modelling geosynthetic-reinforced granular fills over soft soil - IGS ...

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 proposed 

for modelling geosynthetic-reinforced granular fills over soft soils subject to a vertical 

surcharge load. The geosynthetic reinforcement consists of a membrane (geogrid, or 

geotextile) placed horizontally in engineered granular fill, which is constructed over 

soft soil. The proposed model is mainly based on the assumption of a Pasternak shear 

layer. A new approach that consists of incorporating a deformation compatibility condition 

into the model is introduced. The compatibility condition eliminates the need for 

two uncertain model parameters that are required to calculate the shear stresses between 

the top and bottom granular fill layers and the geosynthetic layer. This compatibility 

condition also makes it possible to include the geosynthetic stiffness in the model. The 

results from the proposed model are compared to results from a two-dimensional finite 

element model and three other one-dimensional models reported in the literature. The 

comparison reveals that the proposed model gives similar results with respect to the 

settlement of a geosynthetic-reinforced granular base on soft soft and the mobilised tension 

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 Structural 

Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong 

Kong, Telephone: 852/2766-6065, Telefax: 852/2334-6389, E-mail: 

cejhyin@polyu.edu.hk. 

PUBLICATION: Geosynthetics International is published by the Industrial Fabrics 

Association International, 345 Cedar St., Suite 800, St. Paul, Minnesota 55101-1088, 

USA, Telephone: 1/612-222-2508, Telefax: 1/612-222-8215. Geosynthetics 

International is registered under ISSN 1072-6349. 

DATES: Original manuscript received 16 December 1996, revised version received 

14 March 1997 and accepted 11 April 1997. Discussion open until 1 January 1998. 

REFERENCE: Yin, J.H., 1997, “Modelling Geosynthetic-Reinforced Granular Fills 

Over Soft Soil”, Geosynthetics International, Vol. 4, No. 2, pp. 165-185. 

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

165

YIN D Modelling Geosynthetic-Reinforced Granular Fills Over Soft Soil 

1 INTRODUCTION 

A foundation constructed on soft soils may experience excessive settlement and possible 

bearing capacity failure under a surcharge load. One technique that is used to improve 

the strength of soft foundation soils is the placement of engineered granular fills 

containing geosynthetic reinforcement (e.g. geogrid, or geotextile) on the soft soil. The 

calculation of settlement and the ultimate bearing capacity of the geosynthetic-reinforced 

granular fill over soft soil is an important issue facing design engineers. This paper 

is focussed on the modelling of the load-settlement behaviour of geosynthetic-reinforced 

granular fill over soft soil. Various works have been carried out by many 

researchers in this area of study (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 Madhav 1994; Khing et al. 1994; Shukla 

and Chandra 1994; Shukla and Chandra 1995). In these modelling approaches, a twodimensional 

(2-D) plane strain problem (e.g. a strip footing) is simplified to a one-dimensional 

(1-D) problem. In the 1-D model, the soft soil is represented by Winkler 

springs, the behaviour of which may be linear or nonlinear. The top and the bottom granular 

fill layers are assumed to behave as Pasternak shear layers (Pasternak 1954). The 

geosynthetic layer (“membrane”) buried in the granular fill can withstand tension 

forces only, and the geosynthetic layer behaviour is assumed to be elastic. This 1-D 

model has an inherent problem, as does the Winkler foundation model, because no interaction 

between the springs is considered. However, as pointed out by Shukla and 

Chandra (1995), the interaction of the springs is not very significant for highly compressible 

soft soils. 

In most existing models, the longitudinal stiffness of the geosynthetic layer is assumed 

to be rigid in tension, but does not carry vertical shear forces (Ghosh and Madhav 

1994; Shukla and Chandra 1994; Shukla and Chandra 1995). The shear stresses acting 

on the top and bottom surfaces of the geosynthetic layer are assumed to be related to 

the vertical stresses acting on the top and bottom surfaces by a constant that is defined 

by μ in the following expression: τ = μσ ,whereτ = shear stress and σ = vertical stress 

(Ghosh and Madhav 1994; Shukla and Chandra 1994; Shukla and Chandra 1995). The 

magnitude of the constant μ is not exactly known which may cause difficulties when 

using most existing models. 

