slope evolution.pdf

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EVOLUTION OF NATURAL SLOPES

SUBJECT TO WEATHERING:

AN ANALYTICAL AND NUMERICAL STUDY

Tesi presentata per il

conseguimento del titolo di Dottore di RicercaPolitecnico di Milano

Dipartimento di Ingegneria StrutturaleDottorato in Ingegneria Sismica, Geotecnica

e dellinterazione Ambiente-Struttura - XVI Ciclo

Stefano Utili

April 2004


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EVOLUTION OF NATURAL SLOPES

SUBJECT TO WEATHERING:AN ANALYTICAL AND NUMERICAL STUDY

Ph.D Candidate:Eng. Stefano Utili

Supervisor:Prof. Roberto Nova

April 2004

Dottorato in Ingegneria Sismica, Geotecnicae dellInterazione Ambiente-Struttura del Politecnico di Milano

Scientific Committee:

Prof. Alberto Castellani (Coordinator)Prof. Carlo Andrea CastiglioniProf. Claudio ChesiProf. Annamaria CividiniProf. Claudio di PriscoProf. Ezio FaccioliProf. Cristina JommiProf. Sergio LagomarsinoEng. Paolo NegroProf. Roberto NovaProf. Roberto PaolucciProf. Maria Adelaide Parisi

Prof. Federico PerottiProf. Vincenzo PetriniProf. Giandomenico TonioloProf. Carlo Urbano

(Picture on the front cover: air photo of mudslide corries at Beltinge (UK); after [Hutchinson, 1970])


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One may define the human being, therefore, as the one who seeks thetruth.

[] The capacity to search for truth and to pose questions itselfimplies the rudiments of a response. Human beings would not even begin

to search for something of which they knew nothing or for something

which they thought was wholly beyond them. Only the sense that they can

arrive at an answer leads them to take the first step. This is what

normally happens in scientific research. When scientists, following their

intuition, set out in search of the logical and verifiable explanation of a

phenomenon, they are confident from the first that they will find an

answer, and they do not give up in the face of setbacks. They do not judge

their original intuition useless simply because they have not reachedtheir goal; rightly enough they will say that they have not yet found a

satisfactory answer.(Fides et Ratio)

Giovanni Paolo II


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Acknowledgements

I wish thank my supervisor prof. Roberto Nova who has been

fundamental for my professional training. He taught me a method of

work, knowledge and critical engineering assessment. He also gave me a

very interesting problem to deal with.

Thanks to Riccardo Castellanza, not only for his sincere friendship but

also for his support. His presence and his hints have been fundamental

for me in many circumstances.

Then I would like to thank prof. Claudio di Prisco for all the times I wentto his office asking him something. He always found time to answer me.

Finally, I wish thank all my friends who accompanied me during these

three years with their friendship. In particular, I would like to thank

Dorotea and Antonio.


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Index

i

Index

Introduction

Introduzione

1. Evolution of natural slopes subject to weathering:an introduction .........................................................1

1.1. Introduction ................................................................11.2. Weathering phenomenon.............................................31.3. Slope weathering: models from the literature..............71.3.1. Geomorphologic and geological models............................ 71.3.2. Engineering models.......................................................... 91.4. Objectives of the thesis ............................................. 101.5. Conceptual framework of reference ........................... 111.6. Choice of the methods used in the thesis ..................13


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Index

ii

1.7. Formulation of the problem ......................................161.8. Development of the thesis .........................................181.9. References.................................................................. 19

Part 1: Analytical study

2. Limit equilibrium methods ..................................... 252.1. Introduction .............................................................. 252.2. General formulation ..................................................282.3. Analysis of assumptions which make the problem

determinate ...............................................................35

2.3.1. First group..................................................................... 352.3.2. Second group ................................................................. 392.3.3. Third group ................................................................... 402.3.4. An encompassing algorithm........................................... 442.4. Limit equilibrium solutions: physical admissibility and

optimum .................................................................... 45

2.5. Conclusions and some critical considerations ............ 502.6. References.................................................................. 533. Retrogressive failure analytical law ........................573.1. Introduction .............................................................. 573.2. Mechanism of first failure: formulation based on the

limit equilibrium method........................................... 59

3.2.1. Introduction................................................................... 59


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Index

iii

3.2.2. Determination of the analytical solution........................ 603.2.3. Case of inclined slope..................................................... 643.3. First failure mechanism, formulation by the limit

analysis upper bound method.................................... 66

3.3.1. The limit analysis upper bound theorem ....................... 663.3.2. Determination of the analytical solution........................ 693.3.3. Charts relative to the first mechanism........................... 723.3.4. Numerical solutions by limit analysis ............................ 743.4. Determination of the second failure surface ..............763.4.1. Introduction................................................................... 763.4.2. Determination of the analytical solution........................ 773.4.3. Charts relative to the second mechanism ...................... 843.4.4. Some aspects of the first two mechanisms ..................... 863.4.5. A different problem: which is the best slope profile? .... 903.5. Determination of the successive failure surfaces........933.5.1. Procedure for the determination of the successive failure

surfaces .......................................................................... 93

3.5.2. Discussion of the results ................................................ 953.5.3. Other failure mechanisms .............................................. 993.6. Undrained conditions = 0 .................................... 1003.6.1. First failure surface...................................................... 1003.6.2. Second failure surface .................................................. 1023.7. References................................................................ 103Part 1 Conclusions...105


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Index

iv

Part 2: Numerical study

4. Description of the distinct element method ..........1094.1. Introduction ............................................................ 1094.2. Description of PFC-2D............................................ 1104.2.1. Law of motion.............................................................. 1124.2.2. Force-displacement law................................................ 1134.2.3. Calculation cycle.......................................................... 1184.2.4. Stability of the numerical scheme................................ 1204.2.5. Damping ...................................................................... 1204.2.6. Other discrete element methods .................................. 1214.2.7. PFC-2D parameters..................................................... 1224.3. References................................................................ 1245. Numerical simulations ...........................................1275.1. Introduction ............................................................ 1275.2. Calibration procedure.............................................. 1285.2.1. Calculation of stresses and strains ............................... 1295.2.2. Description of the biaxial test...................................... 1305.2.3. Specimen generation procedure.................................... 1315.2.4.

Biaxial test execution .................................................. 134

5.3. Calibration of micromechanical parameters toreproduce a frictional cohesionless material ............ 139

5.3.1. Influence of particle rotation on ................................ 1395.3.2. Range of confining pressures to be investigated........... 143


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Index

v

5.3.3. The role played by contact stiffness............................. 1455.3.4. Dependence of on confining pressure ........................ 1485.3.5. Choice of a suitable value of contact stiffness.............. 1505.3.6. Determination of

-relationship ............................. 152

5.4. Calibration of micromechanical parameters toreproduce a frictional-cohesive material .................. 154

5.4.1. Modelling of the contact behaviour ............................. 1545.4.2. Calibration of the new micromechanical parameters ... 1625.5. Simulations relative to weathering slope................. 1675.5.1. Features of the slopes analysed.................................... 1675.5.2. Particle radii and micromechanical properties

assigned ....................................................................... 168

5.5.3. Slope generation procedure .......................................... 1685.5.4. Simulation of the retrogressive failure ......................... 1715.5.5. First failure occurrence ................................................ 1715.5.6. Case of strong erosion conditions................................. 1745.5.7. Case of no erosion conditions....................................... 1815.6. References................................................................ 183Part 2 Conclusions.......187

6. Slope weathering: natural time scale .....................1806.1. Introduction ............................................................ 1806.2. Experimental validation of time-weathering laws ... 1816.2.1. Case 1: Warden Point.................................................. 1836.2.2. Case 2: Miramar .......................................................... 186


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Index

vi

6.3. References................................................................ 189Final conclusions..205

Appendix

A. Limit analysis results.....A.1A.1. Tables of results....A.1A.2. Linear interpolation of cohesion - crest retreat

relationships.....A.28


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Introduction

I

Introduction

This thesis is aimed at building a model capable of making

quantitative predictions about the future evolution of natural slopes

which may be significant from an engineering viewpoint (decades).

Slopes subject to weathering undergo a series of landslides, occurring at

different times, which cause the progressive retrogression of the slopefront. In order to make predictions it is necessary to model the discrete

succession of the failures occurring within the slope. To this end,

methods typical of slope stability analysis and the distinct element

method have been used.

