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Operational modal analysis, model updating, and seismic analysis of a cable-stayed bridge McDonald, Steven 2016

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  OPERATIONAL MODAL ANALYSIS, MODEL UPDATING, AND SEISMIC ANALYSIS OF A CABLE-STAYED BRIDGE by  Steve McDonald  B.A.Sc., The University of British Columbia, 2012  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)  January 2016  © Steve McDonald, 2016ii  Abstract  The Port Mann Bridge is currently one of the longest cable-stayed bridges in North America and the second widest bridge in the world. It is a cable-stayed bridge consisting of 288 cables, two approach spans made of concrete box girders and precast deck panels, and a main span consisting of steel girders and cross beams with precast deck panels. This work sets out to accomplish three main goals: study the dynamic behaviour of the Port Mann Bridge, calibrate the finite element model, and study the effects of model updating using a seismic analysis.  The dynamic behaviour of the Port Mann Bridge’s main span is studied using experimental data from field ambient vibration tests and from a structural health monitoring network. A finite element model is created by importing a version of the structural designer’s model and editing it based on design drawings. In order to assess what parameters would be feasible to calibrate, a sensitivity analysis is carried out using various material properties and boundary conditions. The model is then updated to match the experimental analysis results by varying multiple parameters. Finally, the calibrated model is compared to the original model by completing a linear time history analysis. A suite of ground motions were selected and scaled to match specific points on the response spectrum corresponding to the first few periods of the structure. Multiple critical locations are monitored in the time history analysis, and data from these locations are compared before and after calibration to examine the effect of model updating. The study concludes that model updating has a large effect on the predicted seismic behaviour of the bridge, which proves the importance of calibrating finite element models and maintaining physically meaningful parameters. It also shows that having a structural health monitoring program is very important for current and future research endeavours. iii  Preface  The research presented was completed by Steve McDonald under the supervision of Dr. Carlos Ventura. The model was first created by the design engineers, T.Y. Lin International and International Bridge Technologies Inc. using the finite element program ADINA. It was then significantly modified and imported into the computer program SAP2000 to complete the analytical part of the research discussed here.  A preliminary version of Chapter 4 was published in the proceedings of the 11th Canadian Conference on Earthquake Engineering in July, 2015. Most of the data for that publication was analyzed using a different method, and the results are much different than what is being published here. I did the analysis work under the supervision of Dr. Carlos Ventura. The complete findings of this version of Chapter 4 will be published and presented at the 16th World Conference on Earthquake Engineering in January, 2017. iv  Table of Contents Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iii Table of Contents ......................................................................................................................... iv List of Tables .............................................................................................................................. viii List of Figures .................................................................................................................................x Acknowledgements .................................................................................................................... xiv Chapter 1: Introduction ................................................................................................................1 1.1 Computer Modelling ....................................................................................................... 1 1.1.1 Errors in Computer Modelling .................................................................................... 2 1.2 Experimental Testing ...................................................................................................... 2 1.2.1 Frequency Domain Decomposition ............................................................................ 3 1.2.2 Stochastic Subspace Identification ............................................................................. 4 1.2.3 Errors in Experimentation ........................................................................................... 6 1.3 Model Updating Approach ............................................................................................. 8 1.4 Purpose of Model Updating .......................................................................................... 10 1.5 Literature Review of Model Updating .......................................................................... 11 1.6 Research Motivation ..................................................................................................... 14 1.7 Research Goals.............................................................................................................. 15 1.8 Summary of Introduction .............................................................................................. 16 Chapter 2: Background to the Port Mann Bridge ....................................................................18 2.1 Structural Components.................................................................................................. 19 2.2 Finite Element Model ................................................................................................... 21 v  2.2.1 Model Components ................................................................................................... 23 2.2.1.1 Boundary Conditions ........................................................................................ 28 2.3 Summary of Background .............................................................................................. 30 Chapter 3: Experimental and Analytical Results .....................................................................32 3.1 Experimental Modal Analysis ....................................................................................... 32 3.1.1 Field-Acquired Data.................................................................................................. 33 3.1.2 Structural Health Monitoring Data ........................................................................... 38 3.1.2.1 Footing Movement ............................................................................................ 43 3.2 Analytical Results ......................................................................................................... 44 3.3 Summary of Experimental and Analytical Results ....................................................... 46 Chapter 4: Model Updating ........................................................................................................48 4.1 Initial Model Correlation .............................................................................................. 48 4.2 Accounting for Deck Weight ........................................................................................ 51 4.3 Sensitivity Analysis ...................................................................................................... 52 4.3.1 Sensitivity of Supports .............................................................................................. 55 4.3.2 Sensitivity of Boundary Conditions .......................................................................... 57 4.3.2.1 Piers N2 and S2 Connections ............................................................................ 57 4.3.2.1.1 Effect of Longitudinal Release in N2 and S2 Tie-Downs ........................... 58 4.3.2.1.2 Effect of Rotational Stiffness to N2 and S2 Tie-Downs ............................. 59 4.3.2.2 Piers N1 and S1 Connections ............................................................................ 60 4.3.3 Sensitivity of Cable Properties.................................................................................. 62 4.4 Selecting Parameters ..................................................................................................... 64 4.5 Step 1: Modifying N1 and S1 Tower Parameters ......................................................... 66 vi  4.6 Step 2: Modifying Girder Parameters and Adding Counterweight .............................. 67 4.7 Investigation into First Torsional Mode ....................................................................... 69 4.8 Summary of Model Updating ....................................................................................... 70 Chapter 5: Linear Time History Analysis .................................................................................73 5.1 Ground Motions ............................................................................................................ 73 5.2 Responses at Critical Locations .................................................................................... 74 5.2.1 Displacement in Towers N1 and S1.......................................................................... 74 5.2.2 Displacement at Mid-Span ........................................................................................ 76 5.2.3 Base Shears and Moments ........................................................................................ 77 5.3 Summary of Time History Analysis ............................................................................. 79 Chapter 6: Summary and Conclusion........................................................................................80 6.1 Future Research ............................................................................................................ 83 Bibliography .................................................................................................................................86 Appendices ....................................................................................................................................96 Appendix A General Plan and Elevation of Port Mann Bridge ................................................ 96 Appendix B Sensor Setup for Field-Acquired Data ................................................................. 97 Appendix C Channel Numbers and Orientations for SHM-Acquired Data ............................. 98 Appendix D ADINA Modelling Results................................................................................... 99 Appendix E Approximate Pile Group Stiffness Calculations ................................................. 105 E.1 Pile Details .............................................................................................................. 105 E.2 Geometry Calculations............................................................................................ 106 E.3 Vertical Stiffness Calculations ................................................................................ 108 E.4 Vertical Stiffness Plots ............................................................................................ 110 vii  E.5 Pile Group Calculations .......................................................................................... 111 Appendix F Ground Motion Records ..................................................................................... 113 F.1 Crustal Records. ...................................................................................................... 113 F.2 Subcrustal Records.................................................................................................. 116 F.3 Subduction Records ................................................................................................ 119 Appendix G Linear Time History Results .............................................................................. 122 G.1 Crustal Ground Motion Time History Analysis Results ......................................... 122 G.2 Subcrustal Ground Motion Time History Analysis Results ................................... 123 G.3 Subduction Ground Motion Time History Analysis Results .................................. 124 G.4 Envelope of Response Quantities from all Ground Motions .................................. 125  viii  List of Tables Table 2.1: Key Values Used for Modelling Piers N2 and S2 ....................................................... 25 Table 2.2: Range of Key Values Used for Modelling Piers N1 and S1........................................ 26 Table 2.3: Key Values for Modelling Deck Components ............................................................. 27 Table 3.1: ARTeMIS Modal Analysis Results for Field-Acquired Data...................................... 35 Table 3.2: ARTeMIS Modal Analysis Results for SHM-Acquired Data ..................................... 41 Table 3.3: Modal Analysis Results from Original SAP2000 Model ............................................ 44 Table 3.4: Summary of Three Experimental Ambient Vibration Analysis Results ..................... 46 Table 4.1: Mode Shape Pairs for Original SAP2000 Model and Experimental Data................... 50 Table 4.2: Mode Shape Pairs after Adding Deck Weight to SAP2000 Model ............................. 52 Table 4.3: Symmetric Stiffness Matrix for Piers N1 and S1 ........................................................ 55 Table 4.4: Symmetric Stiffness Matrix for Piers N2 and S2 ........................................................ 56 Table 4.5: Mode Shape Difference when Theoretical Pile Stiffness Used ................................... 56 Table 4.6: Mode Shape Pairs after Adding Longitudinal Release in N2 and S2 Tie-Downs ....... 58 Table 4.7 Mode Shape Pairs after Adding Arbitrary Moment Stiffness and Fixing Tie-Down Connections................................................................................................................................... 59 Table 4.8: Mode Shape Comparison for Two Separate Scenarios Modifying Pier N1 and S1 Bearings ........................................................................................................................................ 62 Table 4.9: Mode Shape Pairs after Assigning Cable E Value of 170 GPa and 210 GPa ............. 63 Table 4.10: Mode Shape Pairs after Reducing both E and I2 of Pier N1 and S1 by 10% ............. 67 Table 4.11: Mode Shape Pairs after Modifying Girder Properties and Counterweight in SAP2000 Model ............................................................................................................................................ 69 ix  Table 4.12: MAC Contribution to Torsional Mode from Longitudinal, Transverse, and Vertical Components .................................................................................................................................. 70 Table 4.13: Summary of all Updated FEA Mode Shapes vs. all Test Mode Shapes.................... 71 Table 4.14: Summary of Parameter Modifications ....................................................................... 72 Table 5.1: Summary of Ground Motions Selected for Linear Time History Analysis ................. 73 Table 5.2: Changes in Absolute Maximum Displacement of Main Towers at Deck Level after Model Updating ............................................................................................................................ 75 Table 5.3: Changes in Absolute Maximum Displacement of Deck at Mid-Span after Model Updating ........................................................................................................................................ 77 Table 5.4: Changes in Absolute Maximum Base Shears in Bridge Piers after Model Updating . 78 Table 5.5: Changes in Absolute Maximum Base Moments in Bridge Piers after Model Updating....................................................................................................................................................... 79 Table B.1: Sensor Placement Locations Corresponding to Diagram Labels ................................ 97 Table D.1: ADINA Modal Frequencies Compared to Converted SAP2000 Model..................... 99 Table G.1: Summary of Crustal Ground Motion Results (Units in kN and m) .......................... 122 Table G.2: Summary of Subcrustal Ground Motion Results (Units in kN and m) ..................... 123 Table G.3: Summary of Subduction Ground Motion Results (Units in kN and m) ................... 124 Table G.4: Maximum and Minimum Response Quantities from all Ground Motion (Units in kN and m) ......................................................................................................................................... 125  x  List of Figures Figure 1.1: Example Normalized SVD Plot ................................................................................... 4 Figure 1.2: Example Stabilization Diagram for SSI Method .......................................................... 5 Figure 1.3: Aliasing - Two Waves Producing Same Measured Data (Friswell and Mottershead 1995) ............................................................................................................................................... 6 Figure 1.4: Effect of Sampling Interval and Resulting Leakage when Applying DFT .................. 7 Figure 2.1: Satellite Image Showing Location of Port Mann Bridge ........................................... 18 Figure 2.2: Elevation View of the Port Mann Bridge (Looking South) ....................................... 19 Figure 2.3: Elevation of Port Mann Bridge Showing Pier Labels ................................................ 19 Figure 2.4: Struts Connecting East and West Decks in Main Span .............................................. 20 Figure 2.5: Cross Section of Concrete Box Girders in Approach Spans ...................................... 20 Figure 2.6: Typical Dampers at Approach Piers ........................................................................... 21 Figure 2.7: Port Mann Bridge Model in ADINA Finite Element Software.................................. 21 Figure 2.8: Port Mann Bridge Model in SAP2000 Finite Element Software ............................... 22 Figure 2.9: Modelling of Foundations .......................................................................................... 24 Figure 2.10: Modelling of Pier N2 Compared to Design Drawing............................................... 25 Figure 2.11: Modelling of Pier N1 Compared to Design Drawing............................................... 