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Reliability analysis of base sliding of concrete gravity dams subjected to earthquakes Horyna, Tomáš 1999

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RELIABILITY ANALYSIS OF BASE SLIDING OF CONCRETE GRAVITY DAMS SUBJECTED TO EARTHQUAKES Tomas Horyna Dipl. Engineer, The Czech Technical University in Prague, 1989 M.A.Sc, The University of British Columbia, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Civil Engineering) We accept this thesis as conforming to thej^ uij?erJ~standaid THE UNIVERSITY OF BRITISH COLUMBIA AUGUST 1999 © Tomas Horyna, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of CIVIL EWGIMEEPIN^ The University of British Columbia Vancouver, Canada D a t e AUGUST Sjhf WW DE-6 (2/88) ABSTRACT Concrete gravity dams are typically constructed in blocks separated by vertical contraction joints. The design of straight concrete gravity dams is traditionally performed by assuming each block to be independent, except for gravity dams in valleys with relatively small width to height ratios. Understanding the 2-D behaviour of individual monoliths is thus considered relevant and 2-D models are usually employed in safety evaluations of existing dams. During a strong seismic event, low to medium height concrete gravity dams tend to crack at the base as opposed to tall dams, which attract high stresses and cracking at the level of a slope change on the downstream side of a dam. The state-of-the-practice in the seismic evaluation of concrete gravity dams requires that the failure mode of the dam monolith sliding at its base be considered. This study focused on the post-crack dynamic response of existing concrete gravity dams in order to investigate their safety against sliding considering non-linear effects in the dam-foundation interface. Sliding response of a single monolith of a low to medium height concrete gravity dam at the failure state was studied and, therefore, the monolith separated or unbonded from its foundation was considered. The work included experimental, analytical and reliability studies. During the experimental study, a model of an unbonded concrete gravity dam monolith was developed and tested using a shake table. The model, preloaded by a simulated hydrostatic force, was subjected to a selected variety of base excitations. Other effects, such as hydrodynamic and uplift pressures were not considered in the experiments. A strong influence of amplitude and frequency of the base motions on the sliding response of the model was observed during the tests. ii ABSTRACT Simple and more detailed numerical models to simulate the experiments were developed during the analytical study. It was observed that a simple rigid model could simulate acceptably the tests only in a limited range of excitation frequencies. A finite element (FE) model simulated the experiments satisfactorily over a wider range of dominant frequencies of the base accelerations. The numerical models were used to simulate the seismic response of a 45 m high monolith of a concrete gravity dam subjected to three different earthquake excitations for varying reservoir's water level. The agreement between the results using the simple rigid and the FE models was found acceptable. The results of the numerical simulations were used in a reliability analysis to calculate probabilities of failure of the 45 m high monolith. Probability of failure was defined here as an annual chance of exceeding an allowable amount of the monolith's base sliding during an earthquake. The peak ground acceleration (PGA), the characteristics of the time history, and the reservoir's water level were considered as random parameters during this study. Using the FE model, the annual probabilities of failure ranged from 1. 1E-8 for the mean PGA of 0.2g and 20 cm of allowable sliding to 1.3E-3 for the mean P G A of 0.6g and 1 cm of allowable sliding. The probabilities of failure using the simple rigid model were found close to those using the FE model. It was concluded that the computationally less demanding simple rigid model may be adequately used in reliability calculations of low to medium height concrete gravity dam safety against base sliding. TABLE OF CONTENTS A B S T R A C T ii T A B L E OF CONTENTS iv LIST OF T A B L E S . . . . ix LIST OF FIGURES x LIST OF S Y M B O L S A N D ABBREVIATIONS xiv A C K N O W L E D G E M E N T S xvi DEDICATION - x v i i CHAPTER 1. INTRODUCTION 1 1.1. G E N E R A L 1 1.2. OBJECTIVES A N D SCOPE OF THIS STUDY 4 1.3. THESIS OUTLINE 6 CHAPTER 2. L ITERATURE REVIEW 8 2.1. CONCRETE D A M S DURING PAST E A R T H Q U A K E S 9 2.2. EXPERIMENTS ON A C T U A L CONCRETE G R A V I T Y D A M S 11 2.3. EXPERIMENTS ON S C A L E D MODELS 11 2.3.1. Niwa and Clough (1980) 12 2.3.2. Donlon and Hall (1991) 12 2.3.3. Zadnik and Paskalov (1992) 12 2.3.4. L in et al. (1993) 1 3 2.3.5. Mir and Taylor (1995) 13 2.3.6. Mir and Taylor (1996) 14 2.3.7. Tinawi et al. (1998 b, c) 14 2.4. TRADITIONAL SEISMIC A N A L Y S I S OF CONCRETE G R A V I T Y D A M S 15 2.5. REFINED SEISMIC A N A L Y S I S OF CONCRETE G R A V I T Y D A M S 17 2.6. SEISMIC A N A L Y S I S RECOGNIZING EXPLICITLY THE D A M - F O U N D A T I O N CONTACT P L A N E 2 0 2.6.1. Leger and Katsouli (1989) 2 0 iv TABLE OF CONTENTS 2.6.2. Chopra and Zhang (1991) 21 2.6.3. Danay and Adeghe (1993) 21 2.6.4. Chavez and Fenves (1995 and 1996) 21 2.6.5. Mir and Taylor (1996) 24 2.6.6. Tinawi et al. (1998b, c) 24 CHAPTER 3. PRELIMINARY E X P E R I M E N T A L STUDY OF A D A M M O D E L . . . . 26 3.1. OBJECTIVES A N D SCOPE OF PRELIMINARY TESTS 26 3.2. DESCRIPTION OF TESTING FACILITY 27 3.3. E X P E R I M E N T A L M O D E L 29 3.3.1. Shape of the Model for a Dam Monolith 29 3.3.2. Material for Monolith and Similitude Considerations 31 3.3.3. Sliding Surfaces 34 3.3.3.1. Smooth Surface 36 3.3.3.2. Cracked-ice surface 37 3.3.3.3. Rough surface 38 3.3.4. Simulation of Hydrostatic Load 39 3.4. E X P E R I M E N T A L SETUP A N D TESTING PROCEDURES 39 3.4.1. Static Tests 40 3.4.2. Dynamic Tests 42 3.5. INSTRUMENTATION 45 3.6. S U M M A R Y A N D CONCLUSIONS 46 CHAPTER 4. FURTHER E X P E R I M E N T A L STUDY OF THE D A M M O D E L 49 4.1. OBJECTIVES A N D SCOPE OF FURTHER EXPERIMENTS 49 4.2. E X P E R I M E N T A L M O D E L 50 4.2.1. Sliding Surfaces 50 4.2.2. Hydrostatic Load Simulation 52 4.3. INSTRUMENTATION 54 4.4. TEST SETUP A N D PROCEDURES 56 v TABLE OF CONTENTS 4.4.1. Impact Hammer Tests 56 4.4.2. Static Tests 58 4.4.3. Tests with Harmonic Input 61 4.4.4. Tests with Synthetic Earthquake Input 62 4.5. RESULTS OF EXPERIMENTS 65 4.5.1. Results of Impact Hammer Tests 66 4.5.2. Results of Static Tests 68 4.5.3. Results of Tests with Harmonic Input 71 4.5.4. Results of Tests with Synthetic Earthquake Input 74 4.5.5. Comparison of Results from Harmonic and Earthquake Tests 80 4.6. S U M M A R Y OF E X P E R I M E N T A L WORK. 83 CHAPTER 5. A N A L Y T I C A L STUDY OF THE D A M MONOLITH M O D E L 85 5.1. D E V E L O P M E N T OF N U M E R I C A L MODELS 85 5.1.1. SDOF Numerical Model 86 5.1.2. 3DOF Numerical Model 94 5.1.3. Finite Element Model 96 5.1.3.1. Previous FE Modeling at U B C 96 5.1.3.2. A N S Y S 5.3 Multiphysics/Universify 97 5.1.3.3. Description of the Model 97 5.1.3.4. Plane Element 98 5.1.3.5. Contact Element 99 5.2. CALIBRATION OF THE FE M O D E L 100 5.2.1. Modifications of the Original FE Model 100 5.2.2. Modal Analysis 101 5.3. RESULTS F R O M SIMULATIONS OF S H A K E T A B L E TESTS 107 5.3.1. Parameters of Simulations 107 5.3.2. Comparison of Amounts of Sliding 109 5.3.2.1. Simulations of Harmonic Tests 109 vi TABLE OF CONTENTS 5.3.2.2. Simulations of Synthetic Earthquake (EQ2) Tests 111 5.3.2.3. Summary of Comparisons 111 5.4. A N A L Y S I S OF SLIDING/ROCKING OF 3DOF M O D E L 115 5.5. S U M M A R Y 118 C H A P T E R 6. A N A L Y T I C A L STUDY OF A F U L L - S C A L E D A M MONOLITH 120 6.1. OBJECTIVES A N D SCOPE OF THE A N A L Y T I C A L STUDY ON A F U L L - S C A L E M O D E L 120 6.2. DESCRIPTION OF THE STRUCTURE 121 6.3. DESCRIPTION OF B A S E EXCITATIONS 124 6.4. FINITE E L E M E N T STUDY 125 6.4.1. Description of the FE Model for the Dam Monolith 125 6.4.2. Modal Characteristics of the Dam Monolith 130 6.4.3. Nonlinear Time History Analysis Using FE Model 136 6.5. STUDY USING SDOF M O D E L 137 6.5.1. Description of SDOF Model for the Dam Monolith 137 6.5.2. Nonlinear Time History Analysis Using SDOF Model 138 6.6. COMPARISON OF RESULTS F R O M THE FE A N D SDOF SIMULATIONS 138 C H A P T E R 7. RELIABILITY STUDY OF A F U L L - S C A L E D A M MONOLITH 143 7.1. BASIC CONCEPTS OF RELIABILITY A N A L Y S I S 143 7.2. OBJECTIVES A N D SCOPE OF RELIABILITY A N A L Y S I S 147 7.3. REDUCTION OF THE A N A L Y T I C A L RESULTS 148 7.4. RELIABILITY STUDY WITH TWO R A N D O M V A R I A B L E S 151 7.4.1. Peak Ground Acceleration as a Random Variable 152 7.4.2. Effect of Different Earthquake Records Treated as an Error Random Variable 156 vii TABLE OF CONTENTS 7.4.3. Simplified Reliability Analysis 157 7.4.4. Results of Reliability Analysis with Two Random Variables 159 7.5. RELIABILITY STUDY WITH THREE R A N D O M V A R I A B L E S 163 7.6. RESULTS OF STUDY IN TERMS OF SAFETY FACTORS 168 7.7. INFLUENCE OF FOUNDATION PROPERTIES ON THE PROBABILITIES OF FAILURE 172 7.8. S U M M A R Y 175 CHAPTER 8. S U M M A R Y , CONCLUSIONS, A N D RECOMMENDATIONS 178 8.1. S U M M A R Y A N D CONCLUSIONS 178 8.1.1. Experimental Study 178 8.1.2. Analytical Study 179 8.1.3. Reliability Study 181 8.2. RECOMMENDATIONS FOR FURTHER R E S E A R C H 183 REFERENCES 185 APPENDIX A - PLOTS OF M O D E L MOTIONS F R O M NO-SLIP TESTS A N D SIMULATIONS 190 APPENDIX B - SAMPLES OF INPUT FILES FOR A N S Y S A N A L Y S I S 201 APPENDIX C - PLOTS F R O M FE A N D SDOF SIMULATIONS ON A F U L L - S C A L E D A M MONOLITH 208 APPENDIX D - C A L C U L A T I O N S FOR RELIABILITY STUDY USING RESULTS OF FE SIMULATIONS 215 APPENDIX E - C A L C U L A T I O N S FOR RELIABILITY STUDY USING RESULTS OF SDOF SIMULATIONS 220 viii LIST OF TABLES Table 4.1: List of Instrumentation and Location 56 Table 4.2: Natural Frequencies of the Bonded Model 67 Table 4.3: Natural Frequencies of the Unbonded Model 67 Table 4.4: Friction Coefficients of Rough Surface R l Obtained from Static Tests 70 Table 4.5: Grouping Testing Frequencies According to Response of the Model 81 Table 5.1: Natural Frequencies of the Experimental Model Bonded to the Base 102 Table 5.2: Natural Frequencies of the Unbonded Experimental Model 102 Table 6.1: Natural Frequencies of the Dam Monolith Bonded to the Base 131 Table 6.2: Natural Frequencies of the Unbonded Dam Monolith 131 Table 7.1: Example of Results for a site with a475=0.4g and for an Allowable Sliding of 5 cm 167 Table 7.2: Safety Factors Evaluated for Parameters Used in Reliability Study 170 Table 7.3: Safety Factors Evaluated for the Target Probability of 1E-5 Based on AEP of 1/475 171 Table 7.4: Example of Results for a site with a475=0.4g and for an Allowable Sliding of 5 cm 174 Table 7.5: Results of FE Analyses with Varied Friction Coefficient 175 ix LIST OF FIGURES Figure 2.1: Model of the Dam-Water-Foundation System Used in Program EAGD-SLIDE 22 Figure 3.1: Main Parts of Experimental Model 30 Figure 3.2: Schematic of the Model for a Dam Monolith 31 Figure 3.3: Dimensions of the Upper Surface Plate 35 Figure 3.4: Dimensions of the Lower Surface Plate 36 Figure 3.5: Connection of USP to the Monolith 37 Figure 3.6: Steel Frame for the LSP to Shake Table Connection 38 Figure 3.7: Controlling the Level of Hydrostatic Force during a Dynamic Test 40 Figure 3.8: Experimental Setup for Static Tests 41 Figure 3.9: Photo of Experimental Setup for Static Tests 41 Figure 3.10: Experimental Setup for Dynamic Tests 43 Figure 3.11: Photo of Experimental Setup for Dynamic Tests 43 Figure 3.12: Displacement Sensors Located on Downstream Side of the Monolith 45 Figure 3.13: View of a Load Cell to Measure Force between the Model and the Rigid Arm 46 Figure 4.1: Schematic of the Upper Surface Plate 51 Figure 4.2: Photo Showing High Pressure Jet Preparing Rough Surface 52 Figure 4.3: Schematic of Hydrostatic Load Assembly 53 Figure 4.4: Photo of the Model and the Hydrostatic Load Simulation Assembly 53 Figure 4.5: Location of Instrumentation 55 Figure 4.6: Instrumented Model During Hammer Testing 57 Figure 4.7: View of Downstream Side of the Model Attached to Rigid Floor during Impact Hammer Tests 57 Figure 4.8: Schematic of Strong Arm Used For Static Tests 59 Figure 4.9: Photo of Typical Static Test Setup 59 Figure 4.10: Winch Assembly Used to Pull Model Up-Stream Between Tests 61 Figure 4.11: Example of Harmonic Input Record at 5 Hz 62 Figure 4.12: Typical Harmonic Test Setup 63 Figure 4.13: Example of Synthetic Earthquake Input Record with Dominant Frequency 5 Hz 65 x LIST OF FIGURES Figure 4.14: Example of Signals Recorded during Static Tests with Surface RI 69 Figure 4.15: Example of Signals Recorded during Harmonic Tests with Surface RI 72 Figure 4.16: Table Acceleration vs. Rate of Model Displacement for Harmonic Excitation 73 Figure 4.17: Example of Signals Recorded during Earthquake Tests with Surface RI . . . . 76 Figure 4.18: Table Acceleration vs. Rate of Model Displacement for Excitation EQ1 . . . . 77 Figure 4.19: Table Acceleration vs. Rate of Model Displacement for Excitation EQ2 . . . . 78 Figure 4.20: Table Acceleration vs. Rate of Model Displacement for Excitation EQ3 . . . . 79 Figure 5.1: Schematic of the SDOF Model 87 Figure 5.2: Performance of SDOF Model - Harmonic Excitations at 5 Hz and 0.4g 90 Figure 5.3: Performance of SDOF Model - Harmonic Excitations at 5 Hz and 0.6g 91 Figure 5.4: Performance of SDOF Model - Harmonic Excitations at 5 Hz and l . l g 92 Figure 5.5: Performance of SDOF Model - Harmonic Excitations at 5 Hz and 1.4g 93 Figure 5.6: Comparison of Results from 3DOF and SDOF Models for Harmonic Excitations 95 Figure 5.7: Schematic of the FE Model 98 Figure 5.8: F E Model for Experimental Model of Dam Monolith Bonded to Base 103 Figure 5.9: First Natural Mode of Experimental Model Bonded to Base; f=65.8 Hz 103 Figure 5.10: Second Natural Mode of Experimental Model Bonded to Base; f=l15.4 Hz. 104 Figure 5.11: Third Natural Mode of Experimental Model Bonded to Base; f=173.5 Hz . . 104 Figure 5.12: FE Model for Unbonded Experimental Model of Dam Monolith 105 Figure 5.13: First Natural Mode of Unbonded Experimental Model; f=29.7 Hz 105 Figure 5.14: Second Natural Mode of Unbonded Experimental Model; f=48.9 Hz 106 Figure 5.15: Third Natural Mode of Unbonded Experimental Model; f= 143.3 Hz 106 Figure 5.16: Comparison of Sliding from Tests and Simulations for Harmonic Excitations 110 Figure 5.17: Comparison of Sliding from Tests and Simulations for EQ2 Excitations . . . 112 Figure 5.18: Kinematics of the Dam Monolith Model 116 Figure 6.1: Schematic of the Full-Scale Monolith Structure 122 Figure 6.2: Characteristics of Selected Record from 1985 Nahanni Earthquake 126 Figure 6.3: Characteristics of Selected Record from 1985 Mexico Earthquake 127 xi LIST OF FIGURES Figure 6.4: Characteristics of Selected Record from 1994 Northridge Earthquake 128 Figure 6.5: FE Model for a Full-Scale Dam Monolith Bonded to Base 132 Figure 6.6: First Natural Mode of a Full-Scale Dam Monolith Bonded to Base; f=6.9 Hz. 132 Figure 6.7: Second Natural Mode of a Full-Scale Dam Monolith Bonded to Base; f=16.9 Hz 133 Figure 6.8: Third Natural Mode of a Full-Scale Dam Monolith Bonded to Base; f=18.8 Hz 133 Figure 6.9: FE Model for an Unbonded Full-Scale Dam Monolith 134 Figure 6.10: First Natural Mode of an Unbonded Full-Scale Dam Monolith; f=4.3 Hz . . . 134 Figure 6.11: Second Natural Mode of an Unbonded Full-Scale Dam Monolith; f=9.3 Hz. 135 Figure 6.12: Third Natural Mode of an Unbonded Full-Scale Dam Monolith; f=16.2 Hz . 135 Figure 6.13: Dam Monolith Displacements (Sliding) vs. Reservoir Water Level 140 Figure 6.14: Dam Monolith Displacements (Sliding) vs. Peak Ground Acceleration . . . . 141 Figure 7.1: Values of PGA Necessary to Cause Sliding of 1 mm 155 Figure 7.2: Annual Probabilities of Failure vs. Water Level of the Reservoir for AEP of 1/475 161 Figure 7.3: Annual Probabilities of Failure versus Acceleration 165 Figure 7.4: Forces on the interface zone of the gravity dam monolith 169 Figure 7.5: Safety Factors for a Target Annual Probability of 1E-5 Based on AEP of 1/475 172 Figure A.1: Response of Dam Model to Harmonic Excitation at Frequency of 5 Hz 192 Figure A.2: Response of Dam Model to Harmonic Excitation at Frequency of 7.5 Hz . . . 193 Figure A.3: Response of Dam Model to Harmonic Excitation at Frequency of 10 H z . . . . 194 Figure A.4: Response of Dam Model to Harmonic Excitation at Frequency of 12.5 Hz . . 195 Figure A.5: Response of Dam Model to Harmonic Excitation at Frequency of 15 H z . . . . 196 Figure A.6: Response of Dam Model to Harmonic Excitation at Frequency of 17.5 Hz . . 197 Figure A.7: Response of Dam Model to Harmonic Excitation at Frequency of 20 H z . . . . 198 Figure A.8: Response of Dam Model to Harmonic Excitation at Frequency of 22.5 Hz . . 199 Figure A.9: Response of Dam Model to Harmonic Excitation at Frequency of 25 H z . . . . 200 xii LIST OF FIGURES Figure C.1: Response of Dam Monolith to Nahanni Earthquake Record with PGA 1.04g. 209 Figure C.2; Response of Dam Monolith to Nahanni Earthquake Record with PGA 0.79g. 210 Figure C.3: Response of Dam Monolith to Mexico Earthquake Record with PGA 1.04g . 211 Figure C.4: Response of Dam Monolith to Mexico Earthquake Record with PGA 0.79g .212 Figure C.5: Response of Dam Monolith to Northridge Earthquake Record with PGA 1.04g 213 Figure C.6: Response of Dam Monolith to Northridge Earthquake Record with PGA 0.79g 214 Figure D.1: Sliding vs. PGA Relationships for Different Water Levels Using Results of FE Analyses 216 Figure D.2: Linear Regression of 'a' coefficients for FE Analysis Results 217 Figure D.3: Linear Regression of'b' coefficients for FE Analysis Results 217 Figure D.4: Comparison of Sliding Obtained from FE Simulations and Values Calculated Using Interpolation Function 218 Figure D.5: Distribution of Errors due to Different Earthquake Records vs. PGA's 218 Figure D.6: Annual Probabilities of Exceeding a Given Amount of Sliding from Simplified Analysis Using FE Results 219 Figure E.1: Sliding vs. PGA Relationships for Different Water Levels Using Results of SDOF Analyses .221 Figure E.2: Linear Regression of'a' coefficients for SDOF Analysis Results 222 Figure E.3: Linear Regression of'b' coefficients for SDOF Analysis Results 222 Figure E.4: Comparison of Sliding Obtained from SDOF Simulations and Values Calculated Using Interpolation Function 223 Figure E.5: Distribution of Errors due to Different Earthquake Records vs. PGA's 223 Figure E.6: Annual Probabilities of Exceeding a Given Amount of Sliding from Simplified Analysis Using SDOF Results 224 xiii LIST OF SYMBOLS AND ABBREVIATIONS 3 DOF = 3 Degrees Of Freedom A = constant a, a\, a2 = constants AEP = Annual Exceedence Probability aAEP= P e a k ground acceleration corresponding to a given AEP aG = Peak Ground Acceleration aM = mean value of design acceleration B = constant b, b\,b2= constants C = capacity of a system, cohesion, constant C E A = Canadian Electricity Association CSMIP = California Strong Motion Instrumentation Program D = demand on a system d/s = downstream E = constant FE = Finite Elements FS = Factor of Safety g = acceleration due to gravity G = performance function H= depth of the reservoir L V D T = Linearly Varying Displacement Transformer m, mHD = mass, added mass of water MDE - Maximum Design Earthquake MffD = total added mass of water N = normal force Pa and Pe = annual and event probability of failure Pf = probability of failure PGA = Peak Ground Acceleration (in text) Pr= reliability (probability of survival) 7?N = Standard Normal variable xiv LIST OF SYMBOLS AND ABBREVIATIONS S = sliding of a gravity dam monolith or the experimental model S0 = allowable (limiting) amount of sliding SDOF= Single Degree Of Freedom u/s = upstream USGS = U.S. Geological Survey V= coefficient of variation, dynamic tangential force Vst = coefficient of variation, dynamic tangential force vs. = versus W M = Working Model (software) P = reliability index Y = specific mass of water p = friction coefficient Q> = Standard Normal probability distribution function xv ACKNOWLEDGEMENTS This project was conducted with research funding from the British Columbia Hydro and Power Authority and from the Natural Sciences and Engineering Research Council of Canada. Support of both institutions is gratefully acknowledged. There have been many people who have helped me during the various tasks associated with the completion of this thesis. Their assistance is very much appreciated, and I would not have been able to accomplish as much without their assistance. I would like to thank my supervisors, Dr. Ricardo Foschi and Dr. Carlos Ventura, for the guidance and technical advice they have given me over the years. I am personally indebted to Dr. Ricardo Foschi for financing my graduate studies at UBC. I want to thank both my supervisors for giving me the opportunity to work with them on a number of interesting research and educational projects and for their personal advice and assistance during all those years. Finally, their comments and suggestions during the preparation of this thesis are sincerely appreciated. Professional advice and valuable comments in various stages of the project from several BC Hydro structural engineers is greatly acknowledged. These included Mr. Benedict Fan, Mr. Gilbert Shaw, Dr. Desmond Hartford and others. The experimental section of this thesis would not have been possible without the assistance of the Laboratory technicians. Special thanks go to Howard Nicol for his valuable hints for the testing, and to Doug Hudniuk for his machining and welding help with the experimental model. Several U B C graduate students contributed to the successful completion of this project. Among all these, Jachym Rudolf and Cameron Black contributed the most. Vincent Latendresse, Hong L i , Hugo Armelin and Mahmoud Rezai assisted during the building of the experimental model and shake table testing. I would like to thank Dr. D. L . Anderson of U B C and Mr. B. H. Fan of B C Hydro, who, along with Dr. Foschi and Dr. Ventura, reviewed this thesis. Last, but certainly not the least, I would like to thank my wife Petra and our family. Petra is given special thanks for the patience, care and encouragement she has had for me during the years I spent studying and working at UBC. xvi To Petra and our parents 1 INTRODUCTION 1.1 GENERAL Failures of dams are very infrequent, but in most cases, these would be extremely high consequence events. Therefore, the assessment of dam safety is treated with the utmost care. The good performance of concrete gravity dams in past earthquakes has been used by some to support their claim that these structures are inherently strong in withstanding the effects of earthquakes. Others maintain that this apparently good behaviour should be viewed in the proper perspective: that recorded incidents may not have involved dams experiencing the most severe earthquakes assessed possible for their respective sites and with full reservoirs, and that performance data reporting may not have been complete (Fan and Sled, 1992). For many dams built at the beginning of this century, the design as well as the construction specifications did not correspond to today's requirements. It is not surprising i f many existing dams initially considered safe are now judged unsafe based on up-to date specifications (Tinawi etal., 1998b). Concrete gravity dams are typically constructed in blocks separated by vertical contraction and horizontal construction joints. The vertical joints, extending from the foundation to the crest and from the upstream to the downstream faces, are used to limit the cross-canyon dimension of the blocks, called monoliths. The horizontal joints are provided to divide the structure into convenient building units and to regulate temperature during concrete placement. It is reasonable to assume that joints have a marked weakening effect on concrete gravity dams. This weakening depends, mostly, on the low capacity of the joints to transfer loads when compared to mass concrete. Therefore, the mechanical properties of joints in gravity dams reduces 1 CHAPTER 1 INTRODUCTION significantly their ability to resist severe earthquakes and they play an important role in assessment of dam safety. The design of concrete gravity dams is generally performed by assuming complete bonding in the horizontal joints. However, for a detailed safety evaluation of an existing dam, it is necessary to characterise properly these joints, evaluate the possibility of relative motions, such as sliding or separation, which influence the stability of the structure. Therefore, it is necessary to consider these interfaces with proper characteristics in dam safety analysis. A logical location of one of the important horizontal or near horizontal joints is the interface between the dam body and the foundation rock. Possibility of sliding in this interface has to be evaluated during safety analysis of a concrete gravity dam. The need to accurately predict the seismic stability of concrete gravity dams has led to a proliferation of research activities. A large number of analytical methodologies have been put forward. Material strength parameters, ground motion characteristics and failure mechanisms are represented with varying degrees of rigor (Fan and Sled, 1992). These procedures are expensive and time consuming, and they produce results that are generally only snapshots of the dam performance. Certain procedures, with simplifying assumptions, enable the prediction of the amount of earthquake induced sliding of gravity dams. Such analyses do provide information on damage caused during earthquakes, which is useful the for evaluation of post-earthquake safety of the dam in its damaged state. They do not, however, by themselves enable a credible assessment of dam safety during the earthquake, unless a correlation could be made between the computed damage and probability of dam failure. 2 CHAPTER 1 INTRODUCTION A n alternative to assessing the seismic safety of dams is to assess the existing seismic risks and compare them to acceptable risks. Possible failure mechanisms are first screened and probable ones identified. A range of ground motion characteristics, reservoir conditions, and material strengths are considered. For each combination of parameters, each of the identified probable failure mechanism is checked for its stability. A data set on the fragility of the dam is thus constructed. A relatively large number of such stability checks would have to be made, and it becomes readily apparent that a simple numerical procedure is preferred. Current state-of-the-art analytical tools to evaluate sliding of concrete gravity dams are rarely used for seismic risk evaluations because of time constraints, but they can yield valuable results to verify simpler methods. Experimental data is another source of verification of simple analytical methods. As there is no reported evidence of excessive sliding of any actual concrete gravity dam subjected to severe shaking, it is desirable to conduct shake table tests of dam models to verify analytical predictions. Concrete gravity dams represent about half of British Columbia Water and Power Authority (BC Hydro) portfolio of dams and many of them are located in areas of high and moderate seismicity. They range in age from under 20 years to over 80 years. It is assumed that a probability of a failure of a concrete gravity dam is low, but such an event accompanied by release of the reservoir would have extremely high consequences. Therefore, the problem of dam safety against sliding is worth studying. In an effort to enhance methods of seismic risk assessment of its concrete gravity dams, the B C Hydro has initiated a parametric study on the seismic stability of low to medium height concrete 3 CHAPTER 1 INTRODUCTION gravity dams. In an attempt to calibrate simple analytical methods used in risk assessment, BC Hydro commenced in 1996 a phased experimental and analytical research project with UBC. The studies described in this thesis were partially conducted within the BC Hydro - U B C collaborative project. Some studies were conducted after the final phase of the collaborative project was completed. 1.2 OBJECTIVES AND SCOPE OF THIS STUDY The purpose of this study was to gain improved understanding of certain aspects of the dynamic post-crack response of concrete gravity dams. In particular, the safety against base sliding of existing concrete gravity dams was studied considering the non-linear effects at the dam-foundation interface. A single monolith of a low to medium height concrete gravity dam was considered with a through crack developed at the dam-foundation interface. It was decided to address the problem by a series of studies including experiments, analytical work and reliability analysis. The objectives and scope of the study can be grouped into three main areas: 1) The objective of the experimental work was to develop a scaled model of a single concrete gravity dam monolith, and measure the amount of sliding of the unbonded model when this was preloaded by a simulated hydrostatic force and subjected to different base excitations. It was desired to investigate the response of the model for shake table induced base accelerations of varied characteristics, amplitude and dominant frequency. The model was unbonded at the base and only frictional characteristics of the dam-foundation interface were simulated. The characteristics of base excitations comprised harmonic and synthetic earthquake records with dominant frequencies from 5 to 25 Hz. 4 CHAPTER 1 INTRODUCTION During the experiments, the reservoir's loads were limited to simulated hydrostatic force. Other effects, such as hydrodynamic force or uplift pressure at the foundation, were neglected. It was not the objective of the study to directly apply the test results to predict the seismic performance of a specific prototype. 2) The objective of the analytical study was to develop a simple numerical model to simulate the sliding of a rigid block on a rigid foundation, preloaded by a constant horizontal force and subjected to unidirectional horizontal base excitations. The requirement of the model's simplicity came from the intended applications of the model for reliability analysis, which typically involves a large number of simulations and, therefore, a fast analysis procedure was preferable. Another objective of this study was to develop a flexible model of the experimental setup and use it to simulate some of the shake table tests conducted during the experimental part of the study. This analysis was performed in order to find out how closely the numerical models of varied complexity simulated the response of the experimental setup measured during the shake table tests. The loads considered in the numerical models were limited to those simulated during the shake table tests. The objective of the analytical study of a full-scale monolith was to perform a series of numerical simulations to calculate the response of the monolith with a varying water level of the reservoir, characteristics of the earthquake record, and its peak ground acceleration (PGA). It was the objective to perform these analyses using two different numerical models and compare results. In addition to the forces simulated on the experimental model during the shake table tests, hydrodynamic pressures were considered during this analysis. Other loads, such as uplift or cohesion in the foundation interface were not simulated. Base 5 CHAPTER 1 INTRODUCTION excitations on the monolith were limited to those in a single horizontal direction. During the analysis, only the full-length crack between the dam monolith and the foundation rock was considered. No weak horizontal construction joints in the monoliths were considered and therefore, only the base sliding was calculated in the analyses. Should the sliding stability be evaluated at other location than at the base-foundation interface, it would have to be done in a separate study. 3) The objective of the reliability study was to formulate a procedure to obtain annual probabilities of failure of the full-scale dam monolith using the results obtained from the analytical study. Probability of failure was defined as a chance of exceeding a specified amount of base sliding. The PGA, the error from the characteristics of the earthquake motion and the reservoir's water level were considered as random parameters. Another objective was to relate the results of the reliability study to safety factors against sliding. 1.3 THESIS OUTLINE This section describes the manner in which the remainder of the manuscript is organised. The thesis contains seven main chapters, each divided into several sections and subsections. In addition, it includes five appendices with complementary information. A study of available literature sources on experimental and analytical evaluation of sliding of concrete gravity dams was essential for the project. Chapter 2 provides a comprehensive review of published related research results in the last two decades. Chapters 3 and 4 describe the experimental studies conducted during the project. The 6 CHAPTER 1 INTRODUCTION preliminary experimental study, containing the description of the experimental model and setup, is given in Chapter 3. Further testing including results from several types of experiments is described in Chapter 4. Chapters 5 and 6 deal with the analytical studies conducted within this project. Development, verification and calibration of numerical models, and results of simulations of selected experiments are described in Chapter 5. Chapter 6 contains several sections on the analytical study of a full-scale concrete gravity dam monolith. Chapter 7 describes several reliability studies utilizing the analytical data obtained in Chapter 6. The chapter starts with description of basics of the reliability analysis, which is followed by sections to identify random variables and presentation of results. Chapter 8 provides a summary of the conducted research and a discussion of the conclusions drawn. The need for further studies, both analytical and experimental, is outlined. 7 2 LITERATURE REVIEW The project described in this thesis included three types of studies: experimental, analytical and reliability. This chapter consists of a review of experimental and analytical studies in the area of seismic response of concrete gravity dams. Focus is being placed on the experimental and analytical evaluation of sliding of these dams. The reliability study was limited to using existing methods and procedures and, therefore, no literature review for this study is given here. Instead, fundamentals of the reliability analysis are summarised in the chapter describing this study. The review of previous research conducted in the area of seismic behaviour of concrete gravity dams was limited to the last two decades. The review begins with a section on performance of concrete gravity dams during past earthquakes. This is followed by a section on experimental studies investigating sliding of concrete gravity dams due to base excitations. Since such response has never been measured on an actual dam, the experimental works referenced here included mostly shake table testing of scaled models of single monoliths of concrete gravity dams. The last three sections of this chapter contain a review of related research. In this review, focus is being placed on reported research related to analysis of seismic-induced sliding of concrete gravity dams. It should be mentioned that two publications including a comprehensive literature review were found very useful by the author when developing this literature review. One is the review by Hall (1988), which presented a detailed summary of experimental work conducted on concrete dams and their models before 1988. Analytical findings were included there i f numerical simulations accompanied the experiments in a given study. The other review was performed by Tinawi et al. (1998a) during a project of this research team in the area of structural safety of 8 CHAPTER 2 LITERATURE REVIEW existing concrete dams (Tinawi et al., 1998b and 1998c). This excellent summary of experimental and analytical research addressed a wide range of topics involved in dam safety, such as numerical modelling of lift joints, evaluation of flood and earthquake safety of gravity dams considering lift joints and others. 2.1 CONCRETE DAMS DURING PAST EARTHQUAKES There are several historical records of a concrete gravity dam undergoing shaking caused by an earthquake. None of these cases ended with significant damage or collapse of the affected dam and no base sliding of a dam was observed. The apparent good performance of concrete gravity dams in past earthquakes should be viewed in the proper perspective. No dam on record, with full reservoir, has been subjected to the most severe earthquake assessed possible for its respective site (Fan and Sled, 1992). The most significant reported event experienced by a concrete gravity dam was the 1967 shaking of the 103-m-high Koyna Dam in India with nearly full reservoir (Chopra and Chakrabarti, 1972). Recorded ground motions at the dam from a nearby earthquake of magnitude 6.5 peaked at 0.49g in the stream direction and continued strongly for 4 sec. The dam sustained with significant horizontal cracks through a number of nonoverflow monoliths at an elevation of 36 m below the crest where the downstream face changed slope. Perhaps the strongest shaking experienced by a concrete dam to date was that which acted on the Lower Crystal Springs Dam in California, a curved gravity structure with modest height of 42 m, during the magnitude 8.3 San Francisco earthquake of 1906 (National Research Council, 1990). The dam incurred no damage even though it stood with its reservoir nearly full within 9 CHAPTER 2 LITERATURE REVIEW 350 m of the fault trace at a point where the slip reached 2.4 m. It should be mentioned that the stability of this structure exceeds that of a typical gravity dam due to its curved plan and a cross section that was designed thicker than normal in anticipation of future heightening, which was never completed. Another example of a concrete dam subjected to strong ground shaking is the 103-m-high Pacoima Dam in Southern California during the 1971 San Fernando earthquake (Hall, 1988) and during the 1994 Northridge earthquake (Stewart et al., 1994). This dam, and others like the Ambiesta Dam in Italy during the 6.5 magnitude Friuli earthquake of 1976, the Rapel Dam in Chile during the magnitude 7.8 Chilean earthquake of 1985, and the Gibraltar Dam in California during the magnitude 6.3 Santa Barbara earthquake of 1925, represent cases of concrete arch dams under severe earthquake shaking. However, analysis of such events is beyond the scope of this research. Another two examples of concrete dams subjected to severe base excitations are the 106 m tall Sefi-Rud Dam in Iran hit by a 7.3 magnitude earthquake, which occurred near the dam (Indermaur et al., 1992), and the 105 m high Hsingfengkiang Dam in China affected by a nearby magnitude 6.1 earthquake (Chenet et al., 1974). Both these dams suffered some damage including horizontal cracks at the level of abrupt change of the downstream slope and had to be strenghtened and stabilised. Both these dams are buttress structures and the experience gained from their performance during the earthquakes is not applicable to this study. Although the experience outlined above representing the most significant earthquake events that have acted on concrete dams, is impressive, it falls short of providing a complete confidence that 10 CHAPTER 2 LITERATURE REVIEW concrete gravity dams with full reservoirs are safe against strong seismic shaking. 