Down-Hill Progressive Landslides in Soft Clays

2008:11 RESEARCH REPORT

Bernander Stig Down-hill Progressive Landslides in Soft Clays Triggering Disturbance Agents Slide Propagation over Horizontal or Gently Sloping Ground Sensitivity related to Geometry Down-hillClaysProgressiveSoft in Landslides

qcrit qcrit 2 2 clab = 25 kN/m kN/m 2 cpeak = 30 kN/m J = 16 kN/m2 H = 20 m Slope n:100 gf = 3 % E 25 gel = 1.09 % 5:100

20 tan E = 0.05 6:100

tan E = 0.06 7:100 tan E = 0.07 10 8:100 tan E = 0.08

cR/clab

0.2 0.4 0.6 0.8 1.0

Stig Bernander

Luleå University of Technology Division of Structural Engineering Division of Soil Mechanics and Foundation Engineering Department of Civil, Mining and Environmental Engineering

Universitetstryckeriet, Luleå 2008:11|: -1528|: -fr -- 08 ⁄11 -- 

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 E 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 diagram 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 frequently 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 outcome 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 geotechnical 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 occurred, the confidence in our capacity of identifying potential future slide hazards is seriously 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 horizontal 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 because, 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 condition immediately prior to the final disintegration of the passive zone should not be regarded 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 singular 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 landslide 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

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 investigations have in general not been truly successful in explaining the causes, the mechanisms 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 instance, 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 remained 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 potentially 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 – defined 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 studied soil volume affects the true risk of stability failure. The term previously used by the author for this phenomenon is ‘geometric sensitivity’. ******

v 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 accounted for – i.e. the analysis is not restricted only to varying deformation along the potential 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 developed 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 loading and to ambient drainage conditions, the impact of time may be considered in the analysis. 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 application. 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 progressive 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-

vi 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 horizontal ground.

Chapter 5 and Appendix A of this report are therefore focused on this issue. It is demonstrated there that, considering the combined effects of deformation and deformation softening, 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 hazard in slope stability, it is by no means the only factor conditioning progressive slope failure. 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 effects 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.

vii 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 assessments 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 applying 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 sensitivity 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 already 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 parameter; 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 sensitivity’.

viii 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 sensitivitet 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’.

*********

ix 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 exempel 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 2 hos det 240 000 m 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.

x 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 beroende 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 mellan 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 brottmekanisk 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.

xi 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 vidare 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.

xii 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

xiii 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

xiv Symbols and notations Greek letters

Į Coefficient defining the level of the earth pressure resultant

E, E (x), Ex Slope gradient at coordinate ‘x’

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

Jel Deviator strain, (angular strain) at elastic limit

Jf Deviator strain, (angular strain) at failure stress

M’ = Mi Angle of internal friction, drained conditions

Gx = G(x) Down-slope displacement

GN Displacement in terms of axial deformation generated by force Nx

GW Down-slope displacement in terms of deviator deformation

'G Differential of G

'x Differential of x coordinate

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

GSR Post peak slip in slip surface at ultimate residual shear strength sR

G ave Average down-slope displacement of the soil above the slip surface

DHx Level at which the mean down-slope displacement (G ave) is valid

H Longitudinal strain

Q Poisson’s ratio ȍ Coefficient relating the modulus of elasticity to the un-drained shear strength 3 U , U(z) Soil density (Mg/m )

V1, Major principal stress in axial tests

V3, Minor principal stress in axial tests

Vv Vertical normal stress

Vh Horizontal (down-slope) normal stress

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

Wel Shear stress (deviator stress) at elastic limit

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

W, W(x,o), Wx,o Total shear stress at failure plane (z = 0)

Wo,Wo(x,z), Wox,z In situ shear stress above failure plane

W , W(x,z), Wx,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(Jf) Un-drained shear strength

xv 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 GSR 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 Vh to Vv 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) p p E Rankine= ER Passive Rankine earth pressure resistance A E cr Earth pressure in a Point A, at which the shear stress WA = 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 G = Gcr

ǻLp=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

xvi 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 characteristics 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 deformations 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 inconclusive 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 western 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 sloping or virtually horizontal ground, and how this spread is affected by the degree of deformation-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 develop along firm bottom or weak layers more or less parallel to the inclining ground contour 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 softening soils readily materialize even in horizontal ground. (Cf Bernander, 1981b).

1 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 secondary 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.

2 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 characteristics, 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 additional 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 failure 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.

3 Table 2.1 Examples of down-hill progressive landslides. Locality Year Slide Area Triggering agent length [hec- involved tares] [m] 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, 2004 - - Widening of an up-slope railway Canada 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).

4 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 difficult 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 engineers, 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 explained 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 concept.) b) ….that the features of the finished slide could be related to the structure-mechanics involved – i.e. reasonable confirmation by computational back-analysis of salient slide features 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 – potential future landslides in other similar or slightly differing situations.

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 specific 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 accordance 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 geotechnical 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 cannot 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 simplicity 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 respect 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 studied potentially sliding soil mass. The significance of this specific feature can hardly be exaggerated.

7 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 account 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 ĺ sR*) – is expressed as a time dependent function of deviatory deformation and displacement – i.e. su (Ȗ, į, t) = ĭ(Ȗ,dȖ/dt, į,dį/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 ĮHx from the potential slip surface is considered in the computations. (Cf Figure 3.2.1a).

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

Figure 3.2.1 b Time dependent stress-deformation relationship Ĳ = ĭ(Ȗ, dȖ/dt, į, dį/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 beyond the critical value Ncr, in which case a ‘dynamic’ redistribution of stress and deformation takes place. Note that for x = Linstab , Ncr – ǻN = 0.

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 R Rankine resistance is not overcome, i.e Eo+Nx< Ep , this stage of equilibrium will remain permanent and the visible effects of the slide movement will be limited to minor displacements and cracking in the active zone. From Bernander (1981ĺ 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 respectively. 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 - ǻN = 0) – progressive failure will occur even if the current additional critical load were removed instantaneously. The corresponding parameters are denoted (įinstab) 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 ĺ 2000), it is possible and useful to distinguish between the different phases of a potential slide event as follows:

10 - 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, deformations 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 normally 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 R of the passive zone in terms of exceeding full passive Rankine resistance (EP ). (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- R sures at the foot of the slope are still well below the passive Rankine resistance (EP ). This means that the progressive failure mechanisms in Phase 4 - as in Phase 3 - are in principle 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 pressure distribution in this specific phase of a landslide that actually conditions the degree of disaster. (See Figures 3, 2, 3 & 4.4.3).

R It may be noted that, if the value of EP is not exceeded, Phase 4 will remain a permanent condition, in which the effects of the slide movement will be limited to moderate displacements over the slide area and cracking in the active zone. (Cf The slide at Rävekärr, Bernander, 2000.)

11 - 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 described 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 variation 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 horizontal 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 incidence of potential breakdown of the passive zone in Rankine failure. 3) The FDM-analysis described also underscores the importance of geometry to the brittleness of slope failure. Hence, deformation softening (sensitivity) – while being an important 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.

12 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 particular 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 deformation-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 report.

13 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 hydrological 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 millennia, the formal safety factor may – provided hydrology has remained unaffected by human 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 additional 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: Ĳ = 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 additional load or disturbance may for example be totally inconsequential in a protracted six

14 month scenario, whereas the same change happening in days or weeks may lead to a disastrous 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 human 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.

15 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 distances varying up to 150 m, inhibiting road and railway traffic for 10 and 19 days respectively. 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 section 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 movement. 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 geotechnical assessments.) The thorough field and laboratory investigations made in connection 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 concerned, 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

16 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 potentially 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.

17 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 precipitation 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 artesian 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 sensitive 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 periods 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 indicative 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 activity 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

18 may have been due to some kind of progressive failure process. He then, as it appears, argued 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 pressures 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

19 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 equipment.) Like Jakobson, Löfquist does not present any reason or argumentation as to why the piezometric 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 displacements by eventually blocking the riverbed in a major local slip-circular slide of conventional 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 condition.) 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 important 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 passive 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 initial 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 resulting 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 documented 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

20 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 author, 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 resistance 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 Wel/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).

21 Input data With reference to Figure 4.4.1a and Figure 4.4.1b, the following values of the characteristic parameters have been used in this study. (sR denotes the effective residual shear resistance under current rate of load application and ambient drainage conditions). In situ state condition: 2 sR/s = 1.00 Jel = 3.75 % Jf = 7.5 % Gel,o= Wel/Jel = 480 kN/m 2 2 2 s* = 24 kN/m , Wel = 18 kN/m Eel,o =2(1+Q)Gel,o = 60 su = 1440 kN/m 3 Ug = 15.5 kN/m ko = 0.55 (for horizontal ground) Q = 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 2 sR/su = 0.80 ** Jel = 2 % Jf = 4.0 % G cr = 0.3 m Gel,o = Wel/Jel = 1000 kN/m 2 2 2 su # =30 kN/m Wel = 20 kN/m Eel,o= 3Gel,o = 100su = 3000 kN/m 3 max Ug = 15.5 kN/m ko = 0,594 (computed) Eel,mean = 100 su,mean # Mean values applying to the initiation zone. Disturbance Condition II – failure initiation 2 sR/su = 0.60 ** Jel = 2 % Jf = 4,0 % G cr = 0.3 m Gel,o = Wel/Jel = 1000 kN/m

** The values of sR are to be adapted to the rate of load application and estimated drainage conditions 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: # 2 suR/su = 0.35-0.20 Jel = 1 % Jf = 2 % , G cr = 0.3 m Gel,o = Wel/Jel = 2400 kN/m

2 ## 2 2 su,o = 36 kN/m Wel = 24 kN/m Eel,o= 3Gel,o = 200su, = 7200 kN/m 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 2 constitutive relationship from Jel to Jf is a function of x with vertex at (su,Jf) and connecting tan- gentially at (Wel ,Jel).

22 [kN/m2]W

In situ condition, sR/s = 1.0

Disturbance condition II, sR/su = 0.6

Global failure, sR/su = 0.35

Jf = 7.5% 0.3 0.5 Gcr In situ condition J =4% f Gcr Disturbance condition II Jf = 2% Gcr Global failure [m] 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 Wo = UgHsinE. 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. ǻEo is positive). H(x) Ĳ0 (x,o) = Ȉo UgHsinE - ǻEo(x)/(b(x)ʘǻx ………………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).

23 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

G 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

G 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- 2 plied overload of only qcr | 275/18 = 15.3 kN/m extending 18 m in the slope direction. In 2 disturbance Condition II, the same overload would be qcr | 10,6 kN/m . (Cf Figure 4.4.2 ). By contrast, ideal-plastic failure analysis (I-PFA) based on local slip surfaces such as ABC 2 in Figure 4.4.2 indicates a corresponding value of qcr | 68 kN/m – 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 considerably higher values of qcr.

Condition I: sR/su = 0.80 2 2 2 qcr (Pr FA) = qcr (ABDF) | 15.3 kN/m << qcr (I-PFA) = qcr (ABC) = 68 kN/m < qcr (ABDE) | 118 kN/m

Condition II: sR/su = 0.60 2 2 2 qcr (Pr FA) = qcr (ABDF) | 10.6 kN/m << qcr (I-PFA) = qcr (ABC) = 68 kN/m < qcr (ABDE) | 118 kN/m

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.)

