A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations Accounting for Smear and Well Resistance Effects

Transcription

1 The International Journal of Geomechanics Volume, Number, (00) A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations Accounting for Smear and Well Resistance Effects J. Han and S.L. Ye Department of Civil Engineering, Widener University, One University Place, PA Department of Geotechnical Engineering, Tongji University, 139 Siping Road, Shanghai 0009, P.R. China ABSTRACT. Theoretical and experimental studies have proven that stone columns can be used for accelerating the consolidation rate of soft soil by providing a drainage path and reducing stresses in the soil. In constructing stone columns in fine-grained soils, however, soil zones at the interface between the columns and their surrounding soil can become smeared and the fine-grained soil particles can also be mixed into aggregates in the columns. The smear and well resistance due to aggregates contaminated with the fine-grained soil particles reduce the effectiveness of stone columns in dissipating excess pore water pressures. A theoretical solution is developed in this article for computing the consolidation rates of stone column reinforced foundations accounting for smear and well resistance effects. In the derivations, stone columns and soft soil are both considered deforming one-dimensionally and the stone columns having a higher drained elastic modulus than the surrounding soft soil. A modified coefficient of consolidation is introduced to account for the effect of the stone column-soil modular ratio or stress concentration ratio. A parametric study investigates the influences of six important factors on the rate of consolidation. These influence factors include the diameter ratio of the influence zone to the stone column, the permeability of the stone column, the stress concentration ratio, the size of the smeared zone, the permeability of the smeared zone, and the thickness of the soft soil. To assist geotechnical engineers in utilizing the new solution for the design of stone column reinforced foundations, an illustrative design example is presented at the end of this article. I. Introduction Stone columns, one of the most commonly used soil improvement techniques, have been utilized worldwide to increase bearing capacities and reduce total and differential settlements of Key Words and Phrases. stone column, soft clay, consolidation, stress concentration, pore water pressure, smear, well resistance. 003 ASCE DOI: /(ASCE) (00):(135) ISSN

2 136 J. Han and S.L. Ye superstructures constructed on soft soils. Field observations showed that stone columns could also accelerate the rate of consolidation of soft soils [1, ]. Field pore water pressure measurement under an embankment indicated that a homogeneous clay stratum outside a stone column treated area only completed 5% primary consolidation when the stone column area had reached 100% primary consolidation [1]. Han and Ye [] also reported that the rates of settlement of two similar buildings, one on an unreinforced foundation, and the other on a stone column reinforced foundation on the same site, reached 66% and 95%, respectively, over the same time period (480 days). The acceleration of the rate of consolidation was accredited to stone columns for providing a drainage path and relieving excess pore water pressures by transferring loads from soil to columns. A numerical study demonstrated that an increase of the stone column-soil modular ratio can increase the rate of consolidation of soft clays under a rigid raft but not under a flexible raft [3]. Han and Ye [4] developed a simplified and closed-form solution for estimating the rate of consolidation of stone column reinforced foundations accounting for the stone columnsoil modular ratio with reasonable accuracy as compared with the Balaam and Booker results [3]. The authors previous study also found that the stone column-soil modular ratio is equivalent to the steady stress concentration ratio when the consolidation is complete [4]. The stress concentration ratio is defined as the ratio of the stress on the column to that on the soil, which is commonly adopted in practice. The studies by Balaam and Booker [3] and Han and Ye [5] both assumed that stone columns are free-draining and no smear or soil disturbance exists. Seed and Booker [6] concluded that the permeability of a drain must be 00 times greater than that of the surrounding sand to ensure no buildup in excess pore water pressure within the drain under an earthquake event. Barksdale [7] indicated the possibility of a reduction in the permeability of the stone column due to contamination of aggregates with fine-grained soil during the construction. In other words, the drainage capacity of the stone column may be limited by its permeability. The effect of the limited permeability is commonly referred to as well resistance. In addition, a zone of insitu soil can become smeared, disturbed, and intruded by stone from the column. Field observations indicated that the zone of insitu soil intruded by stone at the interface had about to 5 cm thick when a vibroflotation method was adopted [8]. The combined effects of smear, soil disturbance, and stone intrusion are referred by Barksdale [7] to as smear in general. The intensity of well resistance and smear depends on the method used for installing the stone columns. A casing method apparently can minimize the possibility of fine-grained soil mixed into the stone column as compared with a vibroflotation method, in which the stone column can be easily contaminated by the fine-grained soil in the slurry. Cassagrande and Poulos [9] indicated that the horizontal permeability of the soil smeared by a driven casing was about 10 times less than that by a jetting method. To account for smear and well resistance effects, Barksdale and Bachus [10] suggested that a reduced diameter to 1/ 1/15 of the actual diameter of stone columns should be used along with the solution for drain wells. This suggested fractional reduction in diameter was based on back-calculated results from field data. However, this suggestion lacks a theoretical basis and cannot distinguish different (positive or negative) effects from a variety of influence factors, such as smear, well resistance, and stress concentration, etc. In addition, the reduction in diameter has a wide range difficult for a proper selection. The objective of this article is to develop a simplified theoretical solution for computing the consolidation rates of stone column reinforced foundations by considering their high modular ratio or stress concentration characteristics, smear, and well resistance effects, although this problem can also be solved by using Biot s theory and a finite element method. II. Review of solutions for drain wells and stone columns Reginald A. Barron has been credited as one of the earliest researchers in solving the consolidation problem due to radial flow. In his studies Barron [11] developed solutions for computing