In this paper, a deformation compatibility condition is proposed. The use of this compatibility 

condition eliminates the two uncertain constants defined by μ for the top and 

bottom shear stresses acting on the geosynthetic layer. The tension stiffness modulus 

of the geosynthetic layer is incorporated into the proposed new model. In the first step 

of model development, the behaviour of the granular fill is assumed to be elastic (Shukla 

and Chandra 1995). The nonlinear and elastic-plastic behaviour of the granular fill 

can be considered using the method proposed in this paper. In order to validate the proposed 

model against other models, the results from the proposed model are compared 

to two-dimensional FE modelling results and the results from other one-dimensional 

models. 

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


2 FORMULATION OF THE PROPOSED ONE-DIMENSIONAL MODEL 

Figure 1 shows a simplified foundation model for a geosynthetic (e.g. geogrid, or geotextile) 

reinforced granular fill over soft soil. This simplified model is similar to the 

model proposed by Shukla and Chandra (1995). According to the assumption of a Pasternak 

shear layer, vertical planes in the granular fill layer move relative to each other 

only in the vertical direction without rotation. The interface between the geosynthetic 

layer (“membrane”) and the two granular fill layers is assumed to be fixed (i.e. there 

is rough contact and no slippage). In this study, the behaviour of the granular fill and 

the geosynthetic layer is assumed to be elastic. 

The proposed model can be a simplification of a strip footing (or other similar structures) 

on a geosynthetic-reinforced granular fill over soft soil. The strip footing load is 

simplified as a vertical pressure loading boundary of width 2B (Figure 1). Considering 

a unit length in the direction perpendicular to Figure 1, the vertical force equilibrium 

for Element 1 in Figure 2 (top granular fill layer) leads to the following expression: 

qdx − σ n dl cos θ + τ n dl sin θ + dτ 

dx dxH t = 0 

(1) 

where: q = footing pressure on the top granular fill layer; dx = projected element length 

in the x direction; σ n = average normal stress acting on the bottom of the element; τ n = 

average shear stress acting on the bottom of the element; dl = length of the bottom of 

the element as shown in Figures 2 and 4; θ = angle between the horizontal and the bottom 

of the element; H t = thickness of the top granular fill layer; and τ = average shear 

stress acting on the side of the element. 

According to the assumption of a Pasternak shear layer, the average vertical shear 

stress is τ =G t dw/dx, anddτ/dx =G t d 2 w/dx 2 ,whereG t is the average shear modulus 

of the top granular fill layer, and w is the vertical displacement. Thus, Equation 

1 can be written as follows: 

2B 

q 

Pasternak shear layers 

H t 

G t for top granular fill layer 

E g for geosynthetic layer 

T, T p “membrane” 

T, T p 

H b 

G b for bottom granular fill layer 

k s 

Soft soil 

Bedrock 

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

definition of model parameters. 


167


(a) 

Element 1 

w 

dx 

q 

w +dw 

x 

τ 

τ + (∂τ/∂x)dx 

(b) 

Element 2 

T p ,T 

θ 

σ n 

τ n 

τ n 

σ n 

(c) 

Element 3 

τi n 

σi n 

σi n 

τi n 

θ + dθ 

T p ,T+dT 

y 

τi 

dl 

τi + (∂τi/∂x)dx 

q s =k s w 

Figure 2. Three elements from a vertical segment of a geosynthetic-reinforced granular fill 

of infinitesimal width showing the forces and stresses on each element: (a) top granular fill 

layer; (b) geosynthetic layer; (c) bottom granular fill layer. 

q = σ n − τ n tan θ − H t G t 

d 2 w 

dx 2 

(2) 

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: 

− q s dx + σ ′ n dl cos θ − σ ′ n dl sin θ + dτ′ 

dx dxH b = 0 

(3) 

where: q s = 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 acting 

on the top of the element; H b = thickness of the bottom granular fill layer; and τ′=aver- 



age shear stress acting on the side of the bottom element. Substituting τ′ =G b dw/dx, 

and dτ′/dx =G b d 2 w/dx 2 , Equation 3 becomes: 

q s = σ ′ n − τ ′ n tan θ − H b G b 

d 2 w 

dx 2 

(4) 

where G b is the average shear modulus of the bottom granular fill layer. 

