The Pennsylvania State University
The Graduate School
Department of Civil and Environmental Engineering
IMMEDIATE LOAD-SETTLEMENT RESPONSE OF STRIP FOOTINGS
BEARING ON GEOGRID-REINFORCED CLAY
A Thesis in
Civil Engineering
by
Yuhao Ren
2015 Yuhao Ren
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
May 2015
The thesis of Yuhao Ren was reviewed and approved* by the following:
Prasenjit Basu Assistant Professor of Civil Engineering Thesis Advisor Swagata Banerjee Basu Assistant Professor of Civil Engineering MS Thesis Committee Member Tong Qiu Assistant Professor of Civil Engineering MS Thesis Committee Member Peggy Johnson
Professor of Civil Engineering Head of the Department *Signatures are on file in the Graduate School
ii
ABSTRACT
Footings on reinforced soil have long been considered as an effective solution to
enhance foundation bearing capacity and reduce settlement. Notwithstanding the fact
that footings bearing on reinforced clays can be an economic alternative to other
expensive foundation solutions, studies focusing on such foundations are rather limited
in number. In addition to consolidation and long-term creep settlement, which are most
frequently quantified for footings on clay, immediate settlement (i.e., settlement at a
time shortly after load application) of foundations bearing on reinforced clay should be
studied adequately to establish serviceability criteria that may govern the design.
The present research quantifies immediate settlement of strip footings bearing
on geogrid-reinforced clay. Finite element analyses of a strip footing bearing on
unreinforced and reinforced clay are performed to evaluate and quantify the effects of
several material (e.g., soil and reinforcement properties) and geometric (i.e., those
pertaining to reinforcement layout) parameters on load-settlement response of the
footing. A nonlinear elastic, perfectly plastic constitutive model obeying a non-
associated flow rule is used to represent mechanical behavior of clay. geogrid
reinforcement is modeled as a linear elastic material. Soil-reinforcement interaction is
modeled using cohesive interface elements above and below the reinforcement layers.
The clay constitutive model is also used as the material model for the cohesive interface
element and a contact interaction model is utilized to define the connection between the
cohesive interface and reinforcement layer.
Results show that settlement influence factor, which can be used for calculation
iii
of immediate settlement under varying net stress applied at the footing base, reduces
with increase in number of reinforcement layers and with increase in undrained shear
strength of clay. An increase in plasticity index of clay causes an increase in immediate
settlement and the same can be significant for foundations bearing on clays with high
plasticity. Optimal values of different reinforcement arrangement factors obtained from
this research are in agreement with those reported in literature.
iv
TABLE OF CONTENTS
LIST OF FIGURES.................................................................................................................. vi LIST OF TABLES .................................................................................................................. vii ACKNOWLEDGEMENTS .................................................................................................. viii Chapter 1 Introduction .............................................................................................................. 1
1.1. Background .......................................................................................................... 1 1.2. Motivation and Research Objectives ................................................................... 4 1.3. Thesis Outline ...................................................................................................... 5
Chapter 2 A Review of Existing Literature ............................................................................... 6 2.1. Reinforcing Mechanisms and Bearing Capacity Equations for RSF ................... 6 2.2. Model-scale Experiments .................................................................................. 11 2.3. Numerical Studies .............................................................................................. 17
Chapter 3 Finite Element Modeling ........................................................................................ 23 3.1. Clay Constitutive Model .................................................................................... 23 3.2. Geometric Model for the Soil Domain .............................................................. 25 3.3. Modeling of the Reinforcement Layers ............................................................. 28 3.4. Modeling of the Clay-Reinforcement Interface ................................................. 29 3.5. Convergence Study ............................................................................................ 30 3.6. Validation for the FE Model .............................................................................. 32
Chapter 4 Analyses and Results .............................................................................................. 37 4.1. Influence Factor Iq ............................................................................................. 37 4.2. Effect of Number of Reinforcement Layers N on Influence Factor Iq ............... 39 4.3. Effect of Undrained Shear Strength su on Settlement Influence Factor Iq ......... 45 4.4. Effect of Plasticity Index PI on Settlement Influence Factor Iq ......................... 46 4.5. Optimal Depth for Placement of the Top Reinforcement Layer ........................ 47 4.6. Optimal Reinforcement Width LR for a Single Layer of Reinforcement ........... 48 4.7. Optimal Vertical Spacing between Two Reinforcement Layers ........................ 50 4.8. Effective Total Reinforcement Depth de ............................................................ 51 4.9. Influence Zone ZR below the Footing Base ....................................................... 52 4.10. Optimal Number of Reinforcement/Interspacing for Multi-layer ...................... 55 4.11. Influence of Reinforcement Bending Stiffness .................................................. 56 4.12. Effect of Clay Layer Thickness on Settlement Influence Factor Iq .................... 57
Chapter 5 Discussion and Conclusions ................................................................................... 59 5.1. Comparison with Results Reported in Previous Numerical Studies on Footings on Reinforced Clay ......................................................................................................... 59 5.2. Comparison with Previous Experimental Studies .............................................. 61 5.3. Conclusions ........................................................................................................ 62
References ............................................................................................................................... 64
v
LIST OF FIGURES
Figure 1-1 Representative layout of a reinforced soil foundation ............................................. 4 Figure 3-1 FE mesh for footing on unreinforced clay ............................................................. 27 Figure 3-2 FE mesh for footing on reinfnorced clay ............................................................... 27 Figure 3-3 Initial geostatic vertical effective stress contour ................................................... 28 Figure 3-4 Coulomb friction model (adapted from ABAQUS 6.12 User’s Manual) .............. 30 Figure 3-5 FEA convergence study of strip footing with W/B=20 ......................................... 31 Figure 3-6 FEA convergence study for different mesh densities ............................................ 32 Figure 3-7 Comparison of load-settlement curve reported in Davidson and Chen (1997) with
the one obtained using FE model used in the present research ....................................... 33 Figure 3-8 Comparison of load-settlement data reported in Das and Shin (1994) with that
predicted using the FE modeling scheme employed in the present research: (a) unreinforced clay; (b) reinforced clay ............................................................................. 35
Figure 3-9 Comparison of influence factor reported in Foye et al. (2008) with the one obtained based on FE modeling of the present research ................................................................ 36
Figure 4-1 Variations of settlement influence factor Iq with number of reinforcement layers N for G0/su=200: (c) PI=50, G0=3.2MPa. ........................................................................... 42
Figure 4-2 Variations of settlement influence factor Iq with number of reinforcement layers N for G0/su=100: (d) PI=60 ................................................................................................. 44
Figure 4-3 Variations of settlement influence factor Iq with number of reinforcement layers N for G0/su=50: (b) PI=60, G0=2.2MPa. ............................................................................. 45
Figure 4-4 Variations of settlement influence factor Iq with su based on PI=50, G0=3.2MPa: G0/su=200, 133, 100 and 71, respectively. ....................................................................... 46
Figure 4-5 Variations of settlement influence factor Iq with PI based on su=32kPa: G0=6.9MPa, 4.8MPa, 3.2MPa and 2.2MPa for PI=30, 40, 50 and 60, respectively. ........................... 47
Figure 4-6 Variations of BCR with depth d0 (measured from the footing base) of the top reinforcement layer; based on FEAs with PI =56 (i.e., G0=2.5MPa) and su=50kPa ....... 48
Figure 4-7 Variations of BCR (at different levels of footing settlement) with normalized reinforcement width LR/B for a single layer of reinforcement; based on FEAS with PI = 40 (i.e., G0=5MPa) and su=50kPa. .................................................................................. 50
Figure 4-8 Variation of BCR (for different settlement levels) with normalized vertical spacing h/B between reinforcement layers; based on FEAs with PI = 60 (i.e., G0=2.2MPa) and su=30kPa. ........................................................................................................................ 51
Figure 4-9 Increase in BCR with number of reinforcement layer N; based on FEAs with PI = 60 (i.e., G0=2.2MPa) and su=22kPa. ............................................................................... 52
Figure 4-10 Variation of vertical effective stress change ratio at different depths z below the footing base depth: (c) z = 1.7B and (d) z = 2.9B. .......................................................... 54
Figure 4-11 Variation of BCR (at s/B=3.2%) for different N-h combinations; based on FEAs with PI =50 (i.e., G0=3.2MPa) and su=32kPa.................................................................. 56
Figure 4-12 Influence factors for different reinforcement stiffness; based on FEAs with PI =30 (i.e., G0=6.9MPa), su=34.5kPa, N=4, and E=1.2GPa. ..................................................... 57
Figure 4-13 Influence factor for different clay thickness (N=4) ............................................. 58
vi
LIST OF TABLES
Table 2-1 Summary of finite element analyses of RSFs ......................................................... 19 Table 2-2 Summary of optimum parameters for reinforced soil foundations ......................... 20 Table 3-1 Reinforcement properties ........................................................................................ 29 Table 4-1 Soil input parameters used in the FEAs .................................................................. 40 Table 5-1 Summary of optimal parameters as reported in different numerical studies ........... 60 Table 5-2 Comparison of optimal parameters reported in past experimental studies with those
obtained from the present research.................................................................................. 61
vii
ACKNOWLEDGEMENTS
First of all, I want to thank my mom for the birth and love she gave to me; life is hard,
but yet interesting. There’s still a long way in front of me.
Secondly, I want to thank my father for his early education, I actually started to
understand him after I came to the United States and began to think that pursing a PhD
degree in physics overseas is never an easy thing although he has passed away already
for more than 12 years.
Next, I would like to thank my advisor Dr. Prasenjit Basu for his guiding toward
my research; most of his suggestions are indeed really helpful and may still work for
me in the future. Dr. Swagata Banerjee Basu and Dr. Tong Qiu have asked me some
good questions during the defense and they also offered me some constructive
suggestions that would be definitely beneficial to me, so I really appreciate their time
and help on this research.
Additionally, Mr. Yin Gao helps me a lot as an upperclassman in the department
of civil engineering and I sincerely express my gratitude to him here.
Finally, please allow me to end up with a line from the movie ‘Blackjack’:
“Yesterday is a history and tomorrow is a mystery; it’s all what you do in the moment.”
viii
Chapter 1 Introduction
1.1. Background
Geosynthetics are polymeric products, commonly available in the form of geogrids,
geotextiles, geomembranes and geocells, which are frequently used in civil engineering
practice. The polymeric nature of the material makes different geosynthetics products
durable under different ground and environmental conditions. Common applications of
geosynthetics in the field of geotechnical engineering include improving strength and
stiffness of subsurface soil beneath shallow foundations and pavements, providing
stability to earth retaining structures and slopes, ensuring dam safety, to name a few.