24 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 dependent in a similar way. In consequence, the analysis is not very sensitive to the time factor 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 compressibility 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 (Ginstab), 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 Ginstab | 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 formation. 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 released 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

G instab = 0.289 m Linstab= 228 m

25 4.4.3 Results of the FDM-analysis – Global failure condition The global failure condition, subsequent to the redistribution (related to progressive failure) 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 calculated 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 failure. 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 P Curve D E Rankine = Passive Rankine resistance, kN/m 2 Curve E Wo(x) = In situ shear stress distribution before progressive failure, kN/m Curve F W (x) = Shear stress distribution after progressive failure – i.e. the situation prior 2 to final disintegration in passive Rankine failure, kN/m Curve G G(x) = Displacement, m

26 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 transmitted 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 demonstrated in detail in Chapter 5, the slip surface under the valley floor is already fully developed 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. P Hence, in cases where the resulting maximum earth pressure Emax exceeds E Rankine (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). P If, on the other hand Emax does not exceed E Rankine, then the progressive failure redistribution 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 developed 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 P suR/su = 0.35-0.20 Nmax= 3112 kN/m Emax = 4969 kN/m E 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 P suR/su = 0.40-0.25 Nmax= 2682 kN/m Emax = 4554 kN/m E Rankine (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 P suR/su = 0.40-0.25 Nmax= 2682 kN/m Emax = 4558 kN/ m E Rankine (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 failure, 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

27 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 significant 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 considered as being high. As shown in Figure 4.4.3, the earth pressures resulting from the progressive failure redistribution 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 author, 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 occurred.

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

28 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 deformations inside and outside the soil mass are considered in the analysis. The results are namely remarkably insensitive to moderate variation of properties such as Ȗel,, Ȗf, įcr, 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 disturbance 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.

29 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 deformations 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 2 240000 m 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 definition, 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.

30 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 conventional (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 deformations 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 element 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 tendency 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.

31 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 precisely 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 example to be 20 m. The laboratory shear strength su of the clay at slip surface level is 25 2 2 kN/m , whereas the peak strength during the slide movement is set at 1.2·su = 30 kN/m . Hence, the conditions closely resemble those of the Surte slide.

C 1% 1:100

A H=20 m E J=17kN/m3 B A

W 2 [kN/m ] 30 1.2Su

Su 20

10 1.2Su SuR x m 0 50 100 150 200

(A Lcr [kN/m] 4000

P ER A Ecr Eo x m G [m] 0.3 0.2 0.1 x m

Figure 5.1.1 Data and notations defining conditions under the valley floor. In the figure Eo = A A in situ earth pressure at rest. E cr = E0 + N cr § 3300 kN/m denotes the earth pressure resistance that can be mobilized, when the residual shear strength sR is just attained at Point A. P P ER = E Rankine § 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- A cant. (Note: A precise value of Lcr can only be defined by assigning the ratio of Nx/N cr a limiting minimum value such as for instance 1/10. The x-axis is an asymptote to the curves of Nx, Ĳx and įx). A įAcr = Deformation at a point A corresponding to N cr. P P Eo = in situ earth pressure at rest. ER = E Rankine = Passive Rankine earth pressure.

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

32 A A In Figure 5.1.1, E cr = Eo + N 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 A just started to develop after the peak shear stress has been mobilized. The force N denotes the additional load at A following progressive failure redistribution of stresses and deformations. 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 A P In Figure 5.2.1, it is shown how the ratio of E cr/ER and the values of Lcr and įA,cr vary with different ratios (sR/su) between the actual residual shear resistance (sR) and the laboratory shear strength (su). Interestingly, as may be concluded from the figure, all the computed values are practically independent of the sR/su – ratio. This rather surprising circumstance relates to the fact that the area of the integral EA = Ĳxdx changes very little with decreasing values of sR, since also the location of the peak shear stress (Wmax =1.2·su) is somewhat displaced in the direction of x with falling values A P of sR/su. In fact, the independency (in the example) of the parameters E cr/ER , Lcr and įA,cr of the ratio sR/su is so pronounced that a theoretical justification for this phenomenon is quite possible.

H = 20 m E A B A x Lcr A E P Lcr cr ER A m P G$cr [m] Ecr E R 200 1.0 Lcr 0.8 150 0.6 Gcr 100 0.4 50 0.2

k=sR/su 0.4 0.5 0.60.7 0.8 0.9 1.0

A P Figure 5.2.1 ‘Mobilizable’ earth pressure in Point A in terms of E cr/ER . Displacement įcr and critical length Lcr as functions of the ratio k = sR/su. Denotations as in Figure 5.1.1. In the A P example Eo = 2100 kN/m, E cr =3300 kN/m and ER =4400 kN/m.

33 A P Thus, for all values of sR/su, the ratio of E cr/ER is constant and in the current case equal to A A 0.75. The corresponding deformation įA,cr induced by the pressure E cr = (Eo + N cr) 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). A The crucial question at this stage is then: “What happens when the earth pressure E at A P point A (i.e. where x = 0) increases from E cr = 0.75·ER to full Rankine resistance P (1.00·ER ), as a result of for instance earth pressure redistribution due to progressive failure formation further up-slope?” A P Figure 5.2.2 shows the full deformation curves at a stage, when E has attained 0.95· ER , P i.e. 95 % of the total passive Rankin resistance. In the current example 0.95· ER § 4180 kN/m.

C 1:100

P H=20 m 3 0.95ER J=17 kN/m B A

W [kN/m2] 30

W 20 1.2Su max

Su 10 0.4Su x m

0 50 100 150 200 250

'Lp Lcr

G $ W [m] 2 [kN/m ] 1.2S 3 30 u SSRu/0.4 Su 20 2 SSRu/0.6 1.2S 10 u SR SSRu/0.8 J= 3% G 1 0 0.2

SSRu/1.0 x m 0 50 100 150 200 250

A A P Figure 5.2.2 Computed deformation curves for an earth pressure E = E0 + N = 0.95· ER at

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

34 As opposed to the conditions shown in Figure 5.2.1, the deformation įA at A - as well as the extension of the total failure zone (LP = Lcr +ǻLP) – now exhibit pronounced depend- ence of the current value of the ratio of sR/su. In Table 5.2.1 below, ǻLP denotes the growth A P P of LP related to the increase of E from 0.75· ER to 0.95· ER .

A P Table 5.2.1 Values of įA , LP and ǻLP for E = 0.95·ER § 4180 kN/m

sR/su įA LP ǻLP [m] [m] [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 displacement, the horizontal-bound failure zone A-B will propagate far into level ground. For instance, for sR = 0.4su and įA § 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 deformation 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 horizontal 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 particularly 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 triggering load load qcrit is there studied as a function of the slope E and the residual shear resistance cR/clab. The results are illustrated in Figure 5.2.3.

qcrit

qcrit 2 clab = 25 kN/m 2 kN/m 2 cpeak = 30 kN/m H = 20 m Slope n:100 J = 16 E 25 2 gf = 3 % 5:100 gel = 1.09 %

20 tan E = 0.05 6:100 tan E = 0.06 7:100 tan E = 0.07 10 8:100 tan E = 0.08

cR/clab

0.2 0.4 0.6 0.8 1.0

Figure 5.2.3 Critical up-slope triggering load qcrit as function of slope E 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 developed far out under the valley floor, causing already at this stage substantial horizontal displacement 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 displaced 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 displacements 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.

36 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 under 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 contour 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- OAC PFA) the safety factors Fs for failures along OAC are actually identical for both slopes. (Cf Table 6.2.1). ODE Admittedly, the safety factor Fs 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- ODE face in sloping ground. (Nevertheless, as the safety factor Fs 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 difference in geometry.

37 2 qcr = 34.5 kN/m +41

+21 C F +20 H=20 m Ecr O ±0 G A B x, L x [m] 0 100 200 300 400 500 600 Slope A 2 +41 qcr = 0 kN/m

+21 C +20 E F Ecr O ±0 G H=20 m D A B x, L x [m] 0 100 200 300 400 500 600 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

OAC 2 Slope A Fs = 1.64 Ncr = 1537 kN/m qcr = 77 kN/m OAC 2 Slope B Fs = 1.64 Ncr = 1537 kN/m qcr = 77 kN/m ODE 2 “ (Fs = 1.43) (Ncr = 1030 kN/m) qcr = 52 kN/m

Progressive failure analysis

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

max p max E0 Ncr Ecr qcr Ex ER /Ex Lcr įx=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 --

38 2 Notably, Slope A can sustain a crest load of qcr = 34.5 kN/m roughly corresponding to about 2 meters of earth fill, whereas Slope B cannot carry any additional load at all on its 2 crest, i.e. qcr = 0 kN/m . Disturbance condition - Progressive failure analysis

If, due to disturbance of some sort, sR/su approaches a value of 0.9, both slopes are still p max inherently stable with their respective crest loads but the safety factor in terms of ER /Ex 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.

2 p Table 6.2.3 Disturbance condition (partially drained). sR/su = 0.9 su = 25 kN/m , ER = 4200 kN/m

max p max E0 Ncr Ecr qcr Ex ER /Ex Lcr įx=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 conditions 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- p sive Rankine resistance (ER ) will be exceeded in both slopes causing extensive failure by spreading.

2 Table 6.2.4 Failure condition (un-drained) suR/su = 0.6 su = 30 kN/m p ER =4400 kN/m

max p max * E0 Ncr Ecr qcr Ex ER /Ex Lcr įx=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 p the presence of the considerable crest load in Slope A, the limiting resistance ER is tran- scended for 150 < x < 355 – i.e. over a distance of ‘only’ 205 m, while in Slope B the corresponding 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’

39 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 prediction 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’ (sensitivity > 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 conventionally 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 analyses 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.

40 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 2 this great disaster, destabilizing about 240 000 m of land that had remained stable for several 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 (Wo) 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 differing 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 implication 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 extremely 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

41 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 triggered 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 driving 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 connection 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 undisturbed 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 surveying 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 approach 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 equilibrium. The proper approach to predicting events of this nature is by thorough geotechnical investigation 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 gently sloping ground having collapsed and heaved substantially as a result of exceeding passive 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 formation in the phase immediately prior to disintegration in passive failure (i.e. Phase 4 accord-

42 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 superfluous, since after all, it is the prevailing earth pressure distribution in this phase of an ongoing 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 deformation softening soils. The FDM-analysis outlined in Chapter 3 thus explicitly predicts 2 and explains the remarkable phenomenon of a soil volume of some 160 000 m 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 considering 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 readily 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 concurrent 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 phenomenon 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 deformations 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 –

43 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 situations, where loss of life and serious destruction of economic assets may result from a potential 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 considering 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 consecutive 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 disturbance 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 relationships 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.

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 conventional equilibrium method, based on perfectly plastic behaviour of the clay material.

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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 computer program for the analysis of downward progressive slope failure based on these principles had been applied by the author in practical engineering cases. Yet, the approach to progressive slope failure in view was presented in more detail considerably later (in May 2000) in a licentiate report from Luleå University of Technology denoted 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 horizontal 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).

E= 1:100

x x=L

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

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 constitutive 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 specific likely laws of soil behavior in shear. However, an advantage of the approach described 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, ǻx, dz, ǻz Differentials of the coordinates x and z d(x),N Differential displacement due to axial compression generated by force Nx

Ȉd(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 Ȉd(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

Vx, V(x), Vx 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.

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) A E 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 denotations used in the current context, see Figure A2.a.