The vertical force equilibrium equation for Element 2 in Figure 2b (geosynthetic layer) 

leads to the following expression: 

dT 

dx sin θ =−(T + T p)cosθ dθ 

dx − (σ n − σ ′ n) + (τ n + τ ′ n)tanθ 

(5) 

where: T = mobilised tension in the geosynthetic layer; and T p = pretension force of the 

geosynthetic layer. 

The horizontal force equilibrium equation for Element 2 (geosynthetic layer) is expressed 

as follows: 

dT 

dx cos θ = (T + T p)sinθ dθ 

dx + (σ n − σ ′ n)tanθ + (τ n + τ ′ n) 

(6) 

Figure 3 shows the deformation of the top and bottom granular fill elements due to 

an increase in tension, dT. Assuming no sliding between the geosynthetic layer and the 

top and bottom granular fill layers, the following deformation compatibility condition 

(or relationships) exists: 

u x = u ′ x = u g,x 

(7) 

where u x ,u′ x and u g,x are the horizontal displacements at the top and bottom granular 

fill layers, and the interface of the geosynthetic layer, respectively. 

The values of u x and u′ x are related to the shear strain values, γ x and γ′ x , respectively, 

which are then related to the shear stresses, τ n and τ′ n , as expressed in the following: 

u x = H t γ x = H t 

τ n 

(8) 

u ′ x = H b γ ′ x = H b 

τ ′ n 

G b 

G t 

(9) 

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

τ n = G t H b 

G b H t 

τ ′ n 

(10) 

Figure 4 shows a stretched and rotated element of the geosynthetic layer. Using Equations 

7 to 9, the displacement of the geosynthetic element and the displacement increment 

can be written as follows: 


169


dx 

x 

(a) 

Element 1 

γ x 

G t 

H t 

(b) 

Element 2 

T p ,T 

u x 

σ n 

τ n 

τ 

σ n n 

τi n σi n 

τi n 

T p ,T+dT 

(c) 

Element 3 

ui x 

σi n 

γi x 

G b 

H b 

y 

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. 

dx 

u g,x 

u g,x + du g,x 

θ 

dw 

dl 

Figure 4. 

Stretching and rotation of a geosynthetic element. 



u g,x 

= u ′ x = H b 

G b 

τ ′ n 

du g,x 

= du ′ x = H b 

G b 

dτ ′ n 

(11) 

(12) 

The stretched length, dl, after rotation of the geosynthetic element is given by the following 

expression: 

dl = (dw) 2 + (dx + du g,x ) 2 

(13) 

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

ε g = 

dl − dx 

= dw 

dx 

dx 2 +1 + du 2 

g,x 

− 1 

dx 

(14) 

Using Equations 11, 12 and 14, the mobilised tension, T, in the geosynthetic element 

is given by the following expression: 

T = E g ε g = E g⎪ ⎡ ⎣ 

 

 

2 

G b dx 

dw dx 2 +1 + H bdτ ′ n 

− 1 ⎪ 

⎤ 

⎦ 

where E g 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: 

(15) 

q = σ n − G t H b 

τ ′ n tan θ − H t G 

d 2 w 

t 

G b H t dx 2 

(16) 

Since dθ/dx =(cos 2 θ)d 2 w/dx 2 , Equations 5 and 6 can be written as follows: 

dT 

dx sin θ =−(T + T p)cos 3 θ d2 w 

dx − (σ 2 n − σ ′ n) + (τ n + τ ′ n)tanθ (17) 

dT 

dx cos θ = (T + T p)sinθ cos 2 θ d2 w 

dx + (σ 2 n − σ ′ n)tanθ + (τ n + τ ′ n) (18) 

Since the rotation angle, θ, is related to dw/dx by tanθ = dw/dx, Equations 15 to 18 

and Equation 4 can be used to solve for the five unknowns, i.e. σ n , σ′ n , τ′ n ,wand T. 

