SISTEMATIZACIJA ANALITIČKIH I NUMERIČKIH METODA …

GRAĐEVINSKI MATERIJALI I KONSTRUKCIJE 61 (2018) 1 (129-160)BUILDING MATERIALS AND STRUCTURES 61 (2018) 1 (129-160)

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SISTEMATIZACIJA ANALITIČKIH I NUMERIČKIH METODA PRORAČUNASTABILNOSTI KLIZIŠTA

THE SYSTEMATIZATION OF ANALYTICAL AND NUMERICAL METHODS OFLANDSLIDE STABILITY CALCULATION

Kristina BOŽIĆ TOMIĆNenad ŠUŠIĆMato ULJAREVIĆ

STRUČNI RADPROFESSIONAL PAPER

UDK: 624.131.537doi:10.5937/GRMK1801129B

1 UVOD

S obzirom na kompleksnost geometrije reljefazemljine površi, kosine su među problematičnijimgeološkim formama u geotehnici. Kosine karakterišenagla promena geometrije terena (denivelacija), spredispozicijom promene ove geometrije usled dejstvarazličitih faktora. Najčešći i najsloženiji vid narušavanjatla i geometrije kosine odnosi se na stabilnost terena –bilo prirodnih padina ili veštačkih kosina. Svako naru-šavanje postojeće ravnoteže na padinama ili kosinamaizaziva pomeranja pod uticajem gravitacije: klizanje,odronjavanje ili tečenje površinskog dela tla, ali i dubljihdelova stenske mase. Za ovako uspostavljeno klizanje, ugeološkoj i geotehničkoj terminologiji i nomenklaturi,ustaljen je termin – klizište [11]. Uslovi za nastanak irazvoj klizišta jesu: geotehnički, geološki, geomorfološki,hidrogeološki, meteorološki, vegetacioni, antropogeni,dejstvo zemljotresa, dejstvo akumulacija, vibracije usledsaobraćaja i drugi.

U poslednjih sto godina, zabeležen je znatan brojkatastrofalnih klizišta, nastalih kao posledica dejstvazemljotresa, erupcije vulkana, nagomilavanja snega,višednevnih i intenzivnih kiša i uragana [16]. Zbog formi-ranja ovih klizišta, poginulo je nekoliko stotina hiljada ljudikoji su - u najvećem broju slučajeva - imali sagrađene

Mr Kristina Božić-Tomić, Institut za ispitivanje materijalaIMS, Beograd, Srbija, [email protected] Nenad Šušić, naučni savetnik, Institut za ispitivanjematerijala IMS, Beograd, Srbija, [email protected]. dr Mato Uljarević, Arhitektonsko-građevinsko-geodetski fakultet, Univerzitet u Banjoj Luci, RepublikaSrpska, [email protected]

1 INTRODUCTION

Given the complexity of geometry of the relief of theearth's surface, slopes represent one of the problematicgeological forms in geotechnics. The slopes are charac-terized by a sudden change in geometry of the terrain(denivelation) with a predisposition to the change of thisgeometry due to the effects of various factors. The mostcommon and most complex type of soil disturbance andslope geometry is the stability of the terrain, whethernatural slopes or artificial slopes. Any disturbance of theexisting balance on the slopes causes displacementunder the influence of gravity: sliding, erosion or flowingthe surface of the soil, but also the deeper parts of therock mass. For the established sliding, in the geologicaland geotechnical terminology and nomenclature, theterm "landslide" is established [11]. Conditions for theformation and development of landslides are: geotechni-cal, geological, geo-morphological, hydro-geological,meteorological, vegetation, anthropogenic, earthquakeeffects, accumulation effects, traffic vibrations, etc.

In the last hundred years there has been a significantnumber of catastrophic landslides that have occurred asa result of earthquakes, volcanic eruptions, snowaccumulation, multi-day heavy rainfall and hurricanes[16]. Due to the formation of these landslides several

Mr Kristina Bozic-Tomic, Institute for testing of materialsIMS, Belgrade, Serbia, [email protected] Nenad Susic, Institute for testing of materials IMS,Belgrade, Serbia, [email protected]. dr Mato Uljarevic, Faculty of architecture, civilengineering and geodesy, University of Banja Luka,Republika Srpska, [email protected]


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domove na klizištima ili u njihovoj neposrednoj blizini.Prema [29], u najkatastrofalnija klizišta – zabeležena uposlednjih sto godina – ubrajaju se: Haiyuan landslides u Kini 1920, Vargas tragedy u Venecueli 1999, Nevado delRuiz debris flows u Kolumbiji 1985, Nevados Huascarandebris avalanche u Peru 1970, North India floodmudslides u Indiji 2013, Khait rock slide u USSR 1949 islično. U katastarskom listu evidencije klizišta u Srbiji,zabeleženo je više od 2.200 aktivnih, trenutno umirenih ireaktivnih klizišta [15]. Znatno je manji broj trenutnoumirenih od aktivnih i reaktivnih klizišta. Vrlo često, upraksi se susrećemo s problemima stabilnosti klizišta,kada je nakon izvedenih terenskih istraživanja, labora-torijsko-geomehaničkih ispitivanja, definisanja uzroka iuslova nastanka klizišta potrebno definisati meresanacije. Međutim, da bismo uspešno upravljali svimprojektnim situacijama analize stabilnosti i sanacijeklizišta, potrebno je da imamo kvalitetne matematičkemodele i metode analize stabilnosti klizišta. Dosadašnjaiskustva pokazuju da postoji potreba za implementa-cijom kompleksnijih (realističnijih) matematičkih modelau praktične svrhe, kao i za dodatnim unapređivanjempostojećih metoda analize stabilnosti klizišta.

Jedan od prvih radova u kojem su adekvatno teorijskirazmatrani aspekti nekoliko analitičkih, ali i numeričkihmetoda analize stabilnosti klizišta, jeste rad [9], gde jesprovedena klasifikacija, imajući u vidu: formulacijugraničnog stanja (limit formulation) i formulaciju stanjapomeranja (displacement formulation). Kod formulacijegraničnog stanja, postoje dve opcije: gornja graničnarešenja (upper bound solution) i donja granična rešenja(lower bound solution), pri čemu metoda karakteristikapomeranja (method of characteristics for displacement)pripada grupi gornjih graničnih rešenja, a metodakarakteristika napona (method of characteristics forstress) pripada grupi donjih graničnih rešenja. Metodedefinisane prema formulaciji graničnog stanja, zapravosu metode granične ravnoteže (LEM - Limit EquilibriumMethod), od kojih se najčešće primenjuju gornja gra-nična rešenja. U poređenju s njima, metode definisaneprema formulaciji stanja pomeranja, zapravo su metodeanalize pomeranja (DFM - Displacement FormulationMethod), odnosno numeričke metode. U radu [20] dat jepregled numeričkih metoda stabilnosti klizišta, pri čemuje korišćena formulacija po metodi konačnih razlika(FDM - Finite Difference Method). Studija performansinekoliko različitih metoda stabilnosti klizišta prikazana jeu radu [27], dok su u radu [21] prikazane metodestabilosti klizišta, imajući u vidu deterministički pristup,teoriju pouzdanosti i optimizacije. Primena numeričkihmetoda analize stabilnosti kosina u izmenjenoj serpen-tinskoj stenskoj masi prikazana je u radu [25], pri čemusu, između ostalog, korišćeni i sledeći parametri: geološ-ki indeks čvrstoće i deformabilnosti stenske mase, dok jekao kriterijum sloma primenjen Hoek Brown-ov kriteri-jum. Razmatranje kompleksne problematike stabilnostiklizišta, iz aspekta analize hazarda, analize povred-ljivosti, procene i upravljanja rizikom prikazani su u radu[6], gde je - zasnivajući se na prethodno navedenimteorijama - predložen GIS integralni model za analizuklizišta.

Cilj istraživanja prikazanog u ovom radu jeste da sedetaljnije sistematizuju metode proračuna klizišta ialgoritmi modeliranja, s posebnim osvrtom na numeričkeanalize stabilnosti.

hundred thousand people were killed, who in most caseshad their own homes built on or near the landslide.According to [29], the most catastrophic landslidesrecorded over the last hundred years, were: Haiyuanlandslides in China 1920, Vargas tragedy in Venezuela1999, Nevado del Ruizdebris flows in Colombia 1985,Nevados Huascaran debris avalanche in Peru 1970,North India flood mudslides in India 2013, Khait rockslide in the USSR in 1949 and the like. In the cadastralregister of landslide records in Serbia, more than 2200active, currently calm and reactive landslides have beenrecorded [15]. There is a significantly lower number ofcurrently calm, compared to active and reactive land-slides. Very often, in practice, problems with the stabilityof the landslide are encountered, when after the conduc-ted field investigations, laboratory-geomechanical tests,defining the causes and conditions of landslide forma-tion, it is necessary to define repair measures. However,in order to successfully manage all project situations ofthe landslide stability analysis and landslide repair, it isnecessary to have high quality mathematical models andlandslide methods. Previous experience shows thatthere is a need for the implementation of more complex(more realistic) mathematical models for practicalpurposes and further improvement of existing landslidestability methods.

One of the first papers in which the aspects ofseveral analytical and also numerical methods oflandslide stability are adequately theoretically conside-red is the paper [9], where the classification was carriedout taking into account the limit formulation and thedisplacement formulation. There are two options for thelimit formulation: upper bound solution and lower boundsolution, where the method of characteristics for displa-cement belongs to the group of upper bound solutions,and the method of characteristics for stress belongs tothe group of lower bound solutions. The methodsdefined by the limit formulation are in fact the LimitEquilibrium Method (LEM), of which the upper boundsolutions are most commonly applied. In relation tothem, the methods defined by the displacement formula-tion are in fact Displacement Formulation Methods(DFM), or numerical methods. The paper [20] gives anoverview of the numerical methods of landslide stability,using the Finite Difference Method (FDM) formulation. Astudy of the performances of several different landslidestability methods is presented in [27], while in [21] themethods of landslide stability are presented taking intoaccount the deterministic approach, the theory ofreliability and optimization. The use of numericalmethods for analyzing the stability of slopes in thealternating serpentine rock mass was shown in [25],where, among other things, these parameters wereused: the geological strength index and deformability ofthe rock mass, and for the criterion of failure HoekBrown's criterion was applied. Consideration of thecomplex problem of landslide stability, but also from theaspect of hazard analysis, vulnerability analysis, riskassessment and risk management, are presented in [6],where, based on the aforementioned theories, a GISintegral model for landslide analysis is proposed.

The aim of the research presented in this paper is tofurther systematize the methods of landslide calculationsand modelling algorithms with a special emphasis onnumerical stability analyses.


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2 UOPŠTENO O RAZMATRANJIMA STABILNOSTIKLIZIŠTA I FAKTORIMA BITNIM ZA PRORAČUN

Generalno razmatrajući, kosine se mogu nalaziti ustabilnom ravnotežnom, nestabilnom neravnotežnom iindiferentnom poluravnotežnom stanju. Stabilno ravno-težno stanje karakteriše uspostavljen odnos destabili-zujućih i stabilizujućih sila, tako da – ukoliko je uticajstabilizujućih sila veći - veći je i faktor sigurnosti.Nestabilno neravnotežno stanje karakteriše narušenodnos destabilizujućih i stabilizujućih sila, tako da jeuticaj destabilizujućih sila dominantan. Indiferentno (neo-dređeno) poluravnotežno stanje predstavlja prelaznukategoriju između stabilnog ravnotežnog i nestabilnogneravnotežnog stanja. Odnos destabilizujućih i stabili-zujućih sila indiferentnog poluravnotežnog stanja značaj-nije je narušen u stabilno ravnotežnom stanju i dovoljanje i mali priraštaj destabilizujućih sila, pa da transformišeovo stanje u nestabilno neravnotežno stanje. S obziromna kompleksnost indiferentnog stanja i mogućnostparcijalne promene geometrije kosine, odnosno polufor-miranja klizišta, indiferentno stanje karakteriše skup višerazličitih poluravnotežnih stanja. Ovo je posebno karak-teristično u situacijama kada nastupi narušavanje stabili-tetnog ravnotežnog stanja, pri čemu ne mora doći dopotpunog kretanja klizne mase tla, već se možeuspostaviti novo poluravnotežno stanje. Detaljnijaklasifikacija stabilitetnih i nestabilitetnih stanja kosina,podrazumevajući pritom i prelazne kategorije, prikazanaje u [24]: stabilna kosina, potencijalno nestabilna kosina,rana faza rušenja, srednja faza rušenja, delimično ilitotalno rušenje i potpuno rušenje,dok su mehanizminastanka i razvoja klizišta: rotacioni model, translacionimodel, model formiran iz različitih geometrijskih formiblokova, model s klizanjem, kotrljanjem i padanjemkamena različitih dimenzija, model sa značajnim odvalji-vanjem kliznog tla, klizište formirano usled obimnih kiša,klizište formirano kao obimni bujični tok, klizišteformirano tečenjem tla i klizište formirano puzanjem tlauz pojavu prslina i rascepa u tlu. Jedan od ključnihparametara pri klasifikaciji klizišta jeste brzina kretanjaklizne mase, kao i uticaj površinske i podzemne vode.Uopšte uzev, može se konstatovati da klizišta koja imajuveći nagib spoljašnje konture tla - imaju i veću brzinukretanja klizne mase [18]. Ovo je posledica dejstvagravitacionih sila. Međutim, razmatranje uticaja brzinekretanja klizne mase tla i inkrementalnog povećanjavode u tlu zahteva primenu metoda za proračunstabilnosti klizišta u vremenskom domenu, a što je dostakompleksnije od uobičajenih metoda proračuna.

Metodologija analize potencijalnog klizišta sastoji seiz sledećih nekoliko segmenata: geodetsko osmatranjeterena i prikupljanje podataka, geotehnička in-situispitivanja, analiza fizičko-mehaničkih parametara tla ulaboratoriji i proračun kosine primenom matematičkihmetoda u geotehnici. Metodologija analize formiranogklizišta zasniva se na projektu sanacije klizišta, koji setakođe sastoji iz nekoliko segmenata: geodetskoosmatranje terena i prikupljanje podataka, geotehničkain-situ ispitivanja, analiza fizičko-mehaničkih parametaratla u laboratoriji, rekonstruktivna analiza prethodnogstanja klizišta, analiza faktora koji su doveli do formiranjaklizišta, razmatranje varijantnih rešenja sanacije klizišta,proračuni varijantnih rešenja klizišta primenom matema-tičkih metoda u geotehnici, ekonomska analiza varijant-

2 GENERAL ON LANDSLIDE STABILITYCONSIDERATIONS AND FACTORS RELEVANTTO THE CALCULATION

Generally speaking, the slopes can be found in astable equilibrium state, unstable imbalance state andindifferent semi-equilibrium state. A stable equilibriumstate is characterized by the established relation ofdestabilizing and stabilizing forces, so if the effect ofstabilizing forces is greater, the safety factor is greater.The unstable imbalance state is characterized by adisturbed relation between destabilizing and stabilizingforces, so the influence of destabilizing forces isdominant. The indifferent (indefinite) half-balance staterepresents a transition category between a stableequilibrium and an unstable imbalance state. The ratio ofdestabilizing and stabilizing forces of the indifferentsemi-equilibrium state is significantly more disturbedthan the stable equilibrium state, and it is sufficient thatthe small increment of destabilizing forces transformsthis state into an unstable imbalance state. Given thecomplexity of the indifferent state and the possibility of apartial change in the slope geometry or the semi-formingof the landslide, the indifferent state is characterized bya set of several different half-equilibrium states. This isespecially characteristic in situations where thedisturbance of the stable equilibrium state occurs,without the complete movement of the sliding mass ofthe soil, but a new half-balance state can be established.A more detailed classification of the stable and instablestates of the slopes, taking into account the transitioncategories, is shown in [24]: stable slope, potentiallyunstable slope, early demolition phase, mediumdemolition phase, partial or complete demolition andcomplete demolition, while the mechanisms of formationand development of landslides are: rotational model,translational model, model formed from differentgeometric shapes of blocks, model with sliding, rollingand falling of stone of different dimensions, model withsignificant sliding of the landslide soil, landslide formeddue to heavy rainfall, landslide formed as voluminoustorrential flow, landslide formed by soil flow and landslideformed by soil crawling with the appearance of cracksand clefts in the soil. One of the key parameters inlandslide classification is the velocity of movement of thesliding mass, as well as the level of underground waterin the soil. In general, it can be concluded that thelandslides, which have a higher slope of the outercontour of the soil, have a higher velocity of movementof the sliding mass [18]. This is due to the effect ofgravitational forces. However, the consideration of theinfluence of the rate of movement of the sliding mass ofthe soil and the incremental increase in water in the soilrequires the application of methods for estimating thestability of landslides in the time domain, which isconsiderably more complex than the usual methods ofcalculation.

