perpendicular plane of symmetry.
In the case of isotropic material subjected to pure axial load, there will only be axial strain in the longitudinal and transverse direction without any shear strain. Similarly when the isotropic material is subjected to pure shear stress, there will be pure shear strain without any accompanying normal strain. This signifies that, in the case of
isotropic material the normal and shear stress are independent or uncoupled. Whereas in
the case of general anisotropic material, if they are subjected to pure normal stress or pure
shear stress in any direction, the normal strain and the shearing strain will be coupled.
This mode of response exhibited by the general anisotropic material is termed as shear
coupling effect and is characterized by the factor called shear coupling coefficient. This
clearly depicts the complexity in the material response with the increase in anisotropy
and the necessity of more physical constants to address the complete response of the
particular material (I.M.Daniel, 2006).
The unidirectional FRP rods, which are more pertinent to current study, have
radial symmetry. The unidirectional nature of the reinforcing FRP bars renders them with
so-called transverse isotropy. An orthotropic material can be considered to be
transversely isotropic when one of its principle planes is the plane of symmetry. The
unidirectional FRP material which has generally high fiber volume ratio with fibers
packed in a hexagonal array can be considered to be transversely isotropic (I.M.Daniel,
2006.) (Composites with relatively low fiber volume fraction results in random
18 distribution of the fibers whereas in the case of unidirectional FRP which generally has
higher fiber volume fraction exhibits its inclination to hexagonal packing).
This greatly helps to simplify the analysis of the macromechanical response of
the material. For instance, a general isotropic material requires 21 independent stiffness
or compliance constants to be described completely. When we move to the case of
orthotropic materials, which is a special case of anisotropy, only 9 independent stiffness
constants are needed to specify the material completely. Similarly, in the case of
transversely isotropic materials, the number of independent stiffness constants reduces to
5.The above discussion summarizes the macromechanical formulation of the properties of the FRP material. However, there is also an absolute necessity to obtain the elastic properties of the composite under consideration in terms of the properties of its constituents. This purpose is sufficiently served by micromechanical approach.
The objective of micromechanical approach is to formulate the properties of the
composite in terms of the properties of its constituents. The micromechanical approach
includes mechanics of material approach, numerical approach, variational approach,
semi-empirical approach and experimental approach. It is observed that the mechanics of
materials approach can yield reasonable prediction for longitudinal properties even
though it underestimates the transverse properties of the composite. This necessitates the
analysis of the transverse properties on the basis of approaches like semi-empirical
approach. Halpin-Tsai model, which is a semi-empirical approach, is a very effective
model which provides very good bounds for the composite properties.
The mechanics of material model, considers two models addressing the two
different extremes of the fiber orientation in the matrix. The parallel model is called
19 Voigt model and the series model is called Reuss model. The parallel model considers the parallel orientation of the fibers in the direction of force thus signifying constant strain state. The series model considers the orientation of fibers in direction transverse to the direction of the applied load thus signifying constant stress state. Since the completely parallel orientation of fibers is an analytical idealization, these two approaches can provide us with two extremes or bounds of the respective property.
For the constant strain state or parallel model, the stiffness of the composite is given as
(1)
Where Cc , Cf , Cm = composite, fiber and matrix stiffness respectively.
Similarly, for the constant stress or series model, the stiffness of the composite can be
shown as
(2)
Since in the actual case, the fiber orientation may give rise to no uniform stress and strain
condition, the above two values can be considered to be the two bounds for the value of
stiffness,
(3)
20 The above bounds can be improved by the methods based on energy principles.
By the theorem of least work it can be shown that the stiffness property of the composite
obtained by the condition of constant state of stress (series model) provides the lower bound. Similarly, from the theorem of minimum potential energy it can be shown that the
stiffness obtained from the parallel model is provides the upper bound. Semi-empirical models provided by Halpin-Tsai gives a very good approximation between the lower and
the upper bounds. As per the Halpin-Tsai model, the composite property can be
expressed as
(4)
where (5)
Where ζ = parameter which indicates the reinforcing efficiency
Pf, Pm = fiber and matrix properties respectively
This entails the following generalized relationship for the composite property,
(6)
For the above relation yields the mechanics based method for parallel model
(7)
21 For the above relation yields the mechanics based method for series model,
(8)
The value of ζ can be determined by the following relation if a composite property for a
given fiber is determined experimentally.
(9)
This leads us to the conclusion that the mechanics of materials approach when
used in conjunction with other approaches such as variational and semi-empirical approach as described above, can provide us with the very good insight into the functional correlation between the fiber, matrix and composite properties. The rule of mixture as based on the mechanics of materials approach can yield very good prediction of the material properties like longitudinal modulus of elasticity and longitudinal
Poisson’s ratio. However, they can be significantly improved by the application of semi-
empirical models and variational approach. The variational approach described by Hashin
can provide a more refined approximation of the longitudinal properties (I.M.Daniel,
2006).
The mechanics of material approach underestimates the transverse modulus,
modeling the composite material system as a series model. The Halpin-Tsai model can
provide a better prediction for this case.
22 2.3 Discussion on the Test-Methods for Longitudinal Tensile Properties
The determination of the various physical, mechanical and structural properties of the FRP bar is very important for formalizing the design guidelines and methodologies pertaining to the design of structures employing FRP reinforcement. Physical properties comprise those properties of the given material system which can be attributed to their fundamental microstructure (atoms, molecules). This insinuates the fact that the physical properties of a material system can be derived based on their atomic or molecular structure. The chemical properties, mass properties, geometric properties, thermal properties and transport properties can be categorized as coming under the heading of physical properties (L.C.Bank, 1993). Some of the important physical properties related with the FRP material are volume fractions of constituent elements, specific gravity, coefficient of thermal expansion, glass-transition temperature, electrical conductivity and moisture diffusivity.
Mechanical properties can be defined as the properties of the material system which are exhibited when the material is subjected to certain type of mechanical force.
They are generally dealt in a macroscopic scale, in contrast to the physical properties which are dealt in microscopic scale. They can be better classified based on the continuum mechanics concept which addresses the properties which can be derived from the minimum volume element of the material which is independent of the statistical fluctuations and is capable of yielding average stress and strain fields (L.C.Bank, 1993).
The important mechanical properties of an unidirectional FRP material include the longitudinal tensile strength of the material, modulus of elasticity and the associated
23 elastic constants. P.F.Castro and Carino (1998) has identified tensile strength, modulus of elasticity, development length , in-plane and transverse shear strength as the principal mechanical properties that are required to be specified for the systematic utilization of the
FRP materials (P.F.Castro,1998). ACI has provided the following statistical definition of the important mechanical properties for the FRP material for the systematic design procedures.
Design Tensile strength (10)
Where CE is the environmental reduction factor for various fiber types and various
* exposure conditions as provided by the ACI guidelines. Here f fu is the guaranteed tensile
strength and is given by the relation
Guaranteed tensile strength (11)
Where fu,avg is the mean tensile strength of the sample population and σ is the standard
deviation. The design rupture strain is also defined in a similar manner to the design
tensile strength (i.e. average minus the thrice of standard deviation). The design modulus
of elasticity is taken as the average of the values of modulus of elasticity obtained from
the given test sample population.
The structural properties of the FRP materials include the properties that are
exhibited by the materials when they are used in conjunction with other materials to serve
a particular structural purpose. Hence it includes the functional association of various 24 parameters and their subsequent interactions. Bond strength, flexural stiffness, fatigue and impact properties can be classified as the structural properties. Even though all these
various types of properties seem independent of each other, they are, as a matter of fact,
intrinsically related to each other. Unless the physical properties are known, the
mechanical properties can’t be predicted and unless the mechanical properties are known,
the structural properties can’t be determined.
This inarguably asserts the importance of the determination of the fundamental mechanical properties of the given FRP material to serve the various structural requirements. As already been discussed, for the application of FRP material in structural application, longitudinal tensile strength is one of the most important parameters. Tests used for the determination of longitudinal tensile strength further establishes the stress- strain response of the particular material, modulus of elasticity and the rupture strain, which are the governing design parameters. Since from the micromechanics of the unidirectional FRP material, it can be observed that the transverse stiffness and strength of a FRP material is governed by the matrix properties, it evidently follows that their strength and stiffness in the transverse direction is relatively low in comparison to their longitudinal strength. This imposes a technical limitation on the traditional tensile-test methods using traditional equipments like universal testing method. If the tensile test is performed on the FRP bars in the conventional wedge-shaped frictional gripping system, the high compressive stress in the transverse direction generated by the grips can give rise to local stress-concentration effect ultimately leading to the crushing of the material and the subsequent premature failure (P.F.Castro, 1998). It is generally found that the
25 material fails in the grip itself by the combined effect of shear and crushing accompanied
by tensile stress (S.S.Faza, 1993).
There has been an extensive research on the development of usable ideal gripping system for the testing of the FRP material. The ideal gripping system should be such that it prohibits the premature failure of the material at the grip-zone and it facilitates the determination of the actual tensile strength of the material without interfering the actual material properties. The ideal gripping system should also be easy to use in the laboratory and in the field for pre and post-tensioning application (S.S.Faza,
1993). From the extensive literature review performed by Faza and GangaRao (1993), they concluded that a well-established reusable gripping mechanism is not available.
One of the first gripping systems for the tensile test of the FRP material was developed in West Virginia University. It primarily comprises two steel plates with semicircular groves cut in it to facilitate the accommodation of the FRP bar. The diameter of the grips is increased by 1/8 of an inch to allow for the discrepancy in the nominal diameter of the bar to be tested. Fine wet sand epoxy-sand is used to fill the grooves. Six high strength bolts are used to tie the two plates.
ASTM D 3916 also provides a similar kind of provision for the gripping systems for tensile test of the FRP material. The specification advocates a reusable aluminum tab grip which has a configuration similar to that proposed by Faza and
GangaRao (1993). The tab consists of the semicircular grooves to facilitate the accommodation of the FRP bars to be tested. The two plates are placed around the FRP bar to be tested and clamped with the suitable mechanism with the testing machine. The
26 lateral surface of the semi-circular grooves is sand-blasted with 150-µm Carbide grit to
increase the friction between the grip and the bar.
The related physical dimensions are provided for the particular L/d ratio of the
bar to be tested, where L is the free-length and d is the nominal diameter of the FRP bar.
The schematic of the set up is shown below.
Figure 1: Gripping Arrangement as per ASTM D 1936 for the Tensile-Test
The primary demerit associated with these methods of testing is their incompatibility with the FRP bars with rough surface. Due to the non-uniformity of the production technology and the intrinsic anisotropy of the material itself, there is always a possibility of FRP bars with not perfectly smooth and round surface. This poses a serious question on the
27 universal applicability of these grips for the FRP bars of any surface-geometry. The
various gripping systems developed originally for the anchorage of the end of FRP
prestressing tendons has been modified by many researchers to make them usable for the
tensile test of the FRP bars (P.F.Castro, 1998). The tensile test gripping systems as developed by various researchers are shown in Fig-2.
Figure 2: Different Grip-Arrangements by Various Researchers
The arrangement 2a was used by Bakis for doing the tensile strength tests for the FRP material. It consists of a conical arrangement with a rubber washer glued to the smaller end. The tensile test was then carried out in a manner resembling the pull-out test. Holte modified the previous arrangement by providing the gripping cone with the parabolic profile. Cones with parabolic profile were found to yield better strength results. Fig-2a is the arrangement used by Erki and Rizkalla (1993). It consists of externally screwed metal
28 tube with nut screwed onto its ends. Rahman (1993) used the arrangement as shown in fig
2c as shown above. All these systems are based on the fact that the tensile stress is
transferred from the tube to the FRP rod by the development of shearing stresses in the
bonding medium. All these set up resembles the pull-out test arrangement in its principle.
The demerit associated with this method is the pull-out of the FRP bar if the embedment length is not sufficient to develop the required shear strength in the bonding medium.
P.F.Castro and Carino (1998) conducted a tensile test on FRP bar using an arrangement similar to the arrangement as described above. It consists of a metallic tube where a FRP bar is embedded with the epoxy matrix to provide the shearing stresses. The tubes rest on the plates provided on the machine crosshead and are loaded in tension. The test was first conducted with aluminum tubes when buckling of the tube was observed.
The tube was replaced with steel tubes and the test was successful. However, possibility of buckling of the tubes when loaded is always a detrimental issue. They came up with different arrangement to grip the tube so as to prevent the possibility of buckling. This method comprises embedding both ends of the FRP rod in the tubes and filling it with some strong bonding agents.
This method can now be considered as the most extensively accepted test method for the tensile test of the FRP material. In the tensile test carried out by
B.Benmokrane in (2000) for AFRP and CFRP bars, the similar method is employed. This test consists of 600-mm long potted anchor system at each end. The potted anchor system comprises 600 mm long steel cylinders filled with high performance resin grout for better bonding. Sand can be proportionately mixed with the epoxy resin to increase the bond characteristics. The incompressible nature of sand, when used with epoxy grout, can give
29 rise to dilatancy of cement grout at failure thus increasing the frictional bond resistance
(B.Benmokrane, 2000). The experiment conducted by S.Kocaoz (2004) for the tensile
characterization of the glass FRP bars consists of the the anchorage system comprising
steel pipes filled with expansive cementitious grout for the bonding. For the improvement
of the bond behavior of FRP bars with the grouting, S.Kocaoz (2004) provided threads at
a specific pitch at particular length of the FRP specimens.
One of the issues to be considered in tensile test with the arrangement of two
steel anchorages at its ends is the ratio of free length to the nominal diameter of the bar to
be tested. The free length is the length of the specimen which is not anchored and where
the tensile failure of the specimen is desired. The ACI 4403R-04 (guide test method for
the longitudinal tensile properties of FRP bars) specifies the free length (which is called as length of the test section) to be not less than 200mm, nor should it be less than 40 times the diameter of the FRP bar. For the FRP bars in twisted strand form, the length is also required to be greater than two times the strand pitch.
P.F.Castro (1998) studied the effect of the ratio of free length of the specimen to its diameter in the tensile strength of the FRP bars tested. Tests were undergone with the free length to diameter ratio varying between 40 and 70 for FRP rods with varying geometry and surface characteristics. Based on the results of his experiments he concluded that there is no any observable statistical correlation between the mean tensile strength of the samples tested and the free-length to diameter ratio. This justifies the provision of ACI of free-length to the diameter ratio to be greater than 40.It scan be safely speculated that the provision of free length to diameter ratio of greater than 40 renders the mean tensile strength of the given FRP material independent of the geometry
30 of the specimen including the anchors lengths. His analysis of the correlation between the
modulus of elasticity and tensile strength depicts that the relative scatter of the tensile
strength values is larger than the relative scatter of the modulus values ( P.F.Castro
,1998).
This signifies the sensitive dependence of the tensile strength on the defects in the fiber but at the same time, explains the more robust response of the modulus of elasticity against the inherent defects in the fiber. This provides a good explanation for the ACI statistical definition of the design tensile strength, design rupture strain and design modulus of elasticity, as per equation 11 above. The deduction of three times the standard deviation is therefore not required in the case of modulus of elasticity owing to
the more robust response of the modulus value against the defects in fiber.
The modulus of elasticity value is obtained from the linear portion of the stress- strain curve. The ACI 440 guideline for the longitudinal tensile properties of the FRP bar recommends the calculation of modulus of elasticity from the linear regression of the data points from 20% to 50% of the tensile strength of the bar. It is assumed that the stress- strain relation is completely linear in the range of 20% and 50% of the tensile strength of the material. The guideline allows the use of guaranteed tensile strength as given by equation 12, for the calculation of modulus of elasticity. Once the tensile strength of the respective FRP bar is given, the expression can be used to calculate the modulus value.
(12)
31 where EL is the longitudinal modulus of elasticity, A is the cross sectional area of the
FRP bar, F1 and ε1 are the load and corresponding strain at 50% of the ultimate tensile capacity and F2 and ε2 are the load and the corresponding strain at 20% of the ultimate
tensile capacity of the FRP bar.
Since the tensile strength test of a FRP material consists of an elaborate set up
including metallic anchorages and the bonding or grouting agents like Epoxy and
cementitious grout, there is always a possibility of multifarious interactions between the
various factors associated with the test. The entire test set-up can be regarded to have
more degree of freedom in relative to the tensile test for the convention material like steel
owing to the possibility of different kinds of interactions.Therfore an ideal test method
should be such that the multi-interactive nature of the test setup must not influence the
actual material property that the particular investigation is seeking. The work done by
B.Benmokrane (1999) for the tensile properties of the AFRP and CFRP is particularly
relevant in this issue.
In his research, Benmokrane considered the type of the anchorage tube and the
grouting material as the variables of his study and investigated their effect on the
longitudinal tensile properties of the FRP bar. The actual study is directed to the study of
the effect of the host-media (anchorage tubes) on the bond behavior of the material. Since
the tensile test and bond test relies on the same principle, this can be of significance for
the tensile test also. In his study, he considered three types of cylindrical tubes, steel,
aluminum and PVC. He concluded that the host-media with higher modulus of elasticity increases the radial pressure on the grout, thus increases the efficiency of the test. He also
concluded that the increase in the thickness of the host-media which results in the
32 proportionate increase in its stiffness, give rise to higher radial stiffness at the FRP bar- grout interface ( B.Benmokrane,1999).
Apart from the test methods, one of the major differences between the structural response of the axially loaded FRP bar and conventional metallic bar is the size-factor
associated with the FRP bar. In the case of conventional material like steel, the ultimate
tensile strength is an intrinsic material property and doesn’t depend on the size and
geometry of the specimen. In the case of FRP bars, it is found that the strength of the
material is the function of the size of the FRP bar. This has been attributed to the so-
called shear-lag effect. The shear lag effect arises due to the fact that all the fibers will
not be stressed equally when axial load is applied. The outer fibers will be stressed more
than the core fibers thus decreasing the overall load-carrying capacity of the member.
Therefore, in the case of FRP bar the strength decreases as the size increases and hence
the tensile strength should always be specified for the particular size of the bar
(S.Kocaoz, 2004). The shear-lag phenomenon is associated with the curing related problem of pultruted FRP bars as per the study of S.S.Faza and GangaRao (1993).
As already been discussed, macromechanical approach, micromechanical approach and finite-element modeling are the methods for the analytical predictions of
the FRP bar characteristics. It will be pertinent to discuss the model proposed by
Gangarao, Wu and Prucz developed an analytical model based on the mechanics of
material approach for the tensile characterisization of the FRP bar. The model is more
realistic in the sense that it incorporates the shear-lag effect associated with the FRP bar
and it also considers the effect of the gripping on the structural behavior of the FRP bar.
The primary assumption made in this modeling is that the strain distribution across the
33 section of the circular bar is parabolic and axisymmetric and this is attributed to the radial
stresses induced by the gripping mechanism (S.S.Faza, 1993 ). The model incorporates a
factor designates as ‘c’ which is termed as the thickness of the boundary layer. The
variable thickness of the boundary layer is attributed to the different curing rates
associated with different bar sizes.
This explains the shear lag phenomenon, normally associated with the FRP unidirectional bars. Due to the variation in the curing time history, it will result in different boundary layer thickness in differently sized FRP bars. This finally results in the tensile force resistance differential between the core fibers and the fibers outside the core, which gives rise to shear-lag effect.
All the research conducted in the past decades for the comprehension of the tensile properties of a unidirectional FRP has unequivocally established the point that the
FRP materials are capable of generating very high tensile strength. Another point to be seriously considered is that the stress-strain relationship for this material is always linear upto the failure. The linearity in their constitutive relationship is one of the major demerits associated with the application of FRP bar in structural application. This linearity in their behavior renders them inapplicable for structures requiring large plastic deformation followed by subsequent stress-redistribution. Therefore the improvement in their ductility may be one of the prime concerns for the next generation of scientists and engineers.
The work done by V.Tamuzs (1993) on the hybrid FRP bar may be a subject of interest for this topic. He performed on the tensile test on the braided composite rods instead of unidirectional FRP bars. The braids were made of 8 Aramid fiber strands with
34 central polyurethane foam plastic core. The braids were impregnated with epoxy such
that the fiber volume fraction in relatively low. The stress-strain curve for this composite exhibits a completely different structural response and the ductility of the material was observed to be significant.
35 CHAPTER III
MATERIALS
This section will give the comprehensive description of the materials that were used for the determination of the longitudinal tensile strength of the Basalt FRP bars.
This will include the primary material, that is, the Basalt FRP bar and all the associated subsidiary materials that were used for the particular purpose. It is already been discussed in the literature review that the tensile strength test of FRP materials is an elaborate process in regard to their significantly different mechanical properties in compare to the conventional materials like steel. This subsequently entails the requirements and applications of different type of materials to meet this goal. The detailed description of all the materials is presented below.
3.1 BFRP Bars
The basalt bars which were used for the research were provided by the sponsor
Blackbull. The Basalt bars provided for the research were of the nominal diameter of
3mm,5mm and 7mm.this sizes corresponds to the net size governed by the fibers only.
The method for manufacturing for the basalt rod was reported to be Wet-ley up
36 process. Wet ley-up process is a very simple of producing FRP composite materials and is done manually. The process essentially consists of laying up the fibers and impregnating them with the polymeric resin such that it yields the usable composite material when cured. The fibers for the case of basalt FRP bars were extracted from the igneous rock named Basalt. The primary composition of Basalt rock is generally constituted with various forms of oxides, silica-oxide being the most abundant one. The percentage of silica oxide is generally between 51.6 to 57.5 percent and generally the basalt with the silica-oxide content above 46 percent (acid-basalt) is considered good for fiber-production.
Minerologically, Basalt is primarily constituted of minerals Plagioclase, pyroxene and olivine. When heated at high temperature, Basalt is capable of producing a natural nucleating agent which plays a major role for the thermal stability of the material.
This explains the apparent increased volumetric integrity of basalt in compare to the other materials. The presence of the before mentioned minerals may be the helpful factor for this phenomenon. The polymeric resin used as the matrix for the Basalt fiber is the
Vineylester resin. A vinylester resin is the combination of an epoxy and an unsaturated polyester resin.(L.C.Bank,2006).the advantage of vinylester is that it has the meritorious physical properties of the epoxy and the beneficial processing properties of a polyester resin.
37
Figure 3: Typical Basalt FRP bar Used for the Tensile Strength Test
The actual size of the provided FRP bars were measured in the laboratory with a high precision vernier-calipar. Their gross diameters including the polymeric matrix were
found to be 4.7mm,7mm and 10mm respectively for the bar of sizes 3mm,5mm and
7mm.the resulting fiber-volume fractions were found to be 44%,52% and 41%
respectively. Hence, the average volume fraction was worked out to be 46%.The Basalt
bars will be referred later on with reference to their net diameter. A typical picture of a
Basalt FRP rod used for the Tensile strength test shown in Fig.1.above.The outer surface
of the Basalt bar is provided with the sand alonwith a helical winding along its length to
enhance its bond-properties. This can be observed in Fig-4 shown above.However, the
sanding is not abundantly provided and the helical windings provided were not making
considerable indentations.
38 3.2 Anchorage Tubes
Since basalt is an anisotropic material, the strength of the basalt fiber in the
transverse direction is very low compared to its very high tensile strength in the
longitudinal direction. This necessitates the provision of some kind of anchor at the end
of the specimen so as provide the proper grip while applying the load. To meet this
purpose, two steel tubes, at each end, as dictated by the test methodology were provided.
the steel tube used for the anchor are black-welded steel tubes each eighteen inch long with threads at each end to facilitate the positioning of the cap.
The outside diameter of the tube is 1.315 inch and inside diameter is 1.049 inch.
The wall thickness of the tube is 0.133 inch. The tubes matched all the specifications as
required by ANSI, MSME AND ASTM. The ASTM specifications met are ASTM A53
and ASTM A733.the other end of the pipe is provided with the rubber-cap to hold the
fiber in vertical position. The rubber caps were black soft-rubber grips to hold the Basalt
FRP bar in vertical position at one end. The holes were drilled on the caps in the
laboratory as per required by the size of the bar being tested.
39 A typical picture showing the arrangement of the anchor tube and the steel cap is shown
in Fig-4 below.
Figure 4: Steel Tube and Steel cap for the Anchorage
3.3 Grouting Material-Epoxy and Sand
The Epoxy used for the purpose of grouting the steel tubes with the Basalt FRP bar is a structural epoxy with a commercial designation of AKA-Epoxy system .The mix
ratio for the particular epoxy as per provided by the manufacturer is 1 part resin to 1 part
hardener. The gel time as per specified by the manufacturer was 180 minutes for the
particular epoxy. The viscosity of the epoxy, as per reported by the manufacturer was
around 2300 centipoises. The picture of the epoxy is shown in figure below.
40
Figure 5: Structural Epoxy Used for Grouting
A particular amount of sand was mixed with the structural epoxy as per described in the experimental program section of the tensile strength test. The sand used for the purpose was the river sand obtained from the local supplier.
41 CHAPTER IV
TEST PROCEDURES
This section encompasses the entire tasks that have been performed in relation to the tensile strength test of the basalt FRP bars. The tensile strength test of the FRP bar
was done according to the ASTM standard method for the tensile test of the FRP
pultruded bars. The test essentially consists of the process of applying the tensile load on
the bar in the universal testing machine and loading it up to rupture. However, the
method comprises the various other pertinent activities that were to be undertaken to achieve this objective. The structural issues pertaining to the tensile strength test of the
FRP material has already been sufficiently discussed in the literature review portion of
the respective section.
The different structural response as exhibited by the FRP material in contrast to
the isotropic conventional materials entails the requirement for different ingenious sub-
tasks to make the tensile strength test of the FRP bar successful. The test program
primarily consisted of the preparation of the FRP test specimen which can be regarded as
the most essential part of the program. This further requires the accomplishment of all the
related tasks to make the test successful. All the related works has been described comprehensively in an orderly manner in the following chapters.
42 The schematic of the test set-up for the tensile strength test of the Basalt FRP bar is
shown on Fig-5 below.
Data Acquisition System for strain data
Figure 6: Schematic for the Tensile-Test Set-up for the Basalt Bar
Fig-6 clearly depicts the underlying principle regarding the tensile strength test of the Basalt FRP bar. The tensile specimen provided with the anchorages at its ends for the grips, were loaded to rupture in the universal testing machine as shown in the figure.
The subsidiary activities are described comprehensively in the following chapters.
43 The elevation and the longitudinal cross-section of the test specimen are shown in Fig-7
below.
A Basalt-Rod Rubber-Plug Metal-Cap 0.133"
12" 18" A 2"
Elevation of The Specimen Longitudinal Cross-Sectional Elevation
Figure 7: Longitudinal Details of the Tensile-Test Specimen
Fig-7 above shows the longitudinal details of the Basalt tensile test specimen.
The left hand portion shows the elevation of the specimen and the right hand portion shows the longitudinal cross-sectional details. The cross-section of the tensile test specimen is shown in Fig-8 below. It can be observed that the specimen is primarily equipped with the anchorages provided at the ends of the specimen and the Basalt bar to be tested in embedded in certain type of adhesive with the anchorages. The engineering implication of this procedure is already been explained in the literature review portion of the respective topic.
1" 1.31"
Figure 8:Cross-section Details of the Tensile Test Specimen
44 4.1 Preparation of Wooden Frameworks
The first step undertaken for the accomplishment of the tensile strength test of the Basalt FRP bar was the preparation of the wooden framework to accommodate the tensile specimens to facilitate the grouting of the anchorage tubes with epoxy. Since the basalt bars are required to be grouted with the steel tubes to provide the anchorages while testing, this required the development of a framework mechanism which could hold the tensile specimens vertical and allowed the grouting smoothly. Hence a wooden framework was made on the material lab of the Civil Engineering department using the spare woods. The schematic representation of the framework and its utility is exhibited in fig-10 below. The horizontal members were provided with sufficient size of holes to accommodate the steel anchorage tubes.
Figure 9: Schematic of the Framework Arrangement
45 As shown in the figure, the provided arrangement was able to hold the specimens
completely vertical and horizontal during grouting, thereby avoiding any kind of
eccentricities. A picture showing the exact arrangement of the test specimens in the
framework is shown in Figure below.
Figure 10: Actual Arrangement of Specimens on the Framework
4.2 Preparation of Specimens
The test specimen length includes the length of the test-section (as per
prescribed by the relevant standard) plus the length of the anchors at the ends. The
overall length of the specimen is 52” to account for the length of the test-section and the length of the anchors at its ends. The required length pieces were cut by the hack-saw carefully so as not to disturb the provided sanded roughness of the Bar surface. The basalt bars of the provided sizes and the required length were positioned on the anchorage steel tube and the steel caps were applied at the ends. The next step was the positioning of the 46 specimens on the wooden framework for the process of grouting of the anchorage tubes.
The specimens were sufficient clamped to the wooden framework as shown in Fig-10. A structural epoxy named AKA-Epoxy system was used for the purpose of grouting the steel tubes .This is shown in Figure below.
Figure 11: Structural Epoxy Used for Grouting
The mix-ratio for the used structural epoxy is one part resin to 1 part hardener. This was further mixed with some fixed proportion of sand to increase the bond between the steel tube surface and the epoxy. Sand was dried beforehand in the oven at high-temperature for 24 hours to make it free of any residual moisture.
47 The equal proportioned resin and hardener were mixed with the help of some mechanical means and was finally mixed with fixed proportion of sand. This is shown in
Fig-12 below.
Figure 12: Arrangement for Mixing the Epoxy
For the single batch for the one side of the six specimens , 600ml of resin is mixed with 600ml of hardener for 1 min and finally with 250ml of dry sand ,when the uniformity in the color and desired viscosity is achieved for additional 1 minute. This volume is made in consideration to the require volume to make six samples ready for the test. Fig-12 as shown above clearly depicts the arrangement that was employed for the mixing of the resin and the hardener. As shown in the respective pictures, the mixing was done with the help of the machine-operated mixer in a steel bowl.
48 The Fig-13 as shown below shows the stage where the epoxy of desired and
viscosity was obtained.
.
Figure 13: Epoxy and the Sand Mixture
The mixed epoxy was put in a plastic hole with a small hole at its bottom so as
to ensure the smooth pour of the epoxy inside the tube. This is clearly depicted in Fig-14
below. The tubes were regularly tamped and compacted with a small, clean steel rod to
make sure that the there were no air-voids in the matrix. The rubber plugs were put at the ends when the tubes were completely filled with the epoxy. The holes on the steel caps were applied with fast-setting super-glue along with the putty to make sure that the tubes would not leak .The specimens were kept in the same position for 24 hours to allow for the epoxy to set.
49 The specimens are carefully inverted and the same procedures were employed at the other end.
Figure 14: Arrangement to Pour Epoxy inside the Tube
50
Figure 15: Tubes Filled with Epoxy on One Side
4.3 Experimental Program
The specimens were tested on “BALDWIN” material testing machine, which is
a 300,000lb-capacity, universal testing machine. The type of the machine is UTM, model
300HV-300,000lb capacity, with the serial 300HV-1005.The schematic representation of
the arrangement of the specimens on the testing machine is depicted in Fig-6. The tensile test specimens were mounted on the universal testing machine in such a way that the anchorage tubes would be positioned inside the vee-grip of the testing machine. This is shown in Fig-16 below.
51
Figure 16: Arrangement of Tensile-Test Specimen in Universal Testing Machine
The extensometer is positioned approximately on the centre of the test- length.the extensometer used for the experiment is large-gage length model extensometer
.the model is 3543-SR-0300-200T-ST and the gage length used for the purpose is 2”.the extensometer is hooked up with the data acquisition system, ADMAT. Fig-17, as shown below, shows the positioning of the strain-gage on the tensile test specimen.
52
Figure 17: Positioning of Strain-Gage on the Specimen
The specimen was loaded at a constant rate of 15 pounds per second until it is stresses to the complete Rupture of the specimen. The extracted data by the connected data- acquisition system was used for the subsequent analysis of the tested specimens.
53 CHAPTER V
RESULT AND ANALYSIS
This section will include the comprehensive description of the test results obtained from the tensile test performed on the Basalt FRP bars of three different sizes.
The tensile strength test will provide the idea of the tensile strength of the material, its failure modes, its stress-strain characterization, rupture strain and the modulus of elasticity. The strain data obtained from the strain-gage was used to establish stress-strain curve of the material which further helped to acquire other pertinent characteristics like modulus of elasticity and rupture strain. The results, therefore, primarily consisted of stress-strain curves, their ultimate tensile strength, rupture strain and modulus of elasticity of the material for the respective sizes. One of the important aspects of this test is also the investigation of the failure modes of the material. This test can provide the idea of the qualitative difference between the failure modes of Basalt FRP bar with the other existing FRP materials which may be further helpful for the understanding the properties of the Basalt FRP bar.
54 5.1 Research Objective
The primary objective of this research is the identification of the important mechanical properties of the Basalt FRP bars of the provided sizes by the sponsor. This includes the determination of the tensile strength of the Basalt FRP bars, rupture strain and the modulus of elasticity. The determination of these parameters further ensues the determination of the design strength, design rupture strain and the design modulus of elasticity of the given material. The study, in an indirect way, also intended to give the knowledge of the failure mode of the material and the measure of the applicability of the
ACI440 in the case of testing of the Basalt FRP bar for the longitudinal tensile strength.
5.23 mm Nominal Diameter BFRP Bars
The 3mm Basalt FRP rods as provided by the sponsors were tested in unaxial tension as described in the previous sections. The failure modes observed for all the specimens were tensile-rupture failure.
55 In tension failure mode, the fibers failed in tension, such that the individual fibers got separated and ruptured forming spreaded splinters. A typical failure mode for 3mm basalt bars are shown in figures below.
.
Figure 18: Typical Failure mode for 3 mm Basalt Bar
Table 1: Tensile Test Results for 3mm Basalt Bars
Specimen Max Load,lb Max Load,Kn Rupture Strain Max Stress,psi Max Stress,Mpa Modulus,ksi Modulus,Gpa 3-1 3636.00 16.17 0.024 331.90 2288.00 15271* 105.3* 3-2 3734.00 16.61 0.031 340.90 2350.00 12357.00 85.20 3-3 3615.00 16.08 0.028 329.90 2275.00 12470.00 86.00 3-4 3056.00 13.59 0.022 278.90 1923.00 14658* 101.1* 3-5 3511.00 15.61 0.027 320.50 2210.00 32803* 226.1* 3-6 4037.00 17.95 0.0358* 368.50 2541.00 14067* 978* 3-7 3847.00 17.11 0.031 351.10 2421.00 12107.00 83.40 3-8 3516.00 15.64 0.033 320.90 2213.00 11884.00 81.90 3-9 4092.00 18.20 0.029 374.00 2578.22 12623.00 87.03 3-10 3552.00 15.80 0.028 324.00 2233.50 12780.00 88.14 3-11 3892.00 17.31 0.029 356.00 2454.99 13100.00 90.34 Mean 3680.73 16.37 0.03 336.05 2317.06 12474.43 86.00 Std Dev 289.88 1.29 0.003 26.61 183.42 409.11 2.85
56 The guaranteed tensile strength for the 3 mm Basalt FRP bar as per the equation is found to be 252 ksi. Similarly, the guaranteed rupture strain as per the equation is found to be 0.0216. Similarly, the guaranteed tensile modulus of elasticity, as per the equation, is found to be 12,204 ksi. The typical stress-strain curves for the 3mm sized
Basalt FRP bars are shown in the figures below.
350 "3 mm" Diameter Bar Specimen 6 300
250
200
Stress, ksi 150
100
50 Tensile Modulus = 13782.5 ksi Tensile Strength = 368.5 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 19: Typical Stress-Strain Curve for 3mm Basalt Bar
57 5.35 mm Nominal Diameter BFRP Bar
5mm nominal diameter Basalt FRP bars were tested in the uniaxial tension. The test program included eight specimens for the particular size. The failure modes observed for the 5mm sized specimens were also tensile-failure mode as similar to the 3mm sized
FRP bars. The failure was accompanied by the tensile rupture of the individual fibers,
thus forming a mass of spreaded splinters .A typical failure mode for the 5mm sized
Basalt FRP bar is shown in the figure below.
Figure 20: Typical Failure Mode for 5mm Bar
58 The tensile test results for the 5mm nominal diameter Basalt FRP bars are summarized in table below.
Table 2: Tensile Test Result for 5mm Basalt Bars
Specimen Max Load,lb Max Load,Kn Rupture Strain Max Stress,psi Max Stress,Mpa Modulus,ksi Modulus,Gpa 5-1 10,132 22.8 0.02725 332.90 2,295 12,226 84.30 5-2 10,805 24.30 0.03145 355.00 2,448 11,791 81.30 5-3 8,764 19.70 0.02635 288.00 1,986 12,096 83.40 5-4 9,064 20.40 0.02579 297.80 2,053 11,851 81.10 5-5 9,164 20.60 0.02630 301.10 2,076 11,882 81.90 5-6 9,462 21.30 0.02725 310.90 2,144 11,673 80.50 5-7 10,208 23.00 0.03097 335.40 2,313 14,388 99.20 5-8 9,819 22.10 0.02577 322.60 2,225 13,515 93.20 5-9 9,815 22.09 0.02369 303.43 2,093 13,183 90.91 5-10 10,421 23.45 - 345.30 2,382 12,663 87.32 Mean 9,765 21.97 0.02720 319.24 2,201 12,527 86.31 Std Dev 649 1.47 0.00250 22.36 154 901 6.27
In Table 2, as shown above, all the tests results and their statistical correlations are shown for the case of basalt bar of nominal diameter of 5mm.It can be observed that the standard deviation for the maximum stress in the bar during loading is 7%. Similarly, the standard deviation for the rupture strain is 9.1%. Similary, the standard deviation for the modulus of elasticity is 7.1%. It can be observed that the rupture strain exhibits more variability about the mean value than the other parameters as similar to the 3mm bars.
The guaranteed tensile strength for the 5mm Basalt FRP bar as per the equation is found to be 252 ksi. Similarly, the guaranteed rupture strain as per the equation is found to be 0.02. Similarly, the guaranteed tensile modulus of elasticity, as per the equation, is found to be 12,527 ksi. The typical stress-strain curves for the 5mm sized
Basalt FRP bars are shown in the figures below.
59 350 "5 mm" Diameter Bar Specimen 8 300
250
200
Stress, ksi 150
100
50 Tensile Modulus = 13514.6 ksi Tensile Strength = 322.6 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 21: Typical Stress-Strain Curve for 5mm Basalt Bar
5.4.7mm Nominal Diameter BFRP Bars
7mm nominal diameter Basalt FRP bars were tested in the uniaxial tension. The
test program included fourteen specimens for the particular size. The failure modes
observed for the 7mm sized specimens were, in most of the cases, were the debonding of
the Basalt Bar with the anchorage. Of all the fourteen specimens, only two specimens
underwent pure tensile-rupture failure. Though the specimens showed slip at the ends
with the anchorage tubes, all the specimens were tested to failure. A typical failure mode for 7mm Basalt FRP bar is shown in following figure.
60
Figure 22: Tensile Rupture Failure of 7mm Bar
Figure 23: Debonding of the Basalt bar with the Anchors
The tensile test results for the 7mm nominal diameter Basalt FRP bars are summarized in table below.