Equations 17 and 18 can be simplified by multiplying Equation 17 by cosθ and Equation 

18 by sinθ , and subtracting the former from the latter to give the following expressions: 


171


(σ n − σ ′ n) =−(T + T p )cos 3 θ d2 w 

dx 2 

(τ n + τ ′ n) = dT 

dx 

Substracting Equation 4 from Equation 16 gives the following: 

(19) 

cos θ (20) 

q − q s = (σ n − σ ′ n) − (τ n + τ ′ n)tanθ − (H t G t + H b G b ) d2 w 

dx 2 (21) 

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

q − q s =−(T + T p )cos 3 θ d2 w 

− sin θ 

dT 

dx2 dx − (H t G t + H b G b ) d2 w 

(22) 

dx 2 

where q s =k s w,andk s is a spring constant. The unknowns in Equation 22 are T and w. 

Substituting Equation 10 into Equation 20, the following expression is obtained: 

Differentiating Equation 23: 

dT 

dx = 1 

cos θ (τ n + τ ′ n) = 1 

cos θ G t H b 

G b H t 

+ 1 τ ′ n (23) 

d 2 T 

dx = sin θ G t H b 

+ 2 G b H t 

1 τ ′ d 2 w 

n 

dx + 1 

2 cos θ G t H b 

+ 

G b H t 

1 dτ′ n 

dx 

Rearranging Equation 23: 

τ ′ n = 

cos θ 

G t H b 

G b H t 

+ 1 dT 

(24) 

dx (25) 


T E g 

2 + 1 = dw 

dx 2 +1 + H 2 

b dτ ′ n 

G b dx 


dτ ′ n 

dx = G b 

H b 

⎪ ⎡ ⎣ 

 

T E g 

 

2 + 1 − dw 

dx 2 

− 1 ⎪ 

⎤ 

⎦ 

(26) 

(27) 

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



d 2 T 

dx = sin θ cos θ d2 w dT 

2 dx 2 dx + 1 

cos θ G t 

+ G b 

H t H b 

 

⎡ ⎪ 

⎣ 

 

T E g 

2 + 1 − dw 

dx 2 

− 1 ⎪ 

⎤ 

⎦ (28) 

Equation 22 and Equation 28 are the two basic equations for solving the two unknowns 

T and w. 

3 NORMALISED EQUATIONS AND A NUMERICAL SOLUTION 

3.1 Introduction 

The following normalised parameters were used in the numerical solution (Shukla 

and Chandra 1995): 

X = x B 

, W = w , H * t = H t , 

B 

B 

H * b = H b 

B 

G * t = G t H t 

B 2 k s 

G * b = G b H b 

E * 

B 2 g = E g 

, , 

k s B 2 k s 

q * = 

q , T * 

B 2 p = T p , T * = T 

k s B 2 k s B 2 k s 

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

dimensionless. 

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

q * = W − sin θ dT* 

dX − [(T* + T * p)cos 3 θ + (G * t + G * b)] d2 W 

dX 2 (29) 

d 2 T * 

dX = sin θ cos θ d2 W dT * 

2 dX 2 dX + 1 G* t 

cos θ H *2 

t 

+ G* b 

H *2 

b 

 

⎡ ⎪ 

⎣ 

 

T* 

dX 2 

E * g 2 + 1 − dW 

− 1 ⎪ 

⎤ 

⎦ 

(30) 

where: 

tan θ = dW 

dX 

dW 

, sin θ = 

dX 

, 

 

1 + dW 

dX 2 

cos θ = 

 

1 

1 + dW 

dX 2 

Using the following finite difference scheme: 