2.2 EXPERIMENTS ON ACTUAL CONCRETE GRAVITY DAMS A number of tests on prototypes of concrete gravity dams have been conducted in the last twenty years. However, most of these were associated with low level vibrations, either ambient or shaker-induced. Experimental research of non-linear effects in seismic response of actual gravity dams due to strong ground shaking is very limited. In the absence of prototype data, the need for validating numerical predictions with experimental investigations has long been felt by the engineering community. However, due to the difficulties involved in the modelling of dam-foundation-reservoir systems in the laboratory, the experimental research has not kept pace with the numerical advances in the last two decades. In the last decade, several shake table testing programs have been conducted with the objective to study failure behaviour of scaled models of concrete gravity dam monoliths. Some of these studies are summarised below. 2.3 EXPERIMENTS ON SCALED MODELS In order to assess experimentally different types of non-linear seismic response of concrete gravity dams, researchers conduct shake table tests on failure models. Most of the shaking table tests in the past have been carried out using models of arch dams. In only a few cases have shake table tests been conducted on models of concrete gravity dams. 11 CHAPTER 2 LITERATURE REVIEW 2.3.1 Niwa and Clough (1980) Niwa and Clough (1980) described shake table tests carried out on a single 1:150 model monolith of the 103 m high Koyna Dam. The model of the monolith was developed to maintain similitude with the prototype. A plaster-based material used to build the model permitted the simulation of response of the dam due to ground shaking in linear as well as nonlinear regions. These tests demonstrated that a crack appeared in the model at the level of downstream slope change, which was the location at which a crack appeared in the prototype structure during the 1967 Koyna earthquake. 2.3.2 Donlon and Hall (1991) Another series of shake table tests was reported by Donlon and Hall (1991). These tests were conducted on a 1:115 length-scaled model of the 122 m high single monolith of the Pine Flat Dam. A total of three models were tested, out of which only one was made of a single monolithic plaster based material. The other two models were constructed with composite materials consisting of a weak upper half portion, made of a plaster-based material, and a stronger main body made of a polymer-based material. The tests showed a crack initiation just below the neck of the dam, although owing to the differences in the material properties and the construction, the crack profile in the three models was not consistent and differed significantly from one another. 2.3.3 Zadnik and Paskalov (1992) A series of shake table tests on a concrete model of a single gravity dam monolith was conducted by Zadnik and Paskalov (1992). This testing quantified sliding of the monolith, resting on 12 CHAPTER 2 LITERATURE REVIEW either wooden, concrete or steel floor, due to ground motions generated by the shaking table. Results of these tests were compared with numerical predictions with a good agreement. The experimental model used during these tests was not developed to satisfy similitude with any actual dam. 2.3.4 Lin etal. (1993) Lin et al. (1993) performed a series of shake table tests on five 1:130 scale homogenous models of a concrete gravity dam 195 m high. The models were excited harmonically in the first mode of vibration until the first crack occurred near the neck of the dam. Other cracks appeared later in the middle part of the model. Finally, cracks at the toe and the heel became visible. However, no dam-reservoir interaction was modelled during the tests. If this was taken into consideration, as mentioned in the study, the crack at the heel was likely to appear much sooner. 2.3.5 Mir and Taylor (1995) A series of shake table testing of a gravity dam monolith model was conducted by Mir and Taylor (1995). These tests were conducted to identify crack pattern of a 1:30 scaled model of 30 m high gravity dam. The model has been developed from plaster based material in accordance with similitude laws. The model was fixed securely to the shake table platform, therefore, foundation flexibility was not simulated during these tests. Reservoir effects were simulated by hydrostatic and hydrodynamic (Westergaard added mass approximation) pressures. A total of eight models were tested under different three conditions: no reservoir effects, only hydrostatic pressure, both hydrostatic and hydrodynamic pressures. The models were excited by harmonic ground motion at 60 Hz frequency as well as by earthquake records. 13 CHAPTER 2 LITERATURE REVIEW The amplitudes of shaking were increased until failure occurred. Base cracking was observed to be the main failure mechanism and tendency of the models to slide and rock after the full crack development at the interface was also observed in some cases. 2.3.6 Mir and Taylor (1996) A series of shake table tests complementary to the previous study (Mir and Taylor, 1995) was conducted by Mir and Taylor (1996). Several 1:30 models of a 30 m gravity dam were constructed from concrete since the strength of the model was not the controlling parameter. The interface between the dam and its foundation had only frictional resistance with the static coefficient of friction of 0.72. Under full reservoir condition and with no uplift pressure considered, downstream sliding was identified as the only significant global failure mechanism. For an empty reservoir condition, it was observed that the dam model might not slide, but instead it could overturn upstream about its heel for high interface frictional resistance. 2.3.7 Tinawi etal. (1998 b, c) A n extensive shake table experimental program to study seismic sliding response of three 3.4 m high concrete dam models with a weak lift joint at mid-height was carried out by Tinawi et al. (1998b). The hydrostatic and hydrodynamic forces were approximated with a hanging weight attached to the specimens. The models were subjected to various base excitations including single and double triangular pulses and earthquake accelerograms of varying PGA and different frequency content. It was observed from the measured cumulative sliding displacements that sliding of the specimen due to a single pulse of sufficient duration can be significant. Further, it was concluded that the sliding due to Western North American earthquake records was about 14 CHAPTER 2 LITERATURE REVIEW three times larger compared to Eastern records for a given PGA. Other conclusions, such as the amount of sliding was proportional to the PGA raised to a power of about three, were also made. Most of the experimental studies mentioned in this section were conducted together with numerical simulations of the tests. Some analytical methods to study seismic response of concrete gravity dams will be summarised in the next sections. 2.4 TRADITIONAL SEISMIC ANALYSIS OF CONCRETE GRAVITY DAMS Concrete gravity dams were traditionally designed for static loads up to about mid 1960's. The earthquake forces were treated simply as static equivalent forces and were combined with the hydrostatic pressures and gravity loads. The analysis was concerned with overturning and sliding stability of the monolith treated as a rigid body and with stresses in the monolith, which were calculated by elementary beam theory. In representing the effects of the horizontal upstream/downstream ground motion by static equivalent lateral forces, neither the dynamic response characteristics of the dam-water-foundation system nor the characteristics of the earthquake ground motion were recognized. Two types of static lateral forces were included: • Forces associated with the weight of the dam were expressed as the product of a seismic coefficient, typically constant over the height with a value between 0.05 to 0.1, and the weight of the portion being considered. • Hydrodynamic pressures, in addition to the hydrostatic pressure, were specified in terms of the seismic coefficient and a pressure coefficient, which was based on assumptions of a rigid dam and incompressible water. 15 CHAPTER 2 LITERATURE REVIEW Interaction between the dam and the foundation rock was not considered in computing the aforementioned earthquake forces. The traditional design criteria required that an ample factor of safety be provided against overturning, sliding and overstressing under all loading conditions. Tension was often not permitted, even i f it was, the possibility of cracking of concrete was not seriously considered. It had generally been believed that stresses were not a controlling factor in the design of concrete gravity dams, so that the traditional design procedures were concerned mostly with satisfying the criteria for overturning and sliding stability. It was apparent from observed behaviour of dams subjected to ground shaking, such as Koyna Dam, that stresses thus calculated in gravity dams due to standard design loads have little resemblance to the stresses due to the dynamic response of such dams to earthquake ground motion. The discrepancies between calculated and observed behaviour of the dams were partly the result of not recognizing the dynamic response due to earthquake motions in computing the earthquake forces included in the traditional design methods. The typical values used for the seismic coefficient, ranging from 0.05 to 0.1, were much smaller compared to ordinates of pseudoacceleration response spectra for intense earthquake motions in range of vibration periods up to 1 sec. In addition, traditional analysis and design adopted a uniform distribution of the seismic coefficient, ignoring the vibration properties of the dam. This kind of seismic coefficient lead to underestimating stresses in the upper part of dams and it ignored dynamic influence of the widening of the dam cross-section for non-structural reasons near the dam crest. Another source of discrepancies between calculated and observed behaviour of the dam was not 16 CHAPTER 2 LITERATURE REVIEW recognizing hydrodynamic effects, which were found to be grossly underestimated in the traditional design loading. When the compressibility of water and dam-water interaction resulting from deformations of the dam are included in the analysis, hydrodynamic effects are generally shown to be important in the response of concrete gravity dams. For example, in the analysis of the response of nonoverflow monolith of the Pine Flat Dam, on rigid rock foundation, due to the Kern County, 1952 earthquake (Taft Lincoln School Tunnel record), the tensile stresses in the dam were approximately 30% larger when the hydrodynamic effects were included compared to the case when they were not (Fenves and Chopra, 1985) included. Foundation rock flexibility is another phenomenon, which was not considered in computing earthquake forces in traditional design loading. When dam-foundation rock interaction is properly included in the dynamic analysis, these effects are generally significant and they mostly reduce stresses in the dam (Fenves and Chopra, 1985). The above list of significant phenomena which were not included in the traditional analysis of concrete gravity dams stimulated the development of more advanced tools to analyse response to earthquake loading. 2.5 REFINED SEISMIC ANALYSIS OF CONCRETE GRAVITY DAMS a Realistic analyses of the seismic response were not possible until the development of the finite element method, recent advances in dynamic analysis procedures, and the availability of large-capacity, high-speed computers. Thus, much of the research involving linear time history analyses did not start until the mid-1960's. 17 CHAPTER 2 LITERATURE REVIEW Initially, all nonlinear effects, including those associated with construction-joint opening, concrete cracking and water cavitation, were ignored, and the interaction effects were either neglected or grossly simplified. Subsequently, development of special techniques permitted the incorporation of the interaction effects into linear analyses. These effects included phenomena such as dam-water interaction based on dam flexibility and water compressibility, radiation damping into reservoir bottom, and dam-foundation rock interaction (Fenves and Chopra, 1985a). Based on these techniques, numerical models implemented in computer programs have been developed. Most of the models operated in two dimensions although three-dimensional models have also been developed (Fok et al. 1986). The two-dimensional analysis was recommended whenever it was appropriate for the dam to be analysed, because it is computationally efficient and it could rigorously consider most of the factors significant for the earthquake response. In the three-dimensional analysis, dam-foundation rock interaction and earthquake excitation were usually treated in an overly simplified manner. Nonlinear phenomena in concrete gravity dam seismic analysis began to be addressed in the 1980's. Nonlinear analysis presents a number of problems, such as the modelling of input motions, modelling of horizontal construction joints in the monoliths, nonlinear material behaviour of concrete and rock, and uplift of the dam. Contrary to rigorous elastic solutions, which mostly operate in frequency domain, a nonlinear analysis requires a time stepping solution, and incorporating infinite boundaries in such models is still not reasonably well understood (Taylor, 1996). As most dam engineers and researchers have access to commercial finite element packages such as A B A Q U S , ADINA, ANSYS and others, which have powerful nonlinear capabilities, methodologies to exploit these tools were developed together with the 18 CHAPTER 2 LITERATURE REVIEW development of the specialized codes. One of the most important nonlinearities is cracking of concrete. Cracking can develop, for example in the regions of pre-existing thermal cracks or due to overstressing of concrete during strong shaking by an earthquake. Various post-yield constitutive relationships have been put forward for dam concrete. Cracking was first investigated with smeared crack assumptions and also with plasticity models, and later with the application of fracture mechanics (Reich, Cervenka and Saouma, 1991). Simic and Taylor (1995) compared a smeared cracking model with a plasticity model applied to a gravity dam, which was subjected to a range of hydrostatic loads. This comparison showed how the two models tend to predict very different failure modes. The plasticity model lead to widely distributed yielding, while the smeared cracking model predicted localized cracking that was more consistent with crack propagation through a brittle material. Simic and Taylor (1996) also discussed the application of a typical smeared crack model available in codes such as ADINA and SOLVIA. Possible cracking of low to medium height gravity dams would occur, in contrast to tall dams such as Koyna or Pine Flat Dams, at the dam-foundation interface. This was shown by non-linear analysis for the 56-m-high Russel Dam by Mlakar (1987). This analysis yielded extensive cracking along the base of the dam under strong ground shaking, which was different from the crack pattern in the taller dams considered in the same study. Therefore, base sliding of a dam with a crack through its base has been studied by several researchers in recent years. Several types of nonlinearities could be studied in the seismic response of concrete gravity dams. The study described in this thesis focused on one of them, the base sliding response of 19 CHAPTER 2 LITERATURE REVIEW concrete gravity dams during an earthquake. Only the references related to this type of nonlinearity are summarised in the next section. 2.6 SEISMIC ANALYSIS RECOGNISING EXPLICITLY THE DAM-FOUNDATION CONTACT PLANE Sliding response of concrete gravity dams, defined as a concentrated shear deformation at the interface between the dam and the foundation rock, has been studied in the last decade. This was due to earlier analytical and experimental research, which had identified that the crack pattern in low to medium dams under an earthquake might lead to development of a crack between the dam and its base. 2.6.1 Leger and Katsouli (1989) A study of the seismic stability of a 90 m concrete gravity dam was reported by Leger and Katsouli (1989). Gap friction node-to-node finite elements were used to model the dam-foundation interface. The dam and the foundation were assumed to remain elastic. The contact elements were assumed to have tangential and normal compressive strength but no tensile strength. Thus, the sliding and rocking response of the dam monolith could be simulated. It was found that the maximum base shear values computed at the dam-foundation interface were greater for linear analyses than for nonlinear, especially for sites with flexible foundation conditions, as compared to the dam concrete. A dynamic stability safety factor was defined as a ratio of the specified design PGA to the critical acceleration required to reach a critical safety limit defined as significant sliding displacement or loss of bond over more than 90% of the base. 20 CHAPTER 2 LITERATURE REVIEW 2.6.2 Chopra and Zhang (1991) Simplified analytical formulations to evaluate earthquake-induced sliding of a concrete gravity dam monolith were derived by Chopra and Zhang (1991). The dam monolith was assumed to be supported on a plane surface without bonding. Different formulations were developed for rigid and flexible dams considering only the contribution of the fundamental mode of vibrations. Downstream sliding was found to be the only significant response of the dam. In addition, it was found that the dam flexibility had the effect of increasing the permanent sliding displacement as compared to the rigid case. Vertical ground motions were found to increase the total amount of sliding during an earthquake. 2.6.3 Danay and Adeghe (1993) Empirical formulas to obtain sliding displacement of a concrete gravity dam were derived by Danay and Adeghe (1993). These formulas were derived for the dams of 60 m or less subjected to Eastern North America as well as Western (California) type of earthquake ground motions. The empirical formulas were obtained from statistical regression on the results of a series of analyses performed with a sliding block model with the dam-foundation interface represented by gap-friction elements. The slip induced by horizontal acceleration alone was found to be approximately 60% of that obtained when horizontal and vertical accelerations were applied simultaneously. 2.6.4 Chavez and Fenves (1995 and 1996) The work presented by Chavez and Fenves (1995) included dynamic analysis of sliding of an earthquake-loaded gravity dam and compared the results of this analysis with those from the 21 CHAPTER 2 LITERATURE REVIEW traditional check of sliding stability. This involved computing a factor of safety against sliding, calculated as a function of a Mohr-Coulomb model of concrete and foundation material friction characteristics, self weight of the dam, hydrostatic pressure, uplift force, cohesion force at the dam-foundation interface zone, and equivalent static loads on the dam representing the dynamic effects of an earthquake. Figure 2.1: Model of the Dam-Water-Foundation System Used in Program EAGD-SLIDE The work of Chavez and Fenves (1996) applied a novel modelling and solution algorithm, the hybrid frequency-time domain procedure (Kawamoto, 1983; Darbre and Wolf, 1988). The numerical model developed during this work accounted for the dynamic characteristics of the dam, foundation rock flexibility, compressible water, and Mohr-Coulomb model for base sliding. This numerical model was implemented in computer program EAGD-SLIDE (Chavez 2 2 CHAPTER 2 LITERATURE REVIEW and Fenves, 1994). The idealized dam-foundation-water system used in program E A G D -SLIDE is shown in Fig. 2.1. Using the program, the dam might slide along the rigid interface between the dam base and the foundation rock. Rocking of the dam about the base is not represented in the model. The system is subjected to horizontal and vertical components of free-field earthquake ground motions at the base of the dam. The body of the dam in EAGD-SLIDE is modelled by two-dimensional plane stress finite elements with linear elastic properties. The water impounded in the reservoir is idealized as a two-dimensional domain extending to infinity in the upstream direction. The water is treated as an compressible fluid that produces hydrodynamic pressures depending on the excitation frequency. The foundation rock is idealized as a homogeneous, isotropic and viscoelastic half-plane. The interface between the dam monolith and the foundation rock (foundation interface) is assumed to be straight surface with the resistance to sliding governed by Mohr-Coulomb law. The resistance at the interface depends on a cohesion (bonding) force, the coefficient of friction, and the time varying normal force. The model assumes that sliding occurs along the entire interface. The work of Chavez and Fenves (1996) included a parametric study on a 122-m-high monolith of the Pine Flat Dam. Several combinations of input parameters of dam-foundation-water system were considered. Depending on these parameters, base sliding from 0 to 29 cm was obtained for three different historical earthquake records with PGA of 0.4g. The study identified that the foremost factor influencing the amount of sliding is foundation rock flexibility; rigid foundation rock yielded much larger sliding than flexible foundation rock. The sliding was also sensitive to the coefficient of friction and cohesion of the interface zone. Based on their 23 CHAPTER 2 LITERATURE REVIEW conclusions, the authors proposed a framework for evaluation sliding stability of gravity dams. 2.6.5 Mir and Taylor (1996) A series of finite element analyses were performed together with the shake table tests by Mir and Taylor (1996). The analyses using a nonlinear large displacement contact surface algorithm were carried out in order to establish comparison between the measured and calculated sliding displacement responses. Generally, a good agreement was obtained, even though some discrepancies were observed for the magnitude of displacements under some of the peak pulses. In these cases, it was observed that numerical analyses underestimated sliding displacements under earthquake loads compared to the measured values. Conclusions suggested that the actual interface dynamic friction coefficient was transient in nature and lower than the friction coefficient. Additional analytical work was carried out to assess the validity of simplified sliding analysis methods for dams considered as rigid bodies. 2.6.6 Tinawi et al. (1998b, c) A series of numerical studies were included in the research reported by Tinawi et al. (1998b, c). During these studies, several lift joints constitutive models were developed using thin layer interface finite elements and gap-friction elements. These were implemented into computer programs to perform numerical simulations of the flood and earthquake responses of the three gravity dams (17.9,90 and 116 m high). The finite element seismic response analysis of a 90 m gravity dam model including 8 lift joints along the height lead to several conclusions. The presence of weak joints along the dam's height reduced the shear force and associated residual displacements acting at the dam-foundation 24 CHAPTER 2 LITERATURE REVIEW interface, as compared with the case where only the base joint was modelled. However, reductions in base sliding displacements were associated with significant relative sliding responses of upper lift joints. It was found that the presence of weak lift joints along the height of a gravity dam can affect the peak ground acceleration to induce sliding. A n upper bound estimate of base displacements can be obtained by considering a model with a single joint at the base. However, to ensure the stability of the upper section of the dam, the presence of weak lift joints along the height should be explicitly considered in safety evaluation. The magnitude and spatial distribution of residual sliding displacements were found dependent on the frictional strength, and therefore on the surface preparation and subsequent aging of the joints. An upper bound estimate of residual sliding displacements could be obtained by considering that joints have no tensile or cohesive strength. Static values of friction coefficient, which controls the initiation of sliding, should be selected conservatively i f the analysis is performed to assess the magnitude of residual sliding displacements. That is because for a given magnitude of compressive force, the dynamic frictional strength is likely to be smaller than its corresponding static value. In addition, the analyses included correlations between numerical simulations and experimental seismic sliding responses of the 3.4 m tall dam models tested previously. Good correlations with experimental results were obtained using two numerical models, one based on rigid body dynamics and frictional strength limited by the Mohr-Coulomb criterion, and the other based on finite elements using gap-friction elements. 25 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL Shake table tests are increasingly being used to evaluate seismic performance of structures, their models or various structural and nonstructural components. Shake table testing is a valuable tool especially in cases when the response being studied is associated with failure, such as in the sliding of a model of a concrete gravity dam. In such cases, this type of testing is the only source of experimental data to calibrate numerical models. In December 1996, a series of shake table tests on a scaled model of one monolith of a concrete gravity dam was conducted at the Earthquake Engineering and Structural Dynamic Research Laboratory (Earthquake Laboratory) at the Department of Civil Engineering of the University of British Columbia (UBC). The tests were conducted as the first phase of a collaborative research between U B C and the Dam Safety Program at the Maintenance, Engineering and Projects Division of the British Columbia Hydro and Power Authority (BC Hydro). The goal of this research was to gain an improved understanding on the post-crack dynamic response of concrete gravity dams. These tests were also the first of two series of experiments, which yielded data needed for this study. In this thesis, this experimental program is referred to as preliminary tests. 3.1 OBJECTIVES AND SCOPE OF PRELIMINARY TESTS The objective of the preliminary tests was to measure how much a model of a single monolith of a gravity dam would slide if preloaded by a simulated hydrostatic force and subjected to base excitations. The work focused on frictional characteristics of the dam-foundation interface and 26 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL therefore, the model was unbonded at the base during the tests. Other forces at the interface, such as cohesion, were not simulated. The excitations were limited to harmonic motions only. In addition, there were several other objectives of the tests: • To assess the feasibility of using U B C s Earthquake Laboratory to conduct this type of tests. • To develop sufficient experimental data for the calibration of computer models to simulate sliding of concrete gravity dams unbonded at the base. • To tune-up the experimental setup before a more extensive series of the shake table tests, which were already being planned at the time of the preliminary tests as a second phase of the collaborative U B C - B C Hydro project. The forces acting on the model were limited only to earthquake motions simulated by the shake table and to hydrostatic pressure represented by a single horizontal force on the model. Other effects, such as hydrodynamic force or uplift pressure at the foundation, were neglected. It was not the objective of the study to directly apply the test results to predict the seismic performance of a specific prototype. 3.2 DESCRIPTION OF TESTING FACILITY The Earthquake Laboratory at U B C laboratory is 16.4 m long and 11.5 m wide, providing space for construction, assembly, and handling of relatively large structural models and heavy equipment. Access is available for direct entry of large vehicles. The laboratory is equipped with an advanced, closed-loop, servo-controlled hydraulic seismic 27 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL simulator or shake table. It can accurately reproduce earthquake ground motions in one or more directions. The main element of the table is a 3 m by 3 m platform, which consists of a 0.4 m thick aluminium cellular structure. The weight of the table is about 20.5 kN and it has a grid of 38 mm diameter holes that are used to attach test specimens. The aluminium platform and attached hardware were designed to have a fundamental frequency about 40 Hz, so that it can be considered rigid within the operating frequency range of the shake table tests, which is mostly 1 to 25 Hz. Clearance above the table is 4.2 m and the laboratory is equipped with a 44.5 kN overhead crane for placing models and equipment on the shake table. The shake table has a pay load capacity of 156 kN and it can be configured to produce two types of multi-directional motions. One configuration (called 1 H X 3 V - 1 Horizontal and 3 Vertical) can be used to simulate longitudinal, vertical, pitch and roll motions. The other configuration (called 3H) can be used to simulate longitudinal, lateral, and torsional (yaw) motions. In configuration 3H, horizontal, longitudinal motions are produced by one hydraulic actuator with a maximum peak-to-peak displacement of 15.2 cm (6 inches). This actuator has a main stage area of 80.9 cm 2 . Other two actuators, with 45.2 cm 2 of effective piston area each, could produce horizontal motions in the transverse direction. The actuator force reactions are resisted by a massive pit. The pit is a reinforced concrete foundation extending around the table in the form of an open box with the thickness of 1.5 m on the sides where the actuators are installed. The outside plan dimensions of the wall are 6 m by 5.5 m and it is 2.5 m high, while the inside dimensions are 3.6 m by 3.6 m by 2 m (Rezai, 1999). The displacement of the table is limited by the stroke of the actuators, ±7.6 cm. The flow rate in the servovalves limits the maximum velocities produced in both directions to 100 cm/sec. 28 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL The maximum acceleration is limited by the force limits of the actuators together with the mass of the table-specimen system. The stalling force capacity of the longitudinal actuator is 156 kN, while the other two actuators in the transverse direction have a capacity of 90 kN each. The shake table is controlled by a signal processing subsystem, driven by a replication multi-shaker control software. This software performs a closed-loop control of several shakers which are capable of replicating recorded earthquake shaking and other types of motions with high accuracy. The high performance digital control system can easily replicate earthquake motions for models with different mass-stiffness characteristics. This is a desirable feature for comparative studies of different models or equipment under the same loading conditions. The data acquisition system at the laboratory can record up to 128 channels of instrumentation information from a test specimen. From this number, a total of up to 44 channels can be conditioned by variable gain buffers and cut-off filters, which provide optimal control over signal levels and noise reduction in order to retrieve accurate dynamic testing data. 3.3 EXPERIMENTAL MODEL An experimental model for a single monolith of a concrete gravity dam is described in this section. The model consisted of three parts: a dam monolith, a pair of surfaces plates and hydrostatic load simulation assembly. Description of each part is given here in a separate subsection and the model without the hydrostatic load assembly is shown in Figure 3.1 3.3.1 Shape of the Model for a Dam Monolith The model for a dam monolith was 1500 mm high, 1250 mm wide in the upstream/downstream 29 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL Figure 3.1: Main Parts of Experimental Model direction and 480 mm thick in the cross canyon direction. The shape of the monolith, shown in Figure 3.2 was designed close to a shape, which could have a monolith of a low to of medium height concrete gravity dam. The model was designed with several holes for easier handling. In an optimum case, the experimental model should be of the same dimensions and made from the same material as a prototype. This is mostly not possible for models of large structures due to given dimensions of labs and limited payloads of earthquake simulators. In addition to this, during this testing the model was handled very often because it had to be put back to its original position after each test when it moved. 30 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL 225 1250 o 00 1250 Figure 3.2: Schematic of the Model for a Dam Monolith 3.3.2 Material for Monolith and Similitude Considerations It was considered during the design of the model that it would be moved often back to its original position during the testing. Although the Earthquake Laboratory at U B C is equipped with a crane it would be difficult to handle a heavy model of the monolith. It was decided to keep its weight below 5 kN. The material for the model was designed based on this requirement. 3 1 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL The material used for the monolith was a mix consisting of Portland cement type 10, perlite, styrofoam, silica fume and water. The weight composition of the mix was 42.2% cement, 40% water, 12.5% perlite, 3.6% silica fume and 1.7% Styrofoam (Horyna et al., 1997). Since the strength of the above material was low it was decided to use stronger material at the locations of expected local loads on the model. These locations included about 5 cm thick layer at the bottom of the model, which was in contact with the upper surface plate and also the part of the model where the simulated hydrostatic force was applied. A modified mix without Styrofoam was used for these parts of the model. The total weight of the model including the upper sliding plate and all attachments was 480.6 kg. It was recognised that the test results would be used to study behaviour of the model, but they would not be used to predict response of any prototype dam. However, once the model was built, it was of interest to find the dimensions of a full scale concrete gravity dam monolith which would satisfy similitude with the model. Discussion of the similitude aspects follows. To maintain similitude between inertia and elastic forces requires the quantities of length (Z), mass density ( p), modulus of elasticity (E) and acceleration due to gravity (g) of a prototype satisfy certain relationships with those of the experimental model (Moncarz an Krawinkler, 1981, or Mir and Taylor, 1995). Based on these quantities the following ratios can be defined: LM PM EM * SM where indeces P and M stand for prototype and model, respectively. These scaling factors should satisfy the following condition resulting from Cauchy's requirements for proper simulation of inertial forces and restoring forces: 32 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL (3.2) The material characteristics of the prototype dam monolith made of concrete were considered EP = 27,000 MPa and PP = 2,580 kg/m 3 (Powertech Labs, Inc., 1996). The material for the model had the following characteristics: EM = 500 MPa, which was obtained as a modulus of elasticity measured on a testing cylinder; and oM = 720 kg/m , which was found by weighting the testing cylinder. These values were related only to the basic material of the monolith, but they did not take into account neither rebars in the model nor the parts of the model which were made using stiffer and heavier mix without Styrofoam in it. Therefore, the values taking into account all materials in the model should be used for similitude calculations. Such values were found as: EM= 560 MPa and pM - 800 kg/m 3. The new value for is^was obtained in the analytical part of this study as the modulus of elasticity, which resulted in the best match between natural frequencies of the model obtained experimentally with those from FE study. The new value for p M was obtained after weighing the complete model. It can be concluded that these new values for the characteristics of the model's material are higher than the original one obtained from the testing cylinder. This increase of material properties against those of the testing cylinder is reasonable and can be explained by the fact that the material used in some parts of the model was stiffer and heavier than that of the testing cylinder. Knowing the prototype's and model's material characteristics the ratios 5^=3.22 and SE= 48 were calculated. Then, using the Cauchy's condition and considering S = 1, the scaling factor for length can be calculated as SL = 15. This value indicates that i f the dimensions of the prototype are 15 times as large as those of the model, the similitude between elastic and inertia 33 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL forces would be satisfied. Since the height of the model is 1.5 m such a prototype would be a concrete monolith about 22.5 m tall which is a height of a low concrete gravity dam. Once knowing the length ratio SL, the ratio of prototype to model frequencies S -^can be determined as (Mir and Taylor, 1995): (3.3) The frequency ratio for the model considered can be evaluated as Sf= 0.258, which means that, for example, the frequency of 10 Hz on model scale corresponds to 2.58 Hz for the prototype. The similitude between elastic and inertia forces was discussed in this subsection. Although it was not the objective of the experimental study to simulate response of any specific prototype, this discussion was presented here in order to relate the model to an actual structure. 3.3.3 Sliding Surfaces The model comprised the monolith and a pair of sliding surface plates, which were identified as the Upper Surface Plate (USP) and the Lower Surface Plate (LSP). The experimental model was designed in such a way that sliding occurred between the sliding surface attached to the bottom of the dam and one attached to the shake table. The arrangement of model components provided a convenient and easy way to change pairs of plates. In addition, it was required that the sliding surfaces had stable friction characteristics during many tests. The sliding surfaces were designed according to the following requirements. The dimensions of USP and LSP are given in Figures 3.3 and 3.4, respectively. The USP was clamped to the bottom of the model as can be seen in Figure 3.5. The LSP was bolted to the 34 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL 1150 Plan View 1150 o m Welded "Steel mesh wire 3mm, squares 6' Section A-A 480 Note: all dimensions are in mm, unless otherwise specified o in Section B-B Figure 3.3: Dimensions of the Upper Surface Plate shake table using a steel frame, which is shown in Figure 3.6. A total of three pairs of sliding surfaces were used for testing. These were denoted as: • Smooth surface; • Cracked-ice surface; • Rough surface. It should be noted that these names were given to the surfaces during the fabrication process and independent of how each surface performed during the testing. The plates for sliding surfaces were made of a mortar based mix with fine aggregates only; 100 kg of this mix comprised 10.8 35 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL 1 5 0 0 Plan View 6 6 0 Welded SteefMesh, wire 3 mm, squares 6 o 1 Section C-C 1 5 0 0 o LO 1 z x Section D-D Note: all dimensions are in mm, unless otherwise specified. Figure 3.4: Dimensions of the Lower Surface Plate kg of water, 24.5 kg of Portland cement type 10, and 64.7 kg of fine-grain play sand. The fabrication process of the sliding surfaces for different roughnesses is explained below. 