SLOPE STABILITY Omitted parts: Sections 15.13, 15.14,15.15 Chapter 15
SLOPE STABILITY
Omitted parts:
Sections 15.13, 15.14,15.15
Chapter 15
TOPICS
Introduction
Types of slope movements
Concepts of Slope Stability Analysis
Factor of Safety
Stability of Infinite Slopes
Stability of Finite Slopes with Plane Failure Surface
o Culmann’s Method
Stability of Finite Slopes with Circular Failure Surface
o Mass Method
o Method of Slices
TOPICS
Introduction
Types of slope movements
Concepts of Slope Stability Analysis
Factor of Safety
Stability of Infinite Slopes
Stability of Finite Slopes with Plane Failure Surface
o Culmann’s Method
Stability of Finite Slopes with Circular Failure Surface
o Mass Method
o Method of Slices
What is a Slope?An exposed ground surface that stands at an angle with the horizontal.
Why do we need slope stability?In geotechnical engineering, the topic stability of slopes deals with:
1. The engineering design of slopes of man-made slopes in advance
(a) Earth dams and embankments,(b) Excavated slopes,(c) Deep-seated failure of foundations and retaining walls.
2. The study of the stability of existing or natural slopes of earthworks andnatural slopes.
o In any case the ground not being level results in gravity components of theweight tending to move the soil from the high point to a lower level. Whenthe component of gravity is large enough, slope failure can occur, i.e. the soilmass slide downward.
o The stability of any soil slope depends on the shear strength of the soiltypically expressed by friction angle (f) and cohesion (c).
SLOPE STABILITY
TYPES OF SLOPE
A. Natural slope
• Hill sides
• Mountains
• River banks
B. Man-made slope
• Fill (Embankment)
• Earth dams
• Canal banks
• Excavation sides
• Trenches
• Highway Embankments
Slopes can be categorized into two groups:
• Some of these failure can cause dramatic impact on livesand environment.
Slope failures cost billions of $every year in some countries
Case histories of slope failure
Slope failures cost billions of $every year in some countries
Case histories of slope failure
Bolivia, 4 March 2003, 14 people killed, 400 houses buried
Case histories of slope failure
Brazil, January 2003, 8 people killed
Case histories of slope failure
LaConchita California Slump
Case histories of slope failure
Case histories of slope failure
Case histories of slope failure
Slides: Rotational (slump)
Case histories of slope failure
TOPICS
Introduction
Types of slope movements
Concepts of Slope Stability Analysis
Factor of Safety
Stability of Infinite Slopes
Stability of Finite Slopes with Plane Failure Surface
o Culmann’s Method
Stability of Finite Slopes with Circular Failure Surface
o Mass Method
o Method of Slices
Falls Topples Slides Flows Creep Lateral spreads Complex
o Slope instability (movement) can be classified into sixdifferent types:
Types of Slope Movements
Falls
• Rapidly moving mass of material (rock or soil) that travels mostly through theair with little or no interaction between moving unit and another.
• As they fall, the mass will roll and bounce into the air with great force and thusshatter the material into smaller fragments.
• It typically occurs for rock faces and usually does not provide warning.• Analysis of this type of failure is very complex and rarely done.
• Gravitational effect and shear strength
Gravity has two components of forces:
T driving forces:
N resisting forces (because of friction)
T= W. sin b
N = W. cos bT
S = N tan f
the interface develop its
resistance from friction (f):
b
A = effective Base Area of sliding block
f = friction
N
S
In terms of stresses:
S/A = N/A tan f
tf = s tan f
or
Boulder
Falls
Falls
Topples
This is a forward rotation of soil and/or rock mass about an axis
below the center of gravity of mass being displaced.
Back-Scrap
Bulging at
Toe
Slides
Slides
A. Translational (planar)
o Movements occur along planar failure surfaces that may run more-or lessparallel to the slope. Movement is controlled by discontinuities or weakbedded planes.
A
Weak bedding
plane
(Planar)
Occur when soil of significantly
different strength is presented
Slides
B. Rotational (curved)
This is the downward movement of a soil mass occurring on an almost circular surface of rupture.
B
Back-Scrap
Bulging
Curved escarpment
(Slumps)C. Compound (curved)
Slides
Soil nails
Reinforcement
Slides
شدادات
Reinforcement
Possible failure
surface
Anchors
Slides
o The materials moves like aviscous fluid. The failure planehere does not have a specificshape.
Flows
It can take place in soil withhigh water content or in drysoils. However, this type offailure is common in the QUICKCLAYS, like in Norway.
