Post on 20-Jan-2023
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
123
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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