26 Figure 2.12: Modelling of Deck Components .............................................................................. 27 Figure 2.13: Modelling of Tie-Downs and Transverse Restraints Compared to Design Drawings....................................................................................................................................................... 29 Figure 2.14: Modelling of Pier N1 and S1 Bearings .................................................................... 30 Figure 2.15: Modelling of Pier S1 Longitudinal Restraint Bearings ............................................ 30 Figure 3.1: Sensor Setup Locations .............................................................................................. 33 xi  Figure 3.2: First Six Mode Shapes from Field-Acquired Data for the Main Span of Port Mann Bridge ............................................................................................................................................ 37 Figure 3.3: Sensor Locations used for Modal Analysis ................................................................ 38 Figure 3.4: Geometry and Channel Orientations for ARTeMIS Setup ........................................ 39 Figure 3.5: Frequency Domain Decomposition Peak Picking Plot .............................................. 40 Figure 3.6: Example Stabilization Diagram for SSI Method ........................................................ 40 Figure 3.7: First Six Mode Shapes for SHM-Acquired Data for Main Span of Port Mann Bridge....................................................................................................................................................... 42 Figure 3.8: MAC Value Matrix Comparing SSI Modes vs. EFDD Modes .................................. 43 Figure 3.9: First Six Mode Shapes for SAP2000 Model of the Port Mann Bridge ...................... 45 Figure 4.1: Correlated Mode Shapes for FEM and Test Model ................................................... 49 Figure 4.2: Example of Excellent Correlation Despite Limited Node Spacing ............................ 51 Figure 4.3: Example Distributed Load Added to Simulate Deck Weight .................................... 51 Figure 4.4: Mode 5 Showing Transverse Motion in FEM but not Experimental Model .............. 53 Figure 4.5: Sensitivity Matrix for E and I against Frequency Values for Modes 2-6................... 54 Figure 4.6: Frame End Releases Originally Assigned to N1 and S1 Bearings ............................. 61 Figure 4.7: Sensitivity Contours for Mode 5 and Material Parameter E ...................................... 67 Figure 4.8: Counterweight Force and Moment Applied to West Deck (Units of kN and kNm) .. 68 Figure 5.1: Diagram Showing Location used for Recording Displacements of Towers N1 and S1....................................................................................................................................................... 75 Figure 5.2: Diagram Showing Locations used for Reporting Mid-Span Displacements ............. 76 Figure 5.3: Diagram Showing Locations used for Reporting Base Shears and Moments ............ 78 Figure A.1: Plan and Elevation Design Drawings of Port Mann Bridge ...................................... 96 xii  Figure B.1: Sensor Setup Locations with Placement Labels ........................................................ 97 Figure C.1: Channel Map for Data used in the Analysis of SHM-Acquired Data ....................... 98 Figure D.1: 1st Vertical Mode Shape at 0.2122 Hz ..................................................................... 100 Figure D.2: 1st Torsional Mode at 0.230 Hz ............................................................................... 100 Figure D.3: Second Torsional Mode at 0.2490 Hz ..................................................................... 100 Figure D.4: 2nd Vertical Mode at 0.2716 Hz ............................................................................... 101 Figure D.5: Vertical-Transverse Mode at 0.2879 Hz ................................................................. 101 Figure D.6: Vertical-Transverse Mode at 0.3805 Hz ................................................................. 101 Figure D.7: Vertical Mode at 0.4059 Hz .................................................................................... 102 Figure D.8: 3rd Vertical Mode at 0.4329 Hz ............................................................................... 102 Figure D.9: Vertical Mode at 0.4801 Hz .................................................................................... 102 Figure D.10: Vertical Mode at 0.4923 Hz .................................................................................. 103 Figure D.11: Vertical Mode at 0.5113 Hz .................................................................................. 103 Figure D.12: Vertical Mode at 0.5205 Hz .................................................................................. 103 Figure D.13: Vertical Mode at 0.5456 Hz .................................................................................. 104 Figure D.14: Torsional Mode at 0.5484 Hz ................................................................................ 104 Figure F.1: Plots for CHICHI03_TCU122 Records ................................................................... 113 Figure F.2: Plots for GAZLI_GAZ Records ............................................................................... 114 Figure F.3: Plots for LOMAP_LGP Records ............................................................................. 115 Figure F.4: Plots for GEIYO_EHM0030103241528 Records .................................................... 116 Figure F.5: Plots for Miyagi_Oki_MYG0060508161146 Records ............................................ 117 Figure F.6: Plots for Olympia_OLY0 Records ........................................................................... 118 Figure F.7: Plots for Hokkaido_HKD1270309260450 Records................................................. 119 xiii  Figure F.8: Plots for Tohoku_KNG0041103111446 Records .................................................... 120 Figure F.9: Plots for Tohoku_KNG0061103111446 Records .................................................... 121  xiv  Acknowledgements I would like to acknowledge my supervisor, Dr. Carlos Ventura. I am forever thankful for his support. His encouragement and enthusiasm for engineering inspired me to pursue my master’s degree—I could not have done this without him.  Special thanks are owed to Dr. Martin Turek and Dr. Yavuz Kaya, who both took time out of their work to assist me in getting the data and information I needed for completing my research.  Thanks to my family who has provided me with continuous encouragement. I am very grateful to my mother, father, and grandmother for their moral support. I am also very thankful to my brother, Ryan McDonald, and my good friend, Sean Taylor, who answered my constant questions when I was learning how to use programming to process much of the data.  1  Chapter 1: Introduction Understanding the behaviour of structures is paramount to the field of structural engineering. Engineers designing the structures have been using computer models for decades to predict the response of structures. With advances in computer technology, models are also capable of being extremely large and complex. However, this does not necessarily mean that they have become more accurate—when compared with experimental results it has been found that there are significant differences in modelling results. That being said, the discrepancies are not solely due to the computer model—there are errors associated with the experimental model too, but in application it is most efficient to address the computer model. In order to minimize these differences, it is a common technique to “update” the analytical model by modifying various physical properties. This is a practice called model updating, or model calibration.  1.1 Computer Modelling A structural system is normally analyzed using a set of differential equations that relate the physical properties together. The problem with the differential equation approach is that it is usually impossible to solve them unless the geometry is very simple. To get around this, the mathematical model is broken down into discrete, finite elements and the responses are approximated based on these elements. This is called the finite element method, and it is the most used analysis method in structural design. When modelling a structure in a computer program, it can be difficult to determine how some aspects of the structure should be modelled. With the complexity of structures, many assumptions and simplifications have to be made when modelling. It can be difficult to create a fully representative structure given the amount of uncertainties involved in the process. 2  1.1.1 Errors in Computer Modelling Mottershead et al. (2011) provided a good review of the sources of modelling error. There are many possible sources of error, and some of them cannot be corrected in the model updating process. Errors can be categorized into three types: 1. Errors in idealizing the structure 2. Errors inherent in the finite element method 3. Errors in assumptions for model parameters  The first category deals with factors that are involved in idealizing the structure in certain ways in an attempt to model the structural behaviour. Examples which can cause this type of error can be improper boundary conditions, joint connections, or external loads. The second category encompasses the numerical methods involved in the finite element method, and they are generally unavoidable; these sources of error should be kept in mind, but are not possible to directly compensate for because they are a part of most commercial software packages. The third source of error involves the physical properties assigned to the components of the model, such as the elastic modulus, mass density, or cross sectional properties. These physical properties are the main target for model updating, but still require significant engineering judgement to do so. More information on modelling error can be found in the book by Friswell and Mottershead (1995).  1.2 Experimental Testing There are generally two methods of experimental testing: forced vibration and ambient vibration. Forced vibration involves utilizing a measured input force, such as an impulse hammer or a 3  dynamic shaker, to excite the structure. The structural response is then measured and dynamic characteristics are inferred from them. The second type of experimental testing uses output-only data to determine the dynamic properties of the structure. Ambient vibrations such as traffic, wind, and micro earthquakes excite the structure during its operation, and the response is measured. Ambient vibration testing (AVT), also known as operational modal analysis (OMA), is generally more preferred for testing large civil engineering structures. Forced vibration testing requires large forces to properly excite such structures, and this can be very expensive and may cause damage to the structure. AVT is also usually carried out quicker than forced vibration tests. There are generally two commonly-used methods for analyzing AVT data: frequency domain decomposition and stochastic subspace identification.  1.2.1 Frequency Domain Decomposition The frequency domain decomposition (FDD) technique, introduced in Brincker, Zhang and Andersen (2000), uses the discrete Fourier transform to turn the time-domain data into frequency domain data. The technique essentially decomposes the system response into a set of single degree of freedom systems by decomposing each of the estimated spectral density matrices. The technique first involves estimating the spectral density matrices of the recorded data, and then performing a singular value decomposition of these matrices. The resulting data shows dominating peaks at specific frequencies which usually correspond to the different mode shapes. An example peak picking SVD plot can be shown in Figure 1.1. 4   Figure 1.1: Example Normalized SVD Plot  The biggest advantage associated with the FDD method is its robustness and ease of use. It is very intuitive and simple to use large peaks to indicate the mode shapes. The enhanced frequency domain (EFDD), used in this research, is an extension of the FDD method that gives an improved estimate of the frequencies and mode shapes, and also provides estimates of damping ratios.  1.2.2 Stochastic Subspace Identification The stochastic subspace identification (SSI) technique estimates mode shapes using the data in the time domain. The mathematics can be quite complex, and are outside the scope of this research. The SSI method is the most powerful for time domain modal identification (Brincker and Andersen 2006). The technique essentially fits a parametric model to the time series data by 5  varying a set of parameters in order to reach a better correlation between the predicted response and the measured response. By choosing the number of state space models, you are specifying how many parameters to vary in the model, so it is important to balance the state space to be not too small and not too large. Once a model order is chosen, a stabilization diagram shows all eigenvalues for each state space model. This diagram has an x-axis depicting the frequency values and a y-axis that lists the dimensions of the available state space models. The idea with the stabilization diagram is that a repeated trend across state space models can represent a structural mode if it is located at a resonance frequency. The user must make judgements in setting the stabilization criteria to distinguish between stable, unstable, and noise modes. These criteria generally involve maximum deviation of frequency, damping ratio, or MAC between two state space models. An example of the stochastic subspace identification method’s stabilization diagram is shown in Figure 1.2. For a thorough explanation of the SSI method, see Brincker and Andersen (2006).  Figure 1.2: Example Stabilization Diagram for SSI Method 6   1.2.3 Errors in Experimentation Experimental modal analysis techniques have been used for more than 50 years. They are generally accurate given the right instrumentation, data processing, and testing procedures, but sources of error should still be kept in mind. Errors in signal processing have mostly been minimized with advances in the field. Two main forms of errors involved with signal processing are aliasing and leakage. Aliasing arises when two signals of different frequencies are interpreted as the same. This is possible to determine when the sampling rate is taken into account, and each sample point lies at the same location on both signals, making the two waves seem identical. An example of aliasing is shown in Figure 1.3. To avoid aliasing, you have to filter out any data with frequencies above half of the desired sampling rate, also known as the Nyquist frequency. Most signal processing software today automatically filters this out for you.  Figure 1.3: Aliasing - Two Waves Producing Same Measured Data (Friswell and Mottershead 1995)  The second type of signal processing error is called leakage. To truly know the frequency content of a signal, you would have to measure for an infinite amount of time. However, this is not 7  possible, so the data has to be taken at a finite interval and processed. Choosing this interval and applying the Discrete Fourier Transform (DFT) involves assuming that the interval repeats itself. Depending on what interval of data you choose, the integer multiple can produce different Discrete Fourier Transform results. For example, signal leakage can be seen in Figure 1.4, where the period of the sinusoidal wave perfectly fits the time interval in the first diagram, but does not in the second diagram, causing leakage. The value at the fundamental frequency effectively “leaks” into other frequencies.  Figure 1.4: Effect of Sampling Interval and Resulting Leakage when Applying DFT  (Friswell and Mottershead 1995)  Leakage can be reduced by using window functions such as the well-known Hanning or exponential functions (Harris 1978) in order to force the signal to start and end at zero, removing the discontinuities associated with making the signal periodic.  Aside from signal processing problems, there are also important points to keep in mind when considering location of the sensors in the test setup. In order to fully capture mode shapes of a 8  structure, there has to be a significant number of measured nodes, which is usually not feasible. Often times there are even locations that are not accessible to take measurements, so the data may not properly capture the dynamic behaviour. The number of measured degrees of freedom is usually much smaller than the number of degrees of freedom associated with the finite element model, which should be taken into account. For example, in order to capture higher modes of vibration of a bridge deck, such as a fifth vertical mode, you would need enough closely-spaced sensors to distinguish the difference in excitation between it and lower modes. This is a type of spatial aliasing that needs to be considered.  1.3 Model Updating Approach The idea of model updating is to change specific parameters, such as material properties or boundary conditions, in order to reduce the errors associated with finite element modelling and make the model match more closely to that of the experimental model.  It is very important to recognize that the changes made to the model are physically meaningful; and this is no easy task. Therefore, model updating requires a good amount of engineering judgement, and the parameters that are chosen are very important. Model updating can be done manually by trial and error, or it can be done automatically using one of various optimization algorithms available. There are many different optimization methods used in structural dynamics studies, such as:  Response surface method  Bayesian updating  Nelder-Mead simplex method  Genetic algorithms  Simulated annealing 9   Particle swarm optimization  Hybrid optimization  Artificial neural networks  These methods are based on an iterative approach and they are reviewed in the book by Marwala (2010). In the current study, a manual updating approach is taken. This allows complete control and monitoring of how to update the model, but also requires a significant amount of engineering insight in order to determine potential changes. It is also more difficult to reach a close match when updating a model manually. In the future, a Bayesian-style automatic updating approach may be taken, as this method still allows control over parameter deviation and has the option to put a weighting on the various parameters.   