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Seismic performance of multi-span RC bridge with irregular column heights 2012
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Title | Seismic performance of multi-span RC bridge with irregular column heights |
Creator |
Reza, Samy Muhammad |
Publisher | University of British Columbia |
Date Created | 2012-03-01 |
Date Issued | 2012-03-01 |
Date | 2012 |
Description | Bridges are essential elements in modern transportation network and play a significant role in a country’s economy. However, it has always been a major challenge to keep bridges safe and serviceable. Modern bridge design codes include seismic detailing in order to ensure ductile behavior, which was absent in the pre-1970 codes that made older bridges vulnerable during earthquakes. The main parameters effecting the performance of bridge (tie spacing, concrete and steel properties, amount of reinforcement) varies significantly from old to modern bridges. The presence of irregularity in column heights is one of the common causes of seismic vulnerability and the non-uniform column height is the most common form of irregularity. In this study, a four span RC box-girder bridge has been considered for different column height configurations. Here, a detailed parametric study has been performed to understand the effects of various factors on the limit states of the individual bridge columns using factorial analysis. Static pushover analyses, incremental dynamic analyses and fragility analyses of bridges with irregular column heights have been conducted to identify the seismic vulnerability of bridges in the longitudinal direction due to irregularity in column height. This study also investigated the difference of conventional force-based approach and displacement-based approach in designing a bridge with irregular column heights. Canadian Highway Bridge Design Code (CHBDC) and AASHTO 2007, like other traditional design codes follow force-based design (FBD) method, which is focused at the target force resistance capacity of the structure. On the other hand, displacement-based design approach focuses on a target maximum displacement of the bridge during the earthquake in a specific zone. Seismic performances of the bridges designed in two different methods have been compared by non-linear dynamic analyses in the longitudinal direction in terms of maximum and residual displacements and energy dissipation capacity. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2012-03-01 |
DOI | 10.14288/1.0050683 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2012-05 |
Campus |
UBCO |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/41050 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0050683/source |
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SEISMIC PERFORMANCE OF MULTI-SPAN RC BRIDGE WITH IRREGULAR COLUMN HEIGHTS by Samy Muhammad Reza A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The College of Graduate Studies (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) FEBRUARY 2012 © Samy Muhammad Reza, 2012 ii ABSTRACT Bridges are essential elements in modern transportation network and play a significant role in a country’s economy. However, it has always been a major challenge to keep bridges safe and serviceable. Modern bridge design codes include seismic detailing in order to ensure ductile behavior, which was absent in the pre1970 codes that made older bridges vulnerable during earthquakes. The main parameters effecting the performance of bridge (tie spacing, concrete and steel properties, amount of reinforcement) varies significantly from old to modern bridges. The presence of irregularity in column heights is one of the common causes of seismic vulnerability and the nonuniform column height is the most common form of irregularity. In this study, a four span RC boxgirder bridge has been considered for different column height configurations. Here, a detailed parametric study has been performed to understand the effects of various factors on the limit states of the individual bridge columns using factorial analysis. Static pushover analyses, incremental dynamic analyses and fragility analyses of bridges with irregular column heights have been conducted to identify the seismic vulnerability of bridges in the longitudinal direction due to irregularity in column height. This study also investigated the difference of conventional forcebased approach and displacementbased approach in designing a bridge with irregular column heights. Canadian Highway Bridge Design Code (CHBDC) and AASHTO 2007, like other traditional design codes follow forcebased design (FBD) method, which is focused at the target force resistance capacity of the structure. On the other hand, displacementbased design approach focuses on a target maximum displacement of the bridge during the earthquake in a specific zone. Seismic performances of the bridges designed in two different methods have been compared by nonlinear dynamic analyses in the longitudinal direction in terms of maximum and residual displacements and energy dissipation capacity. iii TABLE OF CONTENTS ABSTRACT………………………………………………………………………………………ii TABLE OF CONTENTS…………………………………………………………………..…….iii LIST OF TABLES……………………………………………………………………………….vii LIST OF FIGURES ................................................................................................................. viii LIST OF NOTATIONS ............................................................................................................ xii ACKNOWLEDGEMENTS ..................................................................................................... xiv DEDICATION…………………………………………………………………………...………xv CHAPTER 1: INTRODUCTION AND THESIS ORGANIZATION .......................................... 1 1.1 GENERAL ........................................................................................................................ 1 1.2 OBJECTIVE OF THE STUDY ......................................................................................... 2 1.3 SCOPE OF THE RESEARCH .......................................................................................... 2 1.4 THESIS ORGANIZATION .............................................................................................. 3 CHAPTER 2: LITERATURE REVIEW ..................................................................................... 5 2.1 GENERAL ........................................................................................................................ 5 2.2 COMPARISON OF OLD AND MODERN BRIDGE COLUMNS .................................... 5 2.2.1 Material Properties...................................................................................................... 5 2.2.2 Detailing of Bridge Column ........................................................................................ 5 2.3 FACTORS AFFECTING THE PROPERTIES OF RC COLUMN .................................... 6 2.3.1 Effect of Material Properties ....................................................................................... 6 2.3.2 Effect of Detailing ...................................................................................................... 7 2.3.3 Geometric Properties .................................................................................................. 7 2.4 FACTORIAL DESIGN ..................................................................................................... 7 2.5 SEISMIC VULNERABILITY OF BRIDGES ................................................................... 8 2.5.1 Irregularity .................................................................................................................. 8 2.5.2 Poor Detailing ............................................................................................................. 9 2.6 SEISMIC DESIGN OF RC BRIDGE COLUMN .............................................................. 9 2.6.1 ForceBased Design (FBD) ......................................................................................... 9 2.6.1.1 Calculation of FBD ............................................................................................ 10 2.6.1.2 Code evolution ................................................................................................... 11 iv 2.6.1.3 Limitations of FBD ............................................................................................ 11 2.6.2 DisplacementBased Design (DBD) .......................................................................... 12 2.6.2.1 Methods of DBD ................................................................................................ 12 2.6.2.2 Direct displacementbased design (DDBD) ........................................................ 12 CHAPTER 3: LATERAL LOAD RESISTANCE OF BRIDGE PIERS UNDER FLEXURE AND SHEAR USING FACTORIAL ANALYSIS .............................................. 14 3.1 GENERAL ...................................................................................................................... 14 3.2 SHEAR CAPACITY OF COLUMNS ............................................................................. 14 3.3 MODELING OF THE BRIDGE COLUMN .................................................................... 16 3.4 PUSHOVER ANALYSIS AND FLEXURAL LIMIT STATES ...................................... 19 3.5 FACTORIAL ANALYSIS FOR DIFFERENT COLUMN PROPERTIES ....................... 20 3.5.1 First Cracking ........................................................................................................... 21 3.5.1.1 Analytical effect of factors at first cracking ........................................................ 21 3.5.1.2 Factorial design at first cracking ......................................................................... 22 3.5.2 First Yielding ............................................................................................................ 25 3.5.2.1 Analytical effect of factors at first yielding ......................................................... 25 3.5.2.2 Factorial design at first yielding.......................................................................... 27 3.5.3 First Crushing ........................................................................................................... 30 3.5.3.1 Analytical effect of factors at first crushing ........................................................ 30 3.5.3.2 Factorial design at first crushing ......................................................................... 32 3.5.4 Factorial Design for Displacement Ductility ............................................................. 37 3.6 PREDICTED EQUATIONS ........................................................................................... 38 3.7 SUMMARY .................................................................................................................... 43 CHAPTER 4: SEISMIC PERFORMANCE EVALUATION OF A MULTISPAN RC BRIDGE WITH IRREGULAR COLUMN HEIGHTS OF VARYING DUCTILITY LEVELS ............................................................................................................. 44 4.1 GENERAL ...................................................................................................................... 44 4.2 BRIDGES WITH DIFFERENT DUCTILITY LEVELS AND NONUNIFORM COLUMN HEIGHTS ........................................................................................................ 45 4.3 MODELING OF THE BRIDGE ..................................................................................... 46 4.4 STATIC PUSHOVER ANALYSIS ................................................................................. 48 v 4.4.1 Effect of Tie Spacing on Bridge Performance ........................................................... 48 4.4.2 Effect of Irregularity on Bridge Performance ............................................................ 50 4.4.3 Ductility ................................................................................................................... 55 4.5 INCREMENTAL DYNAMIC ANALYSIS ..................................................................... 56 4.5.1 Description of Ground Motion Properties ................................................................. 56 4.5.2 Dynamic Pushover Curves ........................................................................................ 58 4.6 FRAGILITY ANALYSIS ............................................................................................... 62 4.6.1 Characterization of Damage States ............................................................................ 63 4.6.2 Fragility Function Methodology................................................................................ 63 4.6.3 Fragility Curves Results ............................................................................................ 67 4.7 SUMMARY .................................................................................................................... 69 CHAPTER 5: COMPARISON OF DIRECT DISPLACEMENTBASED DESIGN AND FORCEBASED DESIGN IN CANADIAN CONTEXTS .................................. 70 5.1 GENERAL ...................................................................................................................... 70 5.2 SAMPLE BRIDGE ......................................................................................................... 70 5.3 BRIDGE COLUMN DESIGN......................................................................................... 71 5.3.1 Direct DisplacementBased Design (DDBD) ............................................................ 73 5.3.2 Force Based Design (FBD) ....................................................................................... 75 5.3.3 Comparison between DDBD and FBD ...................................................................... 77 5.4 NONLINEAR DYNAMIC ANALYSIS......................................................................... 78 5.4.1 Selection and Scaling of Ground Motions ................................................................. 79 5.4.2 Time History Analysis Results .................................................................................. 81 5.5 PERFORMANCE COMPARISON DDBD AND FBD BRIDGE .................................... 82 5.5.1 Maximum Displacement ........................................................................................... 83 5.5.2 Residual Displacement.............................................................................................. 84 5.5.3 Energy Dissipation.................................................................................................... 85 5.5.4 Base Shear Demand .................................................................................................. 86 5.6 DISCUSSION ................................................................................................................. 89 5.7 SUMMARY .................................................................................................................... 90 CHAPTER 6: CONCLUSIONS ............................................................................................... 91 6.1 SUMMARY .................................................................................................................... 91 vi 6.2 LIMITATIONS OF THE STUDY ................................................................................... 91 6.3 CONCLUSIONS ............................................................................................................. 92 6.4 RECOMMENDATIONS FOR FUTURE RESEARCH ................................................... 95 REFERENCES ......................................................................................................................... 96 APPENDIX……………………………………………………………………………………..108 vii LIST OF TABLES Table 31. Slenderness ratio of the columns. ............................................................................. 15 Table 32. Levels of the factors. ................................................................................................ 17 Table 33. Comparison of the cyclic test result of numerical model with experimental result. ...................................................................................................................... 19 Table 34. Coefficient of equations for different limit states of the column. .............................. 41 Table 35. Validation of generated equations for cracking limit state. ....................................... 42 Table 36. Validation of generated equations for first yield limit state. ...................................... 42 Table 37. Validation of generated equation for base shear at first crushing. ............................. 42 Table 38. Validation of generated equations for displacement at crushing/shear failure and ductility. .................................................................................................................. 42 Table 41. Different column height combinations. .................................................................... 46 Table 42. Sectional properties of bridge deck. ......................................................................... 47 Table 43. Yield strength and displacement for different column height combinations. ............. 54 Table 44. Crushing strength and displacement for different column height combinations......... 55 Table 45. Selected earthquake ground motion records. ............................................................ 57 Table 46. PSDMs for bridges with four different column height combinations with two tie spacings. .................................................................................................................. 64 Table 47. Limit states for RC bridge. ....................................................................................... 66 Table 51. Sectional properties of bridge deck. ......................................................................... 71 Table 52. Design shear and moment in columns. ..................................................................... 77 Table 53. Design reinforcement in columns. ............................................................................ 77 Table 54. Earthquake ground motion properties....................................................................... 82 Table 55. Displacement demand of DDBD and FBD bridges from NTHA. ............................. 83 Table 56. Residual displacement of DDBD and FBD bridges from NTHA. ............................. 85 Table 57. Energy dissipation of DDBD and FBD bridges from NTHA. ................................... 86 Table A1. Limit states of columns for factorial analyses. ....................................................... 108 viii LIST OF FIGURES Figure 31. Flow chart of column classification. ..................................................................... 16 Figure 32. (a) Pushover deformation shape of a column, (b) cross section of column. ............ 18 Figure 33. Verification of numerical model with test result. ................................................... 20 Figure 34. Typical pushover curves for 7m column for combinations of the four factors with low levels and high levels. ............................................................................ 21 Figure 35. (a) Range of cracking base shear for different heights of columns, (b) contribution percentage of factors on the change of cracking base shear. ......... 23 Figure 36. Effect of (a) f′c and amount of longitudinal steel, (b) column height and f′c, (c) column height and longitudinal steel of concrete on the cracking base shear. ... 24 Figure 37. (a) Range of cracking displacement for different heights of columns, (b) contribution percentage of factors on the change of cracking displacement. .... 25 Figure 38. Effect of (a) column height and f’c, (b) column height and longitudinal steel ratio on cracking displacement. ............................................................................. 25 Figure 39. (a) Range of yield base shear for different heights of columns, (b) contribution percentage of factors on the change of yielding base shear. ................................... 27 Figure 310. Effect of (a) longitudinal steel ratio and fy of concrete, (b) column height and fy of concrete, (c) column height and longitudinal steel ratio, (d) tie spacing and f’c, (e) column height and f’c, (f) fy and f’c on the yield base shear. .................. 28 Figure 311. (a) Range of yield displacement for different heights of columns, (b) contribution percentage of factors on the change of first yield displacement. ... 