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GEOSYNTHETICS INTERNATIONAL 2001, VOL. 8, NO. 1 27

Technical Paper by C. Zhan and J.H. Yin

ELASTICANALYSIS OF SOIL-GEOSYNTHETICINTERACTION

ABSTRACT: A two-dimensional analytical model based on elasticity theory isdeveloped for modeling soil-geosynthetic interaction for a geosynthetic layer of finitelength embedded horizontally between granular fill and soft ground under a foundationloading. The soil-geosynthetic interaction is analyzed in both the vertical and horizon-tal directions. The vertical soil-geosynthetic interaction is modeled using Winklersprings. The horizontal soil-geosynthetic interaction is simulated using horizontalshear springs. The geosynthetic reinforcement is assumed to be elastic and only expe-riences tension without bending and shearing resistance. The soil mass is treated as anisotropic homogeneous elastic material. The proposed analytical model can be used to

obtain the two-dimensional deformation and stress field in the soil mass and reinforce-ment. The results from the proposed model are compared with rigorous two-dimen-sional finite difference modeling results. Material and geometry parametric analysesare carried out using the proposed model to investigate the effectiveness of soft groundimprovement.

KEYWORDS: Geosynthetic, Reinforcement, Soft ground, Bearing capacity,Settlement, Winkler foundation.

AUTHORS: C. Zhan, Engineer, Binnie Black & Veatch (Hong Kong) Limited, 11/F,New Town Tower, Shatin, NT, Hong Kong, Telephone: 1/852-2601-1000, Telefax: 1/852-2601-3988, E-mail: [email protected]; and J.H. Yin, Professor, Department ofCivil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom,Kowloon, Hong Kong, Telephone: 1/852-2766-6065, Telefax: 1/852-2334-6389, E-

mail: [email protected].

PUBLICATION: Geosynthetics Internationalis published by the Industrial FabricsAssociation International, 1801 County Road B West, Roseville, Minnesota 55113-4061, USA, Telephone: 1/651-222-2508, Telefax: 1/651-631-9334. Geosynthetics

Internationalis registered under ISSN 1072-6349.

DATE: Original manuscript submitted 17 April 2000, revised version received 22August 2000, and accepted 27 August 2000. Discussion open until 1 August 2001.

REFERENCE: Zhan, C. and Yin, J.H., 2001, Elastic Analysis of Soil-GeosyntheticInteraction, Geosynthetics International, Vol. 8, No. 1, pp. 27-48.

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ZHAN AND YIN Elastic Analysis of Soil-Geosynthetic Interaction

28 GEOSYNTHETICS INTERNATIONAL 2001, VOL. 8, NO. 1

1 INTRODUCTION

Geosynthetics have been used for over 70 years; however, its rapid growth is use onlyoccurred in the last 20 years due to the availability of competent geotextiles (Koerner1990). Geosynthetics can be utilized for many different functions, such as separation,reinforcement, filtration, drainage, and as a moisture barrier. Among the many func-tions, geosynthetic reinforcement is widely recognized as an effective means and cost-effective method of improving foundations on soft ground. To increase the bearingcapacity and reduce settlement of a foundation on soft ground, a layer of geosyntheticsis placed on the soft ground and then covered with granular fill. The use of geosyn-thetic reinforcement could effectively confine the soil movement both above andbelow the geosynthetic reinforcement and dissipate the concentrated load to the softground. Hausmann (1987) summarized the following three mechanisms as to how themechanical behavior of soft ground is improved by geosynthetic reinforcement: (1)aggregate restraint; (2) subgrade restraint; and (3) membrane support.

Under an applied foundation load, the soil mass tends to move away from the loadcenter. Due to the large stiffness difference between the geosynthetics and soil massand the deformation compatibility requirement, shear stresses develop at the granularfill (or aggregate)-reinforcement interface and the soft ground (subgrade)-reinforce-ment interface. The shear stress at the granular fill-reinforcement interface provides aconfinement to the granular fill (aggregate) above the reinforcement. This confinementincreases the strength of the fill material and, consequently, leads to a higher load car-rying capacity (aggregate restraint). Similarly, the resulting shear stress at the softground-reinforcement interface also restrains the movement of soft ground (subgrade)below the reinforcement and, consequently, increases the strength of the original softground (subgrade restraint). The shear stresses at the interface along the reinforcementresult in a tensile force in the reinforcement. When the geosynthetic reinforcementdeflects, the vertical component of the reinforcement tensile force reduces the loadtransferred to the soft ground, which is commonly referred as membrane support(Hausmann 1987; Espinoza 1994).

