RESEARCH REPORT

Down-hill Progressive Landslides in Soft Clays Triggering Disturbance Agents

Slide Propagation over Horizontal or Gently Sloping Ground Sensitivity related to Geometry

Universitetstryckeriet, Luleå

Luleå University of TechnologyDivision of Structural Engineering

Division of Soil Mechanics and Foundation EngineeringDepartment of Civil, Mining and Environmental Engineering

2008:11|: 102-1528|: -f -- 08 ⁄11 --

Stig Bernander

2008:11

qcrit kN/m2

clab = 25 kN/m2

cpeak = 30 kN/m2 gf = 3 % gel = 1.09 %

0.2 0.4 0.6 0.8 1.0

cR/clab

10

20

25 H = 20 m Slope n:100

qcrit

tan = 0.05

tan = 0.06

tan = 0.07

tan = 0.08

5:100

6:100

7:100

8:100

= 16 kN/m2

Stig Bernander

Dow

n-hill Progressive Landslides in Soft Clays

Research Report 2008:11

Down-hill Progressive Landslides in Soft Clays Triggering Disturbance Agents

Slide Propagation over Horizontal or Gently Sloping Ground

Sensitivity related to Geometry

Stig Bernander

Division of Structural Engineering

Division of Soil Mechanics and Foundation Engineering Department of Civil, Mining and Environmental Engineering

Luleå University of Technology SE-971 87 Luleå

Phone (+) 46 920 49 10 00 Fax (+) 46 920 49 19 13

http://www.ltu.se

Down-hill Progressive Landslides in Soft Clays Triggering Disturbance Agents

Slide Propagation over Horizontal and Gently Sloping Ground

Sensitivity related to Geometry

Stig Bernander

Research Report 2008:11

ISSN 1402-1528

© Stig Bernander

Division of Structural Engineering

Division of Soil Mechanics and Foundation Engineering

Luleå University of Technology

SE-971 87 Luleå, Sweden

www.ltu.se/shb

The figure on the cover illustrates the critical up-slope triggering load qcrit as function of slope and residual shear resistance cR/clab. The critical load is relatively little affected by the degree of strain-softening especially for steeper values of the slope gradient. The dia-gram also accentuates the acute hazard in respect of progressive failure related to local up-slope fills and embankments. See Chapter 5 and Appendix B.

Preface The quest for sound and valid structure-mechanical explanations of the extensive and fre-quently occurring landslides in the soft sensitive clays of Scandinavia has long challenged geotechnical expertise. This is primarily due to the fact that slides have often presented salient features that were incompatible with the prevailing geotechnical understanding of slide mechanisms.

The conventional ‘ideal-plastic limit equilibrium’ approach to earth movements has for instance time and time again revealed little potential for predicting the incidence and out-come of extended slides in soft clays. Yet, the concept is still widely used in contexts, where the prospects of successful application of this approach must be regarded as highly unsatisfactory.

In the absence of adequate explanatory analytical methods, the incidence of landslides of this kind has often, in the last resort, been ascribed to extreme but undocumented ground water conditions or to the vaguely defined predicament known as presence of ‘quick’ clay.

Landslides in soft clays do raise a number of problematic issues that have puzzled geo-technical engineers charged with the task of explaining - in hindsight - the causes, the mechanisms and the final disastrous features of these slides. This is of course troublesome because, if there are major questions unanswered regarding the mechanisms in slides oc-curred, the confidence in our capacity of identifying potential future slide hazards is seri-ously at stake.

Such questions are for instance:

- How can the local driving of a few prefabricated concrete piles, or the placing of minor local fills, effectively destabilize vast areas of ground up to tens of hectares,

-…and how does it come about that – for instance in the extensive Tuve slide (26 hectares) more than half of the area involved in the initial slide (i.e. about 16 hectares) consisted of almost horizontal ground?

-…and how is it possible that the soil volume (i.e. some 5 600 000 m3) beneath this hori-zontal portion of the slide was plasticized in passive Rankine failure down to a depth of 35 m, thereby raising the ground surface by approximately 5 m?

It may be argued that the final collapse of the passive zone in slides is of little interest be-cause, as one might say: “This phase of a slide is anyway beyond control and just about anything may then happen”.

Nevertheless, it is important to consider that the fracture-mechanical phenomena acting in the final phase are in principle identical to those in the incipient stages of the slope failure, and can therefore also be investigated on a rational basis. In particular, analysis of the con-dition immediately prior to the final disintegration of the passive zone should not be re-garded as superfluous, as after all, it is the earth pressure distribution in this phase of a landslide that actually conditions the degree of disaster.

-…and is it correct, as is usually done, to study landslides of the current type as one singu-lar static event not considering that these slides actually represent a series of different non-simultaneous phases of static and dynamic instability with radically differing types and rates of loading, as well as varying time dependent response of the clay material?

In Sweden, the explanation of slides of great length is generally focused on the presence of so-called ‘quick clays’. However, in the opinion of the author, it is important to recognize that the following issues are also elements of decisive significance namely:

- the geometry of the soil structure and firm bottom,

- the rate of application of the additional disturbing load,

- and the drainage conditions in the incipient failure zone.

Failure modes based on perfectly plastic behaviour of clay, not considering deformations within and outside a longish potentially sliding body of soil, cannot by definition account for the phenomena of the kind mentioned.

The aim of the current report is to focus in some detail on how some incongruities in land-slide formation can be explained and predicted in terms of progressive failure formation.

Acknowledgements In this context, I wish to express my gratitude to Professor Lennart Elfgren, Division of Structural Engineering, Luleå University of Technology, for his encouraging an inspiring support and for his comments on the manuscript and to Tekn Dr Håkan Thun, at the same division, for his editing of the report.

I also want to extend my thanks to Professor Sven Knutsson, Division of Soil Mechanics and Foundation Engineering, Luleå University of Technology, for his positive commit-ment.

Furthermore, I want to express my deep appreciation to former colleagues at Skanska Tek-nik AB for their various contributions far back in the 1980-ties to the research work that has gradually led forward to the present study. I feel particularly obliged to mention the names of the civil engineers Hasse Gustås, Ingvar Olofsson, Jan Olofsson and Ingemar Svensk.

I also take this opportunity to extend my gratitude to Bernt Bernander (former Assistant Administrator of the United Nations Development Program, UNDP) for dedicating time to reading an early version of the manuscript and for valuable editorial advice.

Mölndal, December 2008

Stig Bernander Adjunct Professor Emeritus Division of Structural Engineering Luleå University of Technology Stig Bernander Phone: + 46-31 87 11 04 Mobile: + 46-31-070-9309594 Tegelformsgatan 10 Fax: + 46-31 87 95 32 S-431 36 MÖLNDAL E-mail stig.bernander @ telia.com

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Abstract A numerical model is presented and discussed for down-hill progressive landslides in soft clays. A simple-to-use spread sheet is given and used in order to illustrate the influence of the sensitivity of the clay and the geometry of the ground. The model is also used to discuss the agents and the mechanisms that formed the large landslide in Surte at the Göta River in 1950. Landslides in extensive natural slopes of soft clays constitute a latent threat in many popu-lated areas in Scandinavia and elsewhere. Although some of the slides listed in Table 2.1 in this report have involved loss of human life and serious economic damage, after-slide in-vestigations have in general not been truly successful in explaining the causes, the mecha-nisms and other intriguing features of these slide events – at least not in such a way that geotechnical expertise has been able to predict reliably future landslide hazards in similar, or even identical, geotechnical contexts. It stands to reason that safe prediction of such risk ought to be a paramount objective in post-slide investigations.

For example, the spectacular landslide in Tuve (1977) involving some 26 hectares of ground brought forth about ten different explanations of this slide by as many experienced geotechnical engineers – a predicament indicative of an unsatisfactory State-of-the-Art situation in this field of geotechnical engineering. (See SGI Report No 10, 1981 and SGI Report No 18, 1982).

A conspicuous and vexed issue in many investigations of slides of this kind has, for in-stance, been the fact that the slide movement has involved vast areas of virtually horizontal ground.

Another puzzling phenomenon related to these slides is the apparent triviality of the requi-site disturbance for destabilizing vast areas of virtually stable ground

The main reason why post-slide investigations in soft sensitive clays have frequently re-mained inconclusive, and intrigued many a geotechnical expert is – in the opinion of the author of this report – mainly due to the fact that deformations within and outside the po-tentially sliding soil mass have not been considered in the analyses. Furthermore, there is in Sweden a common tendency to explain slides of the current kind by simply referring to the presence of so called ‘quick clay’, which in Scandinavia is the term for clays with a sensitivity number St = su/sur > 50. Yet, the vague understanding of how sensitivity – de-fined in this way – actually affects incipient slope failure constitutes another complicating factor, contributing to the difficulty of understanding these landslides.

The shear strength of a completely remoulded (stirred) clay specimen (sur), as measured in laboratory, has unclear relevance to the actual residual shear strength in an incipient failure zone. In the present report, this lack of proven compatibility is dealt with by differentiating between the completely remoulded residual shear strength in the laboratory (sur) and the real un-drained residual strength parameter in situ, which is here denoted (suR). However, as the effective residual shear strength in situ will strongly depend on rate of loading and current drainage conditions, it is in this report generally denoted as just sR, thus implying that the effect of time is of crucial importance to this strength parameter.

A third condition, the effect of which eludes the normally used perfectly plastic failure analysis, is in what way geometric variation between the upper and lower limits of a stud-ied soil volume affects the true risk of stability failure. The term previously used by the author for this phenomenon is ‘geometric sensitivity’.

******

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The current report focuses on the issues mentioned above as follows:

Chapters 1, 2 and 3 deal generally with markedly extended landslides in soft clays.

In Chapters 1 and 2 and in Section 3.1, the features and general characteristics of these landslides are defined and discussed in more detail.

Sections 3.2 and 3.3 deal briefly with the specific Finite Difference computational Model (FDM) applied in the different examples of progressive failure formation that are subse-quently dealt with in Chapter 4, 5 and 6 as well as in Appendices A, B and C of this report.

A vital characteristic of this FDM analysis – deviating until very recently from previous approaches to progressive failure formation – is that the effects of differential deformation in the entire shear zone, within and outside the potentially sliding soil volume, are ac-counted for – i.e. the analysis is not restricted only to varying deformation along the poten-tial slip surface or the shear band.

Another salient feature is that the potential failure zone is at all times modelled as consist-ing of two separate components in different states of stress and strain named Stages I and II. Stage I represents the portion of the failure zone where no slip surface has yet devel-oped and Stage II signifies the part where a slip surface is fully established.

Like in conventional analysis, the failure zone is presumed to be known, which means that alternative failure zones and slip surfaces may have to be investigated.

As the assumed shear/deformation properties of the clay can be adapted to the rate of load-ing and to ambient drainage conditions, the impact of time may be considered in the analy-sis. This important feature renders it possible to distinguish between different phases in the development of extensive landslides of the current type.

The kind of slides subject to study in this report cannot - at least not in present State-of-the-Art - be analysed just as one singular mechanically static event considering that such a slide actually represents a series of consecutive - and therefore not simultaneous - phases of static and dynamic instability. These phases are characterized by radically differing conditions in respect of the type of disturbance agent, of loading and rates of load applica-tion. The response of soft clays to time and drainage factors varies radically between the different phases. Five different phases of progressive slides are defined in Section 3.2.

Chapter 4 deals with the spectacular Surte slide (1950).

The great landslide in the township of Surte (September 1950) epitomizes from many points of view the kind of slide in soft clays subject to study in the present report.

Sections 4.1 to 4.3 present a brief description of the actual slide event as well as of post-slide investigations and conclusions regarding the causes of this disaster, which were made at the time. Further, in Section 4.4 and 4.5, the Surte slide is analyzed in terms of progres-sive failure, the primary objective being to evaluate the magnitude of the disturbance effect sufficient to set off progressive failure in the slope. The force required to trigger the slide is found to be astoundingly small – showing also that, once local failure in a sensitive part of the slope had formed, the stability of 240 000 m2 of ground was inexorably lost.

An important issue in almost all the slides listed in Table 2.1 (Chapter 2), and which has intrigued engineers investigating these slides, has been the remarkably extensive propaga-

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tion of the slides over more or less horizontal ground. In the mentioned Surte and Tuve slides, about 50 and 60 % respectively of the initial slide areas consisted of almost horizon-tal ground.

Chapter 5 and Appendix A of this report are therefore focused on this issue. It is demon-strated there that, considering the combined effects of deformation and deformation soften-ing, the progressive failure FDM-analysis explicitly predicts … a) …the extensive spread (in soft clays) of the passive zone over gently sloping – or even horizontal – ground. Furthermore, the analysis explains this phenomenon in terms of static loading and stress change – i.e. without having to resort to the dynamic effects that may arise in the final phase of a slide.

b) …that failure zones and slip surfaces (or shear bands) unavoidably tend to develop far (i.e. hundreds of meters) beyond the foot of a slope under the surface of the valley floor – notably already before the incidence of possible break-down in passive Rankine failure.

c) ....that the wide spread over horizontal ground is not by necessity related only to ‘quick clays’. The clays in the valley proper often exhibit low or medium sensitivity - i.e. su/sur-values in the range of 10 to 15.

d) ….that failure modes based on slip-circles surfacing in the slope have little relevance in the analysis of failure in long slopes of soft clay.

e) ….that although the presence of ‘quick clay’ certainly constitutes an important haz-ard in slope stability, it is by no means the only factor conditioning progressive slope fail-ure. The rate of load application (disturbing load), time and prevailing drainage possibili-ties, also have a most decisive influence.

‘Geometric sensitivity’, is often a determining condition that, depending on the factors mentioned in item e) above, may even eclipse the impact of whether the sensitivity of the clay in a failure zone happens to be, for instance, 20 instead of 50.

Chapter 6 exemplifies – in terms of progressive failure analysis – the significant impact of geometry in two slopes, which are identical but for a slight difference in the shape of the potential slip surface.

Yet, the safety factors as computed on the basis of ideal-plastic equilibrium are identical in both slopes for the same soil volume and slide length.

Chapter 7 summarizes discussion and conclusions.

In this context it may be emphasized that conditions and points made in this report apply to down-hill progressive slope failures in soft clays. They are not necessarily valid for other types of soil instability such as short slides in steep clay slopes, retrogressive slides, slides in permeable cohesion-less soils, mud slides etc. This is in particular true of possible ef-fects of rainfall conditions and climate change.

The issues, on which the current report is focused, are exemplified in terms of FDM-analyses in Appendices A, B and C.

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Main conclusions of the report: Landslide hazards in long natural slopes of soft sensitive clays may – on a strict structure-mechanical basis – only be reliably dealt with in terms of progressive failure analysis. There exists, for instance, no fixed, constant relationship between safety factors based on the conventional limit equilibrium concept and those defining risk of progressive failure. In order to be able to make reasonable predictions of the impact of locally applied disturbance agents – capable of triggering global slope failures – it is imperative to make adequate as-sessments of the effective residual shear resistance (sR) that can be mobilized in a potential zone of local failure. The different phases of progressive landslides should be analyzed separately and not as a singular mechanical event. As already mentioned, the progressive failure analyses made, show that slope failure in sensitive clays develops in direction down-slope rather than along slip circles surfacing in inclining ground. This has the serious implication that a supporting embankment of the kind common, for instance, in road construction can - acting as an effective triggering agent - in itself initiate landslide disaster of much more serious nature than the one meant to be avodided by establishing the embankment. Reliable values of the shear strength sR can only be established if the current rate of apply-ing the destabilizing force (or the disturbance) is considered. In addition, the prevailing drainage conditions in the investigated failure zone must be taken into account. Future research in this field of geotechnical engineering is urgently required if we really aspire to make adequately accurate assessments of landslide hazards in slopes of the kind subject to study in this report. Pending the results from such research, geotechnical engineers will have to resort to sensi-tivity analyses based on existing geotechnical knowledge and available experience. As indicated in Appendices A, B and C, reasonably good prediction of risk can be made al-ready on present State-of -the-Art knowledge. Yet, even if such a procedure should seem imprecise, doing so will in any case provide a better understanding and handling of landslide hazards in long slopes of soft clay than the application of the conventional equilibrium approach, based on perfectly plastic behaviour of the clay material.

Key phrases: Down-hill progressive landslides in soft clays; Deformation softening; Ap-plicability of conventional ideal-plastic failure analysis; Finite difference modelling of progressive failure; Residual shear strength in the incipient failure zone - a decisive pa-rameter; Different phases in down-hill progressive slides; Analysis of the Surte slide in terms of progressive failure; The Surte slide - a ‘time bomb’ ticking through millennia? Triggering disturbance load; Slide propagation over gently sloping ground; Is ‘quick clay’ the only hazard in slopes of soft clay? Are slip-circular failure modes possible in long slopes of soft clays; Brittleness due to nature of loading; Time effects; ‘Geometric sensitiv-ity’.

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Sammanfattning Långsträckta skred i naturliga slänter med lösa leror utgör historiskt en latent risk i många områden i Skandinavien och annorstädes. Då och då har omfattande skred drabbat platser med tät bebyggelse. Några inträffade skred (jfr tabell 2.1 i denna rapport) har medfört för-luster i människoliv och omfattande ekonomisk skada. Dock har utredningarna av dessa skred i allmänhet inte blivit särskilt framgångsrika i den meningen att de gjort det möjligt för geoteknisk expertis att med acceptabel tillförlitlighet förutsäga framtida skredrisk i lik-nande geotekniska situationer - något som ju borde utgöra den kanske viktigaste målsätt-ningen vid utredning av redan inträffade skred.

Det katastrofala och ytterst spektakulära Tuveskredet (1977) frambragte exempelvis ett tiotal olika förklaringsmodeller till skredets uppkomst av lika många erfarna geotekniker. (Jfr SGI Rapport No 10, 1981 och SGI Report No18, 1982). Idérikedom i all ära, men i det aktuella sammanhanget utgör densamma snarast ett predikament som låter ana brister i rådande ’State-of-the-Art’ på ifrågavarande område. Den huvudsakliga orsaken till varför utredningar av skred i lösa leror i egentlig mening ofta förblivit olösta och förbryllat mången geoteknisk expert beror enligt författarens mening huvudsakligen på att man vid analyserna underlåtit att beakta olikformiga deformationer inom och utanför den av skredet berörda jordvolymen.

En iögonenfallande frågeställning som i verkställda utredningar av skred inte fått någon tillfredsställande förklaring är exempelvis varför framåtgripande skred av ifrågavarande slag når en utbredning över praktiskt taget plan mark som kan omfatta 50 à 60 % av hela den yta som omfattas av det initiala skredet.

Som en annan gåtfull omständighet framstår även det faktum att anmärkningsvärt liten lokal belastning (eller annan påverkan) visat sig kunna utlösa skred över stora arealer – något som överensstämmer väl med de i appendix A, B och C genomförda analyserna.

Vidare förekommer i vårt land ofta en tendens att vilja förklara skred av ifrågavarande typ genom att i all enkelhet referera till förekomsten av ’kvicklera’, d v s lera med en sensitivi-tet St = su /sur > 50. Emellertid, hur den i laboratoriet bestämda sensitiviteten egentligen påverkar skeendet vid begynnande skred är i hög grad oklart och bidrar således på ett av-görande sätt till svårigheterna att bedöma skredrisk. Skjuvhållfastheten hos på laboratoriet omrörda lerprover – som i denna rapport betecknas sur – kan rimligen inte – under vilka förhållanden som helst – överensstämma med den odränerade skjuvhållfastheten (här be-tecknad suR) för samma lera vid begynnande brottutveckling i en verklig slänt.

Vidare, eftersom den effektiva resthållfastheten under reala betingelser måste vara starkt beroende av såväl pålastningshastighet som de i brott-zonen lokalt rådande dräneringsför-hållandena, betecknas densamma i det följande som sR – något som, med andra ord, är ett uttryck för tidsfaktorns avgörande betydelse i sammanhanget.

I föreliggande framställning görs således en klar distinktion mellan innebörden av paramet-rarna sur, suR och sR. Ett ytterligare förhållande, vars inverkan undgår konventionell ideal-plastisk jämviktsana-lys är det sätt på vilket geometriska skillnader mellan den övre och nedre avgränsningen av den studerade jordvolymen påverkar verklig brottsäkerhet. En term som tidigare använts av författaren i detta sammanhang är ’geometric sensitivity’.

*********

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Föreliggande rapport fokuserar på ovannämnda problemställningar enligt följande:

Kapitlen 1 till 3 behandlar allmänt långsträckta skred i lösa leror. I kapitel 1, 2 och i avsnitt 3.1 beskrivs och diskuteras närmare egenskaperna hos den typ av skred som behandlas i denna rapport.

I avsnitten 3.2 och 3.3 beskrivs kortfattat den analysmetod, baserad på en Finit Differens-Modell (FDM), som tillämpas i de exempel på progressiv brottbildning som bearbetas i de efterföljande kapitlen 4, 5 och 6 samt i Appendix A, B och C.

En betydelsefull egenskap hos denna FDM-analys i förhållande till tidigare gjorda studier av progressiv brottbildning är att deformationerna i skjuvzonen i dess helhet beaktas – d v s inte endast olikformig förskjutning längs med själva glidytan. Vidare görs i varje läge åtskillnad mellan det skede av brottprocessen då någon glidyta ännu inte utbildats (Stadium I) och det skede (Stadium II) då en glidyta (shear band) är etablerad.

Såsom vid konventionell analys antas den potentiella brottzonens sträckning som varande given – något som innebär att alternativa lägen för densamma kan behöva undersökas. Ef-tersom de antagna skjuv/deformationsegenskaperna hos leran kan anpassas till aktuella belastningshastigheter och dräneringsförhållanden kan också tidsfaktorn på ett allmänt sätt beaktas i FDM-analysen. Detta utgör en viktig omständighet som medför att det är möjligt att särskilja mellan olika faser i långsträckta progressiva skred av aktuell typ.

Skred av ifrågavarande slag bör nämligen – åtminstone vid rådande State-of-the-Art – inte betraktas som en enda sammanhängande statisk brottmekanisk händelse med hänsyn till att dylika skred egentligen utgörs av en serie på varandra följande – och därför inte samtidiga – skilda faser av såväl statisk som dynamisk natur. Dessa faser karakteriseras sinsemellan av radikalt olika förhållanden beträffande störningsmoment, laster, belastningshastigheter samt den sensitiva lerans olikartade respons med avseende på varierande dränerings- och tidsförhållanden. Fem sådana skilda faser definieras i avsnitt 3.2.

Kapitel 4 behandlar skredet i Surte (1950), som ur flera synpunkter utgör ett utmärkt ex-empel på jordskred av just det slag som behandlas i föreliggande rapport.

I avsnitten 4.1 till 4.3 redogörs kortfattat för själva händelsen samt för viktigare punkter be-träffande de vid efterföljande utredningar antagna orsakerna till skredet.

I avsnitten 4.4 och 4.5 analyseras Surte-skredet såsom varande ett exempel på progressiv brottbildning, varvid avsikten i första hand är att söka bestämma storleken av den påverkan i sluttningens känsligaste parti som kunde initiera ett dylikt brottförlopp. Den påverkan i form av tilläggslast som kunde utlösa skredet visar sig vara anmärkningsvärt liten. Vidare framgår att så snart lokalt brott i det brantaste partiet av slänten väl inträffat var stabiliteten hos det 240 000 m2

stora området, och på vilket en stor del av samhället Surte vilade, ohjälpligt förlorad. Problemställningen behandlas även i Appendix B.

I nästan alla av de i Tabell 2.1 upptagna skreden synes den stora utbredningen av desamma över nära nog horisontell mark ha utgjort ett för utredarna förbryllande fenomen. I Surte och Tuve-skreden utgjordes exempelvis ca 50 resp. 60 % av den yta som omfattades av initialskreden av svagt sluttande mark.

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Kapitel 5 och Appendix A har därför inriktats på denna frågeställning. Av kapitlet framgår:

a) ...att utbredningen över plan mark vid skred i lösa leror klart förutsägs genom den an-vända analysmetoden (FDM) samt att detta fenomen kan förklaras med rent statiska be-lastningsförhållanden - d v s utan beaktande av de dynamiska effekter som kan tillkomma i skredets slutskede.

b) ...att brottzon och glidyta (shear band) utbildas långt ut under dalgången, d v s hundra-tals meter bortom släntfoten, redan innan eventuellt sammanbrott av passiv-zonen över huvud taget kan äga rum.

c) ...att den stora utbredningen hos skreden över nästan horisontell mark inte med nödvän-dighet förutsätter förekomst av ‘kvicklera’ över hela skredområdet. Lerorna under dalbott-nen, såväl i Surte som i Tuve, uppvisade normal, låg sensitivitet med su /sur omkring 10 à 15.

d) ...att brottmodeller baserade på cirkulär-cylindriska glidytor (mynnande i slänten) med stor sannolikhet ej har någon relevans vid analyser av skred i långa naturliga lerslänter av den typ som avhandlas i föreliggande rapport.

e) ...att fastän förekomsten av kvicklera givetvis utgör ett uttalat riskmoment, så är den på intet vis den enda viktiga faktorn i samband med bildning av skred. Belastningshastighet och dräneringsförhållanden – d v s tidsfaktorn – är också av avgörande betydelse.

Kapitel 6. Även geometriska betingelser kan vara av avgörande betydelse, mycket beroen-de på de ovannämnda under punkt e) anförda omständigheterna. Ogynnsam geometri kan utan vidare ha större inverkan vid progressiv brottbildning än om sensitiviteten exempelvis skulle råka ha värdet 50 i stället för 20.

I kapitel 6 exemplifieras geometrins påtagliga inverkan vid progressiv brottbildning i två slänter som är identiskt lika förutom med avseende på den potentiella glidytans form mel-lan den övre och nedre begränsningen av den studerade jordvolymen.

Säkerhetsfaktorn, beräknad enligt ideal-plastisk teori, blir däremot i de båda fallen identisk vid samma skredlängd.

Kapitel 7 innehåller diskussion och slutsatser.

I detta sammanhang må särskilt framhållas att de förhållanden som avhandlas rapporten gäller framåtgripande progressiva skred i lösa leror. De är således inte med nödvändighet tillämpliga på andra typer av skred såsom exempelvis korta skred i branta sluttningar, bak-åtgripande skred, skred i permeabla massor, jordflytning (s.k mudslides) o s v. Detta kan i synnerhet vara fallet när det gäller möjlig inverkan av riklig nederbörd och klimatföränd-ringar.

Sammanfattning, slutsats: Skredrisk i långsträckta slänter bestående av lösa sensitiva leror kan – på ett ur brott-mekanisk synpunkt acceptabelt vis – endast fastställas genom analys baserad på progressiv brottbildning. Som framgår av Appendix B finns ingen konstant relation mellan säkerhets-faktorer baserade på konventionell plastisk jämviktsanalys och de säkerhetsfaktorer, som definierar risk för progressiv brottbildning.

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För att kunna göra en rimlig bedömning av storleken hos den lokalt angripande kraft (eller annan störningseffekt) i en slänt, som potentiellt kan utlösa ett omfattande skred, är det nödvändigt att fastställa ett åtminstone ungefärligt värde på den resthållfasthet (sR) som – under aktuella betingelser – kan antas gälla i den zon där det lokala brottet väntas upp-komma. Värdena på skjuvhållfastheten sR måste därvid baseras - dels på den belastningshastighet med vilken störningspåverkan appliceras och - dels på de i denna brottzon rådande dräneringsförhållandena. De olika faserna i progressiva skred bör behandlas som åtskilda fenomen och inte som en enstaka statiskt sammanhängande händelse. Som ovan nämnts visar den genomförda progressiva brottanalysen att brottbildningen i sensitiva leror utvecklas nedåt i släntriktningen och inte längs cirkulär-cylindriska glidytor mynnande i sluttande mark. Detta innebär således att en tryckbank av det slag, som inom vägbyggand ofta användes för stabilisering av en upphöjd vägkropp, i sig kan komma att utgöra en belastning medförande påtagligt ökad risk för längssträckta skred av betydligt allvarligare art än dem man vill undvika med tryckbanken. Framtida forskning på ifrågavarande område inom geotekniken är oundgängligen nödvän-dig därest noggrannare förutsägelser av risken för skred av ifrågavarande art överhuvudta-get skall kunna ske. I avvaktan på resultaten från sådan forskning är geoteknikern i detta sammanhang tills vi-dare hänvisad till att utföra känslighetsanalyser baserade på rådande geoteknisk erfarenhet och kunskap. I vad gäller den i föreliggande rapport aktuella typen av jordskred bör dock en dylik analys under alla förhållanden kunna leda till en fördjupad insikt om hur skred bildas samt en bättre bedömning av potentiell skredrisk än vid tillämpning av konventionell jämviktsana-lys baserad på idealplastiska egenskaper hos förekommande lösa leror. (Se Appendix B)

Nyckel-uttryck: Framåtgripande skred i lösa leror; Deformations-mjuknande; Plasticitets-teorins tillämplighet; Modellering av progressiv brottbildning medelst finita differenser; Resthållfastheten i den begynnande brott-zonen – en avgörande parameter; Skilda faser i framåtgripande skred; Analys an skredet i Surte som ett progressivt brott: Surteskredet - en tidsinställd bomb tickande genom årtusendena? Utlösande störningsfak-tor; Skredens utbredning över svagt sluttande mark – utbildning av brottzon och glidyta under passiv-zonen redan innan risk för passivt sammanbrott föreligger; Är ’kvicklera’ det enda riskmomentet vid skred i sensitiva leror? Har analys av skred i långsträckta slänter baserad på circulär-cylindriska glidytor någon relevans? Inverkan av tilläggslastens natur – koncentration och belastningshastighet; Tidsfaktorns inverkan; Geometrisk sprödhet.

