flexural response of foundation on reinforced beds
DESCRIPTION
Fundatii din pamant armatTRANSCRIPT
FLEXURAL RESPONSE OF FOUNDATIONS ON
REINFORCED BEDS
A Thesis Submitted in Partial Fulfillment of the Requirements
for the degree of
Master of Technology
by
ARINDAM DEY
to the
DEPARTMENT OF CIVIL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY, KANPUR
MAY, 2005
FLEXURAL RESPONSE OF FOUNDATIONS ON
REINFORCED BEDS
by
ARINDAM DEY
DEPARTMENT OF CIVIL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY, KANPUR
MAY, 2005
To Baba, Ma, Didi & Bhaiya
CERTIFICATE
It is certified that the work contained in the thesis titled “FLEXURAL RESPONSE
OF FOUNDATION ON REINFORCED BEDS” by ARINDAM DEY
(Y3103010), has been carried out under my supervision and that this work has not
been submitted elsewhere for a degree.
P. K. BASUDHAR Professor
Department of Civil Engineering Indian Institute of Technology, Kanpur
Kanpur – 208016, India
ABSTRACT
ARINDAM DEY Roll No. Y3103010
Department of Civil Engineering
Indian Institute of Technology, Kanpur
Kanpur – 208016, India
Thesis Supervisor
Prof. P. K. BASUDHAR
May, 2005
FLEXURAL RESPONSE OF FOUNDATIONS ON
REINFORCED BEDS
The thesis pertains to the analysis of foundations on reinforced beds. The
reinforced bed may be of the following type:
i) A compacted sand bed overlying a loose sand bed with the
reinforcement placed at the interface.
ii) A compacted sand bed underlain by a clay strata with the
reinforcement placed at the interface
The analysis developed is based on mechanical model. For the first problem the
compacted and loose sand beds are modeled with Winkler’s springs of different
stiffness values. In the second problem the compacted sand bed is modeled with
Winkler’s spring where as the clay layer is modeled by using Burger’s four element
model consisting of springs and dash pots. The bending stiffness of the reinforcement
and variability of the soil modulus have been considered in the analysis.
Governing differential equations to find the response of the foundation-reinforced bed
systems as stated above have been developed and solved by finite difference scheme.
Convergence studies have been conducted and it has been found the developed
generalized procedure produces convergent solutions. Correctness of the obtained
solutions have been checked and ensured by comparing the results with bench mark
problems reported in literature. Parametric studies have also been carried out to find
the effect of various parameters related to the foundation structure, soils and
reinforcement on the flexural response of the foundation and reinforcement beam.
ACKNOWLEDGEMENTS
I take this opportunity to express my sincere gratitude to Prof. P. K. Basudhar
for his involvement, motivation and encouragement throughout and beyond the thesis
work. His patient hearing, critical comments and approach to the research problem
made me do better every time. His valuable suggestions at all stages of the thesis
work helped me to improvise various sorts of shortcomings of my thesis work. I
express my sincere respect to him for his parental guidance throughout my period of
stay at IIT Kanpur.
I would like to express my sincere tribute to Prof. Sarvesh Chandra and Dr.
Nihar Ranjan Patra for their very friendly nature and treating more than as a student. I
would also like to express my sincere thanks due to their excellent guidance and
teaching during my stay at IIT Kanpur.
I would like to render special thanks to Mr. A. K. Srivastava, Gulab ji, Yadav
ji and Parashuram ji for their kind co-operation and for granting free access to all the
laboratory equipments and accessories as and whenever needed.
I would like to offer my special tribute to Kousik da for his immense help
throughout and beyond my thesis work and for his valuable and critical suggestions
like an elder brother. I am greatly thankful to my classmate, Paritosh Kumar, for his
friendly nature and immense help he rendered me by allowing me to use his PC
during my thesis work. I would like to thank all my friends, especially Abhik,
Pradipta, Bappaditya, Shyam, Meera, Sutapa, Trishikhi, Antara, Samaresh,
Subhotosh, Saikat, Kaustav, Bisu, Dipanjan da, Dip da, Anurag, Deepak, Pradeep,
Brijesh, Col. Saxena, Sourav, Anuj and Waseem and all others who made my stay a
very joyous, pleasant and memorable one and made me feel to be within a family. I
would also like express special thanks to Priti di, Sarat da and Shanker da for their
valuable suggestions regarding my thesis work.
Last but not the least, I would like to offer my cordial homage to my Baba and
Ma for all the hardships and sufferings they had to bear during my distant stay from
home. I like to pay my tribute to them for their blessings, encouragement and
motivation throughout my academic career. I would like to deeply thank my Didi and
Bhaiya for their immense and unlimited moral support and encouragement which
helped me keep up my stamina and will power throughout my stay at IIT Kanpur and
throughout my academic career.
ARINDAM DEY
CONTENTS
LIST OF FIGURES xv
LIST OF TABLES xix
NOTATIONS xxi
CHAPTER 1 INTRODUCTION 1
1.1 General 1
1.2 Reinforced soil system 6
1.3 Motivation and scope of present study 30
CHAPTER 2 MODELING OF FOUNDATIONS ON
REINFORCED SAND BEDS WITH
VARIABLE SUBGRADE MODULUS 33
2.1 Introduction 33
2.2 Statement of the problem 34
2.3 Analysis 36
2.3.1 Assumptions 36
2.3.2 Governing differential equations 36
2.3.3 Non-dimensional form of governing equation 38
2.3.4 Boundary and continuity conditions 39
2.3.5 Method of solution: Finite Difference Method
41
2.4 Results and Discussions 42
2.4.1 Convergence study 42
2.4.2 Correctness of the developed program and the solution
obtained
44
2.4.2.1 Comparison with Hetenyi’s Model 44
2.4.2.2 Comparison with previous research 44
2.4.3 Parametric study 46
2.4.3.1 Effect of depth of placement of reinforcement (H’) 47
2.4.3.2 Effect of relative flexural rigidity of beams (R) 49
2.4.3.3 Effect of relative stiffness of soils (kr) 49
2.43.4 Effect of unit weight of compacted granular layer (γ’) 52
2.4.3.5 Effect of variation of parabolic constants (kn) 53
2.4.3.6 Effect of variation of coefficient of friction (μ) 54
2.4.3.7 Typical variation of normalized bending moment
diagram
56
2.4.3.8 Typical variation of normalized shear force diagram 56
2.5 Conclusion 59
2.6 Scope of further work
62
CHAPTER 3 MODELING OF FOUNDATIONS ON A
COMPACTED SAND BED UNDERLAIN
BY A WEAK CLAY STRATA WITH
REINFORCEMENT PLACED AT THE
INTERFACE
65
3.1 Introduction 65
3.2 Statement of the problem 66
3.3 Analysis 68
3.3.1 Assumptions 68
3.3.2 Governing differential equations 68
3.3.3 Non-dimensional form of governing equation 71
3.3.4 Boundary and continuity conditions 72
3.3.5 Method of solution: Finite Difference Method 74
3.4 Results and Discussions 75
3.4.1 Convergence study 75
3.4.2 Correctness of the developed program and the solution
obtained
77
3.4.3 Parametric study 79
3.4.3.1 Effect of depth of placement of reinforcement (H’) 80
3.4.3.2 Effect of relative flexural rigidity of beams (R) 82
3.4.3.3 Effect of relative stiffness of soils (kr) 84
3.4.3.4 Effect of unit weight of compacted granular layer (γ’) 85
3.4.3.5 Effect of variation of parabolic constants (kn) 87
3.4.3.6 Effect of variation of friction coefficient (μ) 88
3.4.3.7 Effect of variation of relative stiffness coefficients in
Burger model (kk)
90
3.4.3.8 Effect of variation of relative viscous coefficient in
Burger model (ηη)
91
3.4.3.9 Effect of variation of time (t’) 93
3.4.3.10 Typical variation of normalized bending moment
diagram
95
3.4.3.11 Typical variation of normalized shear force diagram 96
3.5 Conclusion 98
3.6 Scope of further work 101
LIST OF FIGURES
Figure Page
1.1 Winkler Model. 3
1.2 Filolenko – Borodich Model. 3
1.3 Pasternak Model. 4
1.4 Hetenyi Model. 4
1.5 Vlasov Model. 5
1.6 Kerr Model. 5
1.7 Madhav and Poorooshasb model (1988). 26
1.8 Ghosh and Madhav model (1994). 26
1.9 Shukla and Chandra model (1994b). 27
1.10 Shukla and Chandra model (1994c). 28
1.11 Yin model (1997a). 28
1.12 Yin model (2000). 29
1.13 Maheshwari model (2004). 29
2.1 Definition Sketch of the problem. 35
2.2 Proposed foundation model. 35
2.3 Convergence study of deflection of footing beam with uniform subgrade modulus. 43
2.4 Convergence study of deflection of footing beam with variable subgrade modulus. 43
2.5 Comparison of settlement profile of footing and reinforcing beam with Hetenyi’s Model. 45
2.6 Comparative study of present solution with previous solutions. 46
2.7 Normalized deflection of footing beam for variation in depth of placement of reinforcement below the footing beam (H’). 48
2.8 Normalized deflection of reinforcing beam for variation in depth of placement of reinforcement below the footing beam (H’). 48
2.9 Normalized deflection of footing beam for variation in relative flexural rigidity of beams (R). 50
2.10 Normalized deflection of reinforcing beam for variation in relative flexural rigidity of beams (R). 50
2.11 Normalized deflection of footing beam for variation in relative stiffness of soils (kr). 51
2.12 Normalized deflection of reinforcing beam for variation in relative stiffness of soils (kr). 51
2.13 Normalized deflection of footing beam for variation in unit weight of compacted granular layer (γ’). 52
2.14 Normalized deflection of reinforcing beam for variation in unit weight of compacted granular layer (γ’). 53
2.15 Normalized deflection of footing beam for variation in parabolic constants (kn). 54
2.16 Normalized deflection of reinforcing beam for variation in parabolic constants (kn). 55
2.17 Normalized deflection of footing beam for variation in coefficient of friction (μ). 55
2.18 Normalized deflection of reinforcing beam for variation in coefficient of friction (μ). 56
2.19 Typical normalized bending moment diagram of footing beam for variation in relative flexural rigidity of beams (R). 57
2.20 Typical normalized bending moment diagram of reinforcing beam for variation in relative flexural rigidity of beams (R). 58
2.21 Typical normalized shear force diagram of footing beam for variation in relative flexural rigidity of beams (R). 58
2.22 Typical normalized shear force diagram of reinforcing beam for variation in relative flexural rigidity of beams (R). 59
2.23 Definition sketch of a railroad track. 62
2.24 Definition sketch of a combined footing. 63
2.25 Definition sketch of a railway tie 63
2.26 Definition sketch of a surface water tank. 64
3.1 Definition Sketch of the problem. 67
3.2 Proposed foundation model for the present study. 67
3.3 Four Element Burger model. 69
3.4 Convergence study of footing beam at time t’ = 0 76
3.5 Convergence study of footing beam at time t’ = 0.5 76
3.6 Comparison of the degenerated cases of elastic and visco-elastic models of the present study. 78
3.7 Time-Settlement (total) plot of reinforcing beam for variation in relative ratio of viscous coefficients in Burger model (ηη) 79
3.8 Normalized deflection of footing beam for variation in depth of placement of reinforcement below the footing beam (H’). 81
3.9 Normalized deflection of reinforcing beam for variation in depth of placement of reinforcement below the footing beam (H’). 81
3.10 Normalized deflection of footing beam for variation in relative flexural rigidity of beams (R). 83
3.11 Normalized deflection of reinforcing beam for variation in relative flexural rigidity of beams (R). 83
3.12 Normalized deflection of footing beam for variation in relative stiffness of soils (kr). 84
3.13 Normalized deflection of reinforcing beam for variation in relative stiffness of soils (kr). 85
3.14 Normalized deflection of footing beam for variation in unit weight of compacted granular layer (γ’). 86
3.15 Normalized deflection of reinforcing beam for variation in unit weight of compacted granular layer (γ’). 86
3.16 Normalized deflection of footing beam for variation in parabolic constants (kn).
87
3.17 Normalized deflection of reinforcing beam for variation in parabolic constants (kn).
88
3.18 Normalized deflection of footing beam for variation in coefficient of friction (μ). 89
3.19 Normalized deflection of reinforcing beam for variation in coefficient of friction (μ). 89
3.20 Normalized deflection of footing beam due to the variation in relative stiffness coefficient in Burger model (kk).
90
3.21 Normalized deflection of reinforcing beam due to the variation in relative stiffness coefficient in Burger model (kk).
91
3.22 Normalized deflection of footing beam due to the variation in relative viscous coefficient in Burger model (ηη).
92
3.23 Normalized deflection of reinforcing beam due to the variation in relative viscous coefficient in Burger model (ηη).
92
3.24 Normalized deflection of footing beam due to the variation in time elapsed (t’). 93
3.25 Normalized deflection of reinforcing beam due to the variation in time elapsed (t’). 94
3.26 Typical time-settlement curve to determine the time lapsed (tpc).
94
3.27 Typical normalized bending moment diagram of footing beam for variation in relative flexural rigidity of beams (R). 95
3.28 Typical normalized bending moment diagram of reinforcing beam for variation in relative flexural rigidity of beams (R). 96
3.29 Typical normalized shear force diagram of footing beam for variation in relative flexural rigidity of beams (R). 97
3.30 Typical normalized shear force diagram of reinforcing beam for variation in relative flexural rigidity of beams (R). 97
A.1 Four Element Burger model. 111
A.2 Representation of applied stress with time. 112
LIST OF TABLES
Table Page
1.1 Summary of literature review related to geosynthetic-reinforced soil system (Theoretical Works). 21
2.1 Range of non-dimensional parameters considered in the study. 47
3.1 Range of non-dimensional parameters considered in the study. 80
NOTATIONS
A0, A1, A2, B1, B2 Coefficients used to determine the stress-strain relation of Burger model
Ci Non-dimensional stiffness coefficients
E1I1 Flexural rigidity of footing beam
E2I2 Flexural rigidity of reinforcing beam
F, Fn Coefficients used to determine the stress-strain relation of Burger model
F1, F2, F3, F4, F5, F6, F7
Non-dimensional functions used to evaluate stiffness matrix
H Height of compacted granular fill, or Depth of placement of reinforcing beam below the footing beam
H’ Normalized depth of placement of reinforcing beam below the footing beam
M Normalized length
Q Concentrated load acting at the centre of the footing beam
R Relative flexural rigidity of footing and reinforcing beam
R1 Characteristic length of footing beam
R2 Characteristic length of reinforcing beam
Rc Characteristic length of beams
Rn Relative characteristic length of footing and reinforcing beam
T Tension force generated on reinforcing arising due to friction
T’ Normalized tension force generated due to friction
Z Relative stiffness coefficients
f1n, f2n, f3n, f4n, f5n Non-dimensional functions used to evaluate stiffness matrix
h Size of mesh segment used for discretization of beams
k1(x) Subgrade modulus of compacted granular fill
k10 Maximum Subgrade modulus of compacted granular fill at
the centre of the footing beam
k11, k12 Parabolic constants to determine the nature of distribution of subgrade modulus of compacted granular layer
k11n, k12
n Normalized parabolic constants to determine the nature of
distribution of subgrade modulus of compacted granular layer
k1n Normalized subgrade modulus of compacted granular layer
k2(x) Subgrade modulus of underlying poor soil
k20 Maximum Subgrade modulus of underlying poor soil at the centre of reinforcing beam
k21, k22 Parabolic constants to determine the nature of distribution of subgrade modulus of underlying poor soil
k21n, k22
n Normalized parabolic constants to determine the nature of
distribution of subgrade modulus of underlying poor soil
k2n Normalized subgrade modulus of underlying poor soil
kb1, kb2 Stiffness coefficients in Burger model
kb2n Relative stiffness coefficients
kk Relative stiffness coefficient of Burger model
kn Ratio of parabolic constants
kr Relative stiffness of soils
l1 Half span of footing beam
l2 Half span of reinforcing beam
ln Relative length of footing and reinforcing beam
nb Number of nodes in half span of footing beam
nr Number of nodes in half span of reinforcing beam
p1 Contact pressure at the base of footing beam
p2 Contact pressure at the base of reinforcing beam
t Time lapsed after application of load
t’ Normalized time lapse after the application of load
tpc Time required for primary compression
x distance from the centre of the beam
xn Normalized distance from the centre of the beam
y1 Deflection coordinates of the footing beam
y1’ Normalized deflection coordinates of footing beam
y2 Deflection coordinates of the reinforcing beam
y2’ Normalized deflection coordinates of reinforcing beam
•
y Time derivative of deflection
γ’ Normalized unit weight of compacted granular fill
γ1 Unit weight of the compacted granular fill
γ1” Normalized unit weight of compacted granular fill
γ2 Unit weight of the underlying poor soil
ε Infinitesimal distance along the length of beams
η1, η2 Viscous coefficients in Burger model
ηη Relative viscous coefficient of Burger model
μ Coefficient of friction
σ’, σ1’, σ2’ Stress applied on Burger elements
Δxn Infinitesimal distance along the length of beam
CHAPTER 1
INTRODUCTION
1.1 General Soil-structure interaction is one of the most interesting and widely studied
topics in geotechnical engineering. Several analytical and experimental studies have
been reported in the literatures on the various aspects of such studies. Since the
introduction of Winkler’s model in 1867, wherein the soil is idealized by a series of
discrete springs, several such lumped parameter models like Pasternak model,
Filolenko-Borodich model, Hetenyi model, Vlasov model, Kerr model etc. have been
developed to remove the deficiency of the above model. Few of the models are shown
in Figure 1.1 to 1.5. Such analyses are very helpful in determining the contact
pressure distribution at the interfaces of the structural members resting on elastic
foundations and its settlement behavior. These models are now available in standard
reference book (Selvadurai, 1973). Since Vidal (Binquet & Lee, 1975) introduced the
concept of soil reinforcement to enhance the performance of soils under various kinds
of loadings, use of reinforcement in construction has attracted the attention of
geotechnical engineers and is increasingly been used in ground engineering. This has
necessitated the development of analytical procedures to predict the behavior of such
foundations. With the above in view, the above models for unreinforced soils are now
being extended to analyze foundations on reinforced earth beds. Modeling of such
foundations are generally based on any one of the following approaches namely limit
equilibrium (a method though does not satisfy all the conditions of plasticity has
acquitted itself quite well in solving such problems and is well known to the
practicing engineers), lumped parameter models (Mechanical models consisting of
springs and dash pots arranged in various configurations) and continuum mechanics
approach (both analytical and numerical). The different approaches has their own
merits and demerits but have served very well the profession in understanding and
predicting the performance of such foundations. These have also enabled the
engineers to evaluate the effect of various parameters on the foundation response. The
basic problem of lumped parameter models (even though quite simple to use) lies in
the fact that the parameters involved are not fundamental parameters like modulus of
elasticity and Poisson ratio (used in the theory of elasticity approach treating the
medium to be a continuum) and are difficult to determine. Continuum mechanics
approach using either theory of elasticity or mathematical plasticity is quite complex
and difficult to use. However, application of finite element method (either direct
formulation or variational formulation) has enabled the engineers to solve very
complicated problems but it is not used as widely as the limit equilibrium and lumped
parameter based models. With the limit equilibrium based methods stability of
foundations can be analyzed but it can not be used to predict displacements, where as,
displacements also can be predicted with the help of lumped parameter models. But
lumped parameter models are incapable of predicting the stresses and displacements
at all place within the concerned medium except at the interface between the soil and
the structural members. In this respect continuum mechanics based models are
superior. However, from the point of simplicity and at the same time due to its
capability in predicting displacements, it has been decided to use a lumped parameter
model in the present study.
