SEISMIC PERFORMANCE EVALUATION OF FRENCH CREEK BRIDGE BASED ON CANADIAN HIGHWAY BRIDGE DESIGN CODE 2015 by Xiang Li B.Sc. Dalian Jiaotong University, 2013 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate and Postdoctoral Studies (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2016 © Xiang Li, 2016 Abstract Built in 1993, the French Creek Bridge is located on highway 19 on Vancouver Island, BC, Canada. The bridge is part of the British Columbia Smart Infrastructure Monitoring System (BCSIMS), funded by the Ministry of Transportation and Infrastructure (MoTI) BC, Canada. The BCSIMS is a real-time seismic monitoring program that continuously assesses the seismic conditions of the selected bridges in BC. As part of this seismic monitoring program, the seismic performance and nonlinear dynamic behavior of the FCB was evaluated by developing the 3D Finite Element (FE) model of the bridge in SAP2000. The model was updated based on the modal properties extracted from an Ambient Vibration (AV) test. The nonlinear behavior of the bridge was modeled by adding plastic hinges on the ductile components. Then the FE model was used to perform the seismic performance evaluation in accordance with the latest Canadian Highway Bridge Design Code 2015. The evaluation result shows that during major earthquake, no primary members of the bridge were damaged, the bridge will maintains repairable and operational, and should be capable of supporting the dead load and live load after earthquake. ii Preface This research was conducted under the supervision of Dr. Carlos Ventura and Dr. Yavuz Kaya. The work of the author includes analyzing the experimental data, establishing the finite element model, preparing the majority of the manuscript and presenting the analysis results in the paper. Dr. Carlos Ventura provided the research proposal and offered feedback over the project during different stages of the research. He also provided the access to hardware and software available in the Earthquake Engineering Research Facility (EERF) to perform the modal identification and finite element time history analysis. The Ambient Vibration test of the bridge was conducted by research team lead by Dr. Yavuz Kaya. Dr. Kaya provided invaluable guidance and discussion throughout the study and helped to prepare and edit the manuscript to this paper and the conference paper. He also provided the structural drawing of the bridge and access to the EERF mainframe to help perform the nonlinear time history analysis. The content presented from Chapters 2 to 5 of this thesis were presented at the 34th International Modal Analysis Conference (IMAC) (Orlando, Florida, 2016), and will be published in the book of“Dynamics of Civil Structures, Volume 2 - proceedings of the 34th IMAC, A Conference and Exposition on Structural Dynamics, 2016”. The title of the paper included in the conference proceedings was: “Finite Element Model Updating of French Creek Bridge”. iii Table of contents Abstract .......................................................................................................................................................... ii Preface ........................................................................................................................................................... iii Table of contents……………………………………………………………………………………………iv List of tables ................................................................................................................................................ viii List of figures ................................................................................................................................................ ix Acknowledgements ..................................................................................................................................... xiii Dedication ................................................................................................................................................... xiv 1. Introduction .......................................................................................................................................... 1 1.1 British Columbia Smart Infrastructure Monitoring System (BCSIMS) ............................................. 1 1.2 The French Creek Bridge ................................................................................................................... 1 1.3 Research needs and objectives ........................................................................................................... 2 2. Description of the French Creek Bridge ............................................................................................. 4 2.1 General description ............................................................................................................................ 4 2.2 Superstructure .................................................................................................................................... 5 2.3 Pier and foundation ............................................................................................................................ 7 2.4 Bearing ............................................................................................................................................... 8 3. Ambient Vibration Test and results .................................................................................................. 10 3.1 Introduction ...................................................................................................................................... 10 3.2 Instrumentation devices .................................................................................................................... 10 3.3 Description of setups ........................................................................................................................ 11 3.4 Data processing ................................................................................................................................ 13 3.5 Modal identification and results ....................................................................................................... 15 3.5.1 Frequency Domain Decomposition .......................................................................................... 15 3.5.2 Modal identification results ...................................................................................................... 16 4. Elastic Finite Element (FE) modeling of the bridge ........................................................................ 18 4.1 Elastic material properties ................................................................................................................ 18 iv 4.2 Modeling of superstructure .............................................................................................................. 19 4.2.1 Superstructure .......................................................................................................................... 19 4.2.2 Expansion joints ....................................................................................................................... 20 4.2.3 Bearings .................................................................................................................................... 21 4.3 Modeling of piers ............................................................................................................................. 23 4.3.1 Columns and cap-beams ........................................................................................................... 23 4.3.2 Pile and pile-cap ....................................................................................................................... 23 4.4 Modeling of boundary condition of the foundation .......................................................................... 24 4.4.1 Abutments and south pile bent .................................................................................................. 24 4.4.2 Pile foundation ......................................................................................................................... 24 4.5 Modal property extraction and results .............................................................................................. 26 5. Finite element model updating .......................................................................................................... 28 5.1 Sensitivity analysis ........................................................................................................................... 28 5.1.1 Sensitivity analysis methodology .............................................................................................. 28 5.1.2 Analysis results ......................................................................................................................... 29 5.2 Model updating ................................................................................................................................ 34 5.3 Results and discussion ...................................................................................................................... 35 6. Selection of ground motion ................................................................................................................ 38 6.1 Introduction ...................................................................................................................................... 38 6.2 Probabilistic Seismic Hazard Analysis ............................................................................................. 38 6.3 Uniform Hazard Spectrum ............................................................................................................... 39 6.4 Seismic hazard deaggregation .......................................................................................................... 41 6.5 Selection and scaling of ground motion ........................................................................................... 43 7. Section analysis of ductile members and nonlinear hinge modeling .............................................. 49 7.1 Introduction ...................................................................................................................................... 49 7.2 Nonlinear material models ............................................................................................................... 50 7.2.1 Expected material properties .................................................................................................... 50 v 7.2.2 Mander concrete model ............................................................................................................ 50 7.2.3 Rebar steel model ..................................................................................................................... 51 7.3 Plastic moment capacity ................................................................................................................... 53 7.3.1 Critical sections of column and cap-beam ............................................................................... 53 7.3.2 Moment-curvature analysis of the column................................................................................ 54 7.3.3 Moment-curvature analysis of the cap-beam ............................................................................ 56 7.4 Bilinear M-φ model .......................................................................................................................... 57 7.5 Shear strength ................................................................................................................................... 58 7.6 Nonlinear hinge model ..................................................................................................................... 60 7.6.1 Nonlinear hinge location and type ........................................................................................... 60 7.6.2 Plastic hinge length .................................................................................................................. 60 7.6.3 Uncoupled Moment Hinge model ............................................................................................. 61 7.6.4 Interaction PMM Hinge model ................................................................................................. 62 7.6.5 Fiber PMM Hinge model .......................................................................................................... 63 8. Seismic evaluation methodology ........................................................................................................ 65 8.1 Introduction ...................................................................................................................................... 65 8.2 Bridge classification and Seismic Performance Category ................................................................ 65 8.3 Analysis requirement ........................................................................................................................ 65 8.4 Performance level and criteria for performance-based seismic evaluation ...................................... 66 8.4.1 Performance levels ................................................................................................................... 66 8.4.2 Performance criteria ................................................................................................................ 67 8.4.3 Performance based assessment method .................................................................................... 68 8.5 Inelastic Static Push-over Analysis (ISPA) ...................................................................................... 72 8.5.1 General consideration .............................................................................................................. 72 8.5.2 Displacement limitation ............................................................................................................ 72 8.5.3 Pushover load cases ................................................................................................................. 74 8.5.4 Force pattern ............................................................................................................................ 75 vi 8.5.5 Geometric nonlinearities .......................................................................................................... 75 8.6 Elastic and Nonlinear Time-history Analysis (ETHA and NTHA) .................................................. 76 8.6.1 General consideration .............................................................................................................. 76 8.6.2 Seismic inputs ........................................................................................................................... 76 8.6.3 Damping ................................................................................................................................... 78 9. Seismic analysis results and discussions ........................................................................................... 81 9.1 Results of Inelastic Static Pushover Analysis................................................................................... 81 9.1.1 Plastic hinges mechanism ......................................................................................................... 81 9.1.2 Pushover curves ........................................................................................................................ 82 9.2 Results of Elastic and Nonlinear Time-history Analysis .................................................................. 85 9.2.1 Columns and cap-beams ........................................................................................................... 85 9.2.2 Bearings .................................................................................................................................. 115 9.2.3 Expansion joints ..................................................................................................................... 120 9.3 Performance based seismic evaluation ........................................................................................... 124 9.3.1 Concrete structures ................................................................................................................ 124 9.3.2 Steel structures ....................................................................................................................... 127 9.3.3 Connections ............................................................................................................................ 127 9.3.4 Bearings and joints ................................................................................................................. 128 9.3.5 Restrainers .............................................................................................................................. 129 9.3.6 Displacement .......................................................................................................................... 129 9.3.7 Foundation ............................................................................................................................. 131 9.3.8 General conclusion ................................................................................................................. 134 10. Conclusion ......................................................................................................................................... 135 References .................................................................................................................................................. 136 Appendix: Selected ground motion details under time and frequency domain ................................... 141 vii List of tables Table 1 Start and end time of each setup....................................................................................................... 12 Table 2 Material property of concrete and steel of initial FE model ............................................................. 19 Table 3 Initial stiffness values of bearings .................................................................................................... 22 Table 4 Natural frequencies before and after updating ................................................................................. 35 Table 5 Parameters before and after updating ............................................................................................... 36 Table 6 Ground motion record selection criteria ........................................................................................... 43 Table 7 Selected ground motion general information ................................................................................... 44 Table 8 Nonlinear material properties of concrete and steel ......................................................................... 53 Table 9 Idealized M-φ curves of pier and cap-beams under different load cases ......................................... 58 Table 10 Shear capacities of piers and cap-beams of all piers ...................................................................... 59 Table 11 Plastic hinge length of ductile members ......................................................................................... 61 Table 12 Input ground motion information ................................................................................................... 78 Table 13 Maximum column shear force of time history analysis ............................................................... 108 Table 14 Maximum cap-beam positive bending moment of time history analysis ...................................... 114 Table 15 Maximum cap-beam negative bending moment of time history analysis ..................................... 114 Table 16 Maximum cap-beam shear force of time history analysis ............................................................. 114 Table 17 Bearing maximum shear loads in transverse (U3) direction ......................................................... 118 Table 18 Bearing maximum shear loads in longitudinal (U2) direction ...................................................... 118 Table 19 Bearing maximum vertical compressive loads .............................................................................. 118 Table 20 Bearing maximum vertical tensile loads ....................................................................................... 118 Table 21 Sliding bearing maximum relative deformation in longitudinal direction (U2) ........................... 120 Table 22 Maximum expansion joint relative movement of time history analysis ....................................... 124 Table 23 Cap-beam maximum positive moment compared with target moment ........................................ 126 Table 24 Cap-beam maximum negative moment compared with target moment ....................................... 127 Table 25 Maximum pier foundation movements ........................................................................................ 133 viii List of figures Fig. 1 Site map and satellite image of the French Creek Bridge ..................................................................... 4 Fig. 2 Elevation (top) and plan (bottom) view of the French Creek Bridge .................................................... 5 Fig. 3 Side view of concrete deck-steel girder system (left) and piers in middle spans (right) ....................... 6 Fig. 4 Location and structural drawing of the expansion joints ...................................................................... 6 Fig. 5 Typical concrete decking cross section ................................................................................................. 6 Fig. 6 Typical diaphragm (top) and diaphragm at piers, bent and abutment (bottom) .................................... 7 Fig. 7 Pier elevation view and detailed layout of pier cap and base ................................................................ 8 Fig. 8 Typical reinforcement detail of column and cap-beam ......................................................................... 