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Lecture 6: Lecture 6: Limit EquilibriumLimit Equilibrium

1 of 38 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Author’s Note:Author’s Note:The lecture slides provided here are taken from the course “Geotechnical Engineering Practice”, which is part of the 4th year Geological Engineering program at the University of British Columbia (V C d ) Th k i i d (Vancouver, Canada). The course covers rock engineering and geotechnical design methodologies, building on those already taken by the students covering Introductory Rock Mechanics and Advanced Rock Mechanics Rock Mechanics.

Although the slides have been modified in part to add context, they of course are missing the detailed narrative that accompanies any l l d h h l lecture. It is also recognized that these lectures summarize, reproduce and build on the work of others for which gratitude is extended. Where possible, efforts have been made to acknowledge th v ri us s urc s ith list f r f r nc s b in pr vid d t th the various sources, with a list of references being provided at the end of each lecture.

Errors, omissions, comments, etc., can be forwarded to the


Errors, omissions, comments, etc., can be forwarded to the author at: [email protected]

Analysis in Geotechnical EngineeringAnalysis in Geotechnical Engineering

(infinite slope, method of slices etc )

LIMIT EQUILIBRIUM

method of slices, etc.)

DISCONTINUUM(distinct element etc )

CONTINUUM(finite element, finite difference, etc.)

(distinct element, etc.)


Limit Equilibrium AnalysisLimit Equilibrium AnalysisTh t id l li d l ti l t h i d i t h i l l i The most widely applied analytical technique used in geotechnical analysis is that of limit equilibrium, whereby force or/and moment equilibrium conditions are examined on the basis of statics. These analyses require information about material strength, but not stress-strain behaviour.g ,

The typical output from a limit equilibrium analysis is the “Factor of Safety”:

FS = resisting forcesdriving forcesdriving forces

DRIVING force

= shear strengthshear stress RESISTING force

FS > 1.0 represents a stable situationFS < 1.0 denotes failure

forceshear stress RESISTING force (i.e.shear strength)


Limit Equilibrium AnalysisLimit Equilibrium Analysis

Although limit equilibrium can be applied to many geotechnical problems, it has been most widely used within the context of slope stability analysis. The analysis of slope stability may be p y y y p y yimplemented at two distinct stages:

Pre-failure analysis – applied to assess safety in a global r fa ur ana ys s app to ass ss saf ty n a g o a sense to ensure that the slope will perform as intended;

Post-failure analysis – also termed back-analysis, should be y yresponsive to the totality of processes which led to failure.

As such analyses are undertaken to provide either a factor of As such, analyses are undertaken to provide either a factor of safety, identify a potential failure surface, or through back-analysis, a range of shear strength parameters at failure.


Analysis in Geotechnical DesignAnalysis in Geotechnical Design

The fundamental requirement for a meaningful analysis should include the following steps of data collection & g pevaluation:

– site characterization (geological di i )conditions);

– groundwater conditions (pore pressure distribution);– geotechnical parameters (strength, deformability, permeability);– primary stability mechanisms (k l f l )

… ideal order of events for a site investigation (Clayton et al., 1995).

(kinematics, potential failure modes).investigation (Clayton et al., 995).


Analysis in Geotechnical DesignAnalysis in Geotechnical DesignG t h i l l i l l t ! Oft fi ld d t i d Geotechnical analyses involve complex systems! Often, field data required for model input (e.g. in situ stresses, material properties, geological structure, etc.) are not available or can never be known completely/exactly. This creates uncertainty, preventing the models from p y y y, p gbeing used to provide design data (e.g. expected displacements).

Such models, however, may prove useful in providing a picture of the mechanisms acting in a particular system. In this role, the model may be g p y yused to aid intuition/judgement providing a series of cause-and-effectexamples.

complicated geology simple geologySituation

complicated geology inaccessible no testing budget

simple geology $$$ spent on site

investigation

Data none

investigation off il h i ( )

complete (?)

predictive(d i )

Approach


failure mechanism(s) (design use)

Limit Equilibrium Limit Equilibrium –– Translational SlidingTranslational SlidingHudson & Harrison (1997)

The solution for translational sliding requires that the strikes of the

Hudson & Harrison (1997)

sliding plane and slope are parallel and that no end restraints are present. Furthermore the solution Furthermore, the solution incorporates the assumptions that the rock mass isimpermeable, the sliding bl k i i id th t th block is rigid, the strength of the slide plane is given by the Mohr-Coulomb shear criterion and that all forces fpass through the centroid of the sliding block.


