Slope Stability Analysis Procedures y y - Cal Poly Pomona

Slope Stability Analysis Proceduresy yPresentation for AEG/GI Short Course

UC Riverside, May 12, 2012UC e s de, ay , 0

William Kitch, Cal Poly Pomona

1 © William A Kitch 2012

Overview

Obj ti f t bilit l i Objectives of stability analysis Measures of stability Available computational methods Available computational methods Limit equilibrium methods Stability analysis processStability analysis process Conclusions & questions


Presentation scope

S il ti k Soil or continuous rock– Does not cover rock behavior governed by jointing (topples, key

wedge, etc)

Translational & rotational modes only– No debris flow or spreading analysis

St ti & d t ti t bilit Static & pseudo static stability– No earthquake deformation analysis


Objectives of Stability analysis

D t i d f i ti l Determine adequacy of an existing slope Evaluate effectiveness of proposed slope remediation Back calculate average shear strength of a slope know Back calculate average shear strength of a slope know

to be in failure Design an engineered slopeg g


Measures of stability

F t f f t Factor of safetysF

– where

shear strength availables ilb i h t

– note

equilbrium shear stress

Ms

f

resisting

driving

MsM

– Definition based on shear strength and shear stress is the only consistent definition


Recommended factors of safety

Cornforth (2005)

Minimal Study Normal StudyLandslide size Borings Acceptable F Borings Acceptable F

Cornforth (2005)

Very Small 1 or none 1.50 1 1.50Small 1 1.50 2 1.35Medium 2 1.40 4 1.25Large 3 1.30 6 1.20Very Large 4 1.20 8 1.15

Uncertainty of analysisCost of failure Small Large

Duncan and Wright (2005)

Cost of failure Small LargeRepair costs y incremental cost of safer design 1.25 1.5Repair costs >> incremental cost of safer design 1.5 2.0 or more6 © William A Kitch 2012

Agency requirements

US Army Corps of Engineers (1970)

Required Factor of safety for given conditionType of slope End of

constructionLong-term steady

state seepageRapid Drawdown

US Army Corps of Engineers (1970)

construction state seepageDams, levees, dikes & other embankments

1.3 1.5 1.0 – 1.2

embankments

Typical Southern California Agency RequirementsStatic Static with pseudo static earthquake load Temporary slopes1.5 1.1 1.25


Limitations of Factor of safety

D t t i i f ti b t th i bilit Does not contain information about the variability or uncertainty of shear strength or mobilized shear stress

Probability of

bilit

y D

ensi

ty

Probability of failure

Pro

bab

Stresss s Same factor of safety can have different reliability Probabilistic methods are available to estimate reliability

Stresss s

Probabilistic methods are available to estimate reliability of slopes


Available computational methods

Li it ilib i th d Limit equilibrium methods– Most common approach– Requires only simple Mohr-Coulomb soil modelq y p– Cannot model progressive failure– Cannot compute displacements

Must search for critical surface– Must search for critical surface

Finite element methods– Do not need to search for critical surface, analysis automatically

finds it– Must have a complete stress-strain model for soil– Can compute displacementsp p– Can model progressive failure


Comparison of limit equilibrium and finite element methods

Limit equilibrium analysis F = 1.75

Finite element analysis F = 1.74


L ti l f il f ith FE l iLocating complex failure surfaces with FE analysis

1 1.0ussu1

su2

2

1.0us

1 0 6s 1

2

0.6u

u

ss

1

2

0.2u

u

ss

Griffiths & Lane (1999)11 © William A Kitch 2012

Limit Equilibrium Approach

G l h f f il f ( l i l i l )1. General shape of failure surface (planar, circle, non-circular) assumed

– Driven by geometry and geology of problemDetermines formulation of the analysis– Determines formulation of the analysis

2. Specific failure surfaced chosen3. Some or all of static equilibrium conditions used to compute

eq ilibri m shear stress on fail re s rfaceequilibrium shear stress on failure surface, 1. Fx = 02. Fy = 0

