A Finite Element Study of Beam on Reinforced Granular Beds with Sand Drains

The 12th International Conference of
International Association for Computer Methods and Advances in Geomechanics (IACMAG)
1-6 October, 2008
Goa, India
A Finite Element Study of Beam on Reinforced Granular Beds with
Sand Drains
Sarvesh Chandra
Professor, Dept. of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur, India
C. S. Upadhyay
Associate Professor, Dept. of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India
Imran Ahmad
Former Post Graduate Student, Dept. of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur, India
Arindam Dey
Research Scholar, Dept. of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur, India
Keywords: Geosynthetic reinforced granular fill-soft soil system, Finite element method, Sand drains
ABSTRACT: This paper presents the settlement analysis of a beam resting on geosynthetic reinforced granular
fill-soft soil system. Each subsystem of the reinforced fill-soil system is idealized by elastic membrane, Pasternak
shear layer, Winkler springs and dashpots, as applicable. The suggested model incorporates various aspects of
the behavior of the geosynthetic-reinforced granular fill-soft soil system such as horizontal stress induced in the
granular fill, the compressibility of the granular fill, and the time-dependent behavior of the subgrade. The
differential equations governing the settlement response of the beam resting on two layered reinforced foundation
soil has been formulated by incorporating deformation compatibility conditions. The numerical solutions are
obtained using Finite Element Method and results are presented in non-dimensional form. The parametric studies
are carried out to enumerate the effects of parameters on the settlement response of the system. Results indicate
that over a large number of various parameters under large deformation the proposed model evaluate the
settlement of the system and horizontal displacement of membrane with reasonable accuracy. It is observed that
compressibility, shear modulus and thickness of granular fill, pre-stressing and tension modulus of reinforcement
have appreciable influence on the settlement of the system and horizontal displacement of membrane. It is
observed that the horizontal displacement of the geosynthetic membrane is negligible as compared to that of the
vertical settlement. The model is also analyzed for the case of a sand drain in the soft soil which indicated that the
elapsed time and the radius of the sand drain significantly affects the settlement response of the system.
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 mostly used nowadays to improve the strength of soft foundation
soils is the placement of engineered granular fills containing geosynthetic reinforcement (e.g. geogrid,
geotextiles) on the soft soil. Model tests carried out by various researchers (Fragaszy and Lawton, 1984; Love et
al. 1987; Mandal and Sah, 1993; Adams and Collin, 1997) exhibited improved behavior of the soft soil system
with respect to their load settlement response whenever a geosynthetic, with or without pretension, is provided
along with the granular fill. The calculation of settlement and ultimate bearing capacity of the geosynthetic
reinforced granular fill over soft soil, along with the requirement of stability in terms of settlement, is an important
issue for the design engineers. Such issues can be, and are commonly taken care of by using the approach of
analyzing the beams on elastic foundations and obtaining the flexural response of the system under different
loading conditions, and by using different solution techniques.
Several concepts have been developed to explain the reinforcing mechanism of the reinforcement used in the
soil. One of the approaches to study the interaction problem is by idealizing the behavior of the soil. The linear
elastic idealization of the supporting soil medium is usually represented by one-parameter or two-parameter
mechanical or mathematical model, such as Winkler model, Filolenko – Borodich model, Pasternak model, Kerr
model (Kerr, 1964) etc. Over times, various combinations of the above simplified models with varying degree of
complexity has come into existence in order to study the influence of the material properties of the soil such as
shape, size, configuration, stress history, soil moisture, and permeability on the behavior of foundation systems.
Several analytical and numerical works have been carried out on this aspect by various researchers (Biot, 1937;
Gazis, 1958; Vesic, 1961; Cheung and Nag, 1968; Rao et al., 1971; Sharma and Dasgupta, 1975; Giroud and
24
Noiray, 1981; Zhaohua and Cook, 1983; Mastuda and Sakiyama, 1987; Razaqpur and Shah, 1991; Gendy and
Saleeb, 1999; Yin, 2000).
