m. samec et al. PROBING ION DYNAMICS IN A CLAY-WATEA SYSTEM WITH DIELECTRIC SPECTROSCOPY h. vrecl-Kojc and s. skrabl DETERMINATION OF PASSIVE EAATH PRESSURE USING THREE-DIMENSIONAL FAILURE MCCHRNISM a. serlanl et al. BEHAVIOUA OF THE SUBWAY TUNNEL IN ALGICAS! PHYSICAL MODEL EXPERIMENTAL STUDY l. vrankar et al. THE USE OF THE MESH FAEE METHODS (AADIflL BASIS FUNCTIONS) IN THE MODELING OF AADIONUCLIDE MIGRATION AND MOVING BOUNDARY VALUE PAOBLEMS nem ggotgchnicfl slovenicfi ISSN: 1854-0171 ustanovLteLjL Founders urednLškL odbor edLtorLaL Board /z Univerza v Mariboru, Fakulteta za gradbeništvo University of Maribor, Faculty of Civil Engineering Univerza v Ljubljani, Fakulteta za gradbeništvo in geodezijo University of Ljubljana, Faculty of Civil and Geodetic Engineering iiSSflKI Univerza v Ljubljani, Naravoslovnotehniška fakulteta University of Ljubljana, Faculty of Natural Sciences and Engineering Slovensko geotehniško društvo Slovenian Geotechnical Society Društvo za podzemne in geotehniške konstrukcije Society for Underground and Geotechnical Constructions ÉIjÉÉÉ ilüiiiíüi LzdajateLj 1pubLLsher Univerza v Mariboru, Fakulteta za gradbeništvo University of Maribor, Faculty of Civil Engineering odgovornL urednLk edLtor-Ln-chLef Ludvik Trauner Univerza v Mariboru urednLka co-edLtors Stanislav Škrabl Univerza v Mariboru Bojan Žlender Univerza v Mariboru TehnLčna urednLka Desk edLtors Bojana Dolinar Univerza v Mariboru Borut Macuh Univerza v Mariboru LektorLča proof-Reader Metka Brkan NakLada cLrčuLatLon 500 izvodov - issues TLsk prLnt Tercia tisk d.o.o. Ptuj Revija redno izhaja dvakrat letno. Članki v reviji so recen-zirani s strani priznanih mednarodnih strokovnjakov. Baze podatkov v katerih je revija indeksirana: ICONDA - The international Construction database, GeoRef Pri financiranju revije sodeluje Javna agencija za raziskovalno dejavnost republike Slovenije. Darinka Battelino Università degli Studi di Trieste József Farkas Budapesti Müszaki és Gazdaságtudományi Egyetem Theodoros Hatzigogos Aristotle University of Thessaloniki Rolf Katzenbach Technische Universität Darmstadt Zlatko Langof Univerzitet u Sarajevu Jakob Likar Univerza v Ljubljani Janko Logar Univerza v Ljubljani Bojan Majes Univerza v Ljubljani Milan Maksimovic Univerzitet u Beogradu Borut Petkovšek Zavod za gradbeništvo Slovenije Mihael Ribičič Univerza v Ljubljani César Sagaseta Universidad de Cantabria Stephan Semprich Technische Universität Graz Abdul-Hamid Soubra Université de Nantes Ivan Vaniček Ceské vysoké učeni technické v Praze Franjo Veric Sveučilište u Zagrebu Address NasLov urednLštva ACTA GEOTECHNICA SLOVENICA Univerza v Mariboru, Fakulteta za gradbeništvo Smetanova ulica 17 2000 Maribor Slovenija Telefon / Telephone: +386 (0)2 22 94 300 Faks / Fax: +386 (0)2 25 24 179 E-pošta / E-mail: ags@uni-mb.s spLetnL nasLov http:IIwww. fg.uni-mb.siIjournal-ags I web Address The journal is published twice a year. Papers are peer reviewed by renowned international experts. Indexation data bases of the journal: ICONDA - The International Construction Database, GeoRef Financially supported also by Slovenian Research Agency. VSEBINA CONTENTS fr Ludvik Trauner UVODNIK Marko samec Ln drugI UGOTRVLJRNJ6 ZNAČILNOSTI DINAMIKE IONOV V MEŠANICI GLINE IN VODE S POMOČJO DIELEKTRIČNE SPEKTROSKOPIJE S Helena VrecL-Kojc Ln Stanislav ŠkrabL DOLOČITEV PASIVNEGA ZEMELJSKEGA PRITISKA Z UPORABO TRIDIMENZIONALNEGA PORUŠNEGA MEHANIZMA [24- AbdelbakL SerLanL Ln drugi OBNAŠANJE AVTOCESTNEGA TUNELA V ALŽIRIJI: EKSPERIMENTALNA ŠTUDIJA FIZIKALNEGA MODELA 53 Leopold Vrankar Ln drugL UPORABA BREZMREŽNIH METOD (radialnih baznih funkcij) za moDELIRANJE MIGRACIJE RADIONUKLIDOV IN PROBLEMOV S PREMIKAJOČO SE MEJO NAVODILA AVTORJEM Ludvik Trauner EDITORIAL A Marko samec et al. PROBING ION DYNAMICS IN A CLAY-WATER SYSTEM WITH DIELECTRIC SPECTROSCOPY E Helena vrecL-Kojc and stanislav ŠkrabL DETERMINATION OF PASSIVE EARTH PRESSURE USING THREE-DIMENSIONAL FAILURE MECHANISM Abdelbaki seriani et al. BEHAVIOUR OF THE SUBWAY TUNNEL IN ALGIERS! PHYSICAL MODEL EXPERIMENTAL STUDY E [Ai Leopold vrankar et al. THE USE OF THE MESH FREE METHODS (radial basis functions) in the MODELING OF RADIONUCLIDE MIGRATION AND MOVING BOUNDARY VALUE PROBLEMS INSTRUCTIONS FOR AUTHORS UVODNIK Mednarodna revija Acta Geotechnica Slovenica postaja vedno bolj prepoznavna, kar kažejo odzivi tako domače kot tuje strokovne javnosti. Zahvala gre predvsem avtorjem, ki s svojimi prispevki seznanjajo bralce z najnovejšimi znanstvenimi in strokovnimi dosežki s širokega področja geotehnike in številnim tujim in domačim recenzentom, ki skrbijo za kvaliteto izbranih člankov. Kljub velikemu trudu vseh sodelujočih pa revija ne bi mogla izhajati brez finančne podpore številnih sponzorjev, ki se jim zato ob tej priliki tudi lepo zahvaljujem. Prva številka četrtega letnika prinaša naslednje štiri zelo zanimive prispevke: M. Samec, D.Korošak in B. Cvikl predstavljajo karakterizacijo mešanic gline in vode na podlagi dielektrične spektroskopije in analize pridobljenih spektrov. Predlagan je teoretičen model dinamike ionov, ki poleg gibanja ionov v porah elektrolita vključuje tudi njihovo ujetost na površinah mineralov. Časovne konstante, ki določajo dinamiko teh pojavov, so povezane s fizikalnimi lastnostmi snovi. Mikroskopsko gibanje ionov v kompleksnem okolju mešanice gline in vode se lahko opiše z dinamiko delcev, ki vodi do subdifuznega obnašanja. H. Vrecl-Kojc in S. Škrabl v prispevku prikazujeta modificiran prostorski porušni mehanizem za določitev 3D koeficienta pasivnega zemeljskega pritiska ob uporabi teorema zgornje vrednosti metode mejne analize. V analizi je upoštevan translatorni kinematično dopustni porušni mehanizem, ki je generaliziran z globino h = 1.0. Mehanizem geometrijsko predstavlja prostorski togi blok sestavljen iz osrednjega dela in dveh stranskih togih blokov, ki ju sestavljajo ovojnice bočnih trenjskih stožcev. Rezultati analize predstavljenega modela so prikazani v grafični obliki v odvisnosti od geometrijskih parametrov in lastnosti temeljnih tal. V prispevku sta tudi predstavljena dva v svetovem merilu priznana prostorska porušna mehanizma, ki prav tako temeljita na metodi mejne analize ter primerjava rezultatov med vsemi tremi porušnimi mehanizmi. A. Seriani, Y. Kismoune Thésard, T. Serradj opisujejo raziskavo, pri kateri so na fizikalnem modelu v razmerju 1:20 eksperimentalno preučili obnašanje avtocestnega tunela v Alžiru ter usipanje okoliških zemljin med izkopavanjem. L. Vrankar, F. Runovc in G. Turk v članku predstavljajo uporabo radialnih baznih funkcij v geostatistični analizi modeliranja migracije radionuklidov. Migracija radionuklidov je simulirana s pomočjo advekcijske-disperzijske enačbe, in sicer v Eulerjevi in Lagrangeovi obliki. V nadaljevanju so predstavljeni tudi Stefanovi problemi oz. problem primikajočih se meja (površin). Položaji primikajoče meje so simulirani s pomočjo metode primikajočih se centrov in nivojne metode. Ludvik Trauner Glavni urednik EDITORIAL The international journal Acta Geotechnica Slovenica has now attained worldwide renown, which has been proved by both domestic and foreign professionals' response to the journal. Thanks are due especially to the authors whose contributions expose their latest scientific and research achievements in the field of geotechnical engineering, and to numerous foreign and home peer-reviewers who assure a quality selection of articles. All efforts of those participating, however, could not have been realized without a solid financial support of numerous sponsors to whom I would also like to extend my warmest thanks. Volume 4, issue 1, brings four very interesting articles: M. Samec, D. Korosak and B. Cvikl present a dielectric spectroscopy characterization of clay-water mixtures and the analysis of the obtained spectra. They propose a theoretical model for ion dynamics in which motion of ions in pore space electrolyte is interrupted by trapping events at the mineral surfaces. Typical time scales for these processes are given in terms of physical properties of the material. It is shown that a microscopic motion of ions in a complex environment of a clay-water system can be described with fractional dynamics leading to sub-diffusive behavior. H. Vrecl-Kojc and S. Skrabl present a modified three-dimensional (3D) failure mechanism for determining the 3D passive earth pressure coefficient using the upper bound theorem within the framework of the limit analysis theory. The translational kinematically admissible failure mechanism generalized with a depth of h = 1.0 is considered in the analysis. The mechanism geometry presents a volume of rigid blocks composed of the central body and two lateral rigid bodies which are connected by a common velocity field. The results are presented in a graphical form depending on the geometrical parameters and soil properties. A brief description of two world-recognized failure mechanisms based on the limit analysis approach, and the comparison of three failure mechanism results are also presented. A. Seriani, Y. Kismoune Thésard and T. Serradj describe the behaviour of a subway tunnel in Algiers and the state of transition of the surrounding ground during digging. These phenomena are studied on an experimental basis using a 1/20 physical model scale. L. Vrankar, F. Runovc and G. Turk have contributed some examples of using RBFs for a geostatistical analysis of modelling radionuclide migration. The migration of radionuclides is simulated using the advection-dispersion equation in Eulerian and Lagrangian forms. Stefan's or moving boundary value problems are also presented. The position of the moving boundary is simulated using the moving data centers method and the level set method. Ludvik Trauner Editor-in-chief UGOTAVLJANJE ZNAČILNOSTI DINAMIKE IONOV V v w w w MEŠANICI GLINE IN VODE S POMOČJO DIELEK-TRIČNE SPEKTROSKOPIJE_ MARKO SAMEC, DEAN KOROŠAK in BRUNO CVIKL o avtorjih Marko Samec Univerza v Mariboru, Fakulteta za gradbeništvo Smetanova ulica 17, 2000 Maribor, Slovenija E-pošta: marko.samec@uni-mb.si Dean Korošak Univerza v Mariboru, Fakulteta za gradbeništvo Smetanova ulica 17, 2000 Maribor, Slovenija E-pošta: dean.korosak@uni-mb.si Bruno Cvikl Univerza v Mariboru, Fakulteta za gradbeništvo Smetanova ulica 17, 2000 Maribor, Slovenija E-pošta: cvikl@uni-mb.si Izvleček V delu je predstavljena karakterizacija mešanic gline in vode na podlagi dielektrične spektroskopije in analiza pridobljenih spektrov. Predlagan je teoretičen model dinamike ionov, ki poleg gibanja ionov v porah elektrolita vključuje tudi njihovo ujetost na površinah mineralov. Časovne konstante, ki določajo dinamiko teh pojavov, so povezane z fizikalnimi lastnostmi snovi. Mikroskopsko gibanje ionov v kompleksnem okolju mešanice gline in vode lahko opišemo z dinamiko delcev, ki vodi do subdifuznega obnašanja. Ključne besede dielektrična spektroskopija, porozna snov, prevodnost, dinamika delca PROBING ION DYNAMICS IN A CLAY-WATER SYSTEM WITH DIELECTRIC SPECTROSCOPY MARKO SAMEC, DEAN KOROŠAK and BRUNO CVIKL About the authors Marko Samec University of Maribor, Faculty of Civil Engineering Smetanova ulica 17, 2000 Maribor, Slovenia E-mail: marko.samec@uni-mb.si Dean Korošak University of Maribor, Faculty of Civil Engineering Smetanova ulica 17, 2000 Maribor, Slovenia E-mail: dean.korosak@uni-mb.si Bruno Cvikl University of Maribor, Faculty of Civil Engineering Smetanova ulica 17, 2000 Maribor, Slovenia E-mail: cvikl@uni-mb.si Abstract Dielectric spectroscopy characterization of clay-water mixtures is presented and the obtained spectra are analysed. A theoretical model for ion dynamics is proposed in which motion of ions in pore space electrolyte is interrupted by trapping events at the mineral surfaces. The typical time scales for these processes are given in terms of the physical properties of the material. It is shown that the microscopic motion of the ions in a complex environment of clay-water system can be described with fractional dynamics leading to subdiffusive behavior. Keywords dielectric spectroscopy, porous media, conductivity, fractional dynamics 1 introduction Characterization of complex heterogeneous materials such as clay minerals turns out to be a complicated issue experimentally as well as theoretically [1, 2]. One of the reasons for the difficulties is the time scale of the ion dynamic processes which are mostly diffusion governed. The most suitable techniques to probe ion dynamics on these intermeditate time scales are NMR and dielectric spectroscopy. Numerous studies have already demonstrated and analysed the anomalous properties of transport in natural porous media caused by its heterogeneous nature [3]. On the other hand, certain anomalous features of transport were also recently discovered studying random walks [4] and dc electrical conductance [5] on complex networks. The main consequence of anomalous diffusion of contaminants in geological formations surrounding a potentially contaminated site is the time dependence of the diffusion coefficient [6]. The microscopic properties of diffusion transport of ions in clayey soils [7] present important issues in migration studies of contaminants in moist soil, for soil remediation problems and engineering