INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS
Int. J. Numer. Anal. Meth. Geomech. (2014)
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2318
A generalized model for geosynthetic reinforced railway tracks
resting on soft clays
S. Rajesh*,†, K. Choudhary and S. Chandra
Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur, India
SUMMARY
The present study pertains to the development of a foundation model for predicting the behavior of
geosynthetic reinforcement railway track system rested on soft clay subgrade. The ballast and sub-ballast
layers have been idealized by Pasternak shear layer. The geosynthetic layer is represented by a stretched
rough elastic membrane. Burger model has been used to characterize the soft clay subgrade. Numerical solutions have been obtained by adopting the finite difference scheme combined with non-dimensioning the
governing equations of the proposed model. The results confirm that the present model is quite capable of
predicting the time-dependent settlement response of geosynthetic reinforcement railway track system placed
on soft clay subgrade. The surface settlement profile and mobilized tensile load of geosynthetics has been
evaluated by considering variation in the wheel load, sleeper width, thickness of ballast and sub-ballast layers
and shear modulus of ballast and sub-ballast layers. It has been observed that an increase in the sleeper width
by 24% results in the reduction in central settlement and mobilized tensile load by 6.5% and 20.1%, respectively. It was found that with a 50% increase in the thickness of the ballast layer, the central settlement has
decreased by 7.3% and the mobilized tension at the zone of maximum curvature has increased by 24.6%.
However, with an increase in the thickness of the sub-ballast layer, a considerable reduction in both central
settlement and the mobilization of tension on geosynthetic has been noticed. The pattern of variation of
settlement and mobilized tension for an increase in the shear modulus of ballast and sub-ballast material
was found to be almost similar. Copyright © 2014 John Wiley & Sons, Ltd.
Received 7 August 2013; Revised 23 June 2014; Accepted 3 July 2014
KEY WORDS:
soft clay; subgrade model; geosynthetics; railway track structure; Pasternak shear layer;
visco-elastic model
1. INTRODUCTION
In recent decades, railway transport infrastructures have been regaining their importance because of
high speed commutation, their efficiency, and environmentally friendly technologies. This has led to
increasing train speeds, higher axle loads, and more frequent train usage. The current track design
techniques are overly simplified by neglecting the complexities of the behavior of the composite
track system consisting of rail, sleeper, ballast, and sub-ballast under repeated traffic loads [1]. The
performance of railway track system primarily depends on distribution of load from rail (high
concentrated wheel load) to the substructure. A well-designed and constructed track would distribute
the loads in a relatively uniform fashion, without overstressing any of the track components: rail,
sleepers, ties, ballast, and subgrade. When a wheel is centered over a sleeper, less than half of the
load is carried by the sleeper directly beneath the wheel, while the rest of the load is distributed
among the neighboring two sleepers [2]. The applied load on the sleeper will be transferred to
ballast layer which in-turn to sub-ballast and subgrade layers. The best design would be to transfer
*Correspondence to: S. Rajesh, Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur, India.
†
E-mail: hsrajesh@iitk.ac.in
Copyright © 2014 John Wiley & Sons, Ltd.
S. RAJESH, K. CHOUDHARY AND S. CHANDRA
lesser load to the subgrade. The reduction in the load transfer to the subgrade can be obtained by
introducing geosynthetics in the railway track structure. The use of geosynthetics in rail tracks has
been studied in the past, and it is proved that geosynthetics can improve the track performance by
reducing deformation and degradation [3]. Even though the substructure components have a major
influence on the cost of track maintenance, less attention has been given to the substructure because
the properties of the substructure are more variable and difficult to define than those of the superstructure. While determining the net effect of wheel loads on the track structure, the track model
should be able to inter-relate the components of the track structure including substructure in a way
such that their complex interactions are properly taken into consideration. Such a model would then
predict the track settlement closer to the realistic value.
