Abstract The paper presents a model for the anal-

ysis of granular foundation beds reinforced with

several geosynthetic layers. Such reinforced granular

beds are often placed on soft soil strata for an effi-

cient and economical transfer of superstructure load.

The granular bed is modeled by the Pasternak shear

layer and the geosynthetic reinforcement layers by

stretched rough elastic membranes. The soft soil is

represented by a series of nonlinear springs. The

reinforcement has been considered to be extensible

and it is assumed that the deformation at the interface

of the reinforcements and soil are same. The non-

linear behavior of the granular bed and the soft soil is

considered. Plane strain conditions are considered for

the loading and reinforced foundation soil system. An

iterative finite difference scheme is applied for

obtaining the solution and results are presented in

nondimensional form. The results from the proposed

model are compared to the results obtained for mul-

tilayer inextensible geosynthetic reinforcement sys-

tem. Significant reduction in the settlement has been

observed when the number of reinforcement layer is

increased. In case of inextensible reinforcements as

the number of reinforcement layer is increased the

settlement is decreased with a decreasing rate, but in

case of extensible reinforcement the reduction rate is

almost constant. Nonlinear behavior of the soft soil

decreases as number of reinforcement layer is in-

creased. The effect of the stiffness of the geosynthetic

layer on the settlement response becomes insignifi-

cant for multilayer reinforced system, but the mobi-

lized tension in the reinforcement layers increases as

the stiffness of the geosynthetic layers increases.

Keywords Extensible Æ Geosynthetic

reinforcement Æ Multilayer Æ Nonlinear Æ Soft soil

Nomenclature

B half width of uniform surcharge load (m)

Eg tension modulus of the geosynthetic layers

(N/m)

Eg* normalized Eg (dimensionless)

Gj0 initial shear modulus of the granular fill

layer 1, 2, 3, 4, respectively (N/m2)

Gj0* normalized Gj0 (dimensionless)

Hj thickness of the granular fill layer 1, 2, 3,

4, respectively (m)

ks0 initial modulus of subgrade reaction for

soft foundation soil (N/m3)

L half width of geosynthetic-reinforced zone

(m)

N number of geosynthetic layer

(dimensionless)

q pressure intensity on the top granular layer

(N/m2)

q* normalized q (dimensionless)

K. Deb (&) Æ S. Chandra Æ P. K. Basudhar

Department of Civil Engineering, Indian Institute of

Technology Kanpur, Kanpur 208016, India

e-mail: kdeb@iitk.ac.in; kousik_deb@rediffmail.com

Geotech Geol Eng (2007) 25:11–23

DOI 10.1007/s10706-006-0002-7

123

ORIGINAL PAPER

Nonlinear analysis of multilayer extensiblegeosynthetic-reinforced granular bed on soft soil

Kousik Deb Æ S. Chandra Æ P. K. Basudhar

Received: 1 June 2005 / Accepted: 12 March 2006 / Published online: 5 October 2006

� Springer Science+Business Media B.V. 2006

qs vertical reaction pressure of the soft

foundation soil (N/m2)

qs* normalized qs (dimensionless)

qu ultimate bearing capacity of the soft soil

(N/m2)

qu* normalized qu (dimensionless)

Tj=1,2,3 mobilized tension in the top, middle and

bottom geosynthetic layer, respectively

(N/m)

Tj=1,2,3* normalized Tj=1,2,3 (dimensionless)

Tp pretension force applied to the

geosynthetic layers (N/m)

Tp* normalized Tp (dimensionless)

w vertical displacement (m)

W normalized w (dimensionless)

x distance from centre of loading (m)

X normalized x (dimensionless)

h slope of the membrane (degree)

sj=1,2,3,4 shear stresses in the granular layer 1, 2, 3,

4, respectively (N/m2)

sj=1,2,3,4* normalized sj=1,2,3 (dimensionless)

suj=1,2,3,4 ultimate shear resistance of the granular

layer 1, 2, 3, 4, respectively (N/m2)

suj=1,2,3,4* normalized suj=1,2,3,4 (dimensionless)

Introduction

Soil reinforcement has become a major part of geo-

technical practice over the last 30 years, and its use is

growing rapidly as worldwide development of infra-

structure poses an increasing demand for land recla-

mation and utilization of soft foundation soils. In

recent years geosynthetics are commonly used in

reinforcing the soil. The use of geosynthetic-rein-

forced granular fill over soft soil effectively reduces

the settlement and increases the bearing capacity of

the soft soil. Many researchers have studied behavior

of such reinforced foundation beds based on lump

parameter modeling (Madhav and Poorooshasb 1988;

