method of the ACI Code 318-83, under a point load to study the flexural behavior of ...... point under consideration can be ignored (ANSYS Manual,2009). 5.2.1.7 ...
Republic of Iraq Ministry of Higher Education and Scientific Research Babylon University College of Engineering Civil Engineering Department
Behavior of R.C. Horizontally Curved Beams with Openings Strengthened by CFRP Laminates A Dissertation Submitted to the College of Engineering in the University of Babylon in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Civil Engineering (Structural)
By
Sadjad Amir Hemzah (B.Sc. Civil Engineering, 2000) (M.Sc. Structural Engineering, 2003)
Supervised By
Prof. Dr. Ammar Yasir Ali 2014
ﺑﺳم ﷲ اﻟرﺣﻣن اﻟرﺣﯾم
ﻧَﺮﻓَﻊ ﺩﺭﺟﺎﺕ ﻣﻦ ﻧﱠﺸﺎﺀ
ﻭﻓَﻮﻕَ ﻛُـﻞﱢ ﺫﻱ ﻋﻠﻢٍ ﻋﻠﻴﻢ ﺻدق ﷲ اﻟﻌﻠﻲ اﻟﻌظﯾم
ﺳﻮﺭﺓ ﻳﻮﺳﻒ /آﻳﺔ 76
Certificate I certify that the preparation of this dissertation titled " Behavior of R.C. Horizontally Curved Beams with Openings Strengthened by CFRP Laminates ", is prepared by " Sadjad Amir Hemzah ", under my supervision at the Department of Civil Engineering in the University of Babylon in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering (Structure).
Signature: Name: Prof. Dr. Ammar Yaser Ali Date: / / 2014
Committee Certification We certify that we, the examining committee, have read the dissertation titled" Behavior of R.C. Horizontally Curved Beams with Openings Strengthened by CFRP Laminates ", which is being submitted by
" Sadjad Amir Hemzah ", and examined the student in its content and in what is connected with it, and that in our opinion, it meets the standard of
a thesis for the degree of Doctor of Philosophy in Structural
Engineering(Structure). Signature:
Signature:
Name: Dr. Ammar Y. Ali
Name: Dr. Ihsan A.S. Al-Shaarbaf
(professor) (Supervisor) Signature:
(Assist. Professor) (Member) Signature:
Name: Dr. Hayder T. Nimnim
Name: Dr. Mustafa B. Dawood
(Assist. Professor) (Member) Signature:
(professor) (Member) Signature:
Name: Dr. Nameer A. Alwash
Name: Dr. Hisham M. Al-Hassani
(professor) (Member)
(professor) (Chairman)
Approved by the Head of the Civil Engineering Department Signature: Name: Asst. Prof. Dr Abdul-Rudha I. Ahmed (The Head of the Civil Engineering Department) Date:
/ / 2014
Approved by the Dean of the College of Engineering Signature: Name: Prof. Dr. Adil A. Al-Moosaway (The Dean of the College of Engineering) Date: / / 2013
ACKNOWLEDGMENT In the name of ALLAH, the most compassionate the most merciful. Praise be to ALLAH and pray and peace be on his prophet Mohammed and his family. First, I would like to express my appreciation and deepest gratitude to my supervisor Prof. Dr. Ammar Y. Ali for his remarkable suggestions, encouragement and guidance through the research; I am really indebted to him. Thanks to all staff of Civil Engineering Department/College of Engineering / University of Babylon for their appreciable support. Special thanks to Dr. Bashar Abid Hamza and my friends Majed, Mohsen, and Bahaa for their help during this study. Thanks also extended to the staff of the Structures Laboratory at College of Engineering / University of Karbala for their help in using the various facilities. A special thank and gratitude to my family for their care, patience and encouragement throughout the research period.
Sadjad Amir Hemzah 2014
i
Abstract This research is devoted to investigate the behavior and performance of reinforced concrete horizontally curved beams with and without openings, unstrengthed and strengthened externally by CFRP laminates or internally by steel reinforcement. The experimental work consisted of fabrication and testing of fourteen reinforced concrete curved beams divided into two group. The first group included ten of semicircular beams (one without opening and nine with openings), while the second group consisted of four full circular reinforced concrete beams (one without opening and three with openings). The variables considered in the experimental program included: geometry of circular beam, location of opening through profile of beam, presence of internal
strengthening
by
reinforcing
steel(stirrups)
and
external
strengthening by CFRP laminates for beam around openings. The beams were tested under effect of point loads at top edge of each midspan . The experimental results showed that the presence of opening at region of maximum bending moment and shear force caused a significant decrease in ultimate load capacity by about 35% for semi-circular beams and 50% for full circular (ring) beams compared with control beams without opening associated with beam shear mode failure. The use of steel reinforcement as internal strengthening and CFRP laminates as external confinement around openings improved the ultimate load capacity of the semi-circular beam by a ratio of (3% to 30%) and (11% to 40%), respectively. While for full circular (ring) beam the increasing in ultimate load reached about 60% and 75% for internal and external strengthening, respectively. Also, both types of strengthening around opening (internally by steel reinforcement or externally by CFRP laminates) enhanced crack patterns, post cracking torsional and flexural stiffnesses. ii
The numerical work included a three-dimensional nonlinear finite element model using the computer program ANSYS version 12.1 suitable for the analysis of the tested reinforced concrete horizontally curved beams with or without openings and unstrengthened or strengthened by(CFRP laminates or reinforcing steel) under incremental loading, as well as a parametric study for many variables. Full bond was assumed between the CFRP and concrete surface and between the steel reinforcement and concrete. In general, a reasonable agreement between the finite element solutions and experimental results has been obtained concerning estimate load-deflection response, mode of failure, cracking and ultimate loads with average difference about 4.1% and 16% for ultimate load and deflection respectively. The numerical results for the analyzed beams show that the ultimate load decreased about 12% when the length increased by 50% and it increased by 14% when the length of opening decreased by 50%. Moreover, an increment in ultimate load about 41% was found for circular opening instead of rectangular opening with the same area. Also, for the same length of the beam, the ultimate load increased about 103%,130% and 172% when the curvature of the curved beam (1/R) decreased (100%,200,and ∞), respectively.
iii
List of Contents Acknowledgments
i
Abstract
ii
List of Contents
iv
List of Tables
xi
List of Figures
x
Notation
xvii
Abbreviations
xviii
CHAPTER ONE:INTRODUCTION 1.1
General
1
1.2
General Guidelines to Selection of the Size and Location of Openings. Reinforced Concrete Curved Beams
5
1.4
Analysis for Ultimate Strength Under Combined Torsion Bending and Shear
7
1.5
Failure Modes Under Combined Torsion, Bending, and Shear with Small Opening
10
1.6
Design for Torsion
17
1.6.1
Design for Beam type Failure
18
1.6.2
Design for Frame Type Failure
21
Fiber Reinforced Polymer FRP
23
1.7.1
Fiber Materials
23
1.2.2
Matrix
25
1.7.3
Properties of FRP
25
1.8
Aims of the Study.
27
1.9
Layout of Thesis
28
1.3
1.7
iv
6
CHAPTER TWO:LITERATURE REVIEW 2.1
Introduction
29
2.2
Experimental and Analytical Studies on Reinforced Concrete Curved Beams.
29
2.3
Experiential and Analytical Works on Reinforced Concrete Beam with Openings.
39
2.4
Concluding Remarks
54
CHAPTER THREE:EXPERIMENTAL WORK 3.1
General
57
3.2
Description of Specimens
57
Semicircular Curved Beams (Group I)
60
3.2.1.1
Semicircular Curved Beams without Opening
60
3.2.1.2
Semicircular Curved Beams with Opening
60
Ring Curved Beams (Group I)
60
3.2.2.1
Ring Curved Beams without Opening
60
3.2.2.2
Ring Curved Beams with Opening
60
Strengthening System
62
3.3.1
Internal Strengthening by Internal Reinforcement
62
3.3.2
External Strengthening by CFRP Straps
62
Material Properties of Tested Specimens
65
Concrete
65
3.4.1.1
Cement
65
3.4.1.2
Coarse Aggregate (Gravel)
66
3.4.1.3
Fine Aggregate (Sand)
66
3.4.1.4
Mixing Water
68
3.2.1
3.2.2
3.3
3.4 3.4.1
v
3.4.1.5
Mix Design
68
3.4.1.6
Mixing Procedure
69
3.4.2
Reinforcing Steel
69
3.4.3
Carbon Fiber (CFRP) Strengthening System
71
Mechanical Properties of Hardened Concrete
72
3.5.1
Compressive Strength (fc')
72
3.5.2
Splitting Tensile Strength (ft)
73
3.6
CFRP Installation
74
3.7
Instrument and Test Procedure
75
3.8
Loading and Support Condition
78
3.9
Steel and wood mold
79
3.5
CHAPTER FOUR:EXPERIMENTAL RESULTS AND DISCUSSION 4.1
General
81
4.2
Semicircular Curved Concrete Beams (Group I)
82
4.2.1
Pilot and Control Semicircular Curved Beam (SCB.P)
82
4.2.2
Semicircular Curved Beam with Opening near External Support (SCB.E)
85
4.2.2.1
Unstrengthened Semicircular Curved Beam (SCB.Eo)
85
4.2.2.2
Internally Strengthened Semicircular Curved Beam (SCB.Esr)
88
4.2.2.3
Externally Strengthened Semicircular Curved Beam (SCB.Ecfrp)
90
4.2.2.4
Summary of tested SCB.E Beams
93
Semicircular Curved Beam with Opening near Midspan (SCB.M)
95
4.2.3
vi
4.2.3.1
Unstrengthened Semicircular Curved Beam (SCB.Mo)
95
4.2.3.2
Internally Strengthened Semicircular Curved Beam (SCB.Msr)
98
4.2.3.3
Externally Strengthened Semicircular Curved Beam (SCB.Mcfrp)
101
4.2.3.4
Summary of tested SCB.M Beams
103
Semicircular Curved Beam with Opening near Interior Support (SCB.I)
105
4.2.4.1
Unstrengthened Semicircular Curved Beam (SCB.Io)
105
4.2.4.2
Internally Strengthened Semicircular Curved Beam (SCB.Isr)
108
4.2.4.3
Externally Strengthened Semicircular Curved Beam (SCB.Icfrp)
110
4.2.4.4
Summary of tested SCB.I Beams
113
Summary of Tested Semicircular Curved Beams
115
Full Circular (Ring) Concrete Beams
115
4.3.1
Control Specimen (FCB.P)
116
4.3.2
Full Circular Beam with Opening (FCB.Mo)
118
4.3.3
Internally Strength Full Circular Beam (FCB.Msr)
121
4.3.4
Externally strengthened Full Circular Beam (FCB.Mcfrp)
123
4.3.5
Summary of Tested FCB.M Beams
126
4.3.6
Summary of The Tested Curved Beams
128
4.2.4
4.2.5 4.3
CAPTER FIVE: FINITE ELEMENTS MODELING AND FORMULATION OF NONLIEAR ANALYSIS OF TESTED CURVED BEAMS 5.1
Introduction
129
5.2
Material Modeling
130
vii
5.2.1
Concrete Modeling
130
5.2.1.1
Uniaxial Compression Behavior for Concrete
131
5.2.1.2
Tensile Behavior of Concrete
134
5.2.1.3
Biaxial Stress Behavior of Concrete
136
5.2.1.4
Triaxial Stress Behavior of Concrete
137
5.2.1.5
Post - Cracking Model (Tension Stiffening Model)
137
5.2.1.6
Crushing Modeling
139
5.2.1.7
Shear Transfer Model
139
5.2.1.8
Cracking Modeling
139
5.2.1.9
Failure Criteria for Concrete
141
5.2.1.9.1
Determination of the Model Parameters
144
5.2.2
Reinforcement Modeling
145
5.2.3
CFRP Composite Modeling
148
Material Idealization
148
5.3.1
Element Types
148
5.3.1.1
Element SOLID65
149
5.3.1.2
Element LINK8
149
5.3.1.3
Element SOLID45
150
5.3.1.4
Element Shell41
151
Nonlinear Finite Element Analysis
151
5.4.1
Numerical Integration
152
5.4.2
Procedure for Solving Nonlinear Finite Element Equations
154
5.4.3
Convergence Criteria
157
5.4.4
Analysis Termination Criteria
158
5.3
5.4
viii
CHAPTER SIX: FINITE ELEMENT ANALYSIS 6.1
Introduction
159
6.2
Description of Specimens in Finite Element
159
6.3
Mesh Refinement
160
6.4
Reinforcing Steel Modeling
162
6.5
Modeling of CFRP Laminates
163
6.6
Loads and Boundary Conditions Representations
164
6.7
Results of Finite Element Analysis
165
6.7.1
Load – Deformations Curves
165
6.7.2
Deflection, Cracking and Ultimate Loads
180
6.7.3
Deflection of Service Loads
180
6.7.4
Crashing and Stresses in Curved Beams.
180
Parametric Study
181
6.8.1
Effect of Curvature
182
6.8.2
Effect of Type and Size of Opening
183
6.8.3
Effect of Opening Length
186
6.8.4
Effect of Opening Height
188
6.8.5
Wrapping Schemes of CFRP (full and U- wrapping )
190
6.8
CHAPTER SEVEN:CONCLUSIONS and RECOMMENDATIONS FOR FUTURE WORKS 7.1
General
193
7.2
Conclusions
193
7.3
Recommendations
196
ix
197
References Analytical Solution and Design of Semicircular
A-1
Appendix A Curved Beam - Control Beam Appendix B
Design of Steel Reinforcement for strengthening of Mid Opening of Semicircular Curved Beams
A-5
Appendix C
Cracks Pattern And Stress In Concrete And CFRP Laminates
A-12
x
List of Tables No.
Title
Page
3.1
Description of Tested Specimens
61
3.2
Chemical and Physical Test Results of the Cement
65
3.3
Grading of Coarse Aggregate
67
3.4
Fine Aggregate Properties
67
3.5
Properties of Concrete Mix
68
3.6
Specification and Test Results of Steel Reinforcing Bar Values
70
3.7
Properties of Sikadur-330 (Impregnating Resin)
71
3.8
Properties of SikaWrap Hex-230C (Carbon Fiber Fabric)
72
3.9
Concrete Compressive Strength of Specimens
73
4.1
Summary of tested SCB.E Beams
93
4.2
Summary of tested SCB.M Beams
104
4.3
Summary of tested SCB.I Beams
113
4.4
Summary of tested FCB Beams
126
5.1
Element Types
148
5.2
Sampling Points Locations and Weighting Factors
154
6.1
Theoretical and Experimental Cracking and Ultimate Loads
180
6.2
Ultimate Load for Different Types of SCB.Io with Variable Radiuses
183
6.3
Ultimate Load Capacity for Different Opening Dimensions of SCB.Io Curved Beam
185
6.4
Ultimate Load Capacity for Different Opening Length of SCB.Io Curved Beam
187
6.5
Ultimate Load Capacity for Different Height of SCB.Io Curved Beam
189
6.6
Ultimate Load Capacity of Wrapping Schemes for SCB.Io and SCB.Mo Curved Beams
192
xi
List of Figures No.
Title
Page
1.1
Applications of Curved Beams
1
1.2
Typical Layout of Pipes for High Rise Building
2
1.3
Collapses Mechanism at Large Opening
4
1.4
Guidelines for Location of Web Opening
6
1.5
Horizontally Curved Beam Loading and Forces
7
1.6
Failure surfaces for a solid beam in Modes 1, 2, and 3
8
1.7
Mode 1 failure surface for beams with a small opening
10
1.8
Geometry of the boundary of failure surface
12
1.9
Mode 2 failure surface for beams with a small opening
13
1.10 Frame-type failure of a beam with a small opening under torsion
18
1.11
22
Idealized free-body diagram at opening of a beam under loading
1.12 Stress-Strain Relationship of Fibers and Steel
24
1.13 a-Stress-Strain Curves of Fibers, FRP and Matrix 26 b-Typical Composition of FRP Material 2.1 yield surfaces for combined bending and torsion, at a section where 31 2.2
a plastic hinge was formed. Geometry of Test Specimens by Jordaan et al (1974)
32
2.3
Geometry of Test Specimens of Badawy et al (1977)
34
2.4
Typical Shear Failure of a Beam with Small Openings Containing no Shear Reinforcement
42
2.5
Shear Failure of a Beam at the Throat Section by Salam
43
2.6
Modes of Failure for Small Opening ,Mansur (1998).
47
2.7
Details of the tested beams by Abdulla (2003)
49
3.1
Details for Group I of Tested Curved Beams
58
3.2
Details for Group II of Tested Ring Beams
59
3.3
External Strengthening around opening with CFRP laminates
63
3.4
Details For Internally Strengthening by Steel Reinforcement
64
3.5
Cage of Steel Reinforcement
70
3.6
CFRP Laminates and Epoxy Resine
71
3.7
Compressive Strength Test and Splitting Tensile Strength Test
73
xii
3.8
Application of CFRP System on Concrete Element
75
3.9
Loading Machine Used in the Testes
76
3.10 Instruments Details
77
3.11 Details of the Typical Support Condition
78
3.12 Semicircular Molds and Tools
79
3.12 Wooden Circular Molds
80
4.1
Control Beam SCB.P
83
4.2
Mode of Failure and Cracks Pattern for Control Beam
83
4.3
Load-Midspan Deflection Curve for Control Beam SCB.P
84
4.4
Load-Midspan Twisting Angle Curve for Control Beam SCB.P
84
4.5
Semicircular Curved Beam SCB.Eo
86
4.6
Mode of Failure and Cracks Pattern for SCB.Eo Curved Beam
86
4.7
Load-Midspan Deflection Curve for Control Beam SCB.Eo
87
4.8
Load-Midspan Twisting Angle Curve for Control Beam SCB.Eo
87
4.9
Semicircular Curved Beam SCB.Esr
88
4.10 Mode of Failure and Cracks Pattern for SCB.Esr Curved Beam
89
4.11 Load-Midspan Deflection Curve for Control Beam SCB.Esr
89
4.12 Load-Midspan Twisting Angle Curve for Control Beam SCB.Esr
90
4.13 Semicircular Curved Beam SCB.Ecfrp
91
4.14 Mode of Failure and Cracks Pattern for SCB.Ecfrp Curved Beam
91
4.15 Load-Midspan Deflection Curve for Control Beam SCB.Ecfrp
92
4.16 Load-Midspan Twisting Angle Curve for Control Beam SCB.Ecfrp
92
4.17 Comparison of Load-Midspan Deflection Curves for SCB.Ep,
94
SCB.Eo, SCB.Esr and SCB.Ecfrp Curved Beams 4.18 Comparison of Twisting Angle Curves at Midspan for SCB.Ep, SCB.Eo, SCB.Esr and SCB.Ecfrp Curved Beams 4.19 Semicircular Curved Beam SCB.Mo
94 96
4.20 Mode of Failure and Cracks Pattern for SCB.Mo Curved Beam
96
4.21 Load-Midspan Deflection Curve for Control Beam SCB.Mo
97
4.22 Load-Midspan Twisting Angle Curve for Control Beam SCB.Mo
97
4.23 Semicircular Curved Beam SCB.Msr
99
xiii
4.24 Mode of Failure and Cracks Pattern for SCB.Msr Curved Beam
99
4.25 Load-Midspan Deflection Curve for Control Beam SCB.Msr
100
4.26 Load-Midspan Twisting Angle Curve for Control Beam SCB.Msr
100
4.27 Semicircular Curved Beam SCB.Mcfrp
101
4.28 Mode of Failure and Cracks Pattern for SCB.Mcfrp Curved Beam
102
4.29 Load-Midspan Deflection Curve for Control Beam SCB.Mcfrp
102
4.30 Load-Midspan Twisting Angle Curve for Control Beam
103
SCB.Mcfrp 4.31 Comparison of Load-Midspan Deflection Curves for SCB.P, SCB. Mo, SCB. Msr and SCB.Mcfrp Curved Beams 4.32 Comparison of Twisting Angle Curves at Midspan for SCB.P, SCB.Mo, SCB.Msr and SCB.Mcfrp Curved Beams 4.33 Semicircular Curved Beam SCB.Io
104 105 106
4.34 Mode of Failure and Cracks Pattern for SCB.Io Curved Beam
106
4.35 Load-Midspan Deflection Curve for Control Beam SCB.Io
107
4.36 Load-Midspan Twisting Angle Curve for Control Beam SCB.Io
107
4.37 Semicircular Curved Beam SCB.Isr
108
4.38 Mode of Failure and Cracks Pattern for SCB.Isr Curved Beam
109
4.39 Load-Midspan Deflection Curve for Control Beam SCB.Isr
109
4.40 Load-Midspan Twisting Angle Curve for Control Beam SCB.Isr
110
4.41 Semicircular Curved Beam SCB.Icfrp
111
4.42 Mode of Failure and Cracks Pattern for SCB.Icfrp Curved Beam
111
4.43 Load-Midspan Deflection Curve for Control Beam SCB.Icfrp
112
4.44 Load-Midspan Twisting Angle Curve for Control Beam SCB.Icfrp
112
4.45 Comparison of Load-Midspan Deflection Curves for SCB.P,
114
SCB.Io, SCB.Isr and SCB.Icfrp Curved Beams 4.46 Comparison of Twisting Angle Curves at Midspan for SCB.P, 4.47 4.48 4.49 4.50 4.51 4.52 4.53
SCB.Io, SCB.Isr and SCB.Icfrp Curved Beams Full Circular Beam FCB.P Mode of Failure and Cracks Pattern for Circular Beam FCB.P Load-Midspan Deflection Curve for FCB.P Circular Beam Load-Midspan Rotation Curve for FCB.P Circular Beam Full Circular Beam FCB.Mo Mode of Failure and Cracks Pattern for Circular Beam FCB.Mo Load-Midspan Deflection Curve for FCB.Mo Circular Beam xiv
114 116 117 117 118 119 119 120
4.54 4.55 4.56 4.57 4.58 4.59 4.60 4.61 4.62 4.63
Load-Midspan Rotation Curve for FCB.Mo Circular Beam Full Circular Beam FCB.Msr Mode of Failure and Cracks Pattern for Circular Beam FCB.Msr Load-Midspan Deflection Curve for FCB.Msr Circular Beam Load-Midspan Rotation Curve for FCB.Msr Circular Beam Full Circular Beam FCB.Mcfrp Mode of Failure and Cracks Pattern for Circular Beam FCB.Mcfrp Load-Midspan Deflection Curve for FCB.Mcfrp Circular Beam Load-Midspan Rotation Curve for FCB.Mcfrp Circular Beam Comparison of Load-Midspan Deflection Curves for FCB.P , FCB.Mo, FCB.Msr and FCB.Mcfrp Curved Beams 4.64 Comparison of Twisting Angle Curves at Midspan for FCB.P, FCB.Mo, FCB.Msr and FCB.Mcfrp Curved Beams
120 121 122 122 123 124 124 125 125 127
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12
132 134 135 136 137 138 140 141 143 145 146 147 148 149 150 150 151 154 155 157 160 161 162 162 163 163 164 166 166 167 167 168
Uniaxial Compressive Strain curve for concrete with different strength Stress-Strain Curve for Concrete in Compression Typical Tensile Stress-Strain Curve for Concrete Biaxial State of Loading Failure Surface of Concrete in 3-D Stress Space Pre and Post-Cracking Behavior of Normal Strength Concrete Cracking representation in discrete cracking modeling approach Smeared Crack Modeling Failure Surface Profile of the Failure Surface as Function of Five Parameters Modeling of Reinforcing Bars . Models for Reinforcement in Reinforced Concrete Schematic Properties of CFRP Composites Geometry of Element SOLID65 Geometry of Element LINK8 Geometry of Element SOLID45 Geometry of Element SHELL41
Distribution of integration points Technique for Solving the Nonlinear Equation Incremental-Iterative Procedures Full Newton-Raphson procedure Adopted Descriptions of Curved Beams Mesh density (Cross Section and Top View) Effect of Number of Elements on Load-Midspan deflection Effect of Number of Elements on Load- Midspan Twisting Angle Reinforcing Steel Bars Modeling CFRP Laminates Arrangement of Tested Concrete Curved Beams Boundary Conditions and Applied Loads Arrangements Load-Midspan Deflection Curves for Control Beam Figure (6.9) Load-Midspan Twisting Angle Curves for Control Beam Load-Midspan Deflection Curves for SCB.Eo Beam Load-Midspan Twisting Angle Curves for SCB.Eo Beam Load-Midspan Deflection Curves for SCB.Esr Beam xv
127
6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33 6.34 6.35 6.36 6.37 6.38 6.39 6.40 6.41 6.42 6.43 6.44 6.45 6.46 6.47
Load-Midspan Twisting Angle Curves for SCB.Esr Beam Load-Midspan Deflection Curves for SCB.Ecfrp Beam Load-Midspan Twisting Angle Curves for SCB.Ecfrp Beam Load-Midspan Deflection Curves for SCB.Mo Beam Load-Midspan Twisting Angle Curves for SCB.Mo Beam Load-Midspan Deflection Curves for SCB.Msr Beam Load-Midspan Twisting Angle Curves for SCB.Msr Beam Load-Midspan Deflection Curves for SCB.Mcfrp Beam Load-Midspan Twisting Angle Curves for SCB.Mcfrp Beam Load-Midspan Deflection Curves for SCB.Io Beam Load-Midspan Twisting Angle Curves for SCB.Io Beam Load-Midspan Deflection Curves for SCB.Isr Beam Load-Midspan Twisting Angle Curves for SCB.Isr Beam Load-Midspan Deflection Curves for SCB.Icfrp Beam Load-Midspan Twisting Angle Curves for SCB.Icfrp Beam Load-Midspan Deflection Curves for FCB.P Beam Load-Midspan Twisting Angle Curves for FCB.P Beam Load-Midspan Deflection Curves for FCB.Mo Beam Load-Midspan Twisting Angle Curves for FCB.Mo Beam Load-Midspan Deflection Curves for FCB.Msr Beam Load-Midspan Twisting Angle Curves for FCB.Msr Beam Load-Midspan Deflection Curves for FCB.Mcfrp Beam Load-Midspan Twisting Angle Curves for FCB.Mcfrp Beam SCB.Io Beams With Variable Radiuses Load-Midspan Deflection Curves for SCB.Io Beams With Variable Radiuses Load-Midspan Deflection Curves for SCB.Io Beams With Variable Opening Dimensions Load-Midspan Twisting Angle Curves for SCB.Io Beams with Variable Opening Dimensions Load-Midspan Deflection Curves for SCB.Io Beams With Variable Opening Length Load-Midspan Twisting Angle Curves for SCB.Io Beams With Variable Opening Length Load-Midspan Deflection Curves for SCB.Io Beams With Variable Opening Height Load-Midspan Twisting Angle Curves for SCB.Io Beams with Variable Opening Height Load-Midspan Deflection Curves for SCB.Io Beams with Different Wrapping Schemes Load- Midspan Twisting Angle Curves for SCB.Io Beams with Different Wrapping Schemes Load-Midspan Deflection Curves for SCB.Mo Beams with Different Wrapping Schemes Load- Midspan Twisting Angle Curves for SCB.Mo Beams with Different Wrapping Schemes
xvi
168 169 169 170 170 171 171 172 172 173 173 174 174 175 175 176 176 177 177 178 178 179 179 182 183 184 185 186 187 188 189 190 191 191 192
Notation Most commonly used symbols are listed below, these and others are defined where they appear in the research; Symbol Description As Area of steel. d Effective depth. Ec Concrete modulus of elasticity. Es Steel modulus of elasticity. ƒc ̀ Nominal concrete compressive strength (cylinder test). ƒy Yield strength of steel reinforcement. ƒs Strength of steel reinforcement fu Ultimate strength of steel reinforcement. fsp Splitting tensile strength of concrete. fyv(or Fvw) yield strength of transverse steel (stirrups) ƒt Concrete Tensile strength of splitting test. ƒr Concrete modulus of rupture. f1 Biaxial crushing stress of concrete under the ambient hydrostatic stress state f2 Uniaxial crushing stress of concrete under the ambient hydrostatic stress state Ultimate biaxial compressive strength of concrete ( f cb ) h/d Depth of opening/ effective depth First stress invariant (I1 ) (J 2 ) Second deviatoric stress invariant n Number of stirrups. Pᵤ Ultimate load Pcr Cracking load R Radius of the arch r Radius of the curve beam Width of the composite laminate wf w/c wc X X1 y y1
Water/cement. Air-dry unit weight of concrete. shorter dimension of a rectangular section shorter center-to-center dimension of closed rectangular stirrup longer dimension of a rectangular section longer center-to-center dimension of closed rectangular stirrup xvii
z α ε ε cu ε˳ x,y,z υ βc,β˳ σ σ xp σ yp σ zp
εf σa h σh
distance between the plastic centroids of the chord members angle of inclination of diagonal bars (stirrups) or web reinforcement to longitudinal axis Strain. Ultimate strain of concrete Strain corresponding to the maximum compressive stress of concrete Cartesian coordinates. Poisson's ratio. Shear transfer coefficient for closed and opened crack. Stress Principal stress in the x – direction Principal stress in the y – direction Principal stress in the z – direction Strain in the fiber Ambient hydrostatic pressure Hydrostatic stresses Abbreviations
Abbreviations
Description
ACI ASTM
American Concrete Institute American Society for Testing and Material Analysis System program Aramid Fiber Reinforced Polymer Carbon Fiber Reinforced Polymer Degrees Of Freedom Experimental Equation Finite Element Method Finite Element Fiber Reinforced Polymer Glass Fiber Reinforced Polymer Hinge – Hinge support system Hinge – Roller supports system High Modulus High Strength Maximum
ANSYS AFRP CFRP DOF Exp. Eq. FEM FE FRP GFRP H.H H.R HM HS Max.
xviii
Min. No. N-R pp. rebar RC 3D L.O.I I.R L.S.F
Minimum Number Newton-Raphson From page … to page … Reinforcing bar Reinforced Concrete Three-Dimensional Loss on ignition Insoluble residue Lime saturation factor
xix
Chapter One
Introduction
CHAPTER ONE INTRODUCTION 1.1 General Reinforced concrete horizontally curved beams are extensively used in many fields, such as in the construction of modern highway intersections, elevated freeways, the rounded corners of buildings, circular balconies,…etc Figure(1-1). In the construction of modern buildings, network of pipes and ducts, is necessary to accommodate essential services like water supply, sewage, air-conditioning, electricity, telephone, and computer network. Usually, these pipes and ducts are placed underneath the beam soffit and, for aesthetic reasons, are covered by a suspended ceiling, thus creating a dead space. Passing these ducts through transverse opening in the floor beams will reduce the dead space and results in a more compact design as shown in Figure (1-2). For small buildings, the saving of dead spaces may not be significant, but for multistory buildings, any saving in story height multiplied by the number of stories can represent a substantial saving in total height,
Figure (1-1) Applications of Curved Beams.
1
Chapter One
Introduction
length of air-conditioning and electrical ducts, plumbing risers, walls and partition surfaces, and overall load on the foundation (Mansur, 2006).
(a)
(b) Figure 1.2:Typical Layout of Pipes for High Rise Building (Mansur and Hasnat, 1979). a-Typical layout of service ducts. b- Alternative arrangement of service ducts A horizontally curved beam, loaded transversely to its plane, is subjected to torsion in addition to bending and shear. Furthermore, it is obvious that inclusion of openings in beams alters the simple beam behavior to a more complex one. Due to abrupt changes in sectional configuration, opening corners are subjecting to high stress concentration that may lead to cracking unacceptable from aesthetic and durability viewpoints. The reduced stiffness of the beam may also give rise to excessive deflection under service load and results in a considerable redistribution of internal forces and moments in a continuous beam. Unless special reinforcement is provided in 2
Chapter One
Introduction
sufficient quantity with proper detailing, the strength and serviceability of such a beam may be seriously affected (Mansur, 2006). In practice, the most common shapes of openings are circular and rectangular. Circular openings are required to accommodate service pipes, such as for plumbing, while rectangular openings provide the passage for airconditioning ducts that are generally rectangular in shape. With regard to the size of openings, many researchers use the terms “small” and “large” without any definition or clear-cut demarcation line. Mansur and Hasnat(1979) have defined small openings as those circular, square or nearly square in shape. Whereas, according to Somes and Corley(1974), a circular opening may be considered as effected when its diameter exceeds 0.25 times the depth of the web because introduction of such openings reduces the strength of the beam. The authors however consider that the essence of classifying an opening either small or large lie in the structural response of the beam. When the opening is small enough to maintain the beam-type behavior or, in other words, if the usual beam theory applies, then the opening may be termed as small opening. In contrast, large openings are those that prevent beam-type behavior to develop(Mansur, 2006; Weng et al, 1999). For small openings, two different failure modes are identified. These types of failure may be labeled as ''beam-type'' failure and ''frame-type'' failure, respectively, and required separate treatment for complete design. In contrast, for large openings the beam-type behavior transforms into a Vierendeel action(Figure 1-3) as the size of opening is increased. Since the behavior of a beam depend on the size of opening, small and large openings need separate treatment in design (Mansur, 1998). The effects of introducing an opening on the overall response of a beam may be summarized as follow (Mansur and Hasnat, 1979):
3
Chapter One
Introduction
Figure (1-3) Collapses Mechanism at Large Opening(Mansur and Hasnat,1979) 1- Introduction of an opening in the web of a beam leads to early diagonal crack, and the load at first crack decreases with an increase in either the length or the depth of opening. 2- Unless additional reinforcement is provided to restrict the growth of cracks, the opening corners are liable to exhibit wide cracking. 3- When the same amount and scheme of reinforcement is used, an increase in the opening size either by increasing the length or the depth of opening decreases the strength as well as stiffness of the beam. The eccentricity of opening, however, has only a marginal effect on both strength and stiffness. 4- The chord members above and below the opening behaves in a manner similar to the chords of a Vierendeel panel with contraflexure points located approximately at mid-span of the chords. Final failure occurs by the formation of a mechanism with four hinges in the chords, one at each corner of the opening as shown in Figure (1-3).