The methodology of the analysis of the potentiallandslide consists of several segments: geodetic surveyof terrain and data collection, geotechnical in-situ testing,the analysis of physico-mechanical parameters of soil inthe laboratory and calculation of slope usingmathematical methods in geo-technics. Themethodology of the analysis of the formed landslide isbased on a landslide repair project consisting of several


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nih rešenja, višekriterijumska optimizacija varijantnihrešenja i detaljna analiza tehnologije sanacije klizišta zaoptimalno izabrano rešenje.

Prilikom analize klizišta, sprovode se prethodnageotehnička in-situ ispitivanja pomoću kojih seprvenstveno formira inženjersko-geološki profil terena.Ključna geotehnička ispitivanja koja se sprovode zaformiranje inženjersko-geološkog profila terena jesuistražne bušotine. One se sprovode tehnikom bušenjajezgrovanjem, prilikom čega se uzorci tla pažljivoklasifikuju radi identifikacije tipa tla po dubini i analizefizičko-mehaničkih karakterstika tla. Izvođenje istražnebušotine potrebno je sprovesti dovoljno duboko, kako bise na inženjersko-geološkom profilu klizišta utvrdilaklizna površ. Broj potrebnih istražnih bušotina u korelacijije s geometrijom klizišta, dimenzijama klizišta, dubinamaklizne površi, promenljivosti geologije i tako dalje.

Za razliku od geotehničkih ispitivanja klizišta,geodetska ispitivanja sprovode se radi utvrđivanjageometrije, dimenzija i monitoringa klizišta. Na osnovusnimljene geometrije klizišta, formira se situacioni planklizišta u 2D koordinatnom sistemu. Identifikacijomvećeg broja kliznih ravni za odgovarajući brojinženjersko-geoloških profila i njihovom integracijom sa2D situacionim planom klizišta, konstruiše se 3D modelklizišta u softveru za geometrijsku prezentaciju (CAD -Computer Added Design). Ovako povezane klizne ravniformiraju kliznu površ. Konstruisan 3D model klizišta,formiran iz oblaka tačaka i linija, može se eksportovati usoftver za numeričku analizu stabilnosti klizišta.Monitoring i analiza pomeranja klizne mase, geodetskimmetodama, sprovodi se radi periodičnog ili kontinualnogpraćenja stanja klizišta: direktno na terenu (geodetskiminstrumentima, primenom radara na zemlji, brzihkamera), primenom radara iz satelita, bespilotnih letelica(dronova), aviona, terestričkog laserskog skeniranja ilikombinovano. Podaci dobijeni monitoringom iz inicijalnihstanica (GPS - Global Positioning System) direktno setransferuju u baznu stanicu, a zatim u kontrolni centar zadalju obradu. S obzirom na to što klizišta karakterišepomeranje klizne mase, prvenstveno se prate horizon-talna i vertikalna površinska pomeranja i horizontalna ivertikalna pomeranja u unutrašnjosti klizišta na odre-đenim dubinama. Takođe, monitoring se sprovodi i zakontrolu varijacije nivoa podzemne vode primenompijezometara i analizu vertikalne i ortogonalnih horizon-talnih akceleracija primenom akcelerometara. Zapisakceleracija prikazuje se akcelerogramom koji senaknadno, u digitalizovanom formatu, procesira:skaliranjem, filtriranjem, korekcijom bazne linije (BLC -base line correction), kompatibilizacijom (SM - spectralmatching) i algoritmom konvolucije/dekonvolucije. Svi ovipodaci – dobijeni geodetskim osmatranjem terena –mogu se interaktivno uključiti u matematički model kojimse sprovodi analiza stabilnosti klizišta, tako da se krozvreme, kontinualnom korekcijom numeričkog modela,upravlja svim aspektima proračuna i dodatno smanjujenivo nepouzadnosti ulaznih parametara (parametriproračunskog modela i parametri spoljašnjih/unutrašnjihdejstava). Ovakav numerički model predstavlja, zapravo,numerički model u realnom vremenu (real timenumerical model).

Prilikom formiranja proračunskog modela klizišta,potrebno je razmotriti sve relevantne parametre i odreditinjihove vrednosti, budući da je konačno rešenje u

segments: geodetic surveying of the terrain and datacollection, geotechnical in-situ testing, analysis ofphysico-mechanical parameters of soil in the laboratory,reconstructive analysis of the previous landslide state,analysis of the factors that led to the formation oflandslides, the consideration of variant solutions forlandslide repair, calculations of variant landslidesolutions using mathematical methods in geotechnics,economic analysis of variant solutions, multicriteriaoptimization of variant solutions, and detailed analysis oflandslide repair technology for the chosen optimalsolution.

During the landslide analysis, preliminarygeotechnical in situ testing are carried out, by which, inthe first place, an engineering-geological profile of theterrain is formed. Key geotechnical investigations carriedout for the formation of the engineering-geological profileof the terrain are exploratory boreholes. Exploratoryboreholes are made using core drilling technique, wheresoil samples are carefully classified for soil typeidentification according to the depth and the analysis ofphysico-mechanical characteristics of the soil. Theexecution of the exploratory borehole must be carriedout deep enough to determine the sliding surface on theengineering-geological profile of the landslide. Thenumber of required exploratory boreholes is incorrelation with: landslide geometry, landslidedimensions, depths of sliding surfaces, geologicalvariations, and the like.

In relation to geotechnical landslides testing,geodetic testing are carried out in order to determine thegeometry, dimensions and monitoring of the landslide.Based on the recorded landslide geometry, thesituational plan of the landslide is formed in the 2Dcoordinate system. By identifying a greater number ofsliding plates for the corresponding number ofengineering-geological profiles and by integrating themwith the 2D situational landslide plan, a 3D model oflandslide in Computer Added Design (CAD) wasconstructed. The associated sliding plane forms a slidingsurface. The constructed 3D landslide model, formedfrom cloud of nodes and lines, can be exported to thesoftware for numerical analysis of landslide stability.Monitoring and analysis of sliding mass movement, bygeodetic methods, are carried out in order to periodicallyor continuously monitor the landslide situation: directlyon the ground (geodetic instruments, using radars onearth, fast cameras), using radar from satellites,unmanned aircraft (drones), planes, terrestrial laserscanning or combined. The data obtained frommonitoring from the initial stations (GPS - GlobalPositioning System) are directly transferred to the basestation, and then to the control centre for furtherprocessing. Since the landslides are characterized bythe movement of the sliding mass, this is primarilyfollowed by horizontal and vertical surface movementsand horizontal and vertical movements in the interior ofthe landslide at certain depths. In addition, monitoring isalso carried out to control the variation of groundwaterlevel by using piezometers and analyzing vertical andorthogonal horizontal acceleration using accelerometers.The acceleration record is displayed with anaccelerometer, which is subsequently processed in adigitized format: scaling, filtering, baseline correction,spectral matching, and convolution/deconvolution


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direktnoj korelaciji sa selekcijom i varijacijom vrednostiparametara. Relevantni parametri proračunskog modelaklizišta mogu se klasifikovati u pet grupa: parametrigeometrije proračunskog modela, parametri fizičko-mehaničkih karakteristika tla, parametri dejstava, poseb-ni tipovi parametara i parametri proračuna. Parametrimageometrije proračunskog modela modelira se kompleks-nost geometrije kosine i višeslojnost tla po dubini. Sobzirom na to što slojevi mogu biti složene geometrije, ane samo horizontalni ili približno horizontalni, to se priproračunu kompleksne višeslojne geometrije tla prime-njuju numeričke metode proračuna klizišta. Takođe, uove parametre ubrajaju se i parametri geometrije kon-strukcija koje se nalaze na klizištu ili u njihovoj blizini ilisu to konstrukcije kojima se sprovodi sanacija klizišta,tako da se i za njih, pri proračunu, uzima u obzir efekatinterakcije konstrukcija-tlo (SSI - soil-structure intera-ction). Pravilan unos ovih parametara zavisi od nivoakvaliteta formiranog inženjersko-geološkog profilaterena. Parametri fizičko-mehaničkih karakteristika tladobijaju se iz laboratorijskog ispitivanja uzoraka, od kojihse izdvajaju: opit jednoaksijalne čvrstoće, opit direktnogsmicanja, triaksijalni opit i edometarski opit stišljivosti. Zaanalize stabilnosti kosina značajni su sledeći parametri:zapreminska težina tla, težina tla u zasićenom stanju,kohezija, ugao unutrašnjeg trenja, Young-ov modulelastičnosti, edometarski modul stišljivosti, modul defor-macije, Poisson-ov koeficijent, referentan modul smica-nja, dilatancija, koeficijent poroznosti i tako dalje. Uzavisnosti od tipa konstitutivnog modela ponašanja tla,definišu se i relevantni parametri, s tim što se kodkonstitutivnih modela kojim se opisuje trodimenzionalnonaponsko stanje znatno povećava broj parametara.Najčešće, kao konstitutivni model ponašanja tla, prianalizi klizišta, primenjuje se Mohr-Coulomb-ov modeltla, dok se – u zavisnosti od specifičnosti tipa tla – mogukoristiti omekšavajući (soft soil model) ili ojačavajući(hardening soil model), Cam-Clay model, Drucker-Prager-ov model i drugi. Postoje i dodatni parametrikojima se unapređuje konstitutivni model ponašanja tla;na primer, parametri kojima se dodatno utiče napromenu čvrstoće i kohezije po dubini tla, uvođenjezatežuće čvrstoće tla i definisanje parametara konso-lidacije. Takođe, dobro je poznavati konzistenciju tla(veoma meka, meka, srednja, kruta, veoma kruta).Prilikom definisanja parametara prema EN 1997-1:2004propisima, potrebno je poznavati parcijalne faktore zaugao unutrašnjeg trenja, efektivnu koheziju i nedreniranusmičuću čvrstoću tla [7]. Parametrima dejstava definišuse: tipovi opterećenja (koncentrisano, linijsko, površin-sko, prostorno), tipovi dejstva opterećenja (stalno,povremeno, incidentno), seizmičko dejstvo (prekoseizmičkih koeficijenata, pri čemu se odgovor klizištarazmatra u domenu analize kapaciteta pomeranja ilipreko akcelerograma, pri čemu se odgovor klizištarazmatra u vremenskom domenu) i projektne situacije(stalna, povremena, incidentna, seizmička). Posebnimtipovima parametara modeliraju se: konturni uslovi(samo komponente krutosti ili komponente krutosti iprigušenja), prelazni uslovi (interface zone), kruta tela(ne uzimaju se u obzir efekti njihovih deformacija, većsamo pomeranja u ukupnim pomeranjima sistema),specifično trenje na relaciji konstrukcija-tlo (zakonstrukcije koje se nalaze na klizištu ili u njegovoj bliziniili su to konstrukcije kojima se sprovodi sanacija klizišta),

algorithm. All these data obtained by geodetic fieldobservation can be interactively included in themathematical model that analyzes the stability of thelandslide so that through time, the continuous correctionof the numerical model is managed by all aspects of thebudget and further decreases the level of inconsistencyof the input parameters (parameters of the budget modeland parameters of the external/internal actions). Thisnumerical model is, in fact, real time numeric model.

When forming the calculated landslide model, it isnecessary to consider all relevant parameters anddetermine their values, since the final solution is in adirect correlation with the selection and variation ofparameter values. The relevant parameters of thecalculated landslide model can be classified into fivegroups: parameters of the geometry of the calculatedmodel, parameters of physical-mechanical charac-teristics of the soil, parameters of actions, special typesof parameters and calculation parameters. Thecomplexity of the slope geometry and the multi-layeredsoil depth are modelled by parameters of the geometryof the calculated landslide model. Since the layers canbe complex geometries, and not only horizontal orapproximately horizontal, the numerical methods ofcalculating the landslide are used in the calculation ofcomplex multilayer soil geometry. In addition, theseparameters include the parameters of the geometry ofthe structures located on or near the landslide, or theyare constructions for which the landslide is beingrepaired, so that for them, the effect of the soil-structureinteraction (SSI) is considered. The correct input ofthese parameters depends on the level of quality of theformed engineering-geological profile of the terrain. Theparameters of the physical-mechanical characteristics ofthe soil are obtained from laboratory testing of samples,of which the following are distinguished: one-axialstrength, direct shear strength, triaxial test andedometric compressibility test. For stability analyzes ofslopes, parameters are important: soil weight, soil weightin saturated state, cohesion, internal friction angle,Young's elastic modulus, edometric modulus of com-pressibility, deformation module, Poisson's coefficient,reference shear modulus, dilatation, coefficient porosityand other. Depending on the type of constitutive modelof soil behaviour, the relevant parameters are defined,whereas for the constitutive models describing the three-dimensional stress state the number of parameters isconsiderably increased. Most often, Mohr-Coulomb's soilmodel is used as a constitutive soil model for analyzinglandslide, while depending on soil type specificity, softsoil model or hardening soil model can be used, Cam-Clay model, Drucker-Prager model and others. Thereare also additional parameters that enhance theconstitutive model of soil behaviour, such as, forexample, parameters that additionally affect the changein strength and cohesion along the depth of the soil, theintroduction of tensile strength of the soil and thedefinition of consolidation parameters. In addition,knowing the soil consistency (very soft, soft, medium,rigid, very rigid) is of significant interest. When definingthe parameters according to EN 1997-1:2004 code, it isnecessary to know the partial factors for: angle ofinternal friction, effective cohesion and undrained shearstrength of soil[7]. The parameters of the actions aredefined: types of loads (concentrated, linear, surface,


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elementi veze kojima se mogu, između ostalog, modeli-rati i specifični konstruktivni elementi (model ponašanjamože biti linearno-elastičan ili nelinearan), podzemnavoda (direktno modeliranje horizontalnog ili promenljivognivoa podzemne vode NPV, modeliranje pornih pritisaka,modeliranje sile uzgona, modeliranje prslina na površinitla ispunjenih vodom i nastalih usled zatezanja) i faznagradnja/sanacija (modeliranje promene geometrije kosi-ne po fazama gradnje/sanacije, modeliranje promene tlapo fazama gradnje/sanacije, modeliranje promene nivoapodzemne vode po fazama gradnje/sanacije, modelira-nje promene opterećenja po fazama gradnje/sanacije,modeliranje promene dejstva zemljotresa po fazamagradnje/sanacije, modeliranje promene projektne situa-cije po fazama gradnje/sanacije). Parametri proračunaumnogome definišu aspekte numeričkih analiza stabilno-sti kosina: broj inkremenata kod inkrementalno-iterativneanalize, broj iteracija kod inkrementalno-iterativne anali-ze, broj korekcija matrice krutosti sistema, vrednostitolerancija (za pomeranje, neizbalansirane/rezidualnesile i energiju) i faktor optimizacije (algoritam pretraži-vanja minimalnog faktora sigurnosti za veći broj kliznihpovrši).

