P E E R U C S D Probabilistic Performance-Based Optimum Seismic Design of (Bridge) Structures PI: Joel. P. Conte Graduate Student: Yong Li Sponsored by the Pacific Earthquake Engineering Research Center 1
(I) PEER PBEE Methodology (Forward PBEE Analysis) 2
F F y SDOF Model and Site Single-Degree-of-Freedom Bridge Model: 1 Site:! SDOF bridge model with the same initial period as an MDOF model of the Middle Channel Humboldt Bay Bridge developed in OpenSees. SDOF Model (Menegotto-Pinto) k 0 U " y 0.075 m 1 bk # 0 U T 1 m k 0 F b y! " " 1.33 sec " " 6,150 tons " 137,200 kn/m " 10,290 kn 0.10 0.02 Uniform Hazard Spectra T 1 =1.33 sec! Site Location: Oakland (37.803N, 122.287W)! Site Condition: Vs30 = 360m/s (NHERP Class C-D) "Uniform Hazard Spectra are obtained for 30 hazard levels from the USGS 2008 Interactive Deaggregation software/website (beta version) 3
Prob. Seismic Hazard Analysis and EQ Selection Seismic Hazard Curve (for single/scalar Intensity Measure IM): N flt * ++ & (im) " & # P $ ( IM ' im m, r %)# f (m) # f (r)# dm # dr i IM i Mi R i" 1 R M i i S! ", T a 1, 5% - Least Square Fitting of 30 Data Points ( IM, MARE ) i i from USGS: Deaggregation (T 1 = 1 sec, 2% in 50 years) Earthquake Record Selection:! 146 Records were selected from NGA database based on fault mechanism, M-R deaggregation, and local site condition (e.g., Vs30) 4
Probabilistic Seismic Demand Hazard Analysis Demand Hazard Curve: & EDP, edp- " P $ EDP ' edp IM % d&, im- + IM ( ) Probabilistic Seismic Demand Analysis conditional on IM: IM Seismic hazard curve!cloud Method"!Convolution"!Deaggregation" PDF (Lognormal) EDP: Displ. Ductility Deaggregation of, - * &, edp - EDP with respect to IM:, imi -, im-, - IM EDP edp " P ( EDP ' edp IM " imi ) #. im i i. i & $ % d& Contribution of bin IM = im i to &, edp EDP - 5
Probabilistic Seismic Demand Hazard Analysis PDF (Lognormal) EDP: Normalized EH!Cloud Method"!Convolution"!Deaggregation" PDF (Lognormal) EDP: Peak Abs. Accel. 6
Probabilistic Capacity (Fragility) Analysis Fragility Curves (postulated & parameterized):! Defined as the probability of the structure/component exceeding k th limit-state given the demand. Damage Measure / ' " 0 P DM ls EDP edp k-th limit state! Developed based on analytical and/or empirical capacity models and experimental data. k Displacement Ductility Normalized E H Dissipated Absolute Acceleration 7
Probabilistic Damage Hazard Analysis Damage Hazard (MAR of limit-state exceedance): & DS k " P $ DM lsk EDP edp% ( ' " ) d& EDP( edp) + EDP Damage Hazard Results: Fragility Analysis Associated EDP Displacement Ductility Normalized E H Dissipated Absolute Acceleration Limit States MARE Return Period PE in 50 yrs I (1 d = 2) 0.0394 26 Years 86% II (1 d = 6) 0.0288 35 Years 76% III (1 d = 8) 0.0231 44 Years 68% I (E H = 5) 0.0171 58Years 57% II (E H = 20) 0.0012 833 Years 6% III (E H = 30) 0.0003 3330 Years 1.5% I (A Abs = 0.1g) 0.0349 29 Years 82% II (A Abs = 0.2g) 0.0104 96 Years 41% III (A Abs = 0.25g) 0.0048 208 Years 21% 8
Probabilistic Loss Hazard Analysis Loss Hazard Curve:! Component-wise Loss Hazard Curve: & ( l) " P[ L ' l DM ] d& " P $ ( L ' l DS " k% )& 3& Number of eqkes in one year N " n + L j DM j DS DS DM k" 1 j k k21! Total Loss Hazard Curve: Ensemble of FE seismic response simulations Poisson model Seismic hazard Conditional probability of collapse given IM L Marginal PDFs and correlation coefficients of EDPs given IM T nls j * # components = å For each eqke For each damaged For each eqke N For each EDP component Collapse? IM " im EDP " 4 DM k Y NATAF joint PDF model of EDPs j= 1 number of limit states for component j L Fragility curves for all damage states of each failure mechanism j Multilayer Monte Carlo Simulation Probability distribution of repair cost For each damaged component " Lj " lj For all eqkes in one year and all damaged components # components LT " * Lj j" 1 9
