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
Outline (I) (II) PEER Performance Based Earthquake Engineering (PBEE) Methodology Probabilistic Seismic Hazard Analysis Probabilistic Seismic Demand Hazard Analysis Probabilistic Seismic Damage Hazard Analysis Probabilistic Seismic Loss Hazard Analysis Parametric PBEE Analysis of SDOF System Yield strength F y as varying parameter Hardening ratio b as varying parameter Both mass m and initial stiffness k 0 as varying parameters (with fixed k 0 /m) Mass m as varying parameter Initial stiffness k 0 as varying parameter (III) Optimization Formulation for Probabilistic Performance Based Optimum Seismic Design 2
(I) PEER PBEE Methodology (Forward PBEE Analysis) 3
SDOF Bridge Model Single-Degree-of-Freedom Bridge Model: SDOF bridge model with the same initial period as an MDOF model of the Middle Channel Humboldt Bay Bridge previously developed in OpenSees. F F y 1 k 0 U y = 0.075 m 1 bk 0 U SDOF Model (Menegotto-Pinto) T1 = 1.33 s m = 6,150 tons k = 137,200 kn/m 0 F y = 10,290 kn b = 0.10 ξ = 0.02 4
Site Description and Prob. Seismic Hazard Analysis Site Location: Site Condition: Oakland (37.803N, 122.287W) V s30 = 360m/s (Class C-D) Uniform Hazard Spectra are obtained for 30 hazard levels from the USGS 2008 Interactive Deaggregation software/website (beta version) Uniform Hazard Spectra and M-R Deaggregation: Uniform Hazard Spectra Deaggregation (T 1 = 1 sec, 2% in 50 years) T 1 =1.33 sec 5
Earthquake Record Selection Earthquake Record Selection: Records were selected from NGA database based on geological and seismological conditions (e.g., fault mechanism), M-R deaggregation of probabilistic seismic hazard results, and local site condition (e.g., V s30 ) NGA Excel Flatfile (3551 records) Mechanism: Strike-slip ( 1004 records) Magnitudes: 5.9 to 7.3 ( 652 records) Distance: 0 40 km ( 193 records) V s30 : C-D range ( 146 records) 146 horizontal ground motion components were selected from NGA database 6
Probabilistic Seismic Hazard Analysis Seismic Hazard Curve (for single/scalar Intensity Measure IM): Data Points: N flt ν (im) = ν P IM > im m,r f (m) f (r) dm dr i IM i Mi R i= 1 R M ( ) x, y = ( IM, PE50 ), i = 1,2,,30 i i i i Fitting Function (lognormal CCDF): i i Least Square Fitting of 30 Data Points ( IM, MARE ) from USGS: i i gx (, θ, θ ) 1 1 2 S T a 1, ξ = 5% ( ) ln x θ 1 = Φ θ2 θ, θ 1 2 np i= 1 [ gx θ θ y] min ln (,, ) ln i 1 2 i 2 7
Probabilistic Seismic Demand Hazard Analysis Demand Hazard Curve: ν EDP Probabilistic Seismic Demand Analysis conditional on IM: Cloud Method based on following assumptions: 1. Linear regression after ln transformation of predictor and response variables: ˆ μ = E[ln EDP IM = im] = aˆ+ bˆ ln IM EDP ln IM 2. Variance of EDP independent of IM: Sˆ ( edp) = P EDP > edp IM dν ( im) = n 2 i= 1 IM Complementary CDF of EDP given IM Seismic hazard curve ( ln edp ( ˆ ˆ ln )) 2 i a + b imi n 2 3. Probability distribution of EDP IM assumed to be Lognormal: EDP IM LN( λ, ς ) where λ = ˆ μ = aˆ+ bˆ ln IM and ln EDP IM ˆ 2 ς = ˆ σ = S IM ln EDP IM 8
Probabilistic Seismic Demand Hazard Analysis EDP: Displacement Ductility Regression analysis results and PDF/CCDF for EDP IM shown at three hazard levels commonly used in Earthquake Engineering: PDF (Lognormal) CCDF (Lognormal) Where the displacement ductility is defined as: μ d ( ) u t = Max 0 < t < t d U y 9