M 03. M = 0

4. Available shear strength, s, along failure surface computed using Mohr-Coulomb failure criteria (c & )

5. Factor of safety computed, F = s/6. Back to step 2, continue until Fmin found


1 unknown, 1 equation, FA = 0

Simple planar failure example for = 0 conditions

2

F 0

2

2 tanHW

H/tanW

FA = 0

sinT W H

2 cos2

H

2 H

H/sin

N

T2 cos sin2

HH

sin cosH N

weak clay seam withundrained strength, su

sin cos2

H

sF 2 us critical surfaceF

sin cos

u

H critical surface


Simple LE methods

M d l i l b t i t t Model simple but important cases Statically determinate problems Can solve directly for F without assumptions about Can solve directly for F without assumptions about

distribution of stress within failure mass Most common and useful methods

– Planar or single wedge– Infinite slope

Swedish slip circle– Swedish slip circle


2 unknowns, F & 2 equations

– FA = 0Infinite slope analysis

FA = 0

FA 0– FB = 0

FA 0

FB = 0D

sinT W sinWl cos sinD

cosW

From Mohr-Coulomb

t

D

cosN W cosWl 2cosD

2DW tan 's c 2cos tan 'c D

2cos tan 'cos sin

s c DFD

EL

ER

For c = 0l

T

2cos tan 'cos sin

DFD

tan 'tan

For = 0, s = su

costl

W tD

NT

cos sinusF

D


1 unknown, F 1 equation, MO = 0

Swedish slip circle for = 0 conditions

M 0O MO = 0

W

a r lr Wa

Wal1

su1 su

Shear strength

W

l

rl

s sl2

su2

us s

sF

us rlWa

l2

resisting

driving

MF

M r s l ui ir s l

FWa


Summary of simple LE methods

Procedure Assumptions Equations Variables solved forocedu e ssu pt o s quat o sused

a ab es so ed o

Infinite Slope • Infinitely long slope• slip surface parallel

• F = 0• F = 0

• Factor of safety• on failure surface

to surfaceSwedish slip circle

• = 0 • Circular slip surface

• MO = 0 • Factor of safety


Methods of slices

Wh 0O When 0

Must determine ' ' tan 's c r

c1, 1

Cannot use simply MO = 0c2, 2

zi Vi

i

zi+1Wi

i

Ei

Vi

Ei+1

Ti

Vi+1

i+1

li

Ni


Equation/unknown count

x Unknowns

F, factor of safety n values of N zi+1

Wi

zi

E

Vi

x

z

n values of Ni

n1 values of Ei

n1 values of Vi

n 1 values of z

i+1Ei

Vi+1

Ei+1

n1 values of zi

Total: 4n2 unknowns Equilibrium equations

1 M

lNi

Ti

1, MO

n, Mi

n, Fx Must make assumptions to

solve problem

li

n, Fz

Total: 3n + 1 equations

solve problem Assumptions made affect

accuracy of solution19 © William A Kitch 2012

1 unknown, F 1 equation, MO = 0

Ordinary method of slices

Assumptions– Circular surface

Ignore all side forces

Wi Ignore side forces

Ignore side forces

– Ignore all side forces Unknown

– FE ti d

forcesHi

Equations used– MO = 0

SolutionNi

Ti

li

Can directly solve for F Simple to implement Generally conservative

' cos tan 'sin

c l W ulF

W

l Generally conservative

Accuracy poor when pore pressure high

cos pore pressure on base of slice

W Hlu


1+n unknowns, F, Ni

1 equation, MO = 0

Simplified bishop method

x

MO 0 n, Fz

Assumptions– Circular surface

Side forces are horizontalzi+1

Wi

zi

Ei

x

z

– Side forces are horizontal Unknown

– 1, FN

Ei+1

– n, Ni

Equations used– MO = 0

li

Ni

Ti

– n, Fz

Solution Requires iterative solution More accurate the OMS

E il i l d i h

i

' cos cos tan 'c l W ul Easily implemented with

spreadsheet

cos sin tan ' /

sinF

FW


Inclusion of external or internal loads

O

rk Wi zi+1

Wi

zi

Ei k Wi

Ri

Ei+1

Ri

Know forces included in

i

ili

Ni

Ti

Know forces included in existing equilibrium equations

Does not increase number of unknowns

Allows for inclusion of– Pseudo static earthquake loads– Forces from pile stabilization