3 A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations 137 the consolidation rates of fine-grained soils by drain wells with or without well resistance or smear. Barron s work extended Terzaghi s one-dimensional vertical flow theory to a radial flow problem. Barron s solution without well resistance and smear as presented in equation (1) has been commonly used in practice for designing sand drains and wick drains under an equal strain condition. U r = 1 e 8 F(N) T r (1) where U r the average rate of consolidation due to radial flow; F(N) = N N 1 ln(n) 3N 1 ; 4N N = d e d c, the diameter ratio; T r = c r t, the time factor in a radial flow; de c r - the coefficient of consolidation due to radial flow; d c and d e - the diameters of a drain well and its influence zone, respectively; t - the time period for consolidation. However, Barron s solution for the rate of consolidation with well resistance or smear effects has not been widely adopted due to its complexity. Hansbo [1] proposed a simplified solution in which a modified factor, F m, is used instead of F(N) in equation (1) and the modified factor can be expressed as: F m = ln N S + k r k s ln S πz(h z) k r q c () where S = d s d c, the diameter ratio of the smeared zone to the drain well; d s the diameter of the smeared zone; k r the radial permeability of the undisturbed surrounding soil; k s the radial permeability of the smeared soil; H the longest drainage distance due to vertical flow; z the depth in the ground at which the rate of consolidation is computed; πdc q c = k c 4 the discharge capacity of the drain well; k c the permeability of the drain well. A similar solution as above was obtained by Zeng and Xia [13], in which the average rate of consolidation was computed in terms of the total thickness of the soft soil rather than that at a certain depth in Hansbo s solution. Stone columns and drained wells have two major differences: (1) stone columns have a larger drained elastic modulus than the surrounding soft soil. The typical elastic modulus ratios of stone column to soft clay range from 10 to 0 [10]. As pointed out by Lane [14] in his discussion, Barron s solution ignored the effect of the stiffness difference between the sand well and the surrounding soil on the consolidation rate. This statement is also true for Hansbo [1]

