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Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
19
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Module 5:
Lecture -1 on Stability of Slopes
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Stability analysis of a slope and finding critical slipsurface;
Sudden Draw down condition, effective stress andtotal stress analysis;
Seismic displacements in marginally stable slopes;
Reliability based design of slopes,
Methods for enhancing stability of unstable slopes.
Contents
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Contents of this lecture
Types of slopes
Failure types
Causes of slope failures
Analysis of slopes by using LE methods
Comparison
Concluding remarks
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Application of shear strength theory
Earth pressure theories
Stability analysis of slopes
Infinite slopes
Slope that extends over along distance and theconditions remain identicalalong some surface orsurfaces for quite somedistance.
Finite slopes
Slope that connect land at oneelevation to land that is not faraway but is at differentelevation.
Can also exist in nature and man-made.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slopes
Natural♦Hill side and valleys
♦Coastal and river cliffs
Man-made♦ Cuttings and embankments for
highways and rail roads
♦ Earth and ash pond dams
♦ Temporary excavations
♦ Waste heaps (landfill slopes)
♦ Landscaping for site development
Type of slopes
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
R
R
Types of slope failure
Circular
Non-circular
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Translational slip
Types of slope failure
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Types of slope failure
Compound slip
Rigid stratum
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
F
N
W
µsN
N
WF1
µkN
Block movement
As long as µsN > F --- block is said to be stationary
Resisting force FR : µsN
Disturbing force FD : F F1
D
R
FFFS =
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Causes of slope failure
Gravity
Seepage
Earthquake
Erosion
Geological features
Construction activities
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Typical slope failures
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Courtesy: Geological natural hazardsSeptember 15, 2004
Typical slope failures
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Landslide damage adjacent to a residential structure
Courtesy: North Carolina Geological Survey
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Typical slope failures
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Highway slope failure at Krishnabhir, Tribhuwan highway Nepal (Aryal, 2003)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
(After Loher et al. 2002)
Typical sacrificial slope failure in highway embankment
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Uttarakand (2013)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Landslide in Chongqing and Hong Kong
After Kwong et al. 2004
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Effect of raising GWT
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Aerial view of Waste slide on March 16, 1996 [USA]
Lateral displacements upto 275 m and vertical displacements upto 61 m
1.2 million m3 of waste
After Eid et al. (2000)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Aerial view of landfill on Feb. 6, 1996
Stark et al. (2000)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slope FailureSlope failures depend on
• Soil type• Soil stratification• Ground water• Seepage• Slope geometry
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Types of Slope Failure
Translational Slide• Failure of a slope along
a weak zone of a soil
• Sliding mass travelslong distances beforecoming to rest.
• Common in coarse-grained soils.
Thin layer of weak soil
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Rotational slide• Common in homogenous fine-grained soil
• It has its point of rotation on an imaginary axis parallel to the slope
• There are three types of rotational failure:– Base slide– Toe slide– Slope slide
Types of Slope Failure
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Base slide
• Occurs by an arc engulfing the whole slope.
• A soft soil layer resting on a stiff layer of soil is prone to base slide
Rotational slide
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Toe slide
• The failure surfacepasses through thetoe of the slope.
Rotational slide
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slope slide
• The failure surface passes through the slope
Rotational slide
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Flow slide• Occurs when internal and
external conditions force asoil to behave as a viscousfluid and flow down,spreading in all directions.
• Multiple failure surfaces occurand change continuously asflow proceeds.
• Occurs in dry and wet soils.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Block and wedge slide• Occurs when a soil mass is shattered along joints, seams,
fissures and weak zones by forces emanating from adjacentsoils.
• The shattered mass moves as blocks and wedges down theslopes.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
FallsSimple detachment of rock mass from its parent body The process is only gravity governed.
Rock falls
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Causes of Slope Failure
• Erosion
Water and windcontinuously erodeslopes.
Erosion changes thegeometry of the slopes,resulting in a slopefailure or a landslide. Steepening of slope by erosion
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
• Erosion:
Rivers and streamscontinuously scour theirbanks undermining theirnatural or man-madeslopes.
Scour by rivers and streams
Causes of Slope Failure
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
• Rainfall:
Long periods of rainfallsaturate, soften, anderode soils.
Water enters intoexisting cracks and mayweaken underlying soillayers, leading to failure,(for example, mud slides)
Causes of Slope Failure
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
• Earthquakes: Earthquakes induce dynamic forces especiallydynamic shear forces that reduce the shear strengthand stiffness of the soil.
Causes Of Slope Failure
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
• Earthquakes:Pore water pressures in saturated coarse-grainedsoils could rise to a value equal to the total meanstress and cause these soils to behave like viscousfluids. This phenomenon is known as dynamicliquefaction. Structures founded on these soils wouldcollapse.
The quickness in which the dynamic forces areinduced prevents even coarse–grained soils fromdraining the excess pore water pressures. Thus, failurein a seismic event often occurs under undrainedconditions.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
• Geological features:
Many failures commonly result from unidentifiedgeological features.
Soil stratification
Causes of Slope Failure
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
•External Loading:
Loads placed on thecrest of a slope add togravitational load andmay cause slopefailure.
