Numerical modeling of liquefaction effects: Development & initial applications of a sand plasticity model

Transcription

1 4 th IASPEI / IAEE International Symposium Santa Barbara, California, Aug 23-26, 2011 Numerical modeling of liquefaction effects: Development & initial applications of a sand plasticity model Ross W. Boulanger Ronnie Kamai Katerina Ziotopoulou

2 Ronnie Kamai Doctoral candidate Katerina Ziotopoulou Doctoral candidate

3 PM4-Sand: A sand plasticity model for nonlinear seismic deformation analyses

4 The challenge for a constitutive model Ø Varied conditions: Loose to dense zones Drained to undrained loading Low to high confining stresses Low to high initial static shear stress ratios Ø Common data: V s, N 60, q c, gradations (N 1 ) 60 = 25 (N 1 ) 60 = 25 (N 1 ) 60 = 15 (N 1 ) 60 = 10

5 The challenge for a constitutive model Ø Calibration to design correlations: Triggering & cyclic mobility/ratcheting G/G max and damping Strengths Others depending on the structure (e.g., volumetric strains) (N 1 ) 60 = 25 (N 1 ) 60 = 25 (N 1 ) 60 = 15 (N 1 ) 60 = 10

6 Triggering Cyclic stress ratio Curve recommended by Idriss & Boulanger (2004) for clean sands Cases with FC 5% Cases with 5 < FC < 15% Cases with 15 < FC < 35% Cases with FC 35% L: Liquefaction; NL: No Liquefaction L NL K σ (N 1 ) 60 =5, q c1n = (N 1 ) 60 =15, q c1n = (N 1 ) 60 =25, q c1n = Cyclic stress ratio Equivalent clean sand corrected standard penetration, (N 1 ) 60cs Shaking table tests conducted by De Alba et al (1976): Relative Density = 90% Relative Density = 82% Relative Density = 68% Relative Density = 54% initial confining pressure = 8 psi (55 kpa) Vertical effective stress, σ' vc /P a Number of cycles to cause initial liquefaction

7 Plasticity model for sand Starting point Ø Starts with framework of Dafalias-Manzari (2004) model Critical state, stress-ratio based Bounding and dilation surfaces rotate with changes in state Fabric tensor used to enhance contraction rates

8 Plasticity model for sand Modifications Ø Modified & calibrated at equation level to approximate design correlations for practice Modified fabric tensor to depend on plastic shear strains Added fabric history, including cumulative fabric term Plastic modulus (K p ), elastic moduli (G), and dilatancy (D) depend on fabric and fabric history D constrained by Bolton's (1986) dilatancy relationship Recast in terms of relative state parameter index (ξ R ) Inclusion of sedimentation effects Modified logic for updating initial back-stress ratio Neglects Lode Angle dependence Ø Implemented as a user-defined material model in FLAC (Itasca 2010)

9 Relative state parameter index Ø Practical means for including critical state framework ξ = D D R R,CS R D R,CS R = 100 ( 1+ 2Ko) σʹ Q ln 3p A vc

10 Stress ratio based Ø Dilatancy & bounding surfaces collapse to M at critical state (ξ R = 0) b = ( b ξ ) d M M exp n ( d ) R M = M exp n ξr q Bounding line Critical line Dilantancy line α + m α Elastic range α - m 0 p

11 Ø Fabric Ø Elastic modulus Ø Plastic modulus Ø Dilatancy d ( ) D = A d α α :n in ( ) D = A α α :n+ C dc in d 2 Fabric effects dε dz n z K p p cz v = max + z D cum 1+ 1 d ( ) ( ) 2z max z 1 cum p 4z max G = G opa CSR pa zcum 1+ C 4z b ( ) max 0.5 ( z ) α α :n = C G ho exp( α α ) :n 1 + C z 1 C z α α :n Kα b ( ) in γ1 + Kp peak max A α α :n+ C = A do d 3 2 z z:n cum D max peak ( C ) zin2 α α :n ( )( )( )( ) 1 Cε Cpzp Cp min Czin1 + 1 z 2 z A dc A DG = do A ( ) dc = f z, ξr,... hp

12 Functionality versus simplicity Ø Simple parts, easy to understand Ø Rides well

13 Parameters Ø Relative density (D R ) Estimate from SPT or CPT; adjusts stress-strain responses Ø Shear modulus coefficient (G o ) Calibrate to in-situ V s data or correlations Ø Contraction rate parameter (h po ) Calibrate to CRR estimated from SPT- or CPT-based liquefaction correlations Ø Secondary parameters 18 secondary parameters with default values chosen to approximate design correlations