Weathering processes within slopes are not object of investigation.

Only the effects of weathering on the stability of slopes have been

studied. Therefore chemical and/or physical weathering processes

occurring in soil are completely disregarded.

The main work hypothesis assumed in the thesis concerns soil

strength: it has been assumed that it may be suitably expressed by two

parameters: c, (cohesion and internal friction angle) according to the

Mohr-Coulomb strength criterion. As regards weathering, it has been

assumed that it influences only cohesion causing its progressive decrease

in time. Moreover, it has been assumed that weathering is uniform within

the slope. Indeed, weathering is not uniform, but this assumption made


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Introduction

II

the problem simpler and numerically treatable. However some

experimental data indicate that, at least in some cases, this assumption

leads to results that are not far from reality.

The first chapter has been conceived to give to the reader a

framework of the work done. In the chapter, the phenomenon studied is

described in detail; the assumptions made and the methods used to tackle

the problem are explained.

As regards the successive chapters, the thesis is divided into two

parts:

1. Analytical study: (Ch. 2 and 3) classical methods have beenused: limit equilibrium methods and limit analysis upper bound

method. An analytical law describing the discrete succession of

retrogressive failures of slopes subject to weathering has been

achieved by the limit analysis upper bound method (Ch. 3). This

law has been obtained as the solution to a mathematical problem.

The law relates the length of crest retreat and the succession of

profiles of the slope front to cohesion decrease. Since the

analytical law determined can be achieved also by limit

equilibrium methods according to a slightly different

formulation, a theoretical study of the limit equilibrium methods(the most widespread and known methods in geotechnical

engineering) has been performed (Ch. 2).

2. Numerical study: (Ch. 4 and 5) the discrete element method hasbeen used. According to this method, soil is modelled as a

discrete assembly of particles (micromechanical approach). An

experimental numerical campaign has been performed in order to

determine the micromechanical parameters which must be

assigned to the system of particles so that it reproduces a c,

continuum. The campaign made possible to achieve simple

relationships between micro and macromechanical parameters.

This is an important result which opens interesting possibilities

concerning the use of the distinct element method in soil

mechanics. Simulations of the retrogressive failure of slopes

subject to weathering have also been run. The results achieved

have been compared with the predictions obtained by the


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Introduction

III

analytical law determined by the limit analysis method (1stpart).

The agreement found between the predictions made by the two

methods was good. This fact corroborates the validity of the

results found since the two methods used (limit analysis and

DEM) are completely different.

The last chapter (Ch. 6) is devoted to the determination of the time

scale relative to the evolution of natural slopes from experimental data.

Simple relationships between time and crest retreat of slopes have been

determined.


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Introduzione

I

Introduzione

Questa tesi si propone di costruire un modello capace di elaborare

predizioni quantitative sullevoluzione di pendii naturali. Linteresse

della tesi rivolto a modellare levoluzione che i pendii subiscono in un

periodo di tempo significativo dal punto di vista ingegneristico (decadi). I

pendii soggetti a degradazione subiscono una serie di eventi franosi cheavvengono in tempi diversi e causano larretramento progressivo del

fronte dei pendii stessi. Per poter elaborare predizioni necessario

modellare una successione discreta di rotture allinterno dei pendii. A tal

fine sono stati usati sia metodi tipici dellanalisi di stabilit dei pendii che

il metodo degli elementi distinti.

Questa tesi non ha lo scopo di studiare i processi responsabili del

degrado del terreno costituente i pendii, ma gli effetti che tale degrado

induce sulla stabilit dei pendii; pertanto i processi chimici e/o fisici

responsabili del degrado del terreno non sono stati presi in alcun modo in

considerazione.

Lipotesi di lavoro principale riguarda la resistenza del suolo: si

ipotizzato che la resistenza del materiale costituente i pendii esaminati sia

caratterizzata da coesione e attrito (c, ) in accordo al criterio di rottura di

Mohr-Coulomb. Si assunto, inoltre che la degradazione influenzi solo

la coesione causando la sua progressiva diminuzione nel tempo. Si


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Introduzione

II

assunto ancora che il degrado sia uniforme allinterno dei pendii.

Questultima ipotesi non certamente verificata nella realt, ma essa

permette di rendere il problema pi semplice da affrontare specialmente

dal punto di vista numerico. Tuttavia, alcuni dati sperimentali indicano

che, in certi casi, lassunzione fatta conduce a risultati non lontani dal

reale.

Il primo capitolo costituisce un inquadramento generale del lavoro

svolto. L sono descritti in dettaglio: il fenomeno studiato, le assunzioni

fatte ed i metodi usati.

La tesi composta da due parti:1. Studio analitico: (Cap. 2 e 3) esso stato basato su metodi

classici dellanalisi di stabilit di pendii: i metodi

dellequilibrio limite e il metodo cinematico dellanalisi limite.

Questultimo ha permesso di determinare una legge analitica che

descrive la successione discreta delle rotture che avvengono nei

pendii soggetti a degradazione (Cap. 3). La legge analitica stata

ottenuta come soluzione ad un problema matematico. La legge

relaziona larretramento del fronte dei pendii (e la successione

dei diversi profili che i pendii via via assumono) alla diminuzione

di coesione. Dato che la legge analitica determinata pu essereanche ricavata, con una formulazione leggermente differente,

mediante i metodi dellequilibrio limite, si svolto un

approfondito studio teorico di questi ultimi (nella pratica

dellingegneria geotecnica i pi conosciuti e diffusi tra i vari

metodi usati per lanalisi di stabilit di pendii).

2. Studio numerico: (Cap. 4 e 5) stato condotto con il metododegli elementi distinti. Coerentemente al metodo, il terreno

stato modellato come un insieme discreto di particelle (approccio

micromeccanico). Per determinare i parametri micromeccanici da

assegnare al sistema di particelle affinch esso riproduca un

continuo caratterizzato da coesione e attrito (c, ) si eseguita

una campagna numerica di prove sperimentali. La serie di prove

eseguite ha permesso di determinare semplici relazioni tra

parametri micro e macromeccanici. Questo risultato apre

interessanti possibilit di sviluppo per limpiego del metodo degli


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Introduzione

III

elementi distinti nella meccanica delle terre. Infine sono state

condotte simulazioni della rottura retrogressiva di pendii soggetti

a degradazione. I risultati ottenuti sono stati confrontati con le

predizioni ottenute con la legge analitica determinata con

lanalisi limite (1a parte). Si cos potuto riscontrare un buon

accordo tra le predizioni elaborate con i due metodi. Questo fatto

corrobora in modo significativo la validit dei risultati trovati

dato che i due metodi usati (analisi limite e DEM) sono

completamente differenti.

Nellultimo capitolo (Cap. 6) si determinato il tempo scala relativoallevoluzione di pendii naturali a partire da dati sperimentali. Si

sono determinate delle semplici relazioni tra tempo e arretramento

del fronte dei pendii.


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Chapter 1 Evolution of natural slopes subject to weathering: an introduction

1

1.

Chapter 1

1. Evolution of natural slopes subject to

weathering: an introduction

1.1. IntroductionThis thesis is aimed at studying the evolution of natural slopes

subject to weathering processes. These processes cause a progressive

irreversible degradation of the slope material manifested as a reduction of

its mechanical properties. Therefore, as time goes on, landslides occur.

The time scale relative to the evolution of natural slopes ranges from

decades to thousands of years depending on soil type and weathering

processes acting on slopes. Weathering processes may have very

different velocities depending on the type of weathering. In the next

paragraph, an overview of such processes will be supplied.

Natural slopes are made of rock (granite, chalk, shale, marl,

limestone, etc.) or cohesive soil (clayey, silty, etc.). According to

[Terzaghi and Peck, 1967] soil means a natural block of grains

separable by a simple physical action like water agitation, whereas rock

defines a natural block of minerals jointed by strong and permanent

bonds. In reality, however, an intact granite and a mature quartzitic


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2

sand deposit are only the extrema of a continuous transformation process

of the material composing the earth crust. There is no distinct threshold

beyond which a geomaterial ceases to be a rock and starts to be a soil

[Nova, 1997]. In fact the earth crust is subject to physical and chemical

processes which continuously transform rocks into soils and viceversa as

shown in Fig. 1.1. These processes can be grouped into four different

types: weathering, erosion, diagenesis and metamorphism. The latter two

are phenomena leading to rock formation, whereas the former two are the

opposite.

