frp-confined rc columns: analysis, · 2020. 6. 29. · a proper design procedure for frp-confined...
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FRP-CONFINED RC COLUMNS: ANALYSIS,
BEHAVIOR AND DESIGN
By
Tao JIANG
A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy
Department of Civil and Structural Engineering The Hong Kong Polytechnic University
July 2008
To my wife, Xiao
II
ABSTRACT
A very popular application of FRP composites is to provide confinement to RC
columns to enhance their load carrying capacity and ductility. This method of
strengthening is based on the well-known phenomenon that the axial compressive
strength and ultimate axial compressive strain of concrete can be significantly
increased through lateral confinement. Despite the increasing popularity of this
strengthening technique, relevant design provisions in most of the existing design
guidelines for external strengthening of RC structures using FRP composites are
only applicable to the design of short columns subjected to concentric compression.
A proper design procedure for FRP-confined RC columns is urgently needed to
facilitate wider practical applications.
Against this background, this thesis is concerned with the development of a rational
design procedure for FRP-confined RC columns to correct the deficiency in
existing design guidelines. The thesis presents a systematic study covering the
behavior and modeling of FRP-confined concrete as well as the analysis and design
of FRP-confined RC columns. A series of axial compression tests on
FRP-confined concrete cylinders was conducted first to gain a good understanding
of the stress-strain behavior of FRP-confined concrete, which is fundamental and
essential to the analysis and design of FRP-confined RC columns. Stress-strain
models for FRP-confined concrete of different levels of sophistication were next
developed as a prerequisite for the analysis of FRP-confined RC columns.
Subsequently, a simple but accurate stress-strain model for FRP-confined concrete
was incorporated into a conventional section analysis procedure to develop design
equations for short FRP-confined RC columns with a negligible slenderness effect.
Finally, two theoretical models of different levels of sophistication were
developed to deal with the slenderness effect in slender FRP-confined RC
columns. The rigorous theoretical model was used to develop a slenderness limit
III
expression to differentiate short columns from slender columns while the simple
theoretical model was used to develop design equations for slender columns. The
results of the present study led to a comprehensive design procedure that includes
a set of design equations for short columns, a simple expression to separate short
columns from slender columns, and a set of design equations for slender columns.
The present study is limited to circular columns, but the framework presented in the
present study can be readily extended to FRP-confined rectangular RC columns
when an accurate stress-strain model for FRP-confined concrete in rectangular
columns becomes available. The present study has been partially motivated by the
need to formulate design provisions for the Chinese Code for the Structural Use of
FRP Composites in Construction, which is currently being finalized. This new code
has been developed within the framework of the current Chinese Code for Design
of Concrete Structures (GB-50010 2002). Therefore, some of the considerations in
the present study follow the specifications given in GB-50010 (2002) and these
considerations are highlighted where appropriate throughout the thesis.
IV
LIST OF PUBLICATIONS
Book Chapter
Teng, J.G. and Jiang, T. “Chapter 6: Strengthening of RC columns with FRP
composites”, Strengthening and Rehabilitation of Civil Infrastructures Using FRP
Composites, Woodhead Publishing Limited, UK.
Refereed Journal Papers
Jiang, T. and Teng, J.G. (2007). “Analysis-oriented models for FRP-confined
concrete”, Engineering Structures, 29(11), 2968-2986.
Teng, J.G., Jiang, T., Lam, L. and Luo, Y.Z. (2008). “Refinement of a
design-oriented stress-strain model for FRP-confined concrete”, Journal of
Composites for Construction, ASCE, submitted.
Conference Papers
Teng, J.G., Jiang, T., Lam, L. and Luo, Y.Z. (2007). “Refinement of Lam and Teng’s
design-oriented stress-strain model for FRP-confined concrete”, Proceedings, 3rd
International Conference on Advanced Composites in Construction (ACIC 2007),
2-4 April 2007, University of Bath, UK, 116-121.
Jiang, T. and Teng, J.G. (2006). “Strengthening of short circular RC columns with
FRP jackets: a design proposal”, Proceedings, 3rd International Conference on FRP
Composites in Civil Engineering, 13-15 December 2006, Miami, Florida, USA,
187-192.
Jiang, T. and Teng, J.G. (2006). “Assessment of analysis-oriented stress-strain
models for FRP-confined concrete under axial compression”, Proceedings, 4th
International Specialty Conference on Fibre Reinforced Materials, 29-31 October
2006, Hong Kong, China, 1-12.
V
ACKNOWLEDGEMENTS
The author would like to express his heartfelt gratitude to his supervisor, Professor
Jin-Guang Teng for his enlightening guidance and enthusiastic support throughout
the course of study. Professor Teng’s rigorous approach to academic research,
breadth and depth of knowledge in structural engineering, and creative and unique
insight into many academic problems have demonstrated the essential qualities
that a good researcher should possess and all of these qualities have greatly
benefited the author. The author would also like to express his warmest thanks to
his co-supervisors, Dr. Lik Lam and Professor Yao-Zhi Luo, for their generous
support and valuable suggestions and advice.
The author is grateful to both the Research Grants Council of Hong Kong Special
Administrative Region and The Hong Kong Polytechnic University for their
financial support. The author is also grateful to The Hong Kong Polytechnic
University for providing the research facilities.
The author wishes to thank the technical support from the Heavy Structures
Laboratory of the Department of Civil and Structural Engineering. The author is
very thankful to the technical staff of the laboratory, in particular, Messrs. K.H.
Wong, W.C. Chan, K. Tam, M.C. Ng, C.F. Cheung and T.T. Wai for their valuable
advice and assistance during the experimental program.
Special thanks go to the author’s friends and colleagues in the Department of Civil
and Structural Engineering of The Hong Kong Polytechnic University, in
particular, Drs. Tao Yu, Lei Zhang, Chi-Lun Ng, Hon-Ting Wong and Yuan-Feng
Duan, Messrs. Yue-Ming Hu, Dilum Fernando, Guang-Ming Chen, Shi-Shun
Zhang and Qiong-Guan Xiao, not only for their constructive discussions but also
VI
for their encouragement in times of difficulty during the course of study. The
author would also like to thank Professor Jostein Hellesland of University of Oslo
for answering the author’s queries about several academic problems.
Last but certainly not the least, the author is greatly indebted to his parents,
parents-in-law, and in particular his wife, Ms. Xiao Chen, for their constant
understanding, encouragement and love. Ms. Xiao Chen has been so considerate,
patient and supportive throughout the author’s candidature and it is she who
shared the most difficult time with the author. The author dedicates this thesis to
her with love.
VII
CONTENTS
CERTIFICATE OF ORIGINALITY I
DEDICATION II
ABSTRACT III
LIST OF PUBLICATIONS V
ACKNOWLEDGEMENTS VI
CONTENTS VIII
NOTATION XIV
CHAPTER 1 INTRODUCTION 1
1.1 BACKGROUND 1
1.2 STRENGTHENING OF RC COLUMNS WITH FRP COMPOSITES 2
1.3 OBJECTIVE AND SCOPE 3
1.4 REFERENCES 8
CHAPTER 2 LITERATURE REVIEW 11
2.1 INTRODUCTION 11
2.2 FRP-CONFINED CONCRETE IN CIRCLUAR COLUMNS UNDER
CONCENTRIC COMPRESSION 11
2.2.1 Confining Action of FRP Jacket 11
2.2.2 Dilation Properties 12
2.2.3 Ultimate Condition 13
2.2.4 Stress-Strain Curves 15
2.2.5 Stress-Strain Models 16
2.2.6 Size Effect 17
2.3 FRP-CONFINED CONCRETE IN RECTANGULAR COLUMNS
UNDER CONCENTRIC COMPRESSION 17
2.3.1 Behavior 17
2.3.2 Stress-Strain Models 18
VIII
2.3.3 Shape Modification 20
2.3.4 Size Effect 21
2.4 FRP-CONFINED CONCRETE UNDER ECCENTRIC
COMPRESSION 21
2.5 FRP-CONFINED RC COLUMNS 23
2.5.1 General 23
2.5.2 Short Columns 23
2.5.3 Slender Columns 24
2.6 ANALYTICAL AND DESIGN METHODS FOR RC COLUMNS 25
2.6.1 General 25
2.6.2 Analytical Methods 26
2.6.3 Design Methods 26
2.7 CONCLUDING REMARKS 27
2.8 REFERENCES 29
CHAPTER 3 ANALYSIS-ORIENTED STRESS-STRAIN MODELS FOR
FRP-CONFINED CONCRETE 40
3.1 INTRODUCTION 40
3.2 TEST DATABASE 42
3.2.1 General 42
3.2.2 Specimens and Instrumentation 43
3.2.3 Test Results 45
3.3 EXISTING ANALYSIS-ORIENTED MODELS FOR
FRP-CONFINED CONCRETE 46
3.3.1 General Concept 46
3.3.2 Peak Axial Stress Point 48
3.3.2.1 Peak axial stress 48
3.3.2.2 Axial strain at peak axial stress 49
3.3.2.3 Stress-strain equation 50
3.3.2.4 Lateral-to-axial strain relationship 51
3.4 ASSESSMENT OF EXISTING MODELS 52
3.4.1 Test Data 52
3.4.2 Dilation Properties 53
IX
3.4.3 Stress-Strain Curves 55
3.4.4 Ultimate Condition 56
3.5 REFINEMENT OF TENG ET AL.’S MODEL 57
3.5.1 General 57
3.5.2 Peak Axial Stress in the Base Model 58
3.5.3 Axial Strain at Peak Axial Stress in the Base Model 59
3.6 CONCLUSIONS 62
3.7 REFERENCES 64
CHAPTER 4 DESIGN-ORIENTED STRESS-STRAIN MODELS FOR
FRP-CONFINED CONCRETE 98
4.1 INTRODUCTION 98
4.2 TEST DATABASE 100
4.2.1 General 100
4.2.2 Stress-Strain Curves 102
4.2.3 Ultimate Condition 103
4.3 LAM AND TENG’S STRESS-STRAIN MODEL FOR
FRP-CONFINED CONCRETE 104
4.4 GENERALIZATION OF EQUATIONS 106
4.5 NEW EQUATIONS FOR THE ULTIMATE CONDITION 106
4.5.1 Ultimate Axial Strain 107
4.5.2 Compressive Strength 107
4.6 MODIFICATION TO LAM AND TENG’S MODEL: VERSION (I)
110
4.7 MODIFICATION TO LAM AND TENG’S MODEL: VERSION (II)
111
4.8 CONCLUSIONS 114
4.9 REFERENCES 116
CHAPTER 5 DESIGN OF SHORT FRP-CONFINED RC COLUMNS 131
5.1 INTRODUCTION 131
5.2 SECTION ANALYSIS 132
5.2.1 The Strength of FRP-confined RC Sections 132
X
5.2.2 Moment-Curvature Curves of FRP-confined RC Sections 135
5.2.3 Comparison with Test Results 135
5.3 EQUIVALENT STRESS BLOCK 138
5.4 DESIGN EQUATIONS 140
5.5 PERFORMANCE OF PROPOSED DESIGN EQUATIONS 142
5.6 CONCLUSIONS 145
5.7 REFERENCES 146
CHAPTER 6 ANALYSIS OF ELASTIC COLUMNS 163
6.1 INTROCUCTION 163
6.2 THE PROLBLEM OF COLUMN DESIGN 164
6.3 EXACT SOLUTIONS 164
6.3.1 General 165
6.3.2 Exact Solution for Restrained Columns 165
6.3.2.1 Deflection caused by the first-order moments 165
6.3.2.2 Deflection caused by the axial load 166
6.3.2.1 Final deflections 169
6.3.3 Exact Solution for Hinged Columns 171
6.4 BEHAVIOR OF RESTRAINED COLUMNS 173
6.5 DESIGN OF RESTRATNED COLUMNS 175
6.5.1 General 175
6.5.2 From a Hinged column to a Standard Hinged Column 175
6.5.3 From a Restrained Column to a Hinged Column 179
6.5.4 Proposed Equations 185
6.6 CONCLUSIONS 187
6.7 REFERENCES 189
CHAPTER 7 THEORETICAL MODEL FOR SLENDER FRP-CONFINED RC
COLUMNS 203
7.1 INTRODUCTION 203
7.2 THEORETICAL MODEL 204
7.2.1 General 204
7.2.2 Construction of Axial Load-Moment-Curvature Curves 205
XI
7.2.3 Numerical Integration for the Column Deflection 205
7.2.4 Generation of the Ascending Branch of the Load-Deflection
Curve 206
7.2.5 Generation of the Descending Branch of the Load-Deflection
Curve 208
7.3 VERIFICATION OF THE THEORETICAL MODEL 209
7.3.1 Comparison with Cranston’s Numerical Results 209
7.3.2 Comparison with Experimental Results 210
7.4 CONCLUSIONS 218
7.5 REFERENCES 220
CHAPTER 8 SLENDERNESS LIMIT FOR SHORT FRP-CONFINED RC
COLUMNS 238
8.1 INTRODUCTION 238
8.2 DEFINITION OF SLENDERNESS LIMIT 240
8.3 PARAMETRIC STUDY 242
8.3.1 Parameters Considered 242
8.3.2 Results for RC Columns 243
8.3.3 Results for FRP-confined RC Columns 245
8.4 SLENDERNESS LIMIT EXPRESSIONS FOR DESIGN USE 246
8.4.1 Slenderness Limit for RC Columns 246
8.4.2 Slenderness Limit for FRP-confined RC Columns 248
8.5 CONCLUSIONS 249
8.6 REFERENCES 251
CHAPTER 9 DESIGN OF SLENDER FRP-CONFINED RC COLUMNS 267
9.1 INTRODUCTION 267
9.2 SIMPLE THEORETICAL MODEL 268
9.2.1 General 268
9.2.2 Method of Analysis 269
9.2.3 Accuracy of the Simple Theoretical Model 270
9.3 LIMITS ON THE USE OF FRP 272
9.4 DESIGN METHOD 275
XII
9.4.1 Review of Current Design Methods for RC Columns 275
9.4.2 Nominal Curvature 276
9.4.3 Axial Load at Balanced Failure 278
9.4.4 Factors 1ξ and 2ξ 279
9.4.5 Proposed Design Equations 282
9.5 CONCLUSIONS 285
9.6 REFERENCES 288
CHAPTER 10 CONCLUSIONS 306
10.1 INTRODUCTION 306
10.2 BEHAVIOR OF FRP-CONFINED CONCRETE 307
10.3 MODELING OF FRP-CONFINED CONCRETE 308
10.4 ANALYSIS AND DESIGN OF FRP-CONFINED RC COLUMNS 309
10.5 FURTHER RESEARCH 310
XIII
NOTATION
A gross area of cross section
cA cross-sectional area of concrete
sA total cross-sectional area of longitudinal steel reinforcement
siA cross-sectional area of the th longitudinal steel bar i
0 0 0 0 0 0, , , , ,a b c d e f coefficients
01 02 03, ,a a a coefficients
b width of a rectangular cross section
01 02 03, ,b b b coefficients
cb width of the section at a distance cλ from the reference axis
mC equivalent uniform moment factor
d diameter of the imaginary steel cylinder
01 02 03, ,d d d coefficients
e load eccentricity
01 02 03, ,e e e coefficients
1e , 2e load eccentricities at column ends
mine minimum eccentricity
D diameter of a concrete cylinder/a circular column
E elastic modulus
2E slope of the second portion of the stress-strain curve of
FRP-confined concrete
cE elastic modulus of unconfined concrete
frpE elastic modulus of FRP
sE elastic modulus of steel
f lateral deflection of a column
XIV
midf lateral deflection at mid-height of a column
lf confining pressure provided by FRP at rupture
nomf nominal lateral deflection
yf yield stress of steel reinforcement
'ccf compressive strength of FRP-confined concrete
'*ccf peak axial stress of concrete under a specific constant
confining pressure '
cof compressive strength of unconfined concrete
cuf characteristic cube strength of unconfined concrete
'cuf axial stress at ultimate axial strain of FRP-confined concrete
1G , 2G column-to-beam stiffness ratios
fcG compressive fracture energy of unconfined concrete
sG the smaller of and 1G 2G
0h effective height of a section
H height of a concrete cylinder
I second moment of area
1k confinement effectiveness coefficient
l length of a column
cl characteristic length of a concrete cylinder
effl effective length of a column
eql equivalent length of a column
N axial load
balN axial load corresponding to balanced failure
cN axial load carried by concrete
crN Euler load of a column
sN axial load carried by longitudinal steel reinforcement
uN axial load capacity of a column
uoN section axial load capacity under concentric compression
XV
,u testN experimental axial load capacity of a column
,u theoN theoretical axial load capacity of a column
M bending moment
1M , 2M first-order moments at column ends
1M , 2M moments at column ends
cM bending moment carried by concrete
1eM , 2eM external moments at column ends
eqM equivalent uniform first-order moment
maxM maximum moment of a column
sM bending moment carried by longitudinal steel reinforcement
uM moment capacity of a column
uoM section bending moment capacity under pure bending
m total number of segments a columns is divided into q constant in Popovics’ stress-strain equation
R radius of a concrete cylinder/a circular column
1R , 2R rotational stiffnesses
sR radius of the imaginary steel cylinder
r radius of gyration
t thickness of an FRP jacket
V shear force caused by unequal 1M and 2M
V shear force caused by unequal 1M and 2M
v lateral deflection of a column due to first-order moment
w lateral deflection of a column due to axial load
x coordinate along column height
nx depth of neutral axis
1α mean stress factor
β ratio of the yield strain of steel reinforcement to the strain of
the extreme compression fiber of concrete
1β block depth factor
XVI
l∆ length of a column segment
cε axial strain of concrete
*ccε axial strain at '*
ccf
coε axial strain at the peak stress of unconfined concrete
cfε axial strain of the extreme compression fiber of concrete
cuε ultimate axial strain of FRP-confined concrete
hε hoop strain of an FRP jacket
,h rupε hoop rupture strain of an FRP jacket
lε lateral strain of concrete
siε axial strain of the th longitudinal steel bar i
tε axial strain at the transition point of the stress-strain curve of
FRP-confined concrete
yε yield strain of longitudinal steel reinforcement
θ ratio of central angle corresponding to the depth of the
equivalent stress block to 2π
0θ ratio of central angle corresponding to the depth of the neutral
axis to 2π
1θ ratio of central angle corresponding to the depth of the yielded
compressive steel reinforcement to 2π '2θ ratio of central angle corresponding to the depth of the yielded
tensile steel reinforcement to 2π
λ slenderness ratio
cλ distance from the reference axis
critλ slenderness limit for short RC columns
maxλ column slenderness above which FRP confinement has little
effect on the load-carrying capacity
φ curvature
balφ curvature corresponding to balanced failure
failφ curvature of the critical section at column failure
midφ curvature at mid-height of a column
XVII
nomφ nominal curvature
secφ maximum curvature that a section can sustain under a given
axial load ϕ moment amplification factor
1ϕ moment amplification factor for standard hinged column
Kρ confinement stiffness ratio
sρ volumetric ratio of longitudinal steel reinforcement
ερ strain ratio
cσ axial stress of concrete
hσ tensile stress in the FRP jacket in the hoop direction
lσ confining pressure
siσ axial stress of the th longitudinal steel bar i
µ effective length factor
sµ secant dilation ratio of concrete
tµ tangent dilation ratio of concrete
1ξ factor in the nominal curvature method that reflects the effect
of axial load level
2ξ factor in the nominal curvature method that reflects the effect
of column slenderness
nξ ratio of the neutral axis depth to the effective height of the
section
XVIII
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND
Fiber reinforced polymer (FRP) composites comprise fibers embedded in a resin
matrix. The fibers are generally carbon, glass and aramid fibers while the resins are
generally epoxy, polyester and vinylester resins. These hi-tech materials can be ten
times as strong as mild steel but only a quarter as heavy as steel, and they are
non-corrosive. Because of these advantages and their ease in site handling derived
from their light weight and the use of adhesive bonding techniques, FRP
composites have a tremendous potential for application in the retrofit of existing
structures as well as the construction of new structures. Nevertheless, it was not
until about two decades ago did engineers and researchers begin to explore the use
of FRP composites in construction, although they have been successfully used in
other industries such as aerospace and automotive industries for many decades.
This was due mainly to their high cost (Teng et al. 2002). With their prices falling
down rapidly in recent years, and the needs for maintaining and upgrading essential
infrastructures all over the world, FRP composites have found their increasingly
wide applications in construction over the last decade.
Nowadays, various forms of FRP products, including bars, sheets, plates and
profiles, among others, are commercially available. These products have been used
in construction in many different ways: from new construction to the retrofit of
existing structures and from internal reinforcing to external strengthening (Bank
2006). As motivated by these applications and in turn to advance these applications
to a new level, related research activities have become very active in recent years.
1
Fig. 1.1 shows the results of SCI database searches using a combination of the
keywords “FRP” and “concrete”. The number of SCI journal publications in the
FRP-concrete area has increased rapidly from zero in 1990 to a total of 234 in 2007.
This exponential growth is clear evidence that intensive research has been and is
being conducted worldwide in this area. It should be noted that the publication
statistics shown in Fig. 1.1 exclude many other papers on other applications of FRP
composites, including the strengthening of steel, masonry and timber structures
using FRP composites, all FRP bridge decks, all FRP pultruded sections as beams
and columns and FRP cables.
The intensity of international research activities in this area may also be reflected
by the following events: 1) in 1996, the American Society of Civil Engineers
(ASCE) launched a new journal, the Journal of Composites for Construction,
dedicated to this new material. The impact factor of this young journal has ranked
among the top journals in the structural engineering field over the last few years; 2)
in 2004, the International Institute for FRP in Construction (IIFC) was established,
with members coming from around the world; and 3) a significant number of
international conference series have been launched and attracted numerous
attendees from both academia and industry. These conference series include the
FRPRCS (Fiber Reinforced Polymer Reinforcement for Concrete Structures) series,
the ACMBS (Advanced Composite Material in Bridges and Structures) series and
the CICE (Composites in Civil Engineering) series, among many others.
1.2 STRENGTHENING OF RC COLUMNS WITH FRP COMPOSITES
Among all the areas in construction involving the use of FRP composites, the
strengthening of reinforced concrete (RC) structures has been the most popular due
to their high strength-to-weight ratio, excellent corrosion resistance and ease of
installation.
Within this area, a very popular application of FRP composites is to provide
confinement to RC columns to enhance their load carrying capacity and ductility.
This method of strengthening is based on the well-known phenomenon that the
axial compressive strength and ultimate axial compressive strain of concrete can be
2
significantly increased through lateral confinement. RC columns may be
strengthened through FRP confinement for two purposes: 1) to increase the axial
load capacity of a column; and 2) to increase the ductility of a column. The former
mainly aims for better performance of the column under static loads such as
increases in dead load or live load while the later mainly aims for better
performance of the column under seismic loads. This thesis is limited to the
analysis, behavior and design of RC columns confined with FRP composites to
achieve the former purpose.
Various methods have been used to provide confinement to columns using FRP
composites (Teng et al. 2002). Among them, in-situ FRP wrapping has been the
most commonly used technique, in which fiber sheets or fabrics are impregnated
with resins and wrapped continuously or discretely around columns in a wet lay-up
process, with the main fibers solely or predominantly oriented in the hoop direction.
This strengthening technique is potentially effective for columns of various section
shapes, but is particularly effective for circular columns. Rectangular columns need
to receive rounding of sharp corners or shape modifications before FRP jacketing to
enhance the effectiveness of confinement. For example, a rectangular section may
be modified into an elliptical section before FRP jacketing (Teng et al. 2002). Fig.
1.2 shows the installation of FRP wraps on RC columns.
1.3 OBJECTIVE AND SCOPE
Several design guidelines (fib 2001; ISIS 2001; ACI-440.2R 2002, 2008; JSCE
2002; CNR-DT200 2004; Concrete Society 2004) for external strengthening of RC
structures using FRP composites have been published as a result of extensive
research and enormous practical needs in this field. Nevertheless, relevant design
provisions for FRP-confined RC columns in these design guidelines are only
applicable to the design of short columns with negligible slenderness effects.
Moreover, Only Concrete Society (2004) and ACI-440.2R (2008) have
recommended a procedure to perform section analysis of short FRP-confined RC
columns so that columns subjected to combined bending and axial compression can
be designed accordingly, but they do not specify the corresponding design
equations. Therefore, a proper design procedure for FRP-confined RC columns is
3
urgently needed.
Against this background, the present study aims to develop a rational design
procedure for FRP-confined RC columns to correct the deficiency in existing
design guidelines. To this end, this thesis presents a systematic study covering the
behavior and modeling of FRP-confined concrete as well as modeling of and
development of design equations for FRP-confined RC columns on a combined
experimental and analytical basis. It should be noted that the present study is
limited to circular columns, strengthened with continuous FRP wraps, with the
fibers solely or predominantly oriented in the hoop direction, but the framework
presented in this study can be readily extended to FRP-confined rectangular RC
columns if an accurate stress-strain model for FRP-confined concrete in rectangular
sections is available. Such a stress-strain model is not yet available as revealed by
the review of existing work on FRP-confined concrete in rectangular sections given
in Chapter 2. The present study has been partially motivated by the need to
formulate design provisions for the Chinese Code for the Structural Use of FRP
Composites in Construction, which is currently being finalized. This new code has
been developed within the framework of the current Chinese Code for Design of
Concrete Structures (GB-50010 2002). Therefore, some considerations in the
present study follow the specifications given in GB-50010 (2002) and they are
highlighted where appropriate throughout the thesis. The topics covered in this
thesis may be summarized as follows.
Chapter 2 presents an extensive literature review of topics related to the present
study. It starts with a brief review of existing stress-strain models for FRP-confined
concrete based on tests on small-scale specimens under concentric compression.
The suitability of applying these models in the analysis of large-scale FRP-confined
RC columns subjected to combined bending and axial compression is then
discussed based on existing experimental and analytical evidence. Existing
analytical methods as well as design methods for RC columns are also reviewed.
Chapter 3 is concerned with analysis-oriented stress-strain models for
FRP-confined concrete, and in particular, those models based on the commonly
accepted approach in which a model for actively-confined concrete is used as the
4
base model. This chapter first provides a critical review and assessment of existing
analysis-oriented models for FRP-confined concrete based on a comprehensive
database of axial compression tests on FRP-confined concrete cylinders recently
conducted at The Hong Kong Polytechnic University. This assessment clarifies
how each of the key elements forming such a model affects its accuracy and
identifies the best-performance model. The chapter then presents a refined version
of this model, which provides more accurate predictions for the stress-strain
behavior of FRP-confined concrete, particularly for weakly-confined concrete.
Chapter 4 deals with design-oriented stress-strain models for FRP-confined
concrete, in particular, the refinement of Lam and Teng’s model. More accurate
expressions for the ultimate axial strain and the compressive strength are proposed
for use in this model. These new expressions are based on experimental results from
the same test database presented in Chapter 3 as well as analytical results from a
parametric study conducted using the refined version of an analysis-oriented model
proposed in Chapter 3. The new expressions account for the effect of confinement
stiffness explicitly and can be easily incorporated into Lam and Teng’s model to
provide more accurate predictions of stress-strain curves. Based on these new
expressions, two modified versions of Lam and Teng’s model are presented. The
first version involves only simple modifications to the original model by updating
the ultimate axial strain and compressive strength equation while the second
version attempts to cater for stress-strain curves with a descending branch, which is
not covered by the original model.
Chapter 5 is concerned with the development of design equations for short
FRP-confined RC columns. Section analysis employing the modified Lam and
Teng model proposed in Chapter 4 is presented for constructing the axial
load-bending moment interaction diagram for FRP-confined RC sections. The
section analysis serves as a basis to develop design equations: the contribution of
the confined concrete to the load capacity of the section is approximated by
transforming the stress profile of concrete into an equivalent stress block; the
contribution of the longitudinal steel reinforcing bars to the load capacity of the
section is approximated by smearing the bars into an equivalent steel ring. The
proposed design equations are in a simple form that is familiar to civil engineers
5
and their performance is shown to be very good through a comprehensive
parametric study.
Chapter 6 deals with the analysis of elastic columns with elastic end restraints to lay
the ground work for the analysis of slender FRP-confined RC columns discussed in
Chapters 7 to 9. The exact solution to the lateral deflection of such columns
induced by combined bending and axial compression is derived first. The rationale
behind column design is next explained: a restrained column with unequal end
eccentricities can be transformed into an equivalent hinged column with equal end
eccentricities. Approximate design equations for elastic columns are also
developed, which represent an improvement to the existing ones.
Chapter 7 presents an analytical model for slender FRP-confined RC columns
subjected to either concentric or eccentric compression. This model allows the
initial eccentricities at the two column ends to be unequal and seeks the deflected
shape of a column in an incremental-iterative manner by making use of the axial
load-bending moment-curvature relationships. The proposed model is described in
great detail first and it is then verified against experimental results of both RC
columns and FRP-confined RC columns, with the latter being emphasized.
Chapter 8 deals with the development of a design expression for the slenderness
limit for short FRP-confined RC columns. With this expression defined, short
columns can readily be differentiated from slender columns so that they can be
designed using the equations proposed in Chapter 5. A comprehensive parametric
study is performed to investigate the effects of various parameters on the
slenderness limit using the analytical model presented in Chapter 7. Based on a
careful interpretation of the parametric study, a simple expression is proposed for
the slenderness limit for FRP-confined RC columns for design use. This expression
accounts for the effects of major parameters and possesses reasonable accuracy for
design purposes. It also reduces to its counterpart for RC columns adopted in
current design codes for RC structures when no FRP confinement is provided.
Chapter 9 is concerned with the development of design equations for slender
FRP-confined RC columns. This chapter completes the design procedure for
6
FRP-confined RC columns. A less sophisticated computer program than the one
presented in Chapter 7 but with sufficient accuracy for the analyses presented in
this chapter is described first. Using this computer program, limits on the use of
FRP are proposed for an effective and economic strengthening scheme. Design
equations are then developed following the framework of the nominal curvature
method, based on the numerical results obtained using the computer program. The
design equations are shown to provide accurate perditions through a comprehensive
parametric study.
The thesis closes with Chapter 10, where the conclusions drawn from previous
chapters are reviewed, and areas in need of further research are highlighted.
7
1.4 REFERENCES
ACI-440.2R (2002). Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, American Concrete Institute, Farmington Hills, Michigan, USA.
ACI-440.2R (2008). Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, American Concrete Institute, Farmington Hills, Michigan, USA.
Bank, L.C. (2006). Composites for Construction: Structural Design with FRP Materials, John Wiley and Sons, Inc., UK.
CNR-DT200 (2004), Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Existing Structures, Advisory Committee on Technical Recommendations For Construction, National Research Council, Rome, Italy.
Concrete Society (2004). Design Guidance for Strengthening Concrete Structures with Fibre Composite Materials, Second Edition, Concrete Society Technical Report No. 55, Crowthorne, Berkshire, UK.
fib (2001). Externally Bonded FRP Reinforcement for RC Structures, International Federation for Structural Concrete, Lausanne, Switzerland.
GB-50010 (2002). Code for Design of Concrete Structures, China Architecture and Building Press, China.
ISIS (2001). Design Manual No. 4: Strengthening Reinforced Concrete Structures with Externally-Bonded Fibre Reinforced Polymers, Intelligent Sensing for Innovative Structures, Canada.
JSCE (2001). Recommendations for Upgrading of Concrete Structures with Use of Continuous Fiber Sheets, Concrete Engineering Series 41, Japan Society of Civil Engineers, Tokyo, Japan.
Teng, J.G., Chen, J.F., Smith, S.T. and Lam, L. (2002). FRP-Strengthened RC Structures, John Wiley and Sons, Inc., UK.
8
90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 070
50
100
150
200
250
Year
Ann
ual N
umbe
r of P
ublic
atio
ns
Fig. 1.1 Growth of number of SCI journal papers on the application of FRP in
concrete structures
9
(a) Circular columns
(b) Rectangular columns
Fig. 1.2 Installation of FRP wraps on RC columns (Bank 2006)
10
CHAPTER 2
LITERATURE REVIEW
2.1 INTRODUCTION
This chapter presents a review of existing knowledge of or related to FRP-confined
RC columns. Although the present study is limited to circular columns, existing
knowledge on rectangular columns is also reviewed in this chapter with the
emphasis on highlighting the uncertainties in the stress-strain behavior of
FRP-confined concrete in rectangular columns.
This chapter starts with a description of the unique behavior of FRP-confined
concrete, including its dilation properties, ultimate condition and stress-strain
behavior, observed from tests on small-scale specimens under concentric
compression. Stress-strain models dedicated to such concrete are next reviewed.
The suitability of applying these models in the analysis of large-scale FRP-confined
RC columns subjected to combined bending and axial compression is then
discussed based on existing experimental and analytical evidence, with particular
attention to the discussion of possible effects of load eccentricity and size. Lastly,
existing analytical methods as well as design methods for RC columns are reviewed
as the starting point for the analysis and design of FRP-confined RC columns.
2.2 FRP-CONFINED CONCRETE IN CIRCLUAR COLUMNS UNDER
CONCENTRIC COMPRESSION
2.2.1 Confining Action of FRP Jacket
11
The lateral confinement provided by an FRP jacket to concrete is passive in nature.
When the concrete is subjected to axial compression, it expands laterally. This
expansion is confined by the FRP jacket which is loaded in tension in the hoop
direction. Different from steel-confined concrete in which the lateral confining
pressure is constant following the yielding of steel, the confining pressure provided
by the FRP jacket increases with the lateral strain of concrete because FRP does not
yield. The confining action in FRP-confined concrete is illustrated in Fig. 2.1. The
lateral confining pressure acting on the concrete core lσ is given by
2 hl
tDσσ = (2.1)
where hσ is the tensile stress in the FRP jacket in the hoop direction, t is the
thickness of the FRP jacket, and is the diameter of the confined concrete core. If
the FRP is loaded in hoop tension only, then the hoop stress in the FRP jacket
D
hσ is
proportional to the hoop strain hε due to the linearity of FRP and is given by
h frpE hσ ε= (2.2)
where frpE is the elastic modulus of FRP in the hoop direction.
The lateral confining pressure reaches its maximum value lf at the rupture of FRP,
with
,2 2h rup frp h rupl
t Ef
D D, tσ ε
= = (2.3)
where ,h rupσ and ,h rupε are the hoop stress and strain of FRP at rupture
respectively, which are generally not the same as the ultimate tensile strength and
the ultimate tensile strain of FRP obtained from material tests as discussed later in
this chapter.
2.2.2 Dilation Properties
12
It has long been accepted that under axial compression, unconfined concrete
experiences a volumetric compaction up to 90% of the peak stress. Thereafter the
concrete shows unstable volumetric dilation due to the rapidly increasing
lateral-to-axial strain ratio. However, this lateral dilation could be effectively
contained by the FRP jacket through the confining action explained in the previous
section. With the consideration of radial displacement compatibility, the lateral
dilation results in a continuously increasing lateral confining pressure provided by
the FRP jacket which gradually reduces the rate of the lateral dilation itself. In
summary, the dilation properties of FRP-confined concrete depend on both force
equilibrium and geometric compatibility, which explains the interaction between
the FRP jacket and the concrete core.
A number of studies have thus been carried out on the dilation properties of
FRP-confined concrete (e.g. Mirmiran and Shahawy 1997; Samaan et al. 1998;
Xiao and Wu 2000; Teng et al. 2007). The secant dilation ratio sµ is commonly
used to characterize the dilation properties. Here, sµ is defined as the absolute
value of the secant slope of the lateral-to-axial strain curve of FRP-confined
concrete and is given by
ls
c
εµε
= (2.4)
where lε and cε are the lateral strain and axial strain of concrete respectively. It
should be noted that lε and hε are equal in magnitude for circular sections. A
typical experimental secant dilation ratio-axial strain curve is shown in Fig. 2.2. At
the initial stage, the secant dilation ratio has a constant value equal to the Poisson’s
ratio, and then it gradually increases as the concrete core begins to dilate. The
confining action is simultaneously activated, especially after the compressive
strength of unconfined concrete is reached. As a result, the secant dilation ratio
eventually decreases due to this continuously increasing confining pressure.
2.2.3 Ultimate Condition
13
The ultimate condition of FRP-confined concrete refers to its compressive strength
and ultimate axial strain. It is evident that the ultimate condition is closely related to
the confining pressure provided by the FRP jacket when it ruptures. Early
researchers (e.g. Samaan et al. 1998; Saffi et al. 1999; Toutanji 1999) commonly
assumed that this confining pressure is equal to the tensile strength of the same FRP
material obtained from tensile coupon tests. However, later experimental evidence
suggested that the material tensile strength of FRP cannot be reached in
FRP-confined concrete as the hoop rupture strains of FRP measured in compression
tests on FRP-confined concrete have been found to be considerably smaller than
those obtained from material tensile tests (e.g. Xiao and Wu 2000; Shahawy et al.
2000, Lam and Teng 2004). The ratio of the FRP rupture strain to the ultimate
material tensile strain is important for a stress-strain model to produce satisfactory
results. Thus, several possible causes that may result in this phenomenon have been
proposed, including the non-uniform deformation of concrete, the effect of
curvature of an FRP jacket, local misalignment or waviness of fibers, residual
strains and a multi-axial stress state, and the existence of the overlapping zone (De
Lorenzis and Tepfers 2003; Lam and Teng 2004).
Lam and Teng (2004) carried out the first carefully designed tests to investigate
these possible causes. Three types of tests, namely, flat coupon tensile tests, ring
splitting tests, and FRP-confined concrete cylinder compression tests were
conducted and compared for both GFRP and CFRP. Based on the fact that the
average FRP hoop rupture strain in all specimens was closer to the ultimate tensile
strain measured from ring splitting tests than that from flat coupon tensile tests, it
was concluded that the curvature of the jacket is a factor that affects the FRP hoop
rupture strain. Another interesting phenomenon is that for the compression tests,
the FRP hoop strains obtained from the overlapping zone fell far below those
measured outside the overlapping zone. The fact that the confining pressure is
basically the same around the whole circumference whereas the FRP jacket is
thicker in the overlapping zone can give rise to this phenomenon. Therefore, it was
concluded that the existence of the overlapping zone is another fact that affects the
average FRP hoop rupture strain. Lastly, it was discovered that even for those
readings from the strain gauges outside the overlapping zone, they fluctuated
around the average value. It was believed that this was caused by the
14
non-uniformity of the deformation of concrete.
More recently, Bisby et al. (2007) extracted the strain readings of FRP jackets
wrapped around concrete cylinders using a digital image correlation technique.
They showed that the FRP hoop strains varied significantly along the specimen
height, even within the mid-height region. Based on the data obtained using their
novel technique, it was shown that the maximum FRP hoop strain could be very
close to that obtained from tensile coupon tests. In addition, in some cases, the
maximum reading was captured at a position close to but not exactly at the
mid-height of a specimen where strain gauges are commonly installed to record the
hoop strain data in most existing tests. As a result, it can be argued that the strain
capacity and thus the material strength of FRP composites may not be significantly
undermined when they are used to confine concrete. The large scatter in the FRP
strain reduction factor reported by existing tests is mainly due to the non-uniform
nature of concrete and the incomplete measurement of FRP strains.
2.2.4 Stress-Strain Curves
It has now become well-known that the stress-strain curve of FRP-confined
concrete features a monotonically ascending bi-linear shape with sharp softening in
a transition zone around the stress level of the unconfined concrete strength (Fig.
2.3a) if the amount of FRP exceeds a certain threshold value. This type of
stress-strain curves (the ascending type) was observed in the vast majority of
existing tests. With this type of stress-strain curves, both the compressive strength
and the ultimate axial strain are reached simultaneously and are significantly
enhanced over those of unconfined concrete. However, existing tests (e.g. Arie
2001; Xiao and Wu 2000, 2003; Harries 2003) have also shown that in some cases
such a bi-linear stress-strain curve cannot be expected. Instead, the stress-strain
curve features a post-peak descending branch and the compressive strength is
reached before the tensile rupture of the FRP jacket (the descending type). It should
be noted that this type of stress-strain curve may end at a stress value (axial stress at
ultimate axial strain) either larger or smaller than the compressive strength of
unconfined concrete, as illustrated in Figs 2.3b and 2.3c respectively.
15
2.2.5 Stress-Strain Models
A significant number of stress-strain models have been proposed for FRP-confined
concrete in circular columns. These models can be classified into two categories
(Teng and Lam 2004): (a) design-oriented models (e.g. Fardis and Khalili 1982;
Karbhari and Gao 1997; Samaan et al. 1998; Miyauchi et al. 1999; Saafi et al. 1999;
Toutanji 1999; Lillistone and Jolly 2000; Xiao and Wu 2000, 2003; Lam and Teng
2003a; Berthet et al. 2006; Harajli 2006; Saenz and Pantelides 2007; Wu et al. 2007;
Youssef et al. 2007), and (b) analysis-oriented models (e.g. Mirmiran and Shahawy
1997; Harmon et al. 1998; Spoelstra and Monti 1999; Fam and Rizkalla 2001; Chun
and Park 2002; Becque et al. 2003; Harries and Kharel 2002; Marques et al. 2004;
Binici 2005; Teng et al. 2007).
Models of the first category generally comprise a closed-form stress-strain equation
and ultimate condition equations derived directly from the interpretation of
experimental results. The accuracy of design-oriented models highly depends on
the definition of the ultimate condition of FRP-confined concrete. The simple form
of design-oriented models makes them convenient for design use. Existing
design-oriented models have been assessed by a number of studies (De Lorenzis
and Tepfers 2003; Teng and Lam 2004; Bisby et al. 2005). Design-oriented models
are dealt with in Chapter 4, with an emphasis on refining Lam and Teng’s (2003a)
model.
In models of the second category, the stress-strain curves of FRP-confined concrete
are generated using an incremental numerical procedure which accounts for the
interaction between the FRP jacket and the concrete core. The accuracy of
analysis-oriented models depends mainly on the modeling of the lateral-to-axial
strain relationship of FRP-confined concrete. Analysis-oriented models are more
suitable for incorporation in computer-based numerical analysis such as nonlinear
finite element analysis. Analysis-oriented models can also be used to produce
numerical results for the development of design-oriented stress-strain models.
Analysis-oriented stress-strain models, particularly those models that employ a
stress-strain model for actively-confined concrete as a base model are assessed in
Chapter 3.
16
2.2.6 Size Effect
It should be noted that the vast majority of existing tests were conducted on
small-scale specimens and these tests served as the experimental basis to develop
various stress-strain models. Therefore, the suitability of applying these
stress-strain models to large-scale columns is still not clear. Recently, a number of
experimental studies have been carried out on large-scale columns (Youssef 2003;
Carey and Harries 2005; Mattys et al. 2005; Rocca et al. 2006; Yeh and Chang
2007). These studies indicated that the behavior of realistically-sized circular
columns could be reasonably well extrapolated from small-scale tests.
2.3 FRP-CONFINED CONCRETE IN RECTANGULAR COLUMNS
UNDER CONCENTRIC COMPRESSION
2.3.1 Behavior
FRP-confined concrete in rectangular columns has also attracted tremendous
research interest recently since rectangular columns are more common in reality. It
is well-known that FRP confinement is less effective for rectangular columns,
because concrete in rectangular columns is non-uniformly confined. As a means to
enhance the confining effect and to reduce the detrimental effect on the tensile
strength of FRP, the sharp corners of rectangular columns should be rounded before
the wrapping of FRP, as shown in Fig. 2.4, in which and are side
dimensions and is the corner radius. However, due to the existence of internal
steel reinforcement, the radius of the corners is limited in practical applications.
b h
cr
A significant number of experimental studies have been carried out on rectangular
columns (e.g. Rochette and Labossiere 2000; Chaallal et al. 2003; Lam and Teng
2003b; Youssef et al. 2007; Wang and Wu 2008). Failure was generally observed to
occur at the corners by FRP tensile rupture, as shown in Fig. 2.5. Besides the
amount of FRP confinement, other key parameters investigated by these studies
include the aspect ratio of the cross section (ratio of the longer side to the shorter
17
side of the cross section) and the radius of the rounded corners. The aspect ratio
investigated generally varied from 1 to 2 while the corner radius was generally kept
at a small value. Recently, Wang and Wu (2008) experimentally investigated the
effect of the corner radius on the confining effect using a series of specimens with
different cross-section shapes varying from square to circular shapes. All these tests
clearly showed that the effectiveness of confinement increases as the amount of
FRP or the corner radius increases and as the aspect ratio of the section reduces. Fig.
2.6 shows two experimental stress-strain curves of FRP-confined concrete in
square columns corresponding to two different amounts of FRP confinement. For
ease of comparison, the axial stress cσ and the axial strain cε are normalized by
the compressive strength of unconfined concrete and its corresponding axial
strain
'cof
coε respectively. Of the two curves, the one for a smaller corner radius and a
smaller amount of confinement features a descending branch with little strength
enhancement. It should be noted that the average axial stress (load divided by
sectional area) is used in Fig. 2.6 due to the fact that the axial stress of concrete in an
FRP-confined rectangular column varies over the section as a result of the
non-uniform confinement.
2.3.2 Stress-Strain Models
Stress-strain models proposed for FRP-confined concrete in circular columns are
not directly applicable to FRP-confined concrete in rectangular columns due to the
non-uniformity of confinement in the latter. Many stress-strain models for
FRP-confined concrete in rectangular columns have been proposed (e.g. Lam and
Teng 2003b; Harajli 2006; Pantelides and Yan 2007; Wu et al. 2007; Youssef et al.
2007). Most of these models are design-oriented, aiming to predict the average
stress-strain curves. Based on the experimental observation that the stress-strain
curves of FRP-confined concrete in rectangular columns feature a very similar
shape to that of the stress-strain curves of its counterpart in circular columns, many
researchers extended their stress-strain models for FRP-confined concrete in
circular columns to FRP-confined concrete in rectangular columns with the original
ultimate condition equations modified to account for the non-uniform confining
effect. The revised ultimate condition equations are generally based on the concepts
18
of effective confinement area and equivalent circular section to transform a
rectangular section into an equivalent circular section so that FRP-confined
concrete in circular columns and rectangular columns can be treated in a unified
manner.
Lam and Teng’s (2003b) model is used as an example herein to illustrate the
commonly accepted approach. This model is an extension of the model proposed by
the same authors (Lam and Teng 2003a) for FRP-confined concrete in circular
columns. In Lam and Teng’s (2003b) model, the effective confinement area is
contained by four parabolas as illustrated in Fig. 2.7, with the initial slopes of the
parabolas being the same as the adjacent diagonal lines. It is assumed that within
the effective confinement area, the distribution of the axial stresses is uniform and
the magnitude of the axial stresses is the average stress of the section being sought.
It is further assumed that the stress state in the effective confinement area is the
same as that in an equivalent circular section. In Lam and Teng’s (2003b) model,
the equivalent circular section circumscribes the original rectangular section (Fig.
2.7).
It is not inaccurate to say that a commonly accepted model has not yet been
identified. This is mainly due to the common and fundamental drawback of all
stress-strain models of this type: all these models have not been based on a rigorous
understanding of the confinement mechanism in rectangular sections. Instead, they
have been based on assumptions made to provide a good fit to the available test data.
As a result, the accuracy of a particular model based on a set of test data in
predicting another set of test data cannot be ensured.
Finite element models are potentially capable of accurately capturing the complex
stress variations in FRP-confined concrete in rectangular columns. Therefore, finite
element modelling offers a powerful tool for studying the confinement mechanism
of FRP-confined concrete. However, the reliability of previous finite element
studies (e.g. Mirmiran et al. 2000; Parvin and Wang 2001) is uncertain due to the
lack of in-depth studies on the constitutive model of concrete and the lack of test
results to verify the finite element model in terms of the distributions of axial stress
over the section and confining pressure around the section perimeter. Yu (2007)
19
recently conducted a critical assessment of existing Drucker-Prager type plasticity
models and proposed a plastic-damage model for concrete based on this assessment.
This plastic-damage model has led to close predictions for concrete in a number of
different stress states, including actively-confined concrete, concrete under biaxial
compression, and FRP-confined concrete in both circular and annular small-scale
columns. This plastic-damage model however needs to be further verified against
tests results of FRP-confined rectangular concrete columns in terms of both the
general behavior (i.e. average stress-strain behavior and lateral expansion behavior),
the axial stress distribution over the section, and the confining stress distribution
around the perimeter. The absence of existing test results of stress variation in
rectangular columns is mainly due to the difficulty in the measurement method. The
emerging technique of pressure films represents a significant development in the
measurement method. These pressure films can record the pressures between two
contacting surfaces and thus capture the stress variation. These films were used by
Yang et al. (2004) to record the pressures between an FRP strip and a steel substrate
with different corner radii to investigate how the distribution of the contacting
stresses vary with the corner radius, however, this technique has not been used in
any experimental work on FRP-confined concrete in the open literature.
2.3.3 Shape Modification
It has been noted that the confining effectiveness of FRP can be significantly
enhanced in rectangular columns if shape modification is implemented before FRP
jacketing (Teng et al. 2002). Among all the possible schemes of shape modification,
the most attractive one is the natural modification of a rectangular section into an
elliptical section. However, FRP-confined concrete in elliptical columns has
received much less research attention so far (Teng and Lam 2002; Yan et al. 2007;
Pantelides and Yan 2007). Yan et al. (2007) and Pantelides and Yan (2007) were
concerned with post-tensioned FRP-confined columns which involved the
application of expansive concrete. Teng and Lam (2002) have presented the only
study concerning the stress-strain behavior of and providing a strength model for
FRP-confined normal strength concrete in elliptical columns, as can be found in
open literature.
20
2.3.4 Size Effect
Again, the existing stress-strain models are mainly based on small-scale tests. Only
a limited number of studies investigated the behavior of large-scale rectangular
columns (Youssef 2003; Rocca et al. 2006). Youssef (2003) did not explicitly
address size effect in rectangular columns. Rocca et al. (2006) stated that it was
difficult to identify size effect in rectangular columns because of the large scatter of
their test results. Therefore, the size effect in rectangular columns is yet to be
clarified.