3 METODE PRORAČUNA STABILNOSTI KLIZIŠTA

3.1 Podela metoda proračuna stabilnosti klizišta

Metode proračuna stabilnosti klizišta generalno semogu podeliti u četiri grupe: analitičke, numeričke,eksperimentalne i hibridne. U zavisnosti od toga koja ćemetoda biti primenjena, dobijaju se rešenja s manjim iliveći stepenom pouzdanosti, s tim što prednost treba datinumeričkim metodama. S obzirom na to što se analitičkei numeričke metode proračuna stabilnosti klizišta najvišeprimenjuju pri projektovanju i sanaciji klizišta, ali i zapotrebe naučnih istraživanja, pregled istraživanja –prikazan u daljem tekstu rada – odnosi se samo na ovemetode. U zavisnosti od načina dobijanja konačnogrešenja ispitivanja stabilnosti klizišta, moguće jesprovesti podelu na metode kojima se rešenje dobijaputem jednog koraka ili jednokoračne analize (one step),putem više koraka ili višekoračne analize (step by step) iinkrementalno-iterativne nelinearne analize. Shodnoprethodno definisanom, uvedena je podela na metodeproračuna klizišta:

spatial), types of load action (permanent, occasional,incidental), seismic effect (through seismic coefficients,where the response of the landslide is considered in thecapacity domain or through the accelerogram, where theresponse of the landslide is considered in the timedomain) and the project situation (permanent,occasional, incidental, seismic). Specific types ofparameters are modelled: contour conditions (onlystiffness or stiffness and damping components),interface zone, rigid bodies (they do not take intoaccount the effects of their deformations, but onlydisplacements in overall system displacements), specificfriction on construction-ground relation (for structureslocated on or near the landslide or structures that areused for landslide repair), link elements which can beused to model specific structural elements (thebehaviour model can be linear-elastic or non-linear),groundwater (direct modelling of horizontal or variablelevel of groundwater NPV, modelling of the pore stress,modelling of the lifting force, modelling of cracks on thesurface of the soil filled with water and caused bytensioning) and phase construction/repair (modelling theslope geometry change by construction/repair phases,modelling the soil change by construction/repair phases,modelling the groundwater level change by construc-tion/repair phases, modelling the load change byconstruction/repair phases, modelling the change of theearthquake effects by construction/repair phases,modelling the change in the project situation by con-struction/repair phases). The calculated parametersdefine, as far as possible, the numerical analysis of theslope stability: number of increments in the incremental-iterative analysis, number of iterations in the incre-mental-iterative analysis, number of corrections of thesystem stiffness matrix, tolerance values (fordisplacement, unbalanced/residual forces and energy)and optimization factor (algorithm for search of theminimal safety factor, for a greater number of slidingsurfaces).

3 METHODS OF LANDSLIDE STABILITYCALCULATION

3.1 Methods of landslide stability calculationdivision

Methods of landslide stability calculation can,generally, be divided into four groups: analytical,numerical, experimental and hybrid. Depending on themethod applied, solutions with a lower or a higherdegree of reliability are obtained, while the priorityshould be given to numerical methods. Since theanalytical and numerical methods of landslide stabilitycalculation are mostly applied in the design and repair oflandslides, but also for the needs of scientificresearches, this exactly is why the overview of theresearches, presented in the following text, applies onlyto these methods. Depending on the method of obtainingthe final solution of the landslide stability test, it ispossible to divide the methods according to whether thesolution is obtained through one step or one-stepanalyses, through several steps or step-by-step analy-ses, and incrementally-iterative nonlinear analyses. Ac-cording to the previously defined, a division of landslidecalculation methods was introduced:


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analitičke jednokoračne; analitičke višekoračne (iteracije kliznih površi); numeričke višekoračne (iteracije kliznih površi); numeričke inkrementalno-iterativne (nelinearne)

analize; numeričke inkrementalno-iterativne (nelinearne)

analize, s primenjivanjem numeričke integracije uvremenskom domenu.

3.2 Analitičke metode proračuna stabilnosti klizišta

Ključni faktor u analizi klizišta jeste proračun stabil-nosti klizišta, tako da se identifikuje da li je klizište ustanju ravnoteže, postoji li opasnost od gubitka ravno-teže ili nije u stanju ravnoteže. U opštem slučaju, kodanalitičkih metoda stabilnosti klizišta, tlo se deli navertikalne blokove, a za svaki blok se određuju odgo-varajuće sile, pri čemu klizna površ može biti kružna ilipoligonalna. U zavisnosti od matematičkog modelaproračuna sila koje deluju između blokova i oblikablokova, postoji veliki broj razvijenih analitičkih metoda,od kojih su se u praksi i u nauci ustalile i izdvojilemetode stabilnosti klizišta prema: Sarma-i, Spencer-u,Janbu, Morgenstern-Price-u, Shahunyants-u, Bishop-u,Fellenius/Petterson-u i tako dalje. Na slici 1 dat ješematski prikaz podele tla na blokove za opštu analizustabilnosti kosine s poligonalnom i kružnom kliznompovrši. Odgovarajuće sile za sve blokove glase: nnormalnih sila Ni – koje deluju na svaki pojedinačanblok, n smičućih sila Ti – koje deluju po ivici klizne površisvakog pojedinačnog bloka, n-1 normalnih sila Ei – kojedeluju između blokova, n-1 smičućih sila Xi – koje delujuizmeđu blokova, n-1 geometrijskih mesta zi – na kojimadeluju sile Ei i n geometrijskih mesta li – na kojima delujusile Ni. Ukupno je 6n-2 nepoznatih koje treba odrediti iz4n jednačina (uslova ravnoteže). Evidentno je da se 2n-2 nepoznatih mora ili aproksimirati ili unapred odrediti.

analytical one-step, analytical step-by-step (iterations of sliding

surfaces), numerical step-by-step (iterations of sliding

surfaces), numerical incremental-iterative (nonlinear)

analysis, numerical incremental-iterative (nonlinear)

analysis, by applying numerical integration in the timedomain.

3.2 Analytical methods of landslide stabilitycalculation

The key factor in landslide analysis is landslidestability calculation, so as to identify whether thelandslide is in the state of balance, whether there is arisk of its losing the balance, or if it is not in the state ofbalance. In general, with analytical landslide stabilitymethods, the ground is divided into vertical blocks, andfor each block corresponding forces are determined,whereby the sliding surface can be circular or polygonal.Depending on the mathematical model of the calculationof the forces acting between the blocks and the shapesof the blocks, there are many analytical methodsdeveloped, and the methods of landslide stability towhich the practice and the science became accustomedwith are those according to: Sarma, Spencer, Janbu,Morgenstern-Price, Shahunyants, Bishop, Fellenius/Pet-terson and the like. Figure 1 gives a schematicpresentation of the ground division into blocks forgeneral analysis of slope stability with a polygonal and acircular slide planes. The corresponding forces for all theblocks are: n normal forces Niacting on each individualblock, ns hear forces Ti which act on the edge of theslide plane of each individual block, n-1 normal forces Eiacting between the blocks, n-1 shear forces Xi which actbetween the blocks, n-1 geometric places zi acted on byEi forces and n the geometric places li where forces Niact. In total, 6n-2 unknowns which should be determinedfrom 4n equations (equilibrium conditions). It is obviousthat 2n-2 unknowns have to be either approximated orpredetermined.

a) b)

Slika 1. Podela tla na blokove za opštu analizu stabilnosti kosine: a) poligonalna klizna površ; b) kružna klizna površ [10]

Figure 1. Division of the ground into blocks for general analysis of slope stability: a) polygonal sliding surface, b) circularsliding surface [10]


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Sarma-ina metoda zasniva se na podeli tla nablokove koji nisu strogo vertikalni, već imaju određeniugao zakošenja, pri čemu su Ei i Xi normalne i smičućesile između blokova, Ni i Ti – normalne i smičuće silekoje deluju po ivici klizne površi svakog pojedinačnogbloka, Wi – sopstvena težina bloka, KhWi – horizontalnasila kojom se obezbeđuje postizanje graničnog stanja[28]. Kh faktor predstavlja odnos horizontalnih igravitacionih ubrzanja. Na slici 2 prikazana je podela tlana blokove za analizu stabilnosti kosine prema Sarma-inoj metodi.

The Sarma method is based on the division of theground into blocks that are not strictly vertical, but ratherhave a certain inclination angle, where Ei and Xi arenormal and shear forces between the blocks, Ni and Tinormal and shear forces acting on the edge of the slidingsurface of each individual block, Wi the block’s selfweight, KhWi horizontal force which ensures reaching thelimit state [28]. The Kh factor represents the ratio ofhorizontal and gravitational accelerations. Figure 2shows the division of the ground into blocks for theanalysis of slope stability according to the Sarmamethod.

Slika 2. Podela tla na blokove za analizu stabilnosti kosine prema Sarma-inoj metodi [28]Figure 2. Division of the ground into blocks for slope stability analysis according to the Sarma method [28]

Algoritam proračuna stabilnosti kosine prema Sarma-inoj metodi zasniva se na jednačinama ravnotežeblokova:

The algorithm of the slope stability calculationaccording to the Sarma method is based on the balanceof the blocks equations:

iiiiiiiii,xihiiii δEδEδXδXFWKαNαT coscossinsinsincos 11 , (1)

iiiiiiiii,yiiiii δEδEδXδXFWαTαN cossincoscossincos 1111 , (2)

ii,giiiiiiiiiiiii xxWzEδααbzEδααbXlN 1i1111i1 sinseccossec

0 i,yi,yi,xi,xii,gih rFrFyyWK , (3)

iiiiii αbcφUNT sectg 1 , iiiiii dcφPWEX tg , (4)

gde su Fx,i i Fy,i komponente horizontalne i vertikalneprojekcije sila, rx,i i ry,i kraci Fx,i i Fy,i sila, respektivno, PWirezultanta sile pornog pritiska na podeljene blokove,

iφ prosečna vrednost ugla unutrašnjeg trenja duž klizne

površine pojedinih blokova, ic prosečna vrednostkohezije duž klizne površine pojedinih blokova. Faktorsigurnosti kosine Fs određuje se iterativno, redukujućiparametre c i tgφ, tako da se dostigne vrednost faktoraKh (nula ili veća od nule).

Spencer-ova metoda zasniva se na graničnojravnoteži kosine, uspostavljanjem ravnoteže sila imomenata koji deluju na pojedine blokove [30]. Na slici 3prikazana je podela tla na blokove za analizu stabilnostikosine prema Spencer-ovoj metodi.

where Fx,i and Fy,i are components of the horizontal andvertical forces projections, rx,i and ry,i arms of the forcesFx,i and Fy,i, respectively, of the PWi resultant of the force

of the pore pressure to the divided blocks, iφ theaverage angle value of the internal friction along the

sliding surface of the individual blocks, ic the averagecohesion value along the sliding surface of individualblocks. The slope safety factor Fs is determined byiteratively reducing the parameters c and tgφ, so as toreach the factor Kh value (zero or greater than zero).

The Spencer method is based on the limit equilibriumof the slope, by reaching the balance of forces andmoments acting on individual blocks [30]. Figure 3shows the division of the ground into blocks for slopestability analysis according to the Spencer method.


137

Slika 3. Podela tla na blokove za analizu stabilnosti kosine prema Spencer-ovoj metodi [30]Figure 3. Division of the ground into blocks for slope stability analysis according to the Spencer method [30]

S ciljem postizanja rešenja problema graničneravnoteže kosine, koja je podeljena na blokove, uvedenesu određene pretpostavke: ravni – kojima su podeljeniblokovi – ostaju vertikalne i tokom proračuna, linijadejstva sopstvene težine bloka Wi prolazi kroz centar i-tog segmenta klizne površi i predstavlja se tačkom M,normalna sila Ni deluje u centru i-tog segmenta kliznepovrši u tački M i ugao dejstva sile Ei, koja deluje izmeđublokova, jeste konstantan za sve blokove i jednak je δ.Algoritam proračuna stabilnosti kosine prema Spencer-ovoj metodi zasniva se na izrazima:

In order to achieve a solution to the problem of thelimit equilibrium of the slope, which is divided into blocks,certain assumptions have been made: the planes, whichdivide the blocks, remain vertical during the calculationsas well, the line of action of the block’s self weight Wipasses through the centre of the i-th segment of thesliding surface and it’s represented as the point M, thenormal force Ni acts in the centre of the i-th segment ofthe slide plane at the point M and the angle of action ofthe force Ei, which acts between the blocks, is constantfor all the blocks and equals δ. The algorithm of theslope stability calculation according to the Spencermethod is based on the following expressions:

iii UNN , (5)

i

iiii

i

iiiii α

bcφNα

bφUNTcos

tgcos

tg , (6)

0sinsinsincossincos 11 iiiiiiii,xii,yiihiiii δαEδαEαFαFαWKαWUN , (7)

11coscossincossincos

tgiiiii,xii,yiihii

is

ii

s

ii δαEαFαFαWKαW

αFbc

FφN

0cos iii δαE , (8)

2

sintg2

cos2

sintg2

cos 11111i

iiii

iiii

iiii

iiibδEαbzδEbδEαbzδE

01 i,gMihi yyWKM , (9)

gde je Ui rezultanta pornog pritiska na za i-ti segmentklizne površi, M1i – momenat sila Fx i Fy oko tačke M.Izraz (5) predstavlja relaciju između efektivne i totalnevrednosti normalnih sila koje deluju duž klizne površi.Izraz (6) predstavlja relaciju između normalnih i smičućihsila segmenta klizne površi (Mohr-Coulomb-ovi uslovi).Preformulacijom izraza (7) i (8) dobija se:

where Ui is the resultant of the pore pressure for thei-th segment of the slide plane, M1i is the moment offorces Fx and Fy around the point M. The expression (5)represents the relation between the effective and thetotal value of the normal forces acting along the slidingsurface. The expression (6) represents the relationbetween the normal and shear forces of the slidingsurface segment (Mohr-Coulomb conditions). Byreformulating the expressions (7) and (8), we get:


138

11

1

cossin

sinsincos

iis

iii

s

iiiiiii,xihii,yi

i

δαFφtgδα

FφtgδαEUαFWKαFW

E

iiiii,xihii,yiis

ii δαEαFWKαFWαF

bc coscossin

cos . (10)

Primenom izraza (10) mogu se odrediti sve sile Eikoje deluju između blokova za date vrednosti δi i Fs.Preformulacijom izraza (9), dobija se:

By applying the expression (10), all the forces Eiacting between the blocks for the given values δi and Fscan be determined. By reformulating expression (9) weget:

11

111

1 cos

1costgcossintgcossin2

ii

i,gMihiiiiiiiiiiiii

i δE

yyWKMδzEαδδEαδδEb

z . (11)

Primenom izraza (11), mogu se odrediti svi kraci silez za date vrednosti ugla δi. Faktor sigurnosti Fs određujese primenom iterativnog algoritma: inicijalna vrednost zaugao δ jeste δ=0, faktor sigurnosti Fs, za datu vrednostugla δ, određuje se prema izrazu (10), imajući u vidu tošto je En+1=0 na kraju klizne površi, ugao δ se određujeiz izraza (11), koristeći vrednosti za silu E – koja jeodređena iz prethodnog koraka analize, pri čemu jevrednost zn+1=0 i prethodna dva koraka analize iterativnose ponavljaju sve dok vrednost ugla δ, u dve uzastopneiteracije, ne postane jednaka. Da bi algoritam iteracijabio dovoljno stabilan, potrebno je intervenisati kako bi seotklonila nestabilna rešenja. Ove nestabilnosti javljaju sekada se u izrazima (10) i (11) pojave situacije deljenja snulom. U izrazu (11) ovakva situacija može se pojaviti zavrednosti ugla δ=π/2 ili δ=-π/2, pa se rešenje mora tražitiza interval ugla δ=[-π/2;π/2]. Deljenje s nulom u izrazu(10) pojavljuje se u slučaju:

By applying the expression (11) all the moment armsof the force z for the given values of the angle δi can bedetermined. The safety factor Fsis determined using aniterative algorithm: the initial value for the angle δ is δ=0,the safety factor Fs for the given value of the angle δ isdetermined according to the expression (10), taking intoaccount that En+1=0 at the end of the sliding surface, theangle δ is determined from the expression (11) using thevalues for the force E, which is determined from theprevious step of the analysis, where the value zn+1=0and the previous two steps of the analysis are repeatediteratively until the value of the angle δ, during twoconsecutive iterations, becomes equal. In order for theiteration algorithm to be stable enough, it is necessary tointervene with the aim of eliminating any unstablesolutions. These instabilities occur when expressions(10) and (11) show the situation of the division by zero.In expression (11) such a situation can occur for thevalues of the angle δ=π/2 or δ=-π/2, so the solutionshould be sought for the interval of the angle δ=[-π/2;π/2]. Division by zero in expression (10) appears inthe case of:

iiis αδφF 1tgtg . (12)

Radi sprečavanja nestabilnosti rešenja, potrebno jesprovesti proveru parametra mα prema izrazu:

In order to prevent the solution instabilities, it isnecessary to perform a parameter check mα according tothe expression:

20tgsincos .F

φααms

iiiα . (13)

Pre nego što se započne sa iterativnom analizom,potrebno je pronaći najveću kritičnu vrednost Fs,min kojazadovoljava prethodne uslove. Vrednosti faktorasigurnosti Fs koje su ispod ove kritične vrednosti Fs,minpripadaju oblasti nestabilnog rešenja. Prva iteracijazapočinje s vrednošću faktora sigurnosti Fs koja je teknešto veća od Fs,min, tako da su i preostale vrednostifaktora sigurnosti Fs – koje se određuju proračunom –uvek veće od Fs,min.

