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Assessment of parameters affecting the resistance of fully restrained reinforced concrete beams or one-way slabs

Permanent Link: http://ufdc.ufl.edu/UFE0045579/00001

Material Information

Title: Assessment of parameters affecting the resistance of fully restrained reinforced concrete beams or one-way slabs
Physical Description: 1 online resource (236 p.)
Language: english
Creator: Goh, Chong Yik M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: beam -- concrete -- deflection -- membrane -- resistance -- restrained -- slab
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The load deflection curve of laterally restrained reinforced concrete slabs or beams is also the resistance curve that is required for a simplified dynamic analysis of the slab as a single-degree-of-freedom model. Hence, the ability to generate the resistance curve is critical to obtaining the dynamic response of laterally restrained reinforced concrete slabs at the midspan. The proposed methodology presented in this study consists of idealizing a three-hinge failure mechanism as a two-spring model that represents the support and the slab as linear springs. A three-spring model is proposed to simulate the behavior of a four-hinge failure mechanism. Using compatibility and equilibrium equations, the membrane force of the slab can be obtained for any displacement of the slab at the midspan. The modified moment capacity of the slab can then be obtained from the load interaction diagram at the critical sections. The load capacity is calculated from the modified moment capacity of the slab and the additional moment from the membrane force. Bond slip is also taken into account in the calculation of the slab deflection.   The proposed methodology was implemented using a computer program and these results were validated by experimental data reported by Woodson and Garner (1985),Woodson (1985) and Yu and Tan (2011). It was found that the proposed model produced reasonable results for slabs that exhibit the three-hinge failure mechanism, the modified three-hinge failure mechanism and the four-hinge failure mechanism. Further study is required for deep slabs, two-way slabs, and partially restrained slabs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chong Yik M Goh.
Thesis: Thesis (M.S.)--University of Florida, 2013.
Local: Adviser: Krauthammer, Theodor.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045579:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045579/00001

Material Information

Title: Assessment of parameters affecting the resistance of fully restrained reinforced concrete beams or one-way slabs
Physical Description: 1 online resource (236 p.)
Language: english
Creator: Goh, Chong Yik M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: beam -- concrete -- deflection -- membrane -- resistance -- restrained -- slab
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The load deflection curve of laterally restrained reinforced concrete slabs or beams is also the resistance curve that is required for a simplified dynamic analysis of the slab as a single-degree-of-freedom model. Hence, the ability to generate the resistance curve is critical to obtaining the dynamic response of laterally restrained reinforced concrete slabs at the midspan. The proposed methodology presented in this study consists of idealizing a three-hinge failure mechanism as a two-spring model that represents the support and the slab as linear springs. A three-spring model is proposed to simulate the behavior of a four-hinge failure mechanism. Using compatibility and equilibrium equations, the membrane force of the slab can be obtained for any displacement of the slab at the midspan. The modified moment capacity of the slab can then be obtained from the load interaction diagram at the critical sections. The load capacity is calculated from the modified moment capacity of the slab and the additional moment from the membrane force. Bond slip is also taken into account in the calculation of the slab deflection.   The proposed methodology was implemented using a computer program and these results were validated by experimental data reported by Woodson and Garner (1985),Woodson (1985) and Yu and Tan (2011). It was found that the proposed model produced reasonable results for slabs that exhibit the three-hinge failure mechanism, the modified three-hinge failure mechanism and the four-hinge failure mechanism. Further study is required for deep slabs, two-way slabs, and partially restrained slabs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chong Yik M Goh.
Thesis: Thesis (M.S.)--University of Florida, 2013.
Local: Adviser: Krauthammer, Theodor.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045579:00001


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1 ASSESSMENT OF PARAME TERS AFFECTING THE R ESISTANCE OF FULLY RESTRAINED R EINFORCED C ONCRETE BEAMS OR ONE WAY SLABS By CHONG YIK M. GOH A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILL MENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013

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2 2013 Chong Yik M. Goh

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3 To my wonderfully loving and supportive wife and family and my adorable children

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4 ACKNOWLEDGMENTS I would like to express my deepest gratitude to my thesis advisor, D r. Theodor Krauthammer, for his guidance on helping me to successfully conduct research on this topic of interest and complete my coursework I would also like to thank my thesis committee member, Dr. Serdar Astarl ioglu for his support in the completion of this research as well as Dr. Long Bui and Corey Astrom for their assistance. I would like to thank my sponsor the Defense Science and Technology Agency (DSTA ) for giving me the opportunity to further my studies I would also like to thank my friends at the Center for Infrastructure Protection and Physical Security (CIPPS) as well as fellow classmates at the University of Florida for their support and friendship. Last but not least, I would like to thank my wife for her unequivocal love sacrifice and understanding and my family for their support.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ 4 LIST OF TABLES ................................ ................................ ................................ ........... 9 LIST OF FIGURES ................................ ................................ ................................ ...... 10 LIST OF ABBREVIATIONS ................................ ................................ .......................... 17 ABSTRACT ................................ ................................ ................................ .................. 24 CHAPTER 1. PROBLEM STATEMENT ................................ ................................ ...................... 26 1.1 Introduction ................................ ................................ ................................ ...... 26 1.2 Objective and Scope ................................ ................................ ........................ 27 1.3 Research Significance ................................ ................................ ..................... 27 2. BACKGROUND AND LITERATURE REVIEW ................................ ...................... 28 2.1 Introduction ................................ ................................ ................................ ...... 28 2.2 SDOF Analysis of Structural Components ................................ ....................... 28 2.3 Load Deflection Behavior of Restrained Concrete Slabs ................................ .. 30 2.3.1 Yield Line Theory ................................ ................................ ................... 30 2.3.2 Membrane Action ................................ ................................ ................... 31 2.3.3 Pa st Research on Membrane Action ................................ ...................... 32 ................................ .................. 40 2.3.4.1 One way slabs ................................ ................................ .............. 40 2.3.4.2 Two way slabs ................................ ................................ .............. 45 ................................ ........................... 46 2.3.6 Other Load Deflect ion Models for One way Slabs ................................ .. 48 2.3.6.1 Krauthammer et al. (1986) ................................ ............................ 48 2.3.6.2 Welch (1999) ................................ ................................ ................. 50 2.3.6.3 Krauthammer et al. (2003) ................................ ............................ 52 2.4 Plastic Hinge Formation ................................ ................................ ................... 53 2.4.1 Idealized Curvature Diagr am ................................ ................................ .. 54 2.4.2 Previous Studies on Plastic Hinge Length ................................ .............. 56 2.5 Moment Area Theorem ................................ ................................ .................... 59 2.6 Limit Analysis of Fixed End Concrete Slabs or Beams ................................ .... 60 2.6.1 Central Hinge Does Not Form ................................ ................................ 61 2.6.2 Full Pl astic Collapse Mechanism Forms ................................ ................. 63 2.7 Bond Slip ................................ ................................ ................................ ......... 64 2.7.1 Proposed Bond Stress slip Model ................................ .......................... 66

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6 2.7.2 Displacements due to Bar Slip ................................ ................................ 68 2.8 Summary ................................ ................................ ................................ ......... 69 3. METHODOLOGY ................................ ................................ ................................ .. 71 3.1 Three Hinge Failure Mechanism ................................ ................................ ...... 71 3.1.1 Material Models ................................ ................................ ...................... 72 3.1.1.1 Concrete in compressio n ................................ .............................. 72 3.1.1.2 Concrete in tension ................................ ................................ ....... 73 3.1.1.3 Reinforcement steel in compression and tension .......................... 74 3.1.2 Simplified Section Analysis for Moment Curvature ................................ 75 3.1.2.1 General procedure ................................ ................................ ........ 75 3.1.2.2 Fir st cracking of concrete ................................ .............................. 77 3.1.2.3 First yielding of reinforcement steel ................................ ............... 77 3.1.2.4 Concrete crushing ................................ ................................ ......... 78 3.1.3 Simplified P M Interaction Diagram ................................ ........................ 78 3.1.3.1 Pure compression ................................ ................................ ......... 79 3.1.3.2 Balance d condition ................................ ................................ ........ 80 3.1.3.3 Pure flexure ................................ ................................ .................. 80 3.1.3.4 Pure tension ................................ ................................ .................. 80 3 .1.4 Load Deflection Curve of Slab without Lateral Restraints under Uniform Load Using Limit Analysis ................................ ............................... 82 3.1.4.1 General equations to calculate load and deflection ....................... 85 3.1.4.2 First cracking ................................ ................................ ................. 89 3.1.4.3 First plastic hinge ................................ ................................ .......... 90 3.1.4.4 Rotation of existing pla stic hinge ................................ ................... 93 3.1.5 Moment displacement (M ) Curve ................................ ........................ 99 3.1.5.1 First cracking ................................ ................................ ............... 100 3.1.5.2 Steel reinforcement yielding ................................ ........................ 100 3.1.5.3 Rotation of plastic hinges ................................ ............................ 100 3.1.6 Two Spring Model ................................ ................................ ................ 101 3.1.6.1 Boundary conditions ................................ ................................ .... 102 3.1.6.2 Transition from CMA to TMA in two spring model ....................... 102 3.1.6.3 Equilibrium ................................ ................................ .................. 104 3.1.6.4 Compatibility ................................ ................................ ............... 105 3.1.6.5 Stiffness of beam spring per unit width ................................ ........ 106 3.1.6.6 Stiffness of support spring per unit width ................................ ..... 108 3.1.6.7 Material properties ................................ ................................ ...... 108 3.1.6.8 Membrane force per unit width (n u ) ................................ ............. 108 3.1.7 New Bending Moment Capacity ................................ ........................... 109 3.1.8 Discussion of To tal Internal Energy ................................ ...................... 111 3.1.9 Total External Work ................................ ................................ .............. 114 3.1.9.1 Uniformly distributed area load (w) ................................ .............. 114 3.1.9.2 Concentrated load (P) at midspan ................................ ............... 115 3.1.10 Load Deflection Curve of Restrained Slab or Beam ............................ 115 3.1.10.1 Uniformly distributed area load (w) ................................ ............ 115 3.1.10.2 Concentrated load (P) at midspan ................................ ............. 115

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7 3.2 Four hinge Fai lure Mechanism ................................ ................................ ...... 116 3.2.1 Introduction ................................ ................................ .......................... 117 3.2.2 Limit Analysis for Plastic Hinges Forming at L ................................ .... 118 3.2.2.1 Load capacity at full plastic collapse mechanism formation ......... 119 3.2.2.2 Midspan deflection at full plastic collapse mechanism formation 119 3.2.3 Three spring Model ................................ ................................ .............. 123 3.2.3.1 Boundary conditions ................................ ................................ .... 124 3.2.3.2 Comp atibility ................................ ................................ ............... 124 3.2.3.3 Equilibrium ................................ ................................ .................. 125 3.2.3.4 Stiffness of Beam Spring 1 per unit width ................................ .... 126 3.2.3.5 Stiffness of Beam Spring 2 per unit width ................................ .... 127 3.2.3.6 Stiffness of support spring per unit width ................................ ..... 128 3.2.3.7 Material properties ................................ ................................ ...... 128 3.2.3.8 Membrane force per unit width (n u ) ................................ ............. 128 3.2.4 Midspan Deflection at which Reinfo rcement Fractures ......................... 129 3.2.4.1 Example from Slab 6 (Woodson and Garner 1985) ..................... 129 3.2.5 Load Deflection Curve ................................ ................................ .......... 130 3.3 Summary ................................ ................................ ................................ ....... 131 4. RESULTS AND DISCUSSION ................................ ................................ ............ 132 4.1 Introduction ................................ ................................ ................................ .... 132 4.2 Validation with Experimental Data from Woodson and Garner (1985) ........... 132 4.2.1 Experimental Procedure ................................ ................................ ....... 132 4.2.2 Slab Details ................................ ................................ .......................... 135 4.2.3 Failure Mechanisms ................................ ................................ ............. 143 4.2.4 Assumed Values of Surround Stiffness ................................ ................ 145 4.2.5 Comparison of Results between Experimental Data and Proposed Model ................................ ................................ ................................ ......... 146 4.2.5.1 Slabs 1 and 2 ................................ ................................ .............. 146 4.2.5.2 Slab 3 ................................ ................................ .......................... 151 4.2.5.3 Slabs 4 and 5 ................................ ................................ .............. 154 4.2.5.4 Slab 6 ................................ ................................ .......................... 159 4.2.5.5 Slabs 7 and 8 ................................ ................................ .............. 161 4.2.5.6 Slabs 9 and 12 ................................ ................................ ............ 165 4.2.5.7 Slab 10 ................................ ................................ ........................ 169 4.2.5.8 Slab 11 ................................ ................................ ........................ 171 4.2.5.9 Slab 13 ................................ ................................ ........................ 173 4.2.5.10 Slab 14 ................................ ................................ ...................... 175 4.2.5.11 Slab 15 ................................ ................................ ...................... 177 4.2.6 Discussion of Results ................................ ................................ ........... 179 4.3 Validation with Experiment al Data from Woodson (1985) .............................. 182 4.3.1 Experimental Procedure ................................ ................................ ....... 182 4.3.2 Slab Details ................................ ................................ .......................... 182 4.3.3 Failure Mechanisms ................................ ................................ ............. 189 4.3.4 Assumed Values of Surround Stiffness ................................ ................ 189

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8 4.3.5 Comparison of Results between Experimental Data and Proposed Model ................................ ................................ ................................ ......... 189 4.3.5.1 Slab 1 ................................ ................................ .......................... 189 4.3.5.2 Slab 2 ................................ ................................ .......................... 191 4.3.5.3 Slab 3 ................................ ................................ .......................... 193 4.3.5.4 Slab 4 ................................ ................................ .......................... 195 4.3.5.5 Slab 5 ................................ ................................ .......................... 197 4.3.5.6 Slab 6 ................................ ................................ .......................... 199 4.3.5.7 Slab 7 ................................ ................................ .......................... 201 4.3.5.8 Slab 8 ................................ ................................ .......................... 203 4.3.5.9 Slab 9 ................................ ................................ .......................... 205 4.3.5.10 Slab 10 ................................ ................................ ...................... 207 4.3.6 Discussion of Results ................................ ................................ ........... 209 4.4 Validation with experimental data from Yu and Tan (2011) ............................ 212 4.4.1 Experimental Procedure ................................ ................................ ....... 212 4.4.2 Beam Sp ecimen Details ................................ ................................ ....... 216 4.4.3 Failure Mechanisms ................................ ................................ ............. 218 4.4.4 Assumptions Used in the Calculations by the Proposed Model ............ 220 4.4.5 Comparison of Results between Experimental Data and Proposed Model ................................ ................................ ................................ ......... 220 4.4.6 Discussion of Results ................................ ................................ ........... 223 4.5 Summary ................................ ................................ ................................ ....... 225 5. CONCLUSIONS AND RECOMMENDATIONS ................................ .................... 228 5.1 Summary ................................ ................................ ................................ ....... 228 5.2 Limitations ................................ ................................ ................................ ..... 2 28 5.3 Recommendations for Further Research ................................ ....................... 230 5.4 Conclusions ................................ ................................ ................................ ... 230 LIST OF REFERENCES ................................ ................................ ............................ 232 BIOGRAPHICAL SKETCH ................................ ................................ ......................... 236

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9 LIST OF TABLES Table page 2 1 Summary of past research on two way slabs (Welch 1999) .............................. 36 2 1 Continued. Summary of past research on two way slabs (Welch 1999) ............ 37 2 2 Summary of past research on one way slabs (Welch 1999) .............................. 38 2 3 Summary of past research on TMA (Welch 1999) ................................ ............. 39 2 4 Continued. Summary of past research on TMA (Welch 1999) ........................... 40 4 1 Slab details (Woodson and Garner 1985) ................................ ........................ 142 4 2 Comparison of experimental results with predicted results (Woodson and Garner 1985) ................................ ................................ ................................ ... 181 4 3 Slab details (Woodson 1985) ................................ ................................ .......... 188 4 4 Comparison of experimental results with predicted results (Woodson 1985) ... 211 4 5 Dimensions of beam specimens (Yu and Tan 2011) ................................ ....... 217 4 6 Reinforcement details of beam specimens (Yu and Tan 2011) ........................ 218 4 7 Comparison of experimental results with predicted results (Yu and Tan 2011) 224

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10 LIST OF FIGURES Figure page 2 1 SDOF model (Krauthammer 1984) ................................ ................................ .... 29 2 2 Load deflection relations hip for two way reinforced concrete slab with edges restrained against lateral movement ( Park and Gamble 2000). ......................... 31 2 3 Comparison between theoretical and experimental load deflection curves on square restrained slabs: A) Brotchie and Holly. B) Wood 1. C) Wood 2. D) Rankin No. S2R (Eyre and Kemp 1994) ................................ ........................... 34 2 4 Plastic hinge locations of restrained one way slab strip (Pa rk and Gamble 20 00). ................................ ................................ ................................ ................ 41 2 5 Portion of strip between yield sections 1 and 2 (Pa rk and Gamble 2000). ......... 41 2 6 Conditions at positive moment yield sec tion (Pa rk and Gamble 2000). .............. 42 2 7 deflection curves (Park and Gamble 1999) ................................ ................................ ................................ ................. 44 2 8 Assum ed yield line pattern for uniformly loaded slab with restrained edges: A) Actual slab; B) Systems of strips (Pa rk and Gamble 2000). .......................... 45 2 9 Uniformly loaded plastic tensile membrane (Pa rk and Ga mble 2000). ............... 46 2 10 Uniformly loaded plastic tensile membrane (Pa rk and Gamble 2000). ............... 47 2 11 Flexural resistance curve (Krauthamm er et al. 1986) ................................ ........ 48 2 12 (1985) experimental results (Welch 1999) ................................ ......................... 51 2 13 Model for deep slabs, intermediate slabs, and slender slabs (Krauthammer et al. 2003) ................................ ................................ ................................ ............ 52 2 14 Load deflection behavior of a flexural member (Pa rk and Paulay 1975). ........... 54 2 15 Curvature distribution along beam at ultimate moment. A) Beam. B) Bending moment diagram. C) Curvature diagram (Pa rk and Paulay 1975). .................... 55 2 16 Comparison of plastic hinge length expressions (Bae and Bayrak 2008). .......... 58 3 1 Stress strain relationship for Collins and Mitchell 1991 model for concrete in compression (Consolazio et al. 2004) ................................ ................................ 72 3 2 Stress strain relationship for concrete in tension ................................ ............... 73

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11 3 3 Stress strain relationship for reinforcement steel in compressio n and tension ... 74 3 4 Stress and strain distribution across section (Chee and Krauthammer 2008) .... 75 3 5 Simplified trilinear moment curvature diagram (Park and Paulay 1975) ............. 76 3 6 Idealized perfectly plastic moment curvature diagram ................................ ....... 76 3 7 Strain and stress dist ribution at concrete first crack ................................ ........... 77 3 8 Strain and stress distribution at tension reinforcement yielding ......................... 77 3 9 Strain and stress distribution at concrete crushing ................................ ............. 78 3 10 Simplified trilinear P M interaction diagram in blue (Chen and Lui 2005) ........... 79 3 11 Example of continuous reinforcement (Woodson and Garner 1985) .................. 81 3 12 Example of bent reinforcement (Woodson and Garner 1985) ............................ 81 3 13 Sta ges of response for Case 1: (Biggs 1964) ........................... 83 3 14 Stages of response for Case 2: (Biggs 1964) ........................... 83 3 15 Stages of response for Case 3: (Biggs 1964) ........................... 85 3 16 Deflected shape of fixed end beam or slab subjected to uniform load ............... 86 3 17 General curvature diagram for one way slab clamped at both ends subjected to uniform load prior to yielding at end supports ................................ ................ 87 3 18 Moment curvature diagram for midspan section ................................ ................ 92 3 19 Moment curvature diagram for midspan section ................................ ................ 92 3 20 Difference in curvature diagram from yielding at end supports to yielding at midspan ................................ ................................ ................................ ............ 94 3 21 Obtaining moment capacity at midspan at the allowable maximum curvature ... 95 3 22 Difference in curvature diagram from yielding at e nd supports to yielding at midspan ................................ ................................ ................................ ............ 97 3 23 Moment displacement curve example ................................ ............................. 101 3 24 Two spring model idealization ................................ ................................ ......... 102 3 25 Transition from CMA to TMA in two spring model ................................ ............ 103

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12 3 26 Equilibrium of horizontal forces ................................ ................................ ........ 104 3 27 Compatibility of deformations in two spring model ................................ ........... 105 3 28 Change in length of reinforcement at start of TMA ................................ .......... 107 3 29 .............................. 107 3 30 Example of finding new bending moment from membrane force ..................... 110 3 31 Rotation of slab ................................ ................................ ............................... 111 3 32 Bending moment contribution from membrane force during TMA .................... 111 3 33 Bending moment contr ibution from membrane force during CMA .................... 112 3 34 Axial force deflection curve of specimens S1 and S2 (Yu and Tan 2011) ........ 114 3 35 Uni form load acting over half span of slab ................................ ....................... 115 3 36 Plastic hinge locations of restrained one way slab strip (Park and Gamble 1999) ................................ ................................ ................................ .............. 117 3 37 Plastic hinge locations of restrained one way slab strip ................................ ... 118 3 38 Moment distribution along slab when plastic hinges form at end supports ....... 120 3 39 Moment distribution along slab when plastic hinges form at L away from the end supports ................................ ................................ ................................ ... 120 3 40 Finding curvature at midspan when full plastic collapse mechanism forms ...... 122 3 41 Three spri ng model idealization ................................ ................................ ....... 124 3 42 Compatibility of deformations in three spring model ................................ ........ 125 3 43 Equilibrium of horizontal forces for t hree spring model ................................ .... 125 3 44 Change in length of reinforcement at start of TMA ................................ .......... 126 3 45 Reinforcement details of Slab 6 under four h inge failure mechanism (Woodson and Garner 1985) ................................ ................................ ........... 130 4 1 Four foot diameter blast load generator (Woodson and Garner 1985) ............ 133 4 2 Cross section of reaction structure (Woodson and Garner 1985) .................... 133 4 3 Cross section of slab clamped at both ends (Woodson and Garner 1985) ...... 134

