Effect of the Smear and Transition Zones around Prefabricated Vertical Drains Installed in a Triangular Pattern on the Rate of Soil Consolidation D. Basu, S.M.ASCE1; and M. Prezzi, M.ASCE2 Abstract: Soil disturbance caused by the installation of prefabricated vertical drains 共PVDs兲 has a large impact on the rate of consolidation. It is essential in design to quantify this effect. In this paper, we investigate the consolidation rate of soils with PVDs installed in a triangular pattern using two-dimensional finite-element analysis with full consideration of disturbance effects. This is done by accounting for the transition zone that exists between the highly disturbed smear zone and the undisturbed zone. The hydraulic conductivity in the transition zone is assumed to increase linearly from a low value in the smear zone to the original in situ value in the undisturbed zone. The actual band shape of the PVD and hexagonal zone of influence around it are used in the analysis. In addition to soil disturbance effects, the influence on consolidation rate of the size of the smear and the transition zones, the PVD spacing, and the mandrel size and shape is also investigated. Guidelines are given for using an equivalent system, where the transition zone is replaced by an expanded smear zone producing the same effect. This equivalent-system approach allows use of existing analytical solutions that consider only the smear zone in analysis and design. DOI: 10.1061/共ASCE兲1532-3641共2007兲7:1共34兲 CE Database subject headings: Drains; Soil consolidation; Ground stabilization; Finite element method. Introduction Thick deposits of soft, saturated clays have low shear strength, high compressibility, and low hydraulic conductivity. Consequently, there is often the need to use techniques such as preloading to increase their strength and stiffness. Preloading is often combined with the installation of vertical drains to speed up the consolidation process and hence increase the strength gain rate 共Holtz 1987; Bergado et al. 1993a兲. Currently, band-shaped prefabricated vertical drains 共PVDs兲 are frequently used in practice. The PVDs are installed at regular intervals in a square, rectangular, or triangular pattern 共Bergado et al. 1996兲, with center to center distance varying from about 1.0 to 3.5 m 共Holtz 1987兲. A typical PVD cross section consists of a plastic core which is wrapped around by a filter sleeve, and has dimensions of 100 mm⫻ 4 mm 共Holtz 1987兲. The excess pore pressure generated due to preloading quickly dissipates due to the easy outflow of water through the PVD. Notwithstanding the successful use of PVDs, there are certain operational problems associated with them 共Akagi 1994; Holtz 1987兲. PVDs are installed using closed-ended mandrels. The in1 Doctoral Candidate, School of Civil Engineering, Purdue Univ., 550 Stadium Mall Dr., West Lafayette, IN 47907. E-mail: dbasu@purdue.edu 2 Assistant Professor, School of Civil Engineering, Purdue Univ., 550 Stadium Mall Dr., West Lafayette, IN 47907. E-mail: mprezzi@ purdue.edu Note. Discussion open until July 1, 2007. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on May 18, 2005; approved on December 16, 2005. This paper is part of the International Journal of Geomechanics, Vol. 7, No. 1, February 1, 2007. ©ASCE, ISSN 1532-3641/2007/1-34–43/$25.00. stallation of the PVDs significantly disturbs the surrounding soil. There is an increase in pore pressure, decrease in strength and water content in the disturbed soil 共Holtz and Holm 1973兲. Also, the hydraulic conductivity of the disturbed soil is reduced below its initial value before the PVD installation 共Holtz and Holm 1973兲. Consequently, the flow of water into the PVD is slowed down and the consolidation process is significantly delayed. The disturbed zone around the PVD consists of basically two zones: the smear zone and the transition zone 共Bergado et al. 1996; Madhav et al. 1993兲. The smear zone is the completely remolded zone of soil immediately adjacent to the drain. The transition zone, which separates the smear zone from the undisturbed zone 共Onoue et al. 1991; Madhav et al. 1993; Indraratna and Redana 1998; Sharma and Xiao 2000兲, is the zone in which there is a gradual transition of soil properties, with the degree of disturbance decreasing with increasing distance from the drain 共Holtz and Holm 1973兲. Many researchers have investigated the size of the smear zone and the degree of disturbance in it. Studies by Holtz and Holm 共1973兲, Jamiolkowski et al. 共1983兲, Hansbo 共1986, 1987, 1997兲, Bergado et al. 共1991, 1993b兲, Onoue et al. 共1991兲, Holtz et al. 共1991兲, Mesri et al. 共1994兲, Madhav et al. 