1
Geotechnics
Marcin Cudny, Lech BałachowskiDepartment of Geotechnics, Geology and Maritime EngineeringCivil and Environmental Engineering Faculty,Gdańsk University of Technology
e-mail: [email protected],web: www.pg.gda.pl/~mcud/phone.: 58 347 2492,room: 302/Hydro,tutorial: Friday 11.15-13.00
LiteratureLiterature
• Geotechnical Engineering Handbook, Editor: Urlich Smotczyk, Ernst & Sohn, Darmstadt 2002.
• Helwany S.: Applied Soil Mechanics with Abaqus Applications. John Wiley & Sons, Inc., USA, 2007.
• Duncan J.M., Wright S.G.: Soil Strength and Slope Stability. John Wiley & Sons, Inc., USA, 2005.
• Material Models Manual – Plaxis version 8, Balkema, The Netherlands, 2006.
• Derski W., Izbicki R., Kisiel I., Mróz Z.: Rock and soil mechanics , PWN, Elsevier, 1988.
• Terzaghi K., Peck R.B., Mesri G.: Soil Mechanics in Engineering Practice, John Wiley & Sons, USA, 1996.
• Muir Wood D.: Geotechnical Modelling, Spon Press, Taylor & Francis Group, 2004.
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On-line resources from our University domain:http://www.bg.pg.gda.pl
Other on-line resources:
Geotechnics, Soil mechanics, Geomechanics, Rock Mechanics ->Geotechnik, Bodenmechanik, Geomechanik, Felsmechanik
Tochnog, Plaxis finite element programs, free and commercial respectivelyhttp://tochnog.sourceforge.net, http://www.plaxis.nl
Andrzej Niemunis web page: Bodenmechanik II, Bodenmechanik III, Numerik in der Geotechnik, Computergestützten Geotechnischen Projektstudien, FE-Berechnungen in der Geotechnik.http://www.rz.uni-karlsruhe.de/~gn99/
keywords:
Arnold Verruijt web page: books and geotechnical programshttp://geo.verruijt.net
Tim Spink web page: Geotechnical & Geoenvironmental Software Directoryhttp://www.ggsd.com
Andrew Schofield web page: interesting articles and links,Book: Critical State Soil Mechanicshttp://www2.eng.cam.ac.uk/~ans
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Magazines:
• Inżynieria Morska i Geotechnika (polish)
• Géotechnique
• ASCE Geotechnical and Environmental Engineering
• Computers and Geotechnics
• Numerical and Analytical Methods in Geomechanics
• Canadian Geotechnical Journal
• Geotechnical Testing Journal
• Soils and Foundations
• Geotechnik (german)
Planned scope of lecturespart of M. Cudny
Planned scope of lecturespart of M. Cudny
1. Shear strength of soils – general rules concerning the application of the Coulomb-Mohr shear strength criterion (drained & undrained conditions, dilatancy).
2. Alternative shear strength criteria for soils.3. Soil slope stability calculations.4. Stiffness of soils: logarithmic and exponential compression laws.5. Soil stiffness at small and intermediate strains: stress and strain dependency
of the stiffness.6. Consolidation of saturated soils under general conditions (Biot theory).7. Secondary consolidation of soils (creep and relaxation).8. Advanced soil constitutive models in practice (Cam-clay, Hardening Soil).
4
Some basics, definitions etc.Some basics, definitions etc.
Stress:
Effective stress (for fully saturated soils):
5
Principal stress space
Strain:
6
Axisymmetric conditions (uniaxial, triaxial and oedometer tests)
Triaxial apparatus
7
Oedometer
True triaxial apparatus
8
(a) direct shear, (b) simple shear, (c) torsional shear
Shearing (plane strain)
Graphs used to illustrate soil material behaviour :
9
Application of soil constitutive models in numerical simulationsof real geotechnical problems
Why we concentrate on the behaviour of small samples ?