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Table of Contents Preface ......................................................................................................................................................................... .. iii Abstract ......................................................................................................................................................................... ....v Sammanfattning............................................................................................................................................................. .. ix Table of Contents ........................................................................................................................................................... xiii Symbols and notations ................................................................................................................................................... . xv

1 Introduction..................................................................................................................... 1

2 Characteristics of down-hill progressive landslides .................................................... 3 2.1 Examples of landslides occurred............................................................................... 3 2.2 Presentation of a typical down-hill progressive landslide......................................... 4

3 Brief description of down-hill progressive failure analysis......................................... 7 3.1 General ...................................................................................................................... 7 3.2 Progressive failure approach used in the current study............................................. 8 3.3 Positive features of the FDM-analysis used............................................................ 12

4 The Surte slide............................................................................................................... 14 4.1 General - history of a slope in the Göta River valley.............................................. 14 4.2 The Surte slide event ............................................................................................... 15 4.3 Investigations and analyses after the slide .............................................................. 16 4.4 Explanation of the Surte landslide in terms of progressive failure ......................... 21

4.4.1 Results of the FDM-analysis – The in situ condition ....................................... 23

4.4.2 Results of the FDM-analysis – Disturbance condition .................................... 23

4.4.3 Results of the FDM-analysis – Global failure condition ................................. 26

4.5 Conclusions from the progressive failure computations ......................................... 28

5 Explanation of the spreading of slides over horizontal ground in soft clays........... 31 5.1 Slide propagation over horizontal or gently sloping ground................................... 31 5.2 Computational analysis - Exemplification .............................................................. 33 5.3 Conclusions ............................................................................................................. 36

6 Brittleness related to geometry – geometric sensitivity............................................. 37 6.1 General .................................................................................................................... 37 6.2 Geometric sensitivity - exemplification .................................................................. 37 6.3 Conclusions ............................................................................................................. 40

7 Final conclusions and discussion ................................................................................. 41 7.1 The Surte slide......................................................................................................... 41 7.2 Slide propagation over horizontal ground............................................................... 42 7.3 Brittleness related to geometry................................................................................ 43

8 References...................................................................................................................... 45

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Appendix A - Exemplification of analysis of landslide spread over practically horizontal ground based on the Finite Difference Method (FDM) ..........................53

Appendix B - Assessment of local up-slope triggering load ...........................................69

Appendix C - Excel spread sheet related to the example in Appendix A......................79

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Symbols and notations Greek letters

Coefficient defining the level of the earth pressure resultant , (x), x Slope gradient at coordinate ‘x’

, (x,z), x,z Deviator strain, (angular strain)

el Deviator strain, (angular strain) at elastic limit

f Deviator strain, (angular strain) at failure stress ’ = i Angle of internal friction, drained conditions

x = (x) Down-slope displacement

N Displacement in terms of axial deformation generated by force Nx

Down-slope displacement in terms of deviator deformation

Differential of x Differential of x coordinate

S, S (x) Post peak slip deformation in the slip surface in relation to the sub-ground

SR Post peak slip in slip surface at ultimate residual shear strength sR ave Average down-slope displacement of the soil above the slip surface Hx Level at which the mean down-slope displacement ( ave) is valid Longitudinal strain Poisson’s ratio Coefficient relating the modulus of elasticity to the un-drained shear

strength , (z) Soil density (Mg/m3)

1, Major principal stress in axial tests

3, Minor principal stress in axial tests

v Vertical normal stress

h Horizontal (down-slope) normal stress

x, (x), x Mean incremental down-slope axial stress corresponding to N

el Shear stress (deviator stress) at elastic limit

o, o(x,o), ox,o In situ shear stress at potential failure plane (z = 0) , (x,o), x,o Total shear stress at failure plane (z = 0)

o, o(x,z), ox,z In situ shear stress above failure plane

, (x,z), x,z Total shear stress (deviator stress) above failure plane Roman letters: b, b(x), bx Width of element s Shear strength of clay su = su( f) Un-drained shear strength

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su,mean Mean un-drained shear strength of the soil mass above the failure plane suR = sR(x) Un-drained shear strength at a post peak slip of SR in failure plain. sur Completely remoulded residual shear strength in laboratory sR, sR(x,t) Residual shear strength at a point (x) at time (t) s’ Drained shear strength g 9.81 m/sec2 ko Ratio of h to v qv(x) Additional vertical surface load qh(x) Additional horizontal load qcr Critical surcharge load initiating local slope failure w Natural water content (%) wL Liquid limit (%) wP Limit of plasticity (%) x Horizontal (or down-slope) coordinate z Vertical coordinate E, E (x), Ex Down-slope earth pressure at point x, i.e. (Ex = Eox + Nx) Eo, Eo(x), Eox In situ earth pressure at point x. DW Submerged depth (when slope borders river or lake) Ep

Rankine= ERp

Passive Rankine earth pressure resistance EA

cr Earth pressure in a Point A, at which the shear stress A = suR or sR Eel,o Elastic modulus of structural element at z = 0 Eel, Eel,mean Mean secant E- modulus in down-slope compression of a vertical struc- tural element H x, i.e. Eel,mean = su,mean Fs Safety factor G, Gel Elastic modulus in shear Go,Gel,o Elastic modulus in shear of structural element at elevation z = 0 H, H(x) Height of element, (from slip surface to ground surface) Lcr Limit length of mobilization of shear stress at Ncr Linstab

Limit length at which slope fails for Ni = 0 and = cr Lp=LE>E(Rankine) Length of the passive Rankine failure zone

N, N(x), Nx= E Earth pressure increment due to progressive failure formation at point x. Ni or NL Additional local up-slope force or load effect Ncr Critical load effect initiating local slope failure Abbreviations: I-PF Ideal-plastic failure I-PFA Ideal-plastic failure analysis. PrFA Progressive failure analysis SGI Swedish Geotechnical Institute NGI Norwegian Geotechnical Institute

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1 Introduction In December 2006 – again – a large landslide, involving some 10 hectares of moderately sloping ground, took place at Småröd about 90 km north of Gothenburg in south-western Sweden. The slide brought about serious destruction of infrastructure, carrying away roughly 500 m of the main road between Oslo and Gothenburg (and the European Union) as well as part of the track of the railway between Gothenburg and the city of Strömstad. Railway traffic was totally blocked and road traffic was seriously obstructed by lengthy detouring on narrow country roads. In the opinion of the author of this study, the slide bore from first sight the evident charac-teristics of progressive failure. The report from the Independent Investigatory Group of the Swedish National Road Administration (I IG-SNRA for short) - chaired by Jan Hartlén – in due course also confirmed that the slide was caused by temporary local stock-piling of earth fill in connection with ongoing road construction. See report by Vägverkets oberoen-de utredningsgrupp; ’Skredet i Småröd, Skredorsak’ (Oct. 2007). This report is referred to below as I IG-SNRA (2007). While not being a member of the investigatory group, the author of the present study – at the request of the chairman of this group – performed an independent analysis of the slide event that essentially confirmed the conclusions made in due course by the investigatory group. (Cf Separate Appendix 6 by CONGEO AB in I IG-SNRA, 2007). In this independent study, the author also addressed some additional issues such as defor-mations at different levels of loading in the incipient failure zone as well as the related propagation of the slide over less sloping ground. The latter issue constitutes a particularly vexed question, which has been the subject of much fruitless contemplation and inconclu-sive discussion over the years in the investigations of many of the slides listed in Table 2.1 below. However, these aspects of the separate study were not included in Appendix 6 of the I IG-SNRA Report. Although the work of the I IG-SNRA – from an analytical and other points of view - is probably the most advanced investigation made to date of slides in the soft clays of west-ern Sweden, the material presented does not focus on certain issues, which in the author’s opinion are of considerable significance in the context of landslides in soft clays. Such issues are for instance: 1) The mechanisms involved in the phenomenon of slide propagation far over gently slop-ing or virtually horizontal ground, and how this spread is affected by the degree of defor-mation-softening? For instance, the remarkable spread of the Surte and the Tuve slides over hundreds of meters of almost horizontal ground occurred despite the fact that the clays in valley proper were not particularly sensitive (or quick) by Scandinavian standards. In fact, as is highlighted in Chapter 5, failure zones and the associated slip surfaces (or shear bands) tend to develop and propagate hundreds of meters beyond the foot of a slope, and that even prior to the possible final incidence of passive Rankine failure. 2) The tendency of failure zones and slip surfaces in deformation softening clays to de-velop along firm bottom or weak layers more or less parallel to the inclining ground con-tour rather than along more or less local slip-circles surfacing in the slope itself. As is demonstrated in Section 4.4.2 as well as in Chapter 5, circular slip-surfaces are in soft clays extremely unlikely to develop in sloping terrain – and do not in deformation soften-ing soils readily materialize even in horizontal ground. (Cf Bernander, 1981b).

2

3) The one-sided attention given to high sensitivity (quick clay) as a universal prerequisite condition for the current type of landslides – i.e. virtually disregarding geometry, rate of loading, drainage conditions as factors of equivalent significance and concern. 4) The seemingly dogmatic notion of the exaggerated role of rainfall conditions in slides of the current type. Extreme rainfall conditions may certainly have a decisive impact on the stability of many slope types under diverse geotechnical conditions, but as regards downward progressive slope failure in soft clays, the effect of heavy precipitation appears to be a factor of secon-dary importance. All of the slides presented in Table 2.1 below – except may be for the Tuve slide – took place while construction activities were actually going on and did not coincide with any documented extreme piezometric levels. In order to identify and highlight the issues mentioned above, a more general discussion of the characteristics related to down-hill progressive landslides in soft clays follows in Chap-ter 2.

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2 Characteristics of down-hill progressive landslides

2.1 Examples of landslides occurred In Table 2.1, some cases of downward progressive landslides in soft clays are presented. A number of other landslides could be added but the author has limited the list to events about which he possesses adequate information. The landslides listed in Table 2.1 are all typical of a kind of spread slope failure frequently occurring in the soft clays of Scandinavia. These slides exhibit the following general char-acteristics, of which some may be deduced directly from items in the table. 1) All of the listed landslides relate somehow to local, and in view of the disastrous conse-quences, seemingly trivial human activities. Except for Tuve slide*, all of the listed events are directly connected with construction activities such as stock-piling of earth or rock de-bris, placing of embankments, pile-driving or the use of vibratory equipment. [* The Tuve slide differs somewhat in this respect but according to SGI-Report No 18 (1982), the causes attributed to the Tuve slide were disturbances generated by high ground water pressures in combination with the weight of an additional road embankment constructed some years before. Man-made changes of the hydrological regime due to urbanization were also believed to have con-tributed to local instability. Hence, also for the Tuve slide, it was concluded that human activity was an important triggering factor.] 2) The actual slide event may or may not coincide with high precipitation – and certainly not by necessity with extreme rainfall conditions. Generally, these slopes had remained stable for thousands of years and in all probability been exposed to highly extreme peak pore pressure situations in the past. In none of the cases presented, precipitation was likely, or was documented, to have been more exceptional than ever before in the history of the slope. The impact of heavy rainfall thus appears to constitute a secondary factor in slides of the current type. Furthermore, in the writer’s opinion, the effect of abundant rainfall in the current context is more likely to be related to destabilizing forces acting in temporarily water-filled cracks (above normal ground water level) in the incipient active zone rather than to the incidence of, for instance, a 1000-year extreme pore water pressure in specific sedimentary layers of the slope. Thus again, the cause of failure is more likely to be related to the man-made ad-ditional critical load immediately connected with the slide event. 3) Slides of the kind reviewed in this study cannot readily be explained by means of the ’ideal plastic limit equilibrium’ approach. In fact, subsequent application of conventional analysis, based on perfectly plastic behaviour of the clay usually indicates ample computed safety against slope failure – a fact suggesting instead the incidence of fracture-mechanical phenomena. 4) A common feature in these slides is the remarkably extensive spread of the passive fail-ure zone over areas of almost horizontal ground, often representing up to 50 to 60 % of the total area involved in the initial slide. This characteristic is significant and indicates progressive failure formation also in the final development of the slide. The phenomenon is explained and exemplified in detail in Chap-ter 5 and in Appendix A of this report. 5) Failure modes based on slip-circles surfacing in the slope itself are not likely to have much relevance.

4

Table 2.1 Examples of down-hill progressive landslides. Locality Year Slide

length involved[m]

Area [hec- tares]

Triggering agent

The Svärta River 1938 160 2 Local road embankment (Cf figure) Surte 1950 600 24 Pile driving for a family house Beckelaget, Norway 1953 160 2 Widening of railway bank up-slope Rollsbo Kungälv, Sweden 1967 2 Driving of pipes for sand drains Rödbo, Kungälv Municipalaty 1968 1 Stock piling of blasted rock Jordbro, V:a Haninge 196x Local up-slope earth fill Rävekärr, Mölndal 1971 300 15 Pile driving for a family house Sem, Norway 1974 120 Local earth fill up-slope Tuve, Göteborg 1977 800 26 Widening of road embankment etc Rissa, Norway, slide C * 1978 800 27 Retrogressive initial slide Kotmale dam site, Sri Lanka 1981 500 9 Stockpiling of concrete aggregate Trestyckevattnet, Uddevalla 1990 400 2 Vibration of road embankment Saint-Fabien, Quebec, Canada

2004 - - Widening of an up-slope railway embankment

Småröd, Munkedal 2006 230 ca 10 Local up-slope earth fill * Slide C in Rissa was initiated by a retrogressive slide caused by human activity, (Gregersen 1981).

Figure 2.1.1 The landslide at the Svärta River (Sweden) 1938 featuring characteristics of slides in Table 2.1. Bygg (1972).

2.2 Presentation of a typical down-hill progressive landslide The landslide in Surte in 1950, involving some 240 000 m2 of ground in the initial slide epitomizes the kind of earth movement dealt with in the present study. The slide stands out as the second biggest of its kind in Sweden in recent times. It is analyzed and explained in terms of progressive failure in Chapter 4 below. The reason for recalling this particular but possibly long forgotten event is that it exhibited two intriguing and until very recently poorly explained characteristics of many Scandina-vian slides, namely: - the apparent triviality of the disturbing agent causing the slide and - that a major portion of the initial slide area consisted of very gently sloping ground, (in Surte about 50 % of the area of the main slide).

5

Another important reason for focusing on the Surte slide in the current context is that the disturbance, which triggered the slide is believed to be more clearly identified than, for instance, is the case in the even greater landslide in Tuve – also in the area just north of Gothenburg. (Area of the initial slide in this case 26 hectares.).* (* In the Tuve slide, the initiating factors were more complex and the impact of each of them diffi-cult to identify. Otherwise, the Tuve slide presents the same typical features as the other slides in this category – e.g. about 60 % of the slide area (i.e. 16 hectares) being practically horizontal ground.) The Surte slide was at the time investigated by at least three renowned geotechnical engi-neers, who however failed to agree between themselves on essential issues. The odd thing about both of the spectacular landslides in Tuve and Surte is that they were not – at least not until very recently (i.e. after a lapse of about 50 years) – adequately ex-plained to the effect…… a) ….that the causes, or the disturbance agents, were analytically related to the incidence of the slide event. (Instead, undefined rainfall conditions, undocumented artesian water pressures were, as is often the case when investigating slides of this type, assumed as last expedients in order to be able to explain the slide applying the prevailing analytical con-cept.) b) ….that the features of the finished slide could be related to the structure-mechanics in-volved – i.e. reasonable confirmation by computational back-analysis of salient slide fea-tures such as length, heave and vast spread over stable ground, etc. c) ….that geotechnical expertise would be able to predict – or evaluate the risk of – poten-tial future landslides in other similar or slightly differing situations.

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3 Brief description of down-hill progressive failure analysis

3.1 General As the subsequent Chapters 4, 5 & 6 (as well as Appendices A, B and C) present the Surte slide and other issues in terms of progressive failure, it may be apt to elaborate briefly on this topic.

Progressive slope failure has many characteristics in common with brittle failures in the discipline of structural mechanics.

Yet, the analysis of stability in long natural slopes of soft clay harbours some rather spe-cific additional complications. The strength parameters to be used are for instance heavily dependent on conditions that, for various reasons, are difficult to define with sufficient accuracy in natural soil deposits. Such conditions are for instance:

- The crucial - but often difficult - task of establishing the in situ states of stress in accor-dance with past geological history, erosion and other contributing agents.

- The loss of available shear resistance on account of past and ongoing deformation and due strain-softening related to the degree of over-consolidation (OCR).

- The time dependent strain-softening, which is strongly related to both the rate of load application and – importantly (in most instances) – on poorly defined drainage conditions in the incipient failure zone near and under the additional, possibly critical load.

In the opinion of the author, the frequently documented lack of success on the part of geo-technical engineers to arrive at satisfactory explanations of landslides stretching to great length is directly linked with the fact that the conventional analytical approach, based on perfectly plastic clay behaviour, disregards differing deformation in the soil mass and can-not therefore, by definition, consider the complicating factors listed above.

Nevertheless, it is only recently that the mainstream of geotechnical research has begun to address the problems and the immediate implications of the factors mentioned above.

This may on one hand largely be due to the high complexity and uncertain character of the issues involved, and on the other hand, to the contrasting tantalizing practicality and sim-plicity of the conventional approach based on ideal-plastic equilibrium analysis. However, simplicity is not a benefit in cases, when it may render totally erroneous results.

A number of researchers have over the years addressed the problems of progressive slope failure in soft clays. Confer e.g. Kjellman (1954), Bjerrum (1954), Skempton (1964), Bishop (1967), Christian & Whitman (1967), Janbu (1979), Aas (1982), Karlsrud et al (1984), and many others.

Yet, in the opinion of the author, much of this early research on progressive failure has lacked adequate stringency from a structure-mechanical point of view – primarily in re-spect of not properly accounting for the deformations within and outside the potentially unstable soil mass.

The crucially important feature of the FDM analysis presented below is that it not only accounts for deformations in, and along, the slip surface (or the shear band) but also for those in the entire shear zone, comprising a considerable portion of the depth of the stud-ied potentially sliding soil mass. The significance of this specific feature can hardly be exaggerated.

8

3.2 Progressive failure approach used in the current study Bernander et al (1981, 1984, 1988, 1989, 2000) applied a model for progressive failure analysis (PrFA), the 1988 version of which is characterized as follows in items a) through h): a) The deformations within and outside the potentially sliding soil mass are taken into ac-count in a 2-dimensional Finite Difference mathematical Model (FDM),

b) The variation of shear strength resulting from deformation-softening of the clay material is considered on the basis of predefined shear stress/deformation relationships, where the peak shear strength – as well as the shear resistance resulting from strain-softening (i.e. from su sR*) – is expressed as a time dependent function of deviatory deformation and displacement – i.e. su ( , , t) = ( ,d /dt, ,d /dt). (*Here sR denotes the effective residual direct shear resistance at large deformation in the local failure zone of an impending progressive slope failure. The value of sR depends importantly on rate of loading and local drainage conditions.)

Figures 3.2.1b and 4.4.1 exemplify shear-deformation relationships of this kind. As the relationships, applying to different phases of a slide, are adaptable to the actual rates of load application and stress change as well as to drainage conditions, also the effect of time can in a general way be considered in this FDM analysis.

c) The deformations in the entire failure zone, and not only those in a discrete shear band, are modelled – i.e. all shear deformation within a distance Hx from the potential slip sur-face is considered in the computations. (Cf Figure 3.2.1a).

Figure 3.2.1 a Soil model – denotations, G= z· x· . From Bernander, Gustås & Olofsson (1988, 1989). A full description is given in Bernander (2000).

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Figure 3.2.1 b Time dependent stress-deformation relationship = ( , d /dt, , d /dt). Labo-ratory test curve compared with the same curve translated to the real dimensions of the soil structure. Note the apparent difference in brittleness. Curves 1 and 2 exemplify stress-deformation relationships for two different rates of loading. From Bernander (1985, 2000).

Figure 3.2.2 below illustrates the effect of increasing the disturbing force Ni beyond the critical value Ncr in which case a ‘dynamic’ redistribution of stress and deformation is bound to take place.

Figure 3.2.2 The figure defines the critical parameters Ncr and Lcr in the disturbance stage, Bernander (1981). It also illustrates the effect of increasing the disturbing force Ni be-yond the critical value Ncr, in which case a ‘dynamic’ redistribution of stress and deformation takes place. Note that for x = Linstab , Ncr – N = 0.

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Figure 3.2.3 Figure illustrating a transitory stage of equilibrium in a potentially fully developed slide prior to the plasticization of the extensive passive zone in Rankine failure. If passive Rankine resistance is not overcome, i.e Eo+Nx< Ep

R, this stage of equilibrium will remain perma-nent and the visible effects of the slide movement will be limited to minor displacements and cracking in the active zone. From Bernander (1981 2000). Cf Chapter 5.

The concepts of the critical parameters Ncr and L cr were introduced already by Bernander & Olofsson (1981a, b), where the equivalent parameters were denoted NSR and LSR respec-tively.

As may be deduced from Figure 3.2.2, a situation may arise when – at a certain value of forced displacement at Point A, (i.e. for Ncr - N = 0) – progressive failure will occur even if the current additional critical load were removed instantaneously. The corresponding parameters are denoted ( instab) and (Linstab). (Cf Bernander (2000), [Chapter 3])

d) The governing basic condition maintained throughout the computational analysis is that down-hill displacements are at all times compatible with the integral shear deformation in every location of the slope. The analysis begins at a point further down the slope (x = 0), where the studied additional load has no effect, and is continued up-hill to the location of the studied additional load (Ni,) at x = L). This constitutes the upper boundary condition.

e) The progressive landslides subject to study in this report cannot, in present State-of-the-Art, be looked upon just as a singular, mechanically static event. This is because a slide event of this kind actually consists of a series of different consecutive phases of static and dynamic instability, which are not simultaneous. In addition, the phases are characterized by radically differing conditions in respect of types and rates of loading, as well as of the response of soft and sensitive clays to time and drainage factors. In the progressive failure FDM-approach, according to Bernander (1984 2000), it is possible and useful to distinguish between the different phases of a potential slide event as follows:

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- Phase 1. The in situ stage. The long-time shear stresses and the earth pressures in situ have to be established. This may be accomplished in different ways – one approach being to regard the in situ condition as a result of a slow progressive creep movement applying the FDM – approach whereby certain stress redistribution takes place.

- Phase 2. The impact of the disturbance force (Ni) susceptible to trigger a progressive failure situation is defined and computed. Then, if the value of Ni exceeds a limiting value Ncr, Phase 2 becomes critical and initiates the next phase – Phase 3.

- Phase 3. This phase represents a virtually dynamic redistribution of stresses, deforma-tions and earth pressures, where unbalanced forces in the steeper parts of the slope are transmitted down-slope, possibly resulting in massive build-up of earth pressures in less inclining ground – especially at the foot of the slope.

The dynamic effects in this intermediate phase are not of great significance, and are nor-mally not part of the study. (Note: The dynamic Phase 3 – necessitating the application of Newton’s equations has only been studied by this author in connection with the Tuve slide (1977). Cf NGM 1984 & Bernander 2000.)

As demonstrated in Chapter 5 below, it is in Phase 3 that the extensive failure zone with its slip surface is generated and tends to progress under the valley floor far beyond the foot of the slope. (Cf Figures 5.1.1 & 5.2.2).

- Phase 4. This phase represents - in fully developed slides - a transitory state of possible equilibrium, in which the soil mass momentarily retains its shape prior to potential collapse of the passive zone in terms of exceeding full passive Rankine resistance (EP

R). (See Fig-ures 3.2.3 & 4.4.3).

The justification for studying the transient condition of equilibrium in Phase 4 is related to the fact that the shear failure and the displacement along the developing ‘slip surface’ are – in deformation-softening clays – not concurrent with the subsequent final collapse and plasticization of the entire soil mass in passive failure. In fact, while the resistance (su) of the clay material in the sheared zone and the residual resistance (sR) along the slip surface are fully mobilized, the clay in the rest of the soil mass is still basically intact – having been subjected to comparatively little deformation – until the very incidence of passive Rankine failure. Consequently, as demonstrated in Chapter 5, and in Appendix A the slip surface under the valley floor must, in deformation-softening clay, form already when earth pres-sures at the foot of the slope are still well below the passive Rankine resistance (EP

R).

This means that the progressive failure mechanisms in Phase 4 - as in Phase 3 - are in prin-ciple identical to those in the incipient disturbance phase and that the conditions in this phase – despite its possibly transient character – can also be investigated.

Therefore, the analysis of the conditions immediately prior to the final disintegration of the passive zone should not be regarded as superfluous, because after all, it is the earth pres-sure distribution in this specific phase of a landslide that actually conditions the degree of disaster. (See Figures 3, 2, 3 & 4.4.3). It may be noted that, if the value of EP

R is not exceeded, Phase 4 will remain a permanent condition, in which the effects of the slide movement will be limited to moderate dis-placements over the slide area and cracking in the active zone. (Cf The slide at Rävekärr, Bernander, 2000.)

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- Phase 5. The final dynamic breakdown and heave of the passive zone in Rankine failure takes place in Phase 5, allowing the soil mass up-slope to move downwards. It is in this phase that the dynamics of the landslide proper actually occur.

f) As the shear/deformation relationship, applying to an investigated phase of a slide, is adaptable to the current rate of load application and stress change, as well as to prevailing drainage conditions, it is possible to consider the effect of time in the analysis.

g) Furthermore, the analysis along the slope distinguishes between

- Stage I, dealing with deformation conditions before any slip surface has developed and

- Stage II, representing the conditions subsequent to the formation of a discrete slip surface or shear band.

h) Finally, it is important to note that the characteristic outcome of the FDM-analysis de-scribed is mainly related to the fact that deformations are considered in combination with the deformation-softening in terms of sR/su*. By contrast, the impact of reasonable varia-tion of other data in the chosen shear stress/deformation relationship is moderate and does not generally alter the nature of the outcome of the analysis. (Cf Appendices A and B, Tables A.III and B.II). * Again, it is important to observe that it is at all times distinguished between the remoulded shear strength (sur) as measured in laboratory tests and the effective residual shear strengths (suR and sR) that are applicable in a real progressive failure scenario in the macro structure. Mainly due to time and drainage effects, both of the parameters suR and sR may exhibit little compatibility with the value of sur.

3.3 Positive features of the FDM-analysis used The FDM analysis (Finite Difference Model) described above has the merit of clarifying certain rather intriguing and un-explained features of large extensive landslides in the soft clays of Scandinavia, such as:

1) How can driving of a few prefabricated concrete piles for a family house, or placing of minor local fills, effectively destabilize areas of ground up to tens of hectares, (e.g. Surte 24, Tuve 26, Rävekärr ca 15 hectares). [1 hectare = 10 000 m2].

2) … and how does it come about that – for instance in the extensive Surte and Tuve slides – more than half of the areas involved in the initial main slides consisted of virtually hori-zontal ground, which was plasticized in passive Rankine failure down to depths as great as 20 and 35 m respectively. (35 m matches about the height of a 10 story building).

The analysis also clearly predicts that the slip surface and the associated zone of rupture tend to develop far beyond the foot of a slope into less inclining ground before the inci-dence of potential breakdown of the passive zone in Rankine failure.

3) The FDM-analysis described also underscores the importance of geometry to the brit-tleness of slope failure. Hence, deformation softening (sensitivity) – while being an impor-tant factor in the current context – by no means stands out as the only significant element leading to progressive slope failure. In fact, adverse geometry may in soft clays sometimes be a more important factor than the vaguely defined degree of strain-softening in terms of sensitivity. The compatibility between the term ‘quick clay’ (with a sensitivity of su/sur > 50 as measured in Sweden), and the deformation-softening properties in actual in situ shear at varying rates of loading (as defined by su/sR), is not well established.

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4) Furthermore, as already mentioned, the approach emphasizes the vital importance of utilising the appropriate degree of strain-softening that is related both to the actual rate of applying the additional load, and to the prevailing drainage conditions. Yet, it is worth not-ing that, as shown in Appendix B, the value of Ncrit – signifying the risk of progressive failure development – is not importantly dependent on the value defining deformation-softening for the steeper slopes studied. (Cf. Figure B.3 in Appendix B). This is of particu-lar interest, as prediction and prevention of progressive slope failure must by necessity focus on defining a relevant value of Ncrit. However, the further development of a slide (i.e. Phase 3) is of course likely to be strongly affected by the degree of strain-softening.

Whatever be the answer to the items 1 through 4, the outcome cannot be a recommendation of continued indiscriminate use of analysis based on limit plastic equilibrium in soft de-formation-softening clays. Indeed, applying the conventional approach, it is for instance easy to show that the extensive slides referred to above should not have taken place – the computed safety factors being well in excess of 1.

For instance, according to SGI Report No 18 (1982) of the Tuve slide (1977), the safety factors – based on current standards for interpretation of data from boring logs, and on normal evaluation of shear strength increase with depth – ranged between 2.0 and 2.6.

By contrast, the Tuve slide was at an early stage explained in terms of progressive failure by Bernander & Olofsson (1981a, b).

The progressive failure model defined in Chapter 3 above was published at international conferences and applied to practical cases already in the late 1980-ies but was presented in more detail in Bernander (2000). For additional information, reference is made to this re-port.

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4 The Surte slide

4.1 General - history of a slope in the Göta River valley The stability conditions in natural slopes are closely related to their geological and hydro-logical history. Clay slopes in western Sweden are made up of glacial and post-glacial sediments that emerged from the regressing sea after the last glacial period. Hence, the sediments deposited in sea and fjords at the end of this period in what was later to become western Sweden, are now found in valleys and plains considerably above present sea level, forming deep layers of soft clays and silty clays.

As the ground gradually rose above the sea level, the strength properties of the soils and the earth pressures in the slope have, by way of consolidation and creep movement, ac-commodated over time to increasing loads and changing conditions. These may have re-sulted from the retreating free water level, falling ground water tables, varying climatic conditions and chemical deterioration.

In consequence, existing slopes are basically stable, as long as they remain undisturbed. Considering the effects of extreme high ground water events during past centuries and mil-lennia, the formal safety factor may – provided hydrology has remained unaffected by hu-man interference – at least be assumed to exceed by some measure the value of 1.0.

Yet, the real safety margin cannot possibly in long slopes of soft sensitive clays be defined in the conventional way on the principle of plastic equilibrium. In fact, the true degree of safety can only be assessed by investigating the response of potential disturbing agents in terms of progressive failure, and will therefore solely depend on the nature of the addi-tional load, to which the slope may be subjected.

Figure 4.1.1 Section through the Surte slide area epitomizing common features in glacial and post-glacial clay deposits in western Sweden. (Vertical scale = 5 x Horizontal scale). Denota-tions in figure: = suv = Vane shear strength; St = Sensitivity = suv/sur. From Jakobson (1952), modified regarding notation.

A vital question when investigating slope stability is then: ‘How will the slope respond to a local additional load or disturbance effect, for which the ‘time horizon’ is measured in terms of days, weeks or months instead of hundreds or thousands of years? A specific addi-tional load or disturbance may for example be totally inconsequential in a protracted six

15

month scenario, whereas the same change happening in days or weeks may lead to a disas-trous slide. In other words, how does the time factor affect slope stability?

For instance, how will climate change affect down-hill progressive failure hazard? On one hand, will extreme and extenuated periods of rainfall generate higher pore water pressures than ever before in the clay and in specific layers of more permeable soil material, or will the clays involved gradually adapt to slowly changing environmental conditions as has, in fact, mostly been the case in both existing slopes and slopes verifiably destabilized by hu-man activity?

And finally, if local failure is conceivable, what manner of disaster is likely to ensue? Will such a failure in a sensitive part of the slope result only in creep movements and minor cracking in the ground up-slope, or will it end up as a major landslide displacing hundreds of meters of horizontal ground over large distances? The basic conditions leading to either of these scenarios may often not be radically different. Aas (1982) discussed the hazards related to such primordial and potentially unfavourable conditions in natural slopes – yet without considering, as is done in this report, the effects differential deformations within the sliding body of soil.

4.2 The Surte slide event

Figure 4.2.1 Aerial view of the Surte landslide in the valley of the Göta River some 10 km North of the city of Gothenburg, Sweden.

The Surte landslide took place soon after 8 a.m. on September 29, 1950. The main slide, involving some 24 hectares of ground, swept away 31 family houses and 10 outhouse units. Due to the favourable hour – most residents being at work elsewhere – the death toll was limited to one person.

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The south-bound branch of the Göta River, which is navigable for heavy shipping transport was blocked for two months. The north-bound railway and highway were displaced dis-tances varying up to 150 m, inhibiting road and railway traffic for 10 and 19 days respec-tively. Transportation and industry incurred serious damage.

Figures 4.2.1 and 4.3.1 are aerial photos of the slide area. Figure 4.3.2 shows plan and sec-tion of the slide.

The actual slide event was observed by a number of people within and outside the slide area, but as is often the case in dramatic circumstances of the current nature, most eye-witnesses only registered incidents that were local in time and space.

However, one witness positioned outside the slide area gave an exceptionally coherent, continuous and time-wise extended description of the main events of the slide. This must be considered to be of great value to anyone who seeks to account for the main events in order to understand the causes and the mechanisms of the slide. The witness Ture Bernts-son sums up his impressions as follows: “The whole ground was moving rather slowly at a speed that can approximately be compared to that of the Bohus ferry. (Estimated speed a few metres per second.) The movement did not proceed at the same speed all the time - the speed increased progressively and the movement finally ceased when the ground piled up against the opposite side of the river. Then the ground rose and folded. However, folding had already begun during the first stage of the movement. House No 13 toppled very slowly when the slide was approaching the opposite side of the river. Water and clay were lifted very high. Cracks of various sizes were formed during the course of the slide. At first, the ground moved straight down towards the river but further down the slide widened, while the main part of the ground continued straight ahead.”

Ture Berntsson’s statement agrees very well with slide development as conceived by the progressive failure analysis used in this study.