8 Fig. 9 Location and structural detail of bearings at Pier No.3 ......................................................................... 9 Fig. 10 Cross section of the central core of a pot bearing and the unidirectional sliding component ............. 9 Fig. 11 TROMINO sensor and specification ................................................................................................. 11 Fig. 12 Setup plans and sensor locations of all setups................................................................................... 13 Fig. 13 High-gain velocity time-history signals of Setup No.1 (N-S direction) ............................................ 14 Fig. 14 Peak-picking of Singular Values of Power Spectral Densities matrix (ARTeMIS) ........................... 16 Fig. 15 Modal identification results from AV test (ARTeMIS) ..................................................................... 17 Fig. 16 Overview of the FE model of French Creek Bridge ......................................................................... 18 Fig. 17 Finite element modeling method of the superstructure system ......................................................... 20 Fig. 18 Superstructure cross section modeled in CSiBridge ......................................................................... 20 Fig. 19 Model detail of expansion joints at south pile bent (left) and pier No.3 ........................................... 21 Fig. 20 Local axis of bearing link element in SAP2000 ................................................................................ 23 Fig. 21 Structural drawing and FE model of Pier No.3 (see Fig. 2) and member cross sections .................. 24 Fig. 22 Winkler Model (left) (Adhikary et al., 2001) and pile discretization (right) ..................................... 25 Fig. 23 Model identification results from AV test (left) and FE model (SAP2000) (right) ........................... 27 Fig. 24 Sensitivity analysis curves of Ksoil (left) and Esteel (right) ................................................................. 29 Fig. 25 Sensitivity analysis curve of mass density of structural steel (ρsteel) ................................................. 30 Fig. 26 Sensitivity analysis curve of elastic modulus of foundation concrete (Efound) ................................... 30 ix Fig. 27 Sensitivity analysis curve of mass density of foundation concrete (ρfound)........................................ 31 Fig. 28 Sensitivity analysis curve of elastic modulus of decking concrete (Edeck) ........................................ 31 Fig. 29 Sensitivity analysis curve of mass density of the decking concrete (ρdeck) ....................................... 32 Fig. 30 Sensitivity curve of translational stiffness of the sliding bearing in U2 direction (KU2) ................... 32 Fig. 31 Sensitivity curve of the rotational stiffness of all bearings in R2 & R3 directions ........................... 33 Fig. 32 Sensitivity curve of the moment of inertial of pier column (Icol) ...................................................... 33 Fig. 33 Sensitivity curve of the moment of inertial of cap-beams (Icap) ........................................................ 34 Fig. 34 Combining hazard curves from individual periods to generate a uniform hazard spectrum ............. 40 Fig. 35 Uniform hazard spectra in FCB site (5% damping ratio) .................................................................. 41 Fig. 36 Magnitude-distance (M-R) deaggregation of hazard level 2% in 50 year at FCB site ...................... 42 Fig. 37 Time history acceleration of unscaled selected ground motions (crustal) ......................................... 45 Fig. 38 Time history acceleration of unscaled selected ground motions (subcrustal) ................................... 45 Fig. 39 Time history acceleration of unscaled selected ground motions (subduction) .................................. 46 Fig. 40 Response spectra of scaled selected ground motions with target spectrum (crustal) ........................ 47 Fig. 41 Response spectra of scaled selected ground motions with target spectrum (subcrustal) ................... 47 Fig. 42 Response spectra of scaled selected ground motions with target spectrum (subduction) ................. 48 Fig. 43 Stress-strain model of confined and unconfined concrete ................................................................. 51 Fig. 44 Steel stress-stain model ..................................................................................................................... 52 Fig. 45 Moment diagram of column and cap-beam at Pier No.2 and 3 ......................................................... 53 Fig. 46 Xtract section model details of columns (left) and cap-beams (right) of all piers ............................ 54 Fig. 47 P-M interaction diagram of columns of all piers ............................................................................... 55 Fig. 48 Moment-curvature diagrams of pier 1 & 2 (top) and pier 3 (bottom) ............................................... 56 Fig. 49 Moment-curvature diagram of cap-beam of all piers ........................................................................ 57 Fig. 50 Actual and idealized M-φ curve relation ........................................................................................... 58 Fig. 51 Expected locations of hinge formation in FCB ................................................................................. 60 Fig. 52 Idealized M-φ relation of Uncoupled Moment Hinge model in SAP2000 ....................................... 62 Fig. 53: Fiber distribution along circular cross-section ................................................................................. 64 x Fig. 54 Bending moment vs. maximum rebar strain of all columns under various levels of axial load ........ 69 Fig. 55 P-M interaction relation of all columns (maximum reinforcement tensile strain 0.015) .................. 70 Fig. 56 Bending moment vs. maximum rebar strain of all cap-beams .......................................................... 71 Fig. 57 Structural configuration of deformation of FCB Pier No.2 and corresponding moment diagram .... 73 Fig. 58 Local displacement capacity of fixed-fixed framed columns............................................................ 74 Fig. 59 Arias Intensity diagram of the Northridge ground motion ................................................................ 78 Fig. 60 Rayleigh damping used for direct-integration time history analysis ................................................. 80 Fig. 61 Hinge formation sequence of nonlinear pushover analysis in transverse direction ........................... 81 Fig. 62 Hinge formation sequence of nonlinear pushover analysis in longitudinal direction ....................... 82 Fig. 63 Location of the monitored node for nonlinear pushover analysis ..................................................... 83 Fig. 64 Pushover curve in transverse direction ............................................................................................. 84 Fig. 65 Pushover curve in longitudinal direction .......................................................................................... 84 Fig. 66 Critical sections of pier columns and cap-beams of all piers ............................................................ 86 Fig. 67 Time history records of axial force of Pier No.1 column sections (ETHA) ...................................... 87 Fig. 68 Time history records of axial force of Pier No.1 column sections (NTHA) ..................................... 88 Fig. 69 Time history records of axial force of Pier No.2 column sections (ETHA) ...................................... 89 Fig. 70 Time history records of axial force of Pier No.2 column sections (NTHA) ..................................... 90 Fig. 71 Time history records of axial force of Pier No.3 column sections (ETHA) ...................................... 91 Fig. 72 Time history records of axial force of Pier No.3 column sections (NTHA) ..................................... 92 Fig. 73 Time history records of bending moment of Pier No.1 column sections (ETHA) ............................ 93 Fig. 74 Time history records of bending moment of Pier No.1 column sections (NTHA)............................ 94 Fig. 75 Time history records of bending moment of Pier No.2 column sections (ETHA) ............................ 95 Fig. 76 Time history records of bending moment of Pier No.2 column sections (NTHA)............................ 96 Fig. 77 Time history records of bending moment of Pier No.3 column sections (ETHA) ............................ 97 Fig. 78 Time history records of bending moment of Pier No.3 column sections (NTHA)............................ 98 Fig. 79 Pier No.1 axial-flexural responses compared with P-M capacity curve ............................................ 99 Fig. 80 Pier No.2 axial-flexural responses compared with P-M capacity curve .......................................... 100 xi Fig. 81 Pier No.3 axial-flexural responses compared with P-M capacity curve .......................................... 100 Fig. 82 Time history records of shear force of Pier No.1 column sections (ETHA) ................................... 102 Fig. 83 Time history records of shear force of Pier No.1 column sections (NTHA) ................................... 103 Fig. 84 Time history records of shear force of Pier No.2 column sections (ETHA) ................................... 104 Fig. 85 Time history records of shear force of Pier No.2 column sections (NTHA) ................................... 105 Fig. 86 Time history records of shear force of Pier No.3 column sections (ETHA) ................................... 106 Fig. 87 Time history records of shear force of Pier No.3 column sections (NTHA) ................................... 107 Fig. 88 Time history records of bending moment of all cap-beams (ETHA) .............................................. 110 Fig. 89 Time history records of bending moment of all cap-beams (NTHA).............................................. 111 Fig. 90 Time history records of shear force of all cap-beams (ETHA) ....................................................... 112 Fig. 91 Time history records of shear force of all cap-beams (NTHA) ....................................................... 113 Fig. 92 Location of monitored bearings and expansion joints on the FCB ................................................. 115 Fig. 93 Time history records of internal forces of link element 430 (ETHA) ............................................. 116 Fig. 94 Time history records of internal forces of link element 430 (NTHA) ............................................. 117 Fig. 95 Time history records of deformation of slide bearing 336 and 514 in U2 direction (ETHA) ......... 119 Fig. 96 Time history records of deformation of slide bearing 336 and 514 in U2 direction (NTHA) ......... 120 Fig. 97 Time history records of deformation of link element of expansion joint along X axis (ETHA) ..... 122 Fig. 98 Time history records of deformation of link element of expansion joint along X axis (ETHA) ..... 123 Fig. 99 Pier No.1 axial-flexural responses compared with P-M performance target curve ......................... 125 Fig. 100 Pier No.2 axial-flexural responses compared with P-M performance target curve ....................... 125 Fig. 101 Pier No.3 axial-flexural responses compared with P-M performance target curve ....................... 126 Fig. 102 Critical beam and column sections of the primary connections .................................................... 128 Fig. 103 Time-history displacement of superstructure midpoint (GM No.1) .............................................. 130 Fig. 104 Time-history displacement of Pier 2 foundation (GM No.1) ........................................................ 131 Fig. 105 Location of superstructure midpoint and monitored point on pier foundations ............................ 132 Fig. 106 Time history records of displacement of Pier 2 foundation in three directions (NTHA) .............. 133 xii Acknowledgements First of all, I would like to thank my supervisor Dr. Carlos Ventura, for your supervising and tutoring for the past two years, and for giving me the opportunity to work on many interesting research projects. My time at UBC would not have been nearly as rewarding without his advice and support throughout my graduate studies at UBC. I am proud of the opportunity I had to be a member of his research team in the Earthquake Engineering Research Facility. I would also like to thank my co-supervisor: Dr. Yavuz Kaya. His guidance and support was invaluable throughout my research. Whenever I need help with my research, he is always available for discussion and question. His kindness and patience presented in every meeting are one of the memories I will cherish the most from my student life at UBC. It was a pleasure to work with him and learn from him. I would like to thank Mr. Felix Yao and Mr. Michael Fairhurst for providing the precious technical supports throughout the research project. I offer my enduring gratitude to the faculty, staff, and my fellow students and teammates at UBC, especially to Yu Feng, Yuxin Pan, Tianci Wang, Xu Xie and Wu Gao who have inspired and motivated me and made my time at UBC much more enjoyable. I would also like to thank British Columbia Ministry of Transportation for provided the funding for this research, as part of the BC Smart Infrastructure System (BCSIMS) project. Special thanks to my parents, for advising and supporting me throughout my years of education. I could never achieve all of this without your nurturing. xiii Dedication To my familyxiv 1. Introduction 1.1 British Columbia Smart Infrastructure Monitoring System (BCSIMS) The west coast line of BC province lies on one of the seismically active regions of North America, known as the Cascadia Subduction Zone. It is a convergent plate boundary that stretches from northern Vancouver Island to northern California. Structures and facilities near this area are vulnerable to seismic impact. Major cities affected by a disturbance in this subduction zone would include Vancouver and Victoria, British Columbia (BC); Seattle, Washington; as well as Portland, Oregon. In the late 1990’s, a program called the British Columbia Smart Infrastructure Monitoring System (BCSIMS) was established by cooperation of Ministry of Transportation and Infrastructure (MoTI) BC and Earthquake Engineering Research Facility (EERF) from University of British Columbia (UBC), aiming at better preparation for possible seismic events and minimizing the damage of provincial Disaster Response Routes of BC (Kaya, 2015). The purpose of the present project is to establish a real-time seismic structural response system to provide instant inspection and rapid damage assessment for the Ministry’s structures in BC province. The BCSIMS monitoring program is developed to integrate the Strong Motion Network (consists of 156 Internet Accelerometers stations which maintained by Geologic Survey of Canada) and the Structural Health Monitoring network (consists of instrumented structures monitored by UBC) of BC. The implementation of the BCSIMS will enable an effective way of inspecting and evaluating the structures by using state-of-the-art sensing technology with efficient techniques of data analysis. 1.2 The French Creek Bridge So far, fourteen bridges were instrumented and monitored in the BCSIMS network. The French Creek Bridge (FCB), which is one of the selected bridges, is located near Parksville, Vancouver Island, BC, Canada. It was designed as a disaster-route bridge, which carries the Inland Island Highway 19 over French Creek from south to north. It was designed in 1993 and opened in 1996. In 1997, it was selected by MoTI BC as one of the seismic monitoring sites. The bridge was instrumented with 12 sensors and monitored since then. In order to identify the structural dynamic characteristic of the bridge, in September 2014, an Ambient Vibration (AV) Testing was conducted on the FCB. 1 1.3 Research needs and objectives As part of the essential information prepared for the BCSIMS bridge database, the seismic performance and nonlinear dynamic behavior of the FCB need to be evaluated. The seismic design of FCB was based on the specification from old version of CSA Standard CAN/CSA-S6-88 and ATC-6 Seismic Design Guideline. However, demands for seismic performance of the highway bridge have changed during the years of development of the seismic and earthquake engineering. In order to find out if the seismic performance of the FCB still satisfies the latest demands of BC area, a full set of performance based seismic evaluation procedure was carried out in accordance with the latest Canadian Highway Bridge Design Code 2015 (CAN/CSA-S6-14). To obtain a relatively realistic and reliable static and dynamic property estimation results, a three dimensional finite element (FE) model was developed in CSiBridge® and SAP2000® (CSI, 2011) in great detail. The data acquired by AV test was used to extract the dynamic characteristics (natural frequencies, mode shapes, and damping ratio) of the FCB. Commercially available software ARTeMIS Modal Pro® (SVIBS) (SVS, 2001) was utilized to carry out the data analysis and modal identification. The established FE model was then manually updated based on the AV test results. Finally, the updated model was used to assess the seismic performance of the bridge in accordance with the CAN/CSA-S6-14. Various types of analysis were employed including the Elastic Dynamic Analysis (EDA), the Inelastic Static Push-over Analysis (ISPA) and the Nonlinear Time-history Analysis (NTHA). The study of the structural characteristic of the FCB gives thorough understanding of the dynamic characteristic as well as nonlinear behavior of the bridge under seismic excitation. Results have shown that the bridge meets all performance requirements of the code. It can be concluded that no primary members were damaged and the bridge will maintain repairable and operational during major earthquake. The evaluation results can be used as reference information by assisting decision-makers in risk management and possible retrofit action of the bridge. The updated FE model will be added as part of the information displayed on the Structure Information Page (SIP) on BCSIMS website, which aims to provide an overview of the status of the structures and detailed results of the various structural assessments. The FE model can also be 2 used for various analysis purposes of future work. 3 2. Description of the French Creek Bridge 2.1 General description The French Creek Bridge is located on N 49°19.36' and W 124°24.71' of geographical coordinates near Parksville, Vancouver Island, British Columbia, Canada (as shown in Fig. 1). It is part of the Inland Island Highway No.19 (yellow line in Google map as shown in Fig. 1) that goes across the Vancouver Island from south to north along the north shore. Built in 1993, the bridge was designed to provide pathway for highway traffic to cross the French Creek, in 1996 the bridge was selected by MoTI BC as one of the three seismic monitoring sites, the bridge was instrumented and monitored up to now. Fig. 1 Site map and satellite image of the French Creek Bridge As shown in Fig. 2, the French Creek Bridge is 212 meters long, 24.2 meters width with four vehicle lanes and two pedestrian lanes. Slope of the bridge floor is 1.6% that makes the elevation descents from south abutment to north abutment. The bridge is composed of five spans and is supported by three piers in the middle, one abutment at each end of the bridge, and one pile bent 10 meters away from the south end. Column heights of three piers are 26.9m, 26.8m and 21.6m respectively, the diameter of all pier columns is 2.44m. 4 Fig. 2 Elevation (top) and plan (bottom) view of the French Creek Bridge 2.2 Superstructure The superstructure of the bridge is composed of reinforced concrete deck casted on six W-shaped steel girders which develop throughout the bridge span (Fig. 3). All girders are supported by fixed bearings and expansion bearings. The fixed bearings are located at all three piers and the expansion bearings are located at the south pile bent and north abutment. There are three expansion joints that divide the deck slab into four parts. They are located at the south pile bent, the north abutment and the pier No.3. Fig. 4 shows the location and structural detail of the expansion joints. The height of the W-shaped steel girder is about 2.2 meters with web thickness about 14mm. The width of the flange varies from 20mm to 60mm. The cross section of the deck is symmetric about the vertical center axis, and develops with a slope of -2% from center to the edges of the slab. Slab thickness is 250mm throughout the superstructure except the expansion joints. Typical concrete deck cross section detail was shown in Fig. 5. At every distance of 6.7m along the steel girders there is a set of steel diaphragm that connect all six girders from side to side, the typical diaphragm is composed of WT-shaped steel frame while diaphragm at south pile bent, north abutment and piers are made of W-shaped steel beam (as shown in Fig. 6). 5 Fig. 3 Side view of concrete deck-steel girder system (left) and piers in middle spans (right) Fig. 4 Location and structural drawing of the expansion joints Fig. 5 Typical concrete decking cross section 6 Fig. 6 Typical diaphragm (top) and diaphragm at piers, bent and abutment (bottom) 2.3 Pier and foundation Each pier is composed of two identical columns and a cap-beam above them, the cap-beam is connected to the bottom of steel girders by fix bearings. Column heights of three piers are 26.9m, 26.8m and 21.6m, the diameter of all columns is 2.44m. All columns are reinforced by 44 longitudinal rebar and spiral transverse stirrups. The cap beams are 24m long, the cross section was rectangle reinforced concrete with dimension of 3m×2.4m. Examples of the reinforcement detail of the column and cap-beam are shown in Fig. 8. At each pier, both columns are supported by twenty-odd pipe piles been driven into the ground, the piles length varies from minimum 7.3m to maximum 10m, the diameter of the all piles is 0.61m, all piles are driven into the bedrock beneath the foundation soil. Layout and dimensional details of pier system are shown in Fig. 7. 