The experimental investigation is not always the best choice for the study of railway track structure
mainly due to the high cost and loading constraints. Comparatively, an accurate analytical/
mathematical modeling and numerical solution of the track structural components reveals distinct
advantage for understanding the response behavior of the track structure. Analytical model used for
idealizing a railways track should be capable of including the effects of sleeper spacing, rail
stiffness, changes in ballast and sub-ballast thickness, and subgrade properties. The simplest
representation of a track structure can be described as a beam resting on an elastic foundation
wherein the substructure is represented by a Winkler spring system. Many geotechnical engineers
use the Winkler model for the analysis of beams on elastic foundations where the vertical
deformation characteristics of the elastic foundation are characterized by means of continuous
closely spaced linear springs; the constant of proportionality of these springs is known as the
modulus of subgrade reaction (k). The Winkler model has been shown to be inconsistent to
represent a continuous medium. In addition, such a system will not be able to quantify the
individual contributions of ballast, sub-ballast, and subgrade in much detail. This made several
researchers to develop various foundation/subgrade models. The evolution of foundation models
along with brief review has been covered in detail previously in Horvath [4]. A more realistic model
for railway track structure was the double beam model in which the upper beam represents the rail,
and the lower beam represents the substructure of the ballasted track. Further, in order to describe
the deformation of a beam on elastic foundation more accurately, Pasternak assumed the existence
of shear interactions between the spring elements in his model [5]. It is accomplished by connecting
the ends of the springs to the beam consisting of incompressible vertical elements which then
deforms only by transverse shear. Kerr [6] proposed a three parameter foundation model in which a
shear layer was introduced in between two Winkler layers. This model removes the inadmissible
jumps encountered in concentrated loads. In another model, the ballast and subgrade are considered
together as a visco-elastic sub-space [7].
In recent past, the inclusion of geosynthetics in the form of geogrid (for improving strength) or
geotextiles (for improving drainage properties) or geocomposites within the subgrade soil has further
complicated the analytical modeling. Shukla and Chandra [8] proposed a mechanical model for
idealizing the settlement response of a geosynthetic reinforced compressible fill-soft soil system, by
representing each sub-system by commonly used mechanical elements such as stretched, rough,
elastic membrane, Pasternak shear layer, Winkler springs, and viscous dashpots. Shukla and
Chandra [8] used Kelvin–Voigt elements to model the soft foundation soil, which has a limitation in
estimating the time—dependent behavior of the soft clays. Fakher and Jones [9] studied the flexural
behavior of geosynthetic reinforced soil considering a single layer of geosynthetic placed between
an overlying homogeneous sand bed and an underlying soft clay layer. Maheshwari et al. [10]
analyzed beams resting on reinforced elastic foundation and obtained the closed form solution for
the same. Deb [11–13] developed a lumped parameter model for single-layer geosynthetic
reinforced granular fill-soft soil with stone columns. Few researchers tried to model the timedependent behavior of soft clays using visco-elastic models. Dey [14] studied the applicability of
various visco-elastic models to study the settlement response of visco-elastic foundation beds and
discussed their efficacy in predicting the time-dependent behavior of soil medium. Dey and
Basudhar [15] used Burger model to simulate the time-dependent flexural behavior of surface strip
footing resting on soft clays. Even though foundation models representing the behavior of
geosynthetics reinforced soil system have been developed by many researchers, the applications of
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
MODEL FOR GEOSYNTHETIC REINFORCED RAILWAY TRACKS RESTING ON SOFT CLAYS
these models for studying railway track behavior has not been considered in the past studies. Hence, in
the present study, a generalized foundation model which idealizes a geosynthetic-based ballasted
railway track resting on soft clay was developed by combining various lumped parameter models.
2. PROBLEM STATEMENT
A typical layout of geosynthetic reinforced railway track is shown in Figure 1. It comprises of rail,
sleepers, ballast layer, geosynthetic layer, sub-ballast layer, and soft subgrade layer. Forces in the
vertical, lateral, and longitudinal directions act on the track structure. These forces can be due to
moving traffic and changing temperature. The static component of vertical force is the weight of the
train while the dynamic component is a function of track conditions, train characteristics, operating
conditions, train speed, and environmental conditions. It is the dynamic component that usually
causes an adverse effect to the track as it can be much larger than the static load. According to Selig
and Waters [16], the magnitude of the dynamic component can be up to 2.4 times the static load. In
the present study, the total load applied on rail considering both static and dynamic component is
termed as Q. In general, static track deflection caused by a single point load Q can be spread over a
total number of between seven and nine sleepers. However, recently, Beranek [2] mentioned that the
total wheel load Q for a unique static position of the train is spread to the sleepers as shown in
Figure 2. In the present analysis, the total wheel load from the train wheel has been appropriately
modified to the stresses intensity q transferred to the sleepers considering suggested load distribution.