Another important witness, Hjördis Svensson, standing in her kitchen (Villaplatsen 2) and facing south, told among other things the following: “She first noticed that a pile driving machine and the ground around it began to subside and that the men engaged in pile driving started to run away. Then she observed that the houses beyond were also moving. …….The pile driving machine did not topple until the last stage of the move-ment. A large number of cracks formed in the ground. The movement was wavelike and smooth. The houses seemed to sail along. “

4.3 Investigations and analyses after the slide The Surte landslide was treated in two comprehensive reports by Jakobson et al (1952a), and by Caldenius & Lundström, (1955). (In the latter report Lundström stood for the geo-technical assessments.) The thorough field and laboratory investigations made in connec-tion with these reports constitute valuable contributions to the knowledge of the behaviour of the types of soft clay involved in the slide.

However, in so far as the causes and the mechanisms that formed the slide event are con-cerned, both of these reports must in the opinion of the writer be regarded as inconclusive, and at least from a stricter engineering standpoint, the Surte slide has remained unex-plained until recently, Bernander (2000).

The reasons for this are as follows:

a) The two official reports from 1952 and 1955 are contradictory on essential issues – e.g. with regard to ground water conditions and piezo-metric levels, to the causes of the initial

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slide, as well as to the sequence of slide events and the failure mechanisms that formed the landslide.

b) In both reports, computational analyses of the various phases of the slide were based on the concept of plastic limit equilibrium - i.e. differential deformations within the poten-tially sliding bodies were not accounted for in any of the back-analyses.

Figure 4.3.1 Aerial photograph taken 13 days after the slide. From Caldenius & Lundström (1955).

In the experience of the author of the present study, gained from comprehensive studies of progressive failure formation, the validity of the ideal-plastic failure (I-PFA) concept may be questionable for many kinds of additional loading in soft and sensitive clays already when potential failure zones are in the range of seventy to a hundred meters in length. This applies of course in particular when, as in Jakobson’s analysis, the slide length exceeds 400 m.

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Lundström, on the other hand, considered effects of strain softening in attempting a verbal description of the spread of what he termed the ‘progressive passive slide’ over gently sloping ground – thereby implying that the slide propagated as a series of consecutive slip circular slides. It should be noted that the term ‘progressive’ is used by him in an alto-gether different sense than that adopted in the present report.

c) Other circumstances present, as discussed below.

Jakobson assumes that the soil volume in the main slide, excluding the serial retrogressive after-slides, moved as a block towards the river. He further finds that the critical cause of the slide was the presence of elevated artesian pore water pressure heads in the order of 7 m in the failure zone, and which he presumes to have been occasioned by high precipita-tion in the years 1949 and 1950.

It is true that high pore water pressures of this magnitude were recorded in deeper clay layers after the slide. The analytical model is of course plausible as such but the serious problem with this theory is that Jakobson presumes – without valid substantiation – that these high piezometric levels had existed prior to the slide event. In fact, no elevated arte-sian pressures of this extraordinary magnitude were registered in undisturbed ground any-where else in the area. Nor were the measured pressure gradients compatible with a stable long-time ground water situation. Thus, Jakobson’s assumptions in respect of elevated ground water heads before the slide were not actually documented and were, incidentally, contested already by Lundström.

In the current context, Jakobson seems to have overlooked the fact that, when soft and sen-sitive clays – i.e. basically collapsible soils – are excessively sheared, high pore water pressures are generated by the very disturbance of the clay and maintained over long peri-ods of time by the weight of the overburden. This phenomenon has been documented in other slides in soft clays. In fact, the few pore pressure measurements actually made in the Surte slide area indicate excess pore water pressures beginning at a depth of some 10 m below the ground level, and from then on rising gradually to maximum values at about 20 m of depth. As this was the level of the slip surface assumed by Jakobson, the excess pore water pressures are indica-tive of disturbance due to shear deformation in the entire zone subject to excessive shear.

Moreover, measurements of excess pore water pressure in the ground immediately outside the slide limits exhibited only about 50 % of the values mentioned above. i.e. roughly 3,5 m. Yet, also these values were most likely induced by the slide itself, considering the close proximity of the location of the pore pressure gauges to the actual boundary of the slide.

Although Jakobson appears to have been aware of the fact that during the thousands of years the slope had existed, more extreme ground water conditions must have prevailed time and time again, he does not present any argument or reason as to why the slide was set off on that particular day in September, 1950.

In the intense discussion of the immediate causes of the slide that followed, Jakobson makes no reference to the fact that prefabricated concrete piles were being driven in a steep part of the sloping ground. (Cf Figure 4.3.2). This is noteworthy since the pile driving ac-tivity was the only abnormal disturbance at the time of the slide event and which, as far as was known, had never taken place before in the steeper portions of the slope. Family houses in the area involved in the slide had been founded without the use of piles. How-ever, it may be noted here that Jakobson, in response to critical comments on his report by Löfquist (1952) in the journal named Teknisk Tidskrift (The Technical Journal), as well as in the animated debate that followed (1953), argued that the immense spread of the slide

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may have been due to some kind of progressive failure process. He then, as it appears, ar-gued that the failure process may have been due to gradual loss of shear resistance as the slide propagated, yet without presenting any supporting analysis or computational docu-mentation.

Figure 4.3.2 Plan of the slide area showing elevation contours and a longitudinal section A-A of the slide. From Jakobson (1952b). The point marked (P) on the plan is the location in the steepest portion of the slope, where piling operations were going on at the time of the slide oc-currence. (This point was not indicated in the source document. Section B-B marks the section analyzed in Figure 4.4.3 and was not either shown on the original plan).

In his review of Jakobson’s report, Löfquist contended that the remarkable spread of the Surte slide must have been due to a near total loss of frictional resistance in a presupposed thin stratum of fine sand, thus effectively implying that failure in sensitive clay was not the decisive factor. He then had to assume that, in this layer, considerably higher artesian pres-sures than even those assumed by Jakobson must have prevailed before the slide. Although also Löfquist’s model for slope failure is viable as a theoretical concept, his approach must be regarded as highly speculative, as artesian pressures of this magnitude had not ever been documented in this area. Nor was any continuous seam of fine sand shown to be present.* (*Comment: In the opinion of the author of this report, liquefaction in sandy or silty layers due to shear deformation is highly unlikely in slopes of this kind owing to the fact that the ground has been subjected to

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shear deformation from creep movements ever since it gradually emerged from the glacial sea. Hence, any discrete seams of cohesion-less material will already long ago have attained a state of constant porosity, in which case liquefaction generated by additional shear deformation is not a probable scenario. However, this does not of course exclude liquefaction from dynamic impact such as pile driving and use of vibratory equip-ment.)

Like Jakobson, Löfquist does not present any reason or argumentation as to why the pie-zometric levels in the sand layer assumed should have been higher than ever before at the time of the slide event.

Contrary to Jakobson, Lundström asserts that the slide developed as a rather complicated and interrelated series of smaller local slides with circular slip surfaces. Hence, he argues that an initial slide first took place in the steeper part of the gradient. From deliberations with regard to the kinetic energy of the local slide, he maintains that this first slide to some extent affected the practically horizontal ground, in displacing it some distance towards the Göta River.

Nevertheless, he concludes that this impact was not sufficient to provoke a continuance of the slide movement all the way to the river. Then, in order explain the further progression of the earth movement, he suggests that inertia forces originating from the retrogressive after-slides acted on the immense soil masses in the almost horizontal part of the valley, completing the passive heave of the ground as far as the river bank. Then in turn, the ground near the riverbank became unstable, thus ending the sequence of ground displace-ments by eventually blocking the riverbed in a major local slip-circular slide of conven-tional type. Lundströms reason for contemplating this final slide event was probably the fact that there was no heave (or subsidence for that matter) over a distance of some 20 m near the riverbed *. (Cf longitudinal section in Figure 4.3.2.) (* For this author’s explanation of this phenomenon, see Section 4.4.3 below – Global failure con-dition.)

Lundström’s reasoning seems complicated, circumstantial and mechanically disputable, but his explanation of the Surte slide has the merit of recognizing inertia forces as an im-portant feature in slide propagation mechanisms.

However, kinetic energies and forces of inertia are time-dependent dynamic phenomena and cannot be added algebraically unless they are perfectly concurrent. Therefore, the main difficulty in accepting Lundström’s failure concept – i.e. when he tries to explain the pas-sive heave of the almost horizontal ground and the riverbed – consists mainly in the way that he compounds the dynamic effects of the retrogressive after-slides to those of the ini-tial slide. These effects were in no way simultaneous.

Lundström, however, ascribes the initiation of the slide to the ongoing pile driving activity in the steepest portion of the slope, i.e. in Point P in Figure 4.3.2.

Conclusions As may be concluded from the discussion above, the explanations of the Surte slide result-ing from the post-slide investigations are, in opinion of the author of this report, not ac-ceptable from several points of view – and that in particular because of not considering differential deformations within and outside the extensive sliding soil mass.

The piezo-metric levels presumed in Jacobson’s and Löfquist’s analyses were not docu-mented and are not likely to have existed before the slide.

Regarding Lundström’s analysis, it is difficult to conceive how any geotechnical engineer, investigating slide hazards in similar geotechnical situations, would be able to predict the

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random sequence of events and the precise risk of potential spread failure on the basis of the complex arbitrary series of slip circles that characterizes his failure concept.

In an article written in Swedish, Lundström (1997) has elaborated somewhat on his 1955 theory regarding the Surte slide events. However, even at this point his presentation does not address the possibility of progressive failure in accordance with concepts that have appeared in soil mechanics literature since 1955. In the absence of a coherent integral analysis in the time domain of the combined static and dynamic forces as well as of the related accelerations and velocities covering the total duration of the slide, his explanation of the Surte slide remains circumstantial and inconclusive.

4.4 Explanation of the Surte landslide in terms of progressive failure Fortunately for the art of slope stability analysis, the issues are, in the opinion of this au-thor, not as erratic or randomly complicated as indicated by the failure concept described above. An investigation of the Surte landslide has been carried out using the progressive failure FDM-approach outlined in Chapter 3 above – i.e. essentially considering among other factors the differential deformations in the potentially sliding volume of soil.

Figure 4.4.1a Constitutive stress/deformation relationship. SR denotes the residual shear resis-tance of the clay in the critical part of the slope when (and where) progressive failure is initi-ated. It may be noted that the ratio el/su is in the current study assumed to be constant as su varies with the coordinate (x). The value of sR is related to the rate of load application and to drainage conditions.

As stated in Section 3.2, the deformations in the failure zone adjacent to the potential slip surface are modelled on the basis of a constitutive shear stress/strain relationship such as the one shown in Figure 4.4.1a.

Deformations in the incipient failure zone are in the current case assumed to be symmetri-cal above and below the potential failure plane only where the slip surface is sufficiently distant from the firm bottom contour. Shear strengths and E-modulus are varied along the slope as interpreted from soils investigation data provided in the report by Jakobson, (1952a).

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Input data With reference to Figure 4.4.1a and Figure 4.4.1b, the following values of the characteris-tic parameters have been used in this study. (sR denotes the effective residual shear resis-tance under current rate of load application and ambient drainage conditions).

In situ state condition:

s R/s = 1.00 el = 3.75 % f = 7.5 % Gel,o= el/ el = 480 kN/m2

s * = 24 kN/m2, el = 18 kN/m2 Eel,o =2(1+ )Gel,o = 60 su = 1440 kN/m2 g = 15.5 kN/m3 ko = 0.55 (for horizontal ground) = 0.5 Eel,mean = 60 su,mean

(* In the current state, s signifies the long time shear resistance – drained conditions.)

Disturbance Condition I – failure initiation

sR/su = 0.80 ** el = 2 % f = 4.0 % cr = 0.3 m Gel,o = el/ el = 1000 kN/m2

su # =30 kN/m2 el = 20 kN/m2 Eel,o= 3Gel,o = 100 su = 3000 kN/m2

g = 15.5 kN/m3 komax = 0,594 (computed) Eel,mean = 100 su,mean

# Mean values applying to the initiation zone.

Disturbance Condition II – failure initiation

sR/su = 0.60 ** el = 2 % f = 4,0 % cr = 0.3 m Gel,o = el/ el = 1000 kN/m2 ** The values of sR are to be adapted to the rate of load application and estimated drainage condi-tions in the failure zone, and are here assumed to correspond to 0.8 (Condition I) and 0.6 (Condi-tion II) of the un-drained shear resistance su. The parameters sR and suR must not be confused with the remoulded shear strength sur as measured in laboratory.

Global failure condition:

suR/su = 0.35-0.20 # el = 1 % f = 2 % , cr = 0.3 m Gel,o = el/ el = 2400 kN/m2

su,o = 36 kN/m2 ## el = 24 kN/m2 Eel,o= 3Gel,o = 200 su, = 7200 kN/m2

ko (as computed in the in situ condition) Eel,mean = 200 su,mean

# The un-drained residual shear resistance suR is in the current case assumed to vary between 0.35 and 0.20·su. ## Mean value applying to the down-slope failure zone.

Note: In all of the calculations in Sections 4.4, 5.2, 6.2 and in Appendices, the curved portion of the constitutive relationship from el to f is a function of x2 with vertex at (su, f) and connecting tan-gentially at ( el , el).

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Figure 4.4.1b Assumed shear/deformation relationships for the three decisive phases of the Surte Landslide.

4.4.1 Results of the FDM-analysis – The in situ condition The results of the in situ FDM computations are given in Table 4.4.1 below.

The in situ state condition In the steepest part of the slope, available shear strengths do not match the in situ shear stress in terms of o = g H sin . This implies that already in the in situ condition, the soil masses were to some extent balanced by elevated earth pressures in less inclined ground further down the slope in accordance with Equation 4.4.1, (i.e. Eo is positive).

0 (x,o) = oH(x) g H sin - Eo(x)/(b(x) x ………………Equ. 4.4.1

Table 4.4.1 Results from FDM- analysis – In situ state condition.

(LN,max = distance to Nmax from the point of application of the additional load)

sR/s = 1.0 Nmax = 138 kN/m LN,max = 120 m Emax = 1673 kN/m ko = 0.594

4.4.2 Results of the FDM-analysis – Disturbance condition The results of the analysis of disturbance Conditions I and II are presented in Table 4.4.2. (LN,max = distance to Nmax from the point of application of the additional load).

In situ condition, sR/s = 1.0

Global failure, sR/su = 0.35

f = 7.5%

f = 4%

f = 2% cr

cr

cr0.5

In situ condition

Disturbance condition II

Global failure [m]

[kN/m2]

Disturbance condition II, sR/su = 0.6

su

0.3

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Table 4.4.2 Results from FDM- analysis – disturbance conditions.

Disturbance Condition I – Force initiated failure

sR/su = 0.80 Ncr = 275 kN/m Lcr = 140 m Emax = 1748 kN/m LN,max = 0 m

Disturbance Condition II – Force initiated failure

sR/su = 0.60 Ncr = 192 kN/m Lcr = 114 m Emax = 1665 kN/m LN,max = 0 m

cr = 0.145 m

Disturbance Condition IIa – Deformation initiated failure

sR/su = 0.60 Ncr = 0 kN/m Emax = 1770 kN/m LN,max= 50 m

instab = 0.292 m Linstab = 162 m (As defined in Figure 3:2.2) The critical load (Ncr), sufficient to initiate local failure in the steepest part of the slope amounts to 275 kN/m in Condition I and 192 kN/m in Condition II.

Although the Surte slide is not believed to have been triggered by the weight of a newly applied fill, it may still be of interest to note that – assuming totally un-drained conditions – the computed value of Ncr in disturbance Condition I would correspond to a rapidly ap-plied overload of only qcr 275/18 = 15.3 kN/m2 extending 18 m in the slope direction. In disturbance Condition II, the same overload would be qcr 10,6 kN/m2. (Cf Figure 4.4.2 ). By contrast, ideal-plastic failure analysis (I-PFA) based on local slip surfaces such as ABC in Figure 4.4.2 indicates a corresponding value of qcr 68 kN/m2 – a difference that can be expressed by a factor of about 4.4 in disturbance Condition I. (Cf. Appendix B) Note: It is important to observe in this context that if the additional load is applied slowly (i.e. if conditions are drained or partially drained), progressive failure analysis would also result in con-siderably higher values of qcr.

Condition I: sR/su = 0.80 qcr (Pr FA) = qcr (ABDF) 15.3 kN/m2 << qcr (I-PFA) = qcr (ABC) = 68 kN/m2 < qcr (ABDE) 118 kN/m2

Condition II: sR/su = 0.60 qcr (Pr FA) = qcr (ABDF) 10.6 kN/m2 << qcr (I-PFA) = qcr (ABC) = 68 kN/m2 < qcr (ABDE) 118 kN/m2

Figure 4.4.2 Comparison between progressive failure analysis (FDM) and ideal-plastic failure analysis (I-PFA) with regard to a local distributed critical load qcr, the extension of which roughly equals the depth to the slip surface. (Un-drained conditions are presumed.)

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This important discrepancy between the results from the ideal-plastic equilibrium approach (I-PFA) on one hand, and analyses considering deformations and deformation softening on the other, stands out as the major reason why downward progressive slides of the type dealt with in this report have long eluded convincing explanation based on sound structural mechanics. In the disturbance condition, the computed resistance is mostly related to Stage I (as defined in Section 3.2), i.e. prior to the formation of a discrete failure band or slip surface. At this stage, both the modulus of elasticity and the shear modulus are time de-pendent in a similar way. In consequence, the analysis is not very sensitive to the time fac-tor considering that the ratio of E/G is largely constant and is not likely to vary widely.

However, in order to establish the sensitivity of the analysis to variation of the compressi-bility of the soil mass in the down-slope direction, the effect of doubling the ratio of E/G has been computed and, other conditions unchanged, found to be as follows in disturbance Condition II: (Compare with Table 4.4.1)

sR /su = 0.60 Ncr = 274 kN/m Lcr = 163 m Emax = 1670 kN/m at LN,max = 0 m

Hence, doubling the ratio of E/G brings about increases of Ncr and Lcr by 43 %, while the value of Emax is virtually unaffected. It may be observed that a 43 % increase of Ncr has little impact on the issue highlighted in Figure 4.4.2. (In fact, this issue would basically remain unchanged even for much higher values of the compression modulus of the soil in the slope).

Deformation- induced failure

As indicated in Figure 3.2.2 (Section 3) and explained in more detail in Bernander (2000), there exists a critical value of forced deformation ( instab), which may result in global slope failure, even in the absence of a sustained external force maintaining the failure process. In reality, such a situation can arise when driving soil-displacing piles, in which case no ex-ternally active sustained force will result.

As already mentioned, the Surte slide is for good reason believed to have been triggered by ongoing pile driving for the foundation of a family house at the time of the slide event. Table 4.4.2 gives a critical deformation value of instab 0.3 m in disturbance Condition IIa. However, as the number of piles in the foundation was not sufficient to generate a down-slope movement of this magnitude, it may be concluded that soil displacement as such was not the sole disturbance initiating the Surte slide.

It is thus very likely that the piling activities also induced locally high pore water pressures and loss of strength in possible local seams of coarser moraine out-wash in the clay forma-tion. Such coarse strata commonly intermix with clay sediments in the vicinity of the an-cient shores of the regressing post-glacial seas. (Cf Broms, 1983).

It may be noted in this context that pile driving is not an unusual agent causing slides in soft clays in Sweden. For instance, driving of only 6 concrete piles for a family house re-leased an earth movement involving roughly 15 hectares of ground south of Gothenburg in 1971. (Cf The Slide at Rävekärr, Bernander, 2000). Other examples of this phenomenon exist.

Lastly, regarding deformation-induced failure in disturbance Condition IIa, doubling of the values of the E/G-ratio has a moderate impact on the issue highlighted in Figure 4.4.2. Thus, although Linstab is increased by 41 %, Emax is only raised by 8 %.

sR /su = 0.60 Ncr = 0 kN/m Lcr = --- m Emax = 1911 kN/m at LN,max = 65 m

instab = 0.289 m Linstab= 228 m

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4.4.3 Results of the FDM-analysis – Global failure condition The global failure condition, subsequent to the redistribution (related to progressive fail-ure) of up-slope unbalanced forces to the less sloping ground further down-slope, is shown in Table 4.4.3 and Figure 4.4.3.

Figure 4.4.3, applying to global failure condition Case I in Table 4.4.3 below, displays cal-culated earth pressures, shear stresses and displacements along the slip surface defined by Jakobson (1952) in the Surte slope. The global failure condition illustrated in the figure

Figure 4.4.3 Static earth pressure distribution in the Surte slide subsequent to the progressive failure phase but prior to the slide proper resulting in disintegration and heave in passive fail-ure. The figure indicates that the static forces developed in the progressive phase of the ground movement suffice to explain the spread of the passive zone over almost horizontal ground. (Cf Chapter 5.)

Global failure condition: Case I, suR/su = 0.35-0.20, Eel = 206 su, mean Curve A Eo(x) = In situ earth pressure prior to local failure, kN/m Curve B N(x) = Earth pressure increment due to Pr F redistribution, kN/m Curve C E(x) = Eo(x) + N(x) = Earth pressure after Pr F redistribution (Phase 4), kN/m Curve D EP

Rankine = Passive Rankine resistance, kN/m Curve E o(x) = In situ shear stress distribution before progressive failure, kN/m2

Curve F (x) = Shear stress distribution after progressive failure – i.e. the situation prior to final disintegration in passive Rankine failure, kN/m2

Curve G (x) = Displacement, m

27

represents the situation at the end of the progressive redistribution in Phase 3 (as defined in Section 3.2), in which unbalanced forces in the steeper parts of the slope have been trans-mitted further down-slope, resulting in massive build-up of earth pressures (Phase 4) in more level ground. It should be observed that the earth pressures in this phase are calculated on the assumption that the potentially sliding soil volume transiently retains its geometrical shape before its possible disintegration in passive Rankine failure. This is justified because, as is demon-strated in detail in Chapter 5, the slip surface under the valley floor is already fully devel-oped far beyond the foot of the slope prior to the potential break-down of the passive zone, and is therefore not concurrent with this final dramatic phase. Cf Chapter 5.

Hence, in cases where the resulting maximum earth pressure Emax exceeds EPRankine

(max), the

computed earth pressure scenario will represent a transitory stage that later merges into the dynamics of the slide proper, (i.e. Phase 5 according to Section 3.2).

If, on the other hand Emax does not exceed EPRankine, then the progressive failure redistribu-

tion of earth pressures will only result in moderate ground displacements such as in the ground movement at Rävekärr referred to above, Bernander (2000). Yet, it is important to note that also in this scenario, the failure zone including the shear band will have devel-oped far beyond the foot of the slope into horizontal or less sloping ground.

Table 4.4.3 Global slope failure – results from FDM- analysis

(LE,max = distance to Emax from the point of application of the additional load)

Global failure condition: - Case I Eel/G = 3 suR/su = 0.35-0.20 Nmax= 3112 kN/m Emax = 4969 kN/m EP

Rankine (max) = 3900 kN/m

ERankine/Emax = 0.785 Eel = 206 cu, mean LE,max = 260 m LE>E(Rankine) = 420 m #

Global failure condition: - Case II Eel/G = 3

suR/su = 0.40-0.25 Nmax= 2682 kN/m Emax = 4554 kN/m EPRankine

(max)= 3900 kN/m

ERankine/Emax = 0.856 Eel = 206 cu, mean LE,max = 260 m LE>E(Rankine) = 234 m #

Global failure condition: - Case III Eel/G = 6

suR/su = 0.40-0.25 Nmax= 2682 kN/m Emax = 4558 kN/ m EPRankine

(max) = 3900 kN/m

ERankine/Emax = 0.855 Eel = 412 cu, mean LE,max = 260 m LE>E(Rankine) = 225 m # # LE>E(Rankine) = The length over which passive Rankine resistance is exceeded. Cf Figure 4.4.3.

The significance of the earth pressure distribution in the transient state of equilibrium in Phase 4 is that it constitutes a measure of the disaster that may result if the critical load in the disturbance condition is exceeded. In other words, will progressive failure lead to a veritable landslide, developing considerable heave over vast areas in passive Rankine fail-ure, or only result in moderate displacements? As the formation of the slip surface is not simultaneous with Rankine failure in the passive zone, the study of the transient condition in Phase 4 also provides information about how far under the valley floor the horizontal

28

failure may have propagated. This may well be a matter of hundreds of metres – e.g in the Surte slide about 400 m.

The calculations in Case I in Table 4.4.3 are based on residual shear strengths roughly in proportion to the magnitude of displacement in the progressive failure phase, varying from suR = 0.35·su to suR = 0.20·su in different locations along the slope. Yet, considering the sig-nificant displacements (1 to 3 metres according to the example in Chapter 5) and the rate of stress change involved already in the progressive phase, these values of suR may be consid-ered as being high.

As shown in Figure 4.4.3, the earth pressures resulting from the progressive failure redis-tribution of forces entail that passive Rankine resistance is exceeded over a distance of some 420 m of gently sloping ground including the riverbed. Thus, even not considering minor dynamic effects in the progressive phase, already the static condition leads to total disintegration and heave in the lower areas of the slope and valley. This inevitably results in the final dynamic phase of the slide proper, which in Section 3.2 is defined as Phase 5. (See also the example in Chapter 5.) Note. In the current context Lundström’s speculation, mentioned in Section 4.3 above, regarding a possible final slip-circular slide near the Göta River, may be of interest. In the opinion of the au-thor, the absence of heave near the river did not result from a local slip-circular slide. It was instead related to the fact that in situ earth pressures were locally considerably lower near the river scarp (i.e. close to active pressure) than elsewhere in the valley, in which case the probability of passive Rankine resistance being exceeded was locally considerably less. However, also this explanation, although different from Lundström’s, relates in a way to reduced stability in the vicinity of the riverbed scarp.

Sensitivity studies The effect of changing the suR/su – ratios from 0.35 – 0.20 to 0.40 – 0.25 is evidenced by the numbers given in Table 4.4.3 above. The maximum earth pressure decreases from 4969 kN/m to 4554 kN/m, i.e. by a factor of 0.92, whereas the length of the potential passive zone is substantially reduced from 420 m to 234 m.

However, for values of suR/su > 0.6, the value of Emax no longer exceeds ERankine implying that global failure with excessive heave of the passive zone would not likely take place under such conditions. Instead a ‘Rävekärr type’ of earth displacement would have oc-curred.

The effect on the global failure condition of doubling the Eel,mean/Go- ratio (i.e. reducing the compressibility of the potentially sliding soil mass) is insignificant as far as the maximum horizontal thrust is concerned. The effect on the length of the passive zone in heave is moderate. Thus for suR/su =0.40-0.25 and Eel,mean/Go= 6 instead of 3, the following values result:

Emax becomes 4558 kN/m instead of 4554 kN/m, and

LE>E(Rankine) becomes 225 m instead of 265 m.

4.5 Conclusions from the progressive failure computations The progressive failure FDM- analysis demonstrates

a) … that the critical force (Ncr) corresponding to full mobilization of the shear capacity in the steep part of the slope was remarkably small. (See Table 4.4.2).

b) … that the corresponding limited length of mobilization of shear stress, defined as Lcr in Figure 3.2.2, is conducive to the formation of progressive failure planes parallel to the

29

sloping ground – i.e. in direction A-B-D-F in Figure 4.4.2 – instead of passive failure planes directly to ground surface such as A-B-C and A-B-D-E in the same figure. Or to put it differently, the example emphasizes the issue dealt with in Chapter 5, namely that – in soft clays – slip-circular failures do not readily develop and surface in sloping ground. Cf Bernander (1981b). The considerable discrepancy between the results of the ideal-plastic equilibrium approach on one hand, and the FDM-analyses considering combined deformation and deformation-softening on the other, clearly stands out as the main reason why downward progressive slides of the type dealt with in this report have long eluded convincing explanations based on the conventional analysis. For instance, with suR/su = 0.80, as in disturbance Condition I, the ratio between qcr (ABDF) and qcr (ABC) only amounts to 0,23. (Cf Figure 4.4.2). Figure B.3 in Appendix B also highlights this issue.

c) ... that merely the static redistribution and build-up of earth pressures in the progressive phase of the initial slide are sufficient to make the slide propagate in due course all the way to the Göta River. (See Table 4.4.3 & Figure 4.4.3). However, obviously dynamic forces in the final break-down phase (Phase 5) tend to extend the passive zone and enhance the heave effect. The analyses also emphasize the fact that the different consecutive phases of progressive landslides must be analyzed separately and not as one singular mechanical event. (Cf Section 3.2 e). The Surte slide can, therefore, readily be explained as a fully developed progressive failure of the kind described in Chapters 2 and 3 above. The dynamic phases (Phases 3 and 5) of the slide events may be understood as having been similar to those depicted in a series of figures related to the Tuve slide 1977 in Bernander (2000). Sensitivity analyses based on reasonable variation of crucial parameters all show that, once the initial local failure at the pile driving site had formed, the stability of the entire slope was inexorably lost.*

* (To any reader, who may find the progressive FDM failure analysis made as somewhat arbitrary in view of the different assumptions made regarding the shear/deformation properties of the clay, it may be emphasized that the outcome of the analysis mainly relates to the very fact that the defor-mations inside and outside the soil mass are considered in the analysis. The results are namely re-markably insensitive to moderate variation of properties such as el,, f, cr, and within reason even to the ratios of sR /s , sR /su and suR /su , See Appendices A and B).

At the time of the slide event, construction activities and driving of concrete piles were, as already stated, going on in the steepest portion of the slope (i.e. in Point P in Figure 4.3.2). It is easy to conceive that activities of this nature may very well generate critical distur-bance effects of the trivial magnitude indicated by the computations. The mentioned investigatory reports by Jakobson (1952a) and Caldenius & Lundström (1955) make no reference to any earth fills on the pile driving site, a situation that nonethe-less is not unlikely as such. As shown in Figure 4.4.2, even a minor fill could have been a contributing factor to the initiation of the slide. Anyhow, the presence of such a fill is not likely to have attracted much attention at the time considering that progressive failure was, as it appears, not an important issue in the after-slide investigations. Yet, the direct cause of the Surte slide must be attributed to the disturbance generated by the ongoing pile driving activities in the steep incline at the upper limit of the initial main slide.

30

However, the primary objective of the FDM analysis of the Surte slide made in this report has been to demonstrate the impact of applying an analysis accounting for differential de-formations in the sliding soil mass, and to highlight how a local minor disturbance in a vulnerable part of the slope could actually result in this great disaster, destabilizing about 240000 m2

of land that had remained stable for thousands of years.

One of the main objectives in regional programs for surveying potential landslide hazards must, accordingly, be to identify in-built, latent disasters of this kind. Already, by defini-tion, this cannot be achieved using the conventional perfectly plastic failure approach.

By applying sensitivity analyses, the FDM approach to progressive failure used in this Sec-tion has the potential to reveal slide hazards in slopes of soft sensitive clays.

31

5 Explanation of the spreading of slides over horizontal ground in soft clays

General. As stated above in Section 4.1, both investigations by Jakobson (1952) and Lundström (1955) explicitly searched for an explanation of the phenomenon related to the vast spread of the Surte slide over some 400 m of practically horizontal ground.

The reason for focusing on this issue was of course that – in conformity with documented data – this phenomenon conspicuously stood out as being incompatible with the conven-tional (I-PFA) approach applied based on full plastic response of the clay. Also, in the af-termath of the Surte slide, discussion (1952-1953) in a technical journal (Teknisk Tidskrift) largely focused on this particular point.

In the opinion of the author, the explanations made at the time were, as already stated in Chapter 4, inconclusive for several reasons – the most important one being that deforma-tions in the sliding soil mass were not accounted for – at least not in any strictly analytical or structure-mechanical sense.

L. Andresen & Jostad (2004), Grimstad (I IG-SNRA, Oct. 2007) have applied finite ele-ment analysis (FEM) to failure in long natural slopes of deformation-softening soils. These studies support the concept that a relatively small load in an adverse position in a slope may trigger progressive failure in soft clays in accordance with the FDM-approach (from 1988 – 89) to progressive failure described in Chapter 3 of this report, and which in the current study has been applied to the Surte slide in Chapter 4.