4
Chapter One
Introduction
1.2 General Guidelines for Selection the Size and Location of Openings. The following guidelines had been proposed to facilitate the selection of the size and location of web opening as shown in figure (1-4): 1. For T-beams, openings should preferably be positioned flush with the flange for ease in construction. In the case of rectangular beams, openings are commonly placed at mid-depth of the section, but they may also be placed eccentrically with respect to depth. Care must be exercised to provide sufficient concrete cover to the reinforcement for the chord members above and below the opening. The compression chord should also have sufficient concrete area to develop the ultimate compression block in flexure and have adequate depth to provide effective shear reinforcement. 2. Openings should not be located closer than one-half the beam depth D to the supports to avoid the critical region for shear failure and reinforcement congestion. Similarly, positioning of an opening closer than 0.5D to any concentrated load should be avoided. 3. Depth of openings should be limited to 0.5 D. 4. The factors that limit the length of an opening are the stability of the chord members, in particular the compression chord, and the serviceability requirement of deflection. When the opening gets bigger, it is preferable to use multiple openings providing the same passageway instead of using a single opening. 5. When multiple openings are used, the post separating two adjacent openings should not be less than 0.5D to insure that each opening behaves independently.
5
Chapter One
Introduction
Figure(1.4) Guidelines for Location of Web Opening for straight beam (Tan and Mansur,1996)
1.3 Reinforced Concrete Curved Beams. A horizontally curved beam may be defined as a curved girder having an out of plane curvature, and supported at its ends by two or more supports. The shape of the curved beam may be circular, elliptical or parabolic and sometimes it is made up by circular arcs of several and different radii or/and centers. It may be subjected to out of plane forces: shear, flexural and torsional forces. Horizontally curved beams are loaded transverse to their planes, and are subjected to torsion, bending, and shear, Figure (1.5). Therefore, a special feature of the analysis and design of such beams is the necessity to include torsional effects. In a reinforced concrete structure, cracks form even at service load. Such cracking causes local reduction in torsional and flexural stiffness’s resulting in redistribution of internal forces. Furthermore openings will increase cracks spatially at their corners because of the sudden change in shear, flexural and torsional stiffness (Mansur and Tan , 1999).
6
Chapter One
Introduction
Figure(1.5) Horizontally Curved Beam Loading and Forces
1.4 Analysis for Ultimate Strength Under Combined Torsion Bending and Shear The analysis of any solid beam subjected to shear, bending and torsional moments are based on the well-known skew-bending theory for torsion in concrete beams. In the case of a solid beam, the theory considers three basic failure modes classified as Mode 1, Mode 2, and Mode 3, according to the location of concrete compression zone near the top, side, and bottom of the beam, respectively, as shown in Figure( 1-6). Aspect ratio of the beam section, relative proportion of top and bottom longitudinal steel, and the ratio of applied torque to bending moment with or without a combination with transverse shear generally govern the failure modes. When a relatively small transverse opening is introduced through the beam web, and the beam is subjected to predominant torsion, one would expect no changes in the mode 7
Chapter One
Introduction
of failure. This has been confirmed experimentally by several investigators (Hasnat and Akhtaruzzaman, 1987; Mansur and Hasnat, 1979; Mansur and Paramasivam, 1984; Mansur et al., 1983b). As a result, an analysis similar to that of a prismatic beam is applicable to beams containing a small opening. With this contention, the analysis is based on three failure modes classified as Mode 1, Mode 2, and Mode 3. Since the opening represents the potential source of weakness in a beam, the failure surface is assumed to be traversed through the center of the opening. In developing strength equations for such beams, each failure mode is considered separately, and the following assumptions are made to simplify the problem (Tan and Mansur,1999):
Figure( 1-6) Failure surfaces for a solid beam in Modes 1, 2, and 3. (Tan and Mansur,1999)
8
Chapter One
Introduction
1) The pattern of reinforcement in the vicinity of the opening consists of longitudinal bars above and below the opening, full-depth stirrups close to either side of the opening, and closed stirrups at the throat section (above and below the opening), in addition to the normal top and bottom reinforcement in the solid section. 2) The spacing of stirrups at the solid cross section as well as at the throat section is uniform along the length of the beam. 3) Failure occurs on a warped plane. The boundaries of the warped plane are defined on the three sides of the beam by a spiral crack and on the fourth side by a compression zone that joins the ends of the spiral crack. 4) The crack defining the failure plane on three sides of the beam consists of three straight lines spiraling around the beam at a constant angle.
5) The concrete outside the compression zone is cracked and carries no tension. 6) All reinforcement crossing the failure plane outside the compression zone yields at failure. 7) Any reinforcement in the compression zone and dowel action of reinforcement is ignored.
9
Chapter One
Introduction
1.5 Failure Modes Under Combined Torsion, Bending, and Shear with Small Opening A structural member is, in general, subjected to torsion, bending, shear, and axial forces, but when the analysis and design of a beam is concerned, the effect of axial load is usually ignored. Combined torsion, bending, and shear, therefore, may be considered as the general loading case for beams, and this loading case is treated first. • Mode 1 failure The assumed failure surface for Mode 1 is shown in Figure(1-7) together with the internal forces developed in steel reinforcement. The skewed compression zone at the top makes an angle θ 1 to the normal cross section. It originates from the tip of the tension crack, which traverses through the opening on one of the side faces of the beam. The resultant of the forces developed in the longitudinal bars intercepted by the tension zone is Fs, and the force in the bottom legs of stirrups that cross the failure surface is F wh · Similarly, F wv and F’ wv are the forces in the vertical legs of the long and short stirrups, respectively. Let the forces Fs and Fwh be located at Ys and Y wh , respectively, below the center of compression and X 1 be the width of stirrups.
Figure (1.7) Mode 1 failure surface for beams with a small opening(Tan and Mansur,1999) 10
Chapter One
Introduction
Take moments about axis A-A, which passes through the center of the compression zone. The moment about A-A of the applied vectors M, T, and V acting at the midsection of the opening (Hasnat and Akhtaruzzaman, 1987) must be equal to the moment of forces in steel reinforcement. The resulting expression may be obtained as follows: 𝑏 2 + 𝑏ℎ 𝑇. 𝑠𝑖𝑛𝜃1 + 𝑀. 𝑐𝑜𝑠𝜃1 + 𝑉. � � 𝑠𝑖𝑛𝜃1 2𝑏 + 4ℎ
1 = 𝐹. 𝑐𝑜𝑠𝜃1. 𝑦𝑠 + 𝐹𝑤ℎ . 𝑠𝑖𝑛𝜃1. 𝑦𝑤ℎ + (𝐹𝑤 2 𝑏(𝑏 + ℎ) + 𝐹 ′ 𝑤) �1 − � 𝑥 . 𝑠𝑖𝑛𝜃1 𝑥1 (𝑏 + 2ℎ) 1
(1-1)
The three expressions on the right-hand side of Eq. (1-1) represent the contributions of longitudinal steel, the bottom legs of stirrups, and the vertical legs of stirrups, respectively. However, forces in the vertical legs of the stirrups have lever arms about A-A so small that their contributions to the resisting moment for Mode 1 failure may be neglected. Dropping the third expression on the right-hand side and dividing the resulting equation by cos θ 1 , eq. (1-1) thus reduces to: 𝑇. 𝑡𝑎𝑛𝜃1 + 𝑀 + 𝑉𝜇. 𝑡𝑎𝑛𝜃1 = 𝐹𝑠. 𝑦𝑠 + 𝐹𝑤ℎ . 𝑦𝑤ℎ 𝑡𝑎𝑛𝜃1
(1-2)
where
𝜇=�
𝑏2 +𝑏ℎ
2𝑏+4ℎ
�
(1-3)
Figure( 1-8) shows a developed diagram of the line PQRS (Figure 1.7)
of the failure surface, where PO is the zone of compression and QR, RS, and SP are the cracks on the other three faces. It may be seen that the number of stirrup legs intersected at the bottom of the beam by the crack is approximately (x 1 tan(β1/ s)), where x 1 is the width of stirrups, s is the stirrup 11
Chapter One
Introduction
spacing, and β1 is the inclination of the failure crack for Mode 1. Referring to figure( 1-7), it can be shown that β1 is related to θ1 by:
𝑡𝑎𝑛𝛽1 =
𝑏.𝑡𝑎𝑛𝜃1 𝑏+2ℎ
=
𝑡𝑎𝑛𝜃1
(1-4)
1+2𝛼
where α = h / b, and b and h are the overall width and depth,
respectively, of the solid cross section of the beam. If A w denotes the area of one leg of the stirrups and f yw is its yield strength, then
𝐹𝑤ℎ =
𝐴𝑤 .𝑓𝑦𝑤 𝑠
𝑥1 𝑡𝑎𝑛𝛽1
(1-5)
Figure (1-8) Geometry of the boundary of failure surface After some modification, assumptions and derivations, the strength in failure Mode 1, which will be denoted by T1, is obtained by the following quadratic equation:
𝑇1 =
1 2𝑀𝑜1 𝐾1 1 1 − �� + � 𝐾1 (𝜓∆)2 𝜓∆ ∆
(1 − 6)
where ψ= T/ M, λ= M/ V, 12
Chapter One
𝐾1 = and
Introduction
1 𝐴𝑤. 𝑓𝑌𝑤 𝑥1 . 𝑦1 � � 1 + 2𝛼 𝑆 𝑀𝑜1
∆= 1 + where :
(1 − 7)
𝜇 𝜓𝜆
(1 − 8)
x 1 is the width of stirrups A w is the area of one leg of the stirrups • Mode 2 failure The failure surface for this mode is illustrated in Figure(1-9). In this failure mode, the compression zone is located along a lateral side of the beam making an angle θ2 with the normal cross section. It is assumed that the longitudinal steel is concentrated at the corners and the lever arm of the force in this steel is x1. The equation of moments about B-B is given by
Figure( 1-9) Mode 2 failure surface for beams with a small opening 13
Chapter One
Introduction
𝑇. 𝑠𝑖𝑛𝜃2 + 𝑉.
𝑥1 = 𝐹𝑠 . 𝑐𝑜𝑠𝜃2. 𝑥1 + 𝐹 ′ 𝑤𝑣 . 𝑠𝑖𝑛𝜃2. 𝑥1 2 𝑏 +𝐹𝑤ℎ � � 𝑥 . 𝑠𝑖𝑛𝜃2 ℎ + 2𝑏 1
(1 − 9)
Similar to Mode 1 failure, the lever arm of the force F wh about B-B is very small, and, hence, its contribution to the resisting moment may be ignored. Also, the first term on the right-hand side of eq. (1.9) represents approximately the lateral flexural strength, M o2 , of the beam at the opening section. Thus, eq. (1.9) reduces to: 𝑥1 �𝑇 + 𝑉. � 𝑡𝑎𝑛𝜃2 = 𝑀𝑜2 + 𝐹′𝑤𝑣 . 𝑡𝑎𝑛𝜃2. 𝑥1 2
(1 − 10)
It may be shown that the inclination, 𝛽2, of the failure crack for Mode 2
is related to the angle θ2 by: 𝑡𝑎𝑛𝛽2 =
and that 𝐹′𝑤𝑣 =
ℎ. 𝑡𝑎𝑛𝜃2 𝑡𝑎𝑛𝜃2 = ℎ + 2𝑏 1 + 2/𝛼
(1 − 11)
𝐴𝑤. 𝑓𝑦𝑤 𝑦′. 𝑡𝑎𝑛𝛽2 𝑆
(1 − 12)
in which y' is the total length of vertical legs of short stirrups at throat section on one face of the beam. Inserting the values of tanθ2 and F' wv from Eqs. (1.12) and (1.11), respectively, into Eq. (1.10), we get
�𝑇 + 𝑉
𝑥1 2 � �1 + � 𝑡𝑎𝑛𝛽2 2 𝛼 𝐴𝑤. 𝑓𝑦𝑤 2 = 𝑀𝑜2 + 𝑥1 . 𝑦′1 . �1 + � 𝑡𝑎𝑛2 𝛽2 𝑆 𝛼
Minimization of T with respect to tanβ2 yields 14
(1 − 13)
Chapter One
�𝑇 + 𝑉
Introduction
𝐴𝑤. 𝑓𝑦𝑤 𝑥1 𝑥1 . 𝑦′1 . 𝑡𝑎𝑛𝛽2 �=2 2 𝑆
(1 − 14)
𝑥1 2 � �1 + � 𝑡𝑎𝑛𝛽2 = 𝑀𝑜2 2 𝛼
(1 − 15)
Substitution of Eq. (1.14) into Eq. (1.13) �𝑇 + 𝑉
The strength equation for Mode 2 is obtained by eliminating tan β2
from Eqs.(1.15) and (1.14) as: 𝑇2 =
2𝑀𝑜1 �𝑅 . 𝐾 1+𝛿 2 2
(1 − 16)
in which δ= x 1 .V/2T , R 2 =M o2 /M 01 , and 𝐾2 =
𝐴𝑤. 𝑓𝑦𝑤 𝑥1 . 𝑦1 � � 2 𝑆 𝑀𝑜1 1+ 1
(1 − 17)
𝛼
• Mode 3 failure
The analysis for Mode 3 is very similar to that for Mode 1. In this case, the skewed compression zone is at the bottom instead of at the top (see figure(1-7)). The equations for Mode 1 can be used to derive the equations for Mode 3 by turning the beam upside down and taking M = -M, V = -V and β1 =β3. eqs. (1.7) and (1.8) thus, respectively, become: 𝑇 − 𝑉𝜇 = 2 and
𝐴𝑤. 𝑓𝑦𝑤 𝑥1 . 𝑦1 . 𝑡𝑎𝑛𝛽3 𝑆
(1 − 18)
𝑇 − 𝑉𝜇 (1 + 2𝛼)𝑡𝑎𝑛𝛽3 − 𝑀 = 𝑀𝑜3 (1 − 19) 2 The term M o3 is the pure flexural strength in negative bending. , the strength in Mode 3 (being denoted by T3) is obtained by as:
15
Chapter One
𝑇3 =
Introduction
2𝑀𝑜1 . 𝐾1 1 𝑅3 1 −� + � � 𝜓∆′ ∆′ 𝐾1 (𝜓∆′)2
(1 − 20)
in which R3 = M o3 / M o1 , and Δ' = μ /(ψ.λ) -1 • Shear-compression mode of failure The three modes of failure, namely Mode 1, Mode 2, and Mode 3, as described above, may be termed as flexural type of torsional failure. Although the failure surface is skewed and warped, it has the general characteristics of the failure surface in pure flexure. As a result, the analysis is very similar to that for bending. These modes of failure usually occur when the beam contains adequate stirrups such that the main steel yields, and the full flexural strength of the beam is reached when loaded to failure. In the case of beams containing inadequate stirrups, the concrete compression zone may shear through prior to yielding of the main steel and this will precipitate failure at a load below the corresponding flexural failure load. This type of failure may be called Shear-compression mode of failure. A detailed analysis to predict the strength of the beam failing in this mode is rather too complex for practical use and is hardly justified because of the limited test data available. For beams without an opening, it was found empirically (Collins et al., 1968) that the possibility of a shear type failure could be checked in a single step by introducing an "equivalent shear" V eq ", as given by 𝑉𝑒𝑞 = 𝑉𝑢 +
1.6 𝑇 𝐵 𝑢
(1 − 21)
In which Vu and Tu are the factored shear and torsion, respectively, at
the section under consideration and B width of section. The shear compression strength of the beam can then be evaluated by means of the 16
Chapter One
Introduction
shear strength equation for a section using Veq instead of Vu to account for torsional effects. This procedure has been found to give results well on the conservative side. A similar equation has been proposed by Hasnat and Akhtaruzzaman (1987) to evaluate the shear compression strength of a beam containing a small opening. It is given as: 𝑉𝑒𝑞 = 𝑉𝑢 +
1.2 𝑇 𝐵 𝑢
(1 − 22)
1.6 Design for Torsion
In the preceding presentation of the skew-bending theory for torsion in concrete beams containing a small opening, the failure surface for a particular mode has been considered to pass through the center of the opening and encroach the solid part of the beam. Observed in many torsion tests (Hasnat and Akhtaruzzaman, 1987; Mansur and Hasnat, 1979; Mansur and Paramasivam, 1984), these failure modes are basically identical to those of a beam without an opening, and, hence, may be termed as "beam-type" failure. A careful examination of the equations derived reveals that only the reinforcement in the solid section outside the opening participates in carrying the external load. When sufficient rebars are used to prevent failure to occur in these modes, then there is a possibility that the failure may precipitate in the members above and below the opening. This type of failure is shown in Figure (1-10). It is similar to the "frame-type" shear failure of a beam with small openings and will be referred as "frame-type'' failure. In this type of failure, the entire applied actions are resisted, independent of the solid part of the beam, by the members framing the opening, and, hence, require a separate treatment in design
17
Chapter One
Introduction
Figure (1-10) Frame-type failure of a beam with a small opening under torsion
1.6.1 Design for Beam type Failure The modes of failure considered in the skew-bending theory are all considered as beam-type failure. These failure modes form the bases of the torsional design provision in the Australian Code, AS 1480 (1974), in which inclusion of the effect of torsion has essentially been reduced to usual flexural and shear design procedures. According to the code, the cross section is first proportioned on the basis of Mode 1 failure. Checks are then made and, if necessary, modifications are introduced to ensure that the beam will not fail in Modes 2 or 3, or the shear-compression mode. Steel percentages are also limited to guard against a primary crushing failure. Since the strength equations for different modes of failure remain basically similar when a small opening is introduced, and they are found to agree very well with reported test data (Hasnat and Akhtaruzzaman, 1987; Mansur and Hasnat, 1979; Mansur and Paramasivam, 1984), the same approach with minor modifications for the inclusion of opening may be used for designing such beams. The design steps involved to account for the various possible modes of failure under predominant torsion are described in sequence as follows. • Mode 1 failure Equations similar to those given in the Australian Code AS 1480 (1974) can be obtained directly from the following equations :
18
Chapter One
Introduction
𝑇 + 𝑉𝜇 = 2 and
𝐴𝑤. 𝑓𝑦𝑤 𝑥1 . 𝑦1 . 𝑡𝑎𝑛𝛽1 𝑆
(1 − 23)
𝑇 − 𝑉𝜇 (1 + 2𝛼)𝑡𝑎𝑛𝛽3 + 𝑀 = 𝑀𝑜1 2
(1 − 24)
According to the Code, the value of tan β1 is chosen as :
𝑡𝑎𝑛𝛽1 =
2
(1 − 25)
√1 + 2𝛼
Substitution of this value into Eqs. (1.23) and (1.24) gives: (𝑇 + 𝑉𝜇)√1 + 2𝛼 = 4
𝐴𝑤. 𝑓𝑦𝑤 𝑥1 . 𝑦1 𝑆
(1 − 26)
(𝑇 + 𝑉𝜇)√1 + 2𝛼 + 𝑀 = 𝑀𝑜1
(1 − 27)
In a design situation, the factored bending moment, torsional moment, and the shear force at the center of the opening are known. Thus, M = Mu, T = Tu, and V = Vu. Designating the first term of Eq. (1.27) as M eq(1) , the equivalent moment due to torsion and shear in Mode 1 failure, that is: 𝑀𝑒𝑞(1) = (𝑇 + 𝑉𝜇)√1 + 2𝛼
the required strength in positive bending becomes
(1 − 28)
𝑀𝑜1 = 𝑀𝑒𝑞(1) + 𝑀𝑢
(1 − 29)
The designer chooses a value of (α=h/b) if the sectional dimensions
are not given or known in advance, evaluates Meq(1), and then finds the value of Mo1. The section and the longitudinal reinforcement must be designed for this moment using the normal flexural design procedure. The transverse reinforcement is obtained from Eq. (1.26). Introducing a capacity reduction factor, Ø, the following equation is obtained:
19
Chapter One
Introduction
𝑀𝑒𝑞(1) 𝐴𝑤 1 = � � 𝜙 4𝑥1 . 𝑦1 . 𝑓𝑦𝑤 𝑠
(1 − 30)
• Mode 2 failure
In the Australian Code (1974), the design equations were derived by assuming that tan β2 is given by the following equation: 𝑡𝑎𝑛𝛽2 =
2
�1 + 2/𝛼 Substitution of Eq. (1.31) for tanβ2 in Eq. (1.15) yields: 𝑀𝑜2 = 𝑀𝑒𝑞(2)
(1 − 31) (1 − 32)
in which M eq(2) , the equivalent moment due to shear and torsion for Mode 2
failure, is given by: 𝑀𝑒𝑞(2) = �𝑇𝑢 + 𝑉𝑢
𝑥1 � 2
(1 − 33)
AS 1480-197 4 suggests that if Mu < 0.5 M eq(2) , the cross section of the beam and the area of longitudinal reinforcement should be such that the beam can withstand an equivalent bending moment, M eq(2) , as given above in lateral bending. Also, the area of web steel should not be less than that given by: 𝑀𝑒𝑞(1) 𝐴𝑤 1 = � � 𝜙 4𝑥1 . 𝑦1 . 𝑓𝑦𝑤 𝑠
(1 − 34)
• Mode 3 failure
Taking tan β3 = tan β1 which is given by Eq. (1.25), upon substitution, Eq. (1.19) then reduces to 𝑀𝑜3 = 𝑀𝑒𝑞(3) − 𝑀𝑢
(1 − 35)
𝑀𝑒𝑞(3) = (𝑇𝑢 + 𝑉𝑢 𝜇)√1 + 2𝛼
(1 − 36)
in which
20
Chapter One
Introduction
is the equivalent moment due to torsion and shear in Mode 3 failure. Thus, if the numerical value of Mu is greater than M eq(3) , there is no possibility of a Mode 3 failure. Physically it means that any tensile stress at the top of the beam induced by M eq(3) is canceled by the compression due to Mu. No top steel is, therefore, required. However, nominal steel comprising at least one bar at each of the top two corners must be provided for anchorage of stirrups. If Mu< M eq(3) , there will be residual tension at the top of the beam, and top steel should be introduced according to the usual flexural theory to withstand a negative bending (that is, one of opposite sign to Mu) of magnitude (M eq(3) -Mu). • Shear-compression mode of failure Similar to the Australian Code (1974) approach, Eq. (2.69) may be used to preclude a shear-compression mode of failure of a beam containing a small opening. Thus, the equivalent shear for this mode is calculated as: 𝑉𝑒𝑞 = 𝑉𝑢 +
1.6 𝑇 𝐵 𝑢
(1 − 37)
The transverse reinforcement is then designed to resist this equivalent shear on the basis of the normal shear design provisions by assuming that the failure plane passes through the center of opening. If the steel area, Aw, thus determined is greater than that already found during the previous design steps, the larger quantity should be adopted.
1.6.2 Design for Frame Type Failure This type of failure occurs when the members above and below the opening are not adequately reinforced for the actions being transmitted through them. In the case of combined bending and shear, the applied shear may be assumed to be shared by the chord members in proportion to their 21
Chapter One
Introduction
cross-sectional areas. Similarly, the applied torque, that produces lateral shear stresses in the chord members, may be assumed to be resisted by the couple formed by the resultant of these stresses, as shown in Figure (1-11).
Figure (1-11) Idealized free-body diagram at opening of a beam under loading
The above assumption may be justified from the work of Mansur et al. (1983). They have assumed that for a beam containing an opening, the applied torque is resisted by torsion in each member and by the couple formed by the lateral shear and shown analytically that the torsional component becomes smaller as the length of opening is decreased and eventually becomes negligible in comparison to the latter component (couple formed by the lateral shear) when the opening reduces to square (or circular) in size. Thus, referring to Figure (1-10), the lateral shear, Vz, may be assumed to be given by (𝑉𝑧 )𝑡 = (𝑉𝑧 )𝑏 =
𝑇𝑢 𝜃𝑡 + 𝜃𝑏
(1 − 38)
With the usual mechanism for applied bending moment, the problem of combined torsion, bending, and shear for frame-type failure thus reduces 22
Chapter One
Introduction
to designing each chord member at the opening for shear in two directions, as shown in Figure (1.10).
1.7 Fiber Reinforced Polymer FRP (Fiber ReinforcedComposite) A composite is a combination of two or more materials (reinforcing laments, fillers, and matrix binder) with different form or composition which, when combined into a material system, exhibit properties which are a combination of its individual components. The matrix can be a ceramic, metal, or polymer. Fillers may be mineral or metallic powders. Reinforcing can be particles, fibers, rods, or bars. On the other hand, reinforced concrete is a composite consisting of steel reinforcement, sand and gravel fillers, and a Portland cement matrix.
1.7.1 Fiber Materials Several materials are available for the fibers, e.g. glass, aramid and carbon. Almost 95 percent of all applications for strengthening purposes in civil engineering are by carbon fibers (Al-Tai, 2010). Figure(1-12), demonstrates some typical response of uniaxial loaded fiber materials and steel. HM and HS are abbreviations of high modulus of elasticity and high strength, respectively. Fibers have a linear elastic behavior until failure which is brittle. The fibers are what make the FRP strong and there are three things that control the mechanical properties of the FRP (Carolin, 2003):
23
Chapter One
Introduction
Figure (1-12): Stress-Strain Relationship of Fibers and Steel (Carolin, 2003) 1. Constituent materials As mentioned earlier there is a wide array of different materials to be used. What is important to remember is that the choice of fiber materials determines, together with choice of polymer, what kind of quality, properties and behavior the FRP finally will obtain. 2. Fiber amount Regarding the amount of fiber used in the FRP it is easy to say that the more fiber used the better properties will be achieved. This is somewhat true but with too high fiber content there will be manufacturing problems. If the fibers are tightly packed the matrix will have problems enclosing the fibers which might deteriorate the FRP. 3. Fiber orientation The FRP will be stiffer and stronger in the fiber direction. For example, a rod with all the fibers as very strong in its fiber direction but in 24
Chapter One
Introduction
perpendicular direction the FRP has not as good properties. A typical FRP product for the construction industry has therefore an anisotropic behavior compared to steel which is isotropic.
1.7.2 Matrix The matrix (the polymer in the composite) is used to bind the fibers together as shown in Figure(1.13-b), transfer the forces between the fibers and to protect the fibers from external mechanical and environmental damage. The shear forces created among the fibers are limited to the properties of the matrix. The matrix is also the limited factor when applying forces perpendicular to the fibers. It is important that the matrix has the capability to take higher strains than the fibers as shown in Figure (1-13-a), if not there will be cracks in the matrix before the fibers fail and fibers will be unprotected.
1.7.3 Properties of FRP 1.Non-Corrosive When dealing with infrastructure, corrosion is a major concern. The high degree of deterioration can be largely attributed to the corrosive property of metals. Composites alleviate the problem of corrosion because they do not rust.
2.Fatigue Life Because of the nature of the loading on bridge structures, the fatigue life is of great concern. Cyclic loading from traffic can lead to the distress of a structure. Most FRPs have an enhanced resistant to fatigue in a certain loading range as compared to conventional materials.
25
Chapter One
Introduction
Figure 1.13: (a) Stress-Strain Curves of Fibers, FRP and Matrix (Bisby, 2004) (b) Typical Composition of FRP Material (Federico, 2001)
3.Non-Magnetic Properties Steel reinforcement and members often inhibit the design of a structure when the interference of electromagnetic waves is of concern. Using FRP in place of the magnetic elements may be a solution to this problem.
4.High Specific Strength and Stiffness Two of the most important advantages of using FRP in civil infrastructure applications are the high specific strength to weight ratio and high stiffness to weight ratio. With the improvement of strength to weight ratio over other conventional materials the designer has the ability to use lower weights and thicknesses. With this in mind, designers are no longer 26
Chapter One
Introduction
limited by the ratios of conventional materials thus allowing new design concepts to be explored. By reducing the weight, installation time is reduced which results in a large reduction in cost. Once, the size of the superstructure is reduced and this will directly affect the required dimensions and capacity of the foundation and substructure.
5.Brittle Material Response FRPs do exhibit definite ductile behavior like metals as shown in Figure (1-13). The ductile behavior is desirable in a structure to provide ample warning to occupants prior to failure. However, FRP components can be designed to exhibit a sequence of damage mechanisms to ensure a relatively slow failure with extensive deformations.
1.8 Aims of the Study. The increase in the use of CFRP composite system for strengthening, repairing and rehabilitation of concrete structure in shear, flexure, torsion and axial lead to use these system in strengthening the web opening in the reinforced concrete curved beam. The present study aims to: 1- Investigate experimentally the behavior of continuous reinforced concrete circular curved beams with and without opening. 2- Investigate experimentally the behavior of reinforced concrete circular curved beams with openings strengthened by CFRP laminates . 3- Verify the adequacy of the design method suggested for straight reinforced concrete beam with opening to utilize for the reinforced concrete curved beam with openings. 4- Carry out finite element technique to analyze the nonlinear behavior of reinforced concrete curved beams with and without openings strengthened by CFRP laminates up to failure by using ANSYS (version 12.1) computer program. 27
Chapter One
Introduction
1.9 Layout of Thesis The present study consists of seven chapters as follows: The current chapter (chapter one) presents a general introduction about applications and effects of the opening in the reinforced concrete beams and their effect on behavior of R/C curved beams and properties of FRP fabrics. Chapter two reviews the most of the previous studies carried out in the field of the present study. In Chapter three an experimental program, detail specimens, material properties and test procedure are presented. Chapter four presents the review and analysis of the experimental results. Chapter five presents the material modeling used in the finite element analysis, types of elements used in idealization, in addition to nonlinear solution technique. Chapter six describes the theoretical results and also a comparison with the experimental results and parametric study of some variables. Chapter seven presents a summary of the conclusions of the present work and recommendations for further work.
28
Chapter Two
Literature Review
CHAPTER TWO LITERATURE REVIEW 2.1 Introduction Horizontally curved reinforced concrete beams occur frequently as members in buildings, bridges, and other structures. Methods for the determination of ultimate loads for straight beams with and without opening have been the subject of many studies and they are well established. Ultimate loads and nonlinear behavior for reinforced concrete curved beams have not been studied in sufficient detail The aim of this chapter is to present a review for the available information concerning the behavior of reinforced concrete (horizontally curved beams and straight beams) with and without opening strengthened and unstrengthened with CFRP laminates, especially those members which are subjected to torsion and shear.
2.2 Experimental and Analytical Studies on Reinforced Concrete Curved Beams. In 1963, Chu and Thelen preformed a plastic analysis for balcony girders of circular curved beam with a subtended angle of not more than 180 degree, with a constant cross section , fixed at both ends and subjected to uniform load perpendicular to the plane of the curved beam. In this analysis, as in any other plastic analysis, to determine the ultimate load the following conditions must be satisfied: (a) static equilibrium, (b) yield condition, (c) flow law, and (d) mechanism and compatibility conditions.
29
Chapter Two
Literature Review
A yield surface for combined bending and torsion, at a section where a plastic hinge was formed, and had been represented by:
m + t = 1 2
2
(2.1)
Fulfillment of the flow law requirement was achieved through the relationship: tαn γ =
t α ⋅ m
(2.2)
where
m = M / M P
(2.3)
t = T / TP
(2.4)
α = TP / M P
(2.5)
M : applied bending moment at cross section. T : applied torsional moment at cross section. M P : plastic bending capacity of cross section. TP : plastic torsional capacity of cross section.