3.3.3.1 Smooth Surface The Smooth surface was formed by casting first the Lower Surface Plate (LSP) placed on a plastic sheet (1.5 mm thick, with smooth surface) at the bottom of the formwork. After curing, the LSP and the plastic sheet were turned upside down and then the Upper Surface Plate (USP) was cast on top. The same 1.5 mm thick plastic sheet was used to separate the two plates. 36 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL t 300 t 600 I 300 _ 1200 b) Plan View Notes: all dimensions are in mm, unless otherwise specified. (1) - steel plate 1/2", size 489*140 mm with holes for bolts(4) (2) - rebars 5 mm, welded to plate (7) (3) - rebars 10 mm, welded to plate (7) (4) - bolts M14 (5) - nuts welded to (1) (6) - spacer - steel plate 1/8" or 1/4", 480*35 mm (7) - steel plate 1/4", 480*300 mm Figure 3.5: Connection of USP to the Monolith 3.3.3.2 Cracked-ice surface The Cracked-ice surface was formed by casting the LSP on a plastic sheet (1.5 mm thick, with a Cracked-ice pattern surface). After curing, the LSP was turned upside down and the USP was cast on top. A very thin plastic foil (wrapping foil with thickness of hundredths of mm) was 37 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL 1720 315 • 200 340 200 415 *4 • I 330 200 , 135 540 530 o o CM o 00 oo o o CM O o 60 u L 100x75x10-1720 mm 1520 L 100x75x10 - 680 mm o CO o L 100x75x10 - 1720 mm L100x75x10, 680 mm / o o Hole, ID 40 mm (Typical) a) Plan View 3/4" Bolt and Welded Nut Plate 1/4" 200x200 mm (Typical) Note: all dimensions are in mm, unless otherwise specified. rop of the Shake Table b) Section F-F Figure 3.6: Steel Frame for the LSP to Shake Table Connection used as separator between LSP-top surface and USP-bottom surface. 3.3.3.3 Rough surface The Rough surface was formed by casting the LSP on a plastic sheet (1.5 mm thick, with smooth surface). After curing, the LSP was turned upside down and the top surface of the LSP was chipped with a chisel. This produced randomly positioned irregularities (about 8 at an area 100 38 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL mm by 100 mm) with depths of 4 to 8 mm. This irregular surface was used as a bottom of the formwork to cast the USP. Thin wrapping plastic foil was used as a separator between the LSP-top surface and the USP-bottom surface. 3.3.4 Simulation of Hydrostatic Load It was the objective of the preliminary test to investigate response of the model due to base excitations and simulated hydrostatic force. This force was provided by a hydrostatic load simulator, which pulled the model in the downstream direction. The force was simulated by a set of long springs stretched between the downstream side of the model and the support fixed to the shake table. The force in the springs was kept relatively constant during the test as the model moved. This was achieved by measuring the force during the test with a load cell and by using a manual valve control to adjust the spring tension as needed. This can be seen in a photo in Figure 3.7. 3.4 EXPERIMENTAL SETUP AND TESTING PROCEDURES The testing program consisted of two types of tests, which were called static and dynamic. Both types of the tests were conducted using the shake table. The objectives of the static tests were to determine friction coefficients of the surfaces as well as the magnitude of static forces necessary to cause sliding of the model preloaded by a specified hydrostatic force. The dynamic tests were conducted to find the amounts of model sliding for varied frequency and amplitude of base acceleration. This was desired to find for three different surfaces (Smooth, Cracked-ice and Rough) and two levels of simulated hydrostatic force (70% and 100% of full reservoir). 39 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL Figure 3.7: Controlling the Level of Hydrostatic Force during a Dynamic Test The 3 H configuration of the shake table was used to conduct all the tests reported here. However, during all the tests conducted within this study, only longitudinal horizontal shake table motions were used. This direction corresponded to the upstream/downstream direction of the model for the dam monolith. 3.4.1 Static Tests The experimental setup for the static tests is shown schematically in Figure 3.8 or in a photo in Figure 3.9. It comprises the dam monolith, a pair of sliding surfaces, a set of springs and a rigid bar. The latter was used to keep the model at the same position when the shake table was moving slowly in the upstream direction (relatively with respect to the model). The springs were used to apply previously computed force, representing the hydrostatic pressure on the model. This force was kept relatively constant during the test as the model moved. During the 40 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL D a m Monol i th, Model" -Support for Spr ings Spr ings to Simulate J Hydrostat ic Force VWWA— r t Load Cell 1 Shake Table ± Direction of Shake Table Mot ion During Static Tests Load Cell 2 -Rigid Bar — i Pinned / Suppor t \ Vert ical Support " (Hoist) Concrete Wear ing Sur faces Figure 3.8: Experimental Setup for Static Tests Figure 3.9: Photo of Experimental Setup for Static Tests 41 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL static tests, the rigid bar pushed against the model through a steel plate with a hinge connection. The force between the model and the rigid bar was measured using another load cell. A typical static shake table test consisted of the following steps: Set up the model on the shake table. • Attach horizontal long springs to downstream face of the model at specified height and pre-stretch them sufficiently to provide a force equivalent to the hydrostatic force due to the reservoir. • Conduct the test by holding the model with the rigid bar attached to its upstream face and slowly moving the table upstream at a constant velocity. Measure the model displacement relative to the shake table and the force between the model and the rigid bar. Maintain hydrostatic force at constant level during the test. • Detach the horizontal springs and lift the model. • Clean the sliding surfaces. 3.4.2 Dynamic Tests The experimental setup for dynamic testing is shown in Figure 3.10. or in a photo in Figure 3.11. It consisted of the dam monolith, USP and LSP and the springs to simulate hydrostatic loads. The excitation of the model during the dynamic tests was due to shake table harmonic motions of prescribed frequencies and amplitudes. A lateral restraining system consisting of three pairs of rollers was used during all test in order to ensure motions of the dam model in the upstream and downstream directions only. The dynamic testing was conducted at frequencies from 5 to 20 Hz with a step of 2.5 Hz. 42 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL Figure 3.10: Experimental Setup for Dynamic Tests Figure 3.11: Photo of Experimental Setup for Dynamic Tests 43 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL During each dynamic test, the shake table input motion time history was consisted of six parts, all with the same frequency, but with successively increasing amplitude of acceleration. In the first part, the amplitude was 75% of the maximum, i.e. 100% level, which was reached in the sixth part. The amplitude difference between adjacent parts was steps 5% of the maximum acceleration. The duration of shaking for each part was 10 seconds followed by 2 seconds of rest to separate each part. A typical dynamic test, for each combination of surfaces, hydrostatic force and frequency, consisted of the following steps: • Set up the model on the shake table. • Attach horizontal long springs to downstream face of the model at specified height and pre-stretch them sufficiently to provide a force equivalent to the hydrostatic force due to the reservoir. • Conduct the test by exciting the model by prescribed motions of the shake table. Measure the shake table acceleration, acceleration of the model at the bottom and at the top, and the model displacement relative to the shake table. Maintain hydrostatic force at constant level during the test. Shut down the excitations if the model relative displacements are too large (about 15 cm) or i f the rate of the model displacement is too high (about 10 cm in 10 sec). • Detach the horizontal springs. • Lift the model and clean the sliding surfaces. At each frequency, the tests started at the peak acceleration levels of about O.lg. The levels 44 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL were increased in the subsequent tests until significant motions of the model were achieved. After significant sliding was achieved an additional test for the same acceleration amplitudes was repeated in order to verify the results. 3.5 INSTRUMENTATION The instrumentation used for the static testing included two displacement transducers (Celesco PT 101 Position Transducer), as shown in a photo in Figure 3.12, to measure relative displacement of the dam with respect to the shake table. The compressive force between the model and the rigid bar was measured by a load cell (Sensotec, model 41/0574-03) shown in a photo in Figure 3.13. In addition, the force in the springs was also measured with a load cell of the same type. The sampling rate for static tests was 20 samples per second. Figure 3.12: Displacement Sensors Located on Downstream Side of the Monolith 45 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL Figure 3.13: View of a Load Cell to Measure Force between the Model and the Rigid Arm The instrumentation used for the dynamic tests included two horizontal accelerometers (IC 3110 Accelerometers) to measure the upstream/downstream accelerations of the model close to the base (see a photo in Figure 3.13), and at the top. The relative displacements of the model with respect to the shake table were measured with two displacement sensors (Celesco PT 101 Position Transducer). Shake table horizontal motions were measured with an LVDT and an accelerometer (Kistler 8304 K-Beam Accelerometer). The simulated hydrostatic force was measured with a load cell (Sensotec, model 41/0574-03). The sampling rate for all dynamic tests was 200 samples per second. 3.6 SUMMARY AND CONCLUSIONS A number of static and dynamic tests were conducted during the preliminary experimental 46 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL study. These tests generated large amount of useful data, which included: • static and kinetic friction coefficients of the surfaces obtained from the static tests, • amplitudes of harmonic base accelerations to initiate sliding of the model at a given frequency of excitations, amplitudes of base accelerations to cause sliding of 3 cm during 10 seconds. These were reduced, analysed and the results of this analysis were presented in a report delivered to BC Hydro (Horyna et al., 1997). However, these results are not presented here since the preliminary tests were superseded in many aspects by the further experiments, which will be described in the next chapter with all results related directly to this thesis. Only a list of general findings from the preliminary experiments is given here: • Combining several dynamic tests in one sequence reduced significantly testing time of the dynamic tests and was found as a very convenient approach. • Response of the model during some tests contained significant undesirable out-of plane rocking motions. These were caused by the fact that some surface plates were not ideally flat and as a result of this the model to foundation contact was limited to only several points. Resting on these points, the model rocked in cross-canyon direction during some tests. The restraining system, which was used during the tests, only reduced the out-of plane rocking. This was considered when the second phase of the tests was planned. • The hydrostatic load simulator used during the tests was able to provide desired nearly constant horizontal force on the model. However, the springs had to be 47 CHAPTER 3 PRELIMINARY EXPERIMENTAL STUDY OF A DAM MODEL detached from the model before every moving it to the original position and after a test and attached again before the next. Considering the number of tests needed to conduct, this procedure was found to time demanding and other hydrostatic load simulator was designed for the tests in the second phase. A n attempt to simulate interlocking phenomena in the foundation interface was done using the pair of plates with the Rough surface. However, this was found very difficult and in order to reduce amount of uncertainties in the testing this was removed as an objective of the second experimental phase. A few experiments with records measured during past earthquakes were conducted at the end of the testing. These tests generated useful information, but it was found that scaling the records in time domain to obtain base excitations of varied dominant frequencies caused complications during data analysis. Therefore, instead of time-scaling records in the second phase it was proposed to develop a series of synthetic records of the same duration but with different dominant frequencies. It was found that the sliding surfaces deteriorated partially during the series of tests with one surface. In order to monitor the influence of this on the response of the model in future tests, it was decided to perform static tests regularly after every few dynamic tests of the next tests. 48 4 F U R T H E R E X P E R I M E N T A L S T U D Y OF THE D A M M O D E L A series of shake table static and dynamic tests was described in the previous chapter. These tests were conducted on the model of a concrete gravity dam monolith unbonded at the base. A new series of shake table was scheduled then with similar goal as the preliminary tests with the experience gained during those initial tests. As a result of this experience, several enhancements of the experimental model, setup and testing procedures were introduced. Result of these enhancements was that more types and a larger number of tests could be scheduled in the new series. These tests were conducted between May and July 1998 also in the Earthquake Laboratory at UBC. 4.1 OBJECTIVES AND SCOPE OF FURTHER EXPERIMENTS A total of four different types of tests were scheduled for this study. They included impact hammer tests, static tests, dynamic tests with harmonic excitations (called harmonic tests) and dynamic tests with synthetic earthquake excitations (called earthquake tests). The objective of the harmonic and earthquake tests was to measure how much the experimental model of a single monolith of a gravity dam slides if preloaded by a simulated hydrostatic force and subjected to base excitations. The focus was put on frictional characteristics of the dam-foundation interface and, therefore, the model was unbonded at the base during the tests. Other forces in a real dam-foundation interface, such as cohesion, were not simulated. Another objective of these tests was to study changes in the response of the model with varying dominant frequency of applied base excitations. In particular, it was of interest to determine if 49 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL the range of testing frequencies can be sorted according to qualitative changes in the response of the model. Along with the dynamic tests, a series of static tests were scheduled. The objective of these tests was to determine the static and kinetic coefficients of friction of the model-foundation interface and static forces necessary to calculate the force ratios. The objective of the impact hammer tests was to determine the natural frequencies of the experimental model. Similarly to the preliminary tests, the further experiments were not conducted in order to simulate response of any specific prototype dam to an earthquake. 4.2 EXPERIMENTAL MODEL The experimental model used during the preliminary tests was also employed during the additional tests, except for two main differences: • Different shape and finishing of the upper surface plates, and • Different hydrostatic load simulator. 4.2.1 Sliding Surfaces The sliding surfaces, fabricated by an external contractor, were designed to be in contact only at the toe and heel zones of the dam monolith model. This was done in order to minimize out-of plane rocking of the model during the shake table tests, which was observed during some of the preliminary tests. The contact between the Upper Surface Plate (USP) and the Lower Surface Plate (LSP) was limited to 150 mm at each end. This type of contact between the sliding 50 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL surfaces was achieved by specifying raised sections at the end of each USP, which can be seen in Figure 4.1. The shape of the LSP was the same as during the preliminary tests. L 1165 . 00 < i Section B-B 1 0 Note: all dimensions are in mm, unless otherwise specified. Figure 4.1: Schematic of the Upper Surface Plate The Smooth surface was created by sanding the cement milk from the friction surface of the plates to give a uniform surface. The Rough surface was developed using an ultra-high pressure water jet to remove the cement matrix leaving an exposed aggregate surface. The water jet had a pressure of 207 MPa and preparation of the Rough surface is shown in Figure 4.2. The finishing was done on both contact surfaces, that is the top of LSP and the bottom of USP. A 51 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL total of four pairs of the surface plates were prepared, two with Smooth and Rough surface. Figure 4.2: Photo Showing High Pressure Jet Preparing Rough Surface 4.2.2 Hydrostatic Load Simulation The hydrostatic load was simulated by applying a pulling force in the downstream direction on the downstream face of the dam model. This force was provided by a steel cable attached to the model at a height corresponding to the resultant of the simulated hydrostatic load. The other end of this cable was attached to a hanging weight equivalent to the required force. Figure 4.3 shows a schematic of the hydrostatic load assembly and Figure 4.4 presents a photo including the experimental model with the applied simulated hydrostatic force. The weight was attached to two vertical rods which constrained it to move only in the vertical direction. Friction between the weight and the sliding rods was minimized by using bronze-oilite bearings, which provided a smooth contact surface. 52 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL During trial shake table harmonic tests with the assembly, the model was subjected to undesirable significant force pulses introduced by the motion of the mass bouncing on the steel cable attached to the model. These pulses manifested themselves when sliding of the model was initiated. Since these prevented a constant level of simulated hydrostatic load, it was decided to minimize their effects. This was accomplished by inserting a 25 cm long piece of rubber, between the model and the cable. This rubber piece, acting as a base-isolator, reduced effectively the pulses on the model to negligible levels. The simulated hydrostatic load was set to 1,040 N , which corresponded to a 95% of full reservoir level considering the maximum water level as 97% of the total height of the model including the USP. When this force was calculated, mass density of water was reduced by factor Sp = 3.22, which corresponded to the ratio of the mass density of concrete to the mass density of the material of the model and was calculated in the previous chapter. Although it was desirable to keep this force constant during all tests, the resulting measured force fluctuated somewhat during the tests as well as between tests. However, these variations did not exceed 5% during any of the tests. 4.3 INSTRUMENTATION During the testing a maximum of ten different time history signals were recorded in each test. The location of the instruments used to record the signals are shown in Figure 4.5. The arrows labelled 6 through 10 show the location of triaxial accelerometers (IC 3110 Accelerometers), which measured the horizontal and vertical accelerations of the model. The relative displacement of the model with respect to the shake table (arrow 3 in Figure 4.5) was measured 54 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Figure 4.5: Location of Instrumentation using a displacement transducer (Celesco PT 101 Position Transducer). The simulated hydrostatic force (arrow 5 in Figure 4.5) was measured with a 11.1 kN load cell (Interface) and the force applied by the arm, for the static-push tests, was measured with a 44.5 kN load cell (Sensotec). Shake table horizontal displacement and acceleration (labelled 1, 2 respectively) were measured with sensors built in the shake table (LVDT and a Kistler 8304 K-Beam accelerometer). A complete list of the sensors used for the testing is given in Table 4.1. A l l recorded signals were filtered with a 30 Hz low-pass filter using a 3 pole Bessel type filter with a 60 dB/decade roll off. 55 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Table 4.1: List of Instrumentation and Location Location Instrument Model Measurement 1 Displacement sensor LVDT Shake table horizontal displacement 2 Accelerometer Kistler 8304 K-Beam Accelerometer Shake table horizontal acceleration 3 Displacement transducer Celesco PT 101 Position Transducer Model displacement relative to shake table 4 10 kip load cell Sensotec Force in rigid arm 5 2.5 kip load cell Interface Simulated hydrostatic force 6 Accelerometer IC3110 Model horizontal accel., u/s face, bottom 7 Accelerometer IC3110 Model vertical accel., u/s face, bottom 8 Accelerometer IC3110 Model horizontal accel., d/s face, bottom 9 Accelerometer IC3110 Model vertical accel., d/s face, bottom 10 Accelerometer IC3110 Model horizontal accel., d/s face, top 4.4 TEST SETUP AND PROCEDURES 4.4.1 Impact Hammer Tests The objective of the impact tests was to obtain the frequencies corresponding to the fundamental modes of vibration of the experimental model. It was desirable to know if any of the frequencies of excitation (harmonic and earthquake tests) was close to one of the natural frequencies of the model. In addition, these tests were conducted in order to generate data for calibration of finite element models developed during the study. It was decided to perform the impact hammer tests for two types of boundary conditions: • unbonded (free-standing) model, and • bonded (fixed at four corners to the base) model. 56 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Figure 4.7: View of Downstream Side of the Model Attached to Rigid Floor during Impact Hammer Tests 57 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Since the floor at the Earthquake Laboratory was not suitable for attaching the model to it, the impact hammer tests were conducted at the UBC's Structures Laboratory. A total of 16 tests were conducted in June, 1998. During half of these tests the model was fixed to the strong floor of the lab at four corners (bonded model). View of the downstream side of the model fixed to the floor is shown in Figure 4.7. The other half of the tests were done with the model free-standing on the floor (unbonded model) in order to monitor changes of its natural frequencies for different boundary conditions. Only one Rough surface (Rl) was used for the testing. A n instrumented sledge hammer (Dytran model 5803A) was used for the testing. The hammer impacts were applied horizontally at the top and mid height of the model, on its upstream side. Hard and soft tips of the hammer were used for the testing. The hydrostatic force was not simulated during these tests. The data measured during the tests included signals from the hammer and four horizontal accelerations measured with accelerometers (IC, model 3110) on the upstream side of the model, at the top, bottom and thirds of the height of the model (see Figure 4.6). Each record had a duration of 10 seconds and the measured signals were recorded with a sampling rate of 500 samples per second. 4.4.2 Static Tests The objectives of the static tests were to determine friction coefficients for each surface and to track the deterioration of the surfaces as testing progressed. For the static tests the model was held stationary while the table moved slowly under it. An arm attached to a strong column provided the restraint necessary to hold the model. Figure 4.8 shows a schematic of the arm and load cell configuration. For the static tests the force in the strong arm, simulating the hydrostatic 58 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Strong Column U 2000 H Figure 4.8: Schematic of Strong Arm Used For Static Tests Figure 4.9: Photo of Typical Static Test Setup 59 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL force, shake table displacement and model displacement were measured (measurements 1, 3,4 and 5 in Figure 4.5) for 160 seconds at a sampling rate of 100 samples per second. Figure 4.9 shows a photo taken during one of the static test. For each of the four surfaces (2 Smooth and 2 Rough), a number of preliminary static tests were conducted. Relatively large changes in the force required to push the model were observed during these first few tests. It was observed that it was during these initial tests that small irregularities on the surface were removed or reduced as sliding between plates occurred. Through the process of wearing down these irregularities it was also observed that the friction coefficients remained stable and thus lifting and cleaning the contact surfaces between tests was deemed unnecessary. A series of preliminary static tests were conducted every time a new pair of plates was used for the first time, and a single test was conducted between each change of excitation frequency during harmonic and earthquake tests. Because the friction values were most stable when the model was not lifted between tests, a test sequence was developed which would allow the model to be pulled back up-stream between tests without lifting it up. This was accomplished by means of a winch attached to the base of the model and the strong column, shown in Figure 4.10. A typical static test consisted of the following steps: • Position the model on the shake table with simulated hydrostatic load applied. • Lower strong arm to horizontal position. • Extend strong arm to make contact with the dam model. • Conduct test (table slowly moves toward strong arm; model held stationary). Pull model back to original position with the winch. 60 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Figure 4.10: Winch Assembly Used to Pull Model Up-Stream Between Tests 4.4.3 Tests with Harmonic Input The objective of the tests with harmonic input was to measure response of the dam monolith model, preloaded by simulated hydrostatic force, due to harmonic base excitation. The parameters varied during these tests included frequency and amplitude of the base acceleration. Nine different frequencies were selected for the harmonic inputs. These frequencies were 5,7.5, 10, 12.5, 15, 17.5, 20, 22.5, and 25 Hz. As an example of an input signal, the 5 Hz harmonic input signal is shown in Figure 4.11. This signal has 6 segments with different amplitudes, in particular ranging from 0.525g to 0.7g, with increments of 0.035g. This record was used as the input signal to the shake table where it was adjusted to the desired amplitude levels. At each particular frequency the simulation was run several times, increasing the amplitude of shaking each time, in order to capture the amplitude of motion which caused the model to slide a 61 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL 0 10 2 0 30 4 0 50 60 70 8 0 90 Time (sec) Figure 4.11: Example of Harmonic Input Record at 5 Hz prescribed amount. Typically, the record for a certain frequency was run 5 to 10 times providing 30 to 60 different ten second long records at different amplitudes. During the harmonic tests, records from locations 1, 2, 3 and 5 through 10 in Figure 4.5 were obtained. The duration of all tests was 90 seconds and each channel was recorded at a sampling rate of 200 samples per second. A typical harmonic setup is shown in Figure 4.12. A typical harmonic test included the following steps: • Position the model on the shake table with simulated hydrostatic load applied. • Run the test using a record with low amplitude (about 0.3g). • Increase amplitude of the record (by about 0.05g). • Continue increasing amplitude - pull back model to start position as needed. • Increase amplitude until model moves full distance in one testing segment. • Run Static Test. 4.4.4 Tests with Synthetic Earthquake Input The objective of the earthquake tests was to measure response of the dam monolith model 62 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Figure 4.12: Typical Harmonic Test Setup preloaded by simulated hydrostatic force to a transient base excitation. The parameters varied during these tests were the same as those for harmonic tests, the experimental setup and data collection for the earthquake tests were also the same. The records used for these tests were generated using the program SIMQKE (Gasparini and Vanmarcke, 1976) from the Power Spectral Densities (PSD) of recorded motions from three selected past earthquakes in California. These records were: • 1992 Landers earthquake, recorded at the Joshua Tree Fire Station, E/W direction (CSMIP, 1992). This is designated as EQ1 for the remainder of this thesis; • 1994 Northridge earthquake, recorded at the Tarzana Nursery Station, E/W direction (CSMIP, 1994), designated as EQ2; • 1979 Imperial Valley earthquake, recorded at E l Centro, Bonds Corner (Highways 98 and 115) Station, SW/NS direction (California Department of Conservation, 63 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL 1983), designated as EQ3. The records used for these tests were developed as follows. First, the PSD from each of the above records was calculated and smoothened using a 50 point moving average. Then, the resulting PSD values were shifted so that the frequency of the maximum PSD value corresponded to the frequency of interest. This frequency was considered to be the dominant frequency fj of the generated record. The shifted PSD values were multiplied by a window function of frequency band (7 -^2) Hz to (/^-4) Hz in order to narrow-band the record being developed into a 6 Hz frequency band. The 6 Hz band was selected after it was observed that all the original PSD's had negligible values out of this band. Finally, program SIMQKE was used to generate a synthetic record with a duration of 10 seconds. This process was repeated over 9 dominant frequencies (from 5 to 25 Hz with a step of 2.5 Hz) using each of the 3 seed earthquake records (EQ1, EQ2 and EQ3). Each of the generated records were identified by reference to the name of the seed earthquake and by the value of the dominant frequency. For example, synthetic earthquake EQ1 with a dominant frequency of 12.5 Hz had components with frequencies from 10.5 Hz to 16.5 Hz, and it was derived from the Landers earthquake seed record. Generated records from the three different seed records and with the same dominant frequency were combined into one time history, which was later used as a driving signal for the shake table. A n example of such driving signal is shown in Figure 4.13, which presents three synthetic earthquake records, each with a dominant frequency of 5 Hz. A typical earthquake test was as follows: • Position the model on the shake table with simulated hydrostatic load applied. • Run the simulated earthquake record with low amplitude (around 0.3g). 64 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL 0 10 2 0 3 0 4 0 T i m e ( s e c ) Figure 4.13: Example of Synthetic Earthquake Input Record with Dominant Frequency 5 Hz • Increase amplitude of record (add 0.05g). • Continue increasing amplitude - pull back model to start position as needed. • Increase amplitude until model moves full distance in one test segment. • Run Static Test. 4.5 RESULTS OF EXPERIMENTS A total of 360 harmonic tests, about the same amount as the earthquake tests, and more than 100 static tests were conducted using the shake table. Nearly 1 GB of data was generated by the tests and it was necessary to reduce, analyse and evaluate all these data. This section presents the main findings obtained from the experimental work. Out of the tests conducted on all four surfaces (Rough surfaces R l and R2, and Smooth Surfaces SI and S2), the experimental results presented here are limited to those from the tests conducted on one surface, the Rough surface R l . This is because all numerical analyses of the experimental 65 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL model described in the next chapter of this thesis were performed using the frictional characteristics of this surface. The test results from other surfaces are not presented here. A complete set of results from all surfaces can be found in Black et al. (1998). 4.5.1 Results of Impact Hammer Tests The impact hammer testing was conducted with the bonded model, that is attached to the strong floor at the UBC' s Structures Laboratory and with the unbonded model, that is free-standing on the floor. The results of the tests in terms of the natural frequencies and characteristics of the corresponding mode shapes are presented in Tables 4.2 and 4.3. The natural frequencies were determined from peaks of the Frequency Response Function (FRF) considering the signals from the hammer as the input and those from the accelerometers as the output of a single degree of freedom system (Bendat and Piersol, 1971). From a comparison of results shown in Tables 4.2 and 4.3, it can be observed that the natural frequencies of the bonded model are quite different from those of the unbonded, free-standing, model. For example, the fundamental natural frequency of the bonded model is about twice as high as that of the free-standing model. It also can be observed from the values in Tables 4.2 and 4.3 that the variability in the frequency values are different for both models. The bonded model exhibited approximately the same natural frequencies for a wide range of tests. In contrast, the unbonded model had natural frequencies dependent on various parameters such as the power of the hammer strike, location of the strike, and the position of the model on the base. These variations in natural frequencies of the unbonded model can be attributed to nonlinearities of the setup, such as rocking of the model and slight changes of the contact characteristics at the foundation interface. 66 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Table 4.2: Natural Frequencies of the Bonded Model Frequency (Hz) Characteristics of Natural Mode 66 the 1st cantilever bending mode (no node along the model height) 174 the 2nd cantilever bending mode (one node along the model height) Table 4.3: Natural Frequencies of the Unbonded Model Frequency (Hz) Characteristics of Natural Mode 27-34 rocking combined with the 1st cantilever bending mode 53-63 character of the mode could not be identified from the tests (such identification is carried out in Chapter 5 with the help of FE model) 140 second cantilever mode (one node along height) combined with rocking The coordinates of the natural modes were determined using amplitudes and phase angles of the FRF calculated from the measured signals. However, because the model was instrumented with four horizontal accelerometers only, it was not possible to identify precisely the characteristics of the natural modes of the free-standing model as these modes exhibited components in the horizontal and vertical directions. Results of a finite element analysis of the model conducted simultaneously by Rudolf (1998) were used as complement to the experimental results to identify the character of the natural modes of the free-standing model. The impact tests were also used to obtain damping of the system. This was determined from the free vibration decay of amplitudes of horizontal acceleration measured at the top of the monolith. The damping was found to be about 5% of critical for the first mode of vibration. 67 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL 4.5.2 Results of Static Tests A number of static tests were conducted on each surface prior to any harmonic or earthquake tests in order to condition the surface before each series of tests. Results from these trial tests were not used for any analysis. Static tests were conducted before every series of the harmonic or earthquake tests with a single dominant frequency, in order to determine the friction coefficients prior to that series. A n example of measured time histories during the static tests is shown in Figure 4.14. The quantities plotted in this figure are identified above each figure. It can be observed from the plot of the total force on the model (part c in the figure) that the friction characteristics of the surface RI were almost uniform along the entire tested path about 8 cm long (see part d). It can be also concluded that no significant drop of the friction force was observed after the sliding started, which means that the difference between the static and kinetic friction was small for surface RI . The results of static tests in terms static and kinetic friction coefficients are given in Table 4.4. The friction coefficients were determined from the signal obtained by adding the measured force time history, obtained from the load cell between the strong arm and strong column, and the measured simulated hydrostatic load. The static coefficient was calculated as the average of the peaks in this signal divided by the weight of the model, while the kinetic coefficient was obtained as the average trough of the signal divided by the weight. 68 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Figure 4.14: Example of Signals Recorded during Static Tests with Surface R l 69 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL Table 4.4: Friction Coefficients of Rough Surface R l Obtained from Static Tests Test# Measurement Taken Prior to: Static Friction Coefficient Kinetic Friction Coefficient 16 Harmonic test at 5 Hz 0.73 0.72 17 Harmonic test at 7.5 Hz 0.72 0.71 18 Harmonic test at 10 Hz 0.74 0.73 19 Harmonic test at 12.5 Hz 0.74 0.73 20 Harmonic test at 15 Hz 0.74 0.74 21 Harmonic test at 17.5 Hz 0.75 0.75 22 Harmonic test at 20 Hz 0.75 0.74 23 Harmonic test at 22.5 Hz 0.76 0.76 24 Harmonic test at 25 Hz 0.76 0.75 28 Earthquake test at 5 Hz 0.75 0.75 29 Earthquake test at 7.5 Hz 0.76 0.75 30 Earthquake test at 10 Hz 0.76 0.75 31 Earthquake test at 12.5 Hz 0.76 0.75 32 Earthquake test at 15 Hz 0.77 0.76 33 Earthquake test at 17.5 Hz 0.76 0.75 34 Earthquake test at 20 Hz 0.77 0.76 35 Earthquake test at 22.5 Hz 0.77 0.77 36 Earthquake test at 25 Hz 0.77 0.77 The differences between the static and kinetic friction coefficients obtained from the same test are very small. No significant fluctuations in the friction properties of this surface were found between the tests. The average static friction and kinetic friction coefficients were determined as mean values of the third and fourth columns of Table 4.4 as 0.75 and 0.74, respectively. However, it can be observed that there was an obvious trend of the friction coefficients increasing as the testing progressed. The increase in the friction coefficients as a result of 70 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL surface changes during the trial tests had been taken into account in further analysis of the experimental data and in the numerical study to simulate some of the shake table tests. 