Concrete contribution to the shear resistance of FRP reinforced concrete beams

M

l

UNIVERSITE DE

m SHERBROOKE
Faculte de genie
Departement de genie civil

CONCRETE CONTRIBUTION TO THE SHEAR RESISTANCE OF
FRP-REINFORCED CONCRETE BEAMS

A Dissertation Submitted in Partial Fulfilment o f the Requirements for the Degree o f
Doctor o f Philosophy
Specialty: Civil Engineering

Ahmed Kamal El-Sayed Ahmed

Sherbrooke (Quebec), Canada

July 2006

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A bstract

ABSTRACT

Corrosion o f steel reinforcement in concrete structures causes deterioration o f concrete,

resulting in costly maintenance and repair. Many steel-reinforced concrete structures
exposed to deicing salts and marine environments require extensive and expensive
maintenance. Recently, the use o f fibre-reinforced polymers (FRP) as an alternative
reinforcing material in reinforced concrete structures has emerged as an innovative
solution to the corrosion problem. However, due to the difference in mechanical
properties between steel and FRP reinforcements, the shear strength o f concrete members
reinforced with FRP longitudinal reinforcement may differ from that o f members
reinforced with steel.
An experimental program including two phases is described. The experimental
program was conducted at the University o f Sherbrooke to investigate the effect o f using
FRP bars as longitudinal reinforcement on the shear strength and behaviour o f concrete
beams without web reinforcement. The first phase included 15 large-scale concrete
slender beams reinforced with glass FRP, carbon FRP, or conventional steel bars. Nine
beams were constructed using normal-strength concrete, whereas six beams were
constructed using high-strength concrete. The test variables were the reinforcement ratio
and the modulus o f elasticity o f the reinforcing bars as well as the concrete compressive
strength. The second experimental phase included 12 large-scale concrete deep beams
reinforced with glass FRP, carbon FRP, or conventional steel bars. The test beams o f this
phase were constructed using normal-strength concrete and the test parameters were the
reinforcement ratio and the modulus o f elasticity o f the reinforcing bars as well as the
shear span-to-depth ratio. The influence o f the considered variables on the shear strength
and behaviour o f the tested beams in the two phases is presented.
An analytical investigation to examine the validity o f the available design
provisions o f concrete contribution to shear strength for members longitudinally
reinforced with FRP bars is reported. For this purpose, the shear strengths o f the tested
beams are analyzed using the shear design provisions o f the different available codes,
manuals, and design guidelines. The results o f the analysis are compared with the
experimental values. Based on the findings o f this investigation, a proposed shear design
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A bstract

equation is presented. The proposed equation is verified against experimental shear
strengths o f 107 specimens tested to date, including the specimens in this investigation. In
addition, the proposed equation is compared to the major design provisions using the
available test data to further evaluate its reliability.
During the course o f the research work, the following related papers have been
published or submitted for publication:

Journal Papers
1. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2006a), “Shear Strength of
FRP-Reinforced Concrete Deep Beams without Web Reinforcement,” Submitted to
ACI Structural Journal.
2. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2006d), “Shear Capacity of
High-Strength Concrete Beams Reinforced with FRP Bars,” ACI Structural Journal,
Vol. 103, No. 3, pp. 383-389.
3. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2006e), “Shear Strength of
FRP-Reinforced Concrete Beams without Transverse Reinforcement,” ACI Structural
Journal, Vol. 103, No. 2, pp. 235-243.

Conference Papers
4. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2005a), “Shear Strength of
Concrete Beams Reinforced with FRP Bars: Design Method,” Proceedings o f the 7th
International Symposium on Fiber Reinforced Polymer Reinforcement for Concrete
Structures (FRPRCS-7), ACI-SP-230, Kansas City, MO., USA, Nov. 5-9, pp. 955-974.
5. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2005b), “Analytical
Modeling o f FRP-Reinforced Concrete Beams Failed in Shear,” Proceeding on CD, 1st

CSCE Specialty Conference on Infrastructure Technologies, Management and Policy,
Toronto, Ontario, Canada, June 2-4, FR-127, lOp.
6. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2005c), “Shear Design o f
Concrete Beams Reinforced with FRP Bars,” Proceeding on CD, 4th Middle East
Symposium on Structural Composites for Infrastructure Applications (MESC-4),
Alexandria, Egypt, May 20-23, 12p.