Early applications of geosynthetics in 1960s were about their use as filters materials in
the United States and as soil reinforcement in Europe.
A geogrid is one of the most common geosynthetic products that are often used
for improving mechanical performance of subsurface soil under external loadings.
Geogrids are widely used as reinforcement layers in mechanically stabilized earth
(MSE) and geosynthetic reinforced soil (GRS) walls, as a measure of slope
stabilization and as reinforcement in subsurface soil below pavements and footings.
Soils are weak in tension; good tensile capacity of geogrids allows the reinforcement
layers to take over a significant part of tensile stresses generated within a soil mass due
to the action of external loading. Thus, geogrids act as “reinforcing” element and
enhance load-deformation behavior of reinforced soil mass. Geogrids are commonly
made of polymers; nowadays different a variety of geogrids are made of polypropylene
1
or high density polypropylene (HDPP).
Based on manufacturing process, geogrids can mainly be categorized in three
distinct types. The first type is commonly known as homogenous or punched geogrid.
Bundles of polyethylene-coated polyester fibers contributing to the flexibility, are used
as reinforcing material in the second type. The third type is made by bonding
polypropylene rods together in a grid-like pattern using laser or ultrasonic technology.
Geogrids can also be classified into three types according to grid structure: uniaxial,
biaxial and triaxial. As the name suggests, while uniaxial geogrids are able to sustain
mainly uniaxial stress (along the direction of longer gird dimension), biaxial and
triaxial geogrids are capable of sustaining loads from two and three directions,
respectively.
Shallow foundations are often used in practice to transfer loads coming from
the structure to the underlying ground at relatively shallow depth (usually less than five
times the width of the foundation). Shallow foundations range from small isolated
foundations, which support load from an individual column, to large foundation
elements that support several columns, or even all the loads from structure. Shallow
foundations are easy to build, requiring little to no specialized equipment. For shallow
foundations, foundation-to-soil load transfer takes place predominantly through the
base of the foundation element, and only a small fraction of the total load can be
transferred through the sides of an embedded shallow foundation element (in most
cases these are reinforced concrete blocks); however, such contribution is often
neglected in design.
2
Shallow foundations are not suitable for subsurface in which weak load bearing
strata exist at shallow depths. Highly compressible soils and uncontrolled fills are not
ideal conditions to transfer structural load through shallow foundations. Under these
conditions, adding reinforcements to the soil layer beneath the footing base is regarded
as a good choice to enhance foundation bearing capacity and reduce foundation
settlement. This type of foundation and reinforced-soil system is called reinforced soil
foundation (RSF). Reinforced soil foundations could be an economical alternative to
conventional shallow foundations with large footing dimensions which in turn increase
foundation settlement due to an increase in the depth of influence zone below the
foundation, or replacement of weak soil layers with competent materials. Geogrids are
widely used to strengthen soil layers below footings because their performance is
generally better than geotextiles and geomembranes.
A schematic of RSF arrangement is shown in Figure 1-1. Several reinforcement
layers can be laid within the soil under the footing base. Main design parameters of
RSF includes: embedment depth of foundation Df, depth of the top reinforcement layer
or top layer spacing d0 measured from the footing base, vertical spacing h between
reinforcement layers, number of reinforcement layers N, total depth of reinforcement
d=d0+ (N-1)h, and width of reinforcement LR.
3
Figure 1-1 Representative layout of a reinforced soil foundation
1.2. Motivation and Research Objectives
With growing interest in employing shallow foundations to support bridges and other
heavy structures, it is important to study and explore all potential combinations ground
improvement and foundation solutions that would allow the use of shallow foundations
even in conditions for which a deep foundation would otherwise be selected. Footing
on reinforced clay is such an alternative foundation solution. Notwithstanding the fact
that footings bearing on reinforced clays can be an effective and economic alternative
to other expensive foundation solutions, studies focusing on such foundations are rather
limited in number. Moreover, in addition to consolidation and long-term creep
settlement (which are most frequently quantified for footings on clay), immediate
settlement (i.e., settlement at a time shortly after load application) of foundations
bearing on reinforced clay should be studied adequately to establish serviceability
4
criteria that may govern the design. Following such an objective, the present study aims
to quantify immediate load-settlement behavior of a strip footing resting on geogrid-
reinforced clay through the influence factor Iq (which will be discussed in chapter 4).
Furthermore, effects of several parameters related to reinforcement arrangement and
properties of soil and reinforcement on foundation performance (i.e., bearing capacity
and settlement) are analyzed.
1.3. Thesis Outline
The content of this thesis is presented in five chapters. Following the background and
introduction to the problem, as presented in this chapter (Chapter 1), Chapter 2 provides
a brief review of existing literature related to the topic. Pertinent details of finite
element modeling including soil and reinforcement constitutive models, geometric
model, interface condition and mesh convergence study are discussed in Chapter 3.
Results from finite element analysis are presented and discussed in Chapter 4. Chapter
5 summarizes important conclusions and findings drawn from this research.
5
Chapter 2 A Review of Existing Literature
This chapter provides a brief summary of relevant literature focusing on shallow
reinforced soil foundations (RSFs); i.e., foundations bearing on reinforced soil.
Following a brief description of foundation-soil-reinforcement interactions reported in
literature, analytical formulations for bearing capacity calculation of shallow
foundations resting on reinforced soil are discussed. Past studies involving physical
model tests and numerical simulations performed to investigate load-displacement
behavior of RSF are summarized in separate sections.
2.1. Reinforcing Mechanisms and Analytical Bearing Capacity Calculation Equations for RSF
2.1.1. Foundation-soil-reinforcement interaction
Three major foundation-soil-reinforcement interaction mechanisms discussed in
literature for analytical solutions are:
(1) Rigid boundary: When the distance from the footing base to the first layer
of reinforcement is larger than a threshold value (usually equal to 0.5B), the
reinforcement performs as a rigid boundary. Failure under such a condition is expected
to happen within the region above the top layer of reinforcement.
(2) Membrane effect: Under the application of external load, soil immediately
below the footing moves down together with the footing causing flexural bending in
the reinforcement layer. The reinforcement layers resist the downward displacement of
reinforced soil foundation (RSF) through a vertical component (acting upward) of the
tensile membrane force. A particular level of displacement is required to mobilize such 6
membrane effect, and the stiffness of the reinforcement needs to be relatively high to
avoid failure.
(3) Confinement effect: It is possible for an interlocking mechanism to be
induced by soil-reinforcement interaction. Such soil-reinforcement interlocking would
increase confinement and reduce lateral deformation of the reinforced soil. An increase
in soil confinement increases shear strength of the reinforced soil, and consequently,
bearing capacity of foundations resting on reinforced soil increases.
2.1.2. Analytical expressions for bearing capacity of RSF
For high values of reinforced-to-unreinforced soil shear strength ratio and for small
values of the ratio of total reinforcement depth d to footing width B, a punching shear
failure in the reinforced zone and a general shear failure in the unreinforced zone are
likely to happen. Meyerhof and Hanna (1978) proposed similar bearing capacity failure
mechanism for footings resting on a strong soil layer underlain by a weaker layer. The
limit bearing capacity of a strip footing bearing on reinforced soil can be obtained by
modifying the Meyerhof and Hanna’s solution to include the effect of additional
confinement provided by the reinforcement layers (Wayne et al. 1998). The limit
bearing capacity qu(R) of reinforced soil foundation can be expressed as:
qu(R) = qu(b) +2cad
B+ γtd
2 �1 +2Df
d�
KstanφtB
+2∑ TitanδN
i=1
B−γtd
(2.1)
where qu(b) is the limit bearing capacity of the same foundation as if it were resting on
unreinforced soil below the reinforced zone; Ca=cad is the adhesive force acting upward,
and ca refers to the unit adhesion; γt is the soil (in the reinforced zone) unit weight; Df 7
is foundation embedment depth (i.e., the distance from the ground surface to the base
of the footing; Ks is the coefficient for punching shear; φt is the soil friction angle
(within the reinforced zone); N is the number of reinforcement layers; Ti is the tensile
force in the ith layer of reinforcement; δ is the friction angle mobilized along footing
sides. The punching shear coefficient Ks depends on friction angle of the reinforced
soil and limit bearing capacity of foundation on both reinforced zone and the following
unreinforced zone.
Similar to equation (2.1), the formulation of limit bearing capacity of square
footings on reinforced soil can be expressed as:
qu(R) = qu(b) +4cad
B+ 2γtd2 �1 +
2Df
d�
KstanφtB
+4∑ TitanδN
i=1
B− γtd (2.2)
Michalowski (2004) performed limit analysis and proposed bearing capacity
calculations for strip footings bearing on reinforced soil. Two failure modes were
considered: slip between soil and reinforcement and rupture of reinforcement.
For the case of slip failure at the soil-reinforcement interface, the limit bearing
capacity for foundation bearing on reinforced with a single layer of reinforcement is
given by Michalowski (2004) as:
𝑞𝑞𝑢𝑢(𝑅𝑅) =1
1 − 𝜇𝜇 𝑑𝑑𝐵𝐵𝑀𝑀𝑝𝑝
�𝑐𝑐(𝑁𝑁𝑐𝑐 + 𝑓𝑓𝑐𝑐𝑀𝑀𝑐𝑐) + 𝑞𝑞�𝑁𝑁𝑞𝑞 + 𝜇𝜇𝑀𝑀𝑞𝑞� + 𝛾𝛾𝐵𝐵(12𝑁𝑁𝛾𝛾 + 𝜇𝜇
𝑑𝑑𝐵𝐵𝑀𝑀𝛾𝛾)� (2.3)
where µ = friction coefficient of soil-reinforcement interface [see Eq. (2.4)]; fc = bond
coefficient [see Eq. (2.5)]; q= overburden pressure at the base of the footing; Nc, Nq
and Nγ are classic bearing capacity factors; Mc, Mq and Mγ are modification factors to
account for the additional bearing capacity induced by the reinforcement; and d/B is
the ratio of total reinforcement depth to footing width. When the soil-reinforcement 8
interface friction becomes equal to zero, Eq. (2.3) represents classic bearing capacity
equation. Coefficient of soil-reinforcement interface friction µ is a fraction fb of tan𝜙𝜙,
and the cohesive shear strength cint of soil-reinforcement interface is a fraction fc of the
apparent cohesion intercept c for the soil. Coefficients fb and fc are called “bond
coefficients”; Michalowski (2004) suggested that the use of fb = fc =0.6 for common
situations.
