Introduction to Slope Stability Analysis

Relevant Course: Geo technical Engineering

Relevant Department : Civil Engineering

Relevant Semester: August-October, 2015

Pre-requisite: Knowledge of Shear Strength of Soil is Preferable

Course Description & Outline :

  • Types of slopes
  • Factors inducing instability
  • Types of instability
  •  Methods of slope stability analysis

             a) Finite Slope Analysis (Total and Effective Stress Analysis)
             b) Infinite Slope Analysis

  • Common Methods of Slope Stabilization.

Schedule for Lecture Delivery

Session 1 : 07-Sep-2015 (10-12 noon)

Session 2 : 08-Sep-2015 (10-12 noon)

Session 3 :14-Sep-2015 (2-4 pm)

Teacher Forum

Outline

  • Types of slopes and slope instability
  • Factors inducing slope failure
  • Aims of slope stability analysis
  •  Slope stability assessment for infinite slopes
  • Slope stability assessment for finite slopes
  • Morgenstern’s Method of Analysis for Rapid Drawdown Condition
  • Methods for seismic analysis of soil slopes
  • Methods for stabilisation of slopes
  • Solved sample problems

Introduction

  • Slope stability: Extremely important in design and construction of various geotechnical structures like embankments, earth dams and many others.
  • Failure of slopes: An age-old phenomenon which has inflicted heavy loss on life and property
  • Slopes exist in states ranging from very stable to marginally stable.
  • Slope Instability: A tendency of the slope to move.
  • Slope Failure: Actual movement of soil mass occurs.
  • The resulting damage may vary from insignificant to catastrophic, depending upon geometry, loading and material characteristic of the slope

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Types Of Slopes

4.1 Based on origin

  • NATURAL SLOPES :Those slopes which exist in nature and were formed by natural phenomenon. These slopes exist in hillside, valleys, coastal and river cliffs. 
  • MAN-MADE SLOPES :These include sides of cuttings, slopes of embankments constructed for roads, railway lines, canals etc. and the slopes of earth dams constructed for storing water.

4.2 Based On Extent 

Slopes whether natural or man-made may be of two types based on extent:

  • FINITE SLOPE :Slopes with limited extent e.g. Slopes of embankments and earth dams.
  • INFINITE SLOPE :Shallow but long slope, failure surface parallel to sloping surface, constant condition (soil properties and stresses remain constant along the slope) e.g. long slope of the face of a mountain.

4.3 Types Of Man-Made Slopes 

They can be further classified into three types [Chowdhury (1978)]:

  • Shallow and deep cuts: Major interest in civil and mining engineering operations is to design a slope with such a height and inclination as to be stable within a reasonable life span and with as much economy as possible.
  • Fill slopes: involving compacted soils include railway and highway embankments, earth dams and levees. The engineering properties of materials used in these structures are controlled by the method of construction and the degree of compaction.
  • Spoil and waste heaps: whose stability, consisting of mining and industrial waste, is being recognized as a problem of major importance.

(Source:http://myhomeimprovement.org/)(Source:http://www.dc-ca.com/site/index.php/projects/dubai-fujairah-freeway)

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Types Of Instability

Broadly classified into three types:

  • SLIDING
  • FLOW
  • FALLS

5.1 Sliding 

  • Moving material remains largely in contact with the parent material   during the movement which takes place along a distinct boundary shear surface – moving material moves like a rigid body.

Mechanisms – Translation, Rotation or Translation Rotation

5.1.1 Translational Sliding

  • Failure along a weak zone of soil,
  • Sliding mass can travel long distances before coming to rest,
  • Common in coarse-grained soils.

5.1.2 Rotational Sliding 

  • Common type of failure in homogeneous fine-grained soils,
  • Can be further classified into three types

1. Base Slide: Failure surface is an arc engulfing the whole slope. Generally occurs in a soft layer of soil resting on a stiffer soil layer.

2. Toe Slide: Slip surface passes through the toe of the slope.

3. Slope Slide: Slip surface passes through the slope. 

5.2 Flow 

  • When the material becomes disintegrated and can move without the concentration of displacements at the boundary shear.
  • The material behave like a viscous fluid and may remain in touch with the failure surface.
  • Failure surface is ill-defined and multiple failure surfaces usually occur which change continuously as the flow proceeds.
  • Flow is predominant in clays.

5.3 Falls

  • Takes place from the steep faces in soils/rocks and involves immediate separation of the falling material from parent material, with movement involving only infrequent or intermittent contacts.
  • Rockfall typically occurs where water enters cracks in rocks, freezes, expands, and breaks the rock apart.
  • Soilfall is generally produced by ocean waves undercutting cliff faces resulting in loss of support.

