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x - SPA Risk LLC

A brief history of PBEE-2 CVEN 5835-02 SP TPS: Nonlinear Structural Analysis; Theory and Applications 17 Feb 2011 Keith Porter, Associate Research Professor Civil, Environmental, and Architectural Engineering University of Colorado at Boulder Today’s objectives Some terminology History and key concepts of LRFD History and main goals of PBEE-1 Origin and main goals of PBEE-2 Overview of how PBEE-2 works Monte Carlo simulation in PBEE-2 Further reading This ppt: http://spot.colorado.edu/~porterka/Porter-2011-CU-PBEE2.pdf Ellingwood, B., Galambos, T.V., MacGregor, J.G., and Cornell, C.A., 1980. Development of a Probability-Based Load Criterion for American National Standard A58, National Bureau of Standards, Washington, DC, 222 pp., http://spot.colorado.edu/~porterka/Ellingwood-1980-LRFD-for-A58.pdf Lays out principles and parameter values for LRFD standards for design codes (ASCE) American Society of Civil Engineers, 2000. FEMA-356: Prestandard and Commentary for the Seismic Rehabilitation of Buildings, Washington, DC, 490 pp., http://www.fema.gov/library/viewRecord.do?id=1427 First-generation PBEE guidelines for assessing future building performance in terms of operability & life safety at multiple hazard levels Porter, K.A. 2000. Assembly-Based Vulnerability and its Uses in Seismic Performance Evaluation and Risk-Management Decision-Making, Report No. 139, John A. Blume Earthquake Engineering Center, Stanford, CA, 214 pp., http://www.sparisk.com/pubs/Porter-2001-ABV-thesis.pdf Lays out (or prefigures) much of PEER- and ATC-58 style PBEE-2, in which future seismic performance is estimated in terms of probabilistic repair costs and repair durations. Porter, K.A., A.S. Kiremidjian, and J.S. LeGrue, 2001. Assembly-based vulnerability of buildings and its use in performance evaluation. Earthquake Spectra, 17 (2), pp. 291-312, http://www.sparisk.com/pubs/Porter-2001-ABV.pdf A brief version of Porter 2000. Porter, K.A., 2003. An overview of PEER’s performance-based earthquake engineering methodology. Proc. Ninth International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP9) July 6-9, 2003, San Francisco, CA. Civil Engineering Risk and Reliability Association (CERRA), 973-980, http://spot.colorado.edu/~porterka/Porter-2003-PEER-Overview.pdf An overview of PEER-style PBEE-2. Linear vs. nonlinear structural analysis ∆ 3 sources of nonlinearity Q Force Q fy Displacement ∆ Small-angle rule may no longer hold Q Stress s Stress s Strain e Static vs. dynamic structural analysis Static analysis: solve Kx = V V Dynamic analysis: solve Mẍ(t) + Cẋ(t) + Kx(t) = -Mẍg(t) Using probability distributions Probability density function PDF, of X: fX(x) Probability mass function PMF, of X: pX(x) Probability that X = x per unit x, for continuous X Probability that X = x, for discrete X Cumulative distribution function CDF, FX(x) Probability that X ≤ x, continuous or discrete X x P X x FX x f z dz Inverse cumulative distribution function, F-1X(p) X Value of X with probability p of not being exceeded x FX1 p Normal distribution Prob density function (PDF) fX(x) = prob that X = x, per unit X Cumulative distr function (CDF) FX(x) = prob that X ≤ x Let μ = mean, a central measure; σ = standard deviation, a measure of dispersion FX(x) = We will use more than 1 uncertain variable, so let us denote fX(x) = probability density function of X evaluated at a particular value x FX(x) = cumulative probability function of X evaluated at a particular value x FX-1(p) = particular value of X such that FX(x) = probability p Monte Carlo Simulation How to produce sample values of arbitrary distribution 1.00 1.00 0.75 FU (u ) 0.75 0.50 0.00 -0.25 0.00 0.50 0.25 0.25 0.25 0.50 0.75 1.00 0.00 -0.25 0.00 1.25 u 0.25 0.50 u 1.00 0.75 FX (x ) Uniform distribution u ~ U(0, 1), e.g., rand() in Excel We want to sample X For each sample, draw a sample u Invert CDF, xu = FX-1(u) Repeat many times with different samples u fU (u ) 1.25 0.50 0.25 0.00 0.00 0.50 1.00 x 1.50 2.00 0.75 1.00 1.25 Joint probability distribution Say X1 and X2 are each normally distributed & independent p[X1 = x1, X2 = x2] = fX1(x1)fX2(x2) Probability density 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 4.5 0.00 4 3.5 0.0 0.5 3 1.0 2.5 1.5 2 2.0 1.5 2.5 X1 3.0 1 3.5 0.5 4.0 4.5 0 X2 Load and Resistance Factor Design Seeks to control failure probability by structural member or connection & load combination Concrete: ACI 318 (1977) Steel: AISC LRFD (1st ed. 1986) General: Ellingwood et al. (1980) http://spot.colorado.edu/~porterka/Ellingwood-1980-LRFD-for-A58.pdf Thumbnail sketch of LRFD Q Q 0.20 Probability density Probability density Let R = resistance, Q = load on a member or connection Say R and Q are independent, normally distributed, means μR and μQ, std deviations σR and σQ. 