2.4 FRP-CONFINED CONCRETE UNDER ECCENTRIC
COMPRESSION
It should be noted that the experimental studies as well as the stress-strain models
reviewed so far are only concerned with FRP-confined concrete in columns
subjected to concentric compression. An important issue that needs to be clarified is
whether these models based on concentric compression tests are directly applicable
in the analysis of columns subjected to combined bending and axial compression.
Most existing studies (e.g. Saadatmanesh et al. 1994; Monti et al. 2001; Yuan and
Mirmiran 2001; Cheng et al. 2002; Teng et al. 2002; Binici 2008; Yuan et al. 2008)
adopted the conventional section analysis approach with the assumption that the
stress profile of FRP-confined concrete in the compression zone of a eccentrically
loaded section can be described using the stress-strain curve obtained from
concentric compression tests, although this assumption has not been experimentally
validated due to the scarcity of such test data and the limitations in the measurement
methods. On the experimental side, Fam et al. (2003) tested eccentrically loaded
circular concrete-filled FRP tubes subjected to a broad range of eccentricities. They
showed that the above assumption led to reasonable prediction of the strength of the
tubes they tested. Hadi (2006) tested five small-scale (150 mm in diameter) circular
normal strength concrete columns wrapped with CFRP and subjected to a fixed end
eccentricity of 42.5 mm. Four specimens were not provided with internal steel
reinforcement so they failed by the cracking of concrete on the tensile face of these
columns. Unfortunately, the hoop strains of the FRP jacket were not reported which
21
makes it difficult to use the test results to verify the above approach. More recently,
Ranger and Bisby (2007) conducted tests on small-scale (152 mm in diameter)
FRP-confined circular RC columns with a fixed height-to-diameter ratio of four.
The load eccentricities covered by their tests ranged from 0 mm to 40 mm. These
specimens were subjected to the coupled effects of load eccentricity and
slenderness and the authors did not compare their experimental results with
analytical predictions to validate the above approach. On the theoretical side, Binici
and Mosalam (2007) proposed an analytical model capable of simulating the
non-uniform hoop strain distribution in the FRP jacket wrapped around circular
columns due to eccentric compression. This model is not subjected to the
assumption given above, but the accuracy of this model depends on a number of
parameters such as the thickness of the adhesive layer which is difficult to
determine. Later, the same researcher (Binici 2008) adopted the above assumption
to perform section analysis of FRP-confined RC columns using a simple bi-linear
stress-strain model for FRP-confined concrete and showed that their section
analysis predicted the test results of Sheikh and Yau (2002) reasonably accurately.
It should be noted that Sheikh and Yau’s (2002) tests were on FRP-confined
circular RC columns subjected to constant axial loading and cyclic lateral loading,
but not on columns subjected to eccentric loading. In summary, it is deemed to be
reasonable to adopt this assumption for circular columns based on the evidence
gained so far, although more tests are needed to fully verify this assumption.
The effect of load eccentricity on the stress-strain behavior of FRP-confined
concrete in rectangular columns is uncertain. Indeed, even the stress-strain behavior
of FRP-confined concrete subjected to concentric compression needs much further
work. There is very limited experimental evidence in this area. Parvin and Wang
(2001) tested 4 small-scale square columns with small load eccentricities while Cao
et al. (2006) tested 5 medium-scale (250mm in side dimension) square columns
with various load eccentricities. These studies were concerned with overall column
behavior, so they did not clarify the effect of load eccentricity on the stress-strain
behavior of the confined concrete. In particular, a rectangular column subjected to
eccentric compression may be bent about either the major axis or the minor axis. It
is unlikely that the same stress-strain model developed from studies on
FRP-confined concrete columns under concentric compression can be used for
22
eccentric loading in both directions. Indeed, such a stress-strain model may not be
applicable to eccentric loading in either direction.
Again, finite element analysis has the potential to simulate the behavior of
FRP-confined concrete subjected to eccentric compression, both in circular and
rectangular columns. However, due to the same reasons as given earlier, existing
finite element models (e.g. Mirmiran et al. 2000; Parvin and Wang 2001; Yu 2007)
have not proven their applicability to the problem under consideration. More tests
with advanced measuring techniques to capture the key results (the distributions of
axial stress over the section and confining pressure around the section perimeter)
are needed for the verification of existing models or the development of a more
convincing finite element model.
2.5 FRP-CONFINED RC COLUMNS
2.5.1 General
Stress-strain models as discussed above are needed in the section analysis of
FRP-confined RC columns which forms an important part of a design procedure for
such columns. Relevant design provisions are now available in existing design
guidelines (fib 2001; ISIS 2001; ACI-440.2R 2002, 2008; JSCE 2002;
CNR-DT200 2004; Concrete Society 2004) for FRP-strengthened RC structures.
However, these provisions are generally concerned with the design of
concentrically loaded short FRP-confined RC columns due to the fact that the effect
of eccentricity on confinement effectiveness and the slenderness effect have not
been well understood. The behavior of FRP-confined RC columns and the relevant
provisions given in the above-mentioned design guidelines are reviewed in this
section.
2.5.2 Short Columns
A short column refers to a column with a negligible slenderness effect. The strength
design of such columns is simply a matter of constructing the axial load-bending
moment interaction diagram for the critical column section using a proper
23
stress-strain model for FRP-confined concrete. However, as the effects of load
eccentricity and size on the stress-strain behavior of FRP-confined concrete are still
uncertain, particularly for concrete in rectangular columns, there are a number of
limitations in existing design guidelines. For circular columns, only Concrete
Society (2004) and ACI-440.2R (2008) give information on the design of
eccentrically loaded columns; other guidelines are only concerned with the design
of concentrically loaded columns. For rectangular columns, only ACI-440.2R
(2008) recommends a procedure to perform section analysis; all the other design
guidelines are only applicable to columns subjected to concentric compression. In
addition, Concrete Society (2004) limits the maximum dimension of a rectangular
section to 250mm. Finally, only ISIS (2001) provides an expression to differentiate
short columns from slender columns. It should be noted that this expression is only
intended for columns with no significant bending (i.e. concentric compression or
slightly eccentric compression). Therefore, designers are faced with a number of
difficulties even when they attempt a design of FRP jackets to strengthen short RC
columns.
2.5.3 Slender Columns
No existing design guidelines have included information for the design of slender
FRP-confined RC columns. This is mainly due to the fact that only a very limited
number of studies have investigated the behavior of slender FRP-confined RC
columns, subjected to either concentric or eccentric loading (Mirmiran et al. 2001;
Yuan and Mirmiran 2001; Tao et al. 2004; Fitzwilliam and Bisby 2006). These
studies have all been concerned with small-scale circular columns.
Mirmiran et al. (2001) tested concentrically loaded fixed-fixed concrete-filled FRP
tubes with length-to-diameter ratios ranging from 2.1 to 18.6. Their test results
showed that these hybrid columns had a more dramatic loss in axial load capacity
than conventional RC columns as the slenderness increased. Yuan and Mirmiran
(2001) further developed an analytical model for such columns subjected to
eccentric compression from which they developed the slenderness limit to
distinguish short columns from slender columns and an approximate expression for
the section flexural stiffness to be used in the moment magnifier method for the
24
design of such columns classified as slender columns. These two studies are beyond
the scope of the present study since the FRP tubes used in these two studies had a
significant longitudinal stiffness.
More recently, Tao et al. (2004) and Fitzwilliam and Bisby (2006) carried out tests
on slender FRP-confined circular RC columns. The columns tested by Tao et al.
(2004) were hinged at both ends, 150mm in diameter and had length-to-diameter
ratios of either 8.4 or 20.4. The nominal end eccentricity of the columns varied from
0 to 150mm. The fibers in the FRP jacket were oriented only in the hoop direction.
In Fitzwilliam and Bisby’s (2006) test series, the columns were 152mm in diameter
and had length-to-diameter ratios up to 8. Some columns received longitudinal FRP
wrapping before hoop FRP wrapping. Both of the above two studies found that
FRP-confined RC columns experienced a larger loss in the axial load capacity than
the corresponding RC columns.
Despite the limited existing research, an important conclusion of the existing work
is that FRP-confined RC columns are subjected to a more profound slenderness
effect than their RC counterparts because FRP confinement can lead to a large
increase in the axial load capacity of an RC section but very limited increase in the
flexural rigidity of the RC section.
2.6 ANALYTICAL AND DESIGN METHODS FOR RC COLUMNS
2.6.1 General
The above review has suggested that a rational design procedure for FRP-confined
RC columns is urgently needed for inclusion in future design guidelines or design
codes. This is particularly true for slender columns where the slenderness effect
must be taken into account. The design of FRP-confined RC columns should follow
the general design procedure for RC columns that is familiar to engineers, so the
current methods dealing with the slenderness effect in RC columns adopted in
existing design codes for RC structures (e.g. ENV-1992-1-1 1992; BS-8110 1997;
GB-50010 2002; ACI-318 2005) are briefly reviewed in this section. Existing
analytical methods for slender RC columns which serve as the analytical basis to
25
develop design equations are also briefly examined.
2.6.2 Analytical Methods
Various analytical methods have been proposed for the analysis of slender columns,
as can be found in many textbooks (e.g. Chen and Atsuta 1976). Despite their
different levels of sophistication, they all aim to predict the lateral deflections of
slender columns, which give rise to the second-order moments. Two
well-established analytical methods are briefly reviewed here.
The first one is only applicable to hinged columns with equal end eccentricities. In
this method, the deflected shape of a column is assumed to be a half sine wave and
equilibrium is only checked at the critical section (the section at the mid-height of
the column) where the maximum lateral deflection of the column takes place. This
method has been widely adopted in the analysis of hinged RC columns with equal
end eccentricities (e.g. Bazant et al. 1991) and has been proven to be very
successful. More details of this method are given in Chapter 9.
The second method is more sophisticated and more versatile than the first one. This
method is generally known as the numerical integration method, in which a column
is divided into a reasonable number of segments and the lateral displacement at
each grid point is found from numerical integration by making use of the axial
load-bending moment-curvature relationship of the column. This method allows
the end eccentricities to be unequal and allows the presence of end restraints. This
method was originally proposed by Newmark (1943) and has been widely adopted
in the analysis of RC columns (e.g. Pfrang and Siess 1961; Cranston 1972), steel
columns (e.g. Shen and Lu 1983) and composite columns (e.g. Choo et al. 2006;
Tikka and Mirza 2006). More details of this method are given in Chapter 8.
2.6.3 Design Methods
The current design approach for RC columns is based on the rationale that a column
with end restraints and unequal end eccentricities can be replaced by an equivalent
hinged column with equal end eccentricities (the rationale of the current design
26
approach is explained in detail through the analysis of elastic columns presented in
Chapter 6). As a result, the design equations are for the equivalent hinged column
with the effects of the end restraints and unequal end eccentricities being dealt with
using the effective length factor and the equivalent moment factor.
There are two major methods in the current design codes for RC structures, namely,
the moment magnifier method and the nominal curvature method. Both methods
approximate the second-order moments by an amplification of the first-order
moments in order to make use of the section strength in design. In other words, the
current design approach transforms the design of a slender column into the design
of a section with an equivalent eccentricity consisting of the end eccentricity of the
slender column and a nominal lateral displacement of the critical section. Therefore,
the key in the current design approach is how to evaluate the lateral displacement of
the critical section.
The moment magnifier method is primarily used in North America (e.g. ACI-318
2005). This method originated from the analysis of elastic columns, where the
lateral deflections can be exactly evaluated provided the section flexural stiffness is
known. When this method is used for the design of RC columns, the key is to find
the equivalent section flexural stiffness that explains the effect of the nonlinearity in
material properties. The nominal curvature method is mainly used in Europe and
also China (e.g. ENV-1992-1-1 1992; BS-8110 1997; GB-50010 2002). The
nominal curvature method was originally proposed by Aas-Jakobsen and
Aas-Jakobsen (1968). This method relates the lateral displacement of the critical
section to the curvature of that section. This method is discussed in greater detail in
Chapter 8.
2.7 CONCLUDING REMARKS
This chapter has presented a review of existing studies on or relevant to
FRP-confined concrete and FRP-confined RC columns. This review has indicated
that the stress-strain behavior of FRP-confined concrete in rectangular columns,
under either concentric or eccentric compression, is still highly uncertain.
Therefore, it seems too early at this stage to carry out detailed theoretical studies on
27
the behavior of FRP-confined rectangular RC columns to develop reliable design
equations for such columns. On the other hand, the stress-strain behavior of
FRP-confined concrete in circular columns is much better understood. Existing
research suggests that the stress-strain behavior of circular columns is not
significantly affected by the possible effects of load eccentricity and size, although
more experimental evidence is still needed. The work presented in the remaining
chapters of this thesis is thus limited to circular columns. Nevertheless, the general
framework presented in this thesis for circular columns can be readily extended to
rectangular columns once an accurate stress-strain model for FRP-confined
concrete in rectangular columns becomes available.
28
2.8 REFERENCES
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29
BS 8110 (1997). Structural Use of Concrete, Part 1. Code of Practice for Design and Construction, British Standards Institution, London, U.K.
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Cheng, H.L., Sotelino, E.D. and Chen, W.F. (2002). “Strength estimation for FRP wrapped reinforced concrete columns”, Steel and Composite Structures, 2(1), 1-20.
Choo, C.C., Harik, I.E. and Gesund, H. (2006). “Strength of rectangular concrete columns reinforced with fiber-reinforced polymer bars”, ACI Structural Journal, 103(3), 452-459.
Chun, S.S. and Park, H.C. (2002). “Load carrying capacity and ductility of RC columns confined by carbon fiber reinforced polymer” Proceedings, 3rd International Conference on Composites in Infrastructure (CD-Rom), San Francisco.
CNR-DT200 (2004), Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Existing Structures, Advisory Committee on Technical Recommendations For Construction, National Research Council, Rome, Italy.
Concrete Society (2004). Design Guidance for Strengthening Concrete Structures with Fibre Composite Materials, Second Edition, Concrete Society Technical Report No. 55, Crowthorne, Berkshire, UK.
Cranston, W.B. (1972). Analysis and Design of Reinforced Concrete Columns, Research Report 20, Cement and Concrete Association, UK.
De Lorenzis, L. and Tepfers, R. (2003). “Comparative study of models on confinement of concrete cylinders with fiber-reinforced polymer composites”, Journal of Composites for Construction, ASCE, 7(3), 219-237.
ENV 1992-1-1 (1992). Eurocode 2: Design of Concrete Structures – Part 1: General Rules and Rules for Buildings, European Committee for Standardization, Brussels.
30
Fam, A.Z. and Rizkalla, S.H. (2001). “Confinement model for axially loaded concrete confined by circular fiber-reinforced polymer tubes”, ACI Structural Journal, 98(4), 451-461.
Fam, A., Flisak, B. and Rizkalla, S. (2003). “Experimental and analytical modeling of concrete-filled fiber-reinforced polymer tubes subjected to combined bending and axial loads”, ACI Structural Journal, 100(4), 499-509.
Fardis, M.N. and Khalili, H. (1982). “FRP-encased concrete as a structural material”, Magazine of Concrete Research, 34(122), 191-202.
fib (2001). Externally Bonded FRP Reinforcement for RC Structures, The International Federation for Structural Concrete, Lausanne, Switzerland.
Fitzwilliam, J. and Bisby, L.A. (2006). “Slenderness effects on circular FRP-wrapped reinforced concrete columns”, Proceedings, 3rd International Conference on FRP Composites in Civil Engineering, December 13-15, Miami, Florida, USA, 499-502.
GB-50010 (2002). Code for Design of Concrete Structures, China Architecture and Building Press, China.
Hadi, M.N.S. (2006). “Behaviour of wrapped normal strength concrete columns under eccentric loading”, Composite Structures, 72(4), 503-511.
Harajli, M.H. (2006). “Axial stress-strain relationship for FRP confined circular and rectangular concrete columns”, Cement & Concrete Composites, 28(10), 938-948.
Harmon, T.G., Ramakrishnan, S. and Wang, E.H. (1998). “Confined concrete subjected to uniaxial monotonic loading.” Journal of Engineering Mechanics, ASCE, 124(12), 1303–1308.
Harries, K.A. and Kharel, G. (2002). “Behavior and modeling of concrete subject to variable confining pressure”, ACI Materials Journal, 99(2), 180-189.
Harries, K.A. and Kharel, G. (2003). “Experimental investigation of the behavior of variably confined concrete”, Cement and Concrete Research, 33(6), 873-880.
ISIS (2001). Design Manual No. 4: Strengthening Reinforced Concrete Structures with Externally-Bonded Fibre Reinforced Polymers, Intelligent Sensing for Innovative Structures, Canada.
JSCE (2001). Recommendations for Upgrading of Concrete Structures with Use of Continuous Fiber Sheets, Concrete Engineering Series 41, Japan Society of Civil Engineers, Tokyo, Japan.
Karbhari, V.M. and Gao, Y. (1997). “Composite jacketed concrete under uniaxial compression–verification of simple design equations”, Journal of Materials in Civil Engineering, ASCE, 9(4), 185-193.
31
Lam, L. and Teng, J.G. (2003a). “Design-oriented stress-strain model for FRP-confined concrete”, Construction and Building Materials,17 (6-7), 471-489.
Lam, L. and Teng, J.G. (2003b). “Design-oriented stress-strain model for FRP-confined concrete in rectangular columns”, Journal of Reinforced Plastics and Composites, 22 (13), 1149-1186.
Lam, L. and Teng, J.G. (2004). “Ultimate condition of fiber reinforced polymer-confined concrete”, Journal of Composites for Construction, ASCE, 8(6), 539-548.
Lillistone, D. and Jolly, C.K. (2000). “An innovative form of reinforcement for concrete columns using advanced composites”, The Structural Engineer, 78(23/24), 20-28.
Marques, S.P.C., Marques, D.C.S.C., da Silva J.L. and Cavalcante, M.A.A. (2004). “Model for analysis of short columns of concrete confined by fiber-reinforced polymer”, Journal of Composites for Construction, ASCE, 8(4), 332-340.
Mattys, S., Toutanji, H., Audenaert, K. and Taerwe, L. (2005). “Axial behavior of large-scale columns confined with fiber-reinforced polymer composites”, ACI Structural Journal, 102(2), 258-267.
Mirmiran, A. and Shahawy, M. (1997). “Dilation characteristics of confined concrete”, Mechanics of Cohesive-Frictional Materials, 2 (3), 237-249.
Mirmiran, A., Zagers, K. and Yuan, W.Q. (2000). “Nonlinear finite element modeling of concrete confined by fiber composites”, Finite Elements in Analysis and Design, 35, 79-96.
Mirmiran, A., Shahawy, M. and Beitleman, T. (2001). “Slenderness limit for hybrid FRP-concrete columns”, Journal of Composites for Construction, ASCE, 5(1), 26-34.
Miyauchi, K., Inoue, S., Kuroda, T. and Kobayashi, (1999). “Strengthening effects of concrete columns with carbon fiber sheet”, Transactions of the Japan Concrete Institute, 21, 143-150.
Monti, G., Nistico, N. and Santini, S. (2001). “Design of FRP jackets for upgrade of circular bridge piers”, Journal of Composites for Construction, ASCE, 5(2), 94-101.
Mosalam, K.M., Talaat, M. and Binici, B. (2007). “A computational model for reinforced concrete members confined with fiber reinforced polymer lamina: implementation and experimental validation”, Composites Part B: Engineering, 38, 598-613.
Newmark, N.M. (1943). “Numerical rocedure for computing deflections, moments, and buckling loads”, ASCE Transactions, 108, 1161-1234.
32
Pantelides, C.P. and Yan, Z.H. (2007). “Confinement model of concrete with externally bonded FRP jackets or posttensioned FRP shells”, Journal of Structrual Engineering, ASCE, 133(9), 1288-1296.
Parvin, A. and Wang, W. (2001). “Behavior of FRP jacketed concrete columns under eccentric loading”, Journal of Composites in Construction, ASCE, 5(3), 146-152.
Pfrang, E.O. and Siess, C.P. (1961). Analytical Study of the Behavior of Long Restrained Reinforced Concrete Columns Subjected to Eccentric Loads, Structural Research Series No. 214, University of Illinois, Urbana, Illinos.
Ranger, M. and Bisby, L.A. (2007). “Effects of load eccentricities on circular FRP-confined reinforced concrete columns”, Proceedings, 8th International Symposium on Fiber Reinforced Polymer Reinforcement for Concrete Structures (FRPRCS-8), University of Patras, Patras, Greece, July 16-18, 2007.
Rocca, S., Galati, N. and Nanni, A. (2006). “Large-size reinforced concrete columns strengthened with carbon FRP: experimental evaluation”, Proceedings, 3rd International Conference on FRP Composites in Civil Engineering, December 13-15 2006, Miami, Florida, USA.
Rochette, P. and Labossiere, P. (2000). “Axial testing of rectangular column models confined with composites”, Journal of composites for Construction, ASCE, 4(3), 129-136.
Saadatmanesh, H., Ehsani, M.R. and Li, M.W. (1994). “Strength and ductility of concrete columns externally reinforced with fiber composites straps”, ACI Structural Journal, 91(4), 434-447.
Saafi, M., Toutanji, H.A. and Li, Z. (1999). “Behavior of concrete columns confined with fiber reinforced polymer tubes”, ACI Materials Journal, 96(4), 500-509.
Saenz, N. and Pantelides, C.P. (2007). “Strain-based confinement model for FRP-confined concrete”, Journal of Structural Engineering, ASCE, 133 (6), 825-833.
Samaan, M., Mirmiran, A. and Shahawy, M. (1998). “Model of concrete confined by fiber composite”, Journal of Structural Engineering, ASCE, 124(9):, 1025-1031.
Shahawy, M., Mirmiran, A. and Beitelman, T. (2000). “Tests and modeling of carbon-wrapped concrete columns”, Composites Part B-Engineering, 31(6-7), 471-480.
Sheikh, S.A. and Yau, G. (2002). “Seismic behavior of concrete columns confined with steel and fiber-reinforced polymers”, ACI Structural Journal, 99(1), 72-80.
Shen, Z.Y. and Lu, L.W. (1983). “Analysis of initially crooked, end restrained steel columns”, Journal of Constructional steel research, 3(1), 10-18.
33
Spoelstra, M.R. and Monti, G. (1999). “FRP-confined concrete model”, Journal of Composites for Construction, ASCE, 3(3), 143-150.
Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, 2nd International Conference on Advanced Polymer Composites in Construction, University of Surrey, UK, 20-22 April 2004, 203-212.
Teng, J.G. and Lam, L. (2002). “Compressive behavior of carbon fiber reinforced polymer-confined concrete in elliptical columns”, Journal of Structural Engineering, ASCE, 128(12), 1535-1543.
Teng, J.G. and Lam, L. (2004). “Behavior and modeling of fiber reinforced polymer-confined concrete”, Journal of Structural Engineering, ASCE, 130(11), 1713-1723.
Teng, J.G., Chen, J.F., Smith, S.T. and Lam. L. (2002) FRP-Strengthened RC Structures, John Wiley and Sons, Inc., UK.
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Tikka, T.M. and Mirza, S.A. (2006). “Nonlinear equation for flexural stiffness of slender composite columns in major axis bending”, Journal of Structural Engineering, ASCE, 132(3), 387-399.
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Wang, L.M. and Wu, Y.F. (2008). “Effect of corner radius on the performance of CFRP-confined square concrete columns: Test”, Engineering Structures, 30(2), 493-505.
Wu, G., Wu, Z.S. and Lu, Z.T. (2007). “Design-oriented stress-strain model for concrete prisms confined with FRP composites”, Construction and Building Materials, 21(5), 1107-1121.
Xiao, Y. and Wu, H. (2000), “Compressive behavior of concrete confined by carbon fiber composite jackets”, Journal of Materials in Civil Engineering, ASCE, 12(2), 139-146.
Xiao, Y. and Wu, H. (2003), “Compressive behavior of concrete confined by various types of FRP composite jackets’, Journal of Reinforced Plastics and Composites, 22(13), 1187-1201.
Yan, Z.H. and Pantelides, C.P. and Reaveley, L.D. (2007). “Posttensioned FRP composite shells for concrete confinement”, Journal of Composites for Construction, ASCE, 11(1), 81-90.
34
Yeh, F.Y. and Chang, K.C. (2007). "Confinement efficiency and size effect of FRP confined circular concrete columns", Structural Engineering and Mechanics, 26(2), 127-150.
Youssef, M.N. (2003). Stress-strain Model for Concrete Confined by FRP Composites, Ph.D. Dissertation, University of California, Irvine.
Youssef, M.N., Feng, M.Q. and Mosallam, A.S. (2007). “Stress-strain model for concrete confined by FRP composites”, Composites Part B – Engineering, 38(5-6), 614-628.
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35
t
frp hE tε frp hE tε
lσ
D
Fig. 2.1 Confining action of FRP jacket
0 0.005 0.01 0.015 0.02 0.025 0.030
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Axial Strain εc
Sec
ant D
ilatio
n R
atio
µs
Fig. 2.2 Typical secant dilation ratio curve
36
cuεAxial Strain εc ε
'ccf
Axi
al S
tress
σc
'cof
(a) Ascending type
cuεccε
'cuf
'ccf
'cof
Axi
al S
tress
σc
Axial Strain εc
(b) Descending type with ' 'cu cof f≥
cuε
'cof
'ccf
'cuf
ccε
Axi
al S
tress
σc
Axial Strain εc
(c) Descending type with ' 'cu cof f<
Fig. 2.3 Classification of stress-strain curves of FRP-confined concrete
37
h
b
cr
Effective confinement area
Fig. 2.4 Effective confinement area of a rectangular section
(a) Square column (b) Rectangular column
Fig. 2.5 Failure of FRP-confined square and rectangular concrete columns with
rounded corners by FRP rupture
38
0 1 2 3 4 5 6 7 80
0.5
1
1.5
2
2.5
Normalized Axial Strain εc/εco
Nor
mal
ized
Axi
al S
tress
σc/
f ′ co
Fig. 2.6 Typical stress-strain curves of FRP-confined concrete in square columns
h
( )arctan b h
b
cr
Effective confinement area
D
Fig. 2.7 Lam and Teng’s model for FRP-confined concrete in rectangular columns
39
CHAPTER 3
ANALYSIS-ORIENTED STRESS-STRAIN MODELS
FOR FRP-CONFINED CONCRETE
3.1 INTRODUCTION
As reviewed in Chapter 2, existing stress-strain models fall into two main
categories: (1) design-oriented models and (2) analysis-oriented models. The
former models are generally in closed-form equations directly derived from test
results, treating FRP-confined concrete as a single “composite” material, and are
thus simple and convenient to apply in design. By contrast, the latter models treat
the FRP jacket and the concrete core separately, and predict the behavior of
FRP-confined concrete by an explicit account of the interaction between the FRP
jacket and the confined concrete core via radial displacement compatibility and
equilibrium conditions. Analysis-oriented models are more versatile and accurate
in general, are often the preferred choice for use in more involved analysis than is
required in design (e.g. nonlinear finite element analysis of concrete structures
with FRP confinement), and are applicable/easily extendible to concrete confined
with materials other than FRP. They can also be employed to generate numerical
results for use in the development of a design-oriented model. This chapter deals
with analysis-oriented models; design-oriented models are discussed in Chapter 4.
The confinement provided by an FRP jacket to a concrete core is passive rather
than active, as the confining pressure from the jacket is induced by and increases
with the expansion of the concrete core. In most existing analysis-oriented models
for FRP-confined concrete, a theoretical model for actively-confined concrete (i.e.
the confining pressure is externally applied and remains constant as the axial
40
stress increases), which is referred to as an active-confinement model for brevity
hereafter, is employed as the base model; the axial stress-axial strain curve
(simply referred to as the axial stress-strain curve or the stress-strain curve
hereafter) of FRP-confined concrete is then generated through an incremental
process, with the resulting stress-strain curve crossing a family of stress-strain
curves for the same concrete under different levels of active confinement (Teng
and Lam 2004). Models of Mirmiran and Shahawy (1996, 1997a), Spoelstra and
Monti (1999), Fam and Rizkalla (2001), Chun and Park (2002), Harries and
Kharel (2002), Marques et al. (2004), Binici (2005) and Teng et al. (2007a) are all
of this type. Two other models (Harmon et al. 1998; Becque et al. 2003) adopted
alternative approaches for modelling the concrete (Teng and Lam 2004). The first
approach, in which an active-confinement model is used, has been much more
popular than the other approaches as it leads to conceptually simple yet effective
models. This chapter is limited to analysis-oriented models developed through this
approach. For a brief discussion of the models by Harmon et al. (1998) and
Becque et al. (2003), the reader is referred to Teng and Lam (2004).
It should be noted that among the models with an active-confinement base model,
the models of Spoelstra and Monti (1999), Fam and Rizkalla (2001), Chun and
Park (2002), and Harries and Kharel (2002) have been briefly summarized and
assessed through comparisons with test stress-strain curves of FRP-confined
concrete with a significant level of confinement (dependent on the hoop
membrane stiffness of the FRP jacket) so that the stress-strain curves are
monotonically ascending. The present chapter extends the work of Teng and Lam
(2004) in the following aspects:
(a) apart from the four models mentioned above, the model of Mirmiran and
Shahawy (1997a) together with three more recent models (Marques et al. 2004;
Binici 2005 and Teng et al. 2007a) are also included in the critical review and
assessment;
(b) a much more thorough assessment is presented for all these models by
considering different levels of confinement covering both the ascending and
descending types of stress-strain curves, with the emphasis being on the accuracy
41
of the key elements of such models, including the lateral-to-axial strain
relationship for FRP-confined concrete, as well as the stress-strain equation and
the peak axial stress point of the active-confinement base model;
(c) new results from recent tests conducted by the author are employed in the
assessment and some of these tests are for FRP-confined concrete with a
descending stress-strain curve (i.e. weakly-confined concrete) which is less well
understood and for concrete with very strong FRP confinement which has
received little attention in previous work.
(d) the lateral-to-axial strain relationship, which takes various forms and is
essential to this type of models (Teng and Lam 2004), is thoroughly examined;
and
(e) finally, a refined version of Teng et al.’s (2007a) model is presented to provide
more accurate predictions, particularly for weakly-confined concrete.
3.2 TEST DATABASE
3.2.1 General
A test database, containing the results of axial compression tests of 48
FRP-confined concrete cylinders (diameter 152D = mm and height
mm), is employed herein to evaluate the performance of existing analysis-oriented
stress-strain models. All these tests have recently been conducted at The Hong
Kong Polytechnic University, so the tests were conducted under standardized
conditions and all information required for evaluating stress-strain models can be
readily and accurately extracted.
305H =
Among these 48 tests, 25 of them have been published [i.e. specimens 01 to 13 in
Lam and Teng (2004); specimens 14 to 19 in Lam et al. (2006) and Specimens 20
to 25 in Teng et al. (2007b)] and were used in developing Teng et al.’s (2007a)
model, the other 23 tests are new tests that have never been published or used in
developing any of the existing stress-strain models. The new tests significantly
42
widened the range of confinement ratios from 0.08-0.46 for the 25 published tests
to 0.07-0.99 for the 48 tests considered in this chapter; some of the test
stress-strain curves feature a descending branch while others are rapidly ascending.
The confinement ratio 'l cof f refers to the ratio of the confinement pressure lf
at jacket rupture to the compressive strength of unconfined concrete 'cof , with
,2 frp h rupl
E tf
Dε
= (3.1)
where frpE and are the elastic modulus and the thickness of FRP jacket
respectively, and
t
,h rupε is the hoop rupture strain of FRP jacket. In the present
database, the maximum increase in concrete strength as a result of FRP
confinement is about 320%. Further details of this test database are available in
Table 3.1.
For ease of discussion, the terms “weakly-confined concrete”,
“moderately-confined concrete” and “heavily-confined concrete” are used herein.
Weakly-confined concrete refers to concrete whose stress-strain curves feature a
descending branch. Moderately-confined concrete and heavily-confined concrete
both refer to concrete whose stress-strain curves are of the bi-linear ascending
type. The latter two are differentiated using the 'cu co
'f f ratio, where 'cuf is the
axial stress at ultimate axial strain of FRP-confined concrete. When ' ' 2cu cof f < ,
the concrete is said to be moderately-confined, while cases with ' ' 2cu cof f ≥ are
said to be heavily-confined.
3.2.2 Specimens and Instrumentation
The preparation of all 48 specimens followed a standard procedure, which is
described below. All the specimens were cast in steel formworks and were cured
at ambient temperature in laboratory. Once the concrete was cured, the steel
formworks were removed and the concrete cylinders were sanded to remove the
attached dust for better bonding with the confining jackets. The FRP jackets were
43
all formed via the wet lay-up process. A layer of resin was first applied to the
surface of the concrete core and followed by the wrapping of the fiber sheets
which had already been saturated by the resin. The wrapping of the fiber sheets
was continuous with the finishing end overlapping the starting end by 150mm. It
should be noted that all the fiber sheets had hoop fibers only. In addition, two FRP
strips of 25mm in width were wrapped at the two ends to prevent possible
premature failure there. A plastic sheet was then used to squeeze out the air voids
as well as the redundant resin to ensure a compact bond between the concrete core
and the confining jacket and also a smooth surface of the FRP for the convenience
of bonding strain gauges. All the specimens were capped to ensure a smooth
loading surface before testing. For each batch of concrete, 2 or 3 control
specimens of the same size were also tested, from which the average values of the
compressive strength of unconfined concrete 'cof and the corresponding axial
strain coε were found.
For each control specimen, two longitudinal strain gauges, with a gauge length of
120 mm covering the mid-height region, were placed at 180° apart to measure the
axial strains. Another two strain gauges, with a gauge length of 60 mm, were
placed at 180° apart to measure the hoop strains. For each FRP-confined specimen,
either six or eight unidirectional strain gauges, with a gauge length of 20 mm,
were installed at mid-height to measure the hoop strains and another two strain
gauges of the same type were installed at mid-height to measure the axial strains.
In addition, axial strains were also measured by two linear variable displacement
transducers (LVDTs) at 180° apart and covering the mid-height region of 120 mm
for both unconfined and confined specimens. All axial strains reported in Table
3.1 are the average values of readings from the two LVDTs. The instrumentation
of control specimens as well as FRP-confined specimens are shown in Fig. 3.1.
Among the hoop strain gauges installed on FRP-confined specimens, five of them
equally spaced at 45 degrees were located outside the 150 mm wide overlapping
zone (SG1 to SG5 in Fig 3.1b), from which the average hoop strain was found.
The hoop strain readings within the overlapping zone are generally smaller than
those elsewhere as the overlapping zone has a larger jacket thickness. These
44
readings therefore reflect neither the actual strain capacity of the confining jacket
nor the actual dilation properties of the confined concrete, and should thus be
excluded when interpreting the behavior of FRP-confined concrete (Lam and Teng
2004). It should be noted that such important processing of the hoop strain
readings is not possible with existing tests reported by other researchers, for
which the precise number and locations of strain gauges for measuring hoop
strains are generally not reported.
Since the 48 tests were conducted within different research projects, there were
some differences in the loading methods employed. Specimens 01 to 13 were
tested with load control at a constant rate of 300kN/min, while all other specimens
were tested with displacement control, at a constant rate of either 0.18 mm/min or
0.6 mm/min; both rates are acceptable for such tests. All test data, including the
readings of the axial load, strain gauges and the LVDTs were collected with a data
logger and simultaneously saved in a computer. Fig. 3.2 shows the test setup of a
FRP-confined specimen before testing.
3.2.3 Test Results
All the FRP-confined specimens were found to fail by the sudden rupture of the
FRP jacket outside the overlapping zone (the typical failure mode is shown in Fig.
3.3). The key test results are given in Table 3.1. The compressive strength of
confined concrete 'ccf was obtained by dividing the maximum load by the
cross-sectional area of the specimen. Stress-strain curves of both the ascending
and the descending types were captured. When the stress-strain curve is of the
descending type, the axial stress at ultimate axial strain 'cuf is also reported in Table
3.1.
The stress-strain curves from all tests of the present database are shown in Fig. 3.4,
where the lateral strains lε are shown on the left and the axial strains cε are
shown on the right. Both the axial strain and the lateral strain are normalized by
the corresponding value of coε , while the axial stress cσ is normalized by the
45
corresponding value of 'cof . The following sign convention is adopted: in the
concrete, compressive stresses and strains are positive, but in the FRP, tensile
stresses and strains are positive. The predictions of the refined version of Teng et
al.’s (2007a) model are also provided in Fig. 3.4. The predicted curves end at a
hoop rupture strain averaged from either two or three physically identical
specimens. The refinement of Teng et al.’s (2007a) model is discussed later in
chapter.
The pair of specimens 20 and 21 as well as the pair of specimens 28 and 29
showed a descending branch in their stress-strain curves and the compressive
strengths of these specimens are only slightly higher than those of the unconfined
specimens. These specimens were all confined with a one-ply GFRP jacket, and
based on the experimental hoop rupture strains, the average confinement ratios of
the two pairs are 0.079 and 0.067 respectively. By contrast, all other
FRP-confined specimens shown in Fig. 3.4 exhibited the well-known bilinear
stress-strain curve of the ascending type. It should be noted that the pair of
specimens 42 and 43, which had a smaller confinement ratio than that of the pair
of specimens 20 and 21, also exhibited the ascending type stress-strain curve. This
is because the elastic modulus of the FRP jacket of the former pair is much larger
than that of the latter pair, indicating that the nature of the stress-strain curve
depends not only on the confinement ratio but also on the stiffness of the jacket.
3.3 EXISTING ANALYSIS-ORIENTED MODELS FOR FRP-CONFINED
CONCRETE
3.3.1 General Concept
The concept of establishing a passive-confinement stress-strain model from an
active-confinement base model through an incremental approach has previously
been employed for steel-confined concrete by Ahmad and Shah (1982) and Madas
and Elnashai (1992). The first documented attempt to extend this approach to
FRP-confined concrete was made by Mirmiran and Shahawy (1996). This model
follows the procedure proposed by Madas and Elnashai (1992). However, as some
46
of its parameters are not clearly defined (Teng and Lam 2004), this model was not
included in the assessment undertaken by Teng and Lam (2004) and is also not
included in the present comparisons. A later version of this model proposed by the
same authors (Mirmiran and Shahawy 1997a) does not specify the
active-confinement base model and was thus also excluded from the assessment
given in Teng and Lam (2004). In the present study, the model of Mander et al.
(1988) is assumed as the active-confinement model for use in this later version, as
this base model is employed in the earlier version of their model (Mirmiran and
Shahawy 1996). Following the work of Mirmiran and Shahawy (1996, 1997a), a
number of models of this kind have been proposed, including Spoelstra and Monti
(1999), Fam and Rizkalla (2001), Chun and Park (2002), Harries and Kharel
(2002), Marques et al. (2004), Binici (2005) and Teng et al. (2007a).
These models are all built on the assumption that the axial stress and the axial
strain of concrete confined with FRP at a given lateral strain are the same as those
of the same concrete actively confined with a constant confining pressure equal to
that supplied by the FRP jacket (Teng et al. 2007a). This assumption is equivalent
to assuming that the stress path of the confined concrete does not affect its
stress-strain behavior. As a result of this assumption, the stress-strain curve of
FRP-confined concrete can be obtained through the following procedure:
(1) for a given axial strain, find the corresponding lateral strain according to the
lateral-to-axial strain relationship;
(2) based on force equilibrium and radial displacement compatibility between the
concrete core and the FRP jacket, calculate the corresponding lateral confining
pressure provided by the FRP jacket;
(3) use the axial strain and the confining pressure obtained from steps (1) and (2)
in conjunction with an active-confinement base model to evaluate the
corresponding axial stress, leading to the identification of one point on the
stress-strain curve of FRP-confined concrete; and
(4) repeat the above steps to generate the entire stress-strain curve. Fig. 3.5
47
illustrates the concept of this incremental approach.
It is not difficult to realise that in the above procedure, the key elements that
determine the accuracy of the predictions are the active-confinement model and
the lateral-to-axial strain relationship. The performance of the active-confinement
model depends on: (a) the peak axial stress (i.e. failure surface) and the
corresponding axial strain; and (b) the stress-strain equation. The lateral-to-axial
strain relationship, which depicts the unique dilation property of FRP-confined
concrete, is either implicitly or explicitly given in existing models. An iterative
procedure is required in steps (1) and (2) to determine the correct lateral strain that
corresponds to the current axial strain.
In the remainder of this section, the existing models are reviewed in terms of the
three key aspects: the stress-strain equation and the peak axial stress point of the
active confinement model, and the lateral-to-axial strain relationship. The existing
models are also summarized in Table 3.2.
3.3.2 Peak Axial Stress Point
3.3.2.1 Peak axial stress
The peak axial stress on the stress-strain curve of actively-confined concrete is the
compressive strength of such concrete and the peak stress equation defines the
failure surface of such concrete. The models of Mirmiran and Shahawy (1997a),
Spoelstra and Monti (1999), Fam and Rizkalla (2001) and Chun and Park (2002)
directly employ the “five parameter” multiaxial failure surface given by Willam
and Warnke (1975) to define the peak axial stress:
'* '' '2.254 1 7.94 2 1.254l l
cc coco co
f ff fσ σ⎛ ⎞
= + − −⎜ ⎟⎜ ⎟⎝ ⎠
(3.2)
where '*ccf is the peak axial stress of concrete under a specific constant confining
pressure lσ .
48
Of the other four models, Harries and Kharel (2002) adopted the following
equation proposed by Mirmiran and Shahawy (1997b):
'* ' 0.5874.269cc co lf f σ= + (3.3)
Marques et al. (2004) adopted the following equation proposed by Razvi and
Saatcioglu (1999):
'* ' 0.836.7cc co lf f σ= + (3.4)
Binici (2005) employed the Leon-Pramono criterion (Pramono and Willam 1989),
which reduces to Eq. 3.5 if the tensile strength of unconfined concrete is taken to
be 0.1 times of its compressive strength.
'* '' '1 9.9 l l
cc coco co
f ff fσ σ⎛ ⎞
= + +⎜⎜⎝ ⎠
⎟⎟ (3.5)
Teng et al. (2007a) proposed the following linear function to define the peak axial
stress:
'* ' 3.5cc co lf f σ= + (3.6)
The above equations for the peak axial stress are compared in Fig. 3.6, showing
large differences between each other.
3.3.2.2 Axial strain at peak axial stress
Except the model of Marques et al. (2004), all existing models employ the
following equation initially proposed by Richart et al. (1928) to define the axial
strain at peak axial stress:
'*
*'1 5 1cc
cc coco
ff
ε ε⎡ ⎤⎛ ⎞
= + −⎢ ⎥⎜⎝ ⎠
⎟⎣ ⎦
(3.7)
49
where coε and *ccε are respectively the axial strains at '
cof and '*ccf . Marques
et al. (2004) used Eq. 3.7 for concrete with ' 40cof ≤ MPa but for concrete of
higher strength, Eq. 3.7 was modified with a factor introduced by Razvi and
Saatcioglu (1999) to account for the reduced effectiveness in the enhancement of
axial strain for high strength concrete. In Fig. 3.7, the predictions from all models
of the axial strain at peak axial stress are compared, where the differences stem
from the different expressions for the peak axial stress.
3.3.2.3 Stress-strain equation
All models except those of Harries and Kharel (2002) and Binici (2005) employ
the following equation originally proposed by Popovics (1973) and later adopted
by Mander et al. (1988) for steel-confined concrete:
( )( )
*
'* *1c ccc
qcc c cc
qf q
ε εσε ε
=− +
(3.8)
where the constant is defined by q
'* *c
c cc cc
EqE f ε
=−
(3.9)
where is the elastic modulus of concrete. cE
In the model of Harries and Kharel (2002), the stress-strain equation of
actively-confined concrete described by Eq. 3.8 is modified by a factor which was
introduced to control the slope of the descending branch. The model of Binici
(2005) employs three separate expressions to describe the full stress-strain curve.
The ascending branch is described using a linear expression followed by Eq. 3.8,
while the descending branch is described using an exponential expression.
50
3.3.2.4 Lateral-to-axial strain relationship
The lateral-to-axial strain relationship, not available in an active-confinement
model, provides the essential connection between the response of the concrete
core and the response of the FRP jacket, in a passive-confinement model for
FRP-confined concrete. This relationship has been established via different
approaches, and is either explicitly stated or implicitly given.
Explicit lateral-to-axial strain relationships are used in the models of Mirmiran
and Shahawy (1997a), Harries and Kharel (2002) and Teng et al. (2007a).
Mirmiran and Shahawy (1997a) used the tangent dilation ratio tµ (the absolute
value of the tangent slope of the lateral-to-axial strain curve of FRP-confined
concrete; i.e. d / dt l cµ ε ε= ) to link the lateral strain and the axial strain. A
fractional function was introduced by these authors to describe the tangent
dilation ratio based on their own test results of FRP-confined concrete. Harries
and Kharel (2002) instead used the secant dilation ratio sµ (the absolute value of
the secant slope of the lateral-to-axial strain curve of FRP-confined concrete; i.e.
/s l cµ ε ε= ), which is also based on their own test results of FRP-confined
concrete, to relate the axial strain to the lateral strain. A simplified tri-linear
equation was used to describe the variation of the secant dilation ratio. It should
be noted that Harries and Kharel (2002) used different equations to predict the
lateral strains of CFRP-confined and GFRP-confined concrete respectively. Based
on a careful interpretation of the dilation properties of confined and unconfined
concrete, Teng et al. (2007a) proposed the following lateral-to-axial strain
equation that is applicable to un-confined, actively confined and FRP-confined
concrete:
0.7
'0.85 1 8 1 0.75 exp 7c l l
co co co cofε σ εε ε
⎧ ⎫⎡ ⎤ ⎡⎛ ⎞ ⎛ ⎞ ⎛−⎪ ⎪= + + − −⎨ ⎬⎢ ⎥ ⎢⎜ ⎟ ⎜ ⎟ ⎜⎝ ⎠ ⎝ ⎠ ⎝⎣ ⎦ ⎣⎪ ⎪⎩ ⎭
lεε
⎤⎞−⎥⎟⎠⎦
(3.10)
where lσ is the confining pressure. In actively confined concrete, the concrete is
subjected to a constant confining pressure lσ throughout the loading process, but
51
in FRP-confined concrete, the passive lateral confining pressure depends on the
stiffness of the FRP jacket and increases continuously with the hoop strain of
FRP hε . For a given hoop/lateral strain, the confining pressure supplied by the
FRP jacket is
frp h frp ll
E t E tR Rε ε
σ = = − (3.11)
Implicit lateral-to-axial strain relationships were adopted by other researchers.
Spoelstra and Monti (1999) adopted the simple constitutive model proposed by
Pantazopoulou and Mills (1995), which describes the decrease of secant modulus
of concrete with an increasing area strain, to determine the lateral strain. Marques
et al. (2004) employed a similar constitutive model, but they argued that the one
used by Spoelstra and Monti (1999) does not ensure ( )' ,co cof ε corresponds to the
peak point on the stress-strain curve of unconfined concrete. A coefficient was
thus introduced to overcome this shortcoming. In the model of Fam and Rizkalla
(2001), an equation representing the variation of secant dilation ratio with
confining pressure was developed based on the results of Gardner (1969) from
triaxial compression tests of concrete to determine the lateral strain. In the model
of Chun and Park (2002), a cubic polynomial equation developed by Elwi and
Murray (1979) based on the results of Kupfer et al. (1969) from uniaxial
compression tests of concrete was used. In the model of Binici (2005), the secant
dilation ratio was set to be a constant in the elastic stage, beyond which it was
assumed to vary with the confining pressure.
3.4 ASSESSMENT OF EXISTING MODELS
3.4.1 Test Data
For the assessment of the accuracy of existing analysis-oriented stress-strain
models in predicting the lateral strain-axial strain curve and the stress-strain curve,
only comparisons with selected tests (specimens 28 and 29; 24 and 25; 34 and 35
in Table 3.1) are reported here for brevity. The observations made below are also
52
supported by comparisons conducted using the remainder of the database not
reported here.
The three sets of specimens correspond to confinement ratios of 0.067, 0.25, and
0.55 and 'cu co
'f f ratios of 0.88, 1.60, and 2.86 respectively to represent
weakly-confined, moderately-confined and heavily-confined concrete. The
comparisons focus on the dilation properties of FRP-confined concrete, although
the full-range axial stress-strain behavior is also examined. Following these
comparisons, the ultimate axial strain and the corresponding axial stress from
analysis-oriented models are also compared with the results of all 48 tests in the
test database.
3.4.2 Dilation Properties
It has been widely accepted that under axial compression, unconfined concrete
experiences volumetric compaction up to 90% of the peak stress. Thereafter the
concrete shows unstable volumetric dilation due to the rapidly increasing
lateral-to-axial strain ratio. However, this lateral dilation can be effectively
constrained by an FRP jacket (Lam and Teng 2003; Teng et al. 2007a). In
FRP-confined concrete, this lateral dilation results in a continuously increasing
lateral confining pressure provided by the FRP jacket. The dilation properties of
FRP-confined concrete are reflected by its lateral-to-axial strain relationship. A
more direct way to investigate the dilation properties of FRP-confined concrete is
to examine the tangent dilation ratio or the secant dilation ratio.
Figs 3.8 to 3.10 show the experimental variations of lateral strain, tangent dilation
ratio and secant dilation ratio as the axial strain increases and those from existing
analysis-oriented stress-strain models. These test data allow the following
observations to be made: (a) in the initial stage of deformation, the tangent
dilation ratio/secant dilation ratio is almost constant and is very close to the secant
dilation ratio of unconfined concrete; (b) afterwards, the tangent dilation
ratio/secant dilation ratio gradually increases, and the FRP jacket is increasingly
mobilized to confine the concrete; (c) a typical lateral-to-axial strain curve of
53
FRP-confined concrete features an inflection point corresponding to the maximum
value of the tangent dilation ratio.
