Calibration of flexural design of concrete members reinforced with FRP bars

Transcription

1 Louisiana State University LSU Digital Commons LSU Masters Theses Graduate Shool 2006 Calibration o lexural design o onrete members reinored with FRP bars Sujata Nandkumar Kulkarni Louisiana State University and Agriultural and Mehanial College, skulka1@lsu.edu Follow this and additional works at: Part o the Civil and Environmental Engineering Commons Reommended Citation Kulkarni, Sujata Nandkumar, "Calibration o lexural design o onrete members reinored with FRP bars" (2006). LSU Masters Theses This Thesis is brought to you or ree and open aess by the Graduate Shool at LSU Digital Commons. It has been aepted or inlusion in LSU Masters Theses by an authorized graduate shool editor o LSU Digital Commons. For more inormation, please ontat gradetd@lsu.edu.

2 CALIBRATION OF FLEXURAL DESIGN OF CONCRETE MEMBERS REINFORCED WITH FRP BARS A Thesis Submitted to the Graduate Faulty o the Louisiana State University and Agriultural and Mehanial College in partial ulillment o the requirements or the degree o Master o Siene in Civil Engineering in The Department o Civil and Environmental Engineering By Sujata Kulkarni B.E., University o Pune, India, 2002 May 2006

3 ACKNOWLEDGEMENTS To, Dr. Ayman M. Okeil, my major proessor and hairman o my advisory ommittee, I express my sinere thanks or making this work possible. No words o gratitude are enough or his guidane, assistane, enouragement and patiene. His experiene and observations helped me a lot to ous on my work. I have learned many things rom him as a researher during last two years. I am very grateul to Dr. R. Rihard Avent and Dr. Steve Cai, other members o my advisory ommittee or taking their time and guiding me. I would like to thank them or their help, patiene, and support. In addition, I would like to thank Hughes Brothers or providing statistial inormation on oupon test results or Fiber Polymer Reinored (FRP) bars. I would like to thank my dearest parents, Pro. Nandkumar Kulkarni, and Mrs. Alka Kulkarni or making me believe in my dreams and or onstantly supporting me to ahieve them. I would like to extend my deepest regards to my beloved sister, Guari or being my soure o inspiration at all times and or being there with me throughout. Anamika and Harshad thanks a lot or all your enouragement and support, this wouldn t have been possible without you. I would like to thank my olleagues Mario, Stanley, Anand and Arhana or their help. I would like to thank my riends Vidya, Jyoti, and Sheela or making my stay here wonderul. Thanks a lot to Hrishikesh and Eashwar or everything. Finally, I would like to thank all riends and olleagues at LSU who have ontributed in numerous ways to make this program an enjoyable one. ii

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS.. ii LIST OF TABLES.....v LIST OF FIGURES.....vii ABSTRACT...x 1. INTRODUCTION General Bakground Objetives Researh Plan Sope o Study Organization LITERATURE REVIEW Introdution History o Fiber Reinored Polymer Composites Properties o Composite Materials Appliations Forms o FRP Reinorement FRP Bars Review o Researh Ativities and Results Analytial Studies Experimental Studies Reliability Based Tehniques Reliability or Bridge Girders Reliability or Bridge Deks ANALYSIS METHODS AND EXPERIMENTAL VERIFICATION Introdution Analysis Methods Method 1: Simpliied Expressions Reommended by ACI 440.1R Method 2: Detailed Closed-Form Expressions Method 3: Layered Setion Analysis Experimental Veriiation Saadatmaneh (1994) Theriault and Bemmokrane (1998) Pee et al. (2000) Aiello and Ombres (2000) Conlusion DEVELOPMENT OF RESISTANCE MODEL Introdution..52 iii

5 4.2 Variables Aeting the Flexural Strength o FRP-RC Members Statistial Charateristis o Random Variables Basi Parameters Basi Funtions Probability Distributions Used in this Study Statistial Properties Sample Design Spae Design Spae or Deks (Slabs) Design Spae or Girders (Beams) Resistane Models or Flexural Capaity o FRP-RC Members Unertainties due to Analysis Method or All FRP-RC members Unertainties due to Material and Fabriation or FRP-RC Slabs Unertainties due to Material and Fabriation or FRP-RC Girders Conlusion RELIABILITY ANALYSIS AND RESULTS Introdution Load Model or Buildings Reliability Analysis Reliability Index, First Order Reliability Method (FORM) Reliability Analysis o FRP-RC Slabs and Beams Results Reliability Indies Obtained or FRP-RC Slabs Reliability Indies Obtained or FRP-RC Beams Conlusions CONCLUSIONS AND RECOMMENDATIONS Summary Conlusions Reommendations.121 REFERENCES APPENDIX A: GENERAL PROCEDURE FOR GENERATING RANDOM NUMBERS FROM ANY ARBITRARY DISTRIBUTION 126 APPENDIX B: CHI-SQUARE STATISTICAL TEST: GOODNESS-OF-FIT TEST 127 APPENDIX C: RELIABILITY INDEX CALCULATIONS 131 VITA iv

6 LIST OF TABLES Table 2.1 Coeiients o Thermal Expansion or Reinoring Bars. 13 Table 2.2 Usual Tensile Properties o Reinoring Bars...13 Table 3.1 Environmental Redution Fators or dierent FRP Bars. 26 Table 3.2 Cross setional properties o Veriiation Beams 1 and 2.37 Table 3.3 Ultimate moment apaities o Veriiation Beams 1 and 2.39 Table 3.4 Cross setional properties o Veriiation Beams 3 and 4.40 Table 3.5 Ultimate moment apaities o Veriiation Beams 3 and 4.42 Table 3.6 Cross setional properties o Veriiation Beams 5 and 6.43 Table 3.7 Ultimate moment apaities o Veriiation Beams 5 and 6.47 Table 3.8 Cross setional properties o Veriiation Beams 7, 8, and Table 3.9 Ultimate moment apaities o Veriiation Beams 7, 8, and 9 50 Table 3.10 Ultimate Moment Capaities...51 Table 4.1 Statistial Properties o b, h, d, A and Table 4.2 Statistial Properties o FRP Bars Table 4.3 Determination o P and V P...69 Table 4.4 Material Properties and Fabriation Desriptors or FRP-RC Deks...72 Table 4.5 Material Properties and Fabriation Desriptors or FRP-RC Girders...76 Table 5.1 Statistial Properties or Dead load and Live load...90 Table 5.2 Relationship between Reliability Index, and Probability o Failure, P...91 Table 5.3 Resistane Fators, φ ACI...99 Table 5.4 Reliability Indies or FRP-RC Slabs Table 5.5 Average Reliability Indies or FRP-RC Slabs v

7 Table 5.6 Reliability Indies or FRP-RC Beams Table 5.7 Average Reliability Indies or FRP-RC Beams Table B1 Chi-Square Test or FRP-RC slab C4.0H8.0R Table B2 CDF o the Chi-Square Distribution Table C1 Statistial Properties or FRP-RC slab C4.0H4.0R Table C2 Iterations or alulating or C4.0H4.0R vi

8 LIST OF FIGURES Figure 1.1 FRP Bars...2 Figure 2.1 FRP Bars Produed by Marshall Industries Composites, In...10 Figure 2.2 SAFPLATE Fiberglass Gritted Plate Produed by Strongwell...10 Figure 2.3 Strutural strengthening with FRP laminates...11 Figure 2.4 Commerially available FRP bar manuatured by V-ROD...12 Figure 2.5 Test setup...19 Figure 3.1 Balaned ailure ondition or Method Figure 3.2 Conrete rushing ailure mode or Method Figure 3.3 FRP rupture ailure mode or Method Figure 3.4 Conrete rushing ailure or Method Figure 3.5 FRP Rupture ailure mode or Method Figure 3.6 Stress-strain urve or onrete...32 Figure 3.7 Shemati o layered setion approah...36 Figure 3.8 Flexural testing o retangular test beams (Saadatamanesh 1994)...37 Figure 3.9 Plot o load-deletion results or Veriiation Beam Figure 3.10 Plot o load-deletion results or Veriiation Beam Figure 3.11 Shemati o test Set-up...40 Figure 3.12 Plot o applied moment-deletion results or Veriiation Beam Figure 3.13 Plot o applied moment-deletion results or Veriiation Beam Figure 3.14 Shemati o test Set-up o Veriiation Beam Figure 3.15 Shemati o test Set-up o Veriiation Beam Figure 3.16 Plot o applied moment-urvature results or Veriiation Beam vii

9 Figure 3.17 Plot o applied moment-urvature results or Veriiation Beam Figure 3.18 Plot o load-deletion results or Veriiation Beam Figure 3.19 Plot o load-deletion results or Veriiation Beam Figure 3.20 Shemati o test Set-up o Veriiation Beams 7, 8, Figure 3.21 Plot o load-deletion results or Veriiation Beam Figure 3.22 Plot o load-deletion results or Veriiation Beam Figure 3.23 Plot o load-deletion results or Veriiation Beam Figure 4.1 PDF o X...54 Figure 4.2 A graphial representation o the relationship between PDF and CDF...55 Figure 4.3 Graphial representation o Normal Distribution...56 Figure 4.4 Graphial representation o Lognormal Distribution...57 Figure 4.5 Graphial representation o Weibull Distribution...58 Figure 4.6 Histogram o tensile strength or #2 FRP bar...61 Figure 4.7 Histogram o tensile strength or #3 FRP bar...61 Figure 4.8 Histogram o tensile strength or #4 FRP bar...62 Figure 4.9 Histogram o tensile strength or #5 FRP bar...62 Figure 4.10 Histogram o tensile strength or #6 FRP bar...63 Figure 4.11 Histogram o tensile strength or #7 FRP bar...63 Figure 4.12 Histogram o tensile strength or #9 FRP bar...64 Figure 4.13 Histogram o tensile strength or #10 FRP bar...64 Figure 4.14 Plot o Bias vs. FRP Reinorement Ratio or FRP-RC slabs...71 Figure 4.15 Plot o COV vs. FRP Reinorement Ratio or FRP-RC slabs...71 Figure 4.16 Plot o Bias, λ MF vs. ρ / ρ b or =4 ksi or FRP-RC slabs...73 viii

10 Figure 4.17 Plot o COV, V MF vs. ρ / ρ b or =4 ksi or FRP-RC slabs...74 Figure 4.18 Plot o Bias, λ MF vs. ρ / ρ b or =5 ksi or FRP-RC slabs...74 Figure 4.19 Plot o COV, V MF vs. ρ / ρ b or =5 ksi or FRP-RC slabs...74 Figure 4.20 Plot o Bias vs. FRP Reinorement Ratio or FRP-RC beams...75 Figure 4.21 Plot o COV vs. FRP Reinorement Ratio or FRP-RC beams...80 Figure 4.22 Plot o Bias, λ MF vs. Width, b or FRP-RC beams...81 Figure 4.23 Plot o Coeiient o Variation, VMF vs. Width, b or FRP-RC beams...82 Figure 4.24 Plot o Bias, λ MF vs. Aspet ratio or FRP-RC beams...84 Figure 4.25 Plot o Coeiient o Variation, VMF vs. Aspet ratio or FRP-RC beams...86 Figure 5.1 Reliability Index deined as the shortest distane in the spae o redued variables...93 Figure 5.2 Reliability Index evaluated at design point (Nowak and Collins 2000)...94 Figure 5.3 Plot o vs. ρ / ρ or dierent thiknesses or FRP-RC slabs b Figure 5.4 Plot o vs. ρ / ρ or dierent b or FRP-RC slabs Figure 5.5 Plot o vs. Figure 5.6 Plot o vs. M / M or L L D M / M or D = 4 ksi or FRP-RC slabs = 5 ksi or FRP-RC slabs Figure 5.7 Plots o vs. R ( ρ / ρ ) or dierent A and B or FRP-RC beams b Figure 5.8 Plots o vs. B or dierent R ( ρ / ρ ) and b or FRP-RC beams Figure 5.9 Plot o vs. M / M or L D = 4 ksi or FRP-RC beams Figure 5.10 Plot o vs. M / M or L D = 5 ksi or FRP-RC beams ix

11 ABSTRACT Fiber Reinored Polymer (FRP) omposites have been inreasingly aepted in the onstrution industry as a promising substitute or steel. Bridge dek deterioration is one o the most ommon deiienies in a bridge system. The use o FRP bars as reinorement or onrete bridge deks provides a potential or inreased servie lie, eonomi, and environmental beneits. This researh presents the development o resistane models or onrete members (beams and slabs) reinored with FRP bars (FRP-RC), or arrying out reliability analysis. The sope o this model is limited to the lexural behavior only; i.e. the ailures are not shear ailure and debonding. Probability o Failure, P, and Reliability Index,, o FRP-RC setions are alulated using the developed resistane model. A wide range o design variables is overed in alibrating the lexural design o FRP-RC members, using First Order Reliability Method (FORM). This study results in the development o resistane models or FRP-RC bridge deks and girders whih an also be used as resistane models or FRP-RC slabs and beams respetively. Also, the lexural reliability study on FRP-RC slabs and beams yielded parameters that aet the Probability o Failure, P, in terms o the Reliability Index,. These results may be used to enhane the urrent reommendations or resistane ators, φ. x

12 1. INTRODUCTION 1.1 General Bakground Fiber Reinored Polymer (FRP) omposite reinorement has been aepted in the onstrution industry as a promising substitute or onventional steel reinorement in the past deade. In the early 1990 s, the deteriorating state o the US inrastruture, partiularly highway bridges due to orrosion (almost 40% o the highway bridges in the US are struturally deiient or untionally no longer in use (ASCE Report ard 2005)), ored strutural engineers to ind alternative reinorement types. The use o FRP omposites as a replaement to steel reinorement has proved to be a promising solution to this problem. FRP omposites possess some outstanding properties suh as: resistane to orrosion, good atigue and damping resistane, high strength to weight ratio, and eletromagneti transpareny. FRP has ound an inreasing number o appliations in onstrution either as internal or as external reinorement or onrete strutures. It is well known that FRP possesses a major advantage over onventional steel in reinoring onrete strutures. Civil strutures made o steel reinored onrete are normally suseptible to environmental attaks that lead to the initiation o an eletrohemial proess whih leads to the orrosion o steel reinorement. Constant maintenane and repairing is needed to enhane the lie yle o those strutures. Bridge dek deterioration due to diret exposure to environment, deiing hemials and everinreasing trai loads is one o the most ommon deiienies in a bridge system. The use o FRP bars as an internal reinorement or onrete bridge deks and also girders provides a potential or inreased servie lie, eonomi, and environmental beneits. 1

13 As the name implies, FRP omposites are materials made o iber reinorements, resin, illers, and additives. The ibers exhibit high tensile strength and stiness and are the main load arrying element. The resin oers high ompressive strength and binds the ibers into a irm matrix. The additives help to improve the mehanial and physial properties as well as the workability o omposites. The most ommon types o ibers used in advaned omposites or strutural appliations are the glass (GFRP), aramid (AFRP), and arbon (CFRP). The GFRP is the least expensive but has lower strength and signiiantly lower stiness ompared to other alternatives. CFRP is the stiest, most durable, and most expensive one. AFRP has improved durability and exellent impat resistane. FRP reinorement is available in dierent orms suh as; bars, grids, prestressing tendons, and laminates to serve a wide range o purposes. This researh work ouses on using FRP bars (Fig. 1.1) as an internal reinoring material or onrete members (FRP-RC). In this present study, the data provided by FRP bar manuaturers is utilized to develop resistane models or FRP-RC strutures. These resistane models are then used to alibrate the design o FRP-RC strutures. Fig. 1.1: FRP bars 2

14 ACI ommittee 440 (ACI 440.1R-05) has developed a doument to assist engineers in designing FRP-RC strutures. Most o the researh onduted so ar on FRP bars as an internal reinorement to onrete has been done in a deterministi manner ignoring statistial variations assoiated with the design variables. Reliability-based tehniques are used to aount or the randomness in variables aeting the lexural strength o FRP-RC members. The development o strutural reliability methods during the last our deades has provided a more logial basis or developing strutural design odes. The overall aim o strutural reliability analysis is to quantiy the reliability o strutures under onsideration o the unertainties assoiated with the resistanes and loads. This researh work ouses on the reliability analysis o lexural behavior o FRP-RC members. Resistanes o strutures shall suiiently surpass the orresponding load eets. Resistanes and load eets are random variables ontaining some degree o unertainty. Thus saety is usually expressed in terms o reliability index,, obtained rom reliability analyses based on the theory o probability. In order to ondut reliability analyses, load and resistane models should be set up, and their statistial parameters suh as means and standard deviations are to be provided. This researh proposes a resistane obtained via Monte Carlo simulations or FRP-RC members. It alibrates the lexural design o FRP-RC members using a reliability-based tehnique. It also studies the inluene o various parameters on the Reliability Index,. 3

15 1.2 Objetives Objetives o this projet are: 1. Calibrating the lexural design o FRP-RC members using a reliability-based tehnique that aounts or the randomness in important design variables. 2. Analyzing the inluene o dierent variables inluded in the lexural design o FRP-RC members on the Reliability Index,. 1.3 Researh Plan Projet objetives are ahieved through the ollowing methodology: 1. Colleting statistial inormation on ommerially available FRP bars and other variables inluening the design proess. 2. Creating a pool o FRP-RC slab/dek and beam/girder designs that over a wide range o design parameters. The pool is omposed o a number o members (slabs and girders) with dierent FRP reinorement ratios, thiknesses, widths, and onrete strengths. 3. Perorming Monte-Carlo simulations on eah o the designed member (slabs and beams) and using the resulting randomly generated data set to develop a Resistane Model or the lexural apaity o FRP-RC members. 4. Establishing a live load model or use in alibrating the lexural design o FRP-RC members. 5. Determining the probability o ailure o the designed setions and the reliability index,, by First Order Reliability Method (FORM) onsidering the Load Model or buildings. 4

16 6. Study the parameters involved in the design o FRP-RC members whih aet the Reliability Index,, and thus the Probability o Failure. 1.4 Sope o Study This researh ouses exlusively on lexural behavior o onrete members (beams and slabs) and assumes that other modes o ailure suh as shear ailure and bond ailure do not ontrol design. 1.5 Organization In this hapter, an introdution to FRP bars and strutural reliability is given. Also the overview o the projet and this report is given. The literature on introduing omposite materials to the onstrution industry and bond between FRP reinorement and onrete is reviewed in Chapter 2. Also the lexural behavior and shear behavior o FRP-RC strutures is reviewed. The design approahes being used or FRP-RC lexural members are studied. Finally, the experimental studies on FRP bars used as the main reinorement are also reviewed in this hapter. The lexural behavior o FRP-RC members is analyzed analytially in Chapter 3 using three methods namely: 1. Simpliied Expressions Reommended by ACI (ACI 440.1R-05), 2. Detailed Closed-Form Expressions, and 3. A Numerial Method based on a Layered-Setion Analysis. The results obtained using these three methods are ompared with the available experimental results with the goal o inding the most suitable analysis method or urther studies in this hapter. 5

