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ABSTRACT
STRESS-STRAIN BEHAVIOR OFHIGH STRENGTH CONCRETE CYLINDERS
by
Danqing Chen
Due to the recent development of high strength concrete in engineering and
construction, it has become more important to study the behavior of materials in
structures.
No results have been previously published in the literature regarding the
influence of aspect ratio in plain high strength concrete cylinders on stress-strain
behavior under uniaxial compression.Thus at present study, compression tests
were conducted on 3x6 in., 3x9 in., and 3x12 in. plain high strength concrete
cylinders to study their stress-strain behavior. Plain normal strength concrete
cylinders were also tested and used for comparative study.
Based on the present investigations, attempting to obtain a complete stress-
strain curve for plain high strength concrete cylinders with the length to diameter
ratio equal to or larger than three ( f c i >12,000 psi ) with axial strain control is
impossible. It can be concluded that the descending branch of a stress-strain
curve of high strength concrete(fc' > 12,000 psi, with l/d> or =3)can no longer be
treated as a material property; rather, it may be treated as a structural property.
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STRESS-STRAIN BEHAVIOR OFHIGH STRENGTH CONCRETE CYLINDERS
byDanqing Chen
A ThesisSubmitted to the Faculty of
New Jersey Institutes of Technologyin Partial Fulfillment of the Requirements for the Degree of
Master of Science in Civil Engineering
Department of Civil and Environmental Engineering
January 1995
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APPROVAL PAGE
STRESS-STRAIN BEHAVIOR OFHIGH STRENGTH CONCRETE CYLINDER
Danqing Chen
Dr. C.l7Thomas Hsu, Thesis Adviser DateProfessor of Civil and Environmental Engineering, NJIT
Dr. William S fliers, Committee Member DateProfessor and Chairman of Civil and Environmental Engineering, NJIT
Dr. Methi Wecharatana, Committee Member DateProfessor of Civil and Environmental Engineering, NJIT
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BIOG PHICAL SKETCH
Author: Danqing Chen
Degree: Master of Science in Civil Engineering
Date: January 1995
Undergraduate and Graduate Education:
• Master of Science in Civil Engineering,New Jersey Institute of Technology,Newark, New Jersey, 1995
• Bachelor of Science in Civil EngineeringJianghan University,Wuhan, People's Republic of China, 1984
Major: Structural Engineering
Working Experience:
Engineer. Wuhan Iron and Steel Design and Research Institute, Wuhan,People's Republic of China. 1985-1993.
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ACKNOWLEDGMENT
The author wishes to express her sincere gratitude to her supervisor,
Professor C.T.Thomas Hsu, for his guidance, friendship, and constant support
throughout this research.
Special thanks to professors William Spillers and Wecharatana Methi for
serving as members of the thesis committee.
The experiments were performed using the MTS testing system which was
purchased under the NSF Grant, No. CEE 8308339.The author is grateful to the
Department of Civil and Environmental Engineering and the Office of Graduate
Studies for the financial support in pursuit of this degree.
The author appreciates the timely help and suggestions from Mr.Allyn Luke,
and from fellow graduate students Mr. Been-jyh Yu and Mr. Rajendar
Navalurkar.
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This thesis is dedicated tomy parent and my husband
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TABLE OF CONTENTS
Chapter Page
INTRODUCTION 1
1.1 General 1
1.2 Scope and Objectives of Research 4
1.2 The Structure of The Thesis 4
2 LITERATURE SURVEY 5
2.1 Behavior of Concrete 5
2.1.1 Introduction 5
2.1.2 Behavior of High Strength Concrete 6
2.1.3 Effect of Interaction Between Testing Machine and Specimenon Behavior of Concrete 8
2.1.4 Effect of Cylinder Size on Concrete Strength and ElasticModulus 9
2.2 Empirical Equations of Complete Stress-Strain Curve for HighStrength Concrete 10
3 MATERIALS AND EXPERIMENTAL METHODS 11
3.1 Specimens, Materials and Mixing Proportion 11
3.1.1 Specimens 11
3.1.2 Materials 11
3.1.3 Mixing Proportions 13
3.2 Experimental Methods 13
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TABLE OF CONTENTS(Continued)
Chapter Page3.2.1 Mixing, Casting, and Curing 13
3.2.2 Experimental Setup 14
4 RESULTS AND DISCUSSIONS 15
4.1 Introduction 15
4.2 Short-Term Uniaxial Compression Test 15
4.2.1 Effect of Specimen Size on Compressive Strength 15
4.2.2 Effect of Specimen Size on Strain Corresponding toCompressive Strength 16
4.2.3 Effect of Specimen Size on Modulus of Elasticity 17
4.2.4 Behavior of Stress-Strain 18
4.2.5 Failure Mode 20
4.2.6 Effect of Interaction of Testing Machine and Specimen onBehavior of Concrete and Failure Mode 22
4.3 Comparison of Experimental Data and Existing Empirical Equations 23
5 SUMMARY AND CONCLUSIONS 26
APPENDIX A THE TABLES AND FIGURES OF RESULTS 28
APPENDIX B COMPUTER PROGRAM FOR PARAMETERS INANALYTICAL CURVES 74
REFERENCES 75
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LIST OF TABLES
Table Page
1 Mixing Proportion for High Strength Concrete 28
2 Mixing Proportion for Normal Strength Concrete 28
3 Average Values of Compressive Strength, Strain at Peak Stress, andElastic of Every Size of Each Batch 29
4 Modulus of Elasticity for Normal and High Strength Concretes 29
5 Parameters Used in Comparing Experimental With Analytical CurveUsing Hsu's Equations 30
6 Parameters Used in Comparing Experimental With Analytical CurveUsing Collins' Equations 30
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LIST OF FIGURES
Figure Page
1 Typical Complete Compressive Stress-Strain Curves(ACI-363) 8
2 Test Specimens with Different Aspect Ratio 31
3 Experimental Setup for Cylinder 3x6-in in Compression Test 31
4 Experimental Setup for Cylinder 3x9-in in Compression Test .. 32
5 Experimental Setup for Cylinder 3x12-in in Compression Test 32
6 Modulus of Elasticity Versus Concrete Strength (ACI-363) 33
7 Failure Mode for High Strength Concrete in Compression Test 34
8 Failure Mode for Normal Strength Concrete in Compression Test 34
9 Stress-Strain Curve for Normal Strength Concrete (N-01, 3x6 in, strainrate:1.67x10-5 strain/sec.) 35
10 Stress-Strain Curve for Normal Strength Concrete (N-02, 3x6 in, strainrate:1.67x10-5 strain/sec.) 35
11 Stress-Strain Curve for Normal Strength Concrete (N-03, 3x6 in, strainrate:1.67x10-5 strain/sec.) 36
12 Stress-Strain Curve for Normal Strength Concrete (N-04, 3x6 in, strainrate:1.67x10-5 strain/sec.) 36
13 Stress-Strain Curve for Normal Strength Concrete (N-05, 3x9 in, strainrate:1.67x10-5 strain/sec.) 37
14 Stress-Strain Curve for Normal Strength Concrete (N-06, 3x9 in, strainrate: 1.67x10-5 strain/sec.) 37
15 Stress-Strain Curve for Normal Strength Concrete (N-07, 3x9 in, strainrate:1.67x10-5 strain/sec.) 38
16 Stress-Strain Curve for Normal Strength Concrete (N-08, 3x12 in, strainrate:1.67x10-5 strain/sec) 38
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LIST OF FIGURES(Continued)
Figure Page
17 Stress-Strain Curve for Normal Strength Concrete (N-09, 3x12 in, strainrate:1.67x10-5 strain/sec.) .. ... 39
18 Stress-Strain Curve for Normal Strength Concrete (N-10, 3x12 in, strainrate:1.67x10-5 strain/sec.) 39
19 Stress-Strain Curve for Normal Strength Concrete (N-11, 3x12 in, strainrate:1.67x10-5 strain/sec.) 40
20 Stress-Strain Curve for High Strength Concrete (H1-1, 3x6 in, strainrate:6.67x10-6 strain/sec.) 40
21 Stress-Strain Curve for High Strength Concrete (H1-2, 3x6 in, strainrate: 3.7x10 -6 strain/sec.) 41
22 Stress-Strain Curve for High Strength Concrete (H1-3, 3x6 in, strainrate: 2.5x10 -6 strain/sec.) 41
23 Stress-Strain Curve for High Strength Concrete (H1-4, 3x6 in, strainrate: 2.5x10-6 strain/sec.) 42
24 Stress-Strain Curve for High Strength Concrete (H1-5, 3x9 in, strainrate: 7x10-7 strain/sec.) 42
25 Stress-Strain Curve for High Strength Concrete (H1-6, 3x9 in, strainrate: 7x10-7 strain/sec.) 43
26 Stress-Strain Curve for High Strength Concrete (H1-7, 3x9 in, strainrate: 2x10-7 strain/sec.) 43
