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Evaluation of seismic performance of a pre-cast concrete block arch system through testing and numerical… Martínez Martínez, Amaia 2016

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EVALUATION OF SEISMIC PERFORMANCE OF A PRE-CAST CONCRETE BLOCK ARCH SYSTEM THROUGH TESTING AND NUMERICAL MODELING by  Amaia Martínez Martínez  Ingeniería de Caminos, Canales y Puertos, Universidad de Cantabria, 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)   May 2016  © Amaia Martínez Martínez, 2016 ii  Abstract  Lock-Block Ltd. from Vancouver, Canada, has developed an arch structural system made from modular pre-cast concrete blocks. The intention of the arch is to provide an easy to construct, cost effective and long lasting structure. This could be achieved with a modular, steel-free system. This study aims to assess the seismic performance of these arches characterizing their seismic behavior using a combination of experimental testing and numerical modeling. Several small scale unreinforced and reinforced arch models were subjected to quasi-static and dynamic testing. For the dynamic testing, a suite of earthquake records was selected of varying magnitudes, types and locations, and applied on a shake-table. From the results of the shake-table testing on the unreinforced models it was found that the arches tend to collapse by the four-hinge mechanism which is typical for these types of structures. For the reinforced arch testing, a steel band was instrumented to provide information on the loads. The reinforced arch performed well when subjected to the same suite of earthquakes. A numerical distinct element model was developed using 3DEC software and calibrated to the quasi-static test. The response of the numerical model matched the experiments with the arch exhibiting the same four-hinge failure mechanism. From numerical analysis, sensitivity studies were performed on various parameters of the arch. This included geometry, material properties and boundary/interface conditions.  It was found that in this configuration, the arches are vulnerable to seismic excitation and at risk of collapse when unreinforced and unconfined. There are several solutions to reduce that risk based on the results of this work: 1) addition of external or internal reinforcement to prevent hinge opening 2) restraint of the bottom courses of blocks and 3) modification of the geometry at the base to improve stability. iii  Preface This thesis is part of the research conducted for the NSERC grant at the ‘Earthquake Engineering Research Facility’ (EERF) at ‘The University of British Columbia’ in collaboration with Lock-Block Ltd. and with Prof. Carlos Ventura as the principal investigator. The author of this thesis, Amaia Martinez Martinez, was responsible for literature review, experimental testing, numerical model development, data processing and presentation of the results. Dr. Martin Turek (Research Associate at the University of British Columbia) was the lead researcher for the project. Lock-Block Ltd. provided all the small scale models used for the experimental testing.  Some of the work included in Chapter 3 and 4 have been presented at the 5th Tongji-UBC Symposium on Earthquake Engineering held in Shanghai, China, in May 2015 and at the 16th Canadian Conference on Earthquake Engineering in Victoria, Canada, in July 2015.  iv  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iii Table of Contents ......................................................................................................................... iv List of Tables ..................................................................................................................................x List of Figures ............................................................................................................................. xiv List of Symbols .............................................................................................................................xx List of Abbreviations ................................................................................................................ xxii Acknowledgements .................................................................................................................. xxiii Chapter 1: Introduction ................................................................................................................1 1.1 Background of the studied arch system .......................................................................... 1 1.2 Goal, objectives and scopes ............................................................................................ 4 1.3 Thesis outline .................................................................................................................. 5 Chapter 2: Literature Review .......................................................................................................6 2.1 Historical seismic performance of arches ....................................................................... 6 2.2 Theory of arch structures ................................................................................................ 8 2.2.1 Stability theory ............................................................................................................ 8 2.2.2 Arch response under seismic loading ....................................................................... 11 2.3 Numerical modeling of arch type structures ................................................................. 14 Chapter 3: Experimental Testing in The Unreinforced Models ..............................................17 3.1 Methodology ................................................................................................................. 17 3.2 Description of the models ............................................................................................. 18 v  3.3 Equipment used for the experimental testing ................................................................ 21 3.3.1 Description of the shake-tables ................................................................................. 21 3.3.1.1 APS electro-seis shaker..................................................................................... 21 3.3.1.2 Multi-axis shake-table (MAST) ........................................................................ 22 3.3.2 Description of the instrumentation............................................................................ 23 3.3.3 Description of the high-speed camera ....................................................................... 24 3.4 Tilt testing ..................................................................................................................... 25 3.4.1 Theory of quasi-static analysis by tilt testing ........................................................... 25 3.4.2 Test procedure and setup .......................................................................................... 26 3.4.3 Tilt testing results ...................................................................................................... 27 3.4.3.1 Collapse mechanism ......................................................................................... 27 3.4.3.2 Approximate collapse acceleration from tilt tests ............................................. 28 3.4.3.3 Effect of arch length on α ................................................................................. 29 3.5 Sine-sweep testing ........................................................................................................ 30 3.5.1 Test procedure and setup .......................................................................................... 30 3.5.2 Sine-sweep results ..................................................................................................... 30 3.6 Shake-table testing ........................................................................................................ 31 3.6.1 Applied scaling ......................................................................................................... 32 3.6.1.1 Applicability of dimensional analysis to this study .......................................... 34 3.6.2 Selection of the earthquakes ..................................................................................... 35 3.6.3 Test procedure and setup .......................................................................................... 35 3.6.4 Results of the unreinforced shake-table testing ........................................................ 37 3.6.4.1 Summary of the shake-table testing .................................................................. 37 vi  3.6.4.1.1 Preliminary study of the response of the arch to pulses .............................. 39 3.6.4.2 Collapse mechanism ......................................................................................... 41 3.6.4.2.1 Critical displacement of the blocks at collapse ........................................... 43 3.6.4.3 Comparison between uniaxial and tri-axial collapse mechanism ..................... 44 3.6.4.3.1 Comparison of collapse mechanism ............................................................ 44 3.6.4.3.2 Comparison of acceleration time histories .................................................. 45 3.7 Concluding remarks of experimental testing of unreinforced arches ........................... 46 Chapter 4: Experimental Testing in The Reinforced Models ..................................................47 4.1 Background in reinforcing of arches............................................................................. 47 4.2 Methodology ................................................................................................................. 49 4.3 Description of the reinforcement used for experiments ................................................ 50 4.4 Tilt Testing .................................................................................................................... 52 4.4.1 Description of the testing .......................................................................................... 52 4.4.2 Tilt testing results ...................................................................................................... 53 4.5 Sine-sweep testing ........................................................................................................ 54 4.5.1 Description of the sine-sweep testing ....................................................................... 54 4.5.2 Sine-sweep results ..................................................................................................... 54 4.6 Shake-table testing ........................................................................................................ 55 4.6.1 Test procedure ........................................................................................................... 55 4.6.2 Shake-table testing results......................................................................................... 56 4.6.2.1 Study of the recorded forces ............................................................................. 56 4.6.2.2 Study of the recorded accelerations .................................................................. 59 4.6.2.3 Study of the effect of the reinforcement ........................................................... 60 vii  4.6.2.4 Study of the effect of vertical motion ............................................................... 64 4.7 Concluding remarks of experimental reinforced testing ............................................... 64 Chapter 5: Numerical Analysis...................................................................................................65 5.1 Numerical model software ............................................................................................ 65 5.2 Description of the modeling.......................................................................................... 68 5.2.1 Modeling assumptions .............................................................................................. 68 5.3 Comparison between numerical and experimental quasi-static analysis ...................... 70 5.3.1 Collapse mechanism ................................................................................................. 71 5.3.2 Equivalent α .............................................................................................................. 72 5.4 Sensitivity analysis results ............................................................................................ 73 5.4.1 Deformability of the blocks ...................................................................................... 74 5.4.2 Length of the arch ..................................................................................................... 74 5.4.3 Thickness of the blocks ............................................................................................. 75 5.4.4 Density of the material .............................................................................................. 78 5.4.5 Joint shear stiffness ................................................................................................... 79 5.4.6 Friction Angle ........................................................................................................... 80 5.4.7 Comparison of the response of numerical models for arches with different scaling 81 5.5 Concluding remarks of the numerical analysis ............................................................. 81 Chapter 6: Discussion of The Results.........................................................................................82 6.1 Unreinforced model ...................................................................................................... 82 6.1.1 Collapse mechanism ................................................................................................. 82 6.1.2 Collapse accelerations ............................................................................................... 82 6.1.3 Acceleration and displacement sensitivity studies .................................................... 83 viii  6.1.4 Effect of the vertical motion on the response of the arch ......................................... 83 6.1.5 Stability of the unreinforced model .......................................................................... 84 6.2 Reinforced model .......................................................................................................... 85 6.2.1 Sensitivity of the reinforced arches ........................................................................... 85 6.2.2 Recorded forces ........................................................................................................ 86 6.2.3 Effect of the pretension ............................................................................................. 86 Chapter 7: Conclusions and Future Work ................................................................................87 7.1 Specific objectives ........................................................................................................ 87 7.2 Important observations from the study ......................................................................... 88 7.3 Potential solutions ......................................................................................................... 89 7.4 Further experimental testing ......................................................................................... 90 7.5 Further development of the numerical model ............................................................... 90 Bibliography .................................................................................................................................91 Appendices ....................................................................................................................................97 Appendix A - Instrumentation used for the experimental testing ............................................. 97 A.1 Specifications for the shake-tables............................................................................ 97 A.2 Control system ........................................................................................................ 101 A.3 Hydraulic system .................................................................................................... 103 A.4 Data acquisition (DAQ) .......................................................................................... 104 A.5 Accelerometers ....................................................................................................... 106 A.6 High-speed camera.................................................................................................. 107 A.7 Strain gauges ........................................................................................................... 108 Appendix B - Experimental unreinforced testing ................................................................... 112 ix  Appendix C - Experimental reinforced testing ....................................................................... 114 Appendix D - Numerical analysis ........................................................................................... 119  x  List of Tables  Table 3.1 Physical parameters of each of the tested models ......................................................... 18 Table 3.2 Collapse tilt angle and the corresponding α for the tested models ............................. 28 Table 3.3 Collapse tilt angle and the corresponding α for the 3 m interior diameter 1/25 scale model with different lengths ......................................................................................................... 29 Table 3.4 Scaling factors for each dimension for different model types (Krawinkler, 1979; Tomaževič & Velechovsky, 1992) ............................................................................................... 33 Table 3.5 Scaled factors based on ‘Complete Model’ for 1/25 and 1/12.5 scale model .............. 34 Table 3.6 Suite of time history records used for the shake-table testing ...................................... 35 Table 3.7 Parameters of the applied time scaled earthquakes for the 1/25 scale model ............... 36 Table 3.8 Parameters of the applied time scaled earthquakes for the 1/12.5 scale model ............ 36 Table 3.10 Summary of shake-table tests on the unreinforced 1/25 scale 6 m arch .................. 38 Table 3.11 Summary of shake-table tests on the unreinforced 1/12.5 scale 6 m arch ............... 38 Table 3.9 Peak displacements (Dmax) of the middle key block relative to the base at collapse TL and at a lower TL for three different earthquakes......................................................................... 43 Table 4.1 Instrumentation of the 1/12.5 small-scale arch model ................................................ 52 Table 4.2 Ratio of the recorded peak forces to the weight of the model in percentage at collapse TL for different earthquakes ......................................................................................................... 57 Table 4.3 Recorded peak forces and FnW ratios for different earthquakes at different TLs. ........ 58 Table 4.4 FtnW ratio for different pretension and test levels during Loma Prieta earthquake....... 61 xi  Table 4.5 Recorded average peak forces at the straps and FnW ratio for uniaxial Northridge shake-table testing at different pretension levels .......................................................................... 62 Table 5.1 Properties of the blocks and of the interfaces between blocks in 3DEC ...................... 70 Table 5.2 Comparison of collapse tilt angle and corresponding αbetween numerical and experimental results ...................................................................................................................... 73 Table 6.1 Recorded IBA-s and IBD-s during shake-table testing for collapse TL in the three directions for the 1/12.5 scale arch model .................................................................................... 83 Table 6.2 Effective thickness and HGF values obtained from the numerical analysis for the calibrated models of the 1/25 and 1/12.5 scale structures ............................................................ 85 Table A.1 Specifications for the APS ELECTRO-SEIS shaker ................................................... 97 Table A.2 Specifications for the MAST ....................................................................................... 99 Table A.3 Jaguar dB table values with their corresponding equivalent intensity and the approximate value used during the study.................................................................................... 102 Table A.4 Specifications of the strain gages used for the reinforced experimental testing ........ 109 Table A.5 Calibration of the straps used for the reinforcing of the 1/25 scale model ................ 111 Table B.1 Obtained collapse tilt angle and the corresponding α for different arch models ....... 112 Table B.2 Obtained angle of collapse and the corresponding α for the 1/25 scale 3 m model with different lengths .......................................................................................................................... 112 Table B.3 Parameters of the selected original time history records ........................................... 113 Table C.1 Measured displacements of the middle key block at 30 and 45 degrees of tilt for different pretension levels for tilt testing in 1/12.5 scale reinforced model ............................... 114 Table C.2 Pretension in the straps at the beginning and end of each sine-sweep test ................ 