γ : angle between the axis of rotation at the plastic hinge and the vector radius.
Figure (2-1) shows the yield surface for combined bending and torsion, at a section where a plastic hinge was formed. The general method of analysis was examined, the various mechanisms of failure were investigated, and the results of the load capacity of curved beams were presented in a chart to facilitate their application.
30
Chapter Two
Literature Review
Figure(2-1) yield surfaces for combined bending and torsion, at a section where a plastic hinge was formed. In 1972, Khalifa (mentioned by Jordaan et al. 1974) modified a plastic analysis for steel girders in the form of a circular arc subjected to single load and uniformly distributed loads to calculate the collapse load for reinforced concrete beams curved in plan. In 1974, Jordaan et al., developed plastic methods of analysis for the determination of collapse load for reinforced concrete curved beams and subtending a central angel less than 180 degree. In this study, the plastic analyses for the cases of a single point load besides uniformly distributed loads were extended to take into account two concentrated loads placed at any point on the beam. A plastic hinge was predicted at a particular cross section of the curved beam if the internal moments at that cross section satisfy the same yield criterion and flow law mentioned in eq. (2.1) and eq. (2.2) respectively. In the experimental part of this study, four curved and six straight reinforced concrete beams were tested. All the beams had constant moment 31
Chapter Two
Literature Review
and torque capacities at any section. The beams were completely fixed at the supports. Each curved beam had a radius of (2.21m) and a subtended angle of (86) degree as shown in figure (2-2). According to amount of reinforcement, the curved beams were divided into two groups. One curved beam from each group was tested under single point load and the other curved beams were tested under two concentrated loads.
B2
B2 6 in
A
A R=2.21m
B1
B1 Machine Head
86ο
Figure(2-2): Geometry of Test Specimens (Jordaan,1974). Straight beams were also divided into two groups. These two groups had similar reinforcement to that of curved beams group. One beam was tested under bending for each group while the other two beams from each group were tested under pure torsion, combined bending and torsion respectively. The purpose of these control tests (for straight beams) was to determine the flexural and torsional strength for each type of cross section and to check the interaction relationship given by yield criterion. 32
Chapter Two
Literature Review
In this paper, the author concluded that the plastic theory used could give satisfactory prediction of the mode of failure (the location and type of plastic hinges), but the prediction value of ultimate load was conservative for some modes and was slightly higher than experimental ultimate load for other modes. In 1977, Badawy et al. formulated two yield criteria to represent the behavior of a reinforced concrete section under the combined action of bending, torsion and shear, and the analysis was modified to include the effect of shear. They tested seven straight and eight curved beams, and the results were compared with the modified analysis. In the test on curved beams, the plastic hinge locations and consequently the modes of failure were recognized from the crack patterns, the deformed shape of the beam and the measured reinforcement strains. The workers observed the plastic hinges either torsion-shear hinges, flexural hinges, or bending-torsion-shear hinges. The analysis of the test results and the comparison with the results predicated by the plastic theory indicates the following conclusions: 1. The methods of plastic analysis can be applied to reinforced concrete curved beams. 2. An analysis using the first criterion gives a good predication of the ultimate load, mode of failure and the internal forces. Whereas an analysis using the second criterion establishes a lower bound for the ultimate load and the internal forces. The dimensionless equations for these two surfaces are : m2 +t2/(1-v2)=1
(2-6)
for the first yield criterion and m2 +t2/(1-v)2=1
(2-7)
for the second yield criterion where : m=M/M p , t=T/T p , v=V/V p ; M, T, and V are the bending moment, torsion and shear respectively; and M p , T p , and V pare the 33
Chapter Two
Literature Review
corresponding plastic capacities of the cross-section in pure bending, torsion, and shear. 3. Two types of redistribution of internal forces occur in a curved beam, one due to cracking and the other due to plastic hinge formation. 4. The effect of modes of failure is accurately predicted by the analysis. In 1977, Badawy et al. presented an experimental study to investigate the applicability of methods of plastic analysis to reinforced concrete horizontally curved beams that including shear effect. Eight curved and seven straight beams were tested. Each curved beam had a radius of (2.21 m) and a subtended angle of (75) degree as shown in Figure (2-3). Each curved beam was designed to be subjected to a single concentrated load. Four curved beams were reinforced identically but were tested under different end conditions. In the other curved beams, both the transverse reinforcement and the end conditions were varied. The straight beams were reinforced according to the reinforcement of the curved beams and were tested under different combinations of bending, torsion and shear.
Figure(2-3): Geometry of Test Specimens of Badawy et al (Badawy,1977). 34
Chapter Two
Literature Review
The results of the straight beams were used to determine the intersection point of the two yield criteria (proposed in an earlier investigation) with the bending, torsion, and shear axes and to check the validity of yield surfaces. This paper concluded that an analysis using the first criterion gives a good predication of the ultimate load, mode of failure, and the internal forces, whereas an analysis using the second criterion might establish a lower bound for the ultimate load and the internal forces. Besides, there were two types of redistribution of internal forces that might occur in a curved beam, one was due to cracking and the other was due to plastic hinge formation. In 1978, Hsu et al. tested seven reinforced concrete horizontally curved beams to investigate torsional and flexural moment redistribution after cracking and to suggest an appropriate design method based on this post-cracking behavior. Each beam had a radius of (2.74 m) and a subtended angle of (90) degree. Each beam was designed to be fixed at both ends and subjected to a concentrated load at mid span. The test program for the seven specimens was as following: the first and second beams were analyzed by the elastic theory based on uncracked sections and the reinforcement at each section was then designed by the ACI Building Code (1971). They were tested to check the applicability of conventional design method. The third and fourth beams were designed for only a portion of the maximum torsion moment calculated by the elastic analysis based on uncracked sections. The fifth and sixth beams were designed by the elastic method based on cracked section. The seventh beam was provided with uniform longitudinal bars and stirrups throughout its length to study the failure mechanism of a uniform beam. For cracked section, the torsional rigidity was determined by the post-cracking torsional 35
Chapter Two
Literature Review
rigidity of reinforced concrete member which was derived from space truss analogy by Hsu (1972). These tests showed that a significant redistribution of torsional and bending moments was observed after cracking. Therefore, the conventional design method based on "Building Code Requirements for Reinforced Concrete (ACI 318-71)" and elastic analysis assuming an uncracked section caused premature yielding at the supports. Besides, this paper recommended that the design of curved beam should be based on cracked section. In 1981, Abul Mansur and Rangan investigated three different methods for design of reinforced concrete horizontally curved beams to study the behavior, strength, mode of collapse, and economy of reinforcing steel. The design methods investigated were: 1) A collapse load method proposed by Badawy et al. (1977) 2) The classic elastic method based on uncracked sections 3) A limit design method proposed by the researchers themselves. The limit design method was based on assuming two additional conditions to obtain the solution for a statically indeterminate system. The first additional condition was provided by assuming an ultimate design torsional moment equal to [ 0.33 f ' x 2 y / 3 ] in(Nmm) at critical sections c 1 1 where x 1 and y 1 are, respectively, the smaller and the larger dimensions of a rectangular section in (mm), and f c' is the compressive strength of concrete in (MPa). The second additional condition was furnished by assuming the position of an inflection point (bending moment is zero) according to elastic analysis. Three curved beams designed by these methods were tested. Each beam had a radius of (2.45 m) and a subtended angle of (90) degree. The beams were completely fixed at the supports and subjected to a vertical point load. According to the results of this study, the authors concluded that: 36
Chapter Two
Literature Review
1) Redistribution of internal forces in a reinforced concrete curved beam occurs at two stages: one after cracking, and the other after the formation of some of the plastic hinges, and 2) The three methods gave satisfactory designs for reinforced concrete curved beams, but the collapse load method proposed by Badawy et al, required significantly more steel (especially hoops) than the conventional elastic design method and the limit design method.
Al-Temeemi (2002) studied the analysis of reinforced concrete horizontally curved beams on elastic foundations by the finite elements method. The material nonlinearity was taken into account for concrete and steel. A twenty-node isoperimetric brick element with sixty degrees of freedom is employed to model the concrete. The reinforcing bars are modeled as axial members embedded within the brick element. Soil is represented by normal and horizontal subgrade reactions. The normal component is represented by Winkler, Kondner, Polynomial models, while the horizontal component is represented by Winkler model. The finite element and the available experimental and numerical results have shown good agreement. Parametric studies have been carried out to examine the influence of some selected parameters (like radius to span-length ratio (R/L), boundary conditions, α1 (the rate of stress release as the crack widens), α2 (the sudden loss of stress at instant of cracking), and the type of soil on the overall behavior of reinforced concrete curved beam on elastic foundations. From the results obtained according the consider examples, it was found that the ultimate load of curved beam on elastic foundation could be increased with increasing the radius to span-length ratio (R/L)[ when (R/L) increases from 1 to 5, the ultimate load increases about 40% . The shear reinforcement bars have a significant effect on decreasing the ultimate load, it was found that 37
Chapter Two
Literature Review
the ultimate load decreases about 51 % when the shear reinforcement was removed. Ali A.Y. (2010) studied the analysis of reinforced concrete horizontally curved deep beams, loaded transversely to its plane, using a threedimensional nonlinear finite element model in the pre and post cracking levels and up to the ultimate load. The 20-node isoperimetric brick element with sixty degrees of freedom was of Ansys Program employed to model the concrete, while the reinforcing bars are modeled as axial members embedded within the concrete brick element. Perfect bond between the concrete and the reinforcing bars was assumed. The effects of some important numerical in addition to material parameters had been investigated to study their influence on the predicted load-deflection curves and the ultimate load capacity. The numerical study was carried out to investigate the effect of the central subtended angle, boundary conditions, amount of transverse reinforcement, and using additional longitudinal bars (horizontal shear reinforcement) on the behavior of reinforced concrete horizontally curved beams with different shear length to effective depth ratios (a/d). It was found that decreasing the central subtended angle causes an increase in the ultimate load resisted by curved beams. Also, it was found that the effect of the internal torsion and the amount of transverse reinforcement on the ultimate load resisted by curved beams was decrease as (a/d) ratio decrease, while the effect of the flexural moment and using additional longitudinal bars as a horizontal shear reinforcement was increase when (a/d) ratio decrease. Al-Mutairee (2013) studied the effects of non-uniform distribution of longitudinal reinforcements on the behavior of reinforced concrete (RC) horizontally curved beams with fixed-ends under static loads to product an optimal strength of these beams without increasing the volume of longitudinal reinforcement. Three dimensional nonlinear finite element analyses done utilizing computer program called NFHCBSL, incorporate 38
Chapter Two
Literature Review
20-node isoperimetric brick element used to represent the concrete elements while reinforcing bars are idealized as axial members embedded within the concrete elements without any relative displacement between them. The results show that the effect of non-uniform distributions of longitudinal reinforcement of RC horizontally curved beams with fixed-ends is effective and can be used to improve the strength of this type of beams and its importance increases with increasing the angle of horizontal curvature (θ).
2.3 Experimental and Analytical Works on Reinforced Concrete Beam with Openings. Many experimental and analytical studies on reinforced concrete beam with openings were made by several researchers. Burton (1965) studied the influence of embedded ducts on strength of continuous reinforced concrete Tbeams. He tested two reinforced concrete T-beams (one with ducts and the other solid). The results showed very little difference in performance of the two beams. He concluded that the design equations for ultimate strength contained in ACI-1963 Building code can safely be applied to wide shallow beams of the type used in his study with opening. In 1967 Nasser reported tests on reinforced concrete beams with openings and made the following assumptions:1. The top and bottom cross member at the opening are assumed to behave similar to the chords of a Vierendeel panel. 2. The cross members of the openings, when they are not subjected to transverse load, have centraflextural points at their mid spans. 3. There is a diagonal stress concentration at the corners induced by the chords shear leading to a force whose magnitude is twice that of the simple shear force.
39
Chapter Two
Literature Review
Depending on the mentioned
assumptions, the researcher tested nine
rectangular beams. His conclusions from the test results of the beams were: a- Large opening in rectangular reinforced concrete beams behave similar to Vierendeel panel. b- Adequately reinforced large opening in rectangular beams do not result in reducing the ultimate capacity of the beams, but it reduces its stiffness. Dalal(1969), studied the behavior and design of reinforced concrete T-beams with an opening in the web, and he proposed an empirical equation for calculating the area of reinforcement around the opening and established linear relationships between the ultimate shear carried by special web reinforcement and the size of opening. • Where:-
𝟏
Av= × 𝑸
𝑽𝒖𝟐 (𝟎.𝟏𝟖+𝟐.𝟓∝) (𝒔𝒊𝒏 𝜽 ×𝟎.𝟔ƒ𝒚)
× (𝟏 −
𝒂� 𝒅−𝟑.𝟕𝟖 ) 𝟓.𝟏𝟔
(2.8)
Av: area of special web reinforcement around the opening. Q = 0.85 Vu2: Ultimate shear force at tie center of the opening. α: Ratio of the length of the opening to the span of the beam. a: Distance between the support and the load for the test specimen. Ɵ: Angle of inclination of web reinforcement with the horizontal. 𝑎� : Shear-span ratio. 𝑑 Vu1= Vu (0.18+ 2.5 α)
(2.9)
where:Vu1= Ultimate shear carried by the web reinforcement. Vu= Total ultimate shear. Hanson(1969), tested a series of longitudinally reinforced T-beams representing a typical joist floor. The specimens contained square openings and were tested to simulate the joist on either side of a continuous support. Many parameters were used in this study, but the main ones were the size and horizontal and vertical locations of the opening. However, as the 40
Chapter Two
Literature Review
opening represents a source of weakness, the failure plane always passes through the opening, except when the opening is very close to the support so as to bypass the potential inclined failure plane. Figure (2-4) shows schematically some typical shear failures of beams containing square and circular openings. In 1974, Somes and Corley reported a similar study of Hanson (1969) but, in this case, the openings were circular in shape. In both cases (square and circular opening), it was found that an opening located adjacent to the center stub (support) produced no reduction in strength. As the opening is moved away from the support, gradual reduction in strength occurs until it levels off to a constant value. Test data suggest that the vertical position of opening has no significant effect, while an increase in the size of opening leads to an almost linear reduction in strength. However, there appears to be a size of opening below which no reduction in shear strength occurs. This size corresponds to about 25% of the beam depth for square openings and 33% of the beam depth for circular openings. They have also noted that the strength of such a longitudinally reinforced beam may be fully restored by providing stirrups on either side of the opening. Figure 2.4 shows schematically some typical shear failures of beams containing square and circular openings. To study the empirical equation for reinforcement around the opening as suggested by Dalal(1969) on the T-beams and to determine the effect of the variation of shear-span ratio, Jindal (1976) , tested seven rectangular beams with opening. He conclude the followings: • The behavior of a beam having an openings provided with special web reinforcement is similar to the solid beams, and • The response of the load deflection and moment rotation are similar in the both cases. 41
Chapter Two
Literature Review
• The beam with openings can be designed by providing special reinforcement around the opening.
Figure(2.4): Typical Shear Failure of a Beam with Small Openings Containing no Shear Reinforcement(Hanson 1969, Somes and Corley 1974)
To determine a reinforcement scheme suitable around the opening to restore the strength of a beam to the level of a corresponding solid beam, Salam (1977), conducted an investigation on beams with opening of rectangular cross section tested under two symmetrical point loads. He found that: • A short stirrup in the members both above and below the opening are necessary to eliminate the weakness due to the provision of opening, in addition to the longitudinal reinforcement above and below the opening and full depth stirrups by its sides. • When sufficient reinforcement is provided to prevent a failure along a diagonal crack passing through the center of the opening and traversing the entire depth, then the failure is precipitated at the minimum section. 42
Chapter Two
Literature Review
In such a case, formation of two independent diagonal cracks in the members above and below the opening split the beam into two separate segments. Figure (2-5) shows the sketch of the cracking pattern of the beam that failed in this manner.
Figure (2.5): Shear Failure of a Beam at the Throat Section by Salam (1977).
In 1978 Jamal, tested eleven reinforced concrete beams with opening. His study concentrated on the examining the influence of an opening in the flexural zone and the shear zone, the position of the opening through the depth of section and the influence of additional reinforcement on the capacity. It was found that an opening in the flexural zone dose not reduce the flexural capacity of the beam up to (h/d) ratio equal to (2/3), where h is the depth of the opening and (d) is the effective depth of the beam. However, the opening did reduce the stiffness of the beam and its energy absorption capacity. When the opening was situated in the shear zone a significant reduction in capacity resulted. Also, if it is not properly reinforced can lead to a very premature type of failure can occur. He presented methods of evaluating the capacity of beams with an opening in any zone and also generalization to the case of uniformly loaded beams. Mansur and Hasnat(1979), tested twenty-two reinforced concrete beams with small openings under torsion. The beams were divided into three groups according to nominal concrete strengths. In the first and the third groups, investigations were made for a 76mm hole size, whereas in the second group four different hole sizes of 51mm, 76mm, 102mm, and 127mm 43
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in diameter were used to study the effect of opening size. All beams were provided with both longitudinal and transverse reinforcement. Torsion load was applied in increments. After application of each load increment the load, rotation, and strains were recorded and cracks, if any, were marked. The theoretical predictions with ACI Code 318-77 are found to be in close agreement with the available test results Mansur and Paramasivam(1984), tested ten beams, each containing a symmetrically placed transverse circular opening in bending and torsion. The beams were divided into two groups designated as CA and CB, according to their cross-sectional dimensions which were 175 × 350mm and 200 × 400mm respectively. The amount and arrangement of reinforcement in each group was kept constant. Group CA consisted of three specimens with different size openings. These specimens were tested under pure torsion. For group CB, opening size was held constant, but the torsion to moment ratio was varied from pure torsion to pure bending. The load was applied in increments. After each increment, steel strains, deflections, twists, and maximum crack width were recorded. Torsion strength and stiffness of a beam decreases with an increase in opening size. It was concluded that, the presence of small amount of bending moment increases the torsion capacity of a beam. But for substantial bending moment the increase in bending moment leads to a drop in beam torsional capacity. This is in agreement with the findings of earlier investigators. Mansur et al.(1985), tested twelve beams designed by the proposed method of the ACI Code 318-83, under a point load to study the flexural behavior of reinforced concrete beams with large rectangular openings that were subjected to bending and shear. The major variables were the length, depth, eccentricity and location of openings, and the amount and arrangement of corner reinforcement. He conclude that at a particular load both the maximum crack width and maximum beam deflection increase with 44
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an increase in opening length, opening depth, or moment shear ratio at the center of the opening. Also, diagonal bars as corner reinforcement were found to be more effective in crack and deflection control than vertical stirrups. Based on test results, a suitable quantity of corner reinforcement was recommended. Mansur et al(1991), tested eight reinforced-concrete continuous beams each containing a large transverse opening. The beams are rectangular in cross section and all contain the same amount and arrangement of longitudinal reinforcement. The number of spans, the size of opening, and its location along the span are considered as major variables. Test results indicate four distinctly different stages of behavior in the load-deflection curve of a continuous beam. Final failure of the beam occurs by the formation of a mechanism, and the two opening ends represent the most vulnerable locations for the development of plastic hinges. Besides early cracking, the strength and stiffness of the beam decrease with an increase either in the length or depth of opening. Similarly, openings located in a high moment region produce larger deflections and result in early collapse of the beam. To study the deflection behavior of reinforced concrete beams with web openings, Mansur et al(1992), tested twenty-two beams (14 simply supported, 3 two-span continuous, and 5 three-span continuous beams), each containing a large rectangular opening. The direct stiffness method were used analyze the beams. The beams were treated as structural members comprising several segments, and an equivalent stiffness had been derived for the segments traversed by the openings. A comparison of the experimental results with the predicted deflections obtained from the direct stiffness method had good agreement. They concluded that, the method also gives a reasonably accurate picture of the redistribution of internal forces and moments in continuous beams due to incorporating opening. 45
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Tan et al(1996) ,studied the flexural behavior of reinforced concrete T-beams with and without large web opening in positive and negative moment regions. Fifteen beams were tested under a point load. The tests had indicated the following : • The presence of web openings leads to a decrease in the cracking and ultimate strength as well as the post-cracking stiffness of continuous beams. • Performance of a beam with multiple opening is more desirable in terms of strength and serviceability. The thickness of the post between the adjacent openings should not less than one-half of the overall beam depth. Tan and Mansur(1996), suggested useful procedure for complete analysis and design of reinforced concrete beams with large web opening. Also they suggested the guidelines to facilitate the selection of the size and location of web openings. Generally, the following should be considered: • Openings should be positioned so that chords have sufficient concrete area to develop the ultimate compression block in flexure and adequate depth to provide effective shear reinforcement. • They should not be deeper than one-half the beam depth • Should be located not closer than one-half the beam depth from supports or concentrated loads. For structural analysis of reinforced concrete beams with openings, they showed that in the case of a statically determinate beam, shear force and bending moment envelopes can be obtained from statics. For continuous beams, they suggested method can be followed, that is, the member containing an opening is considered as a nonprismatic beam with different cross sectional properties: those of a solid section and of the equivalent section for opening segments. They gave the recommended design process 46
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for the opening segment which is based on the observed Vierendeel behavior of chord members at an opening. The behavior and design of a beam containing a transverse opening and subjected to a predominant shear are discussed by Mansur (1998). Based on the observed structural response, some guidelines are suggested to classify the opening as ‘large’ or ‘small’. For small openings, two different failure modes are identified. These types of failure may be labeled as ''beamtype'' failure and ''frame-type'' failure, respectively, and required separate treatment for complete design. In beam-type failure, a an inclined failure plane of angle 45°, similar to solid beam may be assumed, the plane being traversed through the center of the opening, as shown in figure (2-6a). Frame-Type failure occurs due to the formation of two independent diagonal cracks, one in each of the chord members bridging the two solid-beam segments, leads to the failure, as shown in figure (2-6b). In the proposed method, the maximum shear allowed in the section to avoid diagonal compression failure has been assumed to be same as that for solid beam except for considering the net section through the opening.
Figure(2-6): Modes of Failure for Small Opening ,Mansur (1998).
To simulate drilling of holes in an existing beam either for the passage of service ducts or for the determination of in place concrete strength, Weng wei et al (1999) tested nine T-beams were fabricated with circular openings through the web to simulate drilling of holes in an existing beam either for 47
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the passage of service ducts or for the determination of in place concrete strength. The major parameters considered in the study were the size and location of openings. Test results indicated that an opening, when created near the support region of an existing beam, leads to early diagonal cracking and significantly reduces the strength and stiffness of the beam. They concluded that filling an opening by nonshrink grout, as is usually done for openings created by removing concrete cores for the determination of in place concrete strength of an old building, is not adequate to restore the original strength and stiffness, their study reveals that the weakness introduced in terms of cracking, deflection, and ultimate strength by creating an opening in existing beams can be completely eliminated by strengthening the opening region of the beam using a suitable method, like the use of externally bonded FRP plates as used here. In other hand they found that after first shear cracks occurred however, the compression chord tended to carry a larger proportion of the shear force, and this increased with an increase in the applied load. The fraction of the applied shear carried by the compression chord at ultimate ranges is from 65 to 90%. Neff et al(2002), tested six full-type laboratory beams. They were conducted in order to analyze the behavior of reinforced concrete beams with one large rectangular opening. Thus, the shear force distribution and the location of the contra flexure points have been examined. On the basis of the finite element computations, a design concept for beams with one large rectangular opening had developed, which now can be proved by means of the test results. The results of the test series and the finite element calculations have been compared and they have shown that the distribution of the shear force depends on the ratio between the stiffnesses of the chords. Abdulla et al.( 2003), tested ten reinforced concrete beams with openings and with CFRP sheets. The experimental program included strengthening five beams with openings, while four beams kept without 48
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strengthening and the last one was solid beam used as control beam. It was found that, when the opening with height of 0.6 from the beam depth reduce the capacity by 75%. The application of the CFRP sheet around the opening was greatly decreased the beam deflection, controlled the cracks around the openings and increased the ultimate load capacity of the beam. The failure occurred due to a combination of shear cracking of concrete and bond failure of CFRP sheets glued to concrete surface. Figure (2-7) explain internal steel reinforcement and types of external CFRP for strengthening.
Figure(2-7): Details of the tested beams: (a) internal steel reinforcement and (b) types of external CFRP strengthening.(Abdulla,2003) Mohammed(2004), used a three-dimensional nonlinear finite element model to investigate the behavior of reinforced concrete beams with large transverse opening under torsion. The 20-node isoparametric brick elements have been used to model the concrete. The reinforcing bars were idealized as axial members embedded within the concrete element and perfect bond between the concrete and the reinforcement has been assumed to occur. In 49
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general good agreement between the finite element solutions and the experimental results has been obtained. The finite element solutions revealed that the collapse torque and post-cracking tensional stiffness decrease with the increase of length or depth of the opening. Al-Kaisy(2005), investigated the behavior of reinforced concrete beams with large transverse openings subjected to flexure by using a threedimensional nonlinear finite element model. Several parametric studies have been carried out to investigate the effect of some important finite element and material parameters. The effect of concrete compressive strength, amount of longitudinal tensile reinforcement and opening size on the loaddeflection response has been investigated. In general good agreement between the finite element solutions and the experimental results has been obtained. The finite element solutions revealed that the ultimate load and post-cracking stiffness decrease with the increase of length or depth of opening. Akinobu Suzuki et al(2007), tested ten reinforced concrete beams with and without opening for the purpose of comparison. The beams were prepared into presence or absence of web opening, method of reinforcement of beam with web-openings. Difference between beam without shear reinforcement, beam with shear reinforcement only, and beam with shear reinforcement and stirrups, concrete compressive strength: Two levels of concrete design strength (Fc)–36 N/mm2 and 60 N/mm2 and method of load application: Difference between multi-cyclic loading and monotonic loading. They discussed Arakawa’s formula for evaluating shear strength was applied to the beams without opening and Hirosawa’s formula was applied to the beams with an opening. They found that the non-reinforced specimens with openings showed a marked decline in shear resistance compared with the other specimens. In contrast to this, the specimens provided with shear reinforcement and reinforcement near the openings were measured to have a 50
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higher shear resistance than the calculated shear resistance (Hirosawa’s formula).They concluded that provide an additional stirrups around the web opening improved shear resistance Najim(2009), tested eight reinforced concrete deep beams with crosssection of (100x750mm) and total length (1150mm) were tested under twopoint loads. Seven specimens had two rectangular openings with dimensions (100x200 mm), one in each center of the shear span, placed symmetrically about the centerline of the beam. The experimental results indicated that the use of CFRP sheet to upgrade the R.C. deep beams with web openings has significant effect on overall behavior such as the ultimate load, crack width and deflection. The percent of increase in the ultimate load capacity was about (100-190) %. In the other side, three dimensional finite element analysis was used to investigate the performance of the R.C. member strengthened by CFRP laminate. ANSYS computer program (version 9.0, 2004) was utilized through this study. The comparison between the experimental and theoretical results showed reasonable agreement and asserted the validity of the numerical analysis and methodology developed in this study. Bandyapadhya et al (2011), discusse the use of Glass Fiber Reinforced Polymer (GFRP) to strengthen and rehabilitate. In this experiment ten beams with length of 2.0 m for each one, one solid as reference beam and other nine beams categorized as beams with openings (strengthened and rehabilitated beams) are tested. The opening size of these beams is increased from (100x100) mm to (100x300) mm and with different locations. They concluded that GFRP can be used to strengthen and rehabilitate the beams with small opening only. Also, the beams with openings only show the maximum deflection at a point which is in between the middle point of beam and middle point of opening instead of maximum deflection at central point of solid beam. 51
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Ali et al(2011), made an experimental study to clarify the possibilities and drawbacks of using composite materials for the purpose of strengthening or retrofitting concrete structures. There general goal is divided into two aims. First one is to investigate the effect of the shape and dimensions of opening on the behavior, and the second is to examine the effectiveness of CFRP reinforcement in enhancing the flexural capacity of RC beams with opening at the flexural region. Nine beams with length of 1.0 m total length and (150 by 100) mm cross section dimensions were tested. The opening position was fixed at the mid span of the beams for all sizes and shapes. All beams were tested as a simple beam subjected to two concentrated loads. They conclude that the flexural strengthening of R.C. beams with CFRP is effective with increment in deflection of about (310,300 and 360)% for beams with (L/h) 1.8, 5 and 20 respectively, but it is reduce the ultimate load of about (35,16 and 15)% respectively. On the other hand, tests have shown that the CFRP elongation is a major factor that could affect shear crack propagation). Al-Dolaimy(2011),tested eight continuous reinforced concrete beams with cross-section of (150x250mm) and total length (3300mm) with two spans (clear span is 1500 mm) were tested under two-point loads. Six beams, each beam contains one opening, three of which have dimensions (200x100 mm) and the three other have dimensions (140x140mm). The location of opening was in the zone of maximum moment and maximum shear. The experimental results indicated that the use of CFRP sheet to upgrade the R.C. continuous beams with web openings has significant effect on overall behavior such as the ultimate load, crack width and deflection. The percent of increase in the ultimate load capacity was about (60-106) %. While the inclined strengthening model (strengthened continuous beam with opening 200X100mm) shows a stiffer response compared with other strengthened model, which gives 106% increase in ultimate load than (control unstrengthened continuous beam with opening 200X100mm) and 13 52
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% more than (strengthened continuous beam with opening 200X100mm). The inclined stirrups with angle of 70˚ are better than the stirrups with right angle because the stirrups with70˚ is perpendicular on the crack path. Also, the use of CFRP as externally strengthening system decrease the crack width, but, it cannot change the formation of inclined crack. All the tested beams with opening failed due to splitting the beam into two pieces though the inclined crack. In the other hand, three dimensional finite element analysis was used to investigate the performance of the R.C. member strengthened by CFRP laminate. ANSYS computer program (version 9.0, 2004) was utilized through this study. The comparison between the experimental and theoretical results showed reasonable agreement and asserted the validity of the numerical analysis and methodology developed in this study. Hamza (2012), investigates the behavior and performance of reinforced concrete arch beams with and without openings, unstrengthened and strengthened (externally by CFRP laminates or internally by steel reinforcement). The experimental work consists of fabrication and testing of twelve reinforced concrete arch beams. Eight of these beams had rectangular openings. The experimental variables considered in the test program included: curvature forces, location of opening through profile of arch, present of (internal strengthening steel reinforcement or external strengthening CFRP laminates) for opening. The beams were tested under two points loading at top edges with (hinge-roller) supports at bottom. The experimental results showed a significant decrease in ultimate load capacity by about 38 % which were registered for arch beam without confining stirrups to resist forces in radial direction (curvature). The mode of failure is splitting failure, it occurs suddenly with no advance warning of distress. The use of CFRP laminates as external confinement to resist 53
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curvature forces which gives a little increase in ultimate load capacity by about 2.7% , consequently, the ductility ratio decreased rapidly by about 60%, as compared with control arch. The presence of opening at zone of [ pure bending, combined of(bending, shear force and axial compression force) and excessive compressive force ( near the support)] leads to decrease in load capacity about 58%, 60% and 42%, respectively. Patel and Saksena (2013), Investigate the effect of small circular opening on the shear and flexural and ultimate strength of beams. The main factors of the test are the changes of diameter and the position of opening. In this investigation five beams using C20 concrete and Fy415 steel were casted and tested. First beam was solid and was used as reference for comparison with other beams with an opening. Second beam had opening of 110mm (0.55D) at L/8 distance, third beam had opening of 90mm (0.45D) at L/8 distance. Beam number four and beam number five had openings as mentioned above at L/4 distance. The tested beams have been loaded as simple beam with two concentrated and symmetrical load. They conclude that when the diameter of opening increased, the reduction of ultimate strength increased and patterned of cracking as well as mode of failure
of
the
beam changed. They recommended usage of diagonal
reinforcement and stirrups in top and bottom of opening for increasing the ultimate shear strength of the beam. They also concluded that the most critical position of opening to reach the ultimate strength in beams is near the support and also the best place for the location of opening in these beams is in middle of a beam (flexure zone).