173


ΔW 

ΔX = W i − W i−1 

ΔX 

Δ 2 W 

ΔX 2 = W i−1 − 2W i + W i+1 

(ΔX) 2 

Δ 2 T * 

ΔT * 

ΔX = T* i − T * i−1 

ΔX 

ΔX = T* i−1 − 2T * i + T * i+1 

2 (ΔX) 2 

Equations 29 and 30 become: 

q * i = W i −sin θ dT* − [(T * + T * 

dX 

p)cos 3 θ + (G * t + G * b)] i 

 

i 

T * i−1 − 2T * i + T * i+1 

(ΔX) 2 = (sin θ cos θ) i 

+ 1 

(cos θ) i 

 

G* t 

H *2 

t 

W i−1 − 2W i + W i+1 

(31) 

(ΔX) 2 

W i−1 − 2W i + W i+1 

T* i+1 − T * i−1 

(ΔX) 2 

2ΔX 

+ G* b 

H *2 

⎡ 

i⎪ 

b 

⎣ 

 

T * i 

dX 2 i 

E * g 2 + 1 − dW 

− 1 ⎪ 

⎤ 

⎦ 

The 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 differential 

equations (Equations 31 and 32). Refer to Figure 5 for a symmetric geosynthetic-reinforced 

system with a constant pressure loading q of width 2B and a length of 2L. 

At X =0(orx = 0), due to symmetry, the slope, dW/dX, will be zero and the ratio of 

the increase in tension, dT * /dX, will be zero as given in the following expressions: 

dW 

dX = 0 

dT * 

dX = 0 

(32) 

At X = L/B (or x = L), the right end of the geosynthetic layer is free, so that the mobilised 

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 



B 

B 

x 

L 

L 

y 

Figure 5. The one-dimensional model of a geosynthetic-reinforced granular foundation 

under a constant surcharge load. 

the Pasternak shear layer assumes that τ =G t dw/dx, andτ′=G b dw/dx, thendw/dx =0. 

Thus, the boundary conditions at X = L/B are expressed as follows: 

dW 

dX = 0 

T * = 0 

Using the four boundary conditions given above, Equations 31 and 32 can be solved 

to obtain a unique solution for a given load. Since Equations 31 and 32 are both nonlinear 

equations, an iterative computing procedure must be used to obtain a solution. In 

the first iteration, an initial value of zero may be assumed for T i and W i (i = 0, 1, ..., n). 

The iteration is stopped for a relative error of [W i (k+1) - W i (k) ]/W i (k)


B =2m 

Y (m) Y (m) 

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 

5.8 

5.6 

5.4 

5.2 

5.0 

4.8 

4.6 

B =2m 

L =4m 

X (m) 

L =4m 

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

X (m) 

Figure 6. Finite element (FE) model for a surcharge load, q = 52.6 kN/m 2 :(a)FEmodel 

mesh and boundary conditions; (b) deformed FE model mesh. 

Notes: H t = H b =0.3m;H g =0.04m. 

H t 

H g 

H b 

H t 

H g 

H b 

description of the FE program can be found in the SIGMA/W (1995) manual together 

with validation models. 

The height and length of each element in the FE model as shown Figure 6 are 0.02 

m and 0.04 m, respectively. A total of 3000 elements was used. The geosynthetic layer 

was placed in the middle of the granular fill. The geosynthetic layer was simulated as 

a thin layer of thickness H g = 0.04 m using an equivalent Young’s modulus E = E g /0.04. 

The following is a summary of the parameters used in the FE model (∆B is the length 

of the elements supported by springs, and K s is the spring constant acting at each node): 

S Dimensions: B =2m,L =4m,H t = H b =0.3m,H * t = H * b =0.3/2=0.15. 

S Granular fill properties: k s = 263 kN/m 3 , G * t = G * b =0.1,G t =(G * t k s B 2 )/H t = G b =0.1 

× 263 × 4/0.3 = 350.7 kN/m 2 , ν = 0.49, E t = E b =2G t (1+ν) =2× 350.7(1 + 0.49) = 

1045 kN/m 2 (where ν = Poisson’s ratio of the granular fill; and E t and E b = Young’s 

modulus of the top and bottom granular fill layers, respectively). 