27 Stress-Strain Curve for High Strength Concrete (H1-8, 3x9 in, strainrate: 1x10-7 strain/sec.) 44
28 Stress-Strain Curve for High Strength Concrete (H1-9, 3x12 in, strainrate: 1x10-7 strain/sec.) 44
• 29 Stress-Strain Curve for High Strength Concrete (H1-10, 3x12 in, strainrate: 8.9x10-8 strain/sec.) 45
xi
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LIST OF FIGURES(Continued)
Figure Page
30 Stress-Strain Curve for High Strength Concrete (H1-11, 3x12 in, strainrate: 5x10-7 strain/sec.) 45
31 Stress-Strain Curve for High Strength Concrete (H1-12, 3x12 in, strainrate: 2x10-7 strain/sec.) 46
32 Stress-Strain Curve for High Strength Concrete (H2-01, 3x6 in, strainrate: 1.9x10-6 strain/sec.) 46
33 Stress-Strain Curve for High Strength Concrete (H2-02, 3x6 in, strainrate: 1.9x10-6 strain/sec.) 47
34 Stress-Strain Curve for High Strength Concrete (H2-03, 3x6 in, strainrate: 1.9x10-6 strain/sec.). 47
35 Stress-Strain Curve for High Strength Concrete (H2-04, 3x9 in, strainrate: 7x10-7 strain/sec.) 48
36 Stress-Strain Curve for High Strength Concrete (H2-05, 3x9 in, strainrate: 4x10-7 strain/sec.) 48
37 Stress-Strain Curve for High Strength Concrete (H2-06, 3x9 in, strainrate: 3x10-7 strain/sec.) 49
38 Stress-Strain Curve for High Strength Concrete (H2-07, 3x9 in, strainrate: 3x10 -7 strain/sec.) 49
39 Stress-Strain Curve for High Strength Concrete (H2-08, 3x12 in, strainrate: 8.3x10-8 strain/sec.) 50
40 Stress-Strain Curve for High Strength Concrete (H2-09, 3x12 in, strainrate: 8.3x10-8 strain/sec. 1 50
41 Stress-Strain Curve for High Strength Concrete (H2-10, 3x12 in, strainrate: 9.8x10-8 strain/sec.) 51
42 Stress-Strain Curve for High Strength Concrete (H2-11, 3x12 in, strainrate: 9.8x10 -8 strain/sec.) 51
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LIST OF FIGURES(Continued)
Figure Page
43 Comparison of Stress-Strain Curves of Same Size Cylinders for NormalStrength Concrete (N-01, N-02, N-03,, and N-04, 3x6 in) 52
44 Comparison of Stress-Strain Curves of Same Size Cylinders for NormalStrength Concrete (N-05, N-06, and N-07, 3x9 in) 53
45 Comparison of Stress-Strain Curves of Same Size Cylinders for NormalStrength Concrete (N-08, N-10, and N-11, 3x12 in) 54
46 Comparison of Stress-Strain Curves of Same Size Cylinders for HighStrength Concrete (H1-02, H1-03, and H1-04, 3x6 in) 55
47 Comparison of Stress-Strain Curves of Same Size Cylinders for HighStrength Concrete (H1-05, H1-06, H1-07, and H1-08, 3x9 in).... . . ........... 56
48 Comparison of Stress-Strain Curves of Same Size Cylinders for HighStrength Concrete (H1-09, H1-10, H1-11, and H1-12, 3x12 in) 57
49 Comparison of Stress-Strain Curves of Same Size Cylinders for HighStrength Concrete (H2-01, H2-02, and H2-03, 3x6 in) 58
50 Comparison of Stress-Strain Curves of Same Size Cylinders for HighStrength Concrete (H2-04, H2-05, and H2-07, 3x9 in) 59
51 Comparison of Stress-Strain Curves of Same Size Cylinders for HighStrength Concrete (H2-08, H2-09, Hand H2-11, 3x12 in) 60
52 Comparison of Stress-Strain Curves of Different Size Cylinders forNormal Strength Concrete (N-01, N-05, and N-08) 61
53 Comparison of Stress-Strain Curves of Different Size Cylinders for HighStrength Concrete (H1-03, H1-05, and H1-09) 62
54 Comparison of Stress-Strain Curves of Different Size Cylinders for HighStrength Concrete (H2-01, H2-04, and H2-10) 63
55 Experimental and Analytical Curves for Normal Strength Concrete UsingHsu's Equations(N-01,3x6 in) 64
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LIST OF FIGURES(Continued)
Figure Page
56 Experimental and Analytical Curves for Normal Strength Concrete UsingCollins' Equations(N-01,3x6 in) 65
57 Experimental and Analytical Curves for Normal Strength Concrete UsingHsu's Equations(N-06,3x9 in) ....... 66
58 Experimental and Analytical Curves for Normal Strength Concrete UsingCollins' Equations(N-06,3x9 in) 67
59 Experimental and Analytical Curves for Normal Strength Concrete UsingHsu's' Equations(N-08,3x12 in) 68
60 Experimental and Analytical Curves for Normal Strength Concrete UsingCollins' Equations(N-08,3x12 in) 69
61 Experimental and Analytical Curves for High Strength Concrete UsingHsu's Equations(H1-03,3x6 in) 70
62 Experimental and Analytical Curves for High Strength Concrete UsingCollins' Equations(H1-03,3x6 in) 71
63 Experimental and Analytical Curves for High Strength Concrete UsingHsu's Equations(H2-01,3x6 in) 72
64 Experimental and Analytical Curves for High Strength Concrete UsingCollins' Equations(H2-01,3x6 in) 73
xiv
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CHAPTER 1
INTRODUCTION
1.1 General
With its outstanding technological qualities, concrete has been a major structural
material used in buildings and other structures, whatever its forms (plain,
reinforced and prestress concrete). Even though there are high outputs of
structural steel in many developed countries, reinforced and prestressed
concretes are still commonly used in construction. In recent decades, the
development in concrete material science has brought about significant changes
in concrete construction and application. The focus of this development has
been in high strength concrete.
Generally, normal strength concrete is specified as a compressive strength
of 3,000 psi to 6,000 psi. In recent years however, there is a trend to define it as
a compressive strength of 3,000 psi to 8,000 psi. As the development of high
strength concrete has been continued, the definition of high strength concrete
varies in time and on a geographical basis[ACI, 1984] due to lack of the
standard criterion for high strength concrete. Several definitions of high strength
concrete have been made by many researchers. Freedman[Freedman, 1970]
defined the high strength concrete as a concrete with the strength of at least
6,000 psi at 28 days. ACI committee 363[ACI, 1984] defined the high strength
concrete as a concrete with specified compressive strength for design of 6,000
psi(41MPa) or greater. At present, the definition doesn't include concrete made
with exotic materials or techniques.
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The most important properties of high strength concrete used in building
code are high strength and high modulus of elasticity. The strength of high
strength concrete is affected by several main factors:
• Mixing proportioning and the selection of materials
. The method of curing
• The size of specimen
The selection of materials and mixing proportions for high strength concrete are
more crucial than those for normal strength concrete. Concrete is a two-phase
composite material consisting of cement paste and aggregate. The methods of
strength improvement of concrete can be classified into three parts: Strength
improvement of cement matrix, aggregate and bond between cement matrix and
aggregate.
According to ACI Committee 363 (1984), the strength of the cement matrix
is based on the strength of the hydration structure and the porosity of the matrix.
The increase of porosity will reduce the strength of the cement matrix. Because
porosity increases with increasing water to cement ratio, high strength concrete
generally has of a low water-cement ratio. Therefore, reducing the water to
cement ratio and adding other chemical admixtures are methods of attaining
high strength concrete. Chemical admixture, such as superplasticizers and water
reducing products have been widely used in attaining high strength concrete.
The silica fume(SF) has also been used to improve the strength of cement matrix
in a ultra high strength concrete with compressive strength over 20,000 psi.
As a heterogeneous materials, the properties of concrete depend on both
the properties of individual component and their combination. The quality and
type of aggregates have a significant effect on the strength and behavior of high
strength concrete[Aitcin, 1990]. The characteristics of coarse aggregate, such
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as, bond potential with cement paste and low water absorption capacity, are very
important consideration in the production of high strength concrete.