of natural nuclear waste repository barriers. The effective diffusion constant is sometimes considered [8], while the dependence of the diffusion coefficient on the distance from the clay-water interface was studied by [9] showing as much as a ten times increase in the diffusion coefficient near the interface with respect to its value in the bulk. With respect to transport through natural porous media (such as soil or rock) the anomalous features stem from the motion of particles owing to the flow of water in saturated soils and the interaction with the solid matrix. The most important property of the porous media has been identified as its surface conductivity [10] and the structure of the electric double layer (EDL) in general. Within the dynamic Stern layer assumption (where counterions in the Stern layer express a certain mobility), the surface conductivity depends on the contributions from the Stern and the diffusive layer. Recently, a model was proposed for surface electrochemical properties of clay minerals, in which the surface conductivity of the clay was computed based on the dynamic Stern assumption, with the argumentation that the surface conductivity is frequency dependent [11]. However, the analysis of the results of electrical and dielectrical characterization of porous media turns out to be a very complicated issue, specifically in the low frequency region, due to the observed dispersion of the conductivity and the electrode polarization effect [1, 12]. Among the numerous homogenization methods, the Differential Effective Medium (DEM) schemes [13, 14] and the Moment Method (MoM) [15] have shown to be powerful tools for modelling the relationships between microstructural features and electromagnetic parameters (dielectric permittivity and electrical conductivity). In this paper we present the analysis of a low frequency part of the conductivity spectra of kaolinite clay samples prepared at different water contents approximately covering the range between plastic and liquid limits. We show that the measured spectra exhibit an anomalous part in the low frequency region displayed as a power-law frequency dependence before reaching the plateau. We also show that the conductivity dispersion curves for samples with different water contents follow the same master curve signalling the universality as frequently observed in disordered matter [16]. An attempt to explain experimentally observed features is here presented with a model of ion motion in pore space electrolyte interrupted by trapping events at the mineral surface. The typical time scales for these processes are given in terms of the physical properties of the material. The specific features of measured dielectric spectra can be well described by introducing harmonic restoring potential into the ion equation of motion. 2 experimental data Electrical characteristics of the clay-water system were measured using a low frequency impedance analyser at room temperature. The admittance of the sample placed in the measuring cell between two planparallel electrodes (area S=5.5 cm2, distance L=4.5 - 5 mm) was determined from the linear response of the sample to the small oscillating bias on the electrodes of the magnitude of 10 mV. The real (conductance) and imaginary parts (capacitance) of the admittance were measured in the frequency interval of 100 Hz to 100 MHz. The clay sample used was source clay Kaolin KGa-1b purchased from The Clay Minerals Society with mineral kaolinite as the primary constituent, containing also small amounts of metals (Ca, K, Na, Mg). The plastic and liquid limits of the sample were at 25.9 % and 40.1 % water content, respectively. The specific surface of the clay sample was 10 m2/g and its solid specific gravity 2.6. The frequency dependence of the conductance and capacitance was first determined for a dry sample, and then for wet samples, with gravimetric water content ranging from 32 % to 56 %. Wet samples were obtained with the addition of distilled water to the clay. 3 results and discussion The values for real and imaginary parts of frequency dependent conductivity were obtained with recalculating the measured conductance and capacitance. The frequency dependence of the complex conductivity for the samples with different water contents can be well described with the following expression whose physical basis is described below: a (w) = 1 + i^r1 + (i'wr2) (1) where a0, Tp t2 and a are constants determined from substituting the expressions given by Eq. (1). The results for conductivity as a function of frequency are presented in Fig. 1 for the real part of the expression given by Eq. (1) and in Fig. 2 for the imaginary part. The results for the samples with 36 % and 56 % water content are only shown. All the samples show qualitatively similar behaviour. The values obtained for the calculated curves shown in Fig. 1 and Fig. 2 are given in Table 1. Figure 1. Measured (open dots) and calculated (solid lines) real part of frequency dependent conductivity of kaolinite clay samples at 36 % (lower curve) and 56 % (upper curve) water contents. a 0 Figure 2. Measured (open dots) and calculated (solid lines) imaginary part of frequency dependent conductivity of kaolin-ite clay samples at 36 % (lower curve) and 56 % (upper curve) water contents. Table 1. Values of the parameters used to calculate the real and imaginary parts of the conductivity shown in Fig. 1 and Fig. 2. Parametrs 36% Water content 56% oJA/Vm] 0,43 0,725 T, [s] 0,9 * 10" 8 1,45 * 10-8 [s] 3,2 * 10- 4 2,1 * 10-" Here, the static dielectric constant is the low frequency limit of the real part of the dielectric function ss = s'(u ^ 0). The equation of motion for the ion in the external field E(t) is: ^ + 7v(t) + U x(t) = qE(t) at m (5) where the characteristic frequency is defined as: ,2 q 2nm„ U0 =- st Som (6) 0,67 0,67 The physical mechanism of the ion dynamics near the mineral particle surface consists of the succesive trapping and releasing of the particle. After each such event (at the pore/clay particle surface) the particles undergo the Brownian motion in the pore space electrolyte. The time scale for this motion between trapping events is Tr = 1/y , while Tt is set for the intrinsic time scale for the trapping event. We can estimate the characteristic trapping time by considering the linear dimension of the clay particle a (i.e. radius) and thermal velocity of the particle vth m , where T is the temperature and k is the Boltzmann constant: a Let us now discuss the physical basis of the Eq. (1) describing the dynamic response of the ions in a clay-water system. In general the ion current can be given by: j(t) = qnmnv(t) (2) where q is the elementary charge, n 0n is the ion concentration and v(t) is the ion velocity. Without loss of generality, a simple one-particle one-dimension picture is used here. The ions in the clay-water system move in a complex environment consisting of pore space filled with pore electrolyte. During the motion they also interact with the clay mineral surfaces which are charged. Ions moving in the pore space experience drag force which is proportional to ion velocity: Fd = -myv (3) Here m is the mass of the ion, and y is the drag force coefficient. To describe the electrode-pore electrolyte interface an additional harmonic force is introduced acting on the ion in the double layer near the electrode [1]: F =-qnnx (7) The characteristic time for the motion in the pore space depends on the mean free path of the ion and its thermal velocity. For an unsaturated clay-water system we can estimate the mean free path with the average distance between the two adjacent clay particles: <0 (8) Vth a nclayVth In this simplified picture the relation between the characteristic times of ion dynamics can be written as: Tr = PTt , (9) (4) where the details of the clay structure, relative orientation of clay particles and the topology is simply collected within a single parameter P, usually of order one. It was shown [17, 18] that the stochastic motion of the particle experiencing trapping events lead to a memory relation between the mean position and the mean velocity of the particle expressed with a fractional derivative of velocity: a v th 1 Ss S0 t dxr =- D-^) (10) at a(w) = (w) • kT (16) where ( = r^ / rr . The Fourier transform of the velocity leads to: v(L) = 4>(iu)a x(L) (11) After the Fourier transformation of the equation of motion (10) and using j(L) = a(w)E(w) the frequency dependent conductivity is: a (w) = 1 + (w0Tr )2(iwrt) a + iwrr (12) where ^0 = = ^ £o^0iTr (13) m By rewriting the derived expression for the conductivity (13) we can obtain the forms of Eq. (1). Here the notations t2 = Tt (ll0Tr) , and t1 = Tr were introduced. t2 is the effective low frequency relaxation time as detected in the dielectric spectra, determining the peak of the imaginary component of conductivity a2 = Im[a(ll = 1/t2 )) . The value of the static dielectric constant can thus be estimated as: = a2 T2/ £0 (14) Since t1 and t2 are usually several orders of magnitude apart (see Table 1) we also have: a0T2 sin (to /2) £j =—f^- 2e0 (1 + cos(na/2) (15) All the parameters of the model are (Eqs. (13)-(15)) expressed in this way with two free parameters p and a, while all other quantities are material dependent (ion density, temperature, clay particle radius (i.e. specific surface), ion mass). Further investigation into the properties of the ion-surface interaction is necessary to determine the exact values of the remaining parameters. In the low frequency limit, the frequency dependence of the conductivity follows the power-law behaviour: a a ua. The properties of the diffusive motion of the ions near the clay particle-water interface are contained in the frequency dependent diffusion constant D(w) to which the conductivity is connected via the generalized Einstein relation [19]: Recently Revil and Linde [20] have studied the transport of ions and water in charged, deformable porous media, and showed that the relationship between the diffusion coefficient and conductivity was much more complex than the one given in Eq. (16). Since the focus here is on the motion of ions near the clay particle - water interface we consider Eq. (16) as an operational model in which the couplings to other transport mechanisms (such as chemico-osmotic effect) are neglected [21]. The frequency dependent diffusion coefficient is given by [19]: .. TO D(l) = — L f e-iut(x2)dt , (17) n where ^x2^ is the mean-square displacement of the ion. Normal diffusion is characterized by the mean-square displacement that is linear in time (xa t , while anomalously diffusing particles display the mean-square displacement of the (x2 ^ a tfi, where /3 ^ 1. For ¡3 = 1 — a in Eqs. (3) and (4) we obtain the expression for the low frequency limit of the ion conductivity in the clay-water mixture a a La. The ion surface diffusion transport is therefore subdiffusive, and could be characterized with the time-dependent diffusion coefficient D (t) a t—n . This indicates that the experimentally observed dielectric properties result from anomalous ion transport in a clay-water system characterized with the time-dependent diffusion coefficient often observed in complex systems [22, 23]. Recently Dudko et al. have considered the time-dependent diffusion coefficient in periodic porous materials and showed that the diffusion coefficient decreases with time from its value in a free solvent to its effective value which is much smaller [24]. 