A brief history of PBEE-2

CVEN 5835CVEN 5835--02 02 SP TPS: Nonlinear Structural Analysis;SP TPS: Nonlinear Structural Analysis;Theory and ApplicationsTheory and Applications17 Feb 201117 Feb 2011

Keith Porter, Associate Research ProfessorKeith Porter, Associate Research ProfessorCivil, Environmental, and Architectural EngineeringCivil, Environmental, and Architectural EngineeringUniversity of Colorado at BoulderUniversity of Colorado at Boulder

Today’s objectives

Some terminologySome terminology History and key concepts of LRFDHistory and key concepts of LRFD History and main goals of PBEEHistory and main goals of PBEE--11 Origin and main goals of PBEEOrigin and main goals of PBEE--22 Overview of how PBEEOverview of how PBEE--2 works2 works Monte Carlo simulation in PBEEMonte Carlo simulation in PBEE--22

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

Force Q

Stress sStrain e

Displacement ∆

Small-angle rule may no longer hold

Q∆

Stress s

Q

fy

3 sources of nonlinearity

Static vs. dynamic structural analysisStatic analysis: solveStatic analysis: solve

KxKx = V= VDynamic analysis: solve Dynamic analysis: solve

MMẍẍ(t(t) + ) + CCẋẋ(t(t) + ) + Kx(tKx(t) = ) = --MMẍẍgg(t(t))

V

Using probability distributions

Probability density function PDF, of X: Probability density function PDF, of X: ffXX(x(x)) Probability that X = x per unit x, for continuous XProbability that X = x per unit x, for continuous X

Probability mass function PMF, of X: Probability mass function PMF, of X: ppXX(x(x)) Probability that X = x, for discrete XProbability that X = x, for discrete X

Cumulative distribution function CDF, Cumulative distribution function CDF, FFXX(x(x)) Probability that X Probability that X ≤≤ x, continuous or discrete Xx, continuous or discrete X

Inverse cumulative distribution function, FInverse cumulative distribution function, F--11XX(p) (p)

Value of X with probability p of not being exceededValue of X with probability p of not being exceeded

Px

X XxX F x f z dz

1Xx F p

Normal distributionCumulative Cumulative distrdistr function (CDF)function (CDF)

FFXX(x(x) = ) = probprob that X that X ≤≤ xxProbProb density function (PDF)density function (PDF)

ffXX(x(x) = ) = probprob that X = x, per unit Xthat X = x, per unit X

We will use more than 1 uncertain variable, so let us denotefX(x) = probability density function of X evaluated at a particular value xFX(x) = cumulative probability function of X evaluated at a particular value xFX

-1(p) = particular value of X such that FX(x) = probability p

FX(x) =

Let μ = mean, a central measure; σ = standard deviation, a measure of dispersion

Monte Carlo SimulationHow to produce sample values of arbitrary distribution

Uniform distribution Uniform distribution u ~ U(0, 1), e.g., u ~ U(0, 1), e.g.,

rand() in Excelrand() in Excel We want to sample XWe want to sample X For each sample, draw For each sample, draw

a sample ua sample u Invert CDF, Invert CDF, xxuu = = FFXX

--11((uu)) Repeat many times Repeat many times

with different samples with different samples uu

0.00

0.25

0.50

0.75

1.00

-0.25 0.00 0.25 0.50 0.75 1.00 1.25

u

F U(u

)

0.00

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1.00

1.25

-0.25 0.00 0.25 0.50 0.75 1.00 1.25

u

f U(u

)

0.00

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0.50

0.75

1.00

0.00 0.50 1.00 1.50 2.00

x

F X(x

)

0.00

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0.50

0.75

1.00

0.00 0.50 1.00 1.50 2.00

x

F X(x

)

Joint probability distribution

Say XSay X11 and Xand X22 are are each normally each normally distributed & distributed & independentindependent

p[Xp[X1 1 = x= x11, X, X2 2 = = xx22] = ] = ffX1X1(x(x11)f)fX2X2(x(x22))