114 xii  Table C.3 Recorded strains in each channel and the corresponding ratio for different earthquakes at collapse TL using 1/25 scale arch ........................................................................................... 115 Table C.4 Recorded peak forces in each of the channels for different earthquakes at different TL for tri-axial shake-table testing using 1/12.5 scale arch .............................................................. 115 Table C.5 Recorded accelerations at the shake-table and top of the model for different earthquakes at different TL for tri-axial shake-table testing ....................................................... 116 Table C.6 Recorded forces, initial pretension and final pretension at different channels of the straps for Northridge uniaxial reinforced shake-table testing ..................................................... 117 Table C.7 Recorded peak forces in each of the channels, average and FnW values for different pretension levels and different TL for biaxial Loma Prieta record using the 1/12.5 scale arch . 118 Table D.1 Data obtained from quasi-static analysis using 3DEC for totally rigid and deformable blocks with different parameters ................................................................................................. 119 Table D.2 Data obtained from quasi-static analysis using 3DEC for the sensitivity analysis of the thickness of the blocks ................................................................................................................ 119 Table D.3 Data obtained from quasi-static analysis using 3DEC for the sensitivity analysis of the thickness of the blocks for the arch with first two rows of blocks fixed .................................... 119 Table D.4 Data obtained from quasi-static analysis using 3DEC for the sensitivity analysis of the density for the arch with the first row of blocks fixed ................................................................ 119 Table D.5 Data obtained from quasi-static analysis using 3DEC for the sensitivity analysis of the density for the arch with first two rows of blocks fixed ............................................................. 120 Table D.6 Data obtained from the quasi-static analysis using 3DEC for the sensitivity analysis of the shear stiffness ........................................................................................................................ 120 xiii  Table D.7 Data obtained from the quasi-static analysis using 3DEC for the sensitivity analysis of the shear stiffness for the arch with first two rows of blocks fixed ............................................ 120 Table D.8 Data obtained from the quasi-static analysis using 3DEC for the sensitivity analysis of the friction angle ......................................................................................................................... 121  xiv  List of Figures  Figure 1.1 Lock-Block retaining wall ............................................................................................. 2 Figure 1.2 Pre-cast block with cross-shape shear key .................................................................... 2 Figure 1.4 Standard configuration of Lock-Block arch .................................................................. 3 Figure 1.5 Lock-Block 3m arch structure ....................................................................................... 3 Figure 1.5 Construction of a Lock-Block arch structure using the ‘zipper truck’ .......................... 4 Figure 2.1 Failure of the exit vault of the Basilica of Saint Francis of Assisi in 1997 ................... 7 Figure 2.2 Collapse of the dome of a) the Santiago church in Spain during the 2011 earthquake and b) the San Marco church in L’Aquila, Italy, during the 2009 earthquake ............................... 7 Figure 2.3 Hinging and corresponding thrust line (as dashed line) for two different arches under self-weight: (a-b) for catenary shape arch,  (c-d) for 180 degrees circular arch (from Heyman 1982) ............................................................................................................................................... 9 Figure 2.4 Force polygon for equilibrium of a block of an arch (from Saphiro 2012) ................. 10 Figure 2.5 Graphical method of a) thrust line calculation for a 180 degrees arch with b) the corresponding force polygon at the right determining the angle of thrust line for each block (from Saphiro 2012) ................................................................................................................................ 11 Figure 2.6 Four hinge mechanism of an arch introduced by Oppenheim (1992) ......................... 12 Figure 2.7 Failure modes for a) in-plane loading in a circular arch b) out-of-plane loading for a pointed arch (Lemos 1998) ........................................................................................................... 15 Figure 3.1 Specifications for height, interior diameter and voissours number in blue ................. 19 Figure 3.2 1/25 small scale model of the 1.5 m interior diameter arch structure ......................... 19 Figure 3.3 1/25 small scale model of the 3m interior diameter arch structure ............................. 19 xv  Figure 3.4 1/25 small scale model of the 6m interior diameter arch structure ............................. 20 Figure 3.5 1/12.5 small scale model of the 6m interior diameter arch structure .......................... 20 Figure 3.6 Small-scale Lock-Block with the cross-shaped shear key a) showing male connection b) showing female connection of the middle key stone .............................................................. 21 Figure 3.7 APS ELECTRO-SEIS Shaker ..................................................................................... 22 Figure 3.8 MAST with the 1/12.5 scale model on top .................................................................. 23 Figure 3.9 Accelerometer used at the center of the MAST .......................................................... 23 Figure 3.10 Accelerometer used at the top and bottom of the model ........................................... 24 Figure 3.11 Phantom V4.2 High Speed Camera ........................................................................... 24 Figure 3.12 Scheme for the tilting concept of a block .................................................................. 25 Figure 3.13 Setup of the tilt test performed in the unreinforced 1/25 scale models .................. 26 Figure 3.14 Setup of the tilt test performed in the unreinforced 1/12.5 scale models ............... 26 Figure 3.15 Collapse mechanism recorded with the high speed camera for the 1/12.5 scale model a) as the collapse was initiated and b) as the collapse progresses showing the additional fifth hinge in orange .............................................................................................................................. 27 Figure 3.16 Collapse mechanism of the 1/12.5 scale 6 m interior diameter arch model when tilting the structure from the left side ............................................................................................ 28 Figure 3.18 Fourier spectra of the recorded data from the recorded top acceleration for unreinforced sine-sweep testing .................................................................................................... 31 Figure 3.21 Failure domain defined by acceleration and frequency of the main pulse for the 1/25 scale unreinforced model .............................................................................................................. 40 Figure 3.22 Failure domain defined by acceleration and frequency of the main pulse for the 1/12.5 scale unreinforced model ................................................................................................... 40 xvi  Figure 3.19 Different stages of the typical four-hinge failure mechanism taken from the high-speed camera footage for Tokachi-Oki earthquake with the four hinges of the collapse mechanism  in red and the additional fifth hinge in orange .......................................................... 42 Figure 3.20 Different stages of the typical four-hinge failure mechanism taken from the high-speed camera footage for uniaxial Northridge earthquake with the four hinges of the collapse mechanism  in red and the additional fifth hinge in orange .......................................................... 42 Figure 3.23 Different stages of the collapse of the 1/12.5 scale arch taken from the high-speed camera footage subjected to tri-axial Northridge, with red circles for the four hinge collapse mechanism, orange circle for the fifth additional hinge and green circle for the sixth additional hinge .............................................................................................................................................. 44 Figure 3.24 Time histories of bottom and top acceleration in the transversal direction for Northridge tri-axial test at collapse level ...................................................................................... 45 Figure 3.25 Time histories of bottom and top acceleration in the transversal direction for Northridge uniaxial test at collapse level ...................................................................................... 46 Figure 4.1 Interaction forces between reinforcement and arch when the strap is added at the extrados (A) or intrados (B) (from Jurina 2013) ........................................................................... 48 Figure 4.2 Instrumentation of the 1/25 small scale arch model .................................................. 51 Figure 4.3 Instrumentation of the 1/12.5 small-scale arch model ................................................. 51 Figure 4.4 Setup of the tilt test performed in the instrumented reinforced 1/12.5 scale model 53 Figure 4.5 Pretension of the straps versus the measured displacements of the middle key block at 30 and 45 degrees of inclination ................................................................................................... 53 Figure 4.6 Fourier spectra of the recorded accelerations at the top of the model for sine-sweep test #1, #2 and #3in the reinforced model ..................................................................................... 54 xvii  Figure 4.7 FnW ratio for different test levels of the Loma Prieta, Northridge and Parkfield earthquakes ................................................................................................................................... 58 Figure 4.8 Recorded accelerations at the bottom and top of the model for tri-axial Loma Prieta at 100% TL ....................................................................................................................................... 59 Figure 4.9 Recorded accelerations at the bottom and top of the model for tri-axial Northridge at 100% TL ....................................................................................................................................... 59 Figure 4.10 Recorded accelerations at the bottom and top of the model for tri-axial Parkfield at 100% TL ....................................................................................................................................... 60 Figure 4.11 FtnW ratio for different pretension and test levels during Loma Prieta earthquake ... 61 Figure 4.12 FnW ratio for different pretension and test levels during Loma Prieta earthquake .... 62 Figure 4.13 Recorded accelerations at the bottom and top of the model for uniaxial Northridge at 100% TL for 108N of pretension ............................................................................................... 63 Figure 4.14 Recorded accelerations at the bottom and top of the model for uniaxial Northridge at 100% TL for 64 N of pretension ................................................................................................ 63 Figure 5.1 Representation of contacts between blocks as a) joint element, and b) vertex-edge contact (Lemos 2007) ................................................................................................................... 66 Figure 5.2 Representation of contacts between blocks by a) two VF and two EE, and b) an actual EE contact (Lemos 2007) .............................................................................................................. 67 Figure 5.3 Schematic calculation cycle followed by 3DEC (Itasca, 3DEC 2013) ....................... 67 Figure 5.4 Numerical model of the 1/12.5 small scale arch using 3DEC software ...................... 68 Figure 5.5 Vertical and horizontal components of the gravity when tilting a block .................... 71 Figure 5.6 Comparison of four-hinge collapse mechanism during quasi-static analysis (structure tilted from the left side) from a) experimental testing and b) numerical model ........................... 72 xviii  Figure 5.7 Collapse mechanism of the same arch with three different lengths: 1 block, 4 blocks and 20 blocks ................................................................................................................................ 75 Figure 5.8 Percentage of block thickness versus collapse αobtained from quasi-static analysis . 75 Figure 5.9 Percentage of block thickness versus α obtained from the quasi-static analysis for the arch with first two rows of blocks fixed ....................................................................................... 77 Figure 5.10 Normalized density versus α obtained from quasi-static analysis for two numerical models with different boundary conditions .................................................................................. 78 Figure 5.11 Shear stiffness versus α obtained from quasi-static analysis for two numerical models with different boundary conditions .................................................................................. 79 Figure 5.12 Friction angle versus collapse α obtained from the quasi-static analysis .................. 80 Figure A.1 Force envelope with frequency for the APS shaker ................................................... 98  Figure A.2 Velocity envelope with frequency for the APS shaker.............................................. 98 Figure A.3 Velocity in x-axis versus frequency for the MAST .................................................. 100 Figure A.4 Velocity in y-axis versus frequency for the MAST .................................................. 100 Figure A.5 Velocity in z-axis versus frequency for the MAST .................................................. 101 Figure A.6 HPS for the MAST ................................................................................................... 103 Figure A.7 Accumulator bank ..................................................................................................... 103 Figure A.8 Manifolds in room adjacent to table pit .................................................................... 104 Figure A.9 DAQ cabinet with SCXI chassis and connected PC................................................. 105 Figure A.10 Specifications of the seismic, ceramic shear ICP accelerometer ............................ 106 Figure A.11 Specifications of the +/- 10g silicon MEMS top and bottom accelerometer (highlighted inside of the red rectangle) ..................................................................................... 107 Figure A.12 Strain gauge used for the instrumentation of the reinforcement ............................ 109 xix  Figure A.13 Schematic representation of the set-up and instrumentation for the calibration of the straps at the 1/25 scale model ..................................................................................................... 110  xx  List of Symbols α = approximate horizontal acceleration obtained from tilting of a structure acc = acceleration  acc0  =  constant acceleration value for the demand of the limit-state function  Astrap = cross sectional area of the strap β = reliability index Dmax = peak displacement of the center of the middle key in the transversal direction of the arch relative to the base motion ε = error of the approximate fitting function to several data points ɛch = measured strain in  a particular channel of the steel strap used as reinforcement Estrap = modulus of elasticity of the steel strap used as reinforcement obtained from the performed calibrations F = force  f = frequency in Hertz Fch= force in a particular channel of the steel strap used as reinforcement FnW = ratio between peak dynamic forces (without the pretension) recorded at the strap and the weight of the model FtnW = ratio between peak total forces recorded at the strap and the weight of the model 𝐹𝑝𝑒𝑎𝑘 =  average peak force calculated from the recorded strains in the reliable channels i = given quantity  L = length  m = mass  xxi  ӨAB, ӨBC and ӨCD  = link rotations (to the horizontal line) for the displaced configuration of the arch ρ = density of the material 𝑟𝐴𝐵̅̅ ̅̅  = the direction of the distance from point A to the center of gravity of the original (non-displaced) analytical arch configuration Si = scale factor of i parameter T = period t = time  b/r = thickness-interior radius ratio  v = velocity w = frequency W = total weight of the model ψAB = angle between 𝑟𝐴𝐵̅̅ ̅̅  and the displaced AB link xxii  List of Abbreviations Ch: Channel DAQ: Data Acquisition  DEM: Discrete Element Modeling EE: Edge-Edge  EERF: Earthquake Engineering Research Facility EQ: Earthquake Ext: exterior  FEM: Finite Element Modeling FFT: Fast Fourier Transformation  HPS: Hydraulic Power Supply Int: interior  IBA: Instantaneous Base Accelerations  IBD: Instantaneous Base Displacements MAST: Multi-Axis Shake-table  PGA: Peak Ground Acceleration PGD: Peak Ground Displacement TL: Test Level  ULS: Ultimate Limit State  VF: Vertex-Face    xxiii  Acknowledgements  When I first arrived to Vancouver, I would have never thought about pursuing a Master and even less in a topic completely unknown to me as earthquake engineering. However, I could not be more thankful to Prof. Carlos E. Ventura for entering my life and giving me the opportunity of working with him and his team. During these years, he has inspired me to tackle any problem ambitiously and helped me whenever it was needed.  Besides Carlos Ventura, I would like to thank my other supervisor Dr. Martin Turek. All this work would not have been possible without his guidance, patience and expertise. Thank you for making yourself always available for me! I would also like to acknowledge Jay Drew, Arley Drew and Brent Wallace from Lock-Block Ltd. for their excitement about the project, their assistance and their fast responses to any inquire. Their collaboration has been essential for the development of this work. I would also like to acknowledge the financial support for this research from NSERC (National Sciences and Engineering Research Council of Canada). In addition, I would like to thank Mr. Scott Jackson, Mr. Jermin Dela Cruz, Mr. Garrilo Lazio and Mr. Simon Lee from UBC for their contributions during the shake table testing; and Mr. Bryn Wong for helping me with the data analysis and experimental testing.  Furthermore, I would like to thank all those special people that I have met during this journey. Thanks for making Vancouver my second home. And thank you for being always there ready to support me and make me smile.  Last but not least, I would like to thank all my family, especially my parents and sister, for always believing in me and pushing me to give the best of myself. Amaia Martinez Vancouver, April 2016 1  Chapter 1: Introduction The use of masonry arches dates back thousands of years. In British Columbia (Canada), a company called Lock-Block is proposing to use the ancient concept of masonry arches utilizing pre-cast concrete blocks. Prior to approval for construction, an evaluation of their seismic behavior should be performed in order to avoid any catastrophic failure. The Earthquake Engineering Research Facility (EERF) at The University of British Columbia started working in collaboration with Lock-Block Ltd. to perform an engineering seismic assessment of their pre-cast block arch and dome structures. This document will discuss part of the conducted research in support of that assessment. This first section will provide some background about Lock-Block arch structures and set the goals, objectives and scope of this study. A summary of the contents of this thesis will be also given.  1.1 Background of the studied arch system Since 1984 Lock-Block Ltd., from Vancouver (British Columbia), has manufactured pre-cast concrete products including highway lane barriers, retaining walls (see Figure 1.1), pole guards, etc. All of their products can be made using either recycled or virgin concrete.  2   Figure 1.1 Lock-Block retaining wall  Each pre-cast unit has a cross-shape interlocking shear key that allows for assemblage between blocks. This key is shown in Figure 1.2.  Figure 1.2 Pre-cast block with cross-shape shear key More recently, the company has created a system of utilizing these pre-cast blocks in arch and dome structures as a modern approach of the typical masonry arch used by the Romans. These structures could be potentially adopted for short-span highway and railway overpasses, wine cellars, creek crossing and other applications.  The primary objective of these arch systems is to have an easy to build structure, with a long service life by avoiding the use of internal steel reinforcement.  