4 conclusions The conductivity spectra as obtained by dielectric spectroscopy measurements of the clay-water mixtures at different water content levels are analysed. It is shown that the origin for the anomalous conductivity dispersion can be attributed to the motion of the ions near the clay particle-water interface. The ion dynamics is described with the Brownian motion interrupted with trapping events at clay surfaces. The transport of the ions near the clay particle surface is found to be subdiffusive with the effective time-dependent diffusion constant. a 0 0 We have shown that the analysis of the low-frequency part of the dielectric spectra can yield an important insight into certain properties of the diffusion processes of ions in porous media. With respect to contaminant transport it contributes to better understanding and prediction of the long time migration of contaminants in complex environments. references [1] Sanabria, H., and Miller, J. H. (2006). Relaxation processes due to electrode-electrolyte interface in ionic soultions. Phys. Rev. E, Vol. 74, 051505. [2] Rotenberg, B. et al. (2005). An analytical model for probing ion dynamics in clays with broadband dielectric spectroscopy. J. Phys. Chem. B, 109, 15548. [3] Hunt, A. G. (2005). Basic transport properties in natural porous media. Complexity, Vol.10, 22, 22 - 37. [4] Gallos, L. K. (2004). Random walk and trapping processes on scale-free networks. Phys. Rev. E, Vol. 70, 046116. [5] Lopez, E., Buldyrev, S. V., Havlin, S. and Stanley, H. E. (2005). Anomalous transport in scale-free networks. Phys. Rev. Lett., Vol. 94, 248701. [6] Guerrini, I. A., and Swartzendruber, D. (1997). Fractal concepts in relation to soil water diffusivity. Soil Sci. 162, 778. [7] Kozaki T., Inada K., Sato S. and Ohashi, H. (2001). Diffusion mechanisms of chloride ions in sodium montmorillonite. J. Cont. Hyd., Vol. 47, 159-170. [8] Maes, N., Moors, M., Dierckx A., De Canniere, P. and Put, M. (1999). The assessment of electromi-gration as a new technique to study diffusion of radionuclides in clayey soils. J. Cont. Hyd., Vol. 36, 231-247. [9] Yukhnovskii, I. R., Tokarchuk, M. V., Omelyan, I. P. and Zhelem, R. I. (2000). Statistical theory for diffusion of radionuclides in ground and subterranean water. Rad. Phys. Chem., Vol. 59, 361-375. [10] Revil, A., and Glover, P. W. J. (1997). Theory of ionic-surface electrical conduction in porous media. Phys. Rev. B, 55, 1757. [11] Leroy, P., and Revil, A. (2004). A triple-layer model of the surface electrochemical properties of clay minerals. Journal of Colloid and Interface Science, 270, 371-380. [12] Hilfer, R. (1991). Geometric and dielectric characterization of porous media. Phys. Rev. B, Vol 44. [13] Cosenza, Ph., Camerlynck, C. and Tabbagh, A. (2003a). Differential effective medium schemes for investigating the relationship between high frequency relative dielectric permittivity and water content of soils. Water Resources Research, Vol. 39, n°9, 1230 doi:10.1029/2002WR001774. [14] Cosenza, P., Guérin R. and Tabbagh A., (2003b). Relationship between thermal conductivity and water content of soils using numerical modelling. European Journal of Soil Science, Vol. 54, 581-587. [15] Cosenza, P., and Tabbagh, A. (2004). Electromagnetic determination of clay water content: role of the microporosity. Applied Clay Science, Vol. 26, Issues 1-4, 21-36. [16] Dyre, J. C., and Schroeder, T. B. (2000). Universality of ac conduction in disordered solids. Rev. Mod. Phys., Vol. 72, 873-892. [17] Metzler, R., and Klafter, J. (2000). Subdiffusive transport close to thermal equilibrium:From the Langevin equation to fractional diffusion. Phy. Rev. E, Vol. 61, 6308. [18] Metzler, R., Barkai, E. and Klafter J. (1999). Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach. Phys. Rev.Lett. 82, 3563. [19] Scher, H., and Lax, M. (1973). Stochastic transport in a disordered solid. I. Theory, Phys. Rev. B, 7, 4491-4502. [20] Revil, A., and Linde, N. (2006). Chemico-electro-mechanical coupling in microporous media. J. Coll. Int. Sci., Vol. 302, 682-694. [21] Revil, A., Leroy, P. and Titov, K. (2005). Characterization of transport properties of argillaceous sediments. Application to the Callovo-Oxfordian Argillite. J. Geophys. Res., Vol. 110, B06202. [22] Fa, K. S., and Lenzi, E. K. (2005). Time-fractional diffusion equation with time dependent diffusion coefficient. Phys. Rev. E, Vol 72, 011107. [23] Campos, D., Mendez, V. and Fort, J., Description of diffusive and propagative behavior on fractals. Phys. Rev. E, Vol. 69, 2004, 031115. [24] Dudko, O. K., Berezhkovski, A. M. and Weiss, G. H. (2005). Time dependent diffusion coefficients in periodic porous materials. J. Phys. Chem., Vol. 109, 21296-21299. DOLOČITEV PASIVNEGA ZEMELJSKEGA PRITISKA Z UPORABO TRIDIMENZIONALNEGA PORUŠNEGA MEHANIZMA HELENA VRECL-KOJC Ln STANISLAV ŠKRABL o avtorjih Helena Vrecl-Kojc Univerza v Mariboru, Fakulteta za gradbeništvo Smetanova ulica 17, 2000 Maribor, Slovenija E-pošta: helena.vrecl@uni-mb.si Stanislav Škrabl Univerza v Mariboru, Fakulteta za gradbeništvo Smetanova ulica 17, 2000 Maribor, Slovenija E-pošta: stanislav.skrabl@uni-mb.si Izvleček Prispevek prikazuje modificiran prostorski porušni mehanizem za določitev 3D koeficienta pasivnega zemeljskega pritiska ob uporabi teorema zgornje vrednosti metode mejne analize. V analizi je upoštevan translatorni kinematično dopustni porušni mehanizem, ki je generaliziran z globino h = 1.0. Mehanizem geometrijsko predstavlja prostorski togi blok sestavljen iz osrednjega dela in dveh stranskih togih blokov, ki ju sestavljajo ovojnice bočnih trenjskih stožcev. Sprednja ploskev osrednjega dela mehanizma je v interakciji s podporno konstrukcijo, medtem, ko je zgornja ploskev lahko obremenjena z dodatno obtežbo. Bočni segmenti predstavljajo štiri ali tristrani mnogokotnik v prerezni ravnini z osrednjim delom, zaradi tega definirajo poligonalno drsino osrednjega dela. Na zunanji strani je vsak segment sestavljen iz infinitizi-malno oddaljenih togih polstožcev, ki tvorijo ovojnico. Numerični rezultati mejnih vrednosti tridimenzionalnega (3D) pasivnega zemeljskega pritiska so predstavljeni v obliki brez-dimenzijskega koeficienta Kpy kot posledica lastne teže zemljine ter Kpq zaradi dodatne površinske obtežbe. Namen raziskave je bil izboljšanje do sedaj najnižjih vrednosti rešitev pridobljenih po teoriji mejne analize. Rezultati analize predstavljenega modela so prikazani v grafični obliki v odvisnosti od geometrijskih parametrov in lastnosti temeljnih tal. V prispevku sta tudi predstavljena dva v svetovem merilu priznana prostorska porušna mehanizma, ki prav tako temeljita na metodi mejne analize ter primerjava rezultatov med vsemi tremi porušnimi mehanizmi. Ključne besede teorem zgornje vrednosti, optimiranje, pasivni pritisk, tridimenzijski porušni mehanizem DETERMINATION OF PASSIVE GARTH PRESSURE USING THREE-DIMENSIONAL FAILURE MECHANISM HELENA VRECL-KOJC and STANISLAV ŠKRABL About the authors Helena Vrecl-Kojc University of Maribor, Faculty of Civil Engineering Smetanova ulica 17, 2000 Maribor, Slovenia E-mail: helena.vrecl@uni-mb.si Stanislav Škrabl University of Maribor, Faculty of Civil Engineering Smetanova ulica 17, 2000 Maribor, Slovenia E-mail: stanislav. skrabl@uni-mb.si Abstract This paper presents a modified three-dimensional (3D) failure mechanism for determining the 3D passive earth pressure coefficient using the upper bound theorem within the framework of the limit analysis theory. The translational kinematically admissible failure mechanism generalized with a depth of h = 1.0 is considered in the analysis. The mechanism geometry presents a volume of rigid blocks composed of the central body and two lateral rigid bodies, which are connected by a common velocity field. The front surface of the central body interacts with the retaining wall, while the upper surface can be loaded by surcharge loading. The lateral body segments represent four- and three-sided polygons in the cross section of the central body; therefore, they define the polygonal failure surface of the central part. At the outer side, each segment of the lateral body is bounded by infini-tesimally spaced rigid half-cones that describe the envelope of a family of half-cones. The numerical results of 3D passive earth pressure limit values are presented by non-dimensional coefficients of passive earth pressure influenced by the soil weight Kpy and a coefficient of passive earth pressure influenced by the surcharge Kpq. This research was intended to improve the lowest values obtained until now using the limit analysis theory. The results are presented in a graphical form depending on the geometrical parameters and soil properties. A brief description of two world-recognized failure mechanisms based on the limit analysis approach, and the comparison of three failure mechanism results are also presented. Keywords upper-bound theorem, optimization, passive pressure, three-dimensional failure mechanism 1 introduction Passive earth pressure acting on the rigid retaining wall has been widely studied in the past with a stress on refining a 2D analysis. The calculations are based either on the limit-equilibrium method ([3], [13], [14], [16]), the slip line method ([5], [9]), or the limit analysis method ([2], [6], [11]). Three-dimensional (3D) problems of the passive earth pressure were presented by Blum [1] to a restricted extent, by Soubra and Regenass [10] with a multi-block translation failure mechanism using the limit analysis, and by Skrabl and Macuh [12] with a rotational hyperbolical failure mechanism. This paper presents a new modified 3D translational kinematically admissible failure mechanism for determining the passive earth pressure coefficients within the framework of the upper-bound theorem of the limit analysis. The limit analysis theory determines the limit pressures that provide strict lower or upper bounds to the true limit load ([2], [7]). The upper-bound theorem ensures that the rate of work due to the external forces of kinematical systems in equilibrium is smaller than or equal to the rate of dissipated internal energy for all kinematically admissible velocity fields that obey strain velocity compatibility conditions and velocity boundary conditions, as well as the flow rule of the considered materials. This analysis considers a general case of frictional and cohesive soils (0 and c) with the eventual surcharge loading q on the ground surface. The numerical results of the 3D passive earth pressure are presented in the form of dimensionless coefficients Kpy and Kpq, representing the effects of the soil weight and surcharge loading. The coefficient Kpc , which represents the effects of cohesion, can be determined using the coefficient of passive earth pressure due to the surcharge Kpq [5]. In conclusion a brief description of two world-recognized failure mechanisms based on the approach of limit analysis is presented ([10], [12]). The lowest upper-bound solutions of the 3D passive earth pressure coefficient given by a new failure mechanism are compared with the results relating geometrical parameters and soil properties. in the y-z plane, and the envelope of a family of half-cones at both lateral sides, with a horizontal backfill. c) The redistribution of the contact pressures over the entire height h = 1 for the passive pressure due to the soil weight is triangular and is assumed to be inclined at the constant friction angle S at the soil-structure interface. d) The velocity at the soil-structure interface is assumed to be inclined at S to the wall in order to respect normality conditions [8]. e) The work equation is obtained by equating the rate of external work done by external forces to the rate of internal energy dissipation along different velocity discontinuities. f) The resulting value of passive earth pressure is defined by: h2 Pp = KpY • Y • y b + Kpq • q • hb + Kfc • c • hb (1) where y is the unit weight of the soil, q is surcharge loading, and c is cohesion. 2 failure mechanism A new modified translational three-dimensional failure mechanism within the framework of the upper-bound theorem of the limit analysis has been developed in order to optimize the 3D passive earth pressure coefficient [15]. The 3D coefficient is distinguished from the two-dimensional one, by its growing difference, depending on soil properties and geometrical data. Therefore, these coefficients are very useful when analysing different kinds of geotechnical problems, where a 3D state gives more exact and realistic results. For example, it can be applied to retaining pile walls in the case of axially spaced piles, when the resistance of piles along the embedment depth is analysed [15]. 2.1 suppositions and limitations The following suppositions and limitations are applied: a) Soil characteristics present a homogeneous, isotropic Coulomb material using the associative flow rule obeying Hill's maximal work principle [4]. b) The translational failure mechanism is bounded by a polygonal sliding surface in the x-y plane, a rigid block of the dimensions b ■ h (b = width, h = height) 2.2 VELOCITY FIELD FORMULATION The new translational three-dimensional kinematically admissible failure mechanism is shown in Fig. 1, where the cross-section and plane view of the lateral part of the failure mechanism are schematically presented. The Cartesian co-ordinate system is selected with the y-axis along the wall. The optimal polygonal sliding surface in the x-y plane consists of a final number of rigid segments, the mechanism is dimensionless with a height of h'= 1 (see Fig. 2). Figure 1. The scheme of the cross-section and plane view of the three-dimensional failure mechanism lateral plane. A cross-section of the proposed failure mechanism with the velocity field is presented in Fig. 2. The individual segment j has a starting point (x t, yop)j and a final point (0, Yop)j, where the variables are calculated during the optimization procedure (see Fig. 2a). The kinematically-admissible velocity field (see Fig. 2b) is composed of j = 7 rigid segments bounded by the embedment point O (0, -1), and the final point (Xopt, 0). In general, the number of segments can be varied. The kinematics of the segments velocities Vi are inclined at an angle of aj + 0 to the horizontal axis, and the intersegment velocities V..