共1993兲, and Chai and Miura 共1999兲 suggest that the size of the smear zone 共i.e., the distance from the center of a drain to the outer boundary of the smear zone兲 varies between 1 and 4 times the equivalent mandrel radius rm,eq 共the mandrel cross section, if not circular in shape, is converted to an equivalent circle with radius rm,eq that has the same area as that of the actual mandrel cross section兲. Other studies 共by Hansbo 1987; Chai and Miura 1999; Hird and Moseley 2000兲 suggested that the size of the smear zone is in a range from 2 to 3 times rm,eq. These values are based mostly on field investigations, back calculations from case histories, theoretical considerations, and experience 共Hird and Moseley 2000兲. It is important to note that the values proposed by some of the researchers 34 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JANUARY/FEBRUARY 2007 共Jamiolkowski et al. 1983; Bergado et al. 1991, 1993b兲 already incorporate the effect of the transition zone. The degree of disturbance can be expressed in terms of the ratio khs / kho of the horizontal hydraulic conductivity of the smear zone to that of the undisturbed zone. Bergado et al. 共1993a,b兲, Hansbo 共1986, 1997兲, Madhav et al. 共1993兲, and Hird and Moseley 共2000兲 found, based on field data, laboratory tests, model studies, and practical considerations, that khs / kho varies between 0.1 and 0.33. Casagrande and Poulos 共1969兲 suggested a lower value 共0.001兲 for khs / kho based on analyses of case histories, whereas Bergado et al. 共1991兲 and Onoue et al. 共1991兲 suggested higher values 共0.5–0.66兲 based on laboratory tests. The existence of the transition zone was experimentally confirmed by Onoue et al. 共1991兲. They found that the outer boundary of the transition zone, measured from the center of the drain, lies at a distance of about 6 to 7 times the equivalent mandrel radius, and the hydraulic conductivity within the transition zone increases approximately linearly 共from khs to kho兲 as the distance from the drain increases. The approximate linear variation of hydraulic conductivity was also observed in the laboratory experiments by Indraratna and Redana 共1998兲 and Sharma and Xiao 共2000兲. The outer boundary of the transition zone, measured from the center of the drain, extended to a distance of more than 7 times the equivalent mandrel radius in the experiments of Indraratna and Redana 共1998兲, whereas, in the experiments by Sharma and Xiao 共2000兲, it extended to about 10–15 times the equivalent mandrel radius. Madhav et al. 共1993兲 performed laboratory tests on samples collected from a site where PVDs had been installed with a square mandrel and found that the width of the transition zone 共twice the distance from the center of the drain to the outer boundary of the mandrel兲 is about 12 times the width of the mandrel. They further observed that, within the transition zone, the average hydraulic conductivity values had an approximate linear relationship with distance from the center of the drain, even if large scatter was observed. A similar, approximately linear variation was assumed by Miura et al. 共1993兲 as well. Based on studies on pile driving in clay, Jamiolkowski et al. 共1983兲 suggested that the transition zone diameter can be as large as 20 times the equivalent mandrel diameter. Most analyses of vertical drains, accounting for soil disturbance, consider only the smear zone. Barron 共1948兲 developed closed-form solutions for the rate of consolidation considering only radial flow with and without a smear zone around the drain. A simplified analytical solution was later obtained by Hansbo 共1981兲 for PVDs with a smear zone. The solutions of Barron 共1948兲 and Hansbo 共1981兲 assume a circular drain and a circular zone of influence around the drain. Leo 共2004兲 developed an analytical solution by considering vertical flow in addition to radial flow. He observed that the vertical flow has a negligible contribution to the consolidation rate. Analyses using finite difference and finite-element methods accounting for the smear zone have also been performed 共Madhav et al. 1993; Bergado et al. 1993b; Basu and Madhav 2000; Indraratna and Redana 1997兲. The transition zone has been taken into account in a limited number of studies, which include the numerical studies of Madhav et al. 共1993兲 and Hawlader et al. 共2002兲 and an analytical solution by Chai et al. 