A mathematical model to study the soil arching effect in stone column-supported embankment resting on soft foundation soil

Applied Mathematical Modelling 34 (2010) 3871–3883

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A mathematical model to study the soil arching effect in stone column-supported embankment resting on soft foundation soil Kousik Deb * Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721 302, India

a r t i c l e

i n f o

Article history: Received 26 November 2008 Received in revised form 13 March 2010 Accepted 30 March 2010 Available online 11 April 2010 Keywords: Consolidation Embankment Geosynthetic reinforcement Soft soil Soil arching Stone columns

a b s t r a c t Soil arching is a common phenomena in pile or columnar (vibroconcrete columns, soil– cement columns by mixing or grouting, stone columns) supported geosynthetic-reinforced or unreinforced embankments resting on soft soil. Due to soil arching, stress acting on soft soil or geosynthetic reinforcement decreases and stress on piles or columns increases. In this paper, using mechanical elements (such as spring, dashpot), a generalized mathematical model has been developed to study the soil arching effect in stone column-supported geosynthetic-reinforced and unreinforced embankments resting on soft soil. Pasternak model concept has been used to model the embankment soil. The soft soil has been idealized by spring-dashpot system to include the time-dependent behavior. The stone columns and geosynthetic reinforcement are idealized by stiffer nonlinear springs and rough elastic membrane, respectively. The consolidation effect of soft soil due to inclusions of stone columns has also been included in the model to study its effect on soil arching. Plane strain condition has been considered in the analysis. A finite difference scheme has been used to solve the governing differential equations and results are presented in non-dimensional form. It has been observed that the height of embankment, degree of consolidation of soft soil, stiffness of the stone column material, spacing between the stone columns, use of geosynthetic reinforcement and properties of soft and embankment soils (such as ultimate bearing capacity of soft soil, shear modulus and ultimate shearing resistance of embankment soil) significantly influence the degree of soil arching. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Construction of embankments on soft soil is very challenging task for geotechnical engineers due to possible bearing failure, excessive settlement and local and global instability. Use of stone columns below the embankments reduces the excessive settlement, improves the stability and increases the bearing capacity of soft foundation soil with additional advantage of providing a drainage path. The use of geosynthetic reinforcement at the base of embankment along with stone columns provides an economic and effective solution when rapid construction and small deformation are required. Geosynthetic reinforcement reduces the maximum as well as differential settlement and helps to transfers stresses from soft soil to stone columns. The embankment soil in between stone columns has a tendency to settle more due to stiffness difference between stone columns and soft soil. The downward movement of the embankment soil is restricted by shear resistance from the soil above stone columns (as shown in Fig. 1). Due to this shear resistance, the stress coming on the soft soil or geosynthetic layer is reduced and the stress acting on the stone columns is increased. This load transfer phenomenon is called ‘‘soil arching” * Tel.: +91 3222 283434; fax: +91 3222 282254. E-mail addresses: [email protected], [email protected] 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.03.026

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Nomenclature B bw d dc Ec Es Ge0 Ge0 Gf0 Gf 0 He Hf Hc Hs K0 kc0 ks0 L lc Nx n ns q0 qc qc qs qs qt qt qb qb qsu qsu qcu qcu s T T* t U w W x X

a lj ce mc ms /

q seu

seu sfu sfu h

half width of the embankment (m) width of the stone columns (m) depth of the embankment at any position (m) diameter of the stone columns (m) elastic modulus of the stone column material (kN/m2) elastic modulus of the soft soil (kN/m2) initial shear modulus of the embankment soil (kN/m2) normalized Ge0 (non-dimensional) initial shear modulus of the granular layer (kN/m2) normalized Gf0 (non-dimensional) height of the embankment (m) thickness of the granular layer (m) length of the stone columns (m) thickness of the soft soil (m) coefficient of lateral stress (non-dimensional) initial modulus of subgrade reaction for stone column material (kN/m2/m) initial modulus of subgrade reaction for soft foundation soil (kN/m2/m) half width of the granular layer (m) half of the top width of embankment (m) total shear force per unit thickness of the shear layer (kN/m) slope of the embankment (non-dimensional) stress concentration ratio (non-dimensional) uniform surcharge on the embankment (kN/m2) vertical stress acting on stone columns (kN/m2) normalized qc (non-dimensional) vertical stress acting on soft foundation soil (kN/m2) normalized qs (non-dimensional) vertical stress acting on top of the geosynthetic or granular layer (kN/m2) normalized qt (non-dimensional) vertical stress acting on bottom of the geosynthetic layer (kN/m2) normalized qb (non-dimensional) ultimate bearing capacity of the soft soil (kN/m2) normalized qsu (non-dimensional) ultimate bearing capacity of stone column material (kN/m2) normalized qcu (non-dimensional) spacing between stone columns (m) mobilized tension in the geosynthetic layer (kN/m) normalized T (non-dimensional) time (s) degree of consolidation of the soft soil (%) vertical displacement (m) normalized w (non-dimensional) distance from centre of loading (m) normalized x (non-dimensional) spring constant ratio (kc0/ks0) (non-dimensional) interface friction at the interface between the soil and geosynthetic layer (non-dimensional) unit weight of the embankment soil (kN/m3) Poisson’s ratio of the stone column material (non-dimensional) Poisson’s ratio of the soft soil (non-dimensional) angle of shearing resistance (degree) soil arching ratio (non-dimensional) ultimate shear resistance of the embankment soil (kN/m2) normalized seu (non-dimensional) ultimate shear resistance of the granular layer (kN/m2) normalized sfu (non-dimensional) slope of the membrane (non-dimensional)