i

unknowns Solution method the same

– External equipment or structural loads


Uses of non-circular surfaces

Surficial Slide

Weak seam

Weak layer23 © William A Kitch 2012

Non-circular surface methods

A ti f i l f i lifi bl Assumption of circular surface simplifies problem By using MO = 0 number of unknowns substantially

reducedreduced Method of slices works for non-circular surfaces

More unknowns More equilibrium equations required

Two broad groups of solutions availableF ilib i F 0 & F 0 Force equilibrium: uses Fx = 0 & Fz = 0

Full equilibrium: satisfies uses Fx = 0, Fz = 0 & M = 0 All still require assumptions about interslice forcesq p


Force equilibrium methods

A di ti i t li f Assume direction interslice forces– Combined with Fx = 0 & Fz = 0 allows for solution for F

Method Interslice force assumption

Simplified Janbu (Janbu et al.1956) Horizontalp ( )

Lowe and Karafiath (1959) Average of slope of top and bottom of slice

Corps of Engineers’ modified Swedish method (US Army Corps of Engineers, 1970)

Parallel to average slope angle


Force equilibrium solutions sensitive to direction of interslice force

Figure 6.15 Influence of interslice force inclination on the computed factor of safety forFigure 6.15 Influence of interslice force inclination on the computed factor of safety for force equilibrium with parallel interslice forces. (Duncan & Wright, 2005)


Full equilibrium methods

Add t ilib i t & f ilib i Add moment equilibrium to x & y force equilibrium Still requires assumptions Two most common methods

– Spencer (1967) Assumes all interslice forces are parallel Solves for F and

– Morgenstern and Price (1965) Assumes V = f (x) E

– f (x) is an assumed function– is a scaling constant

is a scaling constant Solves for F and

– Morgenstern & Price more general– Spencer easier to implement

f(x)

p p When using any full equilibrium method F is insensitive to

assumptions about interslice forces27 © William A Kitch 2012

Comparison of full equilibrium methods

P d A ti E ti V i bl l d fProcedure Assumptions Equations used

Variables solved for

Spencers • Interslice forces parallel

• Fx = 0• F = 0

• Factor of safety• Interslice angle parallel • Fy = 0

• M = 0• Interslice angle • Interslice force• Location of

interslice force• on failure surface

Morgenstern& Price

• Interslice forces related by V = f (x) EF f f ( )

• Fx = 0• Fy = 0 M 0

• Factor of safety• Scaling factor

I t li f• Form of f (x) • M = 0 • Interslice force• Location of

interslice force• on failure surface


Data available from full equilibrium method


Summary of applicability of LE methodsy pp yProcedure Application

Infinite Slope Homogeneous cohesionless slopes and slopes where the stratigraphy restricts the slip surface to shallow depths and parallelstratigraphy restricts the slip surface to shallow depths and parallel to the slope face. Very accurate where applicable.

Swedish Circle = 0

Undrained analyses in saturated clays, = 0. Relatively thick zones of weaker materials where circular surface is appropriate.

Ordinary Method of Slices

Nonhomogeneous slopes and c– soils where circular surface is appropriate. Convenient for hand calculations. Inaccurate for effective stress analyses with high pore pressures.

Simplified Bishop Nonhomogeneous slopes and c soils where circular surface isSimplified Bishop procedure

Nonhomogeneous slopes and c– soils where circular surface is appropriate. Better than OMS. Calculations feasible by spreadsheet

Force Equilibrium procedures

Applicable to virtually all slopes. Less accurate than complete equilibrium procedures and results sensitive to p q passumed interslice force angles.

Spencer Applicable to virtually all slopes. The simplest full equilibrium procedure for computing the factor of safety.

Morgenstern and Price

Applicable to virtually all slopes. Rigorous, well-established complete equilibrium procedure.