共1997兲. In this paper, we investigate the consolidation rate of soils with PVDs using two-dimensional finite-element 共FE兲 analysis with full consideration of disturbance effects. The Terzaghi-Rendulic theory of consolidation 共Terzaghi 1925; Rendulic 1935, 1936兲 is used. In order to study the effect of soil disturbance on PVD performance, both the smear and transition zones are considered Fig. 1. Hexagonal unit cell with rectangular disturbed zone in the analysis. It is assumed that PVDs with cross-section dimensions of 100 mm⫻ 4 mm are installed in a triangular pattern with a center-to-center spacing equal to s. The resulting area of influence of a PVD 共hereafter referred to as “unit cell”兲 is a regular hexagon with the length of each side equal to s / 冑3 共Fig. 1兲. The smear and transition zones surrounding the PVD are assumed to have rectangular shape. The actual hexagonal shape of the unit cell, the band shape of the PVD, and the rectangular shape of the smear and transition zones are used in the analysis. A method of replacing the transition zone by an equivalent expanded smear zone is outlined so that existing analytical solutions considering only a smear zone can be used in design. Characterization of Disturbed Zone Smear and Transition Zones Since the mandrel displaces and drags down the surrounding soil during PVD installation, the shape and size of the smear and transition zones in plan depend primarily on the shape and size of the mandrel cross section. Accordingly, the values for the size of the smear and transition zones proposed in the literature are mostly related to the mandrel size. However, other factors like mandrel driving speed, mandrel end shoe, and soil properties also affect the size of the disturbed zone 共Holtz et al. 1991; Hird and Moseley 2000兲. Since the values proposed in the literature are mostly based on field studies and analysis of case histories, they incorporate the effects of all these factors. In this paper, the smear and transition zone dimensions are chosen based on the values proposed in the literature. Four rectangular mandrels 共125⫻ 50, 150⫻ 50, 120⫻ 120, and 150⫻ 150, all in millimeters兲 are considered 共Bergado et al. 1993b; Madhav et al. 1993; Saye 2003; American Wick Drain Corporation 2004兲. The mandrels with rectangular cross section are likely to create smear and transition zones that have rectangular or nearly rectangular 共e.g., elliptical兲 shape in plan 共Chai and Miura 1999兲. In this analysis, the smear and transition zones are assumed to be rectangular with dimensions lx ⫻ ly and tx ⫻ ty, respectively 共Figs. 1 and 2兲. When a rectangular mandrel is pushed into soil, the conditions along the larger side of the mandrel 共where the soil was displaced by a value equal to the width d of the mandrel兲 are quite uniform, INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JANUARY/FEBRUARY 2007 / 35 kht共x兲 = khs + 2x − lx 共kho − khs兲 tx − lx 共lx/2 艋 x 艋 tx/2兲 共3兲 A similar expression for the variation of kht in the y direction can be obtained by replacing y for x, ly for lx, and ty for tx. Analysis Consolidation is a time-dependent process that is generally expressed in terms of the time factor T, which is time normalized with respect to the dimensions of the soil volume to be drained and its coefficient of consolidation T= Fig. 2. Smear and transition zone dimensions in terms of mandrel size with deviations in the state of strain around the mandrel appearing only near the corners with the smaller side. Accordingly, the size of the smear zone must depend to a large extent on the smaller dimension 共the width d of the mandrel兲 of the rectangular mandrel and not on its equivalent diameter dm,eq. Assuming that the thickness of the smear zone surrounding the mandrel remains constant along the entire mandrel perimeter 共Fig. 2兲, the dimensions lx ⫻ ly of the smear zone can be obtained from ly = pd 共1兲 lx = a + 共p − 1兲d 共2兲 where p = parameter such that 2 艋 p 艋 3 and a and d = dimensions of the mandrel cross section with a ⬎ d. The transition zone dimensions tx ⫻ ty can likewise be obtained from Eqs. 共1兲 and 共2兲, where ly and lx are replaced by ty and tx, respectively, with p ranging between 6 and 12. The choice of p values for the smear and transition zones is consistent with the values found in the literature. Since the mechanisms determining the shape and size of the disturbed zone are not fully understood, simulations were performed to confirm that the assumption of a rectangular disturbed zone was acceptable. The consolidation rate was studied by varying lx for fixed values of ly. It was observed that, for a given ly, the rate of consolidation does not vary much with lx. It is the thickness ly and not the overall shape of the rectangle that controls the consolidation. Further, this suggests that as long as the thickness ly remains more or less constant, the exact shape of the smear zone 共whether it is rectangular or slightly elliptical兲 will not affect the consolidation rate to a large extent. This observation is consistent with the FE analysis of Chai and Miura 共1999兲, according to which the difference in the consolidation rate due to the assumption of a circular smear zone and a rectangular smear zone with the same area is negligible. 