As of now, according to Andresen & Jostad (2004, Figure 5) and personal communication (L. Andresen, Jan. 2008), it appears that the FEM-analyses have not as yet explained the vast spread over virtually horizontal ground that is explicitly predicted by the FDM-method. (Cf Bernander & Olofsson, Ingvar (1981), Bernander & Gustås, (1984), Bernander et. al (1988) and Bernander (2000)).

It may well be that this discrepancy just emanates from the specific structure of the FEM-model used.

In view of the fact that the vast spread of slides over gently sloping ground was actually explained by the mentioned FDM method already some 25 years ago, it may be of interest in the current context to focus on how the said phenomenon can be demonstrated on the basis of the FDM-approach described in Chapter 3.

Hence, in the next section, an example is presented, which clearly accounts for the ten-dency of slides in strain softening clays to involve large areas of practically horizontal ground. The issue is also dealt with in more detail in Appendices A and C.

5.1 Slide propagation over horizontal or gently sloping ground The FDM approach presented in Chapter 3 clearly indicates potential break-down and vast spread over horizontal ground already as a result of static earth pressure build-up – i.e. even without accounting for the possible dynamic effects in the progressive redistribution phase of the slide. (This phase is defined as Phase 3 in Section 3.2).

As opposed to the moderate displacements and relatively slow dynamic activity in Phase 3, which in this context are not attributed major importance, the dynamic effects in the final phase (Phase 5) may of course significantly aggravate the degree of disaster in a slide.

32

Nevertheless, it is of particular interest to note that the progressive failure FDM-analysis emphatically claims that the vast spread of the tongue of the slide on level ground is pre-cisely what should be expected even if only static conditions are considered.

Figure 5.1.1 depicts the foot of a slope, i.e. the transition between an inclined valley side and a valley floor, dipping only 1:100. The depth to the slip surface is assumed in the ex-ample to be 20 m. The laboratory shear strength su of the clay at slip surface level is 25 kN/m2, whereas the peak strength during the slide movement is set at 1.2·su = 30 kN/m2.

Hence, the conditions closely resemble those of the Surte slide.

1:100 1%

H=20 m

C

A B

x m

x m

x m

50 100 150 2000

AcrE

PRE

[kN/m2]

4000

0.1 0.2 0.3

[m]

Eo

AE

SuR

Su

10

20

30

1.2Su

Lcr

=17kN/m3

A

[kN/m]

1.2Su

Figure 5.1.1 Data and notations defining conditions under the valley floor. In the figure Eo = in situ earth pressure at rest. EA

cr = E0 + NAcr 3300 kN/m denotes the earth pressure resis-

tance that can be mobilized, when the residual shear strength sR is just attained at Point A. ER

P = EPRankine 4400 kN/m is passive Rankine earth pressure resistance.

Denotations: Lcr = denotes a critical length beyond which the effect of the additional force Nx (or Ex - Eo,x) is insignifi-cant. (Note: A precise value of Lcr can only be defined by assigning the ratio of Nx/NA

cr a limiting minimum value such as for instance 1/10. The x-axis is an asymptote to the curves of Nx, x and x).

Acr = Deformation at a point A corresponding to NAcr.

Eo = in situ earth pressure at rest. ERP = EP

Rankine = Passive Rankine earth pressure.

sR = Effective residual shear resistance at current rate of loading and prevailing drainage conditions.

33

In Figure 5.1.1, EAcr = Eo + NA

cr denotes the specific earth pressure at which the computed shear stress in point A attains the residual shear resistance suR and where a slip surface has just started to develop after the peak shear stress has been mobilized. The force NA

denotes the additional load at A following progressive failure redistribution of stresses and defor-mations.

It may be noted that, at this stage, most of the potential failure zone (from point A and on-wards) is still in an intact state – i.e. discrete shear bands have not yet developed.

5.2 Computational analysis - Exemplification In Figure 5.2.1, it is shown how the ratio of EA

cr/ERP and the values of Lcr and A,cr vary

with different ratios (sR/su) between the actual residual shear resistance (sR) and the labora-tory shear strength (su). Interestingly, as may be concluded from the figure, all the com-puted values are practically independent of the sR/su – ratio.

This rather surprising circumstance relates to the fact that the area of the integral EA = xdx changes very little with decreasing values of sR, since also the location of the peak

shear stress ( max =1.2·su) is somewhat displaced in the direction of x with falling values of sR/su. In fact, the independency (in the example) of the parameters EA

cr/ERP, Lcr and A,cr

of the ratio sR/su is so pronounced that a theoretical justification for this phenomenon is quite possible.

H = 20 m

C

A B

0.5 0.6 0.8 0.90.4

AcrE P

RE

AE

0.2

Lcr

0.4

0.6

0.8

1.0

1.0

50

150

100

200 AcrE P

RELcr

cr

Lcrm

0.7

x

cr [m]

k=sR/su

Figure 5.2.1 ‘Mobilizable’ earth pressure in Point A in terms of EAcr/ER

P. Displacement cr and critical length Lcr as functions of the ratio k = sR/su. Denotations as in Figure 5.1.1. In the example Eo = 2100 kN/m, EA

cr =3300 kN/m and ERP=4400 kN/m.

34

Thus, for all values of sR/su, the ratio of EAcr/ER

P is constant and in the current case equal to 0.75. The corresponding deformation A,cr induced by the pressure EA

cr = (Eo + NAcr) is

also constant and amounts to about 0.35 m. This implies, according to the assumed stress/strain (deformation) relationship, that an incipient failure surface (or shear band) must already at this stage have developed behind the peak shear stress in the vicinity of Point A. (Cf Figure 5.1.1).

The crucial question at this stage is then: “What happens when the earth pressure EA at

point A (i.e. where x = 0) increases from EAcr = 0.75·ER

P to full Rankine resistance (1.00·ER

P), as a result of for instance earth pressure redistribution due to progressive fail-ure formation further up-slope?”

Figure 5.2.2 shows the full deformation curves at a stage, when EA has attained 0.95· ERP,

i.e. 95 % of the total passive Rankin resistance. In the current example 0.95· ERP 4180

kN/m. 1:100

H=20 m

C

A B

x m

x m

50 100 150 2000

[kN/m2]

1

2

3

RP0.95E

Su 10

20

30

1.2Su max

0.4Su

250

50 100 150 2000 250

0.2 0

[kN/m2]

Su

10

20

30 1.2Su

SR = 3%

1.2Su

R u/ 0.4S S

=17 kN/m3

R u/ 0.6S S

R u/ 0.8S S

R u/ 1.0S S

Lp Lcr

Lp

[m]

Figure 5.2.2 Computed deformation curves for an earth pressure EA = E0 + NA = 0.95· ERP at

varying degrees of deformation-softening as defined by the sR/su – ratio.

35

As opposed to the conditions shown in Figure 5.2.1, the deformation A at A - as well as the extension of the total failure zone (LP = Lcr + LP) – now exhibit pronounced depend-ence of the current value of the ratio of sR/su. In Table 5.2.1 below, LP denotes the growth of LP related to the increase of EA from 0.75· ER

P to 0.95· ERP.

Table 5.2.1 Values of A , LP and LP for EA = 0.95·ERP 4180 kN/m

sR/su A [m]

LP [m]

LP [m]

0.40 2.4 240 120 0.60 1.9 200 80 0.80 1.4 160 40 1.00 1.2 120 0

It is important to note that already at this point – before full passive Rankine resistance is attained – the horizontal deformation at A is 1.2 m already for sR/su = 1.0, i.e. even in a context, where there is no deformation-softening at all.

However, according to any of the stress/deformation relationships assumed, deformations of this magnitude are, in soft clays of this kind, bound to develop additional strain-softening leading to more deformation that, in turn, produces increased strain-softening and so on. This means that, even in clay with moderate loss of strength at large displace-ment, the horizontal-bound failure zone A-B will propagate far into level ground. For in-stance, for sR = 0.4su and A 2.4 m, the failure zone has according to Table 5.2.1 reached a length of some 240 m at a stage, when full passive earth pressure resistance has not yet been attained.

Or stated somewhat differently, the failure zone with its slip surface (A-B in Fig. 5.2.2) has already developed over a length of 240 m, before there is any possibility of significant de-formation along slip surfaces related to Rankine failure emerging at ground level – i.e. slip surfaces such as A-C in Figures 5.1.1, 5.2.1 and 5.2.2.

Finally, a third way of framing this issue is just to establish that the formation of the hori-zontal failure zone along A-B cannot - in deformation-softening soft clays - in principle ever be simultaneous with the final collapse of passive zone in passive Rankine failure.

Although the outcome of the studied scenario to some extent depends on the prevailing in situ earth pressure situation, the analyzed example clearly indicates that large spread of passive failure zones over gently sloping ground is not related only to so-called ‘quick clays’. Extensive spread of this sort can – once a progressive failure has been released – readily occur in any soft clay of normal sensitivity. This was, for instance, the case both in the Surte and the Tuve slides, where the clays forming the valley floor were not particu-larly sensitive by Scandinavian standards – i.e. St = su/sur = 10 to 15. (See Figure 4.1.1).

Another sensitivity study is presented in Appendix B. The critical size of an up-slope trig-gering load load qcrit is there studied as a function of the slope and the residual shear re-sistance cR/clab. The results are illustrated in Figure 5.2.3.

36

Figure 5.2.3 Critical up-slope triggering load qcrit as function of slope and residual shear resistance cR/clab. Note that the critical load (qcrit) is relatively little affected by the degree of strain-softening - especially for steeper values of the slope gradient. The diagram also accentuates the acute hazard in respect of progressive failure related to local up-slope fills and embankments. See further Appendix B.

5.3 Conclusions The scenario outlined above explains why progressive landslides in soft clays tend to propagate far over gently sloping or horizontal ground as shown both in Figure 4.4.3 with reference to the Surte slide and in Figure 5.2.2.

The formation of a horizontal failure zone cannot - in deformation-softening clays - be concurrent with the collapse of the passive zone in Rankine failure for reasons given.

In other words, before there is any chance of passive failure forming at the foot of a slope, the entire horizontal failure zone (i.e. including the shear band) will already have devel-oped far out under the valley floor, causing already at this stage substantial horizontal dis-placement of the ground, and that even in cases when passive Rankine earth resistance has not been exceeded. The latter phenomenon is clearly evidenced by the slide at Rävekärr (1971) some 10 km south of Gothenburg, where about 15 hectares of ground were dis-placed 0.2 to 0.3 m without the incidence of break-down in passive failure, Cf Bernander (2000). This implies in turn that, in fully completed slides, important horizontal dis-placements are bound to occur also outside the area featuring passive failure and heave, i.e. far outside what is normally conceived as the slide area proper.

Landslide spread over practically horizontal ground is further exemplified in more detail in Appendices A as well as on the Excel spread sheet in Appendix C. The hazard in respect of progressive failure related to local up-slope fills and embankments as illustrated in Figure 5.2.3 is especially discussed in Appendix B.

qcrit

kN/m2 clab = 25 kN/m2

cpeak = 30 kN/m2

gf = 3 %

gel = 1.09 %

0.2 0.4 0.6 0.8 1.0

cR/clab

10

20

25 H = 20 m Slope n:100

qcrit

tan = 0.05

tan = 0.06

tan = 0.07

tan = 0.08

5:100

6:100

7:100

8:100

= 16 2

37

6 Brittleness related to geometry – geometric sensitivity

6.1 General Extensive landslides in the soft clays of Scandinavia are often, in general terms, by Swed-ish geotechnical engineers ascribed to the presence of so-called ‘quick clays’, i.e. clays with a sensitivity number St > 50. The value of St is defined as the ratio of undisturbed shear strength to remoulded (i.e. excessively stirred) shear strength st = su/su,remoulded = su/sur > 50.

However, the implications of sensitivity (or ‘rapidity’) in terms of strain softening in an actual incipient failure zone in situ, are not well established. The laboratory sensitivity numbers (St) are not directly transferable to the true strain softening response of clays un-der arbitrary conditions of loading and drainage. In this report, this lack of compatibility is defined by differentiating between the parameter sur on one hand, and the residual strength parameter sR (or suR) on the other.

Hence, while sensitivity certainly is an important factor in the initiation and development of progressive slope failures, the geometry of the ground surface, and in particular the con-tour of firm bottom in the upper portion of the slope, can be of equal (in cases may be greater) significance to failure formation.

As demonstrated in Chapter 5, spreading over horizontal ground can readily take place in clays of moderate sensitivity. In Section 6.2 below, an example is presented exemplifying the particular effect of geometry on landslide hazards.

6.2 Geometric sensitivity - exemplification Figure 6.2.1 depicts two slopes having different geometry in one singular respect, namely the way in which firm bottom* – and/or the potential failure zone – varies between the foot and the crest of the slope. (*It may be noted that, especially in the upper part of a slide, the failure surface tends to follow firm bottom or some specific sedimentary layer.)

In Slope A, the potential failure zone slopes linearly, whereas in Slope B it varies as a 2nd degree parabola. Otherwise, all relevant data – such as the mean gradient from crest to foot, the depth of the slip surface below ground level – as well as all material parameters are identical.

It may seem obvious that Slope B harbours greater risk potential than Slope A, but it must be born in mind that according to conventional limit state ideal-plastic failure analysis (I-PFA) the safety factors Fs

OAC for failures along OAC are actually identical for both slopes. (Cf Table 6.2.1).

Admittedly, the safety factor FsODE in slope B is lower by a factor of 1.15 but, as pin-

pointed in Chapter 5, failure in deformation-softening clays are extremely unlikely to sur-face in sloping ground. (Nevertheless, as the safety factor Fs

ODE in slope B amounts to 1.43, i.e. considerably more than 1, this issue is in any case not significant in the current context.)

As also the deformation-softening parameters are identical for both slopes, all differences between the slopes shown in Tables 6.2.2 to 6.2.4 are exclusively attributable to the differ-ence in geometry.

38

H=20 m C

A B x [m]

100 200 300 4000

crEF

500 600

x, L

+21

+41

O ±0

+20

qcr = 34.5 kN/m2

Slope A

H=20 m

C

A B x [m]

crEF

x, L

+21

+41

O ±0

+20

qcr = 0 kN/m2

100 200 300 4000 500 600

E

D

Slope B

Figure 6.2.1 Two slopes with varying geometry. Slope A is a linearly descending surface. Slope B is a parabolically descending surface. The slopes A and B are identical in every other respect except as to how the potential failure zone varies between the foot at x = 300 m and at the crest of the slope at x = 0 m.

In situ conditions Ideal-plastic equilibrium (conventional) analysis (I-PFA)

Table 6.2.1 Results in accordance with conventional I-PFA analysis

Slope A FsOAC = 1.64 Ncr = 1537 kN/m qcr = 77 kN/m2

Slope B FsOAC

= 1.64 Ncr = 1537 kN/m qcr = 77 kN/m2

“ (FsODE = 1.43) (Ncr = 1030 kN/m) qcr = 52 kN/m2

Progressive failure analysis

Table 6.2.2 In situ state (drained conditions) s R/su = 1.0

E0 Ncr Ecr qcr Exmax ER

p/Exmax Lcr x=0

kN/m kN/m kN/m kN/m2 kN/m 1 m m

Slope A 24000 690 3090 34.5 3395 1.24 340 --

Slope B 24000 0 2400 0 3159 1.33 340 --

39

Notably, Slope A can sustain a crest load of qcr = 34.5 kN/m2 roughly corresponding to about 2 meters of earth fill, whereas Slope B cannot carry any additional load at all on its crest, i.e. qcr = 0 kN/m2.

Disturbance condition - Progressive failure analysis If, due to disturbance of some sort, sR/su approaches a value of 0.9, both slopes are still inherently stable with their respective crest loads but the safety factor in terms of ER

p/Exmax

is about 1.166 higher in Slope A than in Slope B in spite of its higher crest load. Slope B, on the other hand, is actually on the verge of breaking down in passive Rankine failure.

Table 6.2.3 Disturbance condition (partially drained). sR/su = 0.9 su = 25 kN/m2,

ERp

= 4200 kN/m

E0 Ncr Ecr qcr Exmax ER

p/Exmax Lcr x=0

kN/m kN/m kN/m kN/m2 kN/m 1 m m

Slope A 2400 690 3090 34.5 3466 1.21 377 0.922

Slope B 2400 0 2400 0 4040 1.04 360 0.823

Global failure condition - Progressive failure analysis In the current case, we may for instance assume that, on account of poor drainage condi-tions in the failure zone and/or rapid load application, the ratio of suR/su adopts a value of 0.6 in the incipient failure zone under and near the crest load. Under such conditions pas-sive Rankine resistance (ER

p) will be exceeded in both slopes causing extensive failure by spreading.

Table 6.2.4 Failure condition (un-drained) suR/su = 0.6 su = 30 kN/m2

ERp=4400 kN/m

E0 Ncr Ecr qcr Exmax ER

p/Exmax Lcr x=0

*

kN/m kN/m kN/m kN/m2 kN/m 1 m m

Slope A 2400 690 3090 34.5 5305 0.83 540 3.52

Slope B 2400 0 2400 0 5376 0.82 490 2.86

* Deformation at x = 0 prior to the incidence of passive Rankine failure.

In the failure condition, the two slopes will behave in similar ways. However, regardless of the presence of the considerable crest load in Slope A, the limiting resistance ER

p is tran-scended for 150 < x < 355 – i.e. over a distance of ‘only’ 205 m, while in Slope B the cor-responding values without crest load amount to 75 < x < 320 and 255 m respectively.

Hence, the spread over horizontal ground (i.e. the potential hazard) is more pronounced in Slope B, and that despite the absence of crest load (qcr = 0).

In the example described above, we come up against a troublesome aspect of all brittle and progressive failures mechanisms in clay formations, namely the fact that deformation-softening as such inevitably results in additional deformations, which in turn generate more deformation-softening and so on……… until finally – if the series of ‘cause and effect’

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has no limit – ends in total collapse. Conversely, if such a limit does exist, the disturbance will only result in creep or moderate down-slope displacements, and no proper landslide will occur.

Pending future research on this topic, a way of dealing with the mentioned problem is to perform sensitivity studies based on reasonably realistic or probable disturbance agents and anticipated degrees of deformation softening based on existing geotechnical knowledge and experience.

Then, in case a situation appears to be critical, effective measures designed to improve stability may be considered – in principle as the safety-catch is applied to a gun, when it is not meant to go off. Analyses of cost and benefit are then of course cardinal considera-tions.

By contrast, if only the conventional plastic failure approach is resorted to, no reliable pre-diction at all of risk can actually be made in long slopes of soft sensitive clays.

6.3 Conclusions In the example presented, the same deformation-softening properties have been applied to both Slopes A and B. Resulting considerable deviations in the outcome of the analysis are therefore solely attributed to the effects of geometry.

The salient conclusion that must be drawn from the example is that the geometric features of a slope constitute a most important factor, which – depending on current conditions – may well overshadow the question as to whether a clay is regarded as being ‘quick’ (sensi-tivity > 50) or more normal with sensitivities in the range of e.g. 15-20.

The report from the Independent Investigatory Group of the Swedish Road Administration (I I G RA, 2007, Section 9.4), referred to in Chapter 1 above, recommends that the safety factor computed in the conventional manner should be raised by 10 % when ‘quick’ clays are present.

As may be concluded from the exemplification made above, it appears highly unlikely that the effects of adverse geometry, in combination with marked deformation-softening, can in a universally applicable way be compensated by increasing the normally defined factor of safety by only 10 %.

The fundamental difficulty in defining a credible and reliable additional raise of the con-ventionally determined safety factor in order to cover hazard due to the presence of ‘quick’ clay arises already from the fact that the definitions of risk in progressive failure analysis and that based on conventional plasticity (i.e. the way these safety factors are formulated) are incompatible. Cf. Bernander (2000), Section 3.32.

Appendix B deals with the magnitude of the local up-hill surcharge that is likely to initiate progressive failure under diverse conditions. The results of the analyses are put together in Table B.II and Figure B.3. Table B.III demonstrates clearly that there is no fixed, constant relationship between safety factors based on conventional plastic limit equilibrium analy-ses and those defining risk of progressive failure hazard. Hence, the aim of establishing a generally valid and reliable raise of the conventionally computed safety factor with the intention of compensating jeopardy related to quick clay is not likely to be successful.

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7 Final conclusions and discussion

7.1 The Surte slide The primary objective of the FDM analysis made in Chapter 4 of the Surte slide has been to demonstrate – using an analysis accounting for differential deformations in the sliding soil mass – how a local disturbance in a sensitive part of a slope could actually result in this great disaster, destabilizing about 240 000 m2

of land that had remained stable for sev-eral thousand years.

However, it can be concluded from Chapter 3 that progressive failure formation may not unavoidably result in extensive and catastrophic landslides. The decisive criterion is that the prevailing in situ stress ( o) remains in excess of the residual shear strength sR in the steep portion of the slope during the initial failure process - i.e. during the Phases 2 & 3 of a slide as defined in Chapter 3.

Generally, progressive landslides of the type subject to study in this report cannot be dealt with as one singular, continuous, mechanically static event. This is due to the fact that a slide of this kind actually consists of a series of non-simultaneous consecutive phases of both static and dynamic instability. The different phases are characterized by radically dif-fering conditions in respect of the nature of disturbance load and rates of load application, as well as of the response of soft clays to time and drainage factors. The conclusions from the results of the FDM-computations as regards the Surte slide are presented in more detail in Section 4.5, to which reference is made. Briefly, the analysis indicates - that the critical force (Ncr) corresponding to full mobilization of the shear capacity in the steep part of the slope was strikingly insignificant.

- that the corresponding limited length of mobilization of shear stress, defined as Lcr in Chapter 3, is conducive to the formation of failure planes parallel to the sloping ground, as demonstrated in Figure 4.4.2, rather than to passive circular failure planes directly to the surface of the sloping ground ahead. Failure modes based on circular slip surfaces have little relevance in long slopes of soft sensitive clays. This condition bears the serious im-plication that, what in e.g. road construction is intended to be a supporting embankment, may itself involve risk of causing a far more serious slide than that meant to be avoided by placing the supporting fill.

- that merely the static redistribution and build-up of earth pressures in the progressive failure phase of the initial slide were sufficient to make the slide propagate all the way down to the Göta River. (Figure 4.4.3). - that the different consecutive and non-simultaneous phases of progressive landslides must be analyzed separately and not as one singular mechanical event. ( Section 3.2 e). Hence, the results of the progressive FDM-analysis reveal unambiguously that neither ex-tremely high artesian pressures nor the effects of kinetic energy – as was concluded in post-slide investigations at the time – were necessary prerequisites for the formation of the large passive zone, extending some 400 m over almost horizontal ground.

Slides of this kind in Scandinavian soft clays are mostly triggered by local human activities such as the placing of earth fills, pile driving, heavy vibratory soil compaction, disturbance of prevailing hydrology etc – at times, but not necessarily, in combination with spells of

42

heavy rainfall. The influence of rainfall in this context is most likely due to excess water pressure acting in cracks forming in the active zone - i.e. cracks actually originating from the disturbing load itself.

Thus, large areas of inherently stable ground may be subjected to extensive landslides trig-gered by un-drained response due to some seemingly trivial local disturbance agent.

The main cause of the Surte slide is attributed to the disturbance generated by the pile driv-ing activity in the steep incline at the upper limit of the initial main slide*. A consequence is then that if – hypothetically – there had been no piling operations in 1950, the slope may have remained to this day in the state, in which it had existed during thousands of years. * Driving of prefabricated concrete piles is in Sweden recognized as a notorious triggering agent in connec-tion with slides in soft clays. A most spectacular example of this is the mentioned slide movement at Rävekärr (1971), some 10 km south of Gothenburg, where driving of only six concrete piles for a family house caused a 550 m long and 0.3 m wide crack (local off-set and displacement), involving movements of about 0.2 m in some 150 000 m2 of ground. (Cf Bernander, 2000).

Unprecedented types of loading, inducing un-drained response in soft deformation-softening clays, may thus be conducive to the initiation of progressive failure in a basically stable natural slope, whereas the very same slope may remain permanently stable if undis-turbed or subjected to other less concentrated, or protracted additional load. Thus, although the Surte slope rose from the Littorina Sea some 3000 years ago, pile driving operations for a family house in a sensitive part of the slope – concealing adverse properties in respect of soil strength and ground geometry – triggered the enormous slide event.

A slope in this condition may therefore be thought of as a ‘time bomb’ ticking through the millennia, waiting some day to go off. A main objective in a regional programme for sur-veying potential landslide hazards must, accordingly, be to identify in-built, latent disasters of this kind. Already, by definition, this cannot be achieved using the conventional ap-proach based on full plasticity of soft clays. Evidently, at least in urban areas, the issue calls for more advanced analysis than that based on the concept of ideal-plastic limit equi-librium.

The proper approach to predicting events of this nature is by thorough geotechnical inves-tigation and subsequent application of progressive failure analysis, which has the potential of identifying latent hazards of this kind.

7.2 Slide propagation over horizontal ground Many of the finished landslides in western Sweden and Norway feature large areas of gen-tly sloping ground having collapsed and heaved substantially as a result of exceeding pas-sive Rankine resistance. This phenomenon has been a vexed issue in many post-landslide investigations. As mentioned in Chapter 4, both Jakobson (1952) and Caldenius & Lund-ström (1956) explicitly searched for an explanation of the extensive propagation of the Surte slide over some 400 m of more or less horizontal ground, and the point in question recurred later in the investigations of the Tuve slide (1977). In fact, the crux keeps turning up in more recent studies of slides in soft clays. It is sometimes argued that the spread and final breakdown of the passive zone in these slides is of little interest because, as one might say,….“these phases of a slide are anyway beyond control and practically anything may then happen”. Nevertheless, it is important to recognize that the mechanisms in progressive failure forma-tion in the phase immediately prior to disintegration in passive failure (i.e. Phase 4 accord-

43

ing to Section 3.2), are in principle identical to those in the incipient failure phase, and can therefore also be investigated on a rational basis.

Hence, the analysis of this stage of a landslide (Phase 4) should not be regarded as super-fluous, since after all, it is the prevailing earth pressure distribution in this phase of an on-going slide that finally conditions the degree of disaster. The Surte slide studied in Chapter 4 epitomizes the release of immense potential energy as well as the massive build-up of static and dynamic forces associated with slides in defor-mation softening soils. The FDM-analysis outlined in Chapter 3 thus explicitly predicts and explains the remarkable phenomenon of a soil volume of some 160 000 m2

x 20 m ( 3 200 000 m3) being squeezed and plasticized in passive Rankine failure to the point of raising the ground level by 2 to 5 m over a distance of 400 m – and that even without con-sidering the dynamic effects in the progressive failure phase.

In Chapter 5 as well as in Appendix A, it is demonstrated that already the impact of static forces, generated by progressive failure, is sufficient to explain the propagation of slides over almost horizontal ground. Moreover, the analyses highlight that this spread can read-ily take place in clays of normal sensitivity, and does not – as is often claimed – require the presence of so-called ‘quick clays’.

The examples studied in Chapter 5 and in Appendix A also demonstrate that the failure zone and the slip surface (the shear band) tend to develop far out under the valley floor before any collapse of the passive zone in Rankine failure can take place. The formation of the extensive slip surface and the break-down of the passive zone are simply not concur-rent processes. For instance, in the mentioned slide movement at Rävekärr, the time lapse between the two failure events is infinitely long. (Cf Note * above in Section 7.1). This phe-nomenon also corroborates the conclusion made in Section 7.1 that failure along circular slip surfaces does not readily develop in sloping terrain of strain softening soil.

7.3 Brittleness related to geometry In the example presented in Chapter 6, the same deformation-softening properties have been applied to two slopes, which are identical except in one singular respect – namely the way in which the potential failure zone and the ground surface – vary between the upper and lower slide limits. Resulting deviations in the outcome of the analysis are therefore solely due to the effects of this difference in geometry. Yet, limit equilibrium analysis (I-PFA) based on full plasticity results in the same safety factor for both of the studied slopes.

The salient conclusion that must be drawn from the example is that the geometric features in slopes of soft clays constitute a most important factor, which – depending on current conditions – may well overshadow the question as to whether a clay is ‘quick’ (sensitivity > 50) or not.

The example shows that, as the limit equilibrium I-PFA analysis takes neither deforma-tions nor details in slope geometry into account, it should not be used in the assessment of potential landslide hazards of the kind dealt with in this study.

The report from the Independent Investigatory Group of the Swedish Road Administration (I I G RA, 2007, Section 9.4) referred to in Chapter 1 above, recommends that the safety factor, computed in accordance with the conventional ideal-plastic equilibrium approach, should be raised by 10 %, when ‘quick’ clays are present.

However, the exemplification carried out in Chapter 6, as well as the sensitivity analyses in Appendix B, indicate clearly that it is hardly likely that the effects of adverse geometry –

44

in combination with marked deformation-softening – can be compensated in a general way by increasing the safety factor, as defined conventionally, by only 10 %. (Cf. in particular Table B.III).

Furthermore, the impact of time is a crucial factor to be considered in this context.

Instead, progressive failure analysis is recommended for the investigation of any slope in soft clays, where there is a potential for extensive slides of more than 50 to 100 m in length – depending largely on the depth to probable slip surface. This applies particularly in situa-tions, where loss of life and serious destruction of economic assets may result from a po-tential slide event.

7.4 Final assessments Landslide hazards in long natural slopes of soft sensitive clays may – on a strict structure-mechanical basis – only be reliably dealt with in terms of progressive failure analysis con-sidering deformation and strain softening. The FDM analyses performed in this report pinpoint the fact that, at least in the present State-of-the-Art, a landslide in soft clays cannot be dealt with as just one singular event of static nature. A slide of the current kind actually consists of a series of different consecu-tive phases of static and dynamic instability, which are not simultaneous. In addition, these phases are characterized by markedly different conditions in respect of types and rates of loading, as well as of the differing response of soft clays to time and drainage factors. In order to be able to make fully reliable predictions of the impact of locally applied dis-turbance agents – capable of triggering global slope failures – it is imperative to make adequate assessments of the effective residual shear resistance (sR) that can be mobilized in a potential zone of local failure. Well-founded values of the shear strength sR can then be established if the current rate of applying the destabilizing force (or the disturbance) is defined. In addition, the prevailing drainage conditions in the investigated failure zone must be considered. However, as demonstrated in the Appendices A and B, varying of the constitutive relation-ships does not alter the phenomenological character of the fracture mechanics outcome of the FDM-analysis. The overwhelmingly decisive factor in the current context is the fact that the deformations within the potentially sliding soil mass are taken into account. Future research in this field of geotechnical engineering is urgently required if we really aspire to make adequately accurate assessments of landslide hazards in slopes of the kind subject to study in this report. Pending the results from such research, geotechnical engineers will - using the progressive failure approach - have to resort to sensitivity analyses based on existing geotechnical knowledge and available experience. As shown in Table B.II and Figure B.3 (Appendix B) the critical triggering load is remarkably insensitive to the sR/su-ratio in steeper slopes. This implies that an adequate safety factor, based on the advocated concept, may well cover the hazard related to uncertain knowledge about deformation-softening. Even if such a procedure may seem imprecise, doing so will in any case provide better prediction of landslide hazards in long slopes of soft clay than the application of the con-ventional equilibrium method, based on perfectly plastic behaviour of the clay material.

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8 References In this chapter references are presented to literature dealing with progressive landslides, slope stability and, in some instances, also to fracture mechanics of structures trying to bridge the gap between traditional soil mechanics and structural analysis. Not all of the references are cited in the report.