The configuration of the standard arch proposed by Lock-Block features two rows of straight blocks followed by a half circular arch consisting of identical blocks (see Figure 1.3). The advantage of this configuration is greater 3  interior height and minimizing cost due to repeatability of fabrication. A full scale 3 meter interior diameter structure with the standard configuration using 9 blocks to form the half circular shape is shown in Figure 1.4.  Figure 1.3 Standard configuration of Lock-Block arch  Figure 1.4 Lock-Block 3m arch structure The arches can be built in relatively short time using chains and several extra blocks, and a single excavator. Once all the blocks are in place, the middle keystone is placed and the chains are loosened to bring the structure into place. A more efficient way to assemble the blocks is using the specially designed ‘zipper truck’, shown in Figure 1.5. The arched-rollers on the back of the truck function as the frame for the structure supporting the blocks while they are set in place. The truck is tapered from front to back, so as it moves forward the blocks fall together creating the continuous arch system. 4   Figure 1.5 Construction of a Lock-Block arch structure using the ‘zipper truck’  1.2 Goal, objectives and scopes The main goal of this thesis is to assess the seismic performance of unreinforced Lock-Block arch system in its standard configuration. This work is focused on the following objectives:   Develop observations on the performance of scaled physical models from experimental quasi-static and dynamic testing.   Perform shake-table testing to define the sensitivity of the structure to different ground shaking.  Examine the critical failure mode of the structure for developing possible solutions to prevent collapse.  Evaluate potential reinforcing to improve the performance of the arch system.   Develop a numerical model and calibrate it to the experimental testing for use in more detailed studies.  The scope of the experimental testing is limited to small scale arch models, which size was chosen according to the availability of the models provided by Lock-Block. The experimental testing 5  program was comprised of tilt, sine-sweep and shake-table testing. The scope of the numerical modeling is limited to the creation of the model and calibration to tilt testing. No dynamic analysis was conducted using the numerical model. However, future work will include the calibration and validation of the arch model to time history experimental testing.   1.3 Thesis outline The thesis is divided in seven different Chapters. A brief description of each is provided next. - Chapter 1 includes the introduction, which gives some background about the arch systems of this study and sets the goal, objectives and scope of this work. - Chapter 2 reviews the history of the performance of arch structures and introduces previous research on static and dynamic behavior of arches. Computer modeling used for past historical masonry arch structures is also described.  - Chapter 3 explains the methodology, procedure and setup for the experimental testing performed on the unreinforced small scale arch models and describes the obtained results. - Chapter 4 defines the methodology, procedure and setup for the experimental testing conducted using the reinforced small scale arch models and presents the obtained results.   - Chapter 5 describes the modeling and calibration of a numerical model, which was created using 3DEC discrete element software. It also includes a sensitivity analysis conducted for different parameters of the computer model.   - Chapter 6 compares and discusses some of the results of the experimental testing presented in Chapter 3 and 4 and the numerical modeling in Chapter 5.  - Chapter 7 summarizes the work presented in this document, and presents the final conclusions and recommendations for future research.  6  Chapter 2: Literature Review This Chapter provides a brief review of the historical performance of masonry arches in seismic areas. It also includes a summary of the analytical and numerical studies about arch static and dynamic behavior.   2.1 Historical seismic performance of arches  Arches were used for the first time in the 2nd millennium B.C. as the Mesopotamian brick architecture mainly as underground structures. However, a full understanding and appreciation of the advantages of the arch, vault and domes came with the Romans in the first century B.C. (Robertson 1943). During the Roman Empire, arch structures were used for the erection of bridges, gates, aqueducts and roofing (for large interior spaces).  Although it is rare today to build according to this structural typology, many historical buildings designed with arch elements are still in use and have survived years of settlements and deterioration. Moreover, many of them are located in high seismic risk areas and have performed adequately during seismic past events. Nevertheless, several collapses have been recorded. For instance, the Basilica of Saint Francis of Assisi (see Figure 2.1), in Italy, failed during a Richter magnitude 6 earthquake in 1997 after having survived many stronger earthquakes during previous centuries (Piermarini 2013).  7   Figure 2.1 Failure of the exit vault of the Basilica of Saint Francis of Assisi in 1997 Some other examples are the failure of the dome of the Santiago church in Spain (see Figure 2.2 a) when a magnitude 5.1 earthquake hit the area in 2011, or the collapse of the dome and vault of the San Marco church of L’Aquila in Italy in 2009 (see Figure 2.2 a) caused by a magnitude 6.3 earthquake (Panagiotis et al 2015).           Figure 2.2 Collapse of the dome of a) the Santiago church in Spain during the 2011 earthquake and b) the San Marco church in L’Aquila, Italy, during the 2009 earthquake   a) b) 8  Over the years a good understanding of the load capacity of arch systems has been developed (Pippard et al 1936, Heyman 1982). This knowledge has been complemented with some studies about the dynamic behavior of masonry type arch structures (Oppenheim 1992). However, most of the research in this field has been focused on the assessment of existing buildings and infrastructures.   2.2  Theory of arch structures The idea of a pure compression structure dates back to 1675 when Hooke cited, “As hangs the flexible line, so but inverted will stand the rigid arch”. This hanging line, later named catenary, was mathematically proved by Gregory (1697), who wrote: “…none but the catenaria is the figure of a true legitimate arch. And when an arch of any other figure is supported, it is because in its thickness some catenaria is included.” Since then, the response of arch structures under static and dynamic loading has been studied through several methods, some of which will be briefly described in this section. 2.2.1 Stability theory The stability theory for masonry arches is based on three basic assumptions that Couplet made in 1730 for masonry structures (Heyman 1972): 1) Masonry has no tensile strength 2) Masonry has infinite compressive strength 3) Sliding failure between arch voissours does not occur (friction between blocks is enough to prevent it) These assumptions do not allow the structure to fail due to sliding or crushing of the material. While failure due to sliding has been rarely shown in practice, it has been proved that in some 9  cases strength of the material may need to be considered as a potential critical mechanism for the collapse of the structure. Some of those special cases are: 1) when the structure has been restrained or 2) when mortar with appreciable tensile strength between blocks has been added. However, in general, arches fail by opening between blocks. When the arch can no longer be in equilibrium, a hinge based mechanism is developed and the structure collapses. This failure mechanism was first introduced by Pippard et al (1936), who determined that the collapse mode of the arch depends on three parameters: 1) shape (or geometry), 2) self-weight of the arch, and 3) the position of the applied load. Heyman (1982) extended this theory introducing the line of thrust concept, as a representation of the resultant stresses between voussoirs. The thrust line is calculated through equilibrium of the compression forces at the interface with the self-weight and any applied external forces. If the thrust line is lying within the thickness of the arch, the arch will stand. Otherwise, hinges will be created at the points where the thrust line touches the inner or outer ratio of the arch as shown in Figure 2.3.  Figure 2.3 Hinging and corresponding thrust line (as dashed line) for two different arches under self-weight: (a-b) for catenary shape arch,  (c-d) for 180 degrees circular arch (from Heyman 1982)  10  Several graphical and analytical equilibrium methods have been developed in the past to study this problem. The first ones (introduced by Poleni (1747), Snell (1846) and Karl Cunmann (1866) and recently studied by Huerta (2004) and Allen and Zalewski (2009)) require creating a force polygon. Figure 2.4 shows the force polygon for one of the blocks of the arch where W is the weight of the block, C1 is the initially established horizontal thrust value and C2 is the calculated angle of thrust line for that block.  Figure 2.4 Force polygon for equilibrium of a block of an arch (from Saphiro 2012)  Figure 2.5 shows the application of the graphical force polygon theory to a 180 degrees arch. In the figure, a masonry arch with the minimum necessary thickness to stand its self-weight is shown. The thrust line is highlighted in red. At the right of the arch drawing, the force polygon that determines the angle of the thrust line for each block (C2 of Figure 2.4) is shown. 11   Figure 2.5 Graphical method of a) thrust line calculation for a 180 degrees arch with b) the corresponding force polygon at the right determining the angle of thrust line for each block (from Saphiro 2012)  The main disadvantage of these methods is the tedious graphical calculation. Thus, graphical methods have been replaced by analytical equilibrium methods. Gilbert and Melbourne (1994), Hughes and Blacker (1997) and Boothby (1995) are some of the researches that have applied numerical equilibrium methods for the calculation of the thrust line. In their studies, it is assumed that at collapse level the thrust line is perpendicular to the inner or outer radio in four different locations, where the arch would hinge turning the structure into a mechanism. 2.2.2 Arch response under seismic loading The assessment of masonry type arch structures under seismic loading was first studied by approximating the dynamic effects through a quasi-static analysis. For this analysis, earthquake loading is simplified as a constant horizontal acceleration. Although this analysis provides a good understanding of the seismic performance of the structure, the results tend to be conservative since earthquakes induce instantaneous accelerations and limited duration.  a) b) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 19 10 12  Oppenheim (1992) proposed an analytical model for masonry arches under horizontal ground accelerations. The model assumed fixed hinging points and is based on hinge to hinge rigid-body geometry.  The arch at collapse is described as a single degree of freedom (SDOF) structure with three links forming a four hinge mechanism (see Figure 2.6). In the figure, ӨAB, ӨBC and ӨCD are the rotations of the links (AB, BC and CD) with respect to the horizontal line for the displaced configuration of the arch. The direction of the distance from point A to the center of gravity of the original (non-displaced) configuration is represented as 𝑟𝐴𝐵̅̅ ̅̅ .  Furthermore,  ψAB is the angle between 𝑟𝐴𝐵̅̅ ̅̅  and the displaced AB link.  Figure 2.6 Four hinge mechanism of an arch introduced by Oppenheim (1992) While Oppenheim (1992) solved the required constant horizontal acceleration through Hamilton’s principle and Lagrange equation, Clemente (1998) studied the same problem using the principle of virtual power (similar concept to virtual work but using virtual velocities instead of displacements). Later on, Appleton (1999) and Oschendorf (2002) demonstrated that tilting of the structure can determine the base acceleration required for collapse. In the method, referred to as the tilt test, the arch is lifted from one side until collapse. That collapse tilt angle can simply be determined as a fraction of the downward gravitational acceleration.  13  The quasi-static analytical models described above are convenient and useful tools for understanding dynamic problems. Nevertheless, the dynamic response of simple structures has also been studied through derivation of analytical equations of motions. The dynamic analysis for the rocking of a single rigid block was first introduced by Housner (1963). Yim et al. (1980) studied the same problem and concluded that rocking of blocks is not easy to predict since it depends on the nature of the ground shaking. Since then, research has been conducted to predict the behavior of block structures under harmonic or seismic loading. Oppenheim (1992) and Clemente (1998) were the pioneers of using dynamic analysis for arch structures as an extension of quasi-static analysis. Both of them demonstrated analytically that when a rectangular pulse motion is applied, the response of the arch depends not only in the magnitude but also in the duration of the pulse. The instantaneous maximum ground acceleration that the arch can take reduces as the duration of the pulse increases. When the pulse exceeds a certain duration, a critical ground acceleration is reached; which is equal to the collapse α obtained through tilt test.  Appleton (1999) performed experimental tilt and time history testing based on Oppenheim’s work. However, the main contribution to the validation of analytical with experimental response of arch structures was conducted by DeJong (2009). He performed shake-table testing on scaled models of arches, and compared the results to the analytical model of DeLorenzis et al (2007).  For the experimental program, the excitation was applied as harmonic motions and as earthquake ground motions. Five ground motions with different main impulses were selected to observe a range of behaviors in the arch. The important results that came from this work included: i)    Rocking type failure based on the four-hinge mechanism governs. ii)    Elastic resonance does not occur due to its high frequency (>300Hz) relative to the earthquake input motions. 14  iii)    The analytical model provided accurate failure prediction when using the ‘primary’ impulse from the earthquake record. iv)     80% of the thickness of the blocks needs to be considered when modeling the structure to take into account corner rounding, imperfections and instabilities of the real structure. v)     From harmonic testing it was found that one impulse right after the other have no negative effect, sometimes even helping to stabilize the structure.  vi)     The critical failure mode is collapse in the second half cycle of the impulse vii)     The peak collapse acceleration of the arch decreases asymptotically with the increase of    the impulse duration. viii) A suite of failure curves was created to identify the failure acceleration for a variety of arch geometries; the arch is more stable for higher thickness-radius (b/R) ratios and lower inclusion angles (β). ix)     Scaling effects are important. Arches with same proportions b/R and β, but different R collapse at different acceleration levels. The collapse acceleration level increases by the square root of the ratio of the radius for a larger scale model (as stated by Oppenheim (1992)). x)    Better accuracy of hinging spots is obtained when the number of voissours is increased. DeJong (2009) also proposed an assessment criteria of arch structures based on the failure-domain curves he developed. However, he did not include three directional ground shaking effects.  2.3 Numerical modeling of arch type structures Analytical solutions for arch structures under lateral dynamic loads have been widely studied since Oppenheim (1992). However, these analytical models are based on simple 2D geometries and mostly rectangular type impulses. In order to provide a better insight of the nature of the real 15  seismic behavior of the arch, computer programs that can address correctly the complexity of the problem can be used. In order to capture the discontinuity of masonry type structures, discrete element (DE) programs are preferred to finite element (FE) ones. Among several different DE techniques, 3DEC code (from Itasca Consulting Group) has been used in several studies to capture the response of masonry structures.  In the last decade the validation of block representation for historical masonry structures using 3DEC has progressed considerably due to the increasing available experiment data. For instance, Lemos (2001) compared the deformation of a section of a façade of a monastery in Lisbon using 3DEC with the experiments performed in a full-scale replica. Previous to that in 1998, he studied the in-plane and out-of-plane response of different type of arches under dynamic loading (see Figure 2.7 ).   Figure 2.7 Failure modes for a) in-plane loading in a circular arch b) out-of-plane loading for a pointed arch (Lemos 1998)  Papantonopoulus et al. (2002) and Psycharis et al (2003) also compared numerical models to experimental results in drum columns of a Parthenon. Although qualitatively 3DEC models matched well the overall and peak responses of the tested specimens, the exact location of the joints where sliding or hinging occurred varied and could not be predicted (Lemos 2007).   Toth et al (2006) modeled an existing bridge and conducted a study of the implemented parameters and their effect in the arch response. He found that this software can be a useful and effective tool a) b) 16  with an experimental verification and a development of the necessary input parameters. Sarhosis et al (2014) also performed a sensitivity analysis of the modeling parameters using UDEC software (2D version of 3DEC) for validation to experimental testing of a masonry wall under vertical loading. He also compared the DE model results to FE model ones, and he concluded that in general, UDEC results matched better the collapse mode and crack propagation observed in the laboratory, except when low-strength masonry was used.  17  Chapter 3: Experimental Testing in The Unreinforced Models The first part of the work involved experimental testing, including quasi-static tilt, sine-sweep and shake-table testing of unreinforced arch models. The purpose of the testing was to establish baseline information and to examine the performance of the models when subjected to a variety of appropriately scaled earthquakes. In this Chapter the methodology, the models and the equipment used during the experiments are described, followed by a detailed explanation of the procedure, setup and results obtained for each test.   3.1 Methodology  The objective of this experimental testing was to assess the performance of the unreinforced small scale arches under seismic loading. The experimental program included the following:  Quasi-static tilt testing: an approximation to the maximum ground acceleration that the arch withstands can be obtained by monotonically tilting the models until failure. Additionally, a study of the stability of the models was performed by comparing the response of several arches with varying dimensions.  Sine-sweep testing: the intention of this testing was to obtain the natural frequencies of the unreinforced arch structures.    Shake-table testing: small scale arches were tested at increasing intensity levels of different ground motions to determine if the arch could survive to the 100% level; if not, reinforcement would be required.  In addition, the failure mode, the vulnerability of the 18  arch to different ground shaking and the impact of the vertical motion on the performance of the arch was studied.  3.2 Description of the models Several small scale arches provided by Lock-Block were tested. The material used for all the models was Rockite Cement, which is generally used for patching and anchoring. This material has an average compressive strength of 55 MPa in seven days. This value was appropriate because compressibility has limited effect on the failure of masonry arches. The relevant parameters for each of the tested models are shown in Table 3.1. Figure 3.1 illustrates the height, interior diameter and block thickness of the models. A photo of each model is provided from Figure 3.2, to Figure 3.5.  Table 3.1 Physical parameters of each of the tested models Model Scale Height (cm) Length (cm) Block thickness (cm) Interior Diameter (cm) Voissours Number Weight (kg) Arch  (Figure 3.2) 1.5 m interior diameter 1/25 7.5 3 1.5 6 9 0.16 Arch  (Figure 3.3) 3 m interior diameter 1/25 15 12 3 12 9 2.55 Arch  (Figure 3.4) 6 m interior diameter 1/25 21 24 3 24 15 7.03 Arch  (Figure 3.5) 6 m interior diameter 1/12.5 42 48 6 48 15 63.84  19   Figure 3.1 Specifications for height, interior diameter and voissours number in blue            Figure 3.2 1/25 small scale model of the 1.5 m interior diameter arch structure  Figure 3.3 1/25 small scale model of the 3m interior diameter arch structure 1 2 3 4 5 6 7 8 9 11 10 13 14 12 15 Internal Diameter Height Block Thickness 20   Figure 3.4 1/25 small scale model of the 6m interior diameter arch structure  Figure 3.5 1/12.5 small scale model of the 6m interior diameter arch structure The first course of blocks was oriented at 90 degrees to the rest of the courses (see Figure 3.1). The base blocks were held in place on either side by a block of wood attached to a plywood base; fixing the first row of blocks and preventing spreading of the supports during the testing.  