[1]. Several researches have been carried out to study the soil arching effect in pile-supported embankments resting on soft soil [2,3]. Han and Gabr [4] conducted a numerical analysis on geosynthetic-reinforced and pile-supported earth platform

K. Deb / Applied Mathematical Modelling 34 (2010) 3871–3883

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Fig. 1. Soil arching in stone column-supported embankment.

over soft soil and studied the effects of pile material elastic modulus and geosynthetic stiffness on the degree of soil arching. Han and Gabr [4] observed that the soil arching in the embankment soil was increased with an increase in the height of embankment fill and elastic modulus of the pile material. However, soil arching was decreased as the tensile stiffness of geosynthetic reinforcement was increased. Based on 2-D physical and numerical modeling of pile-supported earth platform over soft soil, Jenck et al. [5] observed that the load transfer onto piles due to soil arching was effectively controlled by shearing mechanisms (development of shearing in the granular fill due to differential settlements at the platform base between the soft soil and the rigid piles, leading to arching which partially transfers the loads onto the piles) and shear strength of the platform or embankment material. Yun-min et al. [6] conducted experimental investigation on reinforced and unreinforced piled embankments to study the effects of pile–subsoil relative displacement, embankment height, cap beam width and clear spacing and geosynthetics (with different tensile strength) on soil arching. It was observed that soil arching and settlements are influenced significantly by embankment height, cap beam width and clear spacing, and reinforcement tensile strength. Moreover, soil arching was strongly dependent on the pile–subsoil relative displacement. Most of the studies related to soil arching were conducted on pile-supported embankments. Very limited studies have been done on stone column-supported embankments resting over soft soil. Based on numerical analysis on geosyntheticencased stone columns, Murugesan and Rajagopal [7] observed that stress on the stone columns was increased as encasement stiffness or height of the embankment was increased and diameter of encased stone columns was decreased. However, effect of the basal reinforcement and soft soil consolidation due to inclusions of stone columns on stress transfer process was not studied. Using the lumped parameter modeling approach and considering the consolidation effect of the soft soil, models were developed for strip footing resting on reinforced and unreinforced granular fill-soft soil system with stone columns [8–10]. However, in the reported lumped parameter models on stone columns, soil arching effect was not considered. In this study, using mechanical elements, a generalized mathematical model has been developed for stone columnsupported geosynthetic-reinforced and unreinforced embankments resting on soft soil to study the effects of various properties related to embankment fill and stone column material on degree of soil arching. The effects of consolidation of soft soil and basal reinforcement on soil arching have also been considered. In addition, the effects of ultimate bearing capacity of the soft soil and ultimate shearing resistance of embankment soil on soil arching have been studied in the present model. 2. Modeling procedure Fig. 2 shows the stone column-supported embankment resting on soft foundation soil. The proposed foundation model as shown in Fig. 3 may idealize the behavior of such system. A granular layer of sand or gravel is usually placed over top of the stone columns [11]. This granular layer or sand bed acts as drainage layer and also distributes the stresses coming from the embankments. A single layer of geosynthetic reinforcement is placed at the base of the embankment as shown in Fig. 1. The geosynthetic reinforcement is placed at top of the granular layer or sand blanket as it has been reported in the literature that a thin granular layer is generally placed over soft soil before placing the geosynthetic reinforcement layer [4]. However, multiple layers of geosynthetic can be placed in the embankment, but for simplicity one layer of geosynthetic is considered in the present study. The embankment has been modeled by Pasternak shear layer with variable thickness as proposed by Sharma [12]. The granular layer and the soft soil have been idealized by the Pasternak shear layer and spring-dashpot system, respectively. Similar visco-elastic model has been considered by Shukla and Chandra [13] to model the time-dependent

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Fig. 2. Embankment resting on stone column-improved soft soil.

Fig. 3. Proposed foundation model.

behavior of soft soil. The stone columns are idealized by stiffer nonlinear springs. The behavior of embankment soil, granular layer, stone column material and soft soil is considered as nonlinear. The geosynthetic layer is assumed to be linearly elastic with stiffness greater than or equal to 4000 kN/m; as beyond this value the influence of geosynthetic tensile stiffness becomes less important on the settlement response [4]. In the present study, the geosynthetic reinforcement is assumed to be inextensible and such geosynthetic reinforcements are relatively stiff (geogrids and relatively stiff geotextile reinforcements can be considered as fairly inextensible [14,15]). Thus, the stiffness of geosynthetic reinforcement is assumed as greater than or equal to 4000 kN/m. However, it has been observed that as stiffness of the geosynthetic reinforcement is increased, the differential settlement is reduced and less soil arching is developed [4]. Plane strain conditions are considered for the loading and the foundation soil system. 2.1. Formulation for unreinforced embankment An embankment of width 2B has been placed on the granular layer or sand blanket of width 2L as shown in Fig. 2. Pasternak model concept has been used to idealize the embankment by considering it a thick shear layer of variable thickness as adopted by Sharma [12]. According to Pasternak model [16], the basic differential equation under 2-D plane strain condition can be written as:

@Nx þ q  qt ¼ 0; @x where Nx is the total shear force per unit thickness of the shear layer and is given by:

ð1Þ

K. Deb / Applied Mathematical Modelling 34 (2010) 3871–3883

Z

Nx ¼

3875

d

sxz dz;

ð2Þ

0

where sxz is the average shear stress acting in the shear layer, q is the loading (including self weight of the embankment and loading due to traffic), qt is the stress coming on top of the granular layer. Considering the proposed nonlinear shear stress– shear strain response of the embankment soil [17], sxz can be represented as:

sxz ¼

Ge0 ð@w=@xÞ 1 þ Ge0 j@w=@xj seu

ð3Þ

;

where Ge0 is the initial shear modulus of the embankment soil; seu is the ultimate shear resistance of the embankment soil (residual shear strength of the soil); w is the vertical displacement; ow/ox is the shear strain; d is the depth of the shear layer (embankment) at any position. Putting the value of sxz in Eq. (2), Nx can be written as:

Nx ¼

Z

d

Ge0 ð@w=@xÞ 1þ

0

Ge0 j@w=@xj

dz ¼

seu

Ge0 ð@w=@xÞ 1 þ Ge0 j@w=@xj seu

d:

ð4Þ

Now

@Nx @w 0 @2w d þ Ge2 d 2 ; ¼ Ge1 @x @x @x where Ge1 ¼

Ge0 Ge0 j@w=@xj seu

and Ge2 ¼ 

1þ

ð5Þ Ge0 Ge0 j@w=@xj 2 seu

1þ

.

The loading q can be expressed as:

q ¼ c e d þ q0 ;

ð6Þ

where ce is the total unit weight of the embankment soil and q0 is uniform surcharge on the embankment due to traffic loading. Putting the value of oNx /ox and q in Eq. (1), one can get,

qt ¼ ce d þ q0 þ Ge1

@w 0 @2w d þ Ge2 d 2 : @x @x

ð7Þ

Using the Pasternak model concept the basic differential equation for the granular layer or sand blanket can be written as:

qt ¼ qs  Gf Hf

@2w ; @x2

ð8Þ

where qs is the vertical stress acting on the soft soil; Hf is the thickness of the granular layer; Gf is the shear modulus of the granular layer and can be expressed as:

Gf ¼ h

1þ

Gf 0 i Gf 0 j@w=@xj 2

;

ð9Þ

sfu

where Gf0 is the initial shear modulus of the granular layer; sfu is the ultimate shear resistance of the granular layer or sand blanket. Considering hyperbolic nonlinear stress–displacement relation proposed by Kondner [18] and consolidation effect of the soft soil, qs (at time t > 0) can be expressed as [8,10]:

qs ¼

ks0 w ; U½1 þ ks0 ðw=qsu Þ

ð10Þ

where ks0 and qsu are initial modulus of subgrade reaction and ultimate bearing capacity of the saturated soft soil, respectively. U is degree of consolidation of the stone column-improved soft soil at any time t. The flow of water into vertical drain is axi-symmetric in nature. The flow of water is in radial as well as vertical direction. The overall rate of consolidation is determined by considering a combining effect of radial and vertical flows. The solution for the vertical flow follows the Terzaghi 1D consolidation solution; while the solution for the radial flow follows the Barron [19] drain well solution (modified by Han and Ye [20] for stone column-improved soil). However, in the present study plane strain condition is considered for stone column-supported embankment. Thus, it is necessary to convert the consolidation equation under axi-symmetric condition to an equivalent plane strain condition (2-D consolidation). Similar 2-D plane strain analysis has been carried out for vertical drain or stone columns beneath embankments or strip footings resting on soft ground [8–10] and [21–23]. The equivalent plane strain condition can be achieved by manipulating either the drain spacing or the horizontal permeability of the soft soil (geometry or permeability matching) as suggested by Hird et al. [21]. The degree of consolidation of the soft soil due to stone column inclusions under plane strain condition can be determined by using the procedure described by Deb

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K. Deb / Applied Mathematical Modelling 34 (2010) 3871–3883

et al. [8,10], where the simplified consolidation equation proposed by Han and Ye [20] for stone column-reinforced soil has been modified to an equivalent plane strain equation by using the procedure suggested by Hird et al. [21]. In such a situation, the width of the plane strain unit cell is taken as equal to the diameter of the unit cell in axi-symmetric condition and the width of the stone column (bw) is taken as equal to the diameter of the stone column (dc). Using similar hyperbolic nonlinear stress–displacement relation [18], the vertical stress coming on the stone column can be expressed as:

qc ¼

kc0 w ; 1 þ kc0 ðw=qcu Þ

ð11Þ

where kc0 and qcu are initial modulus of subgrade reaction and ultimate bearing capacity of the stone column material, respectively. Thus, the stress concentration ratio can be defined as: ns ¼ qc =qs . The modulus of subgrade reaction of soft soil and stone column material is assumed to be constant irrespective of depth and time. The modulus of subgrade reaction (spring stiffness) and shear modulus of the soils can be determined by using the procedures described by Selvadurai [24]. Combining Eqs. (7), (8), (10), and (11), the governing differential equations for unreinforced embankment can be written as (at time t > 0):