0

0.5

1

1.5

2

2.5

3

3.5

44.5

0.00

0.02

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1.01.5

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3.03.5

4.04.5

X2

Prob

ability den

sity

X1

Load and Resistance Factor Design

Seeks to control failure Seeks to control failure probability by structural member probability by structural member or connection & load or connection & load combinationcombination

Concrete: ACI 318 (1977)Concrete: ACI 318 (1977) Steel: AISC LRFD (1Steel: AISC LRFD (1stst ed. 1986)ed. 1986) General: Ellingwood et al. (1980)General: Ellingwood et al. (1980)

http://spot.colorado.edu/~porterka/Ellingwood-1980-LRFD-for-A58.pdf

Thumbnail sketch of LRFD

Let R = resistance, Q = load on a member or connectionLet R = resistance, Q = load on a member or connectionSay R and Q are independent, normally distributed, Say R and Q are independent, normally distributed, means means μμRR and and μμQQ, std deviations , std deviations σσRR and and σσQQ. .

0

0.5

1

1.52

2.53

3.54

4.50.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.00.5

1.01.5

2.02.5

3.03.5

4.04.5

Resistance R

Prob

ability den

sity

Load Q

0

0.5

1

1.52

2.53

3.54

4.50.00

0.02

0.04

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0.00.5

1.01.5

2.02.5

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4.04.5

Resistance R

Prob

ability den

sity

Load Q

Joint probability distribution Probability that R – Q < 0

FailureR < Q

SurvivalR > Q

Q Q

Thumbnail sketch of LRFD Let Let g(R,Qg(R,Q) = R ) = R –– Q (Q (““performance functionperformance function””)) Let Let μμXX= denote mean, = denote mean, σσXX stdevstdev of Xof X μμRR--QQ= = μμRR –– μμQQ

σσRR--QQ = (= (σσRR22 + + σσQQ

22))0.50.5

Let Let ββ = = μμRR--QQ//σσRR--QQ

Failure Failure probprob PPff = = P[gP[g<0] <0] = = ((0 ((0 –– μμRR--QQ)/)/σσRR--QQ))= = ((––μμRR--QQ//σσRR--QQ))= = ((––ββ))

ββ: : ““reliability index,reliability index,”” a measure of the likelihood of failurea measure of the likelihood of failure Bigger Bigger ββ = more reliable= more reliable

Thumbnail sketch of LRFD

A bit of A bit of handwavinghandwaving:: If we fix minimum If we fix minimum ““acceptableacceptable”” ββ, , & know & know μμ//σσ of loads and resistances,of loads and resistances, can calculate the factors by which to increase can calculate the factors by which to increase

loads (load factors, loads (load factors, λλ)) and decrease resistance (resistance factor, and decrease resistance (resistance factor, ϕϕ), so that), so that

If If ϕϕR R ≥≥ ΣλΣλQ Q then then ββ ≥≥ ββminmin

Point of LRFD Establishes load and resistance factors so the (R,Q) Establishes load and resistance factors so the (R,Q)

combination most likely to cause lifecombination most likely to cause life--threatening damage for threatening damage for each member or connection occurs with acceptably low each member or connection occurs with acceptably low probabilityprobability

Limit states always structural, relate to collapse of beams, Limit states always structural, relate to collapse of beams, columns, braces, connectionscolumns, braces, connections……

Limit states are specific to component and load combination. Limit states are specific to component and load combination. Quake: 2/3 x 1/2500Quake: 2/3 x 1/2500--yr shaking yr shaking Provides consistent reliability between materialsProvides consistent reliability between materials Reliability chosen to reflect failure consequencesReliability chosen to reflect failure consequences

Thumbnail sketch of LRFD Hazard analysisHazard analysis

Load combinations from Load combinations from ASCE 7 chapter 2 ASCE 7 chapter 2

E = 2/3 * E = 2/3 * SSaa((TT11,5%) with ,5%) with 2% exceedance 2% exceedance probability in 50 yr probability in 50 yr

Structural analyses: Structural analyses: Linear static analysis Linear static analysis

using elastic EIusing elastic EI Calculate reduced Calculate reduced

plastic capacities plastic capacities ϕϕRRuu

Check Check ϕϕRRuu > > ΣλΣλQQ

fYA/2

fYA/2

Ru = fYAh

Limitations of LRFD Consistency: typ. linear (static) analysis of building (elastic Consistency: typ. linear (static) analysis of building (elastic

