Department of Civil and Environmental Engineering Stanford University STOCHASTIC CHARACTERIZATION AND DECISION BASES UNDER TIME-DEPENDENT AFTERSHOCK RISK IN PERFORMANCE-BASED EARTHQUAKE ENGINEERING by Gee Liek Yeo and C. Allin Cornell Report No. 149 April 2005 The John A. Blume Earthquake Engineering Center was established to promote research and education in earthquake engineering. Through its activities our understanding of earthquakes and their effects on mankind’s facilities and structures is improving. The Center conducts research, provides instruction, publishes reports and articles, conducts seminar and conferences, and provides financial support for students. The Center is named for Dr. John A. Blume, a well-known consulting engineer and Stanford alumnus. Address: The John A. Blume Earthquake Engineering Center Department of Civil and Environmental Engineering Stanford University Stanford CA 94305-4020 (650) 723-4150 (650) 725-9755 (fax) [email protected] http://blume.stanford.edu ©2005 The John A. Blume Earthquake Engineering Center STOCHASTIC CHARACTERIZATION AND DECISION BASES UNDER TIME-DEPENDENT AFTERSHOCK RISK IN PERFORMANCE-BASED EARTHQUAKE ENGINEERING Gee Liek Yeo April 2005 °c Copyright by Gee Liek Yeo 2005 All Rights Reserved ii Preface This thesis addresses the broad role of aftershocks in the Performance-based Earthquake Engineering (PBEE) process. This is an area which has, to date, not received careful scrutiny nor explicit quantitative analysis. I begin by introducing Aftershock Probabilistic Seismic Hazard Analysis (APSHA). APSHA, similar to conventional mainshock PSHA, is a procedure to characterize the time- varying aftershock ground motion hazard at a site. I next show a methodology to quantify, in probabilistic terms, the multi-damage-state capacity of buildings in di®erent post-mainshock damage states. A time-dependent building \tagging" policy (permitting or restricting occu- pancy) is then developed based on the quanti¯cation of life-safety threat in the aftershock environment using the probability of collapse as a proxy for fatality risk. I also develop formal stochastic ¯nancial life-cycle cost models in both the post- and pre-mainshock environment. I include both transition and disruption costs in our model. Transition costs can be attributed to one-time ¯nancial losses due to structural and non- structural damage to the building, and can also include the costs of evacuation of the occupants of a building. Disruption costs can be attributed to the downtime and limited functionality of the damaged building. I begin with the traditional Poisson model for tem- porally homogeneous mainshocks and extend it to nonhomogeneous aftershocks. Further, the model is generalized to include renewal processes for modeling mainshock occurrences and Markov-chain descriptions of the damage states of a building. The analysis procedures are non-homogeneous Markov and semi-Markov decision analysis and stochastic dynamic programming (Howard (1971)). Finally, I introduce a decision analytic framework under improving states of information for both the post- and pre-mainshock environment. I emphasize the role of information in potentially improving our decision-making capability. Decision bases include the expected life-cycle cost and rate of collapse in the aftershock environment. I also introduce the iii concept of the value of information to determine if obtaining more information is ¯nancially desirable, which can potentially improve the quality of the decision. iv Contents Preface iii 1 Introduction 1 1.1 PBEE/PEER . 3 1.2 Signi¯cance of Aftershocks in PBEE . 6 1.3 Challenges of Aftershock Risk Analysis . 8 1.4 Thesis Organization . 15 2 Aftershock Probabilistic Seismic Hazard Analysis 16 2.1 Introduction . 16 2.2 Methodology . 18 2.3 Example . 23 2.3.1 Comparison of Mainshock Hazard to Aftershock Hazard . 24 2.3.2 E®ects of Duration (T ) on Aftershock Hazard . 26 2.3.3 E®ects of Elapsed Time from Initial Rupture (t) on Aftershock Hazard 29 2.3.4 Similarity of APSHA to Mainshock PSHA . 31 2.3.5 E®ects of Mainshock Magnitude (mm) and Site Location on After- shock Hazard . 33 2.3.6 E®ects of Structural Periods (T0) on Aftershock Hazard . 35 2.3.7 Summary and Approximate APSHA . 39 2.4 Correlation between "m and "a ......................... 40 2.5 Conclusion . 44 3 Performance of Mainshock-Damaged Buildings 46 3.1 Methodology . 47 v 3.1.1 Quanti¯cation of aftershock capacity of mainshock-damaged buildings 47 3.1.2 Determination of Transition Probabilities . 49 3.2 Example . 53 3.3 Conclusion . 60 4 Life-safety Based Building Tagging Criteria 62 4.1 Introduction . 63 4.2 Equivalent Constant Rates . 64 4.3 Proposed Building Tagging Methodology . 68 4.4 Tolerable Collapse Rate . 70 4.5 Primary Building Tagging Basis . 