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

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)sinE(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 H(x) to(x,o) = 6o Ȗzdz sinE(x) - ȖwDW(x) sinE(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/>EelH(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.

xn Hence, the total displacement at xn can be written as 6o d(x),Nǻx

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

DH(x) d(shear) = Ȉd(xn),t = 6o >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 xn ĮH 6o d(x),Nǻx = 6o d(xn,z),t ǻz ………A.4:5

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

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

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 displacement 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 compatibility Equ.A.4:5 is substituted for Equation A.4:5a.

xn ĮH el 6o d(x),Nǻx = 6o >(d(xn),t - d (x)@ǻz + Sx ………A.4:5a where to is defined as before H(x) to(x,o) = 6o >Ȗ(z)dz@sinE(x) - Ȗ WDW(x)sinE(x) - 'Eo(x)/b(x)dx and el d = 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 xn ĮH el 6o d(x),Nǻx = 6o >(d(xn,z),t - d (x)@ǻz + 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 Wx,z = I (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 = I1 (tx,z) ……A.4:6a

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 range In 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 = Ggx 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 defined in Figure A.3)

Non-linear range In 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 2 tx,z - tel = 2 (c - tel)>(gx,z- gel)/(gf - gel)@ - (c - tel)>(gx,z-gel)/(gf - gel)@ or ………A.I:2

2 >(gx,z-gel)/(gf -gJel)@ - 2>(gx,z-gel)/(gf -gel)@ + (tx,z- tel)/(c- tel) = 0 …… …A.I:2a

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

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

1/2 gx,z = gf -(gf -gel)>1- (tx,z- tel)/(c- tel)@ ………A.I:2c

Check: For tx,z = tel Æ gx,z = gel and for tx,z = c Æ gx,z = gf Q.E.D.

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 = gftel/(2c- tel) ………A.I:3

2 2 For instance, if gf = 4.0 %, c = 30 kN/m , tel = 20 kN/m then gel = 0,0420/(230-20) = 0.02 = 2.0 %

Range: to < tx,z < c and to > tel The difference in deviator deformation from t = t0(x,z) to t = tx,z according to Equ. AI:2c is 1/2 1/2 'gx,z = (gf -gel)>1- (t0(x,z) - tel)/(c- tel)@ - >1- (tx,z - tel)/(c- tel)@ ………A.I:2d Hence, the total shear displacement in Stage I when to < tx,z < c and to > tel for the soil column the length of which is 'x becomes ĮH Ȉd(xn),t = o6 'gx,z'z ĮH 1/2 1/2 = o6 (gf -gel)(>1- (t0(x,z) - tel)/(c- tel)@ - >1- (tx,z - tel)/(c- tel)@ )'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 1/2 (gf -gel)(1 - >1 - (tx,z- tel)/(c- tel)@ ) ………A.I:2e

Range: to < tx,z < c and to < tel Combining Equations. A.I:1a and A.I:2e, the total shear displacement in Stage I is ĮH Ȉd(xn),t = o6 'gx,z'z ĮH 1/2 =o6 [(tel –t0(x,z))/G+ (gf -gel)(1 - >1 - (tx,z- tel)/(c- tel)@ )]'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 Æ tx,z(max) (peak shear stress) when to > tel ĮH 1/2 1/2 Ȉd(xn),t = o6 (gf -gel)(>1- (t0(x,z) - tel)/(c- tel)@ - >1- (tx,z(max) - tel)/(c- tel)@ )' ..A.I.4(tmax)

Deformation from to Æ tx,z(max) (peak shear stress) when to < tel ĮH 1/2 Ȉd(xn),t = o6 [(tel –t0(x,z))/G+ (gf -gel)(1 - >1 - (tx,z(max) - tel)/(c- tel)@ )]'z .…A.I:4a(tmax)

Elastic rebound – Range: tx,z(max) > tx,z > cR The elastic rebound in an element ('x·'z) el d(d (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 el ĮH d (x,z) = - o6 (t(x,z(max)-tx,z)/G'z. (See Figure A.3) where ĮH is the distance from the failure plane to the resultant force N.

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) Æ 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 Æ tx,z(max) when to > tel ĮH 1/2 1/2 Ȉd(xn),t = o6 >(gf - gel)(>1- (t0(x,z) - tel)/(c- tel)@ - >1- (tx,z(max) - tel)/(c- tel)@ ) –

Rebound tx,z(,max)Æ tx,z Slip deformation cÆ 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 Æ tx,z(max) when to < tel ĮH 1/2 Ȉd(xn),t = o6 [(tel - t0(x,z))/G+(gf - gel)(1 - >1-(tx,z(max) - tel)/(c - tel)@ ) –

Rebound tx,z(max)Æ tx,z Slip deformation cÆ 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 displacement 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 Æ tx,z(max) ĮH 1/2 1/2 Ȉd(xn),t = o6 [gf-gel)(>1- (to(x,z) - tel)/(c- tel)@ – >1- (tx,z(max) - tel)/(c- tel)@ ) –

Rebound tx,z(max)Æ 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 Æ tx,z(max) ĮH 1/2 Ȉd(xn),t = o6 [(tel-t0(x,z))/G+(gf-gel)(1 - >1-(tx,z(max)-tel)/(c - tel)@ )

Rebound tx,z(max)Æ cR Slip deformation when tx,z = cR

– (tx,z(max) - cR)/G]'z + SCR + Sslip ………A.I:5c

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 J = 16 kN/m3 Gradient of ground ahead of the foot of the steeper slope = 1:100 2 Peak shear strength, cpeak 30 kN/m 2 Laboratory shear strength, clab 25 kN/m 2 Elastic limit, tel 18 kN/m 2 Residual shear resistance, cR 0.4 clab = 10 kN/m Slip at residual shear resistance, SCR 0.20 m 2 2 Mean shear strength of soil profile, cmean 24 kN/m (csurface = 18 kN/m ) Deviatory deformation at peak shear, gf 3% = 0.03 Deviatory deformation at elastic limit, gel 1.29% = 0.0129 (from Equ. A.I.3) 2 Gmodulus = tel/gel 1400 kN/m 2 Emean = 2(1+v)Gmean= 2(1+0.5)1120 3360 kN/m Furthermore, the following assumptions are made in the current case: a) The in situ earth pressure Eo is taken to be 2 2 Eo = 0.8 JH /2 = 0.81620 /2 = 2560 kN/m b) The passive Rankine resistance is defined as 2 2 ERankine = ȖH /2 + 2cH = 1620 /2 + 23020 = 3200 + 1200 = 4400 kN/m

E= 1:100

N N H = 20 m x L

x x = L

x1 = L 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 E 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 +

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 (Ȉd(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 Ȉd(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 assuming 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 computerized 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 Ȉd(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 distribution of the parameters (N), (t) and Ȉd(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

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 between x = LRankine= 283 m and x = LF = 454 m, i.e. over a distance of ǻL =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 according to Table A.III.

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 E is in radians.

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) Ȉd(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) Ȉd(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) ǻL = LF-LR 102 m +67 % -52 % (Length of passive Rankine zone) LN = 3000 kN 313 m +45 % -34 % (Length of total failure zone) Ȉd(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. ǻL = LF - LR in Table A.III) varies from ǻL = 49 m in the near plastic condition (when cR/c = 0.9) to ǻL = 171 m when the residual strength only amounts to cR/cpeak = 0.333 - i.e. cR/clab = 0.4.

Conclusions The 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 analysis 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 performed 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.

Table A.III R L F F- 69 98 98 49 98 98 69 98 69 69 98 69 69 N 171 171 171 171 L 102.1

Diff. F N dn = F F N F = N x x=L 312.8 5.49 N 7 R N dn = R Synopsis I - Modified Synopsis 09 03 03 st. c R i R = N R pressure = N x h 179 1.28 248 3.51 194 1.40 292 4.05 213 2.31 311 6.27 283 3.39 454 9.53 158 0.83 206 2.17 210 2.02 308 5.56 282 3.53 453 9.98 208 2.00 306 5.52 184 2.10 254 5.58 205 2.21 303 6.18 182 1.48 251 3.99 262 3.22 433 9.41 181 1.31 250 3.54 269 2.24 440 6.54 209 1.59 307 4.43 180 1.45 249 3.96 183 1.18 252 3.17 x=L 210.8 1.9 1840 kN/m 3000 kN/m N R c tu eart tu i pressure res dn N 0.387 0.218 0.353 0.382 0.267 0.359 0.273 0.405 0.368 0.323 0.367 0.323 0.340 0.351 0.386 0.368 h ng tong 0.336 n s i di R c R ng ng c ne eart ili N 129.3 112.0 132.1 120.3 128.3 119.2 122.8 123.2 125.4 123.2 128.3 158.5 134.5 129.5 135.2 132.8 127.8 x=L ki

Author: Stig BernanderAuthor: correspon

mm m m m m m m R Preva Value of force N when t Ran (R) /

c = = N = = 652 719 118.7 0.246 781 792 797 804 879 886 894 948 950 960 995 N

t=c 1076 1232 1043 1066 k 910.2

Landslide spread, Landslide to

R (Rankine) o

Nc corr. E E x = dn t = c t = c 106.5 0.131 102.6 0.092 orce N f

f o - value x 2800 h

Pa m m engt (s) l o k c

l E a i uenc ry crust ry m fl (Rankine)- mation d r Down-slope displacement In E ) is here taken to be 1840 kN/m. kN/m. to be 1840 ) is here taken

R = ow = = l (R) e c R R b shear slip shear E-mo- corr. corr. corr. corr. corr. corr. corr. corr. corr. corr. kPa N dn x stance i shear mation stance i t Pa % i (el) ear res k t h res 16.3 1.1 16.29 0.3 15.9 3535 104.9 0.119 m l li ua c i 7 ace s id f % 3.0 10 0.75 10 0.30 15 3200 105.4 0.126 3.0 20 2.00 20 0.30 15 2800 101.3 0.135 2.0 10 0.50 10 0.30 15 4800 105.4 0.084 2.0 15 0.86 15 0.30 15 4200 103.1 0.086 2.0 15 0.86 20 0.30 15 4200 103.1 0.086 ast l ur to passive Rankine failure (N Rankine passive to Res S E = = =

Peak shear resistance

results from sensitivity analysis from results Pa 30 3.0 20 1.50 15 0.15 15 3000 103.5 0.137 30 2.0 15 0.67 15 0.30 10 4500 101.3 0.095 30 2.0 21 1.08 27 0.20 15 30 3.0 18 1.29 15 0.20 18 3360 106.5 0.131 30 3.0 15 1.00 15 0.20 15 3375 105.3 0.134 30 3.0 20 1.50 15 0.30 15 3000 103.5 0.137 30 3.0 20 1.50 20 0.20 20 3333 107.3 0.131 30 3.0 10 0.60 10 0.30 10 3333 103.3 0.142 30 3.0 15 1.00 20 0.20 20 3750 108.9 0.127 30 3.0 20 1.50 20 0.30 20 3333 107.3 0.131 30 3.0 15 1.00 20 0.30 20 3750 108.9 0.127 = k (R) (s) (el) 28.53 2. c c t c Input data computations from Results Peakshear shear Assoc. defor-resist- Elas- tic limitance Assoc. defor- Recid. mation resist. Assoc. defor- Surf. resist. Mean dulus to toto to to to to to to to

Legend: Case Case 1 Case 2Case 30 3.0 18 1.29 10 0.20 18 3360 Case 5 Case 4 25 Case 6 Case 3 25 Case 7 Case 8 Case 9 number Case 16 Case 17 Case 10 Case 11 25 Case 12 25 Case 13 Case 14 25 Case 15 reference reference asedonFDM spread sheet accordingto Bernande Table A.III -

Compilationcomputationsfailure progressive of of results b corresponding values Mean N the varying, E in situ and slightly kN/m = 4400 * With E(Rankine)

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 related to the very fact that deformations within the sliding body are considered in the computations.