+1 are inclined at an angle of - 0 to the horizontal axis. The mechanism is defined by 2n-1 angular parameters aj (j = 1, ..., n-1) and Pjj+1 (j = 1, ..., n). The movement of each of the n rigid segments accommodates the movement of the whole failure mechanism soil mass, and its movement accommodates the movement of the retaining structure. y rk=7/ •a7+<|> V! V67 1-067" V» ie- p«- V45 VM /V v 23 V12 Figure 2b. The velocity field of the failure mechanism. Figure 2a. The cross-section of the failure mechanism. The segment velocities Vi and the inter-segment veloci- The kinematically admissible velocity field is consistent ties ViM1 are given by with the normality condition (at the angle 0 to the sliding surface) not only in the x-y plane of the interface sin|, j+1 — 20 — a-} ) between rigid segments (as shown in Fig. 2) but also on Vj+i = Vj 2 (2) the interfaces perpendicular to this plane. sin(n —13j ;+1 — a ;+1 ) sinia ■— a ■ V = V__j (3) Vj, j+i Vj (3) sin(33j, j+i — aj — 29) 2.3 GEOMETRY OF RIGID LATERAL BODIES The geometry of the failure mechanism presents rigid space-segments consisting of a central part and two lateral rigid bodies constituted of the family of half-cone envelopes. Fig. 3a presents the envelope of a family of cones of the first segment of the lateral body where the s-t is the local coordinate system and x-y the global system. R,(s-t) =(t, - t,)• tg$ = (t, -s, • tgft) • tg$ (6) where ft is the angle between the global (x, y) and the local (t, s) coordinate systems. The point on the envelope of a family of cones is defined by dRjds and § (see Fig. 3b): dR- - / - \ —L = -tg /3 •tg $ = sin ^ = asin(tgft •tg $ (7) ds Each point coordinate of the envelope of a family of half-cones in the local coordinate system is defined by: si = k • ti + n ^ k = —-0 A n = sk - k • tk (8) tk - 10 t= tg $ • sin § - k (9) Figure 3a. Geometry of the first segment. Transformation from the local to the global coordinate system leads to: s+ds ZoW+M-ds Figure 3b. The envelope of a family of half-cones. The parametrical equation of a circle in the coordinate system (z, s) is: z{ = Rt • cos ,§i s, = R, • sin (4) (5) where Rl is the radius of the cone, and is the angle of deflection of a tangent to a curve, as a consequence of the differential dR/ds. The radius of the cone Rl in the local coordinate system (t, s) is obtained with: xi = x0, j +1, • cos aj - s, • sin a j y{ = y0,j +1{ • sin a -; + s, • cos a -; (10) where x0,j; y0,j is the distance from the origin of the global coordinate system to the first point on the segment j = 1. The envelope of a family of cones for the first segment (j = 1) is defined by the equations (4) to (10), while for other blocks (j = 2, M) it is based upon the equation: j = asin(tg(5j • tg$) + atg 'dz. 'dn I cos ¡3 j (11) where § j is the deflection angle on the envelope at point i of segment j. Fig. 4 presents the scheme of the envelope of the third segment and its surface plane. After the angle j is known at each of the analysed cones, the envelope of a family of cones can be uniformly defined by Eqs. (8) - (10). Fig. 5 presents the ground plane along the x-axis of the last segment. n Figure 5. The ground plane of the failure mechanism along the On the basis of these results, the volume of the separate block j and finally the volume of the whole failure mechanism needed in the work equation of the upper-bound theorem of the limit analysis can be calculated. The work equation and the background of the limit analysis has been explained in detail ([7], [10], [12]), therefore no attention will be paid to the background of the theory in this paper. Considering the work equation for the condition of equality between the external rate of work and internal rate of dissipation along the velocity interface for no cohesive rigid-plastic material, the coefficients can be written by: K = ^L ; Kp = g pY f pq f J y Jq (12) where f and gy denote the reduced values of the rate of work due to the passive earth pressure and the rate of work due to the unit weight of the ground at Kpy = 1, and fq and gq denote the reduced values of the rate of work due to the passive earth pressure and the rate of work due to the surcharge loading on the backfill surface at Kp = 1. pq The coefficient Kpc , which represents the effects of cohesion, can be determined using the coefficient of passive earth pressure due to the surcharge Kpq [5]: K 1/cosê K — pq_ pc tan (13) 3 numerical results With the numerical analysis, the most critical non-dimensional three-dimensional passive earth pressure coefficient is obtained, where all variables are calculated by considering the scalars and the rigorous system of equality and inequality constraints. The Solver optimization tool of Microsoft Excel, together with the generalized-reduced-gradient method, was used during the numerical process. The scalars, constraints and variables: - geometry scalars are points (0,0) and (0,-1), ratio b/h, - material scalars are soil properties S, Y=1.0, q=c=0, - variable points on x-axis (X p 0)j=7 and on the y-axis (0, Y^j , - variable points on sliding surface (xopt, yopt)j=1 M , - inequality equation of the angles a j < aj+1 — 0.01 and /3j < n/2 — ^ , ' ' - inequality equation of points on the y-axis y] > yj+1 + 0.001. x-axis. The results of three-dimensional passive earth pressures are presented in the form of dimensionless coefficients KpY and Kpq representing the effects of soil weight and surcharge loading, respectively. They are calculated for different soil characteristics for 0 ranging from 15° to 45°, for three values of 8/0 (8/0 = 0, 0.50 and 1.00), and for three values of b/h (b/h = 0.25, 1 and 10). Fig. 6 presents the critical failure mechanism in the x-y plane for different soil characteristics. The following conclusion has been re-established from these results: any increase of the soil friction angle 0 influences the failure mechanism; and while the volume of the failure mechanism increases, the shape of the sliding surface becomes more curved and the length of the last segment on the x-axis increases continuously. Figure 6. Critical failure mechanism in the x-y plane for 7' = 1.0, q' = C = 0, b/h = 0.25 and 0 = 25 ^ 45°. From Fig. 7, it can be established that the length Xopt = L7 and, consequently, the volume of the failure mechanism is maximal at 0 = 45° and 8 = 0 ; these values decrease by lowering the soil friction angle. The friction angle at the soil-structure interface 8 essentially influences the results, and the geometrical factor b/h has the largest influence at the minimum soil friction angle 0 = 25° in the region b/h = 0.25 to b/h = 1; however, at b/h > 1 the influence of the geometry parameters declines. 2,6 - 2,4 - 2,2 / «" 2,0 - /// 1,8- 1,6 $ = 30° b/h T 6 t 8 10 Figure 7. The last segment length along the %-axis Xopt = L7 against b/h for different 8 and f. b/h=0.25 -----1/3 $ .......1/2 $ ----2/3 $ -$ 25 30 35 $ 40 45 120 80 40 0 S -0 b/h=1.00 -----1/3 $ ......1/2 $ ----2/3 $ -$ 25 30 35 $ 40 45 b/h=0.50 -----1/3 $ .......1/2 $ ----2/3 $ $ 30 35 $ 40 45 80 n S b/h=2.00 / 60 0 / -----1/3 $ / ^40- .......1/2 $ / ----2/3 $ Si ' 20 0 - 1 25 30 35 $ 40 45 S -0 - -----1/3 t .......1/2 4 ----2/3 t t b/h=5.00 y / / ✓ -1- —1—1— 25 30 35 t 40 45 S ■0 ■1/3 t b/h=10.00 .......1/2 t ----2/3 t -t 25 30 35 t 40 45 Figure 8 (also on previous page). Non-dimensional coefficients of Kp against f, S and b/h. 25 30 35 t 40 45 25 30 35 t 40 45 60 40 20 0 S b/h=5 s ^ " -0 ■-----1/3 t .......1/2 t ----2/3 t -1 1 1 25 30 35 t 40 45 200 S 150 - 0 -----1/3 t *100 .......1/2 t ----2/3 t 50 -1 _____M" _ b/h=0.50 25 30 80 1 S 60 -0 -----1/3 t £40 - .......1/2 t ----2/3 t 20 -1 0 35 t b/h=2 40 45 25 30 35 t 40 45 25 30 35 t 40 45 Figure 9. Non-dimensional coefficients of Kp against f, S and b/h. Figs. 8 to 9 represent the values of the coefficients Kpy and Kpq for different values of b/h, different shear angles and friction quotients between the retaining structure and the backfill soil. The analysis results show that for the values of Kpy and Kpq , which decrease essentially by increasing the ratio of b/h, the coefficients resemble the value in the 2D state at b/h = 10; likewise, the failure mechanisms of 2D and b/h = 10 have similar shapes [5]. Also the friction angle at the soil-structure interface S plays an important role, as by increasing the ratio of S/0 the coefficients Kpy and Kpq increase essentially. The results from Figs. 8 to 9 can be used in geotechnical practice. brief presentation of these two world-recognized failure mechnisms. 4.1 MULTI-BLOCK FAILURE MECHANISM M nt Soubra and Regenass [10] published a truncated multiblock translational failure mechanism referred to as Mnt , which has been improved from his two previously proposed mechanisms, i.e. the one-block mechanism M1 and the multi-block mechanism Mn. The improvement from Mn has been obtained by a volume reduction of the final block, and from Mj^ by increasing the number of blocks from one to n. 4 comparison with the existing solutions The three-dimensional passive earth pressure acting on a rigid retaining wall has been re-established using a simplified translational failure mechanism [10] and a rotational log spiral failure mechanism [12]. Follows a Fig. 10 presents the cross-section and the plan view of the Mnt mechanism. In this improved mechanism, the lower plane and the lateral planes of the last block of the Mn mechanism are truncated by two portions of right circular cones with vertices at Dn-j and D'n-1, respectively. The right (left) cone is tangential to the lateral plane BD ,D (B'D' .D' ) and the lower plane D D' r n-1 n K n-1 n' r n n D D' . Figure 10. The cross-section and the plane view of the Mnt translational failure mechanism by Soubra and Regenass [10]. Fig. 11 presents the velocity field of the Mnt mechanism. The soil mass of each block moves with the velocity Vi inclined at an angle of fa + 0 to the horizontal direction. The inner-block velocity V-A j is inclined to the inner planes of 0, and to the outer velocities, as shown in Fig. 11. The wall moves with the velocities Vo and Vo t representing the relative velocities at the soil-structure interface. All of these velocities are parallel to the vertical symmetrical plane xOy. A comparison between this failure mechanism and the one presented in this paper can be made while both models are translational, using the same suppositions. The difference from the presented failure mechanism in this paper can be seen from Figs. 2 and 9. The Mnt mechanism of Soubra and Regenass [10] has two major differences from the presented modified failure mechanism, i.e. all blocks have the same starting point, which is the origin of the x-y coordinate system, and just one portion of the right circular cones is used on each side in the lateral plane. 4.2 3D ROTATIONAL HYPERBOLICAL FAILURE MECHANISM Skrabl and Macuh [12] developed this approach within the framework of the limit analysis theory. Similar to the mechanism described before, it is based on a three-dimensional rotational hyperbolical failure mechanism (see Fig. 12). This failure mechanism represents the extension of the plane slip surface in the shape of a log spiral (Fig. 13). The outer sides are laterally bounded by a curved and kinematically-admissible hyperbolic surface, which is defined by enveloping the hyperbolical half cones and part of the case surface of the leading half cone. Every point along the retaining wall height (1-0, see Fig. 12) has an exactly defined hyperbolic friction cone. A common velocity field connects all the bodies. The difference from this mechanism and the presented modified failure mechanism can be seen in Fig. 2 and Figs. 12 to 13. Figure 12. Scheme of the rotational hyperbolical failure mechanism. Figure 13. Log spiral slip surface of the mechanism. 4.3 COMPARISON OF THE RESULTS The results for the dimensionless 3D passive earth pressure coefficients were compared to the results for other types of failure mechanism using the upper-bound theorem ([10], [12]). Table 1 shows the results for Kpy , and Table 2 the results for K for different values pq of 0, S and b/h. The highest differences are at 0 = 45° and b/h = 0.25; by lowering the friction angle 0 and increasing the value b/h the differences decrease, the smallest value being at 0 = 25° and b/h = 10. Table 1. Comparison of Kp results depending on parameters f, 8 and b/h. * (°) b/h 5/* Kpy (Soubra and Regenass) 0.5 1.0 10.0 Kpy (Skrabl and Macuh) 0.5 1.0 10.0 Kpr (proposed mo 0.5 1.0 del) 10.0 25 0.5 7.963 5.779 3.770 7.360 5.539 3.625 7.296 5.027 3.253 1.0 12.776 8.798 5.004 10.985 7.809 4.885 10.431 7.360 4.579 35 0.5 22.855 15.150 8.150 20.779 13.709 7.320 19.847 13.533 7.079 1.0 54.064 33.202 13.730 40.135 25.839 12.857 37.509 24.229 12.131 45 0.5 178.689 99.555 26.684 104.658 61.825 23.293 88.334 53.074 20.652 1.0 379.494 212.364 59.215 239.688 140.857 51.747 207.888 122.938 45.991 Table 2. Comparison of Kpq results depending on parameters f, 8 and b/h. -e- b/h 5/* Kpy (Soubra and Regenass) 0.5 1.0 10.0 Kpr (Skrabl and Macuh) 0.5 1.0 10.0 Kpy (proposed mo 0.5 1.0 del) 10.0 25 0.5 10.135 6.873 3.748 9.733 6.579 3.687 9.368 6.346 3.627 1.0 14.599 9.445 4.620 13.699 9.058 4.782 12.598 8.394 4.495 35 0.5 30.545 19.006 7.951 26.859 16.699 7.426 25.356 15.844 7.209 1.0 57.371 33.627 11.708 49.308 29.814 12.044 43.675 26.561 10.860 45 0.5 176.444 96.248 23.101 107.038 60.919 19.397 86.218 50.080 17.420 1.0 338.705 184.474 43.854 249.114 140.401 42.041 215.048 121.667 36.648 1000 100— 10— 1 1000 KPq 100 10— • 8/4»=l ?. \n • Soubra and Regenass ° Skrabl and Macuh — Proposed method -- ^— 30° 8 s 15° -- 1 1 11 0 2 4 6 8 b/h 10 • 8/<|>=l • Soubra and Regenass ° Skrabl and Macuh \ • \B — Proposed method -a =45° __B__1__ V, 30° V 15° --1-1-1-1- 1000-Kp, 100- 8 b/h 10 • Soubra and Regenass ° Skrabl and Macuh — Proposed method 1 1 0 2 4 1 1 6 8 b/h 10 5/<(>=0 • □ • □ • V • Soubra and Regenass Skrabl and Macuh Proposed method ^—______ • □ • ^--. 