April 26, 2019 Performance-Based Geotechnical Seismic Design Steve - PowerPoint PPT Presentation

Performance-Based Design Uncertainty exists – can’t ignore it • Uncertainty in ground motions varies from location to location • Uncertainty in response varies from site to site • Uncertainty in damage varies from structure to structure • Uncertainty in loss varies with location (material costs, labor costs, …) and time (inflation, interest rates, tweets, …) Ignoring uncertainty, or assuming it is uniform, leads to: • Inaccurate performance predictions • Inconsistent levels of safety from one project to another 33

Performance-Based Design Uncertainty exists – can’t ignore it • Uncertainty in ground motions varies from location to location • Uncertainty in response varies from site to site • Uncertainty in damage varies from structure to structure • Uncertainty in loss varies with location (material costs, labor costs, …) and time (inflation, interest rates, tweets, …) Ignoring uncertainty, or assuming it is uniform, leads to: • Inaccurate performance predictions • Inconsistent levels of safety from one project to another • Inefficient use of resources for seismic retrofit/design 34

Discrete Hazard Level Approach Divide IM s, EDP s, DM s, and DV s into finite number of ranges Consider all combinations Account for conditional probabilities IM 1 IM 2 IM 3 IM 4 IM 5

Discrete Hazard Level Approach Divide IM s, EDP s, DM s, and DV s into finite number of ranges Consider all combinations Account for conditional probabilities EDP 1 IM 1 EDP 2 IM 2 EDP 3 IM 3 EDP 4 IM 4 EDP 5 IM 5

Discrete Hazard Level Approach Divide IM s, EDP s, DM s, and DV s into finite number of ranges Consider all combinations Account for conditional probabilities EDP 1 IM 1 EDP 2 DM 1 IM 2 EDP 3 DM 2 IM 3 EDP 4 DM 3 IM 4 EDP 5 DM 4 IM 5 DM 5

Discrete Hazard Level Approach Divide IM s, EDP s, DM s, and DV s into finite number of ranges Consider all combinations Account for conditional probabilities EDP 1 IM 1 EDP 2 DV 1 DM 1 IM 2 EDP 3 DV 2 DM 2 IM 3 EDP 4 DV 3 DM 3 IM 4 EDP 5 DV 4 DM 4 IM 5 DV 5 DM 5

Discrete Hazard Level Approach Divide IM s, EDP s, DM s, and DV s into finite number of ranges Consider all combinations Account for conditional probabilities EDP 1 IM 1 EDP 2 DV 1 DM 1 IM 2 EDP 3 DV 2 DM 2 IM 3 EDP 4 DV 3 DM 3 IM 4 EDP 5 DV 4 DM 4 IM 5 DV 5 DM 5

Discrete Hazard Level Approach Divide IM s, EDP s, DM s, and DV s into finite number of ranges Consider all combinations Account for conditional probabilities EDP 1 IM 1 EDP 2 DV 1 DM 1 IM 2 EDP 3 DV 2 DM 2 IM 3 EDP 4 DV 3 DM 3 IM 4 EDP 5 DV 4 DM 4 IM 5 DV 5 DM 5 DV 2-4-3-5 = DV 5 P[ IM 2 |eq] P[ EDP 4 | IM 2 ] P[ DM 3 | EDP 4 ] P[ DV 5 | DM 3 ] DV = S DV i-j-k-l Summing over all paths

Discrete Hazard Level Approach Divide IM s, EDP s, DM s, and DV s into finite number of ranges Consider all combinations Account for conditional probabilities EDP 1 IM 1 EDP 2 DV 1 DM 1 IM 2 EDP 3 DV 2 DM 2 IM 3 EDP 4 DV 3 DM 3 IM 4 EDP 5 DV 4 DM 4 IM 5 DV 5 DM 5 For this case, 5 x 5 x 5 x 5 = 625 paths With 100 values for each . . . 100 million paths

Integral Hazard Level Approach Covers entire range of hazard (ground motion) levels Accounts for uncertainty in parameters, relationships PEER framework     ( ) ( | ) ( | ) ( | ) ( ) DV DV DM DM EDP EDP IM IM G dG dG d