Soil-geosynthetic interaction for a geosynthetic placed between soft ground andgranular fill under a foundation load is a two-dimensional (2-D), plane-strain problem.Various analytical models have been developed for this problem (Bourdeau et al. 1982;Love et al. 1987; Madhav and Poorooshasb 1988; Bourdeau 1989; Poorooshasb 1989;Poran et al. 1989; Ghosh 1991; Poorooshasb 1991; Espinoza 1994; Ghosh and Madhav1994a,b,c; Khing et al. 1994; Shukla and Chandra 1994, 1995). Madhav andPoorooshasb (1988) simplify the problem to one dimension and model the soft soilusing Winkler springs. In addition, Madhav and Poorooshasb make the followingassumptions: the granular fill behaves as Pasternak shear layers (Pasternak 1954); thegeosynthetic membrane buried in the granular fills can withstand tensile force only;

and the membrane behavior is elastic. The Madhav and Poorooshasb model was laterextended by Ghosh and Madhav (1994a,b), Shukla and Chandra (1994, 1995), and Yin(1997a,b) to account for the membrane effect, confinement effect, stiffness of thegeomembrane, and compaction. Yin (1999, 2000) noted that the Pasternak model con-siders the shear deformation, but not the bending deformation, while the Winkler

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model considers the bending deformation (i.e., extension or compression along thehorizontal axial direction of the beam), but not the shear deformation (i.e., shearing in

the vertical direction). Yin (1999, 2000) has successfully applied a Timoshenko Beam(TB) model for a reinforced granular base on soft ground. The TB model can considerboth the bending deformation and shear deformation and is a step forward to modelinggeosynthetic-reinforced soils; however, both the Pasternak and TB models are one-dimensional (1-D) models. The true interaction that can be obtained using a two-dimensional (2-D) model has not been fully realized.

Instead of using a Pasternak shear layer for the granular fill, Bourdeau (1989)assumed that the vertical stresses in both the granular fill and soft ground follow a sto-chastic stress distribution. However, this model does not consider the lateral restrainteffect, and it does not satisfy the exact equilibrium equation for the 2-D problem.

To accurately model the soil interaction with a layer of geosynthetic reinforcementof limited length, the present paper views the reinforcement as a disturbance or pertur-bation to the original soil mass. The 2-D problem is modeled: (1) without the presenceof the geosynthetic reinforcement; and (2) as a result of the soil mass-reinforcementinteraction. For the case of no reinforcement, the solution is obtained using the FlamantSolution (Poulos and Davis 1974) for a concentrated line force applied on the surface ofan isotropic homogeneous elastic half-plane (a multi-layer elastic solution can also beused). For the case with reinforcement (soil-geosynthetic interaction case), the solutionis obtained using equilibrium equations for the geosynthetic reinforcement. If the geo-synthetics have the same stiffness as the soil mass, the second solution is zero. The soil-geosynthetic interactions are represented using a Winkler foundation for vertical inter-action and shear springs for horizontal interaction. The proposed model can be used toobtain both the stress and displacement fields of the 2-D problem.

2 SOIL-GEOSYNTHETIC INTERACTION

The soil-geosynthetic interaction is modeled as a 2-D, plane-strain problem (Figure 1).The geosynthetic reinforcement (geotextile, geogrid, geonet, geomembrane, or geo-composite) is placed horizontally over the soft ground and covered with granular fill.The reinforcement is assumed to be an elastic material with only tensile resistance. Forsimplicity, the soil mass, composed of both granular fill and soft ground, is treated asan isotropic homogeneous elastic material. Upon any surface loading, such as founda-tion or traffic loading, both the soil mass and the reinforcement deform. Since the rein-forcement has a larger stiffness compared to the soil mass, the reinforcement deformsless than the soil mass under the same loading. As a result, soil mass-reinforcementinteraction develops. This interaction results in a load transfer from the soft ground tothe strong reinforcement and, consequently, improves the mechanical behavior of thesoft ground.

Under any surface loading, the deformation of a layer of horizontally embeddedreinforcement can be divided into two components: (i) deformation at the reinforce-ment location without the presence of the reinforcement as a result of the surface load-ing; (ii) deformation caused by soil-reinforcement interaction as a result of the

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stiffness difference between the two materials. The horizontal and vertical displace-ments of the reinforcement is given by:

(1)

where: u0(x) and w0(x) = horizontal and vertical displacements as a result of loading with-out geosynthetic reinforcement, respectively (Figure 2); and ui(x) and wi(x) = horizontaland vertical soil-reinforcement interaction displacements, respectively (Figure 2).

The horizontal interaction displacement, ui(x), is related to an interaction shearstress at the soil-reinforcement interface (Figure 2). Using the horizontal shear springconcept, the interaction shear stress, (x), at the soil-reinforcement interface isassumed to be proportional to the horizontal interaction displacement:

(2)

where ks is the interface shear modulus at the soil-reinforcement interface. It should benoted that the interaction shear stress is the sum of both the interaction shear stressesabove and below the geosynthetic reinforcement.

