what is the future of slope stability analysis? is the future of slope stability analysis? (are we...

What is the Future of Slope What is the Future of Slope Stability Analysis?Stability Analysis?

(Are We Approaching the Limits of (Are We Approaching the Limits of Limit Equilibrium Analyses?)Limit Equilibrium Analyses?)

Dr. Delwyn G. FredlundDr. Delwyn G. FredlundUnsaturated Soil Technologies Ltd.Unsaturated Soil Technologies Ltd.

1714 East Heights1714 East HeightsSaskatoon, SK., CanadaSaskatoon, SK., Canada

IntroductionIntroduction Limit Equilibrium methodsLimit Equilibrium methods of slices of slices

have been good for the geotechnical have been good for the geotechnical engineering profession since the engineering profession since the methods have produced financial methods have produced financial benefit benefit

Engineers are often surprised at the Engineers are often surprised at the results they are able to obtain from results they are able to obtain from Limit Equilibrium methodsLimit Equilibrium methods

So Why Change?

There are There are Fundamental LimitationsFundamental Limitations with Limit Equilibrium Methods of with Limit Equilibrium Methods of

SlicesSlices

?

?

The boundaries for a FREE BODY DIAGRAM are not known

-The SHAPE for the slip surface must be assumed-The LOCATION of the critical slip surface must be

found by TRIAL and ERROR

SHAPESHAPE and and LOCATIONLOCATION are the are the driving force for a paradigm shiftdriving force for a paradigm shift

Objectives of this Presentation:Objectives of this Presentation:❑ To show the To show the gradual changegradual change that is that is

emerging in the way that slope emerging in the way that slope stability analyses can be undertakenstability analyses can be undertaken

❑ To illustrate the To illustrate the benefitsbenefits associated associated with improved procedures for the with improved procedures for the assessment of stresses in a slopeassessment of stresses in a slope

Outline of PresentationOutline of Presentation Provide a brief Provide a brief SummarySummary of common of common

Limit Equilibrium methodsLimit Equilibrium methods along with along with their limitations (2-D & 3-D)their limitations (2-D & 3-D)

Take theTake the FIRSTFIRST step forward through step forward through use of an use of an independent stress analysisindependent stress analysis

Take the Take the SECONDSECOND step forward step forward through use of through use of Optimization Optimization TechniquesTechniques

Is a Limit Equilibrium Analysis an Upper Bound or Lower Bound

Solution? Limit Equilibrium Methods primarily

satisfy the requirements of an upper bound type of solution

Reason: the shape of slip surface is selected by the analyst, and thereby a displacement boundary condition is imposed

Limit Formulations from Plasticity A Lower Bound solution satisfies:

Equilibrium equationStress boundary conditionsFailure criterion

An Upper Bound solution satisfies:Velocity (displacement) boundary

conditionsCompatibility conditionsFailure criterion

Limit Equilibrium and Finite Element Based Methods of Analyses

WW

W

WW W W W

N

Limit EquilibriumMethod of Analysis

Sm = τa dl

dlσndl

Finite Element Based Method of Analysis

ldτa dl

QUESTION BEFORE US: How can the Normal Stress at the base of a slice be most accurately computed?

Consider the Free Body Diagrams used to calculate the Normal Stress?

Assumption for all Limit Equilibrium Analysis

Soils behave as Mohr-Coulomb materials (i.e., soil has friction, φ', and cohesion, c')

Factor of safety, Fs, for the cohesive component is equal to the factor of safety for the frictional component

The factor of safety is the same for all slices

( )[ ]m

s

n

s

SF

uF

c =−+ βφσβ 'tan'

Summary of Available Equations Associated with a Limit Equilibrium

Analysis

Equations (knowns): QuantityMoment equilibrium nVertical force equilibrium nHorizontal force equilibrium nMohr-Coulomb failure criterion n

4n

Unknowns: Quantity Total normal force at base of slice n Shear force at the base of slice, Sm n Interslice normal force, E n-1 Interslice shear force, X n-1 Point of application of interslice force, E n-1 Point of application of normal force n Factor of safety, Fs 1