Shear strength of soils Shear strength of soils
*) source: http://ppdem.net/
Numerical simulation of biaxial test with Discrete Element Method (DEM)
10
Numerical simulation of a vertical soil cut with Particle-In-Cell (PIC) method,
*) source: CSIRO Division of Exploration and Mining, Australia
Shear band formation for vertical soil cut
Slope failure – characteristic zones
*) source: Leroueil, 39th Rankine Lecture, Géotechnique 51(3), 2001
11
*) source: Skempton, 1967
Subsequent phases of shear zonemobilization in fine grained(cohesive) soil
Simple description of Coulomb shear strength criterion
12
Coulomb shear strength criterion in different planes
M=
Coulomb shear strength in principal stress space
*) here stress iscompression positive
13
Components of an elasto-plastic constitutivemodel for soils (generally)
1σ−
32σ−
( ) 0ijF σ =
Yield surface
hydrostatic axis
σ 1=σ 2=
σ 3
elasticmodel
e eij ijkl klDσ ε= &&
epij ijkl klDσ ε= &&
Flowrule: lub p p
ij ijij ij
F Gε λ ε λσ σ∂ ∂
= =∂ ∂& && &or
( ) ( ) ( )0 00
0 0
1 2, 1 1 2 2
et etij ijkl kl ijkl ij kl ik jl jk il
Ed D d D νσ ε ν δ δ δ δ δ δν ν
−⎡ ⎤= = + +⎢ ⎥+ − ⎣ ⎦
Coulomb-Mohr
Hooke
0
20
40
60
80
100
0 20 40 60 80 100
√2σ3 [kPa]
σ1 [
kPa]
compression
extension
hydrostatic
axis
+
Coulomb-Mohr model – the most popular elasto-plastic model implemented in geotechnical software
14
Coulomb-Mohr model – simple modifications for better performance
( )( ) ( )⎥⎦⎤
⎢⎣⎡ +
−+
−+== iljkjlikklij
esijkl
etijkl
EDD δδδδνδνδνν 2
21211
Possibilities of improvement:
• introduction of an alternative shearcriterion or yield surface
• introduction of stress and/or straindependent Young’s modulus ex. E(σ) lub E(ε)
• introduction of elastic anisotropyex. cross-anisotropic Hooke’s law
• introduction of hardening andsoftening
*) here stress iscompression positive
Dilatancy and its influence on the soil behaviour
Dilatancy is the observed tendency of a compacted or loose granular material to dilate (expand in volume) or contract (shrink in volume) respectively as it is sheared. This occurs because the soil particles in a compacted state are interlocking and therefore do not have the freedom to move around one another
15
Dilatancy vs. conractancy
*) source: Muir Wood, 2004
Dilatancy and contractancy,drained triaxial test on dense and loose sand samples
at ID=0.5
volumetric strain εv
deviatoric stress q=σ1-σ2
at ID=0.3
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Direction of shearing
Rough joint
shear displacement
shea
r stre
ss(n
orm
aliz
ed to
UC
S)
shear displacement
norm
al d
ispl
acem
ent
(dila
tanc
y/ c
ontra
ctan
cy)
shear stress vs. shear displacement
maximumdilatancyangle
Microcracks from shear failure (GREEN)Microcracks from tensile failure (RED)σn = 0.65 x UCS
vertical displacement vs.shear displacement
Numerical shear box experiment – shearing and volume vhanges(bonded particle model of jointed rock sample)
*) source: Itasca International Inc., PFC2D
Possibilities of stress paths obtained with Coulomb-Mohr modelfor different drainage conditions
φ’
c’ σ ’
τ
σ0 ’
φu=0cuB
cuA A
A – φ’, c’, E’, ν’, undrained
A’
A’ – φ’, c’, E’, ν’, drained
B
B – φu=0, cuB, E’, ν’, undrained
C
C – φu=0, cuB, Eu, νu=0.495,
total stress analysis
17
Pore water changes for undrained triaxial compressionwith Mohr-Coulomb model
p
q
cq
M1
13
Δuun
drai
ned
path
tota
l stre
ss p
athor
dra
ined
path
normal consolidation
overconsolidationor preconsolidation
strengthincrease
void
ratio
e
deposition history
sedi
men
tatio
n
eros
ion
σ´ [kPa]
σ´ [kPa]
τ[k
Pa]
Changes of strength and stiffness observed during deposision history
*) source: Skempton, 1967
18
τ
σ'σc1'
φ'
φs
cu1
c'1c'2
φs
φ'
σc2'
cu2
stressincrease
Krey-Tiedemann shear strength criterion (1933)
φ’, c’,cu – effective friction angle, effective cohesion and undrained cohesionφ’s - total friction angle,σc – consolidation stress (normal to the shearing plane)
! Simple criterion where overconsolidation ratio is taken into account
parameters:
Real undrained behaviour in triaxial compression of overconsolidated and normally consolidated clay sample
*) source: Wehnert PhD, University of Stuttgart, 2006
clay clay
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Stiffness of soil grains Ks, stiffness of soil skeleton (effective) K1and stiffness water Kw
for undrained analysis it is often assumed: Ks= ∞, Kw=∞
deformation of single grains deformation of soil skeleton
*) source: Bodenmechanik II, A. Niemunis
Calculations of pore water pressure
Assumption of incompressibility of water in numerical calculations is not possible, hence stiffness of water and soil skeleton are taken parallely.