Overloading at the crest of the slope
Causes of Slope Failure
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
• Construction activities:Construction activities nearthe toe of an existing slopecan cause failure becauselateral resistance is removed.
Slope failures due toconstruction activities isdivided into two cases:
• Excavated slopes.• Fill slopes.
Excavation at toe of the slope
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
• Excavated slopes:
When excavation occurs, the total stresses arereduced and negative pore pressures aregenerated. With time the negative pore pressuresdissipate, causing a decrease in effective stressesand consequently lowering the shear strength of thesoil. If slope failures occur, they take place afterconstruction is completed.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Fill slopes:Fill slopes are common in embankmentconstruction. If the foundation soil is saturated, thenpositive pore water pressures are generated from theweight of the fill and the compaction process.
The effective stress decrease and consequentlyshear strength decreases. Slope failures in slope arelikely to occur during or immediately afterconstruction.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Factors that contribute to high shear stress:Factors contributing to instability of soil slopes
i) Removal of lateral supporta) Erosion – bank cutting by streams and riversb) Human agencies – cuts, canals, pits, etc.,
ii) Surchargea) Natural agencies – Weight of snow, ice and rain waterb) Human agencies – Fills, buildings, etc.,
iii) Transitory earth stresses – Earthquakesiv) Removal of underlying support
a) Sub aerial weathering – solutioning by ground waterb) Subterranean erosion – pipingc) Human agencies – mining
v) Lateral pressures – water in vertical cracks; freezing water in cracks; root wedging After Gray and Leiser (1982)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Factors contributing to instability of soil slopesFactors that contribute to low shear strengthi) Initial state
a) Composition – inherently weak materialsb) Texture – loose soils, metastable grain structuresc) Gross structure – faults, joining, bedding, planes, varying, etc.,
ii) Changes due to weathering and other physicochemical reactions
- Frost action and thermal expansion, Hydration of clay minerals,Drying and cracking, Leaching
iii) Changes in inter-granular forces due to pore water
- Seepage pressure of percolating ground water, loss in capillarytension upon saturation, buoyancy in saturated state.
ii) Changes in structure – Fissuring of pre-consolidated clays due torelease of lateral restraint; Grain structure collapse upon disturbance.
After Gray and Leiser (1982)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slope Stability AnalysisGeneral Assumptions: The failure can be represented as a two dimensional
problem.
The sliding mass moves as a rigid body and thedeformations of the sliding mass has no significanteffects on the analysis.
The properties of soil mass are isotropic and shearresistance along failure surface remains sameindependent of the orientation of the failure surface.
The analysis is based on limit equilibrium method.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Infinite Slope Stability Analysis
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Examples of slopes which can be infinite slopes
Ore or sand stock piling by dropping from a chute
Embankment formed by end dumping from a truck
Natural slopes formed in granular materials where thecritical failure mechanism is shallow sliding or surfaceravelling
Natural slopes formed in cohesive soils with great extent orweak cohesive material on ledge
Slopes in residual soils where a relatively thin layer ofweathered soil overlies a firmed soil or rock
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Analysis of infinite slopesAssumptions:
Soil is homogenous.
The stress and soilproperties on every verticalplane are identical; On anyplane parallel to the slopestresses and soil properties areidentical.
⇔ Failure in such slope takes place due to sliding of the soilmass along a plane parallel to the slope at a certain depth.
Failure surface
b β
zhw
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Analysis of infinite slopes
Weight of segment ABCD W = γzb(1)z
Db
W
B
C
A
N
T
Tangential stress τ down the slope
ββγββγτ cossin
cos/sin z
bzb
==
Normal stress σ within the segment
βγββγσ 2cos
cos/cos z
bzb
==
Pore water pressure u on the slip surface
( ) βγ 2coswwhzu −=
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Analysis of infinite slopesNormal effective stress σ ′ =
Shearing strength τf at the base of segment
φστ ′′+′= tancf
For the general case:
ττ fFS =
( )( ) βγγγ
βγβγ2
22
cos
coscos
www
ww
hzzhzz
+−=
−−=
Factor of safety can bedefined as:
( )ββγ
γγγβφcossin
costan 2
zhzzcFS www +−′+′
=
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Analysis of infinite slopesSpecial cases
For the critical case FS = 1 ⇔ β = φ′
zhc w ==′ ;0
⇔ FS of an infinite slope with a cohesion-lesssoil is independent of the depth of failureplane.β
φtantan ′
=FS
Case - A Dry Cohesion-less soil
Mohr failure envelope
β
φ′
σ
τ
(σ, τ)
(σf, τf)
For β < φ′ ⇒ τ < τf⇔ Slope is stable
(independent of depth of slope)
For β > φ′ ⇒ τ > τf Slope wouldhave already failed at all depths.
⇒ Slope is just stable
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Analysis of infinite slopesCase – B
βγφγ
tantan ′′
=FS
0;0 ==′ whcSaturated cohesion-less slope
Factor of safety of a saturated cohesion-less slope is about ½ for a slope without saturation.