S Geosynthetic layer: E * g = E g /(k s B 2 ) = 20, E g =20k s B 2 =20× 263 × 4 = 21040 kN/m, 

ν g = 0.49, E = E g /H g = E g /0.04 = 5.26 × 10 5 kN/m 2 (where ν g = Poisson’s ratio of geosynthetic 

layer; and E = Young’s modulus of the geosynthetic layer). 

S Load: q * = q/(k s B) = 0.1, q =0.1× 263 × 2 = 52.6 kN/m 2 , T p =0. 

S Springs: ∆B =0.04m,K S =(∆B)k s =0.04× 263 = 10.52 kN/m 2. 



The principle for selecting the above FE model parameters was to make the normalised 

FE model parameters defined in Section 3 have the same values as the parameters 

in the proposed 1-D model discussed in preceding sections. The actual parameter values 

used 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/ 

m 2 (q * = 0.1) and q = 157.8 kN/m 2 (q * = 0.3). Figure 6 shows the deformed FE model 

mesh for q = 52.6 kN/m 2 . The vertical settlement and mobilised tension values obtained 

using the FE model are compared to the results obtained using the proposed model and 

the model from Shukla and Chandra (1995) (S&C) in Figure 7. It is seen that both the 

proposed model and Shukla and Chandra’s model slightly underestimate the settlement 

of the foundation, but the settlement values obtained using the proposed model are closer 

to the FE modelling results. Shukla and Chandra’s model overestimates the mobilised 

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 close 

to the FE modelling results. Also, the mobilised tension within the footing width, B,(or 

from L/B = 0 to 0.90) increases almost linearly (toward X = 0), which differs from the 

result obtained using the FE model. The proposed model underestimates the mobilised 

tension 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 mobilised 

tension for X > 0.85. For the proposed model, the rate of increase of the mobilised 

tension at X = 0 is zero. The FE model also results in a zero rate of increase of the mobilised 

tension at X = 0. The FE program used (SIGMA/W 1995) is based on a theory of 

small deformation that normally overestimates the settlement and underestimates the 

mobilised 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 are 

compared to model results from Shukla and Chandra (1995), Ghosh (1991), and Madhav 

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 models. 

It is seen that the proposed model generally produces slightly larger settlements 

that 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 layers, 

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 proposed 

model to the results from the other three models, the proposed model generally gives 

larger settlements. 

Figure 10 presents the settlement and mobilised tension values computed using the 

proposed model for different geosynthetic layer stiffnesses (E * g = 2, 20 and 200). It is 

seen that the settlement is slightly reduced within the loaded region and the mobilised 

tension 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 using 

the proposed model. The increase in the pretension force reduces both the settlement 

within and slightly beyond the loaded region, and increases the mobilised tension. 


177


(a) 

-0.1 

0.0 

G * t = G * b =0.1,E * g = 20, L/B =2 

H * t = H * b = 0.15, T * p =0 

Settlement, W = w/B 

0.1 

0.2 

0.3 

0.4 

0.5 

q * =0.1 

Present study 

S & C (1995) 

FE model 

q * =0.3 

Present study 

S & C (1995) 

FE model 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 

Distance from center of loading, X = x/B 

(b) 

Mobilised tension, T * = T/(ks /B 2 ) 

0.40 

0.35 

0.30 

0.25 

0.20 

0.15 

0.10 

0.05 

G * t = G * b =0.1,E * g = 20, L/B =2 

H * q * =0.1 

t = H * b = 0.15, T * p =0 

Present study 

S & C (1995) 

FE model 

q * =0.3 

Present study 

S & C (1995) 

FE model 

0.00 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 


Figure 7. Comparison of the two-dimensional FE model with the one-dimensional models 

proposed by Shukla and Chandra (1995) (S & C) and in the present study: (a) settlement; 

(b) mobilised tension. 