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.

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 seemingly 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 remarkable 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 required 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- 2 slope embankment load (qcrit, kN/m ) - that can set off a downward progressive slide movement, 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

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 ȕ = 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 3 H = 20 m, Density of clay Ȗ = 16 kN/m Slope gradient tan ȕ = 0.05 2 Peak shear strength cpeak = 30 kN/m 2 Laboratory shear strength clab = 25 kN/m 2 Elastic limit tel = 16 kN/m 2 Residual shear resistance cR = 0.4clab = 10 kN/m Slip at residual shear resistance SCR = 0,20 m 2 2 Mean shear strength of soil profile cmean = 22,5 kN/m . (csurface = 15 kN/m ) 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) 2 2 Gmodulus = tel/gel = 1467 kN/m Emean = 2(1+v)Gmean= 2(1+0.5)1120 = 3300 kN/m 2 Emodulus = 2(1+v)1467 = 4400 kN/m

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 (Ȉd(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 = qH) 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 displacement Ȉd(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.

When the effect of a predefined force (NL = qH 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 = qH. (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 Ȉd(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 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= qcritH, when the post-peak shear stress tx,o in the developing shear band precisely attains the locally prevailing in situ shear stress tx,o , which in the 2 2 studied case is equal to Ȗ·H·sin ȕ § 16 kN/m . Hence, qcrit =389.2/20 = 19.46 kN/m

The critical length of influence (Lcrit) is at this point in the order of 100 m and the displacement 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 potential failure zone in the soil mass ahead. If the load qcrit is exceeded, the virtually dynamic redistribution of earth pressures and deformations takes place (i.e. Phase 3, according to the main report), possibly resulting eventually in a total slope failure.

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 (Ȉd(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.

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 (ȕ) is varied from tan ȕ = 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 qcrit 2 2 clab = 25 kN/m [kN/m ] c = 30 kN/m2 peak H = 20 m Slope n:100 gf = 3 % J = 16 E 25 2 qel = 1.09 % 5:100

20 tan E = 0.05 6:100 tan E = 0.06 7:100 tan E = 0.07 10 8:100 tan E = 0.08

cR/clab

0.2 0.4 0.6 0.8 1.0

Figure B.3 Results from variation of key parameters according to Table B.II. It is of considerable 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 accentuates 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 softening 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 difficult to estimate from laboratory tests – depending in reality as it is on poorly defined clay sensitivity, rate of load application and local drainage conditions. Yet, sensitivity is of course likely to strongly affect possible further development of a progressive slide.

The diagram also accentuates the acute hazard related to local up-slope earth fills and embankments 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 2 qcrit = 9.1 kN/m (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 values 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 equilibrium concept.

Conclusions 1. The spread-sheet analysis performed in respect of loading, likely to trigger down-hill progressive 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 insignificant 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 implication that a supporting embankment - in for instance road construction - may actually turn out to be an agent triggering major landslide disaster.

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 below the critical load (qcrit) by some predefined factor of safety, this relative insensitivity to strain-softening constitutes a favourable circumstance considering that residual shear resistance 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 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 slide hazard between the current FDM analysis and that based on perfect plasticity - is to an overwhelmingly 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 present State-of-the-Art in Soil Mechanics.

Table B.II instab instab 446 N = 0 N = 1228 174.3 569.3 179.9 153.5 144.8 135.5 208.7 161.4 151.4 264.0 209.1 L 0

0 at N = 0 at = mm dn N = 0.196 o o o o o o t t t t t t

N instab f > > = = > > at to

00.366 00.243 00.168 00.105 00.420 00.224 00.135 00.492 00.276 R R R o R R R N c c c c c c t ue o kN/m x =L Instability length = length Instability l = va 09 05 10 k crit N File: Triggering load Triggering File: crit crit H = 20.0 at at cm m dn x = L N 0.156 111.8 0.132 112.8 0.098 113.1 0.062 104.1 0.203 118.5 0.168 119.2 0.124 119.4 0.077 110.3 0.175 130.1 0.107 120.8 0 0.226 139.4 0.137 129.7 en post-pea h ) R c crit (t= q 25.65 0.23221.17 121.6 0.227 127.9 q kN/m2 crit orce L w L orce o f t

crit f N t = 389.2 19.46 319.4 15.97 232.1 11.61 140.6 7.03 441.3 22.07 357.8 17.89 258.4 12.92 155.5 7.78 512.9 423.3 304.2 15.21 181.7 9.09 512.9 421.8 344.0 17.20 204.6 10.23 N = ue o l Value of force N when post-peak t Va t = c t =

Bernander Stig Author: = = c rit rit dn x = L c instab instab t = t =

N L

c N t = computations Sensitivity analysis Results from Results load Critical L 0.20 200.7 0.058 95.8 325.0 13.33 0.133 104.9 0.00 0.164 251.6 m kN/m m m kN/m 0.20 295.30.20 0.081 231.30.20 98.0 0.065 171.00.20 97.5 0.051 105.10.20 98.2 0.034 295.30.20 89.6 0.081 231.30.20 98.0 0.065 171.00.20 97.5 0.051 105.10.20 98.2 0.034 295.30.20 89.6 0.081 231.30.20 98.0 0.065 171.00.20 97.5 0.051 105.10.20 98.2 0.034 295.30.20 89.6 0.081 231.30.20 98.0 0.065 171.00.20 97.5 0.051 105.1 98.2 0.034 89.6 orce N orce mation f

f R o Pa c h k 10.0 10.0 10.0 10.0 15.0 15.0 15.0 15.0 20.0 20.0 20.0 20.0 22.5 22.5 22.5 22.5

engt l t

lab i l c a / m i R 0.400 0.400 0.400 0.400 0.600 0.600 0.600 0.600 0.800 0.800 0.800 0.800 0.900 0.900 0.900 0.900 li c shear shear slip corr. corr. corr. corr. or (max) (crit) (instab) corr. c i uenc rit/H rit/H fl ast c l Down-slope displacement at x=L displacement Down-slope In

shear E mation = = = =

rit c (el) (el) 20 m m 20 t q x dn t = h ka 1 kPa% % H stance i Pa 30 3.0 16 1.09 30 3.0 16 1.09 30 3.0 16 1.09 30 3.0 16 1.09 3030 3.030 3.0 16 3.0 16 1.09 16 1.09 1.09 30 3.0 16 1.09 30 3.0 16 1.09 3030 3.030 3.0 1630 3.0 16 1.09 3.0 16 1.09 30 16 1.09 30 3.0 1.09 30 3.0 16 3.0 16 1.09 16 1.09 1.09 k ear stress stress ear res l

ance mation Peak Assoc. shearshear Elas- tic resist- defor- Assoc. limit Recid. defor- Recid. Assoc. resist. resist. defor-to to to to h ear strengt h ua tu s tu id . s o i t Pa b k n s

Res i La

) = = =

Peak shear resistance Peak shear

= (lab) (R) o c c c t Input dataInput Depth to failure surface tan(Gr

Case Gra- In situ Legend: Case 6Case 0.060 19.17 Case 1Case 0.050 15.98 Case 2Case 3Case 0.060 19.17 4Case 0.070 22.35 5Case 0.080 25.52 0.050 15.98 7Case 8Case 0.070 22.35 9Case 0.080 25.52 0.050 15.98 number (Gr) stress Case 13 0.050 15.98 Case 11 0.070 22.35 Case 14 0.060Case 15 19.17 0.070Case 16 22.35 0.080 25.52 Case 10 0.060 19.17 Case 12 0.080 25.52 reference dient shear Table B.II - Downhill progressive slide- triggering loads -

Compilation of results of progressive failure computations failure progressive of results Compilation of based on FDM sheet spread to according Bernander Mean values

Table B.III

c /F 0.49 0.256 0.288 0.346 0.484 0.464 0.385 0.557 0.528 0.433 0.651 0.636 0.526 0.605 0.398 0.235 pr 0.361

F crit at 34.2 -51.5 m N um i x = L r analysis * ilib

crit 7.0 104.0 7.8 110.3 9.1 120.8 6.5 100.0 q 19.5 111.8 16.0 112.7 11.6 112.9 22.1 118.5 17.9 119.2 12.8 119.2 25.6 121.6 21.2 127.9 15.2 130.1 17.2 139.4 10.2 129.7 kN/m2 Progressve f. PrFA c equ i ast Fc l nal Con- 2.23 1.91 1.67 1.53 2.20 1.88 1.65 1.50 2.19 1.85 1.61 1.46 1.58 1.43 1.55 q=18 # ventio- 2009 05 22 File: Triggering load - Synopsis IIFile: Triggering load Synopsis - Stig BernaderAuthor: - p l / L ea h q safety (crit) id kN 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.018.018.0 18.0 25.6 21.1 18.0 engt q l on on /

/ f d = m2 m2 pr crit 0.87 1.87 kN q=18 factor ase q

F b opeo l # f o m a s f 8.36 1.081 7.38 0.887 6.39 0.645 5.41 0.391 8.53 1.226 7.49 0.994 6.43 0.713 5.40 0.432 8.60 1.425 7.63 1.176 6.51 0.845 5.38 0.505 5.56 5.56 6.58 0.956 5.37 0.568 5.41 0.363 actor Max. f s o i ety ety f ys f l q Sa kN/m2

ana IPFA analysis IPFA analysis #

fill Hsin Gr Hsin m1.00 L a m -E p kN/ orce N IPFA analysis IPFA analysis from Results E f

y f o g h 16.0 200016.0 20.0 200016.0 0.0499 20.0 150.43 200016.0 0.0599 20.0 132.88 2000 0.0698 20.0 114.98 16.0 0.0797 2000 97.30 16.0 20.0 200016.0 0.0499 20.0 153.46 200016.0 0.0599 20.0 134.77 200016.0 0.0698 20.0 115.82 200016.0 0.0797 20.0 2000 97.14 16.0 0.0499 20.0 154.84 200016.0 0.0599 20.0 137.31 200016.0 0.0698 20.0 117.26 200016.0 0.0797 20.0 2000 96.87 16.0 0.0499 20.0 100.00 200016.0 0.0599 20.0 100.00 2000 0.0698 20.0 118.50 0.0797 96.64 16.0 2000 20.0 0.0797 97.41 kN/m3 engt t l i

R l Pa c a m 5.0 k i 10.0 10.0 10.0 10.0 15.0 15.0 15.0 15.0 20.0 20.0 20.0 20.0 22.5 22.5 22.5 22.5 li

c i H H / uenc lab rit ast c fl c l / Down-slope displacement at x = L R 0.400 0.400 0.400 0.400 0.400 0.600 0.600 0.600 0.600 0.800 0.800 0.800 0.800 0.900 0.900 0.900 0.900 In