共4兲 where dc,eq = diameter of a circle with the same area as the unit cell; it is a representative drainage path length. dc,eq for the case of a hexagonal domain is given by dc,eq = 冑冑 2 3 s ␲ 共5兲 The degree of consolidation U at any particular time 共or time factor兲 is the ratio of the average excess pore-water pressure already dissipated to its initial value. Mathematically, for twodimensional consolidation, U can be expressed in terms of integrals of the pore-pressure over the unit cell domain as 共Madhav et al. 1993兲 冕冕 冕冕 u共x,y,T兲dxdy U=1− y x y 共6兲 uinidxdy x where u共x , y , T兲 = excess pore pressure at any point with coordinates 共x , y兲 at a time factor T, and uini is the initial excess pore pressure. The integrals in Eq. 共6兲 are evaluated by a Gaussquadrature integration scheme after obtaining the values of pore pressure u at the Gauss points within each element of the finiteelement mesh 共Cook et al. 2002兲. The change in u with time 共or time factor兲 over the entire domain needs to be known. This is described by the TerzaghiRendulic differential equation for two-dimensional consolidation as 冉 ⳵u ⳵ 2u ⳵ 2u + = d2c,eq ⳵T ⳵x2 ⳵ y 2 冊 共7兲 Eq. 共7兲 is valid for a homogeneous and isotropic soil. However, when the soil is disturbed, the homogeneity of soil properties within the unit cell is lost. In such a situation Eq. 共7兲 is modified as 冉 ⳵u khd ⳵2u ⳵2u + = d2c,eq ⳵T kho ⳵x2 ⳵y 2 Hydraulic Conductivity The hydraulic conductivity in the smear and the undisturbed zones are assumed to be constants with values khs and kho, respectively, whereas it increases linearly in the transition zone from khs to kho as the distance from the PVD surface increases 共Fig. 1兲. At any distance within the transition zone, the hydraulic conductivity in the x direction is given by c ht d2c,eq 冊 共8兲 where khd = hydraulic conductivity within any element lying in the disturbed zone. It is assumed that soil disturbance only affects the hydraulic conductivity and not the compressibility of the soil 共Hansbo 1981; Madhav et al. 1993兲. If the element lies within the smear zone, then khd is equal to khs. On the other hand, if the element lies in the transition zone, then khd is equal to kht. 36 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JANUARY/FEBRUARY 2007 As a symmetric flow pattern exists with respect to the coordinate axes 共Fig. 1兲, only the domain within the first quadrant 共within which x 艌 0 and y 艌 0兲 is analyzed. The PVD thickness of 4 mm is neglected. Initial and boundary conditions are needed for the solution of Eqs. 共7兲 and 共8兲. For the boundary that represents the interface between the PVD and soil 共referred to as drain boundary hereafter兲, a value of u = 0 is prescribed 共Dirichlet boundary condition兲 because free flow occurs across it. For the remaining boundaries, the Neumann boundary condition ⳵u / ⳵n = 0 is prescribed because no flow is possible across them 共n denotes the direction of the unit normal vector at the boundary surface兲. The initial condition imposed is uini = u共x , y , 0兲 = 100 within the domain and at all the boundaries except at the drain boundary 共0 艋 x 艋 50 mm, y = 0兲, where the excess pore pressure dissipates instantaneously. Since drainage occurs immediately at the drain boundary, the initial condition prescribed there is uini = 0. Eqs. 共7兲 and 共8兲 were adopted for the FE analysis. The domain was discretized using three-noded triangular elements. The discretized versions of Eqs. 共7兲 and 共8兲 were solved following an implicit 共backward difference兲 numerical time integration scheme. A detailed description of the FE derivation is given in Basu et al. 共2005兲. The FE solution was obtained by writing a FORTRAN program. In order to check convergence, the element size was varied until two successive meshes produced identical results 共difference less than 0.01%兲. For the final accepted mesh, the length of the elements adjacent to the PVD was 25 mm, increasing gradually to up to a maximum of 110 mm at the cell boundaries. The time step ⌬T was taken equal to 0.001T for T 艋 0.1 and equal to 0.01T for T ⬎ 0.1. The accuracy of the FE analysis was checked against the analytical solutions of Barron 共1948兲 and Hansbo 共1981兲. According to Barron 共1948兲, the degree of consolidation U for “equal-strain” consolidation is given by 冉 冊 U = 1 − exp − 8T ␮ 共9兲 with ␮= 冉 冊 3n2 − 1 n2 ln n − n2 − 1 4n2 共10兲 and n= dc dw 共11兲 where dc and dw = circular unit cell and drain diameters, respectively. Hansbo 共1981兲 introduced a circular smear zone 共of diameter ds兲 in the solution which resulted in a modified expression for ␮ 冉冊 ␮ = ln 3 n kho ln m − + 4 m khs 共12兲 with m= ds dw 共13兲 For comparison purposes, the area of the unit cell was converted to an equivalent circle of diameter dc,eq 关Eq. 