S i Normalized joint probabilityI L -...
Transcript of S i Normalized joint probabilityI L -...
Performance Based Earthquake Engineering of Concrete DamsDr. Mohammad A. Hariri-Ardebili and Prof. Victor E. Saouma
Department of Civil Environmental and Architectural Engineering, University of Colorado at Boulder
Fragility
Analysis
Workshop
Thursday, June 18, 2015
1/6: Introduction and Computational Tools (PPACT)
PFMA vs. PBEE• Potential failure mode analysis: is a chain of events leading to
unsatisfactory performance of the dam which could lead to uncontrolledrelease of the water. Can be very qualitative.
• Performance based earthquake engineering: Performance evaluation undercommon and extreme loads based on the diverse needs and objectives ofowners, users and society.
Merlin and PPACT• Merlin is a general purpose finite element software originally developed for
fracture mechanics based static and dynamic analysis of concrete dams.• Probabilistic Performance Assessment of Concrete Dams (PPACT): is a
Matlab-based toolkit works over Merlin.Historical Development• de Araujo and Awruch (1998)• Ellingwood and Tekie (2002)• Lupoi and Callari (2011)• Ghanaat et al. (2011-2015)
0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.2
0.4
0.6
0.8
1
Safety factor
Cum
ulat
ive
prob
abili
ty
ObservationsCDF (normal)CDF (lognormal)
. 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
Sa(T
1) [g]
Lim
it st
ate
prob
abili
ty
LS3: Sliding > 0.1 inLS3: Sliding > 0.5 inLS3: Sliding > 6.0 inLS4: Rel. def. > 0.3 inLS4: Rel. def. > 0.6 in
. 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.2
0.4
0.6
0.8
1
PGA [g]
Pro
babi
lity
of fa
ilure
. 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
PGA [g]
Pro
babi
lity
of s
lidin
g fa
ilure
Dam (Composite)Dam (RTR)Pier (Composite)Pier (RTR)
Hazard analysis
Hazard model
g[IM|D]
Site hazard
g[IM|D]
Structural analysis
Structural model
p[EDP|IM,D]
Structural
response
g[EDP|D]
Damage analysis
Fragility model
p[DM|EDP,D]
Damage
response
g[DM|D]
Loss analysis
Loss model
p[DV|DM,D]
Loss response
g[DV|D]
Facility
definition
D
Facility
informationDecision making
D is OK?
test.bd test.t3d test.ctrl
KumoNoSu
test.inpReferen
ce M
odel
GroundMorion.mat
Block‐10.inp
Block‐80.inpMaterial.mat
Block‐20.inp
Block‐90.inpAging.mat
Loads.mat
...
test‐N.inp
P1.mDetermine N
P0.m
Define Physical Model
User defined input models
Model Initiation
Start
Uncertainties D
efinition
A
2/6: Quantified Potential Failure Mode Analysis
Qualitative vs. Quantitative• One may perform a potential failure mode analysis either
qualitatively (mainly based on site observations) or quantitatively(based on accompanying finite element analyses).
Linear System (Indices)• Demand capacity ratio (DCR)• Cumulative inelastic duration (CID)• Cumulative inelastic area (CIA)• Damage spatial distribution ratio (DSDR)
Linear System• Multiple Strip Analysis (MSA)• Akaike Information Criterion (ACI)• Generate extra EDP based on joint lognormal distribution and Monte
Carlo Simulation• Joint opening/sliding and concrete cracking
Correlation LE vs. NL• Seek a correlation between linear and nonlinear analyses.
(µleR , σ
leR )︸ ︷︷ ︸
Overstressed
=
?︷︸︸︷fm (µnl
R , σnlR )︸ ︷︷ ︸
Cracking
• µleR = 16.5, σle
R = 4.6• µnl
R = 8.6, σnlR = 4.1
DCR
Se
ism
ic In
ten
sity L
eve
l
1.0 1.2 1.4 1.6 1.8 2.01
2
3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
C
(Acceptable)
D (Unacceptable)
1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
DCR
Mea
n C
ID [s
]
Unacceptable
Acceptable
Level 1Level 2Level 3StressStrain
05
1015
2025
0
10
20
30
40
0
0.005
0.01
0.015
DIsliding [mm]DIcracking [%]
Nor
mal
ized
join
t pro
babi
lity
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Damage measure parameter [%]
Cum
ulat
ive
prob
abili
ty
Overstressed
Crack
3/6: Sensitivity and Uncertainty Quantification
Nonlinearity of the system• Fracture mechanics based interface joint model• The generalized failure surface is given by
F = (τ21 + τ2
2 )− 2 c tan(φf )(ft − σ)− tan2(φf )(σ2 − f 2t ) = 0
• A bilinear relationship is used for c(uieff) and σt(uieff).Sensitivity and Uncertainty Analysis• Aleatory vs. epistemic RVs.• Sampling is a key element of an uncertainty analysis.• Monte Carlo Simulation (MCS) vs. Latin Hypercube Sampling
(LHS).• Correlated vs. Uncorrelated RVs.