61
Table 3: Tensile Test Results for the 7mm Basalt Bars
Specimen Max Load,lb Max Load,Kn Rupture Strain Max Stress,psi Max Stress,Mpa Modulus,ksi Modulus,Gpa 7-1 21,380 48.1 0.02991 358.4 2,471 12,942 89.2 7-2 19,974 44.90 0.02867 334.8 2,309 13,023 89.8 7-3 17,542 39.50 0.02424 294.1 2,028 12,857 88.6 7-4 18,145 40.80 0.01801 304.2 2,097 13,387 92.3 7-5 17,767 40.90 297.8 2,054 12,194 84.1 7-6 18,194 37.50 305 2,103 12,106 83.5 7-7 16,869 40.30 0.02351 279.8 1,929 13,193 99.2 7-8 17,928 38.20 0.02701 300.5 2,072 13,241 91.3 7-9 16,968 38.30 0.02372 284.5 1,961 12,643 87.2 7-10 17,031 38.30 0.02417 285.5 1969 12,672 87.4 7-11 18,300 41.20 0.02311 306.8 2,115 13,135 90.6 7-12 17,807 40.10 0.02365 298.5 2,058 12,574 86.7 7-13 17,884 40.27 0.025495 301 2,075 12,410 85.59 7-14 16,123 36.31 0.02236 271.45 1871.51 12,080 83.25 Mean 17,994 40.3 0.02413 301.60 2,079 12,747 88.48 Std Dev 1318.4 3.0 0.00147 22.17 152.94 434.20 4.21
In the Table1,2 and 3, as shown above, all the tests results and their statistical
correlations are shown for the case of basalt bar of nominal diameter of 3mm,5mm and
7mm.It can be observed that the standard deviation for the maximum stress in the bar
during loading is 7.35%.similary, the standard deviation for the rupture strain is
6%.similary, the standard deviation for the modulus of elasticity is 4.75%.it can be
observed that the rupture strain and maximum stress exhibits more variability about the
mean value than the other parameters as similar to the 3mm and 5mm bars.
The guaranteed tensile strength for the 7mm Basalt FRP bar as per the equation
is found to be 235ksi. Similarly, the guaranteed rupture strain as per the equation is found
to be 0.02.similary, the guaranteed tensile modulus of elasticity, as per the equation, is found to be 12,747ksi. The typical stress-strain curves for the 7mm sized Basalt FRP bars are shown in the figures below.
62 350 2500 "7 mm" Diameter Bar Specimen 1 300 2000
250
1500 i 200 a ks P , M s s, e s tr re S 150 St 1000
100
500 50 Tensile Modulus =12941.8 ksi (89.2 GPa) Tensile Strength =358.4 ksi (2471 MPa)
0 0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 24: Typical Stress-Strain curve for 7mm Basalt Bar
In the Table 1, 2 and 3, as shown above, all the tests results and their statistical correlations are shown for the case of basalt bar of different diameters. It can be observed that the standard deviation for the maximum stress in the bar during loading varies from
7.1 to 7.9%.similary, the standard deviation for the rupture strain varies from 6 to 13.7%.
Similarly, the standard deviation for the modulus of elasticity varies from 2.1%.to 5%.it can be observed that the rupture strain exhibits more variability about the mean value than the other parameters. This can be attributed to the fact that it is difficult to obtain the particular point where the material is initiating to rupture. Generally, the strain value is extracted from the stress-strain data for the specimen corresponding to the maximum stress. This may account for the variations on the calculated rupture strain value based on the consideration of the absolute maximum load.
In the case of maximum stress, the variability of this parameter about the mean can be attributed to the difference in the actual size of the tested specimens and the presence of some visible and non-visible imperfections. The presence of any imperfection 63 in the form of air-pockets or dents can surely affect the ultimate strength of the material.Fig-26, as shown below, shows a specimen with a deep dent on its cross-section and the strength of the particular sample was found to be considerably lower than the other samples of same size.
The markedly lesser variability of the moduli of elasticity can be attributed to the lesser sensitivity of the elastic modulus of elasticity on the imperfections. The similar observations were also made by P.F.Castro (1998) regarding the statistical distribution of these parameters in the case of glass fiber FRP bars. He observed that the standard variation in the distribution of tensile strength is varying between 2.2 and 9.1 percent depending on the surface geometry of the specimen. Whereas the moduli of elasticity were found to be varying between 0.95 to 2.03 percent.
64
Figure 25: A Specimen with Imperfection as a Dent
From the tests conducted, it can be concluded that the 7mm diameter bars are not yielding consistent data. From the inference of most of the cases, it is observed that the 7mm bars have not enough bonds with the epoxy-matrix so as to withstand the very high tensile stress in the longitudinal direction. This may be attributed to the fact that the surface of the bar is not sufficiently sanded or not any kind of reasonable surface- roughness is provided to develop the desired bond stresses. For the lesser sized bars
(3mm and 5mm), the results were consistent and the failure modes were typical for the fibers as shown in the related figures above.
65 With the view to more comprehensive investigation, the tensile specimen consisting 7mm
bars were cut along the cross-section.the respective pictures are shown below in the pictures below.
Figure 26:Cross-Section of the Tested Specimens
The study did not reveal any significant information regarding the probable cause for the
slips. There were no any air-pockets in the epoxy matrix and the epoxy system seemed to
be very well compacted. For the further research, the tensile specimens were cut
longitudinally for a length of 8 inches including the cap, which is the bottom position of the specimen while pouring the epoxy. This depicted the deposition of sand at the bottom thereby significantly affecting the local bond strength. The basalt rod seemed to be in good bond with the epoxy for the upper four inches of the longitudinal cut. But for the lower four inches of the cut closer to the cap, there seemed to be local reduction in bond strength due to the deposition of sand. This is shown in the following figures.
66
Fig 27: Longitudinal Section of the 7mm Specimen after Testing
Fig 28: Local Reduction in Bond Strength at the Anchor Ends
67 CHAPTER VI
CONCLUSION AND RECOMMENDATION
The test results obtained were found to be fairly consistent and satisfactory. The
standard deviation for the tensile strength were found to be varying from 7 to 7.9%.the
distribution of rupture strains was found to be more variable ,varying between 6 to
13.74%, maximum being in the case of 3mm bar. For the 3mm bar, it can be assumed
that the strength parameters can be fairly variable owing to the probable non-uniformity of the distribution of fibers in the case of small sized bars. The modulus of elasticity
need not to be adjusted for the standard deviation. The details of the various parameters
for the different bar sizes are tabulated below:
Table 4: Guaranteed Rupture-Srains
Diameter mm (inch) ε u,avg σ ε* fu 3(0.12) 0.0279 0.0038 0.016 5(0.20) 0.0272 0.0025 0.020 7(0.28) 0.0241 0.0015 0.020
68 Table 5: Guaranteed Tensile Strengths
f σ f* Diameter mm(inch) f,avg fu Mpa ksi Mpa ksi Mpa ksi 3(0.12) 2,278 330.32 181 26 1,735 251 5(0.20) 2,201 319.24 154 22 1,739 252 7(0.28) 2,079 301.60 153 22 1,620 235
Table 6: Guaranteed Modulus of Elasticity
f σ f* Diameter mm(inch) f,avg fu Mpa ksi Mpa ksi Mpa ksi 3(0.12) 2,278 330.32 181 26 1,735 251 5(0.20) 2,201 319.24 154 22 1,739 252 7(0.28) 2,079 301.60 153 22 1,620 235
The average modulus of elasticity was found to be 86.3 Gpa (12,493ksi).On
reviewing all the modulus values for the different sizes of basalt bars, a modulus of
elasticity of 84Gpa(12,000ksi) is recommended. In the case of 3mm and 5mm bars, the
results were very satisfactory and consistent. However, in the case of 7mm bars, all the
specimens were found to be slipped except the two specimens. To improve the bond
behavior, the lengths of the anchors for the last six 7mm specimens were increases to 24
inches. It was observed in also most of the cases that the cross-section at the vicinity of the steel-caps was being sheared due to the eccentricity of the specimen during the production process. To counteract this problem, two specimens each for 5mm and 7mm were tested with rubber plugs replaced with steel caps. The results were not significantly
69 affected by the new changes. Hence, for the basalt bar of size 7mm, it is recommended to implement some other methods like threading the bars along the length to increase the bond behavior between the epoxy, steel anchor and the basalt bar.
70 CHAPTER VII
LITERATURE-REVIEW FOR THE BOND-CHARACTERISTICS OF FRPs
This section comprises the overall literature review done in regard to the study of the bond-behavior of the BFRP bars with concrete .Since BFRP bars are relatively new material ,the literature review is based on the similar kind of tests and research done with the other types of FRP materials.
7.1.Basic Introduction to Bond
The basic assumptions in flexure theory for the analysis of reinforced concrete structures can be generally categorized as follows:
1. Bernoulli’s-assumption: plane sections that are perpendicular to the axis of bending before will remain plane after bending. This hypothesis allows the application of linear- strain distribution across the section of the member for the analysis.
2. strain-compatibility: the strain in the reinforcement is equal to the strain in the concrete at the same location in the section.
3. The stresses in the concrete and the reinforcement are computed based on their respective stress-strain curves.
71 The first assumption implies the applicability of the basic elastic analysis for the
reinforced concrete structures which further implies the material to be homogenous and
isotropic. Even though the reinforced concrete structures behave very differently after
cracking, for all practical purposes, this assumption greatly simplifies the analysis of the
reinforced flexural members. The third assumption will facilitate the analysis of the
flexural members based on the mechanical properties of the constituent materials. The
second assumption is very pertinent and significant as it postulates the perfect bond
between the reinforcement and the concrete such that the composite action is guaranteed.
It is very relevant to discuss the basic principle of reinforced concrete structures at this point. Concrete is a material which is very strong in compression compared to its
strength against tension. It is well accepted that the tensile strength of the concrete is
approximately 10 to 15% of its compressive strength. To exploit the strong compressive strength of the concrete, it can be reinforced with the other materials such as steel and various types of FRPs which have high tensile strength.
However, the high tensile strength of the reinforcing material may be a necessary criterion but not necessarily a sufficient one. To extract the complete advantage of the reinforcing materials such as steel and various kinds of FRPs, the complete composite action of the concrete and the reinforcing materials is the fundamental requirement. Unless there is a complete composite action, the use of high-strength materials may not be very useful.
As a matter of fact, in case of reinforced concrete structures, both at serviceability and at ultimate state, the various resisting mechanisms such as bending,
72 shear, and torsion are directly linked with the phenomenon of perfect-bond between the
constituting materials.
Various implications of serviceability limit state, such as control of crack widths
and deformations, are influenced by the phenomenon of tension- stiffening which is directly linked with bond (E.Cosenza, 1997). Another widespread application of the FRPs is their application for the shear/flexural strengthening of the existing structures. For such type application, the respective FRP material is placed in the groove cut on the surface of the structure to be strengthened and is fixed their by some kind of epoxy.
Such application, which includes the multiple bond-interfaces (between FRP and epoxy and between the epoxy and the concrete), may necessitate more detailed
analysis on the bond behavior of the FRP as it may engender more parameters to define
the global joint behavior (R.Tefers, 2003).
Hence, it can be concluded that the bond is the fundamental requirement for
composite action and Effectiveness of the reinforced concrete structures such that its
importance can’t be underestimated. (B.tighihourt, 1997).
7.2 Basic Mechanics of Bond-Stress
It should be noted that bond-strength is not an intrinsic property of a material
but a measure of the composite action between the two materials. For instance, tensile-
strength of a particular material can be attributed to as its intrinsic property as it doesn’t
depend on the interaction with the other materials. On the other hand, bond-strength is a holistic or composite response of a system of materials that are being used.
73 Depending on the nature of the loading, the load-effects such as bending
moment, shear-force, and axial stress may vary along the length of the flexural member.
If we consider the variation of bending moment along the length of the beam, it will
subsequently entail the variation of tensile force along the Reinforcement. For the bar to
be in equilibrium, it necessitates the effectuation of bond-stress along the concrete-
reinforcement interface. Considering the equilibrium of a small length of reinforcement,
the following relation can be written.
(13)
Where (fs2 - f s1) represents the variation of tensile stress over a particular length “L”
being considered. It can be seen from the above equation that the equilibrium of the
particular length considered is Dependent on the bond-stress “µavg” between the concrete
and the provided reinforcement.
In the light of equation 13, it can be concluded that the tensile-force gradient along the length of the reinforcement is proportional to the average bond-strength between the reinforcing bar and the surrounding concrete. If the tensile force-gradient for the particular loading case is high, it will consequently require the proportionate increase in the bond-strength to guarantee equilibrium, hence, the complete composite action.
There can be further implication of this phenomenon. Though the provided reinforcement may be strong enough to withstand high tensile-stress gradient, the
particular system of concrete and the respective material may not be able to develop the
74 sufficient bond between them to effectuate equilibrium. This issue is particularly relevant
for the cases of high strength materials such as FRPs.
7.3 Bond in the Context of FRP Bars
Fiber reinforced plastic rebar are being extensively used as a reinforcing
materials these days owing to their less weight-strength ratio, better corrosion-resistance
,electromagnetic transparency and easy workability. they are generally endowed with high tensile strength in the longitudinal direction .The approximate tensile strength for the common grade of glass fiber (GFRP) is reported to be 475 ksi and for the case of common grades of carbon-fiber (CFRP) to be 504 ksi. It is apparent that they are high- strength materials and hence can withstand high tensile stresses depending on the type of loading.
However, their high tensile strength can only be fully exploited if they can be used as a material for reinforced concrete structures, which can develop adequate bond with the surrounding concrete-matrix. For instance, carbon fiber reinforced beams can undergo failure by interfacial debonding with the concrete due to the high tensile strength of CFRPs (A.Maji, 2005).
In the case of FRPs, this issue becomes more significant since the current methodologies for the analysis of bond behavior in flexural members are based on the
researches done with steel as a reinforcing material. Many experimental studies and researches have shown that the various mechanical and physical properties of the FRPs are significantly different both quantitatively and qualitatively in comparison to the steel
75 bars. (B.tighihourt). Such differences can be attributed to the significant difference in the
mechanical properties between the steel rebars and the FRPs and the difference between
the modes of their interaction with the concrete matrix.
7.4 Difference in Bond-Behavior between Steel and FRPs
The differences in bond behavior between the steel and the FRPs can be broadly classified into the following two topics:
7.4.1 Difference in Material Property
Steel is an isotropic material such that the properties in the transverse and longitudinal direction can be expected to be similar. While in the case of FRPS, it is hugely different. FRPs are anisotropic materials which imply the variation of mechanical properties in longitudinal and transverse direction. The anisotropy of the
FRPs is the result of the fact that the longitudinal properties are governed by the fiber-
properties whereas the transverse and shear properties are governed by the resin-
properties (E.cosenza, 1997).
7.4.2 Difference in Surface Deformations
Based on the study done on the past (E.cosenza, 1997), it can be observed that
the surface deformations play a very prominent role for the development of bond-stress
76 between the concrete and reinforcement –interface. Surface deformations dictate the nature of the mechanism resisting the possible slip of the reinforcement from the concrete and hence can be regarded as a very important parameter governing the bond behavior between concrete and the reinforcing materials. The processes that are developed for improving the bond-behavior of the FRPs can be classified in two different headings
(E.cosenza, 1997).
7.4.3 Deformation of the Outer Surface
The conventional reinforcing steel can be a good example for this category. The surface of the material is ribbed, indented or braided to achieve the good bond behavior.
In addition to the surface-friction and chemical adhesion, these surfaces are able to develop additional bond-strength by developing the mechanism of mechanical- interlocking.
7.4.4 Surface Treatment
Surface treatment is the provision of granular material like sands on the outer surface of the material to improve the bond behavior between the material and the concrete. Generally, their bond resistance is composed of two components, namely chemical-adhesion and surface friction.
It will be very pertinent to discuss the modes of transfer of bond-stress at this juncture. The mode of transfer of bond between FRP rebar and the concrete has been
77 extensively studied in the past by many scientists regarding the anisotropic nature of the
material, variation in the different type of binders used and different type of surface
deformations. They found that the bond- behavior which includes the mode of transfer of
bond-stress, in the case of steel is completely different from the case of conventional
deformed steel.
7.5 Bond-Behavior of Steel
The plain steel bars which were used before in the structures, does not possess the surface deformation as compared to the common steel bars that are being used nowadays. According to the study done by Abrams at the university of Illinois in 1993,
the bond is transferred in plain steel bars by chemical adhesion followed by the sliding
resistance which is developed due to the friction between the reinforcing bar and the
concrete (L.R.Feldman,2007).
According to the CEB-FIP model code, for plain hot –rolled bars that have good
bond conditions, are inclined at 45 to 90 degree to the horizontal during casting, or placed
within 9.8 in from the bottom or at least 11.8 in from top layer of the concrete, the bond-
stress in MPa is given by
(14)
78 In deformed steel bars, the major components of bond resistance include the chemical adhesion, frictional-resistance and the mechanical interlock between the
reinforcement-interface and the concrete.
7.6 Mode of Transfer of Bond in FRPs
As already discussed, the mode of transfer of bond in the case of FRPs is
qualitatively and quantitatively different from that of steel due to the significant
difference in their mechanical and physical properties and their mode of interaction with
the concrete matrix. In contrast to steel, where bond strength is the function of the
compressive strength of the concrete, bond strength of FRP depends on various factors.
They comprise the following:
1. Chemical adhesion between FRP rebar and concrete matrix
2. Friction between the surface of the rebar and the concrete matrix.
3. Mechanical interlock of the FRP against the concrete
4. Hydrostatic pressure against the FRP rods due to the shrinkage of hardened concrete
5. Swelling of the FRP rods due to the temperature change and the moisture absorption
On the basis of the research and studies done on the relevant field, among the
various factors described above, the first three components play the most important role for the transfer of bond-stress. However, the chemical bond between the concrete and the
FRP is generally very low and may not play a very dominant role (E.cosenza, 1997). On
the basis of the research done by Maikitani (1993), however, it was concluded that the
sanding of the surface of the rebar can result in the increase of chemical bond. The other
79 components also play some important role for the transfer of bond-stress, but their effects
are difficult to be quantified and analyzed due to their variable nature.
In accordance with the research conducted by zhang and Benmokrane, the
general mechanisms for the development of bond-strength in the case of FRPs can be attributed to the following factors:
1. Chemical adhesion
2. Friction-action
3. Wedging-effect
For both cases of steel and FRP rebars, when the applied load is relatively low,
chemical adhesion and the frictional resistance plays the major part for the development
of bond-strength. With the further increase in load, the adhesion and the friction decrease
sharply such that the wedging effect comes into play. Due to the inner and outer angle
between the direction of force and the individual wires which are approximately 4 and 12
degrees, respectively, the concrete injected between the wires generates a different kind
of resistance, termed as wedging effect. This can be synonymously used with
mechanical-interlocking also.
Generally, the angle of strands, in case of FRPs, is significantly lower in
comparison to steel, the wedging effect due to the strands is lower than that of ribs of
deformed steel bars. With the further increase in slippage, the strands might distort and
the concrete between the wires might get crushed. This is followed by the decrease of the
bond resistance as the bond-resistance afterwards, depends primarily on the surface
friction only.
80 However, there have been many attempts to classify the nature of bond
mechanism depending on various factors which have significant influence for the
development on bond strength. The historical study done by Abrams at the University of
Illinois in 1913 can be regarded as the cornerstone for the attempt to study bond
mechanism between the reinforcing bar and concrete matrix (l.R.Feldman, 2007).
(E.cosenza), has regarded the work of (Kanakubo) to be the first attempt to classify the nature of bond-mechanisms. On the basis of the various tests and the researches done by the scientists and engineers, the nature of bond transfer can be classified under the following headings:
1. Classification Based on Resistance-Mechanism
From The study done by Abrams at university of Illinois at 1913, for plain steel bars, it was concluded that the bond between the plain steel bars and concrete-matrix was primarily created by the sliding resistance between the reinforcing bar and the surrounding concrete (l.R.Feldman, 2007). This can also be generalized for the case of
FRP reinforcing bars which doesn’t possess the sufficient surface deformation to develop the wedging effect of mechanical interlocking as verified by the various tests done on the respective field.
Kanakubo (1993), on the basis of the research done on the field of FRPs, he identified two types of bond resisting mechanisms. The first one is the friction-resistant type, and second one is the bearing-resistant type. The former type of resisting mechanism will be exhibited by the FRP bars with smooth and strand-shaped bars. The
81 bearing-resistant bond is developed by the ribbed FRP bars. On the basis of the
discussion done above, the bearing resistance seems to be analogous to the resistance
developed by wedging effect of mechanical interlocking.
2. Classification Based on the Nature of Slip
This kind of classification was proposed by Makitani (1993). He identified two kind of bond behavior depending on the surface characteristics of the FRPs. FRP rebars having a typical straight grain-covered rebar exhibited a very small slip (sm) corresponding to the peak bond strength (τm) whereas, the braided rods, exhibited much
larger slip corresponding to the peak bond-stress.
3. Classification Based on the Nature of Pull-Out Failure
Based on the study of Itoh et al, 1989, the sand-covered FRP rebar displays
good bond characteristics initially, but the interface of the sand-cover is removed
abruptly, thus leading to a brittle failure. The braided and ribbed rebar, on the other hand,
owing to the progressive development of mechanical interlock or wedging-effect,
exhibits relatively more ductile failure. Hence this can also be a good criterion for one of
the classification schemes for the FRP bars.
82 4. Classification Based on Reserve Strength Factor
The ratio of load at initial free-end displacement to ultimate load is defined as
the “reserve strength factor”. Reserve strength factor for a smooth rod at failure is unity
for smooth rod. The reserve strength factor for deformed CFRP bars ranges from 3.5 to
4.0 for the embedment length ranging from 12 inch to 18 inch and approaches 9 for the
embedment length of 6 inch (C.V.Jerrett, rilem proceedings 29). Reserve strength factor
can be regarded as the index of the ability of the respective reinforcing bar to sustain the
higher load after the initial free-end slip has occurred; hence, it gives us the idea of the
post-slip behavior of the respective reinforcing bar.
The higher value of reserve strength indicates the ability of the material to carry
more load after the first slip at end has occurred. For instance, the deformed rods exhibit
a higher load at which there is initial free-end displacement (2 to 5 times higher load as compared to the smooth rods) (C.V.Jerrett , rilem proceedings 29). If we consider basalt
FRPs that has been used by us for the determination of the bond-strength, the average
reserve strength factor was found to be 0.87 for the 5mm bars and 1 for 7mm bars. This is
very agreeable as the surface of the used rebar were moderately sanded without
significant indentations and other types of deformations. This indicates the inability of
the particular material to carry significantly higher loads after the first slip has occurred at
the free end.
83 5. Classification Based on the Dependence on Concrete Strength
The bond behavior of reinforcing bars can also be classified based on their dependence on concrete strength.
As per The equation proposed by ACI committee 440, the development length equation for the pull-out controlled failure for FRP rebar is given by
αk (15)
Where ld is the development length for the reinforcement, db is the diameter of the
reinforcement ; ffu is the guaranteed tensile strength of the FRP rebar, in psi; α is the bar
location modification factor ;and k is the cover modification factor. It is evident from
equation 4 that the bond-strength which varies inversely with the embedment length is
independent of the concrete compressive strength.
As per the building code requirements for structural concrete (ACI 318-08), the
development length for the deformed bars and deformed wires in tension, is given by
(16)
From equation-16, it is evident that the bond strength for the steel bars with concrete matrix, which varies inversely with the development length, varies directly with 84 the square root of the concrete compressive strength. This is in sharp contrast with the
case of FRPs where the bond-strength is not the function of the concrete compressive strength.
This difference can be attributed to the many research done in the field of bond-
strength of FRPs in the past few years, which has shown that the bond strength of FRPs
doesn’t increase with the increase of the concrete compressive strength. Unlike steel bars,
the bond strength of GFRP was found to be independent of the concrete compressive
strength (B.tighihourt, 1999).
In the case of FRPs, the bond strength is governed generally by the FRP rebar,
as the pull-out failure is not followed by the crushing of the concrete in the vicinity of the
rebar-concrete, but by the crushing of the reinforcing material in transverse direction and
the gradual removal of the surface-effects such as sanding. As per the pull-out tests
results reported by Nanni (1995), the compressive stress at the vicinity of concrete-rebar
interface, was found to be considerably insignificant, of the order of 0.2 MPa, which is indicative of the fact that the concrete at the vicinity of FRP rebar is stressed very less by the bond action thus preventing the formation of micro cracks (A.Nanni, 1995).
7.7 Constitutive Relation for Bond-Slip Mechanism for FRPs
The relationship between the bond stress and the slip of the reinforcing bars can
be considered to a very important subject of study. It can render us with the significant
insight into the bond behavior of the particular reinforcement and can provide us with the
significant information governing the various structural properties dependent on these
85 parameters. Bond-stress and slip behavior can be particularly employed as a constitutive
relation for FRP-concrete for development length studies as the bond stress varies along
the length of the rebar (A.Maji, 2005).
The study of the bond-slip behavior as a constitutive behavior may also be
important for the numerical analysis for the FRP-concrete interaction. In contrast to the
steel, which has almost gained universality in its method of manufacturing, construction
methods and structural applications, FRPs can be unarguably regarded to be far away
behind this. Owing to the lack of uniformity in the method of manufacturing and the
constituent materials, FRPs are generally available with different surface textures,
different polymeric resins and different surface characteristics. Owing to its inherent
anisotropic and heterogeneous properties, even the bond-slip relation may be variable even for the same type of material. This furthers renders the analysis of bond-slip
behavior more complicated for the case of FRPs. This necessitates the development of the
local-bond stress relationship for the study of the related structural properties such as
development length, anchorage length, and transfer length.
The bond-slip behavior is shown to be important for the study of dynamic
behavior of the reinforced-concrete structures. In the case of cyclic-moment reversals, as
under the circumstance of earthquakes, the interior beam-column joints may be vulnerable in regard to the bond behavior between the reinforcement and the concrete which further necessitates the cognizance of bond-slip relationship (G.Russo, 1990).
86 7.7.1 Local Bond-Slip Relationship
As already discussed, the local bond-slip relation is very important for the study of the various associated structural properties. If the local bond behavior can be mathematically modeled as the bond-stress as the function of slip, the governing differential equation can be solved to yield the corresponding pertinent relations. The governing differential equation can be expressed as (G.Russo, 1990):
(17)
Where
db = diameter of the bar
Ab = cross-sectional area of the bar
Eb = young’s modulus of elasticity of the bar
τ(s) = bond-stress function as the function of slip
s = s(x) = variation of slip along the length of the bar where direction x is referring to the
longitudinal direction of the bar
Equation (17) is valid with the following assumptions (F.Focacci, 2000):
1. FRP has linear stress-strain relationship in longitudinal direction
2. Slip at each point in the FRP bar surface is equal to slip at the FRP-concrete interface
87 If the corresponding local bond-slip relationship can be derived based on the experimental results obtained from the respective bond-slip tests, important structural properties related to the particular FRP can be derived by imposing the suitable boundary conditions on the given governing differential equation. For instance, from the beam-tests done on the CFRP bars by A.Maji and A.l.Orzoco for the study of bond-behavior, a model was assumed which involved the constant shear stress due to bond at the concrete and CFRP interface. This yielded a nonlinear distribution of slip across the length of the bar and linear distribution of normal force which was validated by strain-gauge measurement. However, the assumption of constant shear stress due to the bond at the interface may not be a very realistic model in the case of FRPs (F.Focacci, 2000) in most of the cases owing to its weak strength in transverse direction and other heterogeneity related to its material properties. Since the transverses properties of a FRP rod is governed by the quality of resin, there is every possibility of the development of different interaction mechanisms between the FRP rod and the concrete in contrast to the steel which has a significant strength in the transverse direction also (Cosenza, 1997).
While steel possesses a shear capacity of approximately one third of its tensile strength, some fibers are found to have strength of only five percent of its tensile capacity
(C.W.Dolan, 1993). This difference might be very significant in many cases and
particularly, in the case of bond-slip relationship which the consideration here is. There is
every chance of the development of different interaction mechanisms owing to this
significant difference in their mechanical properties.
However, referring to the research of A.Maji and A.l.Arozoco, where a constant
shear stress condition was assumed to exist at the interface, the model yielded the
88 relationship for slip and normal force distribution along the length of the bar which were
consistent with the experimental results. This can render us with the very important
insight on the very local behavior of the different FRP bars on different situations and the
utmost importance of the universalization and homogenization of the construction
procedures and the development of all-comprehensive and consistent design guidelines
and methodologies. In the absence of the standard production procedures and design
guidelines, it looks judicious to resort to the method of developing the local bond-slip
relationship which may vary for even the same type of material owing to the
unpredictable and variable surface deformations and characteristics.
7.8 Analytical Models of Bond-Slip Relationship
As already been discussed above, the bond-slip relationship can be very useful
constitutive relationship for the study of the various pertinent structural properties of the
particular material. It is evident that bond-slip behavior is not an intrinsic property of the
material itself but the property of the combined system of the material and the matrix
where the material is embedded. Therefore, this relationship can be very important in the
study of various responses of reinforced concrete structures which necessitate a
composite action and give a composite response. For the detailed theoretical non-linear
finite element analysis of the fiber reinforced concrete structures, a bond-slip model is required (W.Xue, 2008).
Due to the lack of extensive amount of research and work in the field of FRPs, there are not completely accepted mathematical models representing bond-slip relations.
89 However, there are some mathematical models which are mostly accepted to be representing the bond-slip behavior of the FRPs. These analytical models already developed to represent the bond behavior of the FRP bar are supposed to generate a general law applicable for any type of FRPs by determining the associated constants by curve fitting (E.cosenza,1997). Malvar can be regarded as the pioneer in the field of analytical bond-slip research for the FRPs and he is considered as the first person to propose the analytical model to represent the bond-slip behavior in the case of FRPs. The commencement of theoretical study of this subject started in the 1990s and the culmination of this study came as the Malvar model in 1994.This is followed by the subsequent research and studies which resulted in the various important results in the corresponding field. Some of the important models are described below:
1. Malvar Model
On the basis of the extensive experimental research carried on GFRP rebar,
Malvar proposed the first mathematical formulation for bond-slip relation for the GFRP bars. The research accounted for the different types of outer surfaces and for different values of confinement pressure and for fixed tensile strength of the concrete (E.cosenza,
1997).The model is represented by the following equation:
(18)
90 Where
τm = peak bond stress
Sm = slip corresponding to peak bond stress
F, G = empirical constants determined for each bar type
2. Eligehausen, Popov and Bertero Model (BPE model)
The bond-slip relationship originally developed for deformed steel bars by
Eligehausen is successfully applied for FRP bars which resulted in BPE model. As per
this model, the bond-slip curve is modeled as curve consisting of four distinct parts. The
first part consists of nonlinearly ascending branch, followed by a branch with constant
shear-stress, followed by a linearly descending branch and finally followed by a
horizontal branch. The analytical law which represents the nonlinearly ascending branch
is expressed as:
(19)
τ1 = maximum bond strength
S1 = corresponding slip
α = curve fitting parameter which can’t be greater than 1
91 The other three branches similarly require the determination of other three independent parameters. These parameters are required to be determined based on experimental results.
3. BPE Modified Model
The modified BPE model was proposed by Cosenza (1996) by modifying the original BPE model. From the experimental results obtained from the tests done with the
FRPs, it was concluded that the bond-slip curve for FRPs lacks the second branch of constant bond stress as proposed by the original model (E.cosenza, 1997). Hence the modified model incorporates the original relation for the nonlinearly ascending branch, a softening branch given by the relation
(20)
Where
τ1 = maximum bond stress s1 = slip corresponding for maximum bond stress p = some parameter related to softening
A horizontal branch to represent the complete dependence of bond on the surface friction only when the slip reaches some critical higher value, s3. Therefore the modified model requires the determination of three parameters: α, related to the
92 nonlinearly ascending branch, parameter P, related to the linearly ascending branch and shear stress τ3, after the critical slip value is reached to represent the horizontal branch.
Modified BEF model seems to be a very good model to represent the bond-slip behavior of FRP as the various branches looks strikingly capable to simulate the corresponding respective processes which will be activated with time, while there is the bond interaction between FRP rod and the concrete. The nonlinearly ascending branch can be regarded to be representing the friction and chemical adhesion for lower values of loads subsequently followed by wedging effect or mechanical interlocking for higher loads. The nonlinearity of the curve can be attributed to the complex response generated by the interaction of FRP bar and the concrete at the considerably lower slip values due to the multiple interplay between the associated processes of chemical adhesion, friction and wedging effect. When the maximum bond stress is reached, which physically represents the stage when concrete between the strands starts to get crushed, there will evidently be a decrease in the bond-strength with increasing slip. There will still be some residual resistance provided by the gradually decreasing mechanical interlocking and the surface friction.
This second stage is represented by the linearly softening branch of the modified
BPE model. When the threshold slip value is reached, the total bond action is totally dependent on the surface friction which is represented by the third horizontal branch representing the constant bond stress.
93 3. CMR Model
Most of the structures are generally designed for serviceability limit-state such
that a more elaborate and detailed model may be required for representing the ascending
branch of the curve only (slip less then Sm) (E.cosenza,1997). The new model for the
ascending branch was proposed by Cosenza (1995).This model is represented as:
(21)
Where
τm = peak bond stress
Sr and β = parameters dependent on the curve fitting of the experimental data
From the CMR model, it is evident that it yields the condition of zero initial
slopes which correctly represents the physical phenomenon of dominance of adhesion
factor at the lower load values. BPE model also has these same characteristics, however,
the MALVAR model has a finite initial slope. (E.cosenza, 1997).
The proposed model for bond-slip behavior of the CFRP bars, by W.Xue
(2008), resembles very closely the BPE model. On the basis of the pull-out carried out on
CFRP bars of diameters ranging from 12.5mm to 15.2mm, he concluded in a similar
fashion that the curve can be divided in three different phases comprising ascending
branch, descending branch and the horizontal branch. He proposed the following
equations:
94
Ascending branch (22)
As per the study of W.Xue, the α value corresponding to a CFRP rod for this particular set of tests is found to be equal to 0.1. Similarly the expression for the descending branch is given as
(23)
Where the constants are given by
(24)
(25)
Horizontal branch (26)
Here τ1 is the maximum bond stress and s1 is the corresponding slip. τ2 is the
point which represents the point from which the descending curve changes to horizontal
branch (W.Xue, 2008). Similarly, from the research carried out by B.Tigihourt, for GFRP bars of diameters ranging from 12.7 to 25.4mm, the experimental data obtained were
calibrated for the CMR model. The proposed bond-slip model is given by the expression
(27)
95 Where
τm = peak bond stress and
S = slip corresponding to particular bond stress. (B.Tigihourt, 1998).
From the above discussion, we can safely conclude that the BPE and CMR models seem to be able to simulate the bond-slip relationship for the FRP bars in most of the cases. They both propose the zero initial slopes, signifying physically the mobilization of adhesion for the development of the bond stress at the earlier phase of the loading, thus more representative of the actual phenomenon in the case of FRPs. They both propose the nonlinearly ascending branch thus signifying the gradual increase in the bond resistance due to the complex interplay of chemical interaction, surface friction and the mechanical interlocking phenomenon.
From the consideration of serviceability limit state, the most important part of the curve for analysis purposes is the ascending branch. Hence, any one of the model can be calibrated for generating the ascending curve for all practical purposes. For the small values of slip, it can be shown that both mBEP and CMR model are effectively equivalent. This observation can be particularly useful for the cases involving the small maximum allowable slip value (E.cosenza, 1997). For FRP bars with granular surface treatment such as the case of sanded surface treatment which generally exhibited a small slip value corresponding to maximum bond stress Makitani, (1993). These models will be particularly significant in the analysis.
96 7.9 Experimental Methods and Analysis
There has been a considerable amount of effort and research conducted in the last few decades to study the bond behavior of steel as well as that of FRPs.The historical study by Abhrams at the university of Illinois in 1913 on the bond behavior of plain steel rebar and concrete can be regarded as the cornerstone of the commencement of the systematic and organized research on the bond behavior. He conducted the number of pull-out tests to investigate the bond behavior of the plain steel and concrete, and proposed a simple two step bond model which is widely accepted even nowadays
(L.R.Feldman, 2007).
Regarding the inherent differences in the different mechanical properties and external geometry of the steel and the FRP bars, it is highly desirable to have some systematic research methods for the bond analysis in the case of FRPs. On the basis of the extensive study on the bond behavior done on the FRPs on the last two decades by the scientists, it can be safely inferred that the common method is to implement the pre- established experimental methods as for the steel bars and to interpret the results in the light of the markedly different mechanical properties and surface characteristics of the respective FRP. The development length equation proposed by B.W.Wambeke and
C.K.Shield for the FRPs which has been incorporated in ACI 440.1R-3, was based on the beam tests done on the GFRP bars, was developed by a similar methodology of
Orangun applied to determine the development length of the steel bar found in ACI 318-
02 (B.W.Wambeke, 2006).
97 The importance of the study of bond behavior for the establishment and standardization of various important structural properties has already been discussed in previous chapters. These studies can provide us with valuable insight into the failure modes of the respective FRP, and subsequently into the development length analysis for the particular material. Development length or anchorage length is one of the most important structural properties as it dictates the applicability of the given material as a high-strength material for the particular purpose and hence requires more detailed scrutiny.
Based on the previous research and work done in the respective field, the general approach for analysis can be divided into the following headings:
1. The data obtained from the experiments are statistically correlated to obtain the desired relationship for the anchorage length.
2. The data obtained from experiments are statistically correlated to define the local bond-slip model and the anchorage length is subsequently derived from mechanics based approach.
The literature search reveals the fact that the first method is the most general and widely used methodology which is being employed for the determination of the anchorage length. The work of Wambake and shield, Ehsani, Saadatmanesh and Tao, and by
B.Benmokrane, B.Tigihourt and O.Challal can be regarded to be falling under this heading. This method is aimed at yielding a more general and global relation which couples the important bond parameters and provide us with more or less unitary approach for the determination of important parameters like maximum bond stress and anchorage length for all the FRP materials or for the particular one. These relations
98 include independent parameters like diameter of the reinforcing bars, compressive strength of the concrete and the ultimate strength of the FRP material such that they can be used for different kind of FRP bars and sizes which render them with more global and
comprehensive applicability.
The second approach, which is more local in its nature and range of
applicability, has its own particular advantages. This method can be used as the method
to check the magnitude of the parameters like anchorage length. This method includes the
calibration of the experimental data to yield a particular bond-slip model for the
particular material. Once the bond-slip model has been defined based on the experimental
data, the boundary conditions can be applied on the governing differential equation and
can be numerically or analytically solved. Defining anchorage length as the minimum
length required to develop the nominal tensile strength of the material within the
particular length with no slip at free end, the following boundary conditions can be
applied:
Boundary condition-1 at (28)
Boundary condition-2 at ) (29)
Where the axis-X is assumed along the embedment length with origin at the unloaded
end. Lan is the desired anchorage length, N(x) as the normal force along the length will be
evidently the nominal tensile strength of the applied FRP bar, EF is the modulus of
elasticity of the respective material and AF is the cross-sectional area of the FRP bar. The
first boundary condition signifies that the slip at the unloaded end is zero and the second
99 boundary condition implies that at the end of the anchorage length, the strain as the first differential of the displacement or slip must be numerically equal to the nominal tensile strength of the applied fiber reinforcement divided by the area of the bar and the modulus of elasticity of the respective material.