Janbu-ova metoda jeste procedura verifikacije

Before beginning iterative analysis, it is necessary tofind the highest critical value of Fs,min that satisfies theprevious conditions. The values of the safety factors Fsbelow this critical value Fs,min belong to the area ofunstable solution. The first iteration starts with the valueof the safety factor Fs, which is just slightly higher thanFs,min, so the remaining values of the safety factors Fs,which are determined by the calculation, are alwayshigher than Fs,min.

The Janbu's method is a procedure of verifying the


139

stabilnosti granične ravnoteže kosina, a zasniva se nauspostavljanju ravnoteže sila i momenata koji deluju napojedine blokove [19]. Na slici 4 prikazana je podela tlana blokove za analizu stabilnosti kosine prema Janbu-ovoj metodi.

stability of the slopes’ limit equilibrium, and it is based onestablishing the balance between forces and momentsacting on individual blocks [19]. Figure 4 shows thedivision of the ground into blocks for slope stabilityanalysis according to Janbu's method.

Slika 4. Podela tla na blokove za analizu stabilnosti kosine prema Janbu-ovoj metodi [19]Figure 4. Division of the ground into blocks for slope stability analysis according to the Janbu's method [19]

Radi postizanja rešenja problema graničneravnoteže kosine koja je podeljena na blokove, uvedenesu određene pretpostavke: ravni – kojima su podeljeniblokovi – ostaju vertikalne i tokom proračuna, linijadejstva sopstvene težine bloka Wi prolazi kroz centar i-tog segmenta klizne površi i predstavlja se tačkom M,normalna sila Ni deluje u centru i-tog segmenta kliznepovrši u tački M i vertikalna pozicija zi dejstva sile Ei,koja deluje između blokova, jednaka je nuli za krajnjetačke klizne površi. Izbor vertikalne pozicije zi dejstvasile Ei ima značajan uticaj na dobijanje konvergentnogrešenja. Ukoliko se loše pretpostave vertikalne pozicijezi, može nastupiti divergencija rešenja, uz prethodnoznatno povećanje vremena proračuna. Vertikalnepozicije zi dejstava sila Ei usvajaju se da su jednakitrećini visine blokova na koje je podeljena kosina.Ukoliko nastupi divergencija rešenja, potrebno jekorigovati vrednosti zi, tako što se one blago povećavajukod blokova pasivne zone (kod nožice kosine) i blagosmanjuju kod blokova aktivne zone (kod vrha kosine).Algoritam proračuna stabilnosti kosine prema Janbu-ovojmetodi zasniva se na izrazima:

In order to reach a solution to the problem of the limitequilibrium of the slope, which is divided into blocks,certain assumptions have been made: the planes whichdivide the blocks, remain vertical during the calculationas well, the line of action of the block’s self weight Wipasses through the centre of the i-th segment of thesliding surface at the point M, the normal force Ni acts inthe centre of the i-th segment of the slide plane at thepoint M and the vertical position zi of the action of theforce Ei, which acts between the blocks, is equal to zerofor the end points of the sliding surface. The choice ofthe vertical position zi of the effect of the force Ei has asignificant influence on obtaining a convergent solution.If the vertical positions of zi are inaccurately assumed,divergence of the solution can occur, with a significantincrease in the calculation time. Vertical positions of ziaction of the forces Ei are assumed to be equal 1/3 ofthe blocks height, to which the slopes are divided. Ifthere a divergence of the solution occurs, it is necessaryto correct the zi values, by slightly increasing them withthe passive zone blocks (at the foot of the slope) andslightly decreasing them with the blocks of the activezone (at the top of the slope). The algorithm of the slopestability calculation according to the Janbu's method isbased on the expressions:

iii UNN , (14)

i

iiii

i

iiiii α

bcφNα

bφUNTcos

tgcos

tg , (15)

0sinsinsincossincos 11 iiiiiiii,xii,yiihiiii δαEδαEαFαFαWKαWUN , (16)

11coscossincossincos

tgiiiii,xii,yiihii

is

ii

s

ii δαEαFαFαWKαW

αFbc

FφN

0cos iii δαE , (17)


140

2

sintg2

cos2

sintg2

cos 11111i

iiii

iiii

iiii

iiibδEαbzδEbδEαbzδE

01 i,gMihi yyWKM . (18)

Preformulacijom izraza (16) i (17) dobija se: Reformulation of the expressions (16) and (17) gives:

11

1

cossin

sinsincos

iis

iii

s

iiiiiii,xihii,yi

i

δαFφtgδα

FφtgδαEUαFWKαFW

E

iiiii,xihii,yiis

ii δαEαFWKαFWαF

bc coscossin

cos. (19)

Preformulacijom izraza (18) dobija se: Reformulation of the expression (18) gives:

22

11

11

22

12

sin2tgcos

arcsintg2arctg

iiiii

ii

iii

iii

ii

ii

bαtgbzE

MbδαbzδEα

bzδ . (20)

Faktor sigurnosti Fs određuje se primenomiterativnog algoritma: inicijalne vrednosti svih uglova suδi=0 i pozicije zi su usvojene da su jednake trećini visineblokova, faktor sigurnosti Fs, za datu vrednost ugla δ,određuje se prema izrazu (19), uzimajući u obzir da jeEn+1=0 na kraju klizne površi, ugao δ se određuje izizraza (20) koristeći vrednosti za silu E, koja je određenaiz prethodnog koraka analize i prethodna dva korakaanalize iterativno se ponavljaju, sve dok vrednost ugla δu dve uzastopne iteracije ne postane jednaka.Otklanjanje nestabilnih rešenja sprovodi se isto kao i uslučaju Spencer-ove metode.

Morgenstern-Price-ova metoda verifikacije stabilnostigranične ravnoteže kosina zasniva se na sličnomprincipu kao i metode Spencer-a i Janbu-a [26], [36]. Naslici 5 prikazana je podela tla na blokove za analizustabilnosti kosine prema Morgenstern-Price-ovoj metodi.

The safety factor Fsis determined using an iterativealgorithm: the initial values of all angles are δi=0 and thepositions zi are assumed to be equal to 1/3 of the blocks’height, the safety factor Fs for the given angle δ value, isdetermined according to the expression (19), taking intoaccount that En+1=0 at the end of the sliding surface, theangle δ is determined from the expression (20) using thevalues for the force E, which is determined from theprevious step of the analysis, and the previous two stepsof the analysis are iteratively repeated until the value ofthe angle δ in two consecutive iterations is equal.Removing any unstable solutions is conducted in thesame way as with Spencer's method.

The Morgenstern-Price's method for verifying thestability of the limit equilibrium of slopes is based on aprinciple similar to Spencer's and Janbu's methods [26],[36]. Figure 5 shows the division of the ground intoblocks for the slope stability analysis according to theMorgenstern-Price's method.

Slika 5. Podela tla na blokove za analizu stabilnosti kosine prema Morgenstern-Price-ovoj metodi [26]Figure 5. Division of the soil into blocks for the slope stability analysis of according to the

Morgenstern-Price's method [26]


141

S ciljem postizanja rešenja problema graničneravnoteže kosine koja je podeljena na blokove, uvedenesu određene pretpostavke (slično Spencer-ovoj metodi):ravni, kojima su podeljeni blokovi, ostaju vertikalne itokom proračuna, linija dejstva sopstvene težine blokaWi prolazi kroz centar i-tog segmenta klizne površi ipredstavlja se tačkom M, normalna sila Ni deluje ucentru i-tog segmenta klizne površi u tački M i ugaodejstva sile Ei (koja deluje između blokova) je različit zasve blokove i jednak je δ=0 za krajnje tačke.Pretpostavka o vrednosti ugla δi uspostavlja seprimenom polusinusne funkcije. Na slici 6 prikazan jespektar polusinusnih funkcija. Izbor oblika funkcije imamanjeg uticaja na kvalitet konačnog rešenja, alioptimalnim izborom oblika funkcije doprinosi sekonvergenciji rešenja. Ugao δi određuje semultiplikacijom vrednosti polusinusne funkcije f(xi) iparametra λ.

In order to reach a solution to the problem of the limitequilibrium of a slope, which is divided into blocks,certain assumptions (similar to the Spencer’s method)have been made: the planes, which divide the blocks,remain vertical during the calculations as well, the line ofaction of the block’s self weight Wi passes through thecentre of the i-th segment of the sliding surface and it’srepresented as the point M, the normal force Ni acts inthe centre of the i-th segment of the sliding surface atthe point M, and the angle of action of the force Ei(acting between the blocks) is different for all the blocksand equals δ=0 for the end points. The assumption ofthe value of the angle δi is established by using the half-sine function. Figure 6 shows a spectrum of half-sinefunctions. The choice of the form of the function has lessinfluence on the quality of the final solution, but with thechoice of an appropriate form of the function, contributesto the convergence of the solution. The angle δi isdetermined by multiplying the value of the half-sinefunction f(xi) and the parameter λ.

Slika 6. Polusinusna funkcija za pretpostavke o vrednosti ugla δi [26]Figure 6. A half-sine function for assumptions about the value of the angle δi [26]

Algoritam proračuna stabilnosti kosine, premaMorgenstern-Price-ovoj metodi, zasniva se na izrazimakoji su identični izrazima (5÷11) kod Spencer-ovemetode. Faktor sigurnosti Fs određuje se primenomiterativnog algoritma: inicijalna vrednost uglova δi jeδi=λf(xi), faktor sigurnosti Fs, za datu vrednost ugla δ,određuje se prema izrazu (10), uzimajući u obzir da jeEn+1=0 na kraju klizne površi,ugao δ se određuje izizraza (11) koristeći vrednosti za silu E, koja je određenaiz prethodnog koraka analize (zn+1=0), pri čemu sevrednost polusinusne funkcije f(xi) zadržava kaokonstantna kroz iteracije, a iterira se parametar λ iprethodna dva koraka analize iterativno se ponavljajusve dok vrednost ugla δ u dve uzastopne iteracije nepostane jednaka. Kako bi se sprečila numeričkanestabilnost rešenja, sprovode se kontrole premaizrazima (12) i (13).

Shahunyants-ova metoda verifikacije stabilnostigranične ravnoteže kosina zasniva se na sličnomprincipu kao i prethodne metode [31]. Na slici 7prikazana je podela tla na blokove za analizu stabilnostikosine prema Shahunyants-ovoj metodi. Radi postizanjarešenja problema granične ravnoteže kosine koja jepodeljena na blokove, uvedene su određenepretpostavke: ravni, kojima su podeljeni blokovi, ostajuvertikalne tokom proračuna i ugao dejstva sile Ei, kojadeluje između blokova, jednak je nuli (sile delujuhorizontalno).

The algorithm of the slope stability calculationaccording to the Morgenstern-Price's method is basedon the expressions that are identical to expressions(5÷11) in the Spencer’s method. The safety factor Fs isdetermined by using an iterative algorithm: the initialvalue of the angles δi is δi=λf(xi), the safety factor Fs forthe given value of the angle δ is determined according tothe expression (10), taking into account that En+1=0 is atthe end of the sliding surface, the angle δ is determinedfrom the expression (11) using the values for the force E,which is determined from the previous step of theanalysis (zn+1=0), while the value of the half-sine functionf(xi) is kept constant through iterations, and theparameter λ is iterated and the previous two steps of theanalysis are iteratively repeated until the value of theangle δ is equal in two consecutive iterations. In order toprevent the numerical instability of the solution, controlsare conducted according to the expressions (12) and(13).

The Shahunyants's method for verifying the stabilityof the limit equilibrium of slopes is based on a similarprinciple as the previous methods [31]. Figure 7 showsthe division of the ground into blocks for slope stabilityanalysis according to the Shahunyants's method. Inorder to reach a solution to the problem of the limitequilibrium of the slope, which is divided into blocks,certain assumptions have been made: the planes, whichdivide the blocks, remain vertical during the calculation,and the angle of action of the force Ei, acting betweenthe blocks, equals zero (the forces act horizontally).


