Rahemi
Rahemi
Rahemi
Negar Rahemi
Submitted to the
Institute of Graduate Studies and Research
in partial fulfillment of the requirements for the Degree of
Master of Science
in
Civil Engineering
Eastern Mediterranean University
July 2012
Gazimağusa, North Cyprus
Approval of the Institute of Graduate Studies and Research
Prof. Dr. Elvan Yılmaz
Director
I certify that this thesis satisfies the requirements as a thesis for the degree of Master
of Science in Civil Engineering.
Asst. Prof. Dr. Murude Çelikağ
Chair, Department of Civil Engineering
We certify that we have read this thesis and that in our opinion it is fully adequate in
scope and quality as a thesis for the degree of Master of Science in Civil
Engineering.
Asst. Prof. Dr. Huriye Bilsel
Supervisor
Examining Committee
1. Assoc. Prof. Dr. Zalihe Nalbantoğlu
2. Asst. Prof. Dr. Giray Ozay
3. Asst. Prof. Dr. Huriye Bilsel
ABSTRACT
In this thesis a 3D nonlinear analysis was performed to study the lateral deflection of
single pile with different slenderness ratios (L/D) under static and dynamic loading.
Different models of pile, soil and loading have been simulated, and the lateral
deflections were studied considering elastic, elasto-plastic and dynamic models of
the soil. For seismic load modeling Ricker wavelet was used. 3D finite element
method was applied for numerical modeling and the ABAQUS program version
6.11was utilized to evaluate the lateral deflection of pile.
It was concluded that one of the most effective parameters on lateral deflection of
pile is slenderness ratio; the pile length has more influence on increasing the lateral
deflection. Furthermore, the lateral deflections versus depth along pile length at the
same slenderness ratio revealed that the piles with bigger diameters exhibited less
lateral deflections, which might be attributed to the increase in the surface area and
hence the skin friction between pile and soil.
Ratio, Ricker Wavelet.
iii
ÖZ
Bu tez çalışmasında 3D nonlinear statik ve dinamik analiz yöntemleri ile yanal
yükler altında kazık temelin narinlik katsayısının (L/D) yanal deplasmanlara etkisi
çalışılmıştır. Farklı kazık, zemin ve yükler modellenmiş ve bunların yanal harekete
edilmiştir. Sismik etkiyi yaratabilmek içinse, sismik modellemede sıklıkla kullanılan
ve band-sınırlı bir frekans içeriği olan Ricker dalgacığı kullanılmıştır. 3D modelleme
sonlu elemanlar yöntemi ile ABAQUS yazılımı kullanılarak değerlendirilmiştir.
Sonuç olarak yanal deplasmanlar üzerinde en büyük unsurun narinlik katsayısı
olduğu ve bu katsayı sabit tutulup kazık boyu ve çapı değiştirildiği zaman, boyunun
artmasının yanal deplasmanları artırırken, çapının artmasının ise yanal deplasmanları
düşürdüğü gözlemlenmiştir. Ayrıca, çapın artması ile kazık yüzey alanının arttığı ve
dolayısıyla sürtünme direncinin de artmasından dolayı yanal deplasmanlarda azalma
olduğu sonucuna varılmıştır.
katsayısı, Ricker dalgacığı.
iv
ACKNOWLEDGMENT
I wish to express my gratitude to a number of people who became involved with this
thesis. I would like to express my profound appreciation to my supervisor Asst. Prof.
Dr. Huriye Bilsel for her continuous caring and valuable guidance in the preparation
of this study.
This thesis could not have been accomplished without Nariman, my husband who is
always with me, I am indebted to him. He always gives me warm encouragement and
love in every situation. He is the center of my universe, and a continuous source of
strength, peace and happiness.
I believe I owe deepest thanks to all people in my entire family, my father, my mom
and my brother for their patience, love, and continuous presence. Their prayer for me
was what sustained me thus far. Words cannot express how grateful I am to
them. My deep appreciations also go to my father-in-law and mother-in-law for their
express my heartfelt gratitude to them.
v
To the four pillars of my life: God, my husband, and my parents.
Without you, my life would fall apart.
I might not know where the life’s road will take me, but walking with
You, God, through this journey has given me strength.
Nariman, you are everything for me, without your love and
understanding I would not be able to make it.
Mom and Daddy, you have given me so much, thanks for your faith in
me, and for teaching me that I should never surrender.
vi
TABLE OF CONTENTS
ABSTRACT ........................................................................................................... iii
ÖZ ........................................................................................................................... iv
ACKNOWLEDGMENT ........................................................................................... v
LIST OF TABLES ................................................................................................... ix
LIST OF FIGURES .................................................................................................. x
1 INTRODUCTION ................................................................................................. 1
1.1 General Observation ........................................................................................1
2 LITERATURE REVIEW ....................................................................................... 5
2.1 Analytical Models for Static Lateral Loading of Piles ......................................5
2.1.1 Beam on Elastic Foundation Method ........................................................5
2.1.2 Elastic Continuum Theory ...................................................................... 13
2.1.3 The Finite Element Theory ..................................................................... 15
2.2 Analysis of Piles under Dynamic Lateral Loads ............................................. 18
2.2.1 Continuum Model ................................................................................... 18
2.2.2 Winkler Model ....................................................................................... 22
2.2.3 Finite Element Method ........................................................................... 26
3 METHODOLOGY .............................................................................................. 32
3.1 Introduction ................................................................................................... 32
3.2 Assumptions and Limitations ........................................................................ 34
3.3 Three Dimensional Finite Element Model ..................................................... 34
3.3.1 Mode1 Simulation .................................................................................. 34
3.3.2 Soil Properties ........................................................................................ 36
3.3.3. Pile Properties ....................................................................................... 48
vii
3.3.4 Soil-Pile Interface ................................................................................... 48
3.4 Boundary Conditions ..................................................................................... 50
3.5 Loading Conditions ....................................................................................... 50
3.5.1 Static Loading ........................................................................................ 50
3.5.2 Dynamic Loading ................................................................................... 52
3.6. Verification of Finite Element Model ........................................................... 53
4 NUMERICAL MODELLING .............................................................................. 56
4.1 Introduction ................................................................................................... 56
4.2 Model Description ......................................................................................... 56
4.2.1 Pile and Soil Properties ........................................................................... 56
4.3 Static Analysis Results .................................................................................. 58
4.3 Dynamic Analysis Results in Elastic Soil ...................................................... 76
5 CONCLUSIONS ................................................................................................. 81
5.1 Conclusions ................................................................................................... 81
5.1.1 Static Analysis results ............................................................................. 81
5.1.2 Dynamic Analysis results ....................................................................... 82
5.2 Future Recommendations .............................................................................. 82
REFERENCES ....................................................................................................... 83
viii
LIST OF TABLES
Table 1: Seismic waves properties (Braile, 2010) ........................................................... 4
Table 2: The advantages and limitations of Mohr-Coulomb and Drucker-Prager
models ......................................................................................................................... 43
Table 3: The relationship between Drucker-Prager material constants and the Mohr-
Coulomb parameters (Serdaroglu, 2010) ...................................................................... 46
Table 4: All model details ............................................................................................ 57
Table 5: Pile and Geotechnical Properties ................................................................... 57
Table 6: Details of Model............................................................................................. 57
ix
LIST OF FIGURES
Figure 1: Typical models of pile foundations (a) Bearing pile, (b) Friction pile, (c)
piles under uplift, (d) piles under lateral load, (e) Batter piles under lateral load
(Prakash & Sharma, 1990) ............................................................................................. 3
Figure 2: Schematic soil-pile structure interaction under seismic excitation.................... 3
Figure 3: Subgrade Reaction Modulus Model, (a) Soil and Pile reaction, (b) Soil
model and the influence of a partial uniform pressure over it (Poulos & Davis, 1980) .... 6
Figure 4: Finite Difference Analysis Model for Laterally Loaded piles (Poulos &
Davis, 1980) .................................................................................................................. 8
Figure 5: P-y Diagram Model for Soil-Pile (FHWA, 1997) ............................................ 9
Figure 6: Contact Stresses Distribution against the Pile Before and After Lateral
Bending (Reese & Van Impe, 2001) ............................................................................. 10
Figure 7: Schematic of Strain Wedge Model for Analyzing Lateral Load Pile
(Ashour and Norris, 2000) ........................................................................................... 12
Figure 8: Distribution of Soil-Pile Interaction along Deflected Pile (Ashour and
Norris, 2000)................................................................................................................ 12
Figure 9: Continuous Analysis Model of Soil-Pile Stress reacting on (a) Pile, (b) Soil
around the Pile (Poulos & Davis, 1980) ....................................................................... 14
Figure 10: A Model of a 3-Dimensional Finite Element Mesh for a Single Pile (Yang
& Jeremic, 2002) ......................................................................................................... 16
Figure 11: A Model of a 3-Dimensional Finite Element Mesh for a Single Pile after
(Shahrour, Ousta, & Sadek, 2001) ................................................................................ 17
Figure 12: Finite Difference Meshing (a) Three Dimensional View, (b) Zoom in
View of Sleeved Region (Ng & Zhang, 2001) .............................................................. 18
x
Figure 13: P-y Method for Hysteretic Damping and Cyclic Degradation in Soil
(Brown et al., 2001) ..................................................................................................... 23
Figure 14: P-y Curve (a) Typical Set of p-y Curves for a given Soil Profile (b) curve
plotted on common axes (Tomlinson & Woodward, 2008) ........................................... 23
Figure 15: Model to Analyze Radiation Damping for a Horizontally Vibrating Pile
(Wang et al., 1998) ...................................................................................................... 24
Figure 16: Beam on Nonlinear Winkler Foundation model with Different Damping
Influences (Nogami & Konagai, 1988) ......................................................................... 26
Figure 17: Hybrid Dynamic Winkler Model for Lateral Pile Response (Nogami &
Konagai, 1988) ............................................................................................................ 26
Figure 18: Finite Element Boundary Elements and Meshing for quarter model, (a)
Top Plan of the Model, (b) Elevation (Maheshwari et al., 2004) ................................... 28
Figure 19: Overall view of the half of pile-soil system with mesh ................................ 35
Figure 20: Schematic of Infinite Elements (a) Three-Dimension Solid Continuum
Element, (b) Axisymmetric Solid Continuum Element (ABAQUS, 2010) .................... 35
Figure 21: Failure surface for sand in the deviator plane (ABAQUS, 2010).................. 38
Figure 22: Deviator plane failure surfaces for two parameter models (ABAQUS,
2010) ........................................................................................................................... 39
Figure 23: Drucker-Prager failure surface (ABAQUS, 2010) ....................................... 41
Figure 24: Yield Surfaces on a Deviator Plane (a) Drucker-Prager and Mohr-
coulomb yield surfaces on deviator plane, (b) the Drucker-Prager failure surface on a
deviator plane, (c) Mohr-Coulomb yield criterion on a deviator plane (Serdaroglu,
2010) ........................................................................................................................... 42
Figure 25: Hardening behavior, (a) Drucker-Prager, (b) common (Jostad et al.