共5兲兴, which was then used in the analytical solution as the diameter dc. The equivalent diameter dw,eq of the band-shaped PVD is obtained by equating the perimeter of the PVD to that of a circle 共Hansbo 1981兲 dw,eq = 2 共db + dt兲 ␲ 共14兲 where db and dt = PVD width and thickness, respectively. dw,eq is taken equal to the diameter dw of the circular drain of the analytical case. For a 100 mm⫻ 4 mm PVD, dw,eq = 66.2 mm. The rectangular smear zone of dimensions lx ⫻ ly is converted to an equivalent circle with a diameter ds,eq given by ds,eq = 冑 4lxly ␲ 共15兲 ds,eq is used as the diameter ds of the circular smear zone in the analytical solution. Comparisons were made for the cases with and without smear for a spacing s = 1 m and a mandrel of 125⫻ 50 mm. The smear zone was assumed to extend to 2d, where d is the smaller dimension of the mandrel cross section. A value of 0.2 was assumed for the khs / kho ratio. The results of the FE analyses were found to compare well with the analytical solutions 共Basu et al. 2005兲. The maximum difference in U, for U ⬎ 50%, was about 3.3%, occurring at T = 0.3 for the case without smear zone, and, for the case with smear zone, the maximum difference in U was 5%, occurring at T = 0.7. Results Effect of Soil Disturbance In design, it is very important to account for the detrimental effects of soil disturbance on PVD-enhanced consolidation. The effects of the smear and transition zones on the degree of consolidation are investigated for two cases: 共1兲 PVD installed with a mandrel of 125⫻ 50 mm at 1-m spacing 共Combination 1兲, and 共2兲 PVD installed with a mandrel of 150⫻ 150 mm at 3-m spacing 共Combination 2兲. The degree of disturbance, quantified in terms of khs / kho, is maintained constant at a value of 0.2. The smear and transition zones are assumed to extend to 2d 共p = 2兲 and 12d 共p = 12兲, respectively. Fig. 3 shows the U versus T curves for Combination 1. It is clear that soil disturbance has a substantial detrimental effect on the effectiveness of PVDs in accelerating consolidation. For Combination 1 共Fig. 3兲, T required for 90% consolidation is equal to 0.65, 1.76, and 2.35 for the following three conditions: 共1兲 absence of a smear zone; 共2兲 presence of a smear zone only; and 共3兲 presence of both smear and transition zones. Compared to the no-disturbance condition, T 共for U = 90%兲 increases by 171% if only the smear zone is considered, and by 262%, if both the smear and transition zones are considered. The difference in T between the cases with only the smear zone and both the smear and transition zones is 34%. For Combination 2 共not plotted兲, T corresponding to 90% consolidation is equal to 0.95 in the absence of a smear zone, 2.98 in the presence of a smear zone only, and 3.72 in the presence of both smear and transition zones. Again, compared with the no-disturbance condition, the increase in T considering the existence of a smear zone only is 214%; if both the smear and transition zones are considered, T increases by 292%. The increase in T for the case where both a smear and a transition zone are present over the case where only a smear zone is present is 25%. INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JANUARY/FEBRUARY 2007 / 37 Fig. 3. Effect of soil disturbance on consolidation rate Disturbed Zone Dimensions In order to evaluate the extent to which the dimensions of the smear zone affect PVD performance, two dimensions of the smear zone, corresponding to ly equal to 2d and 3d, are compared for a fixed transition zone size ty equal to 12d. The same Combinations 1 and 2 described earlier with khs / kho = 0.2 are considered 关Fig. 4共a兲兴. For Combination 1, T required for 90% consolidation is equal to 2.35 and 2.55 for a smear zone size ly equal to 2d and 3d, respectively; the difference between the two cases being 8.5%. For Combination 2, T required for 90% consolidation is equal to 3.73 and 3.89 for ly equal to 2d and 3d, respectively; the difference between the two cases being 4.3%. Thus, the extent of the smear zone has a moderate impact on the rate of consolidation. As the size ty of the transition zone is likely to be in the 6d to 12d range, it is necessary to investigate how the consolidation rate varies for transition zone sizes in this range. Therefore, values of ty equal to 6d and 12d, with a fixed smear zone size ly equal to 2d, were used in the analysis. As before, the same two combinations, with khs / kho = 0.2, are studied 关Fig. 4共b兲兴. For Combination 1, T values corresponding to 90% consolidation are equal to 2.07 and 2.35 for ty equal to 6d and 12d, respectively; the difference being 13.5%. For Combination 2, T values corresponding to U = 90% are equal to 3.34 and 3.73 for ty equal to 6d and 12d, respectively, corresponding to a difference of 11.7%. Thus, the larger the smear and transition zone dimensions, the slower the consolidation rate is. Degree of Disturbance The degree of soil disturbance is accounted for by the ratio khs / kho. Fig. 5 shows the U versus T curves for khs / kho ranging from 0.05 to 0.5 and PVDs installed with a mandrel of 125⫻ 50 mm at 1-m spacing. The smear and the transition zone dimensions ly and ty are taken as 2d and 12d, respectively. The values of T corresponding to U = 90% are 