6n-2

Summary of Unknowns Associated with a Limit Equilibrium Analysis

Definition of Factor of Safety, Fs

“That factor by which the shear strength parameters must be reduced to bring a soil mass into a state of limiting equilibrium along a given slip surface”

ss F'tanand

F'c φ

Forces Acting on Each Slice

Focus on Sm

b

y

x

SmXR

E R

EX L

Slip surface

Ground Ground surfacesurface

W h

R

N = σn ββ

f

N

L

Phreatic linePhreatic line

Mobilized Shear Force, Mobilized Shear Force, SSmm forforSaturated-Unsaturated SoilsSaturated-Unsaturated Soils

( )[ ] ( )s

wa

s

an

s

mF

uuF

uF

cSβφβφσβ btan'tan' −+−+=

Only new variable required for solving saturated-unsaturated soils problems is the shear force mobilized

φφbb = Friction angle with respect to matric suction= Friction angle with respect to matric suction

uuaa = Pore-air pressure = Pore-air pressure

uuww = Pore-water pressure= Pore-water pressure

[ ]

Moment equilibrium, Fm:

Force equilibrium , Ff:

∑ ∑

∑

−

−+

=NfWx

'tanR'tan

tanuNR'cF

b

w

m

φφφββ

∑∑

−+

=α

αφφφβαβ

sin

cos'tan'tan

tancos'

N

uNcF

b

w

f

Pore-air pressures are assumed to be zero gauge

Normal force at base of slice:

Limit Equilibrium methods differ in terms of how (XR-XL) is computed and overall statics satisfied

Reality: The Limit Equilibrium problem is indeterminate: Can apply an assumption (Historical solution) Can utilize additional physics (Future solution)

( )

F

Fu

FcXXW

N

b

wLR

'tansincos

tansinsin'

φαα

φαβαβ

+

+−−−=

σx

b

a

Area = Interslice normal force (E)

width of slice, β

σxτxy

σy

Distance (m)

Ele

v atio

n (m

)

τxy

b

a

Area = Interslice shear force (X)

Vertical slice

Distance (m)

Ele

vati o

n ( m

)

∫=b

axy dyX τ∫=

b

axdyE σ

Stresses on the Boundary Between Slices

Morgenstern & Price, 1965

Summary of Limit Equilibrium Methods and Assumptions

Method Equilibrium Satisfied

Assumptions

Ordinary Moment, to base E and X = 0 Bishop’s Simplified

Vertical, Moment E is horizontal, X = 0

Janbu’s Simplified

Vertical, Horizontal

E is horizontal, X = 0, empirical correction factor, f0 , accounts for interslice shear forces

Janbu’s Generalized

Vertical, Horizontal

E is located by an assumed line of thrust

Spencer Vertical, Horizontal, Moment

Resultant of E and X are of constant slope

Forces Acting on One Slice in Ordinary or Conventional Method

hW

b

β α

Ν = σ β n

Ν

Sm

Forces Acting on

One Slice in Bishop’s

Simplified and Janbu’s Simplified Methods

hW

b

β α

Ν = σ β n

Ν

Sm

ER

EL

Summary of Limit Equilibrium Methods and Assumptions

Direction of X and E is the average of the ground surface slope and the slope at the base of a slice

VerticalHorizontal

Lowe and Karafiath

Direction of X and E is parallel to the ground

VerticalHorizontal

Corps of Engineers

Direction of E and X is defined by an arbitrary function. Percent of the function required to satisfy moment and force equilibrium is called λ

VerticalHorizontalMoment

Morgenstern-Price, GLE

AssumptionsEquilibriumSatisfied

Method

Forces Acting on One Slice in Spencer’s, Morgenstern-Price, andGLE Methods

hW

b

β α

Ν = σ β n

Ν

Sm

ER

EL

XR

XL

Various Various Interslice Interslice

Force Force FunctionsFunctions

Proposed byProposed byMorgenstern & Morgenstern &

Price (1965)Price (1965)

