Simplified Procedures for Estimating Seismic Slope Displacements

Transcription

1 Simplified Procedures for Estimating Seismic Slope Displacements Jonathan D. Bray, Ph.D., P.E., NAE Faculty Chair in Earthquake Engineering Excellence University of California, Berkeley Thanks to Thaleia Travasarou & Others, and to NSF, Packard, & PEER

2 Simplified Procedures for Estimating Seismic Slope Displacements OUTLINE I. Seismic Slope Stability II. Seismic Slope Displacement Analysis III. Simplified Slope Displacement Procedures IV. Pseudostatic Slope Analysis V. Conclusions

3 I. Seismic Slope Stability Two Critical Design Issues 1. Are there materials that will lose significant strength as a result of cyclic loading? Flow Slide 2. If not, will the earth structure or slope undergo significant deformation that may jeopardize performance? Seismic Displacement

4 Las Palmas Gold Mine Tailings Dam Failure Sand ejecta near toe of flow debris View from scarp looking downstream M8.8 Maule, Chile EQ Failure & Flow Caused 4 deaths View across scarp (upstream method) Bray & Frost 2010

5 Fujinuma Dam: 18.5 m high earthfill dam completed 1949 Uncontrolled release of reservoir led to 8 deaths downstream of dam 2011 Tohoku EQ M w = 9.0 R = 102 km PGA = 0.33 g Bray et al. 2011; photographs from M. Yoshizawa

6 LIQUEFACTION EFFECTS Cyclic Mobility strain-hardening limited strain strain-softening large strain Flow Liquefaction Idriss & Boulanger 2008

7 Liquefaction Flow Slides when q c1n < Idriss & Boulanger 2008

8 Post-Liquefaction Residual Strength is a System Property 1971 Lower San Fernando Dam Failure (from H.B. Seed)

9 Seismic Slope Displacement Slip along a distinct surface Distributed shear deformation Newmark-type seismic displacement Add volumetric-induced deformation, when appropriate

10 II. Seismic Slope Displacement Analysis CRITICAL COMPONENTS a. Dynamic Resistance b. Earthquake Ground Motion c. Dynamic Response of Sliding Mass d. Seismic Displacement Calculation

11 a. Dynamic Resistance Yield Coefficient (k y ): seismic coefficient that results in FS=1.0 in pseudostatic stability analysis FS = 1.00 k y = Use method that satisfies all three conditions of equilibrium and focus on unit weight, water pressures, and soil strength

12 Peak Dynamic Strength of Clays Chen, Bray, and Seed (2006) S dynamic, peak = S static, peak (C rate ) (C cyc ) (C prog ) (C def ) Rate of loading: C rate > 1 Number of significant cycles: C cyc < 1 Progressive failure: C prog < 1 Distributed deformation: C def < 1 Typical values often lead to: S dynamic, peak S static, peak (1.4) (0.85 ) (0.9) (0.9) S static, peak

13 Dynamic Strength of Clays Chen, Bray, and Seed (2006) Peak dynamic strength is used for strain-hardening soils or limited displacements As earthquake-induced strain exceeds failure strain, dynamic strength reduces for strain-softening soils Thus, k y is also a function of displacement Shear strength (psf) Peak Strength Residual Strength k y perferential displacement (inches) D

14 b. Earthquake Ground Motion: Acceleration Time History 0.50 acceleration (g) PGA = 0.3 g time (s) Izmit (180 Comp) 1999 Kocaeli EQ (Mw=7.4) scaled to MHA = 0.30 g

15 Acceleration Response Spectrum (provides response of SDOF of different periods at 5% damping, i.e., indicates intensity and frequency content of ground motion) Spectral Acceleration (g) PGA Sa(0.2) Sa(0.5) Sa(1.0) 5% Damping Period (s)

16 c. Dynamic Response of Sliding Mass Seed and Martin 1966

17 Equivalent Acceleration Concept Seed and Martin 1966 accounts for cumulative effect of incoherent motion in deformable sliding mass Horz. Equiv. Accel.: HEA = (τ h /σ v ) g MHEA = max. HEA value k max = MHEA / g H σ vτh