The boundary conditions as specified above alongwith the analytical bond-slip model, will yield the distribution of slip and normal force as the function of the axial coordinate. When the functional relationship between the normal force along the embedment length and the axial coordinate is established, the length of the bar to develop the normal force equal to the nominal tensile strength can be easily interpolated. This kind of approach can be observed in the work of F.Focacci (2000) and A.Maji
(2005).Focacci and Nanni calculated the anchorage length required for the particular FRP bar by the process described above. The modified BEP model was considered to model the bond-slip model and the experimental data were calibrated to yield the required bond parameters. Then the required anchorage length was calculated as per the procedure described above. In the case of A.Maji and A.L.Orozoco,a constant bond-stress model was used. The anchorage length was taken from the bond-test which was finally used to yield the functional relationship between the normal force and slip along the anchorage length and the axial coordinate.
Once the nominal tensile strength of the FRP bar is known, the anchorage length required can be calculated on the basis of the calibrated bond parameters (F.Focacci,
2000). From these studies, the anchorage length can be observed as the function of the bond-parameters which strongly signifies the capacity of this approach to actually simulate the physical situation involving the bond-mechanism. As the anchorage length is
100 the structural property of the given material which is directly linked with the bond
properties, the capacity of this approach to correlate the anchorage length with the bond
parameters can be undoubtly regarded as its inherent capacity to simulate the actual
physical situation involving bond mechanism. However, the shortcomings of this method
is also worth discussing here. One of the foremost drawbacks of this model is the
prediction of the correct bond-slip model. The available analytical models are very less
and they may not be able to capture the characteristic nonhomogenity associated with the
FRPs owing to their variable manufacturing processes and non-standard surface- treatment process. The FRPs are available with different surface characteristics, which is the most important phenomena dictating the consequent bond-behavior of the respective material. Due to the variability associated with this phenomenon, it looks evident that it entails the requirement of further researches and works in this field. Since there are not any globally accepted analytical bond-slip models for the FRPs, an all comprehensive influence of this approach can be surely questioned. If the closed form solution of the governing differential equation is not possible for the proposed bond-slip model, then it may further entail more computational efforts and may demand the numerical computation procedures. This also may not be desirable from different perspectives.
Another methodology ,as listed as number one above, correlates the experimental data obtained from the designated test methods to obtain the anchorage length requirements.
Hence, this methodology comprises the various test methods for the determination of various bond relates properties ,specifically development length. The prevalent tests methods can be broadly classified in the following two types:
101 1. Pull-Out Test
2. Beam-Tests
Although there may be multifarious ramifications of these two test methods like double pull-out or tensile specimen, modified cantilever beam, beam-end specimen
(D.A.Howell,2007), and so on, depending on the nature of the material, experimental constraints and the experimentalist’s individual insight and ingenuity, the majority of the bond-tests can be categorized in the above two headings. Before going into the details of these two methods, it will be very pertinent to describe the various failure modes associated with the bond-tests.the primary failure modes in majority of bond tests comprise of (Ehsani, 1993):
1. Tensile failure of the embedded material
2. pull-out of the embedded material from the concrete-matrix
3. Tension splitting of the concrete
Tensile failure of the material happens when the material reaches its ultimate tensile strength outside the concrete specimen. The second mode of failure, which is the pull-out, happens when the shear stress generated at the concrete-rebar interface is not sufficient to counteract the applied tensile load or the flexural load. If the shear strength generated at the interface is large enough, as in the case of deformed steel bars, the principle tensile stress generated by the shear reaches the tensile strength of the concrete.
This will insinuate the crack-formation and which further entails the development of radial splitting forces at the bar surfaces (R.Tefers, 2003).
102 The tension splitting failure mode is generally observed in structures and is
common with the bars with small concrete cover (Ehsani, 1993). In the case of FRPs,
which are characteristically weak in the transverse direction and doesn’t possess the
sufficient surface deformation to generate strong shear-stresses due to lateral confinement by the rib-area, this failure mode may be delayed. However, on the basis of the works of
R.Tepfers and L.D.Lurenzis, the crack pattern pertaining to the splitting failure in concrete is strongly dependent on whether the surface is with or without lugs, is glossy or
rough. Glossy surface increases the slip and hence give rise to considerable splitting
forces. (R.TEFERS, 2003) .Therefore, there seems to a non-linear relationship between
the pull-out and splitting failure mode. The splitting mode of failure can result in more
slip due to resulting decrease in shear-stress which further increases the splitting forces.
To put the other way round, the insinuation of pull-out failure can also result in the
conditions favorable for splitting forces generation.
However, from the past researches, there seems to a good correlation between
the concrete cover and the mode of failure. The larger concrete cover surely delays the
splitting of the concrete and thus increases bond-performance (Ehsani, 1993).
B.W.Wambeke and C.K.Shield conducted an extensive experimental program to study
the bond-behavior of the GFRP bars which includes 240 beams with the size of
reinforcement ranging from 0.5 in to 1.128 in. Among these, 94 beams resulted in pull- out failure. Approximately fifty percent of these specimens resulting in pull out failure have clear cover of more than three bar diameters. On the other hand, the beams resulting in splitting failure mode have clear cover between one and three bar diameters. Hence there seems to be a strong correlation between clear-cover and the resulting failure mode.
103 Since the failure modes and the physical parameters associated with them have been briefly discussed, we can again resort to the original discussion on the experimental methods. The history of the experimental methods can be successfully traced back to the early investigators like Abrams. He was reported to have conducted pull-out cylinder tests, in which the test set-up consists of bar specimen embedded in the concrete with free end protruding from one end of the cylinder. Specimen was placed on a bearing block and tested to either pull-out or tension-splitting failure (D.A.Howell, 2007). This can be regarded as the early precursor of the cylinder pull-out tests for the FRPs.
This method is still widely used due to its inherent simplicity and its capacity to give a reasonable evaluation of the anchorage length required for the particular material.
However, the alternative methods were sought to remedy the associated shortcomings of this method. The consequent test method is termed as BEAM-TEST as previously mentioned. The method varies greatly in the following manner:
1. Stress-Field Developed in the Tested Specimen
In the pull-out test, the cylinder specimen is mounted on the testing machine and the tensile load is applied at the free end of the specimen. This will induce uniform compressive stress field in the concrete for equilibrium, transferred from the bearing of the specimen on some kind of bearing block. This is in complete contradiction with the stress field existing in the actual situation, such as in the case of the flexural members like beams. In beam members, under the actual loading, both the reinforcement and the concrete will be under the influence of tensile stress field. Hence the major shortcoming
104 of the cylinder pull-out test method is its incapacity to simulate the actual stress field.
Also the boundary conditions of concrete on bearing plate may also be influential on the
resulting bond-stress (D.A.Howell, 2007). As pointed by Leonhard, the placement of the
concrete specimen on the bearing block gives rise to friction and hence the concrete may
be subjected to lateral stresses, thereby providing the upper bound for the measured bond-
stress.in the case of cylinder pull-out test the average bond-stress is calculated as in
equation-13.The tensile load is directly provided by the experimental data. On the other
hand, in the case of beam test, the tensile load on the reinforced and concrete interface is
calculated indirectly from the bending moment requirement. The resulting expression is
(30)
Here T is the tensile force in the reinforcement, P is the applied load and a is the shear
span. J is the lever-arm, which can be directly measured in the particular beam as the centre to centre distance from the hinge-arrangement at the top of the beam to allow free
rotation to the centre of the reinforcement. The calculated value of tensile force from the
above bending moment requirement is equated with the bond force acting along the
embedment length to compute the average bond stress.
2. Position of the Reinforcement while Casting
The characteristic feature that differentiates these two methods of bond-test is the position of the reinforcing bar while casting. It is evident that the position of the
105 reinforcing bar while casting the pull-out cylinder is vertical. In contrast to this, in the
case of beam tests, the position of the reinforcing bars will be horizontal during casting.
This may give rise to different kind of responses in regard to the bond property. One of
the responses of key interest is the top-bar effect. Top bar effect can be defined as the
ratio of ultimate bond strength reached in pulling out the bottom bar to that reached in
pulling out the top bar. Due to the bleeding and air entrapped beneath the top rebars, there
is every likelihood that the top bar is less consolidated than on the bottom. There is the
agreement among the researchers that the top-bar effect adversely affect the bond
strength. (Ehsani, 1993).
Ehsani conducted a test to investigate this phenomenon. This included the first
specimen, which has a rebar casted with a clear cover of 19mm from the bottom of the
specimen. The second specimen consisted of rebar casted with 280mm of concrete cover
below the bar and a clear cover of 19 mm from the top of the specimen. The load-slip
analysis of these specimens indicates that the allowable bond stress in the top bar is 66%
of the bottom bar. This further establishes the fact that the two test methods also have
significant differences between them in this respect also. Since cylinder pull-out test is independent of the top-bar effect, this may provide a considerable advantage over the beam test for a quicker and reliable analysis of the bond-behavior of the FRP rebar.
106 CHAPTER VIII
MATERIALS AND MIXES
Since the primary objective of this investigation is the study of the bond behavior of the Basalt FRP bars and the concrete, the major structural materials that were used for the research purpose is the conventional concrete and basalt FRP bars of specific sizes. The other materials that were used were the auxiliary materials that were required to make the research functional in a smooth and coherent manner.
8.1 Materials
This section includes a brief outline of the materials that were employed for the investigation of the bond behavior of the basalt FRP bars with the concrete. This section is intended to give a comprehensive overview of the various materials that were used, their relevant characteristics, specifications and their associated properties.
107 8.1.1 Basalt FRP Bars
The Blackbull basalt FRP bars which were used for the research were provided
by the sponsor Blackbull. The Basalt bars provided for the research were of net fiber
diameter of 3mm,5mm and 7mm.These sizes corresponds to the net area of fibers only.
The method for manufacturing for the basalt FRP bars is Wet-Lay up process. Wet Lay- up process is a very simple of producing FRP composite materials and is done manually.
The process essentially consists of drawing the fibers and through polymer filled nozzle
such that it yields the usable composite material when hardened and cured. The fibers for
the case of basalt FRP bars were extracted from the igneous rock called Basalt. The
primary composition of Basalt rock is generally constituted with various forms of oxides,
silica-oxide being the most abundant one. The percentage of silica oxide is generally
between 51.6 to 57.5 percent and generally the basalt with the silica-oxide content above
46 percent (acid-basalt) is considered good for fiber-production.
Minerologically, Basalt is primarily constituted of minerals Plagioclase,
pyroxene and olivine. When heated at high temperature, basalt is capable of producing a
natural nucleating agent which plays a major role for the thermal stability of the material.
This explains the apparent increased volumetric integrity of basalt as compared to the
other materials. The presence of the above mentioned minerals may be a helpful factor
for this phenomenon. The polymeric resin used as the matrix for the basalt fiber is the
Vineylester resin. A vinylester resin is the combination of an epoxy and an unsaturated
polyester resin (L.C.Bank, 2006).
108 The advantage of vinyl ester is that it has the meritorious physical properties of the epoxy and the beneficial processing properties of a polyester resin.
Figure 29: Typical Basalt FRP bar used for the Pull-Out Test
The actual sizes of the provided FRP bars were measured in the laboratory with a high precision vernier-calipar. Their gross diameters including the polymeric matrix
were found to be 4.7mm, 7mm and 10mm respectively for the bar of sizes 3mm, 5mm
and 7mm.The resulting fiber-volume fractions were found to be 44%, 52% and 41%
respectively. Hence, the average volume fraction was worked out to be 46%. The Basalt
bars will be referred later on with reference to their net diameter. A typical picture of a
Basalt FRP rod used for the Pull-out test shown in Fig-29. The outer surface of the Basalt
bar is provided with the sand along with a helical winding along its length to enhance its
bond-properties. This can be observed in Figures shown above. However, the sanding is
not abundantly provided and the helical windings provided were not making considerable
indentations.
109 8.1.2 Anchorage Tubes
Since basalt is an anisotropic material, the strength of the basalt fiber in the transverse direction is very low compared to its very high tensile strength in the longitudinal direction. This necessitates the provision of some kind of anchor at the end of the specimen so as to provide proper grip while applying the load during the Pull-out
Test. To meet this purpose, one steel tube, at one end, was provided. The steel tube used for the anchor are black-welded steel tubes each eighteen inch long with threads at each end to facilitate the positioning of the cap.
The outside diameter of the tube is 1.315 inch and inside diameter is 1.049 inch.
The wall thickness of the tube is 0.133 inch. The tubes matched the specifications as required by ANSI, MSME AND ASTM. The ASTM specifications met are ASTM A53 and ASTM A733. The other end of the pipe is provided with the rubber-cap to hold the bars in vertical position. The rubber caps were black soft-rubber grips to hold basalt FRP bar in a vertical position at one end. The holes were drilled on the caps in the laboratory as per required by the size of the bar being tested.
110 A typical picture showing the arrangement of the anchor tube for the Pull-out Test is shown in Fig-30.
Figure 30: Anchor for the Pull-Out Test
8.1.3 PVC Conduits, Hard Rubbers and Steel Plates
A 12 mm diameter commercial PVC pipe was purchased from the local
hardware distributor. The PVC pipe was of the thickness of 2.5mm. The PVC conduits
were provided at the both ends of the cylinder to debond the basalt bar at the edges of the
pull-out cylinder. Fig-31 as shown below clearly shows the PVC conduits at both ends of
the pull-out cylinders.
111
Figure 31: PVC Conduits for the Pull-Out Cylinders at Both Edges
A rectangular steel plate of thickness 1.25 inch was provided to facilitate the development of uniform bearing pressure at the bottom of the test specimen. A hard rubber pad of ¾ inch thickness was also provided to make the distribution of bearing pressure more uniform and to avoid any kind of effects due to the unevenness on the bottom surface of the cylinder. The steel plate and hard rubber pad is shown in Fig-32
below.
Figure 32: Steel Plate and the Hard Rubber
112 8.1.4 Grouting Material-Epoxy and Sand
The epoxy used for the purpose of grouting the steel tubes with the Basalt FRP bar is a structural epoxy with a commercial designation of AKA-Epoxy system. The mix
ratio for the particular epoxy as per provided by the manufacturer is 1 part resin to 1 part
hardener by volume. The gel time as per specified by the manufacturer was 180 minutes for the particular epoxy. The viscosity of the epoxy, as per reported by the manufacturer was around 2300 centipoises. A small amount of sand was mixed with the structural
epoxy as described in the experimental program section of the tensile strength test. The
sand used for the purpose was the river sand obtained from a local supplier.
8.1.5 Coarse Aggregate
The coarse aggregate used for making concrete for the cylinders was crushed
limestone aggregates with the maximum size of ¾ inch (19mm). The aggregates were
angular and free from clay and other impurities.
8.1.6 Fine Aggregate
The fine aggregate was river sand purchased from the local supplier. The sand
was free from clay and other inert impurities.
113 8.1.7 Water
The water used for the concrete mixes was the normal tap water supplied by the city of Akron.
114 CHAPTER IX
TEST PROCEDURE
This section provides a comprehensive outline of the total test procedure for the study of the bond behavior of the basalt FRP bars from cylinder pull-out test. As already been discussed in the literature review portion for the bond test, pull-out is a simple and very useful test method for the investigation of bond behavior. Twelve pull-out specimens were prepared comprising four for each bar size. The net diameters of the available BFRP bars were 3mm, 5mm and 7mm.
0.75" thick steel plate
0.75" thick hard rubber A A 10" 12"
pvc conduits
Figure 33:Cross-Section and Elevation of the Pull-Out Specimen
115 As shown in Fig-34 above, the principle underlying the cylinder pull out test is very simple. Basalt FRP bars to be tested were embedded in the concrete cylinder of height 12 inches and diameter 6 inches. The other end of the bar is grouted with the steel tubes to facilitate the gripping while testing in the universal testing machine. The overall length of the basalt FRP bar was 54 inches.
The total length of the Basalt bar included the embedded length of the bar in cylinder of height 12 inches, steel-tube grips of length 24 inches and free length of 14 inches. It can be seen from the sketches that a one inch protrusion is provided at the bottom of the cylinder to monitor the slip of the bar during testing. The embedded length was provided with one inch long PVC conduits at the ends of the cylinder to eliminate the
“edge-effect”, resulting in an effective embedment length of 10 inches.
The bond effects at the edges of the cylinder may not be truly representative of the actual stress condition and may provide the overestimation of strength. The PVC
conduit inserts can be seen in Fig-34.
Figure 34: PVC Conduits at the Edges 116 In the sectional elevation in Fig-33, we can see that a 0.75 inch thick steel plate
and hard rubber with concentric hole were provided on the top of the cylinder. This is
meant to provide adequate bearing and subsequent distribution of the compressive stress
that is generated on the cylinder surface resting on the cross-head of the universal testing
machine during the pull-out test. In the following sections, the comprehensive description
of the entire procedure is provided.
9.1 Preparation of Specimens
The preparation of specimen is evidently the most important part of any research
study. Only the specimens that have been prepared as per the standard specifications and
based on the previous experiences of the other researchers can provide the right insight
into the property of the material being studied. Needless to say, depending upon the
gravity of the research, this may be the most demanding aspect of the research from
overall perspective. The cylinder pull-out test for the basalt FRP bars was not an exception.
9.1.1 Preparation of Wooden Frameworks
From the sketches of the specimens as shown in Fig-36, it can be observed that the cylinder pull-out test of the FRP bars includes the casting of cylinder at one end of the basalt FRP bar and grouting of the steel-tube at the other end. This was a relatively arduous task as the length of the basalt FRP bar required for the purpose is 54 inch. The
117 provision of the protrusion of 2 inch at the end of the cylinder also posed the problem for
the placement of the specimen in a horizontal plane. This necessitated a need for a
framework which can hold the specimen vertical while grouting the steel tubes and at the
same time can provide some measure for the placement of the specimen in horizontal
plane with the protrusion at its end. To serve the purpose, a wooden framework was
constructed. The sketches of the framework are shown in Fig-36 below. As can be seen from the sketches, the framework was able to provide the perfect verticality and horizontality for making of test.
Figure 35: Wooden Framework
118 Fig-36 as shown below shows the arrangement at the horizontal support of the frame to accommodate the protrusion of the basalt FRP bar at the bottom of the cylinder.
It can be seen in the figure that circular holes had been cut in the horizontal plywood to accommodate the protrusion, thus assuring the horizontality of the specimen. The ply was supported by 3.258*1.25 in wooden blocks as required. These are shown in Fig-36 and
Fig-37.
Figure 36: Horizontal Arrangement to hold the Specimens
9.1.2 Casting of Pull-Out Cylinders for Compressive Strength Test
The next step after making the wooden framework was the casting of the pull- out test cylinders and the cylinders for the compressive strength test. The size of the cylinder for the pull-out specimen was 12*6 in. PVC cylinders were used for casting the 119 specimens. Holes were made at the bottom of the PVC cylinder using a drill-bit of required size to accommodate the protrusion of Basalt bar. Similarly, eleven 8*4 in cylinders were casted the same day for the compressive strength test conforming to the relevant ASTM standard for the hardened concrete test. The plastic molds were cleaned with a clean piece of rag and were subsequently oiled to prevent the sticking of the concrete on the surface of the mould. The concrete used as per the mix-proportion as described above. The basalt bars were positioned vertically on the horizontal ply support with the help of the concentric hole on the cap of the cylinder. Fig-37 and 38 shows the arrangement before casting of the cylinders and the cleaning and oiling of the cylinders.
Figure 37: Arrangement for Specimen Casting
120
Figure 38: Cylinders for Compressive Strength
All the required components for the mixing of the concrete were weighed and stored in a
dry place in the laboratory the day before the day of mixing. The cement was covered with double layer of plastic to avoid the absorbance of moisture. The water was also covered to prevent any kind of vaporization. The concrete was mixed in double batch in a mechanical mixer in the civil engineering materials lab of the university. The mixer is visible in Fig-38 above. Ten percent of the mix was also prepared as the butter mix to prevent any kind of loss of cement during the mixing. The mixer was fed with the butter mix first followed by the actual mix. First the coarse aggregate, sand, cement and one third of the quantity of water was fed to the mixer. The materials were mixed for three minutes and the mixer was stopped for one minute to allow for the hydration of the cement. The remaining water was added and the mixer was run for three more minutes.
The prepared concrete was carefully delivered to a wheel barrow to facilitate the
transportation of concrete. PVC cylinders were filled with fresh concrete and were 121 vibrated on a vibrating table to assure good consolidation of the concrete cylinders. The filling of the cylinders were carefully done so as not to interfere with the positionoing of the PVC conduits. The cylinders were then carefully placed on the horizontal platform with an arrangement to accommodate the protrusion on the bottom.The filled cylinders
are shown in the Fig-39.
Figure 39: Concrete Cylinders
122
Figure 40: Cylinders on Horizontal Platform
The cylinders that were prepared for the pull-out tests and the compressive strength test were left for one hour before finishing the top surface. The rheology of the freshly prepared concrete doesn’t allow the smooth finish of the top surface. After some time, and hence it will be much easier to finish the top surface. The top surfaces of the cylinders were finished with smooth-faced steel trowel and were covered with the plastic for the twelve hours. All the pull-out specimens and the cylinders were then placed in the humid room of the material lab with hundred percent humidity.
9.1.3 Grouting Procedures
The grouting of the steel tube for the provision of the steel tubes was done in exactly the same manner as for the tensile strength test of the basalt FRP bars. The pull- out cylinders were taken out of the humid room after 30 days and were allowed to dry for
123 one day. The dried cylinders were mounted on the framework with the provision to keep the basalt FRP bars vertical during grouting.
AKA-epoxy system, as described in the materials section, was used for grouting
the steel tubes. The mix-ratio for the epoxy was one part resin to 1 part hardener by
volume, mixed with some small proportion of sand. Sand is dried in the oven at high-
temperature for 24 hours to make it free of any residual moisture. The equally
proportioned resin and hardener were mixed with the help of some mechanical means.
600ml of resin is mixed with 600ml of hardener for one minute and finally the sand was
added and was mixed until the uniformity in color and desired viscosity was achieved.
This volume was mixed enough to make 12 samples ready for the test. The mixing was
done in a mechanically operated blender in a steel bowl. All the apparatuses were cleaned with acetone after the mixing was done. The mixed epoxy was poured in a zip-log plastic
bag with a small hole cut at its bottom. The viscous nature of the epoxy allows pouring
the epoxy in the tubes by gently squeezing the bag. The tubes were regularly tamped and
compacted with a small, clean steel rod to make sure that there is no air-voids in the matrix. The holes on the steel caps were applied with quick-set epoxy to make sure that the tubes would not leak.
9.2 Experimental Program
This section includes the comprehensive account of the experimental program pertaining to the cylinder pull-out test for the basalt FRP bars. This included the pull-out test for the twelve cylinders and the compressive strength of the respective cylinders.
124 9.2.1. Pull-Out Test of the Specimens
There is not any standard test method for the pull-out test for the FRP bars.
However, based on the past research and intense investigations done in this particular
field by other researchers, it was decided to resort to the cylinder pull-out test. The
general schematic for the cylinder pull-out test is shown in Fig-41 below.
Dial-Gauge set-up for Slip data
Data Acquisition System for strain data
Figure 41: Test Set-Up for the Pull-Out Test
As it is evident from the Fig-42, the principle underlying this method is similar to that of the tensile strength test for the basalt FRP bars. The specimen was mounted on the universal testing machine with the top surface of the pull-out cylinder resting on the cross-head of the universal testing machine. This test set-up is shown in Fig-42.
125
Figure 42: Specimen Positioning in UTM
From Fig-42, it can be observed that the top surface of the cylinder was resting on the crosshead of the testing machine supported on the steel plate followed by the hard-rubber
plate. The steel plate and hard rubber pad were provided to provide the specimen with
sufficient bearing area to transfer the compressive stress to the top surface of the cylinder
thus preventing any kind of concrete-crushing. The steel tube was anchored at the bottom
with the trapezoidal grip as shown in Fig-44. The protruded end of the basalt FRP rod was instrumented with a dial gauge to record the slip information at the end. A small steel plate was attached with the help of quick-set epoxy at the bottom of the cylinder to facilitate the magnetic placement of the dial gauge. This is shown in Fig-43.
126
Figure 43: Arrangement for Dial-Gauge Set-Up
It is visible from Fig-43 that a steel plate had been attached to the bottom of the cylinder to enable the magnetic placement of the dial gage.
127
Figure 44: Grip for the Steel Tube
The specimens were tested using a “BALDWIN” material testing machine, which is a 300,000 lb-capacity, universal testing machine. The model of the machine is
300HV-300,000 lb capacity (serial 300HV-1005). The whole arrangement of the specimen in the universal testing machine is shown in Fig-45.
128
Figure 45: Specimen Set-Up in the Machine
The extensometer was positioned approximately at the centre of the free length of the
specimen. The extensometer used for the experiment is large-gage length model extensometer .the model is 3543-SR-0300-200T-ST and the gage length used for the purpose is 2”.the extensometer is hooked up with the data acquisition system, ADMAT, as shown schematically in Fig-41 above. The dial-gage used for the purpose of measuring slip was a dial-gage of a brand called FOWLER with the least-count of 0.0005 in. Since it
was not connected to the data acquisition system, the displacement or slip readings for
the different load values were recorded manually. Two people were assigned to record
the displacement data who recorded the data in unison to prevent any sort of mistake in
the data-recording.
129 The specimens were loaded at a constant rate of 15 pounds per second until it is
stressed to complete failure. The modes of failure observed were tensile failure and pull- out failure.
Figure 46: Pull-Out Failure
Figure 47: Tensile Failure
130 Fig-46 and Fig-47 respectively show pull-out and tensile failure modes. In Fig- 46, the slip reading is visible in the dial-gauge display.
Figure 48: Slip Failure of the Specimen
Figure 49: Amount of Slip in One Specimen
131
Figure 50: All of the Tested Pull-Out Specimens
Figure 51: Two Specimens with Different Failure Modes
132 CHAPTER X
RESULT AND ANALYSIS
This section includes comprehensive description of the analysis performed from the test results obtained from the experimental works performed as described in chapter
9. The classes of data obtained from the experiments can be divided into load-slip data and stress-strain data. The load slip data were used for the load-slip analysis of the respective specimen and the stress-strain data were used to acquire the ultimate tensile strength of basalt FRP bar which subsequently allows the determination of the bond strength of the material. The stress-strain data were also used to calculate the modulus of elasticity of the material and to provide the comparison with the stress-strain response obtained from the direct tensile test.
10.1 Research Objective
The primary objective of this part of research is the investigation of the bond
strength of the basalt FRP bar of three different sizes with the concrete matrix. The test
program as already discussed, consisted of four pull-out cylinder specimen of three different sizes. All the sizes were casted with the embedment length of 10 inches. The
133 tests were intended to yield some useful insight into the bond-behavior, load-slip
response and the required embedment length for different bar sizes .Categorically,
research objective comprises the determination of the bond strength of the basalt FRP bar
of different sizes and the load-slip analysis.
10.2 Bond-Slip Analysis of Basalt FRP Bars
The study of the bond-slip was performed for the three differently sized FRP
bars with the same embedment length. This section will comprise the detailed analysis of
the bond behavior exhibited by the three different sizes. The details of the pull-out and
the results associated with the compressive strength of the concrete is summarized in
table 7 below.
10.2.1 Basalt FRP Bar with Nominal Diameter 3mm
The average maximum load carried by the 3mm diameter basalt bar was 3,311 lb (14.7KN). The details of the load carried by the different 3mm bar were shown in the table. Specimens were designates as C5, C6, C7 AND C8.In all of the four cases, the failure mode was tension-rupture failure and no any kind of slip was recorded by the dial- gauge. Hence in can be concluded that the provided embedment length was sufficient to develop the maximum tensile strength of 303 Ksi for the 3mm Basalt FRP bar. The average bond stress was calculated using equation 1 from the chapter on
134 literature review. The tensile strength developed for each was equated with the bond-
stress developed by the bar-concrete interface along the provided embedment length. For
calculating the cross-section and the embedded area of the bar, the nominal diameter of
the bar was used (i.e. including polymer).the embedment length was 10 inches. The bond
stress developed by each specimen is tabulated in table 8 below with an average bond- stress of 595 psi.
Fig-52 shows a 3 mm diameter specimen after the testing. It can be seen from the figure that the specimen has not undergone any slip and hence, failed by tension- rupture.
Figure 52: Specimen C6 after Tension Failure
In Fig-52, it is clearly visible that the slip at the protruded end at the time of
failure is zero. The provided embedment length of 10 inches was sufficient to develop the
135 ultimate tensile strength of the 3mm FRP bar. However, the development length required for the purpose may be smaller than this value. In the absence of sufficient number of tests, optimization of the development length requirement for the Basalt FRP bar was performed.
10.2.2 Basalt FRP Bar with Nominal Diameter 5 mm
The bond response of the 5 mm Basalt FRP bar was found to be a mixed one for the given embedment length of 10 inches. The failure modes exhibited by the specimens consisting of 5mm bars included both tension and slip failure, thus, signifying that the provided embedment length of ten inches was not sufficient to develop the ultimate tensile strength of the particular size. The response of the different specimens was shown in the table 8. Specimen C3 depicted slip-failure mode while C1 depicted slip failure ultimately followed by tension rupture. Specimen C2 failed by pure tension rupture whereas specimen C4 depicted the slip failure mode followed by the tension rupture in similitude with specimen C1. Fig-53 shows the different failure modes exhibited by 5mm sized specimens. The slip undergone by specimen C3 can be seen in Fig-53.similary ,the picture of the dial gauge reading for specimen C2 was not showing any slip which failed by tension-rupture.
The average load carried by 5mm specimen was approximately 300 ksi. Since three of the specimens failed by slipping, the reduced strength exhibited by 5mm bar was sufficiently justified. Among all the specimens that failed by slipping, the slippage started once a particular threshold load was reached. In the case of 5 mm bars, the
136 average threshold load was recorded to be 8100 lb. This can be explained as the
predominant effect of the friction at the interface between the concrete and the FRP for
the bond after the deterioration of the mechanical-interlocking phenomenon at the higher level of load.
Figure 53: Specimen with Same Size but Different Failure Modes
The threshold load causes a tensile stress of 266 psi and a bond stress of 878 psi.
The average bond stress calculated for the 5 mm specimen was 6.7 Mpa. For the 5mm bar, it was evident that the provided embedment length of 10 inches was not sufficient to develop the ultimate tensile strength of the material of the particular size. The analysis of the bond-slip behavior will be discussed in the next chapter.
137 10.2.3 Basalt FRP Bar with Nominal Diameter 7 mm
The bond response of 7 mm Basalt FRP bars was found to be predominantly governed by slip failure, as expected. Among the four specimens, the failure modes exhibited by three specimens were slip-failure mode and the failure mode of one specimen was tensile rupture failure. This signifies that the provided embedment length of ten inches was not sufficient to develop the ultimate tensile strength of the particular size. The response of the different specimens was shown in the table 8. Specimen C9,
C10 and C11 depicted slip-failure mode while specimen C12 depicted tension rupture failure. Fig-54 and Fig-55 shows the amount of slip undergone by specimen C9 relative to its original position.
Figure 54: Specimen C7 at the Start of the Loading
138 Table 7: Details for the Pull-Out test Including the Cylinders for Compressive Strength Test
Size Of The Basalt Fiber Designation Maximum Load(LB) Date Of Casting Date Of Testing Concrete -Strength By Cylinder Test(psi) Remarks C3/BP1 8147 7/7/2008 9/23/2008 4624.5 Pure Slip-Failure C1/BP2 9099 7/7/2008 9/23/2008 4624.5 Slip followed by Tension-Failure 5mm C2/BP3 9265 7/7/2008 9/25/2008 4651.28 Pure Tension-Failure C4/BP4 9429 7/7/2008 9/25/2008 4651.28 Slip followed by Tension-Failure
C5/BP5 3085 7/7/2008 9/25/2008 4651.28 Tension-Failure C6/BP6 3312 7/7/2008 9/25/2008 4651.28 Tension-Failure
3mm C7/BP7 3457 7/7/2008 9/25/2008 4651.28 Tension-Failure C8/BP8 3389 7/7/2008 9/25/2008 4651.28 Tension-Failure 139 C9/BP9 12940 7/7/2008 9/25/2008 4651.28 Pure Slip-Failure C10/BP10 13040 7/7/2008 9/25/2008 4651.28 Pure Slip-Failure 7mm C11/BP11 13003 7/7/2008 9/25/2008 4651.28 Pure Slip-Failure C12/BP12 16378 7/7/2008 9/25/2008 4651.28 Pure Tension-Failure
Figure 55: Specimen C9 with Considerable Slip during Loading
The average load carried by 7 mm specimens was 232 psi. Since three of the specimens
failed by slipping, the reduced strength exhibited by 7 mm bar is sufficiently justified.
Among all the three specimens which failed by slipping, the slippage started once a particular threshold load was reached. In the case of 7 mm bars, the average threshold load was recorded to be 11,580 lb. This can be explained as the predominant effect of the friction at the interface between the concrete and the FRP for the bond after the deterioration of the mechanical-interlocking phenomenon at higher level of loading.
The threshold load causes a tensile stress of 194 psi and a bond stress of 6.1 Mpa.
The average bond stress calculated for the 7mm specimens was 7.3 Mpa. For the 7 mm
bar, it was more prominently evident that the provided embedment length of 10 inches
was not sufficient to develop the ultimate tensile strength of the material of the particular
size. The analysis of the bond-slip behavior of the 7mm size bars is discussed in the next
chapter. 140 Table 8: Summary of the Test Results from the Pull-Out Test
Bar Size specimen Max.Load(Lb) Max Load(Kn) Max Stress(ksi) Max Stress (Mpa) Bond Stress(psi) Bond Stress(Mpa) C5 3085 13.7 281.6 1941 554 3.8 C6 3312 14.7 302.3 2084 595 4.1 3mm C7 3457 15.4 315.5 2175 621 4.3 C8 3389 15.1 309.3 2133 609 4.2
Average 3311 14.7 302.2 2083 595 4.1
C3 8147 36.2 267.7 1846 878 6.1 C1 9099 40.5 299.0 2061 981 6.8 5mm C2 9265 41.2 304.4 2099 999 6.9 C4 9429 41.9 309.8 2136 1016 7.0
Average 8985 40.0 295.2 2036 969 6.7
C9 12940 57.6 216.9 1496 996 6.9 C10 13040 58 218.6 1507 1004 6.9 7mm C11 13003 57.8 218 1503 1001 6.9 C12 16378 72.9 274.6 1893 1261 8.7
Average 13840 61.6 232.0 1600 1066 7.3
141 10.2.4 Bond-Slip Modeling of Basalt FRP Bars
The importance of the establishment of Bond-Slip relationship for any reinforcing
material has already been discussed sufficiently in the literature review section for the
bond behavior of the FRP bars. The prevalent theoretical models for the modeling of
bond-slip behavior of FRP bars are Malvar model, BPE model and CMR model. In this
study, CMR model was chosen to model the bond-slip relationship of the Basalt FRP bar
with the concrete. The various details relevant with the various models and their
effectiveness have already been discussed in the chapter for literature review. The Malvar
model is particularly useful as it tends to provide the model for the ascending portion of
the bond-slip curve which is the important portion relating with the serviceability condition. The Malavar model also yields the condition of zero intial slope at the lower level of loading and in the case of Basalt FRP bars, the similar response was observed in
the experimental bond-slip plot.
The following modified form of CMR model, as proposed by Tighiourt (1998) for
the case of BFRP bars was considered in this analysis.
(31)
Where
τm = peak bond stress and
s = slip corresponding to particular bond stress.
142 The above expression only furnishes the ascending portion of the load-slip curve.
The theoretical load-slip curves were plotted using the above expression for 5 mm and 7
mm diameter specimens which have failed by slippage. In the case of Tighiouart (1998),
the parameter Sr and β are established as 0.25 and 0.5 respectively, as can be observed in
the above expression. In the case of basalt FRP bar, parameter Sr was varied to get the
better curve-fitting with the experimental curve. The parameter was taken as 0.5 which is
similar to the Tighiouart model as it is intended to represent the curvature. By curve
fitting, the value of parameter 1/Sr for the basalt FRP bar was identified to be 60 and the
parameter β was established as 0.5.
Thus, the proposed bond-slip law, as a constitutive relationship, for the case of
basalt FRP bars, is given by the following expression:
(32)
Where
τm = peak bond stress and
S = slip corresponding to particular bond stress.
The plots for the 5mm and 7mm specimens, which have undergone slip-failure, were studied by the above proposed model and they are shown in the following figures.
143 12 Bond-Stress.vs.Slip -5mm-C4
10
8 Experimental Curve 6 Model-Tighiouart Stress(Mpa) -
4 Proposed Bond- Bond Slip Model 2
0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Slip(mm)
Figure 56: Specimen C4 (5mm) Load-Slip Curves
Load-Slip Curve 5mm-C3 12
10
8 Experimental Curve
6 Model-Tighiouart Stress(Mpa) - 4 Proposed Bond-Slip
Bond Model
2
0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Slip(In)
Figure 57: Specimen C3 (5mm) Load-Slip Curves 144 Bond-Stress.vs.Slip-7mm-C11 12
10
8
6 Experimental curve
Stress(mpa) Model-Tigiouart - 4 Proposed Bond-Slip Model Bond
2
0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Slip(mm)
Figure 58: Specimen C11 (7mm) Load-Slip Curves
Bond-Stress.vs.Slip-7mm-C10 12
10
8
6 Experimental Curve Stress(Mpa) - Model-Tighiouart
Bond Proposed Bond-Slip Model 4
2
0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Slip(mm)
Figure 59: Specimen C10 (7mm) Load-Slip Curve 145 Bond-Stress.vs.Slip-7mm-C9 12
10
8
Experimental Curve 6 Stress(Mpa)
- Model-Tighiouart
Proposed Bond-Slip Model Bond 4
2
0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Slip(mm)
Figure 60: Specimen C9 (7mm) Load-Slip Curves
10.3 Stress-Strain Analysis for Basalt FRP bars from Pull-Out Test Data
The stress-strain data acquired from strain-gages were used to calculate the tensile strength of the material during the pull-out test. This further provides us with the required data for modulus of elasticity of the material. Since the modulus of elasticity calculation requires the data pertaining to the fifty percent of the maximum loads as the upper limit, even for the slipped specimens, the data acquired from the Pull-out test can be used for modulus calculation if the specimen has not slipped before the fifty percent of the maximum load has been reached. The modulus of elasticity values obtained from the specimens has been summarized in Table 8. The average modulus of elasticity from all
146 the specimens of different sizes amounts to 12,330 Ksi. The typical Stress-Strain Curve for the different sizes of the Basalt FRP bar from the Pull-Out tests, are shown in the figures below.