Fig. 7 Rockfall

(Source: http://www.irtces.org/isi/WebNews_View-en2.asp?WebNewsID=227)

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Factors Including Slope Failure

6.1 Gravity

  • On a slope, gravitational force can be resolved into two components: a component acting perpendicular to the slope and a component acting tangential to the slope.  
  • The tangential component of gravitational force acts in the direction of probable motion

6.2 Seepage

  • Seepage forces in the sloping direction add to gravity forces and make the slope susceptible to instability.
  • The pore water pressure causes a decrease in shear strength.
  • This condition is critical for the downstream slope.

6.3 Rainfall

  • Addition of water from rainfall makes the slope heavier.
  • Long periods of rainfall saturate, soften and erode soils.
  • Water enters into existing cracks and introduces seepage forces weakening underlying soil layers

Fig. 8 Rainfall fills crack and introduces seepage forces in the thin, weak soil layer

(Source: Muni Budhu, 2008)

6.4 Erosion

  • Water and wind continuously erode natural and man-made slopes.
  • Erosion leads to change of geometry of slope which may lead to steepening of slope.
  • Erosion in the form of undercutting at the toe may increase the height of the slope, or decrease the length of the incipient failure surface, thus decreasing the stability.

Fig. 9(a)Steepening of slope by erosion    Fig. 9(b) Scour by rivers and streams

(Source: Muni Budhu, 2008)

6.5 Earthquake Forces

  • External disturbance in the form of seismic activity (earth tremors or earthquakes) induce dynamic shear forces.
  • In addition there is sudden buildup of pore water pressure that reduces available shear strength.
  • Porewater pressures in saturated coarse-grained soils could rise to a value equal to total mean stress and cause these soils to behave like viscous fluids – a phenomenon known as dynamic liquefaction.

Fig. 10 Gravity and earthquake forces

(Source: Muni Budhu, 2008)

6.6 Geological Features 

Many failures occur due to unidentified geologic features like thin weak layer of soil underlying a thick deposit

6.7 External Loading 

Loads placed at the crest of the slope add to the gravitational load.

6.8 Construction Activities

Construction activities near the toe of an existing slope can cause failure because lateral resistance is removed.

6.9 Rapid Drawdown

  • Reservoirs can be subjected to rapid drawdown.
  • In sudden drawdown lateral resistance provided by water is removed while the excess porewater pressure does not have enough time to dissipate.
  • Thus, the slopes can fail under undrained conditions.
  • Also, if failure did not occur under undrained conditions, seepage of water would result in additional seepage forces at a later stage which may initiate failure.

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Aims of Slope Stability Analysis

  • Safe and economic design of excavations, embankments, earth dams and soil heaps.
  • To assess the stability of different types of slopes under given short term (end of construction) and long term conditions.
  • To assess the possibility of landslides involving natural or existing man made slopes.
  • To analyze slips and landslides that have already occurred and to assist in the understanding of failure mechanisms and the influence of environmental factors.
  • To enable the redesign of failed slopes, and the planning and design of preventive and remedial measures where necessary.
  • To study the effect of exceptional loadings such as earthquake on slopes and embankments.

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Factors Of Safety in Slope Stability Analysis

The factor of safety with respect to shearing strength, Fs, may be written as

The shearing strength mobilized at each point on a failure surface may be written as:

If factors of safety with respect to cohesion and friction are different, 

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Slope Stability Assessment

  • Assess all possible slip mechanisms.
  • Evaluate Fs for each typical surface.
  • The slip surface that gives the lowest factor of safety is called the critical slip surface and the actual factor of safety is this minimum Fs value.
  • If the factor of safety of the critical slip surface < 1, => FAILURE

Fig. 16 Slope stability assessment

Infinite Slope Analysis in Soil

  • c' = 0 and soil is homogeneous throughout.
  • The stresses acting on any vertical plane in the soil are the same as those on any other vertical plane.
  • Side forces X are equal & opposite by symmetry.
  • The stress at any point on a plane EF parallel to the surface at depth will be the same as at every point on this plane. 