Figure 1.1 Winkler Model.
Figure 1.2 Filolenko – Borodich model.
Figure 1.3 Pasternak Model.
Figure 1.4 Hetenyi model
Figure 1.5 Vlasov Model.
Figure 1.6 Kerr Model.
Based on an initial overview of literature it has been decided to work
on the flexural response of beams on reinforced foundation beds. As such, in the
following section a brief review of literature pertaining to the subject under
consideration is presented. Considerable literature on peripheral topics is not included
here. The review refers to the literatures related to the analytical, numerical and other
theoretical works carried out in the reinforced earth to study the response of the beams
on reinforced elastic foundations.
1.2 Reinforced soil system
Maheshwari (2004) presented a detailed literature review on the subject. As
such, the literature covered by her are not reviewed here and only the additional
references are cited and discussed. However, for the sake of completeness all the
papers (including the papers cited by Maheshwari) are summarized subsequently.
Maheshwari (2004) referred to the following papers in her thesis. These are:
Binquet & Lee (1975), Brown & Poulos (1981), Giroud & Noiray (1981), Andrawes
et al. (1982), Madhav & Poorooshasb (1988), Bourdeau (1989), Poran et al. (1989),
Sellmeijer (1990), Poorooshasb (1991), Dixit & Mandal (1993), Ghosh & Madhav
(1994 a, b, c), Shukla & Chandra (1994 a, b, c), Shukla & Chandra (1995), Yin (1997
a, b), Yin (2000), Fakher and Jones (2001), Kotake et al. (2001) .The additional
papers are cited and discussed as follows.
Kondner (1963) proposed a two constant hyperbolic form of stress-strain
response for modeling of soil behavior for load-deformation analysis of foundations.
The response used was such that the ultimate shear strength of the soil is contained
within the general formulation and appeared as a mathematical limit when the stress
became excessive. This represented a remolded cohesive soil tested in consolidated-
undrained triaxial compression. The variables in the hyperbolic stress-strain relation
included the preconsolidation pressure, rebound stress, lateral pressure during the test,
vertical normal stress, strain and the rate of strain. The history effects were included
in terms of overconsolidation ratio. It was observed that the proposed failure relations
degenerated into the conventional Mohr-Coulomb failure envelope in a two-
dimensional stress space. The two constants of hyperbola were designated as a (the
reciprocal of the initial tangent modulus, EI), and b (the reciprocal of the asymptotic
value of stress difference which the stress-strain curve approaches at infinite strain).
Duncan & Chang (1970) developed a simple, practical procedure for
representing the nonlinear, stress-dependent, inelastic stress-strain behavior of soils,
which could be conveniently used for Finite element Analysis. However, the
methodology described suffered loss of accuracy as the stress-strain relationships
were based on standard triaxial tests which employed less general loading conditions
than actually simulating the field conditions. The relationship used six basic soil
stress-strain parameters: c (Cohesion of soil), φ (Angle of internal friction), Rf (The
failure ratio, usually have a value of less than unity; 0.75 to 1.00 for the present case),
K (Modulus number), n (Exponent determining the rate of variation of Ei with σ3) and
Ei (Initial tangent modulus).
Desai (1971) presented the use of a cubic spline equation to simulate stress-
strain curves defined by two measured parameters. The same was extended to three-
dimensional spaces defined by three measured parameters. A bi-cubic spline function
was used for simulation of spaces defined by such groups of parameters as stress
difference, axial strain and confining pressure; radial strain; axial strain and confining
pressure; volumetric strain; mean pressure and relative density; and octahedral stress,
octahedral strain and mean pressure.
Gourc et al. (1982) studied the bearing capacity of a two layer system (a
cohesionless soil sub-base and a clay subgrade) under punching. Quasi-static
punching model analysis was carried out to study the influence of the geotextile
modulus and the setting conditions (free or fixed extremities of the fabric) on the
anchorage design. From the experiments, the membrane effect of the fabric behavior
was interpreted. Lateral sliding of the reinforcement under the axial load was also
studied.
Prakash et al. (1984) proposed a novel analysis to predict the pressure-
settlement characteristics of footings using the hyperbolic stress-strain curves of soils
as constitutive law. The analysis incorporated the effect of shape, base roughness and
flexibility of footings. The analyses were developed both square and strip footings.
Results were presented in terms of ultimate bearing capacity, settlement at failure and
non-dimensional correlations of settlement. The soil mass was assumed to semi-
infinite and isotropic medium. The footing base was assumed to be fully flexible or
fully rigid. The roughness of the footing was assumed to generate uniform tangential
forces at the contact surface, acting inwardly, and zero at the centre. The
mathematical model proposed by Kondner (1963) in the form of a two-constant
hyperbolic model was used to describe the constitutive law of the soil. Though the
parabolic distribution was more realistic, a trapezoidal distribution was followed for
the ease of computations. The whole soil mass was divided into a number of
horizontal layers and the stresses in each layer was calculated by using Boussinesq’s
theory. It was assumed that there is no slippage at the interface of layers of the soil
mass. It was observed that roughness and rigidity of footing had a very negligible
effect o the average pressure-settlement curves. At failure condition, the settlement
was observed to be about 5% and 12.5% for all cases of strip footings and rigid square
footing respectively. It was suggested that the proposed methodology could be
conveniently used to analyze and design shallow foundations and pile foundations in
clays by suitably modifying stress equations. However, the proposed methodology
could not be adopted in cohesionless soils as stress equations based on the theory of
elasticity does not take into account the variation of elastic modulus, E, due to
confining pressure.
Huang & Tatsuoka (1988) predicted the bearing capacity in a level sandy
ground with strip reinforcement by stability analysis by limit equilibrium method, for
both the cases of short and long reinforcement. It was observed that at the peak
footing load, the shear bands developed only in a limited area beneath the footing,
with small strains developed outside the active zone under the footing. Bearing
capacity was also found to be increase markedly by restraining possible strains in the
soil in the zone beneath the footing by means of short strips with the same length as
the footing width. It was also concluded that the prediction of the tensile forces in
reinforcements was essential under practical conditions.
Floss and Gold (1990) performed FEM analysis to predict the bearing and
deformation behavior of the single reinforced two-layer system. The soil continuum
was modeled by eight noded, isoparametric elements with quadratic shape functions
and for the geotextile, isoparametric bar elements were used. Thin layer elements
were used to model the interaction and the potential for the relative movement
between the reinforcement and the soil. Movements of the soil relative to the
movement of the reinforcement under large shear distortions of the thin layer
elements was modeled by supplying a joint parallel to the reinforcement, to limit the
transfer of forces from the soil into the reinforcement by Mohr-Coulomb yield
criterion. The Young’s modulus of the surrounding soil was considered to be one. The
calculations were performed by using elasto-plastic deformation under consideration
of the yield criterion and the flow rule of Mohr-Coulomb. Tension was not allowed
for the soil element, the bar elements were defined for tension. Viscoplastic iteration
algorithm was used to reduce and recontribute the inadmissible stresses. It was
observed that at relatively low load level, the frictional base course began to plastify
in the area of the load application area. With further loading, the plastified area
enlarged until it reached the weak soil. It was also observed that as the subsoil
plastified over large area, the deformations went on increasing over-proportionally till
the yielding of the system occurred. Both the bearing capacity increased and
settlement was reduced on the inclusion of reinforcement in the analysis. This was
attributed t the altered stress distribution due to the placing of the inclusion. It was
observed that the shear stress peaks were lower by 25% compared to unreinforced
system. Because of the reinforcement, the horizontal strains were reduced and the
horizontal stresses were concentrated in the areas with high vertical stresses. The
contribution of the vertical stresses indicated the load spreading effect of
reinforcement.
Murthy et al. (1993) carried out limit equilibrium analysis to study and
evaluate the bearing capacity of a reinforced soil foundation. It was proposed that on
application of the vertical load on a footing, downward movement of soil will take
place along with the lateral flow of soil. It was assumed that the total load carried by
the footing on a reinforced soil bed was carried simultaneously by the soil and the
reinforcements, and that the load carried by the soil alone was responsible for the
settlement of the footing. The boundary of the vertically and the laterally moving soil
mass was assumed to be a vertical plane passing through the edge of the footing.
Right angle kinks were assumed to be formed in the reinforcement along the potential
slip plane resulting in the transfer of tension in the reinforcement as vertical force
required resisting the applied load. Elastic theory was used to determine the stress
distribution inside the soil mass. Failure was observed to occur in the modes of tie
failure or frictional failure, where the frictional failure seemed to be critical for the
evaluation of the mobilized tension in the reinforcement. It was suggested that while
computing the frictional strength of lower layers, the load component carried by the
upper layers had been assumed to be distributed uniformly beyond the loaded area,
and hence not available for the mobilization of frictional strength. This method was
proposed to check the ultimate bearing capacity accurately for both tie pullout and tie
failure conditions.
Burd (1995) presented an analysis concerning the mechanics and design of
unpaved reinforced roads built over soft clay. An analytical design method was
proposed based on the membrane reinforcement mechanism where large surface
deformations were expected to occur. A FEM model was also proposed. However, the
analytical model revealed a lot of discrepancies when compared to the finite element
model, due to non-inclusion of the shear stresses developed very near to the origin of
the reinforcement. This study did not include the effects of elastic soil deformations
and the shear stresses developed at the base of the fill, immediately beneath the load.
Moreover, the model was proposed for a constant fill thickness during the application
of the load, thus leading to an over-stiff response as bearing capacity of the fill was
approached. It was concluded that the load-spread model was very simple enough to
accurately represent the load-spreading mechanism beneath the base layer under the
application of the load; thus further research was needed in the area.
Zhao (1996) presented a failure criterion for the reinforced soil composite. A
slip-line method was described and the failure loads and stress characteristic fields for
reinforced slopes, walls and foundations were calculated using the proposed slip-line
method. The failure criterion presented was anisotropic due to the inclusion of the
geosynthetic reinforcement in preferred direction. It was observed that the
geosynthetic reinforcement enlarges the plastic failure region in a reinforced soil
structure, and significantly increased the load bearing capacity.
Michalowski (1998) presented a kinematic approach of limit analysis in
which a rigorous bound to the required strength of strength is sought. It was suggested
that the required strength of reinforcement was the strength needed to maintain the
stability of the structure. Since limit analysis lead to a rigorous bound on the
reinforcement strength, limit loads or a safety factor, the geometry of the failure
mechanisms considered could be optimized, so that the best bound was obtained (a
solution closest to the exact solution). A dual formulation of kinematic limit analysis
was proposed. The formulation also considered inclination of the reinforcement force.
It was observed that the kinematic approach of limit analysis constituted a convenient
tool for stability analysis of reinforced soil structures. It provided a rigorous solution
(lower bound to the strength of reinforcement, or upper bound to the loads causing
failure). It was concluded that the limit analysis computations should be performed as
if the direction of placement of the reinforcement (typically horizontal) was not
changed, since its inclination did not influence the result for tensile failure mode.
Pitchumani & Madhav (1998) studied the interaction mechanism of one and
more pairs of inextensible reinforcing strips and the soil in reinforced foundation beds
using an elastic continuum approach. Boussinesq’s and Mindlin’s solution wee
integrated to evaluate the lateral soil displacements at the soil-strip interface. The
compatibility of the displacements were satisfied at the of soil-strip interface to obtain
the mobilized shear stresses, based on which the reduction in surface settlements were
computed. The reinforcements were considered inextensible, i.e. they were rigid
longitudinally. Further studies with extensible reinforcements were required. It was
observed that the mobilized shear stresses were affected by the spacing of the
reinforcements. The farther the distance of the reinforcement from the centre of the
loaded area, the lesser is the horizontal displacement, and hence lesser shear stress
was mobilized to counter the displacement. At closer spacing, the interference of the
two strips on each other’s displacement was more and hence lesser mobilized shear
stress was observed. It was also observed that no additional benefit was observed
when the length of the reinforcing strip was greater than twice the footing width. It
was concluded that placing the reinforcing strips at greater depths at farther spacing is
advantageous in achieving maximum settlement reduction due to shear interactions
alone. The ratio relating the combined effect of the strips considered together to the
sum of individual effects was observed to increase with the distances between the
strips.
Saran (1998) proposed an analytical analysis to determine the pressure on a
rectangular footing resting on reinforced sand for a given settlement for which the
pressure on the same footing resting on unreinforced sand is known. A method was
also proposed to obtain the ultimate bearing capacity of the footing on reinforced
sand. The results were presented in non-dimensional form. Good agreement was
observed between the data and the model results. It was observed that the ultimate
bearing capacity of the footing can be increased significantly by adequately
reinforcing the sand bed, which simultaneously resulted in the lowering of settlement
by a significant amount.
Shukla & Chandra (1998) presented a simple mechanical modeling approach
to study the settlement characteristics of geosynthetic-reinforced granular fill-soft soil
system subjected to axi-symmetric load at any stage of consolidation of the soft
subgrade. The salient features of the reinforced soil system were retained to study the
gross behavior of the system. It was concluded that the development of horizontal
stresses in the geosynthetic-reinforced granular fill on soft subgrade under axi-
symmetric load resulted in settlement reduction. Prestressing the geosynthetic
reinforcement in the geosynthetic-reinforced granular fill-soft soil system was found
to be very effective in reducing both the total and differential settlements in the loaded
region. The compressibility of the granular fill had an appreciable influence on the
settlement response of the geosynthetic-reinforced granular fill-soft soil system as
long as the stiffness of the granular fill was less than approximately 50 times that of
the soft soil. Parameters such as interfacial friction coefficients, width of reinforced
zone, prestress in the geosynthetic reinforcement, and lateral stress ratio in the
granular fill etc., showed beneficial effects only at the later stages of consolidation of
soft subgrade.