Stress-Strain Curve 7mm-C12 350
300
250
200
Stress(Ksi) 150
100
50
MODULUSOF ELASTICITY=12541KSI
0 0 0.005 0.01 0.015 0.02 0.025 0.03 Strain(In/(In)
Figure 61: Typical Stress-Strain Curve for 7 mm BFRP Bar from Pull-Out Test
147 Stress-Strain Curve 5mm C-4 250
200
150 Stress(Ksi) 100
50 MODULUSOF ELASTICITY=13,112KSI
0 0 0.005 0.01 0.015 0.02 0.025 Strain(In/In)
Figure 62: Typical Stress-Strain Curve for 5 mm BFRP Bar from Pull-Out Test
Stress-Strain Curve 3mm-C6
300
250
200
150 Stress(Ksi)
100
50 MODULUS OF ELASTICITY=13,879KSI
0 0 0.005 0.01 0.015 0.02 0.025 0.03 Strain(In/In)
Figure 63: Typical Stress-Strain Curve for 3 mm BFRP Bar from Pull-Out Test
148 Table 9: Summary of Modulus of Elasticity from Pull-Out Test
Si ze Specimen M odulus of Elasticity(K si) M odulus of Elasticity(M pa)
C5 12,162 83,854 C6 13,879 95692 3mm C7 12,510 86253 C8 12,700 87563
Average 12,813 88,341
C1 10,613 73174 C2 12,870 88736 5mm C3 13,441 92672 C4 13,112 90404
Average 12,509 86247
C9 11,160 76946 C10 11,670 80462 7mm C11 11,216 77332 C12 12,541 86467
Average 11,647 80302
149 CHAPTER XI
CONCLUSION AND RECOMMENDATION
Among the four specimens with basalt FRP bar of size 3mm, the provided
embedment length of ten inches was sufficient to develop adequate bond for developing
the ultimate tensile strength of the bar. For developing the average tensile strength of 302
ksi, an average bond-stress of magnitude 595 psi was generated. In the case of 5mm sized
specimens, only one specimen underwent pure tensile rupture and the remaining
specimens underwent slippage followed by tensile rupture. For developing the average
tensile strength of 295 ksi, a bond-stress of magnitude 969 psi was generated. In the case of 7mm bars, only one specimen underwent tensile rupture failure while the remaining underwent slip failure. To develop an average tensile strength of 232 ksi, a bond stress of
magnitude 1066 psi was generated.
From the above tests, it was concluded that the embedment length of 10 inches
was sufficient for the 3 mm basalt bar to develop full tensile strength. Comparing with
the results for the 5 mm and 7 mm basalt bars, it can be observed that the embedment
length can be reduced to 7 or 8 inches for the case of 3 mm basalt FRP bars.
150 In the case of 5 and 7 mm bars, it was evident that the provided embedment of 10 inches
was insufficient to develop their full tensile strength. Based on the bond-slip data obtained for the 5 and 7 mm basalt FRP bars, the following equation is proposed to
couple bond-stress and slip.
(32)
Where τ is the bond-stress corresponding to the slip-value of s and τmax is the
maximum bond stress developed for the particular size. The other constants were
obtained from the curve-fitting parameters. The stress-strain data obtained from pull-out
tests were used to calculate the modulus of elasticity of basalt-bar to have a comparison
with the results obtained from the direct tensile strength test. The average modulus of
elasticity from all the specimens obtained was 12,330 ksi. The average modulus of
elasticity proposed from the direct tensile strength test was 12,000 ksi. Hence it can be
observed that the proposed modulus of elasticity is a reasonable and consistent value.
The pull-out tests conducted included just twelve specimens with four
specimens of each size. Even though conducted tests provides a relative measure of the
bond strength of basalt FRP bars of different diameters, it is recommended to conduct
more number of tests to study detailed bond-behavior. Since the pull-out tests only
provide the relative measure of the bond-strength and do not fully simulate the bending
condition, it is recommended to conduct some beam pull-out test along with cylinder
pull-out test.
151 CHAPTER XII
LITERATURE –REVIEW FOR THE STUDY OF BFRP REINFORCED BEAMS
In the previous sections, we concluded that the mechanical properties of basalt
FRP bars which include longitudinal tensile-strength, longitudinal modulus of elasticity and the bond-properties of basalt with concrete. With all the associated mechanical properties, the flexural analysis of the BFRP bar reinforced beams was performed. This section will provide the comprehensive literature review of the past work and research done on the field of FRP reinforced beams.
12.1 General Introduction to Beam Element
A beam can be defined as a structural element which carries load by a resisting
mechanism primarily generated by bending. The loads are applied in the direction
transverse to its longitudinal axis. This further imposes a geometrical constraint on the
definition of beam. Since bending is accompanied by the generation of the normal
stresses, this requires the length to the depth ratio to be larger such that the shear stress
doesn’t play a significant role in developing the resisting mechanisms. This condition
will also allow the decay of the local-concentration effect of the applied load in the
152 vicinity of the load. In accordance with the saint-Venant’s principle, the local effects will disappear in a distance approximately of the order of the depth of the beam; hence the
average region of interest is free of the complex stress trajectories. ACI classified the
family of beam whose span to depth ratio is less than five, as deep beams.
Deep beams are not amenable to the classical analytical methods pertaining to
the beams with larger span to depth ratio as bending is not the factor in developing the
load resisting mechanisms in their case. In our case, the beam we are discussing here is
not deep beams hence the length is much larger than the depth. This entails the
development of normal and shear stresses in the plane of the beam for equilibrium and
they will be most important for all practical purposes. The direct normal stress in the
transverse direction will be negligible as it is well exceeded in magnitude by the resisting
stresses.
One of the major and pivotal hypotheses for beam theory is the so-called
Bernoulli’s assumption. This says that a plane section before bending remains plane after
bending and perpendicular to the midplane after deflection. This constraint is the
structural counterpart of the condition of pure bending and it will ensure that there is the
linear strain distribution across the depth of the beam. For instance, in the case of deep
beams where such assumption can’t be made, the strain distribution is evidently nonlinear
across the depth of the beam. If the Bernoulli’s hypothesis can be made, then this will
also ensure the linear distribution of strain even in the inelastic region. This property will
be particularly very useful in the analysis of reinforced concrete beam which are
inherently nonlinear.
153 Another assumption reads that the beam material is homogenous, isotropic and
elastic. This assumption is not completely applicable in the case of reinforced concrete
beam which doesn’t respond as a classical beam once it is cracked). By imposing the
compatibility conditions (which includes strain-displacement relationships), constitutive- relations (which includes stress-strain equations) and equilibrium requirements in the plane of interest (which includes axial, shear and bending moment equations), the following important relationships for the simple beam theory can be obtained:
Rate of change of shear (33)
Rate of change of moment (34)
The equation (33) equates the rate of change of shear-force(V) along the length
of the beam(along direction X) with the intensity of loading(q) and the second equation
equates the rate of change of moment along the length of the beam with the shear force
acting On the beam. This relationship when furnished with some of the geometrical
relationships associated with the bending of the beam will finally yield the various
relationships for important stress parameters, specifically the normal and shear stress
stresses in the plane of the beam. For an elastic and isotropic beam, that is subjected to pure bending, the following relationship will hold:
(35)
154 Where ρ is the radius of curvature of beam due to the deflection induced by pure bending.
From the geometry of the beam under pure bending it can also be shown that
(36)
From equation (35) and (36), the following important relation called moment-curvature relation can be inferred:
(37)
It should be noted that equation (37) is based on the assumption of small deflection such that linearity-assumption will hold. Equation 37 when combined with the compatibility condition and constitutive relation will yield the following formulation for the bending stress:
(38)
Where σxx is the normal stress due to the bending due to the applied moment M, y is the distance to the layer considered from the neutral axis and I is the moment of inertia of the cross-section of the beam about the longitudinal axis. similarly the relation for the distribution of shear beam across the cross section of the beam along its depth is given by
155 (39)
Where σxz is the shear stress across the section of the beam due to the shear force induced by the transverse loading. Here, Q is the first moment of area about the neutral axis. I is the moment of inertia about the longitudinal axis and b is the width of the beam considered. Now it will be very pertinent to discuss the response of reinforced concrete beam in the light of the above discussion.
12.2 Reinforced Concrete Beams
Reinforced concrete beam can be defined as a structure in which the internal normal stress generated by the bending action of the beam is carried by the different constituent materials. Specifically, the compressive stress is carried by the concrete and the tensile stress is carried by the steel or other FRP rebar. The following important assumptions are made for the analysis of reinforced concrete beams:
1. Bernoulli’s hypothesis is assumed to hold such that the strain distribution is linear across the section.
2. There is a perfect bond between concrete and the reinforcing bar. This will ensure the strain-compatibility between the concrete and the reinforcing bar.
What differentiates a reinforced concrete beam from the homogenous, isotropic and linearly elastic beam as described above is the qualitatively different response of this beam after the cracking of the concrete. After the concrete is cracked when the tensile strength of the concrete is reached, the beam cannot be presumed to be homogenous or 156 elastic, thereby establishing the non-applicability of the elastic-flexural relationship as described above for the homogenous and linearly elastic material. Concrete is in itself a nonlinear material and the complex interaction including both the cracked concrete and reinforcing bar, is therefore, not amenable to theoretical manipulation as described
previously. The flexural behavior of a reinforced concrete beam under transverse loading
can be defined to be composed of the following phases. This phase will also be important
for the analysis of the deflection behavior which is directly related to the sectional
properties of the section pertaining to the corresponding phase.
1. Uncracked Linear Stage
When the applied external moment on the beam is low such that the tensile
stress in the concrete in the extreme fiber below the neutral axis is less than the modulus
of rupture of the concrete, the beam will effectively behave as a homogenous section.
This will allow the linear analysis of the beam by using the transformed sectional
properties and the classical flexural relations for the beam element. Since the bending
moment generally varies along the length of the beam, this will insinuate the variation of
the normal stress along the length of the beam. Once the bending stress at a particular
section, which is proportional to the magnitude of the bending moment at that section, is
reached, the concrete will undergo cracking. The modulus of rupture of the concrete is
given by
157 (40)
Here it can be observed that the modulus of rupture of the concrete is
proportional to the square root of its compressive strength and doesn’t depend on the type
of the reinforcing bar used. The cracking moment is now given as
(41)
Where Ig is the gross moment of inertia of the section and yt is the distance of the extreme
tension fiber from the neutral axis of the beam. For a rectangular beam it is obviously is
half of the depth of the beam.
2. Cracked Linear Stage
When the applied moment increases above the cracking moment, the cracks will
develop along the length of the beam. The neutral axis will shift upward and the cracks
will extend upto the neutral axis level. Since the stress-strain curve for concrete is
’ considered to be linear upto about 0.4fc , if the concrete stress is less then this value,
elastic bending relation for the section will be still applicable.
158 3. Cracked Nonlinear Stage
The stresses will stop to be linear in this regime such that the bending relation
for the homogenous section will not be applicable to compute the stresses. The internal
couple approach, which is based on the equilibrium condition of the section, is to be used.
4. Ultimate Strength Stage
As the applied moment further increases, the strains increase rapidly and the ultimate flexural capacity of the beam is reached. The stress distribution for this stage is highly nonlinear and approaches the stress-strain curve of the concrete. Since reinforced concrete beam is a composite structural element, it’s flexural and shear behavior is unequivocally related with the interaction of the concrete matrix and the provided reinforcement, the mechanical properties of the constituent elements and hence their structural responses. This entails the fact that the failure mechanisms associated with these beams are completely different from the classical linear-elastic, homogenous beams.
One of the basic relationships as stated above for the beam signifies that whenever there is a change in bending moment along the beam, it will give rise to shear stress. Bending moment is generally largest at the mid-span of the simply supported beam and the effect of shear force is relatively low at this region. If we consider the conventional four point bending situation, then the mid-span zone is under pure bending moment and since there is no any variation of bending moment along the length of the
159 beam, this zone is only subjected to normal stresses. Some of the structural response of
these beams can then be explained from the concept of principle stress. For a small beam
element on the flexural zone subjected to uniaxial stress state due to pure bending (σyy =
0, for the beams), the magnitude and direction of principle stress is given by
(42)
(43)
From the above equations, it can be observed that the maximum principal stress
is equal in magnitude to the normal stress across the section generated by pure bending.
(σy = 0, τxy=0). The direction of principle stress can be observed to be vertical as τxy=0.
This fairly explains the appearance of the nearly vertical cracks in the flexural zone in the bottom of the beam subjected to four-point bending. The inclined cracks in the vicinity of the supports are due to the combined shear and flexure; however, the shear stress will be the dominant one. They are generally called inclined cracks, shear cracks or diagonal tension cracks.
The reinforced concrete beams can be classified as under-reinforced, balanced and over-reinforced section depending on the relative reinforcement ratio and the failure mechanism is the function of this parameter. For a particular reinforced concrete beam, there will be a unique amount of reinforcement which will guarantee that the concrete in
160 the compression phase will reach the maximum usable strain of 0.003 and steel in the
tension zone will reach its yield-strain simultaneously.
Similarly for the reinforcement ratio less than the balanced reinforcement ratio,
the steel will reach the yield-strain before concrete in the extreme compression fiber
reaches the strain value of 0.003. With further increase in load the moment strength will
start reducing and the concrete on the compression will start crushing due to the
increasing strain. This type of section is called under-reinforced sections and the mode of
failure is regarded as tension failure as the failure of beam is initiated by the yielding of
the steel. This mode of failure is a ductile-failure and dictates the design-philosophy of
reinforced concrete beams with steel reinforcement almost exclusively. If the section is
over reinforced, the concrete will reach the strain value of 0.003 before the steel yields.
This will instigate the failure of beam by concrete-compression and hence a brittle
failure. This is known as compression failure and is avoided to exploit the inherent
ductility of the steel reinforcement.
12.3 FRP Reinforced Concrete Beams
It will not be an over-exaggeration of the fact that the design philosophy and methodology for the concrete beams reinforced with FRP reinforcement can be significantly different from the traditional steel reinforced beams. The primary factors of interest that play a prominent role for the discrepancy in the structural behavior of
concrete beams reinforced with FRP bars can be categorized as follows:
161 12.3.1 Stress-Strain Behavior of FRPs
Based on extensive research and work done by the scientists and engineers in
the past, it is widely accepted that the FRP rebar are generally elastic and brittle such that
stress-strain relation in axial tension is linearly elastic to failure. This will have a serious
impact on the ensuing design-philosophy. As already discussed, the whole design philosophy of the steel reinforced concrete beams is based on the linear-elastic- plastic behavior of the steel bars such that enough ductility in the total structural response can be ensured. The inability of FRP bars to undergo plastic deformation poses the necessity of different design philosophy in contrast to the conventional steel reinforced beams. Even though there are various design recommendations conforming to the limit state design procedures, ACI design guidelines will be discussed here. ACI 440.1R-06, as reported by
ACI committee 440, recommends the use of conventional methods of strain-compatibility
(which presupposes the perfect bond between FRP bar and concrete) and equilibrium to analyze the FRP concrete sections (L.C.bank, 2007).
Even though the method of finding the internal forces in a FRP reinforced section is similar to the approach being employed for traditional beams, the analysis of the ultimate moment strength of the section is based on different philosophy governed by linear elastic behavior of the FRPs. In the similar lines as described above the balanced reinforcement ratio for the FRP reinforced section is given by the relation as per ACI
440.1R-06 as
162 (44)
where β1 is a factor depending on concrete strength, fc’ is the concrete
compressive strength from the cylinder test, Ef is the guaranteed longitudinal modulus of
the FRP, εcu is the ultimate compressive strain in the concrete (usually taken as 0.003) and ffu is the guaranteed longitudinal tensile strength of the FRP bar. ACI doesn’t allow
to account for the compressive strength of the FRP reinforcement. For instance, in the
case doubly reinforced section, if FRP bars are provided, then their contribution to carry
the compressive stress should be neglected for all practical purposes. The resistance
factors pertaining to flexural design of the FRP reinforced beams do not consider any
desirable or non-desirable effect of the addition of FRP bars in the compression zone
(L.C.bank, 2007).
In the case of FRP reinforced concrete beams , the mode of failure for an under-
reinforced section will be by the rupture of the FRP bars and the mode of failure of the
over reinforced section will be by the crushing of the concrete as in the case of the steel
reinforced beams. The balanced reinforcement ratio as given above will ensure the
simultaneous rupture of the FRP bar and crushing of the concrete. This difference is very
pivotal as in the case of FRP beam where both modes of failure are brittle owing to the
linear elastic behavior of the FRP bar. However, the crushing of the concrete can be
regarded to be less brittle then the rupture of the FRP bar (L.C.bank, 2007). This will
lead us to the conclusion that the FRP reinforced beams are better to be designed as over-
163 reinforced in contrast to the steel beam which is designed as under-reinforced almost exclusively.
This can be regarded as the major upshot from the classical design philosophy of reinforced concrete sections and different novelty associated with the design- philosophy of the FRP reinforced sections. This will also explain why the resistance factor of 0.9 associated with the ductile failure of an under reinforced section when the strain in steel exceeds 0.005 is completely not applicable for the case of fiber-reinforced sections. The resistance factor for the over-reinforced FRP section is provides as 0.65 which is also the resistance factor corresponding to the brittle failure associated with steel reinforced beam due to the crushing of the concrete. However, to account for the indeterminacy associated with the varying degree of over-reinforcement, the factor is allowed to vary from 0.65 to 0.55 (L.C.bank, 2007).
As per the current ACI provision, the nominal moment capacity associated with an over-reinforced FRP section is given by:
(45)
Where
(46)
And
(47)
164 Here ff is the stress in the FRP rebar at concrete compressive failure, a is the depth of the
Whitney stress block, d is the effective depth of the section, b is the width of the beam
and Af the cross-sectional area of the FRP bar, Ef is the longitudinal modulus of elasticity
of the FRP bar, ρf is the reinforcement ratio and εcu is the maximum compressive strain in
concrete.
The preference of the concrete crushing to the FRP rupture is also attributed to the fact that the confined concrete can provide some postpeak large strain capacity, even
at reduced stress level (L.C.bank, 2007). The linear elastic nature of FRP bars also does not allow the formation of plastic hinges and subsequently the moment redistribution.
Where FRP bars are used in layers, the stress in each layer should be calculated separately to calculate the moment capacity of the section in contrast to the steel beams, where it is allowed to assume that the resultant tensile force in the bars acts through the centroid of the bar layers. This has also been verified by the various researches that the anisotropic nature of the material doesn’t significantly the flexural behavior of the section
(Nanni, 1993), though it may be very significant in the bond behavior.
12.3.2 Stiffness of the FRP Bar
Even though the stress-strain relation is very important for the determination of the moment strength of the beam, the stiffness behavior may be equally important for various other structural requirements. This may be particularly important for dictating the various serviceability criteria. The stiffness or longitudinal modulus of elasticity of the
FRP bars is significantly lower than the steel bars. This may be able to give rise to
165 excessive deflection of the beam and hence the larger crack-widths.due to their high strength and low modulus, the FRP material are highly subjected to the condition of high deformation such that serviceability criteria can be a fundamental issue (M.A.Aiello,
2000).
On the basis of the research conducted by Antonio Nanni for the comparative study of the flexural behavior of the Aramid FRP reinforced beams and conventional steel reinforced beams some important observations can be made. The moment-curvature
analysis for both FRP beam and steel was performed .this reveals that the FRP-reinforced
section exhibits the same maximum moment and curvature as in the case of counterpart
steel reinforced beam with a slightly smaller reinforcement ratio, however, the flexural
rigidity of the FRP section is only 38% of the steel reinforced beam. This will lead us to
the fact that the deflection criteria may be as important as flexural strength in the case of
FRP reinforced beams (A.Nanni, 1993). The design for serviceability limit state for the
FRP sections constitutes primarily deflection and crack width. The permissible crack width for the case of the FRP beams is generally greater than for the corresponding steel beam owing to their better corrosion resistance under adverse climatic and environmental conditions.
Prior to 2006, the crack width models were based on the traditional Gregely-
Lutz model and their empirical modification based on the particular FRP type. They include the model proposed by Toutanji and Safi, model by Salib and Abdel-sayel and others. The current ACI provision for the crack-width determination is based on the mechanistic approach developed by Frosch (A.Nanni, 2007). The older relations based on
Gregely-Lutz model were superseded by the Frosch model due to their excessive crack
166 width prediction for the case of concrete beams and slabs with larger covers (A.Nanni,
2007). As per the current ACI provision, the following relation is specified for the crack width prediction.
(48)
Where fr is the reinforcing bar stress, calculated assuming elastic-cracked conditions, Er is the modulus of elasticity of the reinforcement, β is the ratio of the distance from the neutral axis to the tension face of the member to the distance from the neutral axis to the centroid of the tensile reinforcement, dc is the cover thickness from the tension face to the centre of the closest reinforcement ,s is the bar spacing and kb is the factor accounting for the bond characteristics of the reinforcement.
The permissible crack width as per the current ACI provision along with the above equation is 0.020 in for exterior exposure conditions and 0.028 in for interior exposure condition. As already discussed, these values are larger in magnitude for steel reinforced beams for the respective exposure conditions. This is attributed to better corrosion resistance of the FRP bars. Unlike the case of cracking serviceability limit state, deflection is always a hard parameter to calculate and quantify in the case of FRP reinforced concrete beam, even though it is the most important structural response which is invariably a function of the stiffness of the reinforcing bar material.
Even though ACI has provided some guidelines for the theoretical calculation of the deflection of the beam, they have not proved themselves abundant enough to predict
167 the deflection correctly in many cases regarding the qualitatively different mechanical and structural properties of the FRP bars. The current ACI provision for deflection calculation is based on the concept of effective moment of inertia equipped with some modification factor to account for the structural anomalies of FRP reinforced concrete beams. There has been an extensive amount of research in this field in the past and it is still a great subject of interest and investigation. Since deflection-analysis of Basalt FRP reinforced beams is one of the primary objectives of this thesis, this part will be discussed in detail in the following portion.
12.4 Deflection Analysis of Reinforced Concrete Beams
The load-deflection relationship for a reinforced-concrete beam can be observed to be more or less trilinear (E.G.Nawy, 2007). This relationship is also can be generalized for the case of FRP reinforced beams. It comprises of the following parts:
Part 1: Stage Prior to Cracking
This part refers to the part of the load deflection curve when the concrete on the extreme tensile fiber has not cracked yet. Since this part represents completely elastic behavior of the beam, the deflection analysis for this part can be done with classical elastic analysis. The only physical parameter required to be computed is thus the moment of inertia of the section. To account for the additional stiffness provided by reinforcement, the area of reinforcement can be converted to an equivalent concrete area
168 by multiplying the area of reinforcement by the modular ratio. The deflection analysis of
this region may not be very important as one of the fundamental assumptions in the
analysis of reinforced concrete beam is the assumption that the concrete doesn’t carry any
tension. However, all the segments of the beam may not be cracked and this may
necessitates the knowledge of additional stiffness properties for this situation (E.G.Nawy,
2007). This might be particularly important in the case of FRP reinforced beams where
the tension-stiffening plays an important role in the deflection analysis.
Part 2: Postcracking Load Stage
Once the applied moment reaches the cracking moment for a particular beam,
the concrete on the extreme tensile fiber cracks thus significantly reducing the stiffness of
the beam. This is viewed in the load deflection curves as a small discontinuity
(corresponding to the cracking moment) and the subsequent reduction in the slope of the
load-deflection curve signifying the stiffness reduction. In the actual circumstances, the
beams are generally loaded in a way such that the bending moment along the length of
the beam varies. This will entail the variation of normal stresses along the length.
Depending upon the magnitude of stresses developed, not all the sections will be cracked.
Hence the uncracked part will contribute to the stiffness of the beam and the cracked
portion will be neglected. The actual stiffness of the beam can thus be seen to be between
the gross-stiffness (EcIg) and the cracked-stiffness (EcIcr). As the stress in steel reaches the yield stress level, the section stiffness approaches EcIcr.
169 The empirical equation developed by Branson for calculating the effective moment of inertia has been verified to be widely applicable for the case of steel reinforced and prestressed beams. The relation is given as
(49)
Where Ie is the effective moment of inertia, Mcr is the cracking moment, Icr is the cracking moment of inertia, Ig is the gross moment of inertia and Ma is the applied moment. The cracking moment of inertia can be calculated from classical elastic analysis and is given as
(50)
Where c is the depth of the neutral axis at cracking, n is the modular ratio, As is the reinforcement area and d is effective depth of the beam. It can be observed that the
Branson equation is the function of the ratio of the cracking moment to the applied moment. The exponent of this ratio of the Branson’s equation is empirically tuned for the case of steel reinforced beams such that it will assure the transition to the cracking moment of inertia of the section as applied moment reaches the maximum moment.
170 12.5 Deflection of FRP Reinforced Beams
FRP materials have low stiffness owing to its low longitudinal modulus of elasticity. As the deflection of a beam varies inversely with the modulus of elasticity, this can result in considerable amount of deflection compared with the steel beam even for the same value of load. The increased deflection will lead to extensive cracking along the length of the beam thereby significantly reducing its flexural stiffness and hence resulting in more deflection. The Branson’s equation is empirically derived in consideration for steel reinforced beams. Therefore, its applicability for FRP reinforced beams can be seriously considered. FRP reinforced beam can respond qualitatively different from the conventional steel reinforced beams due to the differences in their mechanical properties and also differences in their interaction mechanisms. (M.A.Aiello, 2000). From the past studies and investigations, we can find various instances where the deflection responses calculated for FRP beams from the application of Branson’s equation, were found to be incongruous with the experimental results. ACI has provided some modifications in the original Branson’s equation to account for the differences in mechanical properties of
FRPs and the differences in the interaction mechanisms.
As it is already mentioned, the difference in the mechanical properties can result in different structural responses by the respective materials. One of the important parameter governing the structural responses of a reinforced beam is the phenomenon called Tension-Stiffening. It is one of the basic assumptions of reinforced concrete beam design, that the tensile load is totally carried by the reinforcement once the concrete is cracked. However, the concrete segments between the cracks are able to carry the tension
171 due to the bond with the reinforcement. This phenomenon is technically termed tension- stiffening. Tension stiffening plays an important role to control member-stiffness, deformation characteristics and crack-width properties. This is also useful for the non- linear analysis of the reinforced concrete beams. (P.H.Bischoof, 2004). From the rigorous study, it was found that the GFRP reinforced beams exhibit more tension stiffening then the corresponding steel reinforced beams owing to their lower modulus of elasticity (P.H.Bischoof, 2004).
Tension stiffening is quantified as the difference between the axial member response and the bare bar response as shown in Fig-64 (Bischoff,2004). Though this approach resorts to the analysis of axial members to study the tension-stiffening phenomenon, this can be extended to the case which involves flexural members. Since the structural responses of both axial and flexural members are both related to their respective rigidities, EA (axial stiffness) and EI (flexural stiffness), the tension stiffening associated with the axial member can be correlated with the tension stiffening associated with the flexural members. The fundamental insight which we are endowed by this is that the prediction of the postcracking responses depends primarily on the ratio of the uncracked member rigidity and the cracked member rigidity (P.H.Bischoof, 2004).
172
Figure 64: Comparison of the Member and Bare Response (Bischoff, 2004)
Tension stiffening is generally dealt with in two approaches. The first approach called
tension stiffening strain approach uses a modified relationship for stress-strain behavior of the embedded bar. The second approach called load-sharing approach which considers the contribution of concrete with an average stress-strain relationship for the cracked concrete. (P.H.Bischoff, 2004). The basic concepts regarding the tension-stiffening phenomenon is explained below in the light of tension stiffening strain approach.
From the plot of the relative response of the bare bar and the embedded bar, it can be seen that the axial strain at a particular load is lower in the case of the embedded bar owing to the contribution from the concrete, hence, from tension-stiffening. This effect of tension stiffening in the case of axial member can be quantified as
(51) 173 Where εs is the strain in reinforcement at crack and Δεs is the tension stiffening strain
liable to increase the bare bar stiffness. The tension stiffening effect is most prominent
when the beam cracks first. As the load increases, the tension stiffening gradually
decreases owing to the increase in the proportional increase in cracks. The reduction in
the tension-stiffening strain with increase in load can be attributed to the decrease in the
average tensile strength of the concrete which is due to the redistribution of stresses to
counteract the increasing crack formation (P.H.Bischoff, 2007). The maximum tension
stiffening strain at the onset of cracking, as per Bischoff is defined by the relation
(52)
Where fcr is the cracking strength, Eb is the longitudinal modulus of elasticity of the
reinforcing bar and ρ is the reinforcement ratio. The tension stiffening strain after the
cracking has stabilized is
(53)
Where ϰ ts is the tension stiffening factor and can be viewed as the normalized tension stiffening strain with the maximum tension stiffening strain. The above equation is very
significant for the study of the deflection behavior of FRP reinforced beams and can
provide us with the valuable insight into their structural responses. It can be observed
from the above equation that the tension stiffening strain varies directly with the concrete
174 strength and varies inversely with the stiffness and the reinforcement ratio. For the FRP
reinforced beams which are attributed with lower stiffness and lower reinforcement ratio,
this equations provides explanation for their exhibition of higher tension-stiffening.
The tension stiffening factor is found to be independent of the concrete strength
and reinforcement ratio (P.H.Bischoff, 2004). This factor can be expressed empirically as a function of member strain for the numerical analysis of reinforced concrete beam. This also can be expressed as the ratio of the average tensile stress carried by the concrete to the cracking strength of the concrete from the design perspective. Bischoff gives the general formulation of the effective moment of inertia as follows
(54)
Where (55)
And (56)
is based on the assumption that the stiffening strain varies inversely with bar stress at the
crack location. Since equation is derived from the basic mechanics of the beam, it is
evident that this equation can have more general application for both steel reinforced and
FRP reinforced beam with arbitrary reinforcement ratio. This relation can be seriously
considered particularly for the case of FRP reinforced beams, where the beam with lower
reinforcement ratio makes the applicability of Branson’s equation more questionable.
175 12.6 Branson’s Equation in the Context of FRP Beams
The theoretical analysis procedure of the deflection analysis of the FRP beam
has been classified into two models by M.A.Aiello and L.Ombres. They are termed as
cross-sectional models and block models.cross-sctional models are based on Bernoulli’s hypothesis and the assumption of perfect bond between concrete and reinforcement.
These models correspond to the approach adopted by the codes for deflection analysis. It employs the elastic analysis approach such that effective moment of inertia can be expressed as the linear combination of moment of inertia pertaining to uncracked state and the moment of inertia pertaining to cracked state (M.A.Aiello, 2000). Branson’s
equation provides the suitable envelop to fulfill this criterion.
The block model, as proposed by M.A.Aiello and L.Ombres, studies the
structural responses of reinforced concrete beams by analyzing the member block
between consecutive cracks. This model assumes the local slippage between the FRP
reinforcement and concrete. This may be a more realistic modeling, particularly in the
case of FRPs, where the local slip between the reinforcing bar and the concrete matrix is
a very probable structural issue. With the provision of material constitutive relationship
and a suitable bond-slip relationship, a system of differential equations is obtained by
imposing equilibrium, compatibility and suitable boundary conditions (M.A.Aiello,
2000).
The solution by the numerical procedure will lead to the determination of pertinent factors. From the tests done on the beams reinforced with AFRP bars, it was found that the cross-sectional models predict higher deformability values in comparison
176 to the block models. However, the block model was found to require a lot of computational effort and its effectiveness shows sensitive dependence on the assumed bond-slip law (M.A.Aiello, 2000). As already been said that codes refer to the cross sectional model, we will resort to a more detailed analysis of the deflection based on the linear elastic analysis.
This will simply lead us to further scrutiny of the Branson’s equation in the context of FRP reinforcement. Branson’s equation in the generalized form can be expressed as
(57)
Where for the steel reinforced beam the exponent m is found to be equal to 3.however, the cubic term is based on the concept of average effective moment of inertia and an exponent of 4 was found to giver better approximation of effective moment of inertia for individual sections (Branson, 1977). Branson’s equation was well confirmed for the ratio of gross moment of inertia (Ig) and cracking moment of inertia (Icr) between 1.5 and 4.
(P.H.Bischoff, 2004). FRP beams generally have this value of ratio greater then 5, thus leading to a stiffer response and resulting in underprediction of deflection when used in conjunction with the original Branson’s equation. (P.H.Bischoff, 2007). The exponent m has also the physical significance in the sense that it provides a smooth transition from the gross moment of inertia to the cracking moment of inertia as the load reaches the ultimate value. Hence it can be concluded that the transition of gross moment of inertia of FRP reinforced beam to the cracking moment of inertia is faster which explains the 177 faster reduction in the stiffness of the beam. Hence they are not completely amenable to
the original Branson’s equation, which predicts relatively slower degradation with the
exponent equaling 3.
ACI committee 440.1R-03 recommended a modification on the original
Branson’s equation to account for this anomaly exhibited by FRP reinforced beams. This relation was defined as
(58)
(59)
Where Ef and Es are the values of the FRP and steel modulus of elasticity, and αb is the
bond dependent factor which equals 0.5 for steel bars. A value of 0.5 is also
recommended for FRP bars. However, in the light of extensive studies on the FRP beams,
especially, GFRP it became an established fact that the ACI provision is overestimating
the stiffness of the FRP reinforced beams and thus underpredicting deflection. Ganga,,
Rao and Faza conclude that the deflection predictions for FRP reinforced beam from the
ACI provision are accurate within 5% of experimental data. Nawy reported that the
deflection prediction varies with the quantity of reinforcement. He inferred that the
underestimation of the deflection varies inversely with the reinforcement ratio. Similary
from the tests performed by R.Al.sunna on FRP reinforced beam, similar results were
reported. The deflection calculation based on the original or modified form seems to
178 overestimate the stiffness of the member hence resulting in underpredicting of deflection.
However, the predictions are better as the reinforcement ratio increases.
Similarly, M.M.Rafi (2007) concluded the similar conclusion relating to the over prediction of the stiffness of the CFRP reinforced concrete beams. Based on the past
research and extensive studies conducted by various scientists and engineers, following
generalized plot to depict the relation of the reinforcement ratio with the discrepancy in
theoretical prediction as calculated by Branson’s equation and actual deflection is shown
in the following plot (Toutanji and Safi, 2000).
Figure 65: Effect of Reinforcement Ratio on the Deflection
179 From the above Fig-65, it can be observed that, based on the work of Yost, Masmoudli and Benmokrane on the GFRP reinforced beams, with the lower reinforcement values, the discrepancy in the actual and theoretical deflection is higher and it is getting better with increasing reinforcement ratio. This entire scenario clearly suggested the structural behavior of FRP beams to be significantly different from the conventional steel reinforced beams. Evidently, tension stiffening plays a major role in the analysis of FRP reinforced beams.
As per the equation, as provided by Bischoff, the tension stiffening strain varies inversely with the reinforcement ratio and the longitudinal modulus of elasticity. It signifies that tension-stiffening is larger in the case of FRP reinforced beams with lower longitudinal reinforcement. The original Branson’s equation is not able to incorporate these effects peculiar to FRP reinforced beams .Dolan suggested that the exponent “m” in equation, for the case of FRP reinforced beams to be greater than 3 to assimilate these varying responses (Toutanji and Safi, 2000). Beams reinforced with GFRP in quantity less then 4%, can have an uncracked to cracked ratio of the moment of inertia that varies from about 5 up to 16 which defines the physical significance of the exponent “m”
(Bischoff, 2004).
Bischoff (2004) reported the ratio of gross moment of inertia to the cracking moment of inertia for GFRP beams to be varying from 5 to 25 for which the corresponding steel beams has a ratio of 2 to 3. As already been mentioned, the working range of the Branson’s equation is for the cases which necessitates this ratio to be less than 4. Hence at least 3% reinforcing ratio is required for GFRP beams to be within the working range of Branson’s equation. This issue can be dealt in two ways. One way is in
180 conformity with the spirit of works of Bischoff in which these parameters are studied by developing a theoretical model based on the actual mechanics of the structure including the effect of tension stiffening. This approach can yield the appropriate generalized relationship between these parameters based on the fundamental mechanics of the structure and the suitable assumptions. This can be exemplified by the generalized equation for effective moment of inertia as given by equation (22) proposed by Bischoff, which includes the reinforcement ratio factor, not as a modifying factor, but as an intrinsic parameter which is included by the very nature of the member . The second way as adopted by most of the scientists includes an experimental analysis of the deflection behavior and the formulation of a suitable modification factor that can be included in the original Branson’s equation for the FRP beams. Here it will be pertinent to discuss the works done in the past on this topic.
As it was well established that the stiffness of the cracked FRP beam is also the function of reinforcement ratio, many endeavors by the scientists can be found to incorporate this factor. Toutanji and Safi (2000) identifies the effect of this factor on exponent “m” in the original Branson’s equation as given by equation (25), in the line of thinking similar to Dolan who suggested the varying value for the exponent “m” for FRP beams. They suggested the following limits for the exponent “m”
For (60)
For (61)
181 Where EFRP is the modulus of elasticity of GFRP bars used in the study, ES is the modulus of elasticity of steel and ρFRP is the longitudinal reinforcement ratio.
Similarly, R.AL.Sunna (2005) from his study of the GFRP and CFRP reinforced
beams, concluded that, since the experimental deflection exceeds the ICR limitation on
deflection, there may not be perfect bond between the FRP bars and the concrete. This necessitated the form of Branson’s equation which can provide the transition from IG to a
certain fraction of ICR in the case of FRP beams. Benmokrne (1996) proposed such
modification as
(62)
Where α and β were found to be equal to 0.87 and 7 respectively. Here the factor α accounts for the transition from the gross moment of inertia to the certain fraction of the cracking moment of inertia. R.AL.Sunna, based on his study on GFRP and CFRP
reinforced beams proposed the following relationships:
(62)
Where (63)
182 where the values of α were proposed to be 0.9, 0.85 and 1 for GFRP, CFRP and steel RC beams respectively and are supposed to be related with the bond characteristics of the respective material. A.Maji and Orozco (2005), based on their study on CFRP reinforced concrete beams, concluded that deflection prediction based on the ACI equation as given by equation 23, is dependent on the reinforcement ratio and there is a marked variation in the calculated and actual stiffness properties. Based on their deflection analysis of the
CFRP reinforced beam of length 106.7cm and 10.2cm2 cross-section, and reinforcing bar size 6.35mm in different quantities (1, 2, 3 and 6),they proposed the following modification for the effective moment of inertia
(65)
Where Υ is the modification factor equal to the ratio of modulus of elasticity of the FRP used and modulus of elasticity of steel. η is the factor accounting for the reinforcement ratio and for the particular set of CFRP beams, for the reinforcement ratio ρ, it is given as
(66)
From their analysis, it can be observed that the proposed model is sound enough for the set of beams tested and able to capture the discontinuity in displacement at the moment of cracking. All these discussions lead us to the conclusion that the deflection characteristics of FRP reinforced beams are dependent on the reinforcement ratio and are 183 affected by its low modulus of elasticity. All the studies done in the past, as described above, are very commendable efforts to address this anomaly associated with the FRP reinforced beam. The latest ACI Committee 440.1R-06 proposes the modification on the original Branson’s equation to include the effect of reinforcement ratio on deflection analysis of FRP reinforced beams. The proposed modification is given by
(67)
Where βd is the reduction coefficient to account for the effect of reinforcement ratio. The reduction factor is expressed as
(68)
Where ρfb is the balanced reinforcement ratio and for the FRP beams is given by equation
44.