Poorooshasb 1989, 1991; Ghosh and Madhav 1994;

Shukla and Chandra 1995; Yin 1997a, 1997b;

Maheshwari et al. 2004). However, models reported

in the literature are developed for single layer rein-

forced system. Nogami and Yong (2003) studied the

response of a multilayer geosynthetic reinforced

geomedium subjected to structural loading by con-

sidering each soil layer by a system of infinite num-

ber of closely spaced one-dimensional columns

connected with horizontal springs. Deb et al. (2005)

developed a model for multilayer reinforcement

system, which incorporates the nonlinear behavior of

the granular fill and soft soil as well as the effect of

compressibility of the granular fill. However, in the

existing foundation models with multiple geosyn-

thetic layers, geosynthetics are considered to be

inextensible, but in reality geosynthetics may be

extensible.

In this paper, response of a multilayer extensible

geosynthetic reinforced granular fill-soft soil system

has been studied with incorporates the nonlinear

behavior of the granular fill and soft soil. The

deformation compatibility condition has been incor-

porated as suggested by Yin (1997a). This compati-

bility condition reduces model parameters involved

in the analysis and able to introduce the effect of the

stiffness of the geosynthetic layers in the model.

Fig. 1 Multilayer

extensible geosynthetic-

reinforced granular bed on

soft soil

12 Geotech Geol Eng (2007) 25:11–23

123

Parametric studies have been carried out to assess the

overall behavior of the multilayer extensible geo-

synthetic-reinforced soil as well as that of the unre-

inforced soil. The results are presented in

nondimensional form for the practical range of

parameters.

Statement of the Problem

Figure 1 shows the problem considered in this study.

A granular bed containing several layers of geosyn-

thetic reinforcement is placed on the soft soil. The

behavior of such a system may be idealized by the

proposed foundation model as shown Fig. 2. In this

model, the granular fill and soft soil have been ide-

alized by the Pasternak shear layer and a layer of

nonlinear springs, respectively. Stretched rough

elastic membranes represent the geosynthetic rein-

forcement layers. Nonlinear behavior of soft soil and

granular fill is considered. Plane strain conditions are

considered for the loading and the reinforced foun-

dation soil system. Three geosynthetic layers are

considered in the model. The membranes divide the

shear layer into four parts. General assumptions made

in the present study are (1) geosynthetic reinforce-

ments are linearly elastic with negligible thickness

and rough enough to prevent slippage at the interface

with soil; (2) rotation of the reinforcements is small;

(3) rate of increase in mobilized tension in all rein-

forcement layers is same; (4) deformation at the

interface of the reinforcements and soil are same; (5)

modulus of subgrade reaction of the soft soil has a

constant value irrespective of depth.

Formulation

A footing load of intensity q is applied over a width of

2B on the multilayer extensible geosynthetic-rein-

forced granular bed of width 2L on soft soil is shown

in Fig. 2. Considering a unit length in the direction

perpendicular to Fig. 2, the vertical force equilibrium

for the element of shear layer 1, 2, 3 and 4 (shown in

Fig. 3) leads to the following expressions:

q� rn1 þ sn1 tan hþ G12H1

d2wdx2¼ 0 ð1Þ

r0n1 � rn2 þ ðs0n1 þ sn2Þ tan hþ G22H2

d2wdx2¼ 0 ð2Þ

r0n2 � rn3 þ ðs0n2 þ sn3Þ tan hþ G32H3

d2wdx2¼ 0 ð3Þ

qs � r0n3 þ s0n3 tan hþ G42H4

d2wdx2¼ 0 ð4Þ

where H1, H2, H3 and H4 are thickness of the granular

layer 1, 2, 3 and 4, respectively; q and qs are average

normal stress acting at the top and bottom of the top

most and bottom most shear layers, respectively; rnj

and snj ( j = 1, 2, and 3) are respectively the average

Fig. 2 Proposed

foundation model

Geotech Geol Eng (2007) 25:11–23 13

123

normal and shear stress acting on the top of the ele-

ment of the different shear layers as shown in Fig. 3;

similarly, rnj¢ and snj

¢ (j = 1, 2, and 3) are respectively

the average normal and shear stress acting at the

bottom of the elements; x is distance from centre of

loading; and w is the vertical displacement.