0.0 

0.1 

0.2 

0.3 

q * =0.1 

0.4 

Present study 

M & P (1988) 

0.5 

Ghosh (1991) 

S & C (1995) 

0.6 

q * =0.3 

Present study 

M & P (1988) 

0.7 

Ghosh (1991) 

S & C (1995) 

0.8 

q * =0.8 

Present study 

0.9 

G * t = G * b =0.1,E * g = 20, L/B =2 M & P (1988) 

H * t = H * b = 0.15, T * p =0 

Ghosh (1991) 

1.0 

S & C (1995) 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 


Figure 8. Comparison of the proposed one-dimensional model results with the Madhav 

and Poorooshasb (1988) (M & P), Ghosh (1991), and Shukla and Chandra (1995) (S & C) 

models - normalised settlement values for different normalised loads, q * . 


6 CONCLUSIONS 

A new one-dimensional model is proposed for geosynthetic-reinforced granular fills 

over soft soils subject to a vertical surcharge load. A deformation compatibility condition 

is suggested and implemented in the proposed model. Two uncertain model parameters 

used to calculate the shear stresses between the top and bottom granular fill layers 

and the geosynthetic layer are eliminated. The stiffness of the geosynthetic layer is considered 

in the model. The settlement and mobilised tension computed using the proposed 

model are in a better agreement with the finite element model results than the 


179


0.0 

0.1 

q * =0.8,E * g = 20, L/B =2 

H * t = H * b = 0.15, T * p =0 

0.2 


0.3 

0.4 

0.5 

0.6 

0.7 

G * t = G * b =0.1 

Present study 

M & P (1988) 

Ghosh (1991) 

S & C (1995) 

0.8 

0.9 

1.0 

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

Present study Present study 

M & P (1988) M & P (1988) 

Ghosh (1991) Ghosh (1991) 

S & C (1995) S & C (1995) 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 


Figure 9. Comparison of the proposed one-dimensional model results with the Madhav 

and Poorooshasb (1988) (M & P), Ghosh (1991), and Shukla and Chandra (1995) (S & C) 

models - normalised settlement values for different normalised shear stiffness values at the 

top and bottom of the granular fill layer, G * t and G * b , respectively. 

other one-dimensional models used in the comparison. Nonlinear and elastic-plastic behaviour 

of the granular fill can be considered when using the proposed model. 

ACKNOWLEDGEMENT 

Financial support from the Hong Kong Polytechnic University is acknowledged. 



(a) 


0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

G * t = G * b =0.1,q * =0.8,L/B =2 

H * t = 0.25, H * b = 0.05, T * p =0 

E * g =2 

E * g =20 

E * g = 200 

0.8 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 


(b) 


2.0 

G * t = G * b =0.1,q * =0.8,L/B =2 

E * g =2 

H * t = 0.25, H * b = 0.05, T * p =0 

1.5 

E * g =20 

E * g = 200 

1.0 

0.5 

0.0 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 


Figure 10. The geosynthetic layer stiffness, E * g , results for the proposed 

one-dimensional model of a geosynthetic-reinforced granular fill over soft soil: 

(a) settlement: (b) mobilised tension. 


181


(a) 


0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

G * t = G * b =0.1,q * =0.8,L/B =2 

H * t = 0.25, H * b = 0.05, E * g =20 

T * p =0 

T * p =0.5 

0.7 

T * p =1 

0.8 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 


(b) 


2.0 

G * 

G * t = G * 

t = G * b =0.1,q * =0.8,L/B =2 

H * b =0.1,q * =0.8,L/B 1.5 H * t = 0.25, H * 

t = 0.25, H * b = 0.05, E * 

b = 0.05, T * g =20 

p =0 

T * p =0 

T * p =0.5 

T * p =1 

1.0 

0.5 

0.0 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 


Figure 11. The pretension force analysis, T * p , results, for the proposed one-dimensional 

model of a geosynthetic-reinforced granular fill over soft soil: (a) settlement; (b) mobilised 

tension. 