Flows
Creep
• It is the very slow movement of slope material that occur over a
long period of time
• It is identified by bent post or trees.
o Lateral spreads usually occur on very gentle slopes or essentially flat terrain,especially where a stronger upper layer of rock or soil undergoes extensionand moves above an underlying softer, weaker layer.
weaker layer
Lateral spreads
Complex
1. Falls
2. Topples
3. Slides• A. Translational (planar)
• B. Rotational (slumps)
4. Flow
5. Creep
6. Lateral Spread
7. Complex
Complex movement is by a combination of
one or more of the other principal types of
movement.
Many slope movements are complex,
although one type of movement
generally dominates over the others at
certain areas or at a particular time.
1. Falls
2. Topples
3. Slides• Translational (planar)
• Rotational (curved)
4. Flows
5. Creep
6. Lateral spreads
7. Complex
Slide is the most
common mode of
slope failure, and it will
be our main focus in
this course
Types of Slope Failures
In general, there are six types of slope failures:
Types of Slide Failure Surfaces
• Failure of slopes generally occur along surfaces known as failure surfaces.The main types of surfaces are:
• Planar Surfaces: Occurs in frictional,non cohesive soils
• Rotational surfaces: Occurs in cohesive soils
Circular surface (homogeneous soil)
Non-circular surface (non-homogeneous soil)
Types of Slide Failure Surfaces
• Transitional Slip Surfaces:When there is a hard stratum at arelatively shallow depth
• Compound Slip Surfaces:When there is hard stratum at some depth that intersectswith the failure plane
Slid
es
Translational (planar)
InfiniteLong plane
failure surface
FinitePlane failure
surface
Rotational (curved)
Finite
Above the toe
Through the toe
Deep seated
Failure surface 1
2
3
Types of Failure Surfaces
Stability of infinite slopes
Stability of finite slopes with plane
failure surfaces
Stability of finite slopes with circular
failure surfaces
1
2
3
Types of Failure Surfaces
Types of Failure Surfaces Considered in this Course are
TOPICS
Introduction
Types of slope movements
Concepts of Slope Stability Analysis
Factor of Safety
Stability of Infinite Slopes
Stability of Finite Slopes with Plane Failure Surface
o Culmann’s Method
Stability of Finite Slopes with Circular Failure Surface
o Mass Method
o Method of Slices
In general we need to check
The stability of a given existed slope
Determine the inclination angle for a slope that we want to
build with a given height
The height for a slope that we want to build with a given
inclination
Concepts of Slope Stability Analysis
Methodology of Slope Stability Analysis
It is a method to expresses the relationship between resisting forces anddriving forces
• Driving forces – forces which move earth materials downslope. Downslopecomponent of weight of material including vegetation, fill material, orbuildings.
• Resisting forces – forces which oppose movement. Resisting forces includestrength of material
• Failure occurs when the driving forces (component of thegravity) overcomes the resistance derived from the shearstrength of soil along the potential failure surface.
The analysis involves determining and comparing the shear stress developedalong the most likely rupture surface to the shear strength of soil.
Methodology of Slope Stability Analysis
1. Assume a probable failure surface.
2. Calculate the factor of safety by determining and comparing
the shear stress developed along the most likely rupture
surface to the shear strength of soil.
3. Repeat steps 1 and 2 to determine the most likely failure
surface. The most likely failure surface is the critical surface
that has a minimum factor of safety.
4. Based on the minimum FS, determine whether the slope is
safe or not.
Slope Stability Analysis Procedure
Methods of Slope Stability Analysis
o Limit equilibrium method
o Limit analysis method
o Numerical methods
We will consider only the limit equilibrium method, since it isthe oldest and the mostly used method in practice.
Assumptions of Stability Analysis
o The problem is considered in two-dimensions
o The failure mass moves as a rigid body
o The shear strength along the failure surface is isotropic
o The factor of safety is defined in terms of the average shear
stress and average shear strength along the failure surface
42
TOPICS
Introduction
Types of slope movements
Concepts of Slope Stability Analysis
Factor of Safety
Stability of Infinite Slopes
Stability of Finite Slopes with Plane Failure Surface
o Culmann’s Method
Stability of Finite Slopes with Circular Failure Surface
o Mass Method
o Method of Slices
43
StressShear StrengthShear
Force DrivingForce Resisting safety ofFactor
d
fsF t
t
tf= Avg. Shear strength of soil
td= Avg. Shear stress developed along the failure surface
Factor of Safety
44
Fs is the ratio of resisting forces to the driving forces, or
• The most common analytical methods of slope stability use a
factor of safety FS with respect to the limit equilibrium condition,
FS < 1 unstable
FS ≈ 1 marginal
FS >> 1 stable
Generally, FS ≥ 1.5 is acceptable
for the design of a stable slope
average shear strength of the soil.
average shear stress
developed along the potential
failure surface.