(

ce d þ q0 ¼ Cw  Ge1 d0

) @w @2w for 0  x  B; þ ðGe2 d þ Gf Hf Þ 2 @x @x

ð12aÞ

where

ks0 within soft soil region U½1 þ ks0 ðw=qsu Þ kc0 within stone column region ¼ 1 þ kc0 ðw=qcu Þ

C¼

and

ks0 w @2w ¼ 0 for B < x  L:  Gf Hf U½1 þ ks0 ðw=qsu Þ @x2

ð12bÞ

In Eq. (11), d and d0 are function of x and can be expressed as:

d ¼ He

for 0  x  lc ;

d ¼ He  fðx  lcÞ=ng for lc < x  B and 0

d ¼0 for 0  x  lc ; 0 d ¼ 1=n for lc < x  B; where He is height of the embankment and n is slope of the embankment. Using the non-dimensional parameters as: X ¼ x=B; W ¼ w=B; Ge1 ¼ Ge1 =ks0 B; Ge2 ¼ Ge2 =ks0 B; Gf ¼ Gf =ks0 B; Ge0 ¼ Ge0 =ks0 B; Gf 0 ¼ Gf 0 =ks0 B; qsu ¼ qsu =ks0 B; qcu ¼ qcu =ks0 B; seu ¼ seu =ks0 B; sfu ¼ sfu =ks0 B; a ¼ kc0 =ks0 (where a is the subgrade modulus or spring constant ratio); qs ¼ qs =ks0 B; qc ¼ aW; q0 ¼ q0 =ks0 B; ce ¼ ce =ks0 , the governing differential equations at time t > 0 can be written in non-dimensional form as:

ce

( )   Hf @ 2 W d d 0 @W for 0  X  1; þ q0 ¼ C  W  Ge1 d þ ðGe2 þ Gf Þ B @X B B @X 2

ð13aÞ

where

C ¼ ¼

1 within soft soil region U½1 þ ðW=qsu Þ

a

1 þ ðaW=qcu Þ

within stone column region

and 2 W  Hf @ W ¼ 0 for 1 < X  ðL=BÞ:  G f U½1 þ ðW=qsu Þ B @X 2

ð13bÞ

The subgrade modulus or spring constant ratio (a) can be expressed as [8–10]:

a¼

ð1 þ ms Þð1  2ms Þ Ec ; ð1 þ mc Þð1  2mc Þ Es

ð14Þ

K. Deb / Applied Mathematical Modelling 34 (2010) 3871–3883

3877

where Ec and mc are elastic modulus and Poisson’s ratio of the stone column material, respectively; Es and ms are elastic modulus and Poisson’s ratio of the soft soil, respectively. The ratio Ec/Es is called as modular ratio. 2.2. Formulation for geosynthetic-reinforced embankment In case of geosynthetic-reinforced embankment, the reinforcement layer is analyzed as suggested by Shukla and Chandra [13] for geosynthetic-reinforced granular fill-soft soil system. Simplifying horizontal and vertical force equilibrium equations of the geosynthetic layer, the expressions for normal stresses and mobilized tension are obtained as [13]:

qt ¼ C 1 qb  C 2 T cos h

@2w ; @x2

ð15Þ

@T ¼ D1 qt  D2 qb ; @x

ð16Þ

where

1 þ K 0 tan2 h  ð1  K 0 Þlb tan h ; 1 þ K 0 tan2 h þ ð1  K 0 Þlt tan h 1 C2 ¼ ; 1 þ K 0 tan2 h þ ð1  K 0 Þlt tan h C1 ¼

D1 ¼ lt cos hð1 þ K 0 tan2 hÞ  ð1  K 0 Þ sin h; D2 ¼ lb cos hð1 þ K 0 tan2 hÞ þ ð1  K 0 Þ sin h and T is mobilized tension in the geosynthetic layer; lt and lb are interface friction at top and bottom of the geosynthetic layer, respectively; K0 is coefficient of lateral stress at rest and is assumed to be equal to 1  sin / [25,26]; h is slope of the membrane and tan h = ow/ox; qt and qb are stresses acting at top and bottom of the geosynthetic layer, respectively. Using the Pasternak model concept as described earlier, the qt and qb can be written as:

qt ¼ ce d þ q0 þ Ge1 qb ¼ qs  Gf Hf

@w 0 @2w d þ Ge2 d 2 ; @x @x

ð17Þ

@2w : @x2

ð18Þ

Combining Eqs. (10), (11), (14), (16), and (17), one can get the governing differential equation for reinforced-embankment as:

(

ce d þ q0 ¼ C 1 Cw  Ge1 d0

) @w @2w for 0  x  B: þ ðGe2 d þ C 1 Gf Hf þ C 2 T cos hÞ 2 @x @x

ð19Þ

The expression of mobilized tension in the geosynthetic reinforcement layer at any time, t > 0, can be found out by combining Eqs. (10), (11), (15), (16), and (17) and written as:

! ! @T @2w @2w 0 @w for 0  x  B: ¼ D1 ce d þ q0 þ Ge1 d þ Ge2 d 2  D2 Cw  Gf Hf @x @x @x @x2

ð20Þ

Using the non-dimensional parameters as described earlier, the governing differential equations can be written in nondimensional form as:

ce

(     2 ) Hf d d @ W 0 @W for 0  X  1; þ q0 ¼ C 1 C  W  Ge1 d þ Ge2 þ C 1 Gf þ C 2 T  cos h B @X B B @X 2

@T  d d @2W 0 @W þ Ge2 ¼ D1 ce þ q0 þ Ge1 d B @X B @X 2 @X

! 