The three sets of comparisons are for weakly-confined, moderately-confined, and
heavily-confined concrete specimens respectively (Figs 3.8 to 3.10). Each set of
comparisons includes four types of curves, being the lateral-to-axial strain curves,
the tangent dilation ratio curves, the secant dilation ratio curves and the
stress-strain curves. All curves terminate at the point when the average FRP hoop
rupture strain from the corresponding pair of tests is reached. In plotting the test
tangent dilation ratio, some of the data points were filtered out to remove
disturbances from local fluctuations due to data resolutions of closely spaced
readings.
From the comparisons, it can be seen that the existing models lead to rather
different predictions. The models of Fam and Rizkalla (2001) and Binici (2005)
do not predict an inflection point on the lateral-to-axial strain curve (i.e. the
predicted tangent dilation ratio/secant dilation ratio continuously increases with
the axial strain). For the other models, although they are capable of predicting the
inflection point, the ultimate axial strain and the maximum values of the tangent
dilation ratio and the secant dilation ratio are poorly predicted in most cases.
Mirmiran and Shahawy (1997a) proposed that the maximum value of the tangent
dilation ratio occurs when the axial strain reaches coε , however, the present test
results show that the tangent dilation ratio does not reach its maximum value at
such an early stage. Instead, this point occurs at an axial strain of
approximately 2 coε . In addition, the tangent dilation ratio predicted by Mirmiran
and Shahawy’s (1997a) model can be negative under heavy confinement. The
predictions of Spoelstra and Monti’s (1999) model and Marques et al.’s (2004)
model show some similarity in shape and they predict initial values of the tangent
dilation ratio and the secant dilation ratio which are much smaller than those
obtained from tests. The model of Harries and Kharel (2002) is excluded from Fig.
3.10, since the predicted tangent dilation ratio and secant dilation ratio are
unreasonably small. This shortcoming is due to their logarithmic equation for the
ultimate secant dilation ratio, which results in negative values under heavy
54
confinement. In addition, their definition of the ultimate secant dilation ratio is in
doubt, since it only considers the effect of jacket stiffness, but ignores the effects
of specimen size and compressive strength of unconfined concrete. Chun and
Park’s (2002) model suffers from the same drawback. These authors also used a
logarithmic equation, which may predict negative values, to determine the
ultimate secant dilation ratio. The model of Teng et al. (2007a) is seen to be the
most accurate one. This model provides the most accurate predictions of the
ultimate axial strains in all three situations. Apart from the weakly-confined
concrete specimens, the tangent dilation ratio and the secant dilation ratio are both
accurately predicted by this model.
It may be noted that although some of the models predict the ultimate axial strain
quite accurately in some cases, the shape of the lateral-to-axial strain curve is not
correctly captured. Ideally, both the ultimate point and the shape of the
lateral-to-axial strain curve should be accurately predicted. However, it is found
that given the same ultimate point, the stress-strain curve can be closely predicted
as long as the overall trend of the lateral-to-axial strain ratio can be reasonably but
not necessarily accurately predicted. Taking Teng et al.’s (2007a) model as an
example, Fig. 3.11 demonstrates that for both the ascending and descending types
of stress-strain curves, even if the lateral-to-axial stain curve is simply described
using a straight line, which is obviously incorrect but reasonably close to the
predicted lateral-to-axial strain curve, the predicted stress-strain curves are only
slightly affected. This indicates that local inaccuracy of the lateral-to-axial strain
curve only leads to small deviations in the predicted stress-strain curve. The
parameters used to generate Fig. 3.11 are summarized in Table 3.3.
3.4.3 Stress-Strain Curves
The axial stress-axial strain curves are shown in Figs 3.8d, 3.9d and 3.10d. It can
be seen that for the weakly-confined and moderately-confined concrete specimens,
all examined models overestimate the ultimate axial strain and the corresponding
axial stress. However, for the heavily-confined concrete specimens, the
performance of these models improves. Among them, the model of Teng et al.
(2007a) is again seen to be the most accurate one. A significant deficiency of this
55
model is that it overestimates the axial stress at ultimate axial strain for
weakly-confined concrete and moderately-confined concrete, especially for the
former. In Fig. 3.8d, this model fails to predict the post-peak descending branch.
3.4.4 Ultimate Condition
For brevity, only three sets of typical test results are compared with existing
models in the preceding sub-sections. These comparisons are mainly concerned
with the lateral-to-axial strain relationship, although the axial stress-strain curves
are also discussed. An alternative way to assess the predicted stress-strain
behavior is to compare the predicted ultimate axial strains and the corresponding
axial stresses with the test values, as shown in Figs 3.12 to 3.19, which allows a
much large number of tests to be included in the comparison. It should be noted
that in assessing the model of Harries and Kharel (2002), six specimens with the
largest confinement ratios are not included for the reason mentioned earlier.
On the whole, most models give poor predictions for the ultimate axial strain,
mainly because the accuracy of the predicted ultimate axial strains depends
heavily on the accuracy of the lateral-to-axial strain equation, which needs
improvement as shown earlier. The models of Mirmiran and Shahawy (1997a) and
Teng et al. (2007a) are the better models in predicting the ultimate axial strain. For
the axial stress at ultimate axial strain, the performance of all models except that
of Harries and Kharel (2002) becomes much better, with the models of Marques et
al. (2004), Binici (2005) and Teng et al. (2007a) giving more accurate predictions.
It is interesting to note that the models of Mirmiran and Shahawy (1997a),
Spoelstra and Monti (1999), Fam and Rizkalla (2001) and Chun and Park (2003)
show very similar performance in predicting the axial stress at ultimate axial
strain. A common feature of these four models is that they all employ Eq. 3.2 to
predict the peak axial stress of actively-confined concrete, which suggests that the
use of Eq. 3.2, which is widely accepted for steel-confined concrete, in an
active-confinement base model does not lead to an accurate passive-confinement
model for FRP-confined concrete. The model of Harries and Kharel (2002)
performs worst in predicting the axial stress at ultimate axial strain. This is also
mainly due to the inappropriate definition of the peak axial stress in the base
56
model. If a modified failure surface is used, this model will perform better (Teng
and Lam 2004). Among all these models, Teng et al.’s (2007a) model provides the
most accurate predictions for both the ultimate axial strain and its corresponding
axial stress.
3.5 REFINEMENT OF TENG ET AL.’S MODEL
3.5.1 General
From the comparisons presented in the above section, it is clear that Teng et al.’s
(2007a) model is the most accurate of all examined models. This model correctly
captures the unique dilation properties of FRP-confined concrete, which is central
to models of this kind, and shows much better performance over the other models
in predicting the ultimate condition. Nevertheless, this model still suffers from one
significant deficiency: it overestimates the axial stress at ultimate axial strain for
weakly-confined concrete and to a lesser degree for moderately-confined concrete
(Figs 3.8d and 3.9d). Teng et al. (2007a) noticed that their model overestimated
the responses of some weakly-confined specimens and suggested that any future
improvements to their model should be focused on weakly-confined concrete.
This problem was not resolved by Teng et al. (2007a), primarily due to the
insufficient information available on concrete with weak FRP confinement and the
scatter of the relevant test data available then. This problem becomes more
obvious when the model is compared with the more precise test data of the present
test database. A refinement of this model to eliminate this deficiency is presented
in the present section, so that stress-strain curves of the descending type can also
be accurately predicted by this model.
To identify a way of refining Teng et al.’s (2007a) model, its key elements need to
be screened for possible improvements. The stress-strain equation in the
active-confinement base model of Teng et al. (2007a) is also commonly employed
by other models for FRP-confined concrete and is not expected to be a source of
error. Fig. 3.22 provides some evidence for the accuracy of Eq. 3.8 (Popovics’s
equation). The lateral-to-axial strain relationship proposed by Teng et al. (2007a)
overestimates the axial strain at a given lateral strain for weakly-confined concrete,
57
although it is accurate for moderately-confined and heavily-confined concrete.
However, this overestimation of the axial strain for weakly-confined concrete is
not the cause of inaccuracy in predicting the descending branch. Indeed, if this
overestimation is corrected, the accuracy of the predicted stress-strain curve will
further degrade. Given the limited number of tests available on weakly-confined
concrete and the good overall performance of the lateral-to-axial strain equation
proposed by Teng et al. (2007a) (Figs 3.8 to 3.10), it is difficult to propose
improvements to or to justify any modifications of this equation. The definition of
the peak point of the stress-strain curve in the base model is therefore believed to
be the main source of error. In particular, for the axial strain at peak axial stress,
Eq. 3.7 which was initially proposed by Richart et al. (1928), is employed by all
models without any critical examination, although different equations (Eqs 3.2 to
3.6) have been proposed for the peak axial stress. As a result, the definition of the
peak axial stress and the corresponding axial strain are examined here to develop a
more satisfactory stress-strain model for FRP-confined concrete.
To this end, test results from four recent studies (Imran and Pantazopoulou 1996;
Ansari and Li 1998; Sfer et al. 2002 and Tan and Sun 2004) on actively-confined
concrete were collected and analysed. Since the stress-strain behavior of high
strength concrete under active confinement is known to differ from that of normal
strength concrete (Ansari and Li 1998), only test results of normal strength
concrete ( 'cof = 20 to 50 MPa) reported in the above four studies are included in
the analysis [the 51.8 MPa series of Tan and Sun (2004) is also included]. The
range of confinement ratios of these tests is from 0.04 to 0.91, which is very close
to that of FRP-confined concrete in the current database. These
active-confinement test results are given in Table 3.4.
3.5.2 Peak Axial Stress in the Base Model
The classical work on concrete under active confinement conducted by Richart et
al. (1928) led to the following equation for the peak axial stress:
58
'*
1' 1cc l
co co
f k 'f fσ
= + (3.12)
where is the confinement effectiveness coefficient and = 4.1. While a value of
4.1 for is commonly quoted, a wide range of other values has also been
proposed by different researchers based on their own test data on
actively-confined concrete (see Candappa et al. 2001). In Teng et al.’s (2007a)
model, a value of 3.5 is used for (see Eq. 3.6), which was deduced from test
results of FRP-confined concrete and is within the existing range of proposed
values for (Candappa et al. 2001).
1k
1k
1k
1k
As shown in Fig. 3.20, the dotted line, representing Eq. 3.6 for the peak axial
stress, agrees well with the test results. It is important to note that although Eq. 3.6
was deduced from test results of FRP-confined concrete, it does provide accurate
predictions for the test data of actively-confined normal strength concrete.
3.5.3 Axial Strain at Peak Axial Stress in the Base Model
Richart et al. (1928) also suggested that the effectiveness in the enhancement of
axial strain is around 5 times that in the enhancement of axial stress. Eq. 3.7 was
thus proposed for the axial strain at peak axial stress. Substitution of Eq. 3.12 into
Eq. 3.7 yields
*
1 '1 5cc l
co co
kf
ε σε
= + (3.13)
In the analysis-oriented stress-strain models assessed in the present chapter, Eq.
3.13 is accepted without any modification, except Marques et al. (2004), although
different equations were proposed for the peak axial stress. A more rational
approach to predict the degree of strain enhancement is to separate it from the
definition of the peak axial stress. That is, the relationship between the strain
enhancement ratio *cc coε ε and the confinement ratio '/l cofσ should be directly
59
established from active-confinement test data.
Based on the test data shown in Fig. 3.21, the following nonlinear equation is
proposed for the axial strain at peak axial stress:
1.2*
'1 17.5cc l
co cofε σε
⎛ ⎞= + ⎜ ⎟
⎝ ⎠ (3.14)
Fig. 3.21 shows that Eq. 3.14 provides accurate predictions of *ccε deduced from
the test results of Lam et al. (2006) for specimens confined with a 0.33 mm thick
CFRP jacket (specimens 17-19). In Fig. 3.21, the values of *ccε for specimens
17-19 were deduced from their axial stresses cσ and axial strains cε at
different confining pressures lσ using Eqs 3.6, 3.8 and 3.9. It can also be seen
that Eq. 3.14 provides reasonably close predictions of the test results of
actively-confined concrete (Imran and Pantazopoulou 1996; Ansari and Li 1998;
Sfer et al. 2002 and Tan and Sun 2004), given the wide scatter exhibited by these
test results.
Eq. 3.14 is thus proposed to replace its counterpart in Teng et al.’s (2007a)
original model which can be written as
*
'1 17.5cc l
co cofε σε
⎛ ⎞= + ⎜
⎝ ⎠⎟ (3.15)
Eq. 3.14 is compared with Eqs 3.2 to 3.6 in Fig. 3.7. It is interesting to note that
Eq. 3.14 predicts a trend that is opposite to that of the other equations. Teng et
al.’s (2007a) model with Eq. 3.15 replaced by Eq. 3.14 is referred to as the refined
model.
It should be noted that the incorporation of Eq. 3.14 into the base model shifts the
location of the peak point of the stress-strain curve and slightly modifies the
overall shape of the stress-strain curve. Fig. 3.22 shows the test stress-strain
60
curves of the first three tests by Ansari and Li (1998) in Table 3.4 as well as the
stress-strain curves predicted using the original base model where the peak point
is defined by Eqs 3.6 and 3.15 and using the modified base model where the peak
point is defined by Eqs 3.6 and 3.14. These tests were chosen for comparison as
the curves were clearly reported in the original paper and their peak stresses are
similar to the predictions of Eq. 3.6, which enables a more reliable and direct
comparison. It can be seen that the curves predicted using the modified base
model as well as the original base model are both in reasonably close agreement
with the test curves, considering the large scatter of test values of *ccε shown in
Fig. 3.21. This indicates that the modified definition of the peak point is at least as
valid as the original definition when judged on the basis of these test results.
Fig. 3.23 shows the predictions of the axial stress at ultimate axial strain of the
modified model versus the test data. It can be seen that the overestimation by Teng
et al.’s (2007a) original model of the axial stress at ultimate axial strain for
weakly-confined and moderately-confined concrete (Fig. 3.19b) has been
corrected while the prediction for heavily-confined concrete is only very slightly
affected (Fig. 3.23). The advantage of the refined model over the original model in
predicting the entire stress-strain curve is demonstrated for selected specimens
(Figs 3.4d and 3.4e). In these figures, the end of each curve is provided with a
symbol to indicate the group it belongs to for easy comparisons. The refined
model is seen to perform much better than the original model for weakly-confined
specimens. The difference between the original and refined models reduces as the
hoop membrane stiffness of the FRP jacket increases (Figs 3.4d and 3.4e).
Comparisons shown in Figs 3.4a to 3.4c and 3.4f to 3.4h between the refined
model and the test data show that overall, the refined model provide accurate
predictions.
In all comparisons with test data in this chapter, the test values of 'cof and
coε were used. The elastic modulus and the Poisson’s ratio of unconfined
concrete were either those specified in an individual model or taken to be
cE
'4730 cof (MPa) and 0.2 if they are not specified in the model. For Binici’s
61
(2005) model, the compressive fracture energy was found from
'8.8fc cofG = ( fcG in MPa/mm and 'cof in MPa) (Nakamura and Higai 2001)
and the characteristic length of the specimen in the loading direction was
taken to be the specimen height (i.e. 305 mm).
cl
Teng et al. (2007a) suggested that when test values of coε are not available, a value
of 0.0022 should be used with their model. For more accurate predictions, it is
proposed here that when the refined model is used to predict the behavior of
FRP-confined concrete, the following equation proposed by Popovics (1973)
should be used unless a test value is available:
'40.000937co cofε = ( 'cof in MPa) (3.16)
3.6 CONCLUSIONS
This chapter has presented a thorough assessment of the performance of eight
existing analysis-oriented stress-strain models for FRP-confined concrete which
employ an active-confinement model as the base model, leading to the
identification of Teng et al.’s (2007a) model as the most accurate through this
assessment. A refined version of Teng et al.’s (2007a) model has also been
proposed. The comparisons and discussions presented in this chapter allow the
following conclusions to be drawn:
1) The lateral-to-axial strain relationship, which reflects the unique dilation
properties of FRP-confined concrete, is central to models of this kind. A
successful model should accurately predict this relationship. Nevertheless,
provided the overall trend of this relationship is reasonably well described,
the axial stress-strain curve can be closely predicted, even if local
inaccuracies exist in the lateral-to-axial strain equation;
2) The definitions of the peak axial stress and the corresponding axial strain in
the active-confinement base model are also important to ensure the accuracy
62
of an analysis-oriented model for FRP-confined concrete;
3) Most of the eight models examined in this chapter correctly capture the
dilation properties of FRP-confined concrete, but provide poor predictions of
the ultimate axial strain. Their performance in predicting the axial stress at
ultimate axial strain is however much better. The accuracy of the predicted
ultimate axial strain depends mainly on the accuracy of the lateral-to-axial
strain equation, while that of the predicted axial stress at ultimate axial strain
depends mainly on the definitions of the peak axial stress and its
corresponding strain in the base model.
4) The model of Teng et al. (2007a) performs the best among the eight models
examined in this chapter. This model provides accurate predictions of both the
lateral-axial strain relationship and the ultimate condition, except that it
overestimates the axial stress at ultimate axial strain for weakly-confined and
to a lesser extent for moderately-confined concrete. As a result, its predictions
are likely to be inaccurate for stress-strain curves with a descending branch.
5) The performance of the model of Teng et al. (2007a) for weakly-confined
concrete can be significantly improved by replacing its equation for the strain
at peak axial stress with a nonlinear equation proposed in this chapter.
63
3.7 REFERENCES
Ahmad, S.H. and Shah, S.P. (1982). “Stress-strain curves of concrete confined by spiral reinforcement”, ACI Journal, 79(6), 484-490.
Ansari, F. and Li, Q.B. (1998). “High-strength concrete subjected to triaxial compression”, ACI Materials Journal, 95(6), 747-755.
Becque, J., Patnaik, A.K. and Rizkalla, S.H. (2003). “Analytical models for concrete confined with FRP tubes”, Journal of Composites for Construction, ASCE, 7(1), 31-38.
Binici, B. (2005). “An analytical model for stress–strain behavior of confined concrete”, Engineering Structures, 27(7), 1040-1051.
Candappa, D.C., Sanjayan, J.G. and Setunge, S. (2001). “Complete triaxial stress-strain curves of high-strength concrete”, Journal of Materials in Civil Engineering, ASCE, 13(3), 209-215.
Chun, S.S. and Park, H.C. (2002). “Load carrying capacity and ductility of RC columns confined by carbon fiber reinforced polymer.” Proceedings, 3rd International Conference on Composites in Infrastructure (CD-Rom), University of Arizona, San Francisco, USA.
Elwi, A.A. and Murray, D.W. (1979). “A 3D hypoelastic concrete constitutive relationship”, Journal of the Engineering Mechanics Division, ASCE, 105(4), 623-641.
Fam, A.Z. and Rizkalla, S.H. (2001). “Confinement model for axially loaded concrete confined by circular fiber-reinforced polymer tubes”, ACI Structural Journal, 98(4), 451-461.
Gardner, N.J. (1969). “Triaxial behavior of concrete”, ACI Journal, 66(2), 136-146.
Harmon, T.G., Ramakrishnan S. and Wang, E.H. (1998). “Confined concrete subjected to uniaxial monotonic loading”, Journal of Engineering Mechanics, ASCE, 124(12), 1303-1308.
Harries, K.A. and Kharel, G. (2002). “Behavior and modeling of concrete subject to variable confining pressure”, ACI Materials Journal, 99(2), 180-189.
Imran, I. and Pantazopoulou, S.J. (1996). “Experimental study of plain concrete under triaxial stress”, ACI Materials Journal, 93(6), 589-601.
Kupfer, H.B., Hilsdorf, H.K. and Rusch, H. (1969). “Behavior of concrete under biaxial stresses”, ACI Journal, 66(8), 656-666.
Lam, L. and Teng, J.G. (2003). “Design-oriented stress-strain model for FRP-confined concrete”, Construction and Building Materials, 17(6-7), 471-489.
64
Lam, L. and Teng, J.G. (2004). “Ultimate condition of fiber reinforced polymer-confined concrete”, Journal of Composites for Construction, ASCE, 8(6), 539-548.
Lam, L., Teng, J.G., Cheung, C.H. and Xiao, Y. (2006). “FRP-confined concrete under axial cyclic compression”, Cement and Concrete Composites, 28(10), 949-958.
Madas, P. and Elnashai, A.S. (1992). “A new passive confinement model for the analysis of concrete structures subjected to cyclic and transient dynamic loading”, Earthquake Engineering and Structural Dynamics, 21(5), 409-431.
Mander, J.B., Priestley, M.J.N. and Park, R. (1988). “Theoretical stress-strain model for confined concrete”, Journal of Structural Engineering, ASCE, 114(8), 1804-1826.
Marques, S.P.C. Marques, D.C.S.C., da Silva J.L. and Cavalcante, M.A.A. (2004). “Model for analysis of short columns of concrete confined by fiber-reinforced polymer”, Journal of Composites for Construction, ASCE, 8(4), 332-340.
Mirmiran, A. and Shahawy, M. (1996). “A new concrete-filled hollow FRP composite column”, Composites Part B-Engineering, 27(3-4), 263-268.
Mirmiran, A. and Shahawy, M. (1997a). “Dilation characteristics of confined concrete”, Mechanics of Cohesive-Frictional Materials, 2(3), 237-249.
Mirmiran, A. and Shahawy, M. (1997b). “Behavior of concrete columns confined by fiber composites”, Journal of Structural Engineering, ASCE, 123(5), 583-590.
Nakamura, H. and Higai, T. (2001). “Compressive fracture energy and fracture zone length of concrete.” Modeling of Inelastic Behavior of RC Structures Under Seismic Loads, Edited by P.B. Shing, T. Tanabe, ASCE, 471-487.
Popovics, S. (1973). “Numerical approach to the complete stress-strain relation for concrete”, Cement and Concrete Research, 3(5), 583-599.
Pantazopoulou, S.J. and Mills, R.H. (1995). “Microstructural aspects of the mechanical response of plain concrete”, ACI Materials Journal, 92(6), 605-616.
Pramono, E. and Willam K. (1989). “Fracture-energy based plasticity formulation of plain concrete”, Journal of Engineering Mechanics, ASCE, 115(6), 1183-1204.
Razvi, S. and Saatcioglu, M. (1999). “Confinement model for high-strength concrete”, Cement and Concrete Research, 125(3), 281-289.
Richart, F.E., Brandtzaeg, A. and Brown, R.L. (1928). A Study of the Failure of Concrete under Combined Compressive Stresses, Engineering Experiment Station, University of Illinois, Urbana, U.S.A.
65
Sfer, D., Carol, I., Gettu, R. and Etse, G. (2002). “Study of the behavior of concrete under traxial compression”, Journal of Engineering Mechanics, ASCE, 128(2), 156-163.
Spoelstra, M.R. and Monti, G. (1999). “FRP-confined concrete model”, Journal of Composites for Construction, ASCE, 3(3), 143-150.
Tan, K.H. and Sun, X. (2004). “Failure criteria of concrete under triaxial compression”, Proceedings, International Symposium on Confined Concrete (CD-Rom), 12-14 June, Changsha, China,
Teng, J.G. and Lam, L. (2004). “Behavior and modeling of fiber reinforced polymer-confined concrete”, Journal of Structural Engineering, ASCE, 130(11), 1713-1723.
Teng, J.G., Huang, Y.L. Lam. L and Ye L.P. (2007a). “Theoretical model for fiber reinforced polymer-confined concrete”, Journal of Composites for Construction, ASCE, 11(2).
Teng, J.G., Yu, T. Wong, Y.L. and Dong, S.L. (2007b). “Hybrid FRP-concrete-steel tubular columns: concept and behaviour”, Construction and Building Materials, 21(4), 846-854.
Willam, K.J. and Warnke, E.P. (1975). “Constitutive model for the triaxial behaviour of concrete”, Proceedings, International Association for Bridge and Structural Engineering, 19, 1-30.
66
Table 3.1 Test database of FRP-confined concrete cylinders
Source Specimen
D
(mm)H
(mm)'
cof (MPa)
coε (%)
Fiber type
t (mm)
frpE (GPa)
,h rupε (%)
'ccf ( '
cuf ) (MPa)
cuε (%)
1 152 305 35.9 0.203 Carbon 0.165 250.5 0.969 47.2 1.1062 152 305 35.9 0.203 Carbon 0.165 250.5 0.981 53.2 1.2923 152 305 35.9 0.203 Carbon 0.165 250.5 1.147 50.4 1.2734 152 305 35.9 0.203 Carbon 0.33 250.5 0.949 71.6 1.855 152 305 35.9 0.203 Carbon 0.33 250.5 0.988 68.7 1.6836 152 305 35.9 0.203 Carbon 0.33 250.5 1.001 69.9 1.9627 152 305 34.3 0.188 Carbon 0.495 250.5 0.799 82.6 2.0468 152 305 34.3 0.188 Carbon 0.495 250.5 0.884 90.4 2.4139 152 305 34.3 0.188 Carbon 0.495 250.5 0.968 97.3 2.51610 152 305 38.5 0.223 Glass 1.27 21.8 1.440 51.9 1.31511 152 305 38.5 0.223 Glass 1.27 21.8 1.890 58.3 1.45912 152 305 38.5 0.223 Glass 2.54 21.8 1.670 77.3 2.188
Lam and Teng(2004)
13 152 305 38.5 0.223 Glass 2.54 21.8 1.760 75.7 2.45714 152 305 41.1 0.256 Carbon 0.165 250 0.810 52.6 0.90015 152 305 41.1 0.256 Carbon 0.165 250 1.080 57.0 1.21016 152 305 41.1 0.256 Carbon 0.165 250 1.070 55.4 1.11017 152 305 38.9 0.250 Carbon 0.33 247 1.060 76.8 1.91018 152 305 38.9 0.250 Carbon 0.33 247 1.130 79.1 2.080
Lam et al. (2006)
19 152 305 38.9 0.250 Carbon 0.33 247 0.790 65.8 1.25020 152 305 39.6 0.263 Glass 0.17 80.1 1.869 41.5 (38.8) 0.82521 152 305 39.6 0.263 Glass 0.17 80.1 1.609 40.8 (37.2) 0.94222 152 305 39.6 0.263 Glass 0.34 80.1 2.040 54.6 2.13023 152 305 39.6 0.263 Glass 0.34 80.1 2.061 56.3 1.825
Teng et al. (2007b)
24 152 305 39.6 0.263 Glass 0.51 80.1 1.955 65.7 2.558
67
25 152 305 39.6 0.263 Glass 0.51 80.1 1.667 60.9 1.79226 152 305 33.1 0.309 Glass 0.17 80.1 2.080 42.4 1.30327 152 305 33.1 0.309 Glass 0.17 80.1 1.758 41.6 1.26828 152 305 45.9 0.243 Glass 0.17 80.1 1.523 48.4 (40.5) 0.81329 152 305 45.9 0.243 Glass 0.17 80.1 1.915 46.0 (40.5) 1.06330 152 305 45.9 0.243 Glass 0.34 80.1 1.639 52.8 1.20331 152 305 45.9 0.243 Glass 0.34 80.1 1.799 55.2 1.25432 152 305 45.9 0.243 Glass 0.51 80.1 1.594 64.6 1.55433 152 305 45.9 0.243 Glass 0.51 80.1 1.940 65.9 1.90434 152 305 38.0 0.217 Carbon 0.68 240.7 0.977 110.1 2.55135 152 305 38.0 0.217 Carbon 0.68 240.7 0.965 107.4 2.61336 152 305 38.0 0.217 Carbon 1.02 240.7 0.892 129.0 2.79437 152 305 38.0 0.217 Carbon 1.02 240.7 0.927 135.7 3.08238 152 305 38.0 0.217 Carbon 1.36 240.7 0.872 161.3 3.70039 152 305 38.0 0.217 Carbon 1.36 240.7 0.877 158.5 3.54440 152 305 37.7 0.275 Carbon 0.11 260 0.935 48.5 0.89541 152 305 37.7 0.275 Carbon 0.11 260 1.092 50.3 0.91442 152 305 44.2 0.260 Carbon 0.11 260 0.734 48.1 0.69143 152 305 44.2 0.260 Carbon 0.11 260 0.969 51.1 0.88844 152 305 44.2 0.260 Carbon 0.22 260 1.184 65.7 1.30445 152 305 44.2 0.260 Carbon 0.22 260 0.938 62.9 1.02546 152 305 47.6 0.279 Carbon 0.33 250.5 0.902 82.7 1.30447 152 305 47.6 0.279 Carbon 0.33 250.5 1.130 85.5 1.936
Present study
48 152 305 47.6 0.279 Carbon 0.33 250.5 1.064 85.5 1.821
68
Table 3.2 Summary of analysis-oriented models for FRP-confined concrete
Peak Point Model Stress-strain Equation Stress Strain
Lateral-to-Axial Strain Relationship
Mirmiran and Shahawy (1997a)
Popovics (1973) Eq. 3.2 Eq. 3.7
Explicit, Mirmiran and Shahawy
(1997a)
Spoelstra and Monti (1999)
Popovics (1973) Eq. 3.2 Eq. 3.7
Implicit, Pantazopoulou and Mills
(1995) Fam and Rizkalla (2001)
Popovics (1973) Eq. 3.2 Eq. 3.7 Implicit,
Fam and Rizkalla (2001)
Chun and Park (2002)
Popovics (1973) Eq. 3.2 Eq. 3.7 Implicit,
Elwi and Murray (1979)
Harries and Kharel (2002)
Modified from
Popovics (1973)
Eq. 3.3 Eq. 3.7 Explicit, Harries and Kharel (2002)
Marques et al. (2004)
Popovics (1973) Eq. 3.4
Modified from
Eq. 3.7
Implicit, Pantazopoulou and Mills
(1995)
Binici (2005)
Modified from
Popovics (1973)
Eq. 3.5 Eq. 3.7 Implicit, Binici (2005)
Teng et al. (2007a)
Popovics (1973) Eq. 3.6 Eq. 3.7 Explicit, Eq. 3.10
Teng et al. (2007a)
69
Table 3.3 Summary of parameters used for generating Fig. 3.11
Figure No. D
(mm) '
cof (MPa)
coε (%)
t (mm)
frpE (GPa)
,h rupε (%)
Fig. 3.11a,b 152 40 0.22 0.1 80 1.5 Fig. 3.11c,d 152 40 0.22 0.6 80 1.5
70
Table 3.4 Test results of concrete under active confinement
Source '
cof (MPa)
coε (%)
lf (MPa)
'l cof f '*
ccf (MPa)
*ccε
(%) 28.6 0.260 1.05 0.037 33.6 0.47028.6 0.260 2.1 0.073 36.4 0.67528.6 0.260 4.2 0.147 48.1 1.38528.6 0.260 8.4 0.294 65.2 2.37528.6 0.260 14.7 0.514 92.3 3.42528.6 0.260 21.0 0.734 114.5 4.46047.4 0.280 2.15 0.045 57.7 0.43047.4 0.280 4.3 0.091 67.3 0.69047.4 0.280 8.6 0.181 83.6 1.46047.4 0.280 17.2 0.363 118.1 2.53047.4 0.280 30.1 0.635 161.1 3.600
Imran and Pantazopoulou (1996)
47.4 0.28 43.0 0.907 204.7 4.73047.2 0.202 8.3 0.176 79.7 1.34947.2 0.202 16.5 0.351 109.7 1.56847.2 0.202 24.8 0.527 130.7 2.04947.2 0.202 33.1 0.702 144.2 2.420
Ansari and Li (1998)
47.2 0.202 41.4 0.878 166.9 2.95032.8 0.180 1.5 0.046 45.5 0.26032.8 0.180 4.5 0.137 55.3 0.41032.8 0.180 9.0 0.274 65.7 0.83038.8 0.210 1.5 0.039 47.8 0.34038.8 0.210 4.5 0.116 58.2 0.520
Sfer et al. (2002)
38.8 0.210 9.0 0.232 66.5 0.63027.2 0.182 1.875 0.069 36.2 0.30027.2 0.182 1.875 0.069 35.7 0.28927.2 0.182 7.5 0.276 50.1 0.43527.2 0.182 7.5 0.276 47.5 0.57327.2 0.182 15.0 0.551 72.1 0.74427.2 0.182 15.0 0.551 66.6 0.80251.8 0.238 1.875 0.036 64.8 0.32951.8 0.238 1.875 0.036 66 0.38651.8 0.238 7.5 0.145 86.6 0.45651.8 0.238 7.5 0.145 84.2 0.48951.8 0.238 12.5 0.241 99.3 0.492
Tan and Sun (2004)
51.8 0.238 12.5 0.241 103.3 0.662
71
LVDT2
LVDT1
SG2
SG3
SG1
SG4
(a) Unconfined specimens
LVDT2
LVDT1
SG5
SG4
SG3
SG2
SG1
SG6
45°
overlapping zone
SG8
SG7
(b) FRP-confined specimens
Fig 3.1 Instrumentation of specimens
72
Fig. 3.2 Test setup
Fig. 3.3 Typical failure mode
73
-10 -5 0 5 10 150
1
2
3
Normalized Strain
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimens 01, 02, 03fl /f′co=0.16
Specimens 04, 05, 06fl /f′co=0.30
Specimens 07, 08, 09fl /f′co=0.42
TestRefined Version ofTeng et al.′s (2007a)Model
(a) Specimens 01 to 09
-10 -5 0 5 10 150
0.5
1
1.5
2
2.5
Normalized Strain
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimens 10, 11fl /f′co=0.16
Specimens 12, 13fl /f′co=0.32
TestRefined Version ofTeng et al.′s (2007a)Model
(b) Specimens 10 to 13
74
-5 0 5 100
0.5
1
1.5
2
2.5
Normalized Strain
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimens 14, 15, 16fl /f′co=0.13
Specimens 17, 18, 19fl /f′co=0.27
TestRefined Version ofTeng et al.′s (2007a)Model
(c) Specimens 14 to 19
-10 -5 0 5 100
0.5
1
1.5
2
Normalized Strain
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimens 20, 21fl /f′co=0.079
Specimens 22, 23fl /f′co=0.19
Specimens 24, 25fl /f′co=0.25
TestOriginal Version of Teng et al.′s (2007a) ModelRefined Version of Teng et al.′s (2007a) Model
(d) Specimens 20 to 25
75
-10 -5 0 5 100
0.6
1.2
1.8
Normalized Strain
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimens 28, 29fl /f′co=0.067
Specimens 30, 31fl /f′co=0.13
Specimens 32, 33fl /f′co=0.21
TestOriginal Version of Teng et al.′s (2007a) ModelRefined Version of Teng et al.′s (2007a) Model
(e) Specimens 28 to 33
-8 -4 0 4 80
0.5
1
1.5
2
Normalized Strain
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimens 26, 27fl /f′co=0.10
Specimens 40, 41fl /f′co=0.10
Specimens 46, 47, 48fl /f′co=0.24
TestRefined Version ofTeng et al.′s (2007a)Model
(f) Specimens 34 to 39
76
-6 -4 -2 0 2 4 60
0.4
0.8
1.2
1.6
Normalized Strain
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimens 42, 43fl /f′co=0.073
Specimens 44, 45fl /f′co=0.18
TestRefined Version ofTeng et al.′s (2007a)Model
(g) Specimens 42to 45
-8 -4 0 4 80
0.5
1
1.5
2
Normalized Strain
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimens 26, 27fl /f′co=0.10
Specimens 40, 41fl /f′co=0.10
Specimens 46, 47, 48fl /f′co=0.24
TestRefined Version ofTeng et al.′s (2007a)Model
(h) Specimens 26,27; 40,41; 46,47,48
Fig. 3.4 Experimental stress-strain curves of FRP-confined concrete
77
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Normalized Axial Strain εc /εco
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Active-Confinement ModelPassive-Confinement Model
Fig. 3.5 Generation of a stress-strain curve for FRP-confined concrete
78
0 0.2 0.4 0.6 0.8 11
1.5
2
2.5
3
3.5
4
4.5
5
Confinement Ratio fl /f′co
Nor
mal
ized
Pea
k A
xial
Stre
ss f′ * cc
/f′ c
o
Eq. 3.2Eq. 3.3Eq. 3.4Eq. 3.5Eq. 3.6
Fig. 3.6 Comparison of predictions for the peak axial stress
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
16
18
20
Confinement Ratio fl /f′co
Nor
mal
ized
Axi
al S
train
at P
eak
Axi
al S
tress
ε* cc
/ εco
Eq. 3.2 & Eq. 3.7Eq. 3.3 & Eq. 3.7Eq. 3.4 & Eq. 3.7Eq. 3.5 & Eq. 3.7Eq. 3.6 & Eq. 3.7Eq. 3.14
Fig. 3.7 Comparison of predictions for the axial strain at peak axial stress
79
0 0.005 0.01 0.015 0.02 0.025-0.02
-0.015
-0.01
-0.005
0
Axial Strain εc
Late
ral S
train
εl
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Harries and Kharel (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 28, 29)
(a) Lateral-to-axial strain curves
0 0.005 0.01 0.015 0.02 0.0250
1
2
3
4
5
6
Axial Strain εc
Tan
gent
Dila
tion
Rat
io µ
t
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Harries and Kharel (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 28, 29)
(b) Tangent dilation ratio
80
0 0.005 0.01 0.015 0.02 0.0250
0.5
1
1.5
2
2.5
3
Axial Strain εc
Sec
ant D
ilatio
n R
atio
µs
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Harries and Kharel (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 28, 29)
(c) Secant dilation ratio
0 0.005 0.01 0.015 0.02 0.0250
10
20
30
40
50
60
70
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Harries and Kharel (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 28, 29)
(d) Stress-strain curves
Fig. 3.8 Weakly-confined concrete
81
0 0.01 0.02 0.03 0.04 0.05-0.02
-0.015
-0.01
-0.005
0
Axial Strain εc
Late
ral S
train
εl
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Harries and Kharel (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 24, 25)
(a) Lateral-to-axial strain curves
0 0.01 0.02 0.03 0.04 0.050
0.5
1
1.5
2
2.5
3
Axial Strain εc
Tan
gent
Dila
tion
Rat
io µ
t
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Harries and Kharel (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 24, 25)
(b) Tangent dilation ratio
82
0 0.01 0.02 0.03 0.04 0.050
0.4
0.8
1.2
1.6
Axial Strain εc
Sec
ant D
ilatio
n R
atio
µs
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Harries and Kharel (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 24, 25)
(c) Secant dilation ratio
0 0.01 0.02 0.03 0.04 0.050
10
20
30
40
50
60
70
80
90
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Harries and Kharel (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 24, 25)
(d) Stress-strain curves
Fig. 3.9 Moderately-confined concrete
83
0 0.01 0.02 0.03 0.04 0.05-0.01
-0.008
-0.006
-0.004
-0.002
0
Axial Strain εc
Late
ral S
train
εl
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 34, 35)
(a) Lateral-to-axial strain curves
0 0.01 0.02 0.03 0.04 0.050
0.3
0.6
0.9
1.2
Axial Strain εc
Tan
gent
Dila
tion
Rat
io µ
t
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 34, 35)
(b) Tangent dilation ratio
84
0 0.01 0.02 0.03 0.04 0.050
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Axial Strain εc
Sec
ant D
ilatio
n R
atio
µs
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 34, 35)
(c) Secant dilation ratio
0 0.01 0.02 0.03 0.04 0.050
20
40
60
80
100
120
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 34, 35)
(d) Stress-strain curves
Fig. 3.10 Heavily-confined concrete
85
0 0.002 0.004 0.006 0.008 0.01-0.015
-0.012
-0.009
-0.006
-0.003
0
Axial Strain εc
Late
ral S
train
εl
Linear EquationTeng et al. (2007a)
(a) Descending type: lateral-to-axial strain curves
0 0.002 0.004 0.006 0.008 0.010
10
20
30
40
50
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
Linear EquationTeng et al. (2007a)
(b) Descending type: stress-strain curves
86
0 0.004 0.008 0.012 0.016 0.02-0.015
-0.012
-0.009
-0.006
-0.003
0
Axial Strain εc
Late
ral S
train
εl
Linear EquationTeng et al. (2007a)
(c) Ascending type: lateral-to-axial strain curves
0 0.004 0.008 0.012 0.016 0.020
10
20
30
40
50
60
70
80
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
Linear EquationTeng et al. (2007a)
(d) Ascending type: stress-strain curves
Fig. 3.11 Stress-strain curves predicted by Teng et al.’s (2007a) model: effect of
lateral-to-axial strain equation
87
0 5 10 15 200
5
10
15
20
Normalized Ultimate Axial Strain εcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Model ofMirmiran and Shahawy (1997a)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(a) Ultimate axial strain
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Model ofMirmiran and Shahawy (1997a)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(b) Axial stress at ultimate axial strain
Fig. 3.12 Performance of Mirmiran and Shahawy’s model in predicting the
ultimate condition
88
0 5 10 15 200
5
10
15
20
Normalized Ultimate Axial Strain εcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Model ofSpoelstra and Monti (1999)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(a) Ultimate axial strain
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Model ofSpoelstra and Monti (1999)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(b) Axial stress at ultimate axial strain
Fig. 3.13 Performance of Spoelstra and Monti’s model in predicting the ultimate
condition
89
0 5 10 15 200
5
10
15
20
Normalized Ultimate Axial Strain εcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Model ofFam and Rizkalla (2001)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(a) Ultimate axial strain
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Model ofFam and Rizkalla (2001)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(b) Axial stress at ultimate axial strain
Fig. 3.14 Performance of Fam and Rizkalla’s model in predicting the ultimate
condition
90
0 10 20 30 40 500
10
20
30
40
50
Normalized Ultimate Axial Strain εcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Model ofChun and Park (2002)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(a) Ultimate axial strain
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Model ofChun and Park (2002)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(b) Axial stress at ultimate axial strain
Fig. 3.15 Performance of Chun and Park’s model in predicting the ultimate
condition
91
0 5 10 15 200
5
10
15
20
Normalized Ultimate Axial Strain εcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Model ofHarries and Kharel (2002)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(a) Ultimate axial strain
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Model ofHarries and Kharel (2002)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(b) Axial stress at ultimate axial strain
Fig. 3.16 Performance of Harries and Kharel’s model in predicting the ultimate
condition
92
0 5 10 15 200
5
10
15
20
Normalized Ultimate Axial Strain εcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Model ofMarques et al. (2004)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(a) Ultimate axial strain
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Model ofMarques et al. (2004)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(b) Axial stress at ultimate axial strain
Fig. 3.17 Performance of Marques et al.’s model in predicting the ultimate
condition
93
0 5 10 15 200
5
10
15
20
Normalized Ultimate Axial Strain εcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Model ofBinici (2005)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(a) Ultimate axial strain
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Model ofBinici (2005)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(b) Axial stress at ultimate axial strain
Fig. 3.18 Performance of Binici’s model in predicting the ultimate condition
94
0 5 10 15 20 250
5
10
15
20
25
Normalized Ultimate Axial Strain εcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Model ofTeng et al. (2007a)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(a) Ultimate axial strain
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Model ofTeng et al. (2007a)
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(b) Axial stress at ultimate axial strain
Fig. 3.19 Performance of Teng et al.’s model in predicting the ultimate condition
95
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
Confinement Ratio fl /f′co
Nor
mal
ized
Pea
k A
xial
Stre
ss f′ * cc
/f′ c
o
↑Eq. 3.6
Imran and Pantazopoulou (1996)Ansari and Li (1998)Sfer et al. (2002)Tan and Sun (2004)
Fig. 3.20 Normalized peak axial stress versus confinement ratio
0 0.2 0.4 0.6 0.8 10
5
10
15
20
Confinement Ratio fl /f′co
Nor
mal
ized
Axi
al S
train
at P
eak
Axi
al S
tress
ε* cc
/ εco
← Eq. 3.14
Imran and Pantazopoulou (1996)Ansari and Li (1998)Sfer et al. (2002)Tan and Sun (2004)Specimen 17Specimen 18Specimen 19
Fig. 3.21 Normalized axial strain at peak axial stress versus confinement ratio
96
0 0.01 0.02 0.03 0.04 0.050
20
40
60
80
100
120
140
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
Test (Ansari and Li 1998)Original Base ModelModified Base Model
Fig. 3.22 Comparisons of original base model with modified base model
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Refined Version ofTeng et al.′s (2007a) Model
Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
Fig. 3.23 Performance of refined version of Teng et al.’s model
97
CHAPTER 4
DESIGN-ORIENTED STRESS-STRAIN MODELS FOR
FRP-CONFINED CONCRETE
4.1 INTRODUCTION
Chapter 3 has presented a comprehensive assessment of analysis-oriented models.
Although the complex nature of analysis-oriented models prevents them from
being directly employed for design use, they can be used to produce numerical
results for the development of design-oriented stress-strain models. The refined
version of Teng et al. (2007a)’s model presented in Chapter 3 is used in this
chapter for this purpose.
A great number of design-oriented models have been proposed (Fardis and Khalili
1982; Karbhari and Gao 1997; Samaan et al. 1998; Miyauchi et al. 1999; Saafi et
al. 1999; Toutanji 1999; Lillistone and Jolly 2000; Xiao and Wu 2000, 2003; Lam
and Teng 2003; Berthet et al. 2006; Harajli 2006; Saenz and Pantelides 2007; Wu
et al. 2007; Youssef et al. 2007). Design-oriented models generally comprise a
closed-form stress-strain equation and ultimate condition equations derived
directly from the interpretation of experimental results. The accuracy of design-
oriented models depends highly on the definition of the ultimate condition of
FRP-confined concrete. Existing design-oriented models have been assessed by a
number of studies (De Lorenzis and Tepfers 2003; Teng and Lam 2004; Bisby et
al. 2005). Among these models, the model proposed by Lam and Teng (2003)
appears to be advantageous over other models due to its simplicity and accuracy.
This model, with some modification, has been adopted by the design guidance for
the strengthening of concrete structures using FRP issued by the Concrete Society
98
(Concrete Society 2004) in the UK. More recently, this model has also been
adopted by ACI-440.2R (2008) with only very slight modifications. This model
adopts a simple form that naturally reduces to that for unconfined concrete when
no FRP is provided. Its simple form also caters for easy improvements to the
definition of the ultimate condition (the ultimate axial strain and compressive
strength) of FRP-confined concrete, which are the key to the accurate prediction
of stress-strain curves of FRP-confined concrete by this model.
Although Lam and Teng’s (2003) model was developed on the basis of a large test
database, a number of significant issues could not be readily resolved using the
test database available to them at that time. In particular, there was considerable
uncertainty with the hoop tensile rupture strain reached by the FRP jacket, which
has an important bearing on the definition of the ultimate condition. According to
a subsequent study by the same authors (Lam and Teng 2004), the distribution of
FRP hoop strain is highly non-uniform and the FRP hoop strains measured in the
overlapping zone of the FRP jacket are much lower than those measured
elsewhere. The lower FRP hoop strains in the overlapping zone reduce the
average hoop strain but do not result in lower confining pressure in this zone
because the FRP jacket is thicker there (Lam and Teng 2004). This observation
suggests that hoop strain readings within the overlapping zone have to be
excluded when interpreting the behavior of FRP-confined concrete, as these
readings reflect neither the actual strain capacity of the confining jacket nor the
actual dilation properties of the confined concrete. However, such important
processing of the hoop strain readings is not possible with test data collected by
Lam and Teng (2003) from the existing literature at that time, for which the
precise number and locations of strain gauges for measuring hoop strains are
generally not reported.
In addition to the uncertainty in the FRP hoop strains, the different testing
procedures (particularly the methods for axial strain measurement) adopted by
different researchers also have a bearing on the interpretation of the test data
covered by that test database. To address the deficiencies of that database and
hence those of Lam and Teng’s (2003) stress-strain model based on that database,
a large number of additional tests on FRP-confined concrete cylinders were
99
conducted under standardized testing conditions at the Hong Kong Polytechnic
University. The test database has been presented in Chapter 3 and it comprises test
data reported in Lam and Teng (2004), Lam et al. (2006) and Teng et al. (2007b)
plus test data from new tests conducted by the author (see Chapter 3). In parallel
with the experimental work, theoretical modeling work on FRP-confined concrete
was also carried out (see Chapter 3). With these new test results and the new
understandings from experimental and theoretical work, it then became feasible to
explore the refinement of Lam and Teng’s (2003) stress-strain model.
In this chapter, more accurate expressions for the ultimate axial strain and the
compressive strength of FRP-confined concrete are first proposed on a combined
experimental and analytical basis. Two modified versions of Lam and Teng’s
model based on these new expressions are next presented. The first version
involves only simple modification to the original model by updating the ultimate
condition equations. The second version differs from the first version in that it is
capable of predicting stress-strain curves with a descending branch when the
confinement is weak.
As in Chapter 3, the term “stress-strain” is generally used to refer to “axial stress-
axial strain”. The latter is used only when the axial stress-lateral strain response of
the concrete is also discussed. The sign convention adopted is the same as that of
Chapter 3: in the concrete, compressive stresses and strains are positive, but in the
FRP, tensile stresses and strains are positive.
4.2 TEST DATABASE
4.2.1 General
The test database used in this chapter is exactly the same as the one reported in
Chapter 3. A brief recap is given here. This database contains the results of 48
tests on concrete cylinders (152 mm × 305 mm) confined with varying amounts of
carbon FRP (CFRP) and Glass FRP (GFRP), with the compressive strength of
100
unconfined concrete ranging from 33.1 MPa to 47.6 MPa. The key features of
this test database are summarized as follows
'cof
1) the FRP jackets were all formed via the wet lay-up process and all had hoop
fibers only. For each batch of concrete, two or three control specimens of the
same size were also tested, from which the average values of the compressive
strength of unconfined concrete 'cof and the corresponding axial strain coε
were found;
2) the hoop strain hε of the FRP jacket was found as the average value of the
readings from five hoop strain gauges with a gauge length of 20 mm located
outside the overlapping zone (150 mm in length);
3) the axial strain of concrete cε was found as the average value of the readings
from two linear variable displacement transducers (LVDTs) at 180° apart and
covering the mid-height region of 120 mm. The lateral strain of concrete lε
was assumed to have the same magnitude as but the opposite sign to the
corresponding hε according to the sign convention adopted in this chapter;
and
4) the test database covers a wide range of FRP confinement levels. The most
heavily-confined specimen experienced an increase in the concrete strength of
about 320% while the most weakly-confined specimen exhibited a stress-
strain curve with a post-peak descending branch with negligible strength
enhancement.
