modelling geosynthetic reinforced granular fills over soft soil

165GEOSYNTHETICS INTERNATIONAL S 1997, VOL. 4, NO. 2

Technical Paper by J.H. Yin

MODELLING GEOSYNTHETIC-REINFORCED

GRANULAR FILLS OVER SOFT SOIL

ABSTRACT: In this paper, a new one-dimensional mathematical model is proposedfor modelling geosynthetic-reinforced granular fills over soft soils subject to a verticalsurcharge load. The geosynthetic reinforcement consists of a membrane (geogrid, orgeotextile) placed horizontally in engineered granular fill, which is constructed oversoft soil. The proposed model is mainly based on the assumption of a Pasternak shearlayer. A new approach that consists of incorporating a deformation compatibility condi-tion into the model is introduced. The compatibility condition eliminates the need fortwo uncertain model parameters that are required to calculate the shear stresses betweenthe top and bottom granular fill layers and the geosynthetic layer. This compatibilitycondition also makes it possible to include the geosynthetic stiffness in the model. Theresults from the proposed model are compared to results from a two-dimensional finiteelement model and three other one-dimensional models reported in the literature. Thecomparison reveals that the proposed model gives similar results with respect to thesettlement of a geosynthetic-reinforced granular base on soft soft and the mobilised ten-sion in the geosynthetic layer.

KEYWORDS: Geosynthetic reinforcement, Soil improvement, Foundation model,Settlement, Tension force, Finite element modelling.

AUTHOR: J.H. Yin, Assistant Professor, The Department of Civil and StructuralEngineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, HongKong, Telephone: 852/2766-6065, Telefax: 852/2334-6389, E-mail:[email protected].

PUBLICATION: Geosynthetics International is published by the Industrial FabricsAssociation International, 345 Cedar St., Suite 800, St. Paul, Minnesota 55101-1088,USA, Telephone: 1/612-222-2508, Telefax: 1/612-222-8215. GeosyntheticsInternational is registered under ISSN 1072-6349.

DATES: Original manuscript received 16 December 1996, revised version received14 March 1997 and accepted 11 April 1997. Discussion open until 1 January 1998.

REFERENCE: Yin, J.H., 1997, “Modelling Geosynthetic-Reinforced Granular FillsOver Soft Soil”, Geosynthetics International, Vol. 4, No. 2, pp. 165-185.

YIN D Modelling Geosynthetic-Reinforced Granular Fills Over Soft Soil

166 GEOSYNTHETICS INTERNATIONAL S 1997, VOL. 4, NO. 2

1 INTRODUCTION

A foundation constructed on soft soils may experience excessive settlement and pos-sible bearing capacity failure under a surcharge load. One technique that is used to im-prove the strength of soft foundation soils is the placement of engineered granular fillscontaining geosynthetic reinforcement (e.g. geogrid, or geotextile) on the soft soil. Thecalculation of settlement and the ultimate bearing capacity of the geosynthetic-rein-forced granular fill over soft soil is an important issue facing design engineers. This pap-er is focussed on the modelling of the load-settlement behaviour of geosynthetic-rein-forced granular fill over soft soil. Various works have been carried out by manyresearchers in this area of study (Bourdeau et al. 1982; Love et al. 1987; Madhav andPoorooshasb 1988; Bourdeau 1989; Poorooshasb 1989; Poran et al. 1989; Ghosh 1991;Poorooshasb 1991; Espinoza 1994; Ghosh and Madhav 1994; Khing et al. 1994; Shuklaand Chandra 1994; Shukla and Chandra 1995). In these modelling approaches, a two-dimensional (2-D) plane strain problem (e.g. a strip footing) is simplified to a one-di-mensional (1-D) problem. In the 1-D model, the soft soil is represented by Winklersprings, the behaviour of which may be linear or nonlinear. The top and the bottom gran-ular fill layers are assumed to behave as Pasternak shear layers (Pasternak 1954). Thegeosynthetic layer (“membrane”) buried in the granular fill can withstand tensionforces only, and the geosynthetic layer behaviour is assumed to be elastic. This 1-Dmodel has an inherent problem, as does the Winkler foundation model, because no in-teraction between the springs is considered. However, as pointed out by Shukla andChandra (1995), the interaction of the springs is not very significant for highly com-pressible soft soils.