Case - C For a c - φ soil 0=wh
( )ββγγβφ
cossincostan 2
zzcFS′′+′
=
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Analysis of infinite slopesCase - C For a c - φ soil 0=wh
( )ββγγβφ
cossincostan 2
zzcFS′′+′
=
Assuming FS = 1 z = hc
′′
−
′=
φγγβγ
β
tantan
sec2chc
β
φγγβ
γ 2sec
tantan
′′
−=
′
chc
Stability number
⇔ For c-φ soils there is a limiting depth for stability
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Analysis of infinite slopes
For β1 < φ′ ⇒ τ < τf
For β2 > φ′ ⇒ τ = τf
The depth at which τ = τf is called the critical depth hc
Mohr failure envelope
β1
φ′
σ
τ
(σ, τ)
(σf, τf)
For this depth slope is just stable
β2
τ = τf
Case - C For a c - φ soil
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
20
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Module 5:
Lecture -2 on Stability of Slopes
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Analysis of finite slopes Possible failure surfaces
Planar failure surface: Occurs along a specific plane or weakness
Excavations into stratified deposits (where strata dippingtoward the excavation); In earth dams along sloping cores ofweak material (not likely to occur in homogenous soils)Circular failure surface:
Soils exhibiting cohesion c or c and φ and no specific planes ofweakness or great strength.
Non circular failure surface
When the distribution of shearing resistance within an earthmass is non-uniform, failure can occur along surfaces morecomplex than a circle.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
A finite slope with possible failure surfaces
1VnH
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Typical patterns of rotational slides
a) Spoon-shaped b) cylindrical-shaped
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Effective or Total stress parameters ?Short-term
• Low Permeability Soil, e.g. Clays:At the end of construction the soil is almost stillundrained. Hence total stress analysis making use ofundrained shear strength Cu is adopted.
• Free Draining Materials, e.g. sands/gravels:Drainage takes place immediately and hence effective stress parameters c′ and φ′ are used.
Long-termAfter a relatively long period of time, the fully drainedstage will have been reached, and hence effectivestress parameters c′ and φ′ are used.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The total stress strength is used for short–termconditions in clayey soils, whereas the effectivestress strength is used in long-term conditions inall kinds of soils, or any conditions where thepore pressure is known (Janbu, 1973)
Effective or Total stress parameters ?
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Analysis of finite slopes
The Factor of Safety for finite slope depends on:
Assumed location of centre of rotation for the slip surface
Radius of failure surface
Type of failure (toe failure, base failure and slope failure).
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Various definitionsof the factor ofsafety (FOS)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Review of Stability Analysis MethodsAll limit equilibrium methods utilize the Mohr-Coulombexpression to determine the shear strength τf along thesliding surface.
The available shear strength τf depends on the type of soil andthe effective normal stress, whereas the mobilized shear stress τdepends on the external forces acting on the soil mass.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Summary of LE methods
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Section of unit width assumed for analysis
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
θ
dW
cu
R
R
Circular analysis – un-drained condition or φu = 0 analysis
Analysis in terms of totalstresses and applies to theshort-term condition for acutting or embankmentassuming soil profile tocomprise fully saturated clay.
D
R
MMFS =
( )Wd
RLcFS Au=
Demerit: Determination of W and d
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Calculation of FS
2211 dWdWM D +=
LRcM uR =
( )2211 dWdWLRcFS u
+=
Section of unit width assumed for analysis
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Un-drained analysis stability charts –Taylor’s method
2θ
dW
c
R
R
β DH
H
α
Nomenclature used fordeveloping stability numbers
D = depth factor
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Un-drained analysis stability charts – Taylor’s method
WdcLRFS =
W = f (γ, H, geometry of failure surface)
⇔ Geometry of failure surface can be characterized by the three angles α, β, and θ
(1)
Rewriting (1): ( )θβαγ ,,HfcFSc
r ==
cr = required cohesion to just maintain a stable slope andf (α, β, θ) is pure number, designated as the Stability number Ns
HcN r
s γ=Taylor’s Stability number
FS is lowest Factor of safetyobtained from circular arcanalysis.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Taylor’s curves
β > 53°
Slope inclination β
β [°] Ns
60 0.191
65 0.199
70 0.208
75 0.219
80 0.232
85 0.246
90 0.261
For φ = 0 soils
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Un-drained analysis stability charts – Taylor’s method
This is because for such steep slopes, the critical failure surfacepasses through the toe of the slope and does not go below thetoe.
γcHc
85.3=Critical height
For β < 53° Ns = f (β, D/H)
For β > 53° Ns = f (β) [all critical slip circles pass through toe]
For gentle slopes, the critical failure surface goes below toe andalways restricted above strong layer (hence depends on itslocation).
For a vertical cut (β = 90°) Ns = 0.26 (short-term condition)
Obtained from Taylorstability number
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Un-drained analysis stability charts – Taylor’s method
Position of the critical slip circle ( for FS = 1) may be limited by two factors:
a) The depth of stratum in which sliding can occur
b) The possible distance from the toe of the rupture surface to the toe of the slope.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT BombayDepth factor D/H
Stability numbers for homogeneous simple slopes for φ = 0
After Taylor (1948)
For example:
By knowing D/H and β -- Nsand n can be obtained fromthis chart
n = 0.65
For D/H = 1; β > 53°n = 0
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Un-drained analysis stability charts – Taylor’s methodImportant points: It is necessary to ignore possibility of tensioncracks – otherwise geometrically similar failuresurfaces do not occur on slopes having differentheights --- hc = f (c and γ) and is not proportional to H.