A hyperbolic shear stress–shear strain response of

the granular fill is considered (Ghosh and Madhav

1994) as

sj ¼Gj0ðdw=dxÞ1þ Gj0jdw=dxj

suj

; j ¼ 1; 2; 3 and 4 ð5Þ

where sj (j = 1, 2, 3, and 4) are shear stresses in the

shear layer 1, 2, 3 and 4, respectively; Gj0 (j = 1, 2, 3

and 4) are initial shear modulus of the shear layer 1,

2, 3 and 4, respectively; suj (j=1, 2, 3, and 4) ulti-

mate shear resistance of the shear layer 1, 2, 3 and

4, respectively; dw/dx is shear strain; and G12, G22,

G32, G42 can be expressed as

Gj2 ¼Gj0

1þ Gj0jdw=dxjsuj

h i2ð6Þ

The stress–displacement response of the soft soil

can also be represented by a hyperbolic relation

(Kondner 1963) as

qs ¼ks0w

1þ ks0ðw=quÞð7Þ

where ks0 is initial modulus of the subgrade reaction

(spring constant) and qu is ultimate bearing capacity

of the soft soil. The modulus of subgrade reaction of

the soft soil and shear modulus of the granular fill can

be determined using the procedure described by

Selvadurai (1979).

Considering vertical and horizontal force equi-

librium for element of the geosynthetic layers, the

expressions for the normal stresses and the mobi-

lized tension (shown in Fig. 3) are obtained as fol-

lows:

rnj � r0nj ¼ �ðTp þ TjÞ cos3 hd2wdx2

ð8Þ

snj þ s0nj ¼dTj

dxcos h; j ¼ 1; 2 and 3 ð9Þ

where Tj (j = 1,2 and 3) are mobilized tension in the

top, middle and bottom geosynthetic layer, respec-

tively; Tp is pretension force applied to the geosyn-

thetic layers; and h is slope of the membrane.

Fig. 3 Elements from a vertical segment of infinitesimal width

showing the forces and stress on each element

14 Geotech Geol Eng (2007) 25:11–23

123

It is assumed that the rate of increase in mobilized

tension in all reinforcement layers is same and this is

only possible if ( snj þ s0nj) (j = 1, 2, and 3) is

constant. Using this condition the mobilized tension

for element of the geosynthetic reinforcement layers

can be written as

dT1

dxcos h ¼ dT2

dxcos h ¼ dT3

dxcos h ¼ sn3 þ s0n3 ð10Þ

Combining Equations 1–4, 7, 8 and rearranging

one gets

q ¼ ks0w1þ ks0w=qu

� dT1

dxþ dT2

dxþ dT3

dx

� �sin h

� ½ð3Tp þ T1 þ T2 þ T3Þ cos3 h

þ ðG12H1 þ G22H2 þ G32H3 þ G42H4Þ�d2wdx2

ð11Þ

Figure 4 shows the deformation of the granular fill

elements due to the increase in mobilized tension of

different reinforcement layers. Assuming no slipping

between the geosynthetic layers and the granular fill

layers, the following deformation compatibility con-

dition exists:

ux1 ¼ ugx1; ux2 ¼ ugx2; ux3 ¼ u0x3 ¼ ugx3

where ux1, ux2, ux3 and ux3¢ are the horizontal dis-

placements at the granular fill layer 1, 2, 3 and 4,

respectively and ugx1, ugx2 and ugx3 are the horizontal

displacements at the interface of the top, middle and

bottom geosynthetic layer, respectively.

The horizontal displacements of the granular fill

layers 3 and 4 can be expressed in terms of the shear

strain values, cx and cx¢ , respectively as follows:

ux3 ¼ cx

X3

j¼1

Hj ¼sn3

Ge1

X3

j¼1

Hj since cx ¼sn3

Ge1

� �

ð12Þ

u0x3 ¼ H4c0x ¼ H4

s0n3

G41

since c0x ¼s0n3

G41

� �ð13Þ

where Ge1= equivalent initial shear modulus of the

shear layer 1, 2, 3 and can be expressed as

Ge1 ¼G11H1 þ G21H2 þ G31H3

H1 þ H2 þ H3

¼P3

j¼1 Gj1HjP3j¼1 Hj

ð14Þ

where

Gj1 ¼Gj0

1þ Gj0jdw=dxjsuj

; j ¼ 1; 2; 3 and 4

Using deformation compatibility relation, the

shear stress sn3 can be expressed in terms of sn3¢ as

follows:

sn3 ¼H4

P3j¼1 Gj1Hj

G41

P3j¼1 Hj

� �2s0n3 ð15Þ

The values ux1 and ux2 can be related to ux3¢ by

using the property of similar triangle (shown in

Fig. 4) as

ux1 ¼H1P3j¼1 Hj

u0x3 ð16Þ

ux2 ¼H1 þ H2P3

j¼1 Hju0x3 ð17Þ

Figure 5 shows a stretched and rotate element of

the geosynthetic layer. Using deformation compati-

bility condition, the displacement of the geosynthetic

elements and the displacement increment can be

written as

For the top reinforcement element:

ugx1 ¼ ux1 ¼H1H4P3

j¼1 Hj

s0n3

G41

ð18Þ

dugx1 ¼H1H4P3

j¼1 Hj

ds0n3

G41

ð19Þ

For the middle reinforcement element:

ugx2 ¼ ux2 ¼ðH1 þ H2ÞH4P3

j¼1 Hj

s0n3

G41

ð20Þ

Geotech Geol Eng (2007) 25:11–23 15

123

dugx2 ¼ðH1 þ H2ÞH4P3

j¼1 Hj

ds0n3

G41

ð21Þ

For the bottom reinforcement element:

ugx3 ¼ u0x3 ¼ H4

s0n3

G41

ð22Þ

dugx3 ¼H4

G41

ds0n3 ð23Þ

The stretched length, dl, after rotation of the

geosynthetic elements can be expressed as

dlj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdwÞ2 þ ðdxþ dugxjÞ2

q; j ¼ 1; 2 and 3

ð24Þ

The strain, egj of the geosynthetic elements are

calculated as follows:

egj ¼dlj � dx

dx¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidwdx

� �2

þ 1þ dugxjdx

� �2s

� 1

ð25Þ

The mobilized tension in the geosynthetic elements

are given as

Tj ¼ Egegj ¼ Eg

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidwdx

� �2

þ 1þ dugxj

dx

� �s 2

� 1 ð26Þ

where Eg is tension modulus of the geosynthetic

layers. Putting the value ofdugxj

dx for different rein-

forcement layers in Equation 26 and rearrangingds0n3

dxcan be expressed as

Fig. 4 Shear deformations

due to increase in mobilized

tension in different

geosynthetic layers

Fig. 5 Stretching and rotation of geosynthetic element

16 Geotech Geol Eng (2007) 25:11–23

123

ds0n3

dx¼P3

j¼1 Hj

H1H4

G41

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT1

Eg

þ1

� �2

� dwdx

� �2s

�1

24

35

¼P3

j¼1 Hj

ðH1þH2ÞH4

G41

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2

Eg

þ1

� �2

� dwdx

� �2s

�1

24

35

¼G41

H4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT3

Eg

þ1

� �2

� dwdx

� �2s

�1

24

35

ð27Þ

Substituting Equation 15 into Equation 10 gives

dT1

dx¼ dT2

dx¼ dT3

dx¼ 1

cos hðsn3 þ s0n3Þ

¼ 1

cos h

H4

P3j¼1 Gj1Hj

G41

P3j¼1 Hj

� �2þ 1

264

375s

0

n3

ð28Þ

Rearranging Equation 28:

s0n3 ¼cos h

H4

P3

j¼1Gj1Hj

G41

P3

j¼1Hj

� �2 þ 1

264

375

dTj

dxð29Þ

Differentiating Equation 28 and substituting Equa-

tions 27 and 29 gives the following expressions for

different reinforcement elements:

d2Tj

dx2¼ðsin h cos hþ GGÞ d

2wdx2

dTj

dx

þ GHj

cos h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTj

Eg

þ 1

� �2

� dwdx

� �2s

� 1

24

35

j ¼ 1; 2; and 3

ð30Þ

where

GG ¼ G41H4

G41

P3j¼1 Hj

� �2

þH4

P3j¼1 Gj1Hj

�X3

j¼1

HjGj1

su4

� G41

suj

Gj0

G40

Gj2

G42

� �

GH1 ¼P3

j¼1 Gj1Hj

H1

P3j¼1 Hj

þG41

P3j¼1 Hj

H4H1

GH2 ¼P3

j¼1 Gj1Hj

ðH1 þ H2ÞP3

j¼1 Hjþ

G41

P3j¼1 Hj

H4ðH1 þ H2Þ

GH3 ¼P3

j¼1 Gj1Hj

P3j¼1 Hj

� �2þ G41

H4

Using nondimensional parameters as

X ¼ xB; W ¼ w

B; H�j ¼

Hj

B;

q� ¼ qks0B

; E�g ¼Eg

ks0B2

G�j1 ¼Gj1Hj

ks0B2; G�j2 ¼

Gj2Hj

ks0B2;

