RELIABILITY ASSESSMENT OF STRUCTURAL DYNAMIC SYSTEMS DUE TO EARTHQUAKE-

INDUCED TSUNAMIS

Z. Shao & S.H. Cheung

School of Civil and Environmental Engineering, NTU

07-Nov-2013

1

Motivation

List of 10 COSTLIEST NATURAL HAZARDS from 1980 - 2012

Rank Overall loss [US$ mn] Fatalities Event Year

1. 210,000 15,840 Tohoku earthquake (JP) 2011

2. 125,000 1,322 Hurricane Katrina (US) 2005

3. 100,000 6,430 Kobe earthquake (JP) 1995

4. 85,000 84,000 Sichuan earthquake (CN) 2008

5. 65,000 210 Hurricane Sandy (US) 2012

6. 44,000 61 Northridge earthquake (US) 1994

7. 43,000 813 Thailand flood (SEA) 2011

8. 38,000 170 Hurricane Ike (US) 2008

9. 30,700 4159 China floods (CN) 1998

10. 30,000 520 Chile earthquake (CL) 2010

Source: Munich Re NatCatSERVICE 2

Motivation

List of 10 DEADLIEST TSUNAMIS in the history

Rank Death toll Location Year

1. 350,000 Indonesia (EQ) 2004

2. 100,000+ Ancient Greece (VC) 1410 B.C.

3. 100,000 Portugal (EQ) 1755

4. 100,000 Italy (EQ) 1908

5. 40,000 Taiwan (EQ) 1782

6. 36,500 Indonesia (VC) 1883

7. 30,000 Japan (EQ) 1707

8. 26,360 Japan (EQ) 1896

9. 25,674 Chile (EQ) 1868

10. 15,854 Japan (EQ) 2011

Source: Geist & Parsons 2011 Source: National Geophysical Data Center

EQ: earthquake; VC: volcano eruption 3

General Flowchart

1. Stochastic earthquake source modelling

2. Tsunami generation and propagation

3. Tsunami runup and wave-structure interaction

4. Reliability analysis and structural loss assessment

4

Earthquake Source Mechanism

• Far-field tsunami

– The amplitude of tsunami can be estimated reasonably well based

on the earthquake moment magnitude [Geist, 1999 and references therein]

• Near-field tsunami

– Slip distribution (uniform slip model, k-squared slip model, etc.)

– Fault dimension

– Fault geometry (slip angle, dip angle, strike angle)

– The above are also the important sources of uncertainties in

tsunami risk analysis

5

Tsunami Generation & Propagation

• Tsunami generation & propagation using COMCOT

(Cornell Multi-grid Coupled Tsunami Model)

– Initial water displacement calculation based on Okada’s

analytical solution for surface deformation in vertical

direction due to shear dislocation in a half space.

– Linear shallow water equations (SWE) and nonlinear

shallow water equations (NSWE).

– Explicit leap-frog finite difference method.

6

Tsunami-wave-structure Interaction

• Tsunami force classifications

– Hydrodynamic pressure

– Hydrostatic pressure

– Impact by debris

– etc.

• Simulation methods

– Monolithic & partitioned approaches

– ALE formulation (LS-DYNA)

Monolithic approach

Partitioned approach , ,( ) 0m

i j j i j ij j iv v v v f

7

Probability Model for Tsunami Risk

• Probabilistic methods for evaluating tsunami risk

– Deterministic approach for fast access

– First probabilistic tsunami assessment by Rikitake and Aida (1988)

– FEMA – probabilistic map for Oregon (Gonzalez et al, 2009)

– PBTE (2006-2011)

• No reliability analysis of any systems has been considered and

no model considers the complete physical processes

8

Structural Reliability Analysis

• Structural reliability formulation

– The failure probability can be written in terms of a limit state

function L(θ) given by:

• Simulation methods for reliability analysis for general

nonlinear dynamic system involving high stochastic

dimension

– Monte Carlo

– Subset Simulation (Au and Beck, 2001)

– Competitive S3 and more efficient ADM (by Prof. Cheung and his

collaborators) and a new approach by Prof. Cheung and Sahil

9

Tohoku Tsunami Modelling • Tohoku tsunami (Mw 9.1) modeling: earthquake generation