17 The development o a Resistane Model or FRP-RC lexural member is desribed in Chapter 4. The inormation about statistial harateristis o variables inluening the lexural design o FRP-RC member and sample design spae is given in this hapter. The Monte-Carlo simulation tehnique used to develop these resistane models is also desribed. The Load Models or buildings are disussed in Chapter 5. The reliability o FRP-RC members is studied and the results obtained or dierent ratios o live load and dead load are given. The ontribution o various parameters involved in the lexural design to the Reliability Index, is also studied in this hapter. The researh wok with onlusions and reommendations or uture researh is summarized in Chapter 6. 6

18 2. LITERATURE REVIEW 2.1 Introdution During the last deade, Fiber Reinored Polymer (FRP) omposites have been inreasingly aepted in the onstrution industry as a promising substitute or steel. Civil strutures made o steel reinored onrete are normally suseptible to environmental attaks that lead to eletrohemial orrosion o steel reinorement, whih leads to deiient strutures, and in some ases ailures. Constant maintenane and repair is needed to enhane the design lie o those strutures. Hene as reently as 1990s, an outstanding ombination o the properties o FRP omposites (orrosive resistane, good atigue and damping resistane, high strength to weight ratio, and eletromagneti transpareny) arose an interest in strutural engineers. Many theoretial and experimental studies have been arried out to hek the easibility o using FRP to reinore onrete strutures. 2.2 History o Fiber Reinored Polymer Composites The onept o omposite materials an be traed bak to the use o straw as reinorement in briks used by anient ivilizations (e.g. Israelites and Egyptians in 800 B. C.) (Tang 1997). Reently in the United States, short glass iber reinorement was used in the early 1930 s to reinore onrete. FRPs are the latest version o this very old idea o making better omposite material by ombining two dierent materials (Nanni 1999). Ater World War II, US manuaturers began produing iberglass and polyester resin omposites into vehile hulls and redomes (radar over). The automotive industry irst introdued omposites in early 1950 s and sine then many omponents o today s vehiles are being made out o omposites. The aerospae industry began to use FRP 7

19 omposites as lightweight material with aeptable strength and stiness whih redued the weight o airrat strutures suh as pressure vessels and ontainers. Today s modern jets use large omponents made out o omposites as they are less suseptible to atigue than traditional metals. Other industries like naval, deense and sporting goods have sine used advaned omposite materials on a widespread basis. In the onstrution industry, the irst appliation o omposites was a dome struture built in Benghazi, Libya in 1968 (Tang 1997), and other strutures ollowed. Thus, FRP omposites have emerged as an alternate reinoring material. In the early 1990 s, the deteriorating state o the US inrastruture, partiularly highway bridges due to orrosion (almost 40% o the highway bridges in the US are struturally deiient or untionally no longer in use (ASCE Report ard 2005)) ored strutural engineers to ind alternative reinorement. Parallel researh was also being onduted on FRPs in Europe and Japan. In Europe, onstrution o prestressed FRP Bridge in Germany in 1986 was the beginning o use o FRP (International Federation or Strutural Conrete). More than 100 ommerial projets involving FRP reinorement were undertaken in Japan (ACI Committee 440, 2001). Extended eorts on an international level brought design odes and guidelines or FRP reinored onrete into existene. FRP design provisions were inluded in Reommendation or Design and Constrution o Conrete Strutures Using Continuous Fiber Reinored Materials, Japan Soiety o Civil Engineers (JSCE) The Canadian Standards Assoiation (CSA) published two douments related to the use o FRP, namely, CAN/CSA-S806-02, Design and Constrution o Building Components with Fiber-Reinored Polymers, (CSA 2002), and CAN/CSA-S6_00 Canadian Highway Bridge Design Code (CSA 2000). In Europe, ivil engineers use FIP Task 8

20 Group 9.3 FRP Reinorement or Conrete Strutures (1999). The Amerian Conrete Institute (ACI) presented general design reommendations or lexural onrete elements reinored with FRP reinoring bars in ACI 440.1R-01 (2001), Guide or the Design and Constrution o Conrete Reinored with FRP Bars. ACI also published other douments addressing other omposite-related issues (e.g. Strengthening Using Composites (440.2R-02), Testing o Composite Materials (440.3R-04)). 2.3 Properties o Composite Materials As the name implies, iber reinored polymer omposites is a material made o iber reinorements, resin, illers, and additives. The ibers exhibit very high tensile strength and stiness and are the main load arrying element. The resin oers high ompressive strength and binds the ibers into a irm matrix. The additives help to improve the mehanial and physial properties as well as the workability o omposites. Fibers are seleted based on the strength, stiness, and durability required or the speii appliation. The environment, should the FRP be exposed, and the method by whih the FRP is being manuatured inluenes the hoie o resin. The most ommon types o ibers used in advaned omposites or strutural appliations are the glass (GFRP), aramid (AFRP), and arbon (CFRP) (ACI 440.1R-01). The GFRP is the least expensive but has lower strength and signiiantly lower stiness ompared to other alternatives. One o the problems with GFRP is durability whih means the degradation o FRP due to the environmental onditions in the surroundings o the FRP reinorement. CFRP is the most durable, sti, and expensive one. CFRP an withstand high sustained and atigue onditions. AFRP has improved durability and exellent impat resistane. However, they are the least ommon in the onstrution industry. Resins used in FRP materials are 9

21 either thermosetting or thermoplasti resins. Thermosetting resins are exlusively used in the onstrution industry. Epoxy and vinyl ester are the most ommonly used resins or their durability and adhesion properties (Nanni 1999). 2.4 Appliations Appliations o FRP in ivil engineering an be broadly lassiied into three ategories: appliations or new onstrution, repair and rehabilitation appliations, and arhitetural appliations. Strutural omponents suh as olumns and bridge deks, ompletely reinored by FRP omposites have shown exellent durability, and eetive resistane to the environmental eets. FRP is also ommonly used or the repair and rehabilitation o damaged or deteriorating strutures. To strengthen damaged onrete members, FRP is introdued to the system to beneit rom its superior tensile properties and hene improve their strutural integrity. Arhitets have also been using FRP or many appliations suh as siding or ladding, rooing, looring, and partition walls. 2.5 Forms o FRP Reinorement Conrete reinorement is available in dierent orms to serve a wide range o purposes. In general, FRP bars (Fig. 2.1), grids and prestressing tendons are used or new onstrution appliations. Fig. 2.1: FRP Bars Produed by Marshall Industries Composites, In Fig.: 2.2 SAFPLATE Fiberglass Gritted Plate Produed by Strongwell 10

22 Pre-ured laminates in the orm o plates (Fig. 2.2) and shells are used in onstrution or the repairing o damaged strutures. Sine FRP materials do not orrode, onrete members reinored with FRP are expeted to exhibit longer servie lie and improved durability. FRP in the orm o laminates (Fig. 2.3), externally bonded to reinored onrete is beoming more and more ommon or rehabilitation and strengthening o RC strutures, to solve problems either in servieability or at ultimate onditions (Ceroni 2004). Fig. 2.3: Strutural strengthening with FRP laminates Near Surae Mounted (NSM) FRP rods are also used or repairing and upgrading o reinored onrete strutures. This tehnique beomes partiularly attrative or lexural strengthening in the negative moment regions o slabs and girders where reinorement ould be subjeted to the severe damage due to mehanial and environmental onditions (Lorenzis and Nanni 2002). 11

23 2.5.1 FRP Bars FRP reinoring bars are similar to steel rebars in the shape and deormation patterns. They were not ommerially available until the late 1970 s. Commerially available FRP bars (Fig. 2.4) have various types o ross setional shapes i.e. square, round, solid, dog bone, and hollow. These bars have various deormation systems i.e., exterior wound iber, sand oatings, and separately ormed deormations. FRP bars an be manuatured using various tehniques suh as pultrusion, braiding, and weaving (ACI Committee 440, 2001). Fig. 2.4: Commerially available FRP bar manuatured by V-ROD Mehanial and Physial Properties o FRP Bars (ACI 440.1R-01) The mehanial properties o FRP bars are typially quite dierent rom those o steel bars. The properties are dependent on the iber type, but generally, FRP bars have lower weight, lower Young s Modulus but higher strength than steel. FRP bars have density ranges rom 77.8 to lb/t³, one sixth to one orth that o steel. The redued weight eases the handling o FRP bars on the projet site (ACI Committee 440, 2001). The longitudinal oeiient o thermal expansion is dominated by iber properties, while the transverse oeiient is dominated by the resin. Table 2.1 lists the oeiient o thermal expansion or typial FRP bars and steel. 12

24 Table 2.1: Coeiients o Thermal Expansion or Reinoring Bars CTE, X 10^-6/F ( X 10^-6/C) Diretion Steel GFRP CFRP AFRP Longitudinal 6.5 (11.7) 3.3 to to to -1.1 (6.0 to 10.0) (-9.0 to 0.0) (-6 to -2) Transverse 6.5 (11.7) 11.7 to to to 44.4 (21.0 to 23.0) (74.0 to 104.0) (60.0 to 80.0) The tensile properties o FRP are what make them an attrative alternative to steel reinorement. The tensile strength depends on the iber-volume ration, the rate o uring, and the manuaturing proess. The tensile properties o a partiular FRP bar should be obtained rom the bar manuaturer. When loaded in tension, FRP bars do not exhibit any plasti behavior (yielding) beore rupture. Table 2.2 gives the usual tensile properties o reinoring bars. Table 2.2: Usual Tensile Properties o Reinoring Bars Steel GFRP CFRP AFRP Tensile Strength 70 to to to to 368 ksi ( MPa ) (483 to 690) (483 to 1600) (600 to 3690) (1720 to 2540) Elasti Modulus to to to 18.2 * 10³ ksi ( GPa) (200) (35.0 to 51.0) (120.0 to 580.0) (41.0 to 125.0) The ompressive modulus o elastiity o FRP reinoring bars appears to be smaller than its tensile modulus o elastiity. FRP reinoring bars subjeted to a onstant load over time an suddenly ail. This phenomenon is known as reep rupture. In general, arbon ibers are the least 13

25 suseptible to reep rupture, whereas aramid ibers are moderately suseptible, and the glass ibers are the most suseptible (ACI Committee 440, 2001). 2.6 Review o Researh Ativities and Results Analytial Studies The use o FRP bars as tensile reinorement in onrete strutures has been a pressing issue sine the early 1990s. Researh ativities have been arried out to study the behavior o FRP reinored onrete strutures Composite Materials in the Constrution Industry Ballinger (1990) presented the advantages and disadvantages or reinoring onrete strutures using omposite materials. The advantages inlude high stati and atigue strength, resistane to hemials, and versatility in the abriation proess while the disadvantages inlude high ost o material, low modulus o elastiity and possible plasti deormation under long term loads o omposites in his study. He also gave methods o produing omposites and appliations o omposites. Tegola (1998) ompared the behavior o onrete beams or olumns reinored with non metalli bars to similar members reinored with steel. In this paper some possible statistial distributions o ations or the veriiation at the Ultimate Limit state, and the statistial distribution o the sampled values, to be orretly used or the Servie Limit State veriiations, are evaluated. In another study by Nanni (1999), omposite material properties, FRP orms suitable or onrete reinorement, appliations, installation proedure, and quality ontrol are disussed. The atual FRP system is installed by sandwihing the dry iber sheet between two layers o resin. Ater installing this system, hammer sounding and tap testing are the tehniques used to ind delaminations between the FRP and the substrate. 14

26 For some strengthening projets a inal quality ontrol step is perormed by implementing a load test. Field appliations or FRP omposite bars as reinorement or bridge deks are presented by El-Salakawy et al. (2002). Constrution details and some results o stati and dynami tests or a new girder type onrete bridge, at the Muniipality o Wotton (Quebe, Canada) are presented. They onluded that no obstales are enountered due to the use o the FRP bars and the deletions o the bridge dek are well below AASHTO allowable limits Bond between FRP Reinorement and Conrete A better understanding o the mehanial behavior o FRP reinorements in partiular bond behavior is needed in order to use them or pratial purposes. Numerous tests are analyzed to better understand bond mehanisms and the inluene o iber type, outer surae (shape and type o matrix), and other signiiant parameters on bond perormanes in Cosenza et al. (1997). Furthermore, the study was aimed at estimating the adequay o two analytial models or the onstitutive bond-slip relationship o FRP rebars; (1) the well known model by Malvar (1994), and (2) the model by Eligehausen et al. (1983). In another investigation by Esahani et al. (2005), the results o an experimental study on bond strength o GFRP bars embedded in normal onrete and sel onsolidating onrete are presented. Dierent parameters suh as type o onrete, bar situation in speimen, and over thikness are studied. Based on the experimental results, the author onluded that the splitting bond strength o GFRP reinoring bar was not less than that o steel bars, and also the bond strength o bottom GFRP reinoring bars was almost the same or normal onrete and sel onsolidating onrete. 15

27 Flexural Behavior Flexure has been a widely studied issue. Plevris and Triantaillou (1994) aimed at developing a undamental understanding o the time-dependant (reep and shrinkage) behavior o reinored onrete beams strengthened with FRP laminates. An experimental program is also desribed onirming the analytial analysis. El-Mihilmy and Tedeso (2000) investigated the lexural behavior o reinored onrete beams strengthened with externally bonded FRP laminates. A simple and diret analytial proedure or evaluating the ultimate lexural apaity o FRP strengthened onrete lexural members is given. Another investigation to ind the methods or prediting deletions and rak widths in beams reinored with GFRP bars is presented in Toutanji and Saai (2000). Deletions and rak widths obtained rom experimental results are ompared with those obtained analytially. An experimental and analytial study on behavior o Carbon Fiber-based rods as lexural reinorement was onduted by Thiagarajan (2003). The author onluded rom the pull-out tests that bonding o CF rods is not a major onern and bond quality inreases with bar size. Also CF rods have to undergo high strains to develop high stresses; hene, higher strength onrete is better suited or these beams. It was also onluded that the experimental rak widths are smaller than the rak widths predited using the ACI 440 rak width equation. The study by Yost et al. (2001) evaluates the lexural perormane o simply supported beams reinored with a 2D iber-reinored plasti grid onsidering the main parameter as the amount o longitudinal FRP. Experimental results are ompared with theoretial preditions alulated aording to traditional steel-reinored onrete proedures whih onlude that lexural apaity an 16

28 be aurately predited. In another investigation (Li and Wang 2002) introdued new Engineered Cementitious Composite (ECC) to replae brittle onrete matrix. Experimental results have shown that ECC beams exhibit signiiant inreases in lexural perormane in terms o dutility, load-arrying apaity, shear resistane, and damage tolerane ompared with the ounterpart high-strength onrete beam. Several researhers also studied the use o FRP to reinore masonry strutures. Unreinored masonry (URM) buildings are vulnerable to earthquakes. Bajpai and Duthinh (2003) oused their researh on strengthening and retroitting o existing masonry walls with near mounted, non-orrosive iber reinored polymer bars. The researh established the eetiveness o near surae mounted rods in strengthening onrete masonry walls and provides design guidelines or out-o-plane lexure and in-plane shear Shear Behavior Chajes et al. (1995) oused on the eets o externally applied omposite abris on the shear apaity o onrete beams. In their study, parameters suh as, strength and stiness o a variety o woven omposite abris along with their orientation are studied. The use o NSM, FRP rods is a promising tehnology or inreasing lexural as well as shear strength o deiient reinored onrete strutures. In a study by Lorenzis and Nanni (2002), tensile and bond tests are arried out on arbon FRP deormed rods or appliation as NSM reinorement to strengthen beams in shear and the results are ompared with the predition o a simple design approah. In a reent study by Whitehead and Ibell (2005), an analytial approah to investigate the shear response o FRP reinored and prestressed onrete has been developed based on equilibrium and ompatibility aross a shear disontinuity. Correlation between the analytial and 17

29 experimental results is better than preditions o urrent guideline provisions or prestressed onrete beams ontaining FRP reinorement Design Approahes Triantaillou and Antonopoulos (2000) desribed a simple design model or alulating the ontribution o FRP sheets to the shear apaity o strengthened RC elements. It is demonstrated that the ontribution o FRP sheets to shear apaity is typially ontrolled by the maximum eetive strain. Newhook et al. (2002) onduted a parametri study on retangular and T-setions to show that the design based on allowable strain in the FRP results in setions that exhibit large deormation beore ailure. Amy (2002) presented design methods or FRP strengthened onrete using sheets or NSM bars. Grae and Singh (2003) presented another design approah or bridge beams prestressed with arbon FRP bars. They examined the eet o the reinorement ratio and the level o prestressing ores on the deletions and ultimate load arrying apaity o a double-t beam speimen. ACI Committee 440 (2001) provides reommendations or the design and onstrution o FRP reinored onrete strutures. ACI 440.1R-01 gives general inormation on the history and use o FRP reinorement, material properties o FRP, and ommittee reommendations on the engineering o onrete reinored with FRP bars Experimental Studies There is a wide range o experimental researh pertaining to the behavior o FRP. This review will be limited to researh o lexural behavior o onrete members reinored with FRP bars. It should be noted that omparing experiments by dierent researhers on the lexural behavior o onrete beams reinored with FRP bars is not 18

30 always possible. This is due to the at that eah researh team designed its own program; dierent material systems and reinorement shapes and even the FRP prodution methods were dierent. Nonetheless, general onlusions ould still be drawn. In experimental studies by Brown and Bartholomew (1993) and Tegola (1998), results o pull-out tests o the FRP bars in onrete speimen and bending tests o onrete beams reinored with FRP bars are presented. In bending tests, tension raks are initially ormed in the enter portion o the span and as the load is inreased gradually, the raks widen progressively. In pull-out tests, load was applied at a onstant rate until the bond between the reinoring bar and the onrete is broken. Saadatmanesh (1994) onduted an experimental study on the behavior o onrete beams reinored with GFRP bars. A study enompassed reinorement o GFRP bars and stirrups to two retangular (205 * 450mm) and one T-beam with dierent reinorement ratios. Eah beam was simply supported on a lear span o 3.05 m and was subjeted to our-point lexural testing as shown in Fig Fig. 2.5: Test setup Beam A ailed by rushing o onrete at a load o 380 kn while beam B ailed by FRP rupture at a load o 135 kn. Due to the ully elasti behavior o GFRP bars, substantial deletion reovery was observed on load removal. The elasti behavior o bars resulted in no distint yield point on the load-deletion urve as it would be observed in ase o steel bars. The results o tests perormed on Beams A and B reinored with GFRP bars 19