REFERENCES 

Bourdeau, P.L., 1989, “Modelling of Membrane Action in a Two-Layer Reinforced Soil 

System”, 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 Analytical 

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. thesis, 

Department of Civil Engineering, I.I.T., Kanpur, India, 218 p. 

Ghosh, C. and Madhav, M.R., 1994, “Reinforced Granular Fill-Soft Soil System: Confinement 

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 on 

Strong Sand Underlain by Weak Clay with Geogrid at the Interface”, Geotextiles and 

Geomembranes, Vol. 13, No. 3, pp. 199-206. 

Love, J.P., Burd, H.J., Milligan, G.W.E. and Houlsby, G.T., 1987, “Analytical and Model 

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 Reinforced 

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 by 

Means of Two Foundation Constants”, Gosudarstvennoe Izdatelstro Liberaturi po 

Stroitelsvui Arkhitekture, Moscow. (in Russian) 

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

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

Poorooshasb, H.S., 1991, “On Mechanics of Heavily Reinforced Granular Mats”, Soils 

and Foundations, Vol. 31, No. 2, pp. 134-152. 

Poran, C.J., Herrmann, L.R. and Romastad, K.M., 1989, “Finite Element Analysis of 

Footings 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 Geomembranes, 

Vol. 6, Nos. 1-3, pp. 53-80. 

Shukla, S.K. and Chandra, S., 1994, “A Study of Settlement Response of a Geosynthetic-Reinforced 

Compressible Granular Fill-Soft Soil System”, Geotextiles and Geomembranes, 

Vol. 13, No. 9, pp. 627-639. 

Shukla, S.K. and Chandra, S., 1995, “Modelling of Geosynthetic-Reinforced Engineered 

Granular Fill on Soft Soil”, Geosynthetics International, Vol. 2, No. 3, pp. 

603-617. 


183


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/m 2 ) 

E b = Young’s modulus of bottom granular fill layer (N/m 2 ) 

E g = tension modulus of geosynthetic layer (N/m) 

E t = Young’s modulus of top granular fill layer (N/m 2 ) 

E * g = normalised E g (dimensionless) 

G b = shear modulus of bottom granular fill layer (N/m 2 ) 

G t = shear modulus of top granular fill layer (N/m 2 ) 

G * b = normalised G b (dimensionless) 

G * t = normalised G t (dimensionless) 

H b = thickness of bottom granular fill layer (m) 

H g = thickness of geosynthetic layer (m) 

H t = thickness of top granular fill layer (m) 

H * b = normalised H b (dimensionless) 

H * t = normalised H t (dimensionless) 

i = subscript referring to a nodal point (dimensionless) 

K s = spring constant acting at each node (dimensionless) 

k = iteration index (dimensionless) 

k s = modulus of subgrade reaction for soft foundation soil (N/m 3 ) 

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

n = number of geosynthetic elements (dimensionless) 

q = pressure on the top granular fill layer (N/m 2 ) 

q s = vertical reaction pressure of soft soil (simplified Winkler springs) 

(N/m 2 ) 

q * = normalised q (dimensionless) 

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

T p = pretension force of geosynthetic layer (N/m) 

T * = normalised T (dimensionless) 

T * p = normalised T p (dimensionless) 

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

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

ui x = 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) 

ε g = strain of geosynthetic element (dimensionless) 

γ x = shear strain in geosynthetic element due to horizontal 

displacement u x at top granular fill layer (dimensionless) 

γi x = shear strain in geosynthetic element due to horizontal 

displacement ui x 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 surfaces 

of geosynthetic layer (dimensionless) 

σ n = average normal stress acting on bottom of element (N/m 2 ) 

σi n = average normal stress acting on top of element (N/m 2 ) 

τ = average shear stress acting on the side of element in top granular fill layer 

(N/m 2 ) 

τ n = average shear stress acting on bottom of element (N/m 2 ) 

τi = average shear stress acting on the side of element in bottom granular fill 

layer (N/m 2 ) 

τi n = average shear stress acting on top of element (N/m 2 ) 

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

ν g = Poisson’s ratio of geosynthetic layer (dimensionless) 


185

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