probabilistic damage control approach...

Probabilistic Damage Control Approach (PDCA)

Application to Caltrans Bridge Design

2018 PEER ANNUAL MEETING - BERKELEY, CALIFORNIA 4

California Dep. of TransportationStructure Policy & InnovationOffice of Earthquake Engineering Mark MahanYeo (Tony) YoonSam AtayaAmir Malek

Background

Example

UNR Study

SP&I Study

Future Study


PDCA Application

PDCA Application

Within the context of Performance Based Earthquake Engineering

(PBEE), Probabilistic Damage Control Approach (PDCA) is a

process to quantify the distribution of modern bridge response when

subjected to various seismic events.

PDCA initiated around 2005 at Caltrans (Amir Malek, Mark Mahan,

Abbas Tourzani and Sam Ataya). The research continued at UNR

by Dr. Saiidi. After the PDCA research by UNR, Tony Yoon joined

the PDCA team to work on the application of PDCA.

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PDCA versus Current (Deterministic) Seismic Design

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Current Seismic

Design

PDCA

Target Performance

for 1000-year event

Collapse Prevention

only

Owner’s Choice of

various Damage States

(DS)

Bridge Response

Beyond 1000-year

event?

Not specifically

quantified

Risk1) is calculated for

events ranging from 225

to 1000, to 2500 years.

Design Optimization? No – Yes – Inherent to choice

of DS

Factor of Safety Built on the capacity

side DC

Project specific

involving both demand

and capacity

1) Defined as a probability of exceeding a certain damage condition

Damage States (DSi)?

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Caltrans accepted 6 levels of progressive damage (DS1 thru DS6) that

can be visually observed within the plastic region of columns

Damage Index - Initial Concept of PDCA

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DI = 0, Elastic Limit (No damage) DI = 1, End of Plastic State (DS6 Near Collapse)

What is Damage Index (DI)?

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Definitions:DY = Yield Displacement of the column

DD = Displacement Demand on the column due to various earthquakes

DC = Theoretical Displacement Capacity of the column

DUC = Ultimate Displacement Capacity of the column, Calibrated to test results

DI = 0, Elastic Limit (No damage) DI = 1, End of Plastic State (Extensive Damage, Near Collapse)

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Probability of exceeding a certain Damage State (DSi) is estimated using

the reliability index, bi.

Given the two Damage Indices on the capacity and demand side, (DIR i)

and (DIL), as random variables, respectively, bi is computed:

Determine mean m and coefficient of variance d of DIR i and DIL ?

m and d of DIR i are established by Vosooghi and Saiidi, 2012.

m and d of DIL are from Non-Linear Time History Analyses (NLTHA)

using site specific information

Reliability index, bi. (LRFD formulation, Nowak and Collins 2000).

Correlation between DI and DSi

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Vosooghi and Saiidi, 2012 investigated the correlation between DI and

DSi by constructing the fragility curves of the DSs

The mean of DI within the target DSi range is called “Target DI”. For

example, the mean (50%) DI within the DS3 range is approximately

0.35 and is said to be the “Target DI” for DS3.

PDCA Application

Procedure

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PDCA Basic Procedure (outlined by Saiidi 2014)

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Step 1: Given the 975 yr ARS for specific bridge site (design earthquake),

conduct pushover design of columns. Determine DY , DD , DUC

Step 2: Is Damage Index (DI) 0.35? Calibrated for Target DI, 0.35.

Step 3: Run 51 NLTHA to establish the displacement demand DD set, the

mean m and the coefficient of variation d for the set. These

demand side values are constant for all damage states.

Step 4: Determine the capacity (Resistance) side of DI for Damage State

3 (DS3). Vosooghi and Saiidi 2012 established it (mean m =

0.375, coefficient of variation d = 0.27).

Step 5: Calculate the Reliability Index b3 for DS3.

PDCA Application - Procedure

Reliability Index (b)

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Saini and Saiidi, 2014 proposed an approach to compute the reliability

index (bi) for each DSi using the related variables, DI on the capacity side

(DIR) and DI on the demand side (DIL).

Capacity side mRi and dRi: from fragility curves (Vosooghi and Saiidi, 2012)

Demand Side mL and dL: from NLTHA using site specific information.

PDCA Application - Procedure

Reliability Index (b)

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DS3 DS4 DS5 DS6

mRi 0.375 0.6 0.822 1.0

dRi 0.27 0.2 0.14 0.00

mL and dL: mean and coefficient of variance of DIL This is from NLTHA

using site specific information

from fragility

curves (Vosooghi

and Saiidi, 2012)

PDCA Basic Procedure: Continued

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The basic procedure resulted in the reliability index b3 for exceeding

Damage State 3 (DS3).

Step 6: Repeat Step 4 for b4 , b5 , b6 for DS4, DS5, and DS6 by using

the UNR fragility values and the NLTHA demand values for 975-year

earthquake.

Step 7: The above process can be repeated for the 225- & 2475-year.

Step 8: Using each b3, what is the probability of exceeding DS3 given

any earthquake?

PDCA Application

Example

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PDCA Application - Example

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Design Scenario:

An Ordinary Standard Bridge assumed to be located in downtown LA. The

bridge is a CIP/PT box girder bridge with two spans of 150 feet each. The bent

consists of a single 5’-6” diameter 30 foot tall reinforced concrete column. The

footing is founded on competent rock. The column has a total 18 No. 9

longitudinal reinforcement bars with No. 8 hoop @ 6.5 in. A target damage index

(DI) for this bridge is 0.35 to have DS3. The natural period of Bent 2 is 2 sec.

This

PDCA Application - Example

Step 1: Acceleration Record Generation - Determine Design ARS

from ARS Online or USGS and Obtain Parameters for Ground Motion

Generation.