The vertical interaction displacement, wi(x), is related to an interaction pressure atthe soil-reinforcement interface. It should be mentioned that the interaction pressure isnot the total pressure at the soil-reinforcement interface. This interaction pressure isonly caused by the presence of the reinforcement. If the soil reinforcement has the

same stiffness as the soil mass, the interaction pressure is zero. Using the Winklerfoundation assumption, the interaction pressure,p(x), is given by:

(3)

u x( ) u0 x( ) ui x( )+=

w x( ) w0 x( ) wi x( )+=

x( ) ks ui x( )=

x( ) k w i x( )=

a a orB/2

x

y

L /2

h

L /2

Granular fill

Soft soil

Geosynthetic

Figure 1. Geosynthetic-reinforced soft ground under a foundation loading.

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where kis the elastic foundation modulus for the soil mass. The interaction pressure isalso referred to as an additional bearing capacity due to the membrane effect (Espinoza

1994). The interaction pressure is composed of both the interaction pressures aboveand below the geosynthetic reinforcement.For an isolated, infinite, small reinforcement element (Figure 2), the reinforcement

element is subjected to both shear stress and normal pressure at the interface and ten-sile force within the reinforcement. Equilibrium in the x direction is given by:

(4)

where: and = angles defined in Figure 2; and T= tensile force in the reinforcement.Since the element size is infinitely small, and will be more or less the same value.Therefore, the following approximation can be used:

(5)

Substituting Equation 5 into Equation 4 results in:

T cos ks ui ds +

2-------------

cos+ T dT+( ) cos=

cos cos

+

2-------------

cos

x

y

a

dx T + dT

T

Granular fill

B / 2

Soft soil

Deflectedgeosynthetic

ds

b

pa

Geosynthetic before loading

Deformation underloading without ageosynthetic

u0

u -ui

w0 w

-wi

pb

Notes: a= interaction shear stress above the geosynthetic reinforcement;b= interaction shear stress below the reinforce-ment;pa = interaction pressure above reinforcement;pa = interaction pressure below the reinforcement; T= tension in thereinforcement; = a+b= interaction shear stress; andp =pa +pb = interaction pressure.

Figure 2. Force diagram of an isolated, infinitely small reinforcement element.

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(6)

Equilibrium in they direction gives:

(7)

Since the reinforcement is placed horizontally below ground and the reinforcementdeflection is relatively small, the following approximation can be used

(8)

Substituting Equation 8 into Equation 7 results in:

(9)

Using the equilibrium equation in the x direction (Equation 6), Equation 9becomes:

(10)

Equation 10 is basically the same as that for a thin membrane subjected to a pressureloading.

Since the vertical displacement of the reinforcement is relatively small, the hori-zontal distance of the element shown in Figure 2 is approximately the same as theactual distance between both ends of the element, i.e., ds dx. Therefore, the equilib-

rium equations in both thex andy directions can be simplified as:

(11)

Applying constitutive relationship, t = T/t = E(du/dx), to the reinforcement,whereEand tare the elastic modulus and thickness of the reinforcement, respectively,the equilibrium equations in both the x andy directions can be combined to result inthe following equation in terms of the interaction horizontal displacement, ui(x):

(12)

Under any surface loading, the horizontal displacement of the first component,u0(x), can be determined using the Flamant Solution. With boundary conditions corre-

dTds------ ks ui=

T sin ks ui ds +

2-------------

sin k wi ds+ + T dT+( ) sin=

dwdx------- w tan sin= =

Tw ks ui ds w wdx2------+ k wi ds+ + T dT+( ) w wdx+( )=

wkw i

T--------

dsdx------=

dTdx------ ks ui=

d2w

dx2

---------k wi

T---------=

d2u

i

dx2----------

ks

Et----- ui

d2u

0

dx2-----------=

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sponding to the applied surface load and placement of the reinforcement, Equation 12can be solved and the interaction displacement can be determined. The known interac-

tion displacements can then be used to determine both the interaction shear stress andinteraction pressure at the interface.

3 SOLUTION TO THE 2-D, PLANE-STRAIN PROBLEM

Similar to the solution of reinforcement displacements, the ground movement for the2-D, plane-strain problem is the sum of the movement as a result of surface loadingwithout the presence of reinforcement and the movement resulting from the interactionpressure and interaction shear stress at the soil-reinforcement interface. The grounddeformation (U,V) of geosynthetic-reinforced soft ground can be divided into twocomponents as follows:

(13)

where: (x,y) = coordinate of the 2-D problem; U0(x,y) and V0(x,y) = horizontal and ver-tical displacements at coordinate (x,y) for the 2-D problem under surface loading with-out the presence of reinforcement, respectively; and Ui (x,y) and Vi(x,y) = horizontaland vertical interaction displacements at coordinate (x,y) for the 2-D problem resultingfrom the interaction shear stress and interaction pressure at the soil-reinforcementinterface, respectively.