4.5.3 Results of Tests with Harmonic Input The data measured during the harmonic tests was converted into a form suitable for analysis and some basic signal processing was performed. An example of signals after this analysis is given in Figure 4.15. showing shake table motions and selected response of the experimental model during one of the harmonic tests. Each of plotted quantities is identified above its graph. The plots cover a sequence of six test segments as these were conducted during the test. It can be observed from the plots how the sliding of the model increased with increasing amplitudes of the shake table motions. The plot of the simulated hydrostatic force indicates that it exhibited some minor fluctuations when the sliding of the model occurred, but these fluctuations were found small and they did not exceed 5% during any of the tests. Reduced results of the data obtained from the harmonic tests are given in Figure 4.16. This contains 9 plots of the measured Peak Table Accelerations (PTA) and Root Mean Square (RMS) acceleration versus Rate of Model Displacement (RMD), one plot for each testing frequency. The PTA was obtained as a maximum absolute value of a base acceleration record from every test. The RMS acceleration aRMS was calculated as a Root Mean Square of a vector containing positive and negative peaks of the base acceleration record: aRMS ' N JjZ\Peakj\2 (4.1) N 7=1 where N is the number of the peaks greater than 50% of the maximum absolute acceleration in 71 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL a) Shake Table Displacement -1 2 1 0 -1 -2 2 1 0 -1 -2 1.5 2 1.0 b) Shake Table Acceleration c) Model Base Acceleration 0.5 0.0 e) Displacement of Model Base I . 1 1 1 1 1 , 1 , 1 , 1 • 1 . 1 . 1 d) Simulated Hydrostatic Load I I I ! 1 1 1 , 1 . 0 10 20 30 40 50 60 70 80 90 Time (sec) Figure 4.15: Example of Signals Recorded during Harmonic Tests with Surface R l 72 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL a) 5 Hz b) 7.5 Hz c) 10 Hz 2.0 1.5 o 1.0 < "i—•—r 0.5 0.0 2.0 1.5 o 1.0 < 0.5 0.0 2.0 1.5 8 1.0 o < 0 5 10 15 20 RMD (mm/s) d) 12.5 Hz 0 5 10 15 20 RMD (mm/s) e) 15 Hz 0.5 0.0 1 ! 1 ! 1 n 0 5 10 15 20 RMD (mm/s) f) 17.5 Hz 2.0 o> 1.5 0 o 1.0 < 0.5 0.0 t •4 * Q) O O < 0 5 10 15 20 RMD (mm/s) g) 20 Hz 2.0 1.5 1.0 0.5 0.0 1 1 -2.0 ~ 1 5 h ro a5 1-0 h o o < 0 5 10 15 20 RMD (mm/s) h) 22.5 Hz 0.5 0.0 i 1 r~ 0 5 10 15 20 RMD (mm/s) i) 25 Hz 2.0 _ 1.5 3 1 1.0 < 0.5 0.0 — l ' r~ 2.0 1.5 8 1.0 o < 0 5 10 RMD (mm/s) 0.5 0.0 2.0 1.5 i i o o < 0.5 0.0 -10 -5 0 5 10 RMD (mm/s) + RMS PTA -10 -5 0 5 10 RMD (mm/s) Figure 4.16: Table Acceleration vs. Rate of Model Displacement for Harmonic Excitation 73 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL the signal and peakj is the j-th element of this vector. When calculating the value, the peaks lower than 50% of PTA were disregarded because it was assumed that these do not contribute to the sliding of the dam model. The concept of R M D was introduced from the following reason: During some tests, when the model was approaching the limit available for its sliding, the base excitations had to be suddenly terminated in order to protect certain parts of the model from permanent damage. As a result of this, there were several tests containing valuable information, but the tests were of different duration than the nominal. R M D was introduced to eliminate the duration of the test in further analysis of the data and it was calculated as a ratio of the measured displacement divided by the actual duration of the test. The RMS acceleration values are included together with the PTA values in Figure 4.16. Associating an R M D to a single measured PTA, which in some cases may be uncharacteristically high, may not be the best way to characterise the record. The RMS value may provide better information about the average levels of acceleration that resulted in sliding of the model. Further reduction and analysis of the data from harmonic tests is provided later in this section where these data are compared with that obtained from the earthquake tests. 4.5.4 Results of Tests wi th Synthetic Earthquake Input The tests with the earthquake records yielded significant amount of data, which were analysed and interpreted in several stages. This section describes basic reduction of these data. The records measured during the tests were converted to a format suitable for analysis, conditioned and visually checked for possible errors. An example of time histories describing 74 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL shake table motions simulating three synthetic earthquakes and response of the model due to these motions is shown in Figure 4.17. The plotted quantities are identified above each graph. Analysis of the data obtained during earthquake tests, to determine relationships between shake table accelerations and R M D values, was performed in a way similar to that for the harmonic tests. It was considered that simply taking the measured PTA could be misleading as it might be a single acceleration spike. Obviously this spike should not be the parameter or the effective table acceleration used to characterize the acceleration, which caused certain amount of displacement. To alleviate this problem, two versions of the table acceleration were used: the actual measured peak table acceleration (PTA); and 2) the RMS of the peaks in the acceleration time history. When calculating the RMS value, only those peaks greater than 50% of the PTA were considered, while the peaks lower than 50% of PTA were disregarded as in the case for harmonic excitations. Every test segment from the number of earthquake tests conducted was analysed to obtain the measured R M D corresponding to a given table acceleration (PTA or RMS). These results are given in Figures 4.18 to 4.20, which show the acceleration versus R M D for each earthquake and surface combinations. Similarly to that of the harmonic tests, further reduction and analysis of the data from earthquake tests is not performed independently. Instead, the combined information from Figures 4.17 to 4.20 is analysed in the next subsection in order to get direct comparison from tests with different base excitations. 75 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL a) Shake Table 0 10 20 Time (sec) 30 40 Figure 4.17: Example of Signals Recorded during Earthquake Tests with Surface RI 76 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL a) 5 Hz b) 7.5 Hz i 1 r 4 ++ + J , L_ 2.0 _ 1 . 5 3 8 LO < 0.5 0.0 1 1 r • ^ • ++ + J , I i L 2.0 _ 1 . 5 i LO 3 0.0 c)10Hz ~ i — 1 — i — • * " " + ++ J I I I l_ 0 5 10 15 20 RMD (mm/s) d) 12.5 Hz 0 5 10 15 20 RMD (mm/s) e) 15 Hz 0 5 10 15 20 RMD (mm/s) f) 17.5 Hz 2.0 g 1.5 t % 1.0 0.0 1 1 i 1 r ++ + + i • 3 2.0 1.5 1.0 0.5 0.0 i h J , i , L 2.0 1.5 3 "55 1.0 0 5 10 15 20 RMD (mm/s) g) 20 Hz 2.0 1.5 1.0 0.5 0.0 1 • ' ' ' • : i • 2.0 1.5 8 1 0 0.5 0.0 0 5 10 15 20 RMD (mm/s) h) 22.5 Hz 0-5 0.0 0 5 10 15 20 RMD (mm/s) i) 25 Hz 2.0 1.5 1.0 0.5 1 1 . . . . . . • -k + A i 0 5 10 15 20 RMD (mm/s) -10 -5 0 5 10 RMD (mm/s) + RMS A PTA -10 -5 0 5 10 RMD (mm/s) Figure 4.18: Table Acceleration vs. Rate of Model Displacement for Excitation EQ1 77 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL a) 5 Hz b) 7.5 Hz c) 10 Hz 2.0 _ 1.5 O) § 1 . 0 < 0.5 0.0 " i — 1 — r A * r 2.0 - 1 . 5 < 0.5 0.0 0 5 10 15 20 RMD (mm/s) d) 12.5 Hz 0 5 10 15 20 RMD (mm/s) e) 15 Hz 2.0 _ 1 . 5 3 t 1-0 o < 0.5 0.0 1 ! -A A A *-++ A + + + '4-T , 1 0 5 10 15 20 RMD (mm/s) f) 17.5 Hz 2.0 ra 1.5 Q) " 1.0 < 0.5 0.0 1 1 > i 1 + : r t i 2.0 1.5 1.0 8 o < 0.5 0 5 10 15 20 RMD (mm/s) g) 20 Hz 0.0 1 i 1 A - " -... A.: + + i 2.0 1.5 OT o5 1.0 o o < 0 5 10 15 20 RMD (mm/s) h) 22.5 Hz 0.0 1 1 A 4 A A A + + • 0 5 10 15 20 RMD (mm/s) i) 25 Hz 2.0 ~ 1.5 < 0.5 0.0 A A £4*--i i -10 -5 0 5 10 RMD (mm/s) 2.0 _ 1.5 S 3 1.0 o < 0.5 (• 0.0 AM. . . . + . . ; . . * . . +.A t *t, Acce • r i • 2.0 1.5 -10 -5 0 5 10 RMD (mm/s) 0.5 0.0 L ' A ...... A + A + . HA .. . J . . . 4 i + RMS A PTA -10 -5 0 5 10 RMD (mm/s) Figure 4.19: Table Acceleration vs. Rate of Model Displacement for Excitation EQ2 78 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL a) 5 Hz b) 7.5 Hz c) 10 Hz 2.0 1.5 1.0 0.5 0.0 A A A O) 0 5 10 15 20 RMD (mm/s) 2.0 1.5 1.0 0.5 0.0 ' I 1 I 1 1 p+ I . I . D ) 0 5 10 15 20 RMD (mm/s) e)15Hz 2.0 1-5 1.0 0.5 0.0 ! ' ! ' • A A ! • + ++ ---I I . 0 5 10 15 20 RMD (mm/s) f) 17.5 Hz 1 ! ' ! 1 : A / j i . i . 2.0 1.5 3 ^ 1 . 0 0.5 0 5 10 15 20 RMD (mm/s) g) 20 Hz 0 5 10 15 20 RMD (mm/s) h) 22.5 Hz 0.0 ~ 1 1 1 -i4 - i — i i 0 5 10 15 20 RMD (mm/s) i) 25 Hz 2.0 „ 1.5 3 § 1 0 < 0.5 0.0 i r > O) 0 5 10 RMD (mm/s) 2.0 1.5 1.0 0.5 0.0 1 1 1 A Jk + : A 4-A . . . i . i -10 -5 0 5 10 RMD (mm/s) 1.5 1.0 0.5 0.0 1 1 A ; + A ; A + ; i A + + RMS A PTA -10 -5 0 5 10 RMD (mm/s) Figure 4.20: Table Acceleration vs. Rate of Model Displacement for Excitation EQ3 79 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL 4.5.5 Comparison of Results from Harmonic and Earthquake Tests In the previous two subsections the data from the harmonic and earthquake tests was reduced into the form of R M D to PTA/RMS relationships and these were presented in Figures 4.16 and 4.18 to 4.20. Based in this information and notes and video tapes taken during the tests, the testing frequencies can be divided into three groups depending on the response of the model. These groups can be characterized as follows: • Group 1 - low frequencies: The model slid when the friction was overcome; no in-plane rocking, that is the rocking about an axis in cross canyon direction, was not observed and; resulting sliding of the model after the tests was always downstream - see Figures 4.16 a and b and 4.18 to 4.20 a and b. • Group 2 - medium frequencies: Sliding of the model was affected by flexural behaviour and rocking of the model. As a result of this the sliding of the model was initiated at lower amplitudes than it was for the frequencies from Group 1. The amounts of sliding measured for the frequencies from Group 2 were mostly larger than those for the same amplitude of base acceleration and frequency of excitation from Group 1 - compare Figure 4.16 e and f with a and b, or Figure 4.20 e and f with a and b. • Group 3 - high frequencies: The primary response of the model to the base excitation, which could be clearly observed and heard, was rocking. The resulting sliding of the model depended on both the frequency and amplitude of base excitation. Downstream, upstream as well as almost no motions were observed, however the upstream motions dominated especially for high amplitudes of base accelerations. In some cases, the model started to slide downstream, but at higher amplitudes of base acceleration it slid upstream - see Figure 4.18 or 4.19 h and i. Using the video tapes taken during the tests and from the visual observation of the CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL model's behaviour during the tests, it was concluded that the model, during the tests at frequencies from 20 to 25 Hz combined with certain amplitudes of base accelerations, was not in contact for short periods of time. If the model underwent such motions while the table moved downstream, the model moved slightly upstream relatively to the table. When such relative motions were cumulated they resulted in the residual upstream sliding of the model. Using the above grouping scheme applied on the results from harmonic and earthquake tests on surface RI the information given in Table 4.5 was developed. It can be concluded from this table that the bounds of Groups 1,2, and 3 for each of the four testing surfaces are very similar for all types of excitation. It seems that the effects of nonlinear phenomena, such as rocking, manifested themselves at slightly lower frequencies in case of harmonic excitations than in case of synthetic earthquakes. This can be explained because a single frequency excitation should generate any kind of resonance sooner than more transient synthetic earthquake motion. Table 4.5: Grouping Testing Frequencies According to Response of the Model Excitation Group 1 Group 2 Group 3 Harmonic 5 to 10 Hz 12.5 to 17.5 Hz 20 to 25 Hz EQ1 5 to 12.5 Hz 15 to 20 Hz 22.5 to 25 Hz EQ2 5 to 10 Hz 12.5 to 20 Hz 22.5 to 25 Hz EQ3 5 to 12.5 Hz 17.5 to 20 Hz 22.5 to 25 Hz It can also be observed from Table 4.5 that the groups are, except for a couple of minor irregularities, the same for all four types of excitation considered. Similar conclusions were made from the results of tests using the other surfaces, which are not shown here. This means that for different surfaces, the response of the model was controlled by the same phenomena and 81 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL it also means that the results of the tests were repeatable. It can be concluded from the comparison of plots in Figure 4.16 with the corresponding ones in Figures 4.18 to 4.20 that RMS to R M D relationships show a better agreement over the range of base excitations used than the PTA to R M D relationships. The RMS and PTA values are about the same for the harmonic excitations at each of the testing frequencies, but they vary significantly for all synthetic earthquake inputs. It appears that RMS is a better measure to use than PTA in characterising a base excitation regarding sliding response of a structure. The amount of sliding of such a structure does not depend only on intensity of a single acceleration pulse, but it is a function of the intensity and the number of all acceleration pulses of the considered earthquake. It was shown by the experiments that the model slid upstream at frequencies of base excitation from about 20 to 25 Hz. This is an important finding, but its practical significance should not be overestimated. The frequency range of 20 to 25 Hz on a model frequency scale represents a range of 5.2 to 6.5 Hz on a prototype 15-times larger and it is possible that the base excitation acting on a prototype during an earthquake would have significant components at and above these frequencies. However, it should be remembered that the experimental model used for this study did not simulate several capacity (e.g. cohesion) and demand (e.g. hydrodynamic pressure and uplift) components of a gravity dam-water-foundation system during an earthquake. To reach a conclusion from this upstream motion finding, without confirmation using a model including simulation of all the important phenomena, could be misleading. Such testing is beyond the scope of this study. 82 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL 4.6 SUMMARY OF EXPERIMENTAL WORK Two series of experiments were described in Chapters 3 and 4. These tests had several objectives, which were listed at the beginning of the chapters. In order to satisfy these objectives, an experimental model of a single monolith of a concrete gravity dam was designed and developed. Only a limited number of phenomena involved in the behaviour of a real dam-water-foundation system were modelled. In addition, it was understood that validity of all the tests is limited to the experimental model only and that it was not the objective of the tests to predict performance of any real concrete gravity dam. The model was subjected to a series of preliminary tests, during which important information leading to several enhancements of the model, experimental setup and testing procedures was collected. The second series consisted of the impact hammer, shake table static and shake table dynamic tests. The shake table dynamic tests included those with the harmonic and synthetic earthquake input. Analysis of the data obtained from this series, directly related to the objectives of this thesis, was performed in Chapter 4. This included reduction of data from all tests and in addition to this: • Natural frequencies of the experimental model were extracted from data from the impact hammer tests. These frequencies were summarised in Tables 4.2 for the bonded model and in Table 4.3 for the unbonded model. Different values in these tables indicate large influence of boundary conditions on the natural frequencies. Static and kinetic friction coefficients of the tested model-foundation interface were determined from the data obtained during the static shake table tests. These were summarised in Table 4.4. 83 CHAPTER 4 FURTHER EXPERIMENTAL STUDY OF THE DAM MODEL • Amounts of sliding of the experimental model due to harmonic or synthetic earthquake base excitations were determined from the data gathered during shake table dynamic tests. These results were presented in Figures 4.16 and 4.18 to 4.20. • Results of the shake table harmonic tests were compared with those using synthetic earthquakes. It was found that the model responded in a very similar way during both types of tests as can be seen from grouping the testing frequencies according to the model's behaviour during the tests. This is shown in Table 4.5. It should be mentioned that the amount of reported experimental data and the depth of the data analysis presented here is limited just to ensure the continuity of the studies described in this thesis. A more detailed analysis of the data gathered during the experimental study was presented by Black et al. (1998). Some of the data obtained during the experiments will be used in the next section for verification and calibration of numerical models. 84 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL A series of experiments on a scaled model of a gravity dam monolith was described in the previous chapter. The development of three numerical models to simulate the tests is described in this chapter. A large amount of data collected during the tests is used in this chapter to verify and calibrate numerical models to simulate the shake table tests. The main objective of the analytical study was to develop a simple numerical model to simulate sliding of a rigid block on a rigid foundation, preloaded by a constant horizontal force and subjected to base excitations. The requirement of the model's simplicity came from one of the intended applications of the model. This was its use in reliability analysis, which typically involves a large number of simulations and therefore, a fast analysis procedure is preferable. Another objective of this chapter was to develop a numerical model more complex than the rigid block model, and to use it to simulate some of the shake table tests conducted during the experimental part of the study. Purpose of this analysis was to find out how closely could the numerical models of varied complexity simulate the response of the experimental model measured during the shake table tests. 5.1 DEVELOPMENT OF NUMERICAL MODELS A total of three numerical models were developed during this study. They varied in their complexity and consequently, in their ability to simulate desired phenomena accurately. In addition, the models varied in computational effort necessary for a single simulation. The following numerical models were developed: 85 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL • Single Degree Of Freedom model - SDOF model; • 3-Degree of Freedom model using software Working Model (Knowledge Revolution, 1996) - 3DOF model; and model using finite element software A N S Y S (SAS IP, 1996) - FE model. 5.1.1 SDOF Numerical Model The SDOF model, shown in Figure 5.1, consisted of a block with mass m resting on a rigid foundation. The block was preloaded by a horizontal force F and subjected to a base motions described by a function z(t). In the analysis of the system, it was convenient to describe motions of the block in terms of its absolute displacement function y(t). The solution to the problem was based on the following assumptions: • The block and the foundation are rigid. • Block is constrained to a single horizontal degree of freedom. • Frictional contact between the block and the foundation is assumed, with static and kinetic friction coefficients. The static friction coefficient is applied i f the relative velocity of the block with respect to the foundation is smaller than a specified small number and the kinetic friction coefficient is applied i f the relative velocity is larger or equal to that number. • The block and the base are always in contact, which means that any jumping or rocking motions of the model is not considered. • The horizontal force F is constant and it is smaller than the friction force, which could be transferred by the block-base interface. This force can be expressed as 86 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL mg\i, where m is the mass of the block, p is the friction coefficient, and g is gravity acceleration due to gravity. No cohesion between the block and the foundation is considered. Reference Plane y(t) Block Reference Plane Base Figure 5.1: Schematic of the SDOF Model A closed form solution to a simplified problem was presented by Westermo and Udwadia (1983). They showed solution to a simplified problem without the horizontal force F and considering base motions limited to harmonic in nature. In this study, the solution of Westermo and Udwadia was extended for the case when the horizontal force F is acting on the block and the base excitation is of a general character. This is presented below. It is assumed that the block does not slide at the beginning of the simulation (stick mode). In such a case, the absolute acceleration of the block y(t) is the same as the base acceleration: Rt) = ho (5.i) In the stick mode, the velocities y(t) and z(t) are equal, which means that the relative displacement of the block with respect to the base does not change. The block remains in stick mode until the time when the resultant of the inertia force my(t) and the force F exceeds the 87 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL friction force mg\i. In such a time, sliding of the block is initiated (slip mode) and its motion is controlled by the equation obtained from the equilibrium of the horizontal forces on the block: \y(t)m\ = ±mg\i-F (5.2) Depending on what is the sign of y(t) when the sliding was initiated (at time t0): y\t) = p g - £ if y\t0)>0,and (5.3) y i t ) = - p g - £ i f ;Kf0)<0. (5.4) After the sliding is initiated, the absolute acceleration of the block at any time is given by one of the Eqs (5.3) or (5.4). According to these, y(t) is a constant, which after integrating with respect to time, leads to a linear function for absolute velocity y(t) and a quadratic function for the absolute displacement y(t) of the block. The base velocity z(t) and displacement z(t) have known values at any time. Thus, the relative velocity yr{t) and the relative displacement (sliding) yr(t) can be calculated at any time: yM = An-kO ,and (5.5) yr(t) = y(t)-z(0 (5.6) The sliding will terminate, when the relative velocity of the block yr(t) is equal to zero. At such a time, the block will go to the stick mode. In this mode, the absolute motions of the block will equal to the motions of the base. For base excitations with cyclic character, the above described cycle typically repeats several time as the block is subjected to the base motions. The solution described above implemented into a computer program, which was tested for the block with the following parameters. The mass of the block was 480.6 kg, which corresponded 88 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL to the mass of the experimental dam model used in this study. The static friction coefficient was 0.77 and the kinetic friction coefficient 0.74, which corresponded to the coefficients of the Smooth surface from the preliminary experimental study, as can be seen in Table 7 in Horyna et al. (1997). The magnitude of the force Fwas 1.7 kN, which corresponded to one of the values of this force used during the preliminary experiments. The block was subjected to a 5 Hz harmonic excitation, 1 second long. Four acceleration amplitudes were used for this study: OAg, 0.6g, l.lg, and 1.4g. The solutions obtained using the SDOF model for these four cases are shown in Figures 5.2 to 5.5. Each figure contains four plots: acceleration, velocity, displacement, and sliding. The base motions are plotted using dotted lines, the model absolute motions using solid lines and the model relative motions with dashed lines. The accelerations in Figure 5.2 indicate that sliding of the block was not initiated for OAg harmonic excitations at 5 Hz. The response of the block to the 0.6g harmonic excitations is shown in Figure 5.3. In this case, sliding of the block was initiated only during the negative pulses of base acceleration, when the sense of the inertia and static forces on the model were the same. The response of the block to excitations with amplitudes of 1 Ag, see Figure 5.4, shows that sliding of the model was initiated in both cases, during the positive and the negative parts of the cycle of base acceleration. Figure 5.5 presents the results of SDOF analysis for the excitation with amplitudes of 1 Ag. In this case the sliding of the model was initiated during the first pulse and it did not stop until the end of the excitation. This was due to the fact that the relative velocity of the block with respect to the base never reached zero at the same time when the demand on the system given by combination of the inertia and simulated hydrostatic forces was smaller than the frictional capacity of the system. 89 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL 0.001 •fr 0.000 c -0.001 Block 4E-3 Base Block 2E-1 Block (relative) 5E-1 1.0 time (s) Block Base Figure 5.2: Performance of SDOF Model - Harmonic Excitations at 5 Hz and 0.4g 90 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL 0.010 cn c t75 _ / 0.000 A - Block 2E-2 | 0E-+O 8 ro o. w i5 -2E-2 Base Block 2E-1 Block (relative) 1E40 0.0 0.5 1.0 time (s) 1.5 Block Base 2.0 Figure 5.3: Performance of SDOF Model - Harmonic Excitations at 5 Hz and 0.6g 91 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL 0.200 c jo 0.000 Block 2r>i Base Block 4E-1 Base Block Block (relative) 2 E 4 0 3 •I OE40 0 -2E-K) 0.0 0.5 1.0 time (s) 1.5 2.0 Block Base Figure 5.4: Performance of SDOF Model - Harmonic Excitations at 5 Hz and l . l g 92 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL 0.400-, c T3 0.000 Block 4E-1 OE40 -4E-1 Base Block Figure 5.5: Performance of SDOF Model - Harmonic Excitations at 5 Hz and 1.4g 93 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL It can be concluded that the SDOF model can simulate sliding of a rigid block preloaded by a constant horizontal force due to base excitations. The level of accuracy of the results using this model will be discussed in the next section. 5.1.2 3DOF Numerical Model Working Model (WM) is a tool for engineering simulation. It was developed by Knowledge Revolution of San Mateo, California, USA. For this study, a 2D version 4.0 was used (Knowledge Revolution, 1996). The program's graphical user interface allows the user to define a set of rigid bodies and constraints and simulate its behaviour using a Newtonian mechanics simulation engine. The 3DOF numerical model included the shake table, the dam monolith model and an actuator to simulate the hydrostatic load on the model. The shake table was constrained to move in the horizontal direction only and it was powered by an acceleration controlled actuator. The horizontal actuator simulating the hydrostatic force on the model, was force controlled. It was stretched between the downstream side of the dam model and a support fixed to the shake table. The model of the dam monolith had three degrees of freedom, allowing the centre of gravity of the model to move horizontally and vertically as well as rotate. This means that the 3DOF model could account for sliding and rocking of the model and the frictional contact between the model and its base may not exist at certain times of the numerical simulation. The solution to the system of three equations of motion, corresponding to the three degrees of freedom, was obtained using the WM's simulation engine, using a Kutta-Merson (5th-order Runge-Kutta) time integration scheme in order to increase accuracy of the solution. 94 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL a) Harmonic Excitation at Frequency 5 Hz 1.0 0.8 0.6 < OH 0.4 0.2 0.0 - ; n i ! I - i 1 ; ; ! -i i • 3DOF + SDOF 5 10 15 20 25 30 RMD (mm/sec) b) Harmonic Excitation at Frequency 20 Hz 1.0 0.8 0.6 < CL, 0.4 0.2 0.0 i l ~ i ! | ! . 3DOF + SDOF 0 5 10 15 20 25 30 RMD (mm/s) Figure 5.6: Comparison of Results from 3DOF and SDOF Models for Harmonic Excitations The 3DOF model was developed before the results from the further experiments were available and therefore it was tested against model's parameters corresponding to those used in the preliminary tests (Horyna et. al, 1997), which were also used for the SDOF simulations presented in the previous subsection. The analyses were conducted for harmonic excitations with a duration of 10 seconds. The frequency of excitations of 5 Hz and 20 Hz were used. The excitation was generated in a closed form, using harmonic functions built in the program. The results were reduced to PTA vs R M D plots, which are presented in Figure 5.6. The frequency 95 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL of excitations is indicated at the upper-right corner of each plot. The results from 3DOF and SDOF simulation are identified with legends on the right side of the plots. The results show a very good match between the SDOF and the 3DOF models. The sliding of the block was initiated at the same acceleration level of about 0.4 lg for both frequencies. This is in agreement with assumption that the acceleration to initiate sliding does not depend on the frequency of excitation for rigid body models. It can be concluded that both SDOF and 3DOF numerical models predicted the same amount of sliding of the block. 5.1.3 Finite Element Model 5.1.3.1 Previous FE Modelling at UBC At the same time when the further experiments were under way, a study to simulate the shake table tests using a finite element model was conducted (Rudolf, 1998). The numerical model to simulate the shake table tests was developed during this study using a commercial program ANSYS 5.3 Multiphysics/University (SAS IP, 1996). This model is called the original model in this section. The original model was used to simulate some of the tests and it performed satisfactorily especially at low frequencies of base excitation. The phenomena observed during the tests at higher frequencies, such as combined sliding and rocking, could be simulated with the original model but they did not manifest themselves under the same conditions as it was during the tests. Therefore, the study (Rudolf, 1998) recommended several enhancements to the FE model. These were done by the author and are described in this section. 96 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL 5.1.3.2 ANSYS 5.3 Multiphysics/University ANSYS is a general finite element program with a long history of use for many applications. It is commercially available and has an extensive library of elements and numerous solution options described in detail in the program documentation. The A N S Y S modelling gives the analyst wide control over the input, solution, and output. Data output and postprocessing can be a crucial consideration in studies like this, when many simulations are performed and the amount of generated data is large. The advantage of using ANSYS is that this process can be easily controlled by batch files which can run for several days performing series of analyses. The output data can be partially post-processed and later used as input for other software. The graphical interface allows the user to generate custom plots of the deformed shapes and stress contours at any time step. This option is very valuable during the process of building and debugging the model, which is described next. 5.1.3.3 Description of the Model The experimental setup was modelled as a two-dimensional solid. The model is shown in Figure 5.7. The dam monolith, the Upper Surface Plate (USP), and the shake table were modelled using a simple plane element. The Lower Surface Plate (LSP) was integrated into the shake table to form a single thick concrete plate, called the shake table here. The combined flexibility of the physical shake table, LSP, and the clamping devices between the model and the USP, was modelled through the contact stiffnesses of the contact elements. The contacts between the monolith and the LSP were modelled using a point-to-surface contact element. The elements are described in the next subsection. 97 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL 200 CONCRETE | ^ 1800 J Figure 5.7: Schematic of the FE Model 5.1.3.4 Plane Element The bilinear plane element PLANE42 (SAS IP, 1996a) was used, with the plane-stress option. The geometry of the element is defined by four nodes each with two translational degrees of freedom. The element input further includes thickness and orthotropic material properties. The 98 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL input properties were obtained from the experimental model, which was described in Chapter 3. A total of 64 plane elements were used to describe the geometry. 5.1.3.5 Contact Element The Contact48 2-D Point-To-Surface Contact Element (SAS IP, 1996a) was used to model the friction interface between the block and the LSP (shake table). It is capable of representing contact and sliding of a point to surface in two dimensions. The element consists of three nodes: the contact node and two nodes creating the target surface. In this application, the contact function was assigned to the nodes on the contact projections of the block (heel & toe) and the target function to the nodes on the top surface of the shake table. The algorithm for contact and sliding can be summarized as follows: • Start from a no-contact position, the contact node approaches the target surface. • Contact is made when the contact node penetrates the target surface. • During the contact, the reaction-displacement relationship is governed by two linear stiffnesses: one in the direction normal to the target surface and the other tangential to the target surface. These stiffnesses are input by the analyst. • For subsequent solution, the sticking force is equal to the normal reaction force on the surface times the static coefficient of friction. • If the tangential reaction exceeds the sticking force in the course of contact, friction sliding is initiated with the friction force equal to the normal force times the kinetic coefficient of friction. • As the contact node departs from the target surface, the reactions drop to zero and 99 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL a gap is developed. The input parameters for the element included the normal and tangential stiffnesses entered, respectively, as 20,000 kN/m and 800,000 kN/m. These values were calibrated to match the natural frequencies of the model. The other parameters for the element were friction coefficients, entered as those measured during the static tests. The model had two contact elements on its toe and two on the heel. The shake table top surface included five target surfaces on the downstream side and five on the upstream side. A total 20 of contact elements were used. 5.2 CALIBRATION OF THE FE MODEL 5.2.1 Modifications of the Original FE Model The FE model for the experimental setup, developed by Rudolf (1998), had several known deviations from the physical model. The author tried to remove as many of these as possible. The modifications incorporated in the model included: • The shape of the FE model was modified so that it was closer to that of the experimental setup. • The mass density of the model's material was slightly changed so that the model had the same total mass as that measured during the tests. • The modulus of elasticity of the monolith was slightly updated to match the natural frequencies of the bonded experimental model. • The number of elements of the model was decreased. The reason was that in this study a large number of the analyses needed to be performed and the computational time using the original mesh would not be acceptable. The number of the plane elements decreased from 140 to 64 and contact elements from 42 to 20. After this 100 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL reduction a trial simulation was run with both the original and the reduced models with a very good agreement of sliding. • The normal and tangential springs of the contact elements were updated to match the natural frequencies of the unbonded experimental model. • Damping expressed by stiffness and mass proportional parameters was slightly changed. This change was accepted based on the comparison of the model's response during no-slip tests with the FE simulations. This comparison is described in Appendix A and as a result of it the damping was changed from 5% of critical in the fist two natural modes to 3.5% in the first and the third modes. • Other modifications, such as those to analysis procedures, are described here. 5.2.2 Modal Analysis Modal analysis was performed in order to obtain natural frequencies and modes of the experimental model and compare these with the measured values. In addition, this analysis was one of the tools to calibrate the FE model. The results of this analysis were used to calibrate the stiffness properties of the FE model to match the natural frequencies obtained from the impact hammer testing described in Chapter 4. Two variations of the experimental model approximated by finite elements were subjected to modal analysis: 1) model bonded at the toe and the heel, and 2) unbonded model. The method used to extract modal shapes and corresponding frequencies was the Subspace Iteration Method described in SAS IP (1996b). In modal analysis, the contact elements acted as linear springs. This means that the gap feature of Contact48 element was suppressed and classical problem of linear free vibrations was solved to calculate natural frequencies of the unbonded setup. 101 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL The experimental and calculated natural frequencies of the bonded model are presented in Table 5.1. Two of the four calculated natural frequencies, for which a corresponding experimental value was found, are in a very good agreement with the experimental values. The other two correspond to the natural modes, which were not captured during the impact hammer tests. Figures 5.8 to 5.11 show the FE model for the experimental setup bonded to the base and the first three modes of vibration of the bonded setup. In order to clearly visualize the character of every mode, the deflection shapes in the figures were exaggerated. Table 5.1: Natural Frequencies of the Experimental Model Bonded to the Base No. Experimental Frequency (Hz) Frequency from FE Model (Hz) Character of Natural Mode 1 66 65.8 the 1st cantilever mode, see Figure 5.9 2 not identified 115.4 vertical bending, see Figure 5.10 3 174 173.5 the 2nd cantilever mode, see Figure 5.11 4 not identified 271.7 higher cantilever bending mode Table 5.2: Natural Frequencies of the Unbonded Experimental Model No. Experimental Frequency (Hz) Frequency from FE Model (Hz) Character of Natural Mode 1 27-34 29.7 rocking of the model due to deflections in the vertical springs combined with the 1 st cantilever bending mode, see Figure 5.13 (springs not shown in the figure) 2 53-63 48.9 up/down stretching and shortening of the vertical springs combined with the 1st cantilever bending mode, see Figure 5.14 (springs not shown in the figure) 3 140 143.3 the 2nd cantilever bending mode, see Figure 5.15 4 not identified 226.7 higher cantilever bending mode 102 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL Figure 5.9: First Natural Mode of Experimental Model Bonded to Base; f=65.8 Hz 103 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL H at t i 8 3 3H 4 1 TU 5 n— 6 rz— X' 7 i 76 7» so SI 52 53 54 SO 61 62 63 64 45 k—^ 46 47 r — * 48 i s — « i 49 55 56 5—^ 57 r — ^ 56 59 Figure 5.12: FE Model for Unbonded Experimental Model of Dam Monolith Figure 5.13: First Natural Mode of Unbonded Experimental Model; f=29.7 Hz 105 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL The modal analysis of the unbonded model was performed and its results are presented in Table 5.2. The measured frequencies varied with slightly from test to test. These variations were attributed to nonlinear effects such as rocking or gapping present during the impact hammer tests with the unbonded setup. During the modal analysis the vertical springs in the contact elements were calibrated so that the first calculated natural frequency was very close to the measured frequency. The resulting agreement between the other calculated and measured frequencies was found acceptable. While the first and the third natural frequencies were matched satisfactorily, the calculated frequency corresponding to the second mode of vibration was lower than the measured one. However, the difference of about 15% was considered acceptable because the mode has mostly vertical coordinates and the frequency is high. 5.3 RESULTS FROM SIMULATIONS OF SHAKE TABLE TESTS The numerical models developed in the previous section were tested against the results from selected shake table tests. This verification of the numerical models was limited to the tests with harmonic and EQ2 (PSD shape from the Northridge earthquake record) base excitations. In addition, only the tests with dominant frequencies 5, 10, 15, 20 and 25 Hz were used and only five tests for each combination of the base excitation and dominant frequency were simulated. 