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13 4 4 Example of a strain gage layout (Woodson and Garner 1985) ........................ 134 4 5 Typical instrumentation layout (Woodson and Garner 1985) ........................... 135 4 6 Slab details for Slabs 1 to 5 (Woodson and Garner 1985) ............................... 136 4 7 Slab details for Slab 6 (Woodson and Garner 1985) ................................ ........ 137 4 8 Slab details for Slab 7 to 12 and 15 (Woodson and Garner 1985) ................... 138 4 9 Shear reinforcement details for Slab 9 to 12 (Woodson and Garner 1985) ...... 139 4 10 Slab details for Slab 13 (Woodson and Garner 1985) ................................ ...... 140 4 11 Slab details for Slab 14 (Woodson and Garner 1985) ................................ ...... 141 4 12 Three hinge failure mechanism (Woodson and Garner 1985) ......................... 143 4 13 Modified three hinge failure mechanism (Woodson and Garner 1985) ............ 144 4 14 Plan view of four hinge failure mechanism (Woodson and Garner 1985) ........ 144 4 15 Four hinge failure mechanism (Woodson and Garner 1985) ........................... 145 4 16 Posttest view of Slab 1 (Welch 1999) ................................ .............................. 146 4 17 Posttest close up view of Slab 1 (Woodson and Garner 1985) ........................ 147 4 18 Posttest view of Slab 2 (Woodson and Garner 1985) ................................ ...... 147 4 19 Posttest close up view of Slab 2 (Woodson and Garner 1985) ........................ 148 4 20 Results for proposed model (in red) and experiment (in black) for Slab 1 (Woodson and Garner 1985) ................................ ................................ ........... 149 4 21 Results for proposed model (in red) and experiment (in black) for Slab 2 (Woodson and Garner 1985) ................................ ................................ ........... 150 4 22 Posttest view of Slab 3 (Woodson and Garner 1985) ................................ ...... 151 4 23 Posttest close up view o f Slab 3 (Woodson and Garner 1985) ........................ 152 4 24 Results for proposed model (in red) and experiment (in black) for Slab 3 (Woodson and Garner 1985) ................................ ................................ ........... 153 4 25 Posttest view of Slab 4 (Woodson and Garner 1985) ................................ ...... 154 4 26 Posttest view of Slab 5 (Woodson and Garner 1985) ................................ ...... 155

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14 4 27 Results for proposed model (in red) and experiment (in black) for Slab 4 (Woodson and Garner 1985) ................................ ................................ ........... 156 4 28 Results for proposed model (in red) and experiment (in black) for Slab 5 (W oodson and Garner 1985) ................................ ................................ ........... 157 4 29 Posttest close up view of Slab 5 (Woodson and Garner 1985) ........................ 158 4 30 Posttest view of Slab 6 (Welc h 1999) ................................ .............................. 159 4 31 Results for proposed model (in red) and experiment (in black) for Slab 6 (Woodson and Garner 1985) ................................ ................................ ........... 160 4 32 Postt est view of Slab 7 (Woodson and Garner 1985) ................................ ...... 161 4 33 Posttest view of Slab 8 (Welch 1999) ................................ .............................. 162 4 34 Results for proposed model (in red) and experiment (in black) for Slab 7 (Woodson and Garner 1985) ................................ ................................ ........... 163 4 35 Results for proposed model (in red) and experiment (in black) for Slab 8 (Woodson and Garner 1985) ................................ ................................ ........... 164 4 36 Posttest view of Slab 9 (Woodson and Garner 1985) ................................ ...... 165 4 37 Posttest view of Slab 12 (Woodson and Garner 1985) ................................ .... 166 4 38 Results for proposed model (in red) and experiment (in black) for Slab 9 (Woodson and Garner 1985) ................................ ................................ ........... 167 4 39 Results for proposed model (in red) and experiment (in black) for Slab 12 (Woodson and Garner 1985) ................................ ................................ ........... 168 4 40 Posttest view of Slab 10 (Woodson and Garner 1985) ................................ .... 169 4 41 R esults for proposed model (in red) and experiment (in black) for Slab 10 (Woodson and Garner 1985) ................................ ................................ ........... 170 4 42 Posttest view of Slab 11 (Woodson and Garner 1985) ................................ .... 171 4 43 Results for proposed model (in red) and experiment (in black) for Slab 11 (Woodson and Garner 1985) ................................ ................................ ........... 172 4 44 Posttest view of Slab 13 (Woodson and Garner 1 985) ................................ .... 173 4 45 Results for proposed model (upper bound in red, lower bound in blue) and experiment (in black) for Slab 13 (Woodson and Garner 1985) ....................... 174 4 46 Posttest view of Slab 14 (Woodson and Garner 1985) ................................ .... 175

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15 4 47 Results for proposed model (in red) and experiment (in black) for Slab 14 (Woodson and Garner 1985) ................................ ................................ ........... 176 4 48 Posttest view of Slab 15 (Woodson and Garner 1985) ................................ .... 177 4 49 Results for proposed model (in red) and experiment (in black) for Sl ab 15 (Woodson and Garner 1985) ................................ ................................ ........... 178 4 50 Main and transverse reinforcement details for Slabs 1 to 8 (Woodson 1985) .. 183 4 51 Main a nd transverse reinforcement for Slabs 9 and 10 (Woodson 1985) ......... 184 4 52 Stirrup configurations for Slabs 2 to 9 (Woodson 1985) ................................ ... 185 4 53 Double leg stirrup (Type I) placement details (Woodson 1985) ....................... 186 4 54 Single leg stirrup (Types II and III) placement details (Woodson 1985) ........... 187 4 55 Temperature steel placement (Woodson 1985) ................................ ............... 188 4 56 General three hinge failure mechanism deformation (Woodson 1985) ............ 189 4 57 Results for proposed model (in red) and experiment (in black) for Slab 1 (Woodson 1985) ................................ ................................ .............................. 190 4 58 Posttest view of Slab 1 (Woodson 1985) ................................ ......................... 191 4 59 Results for proposed model (in red) and experiment (in black) for Slab 2 (Woodson 1985) ................................ ................................ .............................. 192 4 60 Posttest view of Slab 2 (Woodson 1985) ................................ ......................... 193 4 61 Results for proposed model (in red) and experiment (in black) for Slab 3 (Woodson 1985) ................................ ................................ .............................. 194 4 62 Posttest view of Slab 3 (Woodson 1985) ................................ ......................... 195 4 63 Results for proposed model (in red) and experiment (in black) for Slab 4 (Woodson 1985) ................................ ................................ .............................. 196 4 64 Posttest view of Slab 4 (Wood son 1985) ................................ ......................... 197 4 65 Results for proposed model (in red) and experiment (in black) for Slab 5 (Woodson 1985) ................................ ................................ .............................. 198 4 66 Posttest view of Slab 5 (Woodson 1985) ................................ ......................... 199 4 67 Results for proposed model (in red) and experiment (in black) for Slab 6 (Woodson 1985) ................................ ................................ .............................. 200

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16 4 68 Posttest view of Slab 6 (Woodson 1985) ................................ ......................... 201 4 69 Results for proposed model (in red) and experiment (in black) for Slab 7 (Woodson 1985) ................................ ................................ .............................. 202 4 70 Posttest view of Slab 7 (Woodson 1985) ................................ ......................... 203 4 71 Results for proposed model (in red) and experiment (in black) for Slab 8 (Woodson 1985) ................................ ................................ .............................. 204 4 72 Posttest view of Slab 8 (Woodson 1985) ................................ ......................... 205 4 73 Results for proposed model (in red) and experiment (in black) for Slab 9 (Woodson 1985) ................................ ................................ .............................. 206 4 74 Posttest view of Slab 9 (Woodson 1985) ................................ ......................... 207 4 75 Results for proposed model (in red) and experiment (in black) for Slab 10 (Woodson 1985) ................................ ................................ .............................. 208 4 76 Posttest view of Slab 10 (Woodson 1985) ................................ ....................... 209 4 77 Experimental setup (Yu and Tan 2011) ................................ ........................... 212 4 78 Vertical support details (Yu and Tan 2011) ................................ ...................... 213 4 79 Horizontal support connected to the steel frame (Yu and Tan 2011) ............... 214 4 80 Horizontal support connected to the reaction wall (Yu and Tan 2011) ............. 214 4 81 Lateral restraint details (Yu and Tan 2011) ................................ ...................... 215 4 82 Instrumentation layout (Yu and Tan 2011) ................................ ....................... 216 4 83 Reinforcement details for beam specimen S1 (Yu and Tan 2011) ................... 217 4 84 Reinforcement details for beam specimen S2 (Yu and Tan 2011) ................... 217 4 85 Three hinge failure mechanism of beam specimens (Yu and Tan 2011) ......... 219 4 86 Failure mode of beam specimen S1 (Yu and Tan 2011) ................................ .. 219 4 87 Failure mode of beam specimen S2 (Yu and Tan 2011) ................................ .. 220 4 88 Load deflection curves for Specimens S1 and S2 (Yu and Tan 2011) ............. 221 4 89 Axial force deflection curves for Specimens S1 and S2 (Yu and Tan 2011) .... 222

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17 LIST OF ABBREVIATION S Gross area of slab or beam section Amount of top reinforcement steel at the end supports Amount of bottom reinforcement steel at the end supports Amount of bottom reinforcement steel at midspan Amount of top reinforcement steel at midspan Total area of all the reinforcement in the slab or beam section Width of the slab or beam Depth of neutral axis at the midspan section of the slab or beam Depth of neutral axis at the end support section of the slab or beam Depth of neutral axis at the balance point Depth of neutral axis at the end supports Depth of neutral axis at midspan Depth of neutral axis at midspan wh en plastic hinge forms at end supports Damping coefficient of SDOF system Compressive force due to concrete Critical damping coefficient of SDOF system Force due to compression steel at midspan section of the slab or beam Force due to compression steel at end support section of the slab or beam Depth of tension steel from the compression face of the slab or beam Depth of compression steel from the compression face of the slab or beam Diameter of reinforc ement bar

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18 Diameter of reinforcement bar at end supports Diameter of reinforcement bar at midspan Depth of tension steel from the compression face of the slab or beam at the end supports Depth of tension steel from the compression face of the slab or beam at midspan Modulus of elasticity for concrete Modulus of elasticity for steel Concrete compressive stress Maximum concrete compressive stress Uniaxial concrete compressive stre ngth under standard test cylinder Cracking stress of plain concrete in psi S tress of steel reinforcement Stress of steel reinforcement at end supports Stress of steel reinforcement at midspan Stress of steel reinforcement at midspan when plastic hinge forms at end supports Ultimate stress of steel reinforcement Yield strength of steel reinforcement Yield strength of steel reinforcement at end supports Yield streng th of steel reinforcement at midspan Force function of SDOF system in terms of time Force in a reinforcement bar Height of the slab or beam section Equivalent stiffness of SDOF system

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19 Development length of reinforcement bar Plastic hinge length with constant ultimate curvature Span of a two way slab in the x direction Span of a two way slab in the y direction Span of the slab or beam Distance between the point of contraflexure to the maxi mum moment Resistance moment at a section of the slab or beam X component of ultimate moment capacity of the slab at the yield line Y component of ultimate moment capacity of the slab at the yield line Bending moment Moment capacity of the slab or beam at balance point Moment capacity of the slab or beam at concrete first cracking Moment capacity at the end supports of the slab or beam at concrete first cracking Moment capacity at the midspan of the slab or beam at concrete first cracking Moment at the end supports of the beam or slab Equivalent mass of SDOF system Moment at midspan of the beam or slab Nominal moment capacity of the slab or beam at concrete crushing Moment capacity of the slab or beam at yield Yield Moment capacity of the slab or beam at end supports Yield Moment capacity of the slab or beam at midspan Membrane force at a section of the slab or beam

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20 Perimeter of steel reinforcement Total axial load on the slab or beam Total axial load capacity of the slab or beam Axial load capacity of the slab or beam at balance point Static r esistance of slab Peak thrust indexed compressive membrane capacity Resistance function of SDOF system Surround stiffness Lateral movement of the supports Force due to tensile reinforcement at the midspan section of the slab Force due to tensile reinforcement at the end support section of the slab Force due to yielded reinforcement in the x direction Force due to yielded reinforcement in the y direction Maximum allowable tensile force of the section per unit width Lateral movement of sup port Uniform bond stress Uniform load on slab Uniformly distributed load on membrane Density of the concrete in lb/ft 3 Uniform load at which slab or beam first cracks Peak load during compressive membrane phase Load capacity at formation of first plastic hinge Load capacity after first plastic hinge rotates Displacement of SDOF system

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21 Velocity of SDOF system Acceleration of SDOF system Distance from point B to the differential element dx Point of inflection along the beam or slab closer to left support Point of inflection along the beam or slab further from left support Distance from the critical section to the point of contraflexure Ratio of distance from the end s upports to the plastic hinge location over the span of the slab or beam Ratio of depth of concrete stress block to the neutral axis depth Downward displacement of slab or beam Midspan deflection of slab or beam at first cracking Midspan deflection of slab or beam due to flexure Corresponding tensile membrane deflection at yield line capacity Midspan deflection of slab or beam due to bond slip Curvature based deflection at center of slab at ulti mate Tangential deviation of point B on the curve from the tangent at point A Displacement due to slip Midspan deflection at formation of first plastic hinge Midspan deflection after first plastic hinge rotates Sum of the elastic creep and shrinkage axial strain C ompressive concrete strain Strain at the extreme compressive section fiber of concrete Cracking strain of concrete in tension Ultimate compressive strain of concret e

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22 Concrete strain at maximum compressive stress Strain of steel reinforcement Strain of steel reinforcement at end supports Strain of steel reinforcement at midspan Strain of steel reinforcement at mids pan when plastic hinge forms at end supports Strain of steel when strain hardening begins Ultimate strain of steel Yield strain of steel Yield strain of steel at end supports Yield strain of steel at midspan Portion of top steel broken in tensile membrane phase Portion of bottom steel broken in tensile membrane phase Portion of top steel broken in compressive membrane phase Portion of bottom steel broken in compress ive membrane phase Undamped natural circular frequency of SDOF system Damped natural circular frequency of SDOF system Curvature at midspan or end supports due to rotation of plastic hinges at end supports or midspan Curva ture at first crack Curvature at end supports at first crack Curvature at midspan at first crack Curvature at end supports Curvature at midspan Curvature at midspan when plastic hinge forms at end sup ports

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23 Nominal curvature at end supports Ultimate curvature Yield curvature Yield curvature at end supports Yield curvature at midspan Ratio of volume of confining steel (including the compression ste el) to the volume of concrete core Total rotation Virtual rotation Rotational capacity of plastic hinge after it has formed Slope at point A Slope at point B Angle between the tangents to the curve at points A and B Elastic rotation Plastic rotation Required rotation of plastic hinge for additional plastic hinges to form elsewhere on the slab or beam Rotation at end support due to bond slip Rotation at bottom of column due to bond slip at bottom of column Rotation at end support due to bond slip at end supports Rotation at midspan due to bond slip at midspan Rotation at top of column due to bond slip at top of column

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24 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ASSESSMENT OF PARAME TERS AFFECTING THE R ESIST ANCE OF FULLY RESTRAINED R EINFORCED C ONCRETE BEAMS OR ONE WAY SLABS 1 By Chong Yik M. Goh May 2013 Chair: Theodore Krauthammer Major: Civil Engineering The load deflection curve of laterally restrained reinforced concrete slabs or beams is also the res istance curve that is required for a simplified dyn amic analysis of the slab or beam as a single degree of f reedom model. Hence, the ability to generate the resistance curve is critical in obtaining the dynamic response of laterally restrained reinforced c oncrete slabs at the midspan The proposed methodology presented in this study consists of idealizing a three hinge failure mechanism as a two spring model that represents the support and the slab as linear springs. A three spring model is proposed to simu late the behavior of a four hinge failure mechanism. Using compatibility and equilibrium equations, the membrane force of the slab can be obtained for any displacement of the slab at the midspan. The modified moment capacity of the slab can then be obtaine d from the load interaction diagram at the critical sections. The load capacity is calculated from the modified moment capacity of the slab and the additional moment from the membrane force. Bond slip is also taken into account in the calculation of the sl ab deflection.

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25 The proposed methodology was implemented using a computer program and these results were validated by experimental data reported by Woodson and Garner (1985), Woodson (1985) and Yu and Tan (2011). It was found that the proposed model prod uced reasonable results for slabs that exhibit th e three hinge failure mechanism, the modified three hinge failure me chanism and the four hinge failure mechanism Further study is required for deep slabs, two way slabs, and partially restrained slabs

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26 1 CHA PTER 1 PROBLEM STATEMENT 1.1 Introduction One way to analyze the dynamic response of stru ctural components is to model the structural component as a n equivalent single degree of freedom (SDOF) spring with a stiffness or resistance function N umerical methods such as the Newmark Beta method can then be used to solve for the nonlinear dynamic response of the structural component for any applied load. In this study, the structural components that are of interest are laterally restrained one way slabs and beams. Ockleston conducted full scale testing on a reinforced concrete building and found that the ultimate capacity of slabs in the building was approximately two and a half times larger than what was predicted based on the Johansen yield line method (Ock leston 1958) This proved that lateral restraints surrounding the slab improve its load carrying capacity significantly due to compressive membrane action Several researchers have since studied the load deflection behavior of restrained reinforced concr ete slabs. However, many of them focus ed on obtaining the ultimate capacity of the slab due to the enhancement by membrane action rather than plotting the load deflection curve, which is also the resistance function. Furthermore, some theories, su ch as th ose based on Park (1964a) require the deflection at which the compressive membrane capacity is at its peak value. Park suggested using a deflection of half the slab thickness for design purposes. The drawback to this approach is that it can only be applied to normal strength concrete. Newer concrete materials, such as ultra high performance concrete (UHPC), may not be able to use this met hodology. For such cases, finite element simulation using accurate models of these new concrete

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27 materials may be able to produce the load deflection curve. However, running the simulation often is a time consuming process. Hence, it would be beneficial to derive a methodology that is computationally efficient reasonably accurate, and can be applied for all types of materials, as long as the material stress strain behavior is known. 1.2 Objective and Scope The objective of this research was to develop an algor ithm that is computationally efficient and sufficiently accurate to generate the resistance function, or load displacement curve, of a laterally restrained reinforced concrete slab and to investigate the parameters that affect the resistance function. The scope of this research entailed conducting a literature review of related past research, developing a methodology to generate the resistance function, implementing the methodology into a computer program and lastly, validating it with past experimental dat a. 1.3 Research Significance Because reinforced concrete structures are monolithic, it is uncommon for reinforced concrete slabs to be considered unrestrained laterally, unless the laterally supports are extremely flexible and do not provide sufficient surrou nd stiffness to the slab. Therefore, this research would apply to most reinforced concrete slabs. If successful, knowing the ultimate load capacity would beneficial for slab designers for static design. Being able to predict the resistance function would a lso be useful for a simplified dynamic design approach using the SDOF methodology that is widely used in dynamic design

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28 2 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW 2.1 Introduction Numerous researchers have investigated the effects of compressive membrane act ion and tensile membrane action on restrained reinforced concrete slabs Some have pursued theoretical work, while others have presented experimental results for one way and two way slabs However, none of the research published thus far has included studi es revealing how a finite plastic hinge length would affect the results of membrane action on the load deflection behavior of restrained slabs. In this chapter, Section 2.2 explains the SDOF analysis of structural components such as slabs or beams. Sectio n 2.3 presents a brief introduction to the typical load and other load deflection models Section 2.4 presents a discussion of plastic hin ge formation and Section 2.5 describes the moment area theorem used to calculate rotations and deflections of the beam or slab. Section 2.6 discusses limit theorem for slabs and beams fixed at both ends, and Section 2.7 introduces a proposed bond slip mod el. 2.2 SDOF Analysis of Structural C omponents Structural components such as slabs or beams may be analyzed by approximating them as SDOF systems. This approach is simple and relatively accurate. The equation of motion for a SDOF is as follows: (2 1 ) where:

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29 Figure 2 1 SDOF model ( Krauthammer 1984)

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30 to obtain the displacement, velocity and accelerati on of the slab with respect to time, as long as the resistance function is known. The resistance function is the lo ad deflection curve of the slab or beam. 2.3 Load Deflection Behavior of Restrained Concrete S labs 2.3.1 Yield Line T heory The yield line theory is an upper bound solution to the limit analysis of reinforced concrete slabs because the assumed collapse mechanism needs only to satisfy compatibility with boundary conditions. If the incorrect collapse mechanism is assumed, the regions of the slab between the yield lines will produce ultimate moment resistance that is higher than the actual material plastic capacity. This leads to a calculated collapse load that is highe r than the actual collapse load, which is obtained from assuming the correct collapse mecha nism. The collapse load is calculated from the balance of internal virtual energy with the external virtual work. To obtain the internal virtual energy, yield criterion (Johansen 1962) is commonly used, and is as follows: The slab is postulated to rotate plastically about the yield line, where the steel reinforcement bars in both directions crossing the yield line are assumed to be yielding. If the yield line is at an angle to the reinforcement bars which are in the x and y directions, the ultimate moment capacity of the s lab at the yield line can be resolved into the x and y components, and respectively. Torsional moment capacities in the x and y directions are ignored at the yield line because it is assumed that and are the princip al moments. There is, however, a torsional

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31 moment along the yield line if is different from but this is not considered in the yield criterion which is based only on the normal moments. The yield line theory assumes that the slab has sufficient shear capacity and fails in a flexural mode. It also assumes that the slab has sufficient ductility for the collapse mechanism to fully develop. Last ly the yield line theory assumes that there are no membrane forces in the slab. 2.3.2 Me mbrane A ction After a slab has cracked, the bottom of the slab will push against the supports and if the slab is laterally restrained by stiff supports, this will create membrane forces in the slab. A typical load deflection curve of a uniformly loaded two way rectang ular reinforced concrete slab with laterally restrained edges is shown in Figure 2 2 (Park and Gamble 1999) Figure 2 2 Load deflection relationship for two way reinforced concrete slab with edges restrained against lateral movement ( Park and Gamble 2000). The slab is initially elastic with a relatively straight slope until it crack s and develops compressive membrane forces that arch between the supports and enhance

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32 the moment capacity of the slab. As load is increased, yield lines develop and maximum load capacity is reached at Point B. The load capacity then decreases because the c ompressive membrane forces are reduced, until Point C is reached. Tension membrane forces start to develop and load capacity is increased again, until failure of the slab occurs at Point D. 2.3.3 Past Research on Membrane A ction There have been several studies o n compressive membrane action (CMA) and tensile mem brane action (TMA) Gvozdez wrote a paper in 1939 citing tests that showed that compressive membrane forces had to occur due to the difference in levels of the neutral axis at the positive and negative yie ld lines (Braestrup 1980) After this, Ockleston carried out a full scale load test on a building and recorded collapse loads that were significantly larger than yield line load capacity (Ockleston 1958) Powell introduced the rigid plastic theory that approximated the slab behavior as rigid except at the plastic hinges (Powell 1956) Wood later developed an analytical procedure for rigid plastic materials and used large deformation theory for circular plates subjected to uni form load (Wood 1961) He determined the interaction curve between membrane forces and bending moments and introduced the plastic potential in the analysis of CMA The plastic potential is the yield locus for total strains and curvatures Christiansen analyzed an elastoplastic slab strip that was laterally restrained by supports (Christiansen 1963) Park extended the rigid plastic theory for rectangular slabs and recommended an estimate of half the slab thickness at which maximum load capacity is reached (Park 1964a) Brotchie et al tested 45 isotropically reinforced concrete slabs and concluded that placing the lateral restraint at the level of the bottom reinforcement provides almost the same capacity as a fully clamped slab (Brotchie et al.