Aas, G. (1966). Special Field Vane Tests for the Investigation of Shear Strength of Marine Clays. Report, Norwegian Geotechnical Institute, Oslo. Aas, G. (1981). Stability of Natural Slopes in Quick Clays. Proc. 10th ICSMFE, Stockholm, 1981. Aas, G. (1982). A Method of Stability Analysis applicable to Natural Slopes in Sensitive and Quick Clays. Proc. Symposium on Landslides. Linköping, Swedish Geotechnical Inst. Report No 17. Alén, Claes (1998). On probability in Geotechnics. Random Calculation Models Exempli-fied on Slope Stability Analysis and Ground - Superstructure Interaction. Doctoral Thesis, Chalmers University of Technology, Gothenburg. Andreasson, L. (1978). Tuveskredet, Väg- &Vattenbyggaren, Nr 1, 1978.

Andresen, L & Jostad H. P. (2002). A constitutive model for anisotropic and strain-softening clay. Proc. Numerical Modelling in Geomechanics – NUMOG VIII, Rome, pp. 79-84.

Andresen, L & Jostad H. P. (2004). Analyses of progressive failure in long naturals slopes. Proc. Num. Mod. Geomech. – NUMOG IX, Ottawa, Canada. Bazant, Zdenek P and Planas, Jaimes (1988): Fracture and Size Effect in Concrete and Other Quasibrittle Materials. 616 pp. CRC, Boca Raton, Fl, ISBN 0-8493-8284-X

De Beer, E. & van Impe, W. (1984). Landvallen in Loop-kleien. Tijdschrift der Openbare Werken van Belgie, Nr 1.(In Flemish) Berg, G. (1981). Tuveskredet 1977-11-30, Inlägg om skredets orsaker. (Comments on the Causes of the Tuve Slide). SGI Rapport nr 10. 1981. (In Swedish.) Bergfelt A. (1981) Tuveskredet 1977-11-30, Inlägg om skredets orsaker. (Comments on the Causes of the Tuve Slide). SGI Rapport nr 10. 1981. (In Swedish) Bernander, Stig (1978). Brittle Failures in Normally Consolidated Soils. Väg- & Vatten-byggaren, No 8- 9, pp 49-52. Bernander, S. (1981a). Active Pressure Build-up – A Trigger Mechanism in Large Land-slides in Sensitive (Quick) Clays. Technical Report 1981:49T, Luleå University of Tech-nology, Sweden. Bernander, S. (1981b). ‘Icke-lineär’ deformationsanalys vid beräkning av släntstabilitet – Är det en nödvändighet eller en onödig komplikation? (Non-linear deformation analysis – Is it a necessity or a just a redundant complication? In Swedish). Teknisk Rapport 1981:68T, Luleå University of Technology, Sweden. Bernander, S. & Olofsson, Ingvar (1981a). Tuveskredet 1977-11-30, Inlägg om skredets orsaker. (Comments on the Causes of the Tuve Slide. In Swedish). SGI Rapport nr 10. 1981. Bernander, S. & Olofsson, Ingvar (1981b). On Formation of Progressive Failures in Slo-pes. Proc. 10th ICSMFE, Stockholm1981.

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Bernander, S. & Olofsson, I H. (1981c). The Landslide at Tuve, Nov. 1977. Technical Re-port 1981:48T, Luleå University of Technology, Sweden. Bernander, S. & Svensk, I. (1982). On the Brittleness of soft Clays and its Effect on Slope Stability. Väg- och Vattenbyggaren No 7 – 8, 1982. (In English). Bernander, S. (1983). Angående Statens Geotekniska Instituts Rapport Nr 18 om Tuvesk-redet 30 nov. 1977. Geotekniknytt, Institutionen för Geoteknik, Chalmers tekniska Högsko-la. (In Swedish). Bernander, S. (1984). Relationship between the Appearence of a Finished Slide and the Mechanisms Acting during the Slide. Proc. Nordic Geotechn. Meeting, Linköping, Bernander, S. & Gustås, H. K. G. (1984a). A Dynamic Study of Downward Progressive Failure in a Natural Slope. Proc. Nordic Geotechn. Meeting, Linköping, Sweden. Bernander, S. & Gustås, H. K. G. (1984b) Consideration of in situ Stresses in Clay Slopes with Special Reference to Progressive Failure Analysis. Proc. IVth Internat. Symposium on Landslides, Toronto. Bernander, S. (1985). On Limit Criteria for Plastic Failure in Strain-rate Softening Soils. Proc. 11th ICSMFE, San Fransisco, Vol. 1/A/2, pp 397 – 400. Bernander, S. & Svensk, I. et al. (1985). Shear strength and deformation properties of clays in direct shear tests at high strain rates. Proc. 11th ICSMFE, San Fransisco, Vol. 2/B/5, pp 987 – 990. Bernander, S., Gustås, H. & Olofsson, Jan (1988). Improved Model for Progressive Failure Analysis of Slope Stability. Proc. Nordic Geotechn. Meeting (NGM), Oslo 1988. Bernander, S., Gustås, H. & Olofsson, Jan (1989). Improved Model for Progressive Failure Analysis of Slope Stability. Proc. 12th ICSMFE, Rio de Janeiro. Bernander, S. (2000). Progressive Landslides in Long Natural Slopes. Formation, potential extension and configuration of finished slides in strain-softening soils. Licentiate Thesis 2000:16, Luleå University of Technology, ISSN: 1402 – 1757. 16+104+17 pp. Available at http://epubl.ltu.se/1402-1757/2000/16/index.html

Berndtsson, J & Lind, G.B. (1981). Tuveskredet 1977-11-30. Inlägg om skredets orsaker. (Comments on the Causes of the Tuve Slide. In Swedish). SGI Rapport nr 10.1981. Berre T. & Bjerrum, L. (1973). Shear Strength of Normally Consolidated Clays. Proceed-ings, 8th ICSMFE, Moscow. Bishop, A.W. (1967). Progressive Failure with Special Reference to the Mechanism Caus-ing It. Geotechnical Conference, Oslo. Vol. 2, pp 3-10. Bjerrum, L. (1954). Stability of Natural Slopes in quick Clay. European Conf. on Stability of Earth Slopes, Stockholm. Also publ. in Geotechnique No 2, 1954, pp 16-40. Bjerrum, L. (1961). The Effective Shear Strength Parameters of Sensitive Clays. Proceed. Vth ICSMFE. Paris. Bjerrum, L. (1966). 3rd Terzaghi Lecture: Progressive failure in slopes of overconsolidated clays and clay shales. Proc. American Soc. of Civil Eng. Journal of the Soil Mech. & Foundations Division.

Bjurström, G. & Broms, B. (1982). The Landslide at Frö-Land, June 5, 1973. Proc. Sym-posium on Land-Slides, Swedish Geotechnical Institute Report No 17. Linköping 1973.

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Broms, B. (1969). Undergrundens bärförmåga och deformation. (Carrying Capacity and Deformation of Soils. In Swedish) Chapter 322 in BYGG, Huvuddel 3, Konstruktionstek-nik, Stockholm,AB Byggmästarens Förlag, pp 73-86. Broms, B. & Stål, T. (1980). Landslides in Sensitive Clays. Proceedings International Symposium on Landslides, New Delhi, Vol. 2 pp 39-66. Broms, B. & Fagerström, H. (1981). Tuveskredet. Comments on the Causes of the Tuve Slide SGI-Rapport nr 10. 1981. (In Swedish) Bygg (1972). Handboken Bygg. Del 1b. Kap 174 av Hans Fagerström. ”Stabilitet och brottproblem” (Handbook on Building, Part 1b, Chapter 174. In Swedish) Stockholm, Byggmästarens Förlag 1972. 603 pp.

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Gregersen, O. (1981) The Quick Clay Slide at Rissa, Norway. 10th ICSMFE , Proc. Vol. 3, pp 421-426. Stockholm. Also in NGI Publication No 135, Oslo.

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Grimstad, G. Thakur, V. & Nordal, S. (2005). Experimental Observation on Formation and Propagation of shear zone in Norwegian Quick Clay, Landslides and Avalanches. 11th ICFL, Trondheim, Norway.

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Hansbo, S. (1957). A New Approach to the Determination of the Shear Strength of Clay by the Fall Cone Test. Proc. No 14, Swedish Geotechnical Institute. Stockholm. Hansbo, S. & Torstensson, B-A. (1978). Tuveskredet. Delrapport, AB Jacobsson & Wid-mark, Lidingö. Hansbo, S. (1984). Soil Mechanics. (Jordmekanik). Chapter G05 in BYGG - Geoteknik, Liber Förlag, Stockholm. (In Swedish) Hansbo, S. Viberg, N-E. Runesson, K. (1984). Stability and Progressive Failure of natural Slopes. Report from Chalmers University of Technology, ISSN 0347-9226. Publ. 84:8. Gothenburg. Hill, R. (1958). A general theory of uniqueness and stability in elastic plastic solids. Journ. of Mech. Phys. Solids, No 6, pp 236 – 249.

Hillerborg, A., Modéer, M. & Pettersson, P-E. (1976). Crack Formation and Crack Growth by Means of Fracture Mechanics and Finite Elements. Cement & Concrete Research. Vol. 6, pp 773-782. Hutchinson, J.N. (1961). A landslide on a thin layer of quick clay at Furre, Central Nor-way. Géotechnique (11) , 2, pp 69-94. Höeg, K. (1972). Finite element analysis of strain softening clay. Journal of Soil Mech. & Foundation Engineering. Div. 98 (SM1), pp. 43-59.

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Jakobson, B. (1952a). The Landslide at Surte on the Göta River. Proc. No 5, Royal Swed-ish Geotechnical Insitute, Stockholm Jakobson, B. (1952b). Surteskredet. Teknisk Tidskrift No:s 43 & 47 in 1952 and No 6 in 1953. Janbu, N. (1973). Shear Strength and Stability of Soils; the Applicability of the Coulom-bian Material 200 years after the ‘ESSAY’. Norwegian Geotechn. Institute. 47 p. Norwe-gian Geotechnical Society. NGF- foredraget 1973. Janbu, N. (1977). State-of-the-Art Report: Slopes and Excavations. Proc. 9th ICSMFE, Tokyo. Janbu, N., Kjekstad, O. & Senneset, K. (1977). Slide in Overconsolidated Clay below Em-bankment. Proc. 9th ICSMFE, Vol. 2 PP 95-102, Tokyo. Janbu, N. (1979). Failure Mechanism in Quick Clays. NGM-79, Nord. Geotekniker mötet, Helsinki.

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Janbu, N. (1979). Mechanisms of Failure in Natural and Artificial Soil Structures. Intern. Symposium, Oaxaca, Mexico. Proc. Vol. 1. Jansson, M. & Stål, T. (1981). The Landslide at Tuve on November 1977. Swedish Geo-technical Institute, SGI-varia No 56. Jostad, H.P. & Andresen, L. (2002). Capacity analysis of anisotropic and strain softening clays. Submitted to NUMOG VIII; Rome, Italy.

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La Rochelle, P. (1981). Causes and Mechanisms of Landslides in Sensitive Clays with Special Reference to Clays in the Québec Area. Proc. 4th Guelp Symposium on Geomor-phology, Ontario. La Rochelle, P. (1981). General Report. Session 11, 10 th ICSMFE, Stockholm. Larsson, R. (1977). Basic Behaviour of Scandinavian Soft Clays. Swedish Geotechnical Institute, Report No 12. Larsson, R. (1981). Drained Behaviour of Swedish Clays. Swedish Geotechnical Institute, report No 4. Larsson, R. & Jansson M. (1982). SGI 1981. Tuveskredet November 30, 1977. Report No 18. Statens Geotekniska Institut, Sweden. (In English). Lefebvre, G. & La Rochelle, P. (1973). The Analysis of Two Slope failures in Cemented Champlain Clays. Canadian Geotechnical Journal, No 11, 1974, pp 89-108. Lefebvre, G. (1982). Use of Post Peak Strength in Slope Stability Analysis. Proc. Sympo-sium on Land- slides. Linköping, Swedish Geotechnical Institute Report No 17. Leonards, G A. (1979). Stability of Slopes in Soft Clays. Special Lecture , 6th Panameri-can Conference on Soil Mechanics and Foundation Engineering. Lima, Peru. Leonards, G A. (1980). The Sixteenth Terzaghi Lecture, Annual Convention, Hollywood Beach, Florida. Leroueil, S., Collins, G. & Tavenas, F. (1982). Total and Effective Stress Analysis of Slopes in Champlain Sea Clays. Proc. Symposium on Landslides in Linköping, Swedish Geotechnical Institute Report No 17.

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Leroueil, S., Tavenas, F. & Le Bihan, J-P. (1983). Propriètés charactéristiques des argiles de l’est du Canada. Revue Canadienne de Géotechnique, Vol. 20, No 4 pp 681-705. Leroueil, S. (1997). Geotechnical characteristics of eastern Canada clays. Workshop on soft clays. Yokosuka, Japan, 30 pp.

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Appendix A - Exemplification of analysis of landslide spread over practically horizontal ground based on the Finite Difference Method (FDM) A.1. General The FDM-analysis used in this appendix in its present form was first presented in theory and principle by Bernander et al (1988, 1989) at the Nordic Geotechnical Meeting in Oslo (1988), and later at the 12th ICSMFE in Rio de Janeiro (1989). Already at this time, a com-puter program for the analysis of downward progressive slope failure based on these prin-ciples had been applied by the author in practical engineering cases. Yet, the approach to progressive slope failure in view was presented in more detail consid-erably later (in May 2000) in a licentiate report from Luleå University of Technology de-noted 2000:16, Bernander (2000). The existing computer software in HP-Basic was in this context transformed – essentially unchanged – into Windows C++. However, the exemplification of progressive failure analysis given below is carried out in an Excel spread-sheet based on the equations given in the conference papers and in the licentiate report mentioned above. Although applying this Excel program is a lengthy exercise compared to using the fully computerized program in Windows C++, it constitutes a valuable tool for research and educational purposes because – although tedious calculations of complex expressions are carried out by the computer – every cognitive step in the analysis is directly controlled by the operator, who therefore masters the analytical procedure having continual full insight in the computational process. Although the Excel spread-sheet is used here for studying a comparatively simple case, it may be observed that it can be adapted to any variation of the input parameters from one section to another. Hence, the software is - within the chosen framework - applicable to any arbitrary slope geometry. A.2 Objectives The main objective in the current exercise is to demonstrate analytically why downward progressive slides in soft clays tend to spread over vast areas of gently sloping or horizon-tal ground as a result of earth pressure increase at the foot of a steeper slope. The additional force NL may result from earth pressure redistribution due to instability further up-slope, which may also be calculated using the current Excel program. (Cf. Appendix B). = 1:100

x

LN

x=L

Figure A.1 Gently and uniformly sloping ground ahead of the foot of a steeper slope sub-jected to an additional slide-induced force NL. Slope angle in radians.

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For the sake of simplicity, easier interpretation and better understanding of the outcome of the analysis, a uniformly and gently inclining ground area ahead of the foot of a steeper slope is subject to study - i.e. such as the one shown in Figure A.1. Another key point of interest is to investigate how variation of the parameters in the consti-tutive relationship applied in the analysis affects the fracture-mechanical issue - i.e. how do the values assumed, characterizing the shear/deformation curve, affect the final spread, and which parameters are most significant for the outcome. The example presented below serves to demonstrate a calculation procedure based on spe-cific likely laws of soil behavior in shear. However, an advantage of the approach de-scribed is that it can, with due modification, accommodate any predefined shear stress/deformation properties of soil that geotechnical engineers may wish to apply to the situation studied. A.3 Denotations The spread-sheet used was prepared in connection with the mentioned licentiate report Bernander (2000). Some of the denotations in this Appendix are therefore different from those applied in the current main report. Symbols and notations applied in the following spread-sheet analysis g el Deviator strain, (angular strain) at elastic limit gf Deviator strain, (angular strain) at peak shear resistance x , z Horizontal (or down-slope) and vertical coordinates dx, x, dz, z Differentials of the coordinates x and z d(x),N Differential displacement due to axial compression generated by force Nx

d(xn),N Total down-slope displacement in terms axial compression at xn d(xn,z),t Differential down-slope displacement due to deviator deformation at xn,z

d(xn),t Total down-slope displacement [d(shear)] in terms of deviator deformation del Elastic rebound in the un-loading stage Sx, S(x) Post peak slip deformation in the slip surface in relation to the sub-ground SCR S(cR) Post peak slip in failure surface at ultimate residual shear strength cR Sslip Additional slip in shear band when SCR is exceeded

x, (x), x Mean incremental down-slope axial stress corresponding to N tel Shear stress (deviator stress) at elastic limit to, to(x,o), to(x,o) In situ shear stress at the potential failure plane (z = 0) t, t(x,o), t(x,o) Total shear stress at failure plane (z = 0) to, to(x,z), to(x,z) In situ shear stresses in the zone above the failure plane t , t(x,z), tx,z Total shear stress (deviator stress) above failure plane c Peak shear resistance of clay clab Tested laboratory strength of clay cmean Mean shear strength of the soil above the failure plane cR Residual resistance at a post peak slip of SCR in failure plane.

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cR(x,t) Residual shear strength at a point (x) at time (t) E, E(x), Ex Down-slope earth pressure resultant at point x, i.e. (Ex = Eox + Nx) EA

cr Earth pressure resultant in a Point A, where the shear stress tA = cuR or cR

ER Earth pressure at failure according to Rankine

Nx, N(x), E(x) Earth pressure increment due to the additional load Nx=L

dN, ( N) Differential of earth pressure increment N (Nx) over dx, (or x) Nx=L, NL Earth pressure increment at x = L due to e.g. progressive failure formation NCR Value of N when tx = cR is just attained LCR Length of shear stress field mobilized by the force NCR

L = LF – LR Extent of zone subject to passive Rankine failure - i.e. when Ex > ERankine E-mean Mean secant E- modulus in down-slope compression. Gmod, Gel Elastic modulus in shear H, H(x) Height of element, (from slip surface to ground surface)

A.4. Basic equations The basic equations used are defined and described in Section 4 of the Licentiate Report (Bernander 2000) mentioned above, and to which reference is made. The equations are briefly recapitulated below with minor necessary modifications in respect of the denota-tions used in the current context, see Figure A2.a.

Figure A2.a Soil model – denotations G= z· x· . From Bernander, Gustås & Olofsson Jan (1988, 1989). A full description is given in Bernander (2000).

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Equilibrium of an element H(x) b(x) x in the down-slope direction requires that, see Figure A2.a, Change of shear stress Vertical load Down-slope load

N = t(x,o) - t o(x,o) b(x) x – qv(x) b(x) sin (x) x – qh(x) b(x) x ………A.4:1 where qv and qh are possible vertical and horizontal surcharge loads. The in situ shear stress in the potential failure zone may be written as: Gravitational load Hydraulic uplift Change of in situ stress to(x,o) = o

H(x)z dz sin (x) - w DW(x) sin (x) - Eo(x)/(b(x) x) ………A.4:2

(Note: x is positive in the up-slope direction, implying that Eo is negative for decreasing earth pressure in the direction of x.) The axial compression of an element in the x direction at xn may be written as

d(x),N = (N+dN/2) x/ Eel H(x) b(x) ………A.4:3 where (xn),N is the incremental mean down-slope displacement due to the compression of an element of length x. Hence, the total displacement at xn can be written as o

xnd(x),N x Compatibility equation in Stage I (i.e. prior to the formation of shear band) The total mean down-slope displacement ( d(xn),N), to which a vertical element is sub-jected, must be compatible with the shear deformation of the same element relative to the ground below the slip surface, i.e.

d(shear) = d(xn),t = oH(x) t(x,z) - to(x,z) /G(x,z,t) z + S(x,o) ………A.4:4

When g(x,z) < gf , then the last term S(x,o) = 0 The compatibility criterion in Stage I (i.e. txz < c) with regard to down-slope displacement demands that

oxnd(x),N x = o

H d(xn,z),t z ………A.4:5 The constitutive relationship defined by the shear stress/deformation curve is expressed as t(x,z) = (g(x,z),Sx,d(Sx)/d(time)) or inversely, ………A.4:6

(g(x,z)),Sx,d(Sx)/d(time) = t(x,z) ………A.4:6a (Note: d(time) is a differential of time).

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Figure A.2b The down-slope displacement of a soil element must be compatible with the shear deformation of the same element in relation to the sub-ground. Thus, the shear stress t (x,z) being a function of the deviatory strain g(x,z) as well as of the dis-placement Sx in the slip surface, the differential Equations A.4:1 to A.4:6 can be integrated numeri-cally yielding the states of stress, strains and displacements for any chosen mode of mobilizing the resistance to failure propagation - and that in any chosen portion of the slope. See Figure A.3 Compatibility equation valid in post peak Stage IIa - i.e. for values of c > txz > cR When the peak shear strength is attained at the level of the potential slip surface, the com-patibility Equ.A.4:5 is substituted for Equation A.4:5a.

oxnd(x),N x = o

H (d(xn),t - del(x) z + Sx ………A.4:5a where to is defined as before to(x,o) = o

H(x) (z) dz sin (x) - W DW(x) sin (x) - Eo(x)/b(x) dx and del = Elastic rebound in the un-loading phase Sx = Post peak slip in slip surface Compatibility equation valid in Stage IIb - i.e. when residual resistance cR is attained

oxnd(x),N x = o

H (d(xn,z),t - del(x) z + SCR + Sslip ………A.4:5b where SCR = Slip in shear band when residual shear strength cR is attained. Sslip = Additional slip displacement beyond SCR A.4. Constitutive shear deformation relationships The general constitutive relationship x,z = (gx,z, dSx , dSx/dt) in equation. A.4:6, may in the range 0 < g < gf be defined by the inverse expression gx,z = 1 (tx,z) ……A.4:6a

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Approximation

Figure A.3 Shear stress/deformation relationship. The ratio tel/c is assumed to be constant as c varies with the ‘z’ coordinate.

Stage I - Elastic and non-linear ranges below peak shear stress

Elastic rangeIn the range 0 < gx,z< gel (i.e. for 0 < txz < tel), the relationship between shear stress and deviator strain is taken to be linear. tx = G gx or gx = tx/G dgx,z = dtx,z/G ………A.I:1a where G = tel/gel (tel and gel denote shear stress and shear strain at the elastic limit as de-fined in Figure A.3)

Non-linear rangeIn the non-linear range, where gel < gx,z< gf (i.e. for tel < txz < c), the relationship between shear stress and deviatory strain is assumed to be a 2 nd power parabolic relationship with its vertex at point (gf,c), as shown in Figure A.3. Hence tx,z - tel = 2 (c - tel) (gx,z- gel)/(gf - gel) - (c - tel) (gx,z-gel)/(gf - gel) 2 or ………A.I:2 (gx,z-gel)/(gf -g el) 2

- 2 (gx,z-gel)/(gf -gel) + (tx,z- tel)/(c- tel) = 0 …… …A.I:2a The solution to Equation A.I:2a is (gx,z-gel)/(gf -gel) = 1- 1- (tx,z- tel)/(c- tel) 1/2 ………A.I:2b

Equation A.1:2b may be transformed to: (gx,z-gel) = (gf -gel) 1- 1- (tx,z- tel)/(c- tel) 1/2 or gx,z = gf -(gf -gel) 1- (tx,z- tel)/(c- tel) 1/2

………A.I:2c Check: For tx,z

= tel gx,z = gel and for tx,z = c gx,z = gf Q.E.D.

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In order to establish continuity between the two curves at the point defining the elastic limit, the following condition must be satisfied: Gel = tel /gel = 2(c- tel)/(gf -gel) i.e. gel = gf tel/(2c- tel) ………A.I:3 For instance, if gf = 4.0 %, c = 30 kN/m2, tel = 20 kN/m2 then gel = 0,04 20/(2 30-20) = 0.02 = 2.0 % Range: to < tx,z < c and to > telThe difference in deviator deformation from t = t0(x,z) to t = tx,z according to Equ. AI:2c is

gx,z = (gf -gel) 1- (t0(x,z) - tel)/(c- tel) 1/2 - 1- (tx,z

- tel)/(c- tel) 1/2 ………A.I:2d Hence, the total shear displacement in Stage I when to < tx,z < c and to > tel for the soil col-umn the length of which is x becomes

d(xn),t = o H gx,z z

= o H (gf -gel) ( 1- (t0(x,z)

- tel)/(c- tel) 1/2 - 1- (tx,z - tel)/(c- tel) 1/2) z ………A.I:4

where t0(x,z) and tx,z denote in situ shear stress and current stress respectively. Combined elastic and non-linear range - i.e. t0 < tel and tel < tx,z < c When the current stress range spans across the transition point between elastic and non-linear behavior, the expression A.I:4a may be derived from equations A.I:1a and A.I:4. However, in the range tel < tx,z < c (i.e. if t0(x,z)

in Equ. A.I.2d = tel), Equ. A.I:2d changes to (gf -gel) (1 - 1 - (tx,z- tel)/(c- tel) 1/2) ………A.I:2e Range: to < tx,z < c and to < telCombining Equations. A.I:1a and A.I:2e, the total shear displacement in Stage I is

d(xn),t = o H gx,z z

=o H [(tel –t0(x,z))/G+ (gf -gel) (1 - 1 - (tx,z- tel)/(c- tel) 1/2)] z ………A.I:4a

(Note: The parabolic relationship to the power of 2, which is used here may of course be replaced by any other relationship considered appropriate by the investigating engineer. However, the issue has little impact on the results of the analysis.)

Stage II - Post peak range - i.e. for tx,z(max) > t x,z > cR (0 < Sx < ScR)

The post peak drop in shear resistance is here assumed to be linear. (Cf Figure A.3) Deformation from to tx,z(max) (peak shear stress) when to > tel

d(xn),t = o H (gf -gel) ( 1- (t0(x,z)

- tel)/(c- tel) 1/2 - 1- (tx,z(max) - tel)/(c- tel) 1/2) ..A.I.4(tmax) Deformation from to tx,z(max) (peak shear stress) when to < tel

d(xn),t = o H [(tel –t0(x,z))/G+ (gf -gel) (1 - 1 - (tx,z(max) - tel)/(c- tel) 1/2)] z .…A.I:4a(tmax) Elastic rebound – Range: tx,z(max) > tx,z > cR The elastic rebound in an element ( x· z) d(del(x,z)) = -(t(x,z(max) - tx,z)/G z Hence, the total elastic rebound at failure plane due to un-loading from tx,z(max) to tx,z del(x,z) = - o H(t(x,z(max)-tx,z)/G z. (See Figure A.3) where H is the distance from the failure plane to the resultant force N.

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Slip in failure plane (shear band) The post peak shear strength (= mobilized shear stress). tx,o = cRx is now set as a function of S(x) according to Figure A.3 Hence for the interval 0 < S(x)< SCR , according to Figure A.3, we derive for the slip in the failure plane – i.e. where z =0 S(x)/SCR = (c - tx,o)/(c-cR) or S(x) = SCR (c - tx,o)/(c-cR) where S(x) = the resulting slip in the failure plain at point x SCR = the slip at which minimum residual shear strength cR is attained (Check: For tx = cR, S(tx) SCR and for tx = c the value of S(x)= 0. Q.E.D) Hence, when to > tel the total displacement in a vertical element x in terms of shear and slip in Stage II using Equ. A.I.4(tmax) above is : Displacement to tx,z(max) when to > tel

d(xn),t = o H (gf - gel) ( 1- (t0(x,z) - tel)/(c- tel) 1/2 - 1- (tx,z(max)

- tel)/(c- tel) 1/2) –

Rebound tx,z(,max) tx,z Slip deformation c tx,z > cR – (tx,z(max) - tx,z)/G] z + SCR (c - tx,o)/(c – cR) ………A:I:5 By contrast, when to < tel, the corresponding total displacement in terms of shear and slip in Stage IIa, using Equ. A.I.4a(tmax), is: Displacement to tx,z(max) when to < tel

d(xn),t = o H [(tel - t0(x,z))/G+(gf - gel) (1 - 1-(tx,z(max) - tel)/(c - tel) 1/2) –

Rebound tx,z(max) tx,z Slip deformation c tx,z > cR – (tx,z(max) - tx,z)/G] z + SCR (c - tx,o)/(c – cR) ………A :I:5a

Stage IIb - Post residual linear range - i.e. when tx,z = cR (residual) resistance and slip>ScR

In this range, the deformation is exclusively controlled by the axial down-slope displace-ment and thus independent of the value of tx,z as indicated by Equation A.I:5b. For to > tel and tx,o = cR Equation A.I:5 changes to: Displacement to tx,z(max)

d(xn),t = o H [gf-gel) ( 1- (to(x,z) - tel)/(c- tel) 1/2– 1- (tx,z(max) - tel)/(c- tel) 1/2) –

Rebound tx,z(max) cR Slip deformation when tx,z = cR – (tx,z(max) - cR))/G] z + SCR + Sslip.. ………A.I:5b Again if to < tel and tx,o = cR Equation A.I:5a changes to: Displacement to tx,z(max)

d(xn),t = o H [(tel-t0(x,z))/G+(gf-gel) (1 - 1-(tx,z(max)-tel)/(c - tel) 1/2)

Rebound tx,z(max) cR Slip deformation when tx,z = cR – (tx,z(max) - cR)/G] z + SCR + Sslip ………A.I:5c

61

A.5 Spread–sheet analysis of example shown in Figures A.I and A.4 (The slope in Figure A.4 is identical to the one depicted on Figure 5.1.1 in Section 5 of the main report). The full spread-sheet analysis is presented in Appendix C. A.5.1 Shear deformation parameters The following parameters apply in the spread-sheet exemplification.