Each scaled block was designed in a 3D CAD program, and then 3D printed to make a master. The blocks (Figure 3.6a) have a cross-shape shear key at the top of it that fits into the bottom of the block above (which has a female connection). The key stone of the arch has a female connection on both sides (Figure 3.6b). 21                Figure 3.6 Small-scale Lock-Block with the cross-shaped shear key a) showing male connection b) showing female connection of the middle key stone  3.3 Equipment used for the experimental testing All of the experimental testing was performed at The University of British Columbia Earthquake Engineering Research Facility (EERF). A description of the shake-tables and instrumentation used is provided in this section. Full details are provided in Appendix A. 3.3.1 Description of the shake-tables The EERF has two large shake-tables and a smaller portable one. Among the large shake-tables, one is capable of six degrees of freedom and the other one, which has higher capacity, is limited to uniaxial motions. For this study, the APS small portable shaker and the larger Multi-Axis Shake-table (MAST) were used.  3.3.1.1 APS electro-seis shaker The APS shaker was used for the dynamic testing program of the 1/25 scale model. It is a portable single degree of freedom electro-dynamic shake-table that can reproduce either horizontal or vertical motion. A photo of the table is shown in Figure 3.7. a) b) 22   Figure 3.7 APS ELECTRO-SEIS Shaker The shaker has a 254mm x 254mm armature with a maximum capacity of 23kg and a maximum displacement of 7.9cm. The operating frequency range is from 0 Hz to 200 Hz.  A Spectral Dynamics Jaguar controller was used to control the shaker. This controller is able to generate multiple type of signals (swept sine, dwell pulses and earthquake records) that are sent to the table through a low level control system.  3.3.1.2 Multi-axis shake-table (MAST) The MAST (Multi-Axis Shake-table) system was used for testing the 1/12.5 scale model. It is a six degree of freedom hydraulic shake-table with a footprint of 4 m x 4 m and a capacity of approximately 10tonnes. A photo of the shake-table with the reinforced model used for this study and the instrumentation is shown in Figure 3.8. The defined test directions are specified as follows: x-axis as the transversal direction, y-axis as the longitudinal direction and z-axis as the vertical direction. 23   Figure 3.8 MAST with the 1/12.5 scale model on top More specifications for the MAST, as well as the control system, hydraulic system and the data acquisition can be found in Appendix A. 3.3.2 Description of the instrumentation Two different type of accelerometers were used during the experimental testing (see Figure 3.8). The accelerations of the MAST in the three directions were recorded using three seismic, ceramic shear ICP accelerometers (1V/g, 0.5 to 2k Hz) (see Figure 3.9).  Figure 3.9 Accelerometer used at the center of the MAST Shake-table accelerometer   z x y Top accelerometer   Bottom accelerometer   24  For recording the accelerations in the transversal direction at the top and bottom of the structure, uniaxial MEMS ‘ICSensors’ accelerometers with +/- 10g range and sensitivity of 3 to 6 mV/g were used (see Figure 3.10).    Figure 3.10 Accelerometer used at the top and bottom of the model 3.3.3 Description of the high-speed camera The high speed camera, shown in Figure 3.11, used for the experimental testing was a ‘Phantom v4.2’. This high speed camera was controlled through the use of ‘Phantom 675’ software package.    Figure 3.11 Phantom V4.2 High Speed Camera  25  3.4 Tilt testing  Tilt testing provided a first understanding of the seismic behavior of the arch system. Several arch models were tilted in order to study the collapse mechanism, and find an estimate of the maximum acceleration value that the structure can stand before it fails. 3.4.1 Theory of quasi-static analysis by tilt testing One method to simulate earthquake loading, as an approximate horizontal ground acceleration, is tilting the structure until it collapses. The critical angle of tilt gives the minimum value of lateral acceleration to cause the arch to collapse. When the structure is tilted one component of gravity acts normal to the tilted surface and the second component acts parallel to the surface (see   Figure 3.12). The approximate collapse acceleration from tilt test (α) is defined as:                                      𝛼 = 𝑡𝑎𝑛 (𝛾) ∗ 𝑔                                                 (1)  where g is acceleration due to gravity and 𝛾 is the tilt angle.     Figure 3.12 Scheme for the tilting concept of a block The lateral acceleration, represented as α, is an approximation of the peak ground acceleration (PGA) which collapses the arch. The α value at which collapse was recorded during this testing is rather conservative but useful for a first approximation of the maximum peak ground acceleration.  26  3.4.2 Test procedure and setup  The models were placed on a tilting platform with the length of the bottom rows (y-axis) parallel to the axis of rotation. As one of the end of the platform was raised, a digital inclinometer recorded the maximum angle of incline before collapse. The tangent of this angle was used to determine the α which causes the collapse of the structure. All of the models listed in Table 3.1 were subjected to tilt testing. For 1/25 scale models, the plywood platform, on which the model was seated, was lifted manually (see Figure 3.13). However, for the 1/12.5 scale model an overhead crane was needed to lift the platform from one of the sides since the model was too heavy for a manual lifting (see Figure 3.14). The tilt test was repeated three times for each of the models, and the average collapse tilt angle and its corresponding α were calculated.   Figure 3.13 Setup of the tilt test performed in the unreinforced 1/25 scale models  Figure 3.14 Setup of the tilt test performed in the unreinforced 1/12.5 scale models 27  3.4.3 Tilt testing results The observed response of the models during tilt testing in terms of collapse mechanism and maximum ground acceleration are discussed in this section. 3.4.3.1 Collapse mechanism  For all of the tilt tests performed, the four-hinge collapse mechanism described in the literature review (Section 2.2.2-Figure 2.6) was observed. The failure mechanism of the arch was initiated by the opening of the four hinges highlighted in red in Figure 3.15a. As the collapse progressed, a fifth additional hinge, highlighted in orange in Figure 3.15b, was developed.    Figure 3.15 Collapse mechanism recorded with the high speed camera for the 1/12.5 scale model a) as the collapse was initiated and b) as the collapse progresses showing the additional fifth hinge in orange  When observing the failure mode along the y-axis, the hinges were seen as a straight line across the entire model. In Figure 3.16 the development of the hinges through the 4 blocks length of the 1/12.5 scale model is shown as white and black dashed lines.  28   Figure 3.16 Collapse mechanism of the 1/12.5 scale 6 m interior diameter arch model when tilting the structure from the left side  3.4.3.2 Approximate collapse acceleration from tilt tests The results obtained from the tilt testing for different models are shown in Table 3.2. The scale, thickness-interior ratio (b/r), recorded collapse tilt angle and the corresponding α ratio of each arch are presented in the table. The collapse tilt angle was calculated as average value from three consecutive tilt tests. A maximum deviation among the tests of 0.0005g was recorded. The recorded data for all the tests can be found in Appendix B (Table B.1). Table 3.2 Collapse tilt angle and the corresponding α for the tested models Model Arch Scale b/r Collapse tilt angle (degrees)  α  (g) 1.5m interior diameter 1/25 0.5 14 0.25 3 m interior diameter 1/25 0.5 15.1 0.27 6 m arch interior diameter 1/25 0.25 3.5 0.06 6 m arch interior diameter 1/12.5 0.25 5.5 0.096 29  The following observations can be made from the results of Table 3.2:  The first two models collapsed at a similar tilt angle, with α values of about 0.25-0.27g. However, the 6m interior diameter 1/25 and 1/12.5 scale arch models collapsed at significantly lower accelerations of 0.06g and 0.096g respectively. This is due to the fact that the first two models have higher block thickness-interior radius ratio (b/r) than the other two; this was shown by DeJong (2009).  A similar α (about 0.25g) was recorded for the first two models of Table 3.2 (1.5 and 3m interior diameter arches). However, 1/25 and 1/12.5 6m arches show different α values. Further discussion about this is provided in Section 5.1. 3.4.3.3 Effect of arch length on α  In order to see the effect of the length of the arch on the α value, several 1/25 scale 3m arches with different lengths (from 0.25 to 1.5 times the diameter of the arch) were tilted until collapse. Each test was repeated three times and their average was calculated. The results are shown in Table 3.3 (see Table B.2 in Appendix B for all the recorded data). Although the α values vary from 0.24g to 0.31g, there was no consistent trend between the length and α.  Table 3.3 Collapse tilt angle and the corresponding α for the 3 m interior diameter 1/25 scale model with different lengths Length of the model (1 block = 6cm) Collapse tilt angle (degrees) α  (g) 1/2 block  17.3 0.31 1 block  13.4 0.24 1 and 1/2 blocks  16.2 0.29 2 blocks  14 0.25 3 blocks 14.7 0.26 30  3.5 Sine-sweep testing  Sine-sweep testing involves subjecting the model to a constant low level acceleration over a range of frequencies. Typically the frequencies are changed at a constant rate; the resonant frequencies of the model are obtained. Several sine-sweep tests were performed on the 1/12.5 6m unreinforced arch model in order to find its natural frequencies. The procedure, setup and analysis of the data are presented in this section. 3.5.1 Test procedure and setup  The 1/12.5 scale model was placed on the MAST and a sine-sweep in the transversal direction was applied. The sine-sweep had a rate of 2 decades per minute and was applied with a constant acceleration of 0.03g from 2 Hz to 40Hz. In total, three sine-sweep tests were conducted for the unreinforced model. The horizontal acceleration at the bottom of the model (on the plywood platform) and at the top of the model were recorded and compared to the input motion (response of the shake-table).  3.5.2 Sine-sweep results No clear resonant frequency was obtained when applying the sine-sweep to the unreinforced models. The Fourier spectra of the recorded signals at the top of the model were computed and are shown for one of the test in Figure 3.17. A moving average was applied.  31   Figure 3.17 Fourier spectra of the recorded data from the recorded top acceleration for unreinforced sine-sweep testing  No clearly defined resonant frequencies of the model were observed. The small peaks (around 2 and 6 Hz) are likely related to friction of the mechanical components of the shake-table at low acceleration levels. Also, this testing was limited from 2 Hz to 40 Hz; and it is possible that resonance could occur at frequency values out of that range.   3.6 Shake-table testing  The 1/25 and 1/12.5 scale 6m arch models were subjected to shake-table testing. Dimensional analysis theory used for establishing the similitude scaling rules between model and full-scale structure, as well as the selection and scaling of the ground motions applied to the models will be explained in this section. Additionally, the setup, procedure and results from the shake-table unreinforced testing will be shown. 32  3.6.1 Applied scaling The goal of experimental testing using small scale models is to learn about the behavior of real structures subjected to real loading conditions and to predict their responses (measured in displacement, forces, etc.). Application of the appropriate scaling laws becomes essential to satisfy similitude requirements and provide conclusions about the performance of the full-scale arch. This scaling is based on dimensional analysis, a mathematical technique that deduces the theoretical relation of variables of a physical problem. In order to apply this technique, some dimensionally independent fundamental quantities have to be identified. Some of the most common that applied in physics are: length (L), mass (m), time (t). All of the physical quantities can be expressed as a product or power of these three basic quantities. For instance, velocity (v) could be described as length divided by time (v= L/t), and inertia force (F) as mass times length divided by square time (F = m*L / t2).  The homologous behavior of the models can be obtained by transforming each quantity of the full-scale structure (𝑖𝑓𝑢𝑙𝑙 𝑠𝑐𝑎𝑙𝑒) to the corresponding quantity of the model (𝑖𝑚𝑜𝑑𝑒𝑙) through multiplication of a constant scale factor (Si) defined as follows:                                                                𝑆𝑖 =  𝑖𝑚𝑜𝑑𝑒𝑙𝑖𝑓𝑢𝑙𝑙 𝑠𝑐𝑎𝑙𝑒                                                                (2) The scaling factors need to remain constant in order to maintain homology between models. After selecting the scale factors for the three fundamental quantities, the remaining variables can be expressed in terms of those selected scale factors. An example of the frequency scale factor (𝑆𝑓) derivation in terms of time scale factor (𝑆𝑡) is as follows:                                            𝑆𝑓 =𝑓𝑚𝑜𝑑𝑒𝑙𝑓𝑓𝑢𝑙𝑙 𝑠𝑐𝑎𝑙𝑒=  1/𝑡𝑚𝑜𝑑𝑒𝑙1/𝑡𝑓𝑢𝑙𝑙 𝑠𝑐𝑎𝑙𝑒=  𝑡𝑓𝑢𝑙𝑙 𝑠𝑐𝑎𝑙𝑒𝑡𝑚𝑜𝑑𝑒𝑙=  1𝑆𝑡                                   (3) 33  Although theoretically the scaling factors can be well defined, building a physical small scale model that fulfills the theory is rather difficult. Hence, several model types have been developed over the years. Some of the most popular ones for seismic testing are shown in Table 3.4. The model types can be divided into two main groups: the first uses modified materials and the second uses the materials of the full-scale structure (or prototype). Even though the first model types better simulate the real physical problem, it is difficult to properly scale the properties of the material in practice. The ‘True Replica Model’ and the ‘Complete Model’ are part of the first group and modify the material properties by the modulus of elasticity or the length scale factors respectively.   ‘Artificial Mass Simulation’, ‘Simple Model’ and ‘Gravity Forces Neglected’ are the model types of the second group shown in Table 3.4. In these models, in order to compensate for the use of the full-scale structure material, either the acceleration or mass scale factors have to be modified. Table 3.4 Scaling factors for each dimension for different model types (Krawinkler, 1979; Tomaževič & Velechovsky, 1992)  34  3.6.1.1 Applicability of dimensional analysis to this study Since material strength is not involved in the critical rocking type failure of arch structures, the blocks could be considered as rigid bodies. Thus, the scaling of the modulus of elasticity is nt needed and the ‘Complete Model’ from Table 3.4 can be used for this study. Based on that model type, the scale factors used are shown in Table 3.5 : Table 3.5 Scaled factors based on ‘Complete Model’ for 1/25 and 1/12.5 scale model Dimension Scale Factor 1/25 scale model 1/12.5 scale model Length SL 1/25 1/12.5 Density Sρ 1 1 Acceleration Sa 1 1 Time St = √𝑺𝑳 1/5 1/√12.5 Frequency Sf = √1/𝑺𝑳 5 √12.5  The length and density scale factors were determined values given by the size and material availability of the physical models provided from Lock-Block. The density of the material used for the models was about 91% of the full-scale structures; so for practical purposes a density scale factor of 1 was chosen. With the assumption of 1:1 acceleration scaling, time is scaled based on the square root of the length scale factor. This results in the records of the ground motions being shorter, while keeping the same acceleration.  For instance, for 1/25 scale model the time interval between the data points of the records is decreased by 1/5, which is equal to the square root of the scale of the models (1/25).  Hence, while the duration of the time histories is reduced to one fifth of the original duration and the frequencies 35  are increased five times, the displacements are 25 times smaller than the displacements of the original records.  3.6.2 Selection of the earthquakes For the shake-table testing, a suite of earthquake records was selected from Pina et al (2013) and the PEER Strong Motion database. Crustal, subcrustal and subduction earthquakes with varying frequency content, maximum acceleration amplitude, maximum displacement amplitude and impulses, were chosen. Table 3.6 lists the records used including the name, location, year, station name and largest recorded acceleration for that station. A full set of the parameters for the selected earthquakes can be found in Table B.3, Appendix B. Table 3.6 Suite of time history records used for the shake-table testing EARTHQUAKE NAME LOCATION YEAR STATION NAME PEAK ACC (g) NISQUALLY Washington 2001 Seattle (BHD) Z 0.16 TOKACHI-OKI Japan 2003 TH2011_FKS031_NS 0.42 LOMA PRIETA California 1989 CDMG 57007 Corralitos 0.64 KOBE Japan 1995 CUE 99999 Nishi-Akashi 0.51 TOHOKU Japan 2011 TH2011_FKS031_NS 0.42 PARKFIELD California 1966 CDMG STATION 1014 0.44 NORTHRIDGE California 1994 RINALDI RECEIVING STA, 228 0.83   3.6.3 Test procedure and setup  Tri-axial and uniaxial shake-table testing using the ground motions from Table 3.6  was conducted in both 1/25 and 1/12.5 scale arch models.  For the 1/25 scale model the APS shaker in the transversal direction of the arch was used. However, the 1/12.5 scale model was tested on the MAST due to weight limitations of the APS shake-table.  36  In Table 3.7, several parameters (peak ground acceleration (PGA), peak ground displacement (PGD), peak ground velocity (PGV), etc.) of the applied six time scaled time histories are shown.  Table 3.7 Parameters of the applied time scaled earthquakes for the 1/25 scale model  EQ PGA (g) PGD (cm) PGV (cm/sec) Duration  (sec) Arias intensity (m/sec) Significant duration  (sec) Predominant period (sec) Nisqually 0.16 0.23 4.71 34.60 0.09 3.55 0.12 Tokachi-oki 0.50 0.48 6.52 48.40 0.74 5.22 0.06 Loma Prieta 0.64 0.43 11.03 7.99 0.65 1.38 0.06 Kobe 0.51 0.38 7.46 8.19 0.67 1.94 0.09 Tohoku 0.42 0.21 6.53 20 0.32 9.32 0.08 Parkfield 0.44 0.02 4.93 8.78 0.17 1.29 0.07  For the 1/12.5 scale model, the Loma Prieta, Northridge and Parkfield earthquake records were used, each with three directions of motion. The component with highest peak acceleration was applied in the transversal direction. These three time scaled records and their parameters are shown in  Table 3.8. Table 3.8 Parameters of the applied time scaled earthquakes for the 1/12.5 scale model  EQ Direction PGA (g) PGD (cm) PGV (cm/sec) Duration (sec) Arias intensity (m/sec) Significant duration  (sec) Predominant period (sec)  Loma Prieta Transversal 0.64 0.87 15.6  11.3 0.91 1.94 0.08 Longitudinal 0.48 0.90 12.77 0.72 2.23 0.16 Vertical 0.46 0.57 5 0.24 2.15 0.06  Northridge Transversal 0.83 2.37 45.29  5.63 2.12 2.05 0.2 Longitudinal 0.49 2.16 21.08 1.20 2.62 0.08 Vertical 0.83 0.80 12.31 1.71 1.87 0.02  Parkfield Transversal 0.44 0.41 6.97  12.42 0.24 1.82 0.10 Longitudinal 0.37 0.31 6.16 0.18 2.10 0.08 Vertical 0.14 0.21 1.93 0.05 3.24 0.02  37  The selected records were applied repeatedly at increasing levels, as a fraction of the original record (referred as “Test Level (TL)”) starting from 40% for the 1/25 scale model and 20% for the 1/12.5 scale model. Once a failure of the model was observed, the test was repeated three times at the same test level. This was done to ensure repeatability. The testing was recorded with two HD video cameras and the high-speed video camera at 400 frames per second. The slow motion videos were helpful determining the collapse mechanism of the model. A set of black dots were used for the tracking of displacements with ProAnalyst software. The relative displacements of the blocks to the ground motion could then be calculated by extracting the motion of the fixed base block to the displacement of any given block. This was used to obtain the critical displacement that caused collapse.  For the 1/12.5 scale model, an additional uniaxial test was performed in order to compare the response of the arch between uniaxial and tri-axial motions. For this purpose, Northridge earthquake was chosen due to the high participation of its vertical motion. 3.6.4 Results of the unreinforced shake-table testing  The results obtained for both 1/25 and 1/12.5 models in terms of collapse mechanism, collapse test level and sensitivity of arches are shown in this section. In addition, the results from tri-axial and uniaxial testing are compared. 3.6.4.1 Summary of the shake-table testing The tables presented in this section summarize the results of the shake-table tests on the unreinforced small scale models. Each applied earthquake and test level are shown; an “O” represents a test where the arch did not collapse, whereas the “X” shows the cases in which collapse occurred. For each applied earthquake the test level was repeated three times. 