15° --1-1-1-1- 8 b/h 10 Figure 14. Comparison of the results for Kp and Kp against b/h for 8/f = 1 and 8/f = 0, f = 15°, 30°, 45°. Fig. 14 presents a comparison of the coefficients Kpy and Kpq for 0 = 15°, 30° and 45°, 5/0 = 1 and 5/0 = 0 against the ratio b/h. The comparison of these results shows that the difference in the coefficient is the greatest at 0 = 45°, at low ratio b/h = 0.25, and 5/0 = 0. 5 conclusions The modified translational failure mechanism presented in this paper was developed for the improvement of the 3D passive earth pressure coefficients. The approach used is based on a new translational three-dimensional failure mechanism within the framework of the upper-bound theorem of the limit analysis. The geometry of the kinematically-admissible failure mechanism presents a rigid space-block consisting of a central part and two lateral rigid parts of a family of cone envelopes. In the past the three-dimensional passive earth pressure was determined by a translational failure mechanism [10] and a rotational hyperbolical failure mechanism [12]. A description of these two failure mechanisms is briefly presented in this paper for a better understanding of differences between all three failure mechanisms. The numerical results for a limit value of 3D passive earth pressure are presented graphically by a non-dimensional coefficient of passive earth pressure influenced by the soil weight Kpy and a coefficient of passive earth pressure influenced by the surcharge Kpq . A comparison of the results for all three mechanisms shows that the difference in the coefficient increases with any improvement in soil properties and lowering the ratio b/h. references [4] Hill, R. (1950). The Mathematical Theory of Plasticity. Clarendon Press, Oxford. [5] Kerisel, J. and Absi, E. (1990). Tables for the calculation of passive pressure, active pressure and bearing capacity of foundations. Gauthier-Villard, Paris. [6] Lysmer, J. (1970). Limit analysis of plane problems in soil mechanics. Journal for Soil Mechanics and Foundation, 96, No. 4, 1311-1334. [7] Michalowski, R.L. (2001). Upper-bound load estimates on square and rectangular footings. Geotechnique, 51, No. 9, 787-798. [8] Mroz, Z. and Drescher, A. (1969). Limit plasticity approach to some cases of flow of bulk solids. Journal of Engineering for Industry, ASME, 91, 357-364. [9] Sokolovski, V.V. (1965). Static of granular media. Pergamon Press, New York. [10] Soubra, A.H. and Regenass, P. (2000). Three-dimensional passive earth pressures by kinematical approach. Journal of Geotechnical and Geoenviron-mental Engineering, ASCE, 2, No.2, 969-978. [11] Soubra, A.H. and Macuh, B. (2002). Active and passive earth pressure coefficients by a kinematical approach. Proceedings of the Institution of Civil Enginees-Geotechnical Engineering, 155, No. 2, str. 119-131. [12] Skrabl, S. and Macuh, B. (2005). Upper-bound solutions of three-dimensional passive earth pressures. Canadian Geotechnical Journal, 42, No.5, 1449-1460. [13] Skrabl, S. (2006). Interactional approach of cantilever pile walls analysis. Acta Geotechnica Slovenica, 3, No. 1, 46-59. [14] Terzaghi, K. (1943). Theoretical soil mechanics. Wiley, New York. [15] Vrecl-Kojc, H. (2006). Upper-bound theorem used for optimizing a geomechanical model of a cantilever retaining structure. PhD thesis, University of Maribor. [16] Zakerzadeh, N., Fredlund, D.G., and Pufahl, D. E. (1999). Interslice force functions for computing active and passive earth force. Can. Geotech. J., Ottawa, 36(6), 1015-1029. [1] Blum, H. (1932). Wirtschaftliche Dalbenformen und den Berechnung. Bautechnik, 10(5), 122-135. [2] Chen, W.F. and Liu, X.L. (1990). Limit analysis in Soil Mechanics. Elsevier, Amsterdam. [3] Coulomb, C.A. (1776). Essai sur une application des règles de maximis et minimis à quelques problèmes de statique relatifs à l'architecture. ACAD. r.Sci. Mém. Math. Phys. par divers savants, 7, 343-382. OBNAŠANJE AVTOCESTNEGA TUNELA V ALŽIRIJI! EKSPERIMENTALNA ŠTUDIJA FIZIKALNEGA MODELA ABDELBAKI SERIAN!, Y. KISMOUNE THéSñRD Ln TAYEB SERRADJ o avtorjih Abdelbaki Seriani Genius University Badji Mokhtar, Faculty of Science of the Ground BP 12, 23000 Annaba, Alžirija E-pošta: seriani7@yahoo.fr Y. Kismoune Thésard Genius University Badji Mokhtar, Faculty of Science of the Ground BP 12, 23000 Annaba, Alžirija Tayeb Serradj Genius University Badji Mokhtar, Faculty of Science of the Ground BP 12, 23000 Annaba, Alžirija izvleček Pred vsakim projektom za izgradnjo tunela je treba zaradi visokih stroškov izdelati manjšo modelno študijo ter jo po potrebi matematično preveriti. S tem se pri realizaciji projekta izognemo nepredvidljivim posledicam. Slednje upošteva tudi predstavljena raziskava, pri kateri smo na fizikalnem modelu v razmerju 1:20 eksperimentalno preučili obnašanje avtocestnega tunela v Alžiru ter usipanje okoliških zemljin med izkopavanjem. Ključne besede tunel, ustrezen material, fizikalen model, podpore, deformacije, prelom, oslabljene cone BEHAVIOUR OF THE SUBWAY TUNNEL IN ALGIERS: PHYSICAL MODEL EXPERIMENTAL STUDY ABDELBAKI SERIANI, Y. KISMOUNE THéSARD and TAYEB SERRADJ About the authors Abdelbaki Seriani Genius University Badji Mokhtar, Faculty of Science of the Ground BP 12, 23000 Annaba, Algeria E-mail: seriani7@yahoo.fr Y. Kismoune Thésard Genius University Badji Mokhtar, Faculty of Science of the Ground BP 12, 23000 Annaba, Algeria Tayeb Serradj Genius University Badji Mokhtar, Faculty of Science of the Ground BP 12, 23000 Annaba, Algier Abstract Tunnel construction projects are too expensive to be approached directly without a preliminary small-scale model study and subsequent verification with a mathematical model, if needed. These studies enable to avoid unforeseen consequences which emerge at the time of the project realization. It is within this framework that we carried out our investigations. The behaviour of a subway tunnel in Algiers and the state of transition of the surrounding ground during digging are studied from an experimental point of view via a 1/20 physical model scale. Keywords tunnel, equivalent material, physical modelling, supports, deformations, rupture, loosened zone 1 goals of the investigations The lack of experimental data on the tunnel in Algiers (Batch 5) deprive us of a comparison base between measurements and observations carried out in situ and the results obtained by the physical model used within the framework of this research. Nevertheless, the work that we present will be highly useful, being the first approximation of the prospects of the subway tunnel in Algiers. The below methods and the reasons were considered in the investigation: - the use of artificial rock medium corresponding perfectly to in situ conditions; - the technology of tunnel construction proposed by the project which will be completely respected during the realization of the tunnel model; - the digging will be carried out under the load which corresponds to actual weight of artificial material. This method of investigation (technology and construction of the tunnel model under constant load) provides a better simulation model, much closer to the in situ conditions. 2 the task to be solved This work is aimed at reporting about the aptitude of physical modelling in tunnel projects considering tender ground conditions. To achieve this, the following is considered: - Measurement of convergence after each stage of construction. - Observation of the ground movement around the tunnel after each phase, owing to the numerous reference marks placed in artificial material. These marks allow us to measure relative displacements and thus to calculate deformations and specify the state of material around the tunnel model. Qualitative verification of combining the supporting systems used. Conception of a physical model to carry out the observations and measurements. A mixture of industrial powder is used as artificial rock material. 3 physical modelling 3.1 CHOiee OF THE PHYSICAL MODEL To simulate the subway tunnel in Algiers (Batch 5), a physical model was used (Fig. 1), which has two essential advantages: - Economic advantage from the point of view of construction, and - Simplicity from the point of view of uses and operation. Figure 1. Physical model. 3.2 CHOICE OF THE SCALES OF SIMULATION We think that it is preferable to choose geometrical scales and scales of equal constraints to avoid complicating the choice of the material equivalent or to have additional loads applied in order to compensate for high values of the voluminal mass in the case of a high scale constraint compared to the 1 : 20 geometrical scale selected. With this scale the width of the tunnel model (50 cm) does not exceed 1/3 of the width of the model (180 cm), as recommended from experience (in order to avoid the influence of borders on the test results). The calculation of the scales is shown in Table 1 (see next page). Physical simulation won large applications since an international symposium in Italy on the physical modelling in geomechanics. Certain researchers approached the physical simulation in tunnelling by using different equivalent materials for the rocks and the systems of supports. Thus, for modelling the behaviour of a tunnel, equivalent materials such as:solids (bentonite, mica baryte, ground cork) and binders (paraffin oil, silicon oil) are used (Sauer, 1979). In order to simulate a tunnel in tender rock a mortar of water, cement , bentonite, and sand is used. Another mixture composed by cement, lime quick, sand, and water (Gajary, 1990) was used for studying experimentally the stability of a roadway. The simulation of the tunnel supports in appropriate conditions with different equivalent materials is summarized below. Authors Type of supports Equivalent material Tazawa.Y Rock bolts Cooper screws of variable length located at convenient relative distances Lining Mortar of cement with a resistance to compression ac = 20-30 MPa Adachi .T. & al. Rock bolts and lining Kint paper of variable thickness and resistance In the light of these and other studies (Stimpson, 1968), and based on the criteria listed below (Baron and Larocque; 1960): - the facility of fabrication of the proposed model, - relative expenses for the material and the equipment, and - environmental and security criteria will be observed too. The author introduces new equivalent materials, such as industrial talk mixed with water. Industrial talk mixed with water possesses plastic properties similar to in situ conditions (clay-marly soil). The mixture in question is simple (talk + % water), which renders this material recoverable, hence reusable, and therefore cheap. Physical and mechanical properties of equivalent materials of the rock and of supporting are presented in table 2 on page 28 (Seriani, 1993). Table 1. Scales of constraints compared to the geometrical scales. Geometrical scale Scale of stresses Insitu Model Insitu Models Dimensions Dimensions Value Names Values according to the scale Names Symbol Units 1/40 1/30 1/25 1/20 1/15 1/4 1/3 1/2 1/1.5 L/m Length L Stress 0 N/m2 Force F NR H/m 20 Depth 20 H Unit weight Accel. gravity Young modulus Poisson coef. Y G Kg/dm3 Kg.m/s2 Model Model Model Model Model Model Model Model Model W/m 10 Width 10 W E V N/m2 N°1 N°2 N°3 N°4 N°5 N°6 N°7 N°8 N°9 b/m 11 Height b Friction angle Cohesion ? C degree N/m2 1/40 0 1/40 1/30 1/25 1/20 1/15 1/4 1/3 1/2 1/1.5 F 1:6400 1:480 1:40000 1:32000 1:24000 1:6400 1:4800 1:3200 1:2400 0.5 Y 1:1 1,33:1 2:1 2:1 2,66:1 10:1 13,3:1 20:1 26,6:1 G 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 0.25 E 1:40 1:30 1:25 1:20 1:15 1:4 1:33 1:2 1:1,5 V 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 0.25 ? 