9.1.1 CASE 1: DRY SLOPE (u = 0)

9.1.2 Generalized Seepage through Infinite Slope for ϕ' soil

9.1.3 CASE 2: θ = 90° i.e., VERTICAL FLOW OF WATER

9.1.4 CASE 3: θ = β i.e., STEADY SEEPAGE OCCURRING PARALLEL TO SLOPE:

9.1.5 CASE 4: SUBMERGED SLOPE WITH NO SEEPAGE

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Infinite Slope Analysis in Soil

9.3 Infinite Slope Analysis in c'-ϕ' Soil

9.3.1 Generalized Seepage through Infinite Slope for c'-ϕ' soil

9.3.2 CASE 1: DRY SOIL (u = 0)

9.3.3 CASE 2: STEADY SEEPAGE ALONG THE SLOPE

9.3.4 CASE 3: FOR VERTICAL FLOW OF WATER i.e., θ = 90°

9.3.5 CASE 4: SUBMERGED SLOPE WITH NO SEEPAGE

Finite Slope Analysis

  • More common problem as compared to infinite slope.
  • Generally failure occurs on curved surfaces.
  • Various methods of analysis includes

             1. Limit Equilibrium Method,

             2. Limit Analysis,

             3. Numerical Analysis [Finite Element Method (FEM), Finite Difference Method (FDM), etc.].

  • Numerical analyses are more flexible and general.
  • However, limit equilibrium is the method of choice and in this section we will be dealing with only limit equilibrium solutions because of its simplicity.
  • Limit equilibrium method normally gives an upper bound solution because a more efficient failure mechanism is possible than that assumed in analysis

9.4.1. Plane Failure Surface (Culmann's Analysis)

  • Failure is assumed to occur along a plane when the average shearing stress is more than the shear strength of the soil.
  • The most critical plane is the one that has a minimum ratio of the shear strength of soil to the average shearing stress developed.
  • The method assumes that the critical surface of failure is a plane surface passing through the toe of the slope.

z

9.4.2 Rotational Failure / Circular Slip Surface

  • Modes of failure are base slide, toe slide and slope slide as discussed earlier.
  • Analysis procedure can be broadly classified into

9.4.2.1 Mass Procedure 

  • The mass of the soil above the surface of sliding is taken as a unit.
  • Useful when the soil is assumed to be homogeneous, although this is not the case in most natural slopes.

Fig. 27 Finite slope stability assessment of finite slope using planar failure surface

9.4.2.2 Method of Slices

  • Soil above the surface of sliding is divided into a number of vertical parallel slices.
  • The stability of each slice is calculated separately.
  • Versatile technique in which the non-homogeneity of the soils and pore water pressure can be taken into consideration.
  • It also accounts for the variation of the normal stress along the potential failure surface.

9.4.3 MASS PROCEDURE

9.4.4 Short Term Stability of cuttings in clay(Total Stress Analysis)

  • Taylor (1937) analyzed the stability of a large number of slopes through a wide range of slope angles, depth factors and angle of internal friction.
  • The figure here shows Taylor’s slope stability chart for undrained conditions.
  • The results were represented in terms of “stability number” (Ns).

9.4.5 Effective Stress Analysis with Circular Slip Surfaces : METHOD OF SLICES

  • Width of base of slice = l
  • F1is resultant of E1&X1
  • F2is resultant of E2and X2
  • T= Shear force on base of slice
  • N= Normal force at base of slice
  • 7 unknowns: E1, E2, X1, X2, N', T and Fs
  • Side forces E1, E2, X1, X2 are required only to calculate N', T – resultants can be grouped as (E1 - E2) and (X1 -X2) – 5 unknowns
  • 4 equations available – horizontal force equilibrium, vertical force equilibrium, moment equilibrium and mobilised shear stress relating N', T and Fs – assumption required to solve this.

9.4.5.1. Swedish Method : Fellenius(1927)

  • Assumptions:
    • Inter-slice forces are equal and opposite i.e., (E1- E2) = (X1- X2) = 0.
  • Resolving forces perpendicular to the assumed slip surface:

9.4.5.2 Bishop's Slimplified Method:Bishop(1955)

  •  Fellenius’ assumption that (E1 - E2) = (X1- X2) = 0 reduces 2 unknowns, whereas reduction of only 1 unknown would make the problem statically determinate.
  • Assumption of Bishop’s simplified method: (X1- X2) = 0, but (E1- E2) ≠ 0 i.e., the resultant of interslice forces do not have any component in the vertical direction.
  • Resolving forces vertically (so that Ecould remain unknown):

9.4.5.3. Spencer's Method : Spencer (1967)

  • Circular slip surface
  • Assumption: the direction of the resultant of the inter-slice forces same for all slices, but not necessarily horizontal.
  • Calculations are insensitive to the angle at which the inter-slice forces act.  So Bishop’s method do not lead to any significant error.

9.4.5.4 Morgenstern and Price's Method : Morgenstern and Price (1965)

  • General solution with any shape of slip surface
  • Functional relationship between side shear force X and normal inter-slice force E was introduced → X= λ.f(x).E, where x is distance along slope.