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33 1965) heory to include elastic and elastic plastic material theories to analyze square slabs (Jacobson 1967) Desayi and Kulkarni tried to define the load deflection curve of restrained slabs by using a semi empirical approach up to yield line capacity, and then with rigid plastic theory to plot the curve up to peak capacity by incrementing the plastic deflection (Desayi and Kulkarni 1977) Meamarian et al modified the compressive membrane equations from Park to include prestress forces and deflections. The modified compression field theory is used to obtain the stress, str ain and curvature from sectional forces (Meamarian et al. 1994) These authors applied deformation theory in their research because they used total strains instead of strain increments and unloading was not taken into account. Other authors used flow theory instead of deformation theory. Morley used strain rates to calculate the energy balance so that elastic unloading could be accounted for as strains decreased after the peak capacity He analyzed polygonal slabs, circular slabs, and square slabs (Morley 1967) Janas provided the flow theory solution for rigid plastic slab strips, circular slabs, and square slabs (Janas 1968) It turns out that neither the deformation theory nor the flow rule should be used exclusively for reinforced concrete where discontinuities, in the form of cracks, occur (Al Hassani 1978; Kemp et al. 1989) When membrane action is increasing, deformation theory, or a total strain flow rule, should be used. Conversely, when membrane action is decreasing, a strain rate flow rule should be used. Eyre presented a mathematical model for a one way axially elastic slab strip between fully plastic hinge that clarified the applicable limits for the total strain and strain rate rules and the conditions at transfer

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34 from one rule to the other (Eyre 1990) However, Eyre and Kemp found that their theory predicted a load deflection behavior that was too stiff (Eyre and Kemp 1994) Figure 2 3 Comparison between theoretical and experimental lo ad deflection curves on square restrained slabs: A) Brotchie and Holly. B) Wood 1. C) Wood 2. D) Rankin N o. S2R (Eyre and Kemp 1994) Furthermore, experimental results show a st eeper decline in load capacity after the peak load capacity than deformation theory or flow rule could predict (Welch 1999)

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35 Some authors have carried out research on partially restrained slabs that accounts fo r the rotation of the supports such as the surrounding beams that the slabs are resting on (Braestrup and Morley 1980; Datta and Ramesh 1975; Guice 1986; Muthu et al. 2007) This, however, is not the focus of this study, which will be limited to the assessment of fully restrained slabs or beams. Several authors have conducted research on tensile membrane behavi or Park used classical membrane theory to estimate the load deflection curve of the slab during the tensile membrane stage (Park 1964b) His assumptions were that concrete did not contribute to the tensile membrane load capacity as it was fully cracked, all the reinforcement reached yield, and that the rein forcement did not strain harden. Keenan ne theory to include the strain hardening effects of the reinforcement and the c ombined effects of flexure and TMA (Keenan 1969) Regan proposed load deflection equations for a bilinear deflected slab and a parabolic deflected slab (Regan 1975) He postulated that yielding of the reinforcement is concentrated at areas where resistance is low. Therefore sufficient ductility and tensile strength are required fo r catenary action to be successful. Ductility can be obtained from good longitudinal reinforcement detailing and is especially critical for precast structures because it can prevent the ties that connect the members from fracturing before catenary action s tarts A summary of past research conducted on one way and two way slabs in CMA and TMA which was compiled by Welch (1999) is presented in the following tables.

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36 Table 2 1 Summary of past research on two wa y slabs (Welch 1999)

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37 Table 2 1. Continued. Summary of past research on two way slabs (Welch 1999)

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38 Table 2 2 Summary of past research on one way slabs (Welch 1999)

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39 Table 2 3 Summary of past research on TMA (W elch 1999)

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40 Table 2 4 Continued. Summary of past research on TMA (Welch 1999) 2.3.4 2.3.4.1 One way slabs Park uses a general case of t wo plastic hinges forming at a distance of L from the fixed end supports to formulate his compressive mem b rane theory for one way slabs (Park 1964a) The supports are assumed to provide full rotational restraint and partial lateral restraint, with a surround stiffness (S) that allows a lateral movement (t) of the supports defined as positive away from the center of the one way slab The top steel at both s upports has to be equal i n quantity, and the area per unit width of bottom steel has to be constant throughout the one way slab, but the amount of top and bottom steel can be different from one another The portions of the one way slab between the plastic hinges are assumed to be straight because deformations of the one way slab are assumed to be concentrated in the rotation of the plastic hinges that lead to a downward displacement ( ) of the one way slab

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41 Figure 2 4 Plastic hinge locations of restrained one way slab strip (Pa rk and Gamble 2000). Figure 2 5 Portion of strip between y ield sections 1 and 2 (Pa rk and Gamble 2000). The middle portion 23 is assumed to shorten by (1 2 L), where is the sum of the elastic creep and shrinkage axial strain. Because of symmetry points 2 and 3 will move toward the center of the one way slab b y 0.5 (1 2 L). Therefore, section 12 will have a new total length of L + 0.5 (1 2 L) + t. From the geometry of deformations, compatibility is satisfied by the following equation: (2 2 )

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42 Figu re 2 6 Conditions at positive moment yield section (Pa rk and Gamble 2000). To satisfy equilibrium, n u has to be equal throughout section 12. Park used the American Concrete Institute ( ACI ) concrete compressi on block to express the concrete compression forces acting on section s 1 and 2 T he following equilibrium equation is obtained: (2 3 )

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43 Solving E quations 2 1 and 2 2 simultaneously, we can obtain the neutral axis respectively as follows: (2 4 ) (2 5 ) With the neutral axis depths known, we can now calculate n u and m u as follows: (2 6 ) (2 7 ) m u from the equation for m u by replacing c with virtual energy from a virtual rotation is then (m u + m u n u ) The load can then be found by equating the external virtual work equation with the internal virtual energy. These equations were used to generate a load deflection cu rve which was compared to experimental results (Roberts 1969) from a series of fixed end concrete strips. cal load deflection curve. After the maximum load, the load capacity decreased more rapidly than predicted by heory. This is because Park used a constant ACI stress block in his theory which does not take into account high extreme fiber concrete c ompression strains and crushing of concrete. These effects should change the stress block parameters of 1 and mean stress in reality. In addition, the theoretical peak capacity occurs at a smaller deflection than the experimental peak capacity and the theoretical capacity starts from model is

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44 based on plastic theory and cannot be used until near peak capacity because the actual slab behavior is initially elastic and not plastic. Figure 2 7 def lection curves (Park and Gamble 1999)

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45 2.3.4.2 Two way slabs Park (1964) assumed that two way slabs can be ideali zed as strips of slab in the x and y direction s where only x direction reinforcement steel is contained in the x direction slab strips and only y direction reinforcement steel is contained in the y direction slab strips. The yield line pattern is assumed to follow from 45 degre e yield lines from the corners of the slab. The yield section at each strip is assumed to be perpendicular to the direction of the strip and torsional moments are zero at the yield sections. The outward lateral displacement (t) is assumed to be the same fo r all strips runni ng in the same direction. Last ly Park assumed that the ultimate load occurs at a displacement that is half the slab thickness. Figure 2 8 Assumed yield line pattern for uniformly loade d slab with restrained edges: A) Actual slab; B) S ystems of strips (Pa rk and Gamble 2000).

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46 The load central deflection curve is obtained by applying a unit virtual displacement at yield line EF to equate the total internal virtual energy from all slab strip s to the total external virtual work done by the load. 2.3.5 Tensile Membrane Theory TMA starts when the boundary of the slab supports starts t o resist inward movement toward the center of the slab. For a two way slab, as central displacement of the slab increases, the compressive ring acting in the outer region of the slab and the supports will decrease in area while the tensile membrane behavior in the center of the s lab will start to spread toward the supports. Park assumed that all the concrete has cr acked in the tensile membrane stage and hence does not contribute to the capacity of the slab. All the reinforcement steel is also assumed to have yielded with no strain hardening and acts as a plastic membrane. Figure 2 9 Uniformly loaded plastic tensile membrane (Pa rk and Gamble 2000).

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47 From equilibrium of forces in the z direction, we obtain the following equation: (2 8 ) By substituting X = x (T y /T x ), we can reduce E quation 2 7 to the standard plastic membrane theory equation as follows: (2 9 ) The solution to E quatio n 2 8 provides the load central deflection equation which is as follows: (2 10 ) This theoretical load deflection curve i s illustrated in the following figure Figure 2 10 Uniformly loaded plastic tensile membrane (Pa rk and Gamble 2000).

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48 Park conducted experiments on uniformly loaded rectangular slabs that had either three or four edges restrained against rotation and translation in an extremely stiff surrounding steel frame. The experimental load deflection curve shown in the figure was for a slab that was restrained on all four edges. The slope of the theoretical curve wa s similar to that of the experimental curve, but the theoretical curve underestimated the capacity of the slab during the tension membrane action phase. This was evidently because d evelop over the entire slab. Instead, at very high deflections, the load was carried by a stronger combined bending and TMA 2.3.6 Other Load D eflection M odel s for One way S labs 2.3.6.1 Krauthammer et al (1986) The model proposed by Krauthammer et al can be seen in t he following figure. Figure 2 11 Flexural resistance curve (Krauthammer et al. 1986)

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49 Point B in the model is found by calculating the maximum load ca pacity ( ) using the Park and Gamble methodology described in Section 2.3.4. The deflection at which the maximum load occurs is taken to be half the slab thickness (h) which is also recommended by Park. The curve that connects Point A to Point B is a par abola with the following formulation: (2 11 ) The load capacity at Point C is taken to be the yield line load capacity and the deflection at which this occurs is taken to be twice the slab thickness. Hence, a straight line connects Point B to Point C. The line that connects Point C to Point D is defined by the following equation: (2 12 ) where: One of the drawbacks of this model is that the deflection at which maximum load occurs is assumed to be half the slab thickness and the deflection at which yield line load capacity occurs is assumed to be twice the slab thickness. These assumptions are derived from experimental results for normal strength concrete slabs. Therefore, it is unclear whether this model is relevant for other concrete materials such as ultra high performance concrete.

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50 2.3.6.2 Welch (1999) the following equation ( Keenan 1969) to model the ascending compressive membrane resistance curve: (2 13 ) where: Another equation from Keenan (1969) was also used to generate t he desc ending portion of the compressive membrane resistance curve as follows: (2 14 ) where: way slabs is as follows: (2 15 ) Welch compared the results obtained from his model with experimental results. The following figure depicts the comparison between his model and experimental results obtained from Woodson and Garner (1985) for Slab 1.

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51 Figure 2 12 Slab 1 from Woodson and Garner (1985) experimental results (Welch 1999) Curve C1 represents the generally parabolic ascending compressive membran e resistance curve. Curve C2 represents the generally parabolic descending compressive membrane resistance curve. Curve T1 represents the tensile membrane resistance which only considers the tensile reinforcement at midspan. Curve T2 represents the tensile membrane resistance which considers all of the reinforcement at midspan. Curve Ten represents the linear tensile membrane resistance that was chosen between Curve T1 and Curve T2. of Curves C1, C2, and Ten.

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52 2.3.6.3 Krauthammer et al (2003) Krauthammer et al (2003) modified the earlier model in 1986 to generate the resistance curve for deep slabs, intermediate slabs, and slender slabs. Figure 2 13 M odel for deep slabs, intermediate slabs, and slender slabs (Krauthammer et al. 2003) The curve that connects Point A to Point B for deep slabs is a parabola with the following formulation : (2 16 ) This equation is similar to the equation for slender slabs in the 1986 model. The only difference is that the coefficients changed because o f the lower deflection to slab thickness ratio that occurred in deep slabs. Instead of using a deflection to slab thickness ratio of 0.5 for slender slabs, a ratio of 0.07 was used for deep slabs. This

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53 was because a much lower deflection to slab thickness ratio was observed for deep slabs (Woodson 1994) These new coefficients were then derived by fitting a parabolic equation to the resistance curve for deep slabs based on the deflection to slab thickness ratio of 0.07 The loads for Points C and D for deep slabs wer e defined in the same way as the 1986 model for sl ender slabs, but the deflection to slab thickness ratios at which they occurred for dee p slabs were changed to 0.17 and 0.35 for Points C and D respectively. Deep slabs were defined as slabs with a span to depth ratio of 6 or less, and slender slabs were defined as slabs with a span to depth ratio of 10 or more. Hence, the span to depth ratio of the intermediate slab will be within 6 to 10. The deflections of the intermediate slab at Points B and C can then be found from linear interpolation of the span to depth ratio between the intermediate slab and either the deep slab or slender slab. Once the deflections for the intermediate slab were found for Points B and C, the resistance at each point can be found f or the intermediate slab by using linear interpolation for the resistances between deep slabs and slender slabs. The model was then calibrated to fit experimental results from Woodson (1994) 2.4 Plastic Hinge F ormation If there is sufficient ductility in a c oncrete structure, it will not fail in a catastrophic manner. Instead, plastic hinges will develop and the load capacity is maintained or increased.

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54 Figure 2 14 Load deflection behavior of a flexural memb er (Pa rk and Paulay 1975). When sufficient plastic hinges have been formed, a collapse mechanism is complete, the structure is unstable, and it fails. Although it is convenient to model the plastic hinge formation in a reinforced concrete section as a line with a point hinge, in reality the concrete member has a finite depth and the plastic hinge requires a finite length to develop rotational capacity (Park a nd Paulay 1975) 2.4.1 Idealized Curvature D iagram Theoretically, a cantilever loaded at its tip by a point load has a linear moment distribution along the beam, and hence, a linear curvature distribution as well. However, t he actual curvature distribution alo ng a beam fluctuates and is not linear, because of local loss in rigidity in the vicinity of the cracks. Nonetheless, the curvature can be idealized as a linear elastic triangle along the length of the beam and a plastic rectangle concentrated at a finite plastic hinge length ( ).

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55 The plastic rectangle of height ( ) and width ( ) approximates the actual inelastic rotation at the plastic hinge, which is the shaded area in the curvature diagram. On the other hand, the linear elastic triangle approximates the elastic rotation. The total rotation is the sum of the elastic and plastic rotations. is the ultimate concrete strain in the extreme compression fib er and c is the neutral axis depth at ultimate curvature. Figure 2 15 Curvature distribution along beam at ultimate moment. A) Beam. B) Bending moment diagram. C) Curvature diagram (Pa rk and Paulay 1975).

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56 2.4.2 Previous Studies on P lastic Hinge L ength A good estimate of the plastic hinge length ( l p ), w here the curvature is constant at u will give a good estimate of the plastic rotation at the plastic hinge Several researchers have conducted studies on this subject and each of them has proposed various empirical equations to estimate the plastic hinge length. Baker : Based on the resul ts of 94 tests on beams and columns, Baker (195 6) proposed the following equation for unconfined concrete : (2 17 ) where = 0.7 for mild steel and 0.9 for cold worked steel, = 1 + P u /P 0 where P u is the axial compressive force in the member and P 0 is the axial compressive strength of the member without bending, and = 0.6 when = 5.1 ksi and 0.9 when = 1.7 ksi, assuming = 0.85 x cube strength of conc rete. z is the distance from the critical section to the point of contraflexure. For most z/d and span/d ratios, was found to be between 0.4d to 2.4d. To calculate the plastic rotation, = 0.0035. Test results for show considerable scatter, mainly due to the variability in Corley : Corley tested 40 simply supported beams subjected to single point loads and proposed the following equation for unconfined concrete (Corley 1966) : (2 18 ) where z and d are in inches and is calculated as follows: (2 19 )

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57 Mattock : Mattock suggeste d that the following equ ations could be used (Mattock 1967) : (2 20 ) (2 21 ) Sawyer : Based on the assumptions that the maximum moment is the ultimate moment, M y = 0.85 M u and the zone of yielding is spread d/4 past the section at which the bending moment is reduced to M y Sawyer proposed the follow ing equation for the plastic hinge length (Sawyer 1964) : (2 22 ) Park, Priestl e y and Gill : Park, Priestl e y and Gill tested four square columns with a shear span to depth (L/h) ratio of 2 and concluded that can be approximated as 0.4h (Park et al. 1982) Priestl e y and Park : Priestl e y and Park later suggested the following equation to take the strength of the steel reinforcement into account (M. Priestley and Park 1987) : (2 23 ) where L s is the shear span which is the distance between the point of contr aflexure to the maximum moment, d b is the longitudinal bar diameter and f y is the grade of steel in ksi. Sheikh and Khoury : Sheikh and Khoury tested six square columns and examined the effects of lateral steel, steel configuration, and level of axial load They concluded that l p can be approximated to be equal to h (Sheikh and Khoury 1993)

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58 Bae and Bayrak : Bae and Bayrak created a graph to compare the different empirical equations for the plastic hinge length ( Bae and Bayrak 2008) Figure 2 16 Comparison of plastic hinge length expressions (Bae and Bayrak 2008). They noted that axial load was not a factor in most equations except for those suggested by Baker. This was because early research concentrated on beams, not columns, and later research produced contradictory results on the effect of axial load on the plastic hinge length. Hence, Bae and Bayrak investigated the effects of axial load, amount of longitudi nal reinforcement and shear span ratio on the plastic hinge length and presented the following equation to ta ke these variables into account:

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59 (2 24 ) This equation was obtained by using the method of least squares to estimate the coefficient for each of these variables using the UW/PEER column da tabase. 2.5 Moment Area T heorem There are two moment area theorems. The first theorem states that the change in slopes between two arbitrary points A ( ) and B ( ) in a beam can be calculated by integrating the area under the curvature diagram from A to B. This is equal to the angle between the tangents to the curve at A and B ( ) as long as the curve is continuous. (2 25 ) If the curve is discontinuous due t o internal hinges, then the moment area theorem would not apply. A positive area under the curvature diagram indicates a counterclockwise angle from the tangent at point A to the tangent at point B, which also means a positive change in slope. The second theorem states that the tangential deviation of point B on the curve from the tangent at point A ( ) is equal to the moment of area under the curvature diagram between points A and B about point B, as long as the curve is continuous between these two points. The infinitesimal deviation between the tangents ( ) from a differential element (dx) on a line perpendicular to the undeformed axis of the beam from point B is approximated as the distance from point B to the differential element ( ) multiplied by the differential angle (2 26 ) Integr ating this expression from points A to B, we obtain the following:

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60 (2 27 ) A positive area under the curvature diagram indicates a positive and point B lies above the tangent to the curve at point A. Figure 2 17 Moment area theorem diagrams (Kassimali 2005) 2.6 Limit Analysis of Fixed End Concrete Slabs or B eams A slab or beam that is fixed at both ends is statically indeterminate as there are more unknown reactions than available equilibrium equations. When the applied load to the slab or beam increases until the critical section reaches its limit capacity, plastic hinges develop and begin to rotate with severe cracking. Each plastic hinge that forms

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61 in the slab or beam reduces the static indeterminacy by one degree. When sufficient plastic hinges form, the slab or beam becomes statically determinate. If an additional hinge forms beyond this point, the slab or beam will form a plastic collapse m echanism (Nawy 2009) In addition to having a sufficient number of plastic hinges for the s lab there must be sufficient rotational capacity at the hinges for the slab or beam to rotate at the hinges for the mechanism to form. Also, equilibrium has to be satisfied between the bending moment distribution and the applied loads and the ultimate moment must not exceed the yield moment at any point on the slab or beam (Hassoun and Al Manaseer 2012) Siddiqi provided two example s for a beam fixed at both ends subjected to a uniformly distributed load (Siddiqi 2009) The first example was for a beam wit h high strength steel reinforcement and there is insufficient ductility for the central hinge to form. T he second example on the other hand, was for a beam with low strength steel reinforcement that had enough rotational capacity for the full plastic coll apse mechanism to form In both examples, t he beam had symmetric reinforcement such that the midspan section and the end support section had the same moment curvature function. 2.6.1 Central Hinge Does Not F orm In the first example, the beam contained Grade 420 steel reinforcement. The first stage that was considered was yielding at the end supports. The load at this stage was calculated as follows: (2 28 )

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62 Figure 2 18 Moment, curvature, and deflection diagrams for fixed end beam (Siddiqi 2009) Figure 2 19 Curvature diagram for first yield at the end supports (Siddiqi 2009) The deflection at this stage can be calculated from the curvature d iagram at first yield b y using the moment area theorem:

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63 (2 29 ) After the first hinge forms, we need to check if there is sufficient rotational capacity at the plastic hinges for the central hinge to form. Siddiqi assumed a plastic hinge length of d/2. The rotat ional capacity can then be calculated as follows: (2 30 ) For the central plastic hinge to form, the curvature at midspan has to be equal to the yield curvature of the midspan section. The requi red rotation is thus as follows: (2 31 ) In this example, the required rotation was more than the rotational capacity. Hence the central hinge did not form. The load capacity at thi s stage can be found be evaluating the curvature at midspan due to rotation at the hinges at the end supports. (2 32 ) The load capacity at this stage is then calculated as follows: (2 33 ) The deflection is then calculated as follows: (2 34 ) 2.6.2 Full Plastic Collapse Mechanism F orms For the second example the load and deflection at first yield is the same as the first example. Because lower streng th steel reinforcement was used in the second example, this provided higher ductility to the beam and allowed the rotational capacity

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64 for the beam in this exampl e to be higher than the required rotation. Hence, full plastic collapse mechanism formed. The beam segments between the plastic hinges were assumed to rotate as rigid bodies until the hinges reached the rotational limit. The load at second yield can be ca lculated using the following equation: (2 35 ) The deflection at second yield can be calculated as follows: (2 36 ) 2.7 Bond S lip When a large crack f orms at the interface between columns and beams, axial strains accumulate in the reinforcement bar crossing the crack. The bond between the reinforcement bar and concrete starts to deteriorate, and the reinforcement bar starts to slip and extend at the int erface between beam and column. This slip can be calculated by integrating the strains along the reinforcement bar Sezen (2002) tested four double curvature columns and found that bond slip deformations contributed between 25% to 40% of total lateral defo rmations. In some cases, bond slip deformations may be as large as flexural deformations. As a result, ignoring bond slip deformations may result in an underestimate of the total deformations and an overestimate of the member stiffness.