29 Figure 312. Effect of (a) fy and f’c, (b) fy and longitudinal reinforcement ratio, (c) fy and tie spacing, (d) fy and column on the yield displacement. ........................................... 30 Figure 313. (a) Range of base shear for crushing or shear capacity for different heights of columns, (b) contribution percentage of factors on the change of base shear due to first crush or to reach shear capacity. .......................................................... 34 Figure 314. Effect of (a) column height and tie spacing, (b) longitudinal reinforcement ratio and tie spacing, (c) column height and fy, (d) column height and f’c on the crushing base shear or shear capacity. ............................................................. 34 ix Figure 315. (a) Range of displacement for crushing or shear capacity for different heights of columns, (b) contribution percentage of factors on the change of displacement due to first crush or to reach shear capacity. .................................... 36 Figure 316. Effect of (a) tie spacing and f’c, (b) fy and longitudinal reinforcement ratio, (c) column height and f’c, (d) column height and tie spacing on the displacement at crushing or shear failure. ............................................................. 36 Figure 317. Range of ductility for different heights of columns, (b) different factors on ductility (displacement of crushing or shear capacity/yield displacement). ............ 37 Figure 318. Effect of (a) tie spacing and f’c, (b) f’c and column height on ductility (displacement of crushing or shear capacity/yield displacement). .......................... 38 Figure 319. Empirical vs. Numerical plots for (a) cracking base shear, (b) cracking displacement, (c) yield base shear, (d) yield displacement, (e) crushing shear/shear capacity, (f) displacement at crushing or shear failure, (g) ductility. .......................................................................................................... 40 Figure 41. Bridges with irregular column height combination. ............................................... 45 Figure 42. Column cross section. ........................................................................................... 47 Figure 43. Results of pushover analysis in longitudinal direction for Case: SSS..................... 49 Figure 44. Results of pushover analysis in longitudinal direction for Case: MMM. ................ 49 Figure 45. Results of pushover analysis in longitudinal direction for Case: LLL. ................... 50 Figure 46. Results of pushover analysis in longitudinal direction for Case: MLL. .................. 51 Figure 47. Results of pushover analysis in longitudinal direction for Case: SLL. ................... 52 Figure 48. Results of pushover analysis in longitudinal direction for Case: MSL. .................. 53 Figure 49. Results of pushover analysis in longitudinal direction for Cases: LLL, SLL, SML and SSS considering 75 mm tie spacing. ............................................. 54 Figure 410. Ductility of the bridge with column height configuration SLL for different tie spacing. ........................................................................................................... 55 Figure 411. Spectral acceleration for the chosen earthquake ground motions. .......................... 57 Figure 412. Dynamic and static pushover curve for LLL with columns of 75 mm tie spacing. ........................................................................................................... 58 Figure 413. Dynamic and static pushover curve for LLL with columns of 200 mm tie spacing. ........................................................................................................... 59 x Figure 414. Dynamic and static pushover curve for SSS with columns of 75 mm tie spacing. ........................................................................................................... 59 Figure 415. Dynamic and static pushover curve for SSS with columns of 200 mm tie spacing. ........................................................................................................... 60 Figure 416. Dynamic and static pushover curve for SLM with columns of 75 mm tie spacing. ........................................................................................................... 60 Figure 417. Dynamic and static pushover curve for SLM with columns of 200 mm tie spacing. ........................................................................................................... 61 Figure 418. Dynamic and static pushover curve for SLL with columns of 75 mm tie spacing. ........................................................................................................... 61 Figure 419. Dynamic and static pushover curve for SLL with columns of 200 mm tie spacing. ........................................................................................................... 62 Figure 420. Comparison of the PSDMs for LLL, SSS, SLM and SLL for 75 mm and 200 mm tie spacing. .............................................................................................. 65 Figure 421. Comparison of fragility curves for bridges with different column height combinations for 75 mm and 200 mm tie spacing for slight damage. .................... 67 Figure 422. Comparison of fragility curves for bridges with different column height combinations for 75 mm and 200 mm tie spacing for moderate damage. ............... 68 Figure 423. Comparison of fragility curves for bridges with different column height combinations for 75 mm and 200 mm tie spacing for extensive damage. .............. 68 Figure 424. Comparison of fragility curves for bridges with different column height combinations for 75 mm and 200 mm tie spacing for collapse damage. ................ 69 Figure 51. Elevation of the bridge with irregular column heights. .......................................... 71 Figure 52. Flowcharts showing step by step procedures: (a) displacementbased and (b) forcebased design........................................................................................... 72 Figure 53. Design displacement spectra for Vancouver. ......................................................... 74 Figure 54. Design acceleration spectra for Vancouver. .......................................................... 76 Figure 55. Spectral acceleration for original earthquake ground motions. .............................. 80 Figure 56. Spectral acceleration for scaled earthquake ground motions. ................................. 81 Figure 57. Maximum displacement demand of bridges designed in displacementbased and forcebased approach. .................................................................................... 84 xi Figure 58. Residual displacement of bridges designed in displacementbased and forcebased approach. ........................................................................................... 85 Figure 59. Dissipated energy of bridges designed in displacementbased and forcebased approach in time history analyses. ........................................................................ 86 Figure 510. Comparison of base shear demand of C1 of bridges designed in displacement based and forcebased approaches through time history analyses. ......................... 88 Figure 511. Comparison of base shear demand of C2 of bridges designed in displacement based and forcebased approaches through time history analyses. ......................... 88 Figure 512. Comparison of base shear demand of C3 of bridges designed in displacement based and forcebased approaches through time history analyses. ......................... 89 xii LIST OF NOTATIONS As Amount of Longitudinal steel b Section width bc Width of core section d Effective depth of column section dc Depth of core section DDBD Direct displacementbased design DPO Dynamic pushover Ec Elastic modulus of concrete EDP Engineering demand parameter f’c Compressive strength of concrete f’cc Confined compressive strength of concrete f’l Effective lateral stress of confined concrete FBD Forcebased Design fy Yield strength of steel H Height of column hp Plastic hinge lenght Ic Cracked moment of inertia IDA Incremental dynamic analysis IM Intensity measures K Elastic lateral stiffness of column Ke Effective lateral stiffness of column Mc Cracking moment My Yield moment NTHA Nonlinear time history analysis PGA Peak ground acceleration PGV Peak ground velocity PSDM Probabilistic seismic demand model Rξ Spectral reduction factor s Tie spacing xiii s’ Clear spacing of transverse reinforcement SPO Static pushover T Time period of structure Te Effective time period of structure V Total design base shear Vc Base shear at first cracking Vi Design base shear of individual column Vshear Shear capacity Vy Base shear at first yileding We Effective weight wi Clear distance of i th longitudinal bar βEDP|IM Dispersion of demand Δc Lateral displacement at first cracking Δcrush Displacement at first concrete crushing Δd Target diplacement Δp Plastic displacement Δy Yield displacement Δy Yield displacement μ Ductiltiy ξeq Equivalent viscous damping ratio ξsys System viscous damping ratio φc Cracking curvature φu Curvature at first concrete crushing φy Yield curvature xiv ACKNOWLEDGEMENTS This is only the mercy of almighty Allah Robbul Izzat that I exist. May the honour and peace of the messenger of Allah (peace be upon him) increase intensely. I express my sincere gratitude to both of my supervisors, Dr. M. Shahria Alam and Dr. Solomon Tesfamariam. They have been instrumental with knowledge, support, placement, mentoring that made my graduate experience at UBC so impeccably productive and rewarding, at any rate. Overall, I attribute my positive experience at the graduate school here at UBC to them. Meanwhile, UBC Okanagan campus, being a unique and generous institution, provided an excellent educational refuge, and that deeply reckoned, in a compelling manner. I really enjoyed the life in here. I also would like to acknowledge Natural Sciences and Engineering Research Council of Canada (NSERC)’s support in this rejuvenating and fulfilling journey. I have had the opportunity to get in touch with an excellent and enthusiastic group of graduate students in the research group who offered technical knowledge and lively discussions. I offer much appreciation to my thoughtful and considerate friends, Muntasir Billah, Nurul Alam, Rafiqul Haque. Most importantly, I would like to pay my innermost respect to my parents, whom I feel to be the key source of inspiration for all my achievements. xv dedicated to my mentors 1 CHAPTER 1: INTRODUCTION AND THESIS ORGANIZATION 1.1 GENERAL Bridges play a vital role in a country’s economic development and bridges are an essential part of the transportation system. They ensure smooth transportation by establishing links between cities and even between countries. The failure of bridges during an earthquake not only costs human lives but also causes a catastrophe to the transport network and the economy. In recent years several earthquakes caused significant damage to many bridges. The bridges in North America are also at high seismic risk, especially which were designed and constructed before 1970. There are more than 3000 highway bridges in British Columbia and many of them were constructed before 1970. Since the earthquakes are getting more frequent these days, it is of prime importance to identify the seismically vulnerable bridges in order to take necessary retrofitting action in order to prevent the bridge collapses during probable seismic events. However, the identification of the seismically vulnerable bridges through numerical analyses is a time consuming and expensive task. Most of the old RC bridges were not seismically detailed, whereas, modern codes ensures ductile detailing. Moreover, the presence of irregularity makes bridges more vulnerable to seismic events. There is a strong need to evolve a quicker and cost effective method to identify the seismically vulnerable bridges by classifying them in terms of their material properties, confinement in columns and the presence of irregularity. 2 It is also necessary to improve and optimize the current design methods for bridges with irregular column heights. Forcebased and displacementbased designs have dissimilarities for designing bridges with irregular column heights. Efforts are needed in order to assess their seismic performance and improve the design methods. 1.2 OBJECTIVE OF THE STUDY The key objectives of the current research include: 1. Determine the effects of different parameters on the limit states of reinforced concrete (RC) bridge column and propose empirical equations for limit states in order to direct determination of limit states of the columns of existing bridges instantly. 2. Determine the effects of irregularity in column heights and spacing of lateral reinforcement in columns on the seismic performance of the RC bridge. 3. Compare the seismic performance of RC bridges with irregular column heights, which are seismically designed for Vancouver region using two different methods, namely: displacementbased approach and forcebased approach. 1.3 SCOPE OF THE RESEARCH In order to achieve the goals of the study, the properties of the old and modern RC bridge piers were obtained from the literature review. This study presents the stateoftheart of material properties and detailing of old and modern bridge piers as well as the seismic design practice of bridge piers. The procedures to achieve the stated objectives are as follows: 1. Finite element models of 1.5 m X 3 m columns of varying heights (7 m, 14 m and 21 m) with different material properties, confinements and amounts of longitudinal steel have 3 been generated in SeismoStruct (2010). Limit states of cracking, first rebar yielding and first concrete crushing have been determined using nonlinear pushover analyses; 34 factorial analyses have been conducted with 243 combinations in order to observe the effect of different parameters on the limit states. 2. Box girder bridges with four equal spans of 50 m have been modeled in SeismoStruct (2010) for different regular and irregular combinations of heights (7 m, 14 m and 21 m) of columns. Limit states of these bridges have been determined using pushover analyses. Effects of irregularity in column height and confinement on the seismic performance of bridge have been determined using fragility analyses. 3. A box girder bridge with irregular column height combination of 7 m, 14 m and 21 m has been designed as per forcebased and displacementbased approach. Nonlinear time history analyses have been conducted in order to compare the two different methods. 1.4 THESIS ORGANIZATION This thesis is arranged in six chapters. In the current chapter a short preface and the objectives and scope are presented. The content of the dissertation is organized into the following chapters: Chapter 2 presents the literature review on the comparison of material and geometric properties of the old and new bridge columns. This study presents the seismic vulnerability of bridges due to irregular columns heights and poor detailing. The design methodologies of force based approach and displacementapproach have also been described. 4 Chapter 3 presents the investigation of the effect of different parameters on the limit states of the bridge pier. The formability of generalized functions of limit states and ductility has also been checked. Chapter 4 investigates the limit states of the bridges with regular and irregular column height configurations and varying tie spacing. This study also evaluates the effect of column height irregularity and tie spacing on the seismic performance of bridges through static pushover and incremental dynamic time history analyses by developing the fragility curves. Chapter 5 presents the design of a bridge with irregular column heights in conventional forcebased and displacementbased approach. Seismic performance of the bridges designed in two different approaches has been evaluated by nonlinear time history analyses. Finally, Chapter 6 presents the key conclusions attained from this research. Few specific recommendations for future research have also been suggested. 5 CHAPTER 2: LITERATURE REVIEW 2.1 GENERAL This chapter comprises the previous works outlining the comparison or old and new RC bridge columns and different factors affecting their performances. The role of the presence of irregularity and poor detailing on the collapse of bridges during earthquakes has been discussed. This study also includes the key features of the traditional and displacementbased seismic design. 2.2 COMPARISON OF OLD AND MODERN BRIDGE COLUMNS 2.2.1 Material Properties The concrete and steel properties have improved over time. Concrete with comparatively higher compressive strength is used in modern bridges whereas in older bridges concrete compressive strength was as low as 28 MPa. Nowadays concrete compressive strength of 69 MPa in bridges is not uncommon (ACI 363R92, HPC Bridge Vies 2010). Yield strength of steel can also vary from 276 MPa to 500 MPa between older and modern bridges (Ranf et al. 2006, Alam et al. 2009). 2.2.2 Detailing of Bridge Column Bridges constructed before 1970 were not designed and detailed according to seismic provisions. Modern code specifies for proper reinforcement detailing with closer transverse reinforcement. Poorly detailed RC columns are susceptible to loss of axial load carrying capacity at drift levels lower than expected during a design level seismic event (Boys et al. 2008). Tie 6 spacing of 300 mm was commonly used in bridge columns before 1970. Ruth and Zhang (1999) conducted a survey of 33 bridges designed from 1957 to 1969 and found that all bridge columns had a tie spacing of 300 mm. Moustafa et al (2011) presented the modern design of an existing bridge pier in Southern California, which was built before 1970, with a pitch of the spiral reinforcement of 100 mm. This pier design according to modern code (California Department of Transportation design standards 1999) resulted in a pitch of 56 mm, which is much smaller than the previous one. The maximum tie bar spacing allowed in CSA Standard S61974 was 16 longitudinal bar diameter, 48 tie bar diameter or the least dimension of the column, however, in CSA Standard 1978 was 300 mm or the least dimension of the member and tie should cover every alternate bar. According to CHBDC 2010 the maximum tie spacing is the smallest of six times the longitudinal bar diameter or 0.25 times the minimum component dimension or 150 mm and tie should cover every longitudinal bar. Therefore, Canadian code of 2010 allows lower tie spacing than that of 1974 and 1978. 2.3 FACTORS AFFECTING THE PROPERTIES OF RC COLUMN A number of studies have been conducted in order to show the effect of different factors on the performance of RC bridge column. 2.3.1 Effect of Material Properties Park and Paulay (1975) discussed the positive or negative effect of the amount of longitudinal steel content, steel yield strength and compressive strength of concrete on the yield point, crushing point and corresponding ductility of the RC column. The improvement of these 7 factors has positive effect on the yield and crushing limit states. However, the relative effects of the change of the factors were not discussed. 2.3.2 Effect of Detailing Several experimental and analytical studies have been conducted in order to observe the effect of confinement on the performance of columns under monotonic and cyclic axial loads (Mander et al.1988 (a and b); Sheikh and Uzumeri 1982; Calderone et al. 2000, Razvi and Saatcioglu 1994, Papanikolaou and Kappos 2009). The previous studies indicate that the shear resistance and flexural behaviour improved with the increased confinement. 2.3.3 Geometric Properties Mo and Nien (2002) concluded that, ductility increases with the increase of axial load. Reduction in displacement ductility and increase of tendency of shear failure rather than flexural failure occur with the decrease in aspect ratio (column height to effective depth ratio) (Stone and Cheok 1989, McDaniel 1997). Zhu et. al. (2007) concluded that column specimens having aspect ratio less than 2.0 fails in shear or flexureshear, and fails in flexure if the aspect ratio is greater than 4.0. 