From Duncan & Wright (2005)30 © William A Kitch 2012

Critical details of LE analysis

S hi f iti l f Searching for critical surface– Check for multiple minima– Special attention required when using non-circular surfacesp q g

Select appropriate shear strength– Progressive failure

P i ti h f– Pre-existing shear surfaces

Check for invalid solutions– Tensile forces near crest – Steep exit slopes– Non-convergence of solutions


Critical surface search: regional minimum


Critical surface search: local minimum


Critical surface search: multiple modes


Progressive failure


Validity of solution: Tension crack at crest

Al h k li f th t Always check line of thrust


Validity of solution: Tension crack at crest

I t t i k t t if d d Insert tension crack at crest if needed


Steep exit angle

C Can cause – Non-convergence of solution– Very high stresses y g– Negative (tensile stresss)

SolutionU Si lifi d Bi h– Use Simplified-Bishop

– For exit slope to be more shallow


Preparing for stability analysis

D t i i d f l i Determine required scope of analysis Assess risk of project and select appropriate F Build subsurface model Build subsurface model Determine drainage conditions which apply

– End-of-construction undrained condition– Long-term drained condition (both?)

Select appropriate soil strength propertiesId tif t f il f t d l t Identify expect failure surface geometry and select analysis procedure– Circular non-cirucular

Select appropriate analysis procedure39 © William A Kitch 2012

Performing stability analysis

I ti t t ti l f il d i i l d l Investigate potential failure modes using simple models– Identify areas where F is low

Adjust subsurface model and analysis method as needed– Soil properties, geometry, computational method

Thoroughly investigate all potential failure modes with rigorous search for critical surface

– Search all area with local minimum– Consider risk of each significant failure mode

Thoroughly examine computations for critical modes – Check line of thrust

Sanity check results– Similar project, hand computation, other methodp j , p ,


Software (a very short list)

St d l t bilit k Standalone stability packages– STABL/STED– Oasysy– UTEXAS4– LimitState

Integrated packages Integrated packages– RocScience– GeoStudio– gINT– SoilVision


Recommended texts

Ab L W (2002) Sl t bilit d t bili ti Abramson, L. W. (2002). Slope stability and stabilization methods. Wiley, New York.

Cornforth D H (2005) Landslides in Practice - Cornforth, D. H. (2005). Landslides in Practice Investigation, Analysis, and Remedial/Preventative Options in Soils. John Wiley & Sons.

Duncan, J. M., and Wright, S. G. (2005). Soil Strength and Slope Stability. John Wiley & Sons, Hoboken, N.J.


References

Abramson L W (2002) Slope stability and stabilization methods Wiley New York Abramson, L. W. (2002). Slope stability and stabilization methods. Wiley, New York. Cornforth, D. H. (2005). Landslides in Practice - Investigation, Analysis, and

Remedial/Preventative Options in Soils. John Wiley & Sons. Duncan, J. M., and Wright, S. G. (2005). Soil Strength and Slope Stability. John Wiley & Sons,

Hoboken N JHoboken, N.J. Griffiths, D. V., and Lane, P. A. (1999). “Slope stability analysis by finite elements.”

Geotechnique, 49(3), 387–403. Janbu, N., Bjerrum, L., and Kjærnsli, B. (1956). Veiledning ved Løsning av

Fundamenteringsoppgaver (Soil Mechanics Applied to Some Engineering Problems), PublicationFundamenteringsoppgaver (Soil Mechanics Applied to Some Engineering Problems), Publication 16, Norwegian Geotechnical Institute, Oslo.

Lowe, J., and Karafiath, L. (1959). Stability of earth dams upon drawdown, Proceedings of the First PanAmerican Conference on Soil Mechanics and Foundation Engineering, Mexico City, Vol. 2, pp. 537–552.

Morgenstern, N. R., and Price, V. E. (1965). “The analysis of the stability of general slip surfaces”, Geotechnique, 15(1), 79–93.

Spencer, E. (1967). “A method of analysis of the stability of embankments assuming parallel inter-slice forces”, Geotechnique, 17(1), 11–26.

U.S. Army Corps of Engineers (1970). Engineering and Design:Stability of Earth and Rock-Fill Dams, Engineer Manual EM 1110-2-1902, Washington, DC, April.


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Slope Stability Analysis Procedures y y - Cal Poly Pomona