In most existing models, the longitudinal stiffness of the geosynthetic layer is as-sumed to be rigid in tension, but does not carry vertical shear forces (Ghosh and Madhav1994; Shukla and Chandra 1994; Shukla and Chandra 1995). The shear stresses actingon the top and bottom surfaces of the geosynthetic layer are assumed to be related tothe vertical stresses acting on the top and bottom surfaces by a constant that is definedby μ in the following expression: τ = μσ , where τ = shear stress and σ = vertical stress(Ghosh and Madhav 1994; Shukla and Chandra 1994; Shukla and Chandra 1995). Themagnitude of the constant μ is not exactly known which may cause difficulties whenusing most existing models.

In this paper, a deformation compatibility condition is proposed. The use of this com-patibility condition eliminates the two uncertain constants defined by μ for the top andbottom shear stresses acting on the geosynthetic layer. The tension stiffness modulusof the geosynthetic layer is incorporated into the proposed new model. In the first stepof model development, the behaviour of the granular fill is assumed to be elastic (Shuk-la and Chandra 1995). The nonlinear and elastic-plastic behaviour of the granular fillcan be considered using the method proposed in this paper. In order to validate the pro-posed model against other models, the results from the proposed model are comparedto two-dimensional FE modelling results and the results from other one-dimensionalmodels.



2 FORMULATION OF THE PROPOSED ONE-DIMENSIONAL MODEL

Figure 1 shows a simplified foundation model for a geosynthetic (e.g. geogrid, or geo-textile) reinforced granular fill over soft soil. This simplified model is similar to themodel proposed by Shukla and Chandra (1995). According to the assumption of a Pas-ternak shear layer, vertical planes in the granular fill layer move relative to each otheronly in the vertical direction without rotation. The interface between the geosyntheticlayer (“membrane”) and the two granular fill layers is assumed to be fixed (i.e. thereis rough contact and no slippage). In this study, the behaviour of the granular fill andthe geosynthetic layer is assumed to be elastic.

The proposed model can be a simplification of a strip footing (or other similar struc-tures) on a geosynthetic-reinforced granular fill over soft soil. The strip footing load issimplified as a vertical pressure loading boundary of width 2B (Figure 1). Consideringa unit length in the direction perpendicular to Figure 1, the vertical force equilibriumfor Element 1 in Figure 2 (top granular fill layer) leads to the following expression:

(1)qdx− σndl cos θ+ τndl sin θ+dτdx

dxHt= 0

where: q = footing pressure on the top granular fill layer; dx = projected element lengthin the x direction; σn = average normal stress acting on the bottom of the element; τn =average shear stress acting on the bottom of the element; dl = length of the bottom ofthe element as shown in Figures 2 and 4; θ = angle between the horizontal and the bot-tom of the element; Ht = thickness of the top granular fill layer; and τ = average shearstress acting on the side of the element.

According to the assumption of a Pasternak shear layer, the average vertical shearstress is τ = Gt dw/dx, and dτ/dx = Gt d2w/dx2, where Gt is the average shear modu-lus of the top granular fill layer, and w is the vertical displacement. Thus, Equation1 can be written as follows:

2Bq

Pasternak shear layers

Soft soil

Bedrock

ks

Hb Gb for bottom granular fill layer

Gt for top granular fill layerHtT, Tp T, TpEg for geosynthetic layer“membrane”

Figure 1. Schematic diagram of the proposed one-dimensional foundation model, and thedefinition of model parameters.



dx

w qx

ydl

σnτn

τ

τ + (∂τ/∂x)dx

τi + (∂τi/∂x)dx

τnσn

τinσin

σinτin

τi

qs = ks w

Tp , T + dT

Tp , T

θ + dθ

θ

Element 1

Figure 2. Three elements from a vertical segment of a geosynthetic-reinforced granular fillof infinitesimal width showing the forces and stresses on each element: (a) top granular filllayer; (b) geosynthetic layer; (c) bottom granular fill layer.