 D2 C W 

Gf

Hf @ 2 W B @X 2

ð21aÞ

! for 0  X  1

ð21bÞ

and

Hf @ 2 W W ¼ 0 for 1 < X  ðL=BÞ;  Gf  U½1 þ ðW=qsu Þ B @X 2

ð21cÞ

where T* = T/ks0B2. The stress acting on the geosynthetic or top of the granular layer can be written in non-dimensional form as:

qt ¼ ce

d @W 0 d @2W þ q0 þ Ge1 d þ Ge2 : B @X B @X 2

ð22Þ

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According to McNulty [27] (see Han and Gabr [4]), the degree of soil arching can be defined as: q = qt/(ceHe + q0), where q is the soil arching ratio; q equal to zero represents complete soil arching and q equal to one represents no soil arching; qt is the vertical stress acting on the granular layer (unreinforced embankment) or on the geosynthetic layer (reinforced embankment); ce is the total unit weight of the embankment soil; He is the height of the embankment; and q0 is uniform surcharge on the embankment due to traffic loading. The soil arching ratio can be expressed in non-dimensional terms as (q0 ¼ 0):

q

q ¼  Ht e  : ce B

ð23Þ

The stresses acting on the soft soil and stone columns can be written in non- dimensional form as:

W ; U½1 þ ðW=qsu Þ aW qc ¼ : 1 þ ðaW=qcu Þ

qs ¼

ð24Þ ð25Þ

3. Method of solution 3.1. Finite difference formulation Finite difference method has been employed to solve the governing differential equations [Eqs. (13) and (21)]. The derivatives o2W/oX2 and oW/oX, have been expressed by central difference scheme. The length L/B is divided into m number of elements with (m + 1) number of node points (i = 1, 2, 3, 4 , . . . , m + 1). Thus, the mesh size (DX) can be written as, DX = (L/B) m. 3.2. Boundary conditions As the problem is symmetric, only half of the system is considered for the analysis. Thus, at the centre of the loaded region X = 0 (or x = 0), due to symmetry, the slope, oW/oX is zero. The width of the granular fill considered in the analysis is sufficiently enough such that at the edge [at X = L/B (or x = L)] the slope, oW/oX of settlement-distance profile is also zero. The continuity at the edge of the stone columns is automatically satisfied. However, the model can be applied for unsymmetrical condition also. In such situation at both the edges slope oW/oX, is zero. As geosynthetic layer is free at the end, the mobilized tension, T* = 0 at X = 1 (or x = B). Sharma and Bolton [28] also observed that at the toe of embankment (i.e. at x = B) the mobilized tension in geosynthetic layer is zero. 4. Results and discussion A computer program based on the formulation as described above has been developed and solutions are obtained using an iterative technique with a tolerance value of 104. The typical values used for this study are, / = 36°; K0 = 0.41; lt = lb = 0.5; n = 2; ms = 0.45 and mc = 0.3. 4.1. Validation To validate the present model, the results of the present analysis have been compared with the results presented by Low et al. [3]. Based on the model tests and theoretical analysis, Low et al. [3] investigated the soil arching in geotextile-reinforced and unreinforced embankments resting on soft ground supported by piles with cap beams. According to Low et al. [3], the vertical stress (qt) acting on the soil midway between the cap beams under 2-D plane strain condition is given by (without geotextile):

qt ¼ ce He

  ðK p  1Þð1  dÞs s s ;  þ ð1  dÞðKp1Þ 1  2He ðK p  2Þ 2He 2He ðK p  2Þ

ð26Þ

where d = bw/s; Kp = (1 + sin /)/(1  sin /); / is angle of shearing resistance of the embankment soil; bw is width of the cap beam (here stone columns); s is spacing between the cap beams (here stone columns); He is height of the embankment; ce is unit weight of the embankment soil. Thus, using Low et al. [3] approach, the soil arching ratio can be expressed as (q0 = 0):

q¼

 qt ðK p  1Þð1  dÞs s s ¼  þ ð1  dÞðKp1Þ 1  : 2He ðK p  2Þ 2He 2He ðK p  2Þ ce He

ð27Þ

Fig. 4 shows the comparison between the results of present model and the results presented by Low et al. [3]. During the comparison, the width of stone column and embankment is taken as 0.7 m (range is 0.6–1 m [11]) and 40 m, respectively. The comparison shows that in both the cases as s/bw ratio increases soil arching ratio also increases. In other words, smaller

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K. Deb / Applied Mathematical Modelling 34 (2010) 3871–3883