4 138 J. Han and S.L. Ye and Zeng and Xia [13] solutions. () Stone columns have a smaller diameter ratio (influence diameter/column diameter) than drain wells. Typical diameter ratios for stone columns range from 1.5 to 5; however, the values for well diameter ratios used by Barron [11] were from 5 to 100. Considering these characteristics of the stone column-reinforced foundation, Han and Ye [5] developed a simplified and closed-form solution for computing the rate of consolidation of this foundation under the conditions of no well resistance and smear. The proposed solution for the rate of consolidation due to radial flow followed Barron s formula in equation (1) with an exception that a modified time factor should be used instead, i. e., T rm = c rmt d e (3) where T rm the modified time factor due to radial flow; c rm the modified consolidation coefficient due to radial flow; ( ) c rm = k r m v,c (1 a s )+m v,s a s 1 γ w m v,s m v,c (1 a s ) = c r 1 + n s ; N 1 m v,c and m v,s the coefficients of volumetric compressibility of stone columns and the surrounding soil, respectively; γ w the unit weight of water; a s the area replacement ratio, the ratio of the replaced area with a stone column to its total influence area; n s the steady stress concentration ratio, the ratio of the stress on the stone column to that on the surrounding soil at the time primary consolidation is complete. The steady stress concentration ratio was typically measured at the elevation of the top of the stone column in the practical range mostly from to 5 [4]. In the authors previous study, it was found that the steady stress concentration ratio is equal to the ratio of the volumetric compressibility coefficient of the soil to that of the stone column, i. e., n s = m v,s m v,c [5]. Similarly, the proposed solution for the rate of consolidation due to vertical flow followed the Terzaghi one-dimensional solution with the same exception that a modified time factor should be used, i. e., T vm = c vmt H (4) where T vm the modified time factor due to vertical flow; c vm the modified consolidation coefficient due to vertical flow; ( ) c vm = k v m v,c (1 a s )+m v,s a s 1 γ w m v,s m v,c (1 a s ) = c v 1 + n s. N 1 For most cases the rate of consolidation due to vertical flow is negligible when compared with that due to radial flow because the distance for radial flow is much shorter than that for vertical flow. In this study, the theoretical solution is developed specially for the rate of consolidation due to radial flow, in other words no vertical flow is considered in the surrounding soil (including the smeared zone and the undisturbed soil).

5 A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations 139 III. Derivation of the theoretical solution In order to obtain a simplified closed-form solution, the following assumptions are made during the development of the solution: 1. No vertical flow occurs within the surrounding soil (smeared and undisturbed soil);. Each stone column has a circular influence zone. The stone column reinforced foundation is fully saturated and water is incompressible; 3. The stone column and the surrounding soil only deform vertically and have the equal strain at any depth; 4. The coefficients of compressibility of the smeared zone and the undisturbed soil are equal; 5. The load is applied instantly and maintained constant during the consolidation; 6. Total vertical stresses within the stone column and the surrounding soil, respectively, are averaged and uniform; 7. The excess pore water pressure within the stone column is averaged and uniform in terms of radius. In addition, the following initial and boundary conditions are assumed in terms of the geometry, the compressibility of the stone column and the soil, and excess pore water pressures defined in Figure 1: where 1. The initial excess pore water pressure within the surrounding soil at the time of instant loading is equal to u 0,i.e.,u t=0 r = u 0 ;. Due to the symmetry of the problem, no flow occurs across the external boundary, i. e., u r r = 0 ; r=re 3. The ground surface is always free-draining, i. e., u z=0 c = 0; 4. Within the stone column, the boundary at z = H is impervious or because of symmetry, no flow occurs across this boundary, i. e., u c z z=h = 0; 5. The quantity of water flowing through the smeared zone into the stone column is equal to that flowing out from the stone column, i. e., πr c dz k s u r γ w r = πrc dz k c u c r=rc γ w ; z 6. The excess pore water pressures at the interface between the smeared zone and the undisturbed in situ soil are equal, i. e., u r r=rs = u r=rs r ; 7. The excess pore water pressures at the interface between the smeared zone and the stone column are equal, i. e., u r r=rc = u c. In any time, both the stone column and the surrounding soil share the applied loads, i. e., σ s A s + σ c A c = pa (5) σ c and σ s - average total stresses within the column and the surrounding soil, respectively; p the average applied pressure on the total influence area; A c,a s, and A cross-section areas of the column, the surrounding soil, and the total influence area, respectively, therefore A = A c + A s.

6 140 J. Han and S.L. Ye FIGURE 1 Definition of terms for modeling consolidation of equivalent cylindrical unit comprising of a stone column and its surrounding fine-grained soil. The assumption of equal strain between the column and the surrounding soil yields the following equation: ε = m v,s σ s = m v,c σ c (6) where σ c and σ s - the average effective stresses within the column and the surrounding soil, respectively; The rate of soil volumetric strain change with time can be expressed as: ε = m σ s v,s. (7) Using the basic soil mechanics principle σ c = σ c u c and the relationship in equation (5) yields the following equation: σ c = p σ s (1 a s) a s u c. (8) From equation (6) and equation (8), the following equation can be obtained: σ s = m ( ) v,c p σ s (1 a s ) u c. (9) m v,s a s