The present study attempts to give an insight to the settlement analysis of a beam resting on a reinforced
granular fill supported on soft clay with sand drain. In this study, the footing is idealized as an elastic beam. A
mechanical foundation model for the reinforced granular fill-soft soil system is considered to incorporate the
compressibility as well as the time-dependent behavior of the foundation system. The various parts of the
geosynthetic reinforced granular fill-soft soil system are idealized by mechanical elements as shear layer, rough
membrane, Winkler springs and viscous dashpots. The equations governing the settlement response of the beam
resting on the geosynthetic reinforced granular fill soft soil system have been derived considering the equilibrium
of shear layer and rough elastic membrane. The numerical solution has been obtained using finite element
method. Since most of the problems have been solved using models like Winkler, Pasternak etc., the same
approach has been adopted and the FEM approach has been incorporated in this study as this is a simpler
approach than finding out equivalent properties of the reinforced layer. The settlement behavior of the system has
been studied for various loading intensities acting over the span of the beam with variation of different soil
parameters, effect of pre-stressing the geosynthetic reinforcement, rigidity of the beam and degree of
consolidation.
2 Definition and formulation of the problem
Figure 1(a) depicts a geosynthetic reinforced granular fill on soft foundation soil supporting a beam. The
foundation consists of a granular fill overlying the soft foundation soil. A layer of geosynthetic has been provided
as reinforcement in the granular fill. An equivalent mechanical model supporting the beam is shown in Figure 1(b)
as proposed by Shukla and Chandra (1994). In this model, a stretched, rough, elastic membrane represents the
geosynthetic reinforcement and the Pasternak shear layer represents the granular fill. The compressibility of the
granular fill is represented by a layer of Winkler springs (spring constant f k ) attached to the bottom of the
Pasternak shear layer. The membrane divides the shear layer into two parts. The saturated soft foundation soil is
idealized by the Terzaghi consolidation model, which has springs and dashpots attached to the bottom of the
Pasternak shear layer. The springs represent the soil skeleton and the dashpots simulate the dissipation of
excess pore water pressure in soil.
(a) (b)
Figure 1. Definition sketch and Schematic diagram of the problem.
The geosynthetic is placed inside the granular fill, for which the granular layer above the reinforcement has a
shear modulus value t G and the granular below the reinforcement has the shear modulus value b G . The
geosynthetic reinforcement has been prestressed with force p T in addition to the mobilized tensile force,T , in the
reinforcement under loadings. The general assumptions made in the study are:
1. Geosynthetic reinforcement is linearly elastic, rough enough to prevent slippage at the soil interface and
has no shear resistance, and thickness of reinforcement is neglected.
2. A rigid perfectly plastic friction model is being adopted to represent the behavior of the soil-geosynthetic
interface in shear.
3. The modulus of subgrade reaction of soil has constant value irrespective of depth and time.
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4. The consolidation characteristics of the soil both in the loaded region and beyond it are considered to be
the same.
5. The rotation of reinforcement is small.
The equations governing the settlement response of the beam on the geosynthetic-reinforced granular fill-soft soil
system at any particular time t > 0 as obtained by Yin (1997) using the following nondimensional parameters:
* * * *
* * * * * *
2 3 2 2 5
, , , , , ,
, , , , ,
t b t b
t b t b
s s
g p
g p
s s s s s s
X xW wV v H HH H G GG G
B B B B B kB kB
E EP P uo uo T TT T EI EI
k B k B k B k B k B K B
= = = = = = =
= = = = = =
(1a)
are written as:
2 2
* * 1 1
g 2 2
T E dV dV dW
dX dX dX
= ⎡⎢ + ⎛⎜ ⎞⎟ + ⎛⎜ ⎞⎟⎤⎥
⎢⎣ ⎝ ⎠ ⎝ ⎠⎥⎦
(1b)
* 2 4 2
* * * * 1
2 4 0 2 ( p) (1 ) 0
dT dW T T d W EI d W W u UC K d W
dX dX dX dX dX
β β
⎡ ⎤
+ + −⎢ + + − ⎥+ =
⎣ ⎦
(1c)
2 2 4 2
* * * * 1
2 2 4 0 2 p (1 ) 0
Td WTd W EId W W u UC Kd W KVdW
dX dX dX dX dX
+ − −β −β − + + ⎛⎜ ⎞⎟=
⎝ ⎠
(1d)
Equations (1a), (1b) and (1c) are the governing differential equations for solving three unknownsT,W,V . Since
the problem to be analyzed is symmetric in terms of loading and geometry, only half of the model needs to be
analyzed.
2.1 Boundary conditions
The solutions are obtained for normalized point load acting over the beam of width 2B . The slope of the
settlement distance profile and mobilized tension distance profile at the centre of the beam is taken as zero. At
the end of the reinforced zone the slope of the settlement distance profile is considered as zero, as observed in
most of the practical cases, whether the membrane is free or fixed. The mobilized tensile force at the edge of the
reinforcement is considered as zero.