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A bstract

7. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2004a), “Concrete
Contribution to the Shear Resistance o f High-Strength Concrete Beams Reinforced
with FRP Bars,” Proceeding on CD, International Conference: Future Vision and
Challenges for Urban Development, Cairo, Egypt, December 20-22, 12p.
8. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2004c), “Evaluation o f
Concrete Shear Strength for Beams Reinforced with FRP Bars” Proceeding on CD, 5th
Structural Specialty Conference o f the CSCE, Saskatoon, Saskatchewan, Canada, June
2-5, ST-224, lOp.

Also the candidate has participated in the following publications during his
doctorate study at the Universite de Sherbrooke:

Journal Papers
9. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2006b), “Mechanical and
Structural Characterization o f New Carbon FRP Stirrups for Concrete Members,”
Accepted for publication in the Journal o f Composites for Construction, ASCE.

10. El-Sayed, A.K., El-Salakawy, E.F., and Benmokrane, B., (2005d), “Shear Strength of
One-way Concrete Slabs Reinforced with FRP Composite Bars,” Journal of
Composites for Construction, ASCE, Vol. 9, No. 2, pp. 147-157.

Conference Papers
11. Ahmed, E. A., El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2006),
“Shear Behaviour o f Concrete Bridge Girders Reinforced with Carbon FRP Stirrups,”
Submitted to the 7th International Conference on Short and Medium Span Bridges,
Montreal, Canada, Aug. 23-25, 10 p.
12. El-Sayed, A. K., El-Salakawy, E. F., and Benmokrane, B., (2006c), “Structural
Behaviour o f Carbon FRP Stirrups Used as Shear Reinforcement for Concrete
Beams.,” Proceedings on CD, 1st International Structural Specialty Conference o f the
Canadian Society for Civil Engineering, CSCE, Calgary, Alberta, Canada, May 2326, 10 p.

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A bstract

13. El-Sayed, A.K., El-Salakawy, E.F., and Benmokrane, B. (2004b), “New Carbon FRP
Stirrups as Shear Reinforcement for Concrete Beams,” Advanced Composites
Materials in Bridges and Structures (IV-ACMBS), Proceedings on CD, Calgary,
Alberta, Canada, July 20-23, 8 p.

Technical Reports
14. Benmokrane, B., El-Sayed, A. K., and El-Salakawy, E. F., (2005e), “Conception de
Poutres de Pont en Beton on Precontraint Renforcees avec des Etriers en Materiaux
Composites,” Technical Report-Phase 2, Submitted to the Ministry o f Transportation

o f Quebec, Quebec, Canada, March, 18p.
15. Benmokrane, B., El-Sayed, A. K., El-Salakawy, E. F., and Massicotte, B., (2004d),
“Conception de Poutres de Pont en Beton on Precontraint Renforcees avec des Etriers
en Materiaux Composites,” Technical Report-Phase 1, Submitted to the Ministry o f
Transportation o f Quebec, Quebec, Canada, January, 21 p.

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Resume

RESUME

La corrosion de l’acier d’armature des structures de beton provoque la deterioration du
beton, ce qui entraine des couts de reparation et d ’entretien importants. De nombreuses
structures de beton arme d ’acier exposees aux sels de degla 9 age o u a u n environnement
marin necessitent des travaux de refection generalises et couteux. Recemment, l’emploi
de polymeres renforces de fibres (PRF) comme materiau de renfort altem atif pour les
structures de beton est apparu comme etant une solution innovatrice aux problemes de
corrosion. Cependant, du fait des differences de proprietes mecaniques entre les
armatures d ’acier et de PRF, la resistance au cisaillement des membrures de beton arme
d’armature de traction en PRF peut differer de celle observee avec l’acier d ’armature.
Un programme experimental incluant deux phases est decrit. Le programme
experimental a ete mene a l’Universite de Sherbrooke afin d ’evaluer l’effet de
l’utilisation des barres de PRF comme armature longitudinale de traction sur la resistance
au cisaillement (ou a 1’effort tranchant) et le comportement des poutres de beton sans
armature d ’ame (sans armature transversale). La premiere etape a porte sur 15 poutres
elancees de beton a grande echelle renforcees avec des barres d ’armature en PRF de

