SIMPLIFIED PREDICTION EQUATION FOR ULTIMATE STRESS IN BEAMS PRESTRESSED WITH HYBRID TENDONS GONCA UNAL

Transcription

1 i SIMPLIFIED PREDICTION EQUATION FOR ULTIMATE STRESS IN BEAMS PRESTRESSED WITH HYBRID TENDONS by GONCA UNAL A thesis submitted to the Graduate School-New Brunswick Rutgers, The State University o New Jersey in partial ulillment o the requirements or the degree o Master o Science Graduate Program in Civil and Environmental Engineering written under the direction o Dr. Hani H. Nassi and approved by New Brunswick, New Jersey October 211 i

2 ii ABSTRACT OF THE THESIS SIMPLIFIED PREDICTION EQUATION FOR ULTIMATE STRESS IN BEAMS PRESTRESSED WITH HYBRID TENDONS By GONCA UNAL Thesis Director: Dr. Hani H. Nassi The use o unbonded tendons is getting more widespread in post-tensioning industry; especially in rehabilitation and strengthening o existing damaged concrete members. The prediction o the stress at ultimate in unbonded tendon is important in calculating the capacity o structural members. This thesis presents a simpliied prediction equation or concrete beams prestressed with hybrid (a combination o bonded and unbonded) tendons. The proposed equation is based on the Generalized Incremental Analysis (GIA) which uses the trussed-beam model developed by Ozkul et al. (28) and Nassi et al. (23). The main objective o this research is to develop a simple, but accurate design equation or the prediction o the stress at ultimate in unbonded tendon. Most important parameters such as loading type, eective prestress o unbonded tendon, concrete strength, area o steel reinorcement and span-to-depth ratio are taken into consideration. The equation is applicable to beams prestressed with unbonded or hybrid, FRP or steel, external or internal tendons. For the validation o the proposed simpliied equation, test ii

3 results available in the literature (199 beams) are collected. The results show that the proposed simpliied equation exhibited very good accuracy or the calculation o stress at ultimate in unbonded tendon. The simpliied equation is easy to use, accurate and applicable to any material type and combination o tendons. iii

4 ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Hani H. Nassi not only or his guidance, assistance and support in producing this thesis, but also or his patience to answer all o my endless questions related to this study. It was an honor to work with him on this topic. I would also like to thank Dr. Husam Najm and Dr. Kaan Ozbay or being in my thesis committee and or their helpul comments and advices. I would like to express my sincerest gratitude to my proessors rom Turkey, Dr. Erhan Karaesmen and Dr. Engin Karaesmen or their valuable guidance. I would like to thank Dan Su or always being helpul on my study. Special thanks to my oice mates Ender Faruk Morgul and Sami Demiroluk or their support and riendship during my study. Special thanks to my close riends Hulya Avci Ozbek and Deniz Velioglu or their support and riendship without considering long distance. My deepest appreciation goes to my dear amily. I would like to thank to my dear mother, or her ininite support, love and to my ather, or his valuable advices and guidance. Finally, I would like to thank to my lovely older sister Iclal Gokce Unal, my best riend at the same time, or her love, help and patience during my study. iv

5 TABLE OF CONTENTS ABSTRACT OF THE THESIS... ii ACKNOWLEDGEMENTS... iv TABLE OF CONTENTS... v LIST OF FIGURES... viii LIST OF TABLES... xviii LIST OF SYMBOLS... xix 1. INTRODUCTION Overview Problem Statement Research Signiicance Objective and Scope o Work Organization o the Thesis LITERATURE REVIEW Introduction Related Studies Studies on Prestressing with Unbonded Tendons Studies on Prestressing with Unbonded and Bonded (Hybrid) Tendons Code Equations SIMPLIFICATION OF PREDICTION EQUATION Introduction Approach I: Finite Element Analysis (FEA) Approach II: Generalized Incremental Analysis (GIA) v

6 3.4 Simpliication o Prediction Equation Quadratic Equation in Terms o Eccentricity Normalized Equation in Terms o Eccentricity Quadratic Equation in Terms o ( d c) Normalized Equation in Terms o ( d c) Simpliication o the Term Evaluation o Prediction Equations Recommended Prediction Equation at Ultimate PARAMETRIC STUDY Introduction Parametric Studies rom Literature Parametric Study RESULTS Introduction Comparison o Results or Prediction o Overall Response o Members Comparison o Results or Estimation o and Results or Proposed Design Equations Results or Equations rom Literature Results or Code Equations Summary o Results SUMMARY AND CONCLUSIONS Summary Conclusions vi

7 7. REFERENCES... vii

8 LIST OF FIGURES Figure 3.1: Trussed-Beam Model (Ozkul et al. 28) Figure 3.2: Summary o Methods o Analysis Figure 3-3: Graphical Solution or the Normalized Equation in terms o ( d c) Figure 3-4: 1 versus 1 L/ d Graph or Dierent Loading Types Figure 4-1: Eect o Span-to-Depth Ratio on (a) and (b) Figure 4-2: Eect o Concrete Compressive Strength on (a) and (b) Figure 4-3: Eect o Area o Non-prestressing Tension Reinorcement on (a) and (b) Figure 4-4: Eect o Yield Stress o Non-prestressing Tension Reinorcement on (a) and (b) Figure 4-5: Eect o Area o Non-prestressing Compression Reinorcement on (a) and (b) Figure 4-6: Eect o Area o Unbonded Prestressing Steel on (a) and (b) Figure 4-7: Eect o Eective Stress o Unbonded Prestressing Steel on (a) and (b) Figure 4-8: Eect o Ultimate Stress o Unbonded Prestressing Steel on (a) and (b) Figure 5-1: Comparison o Load vs. Delection Results o Beams Tested by Ghallab et al. (25) viii