Integral Hazard Level Approach Covers entire range of hazard (ground motion) levels Accounts for uncertainty in parameters, relationships PEER framework     ( ) ( | ) ( | ) ( | ) ( ) DV DV DM DM EDP EDP IM IM G dG dG d Fragility curve – Fragility curve Fragility curve Seismic – crack width – interstory Loss curve repair cost given hazard –  Cost vs curve –  Sa crack width given drift given S a Cost interstory drift vs S a $2.7M $5.8M Damage Response Loss PSHA model model model

PEER Performance-Based Framework Covers entire range of hazard (ground motion) levels Accounts for uncertainty in parameters, relationships PEER framework     ( ) ( | ) ( | ) ( | ) ( ) DV DV DM DM EDP EDP IM IM G dG dG d Alternative design Fragility curve – Fragility curve Fragility curve Seismic – crack width – interstory Risk curve repair cost given hazard –  Cost vs curve –  Sa crack width given drift given S a Cost interstory drift vs S a $1.6M $3.9M Cost Damage Response PSHA model model model

PEER Performance-Based Framework Modular – response, damage, loss components log  EDP log  IM Settlement PGA IM EDP log  DM log  DV Floor slab Repair cracking cost DV DM

PEER Performance-Based Framework Modular – response, damage, loss components log  EDP log  IM Includes - all earthquake magnitudes Settlement PGA - all source-to-site distances - uncertainty in ground motion - uncertainty in response given ground motion IM EDP - uncertainty in damage given response - uncertainty in loss given damage log  DM log  DV Floor slab Repair All levels of shaking are cracking cost cosidered and accounted for, not just shaking at one return period. DV DM

PEER Performance-Based Framework Modular – response, damage, loss components log  EDP log  IM Includes - all earthquake magnitudes Settlement PGA - all source-to-site distances - uncertainty in ground motion - uncertainty in response given ground motion IM EDP - uncertainty in damage given response - uncertainty in loss given damage log  DM log  DV Floor slab Repair Response, damage, and loss cracking cost are all explicitly computed – with explicit consideration of uncertainty in each DV DM

Performance-Based Response Evaluation Closed-form solution  IM ( im ) Assume hazard curve is of power law form  IM ( im ) = k o ( im ) -k im and response is related to intensity as edp = a ( im ) b edp with lognormal conditional uncertainty (ln edp is normally distributed with standard deviation s ln edp|im ) im

Performance-Based Response Evaluation Closed-form solution Then median EDP hazard curve can be expressed in closed form as  k     1 / b   2 edp k         s 2 ( ) exp edp k   ln | edp im EDP o   2    2  a b    IM ( im )  EDP ( edp ) im edp edp im

Performance-Based Response Evaluation Closed-form solution Then median EDP hazard curve can be expressed in closed form as  k     1 / b   2 edp k         s 2 ( ) exp edp k   ln | edp im EDP o   2    2  a b   Based on median  EDP ( edp ) IM and EDP-IM relationship EDP “amplifier” based on uncertainty in EDP | IM relationship edp

Performance-Based Response Evaluation Closed-form solution Example: Slope displacement Combining, with different levels of response model uncertainty …

Performance-Based Response Evaluation Closed-form solution Example: Slope displacement Uncertainty in response prediction accounts for half

Performance-Based Loss Evaluation Closed-form solution Damage Response Loss Extending to DM and DV , with same assumptions, gives model model model  k / b   1 / d      1 /   f     2 1 1 dv k            2 2 2 2 2 2   ( ) exp  dv k d f f 0   DV     2 2 2 R D L   2 a c e b d f     Median relationships Uncertainty amplifier Median relationship (no uncertainty)  DV ( dv ) With response model uncertainty Response and damage model uncertainties Response, damage, and loss model uncertainties 1/T R dv Losses

Implementation of Performance-Based Design Characterization of loading Select IM s – important considerations include: Efficiency – how well does IM predict response? Permanent displacement of shallow slides Travasarou et al. (2003) Displacement (cm) Arias intensity (m/s) PGV 2 (cm/s) 2 PGA (g)

