Edited by: Castorina Silva Vieira, University of Porto, Portugal
Reviewed by: Amarnath M. Hegde, Indian Institute of Technology Patna, India; Sireesh Saride, Indian Institute of Technology Hyderabad, India
This article was submitted to Transportation and Transit Systems, a section of the journal Frontiers in Built Environment
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This research investigates the response of rails on geocellstone column composite reinforced foundation beds under a moving load. Improved earth bed has been considered to respond only to compressive forces. The granular mat below the rail has been idealized as a Pasternak shear layer and geocell reinforcement as an infinite beam with finite bending stiffness. Soft soil and stone columns have been symbolized by Winkler springs of different stiffnesses. Analysis has been carried out with due consideration to viscous damping in the system. The governing differential equations have been established and simplified for general use with the help of dimensionless parameters. These equations have been solved in presence of appropriate boundary conditions by utilizing Finite difference method in combination with iterative GaussSeidel procedure. Inclusion of stone columns has been observed to significantly affect the onset of separation between rail and the soil layer underneath. Various parameters namely, applied load and its velocity, stiffnesses of top, bottom soil layers and stone columns, damping ratio, relative flexural rigidity, depth of placement of geocell, configuration of stone columns have been found to affect the response of soilfoundation system significantly. Improvement in the properties of soil by means of higher value of relative compressibility resulted in typical reduction of 50% in maximum deflection. It has been observed that the region of detachment reduces on increasing the depth of placement of the bottom beam. Sensitivity analysis highlighted the greater sensitivity of upward deflection as compared to the downward deflection of rail with respect to all the parameters except for relative compressibility of the soil and relative stiffness of the stone columns.
With the rapid infrastructural development worldwide, use of ground improvement techniques has increased drastically to enhance the suitability of construction activities over soft soils. Increased speed of trains in case of highspeed rail transportation systems may result in excessive settlement near poor soil strata. In this regard, various case studies have reported the utilization of appropriate ground improvement techniques like geosynthetic reinforcement layer, stone columns, prefabricated vertical drains (PVD) etc. (Arulrajah et al.,
For unreinforced foundation beds, traintracksoil dynamic interaction have been studied by representing the system as an infinite beam resting on one or two parameter foundation system subjected to concentrated moving load (Kenney,
Many researchers have carried out experimental and numerical studies to develop better understanding of these techniques. The experimental study conducted by Raymond (
Nevertheless, all abovementioned studies considered the foundation bed to be in perfect contact with the infinite beam. As the soil essentially reacts only in compression, the above consideration contradicts the actual scenario where the rail is found to show a tendency to lift off the ground at rear as well as in front of applied load. Some of the works that considers this tensionless behavior for unreinforced earth beds include Rao (
Review of literature shows that although analysis of infinite beams subjected to moving load for stone columns has already been carried out, the combined application of it with geocell is yet to be explored for such systems. In view of this, the authors proposed studying the behavior of rails under moving load on stone columngeocell composite earth beds which reacts to compressive forces only. Detailed parametric study and sensitivity analysis has been carried out to understand the influence of spacing, diameter and stiffness of stone columns on response of the system. The impact of other parameters like applied load and its velocity, stiffnesses of top and the bottom soil layers, damping, relative flexural rigidity, and depth of placement of geocell on the proposed system has also been presented in the study.
Few assumptions have been made in modeling and analysis of the system: (i) some components like crossties could not be modeled employing the present approach, (ii) degradation in the properties of geocell and granular material between rail and the geocell with time has not been considered, (iii) quasistationery state has been considered, (iv) smear effect due to installation of stone columns has been neglected. Although, the employed approach has few limitations, however, analysis being simple, it is easier to get an overall picture of the response of soilfoundation system under consideration. Detailed parametric study helps in getting the idea about effect of various parameters and accordingly track design can be carried out.
Longitudinal section of rail resting on geocellstone column composite reinforced earth bed.
The conceptual idealization of physical model (
Idealized representation of the problem.
The governing differential equation of motion based on the idealized model for top and the bottom beam can be expressed as: 
Where, the deflections of top and the bottom beam have been denoted by
The separation between top beam and the geocellstone column composite foundation soil can be mathematically expressed as:
In order to simplify the problem, a new variable ξ has been defined as ξ =
and
The above equations can be rephrased by utilizing the dimensionless parameters mentioned below:
ξ^{*}=
and
Equations (6) and (7) are discretised for an internal node,
and
Where,
Mathematical expressions in Equation (3) can be modified as:
Extent of the beams has been considered such that it behaves as an infinite beam (Selvadurai,
For the top beam
For the bottom beam
Based on the mathematical model established above, a computer code has been developed. The entire extent of trackfoundation system (−
The range of parameters have been assumed as per the Indian railway track conditions and the values considered have been given in
Input parameters.
Applied load  100–250 (Bhatra and Maheshwari, 
kN  
Mass per unit length of the top beam  60 (Bhatra and Maheshwari, 
kg/m  
Mass per unit length of the bottom beam  43 (Indraratna et al., 
kg/m  
Relative compressibility of soil  5–20 (Das, 
–  
Relative stiffness of stone column with respect to surrounding soil  10–100 (Das, 
–  
Diameter of the stone column  0.12–1.2 ( 
m  
Spacing to diameter ratio of the stone columns  2–4 ( 
–  
Relative flexural rigidity of the beams  2,400–5,400 (Shahu et al., 
–  
Damping ratio  ζ  0–25 (Vucetic and Dobry, 
% 
In the absence of experimental data in existing literature for validation purpose, the same has been done by comparing the results with those given by Hussein and Hunt (
Validation.