Table A.I Values of Parameters H = 20 m Density of clay = 16 kN/m3 Gradient of ground ahead of the foot of the steeper slope = 1:100 Peak shear strength, cpeak Laboratory shear strength, clab Elastic limit, tel Residual shear resistance, cR Slip at residual shear resistance, SCR Mean shear strength of soil profile, cmean Deviatory deformation at peak shear, gf Deviatory deformation at elastic limit, gel Gmodulus = tel/gel Emean = 2(1+v)Gmean= 2(1+0.5) 1120

30 kN/m2 25 kN/m2 18 kN/m2 0.4 clab = 10 kN/m2 0.20 m 24 kN/m2 (csurface = 18 kN/m2) 3% = 0.03 1.29% = 0.0129 (from Equ. A.I.3) 1400 kN/m2 3360 kN/m2

Furthermore, the following assumptions are made in the current case: a) The in situ earth pressure Eo is taken to be Eo = 0.8 H2/2 = 0.8 16 202/2 = 2560 kN/m b) The passive Rankine resistance is defined as ERankine = H2/2 + 2c H = 16 202/2 + 2 30 20 = 3200 + 1200 = 4400 kN/m

= 1:100

x

LN

x = L

xN

x1 = L

H = 20 m

L - x1

Figure A. 4 Example applying to soft sensitive clay. Gently inclining ground at the foot of a steeper slope subjected to an additional force NL due to progressive failure further up-slope. (x1 = 0 at x = L, i.e. L = x - x1). Slope angle is in radians. Hence, the additional force N required to reach full Rankine resistance in the studied case is NRankine = 4400 - 2560 = 1840 kN/m (Cf Table A.III) If the force NL at the foot of a steeper slope adopts a value of for instance 3000 kN/m, the total pressure immediately prior to breakdown in passive failure will be Etotal = 2560 +

62

3000 = 5560 kN/m, thus momentarily exceeding ERankine by 5560 – 4400 = 1160 kN/m (Cf Tables A.II and A.III) A.5.2 Computer sequence in the excel spread-sheet analysis. The analysis consists in 2-dimensional integration of the following parameters: a) The shear stresses (tx,z) in soil mass; b) The additional earth pressure resultant Nx due to a force (NL) located at x= L; c) The mean down-slope displacement ( d(x),N) induced by the force NL. The integration process begins at a point defined by x = 0, where the boundary conditions are taken to be known, and where the effects of the force NL are considered negligible. This means that for x = 0, the additional shear stress tx,o = 0, the force Nx = 0 as well as the displacement d(x),N = 0. The 2-dimensional integration then continues up-hill in steps of x (or t), where each individual step involves a ‘trial and error’ process until compatibility between down-slope displacement and shear deformation is attained. The analysis results in the values of the force Nx the shear stress txz and the displacements along the slope corresponding to the additional load NL acting at x = L. However, if the effect of a predefined force (NL = F kN/m), acting in a certain location up-slope at (x1 = 0) is sought, the entire integration procedure will have to be repeated assum-ing alternative locations for the starting point x = 0 - i.e. varying values of the influential length L. Such analysis will therefore normally necessitate repeated trial and error procedures until the correct boundary condition at x = L is attained – i.e. when Nx=L = F. (Note: These computations may appear prohibitively laborious, but using the integrally computer-ized version, a complete trial and error computation may only be a matter of a few seconds in terms of computer time). A.5.3 Results from the example defined in Table A.I In the specific example defined in Figure A.4 and Table A.I the inclinations of the ground surface and the potential failure zone are constant and parallel. Hence, also the depth (H) is constant. The material parameters are also assumed to be identically the same in the area ahead of the foot of the steeper slope. This has the important implication that all values of Nx , tx,z and the displacement d(x),N between 0 < x < L - representing the effects ahead of a force Nx=L, - are independent of the location of the starting point x = 0. Hence, in the diagram for N shown in Figure A.5, every value of Nx indicates the distribu-tion of the parameters (N), (t) and d(x),N, which are valid ahead of an additional external force Nx located at x. The complete Excel spread-sheet based on the parameters given in Table A.I is presented in Appendix C. Some of the results from the spread-sheet computations in Appendix C are summarized in Table A.II below. The diagram for Nx in Figure A.5 indicates for instance a value of NcR = 719 kN/m. (Cf page 87 of the excel spread-sheet). This is the value of NL, when additional slip in the

63

shear band is about to develop, corresponding to the situation when tx,o has just attained the residual shear resistance cR. The length of influence LcR is at this point in the order of 119 m. When Nx = Nx=L = NL adopts a value of 1840 kN/m (=NRankine), the total earth pressure reaches ERankine = 4400 kN/m. The length of influence LRankine before impending passive failure amounts to some 283 m. This distance is therefore a measure of how far the failure zone has propagated already in the progressive Phases 3 and 4 (as defined in Section 3.2 of the main report.) – i.e. prior to the point, when any veritable slide in the normal sense of the word has yet begun. At this stage even maximum displacements are still moderate and in the order of 0.3 m. By contrast, if the force NL adopts a value of for instance 3000 kN/m, the total pressure immediately prior to breakdown in passive failure will amount to Etotal = 2560 + 3000 = 5560 kN/m, thus temporarily exceeding ERankine by 5560 – 4400 = 1160 kN/m. This means according to Figure A.5 that passive Rankine resistance will be exceeded be-tween x = LRankine= 283 m and x = LF = 454 m, i.e. over a distance of L =LF -LR = 171 m resulting in passive heave over at least this length. Cf page 16 in Appendix C. This stage represents Phase 5 (according to Section 3.2 of the main report) forming the actual disastrous slide event. Table A.III below displays the results from a sensitivity study focused on the effect of varying different key parameters in the constitutive shear deformation relationships defined in Figure A.3. For instance, importantly, the residual shear resistance is varied in the table from 33 % to 90 % of the peak value - i.e. from a high degree of strain softening to almost ideal plastic conditions. Table A.II below illustrates how the parameters characterizing spread vary ac-cording to Table A.III.

64

Figure A.5 Diagrams showing the distribution of the parameters N (additional force), tx,o (shear stress in failure zone) and down-slope displacement resulting from the force NL. Cf Pages 9 to 13 in the Excel spread-sheet in Appendix C. The slope angle is in radians.

65

Table A.II Results from variation of key parameters according to Table A.III Parameter Mean value Max. deviation Min. deviation from mean value from mean value a) Situation when t(x,o) = cR - (i.e. when NL =NcR at impending formation of shear band NCR 910 kN/m* +35 % 28 % (Critical load.) LCR 128 m +24 % 12 % (Critical length of failure zone)

d(L),N 0.336 m +20 % 35 %(Critical displacement) b) Situation when NL = NRankine = ERankine – E0 = 1840 kN/m LN = 1840 kN 211 m +34 % -25 % (Length of insipient failure zone)

d(L),N 1.97 m +79 % 58 % (Displacement for NL = 1840 kN/m) c) Extent of zone subject to passive Rankine failure (when NL = 3000 kN/m)

L = LF-LR 102 m +67 % -52 % (Length of passive Rankine zone) LN = 3000 kN 313 m +45 % -34 % (Length of total failure zone)

d(L),N 5.49 m +82 % -60 % (Displacement for NL = 3000 kN/m) * The maximum resistance in terms of NCR, which is the value of NL when the shear stress t(x,o) has just attained the residual shear resistance cR, varies between 726 kN/m and 1238 kN/m. In the situation when NL > NRankine = ERankine- E0 = 3000 kN/m, the minimum length over which passive Rankine failure occurs (i.e. L = LF - LR in Table A.III) varies from

L = 49 m in the near plastic condition (when cR/c = 0.9) to L = 171 m when the residual strength only amounts to cR/cpeak = 0.333 - i.e. cR/clab = 0.4.

ConclusionsThe spread-sheet analyses performed clearly illustrate the fracture mechanics nature of the phenomenon of vast spread of slides in soft clays. From the results of the sensitivity analy-sis compiled in Tables A.II and A.III it is evident that this phenomenon is likely to occur in downward progressive slides for a wide range of material and constitutive clay properties - i.e. even in soft clays, which are not particularly sensitive by Scandinavian standards. The sensitivity of the clays in the valley proper was in Surte and Tuve in the order of 10 to 20 - i.e. well below 50, which is the number usually regarded as characterizing ‘quick clays’. The several hundred meters wide spread over almost horizontal ground occurring in downward progressive landslides in Sweden - such as in Surte ( 400 m) and Tuve ( 300 m) – can thus be rationally explained on the basis of the fracture-mechanical analysis per-formed in accordance with the Excel spread sheet as demonstrated in Table A.III. Importantly, the residual resistance is varied in Table A.III from 33,3 % to 90.0 % of the peak value - i.e. from a high degree of strain softening to almost ideal plastic conditions.

66

Table A.III

Tab

le A

.III -

resu

lts fr

om se

nsiti

vity

ana

lysi

s

0

9 03

03

Com

pila

tion

of re

sults

of p

rogr

essi

ve fa

ilure

com

puta

tions

base

d on

FD

M sp

read

shee

t acc

ordi

ng to

Ber

nand

er

La

ndsl

ide

spre

ad,

Syno

psis

I - M

odifi

ed

A

utho

r: St

ig B

erna

nder

Leg

end:

c =

Peak

shea

r res

ista

nce

dn

=D

own-

slop

e di

spla

cem

ent

Nc R

= V

alue

of f

orce

N w

hen

t =c R

c(

R) =

Res

idua

l res

ista

nce

x

=In

fluen

cial

leng

th o

f for

ce N

E(R

anki

ne) =

Ran

kine

ear

th p

ress

ure

resi

st.

t(e

l) =

Ela

stic

lim

itN

R

= E

(Ran

kine

)- E o

E

o

= P

reva

iling

in si

tu e

arth

pre

ssur

e

c (s) =

Sur

face

shea

r res

ista

nce

belo

w d

ry c

rust

co

rr. t

o

= c

orre

spon

ding

to

Inpu

t dat

a

Res

ults

from

com

puta

tions

Nx=

NR

=N

x=N

F =

Diff

.

1840

kN

/m30

00 k

N/m

Cas

ePe

akA

ssoc

.E

las-

Ass

oc.

Rec

id.

Ass

oc.

Surf

.M

ean

x =

dn

Nc R

x=L

cR

dn

x=L

Rdn

x=

LF

dn

LF-

LR

refe

renc

e sh

ear

shea

rtic

shea

r sh

ear

slip

shea

r E

-mo-

corr

. c

orr.

c

orr.

c

orr.

c

orr.

c

orr.

c

orr.

c

orr.

c

orr.

co

rr.

num

ber

resis

t-

defo

r-lim

itde

for-

resi

st.

defo

r-re

sist

.du

lus

toto

toto

toto

toto

toto

an

cem

atio

nt(e

l)m

atio

nc(

R)

mat

ion

c(s)

t = c

t = c

t=c(

R)

Nc R

NcR

NR

N

R

NF

NF

NF

kPa

% k

Pa%

kPa

m

kPa

mm

kN/m

mm

mm

mm

mC

ase

130

3.0

201.

5015

0.15

1530

0010

3.5

0.13

765

211

2.0

0.21

821

32.

3131

16.

2798

Cas

e 2

303.

018

1.29

100.

2018

3360

106.

50.

131

719

118.

70.

246

283

3.39

454

9.53

171

Cas

e 5

303.

018

1.29

150.

2018

3360

106.

50.

131

781

120.

30.

267

210

2.02

308

5.56

98C

ase

425

3.0

100.

7510

0.30

1532

0010

5.4

0.12

679

212

8.3

0.35

928

23.

5345

39.

9817

1C

ase

630

3.0

151.

0015

0.20

1533

7510

5.3

0.13

479

711

9.2

0.27

320

82.

0030

65.

5298

Cas

e 3

253.

020

2.00

200.

3015

2800

101.

30.

135

804

122.

80.

405

184

2.10

254

5.58

69C

ase

730

3.0

201.

5015

0.30

1530

0010

3.5

0.13

787

912

3.2

0.36

820

52.

2130

36.

1898

Cas

e 8

303.

020

1.50

200.

2020

3333

107.

30.

131

886

125.

40.

323

182

1.48

251

3.99

69C

ase

1030

3.0

100.

6010

0.30

1033

3310

3.3

0.14

289

412

3.2

0.36

726

23.

2243

39.

4117

1C

ase

930

3.0

151.

0020

0.20

2037

5010

8.9

0.12

794

812

8.3

0.32

318

11.

3125

03.

5469

Cas

e 11

252.

010

0.50

100.

3015

4800

105.

40.

084

950

158.

50.

340

269

2.24

440

6.54

171

Cas

e 12

252.

015

0.86

150.

3015

4200

103.

10.

086

960

134.

50.

351

209

1.59

307

4.43

98C

ase

1330

3.0

201.

5020

0.30

2033

3310

7.3

0.13

199

512

9.5

0.38

618

01.

4524

93.

9669

Cas

e 14

252.

015

0.86

200.

3015

4200

103.

10.

086

1043

135.

20.

368

183

1.18

252

3.17

69C

ase

1530

3.0

151.

0020

0.30

2037

5010

8.9

0.12

710

6613

2.8

0.38

717

91.

2824

83.

5169

Cas

e 16

302.

015

0.67

150.

3010

4500

101.

30.

095

1076

129.

30.

353

194

1.40

292

4.05

98C

ase

1730

2.0

211.

0827

0.20

1528

0010

2.6

0.09

212

3213

2.1

0.38

215

80.

8320

62.

1749

Mea

n va

lues

28.5

32.

716

.31.

116

.29

0.3

15.9

3535

104.

90.

119

910.

212

7.8

0.33

621

0.8

1.97

312.

85.

4910

2.1

*

With

E(R

anki

ne) =

440

0 kN

/m a

nd E

in si

tu sl

ight

ly v

aryi

ng, t

he N

x - v

alue

cor

resp

ondi

ng

to p

assi

ve R

anki

ne fa

ilure

(NR) i

s her

e ta

ken

to b

e 18

40 k

N/m

.

67

The results of the spread-sheet analyses clearly identify different conceivable outcomes of downward redistribution of earth pressures triggered by local progressive slope instability further up-slope. (See Appendix B). 1. Thus, if in the studied case, the force NL does not exceed the value ERankine- Eo = 1840 kN//m, the up-hill slope failure will only result in moderate displacements (< 34 cm) and without any heave movement at the foot of the slope. In this instance, no veritable slide takes place although - in fact - the failure zone including the shear band has at this stage already propagated more than 200 m under the valley floor. (Cf for instance the slide at Rävekärr mentioned in Section 3.2 of the main report.) 2. If on the other hand, the force NL = ERankine - Eo is greater than 1840 kN/m, the up-hill slope failure will in the example inevitably generate large displacements and major passive Rankine heave over wide areas of the gently sloping ground ahead of the foot of the slope, and that to an extent depending on how much the value of NL happens to become in excess of (ERankine- Eo), as well as on the currently valid value of the residual shear resistance cR. In the studied case - if NL = 3000 kN/m - the failure zone with its shear band will have developed about 450 m beyond the foot of the slope into level ground. It is important to note that the phenomenological outcome of the analyses performed is not particularly sensitive to the specific assumptions made regarding the constitutive shear/deformation relationships. In fact, the enormous deviation in respect of spread prediction between the current FDM analysis and that based on perfect plasticity is to an overwhelmingly decisive degree re-lated to the very fact that deformations within the sliding body are considered in the com-putations. This implies that reasonable assessments regarding progressive failure issues can be made already on principles, know how and test procedures based on present State-of-the-Art in soil mechanics.

69

Appendix B - Assessment of local up-slope triggering load. Exemplification and sensitivity study. B.1. General Table 2.1 in Section 2 of the main report lists a number of longish down-hill slides having occurred in slopes of soft clays. In all these cases, the landslides have been triggered by seem-ingly trivial local disturbance agents. The disturbance phase of a down-hill progressive slide is defined as Phase 2 in Section 3.2 of the main report. It is therefore of interest in the current context to investigate if this remark-able fracture mechanics issue can be corroborated by means of the FDM analysis applied in the report. Assessment of the up-slope load likely to trigger progressive failure can readily be made using the Excel spread sheet utilized in Appendix A. Again, although applying this Excel program to an arbitrary case is a drawn out exercise compared to using the fully computerized program in Windows C++ referred to in Appendix A, it constitutes a valuable tool for research and educational purposes as every cognitive step in the analysis is directly controlled by the operator. (Although the Excel spread-sheet is used here simply for studying a uniform slope, it can be adapted to any variation of the input parameters from one section to another and is therefore - within the chosen framework - applicable to any arbitrary slope geometry). However, applying the Excel spread sheet to the simple slope configurations studied in this appendix - i.e. with constant gradients and invariable depth to the failure zone - the time re-quired in each case is only a matter of about fifteen minutes. B.2 Objectives The main objective in the current exercise is to define the critical load - in this case an up-slope embankment load (qcrit, kN/m2) - that can set off a downward progressive slide move-ment, and which in due course may develop into a veritable slide event. The additional load (q) can be defined as NL= q·H. For the sake of simplicity, easier interpretation and better understanding of the outcome of the analysis, a uniformly inclining slope is subject to study. See Figure B.1.

However - importantly - as the length of effective shear stress mobilization is considerably less than the distance defined as Lcrit ( 116 m ) in Figure B.2 - say in the order of say 40 to 50 m - the resulting computed hazard will be applicable also to cases, where the inclination studied matches the upper part of longer slopes than 50 m, depending largely on the current degree of strain-softening as well as on the geometry further down-slope.

Another key point of interest is to investigate in what way variation of the parameters used in the analysis affects the fracture-mechanical issue - e.g. how does the degree of deformation-softening relate to the additional up-hill, local load that can be applied prior to the incidence of progressive failure. The example presented below is based on the same calculation procedure as the one used in Appendix A.

B.3. Denotations Confer Appendix A

70

B.4. Basic equations The equations utilized are basically identical to those used in Ap-pendix A. B.5 Spread–sheet analysis of example shown in Figures B.1

Figure B.1 Uniformly sloping ground in a steep slope subjected to a local up-slope load equal to q kN/m2. Slope gradient = arctan (n/100). B.5.1 Shear deformation parameters The following parameters apply in the studied exemplification. (Case 1 in Table B.II).

Table B.I Parameters H = 20 m, Density of clay = 16 kN/m3 Slope gradient tan = 0.05 Peak shear strength cpeak = 30 kN/m2 Laboratory shear strength clab = 25 kN/m2 Elastic limit tel = 16 kN/m2 Residual shear resistance cR = 0.4clab = 10 kN/m2 Slip at residual shear resistance SCR = 0,20 m Mean shear strength of soil profile cmean = 22,5 kN/m2. (csurface = 15 kN/m2) Deviatory deformation at peak shear gf = 3 % = 0.03 Deviatory deformation at elastic limit gel = 1.09 % = 0.0109 (calculated acc. to Equ. A.I.3) Gmodulus = tel/gel = 1467 kN/m2

Emean = 2(1+v)Gmean= 2(1+0.5) 1120 = 3300 kN/m2 Emodulus = 2(1+v) 1467 = 4400 kN/m2 B.5.2 Computer sequence in the excel spread-sheet analysis. The analysis consists in 2-dimensional integration of the following parameters: a) The shear stresses (tx,z) in soil mass; b) The additional earth pressure resultant Nx due to a force (NL) located at x= L; c) The mean down-slope displacement ( d(x),N) induced by the force NL. The computations are in principle identical to those in Appendix A. The integration process begins at a point defined by x = 0, where the boundary conditions are taken to be known, and where the effects of the force (N = q H) are considered negligible. This means that for x = 0, the additional shear stress tx,o = 0, the additional force Nx = 0 as well as the down-hill dis-placement d(x),N = 0. The 2-dimensional integration then continues up-hill in steps of x (or t), where each indi-vidual step involves a ‘trial and error’ process until compatibility between down-slope dis-placement and shear deformation is attained. The analysis results in the values of the force Nx,

71

the shear stress txz and the displacements along the slope corresponding to the additional load NL acting at x = L. When the effect of a predefined force (NL = q H kN/m, acting at x = L) is sought, the entire integration procedure has to be repeated assuming alternative locations for the starting point x = 0 - i.e. varying the values of the influential length L. In a general case, therefore, such analysis will necessitate repeated trial and error procedures until the correct boundary condition at x = L is attained - i.e. when Nx=L = q H. (Note: These computations may appear prohibitively laborious, but using the integrally computerized version, a complete trial and error computation may only be a matter of a few seconds in terms of computer time). B.5.3 Results from the example defined in Table B.I In the specific example defined in Figure B.1 and Table B.I, the inclinations of the ground surface and the potential failure zone are constant and parallel. Hence, also the depth (H) is constant. The material parameters are also presumed to be identical along the slope. This has the crucial implication that all values of Nx , tx,z and the displacement d(x),N be-tween 0 < x < L - representing the effects ahead of a force Nx=L, - are independent of the loca-tion of the starting point x = 0. Hence, in the diagram for N shown in Figure B.2, every value of Nx indicates the distribution of the parameters (N) and (t), which are valid ahead of an additional external force Nx located at x. The studied disturbance stage represents Phase 2 according to Section 3.2 of the main report. The main results from the Excel spread-sheet analysis based on the parameters given in Table B.I are presented in Figure B.2. The diagram for Nx in Figure B.2 indicates for instance a value of Ncrit = 389.2 kN/m. This represents the maximum value of NL= qcrit H, when the post-peak shear stress tx,o in the devel-oping shear band precisely attains the locally prevailing in situ shear stress tx,o , which in the studied case is equal to ·H·sin 16 kN/m2. Hence, qcrit =389.2/20 = 19.46 kN/m2 The critical length of influence (Lcrit) is at this point in the order of 100 m and the displace-ment at the front edge of the load (q), where x = Lcrit, is 195 mm. The distance Lcrit is thus a measure of how far the maximum additional load affects the poten-tial failure zone in the soil mass ahead. If the load qcrit is exceeded, the virtually dynamic redistribution of earth pressures and defor-mations takes place (i.e. Phase 3, according to the main report), possibly resulting eventually in a total slope failure.

72

Figure B.2 Diagrams showing the distribution of the parameters Nx (additional force), tx,o (shear stress in failure zone) and down-hill displacement in the studied example. The parameter denoted Linstab is the influence length, when the value of the active external force NL that can be mobilized may become = 0 owing to the effects of deformation-softening. This situation represents a condition of critical displacement ( d(x),N,crit) at x = L, for which the slope will fail even if, hypothetically, the load (q) - at this point - were to be removed instantaneously. The phenomenon can, for instance, be of practical significance, when ramming soil displacing piles - as was the case when the Surte slide took place.

73

B.6 Sensitivity study Table B.II at the end of this section shows the results of a sensitivity study focused on the effects of varying two key parameters in the context of progressive slide hazard, namely the slope gradient and the residual shear resistance. The residual shear resistance in Table B.II is varied from 40 to 80 % of the laboratory shear strength (clab) - i.e. from a high degree of strain softening to almost plastic conditions. In Fig-ure B.3, the corresponding cR/clab ratios range between 20 and 80 %. The slope gradient ( ) is varied from tan = 0.05 to 0.08. In order to facilitate the understanding of the results regarding the triggering load compiled in Table B.II, they are also presented visually in Figure B.3.

qcrit [kN/m2]

clab = 25 kN/m2

cpeak = 30 kN/m2 gf = 3 % qel = 1.09 %

0.2 0.4 0.6 0.8 1.0

cR/clab

10

20

25 H = 20 m Slope n:100

qcrit

tan = 0.05

tan = 0.06

tan = 0.07

tan = 0.08

5:100

6:100

7:100

8:100

= 16 2

Figure B.3 Results from variation of key parameters according to Table B.II. It is of con-siderable interest to note that the critical load (qcrit) is relatively little affected by the degree of strain-softening - especially for steeper values of the slope gradient. The diagram also ac-centuates the acute hazard in respect of progressive failure related to local up-slope fills and embankments. (This is a condition, which is in good general agreement with Table 2.1 in Sec-tion 2 of the main report listing slides occurred). It should be noted that the critical load (qcrit) is not highly affected by the degree of strain sof-tening for steeper values of the slope gradient. This constitutes a favourable circumstance, as the residual shear resistance (cR) of sensitive (and quick) clays is a parameter, which is diffi-cult to estimate from laboratory tests – depending in reality as it is on poorly defined clay sen-sitivity, rate of load application and local drainage conditions. Yet, sensitivity is of course likely to strongly affect possible further development of a pro-gressive slide.

74

The diagram also accentuates the acute hazard related to local up-slope earth fills and em-bankments in slopes of soft clay. This is in good general agreement with Table 2.1 in Section 2 of the main report, where a number of downward progressive slide events in Scandinavia and Canada are listed. For example, as may, be concluded from Figure B.3 (or table B.II), initiation of progressive failure may be initiated in a slope inclining 8 m per 100 m already for a surcharge load of qcrit = 9.1 kN/m2 (corresponding to about 0,5 m of earth fill), and that even when the residual resistance is reduced by a factor of only 0.8 as a result of deformation-softening.

In Table B.III, the safety factors as computed according to limit plastic equilibrium analysis (Fc) and corresponding safety factors defining risk of progressive failure (Fpr) are compared in 16 cases with different slope gradients and varying degrees of deformation-softening for a specific load of q = 18 kN/m2 As shown in the column for Fpr/Fc in Table B.III, expressing the ratio between the two modes of defining slide hazard, this ratio varies between 0.234 and 0.651 instead of having a target value, which if both modes of analysis were equally relevant, ought to be 1 or at least have a constant value. Expressed in terms of a median value of 0.44 this variation corresponds to a scatter between +47.5 % . Or, to put somewhat differently, the maximum value exceeds the minimum value by a factor of f = (Fpr/Fc)max/(Fpr/Fc) min = 2.77. Considering that Table B.III by no means accounts for all factors affecting the Fpr/Fc - ratio, it is must be expected that the scatter may be much greater, and that the factor (f) can adopt val-ues considerably in excess of 2.77. It is therefore highly unlikely that the effects of the presence of quick clay on slide hazard can be compensated by just raising the safety factor based on the limit plastic equilibrium concept, by small percentages such as 10, 20, 30 % or the like. In the report from the Independent Investigatory Group of the Swedish Road Administration in connection with the slide at Småröd (Cf I I G RA, 2007, Section 9.4), a corresponding raise of the safety factor by only 10 % - due to presence of quick clay - is advocated. The conclusion that must be drawn from Table B.III is that potential slides in quick clay should be based on progressive failure analysis, and decidedly not on the limit plastic equilib-rium concept. Conclusions1. The spread-sheet analysis performed in respect of loading, likely to trigger down-hill pro-gressive failure, clearly illustrates the fracture mechanics nature of slides in slopes of soft clay. From the results of the analyses compiled in Table B.II (and visualized in Figure B.3), it is evident that the hazards related to progressive failure may occur for remarkably insignifi-cant loading effects due, for instance, to minor local earth fills. The exemplifications made on the basis of the FDM analyses are consistent with the experience gained from slide events such as those listed in Table 2.1 of the main report. 2. The analysis made shows that slope failure in sensitive clays develops in direction down-slope rather than along slip circles surfacing in inclining ground. This has the serious implica-tion that a supporting embankment - in for instance road construction - may actually turn out to be an agent triggering major landslide disaster.

75

3. It is important to note that the critical load in the disturbance phase (i.e. Phase 2 according to Section 3.2) - for the steeper values of the slope gradient - is relatively insensitive to the degree of strain softening. (By contrast, high sensitivity plays a more crucial role in the ensu-ing phases of a slide). Yet, in view of the fact that preventing progressive slope failure must imply staying well be-low the critical load (qcrit) by some predefined factor of safety, this relative insensitivity to strain-softening constitutes a favourable circumstance considering that residual shear resis-tance of sensitive (and quick) clays is a parameter difficult to assess correctly – depending as it is on the poorly established relationships between sensitivity numbers (based on completely stirred clay samples) and deformation-softening under actual in situ conditions. Hence, uncer-tainty regarding the residual shear resistance can be compensated by adopting an appropri-ately high local safety factor, based on progressive failure analysis. 4. It may be noted that – from fracture mechanical points of view - the outcome of the analy-ses performed is not particularly sensitive to the specific assumptions made regarding the con-stitutive shear-deformation relationships. In fact, the enormous deviation - in respect of slide hazard between the current FDM analysis and that based on perfect plasticity - is to an over-whelmingly decisive degree related to the fact that deformations within the sliding body are accounted for in the computations. 5. Summing up, the analyses made (Cf Table B.III) clearly indicate that potential slides in quick clays should be based on progressive failure analysis, and decidedly not on the limit plastic equilibrium concept. The circumstances mentioned above imply that reasonable assessments regarding progressive failure issues can be made already on principles, knowledge and test procedures based on pre-sent State-of-the-Art in Soil Mechanics.

76

Table B.II T

able

B.II

- D

ownh

ill p

rogr

essi

ve sl

ide-

trig

geri

ng lo

ads -

Sens

itivi

ty a

naly

sis

09

05 1

0C

ompi

latio

n of

resu

lts o

f pro

gres

sive

failu

re c

ompu

tatio

ns

File

: Trig

gerin

g lo

ad

base

d on

FD

M sp

read

shee

t acc

ordi

ng to

Ber

nand

er

Aut

hor:

Stig

Ber

nand

er

L

egen

d:c

= Pe

ak sh

ear r

esis

tanc

e

dn =

Dow

n-sl

ope

disp

lace

men

t at x

=LN

crit

=

Val

ue o

f for

ce N

whe

n po

st-p

eak

t = t o

c(

lab)

= L

ab. s

hear

stre

ngth

x

= In

fluen

cial

leng

th o

f for

ce N

LL

inst

ab =

Val

ue o

f for

ce L

whe

n po

st-p

eak

valu

e of

N =

0

c(R

) =

Res

idua

l res

ista

nce

q cri

t = N

crit/

H

to

= in

situ

shea

r stre

ss

t(el)

= E

last

ic li

mit

Inpu

t dat

a

Res

ults

from

C

ritic

al lo

adIn

stab

ility

leng

th =

D

epth

to fa

ilure

surf

ace

H =

20

m

com

puta

tions

N =

Ncr

it H

=20

.0x

=Lin

stab

at N

= 0

Cas

eG

ra-

In si

tuPe

akA

ssoc

.E

las-

Ass

oc.

Rec

id.

Rec

id.

Ass

oc.

N

dn

x =

LN

crit

qcri

tdn

x

= L

N

dn

L

inst

ab

refe

renc

e di

ent

shea

r sh

ear

shea

rtic

shea

rsh

ear

shea

rsl

ip c

orr.

c

orr.

co

rr.

cor

r.

or(m

ax)

(cri

t)

(ins

tab)

cor

r.

num

ber

(Gr)

stre

ssre

sist

- de

for-

limit

defo

r-re

sist

.re

sist

.de

for-

toto

toto

q(t=

cR)

atat

at

to

t oan

cem

atio

nt(e

l)m

atio

nc R

/cla

bc R

m

atio

nt =

ct =

ct =

c

t = to

Ncr

it N

crit

N

= 0

N =

0ta

n(G

r) k

Pa k

Pa%

kPa

%1

kPa

m

kN/m

mm

kN/m

kN/m

2cm

mkN

/mm

mC

ase

10.

050

15.9

830

3.0

161.

090.

400

10.0

0.20

295.

30.

081

98.0

389.

219

.46

0.15

611

1.8

00.

366

179.

9C

ase

20.

060

19.1

730

3.0

161.

090.

400

10.0

0.20

231.

30.

065

97.5

319.

415

.97

0.13

211

2.8

00.

243

153.

5C

ase

30.

070

22.3

530

3.0

161.

090.

400

10.0

0.20

171.

00.

051

98.2

232.

111

.61

0.09

811

3.1

00.

168

144.

8C

ase

40.

080

25.5

230

3.0

161.

090.

400

10.0

0.20

105.

10.

034

89.6

140.

67.

030.

062

104.

10

0.10

513

5.5

Cas

e 5

0.05

015

.98

303.

016

1.09

0.60

015

.00.

2029

5.3

0.08

198

.044

1.3

22.0

70.

203

118.

5c R

=

t o56

9.3

Cas

e 6

0.06

019

.17

303.

016

1.09

0.60

015

.00.

2023

1.3

0.06

597

.535

7.8

17.8

90.

168

119.

20

0.42

020

8.7

Cas

e 7

0.07

022

.35

303.

016

1.09

0.60

015

.00.

2017

1.0

0.05

198

.225

8.4

12.9

20.

124

119.

40

0.22

416

1.4

Cas

e 8

0.08

025

.52

303.

016

1.09

0.60

015

.00.

2010

5.1

0.03

489

.615

5.5

7.78

0.07

711

0.3

00.

135

151.

4C

ase

90.

050

15.9

830

3.0

161.

090.

800

20.0

0.20

295.

30.

081

98.0

512.

925

.65

0.23

212

1.6

c R >

t o

1228

Cas

e 10

0.06

019

.17

303.

016

1.09

0.80

020

.00.

2023

1.3

0.06

597

.542

3.3

21.1

70.

227

127.

9c R

>

to44

6C

ase

110.

070

22.3

530

3.0

161.

090.

800

20.0

0.20

171.

00.

051

98.2

304.

215

.21

0.17

513

0.1

00.

492

264.

0C

ase

120.

080

25.5

230

3.0

161.

090.

800

20.0

0.20

105.

10.

034

89.6

181.

79.

090.

107

120.

80

0.19

617

4.3

Cas

e 13

0.05

015

.98

303.

016

1.09

0.90

022

.50.

2029

5.3

0.08

198

.051

2.9

c R >

t o

Cas

e 14

0.06

019

.17

303.

016

1.09

0.90

022

.50.

2023

1.3

0.06

597

.542

1.8

c R >

to

Cas

e 15

0.07

022

.35

303.

016

1.09

0.90

022

.50.

2017

1.0

0.05

198

.234

4.0

17.2

00.

226

139.

4c R

=

t oC

ase

160.

080

25.5

230

3.0

161.

090.

900

22.5

0.20

105.

10.

034

89.6

204.

610

.23

0.13

712

9.7

00.

276

209.

1M

ean

valu

es0.

2020

0.7

0.05

895

.832

5.0

13.3

30.

133

104.

90.

000.

164

251.

6

77

Table B.III

Tab

le B

.III -

Dow

nhill

pro

gres

sive

slid

e - t

rigg

erin

g lo

ads -

2009

05

22

Com

paris

on o

f slo

pe h

azar

d ba

sed

on P

rFA

ana

lysi

s and

File

: Trig

gerin

g lo

ad -

Syno

psis

IIsl

ope

haza

rd b

ased

on

conv

entio

nal (

IPFA

) ana

lysi

s.