𝜇𝜇 = 𝑓𝑓𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝜙𝜙 (2.4)
𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑓𝑓𝑐𝑐𝑐𝑐 (2.5)
Expressions for bearing capacity factors Nc and Nq to be sued in Eq. (2.3) are
the form proposed by Prandtl (1920) and Reissner (1924):
𝑁𝑁𝑐𝑐 = �𝑁𝑁𝑞𝑞 − 1�𝑐𝑐𝑐𝑐𝑡𝑡𝜙𝜙 (2.6)
𝑁𝑁𝑞𝑞 = 𝑡𝑡𝑡𝑡𝑡𝑡2(𝜋𝜋4
+𝜙𝜙2
)𝑒𝑒𝜋𝜋𝑖𝑖𝜋𝜋𝑖𝑖𝜋𝜋 (2.7)
A solution of Nγ was suggested by Michalowski (1997a):
𝑁𝑁𝛾𝛾 = 𝑒𝑒0.66+5.11𝑖𝑖𝜋𝜋𝑖𝑖𝜋𝜋𝑡𝑡𝑡𝑡𝑡𝑡𝜙𝜙 (2.8)
Maximum benefit (in terms of foundation bearing capacity improvement) was
noted when the single reinforcement layer is placed at a depth above the apex off the
triangular elastic soil wedge just below the footing (Michalowski 2004), and in such
cases, Mc = Mq = Mγ=M. For a single layer of reinforcement, M is expressed as:
M = 1.6(1 + 8.5tan1.3ϕ) (2.9)
The coefficient Mp in Eq. (2.3) relates to soil friction angle as:
Mp = 1.5 − 1.25 ∙ 10−2ϕ (2.10)
For two and three reinforcement layers, the bearing capacity equation takes the form
9
1
1 − 𝜇𝜇𝑀𝑀𝑃𝑃 ∑𝑑𝑑𝑖𝑖𝐵𝐵
𝑖𝑖𝑖𝑖=1
�𝑐𝑐(𝑁𝑁𝑐𝑐 + 𝑡𝑡𝑓𝑓𝑐𝑐𝑀𝑀) + 𝑞𝑞�𝑁𝑁𝑞𝑞 + 𝑡𝑡𝜇𝜇𝑀𝑀� + 𝛾𝛾𝐵𝐵(12𝑁𝑁𝛾𝛾 + 𝜇𝜇𝑀𝑀�
𝑑𝑑𝑖𝑖𝐵𝐵
𝑖𝑖
𝑖𝑖=1
)� (2.11)
where n stands for the number of reinforcement layers, di refers to the depth of i th layer
of reinforcement. For the failure mode when all reinforcement layers are placed above
the apex of the rigid elastic triangular wedge below the footing, the coefficient M is
expressed as:
M = 1.1(1 + 10.6tan1.3ϕ) (2.12)
for two layers of reinforcement, and
M = 0.9(1 + 11.9tan1.3ϕ) (2.13)
for three layers of reinforcement. For such cases, the coefficient Mp is proposed to be
Mp = 0.75 − 6.25 ∙ 10−3ϕ (2.14)
for two layers of reinforcement, and
Mp = 0.50 − 6.25 ∙ 10−3ϕ (2.15)
for three layers of reinforcement.
For the case when reinforcement rupture becomes the governing failure mode,
the increment in bearing capacity of a RSF is attributed to work caused by plastic
deformation (or yielding) of reinforced soil mass. The strength of reinforcement layer is
expressed using the tensile strength Tt (per unit width) at some particular strain levels
(e.g., 2%). Based on limit analysis results, Michalowski (2004) proposed the following
form of bearing capacity equation for RSF when the reinforcement reaches the tensile
limit:
𝑞𝑞𝑢𝑢(𝑅𝑅) = cNc + qNq + 12γBNγ + NTt
BMr (2.16)
where N is the number of reinforcement layers, and 10
Mr = 2 cos �π4− ϕ
2� e�
π
4+ϕ2�tanϕ (2.17)
For closely packed reinforcement in clay, 3 reinforcement layers with a
interlayer spacing h=0.2B, the bearing capacity equation takes the form (Michalowski
2004)
𝑞𝑞𝑢𝑢(𝑅𝑅) = c𝑁𝑁𝐶𝐶 + 𝑞𝑞𝑁𝑁𝑞𝑞 + 12𝛾𝛾BNγ + 𝑘𝑘𝑖𝑖𝑀𝑀′𝑟𝑟 (2.18)
where 𝑘𝑘𝑖𝑖 is the tensile strength of reinforcement (per unit width) divided by the
interlayer spacing, and 𝑀𝑀′𝑟𝑟 is expressed as
𝑀𝑀′𝑟𝑟 = (1 + 𝑠𝑠𝑠𝑠𝑡𝑡ϕ)𝑒𝑒�𝜋𝜋2+ϕ�tanϕ (2.19)
2.2. Model-scale Experiments
One of the early experimental studies on RSF was performed by Binquet and Lee (1975)
to estimate the bearing capacity of strip footing on reinforced sand. They did groups of
small-scale model tests in order to simulate three types of subsurface conditions
including a deep homogenous sand foundation, sand over soft clay and sand above a
thin layer of clay. The experiments were performed in a 1.5m-long, 0.51m-wide, and
0.33m-high soil chamber. A model strip footing with width B=0.75m) was used in this
study. Results from this study demonstrated that the bearing capacity could be
enhanced as much as 2 to 4 times of that in case of unreinforced soil through reinforcing
the subsurface soil using metal reinforcement. It was also noted that a minimum
number of reinforcement layers are required for meaningful improvement in
foundation bearing capacity, and more layers of reinforcement could lead to better
foundation load bearing performance. Besides, results indicated that a total
11
reinforcement depth over 2B below the footing base could no longer enhance
foundation bearing capacity. The top layer spacing was found to be another factor
which may affect the bearing capacity. Results from the model tests demonstrated that,
placing the first layer at a distance d0 = 25 mm (i.e., d0/B = 0.3) below footing base
worked best. It was also reported that the broken positions of reinforcement were either
under the edges or along footing center rather than being close to the conventional slip
surface.
Since the early work reported by Binquent and Lee (1975), several model-scale
experiments were performed to estimate bearing capacity of footings resting on
reinforced sand (e.g., Huang and Tatsuoka, 1990; Yetimoglu et al., 1994; Adams and
Collin, 1997; Basudhar et al., 2007) and those on reinforced clay (e.g., Shin et al., 1993;
Das et al. 1994; Das and Shin 1994; Chen et al., 2007). Many scholars evaluated the
profits of using reinforced soil foundations (RSFs) via bearing capacity ratio (BCR),
which is calculated as the ratio of reinforced foundation bearing capacity to the bearing
capacity of unreinforced foundation (Fragaszy and Lawton, 1984; Mandal and Sah,
1992; Otani et al., 1998; Shin et al., 2002). Many studies focus on the parameters that
may contribute to an increase in BCR. Results from experiments reported in the
literature indicated an increase in bearing capacity and reduction in foundation
settlement when the subsurface soil was reinforced with geosynthetics. Better
foundation performance was observed if the reinforcement layers were buried within a
certain distance below the footing base; beyond this depth no significant improvement
was reported.
12
Mandal and Sah (1992) performed load tests for model footings bearing on
geogrid-reinforced (single layer) clay. According to their investigation, the maximum
BCR for a square foundation was about 1.36 for d0/B=0.175. The increase in BCR was
observed for all values of d0/B considered; however, the increase in BCR was
significant for d0/B<0.25. The maximum reduction in the settlement when using
geogrid-reinforcement beneath the compacted and saturated clay layer was found to be
approximately 45%, the corresponding top layer spacing d0 was reported to be around
0.25B.
Another similar study was performed by Omar et al. (1993) for square and strip
footings on sand. The ultimate load was the peak load obtained from the load-
displacement plot. Major conclusions from this study are: (1) for maximum bearing
capacity improvement, the effective total depth of reinforcement is about 2B and 1.4B
for strip and square footings, respectively; (2) maximum width of reinforcement
required for mobilization of maximum BCR is about 8B and 4.5B for strip and square
footings, respectively; (3) the maximum top layer spacing should be less than B for the
reinforcement to be effective in improving foundation load-settlement performance.
Das et al. (1994) studied the influence of total reinforcement depth d, top layer
spacing d0 (below the footing) and the reinforcement layer width LR on performance
of RSF. Two model-scale foundations (304.8mm×76.2 mm) made of aluminum plates
were placed on sand and clay beds prepared within a soil box (1.1m ×304.8mm×
0.91m). A thin layer of sand was cemented at the foundation base to ascertain rough
foundation-soil interface condition and the inner walls of the soil chambers were
13
polished so as to minimize the boundary effects. The optimal total reinforcement depth
(which leads to the maximum bearing capacity) for a given vertical spacing between
reinforcement layers was found to be approximately equal to 2B for sand and 1.75B
for clay, and optimal top layer spacing was reported in the range of 0.3B to 0.4B. The
optimum width of the geogrid-layer was found to be equal to 8B for sand and 5B for
clay. An interesting observation was that there is no significant difference in the
settlement at limit load of a strip footing on reinforced and unreinforced clay. However,
an increase in the limit load for strip footing on reinforced sand was accompanied by
an increase in settlement corresponding to the limit load. Furthermore, the maximum
bearing capacity ratio that could be gained for geogrid-reinforced sand was
significantly higher (approximately 3 times) than that of geogrid-reinforced clay.
Shin et al. (2002) reported a critical total reinforcement depth d for deriving the
maximum benefit from reinforcement is approximately equal to 2B for geogrid-
reinforced sand. For given values of d/B, d0/B, h/B, and LR/B, the value of BCR
(measured at limit load) increased with an increase in foundation embedment ratio Df/B:
Df is the footing depth of embedment of the foundation.
Model tests on circular and ring footings have also been reported in the
literature. Boushehrian and Hataf (2003) performed load tests and numerical analyses
for model circular and ring footings resting on reinforced sand. Model ring and circular
footings were made out of metal and rigid plastics (70mm thick). For the ring footing,
the inner and outer diameters were, respectively, 60 and 150 mm, resulting in an inner-
to-outer diameter ratio of 0.4, which was discovered to be the optimal value (results in
14
highest bearing capacity ratio) from the numerical analysis according to numerical
analyses performed by Boushehrian and Hataf (2003). Speaking of circular footings
bearing on reinforced sand, it was observed that the highest bearing capacity appears
at different values of d0/D0 (D0 is the diameter of circular footing) and h/D0 ratio
depending on the number of reinforcement layers. In terms of ring footings, it was
found out that an increase in the number of reinforcement layers result in the smaller
optimal vertical spacing between adjacent two layers. This phenomenon could be
explained by the depth effect, which generally means that best performance of
reinforcement is achieved when the reinforcement layers are closely placed. It was also
noted that a critical tensile rigidity threshold exists beyond which no significant
improvement in BCR could be achieved for circular and ring footings bearing on
reinforced sand.