Spencer’s

Forces Acting on One Slice in Corps of Engineers, and Lowe - Karafiath Methods

hW

b

β α

Ν = σ β n

Ν

Sm

EREL

XR

XL

Wilson and Wilson and Fredlund (1983)Fredlund (1983)Used a finite element Used a finite element stress analysis (with stress analysis (with gravity switched on) gravity switched on) to determine a shape to determine a shape

for thefor the Interslice Interslice Force FunctionForce Function

Interslice Force Function Interslice Force Function for a Deep-seated Slip for a Deep-seated Slip Surface Through a 1:2 Surface Through a 1:2

SlopeSlope

X = E λ f(x)

Definition of Dimensionless Distance Definition of Dimensionless Distance

f(x) is largest at mid-point

Inflection points near crest & toe

GeneralizedGeneralizedFunctionalFunctional

ShapeShape

where:where:KK = magnitude of function at mid-slope = magnitude of function at mid-slopee = base of natural loge = base of natural logCC = variable to define inflection point = variable to define inflection pointn n = variable to define steepness= variable to define steepnessωω = dimensionless x-position = dimensionless x-position

( ) 2/)(nnCKexf ω−=

Wilson and Fredlund, 1983X = E λ f(x)

Dimensionless DistanceDimensionless Distance

Unique function of “slope angle” for all slip surfaces

““CC” coefficient versus slope angle” coefficient versus slope angle

Unique function of “slope angle” for all slip surfaces

““nn” coefficient versus tangent ” coefficient versus tangent of slope angleof slope angle

Interslice Side Interslice Side Force Ratio, K, Force Ratio, K,

at Mid-slope at Mid-slope versus Depth versus Depth

Factor, DFactor, Dff

Wilson, 1983

X = E λ f(x)

Depth factor, Df

k

Comparison of

Factors of Safety

Circular Slip Surface

0 0.2 0.4 0.61.80

1.85

1.90

1.95

2.00

2.05

2.10

2.15

2.20

2.25

λ

Janbu’s Generalized

Simplified

Bishop

SpencerMorgenstern-Pricef(x) = constant

Ordinary = 1.928

Ff

Fm

Fredlund and Krahn1975

Fact

or o

f saf

ety

Moment and Force Limit Equilibrium Factors of Safety

For a Circular typeslip surface

Moment limit equilibrium analysis

Force limit equilibrium analysis

Fredlund and Krahn, 1975Lambda, λ

Fact

or o

f saf

ety

Force and MomentLimit equilibriumFactors of Safety

for a planar toe slip surface



Lambda, λ

Fact

or o

f saf

ety

Krahn 2003

Force and MomentLimit equilibrium

Factors of Safety for a composite slip surface



Lambda, λ

Fact

or o

f saf

ety

Fredlund and Krahn 1975

Force and MomentLimit equilibrium Factors of Safety

for a “Sliding Block” type slip surface



Lambda, λ

Fact

or o

f saf

ety

Krahn 2003

Extensions of Methods of Extensions of Methods of Slices Slices totoThree-dimensional MethodsThree-dimensional Methods of of ColumnsColumns

Hovland (1977)Hovland (1977) – 3-D of Ordinary – 3-D of Ordinary Chen and Chameau (1982)Chen and Chameau (1982) – 3-D of – 3-D of

Spencer Spencer Cavounidis (1987)Cavounidis (1987) – 3-D F – 3-D Fss > 2-D F > 2-D Fss Hungr (1987)Hungr (1987) – 3-D of Bishop – 3-D of Bishop

SimplifiedSimplified Lam and Fredlund (1993)Lam and Fredlund (1993) – 3-D with – 3-D with

f(x)f(x) on all 3 planes; 3-D of GLE on all 3 planes; 3-D of GLE

ShapeShape and and Location Location Become Even More Become Even More Difficult to Define Difficult to Define in 3-Din 3-D

Two Perpendicular Sections Two Perpendicular Sections Through a Through a 3-D3-D Sliding MassSliding Mass

Section Parallel to Movement Section Perpendicular to Movement

Free Body Diagram of a Free Body Diagram of a Column Column with All Interslice Forceswith All Interslice Forces

Parallel

Perpendicular

Base

Interslice Force FunctionsInterslice Force Functions for for Two of the DirectionsTwo of the Directions

X/E V/P

First Step ForwardFirst Step ForwardQuestion:Question: Is the Is the Normal StressNormal Stress at the base of at the base of

each slice as accurate as can be each slice as accurate as can be obtained?obtained?