For ease of discussion of the test results, some basic ratios are explained here,
namely, the confinement ratio 'l cof f , the confinement stiffness ratio Kρ and the
strain ratio ερ . The confinement ratio is commonly accepted to identify the
amount of FRP confinement with lf being the maximum confining pressure
provided by an FRP jacket. The confinement stiffness ratio represents the stiffness
ratio of the FRP jacket to the concrete core, and the strain ratio is a measure of the
101
strain capacity of the FRP jacket. It should be noted that the confinement ratio can
be taken as the product of the other two. The mathematical definitions of these
three ratios are given below
,' '
2 frp h ruplK
co co
E tff f D ε
ερ ρ= = (4.1a)
( )'
2 frpK
co co
E tf D
ρε
= (4.1b)
,h rup
coε
ερ
ε= (4.1c)
where frpE is the elastic modulus of FRP in the hoop direction, t is the thickness
of the FRP jacket, ,h rupε is the hoop strain of FRP at the rupture of the jacket due
to hoop tensile stresses, and is the diameter of the confined concrete cylinder. D
4.2.2 Stress-Strain Curves
The stress-strain curves as well as the key results from all tests are given in
Chapter 3. Only the results of some typical tests are given here. Eight typical
stress-strain curves are shown in Fig. 4.1, where the lateral strains lε are shown
on the left and the axial strains cε are shown on the right. Both the axial strain and
the lateral strain are normalized by the corresponding value of coε , while the axial
stress cσ is normalized by the corresponding value of 'cof . In Fig. 4.1, both the
ascending type and descending type stress-strain curves are shown. All the
ascending type stress-strain curves exhibit the well-known bi-linear shape, and
both the compressive strength 'ccf and the ultimate axial strain cuε of confined
concrete are reached at the same point representing the rupture of the confining
jacket. Significant enhancement in both the strength and axial strain of concrete is
seen for this type of stress-strain curves. By contrast, 'ccf is reached before the
rupture of the jacket for the descending type stress-strain curves with little
strength enhancement. For this type of stress-strain curves, if the axial stress at
102
ultimate axial strain of confined concrete 'cuf falls below the compressive strength
of unconfined concrete 'cof , the concrete is referred to as insufficiently-confined
concrete in this chapter. On the other hand, concrete whose stress-strain curve
falls into the remainder of the descending type or the ascending type stress-strain
curves is referred to as sufficiently-confined concrete. The key information of
these eight specimens, together with some other specimens that are used for
discussion or comparison in this chapter can be found in Table 3.1. The naming of
the specimens follows that used in Chapter 3.
4.2.3 Ultimate Condition
Using an analysis-oriented model for FRP-confined concrete (Spoelstra and Monti
1999), Lam and Teng (2003) demonstrated that the stiffness of the FRP jacket
affects both the ultimate axial strain and the compressive strength of FRP-
confined concrete. The present tests offer clear experimental evidence on the
effect of jacket stiffness, as illustrated in Fig. 4.2. Fig. 4.2a shows the
experimental stress-strain curves of specimens 31 and 45. The former is confined
with a GFRP jacket with 'cof = 45.9 MPa and a confinement ratio of 0.140, and the
latter is confined with a CFRP jacket with 'cof = 44.2 MPa and a confinement ratio
of 0.141. It can be seen that although the unconfined concrete strength and the
confinement ratio are both very similar for the two cases, the stress-strain curves
deviate from each other significantly because of the difference in the confinement
stiffness. The stress-strain curve of specimen 31 having a smaller confinement
stiffness ratio terminates at a lower axial stress but a larger axial strain. Similarly,
Fig. 4.2b shows the stress-strain curves of specimens 28 and 42. The former had '
cof = 45.9 MPa and a confinement ratio of 0.059 while the latter had 'cof = 44.2
MPa and a confinement ratio of 0.055. It is interesting to note that specimen 28
having a smaller confinement stiffness ratio exhibits a stress-strain curve of the
descending type while specimen 42 exhibits a stress-strain curve of the ascending
type. This observation suggests that the effect of confinement stiffness on the
ultimate condition of FRP-confined concrete should not be neglected when
developing an accurate design-oriented stress-strain model. It further suggests that
103
the confinement stiffness also plays a significant role in determining whether
sufficient confinement is achieved.
4.3 LAM AND TENG’S STRESS-STRAIN MODEL FOR FRP-
CONFINED CONCRETE
Lam and Teng’s design-oriented stress-strain model (Lam and Teng 2003) was
based on the following assumptions: (i) the stress-strain curve consists of a
parabolic first portion and a linear second portion, as given in Fig. 4.3; (ii) the
slope of the parabola at zero axial strain (the initial slope) is the same as the
elastic modulus of unconfined concrete; (iii) the nonlinear part of the first portion
is affected to some degree by the presence of an FRP jacket; (iv) the parabolic
first portion meets the linear second portion smoothly (i.e. there is no change in
slope between the two portions where they meet); (v) the linear second portion
terminates at a point where both the compressive strength and the ultimate axial
strain of confined concrete are reached; and (vi) the linear second portion
intercepts the axial stress axis at a stress level equal to the compressive strength of
unconfined concrete. The justifications for the above assumptions are given in
detail in Lam and Teng (2003) and are thus not discussed herein.
Based on these assumptions, Lam and Teng’s stress-strain model for FRP-
confined concrete is described by the following expressions:
( )22 2
'4c
c c cco
E EE
f cσ ε ε−
= − for 0 c tε ε≤ < (4.2a)
'2c co cf Eσ ε= + for t c cuε ε ε≤ ≤ (4.2b)
where cσ and cε are the axial stress and the axial strain, is the elastic modulus
of unconfined concrete, is the slope of the linear second portion. The parabolic
first portion meets the linear second portion with a smooth transition at
cE
2E
tε which is
given by
104
'
2
2 cot
c
fE E
ε =−
(4.3)
The slope of the linear second portion is given by 2E
' '
2cc co
cu
f fEε−
= (4.4)
This model allows the use of test values or values specified by design codes for
the elastic modulus of unconfined concrete . Lam and Teng (2003) proposed
the following equation to predict the ultimate axial strain
cE
cuε of confined concrete:
1.451.75 12cuK
coε
ε ρ ρε
= + (4.5)
Lam and Teng’s (2003) compressive strength equation takes the following form
'
' '1 3.3cc l
co co
f ff f
= + , if ' 0.07l
co
ff
≥ (4.6a)
'
' 1cc
co
ff
= , if ' 0.07l
co
ff
< (4.6b)
Eqs 4.6a and 4.6b are for sufficiently and insufficiently confined concrete
respectively. A minimum value of 'l cof f = 0.07 for sufficient confinement was
originally suggested by Spoelstra and Monti (1999) and was used in Lam and
Teng’s (2003) model with justification using test data available to them.
A comparison of Lam and Teng’s model with the test data of the present database
is shown in Fig. 4.4. In predicting the compressive strength and the ultimate axial
strain, a constant value of coε = 0.002 and experimental value of ,h rupε were used.
The elastic modulus of unconfined concrete was taken to be '4730c cE f= o (in
MPa). It can be seen from Fig. 4.4 that Eqs 4.5 and 4.6 overestimate the ultimate
105
axial strain of concrete at high levels of confinement and the compressive strength
of concrete at low levels of confinement. In addition, the effect of confinement
stiffness is only accounted for in the ultimate axial strain equation, but not in the
compressive strength equation. Refinement of these equations is therefore
necessary to provide more accurate predictions.
4.4 GENERALIZATION OF EQUATIONS
To take the effect of confinement stiffness into account, the expressions for the
ultimate axial strain and the compressive strength of FRP-confined concrete are
generalized here. In this regard, Eq. 4.5 can be cast into the following form:
( ) ( )cuK
co
C F fε ε ε εε ρ ρε
= + (4.7)
where ( )KFε ρ and ( )fε ερ are functions of these two ratios respectively, and Cε is
a constant.
Since the confinement ratio is the product of the confinement stiffness ratio and
the strain ratio, Eq. 4.6 can be cast into the following general form which is
similar to Eq. 4.7
'
' ( ) (ccK
co
f C F ff
)σ σ σ ερ ρ= + (4.8)
where ( )KFσ ρ and ( )fσ ερ are also functions of the confinement stiffness ratio
and the strain ratio respectively, and Cσ is a constant.
The above generalization allows the effect of confinement stiffness to be
explicitly accounted for in both the ultimate axial strain and the compressive
strength equations.
106
4.5 NEW EQUATIONS FOR THE ULTIMATE CONDITION
4.5.1 Ultimate Axial Strain
Based mainly on the interpretation of the test results in the test database, the
following improved equation for the ultimate axial strain of FRP-confined
concrete is proposed based on the best-fit results
0.8 1.451.75 6.5cuK
coε
ε ρ ρε
= + (4.9)
where the strain at the compressive strength of unconfined concrete coε was taken
to be 0.002 in determining the strain ratio ερ , since this value is commonly
accepted in existing design codes for RC structures. It is therefore suggested that
this value be used when Eq. 4.9 is used in a design specification. The first term on
the right side of Eq. 4.9 was taken to be 1.75 so that it predicts 0.0035cuε = when
no FRP confinement is provided and the coefficient and the exponents of the
second term were determined by the best-fit values of the test results. It should be
noted that 0.0035cuε = is also a commonly accepted value for the ultimate axial
strain of unconfined concrete [e.g. ENV 1992-1-1 (1992); BS 8110 (1997)] and
the constant 1.75 may be adjusted to meet the requirement of a specific design
code. Close agreement between the test results and the predictions of Eq. 4.9 is
seen in Fig. 4.5.
4.5.2 Compressive Strength
The compressive strength equation was refined on a combined experimental and
analytical basis. Fig. 4.1 has shown that both the axial stress-lateral strain curves
and the axial stress-axial strain curves exhibit a clearly bi-linear shape, with the
two portions smoothly connected by a transition zone near the compressive
strength of unconfined concrete. The shape of the second portion (the portion after
the transition zone) is very close to a straight line. It can also be seen that the
second portion of an axial stress-lateral strain curve is closer to a straight line than
107
that of an axial stress-axial strain curve. A careful study showed that the second
portion of the experimental axial stress-lateral strain curves from all tests
intercepted the axial stress axis at a stress level which is very close to the
corresponding value of compressive strength of unconfined concrete when this
portion was approximated using a fitted straight line. Such straight lines can be
expressed using the following equation
' 1c
co co
Kf
lσ εε
= + (4.10)
where constant Cσ is taken to be unity because of the reason given above. K is
the slope of the fitted straight line. It is obvious that the axial stress cσ reaches
'cuf when lε = ,h rupε− , and Eq. 4.10 becomes
'
,' 1 1h rupcu
co co
f Kf
K ε
ερ
ε= − = − (4.11)
Comparing Eq. 4.11 with Eq. 4.8 leads to 1Cσ = and ( )Fε ε ερ ρ= , while the
slope ( )KK Fσ ρ= remains undetermined.
To establish the function ( )KFσ ρ in Eq. 4.11, a parametric study was conducted
using the refined version of the analysis-oriented stress-strain model of Teng et al.
(2007a) for FRP-confined concrete (see Chapter 3). The parametric study covered
concrete cylinders of 152 mm in diameter confined with either CFRP or GFRP,
with 'cof ranging from 20 to 50 MPa and a wide range of jacket thickness to
represent different values of confinement stiffness. The material properties and
parameters examined are given in Table 4.1. In this parametric study, it was
assumed coε = 4 49.37 10 co'f−× ⋅ (Popovics 1973) as recommended in Chapter 3.
The parametric study consisted of three steps: 1) produce a family of axial stress-
lateral strain curves of a concrete cylinder confined with a certain type of FRP
jacket with varying jacket stiffness; 2) fit the second portion of these axial stress-
lateral strain curves using the best-fit straight lines with the point of interception
108
with the vertical axis ( 'c
cofσ ) fixed at unity and find the slopes of the fitted straight
lines; and 3) identify ( )KFσ ρ by finding the relationship between the slope and
ig. 4.6a demonstrates the first two steps for a concrete cylinder with
the confinement stiffness ratio.
F 'cof = 30
MPa confined with a CFRP jacket with varying thicknesses. Each stress-strain
curve in Fig. 4.6a corresponds to a particular value of the confinement stiffness
ratio. For each stress-strain curve with the portion starting from 0.5l coε ε= − was
fitted using a straight line as indicated by a dashed line in Fig. 4.6a. The slope of
the fitted line was then plotted against the confinement stiffness ratio as shown in
Fig. 4.6b, which is the last step of the process. A curve for GFRP-confined
concrete is also shown in Fig. 4.6b. This curve was generated using the numerical
results for the same concrete cylinder used in Fig. 4.6a but confined with a GFRP
jacket. It can be seen that the two curves almost overlap with each other and they
can be closely fitted using the following expression
(4.12)
q. 4.12 also provides accurate predictions for other values of
0.9( ) 3.2 0.06K KK Fσ ρ ρ= = − +
E 'cof studied. With
( )KFσ ρ determined, Eq. 4.11 becomes
( )'
cuf 0.9' 1 3.2 0.06K
cof ερ ρ= + − (4.13)
ig. 4.7 shows that Eq. 4.13 compares well with the test data of the test database. F
In predicting 'cuf for use in Fig. 4.7, experimental values of coε were used, aiming
to verify the validity of Eq. 4.13 experimentally although slightly different coε
values were used in the parametric study.
As the nonlinear relationship between ( )KFσ ρ and Kρ in Eq. 4.13 is slightly
inconvenient for design use, a simple linear equation is proposed as follows
109
( )'
' 1 3.5cuK
co
ff
0.01 ερ ρ= + − (4.14)
It should be noted that Eq. 4.14 predicts the axial stress at the ultimate axial strain,
but not the compressive strength
'ccf of FRP-confined concrete, although they are
nly different from each other when the stress-strain curve has a descending o
branch. Since Lam and Teng’s (2003) model neglects the small strength
enhancement in insufficiently confined concrete and employs a horizontal line to
represent any possible descending branch, Eq. 4.14 can readily be modified so that
it can be incorporated into Lam and Teng’s model (2003) for predicting 'ccf . It is
proposed that
( )'
1 3.5 0.01ccf' K
cof ερ ρ= + − , if 0.01 (4.15a) Kρ ≥
'
' 1cc
co
ff
= , if 0.01Kρ < (4.15b)
The performance of Eq. against
redicting
4.15 the test results is shown in Fig. 4.8. In
p 'ccf , experimental values of the FRP hoop rupture strain was used but
coε = 0.002 was used for the reason mentioned earlier. Eq. 4.15 defines a
minimum confinement stiffness ratio Kρ of 0.01 below which the FRP is assumed
to result in enhancement in the compressive strength of confined concrete.
4.6 MODIFICATION TO LAM AND TENG’S MODEL: VERSION (I)
The newly
no
defined ultimate axial strain and compressive strength equations for
RP-confined concrete as given by Eqs 4.9 and 4.15 can be directly incorporated
on
) of the modified Lam and Teng model. Fig. 4.9 shows a comparison between
F
into Lam and Teng’s (2003) model. The resulting model is referred to as Versi
(I
the predictions from the original model (Lam and Teng 2003) and Version (I) of
the modified model for three sets of specimens. Version (II) is presented in the
110
next section. The first two sets of specimens (specimens 22 and 23 in Fig. 4.9a
and specimens 24 and 25 in Fig. 4.9b) had a compressive strength of unconfined
concrete of 39.6 MPa and were confined with 2 and 3 plies of GFRP, respectively.
The confinement ratio was 0.186 for the 2-ply GFRP jacket and 0.278 for the 3-
ply GFRP jacket, while the confinement stiffness ratio was 0.018 for the former
and 0.027 for the latter. The last set of specimens (specimens 17, 18 and 19 in Fig
8c) had a compressive strength of unconfined concrete of 38.9 MPa and were
confined with 2 plies of CFRP, with a confinement ratio of 0.274 and a
confinement stiffness ratio of 0.055. The confinement stiffness ratios are based on
a constant value of coε = 0.002. This constant value of coε = 0.002 was also used
with experimental values of ,h rupε in determining other ratios and predicting the
stress-strain curves.
It can be seen that the original model gives close predictions for the specimens
with two plies of CFRP (Fig. 4.9c), but not for those with two and three plies of
GFRP (Figs 4.8b and 4.8c). Note that the confinement ratio for the 3-ply GFRP
cket is similar to that for the 2-ply CFRP jacket (0.278 versus 0.274), but the
eng model simply use a horizontal line to represent the post-peak descending
m
nd Teng’s model reduces naturally to the stress-strain model specified in
ja
confinement stiffness ratio for the former (0.027) is only half that for the latter
(0.055). The inaccuracy of the original model in this case is due to the omission of
the effect of confinement stiffness in the compressive strength equation (Eq. 4.5).
It is evident that the use of Eqs 4.9 and 4.15 to replace Eqs 4.5 and 4.6 improves
the performance of the model considerably for cases of low confinement stiffness.
4.7 MODIFICATION TO LAM AND TENG’S MODEL: VERSION (II)
Both the original Lam and Teng model and Version (I) of the modified Lam and
T
branch resulting from weak confinement. This simplification ensures that La
a
Eurocode 2 (ENV 1992-1-1 1992) for unconfined concrete when no FRP
confinement is provided. However, in applications where ductility is the major
concern, for example, in the seismic retrofit of RC columns, a small amount of
FRP confinement may suffice in terms of ductility improvement even though such
111
a small amount of confinement may not be sufficient for a significant
enhancement in the compressive strength. In such a situation, a more precise
definition of the post-peak stress-strain response of FRP-confined concrete is of
interest to structural engineers. In this section, an alternative modification to Lam
and Teng’s model to better cater for such cases is presented. The resulting stress-
strain model is referred to as Version (II) of the modified Lam and Teng model.
In order to cover both the ascedning and descending types of stress-strain curves
of FRP-confined concrete, the stress-strain curve of unconfined concrete for
design use is defined in a form similar to the well-known Hognestad’s (1951)
tress-strain curve. It is assumed that the stress-strain curve of unconfined s
concrete has a linear post-peak descending branch, which terminates at an axial
strain of 0.0035 after a 15% drop from the compressive strength of unconfined
concrete. Note that in Hognestad’s (1951) model the ultimate axial strain of
unconfined concrete is defined as 0.0038, instead of 0.0035. The latter is specified
in some design codes such as BS 8110 (1997) and ENV 1992-1-1 (1992).
For FRP-confined concrete, the parabolic first portion of Lam and Teng’s model
as given by Eq. 4.2a remains unchanged, while the linear second portion as given
by Eq. 4.2b is modified, leading to the following expressions:
( )22
c'
'2
( 0 )4
if 0.01
cc c t
co
c co c K
E EE
f
f E' '
t' ( )( ) if 0.01 c cuco cu
co c co Kcu co
f ff
ε ε ε
σ ε ρε ε ε
ε ε ρε ε
⎪⎪ < ≤⎨ −⎪ − − <⎪⎪ −⎩⎩
(4.16)
where
⎧ −− ≤ ≤⎪
⎪⎪= ⎧⎨ + ≥
2E , cuε and 'ccf are determined by Eqs 4.4, 4.9 and 4.15 respectively, and
'cuf is determined by Eq. 4.14 but is subjected to the following conditions:
0.85 , if 0; or 0.85 , if 0cf f fρ ρ≥ > = = (4.17)
' ' '
K Ku co co
112
Eq. 4.17 limits the axial stress at the ultimate strain of FRP-confined concrete 'cuf
to a minimum value of 0.85 'cof , because the concrete is deem
is stress level, no matter whether the FRP jacket has ruptured or not. Moreover,
ed to have failed at
th
it is suggested that if 'cuf is predicted by Eq. 4.14 to be equal to or smaller than
0.85 'cof , the concrete should be treated as unconfined. In such cases, the values
of 'cuf and cuε in Eq. 4.16 should be taken as 0.85 '
cof and 0.0035, respectively.
It sh ld be noted that Version (II) differs from Version (I) only when the
concrete is insufficiently confined ( 0.01K
ou
ρ < ) and Version (II) is illustrated in
ig. 4.10. The performance of Version (II) of the modified Lam and Teng model
onfined with only 1 ply of GFRP. The first pair (specimens
0 and 21) had a compressive strength of unconfined concrete of 39.6 MPa (Fig.
F
is shown in Fig. 4.11.
Fig. 4.11 is exclusively for two pairs of insufficiently confined specimens. All
these specimens were c
2
4.11a) while this value for the second pair (specimens 28 and 29) was 45.9 MPa
(Fig. 4.11b). The corresponding confinement ratio and confinement stiffness ratio
were 0.079 and 0.0090 for the former, and were 0.067 and 0.0078 for the latter,
respectively. Same as in Fig. 4.7, a constant value of 0.002coε = and
experimental values of ,h rupε were used in predicting the stress-strain curves.
Note that for the specimens covered by Fig. 4.11a, the confinement ratio is greater
than the minimum value of 0.07 for achieving sufficient confinement as suggested
y Lam and Teng (2003), but for the specimens covered by Fig. 4.11b, the
linear second portion for specimens of both Figs 4.10a and 4.10b, which do not
b
confinement ratio is smaller than this minimum value. In both cases, the
confinement stiffness ratios are smaller than the critical value of 0.01 as suggested
in the present study. It can be seen that the original model predicts a
monotonically increasing stress-strain curve for the specimens in Fig. 4.11a, but a
stress-strain curve with a horizontal second portion for the specimens in Fig.
4.11b, although in both cases the test curves are of the decreasing type. The
modified Lam and Teng model (I) predicts stress-strain curves with a horizontal
113
match the test curves well but provide reasonably close approximations. In
comparison, the predictions from the modified Lam and Teng model (II) are in
better agreement with the test stress-strain curves.
4.8 CONCLUSIONS
This paper has presented the results of a recent study aimed at the refinement of a
esign-oriented stress-strain model for FRP-confined concrete developed by Lam
weaknesses of this model have been examined through
omparisons with new test results obtained at The Hong Kong Polytechnic
th a small amount of FRP.
ent stiffness. Version (II) of the modified Lam and Teng model
performs better than Version (I) in predicting stress-strain curves with a
d
and Teng (2003). Some
c
University under standardized test conditions. New equations for predicting the
ultimate axial strain and the compressive strength of FRP-confined concrete have
been developed based on the interpretation of the new test results and results from
a parametric study using a refined version of Teng et al.’s (2007a) recent analysis-
oriented model. Two modified versions of Lam and Teng’s (2003) design-oriented
stress-strain model have been proposed. Version (I) of the modified Lam and
Teng model I) involves only a simple modification to the original model by
updating the ultimate axial strain and compressive strength. Version (II) of the
modified Lam and Teng model attempts to cater for the descending type stress-
strain curves, which are not covered by the original model. The following
conclusions can be drawn:
1) The original Lam and Teng model overestimates the ultimate strain of
concrete confined with a large amount of FRP and the compressive strength
of concrete confined wi
2) The new ultimate strain and compressive strength equations account for the
effect of confinement stiffness explicitly and provide close predictions of test
results.
3) Both modified versions of Lam and Teng’s model provide much closer
predictions of test stress-strain curves than the original model for cases of low
confinem
114
descending branch, although the latter also predicts such stress-strain curves
reasonably well.
115
4.9 REFERENCES
ACI-440.2R (2008). Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, American Concrete Institute, Farmington Hills, Michigan, USA.
Berthet, J.F., Ferrier, E. and Hamelin, P. (2006). “Compressive behavior of concrete externally confined by composite jackets - Part B: modeling”, Construction and Building Materials, 20(5), 338-347.
Binici, B. (2005). “An analytical model for stress–strain behavior of confined concrete”, Engineering Structures, 27(7), 1040-1051.
BS 8110 (1997). Structural Use of Concrete, Part 1. Code of Practice for Design and Construction, British Standards Institution, London, UK.
Concrete Society (2004). Design Guidance for Strengthening Concrete Structures with Fibre Composite Materials, Second Edition, Concrete Society Technical Report No. 55, Crowthorne, Berkshire, UK.
Chun, S.S. and Park, H.C. (2002). “Load carrying capacity and ductility of RC columns confined by carbon fiber reinforced polymer.” Proceedings, 3rd International Conference on Composites in Infrastructure (CD-Rom), San Francisco.
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Mirmiran, A. and Shahawy, M. (1997). “Dilation characteristics of confined concrete”, Mechanics of Cohesive-Frictional Materials, 2 (3), 237-249.
Miyauchi, K., Inoue, S., Kuroda, T. and Kobayashi, (1999). “Strengthening effects of concrete columns with carbon fiber sheet”, Transactions of the Japan Concrete Institute, 21, 143-150.
Monti, G., Nistico, N. and Santini, S. (2001). “Design of FRP jackets for upgrade of circular bridge piers”, Journal of Composites for Construction, ASCE, 5(2), 94-101.
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118
Table 4.1 Parameters used in the parametric study
Concrete 'cof (MPa) 20 to 50 at an interval of 5
frpE (GPa) 230
ruph,ε 0.0075 CFRP t (mm) 0 to 1 at an interval of 0.1 frpE (GPa) 80
ruph,ε 0.015 GFRP t (mm) 0 to 1.5 at an interval of 0.1
119
0 5 10 15 200
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Normalized Axial Strain εc/εco
Nor
mal
ized
Axi
al S
tress
σc/
f ′ co
Specimens 20,21ρK=0.009, ρε=8.7
Specimens 24,25ρK=0.027, ρε=9.1
Specimens 34,35ρK=0.11, ρε=4.9
Specimens 38,39ρK=0.23, ρε=4.4
(a) Axial stress-axial strain curves
-10 -8 -6 -4 -2 00
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Normalized Lateral Strain ε l /εco
Nor
mal
ized
Axi
al S
tress
σc/
f ′ co
Specimens 20,21ρK=0.009, ρε=8.7
Specimens 24,25ρK=0.027, ρε=9.1
Specimens 34,35ρK=0.11, ρε=4.9
Specimens 38,39ρK=0.23, ρε=4.4
(b) Axial stress-lateral strain curves
Fig. 4.1 Typical stress-strain curves of FRP-confined concrete
120
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Normalized Axial Strain εc /εco
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimen 31Specimen 45
(a) Comparison between specimen 31 and specimen 45
0 0.5 1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
Normalized Axial Strain εc /εco
Nor
mal
ized
Axi
al S
tress
σc
/f ′co
Specimen 28Specimen 42
(b) Comparison between specimen 28 and specimen 42
Fig. 4.2 Effect of confinement stiffness on stress-strain behavior of FRP-confined concrete
121
Axial Strain εc
Axi
al S
tress
σc
Unconfined concrete(ENV 1992)FRP-confined concete(Lam and Teng)
f′cc
f′co
εco 0.0035εt εcu
Fig. 4.3 Illustration of Lam and Teng’s model
122
0 5 10 15 20 250
5
10
15
20
25
Normalized Ultimate Axial Strainεcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(a) Ultimate axial strain
0 1 2 3 4 50
1
2
3
4
5
Normalized Compressive Strengthf′cc /f′co - Test
Nor
mal
ized
Com
pres
sive
Stre
ngth
f ′ cc
/f′ co
- P
redi
cted
Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
(b) Compressive strength
Fig. 4.4 Performance of Lam and Teng’ s model against test results
123
0 5 10 15 20 250
5
10
15
20
25
Normalized Ultimate Axial Strainεcu /εco - Test
Nor
mal
ized
Ulti
mat
e A
xial
Stra
inε cu
/ εco
- P
redi
cted
Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
Fig. 4.5 Performance of Eq. 4.9
124
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 00
0.5
1
1.5
2
2.5
3
3.5
4
Normalized Lateral Strain ε l /εco
Nor
mal
ized
Axi
al S
tress
σc /
f′ co
Refined Version ofTeng et al.′s ModelFitted Line
(a) First two steps
0 0.05 0.1 0.15 0.2 0.25-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Confinement Stiffness Ratio ρK
Slo
pe K
CFRP-confinedGFRP-confinedProposed Equation
(b) Last step
Fig. 4.6 Demonstration of the parametric study
125
0 1 2 3 4 50
1
2
3
4
5
Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test
Nor
mal
ized
Axi
al S
tress
at U
ltim
ate
Axi
al S
train
f ′cu
/f′ co
- P
redi
cted
Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
Fig. 4.7 Performance of Eq. 4.13
0 1 2 3 4 50
1
2
3
4
5
Normalized Compressive Strengthf′cc /f′co - Test
Nor
mal
ized
Com
pres
sive
Stre
ngth
f ′ cc
/f′ co
- P
redi
cted
Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP
Fig. 4.8 Performance of Eq. 4.15
126
0 0.005 0.01 0.015 0.02 0.0250
10
20
30
40
50
60
70
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
f′co = 39.6MPa
Efrp = 80100MPa
t = 0.34mmD = 152mmεh,rup = 0.0205
Lam and Teng (2003)Modified Model (I)Test (Specimens 22 and 23)
(a) Specimens 22 and 23
0 0.005 0.01 0.015 0.02 0.025 0.030
10
20
30
40
50
60
70
80
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
f′co = 39.6MPa
Efrp = 80100MPa
t = 0.51mmD = 152mmεh,rup = 0.0181
Lam and Teng (2003)Modified Model (I)Test (Specimens 24 and 25)
(b) Specimens 24 and 25
127
0 0.005 0.01 0.015 0.02 0.0250
10
20
30
40
50
60
70
80
90
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
f′co = 38.9MPa
Efrp = 247000MPa
t = 0.33mmD = 152mmεh,rup = 0.00994
Lam and Teng (2003)Modified Model (I)Test (Specimens 17 to 19)
(c) Specimens 17,18 and 19
Fig. 4.9 Performance of Version (I) of the modified Lam and Teng model
128
Axial Strain εc
Axi
al S
tress
σc
f′cc
f′cof′cu
0.85f′co
εco 0.0035 εcu εcu εcu
ρK>0.01
ρK=0.01
ρK<0.01
Unconfined
Fig. 4.10 Schematic of Version (II) of the modified Lam and Teng model
129
0 0.002 0.004 0.006 0.008 0.01 0.0120
5
10
15
20
25
30
35
40
45
50
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
f′co = 39.6MPa
Efrp = 80100MPa
t = 0.17mmD = 152mmεh,rup = 0.0174
Lam and Teng (2003)Modified Model (I)Modified Model (II)Test (Specimens 20 and 21)
(a) Specimens 20 and 21
0 0.002 0.004 0.006 0.008 0.01 0.0120
5
10
15
20
25
30
35
40
45
50
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
f′co = 45.9MPa
Efrp = 80100MPa
t = 0.17mmD = 152mmεh,rup = 0.0172
Lam and Teng (2003)Modified Model (I)Modified Model (II)Test (Specimens 28 and 29)
(b) Specimens 28 and 29
Fig. 4.11 Prediction of descending type of stress-strain curves
130
CHAPTER 5
DESIGN OF SHORT FRP-CONFINED RC COLUMNS
5.1 INTRODUCTION
The ultimate goal of a design-oriented stress-strain model for FRP-confined
concrete is to facilitate the design of FRP jackets for strengthening RC columns.
Relevant design provisions are now available in various design guidelines (fib
2001; ISIS 2001; ACI-440.2R 2002, 2008; JSCE 2002; CNR-DT 2004; Concrete
Society 2004) for strengthening RC structures. However, current provisions are
only applicable to short columns, where the slenderness effect is negligible.
Moreover, most of the design guidelines are only concerned with short columns
under concentric compression. Only Concrete Society (2004) and ACI-440.2R
(2008) have recommended a general procedure for the section analysis of short
FRP-confined RC columns so that the axial load-bending moment interaction
diagram can be constructed accordingly, but no corresponding design equations
are specified in both design guidelines. Although such a section analysis
procedure can fulfill the design needs, a much simpler method comprising a set of
explicit equations is still highly desirable. This chapter is concerned with the
development of such explicit equations for short FRP-confined RC columns. The
analysis and design of slender FRP-confined RC columns is dealt with in Chapters
7 to 9. It should be noted that the present study has been partially motivated by the
development of the Chinese Code for the Structural Use of FRP Composites in
Construction, which is formulated within the framework of the current Chinese
Code for Design of Concrete Structures (GB-50010 2002). Therefore, some of the
considerations in the present study follow the specifications given in GB-50010
(2002) and these considerations are highlighted where appropriate. The present
131
study is limited to FRP jackets that are continuous over a strengthened region of
the column and possess fibers oriented solely or predominantly in the hoop
direction. Partial safety factors are taken to be unity for convenience of
presentation.
5.2 SECTION ANALYSIS
5.2.1 The Strength of FRP-confined RC sections
Once the stress-strain model for FRP-confined concrete is defined, design of FRP-
confined RC sections can be carried out using the conventional section analysis. It
is necessary to note that the stress-strain models discussed in the preceding two
chapters are all for FRP-confined concrete under concentric compression, but in a
column under combined bending and axial compression, a strain gradient exists.
For conventional RC columns, the assumption that the stress-strain curve of
concrete in an eccentrically-loaded column is the same as that of concrete under
concentric compression is widely used. For FRP-confined RC columns, it has
been concluded in Chapter 2 that it is also reasonable to adopt the same
assumption. Indeed, this assumption has been adopted by previous researchers
(e.g. Saadatmanesh et al. 1994; Mirmiran et al. 2000; Monti et al. 2001; Yuan and
Mirmiran 2001; Cheng et al. 2002; Teng et al. 2002; Binici 2008; Yuan et al.
2008).
When the stress-strain curve of FRP-confined concrete from concentric
compression is directly used in a section analysis, the analysis procedure is similar
to that for conventional RC columns as described in numerous reinforced concrete
textbooks (e.g. Park and Paulay 1975; Kong and Evans 1987). The only difference
in the analysis procedure introduced by the presence of FRP confinement is the
use of a different concrete stress-strain relationship that considers the confinement
effect of the FRP. Numerical integration over the section can still be carried out
using the layer method in which the column section is divided into many small
horizontal layers as shown in Fig. 5.1. In the present study, the modified Lam and
Teng model (I) is used. A small adjustment needs to be made to this model so that
132
it reduces to the stress-strain model for unconfined concrete in GB-50010 (2002).
In GB-50010 (2002), normal strength concrete is assumed to have co 0.002ε =
an ultimate axial strain of 0.0033 and its stress-strain curve consists of
Hognestad’s parabola and a horizontal line. As a result, the original value of 1.75
for the first term on the right hand side of the ultimate axial strain equation (Eq.
4.9) is replaced by 1.65, so that the stress-strain model for FRP-confined concrete
can reduce to that for unconfined normal strength concrete adopted by GB-50010
(2002) when no FRP is provided. Complete composite action between concrete
and FRP is assumed. Compressive stresses are taken to be positive and the tensile
strength of concrete is ignored. Plane sections are assumed to remain plane. Only
columns with limited hoop steel reinforcement are considered so any confinement
effect from the hoop steel reinforcement is ignored. The longitudinal steel
reinforcement is assumed to have an elastic-perfectly plastic stress-strain curve.
The axial load N and the bending moment
and
M at any stage of loading carried by
the section with the reference axis going through the centre of the section are
found by integrating the stresses over the section:
1
( )c n
nR
c c c si c siR xi
N b dλ
σ λ σ σ= −
=
= + −∑∫ A
)
(5.1a)
(1
( )c n
nR
c c c c si c si siR xi
M b d A R dλ
σ λ λ σ σ= −
=
= + −∑∫ − (5.1b)
where R is the radius of the section, is the width of the section at a distance cb cλ
from the reference axis, nx is the depth of the neutral axis, siσ is the stress in the
th layer of longitudinal steel reinforcement, and i siA is the corresponding cross-
sectional area of the longitudinal steel reinforcement. The stress of concrete cσ in
the compression zone can be determined from Eq. 4.2. siσ can be calculated from
si sE siσ ε= if ysi
s
fE
ε < (5.2a)
133
sisi y
si
fεσε
= if ysi
s
fE
ε ≥ (5.2b)
Eq. 5.1 is applicable at any stage of loading. The ultimate limit state of the column
is reached when the strain at the extreme concrete compression fiber reaches the
ultimate strain of FRP-confined concrete, signifying crushing of concrete due to
FRP rupture. This ultimate strain is defined by Eq. 4.9 with the original value of
1.75 for the first term on the right hand side replaced by 1.65.
Section analysis was performed on a reference circular RC column (Fig. 5.2) with
a diameter D 600 mm. Altogether 12 steel bars of 25 mm in diameter are
distributed evenly around the section. The circle defined by the centers of these
bars has a diameter mm. The strengths of concrete and steel
reinforcement were taken to be common values as specified in GB-50010 (2002).
The concrete was assumed to be grade C30, representing a characteristic cube
strength of MPa and a corresponding compressive strength
MPa. The steel was assumed to be grade II with a
characteristic yield strength
=
500d =
30cuf =
' 0.67 20.1co cuf f= =
335yf = MPa and an elastic modulus of 200
GPa. Note that the strengths of unconfined concrete and steel reinforcement given
above are used throughout the present study, unless otherwise stated. The column
was either unconfined or wrapped with a three-ply or six-ply CFRP jacket, with a
nominal ply thickness of 0.165 mm. The elastic modulus and hoop rupture strain
were assumed to be 230 GPa and 0.0075 for the CFRP jacket.
sE =
The interaction curves produced by the section analysis are shown in Fig. 5.3.
These interaction curves are normalized by the axial load capacity (concentric
compression) and moment capacity
uoN
uoM (pure bending) of the reference column
when no FRP confinement is provided. It can be seen that the maximum benefit of
FRP confinement occurs when the section fails in pure compression, but
confinement is much less beneficial when the column fails in pure bending. The
sharp slope change in the interaction curves at high axial loads for the wrapped
134
columns (Fig. 5.3) occurs when the neutral axis begins to fall outside the column
section. As the neutral axis depth moves further away from the column section,
the parabolic portion of the confined concrete stress-strain curve gradually moves
outside the section, with stresses over the section being eventually governed
entirely by the linear second portion of the stress-strain curve.
5.2.2 Moment-Curvature Curves of FRP-confined RC Sections
For a given column section, there exists a unique moment-curvature curve under a
particular axial load . This moment-curvature curve can be readily constructed
using the approach described in the preceding section. For a given axial load ,
the corresponding moment-curvature curve can be generated by specifying a
series of suitable strain values for the extreme compression fiber of concrete
N
N
cfε
up to its ultimate value cuε . For each strain value, the curvature φ is varied until
the resultant axial force acting on the section, calculated from Eq. 5.1a, equals the
applied axial load. Once the neutral axis position has been determined, the
moment can be evaluated using Eq. 5.1b. Fig. 5.4 shows three typical moment
curvature curves for the reference column described in the preceding section with
a 6-ply CFRP jacket. These three curves are for three different levels of axial load,
which represent low, moderate and high axial loads respectively.
5.2.3 Comparison with Test Results
Only a very limited number of experimental studies have been conducted on FRP-
confined circular RC columns subjected to eccentric loading (Tao et al. 2004;
Hadi 2006; Fitzwilliam and Bisby 2006; Ranger and Bisby 2007). The columns
reported in these studies were subjected to the coupled effect of load eccentricity
and slenderness. In addition, these studies were more concerned with the overall
column behavior than the section behavior (e.g. load-deflections curves were
reported in these studies, but none of them reported the curvature distribution of
the columns). As a result, it is difficult to extract experimental data to verify the
section analysis procedure presented above. However, the test data reported in the
135
above studies are used in Chapter 7 to verify a theoretical model capable of
dealing with the slenderness effect in FRP-confined RC columns.
The only two studies that reported the moment-curvature data appear to be the
study by Sheikh and Yau (2002) and Harmon and Gould (2002). Sheikh and Yau
(2002) tested FRP-confined circular RC columns (356 mm in diameter) under
constant axial loading and reversed cyclic lateral loading. The moment-curvature
data measured in the plastic hinge region were reported for all the columns they
tested. The moment-curvature data have recently been used by Binici (2008) to
verify a similar section analysis procedure for FRP-confined RC columns. The
columns used for comparison herein had six evenly distributed longitudinal steel
reinforcing bars with a diameter of 25 mm and only had a very small amount of
transverse steel reinforcement. The geometric and material properties of these
columns are summarized in Table 5.1. The experimental and the theoretical
moment-curvature curves are compared in Fig. 5.5. It can be seen that the
theoretical curves are in good agreement with the experimental envelope curves
for columns ST-2NT and ST-3NT which were tested under a relatively high axial
load (about 50% of ). On the other hand, the ultimate curvatures of columns
ST-4NT and ST-5NT tested under a relatively low axial load (about 25% of )
are considerably underestimated. Binici (2008) made similar observations in his
comparison with the same test data. He argued that the discrepancy might be due
to some unexpected experimental factors, because column ST-5NT showed a
larger deformation capacity than column ST-4NT while it only had about half the
amount of FRP confinement provided to column ST-4NT.
uoN
uoN
Harmon and Gould (2002) conducted similar tests on FRP-confined RC columns
to those reported by Sheikh and Yau (2002). Harmon and Gould’s (2002) columns
were 180 mm in diameter and were either 600 mm or 1200 mm in length. All the
columns were reinforced with six evenly distributed longitudinal steel reinforcing
bars with a diameter of 12.7 mm and had no hoop steel reinforcement. The
concrete strength ranged from 41.1 MPa to 55.2 MPa while the yield strength of
steel was 604 MPa. All the columns were provided with a filament wound GFRP
jacket with the fibers oriented solely in the hoop direction. The thickness of the
136
GFRP jacket was varied to achieve different fiber-to-concrete volume ratios (the
volume of fiber to the volume of confined concrete) of 1%, 3%, and 5%. Some of
the columns were subjected to a constant axial load of 133 kN while others were
subjected to a higher axial load of 578 kN. The elastic modulus of the GFRP
jacket was 68950 MPa (Harmon 2008). The geometric and material properties of
these columns are summarized in Table 5.2. It should be noted that columns 0600-
5-133 and 0600-5-578 were excluded from the present comparisons, because no
curvature data were available due to the failure of the measurement method
initially employed in their study (Gould 1999). Gould and Harmon (2002)
reported that most of their columns failed by the fracture of the steel reinforcing
bars before the rupture of the FRP jacket. FRP rupture was only observed in
column 1200-1-133 which had a length of 1200 mm, a fiber-to-concrete volume
ratio of 1%, and was subjected to a constant axial load of 133 kN. The maximum
FRP hoop strain measured at the critical position of each column is not reported in
Gould (1999) and Gould and Harmon (2002); it is only reported in Gould and
Harmon (2002) that the maximum FRP hoop strain reading recorded in the entire
test series was about 2%. As a result, it is not possible to make precise
comparisons for these columns; an FRP hoop strain of 2% was assumed when
producing the theoretical moment-curvature curves for these columns. Fig. 5.6
shows the comparisons between the experimental and the theoretical results. It
should be noted that in each of Figs 5.6a to 5.6c, a pair of experimental curves are
shown for a pair of columns with the same configuration except their lengths.
Obviously, the theoretical moment-curvature curves are identical for such a pair
of columns. Two theoretical curves corresponding to the lower bound and the
upper bound of concrete strength (41.1 Mpa and 55.2 MPa respectively) are
shown in Fig. 5.6. It can be seen that within this range of concrete strength, the
theoretical curves are only slightly affected by the concrete strength. It should also
be noted that was assumed when producing the theoretical curves as the
exact thickness of the concrete cover is not reported in Gould and Harmon (2002);
the thickness of the concrete cover only has a small effect on the theoretical
moment-curvature curve.
0.8d = D
Fig. 5.6a shows the comparison for columns 0600-1-133 and 1200-1-133. The
theoretical curves are close to the experimental curves. It should be noted that
137
column 1200-1-133 failed by FRP rupture. As a result, the comparison for these
two columns should be more reliable than that for the other columns. It should
also be noted that these two column were subjected to a very small axial load (133
kN), which is only about 7% of . The section analysis procedure presented
above is still reasonably accurate for this case. Fig. 5.6b shows that the theoretical
ultimate curvature is considerably smaller than the experimental value. This might
be due to some unexpected experimental factors, because the columns of Fig. 5.6b
only differed from those of Fig. 5.6a in that the former were subjected to a higher
axial load (578kN). Therefore, the ultimate curvatures of the former are expected
to be smaller than those of the latter, but the excremental ultimate curvatures
shown in Fig. 5.6b contradict this expectation. Besides, a significant discrepancy
also exists between the two experimental curves in Fig. 5.6b although these two
curves are expected to be very similar. Fig 5.7c shows that the theoretical curves
are reasonably close to the experimental curves. By contrast, the theoretical curves
in Figs 5.7d and 5.7e terminate at a much larger ultimate curvature than the
experimental curves. This may be due to the premature occurrence of steel
fracture before the FRP jacket was fully utilized.
uoN
Based on the above comparisons and given the fact that this assumption has been
adopted by many previous researchers, the present section analysis procedure is
deemed to be reasonable. However, more tests are still needed to fully clarify the
possible effect of load eccentricity on the stress-strain behavior of FRP-confined
concrete.
5.3 EQUIVALENT STRESS BLOCK
In the design of RC members, the stress profile of concrete in compression is
generally simplified using an equivalent stress block, over which stresses are
uniformly distributed. This equivalent stress block can be described by two factors,
the magnitude of the stresses over and the depth of this equivalent stress block,
which can be determined from the criterion that the equivalent stress block must
resist the same axial force and bending moment as the original stress profile.
Because of the FRP confinement, existing values of stress block factors adopted
138
for the design of conventional RC members are no longer suitable. This section is
therefore concerned with the development of appropriate stress block factors for
FRP-confined concrete in circular columns. In the present study, the mean stress
factor 1α is defined as the ratio of the uniform stress over the stress block to the
compressive strength of FRP-confined concrete and the block depth factor 1β is
defined as the ratio of the depth of the stress block to that of the neutral axis.
The stress distributions over the compression zone are examined for different
neutral axis positions to find the equivalent stress block factors. The thickness of
the confining CFRP jacket was taken as a variable. The maximum thickness is
such that it leads to a strength enhancement ratio ' ' 2cc cof f = . Fig. 5.7 shows the
variations of the stress block factors against the strength enhancement ratio. It can
be seen that the mean stress factor decreases as the strength enhancement ratio
increases and the block depth factor varies only slightly against the strength
enhancement ratio. It can also be seen that for circular sections, both stress block
factors also depend slightly on the neutral axial depth nx , as the width of
horizontal layers in a circular section varies with its depth. For simplicity, it is
suggested that
1 0.9β = (5.3)
Once 1β is fixed, 1α can then be recalculated according to the criterion that the
equivalent stress block must resist the same axial force as the original stress
profile. The recalculated values of 1α are shown in Fig. 5.8. The following simple
linear equation is suggested
' '
1 1.17 0.2 cc cof fα = − (5.4)
It should be noted that the numerical results shown here are for concrete confined
with a CFRP jacket having a rupture strain of 0.0075. However, Eqs 5.3 and 5.4
are also applicable to concrete confined with other types of FRP jackets. The
139
overall performance of Eqs 5.3 and 5.4 are examined through a comprehensive
parametric study later in the chapter.
5.4 DESIGN EQUATIONS
Using the stress block factors proposed above, design equations based on a
simplified section analysis method are presented herein, following a similar
approach adopted by GB-50010 (2002). This method is only applicable to
columns that have six or more evenly distributed longitudinal steel reinforcing
bars.
As shown in Fig. 5.9, the steel reinforcing bars are smeared into an equivalent
steel cylinder of the same total cross-sectional area and with longitudinal strength
only. 02πθ , 2πθ , 12πθ , and '22πθ are respectively the central angles
corresponding to the depths of the neutral axis, the equivalent stress block, the
yielded compressive steel reinforcement and the yielded tensile steel
reinforcement. 0θ , 1θ , θ and '2θ can be calculated from Eq. 5.5 and it is obvious
that 2θ = 1- '2θ .
