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Seismic Slope Instability and Seismic Slope Instability and Slope Displacement ProceduresSlope Displacement Procedures

Jonathan D. Bray, Ph.D., P.E.

Univ. of California at Berkeley

Thanks to Dr. Thaleia Travasarou, Prof. Ellen Rathje, & others, with support from PEER & the Packard Foundation

Seismic Slope DisplacementSeismic Slope Displacement

EERC Slide Collection

Waste Liner Tear, 1994 Northridge EQ

4th Ave. Slide, 1964 Alaska EQ 1999 Chi-Chi, Taiwan EQ

Solid-Waste Landfill

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Mechanisms Contributing toMechanisms Contributing toSeismic Slope DeformationSeismic Slope Deformation

• Slip along a distinct surface

• Distributed deviatoric shear deformation

• Volumetric deformation

• Combined effects

Use procedures such as Tokimatsu and Seed 1987 for 1D volumetric seismic

compression (e.g., Stewart et al. 2005)

Two Critical Design IssuesTwo Critical Design Issues

• Are there materials that will lose significant strength as a result of cyclic loading? ““Flow SlideFlow Slide””

• If not, will the earth or waste fill system undergo significant deformations that may jeopardize system performance?““Seismically Induced DeformationsSeismically Induced Deformations””

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TerzaghiTerzaghi (1950) commenting on (1950) commenting on pseudostatic slope stability analysis:pseudostatic slope stability analysis:

“Theoretically a value of FS = 1 would mean a slide but in reality a slope may remain stable in spite of FS being smaller than unity and it may fail at a value of FS > 1, depending on the character of the slope forming materials.”

“The most sensitive materials are slightly cemented grain aggregates such as loess and submerged or partly submerged loose sand.”

PseudoPseudo--Static Stability AnalysisStatic Stability Analysis

1. k = seismic coefficient, constant that represents earthquake loading

2. S = dynamic material strengths and geometry give FS

3. Potential sliding mass is rigid

Selection of acceptable combination of S, k, & FS requires calibration through case histories or consistency with more advanced analyses

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Static Slope Stability MethodsStatic Slope Stability Methods• Limit equilibrium methods that satisfy all

conditions of equilibrium give FS within +/- 6% (Duncan 1992)– Morgenstern and Price 1965– Spencer 1967– Generalized Janbu 1968

• Focus on these most important issues: – Defining geometry– Shear strengths– Unit weights– Water pressures

Some Prevalent Pseudostatic methodsSome Prevalent Pseudostatic methods(for embankments that do not lose significant (for embankments that do not lose significant

strength from earthquake shaking)strength from earthquake shaking)• Hynes-Griffen & Franklin (1984)

– 20% strength reduction – k = ½ MHA,rock

– FS > 1.0

• Seed (1979)– “appropriate” dynamic strengths– k = 0.15 – FS > 1.15

BUT these methods were calibrated for earth dams where ~ 1 m of displacement is judged to be acceptable

WHAT about other systems and other levels of acceptable displacement?

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Critical Components of a Critical Components of a Seismic Displacement AnalysisSeismic Displacement Analysis

1. Earthquake Ground Motion

2. Dynamic Resistance of Slope

3. Dynamic Response of Potential Sliding Mass

Earthquake Shaking:Earthquake Shaking:Acceleration Acceleration –– Time HistoryTime History

acce

lera

tion

(g)

-0.50

-0.25

0.00

0.25

0.50

time (s)

0 5 10 15 20 25 30

Izmit (180 Comp) 1999 Kocaeli EQ (Mw=7.4) scaled to MHA = 0.30 g

MHA = 0.3 g Tm = 0.63 s & D5-95 = 15 s

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Acceleration Response SpectrumAcceleration Response Spectrum(provides response of SDOF of different periods at 5% damping,(provides response of SDOF of different periods at 5% damping,

i.e., indicates frequency content of ground motion)i.e., indicates frequency content of ground motion)

0 1 2 3 4 5Period (s)

0.0

0.5

1.0

1.5Sp

ectral

Acc

eler

atio

n (g

)

5% Damping

Sa at T = 0.5 s

Sa at T = 1.0 s

MHA

Dynamic Resistance:Dynamic Resistance:Simplified Estimates of Yield Coefficient (Simplified Estimates of Yield Coefficient (kkyy))

(seismic coefficient that results in FS=1.0 in pseudostatic stab(seismic coefficient that results in FS=1.0 in pseudostatic stability analysis)ility analysis)

H

β

c = cohesionφ = friction angle

kc

Hy = − +⋅ ⋅ ⋅ + ⋅

tan( )cos ( tan tan )

φ βγ β φ β2 1

1 S2

1 H

L

S1

kFS S H

H S S Lystatic=

− ⋅ ⋅ ⋅⋅ + +

⋅( ) cos sin( )

1 22

1 1 1

1 2

θ θ

( )FS

S H L S HS Hstatic =

⋅ ⋅ ⋅ + + ⋅⋅ ⋅ ⋅

tan coscos sin

φ θθ θ

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1 2

1 1 1

2 22

with θ11

11= −tan ( )S

Shallow SlidingDeep Sliding

Bray et al. 1998

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Seed and Martin 1966

Dynamic Response of Potential Sliding Mass

Dynamic Response: Equivalent Acceleration ConceptDynamic Response: Equivalent Acceleration Concept

• accounts for cumulative effect of incoherent motion in deformable sliding block

• In 1-D, HEA = (τh/σv) g

– Calculate shear stress-time history at slide plane depth and divide each value by the total vertical stress acting at that depth

