lesson 5 ADGSE

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Lesson 5
contents
•Basics of soil slope stability
•Basics of rock slope stability
1
Why are we interested in analysing slope
stability?
How a soil or rock slope interacts with a civil
engineering work?
How the stability of a soil or rock slope influences
the design of a civil engineering work?
2
 
A slope instability is a movement of a soil or rock volume down the
slope. On the basis of the mechanism of movement, velocity and
involved volume and material, these phenomena are classified in
different landslide typologies.
The instability event can be triggered by natural phenomena
(rains, earthquakes, etc.) or anthropic activities (overloads,
excavations, etc.).
Landslides geotechnical works are
needed to secure the
elements at risk
Civil engineering works the stability of the soil or rock 
mass has to be guaranteed 
In both cases we need to learn how to analyse the slope stability.
Slope stability
Slope stability analysis
• Simplified and schematic models are necessary to simulate
and predict the behaviour of a (real) system. aspects of
behavior of
• Stability analyses have three main aims:
analysing and studying the phenomenon 
quantifying the problem
designing a stabilization work
• Main analysis methods:
empirical
limit equilibrium
numerical (FEM, FDM, DEM, etc.)
The main phases of a slope stability analysis are:
• defining the shape and the structure of the slope
• identifying the instability typology
• identifying the hydraulic conditions of the site
• characterizing the strength and deformability of the
materials constituting the slope
• choosing the model to interpret the phenomenon
• evaluating the stability condition of the slope
Slope stability analysis
Numerical methods
•Stress-strain analyses are performed on a model discretized in
a certain number of elements
•The progressive failure can be analysed as well as the whole
instability process, from the triggering to the final stable or
unstable configuration
•Complex geometries and the mechanical behaviour of slope
materials (in particular strength and constitutive laws) can be
introduced
•Different phases of excavation and construction can be
simulated, with particular reference to stabilization systems (the
interaction between the reinforcement elements and the slope
material can be simulated)
•Rigorous dynamic analyses can be performed (i.e. seismic
conditions)
The potential sliding surface has to be chosen a priori.
The stability condition of the slope is described by the Factor
of Safety (Fs), that is the ratio between the shear strength
and the acting shear stress along the sliding surface:
The unstable volume is modelled as a rigid block.
Limit Equilibrium Method (LEM)
∫
∫=
τdl
)dl(στ
F nRs
It behaves as a rigid block until the
critical shear stress is reached,
then an irreversible displacement
along the sliding plane direction
occurs (the volume collapses).
•The acting shear stress can be calculated by solving the sliding
equilibrium equation of the unstable volume along the contact
surface between the block and the stable rock mass (sliding
plane)
•The shear strength is represented by an appropriate strength
criterion and is a function of the normal stress acting on the
contact surface
•The LEM assumes that the factor of safety is constant along
the whole sliding surface
•In 2D analyses the thickness of the unstable volume is
considered equal to 1
• The unstable volume is considered in limit equilibrium conditions
if the shear stress is equal to the shear strength (Fs = 1)
• Usually the Mohr-Coulomb strength criterion is assumed along
the sliding plane
• F is an index of the safety level of a slope with respect to the
shear failure along a particular surface
• The acceptable values of F depend on the engineering problem:
reliability of geotechnical parameters used in the analysis,
severity of the consequences of slope collapse
• The limit equilibrium condition is that for which F=1; values of F
lower than 1 are physically impossible
Critical surface in a soil mass
In a soil mass infinite surfaces of potential sliding can be
analysed and a factor of safety calculated for each of them.
The critical sliding surface is assumed to be the one with the
minimum factor of safety.
This minimum factor of safety is assumed to be the global
factor of safety for the slope.
The procedure for the stability analysis is:
• Defining the geometrical characteristics of the volume
formed by the slope and the surface considered
• Defining the acting forces
• Assigning the mechanical characteristics of the soil
• Choosing different surfaces of potential sliding (a huge
number)
• Calculating the factor of safety for each surface
• Searching the surface with the minimum factor of safety
Critical surface in a rock mass
If the slope is modelled as an equivalent continuum medium, the
critical sliding surface is defined as the one characterized by the
lowest factor of safety (like in soils).
If the slope is modelled as a discontinuum, the critical sliding is
defined by the structure of the rock mass (a discontinuity can be a
plane of sliding, two intersecting discontinuities can form a wedge
that can slide, etc.)