The proposed foundation model which idealizes the behavior of geosynthetic reinforced railway
track system (GRRTS) is shown in Figure 3. In the proposed model, the ballast and sub-ballast
layers have been idealized by Pasternak shear layer. Pasternak shear layer is a two parameter model
consisting of a shear layer of unit thickness above the Winkler springs to account for the
compressibility and shear interaction between the spring elements. The compressibility of both
ballast and sub-ballast layer has been incorporated using a layer of stiff Winkler springs connected
to the bottom of the lower Pasternak shear layer. The shear layers are made up of incompressible
vertical elements which deforms only in transverse shear. The geosynthetic layer is represented by a
stretched rough elastic membrane. It is assumed that geosynthetic layer is linearly elastic with
negligible shear resistance and does not allow slippage at the interface due to its roughness while
deriving the settlement response of this system. This modeling approach has been adopted by
several researchers in their analytical and numerical studies [17–20]. A rigid perfectly plastic friction
model has been adopted to represent the soil-geosynthetic interface in shear. Further, it is assumed
that the displacement of the membrane is zero at the instant when the load is applied and the
deformation takes place only after application of the load. The tensile stress mobilized in the
reinforcement has been considered to quantify the effect of geosynthetic layer. These tensile forces
are an outcome of the accumulated frictional forces acting along the entire length of the inclusion.
This whole phenomenon is popularly known as the ‘Rough Membrane Effect’ of the geosynthetic
reinforcement. The induced shear stresses due to the geosynthetic layer also cause confinement of
the soil in the neighboring environment of the inclusion. Soft clay subgrade has been idealized with
Figure 1. Typical layout of a geosynthetic reinforced railway track system.
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
S. RAJESH, K. CHOUDHARY AND S. CHANDRA
Figure 2. Wheel load distribution (modified after Beranek, [2]).
Figure 3. Proposed foundation model for the geosynthetic reinforced railway track system.
four parameter Burger model. The adopted Burger model is a combination of Kelvin–Voigt element
and Maxwell element arranged in series. The Burger model is capable of representing all the
deformation phenomena occurring in a visco-elastic subgrade when subjected to loading and
unloading process.
2.1. Mathematical formulation
The settlement response of the proposed model has been arrived by applying uniformly distributed load
with intensity q, over a length 2B, directly on the top of the ballast layer.
According to Beranek [2], total wheel load Q for a unique static position of the train is spread to five
sleepers. The load intensity q will then be transferred to the corresponding percentages on these five
sleepers as shown in Figure 2. Because the load is transmitted only through sleepers, no load will be
applied on the clear spacing between the sleepers. Hence, for the analysis, total width of the sleeper
(2B) is taken as the summation of the sleeper widths within which the load is being applied for a
unique static position of the train (Figure 2). For example, if the load is applied on sleeper (Sn+3),
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
MODEL FOR GEOSYNTHETIC REINFORCED RAILWAY TRACKS RESTING ON SOFT CLAYS
then influence of load intensity spreads on to the following sleepers Sn+1, Sn+2, Sn+3, Sn+4, and Sn+5. The
wheel load distribution on sleeper Sn+3 will be 40% of the total load intensity q; Sn+2 and Sn+4 will take
20% of q; Sn+1 and Sn+5 will take 10% of q. Plane strain conditions are considered for both the loading
and the foundation soil system. This is due to the fact that the length of the sleeper is quite large in
comparison to the width of the sleeper. As a result, the applied forces would not vary along the
length, that is, loads are uniformly distributed with respect to the larger dimension (in this case
length of the sleeper) and act perpendicular to it as well.
The following assumptions are made while deriving the equations:
• Plane strain conditions exist for both the loading and foundation soil system.
• Downward deflection is positive and loading acting in downward direction is also positive.
• Shear force on the left side of a section is positive in upward direction and consequently on the
right side, it is positive in downward direction.
• Due to symmetric loading conditions, it is sufficient to analyze only one half of the loading.
The governing equations for the settlement response have been derived by considering first an
element of the ballast shear layer and its vertical force equilibrium has been studied. Once this is
done, the geosynthetic reinforcement represented by the rough elastic membrane is considered. Once
the horizontal and vertical forces equilibrium for the membrane element has been established,
equations relating to the vertical force equilibrium of an element of the sub-ballast shear layer have
been established. The stress distribution and the force distribution of the ballast shear layer are given
in Figure 4.
It is assumed that the shear layer parameter, G is isotropic in the x-y plane, which implies
Gx ¼ Gy ¼ GB
τ xz ¼ GB γxz ¼ GB
(1)
∂w
∂x
(2)
Where GB is the shear parameter of the ballast shear layer.
Total shear force Nx is obtained by integrating Equation (2) over the thickness of the ballast layer
H B.
Considering the vertical equilibrium of the forces acting on the ballast shear layer:
q ¼ q B GB H B
∂2 w
∂x2
(3)
where qB is the vertical force interaction between ballast and membrane. The effect of transverse shear
interactions can be seen in the second term on the right hand side of Equation (3).
