Slope InstabilitySlope Instability

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In the history of rock mechanics and rock engineering, more attention has been paid to slope instability considerations than any other topic, and this topic remains one of the most important today.

One of our first considerations must be to identify the basic mechanismsof slope instability. Analysis may then proceed treating the unstable mass as a continuum or discontinuum.

Slope InstabilitySlope Instability

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As a continuum, the failure path passes through the rock mass. As a discontinuum, the failure surface is dictated more directly by the presence of specific pre-existing discontinuities. It is also possible to have intermediate cases where the failure occurs partly along discontinuities and partly through bridges of intact rock.

Although most soil slope instability is of the continuous nature, the majority of rock slopeinstability is caused by individual discontinuities. This is because the strength of the intact rock can be high, with the result that the pre-existing discontinuities are the weakest link.

Slope Instability MechanismsSlope Instability Mechanisms

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The following diagram illustrates what are traditionally regarded as the four basic instability mechanisms for rock slopes. The geometry of the slip in (a) is a function of the slope geometry and the strength of the slope-forming material, but those shown in (b-d) show how the boundaries of the instability are governed by discontinuities.

Curvilinear Slip (Rotational Slides)Curvilinear Slip (Rotational Slides)

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The term ‘curvilinear slip’ is used to describe the group of instabilities where the slip surface is curved and usually terminates at a tension crack at the upper ground surface. The shape and location of the slip surface depends on the strength characteristicsof the ground mass, which in turn depends on the structure of the assumed continuum.

Curvilinear Slip (Rotational Slides)Curvilinear Slip (Rotational Slides)

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… development of curvilinear slips as a function of rock mass structure and characteristics.

Curvilinear Slip Curvilinear Slip –– Limit Equilibrium AnalysisLimit Equilibrium Analysis

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In practice the factor of safety is determined for assumed slip surface locations. The technique used to analyze these failures is a limit equilibriumtechnique called the ‘method of slices’

In analysing the potential for slip, one has to consider:

(a) the location of the slip surface; and (b) determination of the factor of safety

for a given slip surface.

Method of SlicesMethod of Slices

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The method of slices involves the discretization of the slip surface into elements, each of which has normaland shear forces applied to it. Each element has three unknowns associated with it: the normal (N) and shear (S) forces, and the location of the line of action of the normal force relative to the element itself (n).

For the case shown, there is a total of 12 unknown parameters but only three equations of static equilibrium available to solve this problem: ΣFx=0, ΣFy=0 and ΣM=0, where Fx and Fy are the force components in the x and y directions, and M are moments in the x-y plane. Thus to solve the problem, we have to make assumptions which reduce the number of unknowns.

Method of SlicesMethod of Slices

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The reduction of the number of unknowns is usually done by sub-dividing the mass under consideration into ‘slices’, and analysing each slice on the basis that it is in limiting equilibrium, i.e. each N and S is linked through the strength criterion of the slip surface. Examination of a typical slice with the various forces applied to it, and equilibrium analysis leads to:

which enables S to be expressed in terms of the other slice parameters.

Method of SlicesMethod of Slices

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The analysis of the factor of safety, F, of the entire mass then depend on whether the slip surface is generally non-circular, or specifically truly circular.

non-circular:

circular:

Where H is the hydrostatic thrust from the tension crack.

Plane SlidingPlane Sliding

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Planar slides generally involve a major discontinuityof lengthy persistence. The case of planar sliding is unlike that of curvilinear slip, in that it is statistically determinate.

As such, we can calculate the factor of safetydirectly, for kinematicallyfeasible planes, by making suitable assumptions to render the problem two-dimensional.