Implementation of Performance-Based Design Characterization of loading Select IM s Efficiency – how well does IM predict response? Sufficiency – how completely does IM predict response? Deviations from mean excess pore pressure ratio correlation to PGA Kramer and Mitchell (2003) r u under- predicted r u over- predicted Weak trend w/r/t R Systematic trend in pore pressure residuals w/r/t M w Magnitude scaling factor

Implementation of Performance-Based Design Characterization of loading Select IM s Efficiency – how well does IM predict response? Sufficiency – how completely does IM predict response? Predictability – how well can we predict IM ? Standard error, s ln IM Intensity Measure, IM Reference 0.53 – 0.55 PGA Campbell and Bozorgnia, 2008 0.53 – 0.56 PGV Campbell and Bozorgnia, 2008 0.59 – 0.61 S a (0.2 sec) Campbell and Bozorgnia, 2008 0.62 – 0.66 S a (1.0 sec) Campbell and Bozorgnia, 2008 1.0 – 1.3 Arias intensity, I a Travasarou et al. (2003) 0.40 – 0.44 CAV Campbell and Bozorgnia, 2010

Implementation of Performance-Based Design Characterization of loading Select IM s Efficiency – how well does IM predict response? Sufficiency – how completely does IM predict response? Predictability – how well can we predict IM ? Good predictability Poor predictability

Implementation of Performance-Based Design Characterization of loading Select IM s Efficiency – how well does IM predict response? Sufficiency – how completely does IM predict response? Predictability – how well can we predict IM ? Example:

Implementation of Performance-Based Design Characterization of loading Select IM s Efficiency – how well does IM predict response? Sufficiency – how completely does IM predict response? Predictability – how well can we predict IM ? Example: EDP Hazard Curves Typical predictability, typical efficiency Worse predictability, worse efficiency Better predictability, worse efficiency Worse predictability, better efficiency Better predictability, better efficiency

Implementation of Performance-Based Design Characterization of loading Select IM s Efficiency – how well does IM predict response? Sufficiency – how completely does IM predict response? Predictability – how well can we predict IM ? Example: 50-yr exceedance probabilities Predictability and efficiency both affect response for a given return period

Response-Level Implementation Performance characterized in terms of response variables Loss Inferred from Inferred from inferred computed Probabilistic response damage response model needed – must account for uncertainty in EDP | IM

Response-Level Implementation Performance characterized in terms of response variables Site response Rock Soil       hazard hazard ( ) [ | ] ( ) im P IM im im d im IM s S s r IM r S R curve curve 0  Integrating over im      s ( ) [ | ] ( ) im P AF im d im all rock motion IM s r IM r im S R 0 r levels Uncertainty in amplification behavior

Response-Level Implementation Performance characterized in terms of response variables Site response Analytical Empirical

Response-Level Implementation Performance characterized in terms of response variables Liquefaction (Kramer and Mayfield, 2007) FS L hazard curves T R = 50 yrs 6 m Distributions of M at all hazard levels considered T R = 5000 yrs

Response-Level Implementation Lateral Spreading – Franke and Kramer (2014) Reference soil profile Displacement hazard curves ( N 1 ) 60

Response-Level Implementation Performance characterized in terms of response variables Post-liquefaction settlement (Kramer and Huang, 2010) Hypothetical site in Seattle, Washington Settlement hazard curves Considering maximum volumetric strain Assuming soil is susceptible, liquefaction is triggered, and neglecting maximum volumetric strain Considering susceptibility Considering triggering

Response-Level Implementation Performance characterized in terms of response variables Slope instability (Rathje et al., 2013) Displacement hazard curves PGA (g) Displacement (cm)

Response-Level Implementation Performance characterized in terms of response variables Slope instability (Rathje et al., 2013) Displacement hazard curves PGA (g) PGV Displacement (cm) (cm/sec) Vector IM cuts displacement in half

Response-Level Implementation Performance characterized in terms of response variables Uncertainties from different sliding block models PGA and M w PGA and PGV Flexible sliding mass model

Damage-Level Implementation Performance characterized in terms of damage measures Inferred from Requires: damage Characterization of allowable levels of physical damage Damage model How much settlement is required to crack a slab? How much lateral displacement is required to produce hinging in a concrete pile? in a steel pile?