The methods of curing and the curing age are the other important factors in
developing the strength of concrete. Curing is mainly to promote the hydration of
the cement, and extremely important in attaining a high strength concrete. The
strength of concrete is increasing with the increase of its curing age. This has
been proved by many researchers.
Due to the special properties, such as high strength and high elastic
modulus, high strength concrete is often considered in the design of buildings to
achieve greater heights while reducing the mass of concrete needed by reducing
the sizes of column cross sections. Its high elastic modulus can also reduce the
deflections and creep deformations. The brittle property can be partly overcome
by adding fibers and/or tie confinement. It has been shown that addition of fibers
to concrete increased the ductility of concrete[Craig, 1984, Shah, 1976] and
fatigue strength [Kwak, 1991]. As the high content of cement, high strength
concrete is inherently more resistant to chemical deterioration.
With the high strength concrete used more and more commonly in all kinds
of structures, it has become necessary to further study the properties, stress-
strain behaviors, and influence factors of high strength concrete.
1.2 Scope and Objectives of Research
The objectives of this research are to study the behavior of stress-strain of high
strength concrete cylinders with different length to diameter ratio or aspect ratio,
equal to or greater than 3, and to study the interaction between the test
machine and the specimen under a uniaxial compression with the control of axis
strain. This research is divided into three parts:
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To test the normal strength concretes cylinders with different sizes for
comparative study.
• To test the high strength concrete cylinders with different sizes and to
study the influence of cylinder size on the strength of concrete,
modulus of elasticity , failure mode and stress-strain behavior.
• To compare the experimental results with the analytical stress-strain
equations developed by Hsu(1992) and Collins et al (1993),
respectively.
1.3 The Structure of the Thesis
In the following, previous research that is related to the behavior of stress-strain
of concrete, the influence of specimen size on strength of concrete and behavior
of stress-strain will be reviewed. The empirical equations of complete stress-
strain curve for a high strength concrete will also be described in the second
chapter.
An introduction of mixing proportion, the selection of material, and the
properties of several materials, experimental methods used in this research is
presented in chapter 3. Chapter 4 will present and discuss the experimental
results.
The last chapter of this thesis is devoted to the summary and conclusions.
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CHAPTER 2
LITERATURE SURVEY
2.1 Behavior of Concrete
2.1.1 Introduction
One of the most important properties of any structural materials is their
mechanical behavior of stress and strain. Current design practice for reinforced
and prestressed concrete is based on an implicit assumption that plain concrete
subjected to uniaxial compression has a usable descending part. Although the
explicit knowledge of the shape of the descending part is not necessary in most
routine designs; Wang [Wang, 1978] indicated that for an accurate and rational
design of the structures subjected to unusual loading such as earthquakes it is
desirable to know the complete stress-strain curve.
The Stress-strain behavior of concrete is dependent on its strength age at
loading, rate of loading, aggregate and cement properties and type and size of
specimens[Wang,1978, Granholm,1965] The shape of the uniaxial stress-strain
diagram is strongly affected by the several conditions. For testing, the conditions
include stiffness of the testing machine, size and shape of the specimen,
specimen versus machine stiffness, strain rate, type of strain gage, gage length,
and type of loading (preloading, cycling, etc.). For concrete characteristics, the
conditions include water-cement ratio, cement characteristics and content,
concrete unit weight, aggregate characteristics and content, and type of curing
and age of testing [Carreira and Chu, 1985].
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2.1.2 Behavior of High Strength Concrete
There are rather limited literature about the complete stress-strain curve of high
strength concrete. One of the reasons why there are insufficient experimental
results on the complete stress-strain curve is that it is very difficult to measure
the descending portion of the curve. Until recently, only two reports discussed
the complete stress-strain curve for the plain high strength concrete by Shah et
al and Hsu [Shah, 1981, and Hsu, 1992]. In Shah's report, the specimens were
loaded in two different control modes: a constant rate of axial strain and the
controlled rate of circumferential strain. They obtained a complete stress-strain
curve of the high strength concrete for larger sizes of concrete cylinders (4x8
and 3x9 in.). During their experiments, the axial strain rate varied while the
circumferential strain rate was held constant. In Hsu's report, the specimens
were loaded using a constant rate of axial strain. The complete stress-strain
curves for 3x6 in. cylinders were successfully attained.
The shape of the ascending part of the stress-strain curve is more linear
and steeper for a high strength concrete than for a normal concrete. Also, the
slope of the descending branch of the curve is steeper for a high strength
concrete [ACI, 1984]. It is difficult to obtain the descending part of the stress-
strain curve experimentally. Hsu [Hsu,1974] showed experimentally that the
length of strain gages has a definitive effect on the shape of the descending
branch. A similar conclusion is also discussed by Sangha and Dhir [Sangha,
1972]. It is showed experimentally that the shape of the stress-strain diagram is
markedly affected by the following: (1) the confining effect that results from the
end restraint of the specimen by the testing machine platens; (2) as the material
approaches peak stress fc ' and beyond, the influence of end restraint becomes
increasingly important; (3) the maximum size of the aggregate in relation to the
specimen size; (4) the weaker top layers of the specimen; (5) the strain gage
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length compared with the specimen length and slenderness; and (6) the
specimen diameter for a given slenderness ratio. Shah [Shah, 1981] indicated
that the interaction between the testing system and the specimen is more critical.
The stress-strain relationship of high strength concrete in uniaxial
compression has been reported by many researchers. The main differences
between the stress-strain relationships of high strength concrete and normal
strength concrete are that high strength concrete has:
• a more linear stress-strain relationship to a high percentage of the
maximum stress
• a higher strain at maximum stress
• a steeper slope of the descending part of the curve
Carrasquillo et al.[Carrasquillo, 1981] have used X-ray techniques to
demonstrate the relationship between the slope of a stress-strain curve and the
extent of internal microcracking. For normal strength concrete, the unstable
microcracks start to develop at the interface between the paste and the
aggregate at about 65% of the peak stress. Aggregate type also affects the
shape of a stress-strain curve of concrete. While improvements to the quality of
the mortar results in increased compressive strengths, this does not necessarily
lead to a certain increase in the stiffness of the concrete. For a concrete of the
same strength, a stronger aggregate results in a smaller strain corresponding to
the peak stress. For a high strength concrete, unstable microcracks pass
through the aggregates resulting in a sudden release of energy which causes a
sharp descending branch of the stress-strain relation. Typical stress-strain
relationships of normal and high strength concretes are shown below:
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Figure 1 Typical Complete Compressive Stress-Strain Curves (ACI-363)
2.1.3 Effect of Interaction Between Testing Machine and Specimen onBehavior of Concrete
Interaction between the testing machine and specimen mainly includes two
parts: one is the relative stiffness and the other is the end restrain between the
machine and specimen. The effect of end restrain is conventionally assumed to
be negligible for a standard specimen with a length to diameter ratio (lid) of 2
[ASTM, 1992]. Comparing to the end restrain, the stiffness of the testing
machine is a more critical one. The stiffer the testing machine is, the smaller the
amount of released strain energy is. If the stiffness of the machine is greater
than the absolute value of the stiffness of the specimen in the descending part,
then a stable descending part can be experimentally obtained [Hundson, 1972].
Some investigators have developed techniques to increase the stiffness of the
testing machine which is costly, and may not be available in a normal quality
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control laboratory [Ahmed, 1979]. However, Wang and colleagues [Wang, 1978]
introduced a simple method to eliminate the strain energy release of the testing
machine. In their testings, concrete cylinders were loaded in parallel to the
hardened steel tubes. This design ensures that the sum of the load carried by
the steel tubes and the concrete cylinder is always increasing up to a strain of
0.006. Thus, no release of energy comes from the testing machine. During
loading, the strains in the steel tube were measured. Thus, knowing the total
load and the corresponding steel strain, the stress-strain relationship for a
concrete can be obtained. An alternate approach is to use a closed-loop testing
machine [Shah,1981]. In a closed-loop testing machine, specimens can be loads
so as to maintain a constant rate of strain increase and avoid unstable failure.
2.1.4 Effect of Cylinder Size on Concrete Strength and Elastic Modulus
The length-to-diameter ratio of concrete compression test specimens has long
been recognized as another important factor that influences the failure load.
Gonnerman[Gonnerman, 1925] found no significant differences between 6 in.
and small-diameter cylinders having the length to diameter ratio of 2 for normal
strength concrete. Short specimens fail at greater loads because the steel
loading platens of the testing machine restrain lateral expansion throughout the
specimen more effectively [Newman, 1964, and Ottosen, 1984]. The effect of
end restraint is conventionally assumed negligible for a standard specimen with
a length-to-diameter ratio (1/d) of 2 [ASTM, 1992]. By comparing the
compressive strength of 150-mm cylinders with 100-mm cylinders using the
same length/diameter ratio, Baalaki [Baalaki, 1992] found that the larger
specimen has a lower compressive strength and a higher elastic modulus. Up to
now no other results have been published in the literature on the influence of
specimen size and the value of the elastic modulus.