Is the Is the Normal StressNormal Stress only only dependent upon the forces on a dependent upon the forces on a vertical slice?vertical slice?

Improvement of Normal Stress Computations

Fredlund and Scoular1999

Limit equilibrium and finite element

normal stresses for a toe slip surface

From limit equilibrium analysis

From finite element analysis


normal stresses for a deep-seated slip surface




normal stresses for an anchored slope



To illustrate procedures for combining a finite element stress analysis with concepts of limiting equilibrium. (i.e., finite element method of slope stability analysis)

To compare results of a finite element slope stability analysis and conventional limit equilibrium methods

Using Limit Equilibrium Concepts in a Finite Element Slope Stability Analysis

Objective:

The complete stress state from a finite element analysis can be “imported” into a limit equilibrium framework where the normal stress and the actuating shear stress are computed for any selected slip surface

Hypothesis

Assumption: The stresses computed from“switching-on” gravity are more reasonable than

the stresses computed on a vertical slice

Manner of “Importing Stresses” from a Finite Element Analysis into a Limit

Equilibrium Analysis

σn

Finite Element Analysis for Stresses

Limit Equilibrium Analysis

σnτm

Mohr Circleτm

IMPORT:Acting Normal StressActuating Shear Stress

Limit Equilibrium Analysis

Finite Element Analysis for Stresses

Bishop (1952) - stresses from Limit Equilibrium methods do not agree with actual soil stresses

Clough and Woodward (1967) - “meaningful stability analysis can be made only if the stress distribution within the structure can be predicted reliably”

Kulhawy (1969) - used normal and shear stresses from a linear elastic analysis to compute factor of safety “Enhanced Limit Strength Method”

Background to Using Other Stress Analyses

Stress LevelRezendiz 1972

Zienkiewicz et al 1975

Strength & Stress LevelAdikari and Cummins 1985

Enhanced limit methods(finite element analysiswith a limit equilibrium

Finite Element Slope Stability Methods

Direct methods(finite element analysis only)

Strength LevelKulhawy 1969

F [ ]

-

-

Z =

1 3

1 3

∑′ ′

′ ′∑

∆

∆

L

L

f

σ σ

σ σ

{ }F = (c + tan )

-

-

c + tan

A 1 3

1 3

′ ′ ′∑′ ′

′ ′′ ′ ′∑

σ φσ σ

σ σσ φ

∆

∆

L

L

f

*F =

(c + tan )

K

∑ ′ ′ ′

∑

σ φ

τ

∆

∆

L

L

Definition of Factor of Safety

Load increaseto failure

Strength decreaseto failureanalysis)

Differences and Similarities Between the Finite Element Slope Stability and Conventional Limit

Equilibrium

DifferencesSolution is determinateFactor of safety equation is linear

SimilaritiesStill necessary to assume the shape of the

slip surface and search by trial and error to locate the critical slip surface

Why hasn’t Finite Element Slope Stability Method been

extensively used?

Difficulties and perceptions related to the stress analysis

Inability to transfer large amounts of data and find needed information

Now: Microcomputer have dramatically changed our ability to combine Finite Element and Limit Equilibrium analyses

Definition of Factor of Safety

Kulhawy (1969)

where: Sr = resisting shear strength or

Sm = mobilized shear force

∑

∑=m

rFEM S

SF

βφσ }'tan)u('c{S wnr −+=

Actuating ShearActuating Shear

Normal StressNormal Stress

Analysis Study Undertaken by Fredlund and Scoular (1999)

Adopted the Kulhawy (1969) procedure Used Sigma/W and Slope/W Poisson’s ratio range = 0.33 to 0.48 Elastic modulus, E = 20,000 to 200,000 kPa Cohesion, c' = 10 to 40 kPa Friction, φ' = 10 to 30 degrees Compared conventional Limit Equilibrium

results with Finite Element slope stability results

Location of Center of a Section along the Slip Surface within a Finite Element Analysis

x

y

x-Coordinate

y-C

oord

inat

eSlip Surface

Finite Element

(r, s)

s

r

Fictitious slice defined withthe Limit Equilibrium analysis

Center of the base of a slice

(x, y)