31 indiated that this type o bar has good potential as tensile reinorement o onrete strutures. The behavior o onrete beams strengthened with GFRP plates is also demonstrated in the study, and it is onluded that the suess o this strengthening tehnique depends ritially on the perormane o the epoxy used. The eets o FRP reinorement ratio and onrete strength on lexural behavior o onrete beams were investigated by Theriault and Benmokrane (1998). A series o 12 onrete beams (130 mm wide, 180 mm high and 1800 mm long) reinored with ommerially available GFRP C-Bars were designed to ail by onrete rushing. The beams spanning 1500 mm were subjeted to our point lexural testing and were instrumented to monitor deletion, rak width, and strain. The results o the investigation are summarized as ollows. The eet o the onrete strength and the reinorement ratio on rak spaing is negligible. For the same applied moment, a higher reinorement ratio dereases the rak width and height. The stiness o C-BARreinored onrete beam is independent o the onrete strength but inreases with the reinorement ratio. The ultimate moment apaity o the tested beams inreases as the onrete strength and the reinorement ratio inrease, but this is limited by the onrete ompressive ailure strain o over reinored onrete beams. The experimental strain distributions, the raking pattern, the steady stiness, and deletion reovery even ater partial ailure o the beam learly demonstrate a good bond between C-BAR rods and the surrounding onrete. Aiello and Ombres (2000) tested nine onrete beams reinored with ommerially available AFRP bars in lexure to examine rak propagation, beam deletion, and strains. The beams were 150 mm wide, 200 mm high and 3100 mm long. 20

32 Three reinorement ratios were used in the analysis. The span length o 2610 mm was used to test the beams in our-point bending, whih were instrumented to monitor deletions at mid-span. For all tested beams, raks in the lexural span were predominantly vertial. Under high loads, inlined raks propagated towards the load points. A ompression ailure was observed or all tested beams. In this paper, theoretial models were used to alulate deletion and are based on moment-urvature relations. Finally, the servieability behavior o lexural FRP reinored onrete members by theoretial models is ompared with available experimental results to ous on both the eetiveness o theoretial models and their appliability or design purposes. The researh by Pee et al. (2000) proved experimentally that the Bernoulli hypothesis (plane setions remain plane) is valid or onrete beams reinored with FRP bars. The researh was based on the experimental testing o three simply supported onrete beams. The ross-setional dimensions were 500 mm wide, 185 mm high. The beams were tested using our-point loading or a span length o 3400 mm. The beams were reinored by GFRP bars and were designed to have a dominant lexural behavior reduing the shear inluene. Beams with less amount o FRP reinorement showed lower stiness. It is proved that the reinorement amount an be important, not only or inreasing the lexural apaity, but also or servieability purposes. Ferreira et al. (2001) reported a numerial model or the analysis o onrete shell strutures reinored with FRP rebars. Experiments on onrete beams reinored with FRP rebars are perormed to analyze the auray o the preditions obtained by the numerial model. Three point bending tests o onrete beams using dierent reinorement ratios were exeuted. The results obtained show the importane o the 21

33 rebar geometry on the strutural behavior. Dog-bone setions yielded higher ailure loads and delayed raking when ompared with irular setions. Koaoz et al. (2004) studied tensile haraterization o glass FRP bars. This paper reports tensile test results obtained on # 4 GFRP bars with dierent oatings whih might have an eet on the tensile strength o FRP bar. The test data obtained were analyzed using a Statistial Data Analysis (SAS) sotware program ater alulating Standard deviation and mean values or eah bar type. Ater onduting normality tests, the authors onluded that the Gaussian distribution represents the tensile strength o FRP bars. 2.7 Reliability Based Tehniques The development o strutural reliability methods during the last our deades have provided more logial basis or the design o strutures. The overall aim o strutural reliability analysis is to quantiy the reliability o strutures under onsideration o the unertainties assoiated with the resistanes and loads. Resistanes o strutures shall suiiently surpass the orresponding load eets. Resistanes and load eets are random variables ontaining some degree o unertainty. Thus saety is usually expressed in terms o reliability index obtained rom reliability analyses based on the theory o probability. In order to ondut reliability analyses, load and resistane models should be set up, and their statistial parameters suh as means and standard deviations are to be provided. The study by Faber and Sorensen (2002) presents undamental onepts o reliability based ode alibration. It also desribes a proedure or the pratial implementation o reliability based ode alibration o LRFD based design odes. 22

34 2.7.1 Reliability or Bridge Girders The lexural reliability o Reinored Conrete Bridge Girders strengthened with CFRP laminates was investigated by Okeil et al. (2002). Their study relied on non linear analysis model that aounts or material nonlinearities and the ondition at time o CFRP plaement is developed to perorm Monte Carlo simulations. This study showed that the reliability index o the strengthened ross setions is greater than that o the reinored onrete strutures and inreases with CFRP ratio due to the at that the manuaturing proess o omposite materials yields better statistial properties (bias and oeiient o variation) than steel. In another study, El-Tawil and Okeil (2002) used the same model or onduting bridge ross-setion design and analysis. In this study, the design o prestressed onrete bridge girders lexurally strengthened with arbon FRP laminates based on urrent provisions in AASHTO-LRFD is disussed. The reliability index o the designed bridges is alulated using the irst-order reliability method. Monte Carlo simulations are perormed and the resistane models are used to alibrate the lexural resistane ator to ahieve a preset target probability o ailure Reliability or Bridge Deks Bridge deks are one o the main strutural omponents that are most suitable or utilizing the advantages o FRP materials. Atadero et al. (2004) onsidered variation in material properties or FRP strengthened bridge deks in reliability analysis. Statistial analysis is onduted on the data sets o The Watson Wash Bridge (Caliornia) to assess the variation and the goodness-o-it using ommonly used distributions. This showed that lognormal distribution is the best desriptor or modulus, whereas the Weibull distribution is or the tensile strength o FRP. Based on these results, reliability analysis is 23

35 arried out in both the longitudinal and transverse diretions using Monte-Carlo simulations. Jeong et al. (2005) suggested that the target reliability index or FRP bridge deks should be at least 7.0, approximately equal to a saety ator o 5.0 based on the results obtained rom reliability analysis. He also reommended that the deletion limit on FRP bridge dek should be in the range o Span/600 to span/800. Based on this Literature Review, most o the researh onduted so ar on FRP bars as an internal reinorement to onrete has been done in a deterministi manner ignoring statistial variations assoiated with the design variables. In this present study, the oupon test data obtained rom FRP bar manuaturer and available experimental data in the literature are utilized to develop a Resistane Model or FRP-RC members that aount or variability in material properties, abriation and analysis method. Also Live Load Model or buildings is established to arry out the Reliability Analysis. 24

36 3. ANALYSIS METHODS AND EXPERIMENTAL VERIFICATION 3.1 Introdution In this hapter, the behavior o FRP-RC beams is analyzed theoretially. The nominal moment apaity o FRP-RC member is estimated using three analytial methods: 1. Simpliied Expressions Reommended by Guide or the Design and Constrution o Conrete Reinored with FRP Bars by ACI (ACI 440.1R-05) 2. Detailed Closed-Form Expressions 3. Numerial Method based on a Layered-Setion Analysis The results rom these analytial methods were ompared with available experimental results rom studies by Saadatmanesh (1994), Theriault and Benmokrane (1998), Aiello and Ombres (2000), and Pee et al. (2000) or the purpose o determining the most aurate and appropriate analysis model. 3.2 Analysis Methods Method 1: Simpliied Expressions Reommended by ACI 440.1R-05 The design o FRP-RC members or lexure is analogous to the design o steelreinored onrete members. The lexural apaity o onrete members reinored with FRP bars an be alulated based on assumptions similar to those made or members reinored with steel bars. Both onrete rushing and FRP rupture are aeptable ailure modes in governing the design o FRP-RC members provided that strength and servieability riteria are satisied. Assumptions in ACI Method are as ollows: 1. A plane setion beore loading remains plane ater loading 2. The maximum usable ompressive strain in the onrete is assumed to be

37 3. The tensile strength o the onrete is ignored. 4. The tensile behavior o the FRP reinorement is linearly elasti until ailure, and 5. Peret bond exists between onrete and FRP reinorement 6. A retangular stress blok is used or onrete in ompression. The strength design philosophy states that the design lexural apaity o a member must exeed the lexural demand (Eq. 3-1). φ M n M u (3-1) Design apaity reers to the nominal strength o the member multiplied by a strength redution ator, and the demand reers to the load eets alulated rom atored loads (or example, 1.2D+1.6L+ ----). The lexural apaity o an FRP reinored lexural member is dependent on whether the ailure is governed by onrete rushing or FRP rupture. Steps or inding the nominal lexural apaity o FRP-RC beam are as ollows: 1. Long term exposure to various types o environments an redue the tensile strength, reep rupture, and atigue endurane o FRP bars, hene, material properties used in design equations should be redued based on the type and level o environmental exposure. Table 3.1 gives the environmental redution ators. Table 3.1- Environmental Redution Fators or dierent FRP Bars Exposure Condition Fiber Type Environmental Redution Fator CE Carbon 1 Conrete not exposed to earth Glass 0.8 and weather Aramid 0.9 Carbon 0.9 Conrete exposed to earth and Glass 0.7 Weather Aramid

38 The design tensile strength should be determined by, * u = CE u (3-2) where, * u is guaranteed tensile strength o FRP bar, and u is the design tensile strength o FRP. 2. Failure mode an be determined by omparing the FRP reinorement ratio, ρ (Eq. 3-3) to the balaned reinorement ratio, ρ b, whih is the ratio where onrete rushing and FRP rupture ours simultaneously, Fig. 3.1) (Eq. 3-4). Fig Balaned ailure ondition or Method 1 A ρ = bd (3-3) where, A is FRP area, b and d are width and depth o onrete member. Similar to steel-reinored onrete beams, the balaned reinorement ratio is given by, ρ b = u 0.85 β 1 (3-4) u u u E ε E ε + where, is speiied ompressive strength o onrete, ε u is ultimate strain in onrete, E is guaranteed modulus o elastiity o FRP, and β 1 is an equivalent retangular stress blok parameter taken as 0.85 or 4 ksi onrete. 27

39 I ρ ρ b, FRP rupture ailure mode governs. Otherwise, or beams with ρ > ρb onrete rushing governs. ρ > ρ, the ailure o the member is initiated by rushing o the onrete, 3. For b and the stress distribution in the onrete an be approximated with the ACI retangular stress blok (Fig. 3.2). Based on the equilibrium o ores and strain ompatibility, the ollowing equations an be derived. Fig. 3.2: Conrete rushing ailure mode or Method 1 = ( Eεu) 0.85β ρ Eεu 0.5Eεu (3-5) M n = ρ 2 ρ bd (3-6) Where, is stress in the FRP reinorement in tension, and M n is nominal moment apaity o a member. ρ ρ, the ailure o the member is initiated by rupture o FRP bar, 4. For b and the ACI stress blok is not appliable beause the maximum onrete strain 28

40 (0.003) is not attained (Fig. 3.3). The analysis involving unknowns, onrete ompressive strain at ailure,, depth to the neutral axis,, and retangular stress blok parameters 1 and 1 beomes omplex. Fig. 3.3: FRP rupture ailure mode or Method 1 A simpliied and onservative alulation o the nominal lexural apaity o the member an be based on the ollowing Eqs. 3-7 and 3-8. M n = A u d β 1 b 2 (3-7) b = εu εu + ε u d (3-8) Where, b is distane rom extreme ompression iber to the neutral axis, and ε u is design rupture strain o FRP reinorement 5. The ACI guide reommends a onservative resistane ator, or lexural alulations. = 0.55 For ρ ρb = ρ For ρ b < ρ < 1. 4 ρ b ρ b = 0.65 For ρ 1. 4 ρb 29

41 6. The design lexural apaity = φ M n Method 2: Detailed Closed-Form Expressions The expressions reommended by ACI 440 involve many approximations. In this method, more aurate expressions are derived or a retangular setion. The design equations are derived on the basis o the ollowing onventional assumptions: 1. A plane setion beore loading remains plane ater loading 2. The tensile strength o the onrete is ignored. 3. The tensile behavior o the FRP reinorement is linearly elasti until ailure, and 4. Peret bond exists between onrete and FRP reinorement 5. Small deormations and no shear deormations 6. Stress-strain urve or onrete is approximated by a paraboli expression. The analytial method employs strain ompatibility, ore equilibrium. Failure o a lexural member may our by rushing o onrete or rupture o the FRP reinorement Compression Failure The member is said to be over-reinored when ailure is due to rushing o the onrete. Fig. 3.4 shows the strain and stress distribution at ultimate or an overreinored setion. The retangular representation o the ompressive stress blok an be used sine the ultimate strain in the onrete will be reahed. The nominal moment resistane o over-reinored setions is given by: M n = A u y (3-9) CT 30

42 where; yt β 1 = d (3-10) 2 Fig Conrete rushing ailure or Method 2 The stress in the reinorement at ailure, whih has a very smaller value than u, is given by Eq / 2 β 1 = 0.5Eε u1 + 4α 1 1 (3-11) ρeεu Failure by onrete rushing is onsidered to have ourred when the values o equivalent retangular stress blok parameters, 1 and 1, are taken equal to the amiliar values o 0.85 similar to ACI approah (Method 1) FRP Rupture Failure Figure 3.5 shows the stress-strain distributions in a setion at ailure by rupture o the FRP reinorement. Suh a setion is said to be under-reinored. When a setion is under-reinored, the tensile reinorement reahes its ultimate apaity. The strain in the FRP reinorement will be ε u = E u. The orresponding strain ε at the extreme ompressive iber will be less than ultimate ompressive strain, ε u. 31

43 Fig FRP Rupture ailure mode or Method 2 Thus, the analysis inorporates two unknowns, the onrete ompressive strain at ailure,, and the depth o the neutral axis,. The distribution o ompressive stress on the onrete annot be idealized by the traditional retangular blok (Fig. 3.4) and thereore a onrete stress-strain urve must be hosen and used to alulate the equivalent stress blok parameters 1 and 1. The paraboli stress-strain relationship or onrete proposed by Todeshini et al. (1964) and adjusted by MaGregor (1997) (Fig. 3.6) is assumed. Fig. 3.6 Stress-strain urve or onrete 32

44 ε 1. 8 ε o The equation or this relationship is; = 2 ε 1 + ε o (3-12) = E Where, ε o 1.71 (3-13) The mean stress ator, 1, onverts the atual stress-strain relationship o onrete into a retangular stress-strain equivalent (usually a value between ). This parameter is alulated by equating the integral o the area under the stress-strain urve (Fig. 3.6) upto the maximum ompressive strain o onrete, whih may be less than u. The integration is equated to the mean stress ator as ollows: ε d ε α 1 = ε (3-14) 0 By substituting Eq into Eq and solving or mean stress ator, results in: α 1 = 0. 9 ln 1 + ε ε o ε ε o 2 (3-15) The onrete ompressive ore (C) ats at the entroid o the ompressive zone, whih is at a distane o ( 1 /2) rom top extreme iber o beam. The entroid o area under the onrete stress-strain urve and the entroid o the area o the retangular stress blok are same. Taking the irst moment o area (M o ) yields: For the area under onrete stress-strain urve, Mo = (Area under the urve) * (strain at the entroid o the area under the urve) 33

45 34 ( )ε β ε ε 2 / = d M o (3-16) The irst moment o area o the retangular stress blok is, = ε ε ε 0 d M o (3-17) Equating Eqs and 3-17, and solving or the entroid ator, 1 β gives; + = = ln tan o o o o d d ε ε ε ε ε ε ε ε ε ε ε ε β ε ε (3-18) Ater alulating retangular stress blok parameters 1 and 1 β, and the depth o neutral axis, we an alulate the ompressive ore in onrete, C whih is equal to the tensile ore, T, in the FRP reinorement. u A E T ε = (3-19) And the moment resistane o the member an be ound by, = 2 1 d T Mn β (3-20) Steps or inding the nominal lexural apaity o a member: 1. Assume a depth or the neutral axis ( = 0.2d is a reasonable starting point) 2. Calulate the onrete ompressive strain at the top iber, = d εu ε (3-21)

46 3. Calulate the ratio o the depth o the equivalent retangular stress blok to the depth o the neutral axis, 1 using Eq Calulate the ratio o the average onrete stress to the onrete strength, 1 using Eq Calulate depth o neutral axis,, using the equivalent retangular stress blok parameters as; A u = (3-22) ( α 1 )( β 1b) Through iteration, the neutral axis depth is alulated and then veriied against the original assumption until onvergene is ahieved. 6. Knowing the depth o neutral axis, nominal lexural moment an be easily alulated using Eqs and Method 3: Layered Setion Analysis This omputer program, MACS Monotoni Analysis o Composite Setions was developed in the Visual Basi development environment, whih oers the ability to develop a riendly graphial user interae, GUI, to investigate steel-reinored onrete beams that are strengthened with externally-bonded FRP laminates (Okeil et al. 2002). It is apable o handling reinored and prestressed onrete beams o various shapes. The program has been modiied to handle FRP bars or this study. The program stores input data in iles with extension *.bdt and saves the analysis results in two output iles; namely PD.dat and MPhi.dat. The ormer ontains loadmaximum deletion data, while the latter stores the moment-urvature data. The input data inludes ross setion type, geometri and material properties o reinoring FRP bar 35

47 and onrete, span length o a beam and loading onigurations. During the analysis the load-deletion and moment-urvature relationships are plotted and updated as the analysis proeeds. Ater the analysis is ompleted, detailed plots o load-deletion and moment-urvature relationships are displayed. The program also shows the moment and urvature values at raking, FRP rupture, ultimate apaity o the ross setion, ailure type under positive and negative bending and the time spent during the analysis. Displaements, bending moments and shear ore diagrams are also plotted. Conrete Fiber i FRP Fiber i M ε i εi Atual Setion Disretized Setion Strain Distribution Fig. 3.7 Shemati o layered setion approah 3.3 Experimental Veriiation In this setion, results rom experimental studies by other investigators are used to veriy the analytial methods desribed in the previous setion. The veriiation also provides a quantiiation o the unertainty inherent in these methods, whih is o great importane or alibration studies. 36