verre, en PRF de carbone et de Lacier d ’armature conventionnel. N euf poutres ont ete
fabriquees en utilisant du beton normal tandis que les six autres l’ont ete avec du beton a
haute resistance. Les variables d’essai etaient le taux d’armature et le module d ’elasticite
des barres d ’armature de meme que la resistance a la compression du beton. La seconde
phase experim ental portait sur 12 poutres profondes de beton a grande echelle
renforcees avec des barres d ’armature en PRF de verre, en PRF de carbone et de l’acier
d ’armature conventionnel. Les poutres d ’essai etudiees lors de cette phase etaient
fabriquees a partir de beton normal et les parametres d’essai etaient le taux d ’armature et
le module d ’elasticite des barres d’armature de meme que le rapport portee de la zone de
cisaillem ent sur hauteur de la poutre. L ’influence des variables considerees sur la

resistance au cisaillement et le comportement des poutres d ’essai des deux phases est
presentee.
Une etude analytique portant sur la validite des equations de calcul disponibles sur la
contribution du beton a la resistance au cisaillement pour les membrures en beton

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Resume

renforces longitudinalement avec des barres de PRF est aussi rapportee. Dans cet objectif,
les resistances au cisaillement des poutres testees ont ete analysees a l’aide des equations
de calcul concemant le cisaillement a partir des divers codes, manuels et guides de calcul
disponibles. Les resultats de cette analyse sont compares aux valeurs experimentales. Une
equation de calcul pour le cisaillement est proposee et verifiee a partir des valeurs
experimentales de resistance au cisaillement des 107 poutres etudiees a date, incluant les
poutres de la presente etude.

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A c know Iedgem ents

A CKN O W LED GEM EN TS

I would like to express my profound gratitude to my advisors Professor Brahim
Benmokrane and Professor Ehab El-Salakawy for their support, encouragement,
guidance, and valuable advice throughout the research program.

I would like to thank the structural laboratory technical staff in the Department o f Civil
Engineering at the Universite de Sherbrooke, in particular Mr. Francois Ntacorigira and
Mr. Simon Sindayiagaya for their help in my experimental work.

This research program has been carried out through the NSERC Chair o f Professor
Benmokrane on FRP composite reinforcement for concrete structures at the Universite de
Sherbrooke. Thus, the financial support received from the Natural Sciences and
Engineering Research Council of Canada (NSERC), Pultrall Inc. (Thetford Mines,
Quebec), the Ministry o f Transportation o f Quebec, the Network o f Centres o f Excellence
ISIS-Canada, and the Universite de Sherbrooke is greatly acknowledged.

I would like to express my deep appreciation and thanks to my parents, my brother, and
my sisters for their endless love, support, and encouragement. Finally, my words stand
helpless and cannot express my appreciation to my wife and my twin sons for their
patience and support; to them this thesis is dedicated.

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Table o f Contents

TABLE OF CONTENTS

ABSTRACT

ii

ACKNOWLEDGEMENTS

viii

TABLE OF CONTENTS

ix

LIST OF FIGURES

xv

LIST OF TABLES

xx

NOTATION

xxii

1.

INTRODUCTION

1

1.1 General

1

1.2 Objectives and Originality

3

1.3 Methodology

4

1.4 Structure o f the Thesis

5

2.