Chen et al. (2007) investigated benefits for using reinforced foundations to
enhance bearing capacity and reduce immediate settlement in shallow foundations
bearing on clay with low to medium plasticity index. Influence of several parameters
(e.g., top layer spacing, the reinforcement layer number N, interlayer spacing h,
reinforcement tensile strength and the reinforcement type) on performance
enhancement of RSF was investigated. Geogrids and geotextile were both used as
reinforcement. Conclusions drawn from this research are: (1) the best top layer spacing
was around 0.33B in terms of the square footing bearing on geogrid-reinforced clay;
(2) the bearing capacity increased as the number of reinforcement layers increased, but
such increment decreased with an increasing number of layers; (3) the critical total
15
reinforcement depth (beyond which the improvement is insignificant) obtained was
around 1.5B for geogrid-reinforced clay and 1.25B for geotextile-reinforced clay; (4)
the higher bearing capacity could be achieved by shortening the interlayer spacing
between adjacent reinforcement layers; (5) immediate settlement of the footing was
reduced up to 50% by adding reinforcement layers; (6) clays reinforced with higher
stiffness geogrids generally yielded higher bearing capacity ratios than those reinforced
with lower stiffness geogrids, and (7) the effective width of geogrid was observed to
be approximately equal to 6B beyond which the improvement in performance is
insignificant.
Key observations and lessons learnt from model-scale tests on RSF reported in
literature are: (1) the optimum top layer spacing was reported to be ranging from 0.2B
to 0.5B, (2) the optimal interlayer spacing between two adjacent reinforcement layers
was discovered to be ranging from 0.2B-0.5B, (3) the effective reinforcement depth
varied from 1.0B to 2.0B, (4) the critical width of the reinforcement LR was found to
be in the range of 2.0B to 8.0B and (5) the geogrids having higher tensile modulus
performed better than those with lower tensile modulus. However, if we focus just on
the reinforced clay, limited papers (Mandal and Sah, 1992; Das et al., 1994; Shin and
Das, 1998; Chen et al., 2007) have shown that the optimal top layer spacing is from
0.175B to 0.4B, the critical width of reinforcement is around 6B and the effective total
reinforcement depth is in the range of 1.5B-1.75B irrespective of the footing type.
These differences absolutely separate the features of reinforced clay from that of
reinforced sand.
16
An interesting observation is that most of RSF studies focused on reinforced
sand. A possible explanation to this may be that BCR of reinforced sand can be as high
as 5 while BCR for reinforced clay is generally under 2, which makes reinforced-sand
a more attractive foundation subsurface compared to reinforced- clay. Nevertheless,
with increasing interest in utilizing shallow foundations on reinforced ground it is
equally important to characterize and quantify foundation performance on reinforced
clay.
2.3. Numerical Studies
With the advancement of computer technology, numerical analysis (mostly based on
finite element or finite difference method) has become an attractive alternative to
physical model tests for studying performance and characteristics of different earth
work applications. Additionally, carefully executed numerical analysis may save the
time and costs associated with physical model tests. Several researchers investigated
the performance of reinforced soil system using finite element analysis (FEA)
technique (e.g., Love et al. 1987, Kurian et al. 1997, Otani and Yamamoto 1998, Yoo
2001, Maharaj 2003, EI Sawwaf 2007, Ahmed et al. 2008, Alamshahi and Hataf 2009,
Chen and Abu-Farsakh 2011, to name a few). In all previously reported analyses, most
frequently used choices of soil elements can be generally divided into four types: 4-
node bilinear quadrilateral, 6-node quadratic triangle, 8-node biquadratic element and
8-node brick element (3D). Reasons for choosing a particular element type over
another type are not explicitly stated in these papers. Most of the previous studies
17
modeled reinforcement as a linearly elastic material; this is a reasonable assumption
since the induced axial strain in the reinforcement has been observed to be smaller than
about 2%.
The soil constitutive models used in the previous studies mostly include Mohr–
Coulomb model and a hyperbolic model prescribed by Duncan and Chang (1970) for
sand, and Cam-clay (Roscoe et al., 1958) and Drucker-Prager model (1952) for clay.
Some other soil constitutive models such as elastic-perfectly plastic model and Von-
Mises type of constitutive model were also used in studies involving clay. Particularly,
double-node FE was used in the study in terms of Von-Mises constitutive model (Otani
et al., 1998) and it was found to have an excellent accuracy for estimating the limit
bearing capacity of unreinforced clay, which is interesting since the double-node FE is
always used for fracture analysis. The reason why double-node FE is good for
simulating the limit bearing capacity of clay was not explained by the author. Moreover,
another new criterion for the ultimate bearing capacity of the sand foundation was
proposed by Yoo (2001); ultimate (limit) bearing capacity was defined as the pressure
at the footing base at a settlement, equal to 10% of footing width B.
Important details from some of the previous FEA of RSFs are listed in Table
2-1. While most of the previous numerical studies explored characteristics of
foundations on reinforced sand, studies on foundations bearing on reinforced clay is
rather limited in number. Key findings from some of the numerical studies on strip
footings on reinforced sand and clay are reported in Table 2-2.
18
Table 2-1 Summary of finite element analyses of RSFs
References Footing Type Soil Type Soil Model Interface Model Love et al. (1987) Strip Sand Elastic-perfectly Plastic _ Yetimoglu et al.
(1994) Rectangular Sand Modified Duncan
(1980) Friction
Nataraj and McManis (1996)
Strip Sand Duncan Interface Element
Kurian et al. (1997) Square Sand Duncan-Chang Hyperbolic (1970)
Goodman
Otani et al. (1998) Strip Clay Von-Mises No Relative Displacement
Yoo (2001) Strip Sand Duncan (1980) - Boushehrian and
Hataf (2003) Circular and
Ring Sand Mohr-Coulomb Soil &
Reinforcement combined together
Maharaj (2003) Strip Clay Drucker-Prager - Sugimoto (2003) Strip Sand Drucker-Prager Line Interface
Element EI Sawwf (2007) Strip Sand Friction-hardening
Plasticity Friction
Ahmed et al. (2008)
Strip Clay and Sand
Clay: Modified Cam-clay
Sand: Duncan-Chang
Slip Surface
Alamshahi and Hataf (2009)
Strip Sand Non-linear Mohr-Coulomb
Friction
Chen and Abu-Farsakh (2011)
Square Clay Drucker-Prager Hard Contact & Coulomb Friction
Ornek et al. (2012) Circular Sand Elastic-Plastic Mohr Coulomb
-
Raftari et al. (2013) Strip Sand Mohr-Coulomb - Azzam and Nasr
(2014) Strip Sand Mohr-Coulomb Rough Contact
19
Table 2-2 Summary of optimum parameters for reinforced soil foundations
Reference Factor
Maharaj (2003)
Basudhar et al. (2008)
Alamshahi and Hataf
(2009)
Jie (2011)
Raftari (2013)
Soil Type Clayey Soil Sand Sand Clay Sand Reinforcement
Type Geogrid Geotextile Geogrid Geogrid Geotextile
d0/B 0.125 0.6 0.75 0.3-0.6 - h/B - - 0.75 0.25 1
LR/B - - - 4 - d/B - - - 1.5 -
d0:top layer spacing; h:spacing between layers; LR: width of reinforcement layer; d:total reinforcement depth
Ti et al. (2009) reviewed several soil constitutive models including Mohr-
Coulomb, Cam-clay and Duncan-Chang model in geotechnical engineering
applications. According to this research, Mohr-Coulomb model is widely used because
researchers have shown that stress combinations leading to failure in soil samples in
triaxial tests match the failure contour of Mohr-Coulomb criterion (hexagonal shape)
(Goldscheider, 1984). The dilation angle and the friction angle should be set to different
values for most types of sands since non-associated flow rule is always in effect for
sands. However, precise stress-strain behavior on the way of reaching the maximum
shear strength and post-peak conditions (i.e., strain hardening and softening behavior)
cannot be captured using Mohr-Coulomb model. The Drucker-Prager model (Drucker
and Prager, 1952) could be regarded as a simplification of Mohr-Coulomb model
because a simple cone is used in Drucker-Prager model for failure cone instead of a
hexagonal shape in Mohr-Coulomb model. This model shares almost the same
advantages and weaknesses of Mohr-Coulomb model.
Duncan-Chang model is a hyperbolic stress-dependent constitutive model 20
(Duncan and Chang, 1970). This model is constructed according to stress-strain
response (which can be approximated as hyperbolic curve) obtained from drained
triaxial tests on clay and sand (Kondner, 1963). The failure criterion is based on two
strength parameters from Mohr-Coulomb model. Moreover, this model describes three
important constitutive characteristics of soil, including non-linearity, stress-dependent
and plastic behavior. A possible good reason for researchers to select Duncan-Chang
model is that input soil parameters could be easily obtained from traditional triaxial
tests. It is an explicit improvement to the Mohr-Coulomb model. However, this model
is not proper for limit load calculations in fully plastic state because numerical
instability may appear when failure is approaching.
Modified Cam-clay model was used in only a small number of RSF studies for
simulation of clay constitutive behavior. Compared with other soil models mentioned
in Table 2-1, the number of input parameters in the modified Cam-clay model is more,
which makes it harder to use. Five parameters are involved in this model: the isotropic
logarithmic compression index λ, the swelling index κ, stress ratio M, initial yield
surface size a0 and wet yield surface size β. Despite some good modifications to the
original Cam-clay model, Yu (1995, 1998) pointed out some limitations in modified
Cam-clay model. One of them was that an associated flow rule was chosen for this
model. Consequently, it was impossible to predict a peak in the deviatoric stress, which
is noticed in undrained tests on normally consolidated clay (relatively undisturbed) and
loose sand before the critical state is reached (Gens and Potts 1988). Nevertheless,
modified Cam-clay model is still a good choice for predicting the deformation of
21
saturated clay underneath a shallow foundation. In other words, this model is preferred
to express soil deformation over soil failure in terms of normally consolidated clays.
22
Chapter 3 Finite Element Modeling
This chapter presents pertinent details of finite element modeling adopted in the present
research. Material and geometric models used for the numerical analyses are described
in separate sections. Results from mesh convergence study are also presented.