Taylor’s stability numbers were determined froman analysis of total stress only.
Taylor’s method is practically restricted toproblems involving un-drained saturated clays orto much common cases where the pore pressureis everywhere zero.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
γucz 2
0 =
Pw
Vertical tension crack in cohesive soils
( )lzWd
RLcFS
w
cAu
+
=2
021 γ
l
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The ordinary method of slicesIn this method, the potential failure surface is assumedto be a circular arc with centre O and radius r.
The soil mass (ABCD) above a trial surface (AC) isdivided by vertical planes into a series of slices of widthb.
The base of each slice is assumed to be a straight line.
The factor of safety (FS) is defined as the ratio of theavailable shear strength τf to the shear strength τm whichmust be mobilized to maintain a condition of limitingequilibrium.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The Ordinary method (OM) satisfies the momentequilibrium for a circular slip surface, but neglects boththe interslice normal and shear forces. The advantageof this method is its simplicity in solving the FOS, sincethe equation does not require an iteration process.
The ordinary method of slices
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
rsinα
r
r
h
α
b
A B
CD
l
α
X2
X1
E1
E2
α
FBD of slice i
The method of slices
OLA = length of arc AC
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The method of slices
m
fFSττ
=The FS is taken to be the same for eachslice, implying that there must be mutualsupport between slides. i.e. forces must actbetween slices.
1. Total weight of slice W = γbh
2. Total normal force N = σl ( includes N′ = σ′l and U = ul)u = PWP at the centre of the base and l is the length of the base.
3. The shear force on the base, T = τml
4. Total normal forces on sides E1 and E2
5. The shear forces on the sides, X1 and X2
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The method of slices Considering moments about O, the sum of themoments of the shear forces T on the failure arc ACmust be equal the moment of the weight of the soilmass ABCD.
∑ ∑= αsinWrTr
( )∑ ∑= ατ
sinWlFS
f
Using ( ) lFSlT f
m
ττ ==
∑∑=
ατsinW
lFS f
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The method of slicesFor an analysis in terms of effective stress:
∑∑ ′′+′
=αφσ
sin)tan(
Wlc
FS
∑∑ ′′+′
=α
φsin
tanW
NLcFS a
Equation (1) is exact but approximations areintroduced in determining the forces N′.
(1)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The Fellenius (or Swedish ) SolutionIt is assumed that for each slice the resultant of theinterslice forces is zero.
The solution involves resolving the forces on each slicenormal to the base i.e. N′ = Wcosα - ul
∑∑ −′+′
=α
αφsin
)cos(tanW
ulWLcFS a
Rewriting Equation (1):
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
r
r
α4
A
CD
12
34
567
α1
α3
-α5
The Fellenius (or Swedish ) method of slices
The components of Wcosα andWsinα can be determinedgraphically for each slice.
For an analysis in terms of totalstress the parameters cu and φuare used and the value of u = 0
∑∑+
=α
αφsin
)cos(tanW
WLcFS uau
∑=
αsinWLcFS auFor φu = 0
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
In this solution it is assumed that the resultant forces onthe sides of the slices are horizontal. i.e X1 – X2 = 0For equilibrium the shear force on the base of any slice is:
( )φ′′+′= tan1 NlcFS
T
Resolving forces in the vertical direction:αφααα sintansincoscos ′
′+
′++′=
FSN
FSlculNW
After some rearrangement and using l = b secα:
∑∑
′+
′−+′=)/tan(tan1
sec]tan)([sin
1FS
ubWbcW
FSφα
αφα
Bishop simplified Method (BSM)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Bishop (1955) also showed how non-zero values of theresultant forces (X1-X2) could be introduced into theanalysis but refinement has only a marginal effect onthe factor of safety.
The pore water pressure can be related to the total fillpressure at any point by means of dimensionless porepressure ratio ru = u/γh .
For any slice, ru = u/W/b
∑∑
′+
′−+′=)/tan(tan1
sec]tan)1([sin
1FS
rWbcW
FS u φααφ
α
By rewriting:
Bishop simplified Method (BSM)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Bishop simplified Method (BSM)Bishop’s simplified method (BSM) considers theinterslice normal forces but neglects the intersliceshear forces. It further satisfies vertical forceequilibrium to determine the effective base normalforce (N’).
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Janbu’s simplified methodJanbu’s simplified method (JSM) is based on acomposite slip surface (i.e. non-circular) and the FOSis determined by horizontal force equilibrium. As inBSM, the method considers interslice normalforces (E) but neglects the shear forces (T).
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
21
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Module 5:
Lecture -3 on Stability of Slopes
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slope Stability Analysis Methods
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The ordinary method of slicesIn this method, the potential failure surface is assumedto be a circular arc with centre O and radius r.
The soil mass (ABCD) above a trial surface (AC) isdivided by vertical planes into a series of slices of widthb.
The base of each slice is assumed to be a straight line.