N c E

= = = =

rit L c

(el) (el) t dn q

% kPa1 % h 20 m 20 m = Pa H k

ear strengt stance i h Pa % k ear stress res l

ance ance mation Peakshear Lab shear defor- resist- resist- shear Assoc. limit Elas- tic resist. Recid. resist. Recid. shear Densit clay shear of =4H c height h ua tu s id o i oratory s oratory t Pa b k 25.52 30 25 3.0 16 15.98 30 25 3.0 16 19.1722.35 30 30 25 25 3.0 3.0 16 16 19.1722.35 3025.52 30 25 30 25 3.0 25 3.0 16 3.025.52 16 30 16 25 3.0 16 15.98 30 25 3.0 16 15.98 30 25 3.0 16 19.1722.35 3025.52 30 25 30 25 3.0 25 3.019.17 16 3.022.35 30 16 25.52 30 16 25 30 25 3.0 25 3.0 16 3.0 16 16 15.98 30 25 3.0 16 n s

Res i

La ) = = =

Peak shear resistance ent shear

i = (lab) (R) o c c c t Input data data Input Depth to failure surface tan(Gr

Case Gra- In situ Case 9 0.050 Case 2 0.060 Case 3 0.070 Case 4 0.080 Case 6 0.060 Case 7 0.070 Case 8 0.080 Case 1 0.050 Case 5 0.050 number (Gr) stress Legend: Case 4a 0.080 Case 13 0.050 Case 14 0.060 Case 15 0.070 Case 10 0.060 Case 11 0.070 Case 12 0.080 Case 16 0.080 Table B.III - Downhill progressive slide - triggering loads - triggering slide - - Downhill progressive Table B.III reference d Mean values Deviations - max/min Comparison of slopehazard and based on PrFA analysis analysis. slope hazard based on conventional (IPFA)

Appendix C. Excel spread sheet - Exemplification Author: Stig Bernander Page 1 No 5 Data applying to Case 2 in in Table A.III (basic model) 09-02-24 Modified Arketyp Input data: H = 20 m Gradient = tan Gr = 1:100, Gr = arc tan 0,01 = 0,573 Dgr Density = 16 kN/m3 (Residual shear strength) (Surface strength) (Mean shear strength) Cpeak = 30 kN/m2 Clab = 25 kN/m2 CR = 10 kN/m2 CR/Clab = 0,40 Cs = 18 kN/m2 C mean = 24 kN/m2 Shear deformation at failure (gf) = 3 % CR/Cpeak = 0,333 tel = 18 kN/m2 Post-peak deformation at CR = 0,20 m Shear deform. at elastic limit (gel) = 1,29 %

Landslide spread

3.00000 2.50000 2.00000 1.50000 Serie1 1.00000

0.50000Slope surface 0.00000 0.000 50.000 100.000 150.000 200.000 250.000 300.000 Coordinate (x)

79 Equations: t x,z < t el t x,z < t el gx,z(1) = tx/G, gel = gf*tel/(2c - tel) and G = tel/gel Stage I Equ. I:1a S:a d(x,z),t = Integral (gx,z,(1))dz to < txz < c, to > tel t o < t xz < c t o > t el gx,z(4) = (gf-gel)*(1- ROT(1 - (to(x,z) -tel)/(c - tel)) Stage I Equ. I:4 S:a d(x,z),t = Integral (gx,z,(4))dz t o < t el - ROT(1 - (t(x,z) - tel)/(c - tel))) t o < t el gx,z(4a) = (tel -to,(x,z)/G + (gf-gel)*(1-ROT(1 - (t(x,z) -tel)/(c- tel))) Equ. I:4a S:a d(x,z),t = Integral (gx,z,(4a))dz c < t xz < c R , t o > t el c< t xz t el t xz > c R gx,z(5) = (gf-gel)*(ROT(1 - (to(x,z)-tel)/(c - tel)) - Stage IIa Equ. I:5 S:a d(x,z),t = Integral (gx,z,(5a))dz + t o < t el - ROT(1-(t(x,z,max)-tel)/(c - tel))) - (t(x,z,max)-t(x,z))/G t o < t el t xz < c R +ScR(c-tx,z)/(c-cR) gx,z(5a) =(tel-to,(x,z))/G+(gf-gel)(1-ROT(1-(t(x,z,max)-tel)/(c-tel)))- Equ. I:5a S:a d(x,z),t = Integral (gx,z,(5a))dz + t x,o = c R - (t(x,z,max)-t(x,z))/G t x,o = c R +Sc R(c-tx,z)/(c-cR) gx,z(5b) = gx,z(5) (or gx,z(5a)) - (t(x,z,max)-cR)/G Stage IIb Equ. I:5b S:a d(x,z),t = Integral (gx,z,(5b))dz+ScR+Sslip or Equ. I:5c Landslide spread in passive zone - down-hill progressive slide - variable cu Stage I (1) 09-02-24 Page 2 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c (peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 33.78 step No 1 xo 0.00 Stress Mean kN/m2 kN/m2 x1 33.78 n z (n) In situ* In situ* Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:1a m kN/m2 20 to(x,z) t(x,z) dt(x,z) t(x,z)+dt 0.0171 12.00 0 0.00 3.20 3.20 3.20 0.300 3.500 3.350 0.0171 12.00 18.00 0.00021 30.00 0.0171 0.00021 1 1.00 3.04 3.04 3.04 0.285 3.325 3.182 0.0171 11.76 17.64 0.00021 29.40 0.0171 0.00020 2 2.00 2.88 2.88 2.88 0.270 3.150 3.015 0.0171 11.52 17.28 0.00020 28.80 0.0171 0.00020 3 3.00 2.72 2.72 2.72 0.255 2.975 2.847 0.0171 11.28 16.92 0.00019 28.20 0.0171 0.00019

80 4 4.00 2.56 2.56 2.56 0.240 2.800 2.680 0.0171 11.04 16.56 0.00019 27.60 0.0171 0.00018 5 5.00 2.40 2.40 2.40 0.225 2.625 2.512 0.0171 10.80 16.20 0.00018 27.00 0.0171 0.00017 6 6.00 2.24 2.24 2.24 0.210 2.450 2.345 0.0171 10.56 15.84 0.00017 26.40 0.0171 0.00011 6.7 6.67 2.13 2.13 2.13 0.200 2.333 2.233 0.0171 10.40 15.60 0.00016 26.00 20.00 6.00 12 18.00 Author: Stig Bernander ScR 0.20 cR= 10.000 m Calc. S(x) cR/c(peak) 0.33 S:a N x(n) 0.00 S:a d(xn),t 0.00127 dN 5.07 S:a d(xn),N 0.0000 S(x)= m Calc. d(xn,n+1),N 0.00127 S:a Nx(n+1) 5.07 Slip d(slip) = 0.00000 m dx,N-dx(t) 0.00000 S:a dx(n+1),N 0.00127 S:a dx(n+1),t 0.00127 m * In situ shear stress to be modified by the expression dt =(E(n+1) -E(n))/dx(E(n+1) and E(n) are in situ earth pressures) Landslide spread in passive zone - down-hill progressive slide - variable cu Stage I (2) 09-02-24 Page 3 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c (peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 21.09 step No 1 x1 x 2 33.78 Stress x 2 Mean kN/m2 kN/m2 x 2 54.87 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:1a m kN/m2 20 to(x,z) t(x,z) dt(x,z) t(x,z)+dt 0.0171 12.00 0 0.00 3.20 3.20 3.50 1.000 4.500 4.000 0.0171 12.00 18.00 0.00093 30.00 0.0171 0.00091 1 1.00 3.04 3.04 3.32 0.950 4.275 3.800 0.0171 11.76 17.64 0.00090 29.40 0.0171 0.00089 2 2.00 2.88 2.88 3.15 0.900 4.050 3.600 0.0171 11.52 17.28 0.00087 28.80 0.0171 0.00086 3 3.00 2.72 2.72 2.97 0.850 3.825 3.400 0.0171 11.28 16.92 0.00084 28.20 0.0171 0.00082 4 4.00 2.56 2.56 2.80 0.800 3.600 3.200 0.0171 11.04 16.56 0.00081 27.60 81 0.0171 0.00079 5 5.00 2.40 2.40 2.62 0.750 3.375 3.000 0.0171 10.80 16.20 0.00077 27.00 0.0171 0.00076 6 6.00 2.24 2.24 2.45 0.700 3.150 2.800 0.0171 10.56 15.84 0.00074 26.40 0.0171 0.00049 6.7 6.67 2.13 2.13 2.33 0.667 2.999 2.666 0.0171 10.40 15.60 0.00071 26.00

Author: Stig Bernander ScR 0.20 cR= 10.000 m Calc. S(x) cR/c = 0.33 S:a N x(n) 5.07 S:a d(xn),t 0.00551 dN 16.87 S:a d(xn),N 0.0013 S(x)= m Calc. d(xn,n+1),N 0.00424 S:a Nx(n+1) 21.94 Slip d(slip) = 0.00000 m dx,N-dx(t) 0.00000 S:a dx(n+1),N 0.00551 S:a dx(n+1),t 0.00551 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage I (3) 09-02-24 Page 4 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c (peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 14.68 step No 2 x 2 x 3 54.87 Stress x 3 Mean kN/m2 kN/m2 x 3 69.55 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:1a m kN/m2 20 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 12.00 0 0.00 3.20 3.20 4.50 2.000 6.500 5.500 0.0171 12.00 18.00 0.00236 30.00 0.0171 0.00232 1 1.00 3.04 3.04 4.27 1.900 6.175 5.225 0.0171 11.76 17.64 0.00228 29.40 0.0171 0.00225 2 2.00 2.88 2.88 4.05 1.800 5.850 4.950 0.0171 11.52 17.28 0.00221 28.80 0.0171 0.00217 3 3.00 2.72 2.72 3.82 1.700 5.525 4.675 0.0171 11.28 16.92 0.00213 28.20 0.0171 0.00209 4 4.00 2.56 2.56 3.60 1.600 5.200 4.400 0.0171 11.04 16.56 0.00205 27.60 82 0.0171 0.00201 5 5.00 2.40 2.40 3.37 1.500 4.875 4.125 0.0171 10.80 16.20 0.00196 27.00 0.0171 0.00192 6 6.00 2.24 2.24 3.15 1.400 4.550 3.850 0.0171 10.56 15.84 0.00188 26.40 0.0171 0.00124 6.7 6.67 2.13 2.13 3.00 1.333 4.332 3.666 0.0171 10.40 15.60 0.00181 26.00

Author: Stig Bernander ScR 0.20 cR= 10.000 m Calc. S(x) cR/c(peak) 0.33 S:a N x(n) 21.94 S:a d(xn),t 0.01399 dN 33.75 S:a d(xn),N 0.00551 S(x)= m Calc. d(xn,n+1),N 0.00848 S:a Nx(n+1) 55.69 Slip d(slip) = m dx,N-dx(t) 0.00000 S:a dx(n+1),N 0.01399 S:a dx(n+1),t 0.01399 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage I (3) 09-02-24 Page 5 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c (peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 10.55 step No 3 x 3 x 4 69.55 Stress x 4 Mean kN/m2 kN/m2 x 4 80.10 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:1a m kN/m2 20 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 12.00 0 0.00 3.20 3.20 6.50 3.000 9.50 8.000 0.0171 12.00 18.00 0.00450 30.00 0.0171 0.00443 1 1.00 3.04 3.04 6.17 2.850 9.02 7.600 0.0171 11.76 17.64 0.00436 29.40 0.0171 0.00429 2 2.00 2.88 2.88 5.85 2.700 8.55 7.200 0.0171 11.52 17.28 0.00422 28.80 0.0171 0.00414 3 3.00 2.72 2.72 5.52 2.550 8.07 6.800 0.0171 11.28 16.92 0.00407 28.20 0.0171 0.00399 4 4.00 2.56 2.56 5.20 2.400 7.60 6.400 0.0171 11.04 16.56 0.00391 27.60 83 0.0171 0.00383 5 5.00 2.40 2.40 4.87 2.250 7.12 6.000 0.0171 10.80 16.20 0.00375 27.00 0.0171 0.00366 6 6.00 2.24 2.24 4.55 2.100 6.65 5.600 0.0171 10.56 15.84 0.00358 26.40 0.0171 0.00236 6.7 6.67 2.13 2.13 4.33 2.000 6.33 5.332 0.0171 10.40 15.60 0.00346 26.00

