geotechnics - pg.gda.plmcud/teaching/content/gfx/geotechnics_01.pdf · geotechnics marcin cudny,...

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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

16

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


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

19

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

20

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

⎛ ⎞+ += −⎜ ⎟− −⎝ ⎠

21

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 ≈

22

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


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


a)b)

c)


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


Critical slip line

F

critical slip lineassumed centre

of rotation

minimum


44

Standard procedure for searching the critical slip line

grid of centresof rotaion


Local and global slope stability (scale of the failure mechanism)


45

sand

clay

Influence of the soil type for the shape of critical failure mechanism


Effect of a water filled tension crack at the head of a slide


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≈


av

av

47

Method of slices

forces actingon a single slice:

centre of rotation

General scheme


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=

50

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.

geotechnics - pg.gda.plmcud/teaching/content/gfx/geotechnics_01.pdf · geotechnics marcin cudny,...

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