EI), which at design loads EI), which at design loads ΣΣλλQ may be highly nonlinear, and Q may be highly nonlinear, and in fact we check against plastic capacity in fact we check against plastic capacity ϕϕRR

Fidelity: near ultimate capacities there can be much load Fidelity: near ultimate capacities there can be much load redistribution compared with linear elastic caseredistribution compared with linear elastic case

Robust characterization of performance: probability of lifeRobust characterization of performance: probability of life--threatening damage to individual members and connections threatening damage to individual members and connections may not be (all) that the owner or the city cares aboutmay not be (all) that the owner or the city cares about What about moreWhat about more--frequent or morefrequent or more--rare event?rare event? Will the building be operational, Will the building be operational, occupiableoccupiable??

InformativenessInformativeness: no basis for designing above code: no basis for designing above code

Performance-based earthquake engineering11stst generation (1994+) generation (1994+)

Consider nonlinear Consider nonlinear structural responsestructural response

Up to 4 hazard levelsUp to 4 hazard levels Seeks to control Seeks to control

performance at the performance at the wholewhole--building levelbuilding level

Performance in 2Performance in 2--4 4 wholewhole--building building qualitative statesqualitative states

FEMA 310 (ASCE 1998) ASCE/SEI 31 (2003)

FEMA 356 (ASCE 2000) ASCE/SEI 41 (2006)

Performance-based earthquake engineering

Basic safety

objective

11stst generation (1994+) generation (1994+) Considers nonlinear Considers nonlinear

structural responsestructural response Up to 4 Up to 4 ““hazard levelshazard levels”” Seeks to control Seeks to control

performance at the performance at the wholewhole--building levelbuilding level

Performance in 2Performance in 2--4 4 wholewhole--building building qualitative statesqualitative states

PBEE-1 performance levels

PBEE-1 performance levels FEMA 356 gives acceptance criteria (OK or NG) by FEMA 356 gives acceptance criteria (OK or NG) by

analysis procedure, component, and performance analysis procedure, component, and performance level, e.g., steel beams & nonlinear analysis:level, e.g., steel beams & nonlinear analysis:

No gory details of PBEE-1

PBEEPBEE--1 is the state of the practice, the high 1 is the state of the practice, the high end of what many engineering firms are end of what many engineering firms are doingdoing

Well established in nationally accepted Well established in nationally accepted standardsstandards

But academics are always thinking about But academics are always thinking about whatwhat’’s nexts next……

Limitations of PBEE-1 ASCE 31 treats only MCE event (2%/50 yr)ASCE 31 treats only MCE event (2%/50 yr) Limit states are still componentLimit states are still component--based, not truly based, not truly

systemsystem--widewide If 1 component fails LS, is the building unsafe?If 1 component fails LS, is the building unsafe?

Limited treatment of uncertainty & probabilityLimited treatment of uncertainty & probability No expression of the probability of failuresNo expression of the probability of failures Variability in ground motion has a large effect on Variability in ground motion has a large effect on

structural responsestructural response Limited information for designing above codeLimited information for designing above code

What are the benefits?What are the benefits?

Objectives of PBEE-2 Treat multiple levels of hazardTreat multiple levels of hazard Treat and propagate uncertaintyTreat and propagate uncertainty Employ nonlinear dynamic structural analysisEmploy nonlinear dynamic structural analysis Measure performance in systemMeasure performance in system--level losses:level losses:

Repair costs (Repair costs (““dollarsdollars””)) Occupant casualties (Occupant casualties (““deathsdeaths””)) Loss of functionality (Loss of functionality (““downtimedowntime””)) (Catastrophe modelers have been doing this (Catastrophe modelers have been doing this

since 1970s)since 1970s)

Ultimate objectives of PBEE-2With capability to estimate 3Ds, can calculate:With capability to estimate 3Ds, can calculate:

CDF of earthquake repair cost CDF of earthquake repair cost LL given some level given some level of shaking of shaking SSaa((TT11) = ) = ss, i.e., , i.e., FFLL((ll||SSaa((TT11) = ) = ss))

What is the mean number of people killed by What is the mean number of people killed by earthquake damage to this building during the next earthquake damage to this building during the next tt years, years, μμKK((tt))

What is the present value of all future earthquake What is the present value of all future earthquake repair costs to this building, PV? repair costs to this building, PV?