The procedure for the stability analysis is:
•Verifying the kinematic possibility that a rock block forms and
slides along a plane or along the line of intersection between two
planes
•Defining the geometrical characteristics of the volume formed
•Defining the acting forces
•Assigning the mechanical characteristics of the discontinuities
involved in the instability
•Calculating the factor of safety
Basics of soil slope stability
Infinite slope
• Geometry: infinite slope of
inclination α and a potential
sliding surface parallel to the
slope
• Material: soil
• Strength characteristics:
Mohr-Coulomb strength
envelope
• Acting forces: soil weight (W)
and water pressure (U)
Failure surface 
Infinite slope surface 
• Acting forces: W, Ui-1, Ui, Us, Ub, N’, T
• Size of each slice: b×h
• Shear strength parameters: c’≠ 0, ϕ’ ≠ 0
say:
Resolving the forces perpendicular to the slip surface, the
normal force is:
I case: submerged slope
i1i UUU −=Δ −
αΔ−+−α= sinUUUcosW'N sb
Resolving the forces parallel to the slip surface, the shear
force is:
The magnitude of the factor of safety is obtained as the ratio
of the shear force (T) to the shear strength (Tr) :
αΔ+α
α
+ϕαΔ−−−α
==
cosUsinW
cos
b'c'tg)sinU)UU(cosW(
T
TF
sb
r
αΔ+α= cosUsinWT
II case: slope with no hydraulic pressures
• Acting forces: W, N’, T (ΔU = Ub= Us= 0)
• Size of each slice: b×h
• Shear strength parameters: c’≠ 0, ϕ’ ≠ 0
The factor of safety is:
α
α+
α
ϕ
=
α
α
+ϕα
=
sinW
cos
b'c
tg
'tg
sinW
cos
b'c'tgcosW
F
In case of a non cohesive soil (c’ = 0):
That is, in this case the slope stability is always guaranteed if
ϕ’>α ,
whatever the volume of the unstable mass is
α
ϕ
=
tg
'tgF
III case: horizontal water level
• Acting stresses: water pressure U distributed along AB,
BC, CD and DA, N’, T, W
• Size of each slice: b×h
• Shear strength parameters: c’≠ 0, ϕ’ ≠ 0
• Unit weight of the soil: γ
• Unit weight of the soil
below the water table:
γ’= γ- γw
Fattore di sicurezza dell’elemento:
α×××γ−=⎟
⎠
⎞
⎜
⎝
⎛ ×+−
×+
=−− tghb2
h)uu(
2
h)uu(UU wabdci1i
α×α×××γ−α×γ××
⎟
⎠
⎞
⎜
⎝
⎛ ϕα×α×××γ+
α
×××γ−γ×××α+
α=
costgbhsinbh
'tg)sintgbh
cos
1hbhbcos
cos
b'c
F
w
ww
hbW ××γ=
α
ϕ
+
α×α××γ
=
tg
'tg
cossinh'
'c
α
×××γ−=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
α
+
−α
+
=−
cos
1hb
2
cos
b)uu(
2
cos
b)uu(
UU w
bcad
sb
In the case of a non cohesive soil (c’ = 0):
That is, the factor of safety is the same obtained in the case 
of no hydraulic pressures
α
ϕ
=
tg
'tgF
IV case: seepage parallel to the slope
• Acting forces: W, N’, T, Ub (Us=0, ΔU = 0)
• Size of each slice: b×h
• Shear strength parameters: c’≠ 0, ϕ’ ≠ 0
• Unit weight of the soil: γ
• Unit weight of the soil below the water table: γ’= γ- γw
αγα
α
γ
2coscos
cos
××=⇒×=
×=
×=
huhXYXYh
hu
wD
w
wwD
ground surface
If the top flow line coincide with the ground level:
Resolving the forces perpendicular to the slip surface :
Resolving the forces parallel to the slip surface :
α××γ= 2wD coshu
α×××γ=α= sinhbsinWT
α
×α×γ×−α×××γ=−α=
cos
bcoshcoshbUcosW'N 2wb
α×××γ= coshb'
Factor of safety:
α×××γ
ϕ×γ×α××+
α=
sinbh
'tg'cosbh
cos
b'c
F
α
ϕ
×
γ
γ
+
α×α××γ
=
tg
'tg'
sincosh
'c
In the case of non cohesive soil (c’ = 0):
α
ϕ
×
γ
γ
=
tg
'tg'F
Slices methods
• A slip surface is chosen a priori and the so defined
unstable soil volume is divided by a number of slices,
separated by parallel vertical planes
• The equilibrium of the total soil mass is the sum of the
equilibrium of each slice. The forces acting on the sides
of each slice (interslice forces) have to be taken into
account
• A comparison between the forces acting on each slice and
the available equilibrium equations allows to define if the
problem is undefined
Unknown quantities:
Interslice forces Xi (n-1) and
Ei (n-1) and arms hi (n-1)
with respect to the basis of
the slice, for any i slice
Effective forces N’i (n)
normal to the basis of each
slice
Safety factor F
• 4n-2 unknown quantities
• 3n equilibrium equations:
vertical translation (n) for each slice
horizontal translation (n) for each slice
rotation (n) for each slice
n-2 redundant unknown quantities some conditions have
to be introduced.