1

Low et al., 1994 Present Study s / bw

Soil Arching Ratio ( ρ )

0.8

5.0 0.6

4.0 3.0

0.4 2.0 0.2

-2

G e * = G f * = 1.5 x 10 ; E c / E s = 20; -4

0 0.05

-2

U = 100%;τ ue * = τ uf * = 2.5 x 10 ; q us * = 0.5 x 10 0.1

0.15

0.2

0.25

0.3

Height of the Embankment (He /B ) Fig. 4. Validation of the present model.

spacing between the stone columns or piles promotes higher soil arching effect. It has been also observed that as height of the embankment increases soil arching ratio decreases. This is due to the fact that for shallow embankments, the embankment height is not enough to mobilize the arching mechanism. As the height of the embankment increases more shear resistance has been accumulated which helps the development of soil arching [4]. Thus, incase of higher embankment height lower soil arching ratio has been observed. From the experimental studies Yun-min et al. [2] also observed that when embankment height (He) to clear spacing between cap beam (s) ratio is equal or less than 1.4, the embankment height was relatively low and no completed soil arch was formed and the pile–subsoil relative displacement was reflected on the surface of the embankment, and differential settlement on the top of the embankment was occurred. However, when He/s P 1.6, embankment height was relatively high, a completed soil arch was formed and no apparent differential settlement was occurred on the embankment surface. In the comparison, it has been observed that the results of present study are almost similar to the results obtained by Low et al. [3] when s/bw value is 4, whereas for other values of s/bw difference is observed. This is due to the fact that the results of present study depend on the stiffness of the stone column material, properties and degree of consolidation of the soft soil, whereas the results presented by Low et al. [3] are free from the effect of soft soil properties, consolidation and stiffness of pile material. 4.2. Effect of stiffness of stone column material Fig. 5 shows the effect of stiffness of stone columns on soil arching ratio. It has been observed that as the stiffness of stone column material increases soil arching ratio decreases, which is in agreement with the findings by Han and Gabr [4]. Use of stone columns with higher stiffness increases the differential settlement [10] which enhances the soil arching effect. At nondimensional height of the embankment (He/B) equal to 0.25, the soil arching ratio is decreased by 35% as modular ratio (Ec/Es) increases from 5 to 100. 4.3. Effect of consolidation of soft soil Fig. 6 shows the effect of consolidation of soft soil on soil arching ratio. It has been observed that up to 20% degree of consolidation the value of soil arching ratio is 1 and after that as degree of consolidation increases soil arching ratio decreases. At non-dimensional height of the embankment (He/B) equal to 0.25, the soil arching ratio is reduced by 44% as degree of consolidation increases from 20% to 100%. Thus, up to 20% degree of consolidation no soil arching takes place and after that as the consolidation of soft soil progresses higher soil arching effect has been observed. This is due to the fact that up to 20% degree of consolidation the differential settlement is almost negligible, but after that as the degree of consolidation increases differential settlement also increases [10] and more soil arching is occurred. 4.4. Effect of ultimate bearing capacity of soft soil Fig. 7 shows the effect of ultimate bearing capacity of soft soil on soil arching ratio. It has been observed that as the qus value decreases soil arching ratio also decreases. Thus, soft soil with lower ultimate bearing capacity enhances the development of soil arching as more differential settlement has been observed incase of soft soil with lower ultimate bearing capacity [10]. At non-dimensional height of the embankment (He/B) equal to 0.25, as qus decreases from 5  103 to 0.1  103 soil

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1

Ec / Es

Soil Arching Ratio ( ρ )

0.8

5 10 20 30 50 100

0.6

0.4 -2

G e *= G f * = 1.5 x 10 ;

0.2

-3

q us * = 0.5 x 10 ; U = 100%; -4

τ ue * = τ uf * = 2.5 x 10 ; s /b w = 5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Height of the Embankment (He /B ) Fig. 5. Effect of stiffness of the stone columns on soil arching ratio.

U% 20

Soil Arching Ratio ( ρ )

1

40 0.8 60 80

0.6

100 0.4 -2

G e * = G f * = 1.5 x 10 ;

0.2

-3

q us * = 0.5 x 10 ; E c / E s = 20; -4

τ ue * = τ uf * = 2.5 x 10 ; s /b w = 5

0 0

0.05

0.1

0.15

0.2

0.25

0.3

Height of the Embankment (He /B ) Fig. 6. Effect of consolidation of the soft soil on soil arching ratio.

arching ratio is decreased by 70%. However, influence of qus becomes less important on soil arching ratio when qus value exceeds 5  103. 4.5. Effect of properties of embankment soil The effect of the shear modulus of the embankment soil on soil arching ratio has been shown in Fig. 8. It has been observed that as Ge value increases soil arching ratio decreases. At non-dimensional height of the embankment (He/B) equal to 0.25, as Ge increases from 0.01 to 0.5 soil arching ratio is decreased by 37%. It has been further observed that embankment soil with higher ultimate shearing resistance helps to develop more soil arching in the embankment (Fig. 9). At non-dimensional height of the embankment (He/B) equal to 0.25, as sue increases from 1  105 to 5  105 soil arching ratio is decreased by 17%. However, influence of sue becomes less important on soil arching ratio when sue value exceeds 5  105. Embankment soils with higher shear modulus and higher ultimate shear resistance enhance the stress transfer onto the stone columns due to soil arching as the load transfer is effectively controlled by shearing mechanisms and shear strength of the embankment materials [5]. As qus or sue value increases, the soil arching ratio increases up to a limiting value. This is due to fact that as qus or sue increases, the differential settlement decreases up to a limiting vale, whereas beyond this value due to increase in qus or sue value no further reduction in differential settlement is occurred. Thus, beyond the limiting value of qus or sue , the