7.36, 4.13, 2.35, 1.69, and 1.11 for khs / kho equal to 0.05, 0.1, 0.2, 0.3, and 0.5, respectively. For PVDs installed with a mandrel of 150⫻ 150 mm at 3-m spacing 共not plotted兲, the corresponding values of T are 12.43, 6.76, 3.73, 2.62, and 1.7. Interestingly, the variation of T Fig. 4. Effect of disturbed zone dimension on consolidation rate: 共a兲 effect of smear zone dimension; 共b兲 effect of transition zone dimension with khs / kho, for a constant U, follows a power law 共Fig. 6兲. Therefore, for a given value of U, T can be expressed in terms of khs / kho as T = C1 冉 冊 khs kho −C2 共16兲 where C1 and C2 = real positive numbers. Clearly, the degree of disturbance has a significant effect on the consolidation rate. The impact of the degree of disturbance on the consolidation rate is much more pronounced than that of the dimensions of the smear and transition zones. Therefore, the degree of disturbance needs to be predicted with greater accuracy than the extent of the zones of disturbance surrounding the PVD in order to produce an optimal design. PVD Spacing The rate of consolidation decreases with increasing PVD spacing. In order to evaluate the benefit of reducing the PVD spacing, three different spacings 共1, 2, and 3 m兲 are considered for the 125⫻ 50 mm mandrel 共Fig. 7兲. The smear and the transition zone 38 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JANUARY/FEBRUARY 2007 Fig. 7. Effect of PVD spacing on consolidation rate Fig. 5. Effect of degree of disturbance on consolidation rate dimensions ly and ty are fixed at 2d and 12d, respectively, with khs / kho = 0.2. It is found that consolidation occurs at a faster rate for smaller PVD spacing. For U = 90%, T decreases by 4.4% when the spacing is reduced from 3 to 2 m, and by 9%, when it is reduced further from 2 to 1 m. Also plotted in Fig. 7 are the U versus T curves for the corresponding cases where no soil disturbance is considered. Here, a decrease in T of 12.6% 共corresponding to U = 90%兲 occurs when the PVD spacing is reduced from 3 to 2 m; a reduction of 21.7% in T occurs when the spacing is decreased from 2 to 1 m. If no soil disturbance is considered in the calculations, then the estimated increase in the consolidation rate obtained by reducing the PVD spacing is substantially larger than the actual increase that takes place in the presence of the soil disturbance caused by PVD installation. and ty equal to 2d and 12d, respectively, and a ratio khs / kho of 0.2. Four different mandrel sizes: 125⫻ 50, 150⫻ 50, 120⫻ 120 and 150⫻ 150 mm are used in the study. Fig. 8 shows the U versus T curves for the 1-m PVD spacing. The values of T, corresponding to U = 90%, are 2.35, 2.4, 3.22, and 3.23 for the four mandrel sizes mentioned earlier 共in the same order兲. For 3-m spacing 共not plotted兲, the corresponding values of T are 2.7, 2.75, 3.5, and 3.73, respectively. The rate of consolidation decreases with increasing mandrel sizes, although, for practical purposes, the 150⫻ 50 mm mandrel is as effective as the 125⫻ 50 mm mandrel. The same can be stated about the 120⫻ 120 and 150⫻ 150 mm mandrels. However, there is a substantial difference in the consolidation rate when rectangular mandrels and square mandrels are compared. Square mandrels are less effective than rectangular mandrels be- Mandrel Size and Shape The effect of mandrel size is studied for two different PVD spacings 共1 and 3 m兲, with smear and transition zone dimensions ly Fig. 6. Dependence of time factor on the degree of disturbance Fig. 8. Effect of mandrel size on consolidation rate 共the curves for the 125⫻ 25 mm mandrel and the 125⫻ 50 mm mandrel almost coincide; this is also the case for the curves of the 120⫻ 120 mm mandrel and the 150⫻ 150 mm mandrel兲 INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JANUARY/FEBRUARY 2007 / 39 Table 1. Replacement of the Transition Zone by an Expanded Smear Zone khs / kho 0.1 0.2 0.3 Extra length of smear zone per unit length of transition zone 0.13 0.20 0.25 cause they disturb a much larger area. This finding is in agreement with the results of a laboratory study by Hird and Sangtian 共2002兲, which indicates that the smear effect is reduced by replacing circular mandrels with slim rectangular mandrels. Lessons for Design Replacement of Transition Zone by an Equivalent Smear Zone A way of accounting for the transition zone in design is to replace the transition zone and the smear zone with a single equivalent smear zone. In this paper, the extra length of smear zone required to replace the transition zone was determined by studying several combinations of spacings 共1, 2, and 3 m兲 and mandrels 共125⫻ 50, 150⫻ 50, 120⫻ 120, and 150⫻ 150 mm兲 for two different smear zone dimensions 共ly = 2d and 3d兲, three different transition zone dimensions 共ty = 6d, 9d, and 12d兲, and three different values of khs / kho 共0.1, 0.2, and 0.3兲. It