We can operate on:
Interslice forces
Point of application of interslice forces
Normal forces
The different slice methods (Fellenius, Bishop, etc.) differ on
the basis of the conditions introduced in order to make the
problem defined
Fellenius method
• Circular surfaces
• F is calculated by considering the rotational equilibrium of
the whole sliding mass
• To calculate the resultant force N’i perpendicular to the
slice base, the translational equilibrium is considered in
the direction perpendicular to the slice base:
• The global factor of safety is obtained with the rotational
equilibrium about the center of the considered surface:
)sincos(cos' iiiibiiii EXUWN ααα Δ−Δ+−=
( )[ ]
ii
iiiibiiii
sinW
'tgsinEcosXUcosWl'cF
α∑
ϕαΔ−αΔ+−α+Δ∑
=
• Simplifying assumption:
that is, the forces acting at the interfaces between the
slices are neglected
• The factor of safety simplifies in the following linear
expression:
The method significantly underestimate F for deep, circular
surfaces
The estimation of F is more accurate if the surface has a big
radius: the more planar is the surface the more accurate is the
estimation of F
( ) 0'tgsinEcosX iiii =ϕαΔ−αΔ∑
( )[ ]
ii
biiii
sinW
'tgUcosWl'cF
α∑
ϕ−α+Δ∑
=
Bishop method
• Circular surfaces
• F is calculated by considering the rotational equilibrium of the
whole sliding mass
• To calculate the resultant force N’i perpendicular to the slice
base, the translational equilibrium is considered in the vertical
direction:
say:
• The global factor of safety is obtained with the rotational
equilibrium about the center of the considered surface:
iiibiiiiiii αc'Δ'F
αUΔXW'tgα
F
N'αN' sin1cossin1cos −−+=+ ϕ
'tgα
F
αm iiαi ϕsin
1cos +=
[ ]
ii
αi
iibiii
αW
m
'tgΔX')tgαU(Wc'b
F
sin
1cos
∑
+−+∑
=
ϕϕ
• The expression of F is not linear
• Simplifying assumption: the forces acting at the
interfaces between the slices are horizontal ⇒ Xi are null
• The factor of safety simplifies in the following non linear
expression:
For this reason an iterative procedure is needed, as follows:
I. A first attempt value of F is assumed (usually the one
obtained with the Fellenius method, multiplied by 1.2 or 1.1)
II. mαi is calculated
[ ]
ii
i
ibiii
sinW
m
1'tg)cosUW(b'c
F
α∑
ϕα−+∑
= α
III. The correspondent value of F is calculated with the
Bishop formula
IV. With this last value of F, mαi is recalculated
V. The steps from II to IV are repeated until the difference
between two subsequent values of F is lower than a
predefined value
For circular surfaces the Bishop method provides a good
estimation of the global factor of safety.
It is widely used in practice
Janbu simplified method
• Sliding surface of any shape can be considered
• The forces equilibrium is considered. The torques
equilibrium is neglected
• Simplifying assumption: the forces acting at the
interfaces between the slices are horizontal ⇒ Xi are
null
• The global factor of safety is obtained with the horizontal
translational equilibrium about the center of the
considered surface:
• To calculate the resultant force N’i perpendicular to the
slice base, the translational equilibrium is considered in
the vertical direction:
say:
we get:
iiibiiiiii sinl'cF
1cosUW'tgsin
F
1'Ncos'N αΔ−α−=ϕα+α
'tgsin
F
1cosm iii ϕα+α=α
i
iiibii
i m
sinl'c
F
1cosUW
'N
α
⎟
⎠
⎞
⎜
⎝
⎛ αΔ−α−
=
• To calculate the unknown forces ΔEi, the translational
equilibrium of each slice is considered in the horizontal
direction:
( ) ( )
( )
( )
( )
( ) ii
i
iii
iiiiiiii
i
i
iiii
iiii
iiii
i
i
iiii
ibiiiii
tgW
cos
1'tg'Nl'c
F
1E
tgW)sintg(cos'tg'Nl'c
F
1E
sin
cos
sin'tg'Nl'c
F
1W
cos'tg'Nl'c
F
1E
sintg'tg'Nl'c
F
1
cos
Wcos'tg'Nl'c
F
1E
sin)U'N(cosTE
α−
α
ϕ+Δ=Δ
α−αα+αϕ+Δ=Δ
α
⎟⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎝
⎛
α
αϕ+Δ−
−αϕ+Δ=Δ
α⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
αϕ+Δ−
α
−αϕ+Δ=Δ
α+−α=Δ
• The global factor of safety is obtained with the translational
equilibrium of the whole mass in the horizontal direction:
being:
( )
[ ]
iii
αi
ibiii
iii
i
ii
tg αWΔE
n
'tg)αU(Wc'b
F
tg αWΔE
α
'tgN'c' Δ'
F
∑+∑
×−+∑
=
∑+∑
+∑
=
1cos
cos
1
ϕ
ϕ
iαiαi αmn cos=
• The iterative procedure is used again to estimate the
factor of safety
• Simplified Janbu method underestimates the value of F
• The Author suggests to correct the final value of F by
means of a coefficient f0, that is a function of the slope
geometry and the strength characteristics of the soil. In
this case the method is called Janbu corrected method:
where b1 is estimated as follows:
purely cohesive soils: b1 = 0.69
purely frictional soils: b1 = 0.31
cohesive-frictional soils: b1 = 0.5
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛−+=
2
10 4.11 L
d
L
dbf
Basics of rock slope stability
49
Kinematic analysis
The discontinuities divide the rock mass into a number of rock
blocks. They can be stable or unstable with respect to the
orientation of the slope face.