( )0
ar cos 1 1 /n sR Rξθ
π− +⎡ ⎤⎣= ⎦ (5.5a)
( )1ar cos 1 1 /n sR Rβ ξθ
π− +⎡ ⎤⎣= ⎦ (5.5b)
( ) ( )1
ar cos / 1 1 /s n sR R Rβ ξθ
π− − +⎡ ⎤⎣ ⎦=
R (5.5c)
( ) ( )'2
ar cos / 1 1 /s nR R R Rβ ξθ
π− + +⎡ ⎤⎣ ⎦= s (5.5d)
where sR is the radius of the imaginary steel cylinder, 0/n nx hξ = is the ratio of
the neutral axis depth nx to the effective height of the section (0h sR R= + ), and
β is the ratio of the yield strain of steel reinforcement to the strain of the extreme
compressive concrete fiber as given by
140
y
s cu
fE
βε
= (5.6)
The axial load and bending moment carried by concrete can then be calculated
from
'1
sin(2 )12c ccN f A πθα θπθ
⎛= −⎜⎝ ⎠
⎞⎟ (5.7a)
3'
12 sin (3c ccM f AR )πθα
π= (5.7b)
where A is the gross area of the cross section and 2Rπ= . The axial load and
bending moment carried by steel reinforcement can be calculated from
( ) ( )1 2s y s c y s tN f A k f A kθ θ= + − + (5.7c)
1sin( ) sin( )c ts y s s y s s
m mM f A R f A R 2πθπ π
πθ+ += + (5.7d)
where sA is the total cross-sectional area of longitudinal steel bars, and
( ) ( )( )
0 1 0 11 / / sin( ) sin( )1 /
n s sc
n s
R R R Rk
R Rξ π θ θ πθ πθ
πβξ+ − − + −⎡ ⎤⎣ ⎦=
+ (5.8a)
( ) ( )
( )
' '2 0 2 01 / / sin( ) sin(
1 /n s s
tn s
R R R Rk
R R)ξ π θ θ πθ πθ
πβξ
+ − − + −⎡ ⎤⎣ ⎦= −+
(5.8b)
( ) [ ] ( )
( )
0 1 0 10 1
sin(2 ) sin(2 )1 / / sin( ) sin( )2 4
1 /
n s s
cn s
R R R Rm
R R
π θ θ πθ πξ πθ πθ
πβξ
− −+ − − + +⎡ ⎤⎣ ⎦
=+
θ
(5.8c)
( ) ( )
( )
' '2 0' 2 0
2 0sin(2 ) sin(2 )1 / / sin( ) sin( )
2 41 /
n s s
tn s
R R R Rm
R R
π θ θ πθ πξ πθ πθ
πβξ
− −⎡ ⎤+ − − + +⎡ ⎤⎣ ⎦ ⎣ ⎦=
+
θ
(5.8d)
141
Obviously, Eqs 5.7 and 5.8 are too complex for design use. For simplicity, the
following approximate expressions are proposed:
10 1.25 0.125 1c ckθ θ θ≤ = + = − ≤ (5.9a)
20 1.125 1.5 1t tkθ θ≤ = + = − ≤θ (5.9b)
Using Eq. 5.9, the terms 1 ckθ + and 2 tkθ + in Eq. 5.7c can be reasonably well
approximated by cθ and tθ respectively (Fig. 5.10a). In the mean time, the terms
1sin( ) cmπθ + and 2sin( ) tmπθ + in Eq. 5.7d can be approximated by sin( )cπθ and
sin( )tπθ respectively (Fig. 5.10b). It should be noted that Eq. 5.9 was derived
with an assumed value of / sR R =1.16, which was used in developing similar
expressions for GB-50010 (2002). Nevertheless, Eq. 5.9 is still sufficiently
accurate for columns with other values of / sR R , as proven by the parametric
study presented in the next section.
Based on the above discussions, the following expressions are obtained for the
ultimate axial load capacity and bending moment capacity uN uM of an FRP-
confined circular column:
( )'1
sin(2 )12u cc c t yN f A fπθθα θ θπθ
⎡ ⎤= − + −⎢ ⎥⎣ ⎦sA (5.10a)
3
'1
sin( ) sin( )2 sin ( )3
cu cc y sM f AR f A R tπθ ππθα
π+
= +θ
π (5.10b)
5.5 PERFORMANCE OF PROPOSED DESIGN EQUATIONS
In this section, numerical results from a comprehensive parametric study are
presented to assess the performance of Eq. 5.10. In this parametric study, the
column was assumed to have a diameter of 600 mm and to have 12 evenly
distributed longitudinal steel reinforcing bars. The concrete was assumed to be
grade C30 with MPa and the steel was assumed to be grade II with ' 20.1cof =
142
335yf = MPa. With the above parameters fixed, the strength of any given section
can be determined if the values of the following parameters are also specified: 1)
the eccentricity e ; 2) the thickness of the concrete cover, which can be
represented by either d D or sR R . The former ratio is used throughout the
present study and is referred to as the depth ratio; 3) the steel reinforcement ratio
sρ ; 4) the amount and the properties of the confining FRP jacket. The strength
enhancement ratio ' 'cc cof f and the strain ratio ,h rup coε ε were chosen to describe
the effect of FRP confinement. The values of the variables considered in the
parametric study are summarized in Table 5.3.
These parameters lead to 1620 cases of combination. The justifications for the
ranges of the parameters considered are as follows. The lower bound of the
normalized eccentricity is close to the minimum eccentricity , which is
generally specified by existing design codes for RC structures
[ mm in GB-50010 (2002) and
mine
min / 30 20≥e D= min 0.05 20e D= ≤ mm in BS-
8110 (1997)]. On the other hand, MacGregor et al. (1970) stated that the vast
majority of RC columns have a normalized eccentricity less than 0.84 based on
the results of a survey of more than 20,000 columns. The range of sρ studied is
comparable to the allowable range specified by existing design codes for RC
structures [1% to 8% in ACI-318 (2005) and 0.5% to 5% in GB-50010 (2002)].
The depth ratio actually reflects the thickness of the concrete cover. The same
range of the depth ratio (0.7 to 0.9) has been used by Hellesland (2005) for similar
analysis of RC columns and this range is also similar to that given in the ACI
design handbook [ACI 340R-97 (1997)] which provides design charts for circular
RC sections possessing a depth ratio ranging from 0.6 to 0.9. The strength
enhancement ratio goes up to 2, which is believed to cover all the cases of
practical applications. The three values chosen for the strain ratio represent
respectively the typical rupture strains of high modulus CFRP, CFRP and GFRP
jackets when they are used to confine a circular column. However, it should be
noted that the design value of the rupture strain of a given type of FRP composite
is much smaller than the actual material rupture strain of the same type of FRP
composite (i.e. the rupture strain obtained from tensile coupon tests). The reasons
143
for the much smaller design value are: 1) when an FRP jacket is used to confine a
concrete cylinder, its average hoop rupture strain is much smaller than the
material rupture strain due to the non-uniform deformation of concrete and other
factors (see Chapter 2); 2) in limit state design, the characteristic value of the
strain is divided by a partial safety factor to arrive at the design value to ensure a
desired level of safety reserve; and 3) to account for long-term degradation, an
additional reduction factor also needs to be applied to the FRP materials. As a
result, when the values of the FRP rupture strain used in the parametric study are
interpreted as the design values, the maximum value studied (1.5%) actually
corresponds to a material rupture strain of over 3% for a GFRP material.
Therefore, the range covered by the three strain ratios is believed to cover most
commercially available FRP materials.
The axial load capacity of all cases was calculated using both the section analysis
based on Eq. 5.1 and the proposed design equations (Eq. 5.10). The numerical
results from both approaches for all cases are compared in Fig. 5.11, in which the
axial load capacity is normalized by . It can be seen that the majority of cases
fall within the bounds.
uoN
5%±
Fig. 5.12 presents the interaction curves for three selected cases. The column in
Fig. 5.12 is confined with either a high modulus CFRP, a CFRP or a GFRP jacket.
All the information needed to produce these curves is given in Fig. 5.12. It can be
seen that the proposed design equations give results of excellent accuracy except
at high axial load levels. It should be noted that the interaction curves produced by
the design equations terminate at a high axial load level when the neutral axis
starts to fall outside the cross section. This is because the stress block factors are
based on the assumption that the neutral axis stays within the cross section, which
means that the use of Eq. 5.10 leads to errors when the neutral axis falls outside
the cross section. Nevertheless, this limitation of the design equations is
insignificant as such cases are not normally encountered in design due to the need
to include at least the minimum eccentricity for all columns.
144
5.6 CONCLUSIONS
This chapter has been concerned with the development of design equations for
short FRP-confined circular RC columns. Section analysis of such columns
employing Lam and Teng’s design-oriented stress-strain model was first
discussed. Design equations based on a simplified section analysis were next
presented. In this simplified analysis, the contribution of the confined concrete to
the load capacity of the section is approximated by transforming the stress profile
of concrete into an equivalent stress block and the contribution of the longitudinal
steel reinforcing bars to the load capacity of the section is approximated by
smearing the bars into an equivalent steel ring. The proposed design equations are
simple in form and provide an accurate approximation of the results from the
accurate section analysis.
145
5.7 REFERENCES
ACI 340R-97 (1997). ACI Design Handbook: Design of Structural Reinforced Concrete Elements in Accordance with the Strength Design Method of ACI 318-95, ACI Committee 340, American Concrete Institute.
ACI-440.2R (2002). Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, American Concrete Institute, Farmington Hills, Michigan, USA.
ACI-440.2R (2008). Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, American Concrete Institute, Farmington Hills, Michigan, USA.
ACI-318 (2005). Building Code Requirements for Structural Concrete and Commentary, ACI Committee 318, American Concrete Institute, Farmington Hills, Michigan, USA..
Binici, B. (2008). “Design of FRPs in circular bridge column retrofits for ductility enhancement”, Engineering Structures, 30(3), 766-776.
BS 8110 (1997). Structural Use of Concrete, Part 1. Code of Practice for Design and Construction, British Standards Institution, London, UK.
Cheng, H.L., Sotelino, E.D. and Chen, W.F. (2002). “Strength estimation for FRP wrapped reinforced concrete columns”, Steel and Composite Structures, 2(1), 1-20.
CNR-DT 200 (2004), Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Existing Structures, Advisory Committee on Technical Recommendations For Construction, National Research Council, Rome, Italy.
Concrete Society (2004). Design Guidance for Strengthening Concrete Structures with Fibre Composite Materials, Second Edition, Concrete Society Technical Report No. 55, Crowthorne, Berkshire, UK.
fib (2001). Externally Bonded FRP Reinforcement for RC Structures, The International Federation for Structural Concrete, Lausanne, Switzerland.
Fitzwilliam, J. and Bisby, L.A. (2006). “Slenderness effects on circular FRP-wrapped reinforced concrete columns”, Proceedings, 3rd International Conference on FRP Composites in Civil Engineering, December 13-15, Miami, Florida, USA, 499-502.
GB-50010 (2002). Code for Design of Concrete Structures, China Architecture and Building Press, China.
Gould, N.C. (1999). A Mechanistic Model and Design Procedure for Composite-confined Concrete Columns, Ph.D. thesis, Washington University, St. Louis.
146
Gould, N.C. and Harmon, T.G. (2002). “Confined concrete columns subjected to axial load, cyclic shear, and cyclic flexure – part II: experimental program”, ACI Structural Journal, 99(1), 42-50.
Hadi, M.N.S. (2006). “Behaviour of wrapped normal strength concrete columns under eccentric loading”, Composite Structures, 72(4), 503-511.
Harmon, T.G. (2008). Private communication.
ISIS (2001). Design Manual No. 4: Strengthening Reinforced Concrete Structures with Externally-Bonded Fibre Reinforced Polymers, Intelligent Sensing for Innovative Structures, Canada.
JSCE (2001). Recommendations for Upgrading of Concrete Structures with Use of Continuous Fiber Sheets, Concrete Engineering Series 41, Japan Society of Civil Engineers, Tokyo, Japan.
Kong, F.K and Evans, R.H. (1987). Reinforced and Prestressed Concrete, Third Edition. Chapman & Hall, London, UK.
MacGregor, J.G., Breen, J.E. and Pfrang E.O. (1970). “Design of slender concrete columns”, ACI Journal, 67(1), 6-28.
Mirmiran, A., Naguib, W. and Shahawy, M. (2000). “Principle and analysis of concrete filled composite tubes”, Journal of Advanced Materials, 32(4), 16-23.
Monti, G., Nistico, N. and Santini, S. (2001). “Design of FRP jackets for upgrade of circular bridge piers”, Journal of Composites for Construction, ASCE, 5(2), 94-101.
Park, R. and Paulay, T. (1975) Reinforced Concrete Structures, John Wiley & Sons, N.Y., USA.
Ranger, M. and Bisby, L.A. (2007). “Effects of load eccentricities on circular FRP-confined reinforced concrete columns”, Proceedings, 8th International Symposium on Fiber Reinforced Polymer Reinforcement for Concrete Structures (FRPRCS-8), University of Patras, Patras, Greece, July 16-18, 2007.
Saadatmanesh, H., Ehsani, M.R. and Li, M.W. (1994). “Strength and ductility of concrete columns externally reinforced with fiber composites straps”, ACI Structural Journal, 91(4), 434-447.
Sheikh, S.A. and Yau, G. (2002). “Seismic behavior of concrete columns confined with steel and fiber-reinforced polymers”, ACI Structural Journal, 99(1), 72-80.
Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, Second International Conference on Advanced Polymer Composites for Structural Applications in Construction, University of Surrey, Guildford, UK, 203-212.
147
Teng, J.G., Chen, J.F., Smith, S.T. and Lam, L. (2002) FRP-Strengthened RC Structures, John Wiley and Sons, Inc., UK.
Yuan, W. and Mirmiran, A. (2001). “Buckling analysis of concrete-filled FRP tubes”, International Journal of Structural Stability and Dynamics, 1(3):367-383.
Yuan, X.F., Xia, S.H., Lam, L. and Smith, S.T. (2008). “Analysis and behaviour of FRP-confined short concrete columns subjected to eccentric loading”, Journal of Zhejiang University Science A, 9(1), 38-49.
148
Table 5.1 Summary of Sheikh and Yau’s (2002) tests
Specimen D (mm)
H (mm)
d (mm)
frpE (GPa)
t(mm)
,h rupε (%)
yf(Mpa)
'cof
(Mpa) N
(kN) ST-2NT 356 1470 271 21 2.5 2.00 490 40.4 2570 ST-3NT 356 1470 271 75.5 1.0 1.3 490 40.4 2570 ST-4NT 356 1470 271 151 0.5 1.3 490 44.8 1380 ST-5NT 356 1470 271 21 1.25 2.00 490 40.8 1290
Table 5.2 Summary of Gould and Harmon’s (2002) tests
Specimen D (mm)
H (mm)
frpE (GPa)
t(mm)
yf(Mpa)
N(kN)
0600-1-133 180 600 68.95 0.45 604 133 0600-1-578 180 600 68.95 0.45 604 578 0600-3-578 180 600 68.95 1.35 604 578 1200-1-133 180 600 68.95 0.45 604 133 1200-1-578 180 600 68.95 0.45 604 578 1200-3-578 180 600 68.95 1.35 604 578 1200-5-133 180 600 68.95 2.25 604 133 1200-5-578 180 600 68.95 2.25 604 578
Table 5.3 Values of parameters used in the parametric study
Parameter Values e D 0.05,0.1,0.15,0.2,0.25,0.3,0.4,0.6,0.8
sρ 1%, 2%, 3%, 4%, 5% d D 0.7, 0.8, 0.9 ' '
cc cof f 1.25, 1.5, 1.75, 2
,h rup coε ε 1, 3.75, 7.5
149
xn
cu
R
cc'f
d
si
bc
sidsi
D
c
c
Fig. 5.1 Strains and stresses over an FRP-confined circular column section
Dd
Fig. 5.2 Cross section of the reference RC column
150
0 0.5 1 1.5 20
0.5
1
1.5
2
Normalized Moment Capacity Mu/Muo
Nor
mal
ized
Loa
d C
apac
ity N
u/N
uo
Unconfined3-ply CFRP6-ply CFRP
Fig. 5.3 Axial load-bending moment interaction curves for the reference column
0 0.01 0.02 0.03 0.04 0.050
100
200
300
400
500
600
700
800
Curvature φ (1/m)
Mom
ent M
(kN ⋅
m)
D = 600mmd = 500mmf′co = 20.1MPafy = 335MPaρs = 2.08%Efrp = 230GPa
t = 0.495mmεh,rup = 0.0075
1500 kN4500 kN7500 kN
Fig. 5.4 Bending moment-curvature curves for the reference column
151
-100 -50 0 50 100-300
-200
-100
0
100
200
300
Curvature φ (1/km)
Mom
ent
M (k
N ⋅m
)
ST-2NT
(a) Specimen ST-2NT
-100 -50 0 50 100-300
-200
-100
0
100
200
300
Curvature φ (1/km)
Mom
ent
M (k
N ⋅m
)
ST-3NT
(b) Specimen ST-3NT
152
-150 -100 -50 0 50 100 150-300
-200
-100
0
100
200
300
Curvature φ (1/km)
Mom
ent
M (k
N ⋅m
)
ST-4NT
(c) Specimen ST-4NT
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
100
200
300
Curvature φ (1/km)
Mom
ent
M (k
N ⋅m
)
ST-5NT
(d) Specimen ST-5NT
Fig. 5.5 Comparison with Sheikh and Yau’s (2002) tests
153
-400 -300 -200 -100 0 100 200 300 400-60
-40
-20
0
20
40
60
Curvature φ (1/km)
Mom
ent
M (k
N ⋅m
)
Test (0600-1-133)Test (1200-1-133)Analysis (f′co
=41.1 MPa)
Analysis (f′co=55.2 MPa)
(a) Specimens 0600-1-133 and 1200-1-133
-800 -600 -400 -200 0 200 400 600 800-80
-60
-40
-20
0
20
40
60
80
Curvature φ (1/km)
Mom
ent
M (k
N ⋅m
)
Test (0600-1-578)Test (1200-1-578)Analysis (f′co
=41.1 MPa)
Analysis (f′co=55.2 MPa)
(b) Specimens 0600-1-578 and 1200-1-578
154
-600 -400 -200 0 200 400 600-100
-80
-60
-40
-20
0
20
40
60
80
100
Curvature φ (1/km)
Mom
ent
M (k
N ⋅m
)
Test (0600-3-578)Test (1200-3-578)Analysis (f′co
=41.1 MPa)
Analysis (f′co=55.2 MPa)
(c) Specimens 0600-3-578 and 1200-3-578
-1500 -1000 -500 0 500 1000 1500-60
-40
-20
0
20
40
60
Curvature φ (1/km)
Mom
ent
M (k
N ⋅m
)
Test (1200-5-133)Analysis (f′co
=41.1 MPa)
Analysis (f′co=55.2 MPa)
(d) Specimen 1200-5-133
155
-800 -600 -400 -200 0 200 400 600 800-80
-60
-40
-20
0
20
40
60
80
Curvature φ (1/km)
Mom
ent
M (k
N ⋅m
)
Test (1200-5-578)Analysis (f′co
=41.1 MPa)
Analysis (f′co=55.2 MPa)
(e) Specimen 1200-5-578
Fig. 5.6 Comparison with Gould and Harmon’s (2002) tests
156
1 1.2 1.4 1.6 1.8 20.8
0.85
0.9
0.95
1
Strength Enhancement Ratio f′cc / f′co
Mea
n S
tress
Fac
tor α
1
xn = 0.1R
xn = 1R
xn = 1.5R
xn = 2R
(a) Mean stress factor
1 1.2 1.4 1.6 1.8 20.82
0.84
0.86
0.88
0.9
0.92
Strength Enhancement Ratio f′cc / f′co
Blo
ck D
epth
Fac
tor
β 1
xn = 0.1R
xn = 1R
xn = 1.5R
xn = 2R
(b) Block depth factor
Fig. 5.7 Stress block factors for CFRP-confined concrete in circular sections
157
1 1.2 1.4 1.6 1.8 20.75
0.8
0.85
0.9
0.95
1
1.05
←α1 = 1.17-0.2f′cc/f′co
Strength Enhancement Ratio f′cc/f′co
Mea
n S
tress
Fac
tor α
1
xn = 0.1R
xn = 1R
xn = 1.5R
xn = 2R
Fig. 5.8 Mean stress factor for FRP-confined concrete in circular sections
for 1β =0.9
1xnxn
cu
0
R
Rs
yf
yf cc
'
1
2
2
y
y
'f f 'cc
1
Fig. 5.9 Schematic of the simplified section analysis approach
158
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ
θ 1+k
c, θ
2+k t
θ1+kc
θ2+kt
θc=1.25θ-0.125
θt=1.125-1.5θ
(a) Approximation of 1 ckθ + and 2 tkθ + using cθ and tθ
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ
sin(
πθ1)
+mc,
sin
( πθ 2
)+m
t
sin(πθ1)+mc
sin(πθ2)+mt
sin(πθc)
sin(πθt)
(b) Approximation of 1sin( ) cmπθ + and 2sin( ) tmπθ + using sin( )cπθ and sin( )tπθ
Fig. 5.10 Simplifications for strength contributions from steel reinforcement
159
0 0.5 1 1.50
0.5
1
1.5
Normalized Axial Load CapacityNu/Nuo - Section Analysis
Nor
mal
ized
Axi
al L
oad
Cap
acity
Nu/N
uo -
Des
ign
Equ
atio
ns
5%
-5%
Fig. 5.11 Performance of the proposed design equations
160
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
Normalized Moment Capacity Mu/Muo
Nor
mal
ized
Loa
d C
apac
ity N
u/N
uo
D=600mmd=500mmEfrp=700GPaεh,rup=0.002ρs=1%f′cc/f′co=1.35
Section AnalysisDesign Equations
(a) High modulus CFRP jacket
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
Normalized Moment Capacity Mu/Muo
Nor
mal
ized
Loa
d C
apac
ity N
u/N
uo
D=600mmd=500mmEfrp=230GPaεh,rup=0.0075ρs=2%f′cc/f′co=1.5
Section AnalysisDesign Equations
(b) CFRP jacket
161
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Normalized Moment Capacity Mu/Muo
Nor
mal
ized
Loa
d C
apac
ity N
u/N
uo
D=600mmd=450mmEfrp=80GPaεh,rup=0.015ρs=1.5%f′cc/f′co=1.75
Section AnalysisDesign Equations
(c) GFRP jacket
Fig. 5.12 Comparisons between accurate and approximate analyses
162
CHAPTER 6
ANALYSIS OF ELASTIC COLUMNS
6.1 INTROCUCTION
Before developing a design procedure for slender FRP-confined RC columns, it is
desirable to understand the rationale behind the current design approach for
slender RC columns. It is well-known that this approach comprises a first-order
analysis of the frame to determine the first-order moments plus a second-order
analysis of an individual column isolated from the frame with idealized end
restraints representing the adjacent restraining members in the frame to determine
the second-order moments. The second-order moments are approximated by an
amplification of the first-order moments in order for the design to be related to the
section strength. The isolated column with end restraints (referred to as the
restrained column hereafter) is then replaced by an equivalent hinged column
through appropriate treatment of the end restraints of the restrained column.
This approach originated from the analysis of elastic columns. Therefore, the
rationale on which this approach is based is best explained through an analysis of
elastic columns. As a preparation for the analysis of slender FRP-confined RC
columns, this chapter examines all the components forming the current design
approach for slender RC columns through an analysis of non-sway elastic
columns with elastic end restraints. The exact solution for restrained columns is
derived first, followed by a careful investigation into the behavior of such
columns. In addition, new design equations for elastic columns are proposed
based on thorough interpretations of the current design approach.
163
6.2 THE PROLBLEM OF COLUMN DESIGN
As can be seen in later sections of this chapter, the design of a restrained column
can be finally transformed into the design of an equivalent hinged column
subjected to uniform first-order moments. It is advisable to use this equivalent
column to address the problem of column design. The first-order moment is
always enlarged by the interaction between the axial load and the deflection of the
column. For such a column, the maximum moment always occurs at the
mid-height of the column. If the first-order moment and the column properties are
known, the designer needs to determine how much axial load the column can
resist. The section axial load-bending moment diagram offers the answer to this
question. When the combination of the axial load and moment at the critical
section exceeds the interaction curve, it indicates that the column will fail by the
exhaustion of the material strength. It is clear that the key is to determine the
maximum moment of this column and to compare the combination of the axial
load and the maximum moment to the section axial load-bending moment
interaction curve. For elastic columns, the interaction curve features a straight line,
as shown in Fig. 6.1. This straight line can be mathematically described by Eq. 6.1
max 1uo uo
MNN M
+ = (6.1)
where is the maximum axial load the section can take in the absence of
bending moments while
uoN
uoM is the maximum bending moment the section can
take in the absence of the axial load.
The maximum moment of a column can be found using the numerical integration
method developed by Newmark (1943). For elastic columns, an exact expression
for the maximum moment exists and can be derived by solving the governing
differential equations, as presented in the following section.
6.3 EXACT SOLUTIONS
164
6.3.1 General
In this section, the exact solution for restrained columns is derived first and
hinged columns are then treated as a special case of restrained columns with zero
end restraints.
6.3.2 Exact Solution for Restrained Columns
Consider a restrained column as shown in Fig. 6.2a. The length of the column is l
nd the end restraints are represented by two elastic springs with rotational
stiffnesses of
a
1R and 2R respectively. The column is subjected to two external
end moments 1eM and 2eM plus an axial load . For an elastic column, the
final status of the column is the same regardless of whether the loads have been
applied in a proportional manner or not. For ease of discussion in later sections, it
is assumed throughout this chapter that the axial load is applied after the end
moments. The deflections due to the end moments and those due to the axial load
are solved independently and the final deflections of the column are taken as the
sum of the deflections from the two sources.
N
6.3.2.1 Deflection caused by the first-order moments
In Fig. 6.2b, an initially straight column is bent into a deflected shape as the
end moments
v
1M and 2M are applied. A pair of shear forces arises if the
end moments are not equal. It should be noted that the external end moments
V
1eM
and 2eM are shared by the column and the end restraints; 1M and 2M are the
moments transmitted to the ends of the column while the rest is resisted by the
end restraints. 1M and 2M can be considered as the moments at the ends of a
column obtained from a conventional first-order frame analysis and they are used
as input in the design of this column when it is isolated from the frame. In this
chapter, 2M is always assigned a non-negative value; 1M can have a negative
value but its absolute value is always smaller than or equal to that of 2M . The
latter is referred to as the larger first-order end moment or simply the larger end
165
moment for brevity hereafter and the ratio of 1M to 2M is referred to as the
first-order end moment ratio in this chapter.
For the case depicted by Fig. 6.2b, the moment at any given height of the column
can be written as
'' 2 12
M MM EIv M xl−
= − = − (6.2)
where E is the elastic modulus, I is the second moment of area and is any
given height along the column and
x
0x = represents the end where 2M is
applied.
From Eq 6.2 and the boundary conditions that no deflection occurs at either end of
the column, the deflection shape of the column can be easily found to be
(3 22 1 21 2
1 26 2 6
M M M lv x x MEI l
−⎡ ⎤= − + +⎢ ⎥⎣ ⎦)M x (6.3)
6.3.2.2 Deflection caused by the axial load
In Fig. 6.2c, an axial load is applied on the same column where N 1M and
2M are already present. An additional deflection is then introduced by the
application of this axial load. It should be noted that once the axial load is applied,
the end restraints begin to bear additional end moments triggered by the end
rotations as a result of the additional deflection. The end moments are thus no
longer
w
1M and 2M , but are replaced by the end moments defined by the
following equations where the moments from the end restraints are subtracted
from the original end moments
'
2 2 2 (0)M M R w= − (6.4a)
'1 1 1 (l )M M R w= − (6.4b)
166
The pair of shear forces is thus changed to
2 1M MVl−
= (6.5)
As illustrated in Fig. 6.2d, the moment at any given height of the column can then
be written as
( ) ( ''2 )M M V x N v w EI v w= − + + = − + (6.6)
After some rearrangement, Eq. 6.7 can be obtained
( )'' 3 22 1 21 2
' '2 (0) 1 ( )'
2 (0)
26 2 6
l
M M MN lEIw Nw x x M M xEI l
R w R wR w x
l
−⎡ ⎤+ = − − + +⎢ ⎥⎣ ⎦⎡ ⎤−
+ −⎢ ⎥⎢ ⎥⎣ ⎦
(6.7)
If Eq. 6.7 is differentiated four times, it becomes
6 2 4 0w k w+ = (6.8a)
NkEI
= (6.8b)
The general solution to Eq. 6.8 is
3 2
0 0 0 0 0sin cosw a kx b kx c x d x e x f= + + + + 0+ (6.9)
By differentiating Eq. 6.9 twice and comparing it with Eq. 6.7, it can be easily
found that
20 6
1M McEIl−
= − (6.10)
167
and
20 2
MdEI
= (6.11)
From the boundary condition
(0) 0w = (6.12)
it can be found that
0 0f b= − (6.13)
The other three coefficients can be found using the following three boundary
conditions
( ) 0lw = (6.14)
' ''2 (0) (0) 0R w EIw− = (6.15)
' ''1 ( ) ( ) 0l lR w EIw+ = (6.16)
Eqs 6.14 to 6.16 can be expanded and finally written in the following matrix form
0
2 2
1 1
2 21 2
2
21 21 1
sin cos 1
cos sin sin cos
26
2
kl kl l aR k N R b
R k kl N kl R k kl N kl R e
M M k lNMM MM R k l
N
−⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥− − −⎣ ⎦
+⎡ ⎤−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+− −⎢ ⎥⎣ ⎦
0
1 0
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
(6.17)
The expressions for , and are lengthy, but they can be arranged in a
neat and unified form as
0a 0b 0e
168
01 02 030
01 02 03
cos sincos sin
a a kl a klad d kl d kl
+ +=
+ + (6.18)
01 02 030
01 02 03
cos sincos sin
b b kl b klbd d kl d kl
+ +=
+ + (6.19)
01 02 030
01 02 03
cos sincos sin
e e kl e kled d kl d kl
+ +=
+ + (6.20)
where
( ) ( )21 2 1 1 2 11 2 1 1 2
01
26 2
kl M M R kl M M R RNlM M R M Rak k N
+ ++= + + + 2 (6.21a)
( ) ( )21 2 2 1 2 12 2 1 1 2
02
26 2
kl M M R kl M M R RNlM M R M Rak k N
+ ++= − − − − 2 (6.21b)
( )2 21 2 1
03 2 1
26
k l M M R Ra lM R
N+
= − − 2 (6.21c)
( )2 21 2 1
01 1 2
26
k l M M R Rb lM R
N+
= − − 2 (6.22a)
02 03b a= (6.22b)
03 02b a= − (6.22c)
( ) (2
01 1 2 1 2 2 1 1 22k le M M R R M R MN
= − + + − )R (6.23a)
( ) ( ) ( )(2 2 2
02 1 2 1 2 2 1 1 2 1 2 1 222 6k l k le M M R R M R M R M M RN
= + − − + + + )R (6.23b)
( ) ( ) ( )2 2
03 1 2 1 2 1 1 2 1 222 6
N kl kl ke M M M M R M M N Rk N
⎛ ⎞= − − − + − + −⎜
⎝ ⎠R ⎟
R
(6.23c)
01 1 22d R= (6.24a)
( )02 1 2 1 22d R R Nl R R= − − + (6.24b)
( ) 21 2
03 1 2
N R R N ld
k+ +
= klR R− (6.24c)
6.3.2.3 Final deflections
The final deflected shape of the column can be taken as the sum of and v w
169
( )1 20 0 0
2sin cos
6M M l
0f v w a kx b kx e x fEI
+⎡ ⎤= + = + + + +⎢
⎣ ⎦⎥ (6.25)
The moment distribution along the column can then be described by
( )'' 20 0sin cosM EIf EIk a kx b kx= − = + (6.26)
The maximum moment occurs at a location x which is defined by
(30 0cos sin 0dM EIk a k x b k x
dx= − ) = (6.27)
It can be derived from Eq. 6.27 that
0
0
tan ak xb
= (6.28)
By substituting Eq. 6.28 into Eq. 6.26 , the maximum moment can be found as
( )2 2max 0 0 0tan sin cos 2M EIk b k x k x k x N a b= + = + (6.29a)
It should be noted that the solution of Eq. 6.27 may result in a value of x smaller
than zero or larger than , which implies that the maximum moment defined by
Eq. 6.29a may occur outside the column height. A careful examination of the issue
revealed that this special case occurs when and only when , . For
this case, the maximum moment within the column height is either the moment at
or the moment at . For the former case, the maximum moment can be
found from Eq. 6.27 to be
l
0 0a < 0 0b >
0x = x l=
max 0M Nb= (6.29b)
while for the latter case, the maximum moment is
170
max 0 0sin cosM N a kl b kl= + (6.29c)
The absolute value is taken in Eq. 6.29c because the latter case only occurs when
the first-order end moment ratio approaches -1 and the restraint at the end where
the larger first-order end moment acts is much stiffer than that at the opposite end.
The maximum moment has a negative value according to the current sign
convention in such a case.
For all the other cases, the maximum moment is defined by Eq. 6.29a and it
always occurs somewhere between the two ends.
6.3.3 Exact Solution for Hinged Columns
The exact solution for hinged columns can be found in various textbooks (e.g.
Galambos 1968; Chen and Atsuta 1976). Here hinged columns are treated as a
special case of restrained columns with 1 2 0R R= = . With
substituted into the derivation presented in the preceding sub-section, the final
deflection of a hinged column can be written as
1 2 0R R= =
1 1
2 2 2
cos 1sin cos 1
sin
M MklM M Mf kx kxN kl l
⎛ ⎞− −⎜ ⎟⎜ ⎟= + +⎜ ⎟⎜ ⎟⎝ ⎠
x − (6.30)
The moment at any given height of the column is
1
'' 22
cossin cos
sin
M klMM EIf M kx kx
kl
⎛ ⎞−⎜ ⎟⎜ ⎟= − = +⎜ ⎟⎜ ⎟⎝ ⎠
(6.31)
The maximum moment occurs at a height of x which is defined by
171
1
22
coscos sin 0
sin
M klMdM M k k x k x
dx kl
⎛ ⎞−⎜ ⎟⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠
= (6.32)
By rearranging Eq. 6.32, the following equation can be obtained
1
2
costan
sin
M klMk x
kl
−= (6.33)
By substituting Eq. 6.33 into Eq. 6.31 and after some rearrangement, the
maximum moment can be written as
2
1 1
2 2max 2
1 2 cos
sin
cr
cr
M M NM M N
M MN
N
π
π
⎛ ⎞⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝=
⎛ ⎞⎜ ⎟⎝ ⎠
⎠ (6.34a)
where is the Euler load of the column and crN2
2
EIl
π= for a hinged column.
Similar to the situation examined in the previous section for restrained columns,
the maximum moment defined by Eq. 6.34a is only applicable to cases where the
maximum moment does occur within the column height. However, in the absence
of end restraints, x l> is not possible. It can be found from Eq. 6.33 that 0x ≥
is ensured when 1
2
coscr
M NM N
π⎛ ⎞
≥ ⎜⎜⎝ ⎠
⎟⎟ . By contrast, when 1
2
coscr
M NM N
π⎛ ⎞
< ⎜ ⎟⎜ ⎟⎝ ⎠
,
the maximum moment defined by Eq. 6.34a occurs outside the column ( 0x < ).
For such cases, the maximum moment within the column height occurs at ,
thus
0x =
max 2M M= (6.34b)
172
6.4 BEHAVIOR OF RESTRAINED COLUMNS
The behavior of hinged columns is easy to understand; the maximum moment of a
hinged column is either enlarged or remains unaffected as a result of the
interaction between the axial load and the deflection. To be specific, the maximum
moment of a hinged column subjected to uniform first-order moments is always
enlarged by the second-order moments, but the maximum moment of a hinged
column subjected to oblique first-order moments may remain unaffected by the
second-order moments. By contrast, the behavior of restrained columns is more
complicated because the end restraints produce additional moments which do not
exist in hinged columns. It is thus necessary to understand how the end restraints
affect the behavior of restrained columns before investigating the design of such
columns.
A good way to investigate the behavior of a column is to monitor the process of
how its moment distribution changes as the axial load increases. This process is
illustrated in Fig. 6.3, where a series of moment diagrams corresponding to
different loading stages is shown. These moment diagrams were produced using
the exact solution presented in the previous section. In each sub-figure, the
first-order moment diagram is indicated by the shaded area (labeled 0). The area
defined by a dashed line is the moment diagram for a given level of axial loading.
The dashed line denoted by 2 is for the final loading level when the axial load
capacity is reached and the other dashed line dentoed by 1 is for an intermediate
loading level.
It can be seen from Fig. 6.3 that irrespective of the first-order end moment ratio,
the larger end moment 2M decreases as the axial load increases; the direction of
2M may even be reversed if its magnitude is not large enough and the end
restraints are strong enough. The rules of the development of 1M is however
dependent on the first-order end moment ratio, which is discussed in detail below.
173
When 1
2
1 0MM
≥ ≥ 1, M changes in the same manner as 2M . It continuously
decreases as the axial load increases and it may also change direction. When
1
2
0 MM
> ≥ −0.9 , the magnitude of 1M continuously increases as the axial load
increases. It should be noted that the lower limit for 1
2
MM
depends on the
magnitude of the end restraints but it is always close to -1. When 1
2
0.9 1MM
− > ≥ − ,
the magnitude of 1M initially decreases as the axial load increases but then
increases as the axial load further increases.
An important observation is that unlike the hinged column, the maximum moment
of a restrained column can reduce as the axial load increases, as indicated by the
curve denoted by 1 in Fig. 6.3b. This implies that the second-order effect can
enhance the axial load capacity of a restrained column in certain cases. This
phenomenon arises from the existence of the end restraints and is explained in
detail below. The second-order moments can be decomposed into two parts. The
first part is the moments caused by the interaction between the axial load and the
deflection, which can be mathematically described as the product of the axial load
and the total deflection of the column. The second part is the moments produced
by the end rotations. This part of moments can be mathematically described as the
product of the end rotation associated with the additional deflection caused by the
axial load and the stiffness of the end restraints. The increments of end rotations
are always opposite in direction to the increments of end moments induced by
these rotations. As a result, these two parts of moments may combine to yield such
an overall effect that the maximum second-order moment always occurs away
from the column end where 2M is applied and it is always of the same sign as
2M while the second-order moments in the neighborhood of the same column
end are of the opposite sign to 2M . In other words, the maximum moment of a
restrained column subjected to uniform first-order moments is always enlarged by
the second-order moments, but the maximum moment of a restrained column
subjected to oblique first-order moments may however be reduced by the
174
second-order moments.
6.5 DESIGN OF RESTRATNED COLUMNS
6.5.1 General
The current design approach for restrained columns in the existing design codes
(e.g. ENV-1992-1-1 1992; BS-8110 1997; ACI-318 2005) is based on the
philosophy that a restrained column can be replaced by an equivalent hinged
column subjected to symmetrical bending. The latter is referred to as the standard
hinged column in this chapter. The pursuit of the standard hinged column can be
done in two steps. The fist step is to find a hinged column with two end moments
which are equal to the first-order end moments of the restrained column. The key
issue in this step is to determine the length of the hinged column which ensures
that it has a maximum moment equal to that of the restrained column and thus the
same axial load capacity. It is obvious that the hinged column obtained in this step
is subjected to oblique first-order moments unless the column is symmetrically
bent. The second step is to transform the hinged column obtained in the previous
step into the standard hinged column. The latter is of an equal length to the former
and the two also need to have equal maximum moments and axial load capacities.
In other words, an equivalent uniform first-order moment distribution is sought in
step two to replace the oblique first-order moment distribution of the hinged
column obtained in step one. The key issue in this step is to find the relationship
between the two moment distributions. Step two is examined first since it is more
straightforward than step one and it can be independently considered as the
solution to the design of hinged columns.
6.5.2 From a Hinged column to a Standard Hinged Column
Theoretically speaking, the design of hinged columns can be dealt with in an exact
manner provided the maximum moment along the column is found from Eq. 6.34.
However, much simpler design equations are available and they can be found in
many textbooks (e.g. Galambos 1968; Chen and Atsuta 1976). Fig. 6.4 reveals the
philosophy behind the design of hinged columns as explained in the preceding
175
paragraph. In Fig. 6.4, the column on the left hand side is subjected to oblique
first-order moments while the one on the right hand side is subjected to uniform
first-order moments eqM with such a magnitude that the maximum moments in
the two columns are equal. The equivalent moment eqM can be expressed as a
portion of 2M , as 2eq mM C M= , where is the equivalent moment factor and
it can be determined from the derivation given below.
mC
Substituting 1 2 eqM M M= = into Eq. 6.34a, the maximum moment of the
standard hinged column is given by
max 1
cos2
eqeq
cr
MM M
NN
ϕπ
= =⎛ ⎞⎜ ⎟⎝ ⎠
(6.35)
where 1ϕ is the moment amplification factor of the standard hinged column. By
comparing Eq. 6.35 and Eq. 6.34, it can be found that
2
1 1
2 2
1 2 cos
2sin2
crm
cr
M M NM M N
CN
N
π
π
⎛ ⎞⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠=
⎛ ⎞⎜ ⎟⎝ ⎠
, if 1
2
coscr
M NM N
π⎛ ⎞
≥ ⎜⎜⎝ ⎠
⎟⎟ (6.36a)
cos2m
cr
NCN
π⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠, if 1
2
coscr
M NM
π⎛ ⎞
< ⎜⎜⎝ ⎠N ⎟⎟ (6.36b)
Now it is very clear that the maximum moment of any hinged column subjected to
two end moments and an axial load can be written as
max 1 2 2mM C M Mϕ ϕ= = (6.37)
where reflects the effect of moment gradient, mC 1ϕ reflects the moment
amplification of the corresponding standard hinged column and ϕ is the moment
176
amplification factor of the column considered. Conventional design methods are
based on the simplification of and mC 1ϕ .
Fig. 6.5 shows that 1ϕ can be closely approximated by the following expression
proposed by Lai and MacGregor (1983)
1
1 0.25
1cr
cr
NN
NN
ϕ+
=−
(6.38)
In existing design codes, the term 0.25cr
NN
is omitted and 1ϕ is approximated
using the following expression
11
1cr
NN
ϕ =−
(6.39)
The exact expression and the two approximate expressions are compared in Fig.
6.5, which clearly shows that Eq. 6.39 is un-conservative for the full range of the
axial load level while Eq. 6.38 provides a much closer prediction. It is believed
that the term 0.25cr
NN
is omitted in existing design codes as the error so induced
becomes much less significant in the much more complicated design procedure
for columns involving material nonlinearity. The un-conservativeness introduced
by the omission of 0.25cr
NN
can be counterbalanced by other appropriate
considerations in dealing with factors such as material nonlinearity and end
restraints. The design equations in existing design codes for RC structures do not
aim to provide exact answers, but reasonably accurate yet conservative answers in
a reasonably simple manner. However, when material nonlinearity is not involved,
the design problem becomes much more straightforward and design equations
with higher accuracy may be sought. As a result, for elastic columns, the term
177
0.25cr
NN
may be retained.
Eq. 6.36 indicates that is a function of two ratios, that is mC 1
2
MM
and cr
NN
.
The effects of these two ratios on are illustrated in Fig. 6.6. In Fig. 6.6, the
family of thin solid curves is for
mC
cr
NN
increasing from 0 to 1 at an interval of 0.1.
Each of these curves shows how the values of vary with mC 1
2
MM
. Each of them
consists of two portions separated by a small circle which corresponds to
1
2
coscr
M NM N
π⎛ ⎞
= ⎜⎜⎝ ⎠
⎟⎟ . The first portion is a horizontal line and is mathematically
described by Eq. 6.36b while the second portion features an ascending shape and
is mathematically described by Eq. 6.36a. All these curves converge to the point
where when . The shaded area represents cases where the
maximum moment of the column does occur within the column height and is
always larger than
1mC = crN N=
2M while the remaining area represents cases where 2M is
the maximum moment of the column. This indicates that only the values located
in the shaded area need to be reasonably approximated because when the
maximum moment calculated using an approximate expression for is smaller
than
mC
2M , the maximum moment is always taken to be 2M . Many approximate
expressions have been proposed for and a brief discussion and comparison
of these expressions can be found in Austin (1961). Among all these expressions,
the following equation proposed by Austin (1961) is widely accepted because of
its simplicity and accuracy. The values of predicted using Eq. 6.40 is shown
by the dashed line in Fig. 6.6.
mC
mC
1
2
0.6 0.4 0.4mMCM
= + ≥ (6.40)
Eq. 6.40 neglects the effect of the axial load and the value of has a lower mC
178
limit of 0.4 because Eq. 6.40 was originally derived for elastic lateral-torsinal
buckling and this lower limit might be removed for in-plane bending (Chen and
Lui 1991). Lai and MacGregor (1983) however suggested that this lower limit be
retained considering the uncertainties of the behavior of columns with a 1
2
MM
ratio of -0.5 to -1 (MacGregor et al. 1970). It can be seen from Fig. 6.6 that the
dashed line passes through the shaded area, which indicates Eq. 6.40 is
un-conservative for cases falling into the shaded area above the dashed line. This
un-conservativeness is however small as can be seen in Fig. 6.7, where the exact
values of max
2
MM
and the approximate results predicted from Eqs 6.38 and 6.40
are compared. The approximate predictions shown in Fig. 6.7 are generally
conservative but close to the exact results except for 1
2
1MM
= − . When 1
2
1MM
= − ,
the exact curve is a horizontal line since the exact maximum moment is always
2M irrespective of the axial load level. The approximate curve for this case
however overlaps with that for 1
2
0.5MM
= − because 0.4mC = when 1
2
0.5MM
< − .
If this limitation is removed, Eqs 6.38 and 6.40 provide less conservative
predictions for cases where 1
2
0.5MM
< − .
In summary, the proposed approximate equation for the maximum moment of a
hinged column is
max 2 2
1 0.25
1cr
m
cr
NNM C MN
N
+=
−M≥ (6.41)
where Eq. 6.40 is employed to determine . mC
6.5.3 From a Restrained Column to a Hinged Column
179
The final uncertainty in the design of restrained columns is how to determine the
length of the standard hinged column. To sustain the deflected shape of a column,
the deflections, curvatures and moments of this column must maintain certain
relationships as required by the governing differential equation given in the
previous section; these relationships form the basis of the well-known numerical
integration method (e.g. Newmark 1943; Craston 1972). An important implication
of these relationships is that a restrained column can be replaced by an equivalent
hinged column with a length of over which the moment distribution is
exactly the same as that of the restrained column over the same length. It should
be noted that the choice of the equivalent hinged column is not unique since the
equivalent hinged column can be any portion of the restrained column provided
that the end moments of the equivalent hinged column are the same as the
corresponding moments in the restrained column. In practical design, the
first-order end moments
eql
1M and 2M of the real restrained column are always
known so that it is advisable to choose such an equivalent hinged column that it is
subjected to end moments equal to 1M and 2M . The determination of this
equivalent hinged column is illustrated in Fig. 6.8. The length of the equivalent
hinged column is equal to the distance between the two intersection points, where
the moment of the restrained column is equal to the first-order moment of the
adjacent column end. It should be noted that does not exist when the
maximum moment of the restrained column is smaller than the larger first-order
end moment. In such a case, the maximum moment can be taken to be the larger
first-order end moment for a conservative design. Once is known, the
maximum moment in the equivalent hinged column can be readily found from Eq.
6.41 with the use of
eql
eql
2
2creq
EINl
π= .
Now it is clear that the key to a satisfactory design is how to determine . The
exact expression for can be derived from the exact analysis presented in the
previous section, but it is obviously too complicated for design use. The current
design approach adopted in existing design codes (e.g. ACI 2005, EC2) employs
the effective length of the real restrained column as an approximation of .
eql
eql
effl eql
180
The use of the effective length to estimate the equivalent length was originally
proposed by Bijlaard et al. (1953) for restrained elastic columns with equal end
restraints and equal first-order end moments. Bijlaard et al. (1954) extended this
approach to more general cases so that restrained elastic columns with unequal
end restraints and unequal first-order end moments can also be dealt with. The
effective length of a column is the length between the points of inflection of the
buckled shape of a column subjected to pure axial compression. The effective
length can be exactly determined by solving Eq. 6.17 with
imposed. For a non-trivial solution, the determinant of the first matrix on the left
hand side must vanish, which leads to the following eqaution
1 2 0M M= =
( )2
1 2 1 2
2 tan22 1 4 1
tanG G G G
π ππ µ
π πµµ µ
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎛ ⎞⎜ ⎟ ⎜ ⎟ 0µ+ + − + − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(6.42)
where µ is the effective length factor and effll
= . The effective length factor as a
closed-form solution cannot be achieved; it has to be found using a numerical
approach. The current design approach in existing design codes (e.g. ACI-318
2005, ENV-1992-1-1 1992) employs an alternative graphical solution which was
originally proposed by Julian and Lawrence (1959). The value of µ can be
readily found from the alignment charts developed by Julian and Lawrence (1959)
provided that the magnitudes of the end restraints are known. A detailed
derivation of the effective length can be found in and Kavanagh (1960). In Eq.
6.42, and are the column-to-beam stiffness ratios at the two ends. It
should be noted that the conditions
1G 2G
11
2EIRG l
= and 22
2EIRG l
= are used when
deriving Eq. 6.42. These two conditions result from the assumptions made in
deriving Eq. 6.42. Eq. 6.42 is developed for an idealized frame of infinite storeys
and bays with all the columns in the top storey axially loaded. All the columns in
this frame reach their buckling load simultaneously as a result of the idealization.
In real frames, the situation is different but a discussion of the differences between
the idealized frame and a real frame is beyond the scope of this chapter. Interested
181
readers may refer to Johnston (1976) in which attempts were made to take some
of the particular conditions into account. For most columns in real frames, Eq.
6.42 does provide a useful approximation.
From the definitions of the effective length and the equivalent length, it is obvious
that the equivalent length is equal to the effective length when a column subjected
to pure axial compression reaches its buckling load. In the presence of the
first-order end moments, these two lengths become different. Lai and MacGregor
(1983) performed a study to investigate how the equivalent length varies with a
number of parameters. A similar but more comprehensive parametric study was
carried out using the exact solution given in the previous section to see if the use
of the effective length as an approximation of the equivalent length is adequate for
design use.
Following Lai and MacGregor (1983), a careful study revealed that the equivalent
length varies with the following factors: the first-order end moment ratio, the axial
load and the magnitude of the end restraints. Four parameters, namely, 1
2
MM
, 1
2
RR
,
µ and cr
NN
, which completely reflect the effects of the above factors were
varied in the parametric study. From the definitions of the above four parameters,
it is obvious that 1
2
MM
varies from -1 to 1, 1
2
RR
can have any value between zero
and infinity, µ ranges from 0.5 for fixed-fixed columns to 1 for hinged-hinged
columns, and the value of the normalized axial load cr
NN
is in the range of 0 to 1
(here is the buckling load of a restrained column subjected to pure axial
compression and
crN
2
2 2
EIl
πµ
= ).
Figs 6.9 to 6.11 shows the results of the parametric study. In these figures, the
vertical axis is the ratio of the equivalent length to the effective length while the
horizontal axis is the normalized axial load. Figs 6.9 to 6.11 shows respectively
182
the effect of one of the first three parameters as the last parameter varies. It should
be pointed out that the equivalent length is known for the two extreme cases of
loading: when , the equivalent length is equal to the effective length;
when , the equivalent length is equal to the real length of the restrained
column, but it can change dramatically when the axial load approaches zero.