2.2 Method of solution
Virtual displacementsδW in the vertical direction andδV in the horizontal direction are applied. From the principle
of virtual work, the total work done is zero. The unknown functions V,W are obtained approximately using the
finite element method. Since the governing equations are non linear in nature, an iterative technique has to be
employed (here we use direct iteration or Picard’s iteration scheme). An approximation of V andW are
considered as follows:
1 1
,
ne ne
i i i i
j j
V NV W NW
= =
=Σ =Σ (2)
where, {N}e is the Lagrangian shape function vector, and {W}e is elemental vector of vertical displacements.
At (i +1)th iteration, the virtual displacement equations are linearized by using values of
dW and dV
dX dX
from the
ith iteration. For the first iteration, i i 0
i i
W V dW dV
dX dX
= =⎛⎜ ⎞⎟ = ⎛⎜ ⎞⎟ =
⎝ ⎠ ⎝ ⎠
is assumed, and the elemental matrices are
obtained assembling elemental matrices, we get
[ ]{ } { } [ ] [ ] [ ] [ ]
1 1
1 1; 1 1 ; 1 1
ne ne
e e
e e
K W F k K f F
= =
= Σ = Σ = (3)
Equation (3) gives W .
The global finite element form of the above is expressed as follows and used to obtain the solution for each
iteration until the results converge.
26
[ ]{ } { } [ ] [ ] { } { }
[ ]{ } [ ]{ } { } [ ] [ ] [ ] [ ] { } { }
1 1
1 1 1
2 2; 2 2 ; 2 2
3 4 3; 3 3; 4 4; 3 3
ne ne
e e
e e
ne ne ne
e e e
e e e
K W F k K f F
K V K W F k K k K f F
= =
= = =
⎤
= = = ⎥⎥⎥⎥
+ = = = =
⎥⎦
Σ Σ
Σ Σ Σ
(4)
2.3 Convergence Criterion
The solution to the nonlinear problem at an instant of time t has been obtained with the convergence criterion.
2 2
1 1
1 1
2 2
1 1
1 1
( ) ( )
;
( ) ( )
nn nn
i i i i
j j
nn nn
i i
j j
V V W W
V W
ε ε
+ +
= =
+ +
= =
− −
< <
Σ Σ
Σ Σ
(5)
where, (i +1)and i are present and previous iterations respectively; j is the number of nodes used and ε is the
specified tolerance, which in present case is taken as 0.0001. The ranges of various normalized parameters
studied are shown in Table 1.
Table 1. Ranges of various normalized parameters studied.
Sl. No. Normalized Parameters Ranges
1 Load intensity P* 5 – 15
2 Shear modulusGt*,Gb* 0.1 – 1.0
3 Spring constant ratio α 5.0 - ∞
4 Pretension *
Tp 0.1 – 1.0
5 Thickness of the shear layer Ht*,Hb* 0.0 5 – 5.0
6 Degree of consolidation UC 0 – 100 %
7 Tension modulus of geosynthetic membrane *
Eg 1 – 200
8 Radius of sand drain R 0.1 – 0.5
9 Time of consolidation 10 - 25
2.4 Consolidation using sand drains
Figure 2. Definition sketch of geosynthetic reinforced granular fill soft soil system with sand drain
The consolidation process of soft subsoil is a process, which continues for very long time due to its low
permeability. In order to accelerate the process of consolidation settlement, sand drains are installed in the soft
layer. Sand drains are constructed by driving down casing mandrels into the soil. The holes are then filled with
sand, after which casings are pulled out. When a surcharge is applied at ground surface, the pore water pressure
in the clay increases, thus, initiating hydraulic gradient in the vertical and horizontal directions. The horizontal
drainage is induced by sand drains. Hence, the process of dissipation of excess pore water pressure created by
the loading is accelerated. The drainage due to vertical sand drain takes place both horizontally and vertically.
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The horizontal permeability of clay is generally higher than the vertical permeability. The sand drain is assumed to
be incompressible and the ratio of horizontal to vertical permeability of subsoil layer is taken as 2.5.