(1997)) ......................................................................................................................... 44
xi
Figure 26: Linear Drucker-Prager (ABAQUS, 2008) ................................................... 45
Figure 27: Drucker-Prager Yield surface from ABAQUS (2008) ................................. 47
Figure 28:Yield Surface and Plastic Flow direction in the p-t plane from
(ABAQUS,2008) ......................................................................................................... 47
Figure 29: Schematic FEM Soil-Pile interface Elements, (a) No sliding, (b) Sliding ..... 49
Figure 30: Interface between Soil and Pile, (a) Slave surface, (b) Master surface ......... 49
Figure 31: Kelvin Elements (ABAQUS, 2010) ............................................................. 50
Figure 32: Pile schematic as a cantilever beam, a point as a bedrock ............................ 51
Figure 33: Ricker wavelet used in present elastic medium dynamic analysis ................ 53
Figure 34: Schematic of the seismic excitation wave that comes through existing
rigid bedrock ................................................................................................................ 53
Figure 35: Simulated pile with mesh ............................................................................ 54
Figure 36: Plane x-z of soil-pile model shows the mesh density near the pile ............... 54
Figure 37: Top side of soil-pile model shows the mesh density around the pile ............ 55
Figure 38: Comparison between Beam Flexure Theory and ABAQUS result ............... 58
Figure 39: Lateral deflection of pile head in different depth of pile, L=3, D=0.3 .......... 59
Figure 40: Lateral deflection of pile head in different depth of pile, L=5, D=0.5 .......... 60
Figure 41: Lateral deflection of pile head in different depth, L=7.5, D=0.75................. 60
Figure 42: Lateral deflection of pile head in different depth, L=9, D=0.9 ..................... 61
Figure 43: Lateral deflection of pile head in different depth, L=6, D=0.3 ..................... 62
Figure 44: Lateral deflection of pile head in different depth, L=10, D=0.5 ................... 62
Figure 45: Lateral deflection of pile head in different depth, L=15, D=0.75 ................. 63
Figure 46: Lateral deflection of pile head in different depth, L=18, D=0.9 ................... 63
Figure 47: Comparison between max lateral deflections of pile head in different
depth for (L/D) =10...................................................................................................... 64
xii
Figure 48: Comparison between max lateral deflections of pile head in different
depth for (L/D) =20...................................................................................................... 65
Figure 49: Pile head deflection vs. Lateral loading in plastic behavior of soil ............... 66
Figure 50: Lateral deflection of pile head under lateral loading in elasto-plastic
behavior of soil ............................................................................................................ 66
Figure 51: Comparison between Elastic and Elasto-plastic results for (L/D) =10 .......... 67
Figure 52: Comparison between Elastic and Elasto-plastic results for (L/D) =20 .......... 67
Figure 53: Lateral deflection of pile head in different depths, L=3, D=0.3 .................... 68
Figure 54: Lateral deflection of pile head in different depths, L=9, D=0.9 .................... 69
Figure 55: Lateral deflection of pile head in different depths, L=6, D=0.3 .................... 69
Figure 56: Lateral deflection of pile head in different depth, L=18, D=0.9 ................... 70
Figure 57: ABAQUS 3D plot of lateral deflection of pile under static loading for
(L=10m, D=0.5m, P=50 kN) ........................................................................................ 70
Figure 58: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=50 kN in elastic soil ................ 72
Figure 59: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=100 kN in elastic soil .............. 72
Figure 60: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=150 kN in elastic soil .............. 73
Figure 61: Pile lateral displacements along length ( z ) , normal to pile head
displacement along line of loading 0 , under load P=200 kN in elastic soil .............. 73
Figure 62: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=50 kN in elasto-plastic soil ...... 74
xiii
Figure 63: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=100 kN in elasto-plastic soil .... 74
Figure 64: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=150 kN in elasto-plastic soil .... 75
Figure 65: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=200 kN in elasto-plastic soil .... 75
Figure 66: ABAQUS results after applied Ricker wave on the bedrock for the middle
node of soil element on the surface .............................................................................. 76
Figure 67: 3D plot of free field displacement under dynamic loading ........................... 77
Figure 68: Comparison between lateral deflections of pile under 100kN static load
and Dynamic load ........................................................................................................ 78
Figure 69: Comparison between lateral deflections of pile under 100kN static load
and Dynamic load ........................................................................................................ 78
and Dynamic load ........................................................................................................ 79
and Dynamic load ........................................................................................................ 79
Figure 72: 3D plot of lateral deflection of pile under dynamic loading for (L=5m,
D=0.5m) ...................................................................................................................... 80
xiv
Chapter 1
1 INTRODUCTION
Based on recent statistics, many people died in earthquakes worldwide in the last
decade. The majority of deaths occurred in developing countries where urbanization
and population is increasing rapidly without any serious control. The Middle East
region is located at the intersection of main tectonic plates, the African, Arabian and
Eurasian plates, cause of very high tectonic activity. Many earthquake disasters in
the past happened in the Middle East, influencing most countries in the region.
Middle East, extending from Turkey to India is one of the most seismically active
regions of world. It is clear that earthquakes not only damage structures and
buildings but also influence on human lifeline, social and economic losses. As a
result of the high probability of earthquake happening mixed with incremental
population, poor construction standards and the absence of correct mitigation
strategies, Middle East demonstrates one of the most seismically susceptible regions
of the world.
Nowadays, with increasing number of structures and lessening space, the higher and
heavier buildings are built by engineers. The costly and strategic structures like
skyscrapers, offshore platforms and etc. bring out much important risks and new
design problems.
1
Recent destructive earthquakes in Japan, Turkey, and Iran are reminded the
importance of pile foundations and their effect on the response of the supporting
structures. The costs of fixing pile foundations are very expensive regarding time and
cost. In contrast to shallow foundations, the pile foundations can extend to deeper,
stronger soil layers and bedrock to set tolerable resistance. Figure 1 indicated typical
pile foundations for different structures.
Although the static loading is necessary in pile designing, it is the dynamic loading
which presents the important challenge to the design engineer. Dynamic excitation
and lateral loading poses extra forces on the pile foundations. These pile foundations
have to be designed to protect lateral loads cause of earthquakes, wind, and any
impact loads. It is often essential to do a dynamic analysis of the pile for lateral
Figure 2 demonstrates the general description of the problem follow study (Gazetas
& Mylonakis, 1998). It is illustrated in the general case of an embedded foundation
supported with piles but all the final results are logical for any foundation type.
Descriptions of wave characteristics and particle motions for the four wave types are
given in Table 1.
2
Figure 1: Typical models of pile foundations (a) Bearing pile, (b) Friction pile, (c)
piles under uplift, (d) piles under lateral load, (e) Batter piles under lateral load
(Prakash & Sharma, 1990)
Figure 2: Schematic soil-pile structure interaction under seismic excitation
3
Table 1: Seismic waves properties (Braile, 2010)
The main objective of this investigation is to assess the lateral deformation of pile in
sand with elastic behavior and elasto-plastic behavior, under static and dynamic
loading with finite element method. Due to the limited goal of this study, the analysis
is concentrated on a single pile located in homogenous soil deposit, and evaluated as
a semi-infinite space. The pile has a straight axis, circular cross section, and it is
placed vertically.
4
Chapter 2
2 LITERATURE REVIEW
Chapter 2 reviews the literature on soil-pile interaction under lateral loads. The goal
of this chapter is to review past research on pile behavior under static and dynamic
loadings in cohesion-less soils. The simulating and modeling of soil-pile interaction
and analysis of pile behavior under dynamic and static loads are the two main
sections of this study.
There are three main approaches for evaluating pile deflection under lateral loading,
as Poulos and Davis (1980) and Fleming et al. (1992) are illustrated.
In 1867 Winkler proposed this model, which was introduced as Beam on Elastic
Foundation (BEF) and Beam on Winkler Foundation (BWF). This model suggests
that in soil-pile contact at any point along the pile length, there is a linear relationship
independent.