Damage-Level Implementation Performance characterized in terms of damage measures Continuous DM scales Fragility curve approach Some damage states (e.g., collapse) are binary Insufficient data available for others Discrete DM scales Damage probability matrix approach EDP interval Damage Description State, DM edp 1 edp 2 edp 3 edp 4 edp 5 dm 1 Negligible X 11 X 12 X 13 X 14 X 15 Probability that response in EDP interval 2 dm 2 Slight X 21 X 22 X 23 X 24 X 25 produces severe damage dm 3 Moderate X 31 X 32 X 33 X 34 X 35 dm 4 Severe X 41 X 42 X 43 X 44 X 45 dm 5 Catastrophic X 51 X 52 X 53 X 54 X 55

Damage-Level Implementation Performance characterized in terms of damage measures Fragility curve approach Continuous DM scales difficult to quantify Some damage states (e.g., collapse) are binary Insufficient data available for others Damage probability matrix approach Ledezma and Bray, 2010 N = None S = Small Pile-supported M = Moderate bridge founded on L = Large liquefiable soils C = Collapse

Damage-Level Implementation Performance characterized in terms of damage measures Fragility curve approach Continuous DM scales difficult to quantify Some damage states (e.g., collapse) are binary Insufficient data available for others Damage probability matrix approach Ledezma and Bray, 2010

Damage-Level Implementation Performance characterized in terms of damage measures Fragility curve approach Continuous DM scales difficult to quantify Some damage states (e.g., collapse) are binary Insufficient data available for others Damage probability matrix approach Ledezma and Bray, 2010

Loss-Level Implementation Performance characterized in terms of decision variables Example: Caisson quay wall (Iai, 2008) Life cycle cost as decision variable, DV Life-Cycle Costs Options: A: Foundation compaction only Options B: Foundation cementation A: Foundation compaction only Indirect Losses B: Foundation cementation C: Foundation and backfill compaction C: Foundation and backfill compaction (1.8 m spacing) (1.8 m spacing) D: Foundation & backfill compaction (1.6 m spacing) D: Foundation and backfill compaction Direct Losses E: Foundation compaction & structural modification (1.6 m spacing) E: Foundation compaction and Construction Costs structural modification

Loss-Level Implementation Performance characterized in terms of decision variables Example: Expressway embankment widening (Towhata, 2008) Life cycle cost as decision variable, DV 5 m widening with deep mixing

Loss-Level Implementation Performance characterized in terms of decision variables Fragility curve approach – Kramer et al. (2009) Pile-supported bridge on liquefiable soils

Loss-Level Implementation Repair cost losses only Doesn’t include losses due to downtime Doesn’t include losses due to casualties Mean annual rate of exceedance 1 Return period 10 -2 100 yrs 1,000 yrs 10 -4 0.16 0.47 10 -6 0.6 0.0 0.4 1.0 0.2 0.8 Repair cost ratio, RCR

Loss-Level Implementation Repair cost losses only Doesn’t include losses due to downtime Doesn’t include losses due to casualties Mean annual rate of exceedance 1 Return period Liquefaction 10 -2 100 yrs 1,000 yrs 10 -4 No Liquefaction 0.05 0.20 10 -6 0.6 0.0 0.4 1.0 0.2 0.8 Repair cost ratio, RCR

Loss-Level Implementation Impacts on bridge structure PBEE framework allows deaggregation of costs 475-yr losses 'Bar reinforcing steel (footing, retaining wall)' 'Structure 'Other' 'Temporary backfill' support 'Elastomeric (abutment)' bearings' 'Aggregate base (approach slab)' 'Structure excavation' 'Column steel 'Furnish steel casing' pipe pile' 'Joint seal assembly'