7 Fig. 7 Pier elevation view and detailed layout of pier cap and base Fig. 8 Typical reinforcement detail of column and cap-beam 2.4 Bearing There are two types of bearings used in the FCB, the PF Series pot bearing and the PMG Series unidirectional sliding bearing, both types of bearings are produced by manufacturer Goodco Z-Tech®. The pot bearings are located at three piers while the sliding bearings are installed on the south pile bent and the north abutment, location and structural detail are shown in Fig. 9. The pot bearing is designed as a fixed bearing, which restrains horizontal movement, but allows rotational movement of the bridge (Fig. 10 left). On the other hand, the sliding bearing is nothing more than a pot bearing with a sliding component added on the top, the unidirectional sliding 8 bearing allows horizontal movement along the bridge against very low sliding friction contributed by Teflon (Fig. 10 right). Fig. 9 Location and structural detail of bearings at Pier No.3 Fig. 10 Cross section of the central core of a pot bearing (left) and the unidirectional sliding component (right) (Canam Group Inc., 2012) 9 3. Ambient Vibration Test and results 3.1 Introduction AV test is a non-destructive test method preferable for acquisition of modal properties of large structures like the FCB. In AV testing, it is assumed that the input forces are stochastic in nature. This is often the case for civil engineering structures like buildings, towers and bridges, which are mainly loaded by ambient forces like wind, waves, traffic or human activity. By strategically placing numbers of sensors, the velocity and accelerations of the bridge were recorded in time domain. The recorded signals were then processed and converted into frequency domain, modal properties including natural frequencies, corresponding mode shapes as well as their respective modal damping ratios were identified by performing automatic modal analysis. In order to acquire the modal properties of the FCB, an Ambient Vibration (AV) test was conducted on the bridge. In this section, methods and procedures of the AV test on FCB were described in detail. Signals acquired from the test were processed in ARTeMIS and the modal information of the FCB including mode shapes and corresponding modal frequencies as well as damping ratios were extracted. Modal information obtained from the AV test will be used for finite element model updating in the following sections. 3.2 Instrumentation devices Seismic surveys and vibration monitoring instrument TROMINO® (Micromed, 2012) is used to capture the dynamic properties of the bridge. During the AV test of the FCB, the TROMINO sensors are set to record signals in 10 channels. These includes 3 high sensitivity velocity channels, 3 low sensitivity velocity channels and 3 accelerometer channels in all three directions X, Y and Z, plus one radio time stamp channel that used for recording time stamp of each data point. Configuration and main technical specifications of TROMINO are shown in Fig. 11. 10 In each setup, the TROMINOs were time synchronized using radio time stamps. The external radio antennas were used for communication between sensors. Sensor No.1 acts as the “Master Sensor” that controls the start time of other sensors, the time stamp of all sensors are therefore synchronized to the sensor No.1 by radio signal. Fig. 11 TROMINO sensor and specification 3.3 Description of setups The test started at 3:15pm and ended at 8:08pm on the same day on September 6th 2014. Since the fact that there are only limited numbers of sensors available, it is not possible to capture the behavior of the entire bridge with single setup. As a solution, the entire test is divided into several setups and each setup will only cover a small part of the bridge with sensors placed in line along the bridge. Except for the reference one, all other sensors were moved along the bridge from south abutment to north abutment along the edge of the bridge, the sensors first were moved along the southbound from south to north and then moved along the edge of northbound in the same sequence. Details of the locations of all sensors in each setup were illustrated in Fig. 12. Totally eight TROMINO sensors were utilized in every setup of the test, sensor No.1 is selected as reference sensor and was placed at the middle span of southbound between pier 1 and 2, the reference sensor’s location will remain unchanged throughout the entire test. During each setup, sensors were placed closed to the edge of bridge, and the long edge of each sensor was placed parallel to the longitudinal direction of the bridge and pointed to the north abutment as shown in the site photo in Fig.12. As shown in the figure since the vibration 11 characters of both sides of expansion joint are quite different; two sensors were place at both sides with approximately 1 meter distance in between. The acquisition length of each setup is approximately 30 minutes with sampling rate of 128Hz, totally seven setups were measured, Table 1 lists the start and end time of each setup. The site temperature during the test is 28 oC at 6:00pm and 27 oC at 7:50 pm. Table 1 Start and end time of each setup Setup No. 1 2 3 4 5 6 7 Start time (pm) 3:15 3:53 4:32 5:53 6:22 6:59 7:38 End time (pm) 3:45 4:25 5:03 6:14 6:53 7:30 8:08 12 Fig. 12 Setup plans and sensor locations of all setups 3.4 Data processing Time domain signals of all setups were recorded and stored in the built-in memory in TROMINO sensors. The data were stored into ASCII format. Software Grilla was used to extract the test signals from sensors into computer and converted into text format. 13 In order to assure the quality of the data, all test signals were reviewed in time domain. Fig. 13 shows an example of the high-gain velocity time history signals from setup 1, noticed that only signals in North-South direction were plotted as an example. It can be observed that no dead signals, large picks any other inconsistencies were found within the signals. Fig. 13 High-gain velocity time-history signals of Setup No.1 (N-S direction) Further data processing including baseline correction, filtering and modal identification are done using ARTeMIS. No decimation was applied on the original data so that the program could have enough data density to extract the modal information. For a 200 meters concrete bridge like the FCB, the frequency range of interest was lower than approximately 20Hz, therefore the default filtering range of 0-64Hz was utilized, which indicates that any frequency content larger than 64Hz was eliminated from the signal. 14 3.5 Modal identification and results 3.5.1 Frequency Domain Decomposition The Frequency Domain Decomposition (FDD) (Brincker, 2000) is an output-only system identification technique popular in civil engineering, in particular in structural health monitoring. As an output-only algorithm, it is useful when the input data is unknown, and this is exactly the case in an AV test. The relationship between the unknown inputs x(t) and the measured responses y(t) can be expressed as (Bendat, 2011): 𝐺𝐺𝑦𝑦𝑦𝑦(𝑗𝑗𝑗𝑗) = 𝐻𝐻�(𝑗𝑗𝑗𝑗)𝐺𝐺𝑥𝑥𝑥𝑥(𝑗𝑗𝑗𝑗)𝐻𝐻(𝑗𝑗𝑗𝑗)𝑇𝑇 where Gxx(jω) is the (r × r) power spectral density (PSD) matrix of the input, r is the number of inputs, Gyy(jω) is the (m × m) PSD matrix of the responses, m is the number of responses, H(jω) is the (m × r) frequency response function (FRF) matrix and the overbar and superscript T denote the complex conjugate and transpose, respectively. In the FDD identification, the first step is to estimate the PSD matrix. The estimate of the output PSD 𝐺𝐺�𝑦𝑦𝑦𝑦(𝑗𝑗𝑗𝑗𝑖𝑖) known at discrete frequencies ω = ωi is then decomposed by taking the singular value decomposition (SVD) of the matrix: 𝐺𝐺�𝑦𝑦𝑦𝑦(𝑗𝑗𝑗𝑗𝑖𝑖) = 𝑈𝑈𝑖𝑖𝑆𝑆𝑖𝑖𝑈𝑈𝑖𝑖𝐻𝐻 where the matrix Ui = [ui1, ui2, . . . , uim] is a unitary matrix holding the singular vectors uij, and Si is a diagonal matrix holding the scalar singular values sij. After that, the modes can be easily picked by locating the peaks in SVD plots calculated from the spectral density spectra of the responses (as shown in Fig.14). The Enhanced Frequency Domain Decomposition (EFDD) (Brincker, 2001) adds a modal estimation layer which is divided into two steps. The first step is to perform the FDD Peak Picking, and the second step is to use the FDD identified mode shapes to identify the Single-Degree-Of-Freedom (SDOF) Spectral Bell functions and from these SDOF Spectral Bells estimate both the frequency and damping ratio. Compared to FDD, the EFDD gives an improved estimate of the natural frequencies, the mode shapes and the damping ratios. Therefore the 15 EFDD techniques are used to identify the modal properties of the FCB. In ARTeMIS, the EFDD algorism was integrated in the program. 3.5.2 Modal identification results Peak-picking method was performed manually in order to distinguish and select the structural modes of vibration from the rest of the frequency content. Fig. 14 presents the spectral content of the recorded AV test signals and the picked peak values of the natural frequencies estimations of the FCB. Totally eight modes are identified, their mode shapes and corresponding frequencies are summarized in Fig. 15. Fig. 14 Peak-picking of Singular Values of Power Spectral Densities matrix (ARTeMIS) 16 Fig. 15 Modal identification results from AV test (ARTeMIS) 17 4. Elastic Finite Element (FE) modeling of the bridge A FE model of the FCB (as shown in Fig. 16) is created based on the structural drawings provided by BC Ministry of Transportation. First, the model is developed in integrated 3D bridge analysis and design program called CSiBridge (Computers and Structures, Inc.). In CSiBridge, the bridge models are defined using terms that are familiar to bridge engineers. The bridge wizard function of CSiBridge can easily define complex bridge geometries, boundary conditions and load cases. The major components of the FCB including superstructure, bearings, abutments, bents, piers and piles are modeled using frame, shell and link elements created by program, all elements can be automatically updated as the bridge definition parameters are changed. After preliminary modeling of the bridge, the model is converted into program SAP2000 (Computers and Structures, Inc.) for further refinement and modal information extraction. Fig. 16 Overview of the FE model of French Creek Bridge 4.1 Elastic material properties The structural drawings provide the material properties of concrete and structural steel of the bridge. There are only two types of concrete materials used in the FCB: the deck concrete and the foundation concrete. It is specified that the deck concrete is only used to build the concrete decking section. The deck concrete has minimum compressive strength of 35MPa at 28 days. The foundation concrete are used to build abutments, cap-beams, columns and piles, all foundation concrete have a minimum compressive strength of 30 MPa at 28 days. The modulus of elasticity was calculated based on the 28-day compressive strength of concrete using CSA A.23.3-04 (CSA, 2010) with the relationship: Ec = 4500�f′c 18 All steel materials are sorted into the structural steel and the reinforcing steel. All reinforcing rebar in concrete members are made of the reinforcing steel, the reinforcing steel is confirmed to CSA G30.12M Grade 400 (CSA, 1992). The structural steel is used to build girders and diaphragms, it is specified by CSA G40.21M Grade 350 at category 3 (CSA, 2004). Summary of material properties used in initial FE model are provided in Table 2. Table 2 Material property of concrete and steel of initial FE model Material Type F'c (MPa) Ec (MPa) Density (kg/m3) Poisson ratio Foundation Concrete 30 24700 2400 0.2 Deck Concrete 35 26600 2400 0.2 Material Type Fy (MPa) E (MPa) Density (kg/m3) Poisson ratio Structural Steel 350 200000 7900 0.3 Reinforcing Steel 400 200000 7900 0.3 4.2 Modeling of superstructure 4.2.1 Superstructure The superstructure of the bridge is a composite section made of reinforced concrete deck cast on the steel girder. In CSibridge, the concrete decking is modeled by thin shell element and the steel girders are modeled by regular 3D frame element. The bending thickness of the shell element remains identical throughout the bridge, while frame element properties change according to the section changes of the steel girders. The centroid of the girder was located on the node at one-half of the web height (as illustrated in Fig. 17). In order to model the decking and girders rigidly connected as an integral structure, rigid links, through the constraints of DOFs are applied between nodes on decking and girders, the rigid links were also applied between the girders and the bearings as shown in Fig. 17. 19 Fig. 17 Finite element modeling method of the superstructure system For approximately every 6.7 meters along the bridge, there is a set of steel diaphragms that connect all six girders from side to side. In CSibridge, the diaphragms are modeled by frame element. By defining section geometric parameters based on the structural drawing, the program can automatically generate the steel frame members and connect them between nodes on steel girders (as illustrated in Fig. 17). Fig. 18 shows the typical cross section of the superstructure in FE model. Fig. 18 Superstructure cross section modeled in CSiBridge 4.2.2 Expansion joints As shown in Fig. 4, there are three expansion joints located on the superstructure, they disconnect the concrete decking of the superstructure by making gaps with width of maximum 375mm to minimum 60mm in between, therefore when the bridge is subjected to external load, the force can be transferred into the bearings beneath it 20 and finally transferred into the foundation. Since the connection in between the joint is nonstructural, so the stiffness can be ignored. In the FEM, the shell element at the expansion joint is divided in two by removing part of the shell element, the steel girders are also disconnected at the expansion joint. However, in between the gap of every expansion joint, a link element was added to link between the concrete decking. These link elements do not contribute any mass or stiffness. They are only used for monitoring the gap distance during the time history analysis as discussed in Section 9.2.3. The FE model detail of expansion joints at south pile bent and pier No.3 are presented in Fig. 19. Fig. 19 Model detail of expansion joints at south pile bent (left) and pier No.3 4.2.3 Bearings As discussed in Section 2.4, there are two types of bearings used in the FCB, the fixed pot bearings are used in three piers (see Fig. 2) while the unidirectional sliding bearings are applied to the south pile bent and the north abutment. In SAP2000, the bearings are modeled by a set of massless rigid link elements connecting between the superstructure and piers or foundation. Each link element has six uncoupled stiffness value in six DOFs, which includes three translational directions and three rotational directions. It should be noticed that the behavior of all bearings is assumed to remain elastic in this model. 21 The central core of the PF series fixed pot bearing, as shown in Fig. 10, has a piston plate fits in a pot plate and it is in contact with a elastomeric pad, therefore since the pot plate is confined by piston the stiffness in two horizontal directions U2 and U3 (illustrated in Fig. 20) are considered as rigid. On the other hand, the elastomer that been contained in the pot behaves approximately like confined fluid, it allows rotation of the piston but negligible vertical displacement. Therefore the stiffness in vertical direction U1 (see Fig. 20) of the link element is also modeled as rigid, the rotational stiffness of the bearing, however, is calculated by equation suggested by Akogul et al. (2008): Kt = EIt where E is the effective compression and rotational modulus for elastomer, I is the second moment of area of the plan area of the bonded elastomer about its axis of rotation, and t is the thickness of the elastomeric pad. By adding a unidirectional sliding component on the pot bearing, the fixed bearing becomes a mobile bearing that allows horizontal movement along the longitudinal direction of the bridge (as shown in Fig. 10), the sliding component guiding surface is made of PTFE (Teflon) materials, therefore the stiffness in longitudinal direction along the bridge U2 (see Fig. 20) is merely contributed by the friction between PTFE discs. The value of the stiffness in U2 is estimated as the product of axial force due to self-load and coefficient of friction, which is less than 0.03 between the polished stainless steel sheet and the confined PTFE disc. Table 3 summarizes the calculation results of the bearing stiffness. Table 3 Initial stiffness values of bearings Bearing type U1 kN/mm U2 kN/mm U3 kN/mm R1 kN*mm/rad R2 kN*mm/rad R3 kN*mm/rad Fixed bearing rigid rigid rigid 1265 1265 1265 Sliding bearing rigid 423 rigid 1265 1265 1265 22 Fig. 20 Local axis of bearing link element in SAP2000 4.3 Modeling of piers 4.3.1 Columns and cap-beams The pier of FCB is the most important component since it carries most of the self-load and will be subjected to large internal force and deformation. Therefore all components of pier including column, cap-beam, pile and pile-cap were modeled in great detail. 3D beam-column elements are used to model the column and cap beam, section of cap-beams are simplified as rectangular and columns remains as circle section. For elastic modeling stage, reinforcement is not considered in the cross section, description of modeling the nonlinear properties of the column and cap-beam will be provided in Chapter 7. Elevation view of the FE model of Pier No.3 and corresponding cross sections of column and cap-beam are shown in Fig. 21. 4.3.2 Pile and pile-cap According to the borehole data provided by structural drawings and the site classification criteria of CAN/CSA-S6-14, the soil properties at the FCB site is categorized as Site Class C which indicate a relative soft soil condition that may have significant effect on the modal properties. Instead of simply fix the tips of the columns, the pile-foundation interaction was included in the FE model. The pile groups of all three piles are modeled by 3D beam-column element. The pile-caps are modeled by shell element (as shown in Fig. 21). All DOFs of all nodes on one pile-cap were constrained together, which makes the whole pile-cap as a rigid body. 23 Fig. 21 Structural drawing and FE model of Pier No.3 (see Fig. 2) and member cross sections 4.4 Modeling of boundary condition of the foundation Modeling of the boundary condition is one of the most important issues for bridge dynamic analysis. By using the proper element available in SAP2000, the dynamic characters of the boundary condition can be properly simulated. In the FCB, important components including abutment, pile and pile-cap are modeled in detail. 4.4.1 Abutments and south pile bent Since there is not enough information provided regarding the soil condition at the abutment, the soil properties including soil stiffness cannot be correctly estimated. For simplicity of the model, the abutments’ behavior is ignored and considered as rigid. 4.4.2 Pile foundation When modeling the boundary condition of pile group under each pier, the influence of the soil-pile interaction on the dynamic characteristic of the bridge needs to be considered. The pile-foundation system is detailed modeled in the FEM. To properly model the soil-pile interaction effect, the analytical subgrade reaction model, which wildly known as the Winkler Model (Fig. 22) (Fleming et al., 1985) (Selvadurai, 2013), is used for the analysis. The interaction is modeled in terms of: 1.soil to pile vertical stiffness interaction, 2.horizontal stiffness 24 of the soil surrounding the piles and pile caps. Piles are modeled as regular frame element and the pile cap is considered as a rigid body, the surrounding soil is modeled as an array of uncoupled soil spring using linear link element. The elements of 10 meters long piles are evenly discretized into 10 segments, length of each section is 1 meter. Therefore a link element is assigned in every 1 meter along the pile as shown in Fig. 22. Fig. 22 Winkler Model (left) (Adhikary et al., 2001) and pile discretization (right) 1) Horizontal soil model surrounding piles A linear relation between force and displacement is used to idealize the soil strength. Soil spring stiffness varies according to soil types. The horizontal soil spring stiffness at depth Z was obtained by: (Das, 1984) Ksh = ks∆zZ Where Ksh is the equivalent spring stiffness, ks is the coefficient of subgrade reaction given in kN/m3, ∆z is spacing between springs. 2) Horizontal soil model surrounding pile cap The horizontal soil pressure surrounding the pile cap is also considered as passive, the pile cap spring stiffness is calculated as follow expression: Kcap = ks Hcap22 (WcapD∗ ) Where D∗ is the nominal pile diameter and Hcap and Wcap are the height and width of the pile cap, respectively. 25 3) Vertical skin friction resistance stiffness The vertical skin friction resistance stiffness, Kvfi, was given by (Pender 1978; Poulos 1971), K𝑣𝑣𝑣𝑣𝑖𝑖 = 1.8𝐸𝐸𝑠𝑠−𝑡𝑡𝑖𝑖𝑡𝑡𝜂𝜂λ(0.5𝜆𝜆𝜂𝜂)𝛼𝛼 where η is the pile ratio, and λ is the pile-soil stiffness ratio given by, 𝜂𝜂 = 𝐿𝐿𝐷𝐷 λ = 𝐸𝐸𝑡𝑡𝐸𝐸𝑠𝑠−𝑡𝑡𝑖𝑖𝑡𝑡 where Es-tip is the soil young modulus a the pile tip. For distributed vertical springs positioned along the full length the total skin friction resistance given by the piles α is given by, α = 𝑍𝑍𝛥𝛥𝑍𝑍𝐿𝐿2 4) Vertical end bearing stiffness On the other hand, the piles were designed as end-bearing pile instead of friction pile, the tips of the pile was fixed embed into the rock layer beneath the surface. Therefore it is assumed that the tips of the piles are all fixed. 