5.3.1 Parameters of Simulations The simulations with all three numerical models were performed using the parameters of the experimental setup from the further testing program. The simulated hydrostatic force on the model was obtained from the experimental data, which assured that any fluctuations of this force during the tests were reproduced in the simulations. The friction coefficients were considered 107 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL with the values obtained prior to the test being simulated. Base motions were inputted to all models in a form of shake table base accelerations measured during the test being simulated. The FE simulations of the selected shake table tests with harmonic excitations performed by Rudolf (1998) were done using a closed-form harmonic base displacements as driving signals for the numerical model of the shake table. However, in the current FE simulations all the base motions were inputted as measured shake table accelerations. The reason for switching from closed-form to measured base motions was that the shake table did not reproduce exactly the inputted closed-form driving signals during the tests. As a result of this, if the closed-form signals were used, the simulated base motions would not be exactly the same as those which excited the experimental setup during the tests. The reason for switching from the displacement driven to acceleration driven numerical model was that the displacement records measured during the tests at high frequencies (20 and 25 Hz) were not recorded properly by the displacement sensor built in the shake table because of very small amplitudes of the tests at high frequencies. The base acceleration records inputted to the 3DOF simulations had to be resampled to 66% of the sampling rate of those for FE and SDOF simulations. This was because of the limitation, which the software Working Model puts on the number of data points imported during the analysis. As a result of this, some peaks, especially at the records with higher dominant frequencies, were lowered or skipped. This resulted in overall lower amounts of sliding from 3DOF simulations compared to those with SDOF model. Both models yielded the same amounts of sliding when closed-form excitation was used, as shown in Figure 5.6. Therefore, resampling of the input records was the true reason, why SDOF and 3DOF simulations did not 108 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL yield almost identical results during simulations, which will be shown in Figures 5.16 and 5.17. 5.3.2 Comparison of Amounts of Sliding The comparison of measured and simulated amounts of sliding is provided in Figure 5.16 and 5.17, respectively, for the tests with harmonic and synthetic earthquake (EQ2) excitations. Both figures contain five plots, one for each dominant frequency of base excitations. The plots contain four sets of displacement (sliding) to PTA relationships and legend at the lower-right corner is common for all plots in both figures. These relationships were measured or calculated only at five points in each plot and the lines between the points from the same source (experiment, FE model, SDOF model or 3 DOF model) represent only trends reached by each model. The next two paragraphs contain comments on comparison of simulation results with experimental values. 5.3.2.1 Simulations of Harmonic Tests All comments in this paragraph are related to Figure 5.16 and all simulated values are compared with the experimental displacements (sliding). It can be observed from Figure 5.16a that the measured response of the experimental setup at 5 Hz was very well simulated by all numerical models. The situation at 10 Hz (Figure 5.16b) is different. Here, the FE model yielded lower displacements for lower PGA's but larger for higher PGA's. The SDOF and 3DOF results are lower than the experimental. A good agreement between the numerical and experimental results can be observed for the first three simulations at 15 Hz (Figure 5.16c). However, for two high PGA's at this frequency the FE and experimental results are about three times larger than those from SDOF and 3DOF simulations. The agreement of FE and experimental displacements is 109 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL a) Dominant Frequency: 5 Hz b) Dominant Frequency: 10 Hz i 1 r 0.3 0.4 0.5 0.6 0.7 PTA (g) c) Dominant Frequency: 15 Hz i 1 r 0.3 0.4 0.5 0.6 0.7 PTA (g) e) Dominant Frequency: 25 Hz i—1—r 0.4 0.5 0.6 PTA (g) d) Dominant Frequency: 20 Hz 0.4 0.5 PTA (g) Legend for all plots: — Experiment —O— FE Model • SDOF Model A 3DOF Model 0. 0.4 0.5 PTA (g) Figure 5.16: Comparison of Sliding from Tests and Simulations for Harmonic Excitations 110 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL good at 20 Hz (Figure 5.16d), but the SDOF and 3DOF results are lower. However, the trends are the same for all sets of results. The situation is different at 25 Hz (Figure 5.16e) where a very good match between FE and experimental values can be observed. The sliding at first increases with increasing PGA, but for higher PGA's the displacements do not grow. This was caused by dominant in-plane rocking character of the experimental model's response. The SDOF and 3DOF models could not capture this and, consequently, their results do not follow the same trend as the first two. 5.3.2.2 Simulations of Synthetic Earthquake (EQ2) Tests Similarly to the previous paragraph, all simulated results in this paragraph are compared with the experimental displacements (sliding). A l l comments provided here are related to the information presented in Figure 5.17 coming from the shake table tests and simulations with the EQ2 base excitations. The agreement of all numerical results with those from experiments is very good at 5 Hz (Figure 5.17a). Only the FE based displacements are somewhat larger than the rest. The agreement of all results is very good at 10 Hz (Figure 5.17b) even though the SDOF and 3DOF values are lower especially at higher PGA's . The results for 15 Hz (Figure 5.17c) and 20 Hz (Figure 5.17d) are very similar. For these, the experimental and FE displacements match very well, but SDOF and 3DOF simulations resulted in approximately 50% sliding amounts compared to the first two. However, the trends among all results are similar. The results for 25 Hz (Figure 5.17e) show a good agreement between finite elements and experiments, but the SDOF and 3DOF do not exhibit even similar trends. 5.3.2.3 Summary of Comparisons Several conclusions can be drawn from the comparison of measured and calculated amounts of 111 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL a) Dominant Frequency: 5 Hz b) Dominant Frequency: 10 Hz -i i | i | i r 0.6 0.8 1.0 1.2 1.4 1.6 1.8 PTA(g) 15 c) Dominant Frequency: 15 Hz 0.6 0.8 1.0 1.2 1.4 1.6 1.8 PTA (g) , e) Dominant Frequency: 25 Hz TO | 0.6 0.8 1.0 1.2 1.4 1.6 1. PTA (g) 1 c d) Dominant Frequency: 20 Hz 0.6 0.8 1.0 1.2 1.4 1.6 1. PTA (g) Legend for all plots: £ — Experiment -0— FE Model - Q — SDOF Model 3DOF Model 1.0 1.2 1.4 PTA (g) Figure 5.17: Comparison of Sliding from Tests and Simulations for EQ2 Excitations 112 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL sliding of the experimental model during the shake table tests and their simulations. The agreement of all numerical models with the experiment is generally good for frequencies from 5 to 10 Hz. At the frequencies 15 and 20 Hz, the FE model could capture well the behaviour of the experimental setup, while the SDOF and 3DOF models yielded smaller amounts of sliding. However, the trends between the results from the rigid models followed similar trends as those of the experiments and FE simulations. Comparisons at 25 Hz showed that the SDOF and 3DOF results did not follow similar trends. The above generalisation of performance of the three numerical models permits the conclusion: • The response at the frequencies from 5 to 10 Hz, that is about the frequencies from Group 1, which was defined in Chapter 4, can be simulated satisfactorily with all numerical models. • The response at the frequencies of 15 an 20 Hz, that is about the frequencies from Group 2, can be simulated using the FE model. The SDOF and 3DOF models can capture only certain trends in the response. • The response at the frequency of 25 Hz, that is about Group 3, can be simulated using the FE model only. The SDOF and 3DOF models do not simulate response of the model satisfactorily. It can be concluded that the SDOF and the FE models give similar answers as far as the dominant frequency of the base excitation is no more than about 50% of the first natural frequency of the unbonded model. For greater excitation frequencies, the disagreement increases and, in fact, Figure 4.19 shows that at 22.5 Hz and 25 Hz, coupled with larger PTA, 113 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL the experiment showed upstream movement. The SDOF model cannot predict this upstream motion, although the FE could, as shown by Rudolf (1998). Rudolf (1998) showed that the simulated behaviour during the tests at high frequencies and with strong base excitations included significant rocking and upstream sliding of the model. The response simulated by FE model was found to be in an acceptable agreement with the measured and observed response during the tests. The FE model used in this thesis was a modification of the original FE model developed by Rudolf. Due to this and since the results from the lower acceleration level FE simulations presented here matched satisfactorily those from the tests it was assumed that the FE model used here would capture well the rocking and upstream sliding phenomena studied by Rudolf (1998). It follows from the above that the FE model could simulate the response of the experimental setup over wide range of excitation frequencies and PGA's. The range where the rigid models can be used is not as wide and it is limited to the frequencies of base excitations below 15 Hz. No significant improvement in the results was found if the 3DOF model, that is 3-DOF model capable of rocking and jumping, was used compared to the SDOF model. Using the same Pentium II 300 MHz personal computer, one simulation using the FE model took approximately 2.5 hours, with 3DOF model about 6 minutes and with SDOF model about 5 seconds. It can be concluded that the SDOF model can be satisfactorily used for the simulations with the dominant frequencies of the base excitations below 15 Hz. Above these frequencies, the FE model is recommended. The 3DOF did not prove any advantage against either of the other two models. Therefore, only the FE and SDOF models will be used in the 114 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL next chapter to simulate response of a single monolith of a concrete gravity dam. The 3DOF model, which was found useful during the design of the experimental setup, will not be used for any further simulations. A n analysis to find out why the 3DOF model did not simulate rocking of the experimental setup is described in the next section. 5.4 ANALYSIS OF SLIDING/ROCKING OF 3DOF MODEL It was observed during the numerical simulations with the 3DOF and SDOF models that, in contrast with the experiments, none of the numerical simulations yielded upstream sliding of the model. In addition, the 3DOF numerical model, which was capable of simulating rocking, did not show rocking of the model in any of the numerical simulations performed. In an attempt to explain these observations, the following analysis was done. The case of the model resting on the base, as shown in Figure 5.18, is considered. The forces acting on the model include: weight of the model W=mg, where g is acceleration due to gravity and m is mass of the model; • simulated hydrostatic force F; • horizontal inertia force F I N ; • horizontal friction force acting along the foundation interface. A l l the dimensions of the model shown in Figure 5.18 and its mass are considered to be fixed. However, because it was observed that the simulated hydrostatics force changed during all shake table tests, this is given a value from the range 980 N to 1050 N . Also, the friction 115 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL Figure 5.18: Kinematics of the Dam Monolith Model coefficient p of the foundation interface changed as different surfaces were used during the tests and therefore the friction coefficient is considered in the range 0.63 to 0.78. The above two ranges yield four possible combinations of these two input parameters: 1) p = 0.63 andF=980N,2) u = 0.63 andF= 1050 N , 3) p = 0.78 and F = 980 N , and 4) p = 0.78 and F= 1050 N . Two cases will be analysed: a) the inertia force Fin is acting downstream and; b) the inertia force Fin is acting upstream. Case a): If the force Fin is acting downstream, the model can slide downstream or rock about B: • Sliding downstream is governed by the equilibrium equation: 116 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL Fin + F = m\ig or mas+.F = m\ig, (5.7) F which leads to: as = u . g - - (5.8) where as is the acceleration to initiate sliding. For combinations 1) to 4), as has the values: 1) as = 0.42g ;2) as = 0.41g ;3) as = 0.57g \ 4) as = 0.56g • Initiation of rocking about point B is governed by the equilibrium equation: Finy + Fh = mgx' or mary + Fh = mgx' (5.9) mgx' - Fh which leads to: a = (5.10) ' m y v ' where ar is the acceleration to initiate rocking about point B. For combinations 1) to 4), ar has the following values: 1) and 3) ar = 1.17g ; 2) and 4) = 1.16g . It follows from a comparison of as and ar for all the four combinations that ar is always significantly higher than as . This means that for any of the considered combinations the downstream sliding is always initiated before rocking about point B could be initiated. Case b): If the inertia force Fin is acting upstream, the model can slide upstream or rock about A . • Sliding upstream is governed by the equilibrium equation: Fin-F = m\ig or mas-F = m\ig , (5.11) F which leads to: as = \ig+- (5.12) where as is the acceleration to initiate sliding. For combinations 1) to 4), as has the values: 1) as = 0.84g ;2) as = 0.85g ; 3) as = 0.99g ; 4) as = l.OOg • Initiation of rocking about point A is governed by the equilibrium equation: 117 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL Finy - Hh = mgx or mary - Hh = mgx (5.13) which leads to: a r ~ mgx + Fh my (5.14) where ar is the acceleration to initiate rocking about point A . For combinations 1) to 4), ar has the following values: 1) and 3) ar = 1.05g ; 2) and 4) ar = 1.07g . It follows from a comparison of as and ar for all the four combinations that ar is always higher than as , which means that sliding downstream will be initiated before rocking about point B could be initiated. However, i f one closely compares values for combination 4), which correspond to a high value of friction coefficient, it is obvious that sliding upstream and rocking about point A could be initiated because the acceleration to initiate rocking ar = 1.07g is It follows from the above analysis that for the ideal case of a rigid body numerical model with no imperfections, the rocking about A should not be initiated. However, in the experimental analysis dealing with an imperfect and flexible model, or in FE analysis dealing with a flexible model, one can conclude that rocking of the model can be initiated by a pulse of upstream inertia force. The results of the FE simulations lead to the same conclusion. A total of three numerical models to simulate the shake table tests were developed in this chapter. These included: relatively close to that initiate sliding as = 1.00g. 5.5 SUMMARY Single Degree Of Freedom model - SDOF model; 3DOF model using commercial software Working Model - 3DOF model; and 118 CHAPTER 5 ANALYTICAL STUDY OF THE DAM MONOLITH MODEL • model using finite element commercial software A N S Y S - FE model. Parameters of the rigid models, SDOF and 3DOF, were specified according to the characteristics of the experimental setup. Parameters of the FE model were specified similarly, but some additional parameters of this model were obtained from calibration of this model using the impact hammer and no-slip shake table tests. The numerical models were used to simulate selected shake table tests including sliding. The FE model, which is computationally demanding, simulated satisfactorily the majority of the tests over a considered range of the dominant frequencies of base excitations. The SDOF and 3DOF models were successful in the lower range up to 15 Hz. They did not work satisfactorily at higher frequencies. Performance of these two models was about the same and because the SDOF model is not as demanding on computational effort as the 3DOF model, the SDOF model can be recommended for use at the lower frequencies, below 15 Hz. A l l the conclusions given above were related to the unbonded experimental setup with the first natural frequency of about 30 Hz. The SDOF model performed satisfactorily for the frequencies up to 15 Hz, which is a half of the first natural frequency. If an unbonded structure of the same shape, but different dimensions, is to be analysed using the developed numerical models, it can be expected that the SDOF model will yield good results for simulations with the dominant frequencies below one half of the first natural frequency of the structure. This, however, should be verified with some simulations using the FE model. 119 6 ANALYTICAL STUDY OF A FULL -SCALE DAM MONOLITH A n analytical study to simulate the response of the experimental model of a concrete gravity dam monolith was described in the previous chapter. Sliding of the experimental model, unbonded at the base, preloaded by the simulated hydrostatic force, and subjected to base excitations, was obtained using three numerical models. Out of these three, the FE and SDOF models, were selected for another series of numerical simulations. Concrete gravity dams are typically constructed in blocks separated by vertical contraction joints, which may or may not be keyed or grouted. The design and analysis of straight concrete gravity dams, except for those in narrow valleys, is traditionally performed by assuming that each block responds independently. For this reason, understanding 2-D behaviour of individual monoliths is usually considered relevant and 2-D plane-stress models are usually employed to estimate safety for stability and safety. In addition, i f non-linear phenomena are being studied, 2-D models are usually employed in order to reduce computational effort. This chapter describes results from a series of numerical simulations, performed on a single monolith of a full-size 45 m concrete gravity dam. 6.1 OBJECTIVES AND SCOPE OF THE ANALYTICAL STUDY ON A FULL-SCALE MODEL The objectives of this chapter were: • to modify the existing FE and SDOF models to simulate behaviour of the full-scale concrete gravity dam monolith; • to obtain modal characteristics of the full-scale monolith modelled by FE; • to perform a number of FE and SDOF simulations to calculate the response of the full-scale monolith with varying water level of the reservoir, type and peak ground 120 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH acceleration of the base excitation record; • to compare results of numerical simulations obtained in terms of the dam monolith's sliding using the FE model with those using the SDOF model. The analyses in this chapter were limited to those with the FE or SDOF models of a single monolith of a gravity dam. The loads simulated on the monolith included gravity, inertia, hydrostatic and hydrodynamic forces. Other loads were neglected. The dam-foundation interface plane was considered only with friction forces between the monolith and the foundation. The applied base excitations were limited to those in a single upstream/downstream horizontal direction. Other components of earthquake motions were not simulated. 6.2 DESCRIPTION OF THE STRUCTURE The dam-water-foundation system can be considered with different level of complexity in analytical studies. Generally, it can be said that the more time consuming the analysis method is, the simpler model researchers tend to use. This mostly holds even i f the latest computers are employed. During nonlinear analysis of a concrete gravity dam, the analysis of the entire dam is often, in order to reduce computational effort, replaced by analyses of selected independent monolith of the dam. Adopting this concept, boundary conditions and loads on the selected monolith are somewhat different from those on the monolith built-in the dam, but this approximation is accepted by many researchers and practitioners, for example Fenves and Chavez (1995) or Tinawi et al. (1998). A single monolith of a concrete gravity dam of a total height 45 m was selected for analysis described in this chapter. The monolith had the same shape as the experimental model used in this study and therefore, the width of the monolith in the upstream/downstream direction was 36 m. The monolith with a unit thickness of 1 m in the cross-canyon was studied. Concrete with modulus of elasticity equal to 27 GPa, mass density of 2580 kg/m 3 and Poisson ratio of 121 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH 0.22 was considered as mater ial for the mono l i th . These values were obtained f r o m mater ial tests conducted o n one o f B C H y d r o dams o f s imi lar shape and size (Powertech L a b s , 1996). 6.3 m \< H F igure 6 . 1 : Schemat ic o f the F u l l - S c a l e M o n o l i t h Structure The foundat ion interface was selected hor izontal as can be seen i n F igure 6 . 1 . The foundation rock was considered w i t h the modulus o f elasticity o f 15 G P a and Po isson ratio o f 0 .25. The hor izontal forces between the d a m and the foundation were l imi ted to those f r o m f r ic t ion w i t h the static coef f ic ient [is = 1.05 and the k inet ic coeff ic ient \ik = 1.00 . Other effects i n the dam- foundat ion interface p lane, such as cohes ion or in ter lock ing , were not considered. The bedrock was considered without mass i n order to m o d e l on ly its stiffness feature. In other words , the foundat ion was considered to represent on ly the support condit ions for the d a m , but make no attempt at representing jo int dam-foundat ion m o d a l behaviour . The reservoir was considered w i t h the m a x i m u m water leve l o f 9 6 % , that was 43 .2 m , o f the total height o f the mono l i th . The reservoir effects were l imi ted to hydrostatic and hydrodynamic pressures on ly . Other effects, such as upl i f t force, were not considered. 122 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH The hydrodynamic effects of the reservoir were modelled using the added mass approximation (Okamoto, 1973). According to this theory, the hydrodynamic pressurep at a depths can be calculated as: p(y) = ahiw^y (61) where is horizontal acceleration, w is mass density of water, H is the total depth of the reservoir andy is a coordinate from 0 to H, defined in such a way thaty = 0 at the water level andy = //at the reservoir bottom. It follows from Eq. (6.1) that the formula for the added mass mHD a t m e depths is: mHD(y) = i^JHy (6-2) The approximation of hydrodynamic pressures used here assumes the water to be incompressible. Such approach is simple and computationally efficient, but it does not fully capture the behaviour of the water vibrating in interaction with the dam. It would be more accurate to consider the water to be a compressible medium, but such an analysis would have to be done in the frequency domain, since the hydrodynamic pressures from the compressible liquid would be frequency dependent. This approach was used, for example, by Chavez and Fenves (1996) who performed the analysis in time and frequency domains simultaneously. Such analysis is very time consuming and it was considered to be beyond the scope of this study. The approximation adopted here gives larger estimates of hydrodynamic pressures compared to the case when water is considered compressible and, therefore, was on conservative side. In all analyses described in this chapter, the properties of the dam-water-foundation system were assumed constant except for the water level. It was the objective of this study to calculate the response of the base excited dam monolith for varied water levels. These were considered from 60% to 100% of the full reservoir with a step of 5%. The water level of 60% produces a hydrostatic force equal to 36% of the hydrostatic force of the full reservoir and therefore, this 123 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH range was considered sufficient. 6.3 DESCRIPTION OF BASE EXCITATIONS A total of three ground acceleration records were selected for the analyses in this chapter. In order to subject the dam monolith to a wide range of time histories. These included: • an Eastern North American type near source earthquake - the 330 degrees component of the 1985 Nahanni Earthquake, North West Territories, Canada, measured at Slide Mountain (Naeim and Anderson, 1996), P G A of 0.33g, denoted Nahanni earthquake. • a far source subduction earthquake - the East-West degrees component of the 1985 Michoacan, Mexico Earthquake, measured at Villita station (Naeim and Anderson, 1996), P G A of 0.13g, denoted Mexico earthquake. • a Western North American type near source earthquake - the North-South component of the 1994 Northridge, California Earthquake measured at the Sylmar County Hospital parking lot (CSMIP, 1994), P G A 0.84g, denoted Northridge earthquake. The original processed acceleration, velocity and displacement records of the above earthquakes are shown in the parts a to c of Figures 6.2 to 6.4. The time axis, identified in part c of each figure is common for the parts a, b and c. At the bottom parts, the figures also contain acceleration spectra plots for 5% damping. These plots indicate, which SDOF systems would be affected the most by each earthquake. The Nahanni earthquake (Figure 6.2d), which is often considered as a typical Eastern type earthquake motion, would affect the most systems with natural periods of 0.5 second and also those with the natural period from 0.05 to 0.1 second. The systems with natural period of 0.6 second (Figure 6.3d) would be affected the most by the Mexico earthquake. The Northridge record would affect the most the SDOF systems with 124 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH natural periods between 0.3 and 0.4 second, as can be seen in Figure 6.4d. Statistical distribution of the Peak Ground Acceleration (PGA) can be assumed to follow a lognormal (Foschi, 1998) distribution and it obeys a relationship of the form (Madsen et al., 1986): PGA = - ^ e ^ ' n O + F2) ( 6 3 ) For an assumed mean value of this distribution a ^ O . l g and its coefficient of variation F=0.6, a value of PGA can be calculated for a given value of the standard normal variable RN. The value of RN is associated with the probability of the earthquake exceeding the peak ground acceleration equal to PGA. Here, a total of four values of RN equal to 3, 3.5, 4, and 4.5 were considered and using these in Eq. (6.3) yielded the values of PGA equal to 0.453g, 0.597g, 0.788g and 1.040g. Such values of RNwere used in order to cover a wide range of PGA's for known exceedence probabilities. The spectral plots at the bottom parts of Figures 6.2 to 6.4 were calculated for all these four values of PGA. It should be noted that in an analysis of sliding of an actual concrete gravity dam the uplift force and the vertical base motions should be considered. It is recognised that both these loads would have an adverse effect on the overall amount of horizontal sliding. The SDOF model developed for this study was not able to take into account these types of loads. For simplicity, these were not considered also in the finite element analysis in this chapter. The methodology of the reliability study presented in the next section would remain the same should these two loads be considered. 6.4 FINITE ELEMENT STUDY 6.4.1 Description of the FE Model for the Dam Monolith The numerical model for the dam monolith was similar to that in the analytical study of the 125 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH a) Original Record - Ground Acceleration 10 Time (sec) d) Amplified Records - Spectral Acceleration for 5% damping 1.5 Period (sec) Figure 6.2: Characteristics of Selected Record from 1985 Nahanni Earthquake 126 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH a) Original Record - Ground Acceleration i 1 r 20 30 Time (sec) 4 d) Amplified Records - Spectral Acceleration for 5% damping c g ra 2 a o o < . 1.5 Period (sec) Figure 6.3: Characteristics of Selected Record from 1985 Mexico Earthquake 127 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH 1.0 r  v D) 0.5 C g » 2 0.0 <B <D O o -0.5 < -1.0 1.0 10 0.5 0.0 o o 0) > -0.5 -1.0 0.30 0.15 c E 0) 0.00 o ai a. in - 0.15 b 0.30 a) Original Record - Ground Acceleration i 1 r c) Original Record - Ground Displacement 10 Time (sec) d) Amplified Records - Spectral Acceleration for 5% damping 1.5 Period (sec) Figure 6.4: Characteristics of Selected Record from 1994 Northridge Earthquake 128 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH experimental model since the monolith was considered in two variations. These included the models for the bonded and unbonded monolith. Similarly to the FE model of the experimental setup, analysed in the previous chapter, the FE model for the prototype dam monolith was designed to be relatively simple, with a small number of finite elements, in order to keep the computational times of nonlinear time history analyses at an acceptable level. The FE model of the monolith comprised, respectively, 28 and 30 plane stress quadrilateral bilinear elements to simulate the monolith and the foundation rock. The foundation rock was modelled 20 m below and the same distance beyond the toe and the heel of the monolith. In addition, the model contained 7 contact elements distributed along the dam base to simulate the foundation interface zone between the monolith and the foundation rock. Features of the plane stress and the contact elements were described in the previous section and the inputted parameters of these elements corresponded to the material characteristics of the dam monolith and the foundation rock described earlier. In addition to the plane-stress and the contact elements, the FE model contained 6 point mass elements, which were used to account for the added lumped masses at wetted nodal points on the upstream side of the dam monolith. The MASS21 (SAS IP, 1996a) point mass element was used with the option of unidirectional mass in the horizontal upstream/downstream direction only. The added masses corresponding to each wetted node were calculated using Eq. (6.2). Damping characteristics used for the FE model of the full-scale monolith were similar to those for the FE model of the experimental setup. For the full-scale monolith, damping was considered with the value of 2% of critical for the 1st and 3rd natural modes, which is a typical value for concrete structures. It was recognised that damping can be an important parameter for seismic response of a concrete gravity dam, but more complex representation of damping would be beyond the scope of the study. Detailed information on the FE model can be obtained in Appendix B where an example of the input A N S Y S file is given. 129 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH The FE model for the dam monolith was developed in two modifications: • the dam monolith bonded to the base, shown in Figure 6.5; • the unbonded dam monolith freely standing on the base, shown in Figure 6.9. A total of three studies were conducted using the FE model. These included: • modal analysis of the monolith bonded to the base; • modal analysis of the unbonded monolith; • nonlinear time history analysis of the unbonded monolith. 6.4.2 Modal Characteristics of the Dam Monolith The objective of the modal analyses was to obtain natural frequencies and mode shapes of the dam monolith bonded to the base as well as the unbonded one. These were calculated using the Subspace Iteration Method described in SAS IP (1996b) The natural frequencies and the characteristics of the natural modes for the bonded monolith are presented in Table 6.1. The frequencies are listed for the dam monolith with no water in the reservoir and for that with a full reservoir. The natural frequencies of the monolith with the full reservoir are lower than those for the case without water. This is an expected observation. Characteristics of the obtained natural modes are presented in the right column of Table 6.1 where references to the corresponding figures showing the modes are also given. In order to clearly visualize the character of every mode, the deflection shapes in the figures were exaggerated. The natural frequencies of a 3-D model of a similar concrete gravity dam calculated by BC Hydro (1995) corresponding, respectively, to the first mode with the frequency of 5.9 Hz and the second mode with the frequency of 15.2 Hz were 7.5 and 13.9 Hz. This comparison indicates that the 2-D FE model used to simulate behaviour of the bonded dam monolith had natural frequencies close to those of the other model for a similar structure and, therefore the 2-D FE model was found acceptable for further analysis. 130 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH Table 6.1: Natural Frequencies of the Dam Monolith Bonded to the Base No. Frequency (Hz) (no water) Frequency (Hz) (100% water) Character of Natural Mode 1 6.9 5.9 1st cantilever bending mode, see Figure 6.6 2 16.9 15.2 2nd cantilever bending mode, see Figure 6.7 3 18.8 17.4 3rd cantilever bending mode, see Figure 6.8 4 34.1 29.3 higher cantilever bending mode The natural frequencies of the unbonded dam monolith were calculated using the numerical model for this variation. During the modal analysis and the contact elements acted as linear springs. The natural frequencies of the unbonded model are presented in the second and third columns of Table 6.2 for the dam monolith with no water and for that with full reservoir, respectively. Characteristics of the natural modes are given in the right column of the table where references to the corresponding modal plots are also shown. Table 6.2: Natural Frequencies of the Unbonded Dam Monolith No. Frequency (Hz) (no water) Frequency (Hz) (100% water) Character of Natural Mode 1 4.3 3.7 rocking of the model due to deflections in the vertical springs combined with the 1st cantilever bending mode, see Figure 6.10 (springs not shown in the figure) 2 9.3 9.2 up/down stretching and shortening of the vertical springs combined with the 1 st cantilever bending mode, see Figure 6.11 (springs not shown in the figure) 3 16.2 14.2 the 2nd cantilever bending mode, see Figure 6.12 4 27.9 25.5 higher cantilever bending mode 131 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH Figure 6.6: First Natural Mode of a Full-Scale Dam Monolith Bonded to Base; f=6.9 Hz 1 3 2 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH Figure 6.7: Second Natural Mode of a Full-Scale Dam Monolith Bonded to Base; f=16.9 Hz Figure 6.8: Third Natural Mode of a Full-Scale Dam Monolith Bonded to Base; f=18.8 Hz 133 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH Figure 6.10: First Natural Mode of an Unbonded Full-Scale Dam Monolith; f=4.3 Hz 134 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH \\ \ \ Figure 6 . 1 1 : Second Natural Mode of an Unbonded Full-Scale Dam Monolith; f=9.3 Hz 135 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH It can be observed from Table 6.2 that the natural frequencies and the character of the natural modes are quite different compared to the FE model of the bonded dam monolith. The first two natural modes have strong rigid body character and can be characterised by deflections of vertical springs in the contact elements. This can be observed from Figures 6.10 and 6.11. The influence of added mass reduced values of all natural frequencies, but this reduction is small for the second natural mode. This is because this mode has dominant vertical coordinates and the added mass in the horizontal direction did not change significantly the ratio of the stiffness to mass ratio for this mode. 6.4.3 Nonlinear Time History Analysis Using FE Model The FE model of the full-scale unbonded dam monolith was used for nonlinear time history analyses with the earthquake records defined earlier in this chapter. A total of 108 analyses were performed, each taking about 3 hours of computational time of a Pentium II 330 M H z personal computer. During these simulations, three parameters of the load on the dam monolith were varied. These included the type (record) and PGA of the base excitations and the water level. During the A N S Y S simulations, a transient analysis was specified with the Frontal Direct Equation Solver. The Newmark implicit integration procedure was applied with an amplitude decay factor of 0.005 (SAS IP, 1996b). It was found in the course of the study that satisfactory results were obtained with the automatic time-stepping on (SAS IP, 1996c), with bounds on the time step between 0.005 and 0.00005 seconds. The program automatically adjusted the time step to suit the current state of the system. As expected, with an increasingly nonlinear behaviour of the model, the time step became shorter, and the progress of the analysis slower. The results from the analyses were stored at every 0.005 second. The loading on the model was applied in load steps. During the first load step, with the inertia effects turned off (SAS IP, 1996c) the gravity forces were applied. The second step, with inertia 136 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH effects still turned off, included application of hydrostatic pressure. This was superposed to the state of the dam monolith after the first step. The effects of hydrostatic pressure were lumped into hydrostatic forces applied at wetted nodes on the upstream side of the monolith. In the load steps following the second, the earthquake loading in terms of the horizontal acceleration on the dam-foundation system was applied on the monolith at the stage after the second load step. A series of two trial simulations using each earthquake record was performed. The two simulations with the same earthquake record of equal PGA were carried out with the base motions applied in opposite direction. This was done in order to find out i f reversing the record would change significantly the calculated amount of the base sliding. The differences in the sliding response using the records in opposite directions did not exceed 15% for any earthquake. Further analyses were carried out in the direction, for which larger sliding was obtained. Detailed information about the ANSYS commands used during the solution phase of the analysis can be found in Appendix B. The results of the FE simulations are presented together with those from the SDOF study in the last section of this chapter. 