stiffness of water:
w vu K εΔ = Δ or tensorially wij w ij kl klKσ δ δ ε= &&
effective stiffness of soil skeleton
eij ijkl klDσ ε= && (lub )tot w w
ij ij ij ij ij ijσ σ σ σ σ σ′= − = −
total stiffness :
( )tot eij ijkl w ij kl klD Kσ δ δ ε= + &&
or
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Matrix representation for plane stress conditions
0 00 00 0
0 0 0 2 0 0 0 0
totx xw w wtot
yw w wytot
zw w wz
xyxy
A B B K K KB A B K K KB B A K K K
G
σ εεσεσεσ
⎧ ⎫ ⎧ ⎫⎛ ⎞⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎜ ⎟ ⎪ ⎪⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎜ ⎟⎢ ⎥ ⎢ ⎥= + ⋅⎨ ⎬ ⎨ ⎬⎜ ⎟⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎩ ⎭⎩ ⎭
& &
&&
&&&&
( )( ) ( )( ) ( )1 1, ,
1 1 2 1 1 2 2 1A E B E G Eν ν
ν ν ν ν ν−
= = =+ − + − +
For Hooke’s linear elasticity :
How to estimate Kw ?
2 GPawK ≈a)
b) multiplying of the average of effective stiffness normal componentsso-called head (ex. 100 times)
c)
( )0.5,
2 1uEGν
ν≈ =
+
12 13 1 2 1 2
uw
u
GK ν νν ν
⎛ ⎞+ += −⎜ ⎟− −⎝ ⎠
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Settlement of shallow foundation for short time loading,undrained conditions
2
980
wht tk M
γ<< ≈ < 0.01
h=D – hight of the consolidating layer or simply length of drainage path,Ev – stiffness modulus to calculate short time settlement
Skempton parameter B
increment of the total tress:
'ij ij ijuσ σ δΔ = Δ + Δ
isotropic compression:
0 00 00 0
PP P
P
Δ⎡ ⎤⎢ ⎥Δ = Δ ⋅ = Δ⎢ ⎥⎢ ⎥Δ⎣ ⎦
σ 1
( )ruB f SP
Δ= =
Δfor undrained soil: B≈0.999
' 0P u P B PσΔ = Δ − Δ = Δ − Δ ≈
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Skempton parameter A
1
2 3v
q
K QpQ Gq
εε
⎧ ⎫⎡ ⎤⎧ ⎫ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪⎩ ⎭ ⎣ ⎦ ⎩ ⎭
&&
&&
00 3
v
q
p Kq G
εε
⎧ ⎫⎧ ⎫ ⎡ ⎤ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪⎩ ⎭ ⎣ ⎦ ⎩ ⎭
&&
&&
steel : soil :
dilatancy: v
q
d εε
=&
&
Skemptona parameter A
1 3u A A qσ σΔ = Δ − Δ = Δ
stress paths for different values of parameter A
Parameters A and B (undrained behaviour)
( )13
u A q B tr⎡ ⎤Δ = Δ + − Δ⎢ ⎥⎣ ⎦σ
undrained
undrained
drainedregardless A value
23
Dilatancy angle in Mohr-Coulomb model
φ’
c’ σn ’
τ
ψ=φ’
ψ=φ’
F=0
G=const
F=τ - σn’ tanφ - c (yield function),G= τ - σn’ tanψ (plastic potentialfunction)
Influence of dilatancy angle on undrained stress pathin Mohr-Coulomb model
n
24
Influence of stress levelon the behaviour during shearing
*) source: Bolton, 1986
Drucker-Prager shear strength criterionstandard version:
ϕϕ
ϕϕ
σσσ
σσ
sin3cos6 ,
sin3sin6
),2(31 ,
31
, :case icaxisymmetrfor ,23
0
31
31
−=
−=
+==
−==
=−−=−
ccM
pp
qssq
cMpqF
q
kk
ijij
qPD
−σ1
−σ3−σ2
−σ1
−σ3 −σ2
Drucker, Prager (1952)
1
1Me
Mc
cq p
q
Alternative shear strength criteria for soilsAlternative shear strength criteria for soils
σ1=σ2=σ3π surface (deviatoric) Mc=Me
25
Drucker-Prager vs. Mohr-Coulomb, How to choose parameters ?