7 A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations 141 With the assumption that the applied load is maintained constant during the consolidation, i. e., p = 0 and the relationship σ s = σ s u r, equation (9) can be rewritten as: ( 1 as σ s m v,c a s = m v,s a s + m v,c (1 a s ) a s Substituting equation (10) into equation (7) yields: ( ε = m v,s m v,c a s 1 as m v,s a s + m v,c (1 a s ) a s u r u r + u c ). (10) + u ) c. (11) The amount of water expelled from the cylindrical surface πr dz should be equal to the volume change within the soil mass π(re r )dz, i. e.,: πr dz k r u r γ w r = π( re r) dz ε, (r s r r e ) (1) πr dz k s u r γ w r = π( re r) dz ε, (r c r r s ). (13) The above two equations can be simplified as: ( r e ) ε u r r = γ w k r u r r = γ w k s r r ( r e r r, (r s r r e ) (14) ) ε, (r c r r s ). (15) From the boundary condition (5) and equation (15), the following equation can be obtained: u c z = γ w ( N 1 ) ε k c (16) where N = d e d c, the diameter ratio of the influence zone to the stone column. Considering the equal strain assumption, the rate of the volumetric strain has a function of depth, z and time, t but is independent of the radial distance, r. The integrals of equations (14) and (15) with the known boundary conditions are [ ( γw u r = re k ln r r r ) s + γ ( w re r r s k ln S r s )] r c ε s + u c, (r s r r e ) (17) u r = γ w k s ( r e ln r r c r r c ) ε + u c, (r c r r s ) (18) where S = d s d c, the diameter ratio of the smeared zone to the stone column. The average excess pore water pressure at the depth, z, can be computed by ( 1 rs re ) u r = π ( re ) u r r πr dr + u r πr dr. (19) c r c r s Substituting equations (17) and (18) into equation (19) yields the following result: u r = γ wr e F k r ε + u c (0)

8 14 J. Han and S.L. Ye in which F can be expressed as follows: ( ln N S + k r F = N N 1 + k r k s 1 N 1 ) + S ln S 3 ( k s 4 N 1 k ) r )(1 S 1 k s 4N ). (1) ( 1 1 4N Substituting equation (11) into equations (0) and (16) yields the following two equations: u r = γ wre F ( m v,s m v,c a s 1 as u r + u ) c + u c () k r m v,s a s + m v,c (1 a s ) a s u c z = γ w (N ( 1)m v,s m v,c a s 1 as u r + u ) c. (3) k c m v,s a s + m v,c (1 a s ) a s By canceling out the variable, u c or u r from equations () and (3), the following two equations can be obtained: 3 u r z + λm v,sa s + m v,c (1 a s ) u r m v,c (1 a s ) z 1 u r ρ = 0 (4) 1 a s 3 u w z + λm v,sa s + m v,c (1 a s ) u c m v,c (1 a s ) z 1 u c ρ = 0 (5) 1 a s where λ = 8c r de F and ρ = 8k ( r N 1 ) k c de F. Equations (4) and (5) can be further simplified as follows: where λ m = λ m v,sa s + m v,c (1 a s ) m v,c (1 a s ) n s = m v,s and a s = 1 m v,c N 3 u r z + λ u r m z 3 u c z + λ u c m z u r ρ m u c ρ m = 0 (6) = 0 (7) [ ] 1 = λ 1 + n s N in terms of the relationships 1, which are same as those in equation (3) or (4) ; ρm = 1 ρ = ρ N 1 a s N 1. From equations (6) and (7), the following relationship can be set: u r = Cu c (8) where C is a constant to be determined. Apparently, the constant C must be greater or equal to 1 because the dissipation of excess pore water pressure is from the surrounding soil to the column. Equations () and (3) can be further simplified by rearrangement as: u r + a s u c = λ m (u r u c ) 1 a s (9) u c z = ρ (u r u c ). (30)