Limit Equilibrium Limit Equilibrium –– Translational SlidingTranslational Sliding

Hoek & Bray (1981)


Effective StressEffective StressHi h d l ff t th t bilit f l High pore pressures may adversely affect the stability of a slope due to a decrease in effective stresses.

n

W W

Factor = W sin

n’

n f of Safety

Total Normal Stress, n

Pore EffectivePore Pressure

EffectiveStress

’This intergranular stress, or effective stress, may be viewed as the sum of the contact forces di id d b h l


divided by the total area.

Effective StressEffective StressTotal Normal Stress, n

However the total normal stress and

The effective stress cannot be measured; it can only be calculated.constant =

Pore Pressure

EffectiveStress

’

However, the total normal stress and pore pressure can be calculated based on the overburden weight and location of the groundwater table.

’ =n -

W

[n – ]constant

f n

As precipitation infiltrates the ground, the total normal stress remains relatively unchanged but the pore pressure increases decreasing the effective normal stress


unchanged but the pore pressure increases decreasing the effective normal stress acting on the sliding surface (thereby decreasing the frictional strength component).

Planar Analysis Planar Analysis –– Water Pressure Scenarios Water Pressure Scenarios Wyllie & Mah (2004)Wyllie & Mah (2004)

free draining toe

Drainage blocked at toe (uniform pressure on slide plane)

Water table below tension crack (triangular pressure on slide plane)


Limit Equilibrium Limit Equilibrium –– Sensitivity AnalysisSensitivity Analysis

… calculation of factor of safety vs. different depths of water in the tension crack, and vs. horizontal acceleration


acceleration.

Rock Slope Engineering Rock Slope Engineering –– Bench Bench vsvs Pit Wall FailuresPit Wall Failures

A function of scale and joint persistence.

Wyllie & Mah (2004)


Rotational Slip SurfacesRotational Slip SurfacesIn weak materials such as highly weathered or closely fractured rock and In weak materials such as highly weathered or closely fractured rock, and rock fills and soils, a strongly defined structural pattern no longer exists, and the shear failure surface develops along the line of least resistance. These slip surfaces generally take a circular shape.


Limit Equilibrium Limit Equilibrium –– Rotational SlidingRotational SlidingThe fundamental concepts of the limit equilibrium method as applied to rotational/circular slip surfaces are:

slip mechanism results in slope failure;… slip mechanism results in slope failure;

… resisting forces required to equilibrate disturbing mechanisms are found from static

lsolution;

… the shear resistance required for equilibrium is compared with available shear

shallow failureq p

strength in terms of the Factor of Safety;

… the mechanism corresponding to the lowest FS is found by iteration;

deep-seated failure

lowest FS is found by iteration;

… the Factor of Safety is assumed to be constant along the entire slip surface


Morgenstern (1995)

Limit Equilibrium Limit Equilibrium –– Method of SlicesMethod of SlicesThe most commonly used solutions divide the mass above an assumed slip surface into vertical slices. This is to accommodate conditions where the soil accommodate conditions where the soil properties and pore pressures vary with location throughout the slope.

The forces acting on a typical slice, i, are:

W = weight of slicec, = mobilized shear forces at base of slice‘·l = effective normal forces on baseu·l = water pressure force on baseE = side forces exerted by neighboring slices.

ncan

(199

6)


Dun

Method of Slices Method of Slices –– Equations & UnknownsEquations & Unknownsl h Analysing the summation

of forces and/or moments for these slices (i.e. M=0, Fx=0, Fy=0), it M 0, Fx 0, Fy 0), it is soon recognized that there are more unknownsthan equations.

ad (1

992)

As such, the forces involved are statically indeterminate. Various

Brom

hea

methods have therefore been developed to make up the balance between the number of equilibrium number of equilibrium equations and the number of unknowns in the problem. an

(199

6)


Dun

ca

Method of Slices Method of Slices -- AssumptionsAssumptions

can

(199

6)

The various Method of Slices procedures either make assumptions to make the problem determinate (balancing knowns and unknowns), or they do not satisfy all

Dun


p ( g ), y ythe conditions of equilibrium.


The treatment of side forces, is one of the key assumptions that differentiate several of th i M th d f Sli the various Method of Slices procedures.



7

Because different methods use different assumptions to make up the balance between equations and unknowns (to render the problem determinate), some methods do not satisfy all


determinate), some methods do not satisfy all conditions of equilibrium (i.e. force and/or moment).

Ordinary Method Ordinary Method -- ComputationComputation

Th “ di th d” l The “ordinary method” only resolves the forces acting at the base of the slice. This allows for the side forces to b l t d d f th be neglected and for the problem to be easily solved.