18 FACTORS AFFECTING MAXIMUM SEISMIC LOADING MHEA/(MHA,rock*NRF) k max / (MHA rock * NRF / g) Rock site median, and 16th and 84th probability of exceedance lines T Ts-waste/Tm-eq s-sliding mass / T m-eq MHA,rock (g) NRF (Bray & Rathje 1998)

19 k max depends on stiffness and geometry of the sliding mass (i.e., its fundamental period) Ts,1-D = 4 H / Vs Ts, 1-D = Initial Fundamental Period of Sliding Mass H = Height of Sliding Mass Vs = Average Shear Wave Velocity of Sliding Mass

20 k max also depends on the length of the sliding mass k max_2d = C L k max_1d C L = 1.0 for L < 60 m & C L = 0.8 for L > 300 m C L = 1.0 [(L 60 m) / 1200] for 60 m < L < 300 m where L = Length of Potential Sliding Mass (m) MSW 200 ROCK 100 All units in ft This study (all data) Rathje and Bray (2001) Kmax,2D / (1.15*Kmax,1D) Note: Rathje and Bray (2001) data do not include the 1.15 factor because their 1D Kmax values represent the mass-weighted average for several 1D columns, which increases the single column Kmax Length (ft) Rathje & Bray 2008

21 Topographic Amplification of PGA Steep Slope (>60 o ): PGA crest 1.5 PGA 1D (Ashford and Sitar 2002) Moderate Slope: PGA crest 1.3 PGA 1D (Rathje and Bray 2001) Dam Crest: PGA crest (0.5(PGA) ) PGA (Yu et al. 2012) (Yu et al. 2012)

22 Acceleration (g) (a) Topographic Effects on k max Localized shallow sliding near crest k max PGA crest / g Long shallow sliding surface k max 0.5 PGA crest / g Crest Mid Slope Toe Time (s) Normalized Slope Cover Sliding Mass Length (L / L ) s k crest max / PGA 0.6 crest LA Landfills (1.9H:1V to 5.5H:1V) Config. 1 (3H:1V) Config. 2 (2H:1V) Config. 3 (2H:1V) Config. 4 (5H:1V) Best Fit (R = 0.65) 2 MHEA / MHA

23 d. Seismic Displacement Calculation Newmark (1965) Rigid Sliding Block Analysis Assumes: Rigid sliding block Well-defined slip surface develops Slip surface is rigid-perfectly plastic Acceleration-time history defines EQ loading Key Parameters: ky - Yield Coefficient (max. dynamic resistance) k max - Seismic Coefficient (max. seismic loading) ky / kmax (if > 1, D = 0; but if < 1, D > 0)

24 Rigid Sliding Block: uses accel.- time hist. k max = PGA/g Cover Mid-Ht Base Deformable Sliding Block: uses equiv. accel.-time hist. k max = MHEA/g Rock

25 25 Limitations of Rigid Sliding Block Models Displacement (cm) D rigid coupled rigid D coupled (cm) U -U (cm) Damping 15% Ky=0.1 Mag 8 (MHA=0.4g) Ts/Tm = 1.0 Damping 15% Decoupled Coupled Rigid Block Ts/Tm (c) Ky/MHA Unconservative U -U (cm) Rathje and Bray (1999, 2000) D rigid D coupled (cm) rigid coupled Ts/Tm = 4.0 Damping 15% (d) Ky/MHA conservative

26 Seismic Displacement Calculation Deformable Sliding Block Analysis Decoupled Analysis Dynamic Response Flexible System Sliding Response Rigid Block Earth Fill Calculate HEAtime history assuming no sliding along base Double integrate HEA-time history given ky to calculate D Potential Slide Plane Dynamic Response and Sliding Response Coupled Analysis Flexible Flexible System System Max Force at Base = k y W Calculate D