142

Slika 7. Podela tla na blokove za analizu stabilnosti kosine prema Shahunyants-ovoj metodi [31]Figure 7. Division of the ground into blocks for slope stability analysis according to the Shahunyants's method [31]

Algoritam proračuna stabilnosti kosine premaShahunyants-ovoj metodi započinje transformacijom silaPx,i i Py,i u pravcu normale (N) i tangente (T) kliznepovrši:

The algorithm of the slope stability calculationaccording to the Shahunyants's method begins with thetransformation of the forces Px,i and Py,i in the directionof the normal (N) and the tangent (T) of the slidingsurface:

ii,yii,xi,N αPαPP cossin , (21)

ii,xii,yi,Q αPαPP cossin . (22)

Sile koje deluju duž segmenata klizne površiproračunavaju se prema:

The forces acting along the sliding surface segmentsare calculated according to:

iiiiii lcφUNT tg . (23)

Jednačina ravnoteže upravno na ravan segmenta kliznepovrši glasi:

The equation of equilibrium perpendicular to theplane of the sliding surface segment is:

iiiii,Ni αEαEPN sinsin1 , (24)

dok jednačina ravnoteže u ravni segmenta klizne površiglasi:

while the equation of equilibrium in the plane of thesliding surface segment is:

iiiii,Qi αEαEPT coscos 1 . (25)

Uvođenjem izraza (23) u (25) dobija se: By introducing the expression (23) into (25), we get:

iiiii,Qiiiii αEαEPlcφUN coscostg 1 , (26)

dok se uvođenjem izraza (24) u (26) dobija: whereas, by introducing the expression (24) into (26),we get:

iiiii,Qiiiiiiiii,N αEαEPlcφUαEαEP coscostgsinsin 11 . (27)

Nakon sređivanja izraza (27), dobija se: After arranging the expression (27), we get:

iiii,Qiiiiiiiii,N αEEPlcφαEEφUP costgsintg 11 , (28)

odnosno: i.e.:

iiiiii,Qiiiii,N φααEEPlcφUP tgsincostg 1 . (29)

S obzirom na to što je: Taking into the account the following:

ββα

ββαβαβαα

coscos

cossinsincoscostgsincos

, (30)


143

dobija se da je izraz (29): we get that the expression (29) is:

i

iiiii,Qiiiii,N φ

φαEEPlcφUPcos

costg 1

, (31)

a dodatnom modifikacijom izraza (31) dobija se: and the additional modification of the expression (31)gives:

i

iii

i

iiii,Qiiiii,N φ

φαEφφαEPlcφUP

coscos

coscostg 1

. (32)

Primenom izraza (32) sile koje deluju između blokova Eiodređuju se prema:

By applying the expression (32), the forces actionbetween the blocks Ei are determined according to:

1cos

costg

iii

ii,Qiiiii,Ni E

φαφPlcφUP

E . (33)

Sada se u proračun stabilnosti kosine uvodi faktorsigurnosti Fs, dok se PQ,i sile razlažu na sile kojedoprinose klizanju PQ,i,sd (aktivne sile) i sile koje nedoprinose klizanju PQ,i,ud (stabilizujuće sile):

Now, the safety factor Fs is introduced into the slopestability calculation, while the PQ,i forces are brokendown into the forces contributing to the sliding PQ,i,sd(active forces) and the forces that do not contribute tosliding PQ,i,ud (stabilizing forces):

1cos

costg

iii

iud,i,Qsd,i,Qsiiiii,Ni E

φαφPPFlcφUP

E . (34)

PQ,i je pozitivno kada doprinosi klizanju kosine, anegativno kada ne doprinosi klizanju kosine, tako da seizraz (34) može pisati u formi:

PQ,i is positive when it contributes to the sliding of theslope, and negative when it does not contribute to thesliding of the slope, hence, the expression (34) can bewritten in the form:

1cos

costg

i

ii

iud,i,Qsd,i,Qsiiiii,Ni E

φαφPPFlcφUP

E . (35)

Na kliznoj površi vrednost sile E0 jednaka je nula, dok zaE1 važi:

On the sliding surface, the value of the force E0equals zero, whereas the following applies to E1:

11

111111111 cos

costgφα

φPPFlcφUPE ud,,Qsd,,Qs,N

, (36)

a za E2: and to E2:

22

222222222 cos

costgφα

φPPFlcφUPE ud,,Qsd,,Qs,N

11

11111111

coscostg

φαφPPFlcφUP ud,,Qsd,,Qs,N

. (37)

Slično se mogu prikazati i izrazi za sve sile koje delujuizmeđu blokova, pri čemu je En=0:

The expressions for all the forces acting between theblocks can be presented in a similar way, where En=0:

0cos

coscos

costg11

n

i ii

isd,i,Qs

n

i ii

iud,i,Qiiiii,Nn φα

φPFφα

φPlcφUPE , (38)

tako da se iz ovog izraza može direktno prikazati faktorsigurnosti Fs u formi:

so that, from this expression, the safety factor Fs can bedirectly presented in the following form:


144

n

i ii

isd,i,Q

n

i ii

iud,i,Qiiiii,N

s

φαφP

φαφPlcφUP

F

1

1

coscos

coscostg

. (39)

Faktor sigurnosti Fs prema Fellenius/Petterson-ovojmetodi određuje se na osnovu izraza:

The safety factor Fs, according to the Fellenius/Pet-terson's method, is determined on the basis of theexpression:

i

iiiiii

iii

s φluNlcαW

F tgsin1

, (40)

dok se prema Bishop-ovoj metodi određuje na osnovuizraza:

whereas, according to Bishop’s method, it is determinedon the basis of the expression:

i

s

iii

iiiiii

iii

s

Fαφα

φbuWbcαW

F sintgcos

tgsin1

. (41)

3.3 Numeričke metode proračuna stabilnostiklizišta

Proračun stabilnosti klizišta numeričkim metodamazasniva se na metodama diskretizacije domena, kao štosu:

metoda konačnih elemenata (FEM – FiniteElement Method);

proširena metoda konačnih elemenata (XFEM –eXtended Finite Element Method);

metoda graničnih elemenata (BEM – BoundaryElement Method);

metoda diskretnih elemenata (DEM – DiscreteElement Method);

metoda konačnih razlika (FDM – Finite DifferenceMethod).

U ovim metodama, tlo se razmatra kao linearno-elastičan, elasto-plastičan i nelinearan materijal. Metodakonačnih elemenata (FEM) najčešće se upotrebljava zarešavanje problema numeričke analize stabilnostikosina, tako da veliki broj softvera ima implementiranealgoritme zasnovane na ovoj metodi. Na slici 8prikazana je mreža konačnih elemenata diskretnognumeričkog modela kosine i skup tačaka dobijenihoptimizacijom faktora sigurnosti kosine prema metodikonačnih elemenata (FEM). Kosina se modeliraprimenom površinskih konačnih elemenata saintegrisanom matematičkom formulacijom za analizuravnog stanja deformacija (plane strain). Prilikommodeliranja i analize stabilnosti kosina, potrebno je imatiu vidu dva bitna aspekta: diskretizaciju i aproksimaciju.

3.3 Numerical methods of landslide stabilitycalculations

Landslide stability calculation using numericalmethods is based on methods of domain discretization,such as:

Finite Element Method (FEM), eXtended Finite Element Method (XFEM), Boundary Element Method (BEM), Discrete Element Method (DEM), Finite Difference Method (FDM).In these methods, the soil is considered as a linear-

elastic, elasto-plastic and non-linear material. The FiniteElement Method (FEM) is mostly used for solving theproblem of numerical slope stability analysis, so a largenumber of software has implemented algorithms basedon this method. Figure 8 shows the mesh of finiteelements of the discrete numerical model of the slopeand the set of points obtained by optimizing the slopesafety factor according to the Finite Element Method(FEM). The slope is modelled by using surface finiteelements with an integrated mathematical formulation forthe analysis of the plane strain. When modelling andanalyzing slope stability, two important aspects need tobe taken into account: discretization and approximation.Discretization refers to the problem of the grounddomain division into finite elements of sufficiently smalldimensions for which the criteria of the relation betweenthe diagonal and the angles of the quadrangle finiteelement or the relations of the sides of the triangle finiteelement must be respected. In the area of the contact

Diskretizacija se odnosi na problem podele domena tlana konačne elemente dovoljno malih dimenzija za kojese moraju poštovati kriterijumi odnosa dijagonala iuglova četvorougaonog konačnog elementa ili odnosistranica traouganog konačnog elementa. U oblastikontakta tla sa elementima za plitko ili dubokofundiranje, koji se koriste prilikom sanacije klizišta,potrebno je izvršiti progušćenje mreže konačnihelemenata. Takođe, progušćenje se sprovodi i u zoniklizne površi, na mestima diskontinuiteta i otvora u tlu islično.

between the ground and the elements for shallow ordeep foundation, which are used during the landsliderepair, it is necessary to increase the density of themesh of finite elements. In addition, the increase indensity realized in the sliding surface area as well, atdiscontinuity points and in the openings in the ground,and the like.


145

a) b)

Slika 8. 2D numerički model kosine: a) mreža konačnih elemenata diskretnog numeričkog modela kosine prema metodikonačnih elemenata (FEM); b) skup tačaka dobijenih optimizacijom faktora sigurnosti kosine prema metodi konačnih

elemenata (FEM)[12]

Figure 8. 2D numerical model of a slope: a) a finite elements mesh of the discrete numerical model of a slope accordingto the Finite Element Method (FEM); b) a set of points obtained by optimizing the slope safety factor according to the

Finite Element Method (FEM) [12]

Uspostavljanje veze osnovnih konačnih elemenatakoji formiraju domen tla, s progušćenom mrežomkonačnih elemenata, sprovodi se primenom prelaznihelemenata. Kao prelazni elementi, najčešće seprimenjuju trougaoni konačni elementi. Veoma bitanaspekt jeste i uspostavljanje kompatibilnosti čvorovakonačnih elemenata, analizom konformnosti/nekonform-nosti, posebno kod prelaznih konačnih elemenata, pričemu se ne sme dozvoliti da određeni čvorovi, ukombinaciji osnovnih i prelaznih konačnih elemenata,ostanu nepovezani ili parcijalno povezani. Na slici 9prikazani su 2D numerički modeli kosina, s generisanimmrežama konačnih elemenata i progušćenjima poselektovanim domenima.

Establishing a connection between the basic finiteelements, which form the domain of the ground, with theincreased density mesh of finite elements is carried outby using transition elements. As transition elements, themost commonly used are triangular finite elements. Avery important aspect is also establishing the compa-tibility of finite elements nodes through conformity/non-conformity analysis, especially with transition finiteelements, whereby it should not be allowed for certainnodes, in combination of basic and transition finiteelements, to be left unconnected or partially connected.Figure 9 shows the 2D numerical slope models withgenerated finite element mesh and increased densityover selected domains.

Slika 9. 2D numerički modeli kosina s generisanim mrežama konačnih elemenata i progušćenjima po selektovanimdomenima [32]

Figure 9. 2D numerical slope models with generated finite element mesh and increased density over selected domains[32]


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U odnosu na 2D model kosine, koji se i najviše koristiu praktične svrhe, primenom 3D modela kosine mogu semodelirati kompleksniji geometrijski modeli s prostornosloženijom i promenljivijom geologijom na manjemprostoru. Na slici 10 prikazani su 2D i 3D numeričkimodeli kosine, sa izdvojenim prikazom klizne mase tla iprostornim modelom klizne površi. Za modeliranje 3Dmodela kosina koriste se prizmatični (solid) ili tetra-edarski konačni elementi, pri čemu modeliranje domenatla prostornim konačnim elementima zahteva znatnijehardverske kapacitete. Kod prizmatičnih konačnihelemenata, primenjuje se minimalno 2x2x2 numeričkaintegracija preko Gaussian-ovih kvadratura [8].

Compared to the 2D model of the slope, which is theone mostly used for practical purposes, by using the 3Dmodel of the slope, more complex geometric models canbe modelled, with a spatially more complex andsignificantly more variable geology in a smaller area.Figure 10 shows the 2D and 3D numerical models of theslope with a separate representation of the sliding massof the soil and the spatial model of the sliding surface.For the modelling of the 3D model of slopes, solid ortetrahedral finite elements are used, whereby modellingof the ground domain by spatial finite elements requiressignificantly higher hardware capacities. With prismaticfinite elements, a minimum of 2x2x2 numericalintegration is applied over Gaussian quadratures [8].

Slika 10. 2D i 3D numerički modeli kosine sa izdvojenim prikazom klizne mase tla i prostornim modelom klizne površi[23]

Figure 10. 2D and 3D numerical models of the slope with a separate representation of the sliding mass of the soil and aspatial model of the sliding surface [23]

Na slici 11 prikazani su 3D numerički modeli kosina –formirani od tetraedarskih i prizmatičnih konačnihelemenata, dok su na slici 12 prikazani 3D numeričkimodeli kosina formirani od prizmatičnih konačnihelemenata koji za osnovu imaju trougao, kvadrat ičetvorougao s različitim unutrašnjim uglovima.

Figure 11 shows 3D numerical models of slopesformed from tetrahedral and solid finite elements, whileFigure 12 shows 3D numerical models of slopes formedfrom solid finite elements, which have the base in theshape of a triangle, square and quadrangle with differentinner corners.

a) b)

Slika 11. 3D numerički modeli kosina formirani od: a) tetraedarskih konačnih elemenata [33]; b) prizmatičnih konačnihelemenata [14]

Figure 11. 3D numerical models of slopes formed from: a) tetrahedral finite elements [33], b) solid finite elements [14]


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a) b)

Slika 12. 3D numerički modeli kosina formirani od prizmatičnih konačnih elemenata koji za osnovu imaju: a) trougao [1];b) kvadrat i četvorougao s različitim unutrašnjim uglovima [4]

Figure 12. 3D numerical models of slopes formed from solid finite elements that have the base in the shape of a: a)triangle [1], b) square and quadrandgle with different inner angles [4]

U određenim slučajevima, kada je domen tla znatnihdimenzija i kompleksnije geometrije, mreža konačnihelemenata 3D modela kosine može imati i nekolikomiliona konačnih elemenata, pa se u tim slučajevimanajčešće primenjuje tehnika paralelnog procesiranja.Dodatno se kod ovakvih problema optimizuje mrežakonačnih elemenata i numeracija čvorova elemenata, sobzirom na to što se optimizacijom numeracije čvorovakonačnih elemenata redukuje širina trake matricekrutosti sistema i članovi matrice krutosti sistema grupišuoko dijagonale. Na slici 13 prikazani su 3D numeričkimodeli kosina nešto složenije geometrije sa izdvojenomkliznom masom tla. Modeliranje klizne površi – u analizistabilnosti 3D modela kosina – može se sprovesti, kaošto je već prezentovano, primenom 3D prostornihkonačnih elemenata ili čak primenom 2D površinskihkonačnih elementa.

In certain cases, when the ground domain is ofconsiderable dimensions and a slightly complexgeometry, the finite elements mesh of the 3D model ofthe slope can even have a several million finiteelements, so in these cases, the most commonly used isparallel processing technique. With this type ofproblems, the mesh of finite elements and thenumbering of the nodes of the elements are additionallyoptimized, since optimizing the numbering of finiteelement nodes reduces the bandwidth of the systemstiffness matrix and concentrates the members of thesystem stiffness matrix around the diagonal. Figure 13shows 3D numerical slopes models of a slightly complexgeometry with the separate sliding mass of soil.Modelling the sliding surface, when analyzing thestability of 3D slopes models, can be carried out, as ithas already been presented, by using 3D spatial finiteelements or even 2D surface finite elements.

a) b)

Slika 13. 3D numerički modeli kosina složenije geometrije s prikazanom izdvojenom kliznom masom tla [35]Figure 13. 3D numerical models of slopes of a more complex geometry with the sliding mass of the soil separately shown

[35]

Na slici 14 prikazani su 3D numerički modeli kosinanešto složenije geometrije, s prikazanom izdvojenomkliznom masom tla i položajima proračunatih tačakafaktora sigurnosti, dobijenih optimizacijom za konkavnu ikonveksnu kliznu površ. Konkavna klizna površformirana je iz 3D prostornih konačnih elemenata, dok jekonveksna klizna površ formirana kombinacijom 3Dprostornih i 2D površinskih konačnih elemenata.

Figure 14 shows the 3D numerical models of theslopes of a slightly complex geometry with separatelyshown sliding mass of the soil and the locations of thecalculated points of the safety factors, obtained throughoptimization for the concave and convex sliding surface.The concave sliding surface is formed from 3D spatialfinite elements, while the convex sliding surface isformed by combining 3D spatial and 2D surface finiteelements.


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a) b)

Slika 14. 3D numerički modeli kosina složenije geometrije s prikazanom izdvojenom kliznom masom tla i položajimaproračunatih tačaka faktora sigurnosti, dobijenih optimizacijom: a) konkavna klizna površ; b) konveksna klizna površ [35]

Figure 14. 3D numerical models of slopes of a slightly complex geometry with separately shown sliding mass of the soiland the positions of the calculated safety factor points, obtained by optimization: a) concave sliding surface, b) convex

sliding surface [35]

Modeliranje omekšanja i diskontinuiteta u tlusprovodi se korekcijom parametara konstitutivnogmodela ponašanja tla i eliminacijom veze određenihkonačnih elemenata ili čak redukcijom određenog brojakonačnih elemenata koji se nalaze u posebnoj zoniprogušćenja mreže konačnih elemenata. Aproksimacijase odnosi na izbor optimalnog tipa konačnog elementakojim se efikasno modelira polje pomeranja tla u modelukosine. U ovom slučaju, postoji niz razvijenih tipovakonačnih elemenata kod kojih se nepoznate određujuputem sila, pomeranja ili kombinovano (mešovito). Zainterpolacione funkcije koristi se izoparametarskaformulacija, pri čemu su čvorovi za proračun numeričkihintegracija rapoređeni u uglovima, u unutrašnjosti i/ili pokonturi konačnog elementa. Takođe, aspektaproksimacije odnosi se na numeričko modeliranjekonturnih i prelaznih uslova, modeliranje ponašanjamaterijala i modeliranje dejstava – opterećenja.