Author: Stig Bernander ScR 0.20 cR= 10.000 m Calc. S(x) cR/c(peak) 0.33 S:a N x(n) 55.69 S:a d(xn),t 0.02671 dN 50.64 S:a d(xn),N 0.01399 S(x)= m Calc. d(xn,n+1),N 0.01272 S:a Nx(n+1) 106.33 Slip d(slip) = m dx,N-dx(t) -0.00001 S:a dx(n+1),N 0.02671 S:a dx(n+1),t 0.02671 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage I (5) 09-02-24 Page 6 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c (peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 10.26 step No 3 x 4 x 5 80.10 Stress x 5 Mean kN/m2 kN/m2 x 5 90.36 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:1a m kN/m2 20 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 12.00 0 0.00 3.20 3.20 9.50 5.500 15.000 12.250 0.0171 12.00 18.00 0.00843 30.00 0.0171 0.00830 1 1.00 3.04 3.04 9.02 5.225 14.25 11.637 0.0171 11.76 17.64 0.00817 29.40 0.0171 0.00804 2 2.00 2.88 2.88 8.55 4.950 13.50 11.025 0.0171 11.52 17.28 0.00790 28.80 0.0171 0.00776 3 3.00 2.72 2.72 8.07 4.675 12.75 10.412 0.0171 11.28 16.92 0.00762 28.20 0.0171 0.00748 4 4.00 2.56 2.56 7.60 4.400 12.00 9.800 0.0171 11.04 16.56 0.00733 27.60 84 0.0171 0.00718 5 5.00 2.40 2.40 7.12 4.125 11.25 9.187 0.0171 10.80 16.20 0.00702 27.00 0.0171 0.00686 6 6.00 2.24 2.24 6.65 3.850 10.50 8.575 0.0171 10.56 15.84 0.00670 26.40 0.0171 0.00442 6.7 6.67 2.13 2.13 6.33 3.666 10.00 8.165 0.0171 10.40 15.60 0.00648 26.00

Author: Stig Bernander ScR 0.20 cR= 10.000 m Calc. S(x) cR/c(peak) 0.33 S:a N x(n) 106.33 S:a d(xn),t 0.05003 dN 92.85 S:a d(xn),N 0.02671 S(x)= m Calc. d(xn,n+1),N 0.02332 S:a Nx(n+1) 199.18 Slip d(slip) = m dx,N-dx(t) 0.00000 S:a dx(n+1),N 0.05003 S:a dx(n+1),t 0.05003 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage I (6) 09-02-24 Page 7 Density c(s)= g el = Gradient G mod. c (peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 8.30 step No 3 x 5 x 6 90.36 Stress x 6 Mean kN/m2 kN/m2 x 6 98.65 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 m kN/m2 20 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 12.00 Equ.I:4a 0 0.00 3.20 3.20 15.00 7.500 22.500 18.750 0.0171 12.00 18.00 0.01416 30.00 0.0171 0.01389 1 1.00 3.04 3.04 14.25 7.125 21.37 17.812 0.0171 11.76 17.64 0.01362 29.40 0.0171 0.01336 2 2.00 2.88 2.88 13.50 6.750 20.25 16.875 0.0171 11.52 17.28 0.01309 28.80 0.0171 0.01282 3 3.00 2.72 2.72 12.75 6.375 19.12 15.937 0.0171 11.28 16.92 0.01256 28.20 0.0171 0.01229 4 4.00 2.56 2.56 12.00 6.000 18.00 15.000 0.0171 11.04 16.56 0.01203 27.60 85 0.0171 0.01176 5 5.00 2.40 2.40 11.25 5.625 16.87 14.062 0.0171 10.80 16.20 0.01150 27.00 0.0171 Equ.I:1a 0.01123 6 6.00 2.24 2.24 10.50 5.250 15.75 13.125 0.0171 10.56 15.84 0.01097 26.40 0.0171 0.00723 6.7 6.67 2.13 2.13 10.00 4.999 15.00 12.497 0.0171 10.40 15.60 0.01060 26.00

Author: Stig Bernander ScR 0.20 cR= 10.000 m Calc. S(x) cR/c(peak) 0.33 S:a N x(n) 199.18 S:a d(xn),t 0.08258 0.05003 dN 128.99 S:a d(xn),N 0.05003 S(x)= m Calc. d(xn,n+1),N 0.03255 S:a Nx(n+1) 328.17 Slip d(slip) = m dx,N-dx(t) 0.00000 S:a dx(n+1),N 0.08258 S:a dx(n+1),t 0.08258 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage I (7) 09-02-24 Page 8 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c (peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 7.81 step No 3 x 6 x 7 98.65 Stress x 7 Mean kN/m2 kN/m2 x 7 106.46 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 m kN/m2 20 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 12.00 Equ.I:4a 0 0.00 3.20 3.20 22.50 7.500 30.000 26.250 0.0171 12.00 18.00 0.02765 30.00 0.0171 0.02535 1 1.00 3.04 3.04 21.37 7.125 28.50 24.937 0.0171 11.76 17.64 0.02304 29.40 0.0171 0.02206 2 2.00 2.88 2.88 20.25 6.750 27.00 23.625 0.0171 11.52 17.28 0.02108 28.80 0.0171 0.02031 3 3.00 2.72 2.72 19.12 6.375 25.50 22.312 0.0171 11.28 16.92 0.01955 28.20 0.0171 0.01888 4 4.00 2.56 2.56 18.00 6.000 24.00 21.000 0.0171 11.04 16.56 0.01822 27.60 86 0.0171 0.01763 5 5.00 2.40 2.40 16.87 5.625 22.50 19.687 0.0171 10.80 16.20 0.01703 27.00 0.0171 0.01648 6 6.00 2.24 2.24 15.75 5.250 21.00 18.375 0.0171 10.56 15.84 0.01592 26.40 0.0171 0.01043 6.7 6.67 2.13 2.13 15.00 4.999 19.99 17.496 0.0171 10.40 15.60 0.01522 26.00

Author: Stig Bernander ScR 0.20 cR= 10.000 m Calc. S(x) cR/c(peak) 0.33 S:a N x(n) 328.17 S:a d(xn),t 0.13114 dN 179.91 S:a d(xn),N 0.08258 S(x)= m Calc. d(xn,n+1),N 0.04856 S:a Nx(n+1) 508.08 Slip d(slip) = m dx,N-dx(t) 0.00000 S:a dx(n+1),N 0.13114 S:a dx(n+1),t 0.13114 m 0.13114 Landslide spread in passive zone - down-hill progressive slide - variable cu Author: Stig Bernander Example, results 09-02-24 Page 9 Step x t N S:a dn x y H+y Gradient m kN/m2 kN/m m m Gr 0 0.000 0.000 0 0 0.00 0.00000 20.000 0.01 Stage I 1 33.780 3.500 5.1 0.0013 33.78 0.33780 20.338 0.01 " 2 54.870 4.500 21.9 0.0055 54.87 0.54870 20.549 0.01 " 3 69.545 6.500 55.7 0.0140 69.55 0.69545 20.695 0.01 " 4 80.095 9.500 106.3 0.0267 80.10 0.80095 20.801 0.01 " 5 90.355 15.000 199.2 0.0500 90.36 0.90355 20.904 0.01 " 6 98.650 22.500 328.2 0.0826 98.65 0.98650 20.987 0.01 " 7 106.455 30.000 508.1 0.1311 106.46 1.06455 21.065 0.01 " 8 109.870 25.000 591.0 0.1591 109.87 1.09870 21.099 0.01 Stage IIa 9 112.400 21.000 641.1 0.1823 112.40 1.12400 21.124 0.01 " 11 114.762 17.000 678.5 0.2055 114.76 1.14762 21.148 0.01 " 12 117.015 13.000 705.0 0.2286 117.01 1.17015 21.170 0.01 "

87 13 118.655 10.000 718.7 0.2460 118.65 1.18655 21.187 0.01 " 14 128.655 10.000 786.7 0.3580 128.65 1.28655 21.287 0.01 Stage IIb 15 153.655 10.000 956.7 0.6823 153.65 1.53655 21.537 0.01 " 16 211.025 10.000 1346.8 1.6655 211.02 2.11025 22.110 0.01 " 17 283.555 10.000 1840.0 3.3853 283.55 2.83555 22.836 0.01 " 18 454.145 10.000 3000.0 9.5285 454.14 4.54145 24.541 0.01 "

Data applying to Case 2 in in Table A.III (basic model) Example in Appendix Example, shear displacement, variable cu 09-02-24 Page 10 Author: Stig Bernander

Landslide spread

3500 3000 2500 2000 Serie1 1500 N kN/m

88 1000 500

Additional earth pressure, pressure, earth Additional 0 0.000 50.000 100.000 150.000 200.000 250.000 300.000 350.000 400.000 450.000 500.000 Coordinate (x)

Data applying to Case 2 in in Table A.III (basic model) Example in Appendix Example, shear displacement, variable cu 09-02-24 Page 11 Author: Stig Bernander

Landslide spread

35.000 30.000 25.000 20.000 Serie1 15.000 kN/m2 10.000 5.000 Shear stress (t), 0.000 0.000 50.000 100.000 150.000 200.000 250.000 300.000 350.000 400.000 450.000 500.000 Coordinate (x) 89 Data applying to Case 2 in in Table A.III (basic model) Example in Appendix Example, shear displacement, variable cu 09-02-24 Page12 Author: Stig Bernander

Landslide spread

12 10 8 6 Serie1 4

Downslope 2

displacements, m displacements, 0 0.000 50.000 100.000 150.000 200.000 250.000 300.000 350.000 400.000 450.000 500.000 Coordinate (x), m 90 Data applying to Case 2 in in Table A.III (basic model) Example in Appendix Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-03-09 Page 1 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c(peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 3.41 step No7 x7 106.46 Stress x 8 Mean kN/m2 kN/m2 x 8 109.87 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5a m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0 0.00 3.20 3.20 30.00 -5.000 25.000 27.500 0.0171 12.00 18.00 0.02408 30 30.00 Stage IIa S(tx) = dsR*(su -tx)/(su - suR) 0.0171 0.02183 1 1.00 3.04 3.04 28.50 -4.750 23.750 26.125 0.0171 11.76 17.64 0.01958 29.40 28.50 Stage IIb d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.01866 2 2.00 2.88 2.88 27.00 -4.500 22.500 24.750 0.0171 11.52 17.28 0.01773 28.80 27.00 0.0171 0.01702 3 3.00 2.72 2.72 25.50 -4.250 21.250 23.375 0.0171 11.28 16.92 0.01632 28.20 25.50 0.0171 0.01572 4 4.00 2.56 2.56 24.00 -4.000 20.000 22.000 0.0171 11.04 16.56 0.01512 27.60 24.00 0.0171 0.01459 5 5.00 2.40 2.40 22.50 -3.750 18.750 20.625 0.0171 10.80 16.20 0.01405 27.00 22.50 0.0171 0.01357 6 6.00 2.24 2.24 21.00 -3.500 17.500 19.250 0.0171 10.56 15.84 0.01308 26.40 21.00 91 0.0171 0.00768 7 6.60 2.08 2.14 20.10 -3.350 16.750 18.425 0.0171 10.42 15.62 0.01253 26.04 20.10