2.4 Concluding Remarks The experimental and theoretical studies concerning the behavior of concrete beams with opening can be summarized as following: 54
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• Many theoretical studies about the reinforced concrete curved beams were not complete. • Numerous experimental and theoretical studies concerning the behavior of concrete beams with opening have been carried out. The majority of these studies dealt with the straight members like simply supported beams, continuous beams and deep beams with openings. Few studies were about the nonlinear behavior of reinforced concrete curved beams. The overall response of a straight beam and arch beam with opening and horizontally concrete curved beams as reported in the above literature may be summarized as follows: 1.The presence of web opening leads to decrease in both cracking and ultimate load, as well as post cracking stiffness of beams. 2. Unless additional reinforcement is provided to restrict the growth of cracks, the opening corners are liable to exhibit wide cracking. 3. Performance of multiple opening is more desirable in terms of strength and serviceability. The thickness of the post between adjacent openings should not be less than one- half of the overall beam depth, and the post should be adequately reinforced to avoid premature failure. 4. The increase in the opening size either by increasing the length or the depth of opening decreases the strength as well as stiffness of the beam. The eccentricity of opening, however, has only a marginal effect on both strength and stiffness. 5. Specifying the type of the opening (large or small) is depending on the type of structural response of the beam. 6. A large rectangular opening behaves similarly to Vierendeel-panel at the opening segment. Under combined bending and shear, the chord members bend in double curvatures with contraflexure points located approximately at mid span of the chords. Final failure occurs by the formation of a 55
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mechanism with four hinges in the chords, one at each corner of the opening. 7. The application of the CFRP sheet around the opening generally decreased the beam deflection, controlled the cracks around the openings and increased the ultimate load capacity of the beam and changes the failure mode of the beam. It's well-known from the literature mentioned above there is no available technical data about horizontally curved reinforced concrete beam with openings. The present work enhance the knowledge of overall behavior of reinforced concrete circular curved beams with and without opening, strengthened and unstrengthened by internal reinforcing steel or external CFRP laminates through experimental work and numerically by three dimensional finite element method (ANSYS.12.1 package).
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CHAPTER THREE EXPERIMENTAL WORK 3.1 General Although the use of reinforced concrete curved beams leads to several advantages in architectural plans, the fabrication of laboratory models of such type of construction seems to be relatively difficult, costly and time consuming process. In spite of this fact, an experimental investigation was carried out in this study to establish the structural behavior of fourteen specimens of reinforced concrete curved beams with and without opening, strengthened and unstrengthened under effect of two or four point loads. Firstly, the experimental work will be described, which includes the main variables; types of curved (geometry) beam, location of opening through profile of curved beams (effect of combination of internal forces) and, presence of internal strengthening by steel reinforcement and external strengthening by CFRP laminates around opening. Details of casting the specimens, strengthening by internal reinforcement and externally with CFRP, testing procedure and measuring instruments are also presented in this chapter. Then, standard tests are presented in this chapter to determine the properties of concrete and steel reinforcement used in this study.
3.2 Description of Test Specimens The experimental program includes two groups of specimens, the first group consists of ten semicircular continuous curved beams of two spans simply supported at external ends and roller at middle support, with
57
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and without web openings. The second group consists of four closed circular (ring) beams of four roller supports, with and without web openings. Beams of the first group have an inner diameter 2000 mm and outer diameter 2250 mm, and having cross section of dimensions 250 mm overall depth and 125 mm width, as shown in Figure (3-1). These beams were tested under the effect of two point loads located at mid of each span (angle 45°). Steel reinforcement (2Ø12)mm deformed bars were provided for top and bottom for positive and negative moment regions with clear cover of 25 mm. The closed stirrups of Ø6 mm reinforcing bar were placed at angle 4.5o along the beam length (see Appendix A ). P/2
Ø6 @ 4.5°
(125x50x8)mm
2Ø12 (125x50x8)mm
P/2 (125x50x8)mm
250 mm (125x50x8) mm
2Ø12
125 mm (125x50x8)mm
(a)
(b) (c) 1.125 m
Figure(3-1) Details for Group I of Tested Curved Beams a) Geometry and Load , b) Cross Section and Reinforcement , c) Front View
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Beams of second group have an inner diameter 950 mm and outer diameter 1200 mm, and had a cross section of dimensions 250 mm overall depth and 125 mm width as shown in Figure (3-2). These beams were tested under the effect of four point loads located at mid of each span of the beam (of angle 45°) at top surface. Steel reinforcement (2Ø12)mm deformed bars were provided for top and bottom for positive and negative moment regions with clear cover of 25 mm. The closed stirrups of Ø6 mm reinforcing bar were placed at angle 11o along the beam length All curved beams of the two groups had openings with dimensions of (100*200 mm) and each group had a control beam without opening (solid). Table (3.1) illustrates identification, geometry and details of reinforcement around openings. P/4
Ø6 @ 11°
P/4
P/4
2Ø12 250 mm
(125x50x8)mm
2Ø12
P/4
125 mm (a)
(125x50x8)mm
(b) (c)
Figure(3-2) Details for Group II of Tested Ring Beams a) Geometry and Load , b) Cross Section and Reinforcement , c) Front View
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3.2.1 Semicircular Curved Beams (Group I) 3.2.1.1 Semicircular Curved Beams without Opening This type consists of two curved beams. One of them represents the pilot beam of a semicircular curved beam and the other represents a control beam of the same type. Pilot beam is similar to control beam, and it was used to test the devices and loading test system which will be used for the test of other semicircular curved beams.
3.2.1.2 Semicircular Curved Beams with Opening This type consists of nine semicircular curved concrete beams. Three curved beams are with two opening each located at distance ‘ d/2 ‘ from each exterior support Figure (3-3-a), the other three curved beams are with two opening each located at distance ‘ d/2 ‘ from point load toward interior support (3-3-b), and the last three curved beams with two opening each located at distance ‘ d/2 ‘ from both sides of interior support (3-3-c). The first type of opening without any strengthening, the second type of opening was strengthened with internal steel reinforcement stirrups and the last type strengthened with CFRP laminates as illustrated in Table 3.1.
3.2.2 Ring Curved Beams (Group 2) 3.2.2.1 Ring Curved Beams without opening This type consists of one ring beams represents a control beam of the second group (full circular beams). It is used as a reference beam in which results of other ring beams will be compared with it. This ring beam has no opening at any part of its length.
3.2.2.2 Ring Curved Beams with opening This type consists of three full curved concrete beams (ring beams). These full curved beams are with opening spaced at distance ‘d/2 ‘ from applied load. One of them without strengthened; the other one was strengthened internally using steel reinforcement (stirrups), while the last 60
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one strengthened externally using CFRP laminates. Identifications for each of these full curved beams and details of reinforcement around opening are illustrated in Table 3.1.
Group No.
Table 3.1 Descriptions of Tested Specimens Specimen Location of Designation
Details of Reinforcement
External CFRP Laminates
Opening
Around Opening
Around Opening
---
---
---
SCB.P
Near Exterior --Support Near Exterior 3Ø6for each cord, SCB.Esr Support 1Ø6 diagonal bar for each corner, 1Ø6 at each side Near Exterior SCB.Ecfrp --Support Near Applied SCB.Mo --Load Near Applied 6Ø6for each cord, SCB.Msr Load 2Ø6 diagonal bar for each corner, 2Ø6 at each side Near Applied SCB.Mcfrp --Load Near Interior SCB.Io --Support Near Interior 6Ø6for each cord, SCB.Isr Support 2Ø6 diagonal bar for each corner, 2Ø6 at each side SCB.Icfrp Near Interior --Support
Group 1
SCB.Eo
FCB
---
---
--1of 20mm width on each side 3of 20mm for each cord ----1of 25mm width on each side 3of 20mm for each cord ----1of 25mm width on each side 3of 20mm for each cord ---
Near Applied ----Load Near Applied 6Ø6for each cord, FCB.Msr Load 2Ø6 diagonal bar for each corner, -2Ø6 at each side 1of 25mm width on each side FCB.Mcfrp Near Applied --Load 3of 25mm for each cord FCB.Mo
Group 2
---
(*) definitions of symbols SCB = Semi-circular curved beam FCB= Full circular curved beam P = Control beam Ei = Exterior location of opening Mi = middle location of opening Ii= Interior location of opening
Subscript (i) denotes one of the following : o = no strengthening for opening location sr = strengthening using steel reinforcement cfrp = strengthening using cfrp laminates
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3.3 Strengthening System (schemes) Strengthening system was chosen carefully according to crack pattern and failure mode. The method of design adopted for strengthening technique had been suggested by (Mansur, 1998) for straight beam. The design specification of ACI 318-2011 and ACI Committee 440-2002 was satisfied for steel bars reinforcement and CFRP laminates, respectively.
3.3.1 Internal Strengthening by Internal Reinforcement Curved beam SCB.Esr was strengthened by one full depth closed stirrups Ø6 mm on each side of the opening and three pairs of deformed bars of Ø6 mm closed stirrups for top and bottom chords of opening and one diagonal bars Ø6 mm for each corner of opening. Curved beams SCB.Isr, SCB.Msr and FCB.Msr were strengthened by two full depth closed stirrups Ø6 mm on each side of the opening and six pairs closed stirrups of Ø6 mm for top and bottom chords of opening and two diagonal bars Ø6 mm for each corner of opening. Layouts of steel reinforcement scheme for all tested curved beams internally strengthened by steel reinforcement are shown in Figure (3-3).
3.3.2 External Strengthening by CFRP Laminates Beam SCB.Ecfrp was strengthened by two full wrap of CFRP laminates of 0.131 mm thickness and 20 mm width at each side of the opening and three pairs of 20 mm for both top and bottom chords of opening . Beams SCB.Mcfrp and SCB.Icfrp were strengthened by pair of full wrap of CFRP straps of 0.131mm thickness and 25 mm width at each side of the opening and three pairs of 0.131mm thickness and 20mm width for both top and bottom chords of opening. Beam FCB.Mcfrp was strengthened by pair of full wrap of CFRP straps of 0.131mm thickness and 25 mm width on the each side of the opening and three pairs of 20
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mm width for both top and bottom chords of opening. Details of externally strengthening by CFRP laminates for all tested curved beams are shown in Figure (3-4). More details about the method of strengthening of beams are illustrated in Appendices A&B. -1Ø6mm full depth stirrups at each side of opening -1Ø6mm at each corner of opening -3Ø6mm stirrups for each cord of opening
-2Ø6mm at full depth stirrups at each side of opening -2Ø6mm at each corner of opening -6Ø6mm stirrups at each cord of opening
-2Ø6mm at full depth stirrups at each side of opening -2Ø6mm at each corner of opening -6Ø6mm stirrups at each cord of opening
-2Ø6mm at full depth stirrups at each side of opening -2Ø6mm at each corner of opening -6Ø6mm stirrups at each cord of opening
Figure (3-3) Details For Internally Strengthening by Steel Reinforcement Around Opening
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(a
(b
(c
FCB.Mcfrp
(d
Figure (3-4) External Strengthening Around Openings by CFRP Laminates 64
Chapter Three
Experimental Work
3.4 Material Properties of Tested Specimens 3.4.1 Concrete The materials used in producing concrete are locally available materials, which include cement, natural gravel, natural sand and water. These materials were tested in laboratories of Engineering College of Karbala University.
3.4.1.1 Cement Ordinary Portland cement (Iraqi manufacturing) named Tasloga was used throughout this investigation for casting all the specimens. The cement was kept in air-tide plastic containers to avoid exposure to the atmosphere. This conforms to Iraqi Standard Specification No. 5:1984 as illustrated in Table 3.2. Table 3.2: Chemical and Physical Test Results of the Cement Chemical Properties
Test Result
Limit according to IQS
CaO
%
60.21
--------------------
SiO 2
%
19.12
--------------------
Al 2 O 3
%
4.30
--------------------
Fe 2 O 3
%
2.42
-------------------
MgO
%
3.57
≤ 5%
SO 3
%
1.37
≤2.5%ifC 3 A< 5% ≤2.8%ifC 3 A> 5%
Free Lime %
1.07
Loss on Ignition %
1.5
≤4%
Insoluble Resdue %
0.87
≤1.5 %
L.S.F
0.89
0.66-1.02
M.S
2.33
----------------
M.A
1.54
---------------
65
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Table 3-2 : Continue Physical Properties
Test Result
Limit according to IQS
Setting Time, min,
120 Initial
≥45
240 final
≤600
330
≥230
Strength , MPa
21.0
≥15
3days
31.2
≥23
Fineness (Blaine), m2/kg Compressive
7days
3.4.1.2 Coarse Aggregate (Gravel) A maximum size of 19 mm of crushed gravel from Al-Nibaey was used in the current study. The gravel was washed and cleaned by water several times and left to dry in air. Table 3.4 shows the grading of aggregate and the limits specified by the Iraqi Specification No.45/1984.
3.4.1.3 Fine Aggregate (Sand) Natural sand from Karbala of maximum size of 4.75 mm was used in this investigation. Before being ready to use, the sand was washed and cleaned by water several times, later it was spread out and left to dry in air to avoid the humidity saturation which may affect the water content extensively. The grading test results conform to Iraqi specification No. 45/1984 specification. Table 3.5 shows the properties of fine aggregate.
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Chapter Three
Experimental Work
Table 3.3 : Grading of Coarse Aggregate %, Passing
Sieve No.
size (mm)
Coarse Aggregate
Limit of Iraqi Specification No. 45/1984
1
19
99
95-100
2
14
96
-
3
10
48
30-60
4
5
1
0-10
5
2.36
0
-
SO 3 content=0.08% (specification requirements up to 0.1%)
Table 3.4:Fine Aggregate Properties No. Sieve size (mm)
%
Limit of Iraqi Specification
Passing
No.45/1984,zone (2)
1
10
100
100
2
4.75
97
90-100
3
2.36
82
75-100
4
1.18
67
55-90
5
0.6
51
35-59
6
0.3
23
8-30
7
0.15
2
0-10
SO 3 content=0.29% (specification requirements up to 0.5%)
67
Chapter Three
Experimental Work
3.4.1.4 Mixing Water Water used in mixing concrete should be clean and free from injurious amounts of oils, acid, alkalis, salts, organic materials, or other substances deleterious to concrete or reinforcement. In engineering practice, the strength of concrete at a given age and cured in water at a prescribed temperature is assumed to depend primarily on two factors only, the W/C ratio and the degree of compaction. In this study, the tap water has been used for mixing concrete and curing of all spacimens.
3.4.1.5 Mix Design Several trial mixes were made according to the recommendations of the ACI 211.1-97. Reference concrete mixture was designed to achieve normal strength of concrete (about 28 MPa) at (28) days. The mixture was (1 cement: 2.2 sand: 2.42 gravel: 0.55 water cement ratio, by weight), and the slump was approximately (60 mm). Mixture details are given in Table 3.5. It was found that the used mixture produces good workability and uniform mixing of concrete without segregation. Table 3.5: Properties of Concrete Mix Parameter
Normal strength concrete
Water/cement ratio
0.55
Water (kg/m3)
205
Cement (kg/m3)
373
Fine aggregate (kg/m3)
623
Coarse aggregate (kg/m3)
900
68
Chapter Three
Experimental Work
3.4.1.6 Mixing Procedure The mixing procedure was as follows: (1) Before mixing, all quantities were weighted and packed in a clean container. (2) The coarse and fine aggregate are stocked and blended in 2/3 of the required water for 60 seconds in a horizontal rotary mixer with 0.2 m3 capacity. (3) The cement and rest water were added and mixed for 3 minutes. (4) The cylindrical molds were filled with concrete into three layers; each layer was compacted. The concrete was poured into lightly oiled steel mold in two layers, and each layer was vibrated by mechanical vibrator(300 rpm), 5 second for each insertion which gave adequate compaction. The surface of the concrete was leveled off and finished with a trowel. Then, the specimens were covered with a nylon sheet to prevent evaporation of water. (5) All specimens were left in the laboratory until they were removed from their mold after 24 hours, and then burlap sacks were placed over the beams and kept wet for (28) days. The same procedure was followed for the concrete cylinders. After (28) days, they were taken out of the water and left to dry for (24) hours and then tested in accordance with the standard specifications.
3.4.2 Reinforcing Steel Two sizes of reinforcing steel bars were used in the tested beams, deformed bars of size ( Ø12mm) for main longitudinal reinforcement (circumference )and deformed bars of size (Ø6 mm) for closed stirrups and straight pieces around opening as shown in Figure (3-5). Tensile test of steel reinforcement was carried out on at least three specimens,
69
Chapter Three
Experimental Work
prepared for each type of the reinforcing steel bars which were used in the tested beam to determine their tensile properties according to ASTM C370-2005a. The tensile test was performed in the Central Organization for Standardization and Quality Control. The main properties are summarized in Table 3.6 and it is in agree with ASTM A615-86. Table 3.6: Test Results of Steel Reinforcing Bar Nominal
Measured
Yield
Ultimate
Elongation
Modulus of
Diameter
Diameter
Stress(*) f sy
Strength f su
Ratio %
Elasticity(**)
(mm)
(mm)
(MPa)
(MPa)
6
5.72
520
615
9.5
200
12
11.67
550
625
12.5
200
(*)
(GPa)
Each value is an average of three specimens (each 40 cm length). Assumed value.
(**)
Figure (3-5) Cage of Steel Reinforcement 70
Chapter Three
Experimental Work
3.4.3 Strengthening System by (CFRP) Laminates Carbon fiber fabric SikaWrap Hex-230C and epoxy based impregnating resin Sikadur-330 were used for technique of strengthening is shown in Figure (3-6). The main properties of impregnating resin Sikadur-330 and carbon fiber fabric SikaWrap Hex-230C are shown in Tables 3.7 and 3.8, respectively.
Figure (3-6) CFRP Laminates and Epoxy Resine Table 3.7 :Properties of Sikadur-330 (Impregnating Resin) (*) Comp. a: white
Appearance
Comp. b: grey
Density
1.31 kg/l (mixed)
Mixing ratio
A : B = 4 : 1 by weight
Open time
30 min (at + 35◦C)
Viscosity
Pasty, not flowable
Application temperature
+ 15◦C to + 35◦C (ambient and substrate)
Tensile strength Flexural E-modulus
30 MPa (cured 7 days at +23◦C) 3800 MPa (cured 7 days at +23◦C)
(*) Provided by the manufacturer
71
Chapter Three
Experimental Work
Table 3.8:Properties of SikaWrap Hex-230C (Carbon Fiber Fabric) (*) Fiber type
High strength carbon fibers 0◦ (unidirectional). The fabric is equipped
Fiber orientation
with special weft fibers which prevent loosening of the roving (heatset process). 225 g/m2
Areal weight Fabric design thickness
0.131 mm (based on total area of carbon fibers)
Tensile strength of fibers
4300 MPa
Tensile modulus of fibers
238 GPa
Elongation at break
1.8 %
Fabric length/roll
≥ 45.7 m
Fabric width
305 mm
(*) Provided by the manufacturer
3.5 Mechanical Properties of Hardened Concrete 3.5.1 Compressive Strength During casting of each curved beams, three 150×150×150 mm cubes and three 100×200 mm cylinders were made. After cleaning and lubricating the molds, concrete was cast and compacted and then cured under the same conditions. Cube compressive strength and cylinder tensile strength were obtained by standard tests ASTM C39-2001, as shown in Figure (3-7a). The results of each curved beam are listed as an average in Table (3-9).
72
Chapter Three
Experimental Work
3.5.2 Splitting Tensile Strength Splitting tensile strength test was carried out on plain cylindrical concrete specimens (100 mm x 200 mm) in accordance with ASTM C496-1996 as shown in Figure (3-7b). Results are presented in table 3.10.
(a)
(b)
Figure (3-7) compressive Strength and Tensile Strength Test Table 3.9 : Test Results of Compressive and tensile Strength
Beam Symbol
Compressive Strength of Concrete (MPa)(*)
Splitting Tensile
Modulus of
Strength ( fct )
Elasticity Ec
(MPa)
(MPa)(**)
( fcu )
( fc' )
SCB.P
37.63
30.10
3.4
25786
SCB.Eo
39.38
31.50
3.5
26379
SCB.Esr
37.25
29.80
3.4
25657
SCB.Ecfrp
41.38
33.10
3.6
27040
SCB.Mo
39.25
31.40
3.5
26337
SCB.Msr
40.13
32.10
3.5
26629
SCB.Mcfrp
40.50
32.40
3.5
26753
SCB.Io
38.38
30.70
3.4
26042
SCB.Isr
37.75
30.20
3.4
25829
SCB.Icfrp
36.63
29.30
3.1
25440
73
Chapter Three
Experimental Work
Table 3.9 Continue FCB.P
37.13
29.70
3.4
25614
FCB.Mo
37.50
30.00
3.4
25743
FCB.Msr
41.50
33.20
3.6
27081
FCB.Mcfrp
40.38
32.30
3.6
26711
(*)
ƒ̀c= 0.8 ƒ̀cu (Ec=4700 fc' )
(**)
3.6 CFRP Laminates Installation The effectiveness of strengthening or rehabilitation with externally bonded CFRP laminates depends on the bond between the CFRP and concrete. To insure such proper bonding the following steps were followed: (1) The surface of the concrete were grinded using an electrical hand grinder to expose the aggregate and to obtain
a clean sound
surface, free of all contaminants such as cement laitance, and dirt. (2) Specimens were cleaned by washing with water and allowed to dry prior to composite application. This procedure removed loose particles and contaminations from the specimen's surface. (3) The corners of the specimens were rounded (radius of approximately 15 mm) to avoid any stress concentration in the CFRP at corners of the beams. This stress concentration will lead to a rupture failure of CFRP at corners before reaching their ultimate strength. (4) Apply the mixed resin Sikadur-330 to the prepared substrate using a trowel or brush in a quantity of approximately (0.7 to 1.2 kg/m2), depending on roughness of substrate. (5) The SikaWrap Hex-230C fabric was cut by scissors to strips for the required width and length for all the specimens. 74
Chapter Three
Experimental Work
(6) Place the SikaWrap Hex-230C fabric onto the resin with the plastic roller until the resin is squeezed out between the roving. (7) As a covering layer an additional resin layer of approximately (0.5 kg/m2) broadcast with the brush can be added, which will serve as a bonding coat for following cementitious coatings. (8) After allowing the laminate to cure for several days, the specimens will be ready to test. All apparent concrete surface beams were painted white so that crack propagation can be easily detected. Figure (3-8) shows the procedure of application of CFRP laminates on concrete element.
Figure 3.8: Application of CFRP System on Concrete Element
3.7 Instrument and Test Procedure Tests were carried out using 2000 kN hydraulic testing machine which was manufactured for the Civil Engineering Department of Engineering College of Karbala University, as shown in Figure (3-9). This machine was manufactured from a built up steel sections of a thickness 30 mm with a movable braced support (two plates) of thickness
75
Chapter Three
Experimental Work
32 mm that could be moved up or down by a scroll fixed at the top of the machine. These plates are fixed to the frame by two 50 mm steel rods through holes made for this purpose as shown in Figure (3-10). The main characteristics of the structural behavior of the beam specimens were detected at every stage of loading during testing. A dial gage of 0.01 mm accuracy was used at midspan of the beam and at the outer and inner edge of the midspan section to measure the rotation at this section as shown in Figure (3-10). The specimens were placed on the supports of the testing machine, and then the first readings of the gages were recorded. After that, the specimens were loaded with a constant rate of loading. Readings of deflections was recorded at each interval of load as well as recording the first crack load and the ultimate load consequently.
Figure 3.9: Loading Machine Used in the Testes
76
Chapter Three
Experimental Work
(a)
(b)
Loading l
Dial gauge at the edges of midspan section support
(c)
support
Dial gauge at the center of midspan section
(d)
Figure (3-10) Instruments Details a) Semicircular Curved Beam b) Ring Curved Beam c) Dial Gauge d) Dial Gauge Positions
77
Chapter Three
Experimental Work
3.8 Loading and Support Condition The supporting system was hinged at exterior ends and roller at the inner support. The hinged end consists of one smooth stainless steel shaft of diameter 25mm welded to base plate and two shaft, of diameter 25mm welded to steel beam (to make grove). A W-shape steel beam was fixed weld to the supported plates to prevent relative movement and also to provide a rigid reference beam that hold the supports. The roller support be made of two shaft with diameter 25mm welded to the base plate and other steel shaft of diameter 25mm free. Figure (3-11) shows the details of support conditions. The test was done by 2000 kN capacity hydraulic jack. All of the semicircular beams were tested under two point loading, with the load applied at midspan of each panel, while full circular beams were tested under four point load applied at the midspan of each panel.
Figure (3-11) Details of the Typical Support Condition
(a) Hinge Support (b) Roller Support
78
Chapter Three
Experimental Work
3.9 Steel and wood mold Two types of molds were used in this study. The first one used for semicircular beams which is a steel mold from plates of thicknesses (4mm for the main frame, 3mm for stiffeners and 2mm inner cover plate) using a CNC-plasma machine, as shown in Figure (3.12). The second type of molds is a wooden type which was used for the circular ring beams, a 2.44 m by 1.24m plywood blocks of thickness 10 mm were used to form the main frame of this type of mold, while 4mm wooden fibers plates were used to cover the inner surfaces of the molds as shown in Figure (3.13).
a) b)
Figure (3-12) Semicircular Molds and Tools a) Steel Mold Layout b) Steel Mild Photo c) CNC-plasma machine 79
c)
Chapter Three
Experimental Work
a)
b) Figure (3-13) Wooden Circular Molds a) Wood Mold Layout b) Wood Mild Photo
80
Chapter Four
Experimental Results and Discussion
CHAPTER FOUR EXPERIMENTAL RESULTS AND DISCUSSION 4.1 General In this chapter, the general response and observation of the tested horizontally curved concrete beams with and without openings, and strengthened with internal reinforcements or externally with CFRP laminates were reported and discussed. To accomplish this objective, an experimental program has been performed, as described in chapter three. Two groups of circular curved beams were tested, first one includes ten continuous semicircular curved concrete beams with inner radius of 1000 mm and with dimensions of section of 125 mm width and 250 mm height under effect of two midspan point loads. The second group includes four full circular beams with inner radius of 475 mm and section dimensions of 125 mm width and 250 mm height under effect of four midspan point loads. Both of these groups were tested to study the influence of different variables that were considered: opening location, effect of internal reinforcement by steel reinforcements (stirrups) and effect of external strengthening by CFRP laminates. One of each group of the beams represents the control beam, which has no opening and without strengthening scheme at any position. The others have web openings with and without strengthening. Test results are discussed in this chapter based on load-midspan deflection curves and loadtwisting angle curves to explore the influence of internal reinforcement with stirrups and diagonal bars and external strengthening with CFRP laminates on cracking and ultimate loads, crack pattern and failure modes.
81
Chapter Four
Experimental Results and Discussion
4.2 Semicircular Curved Concrete Beams (Group I) This section deals with the overall behavior of ten horizontally circular concrete beams with and without openings, and strengthened with internal reinforcement or with CFRP laminates.
4.2.1 Pilot and Control Semicircular Curved Beam (SCB.P) Two specimens (SCB.P) were constructed, one as a pilot and the other is a control specimen. The pilot beam is used to check the validity of test processes supports and load positions, dial gages and its location, as well as the overall test incremental load applications. The test shows that all devices work properly and could be used for all other specimens. The control specimen is a semicircular curved beam without opening as shown in Figure (4.1). The beam was loaded gradually until the first crack was observed. The first crack (flexural) appeared in the top face of maximum negative moment (internal support) at load of about 62.3 kN. At the same load, a torsional crack appears at both sides with an angle about 45°. As the load was increased further, several flexural and torsional cracks initiated in the positions between interior support and points of loading, spatially at tension zones of the beams. Torsional cracks spread rapidly and began to propagate more than flexural crack with load increments. In the same manner it can be noticed that width of torsional cracks increased more than flexural cracks widths during loading. As the load reaches the value of 117.7 kN the first positive flexural crack at bottom face of midspan appears, while torsional cracks directed to propagate at the compression zone above the interior support. Finally, torsional failure at ultimate load about 147.2 kN occurred. Figure (4.2), show mode of failure and cracks pattern, while Figures (4.3) and (4.4) show load deflection curve and load – angle of twisting curves for midspan, respectively.
82
Chapter Four
Experimental Results and Discussion
Figure (4.1) Control Beam SCB.P
a)
b) Figure (4.2) Mode of Failure and Cracks Pattern for Control Beam at Failure a) Interior Support b)Under Midspan Point Load
83
Chapter Four
Experimental Results and Discussion
Figure (4.3) Load-Midspan Deflection Curve for Control Beam SCB.P
Figure (4.4) Load-Midspan Twisting Angle Curve for Control Beam SCB.P 84
Chapter Four
Experimental Results and Discussion
4.2.2 Semicircular Curved Beam with Opening near External Support (SCB.E) 4.2.2.1 Unstrengthened Semicircular Curved Beam (SCB.Eo) The curved beam SCB.Eo includes opening spaced at distance d/2 = 106.5 mm from external support and without any strengthening as shown in Figure (4.5). The beam deformation was initially within the elastic range at the early stages of loading. The load was increased gradually until the first flexural crack occurred at load about 48.5 kN which was observed in the maximum negative moment position at top face above the interior support. The diagonal crack of opening began to appear from bottom corner of opening that near to support at load of 55.4 kN. Torsional cracks start to appear at load of 58.8 kN at the tension zone and near the maximum torsional force at mid distance between load and internal support As the load was increased further, several flexural and torsional cracks initiated in the positions between the applied loads and internal support. Flexural cracks moved downward and torsional cracks increased in length and numbers at an angle of 45°.Also, diagonal cracks increased in length and width, and start from other corners of opening. As the load was increased further, a torsional conventional failure mode at the opening position was appeared. The curved beam SCB.Eo failed at a load of 128.1 kN by forming two diagonal cracks start from the skew corners of opening, then the cracks growth along top and bottom reinforcement at an angle of 45° until brittle failure occurs of beam type failure, as shown in Figure (4.6). The load–midspan deflection and load–angle of twisting curves are shown Figures (4.7) and (4.8), respectively.
85
Chapter Four
Experimental Results and Discussion
Figure (4.5) Semicircular Curved Beam SCB.Eo
a
b
Figure (4.6) Mode of Failure and Cracks Pattern for SCB.Eo Curved Beam
a) Interior Support b) At Exterior Support
86
Chapter Four
Experimental Results and Discussion
Figure (4.7) Load-Midspan Deflection Curve SCB.Eo Curved Beam
Figure (4.8) Load-Midspan Twisting Angle Curve SCB.Eo Curved Beam
87
Chapter Four
Experimental Results and Discussion
4.2.2.2 Internally Strengthened Semicircular Curved Beam (SCB.Esr) The curved beam SCB.Esr includes opening spaced at distance d/2=106.5 mm from exterior support, as shown in Figure (4.9), and strengthened by three pairs of closed stirrups Ø6 mm for both top and bottom chords of opening and one diagonal bars Ø6 mm for each corner of opening, also one Ø6 mm stirrups at each side of the opening. In this beam, cracks were first observed at the top face of maximum negative moment at a load approximately 50.2 kN. These cracks appeared due to flexural stresses of bending moment. As the load increased to 58.8 kN a torsional and skew cracks at the corners of opening were appeared. After more loads increments flexural and torsional cracks were propagate and widened.
In spite of
forming diagonal corner cracks for curved beam SCB.Esr, the beam fails due to torsional moment at load of 132.0 kN, as shown Figure (4.10). No propagation of cracks at the corners of opening were recorded because of the internal reinforcement which prevent cracks from extended, and the beam failed in a beam type failure mode. The load – midspan deflection and load – angle of twisting curves are shown in Figures (4.11) and (4.12), respectively.
Figure (4.9) Semicircular Curved Beam SCB.Esr
88
Chapter Four
Experimental Results and Discussion
a)
b) Figure (4.10) Mode of Failure and Cracks Pattern for SCB.Esr Beam a) Inner Face of Exterior Support b) Above Interior Support c) Outer Face of Exterior Support
c)
Figure (4.11) Load-Midspan Deflection Curve SCB.Esr Curved Beam 89
Chapter Four
Experimental Results and Discussion
Figure (4.12) Load-Midspan Twisting Angle Curve For SCB.Esr Curved Beam
4.2.2.3 Externally Strengthened Semicircular Curved Beam (SCB.Ecfrp) The curved beam SCB.Ecfrp includes opening spaced at distance d/2=106.5 mm from external support, as shown in Figure (4.13), and strengthened by full wrap of CFRP laminates of 0.131mm thickness (pair of 20mm width on each side of the opening and
three pairs of 20mm width
for both top and bottom chords of opening ). The beam was loaded gradually until the first crack was observed and the deformation was initially within the elastic range at early stages of loading. The first crack was noticed at a load of 41.5 kN at the tension fibers of maximum negative moment above interior support and at the down corner of the opening near the support. Torsional cracks began to appear at the zone between the applied load and the interior support at an angle of 45° at load of 48.5 kN, also flexure cracks
90
Chapter Four
Experimental Results and Discussion
propagated through these increments, while corner cracks at the opening did not propagate or increased because of the confinements of CFRP laminates. At load of about 72.7 kN the crack at corner of opening reaches the CFRP laminates and stop its extension through until failure occurs, while torsional cracks increased and propagated rapidly and cause the failure of the beam. A beam type failure mode is occurred due to torsional moment at load of 141.9 kN, as shown in Figure (4.14). The load – midspan deflection and load – angle of twisting curves are shown in Figures (4.15) and (4.16), respectively.