Shear stress (driving movement)
Shear strength (resisting movement)
(developed)
(Available)
If factor safety Fs equal to or less than 1, the slope is
considered in a state of impending failure
Factor of Safety
45
Causes of slope failure
1. External causes
These which produce increase of shear stress, like steepeningor heightening of a slope, building on the top of the slope
2. Internal causes
These which cause failure without any change in externalconditions, like increase in pore water pressure.
Therefore, slopes fail due either to increase in stress orreduction in strength.
46
Where:c’ = cohesion f’ = angle of internal friction
= cohesion and angle of friction that develop along the potential failure surface
ddc f ,
ff FFFFF csc then When
Factor of safety with respect to cohesion
Factor of safety with respect to friction
When the factor of safety with respect to cohesion is equal to the
factor of safety with respect to friction, it gives the factor of safety
with respect to strength, or
Other aspects of factor of safety
Factor of Safety
47
TOPICS
Introduction
Types of slope movements
Concepts of Slope Stability Analysis
Factor of Safety
Stability of Infinite Slopes
Stability of Finite Slopes with Plane Failure Surface
o Culmann’s Method
Stability of Finite Slopes with Circular Failure Surface
o Mass Method
o Method of Slices
48
What is an Infinite slope?
• Slope that extends for a relatively long distance and hasconsistent subsurface profile can be considered as infinite slope.
• Failure plane parallel to slope surface.
• Depth of the failure surface is small compared to the height ofthe slope.
• For the analysis, forces acting on a single slice of the sliding massalong the failure surface is considered and end effects isneglected.
Stability of Infinite Slopes
49
Force parallel to the plane AB Ta = W sin b = g LH sin b
Infinite slope – no seepage
o we will evaluate the factor of safety against a possible slope failurealong a plane AB located at a depth H below the ground surface.
o Let us consider a slope element abcd that has a unit lengthperpendicular to the plane of the section shown.
o The forces, F, that act on the faces ab and cd are equal and opposite andmay be ignored.
The shear stress at the base of the slope element can be given by
The resistive shear stress is given by
(*)
50
The effective normal stress at the base of the slope element is given by
(**)
For Granular Soil (i.e., c = 0)b
f
tan
tan sF
This means that in case of infinite slope in sand, the value of Fs isindependent of the height H and the slope is stable as long as b < f’
(***)
Equating R.H.S. of Eqs. (*) and (**) gives
Infinite slope – no seepage
51
Case of Granular soil – Derivation From Simple Statics
Equilibrium of forces on a slice:
FSDriving Forces
Resisting Forces
L
L
Extra
52
The depth of plane along which critical equilibrium occurs isobtained by substituting Fs = 1 and H = Hcr into Eq. (***)
Critical Depth, Hcr
Infinite slope – no seepage
53
(*)
(**)
Seepage is assumed to be parallel to the slope
and that the ground water level coincides with
the ground surface.
The resistive shear stress developed at
the base of the element is given by
The shear stress at the base of
the slope element can be given
Infinite slope – with steady state seepage
54
Substituting Eq. (****) Into Eq. (***) and solving for Fs gives
Equating the right-hand sides of Eq. (*) and Eq. (**) yields
b
f
bbg tan
tan
tan cos H
c2
sF
No seepage
(***)
(****)
Recall
Infinite slope – with steady state seepage
55
EXAMPLE
56
EXAMPLE
57
EXAMPLE
58
• Cohesive Soils
With seepage No seepage
tan
'tan'
tan2cos
'
''
'tan'tan
b
f
g
g
bbg
ff
satHsat
csF
sF
cd
c
sFd
)'tan'tan(2cos
'
fgbgb
sat
ccrH
tan
'tan
tan2cos
'
''
'tan'tan
b
f
bbg
ff
H
csF
sF
cd
c
sFd
)'tan(tan2cos
1'
d
ccrH
fbbg
Stability of Infinite Slopes
59
Stability of Infinite Slopes
Granular Soils
With seepage No seepage
Independant of H Slope is stable as long as b < f
tan
'tan'
tan
'tan'
tan2cos
'0.0'
b
f
g
g
b
f
g
g
bbg
satsF
satHsat
csF
c
tan
'tan
tan
'tan
tan2cos
'0.0'
b
f
b
f
bbg
sF
H
csF
c
60
TOPICS
Introduction
Types of slope movements
Concepts of Slope Stability Analysis
Factor of Safety
Stability of Infinite Slopes
Stability of Finite Slopes with Plane Failure Surface
o Culmann’s Method
Stability of Finite Slopes with Circular Failure Surface
o Mass Method
o Method of Slices
61
o For simplicity, when analyzing the stability of a finite slopein a homogeneous soil, we need to make an assumptionabout the general shape of the surface of potential failure.
