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