Figure (4.13) Semicircular Curved Beam SCB.Ecfrp
b)
a)
Figure (4.14) Mode of Failure and Cracks Pattern for SCB.Ecfrp Curved Beam a) Outer Face of Interior Support b) Inner Face of Interior Support 91
Chapter Four
Experimental Results and Discussion
Figure (4.15) Load-Midspan Deflection Curve SCB.Ecfrp Curved Beam
Figure (4.16) Load-Midspan Twisting Angle Curve SCB.Ecfrp Curved Beam 92
Chapter Four
Experimental Results and Discussion
4.2.2.4 Summary of Test Results for SCB.E Beams Table 4.1 shows the cracking load, ultimate load, maximum rotation at and deflection at midspan and failure mode. Also, Figure (4.17) and (4.18) show a comparison of load- deflection and angle of twist curves at midspan for curved beams SCB.P, SCB.Eo, SCB.Esr and SCB.Ecfrp. It can be concluded, that the presence of openings near exterior support reduce the ultimate load capacity (compared with control beam) by about 18%, also no interested changes in deformations were recorded. On the other hand strengthening of the opening by internal reinforcement or external CFRP Laminates will increase ultimate load capacity (compared with SCB.Eo) by about 3% and 11% respectively. Also It could be noticed that torsional deformation of internally strengthened curved beam increased by about 25%, while other deformations had no interested change.Results indicates that the presence of opening in the curve beam reduced its stiffness, while the strengthening of the opening lead to increase the post cracking flexural and torsional stiffnesses and behavior of the curved beam. Table 4.1 Summary of Tested SCB.E Beams Cracking Load, kN Specimen
Corner Flex.
*
Ultimate
Tor. Load, kN
Ultimate Load Diff. %
Max. θ ×10-3 (rad) at midspan
Max. Δ
Failure
(mm) at midspan
Mode
SCB. P
--
62.3
62.3
152.3
18
44.6
20.4
Torsional
SCB. Eo
55.4
48.5
58.8
128.1
--
44.4
18
Shear
SCB. Esr
58.8
50.2
58.8
132.0
3
55.13
20
SCB. Ecfrp
72.7
41.5
48.5
141.9
11
44.2
17.1
Difference = (Pu(Specimen) − Pu(SCB. Eo))/(Pu(SCB. Eo))
93
Torsional Shear Torsional
Chapter Four
Experimental Results and Discussion
Figure (4.17) Comparison of Load-Midspan Deflection Curves for SCB.Ep, SCB.Eo, SCB.Esr and SCB.Ecfrp Curved Beams
Figure (4.18) Comparison of Twisting Angle Curves at Midspan for SCB.Ep, SCB.Eo, SCB.Esr and SCB.Ecfrp Curved Beams 94
Chapter Four
Experimental Results and Discussion
4.2.3 Semicircular Curved Beam with Opening near Midspan (SCB.M) 4.2.3.1 Unstrengthened Semicircular Curved Beam (SCB.Mo) As previously Mentioned the SCB.Mo beam includes opening spaced at distance d/2=106.5 mm from the applied load and without strengthening, as shown in Figure (4.19). The first visible cracks are inclined cracks at the corners of openings at load 34.6 kN directed toward the applied load position. At load step of 41.5 kN another cracks appear at the opposite corners of the opening, and no flexural or torsional cracks appear until this stage. After increasing the load to 48.5 kN, torsional cracks began to appear and propagated rapidly more than corners cracks, while flexural cracks appear at load of 62.3 kN with small length and width at position of maximum positive moment near the applied load. It is well noticed here that there is no flexural cracks at position of tension zone which had maximum moment overall the beam length because of the redistribution of the forces due to existing of the opening at position of maximum positive bending moment and torsional force , and because of the rapid propagation of cracks at opening zone. Figure (4.20) shows the frame type failure mode and the crack pattern of the beam at ultimate load of 83 kN. The load –deflection and load–angle of twisting curves at midspan are shown in Figures (4.21) and (4.22), respectively.
95
Chapter Four
Experimental Results and Discussion
Figure (4.19) Semicircular Curved Beam SCB.Mo
a)
b) Figure (4.20) Mode of Failure and Cracks Pattern for SCB.Mo Curved Beam a) Outer Face of Opening Area b) Inner Face of Opening Area 96
Chapter Four
Experimental Results and Discussion
Figure (4.21) Load-Midspan Deflection Curve SCB.Mo Curved Beam
Figure (4.22) Load-Midspan Twisting Angle Curve SCB.Mo Curved Beam
97
Chapter Four
Experimental Results and Discussion
4.2.3.2 Internally strengthened Semicircular Curved Beam (SCB.Msr) The specimen SCB.Msr semicircular curved beam includes opening spaced at distance d/2=106.5 mm from the applied load as shown in Figure (4.23), and strengthening by six pairs of closed stirrups Ø6 mm for both top and bottom chords of opening and four diagonal bars Ø6 mm, two for each corner of opening, as well as two Ø6 mm full stirrups at each side of the opening spaced at distance 40 mm. In the beam (SCB.Esr) cracks were first observed at the corners of the opening at load of 41.5 kN due to the stress concentration at these positions, these cracks were inclined toward the applied load. As the load was increased to 55.3 kN, torsional cracks start to appear at position of maximum torsional moment between applied load and internal support, while flexural cracks found at a load of 72.7 kN at both positions of maximum positive (under the applied load ) and negative moments (over the internal support). After more load increments a small propagation of diagonal cracks at the corners of opening was noticed, while torsional cracks at top and bottom cords propagated and widen rapidly more than flexural and diagonal cracks. A frame type failure occurs at a load of 100.4 kN by formation of two independent diagonal cracks, one in each cord of the opening as shown in Figure (4.24). The load –deflection and load – angle of twisting curves at midspan are shown in Figures (4.25) and (4.26), respectively.
98
Chapter Four
Experimental Results and Discussion
Figure (4.23) Semicircular Curved Beam SCB.Msr
a)
b) Figure (4.24) Mode of Failure and Cracks Pattern for SCB.Msr Curved Beam a) At Interior Support b) At Opening Region
99
Chapter Four
Experimental Results and Discussion
Figure (4.25) Load-Midspan Deflection Curve SCB.Msr Curved Beam
Figure (4.26) Load-Midspan Twisting Angle Curve SCB.Msr Curved Beam
100
Chapter Four
Experimental Results and Discussion
4.2.3.3 Externally Strengthened Semicircular Curved Beam (SCB.Mcfrp) The curve beam SCB.Mcfrp includes opening spaced at distance d/2=106.5 mm from the applied load, as shown in Figure (4.27), and strengthened by full wrap of CFRP laminates of 0.131mm thickness (pair of 40 mm width on the each side of the opening and
three pairs of 25mm
width for each top and bottom chords of opening ). The beam was loaded gradually until the first crack was observed and the deformation was initially within the elastic range at the early stages of loading. The first crack appears at a load of 34.6 kN at the skew corners of the opening. At a load of 41.5 flexural cracks began to appear at the section of maximum negative above interior support at the top tension position. After more loading increments at load of (72.6 kN) a torsional cracks at an angle of 45° began to appear and propagate rapidly, while flexural cracks stopped until failure occurs, as well as no significant increasing in corner cracks was noticed up to failure. The failure mode occurred through torsional forces (beam type failure) at an ultimate load of 116 kN, as shown in Figure (4.28). The load –deflection and load – angle of twisting curves at midspan are shown in Figures (4.29) and (4.30), respectively.
Figure (4.27) Semicircular Curved Beam SCB.Mcfrp 101
Chapter Four
Experimental Results and Discussion
b)
a)
c) Figure (4.28) Mode of Failure and Cracks Pattern for SCB.Mcfrp Curved Beam a) At Opening Region b) At Interior Support c) At 2nd Opening Region
Figure (4.29) Load-Midspan Deflection Curve SCB.Mcfrp Curved Beam 102
Chapter Four
Experimental Results and Discussion
Figure (4.30) Load-Midspan Twisting Angle Curve SCB.Mcfrp Curved Beam
4.2.3.4 Summary of Test Results for SCB.M Beams Table 4.2 shows the cracking load, ultimate load, maximum rotation at and deflection at midspan. Also, Figure (4.31) and (4.32) show a comparison of load-midspan deflection and angle of twisting curves for curved beams SCB.P, SCB.Mo, SCB.Msr and SCB.Mcfrp. It can be concluded that, the presence the openings near the applied load reduced the ultimate load capacity (compared with control beam) by about 83%, also a significant increase in twisting angle were noticed. On the other hand, strengthening of the opening by internal reinforcement or external CFRP laminates lead to increase in the ultimate load capacity (compared with SCB.Mo) by about 21% and 39% respectively, also both angle of twist and deflection at midspan for both type of strengthening decreased by about 65% and 35% respectively. This is because of the increasing of post cracking stiffness of the beam at the opening which could
103
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Experimental Results and Discussion
be seen clearly in Figures (4.31) and (4.32). Results indicate that the presence of opening in the curve beam will reduce its stiffness as well as ultimate load capacity, while the strengthening of the opening will increase the post cracking stiffness of the curved beam. Table 4.2 Summary of Tested SCB.M Beams Specimen
Cracking Load, kN Ultimate *Ultimate Max. θ ×10-3 Max. Δ at Load
Corner Flex. Tor. Load, kN Diff. %
(rad)
at
Failure
(mm)
midspan
midspan
Mode
SCB. P
--
62.3
62.3
152.3
83
44.6
20.4
Torsional
SCB. Mo
34.6
62.3
48.5
83
--
99.1
17.2
Torsional
SCB. Msr
41.5
72.7
55.3
100.4
21
27
9.7
shear
32
41.5
72.6
116
39
35.2
12.77
SCB. Mcfrp
Torsional and shear
Difference = (Pu(Specimen) − Pu(SCB. Mo))/(Pu(SCB. Mo))
Figure (4.31) Comparison of Load-Midspan Deflection Curves for SCB.P, SCB. Mo, SCB. Msr and SCB.Mcfrp Curved Beams
104
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Experimental Results and Discussion
Figure (4.32) Comparison of Twisting Angle Curves at Midspan for SCB.P, SCB.Mo, SCB.Msr and SCB.Mcfrp Curved Beams
4.2.4 Semicircular Curved Beam with Opening near Interior Support (SCB.I) 4.2.4.1 Unstrengthened Semicircular Curve Beam (SCB.Io) The SCB.Io curved beam includes opening spaced at distance d/2=106.5 mm from face of internal support without any type of strengthening, as shown in Figure (4.33). The beam deformation was initially within the elastic ranges at the early stages of loading, and the load was increased gradually until the first crack occurred which was observed in the top and bottom skew corners of the opening due to concentration of stresses at these corners at load about 34.6 kN. Torsional and flexural cracks start to appear at load of 48.5 kN at the tension zone above interior support and near the maximum torsional moment. As the load was increased further, torsional cracks propagated rapidly. Flexural and corner cracks stop its 105
Chapter Four
Experimental Results and Discussion
propagation until failure occurs, while torsional cracks increased in length and width at an angle of 45°. As the load was increased further, a torsional conventional failure mode at the opening position was appeared. The ultimate load of beam SCB.Io was about 76.2 kN by forming torsional cracks at top and bottom cords of the opening (beam type failure) as shown in Figure (4.34). The load–deflection and load–angle of twisting curves at midspan are shown in Figures (4.35) and (4.36), respectively.
Figure (4.33) Semicircular Curved Beam SCB.Io
b)
a)
Figure (4.34) Mode of Failure and Cracks Pattern for SCB.Io Curved Beam a) At Opening Region b) At Interior Support
106
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Experimental Results and Discussion
Figur (4.35) Load-Midspan Deflection Curve SCB.Io Curved Beam
Figure (4.36) Load-Midspan Twisting Angle Curve SCB.Io Curved Beam 107
Chapter Four
Experimental Results and Discussion
4.2.4.2 Internally Strengthened Semicircular Curve Beam (SCB.Isr) The SCB.Isr curve beam includes opening spaced at distance d/2=106.5 mm from Internal support as shown in Figure (4.37), and strengthened by six pairs closed stirrups Ø6 mm for each top and bottom chords of opening and two diagonal bars Ø6 mm at each corner of opening, also two Ø6 mm stirrups at each side of the opening were used. In the beam (SCB.Isr) cracks were first observed at the corners of the opening at a load of approximately 41.4 kN. As the load increased to 55.9 kN a flexural crack appears at the tension zone of maximum negative moment. After more load increment (at load of 76.7 kN) torsional cracks appear at the location of maximum torsional moment between applied load and internal support. After more load increment, flexural cracks did not propagated in length or width, while a noticeable new torsional cracks were recorded at top and bottom cords of the opening which were the main reason of the beam failure. For SCB.Isr beam, a frame failure mode occurs at load of 99.5 kN as shown in Figure (4.38). The load –deflection and load –angle of twisting curves at midspan are shown in Figures (4.39) and (4.40), respectively.
Figure (4.37) Semicircular Curved Beam SCB.Isr 108
Chapter Four
Experimental Results and Discussion
a)
b)
Figure (4.38) Mode of Failure and Cracks Pattern for SCB.Isr Curved Beam a) At 1st Opening Region b) At 2nd Opening Region
Figure (4.39) Load-Midspan Deflection Curve SCBIsr Curved Beam
109
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Experimental Results and Discussion
Figure (4.40) Load-Midspan Twisting Angle Curve SCB.Isr Curved Beam
4.2.4.3 Externally Strengthened Semicircular Curve Beam (SCB.Icfrp) As mentioned previously, the SCB.Icfrp curved beam included opening spaced at distance d/2=106.5 mm from the internal support as shown in Figure (4.41) and strengthened by full wrap of CFRP laminates of 0.131mm thickness (pair of 40 mm width on each side of the opening and three pairs of 25mm width each both top and bottom chords of opening). The beam was loaded gradually until the first crack was observed at a load of 31 kN at the skew corners of the opening. As load increased a flexural cracks began to appear at the section of maximum negative above interior support and at the tension position at a load of 51.9 kN. After more loading increments torsional cracks at an angle of 45° began to appear and propagate at a load about 76.1kN, while no significant increase in flexural cracks up to
110
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Experimental Results and Discussion
failure. It can be noticed that corner cracks propagation didn’t penetrate CFRP laminates, but it does deeply through cross section and causing failure. A beam failure mode occurred at load of 100.9 kN, as shown in Figure (4.42). The load –deflection and load – angle of twisting curves at midspan are shown in Figures (4.43) and (4.44), respectively.
Figure (4.41) Mode of Failure and Cracks Pattern for SCB.Icfrp Curved Beam
b)
a) Figure (4.42) Mode of Failure and Cracks Pattern for SCB.Icfrp Curved Beam a) 1st Opening Region b) 2nd Opening Region
111
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Figure (4.43) Load-Midspan Deflection Curve SCBIcfrp Curved Beam
Figure (4.44) Load-Midspan Twisting Angle Curve SCB.Icfrp Curved Beam
112
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Experimental Results and Discussion
4.2.4.4 Summary of Test Result for SCB.I Beams Table 4.3 shows the cracking load, ultimate load,
rotation and
deflection at midspan. Also, Figure (4.45) and (4.46) show a comparison of load-midspan deflection and load- angle of twisting curves for curved beams SCB.P, SCB.Io, SCB.Isr and SCB.Icfrp. It can be concluded that, the presence of the openings near interior support reduces the ultimate load capacity (compared with control beam SCB.P) to the half, also a significant reduction in twisting angle and deflection were noticed. On the other hand, strengthening of the opening by internal reinforcement or external CFRP laminates increased ultimate load capacity (compared with SCB.Io) by about 31% and 32% respectively, also angle of twist was increased with an interested ratio because of the confinement of the beam at opening region which postpone failure at opening for both types of strengthening. This is because of the increasing of post cracking stiffness of the beam at the opening which could be seen clearly in Figures (4.45) and (4.46).
Table 4.3 Summery of Tested SCB.I Beams Ultimate Max. θ ×10-3 Max. Δ at
*
Cracking Load , kN
Specimen
Ultimate
Load
Corner Flex. Tor. Load, kN Diff. %
(rad)
at
Failure
(mm)
midspan
midspan
Mode
SCB. P
--
62.3 62.3
152.3
100
44.6
20.4
Torsional
SCB. Io
34.6
48.5 48.5
76.2
--
21.65
13.35
shear
SCB. Isr
41.5
55.9 76.7
99.5
31
30.02
12.1
SCB. Icfrp
31
51.9 76.1
100.9
32
28.01
17.1
Difference = (Pu(Specimen) − Pu(SCB.Io) )/Pu(SCB.Io)
113
Torsional and shear shear
Chapter Four
Experimental Results and Discussion
Figure (4.45) Comparison of Load-Midspan Deflection Curves for SCB.P, SCB.Io, SCB.Isr and SCB.Icfrp Curved Beams
Figure (4.46) Comparison of Twisting Angle Curves at Midspan for SCB.P, SCB.Io, SCB.Isr and SCB.Icfrp Curved Beams 114
Chapter Four
Experimental Results and Discussion
4.2.5 Summary of Tested Semicircular Curved Beams As a summary of test results for full circular beam with strengthened and unstrengthened opening, the following notes were obtained: 1. For the specimens SCB.Eo SCB.Mo and SCB.Io, which were unstrengthened, the decrease in ultimate load capacity was about 18%,83% and 100%, respectively when compared with the corresponding control specimen SCB.P. 2. For internally strengthened specimens SCB.Esr SCB.Msr and SCB.Isr, the ultimate load capacities were enhanced by about 3%,21% and 31%, when compared with unstrengthened specimens FCB.Eo, FCB.Mo and FCB.Io respectively. 3. The use of CFRP laminates as external confinements in spacemens SCB.Ecfrp SCB.Mcfrp and SCB.Icfrp, increased ultimate load capacities by about 11%,39% and 32% when compared with unstrengthened specimens FCB.Eo, FCB.Mo and FCB.Io respectively. 4. A reliable enhancement appears in post-cracking behavior of specimens SCB.Mcfrp, SCB.Msr and SCB.Icfrp, SCB.Isr when compared with unstrengthened specimens SCB.Mo and SCB.Io, respectively, and this is more than that observed from SCB.Ecfrp, SCB.Esr if compared with SCB.Eo. 5. The use of internal confinement change the failure mode from beam type failure to frame type failure, while the use of CFRP laminates retains the failure mode to beam type failure.
4.3 Full Circular (Ring) Concrete Beams (Group II) This section deals with the overall behavior of four horizontally full circular curved (ring) concrete beams with and without openings, and strengthened with internal reinforcements or with external CFRP laminates.
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Experimental Results and Discussion
4.3.1 Control Specimen (FCB.P) A specimen FCB.P is a circular full curved beam and without opening as shown in Figure (4.47). The beam was loaded gradually until the first crack was observed. The first crack (flexural and torsional) appeared in the position of maximum positive moment (under point load) and at the section of maximum torsional moment respectively, at load of approximately 103.8 kN. As the load was increased further, several flexural and torsional cracks initiated in the positions of midspan loads, spatially at tension zones of the beams. Torsional cracks spread and began to propagate more than flexural cracks with load increments. As the load approaches to its ultimate value a negative flexural crack above the support appears, while torsional cracks directed to propagate at the compression zone above the support and under point loads. The value of the ultimate load was 380.8 kN. Figure (4.48) show failure mode and cracks pattern, also Figures (4.49) and (4.50) show load –deflection and load – angle of twisting curves at midspan, respectively.
Figure (4.47) Full Circular Beam FCB.P
116
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Experimental Results and Discussion
a)
b)
Figure (4.48) Mode of Failure and Cracks Pattern for Circular Beam FCB.P a) Overall Crack Pattern b) Point Load Region
Figure (4.49) Load-Midspan Deflection Curve for FCB.P Circular Beam
117
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Experimental Results and Discussion
Figure (4.50) Load-Midspan Twisting Angle Curve for FCB.P Circular Beam
4.3.2 Full Circular Beam with Opening (FCB.Mo) A specimen FCB.Mo is a full circular curved beam with opening spaced at distance d/2=106.5 mm from the applied load as shown in Figure (4.51). The beam was loaded gradually until the first crack was observed. The first crack appears at corners of the opening due to a stress concentration of torsional moment, at load of approximately 58.8 kN. As the load was increased further, and at a load about 95 kN torsional cracks initiated near the corners of opening closed to applied load and support. Also flexural cracks start to appear at a load about 110.7 kN. Torsional cracks spread and began to propagate while no considerable propagation of flexural cracks occurs up to last load increments. As the load approaches failure a negative flexural cracks appears at the top face of maximum negative moment above the support, while torsional cracks directed to propagate at the compression
118
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Experimental Results and Discussion
zone around support and under point loads. Abeam type failure occurs at a load of 190 kN. Figure (4.52), show mode of failure and cracks pattern, while Figures (4.53) and (4.54) show load –midspan deflection and load– midspan angle of twisting curves, respectively.
Figure (4.51) Full Circular Beam FCB.Mo
Figure (4.52) Mode of Failure and Cracks Pattern for FCB.Mo Circular Beam at Different Positions 119
Chapter Four
Experimental Results and Discussion
Figure (4.53) Load-Midspan Deflection Curve FCB.Mo Circular Beam
Figure (4.54) Load-Midspan Twisting Angle Curve FCB.Mo Circular Beam 120
Chapter Four
Experimental Results and Discussion
4.3.3 Internally Strengthened Full Circular Beam (FCB.Msr) The curved beam FCB.Msr includes opening spaced at distance d/2=106.5 mm from applied load as shown in Figure (4.55), and strengthened by six pairs of closed stirrups Ø6 mm for each top and bottom chords of opening and two diagonal bars Ø6 mm for each corner of opening, also two Ø6 mm stirrups at each side of the opening. In this beam cracks were first observed at the corner of the opening at the cord of compression stresses at load of 55.4 kN, while crack in the skew position of the opening at cord of tension stresses appeared at load of 69.2 kN. As the load increased a flexural crack appear at tension zone of maximum negative and torsional moments at a load of approximately 79.8 kN. As the load increased to 89.9 kN, torsional cracks began to appear and propagate and widened more than flexural and corner cracks. In spite of forming diagonal corner and flexural cracks at earlier stages of loading for FCB.Msr, the beam failed due to torsional effect (torsional cracks )at load of 305 kN in a frame type failure mechanism, as shown in Figure (4.56). The extension of cracks at the opening corners had a small propagation because of the internal reinforcement which prevents their extension or widening at opening. The load –deflection and load –angle of twisting curves at midspan are shown in Figures (4.57) and (4.58), respectively
Figure (4.55) Full Circular Beam FCB.Msr 121
Chapter Four
Experimental Results and Discussion
a)
b)
Figure (4.56) Mode of Failure and Cracks Pattern for FCB.Msr Circular Beam a) Load and Support Region b) Exterior Face of Failed Opening
Figure (4.57) Load-Midspan Deflection Curve FCB.Msr Circular Beam
122
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Experimental Results and Discussion
Figure (4.58) Load-Midspan Twisting Angle Curve FCB.Msr Circular Beam
4.3.4 Externally strengthened Full Circular Beam (FCB.Mcfrp) As mentioned previously, the full curved beam FCB.Mcfrp includes opening spaced at distance d/2=106.5 mm from the applied load as shown in Figure (4.59), and strengthened by full wrap of CFRP laminates of 0.131mm thickness (pair of 40 mm width on the each side of the opening and three pairs of 25mm width for both top and bottom chords of opening). The beam was loaded gradually until the first crack was observed. The first crack appears at a load of 83 kN at the corner of the opening. After more loading increments (at load about 100.4 kN) a flexural crack appears at position of maximum positive moment, while torsional cracks at an angle of 45° began to appear and propagate at a load of 128 kN. As load was increased further, corner cracks increased and widened till CFRP laminates, also torsional cracks propagated and spreads in zones that have no CFRP up to failure load.
123
Chapter Four
Experimental Results and Discussion
A beam type failure mode occurred through torsional forces at load of 333.5 kN, as shown in Figure (4.60). The load –deflection and load – angle of twist curves at midspan are shown in Figures (4.61) and (4.62), respectively.
Figure (4.59) Full Circular Beam FCB.Mcfrp
Figure (4.60) Mode of Failure and Cracks Pattern for FCB.Mcfrp Circular Beam at Different Region
124
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Experimental Results and Discussion
Figure (4.61) Load-Midspan Deflection Curve FCB.Mcfrp Circular Beam
Figure (4.62) Load-Midspan Twisting Angle Curve FCB.Mcfrp Circular Beam 125
Chapter Four
Experimental Results and Discussion
4.3.5 Summary of Test Results for FCB.M Beams Table 4.4 shows the cracking load, ultimate load, Percentage of Ultimate Load with Respect to FCB.P. Figure (4.63) and (4.64) show a comparison of load-midspan deflection and angle of twist curves for circular curved beams FCB.P, FCB.Mo, FCB.Msr and FCB.Mcfrp. It can be concluded that, the presence of the openings near the applied load reduce the ultimate load capacity (compared with control full curved beam) to the half, also a significant reduction in twisting angle was noticed because of the ultimate load reduction. On the other hand, strengthening of the opening by internal reinforcement or external CFRP laminates increased ultimate load capacity (compared with FCB.Mo) by about 60% and 75% respectively, also angle of twist and deflection were increased with an interested ratio because of the confinement of the beam at opening region which postpone failure at opening for both types of strengthening. This is because of increasing of post cracking stiffness of the beam at opening which could be seen clearly in Figures (4.63) and (4.64).
Table 4.4 Summery of Tested FCB Beams Cracking Load, kN Specimen
Ultimate Max. θ ×10-3 Max. Δ at
*
Ultimate
Load
Corner Flex.
Tor. Load, kN
103.8 103.8
(rad)
at
(mm)
Failure
Diff. %
midspan
midspan
mode
380.8
100
74.8
14.34
shear
FCB. P
--
FCB. Mo
58.8
110.7
95
190
--
31.7
16.71
FCB. Msr
55.4
79.8
89.9
305
60
46.4
26.04
FCB. Mcfrp
83
100.4
128
333.5
75
27.4
23.9
Difference = (Pu(Specimen) − Pu(FCB.Mo) )/Pu(FCB.Mo)
126
Torsional and shear Torsional and shear Torsional and shear
Chapter Four
Experimental Results and Discussion
Figure (4.63) Comparison of Load-Midspan Deflection Curves for FCB.P , FCB.Mo, FCB.Msr and FCB.Mcfrp Curved Beams
Figure (4.64) Comparison of Twisting Angle Curves at Midspan for FCB.P, FCB.Mo, FCB.Msr and FCB.Mcfrp Curved Beams 127
Chapter Four
Experimental Results and Discussion
4.3.6 Summary of The Tested Curved Beams As a summary of test results for full circular beam with strengthed and unstrengthed opening, the following notes were obtained: 1For the specimen FCB.Mo, which was unstrengthened, the decrease in ultimate load capacity was about 100% when compared with the corresponding control specimen FCB.P.
1. For internally strengthened specimen FCB.Msr, the ultimate load capacities were enhanced by about 75%, when compared with unstrengthened specimens FCB.Mo. 2. The use of CFRP laminates as external confinements in spacemen FCB.Mcfrp, increases ultimate load capacity by about 60% when compared with unstrengthened specimens FCB.Mo, and 15% less than FCB.Mcfrp. 3. A reliable enhancement appears in post-cracking behavior of specimens FCB.Msr and FCB.Mcfrp when compared with unstrengthened specimens FCB.Mo. 4. The use of internal confinement changes the failure mode from beam type failure to frame type failure, while the use of CFRP laminates retain the failure mode to beam type failure. 5. The use of CFRP laminates as external confinement makes the beam more brittle than using internal confinement.
128
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Finite Elements Modeling and Nonlinear Solution Techniques
CAPTER FIVE FINITE ELEMENTS MODELING AND FORMULATION FOR NONLIEAR ANALYSIS OF TESTED CURVED BEAMS 5.1 Introduction It is well known that laboratory tests require a great amount of effort and time, in many cases these tests are very expensive and even impractical. On the other hand, the finite element method has become, in recent years, a powerful and useful tool for the analysis of a wide range of engineering problems. A comprehensive finite element model permits a considerable reduction in the number of experiments. Nevertheless, in a complete investigation of any structural system, the experimental phase is essential . Taking into account that numerical models should be based on reliable test results, experimental and numerical/theoretical analyses complement each other in the investigation of a particular structural phenomenon.. The finite element method is a numerical procedure that can be applied to obtain solutions to a variety of problems in engineering. Steady, transient, linear, or nonlinear problems in stress analysis, heat transfer, fluid flow, and electromagnetism problems may be analyzed by finite element methods (Witte and Kikstra,2002). The Present study applies a nonlinear finite element analysis on continuous horizontally curved concrete beams with web openings and unstrengthen and
strengthened with internal reinforcement or CFRP
laminates and subjected to two points loads at midspan by using ANSYS(version 12.1) computer program. The objective is to explore the
129
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Finite Elements Modeling and Nonlinear Solution Techniques
overall response, such as: load-deformation, ultimate load and cracking load , ……..etc.
5.2 Material Modeling 5.2.1 Concrete Modeling Reinforced concrete is a composite material consisting of steel reinforcement and concrete, these two materials having vastly different properties. The required mechanical properties of reinforcing steel are generally known. However, those for concrete are more difficult to define depending upon the particular condition of mixing, placing, curing, nature, rate of loading and environmental influences. Concrete contains a large number of micro-cracks, especially at interfaces between coarse aggregate and mortar, even before any load has been applied. This property is decisive for the mechanical behavior of concrete. The propagation of these micro-cracks during loading contributes to the nonlinear behavior of concrete at low stress level and cause volume expansion at failure. Many of these cracks are caused by segregation, shrinkage, or thermal expansion in the mortar. Some micro-cracks may be developed during loading because of difference in stiffness between aggregate and mortar. The differences can result in a strain at interface zone several times larger than the average strain. Since the aggregate-mortar interface has a significantly lower tensile strength than the mortar, it constitutes the weakest link in the composite system. This is the primary reason for low tension strength of the concrete material. From the preceding discussion one can expect that the size and texture of aggregate have a significant effect on the mechanical behavior of concrete under various types of loading.
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Many experimental studies of the behavior of concrete under uniaxial and multiaxial loading conditions have been performed. The aims of such investigations have made to understand the complex response of concrete for various imposed stress conditions and to provide the necessary data required to develop accurate numerical models to be used in nonlinear finite element analysis of concrete structures.
5.2.1.1 Uniaxial Compression Behavior for Concrete A typical stress-strain behavior for concrete under uniaxial compression is shown in Figure 5.1. It is nearly linear up to about (0.3-0.5)
( )
times the ultimate strength of concrete f c' . The stress-strain curve shows a gradual increase in curvature that
(
) (
)
occurs up to a stress level of 0.75 f c' to 0.9 f c' , after which the stress-strain
( )
curve bends sharply and approaches the peak point at f c' (Chen,1982, Chen and Saleeb ,1981). Then, the stress-strain curve descends until failure occurs due to the crushing of concrete at the ultimate strain (ε u ) . The shape of the stress-strain curve is similar for concrete of low, normal, and high strengths. High strength concrete behaves in a linear fashion to a relatively higher stress level than the low strength concrete. The strain at the maximum stress is approximately (0.002) (although high strength concretes have somewhat a little higher strain at peak stresses). On the descending portion of the stress-strain curve, higher strength concretes tend to behave in a more brittle manner, with the stress dropping off more sharply than it does for concrete with lower strength (Wischers,1978).
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Finite Elements Modeling and Nonlinear Solution Techniques
Figure 5.1: Uniaxial Compressive Strain curve for concrete with different strength (Wischers,1978).
The modulus of elasticity of concrete (Ec ) is generally taken to be a function of the compressive strength f c' .
For normal weight concrete based on a dry unit weight (2200-2500 kg/m3), (Ec ) can be permitted to be taken as follows (ACI Committee 318M,2011). Ec = 4700 f c' in (MPa)
Where, ( Ec ) is the modulus of elasticity of concrete in (MPa), ( f c' ) is the ultimate strength of concrete in (MPa). The Poison's ratio (ν ) for concrete under uniaxial compression ranges between (0.15 to 0.22). It remains constant up to about 0.8 f c' (Chen,1982,
Mackava and Okamura,1983).