4.5 Modal property extraction and results The natural frequencies and corresponding mode shapes of the French Creek Bridge are evaluated by modal analysis function in SAP2000. The first eight modes extracted were used to compare with the experimental results, summary of the initial finite element results are listed as part of the information provided in Table 4, modal frequencies and corresponding mode shapes are presented and compared with AV test results in Fig. 23. It can be observed in Fig. 23 that the extracted mode shapes from SAP2000 are highly consistent with the mode shapes identified from the AV test, therefore it can be concluded that the assumptions and methods used for the modeling are reasonable, and parameter values evaluated for the material and element properties are relatively realistic. 26 Fig. 23 Model identification results from AV test (left) and FE model (SAP2000) (right) 27 5. Finite element model updating As described above, the finite element model of the French Creek Bridge is developed based on the structural drawing and series of idealized assumptions, which may not be about to represent the real behavior of the actual structure. As shown in Fig. 23, although there is a good match of mode shapes between experimental and analytical results, significant deviation of modal frequencies still exists. In order to achieve an FE model that yields modal properties (frequencies and mode shapes) that more closely match those identified, and more importantly, to achieve a more accurate prediction of the dynamic response of the bridge, a sensitivity-based model updating procedure is conducted on the initial FE model. 5.1 Sensitivity analysis 5.1.1 Sensitivity analysis methodology Sensitivity analysis is the study of how changes of parameters influences the structural responses, it allows an analyst to get a feeling on how structural responses of a model are influenced by modifications of parameters like spring stiffness, material stiffness, material density etc. (Friswell, 1995) In case of updating model with large number of parameters, a primary sensitivity analysis can ease the updating stage by focusing on the sensitive parameters. Not knowing the sensitivity of parameters can result in time being uselessly spent on non-sensitive ones. Therefore in order to select the sensitive parameters and improve the efficiency of optimization of the FE model, the sensitivity of the modal parameter to the modal frequencies need to be studied first. By inspection, several physical properties are selected for the sensitivity analysis. Four different types of parameters were considered for updating: (i) mass density of the concrete and steel, (ii) elastic modulus of the concrete and steel, (iii) translational and rotation stiffness of bearing link and pile soil spring, (ix) moment of inertial of the cap-beam and column sections. For each parameter, the initial value (calculated values based on equations and assumed values based on structural drawings as described in Chapter 4) from the initial FE model is altered by 2, 5, 10, 20, 50 and 100 28 percent at each time while other variables remain unchanged, after each alteration, the corresponding modal frequencies are calculated and recorded. Then the influence curves of each parameter on modal frequencies are plotted, as percentage of modal frequency changed against percentage of parameter value changed. 5.1.2 Analysis results Fig. 24 shows an example of the sensitivity curve of two of the parameters: the elastic modulus of the structural steel Esteel and the stiffness of the soil foundation spring Ksoil. For the Esteel, it can be observed that the values of all modal frequencies are quite sensitive to the value of Esteel: when the value increased 10% mode 1 (see Fig. 23) increased about 1.5% and mode 8 (Fig. 23) get a more significant 3.7% increase. On the other hand, there is less than 0.5% change in all modes even though the Ksoil value has increased 100%. Therefore it can be concluded that Esteel has significant effect on dynamic characteristic of the structure and it is selected as one of the update parameters while the Ksoil is ruled out. Other sensitivity curves including elastic modulus and mass density of the structural steel, deck concrete and foundation concrete; translational and rotational stiffness of bearing; as well as the moment of inertial of pier column and cap-beam were presented through Fig. 25 to 33. Fig. 24 Sensitivity analysis curves of Ksoil (left) and Esteel (right) 29 Fig. 25 Sensitivity analysis curve of mass density of structural steel (ρsteel) Fig. 26 Sensitivity analysis curve of elastic modulus of foundation concrete (Efound) 30 Fig. 27 Sensitivity analysis curve of mass density of foundation concrete (ρfound) Fig. 28 Sensitivity analysis curve of elastic modulus of decking concrete (Edeck) 31 Fig. 29 Sensitivity analysis curve of mass density of the decking concrete (ρdeck) Fig. 30 Sensitivity curve of translational stiffness of the sliding bearing in U2 (see Fig. 20) direction (KU2) 32 Fig. 31 Sensitivity curve of the rotational stiffness of all bearings in R2 & R3 directions (see Fig. 20) Fig. 32 Sensitivity curve of the moment of inertial of pier column (Icol) 33 Fig. 33 Sensitivity curve of the moment of inertial of cap-beams (Icap) By observing the sensitivity curves illustrated above, eight parameters were chosen for the FE model optimization. The eight parameters been considered in model updating are: Edeck, Efound and Esteel for modulus of elasticity of the decking concrete, foundation concrete and structural steel respectively; ρdeck, ρfound and ρsteel for mass density of the three kinds of materials; KU2 for translational stiffness of the sliding bearing in bridge longitudinal direction and Icol for moment of inertial of the pier column. These eight parameters were chosen because the modal property (modal frequency) of the bridge is relatively sensitive to the changes of the values of those parameters. Other parameters including soil spring stiffness, rotational stiffness of the bearing and moment of inertial of the cap-beam will not be considered in further study, since they have very small effect on the modal frequencies of the structure. 5.2 Model updating In the FE model updating, the selected parameters are varied in the way that will improve the match between experimental and analytical results, then the modal results will be evaluated and the parameter will be 34 reassigned with modified value, this “try-and-error” iterative process will be repeated until a good match is obtained. For example it can be easily observed in Fig. 23 that the modal frequencies of the first, second and fourth mode of the initial FE model is significantly small than the AV test results. Therefore as for the “trial” step, one reasonable way would be increasing the modulus of elasticity of the concrete and steel. During the trial if the result is not satisfactory, other factors could be added into consideration such as increasing the moment of inertial of the column section and the procedure repeats. 5.3 Results and discussion Table 4 shows the summary of the results, the natural frequencies before and after updating are listed and the ambient vibration test results are presented again for comparison purpose. And parameters before and after updating is summarized in Table 5. Table 4 Natural frequencies before and after updating Mode Experimental frequency/Hz fFEA/Hz (before) Error/% fFEA /Hz (after) Error/% Frequency changed/% 1 1.438 1.176 -22.28 1.413 -1.77 20.15 2 1.695 1.592 -6.47 1.615 -4.95 1.44 3 1.787 1.852 3.51 1.891 5.50 2.11 4 2.263 1.916 -18.11 2.364 4.27 23.38 5 2.565 2.332 -9.99 2.686 4.50 15.18 6 2.677 2.634 -1.63 2.777 3.60 5.43 7 2.887 2.950 2.14 3.172 8.98 7.53 8 3.317 3.219 -3.04 3.718 10.79 15.50 It can be observed from Table 4 that the higher modes from updated model have bigger errors than initial model. The reason is that in order to match the first mode of the initial model, eight selected parameters were changed to increase the stiffness of the structure. However, when the structure’s stiffness is high enough to match the first mode, the higher modal responses become “over stiffened”. On the other hand, when the higher modes match the test results the structure become “too soft” and the first mode cannot have a good match. Therefore in 35 order to achieve a better overall match in terms of modal frequencies of all eight modes, the low-error matches of the higher modes were sacrificed in certain extent. Table 5 Parameters before and after updating Parameter Unit Initial value Updated value Percentage changed Edeck MPa 26600 31000 16.54 Efound MPa 24700 30000 21.46 Esteel MPa 200000 196000 -2.00 ρdeck kg/m3 2400 2498 4.08 ρfound kg/m3 2400 2396 -0.17 ρsteel kg/m3 7900 7933 0.42 Ku2 kN/mm 118 2500 2172.73 Icol m4 1.74 1.91 9.77 - Edeck, Efound and Esteel: Modulus of elasticity of the decking concrete, foundation concrete and structural steel respectively; - ρdeck, ρfound and ρsteel: Mass density of the three kinds of materials; - KU2 for translational stiffness of the sliding bearing in bridge longitudinal direction U2 (see Fig.20) - Icol: Moment of inertial of the pier column (Icol value is identical for all columns) By comparison in Table 5, it can be observed that: 1) The elastic modulus of the deck and foundation concrete has been underestimated. Probably because the actual concrete strength after construction is larger than the nominal strength grade provided on the structural drawings. 2) The mass density of the deck concrete is increased by 4 percent. Probably due to erroneous assumptions of the extra weight contributed by non-structural component such as asphalt pavement and concrete fence. Since details of the pavement and fence (such as the thickness and mass density of the pavement) were not provided in the structural drawings. 36 3) Translational stiffness of the sliding bearing (KU2) has the most significant change among all parameters, the stiffness is now one-magnitude-order-greater than the initial value. One reasonable explanation is that the stiffness in U2 direction (see Fig. 20) at sliding bearing is not solely contributed by the friction between sliding surface, probably the assumption about the stiffness of the expansion joint above sliding bearing is incorrect, the non-structural connection joint may have a huge among of contribution to the stiffness value. 37 6. Selection of ground motion 6.1 Introduction As specified in Canadian Highway Bridge Design Code 2015 (CAN/CSA-S6-14) Clause 4.4.5, the elastic dynamic analysis (EDA) and the nonlinear time history analysis (NTHA) are required for the seismic analysis of the FCB. According to Clause 4.4.3.6, eleven or more sets of ground motions shall be used for both analyses. In this chapter, eleven sets of ground motions from eight different earthquake events were selected based on the local site condition and tectonic environment. To select the ground motions, first the Probabilistic Seismic Hazard Analysis (PSHA) (see Section 6.2) was conducted using PSHA software EZ-FRISKTM (Risk Engineering Inc., 2011), for the region where FCB is located. Quantitative results including uniform hazard spectrum (UHS) (see Section 6.3) were obtained. Then in order to find out the earthquake magnitude (M) and epicenter distance (R) that contribute most to the seismic hazard on FCB location, the seismic hazard deaggregation (see Section 6.4) procedure was conducted based on certain period and annual frequency of exceedance. Finally the acquired uniform hazard spectrum were used as target spectrums to select and scale ground motions from the UBC Earthquake Engineering Research Facility (EERF) strong motion database using advanced selection algorithm. The methodology described in this section can be used to decide seismic input in any location in BC area. 6.2 Probabilistic Seismic Hazard Analysis To select the ground motions that best represent the seismic hazard according to local geological condition, the Probabilistic Seismic Hazard Analysis (PSHA) (Venture, 2011) (Baker, 2008) was conducted on the target location, the seismic hazard was calculated based on all seismic sources in the fifth generation seismic hazard model provided in the Open File 7576 of Geological Survey of Canada (GSC, 2014), the new model has been adopted by the National Building Code of Canada (NBCC, 2014) 2015. 38 PSHA considers all possible earthquake scenarios on contributing faults near a site to compute exceedance probabilities of spectral quantities. In practice, this is typically computed using the hazard tool provided by the Geological Survey of Canada (http://www.nrcan.gc.ca/hazards/resources/tools-applications/) or with tools that provide more flexibility to the user, such as OpenSHA (http://opensha.org/) or seismic hazard analysis software. In this study, the site-specific earthquake hazard analysis software EZ-FRISKTM was utilized to calculate the seismic hazard. The latest version 7.25 carries the state-of-the-art PSHA method and the newest generation seismic source. Before performing the probabilistic calculations, several site parameters and properties have to be defined first: 1) Site coordinate. The site location was needed for EZ-FRIZK to utilize the seismic source database, the location of the FCB was defined by latitude 49.32 North and longitude -124.41 West, coordinates are given decimally. 2) Soil type. In the area where the FCB is located, the soil type was defined as Site Class C according to the description of the soil type of structural drawing borehole test results. Site Class C represents dense soil and soft rock. 3) Soil shear wave velocity in the upper 30 meters (Vs,30). The Vs,30 value is considered in range of 360-760 m/s as specified in CAN/CSA-S6-14 Table 4.1. With three types of inputs provided above, the PHSA was conducted by EZ-FRISK and the Uniform Hazard Spectrum (UHS) with specific hazard level on target location were calculated, the results will be used as reference to select and scale ground motions. 6.3 Uniform Hazard Spectrum A common goal of PHSA is to identify a design response spectrum to use for structural analysis. As one type of output of PHSA, the Uniform Hazard Spectrum (UHS) has been used as the target spectrum in design practice for the past two decades. This spectrum is called a uniform hazard spectrum because every ordinate has an equal rate of being exceeded. The UHS is developed by performing the PSHA calculations for spectral accelerations at a range of periods as described below. 39 Each PHSA calculation results in a hazard curve for a specified period of interest. Fig. 34 presents examples of the seismic hazard curves for the FCB site, computed for the spectral period of 0.72 second (period T1 of first mode of the FCB as shown in Table 4 and Fig. 23) and 1 second (default setting of the EZ-FRISK). This figure shows that for the 2% in 50 year ground motion, Sa,gm (0.72s) = 0.65g and Sa,gm (1.0s) = 0.52g. After combining all the hazard curves at many periods in the range of interest (0-10s which was the default setting of EZ-FRISK in PHSA), a certain frequency of exceedance (for example 0.0004 which is 2% in 50 years) was selected and for each period the spectral acceleration amplitude corresponding to that rate is identified. The uniform hazard spectrum was created by plotting those spectral acceleration amplitudes versus their periods. Fig. 34 gives an example of how the UHS with hazard level of 2% in 50 years was plotted. Fig. 34 Combining hazard curves from individual periods to generate a uniform hazard spectrum 40 Based on the PHSA results of the FCB area, the UHS were generated in terms of probabilities of exceedance of 2%, 5%, 10% and 50% in 50 years (four hazard levels which are commonly used in seismic analysis and design clauses in CAN/CSA-S6-14), the corresponding return periods are 4975 years, 2475 years, 975 years, 475 years and 72 years. The UHS was acquired considering 5% damping ratio, since the PEER083 report (Aviram et al., 2008) recommend 5% damping ratio of the regular concrete bridge like FCB in seismic analysis. UHS results for different return periods were plotted in Fig. 35. Fig. 35 Uniform hazard spectra in FCB site (5% damping ratio) 6.4 Seismic hazard deaggregation One of the primary advantages of PSHA is that it accounts for all possible earthquake sources in an area when computing seismic hazard, however, this could also be a disadvantage. Because we have aggregated all scenarios together in the PSHA calculations, the answer to the question: "which earthquake scenario is most likely to cause maximum peak ground acceleration (PGA)? ", is not immediately obvious. A process known as deaggregation (Bazzurro and Cornell, 1999) is typically used to calculate the relative contribution of different earthquake sources to the rate of exceedance of a given ground motion intensity (by appointing PGA level). By performing deaggregation on the target location, one can find out which sources 41 (with individual epicenter distance R and earthquake magnitude M) contribute the strongest seismic events to the total seismic hazard. In this study, by using EZ-FRISK, a magnitude-distance (M-R) deaggregation analysis was performed at T=0.72s (approximately the fundamental period of FCB as shown in Fig. 23) and the corresponding spectral acceleration Sa=0.65g (see Fig. 34), hazard level concerned was 2% in 50 years. The hazard level of 2% in 50 years will also be used in further seismic evaluation of the FCB, because it is the most conservative hazard level for seismic design specified in CAN/CSA-S6-14. To give a clear illustration of the distribution of probability density, the 3D graphs of magnitude-distance (M-R) deaggregation were plotted in Fig. 36. Fig. 36 Magnitude-distance (M-R) deaggregation of hazard level 2% in 50 year at FCB site Due to the plate movement between North American Plate and Juan de Fuca Plate, there are three types of earthquake that contribute to the seismic hazard of British Columbia: crustal earthquakes, subcrustal 42 earthquakes and subduction earthquakes. During the PSHA and deaggregation analysis, the probability density integrated the seismic sources of these three types of earthquake. As shown in Fig. 36, three clusters of M-R pairs can be easily observed in the M-R graph, corresponds to three types of earthquake. In the 3D deaggregation graph, higher the probability density is, darker the color of data bar is. High probability means high percentage of contribution to the site seismic hazard. For period of T=0.72s, the hazard is dominated by three events: i) the pink one with short epicenter distance (0-50km) represents crustal earthquake (M=5.0-7.5); ii) the pink one with longer distance (30-120km) represents subcrustal earthquake (M=5.0-7.5) sources; iii) the one with high probability density and longest distance (50-130km) is contributed by subduction earthquakes (M>8.0). According to the deaggregation result, selection criteria for all three types of earthquakes were summarized and listed in Table 6. The ground motions used for seismic evaluation will be selected based on the acquired magnitude-distance ranges. Table 6 Ground motion record selection criteria Source Distance range (km) Magnitude range Crustal 0-50 5.0-7.5 Subcrustal 30-120 5.0-7.5 Subduction 50-130 8.0+ 6.5 Selection and scaling of ground motion Totally eleven ground motions were selected from eight different earthquake events which includes four crustal earthquakes, two subcrustal earthquakes and two subduction earthquakes. All motions are selected from worldwide seismic records saved in EERF strong motions database. A series of Matlab codes are developed by UBC EERF to perform the selecting and scaling based on spectral matching. The method used in the Matlab codes is called the Greedy Selection Algorithm (Jayaram et al., 2011). This procedure starts by selecting certain number of ground motions based on earthquake type and corresponding epicenter distance and earthquake magnitude selection criteria. Then the selected ground motion suite are scaled so that the median of the geometric mean value of selected motions matches certain spectral 43 acceleration of the target spectrum (UHS of 2% in 50 years, see Fig. 35). The spectral acceleration at the period range of interest, namely, 0.2T1 to 1.5T1 (which is 0.28s to 2.1s in this case) is used to match the spectrums, where T1 is period of the first mode (see Fig. 23) of the FCB. After that, one of the ground motions is replaced by the next eligible one in database and the iteration continues until all motions in database are processed and the best matches were saved as final results. Table 7 lists the information of the selected ground motions. Acceleration time history of all unscaled selected ground motions are plotted in Fig. 37 through 39. Table 7 Selected ground motion general information No. Earthquake Year M Distance (km) Station Type Comp. VS30 (m/s) PGA (g) Scale Factor 1 Cape Mendocino 1992 7.0 19.95 FOR000 Crustal NS 457.10 0.141 3.13 2 Northridge 1994 6.7 5.92 LDM064 Crustal NS 629.00 0.511 0.97 3 ChiChi Aftershock 1999 6.3 33.61 CHY028 Crustal NS 542.60 0.141 2.44 4 Irpinia, Italy 1980 6.2 22.69 B-VLT000 Crustal EW 530.00 0.090 3.74 5 Miyagi-Oki, Japan 2005 7.2 118.97 IWT010 Subcrustal EW 565.29 0.173 2.49 6 Miyagi-Oki, Japan 2005 7.2 105.26 MYG013 Subcrustal EW 535.49 0.257 1.53 7 Geiyo, Japan 2001 6.4 59.32 YMG018 Subcrustal EW 499.35 0.229 1.45 8 Geiyo, Japan 2001 6.4 72.06 EHM015 Subcrustal EW 417.15 0.295 1.09 9 Tōhoku, Japan 2011 9.0 100.90 TCG016 Subduction EW 693.17 0.394 1.76 10 Tōhoku, Japan 2011 9.0 59.20 IBR002 Subduction EW 693.20 0.581 0.69 11 Hokkaido, Japan 2003 8.0 110.81 HKD094 Subduction EW 381.10 0.131 3.59 44 Fig. 37 Time history acceleration of unscaled selected ground motions (crustal) Fig. 38 Time history acceleration of unscaled selected ground motions (subcrustal) 45 Fig. 39 Time history acceleration of unscaled selected ground motions (subduction) After selection and scaling of each type of ground motions, the response spectra of each type of ground motions are plotted with the target UHS. The scaled spectra of crustal, subcrustal and subduction motions are illustrated in Fig. 26 to 28. The thin solid lines are scaled ground motions, the dashed line is the geomean response spectra of the scaled motion suite and the solid red line is the 2% in 50 years Site Class C UHS with 5% damping ratio at FCB location. 