Bai Giang Cua Giao Su Tien Si Fredlund (canada)

Where are We Going in the Analysis of Landslides? Dr. Delwyn G. Fredlund Golder Associates Ltd Saskatoon, SK. Canada

Environmental Influences (Infiltration) Often Affect the Stability of a Slope Precipitation

Runoff

MODELING OF LANDSLIDES 1.) Hydrological modeling (including infiltration) is an integral part of analyzing landslides 2.) Slope stability analyses are undergoing a slow evolution that provides a superior assessment of stress conditions

Deep-Seated Landslides, 3-Dimensional in Shape Focus on “Trigger Mechanism” that Precipitates Movement

White Mud Landslide, Edmonton, Canada

Ground Surface Protection These slopes become green with time but the surface hydraulic conductivity and water storage capacity has been changed through use of a concrete covering with holes

Introduction ‰

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Limit Equilibrium methods of slices have been used extensively for analyzing landslides There is a gradual change emerging in the types of slope stability analyses that can be performed There are benefits associated with improved slope stability methodologies

Why Change? There are Limitations with Limit Equilibrium Methods of Slices The SHAPE & LOCATION boundaries for a FREE BODY DIAGRAM are not known ? ?

SHAPE and LOCATION of the critical slip surface are the driving force for a paradigm shift

What is the Best Procedure to Compute the Normal Force at the Base of a Slice? Limit Equilibrium Method of Analysis W

Finite Element Based Method of Analysis

W W W W dl

Sm = τa dl N

W

W

W

dl τa dl σ nd l

Methods of Slices vary in the manner in which the normal force is computed

Summary of Limit Equilibrium Methods and Assumptions Method Ordinary Bishop’s Simplified Janbu’s Simplified Janbu’s Generalized Spencer

Equilibrium Satisfied Moment, to base Vertical, Moment Vertical, Horizontal Vertical, Horizontal Vertical, Horizontal, Moment

Assumptions E and X = 0 E is horizontal, X = 0 E is horizontal, X = 0, empirical correction factor, f0 , accounts for interslice shear forces E is located by an assumed line of thrust Resultant of E and X are of constant slope

X = E λ f(x) Wilson and Fredlund (1983) Used a finite element stress analysis (with gravity switched on) to determine a shape for the Interslice Force Function

2.25 Ff

2.20 Simplified

2.15

Factor of safety

Comparison of Factors of Safety Circular Slip Surface

Bishop

2.10

Fm

2.05 Janbu’s Generalized

2.00 1.95

Ordinary = 1.928

1.90

Spencer Morgenstern-Price f(x) = constant

1.85 1.80 0

0.2

λ

0.4

Fredlund and Krahn 1975

0.6

Descriptor for the Inter-Slice Force Function

Factor of safety

Moment and Force Limit Equilibrium Factors of Safety For a Circular type slip surface Moment limit equilibrium analysis

Force limit equilibrium analysis

Lambda, λ

Fredlund and Krahn, 1975

Factor of safety

Force and Moment Limit equilibrium Factors of Safety for a planar toe slip surface Force limit equilibrium analysis

Moment limit equilibrium analysis Lambda, λ

Force and Moment Limit equilibrium Factors of Safety for a composite slip surface Factor of safety

Moment limit equilibrium analysis

Force limit equilibrium analysis

Lambda, λ

Fredlund and Krahn 1975

Closer Simulation of Soil Conditions in the Field ‰

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Historically, there has been little concern regarding the initial stress state in the soil Little attention has been given to the history of movements in the soil mass Effects of initial stress state and history of movement are accounted for in the selection of soil parameters There is now the possibility to better simulate the onset of instability conditions through use of stress and seepage (infiltration) analysis

Historical Resistance to Computing Stress States Near Failure from Stress Analysis b

b

X = ∫ τ xy dy

E = ∫ σx dy

a

b width of slice,b

sy

txy sx

sx

a Area = Interslice normal force (E) Distance (m)

Elevation ( m )