14 Example responses Shear stress ratio, τ/σ' vc Shear stress ratio, τ/σ' vc Shear strain (%) D R = 35% σ' vo = 100 kpa D R = 55% σ' vo = 100 kpa Shear strain (%)

15 Example responses Cyclic stress ratio D R = 75% 55% 35% σ' vo = 100 kpa, r u > 98% 0.26 Fit with b = Number of uniform cycles Cyclic stress ratio D R = 75% 55% 35% σ' vo = 100 kpa, 3% shear strain Fit with b = Number of uniform cycles

16 Example responses 0.6 Cyclic stress ratio σ' vc = 1 atm D R = 55% 3% shear strain α = Relationships recommended by Boulanger & Idriss (2004): Cyclic stress ratio Number of uniform cycles σ' vc = 1 atm D R = 35% 3% shear strain α = 0.0 K σ Model simulations D R = 35% D R = 55% D R = 75% 35% 55% 75% Vertical effective stress, σ' vc /P a Number of uniform cycles

17 Example responses Shear stress (kpa) D R = 55% σ' vo = 100 kpa G/G max Dashed lines: EPRI (1993) for depths of 0-6 m & m. Shear stress (kpa) Shear strain (%) D R = 55% σ' vo = 400 kpa Shear strain (%) Equiv. damping ratio (%) Shear strain (%) Solid lines: Simulations for σ' vo = 100, 400, & 1,600 kpa Shear strain (%)

18 Site response of Port Island and Wildlife Liquefaction Arrays

19 Wildlife liquefaction array [Data from Bennett et al. 1984, Holzer & Bennett 2010 personal comm.]

20 WLA response in 1987 Superstition Hills Eq. Ø Surface motion

21 Centrifuge test with lateral spreading

22 Centrifuge model SSK01 [NEES test by Kamai, Kano, Conlee, Marinucci, Boulanger, Rathje, Rix, and Howell 2008]

23 Ø V s measured in the model Ø CRR from lab tests Calibration

24 Accelerations Ø Input motion: Sequence of progressively stronger shaking events, each being 20 cycles at 2Hz

25 Excess pore pressures

26 Displacements

27 Strain concentration beneath clay crust

28 Strain concentration beneath clay crust

29 Centrifuge test of slope with silt interlayers

30 Centrifuge test of slope with silt interlayers Ø Nevada sand, D R 35% (Malvick, Kutter, & Boulanger 2008)

31 Initial stresses

32 Accelerations

33 Excess pore pressures

34 Strains & displacements

35 Strain concentration at silt seam Ø Influence of localization scale, permeabilities, re-sedimentation strains and other factors.

36 Strain concentration at silt seam Ø Influence of localization scale, permeabilities, re-sedimentation strains and other factors.

37 Concluding remarks Ø PM4-Sand is a stress-ratio controlled, critical state compatible, bounding surface plasticity model with fabric which was developed and calibrated to approximate trends in design correlations commonly used in the USA. Ø Initial applications of PM4-Sand have been promising, suggesting that it reasonably approximates the principle behaviors of liquefying sands. Ø Numerical analyses of liquefaction effects can provide valuable insights regarding complex mechanisms of behavior, but can have significant bias and dispersion in computed responses depending on the specific problem (and on the numerical procedures & calibration protocols). Ø Dynamic centrifuge model studies provide a valuable basis for systematically evaluating numerical analysis methods.

38 Support Ø U. S. Geological Survey (Award G09AP00121) Ø International Fulbright Science and Technology Award from the Institute of International Education and U.S. Department of State Ø National Science Foundation (NSF) for support for the centrifuge tests

39 4 th IASPEI / IAEE International Symposium Santa Barbara, California, Aug 23-26, 2011 Thank you. Ross W. Boulanger Ronnie Kamai Katerina Ziotopoulou

40 Strains & displacements

41 Localization scales in the field?

42 Localization scales in the field? (modified after Naesgaard et al. 2006) a) Continuous water film b) Venting + collapse of water film c) Undulating surface d) Spatial discontinuity of barriers

43 Example responses 50 φ' = 2[tan -1 ((σ'1 /σ' 3 )0.5 ) - 45 o ] D R = 75% 55% 35% Plane strain compression: φ cv = 33 o, K o = Horizontal effective stress (atm)

44 Example responses Shear stress (kpa) Undrained DSS, D R = 35% σ' vo = 0.25, 1, 4, & 16 atm Shear stress (kpa) Critical state Shear strain (%) Vertical effective stress (kpa)

45 Port Island Array, Kobe [Data from]

46 PIA response in 1995 Kobe Earthquake Ø Surface motion