Figure 5 shows the force equilibrium for the stretched rough elastic membrane. The horizontal force
equilibrium equation for this membrane can be written as
T p þ T þ ΔT cosðθ þ ΔθÞ T p þ T cos θ þ ðμB qB þ μSB qSB ÞΔx
þ K ðqB qSB ÞΔx tan θ ¼ 0
(4)
where qSB is the vertical force interaction between the membrane and the sub-ballast layer, μB and μSB
are the interface friction coefficients at the top and bottom faces of the membrane, respectively, θ is the
slope of the membrane, T(x) is the tensile force per unit length mobilized in the membrane, Tp is the
pretension per unit length applied to the membrane, and K is the coefficient of lateral stress (i.e., ratio
of horizontal to vertical stresses present in the granular fill). If the developed stresses within the fill
are the maximum ever developed in the stress history, the reinforced granular fill is at a Ko
(coefficient of lateral stress at rest) state of stress. However, for an over-consolidated fill, K is greater
than Ko, hence, in such conditions, the relationship suggested by Alpan [21] is used
K ¼ K o Rλc
Copyright © 2014 John Wiley & Sons, Ltd.
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Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
S. RAJESH, K. CHOUDHARY AND S. CHANDRA
Figure 4. Free-body diagrams of ballast shear layer subjected to: (a) stresses and (b) forces.
Figure 5. Forces acting on the membrane element.
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
MODEL FOR GEOSYNTHETIC REINFORCED RAILWAY TRACKS RESTING ON SOFT CLAYS
where Rc is the over consolidation ratio, and λ is rest-rest rebound exponent, which is correlated with
the effective angle of shearing resistance (ϕ′) for sands [16]. Ko is determined by the relation
Ko = 1 sin ϕ′.Now, taking the limit that Δx → 0 in Equation (3), the following equation is obtained:
cos θ
∂T ðxÞ ∂θ
T p þ T sin θ ¼ ðμB qB þ μSB qSB Þ þ K ðqB qSB Þtan θ
∂x
∂x
(6)
The vertical force equilibrium for the stretched rough elastic membrane yields the following equation:
T p þ T þ ΔT sinðθ þ ΔθÞ T p þ T sinθ ðqB qSB ÞΔx þ K ðμB qB þ μSB qSB ÞΔx tan θ ¼ 0 (7)
Taking the limit, Δx → 0, the equation reduces to
sin θ
∂T ðxÞ ∂θ
þ T p þ T cos θ ¼ ðqB qSB Þ K ðμB qB þ μSB qSB Þtan θ
∂x
∂x
(8)
Combining Equations (6) and (8) and simultaneously eliminating the differential term in T(x), and
writing θ in terms of vertical displacement, w(x), the following equation is obtained:
∂2 w
qB ¼ X 1 qSB X 2 T p þ T cos θ 2
∂x
(9)
X1 ¼
1 þ K tan2 θ ð1 K ÞμSB tan θ
1 þ K tan2 θ þ ð1 K ÞμB tan θ
(9a)
X2 ¼
1
1 þ K tan2 θ þ ð1 K ÞμB tan θ
(9b)
where,
and
Figure 6 show the stress distribution for the sub-ballast shear layer element. The same assumption of
isotropic shear parameter is taken here as well, which results in the following governing equation for
the sub-ballast shear layer element:
Figure 6. Stress distribution for sub-ballast shear layer element.
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
S. RAJESH, K. CHOUDHARY AND S. CHANDRA
qSB ¼ qS GSB H SB
∂2 w
∂x2
(10)
The subgrade has been idealized as a Burger element which is a combination of Maxwell element
and Kelvin–Voigt element applied in series. The stress-displacement response of the soft soil
subgrade based on the Burger model has been derived as follows:
qS ¼
w
d2 w
T 2
A
dx
(11)
where T is the tensile force per unit length mobilized in the membrane, and w is the deflection of the
soil. The expression for A is
1
t
1
k2
A¼ þ þ
1 exp t
(11a)
η2
k 1 η1 k2
where k1 is the elastic coefficient of Maxwell model, k2 is the elastic coefficient of Kelvin–Voigt
model, η1 is the viscous coefficient of the Maxwell model, η2 is the viscous coefficient of the
Kelvin–Voigt model, and t is the time.
Substituting qS, qSB, and qB in Equation (3), the following equation is obtained:
d2 w
w q ¼ X 1 GB H B þ X 2 T p þ T cosθ þ X 1 T þ X 1 GSB H SB
(12)
A
dx2
The variation of the mobilized tension in the stretched rough membrane has been obtained by
dividing Equations (6) and (8) by sin θ and cos θ, respectively and substituting the expressions for
qSB and qB to get the final equation given.