Model updating first involves deciding on what response quantities you want to measure your correlation. Usually the responses with which to compare will be the frequency values of each mode. The model can further be calibrated based on the Modal Assurance Criterion (MAC) values which correlate the shapes of each mode (Pastor, Binda and Harcarik 2012). A model calibrated based on frequencies and mode shapes can be considered to be a successfully updated model. When the response quantities are decided upon, then it is important to select a set of parameters that are meaningful in the updating process. The parameters should be chosen by balancing various factors: considering the uncertainty in specific material or section properties, the sensitivity of the model to changes in the parameter, and the computing process involved with too large a selection of parameters.  10  1.4 Purpose of Model Updating Model calibration is becoming more popular in engineering today. Having an accurate finite element model means having the confidence in the results that any analysis of the model produces. This can be very important in engineering. The purpose of model updating can depend on the application. For some models, you may not be interested in creating a physically realistic model, and only want to focus on being able to reproduce the measured frequencies and mode shapes. This can be useful if you wanted to compare specific data which was measured at different times or using different sensors. In most cases, however, having a physically realistic model is a necessity for application. If you manage to update a finite element model to match the frequencies and mode shapes while also improving the parameters, then it can be used for applications such as damage detection or to study the effect of changes in construction, retrofits, etc. Many structures are specifically being monitored with sensors in real-time, allowing data to be streamed and analyzed to determine the current dynamic properties. Structural health monitoring data can be used with updated models to compare behaviour before and after an earthquake, and potentially pinpoint location and severity of damage. Lastly, model updating can be very useful for nonlinear analyses. While experimental data from ambient vibration tests are fully linear, it still allows the model to be calibrated based on the linear behaviour for a more accurate starting point to the nonlinear model. For example, by calibrating parameters involved with the stiffness of elements, you change the point at which an element will undergo nonlinear behaviour in a dynamic analysis.  11  1.5 Literature Review of Model Updating Model updating for structural dynamics has been active since the late 1960s or early 1970s (Mottershead and Friswell 1993). It has been applied for many civil engineering structures such as bridges, dams, masonry structures, steel frame structures, and reinforced concrete structures.   Many papers have been published proposing different optimization techniques. Yang and ZhongDong (2012) proposed a fuzzy finite element method to update a model of a prestressed concrete bridge in China and reached an average error of 5% between modal frequencies. Javier Garcia-Palencia (2008) addressed issues with condition assessments of bridges by using a frequency response function-based model updating algorithm and applied the method to a lab specimen and a full-scale bridge.   Other studies also managed to highlight case studies which demonstrate potential for damage detection applications. Jie, ZhouHong and FuPeng (2015) investigated using response surface model updating and modal strain energy damage index to identify damage in girder bridge structures. The method is applied to a case study to identify cracking locations and severity in the Xinyihe Bridge. Teughels and De Roeck (2004) simulated damage in a prestressed concrete bridge by lowering one of the intermediate piers which acts as a settlement at the foundation. They managed to update a finite element model of the bridge and identify the location and quantity of the simulated damage. Türker and Bayraktar (2014) updated a model of a reinforced concrete building to detect damaged members by varying the moment of inertia and correlating the first three mode shapes. Kharrazi, et al. (2002) managed to identify damage in a one-third scale model of a four-story steel frame by updating the finite element model. 12   Not much has been researched yet for nonlinear effects due to model updating. One notable study by El-Borgi, et al. (2008) proposed a methodology for structural assessment of reinforced concrete bridges using finite element model updating and nonlinear analysis. The method is then demonstrated with a case study on an 8-span concrete bridge.  There are some studies which compare the effect of model updating on a seismic analysis, and seem to be authored by the same research team. Bayraktar, Can Altunisik, et al. (2009) performed finite element model updating on an arch-type steel footbridge and carried out a dynamic analysis using an earthquake record to compare before and after model updating. They found that the maximum displacements tended to increase and the principal stresses tended to decrease after updating. Bayraktar, Sevim, et al. (2010) investigated earthquake behaviour of storage tanks considering fluid-structure interaction. They updated the model using the elastic modulus as a parameter and applied an earthquake motion in the horizontal direction. The displacements were increased and stresses stayed almost the same after updating. Sevim, Bayraktar and Can Altunisik, et al. (2011) compared finite element model predictions of earthquake responses before and after model calibration. They used two different masonry arch bridges built in ANSYS and updated them, then compared the behaviour under an earthquake load. The principal stresses were found to be much lower after updating. Altunisik and Bayraktar (2014) investigated the effects of model updating using a case study on the Birecik Highway Bridge in Turkey. They studied structural performance of the bridge before and after model updating by reporting displacements, internal forces, and stresses. They found that after model updating, the forces and displacements are reduced by 20-30%. What these studies lack are the 13  effects of multiple earthquake ground motions that cover a range of frequency content and source types. With only one ground motion used, it is unclear what effect model updating can really have—it may show a decrease in forces for that ground motion, but still remains unclear if that will be the trend from an envelope of possible earthquake scenarios. Using a suite of scaled ground motions allows a more robust study on the effect of model updating.  Most studies tend to use the material properties such as elastic modulus or mass density as parameters. However, one noteworthy study by Osmancikli, et al. (2015) demonstrated model updating for a precast overpass and a precast production facility by modifying the stiffness coefficients of the connection joints.  There have been other studies that were carried out on large-scale bridges. Turek, Ventura, and Dascotte (2010) updated a model of the Ironworkers Memorial Second Narrows Bridge in several steps using both a manual and automated process. They first updated the stiffness of the longitudinal bearings to match the second frequency, and then they updated the elastic modulus and moment of inertia for various truss components. Another study was carried out by Hao Wang et al. (2010), who analyzed the Runyang suspension bridge, the longest bridge in China, by dividing the model into two phases based on construction process. They then updated the model sequentially for the first and second phases.  Other examples using elastic modulus and density as parameters include research by Sevim, Bayraktar and Altunisik (2009) who manually updated a concrete arch dam using eight mode shapes. Arslan and Durmus (2013) also used the elastic modulus and density for concrete, rebar, 14  and brick to update the model of an in-filled reinforced concrete frame and reduce the frequency errors from 39% to 8%. Alves and Hall (2006) also manually update a concrete arch dam model using two different datasets: one from a recorded earthquake, and one from a forced vibration experiment. They used E for rock and concrete to calibrate their model. Ventura, et al. (2001) updated a model of a 15-storey reinforced concrete shear core building using an automated procedure and modifying the elastic modulus, density, moment of inertia of columns, and cladding thickness.  An important warning by Atamturktur and Laman (2012) states that model updating must be treated with caution until models can truly be physically validated. They performed an in-depth review of model updating for masonry monuments. Lastly, a thorough review of model updating is presented in the survey by Mottershead and Friswell (1993). While it is not as recent, it presents very good background information and state of the art for research up to that point.  1.6 Research Motivation Model updating has been studied in many applications. While research has already been done on the topic in general, each case study that is researched poses its own unique set of problems and goals. The essence of model updating is to reach an accurate, physically representative model through parameter modifications that are accompanied with sound justifications based on engineering insight.  The Port Mann Bridge, a newly-constructed cable-stayed bridge in British Columbia, was once the widest bridge in the world (CTV News Vancouver 2012), until the San Francisco-Oakland 15  Bay Bridge was constructed. It is also one of the main hubs of transportation in the lower mainland which connects the city of Surrey in the south to the city of Coquitlam in the north. As such an important means of transportation, it is very beneficial to have a fully calibrated finite element model with a clear understanding of the dynamic characteristics of the bridge. This enables researchers or engineers to better predict possible scenarios, such as damage caused to the bridge and/or repairs. Ideally, the model will be capable of nonlinear analyses which open up other options for research on the bridge; because a fully linear model can be limited in the amount of information that is obtainable.  1.7 Research Goals The goal of this research is to obtain a physically realistic model that matches closely with the experimental results. This research is carried out with six main objectives: 1. Creation of a benchmark linear finite element model in the program SAP2000 (Computers and Structures Inc. 2014). 2. System identification of the bridge using experimental data and validation using two separate identification techniques. 3. Correlation of the experimental model and the finite element model. 4. Identification of areas of uncertainty and sensitivity of various parameters in the finite element model. 5. Calibration of the finite element model by modifying parameters using proper judgement. 6. Investigation of the effect of model updating using a seismic linear time history analysis.  16  The first step is to have a benchmark finite element model. For the scope of this research, the finite element model will be limited to linear behaviour and will only be composed of the main span. Once the finite element model is prepared, a set of experimental data is needed which identifies the main dynamic properties of the bridge as physically measured. After the two models have been created, a thorough study into the sensitivity and plausibility of various potential parameters can be undertaken. Using the experimental results as a target, and the sensitivity study, parameters will be selected to modify as part of the model calibration process.  After the updating process, a linear time history analysis is to be carried out to provide a good means of comparison between the benchmark model and the updated model.  1.8 Summary of Introduction This chapter reviewed the main aspects of finite element modelling, operational modal analyses, and model updating. Errors prevalent in the modelling processes were also covered, and a literature review was presented which highlighted similar studies and a need for information. The research scope and goals have been presented to provide the purpose of the research undertaken.  Chapter 2 provides a background to the Port Mann Bridge, the main case study for this research. The main structural components of the bridge are highlighted and the finite element model used in the study is reviewed, including boundary and restraint conditions. Chapter 3 covers the material related to the operational modal analysis and the analytical modal analysis that are used together in model updating. It first covers the experimental results from field data acquired in 2012, and then reviews the results from data that was taken from structural health monitoring sensors in 2015. Chapter 4 gives the details of the model updating process, including the 17  experimental and analytical model correlation, sensitivity analysis, parameter and response selection, and then finishes with the results of the manual updating procedure. To study the effect of model updating, the linear time history analysis research is presented in Chapter 5. The suite of ground motions used and the envelope of response changes before and after model updating are documented in this chapter. The chapter concludes by summarizing the differences between the results before and after model updating. Finally, Chapter 6 reviews all of the findings from each chapter and discusses the various aspects of the results. The chapter finishes with a list of future areas of research, which highlights any possible improvements and further topics of research that were outside the scope of the current research. 18  Chapter 2: Background to the Port Mann Bridge The Port Mann Bridge is a 10-lane, 65 m-wide cable-stayed bridge that spans a total of 2020 metres. It is located in British Columbia, Canada, and it is one of the largest cable-stayed bridges in North America as of 2015. By spanning across the Fraser River, the Port Mann Bridge connects the two cities of Surrey and Coquitlam. Figure 2.1 displays the location of the Port Mann Bridge, and Figure 2.2 shows the elevation view. The original Port Mann Bridge was a four-lane steel arch bridge which opened in 1964. It became the most travelled bridge in western Canada, crossed by over 120,000 cars a day. With such a high demand, an extra lane was added and a seismic upgrade was carried out. But only five years later the government decided to replace the bridge entirely and demolish the older one. The new Port Mann Bridge was completed in 2012.   Figure 2.1: Satellite Image Showing Location of Port Mann Bridge 19   Figure 2.2: Elevation View of the Port Mann Bridge (Looking South)  2.1 Structural Components The Port Mann Bridge has three sections: a cable-stayed main span and two approaches. Appendix A gives the general plan and elevation of the entire Port Mann Bridge, and a simplified elevation view is shown in Figure 2.3.    Figure 2.3: Elevation of Port Mann Bridge Showing Pier Labels 20  The cable-stayed span is 850 m long and is made of steel girders and cross beams which support precast concrete deck panels. Two separate decks make up the main span, and they are connected by median struts, as shown in Figure 2.4. In total, there are 288 cables used to connect two 160 m-tall piers to the deck. Each cable has its own properties that vary from one another.  Figure 2.4: Struts Connecting East and West Decks in Main Span  The north and south approaches consist of three concrete box girder sections which can be seen in Figure 2.5. On many of the approach piers there are viscous dampers installed, shown in Figure 2.6. The foundations for each of the piers consist of steel piles of 1.8 m diameter with reinforced concrete.  Figure 2.5: Cross Section of Concrete Box Girders in Approach Spans  21   Figure 2.6: Typical Dampers at Approach Piers  2.2 Finite Element Model The original finite element model was created by the structural designers using the software ADINA version 8.5.5 (ADINA R & D, Inc. 2015), as shown in Figure 2.7. The model is very complex and includes nonlinear properties for the bridge structure and its foundations. For the purpose of this research, only the main span was considered, and only linear properties were used.  Figure 2.7: Port Mann Bridge Model in ADINA Finite Element Software  In order to set up the model for updating, it needed to be transferred into the finite element program SAP2000 because this program allows more flexibility with editing properties, and it is also compatible with the FEMTools program (Dynamic Design Solutions NV 2014) that will be 22  used in the later chapter of this research. Some properties in the ADINA model were not compatible with SAP2000, so they had to be manually recreated. The ADINA model used a value of zero for the mass density of most of the material properties and applies an appropriate dead load to compensate. It was decided to add mass density to all material properties in the SAP2000 model because the original masses were not easily transferable from ADINA to SAP2000. The final converted model, with only the main span, is shown in Figure 2.8.  Figure 2.8: Port Mann Bridge Model in SAP2000 Finite Element Software  In summary, the following changes were made to the original ADINA model:  Removed approach spans  Removed foundations and simplified with “rigid” springs  Created cross sections for piers N2, S2, N1, and S1  Added mass density to material properties  23  2.2.1 Model Components The model consists of many general cross sections with predefined properties. This means that specific cross section geometry is not defined in the model, but instead the section properties such as moment of inertia and cross sectional area are entered. The exceptions to these general cross sections are the four piers that had to be reconstructed as they could not be transferred between the two programs. The concrete for all sections were assumed uncracked with varying values of modulus of elasticity depending on the specified compressive strength in the design drawings. These values were assumed uncracked because they will be compared to ambient vibration measurements which consist of deflections of a very minimal level, and therefore most likely not cracked. This should be kept in mind for any future stress analyses, and an effective cracked section may need to be applied.  