o However, considerable evidence suggests that slopefailures usually occur on curved failure surfaces
,
o The simplest approach is to approximate the surface ofpotential failure as a plane.
o Hence most conventional stability analyses of slopes havebeen made by assuming that the curve of potential slidingis an arc of a circle.
Stability of Finite Slopes with Plane Failure Surface
62Plane Failure Surface
simple wedge
o Culmann’s analysis is based on the assumption that the failure of a slopeoccurs along a plane when the average shearing stress tending to cause theslip is more than the shear strength of the soil.
o Also, the most critical plane is the one thathas a minimum ratio of the averageshearing stress that tends to cause failureto the shear strength of soil.
o The method gives reasonablyaccurate results if the slope isvertical or nearly vertical.
o Culmann’s method assumes that the critical surface of failure is a planesurface passing through the toe.
Culmann’s Method
63
• The forces that act on the mass are shown in the figure, where trial failureplane AC is inclined at angle q with the horizontal.
• A slope of height H and that rises at an angle b is shown below.
The average shear stress on the
plane AC
Ta = W Sin q
Similar procedures as for infinite slope, only different
geometry. Also here we made optimization.
t (*)
Culmann’s Method
64
The average resistive shearing stress (Developed shear strength) developed
along the plane AC also may be expressed as
Culmann’s Method
Na
s’
td (**)
Equating the R.H.S of Eqs. (*) and (**) gives
(***)
65
Culmann’s Method
• To determine the critical failure plane, we must use theprinciple of maxima and minima (for Fs=1 and for given valuesof c’, f’, g, H, b) to find the critical angle q:
Critical failure plane
• The expression in Eq. (***) is derived for the trial failure planeAC.
• Substitution of the value of q = qcr into Eq. (***) yields
(****)
66
• For purely cohesive soils c 0 f = 0.
The maximum height of the slope for which critical equilibrium occurs can
be obtained by substituting iinto into Eq. (****)
Culmann’s Method
67
Culmann’s Method
• Steps for Solution
A. If Fs
is given; H is required
cos(1
cossin'c4 4.
)'d
-
'd H
sF
'tan '
dtan .3
sF
'c '
dc 2.
sFF
cF .1
d
f
f
ff
f
b
b
g
C
g
f‘ H
b
q
68
Culmann’s Method
• Steps for Solution
B. If H is given; Fs
is required
51 stepsRepeat .7
F another try F c
F If 6.
F c
Fs
F F c
F if Check 5.
'd
c
'c
cF .4
'd
)'d
- '
dc .3
sF
'tan '
dtan 2.
F Assume .1
cossin
cos(1
4
H
ff
ff
f
f
ff
f
b
bg
69
A cut is to be made in a soil having properties as shown in the
figure below.
If the failure surface is assumed to be finite plane, determine the
followings:
(a) The angle of the critical failure plane.
(b) The critical depth of the cut slope
(c) The safe (design) depth of the cut slope. Assume the factor of
safety (Fs=3)?
H
45o
g = 20 kN/m3
f’=15o
c’=50 kPa
Given equation:
EXAMPLE
70
(a) The angle of the critical failure plane q
can be calculated from:
(b) The critical depth of the cut slope canbe calculated from:
H g = 20 kN/m3
f’=15o
c’=50 kPa
45o
Key Solution
b 45o
f’ 15o
d
(c) The safe (design) depth of the cutslope.
where: c’d and f’d can be determined from:
71
TOPICS
Introduction
Types of slope movements
Concepts of Slope Stability Analysis
Factor of Safety
Stability of Infinite Slopes
Stability of Finite Slopes with Plane Failure Surface
o Culmann’s Method
Stability of Finite Slopes with Circular Failure Surface
o Mass Method
o Method of Slices
72
Slid
es
Translational (planar)
InfiniteLong plane
failure surface
FinitePlane failure
surface
Rotational (curved)
Finite
Above the toe
Through the toe
Deep seated
Failure surface 1
2
3
Types of Failure Surfaces
73
Modes of Failure
Shallow slope failure
Finite Slopes with Circular Failure Surface
i. Slope failure
• Surface of sliding intersects the slope at or above its toe.