2.4 FACTORIAL DESIGN Like many engineering systems, there are several influencing factors that affect the performance of a bridge column under lateral loads, for example earthquake load. The effect analysis will give misleading results if a single factor is varied at a time, because it will not reflect the interaction with other factors. All the factors need to be varied together in order to examine the effect of various factors including interaction among the factors (Montgomery 2001, Box et al. 1978). Factorial design is a good technique for conducting effect analysis. Padgett and 8 DesRoches (2006) used two level fractional factorial analyses in order to investigate the most important parameter of the seismic performance of retrofitted bridges, however, the system non linearity was not considered. Here, the main effects of steel strength, mass, damping ratio, hinge gap and elastomeric bearing stiffness on the ductility and bearing deformation for different retrofitting options have been observed. The positive and negative effects have been determined to identify the important factors. 2.5 SEISMIC VULNERABILITY OF BRIDGES 2.5.1 Irregularity The presence of unequal span length, skew bent or nonuniform height of columns makes bridge structures irregular. Bridges with unequal span lengths have closely spaced natural periods and will face dynamic amplification, if any of these natural periods matches with the natural period of a vehicle (Senthilvasan et al. 2002). Deck displacement in the transverse direction due to seismic force has been found more for skewed bridges compared to straight bridge (Sevgili and Caner 2009). However, the most common form of irregularity occurs in the nonuniform height of columns of a bridge over a basin (Chen and Duan 2000). If the same section size and reinforcement is provided to the columns of different heights, larger ductility demand will be induced to the shorter columns (Priestley et al. 1996). Bridges with irregular column heights are seismically vulnerable, for example, the shorter columns of Bull Creek Canyon Channel Bridge were damaged in the 1994 Northridge earthquake. This failure was caused by the effects of both irregular column heights and inadequate confinement (Chen and Duan 2000). 9 Several works have been done regarding bridges with irregular column heights. Kappos et. al. (2002) investigated the effects of soilstructure interaction and irregular heights of hollow column on the dynamic behaviour and seismic response of a four span boxgirder bridge. The effect of column height irregularity was found more critical than the effect of inclusion of SSI in the analyses on the bridge dynamic properties and seismic response. Inclusion of SSI effect resulted lower response of structure to ground motion excitation compared to the fixed end analyses. Since, poorly detailed RC columns have lower performance in seismic events (Boys et al. 2008); the effect of ductility capacity of columns should be investigated for bridges with irregular column heights. Saatcioglu and Razvi (2002) developed the design expressions for confinement steel requirement for earthquake resistant concrete columns. 2.5.2 Poor Detailing Poor detailing of the bridge columns has been identified by researchers as one of the most common deficiencies causing the failure of bridges in the past earthquakes (Mitchell et al. 1994). Specifically, bridges which were constructed and designed before 1970s were not seismically detailed (Ruth and Zhang 1999). The failure of columns of an overpass during San Fernando earthquake in 1971 is an example of bridge column failure due to lack of transverse reinforcement (Chen and Duan 2000). 2.6 SEISMIC DESIGN OF RC BRIDGE COLUMN 2.6.1 ForceBased Design (FBD) Seismic design in traditional codes is generally force based. Sebai (2009) demonstrated the difference in seismic design of bridges in different codes for Montreal, Toronto and Vancouver. The seismic design and detailing criteria in Eurocode 8 slightly differs from those of CHBDC 10 2010 and AASHTO 2007 due to the difference in response modificaiton factor. For example, the ductility related response modiciation factor for a single ductile column is 3.0 in CHBDC and AASHTO, whereas, this is 3.5 in Eurocode. 2.6.1.1 Calculation of FBD The seismic design loads in CHBDC and AASHTO are identical (Sebai 2009). The basic steps of the FBD are: Estimation of lateral stiffness of structure Estimation of natural period of structure Estimation of elastic spectral force Selection of forcereduction factor Estimation of seismic force Displacement and member adequacy check The seismic base shear according to NBCC 2005 and NBCC 2010 is determine as = () / [21] Where, S(Ta) is the spectral acceleration, Mv is a factor to account for higher mode effects on the base shear, the ductilityrelated factor Rd and the over strengthrelated factor Ro, IE is the importance factor, and W is the weight of the structure. 11 2.6.1.2 Code evolution Modern codes (AASHTO, CHBDC), based on FBD, have come to this point through different changes and improvements in seismic base shear calculation and detailing over the years in order to ensure safe, ductile and economic design. Previous codes (NRCC 1941, NRCC 1953, NRCC 1960, NRCC 1965) used seismic force coefficients which were not related to the dynamic properties of the structure and thus resulting inaccurate seismic base shear calculation. The codes (NRCC 1970, NRCC 1975, NRCC 1977, NRCC 1980, NRCC 1985, NRCC 1990, NRCC 1995, NRCC 2005, CHBDC 2006, NRCC 2010, CHBDC 2010) gradually improved in seismic base shear calculation by taking time period into account, developing rational importance factor, response modification factor, site coefficient and response spectra. Modern codes also developed better seismic detailing in order to ensure ductile structures. 2.6.1.3 Limitations of FBD Priestley et al. (2007) addressed several problems associated with the forcebased design. The main limitation in this method is that the natural period of the structure is determined from the initial stiffness of the member. However, the stiffness of a structure changes with its deformation. Another limitation is its forcereduction factor that is introduced to scale down the elastic seismic force, which is based on ductility capacity for a given structure type. Here, displacement ductility factor is equal to the forcereduction factor, which is not true for an inelastic system. Traditional forcebased methods ignore the fact that the displacement is more important than the strength for inelastic systems. Thirdly, design seismic force is applied to the structures with initial stiffness, which indicates that the elements of the structure will be 12 subjected to yield point at a time. In reality, seismic force is distributed to the members according to the deformed shape of the structure. Therefore, this assumption is not accurate. 2.6.2 DisplacementBased Design (DBD) 2.6.2.1 Methods of DBD In order to overcome the limitations of FBD, displacement is set as the main criterion for design rather than force (Priestley et al. 2007). Extensive research has been conducted in the effort of developing improved seismic design criteria by using displacement as the main seismic criteria rather than force after the Loma Prieta earthquake in 1989 (Priestley 1993; Caltrans, 2004; ATC, 2003). The main focus is to enhance ductility. There are different approaches of displacementbased design, such as Direct DisplacementBased Design (Priestley 1993), Equal Displacement Approximation (Veletsos and Newmark, 1960), Seismic Design Criteria (Caltran, 2004) and Substitute Structure Method (Shibata and Sozen 1976). Direct DisplacementBased Design (DDBD) has been found to be the most effective among the available displacementbased design methods for design of bridges and other structures (Kowalsky 1995; Calvi and Kingsley 1995; Kowalsky 2002; Ortiz 2006; Suarez and Kowalsky 2006). Priestley et al. (2007) described different stages of modern direct displacementbased design. Bardakis and Fardis (2010) found that the displacementbased design is more cost effective and rational than Eurocode 8. 2.6.2.2 Direct displacement-based design (DDBD) The key difference between the DDBD and FBD is that in the case of DDBD effective time period and lateral stiffness is derived from the target displacement in order to calculate seismic base shear whereas in the case of FBD seismic base shear is calculated from the elastic 13 time period and stiffness without considering the hysteretic damping of the structure. The steps of DDBD include: Determine target displacement from yield displacement and required ductility Determine damping of the structure (material damping + hysteretic damping) from ductility. However, in the case of FBD only material damping is considered Determine the effective time period (Te) of the structure from displacement spectra, whereas, elastic time period of structure is taken Determine the effective lateral stiffness (Ke) of the structure from weight and Te Determine the seismic base shear of the structure The required spacing of lateral reinforcement is limited by an allowable maximum value in the FBD codes, which rule is not present in DDBD. This study also examines the performance of a DDBD bridge which is restrained by the maximum allowable tie spacing and compares with the original DDBD bridge. 14 CHAPTER 3: LATERAL LOAD RESISTANCE OF BRIDGE PIERS UNDER FLEXURE AND SHEAR USING FACTORIAL ANALYSIS 3.1 GENERAL Modern bridge design codes include seismic detailing in order to ensure their ductile behaviour, which was absent in the codes before 1970 that make the older bridges vulnerable during earthquakes. Moreover, there have been significant improvements in the material performance along with its quality control in modern bridge construction. Tie spacing, concrete and steel properties, amount of reinforcement and column height are the main factors which affect the performance of the bridge columns under lateral loads for a constant axial load level. Besides, these parameters differ significantly from old to modern bridges. In this study, nonlinear pushover analyses have been conducted in order to determine the effect of different factors on the limit states of bridge columns. A detailed parametric study has been performed to understand the effect of change of various factors on the limit states, which has been assessed through factorial analysis. The results obtained from the analysis have also been analytically verified. 3.2 SHEAR CAPACITY OF COLUMNS Table 31 shows the aspect ratio and slenderness ratio of the bridge columns considered in this study. The 7 m columns considered in this study has the aspect ratio (column height to effective depth ratio) is 4.83, which is slightly over four; therefore, flexureshear failures are expected (Zhu et al. 2007). The 14 m and 21 m columns have aspect ratios well above four, therefore, flexural failures are expected. According to Hassoun (1998), the effect of slenderness 15 is negligible for slenderness ratio less than 22, therefore, the 21 m column is expected to show long column effect. In this study, the 7 m, 14 m and 21 m represent the shear dominated, flexure dominated and long columns respectively. The shear capacities of the columns have been determined using the Modified Compression Field Theory (Vecchio and Collins 1986). This method is very accurate and can predict the experimentally determined shear failure within 1% error (Bentz et al. 2006). The shear capacity corresponds to certain displacement, which can be found from the pushover curve. Columns, with shear capacity greater than the crushing base shear, are flexure dominated. Ductility of the flexure dominated column is greater than one. If the shear capacity of the column is in between the yield and crushing base shear, the columns is shear dominated with ductility greater than one. However, if the shear capacity is less than the yielding of the column, the column will face shear failure before reaching the flexural yielding. This type of column cannot reach the theoretical yield displacement. The ductility of this column can be determined with respect to the virtual yield point, which will be less than one. In this study, this ductility is defined as virtual ductility. Figure 31 shows the concept of column classification method used in this study. Three column types have been defined: flexure dominated, shear dominated with ductility greater than one and shear dominated with virtual ductility. Table 31. Slenderness ratio of the columns. Column height (m) H/d kH/r 7 4.83 7.75 14 9.65 15.5 21 14.48 23.4 16 Figure 31. Flow chart of column classification. 3.3 MODELING OF THE BRIDGE COLUMN The range of material properties, longitudinal reinforcement ratio and tie spacing considered in this study is given in Table 32. Bridge columns of size 1500 mm x 3000 mm with 92 longitudinal bars have been considered irrespective of the height of columns as shown in Figure 32(a). The bar area has been adjusted in order to consider 2, 3 and 4% longitudinal reinforcement ratio. 16 mm tie bar is used at every alternate longitudinal bar as shown in Figure 32(b). The column is fixed at the bottom and rotationally restrained at top. It has a constant vertical load of 27000 kN. Therefore, the range of the ratio of vertical load to design axial load is from 15.5% (for 60 MPa concrete, 500 MPa steel and 4% longitudinal steel) to 22.4% (for 25 MPa concrete, 300 MPa steel and 2% longitudinal steel). Confinement factor has been calculated according to Park et al. (1982). Columns have been modeled in SeismoStruct (2010). This software is based on the fibre modeling approach. Inelastic displacementbased frame element has been used for modeling the columns in order to consider material nonlinearity. The cross 17 section has been divided into a number of fibres. The uniaxial response of the individual fibre is obtained from the nonlinear stressstrain behavior of the material. These responses of the fibres are integrated in order to get the sectional stressstrain state of the column along the cross sectional area and length of the member. However, this model does not take shear deformation into account. The 25 MPa and 40 MPa concrete have been modeled using the nonlinear constant confinement concrete model. This model was initiated by Madas (1993) following the constitutive relationship proposed by Mander et al. (1988a) and the cyclic rules proposed by MartinezRueda and Elnashai (1997). The 60 MPa concrete has been modeled with nonlinear constant confinement for highstrength concrete model. This model was developed and initiated by Kappos and Konstantinidis (1999) following the constitutive relationship proposed by Nagashima et al. (1992). The confinement effects have been modified by Sheikh and Uzumeri (1982) factor. Steel has been modeled using the model of Monti and Nuti (1992). An additional memory rule (Fragiadakis et al. 2008) has been introduced, for higher numerical stability under transient seismic loading. Table 32. Levels of the factors. Factors Low (1) Medium (0) High (+1) Compressive strength of concrete, f’c 25 MPa 40 MPa 60 MPa Yield strength of steel, fy 300 MPa 400 MPa 500 MPa Longitudinal steel reinforcement ratio, As 2% 3% 4% Tie spacing, s 75 mm 150 mm 300 mm 18 Figure 32. (a) Pushover deformation shape of a column, (b) cross section of column. This model has been verified with a test result of 400x400 mm Type 1 square column for cyclic loading (Takemura and Kawashima, 1997). The effective height of the column was 1245 mm. The compressive strength of concrete and yield strength of the longitudinal reinforcement were 35.9 MPa and 363 MPa respectively. Twenty13D (13 mm diameter) longitudinal reinforcements were used in the column as shown in Figure 33. Tie bar of 6 mm diameter with 368 MPa steel was provided at 70 mm c/c. The applied axial load was 157 kN. The column was free headed and flexure dominated in the cyclic loading test. The base shear vs. displacement graphs for numerical model and test result closely matches, as shown in Figure 33. The cracking, yielding and crushing points are identified in the result of numerical analysis. The numerical model stopped analysis, after reaching the first concrete crushing due to convergence error. However, the data beyond the crushing point is not of interest in this study. Therefore, the test data have been considered up to the same phase as the termination of numerical analysis. The differences between the test result and numerical analysis in predicting maximum base shear 19 and energy dissipation are 0.01% and 14.97%, respectively (Table 33), which is acceptable. SeismoStruct has also been validated with shake table test results of RC columns and RC frames in Alam et al. (2008) and Alam et al. (2009) respectively. 3.4 PUSHOVER ANALYSIS AND FLEXURAL LIMIT STATES The performance of bridge columns can be identified by strain limits. Circular reinforced concrete bridge columns have been investigated in order to establish the curvature relationships for limit states, where simple relationships with curvature and displacement ductility, drift ratio and viscous damping with serviceability and damage control limit state have been established by Kowalsky (2000). Three limit states are considered in this study: first cracking (Vc, Δc), first yielding (Vy, Δy) and first crushing (Vcrush, Δcrush) where V and refers to lateral force and lateral displacement, respectively. Pushover analyses have been conducted by applying lateral load parallel to the shorter dimension of the column. Cracking strain of concrete and yield strain of steel in tension have been assumed to be 0.000133 and 0.025, respectively. Crushing strain of unconfined concrete varies from 0.0025 to 0.006 (MacGregor and Wight, 2005). According to Paulay and Priestley (1992), crushing strain for confined concrete is much higher and it ranges from 0.015 to 0.05. In the present study, crushing strain of confined concrete is taken as 0.015. Table 33. Comparison of the cyclic test result of numerical model with experimental result. Test Numerical Model Difference % Maximum base shear (kN) 153.66 153.65 0.0065 Energy Dissipation (kNm) 6.28 7.22 14.968 20 Figure 33. Verification of numerical model with test result. 3.5 FACTORIAL ANALYSIS FOR DIFFERENT COLUMN PROPERTIES Compressive strength of concrete, yield strength of steel, longitudinal steel reinforcement ratio and tie spacing have been taken as the factors affecting the first cracking, yielding and crushing of 7 m, 14 m and 21 m height columns. The full factorial design of 34 with no confounding patterns has been conducted to account for the effect of these factors and their interactions. The three levels of these four factors are presented in Table 32. The effects estimated from these factorial analyses are valid for the ranges of factors mentioned in this table. Figure 34 shows two typical pushover curves of 7 m column. Case 1111 indicates the f’c, fy, As and s to be 25 MPa, 300 MPa, 2% and 75 mm, respectively. Case +1+1+1+1 indicates the f’c, fy, As and s to be 60 MPa, 500 MPa, 4% and 300 mm respectively. First cracking, yielding and crushing points are marked on these pushover curves as the performance indicators. Each of these performance indicators has two criteria: base shear and displacement. Full factorial analyses of 34 have been conducted for these criteria. 21 Figure 34. Typical pushover curves for 7m column for combinations of the four factors with low levels and high levels. 3.5.1 First Cracking 3.5.1.1 Analytical effect of factors at first cracking The analytical solution for base shear (Vc) at first cracking is given as φ = 2 (31) = φ E I (32) = 2 (33) Where, is the concrete tensile strain at cracking, φc is, the curvature at cracking, b is the section width, Ic is the cracked moment of inertia, varies with the amount of longitudinal reinforcement of steel and Ec is the elastic modulus of concrete, which is proportional to the 22 square root of compressive strength of concrete. Therefore, the change in compressive strength of concrete and longitudinal reinforcement affects the base shear capacity of column of specific height. Column base shear capacity at first cracking is inversely proportional to the height of the column. The lateral deformation at first cracking can be expressed as = (34) The lateral displacement of the column top is proportional to the square of the length of column. 3.5.1.2 Factorial design at first cracking The results obtained from the pushover analyses were compiled in Appendix. Figure 35(a) shows the cracking base shear for different column heights where the variation of cracking base shear is higher in shorter columns compared to longer columns. Column acts elastically before cracking. Figure 35(b) shows the contribution of the four factors and their interactions on the base shear at cracking for different column heights. The results show that the compressive strength, longitudinal reinforcement and tie spacing have significant effect on the base shear at first cracking for 7m height column. The contribution of the interactions between the factors in the change of first cracking is less than 10%. For the 14 m and 21 m column, compressive strength of concrete and longitudinal steel are the controlling factors. f’c has more contribution among these two, which is about 80%. The effect of confinement of concrete on cracking base shear decreases with the increase in length of the column due to the increase of flexural dominance and decrease of shear dominance. These factors have no significant effect on the cracking displacement. 