Element 2

Element 3

(a)

(b)

(c)

w + dw

(2)q= σn− τn tan θ− Ht Gtd2wdx2

where q = 0 outside of the loaded area.The vertical force equilibrium equation for Element 3 (bottom granular fill layer) in

Figure 2c is expressed as follows:

(3)− qs dx+ σ′n dl cos θ− σ′n dl sin θ+dτ ′

dxdx Hb= 0

where: qs = vertical reaction pressure of soft soil (simplified as Winkler springs); σ′n =average normal stress acting on the top of the element; τ′n = average shear stress actingon the top of the element; Hb = thickness of the bottom granular fill layer; and τ′= aver-



age shear stress acting on the side of the bottom element. Substituting τ′ = Gb dw/dx,and dτ′/dx = Gb d2w/dx2, Equation 3 becomes:

(4)qs= σ′n− τ′n tan θ− Hb Gbd2wdx2

where Gb is the average shear modulus of the bottom granular fill layer.The vertical force equilibrium equation for Element 2 in Figure 2b (geosynthetic lay-

er) leads to the following expression:

(5)dTdx

sin θ= − (T+ Tp) cos θdθdx− (σn− σ′n)+ (τn+ τ ′n) tan θ

where: T = mobilised tension in the geosynthetic layer; and Tp = pretension force of thegeosynthetic layer.

The horizontal force equilibrium equation for Element 2 (geosynthetic layer) is ex-pressed as follows:

(6)dTdx

cos θ= (T+ Tp) sin θdθdx+ (σn− σ′n) tan θ+ (τn+ τ′n)

Figure 3 shows the deformation of the top and bottom granular fill elements due toan increase in tension, dT. Assuming no sliding between the geosynthetic layer and thetop and bottom granular fill layers, the following deformation compatibility condition(or relationships) exists:

(7)ux= u′x= ug,x

where ux , u′x and ug,x are the horizontal displacements at the top and bottom granularfill layers, and the interface of the geosynthetic layer, respectively.

The values of ux and u′x are related to the shear strain values, γx and γ′x , respectively,which are then related to the shear stresses, τn and τ′n , as expressed in the following:

(8)ux= Ht γx= Ht

τn

Gt

(9)u ′x= Hb γ′x= Hb

τ′nGb

Using Equations 7 to 9, the shear stress, τn , can be expressed in terms of τ′n as follows:

(10)τn=Gt Hb

Gb Htτ′n

Figure 4 shows a stretched and rotated element of the geosynthetic layer. Using Equa-tions 7 to 9, the displacement of the geosynthetic element and the displacement incre-ment can be written as follows:



dx

ux

x

y

σn τn

τn

τin σin

τin

Tp , T + dTTp , T

uix

γix

γx Gt Ht

HbGb

Figure 3. Shear deformation due to an increase in geosynthetic layer mobilised tension:(a) top granular fill layer; (b) geosynthetic layer; (c) bottom granular fill layer.

(a)

(b)

(c)

Element 1

Element 2

Element 3

σn

σin

dx

ug,x

Figure 4. Stretching and rotation of a geosynthetic element.

dw

dl

ug,x + dug,x

θ



(11)

(12)

ug,x= u′x=Hb

Gbτ ′n

dug,x= du′x=Hb

Gbdτ ′n

The stretched length, dl, after rotation of the geosynthetic element is given by the fol-lowing expression:

(13)dl = (dw)2+ (dx+ dug,x)2

The strain, εg , of the geosynthetic element is calculated as follows:

(14)εg= dl− dxdx

= dwdx2+1+ dug,x

dx2 − 1

Using Equations 11, 12 and 14, the mobilised tension, T, in the geosynthetic elementis given by the following expression:

(15)T= Egεg = Egdw

dx2+1+ Hbdτ ′n

Gbdx2 − 1

where Eg is the tension modulus (kN/m) of the geosynthetic layer.Substituting Equation 10 into Equation 2, the following expression is obtained for

calculating the pressure on top of the granular fill:

(16)q= σn−Gt Hb

Gb Htτ ′n tan θ− Ht Gt

d2wdx2

Since dθ/dx = (cos2θ) d2w/dx2, Equations 5 and 6 can be written as follows:

(17)dTdx

sin θ= − (T+ Tp) cos3 θ d2wdx2 − (σn− σ′n)+ (τn+ τ ′n) tan θ

dTdx

cos θ= (T+ Tp) sin θ cos2 θ d2wdx2 + (σn− σ′n) tan θ+ (τn+ τ ′n) (18)

Since the rotation angle, θ, is related to dw/dx by tanθ = dw/dx, Equations 15 to 18and Equation 4 can be used to solve for the five unknowns, i.e. σn , σ′n , τ′n , w and T.