Additionally, the model represents the soil as a series of unconnected linearly-elastic
beam-column element.
5
Figure 3: Subgrade Reaction Modulus Model, (a) Soil and Pile reaction, (b) Soil
model and the influence of a partial uniform pressure over it (Poulos & Davis,
1980)
Kh (Force/Length3) is the oil modulus of lateral (horizontal) subgrade reaction,
represented by the spring modulus of the model, and is the horizontal pressure
needed to create a unit horizontal displacement. Kh depends on foundation size,
depth and soil type.
The equation of equilibrium of the beam (pile) under the influence of distributed load
(w) is shown in Equation 2.1.
d 4V
E pI p W pd K hV d (2.1)
dZ 4
d 4V
EpI p K hV d 0
dZ 4
where:
E p = pile modulus of elasticity
I p = pile cross section moment of inertia
d = pile diameter
w = soil reaction per unit length over the pile (distributed load)
p = soil pressure over the pile
K h = soil lateral subgrade-reaction modulus
6
V= Lateral displacement of the pile
Analytical solutions, despite the fact that limited concerning practical applications,
present a consequential perception in to the pile reaction and the factors which
impact the soil-pile interaction. These solutions have been acquired for the Equation
1.1 for the case of constant k h with specific boundary condition and depth (for
example, Hetenyi (1946) and Scott (1981), describes the Hetenyi solution in a detail
procedure).
Vesic (1961) obtained accurate elastic solutions of infinite beams on isotropic half
space acted upon by couple and concentrated loads. He suggested values for the
subgrade reaction modulus by comparing these solutions with the Winkler method
solutions. Therefore, the Winkler model provides logically accurate results for the
subgrade reaction modulus under medium and long length beams.
Kagawa (1992) assessed the factors affecting the subgrade reaction modulus K h , and
presented one dimensional analysis, suggesting a protocol to obtain an average value
of K h as a function of the Soil Young’s Modulus which may be used for pile
analysis established upon the BEF.
A logical method of analysis of beam on elastic foundations was improved by Vlasov
& Leont’ev (1966) on the basis of the elastic continuum approach; considering only
to be zero). Finite Difference Method (FDM) has been recommended and carried out
since the early fifties; for instance by Palmer and Thompson (1948), Reese and
Matlock (1956).
7
The basic differential Equation 1, in this model is written in finite-difference form,
and the solution is found at separate points. The general separated model for FDM is
presented in Figure 4. The discretization of the solution with the FDM has the
disadvantages such as the difficulty to define general boundary conditions at the tip
and top of the pile, and that the elements have to be uniform in size.
Foundation engineering books like Bowles (1974) and Bowles (1996) can be used as
a reference for numerical solutions with the Finite Element Method with one-
dimensional elements, or beam elements. This model is mostly mentioned as the
Stiffness Method.
Figure 4: Finite Difference Analysis Model for Laterally Loaded piles (Poulos &
Davis, 1980)
Some authors suggest a complete design procedure for piles have been used the
Beam on Elastic Foundation method (BEF). Possibly, the most familiar of these
approaches is worked by Broms (1964a, 1964b, 1965). For predicting the deflection
under working loads, and for estimating ultimate load resistance supplied by
8
assuming a number of simple ultimate states for the soil-pile system, the BEF
method is used.
The original Beam on Elastic Foundation (BEF) model does not explain the non-
linear reaction of the soil by itself. P-y method is the most common model to
consider the non-linear nature of soil reaction. In this approach the spring stiffness
value is variable, allowing consideration of a non-equivalent relationship between the
soil resistance per unit pile length (p) and the lateral displacement (y).
researchers have presented methods of solution by the FDM method and determined
the p-y curves for different soils and depths based on experimental results, and
obtained information on how to improve a computer program [(Matlock & Reese,
1960), (Matlock, 1970), and (Reese, 1977)]. A schematic of the soil-pile modeling
and the p-y curve for each non-linear spring is shown in Figure 5.
Figure 5: P-y Diagram Model for Soil-Pile (FHWA, 1997)
9
Figure 6 shows a distribution of contact stresses before and after pile lateral
deflection. It is important to mention that (p) is not a contact stress, but the
consequent of the contact stresses and friction along the pile perimeter for a specified
depth. The (p) value depends on soil type and depth, pile shape and type, and the
deflection amount (y) when the reaction is non-linear.
Figure 6: Contact Stresses Distribution against the Pile Before and After Lateral
Bending (Reese & Van Impe, 2001)
For various types of soils, different p-y curves have been improved which are
demonstrating the magnitude of soil pressure as a function of the pile deflection
Brown et al. (1994), (Bransby M. , 1999), (Ashour & Norris, 2000)].
10
Sometimes the p-y method is mentioned as the (BNWF) Beam on Non-linear
Winkler Foundation system to model the soil-pile interaction such as Wang et al.
(1998) and Hutchinson et al. (2004), or as the Load Transfer Method (Basile, 2003).
The p-y method or the finite difference methods in many analyses including the
subgrade reaction method were replaced by Finite Element Method, with
improvement of this model (Hsiung & Chen, 1997), (Sogge, 1981).
In spite of the fact that the concentration the p-y method is not resulted the ultimate
capacity of laterally loaded piles. Nevertheless this method cannot be suitable to
determine the ultimate capacity cause of yielding of the soil. To estimate the ultimate
capacity due to soil yielding, with the assumption of the soil is perfectly plastic; the
Beam on Elastic Foundation method is used.
The Strain Wedge Method, which lets the evaluation of non-linear p-y curve reaction
of laterally loaded piles settled on the three dimensional soil-pile interaction response
through a passive wedge soil developing in front of the pile, improved by Ashour et
al. (1998), Ashour and Norris (2000) as shown in Figure 7 and Figure 8. It should be
noted that the Matlock and Reese (1960) p-y curve was based on the results of field
tests on instrumented piles. This method allows relating the stress-strain-strength
acting of the layered soil in the 3D wedge approach to the 1D BNWF model
parameters. So, the non-linear reaction may be achieved from the analysis that looks
at the actual conditions of soil-pile system (soil layers classification, pile diameter,
etc.).
11
Figure 7: Schematic of Strain Wedge Model for Analyzing Lateral Load Pile
(Ashour and Norris, 2000)
Figure 8: Distribution of Soil-Pile Interaction along Deflected Pile (Ashour and
Norris, 2000)
The Winkler method to predict soil-pile reaction under static lateral loads is used for
BEF analysis which still is a topic of research. Some of the recent developments are
considered here. Shen and Teh (2004) recommended a variation method (same as the
12
Ritz approach) to present the analysis of the laterally loaded pile in a soil with
subgrade reaction modulus increasing by depth. For the maximum bending and
deflection of laterally loaded piles in a soil with uniform subgrade reaction modulus,
(Hsiung, 2003) has proposed the theoretical method. In a site in Korea, Kim et al.
(2004) have managed lateral field tests on instrumented piles, acquiring the p-y
curves and evaluating the effect of the installation method and fixed head conditions
in the soil-pile response.
The modeling of the soil as a homogeneous elastic continuum has been suggested for
the analysis of the soil-pile interaction. For the analysis of limit pile capacity, Plane
Strain Models were developed with some authors like Davis and Booker (1971). For
modeling the 3D system as a series of parallel horizontal planes in plane strain, the
Plane Strain Models are used which are related to the case of shallow-embedded
sheet piling.
Douglas and Davis (1964); Spillers and Stoll (1964); Poulos (1971, 1972), and other
established on Mindlin’s method for the horizontal displacement due to a horizontal
which can be found in various Elasticity handbooks, such as Poulos and Davis
(1974).
Douglas and Davis (1964); Spillers and Stoll (1964) and Basile (2002) recommended
integral solutions over a predefined area, representing a fraction of the pile surface,
corresponding to the point where the load is located. These solutions which define
13
the displacement field due to an assumed loading system (pattern) associated with
the pile-soil interaction, are generally known as Green Functions.
Utilization of the model suggested by Poulos (1971, 1972), was presented by Poulos
and Davis (1980) . The pile is assumed to be a thin rectangular vertical strip divided
in elements in this model, and it is observed that each element is acted upon by
uniform horizontal stresses as it shown in Figure 9 which are related to the element
displacements through the integral solution of Mindlin’s problem.
At last, in which soil pressures over each element are unknown variables, they
realized the differential equation of equilibrium of a beam element on an infinite soil
with the Finite Difference Method (FDM). The displacements are found after
achieving the pressures.
Figure 9: Continuous Analysis Model of Soil-Pile Stress reacting on (a) Pile, (b)
Soil around the Pile (Poulos & Davis, 1980)
The ability to take into consideration the homogeneous nature of soil, the semi-
infinite dimension of the half-space, and the boundary conditions along the unloaded
14
ground surface is the advantage of this model. Although yielding of soil may be
presented by varying the soil elastic modulus, this method does not allow to consider
local yielding and layered soil conditions.
In this way, Spillers and Stoll (1964) suggested the calculation of the maximum
allowable load by any appropriate yielding condition (e.g. a wedge model for the top
part of the soil, where the yielding of soil happens and displacements are larger),
together with the elastic solution and a repetitive procedure to control that the
maximum load is not exceeded at any point.
Two of the disadvantages of the discretization by cooperating with means of the
FDM is that the difficulty to present general boundary conditions at pile top and
bottom, and the needed uniform size of the elements. This soil model was used for
the BEM (Boundary Element Method) analysis of piled foundations, as Basile (2002)
informed.
