# Some basic principles of soil mechanics for landslide analysis

Mechanics

Jul 18, 2012 (6 years and 3 days ago)

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Theo van Asch, Utrecht University
t.vanasch@geo.uu.nl
Stresses on a plane
σ
σ3
σ1
τ
δz
δx
δs
1
(
)
()()
()
)sin(2b)σσ(τ
)cos(2b)σσ()σσ(σ
cos(b)b)sinσ-σ(
cos(b)bsinσ-cos(b)bsinστ
(b)cosσ-bsinσσ
31
2
1
31
2
1
31
2
1
31
31
2
1
2
3
−=
−++=
=
=
=
b
The Mohr’s envelope
)sin(2b)σσ(τ
)cos(2b)σσ()σσ(σ
31
2
1
31
2
1
31
2
1
−=

+
+
=
σ3
σ1
σ
τ
2b
Find by means of the Mohr’s circel an expression for τ
,σ, the radius and the
midpoint of the circle.
Morh’s failure circle
σ3
σ1
σ
τ
The strikeline on the circle gives a point
where τ/σ
has a maximum value
So if failure occurs in a shear plane it
must be that point :
τ/σ
equals thens

= tanφ
according
to Coulomb (if there is no cohesion)
φ
Morh’s envelope
The strikeline along the failure circles
connect the points where failure occurs.
It is a line which can be described by
Coulombs law:
τf=s=c+σftanφ
σ1f
τ1f
τ2f
σ2f
φ
c
Effective stress
=
σ
X
X
σf=effective stress!=σ’
Total area X=1
Total area grains =Xm
Total area water =Xw
Total stress on plain X
w
'
wm
m
mwm
'
wwm
'
σσσσXσσ'
1)X(1
)X(1σXσσ
XσXσσ
+=→+=
≈−
−+=
+=
=
=
=
σw
Effective stress
X
X
Effective stress σ’ determines
shear strength:
''
w
''
tanu)-σ(cs
u-σσ-σσ'
tanσ'cs
φ
φ
+=
==
+=
You can determine
σ
and u but not
σ

Total stress and pore pressure
Total stress σ
is
total weight above
ABCD/area ABCD
The calculation of
pore pressure u is a
hydrological problem
σ
B
A
C
D
Linear strain
Deformation along view line (x)
x1
x2
u2
∆l
l
∆l
∆x
∆u
ttancons
∆x
∆u
δx
δu
δx
δu
∆x
∆u
xx
uu
ε
12
12
=
=
==

=
Constant linear strain
u1
l
Shear strain
Deformation across view line (y)
x1
u2
αtan
δy
δu
∆y
∆u
yy
uu
12
12
===

u1
y1
y2
α
Rheologic properties of material
σ
ε
σ
ε
σ
ε
σ
ε'
σ
ε'
σ
ε'
ε
t
ε
t
ε
t
σ1
σ1
σ0
Elastic
Plastic
Viscous
σ=stress
ε=strain
Rheologic properties of material
Elasto -plastic
Elasto -viscous (Maxwell)
Elasto -viscous (Kelvin)
Visco-plastic
Bingham
Creep behaviour of soils
Initial elastic
strain at
σ1
,σ2,σ3
Strain
Steady creep
Creep failure
Time
Principles of slope equilibrium
Safety
factor F
=
Strength of material=S
Mobilized shear strength=T
Depends on material
properties and equi-
librium of forces in the
ground (N’)
Depends on equilibrium
of forces in the ground
(T)
N
T
W
W
N
T
Principles of slope equilibrium
N
T
W
W
N
T
U
T
tanφN'C
T
tanφU)-(NC
T
S
F
+
=
+
==
Principles of slope equilibrium
•Resolve forces
perpendicular to shear
plane:
•Wcos
α
-N’-U-Vsin
α
=0
•Resolve forces parallel to
shear plane:
•T-Wsin
α
-Vcos
α
=0
U
T
W
N’
V
α
Principles of slope equilibrium
U
T
W
N’
V
VcosαWsinα
)tanφ VsinαU(WcosαC
F
VcosαWsinαT
VsinαUWcosαN'
T
tanφN'C
F
+
−−+
=