The factor of safety (FS) is defined as the ratio of theavailable shear strength τf to the shear strength τm whichmust be mobilized to maintain a condition of limitingequilibrium.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The Ordinary method (OM) satisfies the momentequilibrium for a circular slip surface, but neglects boththe interslice normal and shear forces. The advantageof this method is its simplicity in solving the FOS, sincethe equation does not require an iteration process.
The ordinary method of slices
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
rsinα
r
r
h
α
b
A B
CD
l
α
X2
X1
E1
E2
α
FBD of slice i
The method of slices
OLA = length of arc AC
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The method of slices
m
fFSττ
=The FS is taken to be the same for eachslice, implying that there must be mutualsupport between slides. i.e. forces must actbetween slices.
1. Total weight of slice W = γbh
2. Total normal force N = σl ( includes N′ = σ′l and U = ul)u = PWP at the centre of the base and l is the length of the base.
3. The shear force on the base, T = τml
4. Total normal forces on sides E1 and E2
5. The shear forces on the sides, X1 and X2
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The method of slices Considering moments about O, the sum of themoments of the shear forces T on the failure arc ACmust be equal the moment of the weight of the soilmass ABCD.
∑ ∑= αsinWrTr
( )∑ ∑= ατ
sinWlFS
f
Using ( ) lFSlT f
m
ττ ==
∑∑=
ατsinW
lFS f
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The method of slicesFor an analysis in terms of effective stress:
∑∑ ′′+′
=αφσ
sin)tan(
Wlc
FS
∑∑ ′′+′
=α
φsin
tanW
NLcFS a
Equation (1) is exact but approximations areintroduced in determining the forces N′.
(1)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The Fellenius (or Swedish ) SolutionIt is assumed that for each slice the resultant of theinterslice forces is zero.
The solution involves resolving the forces on each slicenormal to the base i.e. N′ = Wcosα - ul
∑∑ −′+′
=α
αφsin
)cos(tanW
ulWLcFS a
Rewriting Equation (1):
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
r
r
α4
A
CD
12
34
567
α1
α3
-α5
The Fellenius (or Swedish ) method of slices
The components of Wcosα andWsinα can be determinedgraphically for each slice.
For an analysis in terms of totalstress the parameters cu and φuare used and the value of u = 0
∑∑+
=α
αφsin
)cos(tanW
WLcFS uau
∑=
αsinWLcFS auFor φu = 0
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
In this solution it is assumed that the resultant forces onthe sides of the slices are horizontal. i.e X1 – X2 = 0For equilibrium the shear force on the base of any slice is:
( )φ′′+′= tan1 NlcFS
T
Resolving forces in the vertical direction:αφααα sintansincoscos ′
′+
′++′=
FSN
FSlculNW
After some rearrangement and using l = b secα:
∑∑
′+
′−+′=)/tan(tan1
sec]tan)([sin
1FS
ubWbcW
FSφα
αφα
Bishop simplified Method (BSM)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Bishop (1955) also showed how non-zero values of theresultant forces (X1-X2) could be introduced into theanalysis but refinement has only a marginal effect onthe factor of safety.
The pore water pressure can be related to the total fillpressure at any point by means of dimensionless porepressure ratio ru = u/γh .
For any slice, ru = u/W/b
∑∑
′+
′−+′=)/tan(tan1
sec]tan)1([sin
1FS
rWbcW
FS u φααφ
α
By rewriting:
Bishop simplified Method (BSM)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Bishop simplified Method (BSM)Bishop’s simplified method (BSM) considers theinterslice normal forces but neglects the intersliceshear forces. It further satisfies vertical forceequilibrium to determine the effective base normalforce (N’).
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Janbu’s simplified methodJanbu’s simplified method (JSM) is based on acomposite slip surface (i.e. non-circular) and the FOSis determined by horizontal force equilibrium. As inBSM, the method considers interslice normalforces (E) but neglects the shear forces (T).
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Morgenstern-Price method (M-PM)The Morgenstern-Price method (M-PM) also satisfiesboth force and moment equilibriums and assumes theinterslice force function.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Spencer’s method
Spencer’s method (SM) is the same as M-PMexcept the assumption made for interslice forces. Aconstant inclination is assumed for interslice forcesand the FOS is computed for both equilibriums(Spencer 1967)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
A 45 slope is excavated to a depth of 8m in a deeplayer of saturated clay of unit weight 19 kN/m3: therelevant shear strength parameters are cu = 65 kN/m2
and φu = 0. Determine the factor of safety for the trialfailure surface specified in Figure. The cross-sectionalarea ABCD is 70m2.
Example 1
After Craig (2004)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Figure for Example 1
Example 1
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 1
This is the factor of safety for the trial failure surfaceselected and is not necessarily minimum factor ofsafety.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 1
The minimum factor of safety can be estimated by using FS = cu/NsγH.
Using Taylor’s chart for Ns vs Slope inclination β, For β= 45° and assuming that D is large, the value of Ns is0.18.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Taylor’s curves
β > 53°
Slope inclination β
β [°] Ns
60 0.191
65 0.199
70 0.208
75 0.219
80 0.232
85 0.246
90 0.261
For φ = 0 soils
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Example 2
Using the Fellenius method of slices, determine thefactor of safety, in terms of effective stress, of the slopeshown in Figure for the given failure surface usingpeak strength parameters c′ = 10 kPa and φ′ = 29°. Theunit weight of the soil above and below the watertable is 20 kN/m3.