Under certain circumstances, a shallow slopefailure can occur.
1. The failure circle is referred to as a toe circle if itpasses through the toe of the slope
1. The failure circle is referred to as a slope circle ifit passes above the toe of the slope.
ii. Shallow failure
74
Firm Base
H
iii. Base failure
o The surface of sliding passesat some distance below thetoe of the slope.
o The circle is called themidpoint circle because itscenter lies on a vertical linedrawn through the midpoint ofthe slope.
o For b 53o always toe
o For b < 53o could be toe, slope, or midpoint and that depends on depthfunction D where:
Depth function:
Finite Slopes with Circular Failure Surface
75
• Summary
• Toe Circle all circles for soils with f > 3° & b > 53°
• Slope Circle always for D = 0 & b < 53°
• Midpoint Circle always for D > 4 & b < 53°
Finite Slopes with Circular Failure Surface
76
Various procedures of stability analysis may, in general, be divided into twomajor classes:
2. Method of slices
1. Mass procedure
Types of Stability Analysis Procedures
• In this case, the mass of the soil abovethe surface of sliding is taken as a unit.
• Most natural slopes and many man-made slopes consist of more than onsoil with different properties.
• This procedure is useful when the soilthat forms the slope is assumed to behomogeneous.
• In this case the use of mass procedureis inappropriate.
77
• In the method of slices procedure, the soil above the surface of sliding isdivided into a number of vertical parallel slices. The stability of eachslice is calculated separately.
b
2
1
1
2
WV
EE
V T
N'a
h
R
R
O
W
xa
• It is a general method that can be used for analyzing irregular slopes innon-homogeneous slopes in which the values of c’ and f’ are notconstant and pore water pressure can be taken into consideration.
Types of Stability Analysis Procedures
78
TOPICS
Introduction
Types of slope movements
Concepts of Slope Stability Analysis
Factor of Safety
Stability of Infinite Slopes
Stability of Finite Slopes with Plane Failure Surface
o Culmann’s Method
Stability of Finite Slopes with Circular Failure Surface
o Mass Method
o Method of Slices
79
2. Slopes in Homogeneous clay Soil with c 0 , f = 0
Determining factor of safety using equilibrium equations (Case I)
Mdriving = Md = W1l1 – W2l2
W1 = (area of FCDEF) gW2 = (area of ABFEA) g
Mresisting = MR = cd (AED) (1) r= cd r2q
Mass Procedure
1.Slopes in purely cohesionless soil with c = 0, f 0
Failure generally does not take place in the form of a circle. So we will notgo into this analysis.
80
W1
W2
l1l2
Mdriving = Md = W1l1 – W2l2
Mresisting = MR = cd r2q
Mass Procedure
81
• The minimum value of the factor of safety thus obtained is the factor ofsafety against sliding for the slope, and the corresponding circle is thecritical circle.
• The potential curve of sliding, AED, was chosen arbitrarily.
• The critical surface is that for which the ratio of Cu to Cd is a minimum. Inother words, Cd is maximum.
• To find the critical surface for sliding, one must make a number of trialsfor different trial circles.
REMARKS
82
• Fellenius (1927) and Taylor (1937) have analytically solved forthe minimum factor of safety and critical circles.
• They expressed the developed cohesion as
H or
soil oft unit weighγ
slope ofheight H
number Stability m
Where
g
g
d
d
cm
mHc
• We then can calculate the min Fs as
• The critical height (i.e., Fs 1) of the slope can be evaluated by
substituting H = Hcr and cd = cu (full mobilization of the undrained
shear strength) into the preceding equation. Thus,
Finite Slopes with Circular Failure Surface
83
The results of analytical solution to obtain critical circles was representedgraphically as the variation of stability number, m , with slope angle b.
Toe, Midpoint or slope circles Toe slope
Firm Stratum
b
m is obtained from this chart depending on angle b
Finite Slopes with Circular Failure Surface
84
For a slope angle b > 53°, the critical circle is always a toe circle. The
location of the center of the critical toe circle may be found with the aid of
Figure 15.14.
For b < 53°, the critical circle may be a toe, slope, or midpoint circle,
depending on the location of the firm base under the slope. This is called
the depth function, which is defined as
Failure Circle
Finite Slopes with Circular Failure Surface
85
(radius)
Figure 15.13
Location of the center of the critical toe circle
The location of the center of the critical toe circle may be found with the aid of Figure 15.13
86
When the critical circle is a midpoint circle (i.e., the
failure surface is tangent to the firm base), its position
can be determined with the aid of Figure 15.14.