+=
−−=
+
=
α
Infinite slope model
The geometry of the slice is the same along the slope. Xr
and Er
are equal to Xl
and El .along the slope.So X and E can be
cancelled in this equilibrium analysis
For the stability calculation (F) resolve forces perpendicular and
pararallel to shear surfaces
Er
N’
T
Xr
U
El
Xl
W
α
Infinite slope model
Wsinα
U)tanφ(WcosαC
F
T
tanφNC
F
WsinαT0WsinαT
UWcosαN0UNWcosα
'
''
−+
=
+
=
=→=−
−=→=−−
Er
N’
T
Xr
U
El
Xl
W
α
Principals of grounwater flow and pore
pressure calculation
•GW flow generated by difference in energy
between points
•Energy expressed in m (head)
•Energy is sum of (water) pressure head and
elevation head (m) is called total head
•Equi-potential lines connect points with same total
head values
•Groundwater flows perpendicular to equi-potential
lines
Infinite slope model .Principals of grounwater
flow and pore pressure calculation
x
h
Total head
0hElevation head
x0Pressure head
BA
x=h is total head in B
Elevation head in B
=0 so pressure head
in B =h
h=zwcos2α
zw
z
h
γs
α
α
γu
h=zwcos2α
Equipotentiaal lijn
A
B
b
Infinite slope model
Wsinα
U)tanφ(WcosαC
F
cosα
b
αγcoszU
bγz)bγz(zW
c
cosα
b
C
w
2
w
swuw
−+
=
=
+−=
=
zw
z
h
γs
α
α
γu
h=zwcos2
α
Equipotentiaal lijn
A
B
b
Infinite slope model
[]
{
}
{}
sinαbγzb)γz(z
tanφαγcoszcosαbγz)bγz(zc
F
swuw
cosα
b
w
2
wswuw
cosα
b
+−
−+−+
=
zw
z
h
γs
α
α
γu
h=zwcos2α
Equipotentiaal
lijn
A
B
b
Infinite slope model
{}
{}
ws
'
sUU
'
UU
2
w
s
ws
s
w
u
w
w
γγγ
tanαγm(γγ
tanφγm(γγ
αzcos
c
F
z
z
m
tanα
tan
γ
γγ
cosα zsinαγ
c
Fzz
tanα
tan
cosα zsinαγ
c
F0z
tanα
tan
F0c0,z
−=
−−
−−+
=→→→=

+=→→→=
+=→→→=
=→==
ϕ
ϕ
ϕ
General equilibrium analyses: equilibrium of
moment of forces
•The path we follow:
•Expression for T making
use of safety factor
definition
•Expression for P by
resolving forces vertically
•Expression for overall F
m
,
(which includes P) by
resolving moment forces
d
f
R
Er
Xr
El
Xl
T
P
W
α
ul
General equilibrium analyses: equilibrium of
moment of forces
d
f
R
Er
Xr
El
Xl
T
P
W
α
ul
l=length of slipsurface in slice n
{}
''
''
''
''
ul)tanφ(PlcSPσl
ul)tanφ-l (σlcS
l u)tanφ(σcS
u)tanφ(σcs
−+=→=
+=
−+=
−+=
General equilibrium analyses : equilibrium of
moment of forces
d
f
R
Er
Xr
El
Xl
T
P
W
α
ul
{}
ul)tanφ(Plc
F
1
T
T
ul)tanφ(Plc
F
T
S
F
ul)tanφ(PlcSPσl
'
'
''
−+=
−+
=
=
−+=→=
General equilibrium analyses : equilibrium of
forces
d
f
R
Er
Xr
El
Xl
T
P
W
α
ul
{}

+=

−−−−=

−−=+→
−+=
F
tanφ
tanα1 cosαm
/m) sinα'ultanlsinα(c'
F
1
)X(XWP
P?
)X(XWTsinαPcosα:Vertical
ul)tanφ(Plc
F
1
T
α
αlr
lr
'
ϕ
General equilibrium analyses:equilibrium of
moment of forces
d
f
R
Er
Xr
El
Xl
T
P
W
α
ul
l=length of slipsurface in slice n
{}
{}

∑∑∑

−+
=⇒

−+=
+=
=−=−
)Pf(Wd
Rul)tanφ(Plc'
F
replaceT
ul)tanφ(Plc
F
1
T
PfTRWd
!0EEand!0XX
m
'
lrlr
Fm
= overall force equilibrium
General equilibrium analyses :equilibriums of
moment of forces
{
}
αlr
/m) sinα 'ultanlsinα(c'
F
1
)X(XWP
Pf-Wd
Rul)tanφ(Plc'

−−−−=
−+
=

φ
m
F
d
f
R
Er
Xr
El
Xl
T
P
W
α
ul
Problem indeterminate :
Xr-Xl=0 Bishop
Xr=Xl=0 Janbu
X/E=constant Spencer
X/E=λf(x) Morgenstern&Price
From general equilibrium analyses to circular
slip surface
d
f
R
Er
Xr
El
Xl
T
P
W
α
ul
{
}

−+
=
Pf-Wd
Rul)tanφ(Plc'
F
m
{
}

−+
=
Wsinα
ul)tanφ(Plc'
F
m
Circular slipsurface
f=0 d=Rsinα R=constant
From general Equilibrium analyses to circular
slip surface with Bishop assumption
d
f
R
Er
Xr
El
Xl
T
P
W
α
ul
{
}
{}
{}

−+
=

−−−−=
−+
=
Wsinα
1/m
'
ul)tanφ(Wlc'
F
/m) sinα'ultanlsinα(c'
F
1
)X(XWP
Wsinα
ul)tanφ(Plc'
F
α
m
αlr
m
ϕ
According to Bishop (X
r-Xl)=0
for each individual slice

+=
F
tanφ
tanα1 cosαm
'
α