Proširena metoda konačnih elemenata (XFEM), zarazliku od metode konačnih elemenata (FEM), imamogućnost primene poboljšane nelinearne analize iproračuna postnelinearnog ponašanja sistema. Takođe,kod ove metode, prilikom formiranja klizišta, može semodelirati razvoj: prslina, pukotina i raseda u tlu. Prslineu tlu, u opštem slučaju, modeliraju se kao razmazane,dok se kod visokozahtevnih problema formiranja klizištaprimenjuju algoritmi modeliranja diskretnih prslina. Modeldiskretnih prslina u tlu zahteva implementacijualgoritama mehanike kontakta, dok se modelrazmazanih prslina u tlu rešava nelinearnom analizomtrajektorija ekstremnih vrednosti glavnih napona u tlu.Metoda graničnih elemenata (BEM) ima značajnuprimenu u geotehnici, budući da se primenom ovemetode brže dobijaju rešenja, u odnosu na metodukonačnih elemenata (FEM), pri čemu je i nivo kvalitetakonačnog rešenja zadovoljavajući. S obzirom na to štopostoji nekoliko algoritama u okviru metode graničnihelemenata (BEM), oni se – u najvećem broju slučajeva –

Modelling of the softening and discontinuity in thesoil is carried out by correcting the parameters of theconstitutive model of soil behaviour and eliminating theconnection of certain finite elements or even thereducing of a number of finite elements, which arelocated in a special zone of refined finite element mesh.The approximation refers to the choice of the optimaltype of the finite element through which the field of soildisplacement in the slope model is effectively modelled.In this case, there is a number of developed finiteelements types in which unknowns are determined by:force, displacement or combined (mixed). Forinterpolation functions, an isoparametric formulation isused, while the nodes for the numerical integrationcalculation are mapped: in the angles, in the interiorand/or on the contour of the final element. Also, theaspect of approximation refers to: numerical modelling ofcontour and transition conditions, modelling of materialbehaviour and modelling of effects - loads.

The eXtended Finite Element Method (XFEM),compared to the Finite Element Method (FEM), offersthe possibility of applying an improved nonlinear analysisand the post-non-linear system behaviour calculation.Also, with this method, during the formation of thelandslide, it is possible to modelled the development of:cracks, gaps and splits in the soil. In general, cracks inthe soil are modelled as smeared, while with the highlydemanding problems of landslide formation, themodelling algorithms for discrete cracks are applied. Themodel of discrete cracks in the ground requires theimplementation of algorithms of contact mechanics,while the model of smeared cracks in the soil is solvedby nonlinear analysis of the main stress in the soil forextreme values trajectory. The Boundary ElementsMethod (BEM) has a significant application in geo-technics, since the application of this method givessolutions faster than the Finite Elements Method (FEM),while the quality of the final solution is also satisfactory.


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zasnivaju na diskretizaciji granične oblasti (kontura)graničnim elementima. Unutrašnjost oblasti najčešće sene diskretizuje, pa ovakve metode pripadaju grupibezmrežnih metoda. Metoda diskretnih elemenata(DEM) zasniva se na razmatranju ravnotežnog stanjapojedinačno za svaki konačni element. U poređenju smetodom konačnih elemenata (FEM), gde seravnotežno stanje razmatra na globalnom nivou prekokompletnog numeričkog modela, kod metode diskretnihelemenata (DEM) jednačine kretanja definišu seposebno za svaki konačni element, tako da se mogupratiti međusobno nezavisna polja pomeranja konačnihelemenata. Na slici 15 prikazan je 2D numerički modelkosine prema metodi diskretnih elemenata (DEM) saidentifikovanom zonom iniciranja klizišta.

Since there are several algorithms within the BoundaryElements Method (BEM), they are mostly based on thediscretization of the boundary area (contours) by theboundary elements. In most cases, the intrinsic domainis not discretized, so such methods belong to the groupof mesh free methods. The Discrete Element Method(DEM) is based on the analysis of the equilibrium statefor each finite element individually. In comparison to theFinite Element Method (FEM), where the equilibriumstate is considered globally, through a completenumerical model, with the Discrete Element Method(DEM), the motion equations are defined for each finiteelement individually, so that the independent fields offinite elements movement can be traced. Figure 15shows 2D numerical model of the slope according to theDiscrete Elements Method (DEM) with identifiedlandslide initiation zone.

a) b)

Slika 15. 2D numerički model kosine: a) numerički model kosine prema metodi diskretnih elemenata (DEM); b)identifikacija zone iniciranja klizišta prema metodi diskretnih elemenata (DEM)[22]

Figure 15. 2D numerical model of the slope: a) numerical model of the slope according to the Discrete Elements Method(DEM), b) identification of the landslide initiation zone according to the Discrete Elements Method (DEM) [22]

Primenom ove metode, može se pratiti inkrementalnirazvoj klizišta, tako da se kao konačna vrednostproračuna dobija spektar faktora sigurnosti. Takođe, ovametoda primenjuje se i za 3D modeliranje složenih formikosina, pri čemu je razvijen niz algoritama za topologiju ikompaktnost elementa kojima se formira 3D modelkosine. Na slici 16 prikazan je postupak formiranja 3Dnumeričkog modela kosine prema metodi diskretnihelemenata (DEM) i odgovarajuće inkrementalneproračunske faze.

Da bi se ovakav algoritam efikasno primenio upraksi, međusobne veze konačnih elemenata modelirajuse kontaktnim elementima s mogućnošću implemen-tacije različitih nelinearnih ponašanja. Kod kontaktnihelemenata, definišu se komponente krutosti pri pritisku,a naponi zatezanja se takođe mogu definisati ili čakeliminisati. Prilikom modeliranja kontakta dveju tačakamodela, javljaju se dva stanja: aktivno (kontakt jeuspostavljen uz učešće određene krutosti) i neaktivno(kontakt nije uspostavljen uz učešće male krutosti ili bezuvođenja efekata krutosti). Da bi se efikasno modeliraliefekti interakcije kontaktnih elemenata, potrebno jeprimeniti geometrijski nelinearnu inkrementalno-itera-tivnu analizu. Usled nelinearnog ponašanja kontaktnogelementa, gde promenu stanja može pratiti velikapromena krutosti, mogu se javiti ozbiljne teškoće uobezbeđenju konvergencije nelinearnog rešenja. U tomsmislu, može biti povoljnije koristiti proceduru kontroleinkrementalnog priraštaja pomeranja, nego proceduru

Through application of this method, the incrementaldevelopment of the landslide can be traced, so that thespectrum of the safety factors is obtained as the finalvalue of the calculation. Moreover, this method is alsoapplied for 3D modelling of complex slope shapes,where a series of algorithms is developed for thetopology and compactness of the elements which formthe 3D slope model. Figure 16 shows the process offorming the 3D numerical slope model according to theDiscrete Elements Method (DEM) and the correspondingincremental calculation phases.

For this algorithm to be effectively applied in practice,the connections between the finite elements aremodelled by the contact elements with the possibility ofimplementing different nonlinear behaviours. The contactelements define the stiffness components under thepressure, and the tensile stresses can also be defined oreven eliminated. When modelling the contact betweentwo points of the model, two states occur: active (thecontact is established with the involvement of certainstiffness) and inactive (the contact is not established withthe involvement of little stiffness or without theintroduction of stiffness effects). In order to efficientlymodel the effects of contact elements interaction, it isnecessary to apply the geometric nonlinear incremental-iterative analysis. Due to the non-linear behaviour of thecontact element, where the change of the state can befollowed by a major change in stiffness, seriousdifficulties can arise in ensuring the convergence of the


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kontrole inkrementalnog priraštaja silama. nonlinear solution. In that sense, it may be morebeneficial to use the procedure for controlling theincremental increase of displacements, rather than theprocedure for controlling the incremental increase offorces.

а) b)

Slika 16. 3D numerički model kosine: a) postupak formiranja 3D numeričkog modela kosine prema metodi diskretnihelemenata (DEM); b) inkrementalne proračunske faze [3]

Figure 16. 3D numerical slope model: a) the procedure of formation of the 3D numerical slope model according to theDiscrete Elements Method (DEM), b) incremental calculation phases [3]

Uvođenje mehanike kontakta u analizu razvojavelikih plastičnih deformacija i kretanja mase tla klizištasprovodi se i kod proširene metode konačnih elemenata(XFEM), slično kao i kod metode diskretnih elemenata(DEM). U samoj formulaciji problema smatra se da – priinkrementalnim proračunskim fazama – nastupa takvapromena geometrije zone kontakta, da inicijalnojgenerisanoj mreži konačnih elemenata odgovara konfi-guracija mreže konačnih elemenata za bilo kojuinkrementalnu situaciju. Ovim se eliminiše upotrebadodatnih algoritama za pretraživanje povoljne konfigura-cije u povezivanju čvorova mreže u i-toj inkrementalnojanalizi ili čak primena adaptivne metode za korekcijumreže konačnih elemenata sistema [34].

Numeričke inkrementalno-iterativne (nelinearne)analize stabilnosti klizišta zasnivaju se na formulacijinelinearnog problema sistemom nelinearnih algebarskihjednačina oblika [2], [5]:

The introduction of the contact mechanics in theanalysis of the development of large plastic deformationsand the displacement of the landslide soil mass is alsocarried out with the eXtended Finite Element Method(XFEM), similar to the Discrete Element Method (DEM).In the formulation of the problem itself, it is consideredthat during incremental calculation phases occurs such achange in the geometry of the contact zone, that theinitial generated mesh of finite elements iscorresponding to the configuration of the mesh of finiteelements for any incremental situation. This eliminatesthe use of additional algorithms for search for afavourable configuration in connecting the mesh nodesin i-th incremental analysis, or even the use of anadaptive method for correcting the mesh of finiteelements of the system [34].

Numerical incremental-iterative (nonlinear) landslidestability analyses are based on the formulation of a non-linear problem through a system of non-linear algebraicequations of the form [2], [5]:

0 FuK , (42)

odnosno: i.e.:

0 FP , (43)

gde su {u} nepoznati parametri pomeranja, {F}generalisani spoljašnji uticaji (opterećenja) u čvorovimasistema. Jednačine problema (42) umesto za ukupnoopterećenje, rešavaju se za niz posebnih inkrementalnihopterećenja. U okviru svakog inkrementa, pretpostavljase da je sistem jednačina linearan. Na taj način, rešenje

where {u} is the unknown displacement parameters, {F}generalized external effects (loads) in the system nodes.The equations of the problem (42) instead of for the totalload, are solved for a series of specific incrementalloads. Within each increment, it is assumed that theequation system is linear. In that way, the solution of a


151

nelinearnog problema dobija se kao zbir niza linearnih(inkrementalnih) rešenja. Nelinearan problem može dase prikaže izrazom:

nonlinear problem is obtained as the sum of a series oflinear (incremental) solutions. A non-linear problem canbe represented by the expression:

0 FλuΔKt , (44)

odnosno: i.e.: 0 FλP , (45)

gde je {P} vektor unutrašnjih generalisanih sila modelakoje su funkcija vektora generalisanih pomeranja {u}, λparametar inkrementalnog opterećenja (odnosinkrementalnog i kompletnog opterećenja). U skladu skonceptom inkrementalnog rešenja jeste:

where {P} is the vector of the internal generalized modelforces, which are the function of the generalizeddisplacement vector {u}, {λ} the incremental loadingparameter (the ratio of incremental and total load). Inaccordance with the concept of incremental solution, wehave:

FλΔFFFΔλλλΔ

uuuΔFΔKFλΔKuΔ

iiii

iii

iii

iititi

1

1

1

11

. (46)

Iz izraza (46) određuju se inkrementi vektorapomeranja za inkremente opterećenja i tangentnumatricu krutosti modela klizišta, koja se formuliše zareferentno stanje na početku inkrementa. Referentnomstanju na početku prvog inkrementa odgovara linearnamatrica krutosti klizišta (inicijalna matrica krutosti). Opštii-ti korak inkrementalnog postupka obuhvata: formiranjetangentne matrice krutosti [Kt]i numeričkog modelaklizišta, određivanje inkremenata vektora opterećenja{ΔF}i numeričkog modela, određivanje inkremenatavektora generalisanih pomeranja {Δu}i rešavanjem siste-ma linearnih algebarskih jednačina za tangentnu matricukrutosti, određivanje inkremenata uticaja u konačnimelementima (deformacije, naponi), i određivanje ukupnevrednosti generalisanih pomeranja inkrementalnim(kumulativnim) sabiranjem. Pomeranja posle m-toginkrementa određena su izrazom:

From the expression (46), the increments of thedisplacement vector for loading increments of the loadand the tangent stiffness matrix of the landslide modelstiffness are determined, which is formulated for thereference state at the beginning of the increment. Thereference state at the beginning of the first incrementcorresponds to the linear matrix of the landslide stiffness(initial stiffness matrix). The general i-th step of theincremental procedure includes: the formation of atangent stiffness matrix [Kt]i of the numerical landslidemodel, determining the load vector increment {ΔF}i of thenumerical model, determining the vector of generalizeddisplacements increments {Δu}i by solving the system oflinear algebraic equations for the tangent stiffnessmatrix, determining the increments of the impact in thefinite elements (deformations, tensions), and determiningthe total value of generalized displacements byincremental (cumulative) addition. Displacements afterthe m-th increment are defined by the expression:

m

iim uΔuu

10 . (47)

Razlog za pojavu greške inkrementalnog rešenjajeste sprovedena linearizacija u okviru inkrementa.Veličina greške može da se odredi iz uslova ravnotežena kraju inkrementa. Kao posledica linearizacije, javljajuse neuravnotežena (rezidualna) opterećenja koja sumera odstupanja inkrementalnog rešenja od tačnog.Vektor rezidualnog opterećenja može se prikazati kaoodstupanje od ravnoteže:

The reason behind the occurrence of the incrementalsolution error is the linearization conducted within theframework of the increment. The error dimensions canbe determined from the balance conditions at the end ofthe increment. As the linearization consequence,unbalanced (residual) loads occur, that are the measureof deviation of the incremental solution from the exactone. The residual load vector can be represented as adeviation from balance:

iitii uΔKFΔRΔ 1 . (48)

Korekcija greške postiže se dodavanjem rezidualnogopterećenja na spoljašnje opterećenje u sledećeminkrementu:

Error correction is achieved by adding the residualload to the external load in the following increment:

iiRi RΔFΔFΔ 11 . (49)


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Najbolji rezultati postižu se ako se kombinujeinkrementalni i iterativni postupak. U prvoj iteraciji,pojavljuju se rezidualna opterećenja zbogneispunjavanja uslova ravnoteže. Ako se naredneiteracije realizuju samo s rezidualnim opterećenjima, uzkorekciju tangentne matrice krutosti, postupak može dakonvergira uz minimiziranje rezidualnog opterećenja. Priformulisanju iterativne metode, polazi se od izraza zarazvoj u Taylor-ov red vektora rezidualnih sila u okolinipomeranja {u}j:

The best results are achieved if the incremental anditerative processes are combined. In the first iteration,residual loads appear due to unfulfilled balanceconditions. If the following iterations are realized onlywith residual loads, with the correction of the tangentstiffness matrix, the process can converge, with theminimization of the residual load. When formulating theiterative method, it is started with the expression fordevelopment in the Taylor series of the residual forcesvector in the vicinity of the displacement {u}j:

j

j

jjj uΔ

udRd

RR 1 . (50)

Iz uslova da rezidualno opterećenje ispunjava usloveravnoteže {R}j+1=0, važi:

From the condition that the residual load meets thebalance conditions {R}j+1=0, follows:

jtj RKuΔ 1 . (51)

Poslednja dva izraza predstavljaju osnovu iterativnemetode. Kombinacijom inkrementalne i iterativne metodedobija se Newton-Raphson-ova inkrementalno-iterativnametoda (slika 17).