If we can do that, we can calculate Cost per statistical life saved CLSCost per statistical life saved CLS

CLSCLS = C/(= C/(μμKK00(50) (50) –– μμKK11(50))(50)) BenefitBenefit--cost ratio BCRcost ratio BCR

BCRBCR = (PV= (PV00 –– PVPV11)/C)/C Probable maximum loss PMLProbable maximum loss PML

PML = PML = FFLL--11(0.9|(0.9|SSaa((TT11) = s) = s475475))

where where C = cost to strengthenC = cost to strengthen SS0.0020.002 = shaking with 0.002 chance of being = shaking with 0.002 chance of being

exceeded next yearexceeded next year

PBEE-2 Pacific Earthquake Pacific Earthquake

Engineering Research Engineering Research (PEER) Center & others, e.g., (PEER) Center & others, e.g., Porter & Kiremidjian (2001), Porter & Kiremidjian (2001), Porter (2003), Porter (2003), GouletGoulet et al. et al. (2007)(2007)

Applied Technology Council Applied Technology Council ATCATC--58 (in progress)58 (in progress)

PBEE-2 terminology

IM = intensity measure, e.g., IM = intensity measure, e.g., SSaa((TT11)) EDP = engineering demand parameter, e.g., EDP = engineering demand parameter, e.g.,

peak transient drift ratio, story peak transient drift ratio, story nn (now, (now, ““DPDP””)) DM = damage measure, e.g., damage state DM = damage measure, e.g., damage state

of gypsum wallboard partitions at story of gypsum wallboard partitions at story nn DV = decision variable, e.g., repair costDV = decision variable, e.g., repair cost

PBEE-2 terminologyTerm PEER ATC

-58Keith 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 analysisA bit more handA bit more hand--waving:waving:

Calculate fundamental period TCalculate fundamental period T11

Select intensity measure, e.g., Select intensity measure, e.g., SSaa(T(T11)) Get distance R to each nearby faultGet distance R to each nearby fault Know how frequently each produces Know how frequently each produces

earthquakes of magnitude Mearthquakes of magnitude M Calculate shaking S given M, RCalculate shaking S given M, R Integrate over M, R, calculate Integrate over M, R, calculate G(sG(s), ),

mean frequency of shaking mean frequency of shaking SS ≥≥ ss For each of For each of mm values of values of SS

Calculate uniform hazard spectrumCalculate uniform hazard spectrum Collect & scale Collect & scale nn recorded ground motion recorded ground motion

time histories to match spectrumtime histories to match spectrum

Now have Now have mm x x nn recs.recs.

0.0001

0.001

0.01

0.1

1

0.0 0.5 1.0 1.5 2.0Ex

ceed

ance

freq

uenc

y, G

, yr-1

Spectral acceleration, Sa, g

Structural analysis MDOF model, MDOF model, typtyp 2D, 2D,

rarely 3Drarely 3D Choice of element type: Choice of element type:

consider demand levelconsider demand level GouletGoulet et al. (2007): lumped et al. (2007): lumped

plasticity better model for plasticity better model for frame collapse, fiber better frame collapse, fiber better for low demandsfor low demands

Nonlinear dynamic analysis Nonlinear dynamic analysis for each of m x n time for each of m x n time historieshistories

Damage analysisEach of m x n Each of m x n simssims,,

For each component,For each component, Get DPGet DP Calculate CDF of Calculate CDF of

damage state, damage state, FFDD(d(d)) Draw random number u Draw random number u

from U(0,1)from U(0,1) Simulate for damage :Simulate for damage :

d = Fd = FDD--11(u)(u) 0.00

0.25

0.50

0.75

1.00

Failu

re p

roba

bilit

y

Demand, Dd

ln ii

i

dF d

11

1

ln ii

i

dF d

P[i|D=d]