–MHEA = max. HEA value

–Kmax = MHEA/g

H σvτh

Seed and Martin 1966

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9

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MHEA depends on stiffness and geometry of the MHEA depends on stiffness and geometry of the sliding mass (i.e., its fundamental period)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

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Bray & Rathje 1998kmax = MHEA/g

Principal FindingsPrincipal Findings• HEA represents τhf and thus the seismic loading:

HEA = (τh/σv) g; k = HEA/g & kmax = MHEA/g

• MHEA depends primarily on dynamic response of sliding mass (Tfill) and input rock motions (MHA,rock, Tm)

• Development of pseudostatic method has merit due to simplicity; key is the selection of strengths, k & FS

“Use of k = MHEA/g & FS > 1 with conservative strengths is equivalent to calculating no sliding displacement (i.e., max. driving force never exceeds resisting force)”

““Focus on seismically induced permanent displacementsFocus on seismically induced permanent displacements””

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NewmarkNewmark (1965) Rigid Sliding Block Analysis(1965) Rigid Sliding Block Analysis

• Assumes:– Rigid sliding block – Defined slip surface– Material is rigid-perfectly plastic– Material does not lose strength during shaking– Acceleration-time history defines EQ loading

Key Parameters:• Yield Coefficient (ky) (max. dynamic resistance)• Seismic Coefficient (kmax) (max. seismic loading)• ky/kmax (if > 1 = no displ.; but if < 1 = some displ.)

Rigid DeformableSliding SlidingBlock Block(kmax= (kmax =MHA) MHEA)

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Decoupled & Coupled Sliding AnalysisDecoupled & Coupled Sliding Analysis

Earth Fill

Potential Slide Plane

Decoupled Analysis

Coupled Analysis

Flexible System

Dynamic Response

Rigid Block

Sliding Response

Flexible SystemFlexible System

Dynamic Response and Sliding Response

Max Force at Base = ky ·W

Calculate HEA-time history

assuming no sliding along base

Double integrate HEA-time

history given kyto calculate U

Decoupled vs. Coupled AnalysisDecoupled vs. Coupled Analysis

From Rathje and Bray (2000)

• Insignificant difference for Udecoupled < 1 cm

• Conservative for Udecoupled > 1 m

• Between 1 cm and 1 m, could be meaningfully unconservative

0.1 1 10 100 1000U (cm)

-40

-20

0

20

40

60

80

100

Dis

plac

emen

t Diff

eren

ce (c

m):

U

- U

decoupled

deco

uple

d

c

oupl

ed

k = 0.05

k = 0.1

k = 0.2

y

y

y

(b)

Dec

oupl

ed C

onse

rvat

ive

14

(Bray & Rathje 1998)

Bray & Rathje 1998

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SAFE

UNSAFE

??

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Deep Sliding Case: 1D vs. 2DDeep Sliding Case: 1D vs. 2D

• 1D analysis “averages” accelerations over depth to compute HEA-t history

• 2D analysis “averages” over depth andwidth to compute overall HEA-time history

1D Analysis of 2D Geometry

• Common to analyze large slides as 1D– Large areal extent– Relatively flat slopes

• Use representative SHAKE columns

Rock

Base

Cover

I

III

II

Rock

Base

Cover

I

III

II

Calculate HEA-time history for each column at depth of sliding &calculate mass-weighted average in time of overall HEA-t history

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Accounting for 2D Shallow Sliding EffectsAccounting for 2D Shallow Sliding Effects

• Topographic Effects – MHAcrest ~ 1.3 MHA1D (Use MHAcrest ~ 1.5 MHA1D for steep slopes (>60o); Ashford and Sitar 2002)

• Localized shallow sliding near crest– MHEA ~ MHAcrest ~ 1.3 MHA1D

• Long shallow sliding surface– MHEA ~ 0.5 MHAcrest ~ (0.5)(1.3) MHA1D ~

0.6 MHA1D

Sliding Displacement ProgramsSliding Displacement Programs• USGS computer program available

– Rigid and simplified decoupled sliding block displacement calculations

– Degrading Ky vs. U– Large catalog of EQ ground motions– Can import HEA-time histories from other programs

(e.g., SHAKE) for decoupled analysis– Jibson & Jibson (2003) O-F Report 03-005

• New USGS program will have coupled nonlinear sliding block analysis– Jibson, Rathje & Jibson (2007)

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FEA of Shaking Table Test of Clay SlopeFEA of Shaking Table Test of Clay Slope

0 2 4 6 8 10-2.5

-2

-1.5

-1

-0.5

0

0.5

Time (sec)

Dis

plac

emen

t (in

ches

)

Horizontal displacement (D2)

PLAXISRECORD

0 2 4 6 8 10-3

-2

-1

0

1

Dis

plac

emen

t (in

ches

)

Horizontal displacement (D4)

0 2 4 6 8 10-1.5

-1

-0.5

0

0.5

1

Time (sec)

Horizontal displacement (D5)

0 2 4 6 8 10-1.5

-1

-0.5

0

0.5Vertical displacement (D1)

SummarySummary• First question: will earth materials lose strength?

• If not, evaluate seismic slope stability in terms of displacements

• Newmark-type approach with deformable sliding mass captures:

– Earthquake ground motion – Dynamic resistance of slope– Dynamic response of potential sliding mass

• Some Other Issues:

– Decoupled sliding approximation is reasonable, with possible exception of near-fault ground motions case

– 1D analysis is conservative for deep sliding case and can be corrected for topographic effects for shallow sliding case

– Nonlinear FE analysis can lead to better insights but it is difficult to perform well

• With dynamic analyses the full HEA-time history can be calculated for each input rock motion. With ky, the seismic displacement can be calculated.

• Seismic displacement is an index of performance

• Simplified procedures can provide this index of performance