The Finite Element Method (FEM) has been recommended and implemented to
perform a numerical analysis of the soil-pile system to obtain solution for laterally
loaded flexible piles in an elasto-plastic soil mass.
Poulos and Davis (1980) informed that the first tries included two-dimensional finite
element models in the horizontal plane Baguelin and Frank (1979); general 3D finite
element analysis recommended by Desai and Appel (1976), and axis-symmetric
geometries by Banerjee and Davis (1978).
15
Some recent work such as Yang and Jeremic (2002) used 3D Finite Element
Methods of a laterally loaded pile driven in layered and uniform soil profiles so that
numerically achieve p-y curves and compare them to experimental ones as shown in
Figure 10.
Figure 10: A Model of a 3-Dimensional Finite Element Mesh for a Single Pile
(Yang & Jeremic, 2002)
Permitting to account for soil non-linearity by applying appropriate constitutive
models like the Drucker-Prager model is the Finite Element Method ability, [(Ben
Jamma & Shiojiri, 2000); (Yang & Jeremic, 2002)], and using gap-elements to be
able to model soil-pile separation. Capabilities of these modeling are usually
available in strong general purpose objective FEM programs such as ABAQUS and
ANSYS or limited geotechnical engineering software like PLAXIS.
Mostafa and EI Naggar (2002), Wang et al. (1998), EI Naggar and Novak (1996),
Kagawa (1992), and Poulos and Davis (1980) agreed that using FEM analysis is
feasible for the design of large structures only. This is due to the cost of the
specialized software, the time needed for non-linear analysis, the time required for
model creation, the difficulty in explanation of the result in terms of usual pile
16
(beam) variables, and the un-certainties related with soil non-linear modeling in 3D.
Figure 11 presents the finite element meshes utilized for a single micro pile in the
numerical analyzing.
Figure 11: A Model of a 3-Dimensional Finite Element Mesh for a Single Pile after
(Shahrour, Ousta, & Sadek, 2001)
Finally, the practical 3D finite difference method program in recent years called
Flac3D has been used to analyze the complex geotechnical problems, but for pile
analysis it has been rarely used. Ng and Zhang (2001) used 3D finite difference
method to evaluate the behavior of piles placed on a cut slope.
They studied the influence of the sleeving (doughnut of compressible stuff which is
from the buildings to the shallow depths of the slope) on the pile performance, as
shown in Figure 12.
17
Figure 12: Finite Difference Meshing (a) Three Dimensional View, (b) Zoom in
View of Sleeved Region (Ng & Zhang, 2001)
In this section, methods of modeling dynamic behavior of single piles under lateral
loads such as seismic shock have been presented. These are Beam on Non-linear
Winkler Foundation (BNWF), which defines the soil as a series of continuous
springs, Continuum Methods, which makes closed form analysis by assuming soil as
an infinite semi-space, Boundary Element Method (BEM), and Finite Element
Method (FEM), which defines the soil as a homogenous medium. A short
explanation of these approaches is discussed in this section, with special mention on
seismic analysis.
Automatically inclusion of radiation of energy to infinity, called Radiation Damping
through the complicated explanation of pile stiffness, is the main advantage of the
Continuum Method over the FEM and BNWF.
spite of that material damping may be considered by using the complicated form of
the material properties, or Lame’s invariants. Only by changing the elastic modulus
of the full space, the non-linear behavior of the soil can be accounted. This is another
18
disadvantage of the continuum approach. In addition, the soil medium should be
homogeneous or consist of homogeneous layers, with confined boundary conditions.
In spite of this method’s limitations, it is very useful to obtain a better understanding
of the soil-pile interaction, and to acquire the analytical explanation of parameters,
like Subgrade Reaction Modulus (Vesic, 1961), which can be utilized in the Winkler
Model.
The summary of some studies which are related to the aim of the present research
can be summarized as follows:
An approximate continuum model to explain soil-pile interaction proposed by Novak
(1974), where the soil is supposed to made up of a set of independent horizontal
layers of very tiny thickness, extending to infinity.
This model may be observed as a generalized Winkler Method, due to having an
independent plane. The planes are mentioned to be in a plane strain, and those are
isotropic, homogeneous, and linearly elastic.
A differential equation of the damped pile in horizontal motion was formulated by
Novak (1974). For harmonic vibration persuaded through pile ends, he proposed the
Steady State solution, and used it for different boundary conditions to determine
dynamic stiffness of the pile head.
The dimensionless variables which control the soil-pile system reaction, according to
Novak (1974) are:
19
(1) The relation between the shear wave velocity of the soil ( v s ) and the
(2) The relation between specific mass of the soil (ρ) and the specific mass of the
a0 r0 , where G is the shear modulus of the soil),
G
(4) The relation between the pile length (L) and the pile radius ( r0 ),
(slenderness ratio L / r0 ),
By considering a rigid pile cap at the pile heads for a pile group and single pile head,
Novak (1974) recommended the equivalent damping and stiffness constants. He
suggested a numerical example, and compared the reaction of a spread footing with
pile foundations, obtaining the following results:
(1) Pile foundations natural frequencies and resonant amplitudes are more than
spread footings, but their damping are smaller and those are more rigid, (2)
The resonant amplitudes can decrease due to pile (and spread footing)
embedment, (3) The Dynamic Analysis of pile foundations is more important
than shallow foundations because they can’t do away with vibrations,
however the piles can decrease the settlements.
As Klar (2003) reported, the stiffness acquired from the Novak (1974) solution
approaches to zero, as the frequency approaches to zero. Additionally, Novak’s
methods approach to Tajimi’s method (based on a more accurate 3D analysis) by
frequency rising. Because higher frequency wave tends to spread more horizontally,
20
Novak’s model predisposes to get the real behavior. Hence, at very low frequencies
and static states Novak’s solution gives poor results, where for high frequencies it
gives better results.
For comparison with the Winkler model (where the soil is designed as separate
springs and dashpots), Nogami and Novak (1980) studied the coefficient of dynamic
soil response to pile movement, treating the soil as a 3D continuum. They obtained
the followed conclusions by these assumptions:
The Assumptions: (a) Cylindrical elastic pile driven to the bedrock; (b)
Homogeneous soil layer overlying a rigid bedrock; (c) The soil vertical motion is
ignored; (d) Constant hysteretic damping material for linear viscoelastic soil; (e)
Harmonic movement; (f) No relation between soil-pile interface and soil-pile
movement.
The conclusions: (1) The soil damping and local stiffness strongly varies with
frequency, depth, and with respect to soil-pile stiffness; (2) The 3D explanation is the
frequency of the soil layers, as recommended by Novak (1974), and (3) For piles
with stiffer materials and soil deposits with greater depth, it is better to use the
Winkler assumptions. The outcomes of earlier works were not exactly used to
evaluate the soil mass contribution to the dynamic response in a distinct Winkler
model. However these can be used in assessing the pile dynamic stiffness
(Impedance) and modeling the damping and stiffness influences of the soil-pile
dynamic reaction in the Winkler method.
21
2.2.2 Winkler Model
Since the seventies Matlock et al. (1978) the p-y methods for explaining the lateral
stiffness of soil-pile model for seismic analysis has been utilized, taking in to account
that both pile and soil can treat in a nonlinear manner during greatest events. Wang et
al. (1998), Polam et al. (1998), and Boulanger et al. (2004) have worked on this
model. According to p-y model, the cyclic soil degradation should be using. For
executing this analysis, the common linear modal analysis should be replaced by an
iterative nonlinear time-domain analysis, as the expected nonlinear response cannot
be feasible by linear modal analysis Brown et al. (2001).
can define the soil stiffness by a dashpot in parallel with a spring. This is the famous
Kelvin-Voigt model for visco-elastic materials.
Figures 13 and 14 present the hysteretic damping, which is the energy scattering
due to soil nonlinear behavior, which can be explained in p-y method by letting the
unloading path to differ from loading path Brown et al. (2001).As the cyclic loading
proceeds, soil strength decreases which is known as soil degradation.
22
Figure 13: P-y Method for Hysteretic Damping and Cyclic Degradation in Soil
(Brown et al., 2001)
Figure 14: P-y Curve (a) Typical Set of p-y Curves for a given Soil Profile (b)
curve plotted on common axes (Tomlinson & Woodward, 2008)
Radiation damping is the dissipation of energy in the soil-pile system because of the
departing stress waves which transfer from pile-soil interface to infinity. Wang et al.
(1998), Berger et al. (1977), and other researchers recommended an easier approach
by supposing that the pile cross section only creates one dimensional P-waves
23
travelling in the direction of the shaking, and one dimensional S-waves travelling
perpendicular to the pile as it shown in Figure 15a.
Novak et al. (1978) suggested a more accurate model by assuming a plain strain state
for the soil which is linearly elastic, isotropic, and homogeneous. They have
evaluated the pile experiencing uniform harmonic motions in an infinite medium.
The problem become easier for 3D compared to 2D, due to the pile being regarded
as rigid and infinitely long, without mass, like a stiff circular disc vibrating in an
infinite elastic plane as shown in Figure 15b.
Gazetas and Dobry (1984a), (1984b) recommended a simplified model by supposing
that compression-extension waves spread in the two fourth-parts planes along the
direction of shaking, and that S-waves spread in the two fourth-parts perpendicular
to the direction of shaking as presented in Figure 15c. From each of the previous
methods, the coefficient of dashpot (C) can be concluded. A damper like this with
(C) coefficient is settled in parallel with the non-linear spring element.
