Probabilistic Damage Control Approach (PDCA)

Probabilistic Damage Control Approach (PDCA) Application to Caltrans Bridge Design California Dep. of Transportation Structure Policy & Innovation Office of Earthquake Engineering Mark Mahan Yeo (Tony) Yoon Sam Ataya Amir Malek 2018 PEER ANNUAL MEETING - BERKELEY, CALIFORNIA 4

PDCA Application Background Example UNR Study SP&I Study Future Study 2018 PEER ANNUAL MEETING - BERKELEY, CALIFORNIA 5

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. 6

PDCA versus Current (Deterministic) Seismic Design Target Performance for 1000-year event Bridge Response Beyond 1000-year event? Current Seismic Design Collapse Prevention only Not specifically quantified PDCA Owner s Choice of various Damage States (DS) Risk 1) 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 D C Project specific involving both demand and capacity 1) Defined as a probability of exceeding a certain damage condition 7

Damage States (DSi)? Caltrans accepted 6 levels of progressive damage (DS1 thru DS6) that can be visually observed within the plastic region of columns 8

Damage Index - Initial Concept of PDCA DI = 0, Elastic Limit (No damage) DI = 1, End of Plastic State (DS6 Near Collapse) 9

What is Damage Index (DI)? Definitions: D Y = Yield Displacement of the column D D = Displacement Demand on the column due to various earthquakes D C = Theoretical Displacement Capacity of the column D UC = 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) 10

Reliability index, b i. (LRFD formulation, Nowak and Collins 2000). Probability of exceeding a certain Damage State (DS i ) is estimated using the reliability index, b i. Given the two Damage Indices on the capacity and demand side, (DI R i ) and (DI L ), as random variables, respectively, b i is computed: Determine mean m and coefficient of variance d of DI R i and DI L? m and d of DI R i are established by Vosooghi and Saiidi, 2012. m and d of DI L are from Non-Linear Time History Analyses (NLTHA) using site specific information 11

Correlation between DI and DSi 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. 12

PDCA Application Procedure 13

PDCA Basic Procedure (outlined by Saiidi 2014) Step 1: Given the 975 yr ARS for specific bridge site (design earthquake), conduct pushover design of columns. Determine D Y, D D, D UC Step 2: Is Damage Index (DI) 0.35? Calibrated for Target DI, 0.35. Step 3: Run 51 NLTHA to establish the displacement demand D D 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 b 3 for DS3. 14

PDCA Application - Procedure Reliability Index (b) Saini and Saiidi, 2014 proposed an approach to compute the reliability index (b i ) for each DSi using the related variables, DI on the capacity side (DI R ) and DI on the demand side (DI L ). Capacity side m Ri and d Ri : from fragility curves (Vosooghi and Saiidi, 2012) Demand Side m L and d L : from NLTHA using site specific information. 15

PDCA Application - Procedure Reliability Index (b) DS3 DS4 DS5 DS6 m Ri 0.375 0.6 0.822 1.0 d Ri 0.27 0.2 0.14 0.00 from fragility curves (Vosooghi and Saiidi, 2012) m L and d L : mean and coefficient of variance of DI L using site specific information This is from NLTHA 16

PDCA Basic Procedure: Continued The basic procedure resulted in the reliability index b 3 for exceeding Damage State 3 (DS3). Step 6: Repeat Step 4 for b 4, b 5, b 6 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 b 3, what is the probability of exceeding DS3 given any earthquake? 17

PDCA Application Example 18

PDCA Application - Example 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 19

PDCA Application - Example Step 1: Acceleration Record Generation - Determine Design ARS from ARS Online or USGS and Obtain Parameters for Ground Motion Generation. 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 V S30 (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 20

Step 1: Acceleration Record Generation - Generate 51 ground motions (3 x 17 motions from each fault) and Scale the motions to Design ARS linearly Before Linear Scaling After Linear Scaling 21

Step 2: Designer conducts pushover analysis for D Y and D UC f ye : 68 ksi f ce : 5 ksi e u for Hoops: 0.18 L p : 40.3 inches In addition, a displacement demand, D D_ESA against the design ARS. D D_ESA = 15.7 in Yield Displacement, D Y :5.4 in Ultimate Displacement Capacity, D UC : 35 in Step 2a: Compute D D_ESA using ARS for 975 yr EQ Step 3: Check Target DI (D D_ESA - D Y )/(D UC - D Y ) = (15.7-5.4)/(35-5.4) = 0.35 OK is estimated from ESA Reinforcements were properly detailed for Target DI. 22

Step 4: Perform Non-Linear Time History Analyses (NLTHA) with 975 yr motions for m L and d L of DI L Elysian Park (Upper) Elysian Park (Lower CFM) Puente Hills (LA) m L of DI L : 0.39 s L of DI L : 0.20 d L of DI L : 0.20/0.39 =0.51 D D_NLTHA (in) DI L D D_NLTHA (in) DI L D D_NLTHA (in) DI L 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 23

Step 5: Calculate b i, given m Ri and d Ri of DI R from the fragility curve (Vosooghi and Saiidi, 2012) DS3 DS4 DS5 DS6 m Ri 0.375 0.6 0.822 1.0 d Ri 0.27 0.2 0.14 0.00 m L of DI L : 0.39 & d L of DI L : 0.51 from 51 NLTHA DS3 DS4 DS5 DS6 b i 0.08 1.01 1.70 2.19 24

Step 6: Compute probabilities of exceeding each DSi with b i DS3 DS4 DS5 DS6 b i 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 P(EQ) 2.3% 0.78% 0.22% 0.07% 25

Step 7: Repeat for 225 yr and 2475 yr return periods DS3 DS4 DS5 DS6 b i 1.08 1.84 2.37 2.75 14% 3.3% 0.9% 0.3% resulted from b DS3 DS4 DS5 DS6 P(EQ) 14% 3.3% 0.9% 0.3% 20% in 50 yrs 2.8% 0.66% 0.18% 0.06% 26

Step 7: Repeat for 225 yr and 2475 yr return periods DS3 DS4 DS5 DS6 b i -1.72-0.53 0.43 1.18 96% 70% 33% 12% resulted from b DS3 DS4 DS5 DS6 P(EQ) 96% 70% 33% 12% 2% in 50 yrs 1.91% 1.41% 0.67% 0.24% 27

Step 8: Probability of exceeding DS3 given any earthquake (Total Probability) 5 5 DS3 DS4 DS5 DS6 14% 3.3% 0.9% 0.3% 46.9% 15.6% 4.5% 1.4% 96% 70% 33% 12% X P(EQ 225 ) 20% 5 P(EQ 975 ) 5% P(EQ 2475 ) 2% DS3 DS4 DS5 DS6 P(EQ) 2.8% 0.7% 0.2% 0.1% 2.3% 0.8% 0.2% 0.1% 1.9% 1.4% 0.7% 0.2% Sum 7.0% 2.9% 1.1% 0.4% 27% 27% 27% 27% P(DS i ) = 25.9% 10.7% 4.1% 1.5% 28

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. 2018 PEER ANNUAL MEETING - BERKELEY, CALIFORNIA 29