Drucker-Prager criterion is a q=const contour (Mc=Me)and Mohr-Coulomb criterion is a φ=const contour (Mc=Me)Ex. choosing M=Mc(φ=30o) in Drucker-Prager criterion results in very largestrength for axisymmetric extension (ex. passive earth pressure)which is equivalent to the activation of φ=48.6o
Lode angle – influence of the intermediate principal stress component
( ) ( )
( )
33/ 22
* 3 33 3/ 2
2
* *
3 2 2
Dwie popularne definicje kąta Lodego (często mylone):
3 31 arccos ,3 2
27 3 31 1arcsin arcsin ,3 2 3 2
0 60 , 30 30 , 30
gdzie
1det , , 32ij kl kl
JJ
J Jq J
J s J s s q J
θ
θ
θ θ θ θ
⎛ ⎞= −⎜ ⎟⎜ ⎟
⎝ ⎠⎛ ⎞⎛ ⎞
= − = −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= ÷ = − ÷ = − +
= = = =
o o o o o
*
*
32
oraz - dewiator naprężenia
13
Sciskanie trójosiowe: 0 lub 30Rozciąganie trójosiowe: 60 lub 30
ij ij
ij ij ij
kk
s s
s p
p
σ δ
σ
θ θ
θ θ
= +
= −
= =
= = −
o o
o o
-σ1
-σ2-σ3
θ=0°θ∗=30°=0.0b
θ=30°θ ∗=0°=0.5b
θ=60°
θ ∗=-30°
=1.0b
compressionextension
( )( )2 3
1 3
1 1 3 tan 302
b σ σ θσ σ
−= = + −
−o
Two definitions of Lode angle in textbooks (often misleaded)
where
andstress deviator
axisym. compression:
axisym. extension:
26
Stress invariants p, q, θ
σ1
σ3
σ2
p
qRendulic plane
σ
p3
q3/2
Improved version of Drucker-Prager criterion (Abaqus):
qtKqtK
qr
KKqt
rsssr
cMptF
compext
kijkij
qPD
==−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −−+=
−−=⇒=−=
=−−=−
, ; 1778.0
,11112
)( for ,29
0
3
31323
*
σσσσ
K=1.0 K=0.9 K=0.8
27
Matsuoka-Nakai criterion (SMP concept – Spatialy Mobilised Planes):
3213
21313223211
3
321
SMP
SMP
2
23321
2
3
det
,)(21
9
9 9
,3
σσσσ
σσσσσσσσσσ
σττσ
==
++=−=++==
−=⇒
−==
ij
ijijjjiikk
SMPSMP
I
I,σσσσI
IIII
IIIII
II
0or 3
21
3
21 =−== constIIIfconst
III
(1974) ,0sin1sin9tan89 2
2
3
212
3
21 =−−
−=−−=−cm
cmcmNM I
IIIIIF
ϕϕϕ
( )( )( )
31
3
3
1 3
0, Lade i Duncan (1975)
3 sin, det ,
1 sin 1 sin
LD
cmkk ij
cm cm
IFI
I I
κ
ϕσ σ κ
ϕ ϕ
= − =
′− −= = =
′ ′− − − +
Lade-Duncan (empirical criterion):
30cmϕ′ = o 20cmϕ′ = o1σ−
2σ−3σ−
1σ−
2σ−3σ−
0MNF =
0LDF =
( )0.9 0DPF K = =( )1.0 0DPF K = =
28
Lade criterion (empirical):
31 1
3
27 0, Lade (1977)
, , - parameters
m
La
a
I IFI p
p m
η
η
⎛ ⎞⎛ ⎞= − − =⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
1σ−
2 3σ σ− = −
0.5m = 0.8m =!) nonlinear contourin meridian planes
Lade and Duncan (1975) Matsuoka and Nakai (1974)
Modfied Drucker-Prager Lade (1977)
1σ−
2σ−3σ−
30cmϕ′ = o20cmϕ′ = o
40cmϕ′ = o
1σ−
2σ−3σ−
30cmϕ′ = o20cmϕ′ = o
40cmϕ′ = o
1σ−
2σ−3σ−
30cmϕ′ = o
20cmϕ′ = o
40cmϕ′ = o0.778K =
1σ−
2σ−3σ−
28, 0.5, 50 kPaam pη = = =
150 kPap =
200 kPap =
50 kPap =
100 kPap =
Lade (1977) – φ´ depends on p
1σ−
2 3σ σ− = −
0.5m = 0.8m =
Some differences between presented shear strength criteria for soils and rocks
Mohr-Coulomb contouris shown for ϕcm=30°
29
Matsuoka-Nakai, Lade-Duncan and Mohr-Coulomb in principal stress space
Matsuoka-Nakai & Mohr-Coulomb Lade-Duncan & Mohr-Coulomb
0
50
100
150
200
250
0 0.02 0.04 0.06 0.08 0.10
−εyy [-]
t [k
Pa]
CMMNLD
DP
-0.04
-0.03
-0.02
-0.01
0
0 0.02 0.04 0.06 0.08 0.10
−εyy [-]
ε v [-]
CM
MN
LD
DP
Differences between responses of elasto-plastic models built with presentedshear strength criteria for biaxial compression (plane strain)