The main components of the model include:  Foundations  North and south approach piers (N2 and S2, respectively)  North and south main pylons supporting the cables (N1 and S1)  Deck  Cables  In the original ADINA model, there was a significant amount of work that went into modelling the foundations. The detail to which the foundations were modelled was outside the scope of the research, so they were simplified into a point spring, as shown in Figure 2.9. The spring can be assigned its own six-by-six stiffness matrix, but for a baseline, the spring stiffnesses are set to be 24  very high to approximate a rigid foundation. It is the intent that in the model updating process, the validity of the stiffnesses will be investigated.  Figure 2.9: Modelling of Foundations  The two approach piers of the main span are modelled as shown in Figure 2.10, with each colour depicting a different cross section. The elements are frame elements with cross section and material properties according to the design drawings. Rising up from the foundation springs are rigid links that reach a horizontal cross section representing the footing. The three piers then extend up into the cap beam. Attached to the cap beam are the two deck components. A summary of the key properties of the N2 and S2 sections are shown in Table 2.1.  25   Figure 2.10: Modelling of Pier N2 Compared to Design Drawing  Table 2.1: Key Values Used for Modelling Piers N2 and S2 Component I2 (m4) I3 (m4) Cross-Sectional Area, A (m2) E (MPa) Mass Density, ρ (kg/m3) Poisson’s Ratio, ν N2 Columns 47.8 18.4 12 29940.4 2.30 0.17 N2 Infilled 63.0 21.4 21 N2 Cap Beam A 89.0 34.4 25.9 N2 Cap Beam B 70.3 20.3 21.3 S2 Columns 14.9 37.0 8.5 S2 Cap Beam A 85.4 31.0 24.9 S2 Cap Beam B 70.3 20.3 21.3  The main pylons, Pier N1 and S1, consist of foundation springs, the lower, transition, middle, and upper pylon sections, and a post-tensioned stabilizer beam. The modelled pylon is shown in Figure 2.11, with each colour depicting a different section property. Each element is a frame element with properties that are based on the design drawings. Notice that the stabilizer beam 26  consists of 12 different properties on each side, which are represented by a different colour, and they correspond to each precast concrete joint that makes up the stabilizer beam.    Figure 2.11: Modelling of Pier N1 Compared to Design Drawing  Table 2.2: Range of Key Values Used for Modelling Piers N1 and S1 Component I2 (m4) I3 (m4) Cross-Sectional Area, A (m2) E (MPa) Density, ρ (kg/m3) Poisson’s Ratio, ν N1 and S1 Pedestal 1620.0 740.0 60.0 33474.4 2.30 0.17 N1 and S1 Lower 773.3 573.3 40.0 33474.4 N1 and S1 Transition 1408.0 928.0 96.0 36666.0 N1 and S1 Middle 112.0 184.0 24.0 36666.0 N1 and S1 Upper 197.5 258.0 30.0 36666.0 N1 and S1 Stabilizer (Outer – Inner) 5.6 - 14.1 6.8 - 106.1 5.1 - 13.5 29940.4 27   A typical view of the modelled deck is depicted in Figure 2.12, with each colour representing a different section property. There are transverse beam sections supported on the longitudinal girders spanning the length of the bridge. The two inner longitudinal elements represent the floor beam stiffeners.  Figure 2.12: Modelling of Deck Components  Table 2.3: Key Values for Modelling Deck Components Component I2 (m4) I3 (m4) Cross-Sectional Area, A (m2) E (MPa) Mass Density, ρ (kg/m3) Poisson’s Ratio, ν Girders (Outer – Mid-span) 15.375 – 15.374 0.580 – 0.306 0.671 – 0.635 1.998E+05 7.85 0.29 Outer Floor Beam 2.546 0.045 0.211 Interior Floor Beam 2.546 0.060 0.214 Centre Floor Beam 2.546 0.069 0.216 Floor Beam Bracing 0.00018 0.00031 0.017  28  Lastly, the cables are modelled as frame elements using their appropriate cross section and material properties. The cross section for each cable is based off the number of steel strands specified in the design drawings, and each cable material has a slightly varying modulus of elasticity. The frame elements have moment releases on both ends and a torsional release on one end in order to simulate truss behaviour. This assumption was made for cable behaviour because in general practice, the simple truss element is acceptable (Chung and Shuqing 2015). However, alternative modelling techniques for the cables should be explored in the future. In total, there are 288 cables on the bridge. This includes 272 cables anchored to the deck and 16 cables anchored into the stabilizer beams at piers N1 and S1. For the stabilizer cables, there are two closely-spaced cables per anchor location, so the finite element model simplifies these by combining two stabilizer cables into one truss element. For this reason, the finite element model contains 280 cables. The elastic modulus for each cable varies from 1.86E+05 MPa to 1.93E+05 MPa, and the cross-sectional area varies from 3450 mm2 to 10,950 mm2 based on the design drawings.  2.2.1.1 Boundary Conditions At piers N2 and S2, there are four tie-downs per pier that restrict vertical and transverse translation, but allow for rotation and longitudinal translation, as shown in Figure 2.13 (a). These tie-downs are connected to the edge girders of each deck. Along with the tie-downs, there are transverse restraints at the centre of each deck, which can be seen in Figure 2.13 (b). The transverse restraints consist of restraint blocks attached to two multi-directional disk bearings, which allow rotation about the transverse axis. The tie-downs and shear keys are modelled using 29  short, thick steel members with member releases for rotation about the transverse axis, and they finite element model depiction is shown in Figure 2.13 (c).   (a) Tie-Down Elevation (Facing Side)   (b) Transverse Restraint Elevation (Facing Front)  (c) Tie-Down and Transverse Restraint Locations in Piers N2 and S2 Figure 2.13: Modelling of Tie-Downs and Transverse Restraints Compared to Design Drawings  Similar to the end piers, N1 and S1 pylons have bearings which provide restraints in the transverse and vertical directions. The deck is supported on the main tower and stabilizer beam with disk bearings which allow rotation, and restrained transversally with twin disk bearings. The locations of the bearings in the model are shown in Figure 2.14. 30   Figure 2.14: Modelling of Pier N1 and S1 Bearings  Also located on pier S1 only are longitudinal restraint brackets and bearings, which can be seen in Figure 2.15.  Figure 2.15: Modelling of Pier S1 Longitudinal Restraint Bearings  2.3 Summary of Background The Port Mann Bridge, one of the widest bridges in the world, and the case study for this research, was introduced with a brief background to its history. The finite element model, which 31  was originally created by the design engineers, was taken and modified to be transferred into SAP2000. The main structural components of the Port Mann Bridge are replicated in a finite element model. The unique aspects of the model are detailed which describe how the piers, decks, foundations, cables, and connections are modelled. Specific assumptions and simplifications are made for this study: only the main span of the Port Mann Bridge is modelled, and there will be only linear properties defined. With an understanding of the bridge structure and the availability of a finite element model, the next step is to determine the dynamic behaviour from an operational modal analysis and from the finite element model.  32  Chapter 3: Experimental and Analytical Results In this chapter, the results from the experimental data available are analyzed, and a modal analysis is carried out to identify the main mode shapes of the Port Mann Bridge. Following that, the finite element model is analyzed and the first 13 mode shapes are delivered.  3.1 Experimental Modal Analysis There are three data sets that can be used to carry out an operational modal analysis for the main span of the Port Mann Bridge. The first two sets are from ambient vibration tests conducted in September and October, 2012. They were conducted on separate dates because access was restricted on the east side. On September 5 and 6, 2012, the west deck of the main span was measured while the bridge was still being completed, and it was not open to traffic. At this point, the sidewalk had been constructed on the east deck, but the counterweights had not been installed on the west deck, so this should be taken into account in the analysis. On October 10, 2012, the east deck of the main span was measured, and this time there was traffic running along the west deck. The third set of data was collected from the sensors installed on the Port Mann Bridge as part of its structural health monitoring system. This set contains important points along the bridge for capturing dynamic behaviour, but the setup is not as dense as the setups collected from the first two sets. Together, all three data sets can be used to compare dynamic behaviour under different operating conditions as follows: 1. No traffic and no counterweight installed on west side. 2. Traffic only on the east side and counterweight installed on west side. 3. Bridge fully opened to traffic along both decks.  33  In the end, all three data sets were utilized gain an understanding of the full dynamic behaviour of the bridge. For example, the first two data sets contain information on behaviour at the end piers and at the main towers; however, because the tests were conducted at two different times under different operating conditions, they did not encapsulate the behaviour of the structure as a whole. For this reason, the third data set, taken from the structural health monitoring sensors, was used to capture the as-built, final operating behaviour of the entire main span.  3.1.1 Field-Acquired Data The first set of measurements was conducted on the west deck during the days of September 5 and 6, 2012. The second set of measurements was for the east deck on October 10, 2015. Figure 3.1 shows the sensor locations for the setups on both dates. Each blue dot represents a sensor position for the test on the east deck on October 10, and each red dot represents a sensor position for the test on the west deck on September 5 and 6. For a comprehensive summary of sensor placements and setups, see Appendix B.  Figure 3.1: Sensor Setup Locations for Field Testing in 2012  As previously mentioned, the west deck was measured when there was no counterweight installed, so there is expected to be a more mass on the east deck due to the sidewalk. The east deck was measured while there was traffic running along the west deck, which will need to be 34  taken into account when analyzing the dynamic behaviour. The sensors used for data collection were high resolution Tromino sensors (Micromed S.p.A. 2015). These sensors can record high gain velocity, low gain velocity, and acceleration in all three orthogonal directions and synchronized via GPS. Measurements were taken over a period of 30 minutes with a sampling rate of 128 samples per second. The reference sensor was placed at the very centre of the main span on the inner west deck for both days, and roving sensors were placed starting at the south end and sequentially moving across the deck in multiple setups. Each sensor was placed at an intersection of a cable with the bridge deck, which led to an average of 6 m spacing between sensors. The data was later extracted and synchronized using the Tromino’s accompanying software, Grilla (Micromed S.p.A. 2015). The modal analysis was carried out using the software ARTeMIS Extractor version 4.1 (Structural Vibration Solutions, Inc. 2015).  The initial intention when taking measurements along both bridge decks was to combine all of the data to obtain the dynamic behaviour for the whole bridge. However, because of access restrictions resulting in the measurements being taken a month apart, the operating conditions were not the same between the two tests. Therefore, it was decided to carry out the analysis for each deck separately and compare them.  A total of 13 modes were identified using the frequency domain decomposition (FDD) method. The summary of these modes are shown in Table 3.1. From the results, it can be seen that the frequencies in the east deck are all lower than the west deck. This can be explained by the extra mass associated with the sidewalk when compared with the west deck which did not have the counterweight installed. The second notable difference is the transverse motion associated with 35  the torsional modes in the east deck. This can be attributed to the continuous flow of traffic moving along the west deck; the traffic induces transverse motion and there was no traffic on the east deck to balance out this behaviour.  Table 3.1: ARTeMIS Modal Analysis Results for Field-Acquired Data  WEST DECK EAST DECK Mode Frequency [Hz] Damping Ratio (%) Description Frequency [Hz] Damping Ratio (%) Description 1 0.244 1.75 1st Vertical 0.231 1.86 1st Vertical 2 0.250 1.23 1st Torsional 0.244 1.66 Torsional-Transverse 3 0.313 1.26 2nd Vertical 0.300 1.78 2nd Vertical 4 0.350 1.63 Torsional Symmetric 0.338 1.61 Torsional-Transverse 5 0.463 1.02 Torsional Antisymmetric 0.444 0.97 Torsional-Transverse 6 0.500 1.44 3rd Vertical 0.475 0.99 3rd Vertical 7 0.556 1.04 3rd Vertical Antisymmetric 0.531 0.90 3rd Vertical Antisymmetric 8 0.594 0.87 3rd Vertical Symmetric 0.569 0.91 3rd Vertical Symmetric 9 0.650 0.81 Torsional 0.625 0.85 Torsional 10 0.788 0.68 Vertical 0.750 0.66 Vertical 11 0.850 0.74 Vertical 0.800 0.70 Vertical 12 0.969 0.51 Torsional 0.931 0.90 Torsional 13 1.063 0.57 Vertical 1.019 0.71 Vertical  The first six mode shapes are shown in Figure 3.2. For the torsional mode shapes, the two decks are out of phase with each other because they were analyzed separately and their measurements were taken at different times. The transverse motions for the east deck torsional modes are also apparent.  36  Because the test network is quite dense, it is possible to investigate behaviour at important points along the bridge which will aid in configuring the finite element model. In this case, the boundary conditions at the ends of the main span as well as the behaviour of the deck at the two main towers are observed. From the vertical and torsional mode shapes, it can be seen that the end points of the main span are not fixed. This is expected as the deck is pin connected to piers N2 and S2, so it can be assumed that the movement at the ends of the main span is due to the movement in the end piers. When observing the behaviour at the main towers N1 and S1, it is found that for the first torsional mode shape there is a generally uniform movement of the deck over all three spans, even over towers N1 and S1, meaning that the towers either sway with the deck, or there are uncertain boundary conditions at this point. The remaining torsional modes show fixity at the towers, with torsional behaviour being isolated to each of the three spans.   37    (a) 1st Vertical (0.24/0.23 Hz) (b) 1st Torsional (0.25/0.24 Hz)   (c) 2nd Vertical (0.31/0.30 Hz) (d) Torsional Symmetric (0.35/0.34 Hz)   (e) Torsional Antisymmetric (0.46/0.44 Hz) (f) 3rd Vertical (0.50/0.48 Hz) Figure 3.2: First Six Mode Shapes from Field-Acquired Data for the Main Span of Port Mann Bridge 38  3.1.2 Structural Health Monitoring Data The Port Mann Bridge is part of a health monitoring network called BC Smart Infrastructure Monitoring System (BCSIMS). As a result, it is equipped with various sensors that are continuously recording data and transferring it to servers for interested parties to make use of. Acceleration data was downloaded from these sensors and a modal analysis was carried out in order to experimentally determine the dynamic properties of the bridge. The locations of the sensors can be seen in Figure 3.3. The sensor channels and geometry were then created in ARTeMIS for modal analysis purposes, as shown in Figure 3.4, which also denotes the orientation of each channel analyzed. For a full diagram of the channel numbers and orientations used, see Appendix C.  Figure 3.3: Sensor Locations used for Modal Analysis 39   Figure 3.4: Geometry and Channel Orientations for ARTeMIS Setup  The data received from the sensors was acceleration values at a sampling rate of 200 samples per second, and it was taken over a period of 40 minutes.. The data was then imported into ARTeMIS with a sampling decimation of 100 and a spectral density using 512 frequency lines in order to capture the closely-spaced long-period modes. The two methods of system identification used were enhanced frequency domain decomposition (EFDD) and stochastic subspace identification (SSI). The singular value decomposition plot is displayed in Figure 3.5 for FDD peak picking, and the stabilization diagram for the SSI method is shown in Figure 3.6. 40   Figure 3.5: Frequency Domain Decomposition Peak Picking Plot   Figure 3.6: Example Stabilization Diagram for SSI Method  41  The final mode shapes are presented in Table 3.2. Not as many mode shapes could be identified with the SHM data because there are only sensors installed at five locations along the deck, whereas field data was measured at 32 locations. Table 3.2: ARTeMIS Modal Analysis Results for SHM-Acquired Data Mode Number Frequency (Hz) Damping Ratio (%) West Description 1 0.233 0.68 1st Vertical 2 0.251 0.84 1st Torsional 3 0.272 0.64 2nd Torsional 4 0.302 0.66 2nd Vertical 5 0.432 1.14 Mid-span Torsion 6 0.478 0.62 3rd Vertical 7 0.532 0.68 Vertical 8 0.572 0.53 3rd Vertical Symmetric  The first six mode shapes are illustrated in Figure 3.7. With the SHM-acquired data, several points can be observed which were not accessible in the field testing. The SHM data contains information on the towers and foundations, which can provide very useful information when assessing the finite element model. There are also two notable areas which do not have data points, and that is the ends of the main span at piers N2 and S2, and the two decks at the N1 and S1 towers. The advantage of the SHM data is that there is information on the behaviour of the towers and foundations as well as the deck. With this it is possible to have a general idea of the movement of the main towers in combination with the deck movement. The SHM data is also the most current set of data, whereas the field-acquired data was taken during final stages of construction with only one deck open to traffic.   42    (a) 1st Vertical (0.23 Hz) (b) 1st Torsional (0.25 Hz)   (c) 2nd Torsional (0.27 Hz) (d) 2nd Vertical (0.30 Hz)   (e) 3rd Torsional (0.43 Hz) (f) 3rd Vertical (0.48 Hz) Figure 3.7: First Six Mode Shapes for SHM-Acquired Data for Main Span of Port Mann Bridge 43  The SSI method was used to validate the results from the EFDD technique. All mode shapes correlated well between the two methods, with the exception of the seventh mode, which could not be identified clearly in the SSI method. The modal assurance criterion (MAC) values were used to correlate the mode shapes and validate the results. Overall, a value of 0.9 to 1.0 was achieved, as shown in the MAC value matrix in Figure 3.8. The seventh vertical mode had a MAC value of only 0.51, so there is less confidence in this mode shape.  