∂T
∂2 w
w
d2 w
∂2 w
(13)
¼ X 3 q þ GB H B 2 X 4
T 2 GSB H SB 2
∂x
∂x
A
dx
∂x
where
X 3 ¼ μB cosθ 1 þ K tan2 θ ð1 K Þ sinθ
(13a)
X 4 ¼ μSB cosθ 1 þ K tan2 θ þ ð1 K Þ sinθ
(13b)
and
2.2. Non-dimensional form of governing equation
The response of the proposed foundation model is governed by the Equations (12) and (13). The nondimensional form of the governing differential equations is expressed as follows:
o d 2 W W n
q ¼ X 1 GB þ X 2 T p þ T cosθ þ X 1 T þ X 1 GSB
(14)
A
dX 2
2 2 2
∂T W
∂ W
d W
∂ W
¼ X 3 q þ GB 2 X 4 T
GSB 2
∂X
A
∂X
dX 2
∂X
(15)
The non-dimensional parameters are expressed as follows:
X¼
x
w
GB H B GSB H SB
; W ¼ ; GB ¼
; GSB ¼
2
B
B
kS B
kS B2
Copyright © 2014 John Wiley & Sons, Ltd.
(16a)
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
MODEL FOR GEOSYNTHETIC REINFORCED RAILWAY TRACKS RESTING ON SOFT CLAYS
q ¼
q
Tp
T
k1 k2
; T ¼
; T ¼
; A ¼ kS A; kS ¼
k1 þ k2
k S B p k S B2
k S B2
(16b)
where B is the half width of the uniformly distributed loading, and kS represents the equivalent stiffness
of the subgrade for the Maxwell and Kelvin–Voigt system in series.
3. METHOD OF SOLUTION
The governing differential equations so developed are expressed in finite difference form. Central
difference scheme is adopted for writing the double derivative of W with respect to X, while ∂T/∂X
has been expressed by using the forward finite difference scheme. The total length, L can be divided
into ‘n’ parts of equal length with (n + 1) number of nodes. The governing equation in finite
difference form for any ith node is expressed as follows:
o W 2W þ W
Wi n
i1
i
iþ1
qi ¼ X 1i GB þ X 2i T p þ T i cosθi þ X 1i T i þ X 1i GSB
2
A
ðΔX Þ
T i ¼ T iþ1 þ
ΔX
4
"
X 3i þ X 3iþ1
qi þ qiþ1 þ GB
W i þ W iþ1
þ X 4i þ X 4iþ1
T i
A
2
#
2
∂
W
∂
W
þ
GSB
∂X 2 i
∂X 2 iþ1
∂2 W
∂2 W
þ
2
∂X i
∂X 2 iþ1
∂2 W
∂2 W
þ
2
∂X i
∂X 2 iþ1
!
(17)
(18)
In order to minimize the error involved in computations, average values of q*, W, ∂∂XW2 , X 3 , and X 4
have been used in the equations.
2
3.1. Boundary condition and iteration scheme
Because of the symmetry of the system, only half portion is taken into consideration. Taking the origin
to be at the center of the loaded region, then, for the edge of the uniformly distributed loading, X = 1.0
or x = B. At the edge of the soil zone, one can see that X = L/B or x = L. Because of the symmetry of the
problem, the slope of settlement will be zero at X = 0. Also, at X = L/B, shear stress acting on the shear
layer at the edge will be zero due to no confinement. Thus, the boundary conditions can be summarized
as follows:
dW
¼ 0 at X ¼ 0
dX
(19a)
dW
¼0
dX
(19b)
at X ¼ L=B
Also, a third boundary condition arises due to very small magnitude of the mobilized tensile force at
the edge of the reinforcement.
T ¼ 0
Copyright © 2014 John Wiley & Sons, Ltd.
at X ¼ L=B
(19c)
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
S. RAJESH, K. CHOUDHARY AND S. CHANDRA
The loading condition for the uniformly distributed load case can be written as
qi ðX Þ ¼ q
f or jX j ≤ 1:0
(20a)
qi ðX Þ ¼ 0 f or jX j > 1:0
(20b)
3.2. Convergence study
A computer program based on the above formulation has been developed using Matlab. A convergence
study was carried out to obtain optimum number of elements into which the domain shall be divided.
The convergence criterion adopted for obtaining the solutions is
W Ki W K1
i
100% < εs
W Ki
(21)
for all i, where k and k-1 denote the present and the previous iterations, respectively. εs is the specified
tolerance, which has taken to be 0.0001 in the present study. Figure 7 shows the convergence study
carried out for central settlement. It can be inferred from the figure that when the number of nodes
increases beyond 250 (element size is 8 mm), there is negligible change in the central settlement.
Element size of 5 mm has been adopted in the present study considering other factors also into account.