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2.2 Empirical Equations of Complete Stress-Strain Curvefor High Strength Concrete
The study of the stress-strain relationship is an important issue to quantify the
behavior of concrete. A number of empirical equations of complete stress-strain
curve for a normal concrete have been proposed, but few empirical formula are
developed for a high strength concrete. Carreira et al [Carreira, 1985] proposed
an equation to represent the complete stress-strain relationship of a plain normal
concrete. They found the shape of stress-strain curve is strongly affected by
some factors such as strain rate, the relative stiffness between the testing
machine and specimen, size and shape of the specimen, the quality of the
cement matrix, and the aggregate characteristics and their content. The
parameters in the equation are physically significant and can be estimated from
the experiments., Other complete stress-strain equations for a normal concrete
were summarized by Hsu and can be found in his work[Hsu, 1974].Hsu [Hsu,
1992] proposed an equation to represent the complete stress-strain relationship
of plain and fibrous high strength concretes. The parameters of proposed
equation can be predicted by a given single value of maximum compressive
strength, which can be obtained from the experiment. Another equation, first
proposed by Popovics [Popovics, 1973] and modified by other researchers and
summarized by Collins et al [Collins, 1993], can also be predicted by the value of
compressive strength fc. The above two empirical stress-strain equation will be
used in this study to compare with the present experimental test results in
Chapter four.
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CHAPTER 3
MATERIALS AND EXPERIMENTAL METHODS
3.1 Specimens , Materials and Mixing Proportion
3.1.1 Specimens
In this thesis, concretes in normal strength and high strength were studied.
Compressive strength fc' for high strength concrete is designated to be 12,000
psi (84 MPa) at 28 days, and it is 4,000 psi for normal strength concrete at 28
days.
A total of 34 specimens with three different sizes for normal and high
strength concrete cylinders:3x6, 3x9, and 3x12 in.were cast in the plastic molds.
Two batches of high strength concrete cylinders and one batch of normal
strength concrete cylinders were mixed in the concrete laboratory.
They are designated as follows:
Batch H1: plain high strength concrete
Batch H2: plain high strength concrete
Batch N : plain normal strength concrete
A detailed description of the procedures and criteria for the selection of material
and their proportions is given in the following.
3.1.2 Materials
The strength of concrete mainly depends on the strength of the hydration
structure and the porosity of concrete matrix. Porosity increases with increasing
water/cement ratio. Therefore, reducing the water to cement ratio is a method of
attaining higher strength concrete, and adding other chemical products is
another method of improving the workability of concrete.
11
-
12
The materials consisted of Type I cement satisfying with ASTM 150, sand
from local source, basalt with maximum aggregate size of 3/8 in. for high
strength concrete, or limestone aggregate with size of 3/8 in for normal strength
concrete, and tap water. For high strength concrete, silica fume (SF) in a
powder form was used to achieve a higher strength, and the superplasticizers
(SP) was also used to maintain good workability. SF and SP are introduced
below:
silica fume: Silica fume is a by-product resulting from the reduction of high
purity coal, in electric arc furnaces, or in the production of silicon and ferro-
silicon alloys. The silica fume has a high content of amorphous silicon dioxide
and consists of very fine spherical particles with an average particle size about
100 times smaller than a grain of ordinary Portland cement. It is necessary if
very high strength concrete is desired since it permits workability to be
maintained, even for concrete of very low water to cement ratio. Because of its
high Si02 content and fine particle size, silica fume is an extremely reactive
pozzolan for Portland cement concrete [LI,1994].
superplasticizer: superplasticizer is the main chemical admixture used for
attaining high strength concrete, the superplasticizers are used as water
reducers and do not require any significant change in the mix proportioning
(other than reduction in water-cement ratio). It can significantly improve the
dispersibility of cement particles and thus permit a decreasing water - cement
ratio. In addition, it will result in a more flowable concrete so that to enhance
workability [L1,19941.
-
1 3
3.1.3 Mixing Proportions
Table 1 and Table 2 list the mixing proportions of high and normal strength
concretes for current study. With these mixing proportions, the compressive
strength of concrete is approximately 12,000 psi for high strength concrete and
4,000 psi for normal strength at 28 days after casting with curing.
3.2 Experimental Method
3.2.1 Mixing, Casting, and Curing
All specimens followed the same procedures for material mixing, and curing to
minimize scattering of specimen properties.
All materials as described in previous section were mixed by a rotary mixer.
All cylinder specimens molds were prepared and lubricated with oil before the
concrete was poured. The mixing procedures were Firstly, the coarse and fine
aggregates were loaded into the mixer and dry mixed for 2 to 3 minutes.
Secondly, the cement and silica fume were added and mixed for another 1 to 2
minutes. Then the water with the superplasticizers was added to cementitious
materials and mixed for at least 5 minutes until a homogeneous mixture was
achieved. The resulting mixture was then molded into cylindrical molds.
During casting, both table vibrator and steel bar were used to compact the
specimens. All specimens were covered by plastic sheets right after casting and
were left at the room for 24 hours to get hardened. These specimens were then
demolded and cured in a lime saturated water until the day before testing. The
age of the specimens at the time of testing varied from 33 to 36 days.
-
14
3.2.2 Experimental Setup
The experiments are conducted in NWT Concrete and Structures Laboratory
using the Material Testing System (MTS) which is capable of performing a wide
range of experiments. In this study, the load-deformation curves of concrete
specimens are measured by a uniaxial compression.
Prior to testing, each cylinder was capped with sulfur compound at both
ends to ensure parallel and smooth surfaces of the test specimens and to
maintain a constant length for all cylinders. All the cylinder specimens were
tested in an uniaxial compression under servo-controlled, close-loop machine.
The maximum load capacity of the MTS is 100 kips. To obtain a complete stress-
strain curve, a slow strain rate was used. The axial deformations were measured
by two clip-on gages which were mounted to the specimen as shown in Fig.3 to
Fig. 5. In order to record the average axial displacements, the signals from the
two strain gages were averaged and fed back to the controller to constantly
adjust the applied load to maintain a constant strain rate. An IBM electronic data
acquisition system running the Unkelscope program was used to record the
strain values and corresponding loads.
-
CHAPTER 4
RESULTS AND DISCUSSIONS
4.1 Introduction
Twenty three cylinder specimens with size 3x6in., 3x9 in., and 3x12 in. made
with plain high-strength concrete were tested under short-term uniaxial
compression to determine the ultimate strength and to study the stress-strain
behavior with different ratio of length to diameter. Eleven cylinder specimens
made with plain normal strength concrete were also tested (fc '_.7,000 psi). Both
types of the concrete were also studied to understand the effect of specimen
size on their stress-strain behavior. In this chapter, experimental results will be
presented and discussed. These experimental data will also be used to compare
with the empirical equations proposed by other researchers.
4.2 Short-Term Uniaxial Compression Test
Plain high strength concrete and normal strength concrete specimens with
different size were tested. Their behavior of stress-strain, compressive strength,
failure mode, modulus of elasticity and interaction between the testing machine
and specimen end restrain were studied.
4.2.1 Effect of Specimen Size on Compressive Strength
A total of 34 cylinders were tested. Concrete cylinders with three different ratios
of length to diameter (3x6 in., 3x9 in., and 3x12 in.) were shown in Fig 2.
Compressive strengths of two types of concrete, normal and high strengths,
were obtained. These compressive strengths obtained at 33 or 36 days with their
corresponding strain and modulus of elasticity are given in Table 4.
15
-
16
Based on the test results, they show that the values of compressive
strength are slightly influenced by the aspect ratio of cylindrical specimens (the
different ratio of length to diameter). It appears that the compressive strength
and specimen size are inversely proportional. That is, as the ratio of length to
diameter increases, the compressive strength decreases slightly (Fig.52 to
Fig,54). Baalbaki et al [Baalbek', 1992] also obtained the similar results.
There are two possible explanations for this observation:
• The length to diameter ratio of concrete compression test specimens has
long been recognized as a factor that influences the failure load. Shorter
specimens fail at greater loads because the steel loading platens of the
testing machine restrain lateral expansion throughout the specimen more
effectively [Newman, 1964 and Ottosen,1984].
• Compaction is another factor influencing the concrete strength. Smaller
or shorter specimens tend to achieve better compaction, higher density,
and thus higher strength. This is particully true when the same
compaction procedures are used for all concrete specimens.