After Craig (2004)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 2
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Table giving computations (After Craig 2004)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Example 3
2 m 2 m 2 m 2 m 2 m 2 m 1 m
Fellenius method of slices
b =
l = bsecα
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slice W(kN)
c′(kPa)
tanφ′ l (m) N=Wcosα
T=Wsinα
c′l(kN)
ul(kN)
N-ul(kN)
(N-ul) tanφ′
1 27.7 8 0.466 2.3 24.5 -13 18.3 4.8 19.7 9.2
2 96.5 8 0.466 2.1 93.9 -22.5 16.5 14.8 79.1 36.9
3 148 8 0.466 2 148 0 16 22.2 125.8 58.6
4 188.7 8 0.466 2.1 183 44.4 16.5 29 154 72.8
5 199.8 8 0.466 2.3 176.1 94 18.3 34.2 141.9 61.1
6 148 15 0.364 2.8 105.5 103.9 42 31.4 74.1 27
7 37 15 0.364 2 16.5 32.6 30.4 11.4 5.1 1.9
Fellenius method of slices ∑∑∑ −′+′
=T
ulNlcFS
)(tanφ
∑T = 239.4
∑c′ l = 158
∑ = 267.5FS = (158+267.5)/239.4
= 1.78
Solution for Example 2
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Using the Bishop method of slices, determine thefactor of safety in terms of effective stress for the slopedetailed in Figure for the specified failure surface. Thevalue of ru is 0.20 and the unit weight of the soil is 20kN/m3 and the shear strength parameters are c′ = 0kN/m2 and φ′= 33°
Example 3
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Example 3
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 3
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 3
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Comparison of Slope stability analysis methods
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Comparison of LE methods
Grid and radius option used to search for circular CSS
Entry and exit option used to search for circular CSS
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
After Lambe and Whitman, 1969)Schematic diagram slope cross-section
Slope material Properties Value
Unit wt (kN/m3) 19.64
Cohesion (kPa) 4.31
Friction angle (0) 32
Comparison of LE methods
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slice 11 - Ordinary Method
36.661
13.699
29.689
Slope stability analysis (Geo-slope 2012) Slice free body diagram
Ordinary method of slices
1.161
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slice 11 - Bishop Method
36.661
14.727
34.614
40.322
32.668
Slope stability analysis (Geo-slope 2012) Slice free body diagram
1.289
Bishop simplified Method (BSM)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slice 11 - Janbu Method
36.661
15.325
34.2
40.322
32.668
Slope stability analysis (Geo-slope 2012)
Slice free body diagram
Janbu’s simplified method
1.2221.222
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slice 19 - Morgenstern-Price Method
22.893
7.9803
18.608
34.425
17.394
31.108
14.466
Slope stability analysis (Geo-slope 2012) Slice free body diagram
Morgenstern-Price method (M-PM)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Ordinary method of slices (OSM)
Bishop’s simplified method (BSM)
Geo-slope 2012 1.161 1.289
Lambe and Whitman (1969)
1.17 1.3
Comparison of factor of safety
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Aryal (2003)
PLAXIS
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
22
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Module 5:
Lecture -4 on Stability of Slopes
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Sudden drawdown
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Determination of most critical slip surfaceCriteria for most critical slip surface = Minimum factor
of safety
Trial and error approach involves following parameters
a)Center of rotation of the slip surfaceb)Radius of slip surfacec)Distance of intercept of slip surface from the toed)Minimum factor of safety achieved
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Fellenius (1935) proposed empirical approach for cohesive soils (φu = 0)
Slope ratio α Ψ1 : 0.58 29° 40°1 : 1 28° 37°1 : 1.5 26° 35°1 : 2 25° 35°1 : 3 25° 35°1 : 5 25° 37°
α
Ψ
H
β
O1
Draw line through corners of slope at angle α and Ψas per in table.