9 A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations 143 The excess pore water pressure, u c, in the column can be solved by combining equations (8) and (30) and utilizing the boundary conditions: u z=0 c = 0 and u c z z=h = 0, i. e., ( πz ) u c = C 1 sin (31) H and C = 1 + π 4ρ H. (3) By substituting equation (31) into equation (9), the constant, C 1, in equation (31) can be obtained as C 1 = C exp π λ m 4ρ H t π ρ H a s. (33) In terms of depth, the average excess pore water pressure in the surrounding soil can be expressed as u r = 1 H H 0 u r dz. (34) The constant, C, in equation (33) can be obtained by utilizing the initial condition: u t=0 r = u 0, i. e., C = u 0πρ H 4ρ H + π. (35) Therefore, where β rm = where c rm = c r ( 1 + n s 1 u r = πu 0 sin πz H exp( β rmt) (36) u c = πu 0ρ H πz 4ρ H sin + π H exp( β rmt) (37) π ( λ m N 1 ) [ π ( N 1 ) + 4N ρ H ] = 8c rm de N 1 1 F + 3 π G (38) ), the modified consolidation coefficient accounting for stress concentration on the stone column, which is the same as that previously obtained by Han and ( )( ) Ye [5] and defined in equation (3); G = kr Hdc, k c a well resistance factor. Therefore, the average rate of consolidation due to radial flow is 1 H H 0 U r = 1 u r dz = 1 exp( β rm t). (39) u 0 The above formula can be expressed in a general format: U = 1 e 8 F m T rm (40)

10 144 J. Han and S.L. Ye where T rm = c rmt, the modified time factor, which is the same as that in equation (3); de F m = N ( N ln N 1 S + k r + k r k s 1 N 1 ln S 3 ) ( + S k s 4 N 1 k ) r )(1 S 1 k s 4N ) + 3 ( )( ) kr H π. ( 1 1 4N k c d c The format of equation (40) is similar with those of the solutions obtained by Hansbo [1] and Zeng and Xia [13] for drain wells; however, equation (40) takes into account of the characteristics of stone column reinforced foundations. IV. A parametric study and discussion A. Influence factors Six important influence factors have been selected for an investigation in this parametric study. 1. Diameter ratio of the influence zone to the stone column With a typical diameter ratio for stone column reinforced foundations from 1.5 to 5.0 in practice, the reduction of the diameter ratio accelerates the rate of consolidation of the soft soil as shown in Figure. The reduction of the diameter ratio occurs when the spacing of stone columns is reduced and/or the diameter of the stone column is enlarged. The reduction of the diameter ratio can also be considered as the shortening of the drainage path. FIGURE Influence of the diameter ratio on the rate of consolidation.

11 A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations 145. Permeability of stone columns A reduction of the permeability of the stone column, k c, increases the permeability ratio, k r /k c, so that the rate of consolidation is reduced as shown in Figure 3. The reduction of the permeability of the stone column may be attributed by the deposition of fine materials from slurry to the column during the installation. FIGURE 3 Influence of the permeability of stone columns on the rate of consolidation. 3. Stress concentration ratio As discussed in Section II, one of the major differences between stone columns and drain wells is that stone columns have a high modular ratio of the columns to the soil, which results in the stress concentration from the soil to the columns. The effect of the stress concentration ratio on the rate of consolidation for stone columns without smear and well resistance has been investigated by Han and Ye [5]. The same effect is observed for the case with well resistance as shown in Figure 4, that is, the average rate of consolidation accelerated with an increase of the stress concentration ratio. 4. Size of the smeared zone As shown in Figure 5, an increase of the smeared zone reduces the average rate of consolidation. This conclusion is based on the assumption that the permeability of the smeared zone is less than that of the undisturbed soil. In other words, minimizing the size of the smeared zone can enhance the rate of consolidation. Reference [8] indicated that the stone-soil mixture zone could be effectively reduced if the rate of filling aggregates into a hole and the diameter of the column were well controlled without increasing the electrical current during the installation using vibroflotation.

12 146 J. Han and S.L. Ye FIGURE 4 Influence of the stress concentration ratio on the rate of consolidation. FIGURE 5 Influence of the size of smeared zone on the rate of consolidation. 5. Permeability of the smeared zone For most cases, the smearing of soft soil reduces the permeability of the soil because of changing the soil fabric and destroying the existing horizontal drainage fine layers. The reduction of the permeability of the soil due to smearing significantly reduces the average rate of consolidation as shown in Figure 6. It may happen that the smeared zone has a higher permeability when the stones penetrates into the surrounding soil and creates a stone-soil mixture layer.