Presentation of Finite Element Slope Stability Results

Conditions Analyzed: Dry slope Piezometric line at 3/4 height,

exiting at toe Dry slope, partially submerged Piezometric line at 1/2 height and

submerged to mid-height

Selected 2:1 Free-Standing Slope with a Piezometric Line Exiting at the Toe of the Slope

20 40 60 80 100 120

20

40

60

80

0

Crest

Piezometric Line

Toe

21

x - Coordinate (m)

Note: Dry slope with & without piezometric

line

y - C

oord

inat

e (m

)

Selected 2:1 Partially Submerged Slope with a Horizontal Piezometric Line at Mid-Slope

20 40 60 80 100 120

20

40

60

80

0

Crest

Toe

21

x - Coordinate (m)

WaterPiezometric Line

y -

Coo

rdin

ate

(m)

Note: Dry slope with & without piezometric line

050

100

150200

250

300

20 30 40 50 60 70x-Coordinate (m)

Act

ing

and

rest

rict

ing

shea

r st

ress

(kPa

) Crest

Toe

Shear Strength

Shear Force

Poisson Ratio, µ = 0.33

Shear Strength and Shear Force for a 2:1 Dry Slope Calculated Using the Finite Element Slope Stability Method

Local and Global Factors of Safety for a 2:1 Dry Slope

0

1

2

3

4

5

6

7

20 25 30 35 40 45 50 55 60 65 70x-Coordinate

Fact

or o

f Saf

ety

Crest

Toe

Local F (µ Local F (µ= 0.33)

Bishop Method, F = 2.360

= 2.173

Global Factors of SafetyBishop 2.360Janbu 2.173GLE (F.E. function) 2.356Fs (µ = 0.33 ) 2.342Fs (µ = 0.48 ) 2.339Ordinary 2.226

s

Janbu Method, Fs

s

s

Fs = 2.342

Fs = 2.339 = 0.48)

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

0.0

0.5

1.0

1.5

2.0

2.5

0 5 10 15 20 25Stability Number, [γ Htan φ′ /c′ ]

Fact

or o

f Saf

ety c′ = 20kPa

c′ = 10kPa

c = 40kPa

Fs ( µ = 0.33)Fs ( µ = 0.48)

Fs (GLE)

2:1 Dry Slope′

Factor of Safety Versus Stability Coefficient for a 2:1 Dry Slope as a Function of φ′

0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.02 0.04 0.06 0.08 0.10 0.12Stability Coefficient, [c ′/γ H ]

Fact

or o

f Saf

ety

φ ′= 30°

φ′ = 10°

φ′= 20°

2:1 Dry Slope

sFs ( µ = 0.33)F (µ = 0.48)

Fs (GLE)

s

Factor of Safety Versus Stability Coefficient as a Function of φ′ for 2:1 Slope with a

Piezometric Line

0.0

0.4

0.8

1.2

1.6

2.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12Stability Coefficient, [c'/ γ H ]

Fact

or o

f saf

ety

φ′ = 30°

φ′ = 20°φ′ = 10°

2:1 Slope with piezometric line

Fs ( µ = 0.33)Fs ( µ = 0.48)

Fs (GLE)

Location of the Critical Slip Surface for a Slope with a Piezometric Line with Soil Properties of c' = 40 kPa and φ ' = 30°

70

10 20 10060504030 908070 110

50

60

40

10

20

30

x - Coordinate (m)

80

GLE (F.E. function)Fs (µ = 0.33)

Fs (µ = 0.48)

Method X Y R Factor of safetyGLE (F.E. Function) 58.5 56.0 37.9 1.741Fs (µ = 0.33) 57.5 49.5 34.7 1.627Fs (µ = 0.48) 57.5 53.0 37.8 1.661

Y- C

oord

inat

e (m

)

Location of the Critical Slip Surface for a Slope with a Piezometric Line where the