Equations 17 and 18 can be simplified by multiplying Equation 17 by cosθ and Equa-tion 18 by sinθ , and subtracting the former from the latter to give the following expres-sions:



(19)(σn− σ′n)= − (T+ Tp) cos3 θ d2wdx2

(τn+ τ ′n)= dTdx

cos θ (20)

Substracting Equation 4 from Equation 16 gives the following:

q− qs = (σn− σ′n)− (τn+ τ ′n) tan θ− (Ht Gt+ Hb Gb)d2wdx2 (21)

Substituting Equations 19 and 20 into Equation 21 results in the following expression:

q− qs =− (T+ Tp) cos3 θ d2wdx2− sin θ dT

dx− (Ht Gt+ Hb Gb)

d2wdx2 (22)

where qs = ksw, and ks is a spring constant. The unknowns in Equation 22 are T and w.Substituting Equation 10 into Equation 20, the following expression is obtained:

dTdx= 1

cos θ(τn+ τ ′n)= 1

cos θGt Hb

Gb Ht

+ 1 τ ′n (23)

Differentiating Equation 23:

d2Tdx2= sin θ Gt Hb

Gb Ht+ 1 τ ′n d2w

dx2 +1

cos θGt Hb

Gb Ht+ 1 dτ ′n

dx(24)

Rearranging Equation 23:

τ ′n=cos θ

Gt Hb

Gb Ht+ 1

dTdx (25)


TEg2+ 1= dw

dx2+1+ Hb dτ ′n

Gb dx2 (26)


dτ ′ndx=

Gb

Hb T

Eg2+ 1− dw

dx2 − 1

(27)

Substituting Equations 25 and 27 into Equation 24 gives the following expression:



d2Tdx2= sin θ cos θ d2w

dx2dTdx+ 1

cos θGt

Ht+

Gb

Hb

TEg2+ 1− dw

dx2 − 1

(28)

Equation 22 and Equation 28 are the two basic equations for solving the two un-knowns T and w.

3 NORMALISED EQUATIONS AND A NUMERICAL SOLUTION

3.1 Introduction

The following normalised parameters were used in the numerical solution (Shuklaand Chandra 1995):

X =xB

W=wB

H*t =

Ht

BH*

b=Hb

B

G*t =

Gt Ht

B2 ksG*

b=Gb Hb

B2 ks

E*g=

Eg

B2 ks

q*=q

B2 ksT*

p=Tp

B2 ks

T*=T

B2 ks

, ,,

, ,

, ,

where B is half the width of the surcharge load. All of the normalised parameters aredimensionless.

Using the normalisations given above, Equations 22 and 28 can be expressed as:

q*= W− sin θdT*

dX− [(T*+ T*

p) cos3 θ+ (G*t + G*

b)]d2WdX2

(29)

(30)d2T*

dX2= sinθ cosθ d2W

dX2

dT*

dX+ 1

cosθG*

t

H*2t

+G*

b

H*2b

T*

E*g2+ 1−dW

dX2 − 1

where:

tan θ=dWdX

sin θ=

dWdX

1+dWdX2

cos θ= 1

1+dWdX2

, ,

Using the following finite difference scheme:



ΔW

ΔX= Wi−Wi−1

ΔX

Δ2W

ΔX2= Wi−1− 2Wi+Wi+1

(ΔX)2

ΔT*

ΔX=

T*i − T*

i−1

ΔX

Δ2T*

ΔX2=

T*i−1− 2T*

i + T*i+1

(ΔX)2

Equations 29 and 30 become:

q*i = Wi−sin θ dT*

dX

i

− [(T*+ T*p) cos3 θ+ (G*

t + G*b)]i Wi−1− 2Wi+Wi+1

(ΔX)2 (31)

(32)

T*i−1− 2T*

i + T*i+1

(ΔX)2= (sin θ cosθ)i Wi−1− 2Wi+Wi+1

(ΔX)2 T*

i+1− T*i−1

2ΔX

+ 1(cos θ)i

G*t

H*2t

+G*

b

H*2b

i

T*

i

E*g2+ 1−dW

dX2

i

− 1The length L/B may be divided into n elements of the same increment length with (n

+ 1) node points (i = 0, 1, ..., n); thus, ∆X = (L/B)/n.

3.2 Boundary Conditions

A total of four boundary conditions are required for the two second order partial dif-ferential equations (Equations 31 and 32). Refer to Figure 5 for a symmetric geosynthet-ic-reinforced system with a constant pressure loading q of width 2B and a length of 2L.