was found that the extra length of the smear zone required to replace the transition zone depends only on the khs / kho ratio and the size of the transition zone itself, as shown in Table 1. For example, if the original domain consists of a smear zone with ly = 3d and a transition zone with ty = 9d, then 6d is the length of the transition zone that needs to be replaced. Assuming a khs / kho of 0.3 and referring to Table 1, the extra length of smear zone required is 0.25⫻ 6d = 1.5d. Therefore the equivalent smear zone extends to 3d + 1.5d = 4.5d. As can be seen in Fig. 9, which shows the U versus T curves obtained for the original domain considering both smear and transition zones and the equivalent domain Fig. 9. Replacement of original domain with smear and transition zones by an equivalent domain with an expanded smear zone Fig. 10. U versus T curves for square, rectangular, and equivalent circular smear zones 共Curves 1 and 2 are almost coincident; Curves 3–5 are very close to each other, with Curve 3 lying to the right of Curve 4 and to the left of Curve 5 for T ⬎ 0.5兲 with a smear zone only, this procedure works quite well for all the cases considered. However, it is not applicable when there is overlap of adjacent transition zones. Equivalent Circular Smear Zone The procedure outlined in the previous section facilitates the design process to a great extent as a single equivalent smear zone is used in design. Nonetheless, the rectangular smear zone still needs to be converted to an equivalent circle to allow use of the available analytical solutions 共Hansbo 1981兲. Two methods 共A and B兲 can be used to do this conversion. In Method A, two steps need to be followed: 共1兲 Estimate the dimensions of the rectangular smear zone 共lx ⫻ ly兲 from the mandrel dimensions 共a ⫻ d兲 by using Eqs. 共1兲 and 共2兲. 共2兲 Convert the rectangular area of the smear zone into an equivalent circle; Eq. 共15兲, which gives the equivalent diameter ds,eq of the circle, is used for this purpose. Alternatively, in Method B, the procedure is as follows: 共1兲 Convert the rectangular mandrel with dimensions a ⫻ d to an equivalent circle by using Eq. 共15兲 共with lx and ly replaced by a and d, respectively兲 to obtain the equivalent mandrel diameter dm,eq. 共2兲 Multiply dm,eq by the constant p of Eqs. 共1兲 and 共2兲 to obtain the equivalent smear zone diameter ds,eq. For square mandrels, both methods yield the same smear zone diameter. Several combinations of spacings, square mandrel sizes, smear and transition zone dimensions, and degrees of disturbance were studied. The difference between values of U obtained considering square smear zones and the corresponding equivalent circles was negligible. Fig. 10 shows the results for 3-m spacing with a square smear zone of dimensions 600⫻ 600 mm and 40 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JANUARY/FEBRUARY 2007 khs / kho = 0.2. For this particular case, the maximum difference in the resulting U value for the square and equivalent circular smear zones is less than 1%. This illustrates that for square-shaped smear zones, the error in converting the square disturbance zone to equivalent circle is negligible. For rectangular mandrels, the maximum difference in U values obtained for rectangular smear zones and the corresponding equivalent circles is about 5%. In all cases, the U versus T curves obtained for a rectangular smear zone always lie between the curves obtained for the corresponding equivalent circular smear zones 共with the diameter ds,eq obtained by Methods A and B兲. Method B is recommended for use in design as it always produced conservative results 共for large time factors兲. Fig. 10 shows the U versus T curves for a mandrel size of 125⫻ 50 mm and PVD spacing of 1 m 关with p of Eqs. 共1兲 and 共2兲 taken as 4.5兴. The ratio khs / kho is taken as equal to 0.3. Time Factor In order to facilitate the design procedure further, the constants C1 and C2 of Eq. 共16兲 are obtained for U = 50% and U = 90% and s = 1, 2, and 3 m. Equivalent circular unit cells with equivalent circular smear zones are assumed. These constants are given in Table 2 for different ratios of equivalent circular smear zone diameter to equivalent circular unit cell diameter. For the mandrels considered in this paper, it was found that the equivalent diameter of the expanded smear zone ds,eq can range anywhere from 0.15 to 1.0 times the equivalent diameter of the unit cell dc,eq. Hence, the values of C1 and C2 are obtained for ds,eq / dc,eq = 0.1– 1.0; in intervals of 0.1. Using these values, a designer can directly determine T required for 50% or 90% consolidation for a given PVD spacing, equivalent smear zone dimension, and degree of disturbance. Numerical Example A practical example problem is worked out in this section to facilitate the understanding of the method proposed in this paper. A site with ch = 