Kotake et al. (1999) performed FEM simulation on plane strain compression
tests of dense sand reinforced with reinforcements with a wide range of stiffness. The
strain localization was modeled by modeling a shear band of specific thickness and
specific strain softening properties. It was concluded that nonlinear pre-peak stress-
strain characteristics of sand was dependent on both shear strain and confining
pressure level, and that the shear band width and the post-peak stress-strain
characteristics within a shear band influenced the numerical simulation of failure of
reinforced sand. The other factors which influenced the simulation were pointed out to
be tensile and bending rigidities of reinforcement, elastic properties of sand, and the
interaction between the sand and reinforcement at the interface.
Siddiquee et al. (1999) performed FEM simulation of the bearing capacity
characteristics of the strip footings on sand, and explained the scale effects observed
in the model plane strain tests carried out simultaneously. A constitutive model was
developed considering the factors affecting the strength-deformation characteristics of
sand such as confining pressure, anisotropy, nonlinear strain hardening and strain
softening, dilatancy and strain localization into shear bands. The material was
modeled by isotropically hardening, non-associated, elasto-plastic model. Parametric
study revealed that the isotropically perfectly plastic modeling of soil property, as
used in the classical bearing capacity theories was an oversimplified assumption. It
was pointed out that the post-peak strain softening characteristics largely affected the
bearing capacity of footing on dense sand, as the failure of the ground was highly
progressive. It was revealed that the FEM simulation could accurately simulate scale
effects, consisting of the pressure level effect and the particle size effect, which would
generally be over-estimated by the centrifuge modeling test.
Peng et al. (2000) performed large scale plane strain compression tests on
unreinforced and geogrid reinforced specimens. It was observed that when there was
no rupture of the geogrid layer, the covering ratio for each grid layer was a more
dominant and important factor than the tensile stiffness of the geogrid, in terms of the
increment of the bearing capacity of the reinforced ground Numerical analysis was
performed by plane strain nonlinear elasto-plastic FEM considering the strain
localization and the anisotropic stress-strain behavior of sand and interface properties,
in which the geogrid was modeled as a planar reinforcement. Studies were conducted
on the pre-peak stress-strain behavior of the reinforced and unreinforced specimens,
peak strength, post-peak behavior, the dilatancy characteristics, reinforcement rigidity
and the covering ratio of the reinforcement. FEM analysis was used to determine the
relationship between the covering ratio and the equivalent interface friction angle
between the soil and the reinforcement. Mechanism of tensile reinforcing based on the
local stress paths was also analyzed. It was suggested that the classification of
extensible and inextensible reinforcements solely based on the material stiffness was
highly misleading and need further research. It was suggested that the properties of
the interface between the grid layer with a specific cover ratio and the sand layer not
exhibiting slipping with a displacement discontinuity can be well simulated by plastic
flow model in simple or direct shear mode in the sand elements adjacent to the
interface during the analysis, and that the interface friction angle,μ, was uniquely
related to the covering ratio of the geogrid. It was observed from the FEM analysis
that the failure of the reinforced sand specimen was progressive in nature and it was
highly essential to model properly the strain softening behavior of sand associated
with strain localization. However, for proper and more critical simulation of the
reinforced sand including various stress paths, further studies were to be needed in
terms of deformation properties of sand in anisotropic consolidation at stress ratios
between K0 and the failure values.
Madhav and Pitchumani (2000) proposed a method to predict the reduction in
surface settlements due to strip form of reinforcements beneath a rectangular loaded
area. Elastic continuum approach was used to solve the problem. The compatibility of
the vertical displacements at points along the soil-strip interface and the equilibrium
of forces were satisfied to obtain the net normal stress mobilized at the soil-strip
interface. Both the rigid and flexible footing conditions were analyzed. The resulting
uniform translation for a rigid strip footing and the deflection profile for a flexible
strip footing were evaluated. From the same, the reductions in surface settlements due
to the mobilized normal stresses were computed. The results from the parametric
study indicated that for maximum reduction in surface settlements, the strip footing
should be placed close to the surface. Below square loaded areas strips of length
L/B=3 were adequate for maximum settlement reduction. It was observed that the
performance of the flexible strips approached that of the rigid strips as the flexibility
ratio of strip increased, which in turn depended on the flexural stiffness and the length
of the strip.
Dey (2002) modified the dimensionless force curves proposed by (Binquet &
Lee (1975 b)) by considering small intervals of length (Δx) on the length of the tie
breakage (X0) in the hypothetical formula. This resulted in the significant change in
the J (z/B) curve, whereas the curves, I (z/B) & M (z/B) showed minor or no changes
at all. This change reflected the trend of development of tie tension in different layers
of reinforcement as obtained in the model study (Binquet & Lee (1975 a)). However,
this approach was also based on the assumption that the tie force per layer varies
inversely with the number of layers of reinforcement, which required further
investigation and refinement.
Kumar & Saran (2003) presented a method of analysis for calculating the
pressure intensity for calculating the pressure intensity corresponding to a given
settlement for a rectangular footing resting on a reinforced soil foundation. Non-
dimensional charts and an empirical method were suggested to determine the ultimate
bearing capacity of a rectangular footing on rectangular soil. It was pointed out that
inclusion of geosynthetic reinforcement below the footing increased both the ultimate
and allowable bearing stresses at a given settlement. It was concluded that
computation of the pressure ratio or bearing capacity ratio consisted of two essential
steps i.e. computation of the normal force on the reinforcement area and the
estimation of interfacial frictional resistance at different layers of the reinforcement.
However, the method proposed required the pressure-settlement values of the
unreinforced soil as a pre-requisite, which was obtained from the standard methods.
Thus, further researches were required to incorporate the deficiency.
Maharaj (2003) conducted nonlinear two-dimensional finite element analysis
for a strip footing on reinforced clay under plane strain condition. The footing and the
soil was discretized by four noded isoparametric finite elements while the
reinforcement was modeled by four noded one-dimensional finite elements. The soil
was idealized as Drucker-Prager elasto-plastic medium. Investigations were made to
study the effects of embedment depth of first layer of reinforcement, spacing of
reinforcement layers, number of reinforcement layers and the size of the
reinforcement. For case of single layer of reinforcement, optimum embedment depth
of reinforcement resulted in the maximum reduction in settlement. The same was
observed in the case of multi-layer of reinforcement. It was observed that the
increment of tensile stiffness of reinforcement reduced the settlement of footing both
for the cases of single and multi layer of reinforcement up to a critical value, beyond
which the settlement reduction was negligible. It was also observed that the settlement
reduced with the increasing number of layers, only up to a critical value of number of
layers. Closely spaced reinforcement provided a larger bearing capacity. It was
suggested that the load carrying capacity of the reinforced footing was more in the
case where reinforcement of higher tensile rigidity was utilized.
Kumar et al. (2004) proposed a method to obtain the pressure settlement
characteristics of rectangular footings resting on reinforced sand based on constitutive
laws of soils. The analysis incorporated the confining effect of the reinforcement
provided in the soil at different layers by considering the equivalent stresses generated
due to friction at the soil-reinforcement interface. Ultimate bearing capacity value is
needed as a prerequisite to the analysis. The pressure settlement curves provided the
actual settlement of the footing directly for a given pressure intensity. The method
could be used for proportioning of rectangular footing resting on reinforced sand
satisfying the shear failure and settlement criteria. The analysis considers the effect of
the weight of the soil mass in determination of the stresses. Kondner’s two-constant
hyperbolic model had been used in the analysis. It was observed that the predicted and
model test results agreed well up to two-third of the ultimate bearing pressure. It was
suggested that under working stress conditions, the allowable bearing pressure could
be derived from ultimate bearing pressure using a factor of safety 2 to 3.
Maheshwari et al. (2004) presented a model for estimating the flexural
response of beam resting on reinforced beds with reinforcing elements such as
geogrids, which were idealized as beams with smooth surface characteristics. The
lower poor strata and upper dense soil were modeled using Winkler springs of
different stiffness. The effect of depth of placement of had been incorporated by
taking a surcharge load on the reinforcing elements. The governing differential
equations for the response of the beam were derived and closed-form analytical
solutions were obtained subjected to appropriate boundary and continuity conditions.
A particular case of the study identically matched with the solution provided by
Hetenyi for infinite beams on elastic foundation. Practically no change was observed
in the normalized deflection of the upper and lower beams when the normalized
length ratio of beams exceeded 1.5 for the range of parameters considered. The
normalized depth of placement of the lower beam had a significant effect on the
deflection response of beams. The normalized net deflection of the upper and lower
beam increased by 41% and 45% respectively by the increase in normalized depth
from 0.5 to 1.5. For the lower beam, the deflection at the edge of the beam increased
by the same ratio by which the depth of placement increased. The relative flexural
rigidity of the beam, R affected the deflection at the edges of the beam more than at
the centre. For the upper beam, at normalized length of 2.2, the deflection of the beam
behavior reversed. The net normalized deflection of the upper beam decreased by
14% at the centre while it decreased 60% for a reduction of R from 50 to 1. for the
lower beam, the deflection reduced by 63% at the edge of the beam for the same
reduction of R. the relative stiffness of soils, r, had a significant influence on the
normalized deflection of upper and lower beam. Unit weight of the upper granular fill
had a significant influence on the deflection of beams. The normalized deflection of
the beams could be reduced to the extent of 70-75% at the centre and by more than
95% at the edges of the beams. The maximum normalized positive bending moment
occurred at the centre for the upper and lower beams, whereas at the edge it was zero.
The maximum positive bending moment decreased by 46% and 51% for the upper
and lower beams respectively for a decrease in ration r from 20 to 1. The position of
the section of maximum negative bending moment shifted towards the edge of the
beam as the ratio r increased. Similar behavior was observed for the lower beam.
Saran et al. (2004) proposed a mathematical model for soil and reinforced soil
as composite material in a polynomial form which could be easily incorporated in a
nonlinear finite element algorithm. They carried out an investigation to determine the
physical properties of the soil, its stress-strain characteristics and the stress-strain
characteristics of the reinforced soil as a composite material. Triaxial tests on
reinforced soil were carried out to determine its nonlinear characteristics. The
poisson’s ratio of the soil and reinforced soil were also modeled for the finite element
analysis. It was observed that the poisson’s ratio was nearly independent of the
applied confining pressure.
Table 1.1 summarizes the literature presented in this section.
Table 1.1
Summary of literature review related to geosynthetic-reinforced soil system
(Theoretical Works)
Kondner 1963 Proposed a two constant hyperbolic form of stress-strain response for modeling of soil behavior for load-deformation analysis of foundations.
Duncan & Chang 1970
Developed a simple, practical procedure for representing the nonlinear, stress-dependent, inelastic stress strain behavior of soils, which could be conveniently used for Finite Element Analysis.
Desai 1971 Presented the use of a cubic spline equation to simulate stress strain curves defined by two measured parameters.
Binquet & Lee 1975
Carried out the analysis of the bearing capacity problem of a surface strip footing on a granular soil containing horizontal layers of tensile reinforcement, and proposed a failure hypothesis for the first time, which formed the very basis of bearing capacity analysis in the future.
Brown & Poulos 1981
Presented an analytical model to investigate the increase in bearing capacity and stiffness of a foundation due to the placement of reinforcement in the homogeneous soil layer, and soil bed overlying a cavity.
Giroud & Noiray 1981 Presented a method to calculate the required thickness of the aggregate layer and make a proper selection of the geotextile to be used in the unpaved road design.
Andrawes et al. 1982
Analyzed soil-geotextile systems by using finite element method and described the nature of the elements used to represent the soil-geotextile systems for the purpose of predicting the stress-strain behavior.
Gourc et al. 1982 Studied the bearing capacity of a two layer system (a cohesionless soil sub-base and a clay subgrade) under punching.
Prakash et al. 1984
Proposed a novel analysis to predict the pressure-settlement characteristics of footings using the hyperbolic stress-strain curves of soils as constitutive law.
Huang & Tatsuoka 1988
Predicted the bearing capacity in a level sandy ground with strip reinforcement by stability analysis by limit equilibrium method, for both the cases of short and long reinforcement.
Madhav & Poorooshasb 1988 Proposed a new foundation model element – the rough
membrane to represent the response of the geofabric.
Bourdeau 1989 Formulated a numerical model to assess the tensile membrane action in a two layer soil system reinforced by a geotextile.
Poran et al. 1989
Presented a design procedure based on finite element analysis which included a visco-plastic model for soils and special visco-elastic membrane elements to model geogrid behavior.
Floss & Gold 1990 Performed FEM analysis to predict the bearing and deformation behavior of the single reinforced two-layer system.
Sellmeijer 1990 Performed analytical studies to determine the behavior of a soil-geotextile-aggregate (SGA) system, based on combined membrane action and lateral restraint.
Poorooshasb 1991 Proposed a mathematical technique and developed an analytical procedure to predict the performance of heavily reinforced mats by weak subgrades.
Dixit & Mandal 1993 Applied variational method to determine the bearing capacity of the geosynthetic-reinforced soil.
Murthy et al. 1993 Carried out Limit equilibrium Analysis to study and evaluate the bearing capacity of a reinforced soil foundation.
Ghosh & Madhav 1994a
Developed a simple mathematical model to account for the membrane effect of a reinforcement layer on the load settlement response of a granular fill-soft soil foundation system.
Ghosh & Madhav 1994b Proposed a new mathematical model for the analysis of a reinforced foundation bed by incorporating the confinement effect of a single layer of reinforcement.
Ghosh & Madhav 1994c Proposed a new model for a reinforced shallow foundation bed by incorporating the ‘rough membrane’ element for single layer reinforcement.
Shukla & Chandra 1994a Studied the effect of prestressing the geosynthetic reinforcement on the settlement behavior of geosynthetic-reinforced granular fill-soft soil system.
Shukla & Chandra 1994b Proposed a foundation model to incorporate the rate of compressibility of the granular fill by attaching a layer of Winkler springs to the Pasternak shear layer.
Shukla & Chandra 1994c
Described a mechanical model for idealizing the settlement response of a geosynthetic-reinforced compressible fill-soft soil system, by representing each subsystem by commonly used mechanical elements such as stretched, rough, elastic membrane, Pasternak shear layer, Winkler springs and dashpot.
Burd 1995 Presented an analysis concerning the mechanics and design of unpaved reinforced roads built over soft clay.
Shukla & Chandra 1995
Proposed a mechanical foundation model for the analysis of a reinforced granular fill on a soft soil foundation system by the representing the geosynthetic reinforcement, the granular fill, and the soft foundation soil by a stretched rough membrane, a Pasternak shear layer, and Winkler springs respectively.
Zhao 1996
Presented a failure criterion for the reinforced soil composite. A slip-line method was described and the failure loads and stress characteristic fields for reinforced slopes, walls and foundations were calculated using the proposed slip-line method.
Yin 1997a Proposed a new one-dimensional mathematical model for modeling geosynthetic-reinforced granular fills over soft soils subjected to a vertical surcharge load.
Yin 1997b
Extended the developed one-dimensional foundation model for geosynthetic-reinforced granular fills over soft soil using a nonlinear constitutive model for the granular fill and a nonlinear spring model for the soft soil.
Michalowski 1998 Presented a kinematic approach of limit analysis in which a rigorous bound to the required strength of strength is sought.
Pitchumani & Madhav 1998
Studied the interaction mechanism of one and more pairs of inextensible reinforcing strips and the soil in reinforced foundation beds using an elastic continuum approach. Boussinesq’s and Mindlin’s solution wee integrated to evaluate the lateral soil displacements at the soil-strip interface.
Saran 1998
Proposed an analytical analysis to determine the pressure on a rectangular footing resting on reinforced sand for a given settlement for which the pressure on the same footing resting on unreinforced sand is known.
Shukla & Chandra 1998
Presented a simple mechanical modeling approach to study the settlement characteristics of geosynthetic-reinforced granular fill-soft soil system subjected to axi-symmetric load at any stage of consolidation of the soft subgrade.
Kotake et al. 1999 Performed FEM simulation on plane strain compression tests of dense sand reinforced with reinforcements with a wide range of stiffness.
Siddiquee et al. 1999
Performed FEM simulation of the bearing capacity characteristics of the strip footings on sand, and explained the scale effects observed in the model plane strain tests carried out simultaneously.
Peng et al. 2000
Performed FEM simulation of the bearing capacity characteristics of the strip footings on sand, and explained the scale effects observed in the model plane strain tests carried out simultaneously.
Yin 2000
Performed a comparative modeling study of reinforced beam on elastic foundation. Governing ordinary differential equations were derived for a reinforced Timoshenko beam on an elastic foundation.