Figure 3.8: MAC Value Matrix Comparing SSI Modes vs. EFDD Modes  3.1.2.1 Footing Movement The model includes four sensors at each of the footings of piers N1 and S1. Unfortunately the data associated with the footings could not be used because it was unclear how the channels were assigned in the structure based on the data provided, so it was decided to omit the analysis. In the future, it is of interest to investigate the behaviour at these areas which infer characteristics of the 44  foundation that can be useful in model updating processes. Specifically, it is of interest to see how “rigid” each foundation is under operating conditions.  3.2 Analytical Results The finite element model was created in SAP2000 based on design drawings and the designing team’s ADINA model. A modal analysis was carried out and the results for 13 mode shapes are displayed in Table 3.3. For a summary of the original ADINA model results, see the tables and figures in Appendix D, where it also compares the mode shapes to the converted SAP2000 mode shapes. Table 3.3: Modal Analysis Results from Original SAP2000 Model Mode Number Frequency West Description 1 0.263 1st Vertical 2 0.274 1st Torsional 3 0.298 2nd Torsional 4 0.329 2nd Vertical 5 0.431 Vertical-Transverse 6 0.494 Vertical-Transverse 7 0.531 Vertical 8 0.541 Vertical 9 0.557 Mid-span Torsional East 10 0.563 Mid-span Torsional West 11 0.597 Vertical 12 0.641 Vertical 13 0.645 Vertical    45    (a) 1st Vertical (0.26 Hz) (b) 1st Torsional (0.27 Hz)   (c) 2nd Torsional (0.30 Hz) (d) 2nd Vertical (0.33 Hz)   (e) Vertical-Transverse (0.43 Hz) (f) Vertical-Transverse (0.49 Hz) Figure 3.9: First Six Mode Shapes for SAP2000 Model of the Port Mann Bridge     46  3.3 Summary of Experimental and Analytical Results Three sets of experimental data were looked at: two from field-measured data and one from the mounted sensors as part of the structural health monitoring network. All three experimental results are summarized in Table 3.4. The field data managed to identify six modes that could not be found with the SHM data, whereas one mode, the 2nd torsional mode, could be identified only by the SHM data. Table 3.4: Summary of Three Experimental Ambient Vibration Analysis Results Mode West Deck Freq. [Hz] East Deck Freq. [Hz] SHM Data Freq. [Hz] Description 1 0.244 0.231 0.233 1st Vertical 2 0.25 0.244 0.251 1st Torsional  -- -- 0.272 2nd Torsional 3 0.313 0.3 0.302 2nd Vertical 4 0.35 0.338 -- Vertical-Transverse 5 0.463 0.444 0.432 Mid-span Torsion 6 0.5 0.475 0.478 3rd Vertical 7 0.556 0.531 0.532 3rd Vertical Antisymmetric 8 0.594 0.569 0.572 3rd Vertical Symmetric 9 0.65 0.625 -- Torsional 10 0.788 0.75 -- Vertical 11 0.85 0.8 -- Vertical 12 0.969 0.931 -- Torsional 13 1.063 1.019 -- Vertical  The field data identified 13 different mode shapes, but revealed inconsistencies due to the measurements being taken a month apart and being under different operating conditions. For the first and second set of field data, the fundamental frequency was 0.244 and 0.230 Hz, respectively; and damping values for the field data ranged from approximately 0.5% to 1.8%. However, the field data still provides a more robust layout of measurements, so it enables a larger amount of mode shapes to be identified than the SHM data. The structural health 47  monitoring data is consistent and gives clear results. The fundamental frequency for the SHM data was found to be 0.233 Hz, and the damping values of all mode shapes ranged from 0.5% to 1.1%. Unfortunately the data does not provide information at the ends of the bridge, and the foundation data was omitted. It was decided to use the eight mode shapes obtained from the SHM data as the target to which the finite element model will be calibrated. The analytical model results are given up to the 13th mode shape, with the fundamental mode at 0.263 Hz.  The next step in the research is to utilize the experimental and analytical results to carry out the model updating process. The mode shapes from both models will be correlated, a sensitivity analysis carried out, and parameter modifications will be made.  48  Chapter 4: Model Updating The experimental and analytical data have been analyzed, so it is now possible to correlate the data and update the model. First the data is to be compared by checking the difference between frequencies and using the modal assurance criterion. Then a set of possible parameters will be explored and justified based on uncertainty and sensitivity to the dynamic results. The parameters will then be modified manually to obtain an updated model that correlates well with the experimental data.  4.1 Initial Model Correlation FEMTools (Dynamic Design Solutions NV 2014), a commercial product, was used to correlate the finite element model and test model to compare frequencies and mode shapes. The SAP2000 model and the experimental results were read by FEMTools, and the data was correlated using a MAC value threshold. All eight experimental mode shapes were paired with their equivalent finite element modes. Figure 4.1 shows all eight mode shape pairs, with the finite element model in blue superimposed on top of the test model in red. The correlated results are shown in Table 4.1.  It appears that most frequencies in the finite element model are larger than the test model. This could signify a global mass and/or stiffness difference. The most logical reason for this overestimation would be due to a lack of mass. The finite element model accounts for the steel beams, girders, and stiffeners in the deck of the main span, but it does not include the contribution of the concrete panels and asphalt pavement, which will add extra mass and transverse stiffness.  49    (a) 1st Vertical (0.233/0.233 Hz) (b) 1st Torsional (0.253/0.251 Hz)   (c) 2nd Torsional (0.274/0.272 Hz) (d) 2nd Vertical (0.294/0.302 Hz)   (e) Torsional (0.401/0.432 Hz) (f) Vertical (0.475/0.478 Hz)  (g) Vertical (0.535/0.532 Hz)  (h) Vertical (0.579/0.572 Hz) Figure 4.1: Correlated Mode Shapes for FEM and Test Model  50  Table 4.1: Mode Shape Pairs for Original SAP2000 Model and Experimental Data FEA Mode Analytical Freq. (Hz) Test Mode Test Freq. (Hz) Diff. (%) MAC (%) 1 0.263 1 0.233 12.59 99.0 2 0.275 2 0.251 9.31 80.7 3 0.298 3 0.272 9.56 86.2 4 0.329 4 0.302 8.85 97.5 5 0.431 5 0.432 -0.28 64.7 6 0.494 5 0.432 14.40 44.5 7 0.531 6 0.478 11.10 99.3 11 0.597 7 0.532 12.13 97.7 13 0.645 8 0.572 12.79 94.0    AVG = 10.11 84.8  As can be seen in the table, the MAC values are quite high, ranging from 86 to 99%, with the exception of the fifth mode which only has a MAC of 64.3% or possibly 44.5%. The fifth and sixth analytical mode shapes are very similar, with the sixth mode having more relative transverse motion. It was decided to monitor both mode shapes during the updating process. The frequency differences range from 0.3% to 14%. From the table, it can be concluded that generally the mode shapes are in good form, but the difference in frequencies needs to be adjusted. A disparity in frequency, but not MAC, generally indicates a global stiffness or mass discrepancy in the model. One thing that should be kept in mind is that the data from the SHM network contains a limited amount of points on the deck with which to measure the MAC correlation, so a high MAC value only indicates a good correlation at those nodes. It is possible that the behaviour between the nodes is different between the experimental and analytical models. An example demonstrating this concept is shown in Figure 4.2. The test model, with its limited data along the deck, could not capture the vertical mode behaviour between the nodes, 51  but when superimposed with the finite element model, it becomes clear what experimental mode was obtained.  Figure 4.2: Example of Excellent Correlation Despite Limited Node Spacing  4.2 Accounting for Deck Weight As mentioned previously, the model lacks information about the concrete deck panels and asphalt, which contribute to mass and stiffness of the whole deck. Therefore, this was decided as a good starting point for the manual fine tuning of the model. Based on design drawings, an initial value of concrete weight was calculated and added to the model as a uniformly distributed load on the cross beams of the deck (see Figure 4.3). This value was then modified further, based on trial and error, until a final value of 12.25 kN/m was reached.  Figure 4.3: Example Distributed Load Added to Simulate Deck Weight 52  The new results after adding the deck weight are shown in Table 4.2.  Table 4.2: Mode Shape Pairs after Adding Deck Weight to SAP2000 Model FEA Mode FEA Freq. (Hz) Test Mode Test Freq. (Hz) Diff. (%) MAC (%) 1 0.235 1 0.233 0.61 99.0 2 0.267 2 0.251 6.47 82.7 3 0.291 3 0.272 6.75 88.5 4 0.296 4 0.302 -2.26 97.7 5 0.384 5 0.432 -11.1 63.4 6 0.440 5 0.432 2.01 46.1 7 0.472 6 0.478 -1.17 99.4 11 0.531 7 0.532 -0.28 97.6 13 0.572 8 0.572 -0.03 93.7    ABS AVG = 2.45 85.3  The average MAC value stayed the same, as expected; and the average absolute frequency difference was reduced to 2.45%. The second, third, and fifth mode shapes, however, are still quite different between the experimental and analytical models. With the deck weight added, the sixth analytical mode shape has reached a closer frequency to the fifth experimental mode shape, whereas originally it was closer to analytical mode 5. Therefore, analytical mode 6 will be taken as the pair of experimental mode 5.  4.3 Sensitivity Analysis It appears the first and second torsional modes have a lower MAC value and higher frequency difference than most of the vertical modes. The discrepancy in the first and second torsional modes is also visible at the motion of the main towers. By observing the superimposed fifth mode shape of the two models, it is clear that there is significant transverse motion in the finite element model, but not in the test model (see Figure 4.4). 53   Figure 4.4: Mode 5 Showing Transverse Motion in FEM but not Experimental Model  It can be seen that the finite element model is not quite capturing the torsional motion exhibited in the experimental data. The transverse deck motion can be partially attributed to the fact that the concrete deck panels have not been modelled. Including the concrete panels in the model as shell elements would overcomplicate the model and could provide more sources of error than it could be fixing, so in this case it was decided to run a sensitivity analysis to determine what effects small perturbations in various parameters have on the three main torsional modes.  Aside from the three torsional modes, there is still a difference in frequency for the vertical modes 4 and 6. The MAC values are very high, which signifies that the frequency difference is most likely not at a component level, but rather a global discrepancy in a parameter value. These vertical modes were included in the sensitivity analysis to help calibrate frequency values even further.  54  To carry out a sensitivity analysis, the model needed to be broken down into sets based on what properties may vary or be unknown. The main material properties that are of interest are the concrete used in the end piers and main towers, and the steel used for the floor beams and the edge girders. Using FEMTools, the sensitivity is calculated for changes in the elastic modulus and moment of inertia of the main structural components. The resulting sensitivity matrix is shown in Figure 4.5. It shows the six responses (frequency values for modes 2 to 6) plotted against the 25 parameters (E, I1, I2, and I3 for each group of structural components).  Figure 4.5: Sensitivity Matrix for E and I against Frequency Values for Modes 2-6  The sensitivity plot reveals that E, I1, and I2 for the main towers and the edge girders have the largest effect on modes 2 to 6. Because the design documentation states a higher concrete compressive strength for the piers above the deck than the piers below the deck, it is expected 55  that the elastic modulus reflect this difference too. The sensitivity matrix will be used to make the necessary steps in manual model updating.   4.3.1 Sensitivity of Supports One important consideration in modelling large structures is how the foundations are modelled, and if the supports can realistically be represented as “fixed.” Supports can be modelled using a set of springs to represent the stiffness in each direction, and to do this requires geological data and adoption of a theoretical stiffness calculation method. As a preliminary estimation, the calculation method proposed by Vijayvergiya (1977) for pile stiffness was adopted. Material from Lam, Martin and Imbsen (1991) and Mosher and Dawkins (2000) were also used in the theoretical calculations to get an idea of how stiff the foundations were; however, limited information was available so many assumptions were made. For a complete set of pile calculations, see Appendix E. Table 4.3 and Table 4.4 show the symmetrical stiffness matrices which have been roughly calculated based on the aforementioned procedure. Table 4.3: Symmetric Stiffness Matrix for Piers N1 and S1  u1 u2 u3 r1 r2 r3 u1 1.10E+07 kN/m 0 0 0 5.60E+07 kN 0 u2 0 1.10E+07 kN/m 0 5.60E+07 kN 0 0 u3 0 0  1.88E+09 kN/m 0 0 0 r1 0 5.60E+07 kN 0 3.19E+11 mkN/rad 0 0 r2 5.60E+07 kN 0 0 0 3.19E+11 mkN/rad 0 r3 0 0 0 0 0 1.87E+09 mkN/rad   56  Table 4.4: Symmetric Stiffness Matrix for Piers N2 and S2   u1 u2 u3 r1 r2 r3 u1 700507 kN/m 0 0 0 3.56E+06 kN 0 u2 0 700507 kN/m 0 3.56E+06 kN 0 0 u3 0 0 1.19E+08 kN/m 0 0 0 r1 0 3.56E+06 kN 0 3.84E+09 m*kN/rad 0 0 r2 3.56E+06 kN 0 0 0 3.84E+09 m*kN/rad 0 r3 0 0 0 0 0 2.24E+07 m*kN/rad  From the above tables, it is clear that the foundations are extremely stiff and almost rigid. In order to validate the rigid foundation assumption, the finite element model with the extra deck mass was analyzed by assigning springs that have a six-by-six stiffness matrix with the calculated values. The model with the original “rigid” springs is compared with the model with the updated spring stiffnesses based on the above tables, and the results are shown in Table 4.5. Table 4.5: Mode Shape Difference when Theoretical Pile Stiffness Used FEA Mode Original Rigid Model Model with Modified Support Stiffness  Diff. (%) Freq. (Hz) Freq. (Hz) 1 0.235 0.233 -0.16 2 0.267 0.251 -1.05 3 0.291 0.272 -0.64 4 0.296 0.302 -0.17 6 0.440 0.432 -0.76 7 0.472 0.478 -0.06 11 0.531 0.532 -0.10 13 0.572 0.572 -0.72   AVG = -0.46  57  The table shows that using the theoretical pile group stiffness has a minimal effect on the frequency of the mode shapes, with mode 2 being the largest change at 1%. On average, the frequencies were reduced by 0.46% when the calculated stiffness values were used. Based on these results and the experimental data from the structural health monitoring network, it was decided to use the original model with “rigid” springs.  4.3.2 Sensitivity of Boundary Conditions Often discrepancies between finite element models and test models are largely due to improper modelling of boundary conditions. Sometimes a “pinned” connection will still impose a certain amount of rotational stiffness, or an intended “fixed” connection will still exhibit some level of freedom.  4.3.2.1 Piers N2 and S2 Connections At piers N2 and S2, there are tie-downs and transverse restraints that are assumed to act as pinned connections. Judging by the connection of the tie-downs as described in Section 2.2.1.1, there is a high level of confidence that the behaviour will truly act as a pinned connection. However, the connection also allows a longitudinal displacement of approximately 0.5 m in both directions, and there may also be friction in the rotation causing some rotational stiffness. Both of these scenarios and their effects are investigated as part of the sensitivity analysis by comparing the change in frequency to the original SAP2000 model with the modified deck mass.    58  4.3.2.1.1 Effect of Longitudinal Release in N2 and S2 Tie-Downs As mentioned previously, the design of the tie-down is intended to permit a total of approximately 1 m displacement. In order to investigate this effect, a shear release was added to one side of each tie-down element in the model, and the modal analysis was carried out. Table 4.6 summarizes the new results. Table 4.6: Mode Shape Pairs after Adding Longitudinal Release in N2 and S2 Tie-Downs FEA Mode Original Model Model with Longitudinal Tie-Down Release  Diff. (%) Freq. (Hz) Freq. (Hz) 1 0.235 0.229 -2.62 2 0.267 0.265 -0.94 3 0.290 0.288 -0.87 4 0.296 0.286 -3.08 6 0.440 0.439 -0.36 7 0.472 0.467 -0.97 11 0.531 0.524 -1.24 13 0.572 0.568 -0.67   AVG = -1.34  By adding the releases to simulate the longitudinal deck movement allowance, the mode shape frequencies were changed by as much as 3.1%. All frequencies were slightly decreased, which is expected because the longitudinal release will reduce the stiffness. The most affected modes are the first and second vertical modes, with a difference of -2.6% and -3.1% respectively. It should be noted that the higher vertical modes were changed by a much smaller percent, which means a larger longitudinal displacement demand is put on the first two vertical modes at the end piers. On average, the frequencies were reduced by 1.3% with the longitudinal release at the tie-down.  59  If the shear releases are to be used in the model, it is important that the displacement at these locations be monitored during any future linear analyses, as the displacements could exceed that which is physically allowable by the tie-down. For nonlinear analyses, a gap-type nonlinear link element can be created which specifies the allowable displacement before applying a stiffness value.  4.3.2.1.2 Effect of Rotational Stiffness to N2 and S2 Tie-Downs The tie-downs at piers N2 and S2 are designed as pin connections, but it is possible that there is some rotational stiffness present. To examine the effect, two scenarios were analyzed: one where an arbitrary value of 1.0E+06 m*kN/rad rotational stiffness is added to the tie-downs and another where the tie-downs are fixed. The two scenarios are compared to the original model with pinned tie-downs.  Table 4.7 Mode Shape Pairs after Adding Arbitrary Moment Stiffness and Fixing Tie-Down Connections FEA Mode Original Model Model with Partial Moment Stiffness Diff. (%) Model with Fixed Tie-Downs Diff. (%) Freq. (Hz) Freq. (Hz) Freq. (Hz) 1 0.235 0.240 2.34 0.258 9.76 2 0.267 0.268 0.24 0.271 1.31 3 0.290 0.291 0.26 0.294 1.39 4 0.296 0.303 2.38 0.325 9.90 6 0.440 0.442 0.25 0.449 1.88 7 0.472 0.477 1.09 0.492 4.34 11 0.531 0.539 1.48 0.567 6.74 13 0.572 0.577 0.93 0.600 4.89  By adding a partial moment stiffness of 1.0E+06, the frequencies of the vertical mode shapes are changed by as much as 2.3%, and the torsional modes are changed by a negligible amount. The difference between a pinned and fixed tie-down connection at the end piers is quite significant. 