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Consolidation analysis for unsaturated soils Enrico Conte Abstract: This paper

Consolidation analysis for unsaturated soils Enrico Conte Abstract: This paper deals with the multidimensional consolidation of unsaturated soils when both the air phase and water phase are continuous. Following the approach proposed by D.G. Fredlund and his coworkers, the differential equations governing the coupled and uncoupled consolidation are first derived and then solved numerically. The solu- tion is achieved using a procedure that depends on the transformation of the field equations by using the Fourier trans- form. This transformation has the effect of reducing a two- or three-dimensional problem to a problem involving only a single spatial dimension. The transformed equations are solved using a finite element approximation that makes use of simple one-dimensional elements. Once the solution in the transformed domain is obtained, the actual solution is achieved by inversion of the Fourier transform. The time integration process is formulated in a stepwise form. Results are presented to point out some aspects of the consolidation in unsaturated soils. Moreover, it is shown that the results obtained using the simple uncoupled theory are of sufficient accuracy for practical purposes. Key words: coupled consolidation, uncoupled consolidation, unsaturated soils, Fourier transform. Résumé : Cet article traite de la consolidation multidimensionnelle des sols non saturés quand aussi bien la phase li- quide que la phase gazeuse sont continues. A partir de la formulation proposée par D.G. Fredlund et ses collaborateurs, les équations différentielles controlant la consolidation couplée et la consolidation désaccouplée sont d’abord établies et ensuite sont résolues numériquement. La solution est obtenue par une procédure qui peut compter sur la transformation des équations de base au moyen de la transformation de Fourier. Cette transformation a l’effet de réduire un problème à deux ou trois dimensions à un problème unidimensionnel. Les équations transformées sont résolves par la méthode des éléments finis employant de simples éléments unidimensionnels. Une fois que la solution des équations transfor- mées est obtenue, la solution réel est dérivée au moyen de la transformation inverse de Fourier. L’intégration par rap- port au temps est exécutée pas à pas. Des résultats sont présentés pour mettre en évidence quelques caractéristiques de la consolidation des sols non saturés. En outre, on montre que les résultats obtenus par la simple théorie désaccouplée de la consolidation sont suffisamment précis pour leur utilisation pratique. Mots clés : consolidation couplée, consolidation désaccouplée, sols non saturés, transformation de Fourier. Conte 612 Introduction The consolidation of cohesive soils as a result of dissipa- tion of the excess pore pressures generated by external load- ing is a problem of considerable concern for geotechnical engineers. In 1925, Terzaghi presented a simple theory for the analysis of one-dimensional consolidation in saturated soils which is still widely used in practice. Generalization to three dimensions has given rise to two different approaches, which are known as the uncoupled consolidation theory (Rendulic 1936) and the coupled consolidation theory (Biot 1941). The latter is preferable from a theoretical point of view because it provides a coupling between the magnitude and progress of settlement. The uncoupled approach cannot model all the aspects of consolidation in saturated soils (Schiffman et al. 1969), but it has proved to be useful in practice (Davis and Poulos 1972). In the last few decades, there has been a considerable in- crease in the understanding of the behaviour of unsaturated soils; as a result, several theoretical and experimental studies have been conducted to analyse the consolidation processes in such soils (Fredlund and Rahardjo 1993). Unsaturated soils essentially consist of three phases: solid, liquid, and gaseous. As first pointed out by Barden (1965), three differ- ent classes of behaviour may be singled out on the basis of the continuity of the fluid phases: for high values of the de- gree of saturation of the soil, the water phase is continuous and the air phase is discontinuous; for lower values of the degree of saturation, both the air phase and the water phase may be considered as continuous; lastly, when the degree of saturation is low, the air phase is continuous and the water phase is discontinuous. Consolidation should be analysed us- ing a specific approach for each of these three classes. For example, when the soil is close to saturation, the air con- tained in the pores is occluded and cannot flow as a continu- ous fluid. In these circumstances, the air bubbles and pore water behave as a homogeneous compressible fluid flowing under pore water pressure gradients. As a result, the case of occluded air may be analysed using