T �j ¼Tj

ks0B2; T �p ¼

Tp

ks0B2

The normalized form of the Equations 5 and 7 are

s�j ¼G�j0ðdW =dX Þ

1þ G�j0jdW =dX js�uj

and q�s ¼a�W

1þ a�ðW =q�uÞ

where

s�j ¼sjHj

ks0B2G�j0 ¼

Gj0Hj

ks0B2

s�uj ¼sujHj

ks0B2q�s ¼

qs

ks0Bq�u ¼

qu

ks0B

Equations 10 and 30 can be written as

q� ¼ W1þ W =q�u

� dT �1dXþ dT �2

dXþ dT �3

dX

� �sin h

� ½ð3T �p þ T �1 þ T �2 þ T �3 Þ cos3 h

þ ðG�12 þ G�22 þ G�32 þ G�42Þ�d2WdX 2

ð31Þ

d2T �jdX 2

¼ ðsin h cos hþ GG�Þ d2W

dX 2

dT �jdX

þGH �jcos h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT �jE�gþ 1

!2

� dWdX

� �2

vuut � 1

264

375

ð32Þ

Geotech Geol Eng (2007) 25:11–23 17

123

where

GG� ¼ G�41

G�41

P3j¼1 H �j

� �2�

H �4 þ H �4P3

j¼1 G�j1

�X3

j¼1

H�jG�j1s�u4

� G�41

s�uj

G�j0G�40

G�j2G�42

!

GH �1 ¼P3

j¼1 G�j1H�1P3

j¼1 H�jþ

G�41

P3j¼1 H �j

H �1 H �24

GH �2 ¼P3

j¼1 G�j1

ðH�1 þ H �2 ÞP3

j¼1 H �jþ

G�41

P3j¼1 H �j

ðH�1 þ H�2 ÞH�24

GH �3 ¼P3

j¼1 G�j1P3

j¼1 H �j� �2

þ G�41

H �24

and

tan h ¼ dWdX

; sin h ¼dWdXffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ dWdX

2q ;

cos h ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ dW

dX

2q

Method of Solution

Finite difference formulation

Finite difference method has been employed to solve

the differential equations [Equations 30–32]. In these

equations, the derivatives ( d2WdX 2 ;

dWdX ;

d2T �jdX 2 and

dT �jdX Þ are

expressed by central difference scheme. The length

L/B may be divided into n number of equal lengths

with (n+1) number of node points (i=1, 2, 3, 4,...,n);

thus, DX=(L/B)/n. Writing Equations in finite differ-

ence form, for an interior node, i, within the domain,

lead to

q�i ¼Wi

1þ Wi=q�u� dT �1

dXþ dT �2

dXþ dT �3

dX

� �

isin hi

� ½ð3T �p þ T �1i þ T �2i þ T �3iÞ cos3 hi

þ ðG�12 þ G�22 þ G�32 þ G�42Þ�

� Wi�1 � 2Wi þ Wiþ1

ðDX Þ2

( )ð33Þ

T �jðiþ1Þ � 2T �ji þ T �jði�1Þ

ðDX Þ2

¼ ðsin h cos hþ GG�Þid2WdX 2

����i

T �jðiþ1Þ � T �jði�1Þ2DX

� �

þGH �jcos hi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT �jiE�gþ 1

!2

� dWdXji

� �2

vuut � 1

264

375

ð34Þ

Loading and boundary conditions

A total of eight boundary conditions are required to

solve the four-second order differential equations

[Equations 33–34]. Due to symmetry of the system

only half portion is taken into consideration. Thus, at

centre of the loaded region X=0.0 (or x=0.0); at the

edge of the footing X=1.0 (or x=B); and at the edge of

the reinforced zone X=L/B (or x=L).

The loading conditions that are considered are

given as

q�i ðX Þ ¼ q� for Xj j � 1:0

¼ 0:0 for Xj j[1:0 ð35Þ

At X=0, due to symmetry, the slope, dW/dX, will

be zero and the ratio of the increase in tension in

different reinforcement layers, dTj*/dX (j=1, 2, and 3),

will be zero. At X=L/B, the geosynthetic layers are

free, so that the mobilized tension in different geo-

synthetic layers, i.e. Tj*=0. The shear stress acting on

the shear layers at the edge (at X=L/B) will also be

zero since there is no confinement, that is,

s1=s2=s3=s4=0. According to the assumption of the

18 Geotech Geol Eng (2007) 25:11–23

123

Pasternak shear layer sj=Gj2 dW/dX (j=1, 2, 3, and 4),

i.e. dW/dX=0.

