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Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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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