Figure 15.14
Firm base
Finite Slopes with Circular Failure Surface
87
The location of these circles can be
determined with the use of Figure 15.15
and Table 15.1.
Figure 15.15
Note that these critical toe circle are not necessarily the most critical circles that exist.
Critical toe circles for slopes
with b < 53°
88
How to use the stability chart? Given: b 60o, H, g, cu Required: min Fs
m = 0.195
1. Get m from chart
2. Calculate cd from
3. Calculate Fs
mHcd g
d
us
c
cF
89
Given: b 30o, H, g, cu, HD (depth to hard stratum) Required: min. Fs
m = 0.178
1. Calculate D = HD/H
2. Get m from the chart
3. Calculate cd from
4. Calculate Fs
mHcd g
d
us
c
cF
D = Distance from the top surface of slope to firm base
Height of the slope
Note that recent investigation put angle b at 58o instead of the 53o value.
How to use the previous chart?
90Rock layer
EXAMPLE
91
D=1.5m
SOLUTION
92
SOLUTION
93
The results of analytical solution to obtain critical circles was representedgraphically as the variation of stability number, m , with slope angle b.
Toe, Midpoint or slope circles Toe slope
Firm Stratum
b
m is obtained from this chart depending on angle b
Slopes in Homogeneous clay Soil with c 0 , f = 0
94
o Several trails must be made to obtain the most critical sliding surface,minimum factor of safety or along which the developed cohesion is a maximum
numberstabilitymfcd H
fqbag
,,, fqbag ,,,H fcd
o The maximum cohesion developed along the critical surface as
o Determination of the magnitude of described previously is based on atrial surface of sliding.
o The results of analytical solution to obtain minimum Fs was representedgraphically as the variation of stability number, m , with slope angle b forvarious values of f’ (Fig. 15.21).
o Solution to obtain the minimum Fs using this graph is performed by trial-and-error until Fs = Fc’=Ff’
o Since we know the magnitude and direction of W and the direction of Cd
and F we can draw the force polygon to get the magnitude of Cd.
o We can then calculate c’d from
REMARKS
Slopes in clay Soil with f = 0; Cu Increasing with Depth
Slopes in clay Soil with f = 0; Cu Increasing with Depth
EXAMPLE
98
o The Friction Circle method (or the f-Circle Method) is very useful forhomogenous slopes. The method isgenerally used when both cohesive andfrictional components are to be used.
Slopes in Homogeneous C’ – f’ Soils
o Here the situation is more complicatedthan for purely cohesive soils.
o The pore water pressure is assumed to be zero
o F—the resultant of the normal and frictionalforces along the surface of sliding. Forequilibrium, the line of action of F will passthrough the point of intersection of the lineof action of W and Cd.
99
fqbag ,,,H fcd
mHcd g
Friction Circle method
100
1. Assume fd (Generally start with = f’
i.e. full friction is mobilized)
2. Calculate
3. With fd and b Use Chart to get m
4. Calculate
5. Calculate
6. If Fc’ = Ff’ The overall factor of safety
Fs = Fc’ = Ff’
7. If Fc’≠ Ff’ reassume fd and repeat steps 2through 5 until Fc’ = Ff’
OrPlot the calculated points on Fc versus Fφ
coordinates and draw a curve through the points.[see next slide]. Then Draw a line through theorigin that represents Fs= Fc = Fφ
mHcd g
d
cc
cF
Procedures of graphical solutionGiven: H, b, g, c’, f’ Required: Fs
Taylor’s stability
number
101
Note: Similar to Culmann procedure for planar mechanism but here Cd is foundbased on m. In Culmann’s method Cd is found from analytical equation.
Procedures of graphical solutionGiven: H, b, g, c’, f’ Required: Fs
102
Hcr means that Fc’ = Ff’ =Fs = 1.0
1. For the given b and f’, use Chart to get m.
2.Calculate
m
cHcr
g
Given: b, g, C ’, f ’ Required: Hcr
Calculation of Critical Height
103
SUMMARY
f = 0
C f
Mass Procedure – Rotational mechanismneed only the use of Taylor’s chart.