The stress field for the 2-D problem (x , y , xy) can also be divided into two com-ponents as follows:

(14)

where: x = normal stress component in thex direction; y = normal stress componentin they direction; xy = shear stress component; and the superscripts 0 and i representthe response of the soil mass due to the surface loading without the presence of rein-forcement and the response of the soil mass as a result of soil-reinforcement interac-tion, respectively.

Both the stress field and displacement field, as a result of surface loading without thepresence of reinforcement, can be determined using the Flamant Solution. The interactioncomponent can be determined using the Melan Solution (Poulos and Davis 1974) with theknown interaction shear stress and interaction pressure at the reinforcement position.

4 ANALYSIS OF FOOTING ON REINFORCED SOFT GROUND

Equations 1 to 14are applicable to a plane-strain problem with horizontally embedded

U x y

,( )U

0x y

,( )U

ix y

,( )+=

V x y,( ) V0 x y,( ) Vi x y,( )+=

x x y,( ) x0 x y,( ) x

i x y,( )+=

y x y,( ) y0 x y,( ) yi x y,( )+=

xy x y,( ) xy0 x y,( ) xy

i x y,( )+=

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reinforcement under any surface loading. In the following, the above analysis is used tostudy a typical problem of a footing on reinforced soft ground. The footing is simpli-

fied as a uniform distributed surface load (Figure 1). The surface load, Py , is constantover the strip a xa andy = 0, andPy = 0 elsewhere ony = 0. The ground move-ment and stresses as a result of the footing load without reinforcement are given by:

(15)

and

(16)

where:Es and = soil elastic modulus and Poissons ratio, respectively, for both thegranular fill and soft soil; r1 = distance between coordinates (a,0) and (x,y); r2 = dis-tance between coordinates (a,0) and (x,y); l= distance from the loading center wherethe vertical displacement is taken as zero atx = landy = 0; and

(17)

At the reinforcement location, the horizontal and vertical displacements withoutthe presence of reinforcement are given by:

(18)

where h is the reinforcement burial depth:To determine the interaction displacements, the following finite difference method

is used to solve Equations 11 and 12:

U0

x y,( )1 +( )P

y

Es

------------------------ 1 2( ) x a( )1

x a+( )2

a+[ ] 1 ( ) y r12 r

22( )ln+

=

V0

x y,( )1 +( )P

y

Es

------------------------ 1 2( )y 1

2

( ) 1 ( ) x a( ) r12ln x a+( ) r

22ln l a+( ) l a+( )

2l a( ) l a( )

2lnln++

=

x0

Py

------

1 2

y x a( )r12

-------------------y x a+( )

r22-------------------

+=

y0

Py

------ 1 2

y x a( )r1

2-------------------

y x a+( )r2

2-------------------+=

xy0

Py

------ y

2 1

r12

-----1

r22

----- =

1y

x a-----------atan= 2

yx a+------------atan=

r12 x a( )2 y2+= r2

2 x a+( )2 y2+=

u0 x( ) U0 x h,( )=

w0 x( ) V0 x h,( )=

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(19)

where the superscriptj represents the finite difference grid.To solve Equation 19, boundary conditions are required. At the center of the geo-

synthetic reinforcement, the symmetry condition requires that the horizontal interac-tion displacement is zero and the vertical interaction displacement gradient is zero. Atthe far end of the geosynthetic, the tensile force in the geosynthetic and the interactionpressure are zero. The boundary condition can be written as follows:

(20)

whereL is the length of geosynthetic reinforcement.After obtaining the interaction displacements, both the geosynthetic interaction

shear stress and interaction pressure can be calculated from the horizontal and verticalinteraction displacements, respectively. The resulting interaction shear stress and inter-action pressure values can then be used to obtain the stress and displacement fields ofthe soil mass using the Melan Solution.

5 COMPARISON OF THE PROPOSED MODEL WITH RIGOROUS

FINITE DIFFERENCE ANALYSES

The soil-geosynthetic interaction is analyzed using both the proposed mechanicalmodel and a rigorous 2-D finite difference model using the computer program FLAC(FLAC 1998). In the FLAC analyses, the geosynthetics are assumed to be fully bondedto the soil medium. Both the soil mass and the geosynthetics are treated as an elasticmaterial. The geosynthetics are modeled as cable elements. The physical propertyparameters used for the soil mass and the geosynthetics are given in Table 1.