The last two expressions represent the basis of theiterative method. By combining the incremental anditerative methods, Newton-Raphson's incremental-iterative method is obtained (Figure 17).

Slika 17.Newton-Raphson-ova inkrementalno-iterativna metoda [2], [5]Figure 17. Newton-Raphson's incremental-iterative method [2], [5]

Numeričke inkrementalno-iterativne (nelinearne)analize stabilnosti klizišta – u kojima se primenjujenumerička integracija u vremenskom domenu –zasnivaju se na formulaciji nelinearnog problema krozdiferencijalne jednačine kretanja sistema s više stepenislobode u matričnom obliku:

Numerical incremental-iterative (non-linear) landslidestability analyses, in which numerical integration in thetime domain is applied, are based on the formulation of anonlinear problem through the differential equations ofthe motion of the system with several degrees offreedom in the matrix form:

QdKvCaM . (52)

S obzirom na to što se uzimaju u obzir potpuni razvoji geometrijske i materijalne nelinearnosti, ovakva metodau literaturi zove se i potpuna nelinearna dinamičkaanaliza (NDA – Nonlinear Dynamic Analysis). Rešavanjejednačina (52) sprovodi se numeričkom integracijomkorak po korak (step by step) Hilber-Hughes-Taylor-ovim(HHT) postupkom u modifikovanom obliku [13]:

Since the full development and geometric andmaterial non-linearities are taken into account, thismethod is also referred to in the literature as thecomplete Nonlinear Dynamic Analysis (NDA). Solvingthe equations (52) is carried out through step-by-stepnumerical integration by Hilber-Hughes-Taylor (HHT)method in a modified form [13]:

αiiiiii QdKαdKαvCαvCαaM 111 11 , (53)

a za trenutak vremena: and for the moment of time:

tΔtt ii 1 , (54)


153

gde je [M] matrica masa, {a} vektor ubrzanja, [C] matricaprigušenja, {v} vektor brzine, [K] matrica krutosti, {d}vektor pomeranja, {Q} vektor spoljašnjih generalisanihsila. Vektori pomeranja i brzine izražavaju se prema:

where [M] is the mass matrix, {a} acceleration vector, [C]damping matrix, {v} velocity vector, [K] stiffness matrix,{d} displacement vector, {Q} vector of externalgeneralized forces. The displacement and velocityvectors are expressed according to:

1

2

1 2212 iiiii aβaβtΔvtΔdd , (55)

11 1 iiii aγaγtΔvv , (56)

dok za vektor spoljašnjih generalisanih sila važi: while to the vector of external generalized forcesapplies:

αiαi tQQ , (57)

gde je: where:

tΔαttαtαt iiiαi 111 . (58)

HHT postupak postaje bezuslovno stabilan ukolikosu parametri α, β i γ izabrani u skladu s relacijama:

The HHT method becomes unconditionally stable ifthe parameters α, β and γ are selected in accordancewith the relations:

0

31 ,α , 21

41 αβ , αγ

21

. (59)

Vektori brzine {v}i+1 i ubrzanja {a}i+1 u trenutku ti+1izražavaju se preko vektora pomeranja na kraju intervala{d}i+1:

The velocity vector {v}i+1 and the acceleration vector{a}i+1 at the moment ti+1 are expressed by thedisplacement vector at the end of the interval {d}i+1:

iiiii aβγtΔv

βγdd

tΔβγv

1

2111 , (60)

iiiii aβ

vtΔβ

ddtΔβγa

1

211

121 . (61)

Unošenjem ovih izraza u jednačinu (53), dobija seekvivalentna jednačina ravnoteže:

Including these expressions into the equation (53)gives the equivalent equation of equilibrium:

αii QdK 1 , (62)

gde je: where:

CtΔβ

γαMtΔβ

KαK 111 2 , (63)

iiiαiαi a

βv

tΔβd

tΔβMQQ 1

2111

2

iiii dKαaβγαtΔv

βγαd

tΔβγαC

1

21111 . (64)

Ukoliko se vrednosti parametara α, β i γ usvoje dasu:

If the following values are accepted for parameters α,β and γ:

31

α ,94

β ,65

γ , (65)

tada su efektivna matrica krutosti i vektor efektivnogopterećenja:

then the effective stiffness matrix and the effectiveload vector are:


154

CtΔ

MtΔ

KK4

54

932

2 , (66)

iiiαiαi av

tΔd

tΔMQQ

81

49

49

2

iiii dKatΔvdtΔ

C31

241

41

45

, (67)

gde je: where:

tΔttΔtt iiαi 32

31

1 , (68)

odnosno: i.e.:

tΔtQQ iαi 3

2. (69)

Sa određenim pomeranjima na kraju posmatranogintervala, rešavanjem jednačina (62), brzine i ubrzanjana kraju intervala dobijaju se prema izrazima:

With certain shifts at the end of the observed timeinterval by solving the equations (62), velocity andacceleration at the end of the time interval are obtainedaccording to the following expressions:

iiiii atΔvddtΔ

v161

87

815

11 , (70)

iiiii avtΔ

ddtΔ

a81

49

49

121 . (71)

Pre započinjanja algoritma korak po korak, potrebnoje da se početno ubrzanje sistema odredi izdiferencijalne jednačine kretanja prema:

Before starting the step-by-step algorithm, it isnecessary that the initial acceleration of the system isdetermined from the differential equation of motionaccording to:

0001

0 dKvCQMa . (72)

Korekcija matrice krutosti sistema sprovodi se poslesvakog apliciranog koraka vremena, a prema prethodnoprezentovanoj Newton-Raphson-ovoj metodi. PrimenomNDA analize sa HHT postupkom i NR metodom zaproračun 2D i 3D modela klizišta, dobijaju se najpouzda-nija rešenja za procenu nelinearnog odgovora sistema.Primenom ovakve metode, moguće je razmatrati uticajdinamičnosti povećanja nivoa podzemne i površinskevode, a takođe i dejstvo zemljotresa inkrementalnoskalirajući akcelerogram. Odgovor sistema (klizišta)predstavlja se kao funkcija promene faktora sigurnosti Fsu vremenu, a ne samo kao jedinstvena (diskretna)vrednost.

3.4 Kompleksno 3D geometrijsko modeliranje i numeričke metode proračuna stabilnostiklizišta

Standardni pristup u modeliranju terena i klizišta –inkorporiranog u terenu – zasniva se na korišćenjutehnike 2D prezentacije primenom situacionog plana ivertikalnih poprečnih preseka. Na osnovu definisanihtipova slojeva tla po dubini i njihovih fizičko-mehaničkih

The correction of the system stiffness matrix iscarried out after each applied time step, and accordingto the previously presented Newton-Raphson's method.Using the NDA analysis with the HHT method and theNR method for calculating the 2D and 3D landslidemodels, the most reliable solutions for estimating thenonlinear system response are obtained. Using thismethod allows us to consider the influence of the level ofunderground and surface water increase dynamics, aswell as the effect of the earthquake, incrementallyscaling the accelerometer. System (landslide) responseis represented as the function of change of the safetyfactor Fs in time, and not only as a unique (discrete)value.

3.4 Complex 3D geometric modelling andnumerical methods for landslides stabilitycalculations

The standard approach to modelling of the terrainand landslide, incorporated in the terrain, is based on theusage of the 2D presentation technique by applying asituational plan and vertical cross sections. Based on thedefined types of soil layers according to depth and their


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karakteristika, sprovodi se analitički i/ili numeričkiproračun stabilnosti kosina. U slučaju prostorno slože-nijeg modela terena i kompleksnije geometrije klizišta,pitanje 2D modeliranja i pouzdanosti odgovarajućihanaliza može biti diskutabilno. Međutim, i u situacijamakada se pouzdano može tvrditi da je tehnika 2Dprezentacije terena i klizišta, primenom situacionogplana i vertikalnih poprečnih preseka pouzdana, ostajuotvorena neka pitanja – da li se može dodatno poboljšatiprezentacija terena i klizišta u skladu sa savremeniminformacionim tehnologijama i da li se može pouzdanoodrediti zapremina tla koja formira klizište. Odgovori naova pitanja mogu se pronaći u 3D vizuelizaciji terena iklizišta, pri čemu se kao najsofisticiranije rešenje,primenom 4D vizuelizacije (3D + dinamičke simulacije)može predstaviti problem sanacije klizišta, od inicijalnogstanja, preko faznih rešenja, pa sve do finalnog rešenja.3D modeliranje terena i klizišta koristi se za geometrijskuprezentaciju i numeričku analizu primenom površi isolida. Geometrijska 3D prezentacija, u najvećem brojuslučajeva, ima veći stepen vizuelizacije konačnogrešenja, dok je cilj numeričke 3D analize da se prime-nom površi i solida modelira teren i klizište, tako da svakigeometrijsko-numerički element ima u sebi integrisanu imatematičku formulaciju problema. Podrazumeva se dase i prilikom numeričkog modeliranja terena i klizištamože dodatno postići realističan efekat geometrijskeprezentacije, međutim u ovakvim situacijama dodatno sepovećava vreme proračuna, tako da se – u veomasloženim modelima i s veoma velikim brojem konačnihelemenata – proračun svodi na primenu tehnike paralel-nog procesiranja. Međutim, geometrijsko 3D modeliranjeza prezentaciju terena i klizišta dosta je korisnije zaproračune zapremine tla, s obzirom na to što semodeliranjem klizišta kao solida može veoma brzoodrediti odgovarajuća zapremina, čak i u situacijamaveoma složenih solid modela. Postupak kompleksnog3D modeliranja terena i klizišta zasniva se na prethodnojidentifikaciji većeg broja kliznih ravni za odgovarajućibroj inženjersko-geoloških profila, njihovom integracijomsa 2D situacionim planom klizišta i konstrukcijom 3Dmodela klizišta u softveru za geometrijsku prezentaciju(CAD). Za integrisane klizne ravni formira se kliznapovrš u prostoru, dok se za modelirano klizište uprostoru formira solid model klizišta. Modeliranje kliznepovrši u prostoru zasniva se na primeni kompleksnezakrivljene površi koja formira mrežu četvorouglova, dokse solid model klizišta generiše primenom primitiva itehnika za editovanje primitiva: ekstrudiranje, sečenje,proširenje, ujedinjenje, ekstrakcija, intersekcija i slično.Na slici 18 prikazani su generisani kompleksni geome-trijsko-numerički 3D modeli terena za analizu stabilnostiklizišta.

Generalno razmatrajući modeliranje površi u prostorumože se sprovesti primenom matematičkih funkcija,mapiranja i diskretnih vrednosti. Najviše se koristi tehni-ka mapiranja terena s rasterskom mrežom (ortogonalna,poluortogonalna, radijalna i zakrivljena) za formiranjemape terena, ali je primena diskretnih vrednosti iformiranje polilinija, površi i solida u prednosti, pa se zaovakvu grafiku koristi termin vektorska grafika. Izohipseterena i klizne površi, u opštem slučaju, predstavljaju seprimenom polilinija i splajnova. Da bi se geometrijski imatematički modelirao skup tačaka koji formira jednukliznu površ u 2D koordinatnom sistemu, potrebno je

physico-mechanical characteristics, an analytical and/ornumerical calculation of the slope stability is carried out.In the case of a spatially slightly complex terrain modeland slightly complex landslide geometry, the question of2D modelling and the reliability of the correspondinganalyses can be debatable. However, even in situationswhere it can be reliably asserted that the 2Dpresentation of the terrain and landslide by using thesituational plan and vertical cross-sections is reliable, thefollowing questions remain open: can the presentation ofthe terrain and the landslide be further improved inaccordance with modern information technology andwhether the volume of the soil forming the landslide canbe reliably determined? The solution to these issues canbe found in 3D visualization of terrain and landslide,whereby the most sophisticated solution, by using 4Dvisualization (3D + dynamic simulation), can present theproblem of landslide sanation, from the initial state,through phase solutions to the final solution. 3Dmodelling of the terrain and landslide is used forgeometric presentation and numerical analysis throughusing surfaces and solids. Geometric 3D presentation, inmost cases, has a greater degree of visualization of thefinal solution, while numerical 3D analysis aims to usethe surfaces and solids to model the terrain andlandslide, so that each geometric-numerical elementalso has in itself an integrated mathematical formulationof the problem. It is presumed that the realistic effect ofthe geometric presentation can be additionally achievedin numerical modelling of the terrain and landslide,however, in these situations the time of the calculation isfurther increased, so that, in very complex models andwith a very large number of finite elements, thecalculation is reduced to the application of the parallelprocessing technique. However, geometric 3D modellingfor the presentation of terrain and landslides is muchmore useful for soil volume calculations, since modellingthe landslide as a solid can quickly determine theappropriate volume, even in situations of very complexsolid models. The process of complex 3D modelling theterrain and landslide is based on: the previous identifi-cation of a larger number of sliding planes for the cor-responding number of engineering-geological profiles,integration of these with the 2D situational plan of thelandslide and the construction of the landslide 3D modelin the geometric presentation software (CAD). For theintegrated sliding planes, a sliding surface is formed inspace, while for the modelled landslide in space a solidlandslide model is formed. The modelling of the slidingsurfaces in space is based on the application of complexcurved surface that forms a grid of quadrangles, whilethe solid model of the landslide is generated usingprimitives and techniques for editing primitives: extru-sion, cutting, expanding, unifying, extraction, intersec-tion, and the like. Figure 18 shows the generatedcomplex geometric-numerical 3D terrain models forlandslide stability analysis.

In general, modelling the surface in space can beconducted by using mathematical functions, mapping,and discrete values. The technique most widely used isterrain mapping with a raster mesh (orthogonal, semi-orthogonal, radial and curved) to form a map of theterrain, but the application of discrete values and theformation of polylines, surfaces and solids has morebenefits, so the term used for such a graphic is vector


156

sprovesti interpolaciju. Jednostavniji modeli interpolacijazasnivaju se na primeni matematičkih funkcija uzatvorenom obliku. Međutim, interpolacija većeg brojatačaka – koje formiraju jednu kliznu površ u 2D koordi-natnom sistemu – zasniva se na primeni parametarskihfunkcija, gde rešenje nije definisano u zatvorenomobliku, već u skupu funkcija. Povezanost ovih funkcijauspostavlja se uslovima ekvivalencije tangente za krive sleve i desne strane u svakoj tački interpolacije. Na tajnačin, dobija se glatka interpolirana kriva, pa se međunajboljim parametarskim funkcijama pokazala primenasplajna.