Aut

hor:

Stig

Ber

nade

r

L

egen

d:c

= Pe

ak sh

ear r

esis

tanc

e

dn =

Dow

n-sl

ope

disp

lace

men

t at x

= L

c(la

b)=

Labo

rato

ry sh

ear s

treng

th

L

= In

fluen

cial

leng

th o

f for

ce N

L

c(R

) =

Res

idua

l res

ista

nce

q cri

t = N

crit/

H

#

Saf

ety

fact

or b

ased

on

idea

l- pl

astic

equ

ilibr

ium

to

=

in si

tu sh

ear s

tress

t(e

l) =

Ela

stic

lim

it

anal

ysis

of a

slop

eof l

engt

h L

In

put d

ata

IPFA

ana

lysi

s R

esul

ts fr

om

# Con

-Pr

ogre

ssve

f.

Dep

th to

failu

re su

rfac

e H

= 2

0 m

IPFA

ana

lysi

s #ve

ntio

-Pr

FA a

naly

sis *

C

ase

Gra

-In

situ

Peak

Lab

Ass

oc.

Ela

s-R

ecid

.R

ecid

.D

ensi

tyE

p-Ea

Hsi

n G

rqf

M

ax.

Fpr

=na

l qc

rit

x =

L F

pr/F

cre

fere

nce

dien

tsh

ear

shea

r sh

ear

shea

rtic

shea

rsh

ear

of =

4H c

he

ight

qcri

t/ qq

safe

ty

(cri

t)nu

mbe

r(G

r)st

ress

resis

t-

resi

st-

defo

r-lim

itre

sist

.re

sist

.cl

ay

o fq=

18fa

ctor

at

t o

ance

ance

mat

ion

t(el)

c R/c

lab

c R

g

fill

kN /

kN /

FcN

crit

tan(

Gr)

kPa

kPa

kPa

% k

Pa1

kPa

kN/m

3kN

/mm

1.00

kN/m

2m

m

2m

2q=

18kN

/m2

mC

ase

10.

050

15.9

830

253.

016

0.40

010

.016

.020

0020

.00.

0499

150.

438.

361.

081

18.0

2.23

19.5

111.

80.

484

Cas

e 2

0.06

019

.17

3025

3.0

160.

400

10.0

16.0

2000

20.0

0.05

9913

2.88

7.38

0.88

718

.01.

9116

.011

2.7

0.46

4C

ase

30.

070

22.3

530

253.

016

0.40

010

.016

.020

0020

.00.

0698

114.

986.

390.

645

18.0

1.67

11.6

112.

90.

385

Cas

e 4

0.08

025

.52

3025

3.0

160.

400

10.0

16.0

2000

20.0

0.07

9797

.30

5.41

0.39

118

.01.

537.

010

4.0

0.25

6C

ase

4a0.

080

25.5

230

253.

016

0.40

05.

016

.020

0020

.00.

0797

97.4

15.

410.

363

18.0

1.55

6.5

100.

00.

235

Cas

e 5

0.05

015

.98

3025

3.0

160.

600

15.0

16.0

2000

20.0

0.04

9915

3.46

8.53

1.22

618

.02.

2022

.111

8.5

0.55

7C

ase

60.

060

19.1

730

253.

016

0.60

015

.016

.020

0020

.00.

0599

134.

777.

490.

994

18.0

1.88

17.9

119.

20.

528

Cas

e 7

0.07

022

.35

3025

3.0

160.

600

15.0

16.0

2000

20.0

0.06

9811

5.82

6.43

0.71

318

.01.

6512

.811

9.2

0.43

3C

ase

80.

080

25.5

230

253.

016

0.60

015

.016

.020

0020

.00.

0797

97.1

45.

400.

432

18.0

1.50

7.8

110.

30.

288

Cas

e 9

0.05

015

.98

3025

3.0

160.

800

20.0

16.0

2000

20.0

0.04

9915

4.84

8.60

1.42

518

.02.

1925

.612

1.6

0.65

1C

ase

100.

060

19.1

730

253.

016

0.80

020

.016

.020

0020

.00.

0599

137.

317.

631.

176

18.0

1.85

21.2

127.

90.

636

Cas

e 11

0.07

022

.35

3025

3.0

160.

800

20.0

16.0

2000

20.0

0.06

9811

7.26

6.51

0.84

518

.01.

6115

.213

0.1

0.52

6C

ase

120.

080

25.5

230

253.

016

0.80

020

.016

.020

0020

.00.

0797

96.8

75.

380.

505

18.0

1.46

9.1

120.

80.

346

Cas

e 13

0.05

015

.98

3025

3.0

160.

900

22.5

16.0

2000

20.0

0.04

9910

0.00

5.56

18.0

25.6

C

ase

140.

060

19.1

730

253.

016

0.90

022

.516

.020

0020

.00.

0599

100.

005.

5618

.021

.1

Cas

e 15

0.07

022

.35

3025

3.0

160.

900

22.5

16.0

2000

20.0

0.06

9811

8.50

6.58

0.95

618

.01.

5817

.213

9.4

0.60

5C

ase

160.

080

25.5

230

253.

016

0.90

022

.516

.020

0020

.00.

0797

96.6

45.

370.

568

18.0

1.43

10.2

129.

70.

398

Mea

n va

lues

0.

871.

87

0.

49D

evia

tions

- m

ax/m

in%

34.2

-51.

5

0.36

1

App

endi

x C

. E

xcel

spre

ad sh

eet -

Exe

mpl

ifica

tion

Aut

hor:

Stig

Ber

nand

er

Pag

e 1

No

5D

ata

appl

ying

to C

ase

2 in

in T

able

A.II

I (b

asic

mod

el)

09-

02-2

4M

odifi

edA

rket

ypIn

put d

ata:

H =

20

mG

radi

ent =

tan

Gr =

1:1

00,

Gr =

arc

tan

0,01

= 0

,573

Dgr

D

ensi

ty =

16

kN/m

3

(Res

idua

l she

ar st

reng

th)

(Sur

face

stre

ngth

)(M

ean

shea

r st

reng

th)

Cpe

ak=

30 k

N/m

2C

lab

= 25

kN

/m2

CR

= 10

kN

/m2

CR/

Cla

b=

0,4

0 C

s= 1

8 kN

/m2

Cm

ean=

24

kN/m

2Sh

ear d

efor

mat

ion

at fa

ilure

(gf)

= 3

%

CR/

Cpe

ak =

0,3

33tel

= 18

kN

/m2

Post

-pea

k de

form

atio

n at

CR

= 0,

20 m

Shea

r def

orm

. at e

last

ic li

mit

( gel)

= 1,

29 %

Equ

atio

ns:

tx,

z< t

eltx

,z<

tel

gx,z

(1)

= tx

/G,

gel=

gf*te

l/(2c

- tel

)an

dG

= te

l/gel

Stag

e I

Equ

. I:1

a S

:a d

(x,z

),t =

Inte

gral

(gx,

z,(1)

)dz

t o <

txz<

c,

to >

telto

< tx

z< c

t o >

tel

gx,z

(4) =

(gf-g

el)*(

1- R

OT

(1 -

(to(x

,z)-

tel)/(

c- te

l))St

age

IE

qu. I

:4

S:a

d(x

,z),t

= In

tegr

al (g

x,z,(

4))d

zt o

< t

el- R

OT

(1 -

(t(x,

z) -

tel)/(

c - t

el))

)to

< t

elgx

,z(4

a) =

(tel

-to,

(x,z)/G

+ (g

f-gel

)*(1-

RO

T(1

- (t(x

,z)-

tel)/(

c- te

l)))

E

qu. I

:4a

S:a

d(x

,z),t

= In

tegr

al (g

x,z,(

4a))d

zc

< t

xz<

cR

, to

> t

elc<

txz

<cR

to >

tel

txz

>cR

gx,z

(5)

= (g

f-gel

)*(R

OT

(1 -

(to(x

,z)-t

el)/(

c- te

l)) -

Stag

e II

a E

qu. I

:5

S:a

d(x

,z),t

= In

tegr

al (g

x,z,(

5a))d

z +

to<

tel

- RO

T(1

-(t(x

,z,m

ax)-t

el)/(

c - t

el)))

- (t(

x,z,

max

)-t(x

,z))/

Gto

< t

eltx

z < c

R

+Sc

R(c

-tx,z)/(

c-cR

)gx

,z(5

a)=(

tel-t

o,(x,

z))/G

+(gf

-gel

)(1-

RO

T(1-

(t(x,

z,m

ax)-t

el)/(

c-te

l)))-

E

qu. I

:5a

S:a

d(x

,z),t

= In

tegr

al (g

x,z,(

5a))d

z +

tx,o

=cR

- (t(x

,z,m

ax)-t

(x,z

))/G

tx,o

=cR

+

ScR(c

-tx,z)/(

c-cR

)gx

,z(5

b)=

gx,z(

5)(o

r gx,

z(5a)

) - (

t(x,z

,max

)-cR

)/GSt

age

IIb

Equ

. I:5

b S

:a d

(x,z

),t =

Inte

gral

(gx,

z,(5b

))dz+

ScR+S

slip

orE

qu. I

:5c

Land

slid

e sp

read

0.00

000

0.50

000

1.00

000

1.50

000

2.00

000

2.50

000

3.00

000 0.

000

50.0

0010

0.00

015

0.00

020

0.00

025

0.00

030

0.00

0

Coo

rdin

ate

(x)

Slope surface

Ser

ie1

79

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

I(1

) 0

9-02

-24

Pag

e 2

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc (

s)=

g el

=G

radi

ent

Gm

od.

c(p

eak)

E-m

ean

tel=

H =

W

idth

, b1,

0It

er.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

33.7

8st

epN

o 1

xo0.

00St

ress

Mea

nkN

/m2

kN/m

2x1

33.7

8 n

z (n

)In

situ

*In

situ

*Sh

ear

incr

e-Sh

ear

shea

r gf

- ge

lcu

-t el

t el

d(g

)d(

g)*d

zcu

(m

)sh

ear

shea

r st

ress

men

tst

ress

stre

sskN

/m2

kN/m

2kN

/m2

Equ

.I:1a

mkN

/m2

20to(

x,z)

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

12.0

0

00.

003.

203.

203.

200.

300

3.50

03.

350

0.01

7112

.00

18.0

00.

0002

1

30.0

0

0.01

71

0.00

021

1

1.00

3.04

3.04

3.04

0.28

53.

325

3.18

20.

0171

11.7

617

.64

0.00

021

29.4

0

0.01

71

0.00

020

22.

002.

882.

882.

880.

270

3.15

03.

015

0.01

7111

.52

17.2

80.

0002

028

.80

0.

0171

0.

0002

0

33.

002.

722.

722.

720.

255

2.97

52.

847

0.01

7111

.28

16.9

20.

0001

928

.20

0.01

71

0.00

019

44.

002.

562.

562.

560.

240

2.80

02.

680

0.01

7111

.04

16.5

60.

0001

9

27.6

0

0.01

71

0.00

018

5

5.00

2.40

2.40

2.40

0.22

52.

625

2.51

20.

0171

10.8

016

.20

0.00

018

27.0

00.

0171

0.

0001

7

66.

002.

242.

242.

240.

210

2.45

02.

345

0.01

7110

.56

15.8

40.

0001

726

.40

0.

0171

0.

0001

1

6.7

6.67

2.13

2.13

2.13

0.20

02.

333

2.23

30.

0171

10.4

015

.60

0.00

016

26.0

0

20.0

0

6.00

1218

.00

Aut

hor:

Stig

Ber

nand

erSc

R0.

20cR

=10

.000

mC

alc.

S(x)

cR

/c(pe

ak)

0.33

S:a

N x

(n)

0.00

S:a

d(xn

),t0.

0012

7

dN5.

07S:

a d(

xn),N

0.00

00

S(

x)=

m

Cal

c.d(

xn,n

+1),N

0.00

127

S:

a N

x(n+

1)5.

07Sl

ipd(

slip

)=

0.00

000

m

dx,N

-dx(

t)0.

0000

0S:

adx

(n+1

),N0.

0012

7S:

adx

(n+1

),t0.

0012

7 m

*

In si

tu sh

ear s

tress

to b

e m

odifi

ed b

y th

e ex

pres

sion

dt =

(E (n+

1) -E

(n))/

dx(E

(n+1

) and

E(n

) are

in si

tu e

arth

pre

ssur

es)

80

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

I(2

) 0

9-02

-24

Pag

e 3

Aut

hor:

Stig

Ber

nand

er

Den

sit y

c(s)

=g

el =

Gra

dien

tG

mod

.c

(pea

k)E-

mea

nte

l=H

=

Wid

th, b

1,0

Iter

.g

f =0.

030

16.0

0018

0.01

290.

0100

1400

30.0

033

6018

.00

20.0

dx (m

)21

.09

step

No

1x1

x 2

33.7

8St

ress

x2

Mea

nkN

/m2

kN/m

2x

254

.87

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cu

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.I:

1am

kN/m

220

to(x,

z)t(x

,z)

dt(x

,z)

t(x,z

)+dt

0.01

7112

.00

00.

003.

203.

203.

501.

000

4.50

04.

000

0.01

7112

.00

18.0

00.

0009

3

30.0

0

0.

0171

0.

0009

1

11.

003.

043.

043.

320.

950

4.27

53.

800

0.01

7111

.76

17.6

40.

0009

029

.40

0.01

71

0.00

089

22.

002.

882.

883.

150.

900

4.05

03.

600

0.01

7111

.52

17.2

80.

0008

728

.80

0.01

71

0.00

086

33.

002.

722.

722.

970.

850

3.82

53.

400

0.01

7111

.28

16.9

20.

0008

428

.20

0.01

71

0.00

082

4

4.00

2.56

2.56

2.80

0.80

03.

600

3.20

00.

0171

11.0

416

.56

0.00

081

27

.60

0.

0171

0.

0007

9

55.

002.

402.

402.

620.

750

3.37

53.

000

0.01

7110

.80

16.2

00.

0007

727

.00

0.01

71

0.00

076

66.

002.

242.

242.

450.

700

3.15

02.

800

0.01

7110

.56

15.8

40.

0007

426

.40

0.01

71

0.00

049

6.7

6.67

2.13

2.13

2.33

0.66

72.

999

2.66

60.

0171

10.4

015

.60

0.00

071

26.0

0

Aut

hor:

Stig

Ber

nand

erSc

R0.

20cR

=10

.000

mC

alc.

S(x)

cR

/c=

0.33

S:a

N x

(n)

5.07

S:a

d(xn

),t0.

0055

1

dN16

.87

S:a

d(xn

),N0.

0013

S(x)

= m

C

alc.

d(xn

,n+1

),N0.

0042

4

S:a

Nx(

n+1)

21.9

4Sl

ipd(

slip

)=

0.00

000

m

dx,N

-dx(

t)0.

0000

0S:

adx

(n+1

),N0.

0055

1S:

adx

(n+1

),t0.

0055

1 m

81

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

I(3

) 0

9-02

-24

Pag

e 4

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc (

s)=

g el

=G

radi

ent

Gm

od.

c(p

eak)

E-m

ean

tel=

H =

W

idth

, b1,

0It

er.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

14.6

8st

epN

o 2

x2

x 3

54.8

7St

ress

x 3

Mea

nkN

/m2

kN/m

2x

3 69

.55

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cu

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.I:

1am

kN/m

220

to(x,

z)to(

x,z)

+dt

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

12.0

00

0.00

3.20

3.20

4.50

2.00

06.

500

5.50

00.

0171

12.0

018

.00

0.00

236

30

.00

0.

0171

0.

0023

21

1.00

3.04

3.04

4.27

1.90

06.

175

5.22

50.

0171

11.7

617

.64

0.00

228

29.4

00.

0171

0.

0022

52

2.00

2.88

2.88

4.05

1.80

05.

850

4.95

00.

0171

11.5

217

.28

0.00

221

28.8

00.

0171

0.

0021

73

3.00

2.72

2.72

3.82

1.70

05.

525

4.67

50.

0171

11.2

816

.92

0.00

213

28.2

00.

0171

0.

0020

94

4.00

2.56

2.56

3.60

1.60

05.

200

4.40

00.

0171

11.0

416

.56

0.00

205

27

.60

0.

0171

0.

0020

15

5.00

2.40

2.40

3.37

1.50

04.

875

4.12

50.

0171

10.8

016

.20

0.00

196

27.0

00.

0171

0.

0019

2

66.

002.

242.

243.

151.

400

4.55

03.

850

0.01

7110

.56

15.8

40.

0018

826

.40

0.01

71

0.00

124

6.

76.

672.

132.

133.

001.

333

4.33

23.

666

0.01

7110

.40

15.6

00.

0018

126

.00

Aut

hor:

Stig

Ber

nand

erSc

R0.

20cR

=10

.000

m

Cal

c.S(

x)

cR/c(

peak

)0.

33S:

a N

x(n

)21

.94

S:a

d(xn

),t0.

0139

9

dN33

.75

S:a

d(xn

),N0.

0055

1

S(

x)=

mC

alc.

d(xn

,n+1

),N0.

0084

8

S:a

Nx(

n+1)

55.6

9Sl

ipd(

slip

)=

mdx

,N-d

x(t)

0.00

000

S:a

dx(n

+1),N

0.01

399

S:a

dx(n

+1),t

0.01

399

m

82

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

I(3

) 0

9-02

-24

Pag

e 5

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc (

s)=

g el

=G

radi

ent

Gm

od.

c(p

eak)

E-m

ean

tel=

H =

W

idth

, b1,

0It

er.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

10.5

5st

epN

o 3

x 3

x 4

69.5

5St

ress

x 4

Mea

nkN

/m2

kN/m

2x

4 80

.10

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cu

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.I:

1am

kN/m

220

to(x,

z)to(

x,z)

+dt

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

12.0

00

0.00

3.20

3.20

6.50

3.00

09.

508.

000

0.01

7112

.00

18.0

00.

0045

0

30.0

0

0.01

71

0.00

443

1

1.00

3.04

3.04

6.17

2.85

09.

027.

600

0.01

7111

.76

17.6

40.

0043

629

.40

0.01

71

0.00

429

2

2.00

2.88

2.88

5.85

2.70

08.

557.

200

0.01

7111

.52

17.2

80.

0042

228

.80

0.01

71

0.00

414

33.

002.

722.

725.

522.

550

8.07

6.80

00.

0171

11.2

816

.92

0.00

407

28.2

00.

0171

0.

0039

9

44.

002.

562.

565.

202.

400

7.60

6.40

00.

0171

11.0

416

.56

0.00

391

27

.60

0.

0171

0.

0038

35

5.00

2.40

2.40

4.87

2.25

07.

126.

000

0.01

7110

.80

16.2

00.

0037

527

.00

0.

0171

0.

0036

66

6.00

2.24

2.24

4.55

2.10

06.

655.

600

0.01

7110

.56

15.8

40.

0035

826

.40

0.01

71

0.00

236

6.

76.

672.

132.

134.

332.

000

6.33

5.33

20.

0171

10.4

015

.60

0.00

346

26.0

0

Aut

hor:

Stig

Ber

nand

erSc

R0.

20cR

=10

.000

m

C

alc.

S(x)

cR

/c(pe

ak)

0.33

S:a

N x

(n)

55.6

9S:

ad(

xn),t

0.02

671

dN

50.6

4

S:a

d(xn

),N0.

0139

9

S(

x)=

mC

alc.

d(xn

,n+1

),N0.

0127

2

S:a

Nx(

n+1)

106.

33Sl

ipd(

slip

)=

mdx

,N-d

x(t)

-0.0

0001

S:a

dx(n

+1),N

0.02

671

S:a

dx(n

+1),t

0.02

671

m

83

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

I(5

) 0

9-02

-24

Pag

e 6

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc (

s)=

g el

=G

radi

ent

Gm

od.

c(p

eak)

E-m

ean

tel=

H =

W

idth

, b1,

0It

er.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

10.2

6st

epN

o 3

x 4

x 5

80.1

0St

ress

x 5

Mea

nkN

/m2

kN/m

2x

590

.36

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cu

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.I:

1am

kN/m

220

to(x,

z)to(

x,z)

+dt

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

12.0

00

0.00

3.20

3.20

9.50

5.50

015

.000

12.2

500.

0171

12.0

018

.00

0.00

843

30

.00

0.

0171

0.

0083

01

1.00

3.04

3.04

9.02

5.22

514

.25

11.6

370.

0171

11.7

617

.64

0.00

817

29.4

00.

0171

0.

0080

4

22.

002.

882.

888.

554.

950

13.5

011

.025

0.01

7111

.52

17.2

80.

0079

028

.80

0.01

71

0.00

776

3

3.00

2.72

2.72

8.07

4.67

512

.75

10.4

120.

0171

11.2

816

.92

0.00

762

28.2

00.

0171

0.

0074

84

4.00

2.56

2.56

7.60

4.40

012

.00

9.80

00.

0171

11.0

416

.56

0.00

733

27

.60

0.

0171

0.

0071

85

5.00

2.40

2.40

7.12

4.12

511

.25

9.18

70.

0171

10.8

016

.20

0.00

702

27.0

00.

0171

0.

0068

66

6.00

2.24

2.24

6.65

3.85

010

.50

8.57

50.

0171

10.5

615

.84

0.00

670

26.4

0

0.01

71

0.00

442

6.

76.

672.

132.

136.

333.

666

10.0

08.

165

0.01

7110

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15.6

00.

0064

826

.00

A

utho

r: S

tig B

erna

nder

ScR

0.20

cR=

10.0

00

m

C

alc.

S(x)

cR

/c(pe

ak)

0.33

S:a

N x

(n)

106.

33S:

ad(

xn),t

0.05

003

dN

92.8

5S:

a d(

xn),N

0.02

671

S(x)

= m

Cal

c.d(

xn,n

+1),N

0.02

332

S:

a N

x(n+

1)19

9.18

Slip

d(sl

ip)

= m

dx,N

-dx(

t)0.

0000

0S:

adx

(n+1

),N0.

0500

3S:

adx

(n+1

),t0.

0500

3 m

84

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

I(6

) 0

9-02

-24

Pag

e 7

D

ensi

tyc (

s)=

g el

=G

radi

ent

Gm

od.

c(p

eak)

E-m

ean

tel=

H =

W

idth

, b1,

0It

er.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

8.30

step

No

3x

5x

690

.36

Stre

ssx

6M

ean

kN/m

2kN

/m2

x6

98.6

5 n

z (n

)In

situ

In si

tuSh

ear

incr

e-Sh

ear

shea

r gf

- ge

lcu

-t el

t el

d(g

)d(

g)*d

zcu

(m

)sh

ear

shea

r st

ress

men

tst

ress

stre

sskN

/m2

kN/m

2kN

/m2

mkN

/m2

20to(

x,z)

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z)+d

tt(x

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dt(x

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)+dt

0.01

7112

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0171

12.0

018

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0.01

416

30

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0.

0171

0.01

389

1

1.00

3.04

3.04

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57.

125

21.3

717

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0.01

7111

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17.6

40.

0136

229

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0.01

71

0.

0133

6

22.

002.

882.

8813

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6.75

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16.8

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11.5

217

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0.01

309

28.8

00.

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282

33.

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7212

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15.9

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11.2

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256

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0.

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9

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562.

5612

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27

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0.01

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516

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14.0

620.

0171

10.8

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0.01

150

27.0

0

0.01

71

E

qu.I:

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0112

36

6.00

2.24

2.24

10.5

05.

250

15.7

513

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0.01

7110

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15.8

40.

0109

726

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6.

76.

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1310

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4.99

915

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12.4

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0171

10.4

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0.01

060

26.0

0

Aut

hor:

Stig

Ber

nand

erSc

R0.

20cR

=10

.000

m

Cal

c.S(

x)

cR/c(

peak

)0.

33S:

a N

x(n

)19

9.18

S:a

d(xn

),t0.

0825

80.

0500

3

dN12

8.99

S:a

d(xn

),N0.

0500

3

S(

x)=

mC

alc.

d(xn

,n+1

),N0.

0325

5

S:a

Nx(

n+1)

328.

17Sl

ipd(

slip

)=

mdx

,N-d

x(t)

0.00

000

S:a

dx(n

+1),N

0.08

258

S:a

dx(n

+1),t

0.08

258

m

85

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

I(7

) 0

9-02

-24

Pag

e 8

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc (

s)=

g el

=G

radi

ent

Gm

od.

c(p

eak)

E-m

ean

tel=

H =

W

idth

, b1,

0It

er.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

7.81

step

No

3x

6x

7

98.6

5St

ress

x 7

M

ean

kN/m

2kN

/m2

x 7

10

6.46

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cu

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2m

kN/m

220

to(x,

z)to(

x,z)

+dt

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

12.0

0E

qu.I:

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0.00

3.20

3.20

22.5

07.

500

30.0

0026

.250

0.01

7112

.00

18.0

00.

0276

5

30.0

0

0.01

71

0.

0253

51

1.00

3.04

3.04

21.3

77.

125

28.5

024

.937

0.01

7111

.76

17.6

40.

0230

429

.40

0.

0171

0.02

206

22.

002.

882.

8820

.25

6.75

027

.00

23.6

250.

0171

11.5

217

.28

0.02

108

28.8

0

0.01

71

0.

0203

13

3.00

2.72

2.72

19.1

26.

375

25.5

022

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0.01

7111

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16.9

20.

0195

528

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0.01

71

0.01

888

4

4.00

2.56

2.56

18.0

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000

24.0

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.000

0.01

7111

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16.5

60.

0182

2

27.6

0

0.

0171

0.01

763

55.

002.

402.

4016

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5.62

522

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19.6

870.

0171

10.8

016

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0.01

703

27.0

00.

0171

0.01

648

6

6.00

2.24

2.24

15.7

55.

250

21.0

018

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0.01

7110

.56

15.8

40.

0159

226

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0.01

71

0.

0104

3

6.7

6.67

2.13

2.13

15.0

04.

999

19.9

917

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0.01

7110

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15.6

00.

0152

226

.00

A

utho

r: S

tig B

erna

nder

ScR

0.20

cR=

10.0

00

m

C

alc.

S(x)

cR

/c(pe

ak)

0.33

S:a

N x

(n)

328.

17S:

ad(

xn),t

0.13

114

dN

179.

91S:

a d(

xn),N

0.08

258

S(x)

= m

Cal

c.d(

xn,n

+1),N

0.04

856

S:

a N

x(n+

1)50

8.08

Slip

d(sl

ip)

= m

dx,N

-dx(

t)0.

0000

0S:

adx

(n+1

),N0.

1311

4S:

adx

(n+1

),t0.

1311

4 m

0.13

114

86

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

uA

utho

r: St

ig B

erna

nder

Exa

mpl

e, r

esul

ts 0

9-02

-24

Pag

e 9

Step

xt

NS:

a dn

xy

H+y

Gra

dien

tm

kN/m

2kN

/m

mm

Gr

00.

000

0.00

00

00.

000.

0000

020

.000

0.01

Sta

ge I

133

.780

3.50

05.

10.

0013

33.7

80.

3378

020

.338

0.01

"2

54.8

704.

500

21.9

0.00

55

54.8

70.

5487

020

.549

0.01

"3

69.5

45

6.50

055

.70.

0140

69.5

50.

6954

520

.695

0.01

"4

80.0

959.

500

106.

30.

0267

80.1

00.

8009

520

.801

0.01

"5

90.3

5515

.000

199.

20.

0500

90.3

60.

9035

520

.904

0.01

"6

98.6

5022

.500

328.

20.

0826

98.6

50.

9865

020

.987

0.01

"7

106.

455

30.0

0050

8.1

0.13

1110

6.46

1.06

455

21.0

650.

01"

810

9.87

025

.000

591.

00.

1591

109.

871.

0987

021

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0.01

Sta

ge II

a9

112.

400

21.0

0064

1.1

0.18

2311

2.40

1.12

400

21.1

240.

01"

1111

4.76

217

.000

678.

50.

2055

114.

761.

1476

221

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0.01

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117.

015

13.0

0070

5.0

0.22

8611

7.01

1.17

015

21.1

700.

01"

1311

8.65

510

.000

718.

70.

2460

118.

651.

1865

521

.187

0.01

"14

128.

655

10.0

0078

6.7

0.35

8012

8.65

1.28

655

21.2

870.

01 S

tage

IIb

1515

3.65

510

.000

956.

70.

6823

153.

651.

5365

521

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0.01

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211.

025

10.0

0013

46.8

1.66

5521

1.02

2.11

025

22.1

100.

01"

1728

3.55

510

.000

1840

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3853

283.

552.

8355

522

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0.01

"18

454.

145

10.0

0030

00.0

9.52

8545

4.14

4.54

145

24.5

410.

01"

Dat

a ap

plyi

ng to

Cas

e 2

in in

Tab

le A

.III

(bas

ic m

odel

)Ex

ampl

e in

App

endi

x

87

Exa

mpl

e, sh

ear

disp

lace

men

t, va

riab

le c

u 0

9-02

-24

Pag

e 10

Aut

hor:

Stig

Ber

nand

er

Dat

a ap

plyi

ng to

Cas

e 2

in in

Tab

le A

.III

(bas

ic m

odel

)Ex

ampl

e in

App

endi

x

Land

slid

e sp

read

0

500

1000

1500

2000

2500

3000

3500 0.

000

50.0

0010

0.00

015

0.00

020

0.00

025

0.00

030

0.00

035

0.00

040

0.00

045

0.00

050

0.00

0

Coo

rdin

ate

(x)

Additional earth pressure, N kN/m

Ser

ie1

88

Exa

mpl

e, sh

ear

disp

lace

men

t, va

riab

le c

u 0

9-02

-24

Pag

e 11

Aut

hor:

Stig

Ber

nand

er

Dat

a ap

plyi

ng to

Cas

e 2

in in

Tab

le A

.III

(bas

ic m

odel

)Ex

ampl

e in

App

endi

x

Land

slid

e sp

read

0.00

05.

000

10.0

0015

.000

20.0

0025

.000

30.0

0035

.000

0.00

050

.000

100.

000

150.

000

200.

000

250.

000

300.

000

350.

000

400.

000

450.

000

500.

000

Coo

rdin

ate

(x)

Shear stress (t),kN/m2

Ser

ie1

89

Exa

mpl

e, sh

ear

disp

lace

men

t, va

riab

le c

u 0

9-02

-24

Pag

e12

Aut

hor:

Stig

Ber

nand

er

Dat

a ap

plyi

ng to

Cas

e 2

in in

Tab

le A

.III

(bas

ic m

odel

)Ex

ampl

e in

App

endi

x

Land

slid

e sp

read

024681012 0.00

050

.000

100.

000

150.

000

200.

000

250.

000

300.

000

350.

000

400.

000

450.

000

500.

000

Coo

rdin

ate

(x),

m

Downslope displacements, m

Ser

ie1

90

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-03

-09

Pag

e 1

Aut

hor:

Stig

Ber

nand

er

Den

sity

c(s)

=g

el =

Gra

dien

tG

mod

.c(

peak

)E-

mea

nte

l=H

=

Wid

th, b

1,0

Iter

.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

3.41

st

epN

o7x7

106.

46St

ress

x8

Mea

nkN

/m2

kN/m

2x

810

9.87

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cu

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.I:

5am

kN/m

2to(

x,z)

to(x,

z)+d

tt(x

,z)

dt(x

,z)

t(x,z

)+dt

0.01

71 t

(x,z

),pea

k0

0.00

3.20

3.20

30.0

0-5

.000

25.0

0027

.500

0.01

7112

.00

18.0

00.

0240

8

3030

.00

Stag

e II

a

S(tx

) = d

sR*(

su-t

x)/(s

u - s

uR)

0.

0171

0.02

183

11.

003.

043.

0428

.50

-4.7

5023

.750

26.1

250.