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65 Figure 2 20 Comparison of slip, shear and flexure contributions to total deformation (Sezen 2002) For an average uniform bond stress ( ) acting along a development length ( ) of a reinforcement bar that has an area o f a diameter o f and a perimeter of the force equilibrium in the bar is as follows: (2 37 ) Substituting the following equations for the reinforcement area and perimeter, we obtain the following: (2 38 ) (2 39 ) (2 40 )

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66 2.7.1 Proposed B ond S tress slip M odel The proposed bond slip model calculates the member deformations due to longitudinal bar slip for monotonic loads (Sezen and Moehle 2003) The model assumes a bilinear strain distribution along the portion of the reinforcement bar embe dded in the concrete. By integrating this strain along the reinforcement bar, we can obtain the slip as follows: (2 41 ) (2 42 ) (2 43 ) (2 44 ) (2 45 )

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67 Figure 2 21 Proposed bond slip model (Sezen and Moehle 2003) Figure 2 22 Slip rotation (Sezen and Moehle 2003)

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68 2.7.2 Displacements d ue to Bar S lip The rotation due to re inforcement slippage is assumed to be concentrated at the ends of the columns in the form of rigid body rotation, with the neutral axis as the center of rotation. (2 46 ) Once the rotation is known at the top and bottom, the total lateral displacement can be calculated as follows: (2 47 ) Figure 2 23 Displacement due to slip (Sezen and Moehle 2003)

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69 2.8 Summary This chapter gave a brief discussion on th e SDOF method of analysis for structural components such as slabs and beams. The resistance function which is required to solve the equation of motion of the equivalent SDOF system of the restrained slab or beam is the load deflection curve of the restra ined slab or beam T he Johansen yield line theory and the typical load deflection behavior of laterally restrained conc rete slab s were then presented as well as a brief overview on the summary of past research and models used to predict the load deflectio n behavior of laterally restrained slabs, taking into account CMA and TMA T he deformation theor y used by Park and others, as well as the flow theory, produced load deflection curves that were not as steep as recorded by experimental results after the peak load capacity. In addition, during the elastic phase because his theory was based on plastic theory. membrane theory underestimated the slab capacity during the tensile me mbrane stage as it assumed that there was no bending action. The resistance model by Krauthammer et al used empirical data based on normal strength concrete to establish the deflections at which maximum load capacity and yield line capacity occurred in the ir model Hence, it is unclear if this model will be suitable for other concrete materials such as ultra high performance concrete. The idealized curvature distribution along the beam or slab for the plastic hinge formation and empirical expressions to ca lculate the plastic hinge lengths were presented. The moment area theorem and limit analysis used to calculate deflections and load capacity for slabs and beams assuming no membrane action were

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70 discussed. Lastly, a bond slip model was presented and it w as shown that deformation due to bond slip accounts for a significant portion of the total deformation. Chapter 3 aims to address the issues mentioned in Chapter 2 by proposing a model that does not rely on predetermined deflections at peak load capacity and yield line load capacity, such that a ny material with a known stress strain relationship can be used by the proposed model to gen erate the load deflection curve, or resistance function, of the restrained slab or beam.

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71 3 CHAPTER 3 METHODOLOGY 3.1 Three Hing e Failure M echanism This section covers the methodology of generating the load deflection curve of a restrained reinforced concrete beam or one way slab that exhibits the three hinge failure mechanism A brief outline of the procedure is as follows: 1. Provi de input data for the following: a) Slab span b) B readth and depth of section, as well as amount of top and bottom steel reinfor cement, at the critical sections in the slab c) Material models for concrete and steel reinforcement. 2. Carry out simplified mom ent curvature analysis for the critical sections in the slab, such as the midspan and end supports for a three hinge plastic collapse mechanism, for the following points: a) Concrete cracking at extreme tensile fiber. b) Tensile steel yielding. c) Concrete crushing at extreme compression fiber. 3. Construct simplified axial force bending moment (P M) interaction diagram for the following points: a) Pure compression (P 0 ,0). b) Balanced point (P b ,M b ). c) Pure bending (0,M n ). d) Pure tension (T 0 ,0). 4. Perform limit analysis on the slab to investigate the location and sequence of the plastic hinge formation to estimate the deflections and load capacity at each stage. 5. For each midspan displacement of the slab, obtain the slab membrane force n u from the Two Spring Model. 6. For the level of axial force n u obtain the bending moment capacity from the simplified P M interaction diagram. 7. Calculate the total internal energy in the slab strip due to the be nding moment and the membrane force couple over the total rotation. 8. Calculate the total external work by the load over the displacement at midspan of the slab strip. 9. Obtain the load by equating the total internal energy with the total external work. 10. Repeat Step 5 for a different midspan displacement until the membrane force (n u ) reaches the ultimate tensile capacity of the member (T 0 ).

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72 3.1.1 Material M odels In this section the material models that were used in this study are presented 3.1.1.1 Concrete in c ompression The Collins and Mitchell 1991 concrete mode l was used in this study to model the compressive stress strain behavior of concrete (Collins and Mitchell 1991) Figur e 3 1 Stress strain relationship for Collins and Mitchell 1991 model for concrete in compression (Consolazio et al. 2004) The relevant equati ons are as follows: ( 3 1 ) where :

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73 3.1.1.2 Concrete in tension The concrete is assumed to be linear elastic in tension until it cracks Beyond cracking, tension stiffening effects are ignored and concrete has no tensile strength. Figure 3 2 Stress strain relationship for concrete in tension T he equations (ACI Committee 318 2011) are as follows: ( 3 2 ) ( 3 3 ) ( 3 4 )

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74 3.1.1.3 Reinforcement steel in compression and tension The assumed stress strain behavior for the reinforcement steel used in this study w as a trilinear model as shown in Figure 3 3 where is the ultimate strength is the yield strength is the yield strain, is the strain that steel starts to harden, is the ultimate strain us of steel, and is the strain hardening modulus of steel Figure 3 3 Stress strain relationship for reinforcement steel in compression and tension The equation s to calculate the stress in the rei nforcement are as follows: ( 3 5 ) ( 3 6 ) ( 3 7 )

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75 3.1.2 Simplified S ection A nalysis for Moment C urvature 3.1.2.1 General p rocedure The section is divided into layers and a strain distribution is then imposed by setting the strain at one of the layers to be a specified value. The stress at each layer p er unit width is obtained from the stress strain material model. The neutral axis depth is computed iteratively by ensuring the sum of forces is equal to zero. The moment and curvature for that strain distribution are then computed from the neutral axis de pth. Figure 3 4 Stress and strain distribution across section (Chee and Krauthammer 2008) The usual method of computing the moment curvature diagram would be to increment the concrete compression strain at the extreme compression layer of the section from zero to the u ltimate crushing strain. However, this would require several iterations for each increment of strain. To simplify the process and reduce the computation effort, the moment curvature diagram is approximated as a trilinear curve that only requires the cracki ng, yielding and nominal points.

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76 Figure 3 5 Simplified trilinear moment curvature diagram (Park and Pau lay 1975) We will simplify the trilinear moment curvature diagram even further by assuming that the moment capacity remains at the yield moment ( ) instead of increasing to the nominal moment ( ) when we calculate the full plastic collapse load. Figure 3 6 Idealized perfectly plastic moment curvature diagram

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77 3.1.2.2 First c racking of concrete T he extreme tensile layer of the section is assumed to be at the cracking strain of concrete ( ) which was defined in Equation 3 3 Figure 3 7 Strain and stress distribution at concrete first crack 3.1.2.3 First y ielding of reinforcement steel The extreme tensile layer of reinforcement steel is assumed to reach the yield strain of steel ( ) Figure 3 8 Strain and stress distribution at tension reinfo rcement yielding

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78 3.1.2.4 Concrete crushing The extreme compressive layer of the concrete section is assumed to reach the ultimate crushing strain of concrete ( ) which is taken to be 0.003 in this study Figure 3 9 Strain and stress distribution at concrete crushing 3.1.3 Simplified P M Interaction D iagram Constructing a P M interaction diagram numerically is commonly carried out by incrementing the neutral axis depth from a small value to a large value, while keeping the extreme compressive layer at the ultimate crushing concrete strain. Each increment of the neutral axis depth would produce a strain distribution diagram across the section depth, where the stresses in each layer could now be calcul ated and hence the total axial force P and bending moment M. These P and M points per unit width are then plotted in the P M interaction diagram. With the P M interaction diagram, we can now obtain a bending moment capacity for every known axial force on t he reinforced concrete slab or beam.

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79 Figure 3 10 Simplified tri linear P M interaction diagram in blue (Chen and Lui 2005) To simplify the computational process, the P M interaction diagram will be approximated as a trilinear graph consisting of only the pure compression point, the balanced point, the pure bending point and the pure tension point. The approximated P M curve would be an underestimate of the bending moment capacity of the section for a given axial load. 3.1.3.1 Pure compression The strain distribution is uniform with all layers in the section experiencing the same compressive strain of which is the ultimate concrete crushing strain. The total axial load capacity per unit width P 0 is then calculated from the following equation.

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80 ( 3 8 ) w here : = Gross area of section = Total area of steel reinforcement 3.1.3.2 Balanced condition The strain at the extreme compressive layer is at ultimate concrete crushing strain while the strain at the tensile layer of steel reinforcement is at the yield strain of steel. The neutral axis depth at balanced condition can be calculated from similar triangles in the equation as follows: ( 3 9 ) With the strain distribution given, the axial load capacity per unit width and the bending moment capacity per unit width can be computed numerically by summing the forces and moments per unit width for each layer. 3.1.3.3 Pure fl exure The strain distribution is the same as that used in the moment curvature diagram for the concrete crushing stage, where the strain at the extreme compressive layer of the concrete section is assumed to equal the ultimate crushing strain o f concrete With the sum of axial forces equal to zero, the neutral axis depth is found from horizontal equilibrium of forces. The bending moment capacity is then computed from summing the moments for all layers in the section. 3.1.3.4 Pure tension The tension stiffening effect in concrete is negligible compared to the tensile force of the steel reinforcement and hence it is ignored. The maximum allowable tensile force

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81 of the section per unit width ( ) is then dependent on the percentage of reinfor cement not broken during the compression membrane stage and the percentage that breaks during the tension membrane stage. It is defined as follows: ( 3 10 ) ( 3 11 ) Figure 3 11 Example of continuous reinforcement (Woodson and Garner 1985) Figure 3 12 Example of bent reinforcem ent (Woodson and Garner 1985) ( 3 12 ) where: = Percentage of top steel broken during tensile membrane stage = Percentage of bottom steel broken during tensile membrane stage = Percentage of top steel broken during compressive membrane stage = Percentage of bottom steel broken during compressive membrane stage = Amount of top reinforcement steel at midspan

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82 = Amount of bottom reinforcement steel at midspan = Amount of bottom reinforcement steel at end supports = Amount of top reinforcement steel at end supports The assumption is that at ultimate tensile failure of the slab or beam all the reinforcement that did not break during the compressive membrane phase has either broken during the tensile membrane phase at the ultimate stress ( ), or remained at the yield stress ( ) instead of strain hardening. 3.1.4 Load D efle ction Cu rve of S lab without Lateral R estraints under Uniform L oad Using Limit A nalysis Limit analysis can be carried out for any type of loading, but in this study we will focus on the two most common types of loading: uniform load and concentrated point load at m idspan of the slab or beam. In this section, the procedure to obtain the load deflection curve of a laterally unrestrained slab or beam under uniform load using limit analysis with bond slip is discussed. In general, there are three cases that might occur. The first case is when the hogging or negative moment capacity of the section at the end supports ( ) is equal to twice the sagging or positive moment capacity of the section at the midspan ( ) of the one way slab or beam Since the ben ding moment at the support is two times the bending moment at the midspan for any uniform load, this assumption leads to the conclusion that plastic hinges will form at the supports and midspan at the same time, and the full plastic collapse mechanism is c omplete. If this happens, the full plastic load capacity is reached and it is assumed to remain constant as the slab or beam deforms under the plastic collapse mechanism.

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83 Figure 3 13 Stages of response f or Case 1: (Biggs 1964) The second case is if the hogging or negative moment capacity of the section at the end supports ( ) is less than twice the sagging or positive moment capacity of the section at the midspan ( ) of the one way slab or beam. Figure 3 14 Stages of response f or Case 2: (Biggs 1964)

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84 In this case, plastic hinges will form at the end support first because it is weaker than the midspan. The slab or beam will now act as a simply supported member. The rotational capacities of these plastic hinges after they have formed ( ) are dependent on the plastic hinge length ( ) and the difference between the ultimate curvature of the section and the yield curvature of the section ( ). The required rotation for the last plastic hinge to form ( ) depends on the difference in the area under the curvature diagram where the last plastic hinge forms at midspan and the curvature diagram where the plastic hinges form at the end supports. If the rotational capacity of the plastic hinge is more than the required rotation, the last plastic hinge will be able to form in midspan and the full plastic load can be achieved. Otherwise, the final load capacity will be less than the full plastic load. The third and final case is if the hogging or negative mome nt capacity of the section at the end supports ( ) is more than twice the sagging or positive moment capacity of the section at the midspan ( ) of the one way slab or beam. In this case, plastic hinges will form at the midspan first because it is weaker than the end supports. The slab o r beam will now act as if they were two cantilever beams back to back with their tips touching each other If the rotational capacity of the plastic hinge is more than the required rotation, the last plastic hinges will be able to form at the end supports and the plastic collapse mechanism will form. When this happens, the full plastic load can be achieved and it will be maintained as the beam or slab deforms as a plastic collapse mechanism Otherwise, the final load capacity will be less than the full plas tic load

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85 Figure 3 15 Stages of response for Case 3 : (Biggs 1964) 3.1.4.1 General equations to calculate load and deflection To simplify the analysis, we will only focus on a few critical stages in the beam or slab resp onse. These are the load and deflection at which the beam or slab first cracks, the load and deflection at which plastic hinge first forms, and the load and deflection at which the full plastic collapse mechanism is formed. In order to calculate the load a nd deflection for these specific stages, we can use some general equations that will be discussed as follows. The general bending moment equation for a one way slab with a span of L that is clamped at both ends and with a uniform load of w is as follows: ( 3 13 )

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86 Figure 3 16 Deflected shape of fixed end beam or slab subjected to uniform load To find the locations of zero moment, we set the Equation 3 13 to zero and obtain the following results: ( 3 14 ) ( 3 15 ) We can then find the displacement at midspan of the beam or slab by using moment area theorem from the points of inflection to calculate and by taking advantage of the fact that the slope at the midspan and at the end su pports is zero, and that the curvature at the point of inflection is also zero. The curva ture at midspan ( ) will not be equal to half of the curvature at the end support ( ) in general, unless the midspan section has the same moment curvature curve as the end support section. The curvature at midspan ( ) in general can be ob tained by finding the curvature at a

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87 moment of half the moment at the end support ( ) from the moment curvature analysis of the midspan section. Figure 3 17 General curvature diagram for one way s lab clamped at both ends subjected to uniform load prior to yielding at end supports Using the moment area theorem from point C, we calculate and as follows: ( 3 16 ) ( 3 17 ) ( 3 18 )

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88 We can find the uniform load (w) by balancing the internal virtual energy with the external virtual work as follows: ( 3 19 ) ( 3 20 ) ( 3 21 ) The deflection due to slip can be calculated as follows when : ( 3 22 ) ( 3 23 ) ( 3 24 ) where: = Strain of steel reinforcement at end supports = Strain of steel reinforc ement at midspan = Stress of steel reinforcement at end supports ( in psi ) = Stress of steel reinforcement at midspan ( in psi ) = Concrete compression strength ( in psi ) = Diameter of steel reinforcement at end sup ports = Diameter of steel reinforcement at midspan

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89 = Rotation due to slip at end supports = Rotation due to slip at midspan The deflection due to slip can be calculated as follows when : ( 3 25 ) ( 3 26 ) ( 3 27 ) where: = Yield strain of steel reinforcement at midspan = Yield stress of steel reinforcement at midspan (in psi) With these general equations, we can now obtain the load and deflection of the beam or slab at the critical stages of its response. These stages are first cracking of the beam or slab, first plastic hinge forming, and final plastic hinge forming for a full plastic collapse mechanism. 3.1.4.2 Firs t cracking At the uniform load where the moment just reaches the cracking moment of the section, the midspan deflection ( ) and the uniform load ( ) at cracking can be calculated after conducting a moment curvature analysis to obtain cracking moment capacity at the midspan ( ), curvature at the midspan ( ), cracking mom ent capacity at the end supports ( ) and curvature at the end supports ( ). Bond slip is assumed to be negligible at this early stage. The differences in the midspan deflection

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90 and the uniform load at cracking are also assumed to be negligible between t he three cases. ( 3 28 ) ( 3 29 ) 3.1.4.3 First plastic hinge Case 1 : Full collapse mechanism forms If plastic hinge s form at the midspan and at the end supports simultaneously because the applied moment at midspan is half the applied moment at the end support for a beam or slab under uniform load. T he midspan deflection and the uniform load at this point can then be calculated from bending and slip using the general equations as follows: ( 3 30 ) ( 3 31 ) ( 3 32 ) ( 3 33 )

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91 Case 2 : First yield at supports If then the midspan deflection and the uniform load where the moment at the end support just reaches the yield moment of the section can be calculated from bending and slip as follows: ( 3 34 ) ( 3 35 ) ( 3 36 ) ( 3 37 ) where: = S train of steel reinforcement at midspan when plastic hinge forms at end supports = Stress of steel reinforcement at midspan when plastic hinge forms at end supports (in psi) = Depth of neutral axis at midspan when plastic hinge forms at end supports = Curvature at midspan when plastic hinge forms at end supports T he curvature at midspan ( ) can be obtained by finding the curvature at a moment equal to from the moment curvature analysis of the mid span section.

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92 Figure 3 18 Moment curvature diagram for midspan section We can then obtain by iterating incrementally until the moment capacity equals Figure 3 19 Moment curvature diagram for midspan section We then obtain from the stress strain curve by using

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93 Case 3 : First yield at midspan If the calculat ions are similar to Case 2, except now the plastic hinge forms at the midspan first T he midspan deflection and the uniform load whe re the moment at the midspan just reaches the yield moment of the section can then be calculated as follows: ( 3 38 ) ( 3 39 ) ( 3 40 ) ( 3 41 ) 3.1.4.4 Rotation of existing plastic hinge Case 1: Full collapse mechanism forms This section is not relevant for this case because all plastic hinges have already formed C ase 2 : First yield at supports After plastic hinges have formed at the end supports, we can idealize the boundary conditions of the one way slab as being simply supported in flexure. We assume the equival ent plastic hinge length ( ) for a laterally u nrestrained slab to be 0.5d.