27 Decoupled vs. Coupled Analysis Displacement Difference (cm): D decoupled U - D U coupled (cm) decoupled coupled (b) k = 0.05 y k = 0.1 y k = 0.2 y U (cm) decoupled Decoupled Conservative Insignificant difference for D decoupled < 1 cm Conservative for D decoupled > 1 m Between 1 cm and 1 m, could be meaningfully unconservative From Rathje and Bray (2000)

28 Calculated Seismic Displacement Expected k y See programs such as SLAMMER by Jibson et al. (2013) Report

29 Think About It as a Cliff Calculated Seismic Displacement is an Index of Performance SAFE? UNSAFE

30 Evaluate Seismic Performance Given seismic displacement estimates: Minor (e.g., D < 15 cm) Major (e.g., D > 1 m) Evaluate the ability of the earth structure and structures founded on it to accommodate the level of deformation Consider: Consequences of failure and conservatism of hazard assessment and stability analyses Defensive measures that provide redundancy, e.g., crack stoppers, filters, and chimney drain for dams, & robust mat foundations, slip layer, and ductile structure

31 III. Simplified Seismic Slope Procedures Makdisi & Seed (1978) Estimate PGA at crest PGA,top =? MHA at Top vs. Base Rock MHA for Some Solid-Waste Landfills (Bray & Rathje 1998)

32 Makdisi & Seed (1978) Estimate kmax for sliding mass as f (PGA crest & y/h) Kmax varies with Ts

33 Makdisi & Seed (1978) Estimate seismic displacement as f (k y /k max & M w ) WARNING: Do not use figure in original Makdisi & Seed (1978) paper; it is off by an order of magnitude

34 Limitations design curves are derived from a limited number of cases. These curves should be updated and refined as analytical results for more embankments are obtained. Makdisi & Seed (1978) Limited number of earthquake ground motions used Estimating PGA at the crest to estimate k max is difficult Simple shear slice analysis employed Decoupled analysis to calculate seismic displacement Bounds are not true upper and lower bounds Only a few analyses & no estimate of uncertainty

35 Jibson (2007) Used rigid sliding block model with accelerationtime histories (original Newmark 1965 approach) Used several hundred EQ records Estimate seismic displacement as f (k y, a max & M w )

36 Jibson (2007) states: Jibson (2007) Newmark's method treats a landslide as a rigidplastic body: the mass does not deform internally... Therefore, the proposed models are most appropriately applied to thinner landslides in more brittle materials rather than to deeper landslides in softer materials. Only Models Rigid Sliding Mass (i.e., T s = 0)

37 Bray & Travasarou (2007) 1. SLOPE MODEL nonlinear soil response fully coupled deformable stick-slip stiffness (T s ) & strength (k y ) 8 Ts values & 10 ky values T s 2. EARTHQUAKE DATABASE 688 records (41 EQs) Over 55,000 analyses OPTIMAL INTENSITY MEASURES: D k y S a (1.5 T s ) & M w of outcropping motion below slide 3. CALIBRATION 16 case histories

38 EFFICIENCY of Ground Motion Intensity Measures S T R O N G E R Standard Deviation Standard Deviation ky = 0.10 Ts = 0.5 s ky = 0.20 Ts = 0.5 s ky = 0.10 Ts = 1.0 s Sa (1.5T s ) Arias Intensity Housner Spectral Intensity ky = 0.20 Ts = 1.0 s PGA SA SA(1.3Ts) SA(1.5Ts) SA(2.0Ts) arms Arias Ic PGV EPV SI PGD Tm D5_95 Intensity Measure PGA SA SA(1.3Ts) SA(1.5Ts) SA(2.0Ts) arms Arias Ic PGV EPV SI Intensity Measure PGD Tm D5_95 MORE FLEXIBLE

39 Bray & Travasarou (2007): D = f (k y, S a (1.5T s ), T s, M w ) 1) Zero Displacement Estimate ( ln( k ) ln( k ) T 3.52 ln( S (1.5T ))) P ( D = "0") = 1 Φ y y s + a s Φ is the standard normal cumulative distribution function (NORMSDIST in Excel) 2) NonZero Displacement Estimate ln( D) = ln( k 3.04ln( S a (1.5T s y ) ln( )) ( ) 2 k ) ln( k )ln( S (1.5T )) + y 2 ( ln( S (1.5T ))) T ( M 7) ± ε a s s y a s ε is a normally-distributed random variable with zero mean and standard deviation σ = 0.66 *Replace first term (i.e., -1.10) with for cases where Ts < 0.05 s