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1 -3

q us * x 10

Soil Arching Ratio ( ρ )

0.8

10 5 1

0.6

0.5 0.4

0.25 0.1

0.2

-2

G e * = G f * = 1.5 x 10 ; E c / E s = 20; -4

U = 100%;τ ue * = τ uf * = 2.5 x 10 ; s /b w = 5

0 0

0.05

0.1

0.15

0.2

0.25

0.3

Height of the Embankment (He /B ) Fig. 7. Effect of ultimate bearing capacity of the soft soil on soil arching ratio.

1

Soil Arching Ratio ( ρ )

0.8 Ge* 0.6

0.01

0.4

0.025 0.1 0.5 1.0

0.2

-2

-3

G f * = 1.5 x 10 ; E c /E s = 20; q us * = 0.5 x 10 ; -4

U = 100%; τ ue * = τ uf * = 2.5 x 10 ; s /b w = 5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Height of the Embankment (He /B ) Fig. 8. Effect of shear modulus of the embankment soil on soil arching ratio.

influence of qus or sue on soil arching becomes less important. From Figs. 7 and 9 it has been observed that these limiting values are almost independent to the He/B ratio. 4.6. Effect of geosynthetic reinforcement Fig. 10 shows the effect of geosynthetic reinforcement on soil arching ratio. More soil arching has been observed in unreinforced embankment as compared to the geosynthetic-reinforced embankment. This is due to the fact that use of geosynthetic reinforcement reduces the differential settlement [8,9] which causes less soil arching. At non-dimensional height of the embankment (He/B) equal to 0.25, incase of geosynthetic-reinforced embankment the soil arching ratio is increased by 22% as compared to the unreinforced embankment. It has been further observed that that as the height of the embankment increases up to 0.15B, the difference between the soil arching ratio for reinforced and unreinforced embankment also increases, whereas beyond this value the difference is almost constant. This is due to the fact that, as the height of the embankment increases up to 0.15B, the effectiveness of the geosynthetic reinforcement increases [8,29], whereas beyond this height no further effect due to geosynthetic reinforcement has been observed. The effect of geosynthetic reinforcement becomes level off when the embankment height exceeds 0.15B. Thus, as the height of embankment increases differential settlement is effectively reduced due to use of geosynthetic reinforcement as compared to unreinforced case and the development of soil arching is also reduced in reinforced embankment. Although as the height of the embankment increases, the

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1

*

1 2.5 5 25

0.8

Soil Arching Ratio ( ρ )

-5

τ ue x 10

0.6

0.4

0.2

-2

G e * = G f * = 1.5 x 10 ; E c / E s = 20; U = 100%; *

-4

-2

τ uf * = 2.5 x 10 ; s /b w = 5; q us = 0.5 x 10 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Height of the Embankment (He /B ) Fig. 9. Effect of ultimate shear resistance of the embankment soil on soil arching ratio.

1

-2

G e * = G f * = 1.5 x 10 ; E c / E s = 20; *

Soil Arching Ratio ( ρ)

-3

U = 100%; q us = 0.1 x 10 ; s /b w = 5

0.8

-4

τ ue * = τ uf * = 2.5 x 10

0.6

0.4

0.2

Unreinforced Geosynthetic-reinforced 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Height of the Embankment (He /B ) Fig. 10. Effect of geosynthetic reinforcement on soil arching ratio.

differential settlement is increased (even with the use of geosynthetics). However, the increase in differential settlement is more in case of unreinforced embankment as compared to the geosynthetic-reinforced embankment.

5. Conclusion The developed model is very useful to study the soil arching in embankments resting on soft soil improved with stone columns. The effects of the stiffness of the stone column material, spacing between the stone columns, consolidation of the soft soil, use of geosynthetic reinforcement, properties of embankment and soft soil on soil arching have been incorporated in the model. It has been observed that the geosynthetic reinforcement plays a very important role in soil arching. Use of geosynthetic reinforcement reduces the development of soil arching in embankment and the reduction is more incase of higher embankment height. Up to 20% degree of consolidation no soil arching has been observed, whereas beyond that stage as consolidation progresses more soil arching has been occurred. It has been further observed that as the height of the embankment and stiffness of the stone columns increase soil arching also increases. Smaller spacing between the stone columns enhances the development of soil arching. The ultimate bearing capacity of the soft soil has significance influence on the soil arching. Soft soil with lower ultimate bearing capacity causes more soil arching. However, influence of qus becomes less important on soil arching ratio when qus value exceeds 5  103. Incase of embankment soil with higher shear modulus and higher ultimate shearing resistance, more stress is transfer onto the stone columns due to soil arching. However, influence of sue becomes less important on soil arching ratio when sue value exceeds 5  105.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