A 51 51 finite difference grid is used as shown in Figure 3. Only half of the planeis analyzed as a result of the symmetry condition. The vertical boundary on the right-hand side is set 20 m away from the loading center. Both vertical boundaries are

assumed to be free in the vertical direction with restricted horizontal displacement.The bottom horizontal boundary is set 30 m below ground, restricted in the verticaldirection, and assumed to be free in the horizontal direction. A surface load of 100 kPais applied over 1 m x 1 m. Comparisons of the finite difference results from

ui

j 1+ 2ks

Et

----- x2+

ui

ju

i

j 1d

2u0

j

dx2------------ x2+ + 0=

wij 1+ 2

k

Etdu

j

dx--------

---------------- x2+

wij

wij 1 d

2w0

j

dx2------------- x2+ + 0=

uix 0=

0,=d ui u0+( )

dx------------------------

x L 2=0=

dwidx--------

x 0=

0,= wix L 2=

0=

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Table 1. Physical properties of the soil and geosynthetic.

Physical property Value

Elastic modulus of soil,Es 1.0 kPa

Poissons ratio of soil, 0.45

Geosynthetic modulus,E 8 GPa

Geosynthetic thickness, t 5 mm

Geosynthetic length,L 6 m

Interface shear modulus, ks 800 kPa/m

Foundation modulus, k 500 kPa/m

-1.250

-7.50

-2.50

2.50

7.50

12.50

17.50

22.50

-7.50 -2.50 2.50 7.50 12.50 17.50 22.50 27.50

Pressure of 100 kPa

Figure 3. Finite difference mesh used in the present study.

Pressure = 100 kPa

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FLAC with the results from the proposed model are presented in this section.In the proposed mechanical model, the soil-geosynthetic interaction is simulated by

using the foundation modulus and interface shear modulus. The foundation moduluscan be directly estimated from soil modulus and foundation width as Es /B(1

2),whereB is the foundation width. The interface shear modulus can also be evaluatedfrom the soil properties and geometric parameters.

To estimate the interface shear modulus, a finite difference analysis is carried outusing FLAC for a semi-infinite elastic solid subjected to horizontal constant traction,

Px = 100 kPa, over a distance of 0 x 3 and 100 kPa over a distance of3 x 0,z, andy = 1. The soil parameters given in Table 1 are used in the analysis.The finite difference mesh is given in Figure 3. Symmetry boundary conditions areapplied at x = 0. In the proposed mechanical model, the interface shear modulus isdefined as the ratio of shear stress to the corresponding shear deformation at the rein-forcement location. As a result, the ratio between the applied constant traction, Px =100 kPa, and the corresponding horizontal displacement determined from FLAC anal-ysis, is the interface shear modulus.

Figure 4 is a plot of the estimated interface shear modulus profile from FLAC. Theestimated value is between 600 and 1,500 kPa/m with an average value of 800 kPa/m.In the following calculation, the average interface shear modulus of 800 kPa/m is usedfor the proposed mechanical model.

Comparisons of the geosynthetic horizontal displacement using FLAC and the pro-posed model are shown in Figure 5. In general, the two results are similar with onlyminor variations. This variation is mainly attributed to the assumption used in the pro-posed model for modeling the soil-geosynthetic interactions. In the model, the soil-geosynthetic interactions are simplified as Winkler springs in the vertical direction andshear springs in the horizontal direction. The comparison of geosynthetic tensile forcevalues obtained using the model and FLAC is given in Figure 6. As shown in Figure 6,

0

300

600

900

1200

1500

0 0.5 1 1.5 2 2.5 3

Distance from center (m)

Interfaceshearmodulus(kPa/m)

Average value of 802 kPa/m

Interface shear modulus

estimated from FLAC

Figure 4. Variation of interface shear modulus along the geosynthetic reinforcement.

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the proposed model predicts lower tensile force at the center and higher tensile forceclose to edge of the geosynthetic.

0

10

20

30

40

50

60

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Tensileforce(kN) Proposed model

FLAC analysis

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0Distance from center (m)

Horizontaldisplacement(mm)

Proposed model

FLAC analysis

Figure 5. Comparison of the horizontal displacement along the geosynthetic

reinforcement using the proposed model and a FLAC analysis.

Figure 6. Comparison of tensile force along the geosynthetic reinforcement using the

proposed model and a FLAC analysis.

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6 PARAMETRIC ANALYSES

To evaluate the effectiveness of ground improvement using geosynthetics, parametricanalyses were carried out for geosynthetic burial depth, geosynthetic length, interfaceshear modulus, soil elastic modulus, and geosynthetic stiffness. For the remainingparameters, the baseline values given in Table 2 were used. In the following discus-sions, both the vertical and horizontal displacements are the displacements of geosyn-thetic reinforcement.