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94 The amount of rotation al capacity available at the plastic hinge can then be estimated as follows: ( 3 42 ) w here: = Nominal curvature at end supports = Yield curvature at end supports = Depth of tension reinforcement steel at the end supports F or a plastic hinge to form at midspan, the required rotation of the plastic hinges at the supports is the difference in area of curvature diagrams between as follows: ( 3 43 ) where: = Yield curvature at midspan Fig ure 3 20 Difference in curvature diagram from yielding at end supports to yielding at midspan

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95 The actual rotation of the existing plastic hinge ( ) will be the smaller of and ( 3 44 ) T he allowable maximum curvature at midspan ( ) is calculated as f ollows: ( 3 45 ) By using linear interpolation o f the moment curvature diagram for the midspan section we can find the moment capacity at midspan at the allowable maximum curvature ( ) as follows: ( 3 46 ) Figure 3 21 Obtaining moment capacity at midspan at the allowable maximum curvature

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96 The curvature at the end support after the plastic hinge rotates there is as follows: ( 3 47 ) If t hen there is insufficient rotational capacity at the existing plastic hinge for the final plastic hinge to form, the midspan deflec tion due to flexure and slip are as follows: ( 3 48 ) ( 3 49 ) where: = Strain of reinforcement at end supports afte r plastic hinge rotates = Strain of reinforcement at midspan after plastic hinge rotates = Stress of reinforcement at end supports after plastic hinge rotates = Stress of reinforcement at midspan after plastic hinge rotates = Depth of neutral axis at end supports after plastic hinge rotates = Depth of neural axis at midspan after plastic hinge rotates We can obtain and by iterating incrementally until the curvature equals We can then obtain and by iterating incrementally until the moment capacity equals The load capacity ( ) and final midspan deflection ( ) that occurs are as follows: ( 3 50 ) ( 3 51 )

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97 Case 3 : First yield at midspan This is similar to Case 2, except now the plastic hinge has formed at the midspan not at the end supports. W e can idealize the boundary conditions of t he one way slab as two cantilevered beams meeting at their tips in the plastic hinge at the midspan The amount of rotation al capacity can be estimated as follows: ( 3 52 ) where: = Nominal curvature at midspan = Yield curvature at midspan = Depth of tension reinforcement at midspan Figure 3 22 Difference in curvature diagram from yielding at end supports to yielding at midspan F or a plastic hinge to form at the end supports, the required rotation of the plastic hinges at the supports is the difference in area of curvature dia grams between as follows: ( 3 53 )

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98 where: = Yield curvature at end supports = Curvature at end supports whe n end moment The actual rotation of the existing plastic hinge ( ) will be the smaller of and ( 3 54 ) The allowable maximum curvature at the midspan ( ) is calculated as follows: ( 3 55 ) By using linear interpolation of the moment curvature diagram for the end support section, we can find the moment capacity at end supports at the allowable maximum curvature ( ) as follows: ( 3 56 ) The curvature at the midspan after the plastic hinge rotates there is as follows: ( 3 57 ) T he midspan deflection due to flexure and slip are as follows: ( 3 58 )

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99 ( 3 59 ) where: = Strain of reinforcement at midspan after plastic hinge rotates = Strain of reinforcement at end supports after plastic hinge rotates = Stress of reinforcem ent at midspan after plastic hinge rotates = Stress of reinforcement at end supports after plastic hinge rotates = Depth of neutral axis at midspan after plastic hinge rotates = Depth of neural axis at end supports after plastic hinge rotates We can obtain and by iterating incrementally until the curvature equals We can then obtain and by iterating incrementally until the moment capacity equals The load capacity ( ) and final midspan deflection ( ) that occurs are as follows: ( 3 60 ) ( 3 61 ) 3.1.5 Moment displacement (M ) C urve Once we have estimated the deflection at various stages of the load deflection curve, such as concrete cracking ( ), steel reinforcement yielding ( ), and after rotation of plastic hinges ( ), and performed the moment curvature analysis to estimate the cracking moment ( ) and the yield moment ( ), we can construct the moment displacement curve for the end support and midspan sections as follows :

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100 3.1.5.1 First cracking At a displacement of the concrete starts to crack, and t he moments at the end support ( ) and at the midspan ( ) are as follows: ( 3 62 ) ( 3 63 ) 3.1.5.2 Steel reinforcement yielding At a displacement of the steel reinforcement starts to yield, the mo ments at the end support ( ) and at the midspan ( ) are as follows: ( 3 64 ) ( 3 65 ) 3.1.5.3 Rotation of plastic hinges At a displacement of the moments at the end support ( ) and at the midspan ( ) at a midspan displacement of will be as follows: ( 3 66 )

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101 ( 3 67 ) Figure 3 23 Moment displacement curve example 3.1.6 T wo Spring Model The beha vior of the restrained concrete slab and its interaction with the surrounding lateral restraint is modeled in this paper as a two spring model for a three hinge failure mechanism The actual membrane forces due to arching during compressive membrane action (CMA) and due to catenary forces during tensile membrane action (TMA) are assumed to act along a linear elastic spring within the slab that is connected to another linear spring that models the surround stiffness. The initial vertical distance (y) between the compressive forces at the midspan and the support is assumed to be a ratio ( ) of the section depth (h). If we assume the plastic hinge length

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102 ( ) for this case to be equal to the depth of the section (h) as suggested by Sheikh and Khoury ( 1993) the ratio can be found as follows: ( 3 68 ) Figure 3 24 Two spring model idealization 3.1.6.1 Boundary conditions Because of symmetry at the midspan, the only degree of freedom allowed at the midspan is the vertical displacement of the one way slab. The surround supports are assumed to move only laterally. 3.1.6.2 Tran sition from CMA to TMA in two spring model In this study, the transition from CMA to TMA is not defined as the minimum point on the load deflection curve after the peak load. Instead, the transition from CMA to TMA depends on when the membrane force ch anges from compression to tension.

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103 Figure 3 25 Tr ansition from CMA to TMA in two spring model At position 0 where loading has not started, the slab is undeformed and there is no membrane force. After the slab has been loaded beyond its cracking limit, it is now at position 1. At that point, the sum of the bottom surface strain of the slab yields a net gain in length, which causes the ends of the slab to push outwards at the boundary. If the boundary is sti ff enough, it will resist this outward movement of the slab and generate a compressive membrane force. Further loading of the slab causes it to deflect further to position 2. This is the position where the deformation of the beam spring is the maximum. Bec ause we are using the assumption that the springs are linear elastic to simplify the analysis, position 2 is also where the slab has the maximum compressive membrane force in the proposed two spring model. This occurs when Further loading causes the slab to deflect to position 3, which is the mirror image of position 1 about position 2. Another consequence of assuming that the springs are linear elastic is that the compressive membrane force at position 3 is the sa me as the compressive

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104 membrane force at position 1, because the beam spring deformation at the same for both positions 1 and 3. Similarly, the membrane force at position 4 would be the same as the membrane force at position 1, which would be zero. If the s lab is loaded further such that it deflects beyond position 4 to position 5, the beam spring will now be stretched rather than compressed due to geometric compatibility. Hence, the membrane force at position 5 switches to tension. Therefore, position 4 is the transition point between CMA and TMA. This occurs when In reality, if the membrane force in the slab is within the linear elastic range of the concrete stress strain curve, the membrane force deflection curve could be close to symmetric. Ho wever, if the membrane force is beyond the linear elastic limit of the concrete, the concrete will start to soften and the membrane force will be increasingly less than predicted as deflection increase. 3.1.6.3 Equilibrium For equilibrium to be satisfied in the h orizontal direction, the membrane force at the support section has to be equal to the membrane force at the midspan section. Figure 3 26 Equilibrium of horizontal forces

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105 3.1.6.4 Compatibility The original lengt h of the spring (in red) in the slab is When the end of the spring at the midspan section moves downwards by a deflection the spring will contract by a deformation Therefore, the current length of the spring (in blue) wi ll now be The surround spring will also contract by a deformation Figure 3 27 Compa tibility of deformations in two spring model Using the Pythagoras Theorem on t he current configuration ( blue triangle ) we obtain the following compatibility equation: ( 3 69 )

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106 Expanding this equation we obtain the following equ ation : ( 3 70 ) 3.1.6.5 Stiffness of beam spring per unit width negligible com he stiffness of the beam spring per unit width of the beam or slab in compression ( ) during compressive membrane action (CMA) can be estimated by only considering the axial stiffness of the concrete beam or slab as follows: ( 3 71 ) During tensile membrane action (TMA), the converse is assumed to be true. Concrete is assumed to be negligible in terms of tensile streng th compared to the steel reinforcement. Therefore, the stiffness of the beam spring is now governed by the axial stiffness of the steel reinforcement. However, it is pertinent to note that the steel reinforcement had already undergone a certain deformation by the time TMA starts. Therefore, we need to estimate the strain of the steel reinforcement at the slab deflection at which TMA starts. Because TMA starts at a midspan slab deflection of 2y, the strain of the steel reinforcement at this point can be esti mated from the Pythagoras Theorem as follows: ( 3 72 )

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107 Figure 3 28 Change in length of reinforcement at start of TMA ( 3 73 ) Figure 3 29 Secant You It is important to note that some of the reinforcement may have been broken during CMA, and hence does not contribute to the catenary capacity during TMA. Therefore, the secant stiffness of the beam spring dur ing TMA can be estimated as follows: ( 3 74 ) For simplicity, the beam spring stiffness per unit width ( ) is assumed to remain constant at dur ing TMA.

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108 ( 3 75 ) 3.1.6.6 Stiffness of support spring per unit width The stiffness of the support spring per unit width ( ) depends on the type of support th at the beam or slab rests on. For example, if the slab is resting on a wall that is restrained at the top by the slab and at the bottom by the foundation, the support spring stiffness can be estimated as follows: ( 3 76 ) In this example, the support spring stiffness would be the same for both CMA and TMA since the flexural stif fness of the wall should be the same regardless of which direction it bends. 3.1.6.7 Material properties Since we are assuming that both springs are linear elastic to simplify the analysis we obtain the following force deformation relations for the springs: ( 3 77 ) ( 3 78 ) 3.1.6.8 Membrane force per unit width ( n u ) By substituting Equations 3 77 and 3 78 into Equation 3 70 we obtain the following quadrati c equation of : ( 3 79 )

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109 where: We can then solve this quadratic equation to obtain as follows: ( 3 80 ) 3.1.7 New Bending M oment Capacity With the membrane force ( ) calculated, we can use the P M interaction diagram and the m oment displacement diagram constructed in the earlier steps to find the new bending moment capacity per unit width at the end support section ( ) by linear interpolation as follow s: ( 3 81 )

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110 Figure 3 30 Example of finding new bending moment from membrane force Similarly, the new moment cap acity per unit width at the midspan section ( ) can be found as follows: ( 3 82 )

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111 3.1.8 Discussion of T otal Internal E nergy For every midspan deflection ( ) we can calculate the rotation of the slab ( ) over half its span because it is symmetric at midspan as follows: ( 3 83 ) Figure 3 31 Rotation of slab The sum of moments per unit width about the support is as follows : ( 3 84 ) The contribution of the membrane forces to the sum of moments is different between CMA and TMA. This is because d uring TMA, the tensile membrane force is primarily carried by the reinforcement that starts from a horizontal position with zero height. Hence the moment arm for the tensile membrane force s hould be Figure 3 32 Bending moment contribution from membrane force during TMA

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112 On the other hand, the compressive arch force during CMA is assumed to act mostly through the concrete from the bottom of the s urround support to the midspan section at a height of y initially. Therefore, the moment arm for the membrane force should be because it starts from a height of y. Figure 3 33 Bending moment contrib ution from membrane force during CMA At position 1, the compressive membrane force contributes positively to the sum of moments because it is an anticlockwise moment like the resisting moments and in the slab. At position 2, the compre ssive membrane force is at its peak, but contributes nothing to the sum of moments because it is acting through the point of surround support. At position 3, the compressive membrane force is the same as at position 1, but now it is a clockwise moment, whi ch now reduces the sum of moments. Even though the compressive membrane force contributes negatively to the sum of resisting moments after the compressive membrane force reaches its peak value, the resisting moments are still being enhanced from the inter action between axial load and

PAGE 113

113 bending moments in the section. Therefore, the overall sum of resisting moments may still be larger than if there were no membrane force. However, once the enhancement of moments starts to become less than the negative contrib ution of the compressive membrane force to the resisting moment, the load capacity of the slab will start to decrease. Similarly, even though the compressive membrane force starts off small just after position 0, it has a larger moment arm than at positio n 1. Therefore, the positive contribution to the resisting moment will increase or decrease depending on whether the increase in the compressive membrane force is larger than the decrease in its moment arm as the slab deflects. The product of compressive m embrane force and moment arm starts from zero when the membrane force is zero at position 0, reaches a maximum value, and becomes zero at position 2 when the moment arm is zero. On the other hand, the enhancement of resisting moments and reach a maximum value at position 2, since the compressive membrane force is a maximum at position 2. Therefore, if the positive contribution of the compressive membrane force is larger than the positive contribution of the enhanceme nt in moments, the peak load capacity of the slab would occur at a smaller midspan deflection than the midspan deflection at which the peak value of compressive membrane force occurs. This result can be seen from the axial load deflection curve obtained ex perimentally by Yu and Tan (2011). The peak capacity of the slab occurred at a deflection of about 80 mm, but the peak compressive membrane force occurred at a deflection of about 110 mm.

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114 Figure 3 34 Axi al force deflection curve of specimens S1 and S2 (Yu and Tan 2011) With the sum of resisting moments calculated, t he total internal energy can be found as the product of the resisting moments and the angle through which they act. ( 3 85 ) As these equations show, the total internal energy is independent of the type of load s on the structure. 3.1.9 Total External W ork T he total external work however, depends on the type and location of the external loads on the structure Uniform loads and conc entrated loads are studied in this model 3.1.9.1 Uniformly distributed area load (w) The oretically, the total external work requires the integration of external load multiplied by the displaced shape of the slab over the half span of the slab member. To simplify the analysis, we will assume a triangular displaced shape for the slab for all midspan displacements The total external work per unit width will then be as follows:

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115 ( 3 86 ) Figure 3 35 Uniform load acting over half span of slab 3.1.9.2 Concentrated load (P) at midspan For a concentrated load at midspan, the analysis is easy because we only need to multiply the concentrated load with t he midspan displacement because work is equal to force multiplied by distance. The external work per unit width is as follows: ( 3 87 ) 3.1.10 Load Deflection C urve of Restrained Slab or B eam We can obtain the load deflection curv e by calculating either the uniform load (w) or the concentrated load (P) for each deflection ( ) This is done by equating total external work with total internal energy. 3.1.10.1 Uniformly distributed area load (w) The uniformly distributed load (w) per unit are a can be calculated as follows: ( 3 88 ) 3.1.10.2 Concentrated load (P) at midspan Similarly, the concentrated load (P) at midspan can be calculated as follows: ( 3 89 )

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116 3.2 Four hinge Failure M echanism This section covers the methodology of generating the load deflection curve of a restrained reinforced concrete beam or one way slab that exhibits the four hinge failure mechanism. A brief outline of the procedure is as follows: 1. Provide input data for the following: a) Slab span. b) Breadth and depth of section, as well as amount of top and bottom steel reinforcement, at the critical sections in the slab. c) Material models for concrete and steel reinforcement. 2. Carry out simplified moment curvature analysis for the critical sections in the slab, such as the midspan and end supports for a four hinge plastic collapse mechanism, for the following points: a) Concrete cracking at extreme tensile fiber. b) Tensile steel yiel ding. c) Concrete crushing at extreme compression fiber. 3. Construct simplified axial force bending moment (P M) interaction diagram for the following points: a) Pure compression (P 0 ,0). b) Balanced point (P b ,M b ). c) Pure bending (0,M n ). d) Pure tension (T 0 0). 4. Perform limit analysis on the slab to investigate the location and sequence of the plastic hinge formation to estimate the deflections and load capacity at each stage. 5. For each midspan displacement of the slab, obtain the slab membrane force n u from the t hree spring m odel. 6. For the level of axial force n u obtain the bending moment capacity from the simplified P M interaction diagram. 7. Calculate the total internal energy in the slab strip due to the bending moment and the membrane force couple over the total rotation. 8. Calculate the total external work by the load over the displacement at midspan of the slab strip. 9. Obtain the load by equating the total internal energy with the total external work. 10. Repeat Step 5 for a different midspan displacement until the reinforcement reaches ultimate strain This procedure is very similar to the procedure for the three hinge failure mechanism, therefore this section will only cover topics that are different from the three hinge failure mechanism.

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117 3.2.1 Introduction For a four hinge failure mechanism, it is assumed that two of the plastic hinges form at the end supports and the remaining two form at a distance L from the end supports. The portions of the slab or beam between the plastic hinges are assumed to be stra ight because all the deformations are assumed to be concentrated at the plastic hinges. Figure 3 36 Plastic hinge locations of restrained one way slab strip (Park and Gamble 1999) The material models for concrete and steel reinforcement are the same as with the three hinge failure mechanism. The section analyses for moment curvature and axial force bending moment diagram are also the same. The only difference is that instead of doing the section analyses for the midspan section, we have to do it for the section situation a distance L away from the end supports. The limit analysis would also be the same for the first two stages of concrete cracking and first plastic hinges forming at the end supports. However, the limit analysis would be slightly different for the last stage of full plastic collapse mechanism forming because the plastic hinge s now form at a distance of L away from the end supports instead of at the midspan of the beam or slab. Also,

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118 the two spring model cannot be used for the four hinge failure mechanism because the kinematics of the failure mechanism i s different. Therefore, a three spring model is proposed for the four hinge failure mechanism. However, once the membrane for ce is calculated from the three spring model, the rest of the steps in the procedure to calcul ate the load capacity of the slab or beam for a given midspan deflection are similar to that of the three hinge failure mechanism. 3.2.2 Limit A nalysis for Plastic Hinges F orming at L We will focus on Case 2, where plastic hinges form at the end supports first and the loading case of a uniformly distributed area load w The load capacity and deflection at which the slab cracks and later yields at the end supports are the same as for the three hinge failure mechanism. It is only the formation of the plastic collapse mechanism that is different between the two failure mechanisms. Figure 3 37 Plastic hinge locations of restrained one w ay slab strip

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119 3.2.2.1 Load capacity at full plastic collapse mechanism formation We can find the load capacity at which full plastic collapse occurs ( ) for the four hinge failure mechanism by equating the internal energy ( ) with the external work ( ) for any end support rotation ( ) of the slab. ( 3 90 ) ( 3 91 ) ( 3 92 ) ( 3 93 ) 3.2.2.2 Midspan deflection at full plastic collapse mechanism formation We can find the midspan deflection of the slab at which full plastic collapse mechanism occurs by finding the moment at midspan of the slab when plastic hinges form at L away from the e nd supports ( ) First, we need to find the moment at L away from the end supports when plastic hinges start to form at the end supports ( ). ( 3 94 ) ( 3 95 )

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120 Figure 3 38 Moment distribution along slab when plastic hinges form at end supports The slab or beam now acts as a s imply supported structure. Then, when the final plastic hinges form at L away from the end supports the moment capacity at that location is Figure 3 39 Moment distribution a long slab when plastic hinges form at L away from the end supports

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121 Because the bending moment distribution along the simply supported beam or slab is a quadratic equation, we can write it in terms of the following: ( 3 96 ) ( 3 97 ) ( 3 98 ) ( 3 99 ) ( 3 100 ) ( 3 101 ) ( 3 102 ) ( 3 103 ) We can now find the curvature at midspan when full plastic collapse mechanism forms ( ) which correspond s to the moment ( ) from the moment curvature analysis of the mi dspan section. ( 3 104 )

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12 2 Figure 3 40 Finding curvature at midspan when full plastic collapse mechanism forms The area under the curvature diagram is the required rotation for the full plastic c ollapse mechanism to form. ( 3 105 ) Recall that the rotational capacity of the plastic hinges at the end supports are as follows: ( 3 106 ) Therefore, the rotation of the plastic hinges at the end supports will be: ( 3 107 ) We can now find the midspan deflection of the beam and slab at full plastic collapse mechanism as follows: ( 3 108 ) ( 3 109 )

PAGE 123

123 where: = Strain of reinforcement at end supports after plastic hinge rotates = Stress of reinforcement at end supports after plastic hinge rotates = Depth of neutral axis at end supports after plastic hinge rotates = Diameter of reinforcement at L from end supports = Depth of reinforcement at L from end supports = Depth of neural axis at L from end supports at yielding of reinforcement We can obtain and by iterating incrementally until the curvature equals The midspan deflection ( ) when full plastic collapse mechanism occurs is as follows: ( 3 110 ) 3.2.3 Three spring M odel The behavior of the restrained concrete slab and its interaction with the surrounding lateral restraint i s modeled in this paper as a three spring mo del for a four hinge failure mechanism. This is analogous to the two spring model for the three hinge failure mechanism. The only difference is that an additional horizontal beam spring is now required to model the horizontal portion between the two inner plastic hinges in the four hinge failure mechanism. The initial vertical distance (y) between the compressive forces at the midspan and the support is still assumed to be a ratio ( ) of the section depth (h) and the plastic hinge length ( ) is also assumed to be equal to the depth of the section (h) as suggested by Sheikh and Khoury (1993)

PAGE 124

124 T he ratio can be found as follows: ( 3 111 ) Figure 3 41 Three spring model idealization 3.2.3.1 Boundary conditions Because of symmet ry at the midspan, the only degree of freedom allowed at the midspan is the vertical displacement of the one way slab. The surround supports and the inner plastic hinges are assumed to move only laterally. 3.2.3.2 Compatibility The original length of Beam Spring 1 (in red) is When the end of Beam S pring 2 at the midspan section moves downwards by a deflection Beam S pring 1 will contract by a deformation Therefore, the current length of the spring (in blue) will now be The surround spring will also contract by a deformation whereas Beam Spring 2 will contract by a deformation

PAGE 125

125 Figure 3 42 Comp atibility of deformations in three spring model Using the Pythagoras Theorem on the current configuration (blue triangle), we obtain the following compatibility equation: ( 3 112 ) ( 3 113 ) 3.2.3.3 Equilibrium For equilibrium to be satisfied in the horizontal direction, the membrane force at the support section has to be equal to the membrane force at the midspan section. Figure 3 43 Equilibrium of horizontal forces for three spring model Unlike the two spring model, the force in Beam Spring 1 could not be approximated as

PAGE 126

126 3.2.3.4 Stiffness of Beam Spring 1 per unit width The stiffness of Beam Spring 1 per unit width of the beam or slab in compression ( ) during CMA can be estimated by only considering the axial stiffness of the concrete beam or slab as follows: ( 3 114 ) Similarly as with the two spring model, the strain of the steel reinforcement at the slab deflection at which TMA starts at a midspan slab deflection of 2y can be estimated from the Pythagoras Theorem as follows: ( 3 115 ) Figure 3 44 Change in length of reinforcement at start of TMA ment as follows: ( 3 116 ) As some of the reinforcement may have been broken during CMA, they will not contribute to the catenary capacity during TMA. Therefore, th e secant stiffness of B eam S pring 1 during TMA can be estimated as follows: ( 3 117 )

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127 For simplicity, Beam Sp ring 1 stiffnes s per unit width ( ) is assumed to remain constant at during TMA. ( 3 118 ) 3.2.3.5 Stiffness of Beam Spring 2 per unit width Similarly t o Beam Spring 1, the stiffness of Beam Spring 2 per unit width of the beam or slab in compression ( ) during CMA can be estimated by only considering the axial stiffness of the concrete beam or slab as follows: ( 3 119 ) The strain of the steel reinforcement at the slab deflection at which TMA starts at a midspan slab deflection of 2y can be estimated from the Pythagoras Theorem as follows: ( 3 120 ) ( 3 121 ) As some of the reinforcement may have been broken during CMA, they will not contribute to the catenary capacity during TMA. Therefore, the secant stiffness of Beam Spring 2 during TMA can be estimated as follows: ( 3 122 )

PAGE 128

128 For simplicity, Beam Spring 2 stiffness per unit width ( ) is assumed to remain constant at d uring TMA. ( 3 123 ) 3.2.3.6 Stiffness of support spring per unit width The stiffness of the support spring per unit width ( ) is similar to that of the t wo spring model 3.2.3.7 Material properties Because we are assuming that all three springs are linear elastic to simplify the analysis, we obtain the following force deformation relations for the springs: ( 3 124 ) ( 3 125 ) ( 3 126 ) 3.2.3.8 Membrane force per unit width (n u ) By substituting Equations 3 124 3 125 and 3 126 into Equation 3 112 we obtain the following quadratic equation of : ( 3 127 ) where:

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129 We can then solve this quadratic equation to obtain as follows: ( 3 128 ) where: Because most of the deformation occurs at the region between the plastic hinges at the supports and the inner plastic hinges, the strain in most of the reinforcement in the slab are not at as high as in a three hinge failure mechanism. Furthermore, the CMA regime for the four hinge failure mechanism is much shorter. Thus, the TMA regime is much longer and the tensile membrane force in the slab is assu med to remain constant at until the strain of the reinforcement reaches its ultimate strain. 3.2.4 Midspan Deflection at which Reinforcement F ractures The midspan deflection at which reinforcement reaches ultimate strain can be estimated as follows: ( 3 129 ) where: = Original length of reinforcement between plastic hinges 3.2.4.1 Example from Slab 6 (Woodson and Garner 1985) Assuming the drawing is to scale, Therefore,

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130 Figure 3 45 Reinforcement details of Slab 6 under four hinge failur e mechanism (Woodson and Garner 1985) 3.2.5 Load Deflection C urve We can obtain th e load deflection curve by calculating the uniform load (w) for each deflection ( ) by equating total external work with total internal energy. The uniformly distributed load (w) per unit area can be calculated as follows: ( 3 130 )

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131 3.3 Summary This chapter presented the proposed methodology to generate the resistance or load deflection curve of a restrained reinforced con crete slab or beam for both three hinge failure mechanisms and four hinge failure mechanisms After providing the input data, such as the slab dimensions, the amount of reinforcement and the material models for steel and concrete, the next step is to perfo rm a moment curvature analysis at the critical sections. For three hinge failure mechanisms, the critical sections are at the midspan and at the end supports. For four hinge failure mechanisms, the critical sections are at L away from the end supports and at the end supports. A fter which, a simplified linear axial load bending moment interaction diagram is constructed. Limit analysis is then used to obtain the moment deflection curve for the slab or beam if it had been unr estrained laterally. The membrane force is then found from the two spring model for the three hinge failure mechanism or the three spring model for the four hinge failure mechanism. With the membrane force, the modified moments are obtained from the axial load bending moment diagram. Internal work is calculated by multiplying the sum of moments including the effects of the membrane force with the rotation at the end supports. External work is calculated by multiplying the load with the deflection along the slab. T he load for each midspan deflection is finally obtained from equating the internal work with the external work

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132 4 CHAPTER 4 RESULTS AND DISCUSSI ON 4.1 Introduction The proposed procedure to generate the load deflection curve of restrained reinforced co ncrete slabs was implemented using computer programing to ascertain its viability The results generated by the computer program were then compared with past experimental work conducted on restrained reinforced concrete slabs 4.2 Validation with Experimental D ata from Woodson and Garner (1985) Woodson and Garner tested 15 one way slabs which were restrained against rotation and longitudinal expansion, under static uniform water pressure (Woodson and Garner 1985) All slabs were 24 in. wide 36 in. long. Six inches of both ends of the slab were clamped at the supports, thus leaving a clear span of 24 in Only one of the s labs was 2.9 in. thick, which gave a span to depth (L/h) ratio of 8.3. The other fourteen slabs were 2 5/16 inches thick, which corresponds to a span to depth ratio of 10.4. The tensile reinforcement ratio ranged from 0.4% to 1.5 8% whereas the compressive reinforcement ratio ranged from zero to 1.14%. 4.2.1 Experimental P rocedure A reaction structure was placed in a 4 foot diameter Small Blast Load Generator (SBLG). The SBLG was made of 3 foot 10 3/4 inch inner diameter sta cked rings and an elliptical dome top called the bonnet that were bolted together to allow variations in depth.