3.1. Clay Constitutive Model
Mechanical behavior of clay is represented through a nonlinear elastic, perfectly plastic
constitutive model. Foye et al. (2008) used the same constitutive model in finite
element analyses (FEAs) of footings bearing on unreinforced clay. This model can
simulate nonlinear stress-strain response starting from small values of strains. Tresca
yield criterion, which shows a good agreement with experimental data of undrained
yielding of clay in shear (Potts and Zsdravkovic 1999), with a non-associated flow rule
(following Mises plastic potential function) is selected to represent the plastic behavior
of clay at large strains.
Strain-dependent degradation of secant shear modulus G from very early stage
of loading is captured using a two-parameter hyperbolic equation originally proposed
by Kondner (1963) and later modified by Fahey and Carter (1993). The ratio G/G0 is
expressed as:
GG0
= 1 − f �τ
τmax�g (3.1)
where G0 is initial shear modulus (at small strain); τ and τmax are current and
maximum shear stress respectively; f and g are material fitting parameters. While the
asymptotic value of G at large strain is determined through parameter f, the parameter
23
g dictates the rate at which G decreases with increasing shear stress. Lee and Salgado
(1999) proposed a modification of Eq. (3.1) for general three-dimensional stress
condition:
GG0
= 1 − f��J2
�J2,max�g
(3.2)
where J2 and J2, max are second invariant of deviatoric stress tensor and maximum value
of J2, respectively.
The parameters f and g in Eq. (3.1) and Eq. (3.2) are determined by fitting these
equations with laboratory test data. Mayne (2000) suggested that test results from
monotonic loading on clays can be well approximated using f =1 and g =0.3. Foye et
al. (2008) also demonstrated that the use of these values for f and g can successfully
reproduce test data from undrained torsional shear tests on clay specimens (Shibuya
and Mitachi 1994) and load-settlement response of model footing test reported by
Kinner and Ladd (1973).
Realistic estimation of initial shear modulus G0 is important for successful
prediction of nonlinear stress-strain response of clay even at small strain exceeding the
order of 10-6. Viggiani and Atkinson (1995) correlated G0 values for clays with mean
effective stress, stress history and plasticity index of clays as:
G0
pr= A�
σm′
pr�n
OCRm (3.3)
where pr is reference stress(=1kPa); σ’m is mean effective stress at the point of
calculation; OCR is overconsolidation ratio; and A, n and m are function of plasticity
index PI. The following equations are proposed by Viggiani and Atkinson (1995) for
24
parameters A, n and m:
A = 3790 exp(−0.045PI) (3.4)
n = 0.109 ln(PI) + 0.4374 for PI > 5 (3.5)
m = 0.0015PI + 0.1863 for PI > 5 (3.6)
3.2. Geometric Model for the Soil Domain
A commercial finite element analysis (FEA) software ABAQUS is used for the present
study. 6-node plane strain quadratic triangular element with modified and hybrid
formulation (CPE6MH) is used to discretize the soil domain. Reasons for choosing this
particular type of element are: (i) plane strain condition (which means that strain in the
out of plane direction is zero) exists in soil under a strip footing; (ii) quadratic elements
performs better in comparison to linear triangular elements (3-node) in predicting
displacement field below a vertically loaded footing (actually 3-node triangles can only
describe linear displacement field between nodes but 6-node triangles can simulate
nonlinear displacement field between nodes, which is more practical). Note that turns
out that 6-node triangle elements tend to lock volumetrically when used for modeling
incompressible materials (as is the case for undrained loading of clays) and the
displacements are under predicted. To reduce the effect of such volumetric lock, hybrid
formulation was selected so as to calculate displacement field on the element boundary
and stress field inside the element independently. In the hybrid formulation,
displacement field is calculated from external load and stiffness matrix and the stress
field is calculated based on the principle of minimum complementary energy. Modified
25
formulation was also picked to shift integration points away from conventional Gauss
points for mitigating the volumetric locking. Since the focus of the present study is to
simulate short-term foundation load-settlement response under undrained condition,
which associates with zero volume change under shearing, a value of Possion’s ratio
ν equal to 0.5 would be theoretically accurate. However, the use of ν = 0.5 causes
numerical instability and a value of ν = 0.49 is selected in the present study to avoid
such numerical problem. This is a reasonable value that yields minimal error for
undrained FE analysis of soil (Potts and Zdravkovis 1999).
Only half of the footing-soil system is modeled exploiting the symmetry about
the vertical axis passing through the center of strip footing. Two vertical sides of the
analysis domain are modeled as smooth support (i.e., fixed in the horizontal direction
but vertical displacement is allowed) and a fixed displacement boundary condition
(both horizontal and vertical displacements are zero) is used for the bottom boundary
since zero displacement is expected for points at great depth below the footing base.
FE mesh used for modeling of footing on unreinforced clay is shown in Figure 3-1.
Reinforced meshes are almost the same with the unreinforced mesh except that
reinforcement layers are embedded within the soil domain (Figure 3-2).
26
w
Figure 3-1 FE mesh for footing on unreinforced clay
Figure 3-2 FE mesh for footing on reinfnorced clay
An initial geostatic stress field is applied to the analysis domain to simulate in-
situ stress situation before loading. A buoyant unit weight equal to 7kN/m3 is used for
the assignment of geostatic effective stress condition. Figure 3-3 shows the distribution
of vertical geostatic stress within the analysis domain for one of the analyses performed
as part of this research. The coefficient of lateral earth pressure at rest K0 is assumed to
be equal to 0.6.
H
Fixed support
Smooth support
Smooth support
0.5B
27
Figure 3-3 Initial geostatic vertical effective stress contour
The foundation is assumed to be rigid. This is a practical assumption for any
reinforced concrete footing bearing on clay because the rigidity of foundation element
is much greater than that of underlying soil. Uniform displacement is expected at the
base of a rigid foundation and thus displacement-controlled method is used to simulate
the loading process. Analyses are performed for a surface footing and owing to the fact
that the foundation is a rigid one, small displacement increment (in the order of 0.5mm)
was applied at all nodes lying at the foundation base.
3.3. Modeling of the Reinforcement Layers
Most of the geogrids nowadays are made out of Polypropylene (PP) or High Density
Polypropylene (HDPP) that has a Possion’s ratio of around 0.4 and Young’s modulus
about 1.0 GPa (Ashby 2012). The thickness of commonly used geogrid is close to 1
mm and therefore, the mechanical behavior of geogrid layers can be regarded as that
of an Euler–Bernoulli beam instead of a Timoshenko’s beam since the aspect ratio of
the geogird is really high and thus the transverse shear (out-of-plane shear) can be
28
ignored. The related beam type element in ABAQUS is B23. The geogrid
reinforcement layers are modeled as linear elastic material with properties listed in
Table 3-1.
Table 3-1 Reinforcement properties
Material Possion’s
ratio v Young’s
modulus E Thickness
h Moment of Inertia I Element Type
Polypropylene (HDPP)
0.4 1.2GPa 1mm bh3/12
(b=unit out-of-plane thickness)
B23
3.4. Modeling of the Clay-Reinforcement Interface
The soil-reinforcement interface needs particular attention for successful solution of
the problem. Two interface layers are used for each reinforcement layer: one is at the
top of the geogrid and the other one is at the bottom. The interface layers are modeled
using traction-separation type cohesive element (COH2D4) built in ABAQUS. For
such interface element, the thickness of the interface layer is essentially zero. The clay
constitutive model described in section 3.1 is used as the material model for the
cohesive interface element. A contact interaction model is utilized to define the
connection between the cohesive interface and reinforcement layer. The normal
behavior of the interaction is selected as a ‘hard contact’, which implies that the
cohesive element will move with the reinforcement in the vertical direction. For the
transverse behavior, Coulomb friction model with friction coefficient = 0.6 is used with
a shear stress limit equal to the undrained shear strength su of clay. Figure 3-4 shows
the mechanical behavior of the friction model. The sticking friction increases with 29
increase in contact pressure at a constant rate (equal to the constant friction coefficient)
and the sticking friction becomes slipping friction after it reaches the shear stress or
shear stress limit (equal to the undrained shear strength su). The slip displacement limit
was set to be 10% of the geogrid thickness (1mm) beyond which the sticking friction
changes into slipping friction.
Figure 3-4 Coulomb friction model (adapted from ABAQUS 6.12 User’s Manual)
3.4. Convergence Study
A series of convergence study is performed for the unreinforced case in order to
ascertain reasonable mesh size and dimensions of the analysis domain. The right
boundary of the analysis domain is set at a distance w = 20B measured from the footing
centerline. Results from the analysis performed as part of the convergence study
confirmed that vertical displacement at the right boundary is indeed negligible (less
than 5% of the displacement below the footing base) and, therefore, W/B =20 is
selected for all analyses. The vertical distance (clay thickness) plays an important role
in quantification of footing settlement. Analyses are performed with H varying from
30
5B to 20B (H/B=5, 10 and 20). It is clear that the computed bearing stress at the footing
base converges to a constant value and there is no significant difference between
H/B=10 and H/B=20 (based on the mesh for which the number of elements below the
footing base is 20), as is shown in Figure 3-5. Thus, H/B=10 is used for all analyses in
the present study.
Figure 3-5 FE convergence study – effect of clay layer thickness on footing
settlement (for w/B=20)
The mesh density below the footing base is also varied to see the effect of
number of elements Nb below the footing base on foundation load-settlement behavior.
Analyses are performed with three different mesh densities with Nb=10, 20 and 50,
results are plotted in Figure 3-6. A convergence of load-settlement response is observed
for all values of Nb considered. However, the analysis terminates earlier as Nb increases
because element size decreases and mesh distortion becomes more severe with increase
0 1 2 3 4Normalized net stress at the footing base qb,net/su
0
10
20
30
40
Settl
emen
t s (m
m)
H=5BH=10BH=20B
31
in Nb. Based on the convergence study, H/B=20, W/B=20 and Nb=20 were chosen for
the analyses presented in Chapter 4.
Figure 3-6 FE convergence study – effect of different mesh densities on footing
settlement (for w/B = 20, H/B = 10)
3.5. Validation for the FE Model
Davidson and Chen (1977) performed finite element analyses for load-settlement
response of unreinforced clay due to loading from footings under plain-strain condition.