9 Figure 5-2: Comparison o Load vs. Delection Results o Beams Tested by Ghallab et al. (25) Figure 5-3: Comparison o Load vs. Delection Results o Beams Tested by Ghallab et al. (25) Figure 5-4: Comparison o Equation (3.33) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-5: Comparison o Equation (3.33) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-6: Comparison o Equation (3.33) or (e) and () with All Beams Figure 5-7: Comparison o Equation (3.35) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-8: Comparison o Equation (3.35) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendon Figure 5-9: Comparison o Equation (3.35) or (e) and () with All Beams Figure 5-1: Comparison Equation (3.36) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-11: Comparison o Equation (3.36) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendon Figure 5-12: Comparison o Equation (3.36) or (e) and () with All Beams Figure 5-13: Comparison o GIA or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons ix

10 Figure 5-14: Comparison o GIA or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-15: Comparison o GIA or (e) and () with All Beams Figure 5-16: Comparison o Equation by Warwaruk et al. (1962) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-17: Comparison o Equation by Warwaruk et al. (1962) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-18: Comparison o Equation by Warwaruk et al. (1962) or (e) and () with All Beams Figure 5-19: Comparison o Equation by Pannell (1969) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-2: Comparison o Equation by Pannell (1969) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-21: Comparison o Equation by Pannell (1969) or (e) and () with All Beams Figure 5-22: Comparison o Equation by Mattock et al. (1971) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons... 1 Figure 5-23: Comparison o Equation by Mattock et al. (1971) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons... 1 Figure 5-24: Comparison o Equation by Mattock et al. (1971) or (e) and () with All Beams x

11 Figure 5-25: Comparison o Equation by Tam and Pannell (1976) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-26: Comparison o Equation by Tam and Pannell (1976) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-27: Comparison o Equation by Tam and Pannell (1976) or (e) and () with All Beams Figure 5-28: Comparison o Equation by Du and Tao (1985) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-29: Comparison o Equation by Du and Tao (1985) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-3: Comparison o Equation by Du and Tao (1985) or (e) and () with All Beams Figure 5-31: Comparison o Equation by Harajli (199) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-32: Comparison o Equation by Harajli (199) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-33: Comparison o Equation by Harajli (199) or (e) and () with All Beams Figure 5-34: Comparison o Equation by Harajli and Hijazi, Approach I (1991) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-35: Comparison o Equation by Harajli and Hijazi, Approach I (1991) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons xi

12 Figure 5-36: Comparison o Equation by Harajli and Hijazi, Approach I (1991) or (e) and () with All Beams Figure 5-37: Comparison o Equation by Harajli and Hijazi, Approach II (1991) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-38: Comparison o Equation by Harajli and Hijazi, Approach II (1991) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-39: Comparison o Equation by Harajli and Hijazi, Approach II (1991) or (e) and () with All Beams Figure 5-4: Comparison o Equation by Harajli and Kanj (1991) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-41: Comparison o Equation by Harajli and Kanj (1991) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-42: Comparison o Equation by Harajli and Kanj (1991) or (e) and () with All Beams Figure 5-43: Comparison o Equation by Naaman et al. (1991) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-44: Comparison o Equation by Naaman et al. (1991) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-45: Comparison o Equation by Naaman et al. (1991) or (e) and () with All Beams Figure 5-46: Comparison o Equation by Chakrabarti (1995) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons xii

13 Figure 5-47: Comparison o Equation by Chakrabarti (1995) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-48: Comparison o Equation by Chakrabarti (1995) or (e) and () with All Beams Figure 5-49: Comparison o Equation by Lee et al. (1999) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-5: Comparison o Equation by Lee et al. (1999) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-51: Comparison o Equation by Lee et al. (1999) or (e) and () with All Beams Figure 5-52: Comparison o Equation by Allouche et al. (1999) or (a) and (b) with Beams Including Unbonded Tendons Figure 5-53: Comparison o Equation by Au and Du (24) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-54: Comparison o Equation by Au and Du (24) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-55: Comparison o Equation by Au and Du (24) or (e) and () with All Beams Figure 5-56: Comparison o Equation by Roberts-Wollmann et al. (25) or (a) (b) with Beams Including Unbonded Tendons and xiii

14 Figure 5-57: Comparison o Equation by Diep and Niwa (26) or (a) and (b) with Beams Including Unbonded Tendons Figure 5-58: Comparison o Equation by Harajli, Alternative I (26) or (a) and (b) with Beams Including Unbonded Tendons Figure 5-59: Comparison o Equation by Harajli, Alternative II (26) or (a) and (b) with Beams Including Unbonded Tendons Figure 5-6: Comparison o Equation by Harajli, Alternative III (26) or Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-61: Comparison o Equation by Harajli, Alternative III (26) or Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-62: Comparison o Equation by Harajli, Alternative III (26) or (e) with with and () with All Beams Figure 5-63: Comparison o Equation by Ozkul et al. (28) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-64: Comparison o Equation by Ozkul et al. (28) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-65: Comparison o Equation by Ozkul et al. (28) or (e) and () with All Beams Figure 5-66: Comparison o Equation by Du and Au (29) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons xiv

15 Figure 5-67: Comparison o Equation by Du and Au (29) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-68: Comparison o Equation by Du and Au (29) or (e) and () with All Beams Figure 5-69: Comparison o Equation by He and Liu (21) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-7: Comparison o Equation by He and Liu (21) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-71: Comparison o Equation by He and Liu (21) or (e) and () with All Beams Figure 5-72: Comparison o Equation by Zheng et al. (21) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-73: Comparison o Equation by Zheng et al. (21) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-74: Comparison o Equation by Zheng et al. (21) or (e) and () with All Beams Figure 5-75: Comparison o Equation by Yang et al. (211) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-76: Comparison o Equation by Yang et al. (211) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-77: Comparison o Equation by Yang et al. (211) or (e) and () with All Beams xv

16 Figure 5-78: Comparison o Design Equation in ACI (28) or ACI 44-4 (24) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-79: Comparison o Design Equation in ACI (28) or ACI 44-4 (24) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-8: Comparison o Design Equation in ACI (28) or ACI 44-4 (24) or (e) and () with All Beams Figure 5-81: Comparison o Design Equation in AASHTO-LRFD (21) or Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-82: Comparison o Design Equation in AASHTO-LRFD (21) or Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-83: Comparison o Design Equation in AASHTO-LRFD (21) or (e) () with All Beams Figure 5-84: Comparison o Design Equation in Canadian Code (26) or Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-85: Comparison o Design Equation in Canadian Code (26) or (c) with with with and and (d) with All Beams Figure 5-86: Comparison o Design Equation in British Code (21) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons xvi