Plane Sliding Plane Sliding –– KinematicKinematic AnalysisAnalysis

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To consider the kinematic feasibility of plane instability, four necessary but simple criteria are introduced:

(a) The dip of the slope must exceed the dip of the potential slide plane;

(b) The potential slip plane must daylight on the slope plane; (c) The dip of the potential slip plane must be such that the

strength of the plane is reached;(d) The dip direction of the sliding plane should lie approximately

±20° of the dip direction of the slope.

kinematicly feasible not kinematicly feasible

Plane Sliding Plane Sliding ––KinematicKinematic AnalysisAnalysis

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… kinematic analysis of planar stability using stereonetconstructions.

daylightenvelope

frictioncone

slopeface

Plane Sliding Plane Sliding –– Limit Equilibrium AnalysisLimit Equilibrium Analysis

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The solution for plane sliding requires that the strikes of the failure plane and slope are parallel and that no end restraints are present. Furthermore, the solution incorporates the assumptions that the rock mass isimpermeable, the sliding block is rigid, the strength of the slide plane is given by the Mohr-Coulomb criterionand that all forces pass through the centroid of the sliding block.

Plane Sliding Plane Sliding –– Limit Equilibrium AnalysisLimit Equilibrium Analysis

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… calculation of factor of safety vs. different depths of water in the tension crack, indicating a possible significant effect of heavy and prolonged rainfall.

… factor of safety vs. varying effective friction angles along the plane of sliding.

Wedge SlidingWedge Sliding

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Wedge sliding can be extended from plane sliding by considering sliding along two planes simultaneously.

Wedge Sliding Wedge Sliding –– KinematicKinematic AnalysisAnalysis

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To consider the kinematic feasibility of wedge instability, we need to consider three criteria relating to the line of intersection:

(a) The dip of the slope must exceed the dip of the line of intersection of the two wedge forming discontinuity planes;

(b) The same line of intersection must daylight on the slope plane;

(c) The dip of the line of intersection must be such that the strength of the two planes are reached.

Wedge Sliding Wedge Sliding ––KinematicKinematic AnalysisAnalysis

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… kinematic analysis of wedge stability using stereonetconstructions.

Wedge Sliding Wedge Sliding –– Limit Equilibrium AnalysisLimit Equilibrium Analysis

18 - Il numeratore della 2a eq. è N sulla linea di intersezione.

Assuming that the direction of sliding is parallel to the line of intersection of the two sliding planes, forces parallel to this line and perpendicular to the sliding planes can be resolved in order to determine the factor of safety.

Wedge Sliding Wedge Sliding –– Limit Equilibrium AnalysisLimit Equilibrium Analysis

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Consolidating these formulae results in:

A direct insight into the fundamental mechanism of wedge instability is seen in the left-hand side of this equation which shows how the safety factor varies with the sharpness of the wedge (δ) and the verticality of the wedge (β).

Toppling InstabilityToppling Instability

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To final fundamental mechanical mode of structurally controlled slope instability, involves toppling failures. Toppling generally occurs in two modes:

direct toppling – occurs when the centre of gravity of a block lies outside the outline of the base of the block, with the result that a critical overturning moment develops.

flexural toppling – occurs under certain circumstances when a layered rock mass outcrops at a rock slope, and the principal stress parallel to the slope face induces inter-layer slip which causes the intact rock to fracture and the resulting blocks to overturn.

Direct Toppling Direct Toppling –– Kinematic Kinematic AnalysisAnalysis

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The kinematic-based question in the case of direct toppling is whether a block resting on an inclined surface will be stable, or slide, or topple. The nature of direct toppling is determined from considerations of the block geometry and the angle of friction between the block and the surface on which it is resting.

The resulting four categories of equilibrium under these conditions are:

Direct Toppling Direct Toppling ––Kinematic Kinematic AnalysisAnalysis

22; (90°-PSIp) è l'inclinazione del polo del piano

… kinematic analysis of direct toppling instability using stereonetconstructions.

Direct Toppling Direct Toppling ––Limit Equilibrium Limit Equilibrium

AnalysisAnalysis

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… limit equilibrium analysis of direct toppling instability using factor of safety calculations.