BACKGROUND AND REVIEW ON THE SHEAR
BEHAVIOUR OF CONCRETE BEAMS

7

2.1 General

7

2.2 Shear in Reinforced Concrete Beams without Transverse Reinforcement

8

2.2.1 Mechanism o f shear transfer

8

2.2.1.1 Shear stresses in uncracked concrete

8

2.2.1.2 Interface shear transfer

9

2.2.1.3 Dowel action

10

2.2.1.4 Arch action

10

2.2.1.5 Residual tensile stresses across crack

10

2.2.2 Modes of inclined cracking and shear failure

11

2.2.3 Factors affecting shear capacity

15

2.2.3.1 Tensile strength o f concrete

15

2.23.2 Longitudinal reinforcement ratio

15

2.2.3.3 Shear span-to-depth ratio

16

2.2.3.4 Axial force

16

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Table o f Contents

2.2.3.5 Depth o f member

16

2.3 Shear in Reinforced Concrete Beams with Transverse Reinforcement

17

2.3.1 Internal forces in a beam with transverse reinforcement

17

2.3.2 Role o f shear reinforcement in concrete beams

19

2.3.3 Modes o f shear failure

20

2.4 Methods o f Analysis o f the Shear Strength o f Beams

20

2.4.1 Historical background

20

2.4.2 Elastic analysis

22

2.4.3 Equilibrium methods

24

2.4.3.1 The 45° truss model

24

2.4.3.2 Variable angle truss model

26

2.4.3.3 Modified truss model

28

2.4.3.4 Strut and tie model

29

2.4.4 Compression field approaches

37

2.4.4.1 Compression field theory

37

2.4.4.2 Modified compression field theory

39

2.4.4.3 Rotating-angle softened-truss model

45

2.4.4.4 Fixed-angle softened-truss model

48

2.5 Shear Design Procedure in North America

50

2.5.1 Canadian Standard Association, CSA-A23.3-04 code

50

2.5.2 American Concrete Institute, ACI 318-05 code

53

2.5.3 AASHTO LRFD Bridge Design Specifications (2004)

56

2.5.4 Canadian Highway Bridge Design Code (CHBDC), CSA-S6-00

59

3.

BACKGROUND AND REVIEW ON THE SHEAR
BEHAVIOUR OF CONCRETE BEAMS REINFORCED
WITH FRP BARS

61

3.1 General

61

3.2 FRP Composite Materials

62

3.2.1 Reinforcing fibres

62

3.2.1.1 Carbon fibres

62

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Table o f Contents

3.2.1.2 Glass fibres

63

3.2.1.3 Aramid fibres

64

3.2.2 Resins

64

3.2.3 Manufacturing process

65

3.3 General Characteristics o f FRP Reinforcing Bars

66

3.4 Shear Behaviour o f Flexural Members Reinforced with FRP Bars as
Longitudinal Reinforcement

68

3.5 FRP Bars as Shear Reinforcement for Concrete Members

78

3.5.1 Bend radius and tail length o f FRP stirrups

80

3.5.2 Shear behaviour o f concrete beams reinforced with FRP stirrups

91

3.6 Shear Design Provisions for FRP-Reinforced Concrete Members
3.6.1 Japan design guidelines

106
106

3.6.1.1 JSCE design recommendations

106

3.6.1.2 Design recommendations by Building Research Institute

111

3.6.2 Canadian design codes

113

3.6.2.1 Canadian Highway Bridge Design Code (CHBDC), CSAS6-00

114

3.6.2.2 CSA S806-02 shear design provisions

115

3.6.3 ISIS-M03-01 design manual

117

3.6.4 Eurocrete project

118

3.6.5 ACI 440.1R-03 guidelines

118

4.

EXPERIMENTAL PROGRAM

121

4.1 General

121

4.2 Material Properties

122

4.2.1 FRP bars

122

4.2.2 Steel bars

124

4.2.3 Concrete

124

4.3 Experimental Phase I: Concrete SlenderBeams

125

4.3.1 Test specimens

126

4.3.2 Fabrication o f the test beams

127

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Table o f Contents

4.3.3 Instrumentation

13 0

4.3.4 Test setup and procedure

131

4.4 Experimental Phase II: Concrete Deep Beams

133

4.4.1 Test specimens

134

4.4.2 Fabrication of the test beams

136

4.4.3 Instrumentation

138

4.4.4 Test setup and procedure

140

5.