Madhav & Pitchumani 2000
Proposed a method to predict the reduction in surface settlements due to strip form of reinforcements beneath a rectangular loaded area. Elastic continuum approach was used to solve the problem.
Fakher & Jones 2001
Presented a numerical simulation to model a layer of sand overlying a layer of geosynthetic reinforcement and super soft clay. It was suggested that bending stiffness should be considered while having earthworks on super soft clay.
Kotake et al. 2001
Simulated plane strain laboratory model tests using a nonlinear elasto-plastic finite element model (FEM) to investigate the bearing capacity characteristics of both the reinforced and unreinforced footing load and the associated reinforced mechanisms.
Dey 2002 Modified the dimensionless force curves proposed by (Binquet & Lee (1975 b)) by considering small
intervals of length (Δx) on the length of the tie breakage (X0) in the hypothetical formula.
Kumar & Saran 2003
Presented a method of analysis for calculating the pressure intensity for calculating the pressure intensity corresponding to a given settlement for a rectangular footing resting on a reinforced soil foundation.
Maharaj 2003 Conducted nonlinear two-dimensional finite element analysis for a strip footing on reinforced clay under plane strain condition.
Kumar et al. 2004 Proposed a method to obtain the pressure settlement characteristics of rectangular footings resting on reinforced sand based on constitutive laws of soils.
Maheshwari et al. 2004
Presented a model for estimating the flexural response of beam resting on reinforced beds with reinforcing elements such as geogrids, which were idealized as beams with smooth surface characteristics.
Saran et al. 2004
Proposed a mathematical model for soil and reinforced soil as composite material in a polynomial form which could be easily incorporated in a nonlinear finite element algorithm.
Few of the analytical models described above are shown as follows:
Figure 1.7 Madhav and Poorooshasb model (1988).
Figure 1.8 Ghosh and Madhav model (1994).
Figure 1.9 Shukla & Chandra model (1994b).
Figure1.10 Shukla & Chandra model (1994c).
Figure 1.11 Yin model (1997a).
Figure 1.12 Yin model (2000).
Figure 1.13 Maheshwari model (2004).
1.3 Motivation and scope of the present study
From the above literature review, it is observed that the bending stiffness of
the reinforcement is considered only in a few studies (Fakher & Jones, 2001, and
Maheshwari et al., 2004). However, the bending stiffness of the reinforcement may
produce a significant effect on the settlement behavior of the beams on reinforced
elastic foundations. For dealing with geosynthetic reinforced foundation system,
especially reinforced with geogrids, geocells and/or geomats, and if the underlying
soil layer is either a clayey soil with consistency ranging from medium to soft or a
sand layer with low relative density with high compressibility, then the bending
stiffness of the beams plays a significant role in determining the deflection behavior
of the foundation.
It is also noted from the above studies that the soil beneath the footing and
reinforcing beams are considered to be of uniform subgrade modulus, thus neglecting
the confining effect of the underlying soils. But, the soil lying near the centre of the
footing and the reinforcing beam, being subjected to higher confining pressure, would
provide more resistance to deflection in comparison to the soils away from centre.
Thus, it is expected that the discrete springs idealizing the foundation will have
maximum and minimum values of the stiffness respectively at the centre and the edge
of the footing and reinforcing beam and it is needed to be considered in the design.
It is also observed from the literatures that the settlement response of the
beams on reinforced foundation beds with underlying clayey soils had been carried
out as a function of consolidation ratio of the soft soil. The time dependency of the
settlement in such cases is obtained as an indirect effect of the consolidation. Thus, no
analytical work did produce the time-dependent settlement behavior of beams on
elastic foundation. Such an analysis is needed to be carried out to establish design
charts which can predict settlement of a reinforced foundation system as a direct
function of time. As such, in this thesis determination of the flexural behavior of
beams on reinforce sand or clay beds have been taken up. The organization of the
thesis dealing with these problems is detailed out as follows.
Chapter 2 deals with the modeling and analysis of a shallow strip footing
resting on the surface of a densely compacted granular fill and underlain by natural
poor granular deposit. A concentrated load is acting at the centre of the footing beam.
The reinforcing element is considered to offer bending resistance. The reinforcing
layer is placed at the interface of the soil layers. The soil layers are assumed to have a
variable subgrade modulus, maximum at the centre and minimum at the edges of the
beams.
Effects of various parameters on the settlement response of beams such as
depth of placement of reinforcing element, relative flexural rigidity of footing and
reinforcing beam, relative stiffness of compacted and loose granular layer, unit weight
of compacted granular layer, coefficient of friction and the nature of distribution of
subgrade modulus at the beam-soil interface were studied. The results are presented in
non-dimensional forms over a wide range of parameters.
Chapter 3 deals with the modeling and analysis of a shallow strip footing
resting on a sand bed underlain by a weak clayey soil with medium to soft
consistency. A concentrated load acts at the centre of the footing beam. The
reinforcing element is placed at the interface of the soil layers. The compacted
granular layer is idealized as Winkler springs. The weak clayey soil is modeled by
four-parameter Burger model. The granular layer is assumed to have a variable
subgrade modulus. Governing differential equations were derived for the problem,
and a numerical solution is presented by using finite difference technique. Effect of
various parameters on the settlement response of the footing and reinforcing beams
were studied. The parameters studied were depth of reinforcement below the footing,
relative flexural rigidity of footing and reinforcing beam, relative stiffness of soils,
unit weight of compacted granular fill, nature of distribution of subgrade modulus at
the beam-soil interface, friction coefficient, and relative ratio of stiffness coefficients
in Burger model, relative viscous coefficients in Burger model, and the variation in
time. Settlement response of beams with the lapse of time is presented in non-
dimensional form with a wide range of parameters.
The recommendations of the future work are given along with the respected
chapters.
CHAPTER 2
MODELING OF FOUNDATIONS ON REINFORCED SAND BEDS
WITH VARIABLE SUBGRADE MODULUS
2.1 Introduction
If the reinforcing element of a foundation bed is made up of geogrids, geomats
and/or geocells, concrete or metallic strips it may be necessary to take into account
their bending resistance for a realistic analysis of the same. Fakher and Jones (2001),
Maheshwari et al. (2004) made some studies in this direction and obtained solutions
using finite element method and analytical (closed form) technique respectively.
Using finite element method, Fakher and Jones (2001) simulated a layer of
sand overlaying a layer of geosynthetic reinforcement and super soft clay to study the
influence of the bending stiffness (flexural rigidity) of the reinforcement on the
bearing capacity of super soft clay. They assumed the soil layer to be homogeneous. It
was concluded that higher the reinforcement bending stiffness, higher was the bearing
capacity of the system.
Maheshwari et al. (2004) made use of the Hetenyi’s model to analyze the
problem assuming the stiffness of the springs idealizing the soil behavior to be
uniform.
But, the soil lying near the centre of the footing and the reinforcing beam,
being subjected to higher confining pressure, would provide more resistance to
deflection in comparison to the soils away from centre. Thus, it is expected that the
discrete springs idealizing the foundation will have maximum and minimum values of
the stiffness respectively at the centre and the edge of the footing and reinforcing
beam.
As such, in this chapter, the model as proposed by Maheshwari et al. (2004) to
find the response of foundations resting on a granular soil bed reinforced with geogrid
reinforcement is modified to account for the variation of the spring constants along
the length of the footing and the reinforcing beam. With the above modification, a
generalized procedure that has been developed to find the flexural response of the
foundation beam has been reported here.
2.2 Statement of the problem
A shallow strip footing resting on the surface of a densely compacted granular
fill, underlain by natural poor granular deposit is shown in Figure 2.1. A reinforcing
geogrid layer is provided at the interface of the densely compacted and the natural
poor granular media. Both the footing and the reinforcing layer are idealized as elastic
beams of flexural rigidity E1I1 and E2I2 respectively. The lengths of the footing and
reinforcing beams are 2l1 and 2l2 respectively. A concentrated load of magnitude Q
acts at the centre of the footing beam. The unit weights of the upper and lower soil
media are γ1 and γ2 respectively. The soil layers have subgrade modulus k1(x) and k2
(x) respectively. The variation of modulus of subgrade reaction is assumed to be non-
linear. Being deflected by the external loads, the reinforcing beam may experience a
resultant tensile force T (=μγ1H) due to the friction arising from the surrounding
granular media, where μ is the interfacial friction angle. To take care of the effect of
the granular media above the reinforcing layer, a uniform surcharge is considered all
over the length of the reinforcing layer. The primary aim of the present study is to find
the effect of the non-linearity of the spring constant on the deformation response of
the reinforced foundation system.
Figure 2.1 Definition sketch of the problem.
Figure 2.2 Proposed foundation model
2.3 Analysis 2.3.1 Assumptions
a) The problem considered is symmetrical both in terms of geometry and loading
conditions. Hence, only one half of the reinforced foundation system is
analyzed.
b) The subgrade modulus of both the compacted sand layer and the poor soil are
considered to be non-linear, having maximum and minimum values at the
centre and the edges of the beams respectively.
c) The footing and the reinforcing beams were considered to be rough.
d) Both the compacted and the poor granular media are idealized by Winkler
springs.
2.3.2 Governing differential equations
Due to symmetry of the model, only one half (x≥0) is analyzed. The deflection
co-ordinates are denoted as y1 and y2 respectively for the footing and reinforcing
beam. For the loading condition shown in Figure 2.1, the governing differential
equations for the footing and the reinforcing beam can be written as follows,
1121141
4
11 0),()( lxxkyypdx
ydIE ≤≤−−=−= (1)
111221112122
2
42
4
22 0,)()( lxykykkHppHdx
ydT
dxyd
IE ≤≤++−=−−=− γγ (2)
and,
212212122
2
42
4
22 , lxlykHpHdx
ydTdx
ydIE ≤≤−=−=− γγ (3)
where,
111 02 lxHlT ≤≤= μγ (3a)
21121 )(2 lxlllHT ≤≤−= μγ (3b)
The variation of the subgrade modulus of the densely compacted granular fill along
the length of the footing beam is assumed to be of parabolic distribution as,
12
1211101 0)( lxxkxkkxk ≤≤−−= (3c)
Similarly, for the natural loose granular deposit, variation of subgrade modulus along
the length of reinforcement beam is assumed as,
22
2221202 0)( lxxkxkkxk ≤≤−−= (3d)
Equation (1) can be written in the form as
141
4
2121110
1114
14
1
112 )(
ydx
ydxkxkk
IEydx
ydk
IEy +−−
=+= (4)
Equation (2) can be rearranged and modified to be written in the form as
122041
4
110
11211012
22
42
4
22 ykkdx
ydfk
IEkk
kkHdx
ydTdx
ydIE nnr
nn −⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+−=− γ (5)
Substituting equation (4) in equation (5) and after subsequent rearranging results in
H
ykkdx
ydf
kIE
kk
kkdx
ydT
dxydf
ldxydf
ldxydf
kIET
dxydf
klIEIE
dxyd
fldx
ydf
ldxyd
fldx
ydf
kIEIE
nnr
nn
nnnn
nnnn
1
12204
14
110
1121102
12
41
4
322
51
5
22
61
6
110
1141
4
510
42
1122
51
5
432
61
6
322
71
7
22
81
8
110
1122
1366
1
111
γ=
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
+⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛++−
⎥⎦
⎤⎢⎣
⎡++−
⎥⎥⎦
⎤
⎢⎢⎣
⎡++
⎥⎦
⎤⎢⎣
⎡+++
(6)
where, the functions are given as follows
n
n kf
11
1= (6a)
( )
21
12112
24
n
nn
n kxkk
f+
= (6b)
( )31
22121211
21112
33312
n
nnnnn
n kxkxkkkkf +++
= (6c)
( ) ( ){ }
41
3312
22121112
2111212
21111
4
961449624
n
nnnnnnnnn
n kxkxkkxkkkkkk
f+++++
= (6d)
( ) ( )( )
51
4412
331211
221112
212
122
1112112
12122
114
11
5
120240240
120324
n
nnnnnn
nnnnnnnn
n kxkxkkxkkk
xkkkkkkkk
f⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++++
++++
= (6e)
212111 1 xkxkk nn
n −−= (6f)
where,
10
1111 k
kk n = ,10
1212 k
kk n = and 10
11 k
kk n = (6g)
222212 1 xkxkk nn
n −−= (6h)
where,
20
2121 k
kk n = , 20
2222 k
kk n = and 20
22 k
kk n = (6i)
2.3.3 Non-Dimensional form of governing equation Equation (6) can be written in the non-dimensional form as
RlMHyF
dxyd
Fdx
ydF
dxyd
Fdx
ydF
dxyd
Fdx
ydF
n
nn
nnnn 4''1
'172
'1
2
64
'1
4
5
5
'1
5
46
'1
6
37
'1
7
28
'1
8
1
γ=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+++
+++
(7)
where,
The coefficients F1 to F7 are given in Appendix I and
2lxxn = , 3
2
111'1 Ql
IEyy = ,
Ql 2
21'1
γγ = ,
1
'lHH = ,
1
2
Rl
M = , 2
1
llln = and
22
11
IEIE
R =
(7a)
In a similar way, the non-dimensional forms equations (4) and (3) are respectively
'14
'1
4
14
'2
11 yRdx
ydkRM
ynn
+= , 0≤x≤l1 (8)
and,
'2
42
4''12
'2
24'"
'4
'2
44 )1(2 yMkRlH
dxyd
kMllHdx
ydR nnn
nrnn
nn −=−− γμγ , l1≤x≤l2 (9)
where, the non-dimensional parameters are given as,
32
222'2 Ql
IEyy = , 20
10
kk
kr = , 4
10
111 k
IER = , 4
20
222 k
IER = ,
10
1"1 k
γγ = , and
2
1
RRRn = (9a)
2.3.4 Boundary and continuity conditions
For the footing beam, at the point of application of load, i.e. at x=0, slope of
the deflected shape of the beam is zero and the shear force is Q/2. At the edge of the
footing beam, i.e. at x=l1, the bending moment and the shear force are zero, as the
beam end is free. For the reinforcing beam, which is within the foundation soil, at
point x=0, slope of the deflected shape of the beam and the shear force are zero; and at
x=l2, bending moment and shear force are zero. For the reinforcing beam, at x=l1, the
continuity of deflection, slope, bending moment and shear force are duly
incorporated.
The boundary conditions in their non-dimensional form can be presented as
For footing beam,
at 0=x , slope is zero and the shear force is half of the load applied.
i.e. 0'1 =ndx
dy ................................ (10a) 21
3
'1
3
=ndxyd .......................... (10b)
at 1lx = , bending moment and shear force is zero.
i.e. 02
'1
2
=ndxyd ............................. (10c) 0'
'1
3
'1
3
=−nn dx
dyT
dxyd ……...... (10d)
For reinforcement beam,
at 0=x , slope and the shear force is zero.
i.e. 0'2 =ndx
dy ................................ (10e) 03
'2
3
=ndxyd ........................... (10f)
at 2lx = , bending moment and shear force is zero.
02
'2
2
=ndxyd .............................. (10g) 0'
'2
3
'2
3
=−nn dx
dyTdx
yd ........... (10h)
where,
22
22'
IElT
T = (10i)
The settlement response of the reinforcement beam is governed by two
differential equations over the lengths 0≤x≤l1 and l1≤x≤l2. Thus, at the length x=l1 of
the reinforcing beam, continuity is established in terms of deflection, bending moment
and shear force. Thus at an infinitesimal distance ε in the left and right of the distance
x=l1 on the reinforcing beam, the deflection, bending moment and shear force on the
reinforcing beam are equal, and the continuity is established by the following
conditions.
Deflection is equal i.e. εε +=− xyxy 22 (11a)
Slope is equal i.e. εε +
=− xdx
dy
xdxdy 22 (11b)
Bending moment is equal i.e. εε +
=− xdx
yd
xdxyd
22
2
22
2
(11b)
and,
Shear force is equal i.e. εε +
−=−
−xdx
dyIE
Tdx
yd
xdxdy
IET
dxyd 2
223
23
2
223
23
(11c)
2.3.5 Method of solution: Finite Difference Method
The differential equations governing the settlement response of the footing and
the reinforcing beam are discretized using the finite difference technique. The half of
the footing is divided into nb nodes (i.e. i=1, 2, 3, 4………, nb). Thus, using central
difference scheme, equation (7) can be written in the following form as,
RlMhyCyCyCyC
yCyCyCyCyCn
iiiiiiii
iiiiiiiiii 4''1'
44'
33'
22'
11
''11
'22
'33
'44 γ=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++++
++++
++++++++
−−−−−−−− (12)
where,
The coefficients Ci-4 to Ci+4 are given in Appendix I.