Since Equations 33–34 are all nonlinear equations,

an iterative computing procedure has been used for

obtaining solutions. The solutions have been obtained

with a convergence criterion as

W ki � W k�1

i

W ki

��������\e and

T kji � T k�1

ji

T kji

�����

�����\e

for all i, where k and (k)1) are the present and pre-

vious iteration values, respectively and e is the

specified tolerance which has been considered as

10)4 in the present study.

Results and Discussions

Based on the formulation as described above, a

computer program has been developed using finite

difference method and solutions are obtained using

Gauss Elimination technique. Parametric studies have

been carried out to show the effects of various

parameters on the settlement response. The shear

modulus as well as the total thickness of the granular

fill is taken as constant throughout the study. Thick-

ness of the granular fill layer is assumed to be half of

the width of the footing. The number of reinforce-

ment layers is varied from 1 to 3 and it is placed such

that the reinforcement layer equally divides the

granular fill layer.

Comparison of extensible and inextensible

geosynthetic reinforcement models

The settlement responses obtained using the present

model is compared with the results obtained using the

model developed for multilayer inextensible geo-

synthetic reinforcement (Deb et al. 2005) in Fig. 6. It

is observed that settlement reduction is more when

inextensible reinforcements are used compared to

that of extensible reinforcements. For a loading

intensity 0.8, in case of inextensible reinforcement

the maximum settlement at the centre of the loaded

region is reduced by 17.6%, 26% and 32.3% as the

number of reinforcement layer is increased from

unreinforced to one, two and three, respectively.

However, in case of extensible reinforcement the

corresponding settlement reductions are 6.5%, 12.5%

and 18.2%, respectively. From the results it is said

that when inextensible reinforcements are used set-

tlement is decreased with a decreasing rate as the

number of reinforcement layer is increased, whereas

in case of extensible reinforcement the reduction rate

is almost constant.

Figure 7 shows the comparison of the mobilized

tension in the extensible and inextensible reinforce-

ment layers for a particular loading intensity 0.8. It is

noticed that in case of inextensible reinforcements

mobilized tension is higher than that of extensible

reinforcements. For multilayer reinforcement system

the difference of mobilized tension in different rein-

forcement layers is very small in case of inextensible

reinforcement, whereas in case of extensible

Fig. 6 Comparison of the settlement profiles of the multilayer

extensible and inextensible geosynthetic reinforcement system

Fig. 7 Comparison of the mobilized tension in the multilayer

extensible and inextensible geosynthetic reinforcements

Geotech Geol Eng (2007) 25:11–23 19

123

reinforcement the difference is more. In case of three

reinforcement layer system the maximum mobilized

tension in the top inextensible reinforcement layer is

1.15 times that of bottom inextensible reinforcement

layer, whereas in top extensible reinforcement layer

the mobilized tension is 3.1 times that of bottom

extensible reinforcement layer. It is also observed that

the top inextensible geosynthetic layer is subjected to

higher tension up to x/B£ 0.5, whereas beyond that

region the mobilized tension in bottom inextensible

geosynthetic layer is more, but in case of extensible

geosynthetic reinforcement the top geosynthetic layer

is always subjected to higher mobilized tension value.

Effect of loading intensity

Settlement profiles for different loading intensities

with no pre-stress force in the geosynthetic rein-

forcements are shown in Fig. 8. It is noticed that as

the number of geosynthetic reinforcement increases,

the settlement under the loaded region decreases,

whereas the trend is reversed beyond that region. For

loading intensity 0.3, as the number of geosynthetic

layer increases from zero to three, the settlement at

the centre of the loaded region decreases by 6%,

whereas for loading intensities 0.5 and 0.8 this

reduction is 11.4% and 18.2%, respectively. Thus, for

higher loading intensity multiple geosynthetic rein-

forcements reduce the vertical settlement more

effectively than for lower loading intensity. This is

due to the fact that tension mobilization of geosyn-

thetic reinforcements increases with increasing load.

These results are in good agreement with the findings

obtained by Nogami and Yong (2003) and Deb et al.

(2005) for inextensible geosynthetic reinforcement. It

is also observed that for load intensity of 0.8, the

settlement at the centre of the loaded region decreases

by 6.5%, 12.5% and 18.2%, whereas at the edge the

corresponding decreases are 0.6%, 1% and 1.7% as

the number of reinforcement layer increases from

unreinforced to single, double and triple, respec-

tively. Thus, use of multiple geosynthetic reinforce-

ment layers results in reduction in both the total and

differential settlements of the loaded region.