Bishop’s Modified Method Bishop’s Modified Method -- ComputationComputation

The “Bishop’s Modified Method” includes interslice side forces interslice side forces, but requires an iterative procedure to determine the Factor of Safety of Safety.

Duncan (1996)


Bishop’s Modified Method Bishop’s Modified Method -- ComputationComputation

1.43 1.51 1.52

1 51 1 521 521.51 1.521.52

Duncan (1996)


Duncan (1996)

General Limit EquilibriumGeneral Limit Equilibrium

1. Different methods use different assumptions to make up the balance between equations and unknowns to render the problem determinate; orrender the problem determinate; or

2. Some methods, such as the ordinary and Bishop’s modified methods, do not satisfy all conditions of equilibrium satisfy all conditions of equilibrium (i.e. force and/or moment).

The degree to which the force polygon closes i di t h th f ilib i i hi d

General Limit Equilibrium (GLE): Method that encompasses key elements of several Method of Slice solutions calculating one Safety Factor based on

indicates whether force equilibrium is achieved.

several Method of Slice solutions, calculating one Safety Factor based on moment equilibrium and one based on horizontal force equilibrium. The method also allows for a range of interslice shear-normal force conditions, making it the most rigorous of all the methods, satisfying both force and

t ilib i f i l d i l li f


moment equilibrium, for circular and non-circular slip surfaces. Krahn (2003)

ComputerComputer--Aided Limit Equilibrium AnalysisAided Limit Equilibrium AnalysisI h th h f il f i t k it ti i t d In cases where the shear failure surface is not known, its anticipated location can be found from analysis of the whole range of possible surfaces, and taking the actual surface to be that which gives the lowest factor of safety. This procedure can be quickly carried out using y p q y gcomputer-based slip surface search routines.

Hand or spreadsheet calculations can take hours to solve for a can take hours to solve for a single slip surface, whereas a computer requires only seconds to solve for hundreds of potential


slip surfaces.

Critical Slip Surface Search Critical Slip Surface Search


(Rocscience – Slide)

Analysis of NonAnalysis of Non--Circular Slip SurfaceCircular Slip Surface

(Rocscience – Slide)

For a non-circular slip surface, a block search routine is used that analyzes a limited number of slip surfaces relating to the division of the slide mass into an active central and passive slide block


the slide mass into an active, central and passive slide block.

Advanced Limit Equilibrium Analysis Advanced Limit Equilibrium Analysis -- 3D3D

Most limit equilibrium formulations are two-dimensional even though dimensional even though actual slope failures are three-dimensional. However, there are a However, there are a few 3-D limit equilibrium programs employing a “method of columns” approach.

The 3-D analysis program

Hungr et al. (1989)

CLARA divides the sliding mass into columns, rather than slices as used in the 2-D analysis mode


mode.

Limit Equilibrium Analysis Limit Equilibrium Analysis -- LimitationsLimitations

1 Th l f d l b h

Although limit equilibrium methods are very useful in slope analysis, they do have their limitations and weaknesses:

1. The implicit assumptions of ductile stress-strain behaviour for the material (stress-strain relationships are neglected);

2. Most problems are statically indeterminate;

3. The factor of safety is assumed to be constant along the slip surface (an oversimplification, especially if the failure surface passes through different materials);

4. Computational accuracy may vary;

5. Allow only basic loading conditions (do not incorporate in situ stresses);)

6. Provide little insight into slope failure mechanisms (do not consider stress state evolution or progressive failure).


UncertaintyUncertaintyG t h i l i t d l ith t l diti th t l l Geotechnical engineers must deal with natural conditions that are largely unknown and must be inferred from limited and costly observations. The principal uncertainties have to do with the accuracy and completeness with which subsurface conditions are known and with the resistances that the

The uncertainties in geotechnical

fmaterials will be able to mobilize (e.g. strength).

The uncertainties in geotechnical engineering are largely inductive: starting from limited observations, judgment, knowledge of geology, and statistical reasoning are employed to infer the reasoning are employed to infer the behavior of a poorly-defined universe.

In contrast, The uncertainties in structural and mechanical engineering are largely deductive: starting from largely deductive: starting from reasonably well known conditions, models are employed to deduce the behavior of a reasonably well-specified universe.