48 3.3.1 Saadatmanesh (1994) Two retangular beams reinored with GFRP bars and stirrups were tested in this study. Both beams were o idential properties exept or dierent reinorement ratios. Table 3.2 lists the dimensions o both speimens. Eah beam was simply supported on a lear span o 3.05 m and was subjeted to our-point lexural testing as shown in the Fig Table 3.2-Cross setional properties o Veriiation Beams 1 and 2 Veriiation Speimen Width Eetive Depth Height FRP Area Beam b (mm) d (mm) h (mm) A (mm²) 1 A B Fig. 3.8 Flexural testing o retangular test beams (Saadatamanesh 1994) The measured ompressive strength o onrete used in the beams was MPa. The data on the mehanial properties o FRP re-bars were provided by the manuaturer. The mean tensile strength o FRP was 1179 MPa, with a standard deviation o 38.4 MPa. The mean and standard deviation or the modulus o elastiity or the GFRP bars were 54.2 and 8.82 GPa, respetively. Both beams A and B were loaded to ailure in inrements and ater eah load inrement, deletion, strain in the GFRP bar 37

49 and extreme ompression iber o onrete at midspan were measured and plotted against the load. Veriiation Beam 1 ailed by rushing o onrete at a load o 380 kn. Loaddeletion behavior o the beam was initially linear elasti until onrete raked at a load o about 37 kn. Fig. 3.8 shows the experimental and the MACS program urves o loaddeletion behavior. Curve obtained by MACS program orrelates well with the experimental urve in the initial stages o loading, up to 37 kn. As loading inreased beyond 37 kn, the experimental results indiated larger deletions than the results obtained rom MACS program Load, P (kn) Exprimental, beam A (Saadatmaneh 1994) MACS Program Deletion, (mm) Fig 3.9-Plot o load-deletion results or Veriiation Beam 1 (Saadatmanesh 1994) Veriiation Beam 2 ailed by FRP rupture at a load o 135 kn. The behavior o this beam was similar to Veriiation Beam 1. Ater onrete raked at a load o 18 kn, the deletions inreased until the beam ailed. Fig 3.9 presents the omparison between experimental load-deletion urve and the urve obtained by MACS program. 38

50 Load, P (kn) Experimental, beam B (Saadatmaneh 1994) MACS Program Deletion, (mm) Fig Plot o load-deletion results or Veriiation Beam 2 (Saadatmanesh 1994) Table 3.3 lists the results obtained rom dierent theoretial methods and experimental results. Table 3.3-Ultimate moment apaities o Veriiation Beams 1 and 2 Veriiation Failure Experimental Method 1 Method 2 Method 3 Beam Mode Mn, exp Mn, 1 Mn, 2 Mn, 3 (kn.m) (kn.m) (kn.m) (kn.m) 1 Crushing Rupture Theriault and Bemmokrane (1998) Eets o FRP reinorement ratio and onrete strength on lexural behavior o onrete beams were investigated by Theriault and Benmokrane (1998). A series o onrete beams reinored with GFRP C-Bars (manuatured by Marshall Industries 39

51 Composites, Lima, Ohio) were designed to ail by onrete rushing. The beams spanning 1500 mm were subjeted to our point lexural testing as shown in Fig and were instrumented with a linear variable dierential transormer (LVDT) at midspan to monitor deletion, rak width, and strain. Table 3.4 lists the dimensions o speimens. Fig Shemati o test Set-up Table 3.4- Cross setional properties o Veriiation Beams 3 and 4 Veriiation Speimen Width Eetive Depth Height FRP Area Beam b (mm) d (mm) h (mm) A (mm²) 3 BC2NA BC2NB All rebars used were 12.3 mm in diameter. The tensile strength and modulus o elastiity o the rebars are 773 MPa and 38 GPa, respetively. The ompressive strength o onrete was 53.1 MPa or the beams BC2NA and BC2NB. The load was applied inrementaly to the beam at a rate o 20 kn per inrement by means o one 200 kn hyadrauli jak and was measured with a load ell. At the end o eah step, a near midspan rak and midspan deletion is measured. Figure 3.11 shows a omparison between the moment-deletion results obtained experimentally and analytially using MACS or Veriiation Beam 3 (Theriault, Benmokrane 1998). 40

52 25 20 Experimental, beam BC2NA (Thriault, Benmokrane 1998) BMACS Applied Moment, M (kn-m) Deletion, (mm) Fig Plot o applied moment-deletion results or Veriiation Beam 3 (Theriault, Benmokrane 1998) Initially beams were unraked and sti. With urther loading, raking ourred at the midspan. The load-deletion response in the postraking stage was linear until the rushing o onrete. Finally the beam ailed at the applied moment o 21.9 kn-m. Veriiation Beam 3 also showed a similar behavior. The omparison between the experimental and MACS program results are shown in Fig

53 25 20 Applied Moment, M (kn-m) Deletion, (mm) Experimental, beam BC2NB (Theriault, benmokrane 1998) BMACS Fig 3.13-Plot o applied moment-deletion results or Veriiation Beam 4 (Theriault, Benmokrane 1998) Table 3.5 presents a omparison between ultimate moments obtained rom dierent theoretial methods and experimental study by Theriault and Benmokrane (1998). Table 3.5- Ultimate moment apaities o Veriiation Beams 3 and 4 Veriiation Failure Experimental Method 1 Method 2 Method 3 Beam Mode Mn, exp Mn, 1 Mn, 2 Mn, 3 (kn.m) (kn.m) (kn.m) (kn.m) 3 Crushing Crushing

54 3.3.3 Pee et al. (2000) The researh by Pee et al. (2000) proved that the Bernoulli hypothesis (plane setions remain plane) an be experimentally veriied. The researh was based on the experimental testing o three simply supported onrete beams F1, F2, and F3 reinored by GFRP bars where geometrial dimensions were designed to have a dominant lexural behavior reduing the shear inluene. The span lengths o beams F1 and F2 were equal to 3400 mm and were loaded by two equal ores at 1200 mm or support as shown in Fig and Fig Shemati o test Set-up o Veriiation Beam 5 Fig Shemati o test Set-up o Veriiation Beam 6 Table 3.6 lists the dimensions o both speimens. Table 3.6- Cross setional properties o Veriiation Beams 5 and 6 Veriiation Speimen Width Eetive Depth Height FRP Area Beam b (mm) d (mm) h (mm) A (mm²) 5 F F The average onrete ompressive strength is 30 MPa or both speimens. The nominal mehanial average harateristis are tensile strength o 770 MPa and longitudinal elastiity modulus o 42 GPa. 43

55 The tests were arried out using an eletro hydrauli atuator in displaement ontrol up to the ailure o the beams. The applied load was measured by a load ell and deletion at the midspan by a displaement indutive transduer. Figure 3.15 represents omparison between Moment-urvature relationship o midspan setions or veriiation beam 5 obtained experimentally and by MACS program Moment, M (knm) Experimental, beam F1 (Pee et al. 2000) BMACS Curvature, (10^-5/mm) Fig Plot o applied moment-urvature results or Veriiation Beam 5 (Pee et al. 2000) Veriiation Beam 5 ailed at kn.m by rushing o onrete, while the ailure mode or Veriiation beam 6 was FRP rupture at 36.9 kn-m. Figure

56 ompares the moment-urvature relationship or Veriiation Beam 6 obtained experimentally and by MACS program Moment, M (knm) Curvature, (10^-5/mm) Experimental, beam F2 (Pee et al. 2000) BMACS Fig Plot o applied moment-urvature results or Veriiation Beam 6 (Pee et al. 2000) The load-deletion relationship or Veriiation Beams 5 and 6 are given in Fig.3.17 and Fig.3.18, respetively. For Veriiation Beam 5, the load-deletion urve obtained by MACS orrelates well with the experimental urve. For Veriiation Beam 6, the analytial load-deletion relationship obtained rom MACS is stier than those reorded experimentally (see Fig. 3.18). 45

57 Fore, P (kn) Deletion, (mm) Experimental, beam F1 (Pee et al. 2000) BMACS Fig Plot o load-deletion results or Veriiation Beam 5 (Pee et al. 2000) Load, P (kn) Deletion, (mm) Experimental, beam F2 (Pee et al. 2000) BMACS Fig Plot o load-deletion results or Veriiation Beam 6 (Pee et al. 2000) 46

58 Flexural apaities o Veriiation Beams alulated by three methods are tabulated in Table 3.7 Table 3.7- Ultimate moment apaities o Veriiation Beams 5 and 6 Veriiation Failure Experimental Method 1 Method 2 Method 3 Beam Mode Mn, exp Mn, 1 Mn, 2 Mn, 3 (kn.m) (kn.m) (kn.m) (kn.m) 5 Crushing Rupture Aiello and Ombres (2000) Aiello and Ombres (2000) ast nine onrete beams reinored with AFRP rebars (manuatured by Sireg Co., Arore, Italy) or lexural tests to examine rak propagation, beam deletion, and strains. Three reinorement ratios were used in the analysis. The beam spanning 2610 mm was subjeted to our-point bending as shown in Fig and was instrumented with a transduer at midspan to monitor deletions. Fig Shemati o test Set-up o Veriiation Beams 7, 8, 9 Table 3.8 lists the ross-setional properties o speimens. Table 3.8-Cross setional properties o Veriiation Beams 7, 8, and 9 Veriiation Speimen Width Eetive Depth Height FRP Area Beam b (mm) d (mm) h (mm) A (mm²) 7 A B C

59 Average values o the tensile strength and modulus o elastiity o the rebars, determines by standard tensile tests, are 1506 and MPa, respetively. The average tensile strength o the onrete was 46.2 MPa. The load was applied gradually by means o a hydrauli jak and measured with a load ell. Crak ormation and beam deletion were observed and reorded at eah load step. Veriiation Beam 7 ailed by rushing o onrete at a load o kn. Fig shows omparisons between results o the experimental analysis and results obtained rom MACS Load, P (kn) Experimental (Aeillo et al. 2000) BMACS Deletion, (mm) Fig 3.21-Plot o load-deletion results or Veriiation Beam 7 (Aeillo et al. 2000) It should be noted that the reerene did not report deletion results post what is shown in the Fig as the ous o the study was to investigate raking. The deletion at 48

60 ailure was reported and is shown in the igure Load, P (kn) Expeimental (Aeillo et al. 2000) BMACS Deletion, (mm) Fig 3.22-Plot o load-deletion results or Veriiation Beam 8 (Aeillo et al. 2000) Load, P (kn) Deletion, (mm) Experimental (Aeillo et al. 2000) BMACS Fig 3.23-Plot o load-deletion results or Veriiation Beam 9 (Aeillo et al. 2000) 49

61 Analytial results using MACS program are in good agreement with the experimental results. Figs and 3.22 present omparison between experimental loaddeletion urve and urve obtained by MACS or Veriiation Beams 8 and 9, respetively. Table 3.9 show omparisons between results o the experimental analysis and those urnished rom theoretial analysis. Table 3.9- Ultimate moment apaities o Veriiation Beams 7, 8, and 9 Veriiation Failure Experimental Method 1 Method 2 Method 3 Beam Mode Mn, exp Mn, 1 Mn, 2 Mn, 3 (kn.m) (kn.m) (kn.m) (kn.m) 7 Crushing Crushing Crushing Conlusion In this hapter theoretial results obtained using three analytial methods are ompared to available experimental results with the goal o identiying their ability to predit the lexural apaity o FRP-R lexural members. Ater alulating the ultimate moment apaities by Method 1, Method 2, Method 3, olumn b,, and d respetively o Table 3.10, ratios o these moments with available experimental apaities is determined. The mean or all ratios is determined or three methods. For Method 3, the mean o ratios o Mn, 3 and Mn, exp is , or Method 1 and Method 2, the means are and , respetively. The means show learly that results obtained by MACS omputer program are in better agreement with experimental results than other two methods. 50

62 Table 3.10 gives the results obtained by three methods onsidered. Table Ultimate moment apaities Veriiation Ultimate moment apaities (kn.m) Beams a b b/a /a d d/a Experimental Method 1 Method 2 Method 3 Mn,exp Mn,1 Mn,2 Mn, Mean Thereore, the Layered Setion Analysis method will be used in this study to develop lexural resistane models or FRP-RC onrete members. 51

63 4. DEVELOPMENT OF RESISTANCE MODEL 4.1 Introdution In this hapter, the resistane model is developed or FRP-RC slabs and girders. The resistane o a member is typially a untion o material strength, setion geometry, and dimensions. These quantities are oten onsidered to be deterministi, while in reality there is some unertainty assoiated with eah quantity. Aounting or suh unertainties is ahieved in three steps. First, the important variables, whih aet the lexural strength o FRP-RC members, are identiied. Seond, statistial desriptors (mean, standard deviation, and distribution type) or all variables are ound. A sample design spae is then reated onsidering dierent FRP reinorement ratios, thiknesses, widths, and onrete strengths. Finally, Monte-Carlo simulations and omparison with experimental results are arried out to develop a resistane model that aount or variability in material properties, abriation and analysis method. 4.2 Variables Aeting the Flexural Strength o FRP-RC Members As onluded in the previous hapter, Method 3 (MACS omputer program) is hosen as the analysis method to alulate the nominal moment apaity based on the omparison with experimental results (Chapter 3). The input data required or Method 3 inludes ross setional properties, geometri and material properties o reinoring FRP bars, and onrete, whih are the parameters that aet the lexural strength o FRP-RC members. Among all these properties, the member width, b, its height, h, eetive depth, d, area o FRP reinorement, A, onrete ompressive strength,, modulus o elastiity or FRP, E, and FRP rupture strain, ε u are dealt with as the random variables that aet 52

64 the resistane o FRP-RC setions. For FRP-RC deks (slabs) b is treated as deterministi parameter equal to 304.8mm (1 t) sine a unit width is always used in the design. 4.3 Statistial Charateristis o Random Variables Basi Parameters Data desription using maximum and minimum values only is not suiient. Additional parameters are needed to aurately desribe the properties o the variable mathematially. 1. Mean: This is the most likely value o the observations. For a random variable, X, the mean value, µ X, is deined as n 1 µ = X X i (4-1) n i = 1 where, n is number o observations, and X i is the set o observations. 2. Standard deviation: Standard deviation, X, estimates the spread o data rom the mean and is alulated as n 2 2 Xi nµ x i= 1 σ = (4-2) X n 1 3. Coeiient o Variation: Coeiient o variation, V X is alulated as V X σ µ X = (4-3) X 4. Bias: Bias, is the ratio between the mean o the sample to the reported nominal value. µ X λ = (4-4) X X n where, X n is the nominal value o variable. 53

65 In addition to these parameters, knowledge o the probability distributions (4.3.3) is also neessary to desribe a variable Basi Funtions Any random variable is deined by its probability density untion (PDF), X (x) (Fig. 4.1) and umulative distribution untion (CDF), F X (x) (Fig. 4.2). The PDF (Eq. 4-5) is a mathematial expression that desribes a random variable, X, over possible values o the population. The probability o X alling between a and b is obtained by integrating the PDF over this interval: b P ( a < X < b) = ( x) dx (4-5) a X X Fig 4.1 PDF o X A probability density untion is everywhere non-negative and the summation o all probabilities over the entire design spae is equal to 1; i.e. 100% probability. The probability or X alling between and x is reerred to as CDF (Eq. 4-6). x P( < X < x) = ( x) dx F ( x) X = X (4-6) 54

66 The CDF desribes the probability that the set o all random variables takes on a value less than or equal to a number. It is lear rom Eqs. 4-5 and 4-6 that d X ( x) = FX ( x) (4-7) dx In general, PDFs are bell shaped while CDF untions are shaped like the letter S. Fig. 4.2 A graphial representation o the relationship between the PDF and the CDF Probability Distributions Used in this Study Any random variables an be speiied using dierent types o distributions. This setion gives the inormation about various distribution types used in this study. 55

67 Normal or Gaussian Distribution I a variable is Normally distributed then two quantities have to be speiied: the mean, µ, whih oinides with the peak o the PDF urve, and the standard distribution, X X, whih indiates the spread o the bell urve. The PDF or a normal random variable X is given by Eq X ( X ) = 1 1 X µ X exp σ 2 2 X π σ X 2 (4-8) Fig. 4.3 Graphial representation o Normal Distribution There is no losed-orm solution or the CDF o a Normal random variable but tables have been developed to provide values o the CDF or the speial ase in whih 56

68 µ X = 0 and σ X = 1(Nowak and Collins 2000). These tables an be used to obtain values or any general normal distribution Lognormal Distribution The random variable X is a lognormal random variable (Fig. 4.4) i Y = ln(x) is normally distributed. F ( x) F ( y) X = Y The mean and standard deviation or Y are given by Eqs. 4-9 and σ X σ Y = σ ln( X ) = ln + 1 (4-9) µ X 1 2 µ Y = µ ln( X ) = ln( µ X ) σ ln( X ) (4-10) 2 Probability Log-normal PDF X Log-normal CDF Probability Fig. 4.4 Graphial representation o Lognormal Distribution 57

69 Weibull Distribution (Extreme Event Type III) Two variations o the Weibull distribution exist, namely largest value and smallest value. In most ivil engineering appliations, the smallest value is used, where the PDF and CDF or Weibull random variable, X is given by Eqs and 4-12 respetively. X = mσ m o X m 1 X exp σ o m (4-11) F X m X = 1 exp (4-12) σ o Weibull PDF Probability Weibull CDF CDF X Fig. 4.5 Graphial representation o Weibull Distribution 58

70 The relationships between the two weibull parameters m andσ, and the mean and o oeiient o variation, 1+ m µ X = σ oγ σ o m µ X and V X are given by the ollowing Eqs and 4-14: (4-13) 2 + m Γ m 1. 2 V X = 1 (4-14) 2 1+ m m Γ m where, Γ [ ] is the gamma untion. The PDF and CDF or Weibull distribution having m andσ equal to 8 and 950 respetively are shown in Fig o Statistial Properties Geometri Properties (b, h, d, and A ) A review o the literature is perormed to identiy the statistial properties o random variables (Okeil et al. 2002, Nowak et al. 1994). The properties o b, h, d, and A are well studied in the literature. The bias and oeiient o variation or these our variables are ound to be in the range o 1.00 to 1.02 and 0.5 % to 7 % respetively. The bias o 1.0 and oeiient o variation o 3% is adopted in the present study or b, d, and h, while bias o 1.0 and oeiient o variation o 1.5% or A. Based on the survey, these our parameters are assumed to have a Normal distribution Conrete Compressive Strength Statistial properties o onrete are well doumented and have been reently updated (Nowak and Szerszen 2003) in the eort to alibrate the ACI design ode or RC buildings. In the urrent study, onrete statistial properties are adopted rom the values reported in Nowak and Szerszen (2003), whih relets the most reent statistial 59