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Finite Elements Modeling and Nonlinear Solution Techniques
Beyond this level it begins to increase rapidly and values in excess of (1.0) have been measured (Mackava and Okamura,1983) due to the rapidly increasing transverse strain increases rapidly owing to internal cracking parallel to the direction of loading then the longitudinal strain. In this study a value of (0.2) is adopted for analyzing reinforced concrete arches. The multi-linear isotropic material uses Von Mises failure criterion along with (Willam and Warnke, 1974) model to define failure of concrete. Compressive stress-strain relationship for the concrete model was obtained by using the following equations to compute the multi-linear isotropic stressstrain curve for the concrete,( MacGregor, 1992). f =
Ec ε ε 1 + εo
εo =
2 f 'c Ec
E=
f
(5.1)
2
(5.2) (5.3)
ε
where: f: Stress at any strain, ε
ε: Strain of concrete ε 0 : Strain at ultimate compressive strength f' c E: Tangent modulus of elasticity Figure 5.2 shows the stress-strain relationship used in this study. Point 1, defined as 0.30 f c' , is calculated in the linear range Eq.(5.3), Points 2, 3 and 4 are calculated from Eq.(5.1). With ε 0 obtained from Eq.(5.2), strains were selected and stresses were calculated for each strain value. Point 5 is defined at f c' , and point 6 is the termination with ε cu = 0.003 mm/mm indicating traditional crushing strain for unconfined concrete.
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Figure 5.2: Stress-Strain Curve for Concrete in Compression ( MacGregor, 1992)
5.2.1.2 Tensile Behavior of Concrete The general mechanical behavior of concrete under uniaxial tensile loading shows many similarities to the behavior observed in uniaxial compression. Typical stress – strain curves for concrete in uniaxial tension are shown in Figure 5.3. In general, at a stress less than (60 %) of the tensile strength, the creation of new micro-cracks is negligible. So, this stress level will correspond to a limit of elasticity. Above this level of stress, the bond micro-cracks start to grow (Hughes and Chapman,1966). The direction of crack propagation for uniaxial tension is transverse (normal) to the stress direction. The growth of every new crack will reduce the available load - stress capacity and this reduction causes an increase in the stresses at critical crack tips. The failure in tension is caused by a few connected cracks rather than by numerous cracks, as it is for compressive states of stress.
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The ratio between uniaxial tensile strength
( )
( ft )
and compressive
strength f c' may vary considerably but usually ranges from (0.05) to (0.1). The modulus of elasticity under uniaxial tension is somewhat higher and Poisson's ratio somewhat lower than in uniaxial compression. The direct tensile strength of concrete is difficult to measure and is normally taken as (0.3) to o.4 f c' . Many times, either the modulus of
rupture ( f r ) or the split cylinder strength ( f t ) is used to approximate the tensile strength of concrete. The value of the modulus of rupture of concrete
varies quite widely but is normally taken as 0.62
f c' (ACI Committee
318M,2011). The split cylinder tensile strength is usually somewhat lower, at approximately (0.45) to o.55 f c' in (MPa) (Chen,1982).
Figure 5.3:Typical Tensile Stress-Strain Curve for Concrete (Hughes, B. P. and Chapman,1966)
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Chapter Five
Finite Elements Modeling and Nonlinear Solution Techniques
5.2.1.3 Biaxial Stress Behavior Concrete Kupfer and Grestle (1973) had performed extensive experimental tests on concrete plates made with three different concrete strengths (19, 31 and 58 MPa) and loaded in orthogonal directions. The ultimate strength data were reported in terms of a biaxial stress envelope as shown in Figure 5.4. Under biaxial compression, the compressive strength increases approximately (25 %) over that of the uniaxial strength at the stress ratio of
(σ 2
= 0.5 σ 1 )
and this is reduced to about (16 %) at an equal biaxial
compression state (σ 2 = σ 1 ). Under biaxial tension - compression, the compressive strength is decreased almost linearly as the applied tensile stress is increased. Under biaxial tension, the concrete strength is almost the same as that of the uniaxial tensile strength.
Figure 5.4: Biaxial State of Loading( Kupfer and Grestle ,1973)
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5.2.1.4 Triaxial Stress Behavior of Concrete Under triaxial loading, experimental studies indicated that concrete has a consistent failure surface which is a function of the three principal stresses (Chen and Saleeb, 1981). Because of concrete isotropy, the elastic limit and failure limit can be represented as surfaces in three dimensional principal stress space as shown in Figure 5.5. For increasing hydrostatic compressions along the (σ 1 = σ 2 = σ 3 ) axis, the deviatoric sections (planes perpendicular to the axis (σ 1 = σ 2 = σ 3 ) of the failure surface are more or less circular, which indicates that the failure in this region is independent of the third stress invariant. For smaller hydrostatic pressure, these deviatoric cross sections are convex and noncircular. The failure surface can be represented by three stress invariants.
Figure 5.5:Failure Surface of Concrete in 3-D Stress Space ( Chen and Saleeb,1981).
5.2.1.5 Post - Cracking Model (Tension Stiffening Model) Upon cracking, the stresses normal to the cracked plane are released as the cracks propagate. To simulate this behavior in connection with the finite element modeling of reinforced concrete members, the tension stiffening concept is usually used (Banzant, 1983). This concept is based on the fact that some of the tensile stresses can be carried by the concrete between the 137
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cracks due to the bond action between the steel bars and the surrounding concrete. This ability is progressively weakened because of the formation of new cracks. Furthermore, in order to improve the numerical stability of the solution, the tension stiffening effect was introduced in several models. In the present work, the tension stiffening of reinforced concrete after cracking was represented by providing a linearly descending branch as shown in Figure 5.6.
Figure 5.6: Pre and Post-Cracking Behavior of Normal Strength Concrete (Banzant,1983).
This model is given by: (a) For ε cr ≤ ε n ≤ α 1 ε cr α f cr α − ε fn = 2 α 1 − 1 1 ε cr
(5.4)
(b) For ε n α 1 ε cr
fn = 0
(5.5)
where, ( f n , ε n ) is the stress and strain normal to the crack plane, ( f cr , ε cr ) is the cracking stress and strain, (α1 ) is the rate of stress released as the crack widens and (α 2 ) is the sudden loss of stress at the instant of cracking. 138
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5.2.1.6 Crushing Modeling If the material at an integration point fails in uniaxial, biaxial, or triaxial compression, the material is assumed to crush at that point. Under this condition, material strength is assumed to have degraded to an extent such that the contribution to the stiffness of an element at the integration point under consideration can be ignored (ANSYS Manual,2009).
5.2.1.7 Shear Transfer Model The concrete is assumed to behave linearly in tension up to the onset of cracking. When concrete cracks, its shear stiffness is reduced. However, cracked concrete can partially transmit shear across the crack due to aggregate interlock and dowel action of the reinforcement. The shear transfer mechanism depends on the reinforcement ratio, bar size, bar arrangement, the amount of concrete cover, the type of concrete and aggregate size. To estimate such an effect, a shear transfer coefficient (β ) is introduced which represents a shear strength reduction factor for concrete across the crack face. When the crack is formed, only a constant value of a shear transfer
( )
coefficient β 0 for the opened crack is introduced, and if the crack is closed,
( )
the shear transfer coefficient (β c ) is used. The values of β 0 and (β c ) are
(
)
always in the range 1 β c β 0 0 (ANSYS Manual,2007).These values
( )
depend on the texture of the cracked surface. In this study, β 0 is assumed to be (0.18) and (β c ) is assumed to be (0.72).
5.2.1.8 Cracking Modeling The main feature of plain concrete is its low tensile strength compared with the failure stress in compression. In the finite element methods, two
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main approaches have been used for cracked section representation, the discrete cracked model, and the smeared crack model. a) Discrete Crack Model This approach was first used by Ngo and Scordelis (1967) to analyze a simply supported reinforced concrete beam. In this approach, the cracks are restricted to occur at the boundary of the elements by separation of nodal points initially occupying the same position in space, see Figure (5.20). This means that when any crack occurs the topology of the mesh varies. This makes the analysis expensive. These difficulties have resulted in a very limited acceptance of this model in the general structural applications.
( a) (b) Figure (5.7) Cracking representation in discrete cracking modeling approach (a) one directional cracking (b) two directional cracking
b) Smeared Crack Model In this approach, the cracked concrete is assumed to remain a continuum; the cracks are smeared out in a continuous fashion. It is assumed that the concrete becomes orthotropic after first cracking has occurred. In the smeared-cracking model, a crack is not discrete but implies an infinite number of parallel fissures across that part of the finite element, see Figure (5.8). After cracking has occurred, the cracked concrete becomes an orthotropic material and a new relationship must be derived. This is accomplished by modifying the stiffness of the element; the modulus of
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elasticity is reduced to zero in the direction normal to the crack. Further, a reduced shear modulus is assumed on the cracked plane to account for the aggregate interlocking. In the smeared crack simulation, two different models are used for defining the crack direction. The first is the fixed crack model. In this model, the crack is fixed. A perpendicular crack plane is allowed if the tensile stress in this direction exceeds the tensile strength. The direction of crack is held and fixed at all subsequent time steps. The second approach is the rotating crack model, in this approach the cracks are permitted initially to be perpendicular to the principal tensile stress direction when the stress reaches the specified limiting value. With further increment of loading the principal stress changes, the crack is assumed to rotate and orthotropic material axes are set in a new crack direction.
Figure 5.8: Smeared Crack Modeling (Chen and Saleeb,1981).
For ANSYS computer program, crack modeling of concrete, depends on smeared crack. This model is described in terms of shear transfer model and closing and reopening of cracks.
5.2.1.9 Failure Criteria for Concrete The actual behavior and strength of concrete materials are very complex because they depend on many factors such as the physical and
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mechanical properties of the aggregate, cement paste and the nature of loading. No single mathematical model can describe the strength of real concrete materials completely under all conditions, so, simple models or criteria are used to represent the properties that are essential to the problem being considered (Chen,1982). Willam and Warnke (1974) developed a mathematical model capable of predicting failure for concrete materials under multiaxial stress state. Both cracking and crushing failure modes are accounted for. This model is represented by the following equations: F −S ≥0 f c'
(5.6)
where, (F ) is the function of principal state (σ xp , σ yp , σ zp ) , (S ) is the failure surface expressed in terms of principal stresses and five input parameters
(f
' c
) ( )
, f t , f cb , f1 and f 2 , f c' is the ultimate uniaxial compressive strength, ( f t )
is the ultimate uniaxial tensile strength, ( f cb ) is the ultimate biaxial compressive strength, ( f1 ) is the ultimate compressive strength for a state of biaxial compression superimposed on hydrostatic stress state (σ ha ) , ( f 2 ) is the ultimate compressive strength for a state of uniaxial compression superimposed on hydrostatic stress state (σ ha ) , and
(σ ) a h
is the ambient
hydrostatic stress state. The failure surface is separated into hydrostatic (change in volume) and deviatoric (change in shape) sections as shown in Figure 5.9. The hydrostatic section forms a meridianal plane which contains the equisectrix
(σ 1 = σ 2 = σ 3 ) as an axis of revolution. The deviatoric section lies in a plane normal to the equisectrix (dashed line). The deviatoric trace is described by polar coordinate (r , θ ) , where (r ) is the position vector locating the failure surface with angle (θ ) . The failure surface is defined as: 142
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f (σ m , τ m , θ ) =
1 σm 1 τm + −1= 0 2 f c' r (θ ) f c'
(a) Hydrostatic Section (θ = 0)
(5.7)
(b) Deviatoric Section
Figure 5.9:Failure Surface (Chen,1982).
where, (σ m and τ m ) is the average stress components defined as:
1 (σ 1 + σ 2 + σ 3 ) = 1 I1 3 3
(5.8)
2 J2 5
(5.9)
σm = 2 τm =
Where, (I1 ) is the first stress invariant, ( J 2 ) is the second deviatoric stress invariant and (ρ ) is the apex of the surface. The free parameters of failure surface (ρ ) and (r ) are identified from the uniaxial compressive strength ( f c' ) , biaxial compressive strength
( f cb )
and uniaxial tensile strength ( f t ) . If Eq. (5.6) is not satisfied, there is no attendant cracking or crushing. Otherwise, the material will crack if any principal stress is tensile, while crushing will occur if all principal stresses are compressive. 143
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Willam and Warnke (1974) succeeded in finding an expression for the failure cross section, since it can meet not only the conditions of symmetry, smoothness and convexity, but also it degenerates to a circle if (r1 = r2 ) . This means that the cylindrical Von Mises model and the conical Drucker-Prager model are all special cases of Willam and Warnke failure formulation (ANSYS Manual,2009).
5.2.1.9.1 Determination of the Model Parameters The failure surface can be specified with a total of five strength
(
)
parameters f c' , f t , f cb , f1 and f 2 in addition to an ambient hydrostatic
( )
( )
stress state σ ha , as shown in Figure 5.9. f c' and ( f t ) can be specified from two simple tests, and the other constants can be determined from Willam and Warnke (1974): f cb = 1.2 f c'
(5.10)
f1 = 1.45 f c'
(5.11)
f 2 = 1.725 f c'
(5.12)
However, these values are valid only for stress states where the condition stated below is satisfied:
σ h ≤ 3 f c'
(5.13)
where
(σ h ) is the hydrostatic stress state = 1 (σ xp + σ yp + σ zp ) 3
(5.14)
The condition of Eq. (5.7) applies to stress situations with a low hydrostatic stress component.
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Figure 5.10, the lower curve represents all stress states in which
(θ = 0 ), while the upper curve represents stress states for (θ = 60 ). The axis (ξ ) represents hydrostatic length. The failure of concrete is categorized into four domains: 1st domain: (0 ≥ σ 1 ≥ σ 2 ≥ σ 3 ) (compression – compression – compression). 2nd domain: (σ 1 ≥ 0 ≥ σ 2 ≥ σ 3 ) (tension – compression – compression). 3rd domain: (σ 1 ≥ σ 2 ≥ 0 ≥ σ 3 ) (tension – tension – compression). 4th domain: (σ 1 ≥ σ 2 ≥ σ 3 ≥ 0 ) (tension – tension – tension). The concrete will crack if any principal stress is a tensile stress, while crushing will occur if all principal stresses are compressive.
Figure 5.10 Profile of the Failure Surface as Function of Five Parameters (ANSYS Manual,2007)
5.2.2 Reinforcement Modeling Since the reinforcing bars are normally long and relatively slender, they can generally be assumed to be capable of transmitting axial forces only. For the finite element models, the uniaxial stress - strain relation for steel was idealized as a bilinear curve, representing elastic-plastic behavior with strain hardening. This relation is assumed to be identical in tension and in compression as shown in Figure 5.11.
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Figure 5.11 Modeling of Reinforcing Bars.
In the present work, the strain hardening modulus (ET ) is assumed to be (0.01Es).
( )
The value of steel yield strength f y and ultimate tensile strength
( f u ) , corresponding to steel bar, are carried out from tensile test. Table 3.7, gives details about these values. In developing a finite element model, three alternative representations of reinforcement can usually be used (Wolanski,2001, Yousifani,2004), these are: 1. Discrete representation Discrete representation has been widely used. The reinforcement in the discrete model uses one dimensional bar or beam elements that are connected to concrete mesh nodes as shown in Figure 5.12a. Therefore, the concrete and the reinforcement mesh share the same nodes and the same occupied regions. Full displacement compatibility between reinforcement and concrete is a significant advantage of the discrete representation. Their disadvantages are the restriction of the mesh and the increase in the total number of elements.
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2. Embedded representation The embedded representation is often used with high order isoperimetric elements. The bar element is built in a way that keeps reinforcing steel displacements compatible with the surrounding concrete elements as shown in Figure 5.12b. When reinforcement is complex, this model is very advantageous. The stiffness matrix of the reinforcement steel is evaluated separately and then added to that of the concrete to obtain the global stiffness matrix. 3. Smeared (Distributed) representation The smeared model assumes that reinforcement is uniformly spread in a layer throughout the concrete element in a defined region of the finite element mesh as shown in Figure 5.12c. This approach is used for large scale models where the reinforcement does not significantly contribute to the overall response of the structure. In the present study, the steel reinforcements were represented by using 2-node discrete representation (LINK8 in ANSYS) and included within the properties of 8-node brick elements. The reinforcement is assumed to be capable of transmitting axial forces only, and perfect bond is assumed to exist between the concrete and the reinforcing bars. To provide the perfect bond, the link element for the steel reinforcing bar was connected between nodes of each adjacent concrete solid element, so the two materials share the same nodes.
-a-
-b-
-c-
Figure (5.12), Models for reinforcement in R/C (Tavarez 2001). (a) discrete; (b) embedded; and (c) smeared. 147
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5.2.3 CFRP Composite Modeling CFRP composites are materials that consist of two constituents. First constituent is the reinforcement, which is embedded in the second constituent, a continuous polymer called the matrix (Kaw,1997). The reinforcing material is in the form of fiber such as carbon or glass, which are typically stiffer and stronger than the matrix. The CFRP composites are modeled as orthotropic elastic materials; that is, their properties are not the same in both directions. Figure 5.13, shows a schematic of CFRP composites (Gibson,1994).
Figure 5.13:Schematic Properties of CFRP Composites (Gibson,1994).
5.3 Material Idealization 5.3.1 Element Types The elements types shown in Table 5.1 are used to model the tested beams. Element No. 1 2 3 4
Table 5.1: Element Types Used in Modeling of R.C. Beams Element Representation Type SOLID65 Concrete LINK8 Longitudinal (circumference) steel reinforcement (φ 12 mm ) and Radial reinforcement (stirrups) (φ 6 mm ) SOLID45 SHELL41
Steel plate CFRP
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5.3.1.1 Element SOLID65 SOLID65 is used for the 3-D modeling of solids with or without reinforcing bars (rebars). SOLID65 is capable of cracking in tension and crushing in compression. In concrete applications, for example, the solid capability of the element may be used to model the concrete while the rebar capability is available for modeling reinforcement behavior. The element shown in Figure 5.14 is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions. Up to three different rebar specifications may be defined. The most important aspect of this element is treatment of nonlinear material properties. The concrete is capable of cracking (in three orthogonal directions), crushing, plastic deformation, and creep. The rebars are capable of tension and compression, but not shear.
Figure 5.14: Geometry of Element SOLID65 (ANSYS help, 2007)
5.3.1.2 Element LINK8 LINK8 shown in Figure 5.15 is a spar (or truss) element which may be used in a variety of engineering applications. This element can be used to 149
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model trusses, sagging cables, links, springs, etc. The 3-D spar element is a uniaxial tension - compression element with three degrees of freedom at each node: translations in the nodal x, y, and z directions. As in a pin-jointed structure, no bending of the element is considered. Plasticity, creep, swelling, stress stiffening, and large deflection capabilities are included.
Figure 5.15: Geometry of Element LINK8 (ANSYS help, 2007)
5.3.1.3 Element SOLID45 SOLID45 is used for the 3-D modeling of solid structures, the element shown in Figure 5.16 is defined by eight nodes having three degrees of freedom at each node; translations in the nodal x, y, and z directions. The element has plasticity, creep, swelling, stress stiffening, large deflection, and large strain capabilities.
Figure 5.16: Geometry of Element SOLID45. (ANSYS help, 2007) 150
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5.3.1.4 Element SHELL41 SHELL41 shown in Figure 5.17 is a 3-D element having membrane (in-plane) stiffness but no bending (out of plane) stiffness. It is intended for shell structures where bending of the elements is of secondary importance. The element has three degrees of freedom at each node: a translation in the nodal x, y, and z directions. The element has variable thickness, stress stiffening, and large deflection.
Figure 5.17: Geometry of Element SHELL41.(ANSYS help, 2007).
5.4 Nonlinear Finite Element Analysis In any three-dimensional finite element analysis, the performance of any structural member under load depends on the behavior of the type of material used to construct the member. In concrete members which are made of different materials, concrete and reinforcing bars are brought together to behave as a composite system. The steel can be considered as a homogeneous material that exhibits a similar stress-strain relationship in tension and compression. While, the behavior of concrete is monitored to have grossly heterogeneous internal structure because it is dependenting on
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the properties of each of its components; namely, cement mortar, aggregates and air voids. One of the main objectives of the finite element analysis of structures is to determine the response of the structure under loading. A typical loaddeformation response for a monotonically loaded member is essentially linear up to a certain limit of load. Beyond this limit a nonlinear loaddeformation response occurs. Such response is due to a combination of nonlinear material behavior (material nonlinearities), large deformation in the structure (geometric nonlinearities), and interface nonlinearity for composite members (changing status) (Dawlat, 2007). In the analysis of reinforced concrete structures, at nonlinear stage of behavior, it is not possible to solve the governing equilibrium equations directly; therefore, resort has to be made to more sophisticated solution strategies. In the present work, ANSYS computer program (ANalysis SYStem) is used to create the finite element model.
5.4.1 Numerical Integration In most finite element analyses, the element stiffness matrix [Ke] cannot be obtained analytically. Thus, to perform the integration required to evaluate the element stiffness matrix, a suitable scheme of numerical integration has to be used. The application of the three-dimensional finite element analysis in connection with the nonlinear behavior of structures needs a large amount of computation time due to frequent evaluation of the stiffness matrix. Therefore, it is necessary to choose a suitable integration rule that minimizes the computation time but with sufficient accuracy. In the current study, the “Gauss-Legendre quadrature scheme” is used and it has been found to be accurate and efficient (Cervera et al. 1987).
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Hence, the equations of the stiffness matrix element in two dimensions can be expressed as: (5.15) When extended to three-dimensional cases, the equations of the stiffness matrix element are expressed as (ANSYS Theory,2007)
(5.16) Where: n, m, and l are the number of Gaussian points in the ξ, η and ζdirections respectively. In general, "n, m, and l" are taken to be equal in three directions. F (
ξ,η,ζ) is a function which represents the matrix multiplication “
[B]T [D] [B]
J ”.
Wi , Wj and Wk are weighting factors, see Table (5.1) In a similar manner the reinforcing bar element stiffness matrix can be written as: (5.17) The integration rule, which has been used in present work, is 2×2=4 points for the shell element, and 2×2×2 =8 point Gauss quadrature for the brick element and 1x2=2 for reinforced bar. Figure (5.18) shows the distribution of the sampling points over the volume of the 8 node and their weighting factors are given in Table (5.2). The integration points are also the sampling
points
for
stresses
and
(Mottram,1996).
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state
determination
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Figure (5.18) Distribution of integration points (ANSYS Theory,2009) Table 5.2 Sampling Points Locations and Weighting Factors for (2x2x2), (2x2) and (2x1) Gauss Quadrature (ANSYS Theory,2007)
5.4.2 Procedure for Solving Nonlinear Finite Element Equations The finite element discrimination process yields a set of simultaneous equations:
[K ] {U } = {F a }
(5.18)
where, [K ] is the stiffness matrix, {U } is the vector of nodal displacements
{ }
and F a is the vector of applied loads.
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For linear elastic problems Eq. (5.13) is used to find the solution of the unknown displacement {U }. In the case of nonlinear system, the stiffness matrix [K ] is a function of the unknown displacement (or their derivatives). Then the Eq. (5.13) cannot be exactly computed before determination of the unknown displacement {U }. There are several techniques for solving the nonlinear Eq. (5.18); the basic technique can be classified into: (1) Incremental or stepwise procedure, Figure (5.19a). (2) The iterative or Newton-Raphson procedure, Figure (5.19b). (3) Incremental- Iterative procedure, Figure (5.19c).
( (a)
(b)
Figure 5.19:Basic Technique for Solving the Nonlinear Equation (a) Incremental (b) Iterative (c) Incremental–Iterative (Maurizio et al,2002)
In the incremental procedure, the load is applied in several small increments, and the structure is assumed to respond linearly within each
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increment with its stiffness recomputed based on the structural geometry and member end actions at the end of the previous load increment. This is a simple procedure, which requires no iterations, but errors are likely to accumulate after several increments unless very fine increments are used, Figure 5.19a. In the iterative procedure, the total load is applied in one increment at the first iteration, the out of balance forces are then computed and used in the next iteration until the final converged solution would be in equilibrium, such that the internal load vector would equal the applied load vector or within some tolerance. This process can be written as:
[KiT ] {ΔU i }= {F a }− {F nr }
(5.19)
{U i+1}= {U i }+ {∆U i }
(5.20)
[ ] is the stiffness matrix, (i) is the subscript representing the current equilibrium iteration and {F nr } is the internal load vector.
where, K iT
This procedure fails to produce information about the intermediate stage of loading. For structural analysis including path-dependent nonlinearities increments are in equilibrium in order to correctly follow the load path. This can be achieved by using the combined incremental-iterative method. In the combined-iterative procedure, a combination of the incremental and iterative scheme is used. The load is applied incrementally, and iterations are performed in order to obtain converged solution corresponding to the stage of loading under consideration, as shown in Figure 5.19c. The incremental-iterative solution procedures have been used in this study. Full Newton-Raphson procedure is applied. The stiffness matrix is 156
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formed at every iteration. The advantage of this procedure may give more accurate result. The disadvantage of this procedure is that a large amount of computational effort may be required to form and decompose the stiffness matrix, as shown in Figure 5.20.
Figure 5.20: Incremental-Iterative Procedures Full Newton-Raphson procedure (Maurizio et al,2002)
5.4.3 Convergence Criteria For every incremental load the iteration continues until convergence is achieved. The convergence criterion for the nonlinear analysis of structural problems can be classified as: (1) Force criterion. (2) Displacement criterion. (3) Stress criterion. The displacement criterion has been used in this study. In displacement criterion, the incremental displacements at iteration (i) and the total displacements are determined. The solution is considered to be converged when the norm of the incremental displacements is within a given 157
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tolerance of the norm of the total displacements; infinite norm is used and takes the form:
{∆U i } = (max ∆U i ) ≤ Tn
(max ∆U i )
(5.21)
where, U may equal u, v, w or θ z . For force criteria the norm of the residual forces at end of each iteration are checked against the norm of the current applied forces as:
{R} = ∑ Ri2
0 .5
0.5 ≤ Tn ∑ F 2
i
(5.22)
where, {R} is the residual vector:
{R} = {F a }− {F nr }
(5.23)
In this study, the tolerance (Tn ) is taken equal to (1 %) near the ultimate load for loads control.
5.4.4 Analysis Termination Criteria In the physical test under load control, collapse of a structure takes place when no further loading can be sustained. This is usually indicated in the numerical tests by successively increasing iterative displacements and a continuous growth in the dissipated energy. Hence, the convergence of the iterative process cannot be achieved. A maximum number of iterations for each increment are specified to stop the nonlinear solution if the convergence limit has not been achieved for this study. It has been observed that a minimum number about (20) of iterations is generally sufficient to predict the solution divergence or failure. This maximum number of iterations depends on the type of the problem, extent of nonlinearities, and on the specified tolerance. In the present study a maximum number of iterations equal (100) is adopted for load control problems. 158
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CHAPTER SIX FINITE ELEMENT ANALYSIS 6.1 Introduction The aim of the present chapter is to make a comparison between the F.E. analysis results and the experimental results that explore the adequacy of elements type, material modeling, real constants and convergence criteria to model the response of the horizontally curved reinforced concrete beams with and without openings unstrengthened and strengthened by CFRP laminates or internal reinforcement. This chapter includes the numerical analysis of the curved beams tested in chapter four and parametric study of some important variables by using a powerful nonlinear finite element method package (ANSYS 12.1 software).
6-2 Description of The Tested Specimens in Finite Element The actual dimensions of the tested curved beams are shown in Figure (3-1) for semicircular curved beam and Figure (3-2) for full circular beams. By taking advantage of the symmetry for both beams in geometry and loadings, a half of the semicircular curved beam, and a quarter of the full circular beam were used for modeling finite element analysis as shown in Figure (6-1). This approach reduces computational time and the computer disk space requirements significantly.
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Solid45
Brick element of mesh Solid65
Solid 45
θ=90 Solid 45
a)
Brick element of mesh Solid65
b) Figure (6-1) Adopted Descriptions of Curved Beams a) Half of Semicircular b) Quarter ofFull Circular
6.3 Mesh Refinement An important step in finite element modeling is the selection of the mesh density. A convergence of results is obtained when an adequate number of elements are used in a model. This is practically achieved when a 160
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mesh density increases for a reasonable amount to match the results; therefore, a convergence study was carried out for a half part of SCB.P (solid semicircular curved beam) by making use of symmetry. Four types of mesh are used to find the best mesh size for SCB.P as following:
Mesh 1: total number of elements =1200 Mesh 2: total number of elements= 1482 Mesh 3: total number of elements= 2338 Mesh 4: total number of elements= 4017 Mesh 5: total number of elements= 7600 Figure (6.2) shows detail of mesh size for four cases. From the test results, it was found that models with number of elements equal to 4017 element gives the best results compared with experimental Analysis as shown in Figures (6.3) and (6.4). J
Y
Y
X
Mesh(a)1200Element
Y
Y
X
Mesh (b)1482 Element
X
Mesh (c)2338 Element
X
Mesh (d)4017 Element
Mesh (d)7600 Element
Figure (6.2) Mesh density (Cross Section and Top View). 161
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Figure (6.3) Effect of Number of Elements on Load-Midspan deflection
Figure (6.4) Effect of Number of Elements on Load- Midspan Twisting Angle 162
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6.4 Reinforcing Steel Modeling Discrete representation was used to model all types of reinforcement (longitudinal and web reinforcement). Link 8 element was employed to represent the steel reinforcement for group (I) and (II), as shown in Figure (6-5). In this study, a perfect bond between concrete and steel reinforcement is assumed.
b)
a) Figure (6.5) Reinforcing Steel Bars Modeling a) Half of Semicircular Beam b) Quarter of Full Curve Beam
6.5 Modeling of CFRP Laminates No mesh is needed for the CFRP laminates. These elements (shell41) were added by using the existed nodes of concrete. Figure (6.6) shows the CFRP laminates for both semicircular curved and full curved beams.
CFRP Laminates (Shell 41) CFRP Laminates (Shell 41)
Figure (6.6) CFRP Laminates Arrangement of Tested Concrete Curved Beams.
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6.6 Loads and Boundary Conditions Representations Displacement of boundary condition used to constrain the all reinforced concrete beam models for the purpose of getting a unique solution. These models constrained in the z-direction, and y-direction (Uz=Uy= 0 ) at the hinge support, while constrained in the y-direction and x-direction (Uy= Ux=0) at roller support as illustrated in Figure (6.7-a). In the experiment, a steel plate of (125x50x8 mm) is used at the support and loading locations to avoid the situation of stress concentration. The external applied load is represented by the equivalent nodal forces on the top elements face at mid- span points of beam, as shown in Figure (6.7-b). The application of the loads up to failure was done incrementally as required by Newton-Raphson procedure as described previously in details in chapter five of this study. Failure for each of the models is defined when the solution for a minimum load increment still does not converge (convergence fails). The program then gives a message specifying that the models have a significantly load displacement (rigid body motion).
Figure (6.7) Boundary Conditions and Applied Loads Arrangements
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Finite Element Analysis
6.7 Results of Finite Element Analysis All tested curved beams have been analyzed by using ANSYS computer program to determine the validity of this numerical method for the analysis of horizontally reinforced concrete curved beams with web opening strengthened externally with CFRP laminates or internally with steel reinforcement. The overall behavior and specifications illustrated before in chapter three for these strengthened materials have been taken in the consideration during the built up and input data of ANSYS computer program. The load-midspan deflection curves, load-midspan twisting angle curves, cracking and ultimate loads for all analyzed curved beams have been illustrated through the results below .Also, crack pattern and stress distribution through the concrete and CFRP laminates have been listed in appendix C.
6.7.1 Load – Deformations Curves Figures 6.8 to 6.35 include a comparison between the load-midspan deflection and the load-midspan twisting angle curves of the experimental and the numerical results. The variation of mid- span deflection and midspan twisting angle with the step-by-step loads applied for all curved beams are all recorded through these curves. The finite element load-deflection curves for most beams showing a stiffer response rather than the experimental results. Microcracks produced by drying shrinkage and handling are presenting in the concrete; these would reduce the stiffness of the actual beam, while the F.E. does not include the effect of microcracks. The F.E. analysis assumes that concrete is a homogenous material but, the true it is a heterogeneous material. Also, a perfect bond between the concrete and steel or CFRP Laminates is assumed in the F.E. analysis. The comparison shows the validity of the FEM results of the program used in application (ANSYS) by showing a good agreement with the experimental results discussed in chapter four previously.