At X = 0 (or x = 0), due to symmetry, the slope, dW/dX, will be zero and the ratio ofthe increase in tension, dT*/dX, will be zero as given in the following expressions:

dWdX= 0

dT*

dX= 0

At X = L/B (or x = L), the right end of the geosynthetic layer is free, so that the mobi-lised tension, T = 0. The shear stress acting on the geosynthetic layer at the right end(at X = L/B) will also equal zero since there is no confinement, that is, τ = τ′ = 0. Since



Figure 5. The one-dimensional model of a geosynthetic-reinforced granular foundationunder a constant surcharge load.

B B

LL

y

x

the Pasternak shear layer assumes that τ = Gt dw/dx, and τ′=Gb dw/dx, then dw/dx = 0.Thus, the boundary conditions at X = L/B are expressed as follows:

dWdX= 0

T*= 0

Using the four boundary conditions given above, Equations 31 and 32 can be solvedto obtain a unique solution for a given load. Since Equations 31 and 32 are both nonlin-ear equations, an iterative computing procedure must be used to obtain a solution. Inthe first iteration, an initial value of zero may be assumed for Ti and Wi (i = 0, 1, ..., n).The iteration is stopped for a relative error of [Wi

(k+1) -- Wi(k)]/Wi

(k) < 10-4 and [Ti(k+1)

-- Ti(k)]/Ti

(k) < 10-4 (i = 0, 1, ..., n), where k is an iteration index. A computer programwas written to solve Equations 31 and 32 using an iteration technique.

4 FINITE ELEMENT MODELLING

Finite element techniques may be used to study the effects of geosynthetic reinforce-ment in granular fills (Rowe and Soderman 1987). In order to compare the results ofthe proposed 1-D model against 2-D finite element solutions, a finite element (FE) mod-el was constructed as shown in Figure 6. In the FE model, a geosynthetic-reinforcedgranular fill was simulated as a 2-D plane strain problem based on continuum mechan-ics. This 2-D plane strain simulation is more accurate than the simplified 1-D model.The weakness of the 2-D model is that a significant amount of time was required to pre-pare the model, and to execute the numerical solution. As in the 1-D model, the soft soilwas simulated as a spring support so as to validate the 1-D modelling assumptions. Acommercial FE program called SIGMA/W (1995) was used for the FE modelling. The



5.8

5.6

5.4

5.2

5.0

4.84.6

5.8

5.6

5.4

5.2

5.0

4.8

4.6

---0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

---0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

B = 2 m

B = 2 m

Y(m)

Y(m)

L = 4 m

L = 4 m

X (m)

X (m)

Ht

HbHg

Ht

HbHg

Figure 6. Finite element (FE) model for a surcharge load, q = 52.6 kN/m2: (a) FE modelmesh and boundary conditions; (b) deformed FE model mesh.Notes: Ht = Hb = 0.3 m; Hg = 0.04 m.

description of the FE program can be found in the SIGMA/W (1995) manual togetherwith validation models.

The height and length of each element in the FE model as shown Figure 6 are 0.02m and 0.04 m, respectively. A total of 3000 elements was used. The geosynthetic layerwas placed in the middle of the granular fill. The geosynthetic layer was simulated asa thin layer of thickness Hg = 0.04 m using an equivalent Young’s modulus E = Eg/0.04.The following is a summary of the parameters used in the FE model (∆B is the lengthof the elements supported by springs, and Ks is the spring constant acting at each node):

S Dimensions: B = 2 m, L = 4 m, Ht = Hb = 0.3 m, H*t = H*

b = 0.3/2 = 0.15.

S Granular fill properties: ks = 263 kN/m3, G*t = G*

b = 0.1, Gt = (G*t ks B2)/Ht = Gb = 0.1

× 263 × 4/0.3 = 350.7 kN/m2, ν = 0.49, Et = Eb = 2Gt (1+ν) = 2 × 350.7(1 + 0.49) =1045 kN/m2 (where ν = Poisson’s ratio of the granular fill; and Et and Eb = Young’smodulus of the top and bottom granular fill layers, respectively).

S Geosynthetic layer: E*g = Eg /(ks B2) = 20, Eg = 20 ks B2 = 20 × 263 × 4 = 21040 kN/m,

νg = 0.49, E = Eg /Hg = Eg /0.04 = 5.26 × 105 kN/m2 (where νg = Poisson’s ratio of geo-synthetic layer; and E = Young’s modulus of the geosynthetic layer).

S Load: q* = q/(ks B) = 0.1, q = 0.1 × 263 × 2 = 52.6 kN/m2, Tp = 0.