TEST RESULTS AND ANALYSIS

143

5.1 General

143

5.2 Test Results o f Phase I-Slender Beams

144

5.2.1 General behaviour

144

5.2.1.1 Cracking load

145

5.2.1.2 Load-deflection response

147

5.2.1.3 Crack patterns and modes o f failure

151

5.2.1.4 Crack widths

153

5.2.1.5 Load-strain relationship

155

5.2.2 Shear behaviour

158

5.2.2.1 Inclined cracking shear strength

161

5.2.2.2 Ultimate shear strength

161

5.2.2.2.1 Effect o f reinforcement ratio and modulus o f elasticity o f
longitudinal reinforcing bars

162

5.2.2.2.2 Effect o f concrete compressive strength

164

5.3 Code Predictions for Phase I-Test Results

165

5.3.1 Shear strength predictions

165

5.3.2 Deflection predictions

170

5.3.2.1 C A N /C S A -S806-02 code

170

5.3.2.2 ACI 440.1R-03 guide

173

5.3.2.3 CAN/CSA-A23.3-04 and ACI 318-05 codes

177

5.3.3 Crack width predictions

5.3.3.1 CAN/CSA-S806-02 code

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178
178


Table o f Contents

5 3 3 2 ACI 440.1R-03 guide

179

5.3.3.3 CAN/CSA-A23.3-04 code

180

5.3.3.4 ACI 318-05 code

181

5.4 Test Results o f Phase II-Deep Beams

181

5.4.1 General behaviour

182

5.4.1.1 Cracking load

182

5.4.1.2 Load-deflection response

183

5.4.1.3 Crack patterns and modes o f failure

187

5.4.1.4 Crack widths

189

5.4.1.5 Strains in reinforcement and concrete

191

5.4.2 Shear behaviour

195

5.4.2.1 Inclined cracking shear strength

197

5.4.2.2 Ultimate shear strength

197

5.4.2.2.1 Effect o f reinforcement ratio and modulus o f elasticity of
longitudinal reinforcing bars
5.4.2.2.2 Effect o f shear span-to-depth ratio
5.5 Code Predictions for Phase II-Test Results

197
200
202

5.5.1 ACI 318-99 code (1999)

203

5.5.2 ACI 318-05 code (2005)

204

5.6 Comparison between the Shear Behaviour of Slender and Deep Reinforced
Concrete beams

207

5.6.1 Cracking and ultimate shear strengths

207

5.6.2 Arch action factor

209

5.6.3 Effect o f test variables

210

6.

PROPOSED SHEAR DESIGN METHOD

212

6.1 General

212

6.2 Proposed Shear Design Equation

213

6 .3 V erification o f the Proposed Equation

215

6.4 Comparison with Major Design Provisions

216

6.5 Design Considerations

218

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Table o f Contents

7.

SUM M ARY AND CONCLUSIONS

234

7.1 Summary

234

7.2 Conclusions

236

7.2.1 Behaviour o f concrete slender beams

236

7.2.2 Behaviour o f concrete deep beams

237

7.2.3 Code Predictions for slender beam specimens

238

7.2.4 Code Predictions for deep beam specimens

240

7.2.5 Proposed shear design method

240

7.3 Recommendations for Future Work

241

R EFER EN C ES

242

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L ist o f Figures

LIST OF FIGURES

Figure 2.1

Shear transfer mechanism in a cracked reinforced concrete beam
without transverse reinforcement.

9

Figure 2.2

Types o f inclined cracks.

12

Figure 2.3

Typical diagonal tension failure in slender beams (a/d > 2.5).

13

Figure 2.4

Typical shear failures in short beams (a/d - 1.0 to 2.5).

14

Figure 2.5

Modes o f failures o f deep beams (a/d < 1.0) (adapted from
ASCE-ACI 1973).

Figure 2.6

14

Internal forces in a cracked beam with stirrups (adapted from
ASCE-ACI 1973).

Figure 2.7

18

Stress distribution and trajectories o f principal stresses in a
homogeneous rectangular beam.

23

Figure 2.8

Ritter’s truss model.

25

Figure 2.9

Equilibrium considerations for 45° truss (adapted from Collins

25

and Mitchell 1997).
Figure 2.10

Equilibrium considerations for variable-angle truss (adapted from
Collins and Mitchell 1997).

27

Figure 2.11

Examples o f B and D regions (adapted from Ali and White 2001). 30

Figure 2.12

Arch action in a beam (adapted from MacGregor 1997).

Figure 2.13

Examples o f D regions modeled with compressive struts and
tension ties (adapted from ASCE-ACI 1998).

Figure 2.14

32

Strut and tie model for a deep beam (adapted from Collins and
Mitchell 1997).

Figure 2.15

31

32

Predicted and observed strengths o f a series o f reinforced
concrete beams tested by Kani (adapted from Collins and
M itchell 1997).

36

Figure 2.16

Compatibility conditions for cracked web element.

38

Figure 2.17

Stress-strain relationships.

39

Figure 2.18

Equilibrium conditions o f modified compression field theory
(adapted from Collins and Mitchell 1997).

xv

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41

L ist o f Figures

Figure 2.19

Stress-strain relationship for cracked concrete (adapted from
Collins and Mitchell 1997).

Figure 2.20

42

Force transmission across cracks (adapted from Collins and
Mitchell 1997).

Figure 2.21

43

Spacing o f inclined cracks (adapted from Collins and Mitchell
1997).

Figure 2.22

44

Reinforced concrete membrane elements subjected to in-plane
stresses (adapted from Pang and Hsu 1996).

46

Figure 3.1

Pultrusion process o f GFRP rebars.