Models• Mode I: Static POA• Mode II: Static POA)• Mixed-Mode: Dynamic POA
uDmax
γGFIIG
FIφ
dφ
fcf
tk
n
u Dm
ax
kt
γG
FIIG
FIφ d
φ fc
f tk n
k t
uDmax
γGFIIG
FIφ
dφ
fcf
tk
n
u Dm
ax
kt
γG
FIIG
FIφ d
φ fc
f tk n
k t
X1 (Uniform distribution)
X2 (
No
rmal
dis
trib
uti
on
)
X1 (Uniform distribution)X1 (Uniform distribution)
Crude MCS MCS with LHS MCS with LHS and Correlation
0 0.01 0.02 0.03 0.04 0.050
0.1
0.2
Joint opening [mm]
App
lied
disp
lace
men
t [m
m]
0 0.01 0.02 0.03 0.04 0.050
0.1
0.2
Joint opening [mm]
Dis
pers
ion,
β
0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.2
0.4
0.6
0.8
1
Applied displacement [mm]
Pro
babi
lity
of fr
actu
re
η =0.21
β =0.33
Empirical dataFitted curve
0 0.01 0.02 0.03 0.04 0.050
0.1
0.2
Joint opening [mm]
App
lied
disp
lace
men
t [m
m]
0 0.01 0.02 0.03 0.04 0.050
0.1
0.2
0.3
0.4
0.5
Joint opening [mm]
Dis
pers
ion,
β
0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.2
0.4
0.6
0.8
1
Applied displacement [mm]
Pro
babi
lity
of fr
actu
re
η =0.18
β =0.62
Empirical dataFitted curve
0 10 20 30 40 500
0.05
0.1
0.15
0.2
0.25
Crest displacement [mm]
PG
A [g
]
Stepwise curveSmoothed curveCollapse (exact)Collapse (est.)
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
Lcr
/ LT
PG
A [g
]
LS = 0.10LS = 0.30LS = 0.60LS = 0.99
0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
PGA [g]
P[D
I ≥ L
S |
IM]
LS = 0.10LS = 0.30LS = 0.60LS = 0.99
0 10 20 30 40 500
0.05
0.1
0.15
0.2
0.25
Crest displacement [mm]
PG
A [g
]
Stepwise curveSmoothed curveCollapse (exact)Collapse (est.)
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
Lcr
/ LT
PG
A [g
]
LS = 0.10LS = 0.30LS = 0.60LS = 0.99
0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
PGA [g]
P[D
I ≥ L
S |
IM]
LS = 0.10LS = 0.30LS = 0.60LS = 0.99
4/6: Probabilistic Seismic Demand Model
PSDM and Cloud Analysis• PSDM expresses the probability that a system experiences a
certain level of demand for a given IM level.• Cloud analysis is a numerical procedure in which the structure is
subjected to a set of un-scaled ground motions.Criteria for an Optimal IM• Efficiency: Lower dispersion (βEDP|IM)⇒ Higher efficiency.• Practicality: Higher slope (b)⇒ Higher practicality.• Proficiency: Lower ζ ⇒ Higher proficiency.• Sufficiency: Higher p-value⇒ sufficient IM.• Hazard compatibility
Cloud-based fragility function• The results, EDP vs. IM form the so-called cloud response
(arithmetic or logarithmic scale).
ln(ηEDP|IM(IM)
)= b.ln(IM) + ln(a)
P [EDP ≥ edp|IM] = 1− Φ
(ln(edp)− ln(ηEDP|IM)
βEDP|IM
)• Fragility curve vs. fragility surface
4 6 810
1
102
103
M
Rhy
po (
km)
4 6 8
200
400
600
800
1000
1200
1400
M
VS
30 (
m/s
)
500 1000 150010
1
102
103
VS30
(m/s)
Rhy
po (
km)
10−2
10−1
100
101
10−2
100
T [s]
Sa (
T, ξ
= 5
%)
[g]
ln(IM)
ln(EDP)
EDP|IMˆln ln IM lnb a
b
EDP|IM EDP|IMˆln ,N
ED
P|IM
E
DP
|IM
More practical
Less practical
Nonlinear transient analysis
χ (= Mw, Rrup, ε)
ϵEDP|I
M
Smaller p-value
EDP residuals
EDP | IM = , , ,w rupa b M R
Zero slope
Larger p-value
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−5.5
−5
−4.5
−4
−3.5
ln(Sa)
ln(δ
)
Sa(T
1)
Sa(T
2)
Sa(T
3)
Sa(T
4)
Sa(T
5)
−2.5 −2 −1.5 −1 −0.5 0 0.5−5.5
−5
−4.5
−4
−3.5
ln(Sa)
ln(δ
)
Sa1−to−2
Sa1−to−4
Sa1−to−6
Sa1−to−8
Sa1−to−10
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
Sa(T
1) [g]
P[E
DP
≥ e
dp|IM
]
Opening = 2 mmOpening = 5 mmOpening = 8 mmSliding = 2 mmSliding = 5 mmSliding = 8 mm
00.2
0.4 00.01
0.020.03
0
0.5
1
edpIM
P[E
DP
≥ e
dp|IM
]
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
Sa(T
1) [g]
P[D
I ≥ L
S |
IM]
LS = 0.10LS = 0.30LS = 0.60LS = 0.99
IM
LS
0 0.1 0.2 0.3 0.40
0.2
0.4
0.6
0.8
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
5/6: Collapse Fragility Curves for Multiple-Component Ground Motions
Fragility functions• Empirical: From post-earthquake damage data.• Analytical: Transient structural analysis.• Heuristic: Expert opinion.
