1

P

E

E

R

U

C

S

D

Probabilistic Performance-Based

Optimum Seismic Design of

(Bridge) Structures

PI: Joel. P. Conte

Graduate Student: Yong Li

Sponsored by the Pacific Earthquake Engineering Research Center

2

(I) PEER PBEE Methodology

(Forward PBEE Analysis)

3

SDOF Model and Site

• Single-Degree-of-Freedom Bridge Model:

! SDOF bridge model with the same initial period as an MDOF model of

the Middle Channel Humboldt Bay Bridge developed in OpenSees.

• Site:

! Site Location: Oakland (37.803N, 122.287W)

! Site Condition: Vs30 = 360m/s (NHERP Class C-D)

"Uniform Hazard Spectra are obtained for 30 hazard levels from the

USGS 2008 Interactive Deaggregation software/website (beta version)

1

0

1.33 sec

6,150 tons

137,200 kN/m

10,290 kN

0.10

0.02

y

T

m

k

F

b

!

"

"

"

"

"

"

SDOF Model

(Menegotto-Pinto)

0k

F

U

yF

0.075 myU "

1

1 0b k#

Uniform Hazard Spectra

T1=1.33 sec

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Prob. Seismic Hazard Analysis and EQ Selection

• Seismic Hazard Curve (for single/scalar Intensity Measure IM):

• Least Square Fitting of 30 Data Points from USGS:

• Earthquake Record Selection: ! 146 Records were selected from NGA database based on fault mechanism,

M-R deaggregation, and local site condition (e.g., Vs30)

flt

i i

i i

N

IM i M R

i 1 R M

(im) P IM im m,r (m) (r) dm drf f"

$ %& " & # ' # # # #( )* + +

( , )IM MAREi i

, -1, 5%a

TS ! "

Deaggregation (T1 = 1 sec, 2% in 50 years)

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Probabilistic Seismic Demand Hazard Analysis

• Probabilistic Seismic Demand Analysis conditional on IM:

• Deaggregation of with respect to IM:

• Demand Hazard Curve:

, - , -EDP IM

IM

edp P EDP edp IM d im& &" '$ %( )+

Seismic hazard curve

, - , -, -

, -IM i

EDP i ii i

d imedp P EDP edp IM im im

im

&& $ %" ' " #.( ) .*

Contribution of bin

IM = imi to , -EDP edp&

!Cloud Method" !Convolution" !Deaggregation"

, -EDP edp&

PDF (Lognormal)

EDP: Displ. Ductility

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Probabilistic Seismic Demand Hazard Analysis

!Cloud Method" !Convolution" !Deaggregation"

PDF (Lognormal)

EDP: Normalized EH

PDF (Lognormal)

EDP: Peak Abs. Accel.

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Probabilistic Capacity (Fragility) Analysis

• Fragility Curves (postulated & parameterized):

/ 0|kP DM ls EDP edp' "

! Defined as the probability of the structure/component exceeding kth limit-state

given the demand.

! Developed based on analytical and/or empirical capacity models and

experimental data.

Displacement Ductility Absolute Acceleration Normalized EH Dissipated

Damage Measure

k-th limit state

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Probabilistic Damage Hazard Analysis

• Damage Hazard (MAR of limit-state exceedance):

| ( )k

k EDPDS

EDP

P DM ls EDP edp d edp& &$ %" ' "( )+Fragility Analysis

• Damage Hazard Results:

Associated

EDP Limit States MARE

Return

Period PE in 50 yrs

Displacement

Ductility

I (1d = 2) 0.0394 26 Years 86%

II (1d = 6) 0.0288 35 Years 76%

III (1d = 8) 0.0231 44 Years 68%

Absolute

Acceleration

I (AAbs = 0.1g) 0.0349 29 Years 82%

II (AAbs = 0.2g) 0.0104 96 Years 41%

III (AAbs = 0.25g) 0.0048 208 Years 21%

Normalized EH

Dissipated

I (EH = 5) 0.0171 58Years 57%

II (EH = 20) 0.0012 833 Years 6%

III (EH = 30) 0.0003 3330 Years 1.5%

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Probabilistic Loss Hazard Analysis

11

( ) [ | ] | | |j

j k k

nls

L j DM j DS DSkDM

l P L l DM d P L l DS k& & & &2

"

$ %" ' " ' " 3( )*+

• Loss Hazard Curve:

! Component-wise Loss Hazard Curve:

! Total Loss Hazard Curve:

number of limit states for component j

1

# components

T j

j

L L=

= åMultilayer Monte

Carlo Simulation

For each damaged

component

For each damaged

component

For each EDP

For each eqke

Number of eqkes

in one year

N n"

Seismic hazard

Marginal PDFs and

correlation coefficients

of EDPs given IM Fragility curves

for all damage

states of each

failure

mechanism

Probability

distribution of

repair cost

IM im" EDP 4" DM k"j jL l"

For each eqke

For all eqkes in

one year and all

damaged

components

#

1

components

T j

j

L L"

" *Y

N

Poisson model NATAF joint

PDF model

of EDPs

Conditional

probability

of collapse

given IM

Collapse?