46 Fig. 40 Response spectra of scaled selected ground motions with target spectrum (crustal) Fig. 41 Response spectra of scaled selected ground motions with target spectrum (subcrustal) 47 Fig. 42 Response spectra of scaled selected ground motions with target spectrum (subduction) 48 7. Section analysis of ductile members and nonlinear hinge modeling 7.1 Introduction A three-dimensional elastic model of the FCB has been built and updated as described in previous chapters. As discussed in Section 6.1, the elastic and nonlinear time-history analyses are required to be carried out on the FE model. However, under strong ground motion inputs, the linear model will fail to represent certain sources of inelastic response from yielded structural components, therefore gives unrealistic response results. In order to better represent the actual behavior of the bridge, nonlinear behaviors of certain structural components are incorporated in the model. In general, increasing the sophistication of the nonlinear model will also increase the computational effort of the analysis. To make a balance between complexity of the model and calculation time, not all components are modeled as nonlinear, numbers of assumptions has to be made to simplify the nonlinear modeling. It is recommended in PEER803 nonlinear analysis guidelines that for standard bridge structure, structural components including superstructure, foundation spring and soil-structure interaction shall be considered as elastic while pier column, cap-beam and abutment should be modelled as nonlinear. In the FCB case, the detailed soil mechanical characteristics at the abutment and pile foundation was not provided in the drawing, therefore to reduce computational effort, abutments are assumed to be rigid and the soil-foundation interaction is ignored. On the other hand, when exterior load apply to the bridge, the load will transfer from foundation to the piers, and the columns and cap-beams will subjected to huge bending moment and shear forces at both ends of each member, large displacement will occur at plastic region due to the height of the columns, therefore nonlinearity of primary components including piers and cap-beams become the major issue of this model. In order to obtain the nonlinear properties needed for nonlinear modeling of ductile components and what’s more to gather the member capacities for the seismic evaluation, a series of section analyses were conducted on the critical sections of pier column and cap-beam, program Xtract® (Chadwell, 2002) coded by Imbsen Software System was used for the section analysis. Based on code requirement the expected material nonlinear models were developed, the section nonlinear behavior information such as moment-curvature diagrams were 49 obtained by program calculation and the shear capacities of primary components were calculated following the equations specified in the bridge code. Finally based on the nonlinear properties obtained from section analysis, the nonlinear hinge models for piers and cap-beams used in the nonlinear seismic analysis were developed in SAP2000. 7.2 Nonlinear material models 7.2.1 Expected material properties According to CAN/CSA-S6-14 Clause 4.11.8, for seismic evaluation of existing bridges, the expected nominal resistance of the ductile member shall be used as the member capacities that resist all seismic demands. The capacity of concrete components shall be calculated based on the expected material properties to provide a more realistic estimate of design strength. The expected yield strength of reinforcing bars fye and expected compressive strength of concrete f`ce is assumed for the material properties. For the FCB case, the fye shall be taken as 1.2fy and f`ce as 1.25f`c. 7.2.2 Mander concrete model The Mander concrete model (Mander, 1988) was adopted by Xtract to define the stress-strain relationship of confined and unconfined concrete. The confined concrete properties were modified based on unconfined model, taken into consideration of the effect of the confinement by transverse reinforcement. For confined concrete with monotonic loading, the longitudinal compressive concrete stress f is given by, f = f`𝑐𝑐𝑐𝑐𝑥𝑥𝑥𝑥𝑥𝑥 − 1 + 𝑥𝑥𝑟𝑟 where f`cc is the compressive strength of confined concrete. x is defined as εc/εcc, where εc is the longitudinal compressive concrete strain and εcc is given as, ε𝑐𝑐𝑐𝑐 = ε𝑐𝑐𝑐𝑐[1 + 5 �f`𝑐𝑐𝑐𝑐f`𝑐𝑐𝑐𝑐� − 1] 50 where εce is the corresponding strain at expected compressive strength of concrete f`ce, (εce is assumed to be 0.002). r is calculated by, r = 𝐸𝐸𝑐𝑐𝐸𝐸𝑐𝑐 − 𝐸𝐸𝑠𝑠𝑐𝑐𝑐𝑐 where Ec=5000�f`𝑐𝑐𝑐𝑐 and Esec=f`cc/εcc. For the stress-strain curve of unconfined concrete, stress is assumed to reaches zero at spalling strain (εsp) in straight line. The stress-strain models of confined and unconfined concrete are illustrated in Fig. 43, where εcu is the crushing strain of the concrete. Fig. 43 Stress-strain model of confined and unconfined concrete 7.2.3 Rebar steel model For the reinforcement steel material, Kent & Park’s steel model (Kent & Park, 1973) is used to model the yielding and strain hardening behavior of the reinforcement. A typical stress-strain curve (as illustrated in Fig. 44) is divided into three stages: (i) elastic region for strain ε less than expected yield strain εye, (ii) plastic region for ε between εye and strain at strain hardening εsh, (iii) strain-hardening for ε larger than εsh but less than failure strain εsu. The steel stress is given as 51 = E ∗ ε (ε < ε𝑦𝑦𝑐𝑐) f𝑠𝑠 = fye (ε𝑦𝑦𝑐𝑐 < ε < ε𝑠𝑠ℎ) = f𝑢𝑢 − (f𝑢𝑢 − f𝑦𝑦𝑐𝑐) � ε𝑠𝑠𝑠𝑠−εε𝑠𝑠𝑠𝑠−ε𝑠𝑠ℎ�2 (ε𝑠𝑠ℎ < ε < ε𝑠𝑠𝑢𝑢) where fu is the fracture stress and E is the elastic modulus of steel. The stress-strain curve for tension and compression of steel is symmetric. Fig. 44 Steel stress-stain model All elastic material property used to calculate nonlinear material properties are based on the material properties of the updated FE model. By default setting of Xtract, the moment-curvature analysis stops when compression strain of concrete reaches 0.02 or when tensile strain of reinforcement reaches 0.012. Nonlinear material properties for concrete and steel in pier and cap-beam section were summarized in Table 8. The nonlinear material stress-strain relationships obtained will also be used to define the P-M-M interaction fiber hinge and uncoupled moment hinge in SAP2000. 52 Table 8 Nonlinear material properties of concrete and steel Material f`ce (MPa) f`cc (MPa) εcu εsp Confined (Pier) 43.75 54.90 12.60×10-3 -- Confined (Beam) 43.75 55.08 9.26×10-3 -- Unconfined 43.75 -- 4.00×10-3 6.00×10-3 Material fye (MPa) fu (MPa) εsh εsu Rebar steel 480 620.5 8.00×10-3 0.09 7.3 Plastic moment capacity 7.3.1 Critical sections of column and cap-beam Fig. 45 shows the moment diagram of the columns and cap-beams of Pier No.2 and No.3 (see Fig. 2) of the elastic time history analysis. The seismic input was Ground Motion No.1 (see Fig. 37), and the diagram was taken under a random time step, exact value of the moment diagram was not provided. This diagram was simply to give a rough concept of which section takes the largest moment in the columns and cap-beams. It can be easily observed in Fig. 45 that for columns, top and bottom sections were subjected to the largest moment; for the cap-beams, the sections at both ends have the largest moment loading. Therefore, top and bottoms section of the columns and sections at both ends of the cap-beams were considered as the critical sections. Fig. 45 Moment diagram of column and cap-beam at Pier No.2 and 3 53 7.3.2 Moment-curvature analysis of the column The plastic moment capacity of all ductile concrete members of the bridge, particularly column and cap-beam, was calculated by moment-curvature (M−φ) analysis based on expected material properties. Sectional analysis program Xtract was used to obtain the moment-curvature (M-φ) curves as well as the axial-force/biaxial-moment interaction (P-M) surface. Since the column sections are all circular symmetric, calculation about only one axis is conducted for each section. Sections at top and bottom ends of column were considered as critical sections since they sustain the largest bending moment under lateral loads (see Fig. 45). Reinforcement detail of the column sections were defined according to the structural drawing as shown in Fig. 8 (left). The Xtract model details of column cross section were presented in Fig. 46 (left). Fig. 46 Xtract section model details of columns (left) and cap-beams (right) of all piers By performing the moment-curvature analysis, the strength and ductility of the cross section subjected to axial load and incremental bending moment were obtained. The basic axial load used for moment-curvature analysis of each pier section was defined as the dead load, which varies from top to bottom of the pier. For tall columns in the FCB, the variation has significant effect on the flexural strength. Therefore M-φ analyses were conducted on both top and bottom sections for each column. What’s more the axial load will fluctuate when bridge is subjected to static lateral loads or dynamic excitation caused by ground motions. According to PEER803, the maximum range of column axial load is recommended between -0.05Pn in tension and +0.15Pn in compression, where Pn is the nominal bearing capacity of the cross section, estimated as product of gross section area Ag and 54 concrete compressive strength f’c. Therefore for each column section, different M-φ curves were obtained with axial loads of -0.05Pn and +0.15Pn. In Xtract, negative values of axial load represent tension, while positive are for compression. P-M interaction diagram of the columns and M-φ curves were plotted in Fig. 47 and 48 respectively. Axial force due to dead load of pier No.1 and No.2 (see Fig. 2) are the same and section details of all three piers are identical. The P-M diagram result will be used as capacity criteria in the time history analyses as well as input P-M relation of Interaction PMM Hinge. Fig. 47 P-M interaction diagram of columns of all piers 55 Fig. 48 Moment-curvature diagrams of pier 1 & 2 (top) and pier 3 (bottom) 7.3.3 Moment-curvature analysis of the cap-beam Cap-beam in the FCB is another important component that may be damaged due to static and dynamic load. For all cap-beams, the moment capacities of the critical sections, which are at the beam-column connections (see Fig. 45) were calculated by performing the moment-curvature analysis in Xtract. Section reinforcement details were shown in Fig. 8 (right). Since the layout of reinforcement on top and bottom of the cap-beam section is different, positive and negative incrementing moment loads are applied about the 3-3 axis (see Fig. 21) respectively. No initial axial load was applied since the interior axial force of generated by self and exterior loads are negligible, therefore the P-M interaction effect is ignored for the cap-beam. The M-φ 56 curve of cap-beam was plotted in Fig. 49, noticed that only one curve is obtained since the section details at the beam-column connection of all three cap-beams are identical. Fig. 49 Moment-curvature diagram of cap-beam of all piers 7.4 Bilinear M-φ model In order to convert the M-φ results into SAP2000 for nonlinear modeling, the M-φ curve was idealized as bilinear model as illustrated in Fig. 50 (Aviram et al., 2008). The elastic portion of the idealized curve should pass through the point where the first rebar yields at first yield moment My and reach the point marking the effective yield moment MY. The values corresponding to the first yield point (φy, My), effective yield point (φY, MY), ultimate capacity point (φu, Mu) and curvature ductility μφ (μφ=φu/φY), are computed based on M-φ analysis under a certain level of axial load by Xtract, results were summarized in Table 9. These bi-linearized M-φ curves will be used in the definition of the Interaction PMM Hinge for the corresponding level of column axial load, as well as the Uncoupled Moment Hinge for the cap-beams in SAP2000. 57 Fig. 50 Actual and idealized M-φ curve relation Table 9 Idealized M-φ curves of pier and cap-beams under different load cases Member Load case My (107N-m) φy (10-31/m) MY (107N-m) φY (10-31/m) Mu (107N-m) φu (10-21/m) μφ Pier 1 & 2 (Dead)Top 2.14 1.51 2.86 2.02 3.20 3.76 1.86 (Dead)Bot. 2.36 1.55 3.08 2.03 3.38 3.51 1.73 -0.05Pn 0.82 1.21 1.48 2.19 1.93 4.43 2.02 +1.15Pn 3.14 1.54 4.11 2.01 4.24 2.56 1.27 Pier 3 (Dead)Top 1.99 1.48 2.71 2.02 3.07 3.95 1.96 (Dead)Bot. 2.17 1.51 2.89 2.02 3.22 3.73 1.85 -0.05Pn 0.82 1.21 1.48 2.19 1.93 4.43 2.02 +1.15Pn 3.14 1.54 4.11 2.01 4.24 2.56 1.27 Cap-beam +M (P=0) 3.13 1.37 3.80 1.66 4.47 4.35 2.62 -M (P=0) 1.92 1.29 2.41 1.63 2.81 4.28 2.63 7.5 Shear strength In order to estimate the shear capacity of the pier and cap-beam within plastic end region, the formulas specified in CAN/CSA-S6-14 have been used. As specified in Clause 4.7.2, to determine the nominal shear 58 resistance, material resistance factors for concrete Фc and reinforcing bars Фs are assumed as 1.0. In accordance with Clause 8.9.3, the nominal shear resistance Vr of the ductile member shall be calculated as Vc + Vs, where Vc is the shear resistance provided by tensile stresses in concrete, Vc shall be calculated as follows: Vc = 2.5βФcfcrbVdV where β is the factor used to account for shear resistance of cracked concrete and shall equal to 0.18 for section that satisfy the minimum transverse reinforcement, fcr represents the cracking strength of concrete and shall be taken as 0.4�f`ce for normal-density concrete, bV and dV are the effective web width and effective shear depth of section respectively. Vs is the shear resistance provided by shear reinforcement, Vs is given as Vs = ФsfyeAVdVcotθs where θ is angle of inclination of the principal diagonal compressive stresses to the longitudinal axis of a member, θ shall be taken as 42o for non-prestressed components. AV and s are the area and spacing of transverse reinforcement respectively. Equations above are adopted to calculate shear resistance for pier and cap-beam sections, results are summarized in Table 10, those values will be used as shear capacity criteria in the time-history analyses. Table 10 Shear capacities of piers and cap-beams of all piers Member Vc (kN) Vs (kN) Vr (kN) Piers 5987 8836 14820 Cap-beams 7181 9603 16780 59 7.6 Nonlinear hinge model 7.6.1 Nonlinear hinge location and type In the FCB model, inelastic three-dimensional beam-column elements are used to model the column and cap-beam for each of the piers in the bridge. The nonlinearity and hysteretic behavior of column and cap-beam is idealized through discrete plastic hinge models. As discussed in Section 7.3.1, the maximum bending moment shall be expected to occur at both end sections of the column and cap-beam. Therefore the idealized model assumes the formation of plastic hinges at the end of each segment near the point of fixity of the column and cap-beam as shown in Fig. 51. Fig. 51 Expected locations of hinge formation in FCB There are several hinge model options available in SAP2000 to represent the behavior of the plastic hinge which includes the Uncoupled Moment Hinge, the Interaction PMM Hinge and the Fiber PMM Hinge, each of them possess different features. Based on individual capabilities and limitations, these three hinge models were used on different ductile members and analysis types as described in the following sections. 7.6.2 Plastic hinge length The hinge models require an approximate plastic hinge length to convert plastic curvature to plastic rotation. The plastic hinge length Lp was calculated based on equation recommended in Caltrans Seismic Design Criteria (SDC) 2004: 𝐿𝐿𝑡𝑡 = 0.08𝐿𝐿 + 0.022𝑓𝑓𝑦𝑦𝑐𝑐𝑑𝑑𝑏𝑏 ≧ 0.3𝑓𝑓𝑦𝑦𝑐𝑐𝑑𝑑𝑏𝑏 60 where L is the distance from plastic hinge location to location of contraflexure, db is the diameter of longitudinal rebar and fye is the expected yield strength of reinforcing bars. The estimated hinge length of column and cap-beam was summarized in Table 11. Table 11 Plastic hinge length of ductile members Member Pier 1 & 2 Pier 3 Cap-beam Hinge length (mm) 2260 1842 1077 7.6.3 Uncoupled Moment Hinge model 1) General features Uncoupled Moment Hinge model is a lumped plasticity model, this model ignores the coupled behavior of both orthogonal bending directions (M2 and M3) by calculating in the two bending directions separately. The Uncoupled Hinge in SAP2000 includes the P-M interaction effect but only use the axial load from tributary dead load as the input axial load, this limitation makes it fail to adjust the capacity and ductility according to member axial force fluctuation, which is a critical phenomenon expected for column during pushover and dynamic analysis. Therefore this type of hinge is only used to model the nonlinear behavior of cap-beam in which the axial force can be ignored. Further time-history analysis proves that the axial force of cap-beam stays at a very low level, which barely has any effect on the flexural capacity. On the other hand, the Uncoupled Hinge model requires low computational effort but have convergence problems after member yielding in nonlinear time-history analysis. Given the features described above, the use of this model is limited to static analysis. 2) Hinge definition The nonlinear behavior of cap-beam in both positive and negative bending is defined through a normalized M-φ relation with possible degrading behavior. In SAP2000, the definition of the M-φ curve must include the following points, normalized with MY as the scale factor (SF) for the moment and φY as the SF for the curvature: 61 zero load point A (0, 0); yield point B (φY, MY); ultimate capacity point C (φu, Mu); degraded capacity point D, which moment can be taken as 20% of the ultimate capacity Mu while curvature remains as φu; failure point E, which curvature is recommended to have a greater value than point D, 0.1 (1/m) is assumed in this model, all parameters involved were defined in Table 9 . The resulting idealized model for nonlinear analysis is presented in Fig. 52. Fig. 52 Idealized M-φ relation of Uncoupled Moment Hinge model in SAP2000 (Aviram et al., 2008) 7.6.4 Interaction PMM Hinge model 1) General features The applicability and limitations of the Interaction PMM Hinge are similar to those of Uncoupled Moment Hinge, except that it takes into account of the coupled behavior of the column in both orthogonal bending directions. And what’s more, the Interaction PMM Hinge will automatically adjust the capacity and ductility of the member according to fluctuation of the section axial load, in this way it provides more accurate results in pushover and nonlinear time-history analysis, therefore this model was used to model the nonlinear behavior of the columns. Similar to those of the uncoupled hinge, convergence and numerical stability problems occur in SAP2000 after yielding during nonlinear time-history analysis, therefore the use of the Interaction PMM Hinge was also limited to static pushover analysis. 62 2) Hinge definition The M-φ relation model used in the Interaction PMM Hinge is the same one illustrated in Fig. 52. Moment-curvature analysis and interaction diagram results in both bending directions were required to determine the MY and Mu (see Section 7.4) values of the column, as well as the corresponding curvatures. However, in case of the FCB, the column cross sections are circular symmetry, only one M-φ relation and one P-M interaction diagram are required for each level of axial load. As discussed in Section 7.3.1, three M-φ curves with different axial load levels (dead load, maximum compression and maximum tension) were input for definition of each hinge. The SAP2000 program will automatically interpolates the curves between these levels of axial loads. The axial-force/moment interaction effect was defined through a normalized P-M relation based on the relation presented in Fig. 47, the input curve was normalized with the maximum compressive load as the scale factor (SF) for the axial force and maximum bending moment as the SF for the moment. 7.6.5 Fiber PMM Hinge model 1) General features The fiber hinge is a lumped plasticity model with a characteristic length Lp that can be assigned to an elastic element at specific point. The fiber hinge gives more accurate estimation of the nonlinear behavior of ductile members. More advanced than the previous two models, the fiber hinge can calculate the moment-curvature relation in any bending direction as well as automatically adjust the capacity of member the any direction according to fluctuation in axial load. This multiaxial-moment/axial-load interaction and inelastic distribution were obtained by assigning stress-strain relations of corresponding material (unconfined concrete, confined concrete and longitudinal rebar steel) to individual discretized fibers. The fiber hinge model can represent the loss of stiffness caused by concrete cracking, rebar yielding due to flexural yielding and strain hardening. What’s more, it can successfully represent the degradation and softening behavior after yielding without computational problems. Given features described above, the use of Fiber PMM Hinge model was extended to nonlinear static pushover and time-history analysis, and can be used on both column and pier sections. 63 2) Hinge definition The definition of each fiber in the cross section includes the area, centroid coordinates, and material type which uses the stress-strain relations defined in Section 7.2. For example, for the reinforced concrete column, the definition of the stress-strain relation is defined separately for confined concrete (column core), unconfined concrete (column cover), and rebar steel (longitudinal reinforcement). A spreadsheet was used to generate a circular patch that represent the radial distribution of fibers and copied into SAP2000, Fig. 53 illustrate the fiber distribution in the column circular patch. Fig. 53: Fiber distribution along circular cross-section (Aviram et al., 2008) 64 8. Seismic evaluation methodology 8.1 Introduction After completion of the modeling including geometry, mass, material, element, cross section, boundary condition and nonlinear behavior, the FE model must be evaluated by required analysis methods specified in CAN/CSA-S6-14, different analysis methods shall be performed depends on the classification and importance, the level of geometric, structural, and geotechnical irregularity, as well as the performance category of the bridge. In the FCB case, three types of analysis were required to be carried out: the elastic dynamic analysis, the inelastic static push-over analysis and the nonlinear time-history analysis. Each analysis type was used on different analysis purpose based on individual applicability and limitations, which were described in detail in this chapter. This chapter also presents a description on a performance-based evaluation approach to bridge structures. 