φc=30°, c=0 kPa, ψ=5°, E=10000 kPa, ν=0.15; initial stress is isotropic p=100 kPa; symbols: t=(σ1-σ3)/2, εv=ε1+ε3
30
Differences for geotechnical boundary conditions
Bearing capacity problem, shallow foundation
Homogeneous soil: φc=30°, c=1 kPa, ψc=0°, Eoed=80000 kPa, ν=0.2 (E0=72000 kPa), γ=18 kN/m3
shearing extension
Bearing capacity problem, results
yielding zones
force-displacement curves
31
Differences for geotechnical boundary conditions ...
Excavation problem, slurry wall
Homogeneous soil: φc=30°, c=1 kPa, ψc=0°, Eoed=80000 kPa, ν=0.2 (E0=72000 kPa), γ=18 kN/m3
γ12 distribution for Matsuoka-Nakai criterion;values: -5.6% bright to +1.2% dark;
displacement is scaled 20 timesHorizontal displacement vs. overburden pressure
bottom edge of the wall
top edge of the wall
Excavation problem, results
32
Differences for geotechnical boundary conditions ...
Pile bearing capacity problem
*) Eoed=M0
Pile bearing capacity problem, results
33
Soil slope stability calculationsSoil slope stability calculations
Slope failure mechanism is highly dependent on geological layering
*) source: Pouget & Livet, 1988
wys
okość
n.p.
m.
embankmentclayslimestonepowierzchniazniszczenia
disp
lace
men
t rat
e
time
Different stages of slope movements
*) source: Leroueil, 39th Rankine Lecture, Géotechnique 51(3), 2001
pre-failure
firstfailure
post
-fai
lure occasional
reactivation
acttive landslides
34
*) source: geopanorama.rncan.gc.ca
Quick-clay landslides
*) source: geopanorama.rncan.gc.ca
35
St. Jude/ Montreal May 11, 2010
*) http://www.montrealgazette.com
*) Trondheim, 1999
36
begin of observationH
oriz
onta
l dis
plac
emen
tof
the
wal
l, tra
ck le
vel
[ins]
Former ground profile
failure
analysedprobable } slip line
*) source: Skempton, 1967
Landslide has occurred 29 years after retaining wall instalation
Long term landslide, Kensal Green, 1941
General classification of slope stability calculation methods
1. Methods based on the fundamental equations ofcontinuum theory.
2. Methods where a potential failure mechanism is assumed.
37
Methods based on the fundamental equations of continuum theory
Equilibrium (Navier equations):
, 0ij j ifσ + =
Boundary conditions:
0, ij j i i in t v vσ = − =
Plasticity criterion (or constitutive law):
( ) 0, ij ij ijkl klf Dσ σ ε≤ = &&
Strain-displacement compatibility:
( ), ,12kl i j j iv vε = − +&
In practice, very often complicated boundary conditions are far from those which are assumed in the analytical solutions of fundamental equations.
x
y
*) Stability of a road embankment, hight 14.0m, reinforced by geotextiles, soft soil ground piled by jet-grouting columns. At the embankment toe a water reservoir is designed with sheet-pile walls(without anchoring !!!), Poland, Motorway A4, Ruda Śląska, 2004.
38
*) Stability of walls and vaults of historical structure, Wisłoujście Fortress, 2004.
Methods based on the fundamental equations of continuum theory ...
In complex and important engineering cases the fundamental equations ofcontinuum theory can be solved by numerical methods ex. By finite differences method or by finite element method.