O1 will be center of rotation for slip circle.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Jumikis (1962) extended the method for c’- φ’ soil
Possible locations of centers for c’- φ’ soil
P
α
Ψ
H
β
H
4.5 H
O1 Center of rotation of critical circle is assumed to lie on PO1line. Point P is at distance H below the toe in vertical direction and 4.5 H away from toe in horizontal direction
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Comparison of LE methods
Grid and radius option used to search for circular CSS
Entry and exit option used to search for circular CSS
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
After Lambe and Whitman, 1969)Schematic diagram slope cross-section
Slope material Properties Value
Unit wt (kN/m3) 19.64
Cohesion (kPa) 4.31
Friction angle (0) 32
Comparison of LE methods
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slice 11 - Bishop Method
36.661
14.727
34.614
40.322
32.668
Slope stability analysis (Geo-slope 2012) Slice free body diagram
1.289
Bishop simplified Method (BSM)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slices data (Bishop’s method) for Lambe and Whitman problem
0
5
10
15
20
25
30
35
40
0 5 10 15
Distance from toe of the slope (m)
Nor
mal
stre
ss a
t the
bas
e of
slic
es (k
Pa)
0
5
10
15
20
25
0 5 10 15
Distance from toe of the slope (m)
Shea
r stre
ss m
obili
sed
(kPa
)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Impenetrable strata
Embankment
FOS with FEM = 1.29
Finite element modeling with help of Plaxis 2D
Impenetrable strata
Embankment
Possible failure surfaces
Slope stability analysis Lambe and whitman problem
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Method of analysis Factor of safetyLimit Equilibrium
Ordinary method of slices 1.161Bishops method 1.289Janbu’s method 1.222Morgenstern-Price method 1.306
Finite EquilibriumStrength reduction factor 1.29
Comparison of FOS in LEM and FEM
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Aryal (2003)
PLAXIS
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Development of phreatic surfaces within the slope
u/γh
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Comparison of Phreatic surfaces measured and computed from SEEP/W
β = 63.43°
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Variation of FS with u/γh
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
A cutting 9 m deep is to be excavated in a saturatedclay of unit weight 19 kN/m3. The design shear strengthparameters are cu = 30 kN/m2 and φu = 0°. A hardstratum underlies the clay at a depth of 11 m belowground level. Using Taylor’s stability method, determinethe slope angle at which failure would occur. What isthe allowable slope angle if a factor of safety of 1.2 isspecified.
Example 4 for Practice
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Example 5 for Practice
For the given failure surface, determine the factor ofsafety in terms of effective stress for the slope detailedin Figure, using the Fellenius method of slices. The unitweight of the soil is 21 kN/m3 and the characteristicshear strength parameters are c′ = 8 kN/m2 and φ′= 32°.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
After Craig (2004)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Rapid Drawdown Condition
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
After the reservoir or dam has been full for sometime, conditions of steady seepage becomeestablished through the dam with the soil below thetop flow line in the fully saturated state. This conditionmust be analysed in terms of effective stress withvalues of pore pressure being determined from theflow net.
Values of ru up to 0.45 are possible in homogeneousdams but much lower values can be achieved indams having internal drainage. The factor of safety forthis condition should be at least 1.5.
Steady state Seepage
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
After a condition of steady seepage has becomeestablished, a drawdown of the reservoir level willresult in a change in the pore water pressuredistribution.
If the permeability of the soil is low, a drawdownperiod measured in weeks may be ‘rapid’ in relation todissipation time and the change in pore water pressurecan be assumed to take place under undrainedconditions.
Rapid drawdown
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slope stability analysis in drawdown condition
Typical variations in water level during drawdownResponse of slope to rapid drawdown
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
'o w wu (h h h )= γ + −
Pore water pressure before drawdown at a point P on a potential failure surface is given by
Change in total major principal stress = Total or Partial removal of water above the slope on P.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
And the change in pore water pressure is then given by
1 w wh∆σ = −γ
1u B∆ = ∆σ
w wB h= γ
Therefore the pore water pressure at P immediately after rapid drawdown is:
ou u u= + ∆
( ) 'w w(h h 1 B h )= γ + − −
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Hence, pore water pressure ratio
usat
urh
=γ
'w w
usat
h hr 1 (1 B)h h
γ= + − − γ
For a decrease in total stresses, the value of B isslightly greater than 1. An upper bound value of rucould be obtained by assuming B = 1 and neglectingh0.
Typical values of ru immediately after drawdown arewithin the range 0.3–0.4. A minimum factor of safety of1.2 may be acceptable after rapid drawdown.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
The pore water pressure distribution after drawdownin soils of high permeability decreases as pore waterdrains out of the soil above the drawdown level.
The saturation line moves downwards at a ratedepending on the permeability of the soil.
A series of flow nets can be drawn for differentpositions of the saturation line and values of porewater pressure obtained. The factor of safety can thusbe determined, using an effective stress analysis, forany position of the saturation line.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Typical flow net in case of drawdown (After Craig, 2004)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Pore pressure ratio (ru) can be used for stability analysis as explained by Bishop and Morgenstern (1960)This method is based on “effective stress method”. It involves following five parameters:
i) Slope angle, ii) Depth factor, iii) angle of shearing resistance (φ’), iv) non-dimensional parameter (c’/ γH), and v) pore pressure ratio (ru).
Factor of safety can be computed by using charts provided by Bishop-Morgenstern (1960).