6.5 STUDY USING SDOF MODEL 6.5.1 Description of SDOF Model for the Dam Monolith A SDOF model for the unbonded dam monolith was developed based on the theory presented in the previous chapter. Parameters of the SDOF model were obtained from the dimensions and the material characteristics of the dam monolith. The total mass of the block was calculated from the volume of the monolith and the mass density of concrete as 2.46E6 kg, based on the mass density of 2580 kg/m 3. The friction coefficients were used with the same values as those for the FE simulations, static and kinetic 1.05 and 1.00, respectively. The parameters of loads derived from the water level of the reservoir included hydrostatic and 137 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH hydrodynamic forces. The hydrostatic force was calculated as a lumped force from the hydrostatic pressure distribution and it had values from 3.3E6 N for 60% water level to 9.1 E6 N for the 100% water level. The hydrodynamic pressures were simulated using the added mass concept given by Eq. (6.2). This equation was used to calculate the total lumped added mass MHD, which had values from 380E3 kg for 60% water level to 1,069E3 kg for 100% water level: MHD = fmHD{y)dy = f\wjH~ydy = ^zwH2 (6.4) 0 0 o i z 6.5.2 Nonlinear Time History Analysis Using SDOF Model The analyses using the SDOF model were performed with the earthquake records described earlier in this chapter. The time step of the SDOF analyses was 0.001 second, but the results were stored at every 0.005 second. A total of 108 SDOF analyses were performed, each of these took about 8 seconds of computing time of a Pentium II330 M H z I B M compatible computer. 6.6 COMPARISON OF RESULTS FROM THE FE AND SDOF SIMULATIONS A large number of the FE and SDOF analyses were performed in this chapter. Results of these are compared in this section in order to see i f the SDOF model can satisfactorily simulate the response of the dam monolith. Both series of analyses yielded the amounts of sliding of the dam monolith for varied earthquake types, PGA's and water levels. These results were summarised in Figure 6.13 in terms of the monolith displacement (sliding) to water level relationships for varied PGA's and in Figure 6.14 in terms of the monolith displacement (sliding) to PGA relationships for varied water levels. The above means that both figures present the same set of results but in a different way. Both Figures 6.13 and 6.14 contain six parts a to f. These are organised in two columns and three rows in each figure. The results from the SDOF model are presented in the first column 138 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH while those from FE are given in the second. Results from each of the earthquake records are in one row and the name of the earthquake is identified above every plot. Every obtained result, the amount of sliding of the dam monolith for a given combination of PGA and water is indicated with a dot in the figures. To show trends in the results, the dots for the same PGA and water level, respectively, are connected with solid lines in Figures 6.13 and 6.14. In each plot in Figure 6.13, the lowest sliding was obtained for the lowest PGA of 0.453g while the largest amplitude of 1.04g yielded the largest amounts of sliding, for a given water level. Legend shown in part a of the figure also holds for the other parts. In each plot in Figure 6.14, the lowest sliding for a given PGA was obtained for the lowest water level of 60% while the maximum water level yielded the maximum sliding. The results for other water levels are ordered between these for 60% and 100% water levels in steps of 5%. Legend for part a also holds for the rest of the figure. Observations from Figures 6.13 and 6.14 include the following: • Neither model yielded consistently larger amounts of sliding than the other. • The SDOF-based amounts of sliding exhibited greater dependence on the reservoir effects than the FE model (compare Figure 6.13 a to b, c to d, e to f). The amounts of sliding for low water level from the SDOF model were smaller than those from the FE model. The agreement was better for higher water levels. • The SDOF model showed greater dependence on the PGA than the FE model. This can be observed from comparison of Figure 6.14 a to b, c to d, e to f. The SDOF model yielded smaller sliding than the F E model for low P G A ' s . The agreement is better for higher PGA's. • The two observations above lead to a conclusion that the FE model initiated sliding sooner for low water levels and low P G A ' s than the SDOF model. However, the SDOF model, for high water levels and P G A ' s yielded amounts of 139 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH a) SDOF Model - Nahanni Earthquake I 2 Q. b -! 0 5 PGA 1.04g _ PGA 0.79g E 4 - PGA 0.60g menl 3 - • PGA 0.45g I , i 1 1 , ! 1 1— 60 70 80 90 100 Water level (%) c) SDOF Model - Mexico Earthquake b) FE Model - Nahanni Earthquake l 1 i" 70 80 90 Water level (%) d) FE Model - Mexico Earthquake 70 80 90 Water level (%) 1 5 e) SDOF Model - Northridge Earthquake 15 f) FE Model - Northridge Earthquake 60 70 80 90 100 60 70 80 90 100 Water level (%) Water level (%) Figure 6.13: Dam Monolith Displacements (Sliding) vs. Reservoir Water Level 140 CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH a) SDOF Model - Nahanni Earthquake R b) FE Model - Nahanni Earthquake 2Q c) SDOF Model - Mexico Earthquake 2 p d ) F E M o d e l - Mexico Earthquake 0.4 0.6 0.8 1 0 0.4 0.6 0.8 1.0 PGA(g) PGA(g) Figure 6.14: Dam Monolith Displacements (Sliding) vs. Peak Ground Acceleration CHAPTER 6 ANALYTICAL STUDY OF A FULL-SCALE DAM MONOLITH sliding close to those from the FE model. Another comparison of the SDOF and FE results was done for selected simulations in time domain. Results of this comparison are shown in Appendix C where selected time histories from 6 SDOF and 6 FE simulations were plotted. Two simulations for each earthquake record, one with PGA of 1.040g and the other with 0.788g were selected for the comparison, all were for the 100% water level. It follows from data shown in Appendix C that the SDOF and FE friction foundation interfaces did not perform precisely the same way. This was expected due to the different modelling techniques used for each model. However, the time histories of sliding from both models were not found significantly different in any case. A good agreement between obtained base sliding from the FE and SDOF models can be justified using the results from the analytical study on the experimental model. It was observed during this study that both models yielded similar sliding for the base excitation frequencies of up to about 15 Hz, which corresponded to approximately to 50% of the first natural frequency of the unbonded experimental setup (Table 4.3). In the case of the full-scale monolith, the first natural frequency was 4.3 Hz and the dominant frequencies of the earthquake records used were from 1.6 to 1.9 Hz, which is less than 50% of the fundamental frequency of the monolith. If the observation from the analyses on the experimental setup is extended to the full-scale case it can be said that the FE and SDOF models could give similar answers also in this case. It is difficult to make any general conclusion about the performance of the SDOF model without detailed analysis of trends the FE and SDOF results show. The amounts of sliding the SDOF model yielded were very close to those from the FE simulations for some combinations of loading parameters. The agreement was not as good for the others. It is desirable to obtain a simple mathematical relationships between the sliding, PGA and the water level. Such relationships obtained by interpolating the results could be used to observe general trends and differences in performance of the FE and SDOF models. This will be done in the next chapter. 142 7 R E L I A B I L I T Y S T U D Y OF A F U L L - S C A L E D A M M O N O L I T H 7.1 BASIC CONCEPTS OF RELIABILITY ANALYSIS The reliability of an engineering system is the probability that it will perform as required in given conditions within a specified period of time. For example, the reliability level may be expressed as the probability that the maximum deflection of a structure will not exceed a given value in the next 50 years given the load conditions for the site (Madsen et al., 1986). In addition to the design parameters, the performance of the system is controlled by a set of variables, some representing geometric, mechanical, and material characteristics of the structure and other characterizing the external effects such as the load demands. From the probabilistic point of view, the variables are regarded as random and have to be described in probabilistic terms, including their distribution types, mean values, standard deviations and other characteristics. However, those with a small degree of uncertainty can be treated as deterministic parameters represented only by their nominal values. The probabilistic description of a random variable may be achieved by a) experiments, providing statistics and an estimate for the corresponding distribution or; b) engineering experience and judgement when such statistical data are lacking. In addition, in order to reduce computational effort, a pre-sensitivity analysis can be employed to determine the variables that should be treated as random (Foschi etal., 1989). The implementation of the reliability analysis is based on a description of the limit state of interest by a performance function G(X) as follows: 143 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH G(X) = G(xx,x2,...,xN) (7.1) where X = (xh x2,...,x-^)T is an N-dimensional vector of intervening random variables. Some of these may affect the demand on the system, denoted D, while the others influence the system capacity C to withstand the demand. By convention, the performance function G is written as: G = C-D (7.2) The system will then fail i f the combination of the intervening random variables results in the value G < 0. The corresponding probability of such an event (P(G < 0)) is called the probability of failure. Conversely, the combination of the intervening variables resulting in G > 0 will make the system survive and the corresponding probability (P(G > 0)) is called reliability. The situation when the performance function G = 0, is called the limit state failure surface. To calculate G, a simulation model describing the problem of interest is needed. For example, in a deflection problem, G can be expressed as: G = SQ-S(xltx2,...,xN) (7.3) where S0 is the allowable and S is the actual deflection of a structure. In our case, the allowable and the maximum displacements will correspond to amounts of sliding of a gravity dam during an earthquake. The actual displacement is a result of a deterministic calculation using a numerical model. If this model can provide the required answer, is fast, and can be linked with a reliability program, it can then be used directly in the reliability analysis. If getting the result of one calculation takes a significant amount of time, it is convenient to pre-generate results and reduce these into a form representing the response of the dam to varied input variables. Using the response representation, the probability of failure can then be obtained by calculating 144 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH the probability of the event G < 0. As there could be a number of random variables involved in G, the exact calculation could be obtained by integrating the joint probability density function of all random variables over the failure region G < 0 (Madsen et al., 1986). This exact approach can hardly be applied since the joint probability function is unknown and difficult to find. Another possibility to calculate the probability of failure is standard computer simulation, the Monte Carlo Method, which is simple to implement and can converge to the exact solution. However, it could be very computationally demanding, especially when dealing with low probabilities of failure. In such cases, the performance function has to be evaluated a very large number of times to get a single outcome G < 0. In particular, i f each evaluation of the performance function requires a nonlinear dynamic time-domain analysis to be performed, this method can be very lengthy even if the latest computers are used (Li, 1999). A second alternative is the use of approximate methods developed during the last three decades. These include the First and Second Order Reliability Methods (FORM and SORM), which are based on the calculation of the reliability index P . From this index, the probability of failure Pyand the reliability PR can be estimated approximately as follows: PF» <D(-P), and (7.4) PR * 1 - O(-P) = <P(P) (7.5) where <D is the standard normal probability distribution function. The F O R M procedure was developed in the 1970's (Hasofer and Lind, 1974, Rackwitz and Fiessler, 1978) as an extension of the work in the late 1960's (Cornell, 1969) and has been modified and improved since. During the last twenty years, the method has been implemented 145 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH in a number of reliability analysis programs and used in many engineering applications. For an exact estimate of the probability, the method requires the random variables to be normally distributed and uncorrelated, and the function G to be linear. If some of the random variables are correlated or non-normal, transformations are required before the procedure is implemented. The reliability analysis package R E L A N was used for the analysis during this study. This program has been developed at the Department of Civil Engineering of the University of British Columbia (Foschi et al., 1997). The program can be used to solve a number of various reliability problems employing either Monte Carlo or FORM/SORM methods. Two numerical models were used in the analytical study of a full-scale gravity dam monolith to obtain amounts of the monolith's sliding for varied loading parameters. These were the FE and the SDOF models. The SDOF model provides a fast solution to any combination of deterministic input variables and it could be easily linked with a reliability analysis program, in this case, R E L A N . However, the FE model needs a significant amount of time to calculate a single result and, therefore, it is convenient to use a response surface approximation i f this model is employed. Since it was the objective of this chapter to compare the performance of both numerical models, the response surface method was used with both. In the following sections, the reliability analyses conducted within this study will be described. This will include several tasks such as reduction of results obtained from the last chapter, identification of random variables, and several types of reliability analyses. 146 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH 7.2 OBJECTIVES AND SCOPE OF RELIABILITY ANALYSIS Two sets of results were obtained from the analytical study on a full-scale dam monolith, which was described in the previous chapter. The results, obtained using the FE and SDOF models, predicted how much the monolith would slide under three different types of time histories of varying amplitudes and with different reservoir levels. In reality, a concrete gravity dam at a certain site could be hit by various time histories, the characteristics of which would depend on many factors, such as the distance of the dam from the epicentre of the earthquake and the intervening geology. Therefore, it is desirable to reduce the information generated in the previous chapter in order to predict how the dam would perform under a general base motion. The objectives of this chapter is to obtain annual probabilities of failure of the full-scale dam monolith using the FE and SDOF results calculated in the previous chapter. Probability of failure is defined as a probability of exceeding specified amount of sliding. Another objective of this chapter is to express results of the reliability study in terms of safety factors. The methods to achieve these objectives are: • To average the results of the analytical study on a full-scale dam in such a way that the response of the dam is related to an average earthquake and the variability in the time histories is expressed by an error term. The average earthquake is defined as such base excitation which would cause the response of the dam monolith to be the mean obtained from all records used. • To reduce the results by expressing the influence of the reservoir's water level on the amount of sliding in terms of a simple relationship. • To identify intervening random variables and to obtain their statistical parameters. 147 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH • To carry out several reliability-based analyses of varied complexity making use of the averaged and reduced results. • To express the results of the reliability-based studies in terms of factors of safety against sliding. The studies conducted in this chapter were limited to those on a single full-scale monolith of a gravity dam, which was analysed in the previous chapter. It would be incorrect to extrapolate the results of these studies to dams varying significantly from this structure, but it can be assumed that the results are applicable to a concrete gravity dam monolith of similar properties. 7.3 REDUCTION OF THE ANALYTICAL RESULTS The analytical results were obtained in the previous chapter as sets of maximum horizontal displacement of the dam monolith relative to its base. These displacements, or amounts of sliding, were calculated using the FE and SDOF models for: • three different earthquake records - one from each of the Nahanni, Mexico and Northridge earthquakes; four values of PGA - 0.45g, 0.60g, 0.79g, and 1.04g; • nine varied water levels of the reservoir - 60% to 100% with a step of 5%. The reasons for selecting these parameters were given in the previous chapter. In order to reduce the amount of the results the following analyses were performed: • Logarithms of the PGA values and corresponding amounts of sliding were calculated. This was done because the PGA values are assumed to follow a lognormal distribution (Li and Foschi, 1998), which means that their logarithms follow a normal distribution. This conversion to normal variables makes a reliability analysis easier especially i f no 148 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH reliability software is available. Pairs consisting of PGA and amount of sliding for all earthquakes records and PGA's , considered for a given water level, were replaced by a linear approximation. This was obtained by applying linear regression to all the pairs for a given water level. In an obtained equation, the logarithm of sliding (denoted S) is expressed as a linear function of the logarithm of the PGA (denoted aG): In (5) = a + bln(aG) (7.6) a and b are the coefficients calculated by linear regression. These coefficients were calculated for all water levels in Appendix D and E, respectively, for the results from the FE and SDOF analyses. It can be concluded from observation of Figures DI and E l that the linear interpolation used in Eq. (7.6) provided a good fit on the results. The sets of coefficients a and b from Eq. (7.6), obtained for each considered water level of the reservoir, were replaced by linear relationships in terms of this water level: a = a, + a2h , and (7.7) b = bl + b2h (7.8) In the above equations, h is the water level of the reservoir and the coefficients a\, a2, b\, and b2 were obtained using linear regression on the sets of coefficients a and b. The water level was expressed in percent of the maximum level at the reservoir. After the double linearisation, a S to aG relationship was obtained: ln(S) = a{ + a2h + (bx + b2h)\n(aG) (7.9) with one set of the coefficients a\, a2, bx, and b2 for the results from each one of the FE and SDOF studies. This form of the interpolation the S to aG relationships was selected because 149 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH it gives a good fit over the range of water levels from 60 to 100% of the full reservoir as can be seen Figures D2, D3 and E2, E3. In addition, Eq. (7.9) represents a linear function of l n ( a G ) , which is a normal random variable because aG is assumed lognormal. • A set of errors was calculated as difference between the results from the simulations and the values calculated using Eq. (7.9). • Basic statistics, the mean value and the standard deviation, were calculated for the set of the errors. The same statistics were obtained for the set of ln(a G ) values corresponding to all calculated logarithms of sliding. In addition, a correlation coefficient between these two sets was obtained. A series of calculations described above was performed using a commercially available equation editor Mathcad 8 Professional (Mathsoft, 1998). Detailed information on these calculations can be found in Appendices D and E, respectively, for the results from FE and SDOF analyses. The calculation of the logarithms of S and aG, explained in the first of the above described steps, could be carried out directly with the results of the FE study. However, the results of the SDOF study did not allow such direct analysis because their number had to be increased first by the results from analysis with high levels of PGA. This was due to the fact that the SDOF simulations at low PGA levels yielded almost zero displacements, which would make calculation of the coefficients al,a2,bh and b2 difficult. To do that, the amount of sliding equal to 1 mm was selected as a point where the sliding of the dam was considered to be initiated. The SDOF numerical model was used to determine the PGA values needed to cause such amount of sliding. Then, several series of SDOF simulations were performed using all three earthquake 150 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH records with PGA's above the values needed to displace the dam monolith 1 mm. After the data from analytical study was reduced three different reliability analyses were carried out on each set of the results. Before the results of these analyses are presented it is necessary to introduce basic concepts used in such analysis. 7.4 RELIABILITY STUDY WITH TWO RANDOM VARIABLES This chapter of the thesis deals with the problem of evaluating the risk of exceeding a specified amount of sliding for the studied monolith of a concrete gravity dam under an earthquake. Although many uncertainties are involved in the problem, the selection of random variables for the reliability study was based on the parameters varied during the analytical study performed in the previous chapter. A total of three variables were varied in this study: 1) the type of used earthquake record, 2) PGA of the base motions, and 3) the reservoir's water level. These variables were assumed to affect the most the resulting amount of sliding of a gravity dam monolith loaded by an earthquake and therefore their influence was studied in the previous chapter. The reliability analysis performed in this section uses random variables based on the first two of the three above parameters. In addition to the above identified random variables, there would be a great deal of randomness in the characteristics of the foundation rock and the foundation interface. Out of all these parameters, it is assumed that the modulus of elasticity and the friction coefficient of the foundation interface are the most important (Fenves and Chavez, 1996). However, these two parameters were not varied during the deterministic analyses performed in the previous chapter because the structural system was considered deterministic and only the loading was considered 151 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH variable. A sensitivity analysis will be performed in the end of this chapter to assess the influence on the results of the foundation rock modulus of elasticity and the friction coefficient of the foundation interface. 7.4.1 Peak Ground Acceleration as a Random Variable The PGA is widely used as a design parameter in seismic analysis of all types of structures and is one of the major factors controlling the response of a structural system during an earthquake. A total of four PGA's associated with the horizontal motions applied at the base of the dam of interest were used in the analytical part of this study. These acceleration levels were calculated from the distribution of PGA, or aG, which is assumed to be lognormal according to ground motion attenuation laws. Therefore, aG obeys the following relationship of the form (Madsen et al., 1986): a = _ ^ e / W i n ( i + ^) (7.10) where aM is the mean value of the distribution of the peak ground accelerations aG during an event, V is its coefficient of variation and i ? N is a standard normal variable. From typical attenuation laws, the standard deviation of ln(a G ) , expressed as Vln(l + V2) , is assumed here to be equal to 0.55, to which a value of 0.6 is associated. This value, widely used in seismic engineering, is consistent with many soil attenuation relationships (Foschi, 1998). The seismicity of the site and the statistics of the peak ground acceleration can be used to calculate the peak acceleration of Maximum Design Earthquake (MDE) for the site. This peak acceleration usually corresponds to certain Annual Exceedence Probability (AEP) and will be called aAEP here. For buildings, the AEP of 1/475 (NBCC, 1995) is used, and such annual 152 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH risk also corresponds to an exceedence probability of 0.1 in a 50-year window. Conversely, the mean value aM, can be calculated from the probability P E that the peak acceleration aG exceeds the acceleration aAEP during an earthquake, coupled with the assumption that the earthquake occurrences follow a Poisson process. The practice for dams, which are designed and analysed in accordance with the Dam Safety Guidelines (Canadian Dam Association, 1999), is not the same as for buildings. For safety evaluation of dams, the AEP to calculate the value of o:AEP is dependent on the consequence classification of the dam in accordance with the Dam Safety Guidelines. The usual minimum criteria for M D E varies from AEP of 1/100 to 1/1000 for low consequence dams, 1/1000 to 1/ 10,000 for high consequence dams and 1/10,000 for very high consequence dams. Since the consequence classification of the studied concrete gravity dam monolith is not known and the AEP typically used for buildings is within the range for low consequence dams, all analyses described in this section are related to AEP of 1/475. However, it will be shown later in this section that the results can be easily related to an arbitrary desired AEP. Employing the above explained concept and for AEP of 1/475, the relationship for P E and aG can be written: _ L = i_e-vPE(°G>"AEp) ( 7 1 1 ) where v is the Poisson process mean arrival rate. Here, v =0.1, which means that the earthquakes under consideration occur, on average, once every 10 years. For v = 0.1, P E is then P E = 0.0211. Since, from the definition of the probability for a variable lognormally distributed, P E can be expressed as: 153 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH \ - 0 ( R N ) (7.12) the associated normal variable RN is found to be RN - 2.0315. After rewriting Eq. (7.10) in the form: the relationship between the mean value of PGA and the value of aAEP can be obtained. For aM = O.lg the corresponding design P G A is ^AEP ~ 0.265g . The relationship between oAEP and the AEP depends only on the seismicity parameters of a site. That is, given the coefficient of variation V and the Poisson process mean arrival rate v , oAEP can be obtained using Eqs (7.11) to (7.13) with the condition that the mean PGA, aM, be constant. For example, for V=0.6, v=0.1 and a^j=0.\g, the 100 year return acceleration, denoted here as #ioo> is 0.66 times that for the 475 year return period ( # 4 7 5 ) . For AEP of 1/ 1000 and 1/10,000, respectively, the values of # 1000 a n d aioooo are 1.17 and 1.80 times larger than 475. Since the parameters V, v , and aM are fixed for a site, the corresponding aAEP acceleration can be given by the 475 year return or any other return since these are related as discussed above. The corresponding ^AEP f ° r other AEP can be calculated. Here, a site with aM = O.lg, V-0.6 and v=0.1 is assumed. It was observed during the numerical simulations with FE and SDOF models that the sliding of the dam monolith was not initiated until the PGA of the earthquake reached a certain level. If an approximation of the sliding of the dam monolith in terms of Eq. (7.9) is to be used it is necessary to limit the range where Eq. (7.9) is applicable. Since this equation would yield non-zero sliding of the monolith even for very low PGA levels it was decided to use in the reliability a M eRNJln(\ + V1) (7.13) 154 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH analysis only such values of PGA which cause sliding greater than 1 mm. This value was considered as a point where the sliding of the dam monolith was initiated. The PGA values corresponding to this amount of sliding were calculated using Eq. (7.9) and are shown in Figure 7.1. It can be observed from this figure that the PGA's decrease with increasing water level for both models. In addition, it can be said that the SDOF model needs higher PGA's to produce sliding of 1 mm. The above described phenomena was considered when parameters of the P G A as a random variable were being determined. Therefore, it was decided that PGA in the reliability analysis should be considered with a lower bound equal to the acceleration values shown in Figure 7.1. The lower bounds for the PGA distribution had different values for different water levels, in addition to differences resulting from the unequal performance of the FE and SDOF models. 60 70 80 90 Water level (%) 100 Figure 7.1: Values of PGA Necessary to Cause Sliding of 1 mm 155 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH 7 A . 2 Effect of Different Earthquake Records Treated as an Error Random Variable The type of time history is another important parameter in seismic analysis. Three different earthquake records were used during the numerical simulations and the differences among responses due to these were shown in the previous chapter. It was observed that the amount of sliding for the same PGA and water level varied significantly with the type of earthquake record. In order to take this phenomena into account in the reliability analysis a vector of errors was calculated for each of the FE and SDOF result sets. These errors were obtained as a difference between either the FE or SDOF results and the corresponding interpolated values of sliding given by Eq. (7.9). The details of error calculation are presented in Appendices D and E, respectively, for FE and SDOF analyses. It was decided to consider the errors as a normal variable with statistics calculated in the appendices. These included the mean errors, obtained for the FE and SDOF data as numbers very close to zero and the standard deviations calculated as 0.790 for the FE data and 0.791 for the SDOF data. In addition, the correlation coefficient between the PGA's and the errors was calculated. The correlation was found very small for both the FE and SDOF data. The correlation coefficients for the FE and SDOF data were obtained, respectively, as -0.01 and 0.004, as can be seen in Appendices D and E. Moreover, it should be mentioned that a distribution other than normal could have been assumed for the errors. Such distribution should be capable of including positive and negative error values. The normal and uniform distributions would be acceptable. Here, the normal distribution was used. 156 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH Finally, it should be understood that instead of using three historical records, any number of either historical or synthetic base acceleration records consistent with a given response spectra or power spectrum could have been used for this study. As an approximation, it was assumed that the three earthquakes used provide enough variability in the type of base excitations. 7.4.3 Simplified Reliability Analysis The characteristics of the two random variables obtained in the previous subsection permit a simple reliability analysis to estimate probability of failure of the system for a given allowable displacement (sliding). This probability is associated with a constant reservoir level. A reliability program such as R E L A N should be used to perform the analysis and results of such analysis will be presented in the next subsection. Here, another possibility of calculating this probability without a reliability program, in closed-form, will be shown. It was concluded in the previous subsection that the distribution of PGA as a random variable should be considered with a lower bound. However, i f this lower bound is not considered, it is possible to obtain a closed-form solution to calculate the probability of failure. It is desirable to obtain such a solution because it can be used as a check for a RELAN-based analysis. The simplified analysis is based on the procedure explained below. Let us assume that reliability of a system with two random variables is studied and the performance function G is a linear combination of these variables. If the random variables are normally distributed then the reliability of the system P can be calculated (Foschi, 1994): Q P = — (7.14) 157 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH where G is the mean value of the performance function and aG is its standard deviation. If the performance function considered is: G = \n(S0) - [ax + a2h + (bx + b2h)\n(aG) + s] (7.15) where SQ is an allowable displacement, ln(a G ) and £ are normal random variables, and the other parameters in this equation are known, then G can be calculated: G = ln(S 0) - [al + a2h + (bx + b2h)\n(aG) + i ] (7.16) where ln(a G ) and s , respectively, are the means of the logarithms of PGA's and the errors. The standard deviation c G can be obtained from the equation: c G = (bx + b2h)oln{ac) + 2(6, + M)pcr l n ( f l o )a e + ° B (7-17) where a \ n ( a G ) ^ d °"E are, respectively, the standard deviations of the logarithms of PGA's and the errors, and p is the correlation coefficient between these two variables. A l l these quantities were calculated for the FE and SDOF results, respectively, in Appendices D and E. Knowing the reliability P, the event probability of failure PE can be calculated using Eq. (7.4). The desired value of the annual probability of failure can then be calculated from the Poisson process assumption: PA = l - e ~ v / > £ (7.18) The above described procedure was used to calculate the annual probabilities of failure PA for both the FE and SDOF data, for nine water levels from 60% to 100% with a step of 5%, and for six levels of allowable displacement 1, 2, 5, 10, 15 and 20 cm. Sliding limits were selected arbitrarily between 1 cm, which was considered as a significant initiation of sliding, and 20 cm, 158 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH which was considered as a significant amount of sliding. Significant sliding is expected to cause disruption to the functioning of the drainage system in the dam and in the foundation, resulting in dangerous increases in uplift and internal pore pressures. Significant relative sliding between adjacent monoliths could also result in damage to waterstops and pose a serious leakage hazard. Analyses to determine consequences of the dam monolith sliding by a given amount is beyond the scope of this study. The results of the simplified reliability analysis with two random variables are not shown here because the RELAN-based analysis, performed in the next subsection, should supersede them by its accuracy. However, the results are presented in Figures D6 and E6 and when compared against those from the RELAN-based analysis, they exhibit a good agreement. The annual probabilities from the simplified analyses (Figures D6 and E6) are slightly lower than those calculated in the next section and presented in Figure 7.2a and b. The differences are caused by neglecting the lower bound for the peak ground acceleration in the simplified analyses. The differences are larger, but always smaller than one order, for the SDOF-based probabilities than for those from F E . This is because the lower bound was higher for S D O F than for the F E data as it was shown in Figure 7.1. However, in spite o f the differences, it can be concluded that the simplified analysis with two random variables provided a good first approximation to the true solution. 7.4.4 Results of Reliability Analysis with Two Random Variables A simplified reliability analysis with two random variables was described in the previous subsection. Although it can yield valuable results i f a reliability analysis program is not 159 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH available, it is preferable to use the program to obtain reliability of a structure with higher accuracy. Results of such analysis are described in this subsection. The performance function implemented into the program R E L A N had a form according to Eq. (7.15). The two random variables, the PGA and the error from different time histories were used with their statistics determined previously. Since one of the variables, the PGA, had its lower bound given by the PGA to cause sliding of 1 mm, the total event probability of failure was obtained as combined probability of: • The probability that the calculated sliding is greater than the allowable one, assuming that the PGA is greater than its lower bound, and • The probability that the PGA is greater than its lower bound. The above can be written as: PE = (P(S>SQ)\(aG>aG0))P(aG>aG0) (7.19) where S is the amount of sliding calculated using Eq. (7.9), aG0 is a lower bound for PGA obtained from Figure 7.1, and S0 is a given maximum allowable sliding of the dam monolith. The event probabilities PE were calculated from results of R E L A N analyses using Eq. (7.19) and the annual probabilities of failure P A were obtained using Eq. (7.18). The annual probabilities P A , which are the final result of this section of the reliability study are presented in Figure 7.2, a and b, respectively, for the FE and SDOF data. The legend shown in part b of the figure is common for both a and b. The probabilities in both a and b follow very similar trend, and also the values from FE data are close to those using results of SDOF simulations. The SDOF-based probabilities show stronger dependence on the water level 160 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH 1E-4 a) Annual Probabilities of Failure Obtained from Results of FE Simulations 70 80 90 Water Level (%) 100 1E-4 b) Annual Probabilities of Failure Obtained from Results of SDOF Simulations fi 1E-5 TO LL o 1 1E-6 o c < 1E-7 1E-8 Sliding 1 cm Sliding 2 cm Sliding 5 cm Sliding 10 cm Sliding 15 cm Sliding 20 cm 60 70 80 Water Level (%) 90 100 Figure 7.2: Annual Probabilities of Failure vs. Water Level of the Reservoir for A E P of 1/475 161 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH compared to probabilities from FE data. On the other hand, the FE-based results are more sensitive to the sliding criterion than those from SDOF analyses data. The SDOF probabilities of failure are lower than those from FE for low water levels and low sliding. For example the ratio of the FE to SDOF probability for 60% water level and 1 cm sliding is 2.4. Such ratios are close to unity for water levels from 75% to 85% and sliding from 2 to 10 cm. It can be said that in this range both models yielded almost the same results. The ratios are less than unity for high water levels and large amounts of sliding. It can be concluded that the SDOF is unconservative compared to the FE model for low water levels and small allowable sliding, but it is conservative for high water levels and large allowable sliding. These observations are in a good agreement with the conclusions about performance of both models, which were given in the end of the previous chapter. The reliability study with two random variables generated useful information. The probability plots in Figure 7.2 can be used to determine the recommended water level of the reservoir i f a target probability level and allowable sliding are given. For example, i f the annual probability of failure required is 1E-6 and the allowable sliding is 5 cm, the results using FE and SDOF data indicate that the water level should be kept at 79% and 77%, respectively. For the same probability level but allowable sliding increased to 10 cm, the FE-based results allow 93% water level, while the SDOF-based only 87%. The plots can be also used to solve the inverse problem, that is i f a water level is given and the probabilities of failure associated with a given allowable sliding need to be found. The results obtained in this subsection allowed the calculation of the probability of failure i f a water level is given. In practice, the problem to calculate a probability level for a range of water 162 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH level may need to be solved. This is because the water level could vary and it is not known what it would be at the time of the earthquake. In such a case it is necessary to consider water level as another random variable and modify the reliability analysis performed so far. This will be done in the next section. 