Author: Stig Bernander S(cR) 0.20 cR 10.00 m Calc. S(x) 0.05000 cR/c(peak) 0.333 S:a N x(n) 508.08 S:a d(xn),t 0.10906 dN 82.97 S:a d(xn),N 0.13114 S(x)= 0.05000 m Calc. d(xn,n+1),N 0.02792 S:a Nx(n+1) 591.05 Slip d(slip) = m dx,N-dx(t) 0.00000 S:a dx(n+1),N 0.15906 S:a dx(n+1),t 0.15906 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-02-04 Page 2 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c(peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 2.53 step No 9 x 8 x 9 109.87 Stress x 9 Mean kN/m2 kN/m2 x 9 112.40 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5a m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0 0.00 3.20 3.20 25.00 -4.000 21.000 23.000 0.0171 12.00 18.00 0.02122 30 30.00 Stage IIa S(tx) = dsR*(su -tx)/(su - suR) 0.0171 0.01902 1 1.00 3.04 3.04 23.75 -3.800 19.950 21.850 0.0171 11.76 17.64 0.01681 29.40 28.50 Stage IIb d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.01593 2 2.00 2.88 2.88 22.50 -3.600 18.900 20.700 0.0171 11.52 17.28 0.01505 28.80 27.00 0.0171 0.01439 3 3.00 2.72 2.72 21.25 -3.400 17.850 19.550 0.0171 11.28 16.92 0.01373 28.20 25.50 0.0171 0.01318 4 4.00 2.56 2.56 20.00 -3.200 16.800 18.400 0.0171 11.04 16.56 0.01263 27.60 24.00 0.0171 0.01215 5 5.00 2.40 2.40 18.75 -3.000 15.750 17.250 0.0171 10.80 16.20 0.01167 27.00 22.50 0.0171 0.01124

92 6 6.00 2.24 2.24 17.50 -2.800 14.700 16.100 0.0171 10.56 15.84 0.01081 26.40 21.00 0.0171 0.00634 7 6.60 2.08 2.14 16.75 -2.680 14.070 15.410 0.0171 10.42 15.62 0.01033 26.04 20.10

Author: Stig Bernander S(cR) 0.20 cR 10.00 Calc. S(x) 0.09000 cR/c(peak) 0.333 S:a N x(n) 591.05 S:a d(xn),t 0.09226 dN 50.09 S:a d(xn),N 0.15906 S(x)= 0.09000 m Calc. d(xn,n+1),N 0.02320 S:a Nx(n+1) 641.14 Slip d(slip) = m dx,N-dx(t) 0.00000 S:a dx(n+1),N 0.18226 S:a dx(n+1),t 0.18226 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-02-04 Page 3 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c(peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 2.36 step No 10 x 9 x 10 112.40 Stress x 10 Mean kN/m2 kN/m2 x 10 114.76 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5a m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0 0.00 3.20 3.20 21.00 -4.000 17.000 19.000 0.0171 12.00 18.00 0.01837 30 30.00 Stage IIa S(tx) = dsR*(su -tx)/(su - suR) 0.0171 0.01620 1 1.00 3.04 3.04 19.95 -3.800 16.150 18.050 0.0171 11.76 17.64 0.01404 29.40 28.50 Stage IIb d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.01321 2 2.00 2.88 2.88 18.90 -3.600 15.300 17.100 0.0171 11.52 17.28 0.01238 28.80 27.00 0.0171 0.01176 3 3.00 2.72 2.72 17.85 -3.400 14.450 16.150 0.0171 11.28 16.92 0.01115 28.20 25.50 0.0171 0.01065 4 4.00 2.56 2.56 16.80 -3.200 13.600 15.200 0.0171 11.04 16.56 0.01015 27.60 24.00 0.0171 0.00972 5 5.00 2.40 2.40 15.75 -3.000 12.750 14.250 0.0171 10.80 16.20 0.00929 27.00 22.50 0.0171 0.00891

93 6 6.00 2.24 2.24 14.70 -2.800 11.900 13.300 0.0171 10.56 15.84 0.00854 26.40 21.00 0.0171 0.00500 7 6.60 2.08 2.14 14.07 -2.680 11.390 12.730 0.0171 10.42 15.62 0.00812 26.04 20.10

Author: Stig Bernander S(cR) 0.20 cR 10.00 m Calc. S(x) 0.13000 cR/c(peak) 0.333 S:a N x(n) 641.14 S:a d(xn),t 0.07545 dN 37.32 S:a d(xn),N 0.18226 S(x)= 0.13000 m Calc. d(xn,n+1),N 0.02319 S:a Nx(n+1) 678.46 Slip d(slip) = m dx,N-dx(t) 0.0000 S:a dx(n+1),N 0.20545 S:a dx(n+1),t 0.20545 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-02-04 Page 4 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c(peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 2.25 step No 11 x 10 x 11 114.76 Stress x 11 Mean kN/m2 kN/m2 x 11 117.01 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5a m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0 0.00 3.20 3.20 17.00 -4.000 13.000 15.000 0.0171 12.00 18.00 0.01551 30 30.00 Stage IIa S(tx) = dsR*(su -tx)/(su - suR) 0.0171 0.01339 1 1.00 3.04 3.04 16.15 -3.800 12.350 14.250 0.0171 11.76 17.64 0.01127 29.40 28.50 Stage IIb d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.01048 2 2.00 2.88 2.88 15.30 -3.600 11.700 13.500 0.0171 11.52 17.28 0.00970 28.80 27.00 0.0171 0.00913 3 3.00 2.72 2.72 14.45 -3.400 11.050 12.750 0.0171 11.28 16.92 0.00857 28.20 25.50 0.0171 0.00811 4 4.00 2.56 2.56 13.60 -3.200 10.400 12.000 0.0171 11.04 16.56 0.00766 27.60 24.00 0.0171 0.00729 5 5.00 2.40 2.40 12.75 -3.000 9.750 11.250 0.0171 10.80 16.20 0.00691 27.00 22.50 0.0171 0.00659 6 6.00 2.24 2.24 11.90 -2.800 9.100 10.500 0.0171 10.56 15.84 0.00626 26.40 21.00 dN-d(tau) 94 0.0171 0.00365 7 6.60 2.08 2.14 11.39 -2.680 8.710 10.050 0.0171 10.42 15.62 0.00592 26.04 20.10

Author: Stig Bernander S(cR) 0.20 cR 10.00 Calc. S(x) 0.16999 cR/c(peak) 0.333 S:a N x(n) 678.46 S:a d(xn),t 0.05865 m dN 26.59 S:a d(xn),N 0.20545 S(x)= 0.16999 m Calc. d(xn,n+1),N 0.02319 S:a Nx(n+1) 705.05 Slip d(slip) = 0.0000 m dx,N-dx(t) 0.0000 S:a dx(n+1),N 0.22864 S:a dx(n+1),t 0.22864 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-02-04 Page 5 Author: Stig Bernander Density c(s)= g el = Gradient G mod. c(peak) E-mean t el = H = Width, b 1,0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 1.64 step No 11 x 11 x 12 117.01 Stress x 12 Mean kN/m2 kN/m2 x 12 118.65 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu g (x=n) (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5a m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0 0.00 3.20 3.20 13.00 -3.000 10.000 11.500 0.0171 12.00 18.00 0.01337 30 30.00 Stage IIa S(tx) = dsR*(su -tx)/(su - suR) 0.0171 0.01128 1 1.00 3.04 3.04 12.35 -2.850 9.500 10.925 0.0171 11.76 17.64 0.00919 29.40 28.50 Stage IIb d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.00844 2 2.00 2.88 2.88 11.70 -2.700 9.000 10.350 0.0171 11.52 17.28 0.00769 28.80 27.00 0.0171 0.00716 3 3.00 2.72 2.72 11.05 -2.550 8.500 9.775 0.0171 11.28 16.92 0.00663 28.20 25.50 0.0171 0.00621 4 4.00 2.56 2.56 10.40 -2.400 8.000 9.200 0.0171 11.04 16.56 0.00580 27.60 24.00 0.0171 0.00546 5 5.00 2.40 2.40 9.75 -2.250 7.500 8.625 0.0171 10.80 16.20 0.00512 27.00 22.50 0.0171 0.00484 6 6.00 2.24 2.24 9.10 -2.100 7.000 8.050 0.0171 10.56 15.84 0.00456 26.40 21.00 95 0.0171 0.00265 7 6.60 2.08 2.14 8.71 -2.010 6.700 7.705 0.0171 10.42 15.62 0.00426 26.04 20.10

Author: Stig Bernander S(cR) 0.20 cR 10.00 Calc. S(x) 0.20000 cR/c = 0.333 S:a N x(n) 705.05 S:a d(xn),t 0.04604 m dN 13.61 S:a d(xn),N 0.2286 S(x)= 0.20000 m Calc. d(xn,n+1),N 0.01737 S:a Nx(n+1) 718.66 Slip d(slip) = 0.0000 m dx,N-dx(t) 0.0000 S:a dx(n+1),N 0.2460 S:a dx(n+1),t 0.2460 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-02-04 Page 6 Author: Stig Bernander Density c(s)= g el = Gradient E-mod/cu c(peak) E-mean t el = H = Width, b 1.0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 10.00 step No 11 x 12 x 13 118.65 Stress x 13 Mean kN/m2 kN/m2 x 13 128.65 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu g (x=n) (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5c m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0 0.00 3.20 3.20 10.00 0.000 10.000 10.000 0.0171 12.00 18.00 0.01337 30 30.00 Stage IIb d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.01128 1 1.00 3.04 3.04 9.50 0.000 9.500 9.500 0.0171 11.76 17.64 0.00919 29.40 28.50 0.0171 0.00844 2 2.00 2.88 2.88 9.00 0.000 9.000 9.000 0.0171 11.52 17.28 0.00769 28.80 27.00 0.0171 0.00716 3 3.00 2.72 2.72 8.50 0.000 8.500 8.500 0.0171 11.28 16.92 0.00663 28.20 25.50 0.0171 0.00621 4 4.00 2.56 2.56 8.00 0.000 8.000 8.000 0.0171 11.04 16.56 0.00580 27.60 24.00 0.0171 0.00546 5 5.00 2.40 2.40 7.50 0.000 7.500 7.500 0.0171 10.80 16.20 0.00512 27.00 22.50 0.0171 0.00484 6 6.00 2.24 2.24 7.00 0.000 7.000 7.000 0.0171 10.56 15.84 0.00456 26.40 21.00 96 0.0171 0.00265 7 6.60 2.08 2.14 6.70 0.000 6.700 6.700 0.0171 10.42 15.62 0.00426 26.04 20.10