Typical parametric analyses results are given in Figures 7, 8, and 9 using the base-line parameters in Table 2. Figure 7 presents the variation of horizontal displacementalong the length of the geosynthetic. The three curves in Figure 7 represent the hori-zontal displacement without a geosynthetic layer, horizontal displacement with a geo-synthetic layer, and the interaction displacement, respectively. It is evident that thepresence of a geosynthetic layer significantly reduces the horizontal displacement,which consequently provides a restraint to the soil mass both above and below the geo-

synthetic. The restraint provided by the geosynthetic is reflected by the magnitude ofthe tensile force generated in the geosynthetic (Figure 8). As expected, the tensile forceis a maximum at the center and approaches zero at the far end of the geosyntheticlayer. The vertical displacement profile along the geosynthetic reinforcement is givenin Figure 9. The presence of geosynthetics reduces the vertical displacement; however,the reduction is not as significant as the horizontal displacement shown in Figure 7.This is expected because geosynthetics mainly provide ground restraint along thedirection of its placement.

As discussed in Section 1, the improvement of soft ground due to geosynthetics isattributed to three mechanisms: (1) aggregate restraint, (2) subgrade restraint, and (3)membrane support. Mechanisms 1 and 2 are reflected in the magnitude of the tensileforce in the geosynthetic. The larger the tensile force, the greater the restraint to boththe aggregate and subgrade. The membrane support is reflected in the reduction in ver-tical displacement along the geosynthetic, which is defined as the ratio of the interac-

Table 2. Baseline parameters.

Baseline parameter Value

Elastic modulus of soil,Es 1.0 MPa

Poisson's ratio of soil, 0.45

Geosynthetic stiffness,Et 40 MPa-m

Interface shear modulus, ks 1.0 MPa/m

Foundation modulus, k 0.5 MPa/m

Foundation width,B 2.0 m

Geosynthetic burial depth, h 1.0 m

Length of geosynthetic,L 6.0 m

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tion vertical displacement, wi(x), to the vertical displacement without geosynthetics,w0(x). The larger the vertical displacement reduction, the greater the membrane sup-port. In the remainder of this section, the maximum tensile force and the maximumvertical displacement reduction along the geosynthetic are used to evaluate improve-ment to soft ground using geosynthetics.

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Horizontaldisplacement(m)

Displacement with geosynthetics

Displacement without geosynthetics

Interaction displacement

Figure 7. Variation of horizontal displacement along the geosynthetic reinforcement.

0

10

20

30

40

50

60

70

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Tensileforce(kN)

Figure 8. Variation of tensile force along the geosynthetic reinforcement.

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The placement of geosynthetic reinforcement at different depths below groundresults in different magnitudes of ground improvement. Figure 10 shows the depen-dence of the maximum tensile force and the maximum vertical displacement reductionon geosynthetic burial depth. It is evident that the maximum vertical displacementreduction is a maximum when the burial depth is 0.8 m. The optimal location of thegeosynthetic for providing restraint to both the aggregate and subgrade is 1.5 m belowground. Overall, in order to maximize ground improvement, geosynthetics should beplaced 0.35B to 0.7B below ground.

Ground improvement is also affected by geosynthetic length. Figure 11 presentsthe variation of the maximum tensile force and the maximum vertical displacementreduction with geosynthetic length. Both the maximum tensile force and the maximumvertical displacement reduction increase with increased geosynthetic length. However,the increase becomes insignificant when the geosynthetic length is greater than 3B.Consequently, the best and most economical improvement to the soft ground can beachieved when the geosynthetic length is between 2.5B to 3B. It should be noted thatchange in burial depth could also affect the optimal geosynthetic length.

Figure 12 presents the variation of the maximum tensile force and the maximumvertical displacement reduction with interface shear modulus. Figure 12 clearly showsthat the increase in the interface shear modulus increases both the maximum tensileforce and the maximum vertical displacement reduction as expected. Consequently, tobetter improve the strength of soft ground using geosynthetics, the soil-geosynthetic

interface (i.e., the aggregate above the geosynthetic and the subgrade below the geo-synthetic) shear modulus should be maximized.

The effectiveness of soft ground improvement is highly dependent on the stiffnessdifference between the geosynthetics and the soft ground. Figure 13 shows the varia-tion of the maximum tensile force and the maximum vertical displacement reduction

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 0.5 1 1.5 2 2.5 3


Verticaldisplacement(m)

Displacement with geosynthetics

Displacement without geosynthetics

Interaction displacement

Figure 9. Variation of vertical displacement along the geosynthetic.

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with the soil elastic modulus. It is evident from Figure 13 that both the maximum ten-sile force and the maximum vertical displacement reduction decrease with the increase

in the soil elastic modulus. As expected, geosynthetics are most effective for very softground.