Factor of Safety is Closest to 1.0

70

10 20 10060504030 908070

50

60

40

10

20

30

110x - Coordinate (m)

80

Fs (µ = 0.4 8 )Fs (µ = 0.33)

GLE (F.E. function )

s

Method X Y R Factor of safetyGLE (F. E Function . 63.5 59.0 39.6 1.102Fs (µ = 0.33) 63.0 59.0 41.5 1.076F (µ = 0.48) 61.5 59.5 42.3 1.100

y- C

oord

inat

e (m

)

Factor of Safety Versus Stability Coefficient as a Function of φ′ for 2:1 Dry

Slope, 1/2 Submerged

0.0

0.5

1.0

1.5

2.02.5

3.03.5

0.00 0.02 0.04 0.06 0.08 0.10 0.12Stability Coefficient, [c ′ / γ H]

Fact

or o

f Saf

ety

φ′ = 20°

φ ′ = 10°

2:1 Dry slope, one-half submergedφ′ = 30°

Fs ( µ = 0.33)Fs ( µ = 0.48)

Fs (GLE)

Factor of Safety Versus Stability Coefficient as a Function of φ′ for 2:1 Slope Half

Submerged with Piezometric Line

0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.02 0.04 0.06 0.08 0.10 0.12Stability Coefficient , [c′ / H ]

Fact

or o

f Saf

ety

φ′ = 30°

φ′= 10°

φ′ = 20°

2:1 Slope, one-half submerged

γ

Fs ( µ = 0.33)Fs ( µ = 0.48)

Fs (GLE)

10 20 10060504030 908070 110

70

50

60

40

10

20

30

80

Fs (µ = 0.33)

Fs (µ = 0.48) GLE (F.E. Function)s

Method X Y R Factor of safetyGLE (F.E. Function 58.0 58.5 40.2 2.303Fs (µ = 0.33) 52.5 50.5 31.8 2.259F (µ = 0.48) 51.5 51.5 31.0 2.273

Location of the Critical Slip Surface for a Half Submerged Slope where the Soil Properties are c' = 40 kPa and φ ' = 30°

x - Coordinate (m)

y - C

oord

inat

e (m

)

Conclusions from Step 1 Forward Normal and Actuating Shear stresses from a

finite element analysis appears to provide a more accurate representation of the stress state in a slope

The Enhanced Limit method by Kulhawy (1969) appears to open the way to simulate more complex slope stability problems

Enhanced Limit methods can readily be used in routine engineering practice

Global factors of safety appear to be essentially the same for most simple slopes

Selection of Poisson’s ratio has some effect on the Enhanced Limit factor of safety

Factors of Safety appear to differ slightly for:Low cohesion valuesHigh angles of internal friction

How do the Results from Enhanced Limit Methods Compare to Limit

Equilibrium Methods?

Local Factors of Safety can also be computed by the Enhanced Limit Method

Second Step ForwardSecond Step Forward

Question:Question: Is it possible for the computer to Is it possible for the computer to

determine the determine the ShapeShape of the critical of the critical slip surface?slip surface?

Is it possible for the computer to Is it possible for the computer to determine the determine the Location Location of the of the critical slip surface?critical slip surface?

Improvement on Shape and Location

Ha and Fredlund2002

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

Hypothesis

Assumption: The stresses computed from“switching-on” gravity can be used to

represent the stress state in the soil mass

Slope Stability Analysis Using Slope Stability Analysis Using Dynamic ProgrammingDynamic Programming Combined Combined

with a Finite Element Stress Analysiswith a Finite Element Stress Analysis Dynamic Programming (DP) oDynamic Programming (DP) optimization ptimization

techniques ftechniques for slope stability analysis or slope stability analysis (Spencer‘s Method) was introduced by (Spencer‘s Method) was introduced by Baker (1980)Baker (1980)

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

Definition of Factor of SafetyDefinition of Factor of Safety

Smooth curve

Discrete form

(1)

(2)

"Stage"

B "State point"

"n+1"

A

Y

"i""1"

Rii i+1

k

jSi

jk

...i i+1...