60  The vertical modes are the most affected, with the first and second vertical modes being increased by 9.8% and 9.9% respectively.  4.3.2.2 Piers N1 and S1 Connections The connections of the deck to the main piers N1 and S1 are of interest, because they tend to have an important effect on the dynamic behaviour of the bridge. Pier N1 and S1 each consist of six vertical bearings that support the gravity load from the two decks. Each deck is supported by an inner bearing at the tower and two outer bearings that sit on the concrete stabilizer beam. To restrain the transverse movement of the deck, there are transverse bearing assemblies on the sides of the towers. These consist of two disk bearings per assembly. For pier S1 only, a longitudinal restraint assembly exists which also consists of disk bearings with brackets that attach along the edge girder.  To understand the behaviour that is expected at these connections, it is vital to have an understanding of the mechanics of disk bearings. Traditional multi-directional disk bearings consist of an elastomeric material which is sandwiched between two metal plates. The elastomeric material transfers the vertical loads and an internal metal dowel transfers the horizontal loads. Disk bearings are manufactured based on design compression loads, displacement, and rotation expected in the structure. Therefore, it is expected that each bearing allows a small amount of displacements. Currently, the bearings in the finite element model are modelled using frame releases for shear and moment in both lateral and longitudinal directions and also released for torsion. This way the bearing is permitted to rotate and displace in both directions. In actual construction, the bearing may not act perfectly as intended. For example, the 61  model includes shear releases for each bearing, but the metal dowel within the bearing would likely be transmitting shear forces from the deck to the stabilizer beam. The original frame end releases are shown in Figure 4.6, where they are defined by the frame’s local axis which is also displayed.  The effect of various bearing end releases will be investigated for the bearing components to determine sensitivity to overall dynamic behaviour. In the first scenario, the gravity-loaded bearings will be fixed in translation—they will no longer have shear releases. The second scenario involves fixing the transverse restraint bearings from rotation about the transverse axis (M3 in Figure 4.6). Because there are twin disk bearings installed for transverse restraints, it is possible that the two bearings act as force couples to resist rotation. The effects of these two modifications are summarized in Table 4.8.  Figure 4.6: Frame End Releases Originally Assigned to N1 and S1 Bearings   62  Table 4.8: Mode Shape Comparison for Two Separate Scenarios Modifying Pier N1 and S1 Bearings FEA Mode Original Model Fixed V2 and V3 Gravity Bearings Diff. (%) Fixed M3 Transv. Bearings Diff. (%) Freq. (Hz) Freq. (Hz) Freq. (Hz) 1 0.235 0.235 0.10 0.235 0.03 2 0.267 0.268 0.27 0.267 0.01 3 0.290 0.291 0.33 0.290 0.00 4 0.296 0.296 0.04 0.296 0.01 6 0.440 0.442 0.41 0.440 0.00 7 0.472 0.472 0.04 0.472 0.01 11 0.531 0.531 0.00 0.531 0.00 13 0.572 0.572 0.00 0.572 0.00   AVG = 0.15 AVG = 0.01  The changes to the gravity bearing and transverse bearing end releases show a very minimal effect on the overall dynamic behaviour of the structure. The three torsional mode shapes are most affected by fixing the shear releases in the bearings, but only by a small amount.  4.3.3 Sensitivity of Cable Properties The cables comprise a significant portion of the model, and therefore they require due consideration. The model contains 280 different cable sections with their own unique, slightly different, material properties. It is expected that the quality control for cable strands are very high, and as such there is more confidence in the material properties. The elastic modulus of each cable material property in the model varies from 186 GPa to 193 GPa. To investigate the sensitivity of the cable properties, the elastic modulus is varied in two scenarios by assigning a global value of 170 GPa in the first scenario, and 210 GPa in the second scenario. The results of the analysis are shown in Table 4.9.  63   Table 4.9: Mode Shape Pairs after Assigning Cable E Value of 170 GPa and 210 GPa FEA Mode Original Model Model with 170 GPa Cable E Diff. (%) Model with 210 GPa Cable E Diff. (%) Freq. (Hz) Freq. (Hz) Freq. (Hz) 1 0.235 0.226 -3.56 0.243 3.49 2 0.267 0.263 -1.54 0.271 1.20 3 0.290 0.285 -1.76 0.294 1.37 4 0.296 0.286 -3.17 0.304 2.95 6 0.440 0.434 -1.39 0.449 1.93 7 0.472 0.454 -3.77 0.489 3.65 11 0.531 0.509 -4.12 0.550 3.54 13 0.572 0.547 -4.34 0.592 3.61   AVG = -2.96  2.72  A decrease in cable modulus of elasticity results in a decrease in overall frequencies, and the opposite can be said for an increase in modulus of elasticity. An increase of modulus of elasticity of the cables by around 10% causes the vertical mode shapes to be stiffer by an average of 2.7%, and a decrease of 10% results in an average of 3% lower frequencies.  While modifications to the cable material properties do indeed cause a notable change in the frequencies of the mode shapes, it must be approached with significant engineering judgement. In the case of model updating, the properties of the cables can be considered to have a higher level of confidence than other model properties such as the concrete in the piers. This should be taken into consideration when deciding what parameters to include in the model updating process.  64  4.4 Selecting Parameters With a thorough sensitivity analysis carried out on the various properties of the model that can affect the dynamic behaviour, a decision must be made on how to approach the model updating process. FEMTools was used to measure the sensitivity of E, I1, I2, and I3 for the steel girders, steel floor beams, concrete towers N1 and S1, and concrete piers N2 and S2. Following that, a manual sensitivity analysis was conducted to measure the effect of stiffening or releasing the connections for the main towers N1 and S1 and the end piers N2 and S2; and finally, the variation in cable stiffness was analyzed for its overall effect on the dynamic properties.  The decision process involves including enough parameters to allow sufficient calibration for all applicable mode shapes, but not too many where the updating process becomes too convoluted and difficult to reach a solution. Judgements must also be made in determining what properties of the model are more confident than others in order to produce an updated model that is physically realistic.  The boundary conditions for piers N1 and S1 were shown to have little effect on the overall analysis results, but there is a notable difference when changes are made to the end piers N2 and S2. By adding shear releases at the tie-downs of piers N2 and S2, it relaxes the displacement demand at the end piers and results in a large decrease in frequency for the first and second vertical mode shapes. In accounting for this effect, the average differences in overall frequencies becomes more disproportionate than was originally determined; and this indicates that a rigorous component-level model updating approach may be necessary, causing convergence towards a sufficient calibrated model to become much harder to obtain.  65   Cable parameters, while having a large effect on the dynamic behaviour, are deemed to have more confidence than some of the other structural components, and each cable has its own individual parameter. Calibration will require either generalization of the cable properties as a whole, or individual element modifications, which would prove unrealistic or computationally inefficient.   For the concrete towers and piers, there is a larger degree of uncertainty when modelling these elements. There are many aspects of concrete members that may not be captured by the finite element modelling. For example, the reinforcement used in the concrete is not modelled, so the composite action between the two materials is not accounted for. There are also prestressing tendons used in the towers and pier caps which will provide more complex interaction. Other properties can affect the concrete piers such as degree of cracking present or modelling the correct effective length.   The steel girders and floor beams are an integral part of the deck. There can generally be a larger confidence in the detailing of steel components, but uncertainty still remains in modelling these elements, such as joint fixity at lap splices and interaction with the concrete deck panels. The modification of the moment of inertia of the steel girders can be considered as a means of compensating for the interaction between the concrete deck and the steel that is not modelled. But generally the moment of inertia of a steel section is considered accurate, so any parameter change in the moment of inertia should be limited to a small amount.   66  Finally, after the sensitivity analysis and careful consideration of the above factors, a decision was made on the following parameters to use.  E for main piers N1 and S1  I2 (about bridge longitudinal axis) for main piers N1 and S1  I2 (about vertical axis) for steel girders of the main deck  I3 (about bridge transverse axis) for steel girders  E for steel girders  It was originally intended to use FEMTools and its automatic updating tools, but there were problems encountered when trying to converge to a suitable calibrated model. The finite element model is significantly large so automatic calibration was not possible under the time constraints. Therefore, the model will be manually updated and will be carried out in two steps. The first step will focus on calibration of the parameters for piers N1 and S1, which largely affect the response of the second and third mode shapes. The second step will focus on calibrating the steel girders of the deck.  4.5 Step 1: Modifying N1 and S1 Tower Parameters For modes 2 and 3, motion is dominated by the transverse sway of the towers. The frequency of modes 2 and 3 are almost 7% higher than the experimental data, so the tower stiffness in the transverse direction will need to be adjusted. By varying E and I2 of the whole section of the tower, a closer correlation can be achieved, as shown in Table 4.10. The new parameter values ended up both being reduced by 10% to arrive at this result.   67  Table 4.10: Mode Shape Pairs after Reducing both E and I2 of Pier N1 and S1 by 10% FEA Mode FEA Freq. (Hz) Test Mode Test Freq. (Hz) Diff. (%) 1 0.233 1 0.233 -0.03 2 0.251 2 0.251 -0.22 3 0.274 3 0.272 0.66 4 0.293 4 0.302 -3.03 6 0.437 5 0.432 1.29 7 0.470 6 0.478 -1.66 10 0.529 7 0.532 -0.62 13 0.571 8 0.572 -0.15    ABS AVG = 0.96    4.6 Step 2: Modifying Girder Parameters and Adding Counterweight Now that the second and third mode shapes have frequencies that match more closely the experimental data, the fourth, fifth, and sixth mode shapes can be targeted for calibration. The three mode shapes are comprised of mainly deck motion, so calibration will need to be focused on this component. An example plot of the sensitivity contours for material parameter E and the fifth mode is shown in Figure 4.7.  Figure 4.7: Sensitivity Contours for Mode 5 and Material Parameter E 68  The contours show that the edge girders have the highest influence on the frequency of the fifth mode shape. After trial and error, I2, I3, and E for the girders were changed by -10%, +20%, and -5%, respectively. This allowed a closer match to the frequencies of modes 4, 5, and 6. Finally, there exists a counterweight on the actual bridge that was constructed to offset the extra weight provided by the sidewalk on the east deck. To account for this, a counterweight was added to the model at the outside girder of the west deck that equalled the same force and moment given by the sidewalk on the east deck, as shown in Figure 4.8. This equated to a vertical force of 14.5 kN and a moment of 28.3 kNm. The final updated results can be seen in Table 4.11.   Figure 4.8: Counterweight Force and Moment Applied to West Deck (Units of kN and kNm)      69  Table 4.11: Mode Shape Pairs after Modifying Girder Properties and Counterweight in SAP2000 Model FEA Mode FEA Freq. (Hz) Test Mode Test Freq. (Hz) Diff. (%) MAC (%) 1 0.234 1 0.233 0.33 99.0 2 0.251 2 0.251 -0.11 85.7 3 0.274 3 0.272 0.85 88.3 4 0.296 4 0.302 -1.97 97.7 6 0.432 5 0.432 -0.04 65.4 7 0.473 6 0.478 -0.88 99.3 10 0.532 7 0.532 -0.02 97.4 13 0.575 8 0.572 0.60 94.6    ABS AVG = 0.60 90.9  By modifying the girder parameters, the model went from an average frequency difference of 1% to 0.6% from the first step to the second step. The MAC values are all quite high except for the first torsional mode, which is at 65%. It was decided to look further into the first torsional mode to determine what each component contributes to the MAC value.  4.7 Investigation into First Torsional Mode Because the MAC value for the torsional mode is lower at 65%, it was decided to break down the experimental data into separate components: east-west, north-south, and vertical. This component breakdown helps find which orientation is contributing to the low MAC value. A separate modal analysis was carried out for each component, and the torsional mode was identified and paired with the finite element model to determine the correlation. The torsional frequency was identified in the east-west and vertical channels, but it could not be found in the north-south channels. This indicates that there is no north-south behaviour in the experimental torsional mode shape, but the finite element model does indeed exhibit some north-south movement. 70   Table 4.12 shows the MAC values for each component, and it shows that the east-west and vertical components have a high MAC value at 92.6% and 81.7%, respectively. It can be concluded that the lower MAC value is due to the mismatch in the north-south channels.  Table 4.12: MAC Contribution to Torsional Mode from Longitudinal, Transverse, and Vertical Components Component Analytical Freq. (Hz) 1st Torsional Frequency (Hz) MAC (%) East-West 0.432 0.4325 92.6 North-South 0.432 -- -- Vertical 0.432 0.4317 81.7  4.8 Summary of Model Updating In order to determine appropriate parameters to update, a sensitivity analysis was carried out and proper insight into each parameter was exercised. The sensitivity analysis showed that some uncertain properties, such as bearing fixity, had less of an effect on the overall behaviour of the model. It also showed that other parameters, such as cable modulus of elasticity, had a notable effect on the behaviour, but its properties were considered to have higher confidence than others. After the careful consideration of various parameters and experimental results, the model was updated to match very closely to what was measured. Table 4.13 summarizes the finite element model results compared with their respective test results.     71    Table 4.13: Summary of all Updated FEA Mode Shapes vs. all Test Mode Shapes FEA FEA Freq. (Hz) Test Freq. (Hz) Description Diff. (%) MAC (%) 1 0.234 0.233 1st Vertical 0.33 99.0 2 0.251 0.251 1st Torsional -0.11 85.7 3 0.274 0.272 2nd Torsional 0.85 88.3 4 0.296 0.302 2nd Vertical -1.97 97.7 5 0.378 -- Vertical-Transverse -- -- 6 0.432 0.432 Torsional -0.04 65.4 7 0.473 0.478 Vertical -0.88 99.3 8 0.486 -- Vertical (Decks out of Phase) -- -- 9 0.525 -- Mid-span Torsion (Decks out of Phase) -- -- 10 0.528 -- Mid-span Torsion (Decks in Phase) -- -- 11 0.532 0.532 Vertical -0.02 97.4 12 0.571 -- Vertical (Decks out of Phase) -- -- 13 0.575 0.572 Vertical 0.60 94.6    ABS AVG = 0.60 90.9  Overall, the average variation in the frequencies of the mode shapes went from 10.11% to 0.6% after model updating. The average variation in the MAC values went from 84.8% to 90.9%. The four vertical mode shapes (test modes 1, 4, 6, 7, and 8) have a very high modal acceptance criterion, with the lowest MAC value of these being 94.6%, and the highest being 99%. Overall a very close match was obtained.  The three torsional mode shapes (test modes 2, 3, and 5) are not as closely correlated as the vertical mode shapes. Modes 2 and 3 have a good match, with a MAC value of 86% and 88%, respectively. The largest outlier is mode 5, with a MAC value of 65%. Unfortunately, the model updating process was not able to obtain a higher MAC value for mode 5. It was determined that the low MAC value of this mode is mostly due to the north-south component. 72  Parameters were changed according to Table 4.14. The table summarizes the modifications to the properties of piers N1 and S1 and the deck girders, and also lists the additions made for deck weight and sidewalk counterweight. Table 4.14: Summary of Parameter Modifications Parameter Parameter Modifiction Pier N1 and S1 E -10% I2 -10% Deck Girders E -5% I2 -10% I3 +20% Deck Weight Addition W 12.25 kN/m Sidewalk Counterweight Addition W 14.5 kN and 28.3 kNm  73  Chapter 5: Linear Time History Analysis In order to provide a means of comparison between the original model and the updated model, a set of linear time history analyses were carried out utilizing specific ground motions. Output values were tabulated and compared between the two models, and results are discussed. A total of 1000 mode shapes were calculated to reach a cumulative modal mass participation ratio of 100, 100, and 99.9% for UX, UY, and UZ, respectively. For RX, RY, and RZ, a ratio of 96, 97.5, and 98.7% was achieved.   5.1 Ground Motions A diverse set of ground motions were chosen in order to represent a broad range of shaking. Based on the bridge’s experimental period, and using an online tool created by researchers at UBC (S2GM 2015), a set of ground motions were scaled and selected. In total, three from each of crustal, subcrustal, and subduction ground motions were selected. A summary of the ground motions are presented in Table 5.1, and the ground motions along with their elastic acceleration and displacement spectra are plotted in Appendix F. Table 5.1: Summary of Ground Motions Selected for Linear Time History Analysis  Name Magnitude Peak Accel. (g) Scale Factor Crustal: Chi-Chi 1999 6.2 0.15 2.21 Loma Prieta 1989 6.9 0.97 0.60 Gazli 1976 6.8 1.26 0.74 Subcrustal: Miyagi 7.2 0.15 2.60 Olympia 7.1 0.27 2.26 Geiyo 6.4 0.47 3.87 Subduction: Tohoku1 9.0 0.09 2.34 Hokkaido 8.0 0.10 2.51 Tohoku2 9.0 0.09 2.67 74   The two subduction earthquake motions, Tohoku1 and Tohoku2, were taken from the same earthquake source, but with different sensors. Normally a comprehensive and thorough time history analysis would avoid using the same earthquake, but for the purposes of this research, it is sufficient for comparison between two models.  