Figure 15: Model to Analyze Radiation Damping for a Horizontally Vibrating Pile
(Wang et al., 1998)
24
Kagawa and Kraft (1980; 1981), and Badoni and Makris (1996) by using a BNWF
damping and stiffness for a soil-pile system.
The BNWF method is a simple method which can use for nonlinear Soil Pile
and investigate exercises. Various authors have suggested that the Winkler Model
represents a Continuum Model on the assumption that the soil is an isolated
horizontal plane in a plane strain condition of stresses.
As Nogami and Novak (1980) presented these solutions are extremely good
estimations of the real 3D behavior for frequencies higher than basic natural
frequencies of the soil deposit. To compute the linear response of single piles in
bending, established up on the plane strain solutions, Nogami and Konagai (1988)
(presented in Figure 16, 17).
25
Figure 16: Beam on Nonlinear Winkler Foundation model with Different Damping
Influences (Nogami & Konagai, 1988)
The moderated Winkler Model was verified for a large spectrum of frequencies, and
occasionally mentioned as a Hybrid Dynamic Winkler Model (HDWM).
Figure 17: Hybrid Dynamic Winkler Model for Lateral Pile Response (Nogami &
Konagai, 1988)
For static loading, Wolf (1985) defined an imaginary boundary at an adequate
distance from the pile where it was expected that the response disappears from a
feasible point of view, it is introducing as soil with finite domain. For boundary
nodes, pin supports are allocated for restraining displacements, and the finite area is
26
meshed. This imaginary boundary can send back the waves produced by the
vibrating pile due to dynamic loading, in to the defined soil medium. However, in
reality it should be letting the waves propagate into infinity. More attention has to be
paid in putting suitable damping capability at the boundaries of the soil finite element
model. In the following paragraphs, a brief summary of some recent publications is
presented which are used as references for this research. In the following paragraphs,
a brief summary of some recent publications is presented which are used as
references for this research:
A finite element model has been developed by considering the soil nonlinear
behavior and introducing Drucker-Prager yielding criteria, wave scattering by putting
the excitation at the bottom of the model, and discontinuity conditions at the soil-pile
interface by introducing contact elements that enable to slippage (Bentley & El
Naggar , 2000). They used this model for comparing the free-field soil reaction with
the soil-pile system reaction, and dynamic soil-pile response. They applied
at the bedrock meshes, and they realized that the response of piles in elasto-plastic
soil is almost similar to the free-field response to the low frequency seismic
excitation.
Regarding the influence of material nonlinearity in the soil and separation at the soil
pile interface on the dynamic response of a single pile and group piles, Maheshwari
et al. (2004) proposed 3D finite element method to achieve the pile reaction under
loading on the pile cap and seismic excitation. Figure 18 depicts the finite element
meshing which Maheshwari et al. (2004) used.
27
Increasingly the Boundary Element Method (BEM) has been used in the laterally
loaded piles evaluation. Ben Jamma and Shiojiri (2000) utilized a mix of finite
element method and thin layer element for assessing the hybrid soil substructure
system and the dynamic reaction of single pile driven in an infinite half space. Basile
(2003) accounted on the advantages of the boundary element method for soil-pile
interaction modeling and evaluation.
Figure 18: Finite Element Boundary Elements and Meshing for quarter model, (a)
Top Plan of the Model, (b) Elevation (Maheshwari et al., 2004)
the soil, piles, and soil pile structure interfaces are designed simultaneously together.
28
Angelides and Roesset (1980), Randolph (1981), Faruque and Desai (1982),
Trochanis et al. (1991) and Wu and Finn (1997) utilized finite element method for
pile dynamic analysis. Soil is treated as a continuum mass in FEM.
The earliest studies of dynamic response of piles and soil-pile interaction are in a
consequence of Parmele et al. (1964), (Novak, 1974), Novak et al. (1978) to explain
the dynamic elastic stress and displacements of fields by using a nonlinear
viscoelastic bed layer to model the soil and he neglected the vertical part of the soil
movement in his studying of the horizontal response. Novak (1974) supposed
linearity and an elastic layer of soil made-up of independent thin horizontal layers
reaching out to infinity.
Trochanis et al. (1988) utilized a three dimensional nonlinear analyze of piles to gain
some insight of the lateral performance of piles, which led to the improvement of the
simplified medol. In that study the ABAQUS (1987) was utilized. For soil, the
pile gapping at their interface was evaluates for lateral and axial response of piles
due to cyclic and monotonic loading. The piles were simulated as a concrete square
cross-section. The results adapted with experimental field tests. It was carried out
that interaction between neighboring piles was affected by nonlinear reaction, and
neglecting these influences can obviously overestimate the amount of interaction
between piles. It was found for lateral loadings, the plasticity of soil was a
determinant factor influencing the horizontal response.
29
Trochanis et al. (1991) presented that the response of laterally loaded piles predicted
utilized the ABAQUS modeling adapted with static load test data and 3D nonlinear
finite element method. Boulanger et al. (1999) demonstrated that the results of
seismic response of piles utilizing ABAQUS simulation with centrifuge experimental
tests.
The objective is to find a precise solution for a complex problem by substituting it by
a simple problem. The fundamental idea behind any finite element method is to
divide the region, main part or structure being evaluated in to a large number of finite
integrated elements. The key idea of finite element analysis is to (1) discretize
complex region into finite elements and (2) use of interpolating polynomials to
describe the field variable with in an element (Frank, 1985).
Zienkiewicz and Cheung (1967) were the first to present the implementation of the
finite element methods to non-structural problems in the field of conduction heat
transfer, but it was immediately accepted that the procedure was feasible to all
made the finite element analysis as one of the most powerful solution methods in
recent times (Frank, 1985).
Today, the most versatile continuum-based method of analysis available is the finite
element method. Various investigations have done on different forms of the finite
elements coupled with Fourier techniques) to evaluate laterally loaded piles [(Desai
& Appel, 1976), (Randolph, 1981), (Kooijman & Vermeer, 1988), Trochanis et al.
(1991), (Bhowmik & Long, 1991), (Bransby M. F., 1999)].
30
The other available continuum based methods are Baguelin et al. (1977), Pyke &
Beikae (1984), Lee et al. (1987), Lee & Small (1991), Sun (1994), Guo & Lee (2001)
not demonstrate the real field conditions, these are seldom utilized by professionals
due to the analysis include complex mathematics.
Brown and Shie (1991) presented a three dimensional analysis utilizing a simple
elastic-plastic constant yield strength envelope (Von Mises Surface) to simulate a
saturated clay soil and a modified Drucker-Prager model without associated flow rule
for sands.
31
Chapter 3
3 METHODOLOGY
3.1 Introduction
The disastrous damage from recent earthquakes (e.g. Santiago 1985, Whittier
Narrows 1987, Cairo 1992, Kobe 1995, Kozani-Grevena 1995, Yugoslavia 1998,
Athens 1999, Kocaeli 1999, Bingol 2003, and Van 2009) has increased concern
about the current codes and methods utilized for the design of structures and
foundations. Many years ago, free field accelerations, velocities and displacements
have been utilized as input grand motions data for the seismic design of foundations
and structures without mentioning the kinematic interaction of the foundation that
have carried out from the introduction of piles and the soil geology.
underestimate or overestimate real in-situ conditions which as a result, will
fundamentally change design criteria. Kinematic and Inertial are the two basic
loading conditions of earthquake induced loadings. Fan et al. (1991) carried out a
considerable parametric study by utilizing an equivalent linear approach to improve
dimensionless diagrams for pile head deflections versus the free field response for
different soil profiles under perpendicular propagating harmonic waves.
By cooperating frequency-dependent springs and dashpots to analyze the single pile
and group piles response, Maluis and Gazetas (1992) indicated free-field acceleration
32
to a one dimensional Beam-on-Dynamic-Winkler-Foundation model. Both
researches studies that the influences of interaction on kinematic loading are
insignificant. However the interaction effects are significant for pile head loading.
These studies were limited to linear analysis and one-dimensional harmonic loading.
In this thesis a 3D nonlinear dynamic and static analysis were performed to evaluate
the effect of slenderness ratio on lateral deflection of pile under lateral loading and
the input motion (wavelet) on the foundation. The finite element program ABAQUS
was utilized in this analysis.
ABAQUS is a powerful finite element computational simulation tool widely used
both in the academic environment and industry, and its absorbing feature to
researchers and advanced users is the available option to implement user-defined
elements, materials, load and boundary types, etc. through user-defined subroutines.
These subroutines may be written in FORTRAN, C or C++ languages. These
subroutines may be linked to ABAQUS through various ways depending on one’s
preference and the operating system.
The ABAQUS finite element program can solve dynamic response of structural
systems one of the frequency domain or time domain which the time domain was
chosen for current study.
Static analysis process took approximately 45 minutes and more, however the
dynamic analysis needed much more time. For Ricker wavelet with 0.002 second
time intervals, the nonlinear analysis took approximately 1 day on a Pentium 4
33
personal computer with Dual-Core 2.6 GHZ CPU and 3GB RAM, which were
utilized at Eastern Mediterranean University.
The present system consists of a pile foundation supporting a structure. As one-
dimensional horizontal acceleration, the dynamic loading was applied to the
underlying bedrock (X-direction in the model) and the vertical and horizontal
responses were evaluated. Due to boundaries of safety against vertical static forces,
commonly provided sufficient resistance to dynamic forces caused by vertical
accelerations, vertical accelerations were neglected. Wu and Finn (1996), concluded
that deformations in the vertical direction are negligible compared to deformations in
the horizontal direction of shaking by utilizing three dimensional elastic model.