From the recent studies conducted by M.N.Habeeb (2007) for GFRP reinforced beams the equation (67) was used for the deflection analysis. The test program comprised of simply supported and continuous beam reinforced with GFRP rods with sizes varying from 12 to 15.9mm (M.N.Habeeb, 2008). From the tests, it was reported that the ACI equation as given in equation (36), provides good approximation to the actual deflection in the case of simply supported beams. For the case of continuous beams, the same equation yielded the underprediction of deflection with the increase in the magnitude of
184 the applied load. From the test results, it was reported that the underprediction was within
limit upto 50% of the maximum load but it grew larger with increasing load.
This response is congruous to the most of the results that have been gathered
from the research conducted in the past by various scientists and engineers. The latest
provision by the ACI 440.1R-06 for the deflection analysis of the FRP reinforced which
includes the influence of the reinforcement ratio seems to be still overestimating the
stiffness of the member and hence leading to the underprediction of resulting deflection.
M.N.Habbeb proposed the following modification on the equation proposed by ACI
440(2006) for calculating the effective moment of inertia of the cracked FRP reinforced
beam
(69)
The modification is provided by the introduction of the new factor, ΥG. In the tests
conducted by M.N.Habbeb (2007), this factor comes out to be equal to 60%. The factor was introduced to tune the experimental results with the theoretical ones. There is no explanation for the physical significance of the provided modification factor. This renders the need of more researches and tests for the global acceptability of the provided modification for the deflection analysis of the FRP reinforced beams.
From the detailed study of the research conducted in the past, and its ongoing momentum for the FRP reinforced concrete beams, it is observed that the structural responses of FRP reinforced beams is qualitatively different from the conventional steel reinforced beams. In the case of steel reinforced beam, the stiffness of the member can be 185 viewed as the intrinsic property of the material and thus independent of the quantity of the reinforcement. Particularly in the deflection analysis of the beams, this issue becomes certainly more pertinent. The higher stiffness of the steel reinforcement allows the gradual and relatively slower transition of the gross moment of inertia to the cracked moment of inertia at ultimate load and thus assures the acceptable applicability of the
Branson’s equation for the deflection analysis.
In the case of FRP reinforced beam, owing to their lower stiffness and weaker bond strength with the concrete matrix, a different approach is demanded to make the older methodology applicable in their realm. As sufficiently discussed above, the FRP reinforced beam due to their lower stiffness, exhibits completely different behavior when they were analyzed with the established methodology for the steel reinforced beams. It was observed that the application of Branson’s equation for the case of FRP reinforced beams was not totally justified due to some emergent responses generated by the FRP reinforced beams in contrast to the steel reinforced under similar situation. The analytical works by Bischoff in this field was found to be particularly commendable. He quantified the different structural response of the FRP reinforced beam in terms of tension stiffening factor. This factor was found to be inversely proportional to the stiffness of the reinforcement and the reinforcement ratio as given by equation 54 above.
Hence, due to the resulting higher tension stiffening, the FRP reinforced beams depict completely different response in deflection, when they were analyzed with the
Branson’s equation. As already mentioned, the higher tension stiffening allows the faster transition from the gross moment of inertia to the cracking moment of inertia in the case of FRP reinforced beams as it makes the ratio of gross to cracking moment of inertia
186 higher then 4, depending on the relative reinforcement ratio.Therefore, the original
Branson’s equation models the deflection response of the respective beam to be more
stiffer than it actually is.
On the basis of the detailed discussion of the various modifications proposed by
various scienctists, it can be observed that there has surely been a considerable amount
of work to modify the Branson’s equation to make it compatible with the analysis of FRP
beam also.
This issue can be addressed in the following three ways:
1. The derivation of effective moment of inertia on the basis of the fundamental structural
response of the FRP reinforced beams
The works done by the Bischoff, to develop a generalized relationship for the effective moment of inertia including the stiffness and reinforcement-ratio contribution comes under this heading. This approach may be the most effective one as it will provide a global and unified platform for the design of the FRP reinforced beams.However,in absence of the sufficient knowledge of the associated mechanical properties of the various FRPs and the non-uniformity rendered by the manufacturing methods, this may take a long time and effort as well.
2. Tuning of the exponent of Branson’s equation to address the effect of reinforcement ratio
Under this approach, the effective moment of inertia can still be expressed as a linear combination of gross and cracked sectional properties, though, the exponent “m”
187 can be effectively tuned to address the effect of varying reinforcement ratio. This
approach was adopted by Toutanji and Safi.
3. Introduction of the Modification Factor in the Original Branson’s Equation
On the basis of the previous discussion on various modifications, this can be
regarded as the most popular approach to address this issue. This approach expresses the
effective moment of inertia as the linear combination of gross and cracked sectional
properties and uses the original Branson’s equation with exponent 3.however; the various modification factors are incorporated in the original equation to address the effect of reinforcement ratio on the deflection behavior. Most of the researches as discussed above, come under this heading.ACI also have been adopting this method to recommend the suitable relationship for the effective moment of inertia for FRP reinforced beams. This method is simpler and effective, though, may not provide a comprehensive and unified discipline for all FRPs.
This fact can be observed that, there has been a considerable amount of
researches and investigations in the field of the structural behavior of the FRP reinforced
beams, more specifically in the field of deflection behavior. All theses rigorous studies
and researches indicate the need for a unified and global methodology to meet this
objective.
188 CHAPTER XIII
MATERIALS AND MIXES
Since the primary objective of this investigation is the study of the flexural behavior of the BFRP reinforced beams, the major structural materials that were used for the research purpose is the conventional concrete and Basalt bars of specific sizes. The other materials that were used were the auxiliary materials that were required to make the research functional in a smooth and coherent manner.
13.1 Materials
This section includes the brief outline of the materials that were used for the investigation of the flexural behavior of the BFRP reinforced beams. This section is intended to give a comprehensive overview of the various materials that were used, their relevant characteristics, specifications and their associated properties.
189 13.1.1 Basalt FRP Bars
The basalt bars which were used for the research were provided by the sponsor
Blackbull. The Basalt bars provided for the research were of the nominal diameter of
3mm,5mm and 7mm.This sizes corresponds to the net size governed by the fibers only.
The method for manufacturing for the basalt rod was reported to be Wet ley- up process.
Wet ley-up process is a very simple method of producing FRP composite materials and is done manually. The process essentially consists of laying up the fibers and impregnating them with the polymeric resin such that it yields the usable composite material when cured. The fibers for the case of basalt FRP bars were extracted from the igneous rock named Basalt. The primary composition of Basalt rock is generally constituted with various forms of oxides, silica-oxide being the most abundant one. The percentage of silica oxide is generally between 51.6 to 57.5 percent and generally the basalt with the silica-oxide content above 46 percent (acid-basalt) is considered good for fiber- production.
Minerologically, Basalt is primarily constituted of minerals Plagioclase, pyroxene and olivine. When heated at high temperature, Basalt is capable of producing a natural nucleating agent which plays a major role for the thermal stability of the material.
This explains the apparent increased volumetric integrity of basalt in compare to the other materials. The presence of the before mentioned minerals may be a helpful factor for this phenomenon. The polymeric resin used as the matrix for the Basalt fiber is the
Vineylester resin. A vinylester resin is the combination of an epoxy and an unsaturated polyester resin. (L.C.Bank, 2006).The advantage of vinylester is that it has the
190 meritorious physical properties of the epoxy and the beneficial processing properties of a
polyester resin. The BFRP bars of different sizes, used for the reinforcing the concrete
beams were shown in the figure below.
Figure 66: Basalt FRP Bars used in Beams
The actual sizes of the provided FRP bars were measured in the laboratory with a high precision vernier-calipar. Their gross diameters including the polymeric matrix
were found to be 4.7mm, 7mm and 10mm respectively for the bar of sizes 3mm,5mm and
7mm.The resulting fiber-volume fractions were found to be 44%,52% and 41%
respectively. Hence, the average volume fraction was worked out to be 46%.The Basalt
bars will be referred later on with reference to their net diameter.
191 13.1.2 Coarse Aggregate
The coarse aggregate used was the crushed limestone aggregates with the maximum size of ¾ inch (19mm).the aggregates were angular and free from clay and other impurities.
13.1.3 Fine Aggregate
The fine aggregate was the river sand purchased from the local supplier. The sand was free from clay and other inert impurities.
13.1.4 Water
The water used for the concrete mix was the normal tap water supplied by the city of Akron.
13.1.5 Steel Stirrups
For the shear reinforcement, the BFRP reinforced beams were reinforced with no-2 steel stirrups. The steel stirrups were ordered from the local shop to be prefabricated at the particular shape. The mechanical properties were obtained from the tensile coupon test conducted on the Universal Testing machine.
192
Figure 67: Steel-Stirrups Used for Shear-Reinforcement
13.1.6 Steel Bars
For reinforcing the two control steel-beams, no-3 and no-4 steel bars were used as the reinforcing bars. The bars were ordered from the local steel-shop. The mechanical properties were obtained by testing the tensile-coupons in universal testing machine.
13.1.7 Strain-Gages
The various types of strain-gages, which were used to acquire the strain in the reinforcing bars and in the concrete is tabulated in table 11.
193 CHAPTER XIV
TEST PROCEDURES
This section includes the comprehensive description of the test procedures pertinent to the flexural testing of the Basalt FRP reinforced beams. The Test procedures include the systematic summary of the every type of activities involved with the test of the FRP reinforced beams. Needless to say, it includes the complete formulation of the steps which were carefully undertaken to achieve to the particular objective of the study.
The test procedure primarily comprises the casting of the beams followed by their testing.
However, these two most fundamental steps invariably comprise many sub sets which demands clear scrutiny and the lucid description to define the whole process. The whole experimental program consisted of thirteen basalt reinforced concrete beams with varying reinforcement ratios. Two steel beams were also tested simultaneously as the control specimens. The beams were provided with the steel shear stirrups. All the reinforced concrete sections were seven inch by 8 inch and 84 inches long. The cross-sections of the beams are shown in Figures below.
194 8" 8" 8"
2 #2 Steel Bar 2 #2 Steel Bar 2 #2 Steel Bar
7" 7" 7"
2-3 mm Diameter Basalt Bar 2-5mm Diameter Basalt Bar 2-7mm Diameter Basalt Bar
Beam B-1 Cross-Section Beam B-2 Cross-Section Beam B-3 Cross-Section
8" 8" 8"
2 #2 Steel Bar 2 #2 Steel Bar 2 #2 Steel Bar
7" 7" 7"
3-3mm Diameter Basalt Bar 2-3mm Diameter Basalt Bar +1-5MM Diameter Basalt Bar 2-3mm Diameter Basalt Bar +1- 7mm Bar
Beam B-4 Cross-Section Beam B-5 Cross-Section Beam B-6 Cross-Section
8" 8" 8"
2 #2 Steel Bar 2 #2 Steel Bar 2 #2 Steel Bar
7" 7" 7"
2-5mm diameter Basalt Bar+1-3mm Bar 3-5mm Diameter Basalt Bar
Beam B-7 Cross-Section Beam B-8 Cross-Section Beam B-9 Cross-Section
8" 8" 8"
2 #2 Steel Bar 2 #2 Steel Bar 2 #2 Steel Bar
7" 7"
2-7mm Diameter BasalT Bar 3-3mm Diameter Basalt Bar 2-3mm Diameter Basalt Bar +1-5mm Diameter Basalt Bar
7"
Beam B-10 Cross-Section Beam B-11 Cross-Section Beam B-12 Cross-Section
Figure 68:Cross-Section Details of the Beams
195 8" 8" 8"
2 #2 Steel Bar 2 #2 Steel Bar 2 #2 Steel Bar
7" 7" 7"
2-5mm Diameter Basalt Bar+1-3mm Bar -Bottom 2 #3 Steel Rebars 2 #4 Steel Rebar
Beam B-13 Cross-Section Beam B-14 Cross-Section Beam B-15 Cross-Section
Figure 68:Cross-Section Details of the Beams
The reinforcement details for all the Basalt reinforced beams are shown in
Table10 below. The beams are provided with varying reinforcement ratios as the certain percentage of the balanced reinforcement ratio to actualize under reinforced and over reinforced condition. The reinforcement ratios vary between forty three to three hundred and fifty three percent of the balanced reinforcement ratio. It can also be observed that the beam B10, B11, B12 and B13 are exactly congruent to the beams B4, B5, B2 and B7 respectively. This was done with the view to have a check on the variation of the responses of the similar beams under the similar condition of loading. In addition, it can be seen that two steel beams were also tested as the control beams so as to provide a relative comparison with the Basalt FRP reinforced beams.
196 Table 10: Details of the Reinforcement Ratios for the Beams
Beam 2 2 Reinforcement Area(in ) Area(mm ) % of ρb Designation Ratio B1 0.0219 14 0.0004 43% B2 0.0609 39 0.0012 120% B3 0.1193 77 0.0024 235% B4 0.0329 21 0.0007 65% B5 0.0523 34 0.0010 103% B6 0.0816 53 0.0016 161% B7 0.0718 46 0.0014 142% B8 0.0913 59 0.0018 180% B9 0.179 115 0.0036 353% B10 0.0329 21 0.0007 65% B11 0.0523 34 0.0010 103% B12 0.0609 39 0.0012 120% B13 0.0718 46 0.0014 142% B14 0.22 142 Control Steel Beams B15 0.40 258
14.1 Preparation of Specimens
Preparation of specimen includes all the major and minor works relevant to the materialization of the specimens in the form that was amenable for the testing. All the specimens were casted in the university premise .Evidently, it comprised varied types of tasks associated with this steps. The various tasks which had been undertaken under this section are described below:
197 14.1.1 Preparation of Formworks
Preparation of formworks for the beams is undoubtly a very important part of
this procedure. A good efficient testing depends on the geometrical uniformity of the
specimens which on the other hand, depends on the quality of the formwork. All the
formworks were made in the civil-engineering department premise. To have more comfort for the handling, three formworks were made integrally as a single unit as shown in the figure. In total three numbers of formwork-units were made such that it could accommodate nine beams in total.0.75 inch plywood were used for the making of the body of the formworks. Plywood was brought from the local distributor and they were carefully chosen to avoid deformed and knotted pieces. The forms were adequately braced and screwed around the perimeter and mostly on the bottom, to provide lateral stability and stiffening to the formworks during the pour. Alongside the formworks, platforms equipped with the wheel to facilitate the transportation of the poured beams were also made.
14.1.2 Preparation of Cages and the Strain Gages
After the construction of the formworks, the next step which was followed was the making of the reinforcement cages and the fixation of the strain gages. Reinforcement cage included the arrangement of the reinforcing bars and the shear stirrups. The reinforcing bars were simply supported between the two points at the ends of the beams.
This can seen in Fig-64 below. The spacing of the shear stirrups were marked on the
198 main bars. Shear stirrups were provided in the required shape by the local fabrication
shops. The shear stirrups were positioned on the marks on the main reinforcement and
were tied together with the help of six inch long ties. The tie consisted of six inch long
ductile wire with an arrangement to twist at its ends. The wires were placed at the
required spot of assemblage and were twisted manually with the help of a wire-twister.
Table two shows the details of the shear-stirrups for the different beams. The relevant pictures were depicted below.
Figure 69: Reinforcement Cages
199
Figure 70: Formworks and Preparation of Cages
After the preparation of the reinforcement cages, the next step was the fixation of the
strain-gages on the Basalt reinforcing bars. This is unequivocally the most important part
of the whole experimental program. Since the stain-gages are the experimental elements
which were supposed to provide us with the usable data from the tests, the process was
undertaken with much more concentration and diligence. For the fixation of strain gages
the provided information for the concerned brand was strictly followed. First of all the
right place for the positioning of strain-gage was selected, which is approximately the
centre of the longitudinal bar. Then the required length of the rod was filed sufficiently to
render a smooth surface. This was followed by the application of three different types of
sandpapers with the increasing degree of fineness on the surface to eliminate the fine
irregularities. The finished face was applied with conditioner and neutralizer
subsequently. The strain-gage was then attached to the finished surface with the help of the provided adhesive. The attached strain-gage was held between the thumb and the
finger for at least one minute to enable the strain gage to get attached to the surface 200 properly. The strain gages were then covered with wax to eliminate the possibility of moisture engrossment from the concrete. The required length of the wire was then soldered with the terminals. The details of the strain gages were tabulated in the table -11 below. Fig-66 and Fig-67 shows the pertinent pictures.
201 Table 11: Details of Strain gages Used in Beams
beam-designation basalt-reinforcement details steel reinforcement details shear stirrups details strain-gages details number size(mm) number size(mm) Size(number) spacing(in) basalt-rebar basalt-rebar basalt-bar concrete
B1 3 2 2 no-2 no-2(10) 9 CEA-13-250UN-350 CEA-13-250UN-350 EA-06-10CBE-120 B2 5 2 2 no-2 no-2(12) 7.5 CEA-06-375UW-120 CEA-06-375UW-120 EA-06-10CBE-120 B3 7 2 2 no-2 no-2(18) 4.8 CEA-06-375UW-120 CEA-06-375UW-120 EA-06-10CBE-120 B4 3 3 2 no-2 no-2(11) 8.2 CEA-06-250UW-120 CEA-06-250UW-120 EA-06-10CBE-120 B5 2-3mm & 1-5mm 3 & 5 2 no-2 no-2(11) 8.2 CEA-06-250UW-120 CEA-06-250UW-120 EA-06-10CBE-120 B6 2-3mm &1-7mm 3 & 7 2 no-2 no-2(13) 8.2 CEA-06-250UW-120 CEA-06-250UW-120 CEA-06-375UW-120 EA-06-10CBE-120 B7 2-5mm & 1-3mm 3& 5 2 no-2 no-2(11) 8.2 CEA-13-250UN-350 CEA-06-250UW-120 EA-06-10CBE-120 B8 3 5 2 no-2 no-2(11) 8.2 EA-06-10CBE-120 B9 3 7 2 no-2 no-2(13) 6.75 CEA-06-375UW-120 CEA-06-375UW-120 EA-06-10CBE-120 B10 3 3 2 no-2 no-2(13) 6.75 CEA-06-375UW-120 CEA-06-375UW-120 EA-06-10CBE-120 B11 3 3 2 no-2 no-2(13) 6.75 CEA-06-375UW-120 CEA-06-375UW-120 EA-06-10CBE-120 B12 2 5 2 no-2 no-2(13) 6.75 CEA-06-375UW-120 CEA-06-375UW-120 EA-06-10CBE-120 B13 2-5mm & 1-3mm 3& 5 2 no-2 no-2(13) 6.75 CEA-06-375UW-120 CEA-06-375UW-120 EA-06-10CBE-120 202
beam-designation Bottom steel-reinforcement details steel reinforcement details shear stirrups details strain-gages details number size(mm) number size(mm) Size(number) spacing(in) Steel-rebar Steel-rebar Steel-rebar concrete B14 2 no-3 2 no-2 no-2(13) 6.75 CEA-06-375UW-120 CEA-06-375UW-120 EA-06-10CBE-120 B15 2 no-4 2 no-4 no-2(13) 6.75 CEA-06-375UW-120 CEA-06-375UW-120 EA-06-10CBE-120
Figure 71: Students learning to fix the strain gages on the Basalt bar
Figure 72: Students working on the Strain-Gages
203 14.1.3 Casting of Beams and Cylinders
Casting of the beams and cylinders was obviously also one of the most
important part .This part was carefully undertaken to ensure the production of good
concrete and hence the beams. The inside surface of the wooden formworks were
sufficiently applied with the form oil to ensure that the concrete doesn’t stick with the
formwork. The wooden spacers were also provided along the length at reasonable
spacing of the beam so as to provide the sufficient cover at the bottom. The concrete was
delivered to the site of casting by the concerned company. The needle vibrator was used
while casting the beams to ensure the enough compaction. Eleven concrete cylinders
were also casted of the same concrete for the compressive strength test of the provided
concrete. The surfaces of the casted beams were finished with the steel-plates so as to provide the smooth surface for the application of the load. The pertinent pictures of the concrete-works are shown in the figures below.
204
Figure 73: Finishing of the Top Surface of the Beams
205
Figure 74: Concrete Works for the Beams
The beams were then left for 24 hrs to set. After 24 hrs, the beams were demolded .The beams were then covered with the saturated burlap followed by the cover of plastic over it. Beams were then transported to the safe place for storage. The burlap covers were regularly sprayed with water so as to provide the regular supply of moisture. The
cylinders were kept in the humid-room which was supposed to provide the 100 percent humidity to the concrete cylinders.
206 When the beams were ready for testing after 28 days, the surface of the beams were grinded and sandpaper of different grades were applied to obtain the smooth surface for the attachment of strain-gage on the surface of the concrete. The strain-gage were cushioned and strongly clamped to provide the sufficient pressure to let the strain-gage to attach to the concrete surface. The strain gage was fixed at the mid-span of the beam such that the mid-span strain values could be extracted for analysis. The pictures related to the fixation of strain-gages on the concrete surface are shown below.
The details, particularly, the type of strain gages used for the concrete surface is tabulated in table 11.
Figure 75: Application of Strain-Gage on the Concrete Surface
207
Figure 76: Clamping the Strain-Gage on the Surface of the Beam
14.2 Experimental Program
The experimental program primarily consisted of testing basalt FRP bar reinforced beams under four point bending under the simply-supported condition. The beams were positioned in the Universal Testing Machine and the support condition simulating the both supports to be roller supports was provided. The schematic of the experimental program is shown in figure below.
208 7"
12" 60" 12"
Figure 77: Schematic of the Test Set-up for the Beam Test
The first series of beams, which comprised B1 to B9, specimens were tested on
BALDWIN test machine. This testing machine has a 300,000lb-capacity. The type of the machine is UTM (model 300HV-300,000LB capacity). Special I-sections were provided to act as a spreader beam as reaction beam to provide the rigid substratum for the roller supports. This arrangement is supposed to ensure the four point bending condition. This is depicted in the Figure below.
209
Figure 78: Actual Arrangement for Four-Point Bending
The second series of beams, which comprised specimens B10 to B15, were
tested on a 55 Kips capacity, MTS machine, in the Turbine Testing Lab in the premises of the University of Akron. The support arrangement for the beam test in the MTS machine is shown in the Fig-79 below. It consisted of the roller support supported on an I beam. To avoid the fall of the beam during failure and to spread the load over a larger bearing area, a thick steel plate was also provided at the support, as shown.
210
Figure 79: Arrangement at Supports
Even though the two series of beams were tested in different machine, the
underlying principal was similar as it is based on the testing of the beam under four-point
bending. The beams to be tested were lifted with the help of lifting-hoist of sufficient
capacity from the pick-up points where lifting hooks were provided during the casting to
facilitate the lifting. The strain-gage wires were connected to their corresponding channels of the data acquisition system. The roller supports were provided at a distance
of 1’ from both ends such that the effective span of the beam is 5 feet. This is depicted in
the Fig-79. Fig-80 below shows the exact arrangement for the purpose.
211
Figure 80: Position of Beam in UT Machine and MTS Machine
The beams were tested under four-point bending method. This is achieved by applying the total load at two points spaced six inches apart as shown in the picture below.
212
Figure 81: Arrangement for the Load-Application
The first series of beams were loaded at a uniform rate of loading, which generally varied from 10 to 15 lb/s. The deflection data were manually recorded from the dial gage attached to the centre of the beam at the bottom. The dial-gage used for the purpose is a dial-gage of a brand called FOWLER with the least-count of 0.0005 in. In the case of the second series of beams, where the beams were tested in MTS machine, the displacements were recorded by an LVDT. The arrangement of the LVDT is shown in the Figure below.
213
Figure 82: LVDT Positioning for the Deflection Data
After the completion of the tests, the cracks for all the beams were mapped and orthogonal grids lines were drawn on the longitudinal face of the beam to investigate the spatial distribution of cracks and thereby the nature of the failure modes being investigated.
214 CHAPTER XV
RESULTS AND ANALYSIS
This section includes a comprehensive description of the test results obtained from the four-point bending test performed on the reinforced concrete beams reinforced with basalt FRP bars of three different sizes. The results also include corresponding analysis performed on two different steel-reinforced beams for the relative comparisons.
The primary objective of the part of the research is to obtain the flexural characteristics of
BFRP reinforced beams on the basis of the mechanical properties obtained from the longitudinal tensile strength conducted beforehand. The flexural characteristics basically constitutes ,for the limited scope of this particular research, the moment strength analysis of the BFRP reinforced beams with different methods based on the average and guaranteed tensile properties, short term load-deflection analysis based on different relationships for the effective moment of inertia and the study of the crack-propagation during bending.
215 15.1 Research Objective
The primary objective of this part of the research is the identification of the
important flexural properties of Basalt FRP reinforced concrete beams as provided by the
sponsor. The research is intended to have a relative comparison of different methods for the determination of ultimate moment strength of the BFRP reinforced beams. This further helps up to gain some insight into the differing properties of Basalt bars with the
other common FRP bars and to ascertain the limit of applicability of the designated current standard for the analysis of BFRP reinforced beams. For the serviceability limit state, load-deflection analysis is another important subject that was addressed. The
research is also intended to conduct the load-deflection analysis on the BFRP reinforced
beams based on the different methods and to ascertain their limit of applicability in the
case of BFRP reinforced beams.
15.1.1 Moment-Strength of the BFRP Reinforced Beams
The computation of moment-strength of the BFRP reinforced beams is similar to same principle of the conventional steel-reinforced beams. The flexural-strength of a reinforced concrete beam can be defined as the moment of resistance offered by the reinforced section to counteract the moment applied on the section due to the external loads. The moment of resistance, which quantifies the flexural-strength of the particular section is constituted by the couple generated by the internal tension and compression forces acting on the section due to bending under the application of external load.
216 The following two methods can be used for the calculation of the moment-strength computation of the reinforced sections.
15.1.2 ACI 440.0R-06 Method
The method for the calculation of the moment-strength of steel-reinforced-
section is specified in ACI 318-08 and ACI 440 .01R-06 for FRP reinforced concrete sections. They both are based on similar principle of replacing the nonlinear-distribution of the compressive stress across the section in the concrete by an equivalent ACI rectangular stress block for the calculation of the compressive force acting on the section.
For the rectangular stress-block, the strains are assumed to be varying linearly with the depth of the stress-block, the proportionality factor depending on the maximum concrete compressive strain. In case of the steel-reinforced sections, the stress-strain relationship is assumed to be bilinear to simplify the calculation of the tensile forced being carried by the steel bars. Whereas in the case of FRP reinforced section, the actual linear stress- strain relationship is incorporated in the analysis.
15.1.3 Strain Compatibility Method
The computation of the moment-strength of beam sections based on the different models of actual stress-strain curve for concrete constitutes the Strain compatibility method. The stress-strain curve of the concrete is constituted by the initial linear variation of the stress-strain values upto approximately 40-45% of the compressive
217 strength of the concrete, followed by nonlinear variation. The concrete will reach the strain of 0.002 corresponding to the maximum compressive stress reached by the concrete, eventually reaching the ultimate strain of 0.003 at failure (as per the ACI).
There are different models proposed for the modeling the stress-strain curve for concrete.
A parabolic stress-strain curve proposed by Desai and Krishnan (1969) is used for our analysis. The parabolic stress-strain curve proposes the following model for the concrete.
(70)
Where is the compressive strength of the concrete and is the strain in the concrete corresponding to the maximum compressive stress in the concrete, experimentally determined to be equal to 0.002. The parameter yielding the equivalent total compressive stress in the concrete is given in this model as
(71)
Where the variation of the compressive stress is across the depth of the section as the function of the distance from the top compression fiber. Similarly, the expression for the parameter yielding the depth of the resultant compression force from the top of the compression fiber is given as
218 (72)
The strain-compatibility method was applied for the computation of the moment-strength
of the BFRP reinforced beams and the two controlled steel-beams using the above
explained parabolic model for the stress-strain relationship for concrete. For the
controlled steel beams, the moment strengths by strain-compatibility method were computed using the bilinear assumption for the stress-strain curve for steel and the actual stress-strain curve of the steel from the tensile-coupon test. For the BFRP reinforced
beams, the actual stress-strain curve of basalt FRP bars and the above parabolic stress-
strain curve of concrete were used to compute the moment-strength.
Since the BFRP reinforced beams composed of the reinforcing bars of different
diameter and the properties of the FRP bar varies with the size, the equivalent structural
and geometrical properties associated with each beam were calculated from the mean and
the guaranteed properties of the reinforcing bars. They are tabulated below.
15.2 Moment-Strength of the BFRP Reinforced Beams
Based on the average and guaranteed properties associated with each BFRP
reinforced beams as given in the tables above, the moment strengths were calculated
using both ACI and Strain-Compatibility method. The Factored moment strength were
also calculated based on the average mechanical properties using the ACI method to see
the margin of safety associated with the BFRP reinforced beams using the ACI method.
219 The moment-strengths calculated for different beams from different methods are tabulated in Table 12 and 15.
220 Table-12: Compressive-Strength Details for the Beam-Tests
Beam Date of Casting Date of Testing f (psi) f (Mpa) Designation c c B1 5/15/2008 6/3/2008 4743 32.7 B2 5/15/2008 6/3/2008 4743 32.7 B3 5/15/2008 6/3/2008 5107 35.2 B4 5/22/2008 6/11/2008 3735 25.8 B5 5/22/2008 6/11/2008 3735 25.8 B6 5/22/2008 6/11/2008 3735 25.8 B7 5/22/2008 6/11/2008 3665 25.3 B8 5/22/2008 6/11/2008 3770 26.0 B9 5/22/2008 6/11/2008 3665 0.2 B10 6/30/2008 5773 39.8 B11 6/30/2008 8/28/2008 5773 39.8 B12 6/30/2008 9/11/2008 6035 41.6 B13 6/30/2008 9/12/2008 6035 41.6
221 Table 13: Equivalent Beam Properties Based on the Guaranteed Properties of the BFRP Bars
Assumptions: 1.The largest diameter BFRP bar was considered for calculating effective depth 2.For the beams consisting of different bar-size combinations,average rupture strain was considered for relevant calculation
#Table of relavent constants for different BFRP beams based on guarantteed prpperties
spacer- half of the Beam b h d d No of BFRP bars Area of BFRP bars(sq.in) Af ff1 ff2 ff3 ε ε ε E1 E2 E3 ff ε Ef width diameter c f1 f2 f3 f
in in in in in n 4.7(0.185in) 7(0.275in) 10.5(0.413in) Af1(3mm) Af2(5mm) Af3(7 mm) (sq.in) ksi ksi ksi in/in in/in in/in ksi ksi ksi ksi in/in ksi
B-1 8 7 0.50 0.09 0.59 6.41 2 0 0 0.0110 0.0304 0.0597 0.0219 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 336 0.030 12474
B-2 8 7 0.50 0.14 0.64 6.36 0 2 0 0.0110 0.0304 0.0597 0.0609 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 320 0.027 12527
B-3 8 7 0.50 0.21 0.71 6.29 0 0 2 0.0110 0.0304 0.0597 0.1193 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 302 0.024 12747 222 B-4 8 7 0.50 0.09 0.59 6.41 3 0 0 0.0110 0.0304 0.0597 0.0329 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 336 0.030 12474
B-5 8 7 0.50 0.14 0.64 6.36 2 1 0 0.0110 0.0304 0.0597 0.0523 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 327 0.029 12404
B-6 8 7 0.50 0.21 0.71 6.29 2 0 1 0.0110 0.0304 0.0597 0.0816 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 311 0.027 12035
B-7 8 7 0.50 0.14 0.64 6.36 1 2 0 0.0110 0.0304 0.0597 0.0718 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 322 0.029 12092
B-8 8 7 0.50 0.14 0.64 6.36 0 3 0 0.0110 0.0304 0.0597 0.0913 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 320 0.027 12527
B-9 8 7 0.50 0.21 0.71 6.29 0 0 3 0.0110 0.0304 0.0597 0.1790 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 302 0.024 12747
Table 14: Equivalent Beam Properties Based on Average Properties of the BFRP Bars
Assumptions: 1.The largest diameter BFRP bar was considered for calculating effective depth 2.For the beams consisting of different bar-size combinations,average rupture strain was considered for relevant calculation
#Table of relavent constants for different BFRP beams based on average parameters
spacer- half of the Beam b h d d No of BFRP bars Area of BFRP bars(sq.in) A f f f ε ε ε E E E f ε E width diameter c f f1 f2 f3 f1 f2 f3 1 2 3 f f f
in in in in in n 4.7(0.185in) 7(0.275in) 10.5(0.413in) Af1(3mm) Af2(5mm) Af3(7 mm) (sq.in) ksi ksi ksi in/in in/in in/in ksi ksi ksi ksi in/in ksi
B-1 8 7 0.50 0.09 0.59 6.41 2 0 0 0.0110 0.0304 0.0597 0.0219 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 336 0.030 12474
B-2 8 7 0.50 0.14 0.64 6.36 0 2 0 0.0110 0.0304 0.0597 0.0609 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 320 0.027 12527
B-3 8 7 0.50 0.21 0.71 6.29 0 0 2 0.0110 0.0304 0.0597 0.1193 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 302 0.024 12747 223
B-4 8 7 0.50 0.09 0.59 6.41 3 0 0 0.0110 0.0304 0.0597 0.0329 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 336 0.030 12474
B-5 8 7 0.50 0.14 0.64 6.36 2 1 0 0.0110 0.0304 0.0597 0.0523 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 327 0.029 12404
B-6 8 7 0.50 0.21 0.71 6.29 2 0 1 0.0110 0.0304 0.0597 0.0816 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 311 0.027 12035
B-7 8 7 0.50 0.14 0.64 6.36 1 2 0 0.0110 0.0304 0.0597 0.0718 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 322 0.029 12092
B-8 8 7 0.50 0.14 0.64 6.36 0 3 0 0.0110 0.0304 0.0597 0.0913 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 320 0.027 12527
B-9 8 7 0.50 0.21 0.71 6.29 0 0 3 0.0110 0.0304 0.0597 0.1790 336 320 302 0.03 0.0272 0.0242 12474 12527 12747 302 0.024 12747
Table 15: Moment-Strengths of BFRP Reinforced Beams by Different Methods
Experimental Moment Factored Moment Beam Strain-Compatibility ACI % difference Cracking-Load Experimental Load % difference strength(k-ft) strength(k-ft) Moment-Strength based on Moment-Strength based on Moment-Strength based on Moment-Strength based on Average Parameters(k-ft) gurantted parameters(k-ft) Average Parameters(k-ft) gurantted parameters(k-ft) B-1 3.86 2.95 3.78 2.83 4.42 2.08 -12.67 3295 3784 -14.8 B-2 10.00 8.00 10.00 7.70 8.53 5.50 17.23 8753 7444 15.0 B-3 14.20 14.20 10.00 12.80 12.70 8.33 11.81 12486 11140 10.8 B-4 5.70 4.40 5.70 4.30 5.22 3.12 9.20 4931 4506 8.6 B-5 8.20 6.80 7.80 6.70 8.40 4.22 -2.38 7153 7325 -2.4 B-6 9.80 9.90 9.20 9.50 10.93 6.00 -10.34 8575 9579 -11.7 B-7 9.30 9.20 8.75 8.75 10.55 5.70 -11.85 8131 9236 -13.6
B-8 10.60 10.60 10.00 10.00 9.50 6.50 11.58 9286 8296 10.7 B-9 13.80 13.80 13.00 13.00 12.75 8.40 8.24 12131 11191 7.7 B-10 5.80 4.50 5.70 5.15 5.15 3.14 12.62 5019 4435 11.6 224 B-11 8.83 6.92 8.75 6.92 9.04 4.79 -2.30 7716 7900 -2.4 B-12 10.05 8.00 9.35 7.77 10.17 4.28 -1.15 8797 8900 -1.2 B-13 12.18 9.41 11.83 9.18 9.61 6.50 26.80 10694 8400 21.4
The comparison of the moment-strengths by different methods is graphically shown below:
16
14
12
Strain-Compatibility- Average Values 10 - ft) Strain-Compatibility- Gurantted Values
8 ACI-Average Values Strength (K - Strength
ACI-Guarantted Values 6 Moment
Experimental Moment- Strength 4 Factored Moment Strength-Average Values
2
0
B B B B B B B B B B B B B Figure 83: Moment-Strength Comparison for BFRP Beams
225 14000
Failure- Load-Theoritical 12000 Failure-Load-Experimental
10000
8000
6000 Load (lb) -
4000 Cracking 2000
0 B-1 B-2 B-3 B-4 B-5 B-6 B-7 B-8 B-9 B-10 B-11 B-12 B-13
Figure 84: Ultimate Load Comparison for BFRP Reinforced Beams
15.2.1. Cracking-Moment of the BFRP Reinforced Beams
Cracking moment can be defined as the magnitude of bending-moment due to the external load at which the extreme tensile fiber of the reinforced concrete beam reaches the bending tensile strength (modulus of rupture) of the concrete. Cracking moment is primarily a function of the compressive strength of the concrete and the transformed section-properties of the reinforced section. It can be expressed as
(73)
Where is the modulus of rupture of the concrete and is given as per ACI as
(74)
226 and is the transformed Moment of inertia of the reinforced section and is the depth
of the extreme tensile fiber from the neutral-axis. The cracking moments for the BFRP reinforced beams are tabulate in Table-14. Beam B-12 was accidently loaded before the testing and the beam cracked before terminating the test. Hence the cracking-load data for beam B-12 was not considered for comparison.
Table-16: Cracking-Moment Comparison for the BFRP Reinforced Beams
cracking moments(k-ft) EXP CAL % difference B-1 2.75 2.81 -2.14 B-2 2.97 2.81 5.58 B-3 3.19 2.91 9.68 B-4 2.50 2.82 -11.35 B-5 2.00 2.50 -20.00 B-6 2.79 2.50 11.67 B-7 2.52 2.50 0.67 B-8 2.30 2.50 -8.00 B-9 2.70 2.50 8.00 B-10 3.50 3.10 12.82 B-11 2.70 3.10 -12.90 B-12 3.20 B-13 2.90 3.20 -9.38
227 4.0
3.5
3.0
2.5 ft) -
2.0 Cracking-Moment- Experimental Moment(k
- 1.5 Cracking-Moment- Theoritical 1.0 Cracking
0.5
0.0 B-1 B-2 B-3 B-4 B-5 B-6 B-7 B-8 B-9 B-10 B-11 B-12 B-13
Figure 85: Cracking-Moment Comparison for the BFRP Reinforced Beams
Similarly the cracking load can be defined as the magnitude of the external load at which the extreme tensile fiber reaches the stress level corresponding to the modulus of rupture of the concrete. The cracking-load comparison is tabulated in Table-17.