17 Figure 5-87: Comparison o Design Equation in British Code (21) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-88: Comparison o Design Equation in British Code (21) or (e) and () with All Beams Figure 5-89: Comparison o Design Equation in Eurocode (24) or with Beams Including (a)unbonded Tendons (b)unbonded and Bonded Tendons Figure 5-9: Comparison o Design Equation in Eurocode (24) or with Beams Including (c)unbonded Tendons (d)unbonded and Bonded Tendons Figure 5-91: Comparison o Design Equation in Eurocode (24) or (e) and () with All Beams xvii

18 LIST OF TABLES Table 3-1: Properties o Beam Tested by Jerrett et al. (1996) Table 3-2: Deinitions and Equivalences used or Simpliication Table 3-3: Comparison o Prediction Equations Table 4-1: Summary o Parametric Studies rom Literature... 6 Table 4-2: Summary o Parameters used in Parametric Study Table 4-3: Summary o Results Table 5-1: Equations rom Literature to Estimate at Ultimate... 9 Table 5-2: Code Equations to Estimate at Ultimate Table 5-3: Summary o Correlation Factors or Dierent Equations Table 5-4: Summary o Correlation Factors or Dierent Equations (Inclusion o the Term ApsB into A s ) xviii

19 LIST OF SYMBOLS A psb Area o bonded prestressed reinorcement A Area o unbonded prestressed reinorcement A s Area o non-prestressed tensile reinorcement ' A s Area o non-prestressed compressive reinorcement b b w Width o the section Web thickness or langed sections c C Neutral axis depth measured rom extreme top concrete iber Concrete Compressive Force d psb Depth o the bonded prestressed reinorcement measured rom top concrete iber d Depth o the unbonded prestressed reinorcement measured rom top concrete iber d s Depth o the non-prestressed tensile reinorcement measured rom top concrete iber ' d s Depth o the non-prestressed compressive reinorcement measured rom top concrete iber E c Modulus o elasticity o concrete e Eccentricity o unbonded prestressed tendon measured rom the centroid o section E ps Modulus o elasticity o related prestressed tendon Load Geometry Factor xix

20 ' c Compressive strength o concrete rpu Tensile strength o FRP tendon peu Eective prestressing orce in unbonded tendons ater all losses Ultimate stress o unbonded tendons F Stress in unbonded tendons calculated rom orce equations E Stress in unbonded tendons calculated rom energy equilibrium pu Tensile srength o prestressing tendons py Yield stress o prestressing tendons s Stress in non-prestressed tensile reinorcement ' s Stress in non-prestressed compressive reinorcement y Yield stress o non-prestressed reinorcement h h Height o the section Flange thickness o section L L a Span length Theoretical constant moment region length L h Distance rom support to the plastic hinge L L Distance rom support to applied point load L p Plastic hinge length M Bending moment M pl Moment at plastic hinge xx

21 P T Load applied on the beam Tensile Force Total 1 Stress block reduction actor (ACI 318-8) Elongation o unbonded tendon length between end anchorages Midspan delection Strain change in unbonded prestressed tendon Stress change in unbonded prestressed tendon c Strain in the concrete cu Strain in the top iber o concrete at ultimate peu Eective prestrain in unbonded prestressing tendons Ultimate strain in unbonded prestressing tendons Curvature along the beam Angle o rotation at support xxi

22 Subscripts B c E F rp I nor PL pe ps R s u U w Bonded prestressed tendon Concrete External work Flange o a section T section Fiber Reinorced Polymer (FRP) type o tendons Internal energy Normalized equation Point Load Eective stress in prestressing tendon Prestressed tendon Rectangular section Non-prestressed reinorcement Crushing o concrete at ultimate Unbonded prestressed tendon Web o the lange section Superscripts E F Energy equilibrium Force equilibrium xxii

23 1 CHAPTER I INTRODUCTION 1. INTRODUCTION 1.1 Overview Prestressed concrete, in which prestressing reinorcement would put concrete under compressive stresses to overcome its weakness in tension and to improve its response to external loads, has several advantages over traditional reinorced concrete. With prestressing technology, entire section is made eective, longer spans, less delection, better cracking control and corrosion resistance can be accomplished. Two dierent methods were developed or prestressing concrete: pretensioning and post tensioning. In pretensioning, the strands in stressing bed are tensioned prior to casting o the concrete. Ater casting o the concrete, the strands are released rom the tensioning bed and the tensile stress is transerred to concrete by the ull bond between strand and concrete. Most pretensioned concrete elements are preabricated in a actory and transported to the construction site. Pretensioning technology is used in the construction o bridges (precast, pretensioned I girders with spans up to t), parking structures, buildings and cylindrical structures such as water storage tanks. In post-tensioning, ater the casting o the concrete, the tendons are tensioned and then tendons are anchored. Post-tensioning can be applied by using internal bonded and/or unbonded (internal or external) tendons. The bonded tendons are put inside a duct which is then grouted to protect the tendons rom corrosion. In unbonded tendons, the duct is illed with grease. Post-tensioning technology is used in the repair and

24 2 strengthening o the existing structures (use o external tendons), construction o segmental box girder bridges with longer spans, thinner slabs (especially in high rise buildings) and nuclear power plants. The main ocus o this study is to simpliy the equation proposed by Ozkul et al. (28) or use in calculating the stress in combination o bonded and unbonded or hybrid tendons. Hybrid tendons are used in incrementally prestressed concrete girders in which the girder with bonded tendons is pretensioned prior to the casting o the slab. Ater the slab is cast, the girder-slab system with unbonded tendons is post-tensioned (Han et al. 23, Nassi et al. 24). Using hybrid tendons would provide the beneit o achieving longer span lengths with same section depth or shallower section depths can be achieved in bridge technology. In addition, to improve the ductility o the unbonded FRP tendons, bonded steel tendons are added to the beams prestressed with unbonded tendon which is another important use o hybrid tendons. 1.2 Problem Statement In beams prestressed with bonded tendons, the change in strain in the tendon at any section is equal to the change in strain in the adjacent concrete section. Thereore, the stress in the bonded tendons can easily be ound by using orce equilibrium and strain compatibility equations. In comparison to beams prestressed with bonded tendons, or beams including unbonded tendons there is no strain compatibility between the tendon and concrete due to the lack o bond. Thereore, under the eect o applied load, the stress increase in an unbonded tendon depends on the deormation o the whole member. Thereore, the