Elevation (m)

a

b

t xy a Vertical slice

Area = Interslice shear force (X) Distance (m)

First Change: Importing Stresses from a Finite Element Analysis into a Limit Equilibrium Analysis Framework Finite Element Analysis forStresses Stresses Finite Element Analysis for Limit Equilibrium Analysis

Limit Equilibrium Analysis

Mohr Circle

τm

τm

σn σn

IMPORT: Acting Normal Stress Actuating Shear Stress

Finite Element Slope Stability Methods

Direct methods (finite element analysis only) Enhanced Limit methods (finite element analysis with a limit equilibrium analysis)

Load increase to failure

Strength decrease to failure

Definition of Factor of Safety Strength Level Kulhawy 1969

F = K

∑ (c′ + σ ′ tanφ′ )ΔL ∑ τ ΔL

Stress Level Rezendiz 1972 Zienkiewicz et al 1975 F= Z

∑ [ΔL] ⎧ ⎛⎜⎜σ′ - σ′ ⎞⎟⎟ ⎫⎪ ⎪⎝ 1 3⎠ ⎪ ΔL⎪⎬ ∑ ⎪⎪⎨ ⎪⎪⎛⎜σ′ - σ′ ⎞⎟ ⎪⎪ 3 ⎟⎠f ⎪⎭ ⎪⎩⎜⎝ 1

Strength & Stress Level Adikari and Cummins 1985 F= A

∑{(c′ + σ′ tanφ′) ΔL} ⎧⎪⎛⎜⎜σ′ - σ′ ⎞⎟⎟ ⎫⎪ 1 3 ⎝ ⎠ ⎛⎜c′ + σ′ tanφ′⎞⎟ ΔL⎪⎪⎬ ∑⎪⎪⎨ * ⎠ ⎪ ⎞ ⎝ ⎪⎛ ′ ⎪⎪⎜σ - σ′ ⎟ ⎪ 3 ⎠f ⎩⎝ 1 ⎭⎪

Limit equilibrium and finite element normal stresses for a toe slip surface

From finite element analysis

From limit equilibrium analysis

Differences and Similarities Between the Enhanced Slope Stability and Conventional Limit Equilibrium ‰

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Differences z Solution is determinate z Factor of safety equation is linear Similarities z Still necessary to assume the shape of the slip surface and search by trial and error to locate the critical slip surface

Shear Strength and Shear Force for a 2:1 Slope Calculated Using the Finite Element Slope Stability Method Acting and restricting shear stress (kPa)

300 250

Shear Strength

Crest

200 150 Shear Force

100 50

Toe

Poisson Ratio , μ = 0.33

0 20

30

40 50 x-Coordinate (m)

60

70

Local and Global Factors of Safety for a 2:1 Slope Fs = 2.339

7

Factor of Safety

6 Crest 5 4

Global Factors of Safety Local Fs(μ = 0.48) Local F s(μ= 0.33) Bishop 2.360 Janbu 2.173 Fs = 2.342 GLE (FE function)2.356 Fs(m= 0.33) 2.342 Fs(m= 0.48) 2.339 Ordinary 2.226

3

Bishop Method, Fs= 2.360

2 Janbu Method, Fs= 2.173

1 0 20

25

30

35

40

45 50 55 x-Coordinate

60

65

Toe

70

Factors of Safety Versus Stability Number for a 2:1 Slope as a Function of c'

Factor of Safety

2.5

c = 40kPa c = 20kPa

2.0

c = 10kPa

1.5 1.0 Fs(GLE) Fs( μ = 0.33) Fs( μ = 0.48)

0.5 0.0 0

5 10 15 20 Stability Number, [ γ H tan φ /c ]

25

Next Question to Address ‰

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Is it possible for the analysis to determine the Shape of the critical slip surface? Is it possible for the analysis to determine the Location of the Critical Slip Surface? Optimization Techniques (i.e., Dynamic Programming) can be used to find the pathway which minimizes a function of the shear strength available to the actuating shear stress within a soil mass

