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Analytical solutions and design curves for vacuum-assisted consolidation with b

Analytical solutions and design curves for vacuum-assisted consolidation with both vertical and horizontal drainage Cholachat Rujikiatkamjorn and Buddhima Indraratna Abstract: A system of vertical drains combined with vacuum preloading is an effective method for promoting radial flow to accelerate soil consolidation. This study presents the analytical modeling of the consolidation of vertical drains incorporating vacuum preloading considering both vertical and horizontal drainage. The effects of a number of dimensionless parameters involving the drain length, soil permeability, and vacuum pressure are examined through aver- age excess pore pressure, degree of consolidation, associated settlement, and time factor analyses. An analysis of se- lected case histories compliments the use of the proposed solutions. Design charts are also presented for practical use. Key words: analytical solution, consolidation, design charts, vertical drains. Résumé : Un système de drains verticaux combinés avec un préchargement par le vide est une méthode efficace pour favoriser l’écoulement radial dans le but d’accélérer la consolidation du sol. Cette étude présente la modélisation analy- tique d’une consolidation de drains verticaux incorporant un préchargement par le vide et prenant en compte le drai- nage tant horizontal que vertical. Les effets d’un certain nombre de paramètres dimensionnels impliquant la longueur du drain, la perméabilité du sol et la pression de vide sont examinés à la lumière des analyses de l’excédent moyen de pression interstitielle, du degré de consolidation, du tassement associé et du facteur temps. Une analyse d’histoires de cas sélectionnés complète l’utilisation des solutions proposées. On présente aussi des graphiques de conception pour utilisation pratique. Mots-clés : solution analytique, consoliidation, graphiques de conception, drains verticaux. [Traduit par la Rédaction] Rujikiatkamjorn and Indraratna 200 Introduction The vacuum preloading method was first introduced by Kjellman (1952) to improve the strength of soft soil. An in- crease in the effective stress in a soil mass with this method is attributed to applying a vacuum pressure in lieu of a con- ventional surcharge (Qian et al. 1992). This system has been used to achieve a rapid consolidation and reduce the height of surcharge fill by vacuum pressure acting as an additional surcharge load. The advantages of vacuum preloading com- pared to conventional preloading are summarized as follows: (i) the effective stress related to suction pressure increases equiaxially, and the corresponding lateral movement is com- pressive, and consequently the risk of shear failure can be minimized even at a higher rate of embankment construction (Qian et al. 1992); (ii) depending on the vacuum efficiency (e.g., extent of air leaks in the field), the height of surcharge fill can be decreased to achieve the same amount of settle- ment; (iii) at any given time, the maximum excess pore pres- sure prevailing under a vacuum preloading system is less than that from a conventional surcharge; and (iv) with a vac- uum pressure applied through prefabricated vertical drains, the risk of unsaturation at the soil–drain interface due to mandrel withdrawal may be reduced (Indraratna et al. 2004). Prefabricated vertical drains (PVDs) can affect the distri- bution of vacuum pressure to deep subsoil layers and thereby increase the consolidation rate (Holtan 1965; Chu et al. 2000). The effectiveness of vacuum consolidation via PVDs for ground improvement has been verified through various field trials (Choa 1989; Shinsha et al. 1991; Indraratna et al. 2004). In the case of hydraulic fill used in land reclamation projects where the height of surcharge is restricted owing to the low shear strength of soft soil, vacuum-assisted consolidation is an ideal method for ground improvement (Yan and Chu 2003; Song and Kim 2004). The effectiveness of this system depends, however, on (i) the in- tegrity (airtightness) of the membrane, (ii) the effectiveness of the seal between the edges of the membrane and the ground surface, and (iii) soil conditions and the location of the groundwater level (Cognon et al. 1994). The theory of radial drainage and consolidation was ini- tially presented by Carrillo (1942) and Barron (1948). Sub- sequently, Yoshikuni and Nakanodo (1974) proposed a rigorous solution that included well resistance. Hansbo (1981) and Onoue (1988) extended these solutions to take the smear effect into account. In terms of vacuum pre- loading, a rigorous solution for vertical consolidation was Can. Geotech. J. 44: 188–200 (2007) doi:10.1139/T06-111 © 2007 NRC Canada 188 Received 27 June 2005. Accepted 10 October 2006. Published on the NRC Research Press Web site at http://cgj.nrc.ca on 12 March 2007. C. Rujikiatkamjorn and B. Indraratna.1 School of Civil Engineering, Faculty of Engineering, University of Wollongong, Wollongong, NSW 2522, Australia. 1Corresponding author (e-mail: indra@uow.edu.au). proposed by Mohamedelhassan and Shang (2002), and a so- lution for radial consolidation was introduced by Indraratna et al. (2005). To the authors’ knowledge, however, no com- prehensive solution to vacuum-assisted consolidation with both vertical and horizontal drainage including the smear ef- fect is available in the literature. In this paper, the authors present mathematical solutions to the aforementioned problem. The effects of the length of vertical drain, anisotropic soil permeability, and vacuum pressure are considered, and a reduction in consolidation time through vacuum preloading is compared with that through other available methods. Design charts eliminating cumbersome iteration procedures are then developed using the equivalent drain diameter as an independent variable to obtain the relevant drain spacing. Basic equations and solutions To obtain the governing equation for the consolidation of soil with vertical drains, the following assumptions are made: (i) Darcy’s law is valid; (ii) the soil is fully saturated; (iii) water and soil are incompressible; (iv) strains are small; (v) all vertical loads are initially carried by the excess pore pressure, u0; (vi) all compressive strains within the soil mass are assumed to occur vertically, and shear strains are ne- glected because the unit cell is laterally confined and hori- zontal sections remain horizontal during consolidation; and (vii) the coefficients of compressibility and permeability are constant. The schematic representation of the problem under con- sideration where a vertical drain is surrounded by a smear zone is shown in Fig. 1. The basic partial differential equa- tion for excess pore-water pressure by vertical and radial drainage is as follows: [1] ∂ ∂ = ∂ ∂ + ∂ ∂       u r z t t c u r z t r r u r z t r ( , , ) ( , , ) ( , , ) h 2 2 1 + ∂ ∂ c u r z t z v 2 2 ( , , ) where u(r, z, t) is the excess pore pressure, t is the elapsed time, r is the radial distance from the centre of the drain well, z is the vertical distance from the soil surface, ch is the coefficient of consolidation for radial consolidation, and cv is the coefficient of consolidation for vertical consolidation (Barron 1948). The boundary and initial conditions for eq. [1] are as fol- lows: [2] u r t p ( , , ) 0 0 = − for t > 0 (constant vacuum pressure, –p0 at the soil surface) [3] ∂ ∂ = = u r r re 0 for t > 0 (impervious boundary at r = re) [4] u r z t p ( , , ) w = −0 for t > 0 (constant vacuum pressure, –p0 along the drain boundary; however, vacuum pressure at the drain boundary may vary during the consolidation process) [5] ∂ ∂ = = u z z l 0 for t > 0 (impervious boundary at the bottom of the soil layer) and [6] u r z u ( , , ) 0 0 = for t = 0 (initial excess pore pressure due to surcharge load) where re is the radius of the soil cylinder dewatered by a drain; p0 is the applied vacuum pressure at the top soil sur- face and along the drain; l is the soil thickness, which equals the length of vertical drains; u0 is the initial excess pore pressure; and rw is the equivalent radius of the drain, where rw = (a + b)/4, in which a and b are the width and thickness of PVD, respectively (Rixner et al. 1986). Based on the method of separation of variables (Kreyszig 1999), it is appropriate to assume that [7] u r z t u r z u r z u r z u r t u r ( , , ) ( , , ) ( , , ) ( , , ) ( , ) ( , ) − ∞ − ∞ = − ∞ 0 h h u r u r h h ( , ) ( , ) 0 − ∞       × − ∞ − ∞       u z t u z u z u z v v v v ( , ) ( , ) ( , ) ( , ) 0 where uh(r, t) is the excess pore pressure for pure radial con- solidation, and uv(z, t) is the excess pore pressure for pure vertical uploads/Management/ analytical-solutions-and-design-curves-for.pdf