3.3. Validation of the formulation
The formulation made in this study was validated comparing the results with those given by Shukla
and Chandra [8]. Shukla and Chandra [8] model forms a special case of the proposed model when
the clear spacing of the sleepers are neglected (i.e., strip loading). The reason for choosing the strip
loading in place of real track loading for validation is due to the non-availability of real track
response data. Shukla and Chandra [8] developed a formulation in which pre-stressed geosynthetic
layer is sandwiched between two Pasternak shear layers similar to the present study. The loading,
boundary conditions, and the properties of various soils layers were modified according to Shukla
and Chandra [8]. Figure 8 shows the results of the geosynthetic reinforced system obtained from the
present study and Shukla and Chandra [8]. A reasonably good agreement between both the results
can be noticed. The major merits of the present formulation are (i) model uses realistic pressure
distribution of stresses in soils by distributing the load on sleepers and (ii) model can evaluate timedependent settlement. The assumptions made in the formulation forms the basic limitations of the
present study. The formulation could be improved by incorporating dynamic loads in place of static
loads and non-linear response of soils and geosynthetic materials in place of linear response so as to
predict the mobilized tensile load of geosynthetic layer and the settlement response of the GRRTS
accurately. This forms the limitation of the present study.
Figure 7. Variation of central settlement with number of nodes.
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
MODEL FOR GEOSYNTHETIC REINFORCED RAILWAY TRACKS RESTING ON SOFT CLAYS
4. RESULTS AND DISCUSSIONS
An attempt has been made in this study to understand the response of GRRTS, in specific, its surface
settlement and the mobilized tensile load of geosynthetic material considering the variation in the
wheel load, sleeper width, thickness of ballast and sub-ballast layers and shear modulus of ballast
and sub-ballast layers. The material properties used in the analysis is tabulated in Table I. These
guide values are adopted from Shukla and Chandra [8], Shukla [22], and Dey and Basudhar [23].
The parameters for the viscous component of soft clay model can be determined by an inverse
analysis formulation of the four-parameter Burger model [14]. A detailed methodology for obtaining
material properties, that is, guide values can be referred from Dey [14]. The developed model has an
ability to obtain settlement profile of GRRTS at various time intervals, in other words, at different
stages of consolidation. Figure 9 shows the variation of non-dimensional settlement along the width
of the GRRTS with respect to time. The width of the sleeper, clear spacing of the sleeper, and the
effective width of the GRRTS considered are 250, 400, and 625 mm, respectively. It can be noticed
that soon after the application of the wheel load, the settlement is found to increase with time just
below the sleepers. The undulations in the response curves is due to the nature of loading
(i.e., wheel load is transferred to sleepers, while no loads are being applied within the clear spacing
between the sleeper, Figure 2). After 12 h of consolidation, a reduction in the undulations, in other
words, smooth settlement profile covering the entire width of GRRTS can be noticed, which implies
the loads are being evenly distributed during the period of consolidation. With a further increase in
time, settlement profile is found to follow an identical pattern but with an increase in the settlement
value. The predicted surface settlement of GRRTS is the summation of settlement occurred on soft
clay layer, ballast layer and sub-ballast layer. The response of the surface settlement and the
mobilized tensile load of geosynthetic material at 15 days will be considered for the further analysis.
Figure 8. Validation plot for geosynthetic reinforced foundation system.
Table I. Material properties of various components of railway track.
Parameter
Shear modulus of ballast layer, GB, MPa
Thickness of ballast layer, HB, m
Shear modulus of sub-ballast, GSB, MPa
Thickness of sub-ballast layer, HSB, m
Coefficient of friction at geosynthetic interface, μB
Coefficient of friction at geosynthetic interface, μSB
k1, MN/m3
k2, MN/m3
η1, N-days/cm2
η2, N-days/cm2
Copyright © 2014 John Wiley & Sons, Ltd.
Value
21
0.3
28.752
0.15
0.6
0.6
30
3
30000
3000
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
S. RAJESH, K. CHOUDHARY AND S. CHANDRA
Figure 9. Settlement profiles of geosynthetic reinforced track structure at different time instants.
4.1. Influence of magnitude of wheel load
The response of the GRRTS under the influence of wheel load has been studied by various the wheel
load Q from 50 to 500 t. The width of the sleeper and clear spacing of the sleeper considered are 250
and 400 mm, respectively. Figure 10 shows the variation in the settlement profile and the mobilized
tension of geosynthetic for various magnitude of wheel load. It can be clearly noticed that as the
load increases, there is an increase in the settlement. In addition, a sharper curvature can be noticed
with an increase in the wheel load. This sharper curvature can generate higher bending stress within
the width of the GRRTS. This can be clearly noticed with an increase in the mobilized tensile load
of geosynthetic material. When the wheel load is increased 10×, the central settlement and central
mobilized tension are found to increase by 9.5× and 9.7×, respectively. The geosynthetic material
experience higher mobilization within the width of the sleepers mainly because the wheel loads are
applied directly on the sleepers; mobilization reduces slightly away from the sleepers (Figure 3).