5.2 Responses at Critical Locations For the time history analysis, the responses are observed at specific locations of interest in the structure. It was decided to examine the following responses: 1. Maximum displacement for piers N1 and S1 at deck height 2. Maximum displacement at the mid-span 3. Base shear and moments at N1, S1, N2, and S2  A complete summary of response values for each type of ground motion can be found in Appendix G, including maximum and minimum envelope values for each response quantity (note that all units are in kN and metres). The “absolute maximum” specified throughout this chapter refers to the larger of the absolute value of the maximum and minimum response. This section will focus mainly on the changes in responses that were experienced due to the model updating process, rather than the values themselves.  5.2.1 Displacement in Towers N1 and S1 The differences in displacement results for the two main towers were observed. The location at which the results for displacement are reported is at the joint in the tower where the gravity bearings are, as shown in Figure 5.1. A summary of the tower displacement comparison between 75  the original and updated model is shown in Table 5.2, with values highlighted in red indicating the largest change for that response.  Figure 5.1: Diagram Showing Location used for Recording Displacements of Towers N1 and S1  Table 5.2: Changes in Absolute Maximum Displacement of Main Towers at Deck Level after Model Updating  Difference in Transverse Displacement (%) Difference in Longitudinal Displacement (%)  N1 S1 N1 S1 Chi-Chi 23.7 -21.1 22.4 22.7 Gazli -2.0 40.6 43.3 31.1 Loma Prieta 35.0 30.2 12.9 17.4 Geiyo 68.8 -4.2 -17.3 -19.4 Miyagi 42.1 30.8 15.7 11.1 Olympia 22.6 13.8 6.0 26.9 Hokkaido 5.8 -0.6 41.9 40.3 Tohoku1 26.0 22.7 33.4 34.7 Tohoku2 34.7 -2.1 5.2 23.1  The table indicates that quite a large change in expected displacements can occur with the updated model. The highest change was a 69% increase in transverse displacement at pier N1 76  during the Geiyo ground motion analysis. To put it into perspective, the absolute value of the maximum transverse displacement at N1 went from 22.5 mm to 38 mm.  5.2.2 Displacement at Mid-Span Another useful point for measurement is the displacement experienced at the mid span. In this case, the two outer nodes on the deck are used as the point to measure the displacement measurements, as shown in Figure 5.2. Table 5.3 shows the changes in maximum displacement values after model updating, with values highlighted in red indicating the largest change for that response.  Figure 5.2: Diagram Showing Locations used for Reporting Mid-Span Displacements      77  Table 5.3: Changes in Absolute Maximum Displacement of Deck at Mid-Span after Model Updating  Difference in Transverse Displacement (%) Difference in Longitudinal Displacement (%) Difference in Vertical Displacement (%)  West End East End West End East End West End East End Chi-Chi 78.6 73.6 -21.3 25.6 40.6 50.4 Gazli 23.3 18.3 11.2 20.9 43.8 31.1 Loma Prieta 23.4 16.7 -25.2 27.8 34.2 17.5 Geiyo 33.3 26.9 -47.9 38.1 10.4 7.7 Miyagi -5.4 -0.7 -8.8 6.5 -2.9 35.2 Olympia -0.2 1.9 29.2 33.5 -18.1 -17.1 Hokkaido 25.3 25.4 23.4 44.1 5.8 7.3 Tohoku1 24.6 16.8 8.1 45.3 27.2 26.9 Tohoku2 49.3 46.9 -5.8 26.1 28.4 39.9  The mid-span appears to be sensitive to the model updating results, with the largest change occurring during the Chi-Chi ground motion analysis, where the transverse displacement increased by 79% and 74% in the west and east sides of the deck, respectively. Originally this value was 148 mm, and after model updating, the maximum absolute transverse displacement became 265 mm. The longitudinal and vertical displacements also experienced significant changes after updating, with differences of 44 to 50%. Some ground motion analyses were not affected as much by the model updating. For example, the absolute maximum transverse displacement due to the Olympia ground motion only decreased by 0.2% after model updating.  5.2.3 Base Shears and Moments A very important component of time history analyses is to determine the elastic base shear and moment that is expected in an earthquake. The reference points for shear and moment values are displayed in Figure 5.3. Note that, for piers N2 and S2, only the centre column values are reported. The change in base shear values in the longitudinal direction is summarized in Table 5.4. The largest change in each response is highlighted in red in the table. 78   Figure 5.3: Diagram Showing Locations used for Reporting Base Shears and Moments  Table 5.4: Changes in Absolute Maximum Base Shears in Bridge Piers after Model Updating  Difference in Longitudinal Shear (%) Difference in Transverse Shear (%)  N1 S1 N2 S2 N1 S1 N2 S2 Chi-Chi 17.0 8.5 -0.1 -0.3 22.1 -31.5 -18.8 24.0 Gazli 30.5 23.9 -13.2 -4.0 -25.8 -1.0 -24.0 23.1 Loma Prieta 10.7 8.0 -9.1 -26.0 9.2 7.7 -14.6 12.7 Geiyo -24.6 -21.9 -12.1 -28.4 13.0 -20.5 -11.8 -5.5 Miyagi 3.8 9.0 -12.2 -18.2 9.3 -24.0 -9.6 -1.5 Olympia 8.2 17.7 -14.8 11.2 14.6 30.3 -12.7 -9.7 Hokkaido 31.8 23.9 -17.2 3.2 18.4 -2.4 0.6 29.2 Tohoku1 10.4 26.1 0.9 -31.4 -2.0 20.8 10.2 3.0 Tohoku2 5.2 6.4 -15.0 4.9 -14.8 -2.0 3.9 2.0  From the above table, it can be seen that the largest difference in longitudinal shear due to model updating occurred between two of the subduction earthquakes, Hokkaido and Tohoku1. After model updating, the longitudinal shears experienced a maximum difference of +32% in N1, +26% in S1, -17% in N2, and -31% in S2. For transverse shears, the maximum change ranged from -24% to 29%. It should also be noted that each ground motion is affected in different ways by the model updating process. The table demonstrates that the shear forces can increase or 79  decrease depending on the ground motion analyzed. Finally, the changes in base moments are summarized in Table 5.5, with the largest changes being highlighted in red. The largest change in moment occurred at pier N2 from the Chi-Chi analysis: after model updating, the moment about the transverse axis was reduced by 41%. Other moment values also increase or decrease depending on the earthquake motion analyzed. Table 5.5: Changes in Absolute Maximum Base Moments in Bridge Piers after Model Updating  Difference in Moment about Longitudinal Axis (%) Difference in Moment about Transverse Axis (%)  N1 S1 N2 S2 N1 S1 N2 S2 Chi-Chi 6.2 -28.4 -5.2 17.4 9.8 10.2 -40.6 -1.6 Gazli -25.7 -4.9 -24.4 20.8 25.4 21.3 6.4 -5.9 Loma Prieta 0.3 -1.2 -12.6 5.2 9.9 8.6 -5.9 -26.1 Geiyo 18.1 -18.0 -12.3 -20.4 -12.6 -16.3 -16.5 -27.5 Miyagi 4.0 -22.6 -14.8 2.8 5.2 7.5 -16.7 -18.6 Olympia 4.5 21.7 1.2 -10.7 -2.0 11.0 -23.7 12.2 Hokkaido 3.3 -0.3 -5.4 20.6 26.1 17.7 11.6 2.8 Tohoku1 12.3 7.5 17.9 8.6 14.3 23.6 13.9 -31.6 Tohoku2 12.7 6.5 0.1 -4.4 5.4 9.2 -6.1 4.9  5.3 Summary of Time History Analysis The results from the time history analysis show that model updating had a significant effect on the seismic analysis of the bridge. There were large changes by up to 79% in the response values after model updating. Some response values were only affected by a negligible amount, however, and the changes depend on what ground motion was analyzed. In summary, model updating had a large effect on some values for some ground motions, and for others it was minimal. The analysis demonstrates that it is important to have a sufficient amount of ground motions with which to run the analysis, as the impact of model updating could not be appreciated in every result. 80  Chapter 6: Summary and Conclusion The study of the realistic dynamic behaviour of structures is a broad and far-reaching effort. By being able to predict how a structure is going to behave allows engineers and other stakeholders to make important decisions: structural changes or retrofits can be analyzed or potential damage can be identified.  The Port Mann Bridge is one of the largest cable-stayed bridges in North America. The results from two experimental field tests and one dataset from the structural health monitoring (SHM) sensors were analyzed to determine the system behaviour. There were differences in the two experimental field results, as the measurements were taken approximately one month apart with different operating conditions. A total of 13 mode shapes were identified from the field-acquired data, while only eight were found for the SHM data. In the end, the data from the structural health monitoring network was deemed most appropriate for correlation with finite element model results, as the data was more complete and the operating conditions were consistent. From the set of experimental analyses, it was concluded that the dynamic behaviour of the bridge is dependent upon the current operating conditions, and that the spatial density of sensor placement makes a large difference on the number of mode shapes that can be identified.  A finite element model was created in ADINA software by the design engineers, and it was then modified in order to transfer into SAP2000 which was the preferred finite element modelling software. There were issues trying to import the model: for example, all of the concrete piers were defined in ADINA using moment-rotation relationships instead of cross sectional properties, so the piers had to be recreated using design drawings. The foundations also had to be 81  removed and simplified with a set of rigid springs. There were other problems such as the rigid members not being transferable, so they had to be imported using an Excel spreadsheet with the frame labels. The benefits of having the model in SAP2000 are being able to work alongside FEMTools for easier correlation and future model updating. SAP2000 is also more accessible and easy to modify by editing database tables. It was concluded that transferring an ADINA model into SAP2000 is an arduous process, and many components are not supported, but it is worthwhile to have the model in SAP2000.  The model consists of 9707 frame elements which make up the components: the main towers N1 and S1, approach piers N2 and S2, cables, steel deck girders, and steel deck cross beams. The finite element analysis results were compared with the experimental SHM data to find that there were indeed discrepancies between the two results. By using FEMTools as a correlation tool, the model was studied to determine appropriate calibration areas. It was clear at the start that part of the global frequency difference was likely caused by the uncertainty in the deck weight from sources such as concrete topping, barriers, traffic, etc. The first step in the calibration process was applying a load to all deck beams. By accounting for this uncertain weight, a much closer correlation between the analytical and experimental results was achieved. For a more refined calibration, FEMTools was utilized to determine what parameters the model was sensitive to, and also a manual trial-and-error process was undertaken to determine sensitivities of other parameters, such as boundary conditions, cable properties, and foundation stiffnesses. Following the sensitivity analysis, the model was updated manually by changing the modulus of elasticity and moment of inertia of piers N1 and S1 and the deck girders, and finally adding the counterweight to the model which is installed on the west deck and counteracts the weight of the 82  sidewalk on the east deck. The updated model managed to achieve an average frequency difference of 0.6% and average MAC value of 89%. Unfortunately, the third torsional mode was the least correlated, with a MAC value of only 65%. Following the model correlation and updating, some main conclusions can be made:  The finite element model is sensitive to changes in the material and section properties of the deck weight, girders, main towers, cables, and longitudinal stiffness of the tie-downs.  To maintain a realistic model, parameters must be chosen with engineering insight, and limits should be placed on their variation.  The error from the third torsional mode shape is mainly from excessive north-south motion in the finite element model.  In order to provide a means of comparison between the original and updated models, a linear time history analysis was carried out using a suite of ground motions. Nine ground motions were selected and scaled based on the targeted frequency range of the structure. Three ground motions were selected from each of the types: crustal, subcrustal, and subduction. The linear time history analysis was run for all ground motions on the original model and the updated model, and specific response quantities were observed to examine the change after model updating. It was discovered that a significant increase or decrease in response can happen depending on the ground motion and response quantity looked at. Overall, the updated model had some large and some insignificant variations, such as the 79% increase in transverse displacement at mid-span from the Chi-Chi ground motion; but only a 0.2% decrease when looking at the Olympia ground motion result. The linear time history analysis showed that model updating has a significant effect on the response of the Port Mann Bridge, and that the selection of an appropriate suite of 83  ground motions is important. It was also learned that model updating can cause an increase in force or displacement resulting from one ground motion, and a decrease in force or displacement from another ground motion.  It is important to keep in mind that with the current state of the art in model updating, there is still a limitation in the fact that the finite element model cannot be truly validated against the physical data; with proper engineering judgement in the creation of the model and the calibration of the parameters, one is able to achieve a sufficient level of confidence in the model, but a true measure of physical realization does not exist.  6.1 Future Research A very close match was obtained between the experimental and analytical models, and the parameters improved their physical properties owing to the application of engineering insight. Despite the close match, there is still much to be investigated. A review of the assumptions should be made, and an investigation into their appropriateness could prove useful.  This research focused on the main span of the Port Mann Bridge. It was assumed that, because the connections between the approach and the main span connected separately to the cap beam, that they could be analyzed separately with sufficient accuracy. However, there are still forces that will transfer from the approach spans into the pier caps at N2 and S2, thereby affecting the main span. Future research will benefit by investigating the effect of including both approach spans in the model.  84  The cables of the bridge were idealized as truss members and modelled as such. This is because the original ADINA model used truss members to model the cables. However, it is possible for the cable elements to experience compression when modelled like this, which is technically not how a cable should behave. The effect of this assumption was not investigated, so it could be beneficial in future research to attempt a more rigorous approach to modelling the cables that include sag effects and their associated nonlinearities as well as pretensioning.   The application of a calibrated nonlinear model is the main goal of future research. Currently, the calibrated linear model gives a confident estimate of the elastic properties of the bridge, and can be expanded upon to include nonlinear properties in the future.  The experimental data used for the correlation had some limitations: there was no information at the ends of the bridge or on the decks near the main towers, and neither the cable sensors nor foundations were included as part of the analysis. A lack of data for the ends of the bridge means that the behaviour at piers N2 and S2 could not be included in the analysis. This could be important in determining the properties of the end piers or analyzing the effect of the excluded approach spans. Secondly, while there was data included for the main towers N1 and S1, the two decks do not have sensors near this location; therefore, the behaviour at the decks in these locations were not included in the model correlation. It can be assumed to act rigidly with the tower motion, but this is not exactly the case. If sensors were installed at the deck adjacent to the tower, it might be possible to observe any movement within the bearings. Experimental data associated with the cables could also provide meaningful insight for future research. With this data, the cable properties may be selected as a parameter for another set of model updating. 85  Finally, the footings are an important component of the model, so it would be beneficial to finalize the analysis of the experimental data at the footings. Understanding the dynamic behaviour at the footings can offer insight into the stiffness properties of the foundations, so it could contribute to verifying that component of the finite element model.  The finite element model was calibrated by manually adjusting specific parameters. For future research, it may be useful to study the result of a more rigorous automated updating procedure which employs optimization techniques and includes more parameters.  Parameters were, for the most part, calibrated globally. For example, all steel girders of the deck had their properties equally modified, and the same can be said for both piers N1 and S1. This global modification procedure can make it difficult to reach convergence for some mode shapes: a modification to an entire property set could cause an improvement in one mode but further deteriorate the correlation with another mode. A component-level calibration procedure could be explored in the future to investigate the error with the first torsional mode shape that was difficult to match with a high confidence.  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Xianning: IEEE, 2011. 1178-1181.  