4.2.2 Effect of Specimens Size on Strain Corresponding to CompressiveStrength
Based on the present tests on concrete cylinders, no significant difference for
the strain values corresponding to compressive strength of different sizes in this
study was observed between the normal strength concrete cylinders and high
strength concrete cylinders. It is slightly different from the results of other
researchers. Carrasquillo et al [Carrasquillo, 1981] reported that the peak strain
for the high strength concrete is slightly greater than that for the normal strength
concrete within the strength range of 3,000 psi to 6,000 psi.For the strain values
corresponding to compressive strength are influenced by the length/diameter
-
17
ratio. The strain (e.) and the length /diameter are also inversely proportionally
(Table 3)
4.2.3 Effect of Specimen Size on Modulus of Elasticity
The secant modulus at from 25% to 50% of the compressive strength f c ' is
usually considered to be the modulus of elasticity [Wang and Salmon, 1992]. In
this study, the secant modulus of elasticity is defined as the secant slope of the
uniaxial stress-strain curve at a stress level of 50% of compressive strength.
Table 3 shows the values of modulus of elasticity where each elastic modulus
value is the average of three or four measurements of every size specimens.
Fig. 6 shows the relationship of modulus of elasticity and its corresponding
strength as summarized by ACI committee 363 ( ACI-363, 1984 ). The results of
present study are also plotted in Fig.6 for comparison. Based on the results of
Fig. 6, it may be concluded that present ACI recommendation for the modulus of
elasticity is only applicable to the concrete with the compressive strength up to
6,000 psi.
Table 4 summaries the compressive strength ( fc' ), strain ( 6 0 ) at peak
stress, and modulus of elasticity ( E c ) from this study. A comparison of
experimentally determined values for the modulus of elasticity with those
predicted by the expression given in ACI 318-89, is given in Table 4 under
columns 4 and 5, respectively. They show that ACI equation gives higher
modulus of elasticity for high strength concrete than those obtained from present
study. Carrasquillo [Carrasquillo,1981 ] and other reseachers also gave similar
conclusions.
The compressive strength, fc 1 , is inversely proportional to the specimen
size, however, the modulus of elasticity observed from the present experiment
results is proportional to the specimen size. That is, as the ratio of length to
-
18
diameter increases, the modulus of elasticity also increases (Table 3 ). A similar
phenomenon was also observed by Baalbaki et al [ Baalbaki, 1992 on different
sizes of concrete specimens (cylinders with the same ratio of length to diameter
but with different diameter).
Based on present elastic modulus values of total 34 cylinders of normal
strength and high strength concretes tested in this investigation, one may
conclude:
1. For normal strength concrete cylinders: the ratio of moduli of 3x6 in.
cylinders( Eco6 1 ) to 3x9 in. cylinders ( Ecog' ) is 0.83; and the ratio of moduli of
3x6 in. cylinders (E, 06 1 ) to 3x12 in. cylinders ( E c12 ' ) is 0.82. Therefore, the
following relations maybe proposed:
E09 ' = 1.20 Eco6' ( 5 )
E 'c 12 -= 1.21 Ecosi ( 6 )
2. For high strength concrete cylinders: the ratio of moduli of 3x6 in. ( E co6 1 ) to
3x9 in. cylinders ( Ecog' ) is 0.86; and the ratio of moduli of 3x6 in. ( Ec436' ) to
3x12 in. cylinders ( E ci2' ) is 0.83. Therefore, the following relations may be
proposed:
Ecog' = 1.16 Ec06 1
( 7 )
Ec12' 1.22 Eco6 1 ( 8)
where: Eco6r-- Modulus of elasticity of 3x6 in. cylinders
Ecog i Modulus of elasticity of 3x9 in. cylinders
Ec12 1 Modulus of elasticity of 3x12 in. cylinders
4.2.4 Behavior of Stress -Strain
Based on present tests of normal and high strength concrete cylinders, test
results for behavior of stress-strain are exhibited in Fig. 9 through Fig. 42.
-
19
Fig. 52 shows the typical stress-strain curve of normal strength concrete of
compressive strength near 7,000 psi with different length to diameter ( lid ) ratios
in a uniaxial compression. It shows that the compressive strength has the trend
of decreasing with the increasing the ratio of length to diameter. The strain
corresponding to compressive strength fc' has the same trend of decreasing with
the increasing ratio of length to diameter. For normal strength concrete, the
shape of the ascending branch of the stress-strain curve for three different sizes
of specimens shows no significant difference. It may be concluded that the
ascending branch of the stress-strain curve is less influenced by the cylinder
size. However, after reaching the maximum strength, the descending part of the
stress-strain curves for 3x6, 3x9, and 3x12 in. cylinders shows different
behavior. The larger the ratio of length to diameter is, the steeper the
descending rate is. Also the uniaxial strain at maximum strength is larger when
the lid ratio is smaller. The results show that the brittle character exhibited more
in a larger ratio of length to diameter cylinder than that of a smaller ratio.
Different from normal strength concrete cylinder, the ascending branch of
stress-strain curve for high strength concrete cylinder is slightly steeper as the
ratio of length to diameter increases. Comparing the normal and high strength
concrete cylinders with the same size as presented in Fig. 53 and Fig. 54 , the
shape of the ascending branch of the stress-strain curve is more linear and
steeper as the compressive strength increases. This is due to less bond
cracking and high stress-strength ratio to form the continuous crack paterns [
Carrasquillo, 1981 J. The last portion of the descending part of the stress-strain
curves for 3x9 and 3x12 in. at present tests were unable to attain. It may be
attributed to the stability of the concrete cylinders.
From the result of this study, the size of concrete specimen has no
significant influence on the behavior of concrete in pre-peak region. But at post-
-
20
peak strength, the higher the compressive strength is, the steeper the
descending part is. That is, the descending part of the stress-strain curve
exhibits a more brittle characteristic as the compressive strength increases.
For a heterogeneous materials like concrete, cracking is a dominant factor
for its behavior, and the development of microcracking is closely related to the
characteristics of stress-strain relation. The descending branch is attributed to
the extension of continuous cracks, which may be regarded as the reduction of
effective cross-sectional area [Meyer and Okamura, 1986]. The descending
branch of stress-strain curve of concrete can no longer be treated as a material
property. Rather, it has to be referred as a structural property [Mier, 1984].
Previous and present test results show that in pre-peak region, stress-strain
curves have no significant difference regardless of the heights of specimens;
however, they differ considerably in the post-peak region.
4.2.5 Failure Mode
The mode of failure for high strength concrete cylinders observed during testing
was different from that of normal strength concrete cylinder. Generally, the
normal strength concrete gradually fails after reaching its peak load; whereas,
the high strength concrete explodes just passing the peak load. Therefore, it is
difficulty to obtain a complete stress-strain curve for high strength concrete.
However, if the strain rate is sufficiently slow as in the present tests, it is
possible to obtain a gradual failure for high strength concrete cylinders.
Fig. 7 and Fig. 8 show the modes of failure for normal and high strength
concretes under suitable strain rate. The specimens generally fail in axial
splitting parallel to the applied load.
Fig. 8 shows the mode of failure of normal strength concrete cylinders with
the ratio of length to diameter equal to 2, 3 and 4 under the same strain rate
-
21
1.67x10-5 strain/sec. It clearly shows that the larger the ratio of length to
diameter is, the larger the lateral strain is. From the Fig. 8, it shows that the
larger ratio of length to diameter is , the smaller the axial strain is at failure.
Thus, the rate of increase of lateral strain in the descending part is greater than
that of axial strain ; and this rate is larger for larger ratio of length to diameter
cylinders.
For high strength concrete cylinders ( fd=12,000 to 13,000 psi ), similar
results were obtained. But the unstable failure may occur when the length of the
specimens is too long. This may explain why the cylinders suddenly explode at
the peak load for 3x9 and 3x12 in. cylinders even though a sufficient slow strain
rate is used (3x10-7 strain/sec. , 2x10- 7 strain/sec. respectively ).
The crack patterns of both normal and high strength concrete cylinders are
observed to be different. The broken surfaces of high strength concrete cylinder
pass through the aggregate and mortar and are smoother. This indicates that
the strength of the cement-matrix is higher than that of the aggregate and the
crack initiates through the aggregate by forming a much smoother cracked
surface. This phenomenon is different from that seen in the normal strength
concrete. The normal strength concrete exhibits a highly irregular failure
surfaces including bond failure between cement paste and coarse aggregate.