0171

11.7

617

.64

0.01

958

29.4

028

.50

Stag

e II

b

d (s

lip) =

Sum

dN

- Su

m d

(tx)

-Sc R

0.01

71

0.

0186

62

2.00

2.88

2.88

27.0

0-4

.500

22.5

0024

.750

0.01

7111

.52

17.2

80.

0177

328

.80

27.0

00.

0171

0.

0170

23

3.00

2.72

2.72

25.5

0-4

.250

21.2

5023

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0.01

7111

.28

16.9

20.

0163

228

.20

25.5

00.

0171

0.

0157

24

4.00

2.56

2.56

24.0

0-4

.000

20.0

0022

.000

0.01

7111

.04

16.5

60.

0151

2

27.6

024

.00

0.01

71

0.01

459

55.

002.

402.

4022

.50

-3.7

5018

.750

20.6

250.

0171

10.8

016

.20

0.01

405

27.0

022

.50

0.01

71

0.01

357

6

6.00

2.24

2.24

21.0

0-3

.500

17.5

0019

.250

0.01

7110

.56

15.8

40.

0130

826

.40

21.0

00.

0171

0.00

768

7

6.60

2.08

2.14

20.1

0-3

.350

16.7

5018

.425

0.01

7110

.42

15.6

20.

0125

326

.04

20.1

0

Aut

hor:

Stig

Ber

nand

erS(

cR)

0.20

cR10

.00

m

C

alc.

S(x)

0.05

000

cR/c(

peak

)0.

333

S:

a N

x(n

)50

8.08

S:a

d(xn

),t0.

1090

6

dN82

.97

S:a

d(xn

),N0.

1311

4

S(

x)=

0.05

000

m

C

alc.

d (xn

,n+1

),N0.

0279

2

S:a

Nx(

n+1)

591.

05Sl

ip d

(slip

)=

mdx

,N-d

x(t)

0.00

000

S:a

dx(n

+1),N

0.15

906

S:a

dx(n

+1),t

0.15

906

m

91

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-02

-04

Pag

e 2

Aut

hor:

Stig

Ber

nand

er

Den

sity

c(s)

=g

el =

Gra

dien

tG

mod

.c(

peak

)E-

mea

nte

l=H

=

Wid

th, b

1,0

Iter

.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

2.53

st

epN

o 9

x8

x9

109.

87St

ress

x9

Mea

nkN

/m2

kN/m

2x

911

2.40

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cu

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.I:

5am

kN/m

2to(

x,z)

to(x,

z)+d

tt(x

,z)

dt(x

,z)

t(x,z

)+dt

0.01

71 t

(x,z

),pea

k0

0.00

3.20

3.20

25.0

0-4

.000

21.0

0023

.000

0.01

7112

.00

18.0

00.

0212

2

3030

.00

Stag

e II

a

S(tx

) = d

sR*(

su-t

x)/(s

u - s

uR)

0.

0171

0.01

902

11.

003.

043.

0423

.75

-3.8

0019

.950

21.8

500.

0171

11.7

617

.64

0.01

681

29.4

028

.50

Stag

e II

b

d (s

lip) =

Sum

dN

- Su

m d

(tx)

-Sc R

0.01

71

0.

0159

32

2.00

2.88

2.88

22.5

0-3

.600

18.9

0020

.700

0.01

7111

.52

17.2

80.

0150

528

.80

27.0

00.

0171

0.

0143

93

3.00

2.72

2.72

21.2

5-3

.400

17.8

5019

.550

0.01

7111

.28

16.9

20.

0137

328

.20

25.5

00.

0171

0.

0131

84

4.00

2.56

2.56

20.0

0-3

.200

16.8

0018

.400

0.01

7111

.04

16.5

60.

0126

3

27.6

024

.00

0.01

71

0.01

215

55.

002.

402.

4018

.75

-3.0

0015

.750

17.2

500.

0171

10.8

016

.20

0.01

167

27.0

022

.50

0.

0171

0.

0112

46

6.00

2.24

2.24

17.5

0-2

.800

14.7

0016

.100

0.01

7110

.56

15.8

40.

0108

126

.40

21.0

00.

0171

0.00

634

7

6.60

2.08

2.14

16.7

5-2

.680

14.0

7015

.410

0.01

7110

.42

15.6

20.

0103

326

.04

20.1

0

Aut

hor:

Stig

Ber

nand

erS(

cR)

0.20

cR10

.00

Cal

c.S(

x)0.

0900

0cR

/c(pe

ak)

0.33

3S:

a N

x(n

)59

1.05

S:a

d(xn

),t0.

0922

6

dN50

.09

S:a

d(xn

),N0.

1590

6

S(

x)=

0.09

000

m

Cal

c.d (

xn,n

+1),N

0.02

320

S:

a N

x(n+

1)64

1.14

Slip

d(s

lip)

= m

dx,N

-dx(

t)0.

0000

0S:

a dx

(n+1

),N0.

1822

6S:

adx

(n+1

),t0.

1822

6 m

92

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-02

-04

Pag

e 3

Aut

hor:

Stig

Ber

nand

er

Den

sity

c(s)

=g

el =

Gra

dien

tG

mod

.c(

peak

)E-

mea

nte

l=H

=

Wid

th, b

1,0

Iter

.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

2.36

st

epN

o 10

x9

x10

112.

40St

ress

x10

Mea

nkN

/m2

kN/m

2x

1011

4.76

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cu

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.I:

5am

kN/m

2to(

x,z)

to(x,

z)+d

tt(x

,z)

dt(x

,z)

t(x,z

)+dt

0.01

71

t(x

,z),p

eak

00.

003.

203.

2021

.00

-4.0

0017

.000

19.0

000.

0171

12.0

018

.00

0.01

837

30

30.0

0St

age

IIa

S(

tx) =

dsR

*(su

-tx)/

(su -

suR

)

0.01

71

0.

0162

01

1.00

3.04

3.04

19.9

5-3

.800

16.1

5018

.050

0.01

7111

.76

17.6

40.

0140

429

.40

28.5

0St

age

IIb

d

(slip

) =Su

m d

N -

Sum

d(t

x)-S

c R0.

0171

0.01

321

22.

002.

882.

8818

.90

-3.6

0015

.300

17.1

000.

0171

11.5

217

.28

0.01

238

28.8

027

.00

0.01

71

0.01

176

33.

002.

722.

7217

.85

-3.4

0014

.450

16.1

500.

0171

11.2

816

.92

0.01

115

28.2

025

.50

0.01

71

0.01

065

44.

002.

562.

5616

.80

-3.2

0013

.600

15.2

000.

0171

11.0

416

.56

0.01

015

27

.60

24.0

00.

0171

0.

0097

25

5.00

2.40

2.40

15.7

5-3

.000

12.7

5014

.250

0.01

7110

.80

16.2

00.

0092

927

.00

22.5

00.

0171

0.

0089

16

6.00

2.24

2.24

14.7

0-2

.800

11.9

0013

.300

0.01

7110

.56

15.8

40.

0085

426

.40

21.0

00.

0171

0.00

500

7

6.60

2.08

2.14

14.0

7-2

.680

11.3

9012

.730

0.01

7110

.42

15.6

20.

0081

226

.04

20.1

0

Aut

hor:

Stig

Ber

nand

erS(

cR)

0.20

cR10

.00

m

Cal

c.S(

x)0.

1300

0cR

/c(pe

ak)

0.33

3S:

a N

x(n

)64

1.14

S:a

d(xn

),t0.

0754

5

dN37

.32

S:a

d(xn

),N0.

1822

6

S(

x)=

0.13

000

m

Cal

c.d (

xn,n

+1),N

0.02

319

S:

a N

x(n+

1)67

8.46

Slip

d(s

lip)

= m

dx,N

-dx(

t)0.

0000

S:a

dx(n

+1),N

0.20

545

S:a

dx(n

+1),t

0.20

545

m

93

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-02

-04

Pag

e 4

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc(

s)=

g el

=G

radi

ent

Gm

od.

c(pe

ak)

E-m

ean

tel=

H =

W

idth

, b1,

0It

er.

g

f =0.

030

16.0

0018

0.01

290.

0100

1400

30.0

033

6018

.00

20.0

dx (m

)2.

25

step

No

11x

10x

1111

4.76

Stre

ssx

11M

ean

kN/m

2kN

/m2

x11

117.

01 n

z (n

)In

situ

In si

tuSh

ear

incr

e-Sh

ear

shea

r gf

- ge

lcu

-t el

t el

d(g

)d(

g)*d

zcu

(m

)sh

ear

shea

r st

ress

men

tst

ress

stre

sskN

/m2

kN/m

2kN

/m2

Equ

.I:5a

mkN

/m2

to(x,

z)to(

x,z)

+dt

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

t

(x,z

),pea

k0

0.00

3.20

3.20

17.0

0-4

.000

13.0

0015

.000

0.01

7112

.00

18.0

00.

0155

1

3030

.00

Stag

e II

a

S(tx

) = d

sR*(

su-t

x)/(s

u - s

uR)

0.

0171

0.01

339

11.

003.

043.

0416

.15

-3.8

0012

.350

14.2

500.

0171

11.7

617

.64

0.01

127

29.4

028

.50

Stag

e II

b

d (s

lip) =

Sum

dN

- Su

m d

(tx)

-Sc R

0.01

71

0.

0104

82

2.00

2.88

2.88

15.3

0-3

.600

11.7

0013

.500

0.01

7111

.52

17.2

80.

0097

028

.80

27.0

00.

0171

0.

0091

33

3.00

2.72

2.72

14.4

5-3

.400

11.0

5012

.750

0.01

7111

.28

16.9

20.

0085

728

.20

25.5

00.

0171

0.

0081

14

4.00

2.56

2.56

13.6

0-3

.200

10.4

0012

.000

0.01

7111

.04

16.5

60.

0076

6

27.6

024

.00

0.01

71

0.00

729

55.

002.

402.

4012

.75

-3.0

009.

750

11.2

500.

0171

10.8

016

.20

0.00

691

27.0

022

.50

0.01

71

0.00

659

66.

002.

242.

2411

.90

-2.8

009.

100

10.5

000.

0171

10.5

615

.84

0.00

626

26.4

021

.00

dN-d

(tau)

0.01

71

0.

0036

5

76.

602.

082.

1411

.39

-2.6

808.

710

10.0

500.

0171

10.4

215

.62

0.00

592

26.0

420

.10

Aut

hor:

Stig

Ber

nand

erS(

cR)

0.20

cR10

.00

C

alc.

S(x)

0.16

999

cR/c(

peak

)0.

333

S:a

N x

(n)

678.

46S:

ad(

xn),t

0.05

865

m

dN26

.59

S:a

d(xn

),N0.

2054

5

S(

x)=

0.16

999

m

Cal

c.d (

xn,n

+1),N

0.02

319

S:

a N

x(n+

1)70

5.05

Slip

d(s

lip)

=0.

0000

mdx

,N-d

x(t)

0.00

00S:

a dx

(n+1

),N0.

2286

4S:

adx

(n+1

),t0.

2286

4 m

94

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-02

-04

Pag

e 5

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc(

s)=

g el

=G

radi

ent

Gm

od.

c(pe

ak)

E-m

ean

tel=

H =

W

idth

, b1,

0It

er.

g

f =0.

030

16.0

0018

0.01

290.

0100

1400

30.0

033

6018

.00

20.0

dx (m

)1.

64

step

No

11x

11x

1211

7.01

Stre

ssx

12M

ean

kN/m

2kN

/m2

x12

118.

65 n

z (n

)In

situ

In si

tuSh

ear

incr

e-Sh

ear

shea

r gf

- ge

lcu

-t el

t el

d(g

)d(

g)*d

zcu

g (x

=n)

(m

)sh

ear

shea

r st

ress

men

tst

ress

stre

sskN

/m2

kN/m

2kN

/m2

Equ

.I:5a

mkN

/m2

to(x,

z)to(

x,z)

+dt

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

t

(x,z

),pea

k0

0.00

3.20

3.20

13.0

0-3

.000

10.0

0011

.500

0.01

7112

.00

18.0

00.

0133

7

3030

.00

Stag

e II

a

S(tx

) = d

sR*(

su-t

x)/(s

u - s

uR)

0.

0171

0.01

128

11.

003.

043.

0412

.35

-2.8

509.

500

10.9

250.

0171

11.7

617

.64

0.00

919

29.4

028

.50

Stag

e II

b

d (s

lip) =

Sum

dN

- Su

m d

(tx)

-Sc R

0.01

71

0.

0084

42

2.00

2.88

2.88

11.7

0-2

.700

9.00

010

.350

0.01

7111

.52

17.2

80.

0076

928

.80

27.0

00.

0171

0.

0071

63

3.00

2.72

2.72

11.0

5-2

.550

8.50

09.

775

0.01

7111

.28

16.9

20.

0066

328

.20

25.5

00.

0171

0.

0062

14

4.00

2.56

2.56

10.4

0-2

.400

8.00

09.

200

0.01

7111

.04

16.5

60.

0058

0

27.6

024

.00

0.01

71

0.00

546

55.

002.

402.

409.

75-2

.250

7.50

08.

625

0.01

7110

.80

16.2

00.

0051

227

.00

22.5

00.

0171

0.

0048

46

6.00

2.24

2.24

9.10

-2.1

007.

000

8.05

00.

0171

10.5

615

.84

0.00

456

26.4

021

.00

0.01

71

0.

0026

5

76.

602.

082.

148.

71-2

.010

6.70

07.

705

0.01

7110

.42

15.6

20.

0042

626

.04

20.1

0

Aut

hor:

Stig

Ber

nand

erS(

cR)

0.20

cR10

.00

Cal

c.S(

x)0.

2000

0cR

/c=

0.33

3S:

a N

x(n

)70

5.05

S:a

d(xn

),t0.

0460

4 m

dN

13.6

1S:

a d(

xn),N

0.22

86

S(

x)=

0.20

000

mC

alc.

d (xn

,n+1

),N0.

0173

7

S:a

Nx(

n+1)

718.

66Sl

ip d

(slip

)=

0.00

00 m

dx,N

-dx(

t)0.

0000

S:a

dx(n

+1),N

0.24

60S:

adx

(n+1

),t0.

2460

m

95

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-02

-04

Pag

e 6

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc(

s)=

g el

=G

radi

entE

-mod

/cu

c(pe

ak)

E-m

ean

tel=

H =

W

idth

, b1.

0It

er.

g

f =0.

030

16.0

0018

0.01

290.

0100

1400

30.0

033

6018

.00

20.0

dx (m

)10

.00

st

epN

o 11

x12

x13

118.

65St

ress

x13

Mea

nkN

/m2

kN/m

2x

1312

8.65

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cug

(x=n

)

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.I:

5cm

kN/m

2to(

x,z)

to(x,

z)+d

tt(x

,z)

dt(x

,z)

t(x,z

)+dt

0.01

71

t(x

,z),p

eak

00.

003.

203.

2010

.00

0.00

010

.000

10.0

000.

0171

12.0

018

.00

0.01

337

30

30.0

0St

age

IIb

d

(slip

) =Su

m d

N -

Sum

d(t

x)-S

cR

0.01

71

0.

0112

81

1.00

3.04

3.04

9.50

0.00

09.

500

9.50

00.

0171

11.7

617

.64

0.00

919

29.4

028

.50

0.01

71

0.

0084

42

2.00

2.88

2.88

9.00

0.00

09.

000

9.00

00.

0171

11.5

217

.28

0.00

769

28.8

027

.00

0.01

71

0.00

716

33.

002.

722.

728.

500.

000

8.50

08.

500

0.01

7111

.28

16.9

20.

0066

328

.20

25.5

00.

0171

0.

0062

14

4.00

2.56

2.56

8.00

0.00

08.

000

8.00

00.

0171

11.0

416

.56

0.00

580

27

.60

24.0

00.

0171

0.

0054

65

5.00

2.40

2.40

7.50

0.00

07.

500

7.50

00.

0171

10.8

016

.20

0.00

512

27.0

022

.50

0.01

71

0.00

484

66.

002.

242.

247.

000.

000

7.00

07.

000

0.01

7110

.56

15.8

40.

0045

626

.40

21.0

00.

0171

0.00

265

7

6.60

2.08

2.14

6.70

0.00

06.

700

6.70

00.

0171

10.4

215

.62

0.00

426

26.0

420

.10

Aut

hor:

Stig

Ber

nand

erS(

cR)

0.20

cR10

.00

Cal

c.S(

x)0.

2000

0cR

/c=

0.33

3S:

a N

x(n

)71

8.66

S:a

d(xn

),t0.

0460

4

dN68

.00

S:a

d(xn

),N0.

2460

S(x)

=0.

2000

0 m

Cal

c.d (

xn,n

+1),N

0.11

200

S:

a N

x(n+

1)78

6.7

Slip

d(s

lip)

=0.

1120

mS:

a d x

(n+1

),N0.

3580

S:a

dx(n

+1),t

0.35

80 m

96

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-02

-04

Pag

e 7

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc(

s)=

g el

=G

radi

entE

-mod

/cu

c(pe

ak)

E-m

ean

tel=

H =

W

idth

, b1.

0It

er.

g

f =0.

030

16.0

0018

0.01

290.

0100

1400

30.0

033

6018

.00

20.0

dx (m

)25

.00

step

No

11x

13x

1412

8.65

Stre

ssx

14M

ean

kN/m

2kN

/m2

x14

153.

65 n

z (n

)In

situ

In si

tuSh

ear

incr

e-Sh

ear

shea

r gf

- ge

lcu

-t el

t el

d(g

)d(

g)*d

zcu

g (x

=n)

(m

)sh

ear

shea

r st

ress

men

tst

ress

stre

sskN

/m2

kN/m

2kN

/m2

Equ

.I:5c

mkN

/m2

to(x,

z)to(

x,z)

+dt

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

t

(x,z

),pea

k0

0.00

3.20

3.20

10.0

00.

000

10.0

0010

.000

0.01

7112

.00

18.0

00.

0133

7

3030

.00

Stag

e II

b

d (s

lip) =

Sum

dN

- Su

m d

(tx)

-ScR

0.

0171

0.01

128

11.

003.

043.

049.

500.

000

9.50

09.

500

0.01

7111

.76

17.6

40.

0091

929

.40

28.5

00.

0171

0.00

844

22.

002.

882.

889.

000.

000

9.00

09.

000

0.01

7111

.52

17.2

80.

0076

928

.80

27.0

00.

0171

0.

0071

63

3.00

2.72

2.72

8.50

0.00

08.

500

8.50

00.

0171

11.2

816

.92

0.00

663

28.2

025

.50

0.01

71

0.00

621

44.

002.

562.

568.

000.

000

8.00

08.

000

0.01

7111

.04

16.5

60.

0058

0

27.6

024

.00

0.01

71

0.00

546

55.

002.

402.

407.

500.

000

7.50

07.

500

0.01

7110

.80

16.2

00.

0051

227

.00

22.5

00.

0171

0.

0048

46

6.00

2.24

2.24

7.00

0.00

07.

000

7.00

00.

0171

10.5

615

.84

0.00

456

26.4

021

.00

0.01

71

0.

0026

5

76.

602.

082.

146.

700.

000

6.70

06.

700

0.01

7110

.42

15.6

20.

0042

626

.04

20.1

0

Aut

hor:

Stig

Ber

nand

erS(

cR)

0.20

cR10

.00

Cal

c.S(

x)0.

2000

0cR

/c=

0.33

3S:

a N

x(n

)78

6.66

S:a

d(xn

),t0.

0460

4 m

dN

170.

00S:

a d(

xn),N

0.35

80

S(

x)=

0.20

000

mC

alc.

d (xn

,n+1

),N0.

3242

8

S:a

Nx(

n+1)

956.

7Sl

ip d

(slip

)=

0.43

63 m

S:a

d x(n

+1),N

0.68

23S:

adx

(n+1

),t0.

6823

m

97

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-02

-04

Pag

e 8

Aut

hor:

Stig

Ber

nand

er

D

ensi

tyc(

s)=

g el

=G

radi

entE

-mod

/cu

c(pe

ak)

E-m

ean

tel=

H =

W

idth

, b1.

0It

er.

g

f =0.

030

16.0

0018

0.01

290.

0100

1400

30.0

033

6018

.00

20.0

dx (m

)57

.37

step

No

11x

14x

1515

3.65

Stre

ssx

15M

ean

kN/m

2kN

/m2

x15

211.

02 n

z (n

)In

situ

In si

tuSh

ear

incr

e-Sh

ear

shea

r gf

- ge

lcu

-t el

t el

d(g

)d(

g)*d

zcu

g (x

=n)

(m

)sh

ear

shea

r st

ress

men

tst

ress

stre

sskN

/m2

kN/m

2kN

/m2

Equ

.I:5c

mkN

/m2

to(x,

z)to(

x,z)

+dt

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

t

(x,z

),pea

k0

0.00

3.20

3.20

10.0

00.

000

10.0

0010

.000

0.01

7112

.00

18.0

00.

0133

7

3030

.00

Stag

e II

b

d (s

lip) =

Sum

dN

- Su

m d

(tx)

-ScR

0.

0171

0.01

128

11.

003.

043.

049.

500.

000

9.50

09.

500

0.01

7111

.76

17.6

40.

0091

929

.40

28.5

00.

0171

0.00

844

22.

002.

882.

889.

000.

000

9.00

09.

000

0.01

7111

.52

17.2

80.

0076

928

.80

27.0

00.

0171

0.

0071

63

3.00

2.72

2.72

8.50

0.00

08.

500

8.50

00.

0171

11.2

816

.92

0.00

663

28.2

025

.50

0.01

71

0.00

621

44.

002.

562.

568.

000.

000

8.00

08.

000

0.01

7111

.04

16.5

60.

0058

0

27.6

024

.00

0.01

71

0.00

546

55.

002.

402.

407.

500.

000

7.50

07.

500

0.01

7110

.80

16.2

00.

0051

227

.00

22.5

00.

0171

0.

0048

46

6.00

2.24

2.24

7.00

0.00

07.

000

7.00

00.

0171

10.5

615

.84

0.00

456

26.4

021

.00

0.01

71

0.

0026

5

76.

602.

082.

146.

700.

000

6.70

06.

700

0.01

7110

.42

15.6

20.

0042

626

.04

20.1

0

Aut

hor:

Stig

Ber

nand

erS(

cR)

0.20

cR10

.00

cR

Cal

c.S(

x)0.

2000

0cR

/c=

0.33

3S:

a N

x(n

)95

6.66

S:a

d(xn

),t0.

0460

4 m

dN39

0.12

S:a

d(xn

),N0.

6823

S(x)

=0.

2000

0 m

0.00

Cal

c.d (

xn,n

+1),N

0.98

325

S:

a N

x(n+

1)13

46.8

Slip

d(s

lip)

=1.

4195

mS:

a d x

(n+1

),N1.

6655

S:a

dx(n

+1),t

1.66

55 m

0.00

98

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-02

-04

Pag

e 9

0.00

Den

sity

c (s)

=g

el =

Gra

dien

tE-m

od/c

uc(

peak

)E-

mea

nte

l=H

=

Wid

th, b

1.0

Iter

.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

72.5

30.

00st

epN

o 11

x15

x16

211.

02St

ress

x16

Mea

nkN

/m2

kN/m

2x

1628

3.55

nz

(n)

In si

tuIn

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

cug

(x=n

)0.

00

(m)

shea

r sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.I:

5cm

kN/m

2to(

x,z)

to(x,

z)+d

tt(x

,z)

dt(x

,z)

t(x,z

)+dt

0.01

71

t(x

,z),p

eak

0.00

00.

003.

203.

2010

.00

0.00

010

.000

10.0

000.

0171

12.0

018

.00

0.01

337

30

30.0

0

d (s

lip) =

Sum

dN

- Su

m d

(tx)

-Sc R

0.

0171

0.01

128

0.00

11.

003.

043.

049.

500.

000

9.50

09.

500

0.01

7111

.76

17.6

40.

0091

929

.40

28.5

00.

0171

0.00

844

0.00

22.

002.

882.

889.

000.

000

9.00

09.

000

0.01

7111

.52

17.2

80.

0076

928

.80

27.0

00.

0171

0.

0071

63

3.00

2.72

2.72

8.50

0.00

08.

500

8.50

00.

0171

11.2

816

.92

0.00

663

28.2

025

.50

0.01

71

0.00

621

44.

002.

562.

568.

000.

000

8.00

08.

000

0.01

7111

.04

16.5

60.

0058

0

27.6

024

.00

0.01

71

0.00

546

55.

002.

402.

407.

500.

000

7.50

07.

500

0.01

7110

.80

16.2

00.

0051

227

.00

22.5

00.

0171

0.

0048

46

6.00

2.24

2.24

7.00

0.00

07.

000

7.00

00.

0171

10.5

615

.84

0.00

456

26.4

021

.00

0.01

71

0.

0026

5

76.

602.

082.

146.

700.

000

6.70

06.

700

0.01

7110

.42

15.6

20.

0042

626

.04

20.1

0

S(cR

)0.

20cR

10.0

0

C

alc.

S(x)

0.20

000

cR/c(

peak

)0.

333

S:a

N x

(n)

1346

.78

S:a

d(xn

),t0.

0460

4 m

dN

493.

20S:

a d(

xn),N

1.66

55

S(

x)=

0.20

000

mC

alc.

d (xn

,n+1

),N1.

7197

6

S:a

Nx(

n+1)

1840

.0Sl

ip d

(slip

)=

3.13

93 m

S:a

d x(n

+1),N

3.38

53S:

adx

(n+1

),t3.

3853

m

99

Lan

dslid

e sp

read

in p

assi

ve z

one

- dow

n-hi

ll pr

ogre

ssiv

e sl

ide

- va

riab

le c

u S

tage

II 0

9-02

-04

Pag

e 10

Den

sity

c (s)

=g

el =

Gra

dien

tE-m

od/c

uc(

peak

)E-

mea

nte

l=H

=

Wid

th, b

1.0

Iter

.

g f =

0.03

016

.000

180.

0129

0.01

0014

0030

.00

3360

18.0

020

.0dx

(m)

170.

59st

epN

o 11

x16

x17

283.

55St

ress

x17

Mea

nkN

/m2

kN/m

2x

1745

4.14

170.

59 n

z (n

)In

situ

In si

tuSh

ear

incr

e-Sh

ear

shea

r gf

- ge

lcu

-t el

t el

d(g

)d(

g)*d

zcu

g (x

=n)

(m

)sh

ear

shea

r st

ress

men

tst

ress

stre

sskN

/m2

kN/m

2kN

/m2

Equ

.I:5c

mkN

/m2

to(x,

z)to(

x,z)

+dt

t(x,z

)dt

(x,z

)t(x

,z)+

dt0.

0171

t

(x,z

),pea

k0

0.00

3.20

3.20

10.0

00.

000

10.0

0010

.000

0.01

7112

.00

18.0

00.

0133

7

3030

.00

Stag

e II

b

d (s

lip) =

Sum

dN

- Su

m d

(tx)

-ScR

0.

0171

0.01

128

11.

003.

043.

049.

500.

000

9.50

09.

500

0.01

7111

.76

17.6

40.

0091

929

.40

28.5

00.

0171

0.00

844

22.

002.

882.

889.

000.

000

9.00

09.

000

0.01

7111

.52

17.2

80.

0076

928

.80

27.0

00.

0171

0.

0071

63

3.00

2.72

2.72

8.50

0.00

08.

500

8.50

00.

0171

11.2

816

.92

0.00

663

28.2

025

.50

0.01

71

0.00

621

44.

002.

562.

568.

000.

000

8.00

08.

000

0.01

7111

.04

16.5

60.

0058

0

27.6

024

.00

0.01

71

0.00

546

55.

002.

402.

407.

500.

000

7.50

07.

500

0.01

7110

.80

16.2

00.

0051

227

.00

22.5

00.

0171

0.

0048

46

6.00

2.24

2.24

7.00

0.00

07.

000

7.00

00.

0171

10.5

615

.84

0.00

456

26.4

021

.00

0.01

71

0.

0026

5

76.

602.

082.

146.

700.

000

6.70

06.

700

0.01

7110

.42

15.6

20.

0042

626

.04

20.1

0

S(cR

)0.

20cR

10.0

0

C

alc.

S(x)

0.20

000

cR/c(

peak

)0.

333

S:a

N x

(n)

1839

.98

S:a

d(xn

),t0.

0460

4 m

dN

1160

.01

S:a

d(xn

),N3.

3853

S(x)

=0.

2000

0 m

Cal

c.d (

xn,n

+1),N

6.14

324

S:

a N

x(n+

1)30

00.0

Slip

d(s

lip)

=9.

2825

mS:

a d x

(n+1

),N9.

5285

S:a

dx(n

+1),t

9.52

85 m

100

Inte

grat

ion

of sh

ear

disp

lace

men

t in

vert

ical

ele

men

t 9

9-08

-23

Pag

e 6

%

%

kN/m

2kN

/m2

Iter

.g

f =0.

075

g el

=0.

0375

cu=

30.0

0te

l=20

.00

H =

20 m

dx (m

)2.

00st

epN

o 6

x5

57.4

4St

ress

M

ean

x6

59.4

4 n

z (n

)In

situ

Shea

r in

cre-

Shea

r sh

ear

gf -

gel

cu-t

el t

eld(

g )

d(g)

*dz

g (x

=n)

g (x

=n)

(m

)sh

ear

stre

ssm

ent

stre

ssst

ress

kN/m

2kN

/m2

kN/m

2E

qu.9

:1m

Equ

.9:1

to(x,

z)t(x

1,z)

dt(x

,z)

t(x2,

z)t(x

1)+t

(x2)

)/0.

0375

10.0

000

0.00

25.0

030

.00

-0.7

5529

.245

29.6

230.

0375

10.0

020

.00

-0.0

0142

0.

0647

0.04

85

0.03

7510

.00

20.0

0

-0.0

0138

11.

0023

.75

28.5

0-0

.717

27.7

8328

.141

0.03

7510

.00

20.0

0-0

.001

340.

0573

0.04

540.

0375

10.0

020

.00

-0

.001

312

2.00

22.5

027

.00

-0.6

8026

.321

26.6

600.

0375

10.0

020

.00

-0.0

0127

0.05

230.

0425

0.03

7510

.00

20.0

0

-0.0

0124

33.

0021

.25

25.5

0-0

.642

24.8

5825

.179

0.03

7510

.00

20.0

0-0

.001

200.

0481

0.03

990.

0375

10.0

020

.00

-0.0

0117

44.

0020

.00

24.0

0-0

.604

23.3

9623

.698

0.03

7510

.00

20.0

0-0

.001

13

0.04

450.

0375

0.03

7510

.00

20.0

0-0

.001

105

5.00

18.7

522

.50

-0.5

6621

.934

22.2

170.

0375

10.0

020

.00

-0.0

0106

0.04

113

0.03

516

0.03

7510

.00

20.0

0

-0.0

0103

6

6.00

17.5

021

.00

-0.5

2920

.472

20.7

360.

0375

10.0

020

.00

-0.0

0099

0.03

838

0.03

281

0.03

7510

.00

20.0

0

-0.0

0058

76.

6016

.75

20.1

0-0

.506

19.5

9419

.847

0.03

7510

.00

20.0

0-0

.000

950.

0367

40.

0314

1

d cR

0.10

c R15

.000

16.0

000

dx5 =

-0.0

0780

md (

slip

,tota

l)=

Cal

c.S(

x)0.

0101

5

N

(x,n)

508.

08d(

x6) =

0.13

114

0.

0179

5t(x

2,z)

29.2

5dN

9.25

d(xn

)0.

0710

3d(

S,x)

=0.

0101

5 m

C

alc.

t (x2,

z)29

.25

d(x n

,xn+

1)0.

0512

7

N(x

,n+1

)51

7.32

Slip

d(sl

ip.e

l)=

0.00

780

md(

x,n+

1)0.

1223

0d(

shea

r)=

0.14

909

m

101