S Springs: ∆B = 0.04 m, KS = (∆B)ks = 0.04 × 263 = 10.52 kN/m2.



The principle for selecting the above FE model parameters was to make the normal-ised FE model parameters defined in Section 3 have the same values as the parametersin the proposed 1-D model discussed in preceding sections. The actual parameter valuesused for the FE modelling are the values typically used in practice.

Two different vertical pressures were used for the FE modelling, that is, q = 52.6 kN/m2 (q* = 0.1) and q = 157.8 kN/m2 (q* = 0.3). Figure 6 shows the deformed FE modelmesh for q = 52.6 kN/m2. The vertical settlement and mobilised tension values obtainedusing the FE model are compared to the results obtained using the proposed model andthe model from Shukla and Chandra (1995) (S&C) in Figure 7. It is seen that both theproposed model and Shukla and Chandra’s model slightly underestimate the settlementof the foundation, but the settlement values obtained using the proposed model are clos-er to the FE modelling results. Shukla and Chandra’s model overestimates the mobi-lised tension in the region X≲ 0.90, and overestimates the mobilised tension by 200%at X = 0. Beyond this region, the results from Shukla and Chandra’s model are closeto the FE modelling results. Also, the mobilised tension within the footing width, B, (orfrom L/B = 0 to 0.90) increases almost linearly (toward X = 0), which differs from theresult obtained using the FE model. The proposed model underestimates the mobilisedtension by 0 to 15% for q*= 0.1, but overestimates the mobilised tension by 20 to 50%for q*= 0.3 when 0 < X < 0.85. The proposed model generally overestimates the mobi-lised tension for X > 0.85. For the proposed model, the rate of increase of the mobilisedtension at X = 0 is zero. The FE model also results in a zero rate of increase of the mobi-lised tension at X = 0. The FE program used (SIGMA/W 1995) is based on a theory ofsmall deformation that normally overestimates the settlement and underestimates themobilised tension for large deformation problems such as that presented in Figure 7.

5 MODELLING RESULTS, COMPARISON AND DISCUSSION

Additional results from the proposed model are presented in Figures 8 to 11 and arecompared to model results from Shukla and Chandra (1995), Ghosh (1991), and Mad-hav and Poorooshasb (1988) (M&P in Figures 8 and 9) .

Figure 8 shows the computed settlements for q*= 0.1, 0.3 and 0.8 using the four mo-dels. It is seen that the proposed model generally produces slightly larger settlementsthat are closer to the FE modelling results discussed in Section 4.

The effect of the stiffness shear modulus at the bottom and top of the granular fill lay-ers, G*

t and G*b , respectively, can be seen in Figure 9. An increase in the shear modulus

generally reduces foundation settlements. Comparing the results from the proposedmodel to the results from the other three models, the proposed model generally giveslarger settlements.

Figure 10 presents the settlement and mobilised tension values computed using theproposed model for different geosynthetic layer stiffnesses (E*

g = 2, 20 and 200). It isseen that the settlement is slightly reduced within the loaded region and the mobilisedtension slightly increases, due to an increase in the geosynthetic layer stiffness.

Figure 11 presents the effect of the pretension force (T*p = 0, 0.5 and 1) computed us-

ing the proposed model. The increase in the pretension force reduces both the settle-ment within and slightly beyond the loaded region, and increases the mobilised tension.



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Figure 7. Comparison of the two-dimensional FE model with the one-dimensional modelsproposed by Shukla and Chandra (1995) (S & C) and in the present study: (a) settlement;(b) mobilised tension.

G*t = G*b = 0.1, E*g = 20, L/B = 2H*t = H*b = 0.15, T*p = 0

q*= 0.1

q*= 0.3

Present studyS & C (1995)FE model


Distance from center of loading, X = x/B



(a)

(b)

Settlement,W=w/B


Mobilisedtension,T*=T/(ks/B2 )

q*= 0.3

q*= 0.1G*t = G*b = 0.1, E*g = 20, L/B = 2H*t = H*b = 0.15, T*p = 0



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 8. Comparison of the proposed one-dimensional model results with the Madhavand Poorooshasb (1988) (M & P), Ghosh (1991), and Shukla and Chandra (1995) (S & C)models - normalised settlement values for different normalised loads, q*.

q*= 0.1

q*= 0.3

Present studyM & P (1988)Ghosh (1991)

G*t = G*b = 0.1, E*g = 20, L/B = 2H*t = H*b = 0.15, T*p = 0

S & C (1995)