Fig. 1.1: schematic representation of the transformation processes among geomaterials(after [Dobereiner and De Freitas, 1986]).

In literature, weathering is approached essentially in terms of its

description and classification. This phenomenon is very complex since it

depends on both the mineralogy of the intact rock or the decomposed soil

and environmental conditions. Usually different processes act together

and it is difficult to analyse them separately. For this reason, many and

different classification indices and scales have been defined. However, a

common feature of such processes is their capability to cause a

progressive irreversible degradation of the slope material manifested as a

reduction of its mechanical properties.


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3

1.2. Weathering phenomenon

Weathering may be defined as the process of alteration and

breakdown of rock and soil materials at and near the Earthsurface by

chemical decomposition and physical disintegration [Geol. Soc. Eng.

Group Work. Report, 1995].

Weathering reduces hard rocks into soft rocks which maintain the

structure of the intact rocks, but are characterised by higher void ratios

and reduced bond strengths. Soft rocks are transformed into granular

soils generally called residual soils. Residual soils differ from their parentrocks in mineralogical composition and structure.

Weathering processes belong to two categories depending on the

type of process: physical or chemical. Physical weathering causes the

mechanical destructuration of rocks without mineralogical change

whereas chemical weathering is due to chemical reactions leading to the

decomposition of the constituent minerals to stable or metastable

secondary mineral products.

Physical weathering is due to processes such as: freeze-thaw cycles,

temperature variations causing swelling-shrinkage cycles, wetting-drying

cycles, salt crystallisation, wind and rain action, living organism action(see Fig. 1.2).

Former and surface

Erosion

New and surface

PROCESSES:

differential thermal expansion and insulation wet-dry expansion freeze-thaw action rooth growth and burrows wind & rain action

cristilization & expansion

EFFECTS:

Unloading (stress relief) Joints formed Incipient fractures opened Intergranular and rock mass disintegration

PHYSICAL WEATHERING

Fig. 1.2: physical weathering processes (adapted from [Geol. Soc. Eng. Group Work.Report, 1995]).

Freeze and thaw cycles cause rock fracturing (see Fig. 1.3). Water


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thawing process needs a volume expansion (of the order of 10%) to occur

which if constrained by surrounding rock, leads to an increase of pressure

enlarging and developing cracks. Pressure exerted by ice may reach up to

200 MPa.

a) b)

Fig. 1.3: a) rock falls from natural slopes. b) Stages of freeze-thaw cycles (after [Martinati,2003]).

Temperature variations cause variations of the state of stress within

rock mass leading to swelling-shrinkage cycles responsible of rock

fracturing. Temperature variations between night and day may reach up

to 50 Celsius degrees in desert or mountainous areas. Anyway, thermal

excursions may be caused by fires or volcanic eruptions as well.

A similar phenomenon is due to wetting-drying cycles which cause

variations of the state of stress within clayey soils leading to swelling-

shrinkage cycles responsible of cracking development.

Salt crystallisation occurs during rainy seasons and in marine coastal

areas where waters with high salt content go into rock cracks and pores.

Crystals formed within cracks and pores exert pressures which develop

fracturing.

Living organism actions such as tree root growth and burrows

contribute to cracking development as well.

Chemical weathering is mainly due to chemical reactions which


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5

oxygen in the atmosphere, water and weak acids present into it are

involved in.

PROCESS:

solution

Weathered rock(solid product)

Fresh rock

PROCESSES:

Chemical alteration

Volume change

Textural change

EFFECTS:

Alteration of minerals:Feldspar + CO2+ H2O => Clays + Silicas + Cations

(Solid colloids) (Solutions)

Pyrite + O2+ H2O => Iron Oxide-Hydroxide+ (Acid solutions)

Dissolution of limestone:CaCO3+ H2O + CO2=> Ca(H2CO3)2(solute)

products

solution

CHEMICAL WEATHERING

Fig. 1.4: chemical weathering processes (adapted from [Geol. Soc. Eng. Group Work.Report, 1995]).

The reaction of oxygen with minerals gives rise to a process called

oxidation. The chemical element more subject to oxidation is iron. A

typical example is given by the pyrite oxidation:

4FeS2+ 10H2O + 15O2 2Fe2O3.H2O + 8H2SO4

Pyrite Limonite Sulfuric acid

Pyrite is a common rock-forming mineral in soft sedimentary rocks.

Examples of the effects of pyrite oxidation on slopes in Japan, have beenreported by [Chigira and Oyama, 1999].

Another typical example of weathering is due to the chemical

decomposition of granite. When water comes into contact with intact

granite, potassium feldspar and mica are transformed into kaolinite

(clay). This type of weathering is typical of granitic batholiths in the

Hong Kong region. Since long time ago, [Lumb, 1962] made a

systematic study attempting to characterise the mechanical behaviour of

the weathered granite. Such knowledge was requested to satisfy the

demand, particularly strong in that region, of engineering structures such

as high buildings, bridges, tunnels and roads.Usually physical and chemical weathering occur together, in such a

way that one acts to accelerate the other. In fact, the progress of chemical

weathering relies on cracks opened or enlarged as a result of physical

weathering. In the same way, cracks may develop in response to changes

in volume and weakening induced by chemical weathering.

In the light of these observations, a mechanical approach which


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6

neglects the specific weathering processes in order to model only the

weathering effects on the mechanical behaviour of the material subject to

degradation, appears reasonable.

In Fig. 1.5 and in Fig. 1.6 some case examples where it is evident the

need of an engineering study in order to predict the evolution of natural

slopes are shown. Many times civil structures (e. g. roads, railways,

buildings, waste deposits) have been located or designed to be located

nearby slopes subject to strong weathering conditions. In these cases, it is

of fundamental importance to suitably model the retrogression of the

slope front in order to establish if the structure will be reached by theretrogressive slope front and if it will be reached during its design life.

a) b)

Fig. 1.5: photographs of steep slopes in weakly-cemented sand. a) Mogn, Gran Canaria;urban development at the base (after [Delgado, 1991]. b) Daly City (California,USA); urban development on the top (after [Clough et al., 1981]).


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7

a) b)

Fig. 1.6: Esplanade Drive City of Pacifica (California, USA); a) cliff weathering affecting theslope crest (dashed lines indicate the successive failure lines) b) damages caused byslope retrogression. Many houses had to be demolished (after [Snell et al., 2000]).

1.3. Slope weathering: models from the literature

1.3.1. Geomorphologic and geological models

From an historical point of view, the first models aimed at describing

the evolution of natural slopes come from geomorphology and geology.

They have been conceived to describe the evolution of natural slopes

occurred in past eras on a geological time span. Therefore, all these

models are aimed at describing slope evolution in the long term (i.e.

thousands of years). They do not consider a succession of discrete events

(landslides), but a continuous process. In this process the sediment flux

due to landslides represents a long-term average of what is due to

individual events. The use of these average rates assumes that individual

slides are small enough not to change the slope profile significantly[Mills H. and Mills R., 2001]. In the time scale of decades (i.e.

engineering time) this is not true at all. Therefore, the need of a model

predicting the evolution of a slope taking explicitly into account the

single landslides which modify its profile, is underlined.

Slope evolution models were born in 1866 thank to Fishers work

[Fisher, 1866] and successively they have been developed by [Bakker


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8

and Le Heux, 1946, 1952], [Kirkby, 1973, 1984, 1987] and many others.

All these models approach the evolution of slope profiles in 2D and are

based on a mass balance or continuity equation relative to the slope mass:

z S

t x

=

(1.1)

where t is time, x is the Cartesian co-ordinate relative to the horizontal

direction, z the elevation of the slope profile and S the total downslope

flux of sediment. S is given by:

S C W L= + + (1.2)

where 1C K z x= is called creep term, W is called wash term and L

represents the sediment flux due to landslides. Substituting Eq. (1.2) into

(1.1) a differential equation is obtained. This equation is non-linear

because of the complicated expressions relative to W and L.

The simplest models, for instance [Andrews and Hanks, 1985],

neglect the contribution given by W and L terms, so that the substitution

of Eq. (1.2) into (1.1) gives rise to:

2

2

z z zK K

t x x

= =

(1.3)

called linear diffusion equation. This equation is well known to

mathematicians like theFourier equation(widely studied in literature).

Instead, if the wash and landslide contributions are taken into

account a non-linear differential equation is achieved and models are

known as non-linear diffusion models. Non-linear diffusion models have

been widely applied in North-America [Andrews and Hanks, 1987].