Fs = Σ ( Shear Strength) / Σ (Actuating Shear Stress)

∫

∫=

B

A

B

Af

s

dL

dL

Fτ

τ

∑

∑

=

=

∆

∆=

n

iii

n

iiif

sL

LF

1

1

τ

τ

Definition of Definition of “Return Function“;“Return Function“; GG

stage "i+1"stage "i"

l ij

l ij

fτ

τfij

ij

jσijτ

ij

θ k

ijσijτ

Element (ij)

Element (ij)

R = Resisting Shear Strength: S = Actuating Shear Stress

Fs = Σ ( Shear Strength) / Σ (Actuating Shear Stress)

Difficult to Difficult to minimize !minimize ! ∑

=∆−=

n

iiisfi LFG

1)( ττ

dLFG sB

Af )( ττ −= ∫

∑=

−=n

iisi SFRG

1)(

ActuatingActuating Shear Forces and Shear Forces and Resisting Resisting Shear Shear

S = Actuating Shear Stress

R = Resisting Shear Strength

∑∑==

==∆=ne

ijijij

ne

ijijiii l SLS

11

ττ

∑∑==

==∆=ne

ijijf

ne

ijijifi lRLR

iji11

ττ

ijbijwaijaij

ne

ijiji luuucR }tan)(tan)({ '

1

' φ φσ −+−+=∑=

Definition of Definition of “Optimal Function““Optimal Function“ ::Minimum Value of Minimum Value of “Return Function““Return Function“

= the optimal function obtained at point {k} of stage [i+1],

= the optimal function obtained at point {j} in stage [i], and

= the return function calculated when passing from the state

point {j} in stage [i] to the state point {k} in stage [i+1].

where:

Introduce an “optimal function”,

H = Optimal Function

G = Return Function∑=

−==n

iisi SFRGG

1min )(minmin

)( jH i

),()()(1 kjGjHkH iii +=+

)(1 kHi+

)( jHi

),( kjGi

Boundary Conditions of Boundary Conditions of “Optimal Function““Optimal Function“

At the initial stage, (i=1) :

At the final stage, ( i = n+1) :

where:

= the number of state points in the final stage

H = Optimal Function

0)(1 =jH 1...1 NPj =

),()()(1 kjGjHkH nnn +=+

∑=

+ −==n

iisimn SFRGkH

11 ).()( ...1= n+1 NPk

1+nNP

The Minimum The Minimum (or Optimal)(or Optimal) Travelling Time ProblemTravelling Time Problem

DYNAMIC PROGRAMMING SOLUTION

1

1

6

48

7

511

114

12

1H1 (1) = 0

9

2

7

4

7H1(1) =13A

H (2)= 8

123

10

B

567 4STAGE NUMBER

1234567

d=(4, 2)

3G (1,2) = 3

3

10

5 2

43 25

2 827

22441

55

32

B A

THE MINIMUM TRAVELLING TIME PROBLEM

Analytical Scheme of the Analytical Scheme of the Dynamic ProgrammingDynamic Programming Method Method

Entry point

"1"

"Initial

A

B

point"

Y

"State point"

...i i+1...XB

B

"n+1" X...Stage No.

"Exit point"

Si

"Grid element"

boundary""Searching

i i+1k

Searching gridj

Ri

"Final point"j

k

Kinematical RestrictionKinematical Restriction

5S 6

R

S R3

SR5

4

4RS2

23 BR6

S

RS1 1

A

X

Y

R1 1SR 22 S

i SiR

R nSn

...

...

