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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  FX1  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

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