In the case of sand drains, as the process of 3-D consolidation is symmetrical about the vertical axis, the problem
can be simplified as a 2-D problem. In case of equal strain, the differential equation governing the consolidation
process is expressed as:
2 2
2 2
1
vr vz
uC u u C u
t r r r z
∂ ⎡∂ ∂ ⎤ ∂
∂ = ⎢⎣∂ + ∂ ⎥⎦+ ∂
(6)
Using the following normalized parameters
, , ,
.
vr vz
c
Z z R rT C C tU u
H B HB q
= = = = (7)
where, H is the depth of soft foundation soil (length of sand drain), B is the half-width of beam, c T is the
normalized time, yr C and yz C are coefficient of consolidation in radial and vertical direction respectively. Using the
normalized parameters given above, equation (6) is written as,
2 2
2 2
vr 1 vz
c vz vr
U HC U U BC U
T B C R R R H C Z
∂ ⎡∂ ∂ ⎤ ∂
∂ = ⎢⎣∂ + ∂ ⎥⎦+ ∂
(8)
The governing differential equation for this problem is expressed as
* *
*
1 0; vr
A A A
c vz
UvRdRdZ Uv RdRdZ U v RdRdZ H C
T RR Z Z B C
α α
α
∂∂ +⎡⎢⎣ ∂∂ ∂∂ + ∂∂ ∂∂ ⎤⎥⎦= = ∫ ∫ ∫ (9)
The elemental finite element formulation is derived and the global form is expressed as:
[ ]{ } { } [ ] [ ] [ ] { } [ ] ( ) [ ] { }
[ ] [ ] [ ] [ ] ( )
1 1 1
1
1
1 1 1
; ; 1 ;
; ; 1 1
i i i i
ne ne
e e i i
i i
e e i
A U C A M t K C M t K U
m M k K U U U U
t
θ θ
θ
θ
+ + +
+
+
= = +
= = + Δ =⎡⎣ − − Δ ⎤⎦
⎡ − ⎤
= = =⎢⎣ Δ − − ⎥⎦
Σ Σ &
(10)
At time t = 0 we haveU = 1.0 . Equation (10) is the basic equation for solving unknowns { } i 1 U + from the known
values of { }i U . The Crank-Nicholson scheme is used for finite difference for whichθ = 0.5 , the above scheme is
unconditionally stable. The degree of consolidation obtained is a function of radial distance from center of loading,
depth and time, for any particular time the average degree of consolidation at any radial distance is obtained by
averaging the degree of consolidation values at that radial distance along the depth. This average degree of
consolidation is used as an input for the geosynthetic reinforced granular fill soft soil system to study its response
when sand drains are installed in the soft soil.
3 Results and discussions
3.1 Effect of tension modulus of geosynthetic reinforcement
Figure 3(a) shows the effect of variation of tension modulus of geosynthetic membrane on the settlement
response of the geosynthetic reinforced granular fill soft soil system considering other parameters as constant. It
is observed that the variation in the tension modulus of geosynthetic membrane does not appreciably affect the
settlement response of the system under the range of parameters studied. Figure 3(b) shows the horizontal
displacement profiles for various values of tension modulus of geosynthetic reinforcement. It is observed that as
the tension modulus increases, the horizontal displacement decreases.
3.2 Effect of pretension in geosynthetic membrane
Figure 4 shows the effect of pretension in the geosynthetic reinforcement on the settlement behavior of the
system. As pretension in the geosynthetic reinforcement is increased, settlement decreases below the point of
application of load and no appreciable change takes place at the center of system. As the pretension force
increases, heaving near the edge decreases. This results in reduction in differential settlement. It is concluded
that pre-tensioning of geosynthetic reinforcement may prove to be very effective in improving the ground when
very small differential settlement is desired.
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3.3 Effect of thickness of granular fill
Figure 5 shows the effect of variation of thickness of granular fill on the settlement response of the system at 90%
degree of consolidation and without any pretension. It is observed that settlement decreases with increase in the
thickness of granular fill below the point of application of load. There is no appreciable variation in the settlement
at the center of the system while settlement increases at the edge of the beam. It can be seen that for higher
values of thickness, there is very less variation in settlement. It can be stated that increase in the thickness of
granular fill is quite effective to reduce the settlement
3.4 Effect of shear modulus of granular fill
Figure 6 depicts the settlement profiles of the system for various values of shear modulus of granular fill. The
shear modulus of top and bottom granular layer is kept identical. As the magnitude of the shear modulus of fill
increases, settlement decreases below the point of application of load; while it increases at centre of the
reinforcement. Variation in settlement at edge of reinforcement is very small. It is also observed that for higher
values of shear modulus, variation in settlement along the length of the system is very small. From this fact, it
may be concluded that higher values of shear modulus are preferable to avoid differential settlement. It is
observed that increase in the magnitude of shear modulus of fill beyond 0.4 has no effect on the settlement
characteristics of the system.