The reinforcing beam is divided into nr nodes (i=1, 2, 3, ...nb, nb+1, .....nr),
where nb is the number of nodes up to the length l1, and beyond l1 and up to the length
l2, node number ranges from nb to nr. The deflection profile of the reinforcing beam is
governed by the equations (8) and (9), which are subsequently written in the finite
difference form using central difference scheme.
To establish continuity at node nb of reinforcing beam, equation (8) is applied
up to (nb-4)th node using central difference, and thereafter backward difference
scheme is applied to obtain the deflection values up to the node nb
Equation (12) when applied at the nodes of the footing beam, with due
incorporation of the boundary conditions, provided a set of linear equations, which is
solved by the Gauss-Seidel iterative technique to obtain the deflection profile of the
footing beam. Once the deflected shape is determined, from the same, the slope,
bending moment, shear force and contact pressures can be computed.
2.4 Results and Discussions
A computer program had been written in C language and the following studies
were carried out in sequence.
1. Convergence of the numerical solution
2. Correctness of the developed program and the solution obtained
3. Parametric studies
2.4.1 Convergence Study
Convergence study was made by decreasing the size of the element, dividing
the footing beam into a mesh of finite segments. The deflection at the center of the
footing beam was recorded for the decreasing sequence of mesh size to check its’
effect on the solution. The computations were made considering both uniform and
varying subgrade modulus. For the typical combinations of the relative stiffness of the
soil and the relative flexural rigidity of the footing as well as the reinforcing beams,
the effect of the decreasing mesh size ranging from 0.2 to 0.005714 on the normalized
deflection of the footing beam at the mid-span are shown in Figure 2.3 and Figure 2.4
respectively for the uniform and variable subgrade modulus. It is seen that the
numerical solution effectively converges for variable and uniform subgrade modulus
when the mesh size (h/l1) lies in the region of 0.02 to 0.008 and 0.01 to 0.04
respectively.
0.010
0.030
0.050
0.070
0.090
0.110
0.130
0.150
0.170
0.190
0.210
0.001 0.01 0.1 1
Non-dimensional mesh size (h/l1)
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
R=20, kr=5 R=5, kr=5 R=5, kr=20R=10, kr=5 R=10, kr=10 R=5, kr=10
Stable
l n = 0.67γ' = 0.6H' = 0.35μ = 0
Figure 2.3 Convergence study of deflection of footing beam with uniform subgrade modulus.
0.020
0.070
0.120
0.170
0.220
0.270
0.001 0.01 0.1 1
Non-dimensional mesh size (h/l1)
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
R=20, kr=5 R=5, kr=5 R=5, kr=20
R=10, kr=5 R=10, kr=10 R=5, kr=10
Stable
l n = 0.67γ' = 0.6H' = 0.35μ = 0k n = 0.5
Figure 2.4 Convergence study of deflection of footing beam with variable subgrade modulus.
2.4.2 Correctness of the developed program and the solution obtained
2.4.2.1 Comparison with Hetenyi’s Model
Based on the above convergence study, the beam was discretized taking the
mesh size (h/l1) as 0.008. In order to compare the present solution with that of
Hetenyi’s (1946), results were obtained choosing the length of the footing and
reinforcing beams to be equal and neglecting the surcharge on the reinforcing beam.
The comparison is shown in Figure 2.5. Deflection profile of the footing and the
reinforcing beam were found out considering the relative flexural rigidity of beams
and the relative stiffness of the soils to be 10 and 5 respectively. The numerical
solution is observed to be in excellent agreement with the Hetenyi’s solution, the
maximum variation being 3% to 5%. To arrive at the desired solution, several trial
solutions had to be made, adjusting the length of both the beams equal and large
enough so that they may be considered to be long enough to be called as infinite
beams and analogous to that of Hetenyi’s model,. It is observed from the present
study that if the l/Rc (Rc is the characteristic length of the beam) ratio for the beam
exceed 6.7, the beam behaved as a long beam.
2.4.2.2 Comparison with previous research
Figure 2.6 shows the comparison of the deflection profile of the footing
beam to that of the unreinforced case, where the footing beam is considered to be
placed directly on the poor soil. It is observed that incorporation of reinforcement
reduced significantly (by about 16 %) the maximum settlement. The deflection profile
of the footing and reinforcing beam with the degenerated case of uniform subgrade
modulus when compared with the solution reported by Maheshwari et al. (2004)
shows an excellent agreement, the deviation being less than 1%. The figure also
shows the deflection profile of the footing and reinforcing beam using variable
subgrade modulus and significant deviation of 7.5% to 15% is observed in terms of
settlement in comparison to the values obtained with uniform subgrade modulus.
Thus, consideration of variable subgrade reaction has significant influence on the
settlement response of the reinforced foundation bed over and above the same
obtained with uniform modulus of subgrade reaction. As such, its effect should not be
neglected.
-0.001
0
0.001
0.002
0.003
0.004
0.005
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre of beam
Nor
mal
ized
def
lect
ion
Hetenyi (1946) Present study
Footing beam
Reinforcing beam
R = 10kr = 10μ = 0
Figure 2.5 Comparison of settlement profile of footing and reinforcing beam with Hetenyi’s Model.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60
Distance from centre of the beam (m)D
efle
ctio
n (m
m)
Maheshwari et al. footing (2004) Maheshwari et al. reinforcement (2004)
Present study footing (variable) Present study reinforcement (variable)
Unreinforced Present study footing (uniform)
Present study reinforcement (uniform)
Footing (unreinforced case)
Maheshwari and Present
l 1 = 2 m, l 2 = 3m
k 10 = 250 MN/m 3
k 20 = 50 MN/m 3
E 1 I 1 = 50 MNm 2
E 2 I 2 = 2.5 MNm 2
P = 100 kNγ = 15 kN/m 3
μ = 0k n = 0.5H = 0.35 m
Figure 2.6 Comparative study of present solution with previous solutions.
2.4.3 Parametric study
To study the effect of various parameters on the flexural response of the
foundation as well as the reinforcing beam, parametric studies were conducted. The
ranges with in which the various parameters were changed are shown in Table 2.1.
The details of the study are reported under different subsections as follows.
Table 2.1 Range of non-dimensional parameters considered in the study.
Sl. No. Non-dimensional parameters Symbol Range
1 Depth of placement of reinforcement below the footing beam H’ 0.5 – 2.0
2 Relative flexural rigidity of beams R=E1I1/E2I2 5 – 250
3 Relative stiffness of soils kr = k10/k20 5 – 300
4 Unit weight of granular fill γ’ 0.5 – 2.5
5 Non-dimensional shape constants kn 0 - ∞
6 Coefficient of friction μ 0.5 – 1.0
2.4.3.1 Effect of the depth of placement of reinforcement (H’)
Figure 2.7 and 2.8 depicts the effect of the depth of placement of
reinforcement on the normalized deflection profile of footing and reinforcing beam
respectively. The depth of placement of reinforcement is varied from 0.5 to 2l1, the
width of the footing beam. The reinforcement is placed at the interface of the
compacted sand layer and the poor/loose soil deposit. The reinforcement is used to
improve the settlement characteristics of the foundation soil. The settlement of both
the footing and reinforcing beams are seen to continually increase with the increase of
the depth of placement of reinforcement. The normalized settlement of the footing and
reinforcing beam increased by 80 % and 144 % respectively for increase in the
normalized depth of placement from 0.5 to 2.0.
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
H'=0.5 H'=1.0 H'=1.5 H'=2.0
R = 20kr = 5γ' = 0.8μ = 0.5k n = 0.5
Figure 2.7 Normalized deflection of footing beam for variation in depth of placement of reinforcement below the footing beam (H’).
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0.011
0.012
0 0.2 0.4 0.6 0.8 1 1.2Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
H'=0.5 H'=1.0 H'=1.5 H'=2.0
R = 20kr = 5γ' = 0.8μ = 0.5k n = 0.5
Figure 2.8 Normalized deflection of reinforcing beam for variation in depth of placement of reinforcement below the footing beam (H’).
2.4.3.2 Effect of relative flexural rigidity of beams (R)
Figure 2.9 and 2.10 shows the effect of variation of relative flexural rigidity
of beams on the normalized deflection profile of footing and reinforcing beams. The
value of R is varied from 5 to 250. The increase in value of R signifies that the footing
beam is becoming more rigid with respect to the reinforcing beam, and thus would
offer increased resistance to settlement. The figure also shows that the settlement of
the footing beam decreases with the increase in R. It is observed that as the value of R
becomes 150, the variation of settlement of footing beam becomes negligible. Similar
observation is made with the reinforcing beam. As the relative flexural rigidity of
beams increases, the settlement of the reinforcing beam decreases rapidly. As R
becomes 100, the variation in deflection becomes negligible and it barely shows any
deflection throughout the length of the reinforcing beam. The normalized settlement
of the footing and reinforcing beam decreased by 26 % and 100 % due to the increase
the value of relative flexural rigidity of footing and reinforcing beam from 5 to 250.
2.4.3.3 Effect of relative stiffness of soils (kr)
Figure 2.11 and 2.12 shows the effect of variation of relative stiffness of
upper compacted soil layer and the lower poor/loose soil layer on the deflection
profile of the footing and reinforcing beam respectively. The value of kr is varied
from 5 to 300. The increase in the value of kr signifies that the upper compacted
granular layer is becoming stiffer, and that it will offer more resistance to deflection.
The figure also shows that the settlement of the footing beam decreases as the value of
kr increases, and becomes negligible as the kr value becomes 200. Similar observation
is made with the reinforcing beam. The normalized deflection of the footing and
reinforcing beams decreases by 26 % and 28 % respectively as the kr increases from 5
to 300.
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
R=5 R=10 R=20 R=50
R=100 R=150 R=200 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35k n = 0.5
Figure 2.9 Normalized deflection of footing beam for variation in relative flexural rigidity of beams (R).
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
R=5 R=10 R=20 R=50 R=100 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35k n = 0.5
Figure 2.10 Normalized deflection of reinforcing beam for variation in relative flexural rigidity of beams (R).
0.08
0.085
0.09
0.095
0.1
0.105
0.11
0.115
0.12
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
kr=5 kr=10 kr=20 kr=50
kr=100 kr=200 300
R = 20γ' = 0.8H' = 0.35μ = 0.5k n = 0.5
Figure 2.11 Normalized deflection of footing beam for variation in relative stiffness of soils (kr).
0.004
0.0042
0.0044
0.0046
0.0048
0.005
0.0052
0.0054
0.0056
0.0058
0.006
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
kr=5 kr=10 kr=20 kr=50
kr=100 kr=200 kr=300
R = 20γ ' = 0.8μ = 0.5H' = 0.35k n = 0.5
Figure 2.12 Normalized deflection of reinforcing beam for variation in relative stiffness of soils (kr).
2.4.3.4 Effect of unit weight of compacted granular layer (γ’)
Figure 2.13 and 2.14 shows the effect of variation of unit weight of
compacted granular layer on the deflection profile of footing and reinforcing beams
respectively. The non-dimensional unit weight of the compacted granular layer is
varied from 0.5 to 2.5. Higher unit weight produces greater surcharge on the
reinforcing beam. The settlement profile of the footing beam increases with the
increase in unit weight of compacted soil. Similar is the observation with the
reinforcing beam. The normalized settlement of the footing and reinforcing beams
increased by over 100 % each for increase in γ’ from 0.5 to 2.5. Thus surcharge
weight has a profound influence on the settlement of beams on reinforced elastic
foundation.
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
γ'=0.5 γ'=1.5 γ'=2.5
R = 20kr = 5μ = 0.5H' = 0.35k n = 0.5
Figure 2.13 Normalized deflection of footing beam for variation in unit weight of compacted granular layer (γ’).
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
γ'=0.5 γ'=1.5 γ'=2.5
R = 20kr = 5μ = 0.5H' = 0.35k n = 0.5
Figure 2.14 Normalized deflection of reinforcing beam for variation in unit weight of compacted granular layer (γ’).
2.4.3.5 Effect of variation of parabolic constants (kn)
Figure 2.15 and 2.16 shows the effect of the variation of parabolic constants
on the deflection profile of the footing and reinforcing beam respectively. The
subgrade modulus is assumed to vary parabolically along the length of the beam as
mentioned earlier. The ratio kn is defined as the ratio of k11/k12 or k21/k22. The variation
of kn actually signifies the nature of distribution of subgrade modulus along the length
of the beam. kn=∞ indicates linear distribution of subgrade modulus, and succeeding
lower values of kn indicates the variation of subgrade modulus along the length of the
beam with a higher curvature. The ratio kn is varied from 0 to ∞. It is observed that as
the ratio kn increases, the deflection of both the footing and the reinforcing beam
increases. The normalized settlement of the footing and reinforcing beam increased by
37 % and 14 % respectively due to the increase in value of kn from 0 to ∞. Thus, it is
observed that the distribution of subgrade modulus influences the flexural response of
the beams on the reinforced elastic foundations. Thus, it should be considered in the
analysis and design of reinforced foundations.
2.4.3.6 Effect of variation of friction coefficient (μ)
Figure 2.17 and 2.18 depicts the effect of variation of friction coefficient on
the settlement response of the footing and reinforcing beam. It is observed that
variation of μ has a very little effect on the normalized deflection profile of the beams.
Friction coefficients of 0.5 and 1.0 were considered in the study. Maximum deviation
in the normalized deflection at the centre of the footing and reinforcing beam was
observed to be less than 1.0 % for variable subgrade modulus. Thus, it is observed
that the friction have only negligible effect on the flexural response of the beams on
reinforced elastic foundation when the flexural rigidity of the beams are considered in
the analysis.
0.008
0.012
0.016
0.02
0.024
0 0.2 0.4 0.6 0.8 1 1.2Normalized distance from centre of beam
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
kn=0 kn=0.25 kn=1.0
kn=1.5 kn=4.0 kn=9.0
kn=infinity
R = 20kr = 5γ' = 0.8μ = 0.5H' = 0.35
Figure 2.15 Normalized deflection of footing beam for variation in parabolic constants (kn).
0.0017
0.0018
0.0019
0.002
0.0021
0.0022
0.0023
0.0024
0.0025
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre of beam
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
kn=0 kn=0.25 kn=1.0
kn=1.5 kn=4.0 kn=9.0
kn=infinity
R = 20kr = 5γ' = 0.8μ = 0.5H' = 0.35
Figure 2.16 Normalized deflection of reinforcing beam for variation in parabolic constants (kn).
0.102
0.104
0.106
0.108
0.11
0.112
0.114
0.116
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
μ=0.5 μ=1.0
R = 20kr = 5γ' = 0.8H' = 0.35k n = 0.5
Figure 2.17 Normalized deflection of footing beam for variation in coefficient of friction (μ).
0.0051
0.0052
0.0053
0.0054
0.0055
0.0056
0.0057
0.0058
0.0059
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
μ=0.5 μ=1.0
R = 20kr = 5γ' = 0.8H' = 0.35k n = 0.5
Figure 2.18 Normalized deflection of reinforcing beam for variation in
coefficient of friction (μ). 2.4.3.7 Typical variation of normalized bending moment diagram
Figure 2.19 and 2.20 shows the typical variation of normalized bending
moment of the footing and reinforcing beams respectively for the variation in relative
flexural rigidity of footing and reinforcing beams (R). It is observed that the
maximum negative bending moment occurs at the centre of the footing and
reinforcing beam and decreases by 67 % and 58 % respectively with the value of R
increasing from 5 to 250. The bending moment at the edge of the beams is zero.
2.4.3.8 Typical variation in normalized shear force diagram
Figure 2.21 and 2.22 shows the typical variation of normalized shear force
of the footing and reinforcing beams respectively for the variation in relative flexural
rigidity of footing and reinforcing beams (R). It is observed that the maximum
positive shear force occurs at the centre of the footing beam and it has a constant
normalized value of 0.5. As the value of R increases, a positive shear force region
develops in the footing beam. However, the shear force diagram does not show a
major variation with the change of R. In case of reinforcing beam, the shear force at
the centre is zero. The maximum positive shear force occurs near the centre of the
footing beam, and it decreases by 25 % with the increase in value of R from 0 to 250.
The maximum negative shear force occurs at a point below the edge of the footing
beam and it increases by 21 % with the increase in value of R from 0 to 250.
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized distance from centre
Nor
mal
ized
ben
ding
mom
ent
for
foot
ing
beam
R=5 R=20 R=50 R=100
R=150 R=200 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35k n = 0.5
Figure 2.19 Typical normalized bending moment diagram of footing beam for variation in relative flexural rigidity of beams (R).