Recently, a computer code, based on [Kirkby, 1984] non-linear diffusion

model, the most encompassing model attempt according to [Mills and

Mills, 2001], has been implemented by [Kirkby et al., 1992].In all these models, the parameters and variables introduced have

only an apparent physical meaning. They are aimed at taking into

account phenomena such as creep, wash, landsliding, but they are not at

all related to any mechanical quantity characterising the soil slope as well

as to any chemical quantity. The only variables with clear physical

meaning are the geometric ones.


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9

Anyway, parameters and variables without physical meaning, may be

still used to describe a phenomenon if they are able to reproduce the

experimental data relative to it. As stated previously, these models were

born to describe slope evolution in the long-term; therefore they have

been applied to reproduce the evolution undergone by natural slopes in

thousands of years. In order to obtain the succession of slope profiles

formed in the past, the so-called time-location technique or space-time

substitution is used: some slope profiles surveyed in the same area are

taken as the temporal succession of the profile of a slope. [Kirkby, 1984]

took the profiles reported by [Savigear, 1952] (between Laugharne andPendine in Wales, UK) as a temporal sequence of profiles of a slope

which he used to validate his model. Assuming the time-location

technique reliable, Kirkby model matches well the experimental data.

In conclusion, geomorphologic and geological models are valid to

describe the slope evolution occurred in a long term period even if they

have not a solid physical and mechanical background, but they cannot be

used to predict the future evolution of slopes on an engineering time

span. In fact, these models take into account discrete events such as

landslides only by representing them with a variable that contributes

continuously to slope erosion. Moreover, the achieved profiles cannot bein any way actual profiles but only evolution profiles averaged on long

time spans.

1.3.2. Engineering models

Another category of models, belonging to Soil and Rock Mechanics

disciplines, are featured by being related to the mechanical characteristics

of natural slopes. In these models, material characteristics such as unit

weight, strength and deformability, are taken into account. Weathering,

chemical or physical, is not taken into account from a phenomenologicalpoint of view i.e. the specific weathering processes are disregarded.

Weathering is considered as a phenomenon causing variations of the state

variables used to describe soil/rock characteristics. This means that, for

instance in case of chemical weathering, kinetics of chemical reactions,

diffusivity and so on do not appear in the models.

If quantitative predictions about the future evolution of natural slopes


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10

in terms of actual slope profiles and times at which landslides occur,

engineering models have to be used. One of them is that supplied by this

thesis.

1.4. Objectives of the thesis

As already stated in 1.1, a clear distinction between rocks and soils

does not exist in nature. In Geotechnical Engineering, aggregates of

particles (grains) whose chemical bonds are weak and may be broken by

load levels typical of Civil Engineering, are considered soils; whereasmaterials characterised by strong chemical bonds whose mechanical

behaviour is governed by discontinuities (joints and faults) rather than

intact rock mechanical properties, are considered rocks. Many materials

are intermediate between rocks and soils, because they behave as rocks if

subject to low stress levels and as soils if subject to stress levels high

enough to break the chemical bonds among their grains [Nova, 2002].

This thesis is aimed at modelling the evolution of dry natural slopes

made of cohesive soils and soil-like materials i.e. materials whose

discontinuities play a non significant role in determining the conditions

leading to landslide occurrence. According to the definition of soil givenabove, rock slopes whose joints play a negligible role into landslide

occurrence are included in this study. For instance, this is the case of

many coastal cliffs made of chalk (South England, Normandy, Greece,

Sardinia, Abruzzi, etc.).

Slopes are considered dry since this case is the simpler problem to be

tackled upon which extensions to more complex cases (seepage

conditions, earthquake occurrence, consolidation processes) may be dealt

with. This is the first necessary step towards the study of more complex

situations.

As time goes on, natural slopes are subject to a progressive reduction

of their mechanical properties leading to multiple landslides causing the

retrogression of the slope crest. This phenomenon occurs in decades for

coastal cliffs because of high chemical weathering caused by the sea

whereas it needs a longer period to occur (up to thousands of years) for

inland slopes.


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This thesis is aimed at determining a law able to predict times at

which landslides occur. Moreover, the evolution of slope profiles with

particular interest to the retrogression of the slope crest, are object of

investigation (see Fig. 1.7).

t1

t3

t2

t4t5

t6

Fig. 1.7: schematic representation of an initially vertical slope subject to weathering. Eachline represents a failure line at the onset of a landslide at the time ti.

It is pointed out that this law intends to describe the evolution of

natural slopes from a qualitative viewpoint and not from a quantitative

one. This is due to the simplifications introduced to model the problem

(see 1.7) and to the phenomena significant in triggering landslides that

have been neglected on purpose.

1.5. Conceptual framework of reference

Nowadays, the mechanical behaviour of soil-like materials

(according to an engineering classification) at macroscopic level may be

described by constitutive equations with a good level of accuracy. On the

contrary, at microscopic level they present strong heterogeneities as they

are composed of bonded particles with very different shapes and sizes,

bond strength between particles is inhomogeneous, etc.. Moreover, at this

scale, weathering processes are strongly inhomogeneous (see Fig. 1.8).


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a) b) c)

Fig. 1.8: calcarenite (sedimentary soft rock formed by cementation of organogeus calcareousgrains) of Gravina (Puglia, Italy) SEM photos. a) Microfabric (magn. 30). b) Shellsgrains (magn. 130). c) Shell surface (magn. 3500). (after [Castellanza, 2001]).

Therefore, the only available way to tackle the problem is by

considering the slope material at macroscopic level. Within a certain

volume (Representative Elementary Volume) soil is assumed

homogeneous. Hence, the conceptual framework of reference is that of

Continuum Mechanics. Within this framework, soil behaviour is

characterised by constitutive equations relating stresses to strains.

Weathering is modelled as a reduction of soil strength according to a

certain law. This law depends on both space and time. Its dependence on

space has been assumed, whereas its dependence on time has been

determined as result of the study (see Ch. 6).

In this work, weathering has been assumed uniform throughout the

whole slope. This hypothesis may appear very strong. In fact, considering

a coastal cliff subject to chemical weathering (oxidation, dissolution

reactions) and physical weathering (wind and rain actions) it is

reasonable to think that the deeper the soil is, the weaker the attack; or in

other words, that the most weathered soil lies into the most shallow

regions of the cliff. But experimental evidences support this hypothesis.

Yokota and Iwamatsu performed SPT tests into a slope onto Kyushu

Island (Japan) to test the variation of soil (soft pyroclastic rock) hardness

with depth [Yokota and Iwamatsu, 1999]. From Fig. 1.9 emerges that the

hardness is almost constant in the upper part of the slope whereas in the

lower part of the slope (toe) it can be considered constant from a small

depth inwards. According to these data, weathering can be, in first

approximation, considered uniform.


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a)

b) c)Fig. 1.9: Kyushu Island (Japan) hardness distribution in a steep slope: a) and b) data from

the upper part of the slope; c) data from the slope toe (after [Yokota andIwamatsu, 1999]).

1.6. Choice of the methods used in the thesis

The conditions, in terms of level of actual soil strength, at which a

stable slope becomes unstable need to be determined. For this reason,

some methods developed for slope stabilityproblems have been used in

the thesis.

From an historical point of view, the first methods to analyse slopestability are the limit equilibrium methods. Successively, limit analysis

methods have been applied to slope stability problems and later on the

finite element method too.

Limit equilibrium methods are the most widely used methods in

engineering practice for slope stability problems. Usually, they are used


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14

to assess the stability of a slope in terms of a factor of safety. A feature of

these methods is represented by the few parameters needed to describe

the soil properties: unit weight, internal friction angle, cohesion. This is

very important since there is a great lack of data relative to natural

slopes. Moreover, it can be generally stated that if two models manage to

reproduce a phenomenon with the same degree of accuracy, the better is

that which needs the smaller number of parameters.

Limit analysis has been applied in many fields of Engineering. One

of them is represented by slope stability. The limit analysis methods,

lower and upper bound, are much more versatile than limit equilibriummethods and above all, supply solutions which are rigorously lower and

upper bounds on the true collapse load.

The limit analysis methods need to assume an associated flow rule.

As it is well known in literature, soil behaviour does not obey to

associativeness (dilation angle is less than friction angle), but for the

studied problem the influence of dilation is small since soil is not

confined.