Flexural Toppling InstabilityFlexural Toppling Instability

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Remembering that the creation of a new excavation surface results in the principal stresses being parallel and perpendicular to the excavated face, the potential for inter-layer slip is then dependent on the geometryof the layers.

An analysis of instability will include the given geometrical parameters as well as the angle of friction. In limit equilibrium, these parameters can be utilized to define a ‘geometrical factor of safety’:

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Analytical Solutions Analytical Solutions –– Limit EquilibriumLimit Equilibrium

The most widely applied analytical technique used in slope stability 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.

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

FS = =resisting forces

driving forces

shear strength of material

shear stress required for equilibrium

… where FS > 1.0 represents a stable situation and FS < 1.0 denotes failure.

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Limit Equilibrium Limit Equilibrium –– Factor of SafetyFactor of Safety

The factor of safety is an overall measure of the amount by which the strength of the soil would have to fall short of the values described by c and φ in order for the slope to fail.

FS = =resisting forces

driving forces

shear strength of material

shear stress required for equilibrium

FS =c + σ tan φ

τeq

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Limit Equilibrium AnalysisLimit Equilibrium Analysis

The fundamental concepts of the limit equilibrium method for slope stability analysis are (Morgenstern, 1995):

… slip mechanism results in slope failure;

… resisting forces required to equilibrate disturbing mechanismsare found from static solution ;

… the shear resistance required for equilibrium is compared withavailable shear strength in terms of the Factor of Safety;

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

28 - THETA in radianti

Limit Equilibrium Limit Equilibrium –– Circular AnalysisCircular Analysis

… circular failure surface in φ = 0 soil defined by its undrained strength, cu.

Circular failure surfaces are found to be the most critical in soil slopes consisting of homogenous materials. There are two analytical, statically determinate, methods commonly used to calculate the FS for theseslopes: the circular arc (φ=0) and the friction circle method.

WxLRc

FS u=

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Limit Equilibrium Limit Equilibrium –– Method of SlicesMethod of Slices

… this condition is usually analyzed by dividing the failing slope mass into smaller slices and treating each individual slice as a unique sliding block.

The previous methods discussed do not depend on the distribution of the effective normal stresses along the failure surface. However, if the mobilized strength for a c-φ soil is to be calculated, this distribution must be accounted for.

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Method of SlicesMethod of Slices

… forces acting on a typical slice.

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Method of Slices Method of Slices -- Equations & UnknownsEquations & Unknowns

… for this system, there are (6n-2) unknowns but only 4 equations which can be written for the limit equilibrium condition – therefore the solution is statically indeterminate.

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Method of Slices Method of Slices -- Equations & UnknownsEquations & Unknowns

… forces on a typical slice resolved into two known resultant forces and a moment, with the unknown force components shown separately (Bromhead, 1992).

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Method of Slices Method of Slices -- AssumptionsAssumptions

… however, a solution is possible providing the number of unknowns can be reduced by making some simplifying assumptions. It is these assumptions that generally differentiate the many methods of analysis available.

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Limit Equilibrium Limit Equilibrium –– Conditions/FactorsConditions/FactorsDrained and Undrained:… slope failures may occur under drained or undrained conditions depending on the soils that make up the slope and the external loading conditions.

Effective Stress and Total Stress:… in effective stress analyses, the shear strength is related to the effective normal stress on the potential slip surface by means of effective shear stress parameters (cd,φd). Pore pressures must be known and are part of the information required for analysis.

… in total stress analyses, the shear strength is related to the total normal stress on the potential slip surface by means of total shear stress parameters (cu, φu~0). Pore pressures need not be known and are not part of the information required for analysis.

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Method of Slices Method of Slices -- ComputationComputation

… example slope: sand embankment over clay foundation – such problems can be easily solved using spreadsheet calculations.

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Method of Slices Method of Slices -- ComputationComputation

… example computation for ordinary method of slices.