104
• Example
• Given: cu
= 40 kN/m2
& g = 17.5 kN/m3
• Required:
1. Max. Depth
2. Radius r when Fs=1
3. Distance BC
• b = 60 °
> 53 °
from Fig.15.13 m= 0.195
60 °
muccrH 72.11
195.0*5.17 m
40
g
2
sin
H
2
ACDC
DCcr
2
ins
r
a
q
m 9.97 60cot - cot AE - AF EF BC
m 28.17 6.25)3in5)(3 (sin2
72.11
2
in sin2
o5.27 o53o06for 15.14 Fig.
a
qa
qab
crH
ss
crHr
andFrom
EXAMPLE
105
EXAMPLE
106
SOLUTION
107
SOLUTION
108
SOLUTION
109
1.42
SOLUTION
110
15 m
g = 16 kN/m3
C = 40 kN/m2
f = 15o
•Using Taylor’s stability chart determine the factor of safety for the slope shown in Fig.1.•For the same slope height, what slope angle must be used if a factor of safety of 1.5 is required?
50o
10 m
2nd Midterm Fall 1436-1437H QUESTION #2
111
fd Ff=tan f/tan fd m Cd = g H m Fc = C/Cd
15 1 0.092 14.7 2.70
10 1.52 0.116 18.6 2.20
7.5 2.0 0.125 20 2.0
a)
Fs = 2.0
b)
Cd = 40/1.5 =26.7 kpa
mHcd g26.7 = 16 X10 X m
m = 0.167
tan fd = tan f/1.5
fd
fd=10.1o
At m = 0.167 and fd=10.1o from chart b =75o
SOLUTION
112
Method of Slices
• Method of Slices
• Non-homogenous soils (mass procedure is not accurate)
• Soil mass is divided into several vertical Parallel slices
• The width of each slice need not be the same
• It is sometimes called the Swedish method
113
• It is a general method that can be used for analyzing irregular slopes innon-homogeneous slopes in which the values of c’ and f ’ are notconstant.
g1, c’1, f’1
g2, c’2, f’2
g3, c’3, f’3
Non-homogeneous Slope Irregular Slope
g, c’, f’b1
b2
Method of Slices
• Because the SWEDISH GEOTECHNCIAL COMMISION used this methodextensively, it is sometimes referred to as the SWEDISH Method.
• In mass procedure only the moment equilibrium is satisfied. Here attemptis made to satisfy force equilibrium.
114
• The soil mass above the trial slip surface is divided into several vertical parallelslices. The width of the slices need not to be the same (better to have it equal).
• The base of each slice is assumed to be a straight line.
• The inclination of the base to the horizontal is a.
• The height measured in the center line is h.
• The height measured in the center line is h.
• The accuracy of calculation increases if the number of slices is increased.
• The procedure requires that aseries of trial circles are chosenand analyzed in the quest forthe circle with the minimumfactor of safety.
Tr
Method of Slices
115
Method of Slices
• Forces acting on each slice
• Total weight wi=ghb
• Total normal force at the base Nr=s*L
• Shear force at the base Tr=t*L
• Total normal forces on the sides, Pn
and Pn+1
• Shear forces on the sides, Tn
and Tn+1
• 5 unknowns Tr ,P
n ,P
n+1 ,T
n ,T
n+1
• 3 equations SFx=0 , SF
y=0 ,SM=0
• System is statically indeterminate
• Assumptions must be made to solve the problem
• Different assumptions yield different methods
• Two Methods:
• Ordinary Method of Slices (Fellenius Method)
• Bishop’s Simplified Method of Slices
116
Method of Slices
For the whole sliding mass
sin*
)*tan**(
sin*
*
*
sF
sin*
*
sF
*
sin*
0*-sin*r*
0o
M
a
fsa
t
ta
tt
a
a
W
ln
lcFs
W
lf
Fs
lf
W
lf
ld
T
TW
rTW
S
S
S
S
S
SS
SS
S
117
Method of Slices
Slices of Method Simplified sBishop'•
Slices of MethodOrdinary
:Methods Two
N force of valuethe
findingin introduced are ionsapproximatbut exact isEquation
sin*
*tan*
*
sin*
)*tan**(
•
a
f
sa
fs
W
NlcFs
Nln
W
ln
lcFs
S
SS
SS
S
S
118
Fellenius’ Method
Assumption
For each slice, the resultant of the interslice forces is
zero.
The resultants of Pn
and Tn
are equal to the resultants
of Pn+1
and Tn+1
, also their lines of actions coincide.