165
Chapter Six
Finite Element Analysis
Figure (6.8) Load-Midspan Deflection Curves for Control Beam
Figure (6.9) Load-Midspan Twisting Angle Curves for Control Beam 166
Chapter Six
Finite Element Analysis
Figure (6.10) Load-Midspan Deflection Curves for SCB.Eo Beam
Figure (6.11) Load-Midspan Twisting Angle Curves for SCB.Eo Beam 167
Chapter Six
Finite Element Analysis
Figure (6.12) Load-Midspan Deflection Curves for SCB.Esr Beam
Figure (6.13) Load-Midspan Twisting Angle Curves for SCB.Esr Beam 168
Chapter Six
Finite Element Analysis
Figure (6.14) Load-Midspan Deflection Curves for SCB.Ecfrp Beam
Figure (6.15) Load-Midspan Twisting Angle Curves for SCB.Ecfrp Beam 169
Chapter Six
Finite Element Analysis
Figure (6.16) Load-Midspan Deflection Curves for SCB.Mo Beam
Figure (6.17) Load-Midspan Twisting Angle Curves for SCB.Mo Beam 170
Chapter Six
Finite Element Analysis
Figure (6.18) Load-Midspan Deflection Curves for SCB.Msr Beam
Figure (6.19) Load-Midspan Twisting Angle Curves for SCB.Msr Beam 171
Chapter Six
Finite Element Analysis
Figure (6.20) Load-Midspan Deflection Curves for SCB.Mcfrp Beam
Figure (6.21) Load-Midspan Twisting Angle Curves for SCB.Mcfrp Beam 172
Chapter Six
Finite Element Analysis
Figure (6.22) Load-Midspan Deflection Curves for SCB.Io Beam
Figure (6.23) Load-Midspan Twisting Angle Curves for SCB.Io Beam 173
Chapter Six
Finite Element Analysis
Figure (6.24) Load-Midspan Deflection Curves for SCB.Isr Beam
Figure (6.25) Load-Midspan Twisting Angle Curves for SCB.Isr Beam 174
Chapter Six
Finite Element Analysis
Figure (6.26) Load-Midspan Deflection Curves for SCB.Icfrp Beam
Figure (6.27) Load-Midspan Twisting Angle Curves for SCB.Icfrp Beam 175
Chapter Six
Finite Element Analysis
Figure (6.28) Load-Midspan Deflection Curves for FCB.P Beam
Figure (6.29) Load-Midspan Twisting Angle Curves for FCB.P Beam 176
Chapter Six
Finite Element Analysis
Figure (6.30) Load-Midspan Deflection Curves for FCB.Mo Beam
Figure (6.31) Load-Midspan Twisting Angle Curves for FCB.Mo Beam 177
Chapter Six
Finite Element Analysis
Figure (6.32) Load-Midspan Deflection Curves for FCB.Msr Beam
Figure (6.33) Load-Midspan Twisting Angle Curves for FCB.Msr Beam 178
Chapter Six
Finite Element Analysis
Figure (6.34) Load-Midspan Deflection Curves for FCB.Mcfrp Beam
Figure (6.35) Load-Midspan Twisting Angle Curves for FCB.Mcfrp Beam 179
Chapter Six
Finite Element Analysis
6.7.2 Cracking and Ultimate Loads The comparison between the theoretical (FEM by ANSYS) and experimental results of cracking [P cr)theo , P cr)exp ] ,ultimate [P u)theo , P u)exp ] loads is shown in Table 6.1. The table indicates a good agreement between the theoretical and experimental results. The average difference in ultimate loads is about 4.14%.
6.7.3 Deflection of Service Loads Also, Table 6.1 includes the comparison of the theoretical (ANSYS) midspan deflection δ theo and experimental midspan deflection δ exp at service load, the average difference between the theoretical to experimental service load deflections was 16%.
6.7.4 Crushing and Stresses in Curved Beams. The theoretical F.E. analysis by (ANSYS) showed that the overall compressive stress in concrete along the curved beam did not exceed the compressive strength of concrete until failure occurred, except the points of compression zone in touch with internal support and the corners of the openings, which reached the crashing stress. The effect of shear stresses were also found to be significant under the applied load and internal support region. Also, it was found that the maximum stress in CFRP laminates in curved beams with opening near midspan still less than the allowable tensile stress of CFRP laminates. Appendix C shows the local stress distribution (σ x and τ xz ) along the curved beam for concrete and σ x
for CFRP laminates obtained from
theoretical analysis by Ansys for all tested specimens
180
Chapter Six
Finite Element Analysis
Table 6.1: Theoretical and Experimental Cracking and Ultimate Loads Cracking Load
Ultimate Load
(kN)
(kN)
Beam Symbol
P cr)exp P cr)Ansys P u)exp P u)theo 62.3 43.5 147.2 152.3
SCB.P SCB.Eo
55.4
SCB.Esr
58.8
SCB.Ecfrp
72.7
45
SCB.Mo
34
21
SCB.Msr
41.5
30
32
21
116
SCB.Io
34.6
27
SCB.Isr
41.5
1.02
6.01
5.42
δtheo δexp
0.89
128.1
125
0.94
6.40
5.73
0.89
38.5 132.0
140
0.99
6.71
3.96
0.60
141.9
144
0.99
6.6
4.19
0.63
83
84
0.89
7.15
8.89
1.24
0.96
4.05
3.10
0.76
121
0.96
4.75
6.03
1.26
76.2
79
0.92
4.21
5.31
1.26
33
99.5
108
0.92
5.44
3.95
0.72
31
25
100.9
110
0.93
4.91
5.02
1.02
FCB.P
103.8
73
380.8
408
0.99
4.31
3.78
0.87
FCB.Mo
58.8
32
190
192
0.94
6.15
7.4
1.20
FCB.Msr
55.4
36
305
325
0.95
8.19
9.63
1.17
83
45
333.5
350
1.02
7.49
6.23
0.83
SCB.Mcfrp
SCB.Icfrp
FCB.Mcfrp
35
Midspan Defl. at Pu(𝑎𝑛𝑠𝑦𝑠) Service Load+,(kN) Pu(exp) δexp δtheo
100.4 112.5
+Service load =0.70 * Pu ) exp
6.8 Parametric Study A parametric study is performed to assess the influence of several important parameters on the behavior of ultimate load investigation of horizontally curved concrete beams with opening under the effect of concentrated loading. The selected parametric study to be discussed in this chapter can be summarized as follows: • Effect of beam curvature. • Effect of type and size of opening 181
Chapter Six
Finite Element Analysis
• Effect of opening length. • Effect of opening Hight. • Types of wrapping schemes of CFRP (U- wrap and full wrap)
6.8.1 Effect of Curvature To study the effect of curvature on ultimate load capacity of horizontally curved concrete beam with openings, four types of curved R/C beams with different radius were analyzed in the present study (1m, 2m, 3m and ∞). The study carried out on semicircular curved beam with opening near interior support as shown in Figure (6.36). The load- midspan deflection curves of all types are shown in Figure (6.37). A significant difference in general response and ultimate load were noticed and recorded. An increase in ultimate load of (103%, 130%, and 172%) in curved beams of radius (2, 3 and ∞) m respectively, compared with curved beams of radius 1 m. Table 6.2 shows that the ultimate load for these specimens.
Figure (6.36) SCB.Io Beams With Variable Radiuses
182
Chapter Six
Finite Element Analysis
Figure (6.37) Load-Midspan Deflection Curves for SCB.Io Beams With Variable Radiuses Table 6.2 Ultimate Load for Different Types of SCB.Io with Variable Radiuses.
Beam radius
Ultimate Load(kN)
Increase In Ultimate Load %
Exp. R=1 m F.E. R=1 m
76.16 79
---
F.E. R=2 m
160.5
103
F.E. R=3 m
182.1
130
F.E. R=∞ m
215.4
172
6.8.2 Effect of Type and Size of Opening The effect of height/Length ratio of the opening on the load-deflection
curve, load-angle of twist curve and ultimate load capacity of a semicircular curved beam with interior opening (SCB.Io) was also studied here in. In this 183
Chapter Six
Finite Element Analysis
study three aspect ratios of 1:3(80*250),1:2(100*200) and 1:5(66.67*300) were taken with constant area for all openings. A semicircular curved beam with circular opening of radius 79.8mm at interior support which gives the same area of rectangular section was also analyzed. Figures (6.38) and (6.39) show the numerical results of the F.E. analysis with experimental results of load-deflection and load-twisting angle curves for SCB.Io curved beam with different opening type and dimensions. It could be conclude that, as the height/length ratio of the opening decreases, the load carrying capacity increase, furthermore a considerable increasing in load carrying capacity in beam with circular opening was found. A summary of the values of collapse loads obtained from F.E. analysis and experimental test are listed in table (6.3).
Figure (6.38) Load-Midspan Deflection Curves for SCB.Io Beams With Variable Opening Dimensions
184
Chapter Six
Finite Element Analysis
Figure (6.39) Load-Midspan Twisting Angle Curves for SCB.Io Beams with Variable Opening Dimensions
Table 6.3 Ultimate Load Capacity for Different Opening Dimensions of SCB.Io Curved Beam Beam (SCB.Io)
Opening
Opening
Ultimate
Length (mm) Height (mm)
𝑷𝒖
Load(kN)
𝑷𝒖𝑺𝑪𝑩.𝑰𝒐
(Exp.)200
100
76.16
--
200
100
79
1
250
80
85
1.07
300
66.67
85
1.07
79.8
79.8
111.8
1.41
Rectangular
Circular
185
Chapter Six
Finite Element Analysis
6.8.3 Effect of Opening Length The variation in load-deflection curve, load-angle of twist curve and ultimate load capacity of a semicircular curved beam with interior opening (SCB.Io) due to the variation of opening length were numerically carried out herein. The height of the opening was kept constant (100 mm ) while its length was increased from 100 mm to 300 mm by increment of 100mm. Figures (6.40) and (6.41) show the results of the F.E. analysis to gather with experimental results of load-deflection and load-twisting angle curves for SCB.Io curved beam. It could be noticed that decreasing in opening length leads to an increasing in ultimate load capacity of the beam. The collapse loads obtained from this study are listed in table (6.4).
Figure (6.40) Load-Midspan Deflection Curves for SCB.Io Beams With Variable Opening Length
186
Chapter Six
Finite Element Analysis
Figure (6.41) Load-Midspan Twisting Angle Curves for SCB.Io Beams With Variable Opening Length
Table 6.4 Ultimate Load Capacity for Different Opening Length of SCB.Io Curved Beam Opening
Opening
Ultimate
Length (mm) Height (mm)
𝑷𝒖
Load(kN)
𝑷𝒖𝑺𝑪𝑩.𝑰𝒐
(Exp.)200
100
76.16
--
100
100
90
1.14
200
100
79
1
300
100
70
0.88
Beam (SCB.Io)
187
Chapter Six
Finite Element Analysis
6.8.4 Effect of Opening Height The effect of opening height on the response of load-deflection curve, load-angle of twist curve and ultimate load capacity of a semicircular curved beam with interior opening (SCB.Io) was numerically carried out by fixing the length of opening to 200mm and decreasing the depth from 50mm to 100mm by increments of 25mm. Figures (6.42) and (6.42) show the results of the F.E. analysis with experimental results of load-deflection and load-twisting angle curves for SCB.Io curved beam with different opening height. The results show that when opening height decreased, the ultimate load capacity of the beam increased. The predicted values of collapse loads obtained from F.E. analysis and experimental test are listed in table (6.5).
Figure (6.42) Load-Midspan Deflection Curves for SCB.Io Beams With Variable Opening Height
188
Chapter Six
Finite Element Analysis
Figure (6.43) Load-Midspan Twisting Angle Curves for SCB.Io Beams with Variable Opening Height
Table 6.5 Ultimate Load Capacity for Different Height of SCB.Io Curved Beam Opening
Opening
Ultimate
Length (mm) Height (mm) Beam (SCB.Io)
𝑷𝒖
Load(kN)
𝑷𝒖𝑺𝑪𝑩.𝑰𝒐
(Exp.)200
100
76.16
--
200
50
90
1.14
200
75
85
1.07
200
100
79
1.0
189
Chapter Six
Finite Element Analysis
6.8.5 Wrapping Schemes of CFRP (U and Full Wrapping) The effect of wrapping schemes on the behavior of semicircular horizontally curved reinforced concrete beams with opening at midspan or interior support had been studied. A full and U wrapping of CFRP laminates of thickness 0.131 mm were used for specimens SCB.Io and SCB.Mo. Figures (6.44), (6.45), (6.46) and (6.47) show the numerical results of the finite element analyses of load-
deflection and load-twisting angle curves for both SCB.Io and SCB.Mo curved beams. These figures reveal that the full wrapping CFRP laminates gives higher ultimate load capacity than U wrapping CFRP laminates. Also, post-cracking stiffness of curved beam with opening at midspan (SCB.Mo) was increased more than post-cracking stiffness of curved beam with opening at interior support (SCB.Io). This is because torsional forces at midspan region are greater than those at internal support, also full wrapping CFRP laminates can resist higher torsional stresses than U wrapping CFRP laminates. The numerical values of these
specimens are listed in table (6.6)
Figure (6.44) Load-Midspan Deflection Curves for SCB.Io Beams with Different Wrapping Schemes 190
Chapter Six
Finite Element Analysis
Figure (6.45) Load- Midspan Twisting Angle Curves for SCB.Io Beams with Different Wrapping Schemes
Figure (6.46) Load-Midspan Deflection Curves for SCB.Mo Beams with Different Wrapping Schemes 191
Chapter Six
Finite Element Analysis
Figure (6.47) Load- Midspan Twisting Angle Curves for SCB.Mo Beams with Different Wrapping Schemes
Table 6.6 Ultimate Load Capacity of Wrapping Schemes for SCB.Io and SCB.Mo Curved Beams Spacemen
Full Wrapping
U - Wrapping
Name
Ultimate Load(kN)
Ultimate Load(kN)
SCB.Io (Exp.)
100.93
--
SCB.Io
110
96
SCB.Mo(Exp.)
116
--
SCB.Mo
121
110
192
Chapter Seven
Conclusions and Recommendations
CHAPTER SEVEN CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORKS 7.1 General The main objective of this research is to study the behavior of horizontally reinforced concrete curved beams with and without openings, internally strengthened by stirrups or externally strengthened by CFRP laminates. This research included an experimental program as well as a nonlinear finite element analysis by (ANSYS version 12.1 ) in order to predict the ultimate strength and overall behavior of these tested specimen.
In this chapter, conclusions that are obtained from experimental and analytical evidences as well as some recommendations for future extension of the work will be presented.
7.2 Conclusions The main conclusions observed from each phase of investigation (experimental program and finite element analysis) for horizontally reinforced concrete curved beam with and without openings, internally strengthened by steel reinforcement or externally strengthened by CFRP laminates will be presented in this section which represents the summary and benefit of this research: 1- The presence of opening near midspan or interior support reduced the
ultimate load capacity about 35% for semi-circular beams and about 50% for full circular (ring) beam, if compared with control beam without opening.
193
Chapter Seven
Conclusions and Recommendations
2- The presence of opening near exterior support of curved beam had lesser
effect on ultimate load capacity, about 13% , if compared with control beam without opening. 3- The internal strengthening of the opening region by steel reinforcement
(stirrups), increased the ultimate load capacity about (3% to 30%) for semi-circular curved beams, and about 60% for full circular (ring) beam, if compared with unstrengthed beams. 4- The external strengthening (confinement) of the opening region by
CFRP laminates enhanced the ultimate load carrying capacity about (11% to 40%) for semi-circular curved beams, and about 75% for full circular (ring) beam, if compared with unstrengthed beams. 5- The simple method of design suggested by Mansure (1998) for internally
strengthening of opening region by steel reinforcement (stirrups) gave generally a good response of strengthed curved beams in terms of cracking patterns, deflection of service load as well as the mode of failure changed from mode of opening failure to beam type failure 6- The simple method of design proposed here for externally strengthening
with CFRP laminates of opening region gave good result. The mode of failure changed from opening mode failure to a beam type failure. 7- The proposed simple method of design for external strengthening with
CFRP laminates of the opening region, enhanced the general behavior of strengthened curved beams in terms of cracking patterns, deflection of service load as well as the mode of failure changed from mode of opening failure to beam type failure. 8- Both internal strengthened and external confinement of curved beams
with opening near exterior support had a small effect on the maximum deflection and rotation (angle of twist ) of midspan section at ultimate load value. While for beams with opening at midspan a decrease of (33% and 66%) occurred for deflection and rotation of midspan section. 194
Chapter Seven
Conclusions and Recommendations
Furthermore, the deflection and rotation for beams with opening near interior support increased about 28% and 38% respectively. 9- The general response of externally strengthened specimens by CFRP
laminates was approximately in agreement with specimens of internally strengthened by steel reinforcement (stirrups) in terms of deflection and ultimate loads. 10-
The numerical F.E. analysis by ANSYS package is valid for the
analysis of horizontally curved concrete beams with openings, unstrengthened or strengthened by internal reinforcement or CFRP laminates. The general response of load deformation curves (deflection and rotation) of F.E. analysis gave a good approchment with those of experimental curves. The comparison with experimental results confirmed the validity of the analysis with maximum deviation by about 11% in ultimate load capacity. 11- The average difference of deflections at service loads and ultimate load
capacity of all specimens between numarical and experimental results is about 4.8% and 4.1%, respectively, which insure the validity of the F.E. solution. 12- The ultimate load capacity of the curved beam with opening increased
with decreasing the curvature (1/R) for the same length by about (103%,130% and 172%) for curvature ( 0.5,0.33 and 0.0). 13- Circular shape of opening gave an increase in ultimate load capacity
about 41% more than rectangular openings. 14- Opening with length more than height for the same area gave a best
ultimate load capacity than those of length less or equal height by about 7%. 15- Using of U-scheme for CFRP laminates gave a decreasing in load
capacity of the curved beam reached to 12% if compared with full scheme of CFRP laminates. 195
Chapter Seven
Conclusions and Recommendations
7.3 Recommendations for Further Works 1. Investigate the behavior of horizontally curved reinforced concrete beams with and without opening subjected to dynamic and impact loading. 2. Investigate the behavior of horizontally curved reinforced concrete beams with different height/radius (h/R) ratios and/or different shapes (like parabolic or elliptical). 3. Investigate experimentally effect of type of FRP on behavior of horizontally curved reinforced concrete beams without openings. 4. Include effect of bond slip between concrete and reinforcing steel bars. 5. Investigate the behavior of tapered horizontally curved reinforced concrete beams with and without openings. 6. Investigate the behavior of horizontally curved reinforced concrete beams with opening under repeated loads. 7. Investigate experimentally effect of other types of supports of semicircular curved beam.
196
References
References (1) ACI Committee 318, (2011) "Building Code Requirements for Structural Concrete (ACI318M.11) and Commentary", American Concrete Institute, Farmington Hills, Michigan, USA, 473 pp. (2) ACI Committee 440, (2002) "Guide for the design and Construction of externally Bonded FRP Systems for Strengthening of Concrete Structures", American Concrete institute, Michigan, USA. (3) ACI 211.1-97, (Re-approved 1997), "Standard practice for selecting proportions for normal, heavy weight and mass concrete", ACI Manual of Concrete Practice, American Concrete Institute, Detroit, 1997. (4) Ali A.Y., (2010), " Three-Dimensional Nonlinear Finite Element Analysis of Reinforced Concrete Horizontally Curved Deep Beams", Journal of Babylon University, Vol.18, No.1. (5) Ali A.Y. et el , (2011), " Analysis of Reinforced Concrete Beams with Openings and Strengthened by (CFRP)", Journal of Babylon University, Vol.19, No.3. (6) Al-Mutairee H.M , (2013), " Effect of Non-Uniform Distribution of Longitudinal Reinforcement on the Behavior of Reinforced Concrete Horizontally Curved Beams with Fixed-Ends", Journal of Babylon University, Vol.21, No.3. (7) Abdulla, H. A., Torkey A. M., Haggag H. A. and Abu-Amira A. F. , (2003) ”Design Against Crack at The Opening in The Reinforced Concrete Beam Strengthened With Composite Sheet”, Composite Structure Journal, No.60, pp.197-204. (8) Al-Temeemi M.A,(2002) "Nonlinear Analysis of Reinforced Concrete Horizontally Curved Beams on Elastic Foundation", M.Sc. Thesis, University of Babylon. (9) "ANSYS Manual", Version (12.1), USA, 2009. 197
References
(10) American Society of Testing and Material (ASTM), "Standard Specification for Concrete Aggregates"(2002), ASTM C33-02, West Conshohocken, PA. (11) ASTM Designation C370-05a "Standard specification for testing method and definitions for mechanical testing of steel products", (2005) Annual Book of ASTM Standards, American Society for Testing and Material, Philadelphia, Pennsylvania, Section 1, Vol. (1.01), pp 248-287. (12) ASTM Designation C39-01 "Standard specification for testing method for compressive strength of cylindrical concrete specimens", (2001) Annual Book of ASTM Standards, American Society for Testing and Material, Philadelphia, Pennsylvania, Section 4, Vol. (4.02), pp 20-24. (13) ASTM Designation C496-96 "Standard specification for splitting tensile strength of cylindrical concrete specimens", (1996) Annual Book of ASTM Standards, American Society for Testing and Material, Philadelphia, Pennsylvania, Section 4, Vol. (4.02), pp 1-4. (14) Al-Kaisy,
A.I.D,(2005),''Nonlinear
Finite
Element
Analysis
of
Reinforced Concrete Beams with Large Opening under Flexure'' M.Sc. Thesis,University of Technology, Iraq. (15) Al-Dolaimy, A.T., (2011), " Structural Behavior of Continuous Reinforced Concrete Beams with Openings and Strengthened by CFRP Laminates'' M.Sc. Thesis, university of Babylon. (16) Al-Tai, S., L., (2010) "Nonlinear Behavior Of Reinforced Concrete TBeams Strengthened With CFRP Subjected To Shear". M. Sc. Thesis, College of Engineering ,University of Babylon, IRAQ. (17) Badawy, H. E. I, Jordaan, I. J., and McMullen, A. E., "Effect of Shear on Collapse of Curved Beams", Journal of the Structural Division, ASCE, Vol. 103, No. ST9, Proc. Paper 13185, September 1977, pp. 1849-1866.
198
References
(18) Bandyapadhya J.N et el ,(2011), " Strengthening and rehabilitation of reinforced concrete beams with opening", International Journal of Civil and Structural Engineering" , Vol 2, No 1. (19) Banzant, Z. P., (1983)"Comment on Orthotropic Models for Concrete and Geomaterials", Journal of Engineering Mechanics Division, ASCE, Vol. (106), No. 3, pp. 849-865. (20) Burton, K.T,(1965),'' Influence of Embedded Service Ducts on The Strength of Continuous Reinforced Concrete T-Beams''ACI Journal, Vol. 62, October , pp.1327-1344. Cited by (Al-Thabhawee,20012) (21) Bisby, L.A. and Williams, B.K. Strengthening
of
Concrete
(2004) ,"An Introduction to FRP
Structures”
Department
of
Civil
Engineering, Queen’s University, Canada, pp.4. (22) Cervera, M. Hinton., E. and Hassan, O., (1987),''Nonlinear Analysis of Reinforced Concrete Plate and Shell Structures Using 20 – Noded Isoperimetric Brick Elements'', Computers and Structures, Vol.25, no.6, pp.845 – 869 (23) Chen, W. F., (1982) "Plasticity in Reinforced Concrete", McGraw-Hill, pp. 471. (24) Chen, W. F. and Saleeb, A. F., (1981)"Constitutive Equations for Engineering Materials", West Lafayette, Indiana, pp. 580. (25) Chu, K., and Thelen, A., "Plastic Analysis of Circular Balcony Girders", Journal of Structural Division, ASCE, Vol. 89, No. ST6, Proc. Paper 3726, December 1963, pp. 159-185. (26) Carolin, A., (2003)'' Carbon Fiber Reinforced Polymers for Strengthening of Structural Elements '' Ph.D. Thesis, University of Lulea, Sweden. Cited by(Al-Tai,2010).
199
References
(27) Dalal, M.S. , (1969),'' Behavior and Design of Opening Reinforced Concrete T-Beams'' Ph.D. Thesis, University of Taxas Austin, U.S.A. Cited by (Al-Thabhawee,2012). (28) Dawlat, D. A., (2007), "Experimental and Theoretical Investigation of the Behavior of Reinforced Concrete Beams Strengthened by Fiber Reinforced Polymer", Ph. D. Thesis, University of Baghdad/College of Engineering, January (29) Federico M. Mazzolani (2001) "The Use of FRP Materials for The Seismic Upgrading of Existing RC Structures” University degli, Facoltà di Ingegneria (30) Gibson, R. F., (1994) "Principles of Composite Material Mechanics", McGraw-Hill, Inc., New York. (31) Hamzah B.A,(2012) " Behavior of RC Curved Beams with Openings and Strengthened by CFRP Laminates ", P.hd. Thesis, University of Basrah. (32) Hanson, J.M., (1969),''Square openings in webs of continuous joists''. PCA Research and Development Bulletin. RD100 .01 D, Portland Cement Association, Stokie, IL, pp. l-14. (Cited by Mansur, M.A. 1998). (33) Hii, A. K. Y., and Al-Mahaidi, R., (2006) "An Experimental and Numerical Investigation on Torsional Strengthening of Solid and BoxSection RC Beams Using CFRP Laminates", Department of Civil Engineering, Monash University, Melbourne, Vic. 3800, Australia. (34) Hsu, T. T. C., Inan, M., and Fonticiella, L., "Behavior of Reinforced Concrete Horizontally Curved Beams", ACI Journal, Vol. 75, No. 4, April 1978, pp. 112-123.
200
References
(35) Hughes, B. P. and Chapman, G. P., (1966) "The Complete Stress-Strain Curve for Concrete in Direct Tension", Bulletin RILEM, No. 30, pp. 95-97. (36) Iraqi Specification No.5, "Portland Cement",Baghdad,(1984). (37) Iraqi Specification No.45, "Natural Sources for Gravel that is used in concrete and construction", Baghdad,(1984). (38) Jindal, R.L.,(1976),''behavior of reinforced concrete rectangular beam having large openings''zanco ;sul university vol.2, pp.11-25.Cited by (Al-Thabhawee, 2012). (39) Jordaan, I. J., Khalifa, M. M. A., and McMullen, A. E., "Collapse of Curved Reinforced Concrete Beams", Proceedings, ASCE, Vol. 100, ST11, November 1974, pp. 2255-2269. (40) Kaw, A. K., (1997) "Mechanics of Composite Materials", CRC Press LLC, Boca Raton, Florida, USA. (41) Khalifa, M. M. A., "Collapse of Reinforced Concrete Beams Curved in Plan", thesis presented to the University of Calgary, at Calgary, Alberta, Canada, in 1972, in partial fulfillment of requirements for the degree of Master of Science. (42) Kupfer, H., and Grestle, K., (1973) "Behavior of Concrete Under Biaxial Stress", ASCE-Journal of Engineering Mechanics, Vol. (104), No. (EM4). (43) Mackava, K., and Okamura, H., (1983) "The Deformation Behavior and Constitutive Equation of Concrete Using the Elasto-Plastic and Fracture Model", Journal of Faculty of Engineering, University of Tokyo (b), Vol. (xxxvii), No. 2. (44) MacGregor J. G., (1992) "Reinforced Concrete Mechanics and Design", Prentice-Hall, Inc., Englewood Cliffs, NJ.
201
References
(45) Maurizio, G., Kypros, P., and Peter, W., (2002) "Shear Performance of FRP Reinforced Concrete Beam", Journal of Reinforced Plastics and Composites, No. 22, pp. 1389. (46) Mansur, M.A., and Hasnat A. (1979) “Concrete Beams with Small Openings under Torsion” Journal of the Structural Division, ASCE, Vol. 105, No. ST11, November, pp. 2433-2447. (47) Mansur, M.A., and Alwis W.A.M., (1984), ''Reinforced Fiber Concrete Deep Beams with Web Opening'' The International Journal of Cement Composites and Lightweight Concrete, Vol. 6, No. 4, November, pp. 263-271. (48) Mansur, M. A., and Rangan, B. V., "Study of Design Methods for Reinforced Concrete Curved Beams", ACI Journal, Vol. 78. No.3, May-July 1981 pp. 226-231. (49) Mansur, M.A., Tan K.H., and Lee, S.L., (1984) “Collapse Load of Reinforced Concrete Beams with Large Opening” Journal of the Structural Division, ASCE, Vol. 110, No. ST11, November, pp. 26022618. (50) Mansur, M.A., and Paramasivam P., (1984) “Reinforced Concrete Beams with Small Opening in Bending and Torsion” ACI Structural Journal, Vol. 81, No. 2, March-April, pp.180-185. (51) Mansur, M.A., Tan K.H., and Lee S.L., (1985) “Design Method for Reinforced Concrete Beams with Large Openings” ACI Structural Journal, Vol. 82, No. 4, July-August, pp. 517-524. (52) Mansur, M.A., Tan K.H., and Lee, S.L., (1991) “Test on Reinforced Concrete Continuous Beams with Opening” Journal of the Structural Structural, ASCE, Vol. 117, No. ST6, June, pp. 1593-1606. (53) Mansur M.A., Hung L.M., Tan K.H., and Lee S.L., (1992) “Deflections of Reinforced Concrete Beams with Web Openings” ACI Structural Journal, Vol. 89, No. 4, July-August, pp. 391-397. 202
References
(54) Mansur, M.A., (1998) “Effect of Openings on the Behavior and Strength of R/C Beams in Shear” Cement and Concrete Composites, 20, pp. 477-486. (55) Mansur M.A. and Tan K.H., (1998),"Concrete Beam with Opening Analysis and Design'' CRC press LLC, Florida, U.S.A, 33441. (56) Mansur, M.A., (2006) '' Design of Reinforced Concrete Beams with Web Openings '' Proceeding of the 6th Asia-Pacific Structural Engoneering and Construction Conference (APSEC 2006), 5-6 September, Kuala Lumber, Malaysia. (57) Mohammed, T.J., (2004),'' Analysis of Reinforced Concrete Beams with Large Opening Under Torsion ''M. Sc. Thesis, University of Technology , Iraq. (58) Mottram, T., and Shaw, C., (1996), ''Using Finite Element in Mechanical Design'', McGraw-Hill Book Company, London (59) Nasser, K.W., Acavalos, A. and Daniel, H.R., (1967), ''Behavior and Design of Large Opening in Reinforced Concrete Beams'' ACI Journal, Vol. 64, No.1, January, pp.25-33. Cited by (Mansur et al,1985) (60) Najim, W.N., (2009), ''Experimental and Theoretical Investigation of the Behavior of R.C. Deep Beams with Openings Strengthened by CFRP Laminates''M.Sc. Thesis, university of Babylon. (61) Neff, C., Schnellenbach-Held, M., Daus, S., (2002) “RC Beams with one Large Rectangular Opening-Experimental Investigations” DaCon Vol.17, Darmstadt. (62)
Ngo, D. and Scordelis, A.C. (1967). "Finite Element Analysis of Reinforced Concrete Beams," Journal of ACI,Vol. 64, No. 3, pp. 152163.
(63) Patel and Saksena (2013),:" Experimental of Reinforced Concrete Beams With Web Opening", International Journal of Advanced Engineering Research and Studies", Vol. II/Issue III/pp 66-68. 203
References
(64) Salam, S.A., (1977), ''Beams with openings under different stress conditions'', Proceedings of 3rd Conference on Our World in Concrete and Structures, CI-Premier, Singapore, 25-26 Aug., pp. 259-267. (Cited by Mansur, M.A. 1998) (65) Shah, S. P., Swartz, S. E., and Ouyang, C., (1995) " Fracture Mechanics of Concrete ", Jhon Wiley & Sons, Inc., New Yourk. (66) Somes, N.F. and Corley, W.G., (1974), ''Circular openings in webs of continuous beams'', SP-42, American Concrete Institute, Detroit, MI, pp. 359- 398. (Cited by Mansur, M.A. 1998). (67)
Sika CarboDur FRP Composites Constrection for Repair & Strengthening of Structures .. MARCH 2003
(68) Suzuki, A., Masashi, A., Kengo, H. and Shoichi, O., (2007), ''High Strength Deformed Bar-in-Coil “HDC800” for Shear Reinforcement of RC Beam with Web-opening'' Nippon Steel Technical Report, No. 96 July, pp.67-74. (69) Tan K.H., Mansur M.A., and Huang L.M., (1996) “Reinforced Concrete T-Beams with Large Web Opening in Positive and Negative Moment Regions” ACI Structural Journal, Vol. 93, No. 3, May-June, pp. 277-289. (70) Tan K.H., and Mansur M.A., (1996) “Design Procedure for Reinforced Concrete Beams with Large Web Openings” ACI Structural Journal, Vol. 93, No. 3, July-August. (71) Tavarez, F.A., (2001), ''Simulation of Behavior of Composite Grid Reinforced Concrete Beams Using Explicit Finite Element Methods'', M.Sc. Thesis, University of Wisconsin-Madison, Madison, Wisconsin (72) Weng, W., Tan, K.H., and Mansur, M.A., (1999), '' Effects of Creating an Opening in Existing Beams''ACI Structural Journal, V. 96, No. 6, November-December, pp.899-906.