The liquefaction potential is not mentioned in the current analysis, but the dilatation
effect of sands around the piles is considered. Additionally drained conditions were
adopted; hence the excess pore pressures were not taken into account.
Soil-pile systems were illustrated with full 3D geometric models. One half of actual
model was simulated; due to the advantage of symmetry in reducing the computing
time. Figure 19 shows an isometric view of the half of pile-soil system which was
utilized in the system. In both of pile and soil model eight-nodes elements were used.
34
Figure 19: Overall view of the half of pile-soil system with mesh
To avoid the “box effect” (meaning that the waves being reflected back into the
model from the boundaries) through the dynamic loading, transmitting boundaries
were utilized to enable the wave to propagate. Infinite elements were defined for
simulating the transmitting boundary.
Figure 20: Schematic of Infinite Elements (a) Three-Dimension Solid Continuum
Element, (b) Axisymmetric Solid Continuum Element (ABAQUS, 2010)
35
3.3.2 Soil Properties
The soil was simulated as elastic and elasto-plastic. For analyzing the influence of
soil plasticity on the response, a homogeneous elasto-plastic material using Drucker-
Prager failure criteria have been utilized (Chen & Mizumo, 1990). For cases
including plasticity, the dilatation angle was assumed to be unequal to the friction
angle (non-associated flow rule). Excess pore water pressures were not considered
(drained condition).
friction and a small negative value of dilatation angle is realistic for loose sand (Nag
Rao, 2006). A basic equation for dilatation angle and the friction is shown in
Equation 3.1.
Ψ φ 30 0 (3.1)
Utilizing Cam Clay modeling for sand has not achieved any success. The cause was
that the experimental plastic potential was completely antithetic from the
experimental yield locus, as recommended first by Poorooshasb et al. (1967) who
were the first to assess the problem of achieving the soil specimen deformation under
a present stress increase as an incremental elasto-plastic problem.
Fundamental modeling of soil behavior under general loading and different site
conditions is the important factor to achieve accurate numerical outcomes. A linear
researches; but most soil behavior is highly nonlinear. In this chapter, a brief
36
explanation about plastic theory is given and then the extended version of Drucker-
Prager plasticity model is demonstrated in detail.
ABAQUS provides a large number of plasticity models to incorporate soil
nonlinearity in the analysis. These include the Extended and Modified Drucker-
Prager models, the Mohr-Coulomb plasticity model and the Critical state (clay)
plasticity model. These are sophisticated plasticity models that require calibration
based on experimental data. There is a short description of the Drucker-Prager soil
model used in this thesis.
for frictional soils. The plasticity theory and failure criterion model are illustrated in
the following subsections.
The extended Drucker-Prager models are used to model frictional materials, which
are typically granular-like soils and rock, and exhibit pressure-dependent yield (the
material becomes stronger as the pressure increases).
Soil deformation includes elastic and plastic strains relating to loading and unloading
ways. Plastic behavior is taken into account as soil irrecoverable deformation while
elastic is considered the behavior when deformation is recoverable. Incremental
approach of plasticity has been utilized successfully in depicting of a wide range of
materials such as soils.
Many experiments carried out on different sands at various confining pressures
showed the general shape of the failure surface as presented in Figure 21. In
37
numerical soil models the shape of the failure surface should be the same as shown in
Figure 21.
Figure 21: Failure surface for sand in the deviator plane (ABAQUS, 2010)
In Figure 22 four common soil models are demonstrated together with the general
shape of the failure model for sand.
According to overestimating of the tensile strength and the sharp corner which
present singularities, the Extended Tresca does not set a satisfactory shape of failure
compared to the general failure surface. The Mohr-Coulomb accuracy has been well
documented for many soils, and the simplicity of this model is one of the advantages
of it.
38
Figure 22: Deviator plane failure surfaces for two parameter models (ABAQUS,
2010)
Neglecting the effect of the intermediate principal stress ( 2 ), and non-mathematical
adapted in the three dimensional application because of the presence of corners,
which simulate singularities, are the Mohr-Coulomb model primarily disadvantages
(Chen & Baladi, 1985).
Chen and Liu (1990) demonstrated the smooth Drucker-Prager model by improving
the Von-Mises failure criterion. This model is appropriate to use due to the failure
surface being smooth, and with convenient material constants it can adapt with the
Mohr- Coulomb criterion obtained from triaxial tests.
As it can be seen in Figure 22c the Drucker-Prager is circular in the deviator plane
and the general failure surface has a “soft” triangular form. Regarding to the soil
39
strength, it should be expected that the Drucker-Prager model overestimates the
tensile strength with some uncertainty. Instead of this, the Lade failure model could
be used, due to the high accuracy in predicting the general failure surface.
According to the accessibility of the Drucker-Prager model in the present ABAQUS
version, this model is used although there is a lack of accuracy at failure. The
material input data can be obtained from triaxial test.
Equation 3.2 shows the failure function of Drucker-Prager materials involving the
hydrostatic effect on the shearing resistance:
f J 2 . I 1 K 0 (3.2)
and cohesion, consecutively.
In Equation 3.2, when” f” is equal to zero the material will follow the plasticity flow
rule, meaning it goes through both elastic and plastic strain, and when it is less than
zero the material will only go through elastic strain.
f < 0 Ideal elastic behavior.
f = 0 Elasto-plastic behavior.
Chen and Baladi (1985) illustrated that the plastic deformation of Drucker-Prager
material is followed by a volume expansion which represents a uniform dilation. In
Figure 22a it is evident that when the elasto-plastic behavior is mentioned, it is
controlled by plastic flow rule, which again controls the hardening of the material.
40
As it is demonstrated in Figure 22a the Drucker-Prager model utilizes isotropic
hardening behavior.
The Drucker-Prager is linear in the constant J2 , ( I 1 / 3 ) space, as presented in
Figure 23. Since frictional materials are cohesionless by nature, this is not a good
description of the hardening behavior. This is a result of Drucker-Prager needing
cohesion to define the yield stress where elasto-plastic behavior sets in.
Figure 23: Drucker-Prager failure surface (ABAQUS, 2010)
41
Figure 24: Yield Surfaces on a Deviator Plane (a) Drucker-Prager and Mohr-
coulomb yield surfaces on deviator plane, (b) the Drucker-Prager failure surface on
a deviator plane, (c) Mohr-Coulomb yield criterion on a deviator plane (Serdaroglu,
2010)
42
Table 2: The advantages and limitations of Mohr-Coulomb and Drucker-Prager
models
Fundamental
Advantage Limitations
Model
Simple Yield surface has corners
Valid for many soil types Neglect the effects of intermediate
Mohr-Coulomb principal stress
Model parameters can be
obtained easily from soil
experiments
Excessive plastic dilatancy at
Simple to use
yielding
Can be matched with Mohr- Cannot produce the hysteretic
Drucker-Prager
Coulomb behavior within
model the failure surface
Cannot predict the pre pressure
Analysis techniques can be build-up during
implemented an undrained cyclic shear loading
Satisfy the associated flow rule
43
In nature the elasto-plastic behavior exists throughout the loading period, which
implies that the hardening behavior should be defined as illustrated in Figure 25b. By
defining hardening as depicted in the figure, the influence of the cohesion on the
failure surface is excluded, and this is modeled by the “Mobilized Friction Model”
(Jostad et al. (1997).
Figure 25: Hardening behavior, (a) Drucker-Prager, (b) common (Jostad et al.
(1997))
This part includes an explanation of Drucker-Prager model in ABAQUS. The
terminology and sign definition used in this part are exactly the same as the the ones
utilized in the ABAQUS manuals. There are three various models of Drucker-Prager
in ABAQUS, which are, “Linear model”, “Hyperbolic model”, and “Exponential
model”. In this thesis the Linear Drucker-Prager model is chosen. The linear
44
Drucker-Prager criterion is defined in Equation 3.3 in ABAQUS, which is also
depicted in Figure 26 (ABAQUS, 2008).
Figure 26: Linear Drucker-Prager (ABAQUS, 2008)
where “p” is the equivalent pressure stress (mean stress), “β” and “d” are the friction
and cohesion factors, respectively while “t” is the generalized shear stress,
determined as shown in Equation 3.4.
1 1 1 r
t q 1 1 ( ) 3 (3.4)
2 K K q
where “q” is the Von-Mises equivalent stress, “r” is the third constant of the deviator
stress and “K” is a factor that clarifies the ratio between the yield stress in triaxial
compression and tension. The cohesion factor, “d”, is determined from:
1
d 1 . tan . σc if hardening is defined by the uniaxial compression yield
3
stress, c
45
1
d 1 . tan . σ t if hardening is defined by uniaxial tension yield stress,
3
t
The parameters β and d can be adapted with the Mohr-Coulomb parameters for two
different cases (flow associated and non-associated rule) which are summarized in
Table 3.
Table 3: The relationship between Drucker-Prager material constants and the Mohr-
Coulomb parameters (Serdaroglu, 2010)
Flow Rule Drucker-Prager Material Constants
3.sin 3.cos
tan d
Associated Flow Rule, Ф 1 2 1 2
1 sin 1 sin
3 3
Non-Associated Flow Rule,
tan = √3. sin d 3.cos
Ф
The “K” factor defines the ratio between the yield stress in triaxial compression and
tension, and herewith the yield surface is shaped by “K” in the intermediate stresses.