-0.004
-0.0035
-0.003
-0.0025
-0.002
-0.0015
-0.001
-0.0005
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized distance from centre
Nor
mal
ized
ben
ding
mom
ent
for
rein
forc
ing
beam
R=5 R=10 R=20 R=50
R=100 R=150 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35k n = 0.5
Figure 2.20 Typical normalized bending moment diagram of reinforcing beam for variation in relative flexural rigidity of beams (R).
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized distance from centre
Nor
mal
ized
shea
r fo
rce
for
foot
ing
beam
R=5 R=50 R=100 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35k n = 0.5
Figure 2.21 Typical normalized shear force diagram of footing beam for variation in relative flexural rigidity of beams (R).
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized distance from centre
Nor
mal
ized
shea
r fo
rce
for
rein
forc
ing
beam
R=5 R=10 R=20 R=50
R=100 R=150 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35k n = 0.5
Figure 2.22 Typical normalized shear force diagram of reinforcing beam for variation in relative flexural rigidity of beams (R).
2.5 Conclusion
From the above studies, it is seen that the nature of distribution of confining
pressure on the beam-soil interface has a significant effect on the settlement response
of a footing placed on a reinforced soil bed. Observation of the above results indicates
that considering variable subgrade reaction, there is a significant deviation of 5%-15%
over and above the same obtained with uniform subgrade reaction. Parametric studies
indicated that the settlement response of beams is influenced by the shape and nature
of distribution of subgrade reaction at the beam-soil interface. It was observed that
variation of coefficient of friction, μ did not significantly affect the deflection profile
of the beam. Maximum deviation of less than 1 % was observed in the normalized
deflection of the centre of the footing for the values of μ considered in the study. The
numerical solution was found to be in excellent agreement with the closed form
solution using the Hetenyi’s model, and also with the solutions previously reported by
Maheshwari et al. (2004), in both cases the deviation being less than 1%.
Thus, the above study indicate that the effect of distribution of confining
pressure on the settlement response of a footing placed on a reinforced soil bed should
be duly incorporated in the design of a reinforced soil foundation. Based on the
studies conducted above, the following conclusions can be drawn for the range of
parameters considered:
(1) The numerical solution effectively converged for uniform and variable
subgrade modulus when the mesh size (h/l1) lies in the region of 0.02 to 0.008
and 0.01 to 0.04 respectively.
(2) The numerical solution is observed to be in excellent agreement with the
Hetenyi’s solution, the maximum variation being 3% to 5%.
(3) It is observed from the present study that if the l/Rc (Rc is the characteristic
length of the beam) ratio for the beam exceed 6.7, the beam behaved as a long
beam.
(4) The normalized settlement of the footing and reinforcing beam increased by
80 % and 144 % respectively for increase in the normalized depth (H’) of
placement from 0.5 to 2.0.
(5) The normalized settlement of the footing and reinforcing beam decreased by
26 % and 100 % due to the increase the value of relative flexural rigidity of
footing and reinforcing beam (R) from 5 to 250. It is also observed that the
change in normalized settlement becomes negligible beyond a value R = 100
for both the footing and reinforcing beams.
(6) The normalized deflection of the footing and reinforcing beams decreases by
26 % and 28 % respectively as the kr increases from 5 to 300. It is also
observed that the change in normalized settlement becomes negligible beyond
a value kr = 200 for both the footing and reinforcing beams.
(7) The normalized settlement of the footing and reinforcing beams increased by
over 100 % each for increase in γ’ from 0.5 to 2.5. Thus surcharge weight has
a profound influence on the settlement of beams on reinforced elastic
foundation.
(8) The normalized settlement of the footing and reinforcing beam increased by
37 % and 14 % respectively due to the increase in value of kn from 0 to ∞.
(9) Maximum deviation in the normalized deflection at the centre of the footing
and reinforcing beam was observed to be less than 1.0 % for variable
subgrade modulus for the values of μ considered (0.5 & 1.0). Thus, it is
observed that the coefficient of friction have only negligible effect on the
flexural response of the beams on reinforced elastic foundation when the
flexural rigidity of the beams are considered in the analysis.
(10) It is observed that the maximum negative bending moment occurs at the
centre of the footing and reinforcing beam and decreases by 67 % and 58 %
respectively with the value of R increasing from 5 to 250. The bending
moment at the edge of the beams is zero.
(11) The maximum positive shear force occurs near the centre of the footing beam,
and it decreases by 25 % with the increase in value of R from 0 to 250. The
maximum negative shear force occurs at a point below the edge of the footing
beam and it increases by 21 % with the increase in value of R from 0 to 250.
The shear force at the edges of the beams is equal to zero.
2.6 Scope of further work
The model proposed in this section of the thesis can be extended to include
various sorts of geotechnical problems dealing with reinforced foundations. Figure
2.23 shows the sketch of a railroad track where the foundation beam is subjected to a
rolling point load. Figure 2.24 shows the sketch of a combined footing where the
footing beam is subjected to two concentrated loads at the edges. Figure 2.25 shows a
railway tie where the reinforced foundation system is subjected to two rolling point
loads. Figure 2.26 shows the sketch of a surface water tank where the reinforced
foundation system is subjected to two concentrated loads at the edges representing the
wall thrust, a uniformly distributed load representing the water pressure and two
moments at the edges generated due to the water thrust at the walls. Such problems
can be easily dealt with the current model with required modifications.
Figure 2.23 Definition sketch of a railroad track.
Figure 2.24 Definition sketch of a combined footing.
Figure 2.25 Definition sketch of a railway tie.
Figure 2.26 Definition sketch of a surface water tank.
CHAPTER 3
MODELING OF FOUNDATIONS ON A COMPACTED SAND
BED UNDERLAIN BY A WEAK CLAY STRATA WITH
REINFORCEMENT PLACED AT THE INTERFACE
3.1 Introduction
In Chapter 2, a generalized method of analysis of a foundation resting on a
compacted sand bed overlying a natural loose sand bed with the reinforcement placed
at their interface has been developed and presented. In the present chapter this has
further been extended where in the underlying soil strata is composed of soft to
medium clay instead of a loose sand stratum. As such, for accounting the visco-elastic
clay behavior Burger’s 4-element model is adopted in this chapter instead of
Winkler’s model representing elastic behavior of the sand beds as was done in the
previous chapter. It has been pointed out earlier that for analyzing such problems
generally the settlement at any time is estimated as the total consolidation settlement
(as estimated by using Terzaghi’s approach) multiplied by the degree of consolidation
and the same is used in the analysis. Thus the time effect is indirectly taken in to
account. But, by considering Burger’s model to represent the geo-mechanical
behavior of the clay stratum time effect is taken directly in the analysis. The details of
the development of analysis procedure, its’ validation has been presented in this
chapter as follows.
3.2 Statement of the problem
A shallow strip footing resting on the surface of a densely compacted granular
fill, underlain by weak clayey deposit of medium to soft consistency is shown in
Figure 3.1. A reinforcing geogrid layer is provided at the interface of the densely
compacted and the natural weak clay strata. Both the footing and the reinforcing layer
are idealized as elastic beams of flexural rigidity E1I1 and E2I2 respectively. The
lengths of the footing and reinforcing beams are 2l1 and 2l2 respectively. A
concentrated load of magnitude Q acts at the centre of the footing beam. The unit
weights of the upper and lower soil media are γ1 and γ2 respectively. The upper
compacted granular layer is idealized as Winkler springs and has a subgrade modulus
of k1(x). The variation of modulus of subgrade reaction is assumed to be non-linear.
The underlying weak clayey soil of soft to medium consistency is idealized with Four-
element visco-elastic Burger model. Being deflected by the external loads, the
reinforcing beam may experience a resultant tensile force T (=μγ1H) due to the
friction arising from the surrounding granular media, where μ is the interfacial friction
angle. To take care of the effect of the granular media above the reinforcing layer, a
uniform surcharge is considered all over the length of the reinforcing layer. The
primary aim of the present study is to find the effect of the non-linearity of the spring
constant and the time dependent deformation response of the reinforced foundation
system.
Figure 3.1 Definition sketch of the problem.
Figure 3.2 Proposed foundation model for the present study.
3.3 Analysis
3.3.1 Assumptions
e) The problem considered is symmetrical both in terms of geometry and loading
conditions. Hence, only one half of the reinforced foundation system is
analyzed.
f) The subgrade modulus of the compacted sand layer is considered to be non-
linear, having maximum and minimum values at the centre and the edge of the
beam respectively.
g) The footing and the reinforcing beams were considered to be rough.
h) The compacted granular media is idealized by Winkler springs.
i) The underlying weak clay layer with medium to soft consistency is idealized
by Four-element Burger model.
3.3.2 Governing differential equations
Due to symmetry of the model, only one half (x≥0) is analyzed. The deflection
co-ordinates are denoted as y1 and y2 respectively for the footing and reinforcing
beam. For the loading condition shown in Figure 3.1, the governing differential
equations for the footing and the reinforcing beam can be written as follows,
1121141
4
11 0),()( lxxkyypdx
ydIE ≤≤−−=−= (1)
1112112122
2
42
4
22 0,)( lxykpHppHdx
ydTdx
ydIE ≤≤+−=−−=− γγ (2)
and,
212122
2
42
4
22 lxlpHdx
ydTdx
ydIE ≤≤−=− γ (3)
where,
111 02 lxHlT ≤≤= μγ (3a)
21121 )(2 lxlllHT ≤≤−= μγ (3b)
)()( 1211 xkyyp −= (3c)
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
−−+−+
=−−− tBtBtB e
BBtAe
BA
eA
yp
111 111 211
01
12
22 (3d)
and, 1
21
12
1221
21
21
20 ,1,11,
ηηηηηηηkbB
kbAkbAkbA ==++== (3e)
where,
η1, η2, kb1 and kb2 are the viscous and elastic elements of the four-element
Burger model, which are shown as follows.
Figure 3.3 Four Element Burger model
The contact pressure p2 acting at the base of the reinforcing beam is derived
from the Four-element Burger model given in details in Appendix II.
The variation of the subgrade modulus of the densely compacted granular fill
along the length of the footing beam is assumed to be of parabolic distribution as,
12
1211101 0)( lxxkxkkxk ≤≤−−= (4)
Equation (1) can be written in the form as
kb1
kb2 η1
η2
141
4
2121110
1114
14
1
112 )(
ydx
ydxkxkk
IEy
dxyd
kIE
ynn
+−−
=+= (5)
Equation (2) can be rearranged as follows
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−+−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+
−+=−−−− tBtBtB e
BBtAe
BA
eA
ydx
ydk
IE
ykHdx
ydT
dxyd
IE111 111 2
110
1
12
141
4
1
11
11122
2
42
4
22 γ
(6)
Substituting equation (5) in equation (6), we get,
H
ydx
ydk
IEFyk
dxyd
T
dxydf
ldxydf
ldxydf
kIET
dxydf
klIEIE
dxyd
fldx
ydf
ldxyd
fldx
ydf
kIEIE
nnnn
nnnn
1
141
4
1
11112
12
41
4
322
51
5
22
61
6
110
1141
4
510
42
1122
51
5
432
61
6
322
71
7
22
81
8
110
1122
1366
1
111
γ=
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎥⎦
⎤⎢⎣
⎡++−−
⎥⎦
⎤⎢⎣
⎡++−
⎥⎥⎦
⎤
⎢⎢⎣
⎡++
⎥⎦
⎤⎢⎣
⎡+++
(7)
where,
n
n kf
11
1= (7a)
( )
21
12112
24
n
nn
n kxkk
f+
= (7b)
( )31
22121211
21112
33312
n
nnnnn
n kxkxkkkkf +++
= (7c)
( ) ( ){ }
41
3312
22121112
2111212
21111
4
961449624
n
nnnnnnnnn
n kxkxkkxkkkkkk
f+++++
= (7d)
( ) ( )( )
51
4412
331211
221112
212
122
1112112
12122
114
11
5
120240240
120324
n
nnnnnn
nnnnnnnn
n kxkxkkxkkk
xkkkkkkkk
f⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++++
++++
= (7e)
212111 1 xkxkk nn
n −−= (7f)
where,
10
1111 k
kk n = ,10
1212 k
kk n = and 10
11 k
kk n = (7g)
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
−−+−+
=−−− tBtBtB e
BBtAe
BA
eA
F111 111
1
211
01
12
(7h)
3.3.3 Non-Dimensional form of governing equation
Equation (6) can be written in the non-dimensional form as
RlMHyF
dxydF
dxydF
dxydF
dxydF
dxydF
dxydF
n
nn
nnnn 4''1
'172
'1
2
64
'1
4
5
5
'1
5
46
'1
6
37
'1
7
28
'1
8
1
γ=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+++
+++
(8)
where,
The coefficients F1 to F7 are given in Appendix III and
2lxxn = , 3
2
111'1 Ql
IEyy = ,
Ql 2
21'1
γγ = ,
1
'lHH = ,
1
2
Rl
M = , 2
1
llln = and
22
11
IEIE
R =
(8a)
In a similar way, the non-dimensional forms equations (5) and (3) are respectively
'14
'1
4
14
'2
11 yRdx
ydkRM
ynn
+= , 0≤x≤l1 (9)
and,
'2
44''12
'2
24'"
'4
'2
44 1)1(2 yM
FRlH
dxydkMllH
dxydR
nnn
nrnn
nn −=−− γμγ , l1≤x≤l2 (10)
where, the non-dimensional parameters are given as,
32
222'2 Ql
IEyy = ,
2
10
kbk
kr = , 4
10
111 k
IER = , 4
20
222 k
IER = ,
10
1"1 k
γγ = ,
2
1
RR
Rn = , 2
1
kbkb
kk = ,
2
1
ηη
ηη = , tkbkb n2
22 η
= , ηη
nkbZ 2= (10a)
3.3.4 Boundary and continuity conditions
For the footing beam, at the point of application of load, i.e. at x=0, slope of
the deflected shape of the beam is zero and the shear force is Q/2. At the edge of the
footing beam, i.e. at x=l1, the bending moment and the shear force are zero, as the
beam end is free. For the reinforcing beam, which is within the foundation soil, at
point x=0, slope of the deflected shape of the beam and the shear force are zero; and at
x=l2, bending moment and shear force are zero. For the reinforcing beam, at x=l1, the
continuity of deflection, slope, bending moment and shear force are duly
incorporated.
The boundary conditions in their non-dimensional form can be presented as
For footing beam,
at 0=x , slope is zero and the shear force is half of the load applied.
i.e. 0'1 =ndx
dy ................................ (11a) 21
3
'1
3
=ndxyd .......................... (11b)
at 1lx = , bending moment and shear force is zero.
i.e. 02
'1
2
=ndxyd ............................. (11c) 0'
'1
3
'1
3=−
nn dxdyT
dxyd ……...... (11d)
For reinforcement beam,
at 0=x , slope and the shear force is zero.
i.e. 0'2 =ndx
dy ................................ (11e) 03
'2
3
=ndxyd ........................... (11f)
at 2lx = , bending moment and shear force is zero.
02
'2
2
=ndxyd .............................. (11g) 0'
'2
3
'2
3
=−nn dx
dyT
dxyd ........... (11h)
where,
22
22'
IElTT = (11i)
The settlement response of the reinforcement beam is governed by two differential
equations over the lengths 0≤x≤l1 and l1≤x≤l2. Thus, at the length x=l1 of the
reinforcing beam, continuity is established in terms of deflection, bending moment
and shear force. Thus at an infinitesimal distance ε in the left and right of the distance
x=l1 on the reinforcing beam, the deflection, bending moment and shear force on the
reinforcing beam are equal, and the continuity is established by the following
conditions.
Deflection is equal i.e. εε +=− xyxy 22 (12a)
Slope is equal i.e. εε +
=− xdx
dy
xdxdy 22 (12b)
Bending moment is equal i.e. εε +
=− xdx
yd
xdxyd
22
2
22
2
(12b)
and,
Shear force is equal i.e. εε +
−=−
−xdx
dyIE
Tdx
yd
xdxdy
IET
dxyd 2
223
23
2
223
23
(12c)
3.3.5 Method of solution: Finite Difference Method
The differential equations governing the settlement response of the footing and
the reinforcing beam are discretized using the finite difference technique. The half of
the footing is divided into nb nodes (i.e. i=1, 2, 3, 4………, nb). Thus, using central
difference scheme, equation (8) can be written in the following form as,
RlMhyCyCyCyC
yCyCyCyCyCn
iiiiiiii
iiiiiiiiii 4''1'
44'
33'
22'
11
''11
'22
'33
'44 γ=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++++
++++
++++++++
−−−−−−−− (12)
where,
The coefficients Ci-4 to Ci+4 are given in Appendix III.
The reinforcing beam is divided into nr nodes (i=1, 2, 3, ...nb, nb+1, .....nr),
where nb is the number of nodes up to the length l1, and beyond l1 and up to the length
l2, node number ranges from nb to nr. The deflection profile of the reinforcing beam is
governed by the equations (9) and (10), which are subsequently written in the finite
difference form using central difference scheme.