66

Figure 3.2

Typical tensile stress-strain relationships for FRP and steel rebars. 67

Figure 3.3

Relationship between reinforcement ratio and experimental shear
strength (Alkhrdaji et al. 2001).

Figure 3.4

Relationship

between

normalized

73
reinforcement

ratio

and

normalized shear strength (Alkhrdaji et al. 2001).
Figure 3.5

Comparison of results for slabs reinforced with carbon FRP bars
(El-Sayed et al. 2005).

Figure 3.6

79

Comparison o f results for slabs reinforced with No. 16 glass FRP
bars (El-Sayed et al. 2005).

Figure 3.7

73

79

Comparison o f results for slabs reinforced with No. 22 glass FRP
bars (El-Sayed et al. 2005).

80

Figure 3.8

Test setup by Maruyama et al. (1993).

82

Figure 3.9

Relationship between tensile strength and bend radius (Maruyama
etal. 1993)

82

Figure 3.10

General view o f the hooked bar specimens (Ehsani et al. 1993).

83

Figure 3.11

Load versus slip for three No. 6 hooked bars (Ehsani et al. 1993).

83

Figure 3.12

Failure loads o f the hooked bars (Ehsani et al. 1993).

84

Figure 3.13

D etails o f bend specim ens (Shehata et al. 1999).

86

Figure 3.14

Effect o f bend radius, rh, on strength capacity o f the bend,fbend,
(Shehata et al. 1999).

Figure 3.15

88

Stirrups specimens and arrangement o f strain gauges (El-Sayed et
al. 2004).

90

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L ist o f Figures

Figure 3.16

Test setup for bend testing (El-Sayed et al. 2004).

90

Figure 3.17

Fibre-rupture failure mode at the bend (El-Sayed et al. 2004).

92

Figure 3.18

Effect o f bend radius on bend capacity (El-Sayed et al. 2004).

92

Figure 3.19

Details o f specimens by Zhao et al. (1995).

95

Figure 3.20

Stirrup strain distribution proposed by Zhao et al. (1995).

96

Figure 3.21

Effect o f stirrup spacing on effective capacity o f FRP stirrups
(Shehata et al. 1999).

Figure 3.22

Effect o f flexural reinforcement on shear resisting components
(Shehata et al. 1999).

Figure 3.23

103

104

Applied shear versus crack width for beams reinforced with
stirrups spaced at d/2 (Shehata et al. 1999).

105

Figure 4.1

Glass and carbon FRP sand-coated reinforcing bars.

123

Figure 4.2

Typical FRP tension specimens and mode of failure.

123

Figure 4.3

Typical stress-strain relationships o f the reinforcing bars.

125

Figure 4.4

Details o f Phase I-test beams.

129

Figure 4.5

Instrumentation layout o f Phase I-test beams.

130

Figure 4.6

LVDTs used for measuring deflection and crack widths.

131

Figure 4.7

Schematic drawing o f the test setup o f Phase I-test beams.

132

Figure 4.8

A photograph o f the test setup o f Phase I-test beams.

133

Figure 4.9

Details o f Phase II-test beams.

137

Figure 4.10

Instrumentation layout of: (a) all beams o f Phase II except beams
o f Series C; and (b) beams o f Series C.

139

Figure 4.11

Schematic drawing o f the test setup o f Phase II-test beams.

141

Figure 4.12

A photograph o f the test setup o f Phase II-test beams.

142

Figure 5.1

Load-deflection relationships for NSC beams.

149

Figure 5.2

Load-deflection relationships for HSC beams.

149

Figure 5.3

Typical load-deflection relationship (Series 3).

150

Figure 5.4

Load-deflection relationships for beams o f Series 3 and 4.

150

Figure 5.5

Crack patterns for NSC beams.

152

Figure 5.6

Crack patterns for HSC beams.

153

Figure 5.7

Diagonal tension failure mode: (a) associated with no concrete

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L ist o f Figures

splitting; and (b) associated with concrete splitting.

154

Figure 5.8

Load-versus crack widths for NSC beams.

156

Figure 5.9

Load-versus crack widths for HSC beams.

156

Figure 5.10

Typical load-crack widths relationship (Series 3).

157

Figure 5.11

Load- versus crack widths for beams o f Series 3 and 4.

157

Figure 5.12

Load-strain relationships for NSC beams.