When the triaxial tension is equal to the triaxial compression, (K=1) and (t=q), and
the yield surface is equal to the Von-Mises circle in the deviator principal plane. To
ensure the surface is convex, it is required that 0.778 ≤ K ≤ 1. Figure 27 is the proof
of this part.
46
Figure 27: Drucker-Prager Yield surface from ABAQUS (2008)
As pointed out in Figure 28, for granular materials, such as sand, the linear model is
normally used with the non- associated flow, .
Figure 28:Yield Surface and Plastic Flow direction in the p-t plane from
(ABAQUS,2008)
cyclic shear strain laboratory tests (Kramer, 1996). The general equation of the
system is presented by Equation 3.5.
47
where { }̈ is the acceleration vector, { ̇ } is the velocity vector and {u} is the
displacement vector {F(t)} is total force, and [M], [C] and [K] are the global mass,
damping matrice and stiffness matrice, respectively. The damping matrix, [C] = β
2ζ
[K], where β is damping coefficient , and the predominant frequency of the
ω0
loading (rad/sec) is substituted for natural frequency ( 0 ).
Concrete cylindrical section pile with linear elastic properties was used in this study.
The pile was simulated utilizing 8-noded brick elements.
The soil-pile interface modeling is very important due to its influence on the pile
response under lateral loading Trochanis et al. (1988). In the soil-pile interaction, the
surrounding soil and the pile elements are assumed deformable. The surface of pile
elements and soil elements have contact, which the surface of pile elements are
selected as “Master surface” and the surfaces of soil elements are defined as “Slave
surface” . In ABAQUS these surfaces are called the contact pair and they are
depicted in Figure 29. Figure 30 presents the simulated model’s master and slave
surface in this study.
48
Figure 29: Schematic FEM Soil-Pile interface Elements, (a) No sliding, (b) Sliding
Figure 30: Interface between Soil and Pile, (a) Slave surface, (b) Master surface
Following the suggestions made by API (1991) and Gireesha (2011), the coefficient
of friction between concrete and sand relating shear stress to the normal stress was
49
assumed to be 0.7 throughout the analysis. The “penalty function method” was
utilized to represent the contact with normal contact stiffness ( K n ).
The boundary conditions differ according to the type of loading. In static analyzing
the bottom of the model which demonstrates the top of the bedrock layer was fixed in
all directions. However the top face of the model was free to move in all directions in
both static and dynamic analyzing. The symmetry surfaces were free to move on the
surface of the symmetry plane, but fixed against the normal displacement to the
plane. In order to illustrate a horizontally infinite soil medium during static and
dynamic analysis, the elements along the sides of the model were simulated as
Kelvin elements (spring and dashpot), and they were free to move in vertical
direction (Figure 31).
(a) Eight-nodded Element (b) Two-nodded Kelvin Element (c) Five-nodded Contact
Figure 31: Kelvin Elements (ABAQUS, 2010)
head. In this study, because of the symmetric geometry of pile, only half of the load
was applied to the pile in the FEM analysis.
50
In this part the lateral deflection of pile head was evaluated under lateral loading
which was applied on pile head. According to solid mechanics formulations the
amount of lateral deflection can be obtained from Beam Flexure Theory. The pile
which is placed in soil continued up to the bed rock or stiff layer, hence it can be
considered as a cantilever beam (Bentley K. J., 1999), Equation 3.7 and Figure 32
are presented this theory.
PL 3
(3.7)
3EI
where:
δ: Horizontal deflection of pile head
P: Static lateral load
E: Modulus of elasticity
I: Moment of inertia
L: Pile length
This formula is used for verification of the ABAQUS results in soil elasticity.
Figure 32: Pile schematic as a cantilever beam, a point as a bedrock
51
3.5.2 Dynamic Loading
For dynamic part to simulate the seismic wave, the Ricker SV wavelet described by
Equation 3.8, representing the source function and frequency content, is utilized
(Ricker, 1960). Elastic soil treatment is used in this simulation.
2
f t [1 2 .f p . t t 0 ]exp[( .f p .(t t 0 )) 2 ] (3.8)
where:
f (t): amplitude,
f p : predominant frequency in Hz,
t 0 : time parameter of time history in second,
A wavelet is utilized to represent a short time series and simulating a source function.
The wavelets can be described as having amplitude in the frequency domain, and a
time series in the time domain analysis. There are an infinite number of time domain
wavelets for each amplitude spectrum which can be constructed by different type of
phase spectrum. There are 4 typical wavelets named as Ricker, Ormsby, Klauder,
and Butterworth. The Ricker wavelet is one of the particular type of wavelets which
is usually utilized for simulating the excitation function which propagates vertically
and it is distinguished by its dominant frequency. This wavelet is employed because
it is simple to understand and often seems to represent a typical earth response.
The predominant frequency and time interval are considered 2 Hz, 0.002 sec,
respectively in this study, which represent a typical destructive earthquake. Figure 33
presents the Ricker wavelet used in this study. Figure 34 is a schematic diagram of
the seismic wave propagation from bedrock.
52
Figure 33: Ricker wavelet used in present elastic medium dynamic analysis
Figure 34: Schematic of the seismic excitation wave that comes through existing
rigid bedrock
To guarantee that pile, soil, and boundary conditions were distinctly calculated to
minimize error accruement, the verification process was adopted in incremental
steps. Figure 35 is depicts the pile mesh model. The mesh density near the pile was
larger than the other regions because of the severe stress gradients, as shown in
Figures 36 and 37.
53
Figure 35: Simulated pile with mesh
Figure 36: Plane x-z of soil-pile model shows the mesh density near the pile
54
Figure 37: Top side of soil-pile model shows the mesh density around the pile
55
Chapter 4
4 NUMERICAL MODELLING
4.1 Introduction
In this chapter, the extended models for three dimensional conditions are employed
in finite element analysis utilizing ABAQUS / CAE V 6.11. Information about
methods used for modeling was discussed in Chapter 3. This chapter includes the
estimation of parameters and simulation.
The critical location in lateral deflection is the pile head, mainly pile lateral
deflection is considered here. Due to greater deflection of the upper part of piles and
their capacity to carry higher lateral loads than lower parts; under lateral loading, the
most critical part of the pile is the upper part (Poulos & Davis, 1980).
The soil used in this numerical study is standard medium sand, which is simulated as
an elastic medium and elasto-plastic medium by Drucker-Prager material model with
non-associated flow rule, which is explained by angles of friction and dilatation. The
angle of = 2 .
56
Table 4 shows all the model details which were studied in this thesis. A single
and details of the employed materials are summarized in Tables 4-6. Four different
static lateral loads were applied on the pile head for each case.
Table 4: All model details
Models No. Pile Type Soil Type Analyzing Type Load L (m) D (m)
1 Circular / Concrete Sand /Elastic Static 50, 100, 150, 200 kN 3 0.3
2 Circular / Concrete Sand /Elastoplastic Static 50, 100, 150, 200 kN 3 0.3
3 Circular / Concrete Sand /Elastic Static 50, 100, 150, 200 kN 5 0.5
4 Circular / Concrete Sand /Elastic Static 50, 100, 150, 200 kN 7 0.75
5 Circular / Concrete Sand /Elastic Static 50, 100, 150, 200 kN 9 0.9
6 Circular / Concrete Sand /Elastoplastic Static 50, 100, 150, 200 kN 9 0.9
7 Circular / Concrete Sand /Elastic Static 50, 100, 150, 200 kN 6 0.3
8 Circular / Concrete Sand /Elastoplastic Static 50, 100, 150, 200 kN 6 0.3
9 Circular / Concrete Sand /Elastic Static 50, 100, 150, 200 kN 10 0.5
10 Circular / Concrete Sand /Elastic Static 50, 100, 150, 200 kN 15 0.75
11 Circular / Concrete Sand /Elastic Static 50, 100, 150, 200 kN 18 0.9
12 Circular / Concrete Sand /Elastoplastic Static 50, 100, 150, 200 kN 18 0.9
13 Circular / Concrete Sand /Elastic Dynamic Ricker wavelet 5 0.5
14 Circular / Concrete Sand /Elastic Dynamic Ricker wavelet 10 0.5
Table 5: Pile and Geotechnical Properties
Parameters Symbol Soil Pile Unit
Unit Weight γ 1.5 2.3 kN / m 3
Young’s Modulus E 2e+4 2e+7 kPa
Poisson’s Ratio ν 0.45 0.3 -
Friction Angle Ф 32 - Degree
Dilatation Angle 2 - Degree
Table 6: Details of Model
Pile Details Soil Details
Size ( L / D )=10, (L / D Size of Block 50 x 50 m2
)=20
Length & Varied Height 20 m
Diameter
Type of Pile Concrete Type of Soil Sand
57
4.3 Static Analysis Results
Figure 38 presents the comparison between ABAQUS results in elastic behavior of
soil and Beam Flexure Theory, given in Equation 3.7.
Figure 38: Comparison between Beam Flexure Theory and ABAQUS result
Comparing the numerical results in elastic and elasto-plastic behavior of soil, it is
significant that the result of elastic behavior of soil is close to beam flexural theory.
However the lateral deflections are higher in the elasto-plastic medium than the
elastic medium. The beam flexural theory is defined by considering the elastic
condition parameter (E) and geometric properties of pile (L, I). There are some
38. It is because of the lateral soil pressure around the pile which shows that there is
no complete fixity for end of pile.