Parametric PBEE Analysis Varying Parameter: Yield Strength F y Demand Hazard for EDP =! d Demand Hazard for EDP = A Abs. Demand Hazard for EDP = E H Cost Hazard 10
Parametric PBEE Analysis Varying Parameter: Initial Stiffness k 0 Demand Hazard for EDP =! d Demand Hazard for EDP = A Abs. Demand Hazard for EDP = E H Cost Hazard 11
Optimization Formulation of PBEE Optimization Problem Formulation:! Objective (Target/Desired) Loss Hazard Curve: & L () l! Objective function:! Optimization Problem:, - Minimize f k0, Fy, b,! 5k0,! Fy,! b,! 6 subject to : h g, k F b!- 0, k F b!- 0, - Obj 2 0, y,,! & L ( i, 0, y,,!) & ( i ),,, " 0 y,,, 7 0 y Optimization Performed using OpenSees-SNOPT extended Framework Gu, Q., Barbato, M., Conte, J. P., Gill, P. E., and McKenna, F.,!OpenSees-SNOPT Framework for Finite Element-Based Optimization of Structural and Geotechnical Systems," Journal of Structural Engineering, ASCE, in press, 2011. Obj T " * 3 T f k F b l k F b l i Obj & L () l T L T 12
(II) Optimization Tool of OpenSees-SNOPT 13
OpenSees: OpenSees and SNOPT Open source objected-oriented FE analysis software framework, used to model structural, geotechnical and SSI systems to simulate their responses to static and dynamic loads, especially for earthquakes (PEER)! Supports distributed computing capabilities & cloud computing! Includes modules for FE response sensitivity and reliability analysis! Flexibility to incorporate with other software packages SNOPT: General purpose nonlinear optimization code using Sequential Quadratic Programming (SQP) algorithm (Professor Phillip Gill @UCSD)! Advantages of SNOPT as an optimization toolbox for solving structural/geotechnical problems requiring optimization (e.g., structural optimization, FE model updating, design point search problems): # Applies to large scale problems # Tolerates gradient discontinuities # Requires relatively few evaluations of the limit-state (objective) function # Offers a number of options to increase performance for special problems! Open source in Fortran language available for academic use 14
OpenSees-SNOPT Extended Framework OpenSees & SNOPT Link:! After abstract class of Optimization, subclass SNOPTClass is used to encapsulate the SNOPT software package as an interface! Subclass SNOPTReliability for the design point search problem in reliability analysis; while the SNOPTOptimization structural optimization & model updating ObjectiveFunction ConstraintFunction DesignVariable DesignVariablePositioner Optimization Framework in OpenSees Flowchart for OpenSees-SNOPT FE-based Optimization 15
(III) Illustration of Performance- Based Optimum Seismic Design (Inverse PBEE Analysis) 16
k? kn / m 0 " 100, 000 kn/m F "? 14,000 kn kn y Illustrative Example Starting Point, 0 y- Objective function : f k, F * L T 0 y L T Obj 2 =! "# $%& ' (! ) i F F y SDOF Model (Menegotto-Pinto) m " bk # 1 0 6,150 tons PBEE Analysis Obj & L T & L T, k0, Fy- 1 k 0 b "! " U y " 0.10 0.02 0.075 m U k F * 0 * y " 137,200 kn/m " 10,290 kn A priori selected optimum design parameters Expected Optimizer OpenSees- SNOPT 17
Optimization Problem: Starting Point: Objective Function Plot: Objective Function Minimize f k 5k0,! Fy6 subject to :,, F - 0 y 80,000 7k 7187, 200 (kn/m) 0 6,290 7 F 715, 290 ( kn) k F y (0) (0) 0 " 100,000 kn/m, " 14,000kN y Objective Function Zoom Objective Function 18
Optimization Results Optimization Route: v EDP=!D, k "*, F# * - Xstart = [100,000kN/m, 14,000 kn] X* = [137,200kN/m, 10,290 kn] Xend = [135,774kN/m, 10,038 kn] Demand Hazard Curve Objective Function Loss Hazard Curve 19
Current and Future Work Application to base isolated bridge structure:! Using nonlinear MDOF models of increasing complexity.! With increasing number of optimization parameters, e.g., base isolation parameters.! Considering multiple objectives and practical constraints, such as constraints on demand hazard. Marga-Marga Bridge, Vina del Mar in Chile Isolator 20
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