40 Bray & Travasarou (2007): MODEL TRENDS Scenario Event: M 7 at 10 km Soil SS fault SA(1.5Ts) (g) % damping T s (s) P(D="0") ky= 0.3 ky= 0.2 ky= 0.1 ky= 0.05 Displacement (cm) ky= 0.1 ky= 0.2 ky= 0.3 ky= T s (s) T s (s)

41 PROBABILITY OF EXCEEDANCE P(D="0") ky= 0.3 ky= 0.2 ky= 0.1 ky= T s (s) Scenario Event: M 7 at 10 km Soil SS fault Displacement (cm) ky= 0.1 ky= 0.2 ky= 0.3 ky= T s (s) P(D > 30 cm) ky= 0.2 ky= 0.05 ky= 0.1 ky= T s (s)

42 Validation of Bray & Travasarou Simplified Procedure Earth Dam or Landfill EQ Obs. D max Bray & Travasarou 2007 (cm) P (D = 0 ) Est. Disp (cm) Chabot Dam SF Minor Guadalupe LF LP Minor Pacheco Pass LF LP None Austrian Dam LP Lexington Dam LP Sunshine Canyon LF NR OII Section HH LF NR La Villita Dam S La Villita Dam S La Villita Dam S

43 Example: 57 m-high Earth Dam Located 10 km from M = 7.5 EQ No liquefaction or soils that will undergo significant strength loss Undrained Strength: c = 14 kpa and φ = 21 o so k y = 0.14 Triangular Sliding Block with avg. V s ~ 450 m/s & H = 57 m 2.6 H m T = = 0. s s V 450m / s 33 s

44 For fully probabilistic implementation see Travasarou et al Deterministic Analysis: ky = 0.14; Ts = 0.33 s; & M w = 7.5 at R = 10 km; Using 1.5 T s = 1.5 (0.33 s) = 0.5 s & NGA GMPE for rock with V s30 = 600 m/s for strike-slip fault: median Sa(0.5 s) = g Probability of Zero Displacement (i.e., D < 1 cm): ( ln(0.14) 0.484ln(0.14)(0.33) ln(0.483) ) P( D = "0") = 1 Φ = Nonzero Median Seismic Displacement Estimate (ε = 0): ln( D) = ln(0.14) 3.04 ln(0.483) D = exp(ln( D)) = exp(2.28) 10cm Design Seismic Displacement Estimate (16% and 84%): D 0.5 to 2 times median D = 5 cm to 20 cm ( ln(0.14) ) ln(0.14) ln(0.483) + 2 ( ln(0.483) ) (0.33) (7.5 7) ± 0

45 Pseudostatic Slope Stability Analysis 1. k = seismic coefficient; represents earthquake loading 2. S = dynamic material strengths 3. FS = factor of safety Selection of acceptable combination of k, S, & FS requires calibration through case histories or consistency with more advanced analyses

46 A Prevalent Pseudostatic Method Seed (1979) appropriate dynamic strengths k = 0.15 FS > 1.15 BUT this method was calibrated for cases where 1 m of displacement was judged to be acceptable WHAT about other levels of acceptable displacement? IS k = 0.15 reasonable for all sites regardless of M & R? FS > 1.15 does not mean the system is safe!

47 Seismic Coefficient k should depend on level of shaking, i.e., R, M, & dynamic response of earth structure k should also depend on criticality of structure and acceptable level of seismic performance, i.e., amount of allowable seismic displacement How does one then select k?