Therefore these methods have been chosen in order to derive an

analytical law describing the evolution of slopes subject to weathering

(see Ch. 3). Soil strength has been characterised by the Mohr-Coulombfailure criterion (see Fig. 1.11). All the parameters needed by this law to

describe the soil properties are: unit weight, internal friction angle,

cohesion.

In the literature, lower bound solutions for slope stabilityproblems

are a few ([Pastor, 1978]) as the determination of a static stress field is a

very difficult task for a complex slope profile. On the contrary, the

determination of a kinematic admissible velocity field is still an

affordable task even for complex slope profiles. For this reason, only the

upper bound method has been used to determine an analytical law

describing the evolution of slopes subject to weathering (see Ch. 3).

Recently, limit analysis has been incorporated into a numerical

formulation that uses finite elements for the discretisation of the soil

mass [Sloan, 1988, 1989], [Sloan and Kleeman, 1995], [Kim, 1998],

[Kim et al., 1999, 2002]. The lower and upper bound theorems have been

formulated by Sloan [Sloan, 1988, 1989] as linear problems to be solved


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using linear programming technique. Based on finite element

discretisation of the slope the velocity field is optimised to find the

lowest upper bound and the stress field is optimised to obtain the highest

lower bound. Unlike traditional displacement based finite elements, each

node of the finite elements used in limit analysis is unique to a given

element. In order to use linear programming techniques, all the

conditions required for static and kinematic admissibility are formulated

as linear constraints on the nodal variables of the finite element

discretisation.

Some numerical lower and upper bound solutions obtained by[Loukidis et al., 2003] have been used in 3.3.1 for comparison with the

results obtained by the classical upper bound method.

The finite element method (perhaps the most world-wide known

method in Engineering) has been used for slope stability analysis since

1975 [Zienkiewicz et al., 1975].

With this method, it is difficult to determine a failure line since only

a region where deformations localise (shear band) results from analyses.

If soil is modelled within the framework of the classical plasticity theory,

shear band width is mesh dependent. In order to avoid meaningless

results, advanced continuum models are needed: non-local models suchas those in the framework of gradient plasticity theory and viscoplasticity

theory. These continuum models need far more parameters to

characterise soil than the 3 parameters needed by limit analysis and limit

equilibrium methods. Apart the complication due to so many parameters,

often with an obscure physical meaning, their calibration is a very

difficult task.

The evolution of natural slopes is characterised by many landslides.

After each landslide a new slope profile is formed. Remeshing is needed

because of the new geometry. This becomes time-expensive if an

automatic remeshing in the finite element pre-processor is not available.

In conclusion, in comparison with the other methods used in this

thesis, the finite element method appears by far the less convenient.

The distinct element method is the youngest method, introduced by

[Cundall, 1971] and applied to soil mechanics by [Cundall and Strack,

1979]. This method is not based on continuum mechanics but on a


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16

discrete micromechanical approach. In Ch. 4 a detailed description of the

principles of this method will be supplied.

In the numerical code PFC2D (Particle Flow Code 2 Dimensions),

soil is represented by an assembly of rigid disks upon which contact

forces act. Walls apply the boundary conditions relative to displacements.

Dynamic equilibrium equations are imposed for the system of disks.

Explicit schemes of direct integration in time are used to follow the

evolution of the system from a steady state to the successive one (for a

detailed description of the code see Ch. 4).

All the difficulties met by the finite element method, are notencountered by this method. In fact, when a landslide develops, because

of decreasing of soil strength, a part of disks starts to move, giving rise to

a failure line easily detectable. For this reason, this method has been

chosen to numerically simulate the boundary value problem under study

as it will be formulated in the next paragraph. The second part of this

thesis is devoted to this study.

Unfortunately, a micromechanical description of natural slopes is not

available. The method needs parameters which, in general, are not known

for geomaterials. These parameters can be defined as micromechanical

since they rule the mechanical behaviour at a micromechanical scale. Inorder to determine them, calibration is needed. Calibration consists in

determining the micromechanical parameter values such that if a volume

of particles (disks) large enough to be representative is considered, its

mechanical behaviour is identical to that of the continuum described by

known macromechanical constitutive laws and parameters whose

behaviour is intended to be simulated.

Therefore, thank to calibration process, the input parameters needed

to characterise the slope material are the parameters used in the

macromechanical description of it. For this study they are: unit weight,

internal friction angle and cohesion.

1.7. Formulation of the problem

In this paragraph the problem tackled by the analytical and numerical

study of this thesis is formulated.


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17

Let us consider a plane uniform slope made of homogeneous

material, as shown in Fig. 1.10, whose height is H. No loads are present

on the slope. The known soil mechanical properties concern density and

resistance. The former is characterised by unit weight and the latter by

the Mohr-Coulomb failure criterion. According to this criterion, two

parameters are needed to describe the soil strength: , internal friction

angleandc,cohesion. In the literature, this ideal type of soil material is

also known as c- material.

In the limit analysis upper bound method (part 1), the stress-strain

behaviour assumed is rigid perfectly plastic, with associate flow rule, andas a consequence dilation angle equal to internal friction angle . In the

distinct element method (part 2), the constitutive laws are imposed at

micromechanical level since force-displacement laws rule the interaction

between distinct rigid elements at contacts. The force-displacement laws

used are characterised by a micromechanical friction angle 0 and a

micromechanical dilation angle 0 = .

Chemical weathering affects soil mechanical strength causing its

progressive decrease in time. It is assumed that only cohesion decreases

whereas friction remains constant. Cohesion decrease is continuous and

uniform. In Fig. 1.11, the failure surface is represented in the Mohr plane.According to the adopted assumptions, the failure surface evolves

remaining a straight line and lowering with constant inclination until soil

becomes uncohesive, c = 0.

H

Fig. 1.10: uniform slope.


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18

n

fresh materialpartially weathered

materialuncemented soil

time increasing

t

c

Fig. 1.11: failure criterion evolution due to weathering.

According to Fig. 1.11, the tension strength t is equal to cotc . This

assumption is an overestimation of the real material tension strength. But

the simplification is needed by the type of mechanisms assumed in the

limit analysis method (see Ch. 3). Assuming cott c implies the

possibility of the development of tension cracks which cannot be taken

into account by the mechanisms used in the limit analysis method.

Anyway, it is assessed that this assumption should have a small effect on

the predicted slope profiles and the retrogression of the slope crest.

1.8. Development of the thesis

A whole chapter (2) is devoted to limit equilibrium methods showing

their common features and their differences. At this end, all the presented

methods are derived as particular cases of a general formulation. A

classification of the methods according to their features (assumptions and

solution schemes) has been attempted. The performance of these methods

depend on the type of slope and failure surface examined. Therefore an

assessment relative to the various methods for different types of failure

surfaces has been attempted in order to let the reader have an overview as

complete as possible of the capabilities and limitations of the methods.

In chap. 3, an analytical law describing the evolution of the slope

described in 1.7, has been achieved by use of both limit equilibrium

methods and upper bound limit analysis method. It will be shown that

limit equilibrium methods and limit analysis upper bound method lead to

the same equations. Nevertheless, it has been decided to use both


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19

methods since the limit equilibrium formulation is simpler and more

known in engineering practice than the limit analysis formulation

whereas limit analysis upper bound method has a more solid theoretical

background.

In part 2, the evolution of the slope described in 1.7, is studied by

use of the distinct element method. First, an experimental campaign has

been run on numerical specimens in order to assign the micromechanical

properties to the set of rigid disks and contact bonds representing the

slope. Second, the evolution of the slope has been studied by simulating

the progressive weathering of the slope, decreasing the strength ofcontact bonds between disks.

In Ch. 6, simple time-retrogression of the slope front laws have been

determined. At this end, the retrogressive failure analytical law

determined in Ch. 3 and the results obtained in part 2 have been used as

well as experimental data relative to some monitored natural slopes

subject to strong weathering.

1.9. References

Andrews D. J. and Bucknam R. C., 1987. Fitting degradation ofshoreline scarps by a nonlinear diffusion model. J. Geophys. Res., 92,

pp. 12857-12867.

Andrews D. J. and Hanks T. C., 1985. Scarp degraded by linear

diffusion: inverse solution for age. J. Geophys. Res., 90, pp. 10193-

10208.

Castellanza R., 2001. Weathering effects on the mechanical behaviour of

bonded geomaterials: an experimental, theoretical and numerical study.