228 Table 17: Cracking-Load Comparison for BFRP Beams
% Cracking-Load(lb) EXP CAL Difference B-1 2300 2364 -3 B-2 2500 2364 6 B-3 2700 2458 10 B-4 2000 2082 -4 B-5 1500 2082 -28 B-6 2340 2082 12 B-7 2100 2061 2 B-8 1800 2093 -14 B-9 2400 2061 16 B-10 3000 2622 14 B-11 2200 2622 -16 B-12 2683 B-13 2400 2683 -11
14000
Failure- Load-Theoritical 12000 Failure-Load-Experimental
10000
8000
6000 Load (lb) -
4000 Cracking 2000
0 B-1 B-2 B-3 B-4 B-5 B-6 B-7 B-8 B-9 B-10 B-11 B-12 B-13
Figure 86: Graphical Comparison of Cracking-Load for BFRP Beams
229 15.2.2 Moment Strength of the Steel Reinforced Control Beams
As it already been explained before, two steel beams were also casted with the
BFRP beams as the control beams. The reinforcement details for the steel-beams are shown in table. The testing of the steel beams were also intended to see the difference in the prediction of the moment-strength of the steel-beams using the bilinear stress-strain assumption for the steel (ACI method) and also using the actual stress-strain curve of the steel obtained from the tensile-coupon tests. Since the steel-beams were reinforced with
#3 and #4 steel bars, two tensile-coupons for each size were prepared and were tested under the uniaxial loading to obtain the actual stress-strain curve of the steel. The stress- strain curves for steel for the respective sizes from the tests are shown in Fig-87 and 88.
The moment-strengths of the two steel-beams were computed using the following methods for the relative comparison of the various methods.
a. Moment-strength using the ACI method which is based on equivalent rectangular stress-distribution in concrete and the bilinear stress-strain curve for steel.
b. Moment strength using the Strain-compatibility method which is based on the assumption of parabolic stress-strain curve for concrete and bilinear stress-strain curve for steel.
c. Moment strength using the Strain-compatibility method which is based on the assumption of parabolic stress-strain curve for concrete and actual stress-strain curve for steel as shown if figures below.
230
120
100
80
60 Stress (ksi) Stress
40
20
0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Strain (in/in)
Figure 87: Stress-Strain Curve for No-3 Bar
231
120
100
80
60 Stress (ksi) Stress
40
20
0 0 0.05 0.1 0.15 0.2 0.25
Strain (in/in)
Figure 88: Stress-Strain Curve for No-4 Steel Bar
The moment-strengths of the steel beams were computed with the above explained three methods. The relative comparison of the different methods is tabulated below.
Table-18: Moment-Strengths Comparison of the Steel Beams
% bea ACI( SC- SC-actual Experimenta differen m k-in) Bilinear(k-in) curve(k-in) l(k-in) ce B- 12.4 91.50 92.00 120.00 137.00 14 1 B- 166.0 168.00 196.20 207.00 5.22 15 0
232
250
200
ACI in) - 150
SC-Bilinear curve Strength (k Strength - SC-Actual 100 curve
Moment Experimental moment- strength
50
0 1 2
Figure 89: Graphical-Comparison of the Moment-Strengths of the Steel Beams
Where 1 and 2 on the horizontal-axis represents the B-14 and B-15 respectively.B-14 consists of the no-3 steel bars and B-15 consists of no-4 steel bars.
15.3 Load-Deflection Analysis for the BFRP Reinforced Beams
From the serviceability point of view, load-deflection analysis constitutes a very important part of the analysis which importance can’t be underestimated. In the previous sections, we have gone through the ultimate moment –strength analysis of the BFRP reinforced beams. As already been discussed in the literature-review part for the beam- analysis, owing to their relatively lower modulus of elasticity in compare to steel,
233 deflection might be the governing parameter for the design of the FRP reinforced beams.
It can be inferred from the previous researches and investigations that load-deflection analysis of the reinforced concrete beams hinges on the reasonable prediction of the effective moment of inertia after the concrete in the tension-zone is cracked. Once the concrete in the tension-zone is cracked, the flexural-stiffness of the section reduces significantly due to the considerable reduction in the moment of inertia of the section.
Since the concrete is a nonlinear material and the composite action of the reinforced concrete section can not be analyzed from the elastic method once the concrete is cracked, the prediction of effective moment of inertia of the cracked section is always a difficult task. For our research, the following three relations for the effective moment of inertia were used. Here equation (1) is the original Branson’s equation for the reinforced concrete section, equation (2) is the modified Branson’s equation for the FRP reinforced sections as per ACI and equation (3) is the relation proposed by R.AL.Sunna (1996) for the GFRP reinforced bars.
(75)
(77)
Where
234 The load-deflection analysis for the thirteen BFRP reinforced concrete beams were carried out using the above three different relations for effective moment of inertia. The relation, as proposed by R.AL.Sunna, which is equation (77), was selected owing to its more generalized structure. The equation consists of the factors incorporating the reinforcement ratio and the modulus of elasticity of steel which seems to render this equation, a more generic form. On the basis of the load-deflection distribution of the
BFRP beams, the relation proposed by R.AL.Sunna was modified and the following relation was proposed for the effective moment of inertia for the cracked section for
BFRP reinforced beams.
(79)
Where
And
The load-deflection analysis was done using the all the different relations as discussed above. The typical load-deflection curves are shown below:
235 LOAD-DEFLECTION CURVE B-2 5000
4500
4000
3500
3000
test 2500 Branson's Equation
LOAD(LB) 2000 ACI 440 R.L.Sunna Relation 1500 Proposed Relation
1000
500
0 0 0.2 0.4 0.6 0.8 1 1.2 DEFLECTION(IN)
5000 LOAD-DEFLECTION CURVE B-3
4500
4000
3500 Test 3000
2500
2000Load(lb) Branson's Equation 1500 ACI 440
1000
500
0 0 0.2 0.4 0.6 0.8 1 1.2 Deflection(in)
Figure 90: Typical Load-Deflection Curves for BFRP Reinforced Beam
236 Load-Deflection Curve B-8 8000
7000
6000
5000
Test 4000 Branson's Equation
Load(lb) ACI 440 3000 R.L.Sunna Relation Proposed relation 2000
1000
0 0 0.5 1 1.5 2 2.5 Deflection(in)
LOAD-DEFLECTION CURVE B-9 10000
9000
8000
7000 Test 6000 ACI 440
5000 Branson's Equation LOAD(LB) R.L.Sunna 4000 Relation Proposed 3000 Relation
2000
1000
0 0 0.5 1 1.5 2 2.5 3 DEFLECTION(IN)
Figure 91: Typical Load-Deflection Curve for BFRP Reinforced Beam
237 CHAPTER XVI
CRACK-MAP ANALYSIS OF THE BFRP REINFORCED BEAMS
After the flexural tests on the fifteen beams including the control steel-beams, all the beams were arranged and the crack-distribution on the beams was studied. Due to the unavailability of the instrument for the measurement of the crack-width, the theoretical crack-width predicted for the BFRP beams couldn’t be compared with the crack-width obtained from the four-point bending. However, the sample calculation for the crack- width of the BFRP beam was done assuming the service moment to be the sixty percent of the ultimate moment strength following the latest ACI 440 guidelines for the computation of the crack-width of the FRP reinforced beams.
Rectangular grids were drawn on the vertical face of the beams to depict the spatial distribution of the cracks after the tests were completed. It is evident that the crack-map, thus, will not be able to depict the temporal distribution of the crack-width
propagation with the progress of test. However, for the first nine beams, the general
details relating to crack-width, deflection and the corresponding load were noted
observing the progress of test manually. This observation will be helpful to have some
idea of the temporal distribution of crack as the test progressed. The details for the BFRP
reinforced beams from B-1 and B-9 is described below:
238 B-1: First crack appeared at around the load of 2100lb and the corresponding deflection
value was 0.018”. The nature of the cracking didn’t change until the end of the experiment i.e., up to failure. The second crack appeared at the load of 2300 lb and it propagated till the maximum load was attained. The maximum load for the beam was
3784lb.
B-2: First crack appeared at the load value of 2400 lb with the corresponding mid-span deflection of 0.0265”.The second crack was observed at 3500 lb. The beam cracked significantly at around 4300 lb. With the corresponding defection of 0.3085”. Number of cracks observed along the length of the beam was seven.
B-3: First crack appeared at the load value of 2750 lb with the corresponding deflection of 0.047”. The third crack was observed at around 3340 lb at the deflection of 0.119”.
The maximum load for the beam was 11,140lb and maximum deflection at the time of
failure was 1.5”.
B-4: The first crack appeared at around the load value of 2000 lb with the corresponding
deflection of 0.016”. Second crack appeared at the load of 2800 lb and the deflection
value for that load was 0.1685”. Third crack appeared at almost the same spacing from
center-line but on the opposite side of the second crack at the load of 3350lb and the
corresponding deflection of 0.248”. Fourth crack appeared on the same side of the second
crack at the load of 4300 lb with the corresponding deflection of 0.473”. Fifth crack
appeared on the same side of third crack at the load of 3853 lb.
B-5: The first crack appeared at around the load of 1400 lb near to the centerline with the
corresponding deflection of 0.023”. The second crack appeared at the load of 2800 lb
with the corresponding deflection of 0.125”. The third crack appeared at the load of 2600
239 lb with the corresponding deflection of 0.1565”. The fourth and fifth crack appeared at around the load value of 3900lb. The maximum mid-span deflection is 1.5”and the peak load was 7325lb.
B-6: The first crack appeared around the load value of 2305 lb. The second crack at around 1900 lb and the third crack appeared at around the load of 3200 lb. The maximum mid-span deflection was 1.375” and the peak-load for the beam was 9579 lb.
B-7: The maximum load for the beam recorded was 9236 lb and the maximum mid-span deflection as observed, was 1.5”.
B-8: Two cracks appeared simultaneously approximately at the load of 1700 lb with the corresponding mid-span deflection of 0.046”. The peak load was 8296 lb and the maximum deflection as observed was 0.925”.
B-9: The maximum load for the beam was 11,191lb and the maximum deflection at failure was 0.875”.
240 16.1 Crack-Map for the BFRP Reinforced Beams
The pictures of the crack-map distribution with the rectangular grid on their faces were converted to the crack-map sketches to have a more complete and presentable understanding of the crack-distribution. These crack-maps for all the beams are shown below:
B-2
B-1
Figure 92: Crack-Map for BFRP Reinforced Beams B-1 and B-2
241 B-4
B-3
Figure 93: Crack-Map for BFRP Reinforced Beams B-3 and B-4
B-6
B-5
Figure 94: Crack-Map for BFRP Reinforced Beams B-5 and B-6
242 B-9
B-8
B-7
Figure 95: Crack-Map for BFRP Reinforced Beams B-7, B-8 and B-9
2B-2
2B-1
Figure 96: Crack-Map for BFRP Reinforced Beams 2B-1and 2B-2
243
2B-4
2B-3
Figure 97: Crack-Map for BFRP Reinforced Beams 2B-3 and 2B-4
2B-6
2B-5
Figure 98: Crack-Map for BFRP Reinforced Beams 2B-5 and 2B-6
244 CHAPTER XVII
CONCLUSION AND RECOMMENDATION
This section includes the conclusions that have been drawn from the four-point bending tests conducted on the BFRP and steel reinforced beams. This section also includes the recommendations that can be made based on the conclusions drawn from the analysis conducted from the test data obtained from the tests. The conclusions can be subdivided into the following based on the nature of analysis performed.
17.1 Moment Strength of the BFRP Reinforced Beams
From the analysis conducted in the above section, it can be concluded that the
ACI 440.1R-06 can predict the Moment-Strength of BFRP reinforced beams within
reasonable accuracy. The percentage difference in the discrepancy between the actual
strength and predicted strength was tabulated in Table-15. It was also found that the strain-compatibility method can provide a slightly better approximation than the ACI method depending on the reinforcement ratio of the different beams. The percentage difference in the predicted moment-strength based on the average and guaranteed strength parameters using Strain-Compatibility method varies between 0 and 23.6 percent depending on the reinforcement ratio of the particular BFRP reinforced beams.
245
Similarly, the percentage difference in the predicted moment-strength based on the average and guaranteed strength parameters using ACI 440 varies between 0 and 28 percent depending on the reinforcement ratio of the particular BFRP reinforced beams.
The percentage difference between the factored moment-strength using ACI 440 with average strength parameters and the actual moment strength varies between 31.5 percent to 58 percent, yielding an average margin of safety of 42 percent. So it can be concluded that, for the BFRP reinforced beams, using ACI 440 with average strength-parameters, it will provide a 42 percent margin of safety on the ultimate-moment strength.
17.2 Moment Strength of the Steel Beams
For the control steel-beams, the moment-strength was calculated using ACI method, Strain-compatibility with parabolic stress-strain curve for concrete and bilinear stress-strain curve for steel and Strain-Compatibility with parabolic stress-strain curve for concrete and actual stress-strain curve for steel obtained from the tensile-coupon test. The summary of the analysis was presented graphically in Fig-89.It can be observed from the comparison that the moment-strength calculated using the strain-compatibility method with parabolic stress-strain curve for steel and the actual stress-strain curve for steel can provide a very good estimation of the moment-strength of the steel-reinforced beams.
Similarly, it can be concluded that the ACI method and Strain- compatibility method with bilinear stress-strain assumption for steel are yielding a very conservative estimate of the moment-strength. This further clarifies the significance of the post-yielding strength of
246
the steel on the flexural behavior of the steel-reinforced sections and thus dictates the major structural difference in the response of the BFRP and steel-reinforced sections.
17.3 Cracking Moments of the BFRP Reinforced Beams
From the analysis of the cracking moments for the BFRP reinforced beams, it
was observed that due to the lower modulus of elasticity of basalt FRP reinforcing bar, the modular ratio was relatively low which rendered the cracking moments relatively independent of the transformed sectional properties of the BFRP reinforced sections. The cracking-moments were predicted with reasonable accuracy varying between two to
twenty percent. Regarding the fact that the cracking loads were recorded manually, the
prediction can be regarded to be fairly accurate.
17.4. Load-Deflection Analysis of the BFRP Reinforced Beams
From the load-deflection analysis, it was observed that the original Branson’s
equation for the effective moment of inertia for the cracked section predicted larger
stiffness for the BFRP reinforced sections thus leading to underestimation of the
deflection values. Similarly it was observed that the ACI 440 and the relation proposed
by R.AL.Sunna underpredicted the stiffness of the BFRP reinforced beams thus leading
to the overestimation of the deflection values. It was also observed that the degree of
overestimation of the deflection values was reduced with increasing reinforcement ratios.
Hence it can be concluded that reinforcement-ratio should also incorporated into the 247 computation of the stiffness of the BFRP reinforced beams. It was also observed that the
ACI 440 was providing better approximation for the deflection values for about fifty to sixty percent of the maximum load for increasing reinforcement ratios. For our particular set of BFRP beams, Branson’s equation seemed to provide the lower bound and ACI440 and the relation by R.L.Sunna seemed to provide the upper-bound for the flexural- stiffness for the cracked BFRP beams. Hence a relation for the flexural stiffness for the
BFRP beams is proposed which will provide an intermediate stiffness between these two bounds. The proposed relation is given below:
(72)
Where
And
248 REFERENCES
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8. Tensile Properties and Pull-out Behavior of AFRP and CFRP rods for grouted anchor application, B.Benmokrane, B.Zang and A.Chennouf, Construction and Building materials, 14 (2000) 157-170.
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249 12. Behavior and modeling of bond of FRP rebar to concrete, E.Cosenza, G.Manfredi, and R.Realfonzo, Journal of Composite for Construction, vol.1, 1997.
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17. Bond stress along plain steel reinforcing bars in pullout specimens, Lisa R. Feldman and F. Michael Bartlett, ACI Structural Journal, vol.104, no.6, 2007.
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19. Local bond slip relationship for FRP reinforcement in concrete, F.Foccaci, A.Nanni, C.E.Bakis, Journal of Composite for Construction, vol.4, no.1, 2000.
20. Analytical solution for bond-slip of reinforcing bars in R.C. joints, Journal of Structural Engineering vol.116, no.2 G.Russo, 1990.
21. FRP Development in U.S.A., C.E.Bakis, fiber-reinforced-plastic (FRP) reinforcement for concrete structures: properties and applications. A.Nanni (editor) @ 1993, Elsevier publication. . 22. Bond properties of high strength carbon fiber-reinforced polymer strands, Weichen Xue, Xiaohui, ACI Materials Journal, vol.105, no-1, 2008.
23. Investigation of bond in concrete members with fiber reinforced polymer (FRP) bars, Tigihourt, Benmokrane and Gao, construction and building materials, Construction and Building Materials 12 (1998) 453-462.
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250 27. Structural design with FRP materials, Composite for Construction, L.C.Bank, Jhon Willey and son, 2006.
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251
APPENDICES
252 APPENDIX A LIST OF NOTATIONS d Effective diameter of the reinforced beams h Overall depth of the beam b Width of the beam dc Effective concrete cover
’ f c Compressive strength of the concrete
Af Ares of the BFRP reinforcement
εfu Rupture strain of the BFRP bar ffu Tensile strength of the BFRP bar
β1 Concrete factor ff Tensile stress in the BFRP bar a Depth of neutral axis
εcu Ultimate concrete strain
Ef Modulus of elasticity of the BFRP bar
Ec Modulus of elasticity of the concrete
ρf Reinforcement ratio
ηf Modular ratio k Cracked-section neutral axis depth factor
Icr Cracked moment of inertia fr Modulus of rupture for the concrete β Crack-width factor
Ms Service load moment 253 w Crack-width
Mcr Cracking moment of inertia
Ma Applied moment under the external load
Ie Effective moment of inertia
βd ACI reduction factor
ρfb Balanced reinforecement ratio L Total length of the beam l Simply supported span of the beam c Length of the overhangs S Shear-span
IT Transformed moment of inertia of the section
254 Table 19: Stress-Strain Data 3mm Specimen 2
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.004615 45.82 315.9 0.004700 46.91 323.5 0.004780 48.01 331.0 0.014585 185.83 1281.3 0.014660 186.65 1287.0 0.014745 188.11 1297.0 0.014825 188.75 1301.4 0.015735 200.80 1384.5 0.015820 201.89 1392.0 0.015905 203.35 1402.1 0.016010 204.54 1410.3 0.016085 205.91 1419.7 0.016180 206.64 1424.8 0.016280 207.83 1433.0 0.017430 223.34 1539.9 0.018120 232.83 1605.4 0.018205 233.47 1609.8 0.018365 234.57 1617.3 0.018455 235.75 1625.5 0.018540 236.85 1633.1 0.018630 238.13 1641.9 0.018730 239.59 1652.0 0.018825 240.77 1660.1 0.018920 241.69 1666.4 0.019015 242.69 1673.4 0.019110 243.97 1682.2 0.019185 244.97 1689.1 0.019280 246.16 1697.3 0.019295 246.89 1702.3 255 Table 20: Stress-Strain Data 3mm Specimen 3
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.005815 45.27 312.1 0.005850 46.09 317.8 0.010280 101.31 698.5 0.010305 101.68 701.1 0.010340 102.41 706.1 0.022525 257.29 1774.0 0.022610 258.39 1781.6 0.023720 273.63 1886.7 0.023800 275.00 1896.1 0.023880 275.91 1902.4 0.023955 277.01 1910.0 0.024040 277.83 1915.6 0.024115 279.29 1925.7 0.024190 279.93 1930.1 0.024275 281.12 1938.3 0.024345 281.76 1942.7 0.024420 282.94 1950.9 0.024490 284.31 1960.3 0.024565 285.13 1966.0 0.028210 328.76 2266.8 0.028260 329.49 2271.8 0.028300 329.95 2275.0 0.028265 328.94 2268.1 0.028230 328.76 2266.8 0.029590 221.15 1524.8 0.028690 112.08 772.8 256 Table 21: Stress-Strain Data 3mm specimen-4
Strain Stress Stress inch/inch ksi Mpa 0.000 0.00 0.0 0.005 45.27 312.1 0.006 46.64 321.6 0.006 47.64 328.5 0.006 48.65 335.4 0.013 144.30 994.9 0.025 273.91 1888.6 0.025 275.18 1897.4 0.025 275.55 1899.9 0.025 282.94 1950.9 0.026 284.22 1959.7 0.026 284.77 1963.5 0.026 285.68 1969.8 0.026 286.68 1976.7 0.026 287.60 1983.0 0.026 287.87 1984.9 0.026 288.33 1988.0 0.026 290.24 2001.2 0.026 291.16 2007.5 0.026 292.07 2013.8 0.026 292.43 2016.3 0.026 293.89 2026.4 0.026 294.44 2030.2 0.030 339.62 2341.7 0.031 340.53 2348.0 0.031 340.81 2349.9 0.030 320.55 2210.2
257 Table 22: Stress-Strain Curve 3mm Specimen 6
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.004390 45.73 315.3 0.004420 45.18 311.5 0.009070 110.26 760.2 0.009090 110.07 759.0 0.009130 111.17 766.5 0.009165 111.08 765.9 0.022605 288.60 1989.9 0.022635 289.06 1993.0 0.030415 350.66 2417.8 0.030465 350.76 2418.5 0.030505 351.21 2421.6 0.030540 352.03 2427.3 0.030580 351.21 2421.6 0.030625 352.03 2427.3 0.030660 352.13 2427.9 0.030695 352.49 2430.4 0.030735 352.58 2431.1 0.030775 353.04 2434.2 0.030805 353.31 2436.1 0.031330 353.40 2436.7 0.032455 353.68 2438.6 0.033090 354.22 2442.4 0.033320 353.95 2440.5 0.033420 354.41 2443.6 0.033485 354.22 2442.4 0.033960 358.06 2468.8 0.039810 300.83 2074.2 0.040020 300.83 2074.2 0.040475 295.90 2040.2
258 Table 23: Stress-Strain Data 3mm Specimen 7
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.005045 45.54 314.0 0.005050 45.09 310.9 0.009810 101.31 698.5 0.009860 102.04 703.6 0.009910 103.05 710.5 0.009960 103.32 712.4 0.013670 148.13 1021.4 0.013720 149.59 1031.5 0.013785 149.96 1034.0 0.013830 150.51 1037.7 0.013890 151.42 1044.0 0.022145 250.81 1729.4 0.022205 251.54 1734.4 0.027875 316.80 2184.4 0.028315 321.55 2217.1 0.029015 329.22 2269.9 0.029055 330.04 2275.6 0.030350 343.36 2367.5 0.030400 343.91 2371.3 0.030450 344.00 2371.9 0.031080 350.57 2417.2 0.031120 350.85 2419.1 0.031075 351.12 2421.0 0.031140 350.85 2419.1 0.026055 326.30 2249.8 259 Table 24: Stress-Strain Data 5mm Specimen 3
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.001990 16.23 111.9 0.002000 16.63 114.6 0.017250 199.25 1373.8 0.017275 199.38 1374.7 0.017290 199.61 1376.3 0.017320 199.91 1378.4 0.009290 216.11 1490.0 0.009285 216.43 1492.3 0.009275 216.76 1494.6 0.009280 216.66 1493.9 0.009270 216.96 1495.9 0.009265 217.26 1498.0 0.009265 217.45 1499.3 0.009260 217.52 1499.8 0.009275 217.88 1502.3 0.009280 218.01 1503.2 0.009255 217.98 1503.0 0.009265 218.18 1504.3 0.009260 218.27 1505.0
260 Table 25: Stress-Strain Data 5mm Specimen 4
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0
0.007590 92.57 638.3 0.007595 92.71 639.2 0.007610 92.82 640.0
0.007905 96.81 667.5 0.007920 97.00 668.8
0.007930 97.18 670.1 0.007935 97.27 670.6 0.007955 97.48 672.1
0.007965 97.63 673.2 0.007975 97.72 673.8
0.007980 97.85 674.7 0.010340 127.39 878.4 0.010355 127.56 879.5
0.010365 127.81 881.2 0.010375 127.86 881.6
0.010395 128.06 883.0 0.010405 128.31 884.7 0.016230 198.92 1371.6 0.021170 258.50 1782.4
0.021175 258.67 1783.5
0.021185 258.74 1784.0 0.017960 302.71 2087.2 0.017970 302.89 2088.5 0.017990 303.08 2089.7
0.018000 303.40 2091.9 0.018005 303.57 2093.1 261
Table 26: Stress-Strain Curve 5mm Specimen 5
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.001270 16.46 113.5 0.008765 106.46 734.0 0.008780 106.62 735.2 0.008800 106.62 735.2 0.008815 106.98 737.7 0.011580 138.86 957.4 0.011600 139.19 959.7 0.011615 139.19 959.7 0.011625 139.38 961.0 0.011640 139.58 962.4 0.011655 139.61 962.6 0.011685 139.97 965.1 0.011695 140.30 967.4 0.026050 298.48 2058.0 0.026055 298.64 2059.1 0.026065 298.71 2059.6 0.026080 298.84 2060.5 0.026095 299.01 2061.6 0.026105 299.07 2062.1 0.026285 301.01 2075.5 0.026300 301.11 2076.1 0.026250 300.78 2073.9
262 Table 27: Stress-Strain Data 5mm Specimen 6
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.002175 16.33 112.6 0.002185 16.53 114.0 0.006950 72.62 500.7 0.006960 72.94 502.9 0.006975 73.14 504.3 0.006990 73.31 505.4 0.011670 127.26 877.4 0.015385 170.24 1173.8 0.016650 184.50 1272.1 0.018155 201.22 1387.4 0.018185 201.52 1389.5 0.018230 201.91 1392.2 0.018255 202.40 1395.6 0.018285 202.77 1398.1 0.018315 203.13 1400.6 0.018355 203.52 1403.3 0.018390 203.85 1405.5 0.018425 204.34 1408.9 0.018445 204.44 1409.6 0.018485 204.97 1413.2 0.018515 205.39 1416.2 0.022635 251.66 1735.2 0.022675 252.08 1738.1 0.026970 307.25 2118.5 0.026985 307.22 2118.3 0.027005 307.55 2120.5 0.027010 307.71 2121.7 0.027020 307.84 2122.6 263 Table 28: Stress-Strain Data 7mm Specimen 1
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.000290 8.38 57.8 0.000300 8.37 57.7 0.004985 61.26 422.4 0.005030 61.81 426.2 0.005080 62.23 429.1 0.005125 62.73 432.5 0.005430 66.34 457.4 0.005490 66.91 461.3 0.005825 71.06 490.0 0.005885 71.80 495.1 0.005935 72.40 499.2 0.010855 139.71 963.3 0.010920 140.45 968.4 0.010985 141.15 973.3 0.011050 141.91 978.5 0.011115 142.70 983.9 0.011190 143.50 989.4 0.011265 144.32 995.1 0.011335 145.14 1000.8 0.020880 258.86 1784.8 0.020920 259.29 1787.8 0.025845 315.97 2178.6 0.029835 357.58 2465.5 0.029885 358.10 2469.1
264 Table 29: Stress-Strain Curve 7mm Specimen 2
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.000875 8.33 57.4 0.000890 8.52 58.7 0.007525 78.46 541.0 0.007570 79.14 545.7 0.009150 101.98 703.1 0.009190 102.66 707.9 0.016265 191.03 1317.1 0.017610 207.26 1429.0 0.017655 207.74 1432.4 0.017695 208.19 1435.5 0.017735 208.66 1438.7 0.017775 209.00 1441.0 0.017810 209.52 1444.6 0.017855 209.89 1447.2 0.017890 210.46 1451.1 0.018980 223.11 1538.4 0.019575 230.07 1586.3 0.019620 230.69 1590.6 0.020035 235.62 1624.6 0.020085 236.16 1628.3 0.036725 322.36 2222.7 0.036685 323.45 2230.2 0.036750 312.85 2157.1
265 Table 30: Stress-Strain Curve 7mm Specimen 3
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.000565 8.32 57.3 0.005545 63.10 435.1 0.005550 63.20 435.8 0.005560 63.33 436.7 0.005575 63.54 438.1 0.008875 109.02 751.7 0.008895 109.25 753.3 0.009310 114.11 786.8 0.009315 114.23 787.6 0.009320 114.33 788.3 0.009335 114.40 788.8 0.009345 114.58 790.0 0.009360 114.65 790.5 0.009370 114.82 791.7 0.009375 114.93 792.5 0.009395 115.17 794.1 0.009400 115.12 793.7 0.009475 116.07 800.3 0.009495 116.23 801.4 0.009495 116.41 802.6 0.012260 149.23 1029.0 0.021645 261.03 1799.8 0.024265 293.88 2026.3 0.024270 293.93 2026.6 0.024275 293.94 2026.7 0.024275 293.93 2026.6
266 Table 31: Stress-Strain Curve 7mm Specimen 5
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.000675 8.47 58.4 0.000695 8.57 59.1 0.005475 68.67 473.5 0.005495 68.82 474.5 0.005510 69.14 476.7 0.011700 145.50 1003.2 0.011715 145.66 1004.4 0.015670 194.05 1337.9 0.015685 194.30 1339.7 0.015695 194.41 1340.5 0.015915 196.98 1358.2 0.015930 197.10 1359.0 0.015955 197.21 1359.8 0.015955 197.36 1360.8 0.015960 197.45 1361.4 0.015980 197.75 1363.5 0.015990 197.85 1364.2 0.016015 197.98 1365.1 0.016020 198.12 1366.0 0.016040 198.34 1367.5 0.016165 199.91 1378.4 0.016175 199.98 1378.9 0.016675 206.37 1422.9 0.017655 218.54 1506.8 0.017670 218.65 1507.6 0.022055 274.65 1893.7 0.022065 274.78 1894.6 0.022070 274.93 1895.7 0.023505 293.20 2021.6 0.023510 293.36 2022.7 267 Table 32: Stress-Strain Curve 7mm Specimen 6
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.000125 8.38 57.8 0.000120 8.47 58.4 0.002220 77.89 537.0 0.002245 78.14 538.8 0.005420 112.24 773.9 0.005430 112.45 775.4 0.005455 112.72 777.2 0.005470 112.89 778.4 0.005500 113.28 781.0 0.005510 113.44 782.2 0.013270 214.48 1478.8 0.013295 214.68 1480.2 0.013310 214.88 1481.6 0.013310 214.98 1482.3 0.013340 215.20 1483.8 0.013350 215.47 1485.7 0.020035 293.67 2024.9 0.020040 293.81 2025.8 0.020055 293.94 2026.7 0.020070 294.03 2027.3 0.020075 294.16 2028.2
268 Table 33: Stress-Strain Data 5mm Specimen 7
Strain Stress Stress inch/inch ksi Mpa 0.000000 0.00 0.0 0.001725 16.23 111.9 0.001735 16.07 110.8 0.010985 136.89 943.8 0.011015 137.31 946.8 0.011050 137.54 948.4 0.018370 242.33 1670.8 0.018400 242.88 1674.7 0.018430 243.21 1677.0 0.018465 243.28 1677.4 0.018495 243.80 1681.0 0.025095 329.86 2274.4 0.025105 330.09 2276.0 0.025135 330.35 2277.8 0.025165 330.71 2280.3 0.025175 330.84 2281.2 0.025205 330.94 2281.9 0.025225 331.24 2283.9 0.025250 331.53 2285.9 0.025260 331.83 2288.0 0.025285 331.83 2288.0 0.027625 331.83 2288.0 0.027695 332.09 2289.8 0.027725 332.16 2290.2 0.027760 332.52 2292.7
269 Table 34: Load-Slip Data 5mm-C1
Load(lb) Slip(in) Bond-Stress(psi) Bond-Stress(mpa) 0 0 0 0 9000 0 971 6.70 9200 0.0005 993 6.84 5100 0.001 550 3.8
270 Table 35: Load-Slip Data 5mm-C3