Figure 14 is a plot of the variation of the maximum tensile force and the maximumvertical displacement reduction with geosynthetic stiffness and shows that increasedgeosynthetic stiffness can also improve the strength of soft ground. As evident from

20

30

40

50

60

70

80

90

0.5 1.0 1.5 2.0 2.5

Geosynthetic burial depth (m)

Tensileforce(kN)

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

wi

/w0

20

30

40

50

60

70

80

90

100

2 3 4 5 6 7 8 9 10

Geosynthetic length (m)

Te

nsileforce(kN)

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

wi

/w0

Figure 10. Variation of the maximum tensile force and maximum vertical displacement

reduction with geosynthetic burial depth.


reduction with geosynthetic length.

Tensile force

wi(x) / w0(x)

Tensile force

wi(x) / w0(x)

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Figure 14, greater geosynthetic stiffness values increases both the maximum tensileforce and the maximum vertical displacement reduction. However, the increasebecomes negligible when the geosynthetic stiffness exceeds 15 MPa-m.

0

20

40

60

80

100

120

140

0 1 2 3 4 5 6

Soil elastic modulus (MPa)

Tensileforce(kN)

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

wi

/w0

0

50

100

150

200

250

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Interface shear modulus (MPa/m)

Tensileforce(kN)

0%

1%

2%

3%

4%

5%

6%

7%

8%

wi

/w0

Figure 13. Variation of maximum tensile force and maximum vertical displacement

reduction with soil elastic modulus.


reduction with interface shear modulus.

Tensile force

wi(x) / w0(x)

Tensile force

wi(x) / w0(x)

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7 CONCLUSIONS

A two-dimensional analytical model based on elasticity theory is developed for model-ing soil-geosynthetic interaction whereby a layer of geosynthetics of finite length isplaced horizontally between the granular fill and soft ground under foundation load-ing. The mechanical model considers the soil-geosynthetic interactions in both the hor-izontal and vertical directions. The proposed model can be used to obtain both thestress and displacement fields of the two-dimensional problem. The results from theproposed model are compared with the results from the finite difference programFLAC. The two results agree with minor variations due to the assumptions used forboth the horizontal and vertical soil-geosynthetic interactions.

Parametric analyses are carried out for material properties and geometric parame-ters. Based on the parametric analyses, the following conclusions can be made:

The geosynthetic reinforcement is most effective when it is placed 0.35B to 0.7Bbelow ground.

A geosynthetic length of 2.5B to 3B would be economical and effective in improv-ing soft ground; however, it is expected that this could be greatly affected when thegeosynthetic burial depth is changed.

An increase in interface shear modulus greatly improves the soft ground.

Both a decrease in soil modulus and increase in geosynthetic stiffness would resultin an increase in soft ground improvement.

It should be noted that the above conclusions are limited to the cases analyzed. If dif-ferent baseline parameters are used, different results may possibly be obtained.

0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Geosynthetic stiffness (MPa-m)

Tensileforce(kN)

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

wi/w0

Figure 14. Variation of maximum tensile force and maximum vertical displacement

reduction with geosynthetic stiffness.

Tensile forcewi(x) / w0(x)

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ACKNOWLEDGEMENT

The financial support from the Hong Kong Polytechnic University (PolyU ProjectNo.YB68) for the research and preparation of the present paper is gratefully acknowledged.

REFERENCES

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Bourdeau, P.L., Harr, M.E., and Hottz, R.D., 1982, Soil Fabric Interaction: An Ana-lytical Model, Proceedings of the Second International Conference on Geotex-tiles, IFAI, Vol. 2, Las Vegas, Nevada, USA, August 1982, pp. 387-391.

Espinoza, R.D., 1994, Soil-Geotextile Interaction: Evaluation of Membrane Sup-port, Geotextiles and Geomembranes, Vol. 13, No. 5, pp. 281-293.

FLAC, 1998, Software and Manuals, Version 3.4, ITASCA, USA.

Ghosh, C., 1991, Modeling and Analysis of Reinforced Foundation Beds, Ph.D. The-sis, Department of Civil Engineering, I.I.T., Kanpur, India, 218 p.

Ghosh, C. and Madhav, M.R., 1994a, Reinforced Granular Fill-Soft Soil System:Confinement Effect, Geotextiles and Geomembranes, Vol. 13, No. 5, pp.727-741.

Ghosh, C. and Madhav, M.R., 1994b, Reinforced Granular Fill-Soft Soil System:Membrane Effect, Geotextiles and Geomembranes, Vol. 13, No. 5, pp. 743-759.

Ghosh, C. and Madhav, M.R., 1994c, Settlement Response of a Reinforced ShallowEarth Bed, Geotextiles and Geomembranes, Vol. 13, No. 5, pp. 643-656.