To establish continuity at node nb of reinforcing beam, equation (8) is applied
up to (nb-4)th node using central difference, and thereafter backward difference
scheme is applied to obtain the deflection values up to the node nb
Equation (12) when applied at the nodes of the footing beam, with due
incorporation of the boundary conditions, provided a set of linear equations, which is
solved by the Gauss-Seidel iterative technique to obtain the deflection profile of the
footing beam. Once the deflected shape is determined, from the same, the slope,
bending moment, shear force and contact pressures can be computed.
3.4 Results and Discussions
A computer program had been written in C language and the following studies
were carried out in sequence.
4. Convergence of the numerical solution
5. Correctness of the developed program and the solution obtained
6. Parametric studies
3.4.1 Convergence Study
Convergence study was made by decreasing the size of the element, dividing
the footing beam into a mesh of finite segments. The deflection at the center of the
footing beam was recorded for the decreasing sequence of mesh size to check its’
effect on the solution. The computations were made considering normalized times
t’=0 (i.e. at the instant of loading) and t’=0.5 (i.e. at a subsequent later time). For the
typical combinations of the relative stiffness of the soil and the relative flexural
rigidity of the footing as well as the reinforcing beams, the effect of the decreasing
mesh size ranging from 0.1 to 0.0025 on the normalized deflection of the footing
beam at the mid-span are shown in Figure 3.4 and Figure 3.5 respectively for the
normalized times t’=0 and t’=0.5. It is seen that the numerical solution effectively
converges for t’=0 and t’=0.5 when the mesh size (h/l1) lies in the region of 0.01 to
0.00333 in each case.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.001 0.01 0.1 1Non-dimensional mesh size (h/l1)
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
R=20, kr=5 R=5, kr=5 R=5, kr=20
R=10, kr=5 R=10, kr=10 R=10, kr=10
Stable
ln = 0.67γ' = 0.8μ = 0H' = 0.35k n = 0.5t' = 0
Figure 3.4 Convergence study of footing beam at time t’ = 0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.001 0.01 0.1 1Non-dimensional mesh size (h/l1)
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
R=20, kr=5 R=5, kr=5 R=5, kr=20
R=10, kr=5 R=10, kr=10 R=10, kr=10
l n = 0.67γ' = 0.8μ = 0H' = 0.35k n = 0.5t' = 0.5Stable
Figure 3.5 Convergence study of footing beam at time t’ = 0.5
3.4.2 Correctness of the developed program and the solution obtained
As there is no scope of direct comparison of the present study with the
available previous researches, the degenerated case of the current visco-elastic model
is compared with the degenerated case of the elastic model described in chapter 1. For
sake of comparison, in the current visco-elastic model, the viscous parameters are
neglected and the time lapse is considered to be zero, so that the underlying clay bed
is represented only by a combination of Winkler springs of homogeneous stiffness
along the length of the beams. Only the elastic settlement of the clay bed is
considered. The overlying compacted sand layer is also considered of homogeneous
subgrade modulus. The elastic model described in chapter 1 is degenerated to consider
the soil layers of homogeneous subgrade modulus. Figure 3.6 depicts the comparison
of the above mentioned degenerated cases. A deviation of only 2.5 % is observed
between the results, which show that the results are in fair agreement with each other.
It was observed that in the program of visco-elastic model, if the value of ηη was
made to be equal to be zero, the program became unstable. Thus a very low value of
ηη (=1×10-99 to 1×10-299) was considered to make the visco-elastic clay bed to be of
nearly elastic bed with homogeneous subgrade modulus. Beyond the ηη value
considered, the variation in settlement profile of footing beam became extremely
negligible to be considered.
Figure 3.7 shows the plot of time vs. total settlement of reinforcing beam with
the variation in relative viscous coefficients of Burger model. It is observed that as
the ηη increases ten times form 0.1 to 1, the maximum decrease in the total settlement
of the footing beam is 46 %. Again, when the value of ηη increases ten times from 1 to
10, the maximum decrease in settlement becomes 28 %. Thus, it is observed that as
the relative ratio of viscous elements in the Burger model increases by ten times, the
reduction in the total settlement of the beams becomes less. It signifies that as the
viscous elements helps in the larger distribution of stresses, the settlement of the
footing and reinforcing beam decreases, which is identical to the conventional trends.
Thus, it depicts the correctness of the solution obtained.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
Visco-elastic foundation beds
Elastic foundation beds
R = 20kr = 5μ = 0.5t = 0H' = 0.5γ' = 0.8
(ηη=1x10-299)
Figure 3.6 Comparison of the degenerated cases of elastic and visco-elastic models of the present study.
0
0.005
0.01
0.015
0.02
0.025
0 100 200 300 400 500 600 700 800
Time (days)
Nor
mal
ized
def
lect
ion
ofre
info
rcin
g be
am
ηη=0.1 ηη=1 ηη=10
R = 5kr = 20γ' = 0.8μ = 0.5H' = 0.35k k = 1k n = 0.5
Figure 3.7 Time-Settlement (total) plot of reinforcing beam for variation in relative ratio of viscous coefficients in Burger model (ηη).
3.4.3 Parametric study
To study the effect of various parameters on the flexural response of the
foundation as well as the reinforcing beam, parametric studies were conducted. The
various parameters were changes as shown in Table 3.1. The details of the study are
reported under different subsections as follows.
Table 3.1 Range of non-dimensional parameters considered in the study.
Sl. No. Non-dimensional parameters Symbol Range
1 Depth of placement of reinforcement below the footing beam H’ 0.5 – 2.0
2 Relative flexural rigidity of beams R=E1I1/E2I2 5 – 250
3 Relative stiffness of soils kr = k10/kb2 5 – 300
4 Unit weight of granular fill γ’ 0.5 – 2.5
5 Non-dimensional shape constants kn 0 - ∞
6 Coefficient of friction μ 0.5 – 1.0
7 Relative ratio of viscous coefficients in Burger model 2
1
ηη
ηη = 0 – 10
8 Relative ratio of stiffness coefficients in Burger model 2
1
kbkbkk = 0 – 10
9 Time t’ 0 – 25
3.4.3.1 Effect of depth of placement of reinforcement (H’)
Figure 3.8 and 3.9 depicts the effect of the depth of placement of
reinforcement on the normalized deflection profile of footing and reinforcing beam
respectively. The depth of placement of reinforcement is varied from 0.5 to 2l1, the
width of the footing beam. The reinforcement is placed at the interface of the
compacted sand layer and the weak clay deposit with medium to soft consistency. The
reinforcement is used to improve the settlement characteristics of the foundation soil.
The settlement of both the footing and reinforcing beams are seen to continually
increase with the increase of the depth of placement of reinforcement. The normalized
settlement of the footing and reinforcing beam increased by 72 % and 71 %
respectively for increase in the normalized depth of placement from 0.5 to 2.0.
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
H'=0.5 H'=1
H'=1.5 H'=2
R = 20kr = 5γ' = 0.8μ = 0.5ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.8 Normalized deflection of footing beam for variation in depth of placement of reinforcement below the footing beam (H’).
0
0.001
0.002
0.003
0.004
0.005
0.006
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
H'=0.5 H'=1
H'=1.5 H'=2
R = 20kr = 5γ' = 0.8μ = 0.5ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.9 Normalized deflection of reinforcing beam for variation in depth of placement of reinforcement below the footing beam (H’).
3.4.3.2 Effect of relative flexural rigidity of beams (R)
Figure 3.10 and 3.11 shows the effect of variation of relative flexural
rigidity of beams on the normalized deflection profile of footing and reinforcing
beams. The value of R is varied from 5 to 250. The increase in value of R signifies
that the footing beam is becoming more rigid with respect to the reinforcing beam,
and thus would offer increased resistance to settlement. The figure also shows that the
settlement of the footing beam decreases with the increase in R. It is observed that as
the value of R becomes 200, the variation of settlement of footing beam becomes
negligible. It is also observed that at a normalized distance of 0.67 from the centre of
footing beam, the deflection profiles at various R intersect each other. It is observed
that as the R value approaches 250, the settlement of the footing beam tends to be
uniform. Similar observation is made with the reinforcing beam. As the relative
flexural rigidity of beams increases, the settlement of the reinforcing beam decreases
rapidly. As R becomes 100, the variation in deflection becomes negligible and it
barely shows any deflection throughout the length of the reinforcing beam. The
normalized settlement of the footing and reinforcing beam decreased by 40 % and 100
% due to the increase the value of relative flexural rigidity of footing and reinforcing
beam from 5 to 250.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
R=5 R=10 R=20 R=50
R=100 R=200 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.10 Normalized deflection of footing beam for variation in relative flexural rigidity of beams (R).
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
R=5 R=10 R=20
R=50 R=100 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.11 Normalized deflection of reinforcing beam for variation in relative flexural rigidity of beams (R).
3.4.3.3 Effect of relative stiffness of soils (kr)
Figure 3.12 and 3.13 shows the effect of variation of relative stiffness of
upper compacted soil layer and the weak clay layer with medium to soft consistency
on the deflection profile of the footing and reinforcing beam respectively. The value
of kr is varied from 5 to 250. The increase in the value of kr signifies that the upper
compacted granular layer is becoming stiffer, and that it will offer more resistance to
deflection. The figure also shows that the settlement of the footing beam decreases as
the value of kr increases, and becomes negligible as the kr value becomes 150.
Similar observation is made with the reinforcing beam. The normalized deflection of
the footing and reinforcing beams decreases by 67 % and 63 % respectively as the kr
increases from 5 to 250.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
kr=5 kr=50 kr=100
kr=150 kr=250
R = 20γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.12 Normalized deflection of footing beam for variation in relative stiffness of soils (kr).
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0.0018
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
kr=5 kr=50 kr=100
kr=150 kr=250
R = 20γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.13 Normalized deflection of reinforcing beam for variation in relative stiffness of soils (kr).
3.4.3.4 Effect of unit weight of compacted granular layer (γ’)
Figure 3.14 and 3.15 shows the effect of variation of unit weight of
compacted granular layer on the deflection profile of footing and reinforcing beams
respectively. The non-dimensional unit weight of the compacted granular layer is
varied from 0.5 to 2.5. Higher unit weight produces greater surcharge on the
reinforcing beam. The settlement profile of the footing beam increases with the
increase in unit weight of compacted soil. Similar is the observation with the
reinforcing beam. The normalized settlement of the footing and reinforcing beams
increased by 67 % each for increase in γ’ from 0.5 to 2.5. Thus surcharge weight has a
profound influence on the settlement of beams on reinforced elastic foundation.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
γ'=0.5 γ'=1.5 γ'=2.5
R = 20kr = 5μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.14 Normalized deflection of footing beam for variation in unit weight of compacted granular layer (γ’).
0
0.00005
0.0001
0.00015
0.0002
0.00025
0.0003
0.00035
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
γ'=0.5 γ'=1.5 γ'=2.5
R = 20kr = 5μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.15 Normalized deflection of reinforcing beam for variation in unit weight of compacted granular layer (γ’).
3.4.3.5 Effect of variation of parabolic constants (kn)
Figure 3.16 and 3.17 shows the effect of the variation of parabolic constants
on the deflection profile of the footing and reinforcing beam respectively. The
subgrade modulus is assumed to vary parabolically along the length of the beam as
mentioned earlier. The ratio kn is defined as the ratio of k11/k12. The variation of kn
actually signifies the nature of distribution of subgrade modulus along the length of
the beam. kn=∞ indicates linear distribution of subgrade modulus, and succeeding
lower values of kn indicates the variation of subgrade modulus along the length of the
beam with a higher curvature. The value of kn is varied from 0 to ∞. It is observed that
as the ratio kn increases, the deflection of both the footing and the reinforcing beam
increases. The normalized settlement of the footing and reinforcing beam increased by
11 % and 25 % respectively due to the increase in value of kn from 0 to ∞. Thus, it is
observed that the distribution of subgrade modulus influences the flexural response of
the beams on the reinforced elastic foundations.
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
kn=0 kn=0.25
kn=4 kn=infinity
R = 20kr = 5μ = 0.5H' = 0.35γ' = 0.8η η = 1k k = 0.1t' = 0.5
Figure 3.16 Normalized deflection of footing beam for variation in parabolic constants (kn).
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
kn=0 kn=0.25
kn=4 kn=infinity
R = 20kr = 5μ = 0.5H' = 0.35γ' = 0.8ηη = 1k k = 0.1t' = 0.5
Figure 3.17 Normalized deflection of reinforcing beam for variation in parabolic constants (kn).
3.4.3.6 Effect of variation of friction coefficient (μ)
Figure 3.18 and 3.19 depicts the effect of variation of friction coefficient on
the settlement response of the footing and reinforcing beam. It is observed that
variation of μ has a very little effect on the normalized deflection profile of the beams.
Friction coefficients of 0.5 and 1.0 were considered in the study. It is observed that
the normalized deflection of both the footing and the reinforcing beam remains
unaffected by the variation of coefficient of friction. Thus, it is observed that the
coefficient of friction have no effect on the flexural response of the beams on
reinforced elastic foundation when the flexural rigidity of the beams are considered in
the analysis. Due to the identical nature of curves for different values of μ (0.5 and
1.0), the plot co-ordinates became overlapped in both Figures 3.18 and 3.19. Thus, it
was decided to show the alternate points in the plot so that a clear visualization of
both the curves could be obtained.
0.005
0.006
0.007
0.008
0.009
0.01
0.011
0.012
0.013
0.014
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
μ=0.5 μ=1
R = 20kr = 5H' = 0.35γ' = 0.8ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.18 Normalized deflection of footing beam for variation in coefficient of friction (μ).
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
μ=0.5 μ=1
R = 20kr = 5H' = 0.35γ' = 0.8ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.19 Normalized deflection of reinforcing beam for variation in coefficient of friction (μ).
3.4.3.7 Effect of variation of relative stiffness coefficients in Burger model (kk)
Figure 3.20 and 3.21 depicts the effect of variation of relative stiffness
coefficients of Burger model on the settlement response of the footing and reinforcing
beam respectively. The value of kk is varied from 0.1 to 10. It is observed that the
normalized deflection of the footing beam decreases with the increase in the value of
kk. The effect becomes negligible after a value of kk equals to 5. Similar is the
observation with the reinforcing beam. The normalized settlement of the footing and
reinforcing beam decreased by 95 % in each case due to the variation of kk from 0.1 to
10. Both the footing and reinforcing beam shows bare deflection after kk crosses 7.
0
0.02
0.040.06
0.08
0.1
0.12
0.140.16
0.18
0.2
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
kk=0.1 kk=0.2 kk=0.5 kk=1
kk=2 kk=5 kk=7 kk=10
R = 20kr = 5H' = 0.35γ' = 0.8μ = 0.5ηη = 1t' = 0.5k n = 0.5
Figure 3.20 Normalized deflection of footing beam due to the variation in relative stiffness coefficient in Burger model (kk).
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
kk=0.1 kk=0.2 kk=0.5 kk=1
kk=2 kk=5 kk=7 kk=10
R = 20kr = 5H' = 0.35γ' = 0.8μ = 0.5ηη = 1t' = 0.5k n = 0.5
Figure 3.21 Normalized deflection of reinforcing beam due to the variation in relative stiffness coefficient in Burger model (kk).
3.4.3.8 Effect of variation of relative viscous coefficients in Burger model (ηη)
Figure 3.22 and 3.23 shows the effect of variation of relative viscous
coefficients of Burger model on the normalized deflection profile of footing and
reinforcing beam respectively. . The value of ηη is varied from 0.1 to 10. It is observed
that the normalized deflection of the footing beam decreases with the increase in the
value of ηη. The effect becomes negligible after a value of ηη equals to 4. Similar is
the observation with the reinforcing beam. The normalized settlement of the footing
and reinforcing beam decreased by 70 % in each case due to the variation of ηη from
0.1 to 10.
0.025
0.045
0.065
0.085
0.105
0.125
0.145
0.165
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
ηη=0 ηη=0.5 ηη=1 ηη=2 ηη=3
ηη=4 ηη=5 ηη=7 ηη=8 ηη=10
R = 20kr = 5H' = 0.35γ' = 0.8μ = 0.5t' = 0.5k k = 0.1k n = 0.5
Figure 3.22 Normalized deflection of footing beam due to the variation in relative viscous coefficient in Burger model (ηη).
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
ηη=0 ηη=0.5 ηη=1 ηη=2ηη=4 ηη=5 ηη=7 ηη=10
R = 20kr = 5H' = 0.35γ' = 0.8μ = 0.5t' = 0.5k k = 0.1k n = 0.5
Figure 3.23 Normalized deflection of reinforcing beam due to the variation in relative viscous coefficient in Burger model (ηη).