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 C 1:40 1:30 1:25 1:20 1:15 1:4 1:3 1:2 1:1,5 1/30 1:36000 1:27000 1:25500 1:18000 1:1350 1:3600 1:2700 1:1800 1:1350 1:1,33 1:1 1,2:1 1,5:1 2:1 7,5:1 10:1 15:1 20:1 0.66 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:40 1:30 1:25 1:20 1:15 1:4 1:3 1:2 1:1,5 0.33 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 0.35 1:40 1:20 1:25 1:20 1:15 1:4 1:3 1:2 1:1,5 1/25 1:2500 1:18750 1:15625 1:12500 1:9735 1:2500 1:1875 1:1250 1:937,5 0,625:1 0,833:1 1:1 1,253:1 1,66:1 6,25:1 8,3:1 12,5:1 16,6:1 0.8 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 0.4 1:40 1:30 1:25 1:20 1:15 1:4 1:3 1:2 1:1,5 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 0.44 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:40 1:30 1:25 1:20 1:15 1:4 1:3 1:2 1:1,5 1/20 1:16000 1:12000 1:1000 1:8000 1:6000 1:1600 1:1200 1:800 1:600 1:2 1:1,5 0,8:1 1:1 1,33:1 15:1 6,66:1 10:1 13,3:1 1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:40 1:30 1:25 1:20 1:15 1:4 1:3 1:2 1:1,5 0.5 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 0.55 1:40 1:30 1:25 1:20 1:15 1:4 1:3 1:2 1:1,5 1/15 1:9000 1:6750 1:5625 1:4500 1:3375 1:900 1:675 1:450 1:337,5 1:2,66 1:2 0,6:1 1:1,33 1:1 3,75:1 5:1 7,5:1 10:1 1.33 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:40 1:30 1:25 1:20 1:15 1:4 1:3 1:2 1:1,15 0.66 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1 0.73 1:40 1:30 1:25 1:20 1:15 1:4 1:3 1:2 1:1,5 Table 2. Mechanical properties of the mixture. Geomechanical parameters Symbols Units In situ values of marly clay Laboratory tests results Values required by scale (1/20) Compressive strength 0 kPa 270.0-600.0 30.0-50.0 13.5 30.0 Young's modulus E kPpa 9000-40000 368.0-520.0 450-2000 Internal friction angle $ degree 18-30 29.6 30 Cohesion C kPa 100-110 56.8 50-55 Weight unit mass Y kg/m3 1.6 1.6 1.6 Poisson's coefficient v - 0.45 0.40-0.45 0.45 Table 3. Mechanical properties of a support model (scale 1/20). Kind of support Mechanical and geometrical parameters In situ values Laboratory test results Values required by scale Tensile strength 275 MPa 21.49 MPa 13.75 MPa Bolting (soldering wire) Length 3- 4 m 0.15 - 0.20 m Acting air 1.4 m2 0.0035 m2 Shotcrete (mixture of gypsum, quick lime, water and inert material [according to A.M.Kozuna 1957]) Compressive strength 21.0 MPa 1.05 MPa 1.05 MPa Steel arches (very fine aluminium strips transformed in segments with a transversal U section) Moment Transversal section width Distance between two successive steel arches 108750 N.m 180 mm 1.2 m 1.36N.m 1.36 N.m 9.0 mm 0.06 m The materials for the simulation of different types of supports described above are not convenient for the artificial rocks used in our conditions owing to construction technology of the tunnel model (construction under a constant load, in extremely wet conditions, hence in very soft artificial rocks.) The equivalent materials for simulating tunnel supports used in our case and their mechanical properties are shown in Table 3. 3.3 EXPERIMENTAL METHODOLOGY The compacted equivalent material is placed in the model with the following dimensions: - height: 110 cm - extent: 180 cm, and - thickness: 20 cm. Black reference marks are established horizontally in the artificial material, white reference marks are stuck on Plexiglas just above the black reference marks. The black reference marks are mobile and react to artificial rock displacement, while the white reference marks are fixed. These reference marks are photographed after each phase of construction (after it has undergone displacements). After consolidation, the model of tunnel is excavated to the thickness of 7cm representing the distance between the steel arches. The execution time of pickling and installation of the supporting systems is approximately three hours. The nomenclature of different tunnel cutting phases is as follows: - Phase I: Cutting a working place in the excavation roof to a depth equal to rib spacings. - Phase II: Installing rock bolts and a superior rib arch - Phase III: Concrete projection - Phase IV: Cutting a work place in the lower part of the excavation to a depth equal to rib spacings. - Phase V: Installing the inferior rib arch - Phase VI: Concrete projection on the lower part of the excavation. In the model, the various phases of the superior section are represented successively in Figs. 2-7. Figure 3. Appearance of deformation. Figure 4. Installation of bolts and steel fabrics. Figure 5. Installation of steel arches. Figure 6. Projection of shotcrete. Deformations after each construction phase of the first section are illustrated in Fig. 8. All these construction operations are repeated during the digging of other sections. However, it should be noted that a crack appeared in the shotcrete of the first section after the completion of the second section digging. This crack was prolonged after concreting the third section which presupposed the formation of a disturbed zone on the roof of the tunnel model (Fig. 9). Various construction stages in successive diggings of the lower section are presented in Figs. 10-13. Figure 8. Evolution of the deformation at the end of the first section. Figure 9. Appearance of a disturbed zone on the roof of the model. Figure 10. Cutting of the first slice of the inferior section. Figure 11. Installation of steel fabrics and steel arches. Figure 12. Concreting the first slice of the inferior section. Figure 13. Final construction of the model. Time evolution of the deformations is shown in Fig. 14. Figure 14. Convergence of the model according to time. The measurement method of displacements is an optical method leading to misreadings of those reference marks especially which are far away from the centre of the model as shown in Fig. 15. The coefficient of correction is calculated as follows: A mesh of white squares on the model has a width of 10cm, which is 2cm on the photograph. This dimension corresponds to an opening of 19 units on the optical glass objective. These 19 units measure 1.9mm on the photograph, so the scale between the model and the photo is of 1:5 and the scale between the photo and the optical glass is of 10:1. The measurement scale thus becomes of 10:5, therefore 2:1, from which 14 units correspond to 10.5mm. The correction coefficients are presented in Table 4. Table 4. Correction coefficients of measured displacements. Coordinate X Correction Coordinate Y Correction 0 -3.6 0 2.4 10 -3.2 10 2.0 20 -2.8 20 1.6 30 -2.4 30 1.2 40 -2.0 40 0.8 50 -1.6 50 0.4 60 -1.2 60 0.0 70 -0.8 70 -0.4 80 -0.4 80 -0.8 90 0.0 90 -1.2 100 0.4 100 -1.6 110 1.2 110 -2.0 120 1.6 120 -2.4 130 2.0 140 2.4 150 2.8 160 3.2 170 3.6 Figure 15. Concreting the first slice of the inferior section. 4 calculation of the volumetric and angular deformations Many significant problems require constraints and elongations in the x-y plan although the constraints a can exist; a and a have been neglected in our zz ' xz yz o case. If it is supposed that our model is one elastic unit yet sufficiently subjected to constraints which prevent sound movement, no particle displacements are possible inside the model without the model deformation. Small displacements of the model particles are usually presented by the components u and v for a problem of plane deformation. These represent the displacements of the reference marks established in the model. Their measurement method is described later on. Elongations for such a problem are: du dx dv £y = dy O = £rr + (1) (2) (3) du dv Yxy =T~+ (4) dy dx 4 = \ (£* - £y (5) The pictures show the isolines of expansion after digging and supporting each section. The negative sign indicates the areas of extension while the positive sign indicates the areas of compression. That is how two compression zones of low intensity are formed at the end of the first slice in laterally anchored areas.Their value is of 1% . An extension zone starts to appear on the crown of the tunnel (Fig. 16). Fig. 16 reflects the state of material when the higher section of the tunnel is completed. It is evident from Fig. 16 that the zones of high compression located in the previously quoted areas have moved to lower un-anchored areas. In turn, the inside of the stross is compressed. The extension area is amplified to reach a value of 3%. Laterally anchored areas have undergone a compression of 6% just above the crown. The area compressed earlier has slackened and has given rise to a disturbed zone. On the level of stross, compression reached 4% (Fig. 17 on next page). Fig. 18 (on next page) shows a value of a maximum distortion which almost equals the one in the model of an intensity of 6°. Fig. 19 (on page 38) shows a maximum distortion of 4° at the end of the third slice. The distortion concerning this training course is much larger, it is 4°. On the crown of the tunnel model it is 3°. The rocks with the same distortion are similar to concentric boxing rings around the centre of the higher section floor of the tunnel model. The excavation and the supporting of the tunnel model are finished with the completion of the sixth slice. Fig. 19 shows the state of the material around the tunnel model; below the stross, on its walls and at the anchoring level, the material is in a compression state. Above the crown, the extension is always of the same intensity, hence a permanent existence of a disturbed zone (Fig. 20, page 38). Figure 17. State of elongations at the end of the third slice. Figure 18. State of distortions at the end of the first slice. Figure 19. State of distortions at the end of the third slice. Figure 20. State of elongations at the end of the sixth slice. Fig. 20 represents the state of distortions when the tunnel model is fully completed. Laterally anchored areas of the higher section remain subjected to a distor- tion equal to 3.43°, while the value of y max does not exceed 3° on the crown level and it is even lower around the stross (Fig. 21, next page). Figure 21. State of distortions at the end of the sixth slice. 5 evaluation of the plastic zone The fundamental equations are taken from Sakurai S. (1985). From Hook's law s1 • E = a1 - v • a3 s3 • E = a3 - v • a1 (6) we obtain the deviatoric of the deformation 1 + v. E -a3) (7) and the volumetric elongation 1 + v £, + £, = - E (8) According to the Mohr-Coulomb criterion: e1 — e3 =(o"1 + is continuous and bounded on any bounded sub-domain QCRd. Let r denote the Euclidean distance between any pair of points in the domain O. The commonly used radial basis functions are: linear (<(r) = r), cubic (<(r) = r3), thin-plate spline (<(r) = r2 log r), Gaussian (<(r) = e~ar) and multiquadric (MQ) (<(r) = (r2 + c2))3). Commonly used values for ft are -1/2 and 1/2. The parameter c > 0 is a shape parameter controlling the fitting of a smoothing surface to the data. We usually use MQ or inverse MQ RBFs. To introduce RBF collocation methods, we consider a PDE in the form of Lu(x) = f (x), in He Rd, (1) Bu(x) = g (x), on dH, (2) where u is the concentration, d denotes the dimension, 3O is the boundary of the domain O, L is the differential operator on the interior, and B is the operator that specifies the boundary conditions of the Dirichlet, Neumann or mixed type. Using Kansa's asymmetric multiquadric collocation method, the unknown PDE solution u is approximated by RBFs in the form: N M u » U(x) = £a< (x) + ^ Ylp} (x), (3) j=1 l=1 where q> can be any of above mentioned radial basis function, pi,...,pM e nm , is a polynomial of degree m or less. Let (x.)N=1 be the N collocation points in H U dH . We assume the collocation points are arranged in such a way that the first NI points are in O, whereas the last Nb points are on 3O. To evaluate N+M unknown coefficients, N+M linearly independent equations are needed. Ensuring that U(x) satisfies (1) and (2) at the collocation points results in a good approximation of the solution u. The first N equations are given by N M .L0, (12) dt u(x,t) = upart, xenpart(t),t >0, (13) u(x,t) = uso1, x er(t),t > 0, . du (14) (upart - usd)vn(x,t) = D—(x,t), x er(t),t > 0, (15) dn The level set method has gained much popularity for solving moving boundary problems. It was firstly introduced by Osher and Sethian [13]. The level set function captures the interface position as its zero level set, and it is advected by introduction of a hyperbolic equation into the governing set of equations. 4.2.1 The level set formulation In the level set formulation of the moving interface, the interfaces, denoted by r, are represented implicitly through a level set function 0(x, t), where x is a position of the interface, t is a point of time. Usually, the 0 is defined as a signed distance function to the interface. The moving interface is then captured in all times by locating the set of r(t) for which 0 vanishes. The level set function is advected with time by a transport equation which is known as the level set equation: ^ + Vn IWI = 0, dt ¿(x,0) = ¿.(x) (17) du dn (x,t) = 0, xednäp(t)\r(t),t>0, (16) where x is the coordinate vector of a point in O, D means the diffusivity constant, n is the unit normal vector on the interface pointing outward with respect to Opart(t) and vn is the normal component of the velocity of the interface. The initial concentration u(x, 0) inside the diffusive phase is given. 4.1.3 The Numerical solution Method Our interest is to give an accurate discretization of the moving boundary conditions. Here we present an inter-polative moving data centers method by which the data centers are computed for each time step and the solution is interpolated from the old data centers to the new ones. The equations are solved with the collocation method using RBFs. An outline of the algorithm is: I. Compute the concentrations profiles solving Eqs. (12-14) and (16). II. Predict the position of the boundary s1 at the new time-step: s1 (t + At) using Eq. (15). III. Once the boundary is moved, the concentration u can be computed in the new region using Eq. (12). The solution is interpolated from the old point location to the new one. where 0o(x) embeds the initial position of the interface and vn is the normal component of the velocity of the interface: v„ = v V0 'IWI ' (18) where V0/|V0| is the unit normal to the surface N. If we take into account a continuous extension of the interface velocity v, the evolution of the level set function can be done by the hyperbolic equation for the level set equation: dt + v V0 = 0, ¿(x,0) = ¿0(x) (19) In our case the continuous extension of the velocity v is taken as the (steady) solution of the following evolution equation [14]: Ê1 ±dv. N = 0, dr dx (20) where t denotes a fictitious time step not related to the main time step and the sign is determined from the normal direction of the level set function. The RBFs are incorporated into level set methods to construct a more efficient approach. At the initial time, all the time dependent variables should be specified over entire domain. The initial value problem (17) can be considered equivalent to an interpolation problem, and hence the starting point of the use of RBFs to solve partial differential equations is the interpolation problem. Further, the spatial portion is approximated by the RBFs and the temporal variations are approximated by the time dependent expansion coefficients. 