Carrasquillo [Carrasquillo, 1981 ] and Hsu [Hsu, 1992 ] also showed similar
results. The high strength concrete is more likely to pass through the aggregate
due to the greater cement-matrix strength, and the coarse aggregate becomes
the controlling factor for the ultimate strength of concrete. Therefore, the coarse
aggregate strength should be a more important parameter in a high strength
concrete than that in a normal concrete.
-
22
4.2.6 Effect of Interaction of Testing Machine and Specimen on Behavior ofConcrete and Failure Mode
Fig. 52 to Fig. 54 show that after reaching the maximum strength, the
descending part of the stress-strain curve for 3x6, 3x9 and 3x12 in. cylinders
exhibit the phenomenon that the maximum uniaxial strain becomes smaller as
the ratio of length to diameter increases. In contrast, it may be said that the
lateral strain becomes larger as the ratio of length to diameter increases. Thus,
the failure of concrete subjected to uniaxial compression is usually observed by
longitudinal splitting cracks.lt may be concluded that the rate of increase in
lateral strain in the descending part is greater than that of axial strain [Shah,
19811 especially for a longer cylinder. Thus the friction of the end between the
steel loading platens and specimen can not be effectively restrained in lateral
expansion throughout the specimen for longer specimens.
Another important factor affecting the descending part is the stiffness of the
machine.The stiffer the testing machine is, the smaller the amount of released
strain energy is. The servo-controlled closed-looped testing testing machine was
used in this test. But even with the closed-loop system, the relative stiffness of
the testing system may be still critical in deterimining the failure mode.
In this test, the descending part of stress-strain curve was sucessfully
obtained for normal strength concrete ( 3x6, 3x9, 3x12 in.), and high strength
concrete ( 3x6 in.).
For higher strength and higher length to diameter cylinder, the strain energy
stored in the specimen is much greater than that in normal strength
concrete.Thus, the relative stiffness of machine is reduced. When the energy
release rate is greater than the frequency response of the system, then suddenly
failure may occur during testing.
-
23
4.3 Comparison of Experimental Data and Existing Empirical Equations
Two empirical equations were used to compare with the present experimental
data. The first one is Hsu's equation [Hsu, 1992]:
where
f3 and n are the material parameters. fi depends on the shape of the stress-
strain diagram, and n depends on the strength of material. is the normalized
stress; x is the normalized strain; fc is the stress; E is the strain; fc' is the peak
stress of concrete; E. is the strain corresponding to the peak stress fc'; xd is the
strain corresponding to 0.3 fc' in the descending portion of the stress-strain
curve. rid is equal to 0.3 which corresponds to 0.3 fc' in the descending portion
in the dimensional stress-strain curve. kd and a are taken to be 0.8 and 0.5
respectively.
The relationship between f3 and compressive strength fc':
where fc' is the compressive strength of concrete in [ksi].
For 15_ x s x d , then n=1, if, 0(ksi) < f < 9 (ksi)
then n=2, if, 9 (ksi) fc< 11 (ksi)
-
24
Another one is the equation originally proposed by Popovics [Popovics,1973],
later modified by other researchers and summarized by Collins et al [Collins,
1993] :
where n and k are curve fitting factor, as n becomes higher, the rising curve
becomes more linear. fc is the compressive stress , so is the compressive strain,
fc' is the peak stress of concrete, s o' is the strain corresponding to the peak
stress.
A total of five experimental data were compared with the above two
empirical equations. Fig.55 to Fig.64 show the results of comparison.
Experimental data of normal strength concrete cylinders with different
aspect ratio and high strength concrete cylinders of 3x6-in are compared with
the empirical equations presented. Good agreement for the complete stress-
-
25
strain curves of normal and high strength concrete cylinders of 3x6 in. was
obtained using Collins' equations and Hsu's equations, respectively. No
satisfactory agreement for normal strength concrete of aspect ratio larger than
two was achieved. It is because the parameters used in the empirical equations
do not consider the effect of aspect ratios. Thus, good agreement may be
obtained by modifying the parameters in the empirical equations.
-
CHAPTER 5
SUMMARY AND CONCLUSIONS
Based on the present tests on 34 cylinders of three different kinds of
length/diameter ratios, the following conclusions may be made.
1. The compressive strength, uniaxial strain at maximum compressive strength
and modulus of elasticity are influenced by the size of cylindrical specimens.
It is found that the compressive strength is inversely proportional to the
specimen size. The strain at maximum compressive strength is also inversely
proportional to the specimen size. The elastic modulus is proportional to the
specimen size. In this study, it is also found that the ACI equation
overestimates the modulus of elasticity for concrete with compressive
strength greater than 6,000 psi.
2. For normal strength concrete, the shape of the ascending branch of the
stress-strain curve for three different sizes is less influenced by the ratio of
length to diameter ratio. However, the descending part of the stress-strain
curve is steeper for higher ratio of length to diameter.
For high strength concrete, the shape of the ascending part of the stress-
strain curve for higher strength concrete behaves a more linear and steeper
curve. The slope of the descending part of 3x6 in. specimen exhibits a
steeper curve as compared to that of the normal concrete. The descending
part of curves for 3x9 and 3x12 in. are unable to obtain in the experiments
with uniaxial strain control.
It has been shown that in pre-peak region, the stress-strain curve of normal
strength concrete are almost identical regardless of the heights of
specimens. However, that in the post-peak region, they vary considerably.
26
-
27
The stress-strain curves of high strength concrete show similar results as
those in normal strength concrete.
3. The crack patterns for high strength concrete show that the broken surface of
the concrete cylinders exhibits a smoother surface, and its surface passes
through the aggregates.
4. A good agreement for the complete stress-strain curve of high strength
concrete of 3x6 in. was obtained using Collins' equations and Hsu's
equations, respectively.
-
APPENDIX A
THE TABLES AND FIGURES OF RESULTS
Tablel Mixing Proportion for High Strength Concrete
Mixing Proportion
Materials Weight Ratio Weight(Ib/ft3)cement 1.00 40.317water 0.29 11.857sand 0.65 26.027aggregate 1.63 65.65silica fume(SF) 0.11 4.48superp as icizer 1 liter/200 lb
cememtitious0.33 liter
Table 2 Mixing Proportion for High Strength Concrete
Mixing Proportion
materials weight ratio weight(Ib/ft3)water 0.57 14.26cement 1.00 25.02sand 2.08 51.72aggregate 2.01 50.4
28
-
29
Table 3 Average Values of Compressive Strength, Strain at Peak Stress, andModulus of Elastic of Every Size of Each Batch
batch size t (psi) 6 (in/in) Ec(psi)
3x6 7424 0.0040 2525519
batch N 3x9 7120 0.0034 3025885
3x12 6934 0.0034 3076180
3x6 12882 0.0040 3369661
batch H1 3x9 13260 0.0036 4114181
3x12 12272 0.0033 4460820
3x6 12492 0.0036 3913530
batch H2 3x9 12512 0.0034 4328235
3x12 11299 0.0028 4418185
: compressive strength of concreteE 0 : strain at peak stress of concreteE c : modulus of elasticity
Table 4 Modulus of Elasticity For Normal and High Strength Concretes
batch rc (psi) 60 (in/in) E,(106 psi) E:(106 psi)7122 0.0038 2.63 4.91
batch N 7543 0.0041 2.48 4.81(3x6) 7191 0.0038 2.37 4.75
12808 0.0038 3.32 6.37batch H1 13125 0.0037 3.75 6.38(3x6) 13678 0.0041 3.56 6.06
13153 0.0038 3.94 6.47batch H2 11709 0.0032 4.08 6.56(3x6) 12614 0.0038 3.72 6.31
* ACI 318-89E c =57000.yrf c psi
. compressive strength of concrete6 0 : strain at peak stress of concreteE c : modulus of elasticity
-
30
Table 5 Parameters Used in Experimental h Analytical Curve Using Hsu'sEquations
specimen f'c (psi) 0 n xd
N-01 7122.496 3.0167 1 3.0370
N-06 7157.032 3.0230 1 3.0285
N-09 6942.883 2.985 1 3.0760
H1-04 13125.83 5.260 5 1.1797
H2-01 13153.48 5.278 5 1.1802
: compressive strength of concrete in the experiment13: material parameters which are related to the shape of stress-strain diagramn: parameter depending on the strength of materialxd : strain at 0.3 f in the descending portion of stress-strain curve, and it can beobtained using the program shown in APPENDIX B
Table 6 Parameters Used In Comparing Experimental With Analytical CurveUsing Collins' Equations
specimen f n k
N-01 7122.496 3.65 1.461
N-06 7157.032 3.66 1.465
N-09 6942.883 3.58 1.441
H1-04 13125.83 6.05 2.128
H2-01 13153.48 6.06 2.131
fc : compressive strength of concrete in the experimentn : curve fitting factork : curve fitting factor
-
Figure 2 Test Specimens With Different Aspect Ratio
31
Figure 3 Experimental Setup for Cylinder 3x6 in. in Compression Test
-
Figure 4 Experimental Setup for Cylinder 3x9in. in Compression Test
32
Figure 5 Experimental Setup for Cylinder 3x12in. in Compression Test
-
166 4
276 10 207 •
M Pa10
50
ACI 318, E c . 33 wc15 ./F' psi6
Range for which ACI code
formula was derived
40
4s
60 e0 100
MPa
145 1 . 5 3E C ( W ) x30 M Po
•1000 2000. 3000 4000 5000 6000 8000 10000 12000 14000
020 30 40 50 60 70 80 90 100 110
c psi
42 •
).o4«4 4
4 et • 9 4
4• 0 •G •
• 41
44
f 1
•• • Ma, llnot , Ntlson 6 Sla19
IIgAI Kluge, Sp•l•• Tama
- 0 Planar I & J•nson• aighl + PHs• SColdea
• cont. Oa • Hong,
A Hon•.o 51.1•lor
o Mor lint., Milton 0 Slot.• Cort•liqq111o.1411.en d Slot.• near, Monsoon • Cop•Il
Hol real A Potash. Allogor• 001 I Riana,, rata, flo,01saa ft Holtman
COOC1•it
▪
Paws.#
• DO,• 6 Vi•sipsi 4 R1019 , 114 Janson
„-.•• •
A . %• ■ • b
46
▪
.