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 4.5 0.00 4 3.5 0.0 0.5 2.5 1.5 2 2.0 Resistance R 1.5 2.5 1 3.0 3.5 0.5 4.0 4.5 0.18 0.16 0.14 Survival R>Q 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.0 3 1.0 0.20 0.5 1.0 1.5 2.0 2.5 3.0 Failure R<Q Joint probability distribution 3 2.5 2 Resistance R 1.5 1 3.5 0.5 4.0 0 Load Q 4.5 4 3.5 4.5 0 Load Q Probability that R – Q < 0 Thumbnail sketch of LRFD Let g(R,Q) = R – Q (“performance function”) Let μX= denote mean, σX stdev of X μR-Q= μR – μQ σR-Q = (σR2 + σQ2)0.5 Let β = μR-Q/σR-Q Failure prob Pf = P[g<0] = ((0 – μR-Q)/σR-Q) = (–μR-Q/σR-Q) = (–β) β: “reliability index,” a measure of the likelihood of failure Bigger β = more reliable Thumbnail sketch of LRFD A bit of handwaving: If we fix minimum “acceptable” β, & know μ/σ of loads and resistances, can calculate the factors by which to increase loads (load factors, λ) and decrease resistance (resistance factor, ϕ), so that If ϕR ≥ ΣλQ then β ≥ βmin Point of LRFD Establishes load and resistance factors so the (R,Q) combination most likely to cause life-threatening damage for each member or connection occurs with acceptably low probability Limit states always structural, relate to collapse of beams, columns, braces, connections… Limit states are specific to component and load combination. Quake: 2/3 x 1/2500-yr shaking Provides consistent reliability between materials Reliability chosen to reflect failure consequences Thumbnail sketch of LRFD Hazard analysis Load combinations from ASCE 7 chapter 2 E = 2/3 * Sa(T1,5%) with 2% exceedance probability in 50 yr Structural analyses: Linear static analysis using elastic EI Calculate reduced plastic capacities ϕRu Check ϕRu > ΣλQ fYA/2 Ru = fYAh fYA/2 Limitations of LRFD Consistency: typ. linear (static) analysis of building (elastic EI), which at design loads ΣλQ may be highly nonlinear, and in fact we check against plastic capacity ϕR Fidelity: near ultimate capacities there can be much load redistribution compared with linear elastic case Robust characterization of performance: probability of lifethreatening damage to individual members and connections may not be (all) that the owner or the city cares about What about more-frequent or more-rare event? Will the building be operational, occupiable? Informativeness: no basis for designing above code Performance-based earthquake engineering 1st generation (1994+) Consider nonlinear structural response Up to 4 hazard levels Seeks to control performance at the whole-building level Performance in 2-4 whole-building qualitative states FEMA 310 (ASCE 1998) FEMA 356 (ASCE 2000) ASCE/SEI 31 (2003) ASCE/SEI 41 (2006) Performance-based earthquake engineering 1st generation (1994+) Considers nonlinear structural response Up to 4 “hazard levels” Seeks to control performance at the whole-building level Performance in 2-4 whole-building qualitative states Ba s a ob fe sic je ty ct iv e PBEE-1 performance levels PBEE-1 performance levels FEMA 356 gives acceptance criteria (OK or NG) by analysis procedure, component, and performance level, e.g., steel beams & nonlinear analysis: No gory details of PBEE-1 PBEE-1 is the state of the practice, the high end of what many engineering firms are doing Well established in nationally accepted standards But academics are always thinking about what’s next… Limitations of PBEE-1 ASCE 31 treats only MCE event (2%/50 yr) Limit states are still component-based, not truly system-wide If 1 component fails LS, is the building unsafe? Limited treatment of uncertainty & probability No expression of the probability of failures Variability in ground motion has a large effect on structural response Limited information for designing above code What are the benefits? Objectives of PBEE-2 Treat multiple levels of hazard Treat and propagate uncertainty Employ nonlinear dynamic structural analysis Measure performance in system-level losses: Repair costs (“dollars”) Occupant casualties (“deaths”) Loss of functionality (“downtime”) (Catastrophe modelers have been doing this since 1970s) Ultimate objectives of PBEE-2 With capability to estimate 3Ds, can calculate: CDF of earthquake repair cost L given some level of shaking Sa(T1) = s, i.e., FL(l|Sa(T1) = s) What is the mean number of people killed by earthquake damage to this building during the next t years, μK(t) What is the present value of all future earthquake repair costs to this building, PV? If we can do that, we can calculate Cost per statistical life saved CLS CLS = C/(μK0(50) – μK1(50)) Benefit-cost ratio BCR BCR = (PV0 – PV1)/C Probable maximum loss PML PML = FL-1(0.9|Sa(T1) = s475) where C = cost to strengthen S0.002 = shaking with 0.002 chance of being exceeded next year PBEE-2 Pacific Earthquake Engineering Research (PEER) Center & others, e.g., Porter & Kiremidjian (2001), Porter (2003), Goulet et al. (2007) Applied Technology Council ATC-58 (in progress) PBEE-2 terminology IM = intensity measure, e.g., Sa(T1) EDP = engineering demand parameter, e.g., peak transient drift ratio, story n (now, “DP”) DM = damage measure, e.g., damage state of gypsum wallboard partitions at story n DV = decision variable, e.g., repair cost PBEE-2 terminology Term PEER ATC -58 Keith Example Design of the asset: location, design, replacement cost, no. occupants, … N/A N/A A, “asset” Latitude λ, longitude ϕ, replacement cost new RCN Environmental excitation IM “intensity measure” IM S, “shaking” Sa(T1,5%) Structural response EDP “engineering demand parameter” DP R, “response” Axial force in column c, peak transient drift in story n Damage DM “damage measure” DM D, “damage” Cracking in gyp bd partition k Loss DV “decision variable” DV L, “loss” Repair cost, fatalities, or repair duration Hazard analysis A bit more hand-waving: Calculate fundamental period T1 Select intensity measure, e.g., Sa(T1) Get distance R to each nearby fault Know how frequently each produces earthquakes of magnitude M Calculate shaking S given M, R Integrate over M, R, calculate G(s), mean frequency of shaking S ≥ s For each of m values of S Calculate uniform hazard spectrum Collect & scale n recorded ground motion time histories to match spectrum Now have m x n recs. 1 Exceedance frequency, G, yr-1 0.1 0.01 0.001 0.0001 0.0 0.5 1.0 1.5 Spectral acceleration, Sa, g 2.0 Structural analysis MDOF model, typ 2D, rarely 3D Choice of element type: consider demand level Goulet et al. (2007): lumped plasticity better model for frame collapse, fiber better for low demands Nonlinear dynamic analysis for each of m x n time histories Damage analysis Each of m x n sims, For each component, Get DP Calculate CDF of damage state, FD(d) Draw random number u from U(0,1) Simulate for damage : d = FD-1(u) ln d Fi d Failure probability 1.00 i i 0.75 P[i|D=d] 0.50 0.25 ln d Fi 1 d i 1 i 1 0.00 d Demand, D Loss analysis For each of m x n sims, Ea. component class k Each damage state d, Count no. components in that damage state nk,d Get unit cost CDF, FCkd(c) Draw sample u of U(0,1) Calculate ck,d = FCk,d-1(u) Calc repair cost: L = ΣkΣdck,dnk,d ck,d nk,d Recap PEER-style PBEE p[IM|A] A A Hazard G(IM) IM: intensity meas. 1.00 3.00 Undamaged Sa, g Building r Site, Vs30 Cumulative probability Rupture: M, mech 0.75 2.00 Repair T=0 T = 0.3 T = 1.0 0.50 1.00 0.25 Demolish & replace 0.00 0.00 0.000 0.010 0 100.00520 30 40 0.015 50 Peak transient drift Distance r , km Fault 33 Monte Carlo Simulation in PBEE-2 2. 3. 4. 5. Hazard: select a ground-motion timehistory & scale to selected Sa(T1,5%), matching seismic environment Response: sample values of E, I, FY, fc’, etc. & calculate EDPs Damage: feed EDPs into fragility functions, get failure probability pf of each component; draw a sample u; if u ≤ pf, say it failed Loss: count damaged components, sample unit repair costs, multiply & sum. Divide sum by building replacement cost to get damage factor Repeat many times, many levels of Sa(T1,5%) 0.1 0.01 0.001 0.0 0.5 1.0 1.5 2.0 S a (1.0 sec, 5%), g 1.00 Damage factor Y 1. Damage factor Y 1 f Y |S =1g(y ) 0.75 0.50 E[Y |S a =s ] 0.25 0.00 0.0 0.5 1.0 1.5 S a (1.0 sec, 5%), g 2.0 Conclusions Terminology Structural analysis: linear vs. nonlinear, static vs. dynamic Probability: fX(x), FX(x), FX-1(p), μ, σ, σ/μ, ϕ(x), (x) LRFD Load & resistance factors achieve deliberately chosen minimum desired β; sets maximum allowable probability of life-threatening damage Well established national standards beginning around 1977 Prescriptive acceptance criteria (OK, NG), each load combo Component level Conclusions: PBEE-1 New system-level performance objectives, include collapse prevention, occupiability & operability Nonlinear structural analysis, multiple hazard levels, well established national standards ASCE-31 and ASCE-41 Retains prescriptive acceptance criteria (OK, NG) Component based Conclusions: PBEE-2 Nonlinear dynamic analysis, multiple hazard levels, probabilistic treatment of damage and loss, systemlevel losses in terms of direct interest to owner, insurer, etc. (3Ds, PML, BCR…) Rigorous propagation of uncertainty No prescriptive acceptance criteria No national standard yet, many fragility functions & consequence functions left to develop, lots of bookkeeping What you should know about LRFD vs. PBEE-1 vs. PBEE-2 Performance objectives Treatment of hazard Structural analysis approach Treatment of damage & loss State of development & standards Thanks [email protected] (626) 233-9758