38  Table 3.9 shows the results from the uniaxial testing for the 1/25 scale model. It was found that the model could not survive five out of the six applied earthquakes. Furthermore, there was significant variability in the failure level, from 40%, to 120% TL for Nisqually. Table 3.9 Summary of shake-table tests on the unreinforced 1/25 scale 6 m arch EARTHQUAKE TEST LEVEL (TL ) 40% 50% 60% 70% 80% 90% 100% 120% NISQUALLY   O O O O O X TOKACHI-OKI O O O  O X* X  LOMA PRIETA X* X       KOBE X        TOHOKU X        PARKFIELD O  O O X    * the arch did not always collapse for the three repeated tests Same criteria was used to determine the collapse TL of the 1/12.5 arch. Table 3.10 shows that collapse was recorded for 80% of the Loma Prieta and 25-30% of the Northridge earthquakes. However, the model survived the Parkfield earthquake. Table 3.10 Summary of shake-table tests on the unreinforced 1/12.5 scale 6 m arch EARTHQUAKE TEST LEVEL (TL ) 20% 25% 30% 40% 60% 65% 80% 100% PARKFIELD  O  O  O  O LOMA PRIETA  O   O  X  NORTHRIDGE O X* X      * the arch did not always collapse for the three repeated tests   39  3.6.4.1.1 Preliminary study of the response of the arch to pulses A study on the sensitivity of the arches to different parameters of the earthquakes was conducted. The most consistent result agreed with the work done by DeJong et al (2008) in that the impulse-type ground motion has the strongest effect; peak acceleration and duration of the impulse being the governing factors.  A graph of the peak acceleration versus the frequency of the pulse that caused the collapse for each earthquake was plotted following the same approach used by DeJong (2009). The results obtained for the 1/25 and 1/12.5 scale models for each of the applied earthquakes at collapse level are shown in Figure 3.18 and Figure 3.19 respectively.  An estimate of the failure limit is plotted as a dashed red line dividing failure points (solid) and non-failure points (empty). This splits the plots into two domains failure domain and safe domain. If the structure is subjected to a pulse that falls into the upper area of that line, it would be susceptible to collapse. However, for pulses that fall under the failure limit line the structure would be safe. The α value from the tilt test is shown by a dashed black line. It is noted that the failure limit is shifted to the left (towards lower frequencies) when comparing Figure 3.19 to Figure 3.18. This is due to the fact that the frequencies of the earthquakes have been scaled down (by √𝑆𝑙) based on the dimensional analysis explained in Section 3.6.1. 40   Figure 3.18 Failure domain defined by acceleration and frequency of the main pulse for the 1/25 scale unreinforced model       Figure 3.19 Failure domain defined by acceleration and frequency of the main pulse for the 1/12.5 scale unreinforced model    00.050.10.150.20.250.30.350.40.450.50 5 10 15Acceleration (g)Frequency (Hz)SAFE domain FAILURE domain Failure limit Nisqually Tokachi-Oki Loma Prieta Kobe Tohoku Parkfield   α Failure limit FAILURE domain SAFE domain α  Loma Prieta Northridge Parkfield   41  The following observations can be made from the results:  The arches always collapsed at higher ground accelerations than the α value obtained from tilting of the structure. As expected, the failure limit curve approaches the quasi-static collapse acceleration as the frequencies decrease.  The acceleration required to collapse the arch increases as frequency increases.   It was observed that two of the earthquakes (Tokachi-Oki and Nisqually), which are not impulse-type motions, collapsed at accelerations much higher than the failure limit defined by the others.  3.6.4.2 Collapse mechanism The rocking type failure based on the four-hinge mechanism observed during tilt testing, was also seen in all of the tests for both the 1/25 and 1/12.5 scale arches. Failure mode of the arches taken from the high speed camera footage are shown in Figure 3.20 and Figure 3.21, for the 1/25 and 1/12.5 scale models respectively. The collapse was initiated by the four hinge mechanism. In the figures, those four hinges are represented as red circles. As the collapse progressed, an additional fifth hinge (in orange) was developed. In all of the performed tests, the hinges were created at the same points for all the earthquakes with the only variation being the collapse direction. 42   Figure 3.20 Different stages of the typical four-hinge failure mechanism taken from the high-speed camera footage for Tokachi-Oki earthquake with the four hinges of the collapse mechanism  in red and the additional fifth hinge in orange  Figure 3.21 Different stages of the typical four-hinge failure mechanism taken from the high-speed camera footage for uniaxial Northridge earthquake with the four hinges of the collapse mechanism  in red and the additional fifth hinge in orange  Sliding between the blocks was not observed for any of the tests, possibly due to the contribution of the shear keys. Additionally, when the shaking was not strong enough to let the hinges open to 43  a critical displacement, they were closed due to the reversing motion that stabilized the arch. The study of that critical displacement is shown in next section. 3.6.4.2.1 Critical displacement of the blocks at collapse Since the movement of the blocks is very complex, the relative horizontal displacement between the center of the keystone and the base is used here as the reference point. The critical displacement at collapse was determined by tracking of the reference point. In order to do so, the peak relative displacements (Dmax) in the transversal direction for all the earthquakes at collapse level and at lower test level (TL) were recorded and compared. The results obtained for the tracking of the 1/12.5 scale model have been summarized in the following table.  Table 3.11 Peak displacements (Dmax) of the middle key block relative to the base at collapse TL and at a lower TL for three different earthquakes  EQ Test type Before collapse  At collapse TL (%) Dmax (mm)  TL (%) Dmax (mm) Loma Prieta Tri-axial 60 3.7  80 6.3 Northridge Tri-axial 25 2.3  30 8 Uniaxial 20 5.3  25 6.5 Parkfield Tri-axial 100 2.8  - -   From Table 3.11, the minimum recorded peak displacement at collapse TL was 6.3 mm for tri-axial Loma Prieta. In contrast, the maximum peak displacement for among the tests without collapse was 5.3 mm for uniaxial Northridge. Then, it could be argued that for the 1/12.5 scale model the critical displacement of the middle key must be between 5.3 and 6.3 mm; which corresponds to 1.2% and 1.5% of the height of the structure.   44  3.6.4.3 Comparison between uniaxial and tri-axial collapse mechanism In order to study the effect of the vertical motion on the response of the arch, the results from tri-axial and uniaxial testing on the 1/12.5 scale model were compared. For this purpose, Northridge earthquake was used. 3.6.4.3.1 Comparison of collapse mechanism For both uniaxial and tri-axial testing, the failure was initiated by the four-hinge mechanism. As the collapse progressed, two additional hinges were observed for some of the tri-axial tests (see Figure 3.22). While the fifth additional hinge (orange) was also developed during uniaxial testing, the sixth hinge (green) was only observed for tri-axial tests using Loma Prieta and Northridge earthquakes. This is likely due to the fact that the Northridge and Loma Prieta earthquakes have a higher vertical acceleration than Parkfield (see  Table 3.8 ).   Figure 3.22 Different stages of the collapse of the 1/12.5 scale arch taken from the high-speed camera footage subjected to tri-axial Northridge, with red circles for the four hinge collapse mechanism, orange circle for the fifth additional hinge and green circle for the sixth additional hinge   45   During the test with vertical motion, the arch is shown to fall straight down with the blocks at each side of the middle key falling on the outside (as shown in Figure 3.22). However, when the motion was applied just in one direction, all the blocks fell into the same side (see Figure 3.20).  3.6.4.3.2 Comparison of acceleration time histories Figure 3.23 and Figure 3.24 show 1.8 seconds of the recorded acceleration at the bottom and at the top of the model for tri-axial and uniaxial Northridge respectively. The beginning of the development of the four-hinge collapse mechanism is shown as a vertical red line. The yellow line refers to the additional hinges and the black line represents the time at which collapse occurred. After collapse, the response of the top accelerometer becomes saturated and goes beyond the limits of the plots.   Figure 3.23 Time histories of bottom and top acceleration in the transversal direction for Northridge tri-axial test at collapse level  TRI-AXIAL 46   Figure 3.24 Time histories of bottom and top acceleration in the transversal direction for Northridge uniaxial test at collapse level  From the figures, the hinges and the complete collapse were found to occur earlier when the earthquake was applied in three directions than in just one direction. Furthermore, the time between the onset of the four-hinge mechanism to collapse was 0.15 seconds longer for the uniaxial case (which is equivalent to 55% of the collapse duration of the tri-axial).  3.7 Concluding remarks of experimental testing of unreinforced arches Tilt, sine-sweep and shake-table testing was performed using small scale models of the Lock-Block unreinforced arches in the standard configuration. The models collapse when subjected to most of the applied earthquakes, being especially vulnerable to pulse-type ground motions. The PGA that the arches could withstand was reduced as the dominant frequency of the pulse was decreased. In all the tests, failure of the arch was initiated by the four-hinge mechanism. Additionally, the vertical motion was found to have an effect on the response of the arch by changing the collapse time in the record and the hinging mechanism. UNIAXIAL 47  Chapter 4: Experimental Testing in The Reinforced Models From the tests on the unreinforced model, it was found that the model would collapse for most earthquakes. Hence, a reinforcement method was studied to improve the response of these arches under seismic loading and to gather information that could be used in design of full-scale reinforcing. In this section a description of the methodology and the selected reinforcement option for the conducted experimental testing are given. In addition, the setup, procedure and obtained results for all the test (tilt, sine-sweep and shake-table testing) are provided.   4.1 Background in reinforcing of arches The preservation of masonry arches has lead into a wide study of different retrofit possibilities. Traditional retrofit methods used to prevent premature or sudden collapse include (from Day 1995 and Boughton et al 1997): - single or doubled side jacketing with cast in situ reinforced concrete in combination with steel reinforcement - reinforced grouted injections or crack stitching ties - ties between both sides of the arch - internal or external post-tensioning with steel ties to link structural elements together (more recent) External or internal distributed reinforcement could prevent the formation of the hinges by inducing an axial compression between the blocks due to a radial distribution of forces induced by the post-tensioning of the cables (see Figure 4.1). Jurina (2013) introduced the ‘RAM’ (Reinforced Arch Method), which is based on the idea of reinforcing the arch using steel instead of the 48  commonly used poured concrete layer over the arch. This method modifies the load distribution in the arch so that the thrust line can be re-centered.   Figure 4.1 Interaction forces between reinforcement and arch when the strap is added at the extrados (A) or intrados (B) (from Jurina 2013)  Jurina (2013) performed several quasi-static experiments applying distributed horizontal loads in several reinforced masonry arch specimens. The results show an increase on the capacity of the arch with an almost linear relationship between the collapse load and the post tension applied to the cables, as well as an increase in ductility. Oliveira et al (2010) proved that external reinforcement provided a higher ductility behavior than internal reinforcement, which is more effective in increasing the strength. Another experimental testing in partially reinforced arch structure done by Rovero et al. (2013) identifies the migration of the hinges to the un-strengthened areas. Most of these studies are based on restraining the blocks to prevent hinge mechanism failure. However, Rovero et al. (2013) observed that reinforcement could lead into new critical collapse mechanism as (1) compressive failure of the material, (2) sliding of the joints (shear failure) or (3) debonding of the reinforcement from the arch.  49  Although traditional techniques may guarantee capacity increase of the structure (by means of strength and stiffness), they can be tedious for implementation and conservation, with a short life duration and view impact. Hence, several researchers (Foraboschi  2001, Valluzzi et al. 2001, Borri and Castori 2004) have investigated about new retrofit solutions. One of the most interesting of these is the application of fiber reinforced polymer (FRP) bands. The same concept as for the external steel reinforcement is applied since it also behaves as a tensile resistant member.  Some of the advantages of FRP are its low weight, its long durability, high strength and adaptable visibility. However, FRPs can have lower ductility and have higher material cost.  4.2 Methodology The objective of this testing was to examine the behavior of the arch with a simple reinforcement method and to obtain the loads in the reinforcement. Steel straps were utilized as representative reinforcement, which were instrumented with strain gauges. These straps were not intended to be a scaled version of the actual reinforcement, but instead were used to capture the forces due to the movement of the scale models. The following tests were conducted:  Quasi-static tilt testing: the effect of the pretension of the straps on the response of the model was studied by tilting the 1/12.5 scale model at different angles and for different pretension levels.   Sine-sweep testing: the natural frequencies of the reinforced 1/12.5 scale model were determined.   Shake-table testing: reinforced small scale model were tested under different ground shaking in one and three directions at varying intensity levels and with different 50  pretensions. The primary purpose of this test is to characterize the forces at the straps caused by the ground shaking.  4.3 Description of the reinforcement used for experiments The reinforcement was placed along the outer perimeter of the arch, which acted to hold it in compression during the lateral loading through tension. Steel bands were instrumented with strain gauges to measure the forces during the shaking. The material and dimensions of the reinforcing straps were specifically chosen in order to achieve a measureable strain based on the expected levels of force. The material for the reinforcement was 1095 spring steel, with a section 6.35mm wide and 0.127mm thick. These dimensions provided the smallest cross sectional area for the strap out of the available materials. The strain gauges had the ability to register changes in strain of 1µɛ.  Material calibration was performed by adding 1kg of weight equivalent to 9.8N to the steel strap which was oriented in a vertical position. Additional 1kg weights were attached to increase the applied strain linearly. In average, the strain due to 9.8N was around 61µɛ. The elastic modulus of the material was calculated to be approximately 200GPa. The calibration results and more detailed specifications for the strain gauges can be found in Appendix A.7.  The 1/25 scale arch model was reinforced using an external strap in the middle and tightened on either side (see Figure 4.2). The steel band was instrumented using strain gauges in three different locations; shown in the figure as channel #2 (CH2, middle point), channel #3 (CH3, right side of the model) and channel #4 (CH4, left side). 51   Figure 4.2 Instrumentation of the 1/25 small scale arch model The 1/12.5 scale model was also reinforced, but this time using two steel straps (see Figure 4.3). A plywood base was constructed for mounting the model on the shake-table. Holes were cut through the base to allow for the straps to pass vertically through, and connect to an adjustable connection underneath. This allows for tightening at one end to the desirable pretension level. Data from eigth channels was recorded during the testing as shown in Table 4.1 and Figure 4.3. In some locations, strain gauges were added in both the external (ext) and the internal (int) side of the straps (for averaging porpuses). This is the case for CH1-CH2 and CH0-CH3.  Figure 4.3 Instrumentation of the 1/12.5 small-scale arch model CH1 (int) CH2 (ext) CH3 (int) CH0 (ext) CH7   CH6   CH4   CH5   CH4   CH3  CH2  52  Table 4.1 Instrumentation of the 1/12.5 small-scale arch model  Left side Center Right side Strap 1  CH7 CH5 CH3(int)   CH0(ext) Strap2 CH2(ext)  CH1(int) CH4 CH6  The data obtained from the strain gauges of each channel was analyzed and the recorded forces were calculated as follows:                                                           Fch = (Estrap x ɛch ) x Astrap                                                                        (4) Where Fch is the force for that particular channel, Estrap is the modulus of elasticity of the steel strap obtained from the calibrations, Astrap is the cross sectional area of the strap (see Appendix A.7) and ɛch is the measured strain in that channel.  4.4 Tilt Testing Tilt testing was performed using the reinforced 1/12.5 scale arch model. The purpose of this testing was to study of the effect of the pretension on the deformation of the model. 4.4.1 Description of the testing The model was tilted at different angles (30 and 45degrees) and different pretension levels (20, 39, 59, 78, 118 and 157N).  The transversal displacement of the middle key block were visually tracked using a ruler (with millimeter precision) as shown in Figure 4.4. The strains in all the channels were also recorded and the peak forces for each of the tests were computed. The test was repeated at increasing pretension levels until the change in displacement was negligible. 53   Figure 4.4 Setup of the tilt test performed in the instrumented reinforced 1/12.5 scale model 4.4.2 Tilt testing results The measured tangential displacements of the middle key for different tilt angles and different pretension levels are plotted in Figure 4.5. The complete data can be found in Appendix C - Table C.1.  Figure 4.5 Pretension of the straps versus the measured displacements of the middle key block at 30 and 45 degrees of inclination  Figure 4.5 shows that while the pretension in the straps increased from 20N to 157N, the recorded displacements were reduced by approximately 50% for both tilt angles. Since after 117.6 N of 012345670 20 40 60 80 100 120 140 160 180Displacement (cm)Pretension (N)30degrees45degrees54  pretension the recorded displacements changed minimally, this pretension level was chosen to perform shake-table testing.  4.5 Sine-sweep testing The 1/12.5 scale reinforced arch model was subjected to sine-sweep testing in order to obtain the natural frequencies of the arch. The description of the tests as well as the obtained results are included in this section. 4.5.1 Description of the sine-sweep testing The same setup and data analysis procedure applied to the unreinforced model and explained in Section 3.5.1 was followed for the 1/12.5 scale reinforced model.  The data obtained from the strain gauges was analyzed in frequency domain. In total, three tests were performed at 0.1g constant acceleration.  4.5.2 Sine-sweep results The Fourier spectra-s obtained for each of the three sine-sweep tests are presented in Figure 4.6.   Figure 4.6 Fourier spectra of the recorded accelerations at the top of the model for sine-sweep test #1, #2 and #3in the reinforced model 55   Most excited frequencies were found around 12 Hz for the first test and around 7 Hz for the second and third tests. These peaks can be identified as natural frequencies of the system. The shift of the peaks towards lower frequencies is caused by a decrease in the pretension level of the straps. After test #1, the model was subjected to two Northridge time history shake-table tests. The shaking loosened the straps, which were at the beginning at 117N of pretension, up to an average of 64N. This loosening was detected by recording of negative residual forces at the end of each tests (see Appendix C- Table C.2). As the pretension of the straps was reduced, the model became more flexible and its natural frequency decreased considerably. The difference between test #2 and test#3 is very small since the loosening of the straps was almost negligible.  4.6 Shake-table testing Shake-table testing using 1/25 and 1/12.5 scale reinforced 6m arch models was performed in order to characterize the response of the arch when reinforcement was added. The test procedure and the obtained results are explained in this section.  4.6.1 Test procedure For the shake-table testing of the reinforced model, similar setup and procedure used for the unreinforced models were followed. The same selection and scaling of earthquakes from Section 3.6.2 was applied.  The forces recorded at the strain gauges were used for calculation of the ratio FnW. This ratio is a non-dimensional way to express the peak dynamic forces as a percentage of the total weight of the model. This ratio was computed as per the following equation: 56                                                               𝐹𝑛𝑊(%) =  𝐹𝑝𝑒𝑎𝑘𝑊∗ 100                                                    (5) Where 𝐹𝑝𝑒𝑎𝑘 is the average peak force (without accounting for the pretension) of all the channels calculated from the recorded strains, and W is the total weight of the model.  