When top beam is getting lifted from the ground surface due to tensionless nature of the foundation, the deflection in upward direction has been taken as negative deflection for presenting all the results.
The effect of inclusion of stone column on deflection profile of top beam for the parameters:
Deflection of top beam: the effect of stone column inclusion.
Deflection of top beam for perfect contact case and tensionless foundation case.
Deflection of top beam for various magnitudes of applied load.
Bending moment profile of top beam for various magnitudes of applied load.
On further investigation with the same set of input values, it has been found that top beam begins to lift off the ground at a lower value of
Deflection profile of top beam at different values of
Deflection profile of the top beam for different values of α has been presented in
Deflection profile of top beam for various values of α.
The influence of variation in spacing at a specific diameter on deflection profile of the top beam has been presented in
Deflection of top beam at different values of
Deflection of top beam for different depths of placement of the bottom beam.
Deflection of top beam at different velocities of applied load.
At lower velocity, it has been observed that bending of beams remain unaffected by variation in damping coefficients. At higher values of velocity (
Sensitivity analysis:
A practitioner can consider the input values conforming to conditions on site and determine the deflection and bending moment values of the rails. Under the circumstances, where this deformation works out to be more than the allowable values as per required track performance (Beranek,
In addition to this, the sensitiveness of the rail deflection toward variation in different parameters has been highlighted during the sensitivity analysis giving the idea of the impact of that particular parametric variation on the response of rail.
A study has been proposed in order to analyze the combined effect of stone column and geocell improved beds for rails lying over it exposed to moving point load. Tensionless behavior of earth beds has been modeled and included in the analysis. Based on results, the following conclusions can be deduced:
Inclusion of stone columns resulted in 58% reduction in maximum downward deflection of top beam indicating significant improvement from settlement point of view.
Beginning of parting between top beam and the ground surface has been observed at a lower value of
A prominent increase of 48% in maximum downward deflection of top beam has been observed when relative compressibility of soil layers is increased from
Significant reduction in maximum upward and downward deflection has been observed when
Maximum upward deflection of top beam has been observed to reduce by 66% on increasing relative flexural rigidity,
For the top beam, the variation in location of the bottom beam from
Maximum upward deflection of top beam has been found to rise up by 22% compared to 7% initial increment when the velocity is increased up to 80 m/s.
Sensitivity analysis conducted suggested that maximum upward deflection of top beam is exceptionally sensitive toward variation in most of the parameters compared to maximum downward deflection except for the case of relative compressibility of the soil and relative stiffness of the stone columns.
All datasets generated for this study are included in the manuscript/supplementary files.
PM formed the idea and algorithm of the work. SB developed computer program and implemented the algorithm and conducted detailed parametric study. All authors reviewed and accepted the final version.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Coefficients of nodal deflection in the Finite Difference form equation for the top beam.  
Coefficients of nodal deflection in the Finite Difference form equation for the bottom beam  
Viscous damping coefficient of the granular layer  
Nondimensional viscous damping coefficient of the granular layer  
Viscous damping coefficient of the foundation soil  
Nondimensional viscous damping coefficient of the foundation soil  
Diameter of stone columns  
Young's modulus of top beam material  
Young's modulus of bottom beam material  
Shear modulus of granular layer  
Nondimensional shear parameter of granular layer  
Acceleration due to gravity  
Nondimensional thickness of the granular layer  
Thickness of the granular layer  
Second moment of area of the top beam cross section  
Nondimensional modulus of flexural rigidity of the top beam  
Second moment of area of the bottom beam cross section  
Nondimensional modulus of flexural rigidity of the bottom beam  
Subscript referring to nodal points  
Contact function representing tensionless behavior  
Compressibility of the granular layer  
Compressibility of the foundation soil  
Compressibility of foundation soil in stone column region  
Compressibility of foundation soil in soft soil region  
Halflength of beams  
Applied moving load  
Nondimensional applied moving load  
Relative flexural rigidity of the beams  
Relative compressibility of granular layer with respect to soft soil, 

Relative compressibility of granular layer with respect to foundation soil, 

Spacing between the stone columns  
Time  
Velocity of moving load  
Nondimensional selfweight of the top beam  
Nondimensional selfweight of the bottom beam  
Horizontal space coordinate  
Nondimensional top beam deflection  
Nondimensional bottom beam deflection  
Nondimensional deflection of the ground surface  
Deflection of the top beam  
Deflection of the bottom beam  
Deflection of the ground surface  
α  Relative stiffness of stone column with respect to the surrounding soft soil, 
γ_{1}  unit weight of granular mat 
Nondimensional unit weight of granular mat  
Δξ^{*}  Nondimensional distance between Finite Difference nodes 
ζ  Damping ratio 
ξ  Distance from point of action of load at time 
ξ ^{*}  Nondimensional distance from point of action of load at time 
ρ_{1}  Mass per unit length of the top beam 
Nondimensional mass per unit length of the top beam  
ρ_{2}  Mass per unit length of the bottom beam 
Nondimensional mass per unit length of the bottom beam 