23 Figure 35. (a) Range of cracking base shear for different heights of columns, (b) contribution percentage of factors on the change of cracking base shear. The effect of compressive strength of concrete is most significant and the amount of the longitudinal steel has the second largest effect on cracking base shear for each column height. These two factors have positive effect, which means, cracking base shear increases with the increase of these two factors separately. However, no interaction is prominent (Figure 35(a)). Tie spacing along with the interaction between fc’ and tie spacing has little effect (5%) only for 7 m column as shown in Figure 35(b). The effect of column height has been found significant (89%) in combined analysis for all three column heights (Figure 36(b)). Although, the percent contribution of f’c is more in the variation of the cracking base shear for 14 and 21 m column, the total change in cracking base shear with the change in f’c is more in 7 m column, since the variation of cracking base shear is more in shorter column as shown in Figure 36(b). The variation of cracking base shear is less for the change in longitudinal steel ratio as shown in Figure 36(c). 24 Figure 36. Effect of (a) f′c and amount of longitudinal steel, (b) column height and f′c, (c) column height and longitudinal steel of concrete on the cracking base shear. Figure 37(a) illustrates the range of cracking displacement for columns of different heights. The results indicate that the variation of cracking displacement is more in longer column (0.001 m in 7 m column and 0.007 m in 21 m column), and longitudinal reinforcement ratio and concrete compressive strength are the main contributing factors on the change of cracking displacement as shown in Figure 37(b). However, the change in cracking displacement with the change of these factors is not significant as shown in Figure 38. 25 Figure 37. (a) Range of cracking displacement for different heights of columns, (b) contribution percentage of factors on the change of cracking displacement. Figure 38. Effect of (a) column height and f’c, (b) column height and longitudinal steel ratio on cracking displacement. 3.5.2 First Yielding 3.5.2.1 Analytical effect of factors at first yielding The analytical solution for the base shear at first yielding can be expressed as = 2 (35) Where, My is the moment capacity at first yield (a) (b) 7.0 14.0 21.0 25.0 42.5 60.0 0 0.01 0.02 0.03 0.04 C ra c ki n g D is p la ce m e n t (m ) C: f'c (MPa) E: Column Height (m) 7.0 14.0 21.0 2.0 3.0 4.0 0 0.01 0.02 0.03 0.04 C ra ck in g D is p la c e m e n t (m ) D: Longitudinal steel (%) E: Column Height (m) 26 = ( − 2 ) (36) As and fy are the amount of longitudinal reinforcement and yield strength of steel, respectively. The depth of equivalent rectangular stress block, a, is a function of amount of longitudinal steel, yield strength of steel and compressive strength of concrete as shown in equation 37. = 0.85 (37) Therefore, the base shear at yield is a function of As, fy, d and f’c. Since, the value of a is much smaller than d, the amount of longitudinal reinforcement and yield strength of the steel are the main controlling factor for the yield base shear. The concrete compressive strength, f’c has also a little effect on Vy, as, it affects a. The expression for the yield displacement is = (38) Where, φy is the curvature at first yield as shown in equation 39. φ = (39) From the previous discussion, As and fy mainly affect My. Icracked depends on amount of longitudinal reinforcement whereas Ec is proportional to the square root of compressive strength of concrete. As has contribution in both the numerator and the denominator of (39), which causes the decrease in the effect of As on Δy. The change in Δy mainly depends on fy and f’c. 27 3.5.2.2 Factorial design at first yielding It can be observed from Figure 39(a) that yield base shear varies over a wider the range as the column height decreases. Yield strength of steel and reinforcement area play major roles on base shear for yielding as shown in Figure 39(b). No interaction is prominent here. The increase in column height from 7m to 21m does not affect the contribution of factors in the change in base shear for yielding whereas the amount of longitudinal reinforcement has 70% contribution. Similar to the cracking base shear, yield base shear is more dispersed for shorter columns with the changes in parameters [Figure 39(b)]. The effects of longitudinal steel ratio and fy are positive and almost linear in the change of yield base shear as shown in Figure 310(a). The effects of these two factors are more prominent in shorter column as shown in Figures 310(b)3 10(c). Tie spacing has almost no effect on the yield base shear as shown in Figure 3.10(d). Yield base shear increases with the increase of f’c; however, Figures 310(d)310(f) show that its effect is less than the other factors. Figure 39. (a) Range of yield base shear for different heights of columns, (b) contribution percentage of factors on the change of yielding base shear. 28 Figure 310. Effect of (a) longitudinal steel ratio and fy of concrete, (b) column height and fy of concrete, (c) column height and longitudinal steel ratio, (d) tie spacing and f’c, (e) column height and f’c, (f) fy and f’c on the yield base shear. As the column height increases, the variation of yield displacement increases over a wider range as depicted in Figure 311(a). While determining the percent contribution of different factors on the yield displacement it was observed that fy has the major contribution on the displacement for first yield as shown in Figure 311(b). Compressive strength of concrete and (d) (e) (f) (a) (b) (c) 2.0 3.0 4.0 300.0 400.0 500.0 2000 4000 6000 8000 10000 Y e ild B a s e S h e a r (k N ) B: fy (MPa) D: Longitudinal steel (%) 7.0 14.0 21.0 300.0 400.0 500.0 3000 6000 9000 12000 15000 Y e ild B a s e S h e a r (k N ) B: fy (MPa) E: Column Height (m) 25.0 42.5 60.0 75.0 187.5 300.0 6000 6250 6500 6750 7000 Y e il d B a s e S h e a r (k N ) A: Tie spacing (mm) C: f'c (MPa) 7.0 14.0 21.0 2.0 3.0 4.0 0 4000 8000 12000 16000 Y e il d B a s e S h e a r (k N ) D: Longitudinal steel (%) E: Column Height (m) 25.0 42.5 60.0 300.0 400.0 500.0 4000 4800 5600 6400 7200 8000 Y ei ld B as e S h ea r (k N ) B: fy (MPa) C: f'c (MPa) 7.0 14.0 21.025.0 42.5 60.0 4000 8000 12000 16000 Y e ild B a se S h e a r (k N ) C: f'c (MPa) E: Column Height (m) 29 amount of longitudinal reinforcement have also contributions on yield displacement. The effects of interactions among various factors are not significant here. Figure 312(c) demonstrates that the tie spacing or confinement of concrete has no effect on the yield displacement. Figures 3 12(a)312(b) show that the yield displacement increases with the increase in fy; decrease in f’c; and increase in longitudinal reinforcement ratio. The relationship of these three parameters with yield displacement is linear. fy, f’c and longitudinal reinforcement ratio control the section curvature and the yield displacement is obtained by multiplying the square of length with curvature, as shown in Eq. (8). The effects of these factors on yield displacement are more in longer column as shown in Figure 312(d). The relationship between yield displacement and column height is almost parabolic (y = ax2). Figure 311. (a) Range of yield displacement for different heights of columns, (b) contribution percentage of factors on the change of first yield displacement. 30 Figure 312. Effect of (a) fy and f’c, (b) fy and longitudinal reinforcement ratio, (c) fy and tie spacing, (d) fy and column on the yield displacement. 3.5.3 First Crushing 3.5.3.1 Analytical effect of factors at first crushing The curvature ductility for unconfined concrete can be expressed as [Park and Paulay (1975)] = ′ ′ ′′ ′ . ′ ′ ′ . ′ ′ .′ (310) (d) (a) (b) (c) 2.0 3.0 4.0 300.0 400.0 500.0 0.05 0.06 0.07 0.08 0.09 0.1 Y ie ld D is p la c e m e n t (m ) B: fy (MPa) D: Longitudinal steel (%) 25.00 32.00 39.00 46.00 53.00 60.00 300.0 400.0 500.0 0.05 0.06 0.07 0.08 0.09 0.1 Y ie ld D is p la c e m e n t (m ) B: fy (MPa) C: f'c (MPa) 7.0 14.0 21.0 300.0 400.0 500.0 0 0.05 0.1 0.15 0.2 Y ie ld D is p la c e m e n t (m ) B: fy (MPa) E: Column Height (m) 300.0 400.0 500.0 75.0 187.5 300.0 0.05 0.06 0.07 0.08 0.09 0.1 Y ie ld D is p la c e m e n t (m ) A: Tie spacing (mm) B: fy (MPa) 31 Where, ρ and ρ’ are the ratio of longitudinal reinforcement at tension and compression zones respectively; is the concrete compressive strength at crushing. The unconfined compressive strength in Eq. (310) can be replaced by the confined compressive strength. According to Mander et al. (1988b), the confined compressive strength is = −1.254 + 2.254 1 + . − 2 (311) Where, f’c is the unconfined compressive strength of concrete. f’l is the effective lateral confining stress ′= (312) Where, ρs is the volumetric ratio of transverse reinforcement steel to the volume of the core; ke is the confinement effectiveness coefficient, can be defined as [Mander et al. (1984)] = 1 −∑ ( ) (313) Where, ρcc is the ratio of longitudinal reinforcement to area of core section, s’ is the clear vertical spacing of the transverse reinforcement; bc and dc are the width and depth of the core section and w’i is the i th clear distance between adjacent longitudinal bars. Total deformation at plastic zone has two parts = + (314) Δp, the plastic deformation can be derived from equation (39) and (310), estimating φy and φu 32 = ( − ) ℎ ( – ) (315) Where, hp is the plastic hinge length. Therefore, in the plastic deformation zone, all factors have effects on the crushing displacement, and also interactions of different factors are also expected to be prominent. The shear capacity Vshear of the section according to Bentz et al. (2006) is presented as = ′ + (316) Here, b and d are the width and effective depth of the section, s is the center to center distance of transverse reinforcement, Av is the gross area of the tie bars in a single layer, is the angle of shear failure, β is a factor which depends on the average tensile stress in the section. 3.5.3.2 Factorial design at first crushing The range of base shear at crushing or shear failure is wider than that of yield base shear for each column height (Figure 313(a)). Figure 313(b) shows that the tie spacing has significant effect on the crushing base shear of 7 m column, however, very little effect on the 14 and 21 m columns. In most of the cases, 7 m columns are shear critical and most of the combinations of 14 and 21 m columns are flexure dominated. Tie spacing, longitudinal steel ratio and fy play important roles in shear capacity of the column; however, f’c has little effect on shear capacity, hence, has smaller effect on the base shear at crushing or shear failure of 7 m column. The effect of tie spacing is insignificant on the base shear capacity of flexure dominated column; therefore, it has little effect on the crushing base shear for 14 and 21 m column cases as shown in Figure 3 14(a). Longitudinal steel ratio and fy are the most effecting parameters in flexure dominated 14 and 21 m height columns. The effect of fy is more in shorter columns as shown in Figure 314(c). 33 Figure 314(d) shows that the crushing base shear increases with the increase of f’c for 7 m column and decreases with the increase of f’c. The shear capacity of the column increases with the increase of f’c [Eq. (16)]. Since, most of the combinations of 7 m column are shear dominated therefore, effect of increase of f’c on crushing base shear is positive. Due to the P-Δ effect, pushover curve goes downward after reaching the maximum base shear. If the crushing point lies on the downward pushover curve, any increase in crushing displacement will decrease the crushing base shear. The first crushing points of 21 m column combinations lie on the downward pushover curve; i.e. the base shear corresponding to first crushing is less than the maximum base shear capacity due to high slenderness ratio of the 21 m column. Since, the crushing displacement increases with the increase of f’c, the crushing base shear of the combinations of the 21 m columns decreases with the increase of f’c and crushing displacement. Figure 314(d) shows the decrease in crushing base shear with the increase of f’c for 21 m column, which is the effect of slenderness; it does not represent the effect of f’c on the ultimate base shear capacity. Therefore, tie spacing, f’c and fy have significant interaction with column height. Moreover, ANOVA analysis suggests that the interaction between tie spacing and longitudinal reinforcement ratio is also significant as shown in Figure 314(b). The effect of tie spacing is higher for higher reinforcement ratio. 34 Figure 313. (a) Range of base shear for crushing or shear capacity for different heights of columns, (b) contribution percentage of factors on the change of base shear due to first crush or to reach shear capacity. Figure 314. Effect of (a) column height and tie spacing, (b) longitudinal reinforcement ratio and tie spacing, (c) column height and fy, (d) column height and f’c on the crushing base shear or shear capacity. Figure 315(a) shows that the range of displacement at crushing or shear failure gets wider with the increase of column height. Tie spacing, f’c and their interaction mostly affects the (b)(a) B a s e S h e a r a t C ru s h in g / S h e a r fa ilu re Column Height (m) 7.0 14.0 21.0 0 5000 10000 15000 20000 25000 0 20 40 60 80 100 S fy f' c A s S x f y S x f 'c S x A s fy x f 'c fy x A s f' c x A s S x f y x f 'c S x f y x A s S x f 'c x A s fy x f 'c x A s S x f y x f 'c x A s % C o n tr ib u ti o n Factors Base Shear at crushing/ shear failure 7 m 14 m 21 m Column Height ( ) (d) (a) (b) (c) B a se S h e a r a t C ru s h in g / S h e a r fa il u re (k N ) B a se S h e a r a t C ru sh in g / S h e a r fa il u re ( kN ) B a s e S h e a r a t C ru s h in g / S h e a r fa ilu re ( k N ) B a s e S h e a r a t C ru s h in g / S h e a r fa ilu re ( k N ) 7.0 14.0 21.0 75.0 187.5 300.0 2000 4000 6000 8000 10000 12000 14000 16000 C ru s h in g S h e a r/ S h e a r C a p a ci ty ( k N ) A: Tie spacing (mm) E: Column Height (m) 2.00 2.50 3.00 3.50 4.00 75.0 187.5 300.0 4000 6000 8000 10000 12000 C ru sh in g S h e a r/ S h e a r C a p a ci ty ( kN ) A: Tie spacing (mm) D: Longitudinal steel (%) 7.0 14.0 21.0300.0 400.0 500.0 4000 7000 10000 13000 16000 C ru s h in g S h e a r/ S h e a r C a p a c it y ( k N ) B: fy (MPa) E: Column Height (m) 7.0 14.0 21.025.0 42.5 60.0 2000 5000 8000 11000 14000 C: f'c (MPa) E: Column Height (m) 35 crushing displacement for the 7m and 14m height columns. Here, these two factors have significant interaction. The effect of tie spacing decreases with the increase of column height from 7m to 14m and the effect of compressive strength increases. Also, higher order interaction plays some role which is shown in Figure 315(b). The effect of confinement is more significant after the first yielding. The 21m column gets unstable before reaching the first crushing for some combinations of the four factors because of the slenderness effect. Figure 316 shows the contribution of different factors on the displacement for first crushing or for instability. Figures 316(a) and 316(c) show that displacement at crushing or shear failure increases with the increase of f’c, and the effect of f’c is more for lower tie spacing and longer column height. In the same way, tie spacing has higher effect on the displacement for crushing or shear failure for higher compressive strength of concrete. Although, the percent contribution of tie spacing is higher in shorter column, the difference between highest and lowest displacements is smaller in shorter column (7 m column) than in longer column (14 and 21 m column). Therefore, crushing or shear failure displacement decreases with the increase of tie spacing for 14 and 21 m column, however, the effect of tie spacing is very insignificant in the case of 7 m column as shown in Figure 316(d). The displacements at shear failure of most of the combinations of 7 m column are less than the yield displacement. Figure 316(b) shows that fy and longitudinal reinforcement ratio have negligible effect on the displacement at crushing or shear failure. The effects of tie spacing, f’c, column height, tie spacingcolumn interaction; f’c column height interaction and square of f’c have been found significant on the displacement for crushing or shear failure from ANOVA analysis. 36 Figure 315. (a) Range of displacement for crushing or shear capacity for different heights of columns, (b) contribution percentage of factors on the change of displacement due to first crush or to reach shear capacity. Figure 316. Effect of (a) tie spacing and f’c, (b) fy and longitudinal reinforcement ratio, (c) column height and f’c, (d) column height and tie spacing on the displacement at crushing or shear failure. (b)(a) D is p la c e m e n t a t C ru s h in g / S h e a r fa ilu re ( m ) Column Height (m) 2 2 2 2 2 2 2 23 2 2 2 3 2 2 2 2 2 7.0 14.0 21.0 0 1 2 3 4 5 6 0 10 20 30 40 50 60 S fy f' c A s S x f y S x f 'c S x A s fy x f 'c fy x A s f' c x A s S x f y x f 'c S x f y x A s S x f 'c x A s fy x f 'c x A s S x f y x f 'c x A s% C o n tr ib u ti o n Factors Displacement at crushing/ shear failure 7 m 14 m 21 m 7.0 14.0 21.025.0 42.5 60.0 0 0.75 1.5 2.25 3 C: f'c (MPa) E: Column Height (m) D is p la c e m e n t a t C ru s h in g / S h e a r fa ilu re ( m ) D is p la c e m e n t a t C ru s h in g / S h e a r fa ilu re ( m ) (d) (a) (b) (c) D is p la c e m e n t a t C ru s h in g / S h e a r fa ilu re ( m ) D is p la c e m e n t a t C ru s h in g / S h e a r fa ilu re ( m ) 25.0 42.5 60.0 75.0 187.5 300.0 0 0.5 1 1.5 2 2.5 A: Tie spacing (mm) C: f'c (MPa) 2.0 3.0 4.0 300.0 400.0 500.0 0 0.75 1.5 B: fy (MPa) D: Longitudinal steel (%) 7.0 14.0 21.0 75.0 187.5 300.0 0 0.75 1.5 A: Tie spacing (mm) E: Column Height (m) 37 3.5.4 Factorial Design for Displacement Ductility In this study, ductility is defined as the ratio of displacement at crushing or shear failure to the yield displacement, which is never less than one for flexure dominated column. Among 81 combinations of 7 m column, 45 combinations are shear dominated, and 39 combinations are with virtual ductility, i.e. displacement at shear failure less than the yield displacement, among these shear dominated combinations. On the other hand, only one combination has been found to be shear dominated with ductility of 2.43. The range of ductility is more for 14 m height column [Figure 317(a)]. The effects of tie spacing, f’c, column height, tie spacingcolumn interaction; f’c column height interaction and square of f’c have been found significant on the ductility as shown in Figure 317(b). Figure 317. Range of ductility for different heights of columns, (b) different factors on ductility (displacement of crushing or shear capacity/yield displacement). The contributions of factors are almost similar for ductility and displacement at crushing or shear failure. Figure 318(a) shows the interaction between tie spacing and f’c; the effect of f’c is more for lower tie spacing. Figure 318(b) shows that, ductility increases with the increase of column height for 60 MPa concrete, however, ductility increases from 7 m to 14 m column and (b)(a) 0.1 1 10 100 0.0 7.0 14.0 21.0 D u ct il ity ( m /m ) Column Height (m) 0 20 40 60 80 100 S fy f'c A s S x f y S x f 'c S x A s fy x f 'c fy x A s f'c x A s S x f y x f'c S x f y x A s S x f 'c x A s fy x f 'c x A s S x f y x f'c x A s % C o n tr ib u tio n Factors Ductility 7 m 14 m 21 m 38 decreases from 14 m to 21 m for 25 MPa concrete. Thus, the change in ductility does not follow a common trend with the change in column height and f’c. Figure 318. Effect of (a) tie spacing and f’c, (b) f’c and column height on ductility (displacement of crushing or shear capacity/yield displacement). 