25 3 stress increase in the unbonded tendon cannot be estimated simply by using the strain compatibility and orce equilibrium equations; the analysis o the whole member is required. Because o the complexity o the situation, it is diicult to develop an accurate and rational analytical model. In order to develop a deormation-based analysis, it is assumed that the strain is uniorm along the section. 1.3 Research Signiicance In order to compute the ultimate moment resistance o a beam prestressed with unbonded or hybrid tendons, the tendon stress at ultimate is required. In the past, researchers proposed many equations; however, most o these equations are not simple or practical enough to be used as a design equation. Moreover, the code equations are simple, but very conservative compared with experimental data. All code equations do not take into account dierent tendon materials. Thereore, there is a need or an accurate but simple equation or the prediction o the unbonded tendon stress at ultimate. 1.4 Objective and Scope o Work The objective o this research is to develop an accurate, simple and rational equation or the prediction o the stress at ultimate that can be also applicable to beams prestressed with unbonded or hybrid, steel or FRP, external or internal tendons. The scope o work can be explained as our matters: 1. To compare the proposed equation with generalized incremental analysis developed by Ozkul et al. (28) and inite element analysis (FEA).

26 4 2. To perorm a parametric study to investigate the eect o various parameters on the ultimate stress and stress increase in unbonded tendons. 3. To review the existing prediction equations rom literature and also Code equations or the evaluation o the ultimate stress o unbonded tendons and to compare them with the experimental data or determining the accuracy o prediction equations. 4. To veriy the accuracy and applicability o the proposed equation with experimental data. 1.5 Organization o the Thesis The research work is described in six chapters: Chapter 1 presents the introduction which includes the overview o the research, problem statement and objectives o the research. Chapter 2 is the literature review which covers the previous studies done by dierent researches related to the stress in unbonded tendons. In addition, the code provisions or the calculation o the unbonded tendon stress at ultimate are summarized. Chapter 3 presents the simpliication o the equation or calculating the ultimate strength o beams with unbonded tendons. In this chapter, various alternatives or the ultimate strength equation are presented. The most accurate and simplest orm is selected as the inal orm o the equation. Chapter 4 is the parametric study part in which the eects o various parameters on the ultimate stress in unbonded tendons are investigated.

27 5 Chapter 5 is the results part which presents the comparison o proposed equation results with experimental ones. The results are shown or the total loading history and at ultimate or beams prestressed with unbonded or hybrid tendons. Chapter 6 is the conclusion o this study which summarizes the research, observations and conclusions based on the results and inally presents some recommendations or the improvement o the current study.

28 6 CHAPTER II LITERATURE REVIEW 2. LITERATURE REVIEW 2.1 Introduction Since the early 195 s, in order to evaluate the stress at ultimate in unbonded tendons, many experimental and analytical researches have been done. However, very ew researches were implemented about the use o both bonded and unbonded (hybrid) tendons. Due to the increased use o the unbonded tendons there is still a need or more studies on this topic especially about the use o hybrid tendons. In comparison with the analysis o beams and slabs with bonded tendons, the analysis o such members prestressed with unbonded tendons is more diicult. Because in members with unbonded tendons, there is no strain compatibility between the tendon and concrete due to the lack o bond. For that reason, the stress increase in an unbonded tendon is member dependent rather than section dependent. For the beams prestressed with bonded tendons, due to the bond between the concrete and the tendon, the strain increase in tendon is equivalent to the strain increase in the adjacent concrete at that level. Then, or these beams the stress in tendon can be determined by using the strain increase ound rom strain compatibility and the initial strain due to the eective prestress. On the other hand, or the beams prestressed with unbonded tendons, the strain in the tendon is assumed to be constant throughout the span. In order to ind the strain increase in the unbonded tendon, the analysis o the deormation o the entire section is required.

29 7 Thereore, the evaluation o the stress in unbonded tendon at the ultimate limit state is a challenging and complicated process. The stress o the unbonded tendon at the ultimate limit state is needed in order to calculate the moment capacity o the member. Thereore, a simple and accurate equation is required to estimate the stress change in the unbonded tendon. Many researchers have dealt with the design, analysis and the modeling o the prestressed concrete members with unbonded tendons to ind an equation or. In the ollowing section, the studies related to this topic are summarized in two parts. Ater that, the code equations or the calculation o are presented. 2.2 Related Studies The irst part includes the theoretical and experimental investigations related to the beams prestressed with unbonded tendons. Many researches implemented to understand the behavior o members prestressed with unbonded tendons and to propose a design equation or the calculation o. In the second part, the theoretical and experimental studies related to the beams prestressed with bonded and unbonded (hybrid) tendons are summarized. When the amount o researches done related to the unbonded tendons and hybrid tendons are compared, it can be easily observed that the studies about the use o hybrid tendons are very limited. In addition, most o the equations proposed or the beams with unbonded tendons, are not applicable or the beams with hybrid tendons. Moreover, there is no equation proposed particularly or the use o both bonded and unbonded tendons. This situation emphasizes the need or a unique, rational, simple and

30 8 accurate equation to calculate beams prestressed with hybrid tendons. which is applicable or dierent conditions such as Studies on Prestressing with Unbonded Tendons Studies on prestressing with unbonded tendons were examined and summarized by Ozkul (27) and Tanchan (21). Thereore, only the studies ater 25 are summarized in this study. Tanchan (21) carried out an experimental and analytical study to investigate the behavior o high strength concrete (HSC) beams prestressed with unbonded tendons. For that purpose, the author conducted tests on nine rectangular HSC beams prestressed with unbonded tendons. The main variables were area o non-prestressed steel and prestressed steel, prestressing stress, concrete compressive strength and span-to-depth ratio. In order to search the eect o these variables on the stress increase in unbonded tendons at ultimate, the author perormed inite element analysis and parametric study with the use o ABAQUS (2). Naaman (25) proposed a rational method o analysis o beams prestressed or partially prestressed with unbonded tendons in three states: elastic uncracked state, elastic cracked state and ultimate limit state. For this purpose, strain reduction coeicients or bond coeicients, were used to orm a relationship between the case o bonded tendons and unbonded tendons. The values o the bond coeicients or the ultimate limit state u are based on a regression analysis. Thereore, u values should be considered approximate and additional research is needed or the more exact values o strain reduction coeicient at ultimate.