48 Seismic Coefficient from Allowable Displacement Bray &Travasarou (2009) Instead, calculate k as function of D a, S a, T s, M w, & ε Seismic Coefficient M=7.5 Ts= 0.2s M=6 5 cm 15 cm 30 cm Sa(@1.5Ts) (g)

49 Pseudostatic Slope Stability Procedure 1. Can materials lose significant strength? If so, use post-shaking reduced strengths. Otherwise, use strain-compatible dynamic strengths. 2. Select allowable displacement: D a and select exceedance probability (e.g., use ε = 0.66 for 84%) 3. Estimate initial period of sliding mass: T s 4. Characterize seismic demand: S a (1.5T s ) & M w 5. Calculate seismic coefficient: k = f (D a, S a, T s, M w & ε) & perform pseudostatic analysis using k. If FS 1, then D calc < D a at selected exceedance level

50 IV. Conclusions First question: will materials lose strength? If not, evaluate seismic slope stability in terms of seismic displacements Bray & Travasarou (2007) approach with deformable sliding mass captures: a. Dynamic resistance of slope - ky b. Earthquake shaking - Sa(1.5 T s ) & Mw c. Dynamic response of sliding mass - T s d. Coupled seismic displacement - D Earthquake and material characterization are most important

51 References Ashford, S.A. and Sitar, N. (2002) Simplified Method for Evaluating Seismic Stability of Steep Slopes, Journal of Geotechnical and Geoenvironmental Engineering; 128(2): Bray, J.D. Chapter 14: Simplified Seismic Slope Displacement Procedures, Earthquake Geotechnical Engineering, 4th ICEGE - Invited Lectures, in Geotechnical, Geological, and Earthquake Engineering Series, Vol. 6, Pitilakis, Kyriazis D., Ed., Springer, pp , Bray, J.D. and Rathje, E.R. Earthquake-Induced Displacements of Solid-Waste Landfills, Journal of Geotech. & Geoenv. Engrg., ASCE, Vol. 124, No. 3, pp , Bray, J.D. and Travasarou, T., Simplified Procedure for Estimating Earthquake-Induced Deviatoric Slope Displacements, J. of Geotech. & Geoenv. Engrg., ASCE, Vol. 133, No. 4, April 2007, pp Bray, J.D. and Travasarou, T., Pseudostatic Coefficient for Use in Simplified Seismic Slope Stability Evaluation, J. of Geotechnical and Geoenv. Engineering, ASCE, 135(9), 2009, Chen, W.Y., Bray, J.D., and Seed, R.B. Shaking Table Model Experiments to Assess Seismic Slope Deformation Analysis Procedures, Proc. 8 th US Nat. Conf. EQ Engrg., 100 th Anniversary Earthquake Conf. Commemorating the 1906 San Francisco EQ, EERI, April 2006, Paper Harder, L.F., Bray, J.D., Volpe, R.L., and Rodda, K.V., "Performance of Earth Dams During the Loma Prieta Earthquake," The Loma Prieta, California, Earthquake of October 17, Earth Structures and Engineering Characterization of Ground Motion, Performance of the Built Environment, Holzer, T.L., Coord., U.S.G.S. Professional Paper 1552-D, U.S. Gov. Printing Office, Washington D.C., 1998, pp. D3-D26. Makdisi F, and Seed H. (1978) Simplified procedure for estimating dam and embankment earthquake-induced deformations. Journal of Geotechnical Engineering; 104(7): Newmark, N. M. (1965) Effects of earthquakes on dams and embankments, Geotechnique, London, 15(2), Rathje, E. M., and Bray, J.D. (2001) One- and Two-Dimensional Seismic Analysis of Solid-Waste Landfills, Canadian Geotechnical Journal, Vol. 38, No. 4, pp Seed, H.B. (1979) Considerations in the Earthquake-Resistant Design of Earth and Rockfill Dams, Geotechnique, V. 29(3), pp Travasarou, T., Bray, J.D., and Der Kiureghian, A.D. A Probabilistic Methodology for Assessing Seismic Slope Displacements, 13th WCEE, Vancouver, Canada, Paper No. 2326, Aug 1-6, Yu, L., Kong, X., and Xu, B. Seismic Response Characteristics of Earth and Rockfill Dams, 15 th WCEE, Lisbon, Portugal, Paper No.2563, Sept, 2012.