58
Figures 39 to 42 illustrate lateral deflection of pile with slenderness ratio of 10, at
different depths along the pile shaft from the bottom (0.25L, 0.5L, 0.75L, L), under
different lateral loads (50, 100, 150, 200 kN) in soil with elastic behavior.
Figure 39: Lateral deflection of pile head in different depth of pile, L=3, D=0.3
59
Figure 40: Lateral deflection of pile head in different depth of pile, L=5, D=0.5
Figure 41: Lateral deflection of pile head in different depth, L=7.5, D=0.75
60
Figure 42: Lateral deflection of pile head in different depth, L=9, D=0.9
Figures 39 to 42 depict that with constant slenderness ratio under the same lateral
load, increasing pile length and diameter, the lateral deflection of pile head
decreases. It is because of the effect of soil mass and lateral soil pressure. It shows
that the effect of lateral soil pressure and soil mass overcomes the effect of pile
geometry.
Figures 43 to 46 demonstrate lateral deflection of pile with slenderness ratio of 20 at
different depths along the pile shaft from the bottom (0.25L, 0.5L, 0.75L, L) under
different lateral loads (50, 100, 150, 200 kN) in soil with elastic behavior.
61
Figure 43: Lateral deflection of pile head in different depth, L=6, D=0.3
Figure 44: Lateral deflection of pile head in different depth, L=10, D=0.5
62
Figure 45: Lateral deflection of pile head in different depth, L=15, D=0.75
Figure 46: Lateral deflection of pile head in different depth, L=18, D=0.9
63
By comparing the lateral deflection of piles with L / D =20 in Figures 43 to 46 with
piles with L / D =10, it can be observed that the lateral deflection in pile with higher
slenderness ratio is greater than the pile with less slenderness ratio.
Figure 47 shows the comparison between maximum lateral deflection of piles with
slenderness ratio of 10 which are under load P= 200 kN, and Figure 48 shows the
same comparison but for slenderness ratio of 20.
Figure 47: Comparison between max lateral deflections of pile head in different
depth for (L/D) =10
64
Figure 48: Comparison between max lateral deflections of pile head in different
depth for (L/D) =20
Figures 47 and 48 demonstrate that under constant loading and constant slenderness
ratio, the piles with higher length and bigger diameter have less lateral deflection.
This can be attributed to the diameter effect which by increasing the surface area and
skin friction between pile surface and soil increased.
Figure 49 shows the comparison of the lateral deflection of pile head with different
slenderness ratios of 10 and 20 under different loading in soil with elastic behavior.
Figure 50 presents comparison of the lateral deflection of pile head under lateral
loading with different slenderness ratios in soil with elasto-plastic behavior.
65
Figure 49: Pile head deflection vs. Lateral loading in plastic behavior of soil
Figure 50: Lateral deflection of pile head under lateral loading in elasto-plastic
behavior of soil
The comparison between lateral deflection of pile in elastic and elasto-plastic soil for
pile with slenderness ratios of 10 and 20 is presented in Figures 51 and 52
respectively (D=0.9 m, L=9 m, 18 m under P=200 kN).
66
Figure 51: Comparison between Elastic and Elasto-plastic results for (L/D) =10
Figure 52: Comparison between Elastic and Elasto-plastic results for (L/D) =20
According to Figures 51 and 52, it can be concluded that the pile behavior in elasto-
plastic soil is more critical than in elastic soil.
67
Figures 53 and 54 illustrate lateral deflection of pile with slenderness ratio of 10, at
different depths along the pile shaft (0.25L, 0.5L, 0.75L, L) under different lateral
loads (50, 100, 150, 200 kN) in soil with elasto-plastic behavior.
Figures 55 and 56 present lateral deflection of pile with slenderness ratio of 20, at
kN) in soil with elasto-plastic behavior.
Figure 53: Lateral deflection of pile head in different depths, L=3, D=0.3
68
Figure 54: Lateral deflection of pile head in different depths, L=9, D=0.9
Figure 55: Lateral deflection of pile head in different depths, L=6, D=0.3
69
Figure 56: Lateral deflection of pile head in different depth, L=18, D=0.9
Figure 57: ABAQUS 3D plot of lateral deflection of pile under static loading for
(L=10m, D=0.5m, P=50 kN)
70
It is obvious from Figure 57 that the most lateral deflection occurs at the pile head as
shown by red color, which gets more moderate at depths along the pile length.
The normalized graphs are obtained by normalizing the depth parameter “z” with
diameter “D” of the piles and plotting versus the lateral deflection of each depth
along the pie shaft ( z ) with lateral deflection of pile head 0 . These graphs
shown in Figures 58 to 61 are useful for designing a pile with slenderness ratio
between10 to 20 in elastic behavior of soil, by interpolation.
Similarly, Figures 62 to 65 show normalized graphs which are utilizing for pile
design with 10 ≤ L / D ≤ 20 in elasto-plastic behavior of soil, by interpolating a
given values.
71
Figure 58: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=50 kN in elastic soil
Figure 59: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=100 kN in elastic soil
72
Figure 60: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=150 kN in elastic soil
Figure 61: Pile lateral displacements along length ( z ) , normal to pile head
displacement along line of loading 0 , under load P=200 kN in elastic soil
73
Figure 62: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=50 kN in elasto-plastic soil
Figure 63: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=100 kN in elasto-plastic soil
74
Figure 64: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=150 kN in elasto-plastic soil
Figure 65: Pile lateral displacements along length (z ) , normal to pile head
displacement along line of loading 0 , under load P=200 kN in elasto-plastic soil
75
4.3 Dynamic Analysis Results in Elastic Soil
The Ricker wavelet which is described by Equation 3.8 is used to simulate a severe
earthquake by substituting in the predominant frequency of Kocaeli earthquake of
1999 and the time that maximum frequency occurred. By applying this wavelet as a
lateral displacement on the bedrock, and according to the wave propagation theory in
homogeneous and elastic medium, it is expected that the amplitude of the SV wave
which is applied on the stratum, should be double at the surface of the model, as
shown in Figure 66. This figure also demonstrates the maximum displacement of the
node on the middle of the soil surface as approximately 2 times of the maximum
displacement of Ricker wavelet.
Figure 66: ABAQUS results after applied Ricker wave on the bedrock for the
middle node of soil element on the surface
The ABAQUS three dimensional analysis represents this behavior in Figure 67,
which shows that the critical part of free field after applying the seismic wave at the
bedrock is the soil surface which is identified by red color. By moving from bedrock
76
to surface the incremental lateral deflection of soil under dynamic loading is
revealed.
Figure 67: 3D plot of free field displacement under dynamic loading
In Figures 68 to 71 the lateral deflection of pile under wavelet excitation are
presented, which indicate that lateral deflection of pile under dynamic loading is
more than lateral deflection of pile under static loading of 200 kN.
77
Figure 68: Comparison between lateral deflections of pile under 100kN static load
and Dynamic load
Figure 69: Comparison between lateral deflections of pile under 100kN static load
and Dynamic load
78
Figure 70: Comparison between lateral deflections of pile under 200 kN static load
and Dynamic load
Figure 71: Comparison between lateral deflections of pile under 200 kN static load
and Dynamic load
79
Figure 72 shows the ABAQUS three dimensional analysis for lateral deflection of
pile ( L / D = 10) under dynamic loading.
Figure 72: 3D plot of lateral deflection of pile under dynamic loading for (L=5m,
D=0.5m)
It can be seen in this figure that there is less lateral displacement around the pile
which is denoted by green color. However at the corner of the soil and the other side
of the model, where there is no pile, the lateral deflection is higher as shown by red
color.
80
Chapter 5
5 CONCLUSIONS
5.1 Conclusions
The main objective of this chapter is to illustrate the brief summary and general
conclusions of the present research and recommendations for future study topics
related to pile foundations.
Three dimensional finite element methods were utilized to evaluate the lateral
response of elastic pile in elastic and elasto-plastic soil under lateral static and
dynamic loading. The effect of slenderness ratio on lateral deflection of pile under
variable static loading was assessed. For dynamic analysis the Ricker wavelet was
used as a dynamic loading. The following are the main conclusions of this study.
The lateral response of single pile in the soil with elastic behavior by ABAQUS is
too close to the beam flexural theory results.
1) In both elastic and elasto-plastic soil, by increasing the loads from 50 to 200 kN,
the lateral deflection of pile was increased.
2) One of the most effective parameters on lateral deflection of pile is slenderness
ratio and the pile length has more influence on increasing the lateral deflection.
81
3) Comparing the piles with the same slenderness ratio resulted that the piles with
shorter lengths exhibited more lateral deflections. This presents the direct relation
between increasing the pile fixity (lateral soil pressure) and increasing the pile
length.
4) The lateral deflection of pile in elasto-plastic soil is greater than the lateral
deflection of pile in elastic soil.
1) The lateral deflection of pile in elastic soil under dynamic analysis is close to
lateral pile deflection in elasto-plastic soil under static analysis.
2) It is predicted that pile deflection in elasto-plastic soil under dynamic analysis is
more than pile deflection in elasto-plastic soil under static analysis.
The results of this research can be utilized in designing piles with slenderness ratios
between 10 and 20.
The following topics are suggested for future studies.
1) Investigation of lateral deflection of piles in soils with different layers and
properties.
2) Assessment of effect of liquefaction of soil on pile behavior.
3) Consideration of the effect of pore water pressures and degree of saturation of
soil.
4) Evaluation of the effect of group pile on lateral response of piles.
5) Using real earthquake loading for dynamic analysis.
82
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