31 9 Lou and Xiang (26) investigated the eect o the bonded tension reinorcement on the response beams prestressed with unbonded tendons by considering two parameters, namely the area and yield strength o the tension reinorcement. The authors developed a numerical model based on the inite element method to predict the entire nonlinear response o beams including unbonded tendons. In their inite element model, the arc-length method is used instead o conventional incremental load method. In each arc incremental step, the authors utilized the Newton-Raphson method or the iterative procedure. They used experimental test results to veriy the validity o the proposed model. The authors concluded that unbonded beams without tension reinorcement represent very poor behavior. However, with the inclusion o minimum amount o tension reinorcement, the lexural behavior o equivalent beams is greatly improved. Manisekar and Senthil (26) reviewed the previous analytical investigation related to the ultimate stress in unbonded tendons. They examined the existing prediction equations suggested by various researchers and collected six experimental data sets to evaluate the accuracy o each equation. The authors ound out that the stress increase is directly related to the ormation o the plastic hinge or the beams prestressed with internal unbonded tendons. They also made a review o the external tendons. They pointed out that the analysis o external tendons is dierent rom that o the internal unbonded tendons with the inclusion o the second order eects and rictional eects. Thereore, the authors concluded that the prediction procedure or beams with internal unbonded tendons adopted at present is not appropriate or the beams with external tendons.

32 1 Park et al. (26) proposed a numerical model based on the inite element method to analyze the unbonded post-tensioned structures. In addition, they developed another model which represents the riction and bond eect at the interace o tendon and the concrete. The authors presented a numerical procedure or material nonlinear analysis o prestressed concrete structures by using these models. The proposed procedure can be used to predict the response o pretensioned and bonded or unbonded post-tensioned structures such as deormation, cracking throughout their service lie. Diep and Niwa (26) reviewed existing prediction equations or the ultimate unbonded tendon stress and used experimental data o 12 beams or the validation o the equations. They observed that prediction equations recommended in the codes are too conservative and available prediction equations developed by researchers don t provide good accuracy. The authors proposed a new prediction equation or the computation o the unbonded tendon stress at ultimate by using plastic region length concept together with the parametric study. The proposed equation can be observed as ollows: d ps Lo ps pe Eps cu 1 c y L (2.1) where, c y ' ' '.95Aps py As y As y.85 c ( bbw ) h '.851 b c w Harajli (26) presented a comprehensive evaluation o the main parameters that inluence the stress in unbonded tendons at ultimate. The author mentioned that the plastic hinge length has a signiicant eect on the unbonded tendon stress. Thereore, by using a physical model and a large experimental database, the author developed an accurate expression or the equivalent plastic hinge length. Based on this expression,

33 11 three possible design alternatives were proposed or the calculation o the unbonded tendon stress. Based on his investigation, the author concluded that the 25 ACI Code (ACI 318-5) is over conservative or calculating ps and AASHTO-LRFD (21) approach is more rational than the ACI Code. The proposed design alternatives can be observed in the ollowing lines: Alternative I: 1 Q E Q Eps ps 1 K py ps ps ps N 1/ N (2.2) where, dp c 2.7 ps pe cu 1.5 La / n p Then, the stress in the unbonded tendon is calculated using the equation o Menegetto and Pinto (1973) with N=7 and K=1. Alternative II: dp cy 2.7 E 1.5 La / n p ps pe ps cu py (2.3) Alternative III: The stress ps is presented as a combination o Alternatives I and II: ps d pe KoEps cu dp.85 KE d 1 o ps cu p p '.851 c s s y ' 1 c (2.4)

34 12 K o 2.7 np ps 1.5 La where, n p is the number o plastic hinges, and is a unction o loading type which is 1 or one-point loading, 3 or two-point loading and 6 or uniorm loading. Ozkul (27) presented a general method or the analysis o beams prestressed with unbonded tendons. The method is based on orce equilibrium, compatibility o delection and work energy principle. In addition, Ozkul (27) perormed inite element analysis. As a result, an equation or the calculation o the unbonded tendon stress is developed. Thirteen high strength concrete beams prestressed with unbonded tendons were tested. The results o these tests and available test data in literature are used to validate the accuracy o the methodology and the equation. It is concluded that both the general methodology and the equation predict the unbonded tendon stress accurately. Ozkul et al. (28) presented an experimental program and an analytical study or the investigation o the behavior o HSC (high strength concrete) beams prestressed with unbonded tendons. In the experimental program, 25 simply supported HSC beams were tested to ailure. Their analytical study based on the trussed beam model that consists o concrete and unbonded tendon elements. In the analytical study, this model was solved by using two analysis approaches: inite element analysis (FEA) and generalized incremental analysis (GIA). Generalized incremental analysis is based on the orce equilibrium, compatibility o equations and work energy principle and applicable at various loading stages. The authors, proposed the ollowing equation or predicting the stress in beams prestressed with unbonded tendons: E e b k ' ps ps 1 c pe A s y A ps pu py (2.5)

35 13 where, 2 L L h p L L k1 12 ; Lh (.5d.5 Z) L LLL 2 2 Experimental results were used to evaluate the accuracy o the proposed methodology and the equation. According to this evaluation, it was concluded that both the method and the equation presented in this study are accurate compared with experimental results. Ellobody and Bailey (28) conducted tests on two simply supported slabs to investigate the behavior o unbonded post-tensioned one-way concrete slabs. The authors proposed inite element model using ABAQUS (26). Fine mesh 3D solid elements are used to model the unbonded slabs and the model includes the nonlinear material models or the tendon and concrete. Ater the veriication o the model with the two test results, the authors implemented a parametric study with 28 slabs. They compared the design code British Standards, BS811-1 (22) result or ultimate load and delection at midspan with the results obtained rom inite element analysis. It is concluded that BS811-1 (22) results or ultimate load are conservative or unbonded post-tensioned one-way concrete slabs. Du and Au (28) proposed a method or the prediction o the ultimate stress in unbonded steel or FRP external tendons. This method is based on Pannell s deormationbased model. In this method, the ratio o the equivalent plastic hinge length to the neutral axis depth is evaluated by using available three groups o test results. The authors suggested steel 1 or steel tendons or regular purpose design. For FRP tendons, this actor is expressed as a multiplication o the same actor or steel tendons: FRP steel.