96  Appendices Appendix A  General Plan and Elevation of Port Mann Bridge  Figure A.1: Plan and Elevation Design Drawings of Port Mann Bridge97  Appendix B  Sensor Setup for Field-Acquired Data  Figure B.1: Sensor Setup Locations with Placement Labels Table B.1: Sensor Placement Locations Corresponding to Diagram Labels   S1 S2 S3 S4 S5 S6 S7 S8 S9 Sep. 6, 2012 Setup 1 83 67 68 69 70 100 101 102 103 Setup 2 83 71 72 73 74 104 105 106 107 Setup 3 83 75 76 77 78 108 109 110 111 Setup 4 83 79 80 81 82 112 113 114 115 Setup 5 83 84 85 86 87 116 117 118 119 Setup 6 83 88 89 90 91 120 121 122 123 Setup 7 83 92 93 94 95 124 125 126 127 Setup 8 83 96 97 98 99 128 129 130 131 Oct. 10, 2012 Setup 1 83 1 2 3 4 34 35 36 37 Setup 2 83 5 6 7 8 38 39 40 41 Setup 3 83 9 10 11 12 42 43 44 45 Setup 4 83 13 14 15 -- 46 47 48 49 Setup 5 83 17 18 19 -- 50 51 52 16 Setup 6 83 20 21 22 -- 53 54 55 56 Setup 7 83 24 25 26 -- 57 58 59 23 Setup 8 83 27 28 29 -- 60 61 62 63 Setup 9 83 31 32 33 -- 64 65 66 30 98  Appendix C  Channel Numbers and Orientations for SHM-Acquired Data  Figure C.1: Channel Map for Data used in the Analysis of SHM-Acquired Data 99  Appendix D  ADINA Modelling Results  Table D.1: ADINA Modal Frequencies Compared to Converted SAP2000 Model ADINA Mode Number ADINA Frequency SAP2000 Mode Number Corresponding SAP2000 Frequency West Description 1 0.2122 1 0.2626 1st Vertical 2 0.2300 2 0.2745 1st Torsional 3 0.2490 3 0.2977 2nd Torsional 4 0.2716 4 0.3292 2nd Vertical 5 0.2879 5 0.4305 Vertical-Transverse 6 0.3805 6 0.4939 Vertical-Transverse 7 0.3922 -- -- Longitudinal in South Approach 8 0.4059 8 0.5440 Vertical 9 0.4203 -- -- Longitudinal in North Approach 10 0.4329 7 0.5306 Longitudinal-Vertical 11 0.4801 -- -- Longitudinal-Vertical 12 0.4923 12 0.6405 Vertical 13 0.5093 -- -- Longitudinal North Approach 14 0.5113 11 0.5970 Vertical 15 0.5205 14 0.6628 Vertical-Transverse 16 0.5456 15 0.6847 Vertical 17 0.5484 9/10 0.5572/0.5629 Mid-span Torsional 18 0.5515 13 0.6449 Vertical    100   Figure D.1: 1st Vertical Mode Shape at 0.2122 Hz  Figure D.2: 1st Torsional Mode at 0.230 Hz  Figure D.3: Second Torsional Mode at 0.2490 Hz 101   Figure D.4: 2nd Vertical Mode at 0.2716 Hz  Figure D.5: Vertical-Transverse Mode at 0.2879 Hz  Figure D.6: Vertical-Transverse Mode at 0.3805 Hz 102   Figure D.7: Vertical Mode at 0.4059 Hz  Figure D.8: 3rd Vertical Mode at 0.4329 Hz  Figure D.9: Vertical Mode at 0.4801 Hz 103   Figure D.10: Vertical Mode at 0.4923 Hz  Figure D.11: Vertical Mode at 0.5113 Hz  Figure D.12: Vertical Mode at 0.5205 Hz 104   Figure D.13: Vertical Mode at 0.5456 Hz  Figure D.14: Torsional Mode at 0.5484 Hz    105  Appendix E  Approximate Pile Group Stiffness Calculations E.1 Pile Details   106  E.2 Geometry Calculations Bending Stiffness Calculations:        Assuming Section B throughout       r =  0.915 m       Agross =  2.63022 m2       Esteel = 200 Gpa       Econc = 28 GPa (Assumed based on 30 Mpa f'c)   Reinforcement:         No. bars = 17        Dbar = 57.33 mm Abar = 0.0025814 m2    Spiral dia. = 19.05 mm       Tube thickness = 32 mm                Moment of Inertia:        I = π*r^4/4 = 0.550521 m4                Area of Steel:         Steel Tube: ro = 0.915 m       ri = 0.883 m       Atube = 0.180754675 m2      Outer Spiral: ro = 0.813 m       ri = 0.79395 m       Aspiral = 0.096171683 m2      Rebar: Abars = 0.043883644 m2                Asteel = 0.320810002 m2 % Steel = 0.12     Aconc = 2.309409907 m2 % Concrete = 0.88             Net Bending Stiffness:        Enet = 48.97898 Gpa                 EIpile = 26.96397768 GN*m2        26963977.68 kN*m2          9.39615E+12 lb*in2 (Use this value for Vijayvergiya (1977) charts)   107  Selection of f Coefficient:      f = coefficient of variation of soil reaction modulus with depth    Without geotechnical report, major assumptions have to be made.   According to design drawings, all piles are below water table.    φ' = 35 deg (Assume typical)    f =  20 (Approximately taken from Figure 10, Lam et. Al (1991))                                                                                                                                                                           Translational Stiffness: 1000000 lb/in 1000 kip/in (Approximately taken from Figure 7, Lam et. Al (1991)) Rotational Stiffness: 60000000000 in*lb/rad 60000000 kip*lb/rad (Approximately taken from Figure 8, Lam et. Al (1991)) Cross-Coupling Stiffness: 200000000 lb 200000 kip  (Approximately taken from Figure 9, Lam et. Al (1991))    108  E.3 Vertical Stiffness Calculations Vertical Stiffness Calculations:      (Empirical method taken from Vijayvergiya (1977))     (Without geotechnical report, major assumptions have to be made)   F-Z Skin Friction Contribution:       f = fmax(2*sqrt(z/zc) - z/zc)        fmax = maximum skin friction      zc = displacement required for fmax            zc = 0.25 in (Suggested from 0.2 to 0.3, so use 0.25 in) (Vijayvergiya (1977)) γsoil = 120 lb/ft3 (Assumed typical)    φ' = 35 degrees (Assumed typical)    γwater = 62.4 lb/ft3      Lpile = 64.35 m (Value taken as average of (pile tip elevation - elevation to bottom of footing)) Lpile = 211.12205 ft      Lpile = 2533.4646 in      fmax =  0.75 tsf (From Figure 4 of Mosher and Dawkins (2000)  1653.465 lb/ft2       0.0114824 kip/in2      Plotting f-z Curve:        f = fmax(2*sqrt(z/zc) - z/zc)       z (in) f (kips/in2) Load (kips) (Load calculated by multiplying f by circumference and length of pile) 0 0 0      0.01 0.0041337 2370.371         109  Q-Z End-Bearing Contribution:      q = (z/zc)1/3*qmax               zc = 3.6 in (Suggested from 0.04 to 0.06 pile dia., so use 0.05*72") (Vijayvergiya (1977)) Relative Depth = 35  (Length of pile divided by diameter)  qmax = 62.5 tsf (From Figure 17 of Mosher and Dawkins (2000)  137788.75 lb/ft2       0.9568663 kip/in2      Plotting q-z Curve:       q = (z/zc)1/3*qmax       z (in) q (kips/in2) Load (kips) (Load calculated by multiplying q by area of pile) 0 0 0      0.01 0.1345087 548.3723                      Vertical Stiffness: 170005.8 kip/in (Using sum of the two curves together, and the secant stiffness at 0.025 inches = 4250/0.025)    (Mosher (1984) recommends secant tip reaction stiffness corresponding to 0.1 in for side friction stiffness, and 0.025 in for tip reaction stiffness)    110  E.4 Vertical Stiffness Plots F-Z Skin Friction:   Q-Z End Bearing:  Rigid Pile Solution (SUM): r = 36.0236415 in Apile = 4076.853416 in2    Circumference = 226.343215 in        Lpile = 2533.464648 in        zc = 0.25 in zc = 3.6 in    fmax = 0.011482396 kip/in2 qmax = 0.956866319 kip/in2               f = fmax(2*sqrt(z/zc) - z/zc)  q = (z/zc)1/3*qmax  SUM  z (in) f (kips/in2) Load (kips) z (in) q (kips/in2) Load (kips) z (in) Load (kips) 0 0 0 0 0 0 0 0 0.0000001 1.45196E-05 8.32602052 0.0000001 0.002897902 11.8143237 0.0000001 20.1403442 0.0000002 2.05311E-05 11.7732283 0.0000002 0.003651128 14.8851151 0.0000002 26.6583434 0.0000003 2.51429E-05 14.4177511 0.0000003 0.004179499 17.0392032 0.0000003 31.4569543 0.0000004 2.903E-05 16.6467735 0.0000004 0.004600133 18.7540698 0.0000004 35.4008433 0.0000005 3.24542E-05 18.6102683 0.0000005 0.004955344 20.2022093 0.0000005 38.8124776 0.0000006 3.55493E-05 20.3851507 0.0000006 0.005265838 21.4680508 0.0000006 41.8532015 0.0000007 3.83953E-05 22.0171117 0.0000007 0.005543488 22.5999881 0.0000007 44.6170998 0.0000008 4.10439E-05 23.5359216 0.0000008 0.005795805 23.6286473 0.0000008 47.164569 0.0000009 4.35313E-05 24.962259 0.0000009 0.00602788 24.5747835 0.0000009 49.5370426 0.000001 4.58837E-05 26.3111798 0.000001 0.006243342 25.4531887 0.000001 51.7643686 0.000002 6.48624E-05 37.1941992 0.000002 0.007866118 32.0690083 0.000002 69.2632075 0.000003 7.94146E-05 45.5389056 0.000003 0.009004457 36.7098505 0.000003 82.2487561 0.000004 9.16754E-05 52.5696846 0.000004 0.009910687 40.4044186 0.000004 92.9741032 0.000005 0.000102472 58.7607915 0.000005 0.010675964 43.5243405 0.000005 102.285132 0.000006 0.000112228 64.3554535 0.000006 0.011344905 46.2515134 0.000006 110.606967    111  E.5 Pile Group Calculations Single Pile Stiffness Coefficients:          Lateral Translation (k11=k22): 1000 kip/in Rocking Rotation (k44 = k55):   60000000 kip*lb/rad  Vertical Translation (k33):  170006 kip/in Torsional Rotation (k66):   0 in*kip/rad      Cross-Coupling (k15 = k51 = -k24 = -k42):  200000 kip             Pile Group Stiffness Coefficients:          IMPERIAL           Pier N2 and S2:                     SUM(Sn^2):  198450 in2         Number of piles:  4 piles Stiffness Matrix: 4000 0 0 0 8.00E+05 0 Lateral Translation (k11 = k22): = 4*k 4000 kip/in   4000 0 8.00E+05 0 0 Vertical Translation (k33): =4*k 680024 kip/in    6.80E+05 0 0 0 Rocking Rotation (k44 = k55): =4*k44+k33*SUM(Sn^2) 3.398E+10 in*kip/rad     3.40E+10 0 0 Torsional Rotation (k66): =k11*SUM(Sn^2) 1.98E+08 in*kip/rad      3.40E+10 0 Cross-Coupling (k15 = k51 = -k24 = -k42): =4*k 800000 kip       1.98E+08              Pier N1 and S1:            SUM(Sn^2):  16579115.52 in2         Number of piles:  63 piles Stiffness Matrix: 63000 0 0 0 1.26E+07 0 Lateral Translation (k11 = k22): = 63*k 63000 kip/in   63000 0 1.26E+07 0 0 Vertical Translation (k33): =63*k 10710378 kip/in    1.07E+07 0 0 0 Rocking Rotation (k44 = k55): =63*k44+k33*SUM(Sn^2) 2.82233E+12 in*kip/rad     2.82E+12 0 0 Torsional Rotation (k66): =k11*SUM(Sn^2) 16579115520 in*kip/rad      2.82E+12 0 Cross-Coupling (k15 = k51 = -k24 = -k42): =63*k 12600000 kip             1.66E+10  112     METRIC           Pier N2 and S2:                     Number of piles:  12 piles Stiffness Matrix: 700507 0 0 0 3.56E+06 0 Lateral Translation (k11 = k22): = 12*k 700507.3386 kN/m   700507 0 3.56E+06 0 0 Vertical Translation (k33): =12*k 119090450.6 kN/m    1.19E+08 0 0 0 Rocking Rotation (k44 = k55): =12*k44+k33*SUM(Sn^2) 3.839E+09 m*kN/rad     3.84E+09 0 0 Torsional Rotation (k66): =12*k11*SUM(Sn^2) 2.24E+07 m*kN/rad      3.84E+09 0 Cross-Coupling (k15 = k51 = -k24 = -k42): =12*k 3558577.28 kN       2.24E+07                           Pier N1 and S1:            SUM(Sn^2):  16579115.52 in2         Number of piles:  63 piles Stiffness Matrix: 1.10E+07 0 0 0 5.60E+07 0 Lateral Translation (k11 = k22): = 63*k 11032990.58 kN/m   1.10E+07 0 5.60E+07 0 0 Vertical Translation (k33): =63*k 1875674597 kN/m    1.88E+09 0 0 0 Rocking Rotation (k44 = k55): =63*k44+k33*SUM(Sn^2) 3.1888E+11 m*kN/rad     3.19E+11 0 0 Torsional Rotation (k66): =k11*SUM(Sn^2) 1873188526 m*kN/rad      3.19E+11 0 Cross-Coupling (k15 = k51 = -k24 = -k42): =63*k 56047592.16 kN             1.87E+09  113  Appendix F  Ground Motion Records F.1 Crustal Records.   Figure F.1: Plots for CHICHI03_TCU122 Records  114    Figure F.2: Plots for GAZLI_GAZ Records   115    Figure F.3: Plots for LOMAP_LGP Records   116  F.2 Subcrustal Records   Figure F.4: Plots for GEIYO_EHM0030103241528 Records  117    Figure F.5: Plots for Miyagi_Oki_MYG0060508161146 Records   118    Figure F.6: Plots for Olympia_OLY0 Records   119  F.3 Subduction Records   Figure F.7: Plots for Hokkaido_HKD1270309260450 Records  120    Figure F.8: Plots for Tohoku_KNG0041103111446 Records   121    Figure F.9: Plots for Tohoku_KNG0061103111446 Records   122  Appendix G  Linear Time History Results G.1 Crustal Ground Motion Time History Analysis Results Table G.1: Summary of Crustal Ground Motion Results (Units in kN and m)  Chi-Chi Gazli LomaP1989  Original Updated Diff. Original Updated Diff. Original Updated Diff. N1 Base V2: 2.96E+04 3.47E+04 17.0 1.01E+05 1.32E+05 30.5 1.14E+05 1.26E+05 10.7 N1 Base V3: 3.07E+04 3.75E+04 22.1 9.21E+04 6.83E+04 -25.8 1.08E+05 1.18E+05 9.2 S1 Base V2: 4.61E+04 5.00E+04 8.5 1.65E+05 2.05E+05 23.9 2.03E+05 2.19E+05 8.0 S1 Base V3: 4.81E+04 3.30E+04 -31.5 9.21E+04 9.11E+04 -1.0 2.08E+05 2.24E+05 7.7 N1 Base M2: 1.55E+06 1.65E+06 6.2 5.06E+06 3.76E+06 -25.7 5.57E+06 5.59E+06 0.3 N1 Base M3: 1.16E+06 1.27E+06 9.8 3.98E+06 4.99E+06 25.4 4.91E+06 5.40E+06 9.9 S1 Base M2: 1.75E+06 1.25E+06 -28.4 3.94E+06 3.74E+06 -4.9 9.08E+06 8.97E+06 -1.2 S1 Base M3: 1.48E+06 1.63E+06 10.2 5.25E+06 6.37E+06 21.3 6.46E+06 7.02E+06 8.6 N2 Base V2: 5.05E+01 5.04E+01 -0.1 2.06E+02 1.79E+02 -13.2 1.56E+02 1.42E+02 -9.1 N2 Base V3: 7.17E+01 5.82E+01 -18.8 2.43E+02 1.85E+02 -24.0 2.78E+02 2.37E+02 -14.6 S2 Base V2: 1.39E+04 1.39E+04 -0.3 3.60E+04 3.46E+04 -4.0 8.68E+04 6.42E+04 -26.0 S2 Base V3: 1.74E+03 2.15E+03 24.0 5.90E+03 7.26E+03 23.1 7.08E+03 7.98E+03 12.7 N2 Base M2: 4.75E+03 4.51E+03 -5.2 1.72E+04 1.30E+04 -24.4 2.25E+04 1.96E+04 -12.6 N2 Base M3: 4.50E+02 2.67E+02 -40.6 1.29E+03 1.37E+03 6.4 1.15E+03 1.08E+03 -5.9 S2 Base M2: 4.35E+04 5.11E+04 17.4 1.46E+05 1.76E+05 20.8 1.82E+05 1.91E+05 5.2 S2 Base M3: 2.40E+05 2.36E+05 -1.6 6.14E+05 5.78E+05 -5.9 1.50E+06 1.11E+06 -26.1 N1 Mid UX: 4.30E-02 5.32E-02 23.7 1.68E-01 1.64E-01 -2.0 1.62E-01 2.18E-01 35.0 N1 Mid UY: 4.08E-02 5.00E-02 22.4 1.57E-01 2.25E-01 43.3 1.89E-01 2.13E-01 12.9 S1 Mid UX: 3.20E-02 2.52E-02 -21.1 6.74E-02 9.48E-02 40.6 1.48E-01 1.92E-01 30.2 S1 Mid UY: 3.18E-02 3.90E-02 22.7 1.20E-01 1.57E-01 31.1 1.27E-01 1.49E-01 17.4 Mid-span A UX:   1.48E-01 2.65E-01 78.6 2.63E-01 3.25E-01 23.3 7.81E-01 9.64E-01 23.4 Mid-span A UY: 6.15E-02 4.84E-02 -21.3 1.71E-01 1.90E-01 11.2 2.75E-01 2.06E-01 -25.2 Mid-span A UZ: 4.78E-01 6.71E-01 40.6 8.35E-01 1.20E+00 43.8 1.29E+00 1.74E+00 34.2 Mid-span D UX: 1.49E-01 2.59E-01 73.6 2.63E-01 3.12E-01 18.3 7.85E-01 9.16E-01 16.7 Mid-span D UY: 5.98E-02 7.51E-02 25.6 1.89E-01 2.28E-01 20.9 2.58E-01 3.30E-01 27.8 Mid-span D UZ: 3.89E-01 5.86E-01 50.4 3.17E-01 4.15E-01 31.1 1.57E+00 1.85E+00 17.5    123  G.2 Subcrustal Ground Motion Time History Analysis Results  Table G.2: Summary of Subcrustal Ground Motion Results (Units in kN and m)  Geiyo Miyagi Olympia  Original Updated Diff. Original Updated Diff. Original Updated Diff. N1 Base V2: 1.18E+04 8.92E+03 -24.6 2.56E+04 2.65E+04 3.8 3.49E+04 3.78E+04 8.2 N1 Base V3: 2.76E+04 3.12E+04 13.0 1.67E+04 1.82E+04 9.3 2.10E+04 2.40E+04 14.6 S1 Base V2: 1.88E+04 1.47E+04 -21.9 4.04E+04 4.41E+04 9.0 5.78E+04 6.81E+04 17.7 S1 Base V3: 6.05E+04 4.81E+04 -20.5 2.88E+04 2.19E+04 -24.0 2.40E+04 3.12E+04 30.3 N1 Base M2: 9.75E+05 1.15E+06 18.1 9.52E+05 9.90E+05 4.0 9.68E+05 1.01E+06 4.5 N1 Base M3: 4.25E+05 3.72E+05 -12.6 1.06E+06 1.11E+06 5.2 1.53E+06 1.50E+06 -2.0 S1 Base M2: 1.86E+06 1.53E+06 -18.0 1.14E+06 8.79E+05 -22.6 8.02E+05 9.76E+05 21.7 S1 Base M3: 5.76E+05 4.82E+05 -16.3 1.33E+06 1.43E+06 7.5 1.94E+06 2.15E+06 11.0 N2 Base V2: 7.60E+01 6.68E+01 -12.1 3.44E+01 3.02E+01 -12.2 5.78E+01 4.93E+01 -14.8 N2 Base V3: 9.44E+01 8.33E+01 -11.8 4.09E+01 3.70E+01 -9.6 6.29E+01 5.49E+01 -12.7 S2 Base V2: 2.17E+04 1.56E+04 -28.4 9.96E+03 8.15E+03 -18.2 9.34E+03 1.04E+04 11.2 S2 Base V3: 1.17E+03 1.11E+03 -5.5 1.59E+03 1.56E+03 -1.5 2.18E+03 1.97E+03 -9.7 N2 Base M2: 4.47E+03 3.92E+03 -12.3 3.19E+03 2.72E+03 -14.8 4.67E+03 4.72E+03 1.2 N2 Base M3: 1.96E+02 1.64E+02 -16.5 2.82E+02 2.35E+02 -16.7 4.22E+02 3.22E+02 -23.7 S2 Base M2: 1.85E+04 1.47E+04 -20.4 4.20E+04 4.32E+04 2.8 6.02E+04 5.38E+04 -10.7 S2 Base M3: 3.81E+05 2.76E+05 -27.5 1.70E+05 1.39E+05 -18.6 1.59E+05 1.79E+05 12.2 N1 Mid UX: 2.25E-02 3.80E-02 68.8 2.88E-02 4.09E-02 42.1 3.03E-02 3.71E-02 22.6 N1 Mid UY: 1.64E-02 1.36E-02 -17.3 4.34E-02 5.02E-02 15.7 6.09E-02 6.45E-02 6.0 S1 Mid UX: 2.85E-02 2.73E-02 -4.2 1.50E-02 1.97E-02 30.8 1.61E-02 1.84E-02 13.8 S1 Mid UY: 1.46E-02 1.18E-02 -19.4 3.26E-02 3.62E-02 11.1 4.14E-02 5.25E-02 26.9 Mid-span A UX:   1.03E-01 1.37E-01 33.3 7.42E-02 7.02E-02 -5.4 1.23E-01 1.23E-01 -0.2 Mid-span A UY: 2.81E-02 1.47E-02 -47.9 4.74E-02 4.33E-02 -8.8 4.96E-02 6.41E-02 29.2 Mid-span A UZ: 1.31E-01 1.45E-01 10.4 1.78E-01 1.73E-01 -2.9 1.60E-01 1.31E-01 -18.1 Mid-span D UX: 1.05E-01 1.33E-01 26.9 7.40E-02 7.35E-02 -0.7 1.23E-01 1.25E-01 1.9 Mid-span D UY: 2.67E-02 3.68E-02 38.1 4.97E-02 5.29E-02 6.5 4.90E-02 6.54E-02 33.5 Mid-span D UZ: 1.33E-01 1.43E-01 7.7 1.47E-01 1.99E-01 35.2 1.93E-01 1.60E-01 -17.1    124  G.3 Subduction Ground Motion Time History Analysis Results  Table G.3: Summary of Subduction Ground Motion Results (Units in kN and m)  Hokkaido Tohoku1 Tohoku2  Original Updated Diff. Original Updated Diff. Original Updated Diff. N1 Base V2: 3.30E+04 4.34E+04 31.8 2.13E+04 2.35E+04 10.4 1.95E+04 2.05E+04 5.2 N1 Base V3: 1.58E+04 1.87E+04 18.4 1.29E+04 1.27E+04 -2.0 2.00E+04 1.70E+04 -14.8 S1 Base V2: 5.41E+04 6.71E+04 23.9 3.16E+04 3.98E+04 26.1 3.31E+04 3.52E+04 6.4 S1 Base V3: 1.87E+04 1.83E+04 -2.4 1.34E+04 1.62E+04 20.8 2.25E+04 2.20E+04 -2.0 N1 Base M2: 9.04E+05 9.34E+05 3.3 7.37E+05 8.28E+05 12.3 1.06E+06 1.20E+06 12.7 N1 Base M3: 1.38E+06 1.74E+06 26.1 8.69E+05 9.94E+05 14.3 8.19E+05 8.64E+05 5.4 S1 Base M2: 8.32E+05 8.29E+05 -0.3 5.85E+05 6.28E+05 7.5 9.85E+05 1.05E+06 6.5 S1 Base M3: 1.79E+06 2.11E+06 17.7 1.06E+06 1.31E+06 23.6 1.06E+06 1.16E+06 9.2 N2 Base V2: 3.72E+01 3.09E+01 -17.2 1.60E+01 1.62E+01 0.9 1.99E+01 1.69E+01 -15.0 N2 Base V3: 3.33E+01 3.35E+01 0.6 2.33E+01 2.56E+01 10.2 4.31E+01 4.48E+01 3.9 S2 Base V2: 5.74E+03 5.93E+03 3.2 5.41E+03 3.71E+03 -31.4 5.89E+03 6.18E+03 4.9 S2 Base V3: 1.68E+03 2.17E+03 29.2 1.38E+03 1.42E+03 3.0 1.09E+03 1.11E+03 2.0 N2 Base M2: 2.35E+03 2.23E+03 -5.4 1.69E+03 1.99E+03 17.9 3.34E+03 3.34E+03 0.1 N2 Base M3: 3.40E+02 3.80E+02 11.6 2.09E+02 2.37E+02 13.9 2.00E+02 1.88E+02 -6.1 S2 Base M2: 4.92E+04 5.93E+04 20.6 3.79E+04 4.12E+04 8.6 3.37E+04 3.23E+04 -4.4 S2 Base M3: 9.81E+04 1.01E+05 2.8 9.24E+04 6.32E+04 -31.6 1.00E+05 1.05E+05 4.9 N1 Mid UX: 3.45E-02 3.66E-02 5.8 2.16E-02 2.72E-02 26.0 3.35E-02 4.52E-02 34.7 N1 Mid UY: 5.00E-02 7.09E-02 41.9 3.36E-02 4.48E-02 33.4 3.63E-02 3.82E-02 5.2 S1 Mid UX: 1.99E-02 1.98E-02 -0.6 1.37E-02 1.68E-02 22.7 1.76E-02 1.72E-02 -2.1 S1 Mid UY: 3.54E-02 4.96E-02 40.3 2.22E-02 2.99E-02 34.7 2.33E-02 2.87E-02 23.1 Mid-span A UX:   9.30E-02 1.16E-01 25.3 9.30E-02 1.16E-01 24.6 8.82E-02 1.32E-01 49.3 Mid-span A UY: 5.41E-02 6.67E-02 23.4 3.91E-02 4.23E-02 8.1 3.74E-02 3.53E-02 -5.8 Mid-span A UZ: 1.76E-01 1.86E-01 5.8 1.77E-01 2.25E-01 27.2 2.03E-01 2.60E-01 28.4 Mid-span D UX: 9.38E-02 1.18E-01 25.4 9.33E-02 1.09E-01 16.8 8.86E-02 1.30E-01 46.9 Mid-span D UY: 5.15E-02 7.42E-02 44.1 3.72E-02 5.41E-02 45.3 3.63E-02 4.57E-02 26.1 Mid-span D UZ: 1.67E-01 1.80E-01 7.3 1.42E-01 1.81E-01 26.9 1.61E-01 2.25E-01 39.9    125  G.4 Envelope of Response Quantities from all Ground Motions  Table G.4: Maximum and Minimum Response Quantities from all Ground Motion (Units in kN and m)  Original Model Updated Model  Max Min Max Min N1 Base V2: 1.01E+05 -1.14E+05 1.32E+05 -1.26E+05 N1 Base V3: 8.66E+04 -1.08E+05 1.12E+05 -1.18E+05 S1 Base V2: 1.59E+05 -2.03E+05 1.87E+05 -2.19E+05 S1 Base V3: 2.08E+05 -1.77E+05 2.24E+05 -1.65E+05 N1 Base M2: 5.37E+06 -5.57E+06 5.59E+06 -5.30E+06 N1 Base M3: 4.91E+06 -4.13E+06 5.40E+06 -4.96E+06 S1 Base M2: 6.02E+06 -9.08E+06 6.64E+06 -8.97E+06 S1 Base M3: 6.46E+06 -5.08E+06 7.02E+06 -5.93E+06 N2 Base V2: 2.06E+02 -1.56E+02 1.79E+02 -1.42E+02 N2 Base V3: 2.78E+02 -2.53E+02 2.37E+02 -2.27E+02 S2 Base V2: 7.14E+04 -8.68E+04 5.74E+04 -6.42E+04 S2 Base V3: 7.08E+03 -6.82E+03 7.98E+03 -7.43E+03 N2 Base M2: 2.25E+04 -1.91E+04 1.96E+04 -1.77E+04 N2 Base M3: 1.22E+03 -1.29E+03 1.17E+03 -1.37E+03 S2 Base M2: 1.82E+05 -1.68E+05 1.91E+05 -1.88E+05 S2 Base M3: 1.50E+06 -1.23E+06 1.11E+06 -9.98E+05 N1 Mid UX: 0.143 -0.168 0.213 -0.218 N1 Mid UY: 0.189 -0.182 0.213 -0.225 S1 Mid UX: 0.148 -0.120 0.192 -0.130 S1 Mid UY: 0.115 -0.127 0.145 -0.157 Mid-Span A UX:   0.477 -0.781 0.674 -0.964 Mid-Span A UY: 0.225 -0.275 0.190 -0.206 Mid-Span A UZ: 1.294 -0.835 1.737 -1.200 Mid-Span D UX: 0.481 -0.785 0.643 -0.916 Mid-Span D UY: 0.238 -0.258 0.282 -0.330 Mid-Span D UZ: 1.573 -0.970 1.849 -1.591  

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