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Submerged slope of height 7m and slope of 1 V: 3H
Schematic diagram of lope (After Berilgen, 2007)
DH
Drawdown rate (R) = D/H
Seepage and stability analysis for drawdown condition
R1 = 1 m/day (rapid drawdown)R2 = 0.1 m/day (slow drawdown)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Steady state seepage analysis (constant hydraulic boundaries i.e. total head)
Transient seepage analysis (varying hydraulic boundaries i.e. total head)
Stability analysis (consideration of driving forces for failure
i.e. body forces, pore water pressure, etc.)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Property ValueUnit weight (kN/m3) 20Coefficient of permeability (m/sec) 10-6 and 10-8
Cohesion (kPa) 10Internal friction angle (degree) 20
Four cases were studied considering two drawdown rates and two types of soil.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Flow paths during drawdown phenomena
Drawdown
Pore pressure contours at the steady state condition
Steady state seepage analysis using SEEP/W
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
40
45
50
55
60
65
70
75
80
0 5 10 15 20 25 30 35
Time (days)
Pore
wat
er p
ress
ure
(kPa
)
Depletion of phreatic surfaces
P1
Drawdown rate R1 = 1 m/day
Variation of pore water pressure at point “P1”
Pore water pressure dissipation with time
Transient seepage analysis using SEEP/W
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
1
1.25
1.5
1.75
2
2.25
2.5
2.75
0.1 1 10 100
Time (days)
Min
imum
fact
or o
f saf
ety
R = 1 m/day; k = 10-6 m/sec
Critical failure surface at the end of drawdown
Slope stability analysis using SLOPE/W
Factor of safety decreases as drawdown progresses
Critical FOS = 1.497
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Effect of drawdown rate
Transient seepage analysis for R = 1 m/day
Transient seepage analysis for R
Transient seepage analysis for R = 0.1 m/day
More amount of depletion of phreatic surface
At the end of drawdown
At the end of drawdown
K = 10-6 m/sec
K = 10-6 m/sec
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
20
30
40
50
60
70
80
90
0.1 1 10 100 1000
Time (days)
Pore
wat
er p
ress
ure
(kPa
)
R = 1 m/dayR = 0.1 m/day
Variations of pore water pressure with time at the point “P1”
Higher dissipation of pore water pressure in case of slow drawdown
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
0
0.5
1
1.5
2
2.5
3
3.5
4
0.1 1 10 100 1000Time (days)
Min
imum
fact
or o
f saf
ety
R = 1 m/day; k = 10-6 m/secR = 0.1 m/day; k = 10-6 m/sec
Higher factor of safety due to dissipation of pore water pressure
Variations of factor of safety with seepage time
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Effect of coefficient of permeability of soil
Transient seepage analysis for k = 1x 10-6 m/sec
Transient seepage analysis for k = 1x 10-8 m/sec
R = 1m/day
R = 1m/day
Depletion of phreatic surface is marginal for soils with k
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
20
30
40
50
60
70
80
90
0.1 1 10 100
Time (days)
Pore
wat
er p
ress
ure
(kPa
)
k = 10-6 m/seck = 10-8 m/sec
Dissipation of pore water pressure is less for soils with low k
Variations of pore water pressure with time at the point “P1”
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
0
0.5
1
1.5
2
2.5
3
3.5
4
0.1 1 10 100Time (days)
Min
imum
fact
or o
f saf
ety
R = 1 m/day; k = 10-6 m/secR = 1 m/day; k = 10-8 m/sec Critical FOS = 1
Higher FOS for soils having high coefficient of permeability
Variations of factor of safety with seepage time
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Total stress analysis
Requirement Comment
Total stresses in soil mass Common to both methods
Strength of soil when subjected to changes in total stress similar to stress changes in field
Accuracy is doubtful, since strength depends upon induced pore pressures
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Effective stress AnalysisRequirement CommentTotal stresses in soil mass Common to both methodsStrength parameters of soil in relation with effective stress
considerable accuracy, since this is insensitive to test condition
Determination of changes in external loads
Accuracy depends on measurement of pore water pressure
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slopes subject to rainfall
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slope instability is a common problem in manyparts of the world, and cause thousands of deathsand severe infrastructural damage each year.
Rainfall has been identified as a major cause fortriggering landslides and slope failure.
The mechanism leading to slope failure is that thepore water pressure starts increasing when waterinfiltrates the unsaturated soil.
The problem becomes severe if the fill material haslow- permeability, and cannot dissipate the porewater pressure generated due to rainfall.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
To investigate the effect of rainfall on slope stability, alimit equilibrium analysis was carried out by usingSLOPE/W, a product of Geostudio (2012) software.
Two slope configurations (45° and 63° inclination)were selected, and were subjected to rainfall ofvarious intensities (2mm/hr-80 mm/hr) for 24 hrs.
Phreatic surfaces were fed into SLOPE/W, and stabilityanalyses were performed at the onset of rainfall,during rainfall, and upto 24 hours after rainfall.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slope configuration selected (45° inclination)
Applied rainfall intensity
Water table position
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Soil parameters used in SLOPE/W(FOS was computed by Bishop’s modified method of slices)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Note: Slope stability reduces with increasing intensities of rainfall
Effect of rainfall intensity on Slope stability
0.8
1
1.2
1.4
1.6
1.8
2
0 10 20 30 40 50 60
Fact
or o
f saf
ety
Time (hours)
2 mm/hr9 mm/hr22 mm/hr36 mm/hr80 mm/hrLimiting factor of safety
Slope inclination: 45°
Rainfall stopped
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Note: Steeper slopes have lower initial FOS, and the effect of rainfall onsuch slopes is more devastating as compared to flatter ones.
Effect of rainfall intensity on Slope stability
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 10 20 30 40 50 60
Fact
or o
f saf
ety
Time (hours)
2 mm/hr
9 mm/hr
22 mm/hr
36 mm/hr
80 mm/hr
Limiting factor of safety
Slope inclination: 63°
Rainfall stopped