7.5 RELIABILITY STUDY WITH THREE RANDOM VARIABLES In a more complete reliability analysis of sliding of a given concrete gravity dam, it is desirable to obtain annual probabilities of failure if the dam is located in sites with different seismicity and the water level of the reservoir fluctuates between given limits. In the solution of such a problem, the acceleration <*AEP related to a given AEP, can be considered as a parameter. In this section a set of ^AEP w a s considered with the values of 0.2g, 0.265g, 0.3g, 0.4g, 0.5g, and 0.6g. These <*AEP accelerations correspond to AEP of 1/475. As it was explained earlier in this chapter, the corresponding peak accelerations for different AEP, at the same site, can be obtained using Eqs (7.11) to (7.13). For each QAEP> m e distribution of the random variable aG followed, again, a lognormal distribution with the coefficient of variation V = 0.6. For v=0.1, the mean value aM varied for different values of °AEP W A S calculated according to Eq's. (7.11) to (7.13). The water level of the reservoir was considered as a new random variable. It was assumed to fluctuate between 60% and 100% of the full reservoir and it was uniformly distributed with a mean value of 80%. If statistical data for water levels of a certain reservoir were available, some other distribution fitting the data could be used. Since such data were not available, and it was not the objective of this study to analyse any specific concrete gravity dam, the uniform 163 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH distribution was used. The probability of exceeding a given sliding at a given site was calculated as in the previous section with the exception that the variable h in the performance function G, see Eq. (7.15), was now considered random. A series of RELAN-based analyses were carried out and the event probability of failure PE were calculated using Eq. (7.19). Having these, the annual probabilities of failure P A were obtained using Eq. (7.18). Results of these calculations are presented in graphical form in Figure 7.3. This figure shows the annual probabilities of failure PA against the mean peak acceleration aM for the site and the corresponding peak acceleration aAEP for two return periods of 475 and 10,000 years. The probabilities obtained from the FE and SDOF results are shown, respectively, in parts a and b of the figure. The legend shown in part a of the figure is common to both parts. The results presented in Figure 7.3 show how the safety of a dam against base sliding would change if the dam is located in different sites. These results give the annual probability of exceeding a given amount of sliding i f the site seismicity is given. This probability can be obtained based on results either from FE or SDOF simulations. These results follow very similar trends as can be observed from comparison of parts a and b of the figure. Similarly to the previous section, considering allowable sliding as an independent parameter, the FE-based probabilities are more sensitive to the sliding criterion than those from the SDOF data. The results in Figure 7.3 correspond to AEP of 1/475 and 1/10,000. The relationship between the corresponding values of aAEP was derived previously. The results in the figure can be interpreted as follows: Using the FE based probabilities, the annual probability of exceeding a 164 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH 1E-2 1E-3 = 1E-4 o £ 1 E " 5 5 co o 1E-6 CL "ro § 1E-7 < 1E-8 1E-9 a) Annual Probabilities of Failure Obtained from Results of FE Simulations Sliding 1 cm Sliding 2 cm Sliding 5 cm Sliding 10 cm Sliding 15 cm Sliding 20 cm 0.05 0.13 0.24 0.10 0.15 0.26 0.40 0.48 0.71 Acceleration (g) 0.20 0.53 0.95 0.25 0.66 1.19 a 4 7 5 a 10000 1E-2 1E-3 1E-4 ' r o o 1E-5 15 ca jQ 2 1E-6 CL ro 3 C C 1E-7 < 1E-8 1E-9 b) Annual Probabilities of Failure Obtained from Results of SDOF Simulations 0.05 0.13 0.24 0.10 0.15 0.26 0.40 0.48 0.71 Acceleration (g) "1 0.20 0.53 0.95 0.25 a M 0.66 a475 1.19 a-ioooo Figure 7.3: Annual Probabilities of Failure versus Acceleration 165 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH sliding of 10 cm equal to 1E-6 will be obtained in a site with approximately a^O.Wg or a475=0.29g, or a]0000= 0.52g. It can observed from Figure 7.3 that the probabilities obtained from the SDOF-based results are lower than those from the results of FE analyses for low peak acceleration and small allowable sliding, but the differences are diminishing with increasing peak acceleration and allowable sliding. For example, the FE-based probability of failure at the sliding level of 1 cm and a475 of 0.265g is almost 5 times larger than the corresponding probability yielded from SDOF data. The differences in the probabilities are due to the different behaviour of both numerical models and the lower values of SDOF-based probabilities are mainly due to the fact that the accelerations to initiate sliding of 1 mm (see Figure 7.1) are higher than the peak accelerations considered. Therefore, the combined probabilities according to Eq. (7.19) are lower for the SDOF then for the FE data. Ratios of the probabilities from FE to those from SDOF are close to unity from a475 of OAg and sliding of 10 cm higher. In this range, the SDOF and FE models yielded almost identical results. A n example of results from the analysis with three random variables is presented in Table 7.1. This table gives a summary of reliability analyses for a site with a475 of OAg and for an allowable sliding of 5 cm for both response surfaces, one obtained from the FE and the other from the SDOF data. It can be observed from the table that the annual probability of failure using the FE data was almost twice as that using the SDOF results. 166 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH Table 7.1: Example of Results for a site with a475-0Ag and for an Allowable Sliding of 5 cm Reliability Analysis Using FE Data Using SDOF Data Annual Probability - of Failure 2.5E-5 1.3E-5 Random Variable Design Point Sensitivity Factor Design Point Sensitivity Factor PGA (g) 0.79 0.84 0.95 0.81 Water Level (%) 89 0.30 91 0.44 Error due to Different Earthquake Record 0.74 0.45 0.53 0.38 The results in Table 7.1 are given in terms of components of a design point and sensitivity factors. The design point is defined as such combination of the input random variables for which the probability of failure is the highest. It can be seen in the table that the components of design point vary for the FE and SDOF based results. The design point peak ground acceleration from the FE data is lower than the corresponding value from the analysis using the SDOF model. In the contrary, the design point error due to different earthquake record is higher for the FE results than that using the SDOF data. The design point water level is approximately the same for both. The sensitivity factors show how the resulting annual probability of failure depends on every variable. It can be observed in Table 7.1 that the sensitivity factors for the peak ground acceleration are about the same for the FE and SDOF data. However, they vary for both the water level and the error due to different earthquake record. It can be concluded that for this site with a475 of OAg and an allowable sliding of 5 cm, the SDOF model is more sensitive to the water level than the FE model, but it is the opposite for the error due to different time history. Two sets of annual probabilities of failure for varied seismicity of the site and six levels of 167 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH allowable sliding were obtained in this subsection. Both sets yielded low annual probabilities of failure from about 1E-8 to 1E-3, depending on the allowable sliding and the seismicity of the dam site. The probabilities using the FE data are mostly higher, but not more than about 5 times higher, than those from the SDOF data. In spite of these differences it can be concluded, that results from both data sets are in a good agreement and the SDOF model performed satisfactorily in this test. 7.6 RESULTS OF STUDY IN TERMS OF SAFETY FACTORS The reliability analysis with three random variables was carried out in the previous section. One of the intervening parameters was the seismicity of the site. This reliability analysis yielded annual probabilities of exceeding a specified amount of sliding of the same concrete gravity dam monolith at different sites. In this case, the acceleration corresponds to AEP of 1/475, but as shown previously, the acceleration values could be scaled for any other AEP and the same set of results could be used. The concept of the factor of safety is a traditional procedure used in dam safety analysis. It is desirable to express results from the previous section in terms of safety factors in simple design or verification equations. The factor of safety is defined as a fraction between the nominal capacity of a system and the nominal demand. In case of evaluating the stability of a gravity dam monolith, shown if Figure 7.4, against sliding the factor of safety can be written as (Fenves and Chavez, 1996): r c Nominal Capacity c + ^Nst F S = v i • m A o r F S = v , T / ( 7 - 2 0 ) Nominal Demand V+Vst 168 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH Dam Mo S7_ Reservoir Foundation Interface Plane Foundation Rock Figure 7.4: Forces on the interface zone of the gravity dam monolith where Nst is the static normal force (resultant of self weight) and Vst is the static tangential force (hydrostatic) on the interface zone AB, V is the maximum dynamic tangential force (inertia and hydrodynamic). If the plane AB is not horizontal the forces have to be considered with proper components. The properties of the interface zone are represented by the coefficient of friction p and the total cohesion force C. Several assumptions were used in the previous chapter during the development of numerical models for the studied concrete gravity dam monolith. If these assumptions are taken into account Eq. (7.20) can be rewritten in the following form: (7.21) where m = 2,456,000 kg is the mass of the monolith, H = 43.2 m is the maximum water level of the reservoir, mHD = _Ly^_^_/^  is the added mass of water (Okamoto, 1973), h is the relative water level expressed in% of H, g is the acceleration due to gravity, y is specific mass 169 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH of water, and the friction coefficient u, = 1.00. The factor of safety can be calculated using the parameters from the previous section, that is for the mean value for the water level h = 80% and for the six values of the acceleration a475 . Results of this calculation are shown in the first three columns of Table 7.2. Table 7.2: Safety Factors Evaluated for Parameters Used in Reliability Study <*475 (g) Water Level (%) Factor of Safety PA Sliding 1cm PA Sliding 10cm 0.2 80 2.0 2.6E-6 4.6E-8 0.265 80 1.7 1.7E-5 4.4E-7 0.3 80 1.6 3.6E-5 1.1E-6 0.4 80 1.3 1.9E-4 9.0E-6 0.5 80 1.1 6.0E-4 3.8E-5 0.6 80 1.0 1.4E-3 1.1E-4 The results presented in the third column of Table 7.2 can be linked to the results of the reliability study with three random variables as follows. For example, the fourth row of Table 7.2, for a475= OAg shows the factor of safety FS= 1.3. If this is compared with the results using FE-based probabilities shown in Figure 7.3a, it can be concluded that this factor of safety corresponds to the annual probability of exceeding sliding of 1 cm PA = 1.9E-4, or to the annual probability of exceeding sliding of 10 cm PA = 9.0E-6. The values of PA for the other values of a475 were obtained from Figure 7.3a for the allowable sliding of 1 cm and 10 cm. These are shown in the fourth and the fifth columns of Table 7.2. The results presented in the first and the last rows of Table 7.2 can be interpreted as follows. The studied dam, at the site with a475=0.2g has the safety factor against base sliding of 2.0 and the annual probability of sliding more than 1 cm equal to 2.6E-6. The same dam at a site with 170 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH a475=0.6g has the safety factor of 1.0 and the annual probability of sliding more than 1 cm equal to 1.4E-3. Another way in which the results of the reliability analysis can be used is to obtain a relationship between the allowable sliding S0 and a475 for a given target probability of failure. For example, if the target annual probability of failure equal to 10E-5 is considered, the corresponding acceleration a475 can be obtained from Figure 7.3a for each allowable sliding. Then, for each a475, the corresponding factor of safety can be calculated using Eq. (7.21). The above described values are summarised in Table 7.3. Table 7.3: Safety Factors Evaluated for the Target Probability of 1E-5 Based on AEP of 1/475 Allowable Sliding So (cm) Acceleration °475 (g) Factor of Safety 1 0.25 1.79 2 0.28 1.66 5 0.35 1.44 10 0.41 1.31 15 0.45 1.23 20 0.48 1.17 The factors of safety given in the right column of Table 7.3 can be fitted with a quadratic parabola as shown in Figure 7.5. The fitted values can be used in Eq. (7.21) instead of the safety factors from Table 7.3: A + BS0 + C S 0 2 =  ^ h (7.22) aD(mHD + m) + .yg{mH) where A = 1.816, B = -0.073 and C = 0.0021 are constants obtained by fitting the values of safety factors with a least square technique. Eq. (7.22) gives the relationship in between S0 and 171 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH 0 5 10 15 Allowable Sliding (cm) Figure 7.5: Safety Factors for a Target Annual Probability of 1E-5 Based on AEP of 1/475 a475 at a target probability level of IE-5. This relationship can be used to find a475 (or S0) corresponding to S0 (or a475) i f other values than those used in the previous analysis are desired. Similar relationships could be also obtained for other probability levels. When interpreting the information in Figure 7.5, one should keep in mind that the relationship between the safety factor and the allowable sliding is for the same dam, for the constant probability of failure (1E-5) and for the same AEP (1/475). 7.7 INFLUENCE OF FOUNDATION PROPERTIES ON THE PROBABILITIES OF FAILURE The probability of failure of a given gravity dam monolith subjected to base excitations was studied in the previous sections. Loading parameters, such as the type of an earthquake, the PGA of the base excitations and the water level of the reservoir were considered as random variables in those analyses. The parameters of the system itself, such as the foundation rock flexibility, coefficient of friction of the foundation interface, and others, were considered 172 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH deterministic and constant. In some situations it is desirable to analyse a system, which is not exactly defined. In such a case its properties may vary at different locations of the system, and they should be treated as random variables. It was shown (Fenves and Chavez, 1996) that the amounts of sliding of a gravity dam are sensitive to variations in the friction coefficient of the foundation interface and the foundation rock flexibility. Influence of these two parameters on the amount of sliding of a dam monolith is studied in this subsection. The scope of this study was limited to several simulations with the FE model, but the results of these simulations were used to obtain annual probabilities of failure over a wide range of input parameters. The modulus of elasticity of the rock foundation was considered with the value of 15 GPa in the analytical study. For this value, for the Northridge earthquake record and the water level of 80%, the FE model yielded sliding of the dam monolith equal to 3.19 cm. When the modulus of elasticity was changed to 10 and 20 GPa, the obtained sliding was 3.33 and 3.19 cm, respectively. Therefore, it was concluded that the modulus of elasticity of the foundation rock, in the tested range, does not have significant influence on the amount of sliding. The situation was found different when the friction coefficient between the dam monolith and the foundation rock was varied. Therefore, more analyses were performed and the results from these are summarised in Table 7.5. A l l the analyses were performed with the Northridge Earthquake record and with the water level of 80%. The PGA values of 0.60g, 0.78g and 1.04g, were combined with the kinetic friction coefficients of 0.8,1.0 and 1.2. In addition to the results in terms of sliding for varied friction coefficients, Table 7.5 gives ratios of the amounts of sliding against those with p equal to 1.0. These are for p of 0.8 and 1.2, 173 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH respectively, in the third and sixth columns of the table. Using these ratios, the sliding for any obtained from the FE data, one using three and the other four random variables. It can be observed from the table that the annual probability of failure from the analysis with four random variables is about 1.3 times higher than that using three variables. Table 7.4: Example of Results for a site with a475=0Ag and for an Allowable Sliding of 5 cm Reliability Analysis with 3 Random Variables with 4 Random Variables Annual Probability of Failure 2.5E-5 3.2E-5 Random Variable Design Point Sensitivity Factor Design Point Sensitivity Factor PGA (g) 0.79 0.84 0.75 0.80 Water Level (%) 89 0.30 89 0.30 Error due to Different Earthquake Record 0.74 0.45 0.70 0.44 Friction Coefficient not applicable not applicable 0.89 0.26 The results in Table 7.4 are given in terms of components of a design point and sensitivity factors. It can be seen in the table that the components of the design point vary slightly for the peak ground acceleration and the error due to the earthquake record, but are almost the same for the water level. The sensitivity factors for the peak ground acceleration are the largest for both cases, followed by the error due to different earthquake record and the water level. The values of these three sensitivity factors are very similar for both types of analysis. The sensitivity factor for the friction coefficient is relatively small, which indicates that the influence of the friction coefficient on the probability of failure is not as large as, for example, that of the peak ground acceleration. It can be concluded that for the variability considered in the friction coefficient, the annual 174 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH probabilities of sliding of a dam by a specified amount did not change significantly. It would be an interesting study to calculate more sets of FE analyses, each for different friction coefficient, over various earthquake records, PGA's and water levels, and use these results to perform similar analyses as those in the previous subsections. However, time constraints on this project do not permit such analyses. Table 7.5: Results of FE Analyses with Varied Friction Coefficient kinetic friction coefficient p = 0.8 p= 1.0 p= 1.2 Sliding (cm) Ratio to Sliding for p= 1.0 Sliding (cm) Sliding (cm) Ratio to Sliding for p=1.0 PGA (g) 0.60 2.37 2.01 1.18 0.38 0.32 0.79 4.99 1.56 3.19 0.98 0.31 1.04 9.01 1.40 6.43 3.09 0.48 7.8 SUMMARY It was the objective of this chapter to develop a procedure to perform a reliability analysis in" order to find probabilities of sliding of a monolith of a full-scale gravity dam by a given amount. It was also desired to assess if the probabilities obtained from the SDOF-based results are in a good agreement with those from the results of FE analyses. First, the analytical response data previously obtained during this study were reduced and linearised in order to capture general trends in the data. Then, random variables for the reliability analyses were identified and a total of three analyses were performed. Finally, a link between the results of one of the analyses and safety factors against sliding was created. The results of the reliability analysis with two random variables, presented in Figure 7.2, 1 7 5 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH indicate the overall low annual probabilities of sliding of the studied structure. These vary significantly with the water level of the reservoir and with the allowable amount of sliding considered. They range from about 7E-5 for 100% water level and 1 cm of allowable sliding to 2E-8 for the case of 60% water level and 20 cm of sliding allowed. Probabilities obtained from the FE results are in a good agreement with those from the SDOF results. The results of the study with three random variables yielded probabilities of failure i f the acceleration corresponding to a given return period (AEP) and considered as one of the random variables, was a parameter. The obtained probabilities from the FE and SDOF data were presented in Figure 7.3. These results using both sources of data are in a good agreement even though some differences could be noticed. Using the FE data, the annual probabilities of failure range from 1.1E-8 for the a475 acceleration of 0.2g and 20 cm of allowable sliding to 1.3E-3 for 0.6g and 1 cm of sliding. If these results are compared with those from the analysis with two random variables, a very good match is obtained for both the FE and SDOF-based probabilities. This can be concluded if the range of probabilities for water level of 80% from Figure 7.2a is compared with the range for the a475 acceleration of 0.265g in Fig 7.3a. These ranges correspond to each other because the water level of 80% was the mean value water level in the study with three variables and the value of PGA equal to 0.265g was the a475 acceleration considered in the analysis with two variables. Similar conclusion can be made if the corresponding ranges of annual probabilities from Figure 7.2b are compared with those from Fig 7.3b. The results of the reliability study were expressed in terms of safety factors. This provides a 176 CHAPTER 7 RELIABILITY STUDY OF A FULL-SCALE DAM MONOLITH link between the reliability study and the traditional concept used in dam safety analysis. It was also shown, how the results of this study can be used to obtain relationship between the allowable sliding and the seismicity of the dam's site for a given target probability and return period of the earthquake. In the end of the chapter, sensitivity of probabilities of failure against two parameters of the foundation rock was studied. This study showed that small variations in the modulus of elasticity of the foundation rock are not as significant as those in the friction coefficient. It was shown, that if the friction coefficient is considered random with the mean being the same as the deterministic value used previously, the annual probabilities of failure increased slightly but this was not significant. It was observed that the overall results in terms of various annual probabilities of sliding obtained from the SDOF-based data do not vary significantly from those of the FE analyses. Therefore, it can be concluded that the SDOF model is an adequate approximation for the reliability analyses of concrete gravity dams of similar characteristics as the structure studied here. This is a desirable conclusion, since the FE model, due to the longer computational time needed for analysis it requires, makes the reliability analysis more demanding. 177 8 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 8.1 SUMMARY AND CONCLUSIONS This thesis addressed the topic of base sliding of concrete gravity dams in experimental, analytical and reliability studies. The background to the topic was discussed and the pertinent research to-date reviewed. It can be stated that the objectives of all three studies were achieved and the studies yielded ample results, observations and comments. Summary and conclusions from the experiments, the analytical work and the reliability study are presented in the following subsections. 8.1.1 Experimental Study During the preliminary experimental study, a model of a single monolith of a concrete gravity dam was designed and built. The components of the experimental model and setup were described and the similitude aspects were discussed. The enhancements of the experimental setup and testing procedures as a result of the preliminary tests were described. Further experimental study of the model for the dam monolith included static and dynamic shake table tests and impact hammer tests. A large amount of data was gathered during the tests. These data were used for the verification and calibration of the numerical models during the analytical study. Shake table testing manifested itself as a powerful tool to measure sliding of base excited models of concrete gravity dams. The natural frequencies of the experimental model obtained from the impact hammer tests were found strongly dependent on the boundary conditions. The fundamental frequency of the 178 CHAPTER 8 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS unbonded model setup was determined to be in a range from 27 to 34 Hz while the one for the bonded model setup was 66 Hz. The static and kinetic friction coefficients of the interface between the model and its base were extracted using the data from the static shake table tests. Both coefficients were found to increase a little as the tests progressed During the shake table dynamic tests, the response of the experimental unbonded model of a gravity dam monolith was measured in terms of sliding of the model preloaded by a simulated hydrostatic force and subjected to different types of base excitations of varying amplitude and dominant frequency. The results showed that different records, either harmonic motions or synthetic earthquakes, made the model respond in a similar way. The amplitude of base acceleration and its dominant frequency were the parameters controlling the response. In Chapter 4, the dominant frequencies of the base excitations applied to the model were divided into three groups, covering the testing range from 5 to 25 Hz. This grouping corresponded to the changing character of the response of the experimental model in the shake table dynamic tests. This response included downstream sliding (Group 1 - frequencies 5 to 12.5 Hz), sliding combined with in-plane rocking (Group 2 - frequencies 15 to 20 Hz), and dominant rocking with no sliding or sliding either downstream or upstream (Group 3 - frequencies 22.5 to 25 Hz). A n upstream sliding consistently observed at high amplitudes of base accelerations with the frequencies from Group 3 was concluded to be characteristic for the model response only. 8.1.2 Analytical Study A total of three numerical models of varied complexity were developed during the analytical 179 CHAPTER 8 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS study of the scaled dam monolith model. These included SDOF, 3DOF and 2-D FE models. The modal characteristics of the experimental setup obtained from the tests were used to calibrate the FE model. The numerical models were developed in order to simulate the response of the experimental setup preloaded by a simulated hydrostatic force and subjected to specified base excitations. The results using the SDOF, 3DOF and FE numerical models were compared with each other and with the experimental data gathered during the shake table tests. The SDOF and 3DOF models could simulate satisfactorily the response at low frequency the of base excitations, but limitations of these models manifested themselves when the frequencies were increased. It was observed, that the rigid models could simulate acceptably the sliding response of the experimental model up to 15 Hz, which was about a half of its fundamental natural frequency. The FE model could simulate satisfactorily the shake table tests throughout the entire range of tested frequencies and amplitudes. Analysis of combined sliding/rocking response of the 3DOF rigid numerical model was performed during the analytical study. This analysis showed that for the considered geometry of the rigid block and friction coefficient of the block-base interface, the base sliding was always initiated before rocking. Therefore, no rocking was observed in the results from the simulations using the 3DOF rigid model. The results of the 3DOF model using a commercial software were similar to those from the SDOF model, but it was more convenient to use the latter for the simulations during the analytical study on a full-scale monolith. The SDOF and FE models were selected for the numerical analysis of a 45 m high monolith of 180 CHAPTER 8 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS a concrete gravity dam. The models were modified to describe the full-scale structure and to capture the hydrodynamic effects of the reservoir. Modal characteristics of the bonded and unbonded dam monolith were obtained using the FE model. The FE and SDOF models were used to simulate the response of the dam monolith loaded by the hydrostatic and hydrodynamic pressures and subjected to three historic earthquakes of varying PGA. The results using the FE and SDOF models in terms of the base sliding of the monolith were compared. In the analytical study of the full-scale monolith, a set of SDOF simulations yielded sliding values close to those from the FE analyses. The sliding of the monolith using the FE model was initiated at lower base accelerations than it was during SDOF simulations. However, the SDOF model exhibited higher sensitivity to the PGA. A similar trend was observed for the reservoir's water level. The results from the SDOF and FE models were close to each other for the maximum PGA, which was about \g, and for high reservoir's water levels. 8.1.3 Reliability Study The reliability study began with the reduction of data from the numerical simulations. This was done to obtain the response surface describing base sliding of a concrete gravity dam monolith preloaded by the reservoir's effects and subjected to specified base excitations. Then, random variables were identified, described and their basic statistics were obtained. The random variables, including the PGA, the error from the varying characteristics of the earthquake time history, and the reservoir's water level were used in three reliability studies to determine annual probabilities of failure. The failure was defined as exceeding a specified amount of base sliding of the monolith during an earthquake. 181 CHAPTER 8 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS The annual probabilities of failure were obtained considering the three random variables. Using the data from FE analyses, the annual probabilities of failure ranged from LIE-8 for the mean PGA of 0.2g and 20 cm of the allowable sliding to 1.3E-3 for the mean PGA of 0.6g and 1 cm of allowable sliding. It was observed that the probabilities of sliding obtained from the SDOF data did not vary significantly from those of the FE analyses. Therefore, it can be concluded that the SDOF model is an adequate approximation for reliability analyses of concrete gravity dams of similar characteristics as the structure studied here. The results of the study were based on the annual exceedence probability (AEP) of 1/475, which is a typical value used in building design. It was shown how the results can be used for safety calculations of concrete gravity dams, in which different AEP's may be required depending on the consequence classification of the dam. In addition, the results of the reliability study were expressed in terms of the safety factors against sliding. A relationship between the design acceleration and the allowable amount of sliding was established for a selected target probability of failure. This provided a link between the reliability study and the traditional concept used in the dam safety analysis. It was also shown, how the results of this study can be used to obtain relationship between the allowable sliding and the design acceleration for a given target probability. The influence of randomness of the foundation rock on the amount of sliding was also studied. This study showed that small variations in the modulus of elasticity of the foundation rock are not as significant as those in the friction coefficient. It was shown, that i f the friction coefficient 182 CHAPTER 8 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS is considered random with the mean being the same as the deterministic value used previously, the annual probabilities of failure increased only slightly. 8.2 RECOMMENDATIONS FOR FURTHER RESEARCH During this study, a large amount of valuable information was gathered. However, certain simplifying assumptions had to be accepted in the course of the study and some of these could be removed in further studies in order to obtain information applicable to a wider range of problems. In addition, a few problems identified during the project were not addressed since these were beyond the scope of the study. Recommendations for further research resulting from these facts are summarised in this section. During the experimental study, the sliding surfaces simulating the contact between the model and its base, were made of concrete, which was the material of different properties than the rest of the model. This was because the sliding surfaces were designed to last over many tests. After an extensive material research, the sliding surfaces could be designed from some other material, which would be of properties closer to those of the model's material and durable at the same time. In addition, future experiments should address phenomena such as: • simulation of vertical earthquake motions, • simulation of bonding between the dam and its foundation, • simulation of hydrodynamic and uplift forces of the reservoir. The FE model used in this study could be further developed. It is recommended in the A N S Y S documentation (SAS IP, 1996c) that L S - D Y N A explicit integration program should be used for highly geometrically nonlinear problems more exact results. This option should be investigated. 183 CHAPTER 8 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS It is the author's opinion that the solution to the classical mathematical equations (SDOF model) would probably be the most promising research approach when other factors, such as vertical ground motions and uplift, are to be considered. In addition, an oscillator could be used instead of the rigid block to simulate effects of the fundamental mode of vibration. Models other than Coulomb friction could be used during numerical simulations and these could be calibrated using results of cyclic tests, which were conducted as additional experimental study during this project. Data from these tests are available at the Department of Civi l Engineering at U B C . In current practice, uplift in cracks in a concrete gravity dam is typically assumed to remain unchanged during the earthquake. However, following the earthquake, uplift pressure in earthquake-induced cracks would have time to build up. Post-earthquake safety is further jeopardized should there be significant after-shocks with a relatively long time lag after the main shock. Therefore, assessing post-earthquake safety of the dam is as important as assessing its safety during the earthquake. Future research should consider including post-earthquake safety in the numerical model. 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Periodic Response of a Sliding Oscillator System to Harmonic Excitatioa Earthquake Engineering and Structural Dynamics, Vol . 11, pp. 135-146. 189 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SLIP TESTS AND SIMULATIONS A finite element numerical model to simulate response of the experimental model due to shake table motions was developed in Chapter 5. In this chapter, the experimental model unbonded at the base and preloaded by a simulated hydrostatic force was created using commercial package A N S Y S (Swanson Analysis Systems, Inc. 1996). During the experimental part of the study described in Chapter 4, several shake table tests with low level excitations were conducted. The objective of these tests was to generate data for verification of the FE model for response with no sliding of the model. Such verification is described in this appendix. Figures A . l to A.9 show selected motions of the experimental model measured during the shake table tests and those calculated by FE simulations of the same tests. These motions include for each test: • Horizontal acceleration of the model relative to the shake table measured or calculated at the top of the upstream side of the model. This is shown in part a of the figures. Horizontal acceleration of the model relative to the shake table measured or calculated at the base on the upstream side of the model. This is plotted in part b of the figures and the legend shown in this part also holds for part a. The plotted experimental acceleration was reduced by a fraction of the measured acceleration at the top of the model (part a) in order to take into account that the sensor was not located exactly at the base of the dam, but it was attached 18 cm above the plane through foundation interface. • Shake table horizontal acceleration plotted in part c. • Shake table horizontal displacement plotted in part d. Each of the Figures A . l to A.9 presents a 1 second long close-up on results from one test with harmonic base excitations. The amplitudes of shake table motions for all tests were about 0.3g, but the frequency of excitation varied for each tests ranging from 5 Hz to 25 Hz as can be seen in captions at the figures. The observations made from the figures follow. Shapes of the shake table acceleration plots are close but not exactly the same as those of a closed-form harmonic excitation. This distortion is due to the interaction of the shake table with the model during the tests. Since this distortion may be even amplified for tests with high level of excitations, it is necessary to use acceleration records measured during the tests for the numerical simulations of the tests. Using the closed-form base excitations as it was done in the previous study (Rudolf 1998) brings uncertainty to the comparison with the experimental results, which can be eliminated by using measured acceleration records. The plots of experimental relative accelerations at the base of the monolith, presented in part b of Figures A . l to A.9, show that some differential motions with respect to the shake table were measured during no-slip tests in all frequencies. This was probably caused by the fact that the upper surface plate was attached to the experimental model only at the heel and toe of its base, 190 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SUP TESTS AND SIMULATIONS which permitted differential accelerations with amplitudes ranging from 0.05g to O.lg. The observed differential motions are small and most likely did not affect significantly the sliding of the experimental model during the shake table tests including sliding of the model. It can be also observed from the plots that similar response was not found in results from the numerical model. This was due to the fact that in the FE model the upper surface plate was rigidly attached to the monolith along the entire base, which did not allow any differential motions such as those observed in the experimental results. The upper surface plate of the FE model could be attached similarly as it was in the experimental model, but doing this would have required to generate a number of new contact elements between the monolith and the plate and increase the computational effort. Considering this and the fact that the upper surface plate has no meaning for the case of a full-scale concrete gravity dam, the plate was modelled firmly attached to the monolith in the FE study. Comparisons of measured and calculated relative accelerations at the top of the monolith show that these are generally in a good agreement. This conclusion can be made from visual observation of the time histories shown in Figures A.l to A.9, parts a and b. The amplitudes of the measured accelerations are in some cases larger than those calculated during simulations, in other cases in simulations, and in some cases they are about the same. It can be said that the amplitudes are generally in a good agreement. A good agreement between experimental and analytical results from no-slip tests was reached by decreasing the damping of the FE model compared to previous work of Rudolf (1998). Damping in ANSYS model can be approximated using the concept of Rayleigh damping (Chopra 1995) involving stiffness and mass proportional parameters ct and P. In the previous work, these parameters were specified based on the time history results of impact hammer tests on a full experimental setup. The decay of amplitudes of horizontal acceleration at the top of the monolith was measured. Combined with the known natural frequencies, the calculated cc and P were 8.0 and 0.0003, respectively, resulting in a damping ratio of approximately 5% in the first two modes. In the current work, the damping parameters a and P were considered 11.5 and 0.000065, which corresponded to 3.5% of critical damping for the first and third modes. These values of cc and P are very close to those used in the previous work, but a better agreement between experimental and analytical accelerations at the top of the model was achieved if 3.5% of critical damping for the first and third modes was used. Therefore slightly different damping parameters than those obtained from impact hammer test measurement were adopted. It can be concluded from the above observations that the FE model could capture reasonably well the behaviour of the model for selected no-slip tests. A little difference between experimental and FE model was found in the behaviour of the foundation interface, but this difference was not significant and was explained. 