U slučaju 3D modela terena i klizne ravni, tačnijeklizne površi, oni se u prostoru modeliraju primenomNURBS krivih (non-uniform rational basis spline).NURBS krive definisane su kontrolnim čvorovima ivektorom čvora. U opštem slučaju, NURBS krive i iodgovarajuće površi jesu generalizacija B-splajnova iBezier-ovih krivih i površi. Kontrolni čvorovi definišu oblikpovrši, u konkretnom slučaju klizne površi, dok vektorčvora određuje gde i kako površ dodiruje kontrolnečvorove. Međutim, i prilikom primene NURBS površimože se pojaviti problem u interpolaciji, ukoliko se zaodređene kontrolne čvorove – koji su diskretne vrednostiskupa kliznih površi – adekvatno ne izaberu parametriinterpolacije. Mogu se dobiti isuviše velika odstupanja uinterpolaciji, tako da 3D model terena i klizišta može bitiaproksimiran slično kao što se primenjuje princip uregresionim analizama, bilo da su one linearnog ilinelinearnog tipa. Minimiziranje prethodnog problemapostiže se progušćenjem mreže konačnih elemenata,uvođenjem novih međuelemenata. U opštem slučajunajpouzdanija, ali i isto tako i vizuelno grublja rešenjapostižu se primenom četvorouglova čiji čvorovi direktnopovezuju diskretne čvorove (linearna interpolacija)terena i klizišta. Rafiniranost mreže postiže seinterpolacijom trouglovima. Kao što je već prethodnonapisano, prezentacija terena sprovodi se, zapravo,primenom žičanog (wireframe) modela površi sadodavanjem 3D površi, dok se modeliranje klizištasprovodi primenom solida (3D geometrijsko telo).Diferencijacija klizišta u odnosu na ostale delove terenamože se sprovesti izdvajanjem i prikazom samo klizišta,nezavisno od terena, s mogućnošću 4D kontinualnetranslacije i rotacije u prostoru, i renderovanjem, tako dase terenu poveća transparentnost, u odnosu na klizište.

graphics. The terrain isohypse and sliding surfaces, ingeneral, are represented using polylines and splines. Inorder to geometrically and mathematically model the setof points that forms a single sliding surface in the 2Dcoordinate system, interpolation is required. Thosesimpler interpolation models are based on the applica-tion of mathematical functions in closed form. However,interpolation of a large number of points, that form asingle sliding surface in 2D coordinate system, is basedon the application of parametric functions, where thesolution is not defined in a closed form, but in a set offunctions. The connection of these functions isestablished by the conditions of the tangent equivalencefor curves on the left and right at each point ofinterpolation. This way, a smooth interpolated curve isobtained, so the application of the spline has turned outto be among the best parametric functions.

In the case of 3D terrain model and sliding plane,more precisely the sliding surface, they are modelled inthe space using NURBS curves (Non-Uniform RationalBasis Spline). NURBS curves are defined by the controlnodes and the node vector. In general, NURBS curvesand the corresponding surfaces are the generalization ofB-splines and Bezier's curves and surfaces. The controlnodes define the shape of the surface, in particular, thesliding surface, while the node vector determines whereand how the surface touches the control nodes.However, even with the application of NURBS surfaces,a problem may arise in interpolation, if the adequateinterpolation parameters are not selected for certaincontrol nodes, and which are discrete values of a set ofsliding planes. Excessive interpolation deviations canoccur so that the 3D terrain and landslide model can beapproximated in a similar manner as the principle inregression analysis applies, whether they are linear ornonlinear. Minimizing the previous problem is achievedby increase in the density of the mesh of finite elementsthrough the introduction of new inter elements. Ingeneral, the most reliable, but also visually roughersolutions are achieved by applying quadrangles whosenodes directly connect discrete nodes (linearinterpolation) of the terrain and landslide. The meshrefinement is achieved by interpolation by triangles. Aspreviously mentioned, the presentation of the terrain iscarried out, in fact, by using a wireframe plane modelwith the addition of 3D planes, while the landslidemodelling is carried out by using a solid (3D geometricbody). The differentiation of the landslide in relation toother parts of the terrain can be carried out by allocationand display of landslide only, irrespective of the terrain,with the possibility of 4D continuous translation androtation in space, and rendering, so that the terraintransparency is increased in relation to the landslide.


157

Slika 18. Generisani kompleksni geometrijsko-numerički 3D modeli terena za analizu stabilnosti klizišta [17]Figure 18. Generated complex geometric-numerical 3D terrain model for landslide stability analysis [17]


158

4 ZAVRŠNE NAPOMENE

Primenom sprovedene sistematizacije analitičkih inumeričkih metoda proračuna stabilnosti klizišta, možese efikasno razmotriti koji tip metode se može primeniti ufazama preliminarnih i finalnih analiza za naučnaistraživanja i stručne projekte. Autori su napravilisopstvenu sistematizaciju metoda proračuna stabilnostiklizišta, s tim što pojedine metode mogu pripadati iprelaznim kategorijama. Posebno je to slučaj kod onihmetoda koje se zasnivaju na direktnoj analizi stabilnostiza odgovarajuću kliznu površ i kod metoda koje koristeiteracije kliznih površi primenom optimizacionihalgoritama.

Ključni problemi u modeliranju i numeričkoj analiziklizišta današnjice mogli bi se prikazati iz nekolikoaspekata:

generalizacija nedovoljnog broja uzorkovanja idobijanja odgovarajućih kvalitetnih laboratorijskihispitivanja fizičko-mehaničkih karakteristika tla ikonstitutivnih modela ponašanja tla za kompletnoklizište;

primena geometrijsko-numeričke prezentacijeklizišta putem 3D modela (u određenim situacijama,mogu se dobiti i viši faktori sigurnosti usled zaklinjavanjaklizišta pri klizanju tla);

potreba da se dodatno unapredi metodologijaverifikacije stabilnosti klizišta na osnovu matematičkihmodela i analiza inkrementalnog pomeranja klizišta,monitoringom deformacija, a ne sila i momenata;

implementiranje tehnike paralelnog procesiranja upraktične svrhe (povećanje hardverskih kapaciteta –višejezgarnim procesiranjem i resursa – skladištenjemmemorije).

4 FINAL REMARKS

By applying the conducted systematization ofanalytical and numerical methods of landslide stabilitycalculation it can effectively be considered which type ofmethod can be applied in the phases of preliminary andfinal analyzes for scientific research and professionalprojects. The authors have made their ownsystematization of the methods of landslide stabilitycalculation, but some methods can also belong totransition categories. This is especially the case withthose methods that are based on a direct stabilityanalysis for the corresponding sliding surface and formethods using sliding surface iterations by applyingoptimization algorithms.

Key problems in modelling and numerical analysis ofnowadays landslides could be presented through severalaspects:

generalization of insufficient number of samplingand obtaining appropriate quality laboratory tests ofphysical-mechanical characteristics of soil andconstitutive models of soil behaviour for a completelandslide,

the application of the geometric-numericalpresentation of the landslide through 3D models (incertain situations, higher safety factors can be obtaineddue to the wedging of the landslide during the soilsliding),

it is necessary to further improve the methodologyof landslide stability verification based on mathematicalmodels and analysis of incremental displacement of thelandslide, by monitoring the deformations, but not theforces and moments,

implementing parallel processing techniques forpractical purposes (increasing: hardware capacitiesthrough multi-core processing and resources throughstorage of the memory).

5 LITERATURAREFERENCE

[1] Albataineh N.: Slope Stability Analysis Using 2Dand 3D Methods, University of Akron, 2016.

[2] Bathe K.: Finite Element Procedures, Prentice Hall,1037p, Upper Saddle River, USA, 1996.

[3] Bonilla Sierra V.: De la Photogrammetrie a laModelisation 3D: Evaluation Quantitative du Risqued’Eboulement Rocheux, Universite Grenoble Alpes,Docteur de l’Universite Grenoble Alpes, 2006.

[4] Chen X., Wub Y., Yu Y., Liu J., Frank X, Ren J.: ATwo-Grid Search Scheme for Large-Scale 3-DFinite Element Analyses of Slope Stability,Computers and Geotechnics, Vol. 62, 2014, pp.203-2015.

[5] Crisfield M.: Non-Linear Finite Element Analysis ofSolids and Structures, Volume 2: Essentials, JohnWiley & Sons, 345p, New York, USA, 2000.

[6] Dai F., Lee C., Ngai Y.: Landslide Risk Assessmentand Management: An Overview, EngineeringGeology, Vol. 64, No. 1, 2002, pp. 65 87.

[7] EN 1997-1:2004, Eurocode 7: Geotechnical Design– Part 1: General Rules, Brussels, Belgium, 2004.

[8] Fellipa C.: Advanced Finite Element Methods,University of Colorado, Boulder, 2007.

[9] Fredlund D.: Analytical Methods for Slope StabilityAnalysis, State of the Art, The 4th InternationalSymposium on Landslides, Toronto, Canada, 1984,pp. 229-250.

[10] GEO 5, User's Guide, Fine Ltd., 2016.[11] Geološka terminologija i nomenklatura VIII-2,

Inženjerska geologija, Zavod za regionalnu geolo-giju i paleontologiju Rudarsko-geološkog fakulteta,Univerzitet u Beogradu, Beograd, Srbija, 1978.

[12] Gustafsson J., Lindstrom M.: Applicability ofOptimised Slip Surfaces: Evaluation of a Software'sOptimisation Function for Generating CompositeSlip Surfaces, Applied on Stability Analysis of ClaySlopes, Chalmers University of Technology,Gothenburg, Sweden, 2014.

[13] Hilber H., Hughes T., Taylor R.: ImprovedNumerical Dissipation for Time IntegrationAlgorithms in Structural Dynamics, EarthquakeEngineering and Structural Dynamics, Vol. 5, No.3, pp. 283-292, 1977.


159

[14] Ho I-H.: Parametric Studies of Slope StabilityAnalyses Using Three-Dimensional Finite ElementTechnique: Geometric Effect, Journal ofGeoengineering, Vol. 9, No. 1, 2014, pp. 33-43.

[15] http://geoliss.mre.gov.rs/beware/form/guest_page.php

[16] http://landslides.usgs.gov/learn/majorls.php[17] https://www.soilvision.com/products/svoffice5/svslo

pe[18] https://www.studyblue.com/notes/note/n/earth-

science-final-exam/deck/4839996[19] Janbu N.: Slope Stability Computations in

Embankment Dam Engineering, R. Hirschfeld andS. Poulos, eds., John Wiley and Sons, New York,USA, 1973, pp. 47-86.

[20] Kainthola A., Verma D., Thareja R., Singh T.: AReview on Numerical Slope Stability Analysis,International Journal of Science, Engineering andTechnology Research (IJSETR), Vol. 2, No. 6,2013, pp. 1315-1320.

[21] Kaur A., Sharma R.: Slope Stability AnalysisTechniques: A Review, International Journal ofEngineering Applied Sciences and Technology,Vol. 1, No. 4, 2016, pp. 52-57.

[22] Kong Y., Chen P., Yu H.: Analysis of Rock High-Slope Stability Based on a Particle Flow CodeStrength Reduction Method, Electronic Journal ofGeotechnical Engineering, Vol. 20, 2015, pp.13421-13430.

[23] Leong E., Rahardjo H.: Two and Three-Dimensional Slope Stability Reanalyses of BukitBatok Slope, Computers and Geotechnics, Vol. 42,pp. 81-88, 2012.

[24] Maksimović M.: Mehanika tla, Čigoja štampa,Beograd, Srbija, 2001.

[25] Memić M., Folć R., Ibrahimović A.: NumericalModeling and Slope Reparation Methods in anAltered and Unstable Serpentine Rock Mass,Building Materials and Structures, Vol. 55, No. 4,2012, pp. 23-45.

[26] Morgenstern N., Price V.: The Analysis of theStability of General Slip Surfaces. Géotechnique,Vol. 15, No. 1, 1965, pp.79-93.

[27] Pereira T, Robaina A., Peiter M., Braga F., RossoR.: Performance of Analysis Methods of SlopeStability for Different Geotechnical Classes Soil onEarth Dams, Journal of the Brazilian Association ofAgricultural Engineering, Vol. 36, No. 6, 2016,pp.1027-1036.

[28] Sarma S.: Stability Analysis of Embankments andSlopes, Géotechnique, Vol. 23, No. 3, 1973, pp.423-433.

[29] Schuster R.: The 25 Мost Catastrophic Landslidesof the 20th Century, in Chacon, Irigaray andFernandez (eds.), Landslides, Proc. Of the 8thInternational Conf. & Field Trip on Landslides,Granada, Spain, Rotterdam: Balkema, 1996.

[30] Spencer E.: A Method of Analysis of the Stability ofEmbankments Assuming Parallel Inter-SliceForces, Géotechnique, Vol. 17, No. 1, 1967, pp.11-26.

[31] Шахунянц Г.:Железнодорожный путь: учеб. длявузов ж.-д. трансп. /– 3-е изд., перераб. и доп. –М. : Транспорт, 1987.

[32] Tschuchnigg F., Schweiger H., Sloan S.: SlopeStability Analysis by Means of Finite Element LimitAnalysis and Finite Element Strength ReductionTechniques, Part II: Back Analyses of a CaseHistory, Computers and Geotechnics, Vol. 70 ,2015, pp. 178 189.

[33] Usluogullari O., Temugan A., Duman E.:Comparison of Slope Stabilization Methods byThreedimensional Finite Element Analysis, NaturalHazards, Vol. 81, No. 2, 2016, pp. 1027-1050.

[34] Wriggers P.: Computational Contact Mechanics,Springer-Verlag, New York, USA, 2006.

[35] Zhang L., Fredlund M., Fredlund D., Lub H., WilsonG.: The Influence of the Unsaturated Soil Zone on2-D and 3-D Slope Stability Analyses, EngineeringGeology, Vol. 193, 2015, pp. 374 383.

[36] Zhu D., Lee C., Qian Q., Chen G.: A ConciseAlgorithm for Computing the Factor of Safety Usingthe Morgenstern-Price Method, CanadianGeotechnical Journal, Vol. 42, No. 1, 2005,272-278.


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REZIME

SISTEMATIZACIJA ANALITIČKIH I NUMERIČKIHMETODA PRORAČUNA STABILNOSTI KLIZIŠTA

Kristina BOŽIĆ-TOMIĆNenad ŠUŠIĆMato ULJAREVIĆ

Na osnovu analize mnogih naučnih radova, autori sudali prikaz sopstvene originalne sistematizacijeanalitičkih i numeričkih metoda proračuna stabilnostiklizišta, pri čemu mnoge od njih tek treba dodatno da seunaprede, implementiraju i testiraju na kompleksnim 3Dmodelima klizišta. Metode proračuna stabilnosti klizištaklasifikovane su u pet grupa: analitičke jednokoračne,analitičke višekoračne (iteracije kliznih površi),numeričke višekoračne (iteracije kliznih površi),numeričke inkrementalno-iterativne (nelinearne) analize inumeričke inkrementalno-iterativne (nelinearne) analize,uz primenu numeričke integracije u vremenskomdomenu. Primenom sprovedene sistematizacije metodaproračuna stabilnosti klizišta, može se vrlo efikasnorazmotriti koji je tip metode optimalan za analizu klizišta ikoji tip metode je potrebno koristiti u fazi preliminarnih ifinalnih analiza za naučna istraživanja i stručne projekte.

Ključne reči: klizište, sistematizacija, analitičkemetode, numeričke metode, 2D i 3D modeliranje

SUMMАRY

THE SYSTEMATIZATION OF ANALYTICAL ANDNUMERICAL METHODS OF LANDSLIDE STABILITYCALCULATION

Kristina BOZIC-TOMICNenad SUSIC Mato ULJAREVIC

According to the analysis of a large number ofscientific papers, the authors of the paper presentedtheir own original systematization of the analytical andnumerical methods of landslide stability calculation, witha large part of them still to be further improved,implemented and tested on complex 3D landslidemodels. Methods for calculating the stability of thelandslide are classified into five groups: analytical single-step, analytical multi-step (iterations of sliding surfaces),numerical multi-step (iterations of sliding surfaces),numerical incremental-iterative (nonlinear) analysis andnumerical incremental-iterative (nonlinear) analysis,applying numerical integration in the time domain. Byusing the systematization method of calculating thestability of the landslide it can be very effective toconsider which type of method is optimal for landslideanalysis and which type of method should be consideredin the phase of preliminary and final analysis forscientific research and expert projects.

Keywords: landslide, systematization, analyticalmethods, numerical methods, 2D and 3D modelling

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