PhD. thesis, Politecnico di Milano, Milan.

Cendrero A. and Dramis F., 1996. The contribution of landslides to

landscape evolution in Europe. Geomorphology, 15, pp. 191-211.

Chigira M., Oyama T., 1999. Mechanism and effect of chemical

weathering of sedimentary rocks. Engrg. Geol., 55, pp. 3-14.


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Clough G. W., Sitar N., Bachus R. C., Rad N. S., 1981. Cemented sand

under static loading. J. Geotech. Engrg. Div., ASCE, 107(GT6), pp. 799-

817.

Cundall P.A., 1971.A computer model for simulating progressive, large-

scale movements in blocky rock systems. Proc. Symp. Int. Soc. Rock

Mech., Nancy, 2, art. 8.

Cundall P.A. and Strack O. D. L., 1979.A discrete numerical model for

granular assemblies.Gotechnique, 29, pp. 47-65.

Delgado E., 1991. A vista de parajo (Espaa a vuelo). RTVE

Publications, Estagraf.

Dobereiner L., de Freitas M. H., 1986. Geotechnical properties of weak

sandstones.Gotechnique 36(1), pp. 79-94.

Kirkby M. J., 1971. Hillslope process-response models based on the

continuity equation. In: Brunsden D. ed., Slopes: form and process. Inst.

Brit. Geogrs. Special Publication 3, pp. 15-30.

Kirkby M. J., 1973. Landslides and weathering rates. Geol. Appl. eIdrogeol., Bari, 8, pp. 171-183.

Kirkby M. J., 1984. Modelling cliff development in South Wales:

Savigear re-viewed. Zeitschrift fur Geomorphologie., 28, pp. 405-426.

Kirkby M. J., 1987.General models of long-term slope evolution through

mass movement. In: M. G. Anderson and K. S. Richards eds., Slope

Stability. Wiley, pp. 359-379.

Kirkby M. J., Naden P. S., Burt T. P., Butcher D. P., 1992. Computer

simulation in physical geography. Wiley, pp. 85-90.

Martinati S., 2003.Modellazione degli effetti della degradazione chimica

di geomateriali cementati in opere sotterraneee. (in Italian) Degree

thesis, Politecnico di Milano, Milan.

Mills H. H., Mills R. T., 2001. Evolution of undercut slopes on

abandoned incised meanders in the Eastern Highland Rim of Tennessee,


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USA.Geomorphology, 38, pp. 317-336.

Nova R., 1997. On the modelling of the mechanical effects of diagenesis

and weathering. ISRM News Journal, 4(2), pp. 15-20.

Nova R., 2002. Fondamenti di meccanica delle terre. McGraw-Hill

edition.

Pastor J., 1978. Limit analysis: numerical determination of complete

statical solutions. Application to the vertical cut. J. Mecanique applique

2(2), pp. 167-196 (in French).

Savigear R. A. G., 1952. Some observations on slope development in

South Wales. Trans. Inst. Brit. Geogrs., 18, pp. 31-52.

Sloan S. W., 1988.Lower bound limit analysis using finite elements and

linear programming.Int. J. Numer. Anal. Methods Geomech., 12, pp. 61-

77.

Sloan S. W., 1989. Upper bound limit analysis using finite elements and

linear programming. Int. J. Numer. Anal. Methods Geomech., 13, pp.

263-282.

Snell C. B., Lajoie K. R., Medley E. W., 2000. Sea-cliff erosion at

Pacifica, California caused by 1997/98 El Nio storms. In D. V.

Griffiths, G. A. Fenton, T. R. Martin eds., Slope stability 2000. Proc.

Geo-Denver 2000, Denver, Colorado, USA.

Terzaghi K. and Peck R. B., 1967. Soils mechanics in engineering

practice.Wiley, 2ndedition.

Yokota S. and Iwamatsu A., 1999. Weathering distribution in a steep

slope of soft pyroclastic rocks as an indicator of slope instability.Engrg.Geol., 55, pp. 57-68.

Zienkiewicz O. C., Humpheson C., Lewis R. W., 1975. Associated and

non-associated visco-plasticity and plasticity in soil mechanics.

Gotechnique, 25(4), pp. 671-689.


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Part 1

Analytical study

SummaryThis part is devoted to the theoretical study made relative to the

problem tackled: the evolution of natural slopes whose behaviour may be

suitably described by assuming that the slope soil is of c, type.

In chapter two a comprehensive review of limit equilibrium methods

is supplied. The chapter is aimed at showing the main features of such

methods which are the most used in the today geotechnical engineering

practise for the analysis of slope stability.

In chapter three the analytical law achieved by the author to describe

the evolution of the investigated slopes is illustrated. This law can bederived by using two different conceptual frameworks based on two

different theories referring to limit equilibrium methods and to limit

analysis upper bound method respectively.

This part has been called analytical to stress the main result of this

theoretical study: an analytical law describing the retrogression of the

slope front caused by the progression in time of the slope weathering.


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Chapter 2 Limit equilibrium methods

25

2.

Chapter 2

2. Limit equilibrium methods

2.1. IntroductionLimit equilibrium methods have been used in geotechnical

engineering for decades in order to assess the stability of slopes. The idea

of discretizing a potential sliding mass in vertical slices was introduced

early in the 20thcentury. In 1916, Petterson [Petterson, 1955] presented

the stability analysis of the Stigberg Quay in Gothenberg, Sweden where

the slip surface was taken circular. But the first method of slices is

associated to the Felleniuss name [Fellenius, 1927, 1936]. His method

also known as the Ordinary method, the Swedish circle method, the

conventional method and the US Bureau of reclamation method assumes

no interslice forces and the factor of safety is achieved by the overall

moment equilibrium around the centre of a circular slip surface.

In the mid-50s, Janbu [Janbu, 1954] and Bishop [Bishop, 1955] made

advances in the method. Janbu developed his method for generic slip

surfaces whereas Fellenius and Bishop developed their methods for

circular surfaces only (later Bishop extended his method to generic

surfaces).

In the 60s and 70s most methods were invented: some making the


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26

limit equilibrium method a more powerful and refined tool of analysis of

slope stability (Spencer, Morgenstern & Price, Sarma methods) and other

making it more suitable for hand calculations (force equilibrium

methods). Many articles were published in these years on this topic: in

some of them a real contribution to the improvement of the method was

given whereas in others only slight modifications or different

formulations of earlier methods were given.

In the late 50s, these methods began to be implemented in computer

codes. Little and Price (1958) were the first who used a computer to run

stability analyses by the Bishop simplified method.The advent of powerful desktop personal computers in the 1980s

made economically viable to develop commercial software based on limit

equilibrium methods.

Nowadays, these methods are routinely used for stability analyses in

geotechnical engineering practice and many programs are available

(examples will be supplied further on).

Methods of slices can be classified according to different criteria:

1. suitable only for circular failure surfaces or applicable to anyshape of surface.

2. rigorous and simplified: the former satisfy all equilibriumequations whereas the latter satisfy only a part of equilibrium

equations. Some authors developed two versions (simplified and

rigorous) of the same method: Bishop, Sarma, Janbu, etc. Within

simplified methods, a large group is given by the methods of

forces (Lowe & Karafiath, Corps of engineers method, Seed &

Sultan).

3. depending on assumptions made to render the problem staticallydeterminate: 3 groups of methods can be recognised on the basis

of the hypotheses introduced about the interslice forces

[Espinoza et al., 1992a, 1994].

4. based on the parameter used to determine the critical surface: thetraditional factor of safety Fs or other parameters such as the

critical horizontal uniform acceleration [Sarma, 1973, 1979],

[Spencer, 1978].

All methods approximate the bottom boundary of slices with linear


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27

bases. Formulations are based either on differential equations (e.g.

[Janbu, 1954], [Bishop, 1955], [Spencer, 1967]) or algebraic equations

making difficult to compare different methods to the inexperienced

reader. As the factor of safety is calculated by algebraic equations and

limit equilibrium methods are based on a slope division into a discrete

number of slices, here the latter formulation is preferred.