A linear elastic perfectly plastic soil constitutive model with Von-Mises yield criterion
and associated flow rule was used. In one of their analyses, the Young’s modulus E of
the clay was set to 14.4MPa with an undrained shear strength su=144kPa (E/su=100),
Possion’s Ratio v=0.48, effective (submerged) soil unit weight γ = 6kN/m3 and
coefficient of lateral earth pressure at rest K0 = 1. The value of initial Young’s modulus
E (or shear modulus G) reported in Davidson and Chen (1977) is a constant value
0 1 2 3 4Normalized net stress at the footing base qb,net/su
0
10
20
30
40Se
ttlem
ent s
(mm
)
Nb=10Nb=20Nb=50
32
following a linear elastic response (which is in contrast to a nonlinear elastic response
for the constitutive model used in the present study). Thus for a comparison to be
possible, the constant shear modulus G value reported in Davidson and Chen (1977) is
considered as an average shear modulus G during the shear modulus degradation
captured in the constitutive model used in the present research. Therefore, G0=2G is
used in this comparison study. Figure 3-7 shows a comparison of results reported by
Davidson and Chen (1977) and that predicted using FE model developed and used
(both geometric and constitutive model are same) in the present research.
Figure 3-7 Comparison of load-settlement curve reported in Davidson and Chen
(1997) with the one obtained using FE model used in the present research
The load settlement curve predicted using the present FE model matches well
with the initial part of the results reported by Davidson and Chen (1977); however, the
present analysis terminates early because of excessive distortion of the elements near
0 2 4 6Normalized net stress at the footing base qb,net/su
0
0.04
0.08
0.12
Nor
mal
ized
imm
edia
te s
ettle
men
t s/B Davidson and Chen (1977)
Present study
33
the footing base. Such early termination of analysis is certainly a drawback of present
study; the point of analysis termination varies with soil input parameters. Nonetheless,
it is anticipated that the results presented in this thesis can be used for practical range
of qb,net/su ratio that is allowed at the footing base.
Das and Shin (1994) conducted load tests on strip footing resting on
unreinforced and reinforced clay bed prepared within a laboratory-scale soil tank (229
mm wide, 607 mm high and 915 mm long). An average undrained shear strength su =
12 kPa and plasticity index PI = 20 was reported for the clay used in this study. The
footing width B was equal to 76mm. For the reinforced clay, the top layer spacing d0
was set to be equal to 0.4B and number of reinforcement layers N=5 with interlayer
spacing h=0.333B. A FE model was developed using these geometric details and the
soil properties provided in Das and Shin (1994) were used in the soil constitutive model.
However, Das and Shin (1994) does not provide the value of G0 for use in the soil
constitutive model. Comparison results shown in Figure 3-8 (a) (unreinforced clay) and
Figure 3-8 (b) (reinforced clay) are based on a G0 value equal to 250 kPa, which is
significantly low (almost 1/6 times) compared to that calculated (based on Viggiani
and Atkinson 1995) at a representative depth 2B below the footing base. The value of
G0 used in comparisons shown in Figure 3-8 are same as that would be calculated at a
depth (=0.2B) immediately below the footing. It is thus realized that although the
present FE solution scheme can successfully predict results from laboratory-scale
experiments such prediction is subjected to the uncertainty in ascertaining relevant soil
input parameters.
34
(a)
(b)
Figure 3-8 Comparison of load-settlement data reported in Das and Shin (1994) with that predicted using the FE modeling scheme employed in the present research: (a) unreinforced clay; (b) reinforced clay
For strip footings resting on unreinforced clay, the immediate load-settlement
0 1 2 3 4 5Normalized net stress at footing base qb,net/su
0
2
4
6
8
Settl
emen
t s (m
m)
Data from Das and Shin (1994)Present study
0 2 4 6 8Normalized net stress at footing base qb,net/su
0
4
8
12
16
Settl
emen
t s (m
m)
Data from Das and Shin (1994)Present study
35
response is also compared with that reported by Foye et al. (2008) [for PI=20,
G0=10MPa and su=50kPa, G0/su=200]. The coefficient of lateral earth pressure at rest
K0 is varied from 0.3 to 1.2 (K0=0.3, 0.5, 0.7, 0.9 and 1.2). The result of this comparison
is shown in Figure 3-9. Note that the vertical axis in Figure 3-9 represents immediate
settlement influence factor Iq, which is a direct reflection of immediate footing
settlement. The theoretical background for calculation of Iq is and is discussed in detail
in Chapter 4. Figure 3-9 confirms the validity of present FE modeling approach in
reproducing results from Foye et al. (2008) for K0 values lying between 0.7 and 0.9.
Figure 3-9 Comparison of influence factor reported in Foye et al. (2008) with the one
obtained based on FE modeling of the present research
0 1 2 3 4 5Normalized net stress at the footing base qb,net/su
2
4
6
8
Influ
ence
Fac
tor I
q
Foye et al. (2008)Present study K0=0.3Present study K0=0.5Present study K0=0.7Present study K0=0.9Present study K0=1.2
36
Chapter 4 Analyses and Results
Results from a series of finite element analysis (under undrained condition) of a strip
footing bearing on reinforced clay are presented in this chapter. A normalized
settlement influence factor Iq, which is a direct reflection of immediate settlement of
the footing, is introduced. A parametric study is performed to quantify the effects of
important input parameters factors (e.g., the number of reinforcement layers, undrained
shear strength, plasticity index, bending stiffness of reinforcement) that may affect Iq,.
Besides, several parameters related to the reinforcement arrangement and influence
depth beyond which the change in vertical stress becomes insignificant are also studied
and related results are shown in the subsequent sections.
4.1. Influence Factor Iq
Based on elastic FEAs Christian and Carrier (1978) developed design charts to estimate
immediate settlement of footings bearing on clay. These charts suggest that the
immediate settlement ρ can be expressed as:
ρ = 𝐼𝐼1𝐼𝐼0𝑞𝑞𝑏𝑏𝐵𝐵𝐸𝐸𝑢𝑢
(4.1)
where I1 is the influence factor related to footing shape and clay layer thickness beneath
the footing base; I0 is the influence factor related to the embedment depth; qb is unit
load (or stress) at the footing base; B is the footing width; and Eu refers to the
representative Young’s modulus of the foundation soil. However, the correction factor
I0 accounting for the embedment depth of the footing may not be conservative because
the reduction is settlement with increase in embedment depth is insignificant (Christian
37
and Carrier 1978; Burland and Burbidge 1985). Therefore, the factor I0 may be
excluded from Eq. (4.1), and the expression for ρ can be written as (Foye et al. 2008):
ρ = 𝐼𝐼𝑞𝑞𝑞𝑞𝑏𝑏,𝑖𝑖𝑛𝑛𝑖𝑖𝐵𝐵𝐸𝐸0
(4.2)
where Iq is settlement influence factor, E0 is a representative value of initial (small
strain) Young’s modulus of the subsurface soil, qb,net is the net applied stress at the
footing base. Eq. (4.2) can be rearranged to define the settlement influence factor Iq.
𝐼𝐼𝑞𝑞 =𝜌𝜌𝐸𝐸0
𝑞𝑞𝑏𝑏,𝑖𝑖𝑛𝑛𝑖𝑖𝐵𝐵 (4.3)
Note that Iq varies with the level of net load (or stress) applied at the footing base, and
thus quantification of Iq enables calculation of settlement at different levels of working
load. For a given footing dimension, load and subsurface condition a higher influence
factor indicates higher value of immediate settlement. Design charts containing
variations of Iq with qb,net/su would thus allow the designers to choose, without the need
for detailed analyses, different levels of net stress qb,net that can be applied on a footing
and directly obtain associated values of immediate settlement. Several factors may
affect immediate settlement influence factor Iq (and thus immediate settlement) for
footings bearing on reinforced clay. Such factors include the number of reinforcement
layers N, vertical spacing h between reinforcement layers, width LR of reinforcement
measured parallel to the footing width, total depth of reinforcement d, distance d0
between top layer of reinforcement and footing base, bending stiffness of
reinforcement (EI), undrained shear strength su and plasticity index PI of subsurface
clay. Therefore, it is important to quantify the effects of these parameters on the Iq –
qb,net/su variations. A prior knowledge of variation of Iq with reinforcement
38
arrangement parameters and reinforcement and clay properties will enable optimal
design of RSF on clay.
4.2. Effect of Number of Reinforcement Layers N on Settlement Influence Factor Iq
Past studies reported reduction in foundation settlement when the original foundation
soil (both sand and clay) is reinforced and more reinforcement layers are placed in the
soil layer. For all other input parameters being the same, a decrease in settlement
influence factor Iq is expected (from that for a footing on unreinforced soil) when one
and more number of reinforcement layers are placed below the footing base. Analyses
are performed for a strip footing bearing on reinforced normally consolidated clay with
different combinations of undrained shear strength su and plasticity index PI values and
the variations of Iq with N are reported. Following the relationship proposed by
Viggiani and Atkinson (1995), the initial shear modulus of clay G0 changes with change
in PI [see Eq. (3.3)]. For all analyses, G0 is calculated (based on an input value of PI)
at a depth 2B below the footing base. The zone of influence below a strip footing on
unreinforced clay is expected to extend down to a depth of 4B below the footing base
(Foye et al. 2008) and thus representative G0 values used in the present analyses are
calculated at a depth 2B below the footing base. Submerged unit weight for the
saturated normally consolidated clay layer is assumed to be equal to 7kN/m3. su values
used in the FEAs are decided based on assumed values of G0/su ratio. Table 4-1 lists
the soil input parameters used in the analyses.
39
Table 4-1 Soil input parameters used in the FEAs
PI G0 (MPa) G0/su su (kPa)
30 6.9 100 69
200 34.5
40 4.8 100 48
200 24
50 3.2 50 64
100 32
200 16
60 2.2 50 44
100 22
Number of reinforcement layers N are varied from 0 (unreinforced case) to 5
to quantify the reduction in settlement influence factor Iq with N (for a given set of soil
input parameters). The reduction of Iq is no longer significant when N reaches 4 for
most cases, indicating that four layers of reinforcement below the footing base is
perhaps most beneficial for immediate settlement reduction (Figure 4-1, Figure 4-2,
Figure 4-3). It is also observed that the reduction in Iq is more than 10% when the
number of reinforcement layers increases from 0 (unreinforced) to 4 for most cases
according to the figures below.
40
Figure 4-1 (a)
Figure 4-1 (b)
0 1 2 3 4Normalized net stress at footing base qb,net/su
2
2.4
2.8
3.2
3.6
4
Settl
emen
t inf
luen
ce fa
ctor
I q
N=0N=1N=2N=3N=4N=5
0 1 2 3 4Normalized net stress at footing base qb,net/su
2.4
2.8
3.2
3.6
4
Settl
emen
t inf
luen
ce fa
ctor
I q
N=0N=1N=2N=3N=4N=5
41
(c)
Figure 4-1 Variations of settlement influence factor Iq with number of reinforcement layers N for G0/su=200: (a) PI=30, G0=6.9MPa. (b) PI=40, G0=4.8MPa. (c) PI=50,
G0=3.2MPa.
