probabilistic performance-based optimum seismic design of

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PPEEEERR

UUCCSSDD

Probabilistic Performance-Based Optimum Seismic Design of

(Bridge) Structures

PI: Joel. P. ConteGraduate Student: Yong Li

Sponsored by the Pacific Earthquake Engineering Research Center

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(I) PEER Performance Based Earthquake Engineering (PBEE) Methodology

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Demand Hazard AnalysisProbabilistic Seismic Damage Hazard AnalysisProbabilistic Seismic Loss Hazard Analysis

(II) Parametric PBEE Analysis of SDOF SystemYield strength Fy as varying parameterHardening ratio b as varying parameterBoth mass m and initial stiffness k0 as varying parameters (with fixed k0/m)Mass m as varying parameterInitial stiffness k0 as varying parameter

(III) Optimization Formulation for Probabilistic Performance Based Optimum Seismic Design

Outline

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(I) PEER PBEE Methodology(Forward PBEE Analysis)

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SDOF Bridge Model• 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 previously developed in OpenSees.

1

0

1.33 s6,150 tons137,200 kN/m10,290 kN

0.100.02

y

TmkF

bξ

====

==

SDOF Model (Menegotto-Pinto)SDOF Model (Menegotto-Pinto)

0k

F

U

yF

0.075 myU =

1

1 0b k⋅

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Site Description and Prob. Seismic Hazard Analysis• Site Location: Oakland (37.803N, 122.287W)

Uniform Hazard Spectra are obtained for 30 hazard levels from the USGS 2008 Interactive Deaggregation software/website (beta version)

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

• Uniform Hazard Spectra and M-R Deaggregation:

T1=1.33 sec

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

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Earthquake Record Selection

Records were selected from NGA database based on geological and seismological conditions (e.g., fault mechanism), M-R deaggregation of probabilistic seismic hazard results, and local site condition (e.g., Vs30)

NGA Excel Flatfile (3551 records) Mechanism: Strike-slip ( 1004 records)Magnitudes: 5.9 to 7.3 ( 652 records)Distance: 0 – 40 km ( 193 records) Vs30: C-D range ( 146 records)

• Earthquake Record Selection:

146 horizontal ground motion components were selected from NGA database

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

( ), ( , 50 ), 1,2, ,30i i i ix y IM PE i= =Data Points:

Fitting Function (lognormal CCDF):

flt

i i

i i

N

IM i M Ri 1 R M

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

⎡ ⎤ν = ν ⋅ > ⋅ ⋅ ⋅ ⋅⎣ ⎦∑ ∫ ∫

• Least Square Fitting of 30 Data Points from USGS:

[ ]1 2

21 2, 1

min ln ( , , ) lnnp

i ii

g x yθ θ

θ θ=

−∑

11 2

2

ln( , , ) 1 xg x θθ θθ

⎛ ⎞−= −Φ⎜ ⎟

⎝ ⎠

( , )IM MAREi i( )1, 5%a TS ξ =

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Probabilistic Seismic Demand Hazard Analysis

• Probabilistic Seismic Demand Analysis conditional on IM:“Cloud Method” based on following assumptions:

lnˆˆ ˆE[ln | ] ln EDP IM EDP IM im a b IMμ = = = +

( )( )2

2 1

ˆˆ ln ˆ

2

n

i ii

ln edp a b imS

n=

− +=

−

∑

1. Linear regression after ln transformation of predictor and response variables:

2. Variance of EDP independent of IM:

3. Probability distribution of EDP|IM assumed to be Lognormal:| ( , )EDP IM LN λ ς∼

• Demand Hazard Curve:

( ) ( )EDP IMIM

edp P EDP edp IM d imν ν= >⎡ ⎤⎣ ⎦∫Complementary CDF of EDP given IM Seismic hazard curve

ln2

ln

ˆˆ ˆwhere ln and ˆˆ

EDP IM

EDP IM

a b IMS

λ μς σ

= = += =

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Regression analysis results and PDF/CCDF for EDP|IM shown at three hazard levels commonly used in Earthquake Engineering:

PDF (Lognormal) CCDF (Lognormal)


• EDP: Displacement Ductility• EDP: Displacement Ductility

( )0 d

d t ty

u tMax

Uμ

< <

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠Where the displacement ductility is defined as:

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PDF (Lognormal)CCDF (Lognormal)



• EDP: Peak Absolute Acceleration• EDP: Peak Absolute Acceleration

( ). 0 d

Abs t t

u tA Max

g< <

⎛ ⎞= ⎜ ⎟

⎝ ⎠Where the peak absolute acceleration is defined as:

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PDF (Lognormal) CCDF (Lognormal)


• EDP: Normalized Hysteretic Energy Dissipated• EDP: Normalized Hysteretic Energy Dissipated


( ) ( )0

dt

EH

y y

R t du t EE

F U

−= ∫Where the Normalized hysteretic energy is defined as:

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• Demand Hazard Curve:

( ) ( )EDP IMIM

edp P EDP edp IM im d imν ν= > =⎡ ⎤⎣ ⎦∫

( ) ( )

( )( ) ( )

IMEDP

IM

IM ii i

i i

d imedp P EDP edp IM dim

dim

d imP EDP edp IM im im

im

νν

ν

⎡ ⎤= >⎣ ⎦

⎡ ⎤= > = ⋅Δ⎣ ⎦ Δ

∫

∑

• Deaggregation of with respect to IM:( )EDP edpν

Contribution of bin IM = imi to ( )EDP edpν


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MARE

PE50

• Demand Hazard Curve for EDP = Displacement Ductility μd:APE


(RP = 55 years)

Deaggregationwrt IM

(RP = 2500 years)

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• Demand Hazard Curve for EDP = Peak Absolute Acceleration:

MARE

PE50

APE


(RP = 30 years)

(RP = 625 years)

Deaggregationwrt IM

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PE50

APEMARE

• Demand Hazard Curve for EDP = Normalized EH Dissipated:


(RP = 32 years)

(RP = 500 years)

Deaggregationwrt IM

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Probabilistic Capacity (Fragility) Analysis• Fragility Curves (postulated & parameterized):

[ ]|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 Capacity (Fragility) Analysis• Fragility Curve Parameters:

Associated EDP Limit-states

Predicted Capacity

Measured-to-predicted capacity ratio (Normal

distributed) Mean c.o.v

Displacement Ductility

I μ = 2 1.095 0.201II μ = 6 1.124 0.208III μ = 8 1.254 0.200

Peak Absolute Acceleration

[g]

I AAbs.= 0.10 0.934 0.128II AAbs.= 0.20 0.952 0.246III AAbs.= 0.25 0.973 0.265

Normalized Hysteretic

Energy Dissipated

I EH = 5 0.934 0.133II EH = 20 0.965 0.140

III EH = 30 0.983 0.146

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

| ( )k

k EDPDSEDP

P DM ls EDP edp d edpν ν⎡ ⎤= > =⎣ ⎦∫Fragility Analysis

• Deaggregation of Damage Hazard:

with respect to EDP:

with respect to IM:

( ) ( )( )|

k

EDP ik i iDS

i i

d edpP DM ls EDP edp edp

edpν

ν ⎡ ⎤= > = Δ⎣ ⎦ Δ∑

[ ]( )

( )| ( ) || |k

IM ik iDS

i IM i

d im= P DM ls EDP dP EDP edp IM imim

νν ⎡ ⎤> > Δ⎣ ⎦ Δ∑ ∫

Contribution of bin EDP = edpi to

kDSν

Contribution of bin IM = imi to

kDSν

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• Associated EDP = Displacement Ductility:

Probabilistic Damage Hazard Analysis

Associated EDP Limit States MARE RP PE50


I (μd = 2) 0.0394 26 Years 86%II (μd = 6) 0.0288 35 Years 76%III (μd = 8) 0.0231 44 Years 68%

Deaggregationwrt IM

• Deaggregation Results:

Deaggregationwrt EDP

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• Associated EDP = Peak Absolute Acceleration:



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%


Deaggregationwrt IM


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• Associated EDP = Normalized Hysteretic Energy Dissipated:




Energy Dissipated

I (EH = 5) 0.0171 58 Years 57%II (EH = 20) 0.0012 833 Years 6%

III (EH = 30) 0.0003 3330 Years 1.5%



Deaggregationwrt IM

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Probabilistic Loss Hazard Analysis

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( ) [ | ] | | |j

j k k

nls

L j DM j DS DSkDM

l P L l DM d P L l DS kν ν ν ν+

=

⎡ ⎤= > = > = −⎣ ⎦∑∫

• Loss Hazard Curve:

Component-wise Loss Hazard Curve:

Only discrete damage states are used in practice.

= MAR of exceeding the k-th limit-state and

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nlsDSν+=

kDSν

nlsj = number of limit states for component j

P[DS = 0 | EDP = edp]




Limit-State I

Limit-State II

Limit-State III

edp

EDP

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For each damaged component

For each damaged componentFor each EDPFor each eqke

Number of eqkesin 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 δ= DM k= j jL l=

For each eqkeFor all eqkes in one year and all damaged components

#

1

components

T jj

L L=

= ∑Y

N

Poisson model NATAF joint PDF modelof EDPs

Conditional probability of collapse given IM

Collapse?

Ensemble of FE seismic response simulations of bridge-foundation-ground system

Probabilistic Loss Hazard Analysis• Multilayer Monte Carlo Simulation:

Total Loss (Repair/Replacement Cost):

Computation of requires n-fold integration of the joint PDF of the component losses, which is very difficult to obtain.

is estimated using multilayer MCS

costIμ IIμ IIIμ IVμ Vμ

III

IIIIV V

jP L DS⎡ ⎤⎣ ⎦

Probability Distributions of Repair Costs

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# components

T jj

L L=

= ∑( )

TL lν

( )TL lν

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Probabilistic Loss Hazard Analysis• Repair/Replacement Cost Parameters:

Failure Associated

EDPLimit-states

Repair/Replacement Cost (Normal distributed)

Mean ($) c.o.v.


I 146,500 0.12II 246,400 0.25III 350,000 0.32

Peak Absolute Acceleration

[g]

I 55,000 0.11II 100,000 0.20III 500,000 0.28


Energy Dissipated

I 55,650 0.13II 110,000 0.22

III 520,000 0.28

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(II) Parametric PBEE Analysis ofSDOF System

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Parametric PBEE Analysis• Varying Parameter: Yield Strength Fy• Varying Parameter: Yield Strength Fy

Demand Hazard forEDP = μd

Demand Hazard forEDP = AAbs.

Demand Hazard forEDP = EH

Cost Hazard

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Parametric PBEE Analysis• Varying Parameter: Hardening Ratio b• Varying Parameter: Hardening Ratio b




Cost Hazard

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Parametric PBEE Analysis• Varying Parameters: Both Mass m and Initial Stiffness k0 (period fixed):• Varying Parameters: Both Mass m and Initial Stiffness k0 (period fixed):




Cost Hazard

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Parametric PBEE Analysis• Varying Parameter: Mass m• Varying Parameter: Mass m



Cost Hazard


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Parametric PBEE Analysis• Varying Parameter: Initial Stiffness k0• Varying Parameter: Initial Stiffness k0




Cost Hazard

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(III) Optimization Formulation for Probabilistic Performance Based

Optimum Seismic Design(Inverse PBEE Analysis)

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Optimization Formulation of PBEE

{ }( )

( )( )

00

, , ,

0

0

, , ,

:

, , , 0

, , , 0

yy

k F b

y

y

Minimize f k F b

subject to

k F b

k F bg

=

≤

h

• Optimization Problem Formulation:

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

ObjL lν

Objective function:

Optimization Problem:

• Optimization Performed using extended OpenSees-SNOPTFramework (ongoing work).

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, under review, 2011.

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

T T

Objy L i y iL

i

f k F b l k F b lν ν= −∑

( )T

ObjL lν

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Thank you !

Any Questions?

probabilistic performance-based optimum seismic design of

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