9.4.6.  Morgenstern's Method Of Analysis For Rapid Drawdown Condition 

  • In sudden drawdown, lateral resistance provided by water is removed while the excess porewater pressure does not have enough time to dissipate which may result in failure under undrained conditions.
  • Morgenstern (1963) developed this method based on Bishop and Morgenstern method of slices.

1. ASSUMPTIONS

  • Simple slope of homogeneous material,
  • Dam rests on an impermeable base,
  • Slope is completely submerged initially,

2. Morgenstern (1963) provided charts which take into account drawdown ratio, Rwhich is defined as : 

 

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Effect Of Tension Cracks on Stability

  • If a dam is built of cohesive soil, tension cracks are usually present at the crest. The depth of such cracks may be computed from the equation :

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Limitations of Limit Equilibrium Analysis

  • No information on slope deformations can be obtained using this method.
  • Shape of the potential failure surface is highly debated upon by researchers.
  • Limit equilibrium method fails in conditions where soil of very high cohesive strength is present at the top of the slope and/or when the failure surfaces emerge steeply at the base of the slope in case of soils with high frictional strength
  • Since the method depends on the available shear strength of the soil along the failure surface, inaccurate consideration of pore pressure conditions could lead to serious errors in factor of safety calculations.

Seismic Analysis of Soil Slopes

  • All types of slopes which are stable under static conditions may become unstable during earthquakes.
  • Ground shaking induced by earthquakes acts as a destabilizing force, and, hence, reduce the factor of safety against slope failure.
  • During seismic ground shaking, the factor of safety of the slope could be reduced to 1.0 :

            1. For a very short time before a reversal in the direction of the seismic force occurs, or

            2. At the end of the significant duration of the ground motion.

  • In consequence, slopes may fail or may undergo serious damage during earthquakes.

Fig. 34The La Conchita Landslide, Ventura County, Callifornia, 2005

(Photo by U.S. Geological Survey)

  • Seismic slope stability has become one of the most concerned subjects in geotechnical engineering.
  • Seismic slope analyses can be broadly classified into three types

                  1. Pseudo-static analysis,

                  2. Newmark’s sliding block method,

                  3. Dynamic Finite Element Analysis.

In this section only pseudo-static analysis and Newmark’s sliding block method will be discussed because of their simplicity

9.9.1. Pseudo-Statis Analysis of soil slopes

  • Beginning in the 1920’s, the earliest method of seismic slope stability analyses used a pseudo-static approach and considered stability in terms of a simple factor of safety.
  • In this approach, the effects of the earthquake are represented by a constant horizontal acceleration that produces an inertial force (F) that acts through the centroid of a potential failure mass. 
  • This method is very simple, straightforward and still common in practice

  • Fh= Horizontal pseudo-static force acting through the centroid of sliding wedge,
  • Fv= Vertical pseudo-static force acting through the centroid of sliding wedge,
  • m = Total mass of the sliding wedge,
  • W = Total weight of sliding wedge,
  • ah = Horizontal pseudo-static acceleration,
  • av= Vertical pseudo-static acceleration,
  • kh = Horizontal seismic coefficient/horizontal  pseudo-static coefficient  = ah/ g ,
  • kv = Vertical seismic coefficient/vertical pseudo-static coefficient  = av/ g.

Note:Same procedure can be applied for a curvilinear failure surface as well.

  • From the expression of Fs, we can observe that with increase in Fv, both resisting force and driving force decrease. Thus, there is less significant effect of vertical acceleration of factor of safety.
  • However, with increase in Fh, resisting force decreases while driving force decreases resulting in decrease in factor of safety. Thus, horizontal seismic acceleration has pronounced effect on factor of safety.
  • The horizontal seismic coefficient for which factor of safety becomes 1 is known as yield coefficient (ky).
  • Based on comparison of peak ground acceleration (PGA) with yield coefficient (ky), an estimate can be made on the extent of likely damage during a seismic event

9.9.1.1. Limitations of Pseudostatic Analysis 

  • Pseudostatic approach represents the complex, transient, dynamic effects of earthquake by a single constant unidirectional pseudostatic acceleration, which is crude.
  • The slope could be unstable even if the factor of safety computed from pseudostatic approach is > 1.
  • Experiences from earthquakes all over the world showed that the pseudostatic approach is not reliable for soils which develop high pore water pressures and whose strength degrade considerably due to earthquake shaking.
  • This analysis provides an index for the stability but the deformations induced due to earthquake shaking cannot be obtained from this analysis.
  • Since the earthquake induced forces vary with time, the factor of safety varies throughout the duration of the earthquake

9.9.2 Newmark's Rigid Block Analysis 

  • Newmark (1965) developed a simple procedure to estimate permanent slope displacement due to earthquake shaking by extending the pseudo-static approach to a sliding rigid block and considering the acceleration-time history of the sliding mass within the slope.
  • The method assumes that permanent displacement at some well-defined slip surface begins when the inertial forces induced by the seismic excitation exceed the resisting forces.
  • Since the serviceability of the slope after the earthquake depends on the deformations induced in the slope during earthquake, this method is comparatively more useful than the pseudo-static analysis.