However, the application of numerical modelling requires good knowledgeof their basis as well as it requires thorough understanding of continuum mechanics and geomechanics.
39
Examples of Finite Element Method (FEM) applications in geotechnical practice
*) horizontal displacement*) deformation
Examples of Finite Element Method (FEM) applications in geotechnical practice ...
40
*) Pylon foundation of a cable stayed bridge at the highway ring road of Wrocław (A8), 2009.
Examples of Finite Element Method (FEM) applications in geotechnical practice ...
*) pylon, Wrocław (A8) ...
41
*) pylon, Wrocław (A8) ...
a)b)
c)
*) pylon, Wrocław (A8) ...
42
Methods where the potential failure mechanism is assumed.
General assumptions for the methods of slices
1. Analysed boundary problem of slope stability is two dimensional with arbitrary shape of a slip surface. However very often only cylindrical slip surfaces are assumed.
2. Slip occurs simultaneously in all points of the assumed slip surface.
3. In standard calculations inertial forces are neglected.
In the initial phase of slope stability calculations by methods of slices it is very important to choose an appropriate failure mechanism.
Rotational shape of failure line
circular slip line(homogeneous soils)
non-circular slip line(inhomogeneous soils)
Failure mechanism
source: http://www.dur.ac.uk/~des0www4/cal/slopes/
43
Translational mechanism Compound mechanism
source: http://www.dur.ac.uk/~des0www4/cal/slopes/
Critical slip line
F
critical slip lineassumed centre
of rotation
minimum
source: http://www.dur.ac.uk/~des0www4/cal/slopes/
44
Standard procedure for searching the critical slip line
grid of centresof rotaion
source: http://www.dur.ac.uk/~des0www4/cal/slopes/
Local and global slope stability (scale of the failure mechanism)
source: http://www.dur.ac.uk/~des0www4/cal/slopes/
45
sand
clay
Influence of the soil type for the shape of critical failure mechanism
source: http://www.dur.ac.uk/~des0www4/cal/slopes/
Effect of a water filled tension crack at the head of a slide
source: http://www.dur.ac.uk/~des0www4/cal/slopes/
46
Short term and long term slope stability (parameters ϕ′,c′ and ϕu, cu)
excavation
embankment
construction time
construction time
time
time
u
u
+ compression
Shear strength mobilisation
r
r
r
s
s
s
Average value : p sr rτ τ τ> >
slip: sr pτ τ≈
source: http://www.dur.ac.uk/~des0www4/cal/slopes/
av
av
47
Method of slices
forces actingon a single slice:
centre of rotation
General scheme
source: http://www.dur.ac.uk/~des0www4/cal/slopes/
Fellenius method (also called as Swedish or oridinary)
assumptions:
48
Fellenius method- example
Bishop method (simplified)
assumptions:
Fellenius Bishop
( )∑∑ ⎟⎠⎞
⎜⎝⎛ +⋅
+−Δ+=
F
cbubXWW
Fαϕα
ϕα tantan1cos
tansin
1
49
Janbu method
Relates to the Bishop method taking into account lateral forces E.It allows for arbitrary non-rotational slip lines.
αααα
cossinsincos
SNESNXW
−=Δ+=Δ−
( )( )∑∑ ⋅
+−Δ−Δ−
=αα
ϕα m
cbubXWXW
Fcos
tantan
1
Fm αϕαα
sintancos +=
Spencer and Morgenstern-Price methods
constθ = constθ ≠Spencer Morgenstern-Price
( )X f xE
λ=tanXE
θ=
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Non-rotational failure mechanisms
Block mechanism
1. Active pore water pressure based on seepage line
h
ua=ust= h γw
* Very often used in the practice, the most conservative method.
How to take into account the pore water pressure in slope stability calculations ?
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2. Active pore water pressure based on seepage line with Hu reduction
10, ÷== uuwa HHhu γ
equipotential line (seepage)
α2cos=uH
In most cases coefficiant Hu is calculated from seepage line inclination:
3. Active pore water pressure calculated by ru coefficient method
10, ÷== uvua rru σ
Active pore water pressure is estimated as a fraction of the vertical totalstress σv component at the bottom level of analysed slice.
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ϕ-c reduction method
Slope stability safety factor estimated by FE-analysis
Fϕ-c
Strength parameters (tanϕ, c) are reduced in the incremental process up to the loss ofstatic equilibium in the analysed boundary problem. This numerical method falls to the methods based on the fundamental equations of continuum theory.