13 A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations 147 FIGURE 6 Influence of the permeability of smeared zone on the rate of consolidation. 6. Thickness of soft soil Figure 7 demonstrates that an increase of the thickness of soft soil reduces the average rate of consolidation. This reduction can be explained an increase of the drainage path for water draining out after it enters the stone column. FIGURE 7 Influence of the thickness of soft soil on the rate of consolidation.

14 148 J. Han and S.L. Ye B. Comparison of different methods Four different methods have been selected in this study for comparison, which include the Barron solution [11] for drain wells and the authors previous solution [5] for stone columns without smear and well resistance effects, the Hansbo solution [1] for drain wells with smear and well resistance effects, and the solution proposed in this study for stone columns with smear and well resistance effects. As shown in Figure 8, the computed average rate of consolidation by Han and Ye [5] is always higher than that by Barron [11]. This is because the Han and Ye FIGURE 8 Comparison of computed rates of consolidation using different methods. solution [5] considers the modular ratio of stone columns to the surrounding soil or the stress concentration ratio, which accelerated the rate of consolidation. With the same reason due to the modular ratio or the stress concentration ratio effect, the proposed solution in this article always calculates a higher rate of consolidation than the Hansbo solution [1]. Compared with the new solution, the authors previous solution overestimated the rate of consolidation by ignoring the smear and well resistance effects. As shown in Figure 8, the computed rate of consolidation using this proposed method could be less than that using the Barron solution. By using a reduced diameter of drain well, the Barron solution may compute the same rate of consolidation as this proposed method uses the actual diameter of stone column and considers the effects of smear, well resistance, and stress concentration. This implication provides a theoretical explanation for the diameter reduction approximation method used by Barksdale and Bachus [10] for accounting for the smear and well resistance effects. C. Change of vertical stress The change of average vertical effective stress in the soil can be derived from equation (10) as follows: σ s = 1 (N N 1 + 4ρ H ) + n s 1 4ρ H + π u 0 U r. (41)

15 A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations 149 The average vertical total stress in the soil is σ s = u 0 (1 U r ) + σ s. (4) The initial excess pore water pressure can be calculated by considering equal strain in the soil and the column: pn u 0 = N 1 + E (43) cu E su where E cu undrained constrained modulus of the column; E su undrained constrained modulus of the soil. Due to the stone column being free-draining, the undrained constrained modulus of the stone column should be close to its effective constrained modulus. However, the fine-grained soil typically has a much higher undrained constrained modulus than its effective constrained modulus. Equation (43) implies that the initial pore water pressure in the soil can be greater than the average applied pressure if the undrained modulus of the soil is greater than that of the column at the moment of loading. This implication is consistent with the finding obtained by Balaam and Booker [3] using a finite element method. As shown in Figure 9, the average vertical effective stress in the soil increases with the time due to the dissipation of excess pore water pressure. However, the average vertical total stress in the soil decreases with the time due to the stress concentration from the soil to the column. Equations (40) and (41) indicate that the rate of change in the vertical stresses depends on the stress concentration ratio, n s (or modular ratio of the column to the soil). FIGURE 9 Variation of vertical stresses in soil with time.