Probabilistic Seismic Demand Hazard Analysis EDP: Peak Absolute Acceleration Regression analysis results and PDF/CCDF for EDP IM shown at three hazard levels commonly used in Earthquake Engineering: PDF (Lognormal) CCDF (Lognormal) Where the peak absolute acceleration is defined as: A Abs. ( ) u t = Max 0 < t < t d g 10
Probabilistic Seismic Demand Hazard Analysis EDP: Normalized Hysteretic Energy Dissipated Regression analysis results and PDF/CCDF for EDP IM shown at three hazard levels commonly used in Earthquake Engineering: PDF (Lognormal) CCDF (Lognormal) Where the Normalized hysteretic energy is defined as: E H = 0 t d () () R t du t FU y y E E 11
Probabilistic Seismic Demand Hazard Analysis Demand Hazard Curve: ν ν EDP EDP ( ) ( edp) = P EDP > edp IM = im dν ( im) IM i IM Deaggregation of ν ( edp EDP ) with respect to IM: ( im) dν IM edp = P EDP > edp IM dim dim IM ( imi ) ( im) dν IM = P EDP > edp IM = imi Δ im Δ i ( ) i Contribution of bin IM = im i to ν ( edp EDP ) 12
Probabilistic Seismic Demand Hazard Analysis Demand Hazard Curve for EDP = Displacement Ductility μ d : MARE APE PE50 (RP = 55 years) (RP = 2500 years) Deaggregation wrt IM 13
Probabilistic Seismic Demand Hazard Analysis Demand Hazard Curve for EDP = Peak Absolute Acceleration: MARE APE PE50 (RP = 30 years) (RP = 625 years) Deaggregation wrt IM 14
Probabilistic Seismic Demand Hazard Analysis Demand Hazard Curve for EDP = Normalized E H Dissipated: MARE APE PE50 (RP = 32 years) (RP = 500 years) Deaggregation wrt IM 15
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 [ > = ] P DM ls EDP edp k-th limit state Developed based on analytical and/or empirical capacity models and experimental data. k Displacement Ductility Absolute Acceleration Normalized E H Dissipated 16
Probabilistic Capacity (Fragility) Analysis Fragility Curve Parameters: Associated EDP Displacement Ductility Peak Absolute Acceleration [g] Normalized Hysteretic Energy Dissipated Limit-states Predicted Capacity Measured-to-predicted capacity ratio (Normal distributed) Mean c.o.v I μ = 2 1.095 0.201 II μ = 6 1.124 0.208 III μ = 8 1.254 0.200 I A Abs. = 0.10 0.934 0.128 II A Abs. = 0.20 0.952 0.246 III A Abs. = 0.25 0.973 0.265 I E H = 5 0.934 0.133 II E H = 20 0.965 0.140 III E H = 30 0.983 0.146 17
Probabilistic Damage Hazard Analysis Damage Hazard (MAR of limit-state exceedance): ν DS k = P DM lsk EDP edp > = dνedp( edp) EDP Fragility Analysis Deaggregation of Damage Hazard: with respect to EDP: dν EDP( edpi ) ν = P DM > ls EDP = edp Δ edp DS k Δ with respect to IM: i Contribution of bin EDP = edp i to ν DSk ( edp) ( ) k i i i dν IM ( imi ) ν = P DM > lsk EDP dp[ EDP > edp IM ] Δ im DS k Δ ( im) i IM i ( ) i Contribution of bin IM = im i to ν DSk 18
Probabilistic Damage Hazard Analysis Associated EDP = Displacement Ductility: Associated EDP Displacement Ductility Limit States MARE RP PE50 I (μ d = 2) 0.0394 26 Years 86% II (μ d = 6) 0.0288 35 Years 76% III (μ d = 8) 0.0231 44 Years 68% Deaggregation Results: Deaggregation wrt EDP Deaggregation wrt IM 19
Probabilistic Damage Hazard Analysis Associated EDP = Peak Absolute Acceleration: Associated EDP Absolute Acceleration Limit States MARE RP PE50 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% Deaggregation Results: Deaggregation wrt EDP Deaggregation wrt IM 20
Probabilistic Damage Hazard Analysis Associated EDP = Normalized Hysteretic Energy Dissipated: Associated EDP Normalized Hysteretic Energy Dissipated Limit States MARE RP PE50 I (E H = 5) 0.0171 58 Years 57% II (E H = 20) 0.0012 833 Years 6% III (E H = 30) 0.0003 3330 Years 1.5% Deaggregation Results: Deaggregation wrt EDP Deaggregation wrt IM 21