3.4.3.9 Effect of variation of time (t’)
Figure 3.24 and 3.25 shows the effect of variation of time on the normalized
deflection (total settlement including initial, primary and secondary consolidation
settlement) profile of the footing and reinforcing beams respectively. The time is
considered up to t = 700 days. The normalized time (t’) is determined by dividing the
time elapsed (t) by time tpc. The time tpc is determined by the double tangent drawn
from the initial and final linear regions of the time-settlement curve. Figure 3.26
shows a typical case of determination of the time tpc. In the case of footing beam, it is
observed that as the time increases, the settlement of the beam increases, but the rate
of increment of settlement decreases. It becomes nearly constant after the elapse of
500 days. The settlement at t’ = 0 signifies the initial settlement at the instant of load
application. The settlement at consecutive times predicts the consolidation settlement.
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1 1.2
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of fo
otin
g be
am
t'=0 t'=0.32 t'=0.64 t'=1.61
t'=3.22 t'=6.45 t'=9.67 t'=12.90
t'=16.13 t'=19.35 t'=22.58
R = 20kr = 5γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1k n = 0.5
Figure 3.24 Normalized deflection of footing beam due to the variation in time elapsed (t’).
0
0.002
0.004
0.006
0.008
0.01
0.012
0 0.2 0.4 0.6 0.8 1
Normalized distance from centre
Nor
mal
ized
def
lect
ion
of r
einf
orci
ng b
eam
t'=0 t'=0.32 t'=0.64 t'=1.61 t'=3.22
t'=9.67 t'=12.90 t'=16.13 t'=19.35 t'=22.58
R = 20kr = 5γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1k n = 0.5
Figure 3.25 Normalized deflection of reinforcing beam due to the variation in time elapsed (t’).
0
0.002
0.004
0.006
0.008
0.01
0.012
0 50 100
150
200
250
300
350
400
450
500
550
600
650
700
750
Time (Days)
Nor
mal
ized
def
lect
ion
at th
e ce
ntre
of r
einf
orci
ng b
eam
R = 5kr = 20γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1k n = 0.5
Figure 3.26 Typical time-settlement curve to determine the time lapsed (tpc).
tpc
3.4.3.10 Typical variation of normalized bending moment diagram
Figure 3.27 and 3.28 shows the typical variation of normalized bending
moment of the footing and reinforcing beams respectively for the variation in relative
flexural rigidity of footing and reinforcing beams (R). It is observed that the
maximum negative bending moment occurs at the centre of the footing and
reinforcing beam and decreases by 32 % and 80 % respectively with the value of R
increasing from 5 to 250. The bending moment at the edge of the beams is zero. It is
observed that as the R value increases, a positive bending moment develops at the
centre of the beams.
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized distance from centre
Nor
mal
ized
ben
ding
mom
ent
for
foot
ing
beam
R=5 R=20 R=50 R=100
R=150 R=200 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.27 Typical normalized bending moment diagram of footing beam for variation in relative flexural rigidity of beams (R).
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.0010 0.2 0.4 0.6 0.8 1
Normalized distance from centre
Nor
mal
ized
ben
ding
mom
ent
for
rein
forc
ing
beam
R=5 R=20 R=50 R=100
R=150 200 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.28 Typical normalized bending moment diagram of reinforcing beam for variation in relative flexural rigidity of beams (R).
3.4.3.11 Typical variation of normalized shear force diagram
Figure 3.29 and 3.30 shows the typical variation of normalized shear force
of the footing and reinforcing beams respectively for the variation in relative flexural
rigidity of footing and reinforcing beams (R). It is observed that the maximum
positive shear force occurs at the centre of the footing beam and it has a constant
normalized value of 0.5. As the value of R increases, a negative shear force region
develops in the footing beam. In case of reinforcing beam, the shear force at the centre
is zero. The maximum positive shear force occurs near the centre of the footing beam,
and it decreases by 66 % with the increase in value of R from 0 to 250. The maximum
negative shear force occurs at a point below the edge of the footing beam and it
increases by 51 % with the increase in value of R from 0 to 250.
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized distance from centre
Nor
mal
ized
shea
r fo
rce
for
foot
ing
beam
R=5 R=50 R=100 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.29 Typical normalized shear force diagram of footing beam for variation in relative flexural rigidity of beams (R).
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 0.2 0.4 0.6 0.8 1
Normalized distance from centre
Nor
mal
ized
shea
r fo
rce
for
rein
forc
ing
beam
R=5 R=10 R=20 R=50
R=100 R=150 R=250
kr = 5γ' = 0.8μ = 0.5H' = 0.35ηη = 1k k = 0.1t' = 0.5k n = 0.5
Figure 3.30 Typical normalized shear force diagram of reinforcing beam for variation in relative flexural rigidity of beams (R).
3.5 Conclusion
From the above studies, it is seen that the nature of distribution of confining
pressure on the beam-soil interface has a significant effect on the settlement response
of a footing placed on a reinforced soil bed. Parametric studies indicated that the
settlement response of beams is influenced by the shape and nature of distribution of
subgrade reaction at the beam-soil interface. The above parametric studies also
indicate that the choice of the parameters of the Burger model could also significantly
affect the settlement response of the beams. Thus, to obtain a feasible solution from
the settlement response of beams in such cases, a proper choice of the parameters is
necessary. It was observed that variation of coefficient of friction, μ did not
significantly affect the deflection profile of the beam.
The above study indicate that the effect of distribution of confining pressure
on the settlement response of a footing placed on a reinforced soil bed should be duly
incorporated in the design of a reinforced soil foundation. Based on the studies
conducted above, the following conclusions can be drawn for the range of parameters
considered:
(1) The numerical solution effectively converges for t’=0 and t’=0.5 when the
mesh size (h/l1) lies in the region of 0.01 to 0.00333 in each case.
(2) The current visco-elastic model, the viscous parameters are neglected and the
time lapse is considered to be zero, so that the underlying clay bed is
represented only by a combination of Winkler springs of homogeneous
stiffness along the length of the beams. Only the elastic settlement of the clay
bed is considered. The overlying compacted sand layer is also considered of
homogeneous subgrade modulus. The elastic model described in chapter 1 is
degenerated to consider the soil layers of homogeneous subgrade modulus. A
deviation of only 2.5 % is observed between the results, which show that the
results are in fair agreement with each other.
(3) It is observed that as the ηη increases ten times form 0.1 to 1, the maximum
decrease in the total settlement of the footing beam is 46 %. Again, when the
value of ηη increases ten times from 1 to 10, the maximum decrease in
settlement becomes 28 %. Thus, it is observed that as the relative ratio of
viscous elements in the Burger model increases by ten times, the reduction in
the total settlement of the beams becomes less.
(4) The normalized settlement of the footing and reinforcing beam increased by
72 % and 71 % respectively for increase in the normalized depth of placement
from 0.5 to 2.0.
(5) The normalized settlement of the footing and reinforcing beam decreased by
40 % and 100 % due to the increase the value of relative flexural rigidity of
footing and reinforcing beam from 5 to 250. . It is observed that as the value
of R becomes 200, the variation of settlement of footing beam becomes
negligible. It is observed that as the R value approaches 250, the settlement of
the footing beam tends to be uniform. As the relative flexural rigidity of
beams increases, the settlement of the reinforcing beam decreases rapidly. As
R becomes 100, the variation in deflection becomes negligible and it barely
shows any deflection throughout the length of the reinforcing beam.
(6) The normalized deflection of the footing and reinforcing beams decreases by
67 % and 63 % respectively as the kr increases from 5 to 250. The settlement
of both the footing and reinforcing beam decreases as the value of kr
increases, and becomes negligible as the kr value becomes 150.
(7) The normalized settlement of the footing and reinforcing beams increased by
67 % each for increase in γ’ from 0.5 to 2.5. Thus surcharge weight has a
profound influence on the settlement of beams on reinforced elastic
foundation.
(8) The normalized settlement of the footing and reinforcing beam increased by
11 % and 25 % respectively due to the increase in value of kn from 0 to ∞.
Thus, it is observed that the distribution of subgrade modulus influences the
flexural response of the beams on the reinforced elastic foundations. Thus, it
should be considered in the analysis and design of reinforced foundations.
(9) It is observed that the coefficient of friction have no effect on the flexural
response of the beams on reinforced elastic foundation when the flexural
rigidity of the beams are considered in the analysis.
(10) The normalized settlement of the footing and reinforcing beam decreased by
95 % in each case due to the variation of kk from 0.1 to 10. Both the footing
and reinforcing beam shows very small deflection after kk crosses 7. The effect
becomes negligible after a value of kk equals to 5.
(11) The normalized settlement of the footing and reinforcing beam decreased by
70 % in each case due to the variation of ηη from 0.1 to 10. The effect
becomes negligible after a value of ηη equals to 4.
(12) The settlement at t’ = 0 signifies the initial settlement at the instant of load
application. The settlement at consecutive times predicts the total settlement
including the consolidation settlement.
(13) It is observed that the maximum negative bending moment occurs at the centre
of the footing and reinforcing beam and decreases by 32 % and 80 %
respectively with the value of R increasing from 5 to 250. The bending
moment at the edge of the beams is zero. It is observed that as the R value
increases, a positive bending moment develops at the centre of the beams.
(14) It is observed that the maximum positive shear force occurs at the centre of the
footing beam and it has a constant normalized value of 0.5. As the value of R
increases, a positive shear force region develops in the footing beam. In case
of reinforcing beam, the shear force at the centre is zero. The maximum
positive shear force occurs near the centre of the footing beam, and it
decreases by 66 % with the increase in value of R from 0 to 250. The
maximum negative shear force occurs at a point below the edge of the footing
beam and it increases by 51 % with the increase in value of R from 0 to 250.
3.6 Scope of further work
The visco-elastic model proposed in this section of the thesis can be extended
to include various sorts of geotechnical problems dealing with reinforced foundations.
The problem may consist of a railway track subjected to a rolling point load, a
combined footing subjected to two concentrated loads at the edges, a railway tie
subjected to two rolling point loads, or of a surface water tank where the reinforced
foundation is subjected to two concentrated loads at the edges, a uniformly distributed
load along the length of the footing beam and two moments at the edges. The figures
of the problem are same as that shown in chapter 2. Similar problems are analyzed
with the underlying soil consisting of a weak clay layer of soft to medium
consistency.
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APPENDIX I
nfF 11 = (1)
nfF 22 = (2)
nnn fRMlHfF 142'"
133 2μγ−= (3)
nnn fRMlHfF 242'"
144 μγ−= (4)
nnnr
nnn fRMlHf
kk
kRMfMF 342'"
112
14
54
5 31 μγ−⎟⎟
⎠
⎞⎜⎜⎝
⎛+++= (5)
rnn kMRlHF 842'"16 2μγ−= (6)
827 Mk
kRF n
r
= (7)
( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ−
Δ=− 7
28
14 2 nn
i xF
xF
C (8)
( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ−
Δ+
Δ+
Δ−=− 5
46
37
28
13 2
38
nnnni x
FxF
xF
xF
C (9)
( ) ( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ+
Δ−
Δ−
Δ=− 4
55
46
37
28
12
26728
nnnnni x
FxF
xF
xF
xF
C (10)
( ) ( ) ( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ−
Δ−
Δ+
Δ+
Δ−=− 2
64
55
46
37
28
11
45.215756
nnnnnni x
FxF
xF
xF
xF
xF
C (11)
( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡+
Δ−
Δ+
Δ−
Δ= 72
64
56
38
1 262070F
xF
xF
xF
xF
Cnnnn
i (12)
( ) ( ) ( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ−
Δ+
Δ+
Δ−
Δ−=− 2
64
55
46
37
28
11
45.215756
nnnnnni x
FxF
xF
xF
xF
xF
C (13)
( ) ( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ−
Δ−
Δ+
Δ=− 4
55
46
37
28
12
26728
nnnnni x
FxF
xF
xF
xF
C (14)
( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ+
Δ−
Δ−=− 5
46
37
28
13 2
38
nnnni x
FxF
xF
xF
C (15)
( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ=− 7
28
14 2 nn
i xF
xF
C (16)
APPENDIX II
Stress-Strain relation from Four-Element Burger Model
Figure A.1 Four Element Burger model.
For the spring 1: 11 ykb=′σ (1)
For the dashpot 2: 32
•=′ yησ (2)
For the spring 2: 221 ykb=′σ (3)
For the dashpot 1: 212
•=′ yησ (4)
Now,
212212221 )( yDkbyykb ηησσσ +=+=′+′=′•
(5)
From equation (1): 1
1 kby
•• ′
=σ (6)
From equation (2): 2
3 ησ ′
=•y (7)
From equation (3): )( 12
2 Dkby
ησ+′
=
••
(8)
σ′ σ′
σ1′
σ2′
kb1
kb2
y2
y1
η1
η2
y3
Using equations (6), (7) and (8), we get
2121
321 )( ησ
ησσ ′
++′
+′
=++=••••
DkbD
kbDyyyy
σηηηηηη
′⎥⎦
⎤⎢⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛+++=+⇒
21
2
1211
2
1
2
1
22 11)(kb
Dkbkb
kbDyD
kbD (9)
Let,
1,,1,11, 21
21
12
1221
21
21
20 ===++== BkbB
kbAkbAkbA
ηηηηηηη (10)
Thus equation (9) can be written as,
σ ′++=+ )()( 012
212
2 ADADAyDBDB (11)
Figure A.2 Representation of applied stress with time. Considering unit step function as shown above, and taking Laplace transform of both
sides, we get,
( ){ } [ ]sBsBtyLs
AAsA 1
22
0120 +=⎥⎦
⎤⎢⎣⎡ ++′σ (12)
( ){ } ( ) ( ) ( )⎥⎦⎤
⎢⎣
⎡
++
++
+′=⇒
122
0
12
1
12
20 BsBs
ABsBs
ABsBs
sAtyL σ (13)
( ) ( ) ( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
++
++
+′=⇒ −
12
0
1
1
1
20
1
BssA
BssA
BsALty σ (14)
t
σ′(t)
σ0′
σ′(t) = σ0′.Δt
Now,
tBeBs
L 1
1
1 1 −− =⎥⎦
⎤⎢⎣
⎡+
(15)
( ) ( )tBeBBss
L 1111
11
1 −− −=⎥⎦
⎤⎢⎣
⎡+
(16)
( )
( )tBeBB
tBss
L 11112111
21 −− −−=⎥
⎦
⎤⎢⎣
⎡
+ (17)
Thus, the stress-strain relationship from the burger model is obtained as follows
( ) ( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
−−+−+′= −−− tBtBtB eBB
tAeBAeAty 111 111 2
110
1
120σ (18)
APPENDIX III
nfF 11 = (1)
nfF 22 = (2)
nnn fRMlHfF 142'"
133 2μγ−= (3)
nnn fRMlHfF 242'"
144 μγ−= (4)
43
42'"15
45
131 M
kR
FfRMlHfMF
rnnnn +−+= μγ (5)
rnn kMRlHF 842'"16 2μγ−= (6)
⎭⎬⎫
⎩⎨⎧
−= knr
kF
MkRF 18
7 (7)
( ) ( )Zn
Z
k
Zkn ekbe
kekF −−− −−+−⎟⎟
⎠
⎞⎜⎜⎝
⎛+++= 1111 2 ηη ηη (8)
( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ−
Δ=− 7
28
14 2 nn
i xF
xF
C (9)
( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ−
Δ+
Δ+
Δ−=− 5
46
37
28
13 2
38
nnnni x
FxF
xF
xF
C (10)
( ) ( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ+
Δ−
Δ−
Δ=− 4
55
46
37
28
12
26728
nnnnni x
FxF
xF
xF
xF
C (11)
( ) ( ) ( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ−
Δ−
Δ+
Δ+
Δ−=− 2
64
55
46
37
28
11
45.215756
nnnnnni x
FxF
xF
xF
xF
xF
C (12)
( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡+
Δ−
Δ+
Δ−
Δ= 72
64
56
38
1 262070F
xF
xF
xF
xF
Cnnnn
i (13)
( ) ( ) ( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ−
Δ+
Δ+
Δ−
Δ−=− 2
64
55
46
37
28
11
45.215756
nnnnnni x
FxF
xF
xF
xF
xF
C (14)
( ) ( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ−
Δ−
Δ+
Δ=− 4
55
46
37
28
12
26728
nnnnni x
FxF
xF
xF
xF
C (15)
( ) ( ) ( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ+
Δ−
Δ−=− 5
46
37
28
13 2
38
nnnni x
FxF
xF
xF
C (16)
( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
Δ+
Δ=− 7
28
14 2 nn
i xF
xF
C (17)