For most of the tests the data was zeroed at the beginning of each test so that the initial pretension was not included in the measurements of the forces. The residual forces at the end of each test were checked to ensure that constant pretension level was maintained during the study. When the data was not zeroed, the total forces due to the pretension and the dynamics of the mass of the arch were recorded. For these cases, a new ratio, FtnW, was defined as the ratio of the average peak total forces (with pretension of the straps) of all the channels and the total weight of the model. For testing in the 1/25 scale reinforced model, the pretension at the straps was not measured. For the testing in the 1/12.5 scale reinforced model, a pretension consistent to the results obtained from the tilt testing equal to 118N was used (see Section 4.4.2).   4.6.2 Shake-table testing results In all the conducted tests, it was shown that the steel band reinforcement prevented collapse. The results obtained in terms of recorded forces and accelerations, effect of the pretension and effect of the vertical motion are shown. 4.6.2.1 Study of the recorded forces Two different studies were conducted using the 1/25 and the 1/12.5 scale reinforced model. In both studies, an average of the peak forces recorded in all the channels was obtained, and 𝐹𝑛𝑊 ratios for every earthquake were calculated and compared.  57  The purpose of the first study was to measure the dynamic forces in the reinforced model at the TLs that caused the unreinforced arch to collapse. The results for the first round of reinforced testing performed on the 1/25 scale arch model are shown in Table 4.2. The recorded complete data can be found in Appendix C -Table C.3.  Table 4.2 Ratio of the recorded peak forces to the weight of the model in percentage at collapse TL for different earthquakes EARTHQUAKE COLLAPSE TL (%) 𝐅𝐧𝐖 (%) NISQUALLY 120 3.2 TOKACHI-OKI 90 5.47 LOMA PRIETA 40 5.8 KOBE 40 3.86 TOHOKU 40 4.2 PARKFIELD 80 3.6  The FnW ratios at collapse level for all the earthquakes were found to be between 3 to 6%. The highest value was for the Loma Prieta (5.8%), and the lowest one for the Nisqually (3.2%). It was observed that the forces recorded at the straps during collapse of any of the ground motions were very similar.  The purpose of the second study was to obtain the dynamic forces created by different earthquakes in the 1/12.5 scale model. A total of 13 tri-axial shake-table tests were performed varying the intensity level applied to each earthquake (50, 65, 80, 100 and 120%).   Table 4.3 and Figure 4.7 summarize the results obtained using the Loma Prieta, Northridge and Parkfield earthquakes. The recorded data for all the channels can be found in Appendix C - Table C.4. 58    Table 4.3 Recorded peak forces and FnW ratios for different earthquakes at different TLs. EARTHQUAKE TL  (%) Peak Force (N) 𝐅𝐧𝐖 (%)    Northridge 50 97 30.9 65 162 51.9 80 302 96.4 100 371 118.5   Loma Prieta 50 20 6.4 65 40 12.8 80 66 21.1 100 91 29    Parkfield 50 17 5.6 65 15 4.8 80 18 5.6 100 57 18.20 125 59 19   Figure 4.7 FnW ratio for different test levels of the Loma Prieta, Northridge and Parkfield earthquakes It can be observed from the table and figure above that as the intensity increased, the FnW ratio also increased for all the earthquakes. In Figure 4.7, the FnW  ratio for all the three earthquakes seems to increase linearly with the TL.  FnW (%) 59  At the 100% TL, the FnW  value for Loma Prieta (29%) and Parkfield (18.2%) were similar while Northridge (118.5%) was much higher. This is most likely due to the higher horizontal acceleration and the contribution of the vertical motion. 4.6.2.2 Study of the recorded accelerations The accelerations in the transversal direction at the bottom and top of the 1/12.5 scale model for the tri-axial Loma Prieta, Parkfield and Northridge earthquakes at the100% TL were recorded and compared. A five seconds duration of the strong motion part for each earthquake have been plotted in the following figures. The recorded acceleration peaks during all the performed tests can be found in Appendix C - Table C.6.  Figure 4.8 Recorded accelerations at the bottom and top of the model for tri-axial Loma Prieta at 100% TL  Figure 4.9 Recorded accelerations at the bottom and top of the model for tri-axial Northridge at 100% TL  60    Figure 4.10 Recorded accelerations at the bottom and top of the model for tri-axial Parkfield at 100% TL  The following observations were made from the analysis of the data:  For Northridge (Figure 4.9) and Loma Prieta (Figure 4.8), the highest amplification of the top acceleration with respect to the bottom occurred during the second half of the main pulse.   From Figure 4.9 for the Northridge earthquake, the acceleration at the top was found to be double the PGA (from 0.81g to 1.6g). For the Loma Prieta earthquake (Figure 4.8), the accelerations were increased by 60% (from 0.54g to 0.86g).  Parkfield did not experience high amplification since it is not a single pulse motion. Figure 4.10 shows that the recorded PGA was 0.47g, while the maximum acceleration recorded at the top was 0.48g.  4.6.2.3 Study of the effect of the reinforcement This testing focused on the study of the importance of the pretension of the straps at varying intensity levels. Bi-axial testing using the Loma Prieta record in transversal and vertical directions was conducted at different intensity levels (75, 100 and 120%) and using different pretension (20 and 78N). For this test, the data was not zeroed at the beginning of each test; that is, the forces 61  obtained were total forces. The FtnW ratios obtained are summarized in Table 4.4 and Figure 4.11. More detailed data can be found in Appendix C-Table C. 7. Table 4.4 FtnW ratio for different pretension and test levels during Loma Prieta earthquake  Pretension (N) Test Level (%) 𝐅𝐭𝐧𝐖 (%)  20 75 16.4 100 34 120 58  78  75 15 100 36.2 120 63.4   Figure 4.11 FtnW ratio for different pretension and test levels during Loma Prieta earthquake  It can be observed in the figure above that the recorded peak total forces are very similar for the 20 N and 78N pretension levels. That is, the level of pretension did not influence the total forces recorded at the straps.  20 N pretension 78 N pretension  Ft nW (%) 62  For a better comparison with previous test results, the relative dynamic force was calculated as the total force minus the pretension force. Then, the FnW ratios were computed and are plotted in Figure 4.12.   The FnW ratio decreases as the applied pretension increases.   Figure 4.12 FnW ratio for different pretension and test levels during Loma Prieta earthquake While in the graph above the lines for different pretension levels are almost parallel, in Figure 4.11 both lines are approximately the same. It can be argued then, that the difference between the recorded dynamic forces for different pretension levels is equal to the difference on the pretension. An additional uniaxial test performed using Northridge earthquake at 100% TL, confirmed that the dynamic forces increased with the decrease of the pretension level. The summary of the FnW  values obtained for each pretension level are shown in Table 4.5 (for more details go to Appendix C - Table C.6). Table 4.5 Recorded average peak forces at the straps and FnW ratio for uniaxial Northridge shake-table testing at different pretension levels Earthquake TL (%) Pretension  (N) Peak force (N) 𝐅𝐧𝐖   (%) Northridge Transversal direction 100% 108 717 115 64 932 149  20 N pretension 78 N pretension  FnW (%) 63  For this uniaxial testing, the effect of the pretension on the recorded accelerations was studied. The top and bottom accelerations for 5 seconds of strong motion of the Northridge earthquake with 108N and 64N of pretension are plotted in Figure 4.13 and Figure 4.14 respectively. For lower pretension, a small shift (or delay) of the top acceleration to the ground acceleration can be appreciated caused by a higher flexibility in the system. As expected, when higher pretension was applied, the structure became more rigid and followed better the applied ground motion. The recorded peak accelerations were 25% higher for the lower pretension level than for the higher one.  Figure 4.13 Recorded accelerations at the bottom and top of the model for uniaxial Northridge at 100% TL for 108N of pretension  Figure 4.14 Recorded accelerations at the bottom and top of the model for uniaxial Northridge at 100% TL for 64 N of pretension 64  4.6.2.4 Study of the effect of vertical motion  In order to study the effect of the vertical motion in the reinforced arches, uniaxial testing was compared to tri-axial testing using the Northridge earthquake at 100% TL.  The acceleration time histories of the uniaxial testing with 108N of pretension (Figure 4.13) were compared to the tri-axial ones (Figure 4.9). In both cases similar amplification was found during the second half of the first pulse. However, in the following peaks higher amplification values were observed for tri-axial testing. This occurred when a vertical pulse coincided with a horizontal peak. The FnW  ratios for uniaxial testing (Table 4.5) were compared to the ratio obtained for the tri-axial testing (Table 4.3). The recorded ratios were found to be very similar; 115% ratio for uniaxial compared to 118% ratio for tri-axial.   4.7 Concluding remarks of experimental reinforced testing Lock-Block small scale models were externally reinforced using steel straps and tested following the same testing program applied to the unreinforced arches. The reinforced arches performed well under all the applied earthquakes. The straps were instrumented with strain gauges in order to record the forces created by the shaking. The recorded dynamic forces were found to be dependent on the pretension of the straps. At same pretension level, the forces created by different earthquakes at collapse TL of the unreinforced arches were found to be very similar. However, the forces at 100% TL varied between ground motions. By analyzing the acceleration at the top of the model, it was observed that reinforced arches are also sensitive to pulse type motions and vertical component of the records. 65  Chapter 5: Numerical Analysis A discrete element model using 3DEC software was developed and calibrated to the quasi-static experimental testing. In this Chapter the description of the discrete element software used for creating the numerical model is provided, followed by the steps and assumptions made for the creation of the model. Then the experimental data is compared to the results obtained from the computer model. Finally, a sensitivity analysis of the effect of various modeling parameters to the collapse load is provided.   5.1 Numerical model software Based on the behavior of arches, Discrete Element Modeling (DEM) is a desirable choice to create the numerical model due to its ability for allowing large displacements during dynamic analysis and adequately dealing with the discontinuities between the blocks. The software 3DEC Version 5.0 from Itasca was used to model the arch structure. 3DEC is a numerical modelling code based on the distinct element method. This method is a discontinuous analysis technique, which is distinguished from continuous technique by viewing the material as an assembly of distinct bodies interacting along their boundaries. However, with the evolution of these methods, several finite element, boundary element and Lagrangian finite difference programs (adapted from existing continuum programs) have developed interface elements or slide lines that enable them to model also discontinuous elements to some extent. However, Finite Element Modeling (FEM) is still typically limited to small displacements and rotations, to a limited amount of intersecting interfaces, and has restricted capability of recognizing new contacts automatically. A computer program is considered to use discrete element method when it allows finite displacements and rotations of discrete bodies (including complete detachment) and it recognizes 66  new contacts automatically as the calculation progresses. Following this definition, discrete element programs can be divided in four main groups: distinct element, modal, discontinuous deformation analysis and momentum-exchange methods. As mentioned before, 3DEC software falls in the first category using deformable contacts and an explicit time-domain solution of the original equation of motion. The name distinct element method was first used by Cundall and Strack (1979) to refer to a particular discrete element method. This computer program, originally developed to perform stability analysis of jointed rock slopes, is primarily used in studies related to mining engineering. Its capability for simulating the response of discontinuous media subjected to either static or dynamic loading makes it potentially useful for this research. The units can be represented as an assemblage of 3D rigid or deformable blocks, which may take any arbitrary geometry. Joints are represented as interfaces or point vertex-edge contacts (see Figure 5.1), viewed as interaction between the blocks and governed by appropriate stress-displacement constitutive laws. All types of geometric interaction between the blocks can be represented using two type of elementary contacts: vertex-face (VF) and edge-edge (EE), as shown in Figure 5.2.     Figure 5.1 Representation of contacts between blocks as a) joint element, and b) vertex-edge contact (Lemos 2007) 67    Figure 5.2 Representation of contacts between blocks by a) two VF and two EE, and b) an actual EE contact (Lemos 2007)  The software employs explicit in-time solution algorithm allowing large finite displacements (complete detachment and new contact generation included) and rotations of the discrete bodies.  The calculations are made using the force-displacement law at all contacts and the Newton’s second law of motion at all blocks.  The position of the block is obtained by integrating this last law and calculating the displacement increments or the velocities. Then, the force-displacement law is used to obtain the new contact forces, which are applied to the blocks next. The schematic cycle of the calculation that 3DEC follows is shown below.  Figure 5.3 Schematic calculation cycle followed by 3DEC (Itasca, 3DEC 2013)  68  5.2 Description of the modeling A discrete element model of the 1/12.5 scale physical arch was created using 3DEC software. The chosen modeling parameters and the assumptions made during the development of the computer model are explained in this section. 5.2.1 Modeling assumptions As it has been shown experimentally in this study and proved analytically in past studies, this type of arch structure collapses due to the rocking or opening between the block. During the performed experiments, no damage was found in the blocks (cracks were not generated). This suggested that material strength does not affect the failure of the structure; thus, the blocks could be assumed rigid. To confirm this a sensitivity analysis using deformable blocks was conducted and compared to the rigid block assumption (see Section 5.4.1). The results showed that the response of the structure did not change. The geometry of the structure and density of the blocks was obtained from the 1/12.5 small scale arch model, which was used to perform experimental testing. Figure 5.4 shows the geometry of the 3DEC numerical model based on the tested physical arch.  Figure 5.4 Numerical model of the 1/12.5 small scale arch using 3DEC software 69  The cross-shape shear keys, used for assemblage of the blocks, were modeled as simple face interfaces. The extra shear stiffness provided by the keys were assigned to the joints of the numerical model. The zero thickness interfaces between adjacent blocks are modeled as soft (deformable) contacts, following Coulomb slip failure criterion (Itasca C. G. 2013). This criterion requires six parameters: normal and shear stiffness, friction angle, cohesion, tensile strength and dilatation angle. The assigned joint properties should approximate the effect of the roughness of the blocks plus the shear keys. It was observed that in order to avoid penetration between blocks (in compression cases) during calculations, a minimum stiffness of about 0.3 GPa was required.  The modeled structure could only fail by joint sliding or opening between blocks and not due to material failure. However, the experimental tests showed that the shear keys prevent sliding, even when the critical point of collapse is passed. In order to reproduce this behavior, very high joint shear stiffness should have been assumed (around 2000 GPa or higher). However, using lower shear stiffness values of the order of 10GPa reduces computational time and still avoids sliding during most of the collapse. Additional sensitivity analysis (Section 5.4) showed that increasing joint stiffness requires many small time steps and does not change the solution. Hence, 10GPa shear stiffness and 20Gpa normal stiffness were assumed for the joints with the shear keys. Since the real model has more than one block in length, for the lateral interfaces of the blocks, where no shear key is present, different shear stiffness properties had to be assigned. Tilt test experimental results and a sensitivity analysis suggested that the arch is insensitive to the properties of these lateral joints since the hinging occurred along the same line for all the length of the arch. A shear stiffness of 0.1 GPa was chosen for the lateral joints.  70  Friction also plays an important role in preventing failure by sliding. For the numerical model a friction angle typical for masonry arch structures with mortar (Sarhosis et al 2013) of 35 degrees was chosen. This assumption is rather conservative due to the extra equivalent friction that the shear keys provide to the mortar. No cohesion and no tensile strength was assigned to the joints and dilatation angle was neglected. The following table shows the properties considered for both the blocks and the interfaces of the numerical model. Table 5.1 Properties of the blocks and of the interfaces between blocks in 3DEC  Block Properties  Density (kg/m3)  1715    Joint Properties Normal stiffness (GPa) 20 Shear stiffness (GPa) 10 (with shear keys) 0.1(with no shear keys) Friction Angle (degrees) 35 Cohesive strength (GPa) 0 Tensile strength  (GPa) 0 Dilation Angle (degrees) 0  The lower row of blocks which forms the base of the model was fixed in all directions as for the experiments.   5.3  Comparison between numerical and experimental quasi-static analysis As explained in Section 3.4, it is common practice to simulate earthquake loading by an equivalent horizontal force based on the PGA. This acceleration value can be obtained by tilting of the 71  structure. Several tilt tests were performed and were analyzed in Section 3.4.  In order to validate the numerical model with the experimental results, an equivalent tilt test using the 3DEC model was conducted. For this purpose, the tilt angle was increased until failure of the structure was observed. The loading was applied as gravity, divided in its corresponding vertical (in red) and horizontal (in blue) components, in the middle of each of the blocks, as shown in Figure 5.5.   Figure 5.5 Vertical and horizontal components of the gravity when tilting a block From a sensitivity analysis of the length of the arch (see Section 5.4.2) and the experiments (see Section 3.4.3.3), it was found that the response of the arch was the same for the four blocks and one block length structures. For computational purposes, the analysis was conducted using the one block length arch. 5.3.1 Collapse mechanism The numerical model collapsed following the four-hinge mechanism observed in the experimental testing and described in the literature review. Figure 5.6 (b1-b2) shows the failure mode of the computer model. The hinges were created in the same locations for all the cases. In the following figure the collapse mechanism recorded during the experimental tilt test (a) is compared to the one obtained by the computer model (b). The figure shows how the failure was initiated (1) and how it progressed (2). 72   Figure 5.6 Comparison of four-hinge collapse mechanism during quasi-static analysis (structure tilted from the left side) from a) experimental testing and b) numerical model  Most of the hinges occurred at the same locations for the experimental and numerical model, including the secondary hinge (orange color hinge in a2 and b2). However, hinging at the left side of the arch was observed between block number #12 and #13 (Figure 5.6 a) in the experiments instead of between #13 and #14 (Figure 5.6 b). Although, in general, the predicted failure mechanism by 3DEC agreed qualitatively with the hinging behavior shown in the laboratory, the exact location of all the hinges could not be forecasted. This hinging discrepancy, using discrete element modeling, was also observed by Lemos (2007) and DeJong (2009). 5.3.2 Equivalent α The numerical model collapsed at 9.7 degrees of inclination, which is equal to a 0.16g of α. However, DeJong (2009) found that an 80% of the real thickness should be considered in the 73  analytical model to account for the irregularities of the blocks and the instability of the model. A new numerical model was created using blocks with that reduced thickness. For that case, the arch collapses at a lower angle (3.7degrees), which corresponds to 0.07g. Table 5.2 shows the obtained results and compares them to the experimental data.  Table 5.2 Comparison of collapse tilt angle and corresponding α between numerical and experimental results    EXPERIMENTAL  NUMERICAL  Full thickness blocks 80% thickness blocks  Collapse Tilt Angle (degrees)  5.5  9.7  3.7  α (g)  0.1  0.16  0.07  The table above shows that the physical arch model collapsed under a lower acceleration (0.1 g) than the one predicted by the numerical model. But if just 80% of the thickness of the blocks is considered, the experimental data presents higher α values. It can be concluded that the irregularities of the blocks need to be taken into account reducing the thickness of the blocks, but not as much as 80%. In order to calibrate the model to the experimental data, a sensitivity analysis of the thickness of the blocks was conducted and is presented in  Section 5.4.3.  