Effect of ultimate bearing capacity of the soft soil

Figures 9–11 show the settlement profiles of unrein-

forced and multilayer reinforced soil for various

values of ultimate bearing capacity of the soft soil

(qu*). It is observed that the settlement decreases with

the increasing values of qu*. This is depicted by the

difference in settlement decreasing as qu* value in-

creases from 1 to 10. The settlement values become

constant if qu* value is greater than 10. This

observation is valid for unreinforced as well rein-

forced soil and independent of number of geosyn-

thetic layers. This due to the fact that if soft soil has

low value of ultimate bearing capacity and loading

intensity is high, the soil behavior is non linear and in

this situation higher settlements are observed. As the

qu* value is increased and when the loading intensity

is small as compared to the ultimate bearing capacity,

the soil behavior is more close to the linear behavior

and after a certain value of qu* behavior of the soil

becomes linear and no further change in settlements

Fig. 8 Settlement profiles for various loading intensities with

multilayer geosynthetic reinforcementsFig. 9 Settlement profiles of unreinforced soil for various

values of qu*

20 Geotech Geol Eng (2007) 25:11–23

123

is observer due to increase in qu* value. For unrein-

forced soil the maximum settlement at the centre of

the loaded region is decreased by 48% as the qu* in-

creases from 1 to 10, whereas the reduction is 36%

and 31% for two and three layers of geosynthetic

reinforced soil. This indicates that at low value of qu*

the behavior of the soil without reinforcement is

highly nonlinear, but as the number of reinforcement

layer increases the nonlinear behavior of the soil is

decreased. It is also observed that for applied loading

intensity, q*=0.8 and qu*=1.0, as the number of geo-

synthetic reinforcement layer increases from zero to

two, the maximum settlement decreases by 27.2%,

whereas at qu* equal to 2, 5 and 10, the reduction is

19.2%, 14.2% and 12.5%, respectively. In case of

three layers of reinforced soil the reductions are

35.5%, 26.1%, 20.2% and 18.2%, respectively. Thus,

at low qu* values there is a significant reduction in the

maximum settlement from unreinforced soil to rein-

forced soil and settlement reduction rate is also high

as the number of reinforcement layer is increased, but

as the qu* value increases settlement reduction effi-

ciency of the reinforcement layers decreases and

settlement reduction rate is also decreases.

Figures 12 and 13 show the variation of the mobi-

lized tension in the top geosynthetic layer for two and

three layers of reinforced soil, respectively. From

these figures it is observed that mobilized tension va-

lue decreases with the increasing value of qu*. Similar

behavior has been observed for single layer of rein-

forced soil (Yin 1997b). For two layers of reinforced

soil the maximum mobilized tension value in the top

reinforcement layer is decreased by 36% as qu* value

increases from 1 to 10, whereas in case of three layers

of reinforced soil the reduction is 26.5%. Hence, as the

number of reinforcement layer is increased the

reduction rate in the mobilized tension due to increase

in the qu* value decreases. The subsequent reduction in

the mobilized tension is less than 2% as qu* increases

from 10 to 100. This indicates that the effect of qu* on

the mobilized tension in the geosynthetic layers is

negligible if soil is stiff (qu* is greater than 10).

Effect of ultimate shearing resistance

of the granular fill

Table 1 shows the effect of the ultimate shearing

resistance of the granular bed on the maximum

settlement and the maximum mobilized tension in the

Fig. 11 Settlement profiles of reinforced soil for various

values of qu* with three layers of geosynthetic reinforcement

Fig. 12 Mobilized tensions in the top reinforcement layer for

various values of qu* for two layers of geosynthetic reinforced

soil

Fig. 10 Settlement profiles of reinforced soil for various

values of qu* with two layers of geosynthetic reinforcement

Geotech Geol Eng (2007) 25:11–23 21

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top reinforcement layer for three reinforcement layer

system. It is observed that the ultimate shearing

resistance of the granular bed does not significantly

affect the settlement response of the multilayer geo-

synthetic reinforcement system. The difference be-

tween the maximum settlements corresponding to

ultimate shearing resistance (su*) value increasing

from 0.1 to 10 is less than 2%, whereas the mobilized

tension value in the top reinforcement layer is in-

creased by 4.5% as su* increases from 0.1 to 10.

Effect of stiffness of the geosynthetic layers

Table 2 and Fig. 14 present the settlement and the

mobilized tension values in the top geosynthetic layer

(for N=3) computed using the proposed model for

different stiffness of geosynthetic layer, respectively.