16 150 J. Han and S.L. Ye V. A design example An example has been selected to illustrate how the solution developed in this article can be used for actual design. Consider a project in which stone columns with a diameter of 0.85 m and a spacing of 1.5 m (square pattern) are used for treating 13 m soft clay underlain by a dense and permeable sand layer. The soft clay has a coefficient of consolidation in radial flow of cm /sec. Due to the installation, there exists a smeared zone with a thickness of 4.5 mm around stone columns. The radial permeability of the smeared zone is 1/10 times that of the in situ soft clay. Assume the permeability of stone columns is 100 times that of the in situ soft clay. The design is required to calculate the average rate of consolidation of the soft clay due to radial flow after 100 kpa instant loading for 60 days. Considering the square pattern of stone columns, the equivalent influence diameter, d e = m= 1.70 m and the diameter ratio, N = 1.70 m/0.85 m =.0. With the steady stress concentration ratio typically ranging from.0 to 5.0, the steady stress concentration ratio of 3.0 is selected in this design. The modified coefficient of consolidation due to radial flow is calculated as cm /sec and the modified time factor, T rm = using equation (3). Due to the existence of top and bottom drainage surfaces between the soft clay, half the thickness of the soft clay is used for computing the well resistance effect. Considering the well resistance with a permeability ratio of the soil to the column, k r k c = 0.01 and the smearing with a permeability ratio of the undisturbed soil to the smeared zone, k r k s = 10 and a diameter ratio of the smeared zone to the stone column, S = 1.1, the parameter,f m, can be calculated as.73 using equation (40). Therefore, the computed average rate of consolidation due to radial flow reaches (or 40.8%) in 60 days after the loading. If the diameter reduction method is used, the reduction factor of 1/4 should be used along with the Barron solution in order to have an equivalent rate of consolidation, which falls within the range of 1/ to 1/15. VI. Conclusions Considering the characteristics of stone column reinforced foundations, a simplified theoretical closed-form solution has been developed in this article for computing the rate of consolidation accounting for smear and well resistance effects. The parametric study indicates that the reduction of the permeability of the stone column and/or the smeared zone and/or the stress concentration ratio decreases the rate of consolidation. However, the reduction of the diameter ratio (N = d e d c ), the smeared zone size, and/or soft soil thickness increases the rate of the consolidation. Compared with other solutions for drain wells or the authors previous solution for stone columns without smear and well resistance effects, the proposed solution in this article has addressed more comprehensive issues related to the consolidation rates of stone column reinforced foundations. This solution has been developed relying on solid theoretical bases rather than an empirical approximation like the diameter reduction method. References [1] G.A. Munfakah, S.K. Sarkar, and R.J. Castelli, Performance of a test embankment founded on stone columns, Proceedings of the International Conference on Advances in Pilings and Ground Treatment for Foundations, London, pp , (1983). [] J. Han and S.L. Ye, Settlement analysis of buildings on the soft clays stabilized by stone columns, Proceedings of the International Conference on Soil Improvement and Pile Foundations, Nanjing, China, pp , (199). [3] N.P. Balaam and J.R. Booker, Analysis of rigid rafts supported by granular piles, International Journal for Numerical and Analytical Methods in Geomechanics, 5, pp , (1981).

17 A Theoretical Solution for Consolidation Rates of Stone Column-Reinforced Foundations 151 [4] J.K. Mitchell, Soil improvement state of the art report, Proceedings of the 10th ICSMFE, Stockholm, 4, pp , (1981). [5] J. Han and S.L. Ye, A simplified solution for the consolidation rate of stone column reinforced foundations, ASCE Journal of Geotechnical and Geoenvironmental Engineering, 17(7), pp , (001). [6] H.B. Seed and J.R. Booker, Stabilization of potentially liquefiable sand deposits, Report EERC 76-10, University of California, Berkeley, Earthquake Engineering Research Center, Berkeley, California, (1976). [7] R.D. Barksdale, Applications of the state of the art of stone columns liquefaction, local bearing failure, and example calculations, Technical Report REMR-GT-7, The Georgia Institute of Technology, 90, (1987). [8] Nanjing Hydraulic Research Institute, A New Technology in Soil Improvement Vibroflotation, (in Chinese), China Hydraulic Press, (1984). [9] L. Cassagrande and S. Poulos, On the effectiveness of sand drains, Canadian Geotechnical Journal, 6, pp , (1968). [10] R.D. Barksdale and R.C. Bachus, Design and Construction of Stone Columns, FHWA/RD-83/06, 194, (1983). [11] R.A. Barron, Consolidation of fine-grained soils by drain wells, Proceedings, ASCE, 73(6), pp , (1947). [1] S. Hansbo, Consolidation of fine-grained soils by prefabricated drains, Proc. Of 10th Int. Conf. On Soil Mech. and Found. Eng., Stockholm, 3, pp , (1981). [13] G.X. Zeng and K.H. Xia, The influence of well resistance on the rate of consolidation of drain wells, in Chinese, preprint, the First Chinese Soil Improvement Conference, Shanghai, China, (1986). [14] K.S. Lane, Consolidation of fine-grained soils by drain wells discussion, Proceedings, ASCE, 74(1), pp , (1948).