Considering the fact that
crN N=
0N =
0N = is just an idealized situation and for a neater
illustration, the axial load was assigned a small value to represent when
producing Figs 6.9 to 6.11.
0N =
Fig. 6.9 shows the effect of 1
2
MM
. In all the three sub-figures, the effective length
factor is fixed at 0.75, but the end restraint ratio varies. Fig. 6.9a is for a
symmetrically restrained case. It can be seen that when the column is subjected to
uniform first-order moments, the equivalent length is always larger than the
effective length, which means that the effective length approach is
un-conservative. When the column is subjected to oblique first-order moments,
the equivalent length can either be larger or smaller than the effective length and it
should be noted that there is a range of axial load in which the equivalent length
does not exist. This range corresponds to cases where max 2M M< , as discussed in
previous sections and the size of this range depends on the first-order end moment
ratio as well as the magnitude of the end restraints. Fig. 6.9b is for and
. The three curves in the middle converge to a certain value when the axial
load approaches zero. The converged value indicates that when no axial load is
applied, the equivalent length is exactly the same as the real height of the
restrained column. The curve for
1R = ∞
2 0R =
1 2M M= should also converge to this value,
but it remains at a lower position for the reason stated earlier. The curve for
1 2M M= − suddenly drops at a certain axial load level. This is because in this
case the magnitude of 1M is initially reduced by the axial load, as illustrated in
Fig. 6.3c. The equivalent length thus does not exist when the axial load is in the
lower range. Fig. 6.9c is for 1 0R = and 2R = ∞ . The curves for 1 2M M= and
1 2M M= − are the same as those in Fig. 6.9b, because the interchange of the end
restraints does not affect the behavior of the column when the first-order end
183
moments are either symmetrically or anti-symmetrically distributed. The other
three curves however feature a reverse trend compared with those in Fig. 6.9b.
Fig. 6.10 illustrates the effect of the effective length factor. In both sub-figures,
the two end restraints are equally stiff. Fig. 6.10a is for 1 2M M= . It clearly
shows that at a given axial load level, the effective length approach becomes more
un-conservative for stiffer end restraints. A similar observation can be made about
Fig. 6.10b, where only one end of the column is subjected to a first-order end
moment.
Fig. 6.11 shows the effect of the end restraint ratio. Fig. 6.11a is for 1 2M M= .
The lowest curve in Fig. 6.11a is for the symmetrically restrained case and the
other two curves overlap for the reason given earlier. Fig. 6.11b is for cases where
only one end of the column is subjected to a first-order end moment.
Based on the results of the parametric study, it can be concluded that the
equivalent length in most cases is different from the effective length and the
effective length approach can be very un-conservative for certain cases. A possible
solution to overcome the shortcomings of the effective length approach is to
develop a more accurate alternative approximation for the equivalent length.
However, the present parametric study indicates that the equivalent length varies
with a number of equally important parameters. A close approximation must thus
take all these parameters into account, which makes the task difficult, particularly
if a relatively simple approach is sought. An alternative option is to develop a
conservative but simple approximation of the equivalent length but with some
sacrifice in the accuracy. Such a conservative approach can be achieved by
developing an approximation for the effective length that includes an appropriate
degree of overestimation. One such approximation has been proposed by Lai and
MacGregor (1983). Lai and MacGregor (1983) suggested the effective length
factor be taken as the smaller of
( )1 20.7 0.05 1G Gµ = + + ≤ (6.43a)
0.85 0.05 1sGµ = + ≤ (6.43b)
184
where sG is the smaller of and . Eq. 6.43 was originally proposed by
Cranston (1972) based on the numerical results of an elastic analysis similar to the
one presented in this chapter; his original intention was however to provide a
rough estimation of the effective length of restrained RC columns for a safe
design. Eq. 6.43 has been adopted in various design codes (e.g. ACI 2005, BS
1997, EC2). The final approach suggested by Lai and MacGregor (1983) for the
design of a single restrained column was to use Eq. 6.43 for the determination of
in Eq. 6.41, but Lai and MacGregor (1983) did not verify their proposed
approach. A study of the approach proposed by Lai and MacGregor (1983)
showed that their approach could be unnecessarily conservative in some cases,
particularly when the end restraints are strong. A further problem is that in some
other cases, their approach is extremely un-conservative, which must be avoided
in design. Fig. 6.12 gives some evidence of the shortcomings of their approach
and the causes are discussed in detail in the next section.
1G 2G
crN
6.5.4 Proposed Equations
Although the values of the equivalent length can vary with a number of
parameters, they generally fall in the neighborhood of the value of the effective
length. The direct use of the effective length has the advantages of simplicity and
familiarity to engineers. The associated un-conservativeness may be
counterbalanced through other considerations in the design approach. The
following equation is therefore proposed for the design of an isolated column:
( )max 2 2
1 1.25
1cr
m
cr
NNM C N
N
µ+ −=
−M M≥ (6.44)
where is defined by Eq. 6.40 and mC2
2 2crEINl
πµ
= with µ being found from
Eq. 6.42. Eq. 6.44 differs from Eq. 6.41 in that the coefficient 0.25 in the
numerator of Eq. 6.41 is replaced by 1.25 u− in Eq. 6.44. Eq. 6.44 reduces to Eq.
185
6.41 for hinged columns. It can be seen from later comparisons that this
adjustment can effectively eliminates the un-conservativeness associated with the
direct use of the effective length for the majority of cases.
In Figs 6.12 to 6.15, the exact values of the maximum moment are compared with
the values predicted using Eq. 6.44. Figs 6.12 to 6.15 are for four different values
of the first-order end moment ratio ( 1
2
1,0.5,0, 0.5MM
= − ). A comparison for
1
2
1MM
= − is not shown because the predictions of Eq. 6.44 are very conservative
for this case as is assigned a constant value of 0.4. Only selected numerical
results are shown here due to space limitation, and the attention is focused on the
upper bound and the lower bound of the
mC
max
2
MM
ratio. The upper bound and the
lower bound curves are the envelope curves of all curves corresponding to
different magnitudes of the end restraints, but they can be very closely
approximated by the curves for the following four extreme cases: 1) ,
2) , 3) and 4)
1 2 0R R= =
1 2R R= = ∞ 1 20,R R= = ∞ 01 2,R R= ∞ = . For clarity,
comparisons for the first two cases and those for the next two cases are shown in
two separate sub-figures.
Fig. 6.12 is for cases where 1 2M M= . The exact results are shown as the solid
lines while the approximate results from Eq. 6.44 are shown as the dashed lines.
The results from Lai and MacGregor’s (1983) approach are also shown in Fig.
6.12, but not in Figs 6.13 to 6.15 because the performance of their approach is
similar in all cases. Fig. 6.12a shows that when both ends of a column are hinged,
the predictions of Eq. 6.44 and those of Lai and MacGregor’s (1983) approach are
exactly the same and they are both very close to the exact results. When both ends
of a column are fixed, Eq. 6.44 becomes slightly un-conservative while Lai and
MacGregor’s (1983) approach is very un-conservative at lower axial loads and
becomes very conservative at high axial loads. The un-conservativeness of Lai
and MacGregor’s (1983) approach for lower axial loads arises from the
overestimation of the effective length by Eq. 6.43. The minimum value of the
186
effective length factor predicted by Eq. 6.43 is 0.7 while the exact value is 0.5. As
a result of the overestimation of the effective length, the value of in Eq. 6.41
is underestimated by Eq. 6.43 so that the term
crN
1cr
NN
− in Eq. 6.41 can become
negative. When this happens, the maximum moment according to Eq. 6.41 is 2M ,
as indicated by the horizontal line in Fig. 6.12a. Fig. 6.12b shows the comparison
for hinged-fixed columns. Since the results shown in Fig. 6.12b are for columns
with uniform first-order moments, the two solid curves representing the exact
solution overlap with each other. Both approximate approaches neglect the effect
of the end restraint ratio so that both of them predict only a single curve for cases
(3) and (4) in Figs 6.12b to 6.15b. For the cases shown in Fig. 6.12b, Eq. 6.44
provides accurate predictions while the performance of Lai and MacGregor’s
(1983) approach is similar to that revealed by Fig. 6.12a.
Similar observations can be made about Figs 6.13 to 6.15. For case (2), Eq. 6.44 is
always conservative while for case (1), Eq. 6.44 is generally conservative except
when the predicted maximum moment is similar to 2M . Similarly, For case (3)
Eq. 6.44 is always conservative while for case (4), Eq. 6.44 is generally
conservative except when the predicted maximum moment is similar to 2M .For
other cases not shown here, Eq. 6.44 provides consistently conservative
predictions for the majority of them and slightly un-conservative predictions for
the rest of them. In summary, Eq. 6.44 possesses a simple form which makes it
easy to use in design and provides reasonably accurate predictions which are
acceptable for design use.
6.6 CONCLUSIONS
This chapter has been concerned with the analysis of elastic columns with elastic
end restraints as a preparation for the analysis of slender FRP-confined RC
columns presented in Chapters 7 to 9. The exact solution for the lateral deflection
of such columns induced by combined bending and axial compression was
derived first. The rationale behind the current column design approach was next
explained. Finally, approximate design equations for elastic columns were
187
presented, which represent an improvement to existing approaches.
It was shown that the current design approach transforms a restrained column into
an equivalent hinged column with equal end eccentricities. The length of the
equivalent hinged column, referred to as the equivalent length in this chapter, is
generally different from the effective length of the restrained column. However, it
was shown that the equivalent length does not exist in certain cases. In order to
develop a unified design approach, the effective length can be used as a
reasonable estimate of the equivalent length and the errors so introduced may be
counterbalanced by other appropriate considerations in dealing with factors such
as the end restraints and the equivalent uniform moment factor. That is why this
design approach is commonly known as the “effective length approach”. It was
also shown that even for elastic columns the effective length approach is
approximate in nature because it was developed from idealized frame conditions.
The effective length approach is popularly used in the design of inelastic columns,
such as RC columns. With the presence of material nonlinearity, the problem of
column design becomes even more complicated and simple closed-form design
equations with great accuracy are thus not possible. Existing design equations for
RC columns are generally based on the rationale explained in this chapter and
they aim to provide reasonable but conservative predictions through appropriate
treatment of the key elements of the effective length approach: 1) in determining
the effective length of a restrained RC column, some codes (e.g. ACI-318 2005)
use simple charts developed from Eq. 6.42 to relate the effective length to the
column-to-beam stiffness ratio, while others (e.g. BS-8110 1997) allow the
effective length to be roughly estimated according to the end conditions of a
column; 2) in determining the equivalent uniform moment factor, Eq. 6.40 is
generally adopted because of its simplicity. As a result, it must be borne in mind
that although the current design approach has a rational basis, it is approximate in
nature.
188
6.7 REFERENCES
ACI 318-05 (2005). Building Code Requirements for Structural Concrete and Commentary, ACI Committee 318, American Concrete Institute.
Austin, W.J. (1961).”Strength and design of metal beam-columns”, Journal of the Structrual Division, ASCE, 87(ST4), 1-32.
Bijlaard, P.P., Fisher, G.P. and Winter, G. (1953). “Strength of columns elastically restrained and eccentrically loaded”, Proceedings, ASCE, Separate No. 292.
Bijlaard, P.P. (1954). “Buckling of columns with equal and unequal end eccentricites and equal and unequal end restraints”, Procceedings, Second Natl. Congress of Applied Mechanics, Ann Arbor, Mich.
BS 8110 (1997). Structural Use of Concrete, Part 1. Code of Practice for Design and Construction, British Standards Institution, London, UK.
Cranston, W.B. (1972). Analysis and Design of Reinforced Concrete Columns, Research Report 20, Cement and Concrete Association, UK.
Chen, W.F. and Atsuta, T. (1976). Theroy of Beam-Columns, New York, McGraw-Hill.
Chen, W.F. and Liu, E.M. (1991). Stability Design of Steel Frames, Boca Raton, CRC Press.
ENV 1992-1-1 (1992). Eurocode 2: Design of Concrete Structures – Part 1: General Rules and Rules for Buildings, European Committee for Standardization, Brussels.
Galambos, T.V. (1968). Structrual Members and Frames, Englewood Cliffs, N.J., Prentice-Hall.
Johnston, B.G. (1976). Guide to Stability Design Criteria for Metal Structures, New York, Wiley.
Julian, O.G. and Lawrence, L.S. (1959). “Notes on J and L Nomographs for Determination of Effective Lengths, unpublished report.
Kavanagh, T.C. (1960). “Effective length of framed columns”, Proceedings, ASCE, 86(ST2), 81-101.
Lai, S.M.A and MacGregor, J. G. (1983). “Geometric nonlinearities in nonsway frames”, Journal of Structural Engineering, ASCE, 109(12), 2770-2785.
MacGregor, J.G., Breen, J.E. and Pfrang E.O. (1970). “Design of slender concrete columns”, ACI Journal, 67(1), 6-28.
Newmark, N.M. (1943). “Numerical procedure for coputing deflection s, moments, and buckling loads”, Transactions, ASCE, 108, 1161-1234.
189
Winter, G. (1954). “Compression members in trusses and frames”, The Philosophy of Column Design, Proceedings, 4th Technical Session, Column Research Council, Lehigh University.
190
Moment M
Axi
al L
oad
N
Nuo
Muo
Fig. 6.1 Axial load-bending moment diagram of an elastic column
l
N
N
Me1
Me2
M1
M2
V
V
v
M1
v w
N
N
M2
V
V
N
N
v wf
M
f
x
M2
V
V
(a) Restrained column
(b) Column subjected to end moments only
(c) Application of axial load
(d) Moment at a given height of the column
Fig. 6.2 Forces and corresponding deflections of a restrained column
191
0 1 2
0
21
1
2
0
(a) 1 21 0M M≥ ≥ (b) 1 20 0M M> ≥ − .9 (c) 1 20.9 1M M− > ≥ −
Fig. 6.3 Moment distributions of restrained columns with different first-order end
moment ratios
M1
M2
Mmax
Mmax
Meq
Meq
Fig. 6.4 Transformation of a hinged column into a standard hinged column
192
0 0.2 0.4 0.6 0.8 11
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
Exact SolutionEquation 6.38Equation 6.39
Fig. 6.5 Moment amplification factor of the standard hinged column
-1 -0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
End Moment Ratio M1/M2
Equ
ival
ent U
nifo
rm M
omen
t Fac
tor
Cm
N/Ncr = 0 to 1
Eq. 6.40
Fig. 6.6 Equivalent moment factor
193
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
Exact SolutionEquation 6.41
M1/M2 = 1
M1/M2 = 0.5
M1/M2 = 0
M1/M2 = -1
M1/M2 = -0.5
Fig. 6.7 Performance of the proposed equation for the design of hinged columns
M2
M1M1
M2
leq
Fig. 6.8 Transformation of a restrained column into an equivalent hinged column
194
0 0.2 0.4 0.6 0.8 10.9
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
N/Ncr
µ equ
/µ
µ=0.75
R1/R2=1
M1/M2 = 1M1/M2 = 0.5M1/M2 = 0M1/M2 = -0.5M1/M2 = -1
(a)
0 0.2 0.4 0.6 0.8 11
1.05
1.1
1.15
1.2
1.25
1.3
1.35
N/Ncr
µ equ
/µ
µ=0.75R1/R2=Inf
M1/M2 = 1M1/M2 = 0.5M1/M2 = 0M1/M2 = -0.5M1/M2 = -1
(b)
195
0 0.2 0.4 0.6 0.8 10.9
0.95
1
1.05
1.1
1.15
N/Ncr
µ equ
/µ
µ=0.75R1/R2=0
M1/M2 = 1M1/M2 = 0.5M1/M2 = 0M1/M2 = -0.5M1/M2 = -1
(c)
Fig. 6.9 Effect of first-order end moment ratio on equivalent length
196
0 0.2 0.4 0.6 0.8 11
1.05
1.1
1.15
N/Ncr
µ equ
/µ
M1/M2=1
R1/R2=1
µ = 0.6µ = 0.75µ = 0.9
(a)
0 0.2 0.4 0.6 0.8 10.95
1
1.05
1.1
N/Ncr
µ equ
/µ
M1/M2=0
R1/R2=1
µ = 0.6µ = 0.75µ = 0.9
(b)
Fig. 6.10 Effect of effective length factor on equivalent length
197
0 0.2 0.4 0.6 0.8 11
1.01
1.02
1.03
1.04
1.05
1.06
1.07
N/Ncr
µ equ
/µ
M1/M2=1µ=0.75
R1/R2 = 0R1/R2 = 1R1/R2 = Inf
(a)
0 0.2 0.4 0.6 0.8 10.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
N/Ncr
µ equ
/µ
M1/M2=0
µ=0.75
R1/R2 = 0R1/R2 = 1R1/R2 = Inf
(b)
Fig. 6.11 Effect of end restraint ratio on equivalent length
198
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
M1/M2=1
both
end
s hi
nged
both ends f
ixed
Exact SolutionEquation 6.44Lai and MacGregor (1983)
(a) Cases 1 and 2
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
M1/M2=1
one e
nd fix
ed, o
ne e
nd h
inged
Exact SolutionEquation 6.44Lai and MacGregor (1983)
(b) Cases 3 and 4
Fig. 6.12 Comparisons of exact results with approximate results for 1 2 1M M =
199
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
M1/M2=0.5M1/M2=0.5
both
end
s hi
nged
both ends fixed
Exact SolutionEquation 6.44
(a) Cases 1 and 2
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
M1/M2=0.5
end
1 hi
nged
, en
d 2
fixed
end 2 hinged
end 1 fixed,
Exact SolutionEquation 6.44
(b) Cases 3 and 4
Fig. 6.13 Comparisons of exact results with approximate results for 1 2 0.5M M =
200
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
M1/M2=0
both
end
s hi
nged
both
end
s fix
ed
Exact SolutionEquation 6.44
(a) Cases 1 and 2
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
M1/M2=0
end
2 fix
ed
end
1 hi
nged
,
end 1 fixed, end 2 hinged
Exact SolutionEquation 6.44
(b) Cases 3 and 4
Fig. 6.14 Comparisons of exact results with approximate results for 1 2 0M M =
201
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
M1/M2=-0.5
both
end
s hi
nged
both
end
sfix
ed
Exact SolutionEquation 6.44
(a) Cases 1 and 2
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Normalized Axial Load N/Ncr
Nor
mal
ized
Max
imum
Mom
ent
Mm
ax/M
2
M1/M2=-0.5
end
2 fix
ed
end
1 hi
nged
,
end 1 fixed, end
2 hin
ged
Exact SolutionEquation 6.44
(b) Cases 3 and 4
Fig. 6.15 Comparisons of exact results with approximate results for
1 2 0.5M M =−
202
CHAPTER 7
THEORETICAL MODEL FOR
SLENDER FRP-CONFINED RC COLUMNS
7.1 INTRODUCTION
Chapter 5 has presented a set of design equations for FRP-confined RC columns
that are short enough for the slenderness effect to be ignored. Chapter 6 has shown
how the slenderness of an elastic column can undermine its strength through the
interaction between the axial load it sustains and its lateral deflection. The lateral
deflection of an elastic column can be exactly evaluated, but this is not possible
for an FRP-confined RC column due to the nonlinearity of its constituent
materials. As a result, a rational theoretical model based on certain numerical
procedures is required for the analysis of slender FRP-confined RC columns.
Although various theoretical models have been proposed to analyze slender
columns made of different materials (Chen and Atsuta 1976), to the best
knowledge of the author, none of them has been used to model slender FRP-
confined RC columns. A recent study by Yuan and Mirmiran (2001) on the
modeling of slender concrete-filled FRP tubes appears to be the only existing
analytical study addressing the slenderness effect in concrete columns receiving
FRP confinement. In their model, the total eccentricity of a column (i.e. load
eccentricity plus lateral deflection) is assumed to be a portion of a cosine wave
depending on the load eccentricity at either column end. However, as their
columns did not have internal reinforcing steel bars and the FRP tubes had
significant axial stiffness, their model and the results obtained from their study are
not directly applicable to slender FRP-confined RC columns.
203
Against this background, this chapter presents a theoretical model for slender
FRP-confined RC columns. The proposed model is more sophisticated than the
model of Yuan and Mirmiran (2001) in that no assumption is made about the
lateral deflected shape of the column; it is instead sought in an iterative manner.
The theoretical model is described in detail first and its verification using
experimental results of both RC columns and FRP-confined RC columns is next
discussed. It should be noted that only a very limited number of tests on slender
FRP-confined RC columns have been reported in the open literature and all those
available to the author are used herein to verify the proposed model.
7.2 THEORETICAL MODEL
7.2.1 General
The method of analysis used herein is the well-known numerical integration
method. This method was originally proposed by Newmark (1943) and has been
widely adopted in the analysis of RC columns (e.g. Pfrang and Siess 1961;
Cranston 1972), steel columns (e.g. Shen and Lu 1983) and composite columns
(e.g. Choo et al. 2006; Tikka and Mirza 2006). However, to the best knowledge of
the author, this method has not been used in the analysis of FRP-confined RC
columns. In the present study, this method is applied to pin-ended FRP-confined
RC columns subjected to eccentric loads at both ends with the end eccentricities
being and , respectively (Fig. 7.1). It should be noted that is taken to be
the one with a larger absolute value and is always assigned a non-negative value.
This indicates that has a negative value when the column is bent in double
curvature. The column with a length l is equally divided into a desirable number
of segments with a length of
1e 2e 2e
1e
l∆ . The column section at each grid point is divided
into a desirable number of horizontal layers. The lateral displacement at each grid
point at a particular loading stage can be sought in an iterative manner by making
use of the axial load-moment-curvature ( N M φ− − ) relationship of the column
section and the numerical integration function of the column. The full-range axial
load-lateral deflection curve of the column (referred to as the load-deflection
204
curve for brevity hereafter) can then be traced in an incremental manner using
either a force-control or deflection-control technique. For the ascending branch of
the load-deflection curve, it is more convenient to use the force-control technique
(increasing the axial load by small steps). In the case of a stability failure, a
descending branch of the load-deflection curve exists and the deflection-control
technique (increasing the deflection of a particular grid point by small steps)
should be used to trace the descending branch. In summary, the load-moment-
curvature relationship and the numerical integration function are the two key
elements of the numerical integration method and they are discussed in detail in
the following sub-sections, where the procedure for generating the full-range load-
deflection curve is also described.
7.2.2 Construction of Axial Load-Moment-Curvature Curves
The procedures for the construction of the moment-curvature curve under a given
axial load have been presented in detail in Chapter 5, so they are not repeated
herein. The stress profile of confined concrete in compression is described using
the modified Lam and Teng Model (I) with the value of 1.75 for the first term on
the right hand side of Eq. 4.9 replaced by 1.65; the tensile strength of concrete is
ignored. The longitudinal steel reinforcement is assumed to have an elastic-
perfectly plastic stress-strain response and any confinement effect from the hoop
steel reinforcement is ignored. In the present analysis, every section is divided
into 50 horizontal layers and the solution is considered successful once the
difference between the resultant axial force and the applied axial load is within
. Each moment-curvature curve produced in the present analysis consists of
201 points, which ensures sufficient numerical accuracy.
610 N−
7.2.3 Numerical Integration for the Column Deflection
Since the lateral deflections of the column are very small compared to the length
of the column, the curvature can be taken to be the second order derivative of the
lateral deflection of the column. Using the central difference equation, the
relationship between the lateral deflections and the curvatures can be expressed as
205
( )( 1) ( ) ( 1)
( )2
2i i ii
f f f
lφ+ −− +
= −∆
(7.1)
where and ( )if ( )iφ are the lateral displacement and curvature at the grid point
respectively and . i is the index of the grid point and is the
number of segments that the column is equally divided into (Fig. 7.1). is
used in the present analysis. Eq. 7.1 can be rewritten as
thi
2,3, 1i m= − 1m−
31m =
( )2( 1) ( ) ( 1) ( )2i i i if f f lφ+ −= − − ∆ (7.2)
Eq. 7.2 is the numerical integration function used to find the lateral deflection of
the column. The implementation of this equation is described in the following
section.
7.2.4 Generation of the Ascending Branch of the Load-Deflection Curve
The axial load is increased by small increments to generate the ascending branch
of the load-deflection curve. For a given axial load , the first step is to construct
the corresponding moment-curvature curve using the procedure described above.
Under this axial load, the first order moment at each grid point can be easily
calculated
N
1,( ) ( )i iM N e= ⋅ (7.3)
where and are the first order moment and the initial eccentricity at the
grid point respectively. Note that the initial eccentricity follows a linear
distribution with and
1,( )iM ( )ie
thi
(1) 2e e= ( ) 1me e= . If the lateral deflection of the column is
known, the second order moment can be expressed as
2,( ) ( )i iM N f= ⋅ (7.4)
206
and the total moment can then be expressed as
( )( ) 1,( ) 2,( ) ( ) ( )i i i i iM M M N e f= + = ⋅ + (7.5)
Now assume a value for (2)f . The assumed value can be any reasonably small
value or simply zero. The moment at this grid point (2)M can then be evaluated
using Eq. 7.5 and the curvature at this grid point (2)φ can be retrieved from the
moment-curvature curve corresponding to the present axial load. With (2)M and
(2)φ known and note that (1) 0f = , the lateral displacement of the third grid point
(3)f can be evaluated using Eq. 7.2. It is evident that the lateral deflection of the
column can be traced from grid point to grid point by repeating the above
procedure. Once the calculations eventually reach the other end of the column, its
lateral displacement needs to be examined to see if it satisfies . If this
condition is satisfied, then the correct lateral deflection of the column is found.
Otherwise, the assumed value of
( ) 0mf =
(2)f needs to be adjusted until is
satisfied. It is suggested that the new value of
( ) 0mf =
(2)f be taken as the previous value
minus ( )
1mf
m −. In the present analysis, the solution is considered to be acceptable if
the calculated ( )mf has an absolute value smaller than 410− mm.
The above procedure determines the lateral deflection of the column under a
particular axial load. The ascending branch of the load-deflection curve can be
obtained by calculating the deflection of the column for a series of successively
increasing loads. In the present analysis, an initial load increment (i.e. load step
size) of is used, where is the axial load capacity of the column section
under consideration when subjected to concentric compression as given by
10.1 uN 1uN
'
1u cc c yN f A f= + sA (7.6)
207
where and cA sA are the total cross-sectional areas of concrete and longitudinal
steel bars, respectively. After several load increments, the applied load eventually
exceeds the maximum load that the column can sustain. This situation arises when
the moment at any grid point calculated by Eq. 7.5 exceeds the maximum moment
on the moment-curvature curve under the present axial load. When this occurs, the
calculation process needs to restart from the last load level and a smaller load
increment of is used for the subsequent load steps. After some steps, a
similar treatment needs to be adopted with the load step size reduced by a factor
of 10. The same process is repeated until the load step size is eventually reduced
to . The axial load capacity of the column is then taken to be the
maximum load for which a convergent solution of the lateral deflection can be
found. The corresponding value of
10.01 uN
6110 uN−
uN
(2)f is recorded as a reference displacement
value reff for use in the generation of the descending branch of the load-
deflection curve, as discussed in the following sub-section.
7.2.5 Generation of the Descending Branch of the Load-Deflection Curve
If the column is controlled by material failure, its load-deflection curve has no
descending branch. However, if the column is slender enough to trigger stability
failure, a descending branch of the load-deflection curve exists. In this case, the
displacement-control technique should be used to trace the descending branch.
The numerical procedure is similar to that described in the previous sub-section
with the only difference being that the aim is to find the correct axial load under a
prescribed value of (2)f .
A displacement increment of 0.1 reff is initially used (i.e. the prescribed value of
(2)f in the first step is 1.1 reff ). The initial assumed value of the corresponding
axial load can be taken as . The corresponding deflected configuration can
then be calculated using the numerical procedure described in the previous sub-
section. It should be noted that the calculated
uN
( )mf always has a negative value,
because the actual axial load must be smaller than . The assumed axial load is uN
208
thus successively reduced at steps of 0 until the calculated .01 uN ( )mf has a
positive value. The correct axial load can then be determined using the bisection
method by setting the last two assumed values of axial load to be the upper bound
and lower bound respectively in the bisection method. The final axial load derived
from the bisection method is the solution for the current step and it is used as the
initial value in the next step. After several increments of (2)f , the analysis
eventually fails to find a convergent solution. This situation arises when the
moment at any grid point calculated by Eq. 7.5 exceeds the maximum moment on
the moment-curvature curve under the present axial load. When this occurs, the
calculation needs to restart from the prescribed value of (2)f and with a smaller
displacement increment of 0.01 reff . The entire process is repeated and the
analysis stops when the increment is reduced to 610 reff− .
It should be noted that the accuracy of the present analysis is affected by a number
of factors (i.e. the number of segments the column is divided into, the number of
horizontal layers the cross section is divided into, the number of points the
moment-curvature curve consists of, and the tolerances adopted in the analysis). A
convergence study showed that all these factors have been very well looked after
in the present study (i.e. any refinement to these factors will not have any
significant effect on the numerical results). A computer program was developed to
implement the numerical procedure described above using Matlab 7.1.
7.3 VERIFICATION OF THE THEORETICAL MODEL
7.3.1 Comparison with Cranston’s Numerical Results
The validity of the present model is first verified using the numerical results of
Cranston’s (1972) theoretical model for RC columns, which was based on a
similar numerical integration procedure. Cranston’s (1972) model used the same
stress-strain relationships for both concrete and steel as those adopted by the
present study except that the concrete was assumed to have an ultimate axial strain
of 0.0035. This value was also used in the present model for this set of
209
comparisons. Table 7.1 lists the properties of the columns analyzed by Cranston
(1972). These columns were all pin-ended and were eccentrically loaded at only
one end. The cross-sectional shape was rectangular rather than circular and it is
illustrated in Fig. 7.2. The height and the width of the cross section are denoted by
and respectively and the eccentricity is in the height direction. The steel
reinforcement ratio is denoted by
h b
sρ and the distance between the steel
reinforcement is denoted by . 'h cuf is the characteristic cube strength of concrete
and yf is the characteristic yield strength of steel reinforcement. Cranston (1972)
normalized the load-deflection curves (Fig. 7.3) using appropriate reference
values. The lateral deflection at mid-height of the column midf was normalized by
the section height while the axial load was normalized by the design value of the
section axial load capacity under concentric compression , which was
determined from the design values of material strengths (the numbers bracketed in
Table 7.1; the partial safety factors for concrete and steel are 1.5 and 1.15
respectively). However, the characteristic material strengths were used in the
numerical analysis. The predictions of both models are shown in Fig. 7.3. It can
be clearly seen that these predictions are in excellent agreement for both the
material failure and stability failure cases.
uoN
7.3.2 Comparison with Experimental Results
Predictions from the present theoretical model are also compared with exisiting
experimental results of both RC columns and FRP-confined RC columns in this
sub-section. First, Kim and Yang’s (1995) tests on RC columns are used for
comparison. Details of Kim and Yang’s tests are listed in Table 7.2. These
specimens were square in shape and had a wide range of concrete strength. All
these specimens were bent in symmetrical single curvature ( ). and
are the test and theoretical axial load capacities of a column, respectively.
In Kim and Yang’s (1995) tests, two physically identical columns were prepared
for each configuration. Close agreement between the present predictions and the
test results can be seen in the last column of Table 7.2. In addition, the full-range
load-deflection curves were also reported by Kim and Yang (1995), and those of
1 2e e= ,u testN
,u theoN
210
the normal strength series are compared with predictions of the present model in
Fig. 7.4. No further comparisons for RC columns only are discussed herein
because the present method of analysis has long been well-accepted for RC
columns (e.g. Pfrang and Siess 1961; Cranston 1972).
Only a very limited number of experimental studies have been carried out on
FRP-confined circular RC columns. These studies include Tao et al. (2004), Hadi
(2006), Fitzwilliam and Bisby (2006) and Ranger and Bisby (2007). Hadi (2006)
tested five small-scale (150 mm in diameter) circular normal strength concrete
columns wrapped with CFRP and subjected to axial loading with the same end
eccentricity of 42.5 mm. One reference column which received no FRP wrapping
was also tested. Four of the five FRP-confined columns were not provided with
internal steel reinforcement so they failed by the cracking of concrete on the
tension face of these columns. Unfortunately, the hoop strains on the FRP jacket
were not reported which makes it difficult for these test results to be used to verify
the proposed theoretical model.
The two test series of Fitzwilliam and Bisby (2006) and Ranger and Bisby (2007)
were conducted by the same research group and the test configurations of the two
series are similar. Therefore, these two test series are discussed together, although
they had different test objectives. Fitzwilliam and Bisby (2006) varied the column
height but fixed the load eccentricity while Ranger and Bisby (2007) varied the
load eccentricity but fixed the column height. In Ranger and Bisby’s (2007) test
series, all the columns were 152mm in diameter and 600 mm in height, and were
connected to a steel system at both column ends to create the pinned end condition
and the load eccentricity. All the columns were reinforced with four 6.4 mm
diameter steel bars longitudinally and 6.4 mm diameter steel ties spaced at 100
mm transversely with a 25 mm concrete cover to the longitudinal reinforcement.
A total of six load eccentricities were considered: 0, 5, 10, 20, 30, or 40 mm. For
each load eccentricity, a column confined with a single ply of CFRP as well as an
unconfined reference column was tested. The FRP jacket included a 100 mm
overlapping zone with its centerline being at the same circumferential position as
the centerline of the less compressed face of the column. The concrete had a
cylinder strength of 33.2 MPa, the steel reinforcement had an yield strength of 710
211
MPa, and the FRP had an elastic modulus of 90 GPa (based on a nominal
thickness of 0.381 mm) and a rupture strain of 1.12% obtained from tensile
coupon tests. A summary of Ranger and Bisby’s (2007) tests is given in Table 7.3.
In Fitzwilliam and Bisby’s (2006) test series, the column height varied from 300
mm to 1200 mm at an interval of 300 mm and all the columns were tested with a
fixed load eccentricity of 20 mm. The steel reinforcement and FRP used in the
study were the same as those used in Ranger and Bisby (2007). The concrete
cylinder strength averaged from three batches of cylinder tests conducted during
the period of column tests was 35.5 MPa. The other test parameters were similar
to those of Ranger and Bisby’s (2007) tests. It should be noted that some columns
received longitudinal FRP wrapping before hoop FRP wrapping, but these tests
have been excluded from the present comparison. Fitzwilliam and Bisby’s (2006)
tests are summarized in Table 7.4. In both test series, the lateral deflection of the
column was monitored at three different vertical locations with one being located
at the mid-height of the column.
Ancillary tests on standard concrete cylinders under concentric compression were
also conducted in both test series. The concrete cylinders were made from the
same concrete and confined with the same type and amount of FRP as the
columns. The predictions of Lam and Teng’s stress-strain model are first checked
against these cylinder tests before the proposed theoretical model is verified using
the column tests. These cylinder tests are summarized in Table 7.5. The concrete
cylinders in Ranger and Bisby (2007) were only confined with a 1-ply CFRP
jacket while some of the cylinders in Fitzwilliam and Bisby (2006) were confined
with a 2-ply CFRP jacket as some of the columns tested by Fitzwilliam and Bisby
(2006) were also confined with a 2-ply CFRP jacket. The compressive strength of
unconfined concrete varied slightly in Fitzwilliam and Bisby’s (2006) tests
because the concrete cylinders were tested at different ages during column testing.
It should be noted that in Ranger and Bisby (2007), the FRP hoop rupture strain
found from their cylinder tests was reported to be 0.62%, a much smaller value
than that found from Fitzwilliam and Bisby’s (2006) cylinder tests (1.17%),
although the test configurations were almost the same in the two test series. It was
later confirmed by Bisby (2008) that this small value arose from an editorial error
212
and the correct value is 1.15%. The compressive strength 'ccf and the ultimate
axial strain cuε predicted by Lam and Teng’s stress-strain model [modified
Version (I)] are listed in the last two columns of Table 7.5. It can be seen that the
predicted values of the compressive strength are reasonably close to the
experimental values, however, the predicted values of the ultimate axial strain are
much larger than the experimental values, particular for the specimens confined
with a 1-ply FRP jacket. To minimize the errors that might arise from this
discrepancy in modeling the column behavior, the experimental values of 'ccf and
cuε were directly incorporated in Lam and Teng’s stress-strain model [modified
Version (I)] in predicting the behavior of test columns. MPa and ' 44.2ccf =
0.86%cuε = were used in modeling the columns tested by Ranger and Bisby
(2007). For the columns in Fitzwilliam and Bisby (2006), MPa and ' 40.7ccf =
0.788%cuε = were used for the 1-ply jacket while and ' 60.1ccf = 1.443%cuε =
were used for the 2-ply jacket. The predicted values of 'ccf and cuε were also used
in predicting the behavior of test columns as a reference.
The predicted axial load capacities of all columns are compared with the
experimental values in Tables 7.3 and 7.4. It can be noted that for all the
unconfined columns in Ranger and Bisby’s (2007) study, their axial load
capacities are overestimated by some 20% while this overestimation is much
smaller for Fitzwilliam and Bisby’s (2006) tests. The relatively large
overestimation observed for Ranger and Bisby’s (2007) tests may be due to: 1)
additional eccentricities due to geometric/material imperfections and inaccurate
alignment of load; and 2) possible spalling of the concrete cover in unconfined
columns during testing which reduces the effective cross-sectional area. In
particular, it can be shown that the theoretical results are sensitive to an additional
eccentricity, especially when the nominal load eccentricity is small. For example,
if a 10 mm additional eccentricity is assumed, then the predicted axial load
capacity of column U-0 becomes 525 kN, which is much closer to the
experimental value. However, in the comparisons for these two test series, no
additional eccentricity was used except for columns U-0 and C-0 (subjected to
nominally concentric compression) where a small eccentricity of 1 mm was used.
213
For a column subjected to concentric compression, a small load eccentricity (or
other forms of imperfection) needs to be introduced into the theoretical model as
otherwise no lateral deflections can be predicted by the theoretical model. For
FRP-confined columns, in most cases, their axial load capacity is underestimated
by about 5% to 15%. The only exceptions occurred in columns C-30 and C-40
which had relatively large load eccentricities. Their axial load capacities were
overestimated by nearly 20%. However, it is difficult to accept this overestimation
as clear evidence that the load eccentricity has a detrimental effect on confinement
effectiveness because: 1) no such trend can be found for the range of smaller load
eccentricities (0, 5, 10, 15, 20 mm); 2) the theoretical predictions are sensitive to
additional eccentricity whose exact value is unknown; and 3) scatter may exist in
the test results.
The theoretical and experimental full-range load-deflection curves of FRP-
confined columns tested by Ranger and Bisby (2007) and Fitzwilliam and Bisby
(2006) are compared in Fig. 7.6 and Fig. 7.7 respectively. Two theoretical curves
are shown for each column; they were produced using the experimental and
predicted 'ccf and cuε values respectively. All the theoretical curves terminate
when the extreme compression fiber reaches the ultimate axial strain of confined
concrete. It should be noted that column C-0 in Ranger and Bisby’s (2007) test
series and the pair of physically identical columns 300C-1-0A and 300C-1-0B in
Fitzwilliam and Bisby’s (2006) test series are excluded from the present
comparisons, because only very small lateral deflection of these columns were
recorded. This is due to the fact that the former had a relatively small height-to-
diameter ratio of four and was tested under nominally concentric compression
while the latter only had a length-to-diameter ratio of two, so the slenderness
effect in these columns was not significant enough to induce large lateral
deflections. It can be seen in Figs 7.6 and 7.7 that for the same column, the
theoretical curve produced using the 'ccf and cuε values predicted by Lam and
Teng’s stress-strain model [modified Versi (I)] terminates at a larger
deformation value, which is closer to test results, than the other theoretical curve.
This is because the prediction of the deformability capacity of these columns
largely depends on the value used for the ultimate axial strain of confined concrete,
on
214
which is overestimated by Lam and Teng’ stress-strain model as indicated by the
cylinder tests results (see Table 7.5). This observation also implies that
eccentricity might have an effect on the stress-strain behavior of FRP-confined
concrete and this possible effect needs to be fully clarified in the future. It can also
be seen in Figs 7.6 and 7.7 that there exist considerable discrepancies between the
theoretical and experimental stiffnesses of these columns. This may be due to
inaccurate displacement measurements at the initial loading stage, because the
lateral deflections might be too small to be precisely measured (it is worth noting
that even some negative lateral displacements at the mid-height of column C-5
were recorded). It should also be noted that the lateral displacement of column C-
10 was not accurate (see Fig. 7.5b), as confirmed by Bisby (2008). Fig. 7.5b
shows that column C-10 possesses a much larger deformation capacity than
column C-20. This contradicts engineering intuition, because column C-20 should
have a larger deformation capacity than column C-10, given the fact that the
former was loaded with a larger initial load eccentricity and both columns failed at
very similar hoop rupture strains of the FRP jacket (1.07% for column C-10 and
1.15% for column C-20).
Tests on slender FRP-confined circular RC columns performed by Tao et al.
(2004) have also been simulated using the present theoretical model. A total of 16
columns were tested and the properties of these columns are listed in Table 7.6.
All the columns were 150mm in diameter and reinforced with four 12 mm
longitudinal steel bars and 6 mm steel hoops spaced at 100 mm. The columns had
a 21 mm concrete cover to the longitudinal steel reinforcement. The C1 series had
a length-to-diameter ratio of 8.4 while this ratio for the C2 series was 20.4. Each
series included four different load eccentricities (0, 50, 100 and 150 mm) and for
each eccentricity, one unconfined column and one FRP-confined column were
tested. As the load eccentricities adopted were relatively large, all the columns
were cast with corbel ends and capped with a steel plate with V-shaped grooves to
achieve the required load eccentricities and a pinned end condition. A 150 mm
overlapping zone was adopted in forming the FRP jacket and the position of the
overlapping zone is similar to that adopted by Ranger and Bisby (2007). Apart
from the column tests, a series of ancillary cylinder compression tests were also
conducted to determine the material properties. The average cylinder compressive
215
strength of unconfined concrete was found from six standard concrete cylinders to
be 48.2 MPa. The longitudinal steel reinforcement had a yield strength of 388.7
MPa. The CFRP had an elastic modulus of 255 GPa based on a nominal thickness
of 0.17 mm per ply and a rupture strain of 1.67% based on tensile coupon tests. In
addition, three FRP-confined concrete cylinders were tested under concentric
compression. These cylinders were confined with the same type and amount (2-
plies of CFRP) of FRP as the columns. The stress-strain curves predicted using
Lam and Teng’s stress-strain model [modified Version (I)] are compared with the
experimental curves in Fig. 7.7. It should be noted that only two experimental
curves are shown in Fig. 7.7 because the third specimen experienced an
unexpected experimental error during testing. The hoop rupture strain averaged
from the two cylinders was 1.32% and this value was used when generating the
predicted stress-strain curve. It can be seen that the predicted and experimental
curves are in close agreement so Lam and Teng’s model was directly used in the
present theoretical model for the modeling of this series of column tests.
The predicted axial load capacities of all columns are listed in Table 7.6. Again, a
small eccentricity of 1 mm was used when analysing columns tested under
nominally concentric compression (i.e., columns C1-1U, C1-1R, C2-1U, and C2-
1R). It is surprising to note that for all the unconfined columns, the predicted axial
load capacity is considerably larger than the experimental value, particularly for
those columns subjected to nominally concentric loading. By contrast, the
predictions for FRP-confined columns are much more reasonable but the same
trend can still be observed. This might be due to the same reasons as given earlier:
additional eccentricities from geometric/material imperfections, inaccurate
alignment of load, and the possible spalling of the concrete cover in unconfined
columns. In an internal report (Yu et al. 2004), it was suggested that an additional
eccentricity of 15 mm be used for unconfined columns and an additional
eccentricity of 7.5 mm be used for FRP-confined columns when modeling Tao et
al.’s (2004) column tests. When this suggestion is adopted, the predicted values
(bracketed in Table 7.6) become much closer to the excremental values. With the
inclusion of the additional eccentricity in the theoretical model, the average
, ,u theo u testN N ratio for the six FRP-confined columns with a non-zero nominal
216
load eccentr
nother point worth noting is that the theoretical model only predicts a marginal
he theoretical and experimental full-range load-deflection curves of FRP-
curve of column C2-3R is shown in Fig. 7.8c using a dotted line. However, the
icity decreases from 1.11 to 1.01. However, the same ratio for
unconfined columns is still 1.19, indicating significant overestimation.
A
increase in the axial load capacity due to FRP wrapping, which is particularly true
for series C2. This observation indicates that the effectiveness of FRP
confinement decreases as columns become more slender. The same trend can also
be observed in the predictions for Fitzwilliam and Bisby’s (2006) tests. This is
because when columns become more slender, they are more susceptible to
stability failure. When a column is short enough for its load capacity to be
controlled by material failure, failure of the column is predicted when the extreme
compression fiber of the critical section reaches the ultimate axial strain of
confined concrete. By contrast, when the column is slender enough, stability
failure controls and the column reaches its axial load capacity when the extreme
compression fiber has not yet reached its ultimate axial strain.
T
confined columns are also compared (Fig. 7.8). An additional eccentricity of 7.5
mm was added to the nominal load eccentricity when producing the theoretical
curves shown in Fig. 7.8. For specimens C1-1R and C2-1R, two theoretical curves
are shown. The upper one is for an additional eccentricity of 7.5 mm while the
lower one is for an additional eccentricity of 15 mm. These two specimens were
tested under nominally concentric compression, so their behavior is more sensitive
to the additional eccentricity. It can be seen that the use of a 15 mm additional
eccentricity produced closer predictions, but the curves for a 7.5 mm additional
eccentricity are also reasonably close to the experimental curves. It can be seen in
Figs 7.8c and 7.8d that the deformation capacities of columns of series C2 are
significantly overestimated, this is because these columns were very slender and
no FRP rupture was observed even at very large lateral deflection when the tests
stopped. Taking column C2-3R as an example, the ultimate hoop strain recorded
at the compression face of the column was only 0.226%. If this value is used in
Lam and Teng’s stress-strain model, the theoretical deformation capacity of the
column becomes much closer to the experimental value. The new theoretical
217
new theoretical curve almost overlaps with the old theoretical curve so a small
triangle is included to mark the ending point of the new theoretical curve.
7.4 CONCLUSIONS
This chapter has been concerned with the development and verification of a
eoretical model for slender FRP-confined RC columns. The theoretical model
successfully used to model
slender steel columns, RC columns and steel-concrete composite columns.
2) curate in predicting the axial
load capacity of slender FRP-confined RC columns. However, the proposed
3) ith the external FRP strengthening of
RC columns for the enhancement of their axial load capacity. For this purpose,
th
incorporates Version (I) of the modified Lam and Teng model at the section
behavior level and finds the lateral deflection of a column through numerical
integration at the column behavior level. The theoretical model was verified
against a similar model for RC columns through comparison of numerical results
from both models. Besides, the theoretical model was also verified using
experimental results of both RC columns and FRP-confined RC columns, with the
latter being emphasized. The comparisons and discussions presented in this
chapter allow the following conclusions to be drawn:
1) The numerical integration method has long been
However, to the best knowledge of the author, the work presented in this
chapter is the first attempt to extend the numerical integration method to the
analysis of slender FRP-confined RC columns.
The proposed theoretical model is reasonably ac
model performs worse in predicting the deformation capacity of slender FRP-
confined RC columns. One possible reason is that the eccentricity has certain
effect on the ultimate axial strain of FRP-confined concrete, which is the key
to the prediction of deformation capacity of FRP-confined RC columns. More
research is needed to clarify this issue.
The present thesis is only concerned w
the proposed theoretical model is satisfactory as demonstrated by the
comparisons presented in this chapter. However, when RC columns are
218
strengthened with FRP wraps primarily for the enhancement of their
deformation capacity, such as in the case of seismic retrofit of RC columns,
more research is needed to clarify whether the present numerical scheme is
sufficiently accurate for such cases.
219
7.5 REFERENCES
Bisby, L.A. (2008). Private communication.
Chen, W.F. and Atsuta, T. (1976). Theory of Beam-Columns, McGraw-Hill, New York.
Choo, C.C., Harik, I.E. and Gesund, H. (2006). “Strength of rectangular concrete columns reinforced with fiber-reinforced polymer bars”, ACI Structural Journal, 103(3), 452-459.
Cranston, W.B. (1972). Analysis and Design of Reinforced Concrete Columns, Research Report 20, Cement and Concrete Association, UK.
Fitzwilliam, J. and Bisby, L.A. (2006). “Slenderness effects on circular FRP-wrapped reinforced concrete columns”, Proceedings, 3rd International conference on FRP Composites in Civil Engineering, December 13-15, Miami, Florida, USA, 499-502.
Hadi, M.N.S. (2006). “Behaviour of wrapped normal strength concrete columns under eccentric loading”, Composite Structures, 72(4), 503-511.
Kim, J.K. and Yang, J.K. (1995). “Buckling behaviour of slender high-strength concrete columns”, Engineering Structures, 17(1), 39-51.
Newmark, N.M. (1943). “Numerical rocedure for computing deflections, moments, and buckling loads”, ASCE Transactions, 108, 1161-1234.
Pfrang, E.O. and Siess, C.P. (1961). “Analytical study of the behavior of long restrained reinforced concrete columns subjected to eccentric loads”, Structural research series No. 214, University of Illinois, Urbana, Illinos.
Ranger, M. and Bisby, L.A. (2007). “Effects of load eccentricities on circular FRP-confined reinforced concrete columns”, Proceedings, 8th International Symposium on Fiber Reinforced Polymer Reinforcement for Concrete Structures (FRPRCS-8), University of Patras, Patras, Greece, July 16-18, 2007.
Shen, Z.Y. and Lu, L.W. (1983). “Analysis of initially crooked, end restrained steel columns”, Journal of Constructional steel research, 3(1), 10-18.
Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, 2nd International Conference on Advanced Polymer Composites for Structural Applications in Construction, University of Surrey, Guildford, UK, 203-212.