71 4.6 Special Tagging Cases: Emergency Workers . 72 4.7 Tagging Basis with Repair . 74 4.8 Simpli¯ed Building Tagging Basis . 75 4.9 Example . 76 4.10 Conclusion . 81 5 Financial Loss Models 82 5.1 Poisson Loss Model . 84 5.1.1 Discounted Losses due to Homogeneous Poisson Mainshock Process 86 5.1.2 Discounted Losses due to Nonhomogeneous Poisson Aftershock Process 89 5.2 Markov Loss Models . 92 5.2.1 Expected Total Discounted Losses due to Markov Mainshock Process 93 5.2.2 Expected Total Discounted Losses due to Markov Mainshock Process considering no more than 1 event in [0; tmax] . 96 5.2.3 Expected Total Discounted Losses due to Markov Mainshock Process with Repair . 97 5.2.4 Expected Total Discounted Losses due to Semi-Markov Mainshock Process . 98 5.2.5 Expected Total Undiscounted Losses due to Nonhomogeneous Markov Aftershock Process . 103 5.2.6 Expected Total Discounted Losses due to Nonhomogeneous Markov Aftershock Process . 107 5.3 Pre-Mainshock Loss Estimation . 109 vi 5.3.1 Formulation using Homogeneous Poisson Process for Mainshock Oc- currences . 111 5.3.2 Formulation using Renewal Process for Mainshock Occurrences . 112 5.4 Example . 113 5.5 Conclusion . 124 6 Seismic Decision Analysis 125 6.1 Decision Analysis using Decision Trees . 127 6.1.1 Example . 128 6.2 Decision Analysis allowing Damage-State Transitions . 142 6.3 Decision Analysis using Stochastic Dynamic Programming . 145 6.3.1 Methodology . 146 6.3.2 Example . 154 6.4 Conclusion . 161 7 Summary, Limitations and Future Work 162 7.1 Summary . 162 7.2 Conclusions . 164 7.3 Limitations . 165 7.4 Future Work . 167 vii List of Tables 5.1 Notations for Poisson model for a building in initial state i; Capital letters denote random variables . 86 5.2 Potential ¯nancial losses for each building damage state . 113 6.1 Potential ¯nancial losses for each building damage state . 131 6.2 Likelihood functions of \imperfect" engineer . 133 6.3 Case 1: Times in days after mainshock when the optimal decision changes from having evacuation to allowing re-occupancy. The individual life-safety constraint is enforced in this case without explicit minimization of ¯nancial losses. 155 6.4 Typical output of dynamic programming algorithm for the case where the individual life-safety constraint is not imposed, and where a cost per life saved of $2M is used. The optimal policy is shown as a function of elapsed days after the mainshock for all post-mainshock damage states. 158 6.5 Case 2: Times in days after mainshock when the optimal decision changes from having evacuation to allowing re-occupancy. The individual life-safety constraint is not enforced in this case. 158 6.6 Case 3: Times in days after mainshock when the optimal decision changes from having evacuation to allowing re-occupancy. The individual life-safety constraint is also enforced in this case. 160 viii List of Figures 1.1 A schematic plot of the aftershock environment. 2 1.2 Evacuated Occupants in Algeria Earthquake. 3 1.3 Pictures of a building which su®ered slight damage due to the 1999 Turkey Kocaeli mainshock that fully collapsed due to a smaller magnitude aftershock almost one month later. Photographs excerpted from USGS (2000). 10 1.4 Pictures of a column which was only slightly damaged due to the 2002 Italy Molise mainshock that su®ered severe spalling, bar buckling and a resid- ual vertical deformation at the top of the column due to an aftershock one day later. Photographs courtesy of Mucciarelli and Gallipoli, University of Basilicata, Italy and Dr. Paolo Bazzurro. 11 2.1 Schematic of site layout and linear aftershock zone. 24 2.2 Comparison of mainshock and aftershock site hazard curves (as functions of PGA), where aftershock hazard is evaluated at t = 7 days with T = 365 days for aftershocks equally likely to occur at any location on the linear aftershock zone and for aftershocks concentrated at the ends of the linear aftershock zone. The mainshock magnitude is assumed to be 7.0. 25 2.3 Mean number of aftershocks (as a function of elapsed time, t) with site PGA > 0.5g in speci¯ed durations T .......................... 26 2.4 Ratios of expected number of aftershocks resulting in site PGA > 0.5g for T = one week, one month, six months and one year to the expected number of aftershocks resulting in site PGA > 0.5g in one day, as a function of the elapsed time, t, of the ¯rst day in the duration of interest. 28 ix 2.5 Mean number of aftershocks (as a function of elapsed time t) with site PGA > 0.5g. The duration T is constant at one year in the ¯rst case (a), and the end time is constant at one year in the second case (b). 28 2.6 Aftershock hazard as a function of t for site PGA= 0.3g, 0.5g and 0.7g, and T = 365 days. The aftershock hazard is compared to the pre-mainshock hazard of exceeding the respective PGA values.