71 properties or onrete mixes in use today. The bias and oeiient o variation or is and 8.2 % respetively. The random variable desribing the ompressive strength o onrete,, is assumed to be Lognormally distributed. Table 4.1 summarizes the statistial properties o these random variables used in this study. Table 4.1 Statistial Properties o b, h, d, A and Variable Bias COV (%) Distribution Width b 1 3 Normal Depth d, h 1 3 Normal Area o FRP A Normal Conrete strength Lognormal Tensile Strength o FRP Bars Data similar to those reported or other parameters is sare or FRP bars. The study reported by Koaoz et al. (2004) is limited to # 4 GFRP bar with our types o oatings and twelve speimens. This study suggests a mean o 1007 MPa and standard deviation o 47 MPa or the tensile strength o FRP. Thereore, it was neessary to develop suh inormation based on experimental data reported by FRP bar manuaturers. Coupon test results or FRP bars o various diameters were obtained rom an FRP bars manuaturer (Hughes Brothers). The results inluded the tensile strength, u, rupture strain, ε, and modulus o elastiity, E u. The distribution o oupon test results is graphially summarized by histograms that show the spread o the data. Histograms o tensile strength or all bar sizes are shown in Figs. 4.6 to

72 35 Sample = 68 Nominal Strength = 825 MPa Frequeny < >1180 Tensile strength, u ( MPa) Fig. 4.6 Histogram o tensile strength or #2 FRP bar 30 Sample = 108 Nominal Strength = 760 MPa 25 Frequeny < > 1098 Tensile Strength, u (MPa) Fig. 4.7 Histogram o tensile strength or #3 FRP bar 61

73 30 25 Sample = 71 Nominal Strength = 690 MPa Frequeny < > 1102 Tensile strength, u (MPa) Fig. 4.8 Histogram o tensile strength or #4 FRP bar 14 Sample = 73 Nominal Strength = 655 MPa Frequeny < > 725 Tensile strength, u (MPa) Fig. 4.9 Histogram o tensile strength or #5 FRP bar 62

74 20 Sample = 53 Nominal Strength = 620 MPa Frequeny < > 895 Tensile strength. u (MPa) Fig Histogram o tensile strength or #6 FRP bar 14 Sample = 41 Nominal Strength = 586 MPa Frequeny < > 725 Tensile strength, u (MPa) Fig Histogram o tensile strength or #7 FRP bar 63

75 6 Sample = 18 Nominal Strength = 517 MPa 5 4 Frequeny < > 617 Tensile strength, u (MPa) Fig Histogram o tensile strength or #9 FRP bar Sample = 66 Nominal Strength = 480 MPa 12 Frequeny < > 602 Tensile strength, u (MPa) Fig Histogram o tensile strength or #10 FRP bar Sine all three quantities ( u, ε u, and E ) are orrelated, a statistial analysis into the orrelation o the results is needed. The orrelation between the rupture 64

76 strain, ε u, and the tensile strength, u, is investigated and a orrelation oeiient, ρ, is obtained using Eq u ε u ρ u ε u = 1 n 1 n i = 1 u,i ε σ u,i u nµ σ ε u u µ ε u (4-15) where n is the number o samples in the data set or eah bar diameter, µ and µ u ε u are the mean values or the FRP tensile strength and FRP rupture strain or eah data set, respetively, and σ and u ε u σ are the standard deviation o the FRP tensile strength and FRP rupture strain or eah data set, respetively. A oeiient o orrelation was omputed or eah bar size, rom whih an average orrelation oeiient o is determined. This orrelation oeiient is used in generating the random variables or the Monte Carlo simulations disussed later in the study. Generating independent (unorrelated) u and ε u an lead to unintended low or high moduli o elastiity. Based on the guaranteed tensile strength (nominal) as reported by the manuaturer, the bias oeiient, λ, and the oeiient o variation, V u u, or eah bar diameter are omputed using the nominal values and the experimental mean values (Eqs. 4-6 and 4-7). The statistial distribution o the FRP bar material is also investigated. Several distribution types were tested to determine the most appropriate statistial representation o FRP bar material. A Chi-square statistial test revealed that the Weibull distribution (Extreme Event Type III) an adequately represent FRP bar material properties or a signiiane level o 5%. This inding onirms the indings o previous researh (Okeil 65

77 at al. 2000) or omposite laminates. Table 4.2 lists the summary o the statistial properties o the analyzed FRP bars. In addition to the bias and oeiient o variation, the table also lists the two parameters desribing the Weibull distribution, namely the shape parameter, m and the sale parameterσ. Table 4.2 Statistial Properties o FRP Bars Bar Tensile Strength Std Dev COV Size Mean Nominal Bias (Mpa) (%) m σ o MPa MPa Shape ator Sale ator # # # # # # # # # o 4.4 Sample Design Spae Developing the resistane models or FRP-RC deks and girders require investigating a wide range o realisti parameters in the design spae. In this study, suh deks and girders are designed ollowing the reommendations in the guidelines published by ACI Committee 440 (ACI 440.1R-05) Design Spae or Deks (Slabs) As mentioned earlier in this hapter, the width, b, is onsidered as deterministi parameter or deks beause o the way slab is designed. Four slab thiknesses, t, two onrete ompressive strengths,, and ive dierent reinorement ratios, R ρ / ρ ), are overed in the study. Overall 40 deks are designed or this study. The ( b 66

78 designation o eah o the designed deks indiates the parameters involved. For example, C4.0H4.0R0.8 reers to a slab having onrete ompressive strength, C, equal to 4 ksi (27.5 MPa), a thikness, H equal to 4 in (102mm) and an FRP reinorement ratio equal to 80 % o the balaned FRP reinorement ratio Design Spae or Girders (Beams) Two onrete ompressive strengths,, our dierent widths, b, our aspet ratios, A, and ive dierent reinorement ratios, R ( ρ / ρ ), are overed in the study. Overall 160 girders are designed or this study. The designation o eah o the designed girder indiates the parameters involved. For example, C4.0B10.0A1.0R0.8 reers to a beam having onrete ompressive strength, C, equal to 4 ksi (27.5 MPa), a width, B equal to 10 in (254 mm), an aspet ratio, A o 1.0 and an FRP reinorement ratio equal to 80% o the balaned reinorement ratio. It should be noted that the hoie o varying the reinorement ratio rom a value o 0.8 to 1.6 o the balaned reinorement ratio, ρ, is intended to over the range over whih the design resistane ator, φ, is reommended to hange aording to ACI s published guidelines (ACI 440.1R-05). 4.5 Resistane Models or Flexural Capaity o FRP-RC Members As the lexural apaity o an FRP-RC member is a untion o material and ross setional properties, whih are assoiated with some unertainties, the resistane, M R, is also a random variable. The possible soures o unertainty in resistane an be divided into three ategories: Material properties (M): the unertainties assoiated with material properties are unertainties in the strength o the material, the modulus o elastiity, raking b b 67

79 stresses et. Fabriation (F): these are the unertainties in the overall dimensions o the member whih an aet the ross-setional area, moment o inertia et. Analysis (P): the unertainty resulting rom approximate methods o analysis Eah o these unertainty soures has its own statistial properties; i.e. bias, oeiient o variation, and distribution type, whih means that the mean value o the resistane model an be expressed as: µ = (4-16) M M R nµ µ µ M F P where µ M, µ F, µ P are the mean values o M, F, and P, respetively and M n is the nominal apaity o member. Aordingly, the bias ator, λ M, and the oeiient o variation, V R M, desribing the R resistane model o M R M F P M R, are given as: λ = λ λ λ (4-17) V R = V + V + V (4-18) M 2 M 2 F 2 P where M, F, and P are the bias ators and V M, V F, and V P are the oeiients o variation o M, F, and P respetively Unertainties due to Analysis Method or All FRP-RC Members The auray o analysis method (Method 3) used in this study has been veriied by omparing lexural strengths obtained experimentally and reported in the literature to those obtained analytially (Chapter 3). The same data rom the veriiation is used to statistially desribe the unertainty o the analysis model by means o a bias, λ P, and a oeiient o variation, V P as shown in the Table 4.4 using Eqs and It should 68

80 be noted that the ratio values listed in the table are the inverse o those reported in Chapter 3 to onorm to the deinition o bias and oeiient o variation (Eqs. 4-6 and 4-7). M exp λ P = µ (4-19) M Anal V P M = COV M exp Anal (4-20) Based on the results in Table 4.4 and Eqs and 4-20, values o 1.12 and 15.65% are obtained or λ P andv P, respetively. Table 4.3 Determination o P and V P Reerene Speimen Flexural Capaity (kn.m) a b Experimental Method 3 a/b M R, Exp. M n, 3. A Aiello and Ombres (2000) B C Pee et al. (2000) F F Saadatmanesh (1994) A B Theriault and Benmokrane BC2NA (1998) BC2NB Mean, P 1.12 Std. Dev COV, V P 15.67% The eet o model unertainty, P, whih is same or all setions will be ombined with those due to material, M, and abriation, F, (4.5.2 and 4.5.3) using Eqs and 4-18 and λr and V R is alulated or all members. 69

81 4.5.2 Unertainties due to Material and Fabriation or FRP-RC Deks (Slabs) Monte Carlo simulations are perormed to determine λ M, λ F andv M, V F in the same exerise by varying randomly generated values or material properties and dimensions simultaneously. A ombined bias, λ MF, and oeiient o variation, V MF, resulted in rom these simulations. The mean and standard deviation (Eqs. 4-4 and 4-5) o the lexural apaities omputed by Method 3 (Chapter 3) or thousands o randomly generated data sets or eah slab design ase is obtained. The proedure or generating random numbers is explained in Appendix A. The lexural apaity o the same setion omputed using nominal material properties and dimensions is also obtained. The bias, λ MF, and oeiient o variation, V MF deks analyzed in this study. Table 4.3 lists the results o analyzed in this study. The bias, λ MF, is alulated using Eqs. 4-6 and 4-7 or all λ MF and V MF or all deks, and oeiient o variation, V MF are plotted against FRP reinorement ratio, ρ / ρ, as shown in Figs and 4.15, respetively. b A ommon trend an be seen or all ases (Figs and 4.15) and that is or underreinored deks; i.e. reinorement ratio o 0.8, the bias and the oeiient o variation is higher than or all other reinorement ratios (1.0, 1.2, 1.4, and 1.6). Failure o underreinored deks is governed by FRP rupture and hene the higher bias and oeiient o variation or the tensile strength o FRP ompared to onrete ompressive strength are releted in the higher bias and oeiient o variation o the overall setion. 70

82 Bias, MF FRP Reinorement Ratio, / b Fig Plot o Bias vs. FRP Reinorement Ratio or FRP-RC slabs 10 COV, VMF (%) FRP Reinorement Ratio, / b Fig Plot o COV vs. FRP Reinorement Ratio or FRP-RC slabs Investigation o the statistial distribution type o Resistane, M R is also required to deine the resistane model. All distribution types onsidered in this study are tested to determine the most appropriate statistial representation o the Resistane model. A Chisquare statistial test (Appendix B) revealed that the distribution type whih an adequately represent Resistane, M R, depends upon the reinorement ratio onsidered or the slab. 71

83 Table 4.4 Material Properties and Fabriation desriptors or FRP-RC Deks ( λ MF # Slab Flexural Capaity (kn.m.) Bias COV (%) Nominal Mean St Dev λ MF V MF 1 C4.0H4.0R C4.0H4.0R C4.0H4.0R C4.0H4.0R C4.0H4.0R C4.0H6.0R C4.0H6.0R C4.0H6.0R C4.0H6.0R C4.0H6.0R C4.0H8.0R C4.0H8.0R C4.0H8.0R C4.0H8.0R C4.0H8.0R C4.0H10.0R C4.0H10.0R C4.0H10.0R C4.0H10.0R C4.0H10.0R C5.0H4.0R C5.0H4.0R C5.0H4.0R C5.0H4.0R C5.0H4.0R C5.0H6.0R C5.0H6.0R C5.0H6.0R C5.0H6.0R C5.0H6.0R C5.0H8.0R C5.0H8.0R C5.0H8.0R C5.0H8.0R C5.0H8.0R C5.0H10.0R C5.0H10.0R C5.0H10.0R C5.0H10.0R C5.0H10.0R , V MF ) 72

84 For undereinored setions (R = 0.8 and 1.0), ailure is governed FRP rupture and hene Resistane, M R or all these ases an be represented by Weibull distribution. Overreeinored setions ail by onrete rushing and, hene, the major parameters that inluene the apaity o the ross setion are the statistial properties o onrete. Hene or these setions M R is well represented by Lognormal distribution. Chi- Square test results yielded that the M R o overreinored setions is also represented by Normal distribution. Thereore, Normal distribution is hosen or overreinored FRP-RC members as it is the basi and most important distribution in strutural reliability theory. Sample example on the perormane o Chi-Square test or FRP-RC slab, C4.0H8.0R1.6 is solved in Appendix B Parametri Study The plots o bias and oeiient o variation vs. FRP reinorement ratio, ρ / ρ b are drawn to study the inluene o the slab thikness, H, on bias and oeiient o variation, λ MF andv MF as shown in Figs to Bias, MF H = 4 H = 6 H = 8 H = FRP Reinorement Ratio, / b Fig Plot o Bias, λ MF vs. or C=4 ksi or FRP-RC slabs 73

85 COV, VMF (%) FRP Reinorement Ratio, / b H = 4 H = 6 H = 8 H = 10 Fig Plot o COV, V MF vs. or C=4 ksi or FRP-RC slabs Bias, MF H = 4 H = 6 H = 8 H = 10 FRP Reinorement Ratio, / b Fig Plot o Bias, λ MF vs. or C=5 ksi or FRP-RC slabs 9 8 COV, VMF (%) FRP Reinorement Ratio, / b H = 4 H = 6 H = 8 H = 10 Fig Plot o COV, V MF vs. or C=5 ksi or FRP-RC slabs 74

86 It an be learly onluded rom above plots that the thikness o member does not ontribute to the bias and oeiient o variation, λ MF andv MF Unertainties due to Material and Fabriation or FRP-RC Girders (Beams) The same method as used or deks is ollowed to quantiy the unertainties inherent in material and abriation o FRP-RC girders. Table 4.5 lists the results o λ MF and V MF or all girders analyzed in this study. The plots o bias, λ MF, and oeiient o variation, V MF verses FRP reinorement ratio, ρ / ρ, or all girders as shown in Fig and 4.21 respetively onirms the b trend o dereasing λ MF deks., and V MF with inreasing ρ / ρ b obtained in ase o FRP-RC 1.6 B i a s, M F FRP Reinorement Ratio, / b Fig Plot o Bias vs. FRP Reinorement Ratio or FRP-RC beams 75

87 Table 4.5 Material Properties and Fabriation desriptors or FRP-RC Girders ( λ MF # Girder Flexural Capaity (kn.m) COV (%) Bias Nominal Mean StDev V MF λ MF 1 C4.0B10.0A0.5R C4.0B10.0A0.5R C4.0B10.0A0.5R C4.0B10.0A0.5R C4.0B10.0A0.5R C4.0B10.0A0.67R C4.0B10.0A0.67R C4.0B10.0A0.67R C4.0B10.0A0.67R C4.0B10.0A0.67R C4.0B10.0A1.0R C4.0B10.0A1.0R C4.0B10.0A1.0R C4.0B10.0A1.0R C4.0B10.0A1.0R C4.0B10.0A2.0R C4.0B10.0A2.0R C4.0B10.0A2.0R C4.0B10.0A2.0R C4.0B10.0A2.0R C4.0B14.0A0.5R C4.0B14.0A0.5R C4.0B14.0A0.5R C4.0B14.0A0.5R C4.0B14.0A0.5R C4.0B14.0A0.67R C4.0B14.0A0.67R C4.0B14.0A0.67R C4.0B14.0A0.67R C4.0B14.0A0.67R C4.0B14.0A1.0R C4.0B14.0A1.0R C4.0B14.0A1.0R C4.0B14.0A1.0R C4.0B14.0A1.0R C4.0B14.0A2.0R C4.0B14.0A2.0R C4.0B14.0A2.0R C4.0B14.0A2.0R C4.0B14.0A2.0R , V MF ) 76

88 Table 4.5 (ontinued) # Girder Flexural Capaity (kn.m) COV (%) Bias Nominal Mean StDev V MF λ MF 41 C4.0B18.0A0.5R C4.0B18.0A0.5R C4.0B18.0A0.5R C4.0B18.0A0.5R C4.0B18.0A0.5R C4.0B18.0A0.67R C4.0B18.0A0.67R C4.0B18.0A0.67R C4.0B18.0A0.67R C4.0B18.0A0.67R C4.0B18.0A1.0R C4.0B18.0A1.0R C4.0B18.0A1.0R C4.0B18.0A1.0R C4.0B18.0A1.0R C4.0B18.0A2.0R C4.0B18.0A2.0R C4.0B18.0A2.0R C4.0B18.0A2.0R C4.0B18.0A2.0R C4.0B22.0A0.5R C4.0B22.0A0.5R C4.0B22.0A0.5R C4.0B22.0A0.5R C4.0B22.0A0.5R C4.0B22.0A0.67R C4.0B22.0A0.67R C4.0B22.0A0.67R C4.0B22.0A0.67R C4.0B22.0A0.67R C4.0B22.0A1.0R C4.0B22.0A1.0R C4.0B22.0A1.0R C4.0B22.0A1.0R C4.0B22.0A1.0R C4.0B22.0A2.0R C4.0B22.0A2.0R C4.0B22.0A2.0R C4.0B22.0A2.0R C4.0B22.0A2.0R