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133 Figure 4 1 Four foot diameter blast load generator (Woodson and Garner 1985) Figure 4 2 Cross section of reaction stru cture (Woodson and Garner 1985)

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134 The slabs were then placed on the reaction s tructure and a 3/32 inch thick layer of neoprene membrane was placed on top of the slab to prevent pressure loss through and around the slab. Aqua seal putty was placed on the neoprene membrane at the bolts to ensure water tightness. The slab was then bolt ed in place at a torque of approximately 50 ft lb at both its ends by steel plates that were 24 inches long, six inches wide, and 1/2 inch thick. The bonnet was then bolted to the flange of the top ring, which secures the edges of the neoprene membrane and seals the SBLG. Figure 4 3 Cross section of slab clamped at both ends (Woodson and Garner 1985) The strain, displacement, and pressure instruments were then calibrated before the slab was loaded by filling the bonnet with water. For a load that was greater than the commercial wat erline pressure of 70 psi, a pneumatic pump was used. The maximum allowable static water pressure in the SBLG is 500 psi. Figure 4 4 Example of a strain gage layout (Woodson and Garner 1985)

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135 Figure 4 5 Typical instrumentation layo ut (Woodson and Garner 1985) 4.2.2 Slab D etails Slabs 1 and 2 are similar, with the only difference being Slab 2 has symmetric concrete cover for the top and bottom reinforcement whereas Slab 1 has a slightly larger top concrete cover. Both slabs have equal top and bottom steel reinforcement. For Slabs 3, 4 and 5, the top reinforcement was of the total reinforcement in the slab, whereas the remaining of reinforcement was placed at the bottom layer in the slab. This meant that the moment capacity at the end supports were higher than the moment capacity at the midspan for Slab 3. For S labs 4 and 5, dowels were added at the end supports to increase the moment capacity there. The dowels in Slab 5 were extended to the point of zero moment under uniform load, whereas the dowels in Slab 4 were shorter.

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136 Figure 4 6 Slab details for Slabs 1 to 5 (Woodson and Garner 1985)

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137 The reinforcement in Slab 6 was bent such that all the reinforcement was in the tensile zone of the slab and there was no compressive reinforcement. Figure 4 7 Slab details for Slab 6 (Woodson and Garner 1985) Slabs 7 to 12 also had bent reinforcement such that both end su pport sections and midspan sections had the same longitudinal reinforcement ratio of = 1.13% and 0.45%. Slab 7 had no stirrups, whereas Slabs 8 to 12 had single leg stirrups added to them. Slab 8 had a stirrup spacing of three inches, whereas Slabs 9 to 12 had a stirrup spacing of inch. Slabs 9 to 11 had the same reinforcement configuration whereas Slab 12 had the temperature on the outside of the main reinforcement bars.

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138 Figure 4 8 Slab detai ls for Slab 7 to 12 and 15 (Woodson and Garner 1985)

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139 Figure 4 9 Shear reinforcement details for Slab 9 to 12 (Woodson and Garner 1985)

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140 Slab 13 was similar to Slab 7, except instead of having bent bars as in Slab 7, Slab 13 had alternate pairs of full length and cut bars instead. Figure 4 10 Slab detail s for Slab 13 (Woodson and Garner 1985) Slab 14 is the only slab with a secti on depth of 2.9 inches instead of 2 5/16 inches. It has an equal reinforcement ratio of 1.02% for the top and bottom reinforcement for both the end suppor t section and midspan section. One quarter inch diameter deformed wire stirrups were placed at a spaci ng of 1.6 inches center to center.

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141 Figure 4 11 Slab details for Slab 14 (Woodson and Garner 1985) Slab 15 is similar to Slabs 7 and 13, except that with Slab 15, pairs of bent bars were alternated with pairs of straight bars. The summary of these slab details can be found in Table 4 1

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142 Table 4 1 Slab details (Woodson and Garner 1985)

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143 4.2.3 Failure Mechanisms There are primarily three types of failure mechanisms exhibited by the slab specimens. The first type of failure mechanism is the three hinge failure mechanism, where deep cracks occur at the end supports, the concrete is crushed at the midspan and all of the midspan bottom reinforcement is broken, except for Slab 3. This is the expected failure mechanism for beams, where plastic hinges form at the end supports and at the midspan to form a collapse mechanism. Thus, the slabs that exhibit this failure mechanism behave similarly to beams. This is especially so because the crack lines on the slab at the end support and at the midspan are practically concentrated within and parallel to the plastic hinge regions. Figure 4 12 Three hinge failure mechanism (Woodson and Garner 1985) The second type of failure mechanism is the modified three hinge mechanism, where a circular pattern can be seen on the slab. A centra l hinge occurs over a large and often undefined area. Not all of the midspan bottom reinforcement is broken.

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144 Figure 4 13 Modified three hinge failure mechanism (Woodson and Garner 1985) The third type of failure mechanism is the four hinge failure mechanism where no re inforcement r uptured, but a significant amount of concrete was crushed. Figure 4 14 Plan view of four hinge failure mechanism (Woodson and Garner 1985)

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145 Figure 4 15 F our hinge failure mechanism (Woodson and Garner 1985) 4.2.4 Assumed Values of Surround S tiffness The flexural stiffness of the supportin g wall, assuming that bot h ends of the walls were fixed, was estimated from the modulus of the concrete (E), the moment of inertia of the wall (I), and the height of the wall (L). As there were no values of the surround stiffness found in the technical report from Woodson and Garn er (1985), the surround stiffness during the tension membrane action stage was estimated to be equal to the flexural stiffness of the supporting wall, which is equal to 12EI/L 3 During the CMA stage because the slab is pushing outwards against the wall l aterally, and the wall is in turn pushing against the SBLG via the soil backfill, we can idealize the wall as one spring and the rest of the support system as another spring. These two springs are in parallel, and the equivalent stiffness of these two spri ngs is

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146 taken to be the surround stiffness during the CMA stage. The stiffness of the wall remains as (12EI/L 3 ) whereas the stiffness of the rest of the support system is assumed a constant value for all slab tests for consistency. 4.2.5 Comparison of R esults bet we en Experimental Data and Proposed M odel 4.2.5.1 Slabs 1 and 2 Slabs 1 and 2 both experienced complete midspan cracking. Crushing through the s lab thickness spread for about one quarter of the width in Slab 1 and throughout the entire width in Slab 2. There were very little cracking outside of the hinge areas. This is a three hinge failure mechanism Figure 4 16 Posttest view of Slab 1 (Welch 1999)

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147 Figure 4 17 Posttest close up view of Slab 1 (Woodson and Garner 1985) Figure 4 18 Posttest view of Slab 2 (Woodson and Garner 1985)

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148 Figure 4 19 Posttest close up view of Slab 2 (Woodson and Garner 1985) The measured peak pressure ( in black) for Slab 1 was reported to be 66 psi. The calculated peak pressure from the proposed model (in red) was 64 psi. The load deflection behavior from zero to the yield line capacity generated by the proposed model closely resembled the load deflection behavior recorded experimentally. After the peak load, the experimental data exhibited a steeper decline than ge nerated by the proposed model. This could be due to the simplification in the proposed model where t he axial stiffness of the slab was assumed to be a linear spring. In reality, as the comp ressive membrane force increased due to an increase in midspan def l ection, the concrete softened a little and hence lost a little more load capacity than calculated by the proposed mode l. After the load capacity stopped decreasing, the results obtained

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149 from the proposed model matched the experimental data. The final load r ecorded from the experiment wa s 48 psi whereas the final load calculated from the proposed model was 51 psi. All the reinforcement that was broken during the experiment was assumed to have been ruptured during the CMA stage for the analysis to generate t he tension membrane capacity. Figure 4 20 Results for proposed model (in red) and experiment (in black) for Slab 1 (Woodson and Garner 1985) The measured peak pressure (in black) for Slab 2 was reported to be 64 psi. The calculated peak pressure from t he proposed model (in red) was 66 psi. The load

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150 deflection behavior from zero to the peak pressure generated by the proposed model also resembled the load deflection behavior recorded experimentally. Figure 4 21 Results for proposed mo del (in red) and experiment (in black) for Slab 2 (Woodson and Garner 1985) After the peak load, the proposed model produced results that matched the experimental decline in load capacit y. After the load capacity stopped decreasing, the results obtained from the proposed model matched the experimental data. The final load recorded from the experiment was 55 psi whereas the final load calculated from the proposed model was 53 psi. All the reinforcement that was broken during the

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151 experiment was assumed to have been ruptured during the CMA stage for the analysis to generate the tensi on membrane capacity. 4.2.5.2 Slab 3 Unlike Slabs 1 and 2, not all the bottom reinforcement bars in Slab 3 ruptured at midspan. The edge bottom reinforcement did not rupture, and there was some cracking above the edge bars along the length of the slab. The concret e was cracked and crushed at the midspan throughout the entire depth of the slab. This is a three hinge failure mechanism. Figure 4 22 Posttest view of Slab 3 (Woodson and Garner 1985)

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152 Figure 4 23 Posttest close up view of Slab 3 (Woodson and Garner 1985) Two analyses were done for Slab 3. The first analys is (in blue) assumed the plastic hinge length for the slab to be d/2, as with the other slabs. The second analysis allowed the midspan of the slab to form the last p lastic hinge. The second analysis provided a better match with the experimental result than the first analysis, which will subsequently not be discussed. The measured peak pressure (in black) for Slab 3 was reported to be 6 8 psi. The calculated peak pressu re from t he proposed model (in red) was 64 psi. The load deflection behavior from zero to the peak pressure generated by the proposed model resembled the load deflection behavior recorded experimentally. After the peak load, the proposed model produced res ults that matched the slope of the experimental decline in load capacity but produced results that were lower in value

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153 After the load capacity stop ped decreasing, the results obtained from the proposed model matched the experimental data. The final load recorded from the experiment was 58 psi whereas the final load calculated from the proposed model was 60 psi. Figure 4 24 Results for proposed model (in red) and experiment (in black) for Slab 3 (Woodson and Garner 1985) The tensile membrane region produc ed by the proposed model m atched the experimental data well. Thirty five percent of the bottom reinforcement was assumed to have broken during the tensile membrane stage while 45% was assumed to have broken during the compressive membrane stage. The remaining bottom reinforcement

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154 did not break. All the top reinforcement that was broken during the experiment was assumed to break during the compressive membrane stage. 4.2.5.3 Slabs 4 and 5 Slabs 4 and 5 were crushed throughout the thickness at midspan, and most of the bottom reinforcement wa s broken at the midspan. Only a few top reinforcement bars were broken at the supports and none were broken at the midspan. This is a modified three hinge failure mechanism. Figure 4 25 Posttest view of S lab 4 (Woodson and Garner 1985)

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155 Figure 4 26 Posttest view of Slab 5 (Woodson and Garner 1985) The meas ured peak pressure (in black) for Slab 4 was reported to be 68 psi. The calculated peak pressure from the proposed model (in red) was 7 4 psi. The load deflection behavior from zero to the peak pressure generated by the proposed model resembled the load def lection behavior recorded experimentally. After the peak load, the proposed model produced results that matched the load capacity calculated from the experiment However, the proposed model overestimated the peak load capacity After the load capacity stop ped decreasing, the results obtained from the proposed model matched the experimental data. The final load recorded from the experiment was 60 psi whereas the final load calculated from the proposed model was 74 psi. This difference in the actual load cap acity from the proposed model could be because the actual failure mechanism is not a three hinge failure mechanism that the proposed

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156 model assumed. All the reinforcement that was broken during the experiment was assumed to have been ruptured during the CMA stage for the analysis to generate the tension membrane capacity. Figure 4 27 Results for proposed model (in red) and experiment (in black) for Slab 4 (Woodson and Garner 1985) The measured peak pressure (in black) for Slab 5 was reported to be 77 psi. The calculated peak pressure from the proposed model (in red) was 74 psi. The load deflection behavior from zero to the peak pressure generated by the proposed model

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157 was slightly stiffer than the load deflection behavior recorded experimentally. After the peak load, the proposed mode l produced results that matched the slope of the experimental decline in load capacity, but produced results that were slightly lower in value. Figure 4 28 Results for proposed model (in red) and experim ent (in black) for Slab 5 (Woodson and Garner 1985)

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158 After the load capacity stopped decreasing, the results obtained from the proposed model overestimated the minimum point of the load deflection curve from the experimental data. The final load recorded from the experiment was 55 psi whereas the final load calculated from the pr oposed model was 58 psi. This difference in the actual load capacity by the proposed model could be because the actual failure mechanism is not a three hinge failure mechanism that the proposed model assumed. All the reinforcement that was broken during th e experiment was assumed to have been ruptured during the CMA stage for the analysis to generate the tension membrane capacity. Figure 4 29 Posttest close up view of Slab 5 (Woodson and Garner 1985)

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159 4.2.5.4 Slab 6 A four hinge failure mechanism formed, with the plastic hinges forming at the end supports and at the center of the reinforcement bends locations None of the reinforcement ruptured, but concrete was crushed at the hinges and most of the bottom concrete at one end spalled off. Figure 4 30 Posttest view of Slab 6 (Welch 1999) The measured peak pressure (in black) for Slab 6 was reported to be 68 psi. The calculated peak pressure from the proposed model (in red) was 7 0 psi. The load de flection behavior from zero to the peak pressure generated by the proposed model resembled the load deflection behavior recorded experimentally generally in terms of shape After the peak load found experimentally the proposed model produced results that almost the same as the load capacity calculated from the experiment during the

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160 compressive membrane stage However, the l oad deflection curve produced from the model underestimates the minimum load capacity of the slab after peak load. During the tensile membrane stage, the load deflectio n curve produced by the model has almost the same slope as the experimental data, and reached a peak load capacity at a later stage than from the experimental data Figure 4 31 Res ults for proposed model (in red) and experiment (in black) for Slab 6 (Woodson and Garner 1985)

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161 The slope of the curve generated by the proposed model matche d the slope of the experimental curve during the tensile membrane stage It is pertinent to note that the experimenta l curve only reflected the first test done on Slab 6. Th e retest on Slab 6 indicated that the failure load reached was 122 psi. 4.2.5.5 Slabs 7 and 8 Slabs 7 and 8 displayed the modified three hinge failure mechanism, where not all the bottom reinforcement at midspan broke, and no compressive steel broke. Figure 4 32 Posttest view of Slab 7 (Wo odson and Garner 1985)

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162 Figure 4 33 Posttest view of Slab 8 (Welch 1999) The measured peak pressure (in black) for Slab 7 was reported to be 67 psi. The calcu lated peak pressure from the proposed model (in red) was 74 psi. The load deflection behavior from zero to the peak pressure generated by the proposed model was slightly stiffer than the load deflection behavior recorded experimentally. After the peak load the proposed model produced results that were declining at a faster rate than the load capacity calculated from the experimen t. After the load capacity stopped decreasing, the results obtained from the p roposed model matched the experimental data. The fi nal load re corded from the experiment was 63 psi whereas the final load calculated from the proposed mod el was 82 psi. This difference in the actual load capacity by the proposed model could be because the actual failure mechanism is not a three hinge fai lure mechanism that the proposed model assumed. All the reinforcement

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163 that was broken during the experiment was assumed to have been ruptured during the CMA stage for the analysis to generate the tension membrane capacity. Figure 4 34 Results for proposed model (in red) and experiment (in black) for Slab 7 (Woodson and Garner 1985) The measured peak pressure (in black) for Slab 8 was reported to be 68 psi The calculated peak pressure from the proposed model (in red) was 72 psi. The load deflection behavior from zero to the peak pre ssure generated by the proposed model was slightly stiffer than the load deflection behavior recorded experimentally. After the

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164 peak load, the proposed model produced results that were declining at a faster rate than the load capacity calculated from the e xperiment. Figure 4 35 Results for proposed model (in red) and experiment (in black) for Slab 8 (Woodson and Garner 1985) After the load capacity stopped decreasing, the results obtained from the proposed model underestimated the value obtained from the experimental data. The fina l load re corded from the experiment was 70 psi whereas the final load calculated from the proposed model was 71 psi. This difference in the actual load deflection behavior from the proposed model could be because the actual failure mechanism is not a thre e hinge

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165 failure mechanism that the proposed model assumed. Ten percent of the bottom reinforcement was assumed to have broken during the tensile membrane stage while 50% was assumed to have broken during the compressive membrane stage. The remaining bottom reinforcement did not break. All the top reinforcement that was broken during the experiment was assumed to break during the compressive membrane stage. 4.2.5.6 Slabs 9 and 12 Slabs 9 and 1 2 had some additional crushing of concrete outside of the central hinge, w hich might indicate that an additional hinge was forming outside of the modified three hinge failure mechanism Figure 4 36 Posttest view of Slab 9 (Woodson and Garner 1985)

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166 Figure 4 37 Posttest view of Slab 12 (Woodson and Garner 1985) The measured peak pressure (in black) for Slab 9 was reported to be 67 psi. The calculated peak pressure from t he proposed model (in red) was 72 psi. The load deflection behavior from zero to the peak pressure generated by the proposed model was slightly stiffer than the load deflection behavior recorded experimentally. Afte r the peak load, the proposed model produced results that were declining at a faster rate than the load capacity calculated from the experiment. After the load capacity stop ped decreasing, the results obtained from the proposed model underestimated the exp erimental data. The final load re corded from the experiment was 71 psi whereas the final load calculated from the proposed model was 70 psi. This difference in the actual load capacity from the proposed model could be because the actual failure mechanism is not a three hinge failure mechanism that the proposed model assumed. Twenty five percent of the bottom reinforcement was assumed to have broken during the tensile

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167 membrane stage while 20 % was assumed to have broken during the compressive membrane stage The remaining bottom reinforcement did not break. All the top reinforcement that was broken during the experiment was assumed to break during the compressive membrane stage. Figure 4 38 Results for pro posed model (in red) and experiment (in black) for Slab 9 (Woodson and Garner 1985) The measured peak pressure (in black) for Slab 12 was reported to be 71 psi The calculated peak pressure from the proposed model (in red) was 7 3 psi. The load deflection behavior from zero to the peak pressure generated by the proposed model

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168 was sl ightly stiffer than the load deflection behavior recorded by Woodson experimentally. After the peak load, the proposed model produced results that were declining at a faster rate than the load capacity calculated from the experiment. Figure 4 39 Results for proposed model (in red) and experiment (in black) for Slab 12 (Woodson and Garner 1985) After the load capacity stopped decreasing, the results obtained from t he proposed model underestimated the experimental data. The final load recorded from the

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169 experiment was 77 psi whereas the final load calculated from the proposed model was 80 psi. This difference in the actual load capacity by the proposed model could be because the actual failure mechanism is not a three hinge failure mechanism that the proposed model assumed. Ten perce nt of the bottom reinforcement was assumed to have broken during the tensile membrane stage while 40% was assumed to have broken during the compressive membrane stage. The remaining bottom reinforcement did not break. All the top reinforcement that was bro ken during the experiment was assumed to break during the compressive membrane stage. 4.2.5.7 Slab 10 Slab 10 exhibited a failure mode that was similar to the three hinge failure mechanism, where a large crack formed at each end, and the slab was totally crushed t hrough the width at midspan. Figure 4 40 Posttest view of Slab 10 (Woodson and Garner 1985) The measured peak pressure (in black) for Slab 10 was reported to be 7 3 psi. The calculated peak pressure from the proposed model (in red) was 72 psi. The load

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170 deflection behavior from zero to the peak pressure generated by the proposed model was almost the same as the load deflection behavior recorded experimentally. After the peak load, the proposed model produced results that were similar to the load capacity calculated from the experiment Figure 4 41 Results for proposed model (in red) and experiment (in black) for Slab 10 (Woodson and Garner 1985) After the load capacity stopped decreasing, the results obtained from the proposed model overestimated the experimental data. The final load recorded from the experime nt was 62 psi whereas the final load calculated from the proposed model was

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171 61 psi. Twenty percent of the top reinforcement was assumed to be broken during the tensile membrane stage while 2 5 % was assumed to have broken during the compressive membrane sta ge. The remaining top reinforcement did not break. All the bottom reinforcement that was broken during the experiment was assumed to break during the compressive membrane stage. 4.2.5.8 Slab 11 Slab 11 showed the three hinge failure mechanism where concrete crushe d throughout the center and much of the top reinforcement was broken at the supports and at midspan Figure 4 42 Posttest view of Slab 11 (Woodson and Garner 1985) The measured peak pressure (in black) for Slab 11 was reported to be 73 psi. The calculated peak pressure from the pro posed model (in red) was 72 psi. The load deflection behavior from zero to the peak pressure generated by the proposed model was almost the same as the load deflection behavior recorded experimentally.