36 14 According to their indings, i modulus o elasticity o FRP is less than that o steel, FRP is larger than steel and vice versa is also correct. Ater the determination o rom experimental results, the authors combined section analysis with deormation basedapproach and proposed the ollowing ormula which is a modiied version o AASHTO LRFD (1998) equation: dp c ps pe 6 ( py ) steel or ( pu ) FRP [ MPa] le (2.6) Where, l e L (1 N / 2) In the above equations, L is the length o tendon between anchorages, N is the number o support hinges required to orm a ailure mechanism and c is the neutral axis depth. As can be seen rom the representation o the equation, it is both applicable or steel and FRP tendons. Au et al. (29) perormed a parametric study to investigate the eect o 11 various parameters on the lexural ductility o unbonded prestressed concrete beams. For that purpose, the authors developed a numerical method to analyze the behavior o prestressed concrete beams with unbonded tendons. The results obtained rom numerical method are veriied by comparison o the load delection curves with some experimental results. In this study, the lexural ductility actor is expressed in terms o dimensionless curvature ductility actor. The curvature ductility actor is taken as the ratio o the ultimate curvature, u to the curvature at irst yield y. The eect o each parameter on the lexural ductility o beams with unbonded tendons is determined based on the

37 15 curvature ductility actor. The authors concluded that among 11 parameters examined, the depth o the prestressing tendons, depth to non-prestressed tension steel, concrete compressive strength, yield strength o non-prestressed tension steel and partial prestressing ratio have signiicant eect on the curvature ductility. In addition, the authors introduced some conversion actors to code equations used or the calculation o lexural ductility. Vu et al. (21) proposed a model or the calculation o the structural response o post-tensioned beams with unbonded tendons that can be applied at all cracking, serviceability and ultimate loading stages. The authors used non-linear macro inite element (M.F.E) or the computation o prestressed concrete serviceability ater cracking and the concrete tension stiening eect is taken into account in this computation. The M.F.E is a beam inite element deined mainly by the homogenous average inertia. The proposed model is validated by perorming mechanical tests on two beams and using experimental results rom literature. The authors concluded that the calculations o the bearing capacity and the delection at ailure based on the model are accurate when compared with the experimental results. He and Liu (21) proposed a methodology to evaluate the stress in unbonded tendons or externally or internally prestressed beams at the elastic and ultimate states. The methodology is based on a combination o orce equilibrium and deormation compatibility equations. The main idea o the method is determining the relationship between the tendon stress increase, ps and midspan delection, mid in elastic and inelastic states. Based on available experimental results, the authors ound out that the tendon stress increase and midspan delection have a linear relationship in linear elastic

38 16 range o beams. In their analysis, they concluded that ps depends on tendon eccentricity, the neutral axis depth and the second-order eects. For externally prestressed beams, second order eects are taken into account by introducing two reduction actors: the stress reduction actor, R s, and the depth reduction actor, R d. For the plastic hinge length L, the expression proposed by Lee et al. (1999) is used: p 1 1. Based on the methodology, the authors proposed a design ormula which is L d / ps applicable or both external and internal tendons and both steel and FRP tendons: ps, u pe ps, u.8 pu (2.7) e L m 1 ps, u Eps Rs (2.8) c L2 where, [ / ( L/ dps)] or uniorm or two-third-point loads [.3 3 / ( L/ dps )] or one-point loading In Equation (2.8), R s, is the stress increment reduction actor which is equal to 1 or internal tendons, c is the neutral axis depth and L1/ L2is the ratio o the length o the loaded spans to the length o the total length o the tendon. The accuracy o the equation is veriied by using experimental results o 89 internally prestressed and 9 externally prestressed beams. They concluded that the proposed equation does not lose much accuracy in comparison with the proposed methodology. In addition, it is mentioned that the second-order eect can be neglected or the externally prestressed beams with at least one deviator.

39 17 Zheng and Wang (21) perormed a moment-curvature nonlinear analysis method to estimate the stress increase in unbonded tendon at ultimate limit state. The authors considered the ollowing parameters in their investigation: loading type, span to depth ratio L/ h, prestressing reinorcement index p and non-prestressing reinorcement index s. 35 beams rom literature and 38 simulated beams are used to study the eect o these parameters. In conclusion, the authors proposed two sets o equations or simple and continuous beams. For simply supported beams types can be observed as ollows: ps equations or dierent loading ( )( h/ L) or one point ps p s or third point ps p s or uniorm ps p s (2.9) Ater the calculation o ps by using the above equations, the ultimate stress can be calculated rom well-known expression: ps pe ps. The authors compared the predicted values with the experimental results and concluded that the proposed equation has adequate precision. In addition, they included the comparisons o various design code equations and equations o various investigators with experimental results. Yang and Kang (211) proposed a method or the estimation o the stress at ultimate state in simply-supported beams post-tensioned with unbonded tendons. As a basis o their methodology, the authors introduced an equivalent-strain-distribution actor obtained using a nonlinear two-dimensional analysis. The developed nonlinear analysis model can be implemented at each loading stage by using a computer algorithm. Analytical parametric studies and nonlinear regression analysis are carried out to determine the strain reduction actor, which is a unction o unbonded and bonded