89 Table 4.5 (ontinued) # Girder Flexural Capaity (kn.m) COV (%) Bias Nominal Mean StDev V MF λ MF 81 C5.0B10.0A0.5R C5.0B10.0A0.5R C5.0B10.0A0.5R C5.0B10.0A0.5R C5.0B10.0A0.5R C5.0B10.0A0.67R C5.0B10.0A0.67R C5.0B10.0A0.67R C5.0B10.0A0.67R C5.0B10.0A0.67R C5.0B10.0A1.0R C5.0B10.0A1.0R C5.0B10.0A1.0R C5.0B10.0A1.0R C5.0B10.0A1.0R C5.0B10.0A2.0R C5.0B10.0A2.0R C5.0B10.0A2.0R C5.0B10.0A2.0R C5.0B10.0A2.0R C5.0B14.0A0.5R C5.0B14.0A0.5R C5.0B14.0A0.5R C5.0B14.0A0.5R C5.0B14.0A0.5R C5.0B14.0A0.67R C5.0B14.0A0.67R C5.0B14.0A0.67R C5.0B14.0A0.67R C5.0B14.0A0.67R C5.0B14.0A1.0R C5.0B14.0A1.0R C5.0B14.0A1.0R C5.0B14.0A1.0R C5.0B14.0A1.0R C5.0B14.0A2.0R C5.0B14.0A2.0R C5.0B14.0A2.0R C5.0B14.0A2.0R C5.0B14.0A2.0R

90 Table 4.5 (ontinued) # Girder Flexural Capaity (kn.m) COV (%) Bias Nominal Mean StDev V MF λ MF 121 C5.0B18.0A0.5R C5.0B18.0A0.5R C5.0B18.0A0.5R C5.0B18.0A0.5R C5.0B18.0A0.5R C5.0B18.0A0.67R C5.0B18.0A0.67R C5.0B18.0A0.67R C5.0B18.0A0.67R C5.0B18.0A0.67R C5.0B18.0A1.0R C5.0B18.0A1.0R C5.0B18.0A1.0R C5.0B18.0A1.0R C5.0B18.0A1.0R C5.0B18.0A2.0R C5.0B18.0A2.0R C5.0B18.0A2.0R C5.0B18.0A2.0R C5.0B18.0A2.0R C5.0B22.0A0.5R C5.0B22.0A0.5R C5.0B22.0A0.5R C5.0B22.0A0.5R C5.0B22.0A0.5R C5.0B22.0A0.67R C5.0B22.0A0.67R C5.0B22.0A0.67R C5.0B22.0A0.67R C5.0B22.0A0.67R C5.0B22.0A1.0R C5.0B22.0A1.0R C5.0B22.0A1.0R C5.0B22.0A1.0R C5.0B22.0A1.0R C5.0B22.0A2.0R C5.0B22.0A2.0R C5.0B22.0A2.0R C5.0B22.0A2.0R C5.0B22.0A2.0R

91 The statistial distribution o the resistane, M R is investigated by perorming Chi- Square test. Similar to the previous onlusion or FRP-RC deks, it was ound that the resistane o undereinored girders is well represented by Weibull distribution while the resistane o overreinored FRP-RC girders is by Normal distribution COV, VMF (%) FRP Reinorement Ratio, / b Fig Plot o COV vs. FRP Reinorement Ratio or FRP-RC beams Parametri Study The bias and oeiient o variation, λ MF and VMF vs. Aspet Ratio, A, and Width, b, are plotted or dierent FRP reinorement ratios and dierent onrete ompressive strengths. This is used to study whih parameters ontribute to the bias and oeiient o variation obtained through Monte-Carlo simulations. The plots are shown in Figs to bias, λ MF No ommon trend is ound in Fig ((a) to (j)) and 4.23 ((a) to (j)). The, and oeiient o variation, V MF, or FRP-RC beams does not vary muh with the width, B, and hene it is onluded that the parameter, B, does not aet the bias, λ MF, and oeiient o variation, V MF. 80

92 Bias, MF A=0.5 A=0.67 A=1 A= Beam Width, b (in) Bias, MF A=0.5 A=0.67 A=1 A= Beam Width, b (in) (a) R = 0.8 and = 4 ksi (b) R = 0.8 and = 5 ksi Bias, MF A= A= A=1 0.2 A= Bias, MF A=0.5 A=0.67 A=1 A=2 Beam Width, b (in) Beam Width, b (in) () R=1.0 and = 4 ksi (d) R=1.0 and =5 ksi Bias, MF A=0.5 A=0.67 A= Beam Width, b (in) Bias, MF A=2 0.0 A=0.5 A=0.67 A=1 A= Beam Width, b (in) (e) R=1.2 and =4 ksi () R=1.2 and =5 ksi Fig Plot o Bias, λ MF vs. Width, b or FRP-RC beams (ig. ontd.) 81

93 Bias, MF A=0.5 A=0.67 A=1 A= Beam Width, b (in) Bias, MF A=0.5 A=0.67 A=1 A= Beam Width, b (in) (g) R=1.4 and =4 ksi (h) R=1.4 and =5 ksi Bias, MF A= A= A= A= Beam Width, b (in) Bias, MF A=0.5 A=0.67 A=1 A= Beam Width, b (in) (i) R=1.6 and =4 ksi (j) R=1.6 and =5 ksi where, A is the Aspet Ratio, h / b and R is FRP reinorement Ratio, ρ / ρ. b COV, V MF (%) A=0.5 3 A=0.67 A=1 2 A= Beam Width, b (in) COV, V MF (%) A=0.5 A=0.67 A=1 A= Beam Width, b (in) (a) R=0.8 and =4 ksi (b) R=0.8 and =5 ksi Fig Plot o Coeiient o Variation, VMF vs. Width, b or Beams (ig. ontd.) 82

94 COV, V MF (%) A=0.5 A=0.67 A=1 A= Beam Width, b (in) COV, V MF (%) A=0.5 A=0.67 A=1 A= Beam Width, b (in) () R=1.0 and =4 ksi (d) R=1.0 and =5 ksi COV, VMF (%) A=0.5 4 A= A=1 2 A= Beam Width, b (in) COV, VMF (%) A=0.5 A=0.67 A=1 A= Beam Width, b (in) (e) R=1.2 and =4 ksi () R=1.2 and =5 ksi COV, V MF (%) A=0.5 4 A= A=1 2 A= Beam Width, b (in) COV, VMF (%) A=0.5 A=0.67 A=1 A= Beam Width, b (in) (g) R=1.4 and =4 ksi (h) R=1.4 and =5 ksi (Fig ontd.) 83

95 COV, V MF (%) A=0.5 A=0.67 A=1 A= Beam Width, b (in) COV, V MF (%) A=0.5 A=0.67 A=1 A= Beam Width, b (in) (i) R=1.6 and =4 ksi (j) R=1.6 and =5 ksi where, A is the Aspet Ratio, h / b and R is FRP reinorement Ratio, ρ / ρ. As onluded rom the parametri study on FRP-RC slabs, thikness o slab, H does not ontribute to the bias, λ MF the aspet ratio, A (h/b) on the bias, λ MF, and the oeiient o variation, V MF investigated in Figs ((a) to (j)) and 4.25 ((a) to (j))., and the oeiient o variation, V MF b. The eet o, is also Bias,MF b=10 in b=14 in b=18 in Aspet Ratio, h/b Bias,MF b=22 in 0.0 b=10 in b=14 in b=18 in b=22 in Aspet Ratio, h/b (a) R=0.8 and =4 ksi (b) R=0.8 and =5 ksi Fig Plot o Bias, λ MF vs. Aspet ratio or FRP-RC beams (ig. ontd.) 84

96 Bias, MF b=10 in b=10 in 0.4 b=14 in 0.4 b=14 in b=18 in b=18 in b=22 in b=22 in Aspet Ratio, h/b Bias, MF Aspet Ratio, h/b () R=1.0 and =4 ksi (d) R=1.0 and =5 ksi Bias, MF b=10 in b=14 in Bias, MF b=10 in b=14 in b=18 in 0.2 b=22 in b=18 in 0.2 b=22 in Aspet Ratio, h/b Aspet Ratio, h/b (e) R=1.2 and =4 ksi () R=1.2 and =5 ksi Bias, MF b=10 in b=10 in 0.4 b=14 in b=14 in b=18 in 0.2 b=18 in b=22 in b=22 in Aspet Ratio, h/b Bias,MF Aspet Ratio, h/b (g) R=1.4 and =4 ksi (h) R=1.4 and =5 ksi (Fig ontd.) 85

97 Bias, MF b=10 in b=14 in Bias, MF b=10 in b=14 in b=18 in 0.2 b=22 in b=18 in 0.2 b=22 in Aspet Ratio, h/b Aspet Ratio, h/b (i) R=1.6 and =4 ksi (j) R=1.6 and =5 ksi where, R is FRP Reinorement Ratio, ρ / ρ. b COV (MF) (%) b=10 in b=14 in b=18 in b=22 in COV (MF) (%) b=10 in b=14 in b=18 in b=22 in Aspet Ratio, h/b Aspet Ratio, h/b (a) R=0.8 and =4 ksi (b) R=0.8 and =5 ksi COV (MF) (%) Aspet Ratio, h/b b=10 in b=14 in b=18 in b=22 in COV (MF) (%) b=10 in b=14 in b=18 in b=22 in Aspet Ratio, h/b () R=1.0 and =4 ksi (d) R=1.0 and =5 ksi Fig Plot o Coeiient o Variation, VMF vs. Aspet ratio or beams (ig. ontd.) 86

98 COV (MF) (%) b=10 in b=14 in b=18 in b=22 in Aspet Ratio, h/b COV (MF) (%) b=10 in b=14 in b=18 in b=22 in Aspet Ratio, h/b (e) R=1.2 and =4 ksi () R=1.2 and =5 ksi COV (MF) (%) Aspet Ratio, h/b COV (MF) (%) b=10 in 4 b=10 in 3 b=14 in b=14 in 2 b=18 in b=18 in 1 b=22 in b=22 in Aspet Ratio, h/b (g) R=1.4 and =4 ksi (h) R=1.4 and =5 ksi COV(MF) (%) b=10 in b=14 in b=18 in Aspet Ratio, h/b COV (MF) (%) b=22 in 0 5 b=10 in b=14 in b=18 in b=22 in Aspet Ratio, h/b (i) R=1.6 and =4 ksi (j) R=1.6 and =5 ksi 87

99 Thus, it is onluded that the beam ross-setional properties suh as width, b, and aspet ratio, A, do not aet the bias, λ MF 4.6 Conlusion, and the oeiient o variation, V MF. A ommon trend an be seen or all FRP-RC deks and girders (Figs to 4.17) and that is or underreinored onrete members; i.e. reinorement ratio, ρ / ρ b o 0.8, the bias and oeiient o variation is higher than or all other reinorement ratios (1.0, 1.2, 1.4, and 1.6). The Resistane o FRP-RC deks and girders with reinorement ratio less than or equal to balaned reinorement ratio is well represented by Weibull (Extreme Event III) distribution, while the resistane o overeinored FRP-RC members is by Normal distribution. Based on parametri studies over a wide range o variables, it is onluded that the ross-setional properties o FRP-RC slabs and girders have no eet on the bias, λ MF, and oeiient o variation, V MF, obtained through Monte-Carlo simulations. Unertainties inherent in the analysis method or FRP-RC beams and girders are quantiied by λ P andv P equal to 1.12 and 15.65% respetively, based on a omparison o analytial results with experimental results reported in the literature. 88

100 5 RELIABILITY ANALYSIS AND RESULTS 5.1 Introdution The risk o ailure o any struture is oten measured in terms o the Probability o Failure, P. Alternatively; the probability o survival is measured in terms o the Reliability Index,. Both quantities are related as will be seen later in this Chapter. The method or alulating the Reliability index,, is disussed and then reliability analyses are onduted or all FRP-RC slabs and beams used in buildings based on the resistane models developed in Chapters 4 and load model reported in the literature. 5.2 Load Model or Buildings FRP-RC member dead load and live loads oten existing in buildings are the two load ategories onsidered in this study. These models have been developed by other researhers and are reported in the literature. Similar live load models or bridge deks are urrently laking and, hene, this reliability study will be limited to buildings. The dead load onsidered in design is the gravity load due to the sel weight o the struture. The Literature Survey arried out to ind the statistial properties o dead load has revealed that the dead load is typially treated as a normal random variable (Okeil et al. 2002, Nowak and Collins 2000). Beause o the ontrol over onstrution materials, it is assumed that our ability to estimate the total dead load, D, is more aurate than other load types. Most researhers (Okeil et al. 2002, Nowak and Collins 2000) assume a bias and oeiient o variation or dead load equal to 1.0 and 8 % to 10 % respetively. The bias, λ D, o 1.0 and oeiient o variation, V D, o 10 % is adopted in this study. The live load, L, represents the weight o people and their possessions, urniture, movable equipments, and other non permanent objets. Usually, live load is idealized as a 89

101 uniormly distributed load or buildings. The area under onsideration plays an important role in the statistial properties o live load. The magnitude o load intensity derease as the area ontributing to the live load inreases. This is the maximum load intensity o LL and not the average intensity onsidered or the design lie o 50 years. Based on the inluene area (Nowak and Collins 2000) and the Literature Review perormed (Okeil et al. 2002, Nowak et al. 1994), the bias ator, λ L, o 1.0 and oeiient o variation, o 18 % is assumed or this study. The Gumbel distribution (Extreme Event Type I) is assumed or the live load. Table 5.1 lists the biases, oeiient o variations and probability distributions adopted in this study. Table 5.1 Statistial Properties or Dead load and Live load Load Bias COV (%) Distribution Dead 1 10 Normal Live 1 18 Ext. Evt. I V L, 5.3 Reliability Analysis The saety o a strutural omponent depends on its resistane (supply) and load eets (demand), whih an be expressed in terms o a limit state untion; Z. The limit state untion an be as simple as the dierene between the random resistane o the member, R, and the random load eet ating on the member, Q. Z = R Q (5-1) I Z > 0, the struture is sae otherwise it ails. The term ailure does not neessarily mean the atastrophi ailure but is used to indiate that the struture does not meet the intended perormane. Thus, the reliability o a struture is equal the probability that a struture will not ail to perorm its intended untion. The probability o ailure, P, (Eq. 5-2) is equal to, 90

102 ( R Q < 0) = Prob( 0) P = Prob Z < (5-2) Sine R and Q are treated as random variables, the outome Z will also be a random variable. In general, the limit state untion an be a untion o many variables X, X, 1 2 Λ, X n, representing dimensions, material properties, loads and other ators suh as analysis method. Aordingly, Z beomes; ( X X ) Z = g,..., (5-3) 1, 2 X n A diret alulation o the probability o ailure may be very diiult or omplex limit state untions, and thereore, it is onvenient to measure strutural saety in terms o the Reliability Index,. The relationship between the reliability index, and the probability o ailure, P, is ( ) P = Φ β (5-4) where Φ ( ) is the CDF o the limit state untion. Table 5.2 gives a lue o how varies with P. Table 5.2 Relationship between Reliability Index, and Probability o Failure, P P β

103 5.3.1 Reliability Index, To establish a ormal deinition o, random variables will irst be normalized by transorming them into their standard orms whih is a nondimensional orm o a variable. For the simple limit state untion in Eq.5-1, the standard orms o the basi variables R and Q an be expressed as Z R R µ σ R = and R Z Q Q µ Q = (5-5) σ Q where, Z R and Z Q are alled redued variables and and σ Q are standard deviations or variables R and Q, respetively. The limit state untion g( R Q) = R Q as ollows; g ( Z R Z Q ) = ( µ R µ Q ) + Z Rσ R Z Qσ Q µ R and µ Q are the means, and σ R, an be expressed in terms o redued variables, (5-6) For any speii value o g ( Z R, Z Q ), Eq. 5-6 represents a straight line in the spae o redued variables Z and R Z. The line g( ) 0 Q Z R, Z Q = separates the sae and ailure zones in the spae o redued variables as shown in the Fig The reliability index,, is deined (Nowak and Collins 2000) as the shortest distane rom the origin o the redued variables to the line g ( ) 0 Z R, Z Q =. This is the deinition or a limit state untion onsisting o only two variables. The deinition an be generalized or n variables as ollows. For limit state untion g ( X X,..., ) 1, 2 X n, the reliability index is the shortest distane orm the origin in the n-dimensional spae o g Z. 1 Z Z n = redued variables to the n-dimensional surae desribed by (,,..., 2 ) 0 where, Z Z,..., Z X X,..., X 1, 2 n are the redued variables or the random variables 1, 2 n. 92

104 Several methods an be used to ompute the shortest distane in suh a general spae. The First Order Reliability Method is utilized in this study and is desribed next. Fig. 5.1 Reliability Index deined as the shortest distane in the spae o redued variables (Nowak and Collins 2000) First Order Reliability Method (FORM) FORM is based on a irst order Taylor Series expansion o the limit state untion, whih approximates the ailure surae by a tangent plane at the point o interest. It is not always possible to ind a losed orm solution or a non-linear limit state untion or a untion inluding more than two random variables. Hene, to onvert a non-linear limit state untion into simple polynomials, Taylor series (Eq. 5-7) is used. The expansion o a untion, ( X ) at a ertain point a is given by; ( X ) ( a) + ( X a) ( a) ( X a) 2 ( a) ( X a) n = (5-7) 2 n! n ( a) 93

105 FORM uses this expansion to simpliy the limit state untion, g ( Z Z,..., ) 1, 2 Z n by onsidering the expansion o the Taylor series ater trunating terms higher than the irst order. The expansion is done at the atual design point * X. The design point is a point on the ailure surae g ( Z Z,..., ) 5.2 or the ase o two variables in a non-linear limit state untion. 1, 2 Z n as shown in the Fig. Fig 5.2 Reliability Index evaluated at design point (Nowak and Collins 2000) 1 Z Z n = To loate this point on the design spae o (,,..., 2 ) 0 g Z, an iterative proess is needed (Nowak and Collins 2000). For the onvergene o a design point through iterative proedure requires solving o a set o (2n + 1) simultaneous equation * * * with (2n + 1) unknowns, β α, α, Κ, α, Z, Z, Λ Z where, 1 2 n 1 2, n g Z i evaluated at design po int α i = (5-8) 2 n g k = 1 Z k evaluated at design po int 94

106 g Z i g = X i X Z i i g = σ X i X i (5-9) n i= 1 ( ) 2 = 1 α (5-10) i Z * i = βα i (5-11) * * * ( Z, Z,, ) 0 g Λ (5-12) where, 1 2 Z n = α i is a unit vetor in the diretion o a design point rom the origin and * Z i is the design point in transormed spae. Equation 5-12 is a mathematial statement o the requirement that the design point must be on the ailure boundary. This proedure was derived with the assumption that the involved random variables are normally distributed. When the probability distributions or the variables involved in the limit state untion are not normally distributed, it is required to alulate the equivalent normal values o the mean and standard deviation or eah nonnormal random variable. To obtain the equivalent normal mean, µ, and standard deviation, σ, e X e X the CDF and PDF o the atual untion should be equal to normal CDF and normal PDF at the value o the variable * X on the ailure boundary desribed by = 0 g. Mathematially it an be expressed as F X * e X µ X ( X * ) = Φ (5-13) e σ X X * e * ( ) = 1 X µ X φ X e σ e X σ X (5-14) where, X is a random variable with mean µ X and standard deviation σ X and is desribed by a CDF (X ) F X and a PDF X ( X ). And (.) Φ is the CDF or the standard normal 95