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Fault Name Elysian Park (Upper) Elysian Park (Lower CFM) Puente Hills (LA)

Fault Type Non-Strike Non-Strike Non-Strike

R (km) 3.7 5.9 4.7

Mw 6.6 6.7 6.9

VS30 (m/sec) 270 270 270

S (km) 0 5.9 4.7

Dir Angle, q (deg) 4 45 45

Z (km) 3 10 2.1

Step 1: Acceleration Record Generation - Generate 51 ground motions

(3 x 17 motions from each fault) and Scale the motions to Design ARS

linearly

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Before Linear Scaling After Linear Scaling

Step 2: Designer conducts pushover analysis for DY and DUC

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fye: 68 ksi

f’ce: 5 ksi

eu for Hoops: 0.18

Lp: 40.3 inches

Yield Displacement, DY :5.4 in

Ultimate Displacement Capacity,

DUC : 35 in

In addition, a displacement demand, DD_ESA is estimated from ESA

against the design ARS. DD_ESA = 15.7 in

Step 2a: Compute DD_ESA using ARS for 975 yr EQ

Step 3: Check Target DI

(DD_ESA - DY)/(DUC - DY) = (15.7-5.4)/(35-5.4) = 0.35 OK

Reinforcements were properly detailed for Target DI.

Step 4: Perform Non-Linear Time History Analyses (NLTHA) with

975 yr motions for mL and dL of DIL

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Elysian Park (Upper) Elysian Park (Lower CFM) Puente Hills (LA)

DD_NLTHA (in) DIL DD_NLTHA (in) DIL DD_NLTHA (in) DIL

Sim 1 12.84 0.25 12.21 0.23 17.94 0.42

Sim 2 14.24 0.30 19.23 0.47 16.50 0.38

Sim 3 12.95 0.26 21.64 0.55 15.76 0.35

Sim 4 14.04 0.29 11.72 0.21 43.24 1.28

Sim 5 24.87 0.66 17.38 0.40 20.58 0.51

Sim 6 14.24 0.30 16.82 0.39 11.71 0.21

Sim 7 12.28 0.23 11.74 0.21 12.63 0.24

Sim 8 13.98 0.29 20.08 0.50 11.44 0.20

Sim 9 15.90 0.35 46.08 1.37 8.31 0.10

Sim 10 15.91 0.36 15.44 0.34 22.33 0.57

Sim 11 12.83 0.25 19.32 0.47 14.48 0.31

Sim 12 18.11 0.43 12.90 0.25 13.81 0.28

Sim 13 10.49 0.17 17.88 0.42 20.86 0.52

Sim 14 20.82 0.52 14.82 0.32 9.27 0.13

Sim 15 22.68 0.58 28.92 0.79 30.48 0.85

Sim 16 12.02 0.22 11.90 0.22 18.95 0.46

Sim 17 16.64 0.38 19.00 0.46 18.78 0.45

mL of DIL: 0.39

sL of DIL: 0.20

dL of DIL: 0.20/0.39

=0.51

Step 5: Calculate bi, given

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mL of DIL: 0.39 & dL of DIL: 0.51 from 51 NLTHA

DS3 DS4 DS5 DS6

mRi 0.375 0.6 0.822 1.0

dRi 0.27 0.2 0.14 0.00

mRi and dRi of DIR from the fragility curve (Vosooghi and Saiidi, 2012)

DS3 DS4 DS5 DS6

bi 0.08 1.01 1.70 2.19

Step 6: Compute probabilities of exceeding each DSi with bi

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DS3 DS4 DS5 DS6

bi 0.08 1.01 1.70 2.19

46.9% 15.6% 4.5% 1.4%

DS3 DS4 DS5 DS6

46.9% 15.6% 4.5% 1.4%

5% in 50 yrs

2.3% 0.78% 0.22% 0.07%

P(EQ)

DS3 DS4 DS5 DS6

bi 1.08 1.84 2.37 2.75

14% 3.3% 0.9% 0.3%

DS3 DS4 DS5 DS6

14% 3.3% 0.9% 0.3%

20% in 50 yrs

2.8% 0.66% 0.18% 0.06%

P(EQ)

resulted from b

Step 7: Repeat for 225 yr and 2475 yr return periods

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DS3 DS4 DS5 DS6

bi -1.72 -0.53 0.43 1.18

96% 70% 33% 12%

DS3 DS4 DS5 DS6

96% 70% 33% 12%

2% in 50 yrs

1.91% 1.41% 0.67% 0.24%

P(EQ)

resulted from b

Step 7: Repeat for 225 yr and 2475 yr return periods

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DS3 DS4 DS5 DS6

14% 3.3% 0.9% 0.3%

46.9% 15.6% 4.5% 1.4%

96% 70% 33% 12%

P(EQ)

Step 8: Probability of exceeding DS3 given any earthquake (Total Probability)

DS3 DS4 DS5 DS6

2.8% 0.7% 0.2% 0.1%

2.3% 0.8% 0.2% 0.1%

1.9% 1.4% 0.7% 0.2%

P(EQ225) 20%

P(EQ975) 5%

P(EQ2475) 2%

P(DSi) = 25.9% 10.7% 4.1% 1.5%

Sum 7.0% 2.9% 1.1% 0.4%

X

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27% 27% 27% 27%

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Needs

1. Displacement Demand calculations require numerous NLTHA. Can

the mean and covariance be established for zones in CA?

2. Most appropriate scaling method for the ground motions?

3. Mean and covariance of various generations of columns (vintage)? UNR has been working on particular column vintages.

4. Total lifecycle cost analysis: Initial construction cost plus post EQ repair cost is determined for each target damage state. Loss of use cost is difficult to quantify but most likely very large.


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