2 m2 / year is considered, where a 125⫻ 50 mm mandrel is to be used for installation of PVDs in a triangular pattern. The PVD cross-section dimensions are 100 mm⫻ 4 mm 共dw,eq = 66.2 mm兲. The degree of disturbance given by khs / kho is assumed to be equal to 0.2. The smear and transition zone dimensions ly and ty are assumed to be equal to 2d and 12d, respectively. Thus, a transition zone of length 12d − 2d = 10d is to be replaced by an extra length of smear zone. Using Table 1, for khs / kho = 0.2, the additional length of smear zone required is 0.2⫻ 10d = 2d. Therefore, the equivalent smear zone extends to 4d 共2d + 2d兲. For estimating the equivalent circular smear zone diameter, Method B, as described in the previous section, is used. The diameter dm,eq of the equivalent circular mandrel is 89.2 mm 关Eq. 共15兲兴. This means that the diameter ds,eq of the equivalent smear zone is 4 ⫻ 89.2= 356.8 mm and m = 5.39 关Eq. 共13兲兴. For a PVD spacing of 1 m, the diameter dc,eq of the equivalent circular unit cell is 1.05 m 关Eq. 共5兲兴, which results in n = 15.86 关Eq. 共11兲兴. This yields ␮ = 8.75 关Eq. 共12兲兴. Assuming that 90% consolidation is to be achieved, T required is 2.52 关Eq. 共9兲兴. Alternatively, T 共for U = 90%兲 can be obtained from Table 2. For khs / kho = 0.2, s = 1 m 共i.e., dc,eq = 1.05 m兲 and ds,eq / dc,eq = 0.34, values of C1 and C2 are obtained by linear interpolation between ds,eq / dc,eq = 0.3 and 0.4 共C1 = 0.5383 and C2 = 0.9611兲. This yields T = 0.5383共0.2兲−0.9611 = 2.53 关Eq. 共16兲兴, which corresponds to a time equal to 1.4 years 共ch = 2 m2 / year and dc,eq = 1.05 m; 关Eq. 共4兲兴兲. Table 2. Values of the Constants C1 and C2 of Eq. 共16兲 for U = 50% and U = 90% and Various Values of PVD Spacing and ds,eq / dc,eq Spacing 共m兲 C1 ds,eq / dc,eqa 1 U = 50% 0.1 0.1290 0.2 0.1442 0.3 0.1579 0.4 0.1683 0.5 0.1766 0.6 0.1835 0.7 0.1895 0.8 0.1946 0.9 0.1992 1.0 0.2033 2 0.1 0.1823 0.2 0.2035 0.3 0.2178 0.4 0.2283 0.5 0.2367 0.6 0.2436 0.7 0.2495 0.8 0.2546 0.9 0.2591 1.0 0.2632 3 0.1 0.2160 0.2 0.2385 0.3 0.2529 0.4 0.2635 0.5 0.2718 0.6 0.2787 0.7 0.2846 0.8 0.2897 0.9 0.2942 1.0 0.2983 a Ratio of the equivalent smear zone diameter to diameter. U = 90% C2 0.4287 0.6294 0.4790 0.8765 0.5244 0.9467 0.5591 0.9827 0.5867 1.0054 0.6097 1.0215 0.6294 1.0335 0.6465 1.0431 0.6617 1.0509 0.6754 1.0574 0.6056 0.8009 0.6761 0.9181 0.7235 0.9623 0.7585 0.9873 0.7862 1.0041 0.8092 1.0163 0.8287 1.0258 0.8457 1.0334 0.8609 1.0398 0.8744 1.0452 0.7176 0.8429 0.7921 0.9316 0.8401 0.9678 0.8752 0.9891 0.9029 1.0035 0.9258 1.0143 0.9453 1.0227 0.9623 1.0296 0.9774 1.0353 0.9909 1.0402 the equivalent unit cell In order to investigate how much the PVD spacing affects the rate of consolidation in the presence of soil disturbance, the same example is solved for different PVD spacings, as shown in Table 3. It is found that, although the values of the time factor T corresponding to U = 90% for spacings of, say, 3 and 1 m do not differ much, the actual times do differ substantially because time depends not only on T but also on the square of the equivalent cell diameter. Thus, there is a significant benefit from installing PVDs at closer spacing if time is of the essence. Conclusions The paper examined the effect of soil disturbance on the rate of consolidation of a soil deposit engineered with PVDs. A standard PVD cross section of 100 mm⫻ 4 mm and a triangular PVD installation pattern were considered throughout. The analysis was performed using finite elements following the Terzaghi-Rendulic theory of consolidation. PVD installation creates two distinct zones of disturbance: 共1兲 a completely remolded smear zone and 共2兲 a less disturbed transition zone. The hydraulic conductivity was assumed to increase linearly in the transition zone from a low INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / JANUARY/FEBRUARY 2007 / 41 Table 3. Example Showing the Effect of PVD Spacing Spacing 共m兲 0.9 1.0 1.5 2.0 3.0 dc,eq 共mm兲 dw,eq 共mm兲 ds,eq 共mm兲 n m khs / kho T t 共years兲 945.1 1,050.1 1,575.1 2,100.2 3,150.2 66.2 66.2 66.2 66.2 66.2 356.8 356.8 356.8 356.8 356.8 14.28 15.86 23.79 31.73 47.59 5.39 5.39 5.39 5.39 5.39 0.2 0.2 0.2 0.2 0.2 2.49 2.52 2.64 2.72 2.84 1.1 1.4 3.3 6.0 14.1 value in the smear zone to the original in situ value in the undisturbed zone. The actual hexagonal shape of the unit cell, band shape of the drain, and rectangular shape of the smear and transition zones were used in the analysis. It was observed that soil disturbance reduces significantly the PVD consolidation rate, and the transition zone has a definite impact on the consolidation process. Larger smear and transition zones result in slower consolidation rates. However, it is the degree of disturbance, quantified by the decrease in the hydraulic conductivity, that affects the process the most. 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