Advancing Performance-Based Design Improved Characterization of Capacity How should we characterize physical damage? How much ground movement can structures tolerate? Bird et al. (2005; 2006) Analyses of RC frame buildings subjected to ground deformation Four damage states: None to slight – linear elastic response, flexural or shear-type hairline cracks (<1 mm) in some members, no yielding in any critical section LS1 Moderate – member flexural strengths achieved, limited ductility developed, crack widths reach 1 mm, initiation of concrete spalling LS2 Extensive – significant repair required to building, wide flexural or shear cracks, buckling of longitudinal reinforcement may occur LS3 Complete – repair of building not feasible either physically or economically, demolition after earthquake required, could be due to shear failure of vertical elements or excess displacement

Advancing Performance-Based Design Improved Characterization of Capacity How should we characterize physical damage How much ground movement can structures tolerate? Bird et al. (2005) Analyses of structures subjected to ground deformation Horizontal Vertical High uncertainty Rational, quantified fragility curves for R/C frame buildings

Advancing Performance-Based Design Improved Characterization of Capacity Effects of uncertainty in capacity Response hazard curve      N  IM      ( ) | edp P EDP edp IM im P IM im EDP j j i i  1 i Let C = capacity (response corresponding to given damage state)  /   k b   2 1 c k     s 2 ( ) exp   c k ln | | EDP IM EDP C o   2  2  a b Integrating over distribution of capacity  Capacity     ( ) ( ) ( ) edp c f c dc uncertainty | EDP EDP C C 0 amplifier Assuming lognormal capacity distribution     2 2 1 1 k k     s s 2 2 ( ) ( ) exp   exp   c im ln C ln EDP | IM ln C EDP IM 2 2  2   2  b b

Advancing Performance-Based Design Improved Characterization of Capacity Effects of uncertainty in capacity Increasing uncertainty in capacity Accurate characterization of uncertainty in capacity nearly as important as uncertainty in response

Advancing Performance-Based Design Can PBEE concepts be used to develop load and resistance factors associated with predictable rate of limit state exceedance,  LS ? Capacity

Advancing Performance-Based Design Can PBEE concepts be used to develop load and resistance factors associated with predictable rate of limit state exceedance,  LS ? Let LM = load measure = aIM b Capacity  / k b   lm     No uncertainty ( ) lm k  LM ( lm ) 0 LM   a  /   k b   2 Uncertainty lm k      2 ( ) exp   lm k 0 LM L   2 in loading  2  a b lm

Advancing Performance-Based Design Can PBEE concepts be used to develop load and resistance factors associated with predictable rate of limit state exceedance,  LS ? Let LM = load measure = aIM b Capacity  / k b   lm     No uncertainty ( ) lm k  LM ( lm ) 0 LM   a  /   k b   2 Uncertainty lm k      2 ( ) exp   lm k 0 LM L   2 in loading  2  a b  /   k b      2 1 lm k       2  2 exp  k LM o   2 L C  2  a b Uncertainty in loading and capacity lm

Advancing Performance-Based Design Can PBEE concepts be used to develop load and resistance factors associated with predictable rate of limit state exceedance,  LS ? Let LM = load measure Solving previous equations for LM ,  LM ( lm ) b / k       LM LM a   0   k 0 / b k      k     2 LM exp LM a     L L     2  LS k b 0 b / k         k      2  2 LM exp LM a    LC L C     2 k b 0 LM 0 LM L LM LC lm

Advancing Performance-Based Design Can PBEE concepts be used to develop load and resistance factors associated with predictable rate of limit state exceedance,  LS ? Let LM = load measure LM LC can be interpreted as median capacity that will be exceeded every T R years, on average, and LM 0 as the median load. Then  LM ( lm ) LM LM    L LC LM LM LC 0 LM LM 0 L this will occur when LM LM ˆ ˆ  LS    L L C L LM LM LC 0 or ˆ ˆ f     C L LM 0 LM L LM LC lm L C

Advancing Performance-Based Design Can PBEE concepts be used to develop load and resistance factors associated with predictable rate of limit state exceedance,  LS ? Let LM = load measure Load factor So, Resistance LM L LM factor   f  L  LM ( lm ) LM LM 0 LC Substituting closed-form LM expressions,   1 k Uncertainty    2 exp    L  in loading 2 b  LS   1 k Uncertainty f   2 exp    C  in capacity 2 b LM 0 LM L LM LC lm