It is seen that the difference between the maximum

settlements corresponding to the geosynthetic stiff-

ness value (Eg*) changing from 5 to 200 is less than

1%. However, for single layer system the settlement

is slightly reduced within the loaded region due to an

increase in the geosynthetic layer stiffness (Yin,

1997a). Hence, the effect of the stiffness of the

geosynthetic layer on the settlement response be-

comes insignificant for multilayer reinforced system.

Maximum mobilized tension in the top geosynthetic

layer is increased by 9% as the geosynthetic stiffness

value increases from 5 to 20 and subsequent 1%

increase is observed as the geosynthetic stiffness

value is increased from 20 to 200. The results indicate

that for multilayer geosynthetic reinforcement system

the mobilized tension in the reinforcement layers

increases as the stiffness of the geosynthetic layers

increases up to 20, whereas above that value, the

geosynthetic stiffness does not effect the mobilized

tension in the reinforcement layers.

Conclusions

The proposed model is useful in analyzing the

settlement response of the multilayer extensible

Fig. 13 Mobilized tensions in the top reinforcement layer for

various values of qu* for three layers of geosynthetic reinforced

soil

Table 1 Effect of ultimate shearing resistance of the granular

bed

su* 0.1 1 5 10

Maximum

normalized

settlement

0.63123 0.622051 0.621219 0.621147

Maximum

mobilized

tension in top

reinforcement layer

0.22099 0.22987 0.23066 0.23084

q� ¼ 0:8; q�u ¼ 10; E�g ¼ 20; T �p ¼ 0; L=B ¼ 2; N ¼ 3

Table 2 Effect of the geosynthetic stiffness values on the

maximum settlement for three layers of reinforced soil

Eg* 5 10 20 200

Maximum

normalized

settlement

0.2323 0.2314 0.2311 0.2310

q� ¼ 0:3; q�u ¼ 10; s�u ¼ 10; T �p ¼ 0; L=B ¼ 2; N ¼ 3

Fig. 14 Mobilized tension in the top geosynthetic layer for

various values of geosynthetic reinforcement stiffness for three

layers of reinforced soil

22 Geotech Geol Eng (2007) 25:11–23

123

geosynthetic reinforced granular bed overlying soft

soil. Based on the results and discussions presented in

the previous section, the following conclusions can

be drawn:

• Reduction in settlement due to the use of exten-

sible geosynthetic reinforcements is less as com-

pare to the same for inextensible reinforcements.

In case of inextensible reinforcements as the

number of reinforcement layer is increased the

settlement is decreased with a decreasing rate. In

case of extensible reinforcement the reduction

rate is almost constant.

• In general, the mobilized tension in the inexten-

sible geosynthetic reinforcements is more than

that of in the extensible reinforcements. For

multilayer reinforcement system, the mobilized

tension in the top extensible reinforcement layer

is more through out the length of the reinforce-

ment. In case of inextensible reinforcement up to

half of the loaded region from the center of the

loading, the top reinforcement layer is subjected

to higher mobilized tension, whereas beyond that

region the mobilized tension in bottom inexten-

sible geosynthetic layer is more. Difference in the

mobilized tension in the different extensible

reinforcement layers is more, but in case of

inextensible reinforcement the difference is very

small.

• Use of multiple geosynthetic reinforcements re-

duce the total and differential settlements of the

loaded region and are very effective for higher

loading intensity and for very soft soil.

• In case of very soft soil the behavior of the un-

reinforced soil is highly nonlinear, but as the

number of reinforcement layers is increased the

nonlinear behavior of the soil decreases. For

stiffer soil the settlement reduction rate decreases

as the number of reinforcement layer is increased.

• The mobilized tension value in the geosynthetic

layers decreases with the increasing value of

ultimate bearing capacity of the soft soil. How-

ever, for higher values of ultimate bearing

capacity of soft soil (qu* is greater than 10), the

settlement response and the mobilized tension in

the reinforcement layers are not affected.

• The ultimate shearing resistance of the granular

bed does not significantly affect the settlement

response of the multi layer geosynthetic rein-

forcement system, but mobilized tension in the

reinforcement layers slightly increases due to the

decrease of s u*.

• The effect of the stiffness of the geosynthetic

layer on the settlement response becomes insig-

nificant for multilayer reinforced system, but the

mobilized tension in the reinforcement layers in-

creases as the stiffness of the geosynthetic layers

(Eg*) increases up to 20, whereas above that value,

the geosynthetic stiffness does not significantly

effect the mobilized tension in the reinforcement

layers.

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