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172 Figure 4 43 Results for proposed model (in red) and experiment (in black) for Slab 11 (Woodson and Garner 1985) After the peak load, the proposed model produced results that were declining at a slower rate than the load capacity calculated from the experimen t. After the load capacity stopped decreasing, the results obtained from the proposed model overestimated the experimental data. The final load recorded from the experiment was 54 psi whereas the final load calculated from the proposed model was 54 psi Thirty percent of the bottom reinforcement was assumed to have broken during t he tensile membrane stage while 7 0% was assumed to have broken during the compressive

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173 membrane stage. The remaining bottom reinforcement did not break. Twenty percent of the top reinforcement was assumed to be broken during the tensile membrane stage whil e 25% was assumed to be broken during the compressive membrane stage. The remaining top reinforcement did not break. 4.2.5.9 Slab 13 Slab 1 3 was badly cracked throughout the length of the span and displayed a modified three hinge failure mechanism Figure 4 44 Posttest view of Slab 13 (Wood son and Garner 1985) Two analyses were carried out for Slab 13. The first analysis ignored the area of all the reinforcement that was discontinuous and this generated a curve that was the lower bound. The second analysis included all the reinforcement in the slab and this curve constituted the upper bound. The measured peak pressure (in black) for Slab 13 was reported to be 6 4 psi. The calculated peak pressure from the proposed model (in

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174 red) was 72 psi for the upper bound and 54 psi for the lower bound T he load deflection behavior from zero to the peak pressure generated by the proposed model was slightly stiffer than the load deflection behavior recorded experimentally for the upper bound and less stiff for the lower bound Figure 4 45 Results for proposed model ( upper bound in red lower bound in blue ) and experiment (in black) for Slab 13 (Woodson and Garner 1985) The upper bound of the proposed model predicted the load rather well up until yielding of the reinforcement at the ends of the slab. From that point to the peak capac ity of the slab, the experimental curve was almost halfway between the upper and

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175 lower bound curves. After the peak load, the lower bound modeled the experimental curve rather well. The final load recorded from the experiment was 46 psi whereas the final load calculated from the proposed model was 72 psi for the upper bound and 42 psi for the lower bound All the reinforcement that was broken during the experiment was assumed to have been ruptured during the CMA stage for the analysis to generate the tens ion membrane capacity 4.2.5.10 Slab 14 Slab 1 4 displayed a classic three hinge mechanism, where all the tensile reinforcement was broken and there was no cracking between the plastic hinges. Much of the top reinforcement at the supports and the midspan was also b roken Figure 4 46 Posttest view of Slab 14 (Woodson and Garner 1985)

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176 The measured peak pressure (in black) for Slab 14 was reported to be 126 psi. The calculated peak pressure from t he proposed model (in red) was 106 psi. The load deflection behavior from zero to the peak pressur e generated by the proposed model was almost the same as the load deflection behavior recorded experimentally. Figure 4 47 Results for proposed model (in red) and experiment (in black) for Slab 14 (Woodson and Garner 1985) After the peak load, the proposed model produced results t hat were declining at a much faster rate than the load capacity calculated from the experiment. After the load cap acity stopped decreasing, the result obtained from the proposed model greatly overestimated the experimental data. The final load recorded fro m the experiment was

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177 96 psi whereas the final load calculated from the proposed model was 95 psi. This could be due to the simplification in the proposed model where the axial stiffness of the slab is assumed to be a linear spring. In reality, as the comp ressive membrane force increases due to an increase in midspan deflection, the concrete softens a little and hence loses a little more load capacity than calculated by the proposed model Eighty percent of the bottom reinforcement was assumed to have broke n during the tensile membrane stage while 20% was assumed to have broken during the compressive membrane stage. The remaining bottom reinforcement did not break. All the top reinforcement that was broken during the experiment was assumed to break during t he compressive membrane stage. 4.2.5.11 Slab 15 Slab 1 5 was totally crushed at midspan All bottom reinforcement bars were broken and the top reinforcement was pulled free of the concrete. Figure 4 48 Posttest view of Slab 15 (Woodson and Garner 1985)

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178 The measured peak pressure (in black) for Slab 15 was reported to be 52 psi. The calculated peak pressure from t he proposed model was 63 psi, assuming a plastic hinge length of 0.5d (in red) and 53 psi, assuming plastic hinges at the end supports did not rotate (in blue) The blue curve matched the experimental results better. The load deflection behavior from zero to the peak pressure generated by the blue curve was slightly stiffer than the load deflection behavior recorded experimentally. Figure 4 49 Results for proposed model (in red and blue ) and experiment (in black) for Slab 15 (Woodson and Garner 1985) After the peak load, the blue curve was declining at about the same rate than the load capacity calculated from the experiment but the blue curve overestimated the

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179 actual lo ad capacity at the minimum point of the curve The final load recorded from the experiment was 36 psi whereas the final load calculated from the proposed model was 37 psi. Thirty percent of the bottom reinforcement was assumed to have broken during the te nsile membrane stage while 7 0% was assumed to have broken during the compressive membrane stage. All the top reinforcement that was broken during the experiment was assumed to break during the compressive membrane stage. 4.2.6 Discussion of R esults The propose d model was able to model the initial load deflection behavior accurately up until the peak load capacity for Slabs 1, 2, 3, 4, 6, 10, 11, and 14, and predicted a slightly stiffer result for Slabs 5, 7, 8, 9, 12, 13 and 15. The proposed model was able to p redict the maximum compressive membrane peak capacity to within about 10 % for all slabs except Slabs 13, 14 and 15. The proposed model was able to predict the minimum capacity to within about 13% for all slab s except Slabs 5, 6, 10, 11, 14, and 15. The p roposed model was able to predict the maximum tensile membrane peak capacity to within about 10% for all slabs except Slabs 4 and 7 Slab 3 had the lowest tensile reinforcement ratio of all the slabs and hence the proposed model predicted a low ultimate curvature which in turn predicted a lower rotational capacity of the plastic hinges at the end supports. The full plastic load was not predicted to have been reached in this case. However, if the full plastic load was assumed to have been reached, the pr edicted compressive membrane peak capacity would have been different from the experimental value by 6% instead of 26%. On the other hand, the model overestimated the peak load capacity for Slab 15 by 21% but if the plastic hinges at the end supports were assumed to fail to rotate, the model would only overestimate the peak load capacity by 2%.

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180 Slab 6 was the only slab that had no compressive reinforcement at the midspan and at the end supports. Thereafter, a four hinge failure mechanism was formed with t he plastic hinges located at the center o f the bend in the reinforcement The results from the three spring model matched the experimental results reasonably well. Slab 13 was similar to Slab 7, except that it had cut bars instead of bent bars Because th e upper bound curve modeled the experimental curve rather well up unti l the yielding of reinforcement at the ends of the slab, it seems that the cut bars contributed fully to the load capacity up to the point that plastic hinges started forming. After that the cut off bars only contributed about half the load capacity up until the peak load, and almost had no contribution during the tension membrane stage This seems reasonable because Slab 13 had no stirrups to help transfer load from the top cut bars to the bottom cut bars. Therefore, the cut bars could not engage in catenary action. Slab 14 was the only slab with a thicker section depth and hence a lower span to depth ratio. With a thicker section, there could be a more significant error from assuming t hat the springs were linear in the proposed model because progressive damage to the concrete should actually soften the spring when the concrete area was reduced, and the modulus of elasticity of concrete is reduced at a higher stress when the membrane for ce increases. A summary of the comparison of results can be found in Table 4 2. The difference in percentage is the difference between the experimental result and the predicted result. Therefore, a positive difference indicates that the proposed model un derestimated the experimental result. Similarly, a negative difference indicates that the proposed model overestimated the experimental result.

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181 Table 4 2 Comparison of experimental results with predicted res ults (Woodson and Garner 1985)

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182 4.3 Validation with Experimental D ata from Woodson (1985) Woodson tested ten one way slabs, which were restrained against rotation and longitudinal expansion, under static uniform water pressure (Woodson 1985) All slabs were 24 inches wide by 36 inches long and 2 5/16 inches thick Six inches of both ends of the slab were clamped at the supports, thus lea ving a clear span of 24 inches, which correspond s to a spa n to depth (L/h) ratio of 10 .4. 4.3.1 Experimental P rocedure The experimental procedure was similar to that described in Section 4.2.1 (Woodson and Garner 1985) and hence will not be repeated in this section 4.3.2 Slab D etails The tensile reinforcement ratio was 0.72% for Slabs 1 to 8, and the compressive reinforcement ratio was 0.83%. The total tension reinforcement was the same as the total compression reinforcement, but the reinforcement ratios were different because the concrete cover was different on the tension and compression sides of the slab. The differences between Slabs 1 to 8 wer e in the location of the temperature steel and the type of stirrup configuration and spacing. The tensile reinforcement ratio was 0.75% for Slabs 9 and 10, and the compressive reinforc ement ratio was 0.86%. Similar to Slabs 1 to 8, the reinforcement ratios were different because the concrete cover was different on the tension and compression sides of the slab. Unlike Slabs 1 to 8 which use 0.25 inch diameter deformed re inforcement steel at a spacing of 3.75 inches, Slabs 9 and 10 use d D2.5 deformed wire as the principal reinforcement at a spacing of 1.75 inches.

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183 Figure 4 50 Main and transverse reinforcement details for Slabs 1 to 8 (Woodson 1985)

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184 Figure 4 51 Main and transverse reinforcement for Slabs 9 and 10 (Woodson 1985)

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185 Only Slab 1 had no stirrup reinforcement. The s tirrup configuration that was installed for the remaining nine slabs was one of three types of stirrup configurations Type I was a double leg stirrup with 135 degree bends at both ends. Type II was a single leg stirrup with 135 degree bends at both ends. Type III was a single leg stirrup with a 135 degree bend at one end and a 90 degree bend at the other end. Figure 4 52 Stirrup configurations for Slabs 2 to 9 (Woodson 1985)

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186 Figure 4 53 Double leg stirrup (Type I) placement details (Woodson 1985)

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187 Figure 4 54 Single leg stirrup (Types II and III) placement details (Woodson 1985)

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188 Only Slabs 5 and 7 had temperature steel placed outside the principal steel. The other eight slabs had temperature steel placed inside the principal steel Figure 4 55 Temperature steel placement (Woodson 1985) The sum mary of these slab d etails can be found in Table 4 3 Table 4 3 Slab details (Woodson 1985)

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189 4.3.3 Failure Mechanisms Unlike the experiment conducted by Woodson and Garner (1985) where t here were primarily three types of failure mechanisms exhibited by the slab specimens only the three hinge failure mechanism was exhibited by the slabs in the Woodson (1985) experiment Figure 4 56 General t hree hinge failure me chanism deformation (Woodson 1985) 4.3.4 Assumed Values of Surround S tiffness Because the experimental setup was the same as in Woodson and Garner (1985), the surround stiffness for the Woodson ( 1985) experiment was also assumed to be the same to maintain consistency in calculations 4.3.5 Comparison of Results between Experimental Data and Proposed M odel 4.3.5.1 Slab 1 The load deflection behavior generated by the proposed model closely resembled the load defl ection behavior recorded by Woodson experimentally. The peak load generated by the proposed model was within 3% of the experimental value. Sixteen percent of the bottom reinforcement was assumed to have broken during the tensile

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190 membrane stage while 70% was assumed to have broken during the compressive membrane stage. The remaining bottom r einforcement did not break. All the broken top reinforcement was assumed to break during the compressive membrane stage Figure 4 57 Results for proposed model (in red) and experiment (in black) for Slab 1 (Woodson 1985)

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191 Figure 4 58 Posttest view of Slab 1 (Woodson 1985) 4.3.5.2 Slab 2 The load deflection behavior from zero to the peak pressure generated by the proposed model also resembled the load deflection behavior recorded by Woodson experimentally. The peak load capacity that was predicted by the proposed model was within 6% of the experimental value. The minimum load capacity that was predicted by the proposed model was almost exactly the same as the experimental value. Even the final tensile membrane capacity that was predicted by the proposed model was almost exactly the same as the experimenta l value. The load de flection curve that was predicted by the proposed model was slightly stiffer than the experimental load deflection cu rve, but it matched the experimental

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192 curve after the peak load capacity of the slab was reached. It also has almost the same slope as the experimental load deflection curve during the tensile membrane stage. N inety percent of the bottom reinforcement was assumed to have broken during the tensile membrane stage while 10% was assumed to have broken during the compressive membrane stage. The remaining bottom reinforcement did not break. All the broken top reinforcement was assumed to break during the compressive membrane stage. Figure 4 59 Results for proposed model (in red) and experiment (in black) for Slab 2 (Woodson 1985)

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193 Figure 4 60 Posttest view of Slab 2 (Woodson 1985) 4.3.5.3 Slab 3 The load deflection behavior generated by the proposed model resembled the load deflection behavior recorded by Woodson experimen tally in terms of shape, but underestimated the experimental value All the reinforcement that was broken during the experiment was assumed to break during the compressive membrane stage. The peak load capacity that was predicted by the proposed model was within 13% of the experimental value. The minimum load capacity that was predicted by the proposed model u nderestimated the experimental value by about 10 psi However, the p roposed model is able to replicate the behavior of the slab in that there was no tensile membrane stage for this slab.

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194 The load de flection curve that was predicted by the proposed model was slightly stiffer than the experimental load deflection cu rve, but it matched the experimental curve after the peak load c apacity of the slab was reached However, it overestimated the final de f lection of the slab at failure by about 1.5 inches. Figure 4 61 Resul ts for proposed model (in red) and experiment (in black) for Slab 3 (Woodson 1985)

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195 Figure 4 62 Posttest view of Slab 3 (Woodson 1985) 4.3.5.4 Slab 4 The load deflection behavior generated by the proposed model resembled the load deflection behavior recorded by Woodson experimentally in terms of shape, but underestimated the experime ntal value. All the reinforcement that was broken during the experiment was assumed to break during the compressive membrane stage The peak load capacity that was predicted by the proposed model was within 18 % of the experimental value. The minimum load capacity that was predicted by the proposed model u nderestimated the experimental value by about 10 psi However, the p roposed model is able to replicate the behavior of the slab in that there was no tensile membrane stage for this slab

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196 The load de flection curve that was predicted by the proposed model was slightly stiffer than the experimental load deflection cu rve, but it matched the experimental curve after the peak load capacity of the slab was reached. However, it overestimated the final de f lection of the slab at failure by about 1.5 inches Figure 4 63 Results for proposed model (in red) and experiment (in black) for Slab 4 (Woodson 1985)

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197 Figure 4 64 Posttest view of Slab 4 (Woodson 1985) 4.3.5.5 Slab 5 The load deflection behavior generated by the proposed model resembled the load deflection behavior recorded by Woodson experimentally in terms of shape, but underestimated the experimental value. The peak load capacity that was predicted by the proposed model was within 17 % of the experimental value. The minimum load capacity that was predicted by the proposed model underestimated the experimental value by about 17% H owever, the final tensile membrane capacity that was predicted by the proposed model was almost exactly the same as the experimenta l value. The load de flection curve that was predicted by the proposed model was slightly stiffer than the experimental load deflection cu rve, but it matched the experimental

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198 curve after the peak load capacity of the slab was reached. It also has almost the same slope as the experimental load deflection curve during the tensile membrane stage All the reinforcement that was broken during the experiment was assumed to break during the compressive membrane stage. Thirty six percent of the bottom reinforcement was assumed to have broken during the tensile membrane stage while 50% was assumed to have broken during the compressive membrane st age. The remaining bottom reinforcement did not break. All the broken top reinforcement was assumed to break during the compressive membrane stage. Figure 4 65 Results for proposed model (in red) and expe riment (in black) for Slab 5 (Woodson 1985)

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199 Figure 4 66 Posttest view of Slab 5 (Woodson 1985) 4.3.5.6 Slab 6 The load deflection behavior generated by the proposed model resembled the load deflection behavior recorded by Woodson experimentally in terms of shape, but underestimated the experimental value. The peak load capacity that was predicted by the proposed model was within 6% of the experimental value. After the peak load found experimentally, the proposed model produced results that under estimated the load capacity calculated from the experiment A lso, the proposed model predicted a tension membrane behavior that did not happen experimentally. The load de flection curve that was predicted by the proposed model was slightly stiffer than the experimental load deflection cu rve, but it matched the experimental

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200 curve after the peak load capacity of the slab was reached. H owever, it predicted a rise in load capacity during tensile membrane stage but this did not happen experimentally. All the broken reinforcement was a ssumed to break during the compressive membrane stage Figure 4 67 Results for proposed model (in red) and experiment (in black) for Slab 6 (Woodson 1985)

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201 Figure 4 68 Posttest view of Slab 6 (Woodson 1985) 4.3.5.7 Slab 7 The load deflection behavior generated by the proposed model resembled the load deflection behavior recorded by Woodson experimentally in terms of shape, but underest imated the experimental value. The peak load capacity that was predicted by the proposed model was within 6% of the experimental value. The minimum load capacity that was predicted by the proposed model was within 7% as the experimental value. Even the final tensile membrane capacity that was predicted by the proposed model was almost exactly the same as the experimenta l value.

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202 The load de flection curve that was predicted by the proposed model was slightly stiffer than the experimental load deflection cu rve, but it matched the experimental curve after the peak load capacity of the slab was reached. It also has almost the same slope as the experimental load deflection curve during the tensile membrane stage Ten percent of the bottom reinforcement was assumed to have broken during the tensile membrane stage while 76% was ass umed to have broken during the compressive membrane stage. The remaining bottom reinforcement did not break. All the broken top reinforcement was assumed to break during the compressive membrane stage. Figure 4 69 Results for proposed model (in red) and experiment (in black) for Slab 7 (Woodson 1985)

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203 Figure 4 70 Posttest view of Slab 7 (Woodson 1985) 4.3.5.8 Slab 8 The load deflection behavior generated by the proposed model resembled the load deflection behavior recorded by Woodson experimentally in terms of shape, but underestimate d the experimental value. The peak load capacity that was predicted by the proposed model was within 11 % of the experimental value. The minimum load capacity that was pred icted by the proposed model underestimate d the experimental value by 19% H owever the final tensile membrane capacity that was predicted by the proposed model was almost exactly the same as the experimenta l value. The load de flection curve that was predicted by the proposed model was a lmost as stiff as the experimental load deflection cu rve, a nd it matched the experimental curve after the peak load capacity of the slab was reached. It also has almost the same slope

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204 as the experimental load deflection curve during the tensile membrane stage However the predicted deflection at ultimate failure of the slab by the proposed model overestimated the actual failure deflec tion by about 1.5 inches. All the broken reinforcement was assumed to break during the compressive membrane stage. Figure 4 71 Results for proposed model (in red) and experiment (in black) for Slab 8 (Woodson 1985)

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205 Figure 4 72 Posttest view of Slab 8 (Woodson 1985) 4.3.5.9 Slab 9 The load deflection behavior generated by the proposed model resembled the load deflection behavior recorded by Woodson experimentally in terms of shape, but underestimated the experimental value. The peak load capacity that was predicted by the proposed model was within 13 % of the experimental value. The minimum load capacity that was predicted by the proposed model was within 5% as the experimental value. Even the final tensile membrane capacity that was predicted by the proposed model was almost exactly the same as the experimenta l value. The load de flection curve that was predicted by the proposed model was slightly stiffer than the experimental load deflection cu rve, but it matched the experimental

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206 curve after the peak load capacity of the slab was reached. H owever, the proposed model predicted a more flexible s lope than the experimental load deflection curve during the tensile membrane stage Fifty percent of the bottom reinforcement was assumed to ha ve broken during the tensile membrane stage while 50% was assumed to have broken during the compressive membrane stage. The remaining bottom reinforcement did not break. All the broken top reinforcement was assumed to break during the compressive membrane stage. Figure 4 73 Results for proposed model (in red) and experiment (in black) for Slab 9 (Woodson 1985)

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207 Figure 4 74 Posttest view of Slab 9 (Woodson 1985) 4.3.5.10 Slab 10 The load deflection behavior generated by the proposed model resembled the load deflection behavior reco rded by Woodson experimentally in terms of shape, but underestimated the experimental value. The peak load capacity that was predicted by the proposed model was within 1 9 % of the experimental value. The minimum load capacity that was predicted by the proposed model was within 7% as the experimental value. Even the final tensile membrane capacity that was predicted by the proposed model was almost exactly the same as the experimenta l value. The load de flection curve that was predicted by the proposed model was slightly stiffer than the experimental load deflection cu rve, but it matched the experimental

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208 curve after the peak load capacity of the slab was reached. It also has almost the same slope as the experimental load deflection curve during the tensile membrane stage Twenty one percent of the top reinforcement was assumed to have broken during the tensile membrane stage while 50% was assumed to have broken during the compressiv e membrane stage. The remaining top reinforcement did not break. All the broken bottom reinforcement was assumed to break during the tension membrane stage. Figure 4 75 Results for proposed model (in red ) and experiment (in black) for Slab 10 (Woodson 1985)

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209 Figure 4 76 Posttest view of Slab 10 (Woodson 1985) 4.3.6 Discussion of R esults The proposed model was able to model the initial load deflection behavior accurately up until the peak load capacity for Slabs 1, 4, 8 9, and 10, and predict ed a slightly stiffer result for the r emaining slabs The proposed model was able to predict the maximum compressive membrane peak capacity to within 10% for Slabs 1, 2, 6 and 7, and underestimated this value for the remaining slabs to within 20% The proposed model was able to predict the m inimum capacity to within 10 % for Slabs 1, 2, 7, 9 and 10, and to within 20% for Slabs 5 and 8, whereas Slabs 3, 4, and 6 did not exhibit tension membrane action The proposed model was able to predict the maximum tensile membrane peak capacity to within a bout 7 % for all slabs except Slabs 3, 4, and 6 which did not exhibit tension membrane action

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210 A summary of the comparison of results can be found in Table 4 4 The difference in percentage is the difference between the experimental result and the predict ed result. Therefore, a positive difference indicates that the proposed model underestimated the experimental result. Similarly, a negative difference indicates that the proposed model overestimated the experimental result.