Present studyM & P (1988)Ghosh (1991)S & C (1995)

Present studyM & P (1988)Ghosh (1991)S & C (1995)

q*= 0.8


Settlement,W=w/B

6 CONCLUSIONS

A new one-dimensional model is proposed for geosynthetic-reinforced granular fillsover soft soils subject to a vertical surcharge load. A deformation compatibility condi-tion is suggested and implemented in the proposed model. Two uncertain model param-eters used to calculate the shear stresses between the top and bottom granular fill layersand the geosynthetic layer are eliminated. The stiffness of the geosynthetic layer is con-sidered in the model. The settlement and mobilised tension computed using the pro-posed model are in a better agreement with the finite element model results than the



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Present studyM & P (1988)

G*t = G*b = 0.5 G*t = G*b = 1

G*t = G*b = 0.1

S & C (1995)Ghosh (1991)


S & C (1995)Ghosh (1991)


S & C (1995)Ghosh (1991)


Settlement,W=w/B

Figure 9. Comparison of the proposed one-dimensional model results with the Madhavand Poorooshasb (1988) (M & P), Ghosh (1991), and Shukla and Chandra (1995) (S & C)models - normalised settlement values for different normalised shear stiffness values at thetop and bottom of the granular fill layer, G*t and G*b , respectively.

q*= 0.8, E*g = 20, L/B = 2H*t = H*b = 0.15, T*p = 0

other one-dimensional models used in the comparison. Nonlinear and elastic-plastic be-haviour of the granular fill can be considered when using the proposed model.

ACKNOWLEDGEMENT

Financial support from the Hong Kong Polytechnic University is acknowledged.



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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

G*t = G*b = 0.1, q*= 0.8, L/B = 2H*t = 0.25, H*b = 0.05, T*p = 0


Settlement,W=w/B



E*g = 2

E*g = 20

E*g = 200

E*g = 2

E*g = 20

E*g = 200

G*t = G*b = 0.1, q*= 0.8, L/B = 2H*t = 0.25, H*b = 0.05, T*p = 0

(a)

(b)

Figure 10. The geosynthetic layer stiffness, E*g , results for the proposedone-dimensional model of a geosynthetic-reinforced granular fill over soft soil:(a) settlement: (b) mobilised tension.



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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

Figure 11. The pretension force analysis, T*p , results, for the proposed one-dimensionalmodel of a geosynthetic-reinforced granular fill over soft soil: (a) settlement; (b) mobilisedtension.


G*t = G*b = 0.1, q*= 0.8, L/B = 2H*t = 0.25, H*b = 0.05, T*p = 0

(a)

(b)



Settlement,W=w/B

T*p = 0.5

T*p = 1

T*p = 0

T*p = 0.5

T*p = 1

T*p = 0

G*t = G*b = 0.1, q*= 0.8, L/B = 2H*t = 0.25, H*b = 0.05, E*g = 20

G*t = G*b = 0.1, q*= 0.8, L/B = 2H*t = 0.25, H*b = 0.05, E*g = 20



REFERENCES

Bourdeau, P.L., 1989, “Modelling of Membrane Action in a Two-Layer Reinforced SoilSystem”, Computers and Geotechnics, Vol. 7, Nos. 1-2, pp. 19-36.

Bourdeau, P.L., Harr, M.E. and Holtz, R.D., 1982, “Soil-Fabric Interaction - An Analyt-ical Model”, Proceedings of the Second International Conference on Geotextiles ,IFAI, Vol. 2, Las Vegas, Nevada, USA, August 1982, pp. 387-391.

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

Geo-Slope International Ltd., 1995, “SIGMA/W Manual”, Version 3, Calgary, Alberta,Canada.

Ghosh, C., 1991, “Modelling 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., 1994, “Reinforced Granular Fill-Soft Soil System: Con-finement Effect”, Geotextiles and Geomembranes, Vol. 13, No. 5, pp. 727-741.

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”, Geotextiles andGeomembranes, Vol. 13, No. 3, pp. 199-206.

Love, J.P., Burd, H.J., Milligan, G.W.E. and Houlsby, G.T., 1987, “Analytical and Mod-el Studies of Reinforcement of a Layer of Granular Fill on a Soft Clay Subgrade”,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.