40 18 steel reinorcement amount, span to depth ratio and tendon proile. Ater the determination o, the authors developed the ollowing closed orm equation or the calculation o unbonded tendon stress at ultimate state: ps 2 B1 B1 4AC 1 1 py (2.1) 2A 1 where, A 1 1 A p B A A C A E A ' ' s y s y p pe p cu p C E ( A A C.85 b d ) ( A A C ) ' ' ' ' ' 1 p cu s y s y c w 1 p pe s y s y The authors used 239 experimental test results rom literature to veriy the accuracy o the proposed equation. In addition, they compared code equations (ACI and AASHTO, 1996) and various investigators equations with the experimental data. It is concluded that ACI equation is very conservative, while AASHTO equation is unconservative or beams with span to depth ratio o 35 or less Studies on Prestressing with Unbonded and Bonded (Hybrid) Tendons Studies on prestressing with a combination o unbonded and bonded (hybrid) tendons were examined and summarized by Charan (26). Thereore, the studies that were explained beore are not included in here. Ghallab (21) conducted an experimental and analytical study to investigate the behavior o the beams strengthened with external unbonded FRP tendons. For that purpose, the author tested thirteen prestressed beams, one with internal prestressing only, and the rest strengthened externally using Parail Ropes Type G to ailure. The author studied the eect o six actors on the behavior o the tested beams. Namely these actors

41 19 are: the value o the external prestressing orce and its eccentricity, deviator position, previous loading stage beore strengthening, concrete strength and span to depth ratio. In addition, the investigator carried out analytical investigations to propose simple equations that can be used in the analysis o this beam types. Ghallab and Beeby (21) tested our prestressed beams, one with internal bonded prestressing steel only and three with internal bonded prestressing steel and external unbounded post-tensioned FRP-Parail Ropes Type G tendons up to ailure. For all our beams, non-prestressed mild steel was also used. The authors aim was to observe the behavior o the tested beams or examining the beneit o the use o Parail Rope Type G or external prestressing and or evaluating its eect on the ultimate and service load behaviors. All beams were simply supported and two third-point loads were applied. The cracking patterns, load delection responses, ultimate lexural responses, mode o ailures and prestressing orces o the tested beams were compared. In their research, they concluded that using Parail rope is a useul system or strengthening or rehabilitation o prestressed concrete structures. By providing a moderate amount external prestressing orce, they observed a considerable improvement in the stiness, cracking and ultimate lexural strength o the beams without any signiicant reduction in ductility. Nassi et al. (23) developed inite element model using ABAQUS sotware or the analysis o concrete beams prestressed with bonded and unbonded (hybrid) tendons. Their analysis is based on the idealized trussed-beam model which includes the beam element and unbonded tendon element. While developing the inite element model, nonlinear material behavior is considered. The analytical results are compared with the available test results in literature and the authors concluded that the proposed inite

42 2 element model gives accurate results or beams prestressed with bonded, unbonded or bonded and unbonded beams. In addition, the authors observed that ACI code equation or the estimation o stress in unbonded tendons is conservative and should be modiied or the presence o non-prestressed reinorcement. Moreover, ACI equation is conservative and underestimates the ultimate capacity o beams prestressed with hybrid tendons. Han et al. (23) presented results o two ull scale bridge girders prestressed with incrementally prestressed concrete (IPC) design concept. In the multistage prestressing in which prestressing orces applied several times at dierent loading stages, both bonded and unbonded internal tendons were used. The two beams are exactly identical except or the anchorage blocks; one with bracket type o anchorage and the other with coupler type. Both beams were simply supported and were tested under two loads at the third span. Based on the test results, the load delection response, cracking patterns and the load strains o the beams were examined. The authors observed that the use o both bonded and unbonded tendons together in the same member and multistage prestressing o these tendons resulted in the improved overall perormance o that member. With multistage prestressing technique, or the same girder depth longer spans can be achieved or or the same span length shallower girders can be used compared with those o traditional prestressed concrete girders. Thereore, they concluded that this type o bridge girders is an economical alternative to other types o prestressed concrete girders or long span bridges. Ghallab and Beeby (25) aimed to observe the eect o several actors related to prestressing systems, internal bonded prestressed steel, material properties, and beam

43 21 geometry on the increase in the ultimate stress in external Parail ropes as well as external steel tendons in their study. In addition, they examined the accuracy o Eurocode, ACI and British Code (BS811) equations or the ultimate stress in external tendons. For experimental study, they tested nine beams with internal bonded prestressing steel and external unbonded post-tensioned FRP-Parail Ropes Type G tendons up to ailure. All beams were tested under static loads; two loads at the third span or one load at the midspan. Seven beams tested previously by Ghallab (21) were also used in this research. In addition to these beams, several beams externally strengthened were collected rom literature. They ound that the actors which inluenced the ultimate stress in steel tendons had the same eect on the stresses in Parail ropes. The authors also concluded that the Eurocode equation underestimates the ultimate stress in external tendons; the BS 811 equation had good agreement with experimental results or Parail rope, but it was not accurate enough or steel tendons. On the other hand, the ACI equation showed a low accuracy or the prediction o the external prestressing stress. Thereore, they recommended that more studies are needed to include actors such as tendon proile, eective tendon depth, number o deviators and to modiy the ACI equation to be appropriate or high concrete strengths. Du et al. (211) tested nine beams up to ailure under two point loading in order to study the eect o bonded and unbonded FRP tendons on the lexural capacity o the beams. In this investigation, three actors are taken into account: the bonding condition o CFRP tendons, the prestressing ratio and the location o CFRP tendons. Based on the test results, the authors concluded that the lexural capacity o the internally ully bonded beam is the highest and the externally prestressed beam without deviators is the lowest.

44 22 The lexural capacity o the beam with ully bonded tendon is 2% higher than the beam with internal unbonded tendon and 4% higher than the beam with external unbonded tendon without deviators. In addition, they observed that the ductility o the prestressed concrete beams can be improved by using both bonded and unbonded FRP tendons. The authors developed a method to calculate the nominal capacity o the beams with bonded and unbonded FRP tendons. The basis o this method is balanced reinorcement ratio which is used to decide the ailure mode o the beam. The authors used strain reduction coeicient u1 to account or the unbonded tendons and the depth reduction actor Rd and strain reduction actor u 2 to consider the eect o external tendons. Then, these actors are combined with orce equilibrium equations to develop moment capacity equations. The calculated values are compared with the experimental results and good accuracy is achieved.