3.6 PREDICTED EQUATIONS Considering the significant factors found from Ftest in ANOVA analysis, equations have been developed for cracking base shear, cracking displacement, yield base shear, yield displacement, crushing base shear/shear capacity, displacement at crushing/shear failure and ductility respectively. The general formula is X = β + α1 s + α2 fy + α3 f'c + α4 As + α5 H + α6 s * fy + α7 s * f'c + α8 s * As + α9 s * H + α10 fy * f'c + α11 fy * As + α12 fy * H + α13 f'c * As + α14 f'c * H + α15 As * H + α16 s 2 + α17 fy 2 + α18 f'c 2 + α19 As 2 + α20 H 2 + α21 s * f'c * H + α22 s * f'c 2 + α23 s * H 2 + α24 f'c x H2 + α25 s * f'c * H 2 (317) where, β is constant, α’s are the coefficients of the equations, given in Table 34. Combined effect of shear and flexure has been taken intentionally in order to check the (a) (b) 7.0 14.0 21.0 25.0 42.5 60.0 0 5 10 15 20 D u ct ili ty ( m /m ) C: f'c (MPa) E: Column Height (m) 25.0 42.5 60.0 75.0 187.5 300.0 0 5 10 15 20 25 30 D u ct ili ty ( m /m ) A: Tie spacing (mm) C: f'c (MPa) 39 formability of the unified equations for crushing/shear failure limit state and ductility. Factors have been used in their actual unit in these equations. s, f’c, fy, As and H represent the tie spacing in mm, compressive strength of concrete in MPa, yield strength of steel in MPa, longitudinal reinforcement ratio in percentage of concrete gross area and height of column in m, respectively. Figure 317 shows the plots of empirical results derived from these equations versus the numerical results derived from finite element analysis. Four random columns of 7 m, 9 m, 16 m and 21 m heights have been selected in order to validate the equations. Tables 35 to 37 show that these equations can fairly predict the cracking limit state, yield limit state and crushing base shear or shear capacity. However, generated equations make erroneous prediction on the top displacement at first crushing or shear failure and ductility of the column as shown in Table 38. This is due to the presence of high degree of nonlinearity in the displacement at postelastic region or in failure mode of the column. The base shear in the post elastic region, however, is easily predictable, since, the pushover curves flattens after reaching the yield point and there is small difference between the yield and maximum base shear. This study considers three levels for each parameter. Higher level factorial analyses are needed in order to predict the displacement at crushing or shear failure and ductility of a column precisely. 40 Figure 319. Empirical vs. Numerical plots for (a) cracking base shear, (b) cracking displacement, (c) yield base shear, (d) yield displacement, (e) crushing shear/shear capacity, (f) displacement at crushing or shear failure, (g) ductility. Numerical (kN) E m p ir ic a l ( kN ) Cracking Base Shear 0 1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000 9 9 9 9 9 36 9 9 45 9 18 9 18 9 18 63 9 8 9 9 8 9 9 9 Numerical (m) E m p ir ic a l ( m ) Cracking Displacement 0.00 0.01 0.02 0.03 0.04 0.00 0.01 0.02 0.03 0.04 Numerical (kN) E m p ir ic a l (k N ) Yield Base Shear 0 5000 10000 15000 20000 0 5000 10000 15000 20000 3 3 3 3 3 33 3 3 3 2 2 3 3 3 3 2 3 3 3 3 3 3 3 3 3 2 3 2 3 3 2 3 3 2 3 3 3 2 2 2 3 3 2 3 3 2 3 3 2 3 2 2 2 2 2 2 3 2 2 3 2 2 2 3 2 2 2 3 2 2 2 2 Numerical (m) E m p ir ic a l (m ) Yield Displacement 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25 Numerical (kN) E m p ir ic a l ( kN ) Base Shear at Crushing/Shear Failure 0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000 2 2 2 2 2 2 2 2 Numerical (m) E m p ir ic a l ( m ) Displacement at Crushing/Shear Failure 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Numerical (m/m) E m p ir ic a l ( m /m ) Ductility 0 10 20 30 40 50 60 0 10 20 30 40 50 60 (a) (b) (c) (d) (e) (f) (g) R 2 = 0.94 R 2 = 0.997 R2 = 0.99 R2 = 0.999 R 2 = 0.91 R2 = 0.71 R 2 = 0.73 41 Table 34. Coefficient of equations for different limit states of the column. Factor/ Interacti on Cracking Displacem ent (m) R2 = 0.997 Cracki ng Base shear (kN) R2 = 0.94 Yield Displacem ent (m) R2 = 0.999 Yield Base Shear (kN) R2 = 0.99 Displacement at Crushing/shear failure (m) Base Shear Crushing/S hear Failure (kN) R2 = 0.91 Ductili ty (m/m) Cubic R2 = 0.73 Quadratic R2 = 0.71 Cubic R2 = 0.77 Β 8.92E03 6748.5 0.022 7746.9 1.824694 1.662474 6232.05 82.288 α1 0 4.365 0 0 0.00915 0.01711 24.95 0.120 α2 0 0 2.17E06 11.913 0 0.000159 25.622 0 α3 3.40E04 4.547 1.36E04 32.687 0.1426 0.07414 102.4239 3.806 α4 1.76E03 329.43 8.09E04 2433.2 0 0.01857 2679.603 0 α5 3.45E04 588.7 5.84E03 1204.5 0.04553 0.151059 1174.49 6.318 α6 0 0 0 0 0 0 0 0 α7 0 0 0 0 0.00017 0.000871 0 0.0073 α8 0 0 0 0 0 0.0000 6.090 0 α9 0 0 0 0 0.0002 0.000458 2.322 0 α10 0 0 4.24E07 0 0 0 0 0 α11 0 0 9.26E06 3.177 0 0 0 0 α12 0 0 1.55E05 0.834 0 0 0.878 0 α13 0 0 0.00E+00 0 0 0 0 0 α14 8.48E06 1.258 5.87E05 1.19 0.004652 0.0106 7.157 0.2109 α15 3.84E05 12.486 6.23E04 141.30 6 0 0 0 0 α16 0 0.012 0.00E+00 0 4.3E06 0 0 0 α17 0 0 1.28E07 0 0 0 0 0 α18 4.59E06 0.454 7.78E06 0 0.00169 0.000426 0 0.0375 α19 2.96E04 0 1.07E03 0 0 0 0 0 α20 + 6.87E 5 16.675 3.94E04 48.478 0.00111 0 27.46 0.2020 α21 α22 9.6E06 0.0001 α23 α24 0.00021 0.0067 α25 42 Table 35. Validation of generated equations for cracking limit state. Cracking displacement (m) Cracking base shear (kN) H (m) As (%) f'c (Mpa) fy (Mpa) s (mm) Equation Original Error (%) equation original Error (%) 7 2.4 56 310 275 0.003572 0.004 10.71 4408 5110 13.72 9 2.8 42 390 250 0.004637 0.006 22.717 3201 3147 1.739 16 3.2 35 420 130 0.016462 0.017 3.165 1340 1569 14.597 18 3.6 29 460 100 0.02201 0.023 4.306 1207 1432 15.69 Table 36. Validation of generated equations for first yield limit state. Yeilding displacement (m) Yeilding base shear (kN) H (m) As (%) f'c (MPa) fy (MPa) s (mm) Equation Original Error (%) Equation Original Error (%) 7 2.4 56 310 275 0.018019 0.016 12.62 10768 10399 3.54 9 2.8 42 390 250 0.031075 0.031 0.24 10196 9362 8.92 16 3.2 35 420 130 0.108223 0.101 7.15 5583 5497 1.57 18 3.6 29 460 100 0.151834 0.142 6.93 5538 5398 2.59 Table 37. Validation of generated equation for base shear at first crushing. Crushing base shear (kN) H (m) As (%) f'c (Mpa) fy (Mpa) s (mm) Equation Original Error (%) 7 2.4 56 310 275 7811 7101 9.997 9 2.8 42 390 250 8243 8321 0.939 16 3.2 35 420 130 6528 6395 2.091 18 3.6 29 460 100 6807 6562 3.73 Table 38. Validation of generated equations for displacement at crushing/shear failure and ductility. Displacement at crushing/shear failure (m) Ductility (m/m) Quadratic Cubic Cubic H (m) As (%) f'c (Mpa) fy (Mpa) s (mm) Error (%) Error (%) Error (%) 7 2.4 56 310 275 8826 97077 32 9 2.8 42 390 250 1783 11303 1132 16 3.2 35 420 130 134 451 33 18 3.6 29 460 100 21 98 109 43 3.7 SUMMARY In this study, variations in three limit states: cracking, yielding of longitudinal bar and first crushing of concrete, with the change of four parameters (s, f’c, fy and As) have been investigated using design of experiments. Pushover analyses have been performed with 81 numerical models for each three different column heights in order to measure those limit states in terms of base shear and displacement. The effects of the four factors on the limit states have been determined with those 81 combinations by 34 factorial analyses. In order to take the effect of height into the Design of Experiment, the analyses were turned into 35 factorial analyses. The results demonstrated that three level factorial analyses are adequate to predict the limit states except the inelastic displacement and ductility of the bridge piers. 44 CHAPTER 4: SEISMIC PERFORMANCE EVALUATION OF A MULTI- SPAN RC BRIDGE WITH IRREGULAR COLUMN HEIGHTS OF VARYING DUCTILITY LEVELS 4.1 GENERAL Bridges are essential elements in a modern transportation network and play significant roles in a country’s economy. However, it has always been a major challenge to keep bridges safe and serviceable. Due to river geometry and topological profile, different irregularities are introduced in bridges. The most common form of irregularities in bridges is nonuniform height of columns over a basin. This causes earlier failure of shorter columns due to large deformation demand. The objective of this study is to assess the seismic performance a four equal span RC boxgirder bridge monolithically connected to columns of irregular heights. Here, different column height configurations and four ductility levels have been considered where their performance has been assessed through static pushover analyses, incremental dynamic analyses (IDA) and fragility analyses. Static pushover (SPO) analysis gives static pushover curves identifying different limit states and ductility capacity. Dynamic pushover curves have been generated from IDA results and compared with the SPO curves. IDA results have also been utilized in order to develop fragility curves for two regular and two irregular column height combinations of bridges with two different tie spacing. Thus, the effects of column height irregularities and different levels of ductility on the vulnerability of the bridge in longitudinal direction have been examined. 45 4.2 BRIDGES WITH DIFFERENT DUCTILITY LEVELS AND NONUNIFORM COLUMN HEIGHTS Vulnerability of bridges increases with the presence irregularity (Akbari 2010). Displacement ductility demand for short columns in bridges with irregular column heights increases significantly during an earthquake. This study will demonstrate the effect of irregularity in column heights on the performance of bridges using pushover analysis. Here, column of heights 7 m, 14 m and 21 m have been considered to make eight cases with different combinations of column height. Table 41 shows the combinations of column height variations for 7 m, 14 m and 21 m columns, which corresponds to short (S), medium (M) and long (L), respectively. Figure 41 shows one combination case SML case, which can be interpreted as column heights of 7 m, 14 m and 21 m, respectively. Figure 41. Bridges with irregular column height combination. 46 Table 41. Different column height combinations. Current seismic design codes/standards emphasize on providing adequate lateral reinforcement in bridge columns to achieve sufficient ductility during earthquakes. Previously tie spacing of 300 mm was very common in bridge columns designed before 1970 (Ruth and Zhang 1999). However, in CHBDC 2010 (CSAS606), the recommended tie spacing is six times the longitudinal bar diameter where tie should cover every alternate longitudinal bar. This study will assess the seismic performance of fourspan bridges with four ductility levels by considering the tie spacings of 75 mm, 100 mm, 150 mm and 200 mm. Tie spacing is allowed up to 150 mm for compression member in modern code. Confinement factor has been determined according to Park et. al. (1982). 4.3 MODELING OF THE BRIDGE A box girder bridge with four equal spans of length 50 m each has been modeled with SeismoStruct (2010). The description of this software is presented in Chapter 3. Table 42 shows the property of the bridge superstructure. 1500 mm x 3000 mm columns with 2% longitudinal Case C1 C2 C3 SSS 7 m 7 m 7 m MMM 14 m 14 m 14 m LLL 21 m 21 m 21 m MLL 14 m 21 m 21 m SLL 7 m 21 m 21 m SML 7 m 14 m 21 m MSL 14 m 7 m 21 m SLM 7 m 21 m 14 m 47 bars have been considered at all pier locations irrespective of the height of columns. 16 mm tie bar is used at every alternate longitudinal bar (Figure 42). Supports at each abutment are considered fixed in transverse and vertical directions, however, longitudinal direction is considered free. Fixed supports are considered at the base of each column. Columndeck connections are considered fixed. Deck and column have been modeled as inelastic displacementbased elements. 11.1 tonne/m and 20.2 tonne/m uniformly distributed mass have been assigned to the column and deck, respectively. Table 42. Sectional properties of bridge deck. EA (kN) EI (kNm2) EI (kNm2) GJ (kNm2) 2.0187 x 108 1.4926 x 108 2.3198 x 109 1.9747 x 108 Figure 42. Column cross section. The compressive strength of concrete and yield strength of steel are used as 35 MPa and 500 MPa, respectively. Cracking strain of concrete and yield strain of steel in tension are assumed to be 0.000133 and 0.025, respectively. Crushing strain of unconfined concrete varies from 0.0025 to 0.006 (MacGregor and Wight 2005). According to Paulay and Priestley (1992), crushing strain for confined concrete is much higher and it varies from 0.015 to 0.05. In the present study, crushing strain of confined concrete is considered to be 0.015. Static nonlinear 48 pushover analyses have been conducted with SeismoStruct (2010) using the concrete and steel models described in Chapter 3. 4.4 STATIC PUSHOVER ANALYSIS The performance of a bridge can be defined by limit states, related to strain limits. First cracking, first yielding and first crushing have been considered as the limit states of the bridges in this study. Static pushover (SPO) analyses have been conducted in the longitudinal direction in order to identify the limit states for all column combinations (Table 41). Base shears and lateral displacements of the bridges for the each of the limit states have been determined from the pushover curves. The effect of irregularity due to column height will be observed by comparing the performance of bridges with different column height combinations. 4.4.1 Effect of Tie Spacing on Bridge Performance Figures 43 to 45 show the bridge responses with regular column heights of 7 m, 14 m and 21 m, respectively, for different tie spacings of 75 mm, 100 mm, 150 mm and 200 mm obtained from pushover analyses. Effect of confinement has been found significant after the first yielding. The bridge with smaller tie spacing has higher total base shear capacity and can undergo larger deformation before concrete crushing. However, tie spacing does not have significant effect on the yield strength and strain. In the case of SSS, pushover curve remains flat after reaching the first crushing for 75 mm tie spacing whereas it undergoes a sharp fall for larger spacings of tie. The behaviour is the same in the cases of MMM and LLL. The columns of regular bridge did not show uniform pushover curves due to the initial deformation of deck under dead load. In the case of rigid deck, the behaviour of all three columns would be the same. 49 Figure 43. Results of pushover analysis in longitudinal direction for Case: SSS. Figure 44. Results of pushover analysis in longitudinal direction for Case: MMM. 50 Figure 45. Results of pushover analysis in longitudinal direction for Case: LLL. While considering irregular column height combinations, SLL and MLL are the irregular forms of the case LLL. Here, the stiffness of the column increases with the reduction of length and, thus, the shorter column attracts more load than the other two columns, which are shown in Figures 46 and 47. Pushover curves show more ductile behaviour for smaller tie spacing. The pushover curves for tie spacing of 200 mm falls just after it reaches the crushing load, which indicates its brittle behaviour. The ties start comes into action after the start of the concrete crush, therefore, pushover curves falls down and goes up due to strain hardening of tie bars. 4.4.2 Effect of Irregularity on Bridge Performance Pushover analyses are conducted considering irregularities in column height for five different combinations of column heights. The pushover curves of irregular bridges are given in Figures 46 to 48. In the cases of MLL and SLL, they have higher total base shear capacity than 51 that of case LLL, however those reaches to the yield and crush limit in lower strain. For the same deck displacement in longitudinal direction, the shorter column is subjected to more drift and since, it has more stiffness, it attracts more baseshear than the longer columns. Figure 46. Results of pushover analysis in longitudinal direction for Case: MLL. Tables 43 and 44 show the strengths and displacements of bridges with different column height combinations corresponding to yielding and crushing, respectively. For all cases, it is observed that tie spacing has little effect on the first cracking and first yielding limit states, however, it has significant effect on the first crushing limit state. The displacements needed for crushing and yielding of the cases MLL match with that of the cases MMM. The similar results were observed for SLL, SML, SLM and MSL and SSS cases. Therefore, the shortest column of the bridge determines the displacements corresponding to the first yielding and crushing. The 52 shortest column has the highest stiffness in the lateral direction; therefore, it controls the seismic behaviour and capacity of the bridge. The shortest column height in the cases of SML, MSL, SLM is 7m, however, the arrangements of different height of columns are different in these cases and hence the similar performance in terms of first yielding and crushing displacements are observed. The patterns of pushover curves for these three cases are also similar. Therefore, bridges, with of the same column height configuration with different orders, behaves the same in longitudinal direction. Figure 47. Results of pushover analysis in longitudinal direction for Case: SLL. 53 Figure 48. Results of pushover analysis in longitudinal direction for Case: MSL. Figure 49 shows the pushover curves for the cases LLL, SSS, SML and SLL for tie spacing of 75 mm. Although, all cases with a 7m column have almost the same yielding and crushing displacement; SSS possesses base shear capacity almost twice of the capacity of SLL and almost three times the capacity of SML. Since, the main portion of the bridge total mass is lumped in the deck, all four cases in Figure 49 show similar seismic force due to a particular earthquake event. Two cases with regular column height configurations: SSS, with high base shear capacity, and LLL, with high displacement tolerance, are expected to behave better in earthquake excitation. SLL and MLL are of irregular column heights. Among these, SLL is the most irregular, since the stiffness of the side column drastically increases and hence, the worst performance resulted. SML performs better than the Case SLL, because it has the irregularity in lower level with higher displacement tolerance. 54 Table 43. Yield strength and displacement for different column height combinations. Yielding 75 mm spacing 100 mm spacing 150 mm spacing 200 mm spacing Case Displac ement (m) Total base shear (kN) Displacem ent (m) Total base shear (kN) Displace ment (m) Total base shear (kN) Displacem ent (m) Total base shear (kN) SSS 0.02 38185 0.02 38190 0.02 38225 0.02 38265 MMM 0.09 17823 0.09 17800 0.09 17855 0.09 17880 LLL 0.2 11594 0.2 11598 0.2 11218 0.2 1SML6 MLL 0.09 10673 0.09 10678 0.09 10695 0.09 10710 SLL 0.03 16302 0.03 16300 0.03 16307 0.03 16280 SML 0.03 18021 0.03 18018 0.03 18030 0.03 18043 MSL 0.03 17422 0.03 17425 0.03 17444 0.03 17461 SLM 0.03 17851 0.03 17877.55 0.03 17891 0.03 17900 Figure 49. Results of pushover analysis in longitudinal direction for Cases: LLL, SLL, SML and SSS considering 75 mm tie spacing. 55 Table 44. Crushing strength and displacement for different column height combinations. 4.4.3 Ductility Figure 410 shows the ductility ratio at first crushing (CALTRANS, 2004) in the case of SLL for different tie spacings. It is well know that increase in tie spacing causes a decrease in confinement. Consequently, ductility decreases with the increase of tie spacing. Figure 410. Ductility of the bridge with column height configuration SLL for different tie spacing. Crushing 75 mm spacing 100 mm spacing 150 mm spacing 200 mm spacing Case Displa cement (m) Total base shear (kN) Displacem ent (m) Total base shear (kN) Displace ment (m) Total base shear (kN) Displace ment (m) Total base shear (kN) SSS 0.07 46643 0.06 45704 0.05 44066 0.04 43996 MMM 0.26 22174 0.21 21982 0.19 21229 0.14 21445 LLL 0.55 13657 0.43 13784 0.36 13710 0.33 13662 MLL 0.27 16277 0.22 15395 0.18 15395 0.16 13595 SLL 0.08 19661 0.07 19002 0.06 17839 0.05 17542 SML 0.08 22231 0.08 21998 0.06 20806 0.05 18504 MSL 0.09 23792 0.07 MMM3 0.07 22492 0.05 20320 SLM 0.08 22810 0.07 21845 0.06 20639 0.05 19780 56 4.5 INCREMENTAL DYNAMIC ANALYSIS SPO analysis gives the structural response under increasing static lateral load. However, lateral load due to earthquakes are truly dynamic in nature and structural response to the earthquake loads cannot be accurately predicted by SPO analysis. In order to overcome the drawbacks of SPO curves, Luco and Cornell (1998) introduced a technique, which performs a series of nonlinear dynamic analyses of the FE model of the structure for an ensemble of ground motions of increasing intensity resulting a set of dynamic response data corresponding to the lateral load with respect to the increment of ground motion intensity. This technique, called the incremental dynamic analysis (IDA), ensures the response of the structure under lateral load which is dynamic in nature (Vamvatsikos and Cornell 2001). Since, IDA generates a set of structural response data for increasing intensity level of ground motions, a dynamic pushover (DPO) curve can be generated by plotting the maximum displacements and corresponding base shear of the structure. Due to high computational cost of IDA, bridges with four column height combinations of LLL, SSS, SLM and SLL for two tie spacings of 75 mm and 200 mm tie spacing have been selected for IDA in this study. The SPO curves have been compared with DPO curves and to conduct fragility curve results have been generated from IDA results in order to observe the effect of bridge irregularity due to varying column heights and tie spacing of the columns. 