Pasternak, P.L., 1954, “On a New Method of Analysis of an Elastic Foundation byMeans of Two Foundation Constants”, Gosudarstvennoe Izdatelstro Liberaturi poStroitelsvui Arkhitekture, Moscow. (in Russian)

Poorooshasb, H.S., 1989, “Analysis of Geosynthetic Reinforced Soil Using a SimpleTransform Function”, Computers and Geotechnics, Vol. 8, No. 4, pp. 289-309.

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

Poran, C.J., Herrmann, L.R. and Romastad, K.M., 1989, “Finite Element Analysis ofFootings on Geogrid-Reinforced Soil”, Proceedings of Geosynthetics ’89, IFAI, Vol.1, San Diego, California, USA, February 1989, pp. 231-242.

Rowe, R.K. and Soderman, K.L., 1987, “Stabilization of Very Soft Soils Using High-Strength Geosynthetics: the Role of Finite Element Analyses”, Geotextiles and Geo-membranes, Vol. 6, Nos. 1-3, pp. 53-80.

Shukla, S.K. and Chandra, S., 1994, “A Study of Settlement Response of a Geosynthet-ic-Reinforced Compressible Granular Fill-Soft Soil System”, Geotextiles and Geo-membranes, Vol. 13, No. 9, pp. 627-639.

Shukla, S.K. and Chandra, S., 1995, “Modelling of Geosynthetic-Reinforced Engine-ered Granular Fill on Soft Soil”, Geosynthetics International, Vol. 2, No. 3, pp.603-617.



NOTATIONS

Basic SI units are given in parentheses.

B = half width of uniform surcharge load (m)

dl = length of bottom of deformed element (m)

dx = projected element length in the x (horizontal) direction (m)

E = Young’s modulus of geosynthetic layer (N/m2)

Eb = Young’s modulus of bottom granular fill layer (N/m2)

Eg = tension modulus of geosynthetic layer (N/m)

Et = Young’s modulus of top granular fill layer (N/m2)

E*g = normalised Eg (dimensionless)

Gb = shear modulus of bottom granular fill layer (N/m2)

Gt = shear modulus of top granular fill layer (N/m2)

G*b = normalised Gb (dimensionless)

G*t = normalised Gt (dimensionless)

Hb = thickness of bottom granular fill layer (m)

Hg = thickness of geosynthetic layer (m)

Ht = thickness of top granular fill layer (m)

H*b = normalised Hb (dimensionless)

H*t = normalised Ht (dimensionless)

i = subscript referring to a nodal point (dimensionless)

Ks = spring constant acting at each node (dimensionless)

k = iteration index (dimensionless)

ks = modulus of subgrade reaction for soft foundation soil (N/m3)

L = half width of geosynthetic-reinforced zone (m)

n = number of geosynthetic elements (dimensionless)

q = pressure on the top granular fill layer (N/m2)

qs = vertical reaction pressure of soft soil (simplified Winkler springs)(N/m2)

q* = normalised q (dimensionless)

T = mobilised tension in geosynthetic layer (N/m)

Tp = pretension force of geosynthetic layer (N/m)

T* = normalised T (dimensionless)

T*p = normalised Tp (dimensionless)

ux = horizontal displacement at top of granular fill layer (m)

uig,x = horizontal displacement at geosynthetic layer interface (m)

uix = horizontal displacement at bottom of granular fill layer (m)

W = normalised w (dimensionless)



w = vertical displacement (m)

X = normalised x (dimensionless)

x = horizontal distance from centre (Figure 5) (m)

y = vertical distance from centre (Figure 5) (m)

εg = strain of geosynthetic element (dimensionless)

γx = shear strain in geosynthetic element due to horizontaldisplacement ux at top granular fill layer (dimensionless)

γix = shear strain in geosynthetic element due to horizontaldisplacement uix at bottom granular fill layer (dimensionless)

θ = angle between horizontal and bottom of element (_)

μ = constant relating vertical stress to shear stress on top and bottom surfacesof geosynthetic layer (dimensionless)

σn = average normal stress acting on bottom of element (N/m2)

σin = average normal stress acting on top of element (N/m2)

τ = average shear stress acting on the side of element in top granular fill layer(N/m2)

τn = average shear stress acting on bottom of element (N/m2)

τi = average shear stress acting on the side of element in bottom granular filllayer (N/m2)

τin = average shear stress acting on top of element (N/m2)

ν = Poisson’s ratio of granular fill (dimensionless)

νg = Poisson’s ratio of geosynthetic layer (dimensionless)

modelling geosynthetic reinforced granular fills over soft soil

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