Beyond the effective width, geosynthetic material has shown very less mobilization irrespective of
load applied. Even though the variation within and outside the sleeper width for lower wheel load is
quite less, greater variation can be noticed for higher wheel load which is contradictory to the
settlement response.
4.2. Influence of width of sleeper
The influence of width of sleeper on the response of the GRRTS has been studied by varying sleeper
width between 250 and 310 mm. The sleeper width range adopted in the study is based on the
regulations followed by many countries. The clear sleeper spacing used in the analysis is 400 mm,
Figure 10. Effect of magnitude of wheel load on (a) surface settlement and (b) mobilized tension experienced by geosynthetic material.
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
MODEL FOR GEOSYNTHETIC REINFORCED RAILWAY TRACKS RESTING ON SOFT CLAYS
and wheel load applied is 100 t. Figure 11 shows the influence of sleeper width on settlement profile
and mobilized tension of geosynthetics. It can be noticed that for a given wheel load, settlement
decreases with an increase in the sleeper width. The decrease in settlement with an increase in the
sleeper width could be due to the decrease in the load being transferred to the substructure, as
significant amount of load is being distributed on increased sleeper width. Because the lesser load is
transferred to the substructure, the mobilized tension experienced by the geosynthetic material is
also lesser for higher sleeper width, as shown in Figure 11b. The pattern of non-dimensional
mobilized tensile load of geosynthetic material is found to vary within the width of the GRRTS
mainly due to the load distribution adopted in the present study. An increase in the sleeper width by
24% results in the reduction in both central settlement and maximum mobilized tensile load by 6.5%
and 20.1%, respectively.
4.3. Influence of thickness of ballast layer
The response of the GRRTS under the influence of thickness of ballast layer has been studied by
varying the thickness of ballast layer from 200 to 500 mm. The response of the GRRTS under a
nominal wheel load of 100 t is adopted in the study. The sleeper width and the clear sleeper spacing
adopted are 250 and 400 mm, respectively. As noticed from the Equation (3), the thickness of the
ballast layer and the shear modulus of the ballast material have a significant influence on the load
transfer mechanism. Figure 12 shows the effect of thickness of ballast on the surface settlement
profile and the mobilized tension experience by the geosynthetic material along the width of
GRRTS. It can be observed that as the thickness of the ballast layer increases, a significant
reduction in the central settlement can be noticed. However, beyond the effective half-width of the
GRRTS, there is a considerable increase in the settlement with an increase in the thickness of the
ballast. This increase in the settlement could be mainly due to the self weight of the ballast material
and also the percentage of load to be shared by the substructure reduces considerable beyond the
third sleeper for a static position of wheel (Figures 2 and 3). The curvature of the settlement profile
tends to change with an increase in the thickness of the ballast material and interesting almost at the
half-width of the GRRTS, non-dimensional settlement is found to have a unique value. The
geosynthetic material has experienced higher tension at the central loading and gradually decreases
with an increase in the distance from center of loading. The maximum tension experienced by the
geosynthetic material is found to be almost the same, irrespective of the thickness of the ballast
material. However, the mobilized tension experienced in the middle one-third portion of width
shows a significant increase in the mobilized tension with an increase in the thickness of the ballast
layer mainly due to the change in curvature within middle one-third and also possibly due to
Figure 11. Effect of sleeper width on (a) surface settlement and (b) mobilized tension experienced by
geosynthetic material.
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
S. RAJESH, K. CHOUDHARY AND S. CHANDRA
Figure 12. Effect of thickness of ballast on (a) surface settlement and (b) mobilized tension experienced by
geosynthetic material.
increase in the geostatic stress. It can be observed that for a 50% increase in the ballast thickness, the
central settlement has decreased by 7.3%, and the mobilized tension has increased by 24.6%.
4.4. Influence of shear modulus of ballast material
The shear modulus of ballast material is an important design parameter. The shear modulus of ballast
layer has been varied between 10 and 110 MPa. The nominal wheel load, sleeper width, and the clear
sleeper spacing adopted are 100 t, 250 mm, and 400 mm, respectively. Figure 13 shows the settlement
response and the mobilized tension load distribution of geosynthetic material for various values of
shear modulus of ballast material. It can be noticed that the central settlement decreases with an
increase in the shear modulus. Beyond the effective half width of the GRRTS considered, there is a
considerable increase in the settlement with an increase in the shear modulus of ballast material. The
radius of curvature changes with an increase in the shear modulus. The pattern of variation is found
to be similar to the thickness of the ballast layer. Significant reduction in the central settlement can
be noticed when the shear modulus increases from 10 to 50 MPa, with further increase in the shear
modulus, not much significant reduction can be noticed. The mobilized tension experience by the
Figure 13. Effect of shear modulus of ballast material on (a) settlement and (b) mobilized experienced by
geosynthetic material.