Tikka, T.M. and Mirza, S.A. (2006). “Nonlinear equation for flexural stiffness of slender composite columns in major axis bending”, Journal of Structural Engineering, ASCE, 132(3), 387-399.
220
Yuan, W., and Mirmiran, A. (2001). “Buckling analysis of concrete-filled FRP tubes”, International Journal of Structural Stability and Dynamics, 1(3), 367-383.
Yu, Q., Tao, Z., Gao, X., Yang, Y.F., Han, L.H and Zhuang, J.P. (2004). 大轴压比下 FRP 约束混凝土柱抗震性能研究 , Fuzhou University, China (in Chinese).
221
Table 7.1 Properties of columns in Fig. 7.3
Specimen cuf (MPa)
yf (MPa)
sρ (%)
'h h
1 2e e 2e h l h
Column 1 0.5 15 Column 2 0.5 25 Column 3 0.1 40 Column 4
31 (13.8)
414 (360) 6 0.7 0
0.5 40
222
Table 7.2 Summary of Kim and Yang’s (1995) tests
Specimen b
(mm)h
(mm)
'cof
(MPa) yf
(MPa)sρ
(%)l h 'h h 1 2e e 2e h ,u testN
(kN) ,u theoN
(kN) ,
,
u theo
u test
NN
60L2-1 63.7 1.0360L2-2 18 65.7 65.8 1.00 100L2-1 38.2 0.96100L2-2
80 80 25.5 387 1.9830
0.625 1 0.3
35.0 36.6 1.05 60M2-1 102.8 1.0860M2-2 18 113.5 111.1 0.98 100M2-1 45.2 1.22100M2-2
80 80 63.5 387 1.9830
0.625 1 0.3
47.6 55.1 1.16 60H2-1 122.1 1.1060H2-2 18 123.7 134.7 1.09 100H2-1 54.3 1.17100H2-2
80 80 86.2 387 1.9830
0.625 1 0.3
54.9 63.3 1.15
223
Table 7.3 Summary of Ranger and Bisby’s (2007) tests
Specimen D
(mm)l
(mm) 1
2
ee
2e (mm)
ConcreteCover (mm)
frpE (GPa)
t (mm)
'cof
(MPa)yf
(MPa) sρ
(%),u testN
(kN),u theoN
(kN) ,
,
u theo
u test
NN
U-0 90 0 497 641 1.29C-0 152 600 1 0 25 90 0.381 33.2 710 0.71 873 786 0.90U-5 90 0 459 584 1.27C-5 152 600 1 5 25 90 0.381 33.2 710 0.71 770 725 0.94
U-10 90 0 447 525 1.17C-10 152 600 1 10 25 90 0.381 33.2 710 0.71 664 655 0.99U-20 90 0 351 420 1.20C-20 152 600 1 20 25 90 0.381 33.2 710 0.71 579 518 0.89U-30 90 0 253 322 1.27C-30 152 600 1 30 25 90 0.381 33.2 710 0.71 337 402 1.19U-40 90 0 179 242 1.35C-40 152 600 1 40 25 90 0.381 33.2 710 0.71 264 305 1.16
224
Table 7.4 Summary of Fitzwilliam and Bisby’s (2006) tests
Specimen D
(mm) l
(mm) 1
2
ee
2e (mm)
Concrete Cover (mm)
frpE (GPa)
t (mm)
'cof
(MPa)yf
(MPa)sρ
(%),u testN
(kN)
,u theoN
(kN)
,
,
u theo
u test
NN
300U-A 471300U-B 152 300 1 20 25 90 0 35.5 710 0.71 462 458 0.98
300C-1-0-A 675300C-1-0-B 152 300 1 20 25 90 0.381 35.5 710 0.71 679 531 0.78
300C-2-0-B 152 300 1 20 25 90 0.762 35.5 710 0.71 911 684 0.75600U-A 152 600 20 25 90 0 35.5 710 0.71 428 448 1.05
600C-1-0-A 152 600 1 20 25 90 0.381 35.5 710 0.71 563 505 0.90900U-A 152 900 1 20 25 90 0 35.5 710 0.71 398 432 1.09
900C-1-0-A 152 900 1 20 25 90 0.381 35.5 710 0.71 549 468 0.851200U-A 3891200U-B 152 1200 1 20 25 90 0 35.5 710 0.71 411 411 1.03
1200C-1-0-A 4511200C-1-0-B 152 1200 1 20 25 90 0.381 35.5 710 0.71 481 433 0.93
1200C-2-0-A 152 1200 1 20 25 90 0.762 35.5 710 0.71 539 466 0.86
225
Table 7.5 Cylinder tests in Fitzwilliam and Bisby (2006) and Ranger and Bisby (2007)
Source '
cof (MPa)
frpE (GPa)
t (mm)
'ccf
(MPa) cuε
(%) ,h rupε
(%)
'ccf -predicted
(MPa) cuε -predicted
(%) 41 0.804 1.140
39.7 0.768 1.16534.6
90 0.38141.4 0.793 1.192
46.0 1.24
35.8 90 0.762 59.8 0.165 1.111 63.9 1.7658 0.112 1.175
Fitzwilliam and Bisby
(2006) 36.4 90 0.762 62.5 0.156 1.151 65.6 1.83
Ranger and Bisby (2007)
33.2 90 0.381 44.2 0.860 1.15 44.7 1.25
226
Table 7.6 Summary of Tao et al.’s (2004) tests
Specimen D
(mm)l
(mm) 1
2
ee
2e (mm)
ConcreteCover (mm)
'cof
(MPa)yf
(MPa)sρ
(%)frpE
(GPa) t
(mm),u testN
(kN)
,u theoN (kN)
,
,
u theo
u test
NN
C1-1U 455 942(658) 2.07(1.45)C1-1R 150 1260 1 0 21 48.2 388.7 2.56 255 0.34 765 1018(871) 1.33(1.14)C1-2U 149 273(198) 1.83(1.33)C1-2R 150 1260 1 50 21 48.2 388.7 2.56 255 0.34 248 288(243) 1.16(0.98)C1-3U 88 119(101) 1.35(1.15)C1-3R 150 1260 1 100 21 48.2 388.7 2.56 255 0.34 124 131(119) 1.06(0.96)C1-4U 54.5 75.9(68.6) 1.39(1.26)C1-4R 150 1260 1 150 21 48.2 388.7 2.56 255 0.34 77 79.5(75.0) 1.03(0.97)C2-1U 276 668(392) 2.42(1.42)C2-1R 150 3060 1 0 21 48.2 388.7 2.56 255 0.34 386 700(543) 1.81(1.41)C2-2U 108 146(114) 1.35(1.06)C2-2R 150 3060 1 50 21 48.2 388.7 2.56 255 0.34 126 149(131) 1.18(1.04)C2-3U 62.5 76.1(67.4) 1.22(1.08)C2-3R 150 3060 1 100 21 48.2 388.7 2.56 255 0.34 71.5 77.6(72.9) 1.08(1.02)C2-4U 39 54.0(49.7) 1.83(1.27)C2-4R 150 3060 1 150 21 48.2 388.7 2.56 255 0.34 47.5 55.1(52.8) 1.16(1.11)
227
Dl
N
2e
2
l
m-2m-1
N e1
m
d
3
1
Fig. 7.1 Schematic of the theoretical model
228
' hh
b
Fig. 7.2 Cross section of the columns used in Fig. 7.3
229
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.1
0.2
0.3
0.4
0.5
Normalized Lateral Displacement fmid / h
Nor
mal
ized
Axi
al L
oad
N/N
uo
Cranston (1972)Present study
(a) Columns 1 and 2
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Normalized Lateral Displacement fmid / h
Nor
mal
ized
Axi
al L
oad
N/N
uo
Cranston (1972)Present study
(b) Columns 3 and 4
Fig. 7.3 Comparisons with Cranston’s theoretical model
230
0 10 20 30 40 50
0
10
20
30
40
50
60
70
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
60L2-2 (Test)60L2-2 (Predicted)100L2-1 (Test)100L2-1 (Predicted)
Fig. 7.4 Comparisons with Kim and Yang’s tests on RC columns
231
-1 0 1 2 3 4 5 60
100
200
300
400
500
600
700
800
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C-5 (Test)C-5 (Predicted)C-5 (Predicted, Lam and Teng)
(a) Column C-5
0 2 4 6 8 10 120
100
200
300
400
500
600
700
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C-10 (Test)C-10 (Predicted)C-10 (Predicted, Lam and Teng)C-20 (Test)C-20 (Predicted)C-20 (Predicted, Lam and Teng)
(b) Columns C-10 and C-20
232
0 5 10 15 200
50
100
150
200
250
300
350
400
450
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C-30 (Test)C-30 (Predicted)C-30 (Predicted, Lam and Teng)C-40 (Test)C-40 (Predicted)C-40 (Predicted, Lam and Teng)
(c) Columns C-30 and C-40
Fig. 7.5 Comparison with Ranger and Bisby’s tests
233
0 2 4 6 8 100
100
200
300
400
500
600
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
600C-1-0-A (Test)600C-1-0-A (Predicted)600C-1-0-A (Predicted, Lam and Teng)900C-1-0-A (Test)900C-1-0-A (Predicted)900C-1-0-A (Predicted, Lam and Teng)
(a) Columns 600C-1-0-A and 900C-1-0-A
0 10 20 30 40 500
100
200
300
400
500
600
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
1200C-1-0-A (Test)1200C-1-0-B (Test)1200C-1-0 (Predicted)1200C-1-0 (Predicted, Lam and Teng)1200C-2-0-A (Test)1200C-2-0-A (Predicted)1200C-2-0-A (Predicted, Lam and Teng)
(b) Columns 1200C-1-0-A, 1200C-1-0-B and 1200C-2-0-A
Fig. 7.6 Comparison with Fitzwilliam and Bisby’s tests
234
0 0.005 0.01 0.015 0.02 0.0250
10
20
30
40
50
60
70
80
90
100
Axial Strain εc
Axi
al S
tress
σc
(MP
a)
D = 150mmf′co = 48.2MPa
Efrp = 255GPa
t = 0.34mmεh,rup = 0.0132
Test (2 Specimens)Lam and Teng′s Model
Fig. 7.7 Comparison with Tao et al.’s cylinder tests
235
0 10 20 30 40 50 600
100
200
300
400
500
600
700
800
900
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C1-1R (Test)C1-1R (Predicted)C1-2R (Test)C1-2R (Predicted)
e2 = 15 mm
e2 = 7.5 mm
(a) Columns C1-1R and C1-2R
0 10 20 30 40 50 600
20
40
60
80
100
120
140
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C1-3R (Test)C1-3R (Predicted)C1-4R (Test)C1-4R (Predicted)
(b) Columns C1-3R and C1-4R
236
0 50 100 150 2000
100
200
300
400
500
600
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C2-1R (Test)C2-1R (Predicted)C2-2R (Test)C2-2R (Predicted)
e2 = 7.5 mm
e2 = 15 mm
(c) Columns C2-1R and C2-2R
0 50 100 150 200 2500
10
20
30
40
50
60
70
80
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C2-3R (Test)C2-3R (Predicted, old)C2-3R (Predicted, new)C2-4R (Test)C2-4R (Predicted)
(d) Columns C2-3R and C2-4R
Fig. 7.8 Comparisons with Tao et al.’s tests on FRP-confined RC columns
237
CHAPTER 8
SLENDERNESS LIMIT FOR
SHORT FRP-CONFINED RC COLUMNS
8.1 INTRODUCTION
Design equations for short FRP-confined RC columns have been presented in
Chapter 5. However, for the application of these equations, a clear definition of
short columns is needed. In most design codes for RC structures (e.g. ENV-1992-
1-1 1992; BS-8110 1997; ACI-318 2005), a simple check is generally required to
determine whether an RC column is short or slender before applying
corresponding design equations. If the column is classified as a short column, the
design procedure is more straightforward compared to that for slender columns as
the slenderness effect can be ingored. Therefore, a proper definition of the
slenderness value that separates short columns from slender columns is important
in design. In this chapter, this particular slenderness value is referred to as the
slenderness limit for short columns, or simply the slenderness limit for brevity. It
should be noted that although a number of design guidelines for FRP-strengthened
RC structures (fib 2001; ISIS 2001; ACI-440.2R 2002, 2008; JSCE 2002; CNR-
DT200 2004; Concrete Society 2004) have been developed, such a slenderness
limit is only available in ISIS (2001), which is only intended for columns with no
significant bending (i.e. concentric compression or slightly eccentric compression).
This is mainly due to the fact that only a very limited number of studies have
investigated the behavior of slender concrete columns confined with FRP. These
studies include Mirmiran et al. (2001a) and Yuan and Mirmiran (2001) on slender
concrete-filled FRP tubes and Tao et al. (2004), Fitzwilliam and Bisby (2006) and
Ranger and Bisby (2007) on slender FRP-confined RC columns. It has been
238
realized in these studies that when FRP confinement exists, concrete columns tend
to experience more severe slenderness effects. Consequently, existing slenderness
limit expressions for RC columns [Excellent summaries can be found in
Hellesland (2005) and Mari and Hellesland (2005)] are not directly applicable to
concrete columns confined with FRP. Mirmiran et al. (2001a) tested
concentrically-loaded slender concrete-filled FRP tubes and developed a
theoretical model for such columns. Both their experimental and theoretical
results support the above conclusion. Yuan and Mirmiran (2001) further
developed a theoretical model for slender concrete-filled FRP tubes subjected to
eccentric compression (this model has been briefly described in Chapter 7). Their
analysis also led to the same result More recently, Tao et al. (2004), Fitzwilliam
and Bisby (2006) and Ranger and Bisby (2007) carried out tests on FRP-confined
RC columns. These studies found that FRP-confined RC columns experienced a
larger loss in the axial load capacity compared to corresponding RC columns. The
findings of these studies indicate that with FRP confinement, an RC column
which is originally classified as a short column may need to be re-classified as a
slender column, and a proper slenderness limit expression thus needs to be
developed for design use.
Yuan and Mirmiran (2001) recommended that for columns subjected to equal end
eccentricities, the current slenderness limit of 22 for RC columns as specified in
ACI-318 (2005) be reduced to 11 for the slenderness limit for concrete-filled FRP
tubes. Although the slenderness limit value proposed by Yuan and Mirmiran
(2001) is simple and conservative, it should be noted that it only corresponds to a
effective length-to-diameter ratio of 2.75. As a result, most columns would be
classified as slender columns according to their expression. In fact, it is shown in
later sections of this chapter that the magnitudes of the end eccentricities have a
significant effect on the slenderness limit; a more accurate slenderness limit
expression can be developed so that a much wider range of columns can be
classified as short columns.
This chapter is concerned with the development of such a slenderness limit
expression for FRP-confined circular RC columns (RC columns are treated as a
special case where no FRP confinement is provided) in braced frames. It should
239
be noted that existing design codes for RC structures generally specify different
slenderness limit expressions for columns in braced frames and unbraced frames
respectively; the latter are assigned with a smaller slenderness limit (Hellesland
2005). This chapter is only concerned with columns in braced frames. To this end,
a comprehensive parametric study was performed using the theoretical model
developed in Chapter 7 to examine the effects of various parameters on the
slenderness limit. Based on the results of this parametric study, a simple
slenderness limit expression for design use is proposed. This expression considers
all the governing parameters and can reduce to a form that is identical or similar to
the slenderness limit expressions for conventional RC columns in current design
codes. The present study has been partially motivated by the development of the
Chinese Code for the Application of FRP Composites in Construction, which is
formulated within the general framework of the current Chinese Code for Design
of Concrete Structures (GB-50010 2002). Therefore, some of the considerations
herein follow the specifications given in GB-50010 (2002) and they are
highlighted where appropriate.
8.2 DEFINITION OF SLENDERNESS LIMIT
The concept of the slenderness limit can be illustrated by examining the behavior
of a column as its slenderness varies. For simplicity, consider a hinged RC
column under an increasing axial load N bent in symmetrical single curvature
( ). In such a case, the critical section is located at the mid-height of the
column. Fig. 8.1 shows the axial load-bending moment loading paths of the
critical section of the column for three different values of column slenderness; the
load eccentricity is fixed. In the absence of the slenderness effect (i.e. the height
of the columns is very small), the loading path follows the straight line all the
way up until it intersects the cross-section interaction curve at point
1e e= 2
OA
A , which
marks the occurrence of material failure. In such a case, the critical section is only
subjected to the first-order moment 2N e⋅ and the axial load capacity of the
column is defined by point A . Once the column has a certain height, the lateral
displacement of the mid-height section midf induces a second-order moment
and thus causes the loading path to deviate from OA . If the column is not midN f⋅
240
too slender, the loading path is still of an ascending shape when it intersects with
the cross-section interaction curve at point B , which indicates that the column is
still controlled by material failure and the axial load capacity is defined by point
B . By contrast, if the column is very slender, the loading path may exhibit a
descending branch (path ODE ), which indicates that the column is no longer
controlled by material failure, and instead, stability failure occurs prior to material
failure. The axial load capacity of such a column is defined by the peak point D
of the loading path. It is obvious that the larger the column slenderness, the larger
the loss in the axial load capacity due to the second-order moment.
The slenderness limit for short RC columns is commonly defined to ensure that
the slenderness effect leads to only a small amplification of the first-order moment
(i.e. the second-order moment only constitutes a small fraction of the first-order
moment) at the critical section or a small reduction (commonly 5% or 10%) of the
axial load capacity. In this chapter, the criterion of a 5% reduction in the axial
load capacity is adopted to define the slenderness limit for short columns. The
slenderness ratio is defined as
efflr
(8.1) λ =
where is the effective length of a column and r is the radius of gyration
and for circular columns, where is the diameter of the cross section. It
should be noted that the numerical study presented in this chapter is limited to
hinged columns whose effective length is equal to their physical length
effl
/ 4D= D
For restrained columns, it has been concluded in Chapter 6 that the effective
length can be determined following the approximate approach in existing codes
(e.g. ACI-318 2005; ENV-1992-1-1 1992) where simple charts developed from
Eq. 6.42 are provided to relate the effective length to the column-to-beam stiffness
ratio. No further research has been undertaken in the present study on the effective
length of FRP-confined RC columns. It is suggested that the methods given in
241
existing design codes for RC structures be used to estimate the effective length of
FRP-confined RC columns.
.
8.3 PARAMETRIC STUDY
8.3.1 Parameters Considered
A comprehensive parametric study was carried out in order to investigate the
effects of various factors on the slenderness limit for short FRP–confined RC
columns. The parametric study was carried out on the same reference column (Fig.
8.2) as used in Chapter 5. More specifically, the reference RC column had a
diameter of 600 mm and was longitudinally reinforced with 12 evenly distributed
steel bars. The strengths of concrete and steel reinforcement were taken to be
common values as specified in GB-50010 (2002). The concrete was assumed to be
grade C30, representing a characteristic cube strength of MPa and a
corresponding cylinder strength MPa. The steel was assumed
to be grade II with a characteristic yield strength
30cuf =
' '0.67 20.1co cuf f= =
335yf = MPa and an elastic
modulus of 200 GPa. It is evident that all the parameters that influence the
structural behavior of a slender column have some effect on the slenderness limit.
Based on previous studies on slender RC columns (Pfrang and Siess 1961;
MacGregor et al. 1970), the main parameters are identified to be
: the eccentricity ratio
sE =
1 2e e , the normalized eccentricity 2e D , the steel
reinforcement ratio sρ , and the depth ratio d D . When FRP confinement is taken
into account, two additional parameters should be considered, namely, the
stiffness and the strain capacity of the FRP jacket. To keep the slenderness limit
equation as simple as possible, the strength enhancement ratio ' 'cc cof f and the
strain ratio ερ were chosen to reflect the effect of FRP confinement based on a
careful consideration. The values of all the parameters considered in the
parametric study are summarized in Table 8.1.
242
The combinations of these parameters led to about 3000 cases. It should be noted
that most parameters studied herein are the same as those used in the parametric
study presented in Chapter 5. The end eccentricity ratio is the only new parameter.
Theoretically speaking, the end eccentricity ratio ranges from -1 to 1. The use of
instead of is based on computational considerations. Under the
unobtainable idealization of
0.99− 1−
1 2 1e e = − , the lateral deflection of the column is
exactly anti-symmetrical. However, with a slightest disturbance introduced as
always is the case in reality, the column tends to behave in a significantly different
way due to the phenomenon which is commonly known as “unwrap” (Pfrang and
Siess 1961). For the strength enhancement ratio, ' ' 1cc cof f = represents the case of
unconfined concrete, but not cases of insufficiently confined concrete.
The results of the parametric study are presented in two parts. The first part deals
with RC columns while the second part considers the effect of FRP confinement.
In all the figures presenting the results of the parametric study, the vertical axis is
always the slenderness limit while the horizontal axis is one of the six parameters
considered. A family of curves is shown in each figure, showing the variation of
the slenderness limit with another parameter. The values of the other parameters
used to generate these numerical results are also given in each figure.
8.3.2 Results for RC Columns
First, the effect of the end eccentricity ratio is investigated. The numerical results
are shown in Fig. 8.3. Each curve shows the variation of the slenderness limit with
the end eccentricity ratio for a particular normalized eccentricity. It can be clearly
seen that for a given normalized eccentricity, the slenderness limit decreases
almost linearly as the end eccentricity ratio increases. This can be easily
understood, since a column bent in single curvature with always
experiences the largest slenderness effect. It can also be noted that as the
normalized eccentricity increases, the slenderness limit generally increases rapidly,
which is more clearly reflected in Fig. 8.4 and is further discussed later. It should
be noted that these numerical results are for
1e e= 2
0.8d D = and 3%sρ = since the
243
results for other values of these two parameters are similar and are not presented
herein for brevity. The effects of these two parameters are further discussed later.
As stated above, the slenderness limit generally increases with the normalized
eccentricity. The general trend of the curves in Fig. 8.4 can be explained as
follows. It is obvious that the larger the normalized eccentricity, the lower the
axial load level. When approaches infinity, the column approaches the state of
pure bending and the slenderness limit approaches infinity. By contrast, when
, since no lateral deflection would occur at this ideal condition, Fig. 8.4
shows that the end eccentricity ratio hardly has any effect when the end
eccentricity is very small. In Fig. 8.4, the different curves all converge to the same
point obtained using a very small normalized eccentricity of
2e
2 0e =
2 0.001e D = . It
should be noted that this small value of normalized eccentricity was only used in
Fig. 8.4 for illustrative purposes but was not used in the rest of the parametric
study.
Fig. 8.5 shows the effect of the depth ratio. Fig. 8.5a is for a relatively high axial
load level ( 2 0.05e D = ) while Fig. 8.5b is for a relatively low axial load level
( 2 0.8e D = ). It can be seen that for both cases the slenderness limit slightly
increases with the depth ratio. This is because an increase in the depth ratio
increases the lever arm of the steel reinforcement and thus enlarges the load
contribution from the steel reinforcement, which helps the column to resist the
slenderness effect. However, this parameter has a much smaller effect than the
first two parameters discussed above.
Lastly, the effect of the steel reinforcement ratio is investigated. The numerical
results in Fig. 8.6 also cover both a high axial load level and a low axial load level.
Increasing the amount of steel reinforcement increases the slenderness limit
although not very effective. The reason is similar to that given in the preceding
paragraph.
This set of numerical results indicates that the end eccentricity ratio and the
normalized eccentricity are the two primary parameters for RC columns and they
244
should both be taken into account when developing a design expression for the
slenderness limit. The depth ratio and steel reinforcement ratio have negligible
effects on the slenderness limit and they may thus be ignored in the design
expression.
8.3.3 Results for FRP-confined RC Columns
All the results for FRP-confined RC columns presented herein are for
0.7d D = and 1%sρ = since these two parameters are expected to have only
minor effects on the slenderness limit and the numerical results obtained using
these values are expected to be conservative for columns with a higher depth ratio
and a larger steel reinforcement ratio.
The effect of the strength enhancement ratio is investigated first. As stated earlier, ' ' 1cc cof f = represents the case of no FRP confinement. Fig. 8.7 is for 3.75ερ =
which represents the strain capacity of common CFRPs. Fig. 8.7 includes a set of
sub-figures, corresponding to different axial load levels. The effect of the strength
enhancement ratio is substantial. It can be seen that among all the curves shown in
Fig. 8.7, the uppermost curve passing through small circles in Fig. 8.7a has the
sharpest slope. This indicates that the smaller the end eccentricity ratio and the
higher the axial load level, the more significant the effect of the strength
enhancement ratio.
As discussed in Chapter 4, for a given strength enhancement ratio, an FRP jacket
with a larger strain capacity always yields a larger ultimate axial strain for FRP-
confined concrete. As a result, the strength enhancement ratio alone is not
sufficient to reflect the effect of FRP confinement. Fig. 8.8 shows the effect of the
strain capacity of FRP materials. Figs 8.8a and 8.8b show the numerical results for
a relatively high axial load level ( 2 0.05e D = ) and two different strength
enhancement ratios respectively. It is clear that larger strain ratios result in smaller
slenderness limits. This is because for a given strength enhancement ratio, the
second portion of stress-strain curve of FRP-confined concrete defined by a larger
strain ratio is always flatter, which leads to a more severe slenderness effect. It
245
can also be easily understood that the strain ratio has a more significant effect for
a larger amount of confinement. The above observations also hold for Figs 8.8c
and 8.8d, where a relatively low axial load level ( 2 0.8e D = ) is considered.
However, as the axial load level decreases, the strain ratio has a smaller effect.
The above discussions indicate that both the strength enhancement ratio and the
strain ratio have a significant effect on the slenderness limit. Although FRP
confinement enhances the load capacity of an RC column, it also amplifies the
slenderness effect. This is because FRP confinement can substantially increase the
axial lad capacity of a section without significantly enhancing its flexural rigidity.
The effect of the strain ratio varies mainly with the axial load level. At low axial
load levels, its effect may be ignored.
8.4 SLENDERNESS LIMIT EXPRESSIONS FOR DESIGN USE
Based on the results and discussions of the parametric study, the six parameters
examined can be classified into three categories in terms of their significance. The
primary parameters include the strength enhancement ratio, the end eccentricity
ratio, and the normalized eccentricity; these three parameters must be considered
in a slenderness limit expression for use in design. The minor parameters include
the depth ratio and the steel reinforcement ratio; these two parameters may be
ignored in the design expression. The strain ratio has a moderate effect and lies
between the above two categories. In this section, a design expression is first
proposed for RC columns. This expression is then extended to FRP-confined RC
columns by introducing additional terms that reflect the effect of FRP
confinement.
8.4.1 Slenderness Limit for RC Columns
Based on a careful interpretation of the numerical results for RC columns, the
following simple equation is proposed for the slenderness limit for short RC
columns
246
2 1
2
60 (1 ) 20crite eD e
λ = − + (8.2)
Eq. 8.2 accounts for the two primary parameters for RC columns and can reduce
to a form that is similar to the design equations given in existing design codes
where fewer parameters are considered. GB-50010 (2002) specifies the following
expression for the slenderness limit for short RC columns
20critλ = (8.3)
This expression is based on the most conservative condition of 1 2 1e e = and
ignores the effect of load eccentricity. It is easy to see that with these conditions,
Eq. 8.2 reduces to Eq. 8.3. In addition, the current slenderness limit expression in
ACI-318 (2005) is
1
2
34 12critMM
λ = − (8.4)
where 1M and 2M are the first-order moments at the two ends respectively. If Eq.
8.4 is written in terms of the end eccentricity ratio, it becomes
1
2
34 12critee
λ = − (8.5)
Eq. 8.5 considers the end eccentricity ratio but is developed with a fixed load
eccentricity of 2 0.2e D = (Mirmiran et al. 2001b). When 2 0.2e D = , Eq. 8.2
reduces to 1
2
32 12critee
λ = − , which is similar to but slightly more conservative
than Eq. 8.5. The author is aware that the ACI expression was developed using the
moment magnifier method and it was initially based on the same criterion adopted
in this chapter (5% axial load reduction), as documented in MacGregor et al.
(1970). However, in a revisited paper by MacGregor et al. (1993), a new criterion
was adopted (5% first-order moment amplification) but this equation remained
unchanged. Fig. 8.9 shows the performance of Eq. 8.2 and Eq. 8.4. The
247
predictions of both expressions for relatively low values of slenderness limit are
similar. However, the ACI expression is very conservative at high values of
slenderness limit, which occur at low axial load levels, particularly when the
column is bent in double curvature. Both expressions produce a few slightly un-
conservative results at low values of slenderness limit.
8.4.2 Slenderness Limit for FRP-confined RC Columns
Fig. 8.7 shows that the slenderness limit for an FRP-confined RC column can be
roughly approximated by dividing the slenderness limit for the corresponding RC
column without FRP confinement by the strength enhancement ratio (shown as
dashed lines in Fig. 8.7). Thus, the following equation is proposed for the
slenderness limit for short FRP-confined RC columns
2 1
2'
'
60 (1 ) 20
critcc
co
e eD e
ff
λ− +
= (8.6)
This expression has a clear physical meaning: the numerator defines the
slenderness limit for short RC columns without FRP confinement, while the
denominator accounts for the effect of FRP confinement. It should be noted that
the effect of the strain ratio is ignored in this equation. Numerical results for the
slenderness limit corresponding to the 5% axial load reduction criterion are shown
in Fig. 8.10a. These results were generated using the most critical combination of
the depth ratio and the steel reinforcement ratio ( 0.7d D = and 1%sρ = ) since
RC columns with a higher depth ratio and a larger steel reinforcement ratio are
less affected by the slenderness effect. It can be seen that Eq. 8.6 is un-
conservative for a few cases at very low slenderness limit values. Even when the
10% axial load reduction criterion is adopted, Eq. 8.6 still yields un-conservative
results at very low slenderness limit values (Fig. 8.10b). A careful study of the
numerical results showed that all the un-conservative predictions are for columns
bent in symmetrical single curvature ( 1 2 1e e = ). However, in braced frames most
columns are bent in double curvature, which means that most columns have a
248
negative end eccentricity ratio. If the results for 1 2 1e e = are removed from Fig.
8.10a, very few un-conservative predictions rema even for these cases the
predictions are only slightly un-conservative (Fig. 8.10c).
in and
he parametric study has shown that the strain ratio has some effect on the T
slenderness limit in some cases. To account for this effect, the following equation
is proposed:
2 1
2'
,'
60 (1 ) 20
(1 0.06 )crit
h rupcc
co co
e eD e
ff
λεε
− +=
+ (8.7)
umerical results for the slenderness limit based on the 5% axial load reduction N
criterion are shown in Fig. 8.11a. These results are also for 1%sρ = and
0.7d D = . It can be seen from Fig. 8.11a that Eq. 8.7 is conservative for all cases
en the slenderness limit is very small. However, if a 10% loss of axial
load capacity is acceptable, Eq. 8.7 provides a lower bound prediction for all cases,
as shown in Fig. 8.11b. A 10% reduction in the axial load capacity has been
adopted as the criterion for permitted second effects in the existing literature (e.g.
CEB-FIP 1993).
except wh
summary, Eq. 8.6 possesses a simpler form, but may be un-conservative for
.5 CONCLUSIONS
his chapter has been concerned with the development of a slenderness limit
In
low values of slenderness limit. Although Eq. 8.7 takes a slightly more
complicated form, it provides a lower bound prediction for all the cases studied in
this chapter. Eq 8.7 is recommended for design use.
8
T
expression for FRP-confined RC columns for use in design. A comprehensive
parametric study was carried out to investigate the effects of various parameters
on the slenderness limit using the theoretical model developed in Chapter 7.
249
Based on the results of the parametric study, expressions for the slenderness limit
of FRP-confined RC columns in braced frames were proposed. The results and
discussions presented in this chapter allow the following conclusions to be drawn:
1) Although FRP confinement can significantly increase the axial load capacity
) The six parameters examined in the parametric study have different effects on
) The proposed slenderness limit expression has a clear physical meaning: the
of RC columns, it also introduces a greater slenderness effect. This is because
FRP confinement can substantially increase the axial load capacity of an RC
section but affects little the flexural rigidity of the section.
2
the slenderness limit for FRP-confined RC columns. The strength
enhancement ratio, the end eccentricity ratio and the normalized eccentricity
were identified as the primary parameters while the depth ratio and steel
reinforcement ratio were identified as the minor parameters. The strain ratio
was shown to have an effect between the above two categories.
3
numerator defines the slenderness limit for short RC columns without FRP
confinement, while the denominator accounts for the effect of FRP
confinement. The numerator of the proposed expression is slightly more
complex than the slenderness limit expressions for RC columns given in some
of the current design codes for RC structures. This is because an RC column
originally classified as a short column may need to be considered as a slender
column when it is confined with FRP. Therefore, the proposed expression
aims at greater accuracy at the sacrifice of a certain degree of simplicity so
that a much wider range of FRP-confined RC columns can be classified as
short columns to avoid the extra complications involved in the design of
slender columns. It is worth pointing out that existing slenderness limit
expressions for RC columns of different forms can be readily upgraded for
use in the design of FRP-confined RC columns by incorporating the
denominator of the proposed expression.
250
8.6 REFERENCES
ACI-440.2R (2002). Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, ACI Committee 440, American Concrete Institute.
ACI-318 (2005). Building Code Requirements for Structural Concrete and Commentary, ACI Committee 318, American Concrete Institute..
BS 8110 (1997). Structural Use of Concrete, Part 1. Code of Practice for Design and Construction, British Standards Institution, London, UK.
CEB-FIP (1993). Model Code 1990, CEB-Bulletin No. 213/214, Comité Euro-International du Beton.
Concrete Society (2004). Design Guidance for Strengthening Concrete Structures with Fibre Composite Materials, Second Edition, Concrete Society Technical Report No. 55, Crowthorne, Berkshire, UK.
ENV 1992-1-1 (1992). Eurocode 2: Design of Concrete Structures – Part 1: General Rules and Rules for Buildings, European Committee for Standardization, Brussels.
fib (2001). Externally Bonded FRP Reinforcement for RC Structures, The International Federation for Structural Concrete, Lausanne, Switzerland.
Fitzwilliam, J. and Bisby, L.A. (2006). “Slenderness effects on circular FRP-wrapped reinforced concrete columns”, Proceedings, 3rd International conference on FRP Composites in Civil Engineering, December 13-15, Miami, Florida, USA, 499-502.
GB-50010 (2002). Code for Design of Concrete Structures, China Architecture and Building Press, China.
Hellesland, J. (2005). “Nonslender column limits for braced and unbraced reinforced concrete members”, ACI Structural Journal, 102(1), 12-21.
Mari, A.R. and Hellesland, J. (2005). “Lower slenderness limits for rectangular reinforced concrete columns”, Journal of Structural Engineering, ASCE, 131(1), 85-95.
MacGregor, J.G., Breen, J.E. and Pfrang E.O. (1970). “Design of slender concrete columns”, ACI Journal, 67(1), 6-28.
MacGregor, J.G. (1993). “Design of slender concrete columns-revisited”, ACI Structural Journal, 90(3), 302-309.
Mirmiran, A., Shahawy, M. and Beitleman, T. (2001a). “Slenderness limit for hybrid FRP-concrete columns”, Journal of Composites for Construction, 5(1), 26-34.
251
Mirmiran, A., Yuan, W. and Chen, X. (2001b). “Design for slenderness in concrete columns internally reinforced with fiber-reinforced polymer bars”, ACI Structural Journal, 98(1), 116-125.
Pfrang, E.O. and Siess, C.P. (1961). “Analytical study of the behavior of long restrained reinforced concrete columns subjected to eccentric loads”, Structural research series No. 214, University of Illinois, Urbana, Illinos.
Ranger, M. and Bisby, L.A. (2007). “Effects of load eccentricities on circular FRP-confined reinforced concrete columns”, Proceedings, 8th International Symposium on Fiber Reinforced Polymer Reinforcement for Concrete Structures (FRPRCS-8), University of Patras, Patras, Greece, July 16-18, 2007.
Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, 2nd International Conference on Advanced Polymer Composites for Structural Applications in Construction, University of Surrey, Guildford, UK, 203-212.
Yuan, W., and Mirmiran, A. (2001). “Buckling analysis of concrete-filled FRP tubes”, International Journal of Structural Stability and Dynamics, 1(3), 367-383.
252
Table 8.1 Values of parameters used in the parametric study
Parameter Values 1 2e e 1, 0.5, 0, -0.5, -0.99 e D 0.05, 0.1 ,0.2, 0.4, 0.8
sρ 1%, 3%, 5% d D 0.7, 0.8, 0.9 ' '
cc cof f 1, 1.25, 1.5, 1.75, 2
,h rup coε ε 1, 3.75, 7.5
253
Moment M
Axi
al L
oad
N
O
AB
CD
E
e2 = Const
e2
1
N⋅e2
N⋅fmid
Peak Point
Fig. 8.1 Behavior of slender columns
Dl
N
2e
N e1
d
Fig. 8.2 Definition of the reference column
254
-1 -0.5 0 0.5 120
40
60
80
100
120
140
160
180d/D=0.8
ρs=3%
End Eccentricity Ratio e1/e2
Sle
nder
ness
Lim
it λ
crit
e2/D = 0.05e2/D = 0.1e2/D = 0.2e2/D = 0.4e2/D = 0.8
Fig. 8.3 Effect of end eccentricity ratio
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.820
40
60
80
100
120
140
160
180d/D=0.8
ρs=3%
Normalized Eccentricity e2/D
Sle
nder
ness
Lim
it λ
crit
e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
Fig. 8.4 Effect of eccentricity
255
0.7 0.75 0.8 0.85 0.90
20
40
60
80
e2/D=0.05 ρs=1%
Depth Ratio d/D
Sle
nder
ness
Lim
it λ
crit
e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(a) 2 0.05e D =
0.7 0.75 0.8 0.85 0.90
20
40
60
80
100
e2/D=0.8 ρs=1%
Depth Ratio d/D
Sle
nder
ness
Lim
it λ
crit
e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(b) 2 0.8e D =
Fig.8.5 Effect of depth ratio
256
0.01 0.02 0.03 0.04 0.050
20
40
60
80
100
e2/D=0.05 d/D=0.9
Steel Reinforcement Ratio ρs
Sle
nder
ness
Lim
it λ
crit
e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(a) 2 0.05e D =
0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
20
40
60
80
100
120
140
160
180
e2/D=0.8 d/D=0.9
Steel Reinforcement Ratio ρs
Sle
nder
ness
Lim
it λ
crit
e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(b) 2 0.8e D =
Fig. 8.6 Effect of steel reinforcement ratio
257
1 1.25 1.5 1.75 20
10
20
30
40
50
60
70
80
Strength Enhancement Ratio f′cc/f′co
Sle
nder
ness
Lim
it λ
crit
εh,rup/εco=3.75 e2/D=0.05
d/D=0.7 ρs=0.01
e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(a) 2 0.05e D =
1 1.25 1.5 1.75 20
10
20
30
40
50
60
70
80
90
Strength Enhancement Ratio f′cc/f′co
Sle
nder
ness
Lim
it λ
crit
εh,rup/εco=3.75 e2/D=0.1
d/D=0.7 ρs=1%
e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(b) 2 0.1e D =
258
1 1.25 1.5 1.75 20
20
40
60
80
100
Strength Enhancement Ratio f′cc/f′co
Sle
nder
ness
Lim
it λ
crit
εh,rup/εco=3.75 e2/D=0.2
d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(c) 2 0.2e D =
1 1.25 1.5 1.75 20
20
40
60
80
100
120
Strength Enhancement Ratio f′cc/f′co
Sle
nder
ness
Lim
it λ
crit
εh,rup/εco=3.75 e2/D=0.4
d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(d) 2 0.4e D =
259
1 1.25 1.5 1.75 20
20
40
60
80
100
120
140
160
Strength Enhancement Ratio f′cc/f′co
Sle
nder
ness
Lim
it λ
crit
εh,rup/εco=3.75 e2/D=0.8
d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(e) 2 0.8e D =
Fig. 8.7 Effect of strength enhancement ratio
260
0 2 4 6 80
10
20
30
40
50
60
Strain Ratio εh,rup/εco
Sle
nder
ness
Lim
it λ
crit
f′cc/f′co=1.25 e2/D=0.05d/D=0.7 ρs=0.01
e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(a) ' '
2 0.05, 1.25cc coe D f f= =
0 2 4 6 80
10
20
30
40
Strain Ratio εh,rup/εco
Sle
nder
ness
Lim
it λ
crit
f′cc/f′co=2 e2/D=0.05
d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(b) ' '
2 0.05, 2cc coe D f f= =
261
0 2 4 6 80
20
40
60
80
100
120
140
160
180
Strain Ratio εh,rup/εco
Sle
nder
ness
Lim
it λ
crit
f′cc/f′co=1.25 e2/D=0.8
d/D=0.7 ρs=0.01
e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(c) ' '
2 0.8, 1.25cc coe D f f= =
0 2 4 6 80
20
40
60
80
100
120
140
Strain Ratio εh,rup/εco
Sle
nder
ness
Lim
it λ
crit
f′cc/f′co=2 e2/D=0.8
d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1
(d) ' '
2 0.8, 2cc coe D f f= =
Fig. 8.8 Effect of strain ratio
262
0 20 40 60 80 100 120 140 160 1800
20
40
60
80
100
120
140
160
180
Slenderness Limit λcrit - Accurate Analysis
Sle
nder
ness
Lim
it λcr
it - D
esig
n E
quat
ion
5% Axial Load Reduction
(a) Proposed expression
0 20 40 60 80 100 120 140 160 1800
20
40
60
80
100
120
140
160
180
Slenderness Limit λcrit - Accurate Analysis
Sle
nder
ness
Lim
it λcr
it - D
esig
n E
quat
ion
5% Axial Load Reduction
(b) ACI expression
Fig. 8.9 Performance of the proposed expression and the ACI expression for
RC columns
263
0 20 40 60 80 100 120 140 160 1800
20
40
60
80
100
120
140
160
180
Slenderness Limit λcrit - Accurate Analysis
Sle
nder
ness
Lim
it λcr
it - D
esig
n E
quat
ion
5% Axial Load Reduction
(a) Slenderness limits based on a 5% axial load reduction
0 20 40 60 80 100 120 140 160 1800
20
40
60
80
100
120
140
160
180
Slenderness Limit λcrit - Accurate Analysis
Sle
nder
ness
Lim
it λcr
it - D
esig
n E
quat
ion
10% Axial Load Reduction
(b) Slenderness limits based on a 10% axial load reduction
264
0 20 40 60 80 100 120 140 160 1800
20
40
60
80
100
120
140
160
180
Slenderness Limit λcrit - Accurate Analysis
Sle
nder
ness
Lim
it λcr
it - D
esig
n E
quat
ion
5% Axial Load Reduction
(c) Slenderness limits based on a 5% axial load reduction
(results for 1e e2= excluded)
Fig. 8.10 Performance of Eq. 8.6 for FRP-confined RC columns
265
0 20 40 60 80 100 120 140 160 1800
20
40
60
80
100
120
140
160
180
Slenderness Limit λcrit - Accurate Analysis
Sle
nder
ness
Lim
it λcr
it - D
esig
n E
quat
ion
5% Axial Load Reduction
(a) Slenderness limits based on a 5% axial load reduction
0 20 40 60 80 100 120 140 160 1800
20
40
60
80
100
120
140
160
180
Slenderness Limit λcrit - Accurate Analysis
Sle
nder
ness
Lim
it λcr
it - D
esig
n E
quat
ion
10% Axial Load Reduction
(b) Slenderness limits based on a 10% axial load reduction
Fig. 8.11 Performance of Eq. 8.7 for FRP-confined RC columns
266
CHAPTER 9
DESIGN OF SLENDER
FRP-CONFINED RC COLUMNS
9.1 INTRODUCTION
Design equations for short FRP-confined RC columns have been presented in
Chapter 5 and expression for the slenderness limit that differentiates short
columns from slender columns has been given in Chapter 8. The information
given in Chapters 5 and 8 provides a complete procedure for the design of short
FRP-confined RC columns. This chapter deals with the design of slender FRP-
confined RC columns. It has been pointed out in Chapter 2 that no existing design
guidelines include a design procedure for slender FRP-confined RC columns. This
has mainly been due to the fact that only a limited number of studies have
investigated the behavior of such columns (Mirmiran et al. 2001; Yuan and
Mirmiran 2001; Tao et al. 2004; Fitzwilliam and Bisby 2006; Ranger and Bisby
2007). These studies have been reviewed in Chapters 2, 7 and 8, so the same
information is not repeated herein.
In practice, the majority of columns are restrained at both ends and are
eccentrically loaded, with the eccentricities at the two ends often being different.
Nevertheless, it has been shown in Chapter 6 that such a restrained column with
different end eccentricities can be transformed into an equivalent hinged column
where the eccentricities at the two ends are the same through the effective length
approach. Such an equivalent column is referred to as the standard hinged column.
This chapter is therefore limited to the analysis and design of slender FRP-
267
confined circular RC columns in the form of the standard hinged column. The
effective length of a retrained FRP-confined RC column may be estimated using
the methods given in existing design codes for RC structures, as suggested in
Chapter 8. The equivalent uniform moment factor may be evaluated using Eq.
6.40, as done in most existing design codes for the design of RC columns.
This chapter first describes a simple theoretical model for slender FRP-confined
RC columns. This simple model is exclusively for the analysis of standard hinged
columns. Despite its simplicity and higher computational efficiency, the model
leads to accurate predictions. A careful study is then performed to determine the
maximum allowable amount of FRP confinement and the slenderness limit of RC
columns beyond which the use of FRP for strengthening is not recommended.
Finally, design equations are proposed and their performance is assessed through
comprehensive comparisons with the numerical results from the simple theoretical
model. Once again, some of the considerations herein follow the specifications
given in GB-50010 (2002) and they are highlighted where appropriate.
9.2 SIMPLE THEORETICAL MODEL
9.2.1 General
A rigorous theoretical model for the analysis of FRP-confined RC columns has
been presented in Chapter 7. This model allows the eccentricities at the two
column ends to be unequal and finds the lateral deflection of the column at a
particular load level iteratively. It is obvious that this rigorous theoretical model
can be used as an analytical tool for the development of design equations for
slender FRP-confined RC columns. Nevertheless, as only columns with equal end
eccentricities are considered in this chapter, a much simpler model which still
provides accurate predictions was developed and used to produce numerical
results for the development of design equations.
The simple theoretical model differs from the rigorous model in that the deflected
shape of a column is now assumed to be a half-sine wave and equilibrium is only
checked at the critical section (the section at the mid-height of the column) where
268
the maximum lateral deflection of the column takes place.
As a result, the numerical procedure presented in Chapter 7 can be significantly
simplified. The present method of analysis has been widely adopted in similar
studies of hinged columns with equal end eccentricities (e.g. Bazant et al. 1991)
and has proven to be very successful. It is worth noting that the design equations
for slender RC columns specified in existing design codes (e.g. ENV-1992-1-1
1992; GB-50010 2002) are based on the present method of analysis, which is
another reason for the use of this simple model in the development of design
equations for slender FRP-confined RC columns.
9.2.2 Method of Analysis
The assumption that the deflected shape of the columns under consideration can
be closely approximated using a half sine wave can be mathematically expressed
as
sinmidf flπ⎛= − ⎜⎝ ⎠
x ⎞⎟ (9.1)
where midf is the lateral displacement at the critical section and is the distance
from the origin (see Fig. 9.1). Differentiating Eq. 9.1 twice gives
x
2
2 sinmidf xl lπ πφ ⎛= ⎜
⎝ ⎠⎞⎟ (9.2)
so midf is related to the curvature at the critical section midφ through
2
2mid midlf φπ
= (9.3)
The moment acting on the critical section can then be written as
269
( )2
2mid mid midlM N e f N e φπ
⎛ ⎞= + = +⎜
⎝ ⎠⎟ (9.4)
This moment must be balanced by the stresses on the critical section. These
stresses can be determined using the conventional section analysis described in
Chapter 5. For the present purpose, it is more convenient to seek the values of
and for a given value of midM N midφ . That is, for a given value of midφ , assume a
strain value for the extreme compression fiber of concrete to evaluate the resultant
axial force and moment of the critical section and check to see if they satisfy Eq.
9.4. Once Eq. 9.4 is satisfied, the solution for the present value of midφ is found.
Otherwise, adjust the assumed strain value until Eq. 9.4 is satisfied. Once the
solution is found, a point on the full-range load-deflection curve of the column is
identified and the entire curve can be generated by finding successive solutions of
and for increasing values of midM N midφ . The analysis stops when the extreme
compression fiber of concrete reaches its ultimate axial strain. In the present study,
the critical section was divided into 50 horizontal layers and the solution was
considered successful when the difference between the two sides of Eq. 9.4 is
within . It should be noted that equilibrium is only guaranteed at the
critical section within the present theoretical model. A computer program was
written using Matlab 7.1 to fulfill the above numerical procedure.
610 midM−
9.2.3 Accuracy of the Simple Theoretical Model
The present simple model has been well accepted for the analysis of RC columns
(e.g. Bazant et al. 1991). To understand the accuracy of the simple model for
FRP-confined circular RC columns, the predictions of the simple model are
compared herein with the predictions of the rigorous model for the columns tested
by Tao et al. (2004). These tests were used to verify the rigorous model presented
in Chapter 7. The properties of these columns are given in Table 7.6. The
experimental load-deflection curves are compared with the predicted curves of
both the simple model and the rigorous model in Fig. 9.2. All theoretical curves
were obtained with a 7.5 mm additional eccentricity, as was done in Chapter 7.
All theoretical curves terminate when the extreme compression fiber of the critical
270
section reaches the ultimate axial strain of confined concrete based on the hoop
rupture strain of 1.32% as found from the Tao et al.’s (2004) cylinder tests.