K. Terzaghi, Theoretical Soil Mechanics, Wiley, New York, 1943. W.J. Hewlett, M.F. Randolph, Analysis of piled embankments, Ground Eng. 21 (3) (1988) 2–18. B.K. Low, S.K. Tang, V. Choa, Arching in piles embankments, J. Geotech. Eng. (ASCE) 120 (11) (1994) 1917–1938. J. Han, M.A. Gabr, Numerical analysis of geosynthetic-reinforced and pile-supported earth platform over soft soil, J. Geotech. Environ. Eng. (ASCE) 128 (1) (2002) 44–53. O. Jenck, D. Dias, R. Kastner, Two-dimensional physical and numerical modeling of a pile-supported earth platform over soft soil, J. Geotech. Environ. Eng. (ASCE) 133 (3) (2007) 295–305. C. Yun-min, C. Wei-ping, C. Ren-peng, An experimental investigation of soil arching within basal reinforced and unreinforced piles embankments, Geotext. Geomembr. 26 (2) (2008) 164–174. S. Murugesan, K. Rajagopal, Geosynthetic-encased stone columns: numerical evaluation, Geotext. Geomembr. 24 (2006) 349–358. K. Deb, P.K. Basudhar, S. Chandra, Generalized model for geosynthetic-reinforced granular fill-soft soil with stone columns, Int. J. Geomech. (ASCE) 7 (4) (2007) 266–276. K. Deb, S. Chandra, P.K. Basudhar, Response of multi layer geosynthetic-reinforced bed resting on soft soil with stone columns, Comput. Geotech. 35 (3) (2008) 323–330. K. Deb, Modeling of granular bed-stone column-improved soft soil, Int. J. Numer. Anal. Methods Geomech. 32 (10) (2008) 1267–1288. J.K. Mitchell, Soil improvement – state of the art report, in: Proceedings of 10th International Conference on Soil Mechanics and Foundation Engineering, Balkema, Rotterdam, The Netherlands, vol. 4, 1981, pp. 509–565. J.S.N. Sharma, Application of Pasternak Model Concept for Reinforced Earth Walls and Embankments, Ph.D. Thesis, Indian Institute of Technology, Kanpur, 1989. S.K. Shukla, S. Chandra, A generalized mechanical model for geosynthetic-reinforced foundation soil, Geotext. Geomembr. 13 (3) (1994) 813–825. M.R. Madhav, B. Umashankar, Analysis of inextensible sheet reinforcement subject to downward displacement/force: non-linear subgrade response, Geosynth. Int. 10 (2003) 95–102. M.R. Madhav, B. Umashankar, Analysis of inextensible sheet reinforcement subject to downward displacement/force: linear subgrade response, Geotext. Geomembr. 21 (2003) 69–84. P.L. Pasternak, On a new method of analysis of an elastic foundation by means of two foundation constants, Gosudartvennoe Izdatelstro Liberaturi po Stroitelstvui Arkhitekture, Moscow, 1954 (in Russian). C. Ghosh, M.R. Madhav, Settlement response of a reinforced shallow earth bed, Geotext. Geomembr. 13 (9) (1994) 643–656. R.L. Kondner, Hyperbolic stress–strain response: cohesive soils, J. Soil Mech. Foundation Eng. Div. (ASCE) 89 (1) (1963) 115–143. R.A. Barron, Consolidation of fine-grained soils by drain wells, Proc. ASCE 73 (6) (1947) 811–835. J. Han, S.L. Ye, Simplified method for consolidation rate of stone column reinforced foundations, J. Geotech. Environ. Eng. (ASCE) 127 (7) (2001) 597– 603. C.C. Hird, I.C. Pyrah, D. Russell, Finite element modeling of vertical drains beneath embankments on soft ground, Geotehnique 42 (3) (1992) 499–511. J.C. Chai, N. Miura, S. Sakajo, D.T. Bergado, Behavior of vertical drain improved subsoil under embankment loading, Soil Found. 35 (4) (1995) 49–61. B. Indraratna, I.W. Redana, Plane-strain modeling of smear effects associated with vertical drains, J. Geotech. Environ. Eng. (ASCE) 123 (5) (1997) 474– 478. A.P.S. Selvadurai, Elastic Analysis of Soil-foundation Interaction, Elsevier Scientific, Amsterdam, 1979. E.W. Brooker, H.O. Ireland, Earth pressure at rest related to stress history, Can. Geotech. J. 2 (1-2) (1965) 1–15. I. Alpan, The empirical evaluation of the coefficient K0 and K0r, Soil Found. 7 (1) (1967) 31–40. J.W. McNulty, An Experimental Study of Arching in Sand, Rep. No. I-674, US Army Engineer Waterways Experiment Station, Corps of Engineers, Vicksburg, Miss., 1965, 170. J.S. Sharma, M.D. Bolton, Centrifuge modeling of an embankment on soft soil reinforced with geogrid, Geotext. Geomembr. 14 (1996) 1–17. K. Deb, S. Chandra, P.K. Basudhar, Settlement response of a multi layer geosynthetic-reinforced granular fill-soft soil system, Geosynth. Int. 12 (6) (2005) 288–298.