8.2 Bridge classification and Seismic Performance Category According to the CAN/CSA-S6-14 Clause 4.4.4, each bridge shall be assigned to one of the three Seismic Performance Categories, 1 to 3, based on CAN/CSA-S6-14 Table 4.10. The importance category of the FCB is Lifeline Bridge according to MoTI, and Sa(1.0) = 0.52 (see Fig. 34) based on site-specific spectral acceleration under return period of 2475 years (as discussed in Section 6.3). Given information provided above, the seismic performance of the FCB is categorized as level 3. 8.3 Analysis requirement For multi-span bridges the minimum seismic analysis requirement for a probability of exceedance of 2% in 50 years shall be determined based on CAN/CSA-S6-14 Table 4.12. Since the FCB is a lifeline regular bridge which been categorized as seismic performance level 3, the following analysis procedures must be carried out for the seismic analysis: (i) Elastic dynamic analysis (EDA) which includes multi-mode elastic response spectral analysis or elastic time-history analysis; (ii) Inelastic static push-over analysis (ISPA). The ISPA shall account for nonlinear behavior due to plastic hinging in ductile substructure elements, soil-structure interaction, and P-delta effects as appropriate; (iii) Nonlinear time-history analysis (NTHA). The NTHA shall account for 65 nonlinear behavior due to plastic hinging in ductile substructure elements, soil-structure interaction, and P-delta effects as appropriate. The elastic time-history analysis (ETHA) is the most widely used method for EDA and it is very easy to operate. Therefore in this study, the ETHA was chosen for the elastic dynamic analysis. The force and displacement time-history results of all important components from both ETHA and NTHA were gathered, the extreme values were picked and used to check the member capacity as well as compare with the performance criteria for performance design approach. The ISPA results were used to evaluate the bridge strength and deformation capacity as well as discover the failure pattern of the bridge. 8.4 Performance level and criteria for performance-based seismic evaluation As specified in CAN/CSA-S6-14 Clause 4.4.5.3, for Lifeline Bridge with seismic performance category 3 like the FCB, the performance-based design procedure is required. The performance-based design shall meet the minimum requirements specified in CAN/CSA-S6-14 and an explicit demonstration of the performance requirements having been met must be provided. 8.4.1 Performance levels There are two sets of performance levels specified in the CAN/CSA-S6-14 for bridge design and evaluation the Service Performance levels and the Damage Performance levels, for each one there were a set of sub-levels, a structural must be evaluated and classified under a certain performance level. Four discrete service performance levels were defined in CAN/CSA-S6-14 Clause 4.4.6.3: the immediate service level, the limited service level, the service disruption service level and the life safety service level. For each service performance level, there is a corresponding damage performance level that also needs to be satisfied. The corresponding four damage performance levels are: the minimal damage level, the repairable damage level, the extensive damage level and the probable replacement level. Each of the damage level is listed with detailed performance criteria which shall be checked one by one with analysis results. 66 In seismic evaluation of the existing bridge, different performance levels shall be satisfied for different seismic ground motion probability of exceedance in 50 years (or different return periods) as specified in Clause 4.4.6.2. For the FCB case, as a Lifeline Bridge, in case of probability of exceedance of 2% in 50 years (return period of 2475 years), the limited service level of the service performance levels and the repairable performance level of the damage performance levels shall be satisfied. The probability of exceedance of 2% in 50 years was used for the analysis for a more conservative result, because it is the most severe earthquake scenario considered in CAN/CSA-S6-14. 8.4.2 Performance criteria The performance criteria for different performance levels are given in CAN/CSA-S6-14 Table 4.16. Limited service level requires a bridge shall be usable for emergency traffic and be repairable without requiring bridge closure, and at least 50% of the lanes shall remain operational. And it also requires that if the bridge was damaged, normal service shall be restored within a month. The requirements of Service Performance level in the code were rather descriptive therefore cannot be used as clear standards of seismic evaluation. The performance criteria for Damage Performance levels were more explicitly or numerically defined, performance criteria for Repairable Damage level were listed below for further discussion of seismic evaluation purpose: 1) General: There may be some inelastic behavior and moderate damage may occur; however, primary members shall not need to be replaced, shall be repairable in place, and shall be capable of supporting the dead load plus full live load. 2) Concrete structures: Reinforcing steel tensile strains shall not exceed 0.015. 3) Steel structures: Buckling of primary members shall not occur. Secondary members may buckle provided that stability is maintained. Net area rupture of primary members at connections shall not occur. 4) Connections: Primary connections shall not be compromised. 5) Displacements: Permanent offset shall not compromise the service and repair requirements of the bridge. No residual settlement or rotation of main structure shall occur. There may be some movement of wing walls, subject to performance and reparability. 67 6) Bearings and joints: Elastomeric bearings may be replaced. If finger joints are damaged, they shall be repairable. 7) Restrainers: Restraining systems shall not be damaged. 8) Foundations: Foundation movements shall be limited to only slight misalignment of the spans or settlement of some piers or approaches that does not interfere with normal traffic, provided that repairs can bring the structure back to the original operational capacity. 9) Aftershocks: The structure shall retain 90% of seismic capacity for aftershocks and shall have full capacity restored by the repairs. 8.4.3 Performance based assessment method A set of performance criteria has been specified in the code as listed above, however, most performance criteria were very descriptive and were not linked with any engineering parameter. For the proper implementation of performance based seismic evaluation, the performance objective should be evaluated by engineering parameters such as bending moment, shear force, stress, strain etc. Such damage parameters can be better measured at component levels (such as columns, cap-beams, girders, diaphragms etc.) for bridge structure. By comparing the maximum internal force (such as maximum bending moment, shear force, axial load caused by seismic excitation) with the load capacities of the component, it can be easily decided whether that component has meet the performance criteria. For example, as specified in the second requirement of performance criteria, tensile strain in longitudinal reinforcement εs shall be taken as the basis of judgement of whether the ductile components satisfy the performance objective. In general, inelastic action of the FCB is expected to occur in the pier columns and cap-beams (see Fig. 45), therefore the seismic performance of the second criteria is evaluated based on the inelastic capacity of the piers columns and cap-beams. 68 For columns, in order to get the maximum rebar tensile strain, first the axial-bending (P-M) combination which produces the ultimate axial stress in column section shall be identified from the time-history analysis data. However, there is no effective way to pick up the worst combination without taking huge computational effort. In this study, an alternative way is been used by reversing the procedure as described below. First of all, a section analysis of the column was conducted in Xtract. Relation curve of bending moment of the column section versus corresponding maximum rebar strain were plotted by Xtract, different relation curves were obtained under different axial force levels, as presented in Fig. 54 (positive for compression and negative for tension). Then a vertical line was made at the maximum strain value of 0.015 (as specified in second performance criteria), the bending moment values at intersection points and the corresponding axial forces were obtained. In this way, a series of P-M data pairs were collected and plotted in one diagram as shown in Fig 55 (positive for tension and negative for compression). This diagram represents the envelop curve of P-M combination that produce maximum reinforcement tensile strain of 0.015. Any point that falls on the right side of the curve will cause maximum strain greater than 0.015. Notice that the P-M diagram was not complete since maximum tensile strain fail to reach 0.015 when axial load went larger. Fig. 54 Bending moment vs. maximum rebar strain of all columns under various levels of axial load 69 Fig. 55 P-M interaction relation of all columns (maximum reinforcement tensile strain 0.015) For the cap-beams, the procedure was similar to the column analysis. First the relation curves of bending moment versus maximum rebar strain were calculated by Xtract (shown in Fig. 56), but no initial axial load were considered since the P-M interaction effect was ignored for all cap-beams. The bending moment at intersection points at strain of 0.015 were obtained, these values were used as the target moment to determine whether the beams satisfy the performance objective. 70 Fig. 56 Bending moment vs. maximum rebar strain of all cap-beams For the rest components such as bearing, expansion joint, foundation piles and etc., the performance objectives required were mostly very descriptive, therefore a set of target displacement and force capacity were self-defined based on structural and geometric characteristic of the bridge and recommendation of previous research. As specified in CAN/CSA-S6-14 Clause 4.4.6.3, the assessment of damage performance criteria specified above shall be carried using nonlinear time history analysis or by using static pushover analysis up to the design displacement (see Section 8.5.2). In this study, the nonlinear time history analysis results including maximum loads of components, maximum displacement of nodes and maximum deformation of links were summarized and used to compare with the member capacities of force and displacement. Detailed procedure and discussion were presented in Chapter 9. 71 8.5 Inelastic Static Push-over Analysis (ISPA) 8.5.1 General consideration ISPA is a static, nonlinear procedure in which the magnitude of the structural loading is incrementally increased in accordance with a predefined load pattern (see Section 8.5.4). It accounts for the redistribution of internal force as components respond nonlinearly, therefore provides a better measure of behavior than elastic analysis procedures. The pushover analysis can provide an insight into the structural aspects, which control performance during severe earthquakes, with the increase in the magnitude of the loading, weak links, formation of plastic hinges, and redistribution of forces throughout the structure can be observed, thereby capture the failure mechanism of the bridge. The pushover analysis can also evaluate the overall strength, displacement capacity and ductility of the bridge structure measured by base shear, yield displacement and maximum displacement, as well as the ductility capacity of the structure. By pushover analysis, the pushover curve (base shear versus top displacement) of the bridge can be obtained. The softening behavior (typically due to material strength degradation or P-Δ effects) of the structure can be identified on the curve. In order to reduce the computational effort, for the nonlinear pushover analysis of FCB, the PMM Interaction Hinge model was used on column hinges and the Uncoupled Moment Hinge model was used to model the plastic hinges on cap-beams. 8.5.2 Displacement limitation The pushover analysis of the FCB is conducted as a displacement controlled method, the node at the top of the Pier No.2 right column (see Fig. 63) was selected as the point of reference since it is the highest one of all three piers. The node was specified with a specified maximum displacement value and monitored during the pushover analysis, in SAP2000 the pushover analysis will stop once the displacement of the reference point reaches the maximum displacement. 72 Fig. 57 Structural configuration of deformation of FCB Pier No.2 (fixed-fixed column) and corresponding moment diagram It is recommended in PEER083 that for ISPA in SAP2000, the maximum displacement Δmax specified for the reference point shall be taken as Δmax=(1.5–2.0)Δc, where Δc is the ultimate displacement calculated for double cantilever and fixed-fixed columns, the local deformation configuration of all piers of FCB was illustrated in Fig. 57. The local displacement capacity Δc of columns of FCB can be idealized as two cantilever segments presented in equations provided by Section 3.1.3 of SDC 2004: (see Fig. 58 for details) Δ𝑐𝑐 = Δ𝑐𝑐1 + Δ𝑐𝑐2 Δ𝑐𝑐1 = Δ𝑌𝑌1 + Δ𝑡𝑡1 ,Δ𝑐𝑐2 = Δ𝑌𝑌2 + Δ𝑡𝑡2 Δ𝑌𝑌1 = 𝐿𝐿123 ϕ𝑌𝑌1 ,Δ𝑌𝑌 = 𝐿𝐿223 ϕ𝑌𝑌2 Δ𝑡𝑡1 = 𝜃𝜃𝑡𝑡1 �𝐿𝐿1 − 𝐿𝐿𝑡𝑡12 � ,Δ𝑡𝑡2 = 𝜃𝜃𝑡𝑡2(𝐿𝐿2 − 𝐿𝐿𝑡𝑡22 ) 𝜃𝜃𝑡𝑡1 = 𝐿𝐿𝑡𝑡1ϕ𝑡𝑡1 ,𝜃𝜃𝑡𝑡2 = 𝐿𝐿𝑡𝑡2ϕ𝑡𝑡2 ϕ𝑡𝑡1 = ϕ𝑢𝑢1 − ϕ𝑌𝑌1 ,ϕ𝑡𝑡2 = ϕ𝑢𝑢2 − ϕ𝑌𝑌2 where L1 and L2 are the distance from the point of maximum moment to the point of contra-flexure, Lp1 and Lp2 are the analytical plastic hinge length as defined in Section 7.6.2, Δp1 and Δp2 are the idealized plastic displacement capacity due to rotation of the plastic hinge, ΔY1 and ΔY2 are the idealized yield displacement of the column at the formation of the plastic hinge, ϕp1 and ϕp2 are the idealized plastic curvature capacity, ϕY1, ϕY2 73 and ϕY1, ϕY1 are yield curvature and ultimate curvature capacity of the column as defined in Section 7.4, θp1 and θp2 are the plastic rotation capacity. Fig. 58 Local displacement capacity of fixed-fixed framed columns (SDC, 2004) All the lengths and curvature capacities were acquired from the properties of the Pier No. 2. The value of ultimate displacement Δc of Pier 2 column is approximate 1345mm, and the maximum displacement Δmax is taken as 2Δc = 2690mm for pushover analyses in longitudinal and transverse directions. 8.5.3 Pushover load cases In SAP2000, the lateral pushover load case on the bridge structure was specified to start from the final conditions of the gravity pushover, where the dead load of the bridge was fully applied. This means that the lateral load is applied on top of the dead load. The lateral load pushover analysis is conducted in two directions: the longitudinal and transverse directions. 74 8.5.4 Force pattern There are two types of force pattern options available for nonlinear pushover analysis: the Uniform Acceleration Method and the Modal Pushover Method (Chopra, 2001). When the uniform acceleration was applied, the force pattern will be automatically assigned to the entire structure, proportional to the translational mass distribution in the corresponding directions. If modal pushover was used instead, modal analysis results corresponding to the longitudinal and transverse translation of the bridge are to be assigned to the longitudinal or transverse pushover load cases, respectively. Normally the Modal Pushover Method was more accurate in estimating the capacity and ductility of the structure, however, in the FCB case, fundamental modes in the transverse and longitudinal directions (first and fourth mode as shown Fig. 23) were rather combined modes with mode shapes contributed by transverse, vertical and torsional motions, which brings computational problems during pushover analysis in SAP2000. Therefore the uniform acceleration force pattern was applied during the pushover analysis instead to avoid computational problems. 8.5.5 Geometric nonlinearities In SAP2000, there are two types of geometric nonlinearities available for nonlinear static pushover analysis and nonlinear direct-integration time-history analysis: the P-Δ and large displacements effects. While the P-Δ are computed by solving equilibrium equations by taking into partial consideration of the deformed configuration of the structure, the large displacement analysis considers all equilibrium equations in deformed configuration of the structure. However, the large displacement analysis in SAP2000 requires more computation efforts than the P-Δ transformation. As recommended in PEER083 (Aviram et al., 2008), the large displacement option is used for structures undergoing significant deformation and for buckling analysis, therefore it is not recommended for typical bridge analysis. For typical bridges like the FCB, the P-Δ option is adequate. The P-Δ effect or second-order effect refers to the abrupt changes in ground shear, overturning moment, and axial force distribution at the base of a tall structure or structural component when it is subjected to large lateral displacements. Since the piers in FCB were sufficiently tall, consideration of the P-Δ effect is essential to help capturing a more realistic result of strength degradation and relative displacement between top and bottom of 75 the column. In the SAP2000 bridge model, the P-Δ effect was included in pushover and nonlinear direct-integration time-history analysis cases. 8.6 Elastic and Nonlinear Time-history Analysis (ETHA and NTHA) 8.6.1 General consideration Due to the limitations of the ISPA to approximate the dynamic response of a complex three-dimensional structural bridge system, elastic and nonlinear time history analyses are recommended instead. Different from the ISPA, time-history analysis use ground motion acceleration as the input load instead of using the externally applied loads on the structure. Each bridge structure possesses unique predominating mode shapes and frequencies which can be excited by properly selected ground motions, thereby producing the accurate estimation of peak response of the seismic demand on the structure. There are two types of time history analysis, the ETHA and the NTHA. The ETHA is required by CAN/CSA-S6-14, however, the ETHA does not consider the nonlinear behavior of the ductile members which is expected in the major earthquake, therefore it usually overestimate the dynamic response of the members and produce unrealistic analysis results. The NTHA, on the other hand, accounts for the nonlinearities and strength degradation of ductile elements of the bridge as well as the energy dissipation effect caused by hysteresis behavior of the elements, therefore producing more realistic response estimation. In this study, the NTHA results were used as the seismic demand of the FCB (see CAN/CSA-S6-14 Clause 4.4.6.3); however, the ETHA results were also presented for comparison purposes. 8.6.2 Seismic inputs 1) Load factors and load combinations For both ETHA and NTHA, the seismic evaluation of existing bridges shall be based on the load factors and combination of: 1.0D + 1.0EQ (Clause 4.11.6) 76 As specified in CAN/CSA-S6-14 Clause 4.4.9.2, for regular bridges the seismic force from orthogonal loading of horizontal directions shall be combined using 100% of the response in one direction and 30% in the orthogonal direction. In such a method, the ground motion must be applied at a sufficient number of angles to capture the maximum deformation of all critical components. The method requires great computational effort and several runs to establish the critical response. An alternative approach was recommended by PEER083 for the determination of the maximum seismic response. This method use 100% of the maximum input motions in the vertical and two horizontal directions. For each ground motion, only a single analysis is needed to be carried out for this three-directional input method. 