Author: Stig Bernander S(cR) 0.20 cR 10.00 Calc. S(x) 0.20000 cR/c = 0.333 S:a N x(n) 718.66 S:a d(xn),t 0.04604 dN 68.00 S:a d(xn),N 0.2460 S(x)= 0.20000 m Calc. d(xn,n+1),N 0.11200 S:a Nx(n+1) 786.7 Slip d(slip) = 0.1120 m S:a dx(n+1),N 0.3580 S:a dx(n+1),t 0.3580 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-02-04 Page 7 Author: Stig Bernander Density c(s)= g el = Gradient E-mod/cu c(peak) E-mean t el = H = Width, b 1.0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 25.00 step No 11 x 13 x 14 128.65 Stress x 14 Mean kN/m2 kN/m2 x 14 153.65 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu g (x=n) (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5c m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0 0.00 3.20 3.20 10.00 0.000 10.000 10.000 0.0171 12.00 18.00 0.01337 30 30.00 Stage IIb d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.01128 1 1.00 3.04 3.04 9.50 0.000 9.500 9.500 0.0171 11.76 17.64 0.00919 29.40 28.50 0.0171 0.00844 2 2.00 2.88 2.88 9.00 0.000 9.000 9.000 0.0171 11.52 17.28 0.00769 28.80 27.00 0.0171 0.00716 3 3.00 2.72 2.72 8.50 0.000 8.500 8.500 0.0171 11.28 16.92 0.00663 28.20 25.50 0.0171 0.00621 4 4.00 2.56 2.56 8.00 0.000 8.000 8.000 0.0171 11.04 16.56 0.00580 27.60 24.00 0.0171 0.00546 5 5.00 2.40 2.40 7.50 0.000 7.500 7.500 0.0171 10.80 16.20 0.00512 27.00 22.50 0.0171 0.00484 6 6.00 2.24 2.24 7.00 0.000 7.000 7.000 0.0171 10.56 15.84 0.00456 26.40 21.00 97 0.0171 0.00265 7 6.60 2.08 2.14 6.70 0.000 6.700 6.700 0.0171 10.42 15.62 0.00426 26.04 20.10

Author: Stig Bernander S(cR) 0.20 cR 10.00 Calc. S(x) 0.20000 cR/c = 0.333 S:a N x(n) 786.66 S:a d(xn),t 0.04604 m dN 170.00 S:a d(xn),N 0.3580 S(x)= 0.20000 m Calc. d(xn,n+1),N 0.32428 S:a Nx(n+1) 956.7 Slip d(slip) = 0.4363 m S:a dx(n+1),N 0.6823 S:a dx(n+1),t 0.6823 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-02-04 Page 8 Author: Stig Bernander Density c(s)= g el = Gradient E-mod/cu c(peak) E-mean t el = H = Width, b 1.0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 57.37 step No 11 x 14 x 15 153.65 Stress x 15 Mean kN/m2 kN/m2 x 15 211.02 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu g (x=n) (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5c m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0 0.00 3.20 3.20 10.00 0.000 10.000 10.000 0.0171 12.00 18.00 0.01337 30 30.00 Stage IIb d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.01128 1 1.00 3.04 3.04 9.50 0.000 9.500 9.500 0.0171 11.76 17.64 0.00919 29.40 28.50 0.0171 0.00844 2 2.00 2.88 2.88 9.00 0.000 9.000 9.000 0.0171 11.52 17.28 0.00769 28.80 27.00 0.0171 0.00716 3 3.00 2.72 2.72 8.50 0.000 8.500 8.500 0.0171 11.28 16.92 0.00663 28.20 25.50 0.0171 0.00621 4 4.00 2.56 2.56 8.00 0.000 8.000 8.000 0.0171 11.04 16.56 0.00580 27.60 24.00 0.0171 0.00546 5 5.00 2.40 2.40 7.50 0.000 7.500 7.500 0.0171 10.80 16.20 0.00512 27.00 22.50 0.0171 0.00484 6 6.00 2.24 2.24 7.00 0.000 7.000 7.000 0.0171 10.56 15.84 0.00456 26.40 21.00 98 0.0171 0.00265 7 6.60 2.08 2.14 6.70 0.000 6.700 6.700 0.0171 10.42 15.62 0.00426 26.04 20.10

Author: Stig Bernander S(cR) 0.20 cR 10.00 cR Calc. S(x) 0.20000 cR/c = 0.333 S:a N x(n) 956.66 S:a d(xn),t 0.04604 m dN 390.12 S:a d(xn),N 0.6823 S(x)= 0.20000 m 0.00 Calc. d(xn,n+1),N 0.98325 S:a Nx(n+1) 1346.8 Slip d(slip) = 1.4195 m S:a dx(n+1),N 1.6655 S:a dx(n+1),t 1.6655 m 0.00 Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-02-04 Page 9 0.00 Density c(s)= g el = Gradient E-mod/cu c(peak) E-mean t el = H = Width, b 1.0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 72.53 0.00 step No 11 x 15 x 16 211.02 Stress x 16 Mean kN/m2 kN/m2 x 16 283.55 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu g (x=n) 0.00 (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5c m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0.00 0 0.00 3.20 3.20 10.00 0.000 10.000 10.000 0.0171 12.00 18.00 0.01337 30 30.00 d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.01128 0.00 1 1.00 3.04 3.04 9.50 0.000 9.500 9.500 0.0171 11.76 17.64 0.00919 29.40 28.50 0.0171 0.00844 0.00 2 2.00 2.88 2.88 9.00 0.000 9.000 9.000 0.0171 11.52 17.28 0.00769 28.80 27.00 0.0171 0.00716 3 3.00 2.72 2.72 8.50 0.000 8.500 8.500 0.0171 11.28 16.92 0.00663 28.20 25.50 0.0171 0.00621 4 4.00 2.56 2.56 8.00 0.000 8.000 8.000 0.0171 11.04 16.56 0.00580 27.60 24.00 0.0171 0.00546 5 5.00 2.40 2.40 7.50 0.000 7.500 7.500 0.0171 10.80 16.20 0.00512 27.00 22.50 0.0171 0.00484 6 6.00 2.24 2.24 7.00 0.000 7.000 7.000 0.0171 10.56 15.84 0.00456 26.40 21.00 99 0.0171 0.00265 7 6.60 2.08 2.14 6.70 0.000 6.700 6.700 0.0171 10.42 15.62 0.00426 26.04 20.10

S(cR) 0.20 cR 10.00 Calc. S(x) 0.20000 cR/c(peak) 0.333 S:a N x(n) 1346.78 S:a d(xn),t 0.04604 m dN 493.20 S:a d(xn),N 1.6655 S(x)= 0.20000 m Calc. d(xn,n+1),N 1.71976 S:a Nx(n+1) 1840.0 Slip d(slip) = 3.1393 m S:a dx(n+1),N 3.3853 S:a dx(n+1),t 3.3853 m Landslide spread in passive zone - down-hill progressive slide - variable cu Stage II 09-02-04 Page 10 Density c(s)= g el = Gradient E-mod/cu c(peak) E-mean t el = H = Width, b 1.0 Iter. g f = 0.030 16.000 18 0.0129 0.0100 1400 30.00 3360 18.00 20.0 dx (m) 170.59 step No 11 x 16 x 17 283.55 Stress x 17 Mean kN/m2 kN/m2 x 17 454.14 170.59 n z (n) In situ In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz cu g (x=n) (m)shear shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.I:5c m kN/m2 to(x,z) to(x,z)+dt t(x,z) dt(x,z) t(x,z)+dt 0.0171 t(x,z),peak 0 0.00 3.20 3.20 10.00 0.000 10.000 10.000 0.0171 12.00 18.00 0.01337 30 30.00 Stage IIb d (slip) =Sum dN - Sum d(tx)-ScR 0.0171 0.01128 1 1.00 3.04 3.04 9.50 0.000 9.500 9.500 0.0171 11.76 17.64 0.00919 29.40 28.50 0.0171 0.00844 2 2.00 2.88 2.88 9.00 0.000 9.000 9.000 0.0171 11.52 17.28 0.00769 28.80 27.00 0.0171 0.00716 3 3.00 2.72 2.72 8.50 0.000 8.500 8.500 0.0171 11.28 16.92 0.00663 28.20 25.50 0.0171 0.00621 4 4.00 2.56 2.56 8.00 0.000 8.000 8.000 0.0171 11.04 16.56 0.00580 27.60 24.00 0.0171 0.00546 5 5.00 2.40 2.40 7.50 0.000 7.500 7.500 0.0171 10.80 16.20 0.00512 27.00 22.50 0.0171 0.00484

100 6 6.00 2.24 2.24 7.00 0.000 7.000 7.000 0.0171 10.56 15.84 0.00456 26.40 21.00 0.0171 0.00265 7 6.60 2.08 2.14 6.70 0.000 6.700 6.700 0.0171 10.42 15.62 0.00426 26.04 20.10

S(cR) 0.20 cR 10.00 Calc. S(x) 0.20000 cR/c(peak) 0.333 S:a N x(n) 1839.98 S:a d(xn),t 0.04604 m dN 1160.01 S:a d(xn),N 3.3853 S(x)= 0.20000 m Calc. d(xn,n+1),N 6.14324 S:a Nx(n+1) 3000.0 Slip d(slip) = 9.2825 m S:a dx(n+1),N 9.5285 S:a dx(n+1),t 9.5285 m Integration of shear displacement in vertical element 99-08-23 Page 6 % % kN/m2 kN/m2 Iter. g f = 0.075 g el = 0.0375 cu = 30.00 t el = 20.00 H =20 m dx (m) 2.00 step No 6 x 5 57.44 Stress Mean x 6 59.44 n z (n) In situ Shear incre- Shear shear gf - gel cu-t el t el d(g ) d(g)*dz g (x=n) g (x=n) (m)shear stress ment stress stress kN/m2 kN/m2 kN/m2 Equ.9:1 m Equ.9:1 to(x,z) t(x1,z) dt(x,z) t(x2,z) t(x1)+t(x2))/ 0.0375 10.000 0 0.00 25.00 30.00 -0.755 29.245 29.623 0.0375 10.00 20.00 -0.00142 0.0647 0.0485 0.0375 10.00 20.00 -0.00138 1 1.00 23.75 28.50 -0.717 27.783 28.141 0.0375 10.00 20.00 -0.00134 0.0573 0.0454 0.0375 10.00 20.00 -0.00131 2 2.00 22.50 27.00 -0.680 26.321 26.660 0.0375 10.00 20.00 -0.00127 0.0523 0.0425 0.0375 10.00 20.00 -0.00124 101 3 3.00 21.25 25.50 -0.642 24.858 25.179 0.0375 10.00 20.00 -0.00120 0.0481 0.0399 0.0375 10.00 20.00 -0.00117 4 4.00 20.00 24.00 -0.604 23.396 23.698 0.0375 10.00 20.00 -0.00113 0.0445 0.0375 0.0375 10.00 20.00 -0.00110 5 5.00 18.75 22.50 -0.566 21.934 22.217 0.0375 10.00 20.00 -0.00106 0.04113 0.03516 0.0375 10.00 20.00 -0.00103 6 6.00 17.50 21.00 -0.529 20.472 20.736 0.0375 10.00 20.00 -0.00099 0.03838 0.03281 0.0375 10.00 20.00 -0.00058 7 6.60 16.75 20.10 -0.506 19.594 19.847 0.0375 10.00 20.00 -0.00095 0.03674 0.03141

dcR 0.10 cR 15.000 16.0000 dx5 = -0.00780 m d(slip,total) = Calc. S(x) 0.01015 N(x,n) 508.08 d(x6) = 0.13114 0.01795 t(x2,z) 29.25 dN 9.25 d(xn) 0.07103 d(S,x) = 0.01015 m Calc. t(x2,z) 29.25 d(xn,xn+1) 0.05127 N(x,n+1) 517.32 Slip d(slip.el) = 0.00780 m d(x,n+1) 0.12230 d(shear) = 0.14909 m