107 distribution and φ (.) is the PDF or the standard normal distribution. Expressions or e and σ X an be obtained as ollows: e X * e = X σ X 1 * [ Φ ( F ( X )] µ (5-15) * ( X ) X 1 * [ Φ ( F ( X )] e 1 σ X = φ X (5-16) X The basi steps in the iteration proedure (Nowak and Collins 2000) to obtain are as ollows: 1. Formulate the limit state untion. Determine the probability distributions and appropriate parameters or all random variables X i ( i 1,2, Λ, n) = involved. 2. Obtain an initial design point { X * i } by assuming values or n-1 o the random variables. (Mean values are a reasonable hoie.) Solve the limit state equation g =0 or the remaining random variable whih ensures that the design point is on the ailure boundary. 3. Equivalent normal mean, e µ X and standard deviation, e µ X e σ X is determined using Eqs, 5-15 and 5-16 or design values orresponding to a nonnormal distribution. 4. Determine the redued variables { Z * i } orresponding to the design point { * i } X using Z * i = X * i σ µ e X i e X i (5-17) 5. Determine the partial derivatives o the limit state untion with respet to the redued variables.{ G } is a olumn vetor whose elements are the partial derivatives (Eq. 5-9) multiplied by

108 { G} G1 G2 = Μ G n where, G i g = (5-18) Z i evaluated at design point 6. Estimate o is then alulated using the ollowing ormula. T * { G} { Z } * β = where, { Z } T { G} { G} * Z 1 * Z 2 = Μ * Z n (5-19) 7. The diretion osines or the design point to be used in the subsequent iteration are then alulated using { } { G} T { G} { G} α = (5-20) 8. Determine a new design point or n-1 o the variables using Z = * i α i β (5-21) 9. Determine the orresponding design point values in original oordinates or the n-1 values in Step 7 by X * i = µ + Z σ (5-22) e X i * i e X i 10. Determine the value o the remaining random variable by solving the limit state untion g = Repeat Steps 3 to 10 until and { X * i } onverge. Appendix C demonstrates how this proedure is exeuted or one o the slabs (C4.0H4.0R1.2) in this study. 97

109 5.3.3 Reliability Analysis o FRP-RC Slabs and Beams n The LRFD design ode speiies a strength equation in the ollowing ormat. φ R γ Q (5-23) Qi i where, the nominal resistane o a strutural member, ator, φ, while the applied loads, R n, is oten redued by a resistane Q i, are inreased by the load ators, γ Q i. The values o φ and γ Q are set to ensure that members designed aording to this design equation i have a low probability o ailure that is less than a small target value. The basi loads inlude dead load and live load. To evaluate the Reliability Index or the designed FRP- RC slabs and beams, the limit state untion onsists o three random variables, lexural resistane, M R, applied bending moment due to dead load eet, M D, and applied bending moment due to live load eets, M L. ( M M, M ) = M ( M M ) g, + (5-24) R D L R D L Equation 5-24 represents the limit state untion used in this study. The statistial properties o M R are obtained rom the resistane models developed in Chapter 4, while the statistial properties o M D and M L or building loads are disussed earlier in this hapter. The load demands M D and M L are obtained by bak-alulating them rom the design equation in the urrent guidelines (ACI 440.1R-05). FatoredLoads Resistane Fator* Nominal Resistnae γ Q Qi = φ ACI M i n, ACI (5-25) By assuming a ertain LL to DL eet ratio, the load demand an be quantiied. For example, in the ase o equal LL and DL eet, the design equation; 98

110 γ = DM D + γ LM L φ ACI M n, ACI will result into ( γ D γ L ) = M L ( γ D + γ L ) = φ ACI M n ACI M D, + (5-26) or M D = M L φ ACI M = γ + γ D n, ACI L (5-27) Knowing γ D, γ L, φ ACI, and M n, ACI, M D and M L an be alulated using Eq In the urrent study, three LL to DL eet ratios are onsidered. ACI-318 Building Code suggests load ators γ D and γ L equal to 1.2 and 1.6 respetively or building loads. ACI Guidelines (ACI 440.1R-05) reommends a resistane ator, φ ACI reinorement ratio as given in the ollowing Table 5.3. Table 5.3 Resistane Fators, φ ACI, based on the FRP FRP R/F Ratio φ ACI ρ ρ b 0.55 ρ < ρ < 1. 4ρ b b ρ ρ b ρ ρ b where, ρ b is balaned FRP reinorement ratio and ρ is FRP reinorement ratio. The reliability index, is alulated or all FRP-RC slabs and beams ollowing the steps given in Sample alulations or or FRP-RC dek C4.0H4.0R1.2 are given in Appendix C. 99

111 5.4 Results Reliability Indies Obtained or FRP-RC Slabs Reliability indies, alulated or FRP-RC slabs or various LL to DL ratios are tabulated in Table 5.4. A ommon trend an be seen or all FRP-RC slabs and that is the reliability index,, dereases rom a reinorement ratio, ρ / ρ b, o 0.8 to the ases o balaned reinorement ratio, The Reliability Index,, then reahes a maximum value at reinorement ratio, ρ / ρ b, o 1.2 beore dropping at 1.4 b ρ and settling or the ases with higher FRP reinorement ratios: ρ =1.6 ρ. This indiates a non-uniorm b reliability outome i urrent guidelines (ACI 440.1R-05) are used or the design o FRP- RC deks. Furthermore, to avoid FRP rupture mode o ailure, a higher Reliability Index would be desired or FRP reinorement ratios equal to or below Parametri Study The eet o various parameters inluded in the study or FRP-RC slabs on the Reliability Index,, is studied or M / = 1. First parameter onsidered is onrete ompressive strength, ( b L M D. The plot o Reliability Index, vs. FRP Reinorement ratio, R ρ / ρ ) is plotted or all slab thiknesses as shown in Fig It is observed in Fig. 5.3 that the reliability Index,, is slightly higher or = 5 ksi FRP-RC slabs ompared to 4 ksi FRP-RC slabs or all thiknesses. The igure learly shows that the ompressive strength o onrete has an inluene on the reliability o FRP-RC slabs members. 100

112 Table 5.4 Reliability Indies or FRP-RC Slabs # Slab M L / M D = 0.5 M L / M D = 1 M L / M D = 2 1 C4.0H4.0R C4.0H4.0R C4.0H4.0R C4.0H4.0R C4.0H4.0R C4.0H6.0R C4.0H6.0R C4.0H6.0R C4.0H6.0R C4.0H6.0R C4.0H8.0R C4.0H8.0R C4.0H8.0R C4.0H8.0R C4.0H8.0R C4.0H10.0R C4.0H10.0R C4.0H10.0R C4.0H10.0R C4.0H10.0R C5.0H4.0R C5.0H4.0R C5.0H4.0R C5.0H4.0R C5.0H4.0R C5.0H6.0R C5.0H6.0R C5.0H6.0R C5.0H6.0R C5.0H6.0R C5.0H8.0R C5.0H8.0R C5.0H8.0R C5.0H8.0R C5.0H8.0R C5.0H10.0R C5.0H10.0R C5.0H10.0R C5.0H10.0R C5.0H10.0R

113 Reliability Index = 5 ksi = 4 ksi b Reliability Index = 5 ksi = 4 ksi b (a) h = 4 in (b) h = 6 in Reliability Index, = 5 ksi = 4 ksi b Reliability Index, = 5 ksi = 4 ksi b ( C) h = 8 in (d) h = 10 in Fig. 5.3 Plot o vs. ρ / ρ or dierent thiknesses or FRP-RC slabs b The reliability index, is plotted against FRP Reinorement Ratio or dierent thiknesses o slabs and dierent onrete ompressive strengths as shown in Fig As onluded earlier that the thikness, h o the FRP-RC slab does not inluene the bias and oeiient o variation, and thereore it an also be seen that h has no lear eet on (see Fig. 5.4). 102

114 Reliability Index, h = 4 h = h = h = b Reliability Index, 3.9 h = h = h = h = b = 4 ksi = 5 ksi Fig. 5.4 Plot o vs. ρ / ρ or dierent b or FRP-RC slabs Sine, the Reliability Index,, is not aeted by the thikness o slab, h, the average o all values is alulated or all slab thiknesses with dierent oneret ompressive strengths and dierent reinorement ratios. Table 5.5 shows the average reliability indies. To make sure that the reliability study is arried over a realisti range o loading ombinations, three LL to DL ratios, M / M = 0.5, 1.0, and 2.0 are onsidered. L D Table 5.5 Average Reliability Indies or FRP-RC Slabs FRP Reliability Index, ksi R/F Ratio M L / M D = 0. 5 M L / M D = 1 M L / M D =

115 Figures 5.5 and 5.6 show how the ratio M / M aets the Reliability Index,. L D R = 0.8 R = 1 R = 1.2 R = 1.4 R = M L / M D Fig. 5.5 Plot o vs. M / M or L D = 4 ksi or FRP-RC slabs R = 0.8 R = 1 R = 1.2 R = 1.4 R = M L / M D Fig. 5.6 Plot o vs. M / M or L D = 5 ksi or FRP-RC slabs 104

116 A ommon trend o dereasing with inreasing M / M ratio is observed. This L D is primarily due to the higher oeiient o variation, VL or live load ompared to dead load. Higher M / M ratios are unlikely to show a dierent trend and are unommon in L D buildings Reliability Indies Obtained or FRP-RC Beams Using the Resistane models developed in Chapter 4 or FRP-RC beams, reliability indies, are omputed ollowing the steps in or all beams designed in this study. As in the reliability study or FRP-RC beams, three M / M ratios were also L D onsidered or beams. The results are tabulated in Table 5.6. Similar to FRP-RC slabs (Table 5.3), a very ommon trend is observed or all FRP-RC beams. And the trend is rom reinorement level 0.8 ρ b to ρ b (balaned reinorement ratio), dereases, then it reahes maximum value at reinorement level o 1. 2 ρ b and again it starts dereasing rom reinorement ratio o 1.2 ρ b to 1.6 ρ b. This shows a non-onsistent reliability outome i urrent guidelines (ACI 440.1R-05) are used Parametri Study The eet o various parameters suh as, onrete ompressive strength,, aspet ratio, A (h / b), and beam width, b, on the Reliability Index, is studied to ind the ontributing parameter or the ase o M / = 1. Figure 5.7 shows the plots o vs. L M D FRP Reinorement level, R, ( ρ / ρ ) or dierent beam widths and dierent aspet ratios. b 105

117 Table 5.6 Reliability Indies or FRP-RC Beams # Beam M L / M D = 0.5 M L / M D = 1 M L / M D = 2 1 C4.0B10.0A0.5R C4.0B10.0A0.5R C4.0B10.0A0.5R C4.0B10.0A0.5R C4.0B10.0A0.5R C4.0B10.0A0.67R C4.0B10.0A0.67R C4.0B10.0A0.67R C4.0B10.0A0.67R C4.0B10.0A0.67R C4.0B10.0A1.0R C4.0B10.0A1.0R C4.0B10.0A1.0R C4.0B10.0A1.0R C4.0B10.0A1.0R C4.0B10.0A2.0R C4.0B10.0A2.0R C4.0B10.0A2.0R C4.0B10.0A2.0R C4.0B10.0A2.0R C4.0B14.0A0.5R C4.0B14.0A0.5R C4.0B14.0A0.5R C4.0B14.0A0.5R C4.0B14.0A0.5R C4.0B14.0A0.67R C4.0B14.0A0.67R C4.0B14.0A0.67R C4.0B14.0A0.67R C4.0B14.0A0.67R C4.0B14.0A1.0R C4.0B14.0A1.0R C4.0B14.0A1.0R C4.0B14.0A1.0R C4.0B14.0A1.0R C4.0B14.0A2.0R C4.0B14.0A2.0R C4.0B14.0A2.0R C4.0B14.0A2.0R C4.0B14.0A2.0R

118 Table 5.6 (ontd.) # Beam M L / M D = 0.5 M L / M D = 1 M L / M D = 2 41 C4.0B18.0A0.5R C4.0B18.0A0.5R C4.0B18.0A0.5R C4.0B18.0A0.5R C4.0B18.0A0.5R C4.0B18.0A0.67R C4.0B18.0A0.67R C4.0B18.0A0.67R C4.0B18.0A0.67R C4.0B18.0A0.67R C4.0B18.0A1.0R C4.0B18.0A1.0R C4.0B18.0A1.0R C4.0B18.0A1.0R C4.0B18.0A1.0R C4.0B18.0A2.0R C4.0B18.0A2.0R C4.0B18.0A2.0R C4.0B18.0A2.0R C4.0B18.0A2.0R C4.0B22.0A0.5R C4.0B22.0A0.5R C4.0B22.0A0.5R C4.0B22.0A0.5R C4.0B22.0A0.5R C4.0B22.0A0.67R C4.0B22.0A0.67R C4.0B22.0A0.67R C4.0B22.0A0.67R C4.0B22.0A0.67R C4.0B22.0A1.0R C4.0B22.0A1.0R C4.0B22.0A1.0R C4.0B22.0A1.0R C4.0B22.0A1.0R C4.0B22.0A2.0R C4.0B22.0A2.0R C4.0B22.0A2.0R C4.0B22.0A2.0R C4.0B22.0A2.0R

119 Table 5.6 (ontd.) # Slab M L / M D = 0.5 M L / M D = 1 M L / M D = 2 81 C5.0B10.0A0.5R C5.0B10.0A0.5R C5.0B10.0A0.5R C5.0B10.0A0.5R C5.0B10.0A0.5R C5.0B10.0A0.67R C5.0B10.0A0.67R C5.0B10.0A0.67R C5.0B10.0A0.67R C5.0B10.0A0.67R C5.0B10.0A1.0R C5.0B10.0A1.0R C5.0B10.0A1.0R C5.0B10.0A1.0R C5.0B10.0A1.0R C5.0B10.0A2.0R C5.0B10.0A2.0R C5.0B10.0A2.0R C5.0B10.0A2.0R C5.0B10.0A2.0R C5.0B14.0A0.5R C5.0B14.0A0.5R C5.0B14.0A0.5R C5.0B14.0A0.5R C5.0B14.0A0.5R C5.0B14.0A0.67R C5.0B14.0A0.67R C5.0B14.0A0.67R C5.0B14.0A0.67R C5.0B14.0A0.67R C5.0B14.0A1.0R C5.0B14.0A1.0R C5.0B14.0A1.0R C5.0B14.0A1.0R C5.0B14.0A1.0R C5.0B14.0A2.0R C5.0B14.0A2.0R C5.0B14.0A2.0R C5.0B14.0A2.0R C5.0B14.0A2.0R

120 Table 5.6 Contd. # Slab M L / M D = 0.5 M L / M D = 1 M L / M D = C5.0B18.0A0.5R C5.0B18.0A0.5R C5.0B18.0A0.5R C5.0B18.0A0.5R C5.0B18.0A0.5R C5.0B18.0A0.67R C5.0B18.0A0.67R C5.0B18.0A0.67R C5.0B18.0A0.67R C5.0B18.0A0.67R C5.0B18.0A1.0R C5.0B18.0A1.0R C5.0B18.0A1.0R C5.0B18.0A1.0R C5.0B18.0A1.0R C5.0B18.0A2.0R C5.0B18.0A2.0R C5.0B18.0A2.0R C5.0B18.0A2.0R C5.0B18.0A2.0R C5.0B22.0A0.5R C5.0B22.0A0.5R C5.0B22.0A0.5R C5.0B22.0A0.5R C5.0B22.0A0.5R C5.0B22.0A0.67R C5.0B22.0A0.67R C5.0B22.0A0.67R C5.0B22.0A0.67R C5.0B22.0A0.67R C5.0B22.0A1.0R C5.0B22.0A1.0R C5.0B22.0A1.0R C5.0B22.0A1.0R C5.0B22.0A1.0R C5.0B22.0A2.0R C5.0B22.0A2.0R C5.0B22.0A2.0R C5.0B22.0A2.0R C5.0B22.0A2.0R

121 = 5 ksi = 4 ksi R = 5 ksi = 4 ksi R (a) b = 10 in and A = 0.5 (b) b = 10 in and A = = 5 ksi = 5 ksi = 4 ksi 3.5 = 4 ksi R R () b = 10 in and A = 1.0 (d) b = 10 in and A = = 5 ksi = 5 ksi = 4 ksi = 4 ksi R R (e) b = 14 in and A = 0.5 () b = 14 in and A = 0.67 Fig. 5.7 Plots o vs. R ( ρ / ρ ) or dierent A and B or beams (ig. ontd.) b 110

122 = 5 ksi 3.7 = 5 ksi = 4 ksi = 4 ksi R R (g) b = 14 in and A = 1.0 (a) b = 14 in and A = = 5 ksi = 5 ksi = 4 ksi = 4 ksi R R (i) b = 18 in and A = 0.5 (j) b = 18 in and A = = 5 ksi = 5 ksi = 4 ksi 3.5 = 4 ksi R R (k) b = 18 in and A = 1.0 (l) b = 18 in and A = 2.0 (Fig. 5.7 ontd.) 111

123 = 5 ksi = 5 ksi = 4 ksi R = 4 ksi R (m) b = 22 in and A = 0.5 (n) b = 22 in and A = = 5 ksi 3.8 = 5 ksi = 4 ksi R = 4 ksi R (o) b = 22 in and A = 1.0 (p) b = 22 in and A = 2.0 These plots (5.7 (a) to 5.7 (p)) learly show that the Reliability index,, or FRP- RC beams having onrete ompressive strength o 4 ksi is lesser than that or 5 ksi beams. The ontribution by aspet ratio, A (h / b), and beam width, b, is shown in Fig

124 A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) (a) R = 0.8 and = 4 ksi (b) R = 0.8 and = 5 ksi A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) () R = 1.0 and = 4 ksi (d) R = 1.0 and = 5 ksi A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) (e) R = 1.2 and = 4 ksi () R = 1.2 and = 5 ksi Fig. 5.8 Plots o vs. B or dierent R ( ρ / ρ ) and b or FRP-RC beams (ig. ontd.) 113