Lognormal cumulative distribution function (CDF)• CFC
p [C|IM = im] = Φ
(ln(im)− ln(η)
β
)Find the parameters• Method of moments (MM)
η = exp
(1n
n∑i=1
ln(IMci )
), β =
√∑ni=1 (ln (IMc
i ) − ln (η))2
n − 1
• Sum of squared error (SSE){η, β}
= argminη,β
m∑i=1
(nc
ini− Φ
(ln(imi)− ln(η)
β
))2
• Maximum likelihood estimation (MLE){η, β}
= argmaxη,β
∑mi=1
(nc
i ln(
Φ(
ln(imi )−ln(η)β
))+ (ni − nc
i ) ln(
1− Φ(
ln(imi )−ln(η)β
)))
Anatomy of IDA Curves• Capacity limit state in SR-IDA: EDP-based vs IM-based rules.• Hardening in SR-IDA curve based on discrete crack model.• Resurrection in SR-IDA based on smeared crack model.• Summarize the MR-IDA curves to some central values, i.e.
16%, 50% and 84% fractiles. Moreover, analyze theEDP-dependent dispersion.
Epistemic and Aleatory Uncertainties• Epistemic and Aleatory Uncertainties: One may combine the
two uncertainties βRU =√β2
R + β2U assuming independent
uncertainties and fixed median.• Impact of Reservoir Elevation: ηFull
ηEmpty is 0.75 and 0.78 for “H
only” and “H + V” cases. βFull
βEmpty is 1.08 in both cases.
0 5 10 15 20 25 300
0.1
0.2
0.3
EDP
IM
Softening responseMinor hardeningSevere hardening
0 10 20 30 400
0.1
0.2
0.3
EDP (Crest displacement [mm])
IM (
Sa(T
1) [g
])
CapacitypointCapacity
point
CEDP
CIM
Failure IM−based
FailureEDP−based
0 5 10 15 20 25 300
0.1
0.2
0.3
δH [mm]
Sa(T
1) [g
]
H only (Spline)H only (Discrete)H + V (Spline)H + V (Discrete)
EDP
IM
Non-collapse analysisCollapse analysis
PDF CDF
IM IM50%16% and 84%
0.05 0.1 0.15 0.2 0.25 0.30
0.2
0.4
0.6
0.8
1
PGA [g]
Pro
babi
lity
of c
olla
pse
ObservationsLogCDF (MLE)LogCDF (SSE)LogCDF (MM)
0.12 0.14 0.16
0.4
0.5
0.6
vspace*0.1in
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
Sa(T
1) [g]
Pro
babi
lity
of c
olla
pse
RTR Uncert.RTR+Mat.(COV=0.1) Uncert.RTR+Mat.(COV=0.2) Uncert.
0.05 0.1 0.15 0.2 0.25 0.30
0.2
0.4
0.6
0.8
1
PGA [g]
Pro
babi
lity
of c
olla
pse
H only (δH)
H + V (δH)
6/6: Damage Index
• The proposed DI is structural, cumulative, multi-variable,multi-scale. There are three controlling variables
DI = f(
LC,EH ,umax
)• Micro DI for each critical location DIji = β∆ × LC
iLT
ij = A,B,C,D
• Meta DI corresponding to each critical locationDI
j=∑n
i=1 DI ji × ζ
ji ζ j
i =(EH)iEH
• Macro DI for the entire dam or the dam-foundation systemDI =
∑Dj=A DI
j
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Time (s)
Mic
ro D
I at n
eck
ETAF−1ETAF−2ETAF−3
00.25
0.50.75
1 00.1
0.20.3
0.40
0.2
0.4
0.6
0.8
1
PZA, gSa(T
1), g
Mic
ro D
I at
nec
k
Finite element model of the coupled system
Index point
Cra
ck
prop
agat
ion
Dri
ft R
atio
Progressive failure assessment of gravity dam
D
Rt
IM
t
PZA
CAV
PZVIA
Capacity Curve
2 31
t
t
c
c
Damage plasticity
1S
3S2
S
1C
K 23C
K
Dracker-Prager
DI
t
t
DI=1.00
limitDR
4t
5
DI
IM t6
1 2
3
4
5
0.1
0.3
0.6
0.99
DS6
1.00
Intensifying acceleration function
Desired intensity measure parameter ETA Curve
a
t t
,,
aST
t
log T
t