45 Code Equations In the ollowing paragraphs, the code equations presented or the calculation o the unbonded tendons stress are shown. The examined Codes are ACI (28), AASHTO LRFD (21), British Code (21), Canadian Code CSA (26) and Eurocode (24). However, please note that none o these code equations address the design case o using both bonded and unbonded tendons. ACI (28) code equation which is based on the research perormed by Mattock et al. (1971) and then modiied by Mojtahedi and Gamble (1978) can be written as ollows: ps ' c pe 1, [psi] ps L For 35, 1 and 6, and d ps ps pe py L For 35, 3 and 3, and d ps ps pe py (2.11) For beams prestressed with unbonded FRP tendons, ACI 44-4 (24) recommended the ollowing equation based on the work o Naaman et al. (22): d p ps pe uep cu 1 c (2.12) where, 1.5 ( L/ dp ) u 3 ( L/ dp ) or one point loading or two point and uniorm loading

46 24 ACI 44-4 (24) also mentioned that or the beams prestressed with external FRP tendons, the eective depth o the tendon d p should be replaced by d e in Equation(2.12): d R d (2.13) e d p where, R d ( L/ dp).19( Sd / L) 1. or one point loading ( L/ dp).38( Sd / L) 1. or third-point loading and Sd is the spacing o the deviators and L is the span length o the beam. In the same manner, or the beams with external prestressing or a combination o internal and external prestressing, the strain reduction actor Equation(2.12) based on the work o Aravinthan and Mutsuyoshi (1997): u needs to be changed in.21 A ( L/ dp) A ptot u 2.31 A ( L/ dp) A ptot pint.4.4 or one point loading pint.21.6 or third-point loading (2.14) where Ap int is area o internal prestressed reinorcement and A ptot is area o internal and external prestressed reinorcement. AASHTO LRFD (21) recommended the ollowing design equation: dp c 9 le ps pe py (2.15) where, l e =Eective Tendon Length

47 25 l e 2li 2 N s l i = Tendon Length between Anchorages N s = Number o the Hinges at Supports c = Neutral Axis Depth c A A A.85 ( bb ) h ' ' ' ps ps s y s y c 1 w '.85 c1bw The British Code (21) Equation can be seen as ollows: pu A ps ps pe 1.7 pu [ ksi ] L cubd ps d ps (2.16) where, cu ' c Cube Strength o Concrete.8 Candian Code, CSA (26) mentioned that the ultimate stress o unbonded tendons shall be taken as eective prestressing orce unless a detailed deormation based analysis shows that a higher value can be used: ps (2.17) pe In Eurocode (24), it is stated that i no detailed calculation is made, should be taken as 1MPa. As a result, Eurocode equation or ps is: ps 1 [MPa] (2.18) pe ps

48 26 CHAPTER III SIMPLIFICATION OF PREDICTION EQUATION 3. SIMPLIFICATION OF PREDICTION EQUATION 3.1 Introduction In comparison to beams prestressed with bonded tendons, or beams including unbonded tendons there is no strain compatibility between the tendon and concrete due to the lack o bond. For that reason, in an unbonded tendon is member dependent rather than section dependent. It means that the deormation o the whole member and the interaction between the unbonded tendon and concrete beam should be considered or the analysis o beams prestressed with unbonded or hybrid tendons. Thereore, cannot be estimated simply by using the strain compatibility and section analysis; the analysis o the whole member is required. Beore passing to the analysis approaches or the beams prestressed with hybrid tendons, the main methodology behind the approaches is explained in this part. The trussed-beam model which is the structural idealization o a concrete beam prestressed with a straight internal unbonded tendon can be seen in Figure 3.1. This model was beore presented in the researches by Tanchan (21), Charan (26) and Ozkul (27). This model consists o two main elements: the concrete beam and the unbonded tendon. As can be seen rom the igure, two spring elements are used to keep the distance between the concrete beam element and unbonded tendon element constant. At the ends o the beam, rigid connectors are used which remain perpendicular during loading. The bonded tendons are treated as rebar element with initial stress condition.

49 27 P/2 Bonded Prestressed Tendon Unbonded Prestressed Tendon Bonded Non-Prestressed Steel P/ Springs Figure 3.1: Trussed-Beam Model (Ozkul et al. 28) 5 Truss Element Beam Element Rigid Links 3 3 e Element Number Node Number The trussed-beam model is examined and solved at various load levels. For that purpose, two methods o computation are used: 1. Approach I: Finite Element Analysis (FEA): Sotware Program ABAQUS (Version 6.8) is used or inite element analysis. 2. Approach II: Generalized Incremental Analysis (GIA): Mathematical derivation which is based on the nonlinear section analysis, delections and conservation o the work-energy principle (Ozkul et al. 28). 3. Approach III: Closed Form Solution (Ozkul et al. 28) These approaches are explained in the ollowing section as a summary. For more details o the Approach I, Finite Element Analysis, please reer to the studies by Tanchan (21), Charan (26) and Ozkul (27). In the same manner, or details or Approaches II and III Generalized Incremental Analysis and Closed Form Solution, please reer to the study by Ozkul (27) and Ozkul et al. (28), respectively. A summary o the methods o analysis can be seen in the ollowing lowchart, Figure 3.2.

50 28 METHODS OF ANALYSIS (Based on Trussed-Beam Model) Approach I Approach II Finite Element Analysis (FEA) Generalized Incremental Analysis (GIA) Ozkul et al. (28) Nonlinear Analysis o the Model using Sotware ABAQUS An Incremental Loading Procedure used to Estimate Overall Beam Behavior I II I II Determination o Material Models Incremental Loading Procedure- RIKS Algorithm Nonlinear Section Analysis III Beam Delections Work-Energy Principle FEA Results GIA Results Approach III Closed Form Solution or Ozkul et al. (28) Thesis Objective: Simpliy Prediction Equation at Ultimate Limit State Figure 3.2: Summary o Methods o Analysis