Hausmann, M.R., 1987, Geotextiles for Unpaved Roads - A Review of Design Proce-dures, Geotextile and Geomembranes, No. 5, pp. 201-233.

Khing, K.H., Das, B.M., Puri, V.K., Yen, S.C., and Cook, E.E., 1994, Foundation onStrong Sand Underlain by Weak Clay with Geogrid at the Interface, Geotextilesand Geomembranes, Vol. 13, No. 3, pp. 199-206.

Koerner, R.M., 1990, Designing with Geosynthetics, Second Edition, Prentice-HallInc., Englewood Cliffs, New Jersey, USA, 652 p.

Love, J.P., Burd, H.J., Milligan, G.W.E., and Houlsby, G.T., 1987, Analytical andModel Studies of Reinforcement of a Layer of Granular Fill on a Soft Clay Sub-grade, Canadian Geotechnical Journal, Vol. 24, No. 4, pp. 611-622.

Madhav, M.R. and Poorooshasb, H.B., 1988, A New Model for Geosynthetic Rein-forced Soil, Computers and Geotechnics, Vol. 6, No. 4, pp. 277-290.

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Poorooshasb, H.S., 1991, On Mechanics of Heavily Reinforced Granular Mats, Soilsand Foundations, Vol. 31, No. 2, pp. 134-152.

Poorooshasb, H.S., 1989, Analysis of Geosynthetic Reinforced Soil using a Simple

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Transform Function, Computers and Geotechnics, Vol. 8, No. 4, pp. 289-309.

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Footings on Geogrid-Reinforced Soil, Proceedings of Geosynthetics 89, IFAI,Vol. 1, San Diego, California, USA, February 1989, pp. 231-242.

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Shukla, S.K. and Chandra, S., 1995, Modeling Geosynthetic-Reinforced EngineeredGranular Fills on Soft Soil, Geosynthetics International, Vol. 2, No. 3, pp. 603-617.

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NOTATIONS

Basic SI units are given in parentheses.

a = half-width of foundation (m)B = width of foundation (m)

E = elastic modulus of geosynthetic (Pa)

Es = elastic modulus of granular fill and soft soil (Pa)

h = geosynthetic burial depth (m)

k = elastic foundation modulus for soil mass (Pa/m)

ks = interface shear modulus at soil-geosynthetic interface (Pa/m)

L = length of geosynthetic (m)

l = distance from loading center where vertical displacement is taken as

zero atx = 1 and y = 0(m)Px = constant horizontal traction(Pa)

Py = applied surface load (Pa)

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p(x) = interaction pressure at soil-geosynthetic interface (Pa)

pa = soil-geosynthetic interaction pressure above geosynthetic (Pa)pb = soil-geosynthetic interaction pressure below geosynthetic (Pa)

r1 = distance between coordinates (a,0) and (x,y) (m)

r2 = distance between coordinates (a,0) and (x,y) (m)

T = tensile force in geosynthetic (N)

t = thickness of geosynthetic (m)

U(x,y) = horizontal displacement at coordinate within soil mass (m)

Ui (x,y) = horizontal interaction displacement at coordinate within soil mass (m)

U0 (x,y) = horizontal displacement at coordinate within soil mass without the presence

of geosynthetics (m)u(x) = horizontal displacement of geosynthetic (m)

ui(x) = horizontal geosynthetic displacement due to soil-geosyntheticinteraction (m)

u0 (x) = horizontal displacement without geosynthetic at location ofgeosynthetic (m)

V(x,y) = vertical displacement at coordinate within soil mass (m)

Vi(x,y) = vertical interaction displacement at coordinate within soil mass (m)

V0 (x,y) = vertical displacement at coordinate within soil mass without the presenceof geosynthetics (m)

w(x) = vertical displacement of geosynthetic (m)

wi(x) = verticalgeosynthetic displacement due to soil-geosynthetic interaction (m)

w0 (x) = vertical displacement without geosynthetic at location of geosynthetic (m)

x = distance in horizontal direction (m)

y = distance in vertical direction (m)

= gradient of geosynthetic vertical displacement at coordinate (x) ()

= gradient of geosynthetic vertical displacement at coordinate (x + dx) ()

= Poissons ratio of granular fill and soft soil (dimensionless)

1 = angle defined in Equation 17 ()

2 = angle defined in Equation 17 ()

t = tensile stress in geosynthetic (Pa)

x = normal stress component inx direction (Pa)

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y = normal stress component iny direction (Pa)

(x) = soil-geosynthetic interface shear stress (Pa)a = soil-geosynthetic interaction stress above geosynthetic (Pa)

b = soil-geosynthetic interaction stress below geosynthetic (Pa)

xy = shear stress component (Pa)

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