Figure 4-2 (a)
0 1 2 3 4Normalized net stress at footing base qb,net/su
2.8
3.2
3.6
4
4.4
4.8
Settl
emen
t inf
luen
ce fa
ctor
I q
N=0N=1N=2N=3N=4N=5
0 1 2 3 4Normalized net stress at footing base qb,net/su
1.6
2
2.4
2.8
3.2
Settl
emen
t inf
luen
ce fa
ctor
I q
N=0N=1N=2N=3N=4N=5
42
Figure 4-2 (b)
Figure 4-2 (c)
0 1 2 3 4Normalized net stress at footing base qb,net/su
1.6
2
2.4
2.8
3.2
Settl
emen
t inf
luen
ce fa
ctor
I q
N=0N=1N=2N=3N=4N=5
0 1 2 3 4Normalized net stress at footing base qb,net/su
2
2.4
2.8
3.2
3.6
Settl
emen
t inf
luen
ce fa
ctor
I q
N=0N=1N=2N=3N=4N=5
43
(d)
Figure 4-2 Variations of settlement influence factor Iq with number of reinforcement layers N for G0/su=100: (a) PI=30, G0=6.9MPa. (b) PI=40, G0=4.8MPa. (c) PI=50,
G0=3.2MPa. (d) PI=60, G0=2.2MPa.
Figure 4-3 (a)
0 1 2 3 4Normalized net stress at footing base qb,net/su
2
2.4
2.8
3.2
3.6
Settl
emen
t inf
luen
ce fa
ctor
I q
N=0N=1N=2N=3N=4N=5
0 1 2 3Normalized net stress at footing base qb,net/su
1.6
2
2.4
2.8
Settl
emen
t inf
luen
ce fa
ctor
I q
N=0N=1N=2N=3N=4N=5
44
(b) Figure 4-3 Variations of settlement influence factor Iq with number of reinforcement
layers N for G0/su=50: (a) PI=50, G0=3.2MPa. (b) PI=60, G0=2.2MPa.
4.3. Effect of Undrained Shear Strength su on Settlement Influence Factor Iq
Love et al. (1987) reported that immediate settlement of footings on both unreinforced
and reinforced clays reduces with increase in undrained shear strength of undisturbed
soil. This observation could also be explained with the idea that the “stronger” clays
often tend to have stiffer responses in reaction to the load. Therefore, for footings on
clays with same PI but different values of su are expected to experience different levels
of immediate settlement. Figure 4-4 shows that for PI = 50 (corresponding calculated
G0 = 3.2 MPa) and for different levels of net normalized stress qb,net/su applied at the
footing base, Iq reduces by approximately 50% on an average as su increases from 16
kPa to 45 kPa.
0 1 2 3Normalized net stress at footing base qb,net/su
1.6
1.8
2
2.2
2.4
2.6
2.8
Settl
emen
t inf
luen
ce fa
ctor
I q
N=0N=1N=2N=3N=4N=5
45
Figure 4-4 Variations of settlement influence factor Iq with su based on PI=50,
G0=3.2MPa: G0/su=200, 133, 100 and 71, respectively.
4.4. Effect of Plasticity Index PI on Settlement Influence Factor Iq
The plasticity index PI, defined as the difference between liquid limit and plastic limit,
is a very routinely measured index parameter for clays. Foott and Ladd (1981)
concluded that the immediate settlement could be significant for highly plastic clays
(i.e., PI≥50). Thus it is necessary to perform a parametric study concerning the effect
of PI on immediate settlement of strip footings bearing on reinforced clay. Several
FEAs are performed (for N = 4 and su = 32 kPa) as part of the present research to
investigate the effect of PI on immediate settlement influence factor Iq and results are
shown in the Figure 4-5. While the absolute value of immediate settlement ρ increases
with increase in PI value, Iq decrease with an increase in PI (Figure 4-5). The value of
PI affects ρ and E0 in opposite ways. The rate of increase in ρ due to a ceratin increase
0 1 2 3 4Normalized net stress at footing base qb,net/su
1
2
3
4
5
Settl
emen
t inf
luen
ce fa
ctor
I q
Su=16kPaSu=24kPaSu=32kPaSu=45kPa
46
in PI is lower than the rate of decrease in E0 (or G0) for the exact same increase in PI.
Consequently, the net effect is reflected through a decrease in Iq [see equation (4.3)] as
a result of an increase in PI.
Figure 4-5 Variations of settlement influence factor Iq with PI based on su=32kPa: G0=6.9MPa, 4.8MPa, 3.2MPa and 2.2MPa for PI=30, 40, 50 and 60, respectively.
4.5. Optimal Depth for Placement of the Top Reinforcement Layer
The optimal depth d0, measured from the base of the footing, of placement of the top
reinforcement layer is determined through FEAs of a strip footing resting on clay bed
reinforced with a single layer of reinforcement (with width LR = 2B, B is the footing
width). Different combinations of G0/su are used for these analyses. Bearing capacity
ratio (BCR), i.e., the ratio of net vertical stress at the base of the footing on reinforced
soil to that for footing on unreinforced soil, is quantified for different levels of
0 1 2 3 4Normalized net stress at footing base qb,net/su
10
30
50
Settl
emen
t s (m
m)
4 3 2 1 0Normalized net stress at footing base qb,net/su
PI=60PI=50PI=40PI=30
2.4
3
3.6
Settl
emen
t inf
luen
ce fa
ctor
I q
47
immediate settlement (1%B, 2%B, 3%B and 4%B). For PI =56 (corresponding
G0=2.5MPa) and su=50kPa (i.e, G0/su=50), optimal top layer spacing d0 is found to be
around 0.5B (Figure 4-6).
Figure 4-6 Variations of BCR with depth d0 (measured from the footing base) of the
top reinforcement layer; based on FEAs with PI =56 (i.e., G0=2.5MPa) and su=50kPa
4.6. Optimal Reinforcement Width LR for a Single Layer of Reinforcement
FEAs are also performed with different values of reinforcement width (measured
parallel to the footing width) while keeping the top layer spacing d0 (=0.5B) and other
input parameters (PI = 40, G0=5MPa and su=50kPa) constant. Figure 4-7 shows that
for different levels of immediate settlement change in BCR is insignificant beyond LR
= 2B for both clay-reinforcement interface friction angle=30° [Figure 4-7 (a)] and 20°
[Figure 4-7 (b)]. Note that very low values (less than 4%) of increase in BCR are
0 1 2 3 4 5Normalized depth of top reinforcement layer d0/B
1
1.01
1.02
1.03
1.04
Bea
ring
capa
city
ratio
(BC
R)
s/B=4%s/B=3%s/B=2%s/B=1%
48
observed when a single layer of reinforcement is used, and thus, for all practical
purposes it is important to explore the load-settlement response of RSF with multiple
layers of reinforcement.
Figure 4-7 (a)
0 1 2 3 4 5 6Normalized reinforcement width LR/B
1
1.02
1.04
Bea
ring
capa
city
ratio
(BC
R)
s/B=3%s/B=2%s/B=1%
49
(b)
Figure 4-7 Variations of BCR (at different levels of footing settlement) with normalized reinforcement width LR/B for a single layer of reinforcement; based on
FEAS with PI = 40 (i.e., G0=5MPa) and su=50kPa: a) interface angle=30°; b) interface angle=20°.
4.7. Optimal Vertical Spacing between Two Reinforcement Layers
In order to evaluate an optimal spacing between reinforcement layers, FEAs are
performed for footings on reinforced clay with two layers of reinforcement. Different
values of vertical spacing h (=0.1B, 0.2B, 0.3B, 0.5B, 1B) are used for these FEAs and
for all analyses the top layer spacing d0 and width of reinforcement LR are kept at equal
to 0.5B and 2B, respectively. Initial shear modulus G0 (calculated from PI = 60) and
undrained shear strength su values used in these FEAs are 2.2MPa and 30kPa
(G0/Su=73), respectively. Optimal vertical spacing (for which BCR is the maximum) is
observed to be around 0.3-0.5B for settlement levels ranging from 1 to 4% of footing
0 1 2 3 4 5 6Normalized reinforcement width LR/B
1
1.02
1.04B
earin
g ca
paci
ty ra
tio (B
CR
)
s/B=3%s/B=2%s/B=1%
50
width B (Figure 4-8).
Figure 4-8 Variation of BCR (for different settlement levels) with normalized vertical
spacing h/B between reinforcement layers; based on FEAs with PI = 60 (i.e., G0=2.2MPa) and su=30kPa.
4.8. Effective Total Reinforcement Depth de
Performance of multi-layer (with N≥2) reinforced soil-foundation system is examined
by increasing the number of reinforcement N while keeping the other input parameters
constant (d0=0.5B, h=0.3B, PI = 60, G0 = 2.2 MPa, su = 22kPa). Optimal number of
reinforcement is 9, which means that the optimal total depth of reinforcement de below
the footing is around 3B (Figure 4-9).
0 0.2 0.4 0.6 0.8 1Normalized spacing between adjacent layers h/B
1.02
1.04
1.06
1.08
1.1
Bea
ring
capa
city
ratio
(BC
R)
s/B=4%s/B=3%s/B=2%s/B=1%
51
Figure 4-9 Increase in BCR with number of reinforcement layer N; based on FEAs
with PI = 60 (i.e., G0=2.2MPa) and su=22kPa.
4.9. Influence Zone ZR below the Footing Base
The effective zone of stress influence below the footing base is determined using the
recorded values of vertical effective stress increment ∆σ′v0 at different depths below
the footing base. Figure 4-10 shows that the normalized vertical effective stress change
∆σ′v0/ σ′v0 (where σ′v0 is the initial or in-situ vertical effective stress at a given depth
before the footing is loaded) converges to a constant value at a depth z=2.9B. Note that
∆σ′v0/ σ′v0 increases with N at depths closer to the footing base. Figure 4-10 also
signifies that the zone of stress influence ZR below the footing base extends down to a
depth approximately equal to 3B below the footing base.