Load(Lb) Slip(in) Bond-Stress (psi) Bond-Stress (mpa) 0 0.0000 0.00 0.00 6000 0.0005 694.49 4.79 6300 0.0005 679.78 4.69 6600 0.0010 712.15 4.91 6900 0.0015 744.52 5.13 6926 0.0020 747.33 5.15 6960 0.0025 751.00 5.18 7000 0.0035 755.31 5.21 7150 0.0050 771.50 5.32 7200 0.0055 776.89 5.36 7300 0.0065 787.68 5.43 7400 0.0100 798.47 5.51 7450 0.0105 803.87 5.54 7480 0.0200 807.10 5.56 7650 0.0155 825.45 5.69 7670 0.0185 827.61 5.71 7710 0.0205 831.92 5.74 7782 0.0230 839.69 5.79 7850 0.0265 847.03 5.84 7880 0.0300 850.27 5.86 7920 0.0340 854.58 5.89 7980 0.0380 861.06 5.94 8010 0.0400 864.29 5.96 8040 0.0470 867.53 5.98 8070 0.0525 870.77 6.00 8090 0.0580 872.92 6.02 8090 0.0650 872.92 6.02
271 8090 0.0730 872.92 6.02 8090 0.0800 872.92 6.02 8080 0.0900 871.85 6.01 8060 0.1000 869.69 6.00 8050 0.1100 868.61 5.99 8040 0.1200 867.53 5.98 8030 0.1300 866.45 5.97 8010 0.1475 864.29 5.96 8020 0.1575 865.37 5.97 8050 0.1745 868.61 5.99 8070 0.1800 870.77 6.00 8090 0.1875 872.92 6.02 8130 0.1950 877.24 6.05 8140 0.2100 878.32 6.06 8140 0.2215 878.32 6.06 8100 0.2345 874.00 6.03 8065 0.2495 870.23 6.00 8060 0.2575 869.69 6.00 8040 0.2675 867.53 5.98 8020 0.2775 865.37 5.97 7990 0.2875 862.13 5.94 7950 0.2975 857.82 5.91 7840 0.3175 845.95 5.83 7690 0.3415 829.76 5.72 7510 0.3715 810.34 5.59 7400 0.4000 798.47 5.51 7230 0.4175 780.13 5.38 7160 0.4375 772.58 5.33 7120 0.4575 768.26 5.30 7070 0.4775 762.86 5.26
272 7010 0.5000 756.39 5.22 7010 0.5175 756.39 5.22 7020 0.5375 757.47 5.22 7030 0.5575 758.55 5.23 7050 0.5775 760.71 5.25 7050 0.6000 760.71 5.25 7070 0.6225 762.86 5.26 7110 0.6200 767.18 5.29
273 Table 36: Load-Slip Data 5mm-C4
Load (lb) Slip (in) Bond-Stress (psi) Bond-Stress (mpa) 0 0.0000 0.00 0.00 8300 0.0005 895.58 6.18 8780 0.0010 947.38 6.53 9000 0.0015 971.11 6.70 9070 0.0025 978.67 6.75 9170 0.0040 989.46 6.82 9260 0.0060 999.17 6.89 9370 0.0080 1011.04 6.97 8240 0.0100 889.11 7.00 9429 0.0135 1017.40 7.02
274 Table 37: Load-Slip Data 7mm-C9
Bond Bond- Load (lb) Slip (in) Stress Stress (psi) (mpa) 0 0.0000 0.00 0.00 10550 0.0005 813.12 5.61 11150 0.0010 859.36 5.93 11500 0.0020 886.34 6.11 11700 0.0040 901.75 6.22
11900 0.0060 917.16 6.32 12060 0.0080 929.50 6.41 12450 0.0120 959.55 6.62 12600 0.0190 971.11 6.70 12680 0.0230 977.28 6.74 12720 0.0270 980.36 6.76 12760 0.0390 983.45 6.78
12800 0.0500 986.53 6.80 12860 0.0600 991.15 6.83 12900 0.0800 994.24 6.86 12900 0.1000 994.24 6.86 12940 0.1200 997.32 6.88 12770 0.1400 984.22 6.79 12600 0.1600 971.11 6.70 12560 0.1800 968.03 6.67 12430 0.2000 958.01 6.61
12240 0.2500 943.37 6.50 12170 0.2800 937.97 6.47 12140 0.3000 935.66 6.45 12990 0.3500 1001.17 6.41 11940 0.3800 920.25 6.35 11870 0.4000 914.85 6.31 11740 0.4300 904.83 6.24
11500 0.4700 886.34 6.11 11290 0.5000 870.15 6.00 11140 0.5200 858.59 5.92 11000 0.5500 847.80 5.85 10950 0.5700 843.95 5.82 10800 0.6000 832.38 5.74 10650 0.6300 820.82 5.66 10470 0.6600 806.95 5.56 10200 0.7000 786.14 5.42 9970 0.7400 768.41 5.30 275
Table 38: Load-Slip Data 7mm-C9
Load (lb) Slip (in) Bond Stress (psi) Bond-Stress (mpa) 0 0.0000 0.00 0.00 10550 0.0005 813.12 5.61 11150 0.0010 859.36 5.93 11500 0.0020 886.34 6.11 11700 0.0040 901.75 6.22 11900 0.0060 917.16 6.32 12060 0.0080 929.50 6.41 12450 0.0120 959.55 6.62 12600 0.0190 971.11 6.70 12680 0.0230 977.28 6.74 12720 0.0270 980.36 6.76 12760 0.0390 983.45 6.78 12800 0.0500 986.53 6.80 12860 0.0600 991.15 6.83 12900 0.0800 994.24 6.86 12900 0.1000 994.24 6.86 12940 0.1200 997.32 6.88 12770 0.1400 984.22 6.79 12600 0.1600 971.11 6.70 12560 0.1800 968.03 6.67 12430 0.2000 958.01 6.61 12240 0.2500 943.37 6.50 12170 0.2800 937.97 6.47 12140 0.3000 935.66 6.45 12990 0.3500 1001.17 6.41 11940 0.3800 920.25 6.35 276 11870 0.4000 914.85 6.31 11740 0.4300 904.83 6.24 11500 0.4700 886.34 6.11 11290 0.5000 870.15 6.00 11140 0.5200 858.59 5.92 11000 0.5500 847.80 5.85 10950 0.5700 843.95 5.82 10800 0.6000 832.38 5.74 10650 0.6300 820.82 5.66 10470 0.6600 806.95 5.56 10200 0.7000 786.14 5.42 9970 0.7400 768.41 5.30 9820 0.7700 756.85 5.22 9720 0.8000 749.15 5.17 9620 0.8455 741.44 5.11
277 Table 39: Load-Slip Data 7mm-C10
Load (lb) Slip (in) Bond-Stress (psi) Bond-Stress (mpa) 0 0.0000 0.00 0.00 11000 0.0005 1273.24 8.78 11690 0.0010 1353.11 9.33 12150 0.0020 1406.35 9.70 12400 0.0040 1435.29 9.90 12650 0.0070 1464.23 10.10 12850 0.0100 1487.38 10.26 12900 0.0160 1493.16 10.30 13040 0.0250 1509.37 10.41 12700 0.0500 1470.01 10.14 12650 0.0600 1464.23 10.10 12400 0.0955 1435.29 9.90 12250 0.1200 1417.93 9.78 12120 0.1500 1402.88 9.67 12040 0.1800 1393.62 9.61 11860 0.2500 1372.78 9.47 11960 0.3000 1384.36 9.55 11960 0.3500 1384.36 9.55 11700 0.4000 1354.26 9.34 11470 0.4500 1327.64 9.15 11271 0.4800 1304.61 9.00 11132 0.5000 1288.52 8.88 10950 0.5300 1267.45 8.74 10860 0.5700 1257.03 8.67 10800 0.6000 1250.09 8.62 10600 0.6300 1226.94 8.46 10350 0.6700 1198.00 8.26 278 10130 0.7000 1172.54 8.08 9910 0.7300 1147.07 7.91 9650 0.7700 1116.98 7.70 9500 0.8000 1099.62 7.58 9420 0.8300 1090.36 7.52 9320 0.8500 1078.78 7.44 9270 0.8800 1072.99 7.40 9230 0.9000 1068.36 7.37 9180 0.9300 1062.58 7.33 9070 0.9700 1049.84 7.24 9010 0.9815 1042.90 7.19
279 Table 40: Load-Slip Data 7mm-C11
Bond-Stress Bond-Stress Load (lb) Slip (in) (psi) (mpa) 0 0.0000 0.00 0.00 11200 0.0005 863.21 5.95 11900 0.0010 917.16 6.32 12390 0.0020 954.93 6.58 12600 0.0040 971.11 6.70 12800 0.0070 986.53 6.80 12900 0.0100 994.24 6.86 12920 0.0300 995.78 6.87 12690 0.0600 978.05 6.74 12450 0.0900 959.55 6.62 12340 0.1200 951.08 6.56 12350 0.1500 951.85 6.56 12400 0.1800 955.70 6.59 12500 0.2200 963.41 6.64 12550 0.2400 967.26 6.67 12540 0.2805 966.49 6.66 12520 0.3355 964.95 6.65 12500 0.3625 963.41 6.64 12500 0.3805 963.41 6.64 12480 0.4005 961.87 6.63 12470 0.4265 961.10 6.63 12430 0.4469 958.01 6.61 12330 0.4865 950.31 6.55 12290 0.5000 947.22 6.53 12240 0.5215 943.37 6.50 12220 0.5325 941.83 6.49 12190 0.5625 939.52 6.48 12130 0.5875 934.89 6.45 12100 0.6000 932.58 6.43 12050 0.6225 928.72 6.40
280 Table 41: Stress-Strain Data 3mm-C6
Stress Strain 20% of 50% of Stress Max (psi) (in/in) max.stress max.stress 0.0 0 271.0 54.207 135.517
corresponding 45.3 0.00539 0.00622 0.01209 strain corresponding 46.4 0.00549 54.1 135.5 stress 69.2 0.00742
70.5 0.00751
70.9 0.00760
72.9 0.00769 Modulus (ksi) 13879
82.8 0.00849
84.7 0.00859
262.5 0.02234 275.5 0.022
263.7 0.02242
264.2 0.02252
265.1 0.02261 Stress (ksi) Strain (in/in)
266.7 0.02270 0.0 0.002
267.8 0.02278 54.1 0.006
268.3 0.02289 135.5 0.0121
270.0 0.02408 275.5 0.022
271.0 0.02282
281 Table 42: Stress-Strain Data 3mm-C7
Stress Strain 20% of 50% of Stress Max (lb) (in/in) max.stress max.stress 0.00 0.0000 291.2 58.249 145.622 corresponding 45.36 0.0030 0.00428 0.01148 strain corresponding 46.54 0.0031 59.71 149.78 stress
47.18 0.0032
48.28 0.0033
49.56 0.0034
50.66 0.0035 Modulus (ksi) 12510 117.96 0.0093
119.51 0.0094 Discrepancy 6.168 120.43 0.0095
178.49 0.0135 Stress (ksi) Strain (in/in)
208.76 0.0159 6.168 0.0000
285.30 0.0222 59.71 0.0043
286.40 0.0223 149.78 0.0115
287.77 0.0224 282.84 0.0226
288.96 0.0225
290.69 0.0225
291.24 0.0226
282 Table 43: Stress-Strain Curve 3mm-C8
Stress Strain 20% of 50% of (psi) (in/in) Stress Max max.stress max.stress 0 0 287.8597 57.5719 143.930 corresponding 46.18 0.00016 strain 0.00078 0.00879 corresponding 47.09 0.00017 stress 60.809 152.525 48.01 0.00017 49.01 0.00017 125.92 0.00692 128.02 0.00704 Modulus(ksi) 10781.26 225.95 0.01400 227.23 0.01409 228.70 0.01419 230.07 0.01429 Discrepancy 52.40 230.80 0.01438 232.54 0.01448 Stress (ksi) strain (in/in) 248.63 0.01556 51.897 0 249.27 0.01566 60.809 0.00078 250.83 0.01576 152.525 0.00879 252.29 0.01585 234.98 0.016935 252.84 0.01595 254.76 0.01606 256.86 0.01616 257.59 0.01626 258.32 0.01635 279.45 0.01781 280.64 0.01791 282.46 0.01801 282.83 0.01810 285.76 0.01819
283 Table 44: Stress-Strain Data 3mm-C5
Stress Strain 20% of 50% of Stress Max (psi) (in/in) max.stress max.stress 0.0 0.0000 275.6 55.1213 137.803 corresponding 45.8 0.0020 0.00290 0.00986 strain corresponding 46.0 0.0021 56.0540 140.7000 stress 46.8 0.0021
74.7 0.0045
76.4 0.0046
77.5 0.0047 Modulus (ksi) 12162
78.1 0.0048
161.9 0.0114
163.3 0.0115 Discrepancy 20.846
164.0 0.0116
164.6 0.0116
166.3 0.0117 Stress (ksi) Strain (in/in)
167.5 0.0118 20.846 0.000
168.3 0.0118 56.054 0.003
169.2 0.0119 140.700 0.010
185.4 0.0129 225.711 0.017
200.1 0.0139 201.3 0.0140 268.7 0.0187 268.7 0.0188 270.1 0.0189 270.9 0.0189 272.4 0.0190 273.2 0.0190 274.1 0.0191
284 Table 45: Stress-Strain Data 5mm-C3
Stress Strain 20% of 50% of Stress Max (psi) (in/in) max.stress max.stress 0.00 0.0000 226.105 45.22 113.05 corresponding 16.54 0.0026 0.0055 0.0106 strain corresponding 140.33 0.0126 45.44 113.32 stress 140.66 0.0127
154.81 0.0138
155.01 0.0138
155.77 0.0139 Modulus(ksi) 13142
159.15 0.0141
159.59 0.0142
160.15 0.0142 Discrepancy -26.38
160.65 0.0143
161.01 0.0143
161.64 0.0143
161.97 0.0144 Stress (ksi) Strain (in/in)
162.44 0.0144 0.00 0.0020
162.93 0.0145 45.44 0.0055
163.53 0.0145 113.32 0.0106
163.93 0.0145 236.46 0.0200
164.52 0.0146 223.49 0.0198 224.12 0.0198 224.38 0.0199 224.78 0.0450 225.04 0.0894 225.51 0.1617 226.10 0.1841
285 Table 46: Stress-Strain Data 5mm-C1
Stress Strain 20% of 50% of Stress Max (psi) (in/in) max.stress max.stress 0.00 0.0000 279.167 55.83 139.58 corresponding 16.54 0.0003 0.0032 0.0111 strain corresponding 17.47 0.0003 55.88 139.57 stress 17.80 0.0003
78.22 0.0055
79.08 0.0056 Modulus (ksi) 10613
79.44 0.0056
80.04 0.0057
80.77 0.0058 Discrepancy 21.5
174.67 0.0137
175.26 0.0138
175.99 0.0138 Stress (ksi) Strain (in/in)
176.75 0.0139 21.49 0.0000
177.48 0.0139 55.88 0.0032
178.18 0.0140 139.57 0.0111
178.54 0.0140 261.52 0.0226
178.97 0.0140 179.60 0.0141 189.45 0.0149 190.04 0.0149 190.74 0.0150 191.40 0.0150 192.00 0.0151 278.64 0.0226 279.17 0.0225
286 Table 47: Stress-Strain Data 5mm-C2
Stress Strain 20% of 50% of Stress Max (psi) (in/in) max.stress max.stress 0.00 0.0000 312.51 62.50 156.25 corresponding 16.51 0.0050 0.0004 0.008 strain corresponding 16.87 0.0049 62.48 156.30 stress 71.82 0.0006
72.39 0.0007
73.05 0.0007
73.94 0.0008
74.61 0.0009 Modulus (ksi) 12870
104.10 0.0036
104.57 0.0037
105.23 0.0037 Discrepancy 57.65
133.50 0.0059
134.07 0.0059
134.86 0.0060 Stress (ksi) Strain (in/in)
135.46 0.0060 57.65 0.0000
135.99 0.0061 62.48 0.0004
136.75 0.0061 156.30 0.0077
137.21 0.0062 359.72 0.0235
161.64 0.0081 162.07 0.0081 162.97 0.0082 163.76 0.0083 164.29 0.0083 164.92 0.0084 310.22 0.0214 310.79 0.0214 311.08 0.0214 311.61 0.0215 311.95 0.0215 312.28 0.0216
287 Table 48: Stress-Strain Data 5mm-C3
Stress Strain 20% of 50% of (psi) (in/in) Stress Max max.stress max.stress
0.00 0.0000 265.015 53.00 132.51 corresponding 16.54 0.0026 strain 0.01 0.01 corresponding 16.67 0.0027 stress 53.79 134.83
97.97 0.0096
98.44 0.0097
98.77 0.0097 Modulus(ksi) 13461
99.30 0.0097
99.89 0.0097
100.29 0.0098
100.92 0.0098 Discrepancy -29.33
101.58 0.0098
102.25 0.0099
162.44 0.0144
162.93 0.0145 Stress (ksi) strain (in/in)
246.22 0.1462 0.00 0.002
246.55 0.1462 53.79 0.006
250.23 0.1462 134.83 0.012
250.70 0.1462 239.89 0.020
250.93 0.1462
251.33 0.1462
251.82 0.1462
254.34 0.1462
254.81 0.1462
254.87 0.1462
265.02 0.1463
264.85 0.1463
288 Table 49: Stress-Strain Data 7mm-C9
Stress Strain 20% of 50% of Stress Max (psi) (in/in) max.stress max.stress 0.00 0.0000 208.42 41.68 104.21
corresponding 8.40 0.0026 0.01 0.014 strain corresponding 8.86 0.0028 43.89 108.914 stress 153.23 0.0181
153.90 0.0181
185.33 0.0208 Modulus(ksi) 11601
185.86 0.0208
186.44 0.0209
187.17 0.0209 Discrepancy -56.047
187.71 0.0210
188.26 0.0210
188.89 0.0211 Stress (ksi) Strain (in/in)
189.44 0.0211 0.00 0.005
190.18 0.0212 43.89 0.009
190.70 0.0212 108.91 0.014
191.28 0.0213 175.97 0.020
191.88 0.0213
192.56 0.0214
193.11 0.0214
193.57 0.0214
194.27 0.0215
194.66 0.0216
207.99 0.0432
208.42 0.0363
164.02 0.0214
289 Table 50: Stress-Strain Data 7mm-C10
Stress Strain 20% of 50% of (psi) (in/in) Stress Max max.stress max.stress
0.000 0.0000 201.80 40.36 100.90 corresponding 8.384 0.0005 strain 0.002 0.0079 corresponding 8.755 0.0005 stress 43.74 110.26 189.963 0.0140 190.637 0.0140
191.445 0.0141 Modulus (ksi) 11670 192.169 0.0141 192.960 0.0142
193.650 0.0142 Discrepancy 17.60 199.189 0.0148 199.391 0.0148
200.351 0.0148 stress (ksi) Strain (in/in)
200.317 0.0149 17.60 0.0000
200.519 0.0149 43.74 0.0022
200.587 0.0149 110.26 0.0079
200.705 0.0149 192.77 0.0150 200.873 0.0149 201.176 0.0149 201.210 0.0149 201.412 0.0149 201.462 0.0150 201.664 0.0150 201.748 0.0150 201.732 0.0150 201.799 0.0150 201.715 0.0150
290 Table 51: Stress-Strain Data 7mm-C11
Stress Strain 20% of 50% of Stress Max (psi) (in/in) max.stress max.stress 0.00 0.0000 218.92 43.784 109.46 corresponding 8.42 0.0001 0.003 0.009 strain corresponding 144.17 0.0118 40.78 109.42 stress 145.01 0.0119
199.00 0.0157
199.71 0.0158 modulus(ksi) 11216
200.59 0.0158
201.36 0.0159
202.14 0.0159 Discrepancy 7.13
202.78 0.0160
203.70 0.0161
204.39 0.0161 Stress (ksi) Strain (in/in)
213.63 0.0169 7.13 0.0000
214.17 0.0169 40.78 0.0030
214.86 0.0170 109.42 0.0091
215.45 0.0170 231.45 0.0200
215.96 0.0171 216.63 0.0171 217.00 0.0171 217.52 0.0172 217.98 0.0172 218.26 0.0172 218.60 0.0173 218.79 0.0173 218.87 0.0173 218.92 0.0173
291 Table 52: Stress-Strain Data 7mm-C12
Stress Strain 20% of 50% of (psi) (in/in) STRESS MAX max.stress max.stress 0.00 0.0000 275.74 55.15 137.87 corresponding 8.40 0.0016 strain 0.0056 0.0123 corresponding 178.38 0.0155 stress 54.11 137.64 179.05 0.0156 257.17 0.0224 263.66 0.0229 Modulus (ksi) 12541 264.30 0.0230 264.75 0.0230 265.37 0.0231 Discrepancy -16.50 265.83 0.0231 266.43 0.0232 267.06 0.0232 Stress (Ksi) Strain (in/in) 267.59 0.0233 0.00 0.0013 268.10 0.0233 54.11 0.0056 268.55 0.0234 137.64 0.0123 269.13 0.0234 297.04 0.0250 269.65 0.0235 270.22 0.0235 270.81 0.0236 271.37 0.0236 271.87 0.0236 272.36 0.0237 272.92 0.0237 273.42 0.0238 273.91 0.0238 274.41 0.0238 274.90 0.0239 275.52 0.0239
292 Table 53: Load-Deflection Data B-2
Exp- Load Branson's With Reduction R.L.Sunna Proposed Deflection (lb) Equation (in) Coefficients (in) Relation (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 0.0000 300 0.0010 0.0015 0.0024 0.0024 0.0024 600 0.0035 0.0030 0.0039 0.0039 0.0039 900 0.0050 0.0045 0.0054 0.0054 0.0054 1200 0.0065 0.0060 0.0113 0.0069 0.0069 1500 0.0085 0.0075 0.0243 0.0084 0.0084 1800 0.0095 0.0090 0.0450 0.0099 0.0099 2100 0.0120 0.0105 0.0744 0.0114 0.0114 2400 0.0265 0.0138 0.1128 0.1386 0.0693 2500 0.0865 0.0160 0.1276 0.1565 0.0782 2700 0.0995 0.0214 0.1597 0.1953 0.0977 2900 0.1185 0.0279 0.1951 0.2379 0.1189 3100 0.1415 0.0357 0.2331 0.2835 0.1418 3300 0.1695 0.0448 0.2734 0.3317 0.1658 3442 0.1895 0.0523 0.3031 0.3670 0.1835 3500 0.2475 0.0556 0.3155 0.3817 0.1909 3700 0.2585 0.0678 0.3588 0.4332 0.2166 3900 0.2765 0.0818 0.4031 0.4856 0.2428 4100 0.2995 0.0975 0.4479 0.5385 0.2693 4300 0.3085 0.1149 0.4931 0.5917 0.2958
293 Table 54: Load-Deflection Data B-3
Exp- Load Branson's With Reduction R.L.Sunna Proposed Deflection (lb) Equation (in) Coefficients (in) Relation (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 0.0000 300 0.0030 0.0015 0.0024 0.0024 0.0024 600 0.0050 0.0030 0.0039 0.0039 0.0039 900 0.0080 0.0045 0.0054 0.0054 0.0054 1200 0.0100 0.0060 0.0069 0.0069 0.0069 1500 0.0130 0.0075 0.0122 0.0084 0.0084 1800 0.0160 0.0090 0.0227 0.0099 0.0099 2100 0.0195 0.0105 0.0377 0.0114 0.0114 2400 0.0345 0.0137 0.0574 0.1288 0.0644 2700 0.0470 0.0210 0.0816 0.1692 0.0846 2750 0.0600 0.0225 0.0860 0.1761 0.0881 3000 0.0735 0.0306 0.1096 0.2112 0.1056 3200 0.0880 0.0383 0.1301 0.2395 0.1198 3400 0.1025 0.0472 0.1517 0.2678 0.1339 3400 0.1075 0.0472 0.1517 0.2678 0.1339 3340 0.1190 0.0444 0.1451 0.2593 0.1297 3600 0.1285 0.0572 0.1741 0.2958 0.1479 3800 0.1415 0.0684 0.1972 0.3236 0.1618 4000 0.1555 0.0807 0.2208 0.3509 0.1755 4200 0.1660 0.0941 0.2446 0.3779 0.1889 4330 0.1920 0.1035 0.2602 0.3952 0.1976 4400 0.1775 0.1087 0.2686 0.4045 0.2022 4400 0.1895 0.1087 0.2686 0.4045 0.2022 4400 0.1965 0.1087 0.2686 0.4045 0.2022 4600 0.2070 0.1242 0.2926 0.4306 0.2153 4700 0.2135 0.1323 0.3046 0.4435 0.2218 4800 0.2210 0.1406 0.3166 0.4564 0.2282 4900 0.2270 0.1492 0.3285 0.4691 0.2346 5000 0.2330 0.1579 0.3404 0.4818 0.2409 5200 0.2475 0.1760 0.3641 0.5068 0.2534 5300 0.2555 0.1854 0.3758 0.5193 0.2596 5400 0.2625 0.1948 0.3875 0.5316 0.2658 5500 0.2720 0.2045 0.3992 0.5439 0.2719 5600 0.2840 0.2142 0.4108 0.5561 0.2780 294 Table 55: Load-Deflection Curve B-4
Exp- Load Branson's With Reduction R.L.Sunna Proposed Deflection (lb) Equation (in) Coefficients (in) Relation (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 0.0000 200 0.0010 0.0010 0.0019 0.0022 0.0022 300 0.0020 0.0015 0.0024 0.0027 0.0027 500 0.0040 0.0025 0.0034 0.0038 0.0038 700 0.0055 0.0035 0.0044 0.0050 0.0050 900 0.0065 0.0045 0.0078 0.0061 0.0061 1000 0.0080 0.0050 0.0112 0.0066 0.0066 1200 0.0090 0.0060 0.0209 0.0078 0.0078 1400 0.0110 0.0070 0.0355 0.0089 0.0089 1600 0.0125 0.0080 0.0559 0.0100 0.0100 1800 0.0145 0.0090 0.0830 0.0111 0.0111 2000 0.0160 0.0100 0.1172 0.0123 0.0123 1800 0.0220 0.0090 0.0830 0.0111 0.0111 1500 0.0250 0.0075 0.0607 0.0094 0.0094 1440 0.0850 0.0072 0.0530 0.0091 0.0091 1500 0.0415 0.0075 0.0607 0.0094 0.0094 1500 0.0475 0.0075 0.0607 0.0094 0.0094 1600 0.0570 0.0080 0.0559 0.0100 0.0100 1700 0.0655 0.0085 0.0915 0.0106 0.0106 1800 0.0745 0.0090 0.0830 0.0111 0.0111 1900 0.0835 0.0095 0.1305 0.0117 0.0117 2000 0.0195 0.0100 0.1172 0.0123 0.0123 2100 0.1010 0.0105 0.1777 0.1401 0.0700 2200 0.1100 0.0110 0.2041 0.1628 0.0814 2300 0.1205 0.0118 0.2324 0.1876 0.0938 2400 0.1295 0.0138 0.2624 0.2144 0.1072 2500 0.1395 0.0161 0.2940 0.2432 0.1216 2600 0.1490 0.0187 0.3271 0.2739 0.1369 2700 0.1595 0.0216 0.3615 0.3064 0.1532 2800 0.1685 0.0248 0.3970 0.3406 0.1703 2200 0.1760 0.0110 0.2041 0.1628 0.0814 2400 0.1815 0.0138 0.2624 0.2144 0.1072 2500 0.1860 0.0161 0.2940 0.2432 0.1216 2600 0.1895 0.0187 0.3271 0.2739 0.1369 2700 0.1945 0.0216 0.3615 0.3064 0.1532 295 Table 56: Load-Deflection Data B-5
Exp- Load Branson's With Reduction R.L.Sunna Proposed Deflection (lb) Equation (in) Coefficients (in) Relation (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 0.0000 100 0.0010 0.0006 0.0016 0.0016 0.0016 200 0.0020 0.0011 0.0022 0.0022 0.0022 300 0.0030 0.0017 0.0027 0.0027 0.0027 400 0.0045 0.0023 0.0033 0.0033 0.0033 500 0.0060 0.0028 0.0038 0.0038 0.0038 600 0.0065 0.0034 0.0044 0.0044 0.0044 900 0.0105 0.0051 0.0068 0.0061 0.0061 1200 0.0155 0.0067 0.0180 0.0078 0.0078 1300 0.0175 0.0073 0.0236 0.0083 0.0083 1400 0.0230 0.0079 0.0303 0.0089 0.0089 1500 0.0285 0.0084 0.0382 0.0094 0.0094 1600 0.0345 0.0090 0.0473 0.0100 0.0100 1700 0.0400 0.0096 0.0577 0.0106 0.0106 1800 0.0440 0.0101 0.0693 0.0111 0.0111 1900 0.0490 0.0107 0.0823 0.0117 0.0117 2000 0.0530 0.0112 0.0966 0.0123 0.0123 2200 0.0625 0.0159 0.1290 0.1572 0.0786 2300 0.0680 0.0189 0.1470 0.1789 0.0894 2400 0.0730 0.0348 0.1661 0.2019 0.1009 2500 0.0805 0.0258 0.1863 0.2261 0.1130 2600 0.0885 0.0300 0.2074 0.2514 0.1257 2700 0.0965 0.0346 0.2293 0.2776 0.1388 2800 0.1030 0.0396 0.2521 0.3048 0.1524 2650 0.1155 0.0322 0.2183 0.2644 0.1322 2500 0.1145 0.0258 0.1863 0.2261 0.1130 2700 0.1210 0.0346 0.2293 0.2776 0.1388 2800 0.1250 0.0396 0.2521 0.2776 0.1388 2900 0.1290 0.0453 0.2755 0.3327 0.1663 3000 0.1340 0.0849 0.2995 0.3613 0.1806 2500 0.1500 0.0258 0.1863 0.2261 0.1130 2600 0.1565 0.0300 0.2074 0.2514 0.1257 2700 0.1610 0.0346 0.2293 0.2776 0.1388 2800 0.1665 0.0396 0.2521 0.2776 0.1388 2900 0.1725 0.0453 0.2755 0.3327 0.1663 296 3000 0.1790 0.0849 0.2995 0.3613 0.1806 3100 0.1865 0.0581 0.3240 0.3904 0.1952 3200 0.1965 0.0655 0.3490 0.4200 0.2100 3100 0.2065 0.0581 0.3240 0.3904 0.1952 3200 0.2435 0.0655 0.3490 0.4200 0.2100 3300 0.2645 0.0735 0.3743 0.4500 0.2250 3400 0.2865 0.0821 0.3999 0.4803 0.2401 3600 0.3195 0.1015 0.4518 0.5415 0.2707 3700 0.3375 0.1123 0.4779 0.5723 0.2861 3800 0.3505 0.1239 0.5041 0.6031 0.3016 3900 0.3625 0.1363 0.5303 0.6340 0.3170 4000 0.3730 0.1496 0.5565 0.6648 0.3324 3604 0.3855 0.1019 0.4528 0.5427 0.2714 3900 0.3935 0.1363 0.5303 0.6340 0.3170 4000 0.3970 0.1496 0.5565 0.6648 0.3324 4100 0.4030 0.1636 0.5827 0.6955 0.3478 4200 0.4085 0.1786 0.6088 0.7262 0.3631 3830 0.4100 0.1276 0.5119 0.6124 0.3062 3865 0.4195 0.1319 0.5211 0.6232 0.3116 3900 0.4245 0.1363 0.5303 0.6340 0.3170 4000 0.4295 0.1496 0.5565 0.6648 0.3324 4100 0.4375 0.1636 0.5827 0.6955 0.3478 4200 0.4485 0.1786 0.6088 0.7262 0.3631 4300 0.4620 0.1944 0.6349 0.7568 0.3784 4400 0.4770 0.2111 0.6608 0.7872 0.3936 4500 0.4915 0.2288 0.6866 0.8175 0.4087 4600 0.5075 0.2474 0.7123 0.8476 0.4238 4700 0.5255 0.2670 0.7379 0.8775 0.4388
297 Table 57: Load-Deflection Curve B-6
Exp- Load Branson's With Reduction R.L.Sunna Proposed Deflection (lb) Equation (in) Coefficients (in) Relation (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 0.0000 100 0.0005 0.0013 0.0013 0.0013 0.0013 200 0.0010 0.0017 0.0017 0.0017 0.0017 300 0.0015 0.0022 0.0022 0.0022 0.0022 400 0.0025 0.0026 0.0026 0.0026 0.0026 500 0.0030 0.0031 0.0031 0.0031 0.0031 600 0.0035 0.0035 0.0035 0.0035 0.0035 700 0.0045 0.0039 0.0039 0.0039 0.0039 800 0.0050 0.0044 0.0044 0.0044 0.0044 900 0.0055 0.0048 0.0048 0.0048 0.0048 1000 0.0065 0.0053 0.0053 0.0053 0.0053 1200 0.0075 0.0062 0.0062 0.0062 0.0062 1400 0.0090 0.0071 0.0071 0.0071 0.0071 1600 0.0105 0.0080 0.0080 0.0080 0.0080 1800 0.0120 0.0089 0.0089 0.0089 0.0089 2000 0.0135 0.0098 0.0098 0.0098 0.0098 2200 0.0150 0.0106 0.0106 0.0106 0.0106 2400 0.0165 0.0115 0.0115 0.0115 0.0115 2450 0.0740 0.0118 0.0118 0.0118 0.0118 2340 0.0225 0.0113 0.0113 0.0113 0.0113 2384 0.0235 0.0115 0.0115 0.0115 0.0115 2206 0.0275 0.0107 0.0107 0.0107 0.0107 2175 0.0310 0.0105 0.0105 0.0105 0.0105 1945 0.0345 0.0095 0.0095 0.0095 0.0095 1900 0.0410 0.0093 0.0093 0.0093 0.0093 1875 0.0435 0.0092 0.0092 0.0092 0.0092 1900 0.0470 0.0093 0.0093 0.0093 0.0093 2000 0.0535 0.0098 0.0098 0.0098 0.0098 2100 0.0575 0.0102 0.0102 0.0102 0.0102 2200 0.0625 0.0106 0.0106 0.0106 0.0106 2300 0.0675 0.0111 0.0111 0.0111 0.0111 2400 0.0725 0.0115 0.0115 0.0115 0.0115 2500 0.0770 0.0120 0.0120 0.0120 0.0120 2600 0.0815 0.0131 0.0812 0.1637 0.0818 2700 0.0865 0.0150 0.0928 0.2819 0.1409 298 2800 0.0925 0.0172 0.1055 0.3743 0.1872 2900 0.0990 0.0197 0.1193 0.4289 0.2145 3000 0.1065 0.0223 0.1342 0.4608 0.2304 3100 0.1125 0.0253 0.1504 0.4825 0.2413 3200 0.1215 0.0285 0.1677 0.5001 0.2501 3300 0.1325 0.0320 0.1862 0.5161 0.2581 3400 0.1430 0.0358 0.2059 0.5314 0.2657 2900 0.1475 0.0197 0.1193 0.4289 0.2145 3000 0.1535 0.0223 0.1342 0.4608 0.2304 3100 0.1585 0.0253 0.1504 0.4825 0.2413 3200 0.1615 0.0285 0.1677 0.5001 0.2501 3300 0.1685 0.0320 0.1862 0.5161 0.2581 3400 0.1805 0.0358 0.2059 0.5314 0.2657 3500 0.1895 0.0399 0.2269 0.5465 0.2732 3600 0.1985 0.0443 0.2491 0.5614 0.2807 3700 0.2105 0.0491 0.2725 0.5763 0.2881 3800 0.2205 0.0543 0.2971 0.5911 0.2956 3900 0.2385 0.0598 0.3230 0.6060 0.3030 4000 0.2370 0.0657 0.3501 0.6208 0.3104 4100 0.2460 0.0721 0.3783 0.6357 0.3178 4200 0.2560 0.0788 0.4078 0.6505 0.3253 4300 0.2665 0.0860 0.4383 0.6654 0.3327 4400 0.2785 0.0937 0.4700 0.6802 0.3401 4500 0.2990 0.0978 0.4830 0.6950 0.3475 4600 0.3070 0.1061 0.5159 0.7099 0.3549 4700 0.3155 0.1150 0.5497 0.7247 0.3624 4644 0.3185 0.1100 0.5306 0.7164 0.3582 4650 0.3400 0.1105 0.5327 0.7173 0.3587 4700 0.3530 0.1150 0.5497 0.7247 0.3624 4800 0.3685 0.1243 0.5845 0.7396 0.3698 4900 0.3840 0.1342 0.6203 0.7544 0.3772 5000 0.3965 0.1445 0.6569 0.7692 0.3846 5100 0.3980 0.1554 0.6943 0.7841 0.3920
299 Table 59: Load-Deflection Data B-7
Exp- Load Branson's With Reduction R.L.Sunna Proposed Deflection (lb) Equation (in) Coefficients (in) Relation (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 0.0000 300 0.0015 0.0020 0.0020 0.0022 0.0022 600 0.0035 0.0040 0.0040 0.0036 0.0036 900 0.0055 0.0050 0.0050 0.0050 0.0050 1200 0.0075 0.0060 0.0060 0.0063 0.0063 1500 0.0095 0.0080 0.0080 0.0077 0.0077 1800 0.0115 0.0090 0.0090 0.0091 0.0091 2100 0.0115 0.0100 0.0100 0.0105 0.0105 1776 0.0305 0.0090 0.0090 0.0090 0.0090 1900 0.0380 0.0100 0.0100 0.0096 0.0096 2000 0.0435 0.0100 0.0100 0.0100 0.0100 2100 0.0480 0.0100 0.0100 0.0105 0.0105 2400 0.0605 0.0120 0.0120 0.0119 0.0119 2500 0.0660 0.0130 0.0880 0.1335 0.1335 2200 0.0750 0.0110 0.0110 0.0109 0.0109 2400 0.0870 0.0120 0.0120 0.0119 0.0119 2500 0.0195 0.0130 0.0130 0.3855 0.3855 2700 0.1020 0.0170 0.1160 0.4619 0.2310 2800 0.1070 0.0190 0.1330 0.5324 0.2662 3000 0.1175 0.0250 0.1700 0.5500 0.2750 3100 0.1250 0.0280 0.1910 0.5535 0.2768 2600 0.1325 0.0150 0.1020 0.2641 0.1321 2800 0.1265 0.0190 0.1330 0.5324 0.2662 3000 0.1435 0.0250 0.1700 0.5500 0.2750 3100 0.1480 0.0280 0.1910 0.5535 0.2768 3200 0.1515 0.0320 0.2140 0.5720 0.2860 3300 0.1570 0.0360 0.2380 0.5896 0.2948 3700 0.1630 0.0550 0.3550 0.6579 0.3290 2800 0.1670 0.0190 0.1330 0.5324 0.2662 2900 0.1700 0.0220 0.1510 0.5050 0.2525 3000 0.1745 0.0250 0.1700 0.5500 0.2750 3100 0.1785 0.0280 0.1910 0.5535 0.2768 3300 0.1895 0.0360 0.2380 0.5896 0.2948 3400 0.1970 0.0400 0.2650 0.6069 0.3035 3500 0.2045 0.0450 0.2930 0.6239 0.3120 300 3600 0.2115 0.0500 0.3230 0.6409 0.3205 3700 0.2205 0.0550 0.3550 0.6579 0.3290 3800 0.2285 0.0610 0.3890 0.6748 0.3374 3900 0.2375 0.0670 0.4240 0.6918 0.3459 4000 0.2495 0.0740 0.4620 0.7087 0.3544 4100 0.2560 0.0810 0.5020 0.7257 0.3629 3950 0.2695 0.0710 0.4430 0.7003 0.3502 4000 0.2755 0.0740 0.4620 0.7087 0.3544 4100 0.2795 0.0810 0.5020 0.7257 0.3629 4200 0.2875 0.0890 0.5430 0.7426 0.3713 4300 0.2975 0.0970 0.5870 0.7595 0.3798 4400 0.3100 0.1060 0.6320 0.7765 0.3883 4500 0.3235 0.1150 0.6800 0.7934 0.3967 4600 0.3345 0.1250 0.7290 0.8104 0.4052 4700 0.3495 0.1360 0.7800 0.8273 0.4137 4800 0.3625 0.1470 0.8330 0.8443 0.4222 4900 0.3750 0.1590 0.8880 0.8612 0.4306 4560 0.3835 0.1210 0.7090 0.8036 0.4018 4750 0.3900 0.1410 0.8060 0.8358 0.4179 4900 0.3975 0.1590 0.8880 0.8612 0.4306 5000 0.4060 0.1710 0.9440 0.8781 0.4391
301 Table 60: Load-Deflection Data B-8
Exp- Load Branson's With Reduction R.AL.Sunna Proposed Deflection (lb) Equation (in) Coefficients (in) Relation (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 0.0000 100 0.0000 0.0004 0.0004 0.0013 0.0013 200 0.0005 0.0009 0.0009 0.0017 0.0017 300 0.0040 0.0013 0.0013 0.0022 0.0022 400 0.0060 0.0018 0.0018 0.0026 0.0026 500 0.0085 0.0022 0.0022 0.0031 0.0031 600 0.0120 0.0027 0.0027 0.0035 0.0035 700 0.0140 0.0031 0.0031 0.0039 0.0039 800 0.0175 0.0036 0.0036 0.0044 0.0044 900 0.0210 0.0040 0.0040 0.0048 0.0048 1000 0.0235 0.0045 0.0045 0.0053 0.0053 1100 0.0275 0.0049 0.0049 0.0057 0.0057 1200 0.0305 0.0054 0.0054 0.0062 0.0062 1300 0.0335 0.0058 0.0058 0.0066 0.0066 1400 0.0370 0.0063 0.0063 0.0071 0.0071 1500 0.0400 0.0067 0.0067 0.0075 0.0075 1600 0.0430 0.0071 0.0071 0.0080 0.0080 1700 0.0460 0.0076 0.0076 0.0084 0.0084 1800 0.0490 0.0080 0.0080 0.0089 0.0089 2000 0.0530 0.0089 0.0089 0.0098 0.0098 2200 0.0580 0.0098 0.0098 0.0106 0.0106 2300 0.0610 0.0103 0.0103 0.0111 0.0111 2400 0.0635 0.0107 0.0107 0.0115 0.0115 2500 0.0655 0.0112 0.0112 0.0120 0.0120 2600 0.0690 0.0122 0.0686 0.1602 0.0801 2700 0.0715 0.0141 0.0792 0.2668 0.1334 2800 0.0750 0.0162 0.0910 0.3460 0.1730 2900 0.0755 0.0186 0.1040 0.3920 0.1960 3000 0.0805 0.0212 0.1184 0.4192 0.2096 3100 0.0840 0.0240 0.1341 0.4382 0.2191 3200 0.0880 0.0272 0.1513 0.4539 0.2269 3300 0.0915 0.0306 0.1701 0.4682 0.2341 3400 0.0960 0.0344 0.1905 0.4821 0.2410 3500 0.0995 0.0385 0.2127 0.4957 0.2478 3600 0.1040 0.0429 0.2367 0.5092 0.2546 302 3700 0.1085 0.0477 0.2625 0.5227 0.2614 3660 0.1135 0.0457 0.2519 0.5173 0.2587 3430 0.1145 0.0356 0.1970 0.4862 0.2431 3500 0.1165 0.0385 0.2127 0.4957 0.2478 3600 0.1195 0.0429 0.2367 0.5092 0.2546 3700 0.1220 0.0477 0.2625 0.5227 0.2614 3800 0.1250 0.0529 0.2903 0.5362 0.2681 3359 0.1305 0.0328 0.1820 0.4764 0.2382 3110 0.1325 0.0243 0.1358 0.4399 0.2199 3300 0.1415 0.0306 0.1701 0.4682 0.2341 3400 0.1445 0.0344 0.1905 0.4821 0.2410 3550 0.1515 0.0406 0.2244 0.5025 0.2512 3600 0.1545 0.0429 0.2367 0.5092 0.2546 3700 0.1595 0.0477 0.2625 0.5227 0.2614 3800 0.1650 0.0529 0.2903 0.5362 0.2681 3900 0.1710 0.0585 0.3202 0.5497 0.2748 4000 0.1770 0.0645 0.3522 0.5631 0.2816 4100 0.1840 0.0710 0.3865 0.5766 0.2883 4300 0.1905 0.0854 0.4619 0.6035 0.3017 4400 0.2100 0.0934 0.5033 0.6170 0.3085 4250 0.2160 0.0817 0.4422 0.5968 0.2984 4169 0.2175 0.0758 0.4114 0.5859 0.2929 4200 0.2205 0.0780 0.4230 0.5900 0.2950 4300 0.2250 0.0854 0.4619 0.6035 0.3017 4400 0.2325 0.0934 0.5033 0.6170 0.3085 4500 0.2400 0.1019 0.5472 0.6304 0.3152 4000 0.2410 0.0645 0.3522 0.5631 0.2816 4100 0.2475 0.0710 0.3865 0.5766 0.2883 4300 0.2580 0.0854 0.4619 0.6035 0.3017 4400 0.2620 0.0934 0.5033 0.6170 0.3085 4500 0.2710 0.1019 0.5472 0.6304 0.3152 4600 0.2790 0.1110 0.5937 0.6439 0.3219 4700 0.2900 0.1206 0.6428 0.6573 0.3287 4800 0.3000 0.1309 0.6947 0.6708 0.3354 4900 0.3090 0.1418 0.7494 0.6843 0.3421 5000 0.3250 0.1533 0.8070 0.6977 0.3489 4605 0.3255 0.1114 0.5960 0.6446 0.3223 4900 0.3330 0.1418 0.7494 0.6843 0.3421 5000 0.3380 0.1533 0.8070 0.6977 0.3489 5100 0.3445 0.1655 0.8675 0.7112 0.3556
303 5200 0.3515 0.1784 0.9310 0.7246 0.3623 5300 0.3585 0.1921 0.9975 0.7381 0.3691 5400 0.3665 0.2065 1.0671 0.7516 0.3758 5500 0.3750 0.2217 1.1399 0.7650 0.3825 5400 0.3800 0.2065 1.0671 0.7516 0.3758 5600 0.3940 0.2376 1.2158 0.7785 0.3892 5800 0.3990 0.2721 1.3775 0.8054 0.4027 5900 0.4085 0.2906 1.4632 0.8189 0.4094 6000 0.4175 0.3101 1.5523 0.8323 0.4162 6100 0.4265 0.3304 1.6447 0.8458 0.4229 6200 0.4365 0.3518 1.7405 0.8592 0.4296 6330 0.4495 0.3810 1.8701 0.8767 0.4384 6400 0.4555 0.3974 1.9423 0.8862 0.4431 6500 0.4625 0.4217 2.0483 0.8996 0.4498 6600 0.4730 0.4471 2.1577 0.9131 0.4565 6700 0.4860 0.4736 2.2705 0.9266 0.4633