159

Figure 5.13

Load- strain relationships for HSC beams.

159

Figure 5.14

Typical load-strain relationship (Series 1).

160

Figure 5.15

Load-strain relationships for beams o f Series 3 and 4.

160

Figure 5.16

Normalized shear strength versus reinforcement ratio for NSC
beams.

Figure 5.17

163

Experimental shear strength versus reinforcement ratio for HSC
beams.

Figure 5.18

163

(a) Experimental shear strength versus concrete compressive
strength; and (b) Normalized shear strength versus concrete
compressive strength.

Figure 5.19

166

Comparison of experimental and predicted deflections.

172

(cont.)

Comparison o f experimental and predicted deflections.

173

Figure 5.20

Load-deflection relationships for beams having a/d - 1.69.

185

Figure 5.21

Load-deflection relationships for beams having p = 1.24%.

185

Figure 5.22

Crack patterns.

188

(cont.)

Crack patterns.

189

Figure 5.23

Modes o f failure.

190

(cont.)

M odes o f failure.

191

Figure 5.24

Load versus crack widths for beams having a/d = 1.69.

192

Figure 5.25

Load versus crack widths for beams having p = 1.24%.

192

Figure 5.26

Load-strains relationships for beams having a/d = 1.69.

193

Figure 5.27

Load-strains relationships for beams having p = 1.24%.

193

Figure 5.19

Figure 5.22

Figure 5.23

xviii

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L ist o f Figures

Figure 5.28

Strain distribution along the bottom reinforcement layer for
beams o f Series C.

Figure 5.29

Experimental shear strength versus reinforcement ratio for beams

having a/d = 1.69.

Figure 5.30

196

198

Experimental shear strength versus shear span-to-depth ratio for
beams having p = 1.24%.

201

Figure 5.31

Strut and tie model for beams o f Phase II failed in shear.

206

Figure 5.32

Effect o f shear span-to-depth ratio on cracking and ultimate shear
strength.

Figure 6.1

208

Experimental-to-predicted shear strength o f slender beams versus
axial stiffness o f reinforcing bars: (a) ACI 440.1R-03; and

(b) proposed equation.

Figure 6.2

229

Experimental-to-predicted shear strength o f slender beams versus
concrete compressive strength: (a) ACI 440.1R-03; and
(b) proposed equation.

Figure 6.3

230

Experimental-to-predicted shear strength o f slender beams versus
shear span-to-depth ratio: (a) ACI 440.1R-03; and (b) proposed
equation.

Figure 6.4

Experimental-to-predicted shear strength o f slender beams versus
effective depth: (a) ACI 440.1R-03; and (b) proposed equation.

Figure 6.5

231

232

Experimental-to-predicted shear strength o f deep beams versus

axial stiffness of reinforcing bars: (a) ACI 440.1R-03; and
(b) proposed equation.

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233


L ist o f Tables

LIST OF TABLES

Table 2.1

Effective stress levels in nodal zones

Table 2.2

Values o f /? and 8 for sections with transverse reinforcement
(AASHTO LRFD 2004)

Table 2.3

Values o f /? and 9

35

57

for sections with less than minimum

transverse reinforcement (AASHTO LRFD 2004)

58

Table 3.1

Details o f test results o f bend specimens (Shehata et al. 1999)

87

Table 3.2

Test results o f carbon FRP stirrups (El-Sayed et al. 2004)

91

Table 4.1

Properties o f reinforcing bars

124

Table 4.2

Concrete mix proportions

126

Table 4.3

Details o f Phase I-test beams

128

Table 4.4

Details o f Phase II-test beams

136

Table 5.1

Comparison o f theoretical and experimental failure loads for
Phase I-test beams

145

Table 5.2

Test results o f Phase I-test beams

146

Table 5.3

Comparison of predicted and experimental shear capacities for
the FRP-reinforced beams

Table 5.4

Comparison o f predicted and experimental shear capacities for
the steel-reinforced beams

Table 5.5

169

Experimental and predicted service load deflections and crack
widths for the beams reinforced with FRP bars

Table 5.6

168

175

Experimental and predicted service load deflections and crack
widths for the beams reinforced with steel bars
176

Table 5.7

Comparison o f theoretical and experimental failure loads for
Phase II-test beams

183

Table 5.8

Test results o f Phase II-test beams

184

Table 5.9

Comparison o f predicted and experimental ultimate shear
strengths o f Phase II-test beams