A comparison of the theoretical curves reveals the following difference in all
cases: the simple model predicts a slightly quicker ascending branch, a slightly
higher peak load (by about 1% to 2%), and a slightly quicker descending branches
than the rigorous model. Furthermore, the simple model predicts a much longer
descending branch which terminates at a much larger lateral deflection despite
that the same hoop rupture strain was used in both models. This phenomenon is
more pronounced for the C2 series (Figs 9.2c and 9.2d) which has a larger
slenderness ratio than the C1 series (Figs 9.2a and 9.2b). It should be noted that
the theoretical curves for the C2 series terminate at some 700 mm, but only their
early portions are shown in Figs 9.2c and 9.2d. This phenomenon arises from the
half-sine wave assumption and is explained below in detail by taking column C2-
3R as an example. Fig. 9.3a compares the lateral deflection distributions over the
column corresponding to two key deformation states, namely, the peak axial load
and the ultimate lateral deflection, predicted by both models. Figs 9.3b and 9.3c
respectively compare the distributions of the curvature and the moment for the
same two deformation states. The predictions of the simple model at the peak
axial load are shown as thick solid lines while the corresponding predictions from
the rigorous model are shown as thin solid lines; those at the ultimate lateral
deflection are shown as dotted lines. It can be seen that at the peak axial load, the
distributions of lateral deflection, moment and curvature predicted by the two
models are all in close agreement, which indicates that the half-sine wave
provides a good approximation to the deflected shape of the column. This
agreement explains why the two models lead to closely similar predictions for the
ascending branch. It is worth pointing out that the half-sine wave assumption
implies that the curvature is zero at either column end, which is not realistic unless
the initial load eccentricity is zero. After the peak axial load is achieved, a plastic
hinge gradually forms at the mid-height region of the column, as predicted by the
rigorous model (Fig. 9.3b). Therefore, the distributions of the curvature and the
lateral deflection predicted by the two models are no longer similar; the localized
plastic hinge can never be captured using the half-sine wave assumption. As a
result, the simple model predicts a much longer descending branch. Despite this
271
inaccuracy, the peak axial loads predicted by the simple model are in close
agreement with those predicted by the rigorous model. Both sets of theoretical
peak axial loads are close to the experimental results. Therefore, the simple model
is sufficiently accurate as a basis for the development of design equations for the
axial load capacities of slender FRP-confined RC columns.
9.3 LIMITS ON THE USE OF FRP
In the design of FRP-confined RC columns, it is important to ensure that the FRP
material is used in a safe and economical manner. Existing tests on standard FRP-
confined concrete cylinders showed that the concrete strength can be increased by
over three times provided the FRP confinement is strong enough (see Chapter 3).
The failure of strongly-confined specimens can be explosive and is undesirable in
practice. As columns become more slender, the effectiveness of FRP confinement
is reduced (see Chapter 7) and the use of FRP may eventually become
uneconomical. Another concern with FRP-confined RC columns is that the lateral
deflections may exceed an acceptable limit in design. It is thus advisable to
impose certain limits on the use of FRP to ensure that the FRP is used in a safe
and economical manner. To this end, a parametric study was conducted using the
simple theoretical model presented in the preceding section to examine how the
effectiveness of FRP confinement is affected by various parameters.
The studywas carried out on the same reference column that has been used in
Chapter 8 except that the reference column used herein is subjected to equal end
eccentricities , as shown in Fig. 9.1. The parameters studied herein are similar to
those studied in Chapter 8. The end eccentricity ratio is no more a variable but is
fixed at unity. A new and very important parameter is the slenderness ratio
e
λ .
The values used for these parameters are summarized in Table 9.1. These values
are similar to those used in Chapters 5 and 8.
The combinations of these parameters led to about 8,000 cases. The slenderness
ratio goes up to 50 as a preliminary study showed that beyond this slenderness the
confining effect of FRP is very limited. The remaining values used for all
parameters are the same as those used in Chapter 5.
272
Figs 9.4 to 9.9 show the results of the parametric study. In Figs 9.4 to 9.9, the
vertical axis is the axial load capacity enhancement ratio ,u u refN N , where
is the axial load capacity of an RC column before it is confined with FRP. The
horizontal axis is one of the six parameters considered. A series of curves is
provided in each of these figures, showing the variation of another parameter, and
the values of the other four parameters used to generate these numerical results are
also given in each figure.
,u refN
Fig. 9.4 shows the effect of the strength enhancement ratio. This figure is for a
CFRP-confined column with a small amount of steel reinforcement and an
intermediate end eccentricity. It is obvious that the axial load capacity of this
column increases as the amount of FRP confinement increases, but the
effectiveness of FRP confinement drops rapidly as the column becomes slender.
The strain ratio also appears to be an important parameter. It can be seen in Fig.
9.5 that with the same level of concrete strength enhancement, the enhancement of
the axial load capacity is smaller when the confining jacket has a larger strain
capacity. This can be easily understood as an FRP jacket with a larger strain
capacity results in larger lateral deflections. Figs 9.6 and 9.7 respectively show
the effects of the end eccentricity and the slenderness of the column. The
confinement effect is most pronounced in columns with a small slenderness
subjected to a small end eccentricity. It should be noted that the increase in the
axial load capacity may be very limited (in some cases less than 5%) even under
very heavy confinement when the column is slender and is subjected to a large
end eccentricity. In such cases, the use of FRP may not be economical. Compared
with the above four parameters, the depth ratio and the steel reinforcement ratio
are less important. The effects of these two parameters on the axial load capacity
enhancement ratio may be neglected in practice, as suggested by Figs 9.8 and 9.9.
An examination of the numerical results showed that under very heavy
confinement that results in , the lateral deflections at failure exceed the
commonly accepted limit of
' 2cc cof f= '
50l in design (Cranston 1972) in some cases.
273
Therefore, it is suggested that the maximum allowable amount of FRP
confinement be limited to
'
' 1.75cc
co
ff
≤ (9.5)
The present numerical results indicate that with the above limit, the maximum
possible increase in the axial load capacity of an RC column is approximately
50%. For cases where the slenderness and the end eccentricity of the column are
within a reasonable range (e.g. 20λ ≤ and 0.3e D≤ for columns confined with
an FRP jacket possessing a strain capacity of 0.75%), this limit generally ensures
that increases in the axial load capacity of more than 15% can be realized.
Another important conclusion that can be drawn from the parametric study is that
the confinement effect may become very limited when a column is very slender. It
is thus advisable to impose a limit on the slenderness beyond which the use of
FRP should not be recommended. It is suggested herein that a minimum increase
in the axial load capacity of 10% should be achieved as a result of FRP
confinement. The following approximate equation is proposed based on this
criterion
max 50 3 ελ ρ= − (9.6)
where maxλ is the upper bound of the slenderness ratio of a column whose axial
load capacity still benefit significantly from the provision of an FRP jacket. Eq.
9.6 was derived for a reference column with a steel reinforcement ratio of 1%, a
section depth ratio 0.8d D = , and an end eccentricity ratio 0.2e D = . The
concrete strength of the reference column was increased by 75% as a result of
FRP confinement. It should be noted that when the eccentricity ratio is larger than
0.2, Eq. 9.6 may lead to an increase in the axial load capacity smaller than 10%.
Some of the values listed in Table 9.1 do not satisfy the conditions set by Eqs 9.5
and 9.6; these values were thus removed to form a reduced set a parametric study
274
cases for use in later sections of this chapter. The values of the parameters in the
reduced set are listed in Table 9.2. It should be noted that the values of maxλ
defined by Eq. 9.6 for the three different strain ratios considered are 47, 38.75,
and 27.5 respectively. Values of 50, 40, and 30 were however used instead for
simplicity.
9.4 DESIGN METHOD
9.4.1 Review of Current Design Methods for RC Columns
The current design approach adopted in various design codes for RC columns (e.g.
ENV-1992-1-1 1992; BS-8110 1997; GB-50010 2002; ACI-318 2005)
approximate the second-order moments by an amplification of the first-order
moment so that the failure load can be related to the strength of the critical section.
In other words, the current design approach transforms the design of a slender
column into the design of a section with an equivalent eccentricity, which consists
of the initial end eccentricity of the slender column and a an additional
eccentricity equal to the nominal lateral displacement of the critical section nomf .
The concept of this additional eccentricity is illustrated in Fig. 9.10. Fig. 9.10 is
similar to Fig. 8.1 and it shows three loading paths, OA , and , for a
standard hinged column with three different values of column slenderness but a
fixed initial end eccentricity. Graphically, the design approach seeks the straight
loading paths OB and shown as dashed lines in Fig. 9.10 to replace the
original curved loading paths OB and shown as solid lines for material
failure and stability failure respectively. It is obvious that for material failure,
OB ODE
OC
ODE
nomf
is the real lateral displacement of the critical section at failure. However, for
stability failure, nomf is a fictitious lateral displacement. This point must be borne
in mind and is further discussed in a later section. It is now clear that the key
element of the current design approach is to find nomf .
There are two main approaches in the current design codes, namely, the moment
magnifier method and the nominal curvature method. The moment magnifier
275
method has been adopted by ACI-318 (2005) and many of its previous versions,
among others. This approach originated from the elastic analysis of columns,
where the lateral deflections can be exactly determined provided the section
flexural stiffness is known. When this approach is used for the design of RC
columns, the key is to find the equivalent section flexural stiffness that accounts
for the effect of the nonlinearities in the constituent materials. By contrast, the
nominal curvature method was originally proposed by Aas-Jakobsen and Aas-
Jakobsen (1968) has been adopted by ENV-1992-1-1 (1992), BS-8110 (1997),
and GB-50010 (2002), among others. This approach relates the lateral deflection
to the curvature through the relationship defined by Eq. 9.3. It is obvious that the
key to this approach is to determine the nominal curvature nomφ corresponding to
nomf . This chapter follows the framework of the nominal curvature method to
develop a design procedure that is consistent with GB-50010 (2002). Therefore,
the moment magnifier method is not further pursued.
9.4.2 Nominal Curvature
The nominal curvature and the nominal lateral displacement can be related by the
following equation
2
2nom nomlf φπ
= (9.7)
As explained earlier, the nominal curvature sought in the nominal curvature
method for material failure is the real curvature of the critical section at failure
failφ . However, the nominal curvature for stability failure needs some further
explanation, which can be achieved by making use of the moment-curvature
diagram shown in Fig. 9.11. Fig. 9.11 shows the moment-curvature curve of the
critical section when the column reaches its axial load capacity . This curve
shows how the internal moment varies with the curvature of the critical section
under this particular axial load. The curvature at the end of this curve,
uN
secφ , is the
maximum curvature that the critical section can sustain under this particular axial
load. The inclined straight line represents how the external moment varies as the
276
curvature at the critical section increases. This inclined line can be mathematically
described by Eq. 9.4. This line has a slope of 2 2uN el π and intersects the vertical
axis at a value of . At the point where the inclined line meets the moment-
curvature curve, the external moment is equilibrated by the internal moment. It is
obvious that for material failure (Fig. 9.11a), the inclined line must intersect the
moment-curvature curve at the end of the moment-curvature curve
(
uN e
secnom failφ φ φ= = ). However, when stability failure occurs, the above three
curvatures have different values, as illustrated in Fig. 9.11b. When stability failure
occurs, the inclined line must meet the moment-curvature curve at the point of
failure for moment equilibrium. Therefore, the inclined line must be a tangent to
the moment-curvature curve at the point of failure, where the curvature is failφ .
The nominal curvature can be found as the curvature at the intersection point of
the inclined line and the horizontal line since this point corresponds to the point
on the section interaction curve at . Obviously, the nominal curvature always
has a value larger than
uN
failφ but smaller than secφ .
The failure modes of an RC section can be classified into three categories: 1)
balanced failure; 2) compression failure and 3) tension failure. At balanced failure,
the extreme compression fiber of concrete reaches the ultimate compressive strain
of concrete when the most highly-tensioned longitudinal steel bar(s) on the
opposite side of the section reaches its tensile yield strain. The particular axial
load corresponding to balanced failure is denoted by . When compression
failure occurs, the concrete reaches its ultimate compressive strain before the steel
reinforcement yields. Axial loads corresponding to compression failure are always
larger than . Tension failure is the opposite to compression failure and axial
loads corresponding to tension failure are always smaller than . According to
the definitions given above, the curvature at balanced failure can be easily found
to be
balN
balN
balN
2 cu ybal D d
ε εφ
+=
+ (9.8a)
277
balφ is used as the basis to evaluate nomφ in the nominal curvature method. For
material failure, a factor 1ξ is used to reflect the effect of axial load when the axial
load is other than . For stability failure, an additional factor balN 2ξ needs to be
introduced to explain the difference between nomφ and secφ . In summary, nomφ can
be related to balφ using the following equation
1 2nom balφ ξ ξ φ= (9.8b)
It can be concluded from the above discussions that , balN 1ξ and 2ξ are the
essential elements in the nominal curvature method and they are discussed in
detail in the following sub-sections.
9.4.3 Axial Load at Balanced Failure
GB-50010 (2002) specifies the following equation as an estimate of the axial load
at balanced failure for RC sections
'0.5bal coN f= A (9.9)
A careful study showed that Eq. 9.9 is no longer reasonable when used to predict
the axial load at balanced failure of FRP-confined RC sections. Fig. 9.12 shows a
series of interaction curves for an RC section before and after being confined with
various amounts of FRP. Balanced failures at various confinement levels are
indicated by different markers in Fig. 9.12. It is interesting to note that these
markers gradually deviate from the maximum moment point on the interaction
curve as the confinement level increases. The same observation has also been
reported by Cheng et al. (2002). The axial load at balanced failure of an FRP-
confined RC section depends on the following four parameters sρ , d D , ' 'cc cof f
and ερ . To develop an approximate expression for , a simple parametric
study was conducted using the reduced set of parameters given in Table 9.2.
balN
278
Based on the numerical results, the following simple equation is proposed for
FRP-confined RC sections
'0.8bal ccN f= A (9.10)
Fig. 9.13 compares the predictions of Eq. 9.10 with the exact numerical results for
FRP-confined RC sections. It can be seen that Eq. 9.10 only provides a rough
estimation but it will be seen that it is sufficiently accurate for design use. The
predictions of Eq. 9.9 for RC sections are also shown in Fig. 9.13.
9.4.4 Factors 1ξ and 2ξ
GB-50010 (2002) specifies the following equation for 1ξ
1 1bal
u
NN
ξ = ≤ (9.11)
while BS-8110 (1997) and ENV-1992-1-1 (1992) employ the following equation
1 1uo u
uo bal
N NN N
ξ −= ≤
− (9.12)
where is the axial load capacity of an RC section when it is concentrically
compressed. Fig. 9.14 compares the predictions of Eqs 9.11 and 9.12 with the
exact results from section analysis. Fig 9.14a is for an RC section while Fig. 9.14b
is for an FRP-confined RC section with
uoN
' ' 1.5cc cof f = . It should be noted that the
value of used in the comparison was the exact value to eliminate the
discrepancy introduced by the different approximate equations employed in these
codes. The axial loads and curvatures in Fig. 9.14 were normalized using and
balN
balN
balφ respectively. The small circle in Fig. 9.14 represents balanced failure while
the small square represents the case where the section is subjected to the minimum
eccentricity. The definition of the minimum eccentricity in existing design codes
279
for RC structures has been discussed in Chapter 5. The minimum eccentricity was
taken as min 0.05e D= for simplicity in the present study. It can be seen that the
curvature decreases as the axial load increases and in the case of concentric
compression the curvature is zero. Although Eq. 9.12 satisfies the boundary
condition at concentric compression, the overall performance of Eq. 9.11 is better
since the small range beyond the small square is excluded by the minimum
eccentricity and thus does not need to be considered in design. When the FRP
confinement is provided, the range corresponding to compression failure becomes
smaller. It is interesting to note that although the exact results indicate that the
curvatures at tension failure are always larger value balφ , it is limited to balφ in all
the codes mentioned above. This limit on 1ξ is not explained in Aas-Jakobsen and
Aas-Jakobsen (1968) in which the nominal curvature method was originally
proposed. To the best knowledge of the author, this issue has not been clearly
explained so far. An attempt is made below to explain this issue.
When assessing the role played by 1ξ in the nominal curvature method, it is
advisable to combine 1ξ with 2ξ to see the overall effect of these two factors. GB-
50010 (2002) specifies the following equation for 2ξ
2 1.15 0.01 1lD
ξ = − ≤ (9.13)
2ξ is limited to 1 because when a column is relatively short it fails in the mode of
material failure. It has already been explained that for material failure, only 1ξ
needs to be considered and 2ξ always remains unity (Fig. 9.11a). Only when the
column is slender enough to cause stability failure does 2ξ need to be considered.
In such a case, it always has a value smaller than 1, as clearly illustrated in Fig.
9.11b.
A careful study revealed that if the exact values of 1ξ are used, the development
of an expression for 2ξ becomes difficult. This is because in such cases the value
280
of 2ξ depends strongly on the end eccentricity besides the slenderness of the
column. However, if 1ξ is limited to unity, the dependence of 2ξ on the end
eccentricity is much less so that 2ξ may be taken as a function of the slenderness
only. It should be noted that although limiting 1ξ to 1 has the advantage of
simplicity, it can make the nominal curvature method un-conservative in
predicting the axial load capacity of columns subjected to material failure and
with an axial load capacity smaller than . Such a case can happen in a short
column with a large end eccentricity. Such a column has a curvature larger than
balN
balφ but its curvature is forced to be balφ in the above approach, which gives rise to
the un-conservativeness. Nevertheless, as the second order effects in such
columns are limited, the un-conservativeness will be seen to be within a
reasonable range. It should be noted that the above explanation is completely
based on the theoretical results but has not been verified using any test data.
As a result of the above discussions, it is suggested that for FRP-confined RC
columns, the form of Eq. 9.11 be retained but be estimated using Eq. 9.10,
leading to Eq. 9.14a
balN
'
10.8 1bal cc
u u
N f AN N
ξ = = ≤ (9.14a)
Based on the values of 1ξ given by Eq. 9.14a, Eq. 9.14b is proposed for 2ξ for use
in the design of slender FRP-confined circular RC columns
2 (1.15 0.06 ) (0.01 0.012 ) 1lDεξ ρ ρ= + − + ≤ε (9.14b)
It should be noted that Eq. 9.14b ignores the effects of a number of factors, such
as the end eccentricity for simplicity in design. It is shown in a later section of this
chapter that Eq. 9.14b is sufficiently accurate for design use. Eq. 9.14b also
reduces to Eq. 9.13 when no FRP confinement is provided.
281
9.4.5 Proposed Design Equations
With , balN 1ξ and 2ξ determined, the full set of design equations is given below.
( )'1
sin 212u cc c tN f A fπθθα θ θπθ
⎛ ⎞= − + −⎜ ⎟⎝ ⎠
y sA (9.15a)
2 3'
1 2 12
sin sin2 sin3
cu bal cc y s
lN e f AR f A R tπθ ππθξ ξ φ απ π
⎛ ⎞ ++ = +⎜ ⎟
⎝ ⎠
θπ
(9.15b)
' '1 1.17 0.2 cc cof fα = − (9.15c)
0 1.25 0.125 1cθ θ≤ = − ≤ (9.15d)
0 1.125 1.5 1tθ θ≤ = − ≤ (9.15e)
On the right hand side of Eqs 9.15a and 9.15b are the approximate expressions for
the section interaction curve that have been presented in Chapter 5.
When the axial load and the associated initial end eccentricity are known, the
design of the FRP jacket should follow the steps listed below
1) Select the type of FRP and check to see if the slenderness of the column
satisfies Eq. 9.6;
2) Assume a jacket thickness and calculate and 'ccf cuε ;
3) Determine the value of θ through a trial-and-error process until Eqs 9.15a
and 9.15b are both satisfied;
4) Check to see if the axial load capacity calculated from Eq. 9.15a is larger than
the applied axial load;
5) If step 4) is satisfied, the FRP confinement assumed in step 2) is strong
enough to resist the applied axial load; otherwise, increase the jacket
thickness and go through steps 2) to 4) again until step 4) is satisfied;
282
6) If step 4) still cannot be satisfied when the confinement has already been
increased to a very high level that exceeds the limit given in Eq. 9.5, it
indicates that the use of FRP in this case is either inefficient or uneconomical
and other means of strengthening should be used instead (e.g. increase the
cross-sectional area).
It should be noted that in the above procedure, it is assumed that all the geometric
and material properties of the original RC section are known, as is generally the
case in the retrofit of existing RC columns.
Fig. 9.15 compares a series of interaction curves predicted using the proposed
design approach with those produced using the simple theoretical model. The
interaction curves are for a series of CFRP-confined RC columns of the same
section but with a range of slenderness ratios from 10λ = to 40λ = at an
interaval of 10. It should be noted that 40λ = is slighter larger than the maximum
allowable slenderness ratio ( 38.75λ = ) defined by Eq. 9.6. The interaction curves
are cut off by a straight line that represents the minimum eccentricity. The column
was assumed to have a 600 mm diameter. The compressive strength of concrete
and the yield strength of the steel reinforcement were assumed to be 20.1 MPa and
335 MPa respectively. The other parameters used to produce the interaction
curves in Fig. 9.15 are all given in Fig. 9.15. The approximate curve for 10λ =
from the design equation is in excellent agreement with the exact curve as the
second order effect is very limited at this slenderness value. The approximate
curve for 20λ = is un-conservative for a certain range of end eccentricities. This
overestimation mainly stems from the fact that at this slenderness material failure
is still the predominant failure mode and this range of end eccentricities
corresponds to tensile failure. As a result, the exact nominal curvature found using
the simple theoretical model must be larger than balφ but it is forced to be equal to
balφ in the proposed design approach. However, the overestimation is not
significant and is comparable to that found in the current design approach adopted
by GB-50010 (2002) for RC columns.
283
Fig. 9.16 shows the overall performance of the proposed design approach. The
axial load capacities calculated using the simple theoretical model and the
proposed design approach are normalized by . Fig. 9.16 includes the
numerical results of all FRP-confined columns considered in the reduced set of
parametric cases listed in Table 9.2. It can be seen that the majority of cases fall
within the error lines. The maximum overestimation found among the
cases studied is 12.3%.
uoN
10%±
BS-8110 (1997) specifies that for simplicity 1 2 1ξ ξ= = can be used in the design
of RC columns with some sacrifice of accuracy; this simplification is always
conservative. It should be noted that the simplification of 1 1ξ = makes the design
procedure much more straightforward because otherwise 1ξ has to be evaluated
through a trial-and-error process ( 1ξ is a function of as defined in Eqs 9.12,
9.13 and 9.14a). If
uN
1 2 1ξ ξ= = is adopted in the proposed design approach, the
results for the same cases shown in Fig. 9.16 are shown in Fig. 9.17. It can be seen
that with this simplification, the predictions become more conservative but the
majority of cases still fall within the 10%± error lines. The maximum
underestimation found among the cases studied is 18.2%.
When the proposed equations are applied to design FRP jackets for existing RC
columns, it is possible that the axial load capacity for the strengthened column
calculated using the proposed design equations is smaller than that of the original
RC column calculated using the design equations given in GB-50010 (2002). This
situation can occur when the slenderness of the column approaches maxλ because
the increase in the axial load capacity is then very limited and the design
equations in GB-50010 (2002) are slightly un-conservative. Therefore, the
designer must be careful in the design of such columns. The use of more advanced
methods such as the theoretical model presented in this chapter is recommended
for such cases.
Finally, the proposed design equations are used to predict the axial load capacities
of FRP-confined RC columns reported in existing studies (Tao et al. 2004;
284
Fitzwilliam and Bisby 2006; Ranger and Bisby 2007). These column tests have
been described in detail in Chapter 7. The FRP hoop rupture strains used in the
design equations when making predictions for these column tests in these three
studies are 1.32%, 1.16%, and 1.15% respectively, as found from the
corresponding ancillary cylinder tests. It should be noted that some of the columns
do not satisfy the conditions set by Eqs 9.5 and 9.6. The columns of Tao et al.
(2004) and those confined with a 2-ply CFRP jacket in Fitzwilliam and Bisby
(2006) and Ranger and Bisby (2007) have a ' 'cc cof f ratio slightly larger than 1.75.
Besides, the columns of Tao et al. (2004) had a slenderness ratio of 33.6 for series
C1 and 81.6 for series C2 which exceed the maximum allowable slenderness ratio
(30.2) defined by Eq. 9.6. In the present set of comparisons, series C2 in Tao et al.
(2004) was excluded while the remaining columns reported in these three studies
were retained. An additional eccentricity was considered when using the design
equations to predict the axial load capacities, as required by GB-50010 (2002).
GB-50010 (2002) specifies that the additional eccentricity should be taken as the
larger of 20 mm and 1/30 of the diameter for circular columns. Obviously, this
provision is for realistically-sized columns; the use of a 20 mm additional
eccentricity is thus unreasonable for the small-scale columns under consideration.
As a result, an additional eccentricity of 7.5 mm was used for all the cases under
consideration. The predicted axial load capacities using the proposed design
equations and the corresponding experimental values are compared in Fig. 9.18. It
can be seen in Fig. 9.18 that the predictions of the proposed design equations are
reasonably close to the experimental values and are conservative in most cases.
Some slight overestimation can be observed for column C1-1R of Tao et al. (2004)
and columns C-30 and C-40 of Ranger and Bisby (2007). These observations are
consistent with those found in the predictions by the rigorous theoretical model
for the same columns (see Chapter 7).
9.5 CONCLUSIONS
This chapter has been concerned with the development of design equations for
slender FRP-confined circular RC columns. To this end, a simpler theoretical
285
model than the one presented in Chapter 7 was developed for modeling slender
FRP-confined RC columns. This model is exclusively for hinged columns with
equal end eccentricities and assumes the deflected shape of such columns to have
the shape of a half-sine wave. Subsequently, two practical limits on the use of
FRP for the strengthening of RC columns were proposed to ensure a safe and
economical strengthening scheme based on the numerical results produced by the
simple theoretical model. Finally, design equations were developed using the
nominal curvature method. The design equations provide close agreement with the
theoretical results. The results and discussions presented in this chapter allow the
following conclusions to be drawn:
1) The load-deflection curves predicted using the simple theoretical model
presented in this chapter and those predicted using the rigorous theoretical
model presented in Chapter 7 have a very similar ascending branch and a very
similar peak axial load. The simple theoretical model however predicts a
much longer descending branch of the load-deflection curves than the
rigorous model since the half-sine wave assumption fails to capture the
formation of the plastic hinge at the mid-height region of a column. Despite
this deficiency, the simple theoretical model is sufficiently accurate as a basis
for the development of design equations for the axial load capacities of
slender FRP-confined RC columns.
2) The simple theoretical model revealed the same phenomenon as has been
observed in existing tests: the effectiveness of FRP confinement decreases as
the columns become more slender. The theoretical model also indicated that
strong confinement may result in excessive lateral deflection that is not
acceptable in design. As a result, practical limits need to be imposed on the
level of confinement and the column slenderness to ensure a safe and
economical strengthening scheme.
3) The nominal curvature method was fully discussed, including some original
insight into a subtle issue in this method which has not been properly
explained to the best knowledge of the author.
286
4) The proposed design equations have a simple form that is familiar to
engineers and provide predictions that are in close agreement with the
theoretical results. A simplified version of these design equations with a small
sacrifice in accuracy was also presented, which is easier for design use and
leads to more conservative predictions than the original version.
5) The design equations were shown to provide reasonable and generally
conservative predictions for FRP-confined RC columns reported in existing
studies.
287
9.6 REFERENCES
Aas-Jakobsen, A. and Aas-Jakobsen, K. (1968). “Buckling of slender columns”, Bulletin d’ Information, Comite Europeen du Beton, No. 69, 201-270.
ACI-318 (2005). Building Code Requirements for Structural Concrete and Commentary, ACI Committee 318, American Concrete Institue.
Bazant, Z.P., Cedolin, L. and Tabbara, M.R. (1991). “New method of analysis for slender columns”, ACI Structural Journal, 88(4), 391-401.
BS 8110 (1997). Structural Use of Concrete, Part 1. Code of Practice for Design and Construction, British Standards Institution, London, UK.
Cheng, H.L., Sotelino, E.D. and Chen, W.F. (2002). “Strength estimation for FRP wrapped reinforced concrete columns”, Steel and Composite Structures, 2(1), 1-20.
Concrete Society (2004). Design Guidance for Strengthening Concrete Structures with Fibre Composite Materials, Second Edition, Concrete Society Technical Report No. 55, Crowthorne, Berkshire, UK.
Cranston, W.B. (1972). Analysis and Design of Reinforced Concrete Columns, Research Report 20, Cement and Concrete Association, UK.
ENV 1992-1-1 (1992). Eurocode 2: Design of Concrete Structures – Part 1: General Rules and Rules for Buildings, European Committee for Standardization, Brussels.
fib (2001). Externally Bonded FRP Reinforcement for RC Structures, The International Federation for Structural Concrete, Lausanne, Switzerland.
Fitzwilliam, J. and Bisby, L.A. (2006). “Slenderness effects on circular FRP-wrapped reinforced concrete columns”, Proceedings, 3rd International conference on FRP Composites in Civil Engineering, December 13-15, Miami, Florida, USA, 499-502.
GB-50010 (2002). Code for Design of Concrete Structures, China Architecture and Building Press, China.
Mirmiran, A., Shahawy, M. and Beitleman, T. (2001). “Slenderness limit for hybrid FRP-concrete columns”, Journal of Composites for Construction, 5(1), 26-34.
Ranger, M. and Bisby, L.A. (2007). “Effects of load eccentricities on circular FRP-confined reinforced concrete columns”, Proceedings, 8th International Symposium on Fiber Reinforced Polymer Reinforcement for Concrete Structures (FRPRCS-8), University of Patras, Patras, Greece, July 16-18, 2007.
Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, 2nd
288
International Conference on Advanced Polymer Composites for Structural Applications in Construction, University of Surrey, Guildford, UK, 203-212.
Yuan, W., and Mirmiran, A. (2001). “Buckling analysis of concrete-filled FRP tubes”, International Journal of Structural Stability and Dynamics, 1(3), 367-383.
289
Table 9.1 Entire set of parametric study cases
Parameter Values λ 10, 20, 30, 40, 50
e D 0.05,0.1,0.15,0.2,0.25,0.3,0.4,0.6,0.8
sρ 1%, 2%, 3%, 4%, 5% d D 0.7, 0.8, 0.9 ' '
cc cof f 1.25, 1.5, 1.75, 2
,h rup coε ε 1, 3.75, 7.5
Table 9.2 Reduced set of parametric study cases
Parameter Values 10, 20, 30, 40, 50 for , 1h rup coε ε = 10, 20, 30, 40 for , 3.75h rup coε ε = λ
10, 20, 30 for , 7.5h rup coε ε = e D 0.05,0.1,0.15,0.2,0.25,0.3,0.4,0.6,0.8
sρ 1%, 2%, 3%, 4%, 5% d D 0.7, 0.8, 0.9 ' '
cc cof f 1.25, 1.5, 1.75
290
Ddl
N
N
e
e
f
x
fmid
Fig. 9.1 Schematic of the simple theoretical model
291
0 20 40 60 80 1000
100
200
300
400
500
600
700
800
900
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C1-1R (Test)C1-1R (Rigorous Model)C1-1R (Simple Model)C1-2R (Test)C1-2R (Rigorous Model)C1-2R (Simple Model)
(a) Columns C1-1R and C1-2R
0 20 40 60 80 100 1200
20
40
60
80
100
120
140
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C1-3R (Test)C1-3R (Rigorous Model)C1-3R (Simple Model)C1-4R (Test)C1-4R (Rigorous Model)C1-4R (Simple Model)
(b) Columns C1-3R and C1-4R
292
0 50 100 150 2000
100
200
300
400
500
600
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C2-1R (Test)C2-1R (Rigorous Model)C2-1R (Simple Model)C2-2R (Test)C2-2R (Rigorous Model)C2-2R (Simple Model)
(c) Columns C2-1R and C2-2R
0 50 100 150 200 2500
10
20
30
40
50
60
70
80
Lateral Displacement fmid (mm)
Axi
al L
oad
N (k
N)
C2-3R (Test)C2-3R (Rigorous Model)C2-3R (Simple Model)C2-4R (Test)C2-4R (Rigorous Model)C2-4R (Simple Model)
(d) Columns C2-3R and C2-4R
Fig. 9.2 Comparison with Tao’s (2004) tests on FRP-confined circular RC
columns
293
0 100 200 300 400 500 600 700 8000
500
1000
1500
2000
2500
3000
Lateral Deflection (mm)
Col
umn
Hei
ght
(mm
)
Peak axial load
Ultimate lateral deflection
(a) Distribution of lateral deflection
0 1 2 3 4 5 6 7 8
x 10-4
0
500
1000
1500
2000
2500
3000
Curvature (1/mm)
Col
umn
Hei
ght
(mm
)
Ultimate lateral deflection
Peak axial load
(b) Distribution of curvature
294
0 2 4 6 8 10 12 14
x 106
0
500
1000
1500
2000
2500
3000
Moment (N⋅mm)
Col
umn
Hei
ght
(mm
)
Ultimate lateral deflection
Peak axial load
(c) Distribution of moment
Fig. 9.3 Illustration of differences between the two theoretical models
295
1 1.2 1.4 1.6 1.8 21
1.1
1.2
1.3
1.4
1.5
1.6
d/D=0.8ρs=1%
e/D=0.2εh,rup/εco=3.75
Strength Enhancement Ratio f′cc/f′co
Axi
al L
oad
Cap
acity
Enh
ance
men
t Rat
ioN
u/Nu,
ref
λ = 10λ = 20λ = 30λ = 40λ = 50
Fig. 9.4 Effect of strength enhancement ratio
0 2 4 6 81
1.1
1.2
1.3
1.4
1.5
1.6d/D=0.8ρs=1%
e/D=0.05f′cc/f′co=1.75
Strain Ratio εh,rup/εco
Axi
al L
oad
Cap
acity
Enh
ance
men
t Rat
ioN
u/Nu,
ref
λ = 10λ = 20λ = 30λ = 40λ = 50
Fig. 9.5 Effect of strain ratio
296
0 0.2 0.4 0.6 0.81
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
d/D=0.8
ρs=1%
λ=20
εh,rup/εco=3.75
Normalized Eccentricity e/D
Axi
al L
oad
Cap
acity
Enh
ance
men
t Rat
ioN
u/Nu,
ref
f′cc/f′co=1.25
f′cc/f′co=1.5
f′cc/f′co=1.75
f′cc/f′co=2
Fig. 9.6 Effect of end eccentricity
10 20 30 40 501
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
d/D=0.8
ρs=1%
f′cc/f′co=1.75
εh,rup/εco=3.75
Slenderness Ratio λ
Axi
al L
oad
Cap
acity
Enh
ance
men
t Rat
ioN
u/Nu,
ref
e/D=0.1e/D=0.2e/D=0.3e/D=0.4e/D=0.6
Fig. 9.7 Effect of slenderness ratio
297
0.01 0.02 0.03 0.04 0.050.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
d/D=0.8
λ=20e/D=0.2εh,rup/εco=3.75
Steel Reinforcement Ratio λ
Axi
al L
oad
Cap
acity
Enh
ance
men
t Rat
ioN
u/Nu,
ref
f′cc/f′co=1.25
f′cc/f′co=1.5
f′cc/f′co=1.75
f′cc/f′co=2
Fig. 9.8 Effect of steel reinforcement ratio
0.7 0.75 0.8 0.85 0.91
1.05
1.1
1.15
1.2
ρs=1%
λ=30
e/D=0.2
εh,rup/εco=3.75
Depth Ratio d/D
Axi
al L
oad
Cap
acity
Enh
ance
men
t Rat
ioN
u/Nu,
ref
f′cc/f′co=1.25
f′cc/f′co=1.5
f′cc/f′co=1.75
f′cc/f′co=2
Fig. 9.9 Effect of depth ratio
298
Moment M
Axi
al L
oad
N
O
AB
CD
E
e = Const
e
1
NBe
NBfnom
NCe NCfnom
Fig. 9.10 Concept of the fictitious lateral displacement
299
Curvature φ
Mom
ent
Mφnom = φfail = φsec = ξ1φbal
(a) Material failure
Curvature φ
Mom
ent
M φsec =ξ1φbalφnom = ξ2φsec
φfail
N = Const
(b) Stability failure
Fig. 9.11 Determination of the nominal curvature
300
0 0.5 1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Normalized Moment Mu / Muo
Nor
mal
ized
Axi
al L
oad
Nu
/ Nuo
Unconfinedf′cc/f′co=1.25
f′cc/f′co=1.5
f′cc/f′co=1.75
Fig. 9.12 Axial loads at balanced failure
0 2000 4000 6000 8000 100000
2000
4000
6000
8000
10000
Axial Load at Balanced FailureNbal - Analysis (kN)
Axi
al L
oad
at B
alan
ced
Failu
reN
bal -
App
roxi
mat
ion
(kN
)
Unconfinedf′cc/f′co=1.25
f′cc/f′co=1.5
f′cc/f′co=1.75
Fig. 9.13 Performance of Eqs 9.9 and 9.10
301
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
3
Normalized Axial Load Nu/Nbal
Nor
mal
ized
Cur
vatu
re φ u
/ φba
l
RC Section Section AnalysisGB EquationBS Equation
(a) RC section
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
0.5
1
1.5
2
2.5
3
3.5
4
Normalized Axial Load Nu/Nbal
Nor
mal
ized
Cur
vatu
re φ u
/ φba
l
FRP-confined RC Sectionf′cc/f′co=1.5
Section AnalysisGB EquationBS Equation
(b) FRP-confined RC section
Fig. 9.14 Factor 1ξ
302
0 100 200 300 400 500 6000
1000
2000
3000
4000
5000
6000
7000
8000
9000d/D=0.8ρs=1%f′cc/f′co=1.5εh,rup/εco=3.75
Moment M (kN⋅m)
Axi
al L
oad
N (k
N)
Accurate AnalysisDesign Equations
λ=10
λ=20
λ=30
λ=40e=0.05D
Fig. 9.15 Typical interaction curves
303
0 0.2 0.4 0.6 0.8 1 1.20
0.2
0.4
0.6
0.8
1
1.2
Normalized Axial Load CapacityNu/Nuo - Accurate Analysis
Nor
mal
ized
Axi
al L
oad
Cap
acity
Nu/N
uo -
Des
ign
Equ
atio
ns
10%
-10%
Fig. 9.16 Performance of the proposed design equations
0 0.2 0.4 0.6 0.8 1 1.20
0.2
0.4
0.6
0.8
1
1.2
Normalized Axial Load CapacityNu/Nuo - Accurate Analysis
Nor
mal
ized
Axi
al L
oad
Cap
acity
Nu/N
uo -
Des
ign
Equ
atio
ns
10%
-10%
Fig. 9.17 Performance of the proposed design equations with 121 == ξξ
304
0 200 400 600 800 10000
100
200
300
400
500
600
700
800
900
1000
Axial Load Capacity Nu (kN) - Test
Axi
al L
oad
Cap
acity
Nu
(kN
)- D
esig
n E
quat
ions
Ranger and Bisby (2007)Fitzwilliam and Bisby (2006)Tao et al. (2004)
C40C30
C1-1R
Fig. 9.18 Predictions of the proposed design equations against test results
305
CHAPTER 10
CONCLUSIONS
10.1 INTRODUCTION
This thesis has presented a systematic study which aims to develop a rational
design procedure for FRP-confined RC columns. Such a design procedure is not
available in current design guidelines for FRP-strengthened RC structures and is
thus urgently needed for the confident and effective use of FRP wraps to enhance
the axial load capacity of RC columns.
A series of axial compression tests on FRP-confined concrete cylinders was first
presented in the thesis to gain a good understanding of the stress-strain behavior
of FRP-confined concrete, which is fundamental and essential to the analysis and
design of FRP-confined RC columns. Stress-strain models for FRP-confined
concrete of different levels of sophistication and for different purposes were next
developed as a prerequisite for the analysis of FRP-confined RC columns.
Subsequently, a simple but accurate stress-strain model for FRP-confined concrete
was incorporated in a conventional section analysis procedure to develop design
equations for short FRP-confined RC columns with a negligible slenderness effect.
Finally, two theoretical models of different levels of sophistication were
developed to deal with the slenderness effect in slender FRP-confined RC
columns. The rigorous theoretical model was used to develop an expression to
classify columns into short columns and slender columns while the simple
theoretical model was used to develop design equations for slender columns. The
proposed design procedure includes a set of design equations for short columns, a
simple expression to separate short columns from slender columns, and a set of
306
design equations for slender columns.
It should be noted that the present study is limited to circular columns. Therefore,
the conclusions given in this chapter is for circular columns while further research
needs for rectangular columns are highlighted towards the end of this chapter.
10.2 BEHAVIOR OF FRP-CONFINED CONCRETE
A large number of FRP-confined concrete cylinders were tested under
standardized conditions and the test results were reported in Chapters 3 and 4.
These tests confirmed the following observations which have also been made by
previous researchers:
1) The confining action of FRP jackets is passive in nature. The lateral dilation
of concrete results in a continuously increasing lateral confining pressure
provided by the FRP jacket which in turn reduces the rate of the lateral dilation.
2) The hoop rupture strains of FRP measured in compression tests on
FRP-confined concrete are smaller than those obtained from material tensile
tests. Previous researchers have attributed this phenomenon to a number of
factors including the non-uniform deformation of concrete, the effect of jacket
curvature, local misalignment or waviness of fibers, as well as residual strains,
multi-axial stress states, and the existence of an overlapping zone in the jacket.
3) With a sufficient amount of confinement, the stress-strain curves of
FRP-confined concrete feature an ascending bi-linear shape with both the
compressive strength and the ultimate axial strain of concrete significantly
enhanced; the stress-strain curves may exhibit a descending branch with little
strength enhancement when the confinement is weak.
4) The ultimate condition (the compressive strength and the ultimate axial strain)
of FRP-confined concrete depends on both the stiffness and the strain
capacity of the confining jacket. The ultimate condition of concrete subjected
to the same confining pressure at the rupture of the confining jacket can be
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considerably different if the jacket stiffness is different.
10.3 MODELING OF FRP-CONFINED CONCRETE
Analysis-oriented stress-strain models and design-oriented models for
FRP-confined concrete were discussed in Chapters 3 and 4 respectively.
Analysis-oriented models generate the stress-strain curves in an
incremental-iterative manner by accounting for the interaction between the FRP
jacket and the concrete core while design oriented models are in closed-form and
are suitable for design use. The following conclusions can be drawn based on a
comprehensive assessment of analysis-oriented models which employs an active
confinement model as the base model:
1) The lateral-to-axial strain relationship, which reflects the unique dilation
properties of FRP-confined concrete, is central to models of this kind. A
successful model should accurately predict this relationship. Nevertheless,
provided the overall trend of this relationship is reasonably well described, the
axial stress-strain curve can be closely predicted, even if local inaccuracies
exist in the lateral-to-axial strain equation.
2) The definitions of the peak axial stress and the corresponding axial strain in the
active-confinement base model are also important to ensure the accuracy of an
analysis-oriented model for FRP-confined concrete.
3) The analysis-oriented model proposed in Chapter 3 represents an
improvement to the best-performance model identified by the assessment,
particularly for weakly-confined concrete.
The original Lam and Teng’s design-oriented stress-strain model was refined in
Chapter 4 on a combined experimental and theoretical basis. A simple stress-strain
equation and an accurate definition of the ultimate condition of FRP-confined
concrete are central to a successful model of this type. The proposed ultimate
condition equations represent an improvement to existing ones in that they relate
the ultimate condition to both the stiffness and the strain capacity of the confining
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jacket rather than solely to the ultimate confining pressure. The model proposed in
Chapter 4 also naturally reduces to a stress-strain model for unconfined concrete
adopted in existing design codes for RC structures.
10.4 ANALYSIS AND DESIGN OF FRP-CONFINED RC COLUMNS
The proposed design procedure for FRP-confined RC columns follows the
conventional procedure for RC columns: a column needs to be classified to be a
short column or a slender column before applying the corresponding set of design
equations. The slenderness effect is ignored in short columns while it must be
accounted for in slender columns.
The design of short columns is simply a matter of constructing the section axial
load-bending moment interaction diagram. Section analysis incorporating the
refined Lam and Teng’s stress-strain model proposed in Chapter 4 was carried out
in Chapter 5. The section analysis served as a basis to develop design equations: the
contribution of the confined concrete to the load capacity of the section was
approximated by transforming the stress profile of concrete into an equivalent
stress block; the contribution of the longitudinal steel reinforcing bars to the load
capacity of the section was approximated by smearing the bars into an equivalent
steel ring. The proposed design equations are in a simple form that is familiar to
civil engineers and their performance was shown to be very good by a
comprehensive parametric study.
A rigorous theoretical model was developed in Chapter 7 to deal with the
slenderness effect in slender FRP-confined RC columns. This model was shown to
provide reasonably close predictions for existing tests. It was also shown that
FRP-confined RC columns are subjected to a more profound slenderness effect than
their RC counterparts because FRP confinement can substantially increase the axial
load capacity of an RC section but affects little the flexural rigidity of the section.
As a result, a short column may become a slender column when it is confined with
FRP. A simple expression was developed in Chapter 8 to define short
FRP-confined RC columns based on a comprehensive parametric study using the
rigorous theoretical model. This expression provides lower-bound predictions and
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can reduce to an expression that is identical or similar to the slenderness limit
expressions for RC columns given in current design codes for RC structures when
no FRP confinement is provided.
To develop design equations for slender FRP-confined RC columns, the rationale
behind the current design approach (the effective length approach) for slender
columns was first explained through analysis of elastic columns with elastic end
restraints in Chapter 6. A simple theoretical model was then developed in Chapter
9 to analyze standard hinged FRP-confined RC columns. This analysis showed
that some practical limits need to be imposed on the use of FRP to ensure an
effective and safe strengthening scheme: 1) the strength enhancement of FRP
should be limited as strong confinement might lead to excessive lateral deflections;
and 2) the slenderness of columns should be limited as the confinement
effectiveness decreases as the column becomes more slender. Under these two
constraints, approximate design equations were developed using the nominal
curvature method. The key elements in the nominal curvature method were
carefully examined to accommodate necessary adjustments needed to reflect the
effect of FRP confinement. The proposed design equations were shown to be in
close agreement with the theoretical results and provide reasonably accurate yet
generally conservative predictions for existing tests.
10.5 FURTHER RESEARCH
This thesis has primarily been concerned with FRP-confined circular RC columns,
but the framework presented in this thesis can be readily extended to
FRP-confined rectangular RC columns when an accurate stress-strain model for
FRP-confined concrete in rectangular sections becomes available. The common
issues for circular columns and rectangular columns as well as some particular
issues for rectangular columns needing further research are highlighted in this
section.
The behavior of FRP-confined concrete in circular columns under concentric
compression has now been well understood, but much less is known about the
behavior of FRP-confined concrete in rectangular columns under concentric
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compression due to the non-uniform nature of the confinement in such columns.
On the experimental side, more advanced measuring techniques are desirable to
capture the non-uniform stress distribution in such columns to gain a thorough
understanding of the confining mechanism. On the theoretical side, finite element
analysis incorporating a sophisticated constitutive model for concrete represents a
powerful tool to simulate the complex stress state in such concrete.
The theoretical work presented in this thesis has been based on the assumption
that stress-strain models derived from concentric compression tests are directly
applicable in the analysis of columns subject to combined bending and axial
compression. It was noted in Chapter 2 that this assumption had been adopted by
the majority of researchers in the analysis of circular columns. This assumption
was further examined in Chapter 5 using existing test results. Based on the limited
test results, it was deemed reasonable to adopt this assumption for research on
slender column behavior with an emphasis on the load-carrying capacity rather
than the ultimate deformation capacity. Chapter 9 further showed that the design
equations derived on the basis of this assumption are reasonably accurate yet
generally conservative, as demonstrated by existing experimental evidence.
However, more research is needed to clarify the possible effect of load
eccentricity on the stress-strain behavior of FRP-confined concrete in RC columns.
It should be noted that even if further research indicates that the effect of load
eccentricity should be accounted for when defining the ultimate condition (the
compressive strength and the ultimate axial strain) of FRP-confined concrete, the
proposed design equations in the present study will still be valid except that the
ultimate condition of the stress-strain curve needs to be refined. This is because in
the modified Lam and Teng stress-strain model, the effects of the stiffness and the
strain capacity of the confining jacket on the ultimate condition of FRP-confined
concrete in circular columns are already separated from each other. As a result,
when the ultimate condition of FRP-confined concrete is affected by the existence
of eccentricity, an equivalent confinement stiffness ratio and an equivalent strain
ratio which are a function of the eccentricity can be defined. These two equivalent
ratios can then be used in the proposed design procedure without any modification
of the design procedure. On the other hand, the effect of load eccentricity on the
stress-strain behavior of FRP-confined concrete in rectangular columns is much
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vaguer due to: 1) the stress-strain behavior of FRP-confined concrete in
rectangular columns under concentric compression is not clear yet; 2) the
stress-strain behavior of FRP-confined concrete in rectangular columns under
eccentric compression may be different when bent about the major axis and the
minor axis; and 3) only very limited relevant tests have been conducted.
Another important issue that needs to be clarified is the possible effect of size on
the stress-strain behavior of FRP-confined concrete. It was noted in Chapter 2 that
the behavior of large-scale circular columns can be reasonably extrapolated from
the behavior of small-scale circular columns. However, the size effect is much
more uncertain in rectangular columns due mainly to the very limited test data and
the large scatter of test data. With the possible effects of size and eccentricity on
the behavior of FRP-confined concrete in rectangular columns clarified, a reliable
stress-strain model can be developed and incorporated in the theoretical models
presented in this thesis for the analysis of slender columns to develop
corresponding design equations.
Finally, more tests on both circular and rectangular columns with a broad range of
slenderness are needed to verify the design equations proposed in this thesis for
circular columns and the design equations to be developed for rectangular
columns following the framework proposed in this thesis.
In summary, the effects of size and eccentricity on the behavior of FRP-confined
concrete and the effect of slenderness on the behavior of FRP-confined RC
columns are the main issues that need much further research.
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