5.4 Sensitivity analysis results The variation of different parameters of the numerical model could have an important impact on the response of the structure. These parameters are identified as follows: - Joints: shear stiffness, normal stiffness and friction angle 74  - Blocks: deformability and density - Geometry: Length of the arch and thickness of the blocks The results obtained from a sensitivity analysis for the listed parameters are shown in this section. For each parameter a wide range of values was covered by running several quasi-static analyses.  5.4.1 Deformability of the blocks The same arch using deformable blocks was modeled in order to compare the results with the rigid block model. Deformable block models only require three material parameters: mass density, bulk modulus and shear modulus. These blocks are internally discretized into finite difference zone elements (as a coarse mesh) and assumed to behave in a linear elastic manner. Deformable blocks with different bulk and shear modulus were applied to the model covering a wide range of values. From the quasi-static analysis, no variation in the results was observed, obtaining the same collapse tilt angle for all of them (see Appendix D -Table D.1). 5.4.2 Length of the arch Several arches with the same geometry but different lengths (one, four and twenty blocks) were modeled. For all the cases, the same failure mechanism (see Figure 5.7) and same collapse tilt angle were observed. This agrees with the results from the experimental tilt testing using the same model with different lengths (see Table 3.3). Consequently, the arch is insensitive to the length, when the loading is applied in the transversal direction. 75   Figure 5.7 Collapse mechanism of the same arch with three different lengths: 1 block, 4 blocks and 20 blocks  5.4.3 Thickness of the blocks Arches with different block thicknesses (60, 80, 90 and 100% of the actual thickness) were subjected to quasi-static analysis. The thickness of the blocks was reduced equally from the inner and outer radius. The results have been plotted in Figure 5.8 (raw data in Appendix D - Table D.2).  Figure 5.8 Percentage of block thickness versus collapse α obtained from quasi-static analysis  α (g) 76  The α decreases significantly with the reduction of the thickness of the blocks. In fact, at 60% of the original thickness the arch is unstable even under its self-weight.  Heyman (1972) stated that the minimum thickness of the arch required under self-weight could be approximated through the following formulation:                                                        (𝑏𝑅)𝑚𝑖𝑛= 2(𝛽−sin 𝛽)(1−cos 𝛽)𝛽(1+cos 𝛽))                                              (6) Where ‘b’ is the thickness of the blocks, ‘R’ the radius of the arch and ′𝛽′ the angle (with the vertical) in radians at which hinging occurs. For the studied arch this angle is equal to 54 degrees. If this value is substituted in the previous equation, a (𝑏𝑅)𝑚𝑖𝑛 of 0.0736 is obtained; thus, the minimum required thickness for the arch of this study would be 33%. Another study made by Oschendorf (2002) corrects this minimum ratio to 0.1075 for the 180degrees inclusion angle arch; which is equal to a minimum thickness of 50%. The arch was found to be unstable for 60% of thickness as shown in Figure 5.8. This is due to the fact that the half circular arch is supported on top of two rows of vertical blocks, making it more difficult to maintain the thrust line within the thickness of the arch. In order to prove this, a quasi-static analysis was conducted in the same arch but fixing also the second row of blocks. The obtained results are plotted in Figure 5.9 and can be found in Appendix D - Table D.3. 77   Figure 5.9 Percentage of block thickness versus α obtained from the quasi-static analysis for the arch with first two rows of blocks fixed  From  Figure 5.9, the arch with the first two rows fixed was found to be stable for  60% block thickness. This corroborates Heyman (1972) and Oschendorf (2002) calculations for minimum required thickness of the arch.  As shown in both Figure 5.8 and Figure 5.9, the α is decreased as the thickness of the blocks is reduced. For instance, when reducing the effective thickness from 100% to 60%, the α decreases from 0.34g to 0.19 g for the model with the first two rows fixed, and from 0.16g to complete collapse for the original model. Additionally, the α increases significantly when an additional row of blocks was fixed. For 100% of effective thickness, the α was doubled (from 0.16g to 0.34g) when fixing the second row of blocks. In the experiments, the 1/12.5 scale arch collapsed for 0.1g. In order to obtain that value, linear interpolation was applied in Figure 5.8, and the corresponding effective thickness of 92% was obtained. The model was calibrated using that effective thickness. If the blocks get damaged, then a lower effective thickness should be used. α (g) 78  5.4.4 Density of the material  A sensitivity analysis of the arch to the density of the material of the blocks was performed using the numerical models with the first row and the first and second rows of blocks fixed. The results obtained are plotted in Figure 5.10 and can be found in Appendix D - Table D.4 and Table D.5. The density was normalized to the actual density of the blocks (1715 kg/m3).  Figure 5.10 Normalized density versus α obtained from quasi-static analysis for two numerical models with different boundary conditions  In Figure 5.10, the α decreases as the density increases. For the model with the first two rows of blocks fixed, the α drops drastically after the normalized density value of 5. In contrast, the model with the first row of blocks fixed slightly decreases, although it follows the trend of the previous model. It can be argued then that as more rows of blocks are fixed, the arch becomes more stable but also more sensitive to density changes. For the two models, a flat line up to the normalized density value of 4 is shown. In practice, the blocks could be damaged, missing some material, or be made of a denser or lighter mix. However, the density should not vary more than 100% of the actual one; assuming a density increase by a α (g) 79  factor of 4 is unrealistic. The range of interest would fall into the flat line of the graph, meaning no variation in the collapse α due to density changes. 5.4.5 Joint shear stiffness In order to study the influence of the shear stiffness in the response of the arch, a sensitivity analysis of this parameter was conducted. The obtained results are shown in   Figure 5.11 and the data can be found in Appendix D- Table D.6 and D.7.   Figure 5.11 Shear stiffness versus α obtained from quasi-static analysis for two numerical models with different boundary conditions  In the graph above, the α remains almost constant for all the considered shear stiffness range, showing a slight decrease for values lower than 0.1GPa. For that low shear stiffness, sliding was observed to be the dominant failure mechanism. For values higher than 0.1GPa, the arch collapsed always by opening between the blocks and with similar α values. During the experiments, sliding between blocks was not observed. Hence, shear stiffness values higher than 0.1GPa should be considered for a proper characterization of the response of the arch.  α (g) 80  5.4.6 Friction Angle In order to study the effect of the friction angle in the collapse mechanism, tilt testing with a wide range of different values was performed in the numerical model. The results are plotted in Figure 5.12 (values in Appendix D- Table D.8).   Figure 5.12 Friction angle versus collapse α obtained from the quasi-static analysis In Figure 5.12, sliding (red line of the graph) is observed to occur at 23degrees of friction angle. Assumptions of lower values would lead into a failure mechanism governed by the sliding between the blocks. Once sliding is avoided, the value of the friction angle does not influence the α (flat line in the graph). Although a test of the contact between two real blocks to obtain the friction angle was not carried out, past studies regarding to modeling of masonry structures used a friction angle of about 30 to 40 degrees. Lock-Block arch systems should be in that range or even higher since the shear keys provide equivalent extra friction. α (g) 81  5.4.7 Comparison of the response of numerical models for arches with different scaling A full-scale and a 1/25 scale numerical model of the arch structure are built to compare the results with the 1/12.5 scale numerical arch used during this study. The scaling of the dimensions was performed as specified in Section 3.6.1. Quasi-static analysis in all different scale arches showed collapse at the same tilt angle of 9.7 degrees. This comparison confirmed that the scaling rules applied during this study are correct. Failure of arches fundamentally depends on the geometry of the system. If all the dimensions are consistently scaled, the acceleration scale factor remains equal to one.    5.5 Concluding remarks of the numerical analysis A 3DEC distinct element model was created for the 1/12.5 scale arch. The computer model was calibrated to the tilt testing experimental results. The same four-hinge collapse mechanism observed during the experiments was obtained. An effective thickness of the blocks of 92% should be considered to account for the irregularities of the blocks and instability of the physical model. From a sensitivity analysis to different modeling parameters, it was found that the arch is insensitive to most of them. Nevertheless, both thickness of the blocks and boundary conditions seem to have an important impact on its response. The scope of this study was limited to quasi-static analysis. Future work will include dynamic analysis using the calibrated numerical model described in this Chapter.   82  Chapter 6: Discussion of The Results In this section some of the results presented in Chapter 3, 4 and 5 are discussed and compared.   6.1 Unreinforced model 6.1.1 Collapse mechanism In all experimental testing and numerical analysis, the same critical failure mode based on the four-hinge mechanism explained in the literature review was observed. The hinges were always created in the same locations during both tilt and uniaxial shake-table testing.  Since the arch failed due to rocking between the blocks, the shear keys had a negligible impact on preventing the collapse. From a sensitivity analysis of the modeling parameters of the numerical model, it was found that when low joint shear stiffness or friction angle is considered, the arch could fail at lower PGA due to sliding of the blocks. Past studies have shown that this sliding failure mode can also be prevented by mortar or by the roughness of the masonry itself. Therefore, the shear keys may improve performance by minimizing sliding between blocks. 6.1.2 Collapse accelerations  When comparing the α values obtained from tilt testing to the results from the shake-table testing, it is seen that the α values are significantly lower than the collapse PGA for any of the earthquakes. For instance, for the 1/25 scale model, a α of 0.06g was obtained and the lowest collapse PGA recorded was 0.16g (for the Tohoku earthquake). For the 1/12.5 scale model, a α of 0.1g was observed and the lowest PGA was found to be 0.25g (for the Northridge earthquake). In both cases, the PGA-s that the arch could withstand during shake-table testing were at least 2.5 times higher than the α values recorded from tilt testing. This confirms that tilt testing gives a very conservative estimate of the collapse peak acceleration for the selected earthquakes. However, a 83  limited number of earthquakes were used for this study. The level of conservativeness may be affected by: - pulses with very low dominant frequency - records with several pulses  - structure entering into resonance 6.1.3 Acceleration and displacement sensitivity studies The sensitivity of the unreinforced arch to accelerations and displacements was studied. The instantaneous base accelerations (IBA) and displacements (IBD) that caused the collapse of the arch during shake-table testing in the 1/12.5 scale unreinforced arch are compared in Table 6.1.  Table 6.1 Recorded IBA-s and IBD-s during shake-table testing for collapse TL in the three directions for the 1/12.5 scale arch model Earthquake TL IBA (g) IBD (mm)  (%) Transversal Longitudinal Vertical Transversal Longitudinal Vertical Loma Prieta 80 0.47 0.36 0.44 5.46 5.15 2.82 Northridge 30 0.3 0.21 0.33 5.51 5.15 2.75 Parkfield 100 0.47 0.35 0.11 3.9 1.93 0.94  At collapse level, both the Loma Prieta (at 80% TL) and the Northridge (at 30% TL) earthquakes the IBD-s were very similar. The IBD-s recorded for Parkfield, for which no collapse was recorded, were lower than for the other two earthquakes at collapse. However, the IBA-s were higher for Loma Prieta and Parkfield (even if the structure survived to this last one) than for Northridge. This may suggest that Lock-Block arches are displacement sensitive structures. 6.1.4 Effect of the vertical motion on the response of the arch When the ground motions were applied in three directions an additional hinge was created. Furthermore, the structure failed earlier in the record and the time of collapse decreased. This behavior was observed during the Loma Prieta and Northridge earthquakes; for which the PGA of 84  the vertical component is about 73% and 100% of the PGA of the horizontal transversal component respectively. The effect of the vertical motion may be negligible for earthquakes with lower percentage of vertical component. Although in the conducted test, adding vertical motion resulted in an earlier collapse, some other combination of horizontal and vertical pulses could delay the failure of the structure.  This implies that the intensity of the vertical component, as well as the combination of vertical to horizontal pulses could have an effect on the response of the arch.   6.1.5 Stability of the unreinforced model  A study of the stability of the models was performed by comparing some of the results obtained from different tests and numerical analysis. The applied scaling laws were confirmed by numerical analysis; models with different scaling collapsed at same lateral acceleration. However, during both tilt and shake-table testing, the 1/25 scale model collapsed at lower ground acceleration values than the 1/12.5 scale model.  A study using the numerical model showed that this problem could be mainly caused by the high sensitivity of the arch to irregularities, damage and imperfections. In the table below the effective thickness of the calibrated numerical models of the 1/25 and 1/12.5 scale structures and their corresponding α values are shown. Calibration to match the experimental data was obtained using 92% effective thickness for the 1/12.5 scale model, and 79% effective thickness for the 1/25 scale model.  The 79% effective thickness is very close to the 80% effective thickness proposed by DeJong et al (2008) for their analytical model.   85  Table 6.2 Effective thickness and HGF values obtained from the numerical analysis for the calibrated models of the 1/25 and 1/12.5 scale structures  MODEL Calibrated Numerical Model Effective Thickness (%) α(g) 1/25 scale 79 0.06 1/12.5 scale 92 0.1  It can be argued then that a small variation in the effective thickness of the blocks results on a significant lower performance of the arch. Smaller blocks could be affected more by damage since a large percentage of the surface area is involved. For a smaller model, the effective thickness should be lower to properly predict the behavior of the real structure.  6.2 Reinforced model 6.2.1 Sensitivity of the reinforced arches As with unreinforced arches, the externally reinforced arches were also found to be sensitive to pulse shape motions by amplification of the top acceleration. The reinforced arches suffered the highest amplifications at the second half of the main pulse; coinciding with the time when the collapse occurred for the unreinforced arches.  Additionally, vertical motion was observed to amplify the acceleration of the reinforced model. When horizontal and vertical pulses occurred simultaneously, a higher amplification was recorded.  Although failure of the structure was not observed for any of the conducted reinforced experiments, the increase of amplification of the acceleration caused by pulse type ground motions and their vertical component should be taken into account in the design of the reinforcement. 86  6.2.2 Recorded forces  It was observed that the unreinforced arch collapsed at 80%TL for Loma Prieta, 30% TL for Northridge; the arch did not collapse when subjected to Parkfield even beyond 100%. The FnW values were 21% for Loma Prieta, 31% for Northridge (at 40% TL) and 18% for Parkfield (at 100% TL). Since there was a collapse from Loma Prieta and not from Parkfield, this suggests that there is a force level in between which may trigger collapse (~20%). 6.2.3 Effect of the pretension From the experiments, the recorded dynamic forces were found to be dependent on the pretension of the straps. As the pretension was increased, the recorded dynamic forces were reduced (when the strain gauges were zeroed between tests). The total forces recorded at the straps were approximately constant for most of the pretension levels.  In addition, the applied pretension changes the natural frequency of the arches. During sine-sweep testing, 50% of reduction of the pretension resulted on a decrease of about 50% of the natural frequency. This suggests that the structure could be tuned by applying the desired pretension level to reduce the seismic loads.     87  Chapter 7: Conclusions and Future Work Lock-Block Ltd., of Vancouver, Canada has developed a structural arch system consisting of modular concrete blocks. The arch system is intended to be long lasting, easy to construct and have low cost. The main goal of this thesis was to assess the seismic performance of these arch systems and to develop concepts of strengthening if necessary. This was achieved by a program of experimental testing using small scale arches and numerical modelling of an arch in a prototypical configuration. It was found that in this configuration, the arches are vulnerable to seismic excitation and at risk of collapse when unreinforced and unconfined. There are several solutions to reduce that risk based on the results of this work: 1) addition of external or internal reinforcement to prevent hinge opening 2) restraint of the bottom courses of blocks and 3) modification of the geometry at the base to improve stability.  7.1 Specific objectives The main objectives of the study were accomplished and are summarized as follows:  The main observations on the performance of the arches were obtained by performing quasi-static testing, shake-table testing and numerical modelling. It was found that the unreinforced and unconfined arches in the prototypical configuration were vulnerable to collapse due to strong shaking and a possible reinforcement was studied.  Several shake-table tests were performed using a variety of earthquakes, test levels and directions (including vertical). It was found that the most significant parameter on the response of the arch was the impulse-type motion. 88   Based on the results from the experimental testing and the numerical modelling, it was observed that the scale model arches collapsed following the four-hinge mechanism, which is a rocking-type failure and agrees with what is found in the literature.  A simple external reinforcing system was implemented for the experimental tests. It was found that the reinforcing prevented collapse for all tests, and typically had loads of 30% of the weight of the arch for most earthquakes, while in some cases loads in excess of 100% of the weight of the arch.    A numerical model was created and calibrated to the experimental testing with the 3DEC discrete element software. The numerical model agreed with the failure mechanism of the arch system and a sensitivity analysis was performed which showed that restraining multiple rows at the bottom of the arch increased its seismic resistance.  7.2 Important observations from the study The following observations were made from this study:  Although the shear keys prevented sliding between the blocks, they had negligible impact on the critical rocking failure mode.  The α values obtained from tilt testing were confirmed to be conservative estimations when comparing them to the results from the shake-table testing.  The unreinforced arches were more vulnerable to pulses with lower dominant frequency.   Addition of vertical motion does not have an effect on collapse test level as compared to horizontal motion only. It does however change the time of onset and the hinging of the collapse mechanism. 89   The reinforced structures were also found to be sensitive to pulse type motions.  The recorded dynamic forces at the straps were shown to be dependent on the applied pretension.  The forces recorded at the straps were very similar for all the earthquakes at collapse test level of the unreinforced arches.  7.3 Potential solutions  Based on the results of this study, several solutions to improve the seismic performance and reduce the risk of collapse are presented:  Change the geometry: In order to maintain the unreinforced arch, the geometry could be changed to keep the thrust line within the thickness of the blocks. This can be done by increasing the ratio of the block thickness to radius, reducing the inclusion angle, or changing the shape of the arch. This was outside of the scope of this work but the numerical model could be used for this purpose.  