Ensemble of FE seismic response simulations

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Parametric PBEE Analysis

• Varying Parameter: Yield Strength Fy

Demand Hazard for

EDP = !d

Demand Hazard for

EDP = AAbs.

Demand Hazard for

EDP = EH

Cost

Hazard

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Parametric PBEE Analysis

• Varying Parameter: Initial Stiffness k0

Demand Hazard for

EDP = !d

Demand Hazard for

EDP = AAbs.

Demand Hazard for

EDP = EH

Cost

Hazard

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Optimization Formulation of PBEE

5 6, -

, -, -

0

0, , ,

0

0

, , ,

:

, , , 0

, , , 0

y

yk F b

y

y

Minimize f k F b

subject to

k F b

k F bg

"

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!

!

!

!

! !

h

• Optimization Problem Formulation:

! Objective (Target/Desired) Loss Hazard Curve: ( )T

ObjL l&

! Objective function:

! Optimization Problem:

• Optimization Performed using OpenSees-SNOPT extended

Framework

Gu, Q., Barbato, M., Conte, J. P., Gill, P. E., and McKenna, F., !OpenSees-SNOPT Framework for Finite Element-Based

Optimization of Structural and Geotechnical Systems," Journal of Structural Engineering, ASCE, in press, 2011.

, - 20 0, , , | ( , , , , ) ( ) |

T T

Objy L i y iL

i

f k F b l k F b l& &" 3*! !

( )T

ObjL l&

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(II) Optimization Tool of

OpenSees-SNOPT

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OpenSees and SNOPT

• OpenSees:

Open source objected-oriented FE analysis software framework, used to

model structural, geotechnical and SSI systems to simulate their responses to

static and dynamic loads, especially for earthquakes (PEER)

! Supports distributed computing capabilities & cloud computing

! Includes modules for FE response sensitivity and reliability analysis

! Flexibility to incorporate with other software packages

• SNOPT:

General purpose nonlinear optimization code using Sequential Quadratic

Programming (SQP) algorithm (Professor Phillip Gill @UCSD)

! Advantages of SNOPT as an optimization toolbox for solving

structural/geotechnical problems requiring optimization (e.g., structural

optimization, FE model updating, design point search problems):

# Applies to large scale problems

# Tolerates gradient discontinuities

# Requires relatively few evaluations of the limit-state (objective) function

# Offers a number of options to increase performance for special problems

! Open source in Fortran language available for academic use

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OpenSees-SNOPT Extended Framework

Optimization Framework in OpenSees

Flowchart for OpenSees-SNOPT

FE-based Optimization

• OpenSees & SNOPT Link:

! After abstract class of Optimization, subclass

SNOPTClass is used to encapsulate the SNOPT

software package as an interface

! Subclass SNOPTReliability for the design point

search problem in reliability analysis; while the

SNOPTOptimization structural optimization &

model updating

ObjectiveFunction

ConstraintFunction

DesignVariable

DesignVariablePositioner

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(III) Illustration of Performance-

Based Optimum Seismic Design

(Inverse PBEE Analysis)

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Illustrative Example

6,150 tons

0.10

0.02

m

b

!

"

"

"

SDOF Model

(Menegotto-Pinto)

0k

F

U

yF

0.075 myU "

1

1 0b k#

*

0

*

137,200 kN/m

10,290 kNy

k

F

"

"

0 ? /

? y

k kN m

F kN

"

"

PBEE

Analysis

A priori selected optimum

design parameters

Starting Point

0 100,000 kN/m

14,000 kNy

k

F

"

"

, -0, TL yk F&

T

Obj

L&

, -

* T T

0 y

Obj 2L 0 y L

i

Objective function : f k , F

= | ! "# $%& ' ( ! )

Expected Optimizer OpenSees-

SNOPT

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Objective Function

5 6, -

0

0,

0

,

:

80,000 187,200 (kN/m)

6,290 15,290 ( )

y

yk F

y

Minimize f k F

subject to

k

F kN

7 7

7 7

!

(0) (0)

0 100,000 kN/m, 14,000kN yk F" "

Obje

ctiv

e F

un

ctio

n

Zoom

• Optimization Problem:

• Starting Point:

• Objective Function Plot:

Obje

ctiv

e F

unct

ion

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Optimization Results

Loss Hazard Curve

Demand Hazard Curve

• Optimization Route:

Objective Function

, -D

* *

EDP=! " #v k ,F

X* =

[137,200kN/m,

10,290 kN]

Xstart =

[100,000kN/m,

14,000 kN]

Xend =

[135,774kN/m,

10,038 kN]

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Current and Future Work • Application to base isolated

bridge structure:

! Using nonlinear MDOF models of

increasing complexity.

! With increasing number of

optimization parameters, e.g., base

isolation parameters.

! Considering multiple objectives and

practical constraints, such as

constraints on demand hazard.

Marga-Marga Bridge, Vina del Mar in Chile

Isolator

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!"#$%&'()&*&

+),-./($-0&