The most usual types of instabilities are:
• Sliding of a rock block along a discontinuity
• Sliding of a rock wedge along two intersecting discontinuities
In order to study the stability, the kinematic possibility has to be
verified and then a factor of safety has to be calculated.
Be careful: the kinematic possibility has to be verified for each set
of discontinuities with respect to the slope face!!
Kinematic check – sliding along a plane 
Face dip direction: αF
Discontinuity dip direction: αd
αd≅αF
Face dip: ψF
Discontinuity dip: ψF
ψd<ψF
slope face
Kinematic check – sliding of a wedge along the line 
of intersection of two intersecting planes
Face dip direction: αF
Line of intersection dip direction: 
αi
Face dip: ψF
Line of intersection dip: ψi
ψi<ψF
line of intersection i
αi≅αF
Slope face
Plane A
Plane B
Sliding along a plane
General geometrical scheme assuming:
• Presence of a tension crack
• Mohr-Coulomb strength criterion along the discontinuity surface
• Acting forces: block weight (W), hydraulic pressures into the
tension crack and sliding plane interfaces (V e U, respectively),
generic external force (E) with an inclination β with respect to
the sliding plane and stabilizing force (T) with an inclination θ
with respect to the sliding plane
W
U
V
T
E
β
θ
yd
The factor of safety is calculated by solving the equilibrium
equation along the sliding direction:
N: net force of the normal components of the acting forces
S: net force of the components of the acting forces in the
sliding direction
U e V: net forces of the hydraulic pressure distribution into
the tension crack and sliding plane, respectively
A: surface areaat the contact between the block and the
rock mass
c e φ: Coulomb strength parameters
( )[ ]
cosβEcosTcosψVsinψW
tgsinβEsinTsinψVUcosψWAcF
dd
dd
×+×−×+×
××+×+×−−×+×
=
ϑ
ϕϑ
[ ] [ ]
S
ActgN
S
RF ×+×== ϕ
Sliding of a wedge along the line of intersection between 
the two planes
B
O
A
W
C
T
V
1
5
6
3
7
9 8
4
2
If the condition on dip direction is not satisfied, a non-
symmetric wedge can form: in this case the sliding can occur
along one of the two planes.
The condition on dip must always be satisfied.
Input data:
αf , ψf dip direction and dip of the slope face;
αA , ψA dip direction and dip of the plane A;
αB , ψB dip direction and dip of the plane B;
αs , ψs dip direction and dip of the upper plane;
αt , ψt dip direction and dip of the tension crack;
αT , ψT dip direction and dip of the external force (anchoring, 
for example);
T intensity of the external force;
cA ,φA cohesion and friction angle along the plane A;
cB , φB cohesion and friction angle along the plane B;
γ , γw specific weight of rock and water
The equation of the factor of safety is:
where: q,r,s,x,y,z,mW,5, mv,5 and mT,5 are projection coefficients.
( ) ( )
TmVmWm
tgUzTyVxWtgUsTrVqWAcAcF
T,V,W,
BBAABBAA
555 ++
−+++−++++
=
ϕϕ
Some scheme for the hydraulic pressures distribution
2
A)/2]z(H[γU ww ×−×=
U
zw
H
2
)z(zγ
2
)zz)z(zγV
2
A/2)/2z(Hγ
2
A/2)/2]z(Hγ)z(z[γU
2
wwwww
wwwwww
−×
=
−×−×
=
×−×
+
×−×+−×
=
(
zw
z
(H-z)/2 H
U
V
{ }
2
)z(zγ
2
)zz)z(zγV
2
Az-H[γ)]z(z[γU
2
wwwww
wwww
−×
=
−×−×
=
××+−×
=
(
)](
zw
V
U
z
H