4.5.1 Description of Ground Motion Properties The properties of earthquake ground motions vary in terms of predominant period, duration and peak ground acceleration (PGA). Therefore, time history analysis of a structure for a single ground motion may not represent the worst case scenario. Time history analyses with an ensemble of ground motions of varying characteristics gives a fair prediction of structural 57 response under an earthquake. Considering the variation in properties in terms of PGA to PGV ratio 10 earthquake ground motions have been selected for IDA. The magnitudes and epicentral distances of these ground motions are presented in Table 45. The acceleration response spectra (5% damped) for the selected ground motion sets are shown in Figure 411. Figure 411. Spectral acceleration for the chosen earthquake ground motions. Table 45. Selected earthquake ground motion records. No Event Year Record Station M1 R2 (km) PGA (g) PGA/PGV 1 Imperial Valley 1979 Plaster City 6.5 31.7 0.042 1.3 2 Imperial Valley 1979 Plaster City 6.5 31.7 0.057 1.05 3 Imperial Valley 1979 Westmoreland Fire Sta. 6.5 15.1 0.11 0.8 4 Imperial Valley 1979 El Centro Array#13 6.5 21.9 0.139 1.06 5 Loma Prieta 1989 Coyote Lake Dam 6.5 22.3 0.179 0.93 6 Loma Prieta 1989 Anderson Dam 6.9 21.4 0.244 1.2 7 Loma Prieta 1989 Hollister Diff. Array 6.9 25.8 0.279 0.79 8 Imperial Valley 1979 Cucapah 6.9 23.6 0.309 0.85 9 Loma Prieta 1989 16 LGPC 6.9 16.9 0.605 1.19 10 Superstation Hill 1987 Wildlife liquefaction array 6.7 24.4 0.132 1.03 1 Moment Magnitudes 2 Closest Distances to Fault Rupture PEER Strong Motion Database, http://peer.berkeley.edu/svbin 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2 2.5 S p e c tr a l A c c e le ra ti o n ( g ) Period (sec) Ground Motion 1 Ground Motion 2 Ground Motion 3 Ground Motion 4 Ground Motion 5 Ground Motion 6 Ground Motion 7 Ground Motion 8 Ground Motion 9 Ground Motion 10 58 4.5.2 Dynamic Pushover Curves Figures 412 to 419 show the total base shear versus lateral displacement plots for bridges with columns height combinations of LLL, SSS, SLM and SLL and of two different tie spacing of 75 mm and 200 mm based on the results of IDA. Shapes of the IDA curves are also included. DPO points closely coincide with the SPO curves before the first yield of the longitudinal bars. IDA show higher base shear capacity compared to that of SPO between the first yield and first crushing of the concrete, however, the variation of DPO and SPO is within 10% for the cases of SSS, SLM and SLL and is about 30% for the cases of LLL. Therefore, the ductility calculated based on the first yield of rebar and crushing of the concrete is acceptable. The DPO base shear capacity is lower than that of SPO beyond crushing point. Figure 412. Dynamic and static pushover curve for LLL with columns of 75 mm tie spacing. 0 5000 10000 15000 20000 25000 0 0.2 0.4 0.6 0.8 1 1.2 T o ta l B a s e s h e a r (k N ) Displacement (meter) DPO SPO Yield Crushing LLL- 75 mm 59 Figure 413. Dynamic and static pushover curve for LLL with columns of 200 mm tie spacing. Figure 414. Dynamic and static pushover curve for SSS with columns of 75 mm tie spacing. 0 5000 10000 15000 20000 25000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 T o ta l B a s e S h e a r (k N ) Displacement (cm) DPO SPO Yield Crushing LLL- 200 mm 0 10000 20000 30000 40000 50000 60000 0 0.1 0.2 0.3 0.4 0.5 T o ta l B a s e s h e a r (k N ) Displacement (meter) DPO SPO Yield Crushing SSS- 75 60 Figure 415. Dynamic and static pushover curve for SSS with columns of 200 mm tie spacing. Figure 416. Dynamic and static pushover curve for SLM with columns of 75 mm tie spacing. 0 10000 20000 30000 40000 50000 60000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 T o ta l B a s e s h e a r (k N ) Displacement (meter) DPO SPO Yield Crushing SSS- 200 mm 0 5000 10000 15000 20000 25000 30000 35000 0 0.1 0.2 0.3 0.4 0.5 T o ta l B a s e s h e a r (k N ) Displacement (meter) DPO SPO Yield Crushing SLM- 75 mm 61 Figure 417. Dynamic and static pushover curve for SLM with columns of 200 mm tie spacing. Figure 418. Dynamic and static pushover curve for SLL with columns of 75 mm tie spacing. 0 5000 10000 15000 20000 25000 30000 35000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 T o ta l B a s e S h e a r (k N ) Displacement (cm) DPO SPO Yield Crushing SLM- 200 mm 0 5000 10000 15000 20000 25000 30000 35000 0 0.1 0.2 0.3 0.4 0.5 T o ta l B a s e s h e a r (k N ) Displacement (meter) DPO SPO Yield Crushing SLL- 75 mm 62 Figure 419. Dynamic and static pushover curve for SLL with columns of 200 mm tie spacing. 4.6 FRAGILITY ANALYSIS Fragility curves represent the probability of reaching or exceeding a certain damage level of a structure under the earthquake ground motion excitation. Fragility curves can be generated empirically or analytically. The empirical fragility curves are generated by surveying bridge conditions after earthquakes, which is often impractical due to the lack of damage data (Padget and DesRoches 2008). In the analytical method of fragility analysis, linear/nonlinear timehistory analyses can be utilized (Hwang et al. 2001, Choi et al. 2004, Mackie and Stojadinovic, 2004). In this study, the effect of irregularity in column heights and effect of tie spacing have been observed by deriving the analytical fragility curves by probabilistic seismic demand model (PSDM) with IDA results for bridges with two regular and two irregular column height combinations with two different tie spacing of 75 mm and 200 mm. 0 5000 10000 15000 20000 25000 30000 0 0.1 0.2 0.3 0.4 0.5 T o ta l B a s e S h e a r (k N ) Displacement (meter) DPO SPO Yield Crushing SLL- 200 63 4.6.1 Characterization of Damage States In order to develop the fragility curves, the damage states need to be established. Different damage criteria have been established in order to assess the bridge condition, for example, column drift ratio (Dutta and Mander 1999), energy dissipation capacity and ductility demand (Park and Ang 1985, Caltrans 2004) etc. In this study, four damage states have been taken: slight damage, moderate damage, extensive damage and collapse. Ductility has been taken as the engineering demand parameter (EDP) for each damage state as specified in Hwang et al. 2001 for RC bridge column. 4.6.2 Fragility Function Methodology The probabilistic seismic demand model (PSDM) can be obtained by the scaling approach (Zhang and Huo 2009) or by the cloud approach (Choi et al. 2004; Nielson and Desroches 2007). In this study, the cloud method has been used in order to evaluate the seismic fragility functions of the four span bridges with different column height combinations and various tie spacing. Ductility and PGA have been considered as the engineering demand parameters (EDP) and the ground intensity measures (IM), respectively. A correlation between EDP and IM is to be established in the PSDM. Cornell et al. (2002) gives a logarithmic correlation between median EDP and selected IM: EDP = a (IM)b or, ln (EDP) = ln (a) + b ln (IM) (41) where, a and b are regression coefficients as shown in Figure 420. Table 46 shows the values of a and b for bridges with column height combinations of LLL, SSS, SLM and SLL with 75 mm and 200 mm tie spacing. 64 IDA gives the response of the structure for low to high ground motion intensities. It is necessary to obtain intermediate responses in the cloud approach. The dispersion of the demand, β EDP| IM, conditioned upon the IM can be derived as (Baker and Cornell, 2006): β EDP| IM = ∑ (( ) ()) (42) where, N is the number of data points. β EDP| IM for different combinations of bridges are shown in Table 46. β EDP| IM decreases with the increase of the tie spacing. Table 46. PSDMs for bridges with four different column height combinations with two tie spacings. Column combination ln (a) b β EDP| IM LLL 75 0.655 0.538 0.300 LLL 200 0.688 0.583 0.226 SSS 75 1.182 1.111 0.612 SSS 200 1.180 1.140 0.513 SLM 75 2.454 1.074 1.113 SLM 200 2.511 1.102 1.021 SLL 75 2.454 0.905 1.113 SLL 200 2.478 0.936 0.943 65 Figure 420. Comparison of the PSDMs for LLL, SSS, SLM and SLL for 75 mm and 200 mm tie spacing. y = 0.5384x - 0.4231 R² = 0.7246 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -2 -1 0 1 2 3 ln ( D u c ti li ty ) ln(PGA) LLL- 75 mm y = 1.1106x + 0.1675 R² = 0.9448 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1 0 1 2 ln ( D u c ti li ty ) ln (PGA) SSS- 75 mm y = 1.1408x + 0.1658 R² = 0.931 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1 0 1 ln ( D u c ti li ty ) ln (PGA) SSS- 200 mm y = 1.0737x + 0.8978 R² = 0.9054 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2 -1 0 1 2 ln ( D u c ti li ty ) ln (PGA) SLM- 75 mm y = 1.1024x + 0.9205 R² = 0.9147 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2 -1 0 1 2 ln ( D u c ti li ty ) ln (PGA) y = 0.9357x + 0.9076 R² = 0.8456 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 ln ( D u c ti li ty ) ln (PGA) SLL- 200 mm y = 1.0737x + 0.8978 R² = 0.9054 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2 -1 0 1 2 ln ( D u c ti li ty ) ln (PGA) SLL- 75 mm y = 0.5831x - 0.3744 R² = 0.8494 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -2 -1 0 1 2 3 ln ( D u c ti li ty ) ln (PGA) LLL- 200 mm 66 Now, the PSDM can be expressed as (Nielson, 2005): P[LS|IM] = φ[ ()() ] (43) Where, ln(IMn) is the median value of the intensity measure for the chosen damage state (slight, moderate, extensive, collapse). ln(IMn) can be derived as ln(IMn)= () () (44) and βcomp, is the dispersion component, can be determined as βcomp= | (45) where, Sc is the median value and βc is the dispersion of the capacity of the structure. Sc and βc are the lognormal parameters for damage states. Fragility curves are generated by describing the structural capacity and seismic demand to a lognormal distribution, which represent the probability of reaching or exceeding a specific damage state. Sc and βc values are given in Table 47 for the specified damage state of RC bridges (Hwang et al. 2001). Table 47. Limit states for RC bridge. Damage state Sc βc slight 1 0.59 moderate 1.2 0.51 extensive 176 0.64 collapse 4.76 0.65 67 4.6.3 Fragility Curves Results The results of the PSDM for bridges with column height combinations of LLL, SSS, SLM and SLL and of two different tie spacing of 75 mm and 200 mm are presented in Figures 421 to 424 for reaching or exceeding slight, moderate, extensive and collapse damage states. In the case of LLL, the bridge has the lowest probability of damage among all combinations. It can be observed from the fragility curves that the bridges with regular column height combination (SSS, LLL) needs higher intensity of ground motion excitation than the bridges with irregular column heights (SLM, SLL) for reaching the same damage level. Bridges with smaller tie spacing are less vulnerable for all column height combinations, however, the effect of irregularity in column height has been found to have more effect than that of variation in tie spacing from 75mm to 200mm. Figure 421. Comparison of fragility curves for bridges with different column height combinations for 75 mm and 200 mm tie spacing for slight damage. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 P ro b a b il it y o f S li g h t D a m a g e PGA (g) LLL- 75 mm LLL- 200 mm SSS- 75 mm SSS- 200 mm SLM- 75 mm SLM- 200 mm SLL- 75 mm SLL- 200 mm 68 Figure 422. Comparison of fragility curves for bridges with different column height combinations for 75 mm and 200 mm tie spacing for moderate damage. Figure 423. Comparison of fragility curves for bridges with different column height combinations for 75 mm and 200 mm tie spacing for extensive damage. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 P ro b a b il it y o f M o d e ra te D a m a g e PGA (g) LLL- 75 mm LLL- 200 mm SSS- 75 mm SSS- 200 mm SLM- 75 mm SLM- 200 mm SLL- 75 mm SLL- 200 mm 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 P ro b a b il it y o f E x te n s iv e D a m a g e PGA (g) LLL- 75 mm LLL- 200 mm SSS- 75 mm SSS- 200 mm SLM- 75 mm SLM- 200 mm SLL- 75 mm SLL- 200 mm 69 Figure 424. Comparison of fragility curves for bridges with different column height combinations for 75 mm and 200 mm tie spacing for collapse damage. 4.7 SUMMARY In this study, static pushover analyses and incremental dynamic analyses have been conducted for bridges in longitudinal direction considering irregularity in column height with varying tie spacing in bridge column. This study included the comparison of SPO and DPO, and the development of fragility curves show the effect of irregularity in column heights and tie spacing on the seismic performance of bridges. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 P ro b a b il it y o f C o ll a p s e D a m a g e PGA (g) LLL- 75 mm LLL- 200 mm SSS- 75 mm SSS- 200 mm SLM- 75 mm SLM- 200 mm SLL- 75 mm SLL- 200 mm 70 CHAPTER 5: COMPARISON OF DIRECT DISPLACEMENT-BASED DESIGN AND FORCE-BASED DESIGN IN CANADIAN CONTEXTS 5.1 GENERAL Economic bridge design is a great challenge, especially in an earthquake prone zone while ensuring safety. Canadian Highway Bridge Design Code (CHBDC 2010) and AASHTO 2007, like other traditional design codes follow forcebased design (FBD) method, which is focused at the target force resistance capacity of the structure. On the other hand, displacementbased design approach aims to ensure a target maximum displacement of the bridge during the earthquake in a specific zone. In this study, the columns of a bridge with irregular column heights have been designed according to direct displacementbased design (DDBD) method and FBD as per Canadian standards and AASHTO 2007 considering seismic loading. Seismic performances of the bridge designed in two different methods have been compared by nonlinear dynamic analyses in the longitudinal direction in terms of maximum and residual displacements and energy dissipation capacity. This study outlines the necessity of possible modification in the current Canadian seismic design standards as well as displacementbased design for bridges with irregular column heights. 5.2 SAMPLE BRIDGE A box girder highway bridge having four equal spans of 50 m with varying column heights has been considered in this study. The bridge is assumed to be located in Vancouver, BC, Canada. Figure 51 shows the column height irregularity considered in this study. The height of the middle column is 21 m and the heights of the side columns are 7 m and 14 m, respectively as 71 shown in Figure 51. The columns have been designed for the seismic loading in the longitudinal direction. Table 51 shows the property of the bridge superstructure. Supports at the abutments are considered to be fixed in the transverse and vertical directions; however, the longitudinal direction is considered as free. The support at the base as well as columndeck connection of each column is considered as fixed. Uniformly distributed mass of 11.1 tonne/m and 20.2 tonne/m have been assigned to the column and deck, respectively. Compressive strength of concrete and yield strength of steel used in this study are 35 MPa and 400 MPa, respectively. Figure 51. Elevation of the bridge with irregular column heights. Table 51. Sectional properties of bridge deck. EA (kN) EI2 (kNm2) EI3 (kNm2) GJ (kNm2) 2.0187 x 108 1.4926 x 108 2.3198 x 109 1.9748 x 108 5.3 BRIDGE COLUMN DESIGN Design moment and shear of the columns of the sample bridge have been determined according to DDBD and FBD. The design procedures of these two methods are shown in Figure 52. The longitudinal reinforcement has been designed from the bending moment demand. Shear resistance of the column section has been checked using Modified Compression Field Theory 72 (Vecchio and Collins 1986), which predicts the experimentally determined shear failure within 1% error (Bentz et al. 2006). (a) (b) Figure 52. Flowcharts showing step by step procedures: (a) displacementbased and (b) forcebased design. Yes Calculate Yield Displacement, Δy Choose Column Section Calculate Ductility, μ Calculate Target Design Displacement, Δd = Δy * μ Calculate Equivalent viscous damping, ξ Calculate Effective time period, Te from Displacement Response Calculate Effective Lateral Stiffness, Ke Ke < Kelastic Calculate Total Base Shear = Ke * Δd Distribute Base Shear, Vdesign to indivudual column according to 1/H Mdesign < Mcapacity 1.6 Vdesign < Vcapacity Calculate Design Moment, Mdesign Design Finished, Keep the Chosen No Yes Yes No C h a n g e S e c ti o n P ro p e rt ie s In c re a s e S e c ti o n S iz e DDBD Choose Column Section Calculate Elastic Lateral Stiffness, K Calculate Time Period, T Determine Spectral Acceleration, ag from Response Spectra Calculate Total Elastic Base Shear Demand Distribute Base Shear, Velastic to indivudual column according to 1/H3 Select Response Modification Factor, k, and Calculate Vinleastic = Velastic / k Calculate Design Moment, Mdesign, Corresponding to Vinleastic Mdesign < Mcapacity Velastic < Vcapacity Design Finished, Keep the Chosen Column Section Properties No C h a n g e S e c tio n P ro p e rt ie s FBD 73 5.3.1 Direct DisplacementBased Design (DDBD) In displacementbased approach, a target displacement and effective stiffness of the bridge have to be determined in order to calculate the base shear demand. The procedure described in Priestley et al. (2007) has been adopted in this study. In DDBD a number of design solutions are possible by choosing column section size and level of ductility. Therefore, the designer has options in choosing column size and tie spacing. In this study 3m x 1.5m section is chosen. Since column C1 is the shortest column, the yield displacement and target displacement of the bridge are governed by the column, C1. From section size, the yield displacement has been found to be 22.9 mm. Confinement reinforcement in C1 is provided as 12mm tie @ 135 mm c/c, the target displacement for damage control limit state is found to be 111.8 mm. Therefore, the corresponding ductility for C1, C2 and C3 are 4.89, 0.54 and 1.22, respectively. Hence, the equivalent viscous damping ratio of an individual column has been derived as per equation 51 (Priestley et al., 2007). = 0.05 + 0.444 (51) Here, the first part of the equation is for 5% material damping of concrete and the second part is for hysteresis damping, calculated from ductility (μ). The equivalent viscous damping ratios for C1, C2 and C3 have been found to be 0.162, 0.05 and 0.076 respectively. The equivalent viscous damping ratio for the whole system is derived according to equation 52. = ∑ ∑ (52) Where, Vi is the distributed base shear in each column. Base shear distribution in the column is proportional to the inverse of the column height in DDBD. The equivalent viscous 74 damping of the bridge has been found to be 11.83%. The displacement spectra for Vancouver at 11.83% damping has been derived by applying the spectral reduction factor (equation 3) to the 5% damped displacement spectrum, as show in Figure 53. = . . . (53) The effective time period (Te) of the system for the target displacement of 111.83 mm is 1.747 sec, which is determined from the displacement response spectrum of Vancouver. The effective weight (We) of the bridge is 101,043 kN. The effective stiffness of the structure has been determined according to equation 54. = = 133,363 kN/m (54) Figure 53. Design displacement spectra for Vancouver. Total base shear demand has been calculated as 14,912 kN by multiplying the effective stiffness and the target displacement. This base shear is distributed to each column in inverse proportion to the height of the column. Therefore, columns will be subjected to equal bending moments, which leads to equal design longitudinal reinforcement. The design base shear and bending moment in each column is given in Table 52. The design reinforcement is provided in 0 10 20 30 40 0 1 2 3 4 5 D is p la c e m e n t (c m ) Time (sec) Displacement Response Spectra 5% damping 11.83% damping 75 Table 53. Design longitudinal reinforcement for each column is 9235 mm bar. The shear capacity of the column should be greater than 1.6 times the base shear corresponding to the design moment (Wang et al. 2008). Tie bar for C1 is 12mm135 mm c/c as mentioned previously. The required lateral reinforcement in C2 and C3 are 12 mm at 600 mm and 400 mm, respectively. Due to lower ductility demand and lower base shear demand, the required lateral reinforcements in C2 and C3 are low. However, tie spacing more than 300 mm is not very common. The maximum tie spacing allowed in CHBDC 2010 is 150 mm whereas there is no strict guideline for DDBD. Therefore, the comparison between DDBD and FDB are as following: i) Bridge designed as per DDBD where longitudinal and lateral reinforcements are governed by flexure and shear demand ii) Bridge designed as per DDBD for longitudinal rebar; however, with limited tie spacing as specified in CHBDC (2010) iii) Bridge designed as per DDBD for longitudinal rebar; however, an intermediate tie spacing of 144 mm has been selected between 135 mm and 150 mm. 5.3.2 Force Based Design (FBD) The base shear demand has been computed through elastic response spectrum analysis, which has been conducted considering the acceleration response spectrum of Vancouver (NBCC 2010). Elastic response spectrum analysis is performed by calculating the elastic base shear demand for the natural period of the bridge. The response spectrum for Vancouver is shown in Figure 54. The initial stiffness and natural period of the bridge in longitudinal direction have been determined as 576,276 kN/m and 0.84 sec, respectively. The elastic base shear demand has 76 been calculated as 30,102 kN by multiplying the mass of the bridge with the spectral acceleration corresponding to the natural period of the structure. According to CHBDC 2010, the response modification factor (R) is 3. Therefore, the inelastic base shear demand is one third of the elastic base shear demand. Inelastic base shear demand is 10,034 kN. The total base shear has been distributed to each of the three columns according to their initial stiffness, which is given in Table 52. The design reinforcement for the columns is given in Table 53. The shear capacity of the column should be greater than the elastic shear load (CHBDC 2010). The longitudinal reinforcement in column C1 is 9235 mm bars with 12 mm tie bar at 55 mm c/c. Minimum 1% reinforcement governs in columns C2 and C3 for design of longitudinal reinforcement. Maximum 150 mm spacing has been provided for 10 mm tie bar, used for the longitudinal bars of 25 mm, at each alternate longitudinal bar. According to CHBDC 2010 the maximum tie spacing is the smallest of six times the longitudinal bar diameter or 0.25 times the minimum component dimension or 150 mm and tie should cover every longitudinal bar. Figure 54. Design acceleration spectra for Vancouver. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 A c c e le ra ti o n ( g ) Time (sec) Acceleration Response Spectra 5% Damping 11.83% Damping 77 Table 52. Design shear and moment in columns. Design Method C1 C2 C3 V (kN) M (kNm) V (kN) M (kNm) V (kN) M (kNm) DDBD 8606 30122 2869 30122 4303 30122 FBD 25905 30223 960 3358 3237 7556 Table 53. Design reinforcement in columns. DDBD (Direct Displacement Based Design) DDBD (Tie Spacing Limited by CHBDC) DDBD (Equal Tie Spacing in Columns) FBD (Force Based Design) C1 9235 mm Tie bar 12mm135 mm c/c 9235 mm Tie bar 12mm 135 mm c/c 9235 mm Tie bar 12mm 144 mm c/c 9235 mm Tie bar 12mm 55mm c/c C2 9235 mm Tie bar 12mm400 mm c/c 9235 mm Tie bar 12mm 150 mm c/c 9235 mm Tie bar 12mm 144 mm c/c 9225 mm Tie bar 10 mm150mm c/c C3 9235 mm Tie bar 12mm600 mm c/c 9235 mm Tie bar 12mm 150 mm c/c 9235 mm Tie bar 12mm 144 mm c/c 9225 mm Tie bar 10 mm150mm c/c 5.3.3 Comparison between DDBD and FBD For bridges with irregular column heights, the distribution of total base shear demands for flexure and shear design of columns are different in DDBD and FBD. In DDBD the total base shear in an individual column is inversely proportional to the column height, resulting equal bending moment demand in each column, which can be expressed as = ⋯ (55) Where, Vi is the base shear of an individual column and V is the total base shear and Hi is the height of the column. Therefore, the longitudinal reinforcement for each column is the same. 78 However, in FBD, the base shear distribution is inversely proportional to the cube of the column heights (equation 56), resulting design moments in the columns are inversely proportional to the square of the column heights = ⋯ (56) As per FBD, the required amount of longitudinal reinforcement is significantly less in longer column than shorter columns compared to that of DDBD. In case of this bridge, the amount of longitudinal reinforcement in the longer columns is governed by the minimum steel concrete ratio of 1%. Therefore, longitudinal reinforcement in the longer columns is higher than required. The main differences in the two design methods include: a) DDBD considers 1.6 times the base shear corresponding to the design moments and FBD takes the elastic base shear demand according to Canadian Code, which ensures a conservative design for shear resistance; b) In the case of shear design, base shear in the 7 m column has been 88% more for FBD than that of DDBD design, which causes 59% smaller tie spacing in FBD than the DDBD. However, in cases of 14 m and 21 m columns base shears in FBD are 52% and 79% lower, respectively than those in DDBD. The tie spacing in longer columns has been limited by the Canadian code. Since, DDBD does not have any upper limit for tie spacing; the required tie spacing in longer columns have been found even more than 300 mm. 5.4 NONLINEAR DYNAMIC ANALYSIS Nonlinear dynamic time history analysis (NTHA) involves with higher computational cost than the other methods of structural performance evaluation, for example capacity spectrum 79 method (ATC 40). However, this method gives an accurate response of a structure subjected to a particular ground motion. In order to evaluate the dynamic performances of the three DDBD and one FBD bridges finite element models have been generated in SeismoStruct (2010), which is based on the fiber modeling approach. NTHA has been used for the simulation of bridge response to the selected earthquake ground motions. 5.4.1 Selection and Scaling of Ground Motions The structural response depends on the ground motion properties of the earthquake, which can vary in a wide range in terms of predominant period, peak ground acceleration (PGA), peak ground velocity (PGV) and duration. Since, the upcoming earthquake characteristics in any area is truly unpredictable, a set of earthquake ground motions is generally selected containing a different ground motion characteristics in order to predict the worst structural response through NTHA. In this study, an ensemble of seventeen ground motion records has been selected. The properties of these ground motions are provided in Table 54. The predominant period of the structure is varied from 0.09 to 4.55 sec, whereas, the PGA varies from 0.22 to 0.73 g. The acceleration spectra of the ground motions are shown in Figure 55. Since, the bridges have been designed for Vancouver region involving firm soil with 10% probability of exceedance in fifty years, the original earthquake ground motions need to be scaled to fit the design spectra of Vancouver for 5% damping, which is shown in Figure 53. In this study, the earthquake ground motions have been scaled using the method proposed by Shome et al. (1998). In this method, scaling factor for each ground motion is determined so that the spectral acceleration of that ground motion matches to the design spectral acceleration at the firstmode period of the structure. The first modal periods for the bridges designed in DDBD and FBD are 0.692 and 0.696 sec respectively, which are very close. The scale factors have been determined at the 80 spectral period of 0.69 s. Figures 55 and 56 show the acceleration spectra for original and scaled ground motions, respectively. The spectral acceleration of scaled ground motions merge at the period of 0.69 sec. Figure 55. Spectral acceleration for original earthquake ground motions. 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 4 S p e c tr a l A c c e le ra ti o n ( g ) Period (sec) Selected Original Ground Motions EQ 1 EQ 2 EQ 3 EQ 4 EQ 5 EQ 6 EQ 7 EQ 8 EQ 9 EQ 10 EQ 11 EQ 12 EQ 13 EQ 14 EQ 15 EQ 16 EQ 17 81 Figure 56. Spectral acceleration for scaled earthquake ground motions. 5.4.2 Time History Analysis Results Longitudinal responses of the bridge have been simulated by NTHA using SeismoStruct (2010) for 17 scaled ground motions. Collapse of bridge in time history analysis is defined for instability or exceeding the 5% drift (Dutta and Mander 1998) of the column. The FBD bridge collapsed for the scaled ground motion of EQ 2. The DDBD bridge collapsed in EQ 2 and 9. The DDBD bridge with limited tie spacing and equal tie spacing of 144 mm survived all 17 earthquakes. Maximum displacement demand, residual displacement and energy dissipation have been set as the parameters in order to compare the seismic performance of the bridges designed in two methods. In order to compare between the four cases; mean (μ) and standard deviation (σ) have been determined for these parameters excluding EQs 2 and 9, since collapse occurred in some of these cases in these two earthquake ground motions. Mean plus standard deviation and mean plus twice standard deviation represents 68% and 95% data are within the limit. 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 4 S p e c tr a l A c c e le ra ti o n ( g ) Period (sec) Scaled Ground Motions EQ 1 EQ 2 EQ 3 EQ 4 EQ 5 EQ 6 EQ 7 EQ 8 EQ 9 EQ 10 EQ 11 EQ 12 EQ 13 EQ 14 EQ 15 EQ 16 EQ 17 T = .69 82 Table 54. Earthquake ground motion properties. No Earthquake Recording Station Epicentral Distance (km) PGA (g) PGV (cm/s.) Predominant Period (s) M Year Name Name EQ1 6.7 1994 Northridge Beverly Hills Mulhol 13.3 0.42 58.95 0.759 EQ2 7.3 1992 Landers Yermo Fire Station 86 0.24 52 4.551 EQ3 6.7 1994 Northridge Canyon Country WLC 26.5 0.41 42.97 0.63 EQ4 7.3 1992 Landers Coolwater 82.1 0.28 26 0.64 EQ5 7.1 1999 Duzce, Turkey Bolu 41.3 0.73 56.44 1.078 EQ6 6.9 1989 Loma Prieta Capitola 9.8 0.53 35 0.683 EQ7 6.9 1989 Loma Prieta Gilroy Array #3 31.4 0.56 36 0.745 EQ8 7.4 1990 Manjil, Iran Abbar 40.4 0.51 43 1.781 EQ9 6.5 1979 Imperial Valley El Centro Array #11 29.4 0.36 34.44 4.096 EQ10 6.5 1987 Superstition Hills El Centro Imp. Co. 35.8 0.36 46 0.09 EQ11 6.9 1995 Kobe, Japan NishiAkashi 8.7 0.51 37.28 1.28 EQ12 6.7 1994 Nothridge Rinaldi 7.5 0.38 59.7 0.301 EQ13 6.7 1994 Nothridge Olive View 6.4 0.72 120 2.341 EQ14 7.0 1992 Cape Mendocino Rio Dell Overpass 22.7 0.39 44 0.509 EQ15 7.5 1999 Kocaeli, Turkey Duzce 98.2 0.31 59 0.836 EQ16 7.5 1999 Kocaeli, Turkey Arcelik 53.7 0.22 17.69 1.205 EQ17 7.6 1999 ChiChi, Taiwan TCU045 77.5 0.47 37 1.205 5.5 PERFORMANCE COMPARISON DDBD AND FBD BRIDGE The data of maximum and residual displacements and dissipated energy by the bridges have been extracted from the simulated responses to the ground motions, which have been scaled to match the response spectrum of Vancouver. Maximum displacement is the primary indicator of the structural response to an earthquake. Probability of inelastic deformation and damage of the structure increases with the increase of maximum displacement demand. Therefore, the structure with lower maximum displacement is expected to perform better than the structure with higher displacement demand during an earthquake. The residual displacement indicates the level 83 of damage and reusability of the structure after an earthquake. The higher residual displacement is also involved with higher repair and rehabilitation cost. Energy dissipation capacity of the structure is also an important parameter for seismic performance of the structure. Higher energy dissipation capacity indicates the better ductile behaviour of the structure under dynamic loading. The comparison of the bridges based on these performance criteria are as following. 5.5.1 Maximum Displacement Figure 57 shows the displacement demand comparison between the four cases and the maximum, μ, μ + σ and μ + 2σ values for the ground motion excitations, which did not cause any collapse of the bridges, are given in Table 55. Among the three DDBD bridges, the bridge with limited tie spacing has the lowest displacement demand in terms of μ, μ + σ and μ + 2σ. The displacement demand of FBD bridge is lower than that of the DDBD bridges, although, it is very close to the original DDBD bridge and DDBD bridge with limited tie spacing. The DDBD bridge with equal tie spacing is the worst among the four bridges. Table 55. Displacement demand of DDBD and FBD bridges from NTHA. Maximum Displacement (mm) DDBD DDBD (Limited tie spacing) DDBD (Equal Tie spacing of 144 mm) FBD Max 151 152 192 143 μ 77 75 80 76 μ + σ 103 101 115 101 μ + 2σ 128 126 150 125 84 Figure 57. Maximum displacement demand of bridges designed in displacementbased and forcebased approach. 5.5.2 Residual Displacement Figure 58 shows the residual displacement of bridges under ground motion excitations and the maximum, μ, μ + σ and μ + 2σ values for the ground motion excitations, which did not cause any collapse of the bridges, are given in Table 55. The DDBD bridge with limited tie spacing performs the best with respect to residual displacement among the DDBD bridges in terms of μ, μ + σ and μ + 2σ. The residual displacement of DDBD bridge with limited tie spacing is less than half of that of the original DDBD bridge, which indicates that, the performance of DDBD bridge can be significantly improved by imposing the maximum allowable tie spacing rule as per Canadian code. Residual displacement is the lowest in the case of FBD bridge, which implies the better capability of FBD bridge to restore its original position. 0 25 50 75 100 125 150 175 200 225 250 275 D is p la c e m e n t D e m a n d ( m m ) DDBD DDBD (Limited tie spacing) DDBD (Equal Tie spacing of 144 mm) FBD Collapse 85 Figure 58. Residual displacement of bridges designed in displacementbased and forcebased approach. Table 56. Residual displacement of DDBD and FBD bridges from NTHA. Residual Displacement (mm) DDBD DDBD (Limited tie spacing) DDBD (Equal Tie spacing of 144 mm) FBD Max 51 21 31 13 μ 15 8 11 4 μ + σ 27 14 19 8 μ + 2σ 39 21 28 11 5.5.3 Energy Dissipation The energy dissipation under seismic loading is maximum in the case of FBD bridge and and the maximum, μ, μ + σ and μ + 2σ values for the ground motion excitations, which did not cause any collapse of the bridges, are given in Table 55. The minimum energy dissipation in the case of DDBD bridge with limited tie spacing in terms of μ, μ + σ and μ + 2σ, which is shown in Figure 59. The FBD bridge shows the highest energy dissipation capacity. However, variation in energy dissipation among the four cases is very low and is less than 6% for μ, μ + σ and μ + 2σ. 0 20 40 60 80 100 120 140 R e s id u a l D is p la c e m e n t (m m ) DDBD DDBD (Limited tie spacing) DDBD (Equal Tie spacing of 144 mm) FBD Collapse 86 Table 57. Energy dissipation of DDBD and FBD bridges from NTHA. Energy Dissipation (kNm) DDBD DDBD (Limited tie spacing) DDBD (Equal Tie spacing of 144 mm) FBD Max 5430 5363 4978 5782 μ 2978 2945 2996 3031 μ + σ 4200 4081 4121 4257 μ + 2σ 5421 5217 5245 5483 Figure 59. Dissipated energy of bridges designed in displacementbased and forcebased approach in time history analyses. 5.5.4 Base Shear Demand Figures 510 to 512 show the base shear demand in columns C1, C2 and C3, respectively for the ground motion excitations. Base shear demands in each column have been found lower than the shear capacity, therefore, no shear failure has been observed. Since FBD takes the elastic base shear for shear design, the shear capacity of 7 m column in the FBD bridge is more than twice that of the base shear demand; however, the design moment is 23% lower than the 0 2000 4000 6000 8000 10000 12000 E n e rg y D is s ip a ti o n ( k N -m ) DDBD DDBD (Limited tie spacing) DDBD (Equal Tie spacing of 144 mm) FBD Collapse 87 demand. Both of the design base share for shear design and base share corresponding to design moment for 14 m and 21 m column are lower than those of the base shear demands from time history analyses. Since the provided longitudinal and transverse reinforcement in 14 m and 21 m columns are governed by the code specified minimum amount, the actual capacity of these columns are higher than the demand. Therefore, these columns did not fail in the time history analyses. FBD is not accurate in determining the design load in shorter and longer columns. It was observed that the shear capacity of 7 m column in DDBD bridge is 21% higher than the base shear demand. The design moment of 7m column in DDBD is almost equal to that of FBD; however, FBD experienced higher demand by 20% compared to that of DDBD. In longer column, unlike FBD, the design moment as per DDBD is equal to that of shorter column, whereas it is smaller by 24% and 19% in the case of 14 m and 21 m FBD columns compared to that of 7m FBD columns, respectively. The shear capacities of 14 m and 21 m columns in DDBD bridges are 21% and 29% higher than those of the demand base shear, respectively. Hence, the shear capacity for all short and long columns in DDBD is fairly well above the demand, and its moment capacity is slightly lower than the demand, however, this method ensures an even distribution of base shear to the columns of different heights. 88 Figure 510. Comparison of base shear demand of C1 of bridges designed in displacementbased and forcebased approaches through time history analyses. Figure 511. Comparison of base shear demand of C2 of bridges designed in displacementbased and forcebased approaches through time history analyses. 9000 9500 10000 10500 11000 11500 B a s e S h e a r D e m a n d o f C 1 ( k N ) DDBD DDBD (Limited tie spacing) DDBD (Equal Tie spacing of 144 mm) FBD 0 500 1000 1500 2000 2500 3000 3500 4000 4500 B a s e S h e a r D e m a n d o f C 2 ( k N ) DDBD DDBD (Limited tie spacing) DDBD (Equal Tie spacing of 144 mm) FBD 89 Figure 512. Comparison of base shear demand of C3 of bridges designed in displacementbased and forcebased approaches through time history analyses. 5.6 DISCUSSION Results obtained from dynamic time history analysis indicate that the FBD bridge performs better than the DDBD bridges in terms of displacement demand, residual displacement and energy dissipation capacity considering μ, μ + σ and μ + 2σ. However, the original DDBD bridge and the FBD bridge experienced collapse under two and one ground motion excitations respectively among the 17 earthquake records. On the other hand, the DDBD bridges with limited tie spacing and equal tie spacing did not fail in any of the earthquake time history analyses. Therefore, the DDBD with limited tie spacing suggested by CHBDC 2010 has been found to be more balanced compared to the other three design methods. It is also observed that the FBD method highly overestimates the base shear for the shortest column in shear design, however, predicts design moment lower than the demand by the same amount of DDBD case. 0 1000 2000 3000 4000 5000 6000 7000 B a s e S h e a r D e m a n d o f C 3 ( k N ) DDBD DDBD (Limited tie spacing) DDBD (Equal Tie spacing of 144 mm) FBD 90 However, the prediction of base shear for shear design and design moment for the design of longer columns are significantly lower than the demand in the case of FBD method. 5.7 SUMMARY In this study, an irregular bridge with varying column heights has been designed in conventional forcebased approach and displacementbased approach. The limitations of the two methods have been identified by the seismic performance evaluation of the bridges. 91 CHAPTER 6: CONCLUSIONS 6.1 SUMMARY This thesis explores the effects of concrete and steel properties, amount of longitudinal reinforcement and confinement on the limit states of single RC bridge piers, which helps understanding the performance of old and modern bridge pier during earthquake events. The combined effect of piers of different heights has been investigated by analyzing the seismic performance of irregular bridges with varying column heights. The effect of tie spacing has been compared with the effect of column height irregularity, which is the most common form of irregularity in bridges considering seismic performance. This study includes the design of a RC bridge with irregular column height configuration in conventional forcebased approach and displacementbased approach. The dynamic performances of these bridges have been assessed in order to identify the limitations of both approaches while designing the bridges with irregular column height combinations. 