*; • (wc/145)1'5 psi
E c (40,000 A:4-10 x106 1
• present study
120 0
20
10
5
4
E c w
( 145 )1.5x1663psi
Figure 6 Modulus of Elasticity Versus Concrete Strength (ACI-363)
-
Figure 7 Failure Mode for High Strength Concrete in Compression Test
34
Figure 8 Failure Mode for Normal Strength Concrete in Compression Test
-
8000
J5
6000
v) 4000
2000
0.005
0.01
0.015
strain(in/in)
Figure 9 Stress-Strain Curve for Normal Strength Concrete(N-01,3x6 in., strain rate: 1.67x10 -5 strain/sec.)
8000
0 0.005
0.01
0.015
strain(in/in)
Figure 10 Stress-Strain Curve for Normal Strength Concrete(N-02,3x6 in., strain rate: 1.67x10 -5 strain/sec.)
-
8000
6000
rn 4000
2000
36
0
0.005
0.01
0.015
strain(in/in)
Figure 11 Stress-Strain Curve for Normal Strength Concrete(N-03,3x6 in., strain rate: 1.67x10 -5 strain/sec.)
0 0.005 0.01 0.015strain(in/in)
Figure 12 Stress-Strain Curve for Normal Strength Concrete(N-04,3x6 in., strain rate: 1.67x10 -5 strain/sec.)
8000
6000
4000
2000
-
Figure 13 Stress-Strain Curve for Normal Strength Concrete(N-05,3x9 in., strain rate: 1.67x10 -5 strain/sec.)
37
Figure 14 Stress-Strain Curve for Normal Strength Concrete(N-06,3x9 in, strain rate: 1.67x10 -5 strain/sec.)
-
Figure 15 Stress-Strain Curve for Normal Strength Concrete(N-07,3x9 in., strain rate: 1.67x10 -5 strain/sec.)
Figure 16 Stress-Strain Curve for Normal Strength Concrete(N-08,3x12 in., strain rate: 1.67x10 -5 strain/sec.)
-
Figure 17 Stress-Strain Curve for Normal Strength Concrete(N-09,3x12 in., strain rate: 1.67x10 -5 strain/sec.)
(1
Figure 18 Stress-Strain Curve for Normal Strength Concrete(N-10,3x12 in., strain rate: 1.67x10 -5 strain/sec.)
-
Figure 19 Stress-Strain Curve for Normal Strength Concrete(N-11,3x12 in., strain rate: 1.67x10-5 strain/sec.)
Figure 20 Stress-Strain Curve for High Strength Concrete(H1-01,3x6 in., strain rate: 6.67x10 -6 strain/sec.)
-
Figure 21 Stress-Strain Curve for High Strength Concrete(H1-02,3x6 in., strain rate: 3.7x10 -6 strain/sec.)
Figure 22 Stress-Strain Curve for High Strength Concrete(H1-03,3x6 in., strain rate: 2.5x10 -6 strain/sec.)
-
Figure 23 Stress-Strain Curve for Hihg Strength Concrete(H1-04,3x6 in., strain rate: 2.5x10 -6 strain/sec.)
Figure 24 Stress-Strain Curve for High Strength Concrete(H1-05,3x9 in., strain rate: 7x10 -7 strain/sec.)
-
Figure 25 Stress-Strain Curve for High Strength Concrete(H1-06,3x9 in., strain rate: 7x10 -7 strain/sec.)
43
Figure 26 Stress-Strain Curve for High Strength Concrete(H1-07,3x9 in., strain rate: 2x10 -7 strain/sec.)
-
14000
12000 -
10000 -
8000 -sN
6000 -
44
0
0.005 0.01 0.015
strain(in/in)
Figure 27 Stress-Strain Curve for High Strength Concrete(H1-08,3x9 in., strain rate: 1x10 -7 strain/sec.)
14000
12000
10000
8000
6000
4000
2000
0
0 0.005 0.01 0.015
strain(in/in)
Figure 28 Stress-Strain Curve for High Strength Concrete(H1-09,3x12 in., strain rate: lx10 -7 strain/sec.)
-
14000
12000
10000
8000
(ub 6000
4000
2000
0
0 0.005 0.01 0.015strain(in/in)
Figure 29 Stress-Strain Curve for High Strength Concrete(H1-10,3x9 in., strain rate: 8.9x10 -8 strain/sec.)
14000
12000
10000
8000
(1.1b 6000
4000
2000
0 0 0.005 0.01 0.015
strain(in/in)
Figure 30 Stress-Strain Curve for High Strength Concrete(H1 -1 1 ,3x1 2 in., strain rate: 5x1 0 -7 strain/sec.)
45
-
12000
0 0.005 0.01 0.015
strain(in/in)
Figure 31 Stress-Strain Curve for High Strength Concrete(H1-12,3x12 in., strain rate: 2x10 -7 strain/sec.)
14000
12000
10000
8000
451 6000
4000
2000
0 0 0.005 0.01 0.015
strain(inf )
Figure 32 Stress-Strain Curve for High Strength Concrete(H2-01,3x6 in., strain rate: 1.9x10 -7 strain/sec.)
46
-
Figure 33 Stress-Strain Curve for High Strength Concrete(H2-02,3x6 in., strain rate: 1.9x10 -6 strain/sec.)
47
Figure 34 Stress-Strain Curve for High Strength Concrete(H2-03,3x6 in., strain rate: 1.9x10 -6 strain/sec.)
-
Figure 35 Stress-Strain Curve for High Strength Concrete(H2-04,3x9 in , strain rate: 7x10 -7 strain/sec.)
4 8
Figure 36 Stress-Strain Curve for High Strength Concrete(H2-05,3x9 in., strain rate: 4x10 -7 strain/sec.)
-
49
12000
10000
8000
6000
4000
2000 -
0
0 0.005 0.01 0.015
strian(in/in)
Figure 37 Stress-Strain Curve for High Strength Concrete(H2-06,3x9 in., strain rate: 3x10-7 strain/sec.)
14000
12000
10000
8000
N 6000
4000
2000
0 0 0.005 0.01 0.015
strain (in/in)
Figure 38 Stress-Strain Curve for High Strength Concrete(H2-07,3x9 in., strain rate: 3x10 -7 strain/sec.)
-
Figure 39 Stress-Strain Curve for High Strength Concrete(H2-08,3x12 in., strain rate: 8.3x10 -8 strain/sec.
50
=igure 40 Stress-Strain Curve for High Strength Concrete(H2-09,3x12 in., strain rate: 8.3x10 -8 strain/sec.)
-
Figure 41 Stress-Strain Curve for High Strength Concrete(H2-10,3x12 in., strain rate: 9.8x10 -8 strain/sec.)
Figure 42 Stress-Strain Curve for High Strength Concrete(H2-11,3x12 in., strain rate: 9.8x10 -8 strain/sec.)
-
Comparison: Stress-Strain Curve
Figure 43 Comparison of Stress-Strain Curves of Same Size Cylinders for Normal Strength Concrete(N-01,N-02, N-03, and N-04, 3x6 in.)