Securing the bottom rows: Restraining the bottom row(s) of blocks increases the resistance of the arch to lateral loads. This was observed in the numerical modelling where the acceleration to collapse due to the tilting increased as the blocks were fixed. This could be accomplished by attaching the blocks to a rigid foundation or ensuring adequate confinement.  Reinforcement: Reinforcing the arch to ensure that the four hinge mechanism is not allowed to occur would prevent collapse. This could be accomplished by internal or externally applied reinforcement, which maintains the arch in compression and prevents the openings between blocks.  90  7.4 Further experimental testing In order to better characterize the response of the full-scale arches and their response to a wide variety of earthquakes a further experimental testing program could be performed, including:  Study of the natural frequencies of the structure by ambient vibration test in a full-scale structures and sine-sweep testing using small scale models.  Analysis of the response of the arch to different combinations of vertical and horizontal pulses by shake-table testing using wider number of earthquakes.  Comparison between the responses of the full-scale and small scale models by quasi-dynamic testing in the full-scale structure using a cyclic loading protocol.  7.5 Further development of the numerical model In further stages of the project, the calibrated discrete element model will be used to perform dynamic time history analysis. 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Experimental Mechanics Vol. 11, No. 7, (pp. 325-336).  97  Appendices Appendix A  - Instrumentation used for the experimental testing A.1 Specifications for the shake-tables Table A.1 Specifications for the APS ELECTRO-SEIS shaker Size (Armature) 254mm x 254mm (10 in x 10 in) Payload 23 kg (50 lb) Degrees of Freedom 1 Maximum Stroke 158 mm (6.25 in) pk-pk Maximum Velocity 762 mm/s (30 in/s) Frequency Range 0 to 200 Hz Armature Weight 2.27kg (5.0 lb) Shaker Body Weight 35 kg (78 lb)   Force Rating DC to 0.1 Hz 94 N (21 lb) Above 0.1 Hz 133 N peak (30lb) Above 20 Hz Refer to Figure A.1   98   Figure A.1 Force envelope with frequency for the APS shaker   Figure A.2 Velocity envelope with frequency for the APS shaker 99  Table A.2 Specifications for the MAST  Size 4 m x 4 m (13 ft x 13 ft) Payload 10 t (20, 000 lb) Degrees of Freedom 6 X Horizontal Displacement  (max.) static: ±242 mm (±9.5 in) dynamic: ±216 mm (±8.5 in) Y Horizontal Displacement  (max.) static: ±152 mm (±6.0 in) dynamic: ±120 mm (±5.0 in) Z Vertical Displacement  (max.) static: ±84 mm (±3.3 in) dynamic: ±76.2 mm (±3.0 in) X Horizontal Velocity      (400gpm) 1.67 m/s (65.9 in/s) Y Horizontal Velocity      (300gpm) 1.25 m/s (49.4 in/s) Z Vertical Velocity (150gpm) 2.08 m/s (82 in/s) X Horizontal Acceleration 20kip payload: 1.24g (12.2m/s2) Bare table: 2.6g (25.5m/s2) Y Horizontal Acceleration 20kip payload: 2.47g (24.2m/s2) Bare table: 5.2g (51.0m/s2) Z Vertical Acceleration 20kip payload: 0.47g (4.61m/s2) Bare table: 2.1g (20.6m/s2) Operating Frequency (max) 40Hz Table Weight 9 t (18000 lb) X Horizontal Dynamic Force Static: 70,000 lb Dynamic: 47000 lb Y Horizontal Dynamic Force Static: 140,000 lb Dynamic: 94000 lb Z Vertical Dynamic Force Static: 84,000 lb Dynamic: 56 000 lb  100   Figure A.3 Velocity in x-axis versus frequency for the MAST   Figure A.4 Velocity in y-axis versus frequency for the MAST 101    Figure A.5 Velocity in z-axis versus frequency for the MAST  A.2 Control system The actuators, hydraulic power supply, and manifolds are all under direct control of the main control system (low level controller). This comprises of a special computer in the control room with a custom graphic user interface (GUI) which communicates with four hydraulic control computers (Moog MSC 3000’s) in the basement via Ethernet. Using this GUI, a user can control all the hydraulic equipment (pumps, manifolds, valves, etc.) that supplies hydraulic power to the actuators. The operator also controls the operating position of the table, actuator tuning, basic sine wave testing and maximum operating limits using this low level controller. Advanced program signals are generated by a “high level controller” such as the Spectral Dynamics Jaguar or a Dasylab computer. This last one can supply simple control signals such as 102  earthquake records when control demands are not critical. However, the Jaguar was used for the conducted testing since it provides a more precise control of the table systems using feedback signals and sophisticated control techniques.  This controller is able to generate multiple types of program signals (swept sine, dwell pulses and earthquake records) that are sent to the table through the low level control system.   For applying an earthquake record at a certain intensity level, Jaguar controller uses a voltage input (in dB). This way, 0 dB is equal to applying the exact input record at 100%. Following table shows the equivalent intensity values to the input Jaguar level used during experimental testing, as well as the approximate intensity that this study will be referring to.  Table A.3 Jaguar dB table values with their corresponding equivalent intensity and the approximate value used during the study Level in dB Equivalent intensity (%) Approximate intensity used for the study (%) -6 50.1 50 -4 63.1 65 -2 79.4 80 0 100 100 2 125.9 125    103  A.3 Hydraulic system  Figure A.6 HPS for the MAST The HPS (Hydraulic Power Supply) that the MAST uses is located in the basement of the lab and is a custom designed system that consists of an 800gallon reservoir and two 90 GPM hydraulic pumps. The pumps have been taken from an MTS 506.92 HPS unit and the original pump controller has been retained (mounted on the basement wall).  The HPS supplies 3000 PSI at up to 180 GPM to the accumulator bank which is made up of eight 15gallon bladder type accumulators. These accumulators are pre-charged to 2350 psi and store an extra 120 gallons of hydraulic pressure which can be supplied very quickly to the three manifolds as needed.  Figure A.7 Accumulator bank 104  The three 600 GPM manifolds provide low (500 psi) and high pressure (3000 psi) control of the oil being supplied. The manifolds have oil filters and accumulators on them and are generally identical to each other.  Figure A.8 Manifolds in room adjacent to table pit  A.4 Data acquisition (DAQ) Data acquisition was done using a modular National Instruments SCXI chassis (see Figure A.9). The large SCXI chassis is housed in a metal cabinet located in front of the shake-tables. The chassis has room for up to 11 signal conditioning modules, each with multiple input channels. The 1520 module has 8 channels with programmable gain, excitation and low pass filtering and can be used with all types of sensors.  Other modules are more sensor specific but have more channels.   The system can accommodate up to about 100 sensor channels if required. The SCXI chassis is connected to a data acquisition card located in the PC (also in the cabinet). DASYLab software was setup to provide data capture and display while the testing was running.   105  .  Figure A.9 DAQ cabinet with SCXI chassis and connected PC  106  A.5 Accelerometers  Figure A.10 Specifications of the seismic, ceramic shear ICP accelerometer (shake-table accelerometer)  107     Figure A.11 Specifications of the +/- 10g silicon MEMS top and bottom accelerometers (highlighted inside of the red rectangle)  A.6 High-speed camera Phantom v4.2 is a monochromatic camera that operates using a SR-CMOS sensor, which records images with an 8-bit depth. Images stored with a bit depth of 8 can be stored at 256 possible levels. This means that pixels seeing complete darkness are reported as a value of 0 and pixels seeing light at or above the saturation limit of the sensor are reported as a value 255. Light levels between black and saturation are converted linearly to values between 0 and 255. Sampling rates can be up to 2100fps at full 512x512 resolution (which was used for the experiments) and even higher at 108  reduced resolutions. Exposure time can be varied for this camera, from as low as 2μ to the just shy of the inverse of the frame rate. Images are continually stored in the 4 gigabyte DRAM buffer of the camera until triggered, and then a specified number of images in both sides of the trigger point are saved and transferred to a computer via an Ethernet connection. A select number of frames, up to the entire buffer, determined by the user can be saved to the hard drive. The high speed camera was controlled through the use of Phantom 675, a software package provided by the same manufacturer. The software was used to set the frame rate, exposure time, resolution, and to trigger the capturing process. The software also had many post capture image processing capabilities including adjustment of brightness, contrast, and image orientation; determination of distances and speeds; conversion to different file types such as JPEG, AVI, TIFF and Quick Time; and the supply of an image information such as frame rate, exposure time, and date of capture.  A.7 Strain gauges  KFG series gauges were used for the instrumentation of the reinforced models (see Figure A.13). These strain gauges use polymide resin for the base approximately 13 μm thick, ensuring flexibility. The dimensions and specifications of the strain gauges used in this study are shown in Table A.3 and A.4 respectively. 109   Figure A.12 KFG-5-120-C1-11 model strain gauge used for the instrumentation of the reinforcement  Table A.4 Specifications of the strain gauges used for the reinforced experimental testing Type KFG-5-120-C1-11 Grid dimensions (mm) 5 x 1.4 Base dimensions (mm) 9.4 x 2.8 Gauge resistance 119.8 +/- 0.2 Ω Gauge factor (24°C, 50 % RH) 2.11  +/- 1.0 % Adoptable thermal expansion 11.7 PPM/°C Transverse sensitivity (24°C, 50 % RH) 0.40 %    110   Figure A.13 Schematic representation of the setup and instrumentation for the calibration of the straps at the 1/25 scale model          Cross Sectional area = 0.806 mm2 111  Table A.5 Calibration of the straps used for the reinforcing of the 1/25 scale model Run  Added Weight (Kg) CH2(µɛ) CH3(µɛ) CH4(µɛ) 1 1 75 71 62   2 138 132 122   3 196 191 173   4 256 250 230 2 Added Weight (Kg) CH2(µɛ) CH3(µɛ) CH4(µɛ)   1 66 67 60   2 127 127 119   3 186 186 174   4 246 246 235 3 Added Weight (Kg) CH2(µɛ) CH3(µɛ) CH4(µɛ)   1 63 64 57   2 125 126 118   3 184 184 169   4 244 244 232             112   Appendix B  - Experimental unreinforced testing Table B.1 Obtained collapse tilt angle and the corresponding α for different arch models MODEL TEST NUMBER LENGTH (cm) ANGLE OF COLLAPSE (degrees) α (g) AVERAGE α (g) 1/25 scale 1.5 m interior diameter arch 1 3 14.7 0.26 0.25 2 14.4 0.26 3 13 0.23 1/25 scale 6 m interior diameter arch 1 6 2.9 0.05 0.06 2 3.9 0.07 3 3.7 0.06 1/12.5 scale 6 m interior diameter arch 1 48 5.3 0.093 0.096 2 5.6 0.098 3 5.5 0.096  Table B.2 Obtained angle of collapse and the corresponding α for the 1/25 scale 3 m model with different lengths for all the performed tests MODEL (1/25 scale) TEST NUMBER LENGTH (cm) ANGLE OF COLLAPSE (degrees) α (g) AVERAGE α (g) 3 m Arch 1 block width 1 6 13 0.23 0.24 2 12.4 0.22 3 14.8 0.26 3 m Arch 1/2 block width 1 3 17.1 0.31 0.31 2 17 0.31 3 17.7 0.32 3 m Arch 1-1/2 block width 1 9 16.9 0.30 0.29 2 16 0.29 3 15.7 0.28 3 m Arch 2 block width 1 12 16.4 0.29 0.33 2 11.2 0.20 3 12 0.21 4 16.4 0.29 3 m Arch 3 block width 1 18 14.7 0.26 0.26  113  Table B.3 Parameters of the selected original time history records ORIGINAL EARTHQUAKES EARTHQUAKE NAME Component File name Peak acc (g) Peak displ (cm) Peak vel (cm/sec) Duration (sec) Arias intensity (m/sec) Significant duration (sec) Predominant period (sec) NISQUALLY HZ1 BHD-Dir1 0.16 5.65 23.57 172.99 0.429 17.76 0.6 HZ2 BHD-Dir2 0.13 3.49 14.17 172.99 0.247 26.44 0.56 V BHD-Ver 0.087 2.11 8.87 172.99 0.205 24.08 0.58 TOKACHI-OKI HZ1 HKD-EW 0.498 12.1 32.6 241.99 3.72 26.08 0.28 HZ2 HKD-NS 0.298 6.28 20.65 241.99 3.735 27.2 0.24 V HKD-UD 0.245 6.65 12.09 241.99 0.98 31.6 0.1 LOMA PRIETA  HZ1 CLS000 0.64 10.82 55.15 39.94 3.23 6.875 0.3 HZ2 CLS090 0.479 11.29 45.15 39.94 2.53 7.88 0.58 V CLS-UP 0.455 7.108 17.67 39.94 0.86 7.59 0.22 KOBE HZ1 NIS000 0.51 9.53 37.28 40.95 3.35 9.72 0.46 HZ2 NIS090 0.5027 11.26 36.62 40.95 2.269 11.23 0.44 V NIS-UP 0.37 5.63 17.28 40.95 1.32 10.52 0.3 TOHOKU HZ1 MYG0091103111446-EW 0.56 14.57 37.76 299.99 5.81 101.89 0.24 HZ2 MYG0091103111446-NS 0.47 13.52 34.36 299.99 4.96 104.3 0.26 V MYG0091103111446-UD 0.21 7.03 15.98 299.99 1.84 102.76 0.26 PARKFIELD HZ1 CO5085 0.44 5.11 24.63 43.9 0.857 6.45 0.36 HZ2 C05DWN 0.367 3.84 21.77 43.91 0.62 7.44 0.3 V C05355 0.138 2.67 6.84 43.91 0.167 11.47 0.08 NORTHRIDGE HZ1 RRS228 0.825 29.63 160.12 19.9 7.5 7.25 0.7 HZ2 RRS318 0.486 26.97 74.52 19.9 4.23 9.28 0.3 V RRS-UP 0.83 10.05 43.52 19.9 6.04 6.6 0.08 114  Appendix C  - Experimental reinforced testing Table C.1 Measured displacements of the middle key block at 30 and 45 degrees of inclination for different pretension levels for tilt testing in 1/12.5 scale reinforced model  DISPLACEMENTS (mm)                                               Middle point of the key block Tilt Angle (degrees) 0 30 45    Pretension (N) 20 0 5 6.5 39 0 4 6 59 0 3.5 5.5 78 0 3.5 4.5 118 0 2.5 4.5 157 0 2.5 4   Table C.2 Pretension in the straps at the beginning and end of each sine-sweep test TEST NUMBER # Ch0 Ch1 Ch2 Ch3 Ch4 Ch5 Ch6 Ch7 AVERAGE 1             Start Pretension (N) 117.6 117.6 117.6 117.6 117.6 117.6 117.6 117.6 117.6 Residual Pretension (N) -14.7 -9.8 -14.7 -19.6 -9.8 -4.9 0.0 -14.7 -11.0 New Pretension of the straps 102.9 107.8 102.9 98.0 107.8 112.7 117.6 102.9 106.6              Northridge X -100%            2 tests                        2            Start Pretension (N) 34.3 83.3 73.5 29.4 73.5 63.7 117.6 34.3 63.7 Residual Pretension (N) 0 0 0 0 0 0 0 0 0 New Pretension of the straps 34.3 83.3 73.5 29.4 73.5 63.7 117.6 34.3 63.7             Northridge X - 100%                        3            Start Pretension (N) 34.3 83.3 73.5 29.4 73.5 63.7 117.6 34.3 63.7 Residual Pretension (N) 0 0 0 0 0 0 0 0 0 New Pretension of the straps 34.3 83.3 73.5 29.4 73.5 63.7 117.6 34.3 63.7             4            Start Pretension (N) 34.3 83.3 73.5 29.4 73.5 63.7 117.6 34.3 63.7 Residual Pretension (N) 0 0 0 0 0 0 0 0 0 New Pretension of the straps 34.3 83.3 73.5 29.4 73.5 63.7 117.6 34.3 63.7  115  Table C.3 Recorded strains in each channel and the corresponding ratio for different earthquakes at collapse TL using 1/25 scale arch  EARTHQUAKE TL Force (N)  FnW *   Ch2  middle point Ch3 right point Ch4 left point MAX MIN MAX MIN MAX MIN Nisqually 120% 0.99 -3.03 2.19 -2.22 0.97 -2.67 3.18 Tokachi Oki 90% 1.77 -4.40 3.77 -1.58 1.53 -2.09 5.47 Loma Prieta 40% 1.26 -4.09 4.04 -1.41 1.37 -2.25 5.86 Kobe 40% 1.47 -2.37 2.66 -1.25 1.58 -1.79 3.86 Tohoku 40% 2.12 -1.80 2.89 -1.28 2.15 -1.35 4.20 Parkfield 80% 1.42 -2.70 2.50 -1.24 1.45 -1.96 3.62 * FnW ratio was calculated using the recorded maximum forces of Ch3 since the arch failed towards the right side  Table C.4 Recorded peak forces in each of the channels for different earthquakes at different TL for tri-axial shake-table testing using 1/12.5 scale arch  EQ TL % PEAK FORCES (N)  FnW *  Ch0 Ch1 Ch2 Ch3 Ch4 Ch5 Ch6 Ch7 Northridge 50 86 58 110 82 100 126 77 97 15.47 65 151 87 180 147 165 186 153 153 25.93 80 297 137 321 291 292 314 326 270 48.21 100 383 166 391 370 347 386 393 326 59.25 Loma Prieta 50 25 5 18 17 20 14 27 18 3.18 65 42 10 37 32 38 26 65 39 6.38 80 71 18 66 68 67 47 88 55 10.55 100 105 28 88 95 89 79 110 70 14.52 Parkfield 50 14 6 15 13 16 18 20 26 2.80 65 14 6 15 13 15 14 21 14 2.42 80 20 5 19 17 15 14 20 18 2.80 100 51 21 61 53 61 44 67 60 9.10 125 52 17 61 59 69 37 82 55 9.50 * FnW  ratio was calculated using the average of the recorded maximum forces of the channels (except for Ch1 because the data was very off from the rest of the channels)     116  Table C.5 Recorded accelerations at the shake-table and top of the model for different earthquakes at different TL for tri-axial shake-table testing  Earthquake TL (%)  Recorded Acceleration at the middle of the shake-table (g) Acceleration Top of the Model (g)  Amplification of x acceleration* (%)  X Y Z X Northridge 50 Max 0.39 0.22 0.37 0.69 77 Min -0.42 -0.23 -0.34 -0.54 29 65 Max 0.42 0.27 0.46 0.93 121 Min -0.55 -0.27 -0.7 -0.85 55 80 Max 0.63 0.34 0.64 1.4 122 Min -0.67 -0.38 -1.12 -1.32 97 100 Max 0.62 0.42 0.73 1.52 145 Min -0.82 -0.39 -0.75 -1.68 105 Loma Prieta 50 Max 0.23 0.23 0.14 0.21 -9 Min -0.23 -0.24 -0.21 -0.42 83 65 Max 0.43 0.29 0.21 0.34 -21 Min -0.36 -0.24 -0.36 -0.58 61 80 Max 0.6 0.35 0.42 0.35 -42 Min -0.43 -0.36 -0.65 -0.73 70 100 Max 0.53 0.43 0.26 0.5 -6 Min -0.49 -0.37 -0.34 -0.91 86 Parkfield 50 Max 0.14 0.16 0.04 0.12 -14 Min -0.1 -0.11 -0.05 -0.25 150 65 Max 0.19 0.23 0.05 0.19 0 Min -0.15 -0.2 -0.06 -0.33 120 80 Max 0.2 0.33 0.06 0.24 20 Min -0.19 -0.25 -0.08 -0.35 84 100 Max 0.46 0.35 0.1 0.38 -17 Min -0.21 -0.29 -0.11 -0.53 152 125 Max 0.53 0.41 0.13 0.39 -26 Min -0.3 -0.32 -0.15 -0.53 77 *Amplification of the acceleration at the top of the model respect to the bottom acceleration (at the middle of the shake-table) both of them measured in the transversal direction (X direction)    117  Table C.6 Recorded forces, initial pretension and final pretension at different channels of the straps for Northridge uniaxial reinforced shake-table testing  Test# Forces (N)  Ch0 Ch1 Ch2 Ch3 Ch4 Ch5 Ch6 Ch7 Average 1                  Initial Pretension  103 108 103 98 108 113 118 103 107 Peak Forces  324 338 433 328 435 292 438 281 359 Residual Forces  -59 -15 -15 -54 -15 -44 10 -59 -31 New Pretension  44 93 88 44 93 69 127 44 75 2          Initial Pretension  44 93 88 44 93 69 127 44 75 Peak Forces  455 407 503 452 514 400 536 408 459 Residual Forces  -10 -10 -15 -15 -20 -5 -10 -10 -12 New Pretension  34 83 74 29 74 64 118 34 64 3          Initial Pretension  34 83 74 29 74 64 118 34 64 Peak Forces  461 415 517 469 532 417 548 427 473 Residual Forces  0 0 0 0 0 0 0 0 0 New Pretension  34 83 74 29 74 64 118 34 64        118  Table C.7 Recorded peak forces in each of the channels, average and FnW values for different pretension levels and different TL for biaxial Loma Prieta record using the 1/12.5 scale arch  Pretension (N) TL (%) PEAK FORCES (N)  FnW  (%) Ch0 Ch1 Ch2 Ch3 Ch4 Ch5 Ch6 Ch7 Average*          20 75 53 16 37 49 12 48 67 55 51 8.2 100 102 41 105 107 25 85 127 112 106 17.0 120 183 81 221 185 47 119 210 171 181 29.0   78  75 35 16 46 46 14 41 82 30 47 7.5 100 109 41 118 113 28 96 141 102 113 18.1 120 196 78 224 203 52 161 230 176 198 31.7 *Ch1 and Ch4 were not considered for computing the average since the recorded data in those channels was really off from the rest of the channels         119  Appendix D  - Numerical analysis Table D.1 Data obtained from quasi-static analysis using 3DEC for totally rigid and deformable blocks with different parameters Runs Rigidity Collapse Tilt Angle (degrees) ‘α’  (g) K = Bulk modulus (GPa) G = Shear modulus (GPa) 1 Rigid Rigid 9.6 0.169 2 20.8 22.72 9.6 0.169 3 2 2.272 9.6 0.169 4 0.2 0.2272 9.6 0.169  Table D.2 Data obtained from quasi-static analysis using 3DEC for the sensitivity analysis of the thickness of the blocks Percentage % of the real thickness of the blocks Thickness of the blocks (cm) Collapse Tilt Angle (degrees) ‘α’  (g) 100 6 9.6 0.169 90 5.4 5 0.087 80 4.8 3.7 0.064 60 3.6 0 0  Table D.3 Data obtained from quasi-static analysis using 3DEC for the sensitivity analysis of the thickness of the blocks for the arch with first two rows of blocks fixed Percentage % of the real thickness of the blocks Thickness of the blocks (cm) Collapse Tilt Angle (degrees) ‘α’  (g) 100 6 19 0.34 90 5.4 15 0.267 80 4.8 13.5 0.24 60 3.6 11 0.19  Table D.4 Data obtained from quasi-static analysis using 3DEC for the sensitivity analysis of the density for the arch with the first row of blocks fixed Normalized Density (to 1715 kg/m3) Collapse Tilt Angle (degrees) ‘α’  (g) 0.2 9.6 0.169 1 9.6 0.169 2 9.4 0.165 5 9.2 0.1619 10 8.8 0.1548 25 8.5 0.149 50 7.5 0.13 120  Table D.5 Data obtained from quasi-static analysis using 3DEC for the sensitivity analysis of the density for the arch with first two rows of blocks fixed Normalized Density (to 1715 kg/m3) Collapse Tilt Angle (degrees) ‘α’  (g) 1 18.4 0.333 0.2 18.7 0.338 0.5 18.6 0.337 0.8 18.5 0.335 1.2 18.4 0.333 1.5 18.3 0.331 2 18.3 0.331 4 17.6 0.317 8 16.5 0.296 12 15.5 0.277 20 13.5 0.240 30 11.5 0.203 40 6 0.11 50 0 0.00   Table D.6 Data obtained from the quasi-static analysis using 3DEC for the sensitivity analysis of the shear stiffness Shear Stiffness (GPa) Collapse Tilt Angle (degrees) ‘α’ (g) 0.01 9.1 0.16 0.1 9.6 0.332 1 9.6 0.332 10 9.6 0.332 100 9.6 0.332 1000 9.6 0.332  Table D.7 Data obtained from the quasi-static analysis using 3DEC for the sensitivity analysis of the shear stiffness for the arch with first two rows of blocks fixed Shear Stiffness  (GPa) Collapse Tilt Angle (degrees) ‘α’  (g) 0.02 17 0.305 0.128 18.4 0.332 1 18.6 0.336 50 18.7 0.338 200 18.8 0.34 2000 19 0.344 10000 19 0.344  121  Table D.8 Data obtained from the quasi-static analysis using 3DEC for the sensitivity analysis of the friction angle Friction angle (degrees) Collapse Tilt Angle (degrees) Collapse ‘α’ (g) 20 4.5 0.07 23 8.2 0.14 24 9.5 0.16 25 9.6 0.16 30 9.6 0.16 35 9.6 0.16 50 9.6 0.16   

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