Probabilistic Loss Hazard Analysis Loss Hazard Curve: Component-wise Loss Hazard Curve: ν () l = P[ L > l DM] dν = P L > l DS = k ν ν L j DM j DS DS DM k= 1 j k k+ 1 nls j nls j = number of limit states for component j Only discrete damage states are used in practice. = MAR of exceeding ν DSk the k-th limit-state and = 0 ν DS nls + 1 Limit-State I Limit-State II Limit-State III P[DS = 0 EDP = edp] P[DS = 1 EDP = edp] P[DS = 2 EDP = edp] edp P[DS = 3 EDP = edp] EDP 22
Probabilistic Loss Hazard Analysis Multilayer Monte Carlo Simulation: Total Loss (Repair/Replacement Cost): Computation of ν L l requires n-fold integration of the joint PDF of T the component losses, which is very difficult to obtain. ν L l is estimated using multilayer MCS T () () Ensemble of FE seismic response simulations of bridge-foundation-ground system L T # components = j= 1 L P Lj DS I j Probability Distributions of Repair Costs II III IV V Number of eqkes in one year N= n Poisson model Seismic hazard Conditional probability of collapse given IM Marginal PDFs and correlation coefficients of EDPs given IM Fragility curves for all damage states of each failure mechanism For each eqke For each damaged For each eqke N For each EDP component Collapse? IM= im EDP = δ DM k Y NATAF joint PDF model of EDPs μi μii Probability distribution of repair cost For each damaged component L = l = j j μiii μiv μv cost For all eqkes in one year and all damaged components # components LT = Lj j= 1 23
Probabilistic Loss Hazard Analysis Repair/Replacement Cost Parameters: Failure Associated EDP Displacement Ductility Peak Absolute Acceleration [g] Normalized Hysteretic Energy Dissipated Limit-states Repair/Replacement Cost (Normal distributed) Mean ($) c.o.v. I 146,500 0.12 II 246,400 0.25 III 350,000 0.32 I 55,000 0.11 II 100,000 0.20 III 500,000 0.28 I 55,650 0.13 II 110,000 0.22 III 520,000 0.28 24
(II) Parametric PBEE Analysis of SDOF System 25
Parametric PBEE Analysis Varying Parameter: Yield Strength F y Demand Hazard for EDP = μ d Demand Hazard for EDP = A Abs. Demand Hazard for EDP = E H Cost Hazard 26
Parametric PBEE Analysis Varying Parameter: Hardening Ratio b Demand Hazard for EDP = μ d Demand Hazard for EDP = A Abs. Demand Hazard for EDP = E H Cost Hazard 27
Parametric PBEE Analysis Varying Parameters: Both Mass m and Initial Stiffness k 0 (period fixed): Demand Hazard for EDP = μ d Demand Hazard for EDP = A Abs. Demand Hazard for EDP = E H Cost Hazard 28
Parametric PBEE Analysis Varying Parameter: Mass m Demand Hazard for EDP = μ d Demand Hazard for EDP = A Abs. Demand Hazard for EDP = E H Cost Hazard 29
Parametric PBEE Analysis Varying Parameter: Initial Stiffness k 0 Demand Hazard for EDP = μ d Demand Hazard for EDP = A Abs. Demand Hazard for EDP = E H Cost Hazard 30
(III) Optimization Formulation for Probabilistic Performance Based Optimum Seismic Design (Inverse PBEE Analysis) 31
Optimization Formulation of PBEE Optimization Problem Formulation: Objective (Target/Desired) Loss Hazard Curve: ν L () l Objective function: Optimization Problem: ( ) Minimize f k0, Fy, b, { k0, Fy, b, } subject to : h g 2 ( 0,,, ) (, 0,,, ) Obj y = νl ( ) T i y νl T i f k F b l k F b l ( k0 Fy b ) ( k0 Fy b ),,, = 0,,, 0 Optimization Performed using extended OpenSees-SNOPT Framework (ongoing work). 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, under review, 2011. 32 i Obj T Obj ν L () l T
Thank you! Any Questions? 33