-
8000Comparison: Stress-Strain Curce
0
0.005
0.01
0.015strain(in/in)
Figure 44 Comparison of Stress-Strain Curves of Same Size Cylinders for Normal Strength Concrete(N-05,N-06,and N-07, 3x9 in.)
-
Comparison: Stress-Strain Curve
8000
6000
v) 4000cll
2000
0
0.005
0.01
0.015
strain(in/in)
Figure 45 Comparison of Stress-Strain Curves of Same Size Cylinders for Normal Strength Concrete(N-08,N-10,and N-11, 3x12 in.)
-
0 0.005 0.01 0.015
14000
12000
10000
8000"/d40.) 6000
4000
0
Comparison: Stress-Strain Curve
strain(in/in)
Figure 46 Comparison of Stress-Strain Curves of Same Size Cylinders for High Strength Concrete(H1-02,H1-03,and H1-04, 3x6 in.)
-
Comparison: Stress-Strain Curve14000
12000 -
10000 -,-,
8000
4.1 6000
4000 -
2000 -
00 0.005 0.01 0.015
strain(in/in)
Figure 47 Comparison of Stress -Strain Curves of Same Size Cylinders for High Strength Concrete(H1-05,H1-06,H1-07,and H1-08, 3x9 in.)
-
14000
12000
10000
8000
(urn6000
4000
2000
00 0.005 0.01 0.015
Comparison: Stress-Strain Curve
strain(in/in)
Figure 48 Comparison of Stress-Strain Curves of Same Size Cylinders for High Strength Concrete(H1-09,H1-10,H1-11 and H1-12, 3x12 in.)
-
0 0.005 0.01 0.015
14000
12000 -
10000 -
Q., 8000
0.1 6000 -
4000 -
2000 -
0
Comparison: Stress-Strain Curve
strain(in/in)
Figure 49 Comparison of Stress-Strain Curves of Same Size Cylinders for High Strength Concrete(H2-01,H2-02,and H2-03, 3x6 in.)
-
14000
12000 -
10000 -
• 8000
▪ 6000 -
4000 -
2000 -
Comparison: Stress-Strain Curve
0 0 0.005 0.01 0.015
strain(inJin)
Figure 50 Comparison of Stress-Strain Curves of Same Size Cylinders for High Strength Concrete(H2-04,H2-05,and H2-07, 3x9 in.)
-
Comparison: Stress-Strain Curve14000
12000
10000
00„,"7' 8000
I-. 6000
4000
2000
00
0.005 0.01 0.015
strain(in/in)
Figure 51 Comparison of Stress-Strain Curves of Same Size Cylinders for High Strength Concrete(H2-08,H2-09,H2-10 and H2-11, 3x12 in.)
-
8000
3x6 fc' =7122 psi3x9 fc• =7005 psi3x12 fc• =6942 psi
6000
• aCI) 4000
2000
0.005 0.01 0.015
Comparison: Stress-Strain Curve
strain(in/in)
Figure 52 Comparison of Stress-Strain Curves of Different Size Cylinders for Normal Strength Concrete(N-01,N-05,and N-08,)
-
Comparison: Stress-Strain Curve
14000
12000 -
10000 -
8000 -
17, 6000
A: 3x6 fc 1=13125 psiB: 3x9 feL----12870 psiC: 3x12 fc '=12414 psi
4000
2000 -
0
A
0
0.005
0.01
0.015
strain(in/in)
Figure 53 Comparison of Stress-Strain Curves of Different Size Cylinders for High Strength Concrete(H1-03,H1-05,and H1-9)
-
Comparison: Stress-Strain Curve
00 0.005 0.01 0.015
14000
12000 -
10000 -
8000 -ca..CA
1- 6000
4000
2000
A: 3x6 fc 1 =13153 psiB: 3x9 fc 1---13049 psiC: 3x12 feT:=12034 psi
strain(in/in)
Figure 54 Comparison of Stress-Strain Curves of Different Size Cylinders for High Strength Concrete(H2-01,H2-04,and H2-10)
-
542 3normalized strain
--1*- EXPERIMENTAL ANALYTICAL
Figure 55 Experimental and Analytical Curves for Normal Strength Concrete Using Hsu's Equation(N-01, 3x6 in.)
0 I
Comparison: Stress-Strain Curve1 2
1
`4'3 0.8
0.6
2 0.4
0.2
0
-
Comparison: Stress-Strain Curve
0
2 3
4
5normalized strain
--•,- EXPERIMENTAL ANALYTICAL
Figure 56 Experimental and Analytical Curves for Normal Strength Concrete Using Collins' Equation(N-01, 3x6 in.)
-
Comparison: Stress -Strain Curve1.2
1
0.8
. 0.6
E 0.40.2
0 0
0.5
I
1.5 2
2.5
3
3.5normalized strain
--4U- EXPERIMENTAL - ANALYTICAL
Figure 57 Experimental and Analytical Curves for Normal Strength Concrete Using Hsu's Equation(N-06, 3x9 in.)
-
Comparison: Stress-Strain Curve
0
0.5 1 1.5 2 2.5 3 3.5
normalized strain
-so- EXPERIMENTAL ANALYTICAL
Figure 58 Experimental and Analytical Curves for Normal Strength Concrete Using Collins' Equation(N-06, 3x9 in.)
-
10 0.5 2 2.51.5normalized strain
–1m– EXPERIMENTAL — ANAYTICAL
Figure 59 Experimental and Analytical Curves for Normal Strength Concrete Using Hsu's Equation(N-08, 3x12 in.)
3
Comparison: Stress-Strain Curve1 2
1
`,6) 0.8
uN 0.6
0 0.4
0.2
0
-
Comparison: Stress-Strain Curve
1.2
1
0.8
.§ 0.6
0 0.4
0.2
0.5
1
1.5
2
2.5
3
normalized strain
-10.- EXPERIMENTAL — ANALYTICAL
Figure 60 Experimental and Analytical Curves for Normal Strength Concrete Using Collins' Equation(N-08, 3x12 in.)
-
N 0.6
0 0.4
0.2
—ISE NV Iso IN =I in 1. ma a am
Comparison: Stress-Strain Curve1.2
0
1
2 3
4normalized strain
---0— EXPERIMENTAL — ANALYTICAL
Figure 61 Experimental and Analytical Curves for High Strength Concrete Using Hsu's Equation(H1-03, 3x6 in.)
-
10 2 3nomalized strain
4 5
Comparison: Stress-Strain Curve
-RI- EXPERIMENTAL ANALYTICAL
Figure 62 Experimental and Analytical Curves for Highl Strength Concrete Using Collins' Equation(H1-03, 3x6 in.)
-
Comparison: Stress-Strain Curvel 2
V, 0 8
lel 0.6
0.4
0.2
0.5 1 1.5 2
2.5
3
3.5normalized strain
EXPERIMENTAL — ANALYTICAL
Figure 63 Experimental and Analytical Curves for High Strength Concrete Using Hsu's Equation(H2-01, 3x6 in.)
-
Comparison: Stress-Strain Curve
1.2
Z1 0.8
N 0.6
0 0.4
0. 20
0.5
1
1.5 2
2.5
3 3.5
normalized strain
EXPERIMENTAL - ANALYTICAL
Figure 64 Experimental and Analytical Curves for High Strength Concrete Using Collins' Equation(H2-01, 3x6 in.)
-
APPENDIX B
COMPUTER PROGRAM FOR PARAMETERSIN ANALYTICAL CURVE
parameter(a=26.391, b=-87.97, c=25. 391) read(*, *)x1
n=1
10 f=x1 a+b*x1+c
f1=a*(x1**(a-1.))+b
x=x1-f/f1
write(*, 100)n,x1,x
x1 =x
n=n+1
if(abs(x-x1).gt.1E-6) then
xl
n=n+1
goto 10
endif
100 format(1x,'N=1,13,1X1=',f15.7,3x,'X=1,f15.7)
-
!7EFERENCES
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Copyright Warning & RestrictionsPersonal Information StatementAbstractTitle PageApproval PageBiographical SketchAcknowledgment Dedication PageTable of Contents (1 of 2)Table of Contents (2 of 2)Chapter 1: IntroductionChapter 2: Literature SurveyChapter 3: Materials and Experimental MethodsChapter 4: Results and DiscussionChapter 5: Summary and ConclusionsAppendix A: Tables and Figures or ResultsAppendix B: Computer Program for Parameters in Analytical CurveReferences
List of TablesList of Figures (1 of 5)List of Figures (2 of 5)List of Figures (3 of 5)List of Figures (4 of 5)List of Figures (5 of 5)