Epicenter (Lat, Lon) [deg] 38.1, 143.2

Shao et al., 2011

Strike direction (θ) [deg] 199.0

Dip angle (δ) [deg] 10.0

Rake angle (λ) [deg] 92

Focal depth [m] 24400

Length of fault [m] 10^(-2.37+0.57*Mw) Blaster, 2010

Width of fault [m] 10^(-1.86+0.46*Mw) Blaster, 2010

Dislocation (slip) [m] Gallovic, 2004

Fault discretization 30 X 30 -

10

Tohoku Tsunami Modelling

Maximum inundation map (a) COMCOT simulation (b) Post-tsunami survey

observations and inundation lines (Source: Breanyn, Gusman et al. 2013)

(a) (b)

11

Tsunami & Structural Modelling

• Tsunami-wave-structure interaction using LS-DYNA

Geometry of the

structural wall [m]

Height 5

Length 7

Width 0.2

Water element size:

0.25m × 0.25m × 0.25m.

Structural wall

Air/Null Water part

12

Tsunami & Structural Modelling

• Material properties of the reinforced concrete

structural wall

Material parameters for concrete

MAT72 RL3

Mass density 2360 kg/m3

Elastic modulus Ec 27800 Mpa

Poisson ratio 0.2

34.5 MPa

Material parameters for steel rebar

MAT03

Mass density 7800 kg/m3

Young’s modulus Es 200 GPa

Poisson ratio 0.3

Yield’s strength 460 MPa

Percent reinforcement 2%

13

Stochastic Earthquake Parameter Modelling

• Earthquake parameter generation

Parameter Tohoku Region Earthquake Parameter for Mw 8.3

Epicenter (Lat, Lon) [deg] Fixed at 38.1, 143.2

Strike direction (θ) [deg] Fixed at 199.0

Dip angle (δ) [deg] Fixed at 10.0

Rake angle (λ) [deg] Normally distributed with a mean of 90 and a COV of 0.33*

Focal depth [m] Fixed at 24400

Length of fault [m] 10^(-2.37+0.57*Mw)**

Width of fault [m] 10^(-1.86+0.46*Mw)**

Dislocation (slip) [m]

Fault discretization 4 X 4

* Yamamoto & Hori, 2004;

** Blaster, 2010;

*** Gallovic, 2004. 14

Results and Discussion

15

Problem Statement

• Computationally demanding in estimating the failure

probability in terms of a limit state function L(θ):

• MCS: 10/PF e.g. PF =10-3

• Subset Simulation: 500 * (-log(PF))

z(θ1)

z(θ2)

……

z(θNt)

Response

Performance Function

Numerical

simulations

Response Analysis

θ1

θ2

……

θNt

Stochastic

samples

Proposal Distribution

θi ∝ p(θ)

16

Problem Statement

• Computationally demanding in estimating the failure

probability in terms of a limit state function L(θ):

• MCS: 10/PF

• Subset Simulation: 500 * (-log(PF))

z(θ1)

z(θ2)

……

z(θNt)

Response

Performance Function

Response

Surface

Response Analysis

θ1

θ2

……

θNt

Stochastic

samples

Proposal Distribution

θi∝ p(θ)

RSM

RSM

17

Proposed Methodology

• Concept of the proposed method

Subset Simulation Algorithm

Modified Moving Least

Square Response Surface

SS-MLSRSA algorithm

18

Subset Simulation Algorithm

Source: Au & Beck, 2001

*

*

( ) ( | )min 1,

( ) ( | )

j j

j k

j j j

k j k

j j

j

j

p qr

p q

ξ θ ξ

θ ξ θ

|~ ( )k iFpθ θ

*( | )~ j

j k

jq θξ

1

j j

k θ ξ 1 k

j

k θ θ

1

j

kθ

1k iZ F

1k i

FZ

1 ~ ( )ik p Fθ θ |

1 1

k k θ θ 1k k

θ θ

1 1~ ( | )k kZ p Z θ

jr 1

jr

j = 1 : N c

Generate candidate state

Simulate , set with

probability , and with

probability 1-

Accept/reject the proposed samples

w.r.t the limit state function

1 :j

kθ

*( | )~ j

j k

jq θξ 1

j j

k θ ξ

jr 1 k

j

k θ θj

r

11 1 , ,

1

1( | ) ) where( ( | )

i

N

i i i i k i k

k

F iP F F P IN

p F

θ θ θ

19

Modified Moving Least Square Response

Surface Approximation (MLSRSA)

