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
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
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”
Las Palmas Gold Mine Tailings Dam Failure M8.8 Maule, Chile EQ View across scarp Failure & Flow Caused 4 deaths Bray & Frost 2010 (upstream method) Sand ejecta near toe of flow debris View from scarp looking downstream
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
LIQUEFACTION EFFECTS Cyclic Mobility strain-hardening limited strain strain-softening large strain Flow Liquefaction Idriss & Boulanger 2008
Liquefaction Flow Slides when q c1N < 80 80 Idriss & Boulanger 2008
Post-Liquefaction Residual Strength is a System Property 1971 Lower San Fernando Dam Failure (from H.B. Seed)
Seismic Slope Displacement • Slip along a distinct surface Newmark-type seismic displacement • Distributed shear deformation • Add volumetric-induced deformation, when appropriate
II. Seismic Slope Displacement Analysis CRITICAL COMPONENTS a. Dynamic Resistance b. Earthquake Ground Motion c. Dynamic Response of Sliding Mass d. Seismic Displacement Calculation
a. Dynamic Resistance Yield Coefficient (k y ): seismic coefficient that results in FS=1.0 in pseudostatic stability analysis FS = 1.00 k y = 0.105 Use method that satisfies all three conditions of equilibrium and focus on unit weight, water pressures, and soil strength
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
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 60 Peak Strength 50 k y Shear strength (psf) 40 30 20 Residual Strength 10 0 0 1 2 3 4 5 perferential displacement (inches) D
b. Earthquake Ground Motion: Acceleration – Time History 0.50 0.25 acceleration (g) 0.00 -0.25 PGA = 0.3 g -0.50 0 5 10 15 20 25 30 time (s) Izmit (180 Comp) 1999 Kocaeli EQ (M w =7.4) scaled to MHA = 0.30 g
Acceleration Response Spectrum (provides response of SDOF of different periods at 5% damping, i.e., indicates intensity and frequency content of ground motion) 1.5 Sa( 0.2 ) Spectral Acceleration (g) Sa( 0.5 ) 1.0 Sa( 1.0 ) 0.5 PGA 5% Dam ping 0.0 0 1 2 3 4 5 Period (s)
c. Dynamic Response of Sliding Mass Seed and Martin 1966
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 H σ v τ h • k max = MHEA / g
FACTORS AFFECTING MAXIMUM SEISMIC LOADING 2.0 Rock site median, and 16th and 84th 1.8 MHEA/(MHA,rock*NRF) probability of exceedance lines k max / (MHA rock * NRF / g) 1.6 MHA,rock (g) NRF 1.4 0.1 1.35 0.2 1.20 1.2 0.3 1.09 1.0 0.4 1.00 0.5 0.92 0.8 0.6 0.87 0.6 0.7 0.82 0.8 0.78 0.4 0.2 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Ts-waste/Tm-eq T s-sliding mass / T m-EQ (Bray & Rathje 1998)
k max depends on stiffness and geometry of the sliding mass (i.e., its fundamental period) T s ,1-D = 4 H / V s T s , 1-D = Initial Fundamental Period of Sliding Mass H = Height of Sliding Mass V s = Average Shear Wave Velocity of Sliding Mass
k max also depends on the length of the sliding mass 500 k max_2D = C L k max_1D 400 300 MSW 200 ROCK 100 All units in ft 0 0 500 1000 1500 2000 2500 3000 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) 1.5 This study (all data) Rathje and Bray (2001) 1.0 Kmax,2D / (1.15*Kmax,1D) 1 0.8 0.5 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 0 0 200 400 600 800 1000 Rathje & Bray 2008 Length (ft)
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) -0.5 + 1) PGA • (Yu et al. 2012) (Yu et al. 2012)
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 MHEA / MHA crest k max / PGA crest 0.0 0.2 0.4 0.6 0.8 1.0 0.0 LA Landfills (1.9H:1V to 5.5H:1V) Config. 1 (3H:1V) 1.0 0.5 Config. 2 (2H:1V) 0.2 Acceleration (g) Config. 3 (2H:1V) Sliding Mass Length (L / L ) 0.5 s Normalized Slope Cover Config. 4 (5H:1V) 2 Best Fit (R = 0.65) 0.0 0.4 3.5 4 4.5 5 -0.5 Time (s) Crest 0.6 Mid Slope -1.0 Toe (a) 0.8 1.0
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: • k y - Yield Coefficient (max. dynamic resistance) • k max - Seismic Coefficient (max. seismic loading) • k y / k max (if > 1, D = 0; but if < 1, D > 0)
Cover Rigid Deformable ≈ Sliding Sliding Block: Block: Mid-Ht uses uses equiv. accel.-time accel.- hist. time hist. k max = PGA/g k max = MHEA/g Base ≠ Rock
Limitations of Rigid Sliding Block Models 100 Rathje and Bray (1999, 2000) Ky=0.1 Displacement (cm) Mag 8 (MHA=0.4g) 80 Decoupled 60 Coupled Rigid Block 40 20 Damping 15% 0 0 1 2 3 4 Ts/Tm 25 100 D rigid – D coupled (cm) D rigid – D coupled (cm) Ts/Tm = 1.0 Ts/Tm = 4.0 U -U (cm) U -U (cm) 75 0 conservative Unconservative 50 rigid coupled rigid coupled -25 25 -50 0 Damping 15% Damping 15% -75 -25 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (c) Ky/MHA Ky/MHA (d) 25
Seismic Displacement Calculation Deformable Sliding Block Analysis Sliding Response Dynamic Response Decoupled Analysis Rigid Block Flexible System Calculate HEA- Double integrate time history HEA-time assuming no history given ky sliding along base to calculate D Earth Fill Dynamic Response and Potential Slide Plane Sliding Response Flexible System Flexible System Coupled Analysis Max Force at Base = k y ·W Calculate D
Decoupled vs. Coupled Analysis • Insignificant 100 difference for Decoupled Conservative k = 0.05 D decoupled – D coupled (cm) Displacement Difference (cm): y D decoupled < 1 cm 80 k = 0.1 y U - U decoupled coupled k = 0.2 60 y • Conservative for 40 D decoupled > 1 m 20 0 • Between 1 cm and -20 1 m, could be -40 0.1 1 10 100 1000 meaningfully (b) U (cm) decoupled unconservative From Rathje and Bray (2000)
Calculated Seismic Displacement Expected k y See programs such as SLAMMER by Jibson et al. (2013) http://pubs.usgs.gov/tm/12b1/O-F Report 03-005
Think About It as a “Cliff” Calculated Seismic Displacement is an Index of Performance ? SAFE UNSAFE
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
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)
Makdisi & Seed (1978) Estimate k max for sliding mass as f (PGA crest & y/h) K max varies with T s
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