4.2.2 rbf implicit modeling of the Level set function interpolation of the level set function In the present implicit modeling, the MQ RBFs is used to interpolate the scalar implicit level set functions 0(x) with N points by using N MQs centered at these points. The resulting RBF interpolant of the implicit function can be written as N M ) = £ a^j (x) + ^ 7l p} (x), (21) j=1 1=1 Because of the introduction of this polynomial, the RBF interpolant of 0(x) in Eq. (21) must be subject to the side constraints (6). If the interpolation data values f1,—,fN G^ at the point locations x1,—,xN are given, the RBF interpolant of 0(x) in Eq. (21) can be obtained by solving the system of N+2 linear equations for N+2 unknown generalized expansion coefficients: 4>(xi ) = f, i = I,---, N, NN N ^= 0, ^aixi = 0, ^aiyi = 0, which can be re-written in a matrix form as Ha= f, (23) where (22) H= A P PT 0 G^(n+3)> (N+3) (24) A= ^1(xi) ••• ^N (xi) ^1(xN ) - ^N (xN ) G^NxN , (25) P= 1 xi yi 1 xn yN G^Nx3, (26) a = [a1 ••• aN p0 p p2 ] G^Nx3,(27) f = [f • fN 0 0 0 f G^NX3, (28) The generalized expansion coefficients a can be obtained by a = H-1f, (29) The resulting RBF interpolant of the implicit function in Eq. (21) can be re-written compactly as 4>(x) = Y T (x) a, (30) where V(x) = [p(x) ••• ^n(x) 1 * 7 f G^(Nx3)x1, (31) equation of motion Since the Hamilton-Jacobi PDE (17) is time dependent, it is further assumed that all knots are fixed in space and the space and time are separable, and therefore the RBF interpolant of the implicit function in Eq. (30) becomes time dependent as 4>(x ,t ) = V T (x )a(t), (32) Substituting Eq. (32) in (17) yields vn |(W)Ta| = 0, (33) where G,1 = G dt |(W)r a = [(G»2 +(G»2 j1'2, (34) " 010 g^(n+3)x1, dx dx dy dy (35) T The initial value problem can be considered equivalent to the interpolation problem since the expansion coefficients at the initial time are found as a solution of the interpolation problem [15]. Therefore the preliminary starting point of the use of RBFs to solve PDEs is the interpolation problem that is equivalent to solving the initial value problem. The original equation (17) is thus converted into a time-dependent interpolation problem for the initial values of expansion coefficients and the propagation of the front is governed by the time dependent equation (33). For time advance the initial values of a in Eq. (33) we used a collocation formulation of the method of lines. The governing equation of motion of the front (33) is extended to the whole domain O and the normal velocities vn at the front are thus replaced by the extension velocities in O. All nodes of domain are taken as fixed nodes of RBF interpolation. We also take into consideration constraints which must be introduced to guarantee that the generalized coefficients a can be solved. Using the present collocation method for N points and above mentioned constraints, a set of resulting ODEs can be compactly written as: H -a + B(a) = 0, (36) dt where vn (x1)|(v^r K))«! = v'n(Xi)|(Wr(Xi))a eKNx3)xi, (37) 0 0 0 b) the exact explicit time integration. The first order homogeneous ODE of the form d a B The set of ODEs can be solved by several ODE solvers such as the first-order forward Euler's method and higher-order Runge-Kutta, Runge-Kutta-Fehlberg, Adams-Bashforth, or Adams-Moulten methods [16]. We used a) the first-order forward Euler's method, an approximate solution to Eq. (36) is the following: dt + Ea = 0, (39) where E = H-1(ViG,i + v2G,2), (40) has the solution: a(t + dt) = expm(-Edt)a(t - dt). (4i) The expression expm (-Edt) is a MATLAB exponential matrix function represents the series expansion or a rational fraction: expm(—Edt) = I — Edt + (dt2 /2!)E2 — (dt3 /3!)E3 t • • •. (42) E is the coefficient matrix resulting from the application of the Eq. (35). 4.2.3 Numerical example For the simulation we used data from [11]: the concentration inside the part where the material characteristics remain constant upart = 0.53, the concentration on the interface usd = 0 , the initial concentration of the diffusive phase u° = 0.1, the diffusivity constant D = 1, the domain length l = 1 and the initial position of the interface s0 = 0.2. Let N be total number of points, r the number of points that lie inside a constant composition and N — r the number of points that lie inside the diffusive phase. Due to the movement of the interface, the point locations are adapted at each time step. The MQ exponent fi had the values 0.5 and 1.5. In Fig. 8 the movement of the interface positions calculated with a different MQ exponent, fi is presented. In the numerical experiments we compared our numerical solutions with the analytical solutions that exist for the problem presented in Chapter 4. (See [12]). The results are presented in Fig. 8 (see next page). The next example is the rotation of the solid body. Consider the rotation of a circular bubble of the radius r =0.25 centered at (0.5, 0.15) in a vortex flow with the velocity field (v1 , v2) = (-y, x). A half cycle of rotation is presented in Fig. 9. a(tn+i) = a(tn ) - dtH-iB(a(tn ), where dt is the time step, and Movement of the interface 0.45 Time Figure 8. Interface position vs. time. Figure 9. Zero contour of the level set function at different points and time during the rotation of a circle. 5 discussion and conclusion In the case of the radionuclide migration (see also [6], [9] and [12]), two evaluation steps were performed. In the first step the velocities in principal directions were determined from the pressure of the fluid obtained from the Laplace differential equation. In the second step the advection-dispersion equation was solved to find the concentration of the contaminant. In this case the method of evaluation was verified by comparing results with those obtained from the finite difference method. The traditional finite difference scheme was also used for solving the Laplace and advection-dispersion equation. For the approximation of the first derivative second-order central difference or one-sided difference were used. But for the approximation of the second derivatives we used the second-order central second difference. The time dependent part was implemented with the implicit scheme. The discretization grid has actually 12x12 points. Te results show that differences exist between both numerical schemes (Figs. 4 and 5). Different sets of input data yield differences between the schemes. In the simulation, a very large scatter caused by different given values of hydraulic conductivity was observed. Another reason for differences could result from the kriging method applied in the sequential Gaussian methods. The comparison of concentrations calculated with the Eulerian and Lagrangian methods in partly heterogeneous porous (Figs. 6 and 7) media shows that the Lagrangian methods provide a wider concentration cloud in the area of high conductivity. It seems that it shows the influence of non-smooth change between low and high conductivity. In general, the Eulerian approach is more convenient and more frequently used. But if it is important to study sharp changes (in our case between areas of low and high conductivity) of the solutions where important chemistry and physics take place, it is better to use the Lagrangian RBF scheme. The comparison of the moving boundary positions calculated with the moving data centers method and MQ (P=0.5) and MQ (0=1.5) (Fig. 8) shows that MQ (0=1.5) determines the position of the interfaces much more accurately than MQ (0=0.5). The simulations have also shown that the value of the shape parameter c which was computed by the residual error procedure was in the range between 0.01 and 0.09. This confirms the fact that for a fixed number of centers N, smaller shape parameters produce more accurate approximations. The comparison of the moving boundary positions calculated with the moving data centers method (MQ (0=1.5)) and the level set method (Fig. 8) also shows that the moving data centers method gives better results in this case. To achieve better accuracy, the resultant system of the RBF-PDE problem usually becomes badly conditioned. Several different strategies [17] have been somewhat successful in reducing the ill-conditioning problem when using RBF methods in PDE problems. These strategies include: variable shape parameters, domain decomposition, preconditioning of the interpolation matrix, and optimizing the center locations. From chapter 4.2 we can see that RBFs (MQ) can be easily included in the level set formulation. Fig. 9 shows that we can get logical results using MQ in a 2-dimen-sional example. We can conclude that the Kansa method is a valid alternative to the FDM because of its simple implementation and its easy use in the level set formulations. The only geometric properties that are used in the RBF approximation are the pair-wise distances between points. Figs. 4 and 5 show that the RBF solution has far less diffusion than the finite difference method with the included upwinding. In the future work we will use the Gershgorin circle theorem that could be useful a tool for choosing appropriate RBFs. For each value of the shape parameter, eigenvalues and their distribution can be studied to obtain knowledge concerning properties of an approximation matrix and their role being played in finding a better approximation of computed data to the equation solution. The solutions can be improved by using an affine space decomposition that decouples the influence between the interior and boundary collocations. references [1] G. Fairweather, A. and Karageorghis (1998). The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math., 9(1/2), 69-95. [2] Golberg, M.A. and Chen, C. S. (1998). The method of fundamental solutions for potential, Helmholtz and diffusion problems, in Boundary Integral Methods-Numerical and Mathematical Aspects, M. A. Golberg (Ed.). Computational Mechanics Publications, 103-176. [3] Kansa, E. J. (1990). Multiquadrics - A scattered data approximation scheme with applications to computational fluid dynamics - I. Surface approximations and partial derivative estimates. Computers Math. Applic. 19 (8/9), 127-145. [4] Kansa, E. J. (1990). Multiquadrics - A Scattered data approximation scheme with applications to computational fluid dynamics - II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers Math. Applic. 19 (8/9), 147-161. [5] Hardy, R. L. (1971). Multiquadric equation of topography and other irregular surfaces. J. Geophys. Res. Vol. 176, 1905-1915. [6] Vrankar, L., Turk, G. and Runovc, F. (2004). Modelling of radionuclide migration through the geosphere with radial basis function method and geostatistics. Journal of the Chinese Institute of Engineers, Vol. 27, No. 4, 455-462. [7] Fedoseyev, A. I, Friedman, M. J. and Kansa, E. J. (2002). Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. Computers Math. Applic. Vol. 43, 439-455. [8] Smith, G. D. (1978). Numerical solution of partial differential equations: Finite difference methods. Oxford Applied Mathematics and Computing Science Series, Oxford University Press. [9] Vrankar, L., Turk, G. and Runovc, F. (2004). Combining the radial basis function Eulerian and Lagrangian schemes with geostatistics for modelling of radionuclide migration through the Geosphere. Computers. Math. Applic, Vol. 48, No. 10-11, 1517-1529. [10] Crank, J. (1984). Free and Moving Boundary Problems, Clarendon Press, Oxford. [11] Javierre, E., Vuik, C., Vermolen, F. J. and Van der Zwaag, S. (2005). A comparison of numerical models for one-dimensional Stefan problems, Reports of the Delft Institute of Applied Mathematics, Netherlands. [12] Vrankar, L., Turk, G., Runovc, F. and Kansa, E. J. (2006). Solving one-dimensional phase change problems with moving grid method and mesh free radial basis functions. International Conference Nuclear Energy for New Europe, Portorož. 1005.1.1005.8. [13] Osher, S. and Sethian, J. A. (1988). Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics. Vol. 78, 12--49. [14] Li, Z. and Ito, K. (2006). The Immersed Interface Method - Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. Society forIin-dustrial and Applied Mathematics, Philadelphia, USA. [15] Wang, S. Y., Lim, K. M., Khoo, B. C. and Wang, M. Y. (2007). An extended level set method for shape and topology optimization. Journal of Computational Physics. Vol. 221, 395-421. [16] Greenberg, M. D. (1988). Advanced Engineering Mathematics, 2nd ed., Prentice-Hall International Inc., Upper Saddle River, NJ, USA. [17] Kansa, E. J. and Hon, Y. C. (2000). Circumventing the Ill - Conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations, Computers Math. Applic, Vol. 39, No. 7-8, 123-137. NAVODILA AVTORJEM Članki so objavljeni v angleškem jeziku s prevodom izvlečka v slovenski jezik. VSEBINA ČLANKA Članek naj bo napisan v naslednji obliki: - Naslov, ki primerno opisuje vsebino članka in ne presega 80 znakov. - Izvleček, ki naj bo skrajšana oblika članka in naj ne presega 250 besed. Izvleček mora vsebovati osnove, jedro in cilje raziskave, uporabljeno metodologijo dela, povzetek izidov in osnovne sklepe. - Uvod, v katerem naj bo pregled novejšega stanja in zadostne informacije za razumevanje ter pregled izidov dela, predstavljenih v članku. - Teorija. - Eksperimentalni del, ki naj vsebuje podatke o postavitvi preiskusa in metode, uporabljene pri pridobitvi izidov. - Izidi, ki naj bodo jasno prikazani, po potrebi v obliki slik in preglednic. - Razprava, v kateri naj bodo prikazane povezave in posplošitve, uporabljene za pridobitev izidov. Prikazana naj bo tudi pomembnost izidov in primerjava s poprej objavljenimi deli. - Sklepi, v katerih naj bo prikazan en ali več sklepov, ki izhajajo iz izidov in razprave. - Literatura, ki mora biti v besedilu oštevilčena zaporedno in označena z oglatimi oklepaji [1] ter na koncu članka zbrana v seznamu literature. OBLIKA ČLANKA Besedilo naj bo pisano na listih formata A4, z dvojnim presledkom med vrstami in s 3.0 cm širokim robom, da je dovolj prostora za popravke lektorjev. Najbolje je, da pripravite besedilo v urejevalniku Microsoft Word. Hkrati dostavite odtis članka na papirju, vključno z vsemi slikami in preglednicami ter identično kopijo v elektronski obliki. Enačbe naj bodo v besedilu postavljene v ločene vrstice in na desnem robu označene s tekočo številko v okroglih oklepajih. ENOTE IN OKRAJŠAVE V besedilu, preglednicah in slikah uporabljajte le standardne označbe in okrajšave SI. Simbole fizikalnih veličin v besedilu pišite poševno (npr. v, T itn.). Simbole enot, ki sestojijo iz črk, pa pokončno (npr. Pa, m itn.). Vse okrajšave naj bodo, ko se prvič pojavijo, izpisane v celoti. SLIK6 Slike morajo biti zaporedno oštevilčene in označene, v besedilu in podnaslovu, kot sl. 1, sl. 2 itn. Posnete naj bodo v kateremkoli od razširjenih formatov, npr. BMP, JPG, GIF. Za pripravo diagramov in risb priporočamo CDR format (CorelDraw), saj so slike v njem vektorske in jih lahko pri končni obdelavi preprosto povečujemo ali pomanjšujemo. Pri označevanju osi v diagramih, kadar je le mogoče, uporabite označbe veličin (npr. v, T). V diagramih z več krivuljami mora biti vsaka krivulja označena. Pomen oznake mora biti razložen v podnapisu slike. Za vse slike po fotografskih posnetkih je treba priložiti izvirne fotografije ali kakovostno narejen posnetek. PREGLEDNICE Preglednice morajo biti zaporedno oštevilčene in označene, v besedilu in podnaslovu, kot preglednica 1, preglednica 2 itn. V preglednicah ne uporabljajte izpisanih imen veličin, ampak samo ustrezne simbole. K fizikalnim količinam, npr. t (pisano poševno), pripišite enote (pisano pokončno) v novo vrsto brez oklepajev. Vse opombe naj bodo označene z uporabo dvignjene številke1. SEZNAM LITERATURE Vsa literatura mora biti navedena v seznamu na koncu članka v prikazani obliki po vrsti za revije, zbornike in knjige: [1] Feng, T. W. (2000). Fall-cone penetration and water content ralationship of clays. Geotechnique 50, No. 2, 181-187. [2] Ortolan, Ž. and Mihalinec, Z. (1998). Plasticity index-Indicator of shear strength and a major axis of geotechnical modelling. Proceedings of the Elev- enth Danube-European conference on soil mechanics and geotechnical engineering, Poreč, 25 -29 May 1998. [3] Toporišič, J. (1994). Slovenski pravopis. 2nd.ed., DZS, Ljubljana. PODRTKI O AVTORJIH Članku priložite tudi podatke o avtorjih: imena, nazive, popolne poštne naslove, številke telefona in faksa, naslove elektronske pošte. Navedite kontaktno osebo. SPRCJCM ČLANKOV IN AVTORSKE PRAVICE Uredništvo si pridržuje pravico do odločanja o sprejemu članka za objavo, strokovno oceno mednarodnih recenzentov in morebitnem predlogu za krajšanje ali izpopolnitev ter terminološke in jezikovne korekture. Avtor mora predložiti pisno izjavo, da je besedilo njegovo izvirno delo in ni bilo v dani obliki še nikjer objavljeno. Z objavo preidejo avtorske pravice na revijo ACTA GEOTECHNICA SLOVENICA. Pri morebitnih kasnejših objavah mora biti AGS navedena kot vir. Rokopisi člankov ostanejo v arhivu AGS. Vsa nadaljnja pojasnila daje: Uredništvo ACTA GEOTECHNICA SLOVENICA Univerza v Mariboru Fakulteta za gradbeništvo Smetanova ulica 17 2000 Maribor Slovenija E-pošta: ags@uni-mb.si INSTRUCTIONS FOR AUTHORS The papers are published in English with a translation of the abstract into Slovene. FORMAT OF THE PAPER The paper should have the following structure: - A Title that adequately describes the content of the paper and should not exceed 80 characters; - An Abstract, which should be viewed as a mini version of the paper and should not exceed 250 words. The Abstract should state the principal objectives and the scope of the investigation and the methodology employed, it should also summarise the results and state the principal conclusions; - An Introduction, which should provide a review of recent literature and sufficient background information to allow the results of the paper to be understood and evaluated; - A Theoretical section; - An Experimental section, which should provide details of the experimental set-up and the methods used for obtaining the results; - A Results section, which should clearly and concisely present the data using figures and tables where appropriate; - A Discussion section, which should describe the relationships shown and the generalisations made possible by the results and discuss the significance of the results, making comparisons with previously published work; - Conclusions, which should present one or more conclusions that have been drawn from the results and subsequent discussion; - References, which must be numbered consecutively in the text using square brackets [1] and collected together in a reference list at the end of the paper. LAYOUT OF THE TEXT The text should be written in A4 format, with double spacing and margins of 3 cm, to provide editors with space to write in their corrections. Microsoft Word for Windows is the preferred format for submission. One hard copy, including all figures, tables and illustrations and an identical electronic version of the manuscript must be submitted simultaneously. Equations should be on a separate line in the main body of the text and marked on the right-hand side of the page with numbers in round brackets. UNITS AND ABBREVIATIONS Only standard SI symbols and abbreviations should be used in the text, tables and figures. Symbols for physical quantities in the text should be written in Italics (e.g. v, T, etc.). Symbols for units that consist of letters should be in plain text (e.g. Pa, m, etc.). All abbreviations should be spelt out in full on first appearance. FIGURES Figures must be cited in consecutive numerical order in the text and referred to in both the text and the caption as Fig. 1, Fig. 2, etc. Figures may be saved in any common format, e.g. BMP, JPG, GIF. However, the use of CDR format (CorelDraw) is recommended for graphs and line drawings, since vector images can be easily reduced or enlarged during final processing of the paper. When labelling axes, physical quantities (e.g. v, T) should be used whenever possible. Multi-curve graphs should have individual curves marked with a symbol; the meaning of the symbol should be explained in the figure caption. Good quality black-and-white photographs or scanned images should be supplied for illustrations. TABLES Tables must be cited in consecutive numerical order in the text and referred to in both the text and the caption as Table 1, Table 2, etc. The use of names for quantities in tables should be avoided if possible: corresponding symbols are preferred. In addition to the physical quantity, e.g. t (in Italics), units (normal text), should be added on a new line without brackets. Any footnotes should be indicated by the use of the superscript1. [3] Toporišič, J. (1994). Slovenski pravopis. 2nd.ed., DZS, Ljubljana. AUTHOR INFORMATION The following information about the authors should be enclosed with the paper: names, complete postal addresses, telephone and fax numbers and E-mail addresses. Indicate the corresponding person. ACCEPTANCE OF PAPERS AND COPYRIGHT The Editorial Committee of the Slovenian Geotechnical Review reserves the right to decide whether a paper is acceptable for publication, to obtain peer reviews for submitted papers, and if necessary, to require changes in the content, length or language. Authors must also enclose a written statement that the paper is original unpublished work, and not under consideration for publication elsewhere. On publication, copyright for the paper shall pass to the ACTA GEOTECHNICA SLOVENICA. The AGS must be stated as a source in all later publication. Papers will be kept in the archives of the AGS. For further information contact: Editorial Board ACTA GEOTECHNICA SLOVENICA University of Maribor Faculty of Civil Engineering Smetanova ulica 17 2000 Maribor Slovenia E-mail: ags@uni-mb.si LIST OF REFERENCES References should be collected at the end of the paper in the following styles for journals, proceedings and books, respectively: [1] Feng, T. W. (2000). Fall-cone penetration and water content ralationship of clays. Geotechnique 50, No. 2, 181-187. [2] Ortolan, Ž. and Mihalinec, Z. (1998). Plasticity index-Indicator of shear strength and a major axis of geotechnical modelling. Proceedings of the Eleventh Danube-European conference on soil mechanics and geotechnical engineering, Poreč, 25 -29 May 1998. NAMEN REVIJE Namen revije ACTA GEOTECHNICA SLOVENICA je objavljanje kakovostnih teoretičnih člankov z novih pomembnih področij geomehanike in geotehnike, ki bodo dolgoročno vplivali na temeljne in praktične vidike teh področij. ACTA GEOTECHNICA SLOVENICA objavlja članke s področij: mehanika zemljin in kamnin, inženirska geologija, okoljska geotehnika, geosintetika, geotehnične konstrukcije, numerične in analitične metode, računalniško modeliranje, optimizacija geotehničnih konstrukcij, terenske in laboratorijske preiskave. Revija redno izhaja dvakrat letno. AVTORSKE PRAVICE Ko uredništvo prejme članek v objavo, prosi avtorja(je), da prenese(jo) avtorske pravice za članek na izdajatelja, da bi zagotovili kar se da obsežno razširjanje informacij. Naša revija in posamezni prispevki so zaščiteni z avtorskimi pravicami izdajatelja in zanje veljajo naslednji pogoji: Fotokopiranje V skladu z našimi zakoni o zaščiti avtorskih pravic je dovoljeno narediti eno kopijo posameznega članka za osebno uporabo. Za naslednje fotokopije, vključno z večkratnim fotokopiranjem, sistematičnim fotokopiranjem, kopiranjem za reklamne ali predstavitvene namene, nadaljnjo prodajo in vsemi oblikami nedobič-konosne uporabe je treba pridobiti dovoljenje izdajatelja in plačati določen znesek. Naročniki revije smejo kopirati kazalo z vsebino revije ali pripraviti seznam člankov z izvlečki za rabo v svojih ustanovah. Elektronsko shranjevanje Za elektronsko shranjevanje vsakršnega gradiva iz revije, vključno z vsemi članki ali deli članka, je potrebno dovoljenje izdajatelja. ODGOVORNOST Revija ne prevzame nobene odgovornosti za poškodbe in/ali škodo na osebah in na lastnini na podlagi odgovornosti za izdelke, zaradi malomarnosti ali drugače, ali zaradi uporabe kakršnekoli metode, izdelka, navodil ali zamisli, ki so opisani v njej. AIMS AND SCOPE ACTA GEOTECHNICA SLOVENICA aims to play an important role in publishing high-quality, theoretical papers from important and emerging areas that will have a lasting impact on fundamental and practical aspects of geomechanics and geotechnical engineering. ACTA GEOTECHNICA SLOVENICA publishes papers from the following areas: soil and rock mechanics, engineering geology, environmental geotechnics, geosynthetic, geotechnical structures, numerical and analytical methods, computer modelling, optimization of geotechnical structures, field and laboratory testing. The journal is published twice a year. COPYRIGHT Upon acceptance of an article by the Editorial Board, the author(s) will be asked to transfer copyright for the article to the publisher. This transfer will ensure the widest possible dissemination of information. This review and the individual contributions contained in it are protected by publisher's copyright, and the following terms and conditions apply to their use: Photocopying Single photocopies of single articles may be made for personal use, as allowed by national copyright laws. Permission of the publisher and payment of a fee are required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Subscribers may reproduce tables of contents or prepare lists of papers, including abstracts for internal circulation, within their institutions. Electronic storage Permission of the publisher is required to store electronically any material contained in this review, including any paper or part of the paper. RESPONSIBILITY No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of product liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.