0 0.5 1 1.5 2 2.5 3 3.5Normalized total depth of reinforcement d/B
1
1.04
1.08
1.12
1.16
1.2
Bea
ring
capa
city
ratio
(BC
R)
52
Figure 4-10 (a)
Figure 4-10 (b)
0 0.1 0.2 0.3 0.4 0.5Normalized horizontal distance x/B measured from the footing centerline
0
10
20
30
40
50
Nor
mal
ized
ver
tical
effe
ctiv
e st
ress
cha
nge
Depth z=0.5BN=6N=4N=2N=0
0 0.1 0.2 0.3 0.4 0.5Normalized horizontal distance x/B measured from the footing centerline
0
10
20
30
40
50
Nor
mal
ized
ver
tical
effe
ctiv
e st
ress
cha
nge
Depth z=1.1BN=6N=4N=2N=0
53
(c)
(d)
Figure 4-10 Variation of ∆σ′v0/ σ′v0 at different depths z below the footing base depth: (a) z = 0.5B. (b) z = 1.1B. (c) z = 1.7B. (d) z = 2.9B.
0 0.1 0.2 0.3 0.4 0.5Normalized horizontal distance x/B measured from the footing centerline
0
10
20
30
40
50
Nor
mal
ized
ver
tical
effe
ctiv
e st
ress
cha
nge
Depth z=1.7BN=6N=4N=2N=0
0 0.1 0.2 0.3 0.4 0.5Normalized horizontal distance x/B measured from the footing centerline
0
10
20
30
40
50
Nor
mal
ized
ver
tical
effe
ctiv
e st
ress
cha
nge
Depth z=2.9BN=6N=4N=2N=0
54
4.10. Optimal Number of Reinforcement and Interspacing for Multi-layer System
It is important to explore if varying the interlayer spacing h or number of reinforcement
layers N within the same total reinforcement depth d (increasing the interlayer spacing
h would reduce the number of reinforcement layers N within the a constant depth)
would change BCR. In other words, it is important to explore if the reinforcement effect
is independent of interspacing or number of layers and only has something to do with
the total reinforcement depth.
Six combinations of h and N are used (h=0.2B and N=10; h=0.25B and N=8;
h=0.4B and N=5; h=0.5B and N=4; h=1B and N=2; h=2B and N=1) to investigate the
combined effect of h and N on BCR. For all the analyses, the total reinforcement depth
d (i.e., the distance from the footing base to the last reinforcement layer) is equal to
2.5B, G0=3.2MPa and su=32kPa. The results are plotted in Figure 4-11. Significant
increase in BCR (at s/B =3.2%) is observed with increase in N (and corresponding
reduction in h). However, when the interlayer spacing h becomes too small the number
of layers is excessive. Such a configuration is unrealistic because it would be difficult
from a construction point of view to embed too many layers at very small intervals
unless the footing is really wide.
55
Figure 4-11 Variation of BCR (at s/B=3.2%) for different N-h combinations; based on
FEAs with PI =50 (i.e., G0=3.2MPa) and su=32kPa.
4.11. Influence of Reinforcement Bending Stiffness
Past research, both numerical analyses and experiments, have demonstrated that the
flexibility (which is a direct demonstration of bending stiffness) plays important role
in defining failure mode within reinforced soil. Although it is more prominent for
geosynthetics reinforced soil (GRS) walls under ultimate limit (i.e., associated with
collapse) condition, it is of interest to investigate if reinforcement stiffness would play
a significant role in defining serviceability limit state for foundations bearing on
reinforced soil.
Different values of Young’s modulus E is used in FEAs to account for the effect
of reinforcement bending stiffness on foundation load-settlement performance. Figure
4-12 shows that the settlement influence factor Iq decreases with increase in the value
0 2 4 6 8 10Number of reinforcement layers N
1.04
1.08
1.12
1.16
1.2
Bea
ring
capa
city
ratio
(BC
R)
N=10 h=0.2BN=8 h=0.25BN=5 h=0.4BN=4 h=0.5BN=2 h=1BN=1 h=2B
56
of E; the minimum value of Iq is achieved when the value of E approaches to that of
steel and more than 20% reduction in the settlement influence factor Iq could be
achieved with a 7 times increase in the bending stiffness of the reinforcement (E to 8E).
Figure 4-12 Influence factors for different reinforcement stiffness; based on FEAs with PI =30 (i.e., G0=6.9MPa), su=34.5kPa, N=4, and E=1.2GPa.
4.12. Effect of Clay Layer Thickness on Settlement Influence Factor Iq
The clay layer thickness H affects immediate settlement of footing supported on it for
both unreinforced (as shown in the “Convergence Study” section in Chapter 3) and
reinforced cases. To study the influence of clay layer thickness on Iq for the reinforced
case, FEAs are performed with three values of H/B = 5, 10 and 20 (PI=60, G0=2.2MPa,
su=22kPa, N=4, d0=0.5B and h=0.3B. Figure 4-13 demonstrates that beyond H = 10B,
the thickness of the underlying clay layer does not affect immediate settlement of the
strip footing.
0 1 2 3 4 5Normalized net stress at footing base qb,net/su
1.6
2
2.4
2.8
3.2
3.6
4
Settl
emen
t inf
luen
ce fa
ctor
I q
0.25E0.5EE2E4E8ESteel
57
Figure 4-13 Influence factor for different clay thickness (N=4)
0 1 2 3 4Normalized net stress at footing base qb,net/su
1.2
1.6
2
2.4
2.8
3.2
3.6
Settl
emen
t inf
luen
ce fa
ctor
I q
H=20BH=10BH=5B
58
Chapter 5 Discussion and Conclusions
Key findings from the present research are compared to and discussed in light of past
related studies reported in literature. Important conclusions drawn from this study are
summarized.
5.1. Comparison with Results Reported in Previous Numerical Studies on Footings on Reinforced Clay
Comparisons are made for several reinforcement arrangement factors (e.g., top layer
spacing d0, vertical spacing between adjacent two reinforcement layers h, width of the
reinforcement layer LR) reported in literature. While Maharaj (2003) reported an
optimal top layer spacing d0 = 0.125B, this ratio falls in the range of 0.3-0.6B according
to Jie (2011). Based on limit analysis Chakraborty and Kumar (2012) reported the
upper and lower bound solutions for d0 to be equal to 0.22B and 0.64B. The present
research finds the optimal value of d0 to be approximately around 0.5, which is in
general agreement with values reported in literature.
The optimal ratio of interlayer spacing h to B is reported to be equal to 0.25 for
multi-layer within the effective total reinforcement depth (Jie 2011) and in the range of
0.22 to 0.64 for 2-layer case analyzed by Chakraborty and Kumar (2012). Based on
FEA results obtained as part of the present study, the optimal h/B value is in the range
of 0.3-0.5 for 2-layer and 0.2-0.25 for multi-layer system.
As for the optimum reinforcement width LR, the ideal ratio of LR to footing
width B was found to be about 4 according to Jie (2011). The same value obtained from
the present study is equal to 2; this discrepancy might have been caused due to several
59
reasons which may include the difference in the soil constitutive model, the difference
in element type used to model the reinforcement layer, difference in interface model
and finally the difference in settlement level s/B at which BCR was evaluated (while
s/B=10% in Jie 2011, s/B is less than 5% in the present study).
For the effective influence depth of reinforcement d, the critical value of d/B
was reported to be around 1.5 by Jie (2011), however, this value is around 2 to 3 based
on the present study. A brief comparison of results from different numerical studies is
given in Table 5-1.
Table 5-1 Summary of optimal parameters as reported in different numerical studies
References Maharaj (2003)
Jie (2011) Chakraborty and Kumar (2012)
Present study (2015)
Analysis type
Parameters
Limit analysis of strip
footings on reinforced
clay
FEA of strip footings on reinforced
clay
Limit analysis of strip footings on reinforced clay
FEA of strip footings on
reinforced clay
d0/B 0.125 0.3-0.6 0.22-0.64 0.5 h/B
N/A
0.25 0.22-0.64 0.2-0.5 L/B 4 2 2 d/B 1.5 N/A 2-3
BCR 1.1-2 1.13-1.84 1.03-1.3
60
5.2. Comparison with Previous Experimental Studies
Results from past experimental studies on footings bearing on reinforced clay are also
compared with results from the present study in terms of reinforcement arrangement
factors. A summary of such comparison is shown in Table 5-2.
Table 5-2 Comparison of optimal parameters reported in past experimental studies with those obtained from the present research
References Parameters
Mandal and Sah (1992)
Das et al. (1994)
Shin and Das (1998)
Chen et al. (2007)
Present study (2015)
Footing Type Square Strip
Strip with slope
Square Strip
d0/B 0.175 0.3-0.4 0.4 0.33 0.5 h/B
N/A N/A
N/A N/A 0.2-0.5
L/B 5 6 2 d/B 1.75 1.72 1.5 2-3
BCR 1.36 (max) 1.1-1.5 1.4-1.7 1.02-1.6 1.03-1.3
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5.3. Conclusions
Specific conclusions drawn from the present study are:
(1) An increase in the number of reinforcement layers N below the footing would result
in immediate settlement reduction. More than 10% reduction in immediate
settlement influence factor Iq is observed, compared to that for unreinforced case,
when four layers of reinforcement are used (N = 4, h = 0.3B, d0 = 0.5B) below the
footing. The exact amount of such reduction in Iq with increase in N also depends
on soil input parameters.
(2) The settlement influence factor Iq decreases as the undrained shear strength su
increases, approximately 50% decrease is observed when su increases from 16kPa
to 45kPa, indicating lower immediate settlement levels for footings on clays with
higher undrained shear strengths.
(3) The settlement influence factor Iq increases with decrease in plasticity index PI
when initial shear modulus G0 decreases with PI following Viggiani and Atkinson
(1995) relation. Nonetheless, absolute value of immediate settlement of footing
increases with increase in PI.
(4) The optimal depth d0 (below the footing base) of top layer placement is about 0.5B.
(5) The optimal reinforcement width LR is around 2B for soil-reinforcement interface
friction angle equaling 30° and 20°.
(6) The critical value of vertical spacing h between reinforcement layers is in the range
of 0.2B to 0.5B.
62
(7) The effective total depth of reinforcement d below the footing base is about 3B.
(8) The stress influence depth below the footing base (at which the increment in
vertical effective stress is same for reinforced and unreinforced cases) is
approximately equal to 3B.
(9) Bending stiffness of reinforcement layers plays an important role in the
performance of strip footings on geogrid-reinforced clay. According to the results
presented in this thesis, more than 20% reduction in the settlement influence factor
Iq could be achieved with a 7 times increase in the bending stiffness of the
reinforcement.
(10) The clay thickness does affect the influence factor Iq; the impact becomes
insignificant when the ratio of clay thickness to the footing width H/B is higher
than 10.
63
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