Sensitivity AnalysisSensitivity AnalysisS iti it l ll f th d t i ti f th " iti it " f Sensitivity analyses allow for the determination of the "sensitivity" of the safety factor to variation in the input data variables. This is done by varying one variable at a time, while keeping all other variables constant, and plotting a graph of safety factor versus the variable.constant, and plott ng a graph of safety factor versus the var able.


Probability AnalysisProbability AnalysisProbabilistic analyses consider the variability of input parameters and Probabilistic analyses consider the variability of input parameters, and provide the probability of failure based on a given probability distribution function (defined through a known mean and standard deviations).

Probability distribution: A probability density function (PDF) describes the relative likelihood that a random variable will assume a particular value. The area

X

p

under the PDF is always unity.

The normal distribution is the most t f PDF It i d f t

2.5%

common type of PDF. It is used for most probabilistic studies, although for some parameters, a different distribution may be more applicable (e.g. joint spacing).

A small standard deviation indicates a tightly clustered data set while a large standard deviation indicates a large scatter about the mean. For a normal distribution, 68% of the test values will fall within an interval defined by the mean


, y± one standard deviation while 95% will fall within two standard deviations.

Probability Distribution Functions Probability Distribution Functions dd h l d l d b h b In addition to the commonly used normal distribution there are a number

of alternative distributions which are used in probability analyses. Some of the most useful are:


Probabilistic Analysis Probabilistic Analysis –– Monte Carlo Sampling Monte Carlo Sampling h l h d d d d b l The Monte Carlo method uses random or pseudo-random numbers to sample

from the probability distributions and, if sufficiently large numbers of samples are generated and used in a calculation such as that for a factor of safety a distribution of values for the end product will be generated.of safety, a distribution of values for the end product will be generated.

… Monte Carlo sampling (relative frequency) of cohesion taken as a frequency) of cohesion taken as a random variable – 1000 samples, with those producing a factor of safety <1 highlighted in red.


Probability of Failure Probability of Failure Remember that the PF and RI

calculated for the Overall Slope, are not associated with a specific slip surface, but

include the safety factors of all global minimum slip surfaces from the Probabilistic Analysis.

The Probability of Failure is simply equal to the number of analyses with safety factor less than 1, divided by the total Number of Samples.

The Reliability Index is an indication of the number of standard deviations which separate the Mean Safety Factor from th iti l f t f t ( F 1)


the critical safety factor ( F = 1).

ComputerComputer--Aided Probabilistic AnalysisAided Probabilistic AnalysisEberhardt & Stead (2006)

Sensitivity analysis of earthquake loading, followed by probabilistic analysis to account for parameter uncertainty.

FEM groundwater analysis to calculate pore pressures to be used in a stability analysis of a


used in a stability analysis of a rockslide dam.

Lecture ReferencesLecture ReferencesBromhead EM (1992) The Stability of Slopes (2nd Edition) Blackie Academic & Professional:Bromhead, EM (1992). The Stability of Slopes (2nd Edition). Blackie Academic & Professional:London.

Clayton, CRI, Simons, NE & Matthews, MC (1982). Site Investigation: A Handbook for Engineers.Halsted Press: New York.

Duncan, JM (1996). Soil slope stability analysis. In Landslides: Investigation and Mitigation – SpecialReport 247. National Academy Press: Washington, D.C., pp. 337-371.

Eberhardt, E & Stead, D (2004). Incorporating the effects of groundwater and coupledhydromechanical processes in slope stability analysis. Italian Journal of Engineering Geology andy m p p y y f g g gyEnvironment, Special Issue 1: 139-143.

Hoek, E & Bray, JW (1981). Rock Slope Engineering (3rd edition). Institution of Mining andMetallurgy: London.

H d JA & H i JP (1997) E i i R k M h i A I t d ti t th P i i lHudson, JA & Harrison, JP (1997). Engineering Rock Mechanics – An Introduction to the Principles .Elsevier Science: Oxford.

Hungr, O, Salgado, FM & Byrne, PM (1989). Evaluation of a three-dimensional method of slopestability analysis. Canadian Geotechnical Journal 26(4): 679-686.

Krahn, J (2003). The 2001 R.M. Hardy Lecture: The limits of limit equilibrium analyses. CanadianGeotechnical Journal 40: 643-660.

Morgenstern, NR (1995). The role of analysis in the evaluation of slope stability. In Proceedings,Sixth International Symposium on Landslides Christchurch A A Balkema: Rotterdam pp 1615-1629


Sixth International Symposium on Landslides, Christchurch. A.A. Balkema: Rotterdam, pp. 1615 1629.

Wyllie, DC & Mah, CW (2004). Rock Slope Engineering (4th edition). Spon Press: London.

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