Source: Taflanidis, A. A. & Cheung, S.-H., 2012

1

1 1

1

ˆ ( ) ( ) { } { }

[ ( ) ... ( )] [ ( ) ... ( )]

{ } [ ( ; ) ... ( ; ) ]

T T T

T T

NS NS

NS

f

f f

diag w d w d

θ b θ B W θ B B W θ F

B b θ b θ F θ θ

W θ θ θ θ θ

2 2

,

1

( ; )n

I I i i I i

i

d v

vθ θ θ θ θ θ

2

1

ˆ{ } { }[ ( ) ( )]NS

R I IIJ w f f

θ θ θ θ

2 2

2

1

1( ) if ; 0 else

1

k k

k

d

cD c

c

e ew d d D

e

w(d): weight measures depend on

the distance between support points

and interpolation point

d(θ;θI): norm distance with vi

represents importance of each

component of θ vector

20

Modified Moving Least Square Response

Surface Approximation (MLSRSA) (cont’d)

( ) ( )( ) ( ) ( )

( ) ( )

z pz p

z p d

θ θθ θ θ

θ θ θ

Source: Taflanidis, A. A. & Cheung, S.-H., 2012

( )( ( ) || ( )) ( ) log

( )

ire i i i i

i

D q dq

-4 -2 0 2 4 0

0.1

0.2

0.3

0.4

q(θ)

-4 -2 0 2 4 0

0.2

0.4

0.6

0.8

π(θ)

Gaussian kernel density

approximation of PDFs

ˆ( ) θ

21

General procedure of SS-MLSRSA

MCS • Stochastic NS ∝ p(θ)

• 𝜋(θ) ∝ p(θ|F1): Top 10% NS

Construct MLS

• Construct MLS surface

Stochastic sampling

• Stochastic Nmls ∝ p(θ)

• 𝜋(θ) ∝ p(θ|F2): Top 1% (NS+Nmls)

• 𝜋(θ) ∝ p(θ|F3): Top 0.1% (NS+Nmls)

• ……

22

• Naïve combination of Subset Simulation and MLSRSA…

General procedure of SS-MLSRSA

• Results and discussion

Smln Lv1 Lv2

350NS 2.1 -81.0

300NS 15.2 -25.0

250NS 24.0 -63.0

200NS 45.4 -39.0

Fit for the response surface model

based on the error (%) compared to

subset simulations

Threshold levels based on SS simulation:

b1= 0.75cm & b2 = 2.35cm

23

General procedure of SS-MLSRSA

Threshold lv 1

Threshold lv 2

MCS • Stochastic NS ∝ p(θ)

• 𝜋(θ) ∝ p(θ|F1): Top 10% NS

Construct MLS-lv1

• Stochastic Nmls ∝ p(θ)

• 𝜋(θ) ∝ p(θ|F1): Top 10% Nmls

Additional simulation Nsim =10% Nmls

• NS = Nmcs+Nsim

• 𝜋(θ) ∝ p(θ|F2): 10% Nsim+1% NS

Construct MLS-lv2 • Stochastic Nmls ∝ p(θ|F1)

• 𝜋(θ) ∝ p(θ|F2): Top 10% Nmls

24

• A NOVEL integration for them to communicate with each other to

obtain the performance as good as possible.

Results and Discussion

Top 10% of the maximum

displacement computed by

MLS-lv1 (theoretically, θ∝

p(θ|F1)) are selected and their

maximum displacements are computed using the original

physical numerical simulation

instead of using the response

surface.

The basic idea is to gain more

NS samples lying in the failure

domain for a more accurate

MLS-lv2. Failure probability curves for NS350 and SS

25

Results and Discussion

• Comparison of c.o.v. δ for SS and

SS-MLSRSA w.r.t MCS.

PF=10-2 Smln δ

SS 1,000 0.252

MCS 1,000 0.315

SS-MLSRSA 550 0.109

26

PF=10-2 SS MCS

SS-MLSRSA 10 20

SS - 2

• Factor of computational savings

PF=10-5 SS MCS

SS-MLSRSA 10 2500

SS - 250

• Factor of computational savings

Conclusions

• Tsunami-wave-structure interaction process simulation with credible

results using current numerical approaches.

• The proposed SS-MLSRSA approach based on a novel integration of

the SS algorithm and a recently proposed entropy-based MLSRSA

provides comparable failure probability estimates with much fewer

computational efforts compared with the SS without integrating with

the response surface approach.

27

THANK YOU

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