204
References
(73) Witte, F. C. D. and Kikstra, W. P., (2002) "DIANA Finite Element Analysis User's Manual, Release 9.0", TNO Building and Construction research, Netherlands.
(74) Wischers, G., (1978) "Application of Effects of Compressive Loads on Concrete", Betontechnische Berichte, No. 2 and 3, Duesseldorf, Germany. (75) Willam, K., and Warnke, E., (1974) "Constitutive Model for the Triaxial Behavior of Concrete", Proceedings, International Association for Bridge and Structural Engineering, Vol. (19), ISMES, pp. 174, Bergamo, Italy. (76) Wolanski, I. J., (2001) "Flexural Behavior of Reinforced and Prestressed Concrete Beams Using Finite Element Analysis " , M.Sc. Thesis, Marquette University, U.S.A. (77) Yousifani, A. H., (2004) "Investigation of the Behavior of Reinforced Concrete Beams with Construction Joints Using Nonlinear Three – Dimensional Finite Element", M. Sc. Thesis, University of Technology, pp. 123.
205
Appendix – A -
Appendix – A Analytical Solution and Design of Semicircular Curved Beam (Control Beam)
P
y'
c
P
y' Radius of Semicircular Curved Beam = r Applied Load = p …… for each midspan Reaction at A = ra = rb =r1 Reaction at C = r2 Taking moment of forces about y'-y' axis (through support C) : 2×r1×r= 2× p (r - r× sin (45)) r1= r2 = 0.2929 p ƩFy = 0 r2 = 1.4142 p By taking a section before and after the applied load we can find the shear, bending moment and torque equations. •
Shear, Moment and Torsion calculation 1. For θ < 45 V = 0.2929p Mθ = 0.2929p×r×sin (θ) Tθ = 0.2929p×r(1-cos(θ)) 2. For 45 < θ < 90 V= 0.2929p – p = -0.707p Mθ=0.2929p×r×sin (θ) - p×(r×sin(θ-45)) Tθ = 0.2929p×r(1-cos(θ)) - p×r×(1- cos(θ-45))
A1
Appendix – A -
•
Checking flexural capacity of the section
Mu =ρ.b.d2.fy(1-0.59ρ.fy/fc')
Ø6 @ 4.5°
ρ= As/bd = 113×2/(125×213) = 0.008488263 2Ø12
fc'= 31 Mpa fy = 550 Mpa Mu= 24.123 kN.m
250 mm 2Ø12
for θ = 45 the value of the load p could be found from: Mθ = 0.2929p×r×sin (45) p= 119.3 kN
125 mm
also, for θ =90 the value of the load p could be found from: Mθ = 0.2929p×r×sin (90) - p×(r×sin(90-45)) p= 59.64 kN
•
Checking shear capacity of the section
Vc= 0.17× (fc')0.5×b×d =0.17× (31)0.5×125×213/1000 = 25.2 kN Vs= Av.fy.d/s =2×25.5×520×213/90/1000 = 62.7 kN Vu = Vc +Vs = 87.9 kN Maximum shear occurs at interior support = 0,707p 0.707p = 87.9 P = 124.4 kN Therefore , the control ultimate load is the least of the upper three values of ' P ' which equal to 59.64 kN The shear force, bending moment and torque diagrams could be drawn for this load as shown below:
A2
Appendix – A -
Shear Force Diagram For Half of Semicircular kN 30 20 10 0 -10 0
10
20
30
40
50
60
70
80
90
Shear
-20 -30 -40 -50
Moment Diagram For Half of Semicircular kN.m 15.00 10.00 5.00 0.00 -5.00 0
10
20
30
40
50
60
70
80
90
Moment
-10.00 -15.00 -20.00 -25.00 -30.00
Torqu Diagram For Half of Semicircular kN.m 8.00 7.00 6.00 5.00 4.00
Torqu
3.00 2.00 1.00 0.00 0
10
20
30
40
50
A3
60
70
80
90
Appendix – A -
•
Checking torsional reinforcement of the section
The value of maximum torque could be found from the toque diagram which equal to 6.5 kN.m The section properties of the section as required from ACI code -318 are : Acp = 31250 mm2 Pcp = 750 mm2 Xo = 75 mm Yo = 200 mm Aoh = 15000 mm2 Ao = 0.85 Aoh = 12750 mm2 Ph= 550 mm At/s = Tu/(2.Ao.fy) = 6.5×106/(2*12750*550) = 0.423 mm2/mm Vc = 0.17× (31)0.5×125×213/1000 = 25.2 kN Shear force v = 42.17 kN Vs = V-Vc = 42.17 - 25.2 = 16.9 kN Av/s = Vs/fy.d = 16.9×1000/( 520×213) = 0.15 mm2/mm A (v+t) /s = 0.15+0.423 = 0.573 mm2/mm Assume Φ6 mm stirrups Av= 55 mm2 S = 55 / 0.573 = 95 mm So, use Φ6 mm stirrups @ 90 mm Al = At/s × Ph = 0.423× 550 = 232 mm2 So, the use of 4 Φ 12 mm longitudinal reinforcement will satisfy the longitudinal torsion and bending moment requirements which provide area of steel equal to (4*113.0 mm2 = 452 mm2).
•
For ring beam, a Fortran code program was written to get the forces along the beam and then the same procedure used to for the analysis and design
A4
Appendix – B -
Appendix – B 1- Design of Steel Reinforcement for strengthening of Internal Opening of Semicircular Curved Beams R = 1.0625 m fc'= 30.3 Mpa fy= 586 Mpa fyv = 550 Mpa h = 250 mm 250 mm b = 125 mm co = 25 mm d = 250 – 25 – 5.7 – 12/2 = 213 mm dv = 250 – (2*25+2*5.7+10/2+10/2) = 178.6 mm μ = 0.0375
4Ø12
125 mm
2Ø12 mm , As provided =226 mm2 along the curved beam for each top and bottom. Dimension of opening = (100 × 200) mm Vu (at center of opening) = 40.7 kN Mu (at center of opening) = 17.53 kN.m Tu (at center of opening) = 3.76 kN.m • Beam Type Failure • Longitudinal Reinforcement The required longitudinal reinforcement can be found from :
M o(1) = (3.76 +40.7 *0.0375)*�1 + 2/2 + 17.53 = 25.00 Kn.m
→ As = 249.0 mm2
M o3 = (3.76+ 40.7*0.0375)�1 + 2/2 – 17.53 = -10.05 kN.m A5
Appendix – B -
Checking Mode 2 failure from the following equation:
Meq(2) = (3.76+ 40.7*0.075/2)�1 + 2/2 = 7.5 kN.m As = 69 mm2 Required area of steel reinforcement for whole section As total =69+0+249 = 318 mm2 Available area of steel reinforcement for whole section As available = 4*113 = 452 mm2 then no additional reinforcement required. • Design of Stirrups a) For torsion Xo = 125-2×25 = 75 mm Yo = 250 - 2×25 = 200 mm M eq(1) = 25.00 – 17.53 = 7.5 Kn.m 𝑀𝑒𝑞(1) 7.5 × 106 𝐴𝑤 = = = 0.226 𝑚𝑚2 /𝑚𝑚 4𝑋𝑜. 𝑌𝑜. 𝐹𝑦𝑣 4 × 75 × 200 × 550 𝑠 b) For Shear Vc = 1/6 �𝑓𝑐′ bw.d Vc = 1/6 √30 ×125× 213 × 10−3 = 17.5 𝑘𝑁 Vu(sc)= Vu+1.2*Tu/b Vu(sc)=40.7+1.2*3.67/0.125=75.9 kN Vu (eq) = Vu(sc)-Vc = 75.9 -17.5 = 58.4 kN 𝐴𝑤 58.4 × 103 = = 0.498 𝑠 213 × 550 𝑠=
𝐴𝑤
0.498
=
2∗28.3 0.498
= 112.33 𝑚𝑚
A6
Appendix – B -
then use two bar Ø6mm at each side of the opening spaced at a distance of 40 mm which will satisfy the shear requirement at that section. • Frame Type Failure a) Bottom Chord Vertical Shear V ub = Vu / 2 = 40.7 /2 = 20.35 kN 𝑉𝑢𝑏 20.3 × 103 𝐴𝑣 = = = 0.66 𝑚𝑚2/ 𝑚𝑚 2𝑑. 𝐹𝑦𝑣 38 × 550 𝑠 2 ∗ 28.3 = 85.5 𝑚𝑚 𝑆 = 0.66 Lateral Shear
Vuz= Tu/(θt+θb) = 3.67/(0.175) = 20.97 kN 𝑉𝑢𝑧 20.97 × 103 𝐴𝑣 = = = 0.42 𝑚𝑚2/ 𝑚𝑚 2𝑑𝑤. 𝐹𝑦𝑣 91 × 550 𝑠 𝐴𝑤 2∗28.3 𝑠= = = 52.4 𝑚𝑚 0.72+0.66
0.42+0.66
Use Ø6mm @ 40 mm bars for bottom chord, which will satisfy the shear requirement. a) Member above the opening ( compression chord) Since the dimensions of the section of this chord match’s the bottom chord dimensions and forces, but with a compression axial force, then the use of the same spacing of the stirrups will be considered as shown in Figure (B-2). • Design for Crack Control 40.7 × 103 𝑉𝑢 = = 104.65 𝑚𝑚2 𝐴𝑑 = 𝐹𝑦. 𝑠𝑖𝑛𝛼 550 × sin(45)
Use 4Ø6mm diagonal bars in each direction for each corner of the opening
A7
Appendix – B -
2Ø6mm
2Ø6mm 6Ø6mm
2Ø6mm
4Ø6mm
4Ø6mm
6Ø6mm
4Ø6mm Diagonal bar 2Ø12mm
75
100
6Ø6mm Closed Stirrups 2Ø6mm
Opening
75
2Ø12mm 125
A8
Appendix – B -
2- Design of CFRP for External Strengthening of Internal Opening of Semicircular Curved Beams R = 1.0625 m fc'= 30.3 Mpa fy= 586 Mpa fyv = 550 Mpa h = 250 mm 250 mm b = 125 mm co = 25 mm d = 250 – 25 – 5.7 – 12/2 = 213 mm dv = 250 – (2*25+2*5.7+10/2+10/2) = 178.6 mm μ = 0.0375
4Ø12
125 mm
2Ø12 mm , As provided =226 mm2 along the curved beam for each top and bottom. Dimension of opening = (100 × 200) mm Vu (at center of opening) = 40.7 kN Mu (at center of opening) = 17.53 kN.m Tu (at center of opening) = 3.76 kN.m • Beam Type Failure • Longitudinal Reinforcement The required longitudinal reinforcement can be found from :
M o(1) = (3.76 +40.7 *0.0375)*�1 + 2/2 + 17.53 = 25.00 Kn.m
→ As = 249.0 mm2
M o3 = (3.76+ 40.7*0.0375)�1 + 2/2 – 17.53 = -10.05 kN.m Checking Mode 2 failure from the following equation: A9
Appendix – B -
Meq(2) = (3.76+ 40.7*0.075/2)�1 + 2/2 = 7.5 kN.m As = 69 mm2 Required area of steel reinforcement for whole section As total =69+0+249 = 318 mm2 Available area of steel reinforcement for whole section As available = 4*113 = 452 mm2 then no additional CFRP Laminates required. • Design of Stirrups a) For torsion Xo = 125-2×25 = 75 mm Yo = 250 - 2×25 = 200 mm M eq(1) = 25.00 – 17.53 = 7.5 Kn.m Aw/s= 7.5/(4*0.075*0.2*4300)=0.029 mm2/mm s =50*0.131/0.029 = 252.8 mm and this is too much spacing, check shear requirement b) For Shear Vc = 1/6 �𝑓𝑐′ bw.(d-do) Vc = 1/6 √30 ×125× (213 − 100) × 10−3 = 17.5 𝑘𝑁 Vf = Vu – Vc = 40.7 - 17.5 = 23.2 kN 23.2 × 103 = 5.39 𝑚𝑚2 𝐴𝑓 = 4300 𝑏𝑓 =
𝐴𝑓
0.131
=
6
0.131
= 41.8 𝑚𝑚
use bf = 25 mm full wrapping on each side of the opening, which will satisfies torsional requirement too. use bf = 25 mm full wrapping on each side of the opening A10
Appendix – B -
• Frame Type Failure b) Bottom Chord Vertical Shear Vc = 0 because of tension state of bottom chord Vub = Vu / 2 = 40.7 /2 = 20.35 kN Vf = Vu = 20.35 kN Avf = Vf / Ffu Avf = 20.35× 103 / 4300 = 4.7 mm2 bf = (Avf / 2) / tf = (4.7 / 2) / 0.131 = 18.06 mm Lateral Shear Vuz= Tu/(θt+θb) = 3.67/(0.175) = 21 kN Vf = Vuz = 21 kN Avf = Vf / Ffu Avf = 21× 103 / 4300 = 4.877 mm2 bf = (Avf / 2) / tf = (4.877 / 2) / 0.131 = 18.6 mm use three 20 mm CFRP laminates full wrapping for bottom chord
b) Member above the opening ( compression chord) Since the dimensions of the section of this chord match’s the bottom chord dimensions and forces, but with a compression axial force, then the use of the same spacing of the stirrups will be considered as shown in Figure (B-3).
25cm 2 cm
A11
25 cm
Appendix – C -
Cracks Pattern And Stress In Concrete And CFRP Laminates
Figure (C-1) Axial Stress in x-direction of specimen SCB.Eo on outer face
Figure (C-2) Shear Stress τ xz of SCB.Eo Specimen on outer face A12
Appendix – C -
Figure (C-3) Cracks Pattern at Internal Support of SCB.Eo Specimen at Load 125 kN
Crashing
Figure (C-4) Axial Stress in x-direction of specimen SCB.Esr on outer face
A13
Appendix – C -
Figure (C-5) Shear Stress τ xz of SCB.Esr Specimen on outer face
Figure (C-6) Cracks Pattern of SCB.Esr Specimen
A14
Appendix – C -
Figure (C-7) Axial Stress in x-direction of specimen SCB.Ecfrp on outer face
Crashing
Figure (C-8) Shear Stress τ xz of SCB.Ecfrp Specimen on outer face
A15
Appendix – C -
Figure (C-9) Stresses in Y-direction of CFRP for Specimen SCB.Ecfrp
Figure (C-10) Cracks Pattern of SCB.Ecfrp Specimen
A16
Appendix – C -
Figure (C-11) Axial Stress in x-direction of specimen SCB.Mo on outer face
Figure (C-12) Shear Stress τ xz of SCB.Mo Specimen on outer face
A17
Appendix – C -
Crashing
Figure (C-13) Axial Stress in x-direction of specimen SCB.Msr on outer face
Crashing
Figure (C-14) Shear Stress τ xz of SCB.Msr Specimen on outer face
A18
Appendix – C -
Figure (C-15) Cracks Pattern of SCB.Ecfrp Specimen
Figure (C-16) Axial Stress in x-direction of specimen SCB.Mcfrp on outer face
A19
Appendix – C -
Crashing
Figure (C-17) Shear Stress τ xz of SCB.Mcfrp Specimen on outer face
Figure (C-18) Stresses in Y-direction of CFRP for Specimen SCB.Mcfrp
A20
Appendix – C -
Figure (C-19) Cracks Pattern of SCB.Mcfrp Specimen
Figure (C-20) Axial Stress in x-direction of specimen SCB.Io on outer face
A21
Appendix – C -
Crashing
Figure (C-21) Shear Stress τ xz of SCB.Io Specimen on outer face
Figure (C-22) Cracks Pattern of SCB.Io Specimen
A22
Appendix – C -
Figure (C-23) Axial Stress in x-direction of specimen SCB.Isr on outer face
Figure (C-24) Shear Stress τ xz of SCB.Isr Specimen on outer face
A23
Appendix – C -
Figure (C-25) Cracks Pattern of SCB.Isr Specimen
Figure (C-26) Axial Stress in x-direction of specimen SCB.Icfrp on outer face
A24
Appendix – C -
Figure (C-27) Shear Stress τ xz of SCB.Icfrp Specimen on outer face
Figure (C-28) Cracks Pattern of SCB.Icfrp Specimen A25
Appendix – C -
Figure (C-29) Stresses in Y-direction of CFRP for Specimen SCB.Icfrp
Figure (C-30) Axial Stress in x-direction of specimen FCB.Mo on outer face
A26
Appendix – C -
Figure (C-31) Shear Stress τ xz of SCB.Mo Specimen on outer face
Figure (C-32) Cracks Pattern of SCB.Mo Specimen A27
Appendix – C -
Figure (C-33) Axial Stress in x-direction of specimen FCB.Msr on outer face
Figure (C-34) Shear Stress τ xz of SCB.Msr Specimen on outer face A28
Appendix – C -
Figure (C-35) Cracks Pattern of SCB.Msr Specimen
Figure (C-36) Axial Stress in x-direction of specimen FCB.Mcfrp on outer face
A29
Appendix – C -
Figure (C-37) Shear Stress τ xz of SCB.Mcfrp Specimen on outer face
Figure (C-38) Stresses in Y-direction of CFRP for Specimen FCB.Mcfrp
A30
Appendix – C -
Figure (C-39) Cracks Pattern of SCB.Mcfrp Specimen
A31
ΞΎΗϧϟϭ ΓΩΩΣϣϟ έλΎϧόϟ ΔϘϳέρ ϥϣ ΔϠλΣΗϣϟ ΞΎΗϧϟ ϥϳΑ ϕϓϭΗ ϰϠϋ ϝϭλΣϟ ϡΗ ϡΎϋ ϝϛηΑϭ ϭ %4,1ϕέϓϝΩόϣΑϭ ϯϭλϘϟϝΎϣΣϻϭϕϘηΗϟϝΎϣΣϭϝηϔϟρϣϧϭϝϭρϬϠϟ ΓέϓϭΗϣϟΔϳέΑΗΧϣϟ .ϲϟϭΗϟϰϠϋϝϭρϬϟϭϯϭλϘϟϝΎϣΣϼϟ %16 ΓΩΎϳίΑ %12ΔΑγϧΑ ϝϘϳΎϬϠϳϠΣΗϡΗϲΗϟΕΎΑΗόϠϟϰλϗϻϝϣΣϟϥΓΩΩΣϣϟέλΎϧόϟΔϘϳέρΕΩϛ ϥΎϓϙϟΫϛϭ .%50ΔΑγϧΑΔΣΗϔϟϝϭρϝϳϠϘΗΩϧϋ14ΔΑγϧΑΩΩίΗϭ50έΩϘϣΑΔΣΗϔϟϝϭρ ΓΩΎϳί ϥΎϓ ϙϟΫ ϰϟ ΔϓΎο .%50 ΔΑγϧΑ ωΎϔΗέϻ ΓΩΎϳί Ωϧϋ 14 ΔΑγϧΑ ΩΩίϳ ϰλϗϻ ϝϣΣϟ ΔΣΎγϣϟαϔϧΑϭΔόΑέϣϟϝΩΑΔϳέΩΔΣΗϓϡΩΧΗγΩϧϋΕΩΟϭϰλϗϻϝϣΣϟϥϣ41έΩϘϣΑ ϥΎϓ ϭ 200,%100 Ώ (1/R) αϭϘΗϟ ΔΑγϧ ΓΩΎϳί Ωϧϋϭ ΏΗόϟ ϝϭρ αϔϧϟϭ ϪϧΎϓ Ύυϳϭ .%172ϭ130,%103ΕϧΎϛϰλϗϻϝϣΣϟϲϓΓΩΎϳίϟ
اﻟﺧﻼﺻﺔ ث ﺗﻣت دراﺳﺔ ﺗﺻ�رف وأداء اﻻﻋﺗ�ﺎب اﻟﺧرﺳ�ﺎﻧﯾﺔ اﻟﻣﺳ�ﻠﺣﺔ اﻟﻣﻘوﺳ�ﺔ اﻓﻘﯾ�ﺎ وﺑﻧوﻋﯾﮭ�ﺎ ﻓﻲ ھذا اﻟﺑﺣ ِ
اﻟﻐﯾر ﺣﺎوﯾﮫ ﻋﻠﻰ ﻓﺗﺣﺎت واﻟﺣﺎوﯾﺔ ﻋﻠﻰ ﻓﺗﺣﺎت ﻏﯾر ﻣﻘواة او ﻣﻘواة و ﺑﺎﺳﺗﻌﻣﺎل )اﻟﻠداﺋن اﻟﻛرﺑوﻧﯾ�ﺔ اﻟﻣﺳﻠﺣﺔ او ﻗﺿﺑﺎن ﺣدﯾد اﻟﺗﺳﻠﯾﺢ ( وذﻟك ﻣن ﺧﻼل ﺗﻘدﯾم دراﺳﺔ ﻣﺧﺗﺑرﯾﮫ وﺗﺣﻠﯾﻠﯾﺔ. ﯾﺗﺿﻣن اﻟﻌﻣل اﻟﻣﺧﺗﺑري اﻋداد وﻓﺣص ارﺑﻌﺔ ﻋﺷر ﻋﺗﺑﺎ" ﺧرﺳﺎﻧﯾﺎ" ﻣﻘوﺳ�ﺎ" ﺑﺎﻻﺗﺟ�ﺎه اﻻﻓﻘ�ﻲ ﻣﻘﺳ�ﻣﺔ اﻟﻰ ﻣﺟﻣوﻋﺗﯾن .اﻟﻣﺟﻣوﻋﺔ اﻻوﻟﻰ ﺗﺗﺿﻣن ﻋﺷرة ﻧﻣﺎذج ﻋﻠﻰ ﺷﻛل ﻧﺻف داﺋرة )ﻧﻣوذج واﺣد ﺑ�دون ﻓﺗﺣﺎت واﻟﺗﺳﻌﺔ اﻻﺧرى ﺗﺣﺗوي ﻋﻠﻰ ﻓﺗﺣﺎت( ,ﺑﯾﻧﻣﺎ اﻟﻣﺟﻣوﻋﺔ اﻟﺛﺎﻧﯾﺔ ﺗﺗﻛون ﻣ�ن ارﺑﻌ�ﺔ ﻧﻣ�ﺎذج ﻋﻠ�ﻰ ﺷ��ﻛل داﺋ��رة ﻛﺎﻣﻠ��ﺔ )ﻧﻣ��وذج واﺣ��د ﺑ��دون ﻓﺗﺣ��ﺎت واﻟﺛﻼﺛ��ﺔ ﺗﺣﺗ��وي ﻋﻠ��ﻰ ﻓﺗﺣ��ﺎت( .اﻟﻣﺗﻐﯾ��رات اﻟﺗ��ﻲ ﯾﺗﺿﻣﻧﮭﺎ اﻟﺑرﻧﺎﻣﺞ اﻟﻌﻣﻠﻲ ھﻲ ﺗﺄﺛﯾر ﻧ�وع اﻟﻌﺗ�ب وﻛ�ذﻟك ﻣوﻗ�ﻊ اﻟﻔﺗﺣ�ﺔ ﺧ�ﻼل اﻟﻣﻘط�ﻊ اﻟط�وﻟﻲ ﻟﻠﻌﺗ�ب
اﻟﻣﻘوس وﺗﻘوﯾﺔ اﻻﻋﺗﺎب اﻟﻣﻘوﺳﺔ اﻟﺣﺎوﯾﺔ ﻋﻠ�ﻰ ﻓﺗﺣ�ﺎت ﺑﺎﺳ�ﺗﻌﻣﺎل )اﻟﻠ�داﺋن اﻟﻛرﺑوﻧﯾ�ﺔ اﻟﻣﺳ�ﻠﺣﺔ او ﻗﺿﺑﺎن ﺣدﯾد اﻟﺗﺳﻠﯾﺢ ( .ﺗم ﻓﺣص اﻻﻋﺗ�ﺎب ﺗﺣ�ت ﺗ�ﺄﺛﯾر ﺣﻣ�ل ﻣرﻛ�ز ) (points loadﻣﺳ�ﻠط ﻋﻠ�ﻰ اﻟطﺑﻘﺔ اﻟﻌﻠوﯾﺔ ﻟﻣﻧﺗﺻف اﻟﻔﺿﺎء وﺑﺈﺳﻧﺎد ﻣﻔﺻﻠﻲ وﻣﺗدﺣرج. اظﮭرت اﻟﻧﺗﺎﺋﺞ اﻟﻣﺧﺗﺑرﯾﺔ ﻧﻘﺻﺎن ﻣﻠﺣوظ ﻓﻲ ﻗﺎﺑﻠﯾﺔ اﻟﺗﺣﻣل اﻟﻘﺻوى ﻟﻼﻋﺗﺎب ﺑوﺟود اﻟﻔﺗﺣﺔ ﻏﯾر اﻟﻣﻘواة ﺑﺣدﯾد ﺗﺳﻠﯾﺢ او اﻟﯾﺎف ﻛرﺑوﻧﯾﺔ ﺑﺣواﻟﻲ %35ﻟﻠﻌﺗب اﻟﻣﻘوس اﻟﻧﺻف اﻟداﺋري و %50ﻟﻠﻌﺗب اﻟداﺋري اذا ﻣﺎ ﻗورﻧت ﻣﻊ اﻟﻧﻣﺎذج اﻟﻣرﺟﻌﯾﺔ اﻟﻐﯾر ﺣﺎوﯾﺔ ﻋﻠﻰ ﻓﺗﺣﺎت اﺿﺎﻓﺔ اﻟﻰ ﺗﻐﯾﯾر ﻓﻲ ﻧﻣط ﻓﺷل
ﺑﻘوى اﻟﻘص ) .( shear failureان ﻗﺿﺑﺎن ﺣدﯾد اﻟﺗﺳﻠﯾﺢ او اﺳﺗﺧدام اﻟﻠداﺋن اﻟﻛرﺑوﻧﯾﺔ اﻟﻣﺳﻠﺣﺔ ﯾﺣﺳن ﻣن ﻣﻘﺎوﻣﺔ اﻟﺗﺣﻣل اﻻﻗﺻﻰ ﻟﻠﻌﺗب ﺑﻧﺳﺑﺔ ) %3اﻟﻰ (%30و )%11 اﻟﻰ (%40ﻋﻠﻰ اﻟﺗواﻟﻲ ﻟﻸﻋﺗﺎب اﻟﻣﻘوﺳﺔ اﻟﻧﺻف داﺋرﯾﺔ ,ﺑﯾﻧﻣﺎ ﻟﻼﻋﺗﺎب اﻟداﺋرﯾﺔ اﻟﻣﻘوﺳﺔ ﻛﻠﯾﺎ ﻓﺎن ﻧﺳﺑﺔ اﻟزﯾﺎدة ﺗﺻل اﻟﻰ %60و %75ﻟﻠﺗﻘوﯾﺔ اﻟداﺧﻠﯾﺔ واﻟﺧﺎرﺟﯾﺔ ﻋﻠﻰ اﻟﺗواﻟﻲ .ﻛذﻟك ﻓﺎن اﺳﺗﺧدام ﻛﻼ اﻟﻧوﻋﯾن )ﺑﺎﺳﺗﺧدام ﺣدﯾد ﺗﺳﻠﯾﺢ او اﻻﻟﯾﺎف اﻟﻛرﺑوﻧﯾﺔ(
ﯾﻘﻠل ﻣن اﻟﺗﺷﻘﻘﺎت وﯾﺣﺳن ﻣن ﺻﻼدة اﻻﻧﺣﻧﺎء واﻻﻟﺗواء ﻟﻠﻌﺗب ﺑﻌد اﻟﺗﺷﻘق. ﺑﺎﻟﻧﺳﺑﺔ ﻟﻠﺗﺣﻠﯾل اﻟﻧظري ﻓﺎﻧﮫ ﻗدم ﻧﻣوذﺟﺎ ﻻﺧطﯾﺎ ﺛﻼﺛﻲ اﻷﺑﻌﺎد ﻟﻠﻌﻧﺎﺻر اﻟﻣﺣددة ﻣﻼﺋﻣﺎ ً ﻟﺗﺣﻠﯾل اﻻﻋﺗﺎب اﻟﺧرﺳﺎﻧﯾﺔ اﻟﻣﺳﻠﺣﺔ اﻟﻣﻘوﺳﺔ اﻓﻘﯾﺎ اﻟﺣﺎوﯾﺔ ﻋﻠﻰ ﻓﺗﺣﺎت وﻏﯾر اﻟﺣﺎوﯾﮫ ﻋﻠﻰ ﻓﺗﺣﺎت واﻟﻣﻘواة واﻟﻐﯾر ﻣﻘواة ﺑﺎﺳﺗﻌﻣﺎل )اﻟﻠداﺋن اﻟﻛرﺑوﻧﯾﺔ اﻟﻣﺳﻠﺣﺔ او ﺣدﯾد اﻟﺗﺳﻠﯾﺢ ( ﺗﺣت ﺗﺄﺛﯾر أﺣﻣﺎل ﺗزاﯾدﯾﮫ ﺑﺎﺳﺗﺧدام ﺑراﻣﺞ اﻟﺣﺎﺳوب ) (ANSYS 12.1وﻛذﻟك دراﺳﺔ ﺑﻌض اﻟﻣﺗﻐﯾرات اﻻﺿﺎﻓﯾﺔ اﻟﻣﮭﻣﺔ.
ﺠﻤﻬورﯿﺔ اﻟﻌراق وزارة اﻟﺘﻌﻠﯿم اﻟﻌﺎﻟﻲ و اﻟﺒﺤث اﻟﻌﻠﻤﻲ ﺠﺎﻤﻌﺔ ﺒﺎﺒل ﻛﻠﯿﺔ اﻟﻬﻨدﺴﺔ ﻗﺴم اﻟﻬﻨدﺴﺔ اﻟﻤدﻨﯿﺔ
اﻟﺘﺤﻠﯿﻞ اﻟﻼﺧﻄﻲ ﻟﺴﻠﻮك اﻷﻋﺘﺎب اﻟﺨﺮﺳﺎﻧﯿﺔ اﻟﻤﺴﻠﺤﺔ اﻟﻤﻘﻮﺳﺔ اﻓﻘﯿﺎ واﻟﺤﺎوﯾﺔ ﻋﻠﻰ ﻓﺘﺤﺎت ﻣﻘﻮاة ﺑﺄﻟﯿﺎف اﻟﻜﺎرﺑﻮن اﻟﺒﻮﻟﻤﺮﯾﺔ ِ ِ رﺴﺎﻟﺔ ﻤﻘدﻤﺔ ﻟﻛﻠﯿ ِ ِ درﺠﺔ اﻟدﻛﺘوراﻩ اﻟﺤﺼول ﻋﻠﻰ ﺠﺎﻤﻌﺔ ﺒﺎﺒل ﻛﺠزء ِﻤ ْن ﻤﺘطﻠﺒﺎت ﱠﺔ اﻟﻬﻨدﺴﺔ - ِ اﻟﻤدﻨﯿﺔ )اﻹﻨﺸﺎءات( ﻓﻲ اﻟﻔﻠﺴﻔﺔ ﻓﻲ اﻟﻬﻨدﺴﺔ
ﻣﻦ ﻗﺒﻞ
ﺳﺠﺎد ﻋﺎﻣﺮ ﺣﻤﺰة ﺑﻜﺎﻟﻮرﯾﻮس ﻋﻠﻮم ﻓﻲ اﻟﮭﻨﺪﺳﺔ اﻟﻤﺪﻧﯿﺔ ) (2000م ﻣﺎﺟﺴﺘﯿﺮ ھﻨﺪﺳﺔ ﻣﺪﻧﯿﺔ /اﻧﺸﺎءات ) (2003م
ﺃﺷﺮﺍﻑ
اﻷﺳﺘﺎذ اﻟﺪﻛﺘﻮر :ﻋﻤﺎر ﯾﺎﺳﺮ ﻋﻠﻲ
2014