Table 6.1

Verification o f the proposed equation and comparison with

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205


___________________________________________________________________ L ist o f Tables

major design provisions (for slender beams)
Table 6.2

219

Verification o f the proposed equation and comparison with
major design provisions (for deep beams)

xxi

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228


N otation

NOTATION

a

— shear span

a /d

-

shear span-to-depth ratio

■tics

=

cross-sectional area o f strut

=

nominal cross-sectional area o f FRP bars

=

total nominal cross-sectional area o f FRP stirrup within distance s

Af
A ft
Ar

nominal cross-sectional area o f reinforcement

As

nominal cross-sectional area o f steel bars,

A si

-

nominal cross-sectional area o f longitudinal steel bars,

Av

=

total nominal cross-sectional area o f stirrup within distance 5

A v min

=

minimum amount o f shear reinforcement within distance s

bw

=

web width o f the beam

c

=

cracked transformed section neutral axis depth

D

=

resultant of diagonal compression stress

d

=

effective depth o f tensile reinforcement

da

=

maximum aggregate size

db

=

diameter o f FRP stirrup

de

=

effective bar diameter

Ec

=

modulus o f elasticity o f concrete

Ef

=

modulus o f elasticity o f FRP bars

Efl

=

modulus o f elasticity o f longitudinal FRP bars

Efv

=

modulus o f elasticity o f FRP stirrup

Er

=

modulus o f elasticity o f reinforcement

Es

=

modulus o f elasticity o f steel

Esi

=

modulus o f elasticity o f longitudinal steel

F ns

=

nom inal com pressive strength o f strut

Fnt

=

nominal tensile strength of tie

F nn

=

nominal compressive strength o f nodal zone

f

normal stress

ft

principal tensile stress

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N otation

f2

= principal compressive stress

f 2 max

= effective compressive strength o f diagonally cracked concrete

fben d

= tensile strength o f FRP stirrup at the bend

f c

= compressive strength o f concrete

f cd

= design compressive strength o f concrete

f cr

- concrete cracking strength

fjv

= FRP stirrup stress at failure

fjv d

- design tensile strength o f FRP stirrup

ffu

= ultimate tensile strength o f FRP longitudinal reinforcing bars

f/u v

= ultimate longitudinal tensile strength o f FRP stirrup

fv

= stirrup stress

fy

- yield strength of reinforcing steel bars

h

- total depth o f member

I

- moment of inertia o f cross section

ICr

= moment o f inertia o f cracked section

Ie

= effective moment o f inertia o f cross section

Ig

= moment o f inertia o f gross section

jd

= shear depth, defined as the distance between the compressive force and
the tensile force acting on the cross-section

K

= factor representing the beneficial effect o f the prestress force on
concrete diagonal tensile strength

L

- span o f the beam

lt

- tail length

M

= bending moment

Ma

= applied moment

Mcr

= cracking moment

Ma

= design bending moment

Mf

= factored moment at section o f interest

M0

= decompression moment

N d

= design axial compressive force

xxiii

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N otation

Nu

= required force in tension tie

Nv

= tensile force in longitudinal reinforcement due to shear

n

= modular ratio (Er/Ec)

P

= applied load

Q

= statical moment o f cross-sectional area above or below a level about
neutral axis

rb

= bend radius o f FRP stirrup

s

= spacing between FRP stirrups

sz

= crack-spacing parameter for members without shear reinforcement

Smi, smv, sme

= crack spacing in longitudinal, transverse, and inclined directions

V

= shear force

Vay

= shear component resisted by the aggregate interlock along the shear
crack

Vc

= shear-resisting force provided by concrete

Vcf

= shear-resisting force provided by concrete in beams reinforced with
FRP longitudinal reinforcement

Vcr

= inclined cracking shear strength

Vcrexp

= experimental inclined cracking shear strength

Vcz

= shear component resisted by the compression zone

Vd

= shear component resisted by the dowel action

Vf

= factored shear force at section o f interest

Vn

= nominal shear strength

Vs

= shear-resisting force provided by stirrups
=

shear-resisting force provided by FRP stirrups

Vu

= ultimate shear strength

Vuexp

= experimental ultimate shear strength

Eupred

= predicted ultimate shear strength

v

= shear stress, V/bwd

vu

= shear stress at ultimate, VJbwd

w

= crack width

wexp

- experimental crack width

xxiv

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