Rn
Rn+1
Ordinary Method of Slices
119
Ordinary Method of Slices
n
sin*n
W
lu
c
sF
0 u
cc
:condition undrained For
n
sin*n
W
)tann
cos*n
Wn
l*c(
sF
nnr
n
cos*WN
0F )rT fromaway stay (to
aS
f
aS
faS
a
S
n
120
Steps for Ordinary Method of Slices
• Draw the slope to a scale
• Divide the sliding wedge to various slices
• Calculate wn
and an
for each slice
• an
is taken at the middle of the slice
• Calculate the terms in the equation
• Fill the following table
Slice# wn
an
sin an
cos an
ln
wn sin a
nw
n cos a
n
n
sin*n
W
)tann
cos*n
Wn
l*c(
sF
aS
faS
an
-ve
an
+ve
wn
wn
Ordinary Method of Slices
121
Find Fs
against sliding
Use the ordinary method of slices
EXAMPLE
122
Assumption
For each slice, the resultant of the interslice forces is
Horizontal.
i.e. Tn
=Tn+1
Bishop’s Simplified Method of Slices
123
Bishop’s Simplified Method of Slices
n
s
rn
s
nnrn
s
r
s
nr
s
nn
s
nr
n
s
nndr
nrnrn
1n ny
sinF
tanNsin
F
lccos*NW
F
tanN
F
lcT
F
tanl
F
lcT
lF
tancl*T
sin*Tcos*NW
0F )P andP fromaway stay (to
af
a
a
f
fs
fs
t
aa
S
y
124
Bishop’s Simplified Method of Slices
procedure error and Trail
F
sintancos
tanWcb
sinW
1F
cos
bl but
F
sintancos
F
sintancos
sinF
lcW
tancl
F
F
sintancos
sinF
lcW
N
s
nn
nn
nn
s
n
nn
s
nn
s
nn
n
s
nn
n
s
s
nn
n
s
nn
r
afa
f
aS
aafa
afa
a
fS
afa
a
S
125
Bishop’s Simplified Method of Slices
Steps for Bishop’s Simplified Method of Slices
• Draw the slope to a scale
• Divide the sliding wedge to various slices
• Calculate wn
and an
for each slice
• an
is taken at the middle of the slice
• Calculate the terms in the equation
• Fill the following table
Slice# wn
an
sin an
cos an
bn
wn sin a
n
• Assume Fs
and plug it in the right-hand term of the equation
then calculate Fs
• Repeat the previous step until the assumed Fs = the
calculated Fs.
s
nn
nn
nn
s
F
sintancos
tanWcb
sinW
1F
afa
f
aSS
126
Bishop’s Simplified Method of Slices
nn
)n(
nn
s
s
nn)n(
sinW
m
1)tanWcb(
F
F
sintancosm
aS
f
afa
a
a
S
)
tan incos
tan '(
sin
1
n
s
n
nn
nn
s
F
s
Wbc
WF
faa
f
a
127
Bishop’s Simplified Method of Slices
• Example of specialized software: – Geo-Slope,
– Geo5,
– SVSlope
– Many others
128
Determine the safety factor for the given trial rupture surface shown in
Figure 3. Use Bishop's simplified method of slices with first trial factor of
safety Fs = 1.8 and make only one iteration. The following table can be
prepared; however, only needed cells can be generated “filled”.
Final Exam Fall 36-37 QUESTION #4
129
Fs = 1.8
Table 1. “Fill only necessary cell for this particular problem”
SliceNo.(1)
Widthbn
(m)(2)
Heighthl
(m)(3)
Heighth2
(m)(4)
AreaA
(m2)(5)
WeightWn
(kN/m)(6)
α(n)
(7)mα(n)
(8)
Wn sin a(kN/m)
(9)
1 22.4 70
2 294.4 54
3 38
4 435.2 24
5 390 12
6 268.8 0.0
7 66.58 -8
?
SOLUTION
130
Remarks on Method of Slices
o Bishop’s simplified method is probably the most widely used (but it hasto be incorporated into computer programs).
o The ordinary method of slices is presented in this chapter as a learningtool only. It is used rarely now because it is too conservative.
o It yields satisfactory results in most cases.
o Analyses by more refined methods involving consideration of the forces actingon the sides of slices show that the Simplified Bishop Method yields answersfor factors of safety which are very close to the correct answer.
o The Bishop Simplified Method yields factors of safety which are higherthan those obtained with the Ordinary Method of Slices.
o The two methods do not lead to the same critical circle.
o The Fs determined by this method is an underestimate (conservative) butthe error is unlikely to exceed 7% and in most cases is less than 2%.
131
Two Methods:
Ordinary Method of Slices
• Underestimate Fs
(too conservative)
• Error compared to accurate methods (5-20%)
• Rarely used
Bishop’s Simplified Method of Slices
• The most widely used method
• Yields satisfactory results when applying computer
program
Remarks on Method of Slices