2) Time history functions The main disadvantage of the time history analysis method is the high computational and analytical effort required and the large amount of output information produced. In the NTHA, nonlinear capacity evaluation and internal forces redistribution were carried out on every time step, as result the computational time required of NTHA is significantly longer than ETHA. Therefore to reduce the calculation time of NTHA, the length of time history functions of NTHA were reduced based on Arias Intensity (IA) (Arias, 1970) which is a measure of the strength of ground motion, it is defined as the time-integral of the square of the ground acceleration: 𝐼𝐼𝐴𝐴 = 𝜋𝜋2𝑔𝑔� 𝑎𝑎(𝑡𝑡)2𝑡𝑡0 𝑑𝑑𝑡𝑡 It can be observed that the IA of each ground motion increases from 0% to 10% and 90% to 100% in a very low rate, but increases from 10% to 90% in just short amount of time. An example of IA diagram of the ground motion No.2 (Northridge) was shown in Fig. 59, the time history range from 2.5s to 12s covers more than 95% of the total earthquake energy. Therefore to reduce the calculation time, all 11 ground motions were cut into time ranges that cover IA value from approximately 5% to 95%, ground motion inputs information are summarized in Table 12, notice that ground motions remain unchanged for ETHA. 77 Fig. 59 Arias Intensity diagram of the Northridge ground motion Table 12 Input ground motion information No. Time range (cut) Length (cut/uncut) Time interval (s) Time steps (cut/uncut) 1 5s-25s 20s/44s 0.02 1000/2200 2 2s-12s 10s/26.6s 0.005 2000/5315 3 8s-28s 20s/65s 0.005 4000/13012 4 4s-25s 21s/46.4s 0.0029 7241/16003 5 30s-60s 30s/207s 0.01 3000/20700 6 28s-55s 27s/198s 0.01 2700/19800 7 20s-40s 20s/120s 0.01 2000/12000 8 5s-25s 20s/120s 0.01 2000/12000 9 90s-130s 40s/300s 0.01 4000/30000 10 80s-130s 50s/300s 0.01 5000/30000 11 30s-100s 70s/247s 0.01 7000/24700 8.6.3 Damping Damping is an energy-dissipation mechanism that results in decay of motion in vibration. The structural damping in the bridge system includes (i) the material damping in the structural components, (ii) inelastic cyclic 78 behavior of the members and (iii) radiation damping in the soil and abutments. In the FCB case, the radiation damping effect can be ignored due to regular shape and small span, and the material damping coefficients ρ (0<ρ<1) can be specified when defining material properties in SAP2000, default coefficient value of zero was used in the FCB material model. For linear time-history analysis, modal damping ratios are available for the modal time-history method, the modal damping ratios assuming proportional damping and ignoring the modal cross-coupling damping terms. However, the modal damping ratios could significantly overestimate or underestimate the actual damping value. For nonlinear direct-integration time-history analysis, viscous damping is required in SAP2000. As recommended in PEER083, the mass and stiffness proportional damping, also known as Rayleigh damping (Liu & Gorman, 1995) coefficients were used in the linear and nonlinear time-history analysis of the FCB finite element model. The Rayleigh damping matrix C was applied on the entire structure and was calculated as a linear combination of the stiffness and mass matrices: C = 𝛼𝛼0 × m + 𝛼𝛼1 × 𝑘𝑘 where α0 is the mass proportional damping coefficient and α1 is the stiffness proportional damping coefficient. Given coefficients α0 and α1, the damping ratio ξi in mode i can be determined as: 𝜉𝜉𝑖𝑖 = 𝛼𝛼02𝑗𝑗𝑖𝑖 + 𝛼𝛼1𝑗𝑗𝑖𝑖2 Therefore the coefficients α0 and α1 are determined by specifying damping ratios and periods in any two modes, in the FCB case they are the 1st and 2nd modes (see Fig. 23). The Rayleigh damping coefficients in SAP2000 requires the definition of the first two modal periods of the bridge, the same damping ratio of 5%, which is recommended by PEER083 as a typical value for concrete bridge is assumed for both first and second modal. Fig. 60 illustrate how the separate mass and stiffness damping terms contribute to the overall damping ratio by linear combination. 79 Fig. 60 Rayleigh damping used for direct-integration time history analysis 80 9. Seismic analysis results and discussions 9.1 Results of Inelastic Static Pushover Analysis 9.1.1 Plastic hinges mechanism By displaying the deformed shape of pushover analysis step by step, the sequence of hinge formation in the FCB can be obtained. Fig 61 and 62 illustrate the sequence of hinge formation of pushover analysis in transverse and longitudinal directions, respectively. The red dote appears when the bending moment at the hinge zone exceeds the effective yield moment capacity (MY) of the corresponding section. It can be observed that the first yielding happened on the bottom of middle pier column section. In both directions the plastic hinge formed at the bottom of column first then the top. For transverse direction, the cap-beams yield almost at the same time at the left beam-column connections, but in longitudinal direction the cap-beams all remain elastic. Fig. 61 Hinge formation sequence of nonlinear pushover analysis in transverse direction 81 Fig. 62 Hinge formation sequence of nonlinear pushover analysis in longitudinal direction 9.1.2 Pushover curves At the completion of the analysis, the pushover curve which presents monitored node displacement vs. base shear was obtained. The top right beam column connection at Pier No.2 was selected as the monitored node, the displacement of this node will be recorded and used to plot the pushover curves. Location of the monitored node was illustrated in Fig. 63. 82 Fig. 63 Location of the monitored node for nonlinear pushover analysis The resulting pushover curves in transverse and longitudinal directions were presented below in Fig. 64 and 65. Both pushover analyses fail to reach the target displacement limitation (see Section 8.5.2) due to inherent computational problem of SAP2000, however, further push of the structure is not necessary since capturing the full nonlinear behavior of the bridge is not the primary objective of this study, the primary objective of this study is to find out whether the FCB meet the performance criteria in CAN/CSA-S6-14 based on the NTHA results with 11 ground motions as seismic input. The pushover curves represent the global behavior of the frame with stiffness and ductility. Under incrementally increasing lateral load, the structural element may be yield sequentially. 83 Fig. 64 Pushover curve in transverse direction Fig. 65 Pushover curve in longitudinal direction Two pushover curves show different features of nonlinearity of the bridge in transverse and longitudinal directions. For the curve in transverse direction, they are initially linear but start to yield as the beams and the columns undergo inelastic actions, however, instead of losing strength the curves become linear again but with a smaller slope and remain linear until it reaches the target displacement, which shows adequate capacity and 84 ductility of the bridge structure under static lateral load in the transverse direction. In the longitudinal direction, the curve starts to deviate from linearity when the monitored displacement reaches approximate 200mm. The yielding develops rapidly and the whole structure starts to lose strength from the displacement reaches about 400mm, every step that point, the structure experience loss in stiffness, therefore, the slope of the pushover curve is gradually decreasing. The ISPA results were just for reference and will not be used for performance-based seismic evaluation of the bridge. Instead the NTHA results were used for the evaluation in this study. The purpose of ISPA was only limited to providing a general knowledge of the fail mechanism and the global nonlinear behavior in two horizontal directions under incremental lateral load, therefore in this study, no quantitative analysis were presented for the ISPA. 9.2 Results of Elastic and Nonlinear Time-history Analysis After all 11 ground motions were run by linear and nonlinear time history analysis, there were massive amount of time-history data need to be processed. Different analysis results and conclusions were obtained from time-history data for corresponding purposes of seismic evaluation. 9.2.1 Columns and cap-beams First of all, the member capacities of the ductile members were checked with maximum values from time history results. In order to find out the maximum internal forces of ductile members (columns and cap-beams) in the time history, a series of Matlab codes were developed and used to extract maximum internal force from every ground motion load case. The maximum internal forces are compared directly with the member capacities as described below. The results from elastic and nonlinear time history analysis were presented together for comparison purpose. 85 1) Column axial-force and bending-moment time history response After the time history analysis with all 11 ground motions, the time history records of member forces can be extracted from SAP2000 analysis data. Fig. 67 to 78 shows the time history plots of axial force and bending moment of pier columns. For every pier, four critical sections including the top and bottom sections of left column and right column were monitored. An example of the locations of the four sections was illustrated in Fig. 66. Fig. 66 Critical sections of pier columns and cap-beams of all piers Results from all 11 ground motions were presented in time domain, both ETHA and NTHA results were included for comparison purpose. Notice that for the axial force in SAP2000, positive value represents tension and negative are for compression. 86 Fig. 67 Time history records of axial force of Pier No.1 column sections (ETHA) 87 Fig. 68 Time history records of axial force of Pier No.1 column sections (NTHA) 88 Fig. 69 Time history records of axial force of Pier No.2 column sections (ETHA) 89 Fig. 70 Time history records of axial force of Pier No.2 column sections (NTHA) 90 Fig. 71 Time history records of axial force of Pier No.3 column sections (ETHA) 91 Fig. 72 Time history records of axial force of Pier No.3 column sections (NTHA) 92 Fig. 73 Time history records of bending moment of Pier No.1 column sections (ETHA) 93 Fig. 74 Time history records of bending moment of Pier No.1 column sections (NTHA) 94 Fig. 75 Time history records of bending moment of Pier No.2 column sections (ETHA) 95 Fig. 76 Time history records of bending moment of Pier No.2 column sections (NTHA) 96 Fig. 77 Time history records of bending moment of Pier No.3 column sections (ETHA) 97 Fig. 78 Time history records of bending moment of Pier No.3 column sections (NTHA) 98 2) Column axial-force/bending-moment (P-M) combination As discussed in Section 7.3.2, for the columns of FCB, the sections of the column member were subjected to large axial load and bending moment at the same time under the earthquake excitation. Since the change of axial load have significant effect of the bending moment capacity of the column, the axial load and bending moment of the columns were often considered together. By summarizing the bending moment capacity under different levels of axial load, the axial-force/biaxial-moment (P-M) interaction diagram can plotted as shown in Fig. 47. Instead of plotting the axial force and bending moment of column in time domain alone, for a given component section, the axial force and the bending moment at the same time step were considered as a pair of data (P, M). For each pier of the FCB, P-M pairs at all time steps of all ground motions were plotted in one diagram by iteration algorithm in Matlab. The P-M interaction diagram of column sections obtained by Xtract (see Fig. 47) was used as capacity envelop curve. If any data point falls on the right side of the curve then the member capacity does not satisfy the needs of the force demand. The P-M data diagrams of the elastic and nonlinear time history analysis was plotted through Fig. 79 to 81. Fig. 79 Pier No.1 axial-flexural responses compared with P-M capacity curve 99 Fig. 80 Pier No.2 axial-flexural responses compared with P-M capacity curve Fig. 81 Pier No.3 axial-flexural responses compared with P-M capacity curve It can be observed that the response force from the ETHA was 5 to 6 times larger than the NTHA results. In the ETHA, there were a lot of internal force combination cases that exceed the capacity of the column sections, while in NTHA all points fall on the left side of the capacity curve which indicates a sufficient P-M load capacity for all pier columns. Such difference in internal force results shows the FE model significantly 100 overestimates the response force during elastic time history analysis, probably due to lack of energy dissipation during the hysteretic behavior of the plastic hinges. Therefore in the CAN/CSA-S6-14, when assuming all members remain elastic in force based design and evaluation, the resulting elastic force shall be modified by the response modification factor (R-factor), the R factor depends on the ability of the ductile substructure elements. In the FCB case, the R factor shall be 5.0 for the substructure elements according to CAN/CSA-S6-14 Table 4.17, which agree with the time history analysis results presented above. 3) Column shear force time history response After checking the axial force/bending moment capacity of the column, another important type of internal force which is shear force was also monitored and checked. Extreme shear force can cause shear failure at the beam-column connections. The shear failure usually behaves as brittle failure at the beam-column connections which can extremely dangerous for the safety of the structure. Therefore it is very important to make sure that the maximum internal shear force was lower than the shear capacity of the four critical sections (see Fig. 66). Time history records of shear force of all pier column critical sections were extracted and plotted, including response from all 11 ground motions in both ETHA and NTHA. From the time history result it appears that the shear force response of column in the Y direction (shown in Fig. 45) was significantly larger than shear force response in X direction. Since the column is circular symmetric, shear capacities in both directions were identical. Therefore time history response of shear forces in only Y direction were presented for all three piers. See Fig. 82 to 87 for details. 101 Fig. 82 Time history records of shear force of Pier No.1 column sections (ETHA) 102 Fig. 83 Time history records of shear force of Pier No.1 column sections (NTHA) 103 Fig. 84 Time history records of shear force of Pier No.2 column sections (ETHA) 104 Fig. 85 Time history records of shear force of Pier No.2 column sections (NTHA) 105 Fig. 86 Time history records of shear force of Pier No.3 column sections (ETHA) 106 Fig. 87 Time history records of shear force of Pier No.3 column sections (NTHA) 107 The maximum column shear force were picked from time history data using Matlab, Table 13 summarizes the column maximum shear force from all 11 ground motions and compared with the shear capacity of column from Table 10. It appears that the maximum shear force of columns happened at the Pier No.3. What’s more the bottom sections seem to be subjected to larger shear force than the top sections of the columns. By combining all time history results, it can be concluded that shear capacity of all elements meet the demand, no shear failure will happen during earthquake excitation. Table 13 Maximum column shear force of time history analysis Section location Max. Shear (ETHA)(kN) Max. Shear (NTHA)(kN) Shear capacity (kN) Pier1 Top Left 2118.05 1027.31 14820.00 Right 2116.52 964.49 14820.00 Bottom Left 2993.01 1629.08 14820.00 Right 2993.80 1643.08 14820.00 Pier2 Top Left 3215.41 1029.28 14820.00 Right 3218.65 1074.89 14820.00 Bottom Left 4434.27 2143.42 14820.00 Right 4441.42 2020.70 14820.00 Pier3 Top Left 4033.19 1585.83 14820.00 Right 4044.23 1605.78 14820.00 Bottom Left 4650.30 2247.96 14820.00 Right 4659.49 2192.21 14820.00 Max. 4659.49 2247.96 14820.00 108 4) Bending moment and shear force time history response of cap-beam The shear force responses of critical sections of all cap-beams were recorded by SAP2000 as well. Sections at both column-beam connections were selected as the critical section (see Fig. 66), since they were subjected to the largest bending moment as illustrated in Fig. 45. However, since the bridge structure was symmetric about the central axis as shown in Fig. 66. Only the left beam-column section of cap-beam was monitored in the time history analysis. What’s more, the bending moment response of cap-beam about the 3-3 axis (as shown in Fig. 21) was significantly larger than response about the 2-2 axis (also see Fig. 21). And the shear force response in 2-2 direction was also significantly larger than response in the 3-3 axis. Therefore the only bending moment response recorded was about the 3-3 axis, and the only shear force response recorded was in the 2-2 direction Same type of time history response diagrams has been plotted for all cap-beams in both ETHA and NTHA. Fig. 88 to 91 gives the time history response of the bending moment and shear force. Notice that the axial force of the cap-beam was not considered in this study, since time history results shows a very low level of axial force responses throughout both the ETHA and NTHA. 109 Fig. 88 Time history records of bending moment of all cap-beams (ETHA) 110 Fig. 89 Time history records of bending moment of all cap-beams (NTHA) 111 Fig. 90 Time history records of shear force of all cap-beams (ETHA) 112 Fig. 91 Time history records of shear force of all cap-beams (NTHA) Since the longitudinal reinforcement layouts at the top and bottom of the cap-beam section were different (as shown in Fig. 46), the positive and negative bending moment capacities about 3-3 axis (see Fig. 21) were different as well. Therefore the maximum (positive bending) and minimum (negative bending) bending moments were selected and compared with the corresponding moment capacities separately. Notice that the internal axial force in the cap-beam is relatively ignorable, therefore the P-M interaction effect was not considered. Maximum positive and negative bending moment of all 11 ground motions were summarized in Table 14 and 15 respectively, maximum shear forces were summarized in Table 16. Effective 113 yield moment MY calculated in Table 9 was used as moment capacity of the cap-beams, and the shear capacities of cap-beam calculated as shown in Table 10 were used to check for possible shear failure. Table 14 Maximum cap-beam positive bending moment of time history analysis Section Max. M (ETHA)(N-m) Max. M (NTHA)(N-m) Moment capacity (N-m) Capbeam1 1.85E+07 7.97E+06 3.80E+07 Capbeam2 3.49E+07 2.01E+07 3.80E+07 Capbeam3 3.17E+07 1.81E+07 3.80E+07 Max. 3.49E+07 2.01E+07 3.80E+07 Table 15 Maximum cap-beam negative bending moment of time history analysis Section Max. -M (ETHA)(N-m) Max. -M (NTHA)(N-m) Moment capacity (N-m) Capbeam1 2.70E+07 2.11E+07 2.41E+07 Capbeam2 4.32E+07 2.11E+07 2.41E+07 Capbeam3 3.77E+07 2.12E+07 2.41E+07 Max. 4.32E+07 2.12E+07 2.41E+07 Table 16 Maximum cap-beam shear force of time history analysis Section Max. S (ETHA)(kN) Max. S (NTHA)(kN) Shear capacity (kN) Capbeam1 8.15E+03 5.70E+03 1.68E+04 Capbeam2 1.13E+04 6.79E+03 1.68E+04 Capbeam3 1.06E+04 5.99E+03 1.68E+04 Max. 1.13E+04 6.79E+03 1.68E+04 It can be observed from the tables above that there was sufficient strength in both positive and negative bending direction. What’s more the shear capacities of cap-beams also meet the shear force demands. Therefore no cracking of cover concrete or rebar yielding at the cap-beam will happen during the earthquake. And shear failure of the cap-beam will not occur as well. 114 9.2.2 Bearings 1) Maximum axial and shear load of bearing Bearing in the FCB is another type of important component that needs to be checked, there are totally 36 bearings installed in the bridge including 12 sliding bearings and 24 fixed bearings. Both types of bearings were free to rotate about the vertical and horizontal axis but restrain movement in translational (U2 & U3) and vertical (U1) directions (except that slide bearings were free to move in U2 direction). The U1, U2 and U3 directions of bearing link element were illustrated in Fig. 20. Therefore the internal shear force in translational directions and the axial force in vertical direction need to be checked. Since the bridge structure was symmetric about the transverse axis, bearings on only southbound side of the bridge were considered. Fig. 92 illustrates the location of all 36 bearings on the FCB, the red dots were the bearings been monitored. The numbers marked near the bearings were the element numbers of the link elements in SAP2000 (see Fig. 17) which were used to model the bearings. Fig. 92 Location of monitored bearings and expansion joints on the FCB The internal forces of the monitored bearings were recorded in time domain. Forces including translational shear force in transverse (U3) and longitudinal (U2) directions (see Fig. 20) as well as the vertical axial loads were extracted from ETHA and NTHA results. Example of the response results of bearings at the Pier 1 (No. 430) was presented. Fig. 93 and 94 show the translational shear force and vertical axial force time history responses of the link element 430. Time history response records in other directions will not be presented since there will be too many figures involved if time history results of all bearings were presented. Instead the final results of maximum forces were summarized and presented in Table 17 to 20, including response value of all 115 monitored bearings. For the axial load in bearing, the negative values represent compressive loads and positive values are for tensile loads. Fig. 93 Time history records of internal forces of link element 430 (ETHA) 116 Fig. 94 Time history records of internal forces of link element 430 (NTHA) Maximum force values were presented through Table 17 to 20. The bearing service load capacities were based on the data provided in structural drawing. It can be observed that most service loads of bearing did not meet the maximum force demand. Therefore the bearings of FCB may be damaged during the earthquakes. 