IMMEDIATE LOAD-SETTLEMENT RESPONSE OF STRIP FOOTINGS

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

http://en.wikipedia.org/wiki/Laser

http://en.wikipedia.org/wiki/Ultrasonics

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)


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


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)


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


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)


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


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)


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


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


10

30

50

Settl

emen

t s (m

m)


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

10

20

30

40

50

Nor

mal

ized

ver

tical

effe

ctiv

e st

ress

cha

nge


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

10

20

30

40

50

Nor

mal

ized

ver

tical

effe

ctiv

e st

ress

cha

nge



0

10

20

30

40

50

Nor

mal

ized

ver

tical

effe

ctiv

e st

ress

cha

nge


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)


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

61

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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Documents

IMMEDIATE LOAD-SETTLEMENT RESPONSE OF STRIP FOOTINGS