Loss analysisFor each of For each of mm x x nn simssims,, Ea. component class Ea. component class kk

Each damage state Each damage state dd,, Count no. components in Count no. components in

that damage state that damage state nnk,dk,d

Get unit cost CDF, Get unit cost CDF, FFCkdCkd(c(c)) Draw sample Draw sample uu of of UU(0,1)(0,1) Calculate Calculate cck,dk,d = F= FCk,dCk,d

--11((uu))

Calc repair cost:Calc repair cost:L = L = ΣΣkkΣΣddcck,dk,dnnk,dk,d nk,d

ck,d

33

Recap PEER-style PBEE

Site, Vs30

Building

Fault

Rupture:M, mech

0.00

1.00

2.00

3.00

0 10 20 30 40 50Distance r , km

Sa, g

T = 0T = 0.3T = 1.0

r0.00

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0.50

0.75

1.00

0.000 0.005 0.010 0.015

Peak transient drift

Cum

ulat

ive

prob

abili

ty

Repair

Demolish &replace

Undamaged

A

A

p[IM|A]

IM: intensity meas.

Hazard G(IM)

Monte Carlo Simulation in PBEE-21. Hazard: select a ground-motion time-

history & scale to selected Sa(T1,5%), matching seismic environment

2. Response: sample values of E, I, FY, fc’, etc. & calculate EDPs

3. Damage: feed EDPs into fragility functions, get failure probability pf of each component; draw a sample u; if u ≤ pf, say it failed

4. Loss: count damaged components, sample unit repair costs, multiply & sum. Divide sum by building replacement cost to get damage factor

5. Repeat many times, many levels of Sa(T1,5%)

0.001

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1

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S a (1.0 sec, 5%), g

Dam

age

fact

or Y

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0.0 0.5 1.0 1.5 2.0

S a (1.0 sec, 5%), g

Dam

age

fact

or Y

E[Y |S a =s ]

f Y |S =1g(y )

ConclusionsTerminologyTerminology Structural analysis: Structural analysis:

linear vs. nonlinear, static vs. dynamiclinear vs. nonlinear, static vs. dynamic

Probability: Probability: ffXX(x(x), ), FFXX(x(x), F), FXX

--11(p), (p), μμ, , σσ, , σσ//μμ, , ϕϕ(x), (x), (x)(x)

LRFDLRFD Load & resistance factors achieve deliberately chosen minimum deLoad & resistance factors achieve deliberately chosen minimum desired sired

ββ; sets maximum allowable probability of life; sets maximum allowable probability of life--threatening damagethreatening damage Well established national standards beginning around 1977Well established national standards beginning around 1977 Prescriptive acceptance criteria (OK, NG), each load comboPrescriptive acceptance criteria (OK, NG), each load combo Component levelComponent level

Conclusions: PBEE-1

New systemNew system--level performance objectives, level performance objectives, include collapse prevention, occupiability & include collapse prevention, occupiability & operabilityoperability

Nonlinear structural analysis, multiple hazard Nonlinear structural analysis, multiple hazard levels, well established national standards levels, well established national standards ASCEASCE--31 and ASCE31 and ASCE--4141

Retains prescriptive acceptance criteria (OK, Retains prescriptive acceptance criteria (OK, NG)NG)

Component basedComponent based

Conclusions: PBEE-2 Nonlinear dynamic analysis, multiple hazard levels, Nonlinear dynamic analysis, multiple hazard levels,

probabilistic treatment of damage and loss, systemprobabilistic treatment of damage and loss, system--level losses in terms of direct interest to owner, level losses in terms of direct interest to owner, insurer, etc. (3Ds, PML, BCRinsurer, etc. (3Ds, PML, BCR……))

Rigorous propagation of uncertaintyRigorous propagation of uncertainty No prescriptive acceptance criteriaNo prescriptive acceptance criteria No national standard yet, many fragility functions & No national standard yet, many fragility functions &

consequence functions left to develop, lots of consequence functions left to develop, lots of bookkeepingbookkeeping

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

keith.porter@colorado.edukeith.porter@colorado.edu(626) 233(626) 233--97589758