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
MODEL FOR GEOSYNTHETIC REINFORCED RAILWAY TRACKS RESTING ON SOFT CLAYS
geosynthetic material at the center of loading is found to be almost constant, but it tends to decrease
with an increase in the shear modulus.
4.5. Influence of thickness of sub-ballast layer
The thickness of the sub-ballast layer varied from 100 to 250 mm. The nominal wheel load, sleeper
width, and the clear sleeper spacing adopted are 100 t, 250 mm, and 400 mm, respectively. The
settlement profile of GRRTS for various thickness of the sub-ballast layer is shown in Figure 14. It
can be noticed that with an increase in the thickness of the sub-ballast layer, settlement at the central
portion was found to decrease but at a distance 3 m away from the center of loading where there
was a considerable increase in the settlement. The tension experienced by the geosynthetic material
along the width of GRRTS has not shown distinct variation with an increase in the thickness of the
sub-ballast layer; however, a considerable reduction in the mobilized tension can be noticed with an
increase in the thickness of the sub-ballast layer. The mobilized tension is found to be slightly
higher for lesser thickness of the sub-ballast layer. A contrary behavior has been noticed with an
increase in the thickness of the ballast layer. As the geosynthetic layer is placed above the sub-
Figure 14. Effect of thickness of sub-ballast layer on (a) settlement and (b) mobilized tension experienced by
geosynthetic material.
Figure 15. Effect of shear modulus of sub-ballast on (a) settlement and (b) mobilized tension experienced by
geosynthetic material.
Copyright © 2014 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. (2014)
DOI: 10.1002/nag
S. RAJESH, K. CHOUDHARY AND S. CHANDRA
ballast layer, lesser thickness of sub-ballast tends to experience sharper curvature, which makes the
geosynthetic layer mobilize more tension.
4.6. Influence of shear modulus of sub-ballast material
The shear modulus of sub-ballast material is an important design parameter. The shear modulus of subballast layer has been varied between 10 and 110 MPa, similar to the range adopted for the shear
modulus of ballast material. The nominal wheel load, sleeper width, and the clear sleeper spacing
adopted are 100 t, 250 mm, and 400 mm, respectively. The pattern of variation of settlement profile
and mobilized tension along the width of GRRTS for various values of shear modulus of sub-ballast
was found to be similar to that of ballast material, as shown in Figure 15. The magnitude of nondimensional central settlement for an identical value of shear modulus of sub-ballast and ballast
material is found to be much less for sub-ballast case when compared with that of ballast case.
However, not much variation in the response of mobilized tension of geosynthetic for sub-ballast
and ballast can be noticed. The mobilized tension experience by the geosynthetic material at the
center of loading was found to be almost constant, but it tends to decrease with an increase in the
shear modulus.
5. CONCLUSIONS
From the previous discussions, it can be said that the present model is quite capable of predicting the
time-dependent settlement response of GRRTS placed on soft clay subgrade. The formulation made in
this study was validated comparing the results with those given by Shukla and Chandra [8]. A good
agreement between the results is obtained. The central settlement and the mobilized tensile at the
zone of maximum curvature was found to be directly proportional to the magnitude of wheel load
applied on the sleepers and inversely proportional to the width of the sleepers. An increase in the
sleeper width by 24% results in the reduction in both central settlement and maximum mobilized
tensile load by 6.5% and 20.1%, respectively. It can be noticed that for a 50% increase in the ballast
thickness, the central settlement has decreased by 7.3%, and the mobilized tension at the zone of
maximum curvature has increased by 24.6%. However, with an increase in the thickness of the subballast layer, a considerable reduction in the mobilization of tension on geosynthetic has been
noticed. The pattern of variation of settlement and mobilized tension for an increase in the shear
modulus of ballast and sub-ballast material is found to be similar. However, for an identical value of
shear modulus of sub-ballast and ballast material, the magnitude of non-dimensional central
settlement is found to be much less for sub-ballast case compared with that of ballast case. Hence, it
can be concluded that the settlement profile is more sensitive to changes in the thickness of the
ballast layer and the shear modulus of ballast layer when compared with that of sub-ballast cases.
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DOI: 10.1002/nag