191 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SLIP TESTS AND SIMULATIONS Q 4 a) Model Top Acceleration Relative to the Shake Table Tirne^sec) b) Model Base Acceleration Relative to the Shake Table 5.00 5.25 Q ^  c) Shake Table Acceleration 5 50 Time (sec) TO C o CO CD O O < 5.00 5.25 Q 4Q d) Shake Table Displacement 0.20 5.50 Time (sec) 5.50 Time (sec) Figure A. 1: Response of Dam Model to Harmonic Excitation at Frequency of 5 Hz 192 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SLIP TESTS AND SIMULATIONS Q 4 a) Model Top Acceleration Relative to the Shake Table 0.2 5.00 5.25 _ 5.50 x 5.75 Time (sec) n . b) Model Base Acceleration Relative to the Shake Table 0.4 —i— 5.00 5.25 Q 4 c) Shake Table Acceleration 5.50 Time (sec) c CD E 0.00 CD 1-0.10 5.50 Time (sec) Figure A .2: Response of Dam Model to Harmonic Excitation at Frequency of 7 . 5 Hz 193 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SLIP TESTS AND SIMULATIONS a) Model Top Acceleration Relative to the Shake Table 3 c o 2 j D CD O O < Time\sec) _ . b) Model Base Acceleration Relative to the Shake Table 0.4 - r — 3 0 2 c o ro 0.0 JD CD O O < -0.2 -0.4 5.00 5.25 5 50 Time (sec) 5.00 5.25 0 ,|Q d) Shake Table Displacement 0.05 5.50 Time (sec) 5.50 Time (sec) 5.75 Experiment FE Model 6.0 Figure A.3: Response of Dam Model to Harmonic Excitation at Frequency of 10 Hz 194 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SLIP TESTS AND SIMULATIONS 0 4 a) Model Top Acceleration Relative to the Shake Table 3 -0.2 5.00 5.25 _ 5.50 x Time (sec) _ . b) Model Base Acceleration Relative to the Shake Table 0.4 -r— Q) I -0.2 -I -0.4 Experiment FE Model 5.00 5.25 0 4 c) Shake Table Acceleration 5.50 Time (sec) 5.75 6.0 3 c 0.2 -o -« (0 L _ 0.0-d) ccel -0.2-< -04 5.00 5.25 d) Shake Table Displacement 5.50 Time (sec) 5.00 5.25 5.50 Time (sec) 5.75 6.0 Figure A.4: Response of Dam Model to Harmonic Excitation at Frequency of 12.5 Hz 195 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SLIP TESTS AND SIMULATIONS 0 4 a) Model Top Acceleration Relative to the Shake Table _ 5.50 . Time (sec) _ . b) Model Base Acceleration Relative to the Shake Table 0.4 - r — 0.2 2 0.0 0) o 3 -0.2 -0.4 - v w w w w w v w ^ Experiment FE Model 5.00 5.25 0 4 c) Shake Table Acceleration 5 50 Time (sec) 5.75 6.0 5.50 Time (sec) Figure A.5: Response of Dam Model to Harmonic Excitation at Frequency of 15 Hz 196 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SUP TESTS AND SIMULATIONS Q 4 a) Model Top Acceleration Relative to the Shake Table 0.2 5.00 5.25 _ 5.50 . Time (sec) _ . b) Model Base Acceleration Relative to the Shake Table 0.4 -r-1 0.2 0.0 CD O 3 -0.2 -0.4 Experiment FE Model 5.00 5.25 _. 5.50 . Time (sec) 5.75 6.0 0.4 -, "55 0 . 2 -O •4—* 0 . 0 -CD ccel -0.2 -< - 0 4 5.00 5.25 ) 0 2 d) Shake Table Displacement 5.50 Time (sec) 5.50 Time (sec) Figure A.6: Response of Dam Model to Harmonic Excitation at Frequency of 17.5 Hz 197 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SUP TESTS AND SIMULATIONS a) Model Top Acceleration Relative to the Shake Table 5.50 Time (sec) _ . b) Model Base Acceleration Relative to the Shake Table 0.4 -T—^ ^ 0.2 0.0 -0.2 -0.4 : /VW\MA/VWWVWW\M Experiment FE Model 5.00 5.25 0 4 c) Shake Table Acceleration Time (sec) 5.75 6.0 5.50 Time (sec) Figure A.7: Response of Dam Model to Harmonic Excitation at Frequency of 20 Hz 198 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SLIP TESTS AND SIMULATIONS 0 a) Model Top Acceleration Relative to the Shake Table ai 0.5 f. '• *. •*• -1.0 5.00 5.25 5 50 Time (sec) 5.75 6.0 Q b) Model Base Acceleration Relative to the Shake Table 0 c) Shake Table Acceleration Time (sec) ® 0.5 c o £» 0.0 -I 1 -0.5-1 < -1.0 5.00 5.25 d) Shake Table Displacement i 5.50 Time (sec) 5.75 6.0 5.00 5.25 5.50 Time (sec) 5.75 6.0 Figure A.8: Response of Dam Model to Harmonic Excitation at Frequency of 22.5 Hz 199 APPENDIX A - PLOTS OF MODEL MOTIONS FROM NO-SUP TESTS AND SIMULATIONS 0 a) Model Top Acceleration Relative to the Shake Table 0.5 2 0.0 4 -0.5 4 -1.0 \ r.K r.k :.k 5.00 5.25 1.0 ^ 0.5 Time (sec) b) Model Base Acceleration Relative to the Shake Table 5.75 6.0 2 0.0 4 / -0.5 A -1.0 5.00 5.25 1 o c) Shake Table Acceleration 0.5 5.50 Time(sec) 5.00 5.25 Q Q 2 d) Shake Table Displacement 5.50 Time (sec) 5.50 Time (sec) Experiment FE Model 5.75 6.0 Figure A.9: Response of Dam Model to Harmonic Excitation at Frequency of 25 Hz 200 APPENDIX B - SAMPLES OF INPUT F ILES FOR ANSYS ANALYSIS A large number of numerical simulations using ANSYS were performed during this study. In order to optimize computing time the simulations were run in series using batch files. One simulation typically consisted of preprocessing, solution and postprocessing phases, all being controlled by commands from separate files. This appendix shows a sample of the preprocessing file containing description of the studied system. In addition, a sample of the solution file is shown, which contains ANSYS commands to control the nonlinear time-domain dynamics analysis. S A M P L E OF A PREPROCESSING FILE: !-- set f i l t e r s KEYW,PR_SET,1 KEYW,PR_STRUC,1 KEYW,PR_THERM,0 KEYW,PR_ELMAG, 0 KEYW,PR_FLUID,0 KEYW,PR_MULTI, 0 KEYW,PR_CFD,0 KEYW,LSDYNA,0 /PMETH,OFF i preprocessor /PREP7 ! Define PLANE STRESS element ET,1,PLANE42!-- plane q u a d r i l a t e r a l l i n e a r element KEYOPT,1,1,0 KEYOPT,1,2,0 KEYOPT,1,3,3!-- K03=3 plane stress with thickness KEYOPT,1,5,0 KEYOPT,1,6,0 i r e a l constants f o r plane element R , l , l . , !-- thickness i n [m] !__ material properties f o r CONCRETE M P , E X , 1 , 2 7 e 9 ! e l a s t i c modulus i n [Pa] MP,EY,1,27e9!-- e l a s t i c modulus i n [Pa] MP,DENS,1,2580!-- density i n [kg/m3] MP,NUXY,1,0.22!-- Poisson r a t i o i material properties f o r FOUNDATION ROCK MP,EX,2,15e9!-- e l a s t i c modulus MP,EY,2,15e9 MP,DENS,2,10.!-- density i n [kg/m3] - almost massless foundation MP,NUXY,2,0.25!-- Poisson r a t i o i Define CONTACT element ET,2,CONTAC48 !-- node to surface contact element KEYOPT,2,1,0 KEYOPT,2,2,0 !--K03=2 r i g i d coulomb f r i c t i o n !CAUTION: gets unstable! small pivot. KEYOPT,2,3,1!-- K03=l e l a s t i c coulomb f r i c t i o n KEYOPT,2,7,1!-- K07=l use 'reasonable' time-step @ contact time i r e a l constants for contact element R,2,1.8e9,4ell, ,1.05, , , !-- KN, KT, ,FACT ( s t a t i c / k i n e t i c ) 201 APPENDIX B - SAMPLES OF INPUT FILES FOR ANSYS ANALYSIS i material properties for contact element MP,MU,3,1.!-- k i n e t i c f r i c t i o n coeff. i Define concentrated mass element ET,3,MASS21 KEYOPT,3,3,0 ! 6 components of mass - 3 masses, 3 rotatory i n e r t i a KEYOPT,3,2,0 ! element coord system p a r a l e l to global coord system R, 3, 201000.,0,0,0,0,0 R,4,282000.,0,0,0,0,0 R,5,176000.,0,0,0,0,0 R, 6,190000.,0,0,0,0,0 R, 7,173000.,0,0,0,0,0 R,8, 47000.,0,0,0,0,0 added mass added mass added mass added mass added mass added mass i b u i l d model down up - Load Case 1, NHydro 1 - Load Case 1, NHydro 2 - Load Case 1, NHydro 3 - Load Case 1, NHydro 4 - Load Case 1, NHydro 5 Load Case 1, NHydro 6 K, 10,6.3000,45.000,0!--K, 9, 0.000,45.000,0!--K, 8,13.560,34.000,0!--K, 7, 0.000,34.000,0!--K, 6,20.820,23.000,0!--K, 5, 0.000,23.000,0!--K, 4,28.740,11.000,0!--K, 3, 0.000,11.000,0!--K, 2,36.000, 0.000,0!--K, 1, 0.000, 0.000,0!--i key points 9 10 I top \ 8 tx UpStream=U x=0-cg \ DownStream=D \ 4 tx |y=0 i foundation rock K,11,-20.000, 0.000,0!--K,12,-20.000,-20.000,0!-- 11-K,13, 56.000,-20.000,0!-- | 56.000, 0.000,0!-- 12--3.111,-20.000,0 39.111,-20.000,0 39.111, 0.000,0 -3.111, 0.000,0 K,14, K,15, K,16, K,17, K,18, i K,19, 0.000, 17.000,0! K,20, 0.000, 39.500,0! --18-----foundation rock --15 16--17 14 -13 keypoints on US face for added masses for added mass Nhydro3 for added mass Nhydro6 i create horizontal l i n e s L,l,2 !-- create l i n e CM,LineHl,LINE!-- create a named component out of l i n e LSEL,NONE !-- c l e a r s e l e c t i o n L,3,4 CM,LineH2,LINE LSEL,NONE L,5,6 CM,LineH3,LINE LSEL,NONE L,7,8 CM,LineH4,LINE LSEL,NONE L,9,10 CM,LineH5,LINE LSEL,NONE L,ll,18 !-- base l e f t top CM,LineFTl,LINE LSEL,NONE L, 18,17 ! base mid top CM,LineFT2,LINE LSEL,NONE L, 17,14 ! base r i g h t top CM,LineFT3,LINE LSEL,NONE L, 12,15 ! base l e f t bottom CM,LineFBl,LINE LSEL,NONE 202 APPENDIX B - SAMPLES OF INPUT FILES FOR ANSYS ANALYSIS L, 15,16 ! base mid bottom CM,LineFB2,LINE LSEL,NONE L, 16,13 ! base r i g h t bottom CM,LineFB3,LINE LSEL,NONE ; US face v e r t i c a l l i n e s L,l,3 CM,LineUl,LINE LSEL,NONE L,3,5 CM,LineU2,LINE LSEL,NONE L,5,7 CM,LineU3,LINE LSEL,NONE L,7,9 CM,LineU4,LINE LSEL,NONE L,12,11 CM,LineFVl,LINE! base l e f t v e r t i c a l LSEL,NONE L,15,18 CM,LineFV2,LINE! base mid-left v e r t i c a l LSEL,NONE ; DS face v e r t i c a l l i n e s L,2,4 CM,LineDl,LINE LSEL,NONE L,4,6 CM,LineD2,LINE LSEL,NONE L,6,8 CM, LineD3 , LINE LSEL,NONE L,8,10 CM,LineD4,LINE LSEL,NONE L,16,17 CM, LineFV3,LINE! base mid-right v e r t i c a l LSEL,NONE L,13,14 CM,LineFV4,LINE! base r i g h t v e r t i c a l LSEL,NONE i create areas KSEL,ALL A, 1,2,4,3!-- bottom part - t r a n s i t i o n CM,Areal,AREA ASEL,NONE A,3,4,6,5 CM,Area2,AREA ASEL,NONE A, 5,6,8,7!-- second t r a n s i t i o n CM,Area3,AREA AS EL,NONE A,7,8,10,9!-- top CM,Area4,AREA A, 12,15,18,11!-- bottom plate - l e f t part CM,AreaFl,AREA AS EL,NONE A, 15,16,17,18!-- bottom plate - mid part CM,AreaF2,AREA AS EL,NONE A, 16,13,14,17!-- bottom plate - ri g h t part CM,AreaF3,AREA 203 APPENDIX B - SAMPLES OF INPUT FILES FOR ANSYS ANALYSIS ASEL,NONE spe c i f y subdivisions se l e c t a l l components 6!-- sp e c i f y d i v i s i o n horizontal l i n e s 4 4 2 2 ,2 ,7 ,2 ,2 , 7 ,2 1!-- sp e c i f y d i v i s i o n along US face 2 1 2 1!-- sp e c i f y d i v i s i o n along DS face 2 l 2 CMSEL,ALL !-LESIZE,LineHl, LESIZE,LineH2, LESIZE,LineH3, LESIZE,LineH4, LESIZE,LineH5, LESIZE,LineFTl LESIZE,LineFT2 LESIZE,LineFT3 LESIZE,LineFBl LESIZE,LineFB2 LESIZE,LineFB3 LESIZE,LineUl, LESIZE, LineU2, LESIZE,LineU3, LESIZE,LineU4, LESIZE,LineDl, LESIZE,LineD2, LESIZE,LineD3, LESIZE,LineD4, i mesh monolith element type plane material type 1 (concrete) r e a l contants 1 !-- shape f o r c i n g to quads i n uniform areas !-- mesh areas with concrete TYPE,1 MAT, 1 REAL,1 ESHAPE,2 AMESH,Area2 AMESH,Area4 ESHAPE,3 AMESH,Areal AMESH,Area3 i mesh rock foundation -- mesh for t r a n s i t i o n areas !-- mesh t r a n s i t i o n areas use quads i f possible. TYPE,1 MAT, 2 REAL,1 ESHAPE,2 •- element type 1 •- material type 2 (foundation rock) •- r e a l contants 1 !-- shape f o r c i n g to quads i n uniform areas AMESH,AreaFl !-- mesh areas with rock foundation - l e f t part AMESH,AreaF2 !-- mesh areas with rock foundation -mid part AMESH,AreaF3 !-- mesh areas with rock foundation - r i g h t part /PNUM,ELEM,1!-- dis p l a y numbering EPL0T!-- p l o t elements generate elements with added masses TYPE,3 ! -REAL,3!--KMESH,1 TYPE,3 ! -REAL,4 !-KMESH,3 TYPE,3 !-REAL,5 !-KMESH,19 TYPE,3 !-REAL,6 !-KMESH,5 TYPE,3 ! -REAL,7 !-KMESH,7 TYPE,3 ! -REAL,8 !-KMESH,20 • element type MASS21 added mas - Load Case 1, NHydro 1 - keypoint 1 element type MASS21 added mas - Load Case 1, NHydro 2 - keypoint 3 element type MASS21 added mas - Load Case 1, NHydro 3 - keypoint 19 element type MASS21 added mas - Load Case 1, element type MASS21 added mas - Load Case 1, NHydro 4 - keypoint 5 NHydro 5 - keypoint 7 element type MASS21 added mas - Load Case 1, NHydro 6 - keypoint 20 NSEL,ALL NUMMRG,NODES ,0.01! NSEL,NONE merge duplicate nodes tolerance of 0.01 m 204 APPENDIX B- SAMPLES OF INPUT FILES FOR ANSYS ANALYSIS I Setup contacts NSEL,S,NODE,,25,25 CM,Cont1,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,27,27 CM,Cont2,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,28,28 CM,Cont3,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,29,29 CM,Cont4,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,30,30 CM,Cont5,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,31,31 CM,Cont6,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,26,26 CM,Cont7,NODE!-- make selected into a component NSEL,NONE i Setup targets NSEL,S,NODE,,43,43 NSEL,A,NODE,,62,62 CM,Targl,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,62,62 NSEL,A,NODE, ,63,63 CM,Targ2,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE, ,63,63 NSEL,A,NODE,,64,64 CM,Targ3,NODE!-- make selected i n t o a component NSEL,NONE NSEL,S,NODE,,64,64 NSEL,A,NODE, ,65,65 CM,Targ4,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,65,65 NSEL,A,NODE,,66,66 CM,Targ5,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,66,66 NSEL,A,NODE, ,67,67 CM,Targ6,NODE!-- make selected into a component NSEL,NONE NSEL,S,NODE,,67,67 NSEL,A,NODE, ,59,59 CM,Targ7,NODE!-- make selected into a component NSEL,NONE i set props f o r contact element and generate elem's TYPE,2 MAT, 3 REAL,2 GCGEN,Contl,Targl!-- generate contact elements GCGEN,Cont2,Targ2 GCGEN,Cont3,Targ3 GCGEN,Cont4,Targ4 GCGEN,Cont5,Targ5 GCGEN,Cont6,Targ6 GCGEN,Cont7,Targ7 i generate l a b e l s for nodes of i n t e r e s t NSEL,NONE ! points of a p p l i c a t i o n of hydro load - numbered from bottom 205 APPENDIX B - SAMPLES OF INPUT FILES FOR ANSYS ANALYSIS !NSEL,S,LOC,X, 0.000 !NSEL,R,LOC,Y, 0.000 ! at the bottom, keypoint 1 !*GET,NHydrol,NODE,,NUM,MIN!-- assign i t s number to va r i a b l e !NSEL,NONE NSEL,S,LOC,X, 0.000 NSEL,R,LOC,Y, 11.000 ! at 11 m from bottom, keypoint 3 *GET,NHydro2,NODE,,NUM,MIN!-- assign i t s number to va r i a b l e NSEL,NONE NSEL,S,LOC,X, 0.000 NSEL,R,LOC,Y, 17.000 ! at 17 m from bottom, keypoint 19 *GET,NHydro3,NODE,,NUM,MIN!-- assign i t s number to va r i a b l e NSEL,NONE NSEL,S,LOC,X, 0.000 NSEL,R,LOC,Y, 23.000 ! at 23 m from bottom, keypoint 5 *GET,NHydro4,NODE,,NUM,MIN!-- assign i t s number to var i a b l e NSEL,NONE NSEL,S,LOC,X, 0.000 NSEL,R,LOC,Y, 34.000 ! at 34 m from bottom, keypoint 7 *GET,NHydro5,NODE,,NUM,MIN!-- assign i t s number to va r i a b l e NSEL,NONE NSEL,S,LOC,X, 0.000 NSEL,R,LOC,Y, 39.500 ! at 39.5 m from bottom, keypoint 20 *GET,NHydro6,NODE,,NUM,MIN!-- assign i t s number to va r i a b l e NSEL,NONE NSEL,S,LOC,X,-3.Ill ! top of foundation at heel of dam NSEL,R,LOC,Y, 0.000 *GET,NStUs,NODE,,NUM,MIN NSEL,NONE NSEL,S,LOC,X,0.000 !-- top US corner of monolith NSEL,R,LOC,Y,45.000 *GET,NToUs,NODE,,NUM,MIN NSEL,NONE NSEL,S,LOC,X,0.000 !-- bottom US corner of monolith i s also NHydrol node NSEL,R,LOC,Y,0.000 *GET,NBaUs,NODE,,NUM,MIN NSEL,NONE NSEL,S,LOC,X,36.000 !-- bottom DS corner of monolith NSEL,R,LOC,Y,0.000 *GET,NBaDs,NODE,,NUM,MIN NSEL,NONE ALLSEL,ALL,ALL!-- s e l e c t everything i apply boundary conditions KBC,1!-- stepped boundary-conditions a p p l i c a t i o n CMSEL,S,LineFBl!-- s e l e c t l i n e s along underside of bottom p l a t e CMSEL,A,LineFB2!-- add l i n e to s e l e c t i o n CMSEL,A,LineFB3!-- add l i n e to s e l e c t i o n . CMSEL,A,LineFVl!-- add l i n e to s e l e c t i o n CMSEL,A,LineFV4!-- add l i n e to s e l e c t i o n NSLL,S,1 !-- sel e c t nodes along selected l i n e s CM,CNSt,NODE !-- make them into a component D,CNSt,UY,0 !-- r e s t r a i n t r a n s l a t i o n s i n y d i r e c t i o n . D,CNSt,UX,0 !-- r e s t r a i n t r a n s l a t i o n s i n x d i r e c t i o n . ALLSEL,ALL,ALL EPLOT i apply damping ALPHAD,11.5 !-- value of alpha f o r 1st and 3rd modes and 2% of damping BETAD,0.000065 !-- value of beta f o r 1st and 3rd modes and 2% of damping ALLSEL,ALL,ALL!-- s e l e c t everything 206 APPENDIX B - SAMPLES OF INPUT FILES FOR ANSYS ANALYSIS S A M P L E OF A SOLUTION FILE: ! s o l u t i o n f i l e to simulate response of a 45 m t a l l f u l l scale dam monolith i define general s o l u t i o n c r i t e r i a ANTYPE,4 !-- analysis type: transient dynamic TRNOPT,FULL !-- f u l l s o l u t i o n LUMPM,1 !-- lumped masses NLGEOM,1 !-- large deformations SSTIF,0 !-- no stress s t i f f e n n i n g NROPT,AUTO!-- Newton-Rapson n o n l i n e a r i t i e s - auto s e l e c t i o n on EQSLV,FRONT,1E-008,1 !-- solver s e l e c t i o n - f r o n t a l CNVTOL,F,1.,.01,2 ! specif y convergence c r i t e r i a i transient analysis i n t e g r a t i o n parameters TINTP,0.005!-- " A r t i f i c i a l l y Damped" Im p l i c i t Newmark (default) i load step 1: gr a v i t y TIMINT,0!-- ignore i n e r t i a e f f e c t s • TIME,0.005!-- set end time f or g r a v i t y load a p p l i c a t i o n AUT0TS,1!-- automatic time step o f f DELTIM,0.001,0.0001,0.005,ON!-- i n i t i a l time step ACEL,0,9.807,0,!-- apply g r a v i t y acceleration load OUTRES,NSOL,LAST !set output frequency /SOLU!-- switch to solver SOLVE!-- run solver i load step 2: hydro TIMINT,0!-- ignore i n e r t i a e f f e c t s TIME,0.01!-- set end time AUTOTS,l!-- automatic time step o f f DELTIM,0.001,0.0001,0.005,ON!-- i n i t i a l time step !F,NHydrol,FX,2180000.!-- apply hydro load NHydrol (bottom node) F,NBaUs,FX,2180000.!-- apply hydro load NHydrol (bottom node=NBaUs) F,NHydro2,FX,2790000.!-- apply hydro load NHydro2 F,NHydro3,FX,1540000.!-- apply hydro load NHydro3 F,NHydro4,FX,1430000.!-- apply hydro load NHydro4 F,NHydro5,FX,1000000.!-- apply hydro load NHydro5 F,NHydro6,FX, 204000.!-- apply hydro load NHydro6 OUTRES,NSOL,LAST!-- output r e s o l u t i o n /SOLU!-- switch to solver SOLVE!-- run solver i load steps 3,4,5,....: base a c c e l e r a t i o n TIMINT,1!-- include i n e r t i a e f f e c t s KBC,0!-- use ramped BC a p p l i c a t i o n Tm_Sta=0.015!-- s t a r t time (= end time of f i r s t load step) Tm_End=2 0.!-- end time - FULL Tm_lnc=0.005!-- load step duration AUT0TS,1!-- automatic time step on !-- 1,000Hz 20,000Hz 200Hz sol u t i o n frequency l i m i t s DELTIM,0.001,0.00005,0.005,ON!-- time step (try to keep 5 ts / ls) [specify: i n i t i a l , min, max, carry from previous LS OUTRES,NSOL,LAST!-- set output frequency to l a s t substep of each ! load step f o r a t o t a l of 4010 records ! read i n accelerations *DIM,Accel,TABLE,4010,1!-- define acceleration table *VREAD, Accel (0,0) , time200, pm, , 1! - - read time from f i l e (F8.4)!-- format *VREAD,Accel(0,1),eqlt200,prn,,1!-- read accelerations from f i l e (F8.4)!-- format i apply and solve f o r each load step *DO,Tm,Tm_Sta,Tm_End,Tm_Inc!-- f o r a l l load steps TIME,Tm!-- set time AccSca=Accel(Tm,l)*9.807*0.453 ACEL,AccSca,9.807,0!-- apply base acceleration SOLVE!-- solve *ENDDO!-- next load step SAVE 207 APPENDIX C - PLOTS FROM F E AND S D O F SIMULATIONS ON A FULL-SCALE D A M MONOLITH Two numerical models were used in Chapter 6 to simulate response of a full-scale dam monolith due to three different types of earthquake motions. The dam monolith unbonded at the base was subjected to the base excitations of varying PGA. The water level of the reservoir was another parameter varied during the excitations and a total of nine water levels from 60% to 100% of the full reservoir were considered. The amounts of sliding of the monolith obtained from SDOF analyses were compared with those using the F E model. Another way to test the numerical models is to compare the experimental results with those from the simulations in time domain and it is done in this appendix. A total of six simulations were selected for analysis in this appendix. They include three with PGA of 1.04g and three with PGA of 0.79g. These PGA's were used with all earthquake records considered in Chapter 6. The simulated water level was 100% full reservoir for all simulations. The comparison of results is performed in Figures D.l to D6. Each of these figures contains four parts a to d with plots of the obtained results. Parts a present time histories of dam monolith relative displacement (sliding) with respect to the base. It can be observed from these plots that the time histories from SDOF and F E analyses follow very similar trends. However, it cannot be concluded that one of the models would consistently produce more or less sliding than the other. Parts b and c of all figures contain plots of the relative accelerations of the dam monolith with respect to the base. The FE-based acceleration was obtained at the point on the upstream side of the base of the monolith. Only one acceleration record is obtained from the SDOF simulations and it is used for the comparison in this appendix. The character of the acceleration time histories has some similar as well as some different features for the FE and SDOF-based records. The peak relative accelerations occur about at the same time for all the records, but the FE-based peak accelerations are higher in most cases. In addition, the FE accelerations have non-zero values throughout the records. This is in contrast with the SDOF-based accelerations, which show zero relative accelerations except of the point where the dam slid during the analysis. These observations are in agreement with expected behaviour of the models. The FE model yielded non-zero accelerations throughout the records because of the flexibility in horizontal direction of the contact elements used to simulate the dam to base contact. No such flexible elements were used in the SDOF model. The plots in this appendix indicate that foundation interfaces of both numerical models used in the study do not perform the same way. However, it cannot be concluded that this would affect the amount of sliding of the model in such a way that one model would be consistently conservative or unconservative when compared with the other. Therefore, it can be concluded that both models can be used for the analysis if the objective of it is to estimate the amount of sliding of the dam monolith with respect to its base. 208 APPENDIX C - PLOTS FROM FE AND SDOF SIMULATIONS ON A FULL-SCALE DAM MONOLITH a) Dam Downstream Sliding - SDOF and FE Models 6 4 2 0 1 r 4 6 Time (sec) 1 2 b) Dam Base, Relative Acceleration - FE Model T 4 6 Time (sec) 2 c) Dam Base Relative Acceleration - SDOF Model i 1 r 4 6 Time (sec) d) Ground Acceleration FE Model SDOF Model i ' r 0 2 4 , . 6 8 1 Time (sec) Figure C . l : Response of Dam Monolith to Nahanni Earthquake Record with PGA 1.04g 209 APPENDIX C - PLOTS FROM FE AND SDOF SIMULATIONS ON A FULL-SCALE DAM MONOLITH a) Dam Downstream Sliding - SDOF and FE Models 0 2 4 6 Time (sec) 1 2 b) Dam Base Relative Acceleration - FE Model 4 6 Time (sec) 1 2 c) Dam Base Relative Acceleration - SDOF Model 0 2 d) Ground Acceleration 4 6 Time (sec) FE Model SDOF Model n 8 T 0 2 4 _ , . 6 8 1 Time (sec) Figure C.2: Response of Dam Monolith to Nahanni Earthquake Record with PGA 0.79g 210 APPENDIX C - PLOTS FROM FE AND SDOF SIMULATIONS ON A FULL-SCALE DAM MONOLITH E £= <D E o TO D. CO b 20 a) Dam Downstream Sliding - SDOF and FE Models 15 10 5 0 FE Model SDOF Model 10~ Time^sec) 30 1.2 -, 0.6 -c o 2 0.0 -CD <D 8 -0.6 -< -1.2 C 1.2 -. 3 0 . 6 -c o 2 0.0 CD CD 8 - 0 . 6 -< -1 ? Time (sec) 0 10 . _ d) Ground Acceleration i - i Time^sec) Time (sec) Figure C.3: Response of Dam Monolith to Mexico Earthquake Record with PGA 1.04g 211 APPENDIX C - PLOTS FROM FE AND SDOF SIMULATIONS ON A FULL-SCALE DAM MONOLITH 20 15 men 10 <D Q (0 CL 5 V) b 2Q a) Dam Downstream Sliding - SDOF and FE Models 0 -FE Model SDOF Model 10 -ft Time (sec) 1 2 b) Dam Base Relative Acceleration - FE Model 3 0.6 30 Time (sec) 1.2 3 0.6-c o -2 0.0 -8 -0.6-< -1.2 C Time (sec) d) Ground Acceleration Time^sec) Figure C.4: Response of Dam Monolith to Mexico Earthquake Record with PGA 0.79g 30 212 E c E cu o ro o_ w b 15 10 5 0 APPENDIX C - PLOTS FROM FE AND SDOF SIMULATIONS ON A FULL-SCALE DAM MONOLITH a) Dam Downstream Sliding - SDOF and FE Models FE Model SDOF Model Time (sec) b) Dam Base Relative Acceleration - FE Model Time%ec) 1.2 0.6 c o 2 0.0 0) <D 8 -0.6 < -1.2 1.2 s 0.6 c o 2 0.0 03 0) O O -0.6 < 2 4 c) Dam Base Relative Acceleration - SDOF Model 2 4 d) Ground Acceleration Time^sec) I" I 1 I 1 I 1 2 4 -|-. 6. . 8 1 Time (sec) Figure C.5: Response of Dam Monolith to Northridge Earthquake Record with PGA 1.04g 213 APPENDIX C - PLOTS FROM FE AND SDOF SIMULATIONS ON A FULL-SCALE DAM MONOLITH a) Dam Downstream Sliding - SDOF and FE Models 10 5 0 FE Model SDOF Model 1.2 -, 0.6 -c o -E 0.0-a> CD u o -0.6-< -1 ? -Time (sec) b) Dam Base Relative Acceleration - FE Model 1.2 0.6 E 0.0 j D CD 8 -0.6 Time (sec) c) Dam Base Relative Acceleration - SDOF Model -1.2 1.2 -, 0.6 -c o -ro i— 0.0-CD CD O O -0.6-< -1 ? 2 4 d) Ground Acceleration Time (sec) k n i 1 i 1 i Time (sec) Figure C.6: Response of Dam Monolith to Northridge Earthquake Record with PGA 0.79g 214 APPENDIX D - CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF F E SIMULATIONS (Analysis using Mathcad 8 Professional - Mathsoft, 1998) Charac te r i s t i cs o f P e a k Ground A c c e l e r a t i o n ( P G A ) d is t r ibut ion: M e a n v a l u e o f P G A (in g 's) : a M := n.l Ar r iva l rate ( q u a k e s per year ) : v : = n. l Coef f i c ien t o f va r ia t ion : v a G := 0.6 Indexes : i := i . . 4 n := 1.. 3 k := 1. . 10 m := 1.. 12 Rela t ive w a t e r leve ls in % of ful l reservo i r : j := 1.. 9 h. := 60 + ( j - 1 ) 5 Impor t resu l ts f r o m S D O F s imu la t ions : identif := num2st<n) filein := concat (v \ identif, "FElr.prn') FE1 := READPRN( f i l e i i j ) FE2 := READPRN^fileinJ FE3 := READPRN(filein) Put all resu l ts in o n e matr ix : F E . . := F E 1 . . FE. . . := F E 2 . , FE. „ , := F E 3 . , i,k i,k 1 + 4,k i,k 1+8,k i,k D e t e r m i n e P a r a m e t e r s o f L inear In terpo la t ion a . ; = intercept^ FE , FE" (for e a c h w a t e r level ) : J <i> ^\>) b . : = s l o p e ( F E < i > ) F E < J + ' > L inear in te rpo la t ions o f resul ts f r o m S D O F s imu la t ions for e a c h w a t e r leve l , all e a r t h q u a k e s c o n s i d e r e d : 3 2 1 0 -1 - 2 - 3 -4 - 5 a)Sliding vs. PGA Plot - Water Level 6 0 % 1 -0.7 -0.5 -0.3 -0.1 Ln (PGA (g)) 0.1 F E 4 - F E , x : = F E , , . F E , , +—— —..FE "1.1' 1.1 10 '4,1 abso lu te t e r m s : coef f i c ien ts o f l inear t e rms : a = 0.970 5.195 1.100 4.999 1.241 4.907 1.386 4.767 1.518 b = 4.558 1.731 4.440 1.961 4.283 2.055 4.167 2.204 3.950 b)Sliding vs. PGA Plot - Water Level 6 5 % -0.5 -0.3 Ln (PGA (g)) c)Sliding vs. PGA Plot - Water Level 7 0 % -0.9 -0.7 -0.5 -0.3 Ln (PGA (g)) 215 APPENDIX D - CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF FE SIMULATIONS Sliding vs. PGA Plot - Water Level 75% "0.5 -0.3 Ln (PGA (g)) Sliding vs. PGA Plot - Water Level 85% 1 • ' 1 1 +_ - +-- -+ -— ~r + -_ + -1 1 1 -*0.9 "0.7 -0.5 "0.3 Ln (PGA (g)) -0.1 0.1 Sliding vs. PGA Plot - Water Level 95% 1 1 1 1 -t-— jfc^ ^ + + + ~ + 1 1 1 1 -0.9 -0.7 "0.5 "0.3 Ln (PGA (g)) -0.1 c c E u 00 Sliding vs. PGA Plot - Water Level 80% 1 1 1 1 +_ - - +-- -1- --~ + + -1 1 1 — 0.9 "0.7 "0.5 -0.3 Ln (PGA (g)) -0.1 0.1 Sliding vs. PGA Plot - Water Level 90% "0.5 "0.3 Ln (PGA (g)) Sliding vs. PGA Plot - Water Level 100% 1 1 1 + + + + — - + 1 1 1 1 -0.9 "0.7 "0.5 "0.3 Ln (PGA (g)) -0.1 0.1 Figure D1. Sliding vs. PGA Relationship for Different Water Levels Using Data from FE Analyses 216 APPENDIX D - CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF FE SIMULATIONS Linear interpolations for all water levels, all earthquakes considered: Linear interpolation of "a" coefficients: Linear interpolation of "b" coefficients: al := intercepts,a) al = -0.982 bl := intercept,b) bl = 7.000 a2 := slope(h,a) a2 = 0.032 b2 := slope(h,b) b2 =-0.030 Export coefficients: WRITER "FE") := al APPENLX"FE") := a2 APPEND("FE") := bl APPEND("FE") := b2 50 60 70 80 90 100 110 ""50 60 70 80 90 100 110 Water level h (%) Water level h (%) Fig. D2: Linear Regression of 'a' coefficients Fig. D3: Linear Regression of 'b' coefficients Setup the interpolation function as follows: ln(D) = a(h)+b(h).ln(aG) or: ln(D) = a1+a2.h+(b1+b2.h).ln(aG) lnD(lnaG,hh) := al +• a2hhi- (bl + b2hh)lnaG Calculate values of displacements (sliding) using this function: FORM . := FE . m, 1 m, 1 F 0 R M m , j , . : = l n D ( F E m , . - h j , Calculate matrix of errors: error . := FE . . - FORM . , m,j m,j+- 1 m,j+ 1 Convert the matrix into a vector: error vei - v e ( 9 - ( m - i ) + j : error l ) - l -m : = F E m , l Put values of natural logarithms of PGA into a vector: ]naQ v e ( . Statistics for two random variables: natural logarithm of peak ground acceleration (InaG) and the errors between the results from simulations and from interpolation formula: Standard deviation of In(aG) , (distribution considered): ,naG_a := J,n(j + V a G 2 Mean value of In(aG): inaG_m:= ln(aM) - I ln(l + VaG2 lnaG_a= 0.555 InaG m=-2.456 Mean error: E _ m : = m e a n( e r r o r) E_m = -1.336-10' Standard deviation of errors: N : = iast(error vec) .-17 E a=0.790 2-e o= 1.580 E a: N 1 E N ^—l i= 1 error vec - £ m 217 APPENDIX D - CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF FE SIMULATIONS Correlation coefficient between error_vec and distribution of InaG: N 1 p:= — N i= 1 InaG vec - InaG m) • (error vec - e 8_m)] 1 InaG a-E <j p = -9.969-10' - 3 Plot showing comparison result from simulations with those using interpolation formula. The dashed lines represent the range of +/- 2 times standard deviation of the errors. S 1 Q E y ++t±^ _L - 5 - 3 -1 1 3 5 Ln(D) from linear formula Fig. D4: Comparison of Sliding from FE simulations with values using interpolation function Plot showing distribution | of the errors against the w values of logarithms of the peak ground accelerations. Fig. D5: Distribution of errors against values of PGA Reliability analysis with 2 random variables and neglecting the lower bound of Ln(aG) distribution: Setup performance function G: Mean of G: G_m(S0,h) := ln(S0) - (al + a2h + (bl + b2h)lnaG_m+ e_m) Standard deviation of G: G_c(h) := ,J(bl + b2-h)2lnaG_o2 + 2 (b l + b2 • h) • p • InaG jy E_CT +• E C 218 APPENDIX D- CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF FE SIMULATIONS Allowable displacements (sliding) considered: Reliability indexes: S0:=(1 2 5 10 15 20) d:= 1.. last(S0) P j.d G_m(S0.,h.) \ d J/ Event probabilities of failure: G c(h Pe. d := cnorm(-p. Annual Probabilities of failure: p a •= l - exD^-v-Pe j.d ' F \ j.d Pa = 3.61710"6 1.327 4.896-10"6 1.774 6.712-10"6 2.404 9.326-10"6 3.303 1.314 10"5 4.606 1.880 10"5 6.522 2.729-10"5 9.387 4.025 10"5 1.374 [6.034 10"5 2.048 Level of Sliding: 1 10"6 3.253 10"7 1.057 10"7 5.347 10"8 3.26110"8 10' 6 4.262 10"7 1.360 10"7 6.793 10"8 4.104 10"8 10"6 5.660 10"7 1.772 10"7 8.741 10"8 5.231 10"8 10"6 7.623 10"7 2.341 10"7 1.14010"7 6.758 10"8 10"6 1.042 10"6 3.140 10"7 1.510 10"7 8.860 10"8 10"6 1.448 10"6 4.281 IO"7 2.032 10"7 1.180 10"7 6 n A JI i r\~ 6 p r\f> m -t r\" 7 /•» 1 r\~ 7 J ^ 99 l Q~ ^  IO"6 2.047 10"6 5.937 10"7 2.781 10 10 " 5 2.946 10"6 8.388 10"7 3.878 10"7 2.207 10"7 IO"5 4.322 10"6 1.209 10"6 5.51710"7 3.108 10"7 KJ.d Water levels (%) 60% 65% 70% 75% 80% 85% 90% 95% 100% 10 Divide matrix of probabilities into columns for each level of sliding. Sliding 1 cm: paS01.:= Pa j x <i> Sliding 5 cm: paS05. := Pa J v <3> Sliding 15 cm: paS15.:= Pa j <5> 15 20 cm Sliding 2 cm: paS02. Sliding 10 cm: p asi0. Sliding 20 cm: paS20. Pa <2> = Pa <4> := Pa <6> The annual probabilities of failure represent the probability of exceeding the given permanent displacement of the dam. These displacement levels were specified above. The probabilities of failure are shown on logarithmic scale. 110 PaSOl. j PaS02 J 110 PaS05: J1 10 PaSlO. J PaS15j 1 -10 PaS20. — — 110 110 Annual Probabilities of Failure Water Level (%) Fig. D6: Annual Probabilities of Exceeding a Given Amount of Sliding 219 APPENDIX E - CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF S D O F SIMULATIONS (Analysis using Mathcad 8 Professional - Mathsoft, 1998) Characteristics of Peak Ground Acceleration (PGA) distribution: Mean value of PGA (in g's): a M:=0.1 Coefficient of variation: vaG := 0.6 Arrival rate (quakes per year): Relative water levels in % of full reservoir: j := 1.. 9 hj := 60 V (j - 1)5 Import results from SDOF simulations: iden t ip num2st<j) f i l e i p concat("ew", identif, "s.prn" 0.1 SDj := R E A D P R N [ f i l e i n 1 0 _ j}j endj := max^SDxj start := mir^SDXj S D x ^ l n j ^ p ] lengths := SDy j : = l i | [ SD 1<2> rows( SDXj . , . , , max(end) - mir( start) \ x := mir(start), (mir(start) + —-, ].. max(end) Determine Parameters of Linear Interpolation a . : = jntercep( (SDx), fSDy.) 1 b. :=. slope! fSDx.^, fSDy.^' (for each water level): J U V'K  Ji)\ J l l  Linear interpolations of results from SDOF simulations for each water level, all earthquakes considered: a)Sliding vs. P G A Plot - Water Level 60% absolute terms: coefficients of linear terms -0.674 6.628' -0.491 6.949 -0.185 6.970 0.176 7.114 0.578 b = 6.734 1.030 6.479 1.546 5.989 1.665 5.887 2.060 5.648 -0.4 -0.2 0 Ln (PGA (g)) 0.2 0.4 b)Sliding vs. P G A Plot - Water Level 65% C/5 C c)Sliding vs. P G A Plot - Water Level 70% ~ L p B ! — ! r -0.4 -0.2 0 Ln (PGA (g)) 0.2 0.4 -0.4 -0.2 0 Ln (PGA (g)) 0.2 0.4 220 APPENDIX E - CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF SDOF SIMULATIONS Ln (PGA (g)) Ln (PGA (g)) Ln (PGA (g)) Ln (PGA (g)) Figure E1. Sliding vs. PGA Relationship for Different Water Levels Using Data from SDOF Analyses 221 APPENDIX E - CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF SDOF SIMULATIONS Linear interpolations for all water levels, all earthquakes considered: Linear interpolation of "a" coefficients: Linear interpolation of "b" coefficients: al := intercept;h,a) al =-5.158 bl := intercept, b) bl =9.076 a2 := slope(h,a) a2 = 0.072 b2 := slope(h,b) b2 =-0.032 Export coefficients: WRITE("SD") := al APPEND("SD") := a2 APPEND("SD") := bl APPEND("SD") := b2 50 60 70 80 90 100 110 50 60 70 80 90 100 110 Water level h (%) Water level h (%) Fig. E2: Linear Regression of 'a' coefficients Fig. E3: Linear Regression of 'b' coefficients Setup the interpolation function as follows: ln(D) = a(h)+b(h).ln(aG) or: ln(D) = a1+a2.h+(b1+b2.h).ln(aG) lnD(lnaG,hh) := al + a2hh+ (bl + b2hh)lnaG Calculate values of displacements (sliding) using this function: §Df := lnD^SDx.,h.) Stack the values from simulations and those from interpolations into vectors: lnaG_vec:= stack^SDx,, SDxjj lnaG_vec = stack ^ lnaG_vec, SDxjj lnaG_vec:= stack ^ lnaG_vec, SDx4 lnaG_vec:= stack ^ lnaG_vec, SDx^ lnaG_vec = stack (^ lnaG_vec, SDx^ lnaG_vec:= stack ^ lnaG_vec, SDx? lnaG_vec:= stack ^ lnaG_vec, SDx^ lnaG_vec = stack ^ lnaG_vec, SDx^ SDy_vec := stack (SDy,, SDy2) SDy_vec = stack (SDy_vec, SDy3) SDy_vec:= stack (SDy_vec, SDy4 SDy_vec:= stack ^ SDy_vec, SDy5) SDy_vec = stack (SDy_vec, SDy6) SDy_vec:= stack (SDy_vec, SDy? SDy_vec:= stack (SDy_vec, SDyg) SDyvec = stack (SDy_vec,SDy9) SDf_vec:= stack(SDf,, SDfj) SDf_vec: = stack(SDf_vec, SDij) SDf_vec:= stack (SDf_vec, SDf4) SDf_vec:= stack (SDfvecSDfj) SDf_vec: = stack(SDf_vec, SDfg) SDf_vec:= stack (SDf_vec, SDf,) SDf_vec:= stack (SDf.vec.SDfg) SDf_vec: = stack(SDf_vec,SDf9) N := rows(lnaG_vec) n = 1..N 222 APPENDIX E - CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF SDOF SIMULATIONS Statistics for two random variables: natural logarithm of peak ground acceleration (InaG) and the errors between the results from simulations and from interpolation formula: Standard deviation of In(aG) (distribution considered): _rj:= ^ ln(l -h' lnaG_m:= ln(aM) - l - ln( l +• VaG2) lnaG_m= - 2 . 4 5 6 2 InaG a:  Ji  1 i  VaG' Mean value of In(aG): Calculate vector of errors: error_vec := SDf_vec- SDy_vec Mean error: E _ m : = mean(error_vec) Standard deviation of errors: InaG a= 0 . 5 5 5 E m = - 3 . 2 1 2 1 0 " -3 £ a: N N i= 1 error_veCj - £_m E a =0.791 2 E < J = 1 .582 Correlation coefficient between error_vec and distribution of InaG: N InaG vec - InaG m) • (error vec - E m 1 1 InaG O E a p = 4 . 8 3 5 10" 8 Plot showing comparison result from simulations with those using interpolation formula. The dashed lines represent the range of +/- 2 times standard deviation of the errors. - 5 - 3 - 1 1 3 5 Ln(D) from linear formula Fig. E4: Comparison of Sliding from SDOF simulations with values using interpolation function 4r Plot showing distribution of the errors against the values of logarithms of the peak ground accelerations. -0.8 -0.6 0.2 0.4 -0.4 -0.2 0 Ln(aG) values Fig. E5: Distribution of errors against values of PGA Reliability analysis with 2 random variables and neglecting the lower bound of Ln(aG) distribution: Setup performance function G: MeanofG: G_m(S0,h) := ln (S0 ) - (al -i-a2h + (bl + b2-h)-lnaG_m* e_m) Standard deviation of G: G_rj(h) := J(bl + b2h) 2 InaGjj2 + 2-(bl + b2h)plnaG_o£_ai- z_c 223 APPENDIX E - CALCULATIONS FOR RELIABILITY STUDY USING RESULTS OF SDOF SIMULATIONS Allowable displacements (sliding) considered Reliability indexes S0:=(1 2 5 10 15 20) d := 1.. last(S0) G_m(S0.,h., R V " J / Event probabilities of failure: p e •= cnorW-B p j ,d- G ofhA j ' d ' 1 J ' d Annual Probabilities of failure: p a. d := 1 - expf-v-Pe. J.d Pa: 2.785-10'7 1.216 10"7 3.895 10" 4.274-10"7 1.860 10"7 5.919 6.634 10"7 2.879-10"7 9.109 1.042 10"6 4.51210"7 1.420 1.655 10"6 7.158 10"7 2.244 2.661 IO"6 1.15010"6 3.593 4.327 10"6 1.871 IO"6 5.835 7.121 IO"6 3.084 10"6 9.609 1.18510"5 5.148 10"6 1.605 1.592-10"8 9.311 10"9 6.326 10"9 10" 2.405 10" 1.400 10" 9.482 10" 10"8 3.679 10"8 2.134 10"8 1.440-10"8 10"7 5.704 10"8 3.295 10"8 2.21810"8 10"7 8.967 10"8 5.162 10"8 3.464 10"8 10"7 1.430-10"7 8.204 10"8 5.491 10"8 10' 7 2.31410"7 1.324 10"7 8.838 10"8 10"7 3.800 10' 7 2.169 10"7 1.445 10"7 10"6 6.338 10"7 3.610 10"7 2.401 10"7 Allowable Sliding: 1 Divide matrix of probabilities into columns for each level of sliding. 10 15 Water levels (%) 60% 65% 70% 75% 80% 85% 90% 95% 100% Sliding 1 cm: P a S o i . := (pa* <i> Sliding 5 cm: paS05. := Pa J v Sliding 15 cm: p aS!5. := (pa <3> <5> 20 cm Sliding 2 cm: paS02 Sliding 10 cm: pasi0 Sliding 20 cm: pas20 := Pa <2> Pa <4> Pa < 6 > The annual probabilities of failure represent the probability of exceeding the given permanent displacement of the dam. These displacement levels were specified above. The probabilities of failure are shown on logarithmic scale. 110 Annual Probabilities of Failure PaSOl PaS02 J 110 PaSOS _ J l - 1 0 6 PaSlOj PaS15jM0 PaS20 1 1 0 ° h 110 110 Water Level (%) Fig. E6: Annual Probabilities of Exceeding a Given Amount of Sliding 224 

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