125 A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) (g) R = 1.4 and = 4 ksi (h) R = 1.4 and = 5 ksi A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) A = 0.5 A = 0.67 A = 1 A = Beam Width, b (in) (i) R = 1.6 and = 4 ksi (j) R = 1.6 and = 5 ksi Figures 5.8 (a) to 5.8 (j) shown no partiular trends or aspet ratios and beam widths as seen in Fig. 5.7 or. Thereore, it an be onluded that the Aspet ratio, A, and Beam Width, b, are not one o the major ators that aet the Reliability Index,. Table 5.7 lists the average reliability indies or dierent onrete ompressive strengths, and dierent FRP reinorements. Figures 5.9 and 5.10 are the plots o vs. M / M. L D 114

126 Table 5.7 Average Reliability Indies or FRP-RC Beams FRP Reliability Index, ksi R/F Ratio M L / M D = 0. 5 M L / M D = 1 M L / M D = R = 0.8 R = 1 R = 1.2 R = 1.4 R = M L / M D Fig. 5.9 Plot o vs. M / M or L D = 4 ksi or FRP-RC beams A similar trend is ound or FRP-RC beams as ound in FRP-RC slabs. As the live load inreases, the Reliability Index,, dereases due to the higher oeiient o variation o live load. 115

127 R = 0.8 R = 1 R = 1.2 R = 1.4 R = M L / M D Fig Plot o vs. M / M or L D = 5 ksi or FRP-RC beams 5.5 Conlusions FRP-RC members designed using ACI guidelines (ACI 440.1R-05) do not show a onsistent reliability over a wide range o levels o FRP reinorement. The Reliability Index,, varies over the transition zone or the resistane ator with no lear trend o higher or lower values or eah ailure mode. Some ases showed values lower than what is urrently adopted or most o the ode ommittees. The ross-setional properties suh as, member width, member height (or beams) or thikness (or slabs) seem to not be a major ator that aets the Reliability Index,. The reliability study showed that FRP-RC members with higher onrete ompressive strength,., have better reliability than similar members onstruted using lower 116

128 The expeted trend o lower values or higher LL-to-DL ratios was onirmed by this study. However, the derease in values or high LL-to-DL ratio is not substantial over the pratial range o load ombinations or buildings. 117

129 6 CONCLUSIONS AND RECOMMENDATIONS 6.1 Summary The variabilities inherent in the materials, abriation, and analysis method are investigated to arry out a strutural reliability analysis o FRP-RC setions. Coupon test data or ommerially available FRP bars were obtained and analyzed to determine the appropriate statistial distributions and desriptors. Forty FRP-RC slabs (deks) and one hundred and sixty FRP-RC beams (girders) are designed to over a wide range o variables (onrete strengths, ross-setional dimensions and FRP reinorement ratios) and used to develop resistane models or FRP-RC slabs (deks) and beams (girders). Monte-Carlo simulations are perormed to determine the variabilities in material properties and abriation ombined. Experimental data reported in the literature is used to quantiy the variability inherent in the analysis method. A strutural reliability analysis is onduted based on the established resistane models and load model (buildings) obtained rom the literature. The Reliability Index,, is alulated using FORM or all FRP-RC slabs and beams or three ratios o live load moment and dead load moment. A detailed parametri study is arried out to study the parameters whih aet the Reliability Index,. In summary, this study resulted in the development o resistane models or FRP- RC bridge deks and girders whih an also be used as resistane models or FRP-RC slabs and beams respetively. Also, the lexural reliability study on FRP-RC slabs and beams yielded parameters that aet the Probability o Failure, P, in terms o the Reliability Index,. These results may be used to enhane the urrent reommendations or resistane ators, φ. 118

130 6.2 Conlusions 1. To estimate the nominal moment apaity o FRP-RC member, three analytial methods o analysis are studied in this researh. Based on a omparison o results obtained rom these methods and experimental results, it was onluded that the Layered Setion Analysis (MACS omputer program) method is in better agreement with experimental results than other two methods. 2. Statistial analysis o oupon test results or ommerially available FRP bars is perormed in order to ind the appropriate statistial distributions and desriptors or FRP properties. Chi-Square test showed that the tensile strength o FRP, u, an be eetively represented by Weibull Distribution (Extreme Event Type III). This result onirms the indings o previous researh (Okeil et al. 2000) or omposite laminates. 3. The unertainties assoiated with material properties and abriation proess represented by the bias, λ MF Monte-Carlo simulations., and the oeiient o variation, V MF, are ound using 4. The available experimental results on FRP bars are used to statistially desribe the unertainty o the analysis model represented as the bias, λ P, and oeiient o variation, V P, whih were ound to be 1.12, and 15.65% respetively. 5. The bias, λ MF, and oeiient o variation, V MF, obtained through Monte-Carlo simulations have shown a ommon trend or all FRP-RC slabs and beams. For underreinored setions; i.e. reinorement ratio, ρ / ρ, o 0.8, the bias and the oeiient o variation is higher than or all other reinorement ratios. In ase o underreinored FRP-RC members, as the ailure is governed by FRP rupture, b 119

131 statistial properties o FRP are ontrolling. Higher bias and oeiient o variation or u ompared to result in higher λ MF and V MF or FRP-RC underreinored deks. 6. A Chi-square statistial test (Appendix B) revealed that the statistial distribution type whih an adequately represent the nominal lexural resistane, M R, depends on the reinorement ratio onsidered or the FRP-RC member. For undereinored setions (R = 0.8 and 1.0), ailure is governed by FRP rupture whih is distributed as a Weibull distribution, and hene, M R, or all these ases an be represented by Weibull distribution. Overreeinored setions ail by onrete rushing, and hene these setions, M R, are well represented by Lognormal distribution. Aording to Chi-Square test results, M R, o these setions an also be eetively represented by Normal distribution whih is hosen to represent M R o overreinored setions. 7. From parametri studies it is disovered that the ross-setional properties o FRP- RC members do not ontribute to the bias, λ MF obtained through Monte-Carlo simulations., and oeiient o variation, V MF, 8. The Reliability index, alulated or FRP-RC members, whih revealed a ommon trend and that is the Reliability Index,, dereases rom reinorement ratio o 0.8 to balaned reinorement ratio (1.0) and then it reahes maximum value at reinorement ratio o 1. 2 and ater that again it dereases with inrease in FRP reinorement ratio rom 1.2 to 1.6. This indiates a non-uniorm risk level i urrent guidelines (ACI 440.1R-05) are used or the design o FRP-RC members. 9. A parametri study onduted on the Reliability Index,, showed that on one hand, the ross-setional parameters suh as member width, member thikness or height 120

132 have no eet on the Reliability Index,. On the other hand, onrete ompressive strength,, and the ratio o M / M ontribute to. For higher L D, higher values are expeted, while lower values are expeted or higher M / M ratios. L D 6.3 Reommendations 1. The part o unertainty due to the analysis method or resistane models or FRP- RC members an be enhaned by onsidering more experimental results as they beome available in the literature. In the present study, nine FRP beam experimental results are onsidered or the determination o λ P andv P. More experimental results would yield more realisti λ P andv P. 2. With the resistane models now being available through this study or FRP-RC deks, the reliability based alibration to be in onormity with AASHTO-LRFD (2004) an be onduted by developing load models or FRP-RC deks. Existing load models that were used in the alibration o AASHTO-LRFD were developed or the analysis o bridges in longitudinal diretion. Straining ations in bridge deks are aused by a single heaviest axle o the design truk. Thereore, load models should be developed in the transverse diretion based on a ertain oniguration o a group o axles. 3. Alternative ormulations or resistane ators an now be investigated based on the results o this study to yield a uniorm risk level over a wide range o variables. 4. This study ouses exlusively on the lexural behavior o FRP-RC beams and slabs and assumes that the other modes o ailure suh as shear ailure and bond ailure do not ontrol the design. Similar kind o researh an be onduted or other modes o ailure. 121

133 REFERENCES 1. ACI Committee 440, 2001, Guide or the design and onstrution o onrete reinored with FRP bars. ACI R-05, Amerian Conrete Institute, Farmington Hills, Mith. 2. Aiello M., and Ombres L., Load-Deletion Analysis o FRP Reinored Conrete Flexural Members, Journal o Composites or Constrution, Vol.4, No.4, November, Amy K., Design Methods or FRP-Strengthened Conrete using Sheets or Near- Surae Mounted Bars, University o Manitoba, Marh 28, Atadero R., Lee L., and Kambhari V., Materials Variability and Reliability o FRP Rehabilitation o Conerete, Proeedings o the 4 th International Conerene on Advaned Composite Materials in Bridges and Strutures, ACMBS IV, Canada, July Bajpai K., and Duthinh D., Bending Perormane o Masonry Walls Strengthened with Near-Surae Mounted FRP Bars, 9 th North Amerian Masonry Conerene, June 1-4, 2003, Clemson, USA 6. Ballinger C., Strutural FRP Composites, Civil Engineering ASCE, Vol. 60, No. 7, July Brown V., and Bartholomew C., FRP reinoring Bars in Reinored Conrete Members, ACI Materials Journal, V.90, No.1, January-February Chajes, M., Januszka, T., Mertz, D., Thomson, T., and Finh, W., "Shear Strengthening o Reinored Conrete Beams Using Externally Applied Composite Fabris," ACI Strutural Journal, V. 92, No. 3, Cosenza E., Manredi G., Realonzo R., Behavior and Modeling o Bond o FRP Rebars to Conrete, Journal o Composites or Constrution, Vol.1, No.2, May, El-Mihilmy M., and Tedeso J., Analysis o Reinored Conrete Beams Strengthened with FRP Laminates, Journal o Strutural Engineering, Vol. 126, No. 6, June EI-Salakawy E., and Benmokrane B., Design and Testing o a Highway Conrete Bridge Dek Reinored with Glass and Carbon FRP Bars, ACI Speial Publiation, Vol. 215, August,

134 12. El-Tawil S., and Okeil A., LRFD Flexural Provisions or Prestressed Conrete Bridge Girders Strengthened with CFRP Laminates, ACI Strutural Journal, V.99, No. 2, Marh-April Esahani R., Kianoush R., and Lahemi M., Bond strength o Glass Fiber Reinored Polymer Reinoring Bars in Normal and Sel-Consolidating Conrete, Canadian Journal o Civil Engineering, Vol. 32, No. 3, June Faber M., and Sorensen J., Reliability Code Calibration, Paper or the Joint Committee on Strutural Saety Drat, Marh Ferreira A., Camanho P., Marques A., and Fernandes A., Modelling o Conrete Beams Reinored with FRP Rebars, Composite Strutures, Vol. 53, Grae N. and Singh S., Design Approah or Carbon Fiber-Reinored Polymer Prestressed Conrete Bridge Beams, ACI Strutural Journal, V.100, No. 3, May- June, Jeong Sang-Mo, Kim Sang-Jin, Kim Young-Bum, Kim Hyeong-Yeol, and Park Ki- Tae, Reliability Analysis on Flexural Behavior o FRP Bridge Deks, 2003 ECI Conerene on Advaned Materials or Constrution o Bridges, Buildings, an Other Strutures III, Koaoz S., Samaranayake V., and Nanni A., Tesile Charaterization o Glass FRP Bars, Composites, Part B, Vol. 36, Li V., and Wang S., Flexural Behaviors o Glass Fiber-Reinored Polymer (GFRP) Reinored Engineered Cementitious Composite Beams, ACI Materials Journal, V.99, No.1, January-February Lorenzis L., and Nanni A., Bond between Near Surae Mounted FRP Rods and Conrete in Strutural Strengthening, ACI Strutural Journal, V.99, No. 2, Marh- April Nanni A., Flexural Behavior and Design o RC Members Using FRP Reinorement, Journal o Strutural Engineering, Vol. 119, No. 11, November Nanni, A., Composites: Coming on Strong, Conrete Constrution, Vol. 44, Newhook J., Ghali A., and Tadros G., Conrete Flexural Members Reinored with Fiber Reinored Polymer: Design or Craking and Deormability, Canadian Journal o Civil Engineering, 29: , Nowak A. S., and Collins K. R., Reliability o Strutures, MGraw-Hill

135 25. Nowak, A.S. and Szerszen, M.M. (2003). Calibration o Design Code, or Buildings (ACI318): Part 1- Statistial Models or Resistane, Strutural Journal, ACI, Vol. 100, No. 3, pp Okeil A., El-Tawil S, and Shahawy M., Flexural Reliability o Reinored Conrete Bridge Girders Strengthened with CFRP Laminates, Journal o Bridge Engineering, Vol.7, No. 5, September 1, Pee M., Manredi G., and Cosenza E., Experimental Response and Code Modelso GFRP RC Beams in Bending, Journal o Composites or Constrution, Vol.4, No.4, November Plevris N., and Triantaillou T., Time-Dependent Behavior o RC Members Strengthened With FRP Laminates, Journal o Strutural Engineering, Vol. 120, No. 3, Marh Saadatmaneh H., "Fiber Composites or New and existing Strutures, ACI Strutural Journal, V.91, No. 3, May-June Salakawy E., and Benmokrane B., Design and Testing o a Highway Conrete Bridge Dek Reinored with Glass and Carbon FRP Bars, ACI Speial Publiation, Vol. 215, August, Tang B., Fiber Reinored Polymer Composites Appliations in USA, First Korea/U.S.A. Road Workshop Proeedings, January 28-29, Tegola A., Ations or Veriiation o RC Strutures with FRP Bars, Journal o Composites or Constrution, Vol.2, No.3, August, Theriault M., and Benmokrane B., Eets o FRP Reinorement Ratio and Conrete Strength on Flexural Behavior o Conrete Beams, Journal o Composites or Constrution, Vol.2, No.1, February, Thiagarajan G., Experimental and Analytial Behavior o Carbon Fiber-Based Rods as Flexural Reinorement, Journal o Composites or Constrution, Vol. 7, No. 1, February Toutanji H., and Saai M., Flexural Behavior o Conrete Beams Reinored with Glass Fiber-Reinored Polymer (GFRP) Bars, ACI Strutural Journal, V.97, No. 5, September Triantaillou T., and Antonopoulos C., Design o Conrete Flexural Members Strengthened in Shear with FRP, Journal o Composites or Constrution, Vol. 4, No. 4,

136 37. Whitehead P., and Ibell T., Rational Approah to Shear Design in Fiber- Reinored Polymer-Prestressed Conrete Strutures, Journal o Composites or Constrution, Vol. 9, No. 1, February Yost J., Goodspeed C., and Shmekpeper E., Flexural Perormane o Conrete Beams Reinored with FRP Grids, Journal o Composites or Constrution, Vol. 5, No. 1, February

137 APPENDIX A GENERAL PROCEDURE FOR GENERATING RANDOM NUMBERS FROM ANY ARBITRARY DISTRIBUTION The Monte-Carlo simulation method is a speial tehnique to generate some results numerially without atually doing any physial testing. The probability distribution inormation an be eetively used to generate random numerial data. The basis o Monte-Carlo simulations is the generation o random numbers that are uniormly distributed between 0 and 1. The proedure given below is appliable to any type o distribution untion. Consider a random variable X with a CDF F X ( X ). To generate random values x i or the random variable, the ollowing steps should be ollowed. 1. Generate a sample value u i or a uniormly distributed random variable between 0 and Calulate a sample value x i rom the ollowing ormula: x i = F 1 x ( u ) i where, 1 F x is the inverse o F X. Knowing the CDF and basi parameters o the distribution, random numbers an be generated or a partiular variable. 126

138 APPENDIX B CHI-SQUARE STATISTICAL TEST: GOODNESS-OF-FIT TEST The Chi-Square test is oten used to assess the goodness-o-it between an obtained set o requenies in a random sample and what is expeted under a given statistial hypothesis. To be able to deide whih distribution is better or a partiular random variable, the dierene between atual observation values (observed requenies) and theoretial distribution values (theoretial requenies) is quantiied. The steps to be ollowed to determine the probability distribution o a random variable are given below. 1. Divide the observed data range into equal intervals. 2. Find the number o observations (Observed Frequeny, n i ) within eah interval whih do not depend on the distribution type. 3. Assume dierent distribution types that will represent the random variable and ind the theoretial distribution values (Theoretial Frequeny, e i ) within eah interval or the respetive distributions. I a random variable, X, lies in an interval a to b suh that a < X b, then the Theoretial Frequeny, e i, or a ertain distribution type is given by ( a < X b) N e i = P * where, N is the total number o observation (data points), and ( a < X b) = P( X b) P( X a) P < The probability o X less than a or b, P ( X < a) and ( X b) P is ound using the CDF or the respetive distribution. The CDF or dierent distribution types is given in Chapter 4 (4.3.3) 127

139 4. For eah interval, ompute the dierene between n i and e i (squared) as a ratio o e i. 5. Compute the summation o dierenes (squared) as a ratio o e i, m i= 1 ( n e ) i e i i 2 where, m is the total number o intervals. 6. Calulate the degree o reedom,, or Chi-Square test whih is given by = m 1 k where, k is the number o parameters required to desribe a partiular distribution. In this study, Normal, Lognormal, Weibull, and Gumbel distribution types are used and or all these types k = The summation evaluated in Step 5 is ompared to the Chi-Square distribution (reer to page 130 or a ertain signiiane level,α, whih is always taken between 1 % and 10 %. 8. I m i= 1 ( n e ) i enough. e i i 2 < C α,, then the assumed distribution is itting statistial data well Table or C α, values is given on page 132 Sample Example or FRP-RC slab, C4.0H8.0R1.6 Total ive thousand random data sets were reated or Monte-Carlo simulations. Thereore, in this ase, N = Total data set was divided in 14 intervals, m = 14. Sample Mean = kn.m Sample Standard Deviation = 19 kn.m 128

140 Observed requenies and theoretial requenies or Normal, Lognormal and Weibull distributions are shown in Table B. Table B Chi-Square Test or FRP-RC slab C4.0H8.0R1.6 ( ) ni ei Intervals n i e i ei M R M R Normal Lognormal Weibull Normal Lognormal Weibull Total = = 11 and α is assumed 5 % whih gives C α, = (reer to page 130) m i= 1 m i= 1 ( n e ) i e ( n e ) i e i i i i 2 2 C α or Normal distribution = <, = or Lognormal distribution = < C α, = Thereore, it is onluded that M R o FRP-RC slab C4.0H8.0R1.6 is well represented by Normal and Lognormal distribution. 129

141 Table B2 CDF o the Chi-Square Distribution 130