essentially the same for- mulation as that for saturated soils provided that pore fluid compressibility is accounted for. Solutions were proposed by Can. Geotech. J. 41: 599–612 (2004) doi: 10.1139/T04-017 © 2004 NRC Canada 599 Received 15 April 2003. Accepted 2 February 2004. Published on the NRC Research Press Web site at http://cgj@nrc.ca on 20 August 2004. E. Conte. Faculty of Engineering, University of Calabria, 87030 Rende, Cosenza, Italy (e-mail: conte@dds.unical.it). Olson (1986) for one-dimensional consolidation and by other authors for two- or three-dimensional conditions (Biot 1941; Verruijt 1969; Ghaboussi and Wilson 1973; Conte 1998). In this context, Chang and Duncan (1983) developed a modified version of the Cam–Clay model to describe the stress–strain behaviour of unsaturated soils. For lower val- ues of the degree of saturation, consolidation analysis is more complex because air and water may flow simulta- neously and separately through the soil. A general formula- tion for one-dimensional consolidation in which the air and water phases are assumed to be continuous was presented at almost the same time by Fredlund and Hasan (1979) and Lloret and Alonso (1980). This formulation is based on two continuity equations, one for the water phase and one for the air phase, which have to be solved simultaneously to give water and air pressures at any time and elevation. In the method developed by Fredlund and Hasan, the constitutive relations proposed by Fredlund and Morgenstern (1976) were incorporated. Lloret and Alonso used appropriate state surfaces to define the mechanical behaviour of the soil. In both solution procedures, the governing equations were solved using numerical techniques. Ausilio and Conte (1999) showed that the solution for one-dimensional consol- idation may also be expressed in terms of the degree of set- tlement and the average degree of consolidation for both the water phase and the air phase. The three-dimensional con- solidation problem was studied by Dakshanamurthy and Fredlund (1980) using an uncoupled approach and by Dak- shanamurthy et al. (1984), who first presented the differen- tial equations of the coupled theory. Lloret et al. (1987) extended the one-dimensional consolidation model proposed by Lloret and Alonso to three dimensions and developed an iterative procedure for solving the field equations. This pro- cedure was organized into two stages: the first stage solves the stress–strain equations, and the second stage solves the air and water flow equations. An attractive method based on the state surface approach was also developed by Thomas and He (1997), who in addition accounted for the effects of temperature change and water vapour movement. Moreover, Wong et al. (1998) presented a theoretical study on coupled consolidation in unsaturated soils which is based on the three-dimensional formulation proposed by Dakshanamurthy et al. Nevertheless, Wong et al. assumed that pore-air pres- sure is atmospheric and remains unchanged during consoli- dation. As a consequence, the air continuity equation was not considered in the analysis and the solution was obtained by solving simultaneously the equilibrium equation and the water continuity equation via the finite element method. In this paper, following the formulation developed by Dakshanamurthy et al. (1984), a solution is presented for coupled consolidation in unsaturated soils when both the air phase and the water phase are continuous. A numerical pro- cedure is proposed that allows the solution to a coupled con- solidation problem to be obtained using the finite element method without demanding great computational efforts. The procedure makes use of the Fourier transform to formally re- move the dependence of the field variables on the spatial co- ordinates in the horizontal plane. As a result, the soil system can be discretized using simple one-dimensional elements. A similar methodology was previously developed by Booker and Small (1982) to analyse consolidation in saturated soils. Although only soil systems under plane-strain conditions are considered in this paper, the procedure could be extended to three-dimensional problems as well. The method is also used to solve the equations governing the uncoupled consolida- tion in unsaturated soils owing to the application of an exter- nal load. Results are presented to ascertain whether the uncoupled approach is able to provide suitably accurate re- sults, despite its greater simplicity compared with the cou- pled solution. Moreover, some interesting aspects of the consolidation processes in unsaturated soils are pointed out. Coupled consolidation for unsaturated soils When air and water phases are both continuous, the gov- erning equations for the coupled consolidation in unsatu- rated soils are the equilibrium equation and the continuity equations for the two fluid phases. In the present section, these equations are derived on the uploads/Voyage/ consolidation-analysis-for-unsat.pdf