Fig. 37 Forces acting on a rigid block resting on an inclined plane under
(a) Static conditions (b) Dynamic conditions (Source: Kramer, 2013)

  • Assuming that the block’s resistance is purely frictional (c=0),
  • For static case :

9.9.2.1 Computing permanent displacements in Newmark’s rigid block analysis

Fig. 38 Computing permanent displacements for actual earthquake
(Source: Kramer, 2013)

  • The block slides only when |base acceleration| > yield acceleration.
  • Thus, displacement gets accumulated only on such circumstances when block slides i.e, FSd(t) < 1.

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Slope Stabilization

  • Drainage: Drainage helps in reducing the seepage forces and hence increases the stability.
  • Modification of slope profile: Slope flattening can be used wherever possible.
  • Restraining structures: Sheet piles and retaining walls can be installed to provide lateral support and to increase the stability. However, this method is quite expensive.
  • In-situ reinforcement: Soil Nailing, Wire Netting, Anchors, Geosynthetics like Geotextiles, Geogrids, Geomembranes etc.
  •  Chemical Treatment :  
    • Grouting or injection of cement or other compounds in specific zones helps in increasing stability,
    • Electro-osmosis.
  • Erosion control: Water near the surface of the hillside can cause the erosion of surface material due to water runoff. Scouring and slope steepening can be avoided by adopting proper means against erosion.

  • Miscellaneous :

           1. Providing a berm below the toe of a slope increases its stability. It is particularly useful where there is possibility of base failure.

          2.  Vegetations: Natural method synonymous to soil nailing.

         3. Terracing: Graduated terrace steps may be built along slopes on which agriculture, cultivation can be practised.

         4. Use of Piles: Piles can be used to stabilize slopes especially where loads from structures act at the crest of the slope.

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References

  • Arora, K. R. (2011). Soil Mechanics and Foundation Engineering (Geotechnical Engineering). Standard Publishers Distributors.
  • Bishop, A. W. (1955). “The use of the slip circle in the stability analysis of slopes.” Geotechnique, 5(1), 7-17.
  • Budhu, M. (2008). Soil Mechanics and Foundations. Wiley India Pvt. Ltd., John Wiley & Sons, Inc.
  •  Chowdhury, R. N. (1978). Slope Analysis. Elsevier, New York.
  •  Culmann, C. (1875). Die Graphische Statik, Meyer and Zeller, Zurich, 644.
  • Fellenius, W. (1927). Erdstatische berechnungen. W. Ernst und Sohn, Berlin.
  • Kramer, S. L. (2013). Geotechnical Earthquake Engineering. Prentice-Hall International Series in Civil Engineering and Engineering Mechanics, Pearson Education, Inc. and Dorling Kindersley Publishing Inc.
  •  Morgenstern, N.R. (1963). “Stability charts for earth slopes during rapid drawdown.” Geotechnique, 13(2), 121-133.
  •  Morgenstern, N. R., and Price, V. E. (1965). “The analysis of the stability of general slip surfaces.” Geotechnique, 15(1), 79-93.
  • Muir Wood, D. (2009). Soil Mechanics A One-Dimensional Introduction. Cambridge University Press.
  • Murthy, V. N. S. (2009). Textbook of Soil Mechanics and Foundation Engineeering – Geotechnical Engineering Series. CBS Publishers & Distributors Pvt. Ltd.
  • Newmark, N. (1965). “Effects of earthquakes on dams and embankments.” Geotechnique, 15(2), 139-160.
  • Powrie, W. (2009). Soil Mechanics – Concepts & Applications. Spon Press, Taylor and Francis Group, London and New York.
  • Spencer, E. (1967). “A method of analysis of embankments assuming parallel interslices technique.” Geotechnique, 17(1), 11-26.
  • Taylor, D. W. (1937). “Stability of earth slopes.” Journal of the Boston Soc. Of Civil Engineers, 24, 197-246.
  • Terzaghi, K. (1950). “Mechanism of landslides.” Engineering Geology (Berkey) Volume, Geological Society of America.

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