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211 Table 4 4 Comparison of experimental results with predicted results (Woodson 1985)

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212 4.4 Validation with experimental data from Yu and Tan (2011 ) Yu and Tan tested two beam specimens, which consisted of a two bay beam a middle joint, and two e nlarged end column stubs, that were restrained against rotation and longitudinal expansion, under a static point load that was applied at the top of the middle joint under displacement control until specimen failure (Yu and Tan 2011) 4.4.1 Experimental P rocedure Figure 4 77 Experimental setup (Yu and Tan 2011) The beam specimen was connected horizontally using pin connections to a reaction wall on one end of the beam specimen and to a steel frame on the other end of the beam specimen The ends of the beam specimen were supported vert ically by roller supports. Because both horiz ontal and vertical supports can be idealized as roller connections, the horizontal and vertical reaction forces that were measured could be

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213 considered to be independent from each other. The vertical reaction for ces were measured by compression load cells installed under the roller supports. Figure 4 78 Vertical support details (Yu and Tan 2011) The horizontal reaction forces were measured by two tension/compression load cells installed at the end o f the beam specimen which was connected to the steel frame. Strain gages were installed at the horizontal support connected to the reaction wall.

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214 Figure 4 79 Horizontal support connected to the steel fra me (Yu and Tan 2011) Figure 4 80 Horizontal support connected to the reaction wall (Yu and Tan 2011)

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215 A hydraulic actuator that was connected to a portal frame applied load to the middle joint of the beam specimen using displacement control and measured the magnitude of the load that was applied with its built in load cell Lateral restraints were installed along the beam specimen to prevent out of plane failure. Steel rollers between the beam specimen and the lateral restraints reduced friction and allowed the beam specimen to move down along the lateral restraints. Figure 4 81 Lateral restraint details (Yu and Tan 2011)

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216 Other than the load cells and strain gages installed at the supports, there were s ix line displacement transducers and six linear variable diff erential transformers (LVDT) installed along the length of the beam specimen to measure the vertical displacement along the beam span and thus obtain the deflected shape of the beam specimen. These were labeled as L 1 to L 12 In addition, four LVDTs were in stalled at each end of the beam specimen (L 13 to L 16 and L 33 to L 36 ) to measure the axial displacements of the supports. Furthermore, sets of four LVDTs were installed at specified locations (L 17 to L 32 ) to measure the rotation of the beam specimen. Strain gages were also attached to reinforcement bars to measure strains at critical locations. Figure 4 82 Instrumentation layout (Yu and Tan 2011) 4.4.2 Beam S pecimen D etails Both beam specimens were 150 mm wide by 250 mm deep, with a net span of 2750 mm. Specimen S1 had a top reinforcement ratio of 0.9% with one T13 and two T10 reinforcement bars at the ends of the beam specimen and at the middle join t, and a bottom reinforcement ratio of 0.49% with two T10 reinforcement bars throughout the beam specim en. R6 stirrup reinforcement was placed at a spacing of 100 mm throughout the beam specimen, except at regions close to the middle joint and the ends of the beam specimen where the spacing was 50 mm

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217 Figure 4 83 Reinforcement details for beam specimen S1 (Yu and Tan 2011) Spe cimen S2 had a top reinforcement ratio of 0 .73 % with three T10 reinforcement bars at the ends of the beam specimen and at the middle joint, and a bottom reinforcement ratio of 0.49% with two T10 reinforcement bars through out the beam specimen. R6 stirrup reinforcement was placed at a spacing of 100 mm throughout the beam specimen. Figure 4 84 Reinforcement details for beam specimen S2 (Yu and Tan 2011) Table 4 5 Dimensions of beam specimens (Yu and Tan 2011)

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218 Table 4 6 Reinforcement details of beam specimens (Yu and Tan 2011) The yield strength of the T10 reinforcement bars was 511 M Pa, the ultimate strength wa s 731 MP a, and the fracture strain was 12.32%. The yield strength of the T13 reinforcemen t bars was 527 MP a, the ultimate strength was 640 MPa, and the fracture strain was 10.76%. The compressive cylinder strength for concrete was 31.2 MPa. 4.4.3 Failure Mechan isms Both beam specimens S1 and S2 displayed th e three hinge failure mechanism with plastic hinges forming at the ends of the beam specimen and at one side of the middle joint. Plastic hinges did not form on both sides of the middle joint because there wa s no rotational restraint at the middle joint. Therefore, the middle joint was free to rotate and the plastic hinge only formed on one side of the middle joint. There was severe cracking at the ends of the beam specimen and at the middle joint interface These cracks were caused by bending and subsequently tensile forces during catenary action. There was also severe cracking between the curtailment points of the top bars. These cracks were caused mostly by tensile forces during catenary action.

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219 Figure 4 85 Three hinge failure mechanism of beam specimens (Yu and Tan 2011) Figure 4 86 Failure mode of beam specimen S1 (Yu and Tan 2011)

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220 Figure 4 87 Failure mode of beam specimen S2 (Yu and Tan 2011) 4.4.4 Assumptions U s ed in the Calculations by the Proposed M odel As there were no values of the surround sti ffness found in the paper by Yu and Tan (2011 ), the surround stiffness was assumed to be a constant value for both beam specimens S1 and S2 to ensure consistency As the pla stic hinge formed at the interface of the middle joint and the beam specimen, the dimension of the middle joint was ignored in the computation using the proposed model for simplicity. Also, to further simplify the calculations, even though beam specimen S1 had T10 and T13 reinforcement at the top of the beam, the failure strain for all the reinforcement was taken to be the failure strain of the T13 reinforcement in the calculations using the proposed model because it was smaller than the failure strain of t he T10 reinforcement Similarly, the yield strength and failure strength of all reinforcement was taken to be that of the T10 reinforcement as the yield strength of the T10 reinforcement was lower than the T13 reinforcement. 4.4.5 Comparison of R esults bet ween Experimental Data and Proposed M odel The load deflection behavior generated by the proposed model closely resembled the load deflection behavior recorded experimentally for both beam specimens S1 and S2 in terms of the general shape of the curve. The middl e joint displacements

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221 calculated by the proposed model at which the maximum load during the CMA occurred the start of catenary action, and at the maximum load during tension membrane action were similar to the displacement values obtained from experiment. Figure 4 88 Load deflection curves for s pecimens S1 and S2 (Yu and Tan 2011) As expected, the load deflection curve for s pecimen S1 was slightly higher in general for both experimental and proposed model curves than s pecimen S2, which had a slightly lower reinforcement ratio than S1. However, the curves generated by the proposed model underestimated the experimental curves generally after the point where

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222 the reinforcement at the ends of the beam specimen had yielded. Prior to that point, the proposed model slightly overestimated the experimental load deflection curve. Figure 4 89 Axi al force deflection curves for s pecimens S1 and S2 (Yu and Tan 2011) The axial force deflection curves generated by the propos ed model for s pecimens S1 and S2 were very similar, with the values for S1 slightly higher than the values for S2. This is expected since the reinforcement ratio for S1 is slightly higher than for S2. Compared to the experimental results, the axial force deflection curves gener ated by

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223 the proposed model had the same shape in general, but overestimated the actual axial force that was recorded in the experiment after a certain middle joint displacement. 4.4.6 Discussion of R esults The overestimation of axial force by the proposed model is due to the assumption of the proposed model that the springs in the two spring model were linear elastic for simplicity In reality, during the compression membrane stage, as the memb rane force in the beam increased and the compressive stress increased on the concrete, the spring should have start ed to softe n because the slope of the stress strain curve of the concrete reduce d in value as strains increase d Therefore, the proposed model was able to predict the axial force rather well up to the elastic l imit of the concrete. Thereafter, it will over predict the axial force. Similarly, during the tension membrane stage, the proposed model was able to predict the tensile force in the beam up to a certain middle joint displacement. After this point, the rein forcement st eel in the beam specimen softened and load capacity was reduced. This was not captured by the proposed model because it assumed a constant stiffness. A summary of the comparison of results can be found in Table 4 7 The difference in percentag e is the difference between the experimental result and the predicted result. Therefore, a positive difference indicates that the proposed model underestimated the experimental result. Similarly, a negative difference indicates that the proposed model over estimated the experimental result.

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224 Table 4 7 Comparison of experimental results with predicted results (Yu and Tan 2011)

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225 4.5 Summary The methodology that was presented in Chapter 3 was implemented using a comp uter program. The theoretical results from the proposed model were compared against experimental results for 25 laterally restrained slabs tested by Woodson and Garner (1985) and Woodson (1985), as well as two laterally restrained beams tested by Yu and Ta n (2011). The span to depth ratio was 21.9 for the beams, and ranged from 8.3 to 10.4 for the slabs. The theoretical results matched the experimental results reasonably well for an assumed plastic hinge length of 0.5d, with the exception of Slab s 3 and 15 from Woodson and Garner (1985) For Slab 3, a plastic hinge length derived from Priestly produced a better result, whereas for Slab 15, the assumption that the plastic hinges at the end supports did not rotate produced a better result. This illustrated how the plastic hinge length could affect the load deflection curve significantly, because it affects the rotational capacity of the first plastic hinge that formed. This in turn determines whether the full plastic collapse mecha nism can be formed and hence whether the full plastic load capacity can be obtained. This would affect the peak load capacity and the yield line load capacity. The elastic portion of the load deflection curve matched the experimental results reasonably wel l. It was verified that deformation from bond slip indeed contributed significantly to the total deformation of the slab or beam at midspan. Without considering bond slip deformation, the load deflection curve during the elastic stage would be considerably stiffer than what was recorded experimentally. Slab 13 from Woodson and Garner (1985) provide d some insight o n how cut reinforcement affected the resistance function in restrained slabs It was noted that the

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226 load deflection behavior of Slab 13 closely r esembled the theoretical curve which accounted for all reinforcement in the slab, until yielding of reinforcement occurred at the end supports. After the peak load capacity, the load deflection behavior closely resembled the theoretical curve which accoun ted for only the uncut reinforcement in the slab. The load deflection curve was bounded between these two limits after yielding of the reinforcement at the end supports and before the load capacity reached its peak. This indicates that cut reinforcement ma y only be effective up to yielding of the reinforcement at the end supports. After that point, it gradually loses influence on the load deflection curve until it has no contribution after peak load capacity. Slab 6 from Woodson and Garner (1985) indicated that the three spring model could be used for a four hinge failure mechanism in a restrained slab or beam, as the theoretical results matched the experimental results reasonably well. not take into account bending action during the catenary action phase. The theoretical results from the proposed model matched the experimental results for most of the slabs and beams reasonably well during the tensile membrane phase. This indicated that t he proposed model might be able to predict the load deflection behavior reasonably well as long as the amount of reinforcement that was broken during CMA and TMA could be estimated. However, the proposed model has certain limitations due to the assumptions used to derive th e model. The springs in the two spring mo del and the three spring model were assumed to be linear elastic. The comparison between theoretical and experimental results of the axial force curve from the Yu and Tan (2011) experiment

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227 showed t hat the proposed model overestimated the compressive membrane force in the beam during compressive membrane phase and the tensile membrane force in the beam during tensile membrane phase. This is due to the fact that the concrete softens as stress and stra in are increased. Similarly, the secant stiffness of the steel reinforcement is reduced as strains in the reinforcement are increased. These effects are not captured by a linear elastic model for the springs. The theoretical result for Slab 14 from Woodso n and Garner (1985) did not match experimental r esults either Slab 14 had a lower span to depth ratio than the other slabs. After the peak capacity, the slab lost load capacity much quicker than the model predicted. This could also be due to the assumptio n that the springs were linear elastic when in reality, concrete was softening and hence load capacity was not as high as predicted by the model. Also, shear effects were ignored in the proposed model, whereas in reality, shear effects could be very signi ficant.

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228 5 CHAPTER 5 CONCLUSIONS AND RECO MMENDATIONS 5.1 Summary A numerical procedure to generate the load displacement curve of a restrained reinforced concrete slab for three hinge failure mechanisms and four hinge failure mechanisms was presented in t his study. Chapter 2 reviewed the concepts of CMA and TMA as well as plastic hinge concepts moment area theorem, limit analysis, bond slip deformation and past research on membrane action Chapter 3 presented the proposed methodology of generating the lo ad displacement curve of restrained reinforced concrete slabs by conducting moment curvature analyses, followed by load interaction diagrams, limit analysis and lastly, the use of either the proposed two spring model or three spring model to reflect the b ehavior of the restrained slab in both compressive membrane and tensile membrane stages. This proposed methodology was implemented in a computer program and the results were compared with experimental data in Chapter 4. The numerical data compared well wit h the experimental data for some of the slabs but limitations in the proposed model were found for the load deflection curve of deep slabs and the axial force deflection curves of restrained beams 5.2 Limitations This methodology was created with the view that any type of concrete material would be applicable. Hence, the moment curvature analysis and the load interaction diagr am procedure require the stress strain curves of the concrete material and the reinforcement bars to be known Thus, the accuracy of any material model will have to be verified, especially for any new materials, because this would affect the accuracy of the proposed model.

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229 Bond slip was found to be significant in the initial portion of the load displacement curve. Ignoring bond slip w ould result in a curve that would be too stiff compared to actual experimental results. Hence, it is critical to verify the amount of bond slip that can happen for any new concrete material model that has not been tested for bond slip. Plastic hinge lengt h was also found to affect the load deflection curve significantly. H ence it would be important to ensure that the plastic hinge lengths for any new concrete material could be estimated reasonably well Several key parameters that were required in the pro posed methodology were not available in the experimental report and thus had to be assumed. For example, the surround stiffness of the experimental set up that included the test chamber and the soil backfill was difficult to estimate, and a value had to be assumed. Also, the amount of broken reinforcement was reported in the experimental data, but the amount of reinforcement that was ruptured during the compressive membrane stage and the amount of reinforcement that was ruptured during the tensile membrane stage were not reported This information is also needed in the proposed model to estimate the tensile membrane portion of the load displacement curve. The proposed model assumes that the springs are linear, and does not take the progressive damage of the slab into account, which might affect the result of the load displacement curve. The model is also only formulated for one way slabs or beams, and not for two way slabs. The ends of the slab or beam are assumed to be fully clamped and thus the proposed mo del may not be suitable for partially restrained slabs or beams.

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230 5.3 Recommendations for Further R esearch The first recommendation is to conduct experimental studies to further validate the proposed model by recording the amount of reinforcement that ruptured in the compressive membrane stage and in the tensile membrane stage, as well as the surround st iffness of the experimental set up. The second recommendation is to improve the two spring model and the three spring model to take into account the nonlineari ty of the concrete model. This would generalize the proposed model further and it will be applicable to more situations such as for deep slabs The third recommendation is to study how shear affects the load deflection behavior of restrained reinforced con crete slabs during the compressive membrane stage. The proposed methodology currently assumes that there is no shear effect and this is only accurate for slabs with a high enough span to depth ratio. The fourth recommendation is to study whether the propo sed model could be extended to partially restrained slabs or beams. The current model assumed fully clamped conditions at the end supports. The fifth recommendation is to study whether this proposed model could be extended to two way slabs. Currently, the proposed model is only formulated for one way slabs or beams. 5.4 Conclusions Plastic hinge length of 0.5d worked reasonably well with the experimental results for most of the slabs and beams studied in this thesis. The bond slip model helped to produce reaso nable results for most slabs and beams during the elastic stage of the load deflection curve. Based on the results of this present study, it can be concluded

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231 that the proposed model can provide good approximate results for fully restrained beams or one way slabs that are above a certain span to depth limit. However, further study is recommended to expand the limitations of the proposed model to deep slabs or beams, two way slabs, and partially restrained slabs or beams.

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232 LIST OF REFERENCES ACI Committee 318. (2011). Building code requirements for structural concrete (ACI 318 11) and commentary American Concrete Institute, Farmington Hills MI. Al College London (University of London). ACI Structural Journal 105(3), 290 300. Biggs, J. (1964). Introduction to structural dynamics McGraw Hill, New York. Journal of the Structural Division 106(6), 1237 1253. Plastic Journal of the Structural Divis ion 106(6), 1255 1262. Brotchie, J., Jacobson, A., and Holley, M. (1965). Effect of membrane action on slab behavior Dept. of Civil Engineering Massachusetts Institute of Technology, Cambridge Mass. shallow buried reinforced Florida, Gainesville, FL. Chen, W., and Lui, E. M. (2005). Handbook of structural engineering 2nd ed. CRC Press, Boca Raton. Christiansen, K. (1963 The Structural Engineer 41(8), 261 265. Collins, M., and Mitchell, D. (1991). Prestressed concrete structures Prentice Hall, Englewood Cliffs N.J. Consolazio, G., Fung, J., and Ansl ey, M. (200 P Diagrams for Concrete ACI structural journal. 101(1), 114 123. Journal of the Structural Division 92(5), 121 146. Datta, slab Magazine of Concrete Research Thomas Telford, 27(91), 111 120.

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233 Journ al of the Structural Division 103(ST2), 405 419. Journal of Structural Engineering American Society of Civil Engineers, 116(12), 3251 3267. In plane stiffness of reinforced concrete slabs under MAGAZINE OF CONCRETE RESEARCH 46(166), 67 77. Guice, L. (1986). Effects of edge restraint on slab behavior U.S. Army Engineer Waterways Experiment Station, Vicksburg Miss Hassoun, M., and Al Manaseer, A. (2012). Wiley, Hoboken, N.J. Journal of the Structural Division 93(5), 85 112. International Journal of Solids and Structures 4(1), 61 74. Johansen, K. W. (1962). Yield line Theory. Cement & Concrete Assoc., London. Kassimali, A. (2005). Structural analysis 3rd ed. Thomson, Australia. Keenan, W. (1969). Strength and beh avior of restrained reinforced concrete slabs under static and dynamic loadings Naval Civil Engineering Laboratory, Port Hueneme Calif. Kemp, K., Eyre, J., and Al of compressive membrane act R.H.Wood Memorial Conf., Frame and Slab Structures Butterworths, London, UK. Journal of Structural Engineering American Society of Civil Engineers, 110(3), 637 651. Journal of Structural Engineering 112(4), 726 744. Krauthammer, T., Frye, M., Schoedel, R., Seltzer, M., and Astarlioglu, S. (2003). A Single D egree of Freedom (SDOF) Computer Code Development for the Analysis of Structures Subjected to Short Duration Dynamic Loads State College.

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234 Journal of the Structural Division 93(2), 519 522. Laterally Restrained Structural Concrete One ACI Structural Journal 91(6), 719 725. line theory for reinforc ed concrete slabs at moderately large Magazine of Concrete Research 19(61), 211 222. Engineering Structures 29 (5), 663 674. Nawy, E. (2009). Pearson Prentice Hall, Upper Saddle River, NJ. The Structural Engineer 36(6), 197 201. oncrete slabs under short term ICE Proceedings 28(2), 125 150. reinforced concrete slabs with fully restrained e Magazine of Concrete Research Thomas Telford, 16(46), 39 44. Park, R., and Gamble, W. L. (1999). Reinforced Concrete Slabs 2nd Ed. Wiley, New York, 736. Park, R., and Paulay, T. (1975). Reinforced concrete structures Wiley, New York. Park, R., Pr Confined Journal of the Structural Division 108(4), 929 950. Thesis, Cambridge University, Cambridge, UK. ACI Structural Journal (84), 61 76. ACI Special Publication 48, 191 224.

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235 carrying capacity of slab strips restrained against longitudinal Concrete Building Research Station, Watford, 3(9), 369 378. Proceedings of the international symposium on the flexural mechanics of reinforced concrete ASCE ACI, Miami, 405 491. University of California, Berkeley, CA. Concrete Structures in Seismic Regions: Proceedings of the Fib symposium 2003 Athens, 1 10. Sheikh, S. A., and Khoury, S. S. (1 ACI Structural Journal 90(4), 414 431. Siddiqi, Z. (2009). Concrete structures Help Civil Engineering Publisher, Lahore [Pakistan]. Welch, R. (1999). Compressive membrane capacity estimates in laterally edge restrained reinforced concrete one way slabs University of Illinois at Urbana Champaign, Urbana Ill. Wood, R. (1961). Plastic and elastic design of slabs and plates with particular reference to reinforced concrete floor slabs. Ronald Press Co., New Yor k. Woodson, S. (1985). Effects of shear stirrup details on ultimate capacity and tensile membrane behavior of reinforced concrete slabs U.S. Army Engineer Waterways Experiment Station, Vicksburg Miss. Woodson, S. (1994). Shear reinforcement in deep slabs U.S. Army Engineer Waterways Experiment Station, Vicksburg Miss. Woodson, S., and Garner, S. (1985). Effects of reinforcement configuration on reserve capacity of concrete slabs. U.S. Army Engineer Waterways Experiment Station, Vicksburg Miss. Yu, J., an d Tan, K. collapse resistance of reinforced concrete beam column sub Engineering Structures

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236 BIOGRAPHICAL SKETCH Chong Yik Melvin Goh obtained a scholarship from the Defen ce Science & Technology Agenc y (DSTA) to study civil e ngineering in 2000. He graduated with a Bachelor of Science in 2003 and a Master of Science in 2004 from the University of California, Berkeley. After graduation he worked at DSTA for seven years where he was appointed as a Senior Engineer He was awarded the DSTA Postgraduate Scholarship to further his studies. He began graduate school at t he University of Florida in 2011 and worked at the Center for Infrastructure Protection and Physical Security (CI PPS) He received his Master of Science from the University of Florida in 2013, with the Critical Infrastructure Protection Certificate from CIPPS.