Equations Condition

A

B

C

2n

n

n

force equilibrium in two directions for each slice

moment equilibrium for each slice

Mohr-Coulomb failure criterion

4n total number of equations

Unknowns Description

D

E

F

G

HI

L

1

n

n

n

n-1n-1

n-1

factor of safety

normal force at the base of each slice,Pi

location of normal forces at the base of slices

shear force at the base of each slice, Si

interslice horizontal force,Eiinterslice vertical force, Ti

location of interslice forces (line of thrust) hi

6n-2 total number of unknowns

Table 2.1

In table 2.1 the number of equations and unknowns are summarised.

The difference between equations and unknowns gives the number of

assumptions to render the problem statically determinate:

6 2 4 2 2n n n = . All methods make nassumptions on the locations of

Pi. Most methods assume the normal force acting at the base centre(uniform stress distribution), even if other methods assume Pi acting at

the point of intersection between the resultant vertical force and the base

of the slice [Janbu, 1954] or according to a linear stress distribution

[Morgenstern & Price, 1965]. However, the influence of the location of

Pion the safety factor is negligible for a sufficiently large number of thin

slices. It may become an important factor in a wedge type analysis, when,


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28

for instance, only two or three slices are used [Espinoza et al., 1994].

Concerning the remaining n-2 assumptions, methods differ greatly.

Usually, n-1 assumptions are made about interslice forces and an extra-

unknown to be determined together with the factor of safety is

introduced.

2.2. General formulationHere, a general formulation shared by a large number of methods

(the rigorous ones) is given in order to show the main features commonto most methods and their differences. In the following, notation has been

taken according to Fig. 2.1.

Fig. 2.1: notation relative to the forces acting on a slice

In order to achieve the normal and shear base forces, the equilibrium

of forces along two perpendicular directions is written for all the nslices.

Directions may be either vertical and horizontal or parallel and normal tothe slice base. From imposing the equilibrium along the direction normal

to the slice base,Piis achieved:

( )cos sin sini i i i i i i iP W T E Q = + (2.1)

where 1i i iE E E+ = , 1i i iT T T+ = , i si iW W V= + , siW is the slice weight,Vi is the vertical applied load and Qithe horizontal one. Similarly, from


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29

equilibrium along the direction parallel to the slice base, Siis achieved:

( )sin cos cosi i i i i i i iS W T E Q = + . (2.2)

Along the potential slip surface, shear stress is at the limiting of failure;

therefore:

tanf c = + (2.3)

according to the Mohr-Coulomb criterion.

The factor of safety F is defined as that value by which the available

shear strength parameters must be reduced in order to bring the soil massinto a state of limiting equilibrium along a given slip surface. Hence, the

mobilised shear strength is defined by:

tanm

c

F F

= + . (2.4)

Accordingly, the shear force acting along the base of each slice is given

by:

( )1

tani i iS c l P F

= + (2.5)

where cosi i il x = . From Eq. (2.5), Si is substituted into Eq. (2.2),obtaining:

( ) ( )1

sin cos cos tani i i i i i i i iW T E Q c l P F

+ = + (2.6)

From Eq. (2.1),Pi is substituted into Eq. (2.6); rearranging:

( )1

sin cos tanii i i i i i

E W W c l mF

= +

tancos sinii i i i

T m QF

+ +

(2.7)

wheretan

cos sini i i

mF

= + . This parameter was defined first by Janbu

[Janbu et al. 1956].


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30

Summing the increments of horizontal interslice forces throughout the

soil mass gives:

( )1 11 1 1

10 sin cos tan

i i

n n n

i n i i i i i

i i i

E E E W m W c l mF

+= = =

= = = +

1 1

tancos sin

i

n n

i i i i

i i

T m QF

= =

+ +

(2.8)

In fact, according to the assumed notation, 1 1 0nE E+ . The non null

interslice forces range from 2 2,E T to ,n nE T . Rearranging Eq. (2.8), thesafety factor is obtained:

( )1

1 1

cos tan

sin

i

i

n

i i i

iff n n

i i i ff

i i

W c l m

F

W m Q I

=

= =

+ =

+ +

(2.9)

where1

tancos sin

i

n

ff i i i

i ff

I T mF

=

=

.

The subscriptff is used to indicate that the calculated factor of safety

has been achieved by the equilibrium of forces. As ffI and im depend

onFff, thus iterative schemes must be used to determineFff.

From the overall moment equilibrium equation, another factor of

safety Fmm can be achieved, as well. First, two equations must be

imposed for each slice: the equilibrium of forces along the vertical

direction (this direction is chosen so that the normal interslice forces do

not appear) and the Mohr-Coulomb failure criterion. Second, if the

overall equilibrium of moments of all forces about an arbitrary point is

imposed, the factor of safety is obtained.The choice of the pole for the moment equilibrium equation depends

on the slip surface assumed. In case of circular slip, the centre of the

circle is selected as calculations become enormously more simple. In

case of a non circular slip, any point may be chosen, but there are

convergence numerical reasons which determine a criterion of choice. In

fact, there are zones in which the location of the pole can result in


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numerical instabilities due to computer round-off errors [Fredlund et al.

1992]. A good choice, may be the centre of the circle tangent to the

envelope of the slip surface.

In the following, the factor of safety of the moments is derived. First

the equilibrium equations along the vertical direction are imposed for

each slice:

sin

cos

i i i ii

i

W T SP

= (2.10)

then, from Eq. (2.10)Piis substituted into Eq. (2.5) obtaining Si:

( )1

cos tanii i i i i

S c l W T mF

= + . (2.11)

Similarly, from Eq. (2.5) Siis substituted into Eq. (2.10) obtainingPi:

sini

i ii i i

c lP W T m

F

=

. (2.12)

Imposing the overall equilibrium equation of moments leads to:

1 1 1 1

n n n n

i W i i Qi i S i i Pii i i iW b Q b S b Pb= = = =+ = + (2.13)

where bare the arms of the forces. The sign convention adopted refers to

a pole above the toe region of the slip surface (see Fig. 2.2). Substituting

the interslice forces from Eq. (2.11) and (2.12) into Eq. (2.13) leads to:

1 1 1

sini

n n n

i W i i Qi i i i i Pi

i i i

cW b Q b W T l b m

F

= = =

+ = +

1

tan tancos

i

n

i i i i S i

i

c l W T b mF F

=

+ + (2.14)

and rearranging:

1 1 1 1

tani i

n n n n

i W i i Qi i Pi i Pi S i

i i i i

W b Q b W b m T b b mF

= = = =

+ + + =


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( )1

1cos sin tan

i

n

i i S i i Pi i S i

i

c l b b W b mF

=

= + . (2.15)

From Eq. (2.15) the factor of safety is obtained:

( )1

1 1 1

cos sin tani

i

n

i i S i i Pi i S i

imm n n n

i W i i Qi i Pi mm

i i i

c l b b W b m

F

W b Q b W b m I

=

= = =

+ =

+ +

(2.16)

where1

tani

n

mm i Pi S i

i mm

I T b b mF

=

= +

.

Substituting bPiand bSiinto the above Eq. (2.16), leads to:

( )

( )

1

1 1 1

tan cos sin

cos sin

i

i

n

i i i i i i i

imm n n n

i i i Qi i i i i i mm

i i i

c l Y W Y X m

F

W X Q b W X Y m I

=

= = =

+ + =

+ +

(2.17)

with i i OX X X = and i i OY Y Y = . Eq. (2.17) differs from Eq. (2.16)

because the co-ordinates of the centre of the slice bases (Xi;Yi) appear.This is the expression used by computers to perform calculations.

(Xi;Yi)

(Xo;Yo)

Fig. 2.2: slope slices and pole used to calculate the moment equilibrium for a generic slipsuface.


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In case of circular slip surface only Eq. (2.11) is used sincePido not

appear in the moment equilibrium equation. Further, simplified

expressions relative to the force arms are achieved: S ib R= andsinW i ib R = . Therefore the moment equilibrium equation reduces to:

1 1 1

sinn n n

i i i Qi i

i i i

R W Q b R S= = =

+ = (2.18)

Substituting Sifrom Eq. (2.11) into Eq. (2.18) allows to obtain the safety

factor:

[ ]1

1 1

cos tan

sin

i

n

i i i

imm n n

i i i Qi mm

i i

c l W m

F

W Q b I

=

= =

+=

+ +

(2.19)

where1

tani

n

mm i

i

I T mF

=

= .