3.5 Effect of spring constant ratio of granular fill
Figure 7 shows the effect of spring constant ratio on settlement profiles of the system at 90% consolidation with
no pretension in the geosynthetic reinforcement, thus bringing out the effect of the relative compressibility of the
granular fill and the soft soil on the settlement behavior. The settlement profile plotted for infinite spring constant
ratio corresponds to the granular fill being taken as incompressible. It is observed that the settlement at any
location decreases as the spring constant ratio is increased. The variation in settlement for the values of spring
constant more than 10 is very less along the length of system. As the spring constant ratio increases from 5 to 10
and 20, the decrease in settlement is 9.4% and 15.63% respectively. The decrease in settlement is 1.8% when
spring constant increases from 50 to infinity. Therefore it may be concluded that when the granular fill is 50 times
stiffer than soft soil, the compressibility of the granular fill can be ignored.
3.6 Effect of sand drain
Figure 8(a) shows the settlement profiles at various stages of consolidation. It is observed that as consolidation
takes place, the settlement increases at all locations of the system. In the system without sand drains, degree of
consolidation was considered uniform throughout the length of the system; but in the system with sand drains,
degree of consolidation is different at each location along the length. The rate of settlement with time in the
presence of sand drains is greater than without these. This is due to the fact that the consolidation at and near
the center of drain is very fast and a greater degree of consolidation is reached at a small time. Figure 8(b) shows
the effect of radius of sand drain with a constant normalized time factor 20 and all other parameters kept
constant. It is observed that as the radius of sand drain increases, settlement also increases at every point.
Larger radius of sand drain induces greater degree of consolidation, and hence greater settlement.
Fig. 3(a). Settlement profiles for various values of Fig. 3(b). Horizontal dispalcement profiles for tension
modulus of geosynthetic membrane various values of tension modulus
29
Fig. 4. Settlement profiles for various values of Fig. 5. Settlement profiles for various values of
pretension thickness of granular fill
Fig. 6. Settlement profiles for various values of Fig. 7. Settlement profiles for various values of
shear modulus of granular fill spring constant ratio
Fig. 8(a). Settlement profiles for various values of Fig. 8(b). Settlement profiles for various values of
normalized time radius of sand drain
4 Conclusions
The paper presents the settlement analysis of a beam resting on geosynthetic-reinforced granular fill-soft soil
system. Each sub-system of the reinforced fill soft soil system is idealized by the mechanical foundation model
elements, such as rough elastic membrane, Pasternak shear layer, Winkler springs and dashpots. The differential
equations governing the settlement response of the beam resting on two layered reinforced foundation soil have
been formulated by incorporating the deformation compatibility conditions. The numerical solutions are obtained
using finite element method and results are presented in non-dimensional form. It is observed that the variation in
the tension modulus in the geosynthetic membrane does not appreciably affect the settlement response of the
granular fill-soft soil system under the range of parameters studied. Pretension in the geosynthetic membrane
appreciably affects the settlement response of the system. Settlement gets reduced below the point of application
of load; however, no appreciable change is noticed at the centre of the system. A magnitude greater than 0.8 for
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the pretension force results in no-heaving condition at the edges of the beam. Pretension force is found to be
effective in reduction of differential settlement. Increasing the thickness of granular fill resulted in reduction of the
settlement of the system quite effectively. Increment in the shear modulus of the granular fill resulted in the
reduction of settlement below the point of application of load, while the settlement increased at the centre and at
the edge of the beam. It is observed that higher magnitudes of shear modulus are preferable to reduce differential
settlement. Beyond a magnitude of 0.4, the shear modulus of granular fill has no effect on the settlement
characteristics of the system. The spring constant ratio of the granular fill is studied to bring out the effect of
relative compressibility of the granular fill and the soft soil on the settlement behavior. It is observed that when the
stiffness of the granular fill exceeded 50 times that of the soft soil, the effect of the compressibility of the granular
fill is negligible. It is observed that in a system without sand drains, the consolidation is uniform along the length
of the system; however, introduction of sand drains in the system resulted in accelerated and different degrees of
consolidation at each location of the system. It is noticed that the radius of the sand drain has an appreciable
effect on the achieved degree of consolidation; a larger radius induced a higher degree of consolidation and
hence resulted in a higher degree of settlement. In this study, it is observed that the horizontal displacement of
the geosynthetic membrane is negligible as compared to the vertical settlement.
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A Finite Element Study of Beam on Reinforced Granular Beds with Sand Drains 2012-01-23T08:21:00-08:00 Rating: 4.5 Diposkan Oleh: Bona Pasogit