Application to Foundation Design Extension to foundation displacements Let LM = load measure, EDP = response measure Note that LM = { Q , V x , V y , M x , M y } EDP = { w , u , v , q x , q y } Q w q x M y q y M x V x u v V y Loads Deformations ( LM s) ( EDP s)

Application to Foundation Design Extension to foundation displacements Let LM = load measure, EDP = response measure Note that LM = { Q , V x , V y , M x , M y } EDP = { w , u , v , q x , q y }     ( ) ( | ) ( | ) ( ) edp G EDP LM dG LM IM d IM EDP Closed-form assumptions: Loads and moments    LM  EDP  k ( ) ( ) e b im k IM aIM dLM 0 IM Displacements and rotations

Application to Foundation Design     ( ) ( | ) ( | ) ( ) edp G EDP LM dG LM IM d IM EDP Closed-form assumptions:    LM  EDP  k ( ) ( ) b e im k IM aIM dLM 0 IM Solution: Uncertainty in LM | IM Uncertainty in EDP | LM  / k b        1 / e   2 1 edp k          2 2 ( ) exp  edp k e 0 EDP L   2 2 R    2  a d b e   Considering capacity: Uncertainty in capacity  / k b        1 / e   2 1 edp k          2 2   2 ( ) exp  edp k e 0 EDP L R C   2 2    2  a d b e  

Application to Foundation Design     ( ) ( | ) ( | ) ( ) edp G EDP LM dG LM IM d IM EDP  / k b   1 / e  ˆ  1 C       No uncertainty ( ) edp k   EDP 0   a d      EDP ( edp )  Uncertainty in k / b   1 / ˆ e        2 1 C k          2 2 loading and ( ) exp edp k  e   0 EDP   2 2 L R  2  a d b e     response  k / b   1 / e  ˆ       2 1 C k            2 2 2 ( ) exp  edp k e   0 EDP   2 2 L R C  2  a d b e     Uncertainty in loading, response, and capacity edp

Application to Foundation Design     ( ) ( | ) ( | ) ( ) edp G EDP LM dG LM IM d IM EDP Solving previous equations for EDP ,  be / k       EDP EDP d   0 k / b   k a  EDP ( edp ) 0  be / k         k       2 2 EDP exp EDP d e    LR L k / b  R    2 k a be 0  / be k         k        2  2 2  LS EDP exp EDP d e    LRC L k / b  R C    2 k a be 0 EDP 0 EDP LR EDP LRC edp

Application to Foundation Design     ( ) ( | ) ( | ) ( ) edp G EDP LM dG LM IM d IM EDP EDP LRC can be interpreted as median displacement capacity that will be exceeded every T R years, on average, and EDP 0 as the median displacement demand. Then EDP EDP  LM ( lm )    LR LRC EDP EDP 0 LRC EDP EDP 0 LR this will occur when EDP EDP ˆ ˆ    LR LR C D EDP EDP  LS 0 LRC or ˆ ˆ    CF C DF L EDP 0 EDP LR EDP LRC lm

Application to Foundation Design     ( ) ( | ) ( | ) ( ) edp G EDP LM dG LM IM d IM EDP Demand factor So, Capacity factor EDP EDP  CF  LR DF LR EDP EDP 0 LRC  LM ( lm ) Substituting closed-form LM expressions,   1 k     2 2 2 exp ( ) DF e   L R   2 be   1 k  LS   2 exp CF    C  2 be EDP 0 EDP LR EDP LRC lm

Application to Foundation Design Example: 5x5 pile group in sand Closed-form expression helps in understanding Actual problem more complicated Five components of load Five components of displacement Components of both may be correlated Relationships not described by power laws Uncertainty may not be lognormal Numerical integration required – in five dimensions

Application to Foundation Design Computational approach: Decoupled analyses LM | IM

Application to Foundation Design Computational approach: Decoupled analyses p-y t-z LM | IM Q-z