304 Table 61: Load-Deflection Data B-9
Exp- Load Branson's With Reduction R.L.Sunna Proposed Deflection (lb) Equation (in) Coefficients (in) Relation (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 0.0000 300 0.0025 0.0022 0.0022 0.0022 0.0022 600 0.0065 0.0036 0.0036 0.0036 0.0036 900 0.0105 0.0050 0.0050 0.0050 0.0050 1200 0.0155 0.0060 0.0060 0.0063 0.0063 1500 0.0200 0.0080 0.0080 0.0077 0.0077 1800 0.0250 0.0091 0.0091 0.0091 0.0091 2100 0.0290 0.0105 0.0104 0.0105 0.0105 2400 0.0340 0.0119 0.0119 0.0119 0.0119 2700 0.0500 0.0170 0.0470 0.1932 0.0966 2900 0.0550 0.0220 0.0610 0.2197 0.1098 3000 0.0585 0.0250 0.0690 0.2281 0.1141 3300 0.0685 0.0360 0.0990 0.2505 0.1252 3000 0.0750 0.0250 0.0690 0.2281 0.1141 3200 0.0790 0.0320 0.0880 0.2432 0.1216 3500 0.0870 0.0450 0.1230 0.2649 0.1324 3600 0.0910 0.0500 0.1360 0.2721 0.1360 3800 0.0975 0.0610 0.1660 0.2865 0.1432 4000 0.1060 0.0740 0.2010 0.3009 0.1504 3800 0.1135 0.0610 0.1660 0.2865 0.1432 3755 0.1145 0.0580 0.1590 0.2832 0.1416 3800 0.1160 0.0610 0.1660 0.2865 0.1432 3900 0.1185 0.0673 0.1830 0.2937 0.1468 4000 0.1210 0.0740 0.2010 0.3009 0.1504 4200 0.1285 0.0890 0.2400 0.3152 0.1576 4400 0.1385 0.1060 0.2840 0.3296 0.1648 4500 0.1440 0.1160 0.3080 0.3368 0.1684 305 4600 0.1495 0.1256 0.3330 0.3440 0.1720 4700 0.1550 0.1360 0.3590 0.3512 0.1756 4500 0.1650 0.1160 0.3080 0.3368 0.1684 4600 0.1485 0.1256 0.3330 0.3440 0.1720 4700 0.1550 0.1360 0.3590 0.3512 0.1756 4750 0.1730 0.1420 0.3730 0.3548 0.1774 4900 0.1805 0.1590 0.4170 0.3656 0.1828 5000 0.1865 0.1717 0.4470 0.3728 0.1864 5100 0.2030 0.1850 0.4790 0.3800 0.1900 5200 0.2085 0.1990 0.5130 0.3872 0.1936 5300 0.2155 0.2130 0.5480 0.3943 0.1972 5400 0.2215 0.2290 0.5850 0.4015 0.2008 5500 0.2295 0.2450 0.6220 0.4087 0.2044 5600 0.2360 0.2620 0.6620 0.4159 0.2080 5700 0.2430 0.2800 0.7030 0.4231 0.2116 5800 0.2495 0.2980 0.7450 0.4303 0.2152 5700 0.2600 0.2800 0.7030 0.4231 0.2116 5900 0.2665 0.3170 0.7890 0.4375 0.2187 6000 0.2725 0.3380 0.8350 0.4447 0.2223 6050 0.2815 0.3480 0.8580 0.4483 0.2241 6300 0.2955 0.4040 0.9790 0.4663 0.2331 6400 0.3035 0.4270 1.0310 0.4734 0.2367 6500 0.3115 0.4520 1.0830 0.4806 0.2403 6600 0.3195 0.4780 1.1370 0.4878 0.2439 6700 0.3285 0.5040 1.1930 0.4950 0.2475 6800 0.3355 0.5320 1.2500 0.5022 0.2511 6900 0.3445 0.5600 1.3080 0.5094 0.2547 7000 0.3525 0.5900 1.3670 0.5166 0.2583 7100 0.3620 0.6200 1.4280 0.5238 0.2619 7200 0.3715 0.6520 1.4910 0.5310 0.2655 7000 0.3805 0.5900 1.3670 0.5166 0.2583
306 7150 0.3860 0.6360 1.4590 0.5274 0.2637 7200 0.3880 0.6520 1.4910 0.5310 0.2655 7300 0.3955 0.6840 1.5540 0.5382 0.2691 7400 0.4045 0.7180 1.6190 0.5454 0.2727 7500 0.4155 0.7520 1.6850 0.5526 0.2763 7600 0.4270 0.7880 1.7520 0.5597 0.2799 7700 0.4375 0.8240 1.8200 0.5669 0.2835 7800 0.4470 0.8620 1.8890 0.5741 0.2871 7900 0.4565 0.9010 1.9600 0.5813 0.2907 8000 0.4665 0.9400 2.0310 0.5885 0.2943 8100 0.4765 0.9810 2.1040 0.5957 0.2979 8200 0.4860 1.0230 2.1770 0.6029 0.3014 8300 0.4955 1.0660 2.2520 0.6101 0.3050 8400 0.5030 1.1100 2.3270 0.6173 0.3086 8500 0.5125 1.1550 2.4030 0.6245 0.3122 8600 0.5215 1.2010 2.4800 0.6317 0.3158 8700 0.5285 1.2480 2.5580 0.6389 0.3194 8800 0.5380 1.2960 2.6370 0.6460 0.3230 8900 0.5450 1.3450 2.7170 0.6532 0.3266 9000 0.5530 1.3950 2.7970 0.6604 0.3302
307 Table 62: Load-Deflection Data 2B-1
Exp- Load Branson's With Reduction R.L.Sunna Deflection (lb) Equation (in) Coefficients (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 100 0.0011 0.0005 0.0005 0.0013 200 0.0024 0.0009 0.0009 0.0017 400 0.0076 0.0018 0.0018 0.0026 600 0.0204 0.0027 0.0027 0.0035 800 0.0352 0.0036 0.0036 0.0044 1000 0.0520 0.0045 0.0045 0.0053 1200 0.0687 0.0054 0.0054 0.0062 1400 0.0844 0.0063 0.0063 0.0071 1600 0.1023 0.0072 0.0072 0.0081 1800 0.1193 0.0081 0.0081 0.0090 2000 0.1362 0.0090 0.0090 0.0099 2200 0.1531 0.0099 0.0099 0.0108 2400 0.1686 0.0108 0.0108 0.0117 2600 0.1842 0.0127 0.1918 0.2383 2800 0.2013 0.0168 0.2398 0.7650 3000 0.2203 0.0219 0.2927 1.0663 3200 0.3489 0.0279 0.3499 1.1822
308 Table 63: Load-Deflection Data 2B-2
Exp- Load Branson's With Reduction R.L.Sunna Proposed Deflection (lb) Equation (in) Coefficients (in) Relation (in) Relation (in) (in) 0 0.0000 0.0000 0.0000 0.0000 0.0000 200 0.0009 0.0009 0.0009 0.0017 0.0017 400 0.0018 0.0018 0.0018 0.0026 0.0026 600 0.0027 0.0027 0.0027 0.0035 0.0035 800 0.0037 0.0036 0.0036 0.0044 0.0044 1000 0.0051 0.0045 0.0045 0.0053 0.0053 1200 0.0060 0.0054 0.0054 0.0062 0.0062 1400 0.0072 0.0063 0.0063 0.0071 0.0071 1600 0.0086 0.0072 0.0072 0.0081 0.0081 1800 0.0101 0.0081 0.0081 0.0090 0.0090 2000 0.0118 0.0090 0.0090 0.0099 0.0099 2200 0.0147 0.0099 0.0099 0.0108 0.0108 2400 0.0994 0.0108 0.0108 0.0117 0.0117 2600 0.1096 0.0127 0.1225 0.2197 0.1098 2800 0.1688 0.0169 0.1584 0.5525 0.2763 3000 0.2300 0.0219 0.2003 0.7004 0.3502 3200 0.2508 0.0281 0.2483 0.7638 0.3819 3400 0.2745 0.0354 0.3023 0.8122 0.4061 3600 0.3025 0.0440 0.3622 0.8581 0.4291 3800 0.3355 0.0540 0.4278 0.9036 0.4518 4000 0.3913 0.0655 0.4987 0.9490 0.4745 4200 0.4255 0.0788 0.5745 0.9944 0.4972 4400 0.5436 0.0938 0.6547 1.0398 0.5199 4600 0.6376 0.1108 0.7388 1.0851 0.5426 4800 0.6978 0.1298 0.8263 1.1305 0.5652
309 Table 64: Load-Deflection Data 2B-4
Exp- Branson's With Reduction R.L.Sunna Proposed Load (lb) Deflecti Equation Coefficients (in) Relation (in) Relation (in) on (in) (in) 0.00 0.0000 0.0000 0.0000 0.0000 0.0000 94.67 0.0016 0.0012 0.0012 0.0012 0.0012 195.23 0.0034 0.0017 0.0017 0.0017 0.0017 297.33 0.0052 0.0021 0.0021 0.0021 0.0021 396.52 0.0076 0.0026 0.0026 0.0026 0.0026 692.19 0.0175 0.0039 0.0039 0.0039 0.0039 799.49 0.0195 0.0043 0.0043 0.0043 0.0043 996.10 0.0229 0.0052 0.0052 0.0052 0.0052 1201.89 0.0263 0.0061 0.0061 0.0061 0.0061 1402.75 0.0295 0.0070 0.0070 0.0070 0.0070 1597.71 0.0331 0.0079 0.0079 0.0079 0.0079 1792.63 0.0375 0.0087 0.0087 0.0087 0.0087 1998.05 0.1057 0.0096 0.0096 0.0096 0.0096 2197.63 0.1200 0.0105 0.0105 0.0105 0.0105 2404.39 0.1166 0.0114 0.0114 0.0114 0.0114 2597.50 0.1875 0.0124 0.0890 0.1274 0.0637 2801.88 0.2141 0.0165 0.1172 0.3846 0.1923 3001.47 0.2348 0.0214 0.1503 0.5142 0.2571 3393.68 0.2994 0.0341 0.2330 0.6013 0.3007 3598.06 0.3213 0.0426 0.2863 0.6363 0.3181 3801.62 0.3438 0.0525 0.3466 0.6707 0.3353 4009.78 0.4006 0.0643 0.4161 0.7057 0.3529 4198.86 0.4252 0.0765 0.4861 0.7376 0.3688 4398.83 0.4492 0.0912 0.5674 0.7712 0.3856 4602.12 0.4767 0.1082 0.6577 0.8054 0.4027 4806.28 0.5239 0.1275 0.7560 0.8398 0.4199 310 5001.80 0.5595 0.1481 0.8571 0.8727 0.4363 5207.11 0.5956 0.1722 0.9704 0.9072 0.4536 5403.96 0.6445 0.1978 1.0855 0.9404 0.4702 5599.93 0.6880 0.2258 1.2060 0.9734 0.4867 5796.49 0.7202 0.2567 1.3326 1.0064 0.5032 5998.76 0.6880 0.2913 1.4682 1.0405 0.5202 6214.72 0.8046 0.3318 1.6187 1.0768 0.5384 6402.10 0.8447 0.3699 1.7535 1.1084 0.5542 6606.64 0.8837 0.4147 1.9048 1.1428 0.5714 6798.26 0.9224 0.4599 2.0499 1.1750 0.5875 7000.67 0.9588 0.5111 2.2064 1.2091 0.6046
311 Table 65: Stress-Strain Data no-3 Steel Bar
Load (kips) Stress (ksi) Strain (in/in) 0.000 0.00 0.0000 0.013 0.26 0.0000 0.042 0.85 0.0001 4.074 83.00 0.0238 4.314 87.89 0.0309 4.348 88.58 0.0310 4.351 88.64 0.0310 4.355 88.71 0.0310 4.383 89.29 0.0313 4.379 89.21 0.0314 4.397 89.58 0.0315 4.390 89.43 0.0316 4.387 89.38 0.0316 4.403 89.69 0.0317 4.394 89.52 0.0318 4.392 89.48 0.0318 4.411 89.86 0.0319 4.405 89.73 0.0320 4.398 89.60 0.0320 4.418 90.00 0.0321 4.410 89.84 0.0322 4.404 89.73 0.0323 4.425 90.15 0.0323 4.417 89.97 0.0324 4.411 89.87 0.0325 4.430 90.25 0.0326 312 4.423 90.11 0.0326 4.416 89.97 0.0327 4.437 90.40 0.0328 4.431 90.27 0.0328 4.470 91.06 0.0338 4.463 90.91 0.0339 4.457 90.79 0.0340 4.475 91.17 0.0341 4.468 91.02 0.0341 4.464 90.94 0.0342 4.482 91.30 0.0343 4.474 91.15 0.0343 4.470 91.06 0.0344 4.489 91.44 0.0345 4.480 91.26 0.0345 4.477 91.21 0.0346 4.494 91.56 0.0347 4.485 91.37 0.0347 4.484 91.34 0.0348 4.501 91.68 0.0349 4.494 91.55 0.0349 4.490 91.47 0.0350 4.508 91.84 0.0351 4.499 91.66 0.0352 4.493 91.53 0.0352 4.515 91.97 0.0353 4.506 91.79 0.0354 4.502 91.72 0.0355 4.522 92.11 0.0355 4.511 91.90 0.0356
313 4.506 91.80 0.0357 4.526 92.21 0.0357 4.517 92.03 0.0358 4.516 91.99 0.0359 4.532 92.32 0.0360 4.522 92.13 0.0360 4.518 92.04 0.0361 4.540 92.48 0.0362 4.529 92.26 0.0362 4.527 92.21 0.0363 4.543 92.55 0.0364 4.534 92.36 0.0364 4.529 92.26 0.0365 4.550 92.68 0.0366 4.541 92.52 0.0366 4.539 92.47 0.0367 4.554 92.78 0.0368 4.545 92.58 0.0368 4.541 92.51 0.0369 4.559 92.88 0.0370 4.466 90.98 0.0370 4.458 90.82 0.0370 4.505 91.77 0.0370 4.513 91.95 0.0371 4.523 92.15 0.0371 4.555 92.79 0.0372 4.549 92.67 0.0372 4.551 92.72 0.0373 4.571 93.12 0.0373 4.563 92.95 0.0374
314 4.557 92.82 0.0375 4.577 93.24 0.0375 4.568 93.06 0.0376 4.567 93.05 0.0377 4.582 93.34 0.0378 4.574 93.17 0.0378 4.499 91.65 0.0378 4.497 91.60 0.0379 4.498 91.63 0.0379 4.514 91.96 0.0379 4.538 92.45 0.0379 4.539 92.46 0.0380 4.542 92.52 0.0380 4.566 93.01 0.0380 4.562 92.93 0.0381 4.559 92.88 0.0381 4.578 93.26 0.0382 4.572 93.14 0.0382 4.576 93.22 0.0383 4.596 93.63 0.0383 4.588 93.47 0.0384 4.587 93.44 0.0384 4.607 93.85 0.0385 4.597 93.65 0.0386 4.590 93.51 0.0387 4.611 93.93 0.0387 4.600 93.71 0.0388 4.794 97.65 0.0472 4.783 97.44 0.0473 4.779 97.35 0.0474
315 4.796 97.70 0.0475 4.786 97.49 0.0475 4.779 97.35 0.0476 4.801 97.80 0.0477 4.788 97.53 0.0477 4.786 97.51 0.0478 4.801 97.82 0.0479 4.791 97.61 0.0480 4.789 97.55 0.0480 4.805 97.89 0.0481 4.797 97.72 0.0482 4.794 97.66 0.0483 4.809 97.97 0.0483 4.799 97.77 0.0484 4.795 97.68 0.0485 4.815 98.08 0.0486 4.804 97.86 0.0486 4.804 97.87 0.0487 4.818 98.16 0.0488 4.806 97.90 0.0489 4.803 97.84 0.0489 4.819 98.17 0.0490 4.811 98.02 0.0491 4.810 97.98 0.0492 4.826 98.32 0.0492 4.773 97.23 0.0493 4.769 97.15 0.0493 4.805 97.89 0.0493 4.806 97.90 0.0494 4.806 97.91 0.0494
316 4.827 98.34 0.0495 4.713 96.01 0.0495 4.694 95.63 0.0495 5.031 102.48 0.0661 5.022 102.30 0.0662 4.231 86.18 0.1804 4.241 86.40 0.1805 4.228 86.13 0.1806 4.216 85.88 0.1807 4.230 86.18 0.1808 4.216 85.88 0.1808 4.210 85.77 0.1809 4.215 85.88 0.1810 4.204 85.63 0.1811 4.194 85.44 0.1812 4.206 85.69 0.1813 4.194 85.43 0.1814 4.183 85.22 0.1815 4.194 85.45 0.1816 4.182 85.19 0.1817 4.170 84.95 0.1817 4.183 85.22 0.1818 4.170 84.95 0.1819 4.163 84.81 0.1820
317 Table 66: Stress-Strain Data no-4 Steel Bar
Load (kips) Stress (ksi) Strain (in/in) 0.000 0.00 0.0000 0.087 0.79 0.0000 0.161 1.46 0.0001 8.712 78.88 0.0158 8.731 79.05 0.0158 8.714 78.90 0.0159 8.744 79.17 0.0160 8.759 79.30 0.0160 8.604 77.90 0.0160 8.658 78.39 0.0160 8.690 78.68 0.0161 8.682 78.60 0.0161 8.728 79.02 0.0161 8.751 79.24 0.0162 8.705 78.81 0.0162 8.755 79.26 0.0162 11.274 102.07 0.0478 11.250 101.86 0.0479 11.289 102.21 0.0479 11.306 102.37 0.0480 11.286 102.18 0.0481 11.313 102.43 0.0481 11.322 102.51 0.0482 11.294 102.25 0.0483 11.323 102.52 0.0483 11.332 102.60 0.0484 11.307 102.37 0.0485 318 11.330 102.58 0.0485 11.342 102.69 0.0486 11.311 102.41 0.0487 11.341 102.68 0.0487 11.350 102.77 0.0488 11.322 102.51 0.0489 11.327 102.55 0.0490 11.330 102.58 0.0490 11.305 102.36 0.0490 11.347 102.74 0.0491 11.359 102.85 0.0492 11.334 102.62 0.0492 11.359 102.85 0.0493 11.370 102.94 0.0494 11.342 102.69 0.0494 11.309 102.40 0.1724 11.280 102.13 0.1724 11.287 102.20 0.1725 11.294 102.26 0.1726 11.263 101.98 0.1727 11.281 102.14 0.1728 11.289 102.21 0.1729 11.256 101.91 0.1730 11.269 102.03 0.1730 11.276 102.09 0.1731 11.245 101.81 0.1732 11.258 101.93 0.1733 11.267 102.02 0.1734 11.235 101.72 0.1735 11.251 101.86 0.1736
319 11.256 101.91 0.1737 11.218 101.57 0.1737 11.237 101.74 0.1738 11.243 101.80 0.1739 11.213 101.53 0.1740 11.225 101.63 0.1741 11.232 101.69 0.1742 11.197 101.38 0.1742 11.213 101.53 0.1743 11.222 101.60 0.1744 11.192 101.34 0.1745 11.204 101.45 0.1746 11.209 101.49 0.1747 11.174 101.17 0.1748 11.188 101.30 0.1748 11.198 101.39 0.1749 11.170 101.13 0.1750 11.179 101.22 0.1751 11.186 101.28 0.1752 11.153 100.98 0.1753 11.169 101.13 0.1754 11.177 101.20 0.1754 11.144 100.90 0.1755 11.156 101.01 0.1756 11.162 101.07 0.1757 11.128 100.76 0.1758 11.148 100.93 0.1759 11.154 100.99 0.1760 11.120 100.68 0.1760 11.133 100.80 0.1761
320 11.142 100.88 0.1762 11.107 100.56 0.1763 11.126 100.73 0.1764 11.132 100.79 0.1765 11.098 100.48 0.1766 11.110 100.60 0.1767 11.119 100.68 0.1767 11.083 100.35 0.1768 11.100 100.50 0.1769 11.110 100.59 0.1770 11.075 100.28 0.1771 11.085 100.36 0.1772 11.095 100.45 0.1773 11.059 100.13 0.1773 11.076 100.28 0.1774 11.084 100.36 0.1775 11.050 100.05 0.1776 11.063 100.17 0.1777 11.069 100.22 0.1778 11.033 99.90 0.1779 11.053 100.07 0.1780 11.061 100.15 0.1780 11.029 99.86 0.1781 11.040 99.96 0.1782 11.045 100.01 0.1783 11.009 99.68 0.1784 11.031 99.87 0.1785 11.034 99.91 0.1786 10.999 99.59 0.1786 11.016 99.74 0.1787
321 11.023 99.80 0.1788 10.984 99.45 0.1789 11.007 99.66 0.1790 11.012 99.71 0.1791 10.976 99.38 0.1792 10.992 99.52 0.1793 10.996 99.56 0.1793 10.960 99.24 0.1794 10.979 99.41 0.1795 10.986 99.47 0.1796 10.956 99.20 0.1797 10.963 99.26 0.1798 10.971 99.33 0.1799 10.937 99.03 0.1799 10.954 99.18 0.1800 10.961 99.25 0.1801 10.929 98.95 0.1802 10.937 99.02 0.1803 10.946 99.11 0.1804 10.912 98.80 0.1804 10.929 98.95 0.1805 10.936 99.02 0.1806 10.905 98.73 0.1807 10.915 98.83 0.1808 10.920 98.87 0.1809 10.884 98.54 0.1810 10.908 98.76 0.1811 10.913 98.81 0.1811 10.878 98.49 0.1812 10.894 98.63 0.1813
322 10.897 98.67 0.1814 10.860 98.32 0.1815 10.878 98.49 0.1816 10.885 98.55 0.1817 10.854 98.27 0.1817 10.863 98.36 0.1818 10.312 93.37 0.1919 10.273 93.01 0.1920 10.292 93.18 0.1921 10.296 93.22 0.1922 10.261 92.91 0.1922 10.271 92.99 0.1923 10.275 93.04 0.1924 10.240 92.71 0.1925 10.258 92.87 0.1926 10.260 92.90 0.1927 10.223 92.56 0.1928 10.237 92.69 0.1929
323 350 "3 mm" Diameter Bar Specimen 2 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 11388.8 ksi Tensile Strength = 340.8 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 99: Stress-Strain Curve 3mm Specimen 2
350 "3 mm" Diameter Bar Specimen 3 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 12478.4 ksi Tensile Strength = 329.95 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 100: Stress-Strain Curve 3mm Specimen 3
324 350 "3 mm" Diameter Bar Specimen 4 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 14263 ksi Tensile Strength = 278.93 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 101: Stress-Strain Curve 3mm Specimen 4
350 "3 mm" Diameter Bar Specimen 5 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 32482 ksi Tensile Strength = 796.07 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 102: Stress-Strain Curve 3mm Specimen 5
325 350 "3 mm" Diameter Bar Specimen 7 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 12095.7 ksi Tensile Strength = 351.1 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 103: Stress-Strain Curve 3mm Specimen 7
350 "3 mm" Diameter Bar Specimen 8 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 11884.4 ksi Tensile Strength = 320.91 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 104: Stress-Strain Curve 3mm Specimen 8
326 350 "3 mm" Diameter Bar Specimen 6 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 13782.5 ksi Tensile Strength = 368.5 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 105: Stress-Strain Curve 3mm Specimen 6
2500
"3 mm" Diameter Bar Specimen 1
2000
1500
Stress, Mpa 1000
500 Tensile Modulus = 102.4 GPa Tensile Strength = 2288.2 MPa
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, mm/mm
Figure 106: Stress-Strain Curve 3mm Specimen 1
327 350 "5 mm" Diameter Bar Specimen 3 300
250
200
Stress, ksi 150
100
50 Tensile Modulus = 11972.8 ksi Tensile Strength = 288 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 107: Stress-Strain Curve 5mm Specimen 3
2500
"5 mm" Diameter Bar Specimen 4
2000
1500
Stress, Mpa 1000
500 Tensile Modulus = 81.1 GPa Tensile Strength = 2053.5 MPa
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, mm/mm
Figure 18: Stress-Strain Curve 5mm Specimen 4
328 350 "5 mm" Diameter Bar Specimen 5 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 11714.4 ksi Tensile Strength = 301.1 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 109: Stress-Strain Curve 5mm Specimen 5
350 "5 mm" Diameter Bar Specimen 6 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 11592.2 ksi Tensile Strength = 310.9 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 110: Stress-Strain Curve Specimen 6
329 350 "5 mm" Diameter Bar Specimen 7 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 14387.6 ksi Tensile Strength = 335.4 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 111: Stress-Strain Curve 5mm Specimen 7
350 "5 mm" Diameter Bar Specimen 8 300
250
200
Stress, ksi 150
100
50 Tensile Modulus = 13514.6 ksi Tensile Strength = 322.6 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 112: Stress-Strain Curve 5mm Specimen 8
330 Stress-Strain Curve 5mm Specimen 9 350
300
250
200
150 Stress(ksi)
100
MODULUS OF ELASTICITY=13,183KSI 50
0 0 0.005 0.01 0.015 0.02 0.025 Strain (in/in)
Figure 113: Stress-Strain Curve 5mm Specimen 9
Stress-Strain Curve 5mm Specimen 10
350
300
250
200
Stress (ksi) Stress 150
100
50 MODULUS OF ELASTICITY= 12,663KSI
0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Strain (in/in)
Figure 114: Stress-Strain Curve 5mm Specimen 10
331 350 "7 mm" Diameter Bar Specimen 1 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 12941.8 ksi Tensile Strength = 358.4 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 115: Stress-Strain Curve 7mm Specimen 1
350 "7 mm" Diameter Bar Specimen 2 300
250
200
Stress, ksi 150
100
50 Tensile Modulus = 13022.9 ksi Tensile Strength = 334.9 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 2: Stress-Strain Curve 7mm Specimen 2
332 350 "7 mm" Diameter Bar Specimen 3 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 12577.6 ksi Tensile Strength = 294.1 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 117: Stress-Strain Curve 7mm Specimen 3
350 "7 mm" Diameter Bar Specimen 4 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 12998.9ksi Tensile Strength = 304.2 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 118: Stress-Strain Curve 7mm Specimen 4
333 350
300
250
200
Stress, ksi 150
100
Tensile Modulus = 12206.6 ksi 50 Tensile Strength = 297.9 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 119: Stress-Strain Curve 7mm Specimen 5
350 "7 mm" Diameter Bar Specimen 6 300
250
200
Stress, ksi Stress, 150
100
50 Tensile Modulus = 12206.6 ksi Tensile Strength = 297.9 ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, inch/inch
Figure 120: Stress-Strain Curve 7mm Specimen 6
334 2500
"7 mm" Diameter Bar Specimen 8
2000
1500
Stress, Mpa Stress, 1000
500 Tensile Modulus = 91.3 GPa Tensile Strength = 2072.3 MPa
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, mm/mm
Figure 121: Stress-Strain Curve 7mm Specimen 8
2500
"7 mm" Diameter Bar Specimen 7
2000
1500
Stress, Mpa Stress, 1000
500 Tensile Modulus = 90.96 GPa Tensile Strength = 1929.1 MPa
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain, mm/mm
Figure 122: Stress-Strain Curve 7mm Specimen 7
335 Bond-Stress vs. Slip curve 5mm-C3
12
10
8
Stress (mpa) Stress - 6 Experimental
Bond Model by Gao Proposed Model
4
2
0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Slip (in)
Figure 123: Bond-Slip curve 5mm-C3
Bond-Stress.vs.Slip 5mm-C4 12
10
8
6
Experimental Stress (mpa) Stress - Model by Gao
Proposed Model Bond 4
2
0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Slip (mm)
Figure 124: Bond-Slip Curve 5mm-C4
336 Bond-Stress.vs.Slip-7mm-C9 12
10
8
6 experimental Stress (mpa) Stress - model by GAO
Proposed Relation Bond 4
2
0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Slip (mm)
Figure 125: Bond-Slip Curve 7mm-C9
337 Bond-Stress.vs.Slip-C10 12
10
8
6 Experimental Stress (mpa) Stress - MODEL BY GAO
Proposed Relation Bond 4
2
0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Slip(mm)
Figure 126: Bond-Slip Curve 7mm-C9
Bond-Stress.vs.Slip-7mm-c11 12
10
8 )
6 EXPERIMENTAL Stress (mpa Stress - MODEL BY GAO
PROPOSED MODEL Bond 4
2
0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Slip (mm)
Figure 127: Bond-Slip Curve 7mm-C11
338 Stress-Strain Curve 3mm-C6 300
250
200
150 Stress (ksi) Stress
100
50 MODULUS OF ELASTICITY=13,879ksi
0 0.00 0.01 0.01 0.02 0.02 0.03 0.03 Strain (in/in)
Figure 128: Stress-Strain Curve 3mm-C6
Stress-Strain Curve 3mm-C7
350
300
250
200
Stress (ksi) Stress 150
100
50
MODULUS OF ELASTICITY=12,510ksi
0 0.00 0.01 0.01 0.02 0.02 0.03 Strain (in/in)
Figure 129: Stress-Strain Curve 3mm-C7
339 Stress-Strain Curve 3mm C-8
350
300
250
200
150 Stress (ksi) Stress
100
50 MODULUS OF ELASTICITY=12,510ksi
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020
Strain (in/in)
Figure 130: Stress-Strain Curve 3mm-C8
Stress-Strain Curve 3mm-C5 300
250
200
150 Stress (ksi) Stress
100
50 MODULUS OF ELASTICITY=12,162ksi
0 0.000 0.005 0.010 0.015 0.020 0.025 Strain (in/in)
Figure 131: Stress-Strain Curve 3mm-C5
340 Stress-Strain Curve 5mm- C2
400
350
300
250
200 Stress (ksi) Stress 150
100
50 MODULUS OF ELASTICITY=12,870 ksi
0 0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 Strain (in/in)
Figure 132: Stress-Strain Curve 5mm-C2
341 Stress-Strain Curve 5mm-C3 300
250
200
150 Stress (ksi) Stress
100
50 MODULUS OF ELASTICITY=13,441ksi
0 0.000 0.005 0.010 0.015 0.020 Strain (in/in)
Figure 133: Stress-Strain Curve 5mm-C3
342 Stress-StrainCurve 7MM C9 200
180
160
140
120
100 Stress (ksi) Stress 80
60
40 MODULUS OF ELASTICITY=11601KSI
20
0 0.000 0.005 0.010 0.015 0.020 0.025 Strain (in/in)
Figure 134: Stress-Strain Curve 7mm-C9
343 Stress-Strain Curve 7mm-C10 250
200
150 Stress (ksi) Stress 100
50 MODULUS OF ELASTICITY=11670KSI
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 Strain (in/in)
Figure 135: Stress-Strain Curve 7mm-C10
Stress-Strain Curve 7mm C-11 250
200
150 Stress (ksi) Stress 100
50 MODULUS OF ELASTICITY=11216KSI
0 0.000 0.005 0.010 0.015 0.020 0.025 Strain (in/in)
Figure 136: Stress-Strain Curve 7mm-C11
344 Stress-Strain Curve 7mm-C12 350
300
250
200
Stress (ksi) Stress 150
100
50 MODULUSOF ELASTICITY=12541KSI
0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Strain (in/in)
Figure 137: Stress-Strain Curve 7mm-C12
345 Load-Deflection Curve B-2 5000
4500
4000
3500
3000
Experimental 2500 Branson's Equation
Load (lb) Load ACI 440 2000 R.L.Sunna Relation Proposed Relation 1500
1000
500
0 0 0.2 0.4 0.6 0.8 1 1.2 Deflection (in)
Figure 138: Load-Deflection Curve B-2
346 Load-Deflection Curve B-4
5000
4500
4000
3500
3000 Experimental 2500
Branson's Equation Load (lb) Load
2000 ACI 440
1500 R.L. Sunna Relation
Proposed Relation 1000
500
0 0 0.2 0.4 0.6 0.8 1 1.2 Deflection (in)
Figure 139: Load-Deflection Curve B-4
347 Load-Deflection Curve B-6
6000
5000
4000
3000 EXPERIMENTAL WITHOUT COEFF
Load (lb) Load WITH COEFF 2000 modified modo
1000
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Deflection (in)
Figure 140: Load-Deflection Curve B-6
Load-Deflection Curve B-7 6000
5000
4000
Experimental 3000 ACI 440
Load (lb) Load Branson's Equation R.L.Sunna Relation
2000 PROPOSED RELATION
1000
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Deflection (in)
Figure 141: Load-Deflection Curve B-7
348 Load-Deflection Curve B-8 8000
7000
6000
5000
Test 4000 Branson's Equation
Load(lb) ACI 440 3000 R.L.Sunna Relation Proposed relation
2000
1000
0 0 0.5 1 1.5 2 2.5 Deflection(in)
Figure 142: Load-Deflection Curve B-8
Load-Deflection Curve B-9 10000
9000
8000
7000
6000 Test
ACI 440 5000
Branson's Equation Load (lb) Load 4000 R.L.Sunna Relation
Proposed Relation 3000
2000
1000
0 0 0.5 1 1.5 2 2.5 3 Deflection (in)
Figure 143: Load-Deflection Curve B-9
349 Load-Deflection Curve 2B-2
6000
5000
4000
Experimental 3000 Branson's
Equation Load (lb) Load ACI 440
2000 R.L.Sunna Relation Proposed Relation
1000
0 0 0.2 0.4 0.6 0.8 1 1.2 Deflection (in)
Figure 144: Load-Deflection Curve 2B-2
350 Load-DeflectionCurve 2B-3 6000
5000
4000
Branson's Equation 3000 ACI 440
Load (lb) Load R.L.Sunna Proposed Relation Experimental 2000
1000
0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Deflection (in)
Figure 145: Load-Deflection Curve 2B-3
351 Sample Calculation for Moment-Strength
B-2
Associated parameters:
1. Structural Details
2. Geometrical Details
Moment-Strength as per ACI 440
Here
352
Or
Now
Or
Now
Or
353 #Strain-Compatibility
Assumption: ε = 0.03<εcu
Assume depth of neutral-axis c = 0.5”
354
Moment-Strength from the Experiment
355
Sample Calculation for Crack-Width
B-1
Associated parameters:
1. Structural Details
2. Geometrical Details
356
357
358 Sample Calculation for Deflection
B-1
Associated parameters:
1. Structural Details
2. Geometrical Details
359
CALCULATION OF UNCRACKED TRANSFORMED PROPERTIES
360
361
362
Deflection of the beam based on the moment of inertia from original Branson’s equation
Deflection due to self weight
363 Deflection of the beam based on the moment of inertia from ACI440
Deflection of the beam based on the moment of inertia from relation proposed by A.L.Sunna
364 Spreadsheet for the Calculation of Moment-Strength Using Parabolic Stress-Strain Curve for Concrete
width of beam b 8 in Effective depth of beam d 6.4 in ' Concrete strength fc 6035 psi
Designed tensile strength fy 319.24 ksi
Modulus of elasticity of steel Es 12527 ksi Area of reinforcing bars As 0.0609 in2
Crushing concrete strain εcu 0.003 in/in
Concrete strain at peak compressive stress εo 0.002 in/in
Assume srain in concrete extreme fiber εc-trial 0.0032 in/in Check if assumed strain is less then then ultimate strain check not ok
Rupture strain εy 0.027200 in/in
Depth of neutral axis c 0.507 in
Depth of neutal axis for balanced condition cb 2.3000 in Classify the section check UNDER-REINFORCED Strain in steel 0.03716 in/in Stress in steel 319.240 ksi Alfa-factor 0.7936 Total compression C 19 Kips Total tension T 19.4 Kips Goal seek equation EQUATION 0.000 Beta-factor β 0.42134
Internal moment Mu 10 kip-ft Strength reduction factor φ 0.9 Design moment strength φMu 9.020 kip-ft
365 Spreadsheet for the Load-Deflection Analysis as Per the Proposed Model
Total length of the beam L 147.60 in Span of the beam l 108.30 in Overhangs on both sides c 19.70 in Shear-span a 39.40 in Applied load 2P 5058.00 lb P(four point bending) P 2529.00 lb Depth of beam h 4.00 in Effective depth d 3.60 in Width of beam b 39.40 in 2 Area of reinforcement A f 0.62 in
Modulus of elasticity of basalt E f 6382000.00 psi ' Compressive strength of the concrete from cylinder test f c 5547.00 psi
Modulus of elasticity of concrete E c 4245256.53 psi
Modular ratio n f 1.50
Reinforcement ratio ρ f 0.00 K-factor K 0.11 4 Cracking moment of inertia I cr 10.31 in
Modulus of rupture for the concrete f r 558.59 psi 4 Gross moment of inertia of the section Ig 210.13 in Depth of neutral axis y 2.00 in
Cracking moment M cr 58688.81 lb-in
Weight of simply supported part W s 291.66 lb
Weight of overhang Wo 58.33 lb
Moment at the mid-section M c 180510.20 lb-in Numerical Constant 0.08 Reduction factor β 0.11 Alfa-factor α 0.85 4 Effective momemt of inertia I e 9.24 in 4 Effective moment of inertia as per the proposed model I e 4.62 in Unit weight of the concrete γ 150.00 lb/ft3 Total weigt of the beam W 2019.25 lb
Deflection at mid-span due to self weight δ s 0.25 in
Deflection at mid span due to loading δ e 3.07 in Total deflection δ t 3.31 in
366 Spreadsheet for Load-Deflection Analysis Using Branson's Equation and ACI 440 Method
Total length of the beam L 84 in Span of the beam l 60 in Overhangs on both sides c 12 in Shear-span a 28 in Applied load 2P 13007.618 lb P(four point bending) P 6504 lb Depth of beam h 7 in Effective depth d 6.4 in Width of beam b 8.0 in
Number of 3mm bars N 3 2.000
Diameter of reinforcement φ 3 0.500 in
Area of reinforcement A 3f 0.393 sq.in
Number of 5 mm bars N 5 0.000
Diameter of reinforcement φ 5 0.196 in
Area of reinforcement A 5f 0.000 sq.in
Number of 7mm bars N 7 0.000
Diameter of reinforcement φ 7 0.276 in
Area of reinforcement A 7f 0.000 sq.in
Total area A T 0.39 sq.in
Modulus of elasticity of basalt E f 29000 ksi ' Compressive strength of the concrete from cylinder test f c 6035.000 psi
Modulus of elasticity of concrete E c 4428059.959 psi
Modular ratio n f 6.549
Reinforcement ratio ρ f 0.007673 k-factor K 0.271 4 Cracking moment of inertia I cr 111.5321 in
Modulus of rupture for the concrete f r 582.639 psi 4 Gross moment of inertia of the section Ig 228.667 in Unit weight of the concrete γ 150.000 cu.ft
Weight of overhangs W o 58.333 lb
Weight of half of the span W s 145.833 lb Total weigt of the beam W 408.333 lb
Moment at centre due to self-weight M cs 1837.500 lb-in
Moment at centre due to external-load M ce 182106.652 lb-in
Total moment at the centre M c 183944.152 lb-in
Beta-1 factor β 1 0.748
GuarantteD tensile strength of basalt f* fu 70000 psi
Ultimate concrete strain ε cu 0.00312
Balanced fiber ratio ρ fb 0.000070764
Reduction factor β d 1.0000 Depth of neutral axis y 3.500 in
Cracking moment M cr 38065.779 lb-in Ratio of cracking moment to applied moment 0.207 Check if applied moment is equal or greater then cracking -moment NOT OK Counter 1.000 4 Effective momemt of inertia I e 112.570 in
Deflection due to self weight δ s 0.002 in
Deflection due to external load δ e 0.1167 in Total deflection δ t 0.11831 in 367