Slope Stability Analysis Using Dynamic Programming Combined with a Finite Element Stress Analysis ‰

Baker (1980) Dynamic Programming (DP) optimization techniques for slope stability analysis using Spencer‘s method

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Yamagami & Ueta (1988) and Zou et al. (1995) improved on the Baker (1980) solution by coupling Dynamic Programming with a Finite Element stress analysis

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Pham, H.T.V. (2002) Slope Stability Analysis Using Dynamic Programming Method Combined With a Finite Element Stress Analysis

Definition of Factor of Safety Fs = Σ ( Shear Strength) / Σ (Actuating Shear Stress) Smooth curve

Y

B

j

"Stage"

Ri

A

∫ τ f dL

Si i

Fs =

k i+1

A B

(1)

∫ τ dL

A

j

B

k

Discrete form "State point" n

Fs =

∑ τ fi Δ Li

i =1 n

∑ τ i Δ Li

i =1

"1"

...i i+1...

"i"

"n+1"

(2)

Example of a Homogeneous Slope Bishop; M-P = 1.17

Enhanced = 1.13

DYNPROG = 1.02

μ = 0.33

Factor of Safety

Example of a Homogeneous Slope

μ = 0.48

Stability Coefficient, c /(γ H)

The Re-Analysis of the Lodalen Slide

Elevation, m

Actual

Bishop = 1.00 Enhanced = 1.02 Actual

DYNPROG = 0.997

Distance, m

μ = 0.38

Solution of the Concave Slip Surface Problem Using Morgenstern-Price method of slices once the Critical Slip Surface has been defined using the Dynamic Programming method 1.196

30

M-P = 1.196 Elevation, Elevation (m ) m

25

DYNPROG = 1.18

20

15

10

5

0 0

10

20

30

40

Distance (m) Distance, m

50

60

70

80

Comparison of Methodologies Mesh

Seepage Analysis

Linear-Elastic

Elasto-Plastic

Dynamic Programming Search

Compare shape, location, and factor of safety of critical slip surfaces

Homogeneous Dry Slope: FS~1.3

Local Factor of Safety Distributions

Linear Elastic Elasto-Plastic

Homogeneous Dry Slope: FS ~ 1.0

Local Factor of Safety Distributions

Linear Elastic

Elasto-Plastic

Homogeneous Dry Slope: FS < 1.0

Elastic Elastic

Elastic

Elasto-Plastic

Deformed Shape: FS~1.0

Elastic

Benefits from Dynamic Programming – The SHAPE of the slip surface can be made part of the solution – The critical slip surface can be irregular in shape but must be kinematically admissible. – No assumption is required regarding the LOCATION of the critical slip surface which is defined as an assemblage of linear segments – Force and moment equilibrium equations are satisfied through the stress analysis. – Linear factor of safety equation

Is a Need for Closer Simulation of PoreWater Pressures (Positive & Negative) Associated with Field Conditions ‰

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Actual pore-water pressures and pore-water pressure changes are seldom known for the moment of failure Coupled modeling of soil behavior (stress-deformation) and changes in pore-water pressure can provide a better understanding of the failure mechanism The climate gives rise to a moisture flux boundary condition

3-Dimensional Dynamic Programming Analysis FS = ∫ τ f dA τ dA ml 2

FS = ∑ τ fijk Aijk ijk =1

ml 2

∑τ

ijk =1

ml

ijk

ml

Aijk = ∑ Rij

∑S

ij =1

ij =1

ij z

A1,1,1

A1,1, 2 A1, j ,1

A1,m,1

A1, j , 2

A1, m , 2

Ai ,1,1 Ai ,1, 2

x

y

An ,1,1

An ,1, 2

“1” “i”

“ j”

“i+1”

“j+1”

“l+1”

“m+1”

“1”

1.196

30

Elevation (m )

25

20

15

10

5

0 0

10

20

30

40

Distance (m)

Delwyn G. Fredlund

50

60

70

80