6.2 LIMITATIONS OF THE STUDY The main limitations of the current study are Only one column crosssection with fixed column ends has been taken for factorial analysis. This study only considered continuous bridges with fixed columndeck connection. A particular deck property has been taken for bridges. 92 Irregular bridge with only one combination of varying column heights has been considered for comparisons between displacementbased and forcebased design methods. 6.3 CONCLUSIONS Based on the results obtained from the factorial analysis, the following conclusions are drawn: Columns of 7 m height considered in this study are mostly shear dominated. Crushing displacement of flexure dominated column is always greater than its yield displacement resulting ductility greater than one. The ductility of a shear dominated column will be greater than one if the displacement due to shear failure is more than the yield displacement. Otherwise, the displacement at shear failure is less than the yield displacement, yield point cannot be reached and the virtual ductility can be calculated with respect to virtual yield displacement, which is less than one. The tendency of shifting to flexure dominance from shear dominance increases with the increase of the length of the column. Effect of confinement does not have significant effect on the column performance before yielding in flexure dominated cases. However, it has significant effect in plastic zone on crushing displacement and ductility of the column especially for shorter column. 93 The effect of tie spacing or confinement is more in high strength concrete. The amount of longitudinal reinforcement has significant effect on yield and crushing base shear, but, it has little effect on the yield and crushing displacement for flexure dominated columns. Compressive strength of concrete mainly controls the cracking base shear. Cracking displacement is not affected by the four factors. Three level factorial analyses have been found adequate in order to develop generalized formula to predict cracking limit state, yield limit state and base shear at first crushing or shear capacity, however, have been found inadequate to develop formula in order to predict displacement at first crushing or shear failure and ductility. Higher level factorial analysis is needed to predict the post elastic displacement of the column. Based on the results obtained from seismic performance evaluation of irregular bridges with varying column heights, the following conclusions can be drawn: Effect of confinement has little effect on the first cracking and first yielding limit states. Since, the smaller tie spacing increases the effect of confinement; it provides higher base shear capacity for first crushing and increases the ductility of the bridge. The bridge with a moderate change in column height performs better than the bridge, which has rapid change in column height. 94 Smaller tie spacing helps both regular and irregular bridges to perform better. The effect of the order of columns in the bridge with irregular column heights has been found insignificant along longitudinal direction. Static pushover curve closely matches with the dynamic pushover curve until the concrete reaches the first crushing strain. Therefore, cracking, yielding and crushing limit states found from static pushover curves are reliable. Fragility curve results indicate that the presence of irregularity in column heights and larger tie spacing in columns makes bridges more vulnerable to earthquakes. Based on the results obtained from DDBD and FBD design of bridges with irregular column heights and different confinements, the following conclusions can be drawn: The distribution of base shear to columns of different heights is different in DDBD that results in equal design moments in columns of different heights leading to equal longitudinal reinforcement. The distribution of base shear to columns of different heights is different in FBD that predicts higher design base shear in the shortest column, and lower design base shear in longer columns compared to the base shear demands found from non linear time history analyses. However, the provided longitudinal reinforcements in longer columns are higher than the design moments, since, minimum 1% reinforcement governed. 95 FBD takes elastic base shear for shear design, which leads to a large base shear demand and high lateral reinforcement ratio in the shortest column. However, base shear in longer column is small and code specified maximum tie spacing is governed in the longer columns. The required tie spacing in 14 m and 21 m columns in DDBD have been found to be 400 mm and 600 mm, respectively. DDBD bridge with tie spacing of longer columns limited by the maximum allowed tie spacing in CSA Standard S606 (CSA 2010) has performed better than the original DDBD bridge. 6.4 RECOMMENDATIONS FOR FUTURE RESEARCH Since the present study considers columns with the same crosssection and end condition in the factorial analysis of limit states, further research is necessary considering columns of different dimensions and fixity conditions in order to normalize the column dimensions and fixity condition in the formula for computation of the limit states. Further research is required to observe the effect on the unequal span bridges with and without columndeck isolation devices. Further research is necessary to optimize the base shear distribution to shorter and longer column in FBD and to optimize the maximum allowed tie spacing in longer columns in DDBD considering irregular bridges with various column height combinations. 96 REFERENCES Akbari, R. 2010. 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Classification and seismic safety evaluation of existing reinforced concrete columns, Journal of Structural Engineering, ASCE, 133(9): 13161330. 108 APPENDIX Table A1. Limit states of columns for factorial analyses. Column HeightH (m) Longitudinal reinforcement As (%) Compressive Strenght, f'c (MPa) Yield strenght, fy (MPa) Tie spacing, s (mm) Cracking base shear (kN) Cracking displacement (m) Yield base shear (kN) Yield displacement (m) Crushing base shear (kN) Shear Capacity (kN) Displacement at crushing or shear failure (m) DUCTILITY (m/m) 7 2 25 300 75 3366 0.004 8832 0.018 9683 21599 0.084 4.67 7 2 25 300 150 3381 0.004 8832 0.018 9067 12023 0.058 3.22 7 2 25 300 300 3402 0.004 8851 0.018 7909 7235 0.012 0.68 7 2 25 400 75 3364 0.004 10085 0.021 11449 26932 0.092 4.38 7 2 25 400 150 3373 0.004 10065 0.021 10913 14400 0.070 3.33 7 2 25 400 300 3393 0.004 10081 0.021 10062 8134 0.015 0.70 7 2 25 500 75 3366 0.004 10944 0.023 13181 33198 0.098 4.26 7 2 25 500 150 3368 0.004 10892 0.023 12538 17533 0.074 3.22 7 2 25 500 300 3386 0.004 10903 0.023 11795 9701 0.019 0.83 7 2 40 300 75 3951 0.004 9369 0.017 10158 20611 0.097 5.71 7 2 40 300 150 3968 0.004 9394 0.017 9529 11877 0.067 3.94 7 2 40 300 300 3980 0.004 9416 0.017 8720 7510 0.011 0.64 7 2 40 400 75 3944 0.004 10507 0.019 11980 26433 0.110 5.79 7 2 40 400 150 3961 0.004 10538 0.019 11335 14788 0.081 4.26 7 2 40 400 300 3976 0.004 10572 0.019 10466 8966 0.014 0.75 7 2 40 500 75 3939 0.004 11478 0.021 13752 32601 0.120 5.71 7 2 40 500 150 3956 0.004 11514 0.021 13117 17778 0.087 4.14 7 2 40 500 300 3972 0.004 11563 0.021 11984 10366 0.018 0.85 7 2 60 300 75 4967 0.004 9541 0.015 11947 20260 0.360 24.00 7 2 60 300 150 1807 0.004 9551 0.015 10921 12417 0.110 7.33 7 2 60 300 300 4987 0.004 9562 0.015 9803 8495 0.011 0.74 7 2 60 400 75 4952 0.004 10674 0.017 13011 26602 0.341 20.06 109 7 2 60 400 150 4982 0.004 10754 0.017 12537 15371 0.133 7.82 7 2 60 400 300 4987 0.004 10760 0.017 11583 9755 0.014 0.82 7 2 60 500 75 4985 0.004 11618 0.019 13073 32218 0.488 25.68 7 2 60 500 150 1802 0.004 11766 0.019 14944 17513 0.291 15.32 7 2 60 500 300 4987 0.004 11800 0.019 13451 10493 0.016 0.83 7 3 25 300 75 3787 0.004 11130 0.019 12281 19667 0.079 4.16 7 3 25 300 150 3802 0.004 11128 0.019 11681 10609 0.017 0.88 7 3 25 300 300 3821 0.004 11143 0.019 10432 6080 0.075 3.95 7 3 25 400 75 3785 0.004 12937 0.022 14850 21781 0.085 3.86 7 3 25 400 150 3794 0.004 12913 0.022 14193 11862 0.019 0.87 7 3 25 400 300 3813 0.004 12930 0.022 12983 6903 0.009 0.41 7 3 25 500 75 3786 0.004 14493 0.025 17387 26740 0.092 3.68 7 3 25 500 150 3789 0.004 14428 0.025 16666 14342 0.025 0.99 7 3 25 500 300 3807 0.004 14042 0.024 15546 8142 0.011 0.48 7 3 40 300 75 4358 0.004 11716 0.018 12878 19902 0.091 5.06 7 3 40 300 150 4374 0.004 11745 0.018 12057 11168 0.016 0.88 7 3 40 300 300 4385 0.004 11546 0.017 10856 6801 0.008 0.44 7 3 40 400 75 4351 0.004 13338 0.020 15523 25724 0.101 5.05 7 3 40 400 150 4368 0.004 13375 0.020 14731 14079 0.023 1.15 7 3 40 400 300 4381 0.004 13413 0.020 13417 8257 0.010 0.50 7 3 40 500 75 4346 0.004 14603 0.022 18100 32156 0.108 4.91 7 3 40 500 150 4362 0.004 14648 0.022 17765 17059 0.033 1.48 7 3 40 500 300 4378 0.004 14712 0.022 15976 9510 0.012 0.55 7 3 60 300 75 5407 0.004 11840 0.016 14289 19728 0.274 17.13 7 3 60 300 150 4327 0.003 11872 0.016 13633 11601 0.015 0.94 7 3 60 300 300 4327 0.003 11874 0.016 12126 7537 0.007 0.45 7 3 60 400 75 5385 0.004 13770 0.019 16748 25705 0.412 21.68 7 3 60 400 150 4324 0.003 13862 0.019 19038 14474 0.022 1.14 110 7 3 60 400 300 4327 0.003 13870 0.019 14660 8858 0.009 0.49 7 3 60 500 75 4622 0.003 18501 0.022 23913 32093 0.148 6.73 7 3 60 500 150 4321 0.003 15358 0.021 19952 17281 0.029 1.39 7 3 60 500 300 5426 0.004 15413 0.021 17457 10262 0.012 0.56 7 4 25 300 75 4207 0.004 13491 0.020 14836 20620 0.075 3.75 7 4 25 300 150 4221 0.004 13482 0.020 14133 11394 0.015 0.74 7 4 25 300 300 4240 0.004 13489 0.020 12645 6781 0.007 0.37 7 4 25 400 75 4205 0.004 15910 0.023 18194 26771 0.082 3.57 7 4 25 400 150 4214 0.004 15881 0.023 17433 14469 0.020 0.87 7 4 25 400 300 4232 0.004 15897 0.023 15924 8318 0.010 0.42 7 4 25 500 75 4206 0.004 17912 0.026 21504 32922 0.087 3.35 7 4 25 500 150 4209 0.004 17836 0.026 20670 17545 0.025 0.98 7 4 25 500 300 4226 0.004 17840 0.026 19243 9856 0.012 0.47 7 4 40 300 75 3791 0.003 13877 0.018 15528 22247 0.085 4.72 7 4 40 300 150 3803 0.003 13910 0.018 14525 12672 0.015 0.83 7 4 40 300 300 3812 0.003 13940 0.018 13244 7884 0.008 0.44 7 4 40 400 75 3786 0.003 16265 0.021 18971 27427 0.093 4.43 7 4 40 400 150 3799 0.003 16306 0.021 17980 14895 0.015 0.73 7 4 40 400 300 3809 0.003 16352 0.021 16302 8629 0.009 0.42 7 4 40 500 75 3782 0.003 17855 0.023 22341 33693 0.099 4.30 7 4 40 500 150 3794 0.003 17904 0.023 21387 18028 0.023 1.01 7 4 40 500 300 3806 0.003 17977 0.023 19571 9447 0.010 0.44 7 4 60 300 75 5818 0.004 14229 0.017 17629 21966 0.310 18.24 7 4 60 300 150 4666 0.003 14266 0.017 16956 12583 0.013 0.77 7 4 60 300 300 5836 0.004 14268 0.017 14592 8217 0.007 0.41 7 4 60 400 75 4639 0.003 16733 0.020 21573 27139 0.158 7.90 7 4 60 400 150 4664 0.003 16839 0.020 19739 15477 0.017 0.86 7 4 60 400 300 4667 0.003 16860 0.020 18829 9548 0.009 0.44 111 7 4 60 500 75 5803 0.004 18501 0.022 23774 33265 0.144 6.55 7 4 60 500 150 4661 0.003 18731 0.022 23112 18442 0.022 0.98 7 4 60 500 300 5836 0.004 18797 0.022 21534 11030 0.011 0.50 14 2 25 300 75 1572 0.015 4157 0.065 4456 23510 0.084 1.29 14 2 25 300 150 1580 0.015 4211 0.067 4333 13288 0.125 1.87 14 2 25 300 300 1590 0.015 4223 0.067 5194 8410 0.098 1.46 14 2 25 400 75 1572 0.015 4787 0.080 5322 27406 0.230 2.88 14 2 25 400 150 1576 0.015 4778 0.080 5230 15329 0.158 1.98 14 2 25 400 300 1586 0.015 4788 0.080 4466 9290 0.118 1.48 14 2 25 500 75 1572 0.015 5177 0.089 6119 30751 0.261 2.93 14 2 25 500 150 1574 0.015 5190 0.090 5911 16711 0.181 2.01 14 2 25 500 300 1582 0.015 5160 0.089 5547 9692 0.142 1.60 14 2 40 300 75 1673 0.013 4420 0.061 4794 22489 0.183 3.00 14 2 40 300 150 1681 0.013 4435 0.061 4585 13430 0.129 2.11 14 2 40 300 300 1687 0.013 4447 0.061 4772 8901 0.134 2.20 14 2 40 400 75 1670 0.013 4979 0.072 5575 28386 0.251 3.49 14 2 40 400 150 1678 0.013 4996 0.072 5404 15812 0.162 2.25 14 2 40 400 300 1685 0.013 5014 0.072 5721 9773 0.248 3.44 14 2 40 500 75 1668 0.013 5403 0.081 6362 29879 0.299 3.69 14 2 40 500 150 1676 0.013 5378 0.080 6204 16807 0.187 2.34 14 2 40 500 300 1683 0.013 5402 0.080 6069 10270 0.144 1.80 14 2 60 300 75 2127 0.013 4569 0.054 2943 23471 2.339 43.31 14 2 60 300 150 2135 0.013 4585 0.054 4430 14413 1.162 21.52 14 2 60 300 300 2135 0.013 4586 0.054 5026 9884 0.104 1.93 14 2 60 400 75 2118 0.013 5107 0.065 3213 24879 2.361 36.32 14 2 60 400 150 2134 0.013 5023 0.065 5061 16767 1.336 20.55 14 2 60 400 300 2135 0.013 5136 0.065 5748 9737 0.168 2.58 14 2 60 500 75 2107 0.013 5553 0.075 5071 29927 1.911 25.48 112 14 2 60 500 150 2131 0.013 5568 0.074 2829 17308 2.576 34.81 14 2 60 500 300 2135 0.013 5584 0.074 6278 10999 0.654 8.84 14 3 25 300 75 1672 0.014 5320 0.071 5774 18497 0.195 2.75 14 3 25 300 150 1679 0.014 5322 0.071 5563 10369 0.134 1.89 14 3 25 300 300 1688 0.014 5333 0.071 4875 6306 0.097 1.37 14 3 25 400 75 1671 0.014 6185 0.085 6977 24556 0.230 2.71 14 3 25 400 150 1676 0.014 6175 0.085 6755 13324 0.161 1.89 14 3 25 400 300 1685 0.014 6185 0.085 6172 7708 0.120 1.41 14 3 25 500 75 1672 0.014 6831 0.096 8171 30172 0.261 2.72 14 3 25 500 150 1673 0.014 6802 0.096 7921 16132 0.188 1.96 14 3 25 500 300 1682 0.014 6807 0.096 7498 9112 0.144 1.50 14 3 40 300 75 1724 0.012 5561 0.066 6083 17857 0.197 2.98 14 3 40 300 150 1731 0.012 5542 0.065 5745 10548 0.129 1.98 14 3 40 300 300 1714 0.012 5556 0.065 6300 6893 0.262 4.03 14 3 40 400 75 1721 0.012 6400 0.078 7293 22730 0.247 3.17 14 3 40 400 150 1728 0.012 6374 0.077 7041 12984 0.159 2.06 14 3 40 400 300 1717 0.012 6394 0.077 3444 8111 0.125 1.62 14 3 40 500 75 1719 0.012 7048 0.088 8497 29927 0.284 3.23 14 3 40 500 150 1726 0.012 7012 0.087 8229 16382 0.191 2.20 14 3 40 500 300 1733 0.012 7041 0.087 6446 9609 0.145 1.67 14 3 60 300 75 2134 0.012 5653 0.059 4960 18255 2.269 38.46 14 3 60 300 150 2141 0.012 5673 0.059 6092 11197 2.091 35.44 14 3 60 300 300 2141 0.012 5671 0.059 6622 7668 0.301 5.10 14 3 60 400 75 2126 0.012 6570 0.072 5874 22961 2.468 34.28 14 3 60 400 150 2140 0.012 6576 0.072 7189 13550 1.703 23.65 14 3 60 400 300 2141 0.012 6585 0.072 7465 8845 0.163 2.26 14 3 60 500 75 2288 0.012 8784 0.086 9088 28331 2.642 30.72 14 3 60 500 150 2138 0.012 7203 0.081 5805 16148 2.690 33.21 113 14 3 60 500 300 2141 0.012 7225 0.081 8652 10057 1.268 15.65 14 4 25 300 75 1860 0.014 6414 0.073 7015 22634 0.199 2.73 14 4 25 300 150 1867 0.014 6416 0.073 6775 12877 0.134 1.84 14 4 25 300 300 1876 0.014 6428 0.073 6155 7649 0.097 1.33 14 4 25 400 75 1859 0.014 7585 0.088 8614 27145 0.227 2.58 14 4 25 400 150 1863 0.014 8334 0.164 8696 14194 1.965 11.98 14 4 25 400 300 1872 0.014 7584 0.088 7721 9028 0.121 1.38 14 4 25 500 75 1860 0.014 8411 0.099 10197 33198 0.249 2.52 14 4 25 500 150 1861 0.014 8442 0.100 9874 17533 0.174 1.74 14 4 25 500 300 1869 0.014 8446 0.100 9368 9701 0.145 1.45 14 4 40 300 75 1881 0.012 6651 0.068 7365 22347 0.198 2.91 14 4 40 300 150 1887 0.012 6668 0.068 7020 13289 0.130 1.91 14 4 40 300 300 1892 0.012 6684 0.068 7791 8760 0.193 2.84 14 4 40 400 75 1878 0.012 7775 0.081 8983 28268 0.238 2.94 14 4 40 400 150 1885 0.012 7798 0.081 9140 16190 1.422 17.56 14 4 40 400 300 1890 0.012 7823 0.081 8536 10151 0.123 1.52 14 4 40 500 75 1876 0.012 8663 0.092 10587 31459 0.269 2.92 14 4 40 500 150 1883 0.012 8615 0.091 10234 17419 0.188 2.07 14 4 40 500 300 1889 0.012 8648 0.091 8427 10399 0.145 1.59 14 4 60 300 75 2304 0.012 6795 0.063 6235 23298 2.696 42.79 14 4 60 300 150 2311 0.012 6816 0.063 7339 14239 3.368 53.46 14 4 60 300 300 2311 0.012 6811 0.063 7485 9710 0.139 2.21 14 4 60 400 75 2297 0.012 7947 0.076 8097 28729 2.539 33.41 14 4 60 400 150 2310 0.012 7943 0.075 8668 16651 2.229 29.72 14 4 60 400 300 2311 0.012 7951 0.075 8760 10612 0.154 2.05 14 4 60 500 75 2288 0.012 8784 0.086 9170 30718 2.597 30.20 14 4 60 500 150 2308 0.012 8813 0.085 10634 17646 2.252 26.49 14 4 60 500 300 2311 0.012 8842 0.085 11260 11110 0.207 2.44 114 21 2 25 300 75 1025 0.034 2725 0.155 2610 23948 0.500 3.23 21 2 25 300 150 1030 0.034 2721 0.154 2608 13815 0.349 2.27 21 2 25 300 300 1011 0.033 2723 0.153 2500 8748 0.259 1.69 21 2 25 400 75 1024 0.034 3074 0.181 3063 27500 0.603 3.33 21 2 25 400 150 1027 0.034 3068 0.181 3080 15422 0.442 2.44 21 2 25 400 300 1008 0.033 3067 0.180 3018 9383 0.312 1.73 21 2 25 500 75 1025 0.034 3355 0.204 3541 33538 0.675 3.31 21 2 25 500 150 1025 0.034 3307 0.201 3553 18441 0.488 2.43 21 2 25 500 300 1006 0.033 3311 0.201 3505 10893 0.367 1.83 21 2 40 300 75 1054 0.028 2884 0.139 2733 24923 0.557 4.01 21 2 40 300 150 1059 0.028 2887 0.138 2748 14790 0.380 2.75 21 2 40 300 300 1063 0.028 2889 0.137 2082 9723 0.276 2.01 21 2 40 400 75 1052 0.028 3229 0.161 3207 28527 0.638 3.96 21 2 40 400 150 1057 0.028 3232 0.160 3221 16450 0.454 2.84 21 2 40 400 300 1061 0.028 3246 0.160 1 10411 1.884 11.78 21 2 40 500 75 1050 0.028 3487 0.179 3669 34023 0.727 4.06 21 2 40 500 150 1055 0.028 3501 0.179 3699 16447 0.536 2.99 21 2 40 500 300 1060 0.028 3507 0.178 3840 10138 0.364 2.04 21 2 60 300 75 1331 0.028 2985 0.123 23471 21 2 60 300 150 1336 0.003 2990 0.123 14413 21 2 60 300 300 1336 0.028 2990 0.123 1056 9884 1.837 14.93 21 2 60 400 75 1325 0.028 3335 0.146 1 29510 3.675 25.17 21 2 60 400 150 1335 0.028 4064 0.144 14784 3.829 26.59 21 2 60 400 300 1336 0.028 3341 0.144 3305 11394 0.780 5.42 21 2 60 500 75 1318 0.028 3617 0.166 29927 21 2 60 500 150 1381 0.029 3767 0.174 17308 21 2 60 500 300 1336 0.028 3625 0.162 10999 21 3 25 300 75 1102 0.032 3453 0.162 3444 17179 0.509 3.14 115 21 3 25 300 150 1107 0.032 3454 0.162 3438 9869 0.355 2.19 21 3 25 300 300 1113 0.032 3455 0.161 3278 6215 0.257 1.60 21 3 25 400 75 1101 0.032 4003 0.192 4203 22052 0.575 2.99 21 3 25 400 150 1104 0.032 3995 0.192 4167 12306 0.431 2.24 21 3 25 400 300 1110 0.032 4003 0.192 4030 7433 0.311 1.62 21 3 25 500 75 1102 0.032 4400 0.215 4955 29333 0.638 2.97 21 3 25 500 150 1103 0.032 4378 0.215 4895 15788 0.497 2.31 21 3 25 500 300 1108 0.032 4368 0.214 4761 9015 0.365 1.71 21 3 40 300 75 1128 0.027 3615 0.147 3630 17496 0.529 3.60 21 3 40 300 150 1132 0.027 3627 0.147 3572 10438 0.371 2.52 21 3 40 300 300 1135 0.027 3628 0.146 3657 6909 0.311 2.13 21 3 40 400 75 1126 0.027 4156 0.173 4360 22201 0.637 3.68 21 3 40 400 150 1130 0.027 4172 0.173 4353 12790 0.443 2.56 21 3 40 400 300 1134 0.027 4175 0.172 2117 8085 0.316 1.84 21 3 40 500 75 1124 0.027 4561 0.194 5101 27603 0.721 3.72 21 3 40 500 150 1129 0.027 4560 0.193 5090 15421 0.514 2.66 21 3 40 500 300 1133 0.027 4567 0.192 4680 9329 0.369 1.92 21 3 60 300 75 1397 0.027 3711 0.133 18255 21 3 60 300 150 1401 0.027 3727 0.133 11197 21 3 60 300 300 1402 0.027 3726 0.133 1343 7668 2.527 19.00 21 3 60 400 75 1443 0.028 4225 0.156 346 22961 5.388 34.54 21 3 60 400 150 1401 0.027 4291 0.158 1529 13550 3.944 24.96 21 3 60 400 300 1402 0.027 4298 0.158 1633 8845 3.431 21.72 21 3 60 500 75 1499 0.027 5753 0.190 5220 27666 2.203 11.59 21 3 60 500 150 1399 0.027 4604 0.177 1504 15903 4.040 22.82 21 3 60 500 300 1402 0.027 4717 0.177 10021 21 4 25 300 75 1196 0.031 4231 0.172 4292 23045 0.495 2.88 21 4 25 300 150 1201 0.031 4181 0.166 4247 13568 0.348 2.10 116 21 4 25 300 300 1207 0.031 4180 0.165 4021 8596 0.257 1.56 21 4 25 400 75 1196 0.031 4954 0.200 5308 27406 0.566 2.83 21 4 25 400 150 1198 0.031 4945 0.200 5234 15329 0.425 2.13 21 4 25 400 300 1204 0.031 4939 0.199 5017 9290 0.308 1.55 21 4 25 500 75 1196 0.031 5484 0.224 6328 30751 0.621 2.77 21 4 25 500 150 1197 0.031 5459 0.224 6220 16711 0.485 2.17 21 4 25 500 300 1202 0.031 5462 0.224 6002 9692 0.364 1.63 21 4 40 300 75 1233 0.027 4369 0.154 4494 24923 0.531 3.45 21 4 40 300 150 1237 0.027 4370 0.153 4402 14790 0.366 2.39 21 4 40 300 300 1240 0.027 4370 0.152 1943 9723 0.265 1.74 21 4 40 400 75 1231 0.027 5098 0.182 5518 28386 0.611 3.36 21 4 40 400 150 1235 0.027 5099 0.181 5434 16308 0.436 2.41 21 4 40 400 300 1239 0.027 5102 0.180 5020 10269 0.312 1.73 21 4 40 500 75 1229 0.027 5631 0.204 6544 29879 0.675 3.31 21 4 40 500 150 1234 0.027 5648 0.204 6439 16807 0.510 2.50 21 4 40 500 300 1238 0.027 5654 0.203 6133 10270 0.365 1.80 21 4 60 300 75 1510 0.027 4472 0.142 2733 23471 2.635 18.56 21 4 60 300 150 1515 0.027 4477 0.141 2397 14413 4.011 28.45 21 4 60 300 300 1515 0.027 4477 0.141 4628 9884 0.362 2.57 21 4 60 400 75 1505 0.027 5201 0.168 3558 29510 4.860 28.93 21 4 60 400 150 1514 0.027 5230 0.167 5688 17433 1.515 9.07 21 4 60 400 300 1515 0.027 5239 0.167 5766 11394 0.433 2.59 21 4 60 500 75 1499 0.027 5753 0.190 4335 29927 2.884 15.18 21 4 60 500 150 1513 0.027 5769 0.187 6312 17308 1.240 6.63 21 4 60 500 300 1515 0.027 5793 0.187 5941 10999 1.593 8.52
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