117 Table 17 Bearing maximum shear loads in transverse (U3) direction Bearing location Max. Shear (NTHA)(kN) Shear service loads (kN) South pile bent 1105.12 930 Pier 1 573.52 305 Pier 2 1017.96 305 Pier 3 (South) 973.46 950 Pier 3 (North) 902.58 295 North abutment 975.25 545 Table 18 Bearing maximum shear loads in longitudinal (U2) direction Bearing location Max. Shear (NTHA)(kN) Shear service loads (kN) Pier 1 681.42 495 Pier 2 505.97 495 Pier 3 (South) 4468.87 290 Pier 3 (North) 4647.28 290 Table 19 Bearing maximum vertical compressive loads Bearing location Max. Compression (NTHA)(kN) Vertical service loads (Dead+Live load) (kN) South pile bent 1382.10 1130.00 Pier 1 2782.38 3705.00 Pier 2 2430.80 3705.00 Pier 3 (South) 892.52 1190.00 Pier 3 (North) 2294.49 1425.00 North abutment 2075.35 1425.00 Table 20 Bearing maximum vertical tensile loads Bearing location Max. Temsion (NTHA)(kN) Vertical service loads (Live load) (kN) South pile bent 1379.99 105 Pier 1 492.28 90 Pier 2 110.51 90 Pier 3 (South) 625.53 115 Pier 3 (North) 792.56 115 North abutment 1898.52 105 2) Maximum deformation of sliding bearing The standard bearings in FCB are designed for a minimum movement capacity of ± 25 mm. In order to check whether the bearing movements exceed such capacity, the relative horizontal deformation of the link elements 118 which represent the bearings were monitored. Since the translational stiffness assigned in both transverse U3 and longitudinal U2 directions (see Fig. 20) of the bearings was very large (except in U2 direction of sliding bearings), the bearings were considered as rigid in both directions, and the deformation response of bearings in both directions was close to zero. However, for sliding bearings, the translational stiffness in U2 direction was not rigid. Therefore relative displacement will happen between the Teflon plates (see Fig. 10 right) of sliding bearings in U2 direction. In this study the deformation time history response in U2 direction of the slide bearings was studied. Slide bearing No. 336 and No. 514 (see Fig. 92) were monitored to record the time history results. Fig. 95 and 96 present the time history response of slide bearings deformation in U2 direction. Fig. 95 Time history records of deformation of slide bearing 336 and 514 in U2 direction (ETHA) 119 Fig. 96 Time history records of deformation of slide bearing 336 and 514 in U2 direction (NTHA) Table 21 summarized the maximum movement of sliding bearings on the south pile bent (336) and north abutment (514) separately. It can be easily observed in the table that the maximum deformation of the slide bearings was far from reaching the longitudinal travel capacity. Table 21 Sliding bearing maximum relative deformation in longitudinal direction (U2) Bearing location Max. Deformation (NTHA)(mm) Longt. (U2) travel capacity (mm) South pile bent (336) 3.76 175 North abutment (514) 4.21 175 9.2.3 Expansion joints During the earthquake, the expansion joint located on the bridge will open and close repeatedly in longitudinal direction (along the X axis as shown in Fig. 45), possible collision may happen between the concrete decking when the distance come down to zero. If the collision happens, nonlinear gap element needs to be added between the concrete decking to model the sudden change of stiffness. 120 In order to find out if the collision happens during time-history analysis, link elements with no stiffness was added between the gaps of expansion joints as described in Section 4.2.2. These link elements do not represent any component and do not contribute any mass or stiffness. They were only used as the tool of recording the relative movement between concrete decking of the expansion joint. By monitoring the deformation of the link element, the maximum values of the relative longitudinal (in X axis direction) movement between concrete gaps of the expansion joints can be acquired. Fig. 97 and 98 show the time history records of deformation of link elements, results of the expansion joints at all three locations including south pile bent, Pier No. 3 and the north abutment (as shown in Fig. 92) were presented. Notice that the positive deformation represent that the link element gets longer which means the distance of the gap is getting bigger. On the other hand, the negative deformation shows that link element is getting shorter which indicated that the gap is closing. So we are only interested in the negative deformation and the maximum negative values were picked from the time history data. 121 Fig. 97 Time history records of deformation of link element of expansion joint along X axis (ETHA) 122 Fig. 98 Time history records of deformation of link element of expansion joint along X axis (ETHA) Table 22 lists the maximum negative values of time history analyses under all 11 ground motions, they were compared with the gap distances of expansion joints (shown in Fig. 4). It can be concluded that no collision happen during the time history analyses. Therefore including the nonlinear gap element into the FE model is unnecessary. 123 Table 22 Maximum expansion joint relative movement of time history analysis Joint location Max. deformation (ETHA) (mm) Max. deformation (NTHA) (mm) Gap distance (mm) South pile bent 34.65 23.57 300 Pier No.3 16.81 9.30 60 North abutment 26.92 24.76 325 9.3 Performance based seismic evaluation As discussed in Section 8.4.2, in order to perform the performance based seismic evaluation procedure, a set of performance criteria which specified in the code have to be checked one by one, the basic methodology is to evaluate the major components by comparing the extreme response demand with self-defined capacities. Notice that only the NTHA results were used for the performance evaluation, and the maximum values were obtained from the NTHA of 11 ground motions. 9.3.1 Concrete structures 1) Columns Method of determine whether the longitudinal steel tensile strains exceed 0.015 has been explained in Section 8.4.3. For column sections the largest axial stress doesn’t occur under the maximum axial force nor the maximum bending moment. The maximum axial stress of a section occur at the most severe combination of axial force and bending moment, which can only be found when the axial force and bending moment of the column section were plotted together. Therefore for a given column section, the axial force and the bending moment at the same time step were considered as a pair of data (P, M). For every pier, all time history P-M data points of top and bottom column sections were plotted with the target P-M interaction curve (Fig. 55) which represents the envelop curve of P-M combination that produce maximum reinforcement tensile strain of 0.015. Any point that falls on the right side of the curve will cause maximum strain greater than 0.015. 124 The final diagrams of all three piers were presented through Fig. 99 to 101. It can be observed that no point stays on the left side of the P-M target curve. This proves that the maximum reinforcing steel tensile strains of the columns during nonlinear time history analysis did not exceed 0.015. Fig. 99 Pier No.1 axial-flexural responses compared with P-M performance target curve Fig. 100 Pier No.2 axial-flexural responses compared with P-M performance target curve 125 Fig. 101 Pier No.3 axial-flexural responses compared with P-M performance target curve 2) Cap-beams Table 23 and 24 presented the comparison results of bending moment demands with target moments. The target moment was the minimum bending moment that can cause the maximum rebar strain of 0.015. The target moment values were calculated as shown in Fig. 56. It can be concluded that the maximum reinforcement strain of the cap-beam did not exceed 0.015, therefore the performance objective is satisfied. Table 23 Cap-beam maximum positive moment compared with target moment Section Max. +Moment (NTHA)(N-m) Target moment (N-m) Capbeam1 7.97E+06 3.72E+07 Capbeam2 2.01E+07 3.72E+07 Capbeam3 1.81E+07 3.72E+07 126 Table 24 Cap-beam maximum negative moment compared with target moment Section Max. -Moment (NTHA)(N-m) Target moment (N-m) Capbeam1 2.11E+07 2.33E+07 Capbeam2 2.11E+07 2.33E+07 Capbeam3 2.12E+07 2.33E+07 9.3.2 Steel structures Buckling of primary members shall not occur. The primary steel members in the FCB are the W-shaped steel girders under the concrete decking (Fig. 18). The performance criterion requires no bucking shall occur for the steel girders. The buckling of steel members was checked by Euler’s critical load equation: (Timoshenko and Gere, 2009) 𝐹𝐹 = 𝜋𝜋2𝐸𝐸𝐼𝐼(𝐾𝐾𝐿𝐿)2 where F is the expected compressive force on buckling, E is the elastic modulus of the structure steel, I is the moment of inertia of girder section, L is the unsupported length of the element and K is the effective length factor. However, the top flange all steel girders are rigid connected to the bottom of concrete decking with bolts along the entire length of the steel girder (as shown in Fig. 5). This makes the unsupported lengths L of the girders equal to zero, therefore the buckling force of the steel girder will be infinite large. As a result, it can be concluded that the performance requirement of the steel structure was satisfied. 9.3.3 Connections The primary connections, which are the beam-column connections and the column-foundation connections in the FCB case, shall not be compromised during the nonlinear time history analysis. In order to verify that the connections were intact, critical beam and column sections at the plastic hinge zone of all piers (as illustrated in 127 Fig. 102) were examined. Detail method and procedure as well as analysis results were presented in Section 9.2.1. Fig. 102 Critical beam and column sections of the primary connections Both flexural and shear capacities were checked to determine whether there is any possible inelastic behavior which is yielding of the plastic hinge. All data collection and comparison results were presented in Section 9.2.1. It appears that all column sections were able to withhold the maximum moment, axial and shear load demands from earthquake excitation, all seismic responses were kept below the values of performance requirements or expected capacities. The bending moment capacities of the beam section also satisfy the seismic demand as shown in Table 14 and 15. No sign of possible shear failure of beam sections was found during the nonlinear time history analysis according to results in Table 16. Therefore the performance demand related to the connections was satisfied. 9.3.4 Bearings and joints Bearings were allowed to be damaged and replaced under the reparable damage performance level. As discussed in Section 9.2.2, based on the results listed from Table 17 to 20, it can be easily observed that the service loads of bearings were significantly lower than the seismic load demands. Therefore bearing damage due to horizontal shear load, vertical compressive load and vertical tensile load will occur during major earthquake. In horizontal shear failure, the pot plat or piston plat (see Fig. 10) may be damaged by pressing 128 against one another. Under extreme vertical compressive load, the PTFE guide strips of the sliding bearings may be damaged. And under extreme vertical tensile load, uplift of the top part of the bearings may occur since no uplift restrain device was used on the bearing. For the expansion joints of FCB, the analysis results were presented in Section 9.2.3. As shown in Table 22, there is no hazard of possible collision of the expansion joints, therefore the joints will not be damaged during the earthquake. 9.3.5 Restrainers There is no restrainer involved in the bridge structure. 9.3.6 Displacement As required by performance criteria, permanent offset shall not compromise the service and repair requirements of the bridge and no residual settlement of main structure shall occur. In order to confirm that the FCB meets the demand, the midpoint of the superstructure and the southeast corner of Pier No.2 foundation (as shown in Fig. 105) were selected as reference points to monitor the residual displacements of the bridge structure. To capture the residual displacement of the structure, full length time history analysis need to be conducted. However, the NTHA was extremely time-consuming therefore only one of the most severe ground motions was used. In this study, the ground motion No.1 was selected. What’s more, to capture the aftershock behavior, extra 300 seconds were added at the end of the time history ground motion input. The FE model will continue to vibrate after seismic excitation ends, vibration amplitude will reduce gradually until it settles at the final residual displacement. Fig. 103 and 104 present the time history displacement records of the superstructure midpoint and Pier 2 foundation in three orthogonal directions respectively. Both figures show that no obvious permanent offset or residual displacements occur after the earthquake. Therefore it can be concluded that the displacement performance criteria are satisfied. 129 Fig. 103 Time-history displacement of superstructure midpoint (GM No.1) 130 Fig. 104 Time-history displacement of Pier 2 foundation (GM No.1) 9.3.7 Foundation Foundation movements shall be limited to only slight misalignment of the spans or settlement of some piers or approaches that does not interfere with normal traffic. The bridge structure was connected to the foundation at south pile bent, north abutment and three piers. In the FE model, soil-foundation interaction between abutment and soil was ignored, the bridge was fixed at the south pile bent and north abutment. However movement at the pier foundations may have been generated under seismic excitation since they were not fixed to the ground as discussed in Section 4.4.2. 131 To find out the maximum foundation responses at the pier foundations, nodes at southeast corner of the pier base were selected as monitored nodes as shown in Fig. 105. The time history displacement responses in two horizontal directions UX and UY as well as vertical direction UZ (as shown in Fig. 45) of Pier 2 foundations were plotted as an example in Fig 106. Fig. 105 Location of superstructure midpoint and monitored point on pier foundations 132 Fig. 106 Time history records of displacement of Pier 2 foundation in three directions (NTHA) Maximum values were summarized in Table 25. The analysis results show that the misalignment or settlements of the foundations under all piers are ignorable. Therefore, it can be concluded that the foundation movement requirement of slight misalignment was satisfied. Table 25 Maximum pier foundation movements Location Max. UX (NTHA)(mm) Max. UY (NTHA)(mm) Max. UZ (NTHA)(mm) Pier1 3.67 3.28 2.60 Pier2 2.36 2.43 2.80 Pier3 2.60 4.03 3.31 133 9.3.8 General conclusion Given the analysis and discussion presented above, under seismic excitation, inelastic behavior may happen at the pier but no visible damage will happen. Both pier column and cap-beam will have enough strength to resist seismic loads. No sign of possible collision of expansion bearings was noticed and no bucking will happen for primary steel members. The movements and residual displacements at the superstructure and pier foundations were kept in a negligible level. However, there is a high possibility that bearings at all five locations will be damaged due to extreme loads, damage form including horizontal shear failure, vertical compressive failure as well as vertical uplift failure may happen. The structural member connected to the bearing was also under high risk of being damaged. 134 10. Conclusion It can be concluded that during major earthquake: 1) Both moment and shear capacities of ductile components (including all pier columns and cap-beams) meet the demands. 2) The displacement demands of all expansion joints were satisfied, preventing the occurrence of pounding during earthquake. 3) No sign of bucking was noticed for primary steel members. 4) The movements and residual displacements at the superstructure and pier foundations were negligible. 5) The displacement demands on the slide bearings were less than the capacity in longitudinal direction. 6) However, for bearings at all locations, damage types including horizontal shear failure, vertical compressive failure as well as vertical uplift failure may happen. The structural components connected to the bearings were also under high risk of being damaged. To sum up, during major earthquake, no primary members (columns, cap-beams and girders) will be damaged, the bridge will maintain repairable and operational, and should be capable of supporting the dead load plus full live load after major earthquake. But there is a high possibility that severe damage will happen at all bearings, as well as the structural members that were connected to them. In general, the French Creek Bridge meets all requirements of the Service Limited and Repairable damage performance level specified in CAN/CSA-S6-14. 135 References Kaya, Yavuz, et al. British Columbia Smart Infrastructure Monitoring System, EWSHM-7th European Workshop on Structural Health Monitoring, 2014. Ventura C, Kaya Y, Yao F, Huffmann S, Turek ME. Seismic structural health monitoring of bridges in British Columbia, Canada. In Proceedings of the 15 world conference of earthquake engineering, Lisbon 2012 (Vol. 7). Structural Vibration Solutions (SVS), ARTeMIS Extractor: Ambient Response Testing and Modal Identification Software, User’s Manual, 2001. Computers and Structures Inc. (CSI), CSIBridge User’s Manual- version 15, Berkeley, California, 2011. Computers and Structures Inc. (CSI). SAP2000 User's Manual- version 15, Berkeley, California, 2011. CAN/CSA-S6-14. (2015) Canadian Highway Bridge Design Code. Canadian Standards Association, Ottowa. Micromed, Tromino User’s manual, Micromed, 139 pp, 2012. Brincker R, Zhang L, Andersen P. Modal identification from ambient responses using frequency domain decomposition. InProc. of the 18*‘International Modal Analysis Conference (IMAC), San Antonio, Texas 2000 Feb. Brincker R, Zhang L, Andersen P. Modal identification of output-only systems using frequency domain decomposition. Smart materials and structures. 2001 Jun;10(3):441. Bendat JS, Piersol AG. Random data: analysis and measurement procedures. John Wiley & Sons; 2011 Sep. 136 Jacobsen, Niels-Jørgen, Palle Andersen, and Rune Brincker. Using enhanced frequency domain decomposition as a robust technique to harmonic excitation in operational modal analysis, International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, 2006. CAN/CSA-G30.18-M92. (R2007) Billet-Steel Bars for Concrete Reinforcement. Ottawa: Canadian Standards Association, 1992. CAN/CSA-G40.20-04/G40.21-04. (R2009) General Requirements for Rolled or Welded Structural Quality Steel/ Structural Quality Steel. Ottawa: Canadian Standards Association, 2004. CAN/CSA-A23.3-04. (2010) Design of concrete structures. Ottawa: Canadian Standards Association, 2010 Canam Group Inc., Pot Bearings Introduction Manual, 2012. Akogul, Can, and Oguz C. Celik. Effect of elastomeric bearing modeling parameters on the Seismis design of RC highway bridges with precast concrete girders, Proceedings of the 14th World Conference on Earthquake Engineering, 2008. Fleming, Ken, Weltman, A.J., Randolph, M. F., Elson, W. K. Piling engineering. John Wiley & Sons, Inc., New York, 1985. Adhikary S, Singh Y, Paul DK. Modelling of Soil-Foundation-Structure System. Department of Earthquake Engineering, IIT Roorkee., 01. 1893;7. Selvadurai, Antony PS. Elastic analysis of soil-foundation interaction. Elsevier, 2013. Das, Braja. Principles of foundation engineering. Brooks/Cole Engineering Division, 1984. 137 Friswell M, Mottershead JE. Finite element model updating in structural dynamics. Springer Science & Business Media; 1995 Mar 31. Huang M, Guo W, Zhu H, Li L. Dynamic test and finite element model updating of bridge structures based on ambient vibration. Frontiers of Architecture and Civil Engineering in China. 2008 Jun 1;2(2):139-44. S. Halchuk, T.I. Allen, J. Adams, and G.C. Rogers. Open file 7576: Fifth Generation Seismic Hazard Model Input Files as Proposed to Produce Values for the 2015 National Building Code of Canada. Geological Survey of Canada, 2014 National Building Code of Canada 2015, National Research Council of Canada, Ottawa, 2014. EZ-FRISK Version 7.52, User’s Manual. Risk Engineering Inc., Colorado, 2011 Venture NC. Selecting and scaling earthquake ground motions for performing response-history analyses. NIST GCR. 2011:11-917. Baker JW. An introduction to probabilistic seismic hazard analysis (PSHA). White paper, version. 2008;1:72. Bazzurro P, Cornell CA. Disaggregation of seismic hazard. Bulletin of the Seismological Society of America. 1999 Apr 1;89(2):501-20. Jayaram N, Lin T, Baker JW. A computationally efficient ground-motion selection algorithm for matching a target response spectrum mean and variance. Earthquake Spectra. 2011 Aug;27(3):797-815. Aviram A, Mackie KR, Stojadinović B. Guidelines for nonlinear analysis of bridge structures in California. Pacific Earthquake Engineering Research Center; 2008. 138 Chadwell CB, Imbsen RA. 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A modal pushover analysis procedure to estimate seismic demands for buildings: theory and preliminary evaluation. Civil and Environmental Engineering. 2001 Jan 1:55. Arias A. Measure of earthquake intensity. Massachusetts Inst. of Tech., Cambridge. Univ. of Chile, Santiago de Chile; 1970 Jan 1. Liu M, Gorman DG. Formulation of Rayleigh damping and its extensions. Computers & structures. 1995 Oct 17;57(2):277-85. 139 Timoshenko SP, Gere JM. Theory of elastic stability. Courier Corporation; 2009 Jun 22. 140 Appendix: Selected ground motion details under time and frequency domain 141 142 143 144 145 146 147 148 149 150 151
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Seismic performance evaluation of French Creek Bridge based on Canadian Highway Bridge Design Code 2015 Xiang, Li 2016
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Title | Seismic performance evaluation of French Creek Bridge based on Canadian Highway Bridge Design Code 2015 |
Creator |
Xiang, Li |
Publisher | University of British Columbia |
Date Issued | 2016 |
Description | Built in 1993, the French Creek Bridge is located on highway 19 on Vancouver Island, BC, Canada. The bridge is part of the British Columbia Smart Infrastructure Monitoring System (BCSIMS), funded by the Ministry of Transportation and Infrastructure (MoTI) BC, Canada. The BCSIMS is a real-time seismic monitoring program that continuously assesses the seismic conditions of the selected bridges in BC. As part of this seismic monitoring program, the seismic performance and nonlinear dynamic behavior of the FCB was evaluated by developing the 3D Finite Element (FE) model of the bridge in SAP2000. The model was updated based on the modal properties extracted from an Ambient Vibration (AV) test. The nonlinear behavior of the bridge was modeled by adding plastic hinges on the ductile components. Then the FE model was used to perform the seismic performance evaluation in accordance with the latest Canadian Highway Bridge Design Code 2015. The evaluation result shows that during major earthquake, no primary members of the bridge were damaged, the bridge will maintains repairable and operational, and should be capable of supporting the dead load and live load after earthquake. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2016-04-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution 4.0 International |
IsShownAt | 10.14288/1.0300310 |
URI | http://hdl.handle.net/2429/57917 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2016-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by/4.0/ |
AggregatedSourceRepository | DSpace |
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