Kinematical Restriction

Rii i+1

k

jSi

Eliminated> <> <

> <

> <

μ = 0.33DYNPROG = 1.02

Enhanced = 1.13

Bishop; M-P = 1.17

Distance, m

Elev

atio

n, m

Example of a Example of a Homogeneous Homogeneous SlopeSlope

Example of a Example of a HomogeneousHomogeneous Slope Slope

μ = 0.33DYNPROG = 1.02

Bishop; M-P = 1.17

Enhanced = 1.13


μ = 0.33

Fact

or o

f Saf

ety

Fact

or o

f Saf

ety

Stability Coefficient, C/Stability Coefficient, C/γ γ HH

μ = 0.48

Fact

or o

f Saf

ety

Fact

or o

f Saf

ety

Stability Coefficient, C/Stability Coefficient, C/γ γ HH


Example of a Example of a HomogeneousHomogeneous Slope Slope

μ = 0.33

μ = 0.48

Fact

or o

f Saf

ety,

Fact

or o

f Saf

ety,

DYN

PRO

GD

YNPR

OG

Factor of Safety,Factor of Safety, Morgenstern-PriceMorgenstern-Price

μ = 0.33

Distance, m

Elev

atio

n, m

Bishop; M-P = 1.64

Enhanced = 1.62

DYNPROG = 1.49

Example of a Example of a Partially SubmergedPartially Submerged Slope Slope

Example of a Example of a Partially SubmergedPartially Submerged Slope Slope

μ = 0.33

Enhanced = 1.62

Bishop; M-P = 1.64

DYNPROG = 1.49

Example of a Example of a MultilayeredMultilayered Slope Slope

Enhanced = 1.10

M-P = 1.14

DYNPROG = 0.96

Distance, m

Elev

atio

n, m

The Re-Analysis of the The Re-Analysis of the Lodalen SlideLodalen Slide

μ = 0.48DYNPROG = 0.975

Bishop = 1.00

Actual

Actual

Enhanced = 0.997

The Re-Analysis of the The Re-Analysis of the Lodalen SlideLodalen Slide

μ = 0.38

Enhanced = 1.02Bishop = 1.00

DYNPROG = 0.997

Actual

Distance, m

Elev

atio

n, m

Actual

Example Problem Involving the Search Example Problem Involving the Search for a for a Convex Critical Slip SurfaceConvex Critical Slip Surface Along Along

a Weak Clay Layera Weak Clay Layer

40Distance (m)

150 5 10 20 25 30 35 45 50 6055 65 70 75 80

Medium layer20

10

5Hard layer

15Weak layer

Elev

ation

( m) 30

25

35

40

45

50

Hard layer : c' = 100 kPa , ' = 30 , = 0 , = 20 kN/m , = 0.35 , E = 100000 kPa.

Medium layer : c' = 20 kPa , ' = 30 , = 0 , = 15 kN/m , = 0.33 , E = 15000 kPa.Weak layer : c' = 0 kPa , ' = 10 , = 0 , = 18 kN/m , = 0.45 , E = 2000 kPa.

DYNPROG

φoφφ

(1.180)φ

φo bφ

o b

o γb

γγ

o

oµ3

3 µµ3

DYNPROG = 1.18

Distance, m

Elev

atio

n, m

Solution of the Concave Slip Surface Problem Using Slope/W Once the

Critical Slip Surface has been Defined1.196

Distance (m)0 10 20 30 40 50 60 70 80

Elevation (m)

0

5

10

15

20

25

30

Elev

atio

n, m

Distance, m

M-P = 1.196

DYNPROG = 1.18

Conclusions from Conclusions from Step 2Step 2 Forward Forward The Shape of the critical slip surface can

be made part of the solution The critical slip surface can be irregular

in shape but must be kinematically admissible

No assumptions 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

Recommendations for the FutureRecommendations for the Future The normal and shear stresses should be The normal and shear stresses should be

studied using studied using more sophisticated stress-more sophisticated stress-strain nonlinear and elasto-plastic modelsstrain nonlinear and elasto-plastic models including Poisson‘s ratio effectsincluding Poisson‘s ratio effects

Study of Study of “true““true“ 3-dimensional modelling 3-dimensional modelling of slopes and past Case Historiesof slopes and past Case Histories

Dynamic Programming should be applied Dynamic Programming should be applied to to Lateral Earth PressureLateral Earth Pressure and and Bearing Bearing CapacityCapacity problems problems

Delwyn G. FredlundDelwyn G. Fredlund

1.196

Distance (m)0 10 20 30 40 50 60 70 80

Elevation (m)

0

5

10

15

20

25

30

what is the future of slope stability analysis? is the future of slope stability analysis? (are we...

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