If 0iQ = (horizontal external forces absent) and 0iT= , the safetyfactor of the Bishop simplified method, also known as Bishops

modified method [Bishop, 1955], is achieved:

[ ]1

1

cos tan

sin

i

n

i i i

iBishop n

i i

i

c l W m

F

W

=

=

+=

(2.20)

In Bishop simplified method, n-1 assumptions are made ( 0iT= ). As onemore assumption is made than required, one equilibrium condition

cannot be satisfied. Therefore, the horizontal equilibrium of one slice

cannot be satisfied with the computed safety factor.

If 0iE = as well, the vertical equilibrium equations for each slice(2.10) are still the same and therefore Eq. (2.20) does not change.

For this case ( 0iQ = , 0iT= , 0iE= ) FelleniusI derived the factor of

IIn the original formulation, the resultant of interslice forces was assumed to act parallelto slice base. This formulation leads to violate the principle of action and reaction. Here,a more correct formulation is preferred.


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safety of the moments in a different way. He imposed the equilibrium of

forces along the direction normal to the slice base for each slice,

obtaining:

cosi i iN W = . (2.21)

Substituting Eq. (2.21) into Eq. (2.18), a simpler factor of safety is

achieved:

[ ]1

1

cos tan

sin

n

i i i

iFellenius n

i i

i

c l W

FW

=

=

+

=

. (2.22)

No iterations are needed to calculate FFellenius. For this reason, Fellenius

method is also known, in literature, as direct method. Note that the

assumptions made are n+2(n-1): rows E, H, I in table 2.1. Therefore there

are n assumptions more than requested. In fact, this method does not

satisfy equilibrium in the direction parallel to the base of each slice, or

equivalently either horizontal or vertical force equilibrium is not

satisfied.

As in case of force equilibrium, mmI and im depend on the value ofthe safety factor, thus an iterative scheme must be followed to compute

Fmm.

Different expressions have been derived for FffandFmm. Methods of

forces are based on Fffonly, whereas rigorous methods require a factor of

safety satisfying both Eq. (2.9) and (2.17).

In order to solve Fff and Fmm additional hypotheses, concerning

interslice forces, must be introduced. In fact, fI and mmI depend on an

unknown set of forces: iT . n-2 assumptions are required to make theproblem statically determinate. Depending on the method, either n-2

assumptions are made or n-1 assumptions are made and an extra-unknown is introduced. Anyway, the former methods can always be

reformulated as particular cases of the latter ones having assigned a fixed

value to the extra-unknown introduced. According to the hypotheses

introduced, methods of slices may be grouped into three different classes

which will be illustrated in the following paragraph.


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2.3. Analysis of assumptions which make the problemdeterminate

2.3.1. First groupIn the first group, assumptions concern the inclination of the

resultants of interslice forces respect to the horizontal direction:

( ) ( ) ( )1 1T x f x E x= (2.23)

where 1 is a dimensionless scaling parameter to be evaluated with thefactor of safety, and f1(x) a chosen scalar function of the abscissa (x)

representing the distribution of the inclination of the interslice forces.

Morgenstern and Price were the first who proposed this type of

assumption [Morgenstern and Price, 1965]. To solve their method, they

used the Newton-Raphson numerical technique implemented in a

computer program at the University of Alberta (Canada) [Krahn et al.,

1971].

Successively, Fredlund [Fredlund, 1974] at Saskatchewan University

implemented a different numerical procedure (Slope code) based on the

so called method of best fit regression[Fredlund and Krahn, 1977]. It has

been decided to illustrate this technique since it is common to most

methods of slices and let the reader a better understanding of the use of

equilibrium equations into determining the factor of safety than the

Newton-Raphson technique. Moreover, the structure of the algorithm is

almost the same as that implemented, later on, in other computer codes

[Slope/w, 2002].

The procedure can be described as follows: on the first iteration, the

interslice shear forces Tiare set to zero. On subsequent iterations, Eiare

achieved from the set of Eq. (2.7), and then the normal interslice forces

Ei. The shear interslice forces are computed using an assumed 1valuefrom Eq. (2.23). Thus, Tiare computed andIff,Immare calculated. FromEq. (2.9) and Eq. (2.17) the factor of safety of forces and moments

respectively, are calculated. The interslice forces are recomputed for each

iteration. The factors of safety vs. 1are fit by a polynomial regressionand the point of intersection of the two curves satisfies both force and

moment equilibrium.


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Fig. 2.3: vs. factors of safety. The factor of safety F is given by the point of intersection ofthe two curves (Ff and Fm) obtained by best fitting polynomial regression. Theanalysed slip surface is circular and crosses a uniform slope (after [Fredlund and

Krahn, 1977]).

According to Fredlund and Krahn, the sensitivity of FffandFmmupon

the distribution of the inclination of the interslice forces f1(x) is very

different [Fredlund and Krahn, 1977]. In fact, Fff shows a strong

dependence on f1(x), whereas Fmm shows no significant variations with

f1(x) (see Fig. 2.4). However in case of uniform slope, the global factor of

safetyFshows very little dependence onf1(x).


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Fig. 2.4: effect of different assumptions relative to the distribution of the inclination of theinterslice forces (constant, sine, clipped-sine) on the factors of safety. The analysedslip surface is circular and crosses a uniform slope (after [Fredlund and Krahn,1977]).

Spencer proposed a simpler expression than Morgenstern & Price

assumption: ( ) ( )tanT x E x = [Spencer, 1967]. This assumption

corresponds to take ( )1 1f x = and 1 tan = where is the angle betweenthe interslice resultants and the horizontal direction. Therefore it is a

particular case of Morgenstern & Price method. In case of 1 = 0 thefactor of safety coincides withFBishop(see Eq. (2.20)).

According to Spencer static assumption all interslice forces are

parallel. But, a variation of the inclination of the interslice resultants

along slices must be expected since physics suggests that the soil mass

above the slip line is characterised by different stress states: it could be

roughly divided into an active region, a transition region and a passive

region. Therefore, the proposed interslice force distribution is not

realistic.

Lowe and Karafiath proposed to assume the direction of the

resultants of the interslice forces tan equal to the average between the

slope surface and the slip surface [Lowe and Karafiath, 1960]. The U.S.


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Corps of engineers method takes tan equal to either the changing slope

of the ground surface or the average slope of the slip surface between the

two end slices [U.S. Corps of engineers method, 1970]. Both methods do

not introduce an extra unknown. In fact, they compute only Fff,

calculated with the prescribed distribution of tan , assuming the factor

of safety of forces, as the final factor of safety. Of course, this factor does

not satisfy all the equilibrium equations and it can be very distant from

the factor of safety calculated by rigorous methods. In fact, Fff is very

sensitive to the assumptions made (see Fig. 2.3).

Chen and Morgenstern were the first who focused their attention onthe physical admissibility of solutions [Chen and Morgenstern, 1983].

This concept will be treated more in detail in 2.4. Here, the main

interest concerns the new relationship among interslice forces which they

proposed. They took in consideration the slices located at the edges of

slopes (end slices: 1, n). Assuming the slices infinitesimal and

homogeneous, equilibrium considerations together with the Mohr-

Coulomb failure criterion led them to infer that the direction of the

interslice resultants of the end slices must be equal to the slope of the

ground surface above the slices. They concluded that this condition is

necessary to achieve a solution physically admissible.But, this is not true. In fact, their demonstration is based on the

implicit hypothesis of uniform state of stress at failure throughout the end

slices. In case of finite slices, this hypothesis is not acceptable any more.

Therefore, taking the interslice resultants of the end slices equal to the

slope of the ground surface can not be judged a condition of physical

admissibility. Netted out this point, Chen and Morgensterns requirement

on end interslice resultants is reasonable since the stress field relative to

these slices is, of course, well approximated by a uniform active and

passive stress field.

In order to satisfy the new boundary conditions on the interslice force

distribution, they proposed an extension of the Morgenstern & Prices

expression (Eq. (2.23)) that is:

( ) ( ) ( ) ( )0 1 1T x f x f x E x= + (2.24)

where ( ) ( )1 1 0f a f b= = . In this way, the direction of the interslice


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resultants acting on the vertical faces of the end slices does not depend on

any more.Sarma proposed to assume:

( ) ( ) ( ) ( )1 1 avg avgtanT x f x c H x E x = +

where cavg and avg are evaluated on the vertical interslice surfaces and

( )H x is the height of the interslice surfaces [Sarma, 1979]. The term( )1 1f x represents the inverse of the local factor of safety with respect to

shearing on the interslice surface. In the paper, the author suggested to

assume a local factor

slope evolution.pdf

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